Upload
others
View
3
Download
0
Embed Size (px)
Citation preview
MITP20-033
Forward doubly-virtual Compton scattering off the nucleon in
chiral perturbation theory II Spin polarizabilities and moments
of polarized structure functions
Jose Manuel Alarcon
Departamento de Fısica Teorica amp IPARCOS
Universidad Complutense de Madrid 28040 Madrid Spain
Franziska Hagelstein
Albert Einstein Center for Fundamental Physics
Institute for Theoretical Physics University of Bern
Sidlerstrasse 5 CH-3012 Bern Switzerland
Vadim Lensky and Vladimir Pascalutsa
Institut fur Kernphysik amp Cluster of Excellence PRISMA
Johannes Gutenberg-Universitat Mainz D-55128 Mainz Germany
(Dated June 17 2020)
Abstract
We examine the polarized doubly-virtual Compton scattering (VVCS) off the nucleon using chi-
ral perturbation theory (χPT) The polarized VVCS contains a wealth of information on the spin
structure of the nucleon which is relevant to the calculation of the two-photon-exchange effects
in atomic spectroscopy and electron scattering We report on a complete next-to-leading-order
(NLO) calculation of the polarized VVCS amplitudes S1(νQ2) and S2(νQ2) and the corre-
sponding polarized spin structure functions g1(xQ2) and g2(xQ2) Our results for the moments
of polarized structure functions partially related to different spin polarizabilities are compared
to other theoretical predictions and ldquodata-drivenrdquo evaluations as well as to the recent Jefferson
Lab measurements By expanding the results in powers of the inverse nucleon mass we repro-
duce the known ldquoheavy-baryonrdquo expressions This serves as a check of our calculation as well as
demonstrates the differences between the manifestly Lorentz-invariant baryon χPT (BχPT) and
heavy-baryon (HBχPT) frameworks
1
arX
iv2
006
0862
6v1
[he
p-ph
] 1
5 Ju
n 20
20
CONTENTS
I Introduction 3
II Calculation of unpolarized VVCS at NLO 5
A VVCS amplitudes and relations to structure functions 5
B Low-energy expansions and relations to polarizabilities 6
C Details on χPT calculation and uncertainty estimate 8
III Results and discussion 11
A γ0(Q2) mdash generalized forward spin polarizability 11
B δLT (Q2) mdash longitudinal-transverse polarizability 15
C IA(Q2) mdash a generalized GDH integral 16
D Γ1(Q2) and I1(Q
2) mdash the first moment of the structure function g1(xQ2) 20
E d2(Q2) mdash a measure of color polarizability 22
F γ0(Q2) mdash fifth-order generalized forward spin polarizability 24
G Summary 26
IV Conclusions 27
Acknowledgements 28
A Tensor decompositions of the VVCS amplitudes 28
B Photoabsorption cross sections 29
1 πN -production channel 30
2 ∆-production channel 36
C Polarizabilities at Q2 = 0 38
1 πN -loop contribution 38
2 ∆-exchange contribution 40
References 41
2
I INTRODUCTION
In the studies of nucleon structure the forward doubly-virtual Compton scattering
(VVCS) amplitude Fig 1 is playing a central role (see eg Refs [1ndash4] for reviews)
Traditionally its general properties such as unitarity analyticity and crossing are used to
establish various useful sum rules for the nucleon magnetic moment (GerasimovndashDrellndashHearn
[5 6] and Schwinger sum rules [7ndash9]) and polarizabilities (eg Baldin [10] and Gell-Mannndash
GoldbergerndashThirring sum rules [11]) More recently the interest in nucleon VVCS has been
renewed in connection with precision atomic spectroscopy where this amplitude enters in
the form of two-photon exchange (TPE) corrections As the TPE corrections in atomic do-
main are dominated by low-energy VVCS it makes sense to calculate them systematically
using chiral perturbation theory (χPT) which is a low-energy effective-field theory of the
Standard Model
In this paper we present a state-of-the-art χPT calculation of the polarized nucleon
VVCS relevant to TPE corrections to hyperfine structure of hydrogen and muonic hydro-
gen This will extend the leading-order χPT evaluation of the TPE effects in hyperfine
splittings [12ndash17] Here we however do not discuss the TPE evaluation but rather focus on
testing the χPT framework against the available empirical information on low-energy spin
structure of the nucleon
It is especially interesting to confront the χPT predictions with the recent measurements
coming from the ongoing ldquoSpin Physics Programrdquo at Jefferson Laboratory [18ndash27] with
the exception of a recent measurement of the deuteron spin polarizability by the CLAS
Collaboration [28] which does not treat correctly complications due to deuteron spin [29]
Our present calculation is extending Ref [30] to the case of polarized VVCS We therefore
use a manifestly-covariant extension of SU(2) χPT to the baryon sector called Baryon χPT
(BχPT) First attempts to calculate VVCS in the straightforward BχPT framework (rather
than the heavy-baryon expansion or the ldquoinfrared regularizationrdquo) were done by Bernard
et al [31] and our group [32] The two works obtained somewhat different results most
notably for the proton spin polarizability δLT Here we improve on [32] in three important
aspects appreciable at finite Q2 1) inclusion of the Coulomb-quadrupole (C2) N rarr ∆ tran-
sition [33 34] 2) correct inclusion of the elastic form-factor contributions to the integrals
Γ1(Q2) I1(Q2) and IA(Q2) (see Sections III C and III D for details) and 3) cancellations
between different orders in the chiral prediction and their effect on the convergence of the
effective-field-theory calculation and thus the error estimate These improvements how-
ever do not bring us closer to the results of [31] and the source of discrepancies most likely
3
FIG 1 The forward Compton scattering or VVCS in case of virtual photons q2 = minusQ2
lies in the different counting and renormalization of the π∆-loop contributions Bernard
et al [31] use the so-called small-scale expansion [35] for the ∆(1232) contributions whereas
we are using the δ-counting scheme [36] (see also Ref [37 Sec 4] for review)
This paper is organized as follows In Sec II A we introduce the polarized VVCS am-
plitudes and their relations to spin structure functions In Sec II B we introduce the spin
polarizabilities appearing in the low-energy expansion (LEX) of the polarized VVCS am-
plitudes In Sec II C we briefly describe our χPT calculation focusing mainly on the
uncertainty estimate In Sec III we show our predictions for the proton and neutron po-
larizabilities as well as some interesting moments of their structure functions In Sec III G
we summarize the results obtained herein comment on the improvements done with respect
to previous calculations and give an outlook to future applications In App B we discuss
the structure functions in particular we define the longitudinal-transverse response func-
tion discuss the ∆-pole contribution and give analytical results for the tree-level πN - and
∆-production channels of the photoabsorption cross sections In App C we give analytical
expressions for the πN -loop and ∆-exchange contributions to the static values and slopes of
the polarizabilities and moments of structure functions The complete expressions also for
the π∆-loop contributions can be found in the Supplemented material
TABLE I Parameters (fundamental and low-energy constants) [38] at the order they appear first
The πN∆ coupling constant hA is fit to the experimental Delta width and the γlowastN∆ coupling
constants gM gE and gC are taken from the pion photoproduction study of Ref [33]
O(p2) α = 1(13704) MN = Mp = 93827 MeV
O(p3) gA = 1270 fπ = 9221 MeV mπ = 13957 MeV
O(p4∆) M∆ = 1232 MeV hA equiv 2gπN∆ = 285 gM = 297 gE = minus10 gC = minus26
4
II CALCULATION OF UNPOLARIZED VVCS AT NLO
A VVCS amplitudes and relations to structure functions
The polarized part of forward VVCS can be described in terms of two independent
Lorentz-covariant and gauge-invariant tensor structures and two scalar amplitudes [3]
T microν(p q) = minus 1
MN
γmicroναqα S1(νQ2)minus 1
M2N
(γmicroνq2 + qmicroγναqα minus qνγmicroαqα
)S2(νQ2) (1)
Here q and p are the photon and nucleon four-momenta (cf Fig 1) ν is the photon
lab-frame energy Q2 = minusq2 is the photon virtuality and γmicroν = 12
[γmicro γν ] and γmicroνα =12
(γmicroγνγα minus γαγνγmicro) are the usual Dirac matrices Alternatively one can use the following
laboratory-frame amplitudes
gTT (νQ2) =ν
MN
[S1(νQ2)minus Q2
MN νS2(νQ2)
] (2a)
gLT (νQ2) =Q
MN
[S1(νQ2) +
ν
MN
S2(νQ2)] (2b)
introduced in Eq (A2) The optical theorem relates the absorptive parts of the forward
VVCS amplitudes to the nucleon structure functions or the cross sections of virtual pho-
toabsorption
ImS1(νQ2) =4π2α
νg1(xQ2) =
MNνK(νQ2)
ν2 +Q2
[Q
νσLT (νQ2) + σTT (νQ2)
] (3a)
ImS2(νQ2) =4π2αMN
ν2g2(xQ2) =
M2NK(νQ2)
ν2 +Q2
[ν
QσLT (νQ2)minus σTT (νQ2)
] (3b)
with α the fine structure constant andK(νQ2) the photon flux factor Note that the photon
flux factor in the optical theorem and the cross sections measured in electroproduction
processes is a matter of convention and has to be chosen in both quantities consistently In
what follows we use Gilmanrsquos flux factor
K(νQ2) equiv |~q | =radicν2 +Q2 (4)
The helicity-difference photoabsorption cross section is defined as σTT = 12 (σ12 minus σ32)
where the photons are transversely polarized and the subscripts on the right-hand side
indicate the total helicities of the γlowastN states The cross section σLT corresponds to a
simultaneous helicity change of the photon and nucleon spin flip such that the total helicity
5
is conserved A detailed derivation of the connection between this response function and
the photoabsorption cross sections can be found in App B The forward VVCS amplitudes
satisfy dispersion relations derived from the general principles of analyticity and causality1
S1(νQ2) =16παMN
Q2
int 1
0
dxg1(xQ2)
1minus x2(ννel)2 minus i0+(5a)
=2MN
π
int infinνel
dν primeν prime 2[QνprimeσLT + σTT
](ν prime Q2)radic
ν prime 2 +Q2(ν prime 2 minus ν2 minus i0+)
νS2(νQ2) =16παM2
N
Q2
int 1
0
dxg2(xQ2)
1minus x2(ννel)2 minus i0+(5b)
=2M2
N
π
int infinνel
dν primeν prime 2[νprime
QσLT minus σTT
](ν prime Q2)radic
ν prime 2 +Q2(ν prime 2 minus ν2 minus i0+)
with νel = Q22MN the elastic threshold
B Low-energy expansions and relations to polarizabilities
The VVCS amplitudes naturally split into nucleon-pole (Spolei ) and non-pole (Snonpole
i )
parts or Born (SBorni ) and non-Born (Si) parts
Si = Spolei + Snonpole
i = SBorni + Si (6)
The Born amplitudes are given uniquely in terms of the nucleon form factors [1]
SBorn1 (νQ2) =
2πα
M
Q2GM(Q2)F1(Q2)
ν2el minus ν2
minus F 22 (Q2)
(7a)
SBorn2 (νQ2) = minus 2παν
ν2el minus ν2
GM(Q2)F2(Q2) (7b)
The same is true for the nucleon-pole amplitudes which are related to the Born amplitudes
in the following way
Spole1 (νQ2) = SBorn
1 (νQ2) +2πα
MN
F 22 (Q2) (8a)
Spole2 (νQ2) = SBorn
2 (νQ2) (8b)
1 The dispersion relation for νS2(νQ2) is used because it is pole-free in the limit Q2 rarr 0 and then ν rarr 0
cf Eq (7b)
6
Here we used the elastic Dirac and Pauli form factors F1(Q2) and F2(Q2) related to the
electric and magnetic Sachs form factors GE(Q2) and GM(Q2) through
F1(Q2) =GE(Q2) + τGM(Q2)
1 + τ (9a)
F2(Q2) =GM(Q2)minusGE(Q2)
1 + τ (9b)
where τ = Q24M2N
A low-energy expansion (LEX) of Eq (5) in combination with the unitarity relations
given in Eq (3) establishes various sum rules relating the nucleon properties (electromag-
netic moments polarizabilities) to experimentally observable response functions [1 3] The
leading terms in the LEX of the RCS amplitudes are determined uniquely by charge mass
and anomalous magnetic moment as the global properties of the nucleon These lowest-
order terms represent the celebrated low-energy theorem (LET) of Low Gell-Mann and
Goldberger [39 40] The polarizabilities related to the internal structure of the nucleon
enter the LEX at higher orders They make up the non-Born amplitudes and can be related
to moments of inelastic structure functions
The process of VVCS can be realized experimentally in electron-nucleon scattering where
a virtual photon is exchanged between the electron and the nucleon This virtual photon
acts as a probe whose resolution depends on its virtuality Q2 In this way one can access
the so-called generalized polarizabilities which extend the notion of polarizabilities to the
case of response to finite momentum transfer The generalized forward spin polarizability
γ0(Q2) and the longitudinal-transverse polarizability δLT (Q2) are most naturally defined via
the LEX of the non-Born part of the lab-frame VVCS amplitudes [1]
1
4πgnonpoleTT (νQ2) =
2α
M2N
IA(Q2) ν + γ0(Q2)ν3 + γ0(Q2)ν5 + (10a)
1
4πgnonpoleLT (νQ2) =
2α
M2N
I3(Q2)Q+ δLT (Q2)ν2Q+ (10b)
Their definitions in terms of integrals over structure functions are postponed to Eqs (19)
and (22) Here we only give the definition of the moment I3(Q2)
I3(Q2) =M2
N
4π2α
int infinν0
dνK(νQ2)
νQσLT (νQ2) =
2M2N
Q2
int x0
0
dx[g1(xQ2) + g2(xQ2)
] (11)
which is related to the Schwinger sum rule in the real photon limit [7ndash9 41 42] The LEX
of the non-pole part of the covariant VVCS amplitudes can be described entirely in terms
7
of moments of inelastic spin structure functions (up to O(ν4) [43])
1
4πSnonpole
1 (νQ2) =2α
MN
I1(Q2) +
2α
MNQ2
[IA(Q2)minus I1(Q2)
]+MNδLT (Q2)
ν2 (12a)
1
4πνSnonpole
2 (νQ2) = 2αI2(Q2) +2α
Q2
[I1(Q2)minus IA(Q2)
]ν2 (12b)
I1(Q2) and IA(Q2) are generalizations of the famous GerasimovndashDrellndashHearn (GDH) sum
rule [5 6] from RCS to the case of virtual photons [1] Their definitions are given in Eqs (26)
and (32) I2(Q2) is the well-known Burkhardt-Cottingham (BC) sum rule [44]
I2(Q2) equiv 2M2N
Q2
int x0
0
dx g2(x Q2) =1
4F2(Q2)GM(Q2) (13)
which can be written as a ldquosuperconvergence sum rulerdquo
Q2
16παM2N
limνrarr0
νS2(νQ2) =
int 1
0
dx g2 (x Q2) = 0 (14)
The latter is valid for any value of Q2 provided that the integral converges for x rarr 0
Combining Eq (5) with the above LEXs of the VVCS amplitudes we can relate IA(Q2)
I1(Q2) γ0(Q2) and δLT (Q2) to moments of inelastic structure functions see Sec III It is
important to note that only γ0(Q2) and δLT (Q2) are generalized polarizabilities The relation
of the inelastic moments IA(Q2) and I1(Q2) to polarizabilities will be discussed in details in
Secs III C and III D The difference between S1(νQ2) and Snonpole1 (νQ2) cf Eq (8a) will
be important in this context
C Details on χPT calculation and uncertainty estimate
In this work we calculated the NLO prediction of BχPT for the polarized non-Born
VVCS amplitudes This includes the leading pion-nucleon (πN) loops see Ref [32 Fig 1]
as well as the subleading tree-level Delta-exchange (∆-exchange) see Ref [30 Fig 2] and
the pion-Delta (π∆) loops see Ref [32 Fig 2] In the δ-power-counting scheme [36] the
LO and NLO non-Born VVCS amplitudes and polarizabilities are of O(p3) and O(p4∆)
respectively2 The LECs are listed in Table I sorted by the order at which they appear in
our calculation At the given orders there are no ldquonewrdquo LECs that would need to be fitted
from Compton processes For more details on the BχPT formalism we refer to Ref [30]
2 In the full Compton amplitude there is a lower order contribution coming from the Born terms leading
to a shift in nomenclature by one order the LO contribution referred to as the NLO contribution etc
see eg Ref [45]
8
where power counting predictive orders (Sec III A) and the renormalization procedure
(Sec III B) are discussed
A few remarks are in order for the inclusion of the ∆(1232) and the tree-level ∆-exchange
contribution In contrast to Ref [32] we include the Coulomb-quadrupole (C2) N rarr ∆
transition described by the LEC gC The relevant Lagrangian describing the non-minimal
γlowastN∆ coupling [33 34] (note that in these references the overall sign of gC is inconsistent
between the Lagrangian and Feynman rules) reads
L(2)∆ =
3e
2MNM+
N T3
igM F
microν partmicro∆ν minus gEγ5Fmicroν partmicro∆ν (15)
+igCM∆
γ5γα(partα∆ν minus partν∆α)partmicroF
microν
+ Hc
with M+ = MN + M∆ and the dual of the electromagnetic field strength tensor F microν =12εmicroνρλFρλ Even though the Coulomb coupling is subleading compared with the electric
and magnetic couplings (gE and gM) its relatively large magnitude cf Table I makes it
numerically important for instance in γ0(Q2) Furthermore we study the effect of modifying
the magnetic coupling using a dipole form factor
gM rarrgM[
1 + (QΛ)2 ]2 (16)
where Λ2 = 071 GeV2 The inclusion of this Q2 dependence mimics the form expected from
vector-meson dominance It is motivated by observing the importance of this form factor
for the correct description of the electroproduction data [33]
To estimate the uncertainties of our NLO predictions we define
δ(Q2) =
radic(∆
MN
)2
+
(Q2
2MN∆
)2
(17)
such that the neglected next-to-next-to-leading order terms are expected to be of relative
size δ2 [33] The uncertainties in the values of the parameters in Table I have a much
smaller impact compared to the truncation uncertainty and can be neglected Unfortunately
∆IA(Q2) γ0(Q2) and γ0(Q2) ie the sum rules involving the cross section σTT (νQ2)
as well as the polarizability ∆I1(Q2) turn out to be numerically small Their smallness
suggests a cancellation of leading orders (which can indeed be confirmed by looking at
separate contributions as shown below) Therefore an error of δ2(Q2)P (Q2) where P (Q2)
is a generalized polarizability might underestimate the theoretical uncertainty for some of
the NLO predictions To avoid this we estimate the uncertainty of our NLO polarizability
9
predictions by
σP (Q2) =
Max[δ4(0)P (0)2 δ4(0)P LO(0)2 δ2(0)PNLO(0)2
]+Max
[δ4(Q2)
[P (Q2)minus P (0)
]2 δ4(Q2)
[P LO(Q2)minus P LO(0)
]2
δ2(Q2)[PNLO(Q2)minus PNLO(0)
]2]12
(18)
where P LO(Q2) is the πN -loop contribution PNLO(Q2) are the ∆-exchange and π∆-loop
contributions and P (Q2) = P LO(Q2) + PNLO(Q2) This error prescription is similar to the
one used in eg Refs [46ndash49] Here since we are interested in the generalized polarizabil-
ities we added in quadrature the error due to the static piece P (0) and the Q2-dependent
remainder P (Q2) minus P (0) Note that the static values of IA(0) and I1(0) are given by the
elastic Pauli form factor which is not part of our BχPT prediction and is considered to be
exact
Note that our result for the spin polarizabilities (and the unpolarized moments [30]) are
NLO predictions only at low momentum transfers Q mπ At larger values of Q amp ∆ they
become incomplete LO predictions Indeed in this regime the ∆ propagators do not carry
additional suppression compared to the nucleon propagators and the π∆ loops are promoted
to LO In general we only expect a rather small contribution from omitted π∆ loops to the
Q2 dependence of the polarizabilities since π∆ loops show rather weak dependence on Q2
compared with the ∆ exchange or πN loops Nevertheless this issue has to be reflected in
the error estimate Since the static polarizabilities P (0) are not affected it is natural to
separate the error on the Q2-dependent remainder P (Q2) minus P (0) as done in Eq (18) To
accommodate for the potential loss of precision above Q amp ∆ we define the relative error
δ(Q2) as growing with increasing Q2 see Eq (17)
Upon expanding our results in powers of the inverse nucleon mass Mminus1N we are able to
reproduce existing results of heavy-baryon χPT (HBχPT) at LO We however do not see
a rationale to drop the higher-order Mminus1N terms when they are not negligible (ie when
their actual size exceeds by far the natural estimate for the size of higher-order terms)
Comparing our BχPT predictions to HBχPT we will also see a deficiency of HBχPT in
the description of the Q2 behaviour of the polarizabilities Note that the O(p4) HBχPT
results from Ref [50 51] which we use here for comparison do not include the ∆ These
references studied the leading effect of the latter in the HBχPT framework using the small-
scale expansion [35] observing no qualitative improvement in the HBχPT description of the
empirical data [50 51] when including it We therefore choose to use the O(p4) results as
the representative HBχPT curves
10
Another approach used in the literature to calculate the polarizabilities in χPT is the
infrared regularization (IR) scheme introduced in Ref [52] This covariant approach tries to
solve the power counting violation observed in Ref [53] by dropping the regular parts of the
loop integrals that contain the power-counting-breaking terms However this subtraction
scheme modifies the analytic structure of the loop contributions and may lead to unexpected
problems as was shown in Ref [54] As we will see in the next section the IR approach
also fails to describe the Q2 behaviour of the polarizabilities
III RESULTS AND DISCUSSION
We now present the NLO BχPT predictions for the nucleon polarizabilities and selected
moments of the nucleon spin structure functions Our results are obtained from the calcu-
lated non-Born VVCS amplitudes and the LEXs in Eqs (10) and (12) For a cross-check
we used the photoabsorption cross sections described in App B In addition to the full NLO
results we also analyse the individual contributions from the πN loops the ∆ exchange
and the π∆ loops
A γ0(Q2) mdash generalized forward spin polarizability
The forward spin polarizability
γ0(Q2) =1
2π2
int infinν0
dν
radic1 +
Q2
ν2
σTT (νQ2)
ν3(19)
=16αM2
N
Q6
int x0
0
dx x2
[g1(xQ2)minus 4M2
Nx2
Q2g2(xQ2)
]
provides information about the spin-dependent response of the nucleon to transversal pho-
ton probes The RCS analogue of the above generalized forward spin polarizability sum
rule is sometimes referred to as the Gell-Mann Goldberger and Thirring (GGT) sum rule
[11] At Q2 = 0 the forward spin polarizability is expressed through the lowest-order spin
polarizabilities of RCS as γ0 = minus(γE1E1 + γM1M1 + γE1M2 + γM1E2) The forward spin po-
larizability of the proton is relevant for an accurate knowledge of the (muonic-)hydrogen
hyperfine splitting as it controls the leading proton-polarizability correction [16 62]
The πN -loop ∆-exchange and π∆-loop contributions to the NLO BχPT prediction of
11
000 005 010 015 020 025 030
-4
-3
-2
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
4D
proton
000 005 010 015 020 025 030
-4
-3
-2
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
4D
neutron
000 005 010 015 020 025 030
0
1
2
Q2 GeV2D
∆LT
10-
4fm
4D
proton
000 005 010 015 020 025 030
0
1
2
3
Q2 GeV2D
∆LT
10-
4fm
4D
neutron
FIG 2 Upper panel Generalized forward spin polarizability for the proton (left) and neutron
(right) as function of Q2 The result of this work the NLO BχPT prediction is shown by the
blue solid line and the blue band The red line represents the LO BχPT result The purple short-
dashed line is the O(p4) HB result from Ref [50] note that the corresponding proton curve is
outside of the plotted range The black dotted line is the MAID model prediction [55ndash57] which
is taken from Ref [1] (proton) and Ref [20] (neutron) The pink band is the IR+∆ result from
Ref [58] and the gray band is the BχPT+∆ result from Ref [31] Empirical extractions for the
proton Ref [18] (blue dots) Ref [59] (purple square) and Ref [60] (orange triangle uncertainties
added in quadrature) and neutron Ref [20] (blue diamonds) and Ref [61] (green dots statistical
and systematic uncertainties added in quadrature) Lower Panel Longitudinal-transverse spin
polarizability for the proton (left) and neutron (right) The orange dot-dashed and purple short-
dashed lines are the O(p3) and O(p4) HB results from Ref [50] The pink band is the IR result
from Ref [58] and the gray band is the covariant BχPT+∆ result from Ref [31] The black dotted
line is the MAID model prediction [55ndash57] note that for the proton we use the updated estimate
from Ref [1] obtained using the π η ππ channels
12
the static forward spin polarizability amount to in units of 10minus4 fm4
γ0p = minus093(92) asymp 201minus 284minus 010 (20a)
γ0n = 003(92) asymp 298minus 284minus 010 (20b)
while the slope is composed as follows in units of 10minus4 fm6
dγ0p(Q2)
dQ2
∣∣∣∣Q2=0
= minus022(4) asymp minus033 + 011 + 001 (21a)
dγ0n(Q2)
dQ2
∣∣∣∣Q2=0
= minus061(7) asymp minus073 + 011 + 001 (21b)
Figure 2 upper panel shows our NLO prediction as well as the LO πN loops compared
to different experimental and theoretical results For the proton we have one determination
at the real-photon point by the GDH collaboration [19] γ0p = minus100(8)(12)times10minus4 fm4 and
further Jefferson Laboratory data [18 60] at very low Q2 For the neutron only data at finite
Q2 are available [20 61] The experimental data for the proton are fairly well reproduced
in the whole Q2 range considered here while for the neutron the agreement improves with
increasing Q2 The HB limit of our πN -loop contribution reproduces the results published
in Refs [50 63] for arbitrary Q2 In addition our prediction is compared to the MAID
model [1 20] the IR+∆ calculation of Ref [58] and the BχPT+∆ result of Ref [31]
The πN -production channel gives a positive contribution to the photoabsorption cross
section σTT (νQ2) at low Q2 cf Fig 10 Accordingly one observes that the πN loops give
a sizeable positive contribution to γ0(Q2) The Delta on the other hand has a very large
effect by cancelling the πN loops and bringing the result close to the empirical data From
Fig 3 upper panel one can see that it is the ∆ exchange which dominates while π∆
loops are negligible This was expected since the forward spin polarizability sum rule is an
integral over the helicity-difference cross section in which σ32 is governed by the Delta at
low energies (the relevant energy region for the sum rule)
To elucidate the difference between the present calculation and the one from Ref [31] we
note that the two calculations differ in the following important aspects Firstly Ref [31] uses
the small-scale counting [64] that considers ∆ and mπ as being of the same size ∆ sim mπ
In practice this results in a set of π∆-loop graphs which contains graphs with one or two
γ∆∆ couplings and hence two or three Delta propagators Such graphs are suppressed in
the δ-counting and thus omitted from our calculation while present in that of Ref [31]
Secondly the Lagrangians describing the interaction of the Delta are constructed differently
and assume slightly different values for the coupling constants In particular we employ
13
000 005 010 015 020 025 030
-3
-2
-1
0
1
2
Q2GeV
2D
Γ01
0-
4fm
4D
proton
000 005 010 015 020 025 030
-3
-2
-1
0
1
2
3
Q2GeV
2D
Γ01
0-
4fm
4D
neutron
000 005 010 015 020 025 030
00
05
10
15
Q2GeV
2D
∆LT1
0-
4fm
4D
proton
000 005 010 015 020 025 030
00
05
10
15
20
25
Q2GeV
2D
∆LT1
0-
4fm
4D
neutron
FIG 3 Contributions of the different orders to the chiral predictions of γ0(Q2) upper paneland δLT (Q2) lower panel for the proton (left) and neutron (right) Red solid line πN -loop
contribution green dot-dashed line ∆-exchange contribution orange dotted line π∆-loop con-
tribution blue long-dashed line total result purple dot-dot-dashed line total result without gCcontribution black short-dashed line total result without gM dipole
(where possible) the so-called ldquoconsistentrdquo couplings to the Delta field ie those couplings
that project out the spurious degree of freedom see Refs [37 65 66] The authors of
Ref [31] on the other hand use couplings where the consistency in this sense is not enforced
The effects of these differences are of higher order in the δ-counting expansion and their
contribution to the Q2 dependence of the considered polarizabilities is expected to be rather
small however the differences at Q2 = 0 could be noticeable [67]
Finally as mentioned in Sec II C the inclusion of the dipole form factor in the magnetic
coupling gM is expected to be important to generate the correct Q2 behaviour of the polar-
izabilities Comparing our predictions for the forward spin polarizability with and without
inclusion of the form factor see Fig 3 upper panel confirms this Without the dipole
our results for the proton and neutron are closer to the ones from Ref [31] where the form
factor is not included For the neutron our prediction without the dipole is able to describe
14
the experimental points at very low Q2 deviating from the data with increasing Q2 The
π∆-loop contribution does not modify the Q2 behavior of γ0(Q2) and only differs from
Ref [31] by a small global shift Note also the relatively large effect of gC which generates
a sign change for virtualities above sim 02 GeV2 see Fig 3 upper panel
B δLT (Q2) mdash longitudinal-transverse polarizability
The longitudinal-transverse spin polarizability
δLT (Q2) =1
2π2
int infinν0
dν
radic1 +
Q2
ν2
σLT (νQ2)
Qν2(22)
=16αM2
N
Q6
int x0
0
dx x2[g1(xQ2) + g2(xQ2)
]
contains information about the spin structure of the nucleon and is another important
input in the determination of the (muonic-)hydrogen hyperfine splitting [16 62] It is also
relevant in studies of higher-twist corrections to the structure function g2(xQ2) given by
the moment d2(Q2) [51] see Section III E The peculiarity of the response encoded in this
polarizability is that it involves a spin flip of the nucleon and a polarization change of the
photon see App B and Fig 11
It is expected that the Delta isobar gives only a small contribution to δLT (Q2) what makes
this polarizability a potentially clean test case for chiral calculations Consequently there are
relatively many different theoretical calculations of δLT (Q2) coming from different versions
of χPT with baryons (HB IR and covariant) Ref [50] found a systematic deviation of the
HB result for δLTn(Q2) from the MAID model prediction This disagreement was identified
by the authors of Ref [68] as a puzzle involving the neutron δLT polarizabilitymdashthe δLT
puzzle The IR calculation in Ref [58] also showed a deviation from the data and predicted
a rapid rise of δLT (Q2) with growing Q2 The problem is solved by keeping the relativistic
structure of the theory as the BχPT+∆ result of Ref [31] showed
As expected already the leading πN loops provide a reasonable agreement with the
experimental data cf Fig 2 lower panel Since the ∆-exchange contribution to δLT (Q2)
is small the effect of the gM form factor is negligible in this polarizability as is that of the gC
coupling cf Fig 3 lower panel In fact we predict both the ∆-exchange and the π∆-loop
contributions to be small and negative This is in agreement with the MAID model which
predicts a small and negative contribution of the P33 wave to δLT (Q2) However in the
calculation of Ref [31] which is different from the one presented here only in the way the
∆(1232) is included the contribution of this resonance to δLTp(Q2) is sizeable and positive
15
The authors of that work attributed this large contribution to diagrams where the photons
couple directly to the Delta inside a loop As mentioned in Sec III F the effect of such loop
diagrams does not change the Q2 behaviour of the polarizabilities On the other hand it
can produce a substantial shift of the δLT (Q2) as a whole A higher-order calculation should
resolve the discrepancy between the two covariant approaches however it will partially lose
the predictive power since the LECs appearing at higher orders will have to be fitted to
experimental data
The πN -loop ∆-exchange and π∆-loop contributions to the NLO BχPT prediction of
the static longitudinal-transverse polarizability are in units of 10minus4 fm4
δLTp = 132(15) asymp 150minus 016minus 002 (23a)
δLTn = 218(23) asymp 235minus 016minus 002 (23b)
while the slopes are in units of 10minus4 fm6
dδLTp(Q2)
dQ2
∣∣∣∣Q2=0
= minus085(8) asymp minus080minus 004minus 001 (24a)
dδLTn(Q2)
dQ2
∣∣∣∣Q2=0
= minus124(12) asymp minus119minus 004minus 001 (24b)
C IA(Q2) mdash a generalized GDH integral
The helicity-difference cross section σTT exhibits a faster fall-off in ν than its spin-averaged
counterpart σT This is due to a cancellation between the leading (constant) terms of σ12
and σ32 at large ν3 The resulting 1ν fall-off of the helicity-difference cross section allows
one to write an unsubtracted dispersion relation for the VVCS amplitude gnonpoleTT (νQ2) cf
Eq (10a) This is the origin of the GDH sum rule [5 6]
minus α
2M2N
κ2 =1
2π2
int infinν0
dνσTT (ν)
ν (25)
which establishes a relation to the anomalous magnetic moment κ It is experimentally
verified for the nucleon by MAMI (Mainz) and ELSA (Bonn) [71 72]
There are two extensions of the GDH sum rule to finite Q2 the generalized GDH integrals
3 Notice that a constant term in σTT at ν rarrinfin is forbidden by crossing symmetry
16
000 005 010 015 020 025 030
-30
-25
-20
-15
-10
-05
00
Q2 GeV2D
I Aproton
000 005 010 015 020 025 030
-25
-20
-15
-10
-05
00
Q2 GeV2D
I A
neutron
000 005 010 015 020 025 030
-05
00
05
Q2 GeV2D
I 1
proton
000 005 010 015 020 025 030
-10
-08
-06
-04
-02
00
02
Q2 GeV2D
I 1
neutron
FIG 4 Upper panel The generalized GDH integral IA(Q2) for the proton (left) and neutron
(right) as function of Q2 The result of this work the NLO BχPT prediction is shown by the blue
solid line and the blue band The red line represents the LO BχPT result The purple short-dashed
line is the O(p4) HB result from Ref [50 51] The gray band is the BχPT+∆ result from Ref [31]
The black dotted line is the MAID model prediction [69] Experimental extractions for the proton
Ref [60] (orange triangle uncertainties added in quadrature) and neutron Refs [21][27] where
magenta dotsorange diagonal crosses correspond to data and red squareslilac crosses correspond
to data plus extrapolation to unmeasured energy regions The green stars at the real-photon point
are derived from the anomalous magnetic moments κp asymp 1793 and κn asymp minus1913 [70] Lower
panel The generalized GDH integral I1(Q2) for the proton (left) and neutron (right) as function
of Q2 The purple short-dashed line is the HB result from Ref [51] Experimental extractions for
the proton Ref [18] (blue dots) and Ref [60] (orange triangle uncertainties added in quadrature)
and neutron Ref [61][27] (uncertainties added in quadrature) where green dotsorange diagonal
crosses correspond to data and gray squareslilac crosses correspond to data plus extrapolation to
unmeasured energy regions
17
IA(Q2) and I1(Q2) The latter will be discussed in Sec III D The former is defined as4
minus α
2M2N
IA(Q2) = minus 1
8π2
int infinν0
dν
radic1 +
Q2
ν2
σTT (νQ2)
ν(26)
=α
Q2
int x0
0
dx[4M2
Nx2
Q2g2(xQ2)minus g1(xQ2)
]
Due to its energy weighting the integral in Eq (26) converges slower than the one in the
generalized forward spin polarizability sum rule (19) Therefore knowledge of the cross
section at higher energies is required and the evaluation of the generalized GDH integral
IA(Q2) is not as simple as the evaluation of γ0(Q2)
The generalized GDH integral IA(Q2) is directly related to the non-pole amplitude
gnonpoleTT (νQ2) which differs from non-Born amplitude gTT (νQ2) by a term involving the
elastic Pauli form factor
gnonpoleTT (νQ2) = gTT (νQ2)minus 2παν
M2N
F 22 (Q2) (27)
cf Eqs (2a) and (8a) Consequently IA(Q2) is not a pure polarizability but also contains
an elastic contribution The ldquonon-polarizabilityrdquo or the Born part of IA(Q2) is given by
IBornA (Q2) = IA(Q2)minus∆IA(Q2) = minus1
4F 2
2 (Q2) (28)
where we refer to the polarizability part as ∆IA(Q2) The same is true for the generalized
GDH integral I1(Q2) which is directly related to Snonpole1 (νQ2)
IBorn1 (Q2) = I1(Q2)minus∆I1(Q2) = minus1
4F 2
2 (Q2) (29)
In the following we will add the Born parts to our LO and NLO BχPT predictions for
the polarizabilities ∆IA(Q2) and ∆I1(Q2) employing an empirical parametrization for the
elastic Pauli form factor [73] This allows us to compare to the experimental results for
IA(Q2) and I1(Q2) cf Fig 4 Note that the blue error bands only describe the uncertainties
of our BχPT predictions of the polarizabilities while the elastic contributions are considered
to be exact as explained in Sec II C The uncertainties of the polarizability predictions are
therefore better reflected in Fig 5 where we show the contributions of the different orders
to the BχPT predictions of ∆IA(Q2) and ∆I1(Q2) as well as the total results with error
bands
4 Note that IA(Q2) is sometimes called ITT (Q2)
18
000 005 010 015 020 025 030
-08
-06
-04
-02
00
02
Q2 GeV2D
DI A
proton
000 005 010 015 020 025 030
-10
-08
-06
-04
-02
00
02
Q2 GeV2D
DI A
neutron
000 005 010 015 020 025 030-005
000
005
010
015
020
025
Q2 GeV2D
DI 1
proton
000 005 010 015 020 025 030-015
-010
-005
000
005
010
015
Q2 GeV2D
DI 1
neutron
FIG 5 Contributions of the different orders to the chiral predictions of ∆IA(Q2) upper paneland ∆I1(Q2) lower panel for the proton (left) and neutron (right) Red solid line πN -loop
contribution green dot-dashed line ∆-exchange contribution orange dotted line π∆-loop contri-
bution blue solid line and blue band total result purple dot-dot-dashed line total result without
gC contribution black short-dashed line total result without gM dipole
The E97-110 experiment at Jefferson Lab has recently published their data for IAn(Q2) in
the region of 0035 GeV2 lt Q2 lt 024 GeV2 [27] In addition there are results for IAn(Q2)
from the earlier E94-010 experiment [21] and for IAp(Q2) from the E08-027 experiment [60]
The O(p4) HB calculation gives a large negative effect [51] which does not describe the data
The BχPT+∆ result from Ref [31] which mainly differs from our work by the absence of
the dipole form factor in gM looks similar to this HB result and only describes the data
points at lowest Q2 Our NLO prediction however follows closely the Q2 evolution of the
data In Fig 5 upper panel we show the polarizability ∆IA(Q2) whose Q2 evolution is
clearly dominated by the ∆ exchange Similar to the case of γ0p(Q2) inclusion of the dipole
in gM and the Coulomb coupling gC is very important in order to describe the experimental
data The LO prediction on the other hand slightly overestimates the data cf Fig 4
upper panel
19
At the real-photon point IA(0) = minusκ2
4and ∆IA(0) = 0 Therefore we give only the
slope of the polarizability ∆IA(Q2) [showing also the separate contributions from πN loops
∆ exchange and π∆ loops] in units of GeVminus2
d∆IAp(Q2)
dQ2
∣∣∣∣Q2=0
= minus858(343) asymp 238minus 1121 + 025 (30a)
d∆IAn(Q2)
dQ2
∣∣∣∣Q2=0
= minus955(343) asymp 141minus 1121 + 025 (30b)
Including the empirical Pauli form factor [73] we find in units of GeVminus2
dIAp(Q2)
dQ2
∣∣∣∣Q2=0
= minus318dIAn(Q2)
dQ2
∣∣∣∣Q2=0
= minus300 (31)
D Γ1(Q2) and I1(Q2) mdash the first moment of the structure function g1(xQ2)
The second variant for a generalization of the GDH sum rule to finite Q2 is defined as
minus α
2M2N
I1(Q2) = minus 1
8π2
int infinν0
dν1radic
ν2 +Q2
[σTT (νQ2) +
Q
νσLT (νQ2)
](32)
= minus α
Q2
int x0
0
dx g1(xQ2)
where I1(0) = minusκ2
4 This generalized GDH integral directly stems from the amplitude
Snonpole1 (νQ2) with the LEX from Eq (12a) It is given by the first moment of the struc-
ture function g1(xQ2) Γ1(Q2) =int x0
0dx g1(xQ2) as follows I1(Q2) =
2M2N
Q2 Γ1(Q2) The
isovector combination
Γ1(pminusn)(Q2) =
int x0
0
dx[g1p(xQ
2)minus g1n(xQ2)] (33)
is related to the axial coupling of the nucleon through the Bjorken sum rule [74 75]
limQ2rarrinfin
Γ1(pminusn)(Q2) =
gA6 (34)
As explained in Eq (28) the moment I1(Q2) splits into a polarizability part ∆I1(Q2) and
a Born part IBorn1 (Q2) Figure 4 lower panel shows the Q2 dependence of I1(Q2) which in
contrast to IA(Q2) shown in Figure 4 upper panel is clearly dominated by its Born part
and the elastic Pauli form factor The πN -loop ∆-exchange and π∆-loop contributions to
the polarizability ∆I1(Q2) are shown in Fig 5 lower panel Comparing to Fig 5 upper
20
000 005 010 015 020 025 030
-006
-004
-002
000
002
004
Q2 GeV2D
G1
proton
000 005 010 015 020 025 030
-006
-005
-004
-003
-002
-001
000
001
Q2 GeV2D
G1
neutron
FIG 6 First moment of the structure function g1(xQ2) for the proton (left) and neutron (right)
as function of Q2 The legend is the same as in Fig 4
panel one sees that ∆I1(Q2) is less sensitive to gC and the dipole form factor in gM than
∆IA(Q2)
For the proton our NLO BχPT prediction gives a very good description of the ex-
perimental data [18 60] and is in reasonable agreement with the MAID prediction [69]
For the neutron one observes good agreement with the empirical evaluations including ex-
trapolations to unmeasured energy regions starting from Q2 gt 01 GeV2 [27 61] In the
region of Q2 lt 005 GeV2 one observes an interesting tension between the recent E97-
110 experiment [27] and the data from CLAS [61] While the newest measurement finds
I1n(0035 GeV2) lt κ2n4 thus suggesting a negative slope at low Q the older measurement
found a rather large value for I1n(00496 GeV2) A similar but milder behaviour is seen in
the E97-110 [27] and E94-010 [21] data for IAn The MAID predictions do not agree with
the CODATA recommended values for the anomalous magnetic moments of the proton and
neutron [70] which in our work are imposed by using empirical parametrizations for the
elastic Pauli form factors [73] The slope of the HB result from Ref [51] is too large and
therefore only reproduces the data at very low Q2
Figure 6 shows the moment Γ1(Q2) for the proton and neutron while Fig 7 shows the
isovector combination Γ1 pminusn(Q2) The LO and NLO BχPT predictions are identical be-
cause our calculation produces the same Delta contributions for the proton and the neutron
For the isovector combination the MAID model only agrees with the data at very low
Q2 lt 010 GeV2 The same is true for the IR result [58 76] while all other chiral results
describe the data NLO BχPT (this work) BχPT+∆ [31] and HBχPT [51]
At the real-photon point I1(0) = minusκ2
4and ∆I1(0) = 0 Therefore we give only the
slope of the polarizability ∆I1(Q2) [showing also the separate contributions from πN loops
21
000 005 010 015 020 025 030-002
000
002
004
006
008
010
Q2 GeV2D
G1Hp-
nL
FIG 7 Isovector combination of Γ1(Q2) as function of Q2 The legend is the same as in Fig 6
The pink curve is the IR result from Ref [58 76] The experimental points are from Ref [77]
(brown dots) and Ref [23] (orange squares)
∆ exchange and π∆ loops] in units of GeVminus2
d∆I1p(Q2)
dQ2
∣∣∣∣Q2=0
= 039(4) asymp 034minus 053 + 058 (35a)
d∆I1n(Q2)
dQ2
∣∣∣∣Q2=0
= minus101(10) asymp minus107minus 053 + 058 (35b)
Including the empirical Pauli form factor [73] we find in units of GeVminus2
dI1p(Q2)
dQ2
∣∣∣∣Q2=0
= 580dI1n(Q2)
dQ2
∣∣∣∣Q2=0
= 553 (36)
E d2(Q2) mdash a measure of color polarizability
Another interesting moment to consider is d2(Q2) which is related to the twist-3 part of
the spin structure function g2(xQ2) [79 80]
d2(Q2) equiv 3
int 1
0
dx x2[g2(xQ2)minus gWW2 (xQ2)] (37)
where gWW2 (xQ2) is the twist-2 part of g2(xQ2) Using the Wandzura-Wilczek relation [81]
one can relate d2(Q2) to moments of the spin structure functions g1(xQ2) and g2(xQ2)
d2(Q2) =
int 1
0
dx x2 [3g2(xQ2) + 2g1(xQ2)] (38)
22
000 005 010 015 020 025 030
0000
0001
0002
0003
0004
0005
0006
0007
Q2 GeV2D
d2
proton
000 005 010 015 020 025 030
0000
0002
0004
0006
0008
0010
0012
Q2 GeV2D
d2
neutron
000 005 010 015 020 025 030
-1
0
1
2
Q2 GeV2D
Γ 010
-4
fm6
D
proton
000 005 010 015 020 025 030-2
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
6D
neutron
FIG 8 Upper panel The inelastic moment d2(Q2) for the proton (left) and neutron (right) as
function of Q2 The result of this work the NLO BχPT prediction is shown by the blue solid line
and the blue band The red line represents the LO BχPT result The purple short-dashed line is
the O(p4) HB result from Ref [50 51] The black dotted line is the MAID model prediction [69]
The experimental points for the neutron (cyan dots) are from Ref [22] Lower panel Fifth-order
generalized forward spin polarizability γ0(Q2) for the proton (left) and neutron (right) as function
of Q2 The experimental points for the proton are from Ref [59] (purple square) and Ref [78]
(orange dot)
This relation however only holds for asymptotically large Q2 It is also in the high-Q2
region where d2(Q2) is a measure of color polarizability [82 83] through its relation to the
gluon field strength tensor [80] We refer to Ref [84] for a recent review on the spin structure
of the nucleon including a discussion of sum rules for deep inelastic scattering and color
polarizabilities
What we consider in the following is the inelastic part of d2(Q2) defined as the moment
of g1(xQ2) and g2(xQ2) spin structure functions cf Eq (38)
d2(Q2) =
int x0
0
dx x2 [3g2(xQ2) + 2g1(xQ2)] (39)
23
This moment provides another testing ground for our BχPT predictions through comparison
with experiments on the neutron [22] Going towards the low-Q2 region the interpretation
of d2(Q2) in terms of color polarizabilities will fade out The above definition however
implies it is related to other VVCS polarizabilities
d2(Q2) =Q4
8M4N
[M2
NQ2
αδLT (Q2) + I1(Q2)minus IA(Q2)
] (40)
Note that d2(Q2) and its first two derivatives with respect to Q2 vanish at Q2 = 0 The
considerations in Eqs (28) and (29) have no effect on d2(Q2) since the Born contribution
from IA(Q2) and I1(Q2) cancel out Therefore d2(Q2) is a pure polarizability
In Fig 8 upper panel we show our NLO BχPT prediction and other results for d2(Q2)
While MAID [69] and BχPT describe the experimental data for the neutron [22] very well
the HB limit [50 51] is showing a fast growth with Q2 This illustrates the importance
of keeping the relativistic result Note also that even though the πN -loop contribution
is dominant both gC and the form factor in gM are essential to obtain a curvature that
reproduces the data cf Fig 9 upper panel For the proton there are to our knowledge
no experimental results to compare with However the agreement between the NLO BχPT
prediction and the MAID prediction at low energies is reasonable
F γ0(Q2) mdash fifth-order generalized forward spin polarizability
It is interesting to compare the generalized fifth-order forward spin polarizability sum
rule
γ0(Q2) =1
2π2
int infinν0
dν
radic1 +
Q2
ν2
σTT (νQ2)
ν5(41)
=64αM4
N
Q10
int x0
0
dx x4
[g1(xQ2)minus 4M2
Nx2
Q2g2(xQ2)
]
to the sum rule integrals for IA(Q2) and γ0(Q2) since they differ merely by their energy
weighting of σTT (νQ2) and a constant prefactor cf Eqs (19) (26) and (41) From IA(Q2)
to γ0(Q2) to γ0(Q2) the energy suppression is increasing by a factor of νminus2 respectively
Therefore the description of γ0(Q2) should be easiest in a low-energy effective-field theory
such as χPT whereas γ0(Q2) and IA(Q2) receive larger contributions from higher energies
In Fig 8 lower panel we show our LO and NLO BχPT predictions for γ0(Q2) One
can see that the πN -loop contribution is positive (in accordance to what we see for the cross
section σTT see Fig 10) The Delta shifts it substantially especially in the low Q2 region
24
000 005 010 015 020 025 030
-0004
-0002
0000
0002
0004
0006
0008
0010
Q2GeV
2D
d2
proton
000 005 010 015 020 025 030
-0005
0000
0005
0010
0015
Q2GeV
2D
d2
neutron
000 005 010 015 020 025 030
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
6D
proton
000 005 010 015 020 025 030
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
6D
neutron
FIG 9 Contributions of the different orders to the chiral predictions of d2(Q2) upper panel and
γ0(Q2) lower panel for the proton (left) and neutron (right) Red solid line πN -loop contribu-
tion green dot-dashed line ∆-exchange contribution orange dotted line π∆-loop contribution
blue long-dashed line total result purple dot-dot-dashed line total result without gC contribution
black short-dashed line total result without gM dipole
bringing it into a better agreement with data In general the BχPT curves start above the
empirical data points at the real-photon point and then decrease asymptotically to zero
above Q2 gt 01 GeV2 On the other hand the MAID prediction reproduces the empirical
data at the real-photon point then decreases to negative values until about Q2 gt 006 GeV2
from where it also starts to asymptotically approach zero Consequently our NLO BχPT
prediction of γ0(Q2) is consistently above the MAID prediction This is very different to
what we saw for IA(Q2) in Fig 4 upper panel where the MAID prediction at the real-
photon point is above the experimental value While the agreement of our predictions with
the empirical data is in general quite good for all moments of σTT (νQ2) one should point
out that both for γ0n(Q2) and γ0p(Q2) we overestimate the data at low Q2 For IA(Q2) such
observation cannot be made because ∆IA(0) = 0 and thus IA(0) is given by the empirical
Pauli form factor only From IA(Q2) γ0(Q2) and γ0(Q2) the latter has the smallest however
25
non-negligible dependence on gC and the dipole in gM cf Fig 9 lower panel
The πN -loop ∆-exchange and π∆-loop contributions to the NLO BχPT prediction of
the static fifth-order forward spin polarizability amount to in units of 10minus4 fm6
γ0p = 112(30) asymp 208minus 096minus 001 (42a)
γ0n = 195(30) asymp 292minus 096minus 001 (42b)
while the slope is composed as follows in units of 10minus4 fm8
dγ0p(Q2)
dQ2
∣∣∣∣Q2=0
= minus084(10) asymp minus100 + 016 + 000 (43a)
dγ0n(Q2)
dQ2
∣∣∣∣Q2=0
= minus142(15) asymp minus158 + 016 + 000 (43b)
Note that the HB prediction of the static polarizability γ0p (423 at O(p3) and 365 at O(ε3)
[78 85]) is almost one order of magnitude larger than the empirical value and therefore not
shown in Fig 8
G Summary
Our results are summarized in Table II where we give the contributions of the different
orders to the chiral predictions of the polarizabilities and their slopes at the real-photon
point A quantitative comparison of our predictions for the static spin polarizabilities to
the work of Bernard et al [31] and different empirical evaluations is shown in Table III We
can see that the inclusion of the Delta turns out to be very important for all moments of
the helicity-difference cross section To describe the Q2 behavior of the polarizabilities the
magnetic coupling of the N rarr ∆ transition should be modified by a dipole form factor as
has been observed previously in the description of electroproduction data [33] This dipole
form factor effectively takes account of vector-meson exchanges The Coulomb-quadrupole
N rarr ∆ transition despite its subleading order is important in the description of some
moments of spin structure functions This is contrary to what we saw for the moments of
unpolarized structure functions [30] where the Coulomb coupling had a negligible effect
The π∆ loops are mainly relevant for the generalized GDH integrals
26
TABLE II The NLO BχPT predictions for the forward VVCS polarizabilities and their slopes at
Q2 = 0 The contributions of the πN loops the ∆ exchange and the π∆ loops are shown together
with the combined total result Note that IA(0) = I1(0) = d2(0) = 0 and (d2)prime = 0
πN loops ∆ exchange π∆ loops Total
γ0 p 201 minus284 minus010minus093(92)
(10minus4 fm4) n 298 003(92)
δLT p 150 minus016 minus002132(15)
(10minus4 fm4) n 235 218(23)
γ0 p 208 minus096 minus001112(30)
(10minus4 fm6) n 292 195(30)
(γ0)prime p minus033011 001
minus022(4)
(10minus4 fm6) n minus073 minus061(7)
(δLT )prime p minus080 minus004 minus001minus085(8)
(10minus4 fm6) n minus119 minus124(12)
(γ0)prime p minus100016 000
minus084(10)
(10minus4 fm8) n minus158 minus142(15)
(∆IA)prime p 238 minus1121 025minus858(343)
(GeVminus2) n 141 minus955(343)
(∆I1)prime p 034 minus053 058039(4)
(GeVminus2) n minus107 minus101(10)
TABLE III Our NLO BχPT predictions for the spin polarizabilities at Q2 = 0 compared with
the BχPT+∆ predictions from Bernard et al [31] and the available empirical information Where
the reference is not given the empirical number is provided by the MAID analysis [55 69] with
unspecified uncertainty
Proton Neutron
This work BχPT+∆ Empirical This work BχPT+∆ Empirical
γ0 minus093(92) minus174(40) minus100(8)(12) [19] 003(92) minus077(40) minus0005
(10minus4 fm4) minus090(8)(11) [78] [MAID]
minus0929(105) [59]
δLT 132(15) 240(1) 134 218(23) 238(3) 203
(10minus4 fm4) [MAID] [MAID]
γ0 112(30) 060(7)(7) [78] 195(30) 123
(10minus4 fm6) 0484(82) [59] [MAID]
IV CONCLUSIONS
We have presented a complete NLO calculation of the polarized non-Born VVCS ampli-
tudes in covariant BχPT with pion nucleon and ∆(1232) fields The dispersion relations
between the VVCS amplitudes and the tree-level photoabsorption cross sections served as a
27
cross-check of these calculations
The obtained moments of the proton and neutron spin structure functions related to gen-
eralized polarizabilities and GDH-type integrals agree well with the available experimental
data The description of their Q2 evolution is improved compared to the previous χPT pre-
dictions In particular the NLO BχPT predictions obtained here give a better description
of the empirical data (eg from the Jefferson Laboratory ldquoSpin Physics Programrdquo) than
the HB [50 51] and IR [58] calculations
The demonstrated predictive power of the χPT framework amplitudes makes it well
suited for extending the χPT evaluation of the TPE effect in the hyperfine structure of
(muonic-)hydrogen [15ndash17] to next-to-leading order
ACKNOWLEDGEMENTS
We thank Lothar Tiator and Marc Vanderhaeghen for helpful discussions This work
is supported by the Deutsche Forschungsgemeinschaft (DFG) through the Collaborative
Research Center [The Low-Energy Frontier of the Standard Model (SFB 1044)] JMA ac-
knowledges support from the Community of Madrid through the ldquoPrograma de atraccion de
talento investigador 2017 (Modalidad 1)rdquo and the Spanish MECD grants FPA2016-77313-P
FH gratefully acknowledges financial support from the Swiss National Science Foundation
Appendix A Tensor decompositions of the VVCS amplitudes
In this appendix we review the decomposition of the forward VVCS process into tensor struc-
tures and scalar amplitudes In particular we consider the connection between the covariant and
the semi-relativistic decomposition in the lab frame that is defined in terms of the conventional
transverse longitudinal transverse-transverse and transverse-longitudinal amplitudes
As explained in Sec II A the process of forward VVCS off the nucleon can be described in
terms of four explicitly covariant amplitudes S1 2 and T1 2 [3]
T (νQ2) =
(minusgmicroν +
qmicroqν
q2
)T1(νQ2) +
1
M2N
(pmicro minus p middot q
q2qmicro)(
pν minus p middot qq2
qν)T2(νQ2) (A1)
minus 1
MNγmicroναqα S1(νQ2)minus 1
M2N
(γmicroνq2 + qmicroγναqα minus qνγmicroαqα
)S2(νQ2)
εprimelowastmicro εν
where εmicro (εprimelowastmicro ) are the incoming (outgoing) photon polarization vectors ν is the photon lab-frame
energy and Q2 is the photon virtuality Alternatively the decomposition in the laboratory frame
28
(which in the forward case coincides with the Breit frame) is parametrized in terms of the nucleon
Pauli matrices ~σ and the four scalar functions fL fT gTT and gLT
T (νQ2) = ε0 εprimelowast0 fL(νQ2) + (~ε primelowast middot ~ε ) fT (νQ2) + i~σ middot (~ε primelowast times ~ε ) gTT (νQ2) (A2)
minusi~σ middot [(ε0~εprimelowast minus ~ε ε primelowast0 )times q] gLT (νQ2)
Here ~q and q = ~q|~q | are the photon three-momentum in the lab system and its unit vector The
modified polarization vector components are given by
ε0 =
[ε0 minus
ν
|~q | (~ε middot q )
] |~q |Q ~ε = ~εminus q (~ε middot q ) (A3)
where ε = (ε0~ε ) is the usual incoming photon polarization vector and εprimelowast the outgoing polarization
vector The LEX of the lab frame amplitudes [Eq (10)] can serve in particular as the definition
of the generalized polarizabilities The lab frame amplitudes are also conveniently used for the
definition of the response functions see the example of the scalar amplitude gLT (νQ2) and the
corresponding response function σLT (νQ2) below in App B
Appendix B Photoabsorption cross sections
In the forward kinematics the spin-dependent VVCS amplitudes and the spin polarizabilities
can be described in terms of the polarized structure functions g1(xQ2) and g2(xQ2) or equiv-
alently the helicity-difference cross section σTT (νQ2) and the longitudinal-transverse response
function σLT (νQ2) with the help of dispersion relations (5) and the optical theorem (3) In this
way the photoabsorption cross sections measured in electroproduction processes form the basis
for most empirical evaluations shown throughout Sec III In the following we present the BχPT
predictions for the tree-level cross sections of πN - π∆- and ∆-production through photoabsorption
on the nucleon cf Figs 8 9 and 10 in Ref [30] In Secs B 1 and B 2 we will discuss the leading
πN -production channel and the ∆-production channel respectively We used these cross sections
to verify the polarizability predictions obtained otherwise from the calculated non-Born VVCS
amplitudes Due to the bad high-energy behavior of the π∆-production cross sections in BχPT
cf Fig 10 the dispersion relations in Eq (5) require further subtractions for a reconstruction of
the π∆-loop contribution to the spin-dependent VVCS amplitudes Therefore not all polarizabil-
ities could be verified but only those appearing as higher-order terms in the LEX of the VVCS
amplitudes such as γ0 [16]
29
proton neutron
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Neutron
02 04 06 08 100
50
100
150
200
250
[GeV]
T [b]
02 04 06 08 10
-50
0
50
100
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Neutron
02 04 06 08 100
50
100
150
200
250
[GeV]
T [b]
02 04 06 08 10
-50
0
50
100
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02040608100
50
100
150
200
250
300
[GeV]
T[b]
0204060810
-50
0
50
[GeV]
TT[b]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
0204060810
00
02
04
06
[GeV]
LTQ
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
0204060810
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
FIG 10 Photoabsorption cross sections for πN (red) and π∆ production (orange) with Q2 = 0
(solid) and Q2 = 01 GeV2 (dashed for πN and dotted for π∆ channel)
1 πN-production channel
In order to extract the response function σLT (νQ2) we have developed a method similar to
the one used to calculate σTT (νQ2) see for example Ref [86] For σLT (νQ2) however the
calculation is more complicated because one has to take into account that the associated Compton
process involves a spin-flip of the nucleon as illustrated in Fig 11 When calculating the cross
section the product of the incoming nucleon spinors has to reflect this flip
The forward VVCS amplitude related to σLT (νQ2) mdash and δLT (Q2) mdash is gLT (νQ2) It can
be extracted from Eq (A2) if one takes the modified polarization vector components in Eq (A3)
with ε = εL and εprimelowast = εlowastplusmn as input where εL = 1Q(|~q | 0 0 q0) and εplusmn = ∓ 1radic
2(0 1plusmni 0) are the
standard longitudinal and transverse polarization vectors respectively For εL and εlowastplusmn only the
choice of helicities hprime = plusmn12 and h = ∓12 gives a non-zero contribution and one obtains
χdaggerhprime T (νQ2)χh = χdaggerhprimeminusi~σ middot [(ε0~εprimelowast minus ~ε ε primelowast0 )times q] gLT (νQ2)χh =
radic2 gLT (νQ2) (B1)
where χh and χdaggerhprime are two-component Pauli spinors with opposite helicities or here spins
Let us now consider the related photoabsorption process and in particular the tree-level γlowastN rarr
30
s = 12
sprime = minus12
λ = 0 λprime = 1
rprime rprime
s=12sprime=minus1
2
λ=0λprime=1
rprimerprime
s=12
sprime=minus12
λ=0λprime=1
rprimerprime hacute=frac12h=-frac12
λacute=1λ=0
FIG 11 Relation between the forward Compton process and the photoabsorption process given
by the optical theorem In particular we show the longitudinal-transverse contribution The
double-line arrows represent the spin of the external particles while the dot represents the scalar
(longitudinal) polarization of the incoming photon Inside the blob the intermediate states are
represented eg nucleons with spins rprime (which are averaged in the calculation of the cross section)
and pions
πN channel see diagrams in Fig 8 of Ref [30] We define the πN -production amplitude as
T = uhB (PB)sumi
Ai(s t)Γi uhA(PA) (B2)
with the Dirac structures
Γ1 = γ5 (B3a)
Γ2 =1
2
[qA ε
]γ5 (B3b)
where uhA(PA) and udaggerhB (PB) are the Dirac spinors and PA and PB are the four-momenta of
the incoming and outgoing nucleons respectively When calculating the photoabsorption cross
section related to the VVCS amplitude in Eq (B1) the nucleon spin flip should be implemented
by uhprime(PA) in T dagger and uh(PA) in T together with the appropriate transverse and longitudinal
photon polarization vectors εlowastplusmn and εL
However if one wants to use the properties of the Dirac matrices it is more useful to construct
an operator to produce this spin flip in the external nucleons of Fig 11 This is accomplished by
introducing the projector ΓLT equiv 12radic
2(γ1 + iγ2)γ5 which also takes into account the extra factor
radic2 in Eq (B1) We checked that with this projector one correctly extracts δLT by comparing the
HB limit of our result to the HB result of Ref [50] where the authors calculate this polarizability
from the Compton amplitude directly With all those ingredients the longitudinal-transverse cross
section is calculated in the following way
σLT (νQ2) =1
64π2 s
|~pf |cm
|~pi|cm
int 1
minus1dcos θ
sumij
AiAdaggerjXij (B4)
31
with
Xij = Tr[(PB +MN )Γi(PA +MN )ΓLTγ0Γdaggerjγ
0] (B5)
where θ is the scattering angle in the center-of-mass (cm) frame and |~pi|cm (|~pf |cm) is the three-
momentum of an incoming (outgoing) particle in the cm frame An explicit calculation of the
matrix Xij leads to
X = MNQ
0 2(PB minus PA) middot εLminusradic
2|~qf |cm sin θ (sminus u)
(B6)
where |~qi|cm (|~qf |cm) is the relative three-momentum of the incoming (outgoing) particles in the
cm frame Here s t and u are the usual Mandelstam variables For the different γlowastN rarr πN
channels we obtain the following amplitudes Ai where we introduce qA as the four-momentum of
the incoming photon and qB as the four-momentum of the outgoing pion
bull γlowastprarr π0p
A1 =e gAMN
fπ
[2PA middot ε+ qA middot ε
sminusM2N
+2PB middot εminus qA middot ε
uminusM2N
] (B7a)
A2 =e gAMN
fπ
[1
sminusM2N
+1
uminusM2N
] (B7b)
bull γlowastprarr π+n
A1 =
radic2 e gAMN
fπ
[2PA middot ε+ qA middot ε
sminusM2N
+2(PA minus PB) middot ε+ qA middot ε
tminusm2π
] (B8a)
A2 =
radic2 e gAMN
fπ(sminusM2N )
(B8b)
bull γlowastnrarr π0n
A1 = 0 (B9a)
A2 = 0 (B9b)
bull γlowastnrarr πminusp
A1 =
radic2 e gAMN
fπ
[2PB middot εminus qA middot ε
uminusM2N
minus 2(PA minus PB) middot ε+ qA middot εtminusm2
π
] (B10a)
A2 =
radic2 e gAMN
fπ(uminusM2N )
(B10b)
The analytical expressions shown above were checked with the amplitudes given in Ref [87] An-
alytical expressions for the tree-level γlowastN rarr πN channel of the σLT (νQ2) and σTT (νQ2) cross
32
sections are given below (proton channels π+n and π0p neutron channel πminusp) We checked that
they reproduce the known results in the real-photon limit [45 86] To shorten the final expressions
for the cross sections which are considerably longer for finite Q2 than in the real-photon limit we
define the following dimensionless kinematic variables
αγ = (ENi )cmradics =
s+M2N +Q2
2s (B11)
απ = (ENf )cmradics =
s+M2N minusm2
π
2s (B12)
βγ = Eγcmradics =
sminusM2N minusQ2
2s (B13)
βπ = Eπcmradics =
sminusM2N +m2
π
2s (B14)
λγ = |~qi|cmradics =
radic(sminusM2
N minusQ2)2 + 4sQ2
2s (B15)
λπ = |~qf |cmradics =
radic(sminusM2
N +m2π)2 minus 4sm2
π
2s (B16)
Here (ENi )cm and (ENf )cm are the energies of the incoming and outgoing nucleon Eγcm is the energy
of the incoming photon Eπcm is the energy of the outgoing pion all in the cm frame
σ(π+n)TT = minus e2g2
AM2N
64πf2πs
3(sminusM2N )2λ4
γ
4sλπλγ
[(M2
N minus s)(M2N minusQ2 minus s)(Q2 + 2sβγβπ)
+ 2s(m2πQ
2 minus (M2N minus s)
(M2N + s(minus1 + 2βγβπ)
))λ2γ
]+ (M2
N minus s)(M2N minusQ2 minus s)
times (Q2 + 2sβγβπ minus 2sλπλγ)(Q2 + 2s(βγβπ + λπλγ)
)times log
(Q2 + 2sβγβπ minus 2sλγλπQ2 + 2sβγβπ + 2sλγλπ
) (B17)
33
σ(π0p)TT =
e2g2AM
2Nλπ
64πf2πs(sminusM2
N )2λγ
1
s2λπλ3γ
[(M2
N minus s)(M2N minusQ2 minus s)
(3M2
N + s(2βγβπ minus 3))(M2N
+ s(2βγβπ minus 1))
+ 2s
((2m2
πQ2 + (M2
N minus s)2)(M2
N minus s)minus 2(minusm2
πQ2 + (sminusM2
N )2)
times sβγβπ + 4s2(sminusM2N )β2
γβ2π
)λ2γ
]arctan
(2sλπλγ
M2N + s(2βγβπ minus 1)
)+ 2
[minus 2m2
πQ2
+ (M2N minus s)
(2(sminusM2
N ) + 4sβγβπ +(sminusM2
N +Q2)(3M2N minus 3s+ 2sβγβπ)
sλ2γ
+m2πQ
2(sminusM2N )
(M2N minus s+ 2sβγβπ)(M2
N + s(minus1 + 2βγβπ minus 2λπλγ))
+m2πQ
2(sminusM2N )
(M2N minus s+ 2sβγβπ)(M2
N + s(minus1 + 2βγβπ + 2λπλγ))
)
+1
λπλγ
(βγβπ
(minusm2
πQ2 + (M2
N minus s)(M2N minus s+ 2sβγβπ)
)minus (M2
N minus s)(M2N minusQ2 minus s)λ2
π
)log
(M2N + s(minus1 + 2βγβπ + 2λπλγ)
M2N + s(minus1 + 2βγβπ minus 2λπλγ)
)] (B18)
σ(πminusp)TT =
e2g2AM
2N
64πf2πs
3λ4γ(M2
N + s(2βπβγ minus 1))2 minus 4λ2πs
2λ2γ)
4sλπλγ
((minusM2
N +Q2 + s)(M2N
+ s(2βπβγ minus 1))
2 minus 2sλ2γ
(2λ2
πs(minusM2N +Q2 + s) +m2
πQ2))
+((M2N + s(2βπβγ minus 1)
)2
minus 4λ2πs
2λ2γ
)((M2N + s(2βπβγ minus 1)
)(M2N minus 2Q2 minus s(2βπβγ + 1)
)+ 4λ2
πs2λ2
γ
)times log
(minusM2
N minus 2βπsβγ minus 2λπsλγ + s)minus(M2N + s(2βπβγ minus 2λπλγ minus 1)
)(M2N + s(minus1
+ 2βπβγ + 2λπλγ))[((
M2N + s(2βπβγ minus 1)
)(M2N minus 2Q2 minus s(2βπβγ + 1)
)+ 4λ2
πs2λ2
γ
)times log
(minusM2
N minus 2βπsβγ + 2λπsλγ + s)
+((Q2 + 2βπsβγ)2 minus 4λ2
πs2λ2
γ
)times log
(Q2 + 2βπsβγ minus 2λπsλγQ2 + 2sβπβγ + 2λπsλγ
)]+ 4sλ2
γ
((M2
N + s(2βπβγ minus 1))2 minus 4λ2πs
2λ2γ
)times (M2
N minus s) arctanh
(2λπsλγ
M2N + 2βπsβγ minus s
) (B19)
34
σ(π+n)LT =
e2g2AM
3Nλπ
32πf2πQs
3(sminusM2N )2λ4
γ
2sλγ
[(M2
N minus s)(Q2 + 2sβ2γ)(Q2 + 2sβγβπ)
minus 4s
((M2
N minus s)(Q2 minus 2s(απ minus 1)βγ
)+Q2sβγβπ
)λ2γ + 8s3(minus1 + απ)λ4
γ
]minus M2
N minus sλπ
[(Q2 + 2sβ2
γ)(Q2 + 2sβγβπ)2 + 4s2(2(απ minus 1)βγ(Q2 + 2sβγβπ)
minusQ2λ2π
)λ2γ + 8s3(απ minus 1)2λ4
γ
]arctan
(2sλπλγ
Q2 + 2sβγβπ
) (B20)
σ(π0p)LT =
e2g2AM
3Nλπ
16πf2πQs(sminusM2
N )2λγ
1
minus2s(M2N + s(minus1 + 2βγβπ))2λ2
γ + 8s3λ2πλ
4γ
[minus 3M8
N (Q2
+ 2sβ2γ) + 2M4
Ns2
(minus (Q2 + 2sβ2
γ)(minus3 + 2βγβπ) +(Q2(minus3 + 2βγβπ + 6λ2
π) + 2sβγ
times (minus3 + 12απ + 2βγβπ minus 12απβγβπ + 4βγλ2π))λ2γ minus 4s(α2
π minus 1)λ4π
)+ 2M2
Ns3
[minus (Q2
+ 2sβ2γ)(minus1 + 2βγβπ)
(6 + βγβπ(minus9 + 2βγβπ)
)+
(Q2(3minus 12λ2
π + 4βγβπ(minus1 + 2βγβπ
+ λ2π))minus 2sβγ
(minus 3 + 4απ(3 + 2απ(3 + 2βγβπ(minus3 + βγβπ)) + 4βγ(βπ + (2minus βγβπ)
times λ2π)))λ2γ + 8s
((απ minus 1)(1 + απ minus 2βγβπ) + 2απβγλ
2π
)λ4γ
)+ s4
((Q2 + 2sβ2
γ)
times (1minus 2βγβπ)2(2βγβπ minus 3) + 2(Q2(minus1 + 6λ2
π + 2βγβπ((1minus 2βγβπ)2 minus 2λ2π)) + 2sβγ
times (minus1 + 2βγβπ + 4(minus1 + βγβπ)(minusαπ + 2απβγβπ minus βγλ2π)))λ2γ minus 8
(s(απ minus 1)(απ
+ (1minus 2βγβπ)2) + 2βγ(2sαπ +Q2βγ)λ2π
)λ4γ + 32s(απ minus 1)λ2
πλ6γ
)+ 2M6
Ns
(Q2(6
minus 7βγβπ + λ2γ) + 2sβγ(βγ(6minus 7βγβπ) + (1minus 4απ)λ2
γ)
)]+sminusM2
N
4s2λπλ3γ
[(Q2 + 2sβ2
γ)
times(3M2
N + s(2βγβπ minus 3))(M2N + s(2βγβπ minus 1)
)+ 2s
(minusM2
N (Q2 + 2sβγ minus 8sαπβγ)
+ s(2sβγ(1minus 4απ + 4απβγβπ) +Q2(1minus 2λ2π)))λ2γ + 8s3(απ minus 1)λ4
γ
]
times arctan
(2sλπλγ
M2N + s(2βγβπ minus 1)
) (B21)
35
σ(πminusp)LT =
e2g2AM
3N
64πf2πQs
3λ4γ(M2
N minusQ2 minus s)(M2N + s(2βπβγ + 2λπλγ minus 1))
times
1
M2N + s(2βπβγ minus 2λπλγ minus 1)
[4λπsλγ(M2
N minusQ2 minus s)(
2sλ2γ
(s((1minus 2λ2
π)Q2
+ 2(2απ minus 1)sβγ(2βπβγ minus 1))minusM2N (Q2 + 2(1minus 2απ)sβγ)
)+ (Q2 + 2sβ2
γ)(M2N + s(2βπβγ minus 1)
)2 + 8(απ minus 1)απs
3λ4γ
)]minus(M2N + s(2βπβγ + 2λπλγ minus 1)
)times[
log
(minusM2N minus 2βπsβγ minus 2λπsλγ + s
minusM2N minus 2βπsβγ + 2λπsλγ + s
)(minus (Q2 + 2sβ2
γ)(M2N + s(2βπβγ minus 1)
)(minusM2
N + 2Q2 + 2βπsβγ + s)minus 2sλ2γ
(minusM2
N (Q2 + 2sβγ) +Q4 +Q2s(4απβγ
minus 2βγ minus 2λ2π + 1) + 2s2βγ(4(απ minus 1)βπβγ + 1)
)minus 8(απ minus 1)2s3λ4
γ
)+ 2
(4s2λ2
γ
(2(απ minus 1)βγ(Q2 + 2βπsβγ)minus λ2
πQ2)
+ (Q2 + 2βπsβγ)2(Q2 + 2sβ2γ)
+ 8(απ minus 1)2s3λ4γ
)arctanh
(2λπsλγ
Q2 + 2βπsβγ
)] (B22)
2 ∆-production channel
The tree-level ∆-exchange diagram in Fig 2 of Ref [30] contributes to the non-Born part of
the VVCS amplitudes The contribution of the ∆ exchange to the VVCS amplitudes can be split
into [17]
S∆-exch1 (νQ2) = S∆-pole
1 (νQ2) + S∆-exch1 (νQ2) (B23a)
νS∆-exch2 (νQ2) = νS∆-pole
2 (νQ2) + νS2∆-exch
(νQ2) (B23b)
and similarly for the unpolarized VVCS amplitudes discussed in Ref [30] Here we introduced the
∆-pole contributions S∆-polei and the ∆-non-pole contributions S∆-exch
i The former amplitudes
feature a pole at the ∆(1232)-production threshold and thus are proportional to
1
[sminusM2∆][uminusM2
∆]=
1
4M2N
1
ν2∆ minus ν2
(B24)
36
They can be reconstructed from the dispersion relations in Eq (5) with the tree-level ∆-production
cross sections as input cf Fig 10 in Ref [30]
σTT (νQ2) =π2α
M2NM
2+|~q |
minus g2
MMN (M+ + ν)|~q |2 +g2E(∆minus ν)(Q2 minusMNν)2
MN(B25a)
+g2CQ
4s(∆minus ν)
MNM2∆
minus 4gMgE(Q2 minusMNν)|~q |2 minus 4gMgCQ2|~q |2
+2gEgCQ
2[minusMNM∆ |~q |2 + s(Q2 +∆ν)
]MNM∆
δ(ν minus ν∆)
σLT (νQ2) =Qπ2α
M2NM
2+|~q |
g2E(MNν minusQ2) [M∆(MN + ν)minus s]
MN(B25b)
+g2CQ
2[MNM∆|~q |2 minus s(Q2 +∆ν)
]MNM2
∆
+ gMgEM∆|~q |2 minusgMgC(Q2 minusMNν)|~q |2
M∆
+gEgC(ν minus∆)(M2
N |~q |2 minus 2Q2s)
MNM∆
δ(ν minus ν∆)
with ∆ = M∆ minusMN M+ = M∆ + MN and the Mandelstam variable s = M2N + 2MNν minus Q2
Analytical expressions for the spin structure functions g1(xQ2) and g2(xQ2) can be constructed
from Eq (3) with the flux factor K(νQ2) = |~q | =radicν2 +Q2
In the ∆-non-pole contributions to S1(νQ2) and νS2(νQ2) the pole in ν at the ∆(1232)-
production threshold has canceled out
S∆-exch1 (νQ2) =
πα
MNM2+
[g2MQ
2+ + g2
E
(∆2 minus 3Q2
)+
4g2CQ
4
M2∆
minus 8gMgEM∆ωminus (B26a)
minus2gMgCQ2(MN minus 4M∆)
M∆+
2gEgCQ2(3MN minus 2M∆)
M∆
]
νS2∆-exch
(νQ2) =2πα
MNM2+
[g2EM∆∆ωminus +
g2M MNQ
2+
2+g2C Q
2(Q2 minus∆2)
2M∆(B26b)
+gEgM M∆(M∆ω+ minus 4MNωminus)minus gEgC ∆(2Q2 +MNω+)
+gMgC Q2(4MN minus ω+)
]+S∆-exch
2 (νQ2)
ν
[M2
∆ ω2+
M2N
+ ν2
]
with Q+ =radic
(M∆ +MN )2 +Q2 and ωplusmn = (M2∆minusM2
N plusmnQ2)2M∆ and the non-pole contribution
to S2(νQ2)
S∆-exch2 (νQ2) = minus2παMNν
M∆M2+
[gM + gE
]gC (B27)
These amplitudes to the contrary are not described by the tree-level ∆-production cross sections
in the standard dispersive approach [17] This peculiarity has been previously missed e g in the
calculation of the ∆-exchange contribution to the hydrogen hyperfine splitting in Ref [88] The
importance of including the ∆-non-pole contribution is also evident when considering the BC sum
37
rule in Eq (14) The ∆-pole terms by themselves violate the BC sum rule but cancel exactly with
the ∆-non-pole terms
limνrarr0
νS∆-pole2 (νQ2) + lim
νrarr0νS2
∆-exch(νQ2) = 0 (B28)
Appendix C Polarizabilities at Q2 = 0
In this section we give analytical expressions for the static values and slopes at Q2 = 0 of
the spin polarizabilities and moments of polarized structure functions In particular we give the
HB expansion of the πN -loop contributions and the ∆-exchange contributions The complete
expressions also for the π∆-loop contributions can be found in the Supplemented material Recall
that IA(0) = I1(0) = d2(0) = 0 and dd2(Q2)dQ2
∣∣∣Q2=0
= 0
1 πN-loop contribution
Here we give analytical expressions for the πN -loop contributions to the proton and neutron
spin polarizabilities expanded in powers of micro = mπMN viz the HB expansion Note that we
choose to expand here to a high order in micro the strict HB expansion would only retain the leading
term in an analogous NLO calculation
bull Static polarizabilities (Q2 = 0)
γ0p =e2g2
A
96π3f2πm
2π
1minus 21πmicro
8minus(
59
2+ 26 logmicro
)micro2 +
1875πmicro3
64
+3
(3
2+ 26 logmicro
)micro4 +
(C1)
γ0n =e2g2
A
48π3f2πm
2π
1
2minus 9πmicro
16minus 2micro2 logmicro+
75πmicro3
128minus 3micro4
4+
(C2)
δLTp =e2g2
A
192π3f2πm
2π
1minus 9πmicro
8+
(13
2minus 2 logmicro
)micro2 minus 465πmicro3
64
minus(
47
2+ 42 logmicro
)micro4 +
(C3)
δLTn =e2g2
A
96π3f2πm
2π
1
2+
3πmicro
16+ (1 + 2 log micro)micro2 minus 105πmicro3
128+
5micro4
4+
(C4)
38
δLTn =e2g2
A
96π3f2πm
2π
1
2+
3πmicro
16+ (1 + 2 log micro)micro2 minus 105πmicro3
128+
5micro4
4+
(C5)
γ0p =e2g2
A
16π3f2πm
4π
4
45minus 3πmicro
16+
14micro2
5minus 1813πmicro3
384minus 192
5(1 + log micro)micro4
+80703πmicro5
2048+
(C6)
γ0n =e2g2
A
16π3f2πm
4π
4
45minus 5πmicro
48+
4micro2
5minus 245πmicro3
384minus 32micro4 logmicro
15
+1323πmicro5
2048+
(C7)
bull Slopes of polarizabilities at Q2 = 0
dγ0p(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
2minus 45πmicro
4+ 223micro2 minus 28515πmicro3
64
minus9
(1953
4+ 449 logmicro
)micro4 +
570255πmicro5
128+
(C8)
dγ0n(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
2minus 81πmicro
8+ 94micro2 minus 2535πmicro3
32
minus3 (1 + 90 logmicro)micro4 +84315πmicro5
1024+
(C9)
dδLTp(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
2880π3f2πm
4π
minus5
2minus 27πmicro
32+ 20micro2 minus 5865πmicro3
256
+3
(617
4+ 36 logmicro
)micro4 minus 2056845πmicro5
4096+
(C10)
dδLTn(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
minus5
4minus 81πmicro
64minus 11micro2 +
10005πmicro3
512
+15
8(11 + 48 logmicro)micro4 minus 267015πmicro5
8192+
(C11)
dIAp(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
96π2f2πmicro
2
1minus 15πmicro
4minus 1
2(115 + 88 logmicro)micro2 +
1839πmicro3
32
+5 (5 + 34 logmicro)micro4 +
(C12)
dIAn(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
48π2f2πmicro
2
1
2minus 11πmicro
8minus 1
4(1 + 20 logmicro)micro2 +
99πmicro3
64
minus25micro4
12+
(C13)
dI1p(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
96π2f2πmicro
3π
8+ 2 (4 + 3 logmicro)microminus 537πmicro2
64
minus1
2(15 + 56 logmicro)micro3 +
(C14)
39
dI1n(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
48π2f2πmicro
minus π
16+
1
4(3 + 4 logmicro)microminus 57πmicro2
128+
2micro3
3+
(C15)
γ0p(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2g2
A
16π3f2πm
6π
1
105minus 23πmicro
256+
377micro2
210minus 15551πmicro3
6144+
3371micro4
105
minus1640457πmicro5
32768+
(C16)
γ0n(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2g2
A
16π3f2πm
6π
1
105minus 153πmicro
1792+
69micro2
70minus 4615πmicro3
6144+
172micro4
35
minus120897πmicro5
32768+
(C17)
2 ∆-exchange contribution
Here we give analytical expressions for the tree-level ∆-exchange contributions to the nucleon
spin polarizabilities and their slopes at Q2 = 0 Note that the ∆-exchange contributes equally to
proton and neutron polarizabilities Recall that for the magnetic γlowastN∆ coupling we introduced a
dipole form factor to mimic vector-meson dominance gM rarr gM(1 +Q2Λ2)2
bull Static polarizabilities (Q2 = 0)
γ0 = minus e2
4πM2+
(g2M
∆2+
g2E
M2+
minus 4gMgEM+∆
) (C18)
δLT =e2M∆
4πM3+
(g2E
MNM++gMgE∆MN
minus gEgCM2
∆
) (C19)
γ0 =e2M2
N
π∆2M4+
(minusg
2M
∆2+
g2E
M2+
+4gMgE∆M+
) (C20)
bull Slopes of polarizabilities at Q2 = 0
dγ0(Q2)
dQ2
∣∣∣∣∣Q2=0
= minus e2
πM2+∆
(g2M
∆
[1
4∆2minus 1
∆M++
1
2M2+
]minus 1
Λ2
g2M
∆+
g2E
2M2+
[1
2∆minus 3
M+
]minusgMgE
M+
[1
∆2minus 5
∆M++
1
M2+
]+
1
Λ2
2gMgEM+
+2gMgC∆M2
+
minus gEgCM3
+
) (C21)
40
d δLT (Q2)
dQ2
∣∣∣∣∣Q2=0
=e2M∆∆
4πMNM2+
(g2E
∆2M2+
[1
∆minus 4
M+
]minus g2
C
∆M2∆M
2+
+gMgE∆2M+
[1
∆2
minus 3
∆M++
1
M2+
]minus 2
Λ2
gMgE∆2M+
+gMgC∆M2
∆
[1
2∆2minus 2
∆M++
1
2M2+
]minus gEgC
2M2∆M
2+
[7
∆+
1
M+
]) (C22)
dIA(Q2)
dQ2
∣∣∣∣∣Q2=0
= minusM2N
M2+
(g2M
2∆2+
g2E
MNM+minus 2gMgE
∆M+minus gEgCM∆M+
) (C23)
dI1(Q2)
dQ2
∣∣∣∣∣Q2=0
= minusM∆M2N
2M3+
(g2E
MNM∆minus gMgE∆MN
minus gEgCM2
∆
) (C24)
γ0(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2M2
N
π∆3M6+
(g2M
∆
[3M2
∆ + 2M∆MN minus 9M2N
∆2+
4M2+
Λ2
]
minus4gMgE
[5M2
∆ minus 9M2N
M+∆2+
2M+
Λ2
]+
g2E
∆M+[7M∆ minus 9MN ]
minus8gMgC∆
+4gEgCM+
) (C25)
[1] D Drechsel B Pasquini and M Vanderhaeghen Dispersion relations in real and virtual
Compton scattering Phys Rept 378 99 (2003) hep-ph0212124
[2] S E Kuhn J-P Chen and E Leader Spin structure of the nucleon mdash status and recent
results Prog Part Nucl Phys 63 1 (2009) arXiv08123535 [hep-ph]
[3] F Hagelstein R Miskimen and V Pascalutsa Nucleon polarizabilities from Compton scat-
tering to hydrogen atom Prog Part Nucl Phys 88 29 (2016) arXiv151203765 [nucl-th]
[4] B Pasquini and M Vanderhaeghen Dispersion theory in electromagnetic interactions Ann
Rev Nucl Part Sci 68 75 (2018) arXiv180510482 [hep-ph]
[5] S Gerasimov A Sum rule for magnetic moments and the damping of the nucleon magnetic
moment in nuclei Sov J Nucl Phys 2 430 (1966)
[6] S Drell and A C Hearn Exact sum rule for nucleon magnetic moments Phys Rev Lett
16 908 (1966)
[7] J S Schwinger Source theory viewpoints in deep inelastic scattering Proc Natl Acad Sci
USA 72 1 (1975)
[8] J S Schwinger Source theory viewpoints in deep inelastic scattering Electromagnetic In-
teractions and Field Theory Proceedings 14 Internationale Universitatswochen Schladming
Austria February 24-March 7 1975 Acta Phys Austriaca Suppl 14 471 (1975)
41
[9] J Schwinger Source theory discussion of deep inelastic scattering with polarized particles
Proc Natl Acad Sci USA 72 1559 (1975)
[10] A M Baldin Polarizability of nucleons Nucl Phys 18 310 (1960)
[11] M Gell-Mann M L Goldberger and W E Thirring Use of causality conditions in quantum
theory Phys Rev 95 1612 (1954)
[12] A Pineda Leading chiral logarithms to the hyperfine splitting of the hydrogen and muonic
hydrogen Phys Rev C 67 025201 (2003)
[13] C Peset and A Pineda The two-photon exchange contribution to muonic hydrogen from
chiral perturbation theory Nucl Phys B 887 69 (2014) arXiv14064524 [hep-ph]
[14] C Peset and A Pineda Model-independent determination of the two-photon exchange contri-
bution to hyperfine splitting in muonic hydrogen JHEP 04 060 arXiv161205206 [nucl-th]
[15] F Hagelstein and V Pascalutsa Proton structure in the hyperfine splitting of muonic hydro-
gen PoS CD15 077 (2016) arXiv151104301 [nucl-th]
[16] F Hagelstein Exciting Nucleons in Compton Scattering and Hydrogen-Like Atoms PhD
thesis Mainz U Inst Kernphys (2017) arXiv171000874 [nucl-th]
[17] F Hagelstein ∆(1232)-Resonance in the hydrogen spectrum Proceedings 11th International
Workshop on the Physics of Excited Nucleons (NSTAR 2017) Columbia SC USA August
20-23 2017 Few Body Syst 59 93 (2018) arXiv180109790 [nucl-th]
[18] Y Prok et al (CLAS) Moments of the spin structure functions gp1 and gd1 for 005 lt Q2 lt 30
GeV2 Phys Lett B 672 12 (2009) arXiv08022232 [nucl-ex]
[19] H Dutz et al (GDH) First measurement of the Gerasimov-Drell-Hearn sum rule for 1H from
07 GeV to 18 GeV at ELSA Phys Rev Lett 91 192001 (2003)
[20] M Amarian et al (Jefferson Lab E94010) Measurement of the generalized forward spin po-
larizabilities of the neutron Phys Rev Lett 93 152301 (2004) arXivnucl-ex0406005
[21] M Amarian et al The Q2 evolution of the generalized Gerasimov-Drell-Hearn integral for
the neutron using a 3He target Phys Rev Lett 89 242301 (2002) arXivnucl-ex0205020
[22] M Amarian et al (Jefferson Lab E94-010)Q2 evolution of the neutron spin structure moments
using a 3He target Phys Rev Lett 92 022301 (2004) arXivhep-ex0310003
[23] A Deur et al Experimental determination of the evolution of the Bjorken integral at low Q2
Phys Rev Lett 93 212001 (2004) arXivhep-ex0407007
[24] K Slifer Low Q2 measurement of gp2 and the δLT spin polarizability Spin structure at long dis-
tance Proceedings Workshop Newport News USA March 12ndash13 2009 nucl-ex09064775
(2009) AIP Conf Proc 1155 10106313203293 (2009) arXiv09064775 [nucl-ex]
[25] P Solvignon et al (E01-012) Moments of the neutron g2 structure function at intermediate
42
Q2 Phys Rev C 92 015208 (2015) arXiv13044497 [nucl-ex]
[26] A Deur Experimental studies at low Q2 of the spin structure of the nucleon at Jefferson Lab
in 9th International Workshop on Chiral Dynamics (CD18) Durham NC USA September
17-21 2018 (2019) arXiv190305661 [nucl-ex]
[27] V Sulkosky et al (Jefferson Lab E97-110) Measurement of the 3He spin-structure functions
and of neutron (3He) spin-dependent sum rules at 0035 le Q2 le 024 GeV2 Phys Lett B
805 135428 (2020) arXiv190805709 [nucl-ex]
[28] K Adhikari et al (CLAS) Measurement of the Q2 Dependence of the Deuteron Spin Structure
Function g1 and its Moments at Low Q2 with CLAS Phys Rev Lett 120 062501 (2018)
arXiv171101974 [nucl-ex]
[29] V Lensky F Hagelstein A Hiller Blin and V Pascalutsa Comment on rdquoMeasurement of
the Q2 Dependence of the Deuteron Spin Structure Function g1 and its Moments at Low Q2
with CLASrdquo (2018) arXiv180603219 [nucl-th]
[30] J M Alarcn F Hagelstein V Lensky and V Pascalutsa Forward doubly-virtual Compton
scattering off the nucleon in chiral perturbation theory at NLO the subtraction function and
moments of unpolarized structure functions (2020) arXiv200509518 [hep-ph]
[31] V Bernard E Epelbaum H Krebs and U-G Meiszligner New insights into the spin structure
of the nucleon Phys Rev D 87 054032 (2013) arXiv12092523 [hep-ph]
[32] V Lensky J M Alarcon and V Pascalutsa Moments of nucleon structure functions at
next-to-leading order in baryon chiral perturbation theory Phys Rev C 90 055202 (2014)
arXiv14072574 [hep-ph]
[33] V Pascalutsa and M Vanderhaeghen Chiral effective-field theory in the ∆(1232) region
I Pion electroproduction on the nucleon Phys Rev D 73 034003 (2006) arXivhep-
ph0512244
[34] V Pascalutsa and M Vanderhaeghen Electromagnetic nucleon-to-Delta transition in chiral
effective field theory Phys Rev Lett 95 232001 (2005) arXivhep-ph0508060
[35] T R Hemmert B R Holstein and J Kambor Systematic 1M expansion for spin 32
particles in baryon chiral perturbation theory Phys Lett B 395 89 (1997) arXivhep-
ph9606456
[36] V Pascalutsa and D R Phillips Effective theory of the ∆(1232) in Compton scattering off
the nucleon Phys Rev C 67 055202 (2003) arXivnucl-th0212024
[37] V Pascalutsa M Vanderhaeghen and S N Yang Electromagnetic excitation of the ∆(1232)-
resonance Phys Rept 437 125 (2007) arXivhep-ph0609004
[38] K A Olive et al (Particle Data Group) Review of Particle Physics Chin Phys C 38 090001
43
(2014)
[39] F E Low Scattering of light of very low frequency by systems of spin 12 Phys Rev 96
1428 (1954)
[40] M Gell-Mann and M L Goldberger Scattering of low-energy photons by particles of spin
12 Phys Rev 96 1433 (1954)
[41] A M Harun ar-Rashid A simple derivation of Schwingerrsquos sum rule for spin dependent
structure functions Nuovo Cim A 33 447 (1976)
[42] F Hagelstein and V Pascalutsa Dissecting the hadronic contributions to (g minus 2)micro by
Schwingerrsquos sum rule Phys Rev Lett 120 072002 (2018) arXiv171004571 [hep-ph]
[43] V Lensky V Pascalutsa M Vanderhaeghen and C Kao Spin-dependent sum rules con-
necting real and virtual Compton scattering verified Phys Rev D 95 074001 (2017)
arXiv170101947 [hep-ph]
[44] H Burkhardt and W N Cottingham Sum rules for forward virtual Compton scattering
Annals Phys 56 453 (1970)
[45] V Lensky and V Pascalutsa Predictive powers of chiral perturbation theory in Compton
scattering off protons Eur Phys J C 65 195 (2010) arXiv09070451 [hep-ph]
[46] H Grieszlighammer J McGovern D Phillips and G Feldman Using effective field theory to
analyse low-energy Compton scattering data from protons and light nuclei Prog Part Nucl
Phys 67 841 (2012) arXiv12036834 [nucl-th]
[47] H W Grieszlighammer J A McGovern and D R Phillips Nucleon polarisabilities at and
beyond physical pion masses Eur Phys J A 52 139 (2016) arXiv151101952 [nucl-th]
[48] E Epelbaum H Krebs and U-G Meiszligner Improved chiral nucleon-nucleon potential up to
next-to-next-to-next-to-leading order Eur Phys J A 51 53 (2015) arXiv14120142 [nucl-
th]
[49] E Epelbaum H Krebs and U-G Meiszligner Precision nucleon-nucleon potential at fifth order
in the chiral expansion Phys Rev Lett 115 122301 (2015) arXiv14124623 [nucl-th]
[50] C W Kao T Spitzenberg and M Vanderhaeghen Burkhardt-Cottingham sum rule and
forward spin polarizabilities in heavy baryon chiral perturbation theory Phys Rev D 67
016001 (2003) arXivhep-ph0209241
[51] C-W Kao D Drechsel S Kamalov and M Vanderhaeghen Higher moments of nucleon
spin structure functions in heavy baryon chiral perturbation theory and in a resonance model
Phys Rev D 69 056004 (2004) arXivhep-ph0312102
[52] T Becher and H Leutwyler Baryon chiral perturbation theory in manifestly Lorentz invariant
form Eur Phys J C 9 643 (1999) arXivhep-ph9901384
44
[53] J Gasser M E Sainio and A Svarc Nucleons with chiral loops Nucl Phys B 307 779
(1988)
[54] L S Geng J Martin Camalich L Alvarez-Ruso and M J Vicente Vacas Leading SU(3)-
breaking corrections to the baryon magnetic moments in chiral perturbation theory Phys
Rev Lett 101 222002 (2008) arXiv08051419 [hep-ph]
[55] D Drechsel S S Kamalov and L Tiator The GDH sum rule and related integrals Phys
Rev D 63 114010 (2001) arXivhep-ph0008306
[56] D Drechsel O Hanstein S S Kamalov and L Tiator A unitary isobar model for pion
photo- and electroproduction on the proton up to 1 GeV Nucl Phys A 645 145 (1999)
arXivnucl-th9807001
[57] L Tiator private communication (2020)
[58] V Bernard T R Hemmert and U-G Meiszligner Spin structure of the nucleon at low energies
Phys Rev D 67 076008 (2003) arXivhep-ph0212033
[59] O Gryniuk F Hagelstein and V Pascalutsa Evaluation of the forward Compton scattering
off protons II Spin-dependent amplitude and observables Phys Rev D 94 034043 (2016)
arXiv160400789 [nucl-th]
[60] R Zielinski The g2p Experiment A Measurement of the Protonrsquos Spin Structure Functions
PhD thesis New Hampshire U (2010) arXiv170808297 [nucl-ex]
[61] N Guler et al (CLAS) Precise determination of the deuteron spin structure at low to mod-
erate Q2 with CLAS and extraction of the neutron contribution Phys Rev C 92 055201
(2015) arXiv150507877 [nucl-ex]
[62] C E Carlson V Nazaryan and K Griffioen Proton structure corrections to electronic
and muonic hydrogen hyperfine splitting Phys Rev A 78 022517 (2008) arXiv08052603
[physicsatom-ph]
[63] V Bernard N Kaiser and U-G Meiszligner Chiral dynamics in nucleons and nuclei Int J
Mod Phys E 4 193 (1995) arXivhep-ph9501384
[64] T R Hemmert B R Holstein and J Kambor ∆(1232) and the polarizabilities of the nucleon
Phys Rev D 55 5598 (1997) arXivhep-ph9612374
[65] V Pascalutsa and R Timmermans Field theory of nucleon to higher-spin baryon transitions
Phys Rev C 60 042201 (1999) arXivnucl-th9905065
[66] V Pascalutsa Quantization of an interacting spin-32 field and the Delta isobar Phys Rev
D 58 096002 (1998) arXivhep-ph9802288
[67] H Krebs Double Virtual Compton Scattering and SpinStructure of the Nucleon PoS
CD2018 031 (2019)
45
[68] N Kochelev and Y Oh Axial anomaly and the δLT puzzle Phys Rev D 016012 (2012)
arXiv11034892 [hep-ph]
[69] D Drechsel S Kamalov and L Tiator Unitary isobar model ndash MAID2007 Eur Phys J A
69 (2007) available at httpsmaidkphuni-mainzde arXiv07100306 [nucl-th]
[70] P J Mohr B N Taylor and D B Newell CODATA recommended values of the fundamental
physical constants 2010 Rev Mod Phys 84 1527 (2012)
[71] J Ahrens et al (GDH A2) First measurement of the Gerasimov-Drell-Hearn integral for 1H
from 200 to 800 MeV Phys Rev Lett 87 022003 (2001) arXivhep-ex0105089 [hep-ex]
[72] K Helbing (GDH) Experimental verification of the GDH sum rule at ELSA and MAMI Nucl
Phys Proc Suppl 105 113 (2002)
[73] R Bradford A Bodek H S Budd and J Arrington A New parameterization of the nucleon
elastic form-factors NuInt05 proceedings of the 4th International Workshop on Neutrino-
Nucleus Interactions in the Few-GeV Region Okayama Japan 26-29 September 2005 Nucl
Phys Proc Suppl 159 127 (2006) arXivhep-ex0602017
[74] J D Bjorken Applications of the chiral U(6)timesU(6) algebra of current densities Phys Rev
148 1467 (1966)
[75] J D Bjorken Inelastic scattering of polarized leptons from polarized nucleons Phys Rev D
1 1376 (1970)
[76] V Bernard T R Hemmert and U-G Meiszligner Novel analysis of chiral loop effects in
the generalized Gerasimov-Drell-Hearn sum rule Phys Lett B 545 105 (2002) arXivhep-
ph0203167
[77] A Deur et al Experimental study of isovector spin sum rules Phys Rev D 78 032001
(2008) arXiv08023198 [nucl-ex]
[78] B Pasquini P Pedroni and D Drechsel Higher order forward spin polarizability Phys Lett
B 687 160 (2010) arXiv10014230 [hep-ph]
[79] R Jaffe g2ndashThe nucleonrsquos other spin-dependent structure function Comments Nucl Part
Phys 19 239 (1990)
[80] E V Shuryak and A Vainshtein Theory of power corrections to deep inelastic scattering in
quantum chromodynamics (II) Qminus4 effects polarized target Nucl Phys B 201 141 (1982)
[81] S Wandzura and F Wilczek Sum rules for spin dependent electroproduction Test of rela-
tivistic constituent quarks Phys Lett B 72 195 (1977)
[82] B W Filippone and X-D Ji The spin structure of the nucleon Adv Nucl Phys 26 1
(2001) arXivhep-ph0101224
[83] M Burkardt The g2 structure function Proceedings Workshop on Spin structure at long
46
distance Newport News USA March 12-13 2009 AIP Conf Proc 1155 26 (2009)
arXiv09054079 [hep-ph]
[84] A Deur S J Brodsky and G F de Teramond The spin structure of the nucleon Rept
Prog Phys 82 076201 (2019) arXiv180705250 [hep-ph]
[85] B R Holstein D Drechsel B Pasquini and M Vanderhaeghen Higher order polarizabilities
of the proton Phys Rev C 61 034316 (2000) arXivhep-ph9910427
[86] B R Holstein V Pascalutsa and M Vanderhaeghen Sum rules for magnetic moments and
polarizabilities in QED and chiral effective-field theory Phys Rev D 72 094014 (2005)
arXivhep-ph0507016
[87] B Pasquini D Drechsel and L Tiator Invariant amplitudes for pion electroproduction Eur
Phys J A 34 387 (2007) arXiv07122327 [hep-ph]
[88] A J Buchmann Non-spherical proton shape and hydrogen hyperfine splitting Proceedings
International Workshop on Precision Physics of Simple Atomic Systems (PSAS 2008) Wind-
sor Ontario Canada July 21-26 2008 Can J Phys 87 773 (2009) arXiv09104747
[physicsatom-ph]
47
CONTENTS
I Introduction 3
II Calculation of unpolarized VVCS at NLO 5
A VVCS amplitudes and relations to structure functions 5
B Low-energy expansions and relations to polarizabilities 6
C Details on χPT calculation and uncertainty estimate 8
III Results and discussion 11
A γ0(Q2) mdash generalized forward spin polarizability 11
B δLT (Q2) mdash longitudinal-transverse polarizability 15
C IA(Q2) mdash a generalized GDH integral 16
D Γ1(Q2) and I1(Q
2) mdash the first moment of the structure function g1(xQ2) 20
E d2(Q2) mdash a measure of color polarizability 22
F γ0(Q2) mdash fifth-order generalized forward spin polarizability 24
G Summary 26
IV Conclusions 27
Acknowledgements 28
A Tensor decompositions of the VVCS amplitudes 28
B Photoabsorption cross sections 29
1 πN -production channel 30
2 ∆-production channel 36
C Polarizabilities at Q2 = 0 38
1 πN -loop contribution 38
2 ∆-exchange contribution 40
References 41
2
I INTRODUCTION
In the studies of nucleon structure the forward doubly-virtual Compton scattering
(VVCS) amplitude Fig 1 is playing a central role (see eg Refs [1ndash4] for reviews)
Traditionally its general properties such as unitarity analyticity and crossing are used to
establish various useful sum rules for the nucleon magnetic moment (GerasimovndashDrellndashHearn
[5 6] and Schwinger sum rules [7ndash9]) and polarizabilities (eg Baldin [10] and Gell-Mannndash
GoldbergerndashThirring sum rules [11]) More recently the interest in nucleon VVCS has been
renewed in connection with precision atomic spectroscopy where this amplitude enters in
the form of two-photon exchange (TPE) corrections As the TPE corrections in atomic do-
main are dominated by low-energy VVCS it makes sense to calculate them systematically
using chiral perturbation theory (χPT) which is a low-energy effective-field theory of the
Standard Model
In this paper we present a state-of-the-art χPT calculation of the polarized nucleon
VVCS relevant to TPE corrections to hyperfine structure of hydrogen and muonic hydro-
gen This will extend the leading-order χPT evaluation of the TPE effects in hyperfine
splittings [12ndash17] Here we however do not discuss the TPE evaluation but rather focus on
testing the χPT framework against the available empirical information on low-energy spin
structure of the nucleon
It is especially interesting to confront the χPT predictions with the recent measurements
coming from the ongoing ldquoSpin Physics Programrdquo at Jefferson Laboratory [18ndash27] with
the exception of a recent measurement of the deuteron spin polarizability by the CLAS
Collaboration [28] which does not treat correctly complications due to deuteron spin [29]
Our present calculation is extending Ref [30] to the case of polarized VVCS We therefore
use a manifestly-covariant extension of SU(2) χPT to the baryon sector called Baryon χPT
(BχPT) First attempts to calculate VVCS in the straightforward BχPT framework (rather
than the heavy-baryon expansion or the ldquoinfrared regularizationrdquo) were done by Bernard
et al [31] and our group [32] The two works obtained somewhat different results most
notably for the proton spin polarizability δLT Here we improve on [32] in three important
aspects appreciable at finite Q2 1) inclusion of the Coulomb-quadrupole (C2) N rarr ∆ tran-
sition [33 34] 2) correct inclusion of the elastic form-factor contributions to the integrals
Γ1(Q2) I1(Q2) and IA(Q2) (see Sections III C and III D for details) and 3) cancellations
between different orders in the chiral prediction and their effect on the convergence of the
effective-field-theory calculation and thus the error estimate These improvements how-
ever do not bring us closer to the results of [31] and the source of discrepancies most likely
3
FIG 1 The forward Compton scattering or VVCS in case of virtual photons q2 = minusQ2
lies in the different counting and renormalization of the π∆-loop contributions Bernard
et al [31] use the so-called small-scale expansion [35] for the ∆(1232) contributions whereas
we are using the δ-counting scheme [36] (see also Ref [37 Sec 4] for review)
This paper is organized as follows In Sec II A we introduce the polarized VVCS am-
plitudes and their relations to spin structure functions In Sec II B we introduce the spin
polarizabilities appearing in the low-energy expansion (LEX) of the polarized VVCS am-
plitudes In Sec II C we briefly describe our χPT calculation focusing mainly on the
uncertainty estimate In Sec III we show our predictions for the proton and neutron po-
larizabilities as well as some interesting moments of their structure functions In Sec III G
we summarize the results obtained herein comment on the improvements done with respect
to previous calculations and give an outlook to future applications In App B we discuss
the structure functions in particular we define the longitudinal-transverse response func-
tion discuss the ∆-pole contribution and give analytical results for the tree-level πN - and
∆-production channels of the photoabsorption cross sections In App C we give analytical
expressions for the πN -loop and ∆-exchange contributions to the static values and slopes of
the polarizabilities and moments of structure functions The complete expressions also for
the π∆-loop contributions can be found in the Supplemented material
TABLE I Parameters (fundamental and low-energy constants) [38] at the order they appear first
The πN∆ coupling constant hA is fit to the experimental Delta width and the γlowastN∆ coupling
constants gM gE and gC are taken from the pion photoproduction study of Ref [33]
O(p2) α = 1(13704) MN = Mp = 93827 MeV
O(p3) gA = 1270 fπ = 9221 MeV mπ = 13957 MeV
O(p4∆) M∆ = 1232 MeV hA equiv 2gπN∆ = 285 gM = 297 gE = minus10 gC = minus26
4
II CALCULATION OF UNPOLARIZED VVCS AT NLO
A VVCS amplitudes and relations to structure functions
The polarized part of forward VVCS can be described in terms of two independent
Lorentz-covariant and gauge-invariant tensor structures and two scalar amplitudes [3]
T microν(p q) = minus 1
MN
γmicroναqα S1(νQ2)minus 1
M2N
(γmicroνq2 + qmicroγναqα minus qνγmicroαqα
)S2(νQ2) (1)
Here q and p are the photon and nucleon four-momenta (cf Fig 1) ν is the photon
lab-frame energy Q2 = minusq2 is the photon virtuality and γmicroν = 12
[γmicro γν ] and γmicroνα =12
(γmicroγνγα minus γαγνγmicro) are the usual Dirac matrices Alternatively one can use the following
laboratory-frame amplitudes
gTT (νQ2) =ν
MN
[S1(νQ2)minus Q2
MN νS2(νQ2)
] (2a)
gLT (νQ2) =Q
MN
[S1(νQ2) +
ν
MN
S2(νQ2)] (2b)
introduced in Eq (A2) The optical theorem relates the absorptive parts of the forward
VVCS amplitudes to the nucleon structure functions or the cross sections of virtual pho-
toabsorption
ImS1(νQ2) =4π2α
νg1(xQ2) =
MNνK(νQ2)
ν2 +Q2
[Q
νσLT (νQ2) + σTT (νQ2)
] (3a)
ImS2(νQ2) =4π2αMN
ν2g2(xQ2) =
M2NK(νQ2)
ν2 +Q2
[ν
QσLT (νQ2)minus σTT (νQ2)
] (3b)
with α the fine structure constant andK(νQ2) the photon flux factor Note that the photon
flux factor in the optical theorem and the cross sections measured in electroproduction
processes is a matter of convention and has to be chosen in both quantities consistently In
what follows we use Gilmanrsquos flux factor
K(νQ2) equiv |~q | =radicν2 +Q2 (4)
The helicity-difference photoabsorption cross section is defined as σTT = 12 (σ12 minus σ32)
where the photons are transversely polarized and the subscripts on the right-hand side
indicate the total helicities of the γlowastN states The cross section σLT corresponds to a
simultaneous helicity change of the photon and nucleon spin flip such that the total helicity
5
is conserved A detailed derivation of the connection between this response function and
the photoabsorption cross sections can be found in App B The forward VVCS amplitudes
satisfy dispersion relations derived from the general principles of analyticity and causality1
S1(νQ2) =16παMN
Q2
int 1
0
dxg1(xQ2)
1minus x2(ννel)2 minus i0+(5a)
=2MN
π
int infinνel
dν primeν prime 2[QνprimeσLT + σTT
](ν prime Q2)radic
ν prime 2 +Q2(ν prime 2 minus ν2 minus i0+)
νS2(νQ2) =16παM2
N
Q2
int 1
0
dxg2(xQ2)
1minus x2(ννel)2 minus i0+(5b)
=2M2
N
π
int infinνel
dν primeν prime 2[νprime
QσLT minus σTT
](ν prime Q2)radic
ν prime 2 +Q2(ν prime 2 minus ν2 minus i0+)
with νel = Q22MN the elastic threshold
B Low-energy expansions and relations to polarizabilities
The VVCS amplitudes naturally split into nucleon-pole (Spolei ) and non-pole (Snonpole
i )
parts or Born (SBorni ) and non-Born (Si) parts
Si = Spolei + Snonpole
i = SBorni + Si (6)
The Born amplitudes are given uniquely in terms of the nucleon form factors [1]
SBorn1 (νQ2) =
2πα
M
Q2GM(Q2)F1(Q2)
ν2el minus ν2
minus F 22 (Q2)
(7a)
SBorn2 (νQ2) = minus 2παν
ν2el minus ν2
GM(Q2)F2(Q2) (7b)
The same is true for the nucleon-pole amplitudes which are related to the Born amplitudes
in the following way
Spole1 (νQ2) = SBorn
1 (νQ2) +2πα
MN
F 22 (Q2) (8a)
Spole2 (νQ2) = SBorn
2 (νQ2) (8b)
1 The dispersion relation for νS2(νQ2) is used because it is pole-free in the limit Q2 rarr 0 and then ν rarr 0
cf Eq (7b)
6
Here we used the elastic Dirac and Pauli form factors F1(Q2) and F2(Q2) related to the
electric and magnetic Sachs form factors GE(Q2) and GM(Q2) through
F1(Q2) =GE(Q2) + τGM(Q2)
1 + τ (9a)
F2(Q2) =GM(Q2)minusGE(Q2)
1 + τ (9b)
where τ = Q24M2N
A low-energy expansion (LEX) of Eq (5) in combination with the unitarity relations
given in Eq (3) establishes various sum rules relating the nucleon properties (electromag-
netic moments polarizabilities) to experimentally observable response functions [1 3] The
leading terms in the LEX of the RCS amplitudes are determined uniquely by charge mass
and anomalous magnetic moment as the global properties of the nucleon These lowest-
order terms represent the celebrated low-energy theorem (LET) of Low Gell-Mann and
Goldberger [39 40] The polarizabilities related to the internal structure of the nucleon
enter the LEX at higher orders They make up the non-Born amplitudes and can be related
to moments of inelastic structure functions
The process of VVCS can be realized experimentally in electron-nucleon scattering where
a virtual photon is exchanged between the electron and the nucleon This virtual photon
acts as a probe whose resolution depends on its virtuality Q2 In this way one can access
the so-called generalized polarizabilities which extend the notion of polarizabilities to the
case of response to finite momentum transfer The generalized forward spin polarizability
γ0(Q2) and the longitudinal-transverse polarizability δLT (Q2) are most naturally defined via
the LEX of the non-Born part of the lab-frame VVCS amplitudes [1]
1
4πgnonpoleTT (νQ2) =
2α
M2N
IA(Q2) ν + γ0(Q2)ν3 + γ0(Q2)ν5 + (10a)
1
4πgnonpoleLT (νQ2) =
2α
M2N
I3(Q2)Q+ δLT (Q2)ν2Q+ (10b)
Their definitions in terms of integrals over structure functions are postponed to Eqs (19)
and (22) Here we only give the definition of the moment I3(Q2)
I3(Q2) =M2
N
4π2α
int infinν0
dνK(νQ2)
νQσLT (νQ2) =
2M2N
Q2
int x0
0
dx[g1(xQ2) + g2(xQ2)
] (11)
which is related to the Schwinger sum rule in the real photon limit [7ndash9 41 42] The LEX
of the non-pole part of the covariant VVCS amplitudes can be described entirely in terms
7
of moments of inelastic spin structure functions (up to O(ν4) [43])
1
4πSnonpole
1 (νQ2) =2α
MN
I1(Q2) +
2α
MNQ2
[IA(Q2)minus I1(Q2)
]+MNδLT (Q2)
ν2 (12a)
1
4πνSnonpole
2 (νQ2) = 2αI2(Q2) +2α
Q2
[I1(Q2)minus IA(Q2)
]ν2 (12b)
I1(Q2) and IA(Q2) are generalizations of the famous GerasimovndashDrellndashHearn (GDH) sum
rule [5 6] from RCS to the case of virtual photons [1] Their definitions are given in Eqs (26)
and (32) I2(Q2) is the well-known Burkhardt-Cottingham (BC) sum rule [44]
I2(Q2) equiv 2M2N
Q2
int x0
0
dx g2(x Q2) =1
4F2(Q2)GM(Q2) (13)
which can be written as a ldquosuperconvergence sum rulerdquo
Q2
16παM2N
limνrarr0
νS2(νQ2) =
int 1
0
dx g2 (x Q2) = 0 (14)
The latter is valid for any value of Q2 provided that the integral converges for x rarr 0
Combining Eq (5) with the above LEXs of the VVCS amplitudes we can relate IA(Q2)
I1(Q2) γ0(Q2) and δLT (Q2) to moments of inelastic structure functions see Sec III It is
important to note that only γ0(Q2) and δLT (Q2) are generalized polarizabilities The relation
of the inelastic moments IA(Q2) and I1(Q2) to polarizabilities will be discussed in details in
Secs III C and III D The difference between S1(νQ2) and Snonpole1 (νQ2) cf Eq (8a) will
be important in this context
C Details on χPT calculation and uncertainty estimate
In this work we calculated the NLO prediction of BχPT for the polarized non-Born
VVCS amplitudes This includes the leading pion-nucleon (πN) loops see Ref [32 Fig 1]
as well as the subleading tree-level Delta-exchange (∆-exchange) see Ref [30 Fig 2] and
the pion-Delta (π∆) loops see Ref [32 Fig 2] In the δ-power-counting scheme [36] the
LO and NLO non-Born VVCS amplitudes and polarizabilities are of O(p3) and O(p4∆)
respectively2 The LECs are listed in Table I sorted by the order at which they appear in
our calculation At the given orders there are no ldquonewrdquo LECs that would need to be fitted
from Compton processes For more details on the BχPT formalism we refer to Ref [30]
2 In the full Compton amplitude there is a lower order contribution coming from the Born terms leading
to a shift in nomenclature by one order the LO contribution referred to as the NLO contribution etc
see eg Ref [45]
8
where power counting predictive orders (Sec III A) and the renormalization procedure
(Sec III B) are discussed
A few remarks are in order for the inclusion of the ∆(1232) and the tree-level ∆-exchange
contribution In contrast to Ref [32] we include the Coulomb-quadrupole (C2) N rarr ∆
transition described by the LEC gC The relevant Lagrangian describing the non-minimal
γlowastN∆ coupling [33 34] (note that in these references the overall sign of gC is inconsistent
between the Lagrangian and Feynman rules) reads
L(2)∆ =
3e
2MNM+
N T3
igM F
microν partmicro∆ν minus gEγ5Fmicroν partmicro∆ν (15)
+igCM∆
γ5γα(partα∆ν minus partν∆α)partmicroF
microν
+ Hc
with M+ = MN + M∆ and the dual of the electromagnetic field strength tensor F microν =12εmicroνρλFρλ Even though the Coulomb coupling is subleading compared with the electric
and magnetic couplings (gE and gM) its relatively large magnitude cf Table I makes it
numerically important for instance in γ0(Q2) Furthermore we study the effect of modifying
the magnetic coupling using a dipole form factor
gM rarrgM[
1 + (QΛ)2 ]2 (16)
where Λ2 = 071 GeV2 The inclusion of this Q2 dependence mimics the form expected from
vector-meson dominance It is motivated by observing the importance of this form factor
for the correct description of the electroproduction data [33]
To estimate the uncertainties of our NLO predictions we define
δ(Q2) =
radic(∆
MN
)2
+
(Q2
2MN∆
)2
(17)
such that the neglected next-to-next-to-leading order terms are expected to be of relative
size δ2 [33] The uncertainties in the values of the parameters in Table I have a much
smaller impact compared to the truncation uncertainty and can be neglected Unfortunately
∆IA(Q2) γ0(Q2) and γ0(Q2) ie the sum rules involving the cross section σTT (νQ2)
as well as the polarizability ∆I1(Q2) turn out to be numerically small Their smallness
suggests a cancellation of leading orders (which can indeed be confirmed by looking at
separate contributions as shown below) Therefore an error of δ2(Q2)P (Q2) where P (Q2)
is a generalized polarizability might underestimate the theoretical uncertainty for some of
the NLO predictions To avoid this we estimate the uncertainty of our NLO polarizability
9
predictions by
σP (Q2) =
Max[δ4(0)P (0)2 δ4(0)P LO(0)2 δ2(0)PNLO(0)2
]+Max
[δ4(Q2)
[P (Q2)minus P (0)
]2 δ4(Q2)
[P LO(Q2)minus P LO(0)
]2
δ2(Q2)[PNLO(Q2)minus PNLO(0)
]2]12
(18)
where P LO(Q2) is the πN -loop contribution PNLO(Q2) are the ∆-exchange and π∆-loop
contributions and P (Q2) = P LO(Q2) + PNLO(Q2) This error prescription is similar to the
one used in eg Refs [46ndash49] Here since we are interested in the generalized polarizabil-
ities we added in quadrature the error due to the static piece P (0) and the Q2-dependent
remainder P (Q2) minus P (0) Note that the static values of IA(0) and I1(0) are given by the
elastic Pauli form factor which is not part of our BχPT prediction and is considered to be
exact
Note that our result for the spin polarizabilities (and the unpolarized moments [30]) are
NLO predictions only at low momentum transfers Q mπ At larger values of Q amp ∆ they
become incomplete LO predictions Indeed in this regime the ∆ propagators do not carry
additional suppression compared to the nucleon propagators and the π∆ loops are promoted
to LO In general we only expect a rather small contribution from omitted π∆ loops to the
Q2 dependence of the polarizabilities since π∆ loops show rather weak dependence on Q2
compared with the ∆ exchange or πN loops Nevertheless this issue has to be reflected in
the error estimate Since the static polarizabilities P (0) are not affected it is natural to
separate the error on the Q2-dependent remainder P (Q2) minus P (0) as done in Eq (18) To
accommodate for the potential loss of precision above Q amp ∆ we define the relative error
δ(Q2) as growing with increasing Q2 see Eq (17)
Upon expanding our results in powers of the inverse nucleon mass Mminus1N we are able to
reproduce existing results of heavy-baryon χPT (HBχPT) at LO We however do not see
a rationale to drop the higher-order Mminus1N terms when they are not negligible (ie when
their actual size exceeds by far the natural estimate for the size of higher-order terms)
Comparing our BχPT predictions to HBχPT we will also see a deficiency of HBχPT in
the description of the Q2 behaviour of the polarizabilities Note that the O(p4) HBχPT
results from Ref [50 51] which we use here for comparison do not include the ∆ These
references studied the leading effect of the latter in the HBχPT framework using the small-
scale expansion [35] observing no qualitative improvement in the HBχPT description of the
empirical data [50 51] when including it We therefore choose to use the O(p4) results as
the representative HBχPT curves
10
Another approach used in the literature to calculate the polarizabilities in χPT is the
infrared regularization (IR) scheme introduced in Ref [52] This covariant approach tries to
solve the power counting violation observed in Ref [53] by dropping the regular parts of the
loop integrals that contain the power-counting-breaking terms However this subtraction
scheme modifies the analytic structure of the loop contributions and may lead to unexpected
problems as was shown in Ref [54] As we will see in the next section the IR approach
also fails to describe the Q2 behaviour of the polarizabilities
III RESULTS AND DISCUSSION
We now present the NLO BχPT predictions for the nucleon polarizabilities and selected
moments of the nucleon spin structure functions Our results are obtained from the calcu-
lated non-Born VVCS amplitudes and the LEXs in Eqs (10) and (12) For a cross-check
we used the photoabsorption cross sections described in App B In addition to the full NLO
results we also analyse the individual contributions from the πN loops the ∆ exchange
and the π∆ loops
A γ0(Q2) mdash generalized forward spin polarizability
The forward spin polarizability
γ0(Q2) =1
2π2
int infinν0
dν
radic1 +
Q2
ν2
σTT (νQ2)
ν3(19)
=16αM2
N
Q6
int x0
0
dx x2
[g1(xQ2)minus 4M2
Nx2
Q2g2(xQ2)
]
provides information about the spin-dependent response of the nucleon to transversal pho-
ton probes The RCS analogue of the above generalized forward spin polarizability sum
rule is sometimes referred to as the Gell-Mann Goldberger and Thirring (GGT) sum rule
[11] At Q2 = 0 the forward spin polarizability is expressed through the lowest-order spin
polarizabilities of RCS as γ0 = minus(γE1E1 + γM1M1 + γE1M2 + γM1E2) The forward spin po-
larizability of the proton is relevant for an accurate knowledge of the (muonic-)hydrogen
hyperfine splitting as it controls the leading proton-polarizability correction [16 62]
The πN -loop ∆-exchange and π∆-loop contributions to the NLO BχPT prediction of
11
000 005 010 015 020 025 030
-4
-3
-2
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
4D
proton
000 005 010 015 020 025 030
-4
-3
-2
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
4D
neutron
000 005 010 015 020 025 030
0
1
2
Q2 GeV2D
∆LT
10-
4fm
4D
proton
000 005 010 015 020 025 030
0
1
2
3
Q2 GeV2D
∆LT
10-
4fm
4D
neutron
FIG 2 Upper panel Generalized forward spin polarizability for the proton (left) and neutron
(right) as function of Q2 The result of this work the NLO BχPT prediction is shown by the
blue solid line and the blue band The red line represents the LO BχPT result The purple short-
dashed line is the O(p4) HB result from Ref [50] note that the corresponding proton curve is
outside of the plotted range The black dotted line is the MAID model prediction [55ndash57] which
is taken from Ref [1] (proton) and Ref [20] (neutron) The pink band is the IR+∆ result from
Ref [58] and the gray band is the BχPT+∆ result from Ref [31] Empirical extractions for the
proton Ref [18] (blue dots) Ref [59] (purple square) and Ref [60] (orange triangle uncertainties
added in quadrature) and neutron Ref [20] (blue diamonds) and Ref [61] (green dots statistical
and systematic uncertainties added in quadrature) Lower Panel Longitudinal-transverse spin
polarizability for the proton (left) and neutron (right) The orange dot-dashed and purple short-
dashed lines are the O(p3) and O(p4) HB results from Ref [50] The pink band is the IR result
from Ref [58] and the gray band is the covariant BχPT+∆ result from Ref [31] The black dotted
line is the MAID model prediction [55ndash57] note that for the proton we use the updated estimate
from Ref [1] obtained using the π η ππ channels
12
the static forward spin polarizability amount to in units of 10minus4 fm4
γ0p = minus093(92) asymp 201minus 284minus 010 (20a)
γ0n = 003(92) asymp 298minus 284minus 010 (20b)
while the slope is composed as follows in units of 10minus4 fm6
dγ0p(Q2)
dQ2
∣∣∣∣Q2=0
= minus022(4) asymp minus033 + 011 + 001 (21a)
dγ0n(Q2)
dQ2
∣∣∣∣Q2=0
= minus061(7) asymp minus073 + 011 + 001 (21b)
Figure 2 upper panel shows our NLO prediction as well as the LO πN loops compared
to different experimental and theoretical results For the proton we have one determination
at the real-photon point by the GDH collaboration [19] γ0p = minus100(8)(12)times10minus4 fm4 and
further Jefferson Laboratory data [18 60] at very low Q2 For the neutron only data at finite
Q2 are available [20 61] The experimental data for the proton are fairly well reproduced
in the whole Q2 range considered here while for the neutron the agreement improves with
increasing Q2 The HB limit of our πN -loop contribution reproduces the results published
in Refs [50 63] for arbitrary Q2 In addition our prediction is compared to the MAID
model [1 20] the IR+∆ calculation of Ref [58] and the BχPT+∆ result of Ref [31]
The πN -production channel gives a positive contribution to the photoabsorption cross
section σTT (νQ2) at low Q2 cf Fig 10 Accordingly one observes that the πN loops give
a sizeable positive contribution to γ0(Q2) The Delta on the other hand has a very large
effect by cancelling the πN loops and bringing the result close to the empirical data From
Fig 3 upper panel one can see that it is the ∆ exchange which dominates while π∆
loops are negligible This was expected since the forward spin polarizability sum rule is an
integral over the helicity-difference cross section in which σ32 is governed by the Delta at
low energies (the relevant energy region for the sum rule)
To elucidate the difference between the present calculation and the one from Ref [31] we
note that the two calculations differ in the following important aspects Firstly Ref [31] uses
the small-scale counting [64] that considers ∆ and mπ as being of the same size ∆ sim mπ
In practice this results in a set of π∆-loop graphs which contains graphs with one or two
γ∆∆ couplings and hence two or three Delta propagators Such graphs are suppressed in
the δ-counting and thus omitted from our calculation while present in that of Ref [31]
Secondly the Lagrangians describing the interaction of the Delta are constructed differently
and assume slightly different values for the coupling constants In particular we employ
13
000 005 010 015 020 025 030
-3
-2
-1
0
1
2
Q2GeV
2D
Γ01
0-
4fm
4D
proton
000 005 010 015 020 025 030
-3
-2
-1
0
1
2
3
Q2GeV
2D
Γ01
0-
4fm
4D
neutron
000 005 010 015 020 025 030
00
05
10
15
Q2GeV
2D
∆LT1
0-
4fm
4D
proton
000 005 010 015 020 025 030
00
05
10
15
20
25
Q2GeV
2D
∆LT1
0-
4fm
4D
neutron
FIG 3 Contributions of the different orders to the chiral predictions of γ0(Q2) upper paneland δLT (Q2) lower panel for the proton (left) and neutron (right) Red solid line πN -loop
contribution green dot-dashed line ∆-exchange contribution orange dotted line π∆-loop con-
tribution blue long-dashed line total result purple dot-dot-dashed line total result without gCcontribution black short-dashed line total result without gM dipole
(where possible) the so-called ldquoconsistentrdquo couplings to the Delta field ie those couplings
that project out the spurious degree of freedom see Refs [37 65 66] The authors of
Ref [31] on the other hand use couplings where the consistency in this sense is not enforced
The effects of these differences are of higher order in the δ-counting expansion and their
contribution to the Q2 dependence of the considered polarizabilities is expected to be rather
small however the differences at Q2 = 0 could be noticeable [67]
Finally as mentioned in Sec II C the inclusion of the dipole form factor in the magnetic
coupling gM is expected to be important to generate the correct Q2 behaviour of the polar-
izabilities Comparing our predictions for the forward spin polarizability with and without
inclusion of the form factor see Fig 3 upper panel confirms this Without the dipole
our results for the proton and neutron are closer to the ones from Ref [31] where the form
factor is not included For the neutron our prediction without the dipole is able to describe
14
the experimental points at very low Q2 deviating from the data with increasing Q2 The
π∆-loop contribution does not modify the Q2 behavior of γ0(Q2) and only differs from
Ref [31] by a small global shift Note also the relatively large effect of gC which generates
a sign change for virtualities above sim 02 GeV2 see Fig 3 upper panel
B δLT (Q2) mdash longitudinal-transverse polarizability
The longitudinal-transverse spin polarizability
δLT (Q2) =1
2π2
int infinν0
dν
radic1 +
Q2
ν2
σLT (νQ2)
Qν2(22)
=16αM2
N
Q6
int x0
0
dx x2[g1(xQ2) + g2(xQ2)
]
contains information about the spin structure of the nucleon and is another important
input in the determination of the (muonic-)hydrogen hyperfine splitting [16 62] It is also
relevant in studies of higher-twist corrections to the structure function g2(xQ2) given by
the moment d2(Q2) [51] see Section III E The peculiarity of the response encoded in this
polarizability is that it involves a spin flip of the nucleon and a polarization change of the
photon see App B and Fig 11
It is expected that the Delta isobar gives only a small contribution to δLT (Q2) what makes
this polarizability a potentially clean test case for chiral calculations Consequently there are
relatively many different theoretical calculations of δLT (Q2) coming from different versions
of χPT with baryons (HB IR and covariant) Ref [50] found a systematic deviation of the
HB result for δLTn(Q2) from the MAID model prediction This disagreement was identified
by the authors of Ref [68] as a puzzle involving the neutron δLT polarizabilitymdashthe δLT
puzzle The IR calculation in Ref [58] also showed a deviation from the data and predicted
a rapid rise of δLT (Q2) with growing Q2 The problem is solved by keeping the relativistic
structure of the theory as the BχPT+∆ result of Ref [31] showed
As expected already the leading πN loops provide a reasonable agreement with the
experimental data cf Fig 2 lower panel Since the ∆-exchange contribution to δLT (Q2)
is small the effect of the gM form factor is negligible in this polarizability as is that of the gC
coupling cf Fig 3 lower panel In fact we predict both the ∆-exchange and the π∆-loop
contributions to be small and negative This is in agreement with the MAID model which
predicts a small and negative contribution of the P33 wave to δLT (Q2) However in the
calculation of Ref [31] which is different from the one presented here only in the way the
∆(1232) is included the contribution of this resonance to δLTp(Q2) is sizeable and positive
15
The authors of that work attributed this large contribution to diagrams where the photons
couple directly to the Delta inside a loop As mentioned in Sec III F the effect of such loop
diagrams does not change the Q2 behaviour of the polarizabilities On the other hand it
can produce a substantial shift of the δLT (Q2) as a whole A higher-order calculation should
resolve the discrepancy between the two covariant approaches however it will partially lose
the predictive power since the LECs appearing at higher orders will have to be fitted to
experimental data
The πN -loop ∆-exchange and π∆-loop contributions to the NLO BχPT prediction of
the static longitudinal-transverse polarizability are in units of 10minus4 fm4
δLTp = 132(15) asymp 150minus 016minus 002 (23a)
δLTn = 218(23) asymp 235minus 016minus 002 (23b)
while the slopes are in units of 10minus4 fm6
dδLTp(Q2)
dQ2
∣∣∣∣Q2=0
= minus085(8) asymp minus080minus 004minus 001 (24a)
dδLTn(Q2)
dQ2
∣∣∣∣Q2=0
= minus124(12) asymp minus119minus 004minus 001 (24b)
C IA(Q2) mdash a generalized GDH integral
The helicity-difference cross section σTT exhibits a faster fall-off in ν than its spin-averaged
counterpart σT This is due to a cancellation between the leading (constant) terms of σ12
and σ32 at large ν3 The resulting 1ν fall-off of the helicity-difference cross section allows
one to write an unsubtracted dispersion relation for the VVCS amplitude gnonpoleTT (νQ2) cf
Eq (10a) This is the origin of the GDH sum rule [5 6]
minus α
2M2N
κ2 =1
2π2
int infinν0
dνσTT (ν)
ν (25)
which establishes a relation to the anomalous magnetic moment κ It is experimentally
verified for the nucleon by MAMI (Mainz) and ELSA (Bonn) [71 72]
There are two extensions of the GDH sum rule to finite Q2 the generalized GDH integrals
3 Notice that a constant term in σTT at ν rarrinfin is forbidden by crossing symmetry
16
000 005 010 015 020 025 030
-30
-25
-20
-15
-10
-05
00
Q2 GeV2D
I Aproton
000 005 010 015 020 025 030
-25
-20
-15
-10
-05
00
Q2 GeV2D
I A
neutron
000 005 010 015 020 025 030
-05
00
05
Q2 GeV2D
I 1
proton
000 005 010 015 020 025 030
-10
-08
-06
-04
-02
00
02
Q2 GeV2D
I 1
neutron
FIG 4 Upper panel The generalized GDH integral IA(Q2) for the proton (left) and neutron
(right) as function of Q2 The result of this work the NLO BχPT prediction is shown by the blue
solid line and the blue band The red line represents the LO BχPT result The purple short-dashed
line is the O(p4) HB result from Ref [50 51] The gray band is the BχPT+∆ result from Ref [31]
The black dotted line is the MAID model prediction [69] Experimental extractions for the proton
Ref [60] (orange triangle uncertainties added in quadrature) and neutron Refs [21][27] where
magenta dotsorange diagonal crosses correspond to data and red squareslilac crosses correspond
to data plus extrapolation to unmeasured energy regions The green stars at the real-photon point
are derived from the anomalous magnetic moments κp asymp 1793 and κn asymp minus1913 [70] Lower
panel The generalized GDH integral I1(Q2) for the proton (left) and neutron (right) as function
of Q2 The purple short-dashed line is the HB result from Ref [51] Experimental extractions for
the proton Ref [18] (blue dots) and Ref [60] (orange triangle uncertainties added in quadrature)
and neutron Ref [61][27] (uncertainties added in quadrature) where green dotsorange diagonal
crosses correspond to data and gray squareslilac crosses correspond to data plus extrapolation to
unmeasured energy regions
17
IA(Q2) and I1(Q2) The latter will be discussed in Sec III D The former is defined as4
minus α
2M2N
IA(Q2) = minus 1
8π2
int infinν0
dν
radic1 +
Q2
ν2
σTT (νQ2)
ν(26)
=α
Q2
int x0
0
dx[4M2
Nx2
Q2g2(xQ2)minus g1(xQ2)
]
Due to its energy weighting the integral in Eq (26) converges slower than the one in the
generalized forward spin polarizability sum rule (19) Therefore knowledge of the cross
section at higher energies is required and the evaluation of the generalized GDH integral
IA(Q2) is not as simple as the evaluation of γ0(Q2)
The generalized GDH integral IA(Q2) is directly related to the non-pole amplitude
gnonpoleTT (νQ2) which differs from non-Born amplitude gTT (νQ2) by a term involving the
elastic Pauli form factor
gnonpoleTT (νQ2) = gTT (νQ2)minus 2παν
M2N
F 22 (Q2) (27)
cf Eqs (2a) and (8a) Consequently IA(Q2) is not a pure polarizability but also contains
an elastic contribution The ldquonon-polarizabilityrdquo or the Born part of IA(Q2) is given by
IBornA (Q2) = IA(Q2)minus∆IA(Q2) = minus1
4F 2
2 (Q2) (28)
where we refer to the polarizability part as ∆IA(Q2) The same is true for the generalized
GDH integral I1(Q2) which is directly related to Snonpole1 (νQ2)
IBorn1 (Q2) = I1(Q2)minus∆I1(Q2) = minus1
4F 2
2 (Q2) (29)
In the following we will add the Born parts to our LO and NLO BχPT predictions for
the polarizabilities ∆IA(Q2) and ∆I1(Q2) employing an empirical parametrization for the
elastic Pauli form factor [73] This allows us to compare to the experimental results for
IA(Q2) and I1(Q2) cf Fig 4 Note that the blue error bands only describe the uncertainties
of our BχPT predictions of the polarizabilities while the elastic contributions are considered
to be exact as explained in Sec II C The uncertainties of the polarizability predictions are
therefore better reflected in Fig 5 where we show the contributions of the different orders
to the BχPT predictions of ∆IA(Q2) and ∆I1(Q2) as well as the total results with error
bands
4 Note that IA(Q2) is sometimes called ITT (Q2)
18
000 005 010 015 020 025 030
-08
-06
-04
-02
00
02
Q2 GeV2D
DI A
proton
000 005 010 015 020 025 030
-10
-08
-06
-04
-02
00
02
Q2 GeV2D
DI A
neutron
000 005 010 015 020 025 030-005
000
005
010
015
020
025
Q2 GeV2D
DI 1
proton
000 005 010 015 020 025 030-015
-010
-005
000
005
010
015
Q2 GeV2D
DI 1
neutron
FIG 5 Contributions of the different orders to the chiral predictions of ∆IA(Q2) upper paneland ∆I1(Q2) lower panel for the proton (left) and neutron (right) Red solid line πN -loop
contribution green dot-dashed line ∆-exchange contribution orange dotted line π∆-loop contri-
bution blue solid line and blue band total result purple dot-dot-dashed line total result without
gC contribution black short-dashed line total result without gM dipole
The E97-110 experiment at Jefferson Lab has recently published their data for IAn(Q2) in
the region of 0035 GeV2 lt Q2 lt 024 GeV2 [27] In addition there are results for IAn(Q2)
from the earlier E94-010 experiment [21] and for IAp(Q2) from the E08-027 experiment [60]
The O(p4) HB calculation gives a large negative effect [51] which does not describe the data
The BχPT+∆ result from Ref [31] which mainly differs from our work by the absence of
the dipole form factor in gM looks similar to this HB result and only describes the data
points at lowest Q2 Our NLO prediction however follows closely the Q2 evolution of the
data In Fig 5 upper panel we show the polarizability ∆IA(Q2) whose Q2 evolution is
clearly dominated by the ∆ exchange Similar to the case of γ0p(Q2) inclusion of the dipole
in gM and the Coulomb coupling gC is very important in order to describe the experimental
data The LO prediction on the other hand slightly overestimates the data cf Fig 4
upper panel
19
At the real-photon point IA(0) = minusκ2
4and ∆IA(0) = 0 Therefore we give only the
slope of the polarizability ∆IA(Q2) [showing also the separate contributions from πN loops
∆ exchange and π∆ loops] in units of GeVminus2
d∆IAp(Q2)
dQ2
∣∣∣∣Q2=0
= minus858(343) asymp 238minus 1121 + 025 (30a)
d∆IAn(Q2)
dQ2
∣∣∣∣Q2=0
= minus955(343) asymp 141minus 1121 + 025 (30b)
Including the empirical Pauli form factor [73] we find in units of GeVminus2
dIAp(Q2)
dQ2
∣∣∣∣Q2=0
= minus318dIAn(Q2)
dQ2
∣∣∣∣Q2=0
= minus300 (31)
D Γ1(Q2) and I1(Q2) mdash the first moment of the structure function g1(xQ2)
The second variant for a generalization of the GDH sum rule to finite Q2 is defined as
minus α
2M2N
I1(Q2) = minus 1
8π2
int infinν0
dν1radic
ν2 +Q2
[σTT (νQ2) +
Q
νσLT (νQ2)
](32)
= minus α
Q2
int x0
0
dx g1(xQ2)
where I1(0) = minusκ2
4 This generalized GDH integral directly stems from the amplitude
Snonpole1 (νQ2) with the LEX from Eq (12a) It is given by the first moment of the struc-
ture function g1(xQ2) Γ1(Q2) =int x0
0dx g1(xQ2) as follows I1(Q2) =
2M2N
Q2 Γ1(Q2) The
isovector combination
Γ1(pminusn)(Q2) =
int x0
0
dx[g1p(xQ
2)minus g1n(xQ2)] (33)
is related to the axial coupling of the nucleon through the Bjorken sum rule [74 75]
limQ2rarrinfin
Γ1(pminusn)(Q2) =
gA6 (34)
As explained in Eq (28) the moment I1(Q2) splits into a polarizability part ∆I1(Q2) and
a Born part IBorn1 (Q2) Figure 4 lower panel shows the Q2 dependence of I1(Q2) which in
contrast to IA(Q2) shown in Figure 4 upper panel is clearly dominated by its Born part
and the elastic Pauli form factor The πN -loop ∆-exchange and π∆-loop contributions to
the polarizability ∆I1(Q2) are shown in Fig 5 lower panel Comparing to Fig 5 upper
20
000 005 010 015 020 025 030
-006
-004
-002
000
002
004
Q2 GeV2D
G1
proton
000 005 010 015 020 025 030
-006
-005
-004
-003
-002
-001
000
001
Q2 GeV2D
G1
neutron
FIG 6 First moment of the structure function g1(xQ2) for the proton (left) and neutron (right)
as function of Q2 The legend is the same as in Fig 4
panel one sees that ∆I1(Q2) is less sensitive to gC and the dipole form factor in gM than
∆IA(Q2)
For the proton our NLO BχPT prediction gives a very good description of the ex-
perimental data [18 60] and is in reasonable agreement with the MAID prediction [69]
For the neutron one observes good agreement with the empirical evaluations including ex-
trapolations to unmeasured energy regions starting from Q2 gt 01 GeV2 [27 61] In the
region of Q2 lt 005 GeV2 one observes an interesting tension between the recent E97-
110 experiment [27] and the data from CLAS [61] While the newest measurement finds
I1n(0035 GeV2) lt κ2n4 thus suggesting a negative slope at low Q the older measurement
found a rather large value for I1n(00496 GeV2) A similar but milder behaviour is seen in
the E97-110 [27] and E94-010 [21] data for IAn The MAID predictions do not agree with
the CODATA recommended values for the anomalous magnetic moments of the proton and
neutron [70] which in our work are imposed by using empirical parametrizations for the
elastic Pauli form factors [73] The slope of the HB result from Ref [51] is too large and
therefore only reproduces the data at very low Q2
Figure 6 shows the moment Γ1(Q2) for the proton and neutron while Fig 7 shows the
isovector combination Γ1 pminusn(Q2) The LO and NLO BχPT predictions are identical be-
cause our calculation produces the same Delta contributions for the proton and the neutron
For the isovector combination the MAID model only agrees with the data at very low
Q2 lt 010 GeV2 The same is true for the IR result [58 76] while all other chiral results
describe the data NLO BχPT (this work) BχPT+∆ [31] and HBχPT [51]
At the real-photon point I1(0) = minusκ2
4and ∆I1(0) = 0 Therefore we give only the
slope of the polarizability ∆I1(Q2) [showing also the separate contributions from πN loops
21
000 005 010 015 020 025 030-002
000
002
004
006
008
010
Q2 GeV2D
G1Hp-
nL
FIG 7 Isovector combination of Γ1(Q2) as function of Q2 The legend is the same as in Fig 6
The pink curve is the IR result from Ref [58 76] The experimental points are from Ref [77]
(brown dots) and Ref [23] (orange squares)
∆ exchange and π∆ loops] in units of GeVminus2
d∆I1p(Q2)
dQ2
∣∣∣∣Q2=0
= 039(4) asymp 034minus 053 + 058 (35a)
d∆I1n(Q2)
dQ2
∣∣∣∣Q2=0
= minus101(10) asymp minus107minus 053 + 058 (35b)
Including the empirical Pauli form factor [73] we find in units of GeVminus2
dI1p(Q2)
dQ2
∣∣∣∣Q2=0
= 580dI1n(Q2)
dQ2
∣∣∣∣Q2=0
= 553 (36)
E d2(Q2) mdash a measure of color polarizability
Another interesting moment to consider is d2(Q2) which is related to the twist-3 part of
the spin structure function g2(xQ2) [79 80]
d2(Q2) equiv 3
int 1
0
dx x2[g2(xQ2)minus gWW2 (xQ2)] (37)
where gWW2 (xQ2) is the twist-2 part of g2(xQ2) Using the Wandzura-Wilczek relation [81]
one can relate d2(Q2) to moments of the spin structure functions g1(xQ2) and g2(xQ2)
d2(Q2) =
int 1
0
dx x2 [3g2(xQ2) + 2g1(xQ2)] (38)
22
000 005 010 015 020 025 030
0000
0001
0002
0003
0004
0005
0006
0007
Q2 GeV2D
d2
proton
000 005 010 015 020 025 030
0000
0002
0004
0006
0008
0010
0012
Q2 GeV2D
d2
neutron
000 005 010 015 020 025 030
-1
0
1
2
Q2 GeV2D
Γ 010
-4
fm6
D
proton
000 005 010 015 020 025 030-2
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
6D
neutron
FIG 8 Upper panel The inelastic moment d2(Q2) for the proton (left) and neutron (right) as
function of Q2 The result of this work the NLO BχPT prediction is shown by the blue solid line
and the blue band The red line represents the LO BχPT result The purple short-dashed line is
the O(p4) HB result from Ref [50 51] The black dotted line is the MAID model prediction [69]
The experimental points for the neutron (cyan dots) are from Ref [22] Lower panel Fifth-order
generalized forward spin polarizability γ0(Q2) for the proton (left) and neutron (right) as function
of Q2 The experimental points for the proton are from Ref [59] (purple square) and Ref [78]
(orange dot)
This relation however only holds for asymptotically large Q2 It is also in the high-Q2
region where d2(Q2) is a measure of color polarizability [82 83] through its relation to the
gluon field strength tensor [80] We refer to Ref [84] for a recent review on the spin structure
of the nucleon including a discussion of sum rules for deep inelastic scattering and color
polarizabilities
What we consider in the following is the inelastic part of d2(Q2) defined as the moment
of g1(xQ2) and g2(xQ2) spin structure functions cf Eq (38)
d2(Q2) =
int x0
0
dx x2 [3g2(xQ2) + 2g1(xQ2)] (39)
23
This moment provides another testing ground for our BχPT predictions through comparison
with experiments on the neutron [22] Going towards the low-Q2 region the interpretation
of d2(Q2) in terms of color polarizabilities will fade out The above definition however
implies it is related to other VVCS polarizabilities
d2(Q2) =Q4
8M4N
[M2
NQ2
αδLT (Q2) + I1(Q2)minus IA(Q2)
] (40)
Note that d2(Q2) and its first two derivatives with respect to Q2 vanish at Q2 = 0 The
considerations in Eqs (28) and (29) have no effect on d2(Q2) since the Born contribution
from IA(Q2) and I1(Q2) cancel out Therefore d2(Q2) is a pure polarizability
In Fig 8 upper panel we show our NLO BχPT prediction and other results for d2(Q2)
While MAID [69] and BχPT describe the experimental data for the neutron [22] very well
the HB limit [50 51] is showing a fast growth with Q2 This illustrates the importance
of keeping the relativistic result Note also that even though the πN -loop contribution
is dominant both gC and the form factor in gM are essential to obtain a curvature that
reproduces the data cf Fig 9 upper panel For the proton there are to our knowledge
no experimental results to compare with However the agreement between the NLO BχPT
prediction and the MAID prediction at low energies is reasonable
F γ0(Q2) mdash fifth-order generalized forward spin polarizability
It is interesting to compare the generalized fifth-order forward spin polarizability sum
rule
γ0(Q2) =1
2π2
int infinν0
dν
radic1 +
Q2
ν2
σTT (νQ2)
ν5(41)
=64αM4
N
Q10
int x0
0
dx x4
[g1(xQ2)minus 4M2
Nx2
Q2g2(xQ2)
]
to the sum rule integrals for IA(Q2) and γ0(Q2) since they differ merely by their energy
weighting of σTT (νQ2) and a constant prefactor cf Eqs (19) (26) and (41) From IA(Q2)
to γ0(Q2) to γ0(Q2) the energy suppression is increasing by a factor of νminus2 respectively
Therefore the description of γ0(Q2) should be easiest in a low-energy effective-field theory
such as χPT whereas γ0(Q2) and IA(Q2) receive larger contributions from higher energies
In Fig 8 lower panel we show our LO and NLO BχPT predictions for γ0(Q2) One
can see that the πN -loop contribution is positive (in accordance to what we see for the cross
section σTT see Fig 10) The Delta shifts it substantially especially in the low Q2 region
24
000 005 010 015 020 025 030
-0004
-0002
0000
0002
0004
0006
0008
0010
Q2GeV
2D
d2
proton
000 005 010 015 020 025 030
-0005
0000
0005
0010
0015
Q2GeV
2D
d2
neutron
000 005 010 015 020 025 030
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
6D
proton
000 005 010 015 020 025 030
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
6D
neutron
FIG 9 Contributions of the different orders to the chiral predictions of d2(Q2) upper panel and
γ0(Q2) lower panel for the proton (left) and neutron (right) Red solid line πN -loop contribu-
tion green dot-dashed line ∆-exchange contribution orange dotted line π∆-loop contribution
blue long-dashed line total result purple dot-dot-dashed line total result without gC contribution
black short-dashed line total result without gM dipole
bringing it into a better agreement with data In general the BχPT curves start above the
empirical data points at the real-photon point and then decrease asymptotically to zero
above Q2 gt 01 GeV2 On the other hand the MAID prediction reproduces the empirical
data at the real-photon point then decreases to negative values until about Q2 gt 006 GeV2
from where it also starts to asymptotically approach zero Consequently our NLO BχPT
prediction of γ0(Q2) is consistently above the MAID prediction This is very different to
what we saw for IA(Q2) in Fig 4 upper panel where the MAID prediction at the real-
photon point is above the experimental value While the agreement of our predictions with
the empirical data is in general quite good for all moments of σTT (νQ2) one should point
out that both for γ0n(Q2) and γ0p(Q2) we overestimate the data at low Q2 For IA(Q2) such
observation cannot be made because ∆IA(0) = 0 and thus IA(0) is given by the empirical
Pauli form factor only From IA(Q2) γ0(Q2) and γ0(Q2) the latter has the smallest however
25
non-negligible dependence on gC and the dipole in gM cf Fig 9 lower panel
The πN -loop ∆-exchange and π∆-loop contributions to the NLO BχPT prediction of
the static fifth-order forward spin polarizability amount to in units of 10minus4 fm6
γ0p = 112(30) asymp 208minus 096minus 001 (42a)
γ0n = 195(30) asymp 292minus 096minus 001 (42b)
while the slope is composed as follows in units of 10minus4 fm8
dγ0p(Q2)
dQ2
∣∣∣∣Q2=0
= minus084(10) asymp minus100 + 016 + 000 (43a)
dγ0n(Q2)
dQ2
∣∣∣∣Q2=0
= minus142(15) asymp minus158 + 016 + 000 (43b)
Note that the HB prediction of the static polarizability γ0p (423 at O(p3) and 365 at O(ε3)
[78 85]) is almost one order of magnitude larger than the empirical value and therefore not
shown in Fig 8
G Summary
Our results are summarized in Table II where we give the contributions of the different
orders to the chiral predictions of the polarizabilities and their slopes at the real-photon
point A quantitative comparison of our predictions for the static spin polarizabilities to
the work of Bernard et al [31] and different empirical evaluations is shown in Table III We
can see that the inclusion of the Delta turns out to be very important for all moments of
the helicity-difference cross section To describe the Q2 behavior of the polarizabilities the
magnetic coupling of the N rarr ∆ transition should be modified by a dipole form factor as
has been observed previously in the description of electroproduction data [33] This dipole
form factor effectively takes account of vector-meson exchanges The Coulomb-quadrupole
N rarr ∆ transition despite its subleading order is important in the description of some
moments of spin structure functions This is contrary to what we saw for the moments of
unpolarized structure functions [30] where the Coulomb coupling had a negligible effect
The π∆ loops are mainly relevant for the generalized GDH integrals
26
TABLE II The NLO BχPT predictions for the forward VVCS polarizabilities and their slopes at
Q2 = 0 The contributions of the πN loops the ∆ exchange and the π∆ loops are shown together
with the combined total result Note that IA(0) = I1(0) = d2(0) = 0 and (d2)prime = 0
πN loops ∆ exchange π∆ loops Total
γ0 p 201 minus284 minus010minus093(92)
(10minus4 fm4) n 298 003(92)
δLT p 150 minus016 minus002132(15)
(10minus4 fm4) n 235 218(23)
γ0 p 208 minus096 minus001112(30)
(10minus4 fm6) n 292 195(30)
(γ0)prime p minus033011 001
minus022(4)
(10minus4 fm6) n minus073 minus061(7)
(δLT )prime p minus080 minus004 minus001minus085(8)
(10minus4 fm6) n minus119 minus124(12)
(γ0)prime p minus100016 000
minus084(10)
(10minus4 fm8) n minus158 minus142(15)
(∆IA)prime p 238 minus1121 025minus858(343)
(GeVminus2) n 141 minus955(343)
(∆I1)prime p 034 minus053 058039(4)
(GeVminus2) n minus107 minus101(10)
TABLE III Our NLO BχPT predictions for the spin polarizabilities at Q2 = 0 compared with
the BχPT+∆ predictions from Bernard et al [31] and the available empirical information Where
the reference is not given the empirical number is provided by the MAID analysis [55 69] with
unspecified uncertainty
Proton Neutron
This work BχPT+∆ Empirical This work BχPT+∆ Empirical
γ0 minus093(92) minus174(40) minus100(8)(12) [19] 003(92) minus077(40) minus0005
(10minus4 fm4) minus090(8)(11) [78] [MAID]
minus0929(105) [59]
δLT 132(15) 240(1) 134 218(23) 238(3) 203
(10minus4 fm4) [MAID] [MAID]
γ0 112(30) 060(7)(7) [78] 195(30) 123
(10minus4 fm6) 0484(82) [59] [MAID]
IV CONCLUSIONS
We have presented a complete NLO calculation of the polarized non-Born VVCS ampli-
tudes in covariant BχPT with pion nucleon and ∆(1232) fields The dispersion relations
between the VVCS amplitudes and the tree-level photoabsorption cross sections served as a
27
cross-check of these calculations
The obtained moments of the proton and neutron spin structure functions related to gen-
eralized polarizabilities and GDH-type integrals agree well with the available experimental
data The description of their Q2 evolution is improved compared to the previous χPT pre-
dictions In particular the NLO BχPT predictions obtained here give a better description
of the empirical data (eg from the Jefferson Laboratory ldquoSpin Physics Programrdquo) than
the HB [50 51] and IR [58] calculations
The demonstrated predictive power of the χPT framework amplitudes makes it well
suited for extending the χPT evaluation of the TPE effect in the hyperfine structure of
(muonic-)hydrogen [15ndash17] to next-to-leading order
ACKNOWLEDGEMENTS
We thank Lothar Tiator and Marc Vanderhaeghen for helpful discussions This work
is supported by the Deutsche Forschungsgemeinschaft (DFG) through the Collaborative
Research Center [The Low-Energy Frontier of the Standard Model (SFB 1044)] JMA ac-
knowledges support from the Community of Madrid through the ldquoPrograma de atraccion de
talento investigador 2017 (Modalidad 1)rdquo and the Spanish MECD grants FPA2016-77313-P
FH gratefully acknowledges financial support from the Swiss National Science Foundation
Appendix A Tensor decompositions of the VVCS amplitudes
In this appendix we review the decomposition of the forward VVCS process into tensor struc-
tures and scalar amplitudes In particular we consider the connection between the covariant and
the semi-relativistic decomposition in the lab frame that is defined in terms of the conventional
transverse longitudinal transverse-transverse and transverse-longitudinal amplitudes
As explained in Sec II A the process of forward VVCS off the nucleon can be described in
terms of four explicitly covariant amplitudes S1 2 and T1 2 [3]
T (νQ2) =
(minusgmicroν +
qmicroqν
q2
)T1(νQ2) +
1
M2N
(pmicro minus p middot q
q2qmicro)(
pν minus p middot qq2
qν)T2(νQ2) (A1)
minus 1
MNγmicroναqα S1(νQ2)minus 1
M2N
(γmicroνq2 + qmicroγναqα minus qνγmicroαqα
)S2(νQ2)
εprimelowastmicro εν
where εmicro (εprimelowastmicro ) are the incoming (outgoing) photon polarization vectors ν is the photon lab-frame
energy and Q2 is the photon virtuality Alternatively the decomposition in the laboratory frame
28
(which in the forward case coincides with the Breit frame) is parametrized in terms of the nucleon
Pauli matrices ~σ and the four scalar functions fL fT gTT and gLT
T (νQ2) = ε0 εprimelowast0 fL(νQ2) + (~ε primelowast middot ~ε ) fT (νQ2) + i~σ middot (~ε primelowast times ~ε ) gTT (νQ2) (A2)
minusi~σ middot [(ε0~εprimelowast minus ~ε ε primelowast0 )times q] gLT (νQ2)
Here ~q and q = ~q|~q | are the photon three-momentum in the lab system and its unit vector The
modified polarization vector components are given by
ε0 =
[ε0 minus
ν
|~q | (~ε middot q )
] |~q |Q ~ε = ~εminus q (~ε middot q ) (A3)
where ε = (ε0~ε ) is the usual incoming photon polarization vector and εprimelowast the outgoing polarization
vector The LEX of the lab frame amplitudes [Eq (10)] can serve in particular as the definition
of the generalized polarizabilities The lab frame amplitudes are also conveniently used for the
definition of the response functions see the example of the scalar amplitude gLT (νQ2) and the
corresponding response function σLT (νQ2) below in App B
Appendix B Photoabsorption cross sections
In the forward kinematics the spin-dependent VVCS amplitudes and the spin polarizabilities
can be described in terms of the polarized structure functions g1(xQ2) and g2(xQ2) or equiv-
alently the helicity-difference cross section σTT (νQ2) and the longitudinal-transverse response
function σLT (νQ2) with the help of dispersion relations (5) and the optical theorem (3) In this
way the photoabsorption cross sections measured in electroproduction processes form the basis
for most empirical evaluations shown throughout Sec III In the following we present the BχPT
predictions for the tree-level cross sections of πN - π∆- and ∆-production through photoabsorption
on the nucleon cf Figs 8 9 and 10 in Ref [30] In Secs B 1 and B 2 we will discuss the leading
πN -production channel and the ∆-production channel respectively We used these cross sections
to verify the polarizability predictions obtained otherwise from the calculated non-Born VVCS
amplitudes Due to the bad high-energy behavior of the π∆-production cross sections in BχPT
cf Fig 10 the dispersion relations in Eq (5) require further subtractions for a reconstruction of
the π∆-loop contribution to the spin-dependent VVCS amplitudes Therefore not all polarizabil-
ities could be verified but only those appearing as higher-order terms in the LEX of the VVCS
amplitudes such as γ0 [16]
29
proton neutron
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Neutron
02 04 06 08 100
50
100
150
200
250
[GeV]
T [b]
02 04 06 08 10
-50
0
50
100
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Neutron
02 04 06 08 100
50
100
150
200
250
[GeV]
T [b]
02 04 06 08 10
-50
0
50
100
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02040608100
50
100
150
200
250
300
[GeV]
T[b]
0204060810
-50
0
50
[GeV]
TT[b]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
0204060810
00
02
04
06
[GeV]
LTQ
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
0204060810
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
FIG 10 Photoabsorption cross sections for πN (red) and π∆ production (orange) with Q2 = 0
(solid) and Q2 = 01 GeV2 (dashed for πN and dotted for π∆ channel)
1 πN-production channel
In order to extract the response function σLT (νQ2) we have developed a method similar to
the one used to calculate σTT (νQ2) see for example Ref [86] For σLT (νQ2) however the
calculation is more complicated because one has to take into account that the associated Compton
process involves a spin-flip of the nucleon as illustrated in Fig 11 When calculating the cross
section the product of the incoming nucleon spinors has to reflect this flip
The forward VVCS amplitude related to σLT (νQ2) mdash and δLT (Q2) mdash is gLT (νQ2) It can
be extracted from Eq (A2) if one takes the modified polarization vector components in Eq (A3)
with ε = εL and εprimelowast = εlowastplusmn as input where εL = 1Q(|~q | 0 0 q0) and εplusmn = ∓ 1radic
2(0 1plusmni 0) are the
standard longitudinal and transverse polarization vectors respectively For εL and εlowastplusmn only the
choice of helicities hprime = plusmn12 and h = ∓12 gives a non-zero contribution and one obtains
χdaggerhprime T (νQ2)χh = χdaggerhprimeminusi~σ middot [(ε0~εprimelowast minus ~ε ε primelowast0 )times q] gLT (νQ2)χh =
radic2 gLT (νQ2) (B1)
where χh and χdaggerhprime are two-component Pauli spinors with opposite helicities or here spins
Let us now consider the related photoabsorption process and in particular the tree-level γlowastN rarr
30
s = 12
sprime = minus12
λ = 0 λprime = 1
rprime rprime
s=12sprime=minus1
2
λ=0λprime=1
rprimerprime
s=12
sprime=minus12
λ=0λprime=1
rprimerprime hacute=frac12h=-frac12
λacute=1λ=0
FIG 11 Relation between the forward Compton process and the photoabsorption process given
by the optical theorem In particular we show the longitudinal-transverse contribution The
double-line arrows represent the spin of the external particles while the dot represents the scalar
(longitudinal) polarization of the incoming photon Inside the blob the intermediate states are
represented eg nucleons with spins rprime (which are averaged in the calculation of the cross section)
and pions
πN channel see diagrams in Fig 8 of Ref [30] We define the πN -production amplitude as
T = uhB (PB)sumi
Ai(s t)Γi uhA(PA) (B2)
with the Dirac structures
Γ1 = γ5 (B3a)
Γ2 =1
2
[qA ε
]γ5 (B3b)
where uhA(PA) and udaggerhB (PB) are the Dirac spinors and PA and PB are the four-momenta of
the incoming and outgoing nucleons respectively When calculating the photoabsorption cross
section related to the VVCS amplitude in Eq (B1) the nucleon spin flip should be implemented
by uhprime(PA) in T dagger and uh(PA) in T together with the appropriate transverse and longitudinal
photon polarization vectors εlowastplusmn and εL
However if one wants to use the properties of the Dirac matrices it is more useful to construct
an operator to produce this spin flip in the external nucleons of Fig 11 This is accomplished by
introducing the projector ΓLT equiv 12radic
2(γ1 + iγ2)γ5 which also takes into account the extra factor
radic2 in Eq (B1) We checked that with this projector one correctly extracts δLT by comparing the
HB limit of our result to the HB result of Ref [50] where the authors calculate this polarizability
from the Compton amplitude directly With all those ingredients the longitudinal-transverse cross
section is calculated in the following way
σLT (νQ2) =1
64π2 s
|~pf |cm
|~pi|cm
int 1
minus1dcos θ
sumij
AiAdaggerjXij (B4)
31
with
Xij = Tr[(PB +MN )Γi(PA +MN )ΓLTγ0Γdaggerjγ
0] (B5)
where θ is the scattering angle in the center-of-mass (cm) frame and |~pi|cm (|~pf |cm) is the three-
momentum of an incoming (outgoing) particle in the cm frame An explicit calculation of the
matrix Xij leads to
X = MNQ
0 2(PB minus PA) middot εLminusradic
2|~qf |cm sin θ (sminus u)
(B6)
where |~qi|cm (|~qf |cm) is the relative three-momentum of the incoming (outgoing) particles in the
cm frame Here s t and u are the usual Mandelstam variables For the different γlowastN rarr πN
channels we obtain the following amplitudes Ai where we introduce qA as the four-momentum of
the incoming photon and qB as the four-momentum of the outgoing pion
bull γlowastprarr π0p
A1 =e gAMN
fπ
[2PA middot ε+ qA middot ε
sminusM2N
+2PB middot εminus qA middot ε
uminusM2N
] (B7a)
A2 =e gAMN
fπ
[1
sminusM2N
+1
uminusM2N
] (B7b)
bull γlowastprarr π+n
A1 =
radic2 e gAMN
fπ
[2PA middot ε+ qA middot ε
sminusM2N
+2(PA minus PB) middot ε+ qA middot ε
tminusm2π
] (B8a)
A2 =
radic2 e gAMN
fπ(sminusM2N )
(B8b)
bull γlowastnrarr π0n
A1 = 0 (B9a)
A2 = 0 (B9b)
bull γlowastnrarr πminusp
A1 =
radic2 e gAMN
fπ
[2PB middot εminus qA middot ε
uminusM2N
minus 2(PA minus PB) middot ε+ qA middot εtminusm2
π
] (B10a)
A2 =
radic2 e gAMN
fπ(uminusM2N )
(B10b)
The analytical expressions shown above were checked with the amplitudes given in Ref [87] An-
alytical expressions for the tree-level γlowastN rarr πN channel of the σLT (νQ2) and σTT (νQ2) cross
32
sections are given below (proton channels π+n and π0p neutron channel πminusp) We checked that
they reproduce the known results in the real-photon limit [45 86] To shorten the final expressions
for the cross sections which are considerably longer for finite Q2 than in the real-photon limit we
define the following dimensionless kinematic variables
αγ = (ENi )cmradics =
s+M2N +Q2
2s (B11)
απ = (ENf )cmradics =
s+M2N minusm2
π
2s (B12)
βγ = Eγcmradics =
sminusM2N minusQ2
2s (B13)
βπ = Eπcmradics =
sminusM2N +m2
π
2s (B14)
λγ = |~qi|cmradics =
radic(sminusM2
N minusQ2)2 + 4sQ2
2s (B15)
λπ = |~qf |cmradics =
radic(sminusM2
N +m2π)2 minus 4sm2
π
2s (B16)
Here (ENi )cm and (ENf )cm are the energies of the incoming and outgoing nucleon Eγcm is the energy
of the incoming photon Eπcm is the energy of the outgoing pion all in the cm frame
σ(π+n)TT = minus e2g2
AM2N
64πf2πs
3(sminusM2N )2λ4
γ
4sλπλγ
[(M2
N minus s)(M2N minusQ2 minus s)(Q2 + 2sβγβπ)
+ 2s(m2πQ
2 minus (M2N minus s)
(M2N + s(minus1 + 2βγβπ)
))λ2γ
]+ (M2
N minus s)(M2N minusQ2 minus s)
times (Q2 + 2sβγβπ minus 2sλπλγ)(Q2 + 2s(βγβπ + λπλγ)
)times log
(Q2 + 2sβγβπ minus 2sλγλπQ2 + 2sβγβπ + 2sλγλπ
) (B17)
33
σ(π0p)TT =
e2g2AM
2Nλπ
64πf2πs(sminusM2
N )2λγ
1
s2λπλ3γ
[(M2
N minus s)(M2N minusQ2 minus s)
(3M2
N + s(2βγβπ minus 3))(M2N
+ s(2βγβπ minus 1))
+ 2s
((2m2
πQ2 + (M2
N minus s)2)(M2
N minus s)minus 2(minusm2
πQ2 + (sminusM2
N )2)
times sβγβπ + 4s2(sminusM2N )β2
γβ2π
)λ2γ
]arctan
(2sλπλγ
M2N + s(2βγβπ minus 1)
)+ 2
[minus 2m2
πQ2
+ (M2N minus s)
(2(sminusM2
N ) + 4sβγβπ +(sminusM2
N +Q2)(3M2N minus 3s+ 2sβγβπ)
sλ2γ
+m2πQ
2(sminusM2N )
(M2N minus s+ 2sβγβπ)(M2
N + s(minus1 + 2βγβπ minus 2λπλγ))
+m2πQ
2(sminusM2N )
(M2N minus s+ 2sβγβπ)(M2
N + s(minus1 + 2βγβπ + 2λπλγ))
)
+1
λπλγ
(βγβπ
(minusm2
πQ2 + (M2
N minus s)(M2N minus s+ 2sβγβπ)
)minus (M2
N minus s)(M2N minusQ2 minus s)λ2
π
)log
(M2N + s(minus1 + 2βγβπ + 2λπλγ)
M2N + s(minus1 + 2βγβπ minus 2λπλγ)
)] (B18)
σ(πminusp)TT =
e2g2AM
2N
64πf2πs
3λ4γ(M2
N + s(2βπβγ minus 1))2 minus 4λ2πs
2λ2γ)
4sλπλγ
((minusM2
N +Q2 + s)(M2N
+ s(2βπβγ minus 1))
2 minus 2sλ2γ
(2λ2
πs(minusM2N +Q2 + s) +m2
πQ2))
+((M2N + s(2βπβγ minus 1)
)2
minus 4λ2πs
2λ2γ
)((M2N + s(2βπβγ minus 1)
)(M2N minus 2Q2 minus s(2βπβγ + 1)
)+ 4λ2
πs2λ2
γ
)times log
(minusM2
N minus 2βπsβγ minus 2λπsλγ + s)minus(M2N + s(2βπβγ minus 2λπλγ minus 1)
)(M2N + s(minus1
+ 2βπβγ + 2λπλγ))[((
M2N + s(2βπβγ minus 1)
)(M2N minus 2Q2 minus s(2βπβγ + 1)
)+ 4λ2
πs2λ2
γ
)times log
(minusM2
N minus 2βπsβγ + 2λπsλγ + s)
+((Q2 + 2βπsβγ)2 minus 4λ2
πs2λ2
γ
)times log
(Q2 + 2βπsβγ minus 2λπsλγQ2 + 2sβπβγ + 2λπsλγ
)]+ 4sλ2
γ
((M2
N + s(2βπβγ minus 1))2 minus 4λ2πs
2λ2γ
)times (M2
N minus s) arctanh
(2λπsλγ
M2N + 2βπsβγ minus s
) (B19)
34
σ(π+n)LT =
e2g2AM
3Nλπ
32πf2πQs
3(sminusM2N )2λ4
γ
2sλγ
[(M2
N minus s)(Q2 + 2sβ2γ)(Q2 + 2sβγβπ)
minus 4s
((M2
N minus s)(Q2 minus 2s(απ minus 1)βγ
)+Q2sβγβπ
)λ2γ + 8s3(minus1 + απ)λ4
γ
]minus M2
N minus sλπ
[(Q2 + 2sβ2
γ)(Q2 + 2sβγβπ)2 + 4s2(2(απ minus 1)βγ(Q2 + 2sβγβπ)
minusQ2λ2π
)λ2γ + 8s3(απ minus 1)2λ4
γ
]arctan
(2sλπλγ
Q2 + 2sβγβπ
) (B20)
σ(π0p)LT =
e2g2AM
3Nλπ
16πf2πQs(sminusM2
N )2λγ
1
minus2s(M2N + s(minus1 + 2βγβπ))2λ2
γ + 8s3λ2πλ
4γ
[minus 3M8
N (Q2
+ 2sβ2γ) + 2M4
Ns2
(minus (Q2 + 2sβ2
γ)(minus3 + 2βγβπ) +(Q2(minus3 + 2βγβπ + 6λ2
π) + 2sβγ
times (minus3 + 12απ + 2βγβπ minus 12απβγβπ + 4βγλ2π))λ2γ minus 4s(α2
π minus 1)λ4π
)+ 2M2
Ns3
[minus (Q2
+ 2sβ2γ)(minus1 + 2βγβπ)
(6 + βγβπ(minus9 + 2βγβπ)
)+
(Q2(3minus 12λ2
π + 4βγβπ(minus1 + 2βγβπ
+ λ2π))minus 2sβγ
(minus 3 + 4απ(3 + 2απ(3 + 2βγβπ(minus3 + βγβπ)) + 4βγ(βπ + (2minus βγβπ)
times λ2π)))λ2γ + 8s
((απ minus 1)(1 + απ minus 2βγβπ) + 2απβγλ
2π
)λ4γ
)+ s4
((Q2 + 2sβ2
γ)
times (1minus 2βγβπ)2(2βγβπ minus 3) + 2(Q2(minus1 + 6λ2
π + 2βγβπ((1minus 2βγβπ)2 minus 2λ2π)) + 2sβγ
times (minus1 + 2βγβπ + 4(minus1 + βγβπ)(minusαπ + 2απβγβπ minus βγλ2π)))λ2γ minus 8
(s(απ minus 1)(απ
+ (1minus 2βγβπ)2) + 2βγ(2sαπ +Q2βγ)λ2π
)λ4γ + 32s(απ minus 1)λ2
πλ6γ
)+ 2M6
Ns
(Q2(6
minus 7βγβπ + λ2γ) + 2sβγ(βγ(6minus 7βγβπ) + (1minus 4απ)λ2
γ)
)]+sminusM2
N
4s2λπλ3γ
[(Q2 + 2sβ2
γ)
times(3M2
N + s(2βγβπ minus 3))(M2N + s(2βγβπ minus 1)
)+ 2s
(minusM2
N (Q2 + 2sβγ minus 8sαπβγ)
+ s(2sβγ(1minus 4απ + 4απβγβπ) +Q2(1minus 2λ2π)))λ2γ + 8s3(απ minus 1)λ4
γ
]
times arctan
(2sλπλγ
M2N + s(2βγβπ minus 1)
) (B21)
35
σ(πminusp)LT =
e2g2AM
3N
64πf2πQs
3λ4γ(M2
N minusQ2 minus s)(M2N + s(2βπβγ + 2λπλγ minus 1))
times
1
M2N + s(2βπβγ minus 2λπλγ minus 1)
[4λπsλγ(M2
N minusQ2 minus s)(
2sλ2γ
(s((1minus 2λ2
π)Q2
+ 2(2απ minus 1)sβγ(2βπβγ minus 1))minusM2N (Q2 + 2(1minus 2απ)sβγ)
)+ (Q2 + 2sβ2
γ)(M2N + s(2βπβγ minus 1)
)2 + 8(απ minus 1)απs
3λ4γ
)]minus(M2N + s(2βπβγ + 2λπλγ minus 1)
)times[
log
(minusM2N minus 2βπsβγ minus 2λπsλγ + s
minusM2N minus 2βπsβγ + 2λπsλγ + s
)(minus (Q2 + 2sβ2
γ)(M2N + s(2βπβγ minus 1)
)(minusM2
N + 2Q2 + 2βπsβγ + s)minus 2sλ2γ
(minusM2
N (Q2 + 2sβγ) +Q4 +Q2s(4απβγ
minus 2βγ minus 2λ2π + 1) + 2s2βγ(4(απ minus 1)βπβγ + 1)
)minus 8(απ minus 1)2s3λ4
γ
)+ 2
(4s2λ2
γ
(2(απ minus 1)βγ(Q2 + 2βπsβγ)minus λ2
πQ2)
+ (Q2 + 2βπsβγ)2(Q2 + 2sβ2γ)
+ 8(απ minus 1)2s3λ4γ
)arctanh
(2λπsλγ
Q2 + 2βπsβγ
)] (B22)
2 ∆-production channel
The tree-level ∆-exchange diagram in Fig 2 of Ref [30] contributes to the non-Born part of
the VVCS amplitudes The contribution of the ∆ exchange to the VVCS amplitudes can be split
into [17]
S∆-exch1 (νQ2) = S∆-pole
1 (νQ2) + S∆-exch1 (νQ2) (B23a)
νS∆-exch2 (νQ2) = νS∆-pole
2 (νQ2) + νS2∆-exch
(νQ2) (B23b)
and similarly for the unpolarized VVCS amplitudes discussed in Ref [30] Here we introduced the
∆-pole contributions S∆-polei and the ∆-non-pole contributions S∆-exch
i The former amplitudes
feature a pole at the ∆(1232)-production threshold and thus are proportional to
1
[sminusM2∆][uminusM2
∆]=
1
4M2N
1
ν2∆ minus ν2
(B24)
36
They can be reconstructed from the dispersion relations in Eq (5) with the tree-level ∆-production
cross sections as input cf Fig 10 in Ref [30]
σTT (νQ2) =π2α
M2NM
2+|~q |
minus g2
MMN (M+ + ν)|~q |2 +g2E(∆minus ν)(Q2 minusMNν)2
MN(B25a)
+g2CQ
4s(∆minus ν)
MNM2∆
minus 4gMgE(Q2 minusMNν)|~q |2 minus 4gMgCQ2|~q |2
+2gEgCQ
2[minusMNM∆ |~q |2 + s(Q2 +∆ν)
]MNM∆
δ(ν minus ν∆)
σLT (νQ2) =Qπ2α
M2NM
2+|~q |
g2E(MNν minusQ2) [M∆(MN + ν)minus s]
MN(B25b)
+g2CQ
2[MNM∆|~q |2 minus s(Q2 +∆ν)
]MNM2
∆
+ gMgEM∆|~q |2 minusgMgC(Q2 minusMNν)|~q |2
M∆
+gEgC(ν minus∆)(M2
N |~q |2 minus 2Q2s)
MNM∆
δ(ν minus ν∆)
with ∆ = M∆ minusMN M+ = M∆ + MN and the Mandelstam variable s = M2N + 2MNν minus Q2
Analytical expressions for the spin structure functions g1(xQ2) and g2(xQ2) can be constructed
from Eq (3) with the flux factor K(νQ2) = |~q | =radicν2 +Q2
In the ∆-non-pole contributions to S1(νQ2) and νS2(νQ2) the pole in ν at the ∆(1232)-
production threshold has canceled out
S∆-exch1 (νQ2) =
πα
MNM2+
[g2MQ
2+ + g2
E
(∆2 minus 3Q2
)+
4g2CQ
4
M2∆
minus 8gMgEM∆ωminus (B26a)
minus2gMgCQ2(MN minus 4M∆)
M∆+
2gEgCQ2(3MN minus 2M∆)
M∆
]
νS2∆-exch
(νQ2) =2πα
MNM2+
[g2EM∆∆ωminus +
g2M MNQ
2+
2+g2C Q
2(Q2 minus∆2)
2M∆(B26b)
+gEgM M∆(M∆ω+ minus 4MNωminus)minus gEgC ∆(2Q2 +MNω+)
+gMgC Q2(4MN minus ω+)
]+S∆-exch
2 (νQ2)
ν
[M2
∆ ω2+
M2N
+ ν2
]
with Q+ =radic
(M∆ +MN )2 +Q2 and ωplusmn = (M2∆minusM2
N plusmnQ2)2M∆ and the non-pole contribution
to S2(νQ2)
S∆-exch2 (νQ2) = minus2παMNν
M∆M2+
[gM + gE
]gC (B27)
These amplitudes to the contrary are not described by the tree-level ∆-production cross sections
in the standard dispersive approach [17] This peculiarity has been previously missed e g in the
calculation of the ∆-exchange contribution to the hydrogen hyperfine splitting in Ref [88] The
importance of including the ∆-non-pole contribution is also evident when considering the BC sum
37
rule in Eq (14) The ∆-pole terms by themselves violate the BC sum rule but cancel exactly with
the ∆-non-pole terms
limνrarr0
νS∆-pole2 (νQ2) + lim
νrarr0νS2
∆-exch(νQ2) = 0 (B28)
Appendix C Polarizabilities at Q2 = 0
In this section we give analytical expressions for the static values and slopes at Q2 = 0 of
the spin polarizabilities and moments of polarized structure functions In particular we give the
HB expansion of the πN -loop contributions and the ∆-exchange contributions The complete
expressions also for the π∆-loop contributions can be found in the Supplemented material Recall
that IA(0) = I1(0) = d2(0) = 0 and dd2(Q2)dQ2
∣∣∣Q2=0
= 0
1 πN-loop contribution
Here we give analytical expressions for the πN -loop contributions to the proton and neutron
spin polarizabilities expanded in powers of micro = mπMN viz the HB expansion Note that we
choose to expand here to a high order in micro the strict HB expansion would only retain the leading
term in an analogous NLO calculation
bull Static polarizabilities (Q2 = 0)
γ0p =e2g2
A
96π3f2πm
2π
1minus 21πmicro
8minus(
59
2+ 26 logmicro
)micro2 +
1875πmicro3
64
+3
(3
2+ 26 logmicro
)micro4 +
(C1)
γ0n =e2g2
A
48π3f2πm
2π
1
2minus 9πmicro
16minus 2micro2 logmicro+
75πmicro3
128minus 3micro4
4+
(C2)
δLTp =e2g2
A
192π3f2πm
2π
1minus 9πmicro
8+
(13
2minus 2 logmicro
)micro2 minus 465πmicro3
64
minus(
47
2+ 42 logmicro
)micro4 +
(C3)
δLTn =e2g2
A
96π3f2πm
2π
1
2+
3πmicro
16+ (1 + 2 log micro)micro2 minus 105πmicro3
128+
5micro4
4+
(C4)
38
δLTn =e2g2
A
96π3f2πm
2π
1
2+
3πmicro
16+ (1 + 2 log micro)micro2 minus 105πmicro3
128+
5micro4
4+
(C5)
γ0p =e2g2
A
16π3f2πm
4π
4
45minus 3πmicro
16+
14micro2
5minus 1813πmicro3
384minus 192
5(1 + log micro)micro4
+80703πmicro5
2048+
(C6)
γ0n =e2g2
A
16π3f2πm
4π
4
45minus 5πmicro
48+
4micro2
5minus 245πmicro3
384minus 32micro4 logmicro
15
+1323πmicro5
2048+
(C7)
bull Slopes of polarizabilities at Q2 = 0
dγ0p(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
2minus 45πmicro
4+ 223micro2 minus 28515πmicro3
64
minus9
(1953
4+ 449 logmicro
)micro4 +
570255πmicro5
128+
(C8)
dγ0n(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
2minus 81πmicro
8+ 94micro2 minus 2535πmicro3
32
minus3 (1 + 90 logmicro)micro4 +84315πmicro5
1024+
(C9)
dδLTp(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
2880π3f2πm
4π
minus5
2minus 27πmicro
32+ 20micro2 minus 5865πmicro3
256
+3
(617
4+ 36 logmicro
)micro4 minus 2056845πmicro5
4096+
(C10)
dδLTn(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
minus5
4minus 81πmicro
64minus 11micro2 +
10005πmicro3
512
+15
8(11 + 48 logmicro)micro4 minus 267015πmicro5
8192+
(C11)
dIAp(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
96π2f2πmicro
2
1minus 15πmicro
4minus 1
2(115 + 88 logmicro)micro2 +
1839πmicro3
32
+5 (5 + 34 logmicro)micro4 +
(C12)
dIAn(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
48π2f2πmicro
2
1
2minus 11πmicro
8minus 1
4(1 + 20 logmicro)micro2 +
99πmicro3
64
minus25micro4
12+
(C13)
dI1p(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
96π2f2πmicro
3π
8+ 2 (4 + 3 logmicro)microminus 537πmicro2
64
minus1
2(15 + 56 logmicro)micro3 +
(C14)
39
dI1n(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
48π2f2πmicro
minus π
16+
1
4(3 + 4 logmicro)microminus 57πmicro2
128+
2micro3
3+
(C15)
γ0p(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2g2
A
16π3f2πm
6π
1
105minus 23πmicro
256+
377micro2
210minus 15551πmicro3
6144+
3371micro4
105
minus1640457πmicro5
32768+
(C16)
γ0n(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2g2
A
16π3f2πm
6π
1
105minus 153πmicro
1792+
69micro2
70minus 4615πmicro3
6144+
172micro4
35
minus120897πmicro5
32768+
(C17)
2 ∆-exchange contribution
Here we give analytical expressions for the tree-level ∆-exchange contributions to the nucleon
spin polarizabilities and their slopes at Q2 = 0 Note that the ∆-exchange contributes equally to
proton and neutron polarizabilities Recall that for the magnetic γlowastN∆ coupling we introduced a
dipole form factor to mimic vector-meson dominance gM rarr gM(1 +Q2Λ2)2
bull Static polarizabilities (Q2 = 0)
γ0 = minus e2
4πM2+
(g2M
∆2+
g2E
M2+
minus 4gMgEM+∆
) (C18)
δLT =e2M∆
4πM3+
(g2E
MNM++gMgE∆MN
minus gEgCM2
∆
) (C19)
γ0 =e2M2
N
π∆2M4+
(minusg
2M
∆2+
g2E
M2+
+4gMgE∆M+
) (C20)
bull Slopes of polarizabilities at Q2 = 0
dγ0(Q2)
dQ2
∣∣∣∣∣Q2=0
= minus e2
πM2+∆
(g2M
∆
[1
4∆2minus 1
∆M++
1
2M2+
]minus 1
Λ2
g2M
∆+
g2E
2M2+
[1
2∆minus 3
M+
]minusgMgE
M+
[1
∆2minus 5
∆M++
1
M2+
]+
1
Λ2
2gMgEM+
+2gMgC∆M2
+
minus gEgCM3
+
) (C21)
40
d δLT (Q2)
dQ2
∣∣∣∣∣Q2=0
=e2M∆∆
4πMNM2+
(g2E
∆2M2+
[1
∆minus 4
M+
]minus g2
C
∆M2∆M
2+
+gMgE∆2M+
[1
∆2
minus 3
∆M++
1
M2+
]minus 2
Λ2
gMgE∆2M+
+gMgC∆M2
∆
[1
2∆2minus 2
∆M++
1
2M2+
]minus gEgC
2M2∆M
2+
[7
∆+
1
M+
]) (C22)
dIA(Q2)
dQ2
∣∣∣∣∣Q2=0
= minusM2N
M2+
(g2M
2∆2+
g2E
MNM+minus 2gMgE
∆M+minus gEgCM∆M+
) (C23)
dI1(Q2)
dQ2
∣∣∣∣∣Q2=0
= minusM∆M2N
2M3+
(g2E
MNM∆minus gMgE∆MN
minus gEgCM2
∆
) (C24)
γ0(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2M2
N
π∆3M6+
(g2M
∆
[3M2
∆ + 2M∆MN minus 9M2N
∆2+
4M2+
Λ2
]
minus4gMgE
[5M2
∆ minus 9M2N
M+∆2+
2M+
Λ2
]+
g2E
∆M+[7M∆ minus 9MN ]
minus8gMgC∆
+4gEgCM+
) (C25)
[1] D Drechsel B Pasquini and M Vanderhaeghen Dispersion relations in real and virtual
Compton scattering Phys Rept 378 99 (2003) hep-ph0212124
[2] S E Kuhn J-P Chen and E Leader Spin structure of the nucleon mdash status and recent
results Prog Part Nucl Phys 63 1 (2009) arXiv08123535 [hep-ph]
[3] F Hagelstein R Miskimen and V Pascalutsa Nucleon polarizabilities from Compton scat-
tering to hydrogen atom Prog Part Nucl Phys 88 29 (2016) arXiv151203765 [nucl-th]
[4] B Pasquini and M Vanderhaeghen Dispersion theory in electromagnetic interactions Ann
Rev Nucl Part Sci 68 75 (2018) arXiv180510482 [hep-ph]
[5] S Gerasimov A Sum rule for magnetic moments and the damping of the nucleon magnetic
moment in nuclei Sov J Nucl Phys 2 430 (1966)
[6] S Drell and A C Hearn Exact sum rule for nucleon magnetic moments Phys Rev Lett
16 908 (1966)
[7] J S Schwinger Source theory viewpoints in deep inelastic scattering Proc Natl Acad Sci
USA 72 1 (1975)
[8] J S Schwinger Source theory viewpoints in deep inelastic scattering Electromagnetic In-
teractions and Field Theory Proceedings 14 Internationale Universitatswochen Schladming
Austria February 24-March 7 1975 Acta Phys Austriaca Suppl 14 471 (1975)
41
[9] J Schwinger Source theory discussion of deep inelastic scattering with polarized particles
Proc Natl Acad Sci USA 72 1559 (1975)
[10] A M Baldin Polarizability of nucleons Nucl Phys 18 310 (1960)
[11] M Gell-Mann M L Goldberger and W E Thirring Use of causality conditions in quantum
theory Phys Rev 95 1612 (1954)
[12] A Pineda Leading chiral logarithms to the hyperfine splitting of the hydrogen and muonic
hydrogen Phys Rev C 67 025201 (2003)
[13] C Peset and A Pineda The two-photon exchange contribution to muonic hydrogen from
chiral perturbation theory Nucl Phys B 887 69 (2014) arXiv14064524 [hep-ph]
[14] C Peset and A Pineda Model-independent determination of the two-photon exchange contri-
bution to hyperfine splitting in muonic hydrogen JHEP 04 060 arXiv161205206 [nucl-th]
[15] F Hagelstein and V Pascalutsa Proton structure in the hyperfine splitting of muonic hydro-
gen PoS CD15 077 (2016) arXiv151104301 [nucl-th]
[16] F Hagelstein Exciting Nucleons in Compton Scattering and Hydrogen-Like Atoms PhD
thesis Mainz U Inst Kernphys (2017) arXiv171000874 [nucl-th]
[17] F Hagelstein ∆(1232)-Resonance in the hydrogen spectrum Proceedings 11th International
Workshop on the Physics of Excited Nucleons (NSTAR 2017) Columbia SC USA August
20-23 2017 Few Body Syst 59 93 (2018) arXiv180109790 [nucl-th]
[18] Y Prok et al (CLAS) Moments of the spin structure functions gp1 and gd1 for 005 lt Q2 lt 30
GeV2 Phys Lett B 672 12 (2009) arXiv08022232 [nucl-ex]
[19] H Dutz et al (GDH) First measurement of the Gerasimov-Drell-Hearn sum rule for 1H from
07 GeV to 18 GeV at ELSA Phys Rev Lett 91 192001 (2003)
[20] M Amarian et al (Jefferson Lab E94010) Measurement of the generalized forward spin po-
larizabilities of the neutron Phys Rev Lett 93 152301 (2004) arXivnucl-ex0406005
[21] M Amarian et al The Q2 evolution of the generalized Gerasimov-Drell-Hearn integral for
the neutron using a 3He target Phys Rev Lett 89 242301 (2002) arXivnucl-ex0205020
[22] M Amarian et al (Jefferson Lab E94-010)Q2 evolution of the neutron spin structure moments
using a 3He target Phys Rev Lett 92 022301 (2004) arXivhep-ex0310003
[23] A Deur et al Experimental determination of the evolution of the Bjorken integral at low Q2
Phys Rev Lett 93 212001 (2004) arXivhep-ex0407007
[24] K Slifer Low Q2 measurement of gp2 and the δLT spin polarizability Spin structure at long dis-
tance Proceedings Workshop Newport News USA March 12ndash13 2009 nucl-ex09064775
(2009) AIP Conf Proc 1155 10106313203293 (2009) arXiv09064775 [nucl-ex]
[25] P Solvignon et al (E01-012) Moments of the neutron g2 structure function at intermediate
42
Q2 Phys Rev C 92 015208 (2015) arXiv13044497 [nucl-ex]
[26] A Deur Experimental studies at low Q2 of the spin structure of the nucleon at Jefferson Lab
in 9th International Workshop on Chiral Dynamics (CD18) Durham NC USA September
17-21 2018 (2019) arXiv190305661 [nucl-ex]
[27] V Sulkosky et al (Jefferson Lab E97-110) Measurement of the 3He spin-structure functions
and of neutron (3He) spin-dependent sum rules at 0035 le Q2 le 024 GeV2 Phys Lett B
805 135428 (2020) arXiv190805709 [nucl-ex]
[28] K Adhikari et al (CLAS) Measurement of the Q2 Dependence of the Deuteron Spin Structure
Function g1 and its Moments at Low Q2 with CLAS Phys Rev Lett 120 062501 (2018)
arXiv171101974 [nucl-ex]
[29] V Lensky F Hagelstein A Hiller Blin and V Pascalutsa Comment on rdquoMeasurement of
the Q2 Dependence of the Deuteron Spin Structure Function g1 and its Moments at Low Q2
with CLASrdquo (2018) arXiv180603219 [nucl-th]
[30] J M Alarcn F Hagelstein V Lensky and V Pascalutsa Forward doubly-virtual Compton
scattering off the nucleon in chiral perturbation theory at NLO the subtraction function and
moments of unpolarized structure functions (2020) arXiv200509518 [hep-ph]
[31] V Bernard E Epelbaum H Krebs and U-G Meiszligner New insights into the spin structure
of the nucleon Phys Rev D 87 054032 (2013) arXiv12092523 [hep-ph]
[32] V Lensky J M Alarcon and V Pascalutsa Moments of nucleon structure functions at
next-to-leading order in baryon chiral perturbation theory Phys Rev C 90 055202 (2014)
arXiv14072574 [hep-ph]
[33] V Pascalutsa and M Vanderhaeghen Chiral effective-field theory in the ∆(1232) region
I Pion electroproduction on the nucleon Phys Rev D 73 034003 (2006) arXivhep-
ph0512244
[34] V Pascalutsa and M Vanderhaeghen Electromagnetic nucleon-to-Delta transition in chiral
effective field theory Phys Rev Lett 95 232001 (2005) arXivhep-ph0508060
[35] T R Hemmert B R Holstein and J Kambor Systematic 1M expansion for spin 32
particles in baryon chiral perturbation theory Phys Lett B 395 89 (1997) arXivhep-
ph9606456
[36] V Pascalutsa and D R Phillips Effective theory of the ∆(1232) in Compton scattering off
the nucleon Phys Rev C 67 055202 (2003) arXivnucl-th0212024
[37] V Pascalutsa M Vanderhaeghen and S N Yang Electromagnetic excitation of the ∆(1232)-
resonance Phys Rept 437 125 (2007) arXivhep-ph0609004
[38] K A Olive et al (Particle Data Group) Review of Particle Physics Chin Phys C 38 090001
43
(2014)
[39] F E Low Scattering of light of very low frequency by systems of spin 12 Phys Rev 96
1428 (1954)
[40] M Gell-Mann and M L Goldberger Scattering of low-energy photons by particles of spin
12 Phys Rev 96 1433 (1954)
[41] A M Harun ar-Rashid A simple derivation of Schwingerrsquos sum rule for spin dependent
structure functions Nuovo Cim A 33 447 (1976)
[42] F Hagelstein and V Pascalutsa Dissecting the hadronic contributions to (g minus 2)micro by
Schwingerrsquos sum rule Phys Rev Lett 120 072002 (2018) arXiv171004571 [hep-ph]
[43] V Lensky V Pascalutsa M Vanderhaeghen and C Kao Spin-dependent sum rules con-
necting real and virtual Compton scattering verified Phys Rev D 95 074001 (2017)
arXiv170101947 [hep-ph]
[44] H Burkhardt and W N Cottingham Sum rules for forward virtual Compton scattering
Annals Phys 56 453 (1970)
[45] V Lensky and V Pascalutsa Predictive powers of chiral perturbation theory in Compton
scattering off protons Eur Phys J C 65 195 (2010) arXiv09070451 [hep-ph]
[46] H Grieszlighammer J McGovern D Phillips and G Feldman Using effective field theory to
analyse low-energy Compton scattering data from protons and light nuclei Prog Part Nucl
Phys 67 841 (2012) arXiv12036834 [nucl-th]
[47] H W Grieszlighammer J A McGovern and D R Phillips Nucleon polarisabilities at and
beyond physical pion masses Eur Phys J A 52 139 (2016) arXiv151101952 [nucl-th]
[48] E Epelbaum H Krebs and U-G Meiszligner Improved chiral nucleon-nucleon potential up to
next-to-next-to-next-to-leading order Eur Phys J A 51 53 (2015) arXiv14120142 [nucl-
th]
[49] E Epelbaum H Krebs and U-G Meiszligner Precision nucleon-nucleon potential at fifth order
in the chiral expansion Phys Rev Lett 115 122301 (2015) arXiv14124623 [nucl-th]
[50] C W Kao T Spitzenberg and M Vanderhaeghen Burkhardt-Cottingham sum rule and
forward spin polarizabilities in heavy baryon chiral perturbation theory Phys Rev D 67
016001 (2003) arXivhep-ph0209241
[51] C-W Kao D Drechsel S Kamalov and M Vanderhaeghen Higher moments of nucleon
spin structure functions in heavy baryon chiral perturbation theory and in a resonance model
Phys Rev D 69 056004 (2004) arXivhep-ph0312102
[52] T Becher and H Leutwyler Baryon chiral perturbation theory in manifestly Lorentz invariant
form Eur Phys J C 9 643 (1999) arXivhep-ph9901384
44
[53] J Gasser M E Sainio and A Svarc Nucleons with chiral loops Nucl Phys B 307 779
(1988)
[54] L S Geng J Martin Camalich L Alvarez-Ruso and M J Vicente Vacas Leading SU(3)-
breaking corrections to the baryon magnetic moments in chiral perturbation theory Phys
Rev Lett 101 222002 (2008) arXiv08051419 [hep-ph]
[55] D Drechsel S S Kamalov and L Tiator The GDH sum rule and related integrals Phys
Rev D 63 114010 (2001) arXivhep-ph0008306
[56] D Drechsel O Hanstein S S Kamalov and L Tiator A unitary isobar model for pion
photo- and electroproduction on the proton up to 1 GeV Nucl Phys A 645 145 (1999)
arXivnucl-th9807001
[57] L Tiator private communication (2020)
[58] V Bernard T R Hemmert and U-G Meiszligner Spin structure of the nucleon at low energies
Phys Rev D 67 076008 (2003) arXivhep-ph0212033
[59] O Gryniuk F Hagelstein and V Pascalutsa Evaluation of the forward Compton scattering
off protons II Spin-dependent amplitude and observables Phys Rev D 94 034043 (2016)
arXiv160400789 [nucl-th]
[60] R Zielinski The g2p Experiment A Measurement of the Protonrsquos Spin Structure Functions
PhD thesis New Hampshire U (2010) arXiv170808297 [nucl-ex]
[61] N Guler et al (CLAS) Precise determination of the deuteron spin structure at low to mod-
erate Q2 with CLAS and extraction of the neutron contribution Phys Rev C 92 055201
(2015) arXiv150507877 [nucl-ex]
[62] C E Carlson V Nazaryan and K Griffioen Proton structure corrections to electronic
and muonic hydrogen hyperfine splitting Phys Rev A 78 022517 (2008) arXiv08052603
[physicsatom-ph]
[63] V Bernard N Kaiser and U-G Meiszligner Chiral dynamics in nucleons and nuclei Int J
Mod Phys E 4 193 (1995) arXivhep-ph9501384
[64] T R Hemmert B R Holstein and J Kambor ∆(1232) and the polarizabilities of the nucleon
Phys Rev D 55 5598 (1997) arXivhep-ph9612374
[65] V Pascalutsa and R Timmermans Field theory of nucleon to higher-spin baryon transitions
Phys Rev C 60 042201 (1999) arXivnucl-th9905065
[66] V Pascalutsa Quantization of an interacting spin-32 field and the Delta isobar Phys Rev
D 58 096002 (1998) arXivhep-ph9802288
[67] H Krebs Double Virtual Compton Scattering and SpinStructure of the Nucleon PoS
CD2018 031 (2019)
45
[68] N Kochelev and Y Oh Axial anomaly and the δLT puzzle Phys Rev D 016012 (2012)
arXiv11034892 [hep-ph]
[69] D Drechsel S Kamalov and L Tiator Unitary isobar model ndash MAID2007 Eur Phys J A
69 (2007) available at httpsmaidkphuni-mainzde arXiv07100306 [nucl-th]
[70] P J Mohr B N Taylor and D B Newell CODATA recommended values of the fundamental
physical constants 2010 Rev Mod Phys 84 1527 (2012)
[71] J Ahrens et al (GDH A2) First measurement of the Gerasimov-Drell-Hearn integral for 1H
from 200 to 800 MeV Phys Rev Lett 87 022003 (2001) arXivhep-ex0105089 [hep-ex]
[72] K Helbing (GDH) Experimental verification of the GDH sum rule at ELSA and MAMI Nucl
Phys Proc Suppl 105 113 (2002)
[73] R Bradford A Bodek H S Budd and J Arrington A New parameterization of the nucleon
elastic form-factors NuInt05 proceedings of the 4th International Workshop on Neutrino-
Nucleus Interactions in the Few-GeV Region Okayama Japan 26-29 September 2005 Nucl
Phys Proc Suppl 159 127 (2006) arXivhep-ex0602017
[74] J D Bjorken Applications of the chiral U(6)timesU(6) algebra of current densities Phys Rev
148 1467 (1966)
[75] J D Bjorken Inelastic scattering of polarized leptons from polarized nucleons Phys Rev D
1 1376 (1970)
[76] V Bernard T R Hemmert and U-G Meiszligner Novel analysis of chiral loop effects in
the generalized Gerasimov-Drell-Hearn sum rule Phys Lett B 545 105 (2002) arXivhep-
ph0203167
[77] A Deur et al Experimental study of isovector spin sum rules Phys Rev D 78 032001
(2008) arXiv08023198 [nucl-ex]
[78] B Pasquini P Pedroni and D Drechsel Higher order forward spin polarizability Phys Lett
B 687 160 (2010) arXiv10014230 [hep-ph]
[79] R Jaffe g2ndashThe nucleonrsquos other spin-dependent structure function Comments Nucl Part
Phys 19 239 (1990)
[80] E V Shuryak and A Vainshtein Theory of power corrections to deep inelastic scattering in
quantum chromodynamics (II) Qminus4 effects polarized target Nucl Phys B 201 141 (1982)
[81] S Wandzura and F Wilczek Sum rules for spin dependent electroproduction Test of rela-
tivistic constituent quarks Phys Lett B 72 195 (1977)
[82] B W Filippone and X-D Ji The spin structure of the nucleon Adv Nucl Phys 26 1
(2001) arXivhep-ph0101224
[83] M Burkardt The g2 structure function Proceedings Workshop on Spin structure at long
46
distance Newport News USA March 12-13 2009 AIP Conf Proc 1155 26 (2009)
arXiv09054079 [hep-ph]
[84] A Deur S J Brodsky and G F de Teramond The spin structure of the nucleon Rept
Prog Phys 82 076201 (2019) arXiv180705250 [hep-ph]
[85] B R Holstein D Drechsel B Pasquini and M Vanderhaeghen Higher order polarizabilities
of the proton Phys Rev C 61 034316 (2000) arXivhep-ph9910427
[86] B R Holstein V Pascalutsa and M Vanderhaeghen Sum rules for magnetic moments and
polarizabilities in QED and chiral effective-field theory Phys Rev D 72 094014 (2005)
arXivhep-ph0507016
[87] B Pasquini D Drechsel and L Tiator Invariant amplitudes for pion electroproduction Eur
Phys J A 34 387 (2007) arXiv07122327 [hep-ph]
[88] A J Buchmann Non-spherical proton shape and hydrogen hyperfine splitting Proceedings
International Workshop on Precision Physics of Simple Atomic Systems (PSAS 2008) Wind-
sor Ontario Canada July 21-26 2008 Can J Phys 87 773 (2009) arXiv09104747
[physicsatom-ph]
47
I INTRODUCTION
In the studies of nucleon structure the forward doubly-virtual Compton scattering
(VVCS) amplitude Fig 1 is playing a central role (see eg Refs [1ndash4] for reviews)
Traditionally its general properties such as unitarity analyticity and crossing are used to
establish various useful sum rules for the nucleon magnetic moment (GerasimovndashDrellndashHearn
[5 6] and Schwinger sum rules [7ndash9]) and polarizabilities (eg Baldin [10] and Gell-Mannndash
GoldbergerndashThirring sum rules [11]) More recently the interest in nucleon VVCS has been
renewed in connection with precision atomic spectroscopy where this amplitude enters in
the form of two-photon exchange (TPE) corrections As the TPE corrections in atomic do-
main are dominated by low-energy VVCS it makes sense to calculate them systematically
using chiral perturbation theory (χPT) which is a low-energy effective-field theory of the
Standard Model
In this paper we present a state-of-the-art χPT calculation of the polarized nucleon
VVCS relevant to TPE corrections to hyperfine structure of hydrogen and muonic hydro-
gen This will extend the leading-order χPT evaluation of the TPE effects in hyperfine
splittings [12ndash17] Here we however do not discuss the TPE evaluation but rather focus on
testing the χPT framework against the available empirical information on low-energy spin
structure of the nucleon
It is especially interesting to confront the χPT predictions with the recent measurements
coming from the ongoing ldquoSpin Physics Programrdquo at Jefferson Laboratory [18ndash27] with
the exception of a recent measurement of the deuteron spin polarizability by the CLAS
Collaboration [28] which does not treat correctly complications due to deuteron spin [29]
Our present calculation is extending Ref [30] to the case of polarized VVCS We therefore
use a manifestly-covariant extension of SU(2) χPT to the baryon sector called Baryon χPT
(BχPT) First attempts to calculate VVCS in the straightforward BχPT framework (rather
than the heavy-baryon expansion or the ldquoinfrared regularizationrdquo) were done by Bernard
et al [31] and our group [32] The two works obtained somewhat different results most
notably for the proton spin polarizability δLT Here we improve on [32] in three important
aspects appreciable at finite Q2 1) inclusion of the Coulomb-quadrupole (C2) N rarr ∆ tran-
sition [33 34] 2) correct inclusion of the elastic form-factor contributions to the integrals
Γ1(Q2) I1(Q2) and IA(Q2) (see Sections III C and III D for details) and 3) cancellations
between different orders in the chiral prediction and their effect on the convergence of the
effective-field-theory calculation and thus the error estimate These improvements how-
ever do not bring us closer to the results of [31] and the source of discrepancies most likely
3
FIG 1 The forward Compton scattering or VVCS in case of virtual photons q2 = minusQ2
lies in the different counting and renormalization of the π∆-loop contributions Bernard
et al [31] use the so-called small-scale expansion [35] for the ∆(1232) contributions whereas
we are using the δ-counting scheme [36] (see also Ref [37 Sec 4] for review)
This paper is organized as follows In Sec II A we introduce the polarized VVCS am-
plitudes and their relations to spin structure functions In Sec II B we introduce the spin
polarizabilities appearing in the low-energy expansion (LEX) of the polarized VVCS am-
plitudes In Sec II C we briefly describe our χPT calculation focusing mainly on the
uncertainty estimate In Sec III we show our predictions for the proton and neutron po-
larizabilities as well as some interesting moments of their structure functions In Sec III G
we summarize the results obtained herein comment on the improvements done with respect
to previous calculations and give an outlook to future applications In App B we discuss
the structure functions in particular we define the longitudinal-transverse response func-
tion discuss the ∆-pole contribution and give analytical results for the tree-level πN - and
∆-production channels of the photoabsorption cross sections In App C we give analytical
expressions for the πN -loop and ∆-exchange contributions to the static values and slopes of
the polarizabilities and moments of structure functions The complete expressions also for
the π∆-loop contributions can be found in the Supplemented material
TABLE I Parameters (fundamental and low-energy constants) [38] at the order they appear first
The πN∆ coupling constant hA is fit to the experimental Delta width and the γlowastN∆ coupling
constants gM gE and gC are taken from the pion photoproduction study of Ref [33]
O(p2) α = 1(13704) MN = Mp = 93827 MeV
O(p3) gA = 1270 fπ = 9221 MeV mπ = 13957 MeV
O(p4∆) M∆ = 1232 MeV hA equiv 2gπN∆ = 285 gM = 297 gE = minus10 gC = minus26
4
II CALCULATION OF UNPOLARIZED VVCS AT NLO
A VVCS amplitudes and relations to structure functions
The polarized part of forward VVCS can be described in terms of two independent
Lorentz-covariant and gauge-invariant tensor structures and two scalar amplitudes [3]
T microν(p q) = minus 1
MN
γmicroναqα S1(νQ2)minus 1
M2N
(γmicroνq2 + qmicroγναqα minus qνγmicroαqα
)S2(νQ2) (1)
Here q and p are the photon and nucleon four-momenta (cf Fig 1) ν is the photon
lab-frame energy Q2 = minusq2 is the photon virtuality and γmicroν = 12
[γmicro γν ] and γmicroνα =12
(γmicroγνγα minus γαγνγmicro) are the usual Dirac matrices Alternatively one can use the following
laboratory-frame amplitudes
gTT (νQ2) =ν
MN
[S1(νQ2)minus Q2
MN νS2(νQ2)
] (2a)
gLT (νQ2) =Q
MN
[S1(νQ2) +
ν
MN
S2(νQ2)] (2b)
introduced in Eq (A2) The optical theorem relates the absorptive parts of the forward
VVCS amplitudes to the nucleon structure functions or the cross sections of virtual pho-
toabsorption
ImS1(νQ2) =4π2α
νg1(xQ2) =
MNνK(νQ2)
ν2 +Q2
[Q
νσLT (νQ2) + σTT (νQ2)
] (3a)
ImS2(νQ2) =4π2αMN
ν2g2(xQ2) =
M2NK(νQ2)
ν2 +Q2
[ν
QσLT (νQ2)minus σTT (νQ2)
] (3b)
with α the fine structure constant andK(νQ2) the photon flux factor Note that the photon
flux factor in the optical theorem and the cross sections measured in electroproduction
processes is a matter of convention and has to be chosen in both quantities consistently In
what follows we use Gilmanrsquos flux factor
K(νQ2) equiv |~q | =radicν2 +Q2 (4)
The helicity-difference photoabsorption cross section is defined as σTT = 12 (σ12 minus σ32)
where the photons are transversely polarized and the subscripts on the right-hand side
indicate the total helicities of the γlowastN states The cross section σLT corresponds to a
simultaneous helicity change of the photon and nucleon spin flip such that the total helicity
5
is conserved A detailed derivation of the connection between this response function and
the photoabsorption cross sections can be found in App B The forward VVCS amplitudes
satisfy dispersion relations derived from the general principles of analyticity and causality1
S1(νQ2) =16παMN
Q2
int 1
0
dxg1(xQ2)
1minus x2(ννel)2 minus i0+(5a)
=2MN
π
int infinνel
dν primeν prime 2[QνprimeσLT + σTT
](ν prime Q2)radic
ν prime 2 +Q2(ν prime 2 minus ν2 minus i0+)
νS2(νQ2) =16παM2
N
Q2
int 1
0
dxg2(xQ2)
1minus x2(ννel)2 minus i0+(5b)
=2M2
N
π
int infinνel
dν primeν prime 2[νprime
QσLT minus σTT
](ν prime Q2)radic
ν prime 2 +Q2(ν prime 2 minus ν2 minus i0+)
with νel = Q22MN the elastic threshold
B Low-energy expansions and relations to polarizabilities
The VVCS amplitudes naturally split into nucleon-pole (Spolei ) and non-pole (Snonpole
i )
parts or Born (SBorni ) and non-Born (Si) parts
Si = Spolei + Snonpole
i = SBorni + Si (6)
The Born amplitudes are given uniquely in terms of the nucleon form factors [1]
SBorn1 (νQ2) =
2πα
M
Q2GM(Q2)F1(Q2)
ν2el minus ν2
minus F 22 (Q2)
(7a)
SBorn2 (νQ2) = minus 2παν
ν2el minus ν2
GM(Q2)F2(Q2) (7b)
The same is true for the nucleon-pole amplitudes which are related to the Born amplitudes
in the following way
Spole1 (νQ2) = SBorn
1 (νQ2) +2πα
MN
F 22 (Q2) (8a)
Spole2 (νQ2) = SBorn
2 (νQ2) (8b)
1 The dispersion relation for νS2(νQ2) is used because it is pole-free in the limit Q2 rarr 0 and then ν rarr 0
cf Eq (7b)
6
Here we used the elastic Dirac and Pauli form factors F1(Q2) and F2(Q2) related to the
electric and magnetic Sachs form factors GE(Q2) and GM(Q2) through
F1(Q2) =GE(Q2) + τGM(Q2)
1 + τ (9a)
F2(Q2) =GM(Q2)minusGE(Q2)
1 + τ (9b)
where τ = Q24M2N
A low-energy expansion (LEX) of Eq (5) in combination with the unitarity relations
given in Eq (3) establishes various sum rules relating the nucleon properties (electromag-
netic moments polarizabilities) to experimentally observable response functions [1 3] The
leading terms in the LEX of the RCS amplitudes are determined uniquely by charge mass
and anomalous magnetic moment as the global properties of the nucleon These lowest-
order terms represent the celebrated low-energy theorem (LET) of Low Gell-Mann and
Goldberger [39 40] The polarizabilities related to the internal structure of the nucleon
enter the LEX at higher orders They make up the non-Born amplitudes and can be related
to moments of inelastic structure functions
The process of VVCS can be realized experimentally in electron-nucleon scattering where
a virtual photon is exchanged between the electron and the nucleon This virtual photon
acts as a probe whose resolution depends on its virtuality Q2 In this way one can access
the so-called generalized polarizabilities which extend the notion of polarizabilities to the
case of response to finite momentum transfer The generalized forward spin polarizability
γ0(Q2) and the longitudinal-transverse polarizability δLT (Q2) are most naturally defined via
the LEX of the non-Born part of the lab-frame VVCS amplitudes [1]
1
4πgnonpoleTT (νQ2) =
2α
M2N
IA(Q2) ν + γ0(Q2)ν3 + γ0(Q2)ν5 + (10a)
1
4πgnonpoleLT (νQ2) =
2α
M2N
I3(Q2)Q+ δLT (Q2)ν2Q+ (10b)
Their definitions in terms of integrals over structure functions are postponed to Eqs (19)
and (22) Here we only give the definition of the moment I3(Q2)
I3(Q2) =M2
N
4π2α
int infinν0
dνK(νQ2)
νQσLT (νQ2) =
2M2N
Q2
int x0
0
dx[g1(xQ2) + g2(xQ2)
] (11)
which is related to the Schwinger sum rule in the real photon limit [7ndash9 41 42] The LEX
of the non-pole part of the covariant VVCS amplitudes can be described entirely in terms
7
of moments of inelastic spin structure functions (up to O(ν4) [43])
1
4πSnonpole
1 (νQ2) =2α
MN
I1(Q2) +
2α
MNQ2
[IA(Q2)minus I1(Q2)
]+MNδLT (Q2)
ν2 (12a)
1
4πνSnonpole
2 (νQ2) = 2αI2(Q2) +2α
Q2
[I1(Q2)minus IA(Q2)
]ν2 (12b)
I1(Q2) and IA(Q2) are generalizations of the famous GerasimovndashDrellndashHearn (GDH) sum
rule [5 6] from RCS to the case of virtual photons [1] Their definitions are given in Eqs (26)
and (32) I2(Q2) is the well-known Burkhardt-Cottingham (BC) sum rule [44]
I2(Q2) equiv 2M2N
Q2
int x0
0
dx g2(x Q2) =1
4F2(Q2)GM(Q2) (13)
which can be written as a ldquosuperconvergence sum rulerdquo
Q2
16παM2N
limνrarr0
νS2(νQ2) =
int 1
0
dx g2 (x Q2) = 0 (14)
The latter is valid for any value of Q2 provided that the integral converges for x rarr 0
Combining Eq (5) with the above LEXs of the VVCS amplitudes we can relate IA(Q2)
I1(Q2) γ0(Q2) and δLT (Q2) to moments of inelastic structure functions see Sec III It is
important to note that only γ0(Q2) and δLT (Q2) are generalized polarizabilities The relation
of the inelastic moments IA(Q2) and I1(Q2) to polarizabilities will be discussed in details in
Secs III C and III D The difference between S1(νQ2) and Snonpole1 (νQ2) cf Eq (8a) will
be important in this context
C Details on χPT calculation and uncertainty estimate
In this work we calculated the NLO prediction of BχPT for the polarized non-Born
VVCS amplitudes This includes the leading pion-nucleon (πN) loops see Ref [32 Fig 1]
as well as the subleading tree-level Delta-exchange (∆-exchange) see Ref [30 Fig 2] and
the pion-Delta (π∆) loops see Ref [32 Fig 2] In the δ-power-counting scheme [36] the
LO and NLO non-Born VVCS amplitudes and polarizabilities are of O(p3) and O(p4∆)
respectively2 The LECs are listed in Table I sorted by the order at which they appear in
our calculation At the given orders there are no ldquonewrdquo LECs that would need to be fitted
from Compton processes For more details on the BχPT formalism we refer to Ref [30]
2 In the full Compton amplitude there is a lower order contribution coming from the Born terms leading
to a shift in nomenclature by one order the LO contribution referred to as the NLO contribution etc
see eg Ref [45]
8
where power counting predictive orders (Sec III A) and the renormalization procedure
(Sec III B) are discussed
A few remarks are in order for the inclusion of the ∆(1232) and the tree-level ∆-exchange
contribution In contrast to Ref [32] we include the Coulomb-quadrupole (C2) N rarr ∆
transition described by the LEC gC The relevant Lagrangian describing the non-minimal
γlowastN∆ coupling [33 34] (note that in these references the overall sign of gC is inconsistent
between the Lagrangian and Feynman rules) reads
L(2)∆ =
3e
2MNM+
N T3
igM F
microν partmicro∆ν minus gEγ5Fmicroν partmicro∆ν (15)
+igCM∆
γ5γα(partα∆ν minus partν∆α)partmicroF
microν
+ Hc
with M+ = MN + M∆ and the dual of the electromagnetic field strength tensor F microν =12εmicroνρλFρλ Even though the Coulomb coupling is subleading compared with the electric
and magnetic couplings (gE and gM) its relatively large magnitude cf Table I makes it
numerically important for instance in γ0(Q2) Furthermore we study the effect of modifying
the magnetic coupling using a dipole form factor
gM rarrgM[
1 + (QΛ)2 ]2 (16)
where Λ2 = 071 GeV2 The inclusion of this Q2 dependence mimics the form expected from
vector-meson dominance It is motivated by observing the importance of this form factor
for the correct description of the electroproduction data [33]
To estimate the uncertainties of our NLO predictions we define
δ(Q2) =
radic(∆
MN
)2
+
(Q2
2MN∆
)2
(17)
such that the neglected next-to-next-to-leading order terms are expected to be of relative
size δ2 [33] The uncertainties in the values of the parameters in Table I have a much
smaller impact compared to the truncation uncertainty and can be neglected Unfortunately
∆IA(Q2) γ0(Q2) and γ0(Q2) ie the sum rules involving the cross section σTT (νQ2)
as well as the polarizability ∆I1(Q2) turn out to be numerically small Their smallness
suggests a cancellation of leading orders (which can indeed be confirmed by looking at
separate contributions as shown below) Therefore an error of δ2(Q2)P (Q2) where P (Q2)
is a generalized polarizability might underestimate the theoretical uncertainty for some of
the NLO predictions To avoid this we estimate the uncertainty of our NLO polarizability
9
predictions by
σP (Q2) =
Max[δ4(0)P (0)2 δ4(0)P LO(0)2 δ2(0)PNLO(0)2
]+Max
[δ4(Q2)
[P (Q2)minus P (0)
]2 δ4(Q2)
[P LO(Q2)minus P LO(0)
]2
δ2(Q2)[PNLO(Q2)minus PNLO(0)
]2]12
(18)
where P LO(Q2) is the πN -loop contribution PNLO(Q2) are the ∆-exchange and π∆-loop
contributions and P (Q2) = P LO(Q2) + PNLO(Q2) This error prescription is similar to the
one used in eg Refs [46ndash49] Here since we are interested in the generalized polarizabil-
ities we added in quadrature the error due to the static piece P (0) and the Q2-dependent
remainder P (Q2) minus P (0) Note that the static values of IA(0) and I1(0) are given by the
elastic Pauli form factor which is not part of our BχPT prediction and is considered to be
exact
Note that our result for the spin polarizabilities (and the unpolarized moments [30]) are
NLO predictions only at low momentum transfers Q mπ At larger values of Q amp ∆ they
become incomplete LO predictions Indeed in this regime the ∆ propagators do not carry
additional suppression compared to the nucleon propagators and the π∆ loops are promoted
to LO In general we only expect a rather small contribution from omitted π∆ loops to the
Q2 dependence of the polarizabilities since π∆ loops show rather weak dependence on Q2
compared with the ∆ exchange or πN loops Nevertheless this issue has to be reflected in
the error estimate Since the static polarizabilities P (0) are not affected it is natural to
separate the error on the Q2-dependent remainder P (Q2) minus P (0) as done in Eq (18) To
accommodate for the potential loss of precision above Q amp ∆ we define the relative error
δ(Q2) as growing with increasing Q2 see Eq (17)
Upon expanding our results in powers of the inverse nucleon mass Mminus1N we are able to
reproduce existing results of heavy-baryon χPT (HBχPT) at LO We however do not see
a rationale to drop the higher-order Mminus1N terms when they are not negligible (ie when
their actual size exceeds by far the natural estimate for the size of higher-order terms)
Comparing our BχPT predictions to HBχPT we will also see a deficiency of HBχPT in
the description of the Q2 behaviour of the polarizabilities Note that the O(p4) HBχPT
results from Ref [50 51] which we use here for comparison do not include the ∆ These
references studied the leading effect of the latter in the HBχPT framework using the small-
scale expansion [35] observing no qualitative improvement in the HBχPT description of the
empirical data [50 51] when including it We therefore choose to use the O(p4) results as
the representative HBχPT curves
10
Another approach used in the literature to calculate the polarizabilities in χPT is the
infrared regularization (IR) scheme introduced in Ref [52] This covariant approach tries to
solve the power counting violation observed in Ref [53] by dropping the regular parts of the
loop integrals that contain the power-counting-breaking terms However this subtraction
scheme modifies the analytic structure of the loop contributions and may lead to unexpected
problems as was shown in Ref [54] As we will see in the next section the IR approach
also fails to describe the Q2 behaviour of the polarizabilities
III RESULTS AND DISCUSSION
We now present the NLO BχPT predictions for the nucleon polarizabilities and selected
moments of the nucleon spin structure functions Our results are obtained from the calcu-
lated non-Born VVCS amplitudes and the LEXs in Eqs (10) and (12) For a cross-check
we used the photoabsorption cross sections described in App B In addition to the full NLO
results we also analyse the individual contributions from the πN loops the ∆ exchange
and the π∆ loops
A γ0(Q2) mdash generalized forward spin polarizability
The forward spin polarizability
γ0(Q2) =1
2π2
int infinν0
dν
radic1 +
Q2
ν2
σTT (νQ2)
ν3(19)
=16αM2
N
Q6
int x0
0
dx x2
[g1(xQ2)minus 4M2
Nx2
Q2g2(xQ2)
]
provides information about the spin-dependent response of the nucleon to transversal pho-
ton probes The RCS analogue of the above generalized forward spin polarizability sum
rule is sometimes referred to as the Gell-Mann Goldberger and Thirring (GGT) sum rule
[11] At Q2 = 0 the forward spin polarizability is expressed through the lowest-order spin
polarizabilities of RCS as γ0 = minus(γE1E1 + γM1M1 + γE1M2 + γM1E2) The forward spin po-
larizability of the proton is relevant for an accurate knowledge of the (muonic-)hydrogen
hyperfine splitting as it controls the leading proton-polarizability correction [16 62]
The πN -loop ∆-exchange and π∆-loop contributions to the NLO BχPT prediction of
11
000 005 010 015 020 025 030
-4
-3
-2
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
4D
proton
000 005 010 015 020 025 030
-4
-3
-2
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
4D
neutron
000 005 010 015 020 025 030
0
1
2
Q2 GeV2D
∆LT
10-
4fm
4D
proton
000 005 010 015 020 025 030
0
1
2
3
Q2 GeV2D
∆LT
10-
4fm
4D
neutron
FIG 2 Upper panel Generalized forward spin polarizability for the proton (left) and neutron
(right) as function of Q2 The result of this work the NLO BχPT prediction is shown by the
blue solid line and the blue band The red line represents the LO BχPT result The purple short-
dashed line is the O(p4) HB result from Ref [50] note that the corresponding proton curve is
outside of the plotted range The black dotted line is the MAID model prediction [55ndash57] which
is taken from Ref [1] (proton) and Ref [20] (neutron) The pink band is the IR+∆ result from
Ref [58] and the gray band is the BχPT+∆ result from Ref [31] Empirical extractions for the
proton Ref [18] (blue dots) Ref [59] (purple square) and Ref [60] (orange triangle uncertainties
added in quadrature) and neutron Ref [20] (blue diamonds) and Ref [61] (green dots statistical
and systematic uncertainties added in quadrature) Lower Panel Longitudinal-transverse spin
polarizability for the proton (left) and neutron (right) The orange dot-dashed and purple short-
dashed lines are the O(p3) and O(p4) HB results from Ref [50] The pink band is the IR result
from Ref [58] and the gray band is the covariant BχPT+∆ result from Ref [31] The black dotted
line is the MAID model prediction [55ndash57] note that for the proton we use the updated estimate
from Ref [1] obtained using the π η ππ channels
12
the static forward spin polarizability amount to in units of 10minus4 fm4
γ0p = minus093(92) asymp 201minus 284minus 010 (20a)
γ0n = 003(92) asymp 298minus 284minus 010 (20b)
while the slope is composed as follows in units of 10minus4 fm6
dγ0p(Q2)
dQ2
∣∣∣∣Q2=0
= minus022(4) asymp minus033 + 011 + 001 (21a)
dγ0n(Q2)
dQ2
∣∣∣∣Q2=0
= minus061(7) asymp minus073 + 011 + 001 (21b)
Figure 2 upper panel shows our NLO prediction as well as the LO πN loops compared
to different experimental and theoretical results For the proton we have one determination
at the real-photon point by the GDH collaboration [19] γ0p = minus100(8)(12)times10minus4 fm4 and
further Jefferson Laboratory data [18 60] at very low Q2 For the neutron only data at finite
Q2 are available [20 61] The experimental data for the proton are fairly well reproduced
in the whole Q2 range considered here while for the neutron the agreement improves with
increasing Q2 The HB limit of our πN -loop contribution reproduces the results published
in Refs [50 63] for arbitrary Q2 In addition our prediction is compared to the MAID
model [1 20] the IR+∆ calculation of Ref [58] and the BχPT+∆ result of Ref [31]
The πN -production channel gives a positive contribution to the photoabsorption cross
section σTT (νQ2) at low Q2 cf Fig 10 Accordingly one observes that the πN loops give
a sizeable positive contribution to γ0(Q2) The Delta on the other hand has a very large
effect by cancelling the πN loops and bringing the result close to the empirical data From
Fig 3 upper panel one can see that it is the ∆ exchange which dominates while π∆
loops are negligible This was expected since the forward spin polarizability sum rule is an
integral over the helicity-difference cross section in which σ32 is governed by the Delta at
low energies (the relevant energy region for the sum rule)
To elucidate the difference between the present calculation and the one from Ref [31] we
note that the two calculations differ in the following important aspects Firstly Ref [31] uses
the small-scale counting [64] that considers ∆ and mπ as being of the same size ∆ sim mπ
In practice this results in a set of π∆-loop graphs which contains graphs with one or two
γ∆∆ couplings and hence two or three Delta propagators Such graphs are suppressed in
the δ-counting and thus omitted from our calculation while present in that of Ref [31]
Secondly the Lagrangians describing the interaction of the Delta are constructed differently
and assume slightly different values for the coupling constants In particular we employ
13
000 005 010 015 020 025 030
-3
-2
-1
0
1
2
Q2GeV
2D
Γ01
0-
4fm
4D
proton
000 005 010 015 020 025 030
-3
-2
-1
0
1
2
3
Q2GeV
2D
Γ01
0-
4fm
4D
neutron
000 005 010 015 020 025 030
00
05
10
15
Q2GeV
2D
∆LT1
0-
4fm
4D
proton
000 005 010 015 020 025 030
00
05
10
15
20
25
Q2GeV
2D
∆LT1
0-
4fm
4D
neutron
FIG 3 Contributions of the different orders to the chiral predictions of γ0(Q2) upper paneland δLT (Q2) lower panel for the proton (left) and neutron (right) Red solid line πN -loop
contribution green dot-dashed line ∆-exchange contribution orange dotted line π∆-loop con-
tribution blue long-dashed line total result purple dot-dot-dashed line total result without gCcontribution black short-dashed line total result without gM dipole
(where possible) the so-called ldquoconsistentrdquo couplings to the Delta field ie those couplings
that project out the spurious degree of freedom see Refs [37 65 66] The authors of
Ref [31] on the other hand use couplings where the consistency in this sense is not enforced
The effects of these differences are of higher order in the δ-counting expansion and their
contribution to the Q2 dependence of the considered polarizabilities is expected to be rather
small however the differences at Q2 = 0 could be noticeable [67]
Finally as mentioned in Sec II C the inclusion of the dipole form factor in the magnetic
coupling gM is expected to be important to generate the correct Q2 behaviour of the polar-
izabilities Comparing our predictions for the forward spin polarizability with and without
inclusion of the form factor see Fig 3 upper panel confirms this Without the dipole
our results for the proton and neutron are closer to the ones from Ref [31] where the form
factor is not included For the neutron our prediction without the dipole is able to describe
14
the experimental points at very low Q2 deviating from the data with increasing Q2 The
π∆-loop contribution does not modify the Q2 behavior of γ0(Q2) and only differs from
Ref [31] by a small global shift Note also the relatively large effect of gC which generates
a sign change for virtualities above sim 02 GeV2 see Fig 3 upper panel
B δLT (Q2) mdash longitudinal-transverse polarizability
The longitudinal-transverse spin polarizability
δLT (Q2) =1
2π2
int infinν0
dν
radic1 +
Q2
ν2
σLT (νQ2)
Qν2(22)
=16αM2
N
Q6
int x0
0
dx x2[g1(xQ2) + g2(xQ2)
]
contains information about the spin structure of the nucleon and is another important
input in the determination of the (muonic-)hydrogen hyperfine splitting [16 62] It is also
relevant in studies of higher-twist corrections to the structure function g2(xQ2) given by
the moment d2(Q2) [51] see Section III E The peculiarity of the response encoded in this
polarizability is that it involves a spin flip of the nucleon and a polarization change of the
photon see App B and Fig 11
It is expected that the Delta isobar gives only a small contribution to δLT (Q2) what makes
this polarizability a potentially clean test case for chiral calculations Consequently there are
relatively many different theoretical calculations of δLT (Q2) coming from different versions
of χPT with baryons (HB IR and covariant) Ref [50] found a systematic deviation of the
HB result for δLTn(Q2) from the MAID model prediction This disagreement was identified
by the authors of Ref [68] as a puzzle involving the neutron δLT polarizabilitymdashthe δLT
puzzle The IR calculation in Ref [58] also showed a deviation from the data and predicted
a rapid rise of δLT (Q2) with growing Q2 The problem is solved by keeping the relativistic
structure of the theory as the BχPT+∆ result of Ref [31] showed
As expected already the leading πN loops provide a reasonable agreement with the
experimental data cf Fig 2 lower panel Since the ∆-exchange contribution to δLT (Q2)
is small the effect of the gM form factor is negligible in this polarizability as is that of the gC
coupling cf Fig 3 lower panel In fact we predict both the ∆-exchange and the π∆-loop
contributions to be small and negative This is in agreement with the MAID model which
predicts a small and negative contribution of the P33 wave to δLT (Q2) However in the
calculation of Ref [31] which is different from the one presented here only in the way the
∆(1232) is included the contribution of this resonance to δLTp(Q2) is sizeable and positive
15
The authors of that work attributed this large contribution to diagrams where the photons
couple directly to the Delta inside a loop As mentioned in Sec III F the effect of such loop
diagrams does not change the Q2 behaviour of the polarizabilities On the other hand it
can produce a substantial shift of the δLT (Q2) as a whole A higher-order calculation should
resolve the discrepancy between the two covariant approaches however it will partially lose
the predictive power since the LECs appearing at higher orders will have to be fitted to
experimental data
The πN -loop ∆-exchange and π∆-loop contributions to the NLO BχPT prediction of
the static longitudinal-transverse polarizability are in units of 10minus4 fm4
δLTp = 132(15) asymp 150minus 016minus 002 (23a)
δLTn = 218(23) asymp 235minus 016minus 002 (23b)
while the slopes are in units of 10minus4 fm6
dδLTp(Q2)
dQ2
∣∣∣∣Q2=0
= minus085(8) asymp minus080minus 004minus 001 (24a)
dδLTn(Q2)
dQ2
∣∣∣∣Q2=0
= minus124(12) asymp minus119minus 004minus 001 (24b)
C IA(Q2) mdash a generalized GDH integral
The helicity-difference cross section σTT exhibits a faster fall-off in ν than its spin-averaged
counterpart σT This is due to a cancellation between the leading (constant) terms of σ12
and σ32 at large ν3 The resulting 1ν fall-off of the helicity-difference cross section allows
one to write an unsubtracted dispersion relation for the VVCS amplitude gnonpoleTT (νQ2) cf
Eq (10a) This is the origin of the GDH sum rule [5 6]
minus α
2M2N
κ2 =1
2π2
int infinν0
dνσTT (ν)
ν (25)
which establishes a relation to the anomalous magnetic moment κ It is experimentally
verified for the nucleon by MAMI (Mainz) and ELSA (Bonn) [71 72]
There are two extensions of the GDH sum rule to finite Q2 the generalized GDH integrals
3 Notice that a constant term in σTT at ν rarrinfin is forbidden by crossing symmetry
16
000 005 010 015 020 025 030
-30
-25
-20
-15
-10
-05
00
Q2 GeV2D
I Aproton
000 005 010 015 020 025 030
-25
-20
-15
-10
-05
00
Q2 GeV2D
I A
neutron
000 005 010 015 020 025 030
-05
00
05
Q2 GeV2D
I 1
proton
000 005 010 015 020 025 030
-10
-08
-06
-04
-02
00
02
Q2 GeV2D
I 1
neutron
FIG 4 Upper panel The generalized GDH integral IA(Q2) for the proton (left) and neutron
(right) as function of Q2 The result of this work the NLO BχPT prediction is shown by the blue
solid line and the blue band The red line represents the LO BχPT result The purple short-dashed
line is the O(p4) HB result from Ref [50 51] The gray band is the BχPT+∆ result from Ref [31]
The black dotted line is the MAID model prediction [69] Experimental extractions for the proton
Ref [60] (orange triangle uncertainties added in quadrature) and neutron Refs [21][27] where
magenta dotsorange diagonal crosses correspond to data and red squareslilac crosses correspond
to data plus extrapolation to unmeasured energy regions The green stars at the real-photon point
are derived from the anomalous magnetic moments κp asymp 1793 and κn asymp minus1913 [70] Lower
panel The generalized GDH integral I1(Q2) for the proton (left) and neutron (right) as function
of Q2 The purple short-dashed line is the HB result from Ref [51] Experimental extractions for
the proton Ref [18] (blue dots) and Ref [60] (orange triangle uncertainties added in quadrature)
and neutron Ref [61][27] (uncertainties added in quadrature) where green dotsorange diagonal
crosses correspond to data and gray squareslilac crosses correspond to data plus extrapolation to
unmeasured energy regions
17
IA(Q2) and I1(Q2) The latter will be discussed in Sec III D The former is defined as4
minus α
2M2N
IA(Q2) = minus 1
8π2
int infinν0
dν
radic1 +
Q2
ν2
σTT (νQ2)
ν(26)
=α
Q2
int x0
0
dx[4M2
Nx2
Q2g2(xQ2)minus g1(xQ2)
]
Due to its energy weighting the integral in Eq (26) converges slower than the one in the
generalized forward spin polarizability sum rule (19) Therefore knowledge of the cross
section at higher energies is required and the evaluation of the generalized GDH integral
IA(Q2) is not as simple as the evaluation of γ0(Q2)
The generalized GDH integral IA(Q2) is directly related to the non-pole amplitude
gnonpoleTT (νQ2) which differs from non-Born amplitude gTT (νQ2) by a term involving the
elastic Pauli form factor
gnonpoleTT (νQ2) = gTT (νQ2)minus 2παν
M2N
F 22 (Q2) (27)
cf Eqs (2a) and (8a) Consequently IA(Q2) is not a pure polarizability but also contains
an elastic contribution The ldquonon-polarizabilityrdquo or the Born part of IA(Q2) is given by
IBornA (Q2) = IA(Q2)minus∆IA(Q2) = minus1
4F 2
2 (Q2) (28)
where we refer to the polarizability part as ∆IA(Q2) The same is true for the generalized
GDH integral I1(Q2) which is directly related to Snonpole1 (νQ2)
IBorn1 (Q2) = I1(Q2)minus∆I1(Q2) = minus1
4F 2
2 (Q2) (29)
In the following we will add the Born parts to our LO and NLO BχPT predictions for
the polarizabilities ∆IA(Q2) and ∆I1(Q2) employing an empirical parametrization for the
elastic Pauli form factor [73] This allows us to compare to the experimental results for
IA(Q2) and I1(Q2) cf Fig 4 Note that the blue error bands only describe the uncertainties
of our BχPT predictions of the polarizabilities while the elastic contributions are considered
to be exact as explained in Sec II C The uncertainties of the polarizability predictions are
therefore better reflected in Fig 5 where we show the contributions of the different orders
to the BχPT predictions of ∆IA(Q2) and ∆I1(Q2) as well as the total results with error
bands
4 Note that IA(Q2) is sometimes called ITT (Q2)
18
000 005 010 015 020 025 030
-08
-06
-04
-02
00
02
Q2 GeV2D
DI A
proton
000 005 010 015 020 025 030
-10
-08
-06
-04
-02
00
02
Q2 GeV2D
DI A
neutron
000 005 010 015 020 025 030-005
000
005
010
015
020
025
Q2 GeV2D
DI 1
proton
000 005 010 015 020 025 030-015
-010
-005
000
005
010
015
Q2 GeV2D
DI 1
neutron
FIG 5 Contributions of the different orders to the chiral predictions of ∆IA(Q2) upper paneland ∆I1(Q2) lower panel for the proton (left) and neutron (right) Red solid line πN -loop
contribution green dot-dashed line ∆-exchange contribution orange dotted line π∆-loop contri-
bution blue solid line and blue band total result purple dot-dot-dashed line total result without
gC contribution black short-dashed line total result without gM dipole
The E97-110 experiment at Jefferson Lab has recently published their data for IAn(Q2) in
the region of 0035 GeV2 lt Q2 lt 024 GeV2 [27] In addition there are results for IAn(Q2)
from the earlier E94-010 experiment [21] and for IAp(Q2) from the E08-027 experiment [60]
The O(p4) HB calculation gives a large negative effect [51] which does not describe the data
The BχPT+∆ result from Ref [31] which mainly differs from our work by the absence of
the dipole form factor in gM looks similar to this HB result and only describes the data
points at lowest Q2 Our NLO prediction however follows closely the Q2 evolution of the
data In Fig 5 upper panel we show the polarizability ∆IA(Q2) whose Q2 evolution is
clearly dominated by the ∆ exchange Similar to the case of γ0p(Q2) inclusion of the dipole
in gM and the Coulomb coupling gC is very important in order to describe the experimental
data The LO prediction on the other hand slightly overestimates the data cf Fig 4
upper panel
19
At the real-photon point IA(0) = minusκ2
4and ∆IA(0) = 0 Therefore we give only the
slope of the polarizability ∆IA(Q2) [showing also the separate contributions from πN loops
∆ exchange and π∆ loops] in units of GeVminus2
d∆IAp(Q2)
dQ2
∣∣∣∣Q2=0
= minus858(343) asymp 238minus 1121 + 025 (30a)
d∆IAn(Q2)
dQ2
∣∣∣∣Q2=0
= minus955(343) asymp 141minus 1121 + 025 (30b)
Including the empirical Pauli form factor [73] we find in units of GeVminus2
dIAp(Q2)
dQ2
∣∣∣∣Q2=0
= minus318dIAn(Q2)
dQ2
∣∣∣∣Q2=0
= minus300 (31)
D Γ1(Q2) and I1(Q2) mdash the first moment of the structure function g1(xQ2)
The second variant for a generalization of the GDH sum rule to finite Q2 is defined as
minus α
2M2N
I1(Q2) = minus 1
8π2
int infinν0
dν1radic
ν2 +Q2
[σTT (νQ2) +
Q
νσLT (νQ2)
](32)
= minus α
Q2
int x0
0
dx g1(xQ2)
where I1(0) = minusκ2
4 This generalized GDH integral directly stems from the amplitude
Snonpole1 (νQ2) with the LEX from Eq (12a) It is given by the first moment of the struc-
ture function g1(xQ2) Γ1(Q2) =int x0
0dx g1(xQ2) as follows I1(Q2) =
2M2N
Q2 Γ1(Q2) The
isovector combination
Γ1(pminusn)(Q2) =
int x0
0
dx[g1p(xQ
2)minus g1n(xQ2)] (33)
is related to the axial coupling of the nucleon through the Bjorken sum rule [74 75]
limQ2rarrinfin
Γ1(pminusn)(Q2) =
gA6 (34)
As explained in Eq (28) the moment I1(Q2) splits into a polarizability part ∆I1(Q2) and
a Born part IBorn1 (Q2) Figure 4 lower panel shows the Q2 dependence of I1(Q2) which in
contrast to IA(Q2) shown in Figure 4 upper panel is clearly dominated by its Born part
and the elastic Pauli form factor The πN -loop ∆-exchange and π∆-loop contributions to
the polarizability ∆I1(Q2) are shown in Fig 5 lower panel Comparing to Fig 5 upper
20
000 005 010 015 020 025 030
-006
-004
-002
000
002
004
Q2 GeV2D
G1
proton
000 005 010 015 020 025 030
-006
-005
-004
-003
-002
-001
000
001
Q2 GeV2D
G1
neutron
FIG 6 First moment of the structure function g1(xQ2) for the proton (left) and neutron (right)
as function of Q2 The legend is the same as in Fig 4
panel one sees that ∆I1(Q2) is less sensitive to gC and the dipole form factor in gM than
∆IA(Q2)
For the proton our NLO BχPT prediction gives a very good description of the ex-
perimental data [18 60] and is in reasonable agreement with the MAID prediction [69]
For the neutron one observes good agreement with the empirical evaluations including ex-
trapolations to unmeasured energy regions starting from Q2 gt 01 GeV2 [27 61] In the
region of Q2 lt 005 GeV2 one observes an interesting tension between the recent E97-
110 experiment [27] and the data from CLAS [61] While the newest measurement finds
I1n(0035 GeV2) lt κ2n4 thus suggesting a negative slope at low Q the older measurement
found a rather large value for I1n(00496 GeV2) A similar but milder behaviour is seen in
the E97-110 [27] and E94-010 [21] data for IAn The MAID predictions do not agree with
the CODATA recommended values for the anomalous magnetic moments of the proton and
neutron [70] which in our work are imposed by using empirical parametrizations for the
elastic Pauli form factors [73] The slope of the HB result from Ref [51] is too large and
therefore only reproduces the data at very low Q2
Figure 6 shows the moment Γ1(Q2) for the proton and neutron while Fig 7 shows the
isovector combination Γ1 pminusn(Q2) The LO and NLO BχPT predictions are identical be-
cause our calculation produces the same Delta contributions for the proton and the neutron
For the isovector combination the MAID model only agrees with the data at very low
Q2 lt 010 GeV2 The same is true for the IR result [58 76] while all other chiral results
describe the data NLO BχPT (this work) BχPT+∆ [31] and HBχPT [51]
At the real-photon point I1(0) = minusκ2
4and ∆I1(0) = 0 Therefore we give only the
slope of the polarizability ∆I1(Q2) [showing also the separate contributions from πN loops
21
000 005 010 015 020 025 030-002
000
002
004
006
008
010
Q2 GeV2D
G1Hp-
nL
FIG 7 Isovector combination of Γ1(Q2) as function of Q2 The legend is the same as in Fig 6
The pink curve is the IR result from Ref [58 76] The experimental points are from Ref [77]
(brown dots) and Ref [23] (orange squares)
∆ exchange and π∆ loops] in units of GeVminus2
d∆I1p(Q2)
dQ2
∣∣∣∣Q2=0
= 039(4) asymp 034minus 053 + 058 (35a)
d∆I1n(Q2)
dQ2
∣∣∣∣Q2=0
= minus101(10) asymp minus107minus 053 + 058 (35b)
Including the empirical Pauli form factor [73] we find in units of GeVminus2
dI1p(Q2)
dQ2
∣∣∣∣Q2=0
= 580dI1n(Q2)
dQ2
∣∣∣∣Q2=0
= 553 (36)
E d2(Q2) mdash a measure of color polarizability
Another interesting moment to consider is d2(Q2) which is related to the twist-3 part of
the spin structure function g2(xQ2) [79 80]
d2(Q2) equiv 3
int 1
0
dx x2[g2(xQ2)minus gWW2 (xQ2)] (37)
where gWW2 (xQ2) is the twist-2 part of g2(xQ2) Using the Wandzura-Wilczek relation [81]
one can relate d2(Q2) to moments of the spin structure functions g1(xQ2) and g2(xQ2)
d2(Q2) =
int 1
0
dx x2 [3g2(xQ2) + 2g1(xQ2)] (38)
22
000 005 010 015 020 025 030
0000
0001
0002
0003
0004
0005
0006
0007
Q2 GeV2D
d2
proton
000 005 010 015 020 025 030
0000
0002
0004
0006
0008
0010
0012
Q2 GeV2D
d2
neutron
000 005 010 015 020 025 030
-1
0
1
2
Q2 GeV2D
Γ 010
-4
fm6
D
proton
000 005 010 015 020 025 030-2
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
6D
neutron
FIG 8 Upper panel The inelastic moment d2(Q2) for the proton (left) and neutron (right) as
function of Q2 The result of this work the NLO BχPT prediction is shown by the blue solid line
and the blue band The red line represents the LO BχPT result The purple short-dashed line is
the O(p4) HB result from Ref [50 51] The black dotted line is the MAID model prediction [69]
The experimental points for the neutron (cyan dots) are from Ref [22] Lower panel Fifth-order
generalized forward spin polarizability γ0(Q2) for the proton (left) and neutron (right) as function
of Q2 The experimental points for the proton are from Ref [59] (purple square) and Ref [78]
(orange dot)
This relation however only holds for asymptotically large Q2 It is also in the high-Q2
region where d2(Q2) is a measure of color polarizability [82 83] through its relation to the
gluon field strength tensor [80] We refer to Ref [84] for a recent review on the spin structure
of the nucleon including a discussion of sum rules for deep inelastic scattering and color
polarizabilities
What we consider in the following is the inelastic part of d2(Q2) defined as the moment
of g1(xQ2) and g2(xQ2) spin structure functions cf Eq (38)
d2(Q2) =
int x0
0
dx x2 [3g2(xQ2) + 2g1(xQ2)] (39)
23
This moment provides another testing ground for our BχPT predictions through comparison
with experiments on the neutron [22] Going towards the low-Q2 region the interpretation
of d2(Q2) in terms of color polarizabilities will fade out The above definition however
implies it is related to other VVCS polarizabilities
d2(Q2) =Q4
8M4N
[M2
NQ2
αδLT (Q2) + I1(Q2)minus IA(Q2)
] (40)
Note that d2(Q2) and its first two derivatives with respect to Q2 vanish at Q2 = 0 The
considerations in Eqs (28) and (29) have no effect on d2(Q2) since the Born contribution
from IA(Q2) and I1(Q2) cancel out Therefore d2(Q2) is a pure polarizability
In Fig 8 upper panel we show our NLO BχPT prediction and other results for d2(Q2)
While MAID [69] and BχPT describe the experimental data for the neutron [22] very well
the HB limit [50 51] is showing a fast growth with Q2 This illustrates the importance
of keeping the relativistic result Note also that even though the πN -loop contribution
is dominant both gC and the form factor in gM are essential to obtain a curvature that
reproduces the data cf Fig 9 upper panel For the proton there are to our knowledge
no experimental results to compare with However the agreement between the NLO BχPT
prediction and the MAID prediction at low energies is reasonable
F γ0(Q2) mdash fifth-order generalized forward spin polarizability
It is interesting to compare the generalized fifth-order forward spin polarizability sum
rule
γ0(Q2) =1
2π2
int infinν0
dν
radic1 +
Q2
ν2
σTT (νQ2)
ν5(41)
=64αM4
N
Q10
int x0
0
dx x4
[g1(xQ2)minus 4M2
Nx2
Q2g2(xQ2)
]
to the sum rule integrals for IA(Q2) and γ0(Q2) since they differ merely by their energy
weighting of σTT (νQ2) and a constant prefactor cf Eqs (19) (26) and (41) From IA(Q2)
to γ0(Q2) to γ0(Q2) the energy suppression is increasing by a factor of νminus2 respectively
Therefore the description of γ0(Q2) should be easiest in a low-energy effective-field theory
such as χPT whereas γ0(Q2) and IA(Q2) receive larger contributions from higher energies
In Fig 8 lower panel we show our LO and NLO BχPT predictions for γ0(Q2) One
can see that the πN -loop contribution is positive (in accordance to what we see for the cross
section σTT see Fig 10) The Delta shifts it substantially especially in the low Q2 region
24
000 005 010 015 020 025 030
-0004
-0002
0000
0002
0004
0006
0008
0010
Q2GeV
2D
d2
proton
000 005 010 015 020 025 030
-0005
0000
0005
0010
0015
Q2GeV
2D
d2
neutron
000 005 010 015 020 025 030
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
6D
proton
000 005 010 015 020 025 030
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
6D
neutron
FIG 9 Contributions of the different orders to the chiral predictions of d2(Q2) upper panel and
γ0(Q2) lower panel for the proton (left) and neutron (right) Red solid line πN -loop contribu-
tion green dot-dashed line ∆-exchange contribution orange dotted line π∆-loop contribution
blue long-dashed line total result purple dot-dot-dashed line total result without gC contribution
black short-dashed line total result without gM dipole
bringing it into a better agreement with data In general the BχPT curves start above the
empirical data points at the real-photon point and then decrease asymptotically to zero
above Q2 gt 01 GeV2 On the other hand the MAID prediction reproduces the empirical
data at the real-photon point then decreases to negative values until about Q2 gt 006 GeV2
from where it also starts to asymptotically approach zero Consequently our NLO BχPT
prediction of γ0(Q2) is consistently above the MAID prediction This is very different to
what we saw for IA(Q2) in Fig 4 upper panel where the MAID prediction at the real-
photon point is above the experimental value While the agreement of our predictions with
the empirical data is in general quite good for all moments of σTT (νQ2) one should point
out that both for γ0n(Q2) and γ0p(Q2) we overestimate the data at low Q2 For IA(Q2) such
observation cannot be made because ∆IA(0) = 0 and thus IA(0) is given by the empirical
Pauli form factor only From IA(Q2) γ0(Q2) and γ0(Q2) the latter has the smallest however
25
non-negligible dependence on gC and the dipole in gM cf Fig 9 lower panel
The πN -loop ∆-exchange and π∆-loop contributions to the NLO BχPT prediction of
the static fifth-order forward spin polarizability amount to in units of 10minus4 fm6
γ0p = 112(30) asymp 208minus 096minus 001 (42a)
γ0n = 195(30) asymp 292minus 096minus 001 (42b)
while the slope is composed as follows in units of 10minus4 fm8
dγ0p(Q2)
dQ2
∣∣∣∣Q2=0
= minus084(10) asymp minus100 + 016 + 000 (43a)
dγ0n(Q2)
dQ2
∣∣∣∣Q2=0
= minus142(15) asymp minus158 + 016 + 000 (43b)
Note that the HB prediction of the static polarizability γ0p (423 at O(p3) and 365 at O(ε3)
[78 85]) is almost one order of magnitude larger than the empirical value and therefore not
shown in Fig 8
G Summary
Our results are summarized in Table II where we give the contributions of the different
orders to the chiral predictions of the polarizabilities and their slopes at the real-photon
point A quantitative comparison of our predictions for the static spin polarizabilities to
the work of Bernard et al [31] and different empirical evaluations is shown in Table III We
can see that the inclusion of the Delta turns out to be very important for all moments of
the helicity-difference cross section To describe the Q2 behavior of the polarizabilities the
magnetic coupling of the N rarr ∆ transition should be modified by a dipole form factor as
has been observed previously in the description of electroproduction data [33] This dipole
form factor effectively takes account of vector-meson exchanges The Coulomb-quadrupole
N rarr ∆ transition despite its subleading order is important in the description of some
moments of spin structure functions This is contrary to what we saw for the moments of
unpolarized structure functions [30] where the Coulomb coupling had a negligible effect
The π∆ loops are mainly relevant for the generalized GDH integrals
26
TABLE II The NLO BχPT predictions for the forward VVCS polarizabilities and their slopes at
Q2 = 0 The contributions of the πN loops the ∆ exchange and the π∆ loops are shown together
with the combined total result Note that IA(0) = I1(0) = d2(0) = 0 and (d2)prime = 0
πN loops ∆ exchange π∆ loops Total
γ0 p 201 minus284 minus010minus093(92)
(10minus4 fm4) n 298 003(92)
δLT p 150 minus016 minus002132(15)
(10minus4 fm4) n 235 218(23)
γ0 p 208 minus096 minus001112(30)
(10minus4 fm6) n 292 195(30)
(γ0)prime p minus033011 001
minus022(4)
(10minus4 fm6) n minus073 minus061(7)
(δLT )prime p minus080 minus004 minus001minus085(8)
(10minus4 fm6) n minus119 minus124(12)
(γ0)prime p minus100016 000
minus084(10)
(10minus4 fm8) n minus158 minus142(15)
(∆IA)prime p 238 minus1121 025minus858(343)
(GeVminus2) n 141 minus955(343)
(∆I1)prime p 034 minus053 058039(4)
(GeVminus2) n minus107 minus101(10)
TABLE III Our NLO BχPT predictions for the spin polarizabilities at Q2 = 0 compared with
the BχPT+∆ predictions from Bernard et al [31] and the available empirical information Where
the reference is not given the empirical number is provided by the MAID analysis [55 69] with
unspecified uncertainty
Proton Neutron
This work BχPT+∆ Empirical This work BχPT+∆ Empirical
γ0 minus093(92) minus174(40) minus100(8)(12) [19] 003(92) minus077(40) minus0005
(10minus4 fm4) minus090(8)(11) [78] [MAID]
minus0929(105) [59]
δLT 132(15) 240(1) 134 218(23) 238(3) 203
(10minus4 fm4) [MAID] [MAID]
γ0 112(30) 060(7)(7) [78] 195(30) 123
(10minus4 fm6) 0484(82) [59] [MAID]
IV CONCLUSIONS
We have presented a complete NLO calculation of the polarized non-Born VVCS ampli-
tudes in covariant BχPT with pion nucleon and ∆(1232) fields The dispersion relations
between the VVCS amplitudes and the tree-level photoabsorption cross sections served as a
27
cross-check of these calculations
The obtained moments of the proton and neutron spin structure functions related to gen-
eralized polarizabilities and GDH-type integrals agree well with the available experimental
data The description of their Q2 evolution is improved compared to the previous χPT pre-
dictions In particular the NLO BχPT predictions obtained here give a better description
of the empirical data (eg from the Jefferson Laboratory ldquoSpin Physics Programrdquo) than
the HB [50 51] and IR [58] calculations
The demonstrated predictive power of the χPT framework amplitudes makes it well
suited for extending the χPT evaluation of the TPE effect in the hyperfine structure of
(muonic-)hydrogen [15ndash17] to next-to-leading order
ACKNOWLEDGEMENTS
We thank Lothar Tiator and Marc Vanderhaeghen for helpful discussions This work
is supported by the Deutsche Forschungsgemeinschaft (DFG) through the Collaborative
Research Center [The Low-Energy Frontier of the Standard Model (SFB 1044)] JMA ac-
knowledges support from the Community of Madrid through the ldquoPrograma de atraccion de
talento investigador 2017 (Modalidad 1)rdquo and the Spanish MECD grants FPA2016-77313-P
FH gratefully acknowledges financial support from the Swiss National Science Foundation
Appendix A Tensor decompositions of the VVCS amplitudes
In this appendix we review the decomposition of the forward VVCS process into tensor struc-
tures and scalar amplitudes In particular we consider the connection between the covariant and
the semi-relativistic decomposition in the lab frame that is defined in terms of the conventional
transverse longitudinal transverse-transverse and transverse-longitudinal amplitudes
As explained in Sec II A the process of forward VVCS off the nucleon can be described in
terms of four explicitly covariant amplitudes S1 2 and T1 2 [3]
T (νQ2) =
(minusgmicroν +
qmicroqν
q2
)T1(νQ2) +
1
M2N
(pmicro minus p middot q
q2qmicro)(
pν minus p middot qq2
qν)T2(νQ2) (A1)
minus 1
MNγmicroναqα S1(νQ2)minus 1
M2N
(γmicroνq2 + qmicroγναqα minus qνγmicroαqα
)S2(νQ2)
εprimelowastmicro εν
where εmicro (εprimelowastmicro ) are the incoming (outgoing) photon polarization vectors ν is the photon lab-frame
energy and Q2 is the photon virtuality Alternatively the decomposition in the laboratory frame
28
(which in the forward case coincides with the Breit frame) is parametrized in terms of the nucleon
Pauli matrices ~σ and the four scalar functions fL fT gTT and gLT
T (νQ2) = ε0 εprimelowast0 fL(νQ2) + (~ε primelowast middot ~ε ) fT (νQ2) + i~σ middot (~ε primelowast times ~ε ) gTT (νQ2) (A2)
minusi~σ middot [(ε0~εprimelowast minus ~ε ε primelowast0 )times q] gLT (νQ2)
Here ~q and q = ~q|~q | are the photon three-momentum in the lab system and its unit vector The
modified polarization vector components are given by
ε0 =
[ε0 minus
ν
|~q | (~ε middot q )
] |~q |Q ~ε = ~εminus q (~ε middot q ) (A3)
where ε = (ε0~ε ) is the usual incoming photon polarization vector and εprimelowast the outgoing polarization
vector The LEX of the lab frame amplitudes [Eq (10)] can serve in particular as the definition
of the generalized polarizabilities The lab frame amplitudes are also conveniently used for the
definition of the response functions see the example of the scalar amplitude gLT (νQ2) and the
corresponding response function σLT (νQ2) below in App B
Appendix B Photoabsorption cross sections
In the forward kinematics the spin-dependent VVCS amplitudes and the spin polarizabilities
can be described in terms of the polarized structure functions g1(xQ2) and g2(xQ2) or equiv-
alently the helicity-difference cross section σTT (νQ2) and the longitudinal-transverse response
function σLT (νQ2) with the help of dispersion relations (5) and the optical theorem (3) In this
way the photoabsorption cross sections measured in electroproduction processes form the basis
for most empirical evaluations shown throughout Sec III In the following we present the BχPT
predictions for the tree-level cross sections of πN - π∆- and ∆-production through photoabsorption
on the nucleon cf Figs 8 9 and 10 in Ref [30] In Secs B 1 and B 2 we will discuss the leading
πN -production channel and the ∆-production channel respectively We used these cross sections
to verify the polarizability predictions obtained otherwise from the calculated non-Born VVCS
amplitudes Due to the bad high-energy behavior of the π∆-production cross sections in BχPT
cf Fig 10 the dispersion relations in Eq (5) require further subtractions for a reconstruction of
the π∆-loop contribution to the spin-dependent VVCS amplitudes Therefore not all polarizabil-
ities could be verified but only those appearing as higher-order terms in the LEX of the VVCS
amplitudes such as γ0 [16]
29
proton neutron
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Neutron
02 04 06 08 100
50
100
150
200
250
[GeV]
T [b]
02 04 06 08 10
-50
0
50
100
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Neutron
02 04 06 08 100
50
100
150
200
250
[GeV]
T [b]
02 04 06 08 10
-50
0
50
100
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02040608100
50
100
150
200
250
300
[GeV]
T[b]
0204060810
-50
0
50
[GeV]
TT[b]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
0204060810
00
02
04
06
[GeV]
LTQ
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
0204060810
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
FIG 10 Photoabsorption cross sections for πN (red) and π∆ production (orange) with Q2 = 0
(solid) and Q2 = 01 GeV2 (dashed for πN and dotted for π∆ channel)
1 πN-production channel
In order to extract the response function σLT (νQ2) we have developed a method similar to
the one used to calculate σTT (νQ2) see for example Ref [86] For σLT (νQ2) however the
calculation is more complicated because one has to take into account that the associated Compton
process involves a spin-flip of the nucleon as illustrated in Fig 11 When calculating the cross
section the product of the incoming nucleon spinors has to reflect this flip
The forward VVCS amplitude related to σLT (νQ2) mdash and δLT (Q2) mdash is gLT (νQ2) It can
be extracted from Eq (A2) if one takes the modified polarization vector components in Eq (A3)
with ε = εL and εprimelowast = εlowastplusmn as input where εL = 1Q(|~q | 0 0 q0) and εplusmn = ∓ 1radic
2(0 1plusmni 0) are the
standard longitudinal and transverse polarization vectors respectively For εL and εlowastplusmn only the
choice of helicities hprime = plusmn12 and h = ∓12 gives a non-zero contribution and one obtains
χdaggerhprime T (νQ2)χh = χdaggerhprimeminusi~σ middot [(ε0~εprimelowast minus ~ε ε primelowast0 )times q] gLT (νQ2)χh =
radic2 gLT (νQ2) (B1)
where χh and χdaggerhprime are two-component Pauli spinors with opposite helicities or here spins
Let us now consider the related photoabsorption process and in particular the tree-level γlowastN rarr
30
s = 12
sprime = minus12
λ = 0 λprime = 1
rprime rprime
s=12sprime=minus1
2
λ=0λprime=1
rprimerprime
s=12
sprime=minus12
λ=0λprime=1
rprimerprime hacute=frac12h=-frac12
λacute=1λ=0
FIG 11 Relation between the forward Compton process and the photoabsorption process given
by the optical theorem In particular we show the longitudinal-transverse contribution The
double-line arrows represent the spin of the external particles while the dot represents the scalar
(longitudinal) polarization of the incoming photon Inside the blob the intermediate states are
represented eg nucleons with spins rprime (which are averaged in the calculation of the cross section)
and pions
πN channel see diagrams in Fig 8 of Ref [30] We define the πN -production amplitude as
T = uhB (PB)sumi
Ai(s t)Γi uhA(PA) (B2)
with the Dirac structures
Γ1 = γ5 (B3a)
Γ2 =1
2
[qA ε
]γ5 (B3b)
where uhA(PA) and udaggerhB (PB) are the Dirac spinors and PA and PB are the four-momenta of
the incoming and outgoing nucleons respectively When calculating the photoabsorption cross
section related to the VVCS amplitude in Eq (B1) the nucleon spin flip should be implemented
by uhprime(PA) in T dagger and uh(PA) in T together with the appropriate transverse and longitudinal
photon polarization vectors εlowastplusmn and εL
However if one wants to use the properties of the Dirac matrices it is more useful to construct
an operator to produce this spin flip in the external nucleons of Fig 11 This is accomplished by
introducing the projector ΓLT equiv 12radic
2(γ1 + iγ2)γ5 which also takes into account the extra factor
radic2 in Eq (B1) We checked that with this projector one correctly extracts δLT by comparing the
HB limit of our result to the HB result of Ref [50] where the authors calculate this polarizability
from the Compton amplitude directly With all those ingredients the longitudinal-transverse cross
section is calculated in the following way
σLT (νQ2) =1
64π2 s
|~pf |cm
|~pi|cm
int 1
minus1dcos θ
sumij
AiAdaggerjXij (B4)
31
with
Xij = Tr[(PB +MN )Γi(PA +MN )ΓLTγ0Γdaggerjγ
0] (B5)
where θ is the scattering angle in the center-of-mass (cm) frame and |~pi|cm (|~pf |cm) is the three-
momentum of an incoming (outgoing) particle in the cm frame An explicit calculation of the
matrix Xij leads to
X = MNQ
0 2(PB minus PA) middot εLminusradic
2|~qf |cm sin θ (sminus u)
(B6)
where |~qi|cm (|~qf |cm) is the relative three-momentum of the incoming (outgoing) particles in the
cm frame Here s t and u are the usual Mandelstam variables For the different γlowastN rarr πN
channels we obtain the following amplitudes Ai where we introduce qA as the four-momentum of
the incoming photon and qB as the four-momentum of the outgoing pion
bull γlowastprarr π0p
A1 =e gAMN
fπ
[2PA middot ε+ qA middot ε
sminusM2N
+2PB middot εminus qA middot ε
uminusM2N
] (B7a)
A2 =e gAMN
fπ
[1
sminusM2N
+1
uminusM2N
] (B7b)
bull γlowastprarr π+n
A1 =
radic2 e gAMN
fπ
[2PA middot ε+ qA middot ε
sminusM2N
+2(PA minus PB) middot ε+ qA middot ε
tminusm2π
] (B8a)
A2 =
radic2 e gAMN
fπ(sminusM2N )
(B8b)
bull γlowastnrarr π0n
A1 = 0 (B9a)
A2 = 0 (B9b)
bull γlowastnrarr πminusp
A1 =
radic2 e gAMN
fπ
[2PB middot εminus qA middot ε
uminusM2N
minus 2(PA minus PB) middot ε+ qA middot εtminusm2
π
] (B10a)
A2 =
radic2 e gAMN
fπ(uminusM2N )
(B10b)
The analytical expressions shown above were checked with the amplitudes given in Ref [87] An-
alytical expressions for the tree-level γlowastN rarr πN channel of the σLT (νQ2) and σTT (νQ2) cross
32
sections are given below (proton channels π+n and π0p neutron channel πminusp) We checked that
they reproduce the known results in the real-photon limit [45 86] To shorten the final expressions
for the cross sections which are considerably longer for finite Q2 than in the real-photon limit we
define the following dimensionless kinematic variables
αγ = (ENi )cmradics =
s+M2N +Q2
2s (B11)
απ = (ENf )cmradics =
s+M2N minusm2
π
2s (B12)
βγ = Eγcmradics =
sminusM2N minusQ2
2s (B13)
βπ = Eπcmradics =
sminusM2N +m2
π
2s (B14)
λγ = |~qi|cmradics =
radic(sminusM2
N minusQ2)2 + 4sQ2
2s (B15)
λπ = |~qf |cmradics =
radic(sminusM2
N +m2π)2 minus 4sm2
π
2s (B16)
Here (ENi )cm and (ENf )cm are the energies of the incoming and outgoing nucleon Eγcm is the energy
of the incoming photon Eπcm is the energy of the outgoing pion all in the cm frame
σ(π+n)TT = minus e2g2
AM2N
64πf2πs
3(sminusM2N )2λ4
γ
4sλπλγ
[(M2
N minus s)(M2N minusQ2 minus s)(Q2 + 2sβγβπ)
+ 2s(m2πQ
2 minus (M2N minus s)
(M2N + s(minus1 + 2βγβπ)
))λ2γ
]+ (M2
N minus s)(M2N minusQ2 minus s)
times (Q2 + 2sβγβπ minus 2sλπλγ)(Q2 + 2s(βγβπ + λπλγ)
)times log
(Q2 + 2sβγβπ minus 2sλγλπQ2 + 2sβγβπ + 2sλγλπ
) (B17)
33
σ(π0p)TT =
e2g2AM
2Nλπ
64πf2πs(sminusM2
N )2λγ
1
s2λπλ3γ
[(M2
N minus s)(M2N minusQ2 minus s)
(3M2
N + s(2βγβπ minus 3))(M2N
+ s(2βγβπ minus 1))
+ 2s
((2m2
πQ2 + (M2
N minus s)2)(M2
N minus s)minus 2(minusm2
πQ2 + (sminusM2
N )2)
times sβγβπ + 4s2(sminusM2N )β2
γβ2π
)λ2γ
]arctan
(2sλπλγ
M2N + s(2βγβπ minus 1)
)+ 2
[minus 2m2
πQ2
+ (M2N minus s)
(2(sminusM2
N ) + 4sβγβπ +(sminusM2
N +Q2)(3M2N minus 3s+ 2sβγβπ)
sλ2γ
+m2πQ
2(sminusM2N )
(M2N minus s+ 2sβγβπ)(M2
N + s(minus1 + 2βγβπ minus 2λπλγ))
+m2πQ
2(sminusM2N )
(M2N minus s+ 2sβγβπ)(M2
N + s(minus1 + 2βγβπ + 2λπλγ))
)
+1
λπλγ
(βγβπ
(minusm2
πQ2 + (M2
N minus s)(M2N minus s+ 2sβγβπ)
)minus (M2
N minus s)(M2N minusQ2 minus s)λ2
π
)log
(M2N + s(minus1 + 2βγβπ + 2λπλγ)
M2N + s(minus1 + 2βγβπ minus 2λπλγ)
)] (B18)
σ(πminusp)TT =
e2g2AM
2N
64πf2πs
3λ4γ(M2
N + s(2βπβγ minus 1))2 minus 4λ2πs
2λ2γ)
4sλπλγ
((minusM2
N +Q2 + s)(M2N
+ s(2βπβγ minus 1))
2 minus 2sλ2γ
(2λ2
πs(minusM2N +Q2 + s) +m2
πQ2))
+((M2N + s(2βπβγ minus 1)
)2
minus 4λ2πs
2λ2γ
)((M2N + s(2βπβγ minus 1)
)(M2N minus 2Q2 minus s(2βπβγ + 1)
)+ 4λ2
πs2λ2
γ
)times log
(minusM2
N minus 2βπsβγ minus 2λπsλγ + s)minus(M2N + s(2βπβγ minus 2λπλγ minus 1)
)(M2N + s(minus1
+ 2βπβγ + 2λπλγ))[((
M2N + s(2βπβγ minus 1)
)(M2N minus 2Q2 minus s(2βπβγ + 1)
)+ 4λ2
πs2λ2
γ
)times log
(minusM2
N minus 2βπsβγ + 2λπsλγ + s)
+((Q2 + 2βπsβγ)2 minus 4λ2
πs2λ2
γ
)times log
(Q2 + 2βπsβγ minus 2λπsλγQ2 + 2sβπβγ + 2λπsλγ
)]+ 4sλ2
γ
((M2
N + s(2βπβγ minus 1))2 minus 4λ2πs
2λ2γ
)times (M2
N minus s) arctanh
(2λπsλγ
M2N + 2βπsβγ minus s
) (B19)
34
σ(π+n)LT =
e2g2AM
3Nλπ
32πf2πQs
3(sminusM2N )2λ4
γ
2sλγ
[(M2
N minus s)(Q2 + 2sβ2γ)(Q2 + 2sβγβπ)
minus 4s
((M2
N minus s)(Q2 minus 2s(απ minus 1)βγ
)+Q2sβγβπ
)λ2γ + 8s3(minus1 + απ)λ4
γ
]minus M2
N minus sλπ
[(Q2 + 2sβ2
γ)(Q2 + 2sβγβπ)2 + 4s2(2(απ minus 1)βγ(Q2 + 2sβγβπ)
minusQ2λ2π
)λ2γ + 8s3(απ minus 1)2λ4
γ
]arctan
(2sλπλγ
Q2 + 2sβγβπ
) (B20)
σ(π0p)LT =
e2g2AM
3Nλπ
16πf2πQs(sminusM2
N )2λγ
1
minus2s(M2N + s(minus1 + 2βγβπ))2λ2
γ + 8s3λ2πλ
4γ
[minus 3M8
N (Q2
+ 2sβ2γ) + 2M4
Ns2
(minus (Q2 + 2sβ2
γ)(minus3 + 2βγβπ) +(Q2(minus3 + 2βγβπ + 6λ2
π) + 2sβγ
times (minus3 + 12απ + 2βγβπ minus 12απβγβπ + 4βγλ2π))λ2γ minus 4s(α2
π minus 1)λ4π
)+ 2M2
Ns3
[minus (Q2
+ 2sβ2γ)(minus1 + 2βγβπ)
(6 + βγβπ(minus9 + 2βγβπ)
)+
(Q2(3minus 12λ2
π + 4βγβπ(minus1 + 2βγβπ
+ λ2π))minus 2sβγ
(minus 3 + 4απ(3 + 2απ(3 + 2βγβπ(minus3 + βγβπ)) + 4βγ(βπ + (2minus βγβπ)
times λ2π)))λ2γ + 8s
((απ minus 1)(1 + απ minus 2βγβπ) + 2απβγλ
2π
)λ4γ
)+ s4
((Q2 + 2sβ2
γ)
times (1minus 2βγβπ)2(2βγβπ minus 3) + 2(Q2(minus1 + 6λ2
π + 2βγβπ((1minus 2βγβπ)2 minus 2λ2π)) + 2sβγ
times (minus1 + 2βγβπ + 4(minus1 + βγβπ)(minusαπ + 2απβγβπ minus βγλ2π)))λ2γ minus 8
(s(απ minus 1)(απ
+ (1minus 2βγβπ)2) + 2βγ(2sαπ +Q2βγ)λ2π
)λ4γ + 32s(απ minus 1)λ2
πλ6γ
)+ 2M6
Ns
(Q2(6
minus 7βγβπ + λ2γ) + 2sβγ(βγ(6minus 7βγβπ) + (1minus 4απ)λ2
γ)
)]+sminusM2
N
4s2λπλ3γ
[(Q2 + 2sβ2
γ)
times(3M2
N + s(2βγβπ minus 3))(M2N + s(2βγβπ minus 1)
)+ 2s
(minusM2
N (Q2 + 2sβγ minus 8sαπβγ)
+ s(2sβγ(1minus 4απ + 4απβγβπ) +Q2(1minus 2λ2π)))λ2γ + 8s3(απ minus 1)λ4
γ
]
times arctan
(2sλπλγ
M2N + s(2βγβπ minus 1)
) (B21)
35
σ(πminusp)LT =
e2g2AM
3N
64πf2πQs
3λ4γ(M2
N minusQ2 minus s)(M2N + s(2βπβγ + 2λπλγ minus 1))
times
1
M2N + s(2βπβγ minus 2λπλγ minus 1)
[4λπsλγ(M2
N minusQ2 minus s)(
2sλ2γ
(s((1minus 2λ2
π)Q2
+ 2(2απ minus 1)sβγ(2βπβγ minus 1))minusM2N (Q2 + 2(1minus 2απ)sβγ)
)+ (Q2 + 2sβ2
γ)(M2N + s(2βπβγ minus 1)
)2 + 8(απ minus 1)απs
3λ4γ
)]minus(M2N + s(2βπβγ + 2λπλγ minus 1)
)times[
log
(minusM2N minus 2βπsβγ minus 2λπsλγ + s
minusM2N minus 2βπsβγ + 2λπsλγ + s
)(minus (Q2 + 2sβ2
γ)(M2N + s(2βπβγ minus 1)
)(minusM2
N + 2Q2 + 2βπsβγ + s)minus 2sλ2γ
(minusM2
N (Q2 + 2sβγ) +Q4 +Q2s(4απβγ
minus 2βγ minus 2λ2π + 1) + 2s2βγ(4(απ minus 1)βπβγ + 1)
)minus 8(απ minus 1)2s3λ4
γ
)+ 2
(4s2λ2
γ
(2(απ minus 1)βγ(Q2 + 2βπsβγ)minus λ2
πQ2)
+ (Q2 + 2βπsβγ)2(Q2 + 2sβ2γ)
+ 8(απ minus 1)2s3λ4γ
)arctanh
(2λπsλγ
Q2 + 2βπsβγ
)] (B22)
2 ∆-production channel
The tree-level ∆-exchange diagram in Fig 2 of Ref [30] contributes to the non-Born part of
the VVCS amplitudes The contribution of the ∆ exchange to the VVCS amplitudes can be split
into [17]
S∆-exch1 (νQ2) = S∆-pole
1 (νQ2) + S∆-exch1 (νQ2) (B23a)
νS∆-exch2 (νQ2) = νS∆-pole
2 (νQ2) + νS2∆-exch
(νQ2) (B23b)
and similarly for the unpolarized VVCS amplitudes discussed in Ref [30] Here we introduced the
∆-pole contributions S∆-polei and the ∆-non-pole contributions S∆-exch
i The former amplitudes
feature a pole at the ∆(1232)-production threshold and thus are proportional to
1
[sminusM2∆][uminusM2
∆]=
1
4M2N
1
ν2∆ minus ν2
(B24)
36
They can be reconstructed from the dispersion relations in Eq (5) with the tree-level ∆-production
cross sections as input cf Fig 10 in Ref [30]
σTT (νQ2) =π2α
M2NM
2+|~q |
minus g2
MMN (M+ + ν)|~q |2 +g2E(∆minus ν)(Q2 minusMNν)2
MN(B25a)
+g2CQ
4s(∆minus ν)
MNM2∆
minus 4gMgE(Q2 minusMNν)|~q |2 minus 4gMgCQ2|~q |2
+2gEgCQ
2[minusMNM∆ |~q |2 + s(Q2 +∆ν)
]MNM∆
δ(ν minus ν∆)
σLT (νQ2) =Qπ2α
M2NM
2+|~q |
g2E(MNν minusQ2) [M∆(MN + ν)minus s]
MN(B25b)
+g2CQ
2[MNM∆|~q |2 minus s(Q2 +∆ν)
]MNM2
∆
+ gMgEM∆|~q |2 minusgMgC(Q2 minusMNν)|~q |2
M∆
+gEgC(ν minus∆)(M2
N |~q |2 minus 2Q2s)
MNM∆
δ(ν minus ν∆)
with ∆ = M∆ minusMN M+ = M∆ + MN and the Mandelstam variable s = M2N + 2MNν minus Q2
Analytical expressions for the spin structure functions g1(xQ2) and g2(xQ2) can be constructed
from Eq (3) with the flux factor K(νQ2) = |~q | =radicν2 +Q2
In the ∆-non-pole contributions to S1(νQ2) and νS2(νQ2) the pole in ν at the ∆(1232)-
production threshold has canceled out
S∆-exch1 (νQ2) =
πα
MNM2+
[g2MQ
2+ + g2
E
(∆2 minus 3Q2
)+
4g2CQ
4
M2∆
minus 8gMgEM∆ωminus (B26a)
minus2gMgCQ2(MN minus 4M∆)
M∆+
2gEgCQ2(3MN minus 2M∆)
M∆
]
νS2∆-exch
(νQ2) =2πα
MNM2+
[g2EM∆∆ωminus +
g2M MNQ
2+
2+g2C Q
2(Q2 minus∆2)
2M∆(B26b)
+gEgM M∆(M∆ω+ minus 4MNωminus)minus gEgC ∆(2Q2 +MNω+)
+gMgC Q2(4MN minus ω+)
]+S∆-exch
2 (νQ2)
ν
[M2
∆ ω2+
M2N
+ ν2
]
with Q+ =radic
(M∆ +MN )2 +Q2 and ωplusmn = (M2∆minusM2
N plusmnQ2)2M∆ and the non-pole contribution
to S2(νQ2)
S∆-exch2 (νQ2) = minus2παMNν
M∆M2+
[gM + gE
]gC (B27)
These amplitudes to the contrary are not described by the tree-level ∆-production cross sections
in the standard dispersive approach [17] This peculiarity has been previously missed e g in the
calculation of the ∆-exchange contribution to the hydrogen hyperfine splitting in Ref [88] The
importance of including the ∆-non-pole contribution is also evident when considering the BC sum
37
rule in Eq (14) The ∆-pole terms by themselves violate the BC sum rule but cancel exactly with
the ∆-non-pole terms
limνrarr0
νS∆-pole2 (νQ2) + lim
νrarr0νS2
∆-exch(νQ2) = 0 (B28)
Appendix C Polarizabilities at Q2 = 0
In this section we give analytical expressions for the static values and slopes at Q2 = 0 of
the spin polarizabilities and moments of polarized structure functions In particular we give the
HB expansion of the πN -loop contributions and the ∆-exchange contributions The complete
expressions also for the π∆-loop contributions can be found in the Supplemented material Recall
that IA(0) = I1(0) = d2(0) = 0 and dd2(Q2)dQ2
∣∣∣Q2=0
= 0
1 πN-loop contribution
Here we give analytical expressions for the πN -loop contributions to the proton and neutron
spin polarizabilities expanded in powers of micro = mπMN viz the HB expansion Note that we
choose to expand here to a high order in micro the strict HB expansion would only retain the leading
term in an analogous NLO calculation
bull Static polarizabilities (Q2 = 0)
γ0p =e2g2
A
96π3f2πm
2π
1minus 21πmicro
8minus(
59
2+ 26 logmicro
)micro2 +
1875πmicro3
64
+3
(3
2+ 26 logmicro
)micro4 +
(C1)
γ0n =e2g2
A
48π3f2πm
2π
1
2minus 9πmicro
16minus 2micro2 logmicro+
75πmicro3
128minus 3micro4
4+
(C2)
δLTp =e2g2
A
192π3f2πm
2π
1minus 9πmicro
8+
(13
2minus 2 logmicro
)micro2 minus 465πmicro3
64
minus(
47
2+ 42 logmicro
)micro4 +
(C3)
δLTn =e2g2
A
96π3f2πm
2π
1
2+
3πmicro
16+ (1 + 2 log micro)micro2 minus 105πmicro3
128+
5micro4
4+
(C4)
38
δLTn =e2g2
A
96π3f2πm
2π
1
2+
3πmicro
16+ (1 + 2 log micro)micro2 minus 105πmicro3
128+
5micro4
4+
(C5)
γ0p =e2g2
A
16π3f2πm
4π
4
45minus 3πmicro
16+
14micro2
5minus 1813πmicro3
384minus 192
5(1 + log micro)micro4
+80703πmicro5
2048+
(C6)
γ0n =e2g2
A
16π3f2πm
4π
4
45minus 5πmicro
48+
4micro2
5minus 245πmicro3
384minus 32micro4 logmicro
15
+1323πmicro5
2048+
(C7)
bull Slopes of polarizabilities at Q2 = 0
dγ0p(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
2minus 45πmicro
4+ 223micro2 minus 28515πmicro3
64
minus9
(1953
4+ 449 logmicro
)micro4 +
570255πmicro5
128+
(C8)
dγ0n(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
2minus 81πmicro
8+ 94micro2 minus 2535πmicro3
32
minus3 (1 + 90 logmicro)micro4 +84315πmicro5
1024+
(C9)
dδLTp(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
2880π3f2πm
4π
minus5
2minus 27πmicro
32+ 20micro2 minus 5865πmicro3
256
+3
(617
4+ 36 logmicro
)micro4 minus 2056845πmicro5
4096+
(C10)
dδLTn(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
minus5
4minus 81πmicro
64minus 11micro2 +
10005πmicro3
512
+15
8(11 + 48 logmicro)micro4 minus 267015πmicro5
8192+
(C11)
dIAp(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
96π2f2πmicro
2
1minus 15πmicro
4minus 1
2(115 + 88 logmicro)micro2 +
1839πmicro3
32
+5 (5 + 34 logmicro)micro4 +
(C12)
dIAn(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
48π2f2πmicro
2
1
2minus 11πmicro
8minus 1
4(1 + 20 logmicro)micro2 +
99πmicro3
64
minus25micro4
12+
(C13)
dI1p(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
96π2f2πmicro
3π
8+ 2 (4 + 3 logmicro)microminus 537πmicro2
64
minus1
2(15 + 56 logmicro)micro3 +
(C14)
39
dI1n(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
48π2f2πmicro
minus π
16+
1
4(3 + 4 logmicro)microminus 57πmicro2
128+
2micro3
3+
(C15)
γ0p(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2g2
A
16π3f2πm
6π
1
105minus 23πmicro
256+
377micro2
210minus 15551πmicro3
6144+
3371micro4
105
minus1640457πmicro5
32768+
(C16)
γ0n(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2g2
A
16π3f2πm
6π
1
105minus 153πmicro
1792+
69micro2
70minus 4615πmicro3
6144+
172micro4
35
minus120897πmicro5
32768+
(C17)
2 ∆-exchange contribution
Here we give analytical expressions for the tree-level ∆-exchange contributions to the nucleon
spin polarizabilities and their slopes at Q2 = 0 Note that the ∆-exchange contributes equally to
proton and neutron polarizabilities Recall that for the magnetic γlowastN∆ coupling we introduced a
dipole form factor to mimic vector-meson dominance gM rarr gM(1 +Q2Λ2)2
bull Static polarizabilities (Q2 = 0)
γ0 = minus e2
4πM2+
(g2M
∆2+
g2E
M2+
minus 4gMgEM+∆
) (C18)
δLT =e2M∆
4πM3+
(g2E
MNM++gMgE∆MN
minus gEgCM2
∆
) (C19)
γ0 =e2M2
N
π∆2M4+
(minusg
2M
∆2+
g2E
M2+
+4gMgE∆M+
) (C20)
bull Slopes of polarizabilities at Q2 = 0
dγ0(Q2)
dQ2
∣∣∣∣∣Q2=0
= minus e2
πM2+∆
(g2M
∆
[1
4∆2minus 1
∆M++
1
2M2+
]minus 1
Λ2
g2M
∆+
g2E
2M2+
[1
2∆minus 3
M+
]minusgMgE
M+
[1
∆2minus 5
∆M++
1
M2+
]+
1
Λ2
2gMgEM+
+2gMgC∆M2
+
minus gEgCM3
+
) (C21)
40
d δLT (Q2)
dQ2
∣∣∣∣∣Q2=0
=e2M∆∆
4πMNM2+
(g2E
∆2M2+
[1
∆minus 4
M+
]minus g2
C
∆M2∆M
2+
+gMgE∆2M+
[1
∆2
minus 3
∆M++
1
M2+
]minus 2
Λ2
gMgE∆2M+
+gMgC∆M2
∆
[1
2∆2minus 2
∆M++
1
2M2+
]minus gEgC
2M2∆M
2+
[7
∆+
1
M+
]) (C22)
dIA(Q2)
dQ2
∣∣∣∣∣Q2=0
= minusM2N
M2+
(g2M
2∆2+
g2E
MNM+minus 2gMgE
∆M+minus gEgCM∆M+
) (C23)
dI1(Q2)
dQ2
∣∣∣∣∣Q2=0
= minusM∆M2N
2M3+
(g2E
MNM∆minus gMgE∆MN
minus gEgCM2
∆
) (C24)
γ0(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2M2
N
π∆3M6+
(g2M
∆
[3M2
∆ + 2M∆MN minus 9M2N
∆2+
4M2+
Λ2
]
minus4gMgE
[5M2
∆ minus 9M2N
M+∆2+
2M+
Λ2
]+
g2E
∆M+[7M∆ minus 9MN ]
minus8gMgC∆
+4gEgCM+
) (C25)
[1] D Drechsel B Pasquini and M Vanderhaeghen Dispersion relations in real and virtual
Compton scattering Phys Rept 378 99 (2003) hep-ph0212124
[2] S E Kuhn J-P Chen and E Leader Spin structure of the nucleon mdash status and recent
results Prog Part Nucl Phys 63 1 (2009) arXiv08123535 [hep-ph]
[3] F Hagelstein R Miskimen and V Pascalutsa Nucleon polarizabilities from Compton scat-
tering to hydrogen atom Prog Part Nucl Phys 88 29 (2016) arXiv151203765 [nucl-th]
[4] B Pasquini and M Vanderhaeghen Dispersion theory in electromagnetic interactions Ann
Rev Nucl Part Sci 68 75 (2018) arXiv180510482 [hep-ph]
[5] S Gerasimov A Sum rule for magnetic moments and the damping of the nucleon magnetic
moment in nuclei Sov J Nucl Phys 2 430 (1966)
[6] S Drell and A C Hearn Exact sum rule for nucleon magnetic moments Phys Rev Lett
16 908 (1966)
[7] J S Schwinger Source theory viewpoints in deep inelastic scattering Proc Natl Acad Sci
USA 72 1 (1975)
[8] J S Schwinger Source theory viewpoints in deep inelastic scattering Electromagnetic In-
teractions and Field Theory Proceedings 14 Internationale Universitatswochen Schladming
Austria February 24-March 7 1975 Acta Phys Austriaca Suppl 14 471 (1975)
41
[9] J Schwinger Source theory discussion of deep inelastic scattering with polarized particles
Proc Natl Acad Sci USA 72 1559 (1975)
[10] A M Baldin Polarizability of nucleons Nucl Phys 18 310 (1960)
[11] M Gell-Mann M L Goldberger and W E Thirring Use of causality conditions in quantum
theory Phys Rev 95 1612 (1954)
[12] A Pineda Leading chiral logarithms to the hyperfine splitting of the hydrogen and muonic
hydrogen Phys Rev C 67 025201 (2003)
[13] C Peset and A Pineda The two-photon exchange contribution to muonic hydrogen from
chiral perturbation theory Nucl Phys B 887 69 (2014) arXiv14064524 [hep-ph]
[14] C Peset and A Pineda Model-independent determination of the two-photon exchange contri-
bution to hyperfine splitting in muonic hydrogen JHEP 04 060 arXiv161205206 [nucl-th]
[15] F Hagelstein and V Pascalutsa Proton structure in the hyperfine splitting of muonic hydro-
gen PoS CD15 077 (2016) arXiv151104301 [nucl-th]
[16] F Hagelstein Exciting Nucleons in Compton Scattering and Hydrogen-Like Atoms PhD
thesis Mainz U Inst Kernphys (2017) arXiv171000874 [nucl-th]
[17] F Hagelstein ∆(1232)-Resonance in the hydrogen spectrum Proceedings 11th International
Workshop on the Physics of Excited Nucleons (NSTAR 2017) Columbia SC USA August
20-23 2017 Few Body Syst 59 93 (2018) arXiv180109790 [nucl-th]
[18] Y Prok et al (CLAS) Moments of the spin structure functions gp1 and gd1 for 005 lt Q2 lt 30
GeV2 Phys Lett B 672 12 (2009) arXiv08022232 [nucl-ex]
[19] H Dutz et al (GDH) First measurement of the Gerasimov-Drell-Hearn sum rule for 1H from
07 GeV to 18 GeV at ELSA Phys Rev Lett 91 192001 (2003)
[20] M Amarian et al (Jefferson Lab E94010) Measurement of the generalized forward spin po-
larizabilities of the neutron Phys Rev Lett 93 152301 (2004) arXivnucl-ex0406005
[21] M Amarian et al The Q2 evolution of the generalized Gerasimov-Drell-Hearn integral for
the neutron using a 3He target Phys Rev Lett 89 242301 (2002) arXivnucl-ex0205020
[22] M Amarian et al (Jefferson Lab E94-010)Q2 evolution of the neutron spin structure moments
using a 3He target Phys Rev Lett 92 022301 (2004) arXivhep-ex0310003
[23] A Deur et al Experimental determination of the evolution of the Bjorken integral at low Q2
Phys Rev Lett 93 212001 (2004) arXivhep-ex0407007
[24] K Slifer Low Q2 measurement of gp2 and the δLT spin polarizability Spin structure at long dis-
tance Proceedings Workshop Newport News USA March 12ndash13 2009 nucl-ex09064775
(2009) AIP Conf Proc 1155 10106313203293 (2009) arXiv09064775 [nucl-ex]
[25] P Solvignon et al (E01-012) Moments of the neutron g2 structure function at intermediate
42
Q2 Phys Rev C 92 015208 (2015) arXiv13044497 [nucl-ex]
[26] A Deur Experimental studies at low Q2 of the spin structure of the nucleon at Jefferson Lab
in 9th International Workshop on Chiral Dynamics (CD18) Durham NC USA September
17-21 2018 (2019) arXiv190305661 [nucl-ex]
[27] V Sulkosky et al (Jefferson Lab E97-110) Measurement of the 3He spin-structure functions
and of neutron (3He) spin-dependent sum rules at 0035 le Q2 le 024 GeV2 Phys Lett B
805 135428 (2020) arXiv190805709 [nucl-ex]
[28] K Adhikari et al (CLAS) Measurement of the Q2 Dependence of the Deuteron Spin Structure
Function g1 and its Moments at Low Q2 with CLAS Phys Rev Lett 120 062501 (2018)
arXiv171101974 [nucl-ex]
[29] V Lensky F Hagelstein A Hiller Blin and V Pascalutsa Comment on rdquoMeasurement of
the Q2 Dependence of the Deuteron Spin Structure Function g1 and its Moments at Low Q2
with CLASrdquo (2018) arXiv180603219 [nucl-th]
[30] J M Alarcn F Hagelstein V Lensky and V Pascalutsa Forward doubly-virtual Compton
scattering off the nucleon in chiral perturbation theory at NLO the subtraction function and
moments of unpolarized structure functions (2020) arXiv200509518 [hep-ph]
[31] V Bernard E Epelbaum H Krebs and U-G Meiszligner New insights into the spin structure
of the nucleon Phys Rev D 87 054032 (2013) arXiv12092523 [hep-ph]
[32] V Lensky J M Alarcon and V Pascalutsa Moments of nucleon structure functions at
next-to-leading order in baryon chiral perturbation theory Phys Rev C 90 055202 (2014)
arXiv14072574 [hep-ph]
[33] V Pascalutsa and M Vanderhaeghen Chiral effective-field theory in the ∆(1232) region
I Pion electroproduction on the nucleon Phys Rev D 73 034003 (2006) arXivhep-
ph0512244
[34] V Pascalutsa and M Vanderhaeghen Electromagnetic nucleon-to-Delta transition in chiral
effective field theory Phys Rev Lett 95 232001 (2005) arXivhep-ph0508060
[35] T R Hemmert B R Holstein and J Kambor Systematic 1M expansion for spin 32
particles in baryon chiral perturbation theory Phys Lett B 395 89 (1997) arXivhep-
ph9606456
[36] V Pascalutsa and D R Phillips Effective theory of the ∆(1232) in Compton scattering off
the nucleon Phys Rev C 67 055202 (2003) arXivnucl-th0212024
[37] V Pascalutsa M Vanderhaeghen and S N Yang Electromagnetic excitation of the ∆(1232)-
resonance Phys Rept 437 125 (2007) arXivhep-ph0609004
[38] K A Olive et al (Particle Data Group) Review of Particle Physics Chin Phys C 38 090001
43
(2014)
[39] F E Low Scattering of light of very low frequency by systems of spin 12 Phys Rev 96
1428 (1954)
[40] M Gell-Mann and M L Goldberger Scattering of low-energy photons by particles of spin
12 Phys Rev 96 1433 (1954)
[41] A M Harun ar-Rashid A simple derivation of Schwingerrsquos sum rule for spin dependent
structure functions Nuovo Cim A 33 447 (1976)
[42] F Hagelstein and V Pascalutsa Dissecting the hadronic contributions to (g minus 2)micro by
Schwingerrsquos sum rule Phys Rev Lett 120 072002 (2018) arXiv171004571 [hep-ph]
[43] V Lensky V Pascalutsa M Vanderhaeghen and C Kao Spin-dependent sum rules con-
necting real and virtual Compton scattering verified Phys Rev D 95 074001 (2017)
arXiv170101947 [hep-ph]
[44] H Burkhardt and W N Cottingham Sum rules for forward virtual Compton scattering
Annals Phys 56 453 (1970)
[45] V Lensky and V Pascalutsa Predictive powers of chiral perturbation theory in Compton
scattering off protons Eur Phys J C 65 195 (2010) arXiv09070451 [hep-ph]
[46] H Grieszlighammer J McGovern D Phillips and G Feldman Using effective field theory to
analyse low-energy Compton scattering data from protons and light nuclei Prog Part Nucl
Phys 67 841 (2012) arXiv12036834 [nucl-th]
[47] H W Grieszlighammer J A McGovern and D R Phillips Nucleon polarisabilities at and
beyond physical pion masses Eur Phys J A 52 139 (2016) arXiv151101952 [nucl-th]
[48] E Epelbaum H Krebs and U-G Meiszligner Improved chiral nucleon-nucleon potential up to
next-to-next-to-next-to-leading order Eur Phys J A 51 53 (2015) arXiv14120142 [nucl-
th]
[49] E Epelbaum H Krebs and U-G Meiszligner Precision nucleon-nucleon potential at fifth order
in the chiral expansion Phys Rev Lett 115 122301 (2015) arXiv14124623 [nucl-th]
[50] C W Kao T Spitzenberg and M Vanderhaeghen Burkhardt-Cottingham sum rule and
forward spin polarizabilities in heavy baryon chiral perturbation theory Phys Rev D 67
016001 (2003) arXivhep-ph0209241
[51] C-W Kao D Drechsel S Kamalov and M Vanderhaeghen Higher moments of nucleon
spin structure functions in heavy baryon chiral perturbation theory and in a resonance model
Phys Rev D 69 056004 (2004) arXivhep-ph0312102
[52] T Becher and H Leutwyler Baryon chiral perturbation theory in manifestly Lorentz invariant
form Eur Phys J C 9 643 (1999) arXivhep-ph9901384
44
[53] J Gasser M E Sainio and A Svarc Nucleons with chiral loops Nucl Phys B 307 779
(1988)
[54] L S Geng J Martin Camalich L Alvarez-Ruso and M J Vicente Vacas Leading SU(3)-
breaking corrections to the baryon magnetic moments in chiral perturbation theory Phys
Rev Lett 101 222002 (2008) arXiv08051419 [hep-ph]
[55] D Drechsel S S Kamalov and L Tiator The GDH sum rule and related integrals Phys
Rev D 63 114010 (2001) arXivhep-ph0008306
[56] D Drechsel O Hanstein S S Kamalov and L Tiator A unitary isobar model for pion
photo- and electroproduction on the proton up to 1 GeV Nucl Phys A 645 145 (1999)
arXivnucl-th9807001
[57] L Tiator private communication (2020)
[58] V Bernard T R Hemmert and U-G Meiszligner Spin structure of the nucleon at low energies
Phys Rev D 67 076008 (2003) arXivhep-ph0212033
[59] O Gryniuk F Hagelstein and V Pascalutsa Evaluation of the forward Compton scattering
off protons II Spin-dependent amplitude and observables Phys Rev D 94 034043 (2016)
arXiv160400789 [nucl-th]
[60] R Zielinski The g2p Experiment A Measurement of the Protonrsquos Spin Structure Functions
PhD thesis New Hampshire U (2010) arXiv170808297 [nucl-ex]
[61] N Guler et al (CLAS) Precise determination of the deuteron spin structure at low to mod-
erate Q2 with CLAS and extraction of the neutron contribution Phys Rev C 92 055201
(2015) arXiv150507877 [nucl-ex]
[62] C E Carlson V Nazaryan and K Griffioen Proton structure corrections to electronic
and muonic hydrogen hyperfine splitting Phys Rev A 78 022517 (2008) arXiv08052603
[physicsatom-ph]
[63] V Bernard N Kaiser and U-G Meiszligner Chiral dynamics in nucleons and nuclei Int J
Mod Phys E 4 193 (1995) arXivhep-ph9501384
[64] T R Hemmert B R Holstein and J Kambor ∆(1232) and the polarizabilities of the nucleon
Phys Rev D 55 5598 (1997) arXivhep-ph9612374
[65] V Pascalutsa and R Timmermans Field theory of nucleon to higher-spin baryon transitions
Phys Rev C 60 042201 (1999) arXivnucl-th9905065
[66] V Pascalutsa Quantization of an interacting spin-32 field and the Delta isobar Phys Rev
D 58 096002 (1998) arXivhep-ph9802288
[67] H Krebs Double Virtual Compton Scattering and SpinStructure of the Nucleon PoS
CD2018 031 (2019)
45
[68] N Kochelev and Y Oh Axial anomaly and the δLT puzzle Phys Rev D 016012 (2012)
arXiv11034892 [hep-ph]
[69] D Drechsel S Kamalov and L Tiator Unitary isobar model ndash MAID2007 Eur Phys J A
69 (2007) available at httpsmaidkphuni-mainzde arXiv07100306 [nucl-th]
[70] P J Mohr B N Taylor and D B Newell CODATA recommended values of the fundamental
physical constants 2010 Rev Mod Phys 84 1527 (2012)
[71] J Ahrens et al (GDH A2) First measurement of the Gerasimov-Drell-Hearn integral for 1H
from 200 to 800 MeV Phys Rev Lett 87 022003 (2001) arXivhep-ex0105089 [hep-ex]
[72] K Helbing (GDH) Experimental verification of the GDH sum rule at ELSA and MAMI Nucl
Phys Proc Suppl 105 113 (2002)
[73] R Bradford A Bodek H S Budd and J Arrington A New parameterization of the nucleon
elastic form-factors NuInt05 proceedings of the 4th International Workshop on Neutrino-
Nucleus Interactions in the Few-GeV Region Okayama Japan 26-29 September 2005 Nucl
Phys Proc Suppl 159 127 (2006) arXivhep-ex0602017
[74] J D Bjorken Applications of the chiral U(6)timesU(6) algebra of current densities Phys Rev
148 1467 (1966)
[75] J D Bjorken Inelastic scattering of polarized leptons from polarized nucleons Phys Rev D
1 1376 (1970)
[76] V Bernard T R Hemmert and U-G Meiszligner Novel analysis of chiral loop effects in
the generalized Gerasimov-Drell-Hearn sum rule Phys Lett B 545 105 (2002) arXivhep-
ph0203167
[77] A Deur et al Experimental study of isovector spin sum rules Phys Rev D 78 032001
(2008) arXiv08023198 [nucl-ex]
[78] B Pasquini P Pedroni and D Drechsel Higher order forward spin polarizability Phys Lett
B 687 160 (2010) arXiv10014230 [hep-ph]
[79] R Jaffe g2ndashThe nucleonrsquos other spin-dependent structure function Comments Nucl Part
Phys 19 239 (1990)
[80] E V Shuryak and A Vainshtein Theory of power corrections to deep inelastic scattering in
quantum chromodynamics (II) Qminus4 effects polarized target Nucl Phys B 201 141 (1982)
[81] S Wandzura and F Wilczek Sum rules for spin dependent electroproduction Test of rela-
tivistic constituent quarks Phys Lett B 72 195 (1977)
[82] B W Filippone and X-D Ji The spin structure of the nucleon Adv Nucl Phys 26 1
(2001) arXivhep-ph0101224
[83] M Burkardt The g2 structure function Proceedings Workshop on Spin structure at long
46
distance Newport News USA March 12-13 2009 AIP Conf Proc 1155 26 (2009)
arXiv09054079 [hep-ph]
[84] A Deur S J Brodsky and G F de Teramond The spin structure of the nucleon Rept
Prog Phys 82 076201 (2019) arXiv180705250 [hep-ph]
[85] B R Holstein D Drechsel B Pasquini and M Vanderhaeghen Higher order polarizabilities
of the proton Phys Rev C 61 034316 (2000) arXivhep-ph9910427
[86] B R Holstein V Pascalutsa and M Vanderhaeghen Sum rules for magnetic moments and
polarizabilities in QED and chiral effective-field theory Phys Rev D 72 094014 (2005)
arXivhep-ph0507016
[87] B Pasquini D Drechsel and L Tiator Invariant amplitudes for pion electroproduction Eur
Phys J A 34 387 (2007) arXiv07122327 [hep-ph]
[88] A J Buchmann Non-spherical proton shape and hydrogen hyperfine splitting Proceedings
International Workshop on Precision Physics of Simple Atomic Systems (PSAS 2008) Wind-
sor Ontario Canada July 21-26 2008 Can J Phys 87 773 (2009) arXiv09104747
[physicsatom-ph]
47
FIG 1 The forward Compton scattering or VVCS in case of virtual photons q2 = minusQ2
lies in the different counting and renormalization of the π∆-loop contributions Bernard
et al [31] use the so-called small-scale expansion [35] for the ∆(1232) contributions whereas
we are using the δ-counting scheme [36] (see also Ref [37 Sec 4] for review)
This paper is organized as follows In Sec II A we introduce the polarized VVCS am-
plitudes and their relations to spin structure functions In Sec II B we introduce the spin
polarizabilities appearing in the low-energy expansion (LEX) of the polarized VVCS am-
plitudes In Sec II C we briefly describe our χPT calculation focusing mainly on the
uncertainty estimate In Sec III we show our predictions for the proton and neutron po-
larizabilities as well as some interesting moments of their structure functions In Sec III G
we summarize the results obtained herein comment on the improvements done with respect
to previous calculations and give an outlook to future applications In App B we discuss
the structure functions in particular we define the longitudinal-transverse response func-
tion discuss the ∆-pole contribution and give analytical results for the tree-level πN - and
∆-production channels of the photoabsorption cross sections In App C we give analytical
expressions for the πN -loop and ∆-exchange contributions to the static values and slopes of
the polarizabilities and moments of structure functions The complete expressions also for
the π∆-loop contributions can be found in the Supplemented material
TABLE I Parameters (fundamental and low-energy constants) [38] at the order they appear first
The πN∆ coupling constant hA is fit to the experimental Delta width and the γlowastN∆ coupling
constants gM gE and gC are taken from the pion photoproduction study of Ref [33]
O(p2) α = 1(13704) MN = Mp = 93827 MeV
O(p3) gA = 1270 fπ = 9221 MeV mπ = 13957 MeV
O(p4∆) M∆ = 1232 MeV hA equiv 2gπN∆ = 285 gM = 297 gE = minus10 gC = minus26
4
II CALCULATION OF UNPOLARIZED VVCS AT NLO
A VVCS amplitudes and relations to structure functions
The polarized part of forward VVCS can be described in terms of two independent
Lorentz-covariant and gauge-invariant tensor structures and two scalar amplitudes [3]
T microν(p q) = minus 1
MN
γmicroναqα S1(νQ2)minus 1
M2N
(γmicroνq2 + qmicroγναqα minus qνγmicroαqα
)S2(νQ2) (1)
Here q and p are the photon and nucleon four-momenta (cf Fig 1) ν is the photon
lab-frame energy Q2 = minusq2 is the photon virtuality and γmicroν = 12
[γmicro γν ] and γmicroνα =12
(γmicroγνγα minus γαγνγmicro) are the usual Dirac matrices Alternatively one can use the following
laboratory-frame amplitudes
gTT (νQ2) =ν
MN
[S1(νQ2)minus Q2
MN νS2(νQ2)
] (2a)
gLT (νQ2) =Q
MN
[S1(νQ2) +
ν
MN
S2(νQ2)] (2b)
introduced in Eq (A2) The optical theorem relates the absorptive parts of the forward
VVCS amplitudes to the nucleon structure functions or the cross sections of virtual pho-
toabsorption
ImS1(νQ2) =4π2α
νg1(xQ2) =
MNνK(νQ2)
ν2 +Q2
[Q
νσLT (νQ2) + σTT (νQ2)
] (3a)
ImS2(νQ2) =4π2αMN
ν2g2(xQ2) =
M2NK(νQ2)
ν2 +Q2
[ν
QσLT (νQ2)minus σTT (νQ2)
] (3b)
with α the fine structure constant andK(νQ2) the photon flux factor Note that the photon
flux factor in the optical theorem and the cross sections measured in electroproduction
processes is a matter of convention and has to be chosen in both quantities consistently In
what follows we use Gilmanrsquos flux factor
K(νQ2) equiv |~q | =radicν2 +Q2 (4)
The helicity-difference photoabsorption cross section is defined as σTT = 12 (σ12 minus σ32)
where the photons are transversely polarized and the subscripts on the right-hand side
indicate the total helicities of the γlowastN states The cross section σLT corresponds to a
simultaneous helicity change of the photon and nucleon spin flip such that the total helicity
5
is conserved A detailed derivation of the connection between this response function and
the photoabsorption cross sections can be found in App B The forward VVCS amplitudes
satisfy dispersion relations derived from the general principles of analyticity and causality1
S1(νQ2) =16παMN
Q2
int 1
0
dxg1(xQ2)
1minus x2(ννel)2 minus i0+(5a)
=2MN
π
int infinνel
dν primeν prime 2[QνprimeσLT + σTT
](ν prime Q2)radic
ν prime 2 +Q2(ν prime 2 minus ν2 minus i0+)
νS2(νQ2) =16παM2
N
Q2
int 1
0
dxg2(xQ2)
1minus x2(ννel)2 minus i0+(5b)
=2M2
N
π
int infinνel
dν primeν prime 2[νprime
QσLT minus σTT
](ν prime Q2)radic
ν prime 2 +Q2(ν prime 2 minus ν2 minus i0+)
with νel = Q22MN the elastic threshold
B Low-energy expansions and relations to polarizabilities
The VVCS amplitudes naturally split into nucleon-pole (Spolei ) and non-pole (Snonpole
i )
parts or Born (SBorni ) and non-Born (Si) parts
Si = Spolei + Snonpole
i = SBorni + Si (6)
The Born amplitudes are given uniquely in terms of the nucleon form factors [1]
SBorn1 (νQ2) =
2πα
M
Q2GM(Q2)F1(Q2)
ν2el minus ν2
minus F 22 (Q2)
(7a)
SBorn2 (νQ2) = minus 2παν
ν2el minus ν2
GM(Q2)F2(Q2) (7b)
The same is true for the nucleon-pole amplitudes which are related to the Born amplitudes
in the following way
Spole1 (νQ2) = SBorn
1 (νQ2) +2πα
MN
F 22 (Q2) (8a)
Spole2 (νQ2) = SBorn
2 (νQ2) (8b)
1 The dispersion relation for νS2(νQ2) is used because it is pole-free in the limit Q2 rarr 0 and then ν rarr 0
cf Eq (7b)
6
Here we used the elastic Dirac and Pauli form factors F1(Q2) and F2(Q2) related to the
electric and magnetic Sachs form factors GE(Q2) and GM(Q2) through
F1(Q2) =GE(Q2) + τGM(Q2)
1 + τ (9a)
F2(Q2) =GM(Q2)minusGE(Q2)
1 + τ (9b)
where τ = Q24M2N
A low-energy expansion (LEX) of Eq (5) in combination with the unitarity relations
given in Eq (3) establishes various sum rules relating the nucleon properties (electromag-
netic moments polarizabilities) to experimentally observable response functions [1 3] The
leading terms in the LEX of the RCS amplitudes are determined uniquely by charge mass
and anomalous magnetic moment as the global properties of the nucleon These lowest-
order terms represent the celebrated low-energy theorem (LET) of Low Gell-Mann and
Goldberger [39 40] The polarizabilities related to the internal structure of the nucleon
enter the LEX at higher orders They make up the non-Born amplitudes and can be related
to moments of inelastic structure functions
The process of VVCS can be realized experimentally in electron-nucleon scattering where
a virtual photon is exchanged between the electron and the nucleon This virtual photon
acts as a probe whose resolution depends on its virtuality Q2 In this way one can access
the so-called generalized polarizabilities which extend the notion of polarizabilities to the
case of response to finite momentum transfer The generalized forward spin polarizability
γ0(Q2) and the longitudinal-transverse polarizability δLT (Q2) are most naturally defined via
the LEX of the non-Born part of the lab-frame VVCS amplitudes [1]
1
4πgnonpoleTT (νQ2) =
2α
M2N
IA(Q2) ν + γ0(Q2)ν3 + γ0(Q2)ν5 + (10a)
1
4πgnonpoleLT (νQ2) =
2α
M2N
I3(Q2)Q+ δLT (Q2)ν2Q+ (10b)
Their definitions in terms of integrals over structure functions are postponed to Eqs (19)
and (22) Here we only give the definition of the moment I3(Q2)
I3(Q2) =M2
N
4π2α
int infinν0
dνK(νQ2)
νQσLT (νQ2) =
2M2N
Q2
int x0
0
dx[g1(xQ2) + g2(xQ2)
] (11)
which is related to the Schwinger sum rule in the real photon limit [7ndash9 41 42] The LEX
of the non-pole part of the covariant VVCS amplitudes can be described entirely in terms
7
of moments of inelastic spin structure functions (up to O(ν4) [43])
1
4πSnonpole
1 (νQ2) =2α
MN
I1(Q2) +
2α
MNQ2
[IA(Q2)minus I1(Q2)
]+MNδLT (Q2)
ν2 (12a)
1
4πνSnonpole
2 (νQ2) = 2αI2(Q2) +2α
Q2
[I1(Q2)minus IA(Q2)
]ν2 (12b)
I1(Q2) and IA(Q2) are generalizations of the famous GerasimovndashDrellndashHearn (GDH) sum
rule [5 6] from RCS to the case of virtual photons [1] Their definitions are given in Eqs (26)
and (32) I2(Q2) is the well-known Burkhardt-Cottingham (BC) sum rule [44]
I2(Q2) equiv 2M2N
Q2
int x0
0
dx g2(x Q2) =1
4F2(Q2)GM(Q2) (13)
which can be written as a ldquosuperconvergence sum rulerdquo
Q2
16παM2N
limνrarr0
νS2(νQ2) =
int 1
0
dx g2 (x Q2) = 0 (14)
The latter is valid for any value of Q2 provided that the integral converges for x rarr 0
Combining Eq (5) with the above LEXs of the VVCS amplitudes we can relate IA(Q2)
I1(Q2) γ0(Q2) and δLT (Q2) to moments of inelastic structure functions see Sec III It is
important to note that only γ0(Q2) and δLT (Q2) are generalized polarizabilities The relation
of the inelastic moments IA(Q2) and I1(Q2) to polarizabilities will be discussed in details in
Secs III C and III D The difference between S1(νQ2) and Snonpole1 (νQ2) cf Eq (8a) will
be important in this context
C Details on χPT calculation and uncertainty estimate
In this work we calculated the NLO prediction of BχPT for the polarized non-Born
VVCS amplitudes This includes the leading pion-nucleon (πN) loops see Ref [32 Fig 1]
as well as the subleading tree-level Delta-exchange (∆-exchange) see Ref [30 Fig 2] and
the pion-Delta (π∆) loops see Ref [32 Fig 2] In the δ-power-counting scheme [36] the
LO and NLO non-Born VVCS amplitudes and polarizabilities are of O(p3) and O(p4∆)
respectively2 The LECs are listed in Table I sorted by the order at which they appear in
our calculation At the given orders there are no ldquonewrdquo LECs that would need to be fitted
from Compton processes For more details on the BχPT formalism we refer to Ref [30]
2 In the full Compton amplitude there is a lower order contribution coming from the Born terms leading
to a shift in nomenclature by one order the LO contribution referred to as the NLO contribution etc
see eg Ref [45]
8
where power counting predictive orders (Sec III A) and the renormalization procedure
(Sec III B) are discussed
A few remarks are in order for the inclusion of the ∆(1232) and the tree-level ∆-exchange
contribution In contrast to Ref [32] we include the Coulomb-quadrupole (C2) N rarr ∆
transition described by the LEC gC The relevant Lagrangian describing the non-minimal
γlowastN∆ coupling [33 34] (note that in these references the overall sign of gC is inconsistent
between the Lagrangian and Feynman rules) reads
L(2)∆ =
3e
2MNM+
N T3
igM F
microν partmicro∆ν minus gEγ5Fmicroν partmicro∆ν (15)
+igCM∆
γ5γα(partα∆ν minus partν∆α)partmicroF
microν
+ Hc
with M+ = MN + M∆ and the dual of the electromagnetic field strength tensor F microν =12εmicroνρλFρλ Even though the Coulomb coupling is subleading compared with the electric
and magnetic couplings (gE and gM) its relatively large magnitude cf Table I makes it
numerically important for instance in γ0(Q2) Furthermore we study the effect of modifying
the magnetic coupling using a dipole form factor
gM rarrgM[
1 + (QΛ)2 ]2 (16)
where Λ2 = 071 GeV2 The inclusion of this Q2 dependence mimics the form expected from
vector-meson dominance It is motivated by observing the importance of this form factor
for the correct description of the electroproduction data [33]
To estimate the uncertainties of our NLO predictions we define
δ(Q2) =
radic(∆
MN
)2
+
(Q2
2MN∆
)2
(17)
such that the neglected next-to-next-to-leading order terms are expected to be of relative
size δ2 [33] The uncertainties in the values of the parameters in Table I have a much
smaller impact compared to the truncation uncertainty and can be neglected Unfortunately
∆IA(Q2) γ0(Q2) and γ0(Q2) ie the sum rules involving the cross section σTT (νQ2)
as well as the polarizability ∆I1(Q2) turn out to be numerically small Their smallness
suggests a cancellation of leading orders (which can indeed be confirmed by looking at
separate contributions as shown below) Therefore an error of δ2(Q2)P (Q2) where P (Q2)
is a generalized polarizability might underestimate the theoretical uncertainty for some of
the NLO predictions To avoid this we estimate the uncertainty of our NLO polarizability
9
predictions by
σP (Q2) =
Max[δ4(0)P (0)2 δ4(0)P LO(0)2 δ2(0)PNLO(0)2
]+Max
[δ4(Q2)
[P (Q2)minus P (0)
]2 δ4(Q2)
[P LO(Q2)minus P LO(0)
]2
δ2(Q2)[PNLO(Q2)minus PNLO(0)
]2]12
(18)
where P LO(Q2) is the πN -loop contribution PNLO(Q2) are the ∆-exchange and π∆-loop
contributions and P (Q2) = P LO(Q2) + PNLO(Q2) This error prescription is similar to the
one used in eg Refs [46ndash49] Here since we are interested in the generalized polarizabil-
ities we added in quadrature the error due to the static piece P (0) and the Q2-dependent
remainder P (Q2) minus P (0) Note that the static values of IA(0) and I1(0) are given by the
elastic Pauli form factor which is not part of our BχPT prediction and is considered to be
exact
Note that our result for the spin polarizabilities (and the unpolarized moments [30]) are
NLO predictions only at low momentum transfers Q mπ At larger values of Q amp ∆ they
become incomplete LO predictions Indeed in this regime the ∆ propagators do not carry
additional suppression compared to the nucleon propagators and the π∆ loops are promoted
to LO In general we only expect a rather small contribution from omitted π∆ loops to the
Q2 dependence of the polarizabilities since π∆ loops show rather weak dependence on Q2
compared with the ∆ exchange or πN loops Nevertheless this issue has to be reflected in
the error estimate Since the static polarizabilities P (0) are not affected it is natural to
separate the error on the Q2-dependent remainder P (Q2) minus P (0) as done in Eq (18) To
accommodate for the potential loss of precision above Q amp ∆ we define the relative error
δ(Q2) as growing with increasing Q2 see Eq (17)
Upon expanding our results in powers of the inverse nucleon mass Mminus1N we are able to
reproduce existing results of heavy-baryon χPT (HBχPT) at LO We however do not see
a rationale to drop the higher-order Mminus1N terms when they are not negligible (ie when
their actual size exceeds by far the natural estimate for the size of higher-order terms)
Comparing our BχPT predictions to HBχPT we will also see a deficiency of HBχPT in
the description of the Q2 behaviour of the polarizabilities Note that the O(p4) HBχPT
results from Ref [50 51] which we use here for comparison do not include the ∆ These
references studied the leading effect of the latter in the HBχPT framework using the small-
scale expansion [35] observing no qualitative improvement in the HBχPT description of the
empirical data [50 51] when including it We therefore choose to use the O(p4) results as
the representative HBχPT curves
10
Another approach used in the literature to calculate the polarizabilities in χPT is the
infrared regularization (IR) scheme introduced in Ref [52] This covariant approach tries to
solve the power counting violation observed in Ref [53] by dropping the regular parts of the
loop integrals that contain the power-counting-breaking terms However this subtraction
scheme modifies the analytic structure of the loop contributions and may lead to unexpected
problems as was shown in Ref [54] As we will see in the next section the IR approach
also fails to describe the Q2 behaviour of the polarizabilities
III RESULTS AND DISCUSSION
We now present the NLO BχPT predictions for the nucleon polarizabilities and selected
moments of the nucleon spin structure functions Our results are obtained from the calcu-
lated non-Born VVCS amplitudes and the LEXs in Eqs (10) and (12) For a cross-check
we used the photoabsorption cross sections described in App B In addition to the full NLO
results we also analyse the individual contributions from the πN loops the ∆ exchange
and the π∆ loops
A γ0(Q2) mdash generalized forward spin polarizability
The forward spin polarizability
γ0(Q2) =1
2π2
int infinν0
dν
radic1 +
Q2
ν2
σTT (νQ2)
ν3(19)
=16αM2
N
Q6
int x0
0
dx x2
[g1(xQ2)minus 4M2
Nx2
Q2g2(xQ2)
]
provides information about the spin-dependent response of the nucleon to transversal pho-
ton probes The RCS analogue of the above generalized forward spin polarizability sum
rule is sometimes referred to as the Gell-Mann Goldberger and Thirring (GGT) sum rule
[11] At Q2 = 0 the forward spin polarizability is expressed through the lowest-order spin
polarizabilities of RCS as γ0 = minus(γE1E1 + γM1M1 + γE1M2 + γM1E2) The forward spin po-
larizability of the proton is relevant for an accurate knowledge of the (muonic-)hydrogen
hyperfine splitting as it controls the leading proton-polarizability correction [16 62]
The πN -loop ∆-exchange and π∆-loop contributions to the NLO BχPT prediction of
11
000 005 010 015 020 025 030
-4
-3
-2
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
4D
proton
000 005 010 015 020 025 030
-4
-3
-2
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
4D
neutron
000 005 010 015 020 025 030
0
1
2
Q2 GeV2D
∆LT
10-
4fm
4D
proton
000 005 010 015 020 025 030
0
1
2
3
Q2 GeV2D
∆LT
10-
4fm
4D
neutron
FIG 2 Upper panel Generalized forward spin polarizability for the proton (left) and neutron
(right) as function of Q2 The result of this work the NLO BχPT prediction is shown by the
blue solid line and the blue band The red line represents the LO BχPT result The purple short-
dashed line is the O(p4) HB result from Ref [50] note that the corresponding proton curve is
outside of the plotted range The black dotted line is the MAID model prediction [55ndash57] which
is taken from Ref [1] (proton) and Ref [20] (neutron) The pink band is the IR+∆ result from
Ref [58] and the gray band is the BχPT+∆ result from Ref [31] Empirical extractions for the
proton Ref [18] (blue dots) Ref [59] (purple square) and Ref [60] (orange triangle uncertainties
added in quadrature) and neutron Ref [20] (blue diamonds) and Ref [61] (green dots statistical
and systematic uncertainties added in quadrature) Lower Panel Longitudinal-transverse spin
polarizability for the proton (left) and neutron (right) The orange dot-dashed and purple short-
dashed lines are the O(p3) and O(p4) HB results from Ref [50] The pink band is the IR result
from Ref [58] and the gray band is the covariant BχPT+∆ result from Ref [31] The black dotted
line is the MAID model prediction [55ndash57] note that for the proton we use the updated estimate
from Ref [1] obtained using the π η ππ channels
12
the static forward spin polarizability amount to in units of 10minus4 fm4
γ0p = minus093(92) asymp 201minus 284minus 010 (20a)
γ0n = 003(92) asymp 298minus 284minus 010 (20b)
while the slope is composed as follows in units of 10minus4 fm6
dγ0p(Q2)
dQ2
∣∣∣∣Q2=0
= minus022(4) asymp minus033 + 011 + 001 (21a)
dγ0n(Q2)
dQ2
∣∣∣∣Q2=0
= minus061(7) asymp minus073 + 011 + 001 (21b)
Figure 2 upper panel shows our NLO prediction as well as the LO πN loops compared
to different experimental and theoretical results For the proton we have one determination
at the real-photon point by the GDH collaboration [19] γ0p = minus100(8)(12)times10minus4 fm4 and
further Jefferson Laboratory data [18 60] at very low Q2 For the neutron only data at finite
Q2 are available [20 61] The experimental data for the proton are fairly well reproduced
in the whole Q2 range considered here while for the neutron the agreement improves with
increasing Q2 The HB limit of our πN -loop contribution reproduces the results published
in Refs [50 63] for arbitrary Q2 In addition our prediction is compared to the MAID
model [1 20] the IR+∆ calculation of Ref [58] and the BχPT+∆ result of Ref [31]
The πN -production channel gives a positive contribution to the photoabsorption cross
section σTT (νQ2) at low Q2 cf Fig 10 Accordingly one observes that the πN loops give
a sizeable positive contribution to γ0(Q2) The Delta on the other hand has a very large
effect by cancelling the πN loops and bringing the result close to the empirical data From
Fig 3 upper panel one can see that it is the ∆ exchange which dominates while π∆
loops are negligible This was expected since the forward spin polarizability sum rule is an
integral over the helicity-difference cross section in which σ32 is governed by the Delta at
low energies (the relevant energy region for the sum rule)
To elucidate the difference between the present calculation and the one from Ref [31] we
note that the two calculations differ in the following important aspects Firstly Ref [31] uses
the small-scale counting [64] that considers ∆ and mπ as being of the same size ∆ sim mπ
In practice this results in a set of π∆-loop graphs which contains graphs with one or two
γ∆∆ couplings and hence two or three Delta propagators Such graphs are suppressed in
the δ-counting and thus omitted from our calculation while present in that of Ref [31]
Secondly the Lagrangians describing the interaction of the Delta are constructed differently
and assume slightly different values for the coupling constants In particular we employ
13
000 005 010 015 020 025 030
-3
-2
-1
0
1
2
Q2GeV
2D
Γ01
0-
4fm
4D
proton
000 005 010 015 020 025 030
-3
-2
-1
0
1
2
3
Q2GeV
2D
Γ01
0-
4fm
4D
neutron
000 005 010 015 020 025 030
00
05
10
15
Q2GeV
2D
∆LT1
0-
4fm
4D
proton
000 005 010 015 020 025 030
00
05
10
15
20
25
Q2GeV
2D
∆LT1
0-
4fm
4D
neutron
FIG 3 Contributions of the different orders to the chiral predictions of γ0(Q2) upper paneland δLT (Q2) lower panel for the proton (left) and neutron (right) Red solid line πN -loop
contribution green dot-dashed line ∆-exchange contribution orange dotted line π∆-loop con-
tribution blue long-dashed line total result purple dot-dot-dashed line total result without gCcontribution black short-dashed line total result without gM dipole
(where possible) the so-called ldquoconsistentrdquo couplings to the Delta field ie those couplings
that project out the spurious degree of freedom see Refs [37 65 66] The authors of
Ref [31] on the other hand use couplings where the consistency in this sense is not enforced
The effects of these differences are of higher order in the δ-counting expansion and their
contribution to the Q2 dependence of the considered polarizabilities is expected to be rather
small however the differences at Q2 = 0 could be noticeable [67]
Finally as mentioned in Sec II C the inclusion of the dipole form factor in the magnetic
coupling gM is expected to be important to generate the correct Q2 behaviour of the polar-
izabilities Comparing our predictions for the forward spin polarizability with and without
inclusion of the form factor see Fig 3 upper panel confirms this Without the dipole
our results for the proton and neutron are closer to the ones from Ref [31] where the form
factor is not included For the neutron our prediction without the dipole is able to describe
14
the experimental points at very low Q2 deviating from the data with increasing Q2 The
π∆-loop contribution does not modify the Q2 behavior of γ0(Q2) and only differs from
Ref [31] by a small global shift Note also the relatively large effect of gC which generates
a sign change for virtualities above sim 02 GeV2 see Fig 3 upper panel
B δLT (Q2) mdash longitudinal-transverse polarizability
The longitudinal-transverse spin polarizability
δLT (Q2) =1
2π2
int infinν0
dν
radic1 +
Q2
ν2
σLT (νQ2)
Qν2(22)
=16αM2
N
Q6
int x0
0
dx x2[g1(xQ2) + g2(xQ2)
]
contains information about the spin structure of the nucleon and is another important
input in the determination of the (muonic-)hydrogen hyperfine splitting [16 62] It is also
relevant in studies of higher-twist corrections to the structure function g2(xQ2) given by
the moment d2(Q2) [51] see Section III E The peculiarity of the response encoded in this
polarizability is that it involves a spin flip of the nucleon and a polarization change of the
photon see App B and Fig 11
It is expected that the Delta isobar gives only a small contribution to δLT (Q2) what makes
this polarizability a potentially clean test case for chiral calculations Consequently there are
relatively many different theoretical calculations of δLT (Q2) coming from different versions
of χPT with baryons (HB IR and covariant) Ref [50] found a systematic deviation of the
HB result for δLTn(Q2) from the MAID model prediction This disagreement was identified
by the authors of Ref [68] as a puzzle involving the neutron δLT polarizabilitymdashthe δLT
puzzle The IR calculation in Ref [58] also showed a deviation from the data and predicted
a rapid rise of δLT (Q2) with growing Q2 The problem is solved by keeping the relativistic
structure of the theory as the BχPT+∆ result of Ref [31] showed
As expected already the leading πN loops provide a reasonable agreement with the
experimental data cf Fig 2 lower panel Since the ∆-exchange contribution to δLT (Q2)
is small the effect of the gM form factor is negligible in this polarizability as is that of the gC
coupling cf Fig 3 lower panel In fact we predict both the ∆-exchange and the π∆-loop
contributions to be small and negative This is in agreement with the MAID model which
predicts a small and negative contribution of the P33 wave to δLT (Q2) However in the
calculation of Ref [31] which is different from the one presented here only in the way the
∆(1232) is included the contribution of this resonance to δLTp(Q2) is sizeable and positive
15
The authors of that work attributed this large contribution to diagrams where the photons
couple directly to the Delta inside a loop As mentioned in Sec III F the effect of such loop
diagrams does not change the Q2 behaviour of the polarizabilities On the other hand it
can produce a substantial shift of the δLT (Q2) as a whole A higher-order calculation should
resolve the discrepancy between the two covariant approaches however it will partially lose
the predictive power since the LECs appearing at higher orders will have to be fitted to
experimental data
The πN -loop ∆-exchange and π∆-loop contributions to the NLO BχPT prediction of
the static longitudinal-transverse polarizability are in units of 10minus4 fm4
δLTp = 132(15) asymp 150minus 016minus 002 (23a)
δLTn = 218(23) asymp 235minus 016minus 002 (23b)
while the slopes are in units of 10minus4 fm6
dδLTp(Q2)
dQ2
∣∣∣∣Q2=0
= minus085(8) asymp minus080minus 004minus 001 (24a)
dδLTn(Q2)
dQ2
∣∣∣∣Q2=0
= minus124(12) asymp minus119minus 004minus 001 (24b)
C IA(Q2) mdash a generalized GDH integral
The helicity-difference cross section σTT exhibits a faster fall-off in ν than its spin-averaged
counterpart σT This is due to a cancellation between the leading (constant) terms of σ12
and σ32 at large ν3 The resulting 1ν fall-off of the helicity-difference cross section allows
one to write an unsubtracted dispersion relation for the VVCS amplitude gnonpoleTT (νQ2) cf
Eq (10a) This is the origin of the GDH sum rule [5 6]
minus α
2M2N
κ2 =1
2π2
int infinν0
dνσTT (ν)
ν (25)
which establishes a relation to the anomalous magnetic moment κ It is experimentally
verified for the nucleon by MAMI (Mainz) and ELSA (Bonn) [71 72]
There are two extensions of the GDH sum rule to finite Q2 the generalized GDH integrals
3 Notice that a constant term in σTT at ν rarrinfin is forbidden by crossing symmetry
16
000 005 010 015 020 025 030
-30
-25
-20
-15
-10
-05
00
Q2 GeV2D
I Aproton
000 005 010 015 020 025 030
-25
-20
-15
-10
-05
00
Q2 GeV2D
I A
neutron
000 005 010 015 020 025 030
-05
00
05
Q2 GeV2D
I 1
proton
000 005 010 015 020 025 030
-10
-08
-06
-04
-02
00
02
Q2 GeV2D
I 1
neutron
FIG 4 Upper panel The generalized GDH integral IA(Q2) for the proton (left) and neutron
(right) as function of Q2 The result of this work the NLO BχPT prediction is shown by the blue
solid line and the blue band The red line represents the LO BχPT result The purple short-dashed
line is the O(p4) HB result from Ref [50 51] The gray band is the BχPT+∆ result from Ref [31]
The black dotted line is the MAID model prediction [69] Experimental extractions for the proton
Ref [60] (orange triangle uncertainties added in quadrature) and neutron Refs [21][27] where
magenta dotsorange diagonal crosses correspond to data and red squareslilac crosses correspond
to data plus extrapolation to unmeasured energy regions The green stars at the real-photon point
are derived from the anomalous magnetic moments κp asymp 1793 and κn asymp minus1913 [70] Lower
panel The generalized GDH integral I1(Q2) for the proton (left) and neutron (right) as function
of Q2 The purple short-dashed line is the HB result from Ref [51] Experimental extractions for
the proton Ref [18] (blue dots) and Ref [60] (orange triangle uncertainties added in quadrature)
and neutron Ref [61][27] (uncertainties added in quadrature) where green dotsorange diagonal
crosses correspond to data and gray squareslilac crosses correspond to data plus extrapolation to
unmeasured energy regions
17
IA(Q2) and I1(Q2) The latter will be discussed in Sec III D The former is defined as4
minus α
2M2N
IA(Q2) = minus 1
8π2
int infinν0
dν
radic1 +
Q2
ν2
σTT (νQ2)
ν(26)
=α
Q2
int x0
0
dx[4M2
Nx2
Q2g2(xQ2)minus g1(xQ2)
]
Due to its energy weighting the integral in Eq (26) converges slower than the one in the
generalized forward spin polarizability sum rule (19) Therefore knowledge of the cross
section at higher energies is required and the evaluation of the generalized GDH integral
IA(Q2) is not as simple as the evaluation of γ0(Q2)
The generalized GDH integral IA(Q2) is directly related to the non-pole amplitude
gnonpoleTT (νQ2) which differs from non-Born amplitude gTT (νQ2) by a term involving the
elastic Pauli form factor
gnonpoleTT (νQ2) = gTT (νQ2)minus 2παν
M2N
F 22 (Q2) (27)
cf Eqs (2a) and (8a) Consequently IA(Q2) is not a pure polarizability but also contains
an elastic contribution The ldquonon-polarizabilityrdquo or the Born part of IA(Q2) is given by
IBornA (Q2) = IA(Q2)minus∆IA(Q2) = minus1
4F 2
2 (Q2) (28)
where we refer to the polarizability part as ∆IA(Q2) The same is true for the generalized
GDH integral I1(Q2) which is directly related to Snonpole1 (νQ2)
IBorn1 (Q2) = I1(Q2)minus∆I1(Q2) = minus1
4F 2
2 (Q2) (29)
In the following we will add the Born parts to our LO and NLO BχPT predictions for
the polarizabilities ∆IA(Q2) and ∆I1(Q2) employing an empirical parametrization for the
elastic Pauli form factor [73] This allows us to compare to the experimental results for
IA(Q2) and I1(Q2) cf Fig 4 Note that the blue error bands only describe the uncertainties
of our BχPT predictions of the polarizabilities while the elastic contributions are considered
to be exact as explained in Sec II C The uncertainties of the polarizability predictions are
therefore better reflected in Fig 5 where we show the contributions of the different orders
to the BχPT predictions of ∆IA(Q2) and ∆I1(Q2) as well as the total results with error
bands
4 Note that IA(Q2) is sometimes called ITT (Q2)
18
000 005 010 015 020 025 030
-08
-06
-04
-02
00
02
Q2 GeV2D
DI A
proton
000 005 010 015 020 025 030
-10
-08
-06
-04
-02
00
02
Q2 GeV2D
DI A
neutron
000 005 010 015 020 025 030-005
000
005
010
015
020
025
Q2 GeV2D
DI 1
proton
000 005 010 015 020 025 030-015
-010
-005
000
005
010
015
Q2 GeV2D
DI 1
neutron
FIG 5 Contributions of the different orders to the chiral predictions of ∆IA(Q2) upper paneland ∆I1(Q2) lower panel for the proton (left) and neutron (right) Red solid line πN -loop
contribution green dot-dashed line ∆-exchange contribution orange dotted line π∆-loop contri-
bution blue solid line and blue band total result purple dot-dot-dashed line total result without
gC contribution black short-dashed line total result without gM dipole
The E97-110 experiment at Jefferson Lab has recently published their data for IAn(Q2) in
the region of 0035 GeV2 lt Q2 lt 024 GeV2 [27] In addition there are results for IAn(Q2)
from the earlier E94-010 experiment [21] and for IAp(Q2) from the E08-027 experiment [60]
The O(p4) HB calculation gives a large negative effect [51] which does not describe the data
The BχPT+∆ result from Ref [31] which mainly differs from our work by the absence of
the dipole form factor in gM looks similar to this HB result and only describes the data
points at lowest Q2 Our NLO prediction however follows closely the Q2 evolution of the
data In Fig 5 upper panel we show the polarizability ∆IA(Q2) whose Q2 evolution is
clearly dominated by the ∆ exchange Similar to the case of γ0p(Q2) inclusion of the dipole
in gM and the Coulomb coupling gC is very important in order to describe the experimental
data The LO prediction on the other hand slightly overestimates the data cf Fig 4
upper panel
19
At the real-photon point IA(0) = minusκ2
4and ∆IA(0) = 0 Therefore we give only the
slope of the polarizability ∆IA(Q2) [showing also the separate contributions from πN loops
∆ exchange and π∆ loops] in units of GeVminus2
d∆IAp(Q2)
dQ2
∣∣∣∣Q2=0
= minus858(343) asymp 238minus 1121 + 025 (30a)
d∆IAn(Q2)
dQ2
∣∣∣∣Q2=0
= minus955(343) asymp 141minus 1121 + 025 (30b)
Including the empirical Pauli form factor [73] we find in units of GeVminus2
dIAp(Q2)
dQ2
∣∣∣∣Q2=0
= minus318dIAn(Q2)
dQ2
∣∣∣∣Q2=0
= minus300 (31)
D Γ1(Q2) and I1(Q2) mdash the first moment of the structure function g1(xQ2)
The second variant for a generalization of the GDH sum rule to finite Q2 is defined as
minus α
2M2N
I1(Q2) = minus 1
8π2
int infinν0
dν1radic
ν2 +Q2
[σTT (νQ2) +
Q
νσLT (νQ2)
](32)
= minus α
Q2
int x0
0
dx g1(xQ2)
where I1(0) = minusκ2
4 This generalized GDH integral directly stems from the amplitude
Snonpole1 (νQ2) with the LEX from Eq (12a) It is given by the first moment of the struc-
ture function g1(xQ2) Γ1(Q2) =int x0
0dx g1(xQ2) as follows I1(Q2) =
2M2N
Q2 Γ1(Q2) The
isovector combination
Γ1(pminusn)(Q2) =
int x0
0
dx[g1p(xQ
2)minus g1n(xQ2)] (33)
is related to the axial coupling of the nucleon through the Bjorken sum rule [74 75]
limQ2rarrinfin
Γ1(pminusn)(Q2) =
gA6 (34)
As explained in Eq (28) the moment I1(Q2) splits into a polarizability part ∆I1(Q2) and
a Born part IBorn1 (Q2) Figure 4 lower panel shows the Q2 dependence of I1(Q2) which in
contrast to IA(Q2) shown in Figure 4 upper panel is clearly dominated by its Born part
and the elastic Pauli form factor The πN -loop ∆-exchange and π∆-loop contributions to
the polarizability ∆I1(Q2) are shown in Fig 5 lower panel Comparing to Fig 5 upper
20
000 005 010 015 020 025 030
-006
-004
-002
000
002
004
Q2 GeV2D
G1
proton
000 005 010 015 020 025 030
-006
-005
-004
-003
-002
-001
000
001
Q2 GeV2D
G1
neutron
FIG 6 First moment of the structure function g1(xQ2) for the proton (left) and neutron (right)
as function of Q2 The legend is the same as in Fig 4
panel one sees that ∆I1(Q2) is less sensitive to gC and the dipole form factor in gM than
∆IA(Q2)
For the proton our NLO BχPT prediction gives a very good description of the ex-
perimental data [18 60] and is in reasonable agreement with the MAID prediction [69]
For the neutron one observes good agreement with the empirical evaluations including ex-
trapolations to unmeasured energy regions starting from Q2 gt 01 GeV2 [27 61] In the
region of Q2 lt 005 GeV2 one observes an interesting tension between the recent E97-
110 experiment [27] and the data from CLAS [61] While the newest measurement finds
I1n(0035 GeV2) lt κ2n4 thus suggesting a negative slope at low Q the older measurement
found a rather large value for I1n(00496 GeV2) A similar but milder behaviour is seen in
the E97-110 [27] and E94-010 [21] data for IAn The MAID predictions do not agree with
the CODATA recommended values for the anomalous magnetic moments of the proton and
neutron [70] which in our work are imposed by using empirical parametrizations for the
elastic Pauli form factors [73] The slope of the HB result from Ref [51] is too large and
therefore only reproduces the data at very low Q2
Figure 6 shows the moment Γ1(Q2) for the proton and neutron while Fig 7 shows the
isovector combination Γ1 pminusn(Q2) The LO and NLO BχPT predictions are identical be-
cause our calculation produces the same Delta contributions for the proton and the neutron
For the isovector combination the MAID model only agrees with the data at very low
Q2 lt 010 GeV2 The same is true for the IR result [58 76] while all other chiral results
describe the data NLO BχPT (this work) BχPT+∆ [31] and HBχPT [51]
At the real-photon point I1(0) = minusκ2
4and ∆I1(0) = 0 Therefore we give only the
slope of the polarizability ∆I1(Q2) [showing also the separate contributions from πN loops
21
000 005 010 015 020 025 030-002
000
002
004
006
008
010
Q2 GeV2D
G1Hp-
nL
FIG 7 Isovector combination of Γ1(Q2) as function of Q2 The legend is the same as in Fig 6
The pink curve is the IR result from Ref [58 76] The experimental points are from Ref [77]
(brown dots) and Ref [23] (orange squares)
∆ exchange and π∆ loops] in units of GeVminus2
d∆I1p(Q2)
dQ2
∣∣∣∣Q2=0
= 039(4) asymp 034minus 053 + 058 (35a)
d∆I1n(Q2)
dQ2
∣∣∣∣Q2=0
= minus101(10) asymp minus107minus 053 + 058 (35b)
Including the empirical Pauli form factor [73] we find in units of GeVminus2
dI1p(Q2)
dQ2
∣∣∣∣Q2=0
= 580dI1n(Q2)
dQ2
∣∣∣∣Q2=0
= 553 (36)
E d2(Q2) mdash a measure of color polarizability
Another interesting moment to consider is d2(Q2) which is related to the twist-3 part of
the spin structure function g2(xQ2) [79 80]
d2(Q2) equiv 3
int 1
0
dx x2[g2(xQ2)minus gWW2 (xQ2)] (37)
where gWW2 (xQ2) is the twist-2 part of g2(xQ2) Using the Wandzura-Wilczek relation [81]
one can relate d2(Q2) to moments of the spin structure functions g1(xQ2) and g2(xQ2)
d2(Q2) =
int 1
0
dx x2 [3g2(xQ2) + 2g1(xQ2)] (38)
22
000 005 010 015 020 025 030
0000
0001
0002
0003
0004
0005
0006
0007
Q2 GeV2D
d2
proton
000 005 010 015 020 025 030
0000
0002
0004
0006
0008
0010
0012
Q2 GeV2D
d2
neutron
000 005 010 015 020 025 030
-1
0
1
2
Q2 GeV2D
Γ 010
-4
fm6
D
proton
000 005 010 015 020 025 030-2
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
6D
neutron
FIG 8 Upper panel The inelastic moment d2(Q2) for the proton (left) and neutron (right) as
function of Q2 The result of this work the NLO BχPT prediction is shown by the blue solid line
and the blue band The red line represents the LO BχPT result The purple short-dashed line is
the O(p4) HB result from Ref [50 51] The black dotted line is the MAID model prediction [69]
The experimental points for the neutron (cyan dots) are from Ref [22] Lower panel Fifth-order
generalized forward spin polarizability γ0(Q2) for the proton (left) and neutron (right) as function
of Q2 The experimental points for the proton are from Ref [59] (purple square) and Ref [78]
(orange dot)
This relation however only holds for asymptotically large Q2 It is also in the high-Q2
region where d2(Q2) is a measure of color polarizability [82 83] through its relation to the
gluon field strength tensor [80] We refer to Ref [84] for a recent review on the spin structure
of the nucleon including a discussion of sum rules for deep inelastic scattering and color
polarizabilities
What we consider in the following is the inelastic part of d2(Q2) defined as the moment
of g1(xQ2) and g2(xQ2) spin structure functions cf Eq (38)
d2(Q2) =
int x0
0
dx x2 [3g2(xQ2) + 2g1(xQ2)] (39)
23
This moment provides another testing ground for our BχPT predictions through comparison
with experiments on the neutron [22] Going towards the low-Q2 region the interpretation
of d2(Q2) in terms of color polarizabilities will fade out The above definition however
implies it is related to other VVCS polarizabilities
d2(Q2) =Q4
8M4N
[M2
NQ2
αδLT (Q2) + I1(Q2)minus IA(Q2)
] (40)
Note that d2(Q2) and its first two derivatives with respect to Q2 vanish at Q2 = 0 The
considerations in Eqs (28) and (29) have no effect on d2(Q2) since the Born contribution
from IA(Q2) and I1(Q2) cancel out Therefore d2(Q2) is a pure polarizability
In Fig 8 upper panel we show our NLO BχPT prediction and other results for d2(Q2)
While MAID [69] and BχPT describe the experimental data for the neutron [22] very well
the HB limit [50 51] is showing a fast growth with Q2 This illustrates the importance
of keeping the relativistic result Note also that even though the πN -loop contribution
is dominant both gC and the form factor in gM are essential to obtain a curvature that
reproduces the data cf Fig 9 upper panel For the proton there are to our knowledge
no experimental results to compare with However the agreement between the NLO BχPT
prediction and the MAID prediction at low energies is reasonable
F γ0(Q2) mdash fifth-order generalized forward spin polarizability
It is interesting to compare the generalized fifth-order forward spin polarizability sum
rule
γ0(Q2) =1
2π2
int infinν0
dν
radic1 +
Q2
ν2
σTT (νQ2)
ν5(41)
=64αM4
N
Q10
int x0
0
dx x4
[g1(xQ2)minus 4M2
Nx2
Q2g2(xQ2)
]
to the sum rule integrals for IA(Q2) and γ0(Q2) since they differ merely by their energy
weighting of σTT (νQ2) and a constant prefactor cf Eqs (19) (26) and (41) From IA(Q2)
to γ0(Q2) to γ0(Q2) the energy suppression is increasing by a factor of νminus2 respectively
Therefore the description of γ0(Q2) should be easiest in a low-energy effective-field theory
such as χPT whereas γ0(Q2) and IA(Q2) receive larger contributions from higher energies
In Fig 8 lower panel we show our LO and NLO BχPT predictions for γ0(Q2) One
can see that the πN -loop contribution is positive (in accordance to what we see for the cross
section σTT see Fig 10) The Delta shifts it substantially especially in the low Q2 region
24
000 005 010 015 020 025 030
-0004
-0002
0000
0002
0004
0006
0008
0010
Q2GeV
2D
d2
proton
000 005 010 015 020 025 030
-0005
0000
0005
0010
0015
Q2GeV
2D
d2
neutron
000 005 010 015 020 025 030
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
6D
proton
000 005 010 015 020 025 030
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
6D
neutron
FIG 9 Contributions of the different orders to the chiral predictions of d2(Q2) upper panel and
γ0(Q2) lower panel for the proton (left) and neutron (right) Red solid line πN -loop contribu-
tion green dot-dashed line ∆-exchange contribution orange dotted line π∆-loop contribution
blue long-dashed line total result purple dot-dot-dashed line total result without gC contribution
black short-dashed line total result without gM dipole
bringing it into a better agreement with data In general the BχPT curves start above the
empirical data points at the real-photon point and then decrease asymptotically to zero
above Q2 gt 01 GeV2 On the other hand the MAID prediction reproduces the empirical
data at the real-photon point then decreases to negative values until about Q2 gt 006 GeV2
from where it also starts to asymptotically approach zero Consequently our NLO BχPT
prediction of γ0(Q2) is consistently above the MAID prediction This is very different to
what we saw for IA(Q2) in Fig 4 upper panel where the MAID prediction at the real-
photon point is above the experimental value While the agreement of our predictions with
the empirical data is in general quite good for all moments of σTT (νQ2) one should point
out that both for γ0n(Q2) and γ0p(Q2) we overestimate the data at low Q2 For IA(Q2) such
observation cannot be made because ∆IA(0) = 0 and thus IA(0) is given by the empirical
Pauli form factor only From IA(Q2) γ0(Q2) and γ0(Q2) the latter has the smallest however
25
non-negligible dependence on gC and the dipole in gM cf Fig 9 lower panel
The πN -loop ∆-exchange and π∆-loop contributions to the NLO BχPT prediction of
the static fifth-order forward spin polarizability amount to in units of 10minus4 fm6
γ0p = 112(30) asymp 208minus 096minus 001 (42a)
γ0n = 195(30) asymp 292minus 096minus 001 (42b)
while the slope is composed as follows in units of 10minus4 fm8
dγ0p(Q2)
dQ2
∣∣∣∣Q2=0
= minus084(10) asymp minus100 + 016 + 000 (43a)
dγ0n(Q2)
dQ2
∣∣∣∣Q2=0
= minus142(15) asymp minus158 + 016 + 000 (43b)
Note that the HB prediction of the static polarizability γ0p (423 at O(p3) and 365 at O(ε3)
[78 85]) is almost one order of magnitude larger than the empirical value and therefore not
shown in Fig 8
G Summary
Our results are summarized in Table II where we give the contributions of the different
orders to the chiral predictions of the polarizabilities and their slopes at the real-photon
point A quantitative comparison of our predictions for the static spin polarizabilities to
the work of Bernard et al [31] and different empirical evaluations is shown in Table III We
can see that the inclusion of the Delta turns out to be very important for all moments of
the helicity-difference cross section To describe the Q2 behavior of the polarizabilities the
magnetic coupling of the N rarr ∆ transition should be modified by a dipole form factor as
has been observed previously in the description of electroproduction data [33] This dipole
form factor effectively takes account of vector-meson exchanges The Coulomb-quadrupole
N rarr ∆ transition despite its subleading order is important in the description of some
moments of spin structure functions This is contrary to what we saw for the moments of
unpolarized structure functions [30] where the Coulomb coupling had a negligible effect
The π∆ loops are mainly relevant for the generalized GDH integrals
26
TABLE II The NLO BχPT predictions for the forward VVCS polarizabilities and their slopes at
Q2 = 0 The contributions of the πN loops the ∆ exchange and the π∆ loops are shown together
with the combined total result Note that IA(0) = I1(0) = d2(0) = 0 and (d2)prime = 0
πN loops ∆ exchange π∆ loops Total
γ0 p 201 minus284 minus010minus093(92)
(10minus4 fm4) n 298 003(92)
δLT p 150 minus016 minus002132(15)
(10minus4 fm4) n 235 218(23)
γ0 p 208 minus096 minus001112(30)
(10minus4 fm6) n 292 195(30)
(γ0)prime p minus033011 001
minus022(4)
(10minus4 fm6) n minus073 minus061(7)
(δLT )prime p minus080 minus004 minus001minus085(8)
(10minus4 fm6) n minus119 minus124(12)
(γ0)prime p minus100016 000
minus084(10)
(10minus4 fm8) n minus158 minus142(15)
(∆IA)prime p 238 minus1121 025minus858(343)
(GeVminus2) n 141 minus955(343)
(∆I1)prime p 034 minus053 058039(4)
(GeVminus2) n minus107 minus101(10)
TABLE III Our NLO BχPT predictions for the spin polarizabilities at Q2 = 0 compared with
the BχPT+∆ predictions from Bernard et al [31] and the available empirical information Where
the reference is not given the empirical number is provided by the MAID analysis [55 69] with
unspecified uncertainty
Proton Neutron
This work BχPT+∆ Empirical This work BχPT+∆ Empirical
γ0 minus093(92) minus174(40) minus100(8)(12) [19] 003(92) minus077(40) minus0005
(10minus4 fm4) minus090(8)(11) [78] [MAID]
minus0929(105) [59]
δLT 132(15) 240(1) 134 218(23) 238(3) 203
(10minus4 fm4) [MAID] [MAID]
γ0 112(30) 060(7)(7) [78] 195(30) 123
(10minus4 fm6) 0484(82) [59] [MAID]
IV CONCLUSIONS
We have presented a complete NLO calculation of the polarized non-Born VVCS ampli-
tudes in covariant BχPT with pion nucleon and ∆(1232) fields The dispersion relations
between the VVCS amplitudes and the tree-level photoabsorption cross sections served as a
27
cross-check of these calculations
The obtained moments of the proton and neutron spin structure functions related to gen-
eralized polarizabilities and GDH-type integrals agree well with the available experimental
data The description of their Q2 evolution is improved compared to the previous χPT pre-
dictions In particular the NLO BχPT predictions obtained here give a better description
of the empirical data (eg from the Jefferson Laboratory ldquoSpin Physics Programrdquo) than
the HB [50 51] and IR [58] calculations
The demonstrated predictive power of the χPT framework amplitudes makes it well
suited for extending the χPT evaluation of the TPE effect in the hyperfine structure of
(muonic-)hydrogen [15ndash17] to next-to-leading order
ACKNOWLEDGEMENTS
We thank Lothar Tiator and Marc Vanderhaeghen for helpful discussions This work
is supported by the Deutsche Forschungsgemeinschaft (DFG) through the Collaborative
Research Center [The Low-Energy Frontier of the Standard Model (SFB 1044)] JMA ac-
knowledges support from the Community of Madrid through the ldquoPrograma de atraccion de
talento investigador 2017 (Modalidad 1)rdquo and the Spanish MECD grants FPA2016-77313-P
FH gratefully acknowledges financial support from the Swiss National Science Foundation
Appendix A Tensor decompositions of the VVCS amplitudes
In this appendix we review the decomposition of the forward VVCS process into tensor struc-
tures and scalar amplitudes In particular we consider the connection between the covariant and
the semi-relativistic decomposition in the lab frame that is defined in terms of the conventional
transverse longitudinal transverse-transverse and transverse-longitudinal amplitudes
As explained in Sec II A the process of forward VVCS off the nucleon can be described in
terms of four explicitly covariant amplitudes S1 2 and T1 2 [3]
T (νQ2) =
(minusgmicroν +
qmicroqν
q2
)T1(νQ2) +
1
M2N
(pmicro minus p middot q
q2qmicro)(
pν minus p middot qq2
qν)T2(νQ2) (A1)
minus 1
MNγmicroναqα S1(νQ2)minus 1
M2N
(γmicroνq2 + qmicroγναqα minus qνγmicroαqα
)S2(νQ2)
εprimelowastmicro εν
where εmicro (εprimelowastmicro ) are the incoming (outgoing) photon polarization vectors ν is the photon lab-frame
energy and Q2 is the photon virtuality Alternatively the decomposition in the laboratory frame
28
(which in the forward case coincides with the Breit frame) is parametrized in terms of the nucleon
Pauli matrices ~σ and the four scalar functions fL fT gTT and gLT
T (νQ2) = ε0 εprimelowast0 fL(νQ2) + (~ε primelowast middot ~ε ) fT (νQ2) + i~σ middot (~ε primelowast times ~ε ) gTT (νQ2) (A2)
minusi~σ middot [(ε0~εprimelowast minus ~ε ε primelowast0 )times q] gLT (νQ2)
Here ~q and q = ~q|~q | are the photon three-momentum in the lab system and its unit vector The
modified polarization vector components are given by
ε0 =
[ε0 minus
ν
|~q | (~ε middot q )
] |~q |Q ~ε = ~εminus q (~ε middot q ) (A3)
where ε = (ε0~ε ) is the usual incoming photon polarization vector and εprimelowast the outgoing polarization
vector The LEX of the lab frame amplitudes [Eq (10)] can serve in particular as the definition
of the generalized polarizabilities The lab frame amplitudes are also conveniently used for the
definition of the response functions see the example of the scalar amplitude gLT (νQ2) and the
corresponding response function σLT (νQ2) below in App B
Appendix B Photoabsorption cross sections
In the forward kinematics the spin-dependent VVCS amplitudes and the spin polarizabilities
can be described in terms of the polarized structure functions g1(xQ2) and g2(xQ2) or equiv-
alently the helicity-difference cross section σTT (νQ2) and the longitudinal-transverse response
function σLT (νQ2) with the help of dispersion relations (5) and the optical theorem (3) In this
way the photoabsorption cross sections measured in electroproduction processes form the basis
for most empirical evaluations shown throughout Sec III In the following we present the BχPT
predictions for the tree-level cross sections of πN - π∆- and ∆-production through photoabsorption
on the nucleon cf Figs 8 9 and 10 in Ref [30] In Secs B 1 and B 2 we will discuss the leading
πN -production channel and the ∆-production channel respectively We used these cross sections
to verify the polarizability predictions obtained otherwise from the calculated non-Born VVCS
amplitudes Due to the bad high-energy behavior of the π∆-production cross sections in BχPT
cf Fig 10 the dispersion relations in Eq (5) require further subtractions for a reconstruction of
the π∆-loop contribution to the spin-dependent VVCS amplitudes Therefore not all polarizabil-
ities could be verified but only those appearing as higher-order terms in the LEX of the VVCS
amplitudes such as γ0 [16]
29
proton neutron
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Neutron
02 04 06 08 100
50
100
150
200
250
[GeV]
T [b]
02 04 06 08 10
-50
0
50
100
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Neutron
02 04 06 08 100
50
100
150
200
250
[GeV]
T [b]
02 04 06 08 10
-50
0
50
100
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02040608100
50
100
150
200
250
300
[GeV]
T[b]
0204060810
-50
0
50
[GeV]
TT[b]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
0204060810
00
02
04
06
[GeV]
LTQ
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
0204060810
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
FIG 10 Photoabsorption cross sections for πN (red) and π∆ production (orange) with Q2 = 0
(solid) and Q2 = 01 GeV2 (dashed for πN and dotted for π∆ channel)
1 πN-production channel
In order to extract the response function σLT (νQ2) we have developed a method similar to
the one used to calculate σTT (νQ2) see for example Ref [86] For σLT (νQ2) however the
calculation is more complicated because one has to take into account that the associated Compton
process involves a spin-flip of the nucleon as illustrated in Fig 11 When calculating the cross
section the product of the incoming nucleon spinors has to reflect this flip
The forward VVCS amplitude related to σLT (νQ2) mdash and δLT (Q2) mdash is gLT (νQ2) It can
be extracted from Eq (A2) if one takes the modified polarization vector components in Eq (A3)
with ε = εL and εprimelowast = εlowastplusmn as input where εL = 1Q(|~q | 0 0 q0) and εplusmn = ∓ 1radic
2(0 1plusmni 0) are the
standard longitudinal and transverse polarization vectors respectively For εL and εlowastplusmn only the
choice of helicities hprime = plusmn12 and h = ∓12 gives a non-zero contribution and one obtains
χdaggerhprime T (νQ2)χh = χdaggerhprimeminusi~σ middot [(ε0~εprimelowast minus ~ε ε primelowast0 )times q] gLT (νQ2)χh =
radic2 gLT (νQ2) (B1)
where χh and χdaggerhprime are two-component Pauli spinors with opposite helicities or here spins
Let us now consider the related photoabsorption process and in particular the tree-level γlowastN rarr
30
s = 12
sprime = minus12
λ = 0 λprime = 1
rprime rprime
s=12sprime=minus1
2
λ=0λprime=1
rprimerprime
s=12
sprime=minus12
λ=0λprime=1
rprimerprime hacute=frac12h=-frac12
λacute=1λ=0
FIG 11 Relation between the forward Compton process and the photoabsorption process given
by the optical theorem In particular we show the longitudinal-transverse contribution The
double-line arrows represent the spin of the external particles while the dot represents the scalar
(longitudinal) polarization of the incoming photon Inside the blob the intermediate states are
represented eg nucleons with spins rprime (which are averaged in the calculation of the cross section)
and pions
πN channel see diagrams in Fig 8 of Ref [30] We define the πN -production amplitude as
T = uhB (PB)sumi
Ai(s t)Γi uhA(PA) (B2)
with the Dirac structures
Γ1 = γ5 (B3a)
Γ2 =1
2
[qA ε
]γ5 (B3b)
where uhA(PA) and udaggerhB (PB) are the Dirac spinors and PA and PB are the four-momenta of
the incoming and outgoing nucleons respectively When calculating the photoabsorption cross
section related to the VVCS amplitude in Eq (B1) the nucleon spin flip should be implemented
by uhprime(PA) in T dagger and uh(PA) in T together with the appropriate transverse and longitudinal
photon polarization vectors εlowastplusmn and εL
However if one wants to use the properties of the Dirac matrices it is more useful to construct
an operator to produce this spin flip in the external nucleons of Fig 11 This is accomplished by
introducing the projector ΓLT equiv 12radic
2(γ1 + iγ2)γ5 which also takes into account the extra factor
radic2 in Eq (B1) We checked that with this projector one correctly extracts δLT by comparing the
HB limit of our result to the HB result of Ref [50] where the authors calculate this polarizability
from the Compton amplitude directly With all those ingredients the longitudinal-transverse cross
section is calculated in the following way
σLT (νQ2) =1
64π2 s
|~pf |cm
|~pi|cm
int 1
minus1dcos θ
sumij
AiAdaggerjXij (B4)
31
with
Xij = Tr[(PB +MN )Γi(PA +MN )ΓLTγ0Γdaggerjγ
0] (B5)
where θ is the scattering angle in the center-of-mass (cm) frame and |~pi|cm (|~pf |cm) is the three-
momentum of an incoming (outgoing) particle in the cm frame An explicit calculation of the
matrix Xij leads to
X = MNQ
0 2(PB minus PA) middot εLminusradic
2|~qf |cm sin θ (sminus u)
(B6)
where |~qi|cm (|~qf |cm) is the relative three-momentum of the incoming (outgoing) particles in the
cm frame Here s t and u are the usual Mandelstam variables For the different γlowastN rarr πN
channels we obtain the following amplitudes Ai where we introduce qA as the four-momentum of
the incoming photon and qB as the four-momentum of the outgoing pion
bull γlowastprarr π0p
A1 =e gAMN
fπ
[2PA middot ε+ qA middot ε
sminusM2N
+2PB middot εminus qA middot ε
uminusM2N
] (B7a)
A2 =e gAMN
fπ
[1
sminusM2N
+1
uminusM2N
] (B7b)
bull γlowastprarr π+n
A1 =
radic2 e gAMN
fπ
[2PA middot ε+ qA middot ε
sminusM2N
+2(PA minus PB) middot ε+ qA middot ε
tminusm2π
] (B8a)
A2 =
radic2 e gAMN
fπ(sminusM2N )
(B8b)
bull γlowastnrarr π0n
A1 = 0 (B9a)
A2 = 0 (B9b)
bull γlowastnrarr πminusp
A1 =
radic2 e gAMN
fπ
[2PB middot εminus qA middot ε
uminusM2N
minus 2(PA minus PB) middot ε+ qA middot εtminusm2
π
] (B10a)
A2 =
radic2 e gAMN
fπ(uminusM2N )
(B10b)
The analytical expressions shown above were checked with the amplitudes given in Ref [87] An-
alytical expressions for the tree-level γlowastN rarr πN channel of the σLT (νQ2) and σTT (νQ2) cross
32
sections are given below (proton channels π+n and π0p neutron channel πminusp) We checked that
they reproduce the known results in the real-photon limit [45 86] To shorten the final expressions
for the cross sections which are considerably longer for finite Q2 than in the real-photon limit we
define the following dimensionless kinematic variables
αγ = (ENi )cmradics =
s+M2N +Q2
2s (B11)
απ = (ENf )cmradics =
s+M2N minusm2
π
2s (B12)
βγ = Eγcmradics =
sminusM2N minusQ2
2s (B13)
βπ = Eπcmradics =
sminusM2N +m2
π
2s (B14)
λγ = |~qi|cmradics =
radic(sminusM2
N minusQ2)2 + 4sQ2
2s (B15)
λπ = |~qf |cmradics =
radic(sminusM2
N +m2π)2 minus 4sm2
π
2s (B16)
Here (ENi )cm and (ENf )cm are the energies of the incoming and outgoing nucleon Eγcm is the energy
of the incoming photon Eπcm is the energy of the outgoing pion all in the cm frame
σ(π+n)TT = minus e2g2
AM2N
64πf2πs
3(sminusM2N )2λ4
γ
4sλπλγ
[(M2
N minus s)(M2N minusQ2 minus s)(Q2 + 2sβγβπ)
+ 2s(m2πQ
2 minus (M2N minus s)
(M2N + s(minus1 + 2βγβπ)
))λ2γ
]+ (M2
N minus s)(M2N minusQ2 minus s)
times (Q2 + 2sβγβπ minus 2sλπλγ)(Q2 + 2s(βγβπ + λπλγ)
)times log
(Q2 + 2sβγβπ minus 2sλγλπQ2 + 2sβγβπ + 2sλγλπ
) (B17)
33
σ(π0p)TT =
e2g2AM
2Nλπ
64πf2πs(sminusM2
N )2λγ
1
s2λπλ3γ
[(M2
N minus s)(M2N minusQ2 minus s)
(3M2
N + s(2βγβπ minus 3))(M2N
+ s(2βγβπ minus 1))
+ 2s
((2m2
πQ2 + (M2
N minus s)2)(M2
N minus s)minus 2(minusm2
πQ2 + (sminusM2
N )2)
times sβγβπ + 4s2(sminusM2N )β2
γβ2π
)λ2γ
]arctan
(2sλπλγ
M2N + s(2βγβπ minus 1)
)+ 2
[minus 2m2
πQ2
+ (M2N minus s)
(2(sminusM2
N ) + 4sβγβπ +(sminusM2
N +Q2)(3M2N minus 3s+ 2sβγβπ)
sλ2γ
+m2πQ
2(sminusM2N )
(M2N minus s+ 2sβγβπ)(M2
N + s(minus1 + 2βγβπ minus 2λπλγ))
+m2πQ
2(sminusM2N )
(M2N minus s+ 2sβγβπ)(M2
N + s(minus1 + 2βγβπ + 2λπλγ))
)
+1
λπλγ
(βγβπ
(minusm2
πQ2 + (M2
N minus s)(M2N minus s+ 2sβγβπ)
)minus (M2
N minus s)(M2N minusQ2 minus s)λ2
π
)log
(M2N + s(minus1 + 2βγβπ + 2λπλγ)
M2N + s(minus1 + 2βγβπ minus 2λπλγ)
)] (B18)
σ(πminusp)TT =
e2g2AM
2N
64πf2πs
3λ4γ(M2
N + s(2βπβγ minus 1))2 minus 4λ2πs
2λ2γ)
4sλπλγ
((minusM2
N +Q2 + s)(M2N
+ s(2βπβγ minus 1))
2 minus 2sλ2γ
(2λ2
πs(minusM2N +Q2 + s) +m2
πQ2))
+((M2N + s(2βπβγ minus 1)
)2
minus 4λ2πs
2λ2γ
)((M2N + s(2βπβγ minus 1)
)(M2N minus 2Q2 minus s(2βπβγ + 1)
)+ 4λ2
πs2λ2
γ
)times log
(minusM2
N minus 2βπsβγ minus 2λπsλγ + s)minus(M2N + s(2βπβγ minus 2λπλγ minus 1)
)(M2N + s(minus1
+ 2βπβγ + 2λπλγ))[((
M2N + s(2βπβγ minus 1)
)(M2N minus 2Q2 minus s(2βπβγ + 1)
)+ 4λ2
πs2λ2
γ
)times log
(minusM2
N minus 2βπsβγ + 2λπsλγ + s)
+((Q2 + 2βπsβγ)2 minus 4λ2
πs2λ2
γ
)times log
(Q2 + 2βπsβγ minus 2λπsλγQ2 + 2sβπβγ + 2λπsλγ
)]+ 4sλ2
γ
((M2
N + s(2βπβγ minus 1))2 minus 4λ2πs
2λ2γ
)times (M2
N minus s) arctanh
(2λπsλγ
M2N + 2βπsβγ minus s
) (B19)
34
σ(π+n)LT =
e2g2AM
3Nλπ
32πf2πQs
3(sminusM2N )2λ4
γ
2sλγ
[(M2
N minus s)(Q2 + 2sβ2γ)(Q2 + 2sβγβπ)
minus 4s
((M2
N minus s)(Q2 minus 2s(απ minus 1)βγ
)+Q2sβγβπ
)λ2γ + 8s3(minus1 + απ)λ4
γ
]minus M2
N minus sλπ
[(Q2 + 2sβ2
γ)(Q2 + 2sβγβπ)2 + 4s2(2(απ minus 1)βγ(Q2 + 2sβγβπ)
minusQ2λ2π
)λ2γ + 8s3(απ minus 1)2λ4
γ
]arctan
(2sλπλγ
Q2 + 2sβγβπ
) (B20)
σ(π0p)LT =
e2g2AM
3Nλπ
16πf2πQs(sminusM2
N )2λγ
1
minus2s(M2N + s(minus1 + 2βγβπ))2λ2
γ + 8s3λ2πλ
4γ
[minus 3M8
N (Q2
+ 2sβ2γ) + 2M4
Ns2
(minus (Q2 + 2sβ2
γ)(minus3 + 2βγβπ) +(Q2(minus3 + 2βγβπ + 6λ2
π) + 2sβγ
times (minus3 + 12απ + 2βγβπ minus 12απβγβπ + 4βγλ2π))λ2γ minus 4s(α2
π minus 1)λ4π
)+ 2M2
Ns3
[minus (Q2
+ 2sβ2γ)(minus1 + 2βγβπ)
(6 + βγβπ(minus9 + 2βγβπ)
)+
(Q2(3minus 12λ2
π + 4βγβπ(minus1 + 2βγβπ
+ λ2π))minus 2sβγ
(minus 3 + 4απ(3 + 2απ(3 + 2βγβπ(minus3 + βγβπ)) + 4βγ(βπ + (2minus βγβπ)
times λ2π)))λ2γ + 8s
((απ minus 1)(1 + απ minus 2βγβπ) + 2απβγλ
2π
)λ4γ
)+ s4
((Q2 + 2sβ2
γ)
times (1minus 2βγβπ)2(2βγβπ minus 3) + 2(Q2(minus1 + 6λ2
π + 2βγβπ((1minus 2βγβπ)2 minus 2λ2π)) + 2sβγ
times (minus1 + 2βγβπ + 4(minus1 + βγβπ)(minusαπ + 2απβγβπ minus βγλ2π)))λ2γ minus 8
(s(απ minus 1)(απ
+ (1minus 2βγβπ)2) + 2βγ(2sαπ +Q2βγ)λ2π
)λ4γ + 32s(απ minus 1)λ2
πλ6γ
)+ 2M6
Ns
(Q2(6
minus 7βγβπ + λ2γ) + 2sβγ(βγ(6minus 7βγβπ) + (1minus 4απ)λ2
γ)
)]+sminusM2
N
4s2λπλ3γ
[(Q2 + 2sβ2
γ)
times(3M2
N + s(2βγβπ minus 3))(M2N + s(2βγβπ minus 1)
)+ 2s
(minusM2
N (Q2 + 2sβγ minus 8sαπβγ)
+ s(2sβγ(1minus 4απ + 4απβγβπ) +Q2(1minus 2λ2π)))λ2γ + 8s3(απ minus 1)λ4
γ
]
times arctan
(2sλπλγ
M2N + s(2βγβπ minus 1)
) (B21)
35
σ(πminusp)LT =
e2g2AM
3N
64πf2πQs
3λ4γ(M2
N minusQ2 minus s)(M2N + s(2βπβγ + 2λπλγ minus 1))
times
1
M2N + s(2βπβγ minus 2λπλγ minus 1)
[4λπsλγ(M2
N minusQ2 minus s)(
2sλ2γ
(s((1minus 2λ2
π)Q2
+ 2(2απ minus 1)sβγ(2βπβγ minus 1))minusM2N (Q2 + 2(1minus 2απ)sβγ)
)+ (Q2 + 2sβ2
γ)(M2N + s(2βπβγ minus 1)
)2 + 8(απ minus 1)απs
3λ4γ
)]minus(M2N + s(2βπβγ + 2λπλγ minus 1)
)times[
log
(minusM2N minus 2βπsβγ minus 2λπsλγ + s
minusM2N minus 2βπsβγ + 2λπsλγ + s
)(minus (Q2 + 2sβ2
γ)(M2N + s(2βπβγ minus 1)
)(minusM2
N + 2Q2 + 2βπsβγ + s)minus 2sλ2γ
(minusM2
N (Q2 + 2sβγ) +Q4 +Q2s(4απβγ
minus 2βγ minus 2λ2π + 1) + 2s2βγ(4(απ minus 1)βπβγ + 1)
)minus 8(απ minus 1)2s3λ4
γ
)+ 2
(4s2λ2
γ
(2(απ minus 1)βγ(Q2 + 2βπsβγ)minus λ2
πQ2)
+ (Q2 + 2βπsβγ)2(Q2 + 2sβ2γ)
+ 8(απ minus 1)2s3λ4γ
)arctanh
(2λπsλγ
Q2 + 2βπsβγ
)] (B22)
2 ∆-production channel
The tree-level ∆-exchange diagram in Fig 2 of Ref [30] contributes to the non-Born part of
the VVCS amplitudes The contribution of the ∆ exchange to the VVCS amplitudes can be split
into [17]
S∆-exch1 (νQ2) = S∆-pole
1 (νQ2) + S∆-exch1 (νQ2) (B23a)
νS∆-exch2 (νQ2) = νS∆-pole
2 (νQ2) + νS2∆-exch
(νQ2) (B23b)
and similarly for the unpolarized VVCS amplitudes discussed in Ref [30] Here we introduced the
∆-pole contributions S∆-polei and the ∆-non-pole contributions S∆-exch
i The former amplitudes
feature a pole at the ∆(1232)-production threshold and thus are proportional to
1
[sminusM2∆][uminusM2
∆]=
1
4M2N
1
ν2∆ minus ν2
(B24)
36
They can be reconstructed from the dispersion relations in Eq (5) with the tree-level ∆-production
cross sections as input cf Fig 10 in Ref [30]
σTT (νQ2) =π2α
M2NM
2+|~q |
minus g2
MMN (M+ + ν)|~q |2 +g2E(∆minus ν)(Q2 minusMNν)2
MN(B25a)
+g2CQ
4s(∆minus ν)
MNM2∆
minus 4gMgE(Q2 minusMNν)|~q |2 minus 4gMgCQ2|~q |2
+2gEgCQ
2[minusMNM∆ |~q |2 + s(Q2 +∆ν)
]MNM∆
δ(ν minus ν∆)
σLT (νQ2) =Qπ2α
M2NM
2+|~q |
g2E(MNν minusQ2) [M∆(MN + ν)minus s]
MN(B25b)
+g2CQ
2[MNM∆|~q |2 minus s(Q2 +∆ν)
]MNM2
∆
+ gMgEM∆|~q |2 minusgMgC(Q2 minusMNν)|~q |2
M∆
+gEgC(ν minus∆)(M2
N |~q |2 minus 2Q2s)
MNM∆
δ(ν minus ν∆)
with ∆ = M∆ minusMN M+ = M∆ + MN and the Mandelstam variable s = M2N + 2MNν minus Q2
Analytical expressions for the spin structure functions g1(xQ2) and g2(xQ2) can be constructed
from Eq (3) with the flux factor K(νQ2) = |~q | =radicν2 +Q2
In the ∆-non-pole contributions to S1(νQ2) and νS2(νQ2) the pole in ν at the ∆(1232)-
production threshold has canceled out
S∆-exch1 (νQ2) =
πα
MNM2+
[g2MQ
2+ + g2
E
(∆2 minus 3Q2
)+
4g2CQ
4
M2∆
minus 8gMgEM∆ωminus (B26a)
minus2gMgCQ2(MN minus 4M∆)
M∆+
2gEgCQ2(3MN minus 2M∆)
M∆
]
νS2∆-exch
(νQ2) =2πα
MNM2+
[g2EM∆∆ωminus +
g2M MNQ
2+
2+g2C Q
2(Q2 minus∆2)
2M∆(B26b)
+gEgM M∆(M∆ω+ minus 4MNωminus)minus gEgC ∆(2Q2 +MNω+)
+gMgC Q2(4MN minus ω+)
]+S∆-exch
2 (νQ2)
ν
[M2
∆ ω2+
M2N
+ ν2
]
with Q+ =radic
(M∆ +MN )2 +Q2 and ωplusmn = (M2∆minusM2
N plusmnQ2)2M∆ and the non-pole contribution
to S2(νQ2)
S∆-exch2 (νQ2) = minus2παMNν
M∆M2+
[gM + gE
]gC (B27)
These amplitudes to the contrary are not described by the tree-level ∆-production cross sections
in the standard dispersive approach [17] This peculiarity has been previously missed e g in the
calculation of the ∆-exchange contribution to the hydrogen hyperfine splitting in Ref [88] The
importance of including the ∆-non-pole contribution is also evident when considering the BC sum
37
rule in Eq (14) The ∆-pole terms by themselves violate the BC sum rule but cancel exactly with
the ∆-non-pole terms
limνrarr0
νS∆-pole2 (νQ2) + lim
νrarr0νS2
∆-exch(νQ2) = 0 (B28)
Appendix C Polarizabilities at Q2 = 0
In this section we give analytical expressions for the static values and slopes at Q2 = 0 of
the spin polarizabilities and moments of polarized structure functions In particular we give the
HB expansion of the πN -loop contributions and the ∆-exchange contributions The complete
expressions also for the π∆-loop contributions can be found in the Supplemented material Recall
that IA(0) = I1(0) = d2(0) = 0 and dd2(Q2)dQ2
∣∣∣Q2=0
= 0
1 πN-loop contribution
Here we give analytical expressions for the πN -loop contributions to the proton and neutron
spin polarizabilities expanded in powers of micro = mπMN viz the HB expansion Note that we
choose to expand here to a high order in micro the strict HB expansion would only retain the leading
term in an analogous NLO calculation
bull Static polarizabilities (Q2 = 0)
γ0p =e2g2
A
96π3f2πm
2π
1minus 21πmicro
8minus(
59
2+ 26 logmicro
)micro2 +
1875πmicro3
64
+3
(3
2+ 26 logmicro
)micro4 +
(C1)
γ0n =e2g2
A
48π3f2πm
2π
1
2minus 9πmicro
16minus 2micro2 logmicro+
75πmicro3
128minus 3micro4
4+
(C2)
δLTp =e2g2
A
192π3f2πm
2π
1minus 9πmicro
8+
(13
2minus 2 logmicro
)micro2 minus 465πmicro3
64
minus(
47
2+ 42 logmicro
)micro4 +
(C3)
δLTn =e2g2
A
96π3f2πm
2π
1
2+
3πmicro
16+ (1 + 2 log micro)micro2 minus 105πmicro3
128+
5micro4
4+
(C4)
38
δLTn =e2g2
A
96π3f2πm
2π
1
2+
3πmicro
16+ (1 + 2 log micro)micro2 minus 105πmicro3
128+
5micro4
4+
(C5)
γ0p =e2g2
A
16π3f2πm
4π
4
45minus 3πmicro
16+
14micro2
5minus 1813πmicro3
384minus 192
5(1 + log micro)micro4
+80703πmicro5
2048+
(C6)
γ0n =e2g2
A
16π3f2πm
4π
4
45minus 5πmicro
48+
4micro2
5minus 245πmicro3
384minus 32micro4 logmicro
15
+1323πmicro5
2048+
(C7)
bull Slopes of polarizabilities at Q2 = 0
dγ0p(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
2minus 45πmicro
4+ 223micro2 minus 28515πmicro3
64
minus9
(1953
4+ 449 logmicro
)micro4 +
570255πmicro5
128+
(C8)
dγ0n(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
2minus 81πmicro
8+ 94micro2 minus 2535πmicro3
32
minus3 (1 + 90 logmicro)micro4 +84315πmicro5
1024+
(C9)
dδLTp(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
2880π3f2πm
4π
minus5
2minus 27πmicro
32+ 20micro2 minus 5865πmicro3
256
+3
(617
4+ 36 logmicro
)micro4 minus 2056845πmicro5
4096+
(C10)
dδLTn(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
minus5
4minus 81πmicro
64minus 11micro2 +
10005πmicro3
512
+15
8(11 + 48 logmicro)micro4 minus 267015πmicro5
8192+
(C11)
dIAp(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
96π2f2πmicro
2
1minus 15πmicro
4minus 1
2(115 + 88 logmicro)micro2 +
1839πmicro3
32
+5 (5 + 34 logmicro)micro4 +
(C12)
dIAn(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
48π2f2πmicro
2
1
2minus 11πmicro
8minus 1
4(1 + 20 logmicro)micro2 +
99πmicro3
64
minus25micro4
12+
(C13)
dI1p(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
96π2f2πmicro
3π
8+ 2 (4 + 3 logmicro)microminus 537πmicro2
64
minus1
2(15 + 56 logmicro)micro3 +
(C14)
39
dI1n(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
48π2f2πmicro
minus π
16+
1
4(3 + 4 logmicro)microminus 57πmicro2
128+
2micro3
3+
(C15)
γ0p(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2g2
A
16π3f2πm
6π
1
105minus 23πmicro
256+
377micro2
210minus 15551πmicro3
6144+
3371micro4
105
minus1640457πmicro5
32768+
(C16)
γ0n(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2g2
A
16π3f2πm
6π
1
105minus 153πmicro
1792+
69micro2
70minus 4615πmicro3
6144+
172micro4
35
minus120897πmicro5
32768+
(C17)
2 ∆-exchange contribution
Here we give analytical expressions for the tree-level ∆-exchange contributions to the nucleon
spin polarizabilities and their slopes at Q2 = 0 Note that the ∆-exchange contributes equally to
proton and neutron polarizabilities Recall that for the magnetic γlowastN∆ coupling we introduced a
dipole form factor to mimic vector-meson dominance gM rarr gM(1 +Q2Λ2)2
bull Static polarizabilities (Q2 = 0)
γ0 = minus e2
4πM2+
(g2M
∆2+
g2E
M2+
minus 4gMgEM+∆
) (C18)
δLT =e2M∆
4πM3+
(g2E
MNM++gMgE∆MN
minus gEgCM2
∆
) (C19)
γ0 =e2M2
N
π∆2M4+
(minusg
2M
∆2+
g2E
M2+
+4gMgE∆M+
) (C20)
bull Slopes of polarizabilities at Q2 = 0
dγ0(Q2)
dQ2
∣∣∣∣∣Q2=0
= minus e2
πM2+∆
(g2M
∆
[1
4∆2minus 1
∆M++
1
2M2+
]minus 1
Λ2
g2M
∆+
g2E
2M2+
[1
2∆minus 3
M+
]minusgMgE
M+
[1
∆2minus 5
∆M++
1
M2+
]+
1
Λ2
2gMgEM+
+2gMgC∆M2
+
minus gEgCM3
+
) (C21)
40
d δLT (Q2)
dQ2
∣∣∣∣∣Q2=0
=e2M∆∆
4πMNM2+
(g2E
∆2M2+
[1
∆minus 4
M+
]minus g2
C
∆M2∆M
2+
+gMgE∆2M+
[1
∆2
minus 3
∆M++
1
M2+
]minus 2
Λ2
gMgE∆2M+
+gMgC∆M2
∆
[1
2∆2minus 2
∆M++
1
2M2+
]minus gEgC
2M2∆M
2+
[7
∆+
1
M+
]) (C22)
dIA(Q2)
dQ2
∣∣∣∣∣Q2=0
= minusM2N
M2+
(g2M
2∆2+
g2E
MNM+minus 2gMgE
∆M+minus gEgCM∆M+
) (C23)
dI1(Q2)
dQ2
∣∣∣∣∣Q2=0
= minusM∆M2N
2M3+
(g2E
MNM∆minus gMgE∆MN
minus gEgCM2
∆
) (C24)
γ0(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2M2
N
π∆3M6+
(g2M
∆
[3M2
∆ + 2M∆MN minus 9M2N
∆2+
4M2+
Λ2
]
minus4gMgE
[5M2
∆ minus 9M2N
M+∆2+
2M+
Λ2
]+
g2E
∆M+[7M∆ minus 9MN ]
minus8gMgC∆
+4gEgCM+
) (C25)
[1] D Drechsel B Pasquini and M Vanderhaeghen Dispersion relations in real and virtual
Compton scattering Phys Rept 378 99 (2003) hep-ph0212124
[2] S E Kuhn J-P Chen and E Leader Spin structure of the nucleon mdash status and recent
results Prog Part Nucl Phys 63 1 (2009) arXiv08123535 [hep-ph]
[3] F Hagelstein R Miskimen and V Pascalutsa Nucleon polarizabilities from Compton scat-
tering to hydrogen atom Prog Part Nucl Phys 88 29 (2016) arXiv151203765 [nucl-th]
[4] B Pasquini and M Vanderhaeghen Dispersion theory in electromagnetic interactions Ann
Rev Nucl Part Sci 68 75 (2018) arXiv180510482 [hep-ph]
[5] S Gerasimov A Sum rule for magnetic moments and the damping of the nucleon magnetic
moment in nuclei Sov J Nucl Phys 2 430 (1966)
[6] S Drell and A C Hearn Exact sum rule for nucleon magnetic moments Phys Rev Lett
16 908 (1966)
[7] J S Schwinger Source theory viewpoints in deep inelastic scattering Proc Natl Acad Sci
USA 72 1 (1975)
[8] J S Schwinger Source theory viewpoints in deep inelastic scattering Electromagnetic In-
teractions and Field Theory Proceedings 14 Internationale Universitatswochen Schladming
Austria February 24-March 7 1975 Acta Phys Austriaca Suppl 14 471 (1975)
41
[9] J Schwinger Source theory discussion of deep inelastic scattering with polarized particles
Proc Natl Acad Sci USA 72 1559 (1975)
[10] A M Baldin Polarizability of nucleons Nucl Phys 18 310 (1960)
[11] M Gell-Mann M L Goldberger and W E Thirring Use of causality conditions in quantum
theory Phys Rev 95 1612 (1954)
[12] A Pineda Leading chiral logarithms to the hyperfine splitting of the hydrogen and muonic
hydrogen Phys Rev C 67 025201 (2003)
[13] C Peset and A Pineda The two-photon exchange contribution to muonic hydrogen from
chiral perturbation theory Nucl Phys B 887 69 (2014) arXiv14064524 [hep-ph]
[14] C Peset and A Pineda Model-independent determination of the two-photon exchange contri-
bution to hyperfine splitting in muonic hydrogen JHEP 04 060 arXiv161205206 [nucl-th]
[15] F Hagelstein and V Pascalutsa Proton structure in the hyperfine splitting of muonic hydro-
gen PoS CD15 077 (2016) arXiv151104301 [nucl-th]
[16] F Hagelstein Exciting Nucleons in Compton Scattering and Hydrogen-Like Atoms PhD
thesis Mainz U Inst Kernphys (2017) arXiv171000874 [nucl-th]
[17] F Hagelstein ∆(1232)-Resonance in the hydrogen spectrum Proceedings 11th International
Workshop on the Physics of Excited Nucleons (NSTAR 2017) Columbia SC USA August
20-23 2017 Few Body Syst 59 93 (2018) arXiv180109790 [nucl-th]
[18] Y Prok et al (CLAS) Moments of the spin structure functions gp1 and gd1 for 005 lt Q2 lt 30
GeV2 Phys Lett B 672 12 (2009) arXiv08022232 [nucl-ex]
[19] H Dutz et al (GDH) First measurement of the Gerasimov-Drell-Hearn sum rule for 1H from
07 GeV to 18 GeV at ELSA Phys Rev Lett 91 192001 (2003)
[20] M Amarian et al (Jefferson Lab E94010) Measurement of the generalized forward spin po-
larizabilities of the neutron Phys Rev Lett 93 152301 (2004) arXivnucl-ex0406005
[21] M Amarian et al The Q2 evolution of the generalized Gerasimov-Drell-Hearn integral for
the neutron using a 3He target Phys Rev Lett 89 242301 (2002) arXivnucl-ex0205020
[22] M Amarian et al (Jefferson Lab E94-010)Q2 evolution of the neutron spin structure moments
using a 3He target Phys Rev Lett 92 022301 (2004) arXivhep-ex0310003
[23] A Deur et al Experimental determination of the evolution of the Bjorken integral at low Q2
Phys Rev Lett 93 212001 (2004) arXivhep-ex0407007
[24] K Slifer Low Q2 measurement of gp2 and the δLT spin polarizability Spin structure at long dis-
tance Proceedings Workshop Newport News USA March 12ndash13 2009 nucl-ex09064775
(2009) AIP Conf Proc 1155 10106313203293 (2009) arXiv09064775 [nucl-ex]
[25] P Solvignon et al (E01-012) Moments of the neutron g2 structure function at intermediate
42
Q2 Phys Rev C 92 015208 (2015) arXiv13044497 [nucl-ex]
[26] A Deur Experimental studies at low Q2 of the spin structure of the nucleon at Jefferson Lab
in 9th International Workshop on Chiral Dynamics (CD18) Durham NC USA September
17-21 2018 (2019) arXiv190305661 [nucl-ex]
[27] V Sulkosky et al (Jefferson Lab E97-110) Measurement of the 3He spin-structure functions
and of neutron (3He) spin-dependent sum rules at 0035 le Q2 le 024 GeV2 Phys Lett B
805 135428 (2020) arXiv190805709 [nucl-ex]
[28] K Adhikari et al (CLAS) Measurement of the Q2 Dependence of the Deuteron Spin Structure
Function g1 and its Moments at Low Q2 with CLAS Phys Rev Lett 120 062501 (2018)
arXiv171101974 [nucl-ex]
[29] V Lensky F Hagelstein A Hiller Blin and V Pascalutsa Comment on rdquoMeasurement of
the Q2 Dependence of the Deuteron Spin Structure Function g1 and its Moments at Low Q2
with CLASrdquo (2018) arXiv180603219 [nucl-th]
[30] J M Alarcn F Hagelstein V Lensky and V Pascalutsa Forward doubly-virtual Compton
scattering off the nucleon in chiral perturbation theory at NLO the subtraction function and
moments of unpolarized structure functions (2020) arXiv200509518 [hep-ph]
[31] V Bernard E Epelbaum H Krebs and U-G Meiszligner New insights into the spin structure
of the nucleon Phys Rev D 87 054032 (2013) arXiv12092523 [hep-ph]
[32] V Lensky J M Alarcon and V Pascalutsa Moments of nucleon structure functions at
next-to-leading order in baryon chiral perturbation theory Phys Rev C 90 055202 (2014)
arXiv14072574 [hep-ph]
[33] V Pascalutsa and M Vanderhaeghen Chiral effective-field theory in the ∆(1232) region
I Pion electroproduction on the nucleon Phys Rev D 73 034003 (2006) arXivhep-
ph0512244
[34] V Pascalutsa and M Vanderhaeghen Electromagnetic nucleon-to-Delta transition in chiral
effective field theory Phys Rev Lett 95 232001 (2005) arXivhep-ph0508060
[35] T R Hemmert B R Holstein and J Kambor Systematic 1M expansion for spin 32
particles in baryon chiral perturbation theory Phys Lett B 395 89 (1997) arXivhep-
ph9606456
[36] V Pascalutsa and D R Phillips Effective theory of the ∆(1232) in Compton scattering off
the nucleon Phys Rev C 67 055202 (2003) arXivnucl-th0212024
[37] V Pascalutsa M Vanderhaeghen and S N Yang Electromagnetic excitation of the ∆(1232)-
resonance Phys Rept 437 125 (2007) arXivhep-ph0609004
[38] K A Olive et al (Particle Data Group) Review of Particle Physics Chin Phys C 38 090001
43
(2014)
[39] F E Low Scattering of light of very low frequency by systems of spin 12 Phys Rev 96
1428 (1954)
[40] M Gell-Mann and M L Goldberger Scattering of low-energy photons by particles of spin
12 Phys Rev 96 1433 (1954)
[41] A M Harun ar-Rashid A simple derivation of Schwingerrsquos sum rule for spin dependent
structure functions Nuovo Cim A 33 447 (1976)
[42] F Hagelstein and V Pascalutsa Dissecting the hadronic contributions to (g minus 2)micro by
Schwingerrsquos sum rule Phys Rev Lett 120 072002 (2018) arXiv171004571 [hep-ph]
[43] V Lensky V Pascalutsa M Vanderhaeghen and C Kao Spin-dependent sum rules con-
necting real and virtual Compton scattering verified Phys Rev D 95 074001 (2017)
arXiv170101947 [hep-ph]
[44] H Burkhardt and W N Cottingham Sum rules for forward virtual Compton scattering
Annals Phys 56 453 (1970)
[45] V Lensky and V Pascalutsa Predictive powers of chiral perturbation theory in Compton
scattering off protons Eur Phys J C 65 195 (2010) arXiv09070451 [hep-ph]
[46] H Grieszlighammer J McGovern D Phillips and G Feldman Using effective field theory to
analyse low-energy Compton scattering data from protons and light nuclei Prog Part Nucl
Phys 67 841 (2012) arXiv12036834 [nucl-th]
[47] H W Grieszlighammer J A McGovern and D R Phillips Nucleon polarisabilities at and
beyond physical pion masses Eur Phys J A 52 139 (2016) arXiv151101952 [nucl-th]
[48] E Epelbaum H Krebs and U-G Meiszligner Improved chiral nucleon-nucleon potential up to
next-to-next-to-next-to-leading order Eur Phys J A 51 53 (2015) arXiv14120142 [nucl-
th]
[49] E Epelbaum H Krebs and U-G Meiszligner Precision nucleon-nucleon potential at fifth order
in the chiral expansion Phys Rev Lett 115 122301 (2015) arXiv14124623 [nucl-th]
[50] C W Kao T Spitzenberg and M Vanderhaeghen Burkhardt-Cottingham sum rule and
forward spin polarizabilities in heavy baryon chiral perturbation theory Phys Rev D 67
016001 (2003) arXivhep-ph0209241
[51] C-W Kao D Drechsel S Kamalov and M Vanderhaeghen Higher moments of nucleon
spin structure functions in heavy baryon chiral perturbation theory and in a resonance model
Phys Rev D 69 056004 (2004) arXivhep-ph0312102
[52] T Becher and H Leutwyler Baryon chiral perturbation theory in manifestly Lorentz invariant
form Eur Phys J C 9 643 (1999) arXivhep-ph9901384
44
[53] J Gasser M E Sainio and A Svarc Nucleons with chiral loops Nucl Phys B 307 779
(1988)
[54] L S Geng J Martin Camalich L Alvarez-Ruso and M J Vicente Vacas Leading SU(3)-
breaking corrections to the baryon magnetic moments in chiral perturbation theory Phys
Rev Lett 101 222002 (2008) arXiv08051419 [hep-ph]
[55] D Drechsel S S Kamalov and L Tiator The GDH sum rule and related integrals Phys
Rev D 63 114010 (2001) arXivhep-ph0008306
[56] D Drechsel O Hanstein S S Kamalov and L Tiator A unitary isobar model for pion
photo- and electroproduction on the proton up to 1 GeV Nucl Phys A 645 145 (1999)
arXivnucl-th9807001
[57] L Tiator private communication (2020)
[58] V Bernard T R Hemmert and U-G Meiszligner Spin structure of the nucleon at low energies
Phys Rev D 67 076008 (2003) arXivhep-ph0212033
[59] O Gryniuk F Hagelstein and V Pascalutsa Evaluation of the forward Compton scattering
off protons II Spin-dependent amplitude and observables Phys Rev D 94 034043 (2016)
arXiv160400789 [nucl-th]
[60] R Zielinski The g2p Experiment A Measurement of the Protonrsquos Spin Structure Functions
PhD thesis New Hampshire U (2010) arXiv170808297 [nucl-ex]
[61] N Guler et al (CLAS) Precise determination of the deuteron spin structure at low to mod-
erate Q2 with CLAS and extraction of the neutron contribution Phys Rev C 92 055201
(2015) arXiv150507877 [nucl-ex]
[62] C E Carlson V Nazaryan and K Griffioen Proton structure corrections to electronic
and muonic hydrogen hyperfine splitting Phys Rev A 78 022517 (2008) arXiv08052603
[physicsatom-ph]
[63] V Bernard N Kaiser and U-G Meiszligner Chiral dynamics in nucleons and nuclei Int J
Mod Phys E 4 193 (1995) arXivhep-ph9501384
[64] T R Hemmert B R Holstein and J Kambor ∆(1232) and the polarizabilities of the nucleon
Phys Rev D 55 5598 (1997) arXivhep-ph9612374
[65] V Pascalutsa and R Timmermans Field theory of nucleon to higher-spin baryon transitions
Phys Rev C 60 042201 (1999) arXivnucl-th9905065
[66] V Pascalutsa Quantization of an interacting spin-32 field and the Delta isobar Phys Rev
D 58 096002 (1998) arXivhep-ph9802288
[67] H Krebs Double Virtual Compton Scattering and SpinStructure of the Nucleon PoS
CD2018 031 (2019)
45
[68] N Kochelev and Y Oh Axial anomaly and the δLT puzzle Phys Rev D 016012 (2012)
arXiv11034892 [hep-ph]
[69] D Drechsel S Kamalov and L Tiator Unitary isobar model ndash MAID2007 Eur Phys J A
69 (2007) available at httpsmaidkphuni-mainzde arXiv07100306 [nucl-th]
[70] P J Mohr B N Taylor and D B Newell CODATA recommended values of the fundamental
physical constants 2010 Rev Mod Phys 84 1527 (2012)
[71] J Ahrens et al (GDH A2) First measurement of the Gerasimov-Drell-Hearn integral for 1H
from 200 to 800 MeV Phys Rev Lett 87 022003 (2001) arXivhep-ex0105089 [hep-ex]
[72] K Helbing (GDH) Experimental verification of the GDH sum rule at ELSA and MAMI Nucl
Phys Proc Suppl 105 113 (2002)
[73] R Bradford A Bodek H S Budd and J Arrington A New parameterization of the nucleon
elastic form-factors NuInt05 proceedings of the 4th International Workshop on Neutrino-
Nucleus Interactions in the Few-GeV Region Okayama Japan 26-29 September 2005 Nucl
Phys Proc Suppl 159 127 (2006) arXivhep-ex0602017
[74] J D Bjorken Applications of the chiral U(6)timesU(6) algebra of current densities Phys Rev
148 1467 (1966)
[75] J D Bjorken Inelastic scattering of polarized leptons from polarized nucleons Phys Rev D
1 1376 (1970)
[76] V Bernard T R Hemmert and U-G Meiszligner Novel analysis of chiral loop effects in
the generalized Gerasimov-Drell-Hearn sum rule Phys Lett B 545 105 (2002) arXivhep-
ph0203167
[77] A Deur et al Experimental study of isovector spin sum rules Phys Rev D 78 032001
(2008) arXiv08023198 [nucl-ex]
[78] B Pasquini P Pedroni and D Drechsel Higher order forward spin polarizability Phys Lett
B 687 160 (2010) arXiv10014230 [hep-ph]
[79] R Jaffe g2ndashThe nucleonrsquos other spin-dependent structure function Comments Nucl Part
Phys 19 239 (1990)
[80] E V Shuryak and A Vainshtein Theory of power corrections to deep inelastic scattering in
quantum chromodynamics (II) Qminus4 effects polarized target Nucl Phys B 201 141 (1982)
[81] S Wandzura and F Wilczek Sum rules for spin dependent electroproduction Test of rela-
tivistic constituent quarks Phys Lett B 72 195 (1977)
[82] B W Filippone and X-D Ji The spin structure of the nucleon Adv Nucl Phys 26 1
(2001) arXivhep-ph0101224
[83] M Burkardt The g2 structure function Proceedings Workshop on Spin structure at long
46
distance Newport News USA March 12-13 2009 AIP Conf Proc 1155 26 (2009)
arXiv09054079 [hep-ph]
[84] A Deur S J Brodsky and G F de Teramond The spin structure of the nucleon Rept
Prog Phys 82 076201 (2019) arXiv180705250 [hep-ph]
[85] B R Holstein D Drechsel B Pasquini and M Vanderhaeghen Higher order polarizabilities
of the proton Phys Rev C 61 034316 (2000) arXivhep-ph9910427
[86] B R Holstein V Pascalutsa and M Vanderhaeghen Sum rules for magnetic moments and
polarizabilities in QED and chiral effective-field theory Phys Rev D 72 094014 (2005)
arXivhep-ph0507016
[87] B Pasquini D Drechsel and L Tiator Invariant amplitudes for pion electroproduction Eur
Phys J A 34 387 (2007) arXiv07122327 [hep-ph]
[88] A J Buchmann Non-spherical proton shape and hydrogen hyperfine splitting Proceedings
International Workshop on Precision Physics of Simple Atomic Systems (PSAS 2008) Wind-
sor Ontario Canada July 21-26 2008 Can J Phys 87 773 (2009) arXiv09104747
[physicsatom-ph]
47
II CALCULATION OF UNPOLARIZED VVCS AT NLO
A VVCS amplitudes and relations to structure functions
The polarized part of forward VVCS can be described in terms of two independent
Lorentz-covariant and gauge-invariant tensor structures and two scalar amplitudes [3]
T microν(p q) = minus 1
MN
γmicroναqα S1(νQ2)minus 1
M2N
(γmicroνq2 + qmicroγναqα minus qνγmicroαqα
)S2(νQ2) (1)
Here q and p are the photon and nucleon four-momenta (cf Fig 1) ν is the photon
lab-frame energy Q2 = minusq2 is the photon virtuality and γmicroν = 12
[γmicro γν ] and γmicroνα =12
(γmicroγνγα minus γαγνγmicro) are the usual Dirac matrices Alternatively one can use the following
laboratory-frame amplitudes
gTT (νQ2) =ν
MN
[S1(νQ2)minus Q2
MN νS2(νQ2)
] (2a)
gLT (νQ2) =Q
MN
[S1(νQ2) +
ν
MN
S2(νQ2)] (2b)
introduced in Eq (A2) The optical theorem relates the absorptive parts of the forward
VVCS amplitudes to the nucleon structure functions or the cross sections of virtual pho-
toabsorption
ImS1(νQ2) =4π2α
νg1(xQ2) =
MNνK(νQ2)
ν2 +Q2
[Q
νσLT (νQ2) + σTT (νQ2)
] (3a)
ImS2(νQ2) =4π2αMN
ν2g2(xQ2) =
M2NK(νQ2)
ν2 +Q2
[ν
QσLT (νQ2)minus σTT (νQ2)
] (3b)
with α the fine structure constant andK(νQ2) the photon flux factor Note that the photon
flux factor in the optical theorem and the cross sections measured in electroproduction
processes is a matter of convention and has to be chosen in both quantities consistently In
what follows we use Gilmanrsquos flux factor
K(νQ2) equiv |~q | =radicν2 +Q2 (4)
The helicity-difference photoabsorption cross section is defined as σTT = 12 (σ12 minus σ32)
where the photons are transversely polarized and the subscripts on the right-hand side
indicate the total helicities of the γlowastN states The cross section σLT corresponds to a
simultaneous helicity change of the photon and nucleon spin flip such that the total helicity
5
is conserved A detailed derivation of the connection between this response function and
the photoabsorption cross sections can be found in App B The forward VVCS amplitudes
satisfy dispersion relations derived from the general principles of analyticity and causality1
S1(νQ2) =16παMN
Q2
int 1
0
dxg1(xQ2)
1minus x2(ννel)2 minus i0+(5a)
=2MN
π
int infinνel
dν primeν prime 2[QνprimeσLT + σTT
](ν prime Q2)radic
ν prime 2 +Q2(ν prime 2 minus ν2 minus i0+)
νS2(νQ2) =16παM2
N
Q2
int 1
0
dxg2(xQ2)
1minus x2(ννel)2 minus i0+(5b)
=2M2
N
π
int infinνel
dν primeν prime 2[νprime
QσLT minus σTT
](ν prime Q2)radic
ν prime 2 +Q2(ν prime 2 minus ν2 minus i0+)
with νel = Q22MN the elastic threshold
B Low-energy expansions and relations to polarizabilities
The VVCS amplitudes naturally split into nucleon-pole (Spolei ) and non-pole (Snonpole
i )
parts or Born (SBorni ) and non-Born (Si) parts
Si = Spolei + Snonpole
i = SBorni + Si (6)
The Born amplitudes are given uniquely in terms of the nucleon form factors [1]
SBorn1 (νQ2) =
2πα
M
Q2GM(Q2)F1(Q2)
ν2el minus ν2
minus F 22 (Q2)
(7a)
SBorn2 (νQ2) = minus 2παν
ν2el minus ν2
GM(Q2)F2(Q2) (7b)
The same is true for the nucleon-pole amplitudes which are related to the Born amplitudes
in the following way
Spole1 (νQ2) = SBorn
1 (νQ2) +2πα
MN
F 22 (Q2) (8a)
Spole2 (νQ2) = SBorn
2 (νQ2) (8b)
1 The dispersion relation for νS2(νQ2) is used because it is pole-free in the limit Q2 rarr 0 and then ν rarr 0
cf Eq (7b)
6
Here we used the elastic Dirac and Pauli form factors F1(Q2) and F2(Q2) related to the
electric and magnetic Sachs form factors GE(Q2) and GM(Q2) through
F1(Q2) =GE(Q2) + τGM(Q2)
1 + τ (9a)
F2(Q2) =GM(Q2)minusGE(Q2)
1 + τ (9b)
where τ = Q24M2N
A low-energy expansion (LEX) of Eq (5) in combination with the unitarity relations
given in Eq (3) establishes various sum rules relating the nucleon properties (electromag-
netic moments polarizabilities) to experimentally observable response functions [1 3] The
leading terms in the LEX of the RCS amplitudes are determined uniquely by charge mass
and anomalous magnetic moment as the global properties of the nucleon These lowest-
order terms represent the celebrated low-energy theorem (LET) of Low Gell-Mann and
Goldberger [39 40] The polarizabilities related to the internal structure of the nucleon
enter the LEX at higher orders They make up the non-Born amplitudes and can be related
to moments of inelastic structure functions
The process of VVCS can be realized experimentally in electron-nucleon scattering where
a virtual photon is exchanged between the electron and the nucleon This virtual photon
acts as a probe whose resolution depends on its virtuality Q2 In this way one can access
the so-called generalized polarizabilities which extend the notion of polarizabilities to the
case of response to finite momentum transfer The generalized forward spin polarizability
γ0(Q2) and the longitudinal-transverse polarizability δLT (Q2) are most naturally defined via
the LEX of the non-Born part of the lab-frame VVCS amplitudes [1]
1
4πgnonpoleTT (νQ2) =
2α
M2N
IA(Q2) ν + γ0(Q2)ν3 + γ0(Q2)ν5 + (10a)
1
4πgnonpoleLT (νQ2) =
2α
M2N
I3(Q2)Q+ δLT (Q2)ν2Q+ (10b)
Their definitions in terms of integrals over structure functions are postponed to Eqs (19)
and (22) Here we only give the definition of the moment I3(Q2)
I3(Q2) =M2
N
4π2α
int infinν0
dνK(νQ2)
νQσLT (νQ2) =
2M2N
Q2
int x0
0
dx[g1(xQ2) + g2(xQ2)
] (11)
which is related to the Schwinger sum rule in the real photon limit [7ndash9 41 42] The LEX
of the non-pole part of the covariant VVCS amplitudes can be described entirely in terms
7
of moments of inelastic spin structure functions (up to O(ν4) [43])
1
4πSnonpole
1 (νQ2) =2α
MN
I1(Q2) +
2α
MNQ2
[IA(Q2)minus I1(Q2)
]+MNδLT (Q2)
ν2 (12a)
1
4πνSnonpole
2 (νQ2) = 2αI2(Q2) +2α
Q2
[I1(Q2)minus IA(Q2)
]ν2 (12b)
I1(Q2) and IA(Q2) are generalizations of the famous GerasimovndashDrellndashHearn (GDH) sum
rule [5 6] from RCS to the case of virtual photons [1] Their definitions are given in Eqs (26)
and (32) I2(Q2) is the well-known Burkhardt-Cottingham (BC) sum rule [44]
I2(Q2) equiv 2M2N
Q2
int x0
0
dx g2(x Q2) =1
4F2(Q2)GM(Q2) (13)
which can be written as a ldquosuperconvergence sum rulerdquo
Q2
16παM2N
limνrarr0
νS2(νQ2) =
int 1
0
dx g2 (x Q2) = 0 (14)
The latter is valid for any value of Q2 provided that the integral converges for x rarr 0
Combining Eq (5) with the above LEXs of the VVCS amplitudes we can relate IA(Q2)
I1(Q2) γ0(Q2) and δLT (Q2) to moments of inelastic structure functions see Sec III It is
important to note that only γ0(Q2) and δLT (Q2) are generalized polarizabilities The relation
of the inelastic moments IA(Q2) and I1(Q2) to polarizabilities will be discussed in details in
Secs III C and III D The difference between S1(νQ2) and Snonpole1 (νQ2) cf Eq (8a) will
be important in this context
C Details on χPT calculation and uncertainty estimate
In this work we calculated the NLO prediction of BχPT for the polarized non-Born
VVCS amplitudes This includes the leading pion-nucleon (πN) loops see Ref [32 Fig 1]
as well as the subleading tree-level Delta-exchange (∆-exchange) see Ref [30 Fig 2] and
the pion-Delta (π∆) loops see Ref [32 Fig 2] In the δ-power-counting scheme [36] the
LO and NLO non-Born VVCS amplitudes and polarizabilities are of O(p3) and O(p4∆)
respectively2 The LECs are listed in Table I sorted by the order at which they appear in
our calculation At the given orders there are no ldquonewrdquo LECs that would need to be fitted
from Compton processes For more details on the BχPT formalism we refer to Ref [30]
2 In the full Compton amplitude there is a lower order contribution coming from the Born terms leading
to a shift in nomenclature by one order the LO contribution referred to as the NLO contribution etc
see eg Ref [45]
8
where power counting predictive orders (Sec III A) and the renormalization procedure
(Sec III B) are discussed
A few remarks are in order for the inclusion of the ∆(1232) and the tree-level ∆-exchange
contribution In contrast to Ref [32] we include the Coulomb-quadrupole (C2) N rarr ∆
transition described by the LEC gC The relevant Lagrangian describing the non-minimal
γlowastN∆ coupling [33 34] (note that in these references the overall sign of gC is inconsistent
between the Lagrangian and Feynman rules) reads
L(2)∆ =
3e
2MNM+
N T3
igM F
microν partmicro∆ν minus gEγ5Fmicroν partmicro∆ν (15)
+igCM∆
γ5γα(partα∆ν minus partν∆α)partmicroF
microν
+ Hc
with M+ = MN + M∆ and the dual of the electromagnetic field strength tensor F microν =12εmicroνρλFρλ Even though the Coulomb coupling is subleading compared with the electric
and magnetic couplings (gE and gM) its relatively large magnitude cf Table I makes it
numerically important for instance in γ0(Q2) Furthermore we study the effect of modifying
the magnetic coupling using a dipole form factor
gM rarrgM[
1 + (QΛ)2 ]2 (16)
where Λ2 = 071 GeV2 The inclusion of this Q2 dependence mimics the form expected from
vector-meson dominance It is motivated by observing the importance of this form factor
for the correct description of the electroproduction data [33]
To estimate the uncertainties of our NLO predictions we define
δ(Q2) =
radic(∆
MN
)2
+
(Q2
2MN∆
)2
(17)
such that the neglected next-to-next-to-leading order terms are expected to be of relative
size δ2 [33] The uncertainties in the values of the parameters in Table I have a much
smaller impact compared to the truncation uncertainty and can be neglected Unfortunately
∆IA(Q2) γ0(Q2) and γ0(Q2) ie the sum rules involving the cross section σTT (νQ2)
as well as the polarizability ∆I1(Q2) turn out to be numerically small Their smallness
suggests a cancellation of leading orders (which can indeed be confirmed by looking at
separate contributions as shown below) Therefore an error of δ2(Q2)P (Q2) where P (Q2)
is a generalized polarizability might underestimate the theoretical uncertainty for some of
the NLO predictions To avoid this we estimate the uncertainty of our NLO polarizability
9
predictions by
σP (Q2) =
Max[δ4(0)P (0)2 δ4(0)P LO(0)2 δ2(0)PNLO(0)2
]+Max
[δ4(Q2)
[P (Q2)minus P (0)
]2 δ4(Q2)
[P LO(Q2)minus P LO(0)
]2
δ2(Q2)[PNLO(Q2)minus PNLO(0)
]2]12
(18)
where P LO(Q2) is the πN -loop contribution PNLO(Q2) are the ∆-exchange and π∆-loop
contributions and P (Q2) = P LO(Q2) + PNLO(Q2) This error prescription is similar to the
one used in eg Refs [46ndash49] Here since we are interested in the generalized polarizabil-
ities we added in quadrature the error due to the static piece P (0) and the Q2-dependent
remainder P (Q2) minus P (0) Note that the static values of IA(0) and I1(0) are given by the
elastic Pauli form factor which is not part of our BχPT prediction and is considered to be
exact
Note that our result for the spin polarizabilities (and the unpolarized moments [30]) are
NLO predictions only at low momentum transfers Q mπ At larger values of Q amp ∆ they
become incomplete LO predictions Indeed in this regime the ∆ propagators do not carry
additional suppression compared to the nucleon propagators and the π∆ loops are promoted
to LO In general we only expect a rather small contribution from omitted π∆ loops to the
Q2 dependence of the polarizabilities since π∆ loops show rather weak dependence on Q2
compared with the ∆ exchange or πN loops Nevertheless this issue has to be reflected in
the error estimate Since the static polarizabilities P (0) are not affected it is natural to
separate the error on the Q2-dependent remainder P (Q2) minus P (0) as done in Eq (18) To
accommodate for the potential loss of precision above Q amp ∆ we define the relative error
δ(Q2) as growing with increasing Q2 see Eq (17)
Upon expanding our results in powers of the inverse nucleon mass Mminus1N we are able to
reproduce existing results of heavy-baryon χPT (HBχPT) at LO We however do not see
a rationale to drop the higher-order Mminus1N terms when they are not negligible (ie when
their actual size exceeds by far the natural estimate for the size of higher-order terms)
Comparing our BχPT predictions to HBχPT we will also see a deficiency of HBχPT in
the description of the Q2 behaviour of the polarizabilities Note that the O(p4) HBχPT
results from Ref [50 51] which we use here for comparison do not include the ∆ These
references studied the leading effect of the latter in the HBχPT framework using the small-
scale expansion [35] observing no qualitative improvement in the HBχPT description of the
empirical data [50 51] when including it We therefore choose to use the O(p4) results as
the representative HBχPT curves
10
Another approach used in the literature to calculate the polarizabilities in χPT is the
infrared regularization (IR) scheme introduced in Ref [52] This covariant approach tries to
solve the power counting violation observed in Ref [53] by dropping the regular parts of the
loop integrals that contain the power-counting-breaking terms However this subtraction
scheme modifies the analytic structure of the loop contributions and may lead to unexpected
problems as was shown in Ref [54] As we will see in the next section the IR approach
also fails to describe the Q2 behaviour of the polarizabilities
III RESULTS AND DISCUSSION
We now present the NLO BχPT predictions for the nucleon polarizabilities and selected
moments of the nucleon spin structure functions Our results are obtained from the calcu-
lated non-Born VVCS amplitudes and the LEXs in Eqs (10) and (12) For a cross-check
we used the photoabsorption cross sections described in App B In addition to the full NLO
results we also analyse the individual contributions from the πN loops the ∆ exchange
and the π∆ loops
A γ0(Q2) mdash generalized forward spin polarizability
The forward spin polarizability
γ0(Q2) =1
2π2
int infinν0
dν
radic1 +
Q2
ν2
σTT (νQ2)
ν3(19)
=16αM2
N
Q6
int x0
0
dx x2
[g1(xQ2)minus 4M2
Nx2
Q2g2(xQ2)
]
provides information about the spin-dependent response of the nucleon to transversal pho-
ton probes The RCS analogue of the above generalized forward spin polarizability sum
rule is sometimes referred to as the Gell-Mann Goldberger and Thirring (GGT) sum rule
[11] At Q2 = 0 the forward spin polarizability is expressed through the lowest-order spin
polarizabilities of RCS as γ0 = minus(γE1E1 + γM1M1 + γE1M2 + γM1E2) The forward spin po-
larizability of the proton is relevant for an accurate knowledge of the (muonic-)hydrogen
hyperfine splitting as it controls the leading proton-polarizability correction [16 62]
The πN -loop ∆-exchange and π∆-loop contributions to the NLO BχPT prediction of
11
000 005 010 015 020 025 030
-4
-3
-2
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
4D
proton
000 005 010 015 020 025 030
-4
-3
-2
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
4D
neutron
000 005 010 015 020 025 030
0
1
2
Q2 GeV2D
∆LT
10-
4fm
4D
proton
000 005 010 015 020 025 030
0
1
2
3
Q2 GeV2D
∆LT
10-
4fm
4D
neutron
FIG 2 Upper panel Generalized forward spin polarizability for the proton (left) and neutron
(right) as function of Q2 The result of this work the NLO BχPT prediction is shown by the
blue solid line and the blue band The red line represents the LO BχPT result The purple short-
dashed line is the O(p4) HB result from Ref [50] note that the corresponding proton curve is
outside of the plotted range The black dotted line is the MAID model prediction [55ndash57] which
is taken from Ref [1] (proton) and Ref [20] (neutron) The pink band is the IR+∆ result from
Ref [58] and the gray band is the BχPT+∆ result from Ref [31] Empirical extractions for the
proton Ref [18] (blue dots) Ref [59] (purple square) and Ref [60] (orange triangle uncertainties
added in quadrature) and neutron Ref [20] (blue diamonds) and Ref [61] (green dots statistical
and systematic uncertainties added in quadrature) Lower Panel Longitudinal-transverse spin
polarizability for the proton (left) and neutron (right) The orange dot-dashed and purple short-
dashed lines are the O(p3) and O(p4) HB results from Ref [50] The pink band is the IR result
from Ref [58] and the gray band is the covariant BχPT+∆ result from Ref [31] The black dotted
line is the MAID model prediction [55ndash57] note that for the proton we use the updated estimate
from Ref [1] obtained using the π η ππ channels
12
the static forward spin polarizability amount to in units of 10minus4 fm4
γ0p = minus093(92) asymp 201minus 284minus 010 (20a)
γ0n = 003(92) asymp 298minus 284minus 010 (20b)
while the slope is composed as follows in units of 10minus4 fm6
dγ0p(Q2)
dQ2
∣∣∣∣Q2=0
= minus022(4) asymp minus033 + 011 + 001 (21a)
dγ0n(Q2)
dQ2
∣∣∣∣Q2=0
= minus061(7) asymp minus073 + 011 + 001 (21b)
Figure 2 upper panel shows our NLO prediction as well as the LO πN loops compared
to different experimental and theoretical results For the proton we have one determination
at the real-photon point by the GDH collaboration [19] γ0p = minus100(8)(12)times10minus4 fm4 and
further Jefferson Laboratory data [18 60] at very low Q2 For the neutron only data at finite
Q2 are available [20 61] The experimental data for the proton are fairly well reproduced
in the whole Q2 range considered here while for the neutron the agreement improves with
increasing Q2 The HB limit of our πN -loop contribution reproduces the results published
in Refs [50 63] for arbitrary Q2 In addition our prediction is compared to the MAID
model [1 20] the IR+∆ calculation of Ref [58] and the BχPT+∆ result of Ref [31]
The πN -production channel gives a positive contribution to the photoabsorption cross
section σTT (νQ2) at low Q2 cf Fig 10 Accordingly one observes that the πN loops give
a sizeable positive contribution to γ0(Q2) The Delta on the other hand has a very large
effect by cancelling the πN loops and bringing the result close to the empirical data From
Fig 3 upper panel one can see that it is the ∆ exchange which dominates while π∆
loops are negligible This was expected since the forward spin polarizability sum rule is an
integral over the helicity-difference cross section in which σ32 is governed by the Delta at
low energies (the relevant energy region for the sum rule)
To elucidate the difference between the present calculation and the one from Ref [31] we
note that the two calculations differ in the following important aspects Firstly Ref [31] uses
the small-scale counting [64] that considers ∆ and mπ as being of the same size ∆ sim mπ
In practice this results in a set of π∆-loop graphs which contains graphs with one or two
γ∆∆ couplings and hence two or three Delta propagators Such graphs are suppressed in
the δ-counting and thus omitted from our calculation while present in that of Ref [31]
Secondly the Lagrangians describing the interaction of the Delta are constructed differently
and assume slightly different values for the coupling constants In particular we employ
13
000 005 010 015 020 025 030
-3
-2
-1
0
1
2
Q2GeV
2D
Γ01
0-
4fm
4D
proton
000 005 010 015 020 025 030
-3
-2
-1
0
1
2
3
Q2GeV
2D
Γ01
0-
4fm
4D
neutron
000 005 010 015 020 025 030
00
05
10
15
Q2GeV
2D
∆LT1
0-
4fm
4D
proton
000 005 010 015 020 025 030
00
05
10
15
20
25
Q2GeV
2D
∆LT1
0-
4fm
4D
neutron
FIG 3 Contributions of the different orders to the chiral predictions of γ0(Q2) upper paneland δLT (Q2) lower panel for the proton (left) and neutron (right) Red solid line πN -loop
contribution green dot-dashed line ∆-exchange contribution orange dotted line π∆-loop con-
tribution blue long-dashed line total result purple dot-dot-dashed line total result without gCcontribution black short-dashed line total result without gM dipole
(where possible) the so-called ldquoconsistentrdquo couplings to the Delta field ie those couplings
that project out the spurious degree of freedom see Refs [37 65 66] The authors of
Ref [31] on the other hand use couplings where the consistency in this sense is not enforced
The effects of these differences are of higher order in the δ-counting expansion and their
contribution to the Q2 dependence of the considered polarizabilities is expected to be rather
small however the differences at Q2 = 0 could be noticeable [67]
Finally as mentioned in Sec II C the inclusion of the dipole form factor in the magnetic
coupling gM is expected to be important to generate the correct Q2 behaviour of the polar-
izabilities Comparing our predictions for the forward spin polarizability with and without
inclusion of the form factor see Fig 3 upper panel confirms this Without the dipole
our results for the proton and neutron are closer to the ones from Ref [31] where the form
factor is not included For the neutron our prediction without the dipole is able to describe
14
the experimental points at very low Q2 deviating from the data with increasing Q2 The
π∆-loop contribution does not modify the Q2 behavior of γ0(Q2) and only differs from
Ref [31] by a small global shift Note also the relatively large effect of gC which generates
a sign change for virtualities above sim 02 GeV2 see Fig 3 upper panel
B δLT (Q2) mdash longitudinal-transverse polarizability
The longitudinal-transverse spin polarizability
δLT (Q2) =1
2π2
int infinν0
dν
radic1 +
Q2
ν2
σLT (νQ2)
Qν2(22)
=16αM2
N
Q6
int x0
0
dx x2[g1(xQ2) + g2(xQ2)
]
contains information about the spin structure of the nucleon and is another important
input in the determination of the (muonic-)hydrogen hyperfine splitting [16 62] It is also
relevant in studies of higher-twist corrections to the structure function g2(xQ2) given by
the moment d2(Q2) [51] see Section III E The peculiarity of the response encoded in this
polarizability is that it involves a spin flip of the nucleon and a polarization change of the
photon see App B and Fig 11
It is expected that the Delta isobar gives only a small contribution to δLT (Q2) what makes
this polarizability a potentially clean test case for chiral calculations Consequently there are
relatively many different theoretical calculations of δLT (Q2) coming from different versions
of χPT with baryons (HB IR and covariant) Ref [50] found a systematic deviation of the
HB result for δLTn(Q2) from the MAID model prediction This disagreement was identified
by the authors of Ref [68] as a puzzle involving the neutron δLT polarizabilitymdashthe δLT
puzzle The IR calculation in Ref [58] also showed a deviation from the data and predicted
a rapid rise of δLT (Q2) with growing Q2 The problem is solved by keeping the relativistic
structure of the theory as the BχPT+∆ result of Ref [31] showed
As expected already the leading πN loops provide a reasonable agreement with the
experimental data cf Fig 2 lower panel Since the ∆-exchange contribution to δLT (Q2)
is small the effect of the gM form factor is negligible in this polarizability as is that of the gC
coupling cf Fig 3 lower panel In fact we predict both the ∆-exchange and the π∆-loop
contributions to be small and negative This is in agreement with the MAID model which
predicts a small and negative contribution of the P33 wave to δLT (Q2) However in the
calculation of Ref [31] which is different from the one presented here only in the way the
∆(1232) is included the contribution of this resonance to δLTp(Q2) is sizeable and positive
15
The authors of that work attributed this large contribution to diagrams where the photons
couple directly to the Delta inside a loop As mentioned in Sec III F the effect of such loop
diagrams does not change the Q2 behaviour of the polarizabilities On the other hand it
can produce a substantial shift of the δLT (Q2) as a whole A higher-order calculation should
resolve the discrepancy between the two covariant approaches however it will partially lose
the predictive power since the LECs appearing at higher orders will have to be fitted to
experimental data
The πN -loop ∆-exchange and π∆-loop contributions to the NLO BχPT prediction of
the static longitudinal-transverse polarizability are in units of 10minus4 fm4
δLTp = 132(15) asymp 150minus 016minus 002 (23a)
δLTn = 218(23) asymp 235minus 016minus 002 (23b)
while the slopes are in units of 10minus4 fm6
dδLTp(Q2)
dQ2
∣∣∣∣Q2=0
= minus085(8) asymp minus080minus 004minus 001 (24a)
dδLTn(Q2)
dQ2
∣∣∣∣Q2=0
= minus124(12) asymp minus119minus 004minus 001 (24b)
C IA(Q2) mdash a generalized GDH integral
The helicity-difference cross section σTT exhibits a faster fall-off in ν than its spin-averaged
counterpart σT This is due to a cancellation between the leading (constant) terms of σ12
and σ32 at large ν3 The resulting 1ν fall-off of the helicity-difference cross section allows
one to write an unsubtracted dispersion relation for the VVCS amplitude gnonpoleTT (νQ2) cf
Eq (10a) This is the origin of the GDH sum rule [5 6]
minus α
2M2N
κ2 =1
2π2
int infinν0
dνσTT (ν)
ν (25)
which establishes a relation to the anomalous magnetic moment κ It is experimentally
verified for the nucleon by MAMI (Mainz) and ELSA (Bonn) [71 72]
There are two extensions of the GDH sum rule to finite Q2 the generalized GDH integrals
3 Notice that a constant term in σTT at ν rarrinfin is forbidden by crossing symmetry
16
000 005 010 015 020 025 030
-30
-25
-20
-15
-10
-05
00
Q2 GeV2D
I Aproton
000 005 010 015 020 025 030
-25
-20
-15
-10
-05
00
Q2 GeV2D
I A
neutron
000 005 010 015 020 025 030
-05
00
05
Q2 GeV2D
I 1
proton
000 005 010 015 020 025 030
-10
-08
-06
-04
-02
00
02
Q2 GeV2D
I 1
neutron
FIG 4 Upper panel The generalized GDH integral IA(Q2) for the proton (left) and neutron
(right) as function of Q2 The result of this work the NLO BχPT prediction is shown by the blue
solid line and the blue band The red line represents the LO BχPT result The purple short-dashed
line is the O(p4) HB result from Ref [50 51] The gray band is the BχPT+∆ result from Ref [31]
The black dotted line is the MAID model prediction [69] Experimental extractions for the proton
Ref [60] (orange triangle uncertainties added in quadrature) and neutron Refs [21][27] where
magenta dotsorange diagonal crosses correspond to data and red squareslilac crosses correspond
to data plus extrapolation to unmeasured energy regions The green stars at the real-photon point
are derived from the anomalous magnetic moments κp asymp 1793 and κn asymp minus1913 [70] Lower
panel The generalized GDH integral I1(Q2) for the proton (left) and neutron (right) as function
of Q2 The purple short-dashed line is the HB result from Ref [51] Experimental extractions for
the proton Ref [18] (blue dots) and Ref [60] (orange triangle uncertainties added in quadrature)
and neutron Ref [61][27] (uncertainties added in quadrature) where green dotsorange diagonal
crosses correspond to data and gray squareslilac crosses correspond to data plus extrapolation to
unmeasured energy regions
17
IA(Q2) and I1(Q2) The latter will be discussed in Sec III D The former is defined as4
minus α
2M2N
IA(Q2) = minus 1
8π2
int infinν0
dν
radic1 +
Q2
ν2
σTT (νQ2)
ν(26)
=α
Q2
int x0
0
dx[4M2
Nx2
Q2g2(xQ2)minus g1(xQ2)
]
Due to its energy weighting the integral in Eq (26) converges slower than the one in the
generalized forward spin polarizability sum rule (19) Therefore knowledge of the cross
section at higher energies is required and the evaluation of the generalized GDH integral
IA(Q2) is not as simple as the evaluation of γ0(Q2)
The generalized GDH integral IA(Q2) is directly related to the non-pole amplitude
gnonpoleTT (νQ2) which differs from non-Born amplitude gTT (νQ2) by a term involving the
elastic Pauli form factor
gnonpoleTT (νQ2) = gTT (νQ2)minus 2παν
M2N
F 22 (Q2) (27)
cf Eqs (2a) and (8a) Consequently IA(Q2) is not a pure polarizability but also contains
an elastic contribution The ldquonon-polarizabilityrdquo or the Born part of IA(Q2) is given by
IBornA (Q2) = IA(Q2)minus∆IA(Q2) = minus1
4F 2
2 (Q2) (28)
where we refer to the polarizability part as ∆IA(Q2) The same is true for the generalized
GDH integral I1(Q2) which is directly related to Snonpole1 (νQ2)
IBorn1 (Q2) = I1(Q2)minus∆I1(Q2) = minus1
4F 2
2 (Q2) (29)
In the following we will add the Born parts to our LO and NLO BχPT predictions for
the polarizabilities ∆IA(Q2) and ∆I1(Q2) employing an empirical parametrization for the
elastic Pauli form factor [73] This allows us to compare to the experimental results for
IA(Q2) and I1(Q2) cf Fig 4 Note that the blue error bands only describe the uncertainties
of our BχPT predictions of the polarizabilities while the elastic contributions are considered
to be exact as explained in Sec II C The uncertainties of the polarizability predictions are
therefore better reflected in Fig 5 where we show the contributions of the different orders
to the BχPT predictions of ∆IA(Q2) and ∆I1(Q2) as well as the total results with error
bands
4 Note that IA(Q2) is sometimes called ITT (Q2)
18
000 005 010 015 020 025 030
-08
-06
-04
-02
00
02
Q2 GeV2D
DI A
proton
000 005 010 015 020 025 030
-10
-08
-06
-04
-02
00
02
Q2 GeV2D
DI A
neutron
000 005 010 015 020 025 030-005
000
005
010
015
020
025
Q2 GeV2D
DI 1
proton
000 005 010 015 020 025 030-015
-010
-005
000
005
010
015
Q2 GeV2D
DI 1
neutron
FIG 5 Contributions of the different orders to the chiral predictions of ∆IA(Q2) upper paneland ∆I1(Q2) lower panel for the proton (left) and neutron (right) Red solid line πN -loop
contribution green dot-dashed line ∆-exchange contribution orange dotted line π∆-loop contri-
bution blue solid line and blue band total result purple dot-dot-dashed line total result without
gC contribution black short-dashed line total result without gM dipole
The E97-110 experiment at Jefferson Lab has recently published their data for IAn(Q2) in
the region of 0035 GeV2 lt Q2 lt 024 GeV2 [27] In addition there are results for IAn(Q2)
from the earlier E94-010 experiment [21] and for IAp(Q2) from the E08-027 experiment [60]
The O(p4) HB calculation gives a large negative effect [51] which does not describe the data
The BχPT+∆ result from Ref [31] which mainly differs from our work by the absence of
the dipole form factor in gM looks similar to this HB result and only describes the data
points at lowest Q2 Our NLO prediction however follows closely the Q2 evolution of the
data In Fig 5 upper panel we show the polarizability ∆IA(Q2) whose Q2 evolution is
clearly dominated by the ∆ exchange Similar to the case of γ0p(Q2) inclusion of the dipole
in gM and the Coulomb coupling gC is very important in order to describe the experimental
data The LO prediction on the other hand slightly overestimates the data cf Fig 4
upper panel
19
At the real-photon point IA(0) = minusκ2
4and ∆IA(0) = 0 Therefore we give only the
slope of the polarizability ∆IA(Q2) [showing also the separate contributions from πN loops
∆ exchange and π∆ loops] in units of GeVminus2
d∆IAp(Q2)
dQ2
∣∣∣∣Q2=0
= minus858(343) asymp 238minus 1121 + 025 (30a)
d∆IAn(Q2)
dQ2
∣∣∣∣Q2=0
= minus955(343) asymp 141minus 1121 + 025 (30b)
Including the empirical Pauli form factor [73] we find in units of GeVminus2
dIAp(Q2)
dQ2
∣∣∣∣Q2=0
= minus318dIAn(Q2)
dQ2
∣∣∣∣Q2=0
= minus300 (31)
D Γ1(Q2) and I1(Q2) mdash the first moment of the structure function g1(xQ2)
The second variant for a generalization of the GDH sum rule to finite Q2 is defined as
minus α
2M2N
I1(Q2) = minus 1
8π2
int infinν0
dν1radic
ν2 +Q2
[σTT (νQ2) +
Q
νσLT (νQ2)
](32)
= minus α
Q2
int x0
0
dx g1(xQ2)
where I1(0) = minusκ2
4 This generalized GDH integral directly stems from the amplitude
Snonpole1 (νQ2) with the LEX from Eq (12a) It is given by the first moment of the struc-
ture function g1(xQ2) Γ1(Q2) =int x0
0dx g1(xQ2) as follows I1(Q2) =
2M2N
Q2 Γ1(Q2) The
isovector combination
Γ1(pminusn)(Q2) =
int x0
0
dx[g1p(xQ
2)minus g1n(xQ2)] (33)
is related to the axial coupling of the nucleon through the Bjorken sum rule [74 75]
limQ2rarrinfin
Γ1(pminusn)(Q2) =
gA6 (34)
As explained in Eq (28) the moment I1(Q2) splits into a polarizability part ∆I1(Q2) and
a Born part IBorn1 (Q2) Figure 4 lower panel shows the Q2 dependence of I1(Q2) which in
contrast to IA(Q2) shown in Figure 4 upper panel is clearly dominated by its Born part
and the elastic Pauli form factor The πN -loop ∆-exchange and π∆-loop contributions to
the polarizability ∆I1(Q2) are shown in Fig 5 lower panel Comparing to Fig 5 upper
20
000 005 010 015 020 025 030
-006
-004
-002
000
002
004
Q2 GeV2D
G1
proton
000 005 010 015 020 025 030
-006
-005
-004
-003
-002
-001
000
001
Q2 GeV2D
G1
neutron
FIG 6 First moment of the structure function g1(xQ2) for the proton (left) and neutron (right)
as function of Q2 The legend is the same as in Fig 4
panel one sees that ∆I1(Q2) is less sensitive to gC and the dipole form factor in gM than
∆IA(Q2)
For the proton our NLO BχPT prediction gives a very good description of the ex-
perimental data [18 60] and is in reasonable agreement with the MAID prediction [69]
For the neutron one observes good agreement with the empirical evaluations including ex-
trapolations to unmeasured energy regions starting from Q2 gt 01 GeV2 [27 61] In the
region of Q2 lt 005 GeV2 one observes an interesting tension between the recent E97-
110 experiment [27] and the data from CLAS [61] While the newest measurement finds
I1n(0035 GeV2) lt κ2n4 thus suggesting a negative slope at low Q the older measurement
found a rather large value for I1n(00496 GeV2) A similar but milder behaviour is seen in
the E97-110 [27] and E94-010 [21] data for IAn The MAID predictions do not agree with
the CODATA recommended values for the anomalous magnetic moments of the proton and
neutron [70] which in our work are imposed by using empirical parametrizations for the
elastic Pauli form factors [73] The slope of the HB result from Ref [51] is too large and
therefore only reproduces the data at very low Q2
Figure 6 shows the moment Γ1(Q2) for the proton and neutron while Fig 7 shows the
isovector combination Γ1 pminusn(Q2) The LO and NLO BχPT predictions are identical be-
cause our calculation produces the same Delta contributions for the proton and the neutron
For the isovector combination the MAID model only agrees with the data at very low
Q2 lt 010 GeV2 The same is true for the IR result [58 76] while all other chiral results
describe the data NLO BχPT (this work) BχPT+∆ [31] and HBχPT [51]
At the real-photon point I1(0) = minusκ2
4and ∆I1(0) = 0 Therefore we give only the
slope of the polarizability ∆I1(Q2) [showing also the separate contributions from πN loops
21
000 005 010 015 020 025 030-002
000
002
004
006
008
010
Q2 GeV2D
G1Hp-
nL
FIG 7 Isovector combination of Γ1(Q2) as function of Q2 The legend is the same as in Fig 6
The pink curve is the IR result from Ref [58 76] The experimental points are from Ref [77]
(brown dots) and Ref [23] (orange squares)
∆ exchange and π∆ loops] in units of GeVminus2
d∆I1p(Q2)
dQ2
∣∣∣∣Q2=0
= 039(4) asymp 034minus 053 + 058 (35a)
d∆I1n(Q2)
dQ2
∣∣∣∣Q2=0
= minus101(10) asymp minus107minus 053 + 058 (35b)
Including the empirical Pauli form factor [73] we find in units of GeVminus2
dI1p(Q2)
dQ2
∣∣∣∣Q2=0
= 580dI1n(Q2)
dQ2
∣∣∣∣Q2=0
= 553 (36)
E d2(Q2) mdash a measure of color polarizability
Another interesting moment to consider is d2(Q2) which is related to the twist-3 part of
the spin structure function g2(xQ2) [79 80]
d2(Q2) equiv 3
int 1
0
dx x2[g2(xQ2)minus gWW2 (xQ2)] (37)
where gWW2 (xQ2) is the twist-2 part of g2(xQ2) Using the Wandzura-Wilczek relation [81]
one can relate d2(Q2) to moments of the spin structure functions g1(xQ2) and g2(xQ2)
d2(Q2) =
int 1
0
dx x2 [3g2(xQ2) + 2g1(xQ2)] (38)
22
000 005 010 015 020 025 030
0000
0001
0002
0003
0004
0005
0006
0007
Q2 GeV2D
d2
proton
000 005 010 015 020 025 030
0000
0002
0004
0006
0008
0010
0012
Q2 GeV2D
d2
neutron
000 005 010 015 020 025 030
-1
0
1
2
Q2 GeV2D
Γ 010
-4
fm6
D
proton
000 005 010 015 020 025 030-2
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
6D
neutron
FIG 8 Upper panel The inelastic moment d2(Q2) for the proton (left) and neutron (right) as
function of Q2 The result of this work the NLO BχPT prediction is shown by the blue solid line
and the blue band The red line represents the LO BχPT result The purple short-dashed line is
the O(p4) HB result from Ref [50 51] The black dotted line is the MAID model prediction [69]
The experimental points for the neutron (cyan dots) are from Ref [22] Lower panel Fifth-order
generalized forward spin polarizability γ0(Q2) for the proton (left) and neutron (right) as function
of Q2 The experimental points for the proton are from Ref [59] (purple square) and Ref [78]
(orange dot)
This relation however only holds for asymptotically large Q2 It is also in the high-Q2
region where d2(Q2) is a measure of color polarizability [82 83] through its relation to the
gluon field strength tensor [80] We refer to Ref [84] for a recent review on the spin structure
of the nucleon including a discussion of sum rules for deep inelastic scattering and color
polarizabilities
What we consider in the following is the inelastic part of d2(Q2) defined as the moment
of g1(xQ2) and g2(xQ2) spin structure functions cf Eq (38)
d2(Q2) =
int x0
0
dx x2 [3g2(xQ2) + 2g1(xQ2)] (39)
23
This moment provides another testing ground for our BχPT predictions through comparison
with experiments on the neutron [22] Going towards the low-Q2 region the interpretation
of d2(Q2) in terms of color polarizabilities will fade out The above definition however
implies it is related to other VVCS polarizabilities
d2(Q2) =Q4
8M4N
[M2
NQ2
αδLT (Q2) + I1(Q2)minus IA(Q2)
] (40)
Note that d2(Q2) and its first two derivatives with respect to Q2 vanish at Q2 = 0 The
considerations in Eqs (28) and (29) have no effect on d2(Q2) since the Born contribution
from IA(Q2) and I1(Q2) cancel out Therefore d2(Q2) is a pure polarizability
In Fig 8 upper panel we show our NLO BχPT prediction and other results for d2(Q2)
While MAID [69] and BχPT describe the experimental data for the neutron [22] very well
the HB limit [50 51] is showing a fast growth with Q2 This illustrates the importance
of keeping the relativistic result Note also that even though the πN -loop contribution
is dominant both gC and the form factor in gM are essential to obtain a curvature that
reproduces the data cf Fig 9 upper panel For the proton there are to our knowledge
no experimental results to compare with However the agreement between the NLO BχPT
prediction and the MAID prediction at low energies is reasonable
F γ0(Q2) mdash fifth-order generalized forward spin polarizability
It is interesting to compare the generalized fifth-order forward spin polarizability sum
rule
γ0(Q2) =1
2π2
int infinν0
dν
radic1 +
Q2
ν2
σTT (νQ2)
ν5(41)
=64αM4
N
Q10
int x0
0
dx x4
[g1(xQ2)minus 4M2
Nx2
Q2g2(xQ2)
]
to the sum rule integrals for IA(Q2) and γ0(Q2) since they differ merely by their energy
weighting of σTT (νQ2) and a constant prefactor cf Eqs (19) (26) and (41) From IA(Q2)
to γ0(Q2) to γ0(Q2) the energy suppression is increasing by a factor of νminus2 respectively
Therefore the description of γ0(Q2) should be easiest in a low-energy effective-field theory
such as χPT whereas γ0(Q2) and IA(Q2) receive larger contributions from higher energies
In Fig 8 lower panel we show our LO and NLO BχPT predictions for γ0(Q2) One
can see that the πN -loop contribution is positive (in accordance to what we see for the cross
section σTT see Fig 10) The Delta shifts it substantially especially in the low Q2 region
24
000 005 010 015 020 025 030
-0004
-0002
0000
0002
0004
0006
0008
0010
Q2GeV
2D
d2
proton
000 005 010 015 020 025 030
-0005
0000
0005
0010
0015
Q2GeV
2D
d2
neutron
000 005 010 015 020 025 030
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
6D
proton
000 005 010 015 020 025 030
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
6D
neutron
FIG 9 Contributions of the different orders to the chiral predictions of d2(Q2) upper panel and
γ0(Q2) lower panel for the proton (left) and neutron (right) Red solid line πN -loop contribu-
tion green dot-dashed line ∆-exchange contribution orange dotted line π∆-loop contribution
blue long-dashed line total result purple dot-dot-dashed line total result without gC contribution
black short-dashed line total result without gM dipole
bringing it into a better agreement with data In general the BχPT curves start above the
empirical data points at the real-photon point and then decrease asymptotically to zero
above Q2 gt 01 GeV2 On the other hand the MAID prediction reproduces the empirical
data at the real-photon point then decreases to negative values until about Q2 gt 006 GeV2
from where it also starts to asymptotically approach zero Consequently our NLO BχPT
prediction of γ0(Q2) is consistently above the MAID prediction This is very different to
what we saw for IA(Q2) in Fig 4 upper panel where the MAID prediction at the real-
photon point is above the experimental value While the agreement of our predictions with
the empirical data is in general quite good for all moments of σTT (νQ2) one should point
out that both for γ0n(Q2) and γ0p(Q2) we overestimate the data at low Q2 For IA(Q2) such
observation cannot be made because ∆IA(0) = 0 and thus IA(0) is given by the empirical
Pauli form factor only From IA(Q2) γ0(Q2) and γ0(Q2) the latter has the smallest however
25
non-negligible dependence on gC and the dipole in gM cf Fig 9 lower panel
The πN -loop ∆-exchange and π∆-loop contributions to the NLO BχPT prediction of
the static fifth-order forward spin polarizability amount to in units of 10minus4 fm6
γ0p = 112(30) asymp 208minus 096minus 001 (42a)
γ0n = 195(30) asymp 292minus 096minus 001 (42b)
while the slope is composed as follows in units of 10minus4 fm8
dγ0p(Q2)
dQ2
∣∣∣∣Q2=0
= minus084(10) asymp minus100 + 016 + 000 (43a)
dγ0n(Q2)
dQ2
∣∣∣∣Q2=0
= minus142(15) asymp minus158 + 016 + 000 (43b)
Note that the HB prediction of the static polarizability γ0p (423 at O(p3) and 365 at O(ε3)
[78 85]) is almost one order of magnitude larger than the empirical value and therefore not
shown in Fig 8
G Summary
Our results are summarized in Table II where we give the contributions of the different
orders to the chiral predictions of the polarizabilities and their slopes at the real-photon
point A quantitative comparison of our predictions for the static spin polarizabilities to
the work of Bernard et al [31] and different empirical evaluations is shown in Table III We
can see that the inclusion of the Delta turns out to be very important for all moments of
the helicity-difference cross section To describe the Q2 behavior of the polarizabilities the
magnetic coupling of the N rarr ∆ transition should be modified by a dipole form factor as
has been observed previously in the description of electroproduction data [33] This dipole
form factor effectively takes account of vector-meson exchanges The Coulomb-quadrupole
N rarr ∆ transition despite its subleading order is important in the description of some
moments of spin structure functions This is contrary to what we saw for the moments of
unpolarized structure functions [30] where the Coulomb coupling had a negligible effect
The π∆ loops are mainly relevant for the generalized GDH integrals
26
TABLE II The NLO BχPT predictions for the forward VVCS polarizabilities and their slopes at
Q2 = 0 The contributions of the πN loops the ∆ exchange and the π∆ loops are shown together
with the combined total result Note that IA(0) = I1(0) = d2(0) = 0 and (d2)prime = 0
πN loops ∆ exchange π∆ loops Total
γ0 p 201 minus284 minus010minus093(92)
(10minus4 fm4) n 298 003(92)
δLT p 150 minus016 minus002132(15)
(10minus4 fm4) n 235 218(23)
γ0 p 208 minus096 minus001112(30)
(10minus4 fm6) n 292 195(30)
(γ0)prime p minus033011 001
minus022(4)
(10minus4 fm6) n minus073 minus061(7)
(δLT )prime p minus080 minus004 minus001minus085(8)
(10minus4 fm6) n minus119 minus124(12)
(γ0)prime p minus100016 000
minus084(10)
(10minus4 fm8) n minus158 minus142(15)
(∆IA)prime p 238 minus1121 025minus858(343)
(GeVminus2) n 141 minus955(343)
(∆I1)prime p 034 minus053 058039(4)
(GeVminus2) n minus107 minus101(10)
TABLE III Our NLO BχPT predictions for the spin polarizabilities at Q2 = 0 compared with
the BχPT+∆ predictions from Bernard et al [31] and the available empirical information Where
the reference is not given the empirical number is provided by the MAID analysis [55 69] with
unspecified uncertainty
Proton Neutron
This work BχPT+∆ Empirical This work BχPT+∆ Empirical
γ0 minus093(92) minus174(40) minus100(8)(12) [19] 003(92) minus077(40) minus0005
(10minus4 fm4) minus090(8)(11) [78] [MAID]
minus0929(105) [59]
δLT 132(15) 240(1) 134 218(23) 238(3) 203
(10minus4 fm4) [MAID] [MAID]
γ0 112(30) 060(7)(7) [78] 195(30) 123
(10minus4 fm6) 0484(82) [59] [MAID]
IV CONCLUSIONS
We have presented a complete NLO calculation of the polarized non-Born VVCS ampli-
tudes in covariant BχPT with pion nucleon and ∆(1232) fields The dispersion relations
between the VVCS amplitudes and the tree-level photoabsorption cross sections served as a
27
cross-check of these calculations
The obtained moments of the proton and neutron spin structure functions related to gen-
eralized polarizabilities and GDH-type integrals agree well with the available experimental
data The description of their Q2 evolution is improved compared to the previous χPT pre-
dictions In particular the NLO BχPT predictions obtained here give a better description
of the empirical data (eg from the Jefferson Laboratory ldquoSpin Physics Programrdquo) than
the HB [50 51] and IR [58] calculations
The demonstrated predictive power of the χPT framework amplitudes makes it well
suited for extending the χPT evaluation of the TPE effect in the hyperfine structure of
(muonic-)hydrogen [15ndash17] to next-to-leading order
ACKNOWLEDGEMENTS
We thank Lothar Tiator and Marc Vanderhaeghen for helpful discussions This work
is supported by the Deutsche Forschungsgemeinschaft (DFG) through the Collaborative
Research Center [The Low-Energy Frontier of the Standard Model (SFB 1044)] JMA ac-
knowledges support from the Community of Madrid through the ldquoPrograma de atraccion de
talento investigador 2017 (Modalidad 1)rdquo and the Spanish MECD grants FPA2016-77313-P
FH gratefully acknowledges financial support from the Swiss National Science Foundation
Appendix A Tensor decompositions of the VVCS amplitudes
In this appendix we review the decomposition of the forward VVCS process into tensor struc-
tures and scalar amplitudes In particular we consider the connection between the covariant and
the semi-relativistic decomposition in the lab frame that is defined in terms of the conventional
transverse longitudinal transverse-transverse and transverse-longitudinal amplitudes
As explained in Sec II A the process of forward VVCS off the nucleon can be described in
terms of four explicitly covariant amplitudes S1 2 and T1 2 [3]
T (νQ2) =
(minusgmicroν +
qmicroqν
q2
)T1(νQ2) +
1
M2N
(pmicro minus p middot q
q2qmicro)(
pν minus p middot qq2
qν)T2(νQ2) (A1)
minus 1
MNγmicroναqα S1(νQ2)minus 1
M2N
(γmicroνq2 + qmicroγναqα minus qνγmicroαqα
)S2(νQ2)
εprimelowastmicro εν
where εmicro (εprimelowastmicro ) are the incoming (outgoing) photon polarization vectors ν is the photon lab-frame
energy and Q2 is the photon virtuality Alternatively the decomposition in the laboratory frame
28
(which in the forward case coincides with the Breit frame) is parametrized in terms of the nucleon
Pauli matrices ~σ and the four scalar functions fL fT gTT and gLT
T (νQ2) = ε0 εprimelowast0 fL(νQ2) + (~ε primelowast middot ~ε ) fT (νQ2) + i~σ middot (~ε primelowast times ~ε ) gTT (νQ2) (A2)
minusi~σ middot [(ε0~εprimelowast minus ~ε ε primelowast0 )times q] gLT (νQ2)
Here ~q and q = ~q|~q | are the photon three-momentum in the lab system and its unit vector The
modified polarization vector components are given by
ε0 =
[ε0 minus
ν
|~q | (~ε middot q )
] |~q |Q ~ε = ~εminus q (~ε middot q ) (A3)
where ε = (ε0~ε ) is the usual incoming photon polarization vector and εprimelowast the outgoing polarization
vector The LEX of the lab frame amplitudes [Eq (10)] can serve in particular as the definition
of the generalized polarizabilities The lab frame amplitudes are also conveniently used for the
definition of the response functions see the example of the scalar amplitude gLT (νQ2) and the
corresponding response function σLT (νQ2) below in App B
Appendix B Photoabsorption cross sections
In the forward kinematics the spin-dependent VVCS amplitudes and the spin polarizabilities
can be described in terms of the polarized structure functions g1(xQ2) and g2(xQ2) or equiv-
alently the helicity-difference cross section σTT (νQ2) and the longitudinal-transverse response
function σLT (νQ2) with the help of dispersion relations (5) and the optical theorem (3) In this
way the photoabsorption cross sections measured in electroproduction processes form the basis
for most empirical evaluations shown throughout Sec III In the following we present the BχPT
predictions for the tree-level cross sections of πN - π∆- and ∆-production through photoabsorption
on the nucleon cf Figs 8 9 and 10 in Ref [30] In Secs B 1 and B 2 we will discuss the leading
πN -production channel and the ∆-production channel respectively We used these cross sections
to verify the polarizability predictions obtained otherwise from the calculated non-Born VVCS
amplitudes Due to the bad high-energy behavior of the π∆-production cross sections in BχPT
cf Fig 10 the dispersion relations in Eq (5) require further subtractions for a reconstruction of
the π∆-loop contribution to the spin-dependent VVCS amplitudes Therefore not all polarizabil-
ities could be verified but only those appearing as higher-order terms in the LEX of the VVCS
amplitudes such as γ0 [16]
29
proton neutron
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Neutron
02 04 06 08 100
50
100
150
200
250
[GeV]
T [b]
02 04 06 08 10
-50
0
50
100
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Neutron
02 04 06 08 100
50
100
150
200
250
[GeV]
T [b]
02 04 06 08 10
-50
0
50
100
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02040608100
50
100
150
200
250
300
[GeV]
T[b]
0204060810
-50
0
50
[GeV]
TT[b]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
0204060810
00
02
04
06
[GeV]
LTQ
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
0204060810
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
FIG 10 Photoabsorption cross sections for πN (red) and π∆ production (orange) with Q2 = 0
(solid) and Q2 = 01 GeV2 (dashed for πN and dotted for π∆ channel)
1 πN-production channel
In order to extract the response function σLT (νQ2) we have developed a method similar to
the one used to calculate σTT (νQ2) see for example Ref [86] For σLT (νQ2) however the
calculation is more complicated because one has to take into account that the associated Compton
process involves a spin-flip of the nucleon as illustrated in Fig 11 When calculating the cross
section the product of the incoming nucleon spinors has to reflect this flip
The forward VVCS amplitude related to σLT (νQ2) mdash and δLT (Q2) mdash is gLT (νQ2) It can
be extracted from Eq (A2) if one takes the modified polarization vector components in Eq (A3)
with ε = εL and εprimelowast = εlowastplusmn as input where εL = 1Q(|~q | 0 0 q0) and εplusmn = ∓ 1radic
2(0 1plusmni 0) are the
standard longitudinal and transverse polarization vectors respectively For εL and εlowastplusmn only the
choice of helicities hprime = plusmn12 and h = ∓12 gives a non-zero contribution and one obtains
χdaggerhprime T (νQ2)χh = χdaggerhprimeminusi~σ middot [(ε0~εprimelowast minus ~ε ε primelowast0 )times q] gLT (νQ2)χh =
radic2 gLT (νQ2) (B1)
where χh and χdaggerhprime are two-component Pauli spinors with opposite helicities or here spins
Let us now consider the related photoabsorption process and in particular the tree-level γlowastN rarr
30
s = 12
sprime = minus12
λ = 0 λprime = 1
rprime rprime
s=12sprime=minus1
2
λ=0λprime=1
rprimerprime
s=12
sprime=minus12
λ=0λprime=1
rprimerprime hacute=frac12h=-frac12
λacute=1λ=0
FIG 11 Relation between the forward Compton process and the photoabsorption process given
by the optical theorem In particular we show the longitudinal-transverse contribution The
double-line arrows represent the spin of the external particles while the dot represents the scalar
(longitudinal) polarization of the incoming photon Inside the blob the intermediate states are
represented eg nucleons with spins rprime (which are averaged in the calculation of the cross section)
and pions
πN channel see diagrams in Fig 8 of Ref [30] We define the πN -production amplitude as
T = uhB (PB)sumi
Ai(s t)Γi uhA(PA) (B2)
with the Dirac structures
Γ1 = γ5 (B3a)
Γ2 =1
2
[qA ε
]γ5 (B3b)
where uhA(PA) and udaggerhB (PB) are the Dirac spinors and PA and PB are the four-momenta of
the incoming and outgoing nucleons respectively When calculating the photoabsorption cross
section related to the VVCS amplitude in Eq (B1) the nucleon spin flip should be implemented
by uhprime(PA) in T dagger and uh(PA) in T together with the appropriate transverse and longitudinal
photon polarization vectors εlowastplusmn and εL
However if one wants to use the properties of the Dirac matrices it is more useful to construct
an operator to produce this spin flip in the external nucleons of Fig 11 This is accomplished by
introducing the projector ΓLT equiv 12radic
2(γ1 + iγ2)γ5 which also takes into account the extra factor
radic2 in Eq (B1) We checked that with this projector one correctly extracts δLT by comparing the
HB limit of our result to the HB result of Ref [50] where the authors calculate this polarizability
from the Compton amplitude directly With all those ingredients the longitudinal-transverse cross
section is calculated in the following way
σLT (νQ2) =1
64π2 s
|~pf |cm
|~pi|cm
int 1
minus1dcos θ
sumij
AiAdaggerjXij (B4)
31
with
Xij = Tr[(PB +MN )Γi(PA +MN )ΓLTγ0Γdaggerjγ
0] (B5)
where θ is the scattering angle in the center-of-mass (cm) frame and |~pi|cm (|~pf |cm) is the three-
momentum of an incoming (outgoing) particle in the cm frame An explicit calculation of the
matrix Xij leads to
X = MNQ
0 2(PB minus PA) middot εLminusradic
2|~qf |cm sin θ (sminus u)
(B6)
where |~qi|cm (|~qf |cm) is the relative three-momentum of the incoming (outgoing) particles in the
cm frame Here s t and u are the usual Mandelstam variables For the different γlowastN rarr πN
channels we obtain the following amplitudes Ai where we introduce qA as the four-momentum of
the incoming photon and qB as the four-momentum of the outgoing pion
bull γlowastprarr π0p
A1 =e gAMN
fπ
[2PA middot ε+ qA middot ε
sminusM2N
+2PB middot εminus qA middot ε
uminusM2N
] (B7a)
A2 =e gAMN
fπ
[1
sminusM2N
+1
uminusM2N
] (B7b)
bull γlowastprarr π+n
A1 =
radic2 e gAMN
fπ
[2PA middot ε+ qA middot ε
sminusM2N
+2(PA minus PB) middot ε+ qA middot ε
tminusm2π
] (B8a)
A2 =
radic2 e gAMN
fπ(sminusM2N )
(B8b)
bull γlowastnrarr π0n
A1 = 0 (B9a)
A2 = 0 (B9b)
bull γlowastnrarr πminusp
A1 =
radic2 e gAMN
fπ
[2PB middot εminus qA middot ε
uminusM2N
minus 2(PA minus PB) middot ε+ qA middot εtminusm2
π
] (B10a)
A2 =
radic2 e gAMN
fπ(uminusM2N )
(B10b)
The analytical expressions shown above were checked with the amplitudes given in Ref [87] An-
alytical expressions for the tree-level γlowastN rarr πN channel of the σLT (νQ2) and σTT (νQ2) cross
32
sections are given below (proton channels π+n and π0p neutron channel πminusp) We checked that
they reproduce the known results in the real-photon limit [45 86] To shorten the final expressions
for the cross sections which are considerably longer for finite Q2 than in the real-photon limit we
define the following dimensionless kinematic variables
αγ = (ENi )cmradics =
s+M2N +Q2
2s (B11)
απ = (ENf )cmradics =
s+M2N minusm2
π
2s (B12)
βγ = Eγcmradics =
sminusM2N minusQ2
2s (B13)
βπ = Eπcmradics =
sminusM2N +m2
π
2s (B14)
λγ = |~qi|cmradics =
radic(sminusM2
N minusQ2)2 + 4sQ2
2s (B15)
λπ = |~qf |cmradics =
radic(sminusM2
N +m2π)2 minus 4sm2
π
2s (B16)
Here (ENi )cm and (ENf )cm are the energies of the incoming and outgoing nucleon Eγcm is the energy
of the incoming photon Eπcm is the energy of the outgoing pion all in the cm frame
σ(π+n)TT = minus e2g2
AM2N
64πf2πs
3(sminusM2N )2λ4
γ
4sλπλγ
[(M2
N minus s)(M2N minusQ2 minus s)(Q2 + 2sβγβπ)
+ 2s(m2πQ
2 minus (M2N minus s)
(M2N + s(minus1 + 2βγβπ)
))λ2γ
]+ (M2
N minus s)(M2N minusQ2 minus s)
times (Q2 + 2sβγβπ minus 2sλπλγ)(Q2 + 2s(βγβπ + λπλγ)
)times log
(Q2 + 2sβγβπ minus 2sλγλπQ2 + 2sβγβπ + 2sλγλπ
) (B17)
33
σ(π0p)TT =
e2g2AM
2Nλπ
64πf2πs(sminusM2
N )2λγ
1
s2λπλ3γ
[(M2
N minus s)(M2N minusQ2 minus s)
(3M2
N + s(2βγβπ minus 3))(M2N
+ s(2βγβπ minus 1))
+ 2s
((2m2
πQ2 + (M2
N minus s)2)(M2
N minus s)minus 2(minusm2
πQ2 + (sminusM2
N )2)
times sβγβπ + 4s2(sminusM2N )β2
γβ2π
)λ2γ
]arctan
(2sλπλγ
M2N + s(2βγβπ minus 1)
)+ 2
[minus 2m2
πQ2
+ (M2N minus s)
(2(sminusM2
N ) + 4sβγβπ +(sminusM2
N +Q2)(3M2N minus 3s+ 2sβγβπ)
sλ2γ
+m2πQ
2(sminusM2N )
(M2N minus s+ 2sβγβπ)(M2
N + s(minus1 + 2βγβπ minus 2λπλγ))
+m2πQ
2(sminusM2N )
(M2N minus s+ 2sβγβπ)(M2
N + s(minus1 + 2βγβπ + 2λπλγ))
)
+1
λπλγ
(βγβπ
(minusm2
πQ2 + (M2
N minus s)(M2N minus s+ 2sβγβπ)
)minus (M2
N minus s)(M2N minusQ2 minus s)λ2
π
)log
(M2N + s(minus1 + 2βγβπ + 2λπλγ)
M2N + s(minus1 + 2βγβπ minus 2λπλγ)
)] (B18)
σ(πminusp)TT =
e2g2AM
2N
64πf2πs
3λ4γ(M2
N + s(2βπβγ minus 1))2 minus 4λ2πs
2λ2γ)
4sλπλγ
((minusM2
N +Q2 + s)(M2N
+ s(2βπβγ minus 1))
2 minus 2sλ2γ
(2λ2
πs(minusM2N +Q2 + s) +m2
πQ2))
+((M2N + s(2βπβγ minus 1)
)2
minus 4λ2πs
2λ2γ
)((M2N + s(2βπβγ minus 1)
)(M2N minus 2Q2 minus s(2βπβγ + 1)
)+ 4λ2
πs2λ2
γ
)times log
(minusM2
N minus 2βπsβγ minus 2λπsλγ + s)minus(M2N + s(2βπβγ minus 2λπλγ minus 1)
)(M2N + s(minus1
+ 2βπβγ + 2λπλγ))[((
M2N + s(2βπβγ minus 1)
)(M2N minus 2Q2 minus s(2βπβγ + 1)
)+ 4λ2
πs2λ2
γ
)times log
(minusM2
N minus 2βπsβγ + 2λπsλγ + s)
+((Q2 + 2βπsβγ)2 minus 4λ2
πs2λ2
γ
)times log
(Q2 + 2βπsβγ minus 2λπsλγQ2 + 2sβπβγ + 2λπsλγ
)]+ 4sλ2
γ
((M2
N + s(2βπβγ minus 1))2 minus 4λ2πs
2λ2γ
)times (M2
N minus s) arctanh
(2λπsλγ
M2N + 2βπsβγ minus s
) (B19)
34
σ(π+n)LT =
e2g2AM
3Nλπ
32πf2πQs
3(sminusM2N )2λ4
γ
2sλγ
[(M2
N minus s)(Q2 + 2sβ2γ)(Q2 + 2sβγβπ)
minus 4s
((M2
N minus s)(Q2 minus 2s(απ minus 1)βγ
)+Q2sβγβπ
)λ2γ + 8s3(minus1 + απ)λ4
γ
]minus M2
N minus sλπ
[(Q2 + 2sβ2
γ)(Q2 + 2sβγβπ)2 + 4s2(2(απ minus 1)βγ(Q2 + 2sβγβπ)
minusQ2λ2π
)λ2γ + 8s3(απ minus 1)2λ4
γ
]arctan
(2sλπλγ
Q2 + 2sβγβπ
) (B20)
σ(π0p)LT =
e2g2AM
3Nλπ
16πf2πQs(sminusM2
N )2λγ
1
minus2s(M2N + s(minus1 + 2βγβπ))2λ2
γ + 8s3λ2πλ
4γ
[minus 3M8
N (Q2
+ 2sβ2γ) + 2M4
Ns2
(minus (Q2 + 2sβ2
γ)(minus3 + 2βγβπ) +(Q2(minus3 + 2βγβπ + 6λ2
π) + 2sβγ
times (minus3 + 12απ + 2βγβπ minus 12απβγβπ + 4βγλ2π))λ2γ minus 4s(α2
π minus 1)λ4π
)+ 2M2
Ns3
[minus (Q2
+ 2sβ2γ)(minus1 + 2βγβπ)
(6 + βγβπ(minus9 + 2βγβπ)
)+
(Q2(3minus 12λ2
π + 4βγβπ(minus1 + 2βγβπ
+ λ2π))minus 2sβγ
(minus 3 + 4απ(3 + 2απ(3 + 2βγβπ(minus3 + βγβπ)) + 4βγ(βπ + (2minus βγβπ)
times λ2π)))λ2γ + 8s
((απ minus 1)(1 + απ minus 2βγβπ) + 2απβγλ
2π
)λ4γ
)+ s4
((Q2 + 2sβ2
γ)
times (1minus 2βγβπ)2(2βγβπ minus 3) + 2(Q2(minus1 + 6λ2
π + 2βγβπ((1minus 2βγβπ)2 minus 2λ2π)) + 2sβγ
times (minus1 + 2βγβπ + 4(minus1 + βγβπ)(minusαπ + 2απβγβπ minus βγλ2π)))λ2γ minus 8
(s(απ minus 1)(απ
+ (1minus 2βγβπ)2) + 2βγ(2sαπ +Q2βγ)λ2π
)λ4γ + 32s(απ minus 1)λ2
πλ6γ
)+ 2M6
Ns
(Q2(6
minus 7βγβπ + λ2γ) + 2sβγ(βγ(6minus 7βγβπ) + (1minus 4απ)λ2
γ)
)]+sminusM2
N
4s2λπλ3γ
[(Q2 + 2sβ2
γ)
times(3M2
N + s(2βγβπ minus 3))(M2N + s(2βγβπ minus 1)
)+ 2s
(minusM2
N (Q2 + 2sβγ minus 8sαπβγ)
+ s(2sβγ(1minus 4απ + 4απβγβπ) +Q2(1minus 2λ2π)))λ2γ + 8s3(απ minus 1)λ4
γ
]
times arctan
(2sλπλγ
M2N + s(2βγβπ minus 1)
) (B21)
35
σ(πminusp)LT =
e2g2AM
3N
64πf2πQs
3λ4γ(M2
N minusQ2 minus s)(M2N + s(2βπβγ + 2λπλγ minus 1))
times
1
M2N + s(2βπβγ minus 2λπλγ minus 1)
[4λπsλγ(M2
N minusQ2 minus s)(
2sλ2γ
(s((1minus 2λ2
π)Q2
+ 2(2απ minus 1)sβγ(2βπβγ minus 1))minusM2N (Q2 + 2(1minus 2απ)sβγ)
)+ (Q2 + 2sβ2
γ)(M2N + s(2βπβγ minus 1)
)2 + 8(απ minus 1)απs
3λ4γ
)]minus(M2N + s(2βπβγ + 2λπλγ minus 1)
)times[
log
(minusM2N minus 2βπsβγ minus 2λπsλγ + s
minusM2N minus 2βπsβγ + 2λπsλγ + s
)(minus (Q2 + 2sβ2
γ)(M2N + s(2βπβγ minus 1)
)(minusM2
N + 2Q2 + 2βπsβγ + s)minus 2sλ2γ
(minusM2
N (Q2 + 2sβγ) +Q4 +Q2s(4απβγ
minus 2βγ minus 2λ2π + 1) + 2s2βγ(4(απ minus 1)βπβγ + 1)
)minus 8(απ minus 1)2s3λ4
γ
)+ 2
(4s2λ2
γ
(2(απ minus 1)βγ(Q2 + 2βπsβγ)minus λ2
πQ2)
+ (Q2 + 2βπsβγ)2(Q2 + 2sβ2γ)
+ 8(απ minus 1)2s3λ4γ
)arctanh
(2λπsλγ
Q2 + 2βπsβγ
)] (B22)
2 ∆-production channel
The tree-level ∆-exchange diagram in Fig 2 of Ref [30] contributes to the non-Born part of
the VVCS amplitudes The contribution of the ∆ exchange to the VVCS amplitudes can be split
into [17]
S∆-exch1 (νQ2) = S∆-pole
1 (νQ2) + S∆-exch1 (νQ2) (B23a)
νS∆-exch2 (νQ2) = νS∆-pole
2 (νQ2) + νS2∆-exch
(νQ2) (B23b)
and similarly for the unpolarized VVCS amplitudes discussed in Ref [30] Here we introduced the
∆-pole contributions S∆-polei and the ∆-non-pole contributions S∆-exch
i The former amplitudes
feature a pole at the ∆(1232)-production threshold and thus are proportional to
1
[sminusM2∆][uminusM2
∆]=
1
4M2N
1
ν2∆ minus ν2
(B24)
36
They can be reconstructed from the dispersion relations in Eq (5) with the tree-level ∆-production
cross sections as input cf Fig 10 in Ref [30]
σTT (νQ2) =π2α
M2NM
2+|~q |
minus g2
MMN (M+ + ν)|~q |2 +g2E(∆minus ν)(Q2 minusMNν)2
MN(B25a)
+g2CQ
4s(∆minus ν)
MNM2∆
minus 4gMgE(Q2 minusMNν)|~q |2 minus 4gMgCQ2|~q |2
+2gEgCQ
2[minusMNM∆ |~q |2 + s(Q2 +∆ν)
]MNM∆
δ(ν minus ν∆)
σLT (νQ2) =Qπ2α
M2NM
2+|~q |
g2E(MNν minusQ2) [M∆(MN + ν)minus s]
MN(B25b)
+g2CQ
2[MNM∆|~q |2 minus s(Q2 +∆ν)
]MNM2
∆
+ gMgEM∆|~q |2 minusgMgC(Q2 minusMNν)|~q |2
M∆
+gEgC(ν minus∆)(M2
N |~q |2 minus 2Q2s)
MNM∆
δ(ν minus ν∆)
with ∆ = M∆ minusMN M+ = M∆ + MN and the Mandelstam variable s = M2N + 2MNν minus Q2
Analytical expressions for the spin structure functions g1(xQ2) and g2(xQ2) can be constructed
from Eq (3) with the flux factor K(νQ2) = |~q | =radicν2 +Q2
In the ∆-non-pole contributions to S1(νQ2) and νS2(νQ2) the pole in ν at the ∆(1232)-
production threshold has canceled out
S∆-exch1 (νQ2) =
πα
MNM2+
[g2MQ
2+ + g2
E
(∆2 minus 3Q2
)+
4g2CQ
4
M2∆
minus 8gMgEM∆ωminus (B26a)
minus2gMgCQ2(MN minus 4M∆)
M∆+
2gEgCQ2(3MN minus 2M∆)
M∆
]
νS2∆-exch
(νQ2) =2πα
MNM2+
[g2EM∆∆ωminus +
g2M MNQ
2+
2+g2C Q
2(Q2 minus∆2)
2M∆(B26b)
+gEgM M∆(M∆ω+ minus 4MNωminus)minus gEgC ∆(2Q2 +MNω+)
+gMgC Q2(4MN minus ω+)
]+S∆-exch
2 (νQ2)
ν
[M2
∆ ω2+
M2N
+ ν2
]
with Q+ =radic
(M∆ +MN )2 +Q2 and ωplusmn = (M2∆minusM2
N plusmnQ2)2M∆ and the non-pole contribution
to S2(νQ2)
S∆-exch2 (νQ2) = minus2παMNν
M∆M2+
[gM + gE
]gC (B27)
These amplitudes to the contrary are not described by the tree-level ∆-production cross sections
in the standard dispersive approach [17] This peculiarity has been previously missed e g in the
calculation of the ∆-exchange contribution to the hydrogen hyperfine splitting in Ref [88] The
importance of including the ∆-non-pole contribution is also evident when considering the BC sum
37
rule in Eq (14) The ∆-pole terms by themselves violate the BC sum rule but cancel exactly with
the ∆-non-pole terms
limνrarr0
νS∆-pole2 (νQ2) + lim
νrarr0νS2
∆-exch(νQ2) = 0 (B28)
Appendix C Polarizabilities at Q2 = 0
In this section we give analytical expressions for the static values and slopes at Q2 = 0 of
the spin polarizabilities and moments of polarized structure functions In particular we give the
HB expansion of the πN -loop contributions and the ∆-exchange contributions The complete
expressions also for the π∆-loop contributions can be found in the Supplemented material Recall
that IA(0) = I1(0) = d2(0) = 0 and dd2(Q2)dQ2
∣∣∣Q2=0
= 0
1 πN-loop contribution
Here we give analytical expressions for the πN -loop contributions to the proton and neutron
spin polarizabilities expanded in powers of micro = mπMN viz the HB expansion Note that we
choose to expand here to a high order in micro the strict HB expansion would only retain the leading
term in an analogous NLO calculation
bull Static polarizabilities (Q2 = 0)
γ0p =e2g2
A
96π3f2πm
2π
1minus 21πmicro
8minus(
59
2+ 26 logmicro
)micro2 +
1875πmicro3
64
+3
(3
2+ 26 logmicro
)micro4 +
(C1)
γ0n =e2g2
A
48π3f2πm
2π
1
2minus 9πmicro
16minus 2micro2 logmicro+
75πmicro3
128minus 3micro4
4+
(C2)
δLTp =e2g2
A
192π3f2πm
2π
1minus 9πmicro
8+
(13
2minus 2 logmicro
)micro2 minus 465πmicro3
64
minus(
47
2+ 42 logmicro
)micro4 +
(C3)
δLTn =e2g2
A
96π3f2πm
2π
1
2+
3πmicro
16+ (1 + 2 log micro)micro2 minus 105πmicro3
128+
5micro4
4+
(C4)
38
δLTn =e2g2
A
96π3f2πm
2π
1
2+
3πmicro
16+ (1 + 2 log micro)micro2 minus 105πmicro3
128+
5micro4
4+
(C5)
γ0p =e2g2
A
16π3f2πm
4π
4
45minus 3πmicro
16+
14micro2
5minus 1813πmicro3
384minus 192
5(1 + log micro)micro4
+80703πmicro5
2048+
(C6)
γ0n =e2g2
A
16π3f2πm
4π
4
45minus 5πmicro
48+
4micro2
5minus 245πmicro3
384minus 32micro4 logmicro
15
+1323πmicro5
2048+
(C7)
bull Slopes of polarizabilities at Q2 = 0
dγ0p(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
2minus 45πmicro
4+ 223micro2 minus 28515πmicro3
64
minus9
(1953
4+ 449 logmicro
)micro4 +
570255πmicro5
128+
(C8)
dγ0n(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
2minus 81πmicro
8+ 94micro2 minus 2535πmicro3
32
minus3 (1 + 90 logmicro)micro4 +84315πmicro5
1024+
(C9)
dδLTp(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
2880π3f2πm
4π
minus5
2minus 27πmicro
32+ 20micro2 minus 5865πmicro3
256
+3
(617
4+ 36 logmicro
)micro4 minus 2056845πmicro5
4096+
(C10)
dδLTn(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
minus5
4minus 81πmicro
64minus 11micro2 +
10005πmicro3
512
+15
8(11 + 48 logmicro)micro4 minus 267015πmicro5
8192+
(C11)
dIAp(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
96π2f2πmicro
2
1minus 15πmicro
4minus 1
2(115 + 88 logmicro)micro2 +
1839πmicro3
32
+5 (5 + 34 logmicro)micro4 +
(C12)
dIAn(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
48π2f2πmicro
2
1
2minus 11πmicro
8minus 1
4(1 + 20 logmicro)micro2 +
99πmicro3
64
minus25micro4
12+
(C13)
dI1p(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
96π2f2πmicro
3π
8+ 2 (4 + 3 logmicro)microminus 537πmicro2
64
minus1
2(15 + 56 logmicro)micro3 +
(C14)
39
dI1n(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
48π2f2πmicro
minus π
16+
1
4(3 + 4 logmicro)microminus 57πmicro2
128+
2micro3
3+
(C15)
γ0p(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2g2
A
16π3f2πm
6π
1
105minus 23πmicro
256+
377micro2
210minus 15551πmicro3
6144+
3371micro4
105
minus1640457πmicro5
32768+
(C16)
γ0n(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2g2
A
16π3f2πm
6π
1
105minus 153πmicro
1792+
69micro2
70minus 4615πmicro3
6144+
172micro4
35
minus120897πmicro5
32768+
(C17)
2 ∆-exchange contribution
Here we give analytical expressions for the tree-level ∆-exchange contributions to the nucleon
spin polarizabilities and their slopes at Q2 = 0 Note that the ∆-exchange contributes equally to
proton and neutron polarizabilities Recall that for the magnetic γlowastN∆ coupling we introduced a
dipole form factor to mimic vector-meson dominance gM rarr gM(1 +Q2Λ2)2
bull Static polarizabilities (Q2 = 0)
γ0 = minus e2
4πM2+
(g2M
∆2+
g2E
M2+
minus 4gMgEM+∆
) (C18)
δLT =e2M∆
4πM3+
(g2E
MNM++gMgE∆MN
minus gEgCM2
∆
) (C19)
γ0 =e2M2
N
π∆2M4+
(minusg
2M
∆2+
g2E
M2+
+4gMgE∆M+
) (C20)
bull Slopes of polarizabilities at Q2 = 0
dγ0(Q2)
dQ2
∣∣∣∣∣Q2=0
= minus e2
πM2+∆
(g2M
∆
[1
4∆2minus 1
∆M++
1
2M2+
]minus 1
Λ2
g2M
∆+
g2E
2M2+
[1
2∆minus 3
M+
]minusgMgE
M+
[1
∆2minus 5
∆M++
1
M2+
]+
1
Λ2
2gMgEM+
+2gMgC∆M2
+
minus gEgCM3
+
) (C21)
40
d δLT (Q2)
dQ2
∣∣∣∣∣Q2=0
=e2M∆∆
4πMNM2+
(g2E
∆2M2+
[1
∆minus 4
M+
]minus g2
C
∆M2∆M
2+
+gMgE∆2M+
[1
∆2
minus 3
∆M++
1
M2+
]minus 2
Λ2
gMgE∆2M+
+gMgC∆M2
∆
[1
2∆2minus 2
∆M++
1
2M2+
]minus gEgC
2M2∆M
2+
[7
∆+
1
M+
]) (C22)
dIA(Q2)
dQ2
∣∣∣∣∣Q2=0
= minusM2N
M2+
(g2M
2∆2+
g2E
MNM+minus 2gMgE
∆M+minus gEgCM∆M+
) (C23)
dI1(Q2)
dQ2
∣∣∣∣∣Q2=0
= minusM∆M2N
2M3+
(g2E
MNM∆minus gMgE∆MN
minus gEgCM2
∆
) (C24)
γ0(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2M2
N
π∆3M6+
(g2M
∆
[3M2
∆ + 2M∆MN minus 9M2N
∆2+
4M2+
Λ2
]
minus4gMgE
[5M2
∆ minus 9M2N
M+∆2+
2M+
Λ2
]+
g2E
∆M+[7M∆ minus 9MN ]
minus8gMgC∆
+4gEgCM+
) (C25)
[1] D Drechsel B Pasquini and M Vanderhaeghen Dispersion relations in real and virtual
Compton scattering Phys Rept 378 99 (2003) hep-ph0212124
[2] S E Kuhn J-P Chen and E Leader Spin structure of the nucleon mdash status and recent
results Prog Part Nucl Phys 63 1 (2009) arXiv08123535 [hep-ph]
[3] F Hagelstein R Miskimen and V Pascalutsa Nucleon polarizabilities from Compton scat-
tering to hydrogen atom Prog Part Nucl Phys 88 29 (2016) arXiv151203765 [nucl-th]
[4] B Pasquini and M Vanderhaeghen Dispersion theory in electromagnetic interactions Ann
Rev Nucl Part Sci 68 75 (2018) arXiv180510482 [hep-ph]
[5] S Gerasimov A Sum rule for magnetic moments and the damping of the nucleon magnetic
moment in nuclei Sov J Nucl Phys 2 430 (1966)
[6] S Drell and A C Hearn Exact sum rule for nucleon magnetic moments Phys Rev Lett
16 908 (1966)
[7] J S Schwinger Source theory viewpoints in deep inelastic scattering Proc Natl Acad Sci
USA 72 1 (1975)
[8] J S Schwinger Source theory viewpoints in deep inelastic scattering Electromagnetic In-
teractions and Field Theory Proceedings 14 Internationale Universitatswochen Schladming
Austria February 24-March 7 1975 Acta Phys Austriaca Suppl 14 471 (1975)
41
[9] J Schwinger Source theory discussion of deep inelastic scattering with polarized particles
Proc Natl Acad Sci USA 72 1559 (1975)
[10] A M Baldin Polarizability of nucleons Nucl Phys 18 310 (1960)
[11] M Gell-Mann M L Goldberger and W E Thirring Use of causality conditions in quantum
theory Phys Rev 95 1612 (1954)
[12] A Pineda Leading chiral logarithms to the hyperfine splitting of the hydrogen and muonic
hydrogen Phys Rev C 67 025201 (2003)
[13] C Peset and A Pineda The two-photon exchange contribution to muonic hydrogen from
chiral perturbation theory Nucl Phys B 887 69 (2014) arXiv14064524 [hep-ph]
[14] C Peset and A Pineda Model-independent determination of the two-photon exchange contri-
bution to hyperfine splitting in muonic hydrogen JHEP 04 060 arXiv161205206 [nucl-th]
[15] F Hagelstein and V Pascalutsa Proton structure in the hyperfine splitting of muonic hydro-
gen PoS CD15 077 (2016) arXiv151104301 [nucl-th]
[16] F Hagelstein Exciting Nucleons in Compton Scattering and Hydrogen-Like Atoms PhD
thesis Mainz U Inst Kernphys (2017) arXiv171000874 [nucl-th]
[17] F Hagelstein ∆(1232)-Resonance in the hydrogen spectrum Proceedings 11th International
Workshop on the Physics of Excited Nucleons (NSTAR 2017) Columbia SC USA August
20-23 2017 Few Body Syst 59 93 (2018) arXiv180109790 [nucl-th]
[18] Y Prok et al (CLAS) Moments of the spin structure functions gp1 and gd1 for 005 lt Q2 lt 30
GeV2 Phys Lett B 672 12 (2009) arXiv08022232 [nucl-ex]
[19] H Dutz et al (GDH) First measurement of the Gerasimov-Drell-Hearn sum rule for 1H from
07 GeV to 18 GeV at ELSA Phys Rev Lett 91 192001 (2003)
[20] M Amarian et al (Jefferson Lab E94010) Measurement of the generalized forward spin po-
larizabilities of the neutron Phys Rev Lett 93 152301 (2004) arXivnucl-ex0406005
[21] M Amarian et al The Q2 evolution of the generalized Gerasimov-Drell-Hearn integral for
the neutron using a 3He target Phys Rev Lett 89 242301 (2002) arXivnucl-ex0205020
[22] M Amarian et al (Jefferson Lab E94-010)Q2 evolution of the neutron spin structure moments
using a 3He target Phys Rev Lett 92 022301 (2004) arXivhep-ex0310003
[23] A Deur et al Experimental determination of the evolution of the Bjorken integral at low Q2
Phys Rev Lett 93 212001 (2004) arXivhep-ex0407007
[24] K Slifer Low Q2 measurement of gp2 and the δLT spin polarizability Spin structure at long dis-
tance Proceedings Workshop Newport News USA March 12ndash13 2009 nucl-ex09064775
(2009) AIP Conf Proc 1155 10106313203293 (2009) arXiv09064775 [nucl-ex]
[25] P Solvignon et al (E01-012) Moments of the neutron g2 structure function at intermediate
42
Q2 Phys Rev C 92 015208 (2015) arXiv13044497 [nucl-ex]
[26] A Deur Experimental studies at low Q2 of the spin structure of the nucleon at Jefferson Lab
in 9th International Workshop on Chiral Dynamics (CD18) Durham NC USA September
17-21 2018 (2019) arXiv190305661 [nucl-ex]
[27] V Sulkosky et al (Jefferson Lab E97-110) Measurement of the 3He spin-structure functions
and of neutron (3He) spin-dependent sum rules at 0035 le Q2 le 024 GeV2 Phys Lett B
805 135428 (2020) arXiv190805709 [nucl-ex]
[28] K Adhikari et al (CLAS) Measurement of the Q2 Dependence of the Deuteron Spin Structure
Function g1 and its Moments at Low Q2 with CLAS Phys Rev Lett 120 062501 (2018)
arXiv171101974 [nucl-ex]
[29] V Lensky F Hagelstein A Hiller Blin and V Pascalutsa Comment on rdquoMeasurement of
the Q2 Dependence of the Deuteron Spin Structure Function g1 and its Moments at Low Q2
with CLASrdquo (2018) arXiv180603219 [nucl-th]
[30] J M Alarcn F Hagelstein V Lensky and V Pascalutsa Forward doubly-virtual Compton
scattering off the nucleon in chiral perturbation theory at NLO the subtraction function and
moments of unpolarized structure functions (2020) arXiv200509518 [hep-ph]
[31] V Bernard E Epelbaum H Krebs and U-G Meiszligner New insights into the spin structure
of the nucleon Phys Rev D 87 054032 (2013) arXiv12092523 [hep-ph]
[32] V Lensky J M Alarcon and V Pascalutsa Moments of nucleon structure functions at
next-to-leading order in baryon chiral perturbation theory Phys Rev C 90 055202 (2014)
arXiv14072574 [hep-ph]
[33] V Pascalutsa and M Vanderhaeghen Chiral effective-field theory in the ∆(1232) region
I Pion electroproduction on the nucleon Phys Rev D 73 034003 (2006) arXivhep-
ph0512244
[34] V Pascalutsa and M Vanderhaeghen Electromagnetic nucleon-to-Delta transition in chiral
effective field theory Phys Rev Lett 95 232001 (2005) arXivhep-ph0508060
[35] T R Hemmert B R Holstein and J Kambor Systematic 1M expansion for spin 32
particles in baryon chiral perturbation theory Phys Lett B 395 89 (1997) arXivhep-
ph9606456
[36] V Pascalutsa and D R Phillips Effective theory of the ∆(1232) in Compton scattering off
the nucleon Phys Rev C 67 055202 (2003) arXivnucl-th0212024
[37] V Pascalutsa M Vanderhaeghen and S N Yang Electromagnetic excitation of the ∆(1232)-
resonance Phys Rept 437 125 (2007) arXivhep-ph0609004
[38] K A Olive et al (Particle Data Group) Review of Particle Physics Chin Phys C 38 090001
43
(2014)
[39] F E Low Scattering of light of very low frequency by systems of spin 12 Phys Rev 96
1428 (1954)
[40] M Gell-Mann and M L Goldberger Scattering of low-energy photons by particles of spin
12 Phys Rev 96 1433 (1954)
[41] A M Harun ar-Rashid A simple derivation of Schwingerrsquos sum rule for spin dependent
structure functions Nuovo Cim A 33 447 (1976)
[42] F Hagelstein and V Pascalutsa Dissecting the hadronic contributions to (g minus 2)micro by
Schwingerrsquos sum rule Phys Rev Lett 120 072002 (2018) arXiv171004571 [hep-ph]
[43] V Lensky V Pascalutsa M Vanderhaeghen and C Kao Spin-dependent sum rules con-
necting real and virtual Compton scattering verified Phys Rev D 95 074001 (2017)
arXiv170101947 [hep-ph]
[44] H Burkhardt and W N Cottingham Sum rules for forward virtual Compton scattering
Annals Phys 56 453 (1970)
[45] V Lensky and V Pascalutsa Predictive powers of chiral perturbation theory in Compton
scattering off protons Eur Phys J C 65 195 (2010) arXiv09070451 [hep-ph]
[46] H Grieszlighammer J McGovern D Phillips and G Feldman Using effective field theory to
analyse low-energy Compton scattering data from protons and light nuclei Prog Part Nucl
Phys 67 841 (2012) arXiv12036834 [nucl-th]
[47] H W Grieszlighammer J A McGovern and D R Phillips Nucleon polarisabilities at and
beyond physical pion masses Eur Phys J A 52 139 (2016) arXiv151101952 [nucl-th]
[48] E Epelbaum H Krebs and U-G Meiszligner Improved chiral nucleon-nucleon potential up to
next-to-next-to-next-to-leading order Eur Phys J A 51 53 (2015) arXiv14120142 [nucl-
th]
[49] E Epelbaum H Krebs and U-G Meiszligner Precision nucleon-nucleon potential at fifth order
in the chiral expansion Phys Rev Lett 115 122301 (2015) arXiv14124623 [nucl-th]
[50] C W Kao T Spitzenberg and M Vanderhaeghen Burkhardt-Cottingham sum rule and
forward spin polarizabilities in heavy baryon chiral perturbation theory Phys Rev D 67
016001 (2003) arXivhep-ph0209241
[51] C-W Kao D Drechsel S Kamalov and M Vanderhaeghen Higher moments of nucleon
spin structure functions in heavy baryon chiral perturbation theory and in a resonance model
Phys Rev D 69 056004 (2004) arXivhep-ph0312102
[52] T Becher and H Leutwyler Baryon chiral perturbation theory in manifestly Lorentz invariant
form Eur Phys J C 9 643 (1999) arXivhep-ph9901384
44
[53] J Gasser M E Sainio and A Svarc Nucleons with chiral loops Nucl Phys B 307 779
(1988)
[54] L S Geng J Martin Camalich L Alvarez-Ruso and M J Vicente Vacas Leading SU(3)-
breaking corrections to the baryon magnetic moments in chiral perturbation theory Phys
Rev Lett 101 222002 (2008) arXiv08051419 [hep-ph]
[55] D Drechsel S S Kamalov and L Tiator The GDH sum rule and related integrals Phys
Rev D 63 114010 (2001) arXivhep-ph0008306
[56] D Drechsel O Hanstein S S Kamalov and L Tiator A unitary isobar model for pion
photo- and electroproduction on the proton up to 1 GeV Nucl Phys A 645 145 (1999)
arXivnucl-th9807001
[57] L Tiator private communication (2020)
[58] V Bernard T R Hemmert and U-G Meiszligner Spin structure of the nucleon at low energies
Phys Rev D 67 076008 (2003) arXivhep-ph0212033
[59] O Gryniuk F Hagelstein and V Pascalutsa Evaluation of the forward Compton scattering
off protons II Spin-dependent amplitude and observables Phys Rev D 94 034043 (2016)
arXiv160400789 [nucl-th]
[60] R Zielinski The g2p Experiment A Measurement of the Protonrsquos Spin Structure Functions
PhD thesis New Hampshire U (2010) arXiv170808297 [nucl-ex]
[61] N Guler et al (CLAS) Precise determination of the deuteron spin structure at low to mod-
erate Q2 with CLAS and extraction of the neutron contribution Phys Rev C 92 055201
(2015) arXiv150507877 [nucl-ex]
[62] C E Carlson V Nazaryan and K Griffioen Proton structure corrections to electronic
and muonic hydrogen hyperfine splitting Phys Rev A 78 022517 (2008) arXiv08052603
[physicsatom-ph]
[63] V Bernard N Kaiser and U-G Meiszligner Chiral dynamics in nucleons and nuclei Int J
Mod Phys E 4 193 (1995) arXivhep-ph9501384
[64] T R Hemmert B R Holstein and J Kambor ∆(1232) and the polarizabilities of the nucleon
Phys Rev D 55 5598 (1997) arXivhep-ph9612374
[65] V Pascalutsa and R Timmermans Field theory of nucleon to higher-spin baryon transitions
Phys Rev C 60 042201 (1999) arXivnucl-th9905065
[66] V Pascalutsa Quantization of an interacting spin-32 field and the Delta isobar Phys Rev
D 58 096002 (1998) arXivhep-ph9802288
[67] H Krebs Double Virtual Compton Scattering and SpinStructure of the Nucleon PoS
CD2018 031 (2019)
45
[68] N Kochelev and Y Oh Axial anomaly and the δLT puzzle Phys Rev D 016012 (2012)
arXiv11034892 [hep-ph]
[69] D Drechsel S Kamalov and L Tiator Unitary isobar model ndash MAID2007 Eur Phys J A
69 (2007) available at httpsmaidkphuni-mainzde arXiv07100306 [nucl-th]
[70] P J Mohr B N Taylor and D B Newell CODATA recommended values of the fundamental
physical constants 2010 Rev Mod Phys 84 1527 (2012)
[71] J Ahrens et al (GDH A2) First measurement of the Gerasimov-Drell-Hearn integral for 1H
from 200 to 800 MeV Phys Rev Lett 87 022003 (2001) arXivhep-ex0105089 [hep-ex]
[72] K Helbing (GDH) Experimental verification of the GDH sum rule at ELSA and MAMI Nucl
Phys Proc Suppl 105 113 (2002)
[73] R Bradford A Bodek H S Budd and J Arrington A New parameterization of the nucleon
elastic form-factors NuInt05 proceedings of the 4th International Workshop on Neutrino-
Nucleus Interactions in the Few-GeV Region Okayama Japan 26-29 September 2005 Nucl
Phys Proc Suppl 159 127 (2006) arXivhep-ex0602017
[74] J D Bjorken Applications of the chiral U(6)timesU(6) algebra of current densities Phys Rev
148 1467 (1966)
[75] J D Bjorken Inelastic scattering of polarized leptons from polarized nucleons Phys Rev D
1 1376 (1970)
[76] V Bernard T R Hemmert and U-G Meiszligner Novel analysis of chiral loop effects in
the generalized Gerasimov-Drell-Hearn sum rule Phys Lett B 545 105 (2002) arXivhep-
ph0203167
[77] A Deur et al Experimental study of isovector spin sum rules Phys Rev D 78 032001
(2008) arXiv08023198 [nucl-ex]
[78] B Pasquini P Pedroni and D Drechsel Higher order forward spin polarizability Phys Lett
B 687 160 (2010) arXiv10014230 [hep-ph]
[79] R Jaffe g2ndashThe nucleonrsquos other spin-dependent structure function Comments Nucl Part
Phys 19 239 (1990)
[80] E V Shuryak and A Vainshtein Theory of power corrections to deep inelastic scattering in
quantum chromodynamics (II) Qminus4 effects polarized target Nucl Phys B 201 141 (1982)
[81] S Wandzura and F Wilczek Sum rules for spin dependent electroproduction Test of rela-
tivistic constituent quarks Phys Lett B 72 195 (1977)
[82] B W Filippone and X-D Ji The spin structure of the nucleon Adv Nucl Phys 26 1
(2001) arXivhep-ph0101224
[83] M Burkardt The g2 structure function Proceedings Workshop on Spin structure at long
46
distance Newport News USA March 12-13 2009 AIP Conf Proc 1155 26 (2009)
arXiv09054079 [hep-ph]
[84] A Deur S J Brodsky and G F de Teramond The spin structure of the nucleon Rept
Prog Phys 82 076201 (2019) arXiv180705250 [hep-ph]
[85] B R Holstein D Drechsel B Pasquini and M Vanderhaeghen Higher order polarizabilities
of the proton Phys Rev C 61 034316 (2000) arXivhep-ph9910427
[86] B R Holstein V Pascalutsa and M Vanderhaeghen Sum rules for magnetic moments and
polarizabilities in QED and chiral effective-field theory Phys Rev D 72 094014 (2005)
arXivhep-ph0507016
[87] B Pasquini D Drechsel and L Tiator Invariant amplitudes for pion electroproduction Eur
Phys J A 34 387 (2007) arXiv07122327 [hep-ph]
[88] A J Buchmann Non-spherical proton shape and hydrogen hyperfine splitting Proceedings
International Workshop on Precision Physics of Simple Atomic Systems (PSAS 2008) Wind-
sor Ontario Canada July 21-26 2008 Can J Phys 87 773 (2009) arXiv09104747
[physicsatom-ph]
47
is conserved A detailed derivation of the connection between this response function and
the photoabsorption cross sections can be found in App B The forward VVCS amplitudes
satisfy dispersion relations derived from the general principles of analyticity and causality1
S1(νQ2) =16παMN
Q2
int 1
0
dxg1(xQ2)
1minus x2(ννel)2 minus i0+(5a)
=2MN
π
int infinνel
dν primeν prime 2[QνprimeσLT + σTT
](ν prime Q2)radic
ν prime 2 +Q2(ν prime 2 minus ν2 minus i0+)
νS2(νQ2) =16παM2
N
Q2
int 1
0
dxg2(xQ2)
1minus x2(ννel)2 minus i0+(5b)
=2M2
N
π
int infinνel
dν primeν prime 2[νprime
QσLT minus σTT
](ν prime Q2)radic
ν prime 2 +Q2(ν prime 2 minus ν2 minus i0+)
with νel = Q22MN the elastic threshold
B Low-energy expansions and relations to polarizabilities
The VVCS amplitudes naturally split into nucleon-pole (Spolei ) and non-pole (Snonpole
i )
parts or Born (SBorni ) and non-Born (Si) parts
Si = Spolei + Snonpole
i = SBorni + Si (6)
The Born amplitudes are given uniquely in terms of the nucleon form factors [1]
SBorn1 (νQ2) =
2πα
M
Q2GM(Q2)F1(Q2)
ν2el minus ν2
minus F 22 (Q2)
(7a)
SBorn2 (νQ2) = minus 2παν
ν2el minus ν2
GM(Q2)F2(Q2) (7b)
The same is true for the nucleon-pole amplitudes which are related to the Born amplitudes
in the following way
Spole1 (νQ2) = SBorn
1 (νQ2) +2πα
MN
F 22 (Q2) (8a)
Spole2 (νQ2) = SBorn
2 (νQ2) (8b)
1 The dispersion relation for νS2(νQ2) is used because it is pole-free in the limit Q2 rarr 0 and then ν rarr 0
cf Eq (7b)
6
Here we used the elastic Dirac and Pauli form factors F1(Q2) and F2(Q2) related to the
electric and magnetic Sachs form factors GE(Q2) and GM(Q2) through
F1(Q2) =GE(Q2) + τGM(Q2)
1 + τ (9a)
F2(Q2) =GM(Q2)minusGE(Q2)
1 + τ (9b)
where τ = Q24M2N
A low-energy expansion (LEX) of Eq (5) in combination with the unitarity relations
given in Eq (3) establishes various sum rules relating the nucleon properties (electromag-
netic moments polarizabilities) to experimentally observable response functions [1 3] The
leading terms in the LEX of the RCS amplitudes are determined uniquely by charge mass
and anomalous magnetic moment as the global properties of the nucleon These lowest-
order terms represent the celebrated low-energy theorem (LET) of Low Gell-Mann and
Goldberger [39 40] The polarizabilities related to the internal structure of the nucleon
enter the LEX at higher orders They make up the non-Born amplitudes and can be related
to moments of inelastic structure functions
The process of VVCS can be realized experimentally in electron-nucleon scattering where
a virtual photon is exchanged between the electron and the nucleon This virtual photon
acts as a probe whose resolution depends on its virtuality Q2 In this way one can access
the so-called generalized polarizabilities which extend the notion of polarizabilities to the
case of response to finite momentum transfer The generalized forward spin polarizability
γ0(Q2) and the longitudinal-transverse polarizability δLT (Q2) are most naturally defined via
the LEX of the non-Born part of the lab-frame VVCS amplitudes [1]
1
4πgnonpoleTT (νQ2) =
2α
M2N
IA(Q2) ν + γ0(Q2)ν3 + γ0(Q2)ν5 + (10a)
1
4πgnonpoleLT (νQ2) =
2α
M2N
I3(Q2)Q+ δLT (Q2)ν2Q+ (10b)
Their definitions in terms of integrals over structure functions are postponed to Eqs (19)
and (22) Here we only give the definition of the moment I3(Q2)
I3(Q2) =M2
N
4π2α
int infinν0
dνK(νQ2)
νQσLT (νQ2) =
2M2N
Q2
int x0
0
dx[g1(xQ2) + g2(xQ2)
] (11)
which is related to the Schwinger sum rule in the real photon limit [7ndash9 41 42] The LEX
of the non-pole part of the covariant VVCS amplitudes can be described entirely in terms
7
of moments of inelastic spin structure functions (up to O(ν4) [43])
1
4πSnonpole
1 (νQ2) =2α
MN
I1(Q2) +
2α
MNQ2
[IA(Q2)minus I1(Q2)
]+MNδLT (Q2)
ν2 (12a)
1
4πνSnonpole
2 (νQ2) = 2αI2(Q2) +2α
Q2
[I1(Q2)minus IA(Q2)
]ν2 (12b)
I1(Q2) and IA(Q2) are generalizations of the famous GerasimovndashDrellndashHearn (GDH) sum
rule [5 6] from RCS to the case of virtual photons [1] Their definitions are given in Eqs (26)
and (32) I2(Q2) is the well-known Burkhardt-Cottingham (BC) sum rule [44]
I2(Q2) equiv 2M2N
Q2
int x0
0
dx g2(x Q2) =1
4F2(Q2)GM(Q2) (13)
which can be written as a ldquosuperconvergence sum rulerdquo
Q2
16παM2N
limνrarr0
νS2(νQ2) =
int 1
0
dx g2 (x Q2) = 0 (14)
The latter is valid for any value of Q2 provided that the integral converges for x rarr 0
Combining Eq (5) with the above LEXs of the VVCS amplitudes we can relate IA(Q2)
I1(Q2) γ0(Q2) and δLT (Q2) to moments of inelastic structure functions see Sec III It is
important to note that only γ0(Q2) and δLT (Q2) are generalized polarizabilities The relation
of the inelastic moments IA(Q2) and I1(Q2) to polarizabilities will be discussed in details in
Secs III C and III D The difference between S1(νQ2) and Snonpole1 (νQ2) cf Eq (8a) will
be important in this context
C Details on χPT calculation and uncertainty estimate
In this work we calculated the NLO prediction of BχPT for the polarized non-Born
VVCS amplitudes This includes the leading pion-nucleon (πN) loops see Ref [32 Fig 1]
as well as the subleading tree-level Delta-exchange (∆-exchange) see Ref [30 Fig 2] and
the pion-Delta (π∆) loops see Ref [32 Fig 2] In the δ-power-counting scheme [36] the
LO and NLO non-Born VVCS amplitudes and polarizabilities are of O(p3) and O(p4∆)
respectively2 The LECs are listed in Table I sorted by the order at which they appear in
our calculation At the given orders there are no ldquonewrdquo LECs that would need to be fitted
from Compton processes For more details on the BχPT formalism we refer to Ref [30]
2 In the full Compton amplitude there is a lower order contribution coming from the Born terms leading
to a shift in nomenclature by one order the LO contribution referred to as the NLO contribution etc
see eg Ref [45]
8
where power counting predictive orders (Sec III A) and the renormalization procedure
(Sec III B) are discussed
A few remarks are in order for the inclusion of the ∆(1232) and the tree-level ∆-exchange
contribution In contrast to Ref [32] we include the Coulomb-quadrupole (C2) N rarr ∆
transition described by the LEC gC The relevant Lagrangian describing the non-minimal
γlowastN∆ coupling [33 34] (note that in these references the overall sign of gC is inconsistent
between the Lagrangian and Feynman rules) reads
L(2)∆ =
3e
2MNM+
N T3
igM F
microν partmicro∆ν minus gEγ5Fmicroν partmicro∆ν (15)
+igCM∆
γ5γα(partα∆ν minus partν∆α)partmicroF
microν
+ Hc
with M+ = MN + M∆ and the dual of the electromagnetic field strength tensor F microν =12εmicroνρλFρλ Even though the Coulomb coupling is subleading compared with the electric
and magnetic couplings (gE and gM) its relatively large magnitude cf Table I makes it
numerically important for instance in γ0(Q2) Furthermore we study the effect of modifying
the magnetic coupling using a dipole form factor
gM rarrgM[
1 + (QΛ)2 ]2 (16)
where Λ2 = 071 GeV2 The inclusion of this Q2 dependence mimics the form expected from
vector-meson dominance It is motivated by observing the importance of this form factor
for the correct description of the electroproduction data [33]
To estimate the uncertainties of our NLO predictions we define
δ(Q2) =
radic(∆
MN
)2
+
(Q2
2MN∆
)2
(17)
such that the neglected next-to-next-to-leading order terms are expected to be of relative
size δ2 [33] The uncertainties in the values of the parameters in Table I have a much
smaller impact compared to the truncation uncertainty and can be neglected Unfortunately
∆IA(Q2) γ0(Q2) and γ0(Q2) ie the sum rules involving the cross section σTT (νQ2)
as well as the polarizability ∆I1(Q2) turn out to be numerically small Their smallness
suggests a cancellation of leading orders (which can indeed be confirmed by looking at
separate contributions as shown below) Therefore an error of δ2(Q2)P (Q2) where P (Q2)
is a generalized polarizability might underestimate the theoretical uncertainty for some of
the NLO predictions To avoid this we estimate the uncertainty of our NLO polarizability
9
predictions by
σP (Q2) =
Max[δ4(0)P (0)2 δ4(0)P LO(0)2 δ2(0)PNLO(0)2
]+Max
[δ4(Q2)
[P (Q2)minus P (0)
]2 δ4(Q2)
[P LO(Q2)minus P LO(0)
]2
δ2(Q2)[PNLO(Q2)minus PNLO(0)
]2]12
(18)
where P LO(Q2) is the πN -loop contribution PNLO(Q2) are the ∆-exchange and π∆-loop
contributions and P (Q2) = P LO(Q2) + PNLO(Q2) This error prescription is similar to the
one used in eg Refs [46ndash49] Here since we are interested in the generalized polarizabil-
ities we added in quadrature the error due to the static piece P (0) and the Q2-dependent
remainder P (Q2) minus P (0) Note that the static values of IA(0) and I1(0) are given by the
elastic Pauli form factor which is not part of our BχPT prediction and is considered to be
exact
Note that our result for the spin polarizabilities (and the unpolarized moments [30]) are
NLO predictions only at low momentum transfers Q mπ At larger values of Q amp ∆ they
become incomplete LO predictions Indeed in this regime the ∆ propagators do not carry
additional suppression compared to the nucleon propagators and the π∆ loops are promoted
to LO In general we only expect a rather small contribution from omitted π∆ loops to the
Q2 dependence of the polarizabilities since π∆ loops show rather weak dependence on Q2
compared with the ∆ exchange or πN loops Nevertheless this issue has to be reflected in
the error estimate Since the static polarizabilities P (0) are not affected it is natural to
separate the error on the Q2-dependent remainder P (Q2) minus P (0) as done in Eq (18) To
accommodate for the potential loss of precision above Q amp ∆ we define the relative error
δ(Q2) as growing with increasing Q2 see Eq (17)
Upon expanding our results in powers of the inverse nucleon mass Mminus1N we are able to
reproduce existing results of heavy-baryon χPT (HBχPT) at LO We however do not see
a rationale to drop the higher-order Mminus1N terms when they are not negligible (ie when
their actual size exceeds by far the natural estimate for the size of higher-order terms)
Comparing our BχPT predictions to HBχPT we will also see a deficiency of HBχPT in
the description of the Q2 behaviour of the polarizabilities Note that the O(p4) HBχPT
results from Ref [50 51] which we use here for comparison do not include the ∆ These
references studied the leading effect of the latter in the HBχPT framework using the small-
scale expansion [35] observing no qualitative improvement in the HBχPT description of the
empirical data [50 51] when including it We therefore choose to use the O(p4) results as
the representative HBχPT curves
10
Another approach used in the literature to calculate the polarizabilities in χPT is the
infrared regularization (IR) scheme introduced in Ref [52] This covariant approach tries to
solve the power counting violation observed in Ref [53] by dropping the regular parts of the
loop integrals that contain the power-counting-breaking terms However this subtraction
scheme modifies the analytic structure of the loop contributions and may lead to unexpected
problems as was shown in Ref [54] As we will see in the next section the IR approach
also fails to describe the Q2 behaviour of the polarizabilities
III RESULTS AND DISCUSSION
We now present the NLO BχPT predictions for the nucleon polarizabilities and selected
moments of the nucleon spin structure functions Our results are obtained from the calcu-
lated non-Born VVCS amplitudes and the LEXs in Eqs (10) and (12) For a cross-check
we used the photoabsorption cross sections described in App B In addition to the full NLO
results we also analyse the individual contributions from the πN loops the ∆ exchange
and the π∆ loops
A γ0(Q2) mdash generalized forward spin polarizability
The forward spin polarizability
γ0(Q2) =1
2π2
int infinν0
dν
radic1 +
Q2
ν2
σTT (νQ2)
ν3(19)
=16αM2
N
Q6
int x0
0
dx x2
[g1(xQ2)minus 4M2
Nx2
Q2g2(xQ2)
]
provides information about the spin-dependent response of the nucleon to transversal pho-
ton probes The RCS analogue of the above generalized forward spin polarizability sum
rule is sometimes referred to as the Gell-Mann Goldberger and Thirring (GGT) sum rule
[11] At Q2 = 0 the forward spin polarizability is expressed through the lowest-order spin
polarizabilities of RCS as γ0 = minus(γE1E1 + γM1M1 + γE1M2 + γM1E2) The forward spin po-
larizability of the proton is relevant for an accurate knowledge of the (muonic-)hydrogen
hyperfine splitting as it controls the leading proton-polarizability correction [16 62]
The πN -loop ∆-exchange and π∆-loop contributions to the NLO BχPT prediction of
11
000 005 010 015 020 025 030
-4
-3
-2
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
4D
proton
000 005 010 015 020 025 030
-4
-3
-2
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
4D
neutron
000 005 010 015 020 025 030
0
1
2
Q2 GeV2D
∆LT
10-
4fm
4D
proton
000 005 010 015 020 025 030
0
1
2
3
Q2 GeV2D
∆LT
10-
4fm
4D
neutron
FIG 2 Upper panel Generalized forward spin polarizability for the proton (left) and neutron
(right) as function of Q2 The result of this work the NLO BχPT prediction is shown by the
blue solid line and the blue band The red line represents the LO BχPT result The purple short-
dashed line is the O(p4) HB result from Ref [50] note that the corresponding proton curve is
outside of the plotted range The black dotted line is the MAID model prediction [55ndash57] which
is taken from Ref [1] (proton) and Ref [20] (neutron) The pink band is the IR+∆ result from
Ref [58] and the gray band is the BχPT+∆ result from Ref [31] Empirical extractions for the
proton Ref [18] (blue dots) Ref [59] (purple square) and Ref [60] (orange triangle uncertainties
added in quadrature) and neutron Ref [20] (blue diamonds) and Ref [61] (green dots statistical
and systematic uncertainties added in quadrature) Lower Panel Longitudinal-transverse spin
polarizability for the proton (left) and neutron (right) The orange dot-dashed and purple short-
dashed lines are the O(p3) and O(p4) HB results from Ref [50] The pink band is the IR result
from Ref [58] and the gray band is the covariant BχPT+∆ result from Ref [31] The black dotted
line is the MAID model prediction [55ndash57] note that for the proton we use the updated estimate
from Ref [1] obtained using the π η ππ channels
12
the static forward spin polarizability amount to in units of 10minus4 fm4
γ0p = minus093(92) asymp 201minus 284minus 010 (20a)
γ0n = 003(92) asymp 298minus 284minus 010 (20b)
while the slope is composed as follows in units of 10minus4 fm6
dγ0p(Q2)
dQ2
∣∣∣∣Q2=0
= minus022(4) asymp minus033 + 011 + 001 (21a)
dγ0n(Q2)
dQ2
∣∣∣∣Q2=0
= minus061(7) asymp minus073 + 011 + 001 (21b)
Figure 2 upper panel shows our NLO prediction as well as the LO πN loops compared
to different experimental and theoretical results For the proton we have one determination
at the real-photon point by the GDH collaboration [19] γ0p = minus100(8)(12)times10minus4 fm4 and
further Jefferson Laboratory data [18 60] at very low Q2 For the neutron only data at finite
Q2 are available [20 61] The experimental data for the proton are fairly well reproduced
in the whole Q2 range considered here while for the neutron the agreement improves with
increasing Q2 The HB limit of our πN -loop contribution reproduces the results published
in Refs [50 63] for arbitrary Q2 In addition our prediction is compared to the MAID
model [1 20] the IR+∆ calculation of Ref [58] and the BχPT+∆ result of Ref [31]
The πN -production channel gives a positive contribution to the photoabsorption cross
section σTT (νQ2) at low Q2 cf Fig 10 Accordingly one observes that the πN loops give
a sizeable positive contribution to γ0(Q2) The Delta on the other hand has a very large
effect by cancelling the πN loops and bringing the result close to the empirical data From
Fig 3 upper panel one can see that it is the ∆ exchange which dominates while π∆
loops are negligible This was expected since the forward spin polarizability sum rule is an
integral over the helicity-difference cross section in which σ32 is governed by the Delta at
low energies (the relevant energy region for the sum rule)
To elucidate the difference between the present calculation and the one from Ref [31] we
note that the two calculations differ in the following important aspects Firstly Ref [31] uses
the small-scale counting [64] that considers ∆ and mπ as being of the same size ∆ sim mπ
In practice this results in a set of π∆-loop graphs which contains graphs with one or two
γ∆∆ couplings and hence two or three Delta propagators Such graphs are suppressed in
the δ-counting and thus omitted from our calculation while present in that of Ref [31]
Secondly the Lagrangians describing the interaction of the Delta are constructed differently
and assume slightly different values for the coupling constants In particular we employ
13
000 005 010 015 020 025 030
-3
-2
-1
0
1
2
Q2GeV
2D
Γ01
0-
4fm
4D
proton
000 005 010 015 020 025 030
-3
-2
-1
0
1
2
3
Q2GeV
2D
Γ01
0-
4fm
4D
neutron
000 005 010 015 020 025 030
00
05
10
15
Q2GeV
2D
∆LT1
0-
4fm
4D
proton
000 005 010 015 020 025 030
00
05
10
15
20
25
Q2GeV
2D
∆LT1
0-
4fm
4D
neutron
FIG 3 Contributions of the different orders to the chiral predictions of γ0(Q2) upper paneland δLT (Q2) lower panel for the proton (left) and neutron (right) Red solid line πN -loop
contribution green dot-dashed line ∆-exchange contribution orange dotted line π∆-loop con-
tribution blue long-dashed line total result purple dot-dot-dashed line total result without gCcontribution black short-dashed line total result without gM dipole
(where possible) the so-called ldquoconsistentrdquo couplings to the Delta field ie those couplings
that project out the spurious degree of freedom see Refs [37 65 66] The authors of
Ref [31] on the other hand use couplings where the consistency in this sense is not enforced
The effects of these differences are of higher order in the δ-counting expansion and their
contribution to the Q2 dependence of the considered polarizabilities is expected to be rather
small however the differences at Q2 = 0 could be noticeable [67]
Finally as mentioned in Sec II C the inclusion of the dipole form factor in the magnetic
coupling gM is expected to be important to generate the correct Q2 behaviour of the polar-
izabilities Comparing our predictions for the forward spin polarizability with and without
inclusion of the form factor see Fig 3 upper panel confirms this Without the dipole
our results for the proton and neutron are closer to the ones from Ref [31] where the form
factor is not included For the neutron our prediction without the dipole is able to describe
14
the experimental points at very low Q2 deviating from the data with increasing Q2 The
π∆-loop contribution does not modify the Q2 behavior of γ0(Q2) and only differs from
Ref [31] by a small global shift Note also the relatively large effect of gC which generates
a sign change for virtualities above sim 02 GeV2 see Fig 3 upper panel
B δLT (Q2) mdash longitudinal-transverse polarizability
The longitudinal-transverse spin polarizability
δLT (Q2) =1
2π2
int infinν0
dν
radic1 +
Q2
ν2
σLT (νQ2)
Qν2(22)
=16αM2
N
Q6
int x0
0
dx x2[g1(xQ2) + g2(xQ2)
]
contains information about the spin structure of the nucleon and is another important
input in the determination of the (muonic-)hydrogen hyperfine splitting [16 62] It is also
relevant in studies of higher-twist corrections to the structure function g2(xQ2) given by
the moment d2(Q2) [51] see Section III E The peculiarity of the response encoded in this
polarizability is that it involves a spin flip of the nucleon and a polarization change of the
photon see App B and Fig 11
It is expected that the Delta isobar gives only a small contribution to δLT (Q2) what makes
this polarizability a potentially clean test case for chiral calculations Consequently there are
relatively many different theoretical calculations of δLT (Q2) coming from different versions
of χPT with baryons (HB IR and covariant) Ref [50] found a systematic deviation of the
HB result for δLTn(Q2) from the MAID model prediction This disagreement was identified
by the authors of Ref [68] as a puzzle involving the neutron δLT polarizabilitymdashthe δLT
puzzle The IR calculation in Ref [58] also showed a deviation from the data and predicted
a rapid rise of δLT (Q2) with growing Q2 The problem is solved by keeping the relativistic
structure of the theory as the BχPT+∆ result of Ref [31] showed
As expected already the leading πN loops provide a reasonable agreement with the
experimental data cf Fig 2 lower panel Since the ∆-exchange contribution to δLT (Q2)
is small the effect of the gM form factor is negligible in this polarizability as is that of the gC
coupling cf Fig 3 lower panel In fact we predict both the ∆-exchange and the π∆-loop
contributions to be small and negative This is in agreement with the MAID model which
predicts a small and negative contribution of the P33 wave to δLT (Q2) However in the
calculation of Ref [31] which is different from the one presented here only in the way the
∆(1232) is included the contribution of this resonance to δLTp(Q2) is sizeable and positive
15
The authors of that work attributed this large contribution to diagrams where the photons
couple directly to the Delta inside a loop As mentioned in Sec III F the effect of such loop
diagrams does not change the Q2 behaviour of the polarizabilities On the other hand it
can produce a substantial shift of the δLT (Q2) as a whole A higher-order calculation should
resolve the discrepancy between the two covariant approaches however it will partially lose
the predictive power since the LECs appearing at higher orders will have to be fitted to
experimental data
The πN -loop ∆-exchange and π∆-loop contributions to the NLO BχPT prediction of
the static longitudinal-transverse polarizability are in units of 10minus4 fm4
δLTp = 132(15) asymp 150minus 016minus 002 (23a)
δLTn = 218(23) asymp 235minus 016minus 002 (23b)
while the slopes are in units of 10minus4 fm6
dδLTp(Q2)
dQ2
∣∣∣∣Q2=0
= minus085(8) asymp minus080minus 004minus 001 (24a)
dδLTn(Q2)
dQ2
∣∣∣∣Q2=0
= minus124(12) asymp minus119minus 004minus 001 (24b)
C IA(Q2) mdash a generalized GDH integral
The helicity-difference cross section σTT exhibits a faster fall-off in ν than its spin-averaged
counterpart σT This is due to a cancellation between the leading (constant) terms of σ12
and σ32 at large ν3 The resulting 1ν fall-off of the helicity-difference cross section allows
one to write an unsubtracted dispersion relation for the VVCS amplitude gnonpoleTT (νQ2) cf
Eq (10a) This is the origin of the GDH sum rule [5 6]
minus α
2M2N
κ2 =1
2π2
int infinν0
dνσTT (ν)
ν (25)
which establishes a relation to the anomalous magnetic moment κ It is experimentally
verified for the nucleon by MAMI (Mainz) and ELSA (Bonn) [71 72]
There are two extensions of the GDH sum rule to finite Q2 the generalized GDH integrals
3 Notice that a constant term in σTT at ν rarrinfin is forbidden by crossing symmetry
16
000 005 010 015 020 025 030
-30
-25
-20
-15
-10
-05
00
Q2 GeV2D
I Aproton
000 005 010 015 020 025 030
-25
-20
-15
-10
-05
00
Q2 GeV2D
I A
neutron
000 005 010 015 020 025 030
-05
00
05
Q2 GeV2D
I 1
proton
000 005 010 015 020 025 030
-10
-08
-06
-04
-02
00
02
Q2 GeV2D
I 1
neutron
FIG 4 Upper panel The generalized GDH integral IA(Q2) for the proton (left) and neutron
(right) as function of Q2 The result of this work the NLO BχPT prediction is shown by the blue
solid line and the blue band The red line represents the LO BχPT result The purple short-dashed
line is the O(p4) HB result from Ref [50 51] The gray band is the BχPT+∆ result from Ref [31]
The black dotted line is the MAID model prediction [69] Experimental extractions for the proton
Ref [60] (orange triangle uncertainties added in quadrature) and neutron Refs [21][27] where
magenta dotsorange diagonal crosses correspond to data and red squareslilac crosses correspond
to data plus extrapolation to unmeasured energy regions The green stars at the real-photon point
are derived from the anomalous magnetic moments κp asymp 1793 and κn asymp minus1913 [70] Lower
panel The generalized GDH integral I1(Q2) for the proton (left) and neutron (right) as function
of Q2 The purple short-dashed line is the HB result from Ref [51] Experimental extractions for
the proton Ref [18] (blue dots) and Ref [60] (orange triangle uncertainties added in quadrature)
and neutron Ref [61][27] (uncertainties added in quadrature) where green dotsorange diagonal
crosses correspond to data and gray squareslilac crosses correspond to data plus extrapolation to
unmeasured energy regions
17
IA(Q2) and I1(Q2) The latter will be discussed in Sec III D The former is defined as4
minus α
2M2N
IA(Q2) = minus 1
8π2
int infinν0
dν
radic1 +
Q2
ν2
σTT (νQ2)
ν(26)
=α
Q2
int x0
0
dx[4M2
Nx2
Q2g2(xQ2)minus g1(xQ2)
]
Due to its energy weighting the integral in Eq (26) converges slower than the one in the
generalized forward spin polarizability sum rule (19) Therefore knowledge of the cross
section at higher energies is required and the evaluation of the generalized GDH integral
IA(Q2) is not as simple as the evaluation of γ0(Q2)
The generalized GDH integral IA(Q2) is directly related to the non-pole amplitude
gnonpoleTT (νQ2) which differs from non-Born amplitude gTT (νQ2) by a term involving the
elastic Pauli form factor
gnonpoleTT (νQ2) = gTT (νQ2)minus 2παν
M2N
F 22 (Q2) (27)
cf Eqs (2a) and (8a) Consequently IA(Q2) is not a pure polarizability but also contains
an elastic contribution The ldquonon-polarizabilityrdquo or the Born part of IA(Q2) is given by
IBornA (Q2) = IA(Q2)minus∆IA(Q2) = minus1
4F 2
2 (Q2) (28)
where we refer to the polarizability part as ∆IA(Q2) The same is true for the generalized
GDH integral I1(Q2) which is directly related to Snonpole1 (νQ2)
IBorn1 (Q2) = I1(Q2)minus∆I1(Q2) = minus1
4F 2
2 (Q2) (29)
In the following we will add the Born parts to our LO and NLO BχPT predictions for
the polarizabilities ∆IA(Q2) and ∆I1(Q2) employing an empirical parametrization for the
elastic Pauli form factor [73] This allows us to compare to the experimental results for
IA(Q2) and I1(Q2) cf Fig 4 Note that the blue error bands only describe the uncertainties
of our BχPT predictions of the polarizabilities while the elastic contributions are considered
to be exact as explained in Sec II C The uncertainties of the polarizability predictions are
therefore better reflected in Fig 5 where we show the contributions of the different orders
to the BχPT predictions of ∆IA(Q2) and ∆I1(Q2) as well as the total results with error
bands
4 Note that IA(Q2) is sometimes called ITT (Q2)
18
000 005 010 015 020 025 030
-08
-06
-04
-02
00
02
Q2 GeV2D
DI A
proton
000 005 010 015 020 025 030
-10
-08
-06
-04
-02
00
02
Q2 GeV2D
DI A
neutron
000 005 010 015 020 025 030-005
000
005
010
015
020
025
Q2 GeV2D
DI 1
proton
000 005 010 015 020 025 030-015
-010
-005
000
005
010
015
Q2 GeV2D
DI 1
neutron
FIG 5 Contributions of the different orders to the chiral predictions of ∆IA(Q2) upper paneland ∆I1(Q2) lower panel for the proton (left) and neutron (right) Red solid line πN -loop
contribution green dot-dashed line ∆-exchange contribution orange dotted line π∆-loop contri-
bution blue solid line and blue band total result purple dot-dot-dashed line total result without
gC contribution black short-dashed line total result without gM dipole
The E97-110 experiment at Jefferson Lab has recently published their data for IAn(Q2) in
the region of 0035 GeV2 lt Q2 lt 024 GeV2 [27] In addition there are results for IAn(Q2)
from the earlier E94-010 experiment [21] and for IAp(Q2) from the E08-027 experiment [60]
The O(p4) HB calculation gives a large negative effect [51] which does not describe the data
The BχPT+∆ result from Ref [31] which mainly differs from our work by the absence of
the dipole form factor in gM looks similar to this HB result and only describes the data
points at lowest Q2 Our NLO prediction however follows closely the Q2 evolution of the
data In Fig 5 upper panel we show the polarizability ∆IA(Q2) whose Q2 evolution is
clearly dominated by the ∆ exchange Similar to the case of γ0p(Q2) inclusion of the dipole
in gM and the Coulomb coupling gC is very important in order to describe the experimental
data The LO prediction on the other hand slightly overestimates the data cf Fig 4
upper panel
19
At the real-photon point IA(0) = minusκ2
4and ∆IA(0) = 0 Therefore we give only the
slope of the polarizability ∆IA(Q2) [showing also the separate contributions from πN loops
∆ exchange and π∆ loops] in units of GeVminus2
d∆IAp(Q2)
dQ2
∣∣∣∣Q2=0
= minus858(343) asymp 238minus 1121 + 025 (30a)
d∆IAn(Q2)
dQ2
∣∣∣∣Q2=0
= minus955(343) asymp 141minus 1121 + 025 (30b)
Including the empirical Pauli form factor [73] we find in units of GeVminus2
dIAp(Q2)
dQ2
∣∣∣∣Q2=0
= minus318dIAn(Q2)
dQ2
∣∣∣∣Q2=0
= minus300 (31)
D Γ1(Q2) and I1(Q2) mdash the first moment of the structure function g1(xQ2)
The second variant for a generalization of the GDH sum rule to finite Q2 is defined as
minus α
2M2N
I1(Q2) = minus 1
8π2
int infinν0
dν1radic
ν2 +Q2
[σTT (νQ2) +
Q
νσLT (νQ2)
](32)
= minus α
Q2
int x0
0
dx g1(xQ2)
where I1(0) = minusκ2
4 This generalized GDH integral directly stems from the amplitude
Snonpole1 (νQ2) with the LEX from Eq (12a) It is given by the first moment of the struc-
ture function g1(xQ2) Γ1(Q2) =int x0
0dx g1(xQ2) as follows I1(Q2) =
2M2N
Q2 Γ1(Q2) The
isovector combination
Γ1(pminusn)(Q2) =
int x0
0
dx[g1p(xQ
2)minus g1n(xQ2)] (33)
is related to the axial coupling of the nucleon through the Bjorken sum rule [74 75]
limQ2rarrinfin
Γ1(pminusn)(Q2) =
gA6 (34)
As explained in Eq (28) the moment I1(Q2) splits into a polarizability part ∆I1(Q2) and
a Born part IBorn1 (Q2) Figure 4 lower panel shows the Q2 dependence of I1(Q2) which in
contrast to IA(Q2) shown in Figure 4 upper panel is clearly dominated by its Born part
and the elastic Pauli form factor The πN -loop ∆-exchange and π∆-loop contributions to
the polarizability ∆I1(Q2) are shown in Fig 5 lower panel Comparing to Fig 5 upper
20
000 005 010 015 020 025 030
-006
-004
-002
000
002
004
Q2 GeV2D
G1
proton
000 005 010 015 020 025 030
-006
-005
-004
-003
-002
-001
000
001
Q2 GeV2D
G1
neutron
FIG 6 First moment of the structure function g1(xQ2) for the proton (left) and neutron (right)
as function of Q2 The legend is the same as in Fig 4
panel one sees that ∆I1(Q2) is less sensitive to gC and the dipole form factor in gM than
∆IA(Q2)
For the proton our NLO BχPT prediction gives a very good description of the ex-
perimental data [18 60] and is in reasonable agreement with the MAID prediction [69]
For the neutron one observes good agreement with the empirical evaluations including ex-
trapolations to unmeasured energy regions starting from Q2 gt 01 GeV2 [27 61] In the
region of Q2 lt 005 GeV2 one observes an interesting tension between the recent E97-
110 experiment [27] and the data from CLAS [61] While the newest measurement finds
I1n(0035 GeV2) lt κ2n4 thus suggesting a negative slope at low Q the older measurement
found a rather large value for I1n(00496 GeV2) A similar but milder behaviour is seen in
the E97-110 [27] and E94-010 [21] data for IAn The MAID predictions do not agree with
the CODATA recommended values for the anomalous magnetic moments of the proton and
neutron [70] which in our work are imposed by using empirical parametrizations for the
elastic Pauli form factors [73] The slope of the HB result from Ref [51] is too large and
therefore only reproduces the data at very low Q2
Figure 6 shows the moment Γ1(Q2) for the proton and neutron while Fig 7 shows the
isovector combination Γ1 pminusn(Q2) The LO and NLO BχPT predictions are identical be-
cause our calculation produces the same Delta contributions for the proton and the neutron
For the isovector combination the MAID model only agrees with the data at very low
Q2 lt 010 GeV2 The same is true for the IR result [58 76] while all other chiral results
describe the data NLO BχPT (this work) BχPT+∆ [31] and HBχPT [51]
At the real-photon point I1(0) = minusκ2
4and ∆I1(0) = 0 Therefore we give only the
slope of the polarizability ∆I1(Q2) [showing also the separate contributions from πN loops
21
000 005 010 015 020 025 030-002
000
002
004
006
008
010
Q2 GeV2D
G1Hp-
nL
FIG 7 Isovector combination of Γ1(Q2) as function of Q2 The legend is the same as in Fig 6
The pink curve is the IR result from Ref [58 76] The experimental points are from Ref [77]
(brown dots) and Ref [23] (orange squares)
∆ exchange and π∆ loops] in units of GeVminus2
d∆I1p(Q2)
dQ2
∣∣∣∣Q2=0
= 039(4) asymp 034minus 053 + 058 (35a)
d∆I1n(Q2)
dQ2
∣∣∣∣Q2=0
= minus101(10) asymp minus107minus 053 + 058 (35b)
Including the empirical Pauli form factor [73] we find in units of GeVminus2
dI1p(Q2)
dQ2
∣∣∣∣Q2=0
= 580dI1n(Q2)
dQ2
∣∣∣∣Q2=0
= 553 (36)
E d2(Q2) mdash a measure of color polarizability
Another interesting moment to consider is d2(Q2) which is related to the twist-3 part of
the spin structure function g2(xQ2) [79 80]
d2(Q2) equiv 3
int 1
0
dx x2[g2(xQ2)minus gWW2 (xQ2)] (37)
where gWW2 (xQ2) is the twist-2 part of g2(xQ2) Using the Wandzura-Wilczek relation [81]
one can relate d2(Q2) to moments of the spin structure functions g1(xQ2) and g2(xQ2)
d2(Q2) =
int 1
0
dx x2 [3g2(xQ2) + 2g1(xQ2)] (38)
22
000 005 010 015 020 025 030
0000
0001
0002
0003
0004
0005
0006
0007
Q2 GeV2D
d2
proton
000 005 010 015 020 025 030
0000
0002
0004
0006
0008
0010
0012
Q2 GeV2D
d2
neutron
000 005 010 015 020 025 030
-1
0
1
2
Q2 GeV2D
Γ 010
-4
fm6
D
proton
000 005 010 015 020 025 030-2
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
6D
neutron
FIG 8 Upper panel The inelastic moment d2(Q2) for the proton (left) and neutron (right) as
function of Q2 The result of this work the NLO BχPT prediction is shown by the blue solid line
and the blue band The red line represents the LO BχPT result The purple short-dashed line is
the O(p4) HB result from Ref [50 51] The black dotted line is the MAID model prediction [69]
The experimental points for the neutron (cyan dots) are from Ref [22] Lower panel Fifth-order
generalized forward spin polarizability γ0(Q2) for the proton (left) and neutron (right) as function
of Q2 The experimental points for the proton are from Ref [59] (purple square) and Ref [78]
(orange dot)
This relation however only holds for asymptotically large Q2 It is also in the high-Q2
region where d2(Q2) is a measure of color polarizability [82 83] through its relation to the
gluon field strength tensor [80] We refer to Ref [84] for a recent review on the spin structure
of the nucleon including a discussion of sum rules for deep inelastic scattering and color
polarizabilities
What we consider in the following is the inelastic part of d2(Q2) defined as the moment
of g1(xQ2) and g2(xQ2) spin structure functions cf Eq (38)
d2(Q2) =
int x0
0
dx x2 [3g2(xQ2) + 2g1(xQ2)] (39)
23
This moment provides another testing ground for our BχPT predictions through comparison
with experiments on the neutron [22] Going towards the low-Q2 region the interpretation
of d2(Q2) in terms of color polarizabilities will fade out The above definition however
implies it is related to other VVCS polarizabilities
d2(Q2) =Q4
8M4N
[M2
NQ2
αδLT (Q2) + I1(Q2)minus IA(Q2)
] (40)
Note that d2(Q2) and its first two derivatives with respect to Q2 vanish at Q2 = 0 The
considerations in Eqs (28) and (29) have no effect on d2(Q2) since the Born contribution
from IA(Q2) and I1(Q2) cancel out Therefore d2(Q2) is a pure polarizability
In Fig 8 upper panel we show our NLO BχPT prediction and other results for d2(Q2)
While MAID [69] and BχPT describe the experimental data for the neutron [22] very well
the HB limit [50 51] is showing a fast growth with Q2 This illustrates the importance
of keeping the relativistic result Note also that even though the πN -loop contribution
is dominant both gC and the form factor in gM are essential to obtain a curvature that
reproduces the data cf Fig 9 upper panel For the proton there are to our knowledge
no experimental results to compare with However the agreement between the NLO BχPT
prediction and the MAID prediction at low energies is reasonable
F γ0(Q2) mdash fifth-order generalized forward spin polarizability
It is interesting to compare the generalized fifth-order forward spin polarizability sum
rule
γ0(Q2) =1
2π2
int infinν0
dν
radic1 +
Q2
ν2
σTT (νQ2)
ν5(41)
=64αM4
N
Q10
int x0
0
dx x4
[g1(xQ2)minus 4M2
Nx2
Q2g2(xQ2)
]
to the sum rule integrals for IA(Q2) and γ0(Q2) since they differ merely by their energy
weighting of σTT (νQ2) and a constant prefactor cf Eqs (19) (26) and (41) From IA(Q2)
to γ0(Q2) to γ0(Q2) the energy suppression is increasing by a factor of νminus2 respectively
Therefore the description of γ0(Q2) should be easiest in a low-energy effective-field theory
such as χPT whereas γ0(Q2) and IA(Q2) receive larger contributions from higher energies
In Fig 8 lower panel we show our LO and NLO BχPT predictions for γ0(Q2) One
can see that the πN -loop contribution is positive (in accordance to what we see for the cross
section σTT see Fig 10) The Delta shifts it substantially especially in the low Q2 region
24
000 005 010 015 020 025 030
-0004
-0002
0000
0002
0004
0006
0008
0010
Q2GeV
2D
d2
proton
000 005 010 015 020 025 030
-0005
0000
0005
0010
0015
Q2GeV
2D
d2
neutron
000 005 010 015 020 025 030
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
6D
proton
000 005 010 015 020 025 030
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
6D
neutron
FIG 9 Contributions of the different orders to the chiral predictions of d2(Q2) upper panel and
γ0(Q2) lower panel for the proton (left) and neutron (right) Red solid line πN -loop contribu-
tion green dot-dashed line ∆-exchange contribution orange dotted line π∆-loop contribution
blue long-dashed line total result purple dot-dot-dashed line total result without gC contribution
black short-dashed line total result without gM dipole
bringing it into a better agreement with data In general the BχPT curves start above the
empirical data points at the real-photon point and then decrease asymptotically to zero
above Q2 gt 01 GeV2 On the other hand the MAID prediction reproduces the empirical
data at the real-photon point then decreases to negative values until about Q2 gt 006 GeV2
from where it also starts to asymptotically approach zero Consequently our NLO BχPT
prediction of γ0(Q2) is consistently above the MAID prediction This is very different to
what we saw for IA(Q2) in Fig 4 upper panel where the MAID prediction at the real-
photon point is above the experimental value While the agreement of our predictions with
the empirical data is in general quite good for all moments of σTT (νQ2) one should point
out that both for γ0n(Q2) and γ0p(Q2) we overestimate the data at low Q2 For IA(Q2) such
observation cannot be made because ∆IA(0) = 0 and thus IA(0) is given by the empirical
Pauli form factor only From IA(Q2) γ0(Q2) and γ0(Q2) the latter has the smallest however
25
non-negligible dependence on gC and the dipole in gM cf Fig 9 lower panel
The πN -loop ∆-exchange and π∆-loop contributions to the NLO BχPT prediction of
the static fifth-order forward spin polarizability amount to in units of 10minus4 fm6
γ0p = 112(30) asymp 208minus 096minus 001 (42a)
γ0n = 195(30) asymp 292minus 096minus 001 (42b)
while the slope is composed as follows in units of 10minus4 fm8
dγ0p(Q2)
dQ2
∣∣∣∣Q2=0
= minus084(10) asymp minus100 + 016 + 000 (43a)
dγ0n(Q2)
dQ2
∣∣∣∣Q2=0
= minus142(15) asymp minus158 + 016 + 000 (43b)
Note that the HB prediction of the static polarizability γ0p (423 at O(p3) and 365 at O(ε3)
[78 85]) is almost one order of magnitude larger than the empirical value and therefore not
shown in Fig 8
G Summary
Our results are summarized in Table II where we give the contributions of the different
orders to the chiral predictions of the polarizabilities and their slopes at the real-photon
point A quantitative comparison of our predictions for the static spin polarizabilities to
the work of Bernard et al [31] and different empirical evaluations is shown in Table III We
can see that the inclusion of the Delta turns out to be very important for all moments of
the helicity-difference cross section To describe the Q2 behavior of the polarizabilities the
magnetic coupling of the N rarr ∆ transition should be modified by a dipole form factor as
has been observed previously in the description of electroproduction data [33] This dipole
form factor effectively takes account of vector-meson exchanges The Coulomb-quadrupole
N rarr ∆ transition despite its subleading order is important in the description of some
moments of spin structure functions This is contrary to what we saw for the moments of
unpolarized structure functions [30] where the Coulomb coupling had a negligible effect
The π∆ loops are mainly relevant for the generalized GDH integrals
26
TABLE II The NLO BχPT predictions for the forward VVCS polarizabilities and their slopes at
Q2 = 0 The contributions of the πN loops the ∆ exchange and the π∆ loops are shown together
with the combined total result Note that IA(0) = I1(0) = d2(0) = 0 and (d2)prime = 0
πN loops ∆ exchange π∆ loops Total
γ0 p 201 minus284 minus010minus093(92)
(10minus4 fm4) n 298 003(92)
δLT p 150 minus016 minus002132(15)
(10minus4 fm4) n 235 218(23)
γ0 p 208 minus096 minus001112(30)
(10minus4 fm6) n 292 195(30)
(γ0)prime p minus033011 001
minus022(4)
(10minus4 fm6) n minus073 minus061(7)
(δLT )prime p minus080 minus004 minus001minus085(8)
(10minus4 fm6) n minus119 minus124(12)
(γ0)prime p minus100016 000
minus084(10)
(10minus4 fm8) n minus158 minus142(15)
(∆IA)prime p 238 minus1121 025minus858(343)
(GeVminus2) n 141 minus955(343)
(∆I1)prime p 034 minus053 058039(4)
(GeVminus2) n minus107 minus101(10)
TABLE III Our NLO BχPT predictions for the spin polarizabilities at Q2 = 0 compared with
the BχPT+∆ predictions from Bernard et al [31] and the available empirical information Where
the reference is not given the empirical number is provided by the MAID analysis [55 69] with
unspecified uncertainty
Proton Neutron
This work BχPT+∆ Empirical This work BχPT+∆ Empirical
γ0 minus093(92) minus174(40) minus100(8)(12) [19] 003(92) minus077(40) minus0005
(10minus4 fm4) minus090(8)(11) [78] [MAID]
minus0929(105) [59]
δLT 132(15) 240(1) 134 218(23) 238(3) 203
(10minus4 fm4) [MAID] [MAID]
γ0 112(30) 060(7)(7) [78] 195(30) 123
(10minus4 fm6) 0484(82) [59] [MAID]
IV CONCLUSIONS
We have presented a complete NLO calculation of the polarized non-Born VVCS ampli-
tudes in covariant BχPT with pion nucleon and ∆(1232) fields The dispersion relations
between the VVCS amplitudes and the tree-level photoabsorption cross sections served as a
27
cross-check of these calculations
The obtained moments of the proton and neutron spin structure functions related to gen-
eralized polarizabilities and GDH-type integrals agree well with the available experimental
data The description of their Q2 evolution is improved compared to the previous χPT pre-
dictions In particular the NLO BχPT predictions obtained here give a better description
of the empirical data (eg from the Jefferson Laboratory ldquoSpin Physics Programrdquo) than
the HB [50 51] and IR [58] calculations
The demonstrated predictive power of the χPT framework amplitudes makes it well
suited for extending the χPT evaluation of the TPE effect in the hyperfine structure of
(muonic-)hydrogen [15ndash17] to next-to-leading order
ACKNOWLEDGEMENTS
We thank Lothar Tiator and Marc Vanderhaeghen for helpful discussions This work
is supported by the Deutsche Forschungsgemeinschaft (DFG) through the Collaborative
Research Center [The Low-Energy Frontier of the Standard Model (SFB 1044)] JMA ac-
knowledges support from the Community of Madrid through the ldquoPrograma de atraccion de
talento investigador 2017 (Modalidad 1)rdquo and the Spanish MECD grants FPA2016-77313-P
FH gratefully acknowledges financial support from the Swiss National Science Foundation
Appendix A Tensor decompositions of the VVCS amplitudes
In this appendix we review the decomposition of the forward VVCS process into tensor struc-
tures and scalar amplitudes In particular we consider the connection between the covariant and
the semi-relativistic decomposition in the lab frame that is defined in terms of the conventional
transverse longitudinal transverse-transverse and transverse-longitudinal amplitudes
As explained in Sec II A the process of forward VVCS off the nucleon can be described in
terms of four explicitly covariant amplitudes S1 2 and T1 2 [3]
T (νQ2) =
(minusgmicroν +
qmicroqν
q2
)T1(νQ2) +
1
M2N
(pmicro minus p middot q
q2qmicro)(
pν minus p middot qq2
qν)T2(νQ2) (A1)
minus 1
MNγmicroναqα S1(νQ2)minus 1
M2N
(γmicroνq2 + qmicroγναqα minus qνγmicroαqα
)S2(νQ2)
εprimelowastmicro εν
where εmicro (εprimelowastmicro ) are the incoming (outgoing) photon polarization vectors ν is the photon lab-frame
energy and Q2 is the photon virtuality Alternatively the decomposition in the laboratory frame
28
(which in the forward case coincides with the Breit frame) is parametrized in terms of the nucleon
Pauli matrices ~σ and the four scalar functions fL fT gTT and gLT
T (νQ2) = ε0 εprimelowast0 fL(νQ2) + (~ε primelowast middot ~ε ) fT (νQ2) + i~σ middot (~ε primelowast times ~ε ) gTT (νQ2) (A2)
minusi~σ middot [(ε0~εprimelowast minus ~ε ε primelowast0 )times q] gLT (νQ2)
Here ~q and q = ~q|~q | are the photon three-momentum in the lab system and its unit vector The
modified polarization vector components are given by
ε0 =
[ε0 minus
ν
|~q | (~ε middot q )
] |~q |Q ~ε = ~εminus q (~ε middot q ) (A3)
where ε = (ε0~ε ) is the usual incoming photon polarization vector and εprimelowast the outgoing polarization
vector The LEX of the lab frame amplitudes [Eq (10)] can serve in particular as the definition
of the generalized polarizabilities The lab frame amplitudes are also conveniently used for the
definition of the response functions see the example of the scalar amplitude gLT (νQ2) and the
corresponding response function σLT (νQ2) below in App B
Appendix B Photoabsorption cross sections
In the forward kinematics the spin-dependent VVCS amplitudes and the spin polarizabilities
can be described in terms of the polarized structure functions g1(xQ2) and g2(xQ2) or equiv-
alently the helicity-difference cross section σTT (νQ2) and the longitudinal-transverse response
function σLT (νQ2) with the help of dispersion relations (5) and the optical theorem (3) In this
way the photoabsorption cross sections measured in electroproduction processes form the basis
for most empirical evaluations shown throughout Sec III In the following we present the BχPT
predictions for the tree-level cross sections of πN - π∆- and ∆-production through photoabsorption
on the nucleon cf Figs 8 9 and 10 in Ref [30] In Secs B 1 and B 2 we will discuss the leading
πN -production channel and the ∆-production channel respectively We used these cross sections
to verify the polarizability predictions obtained otherwise from the calculated non-Born VVCS
amplitudes Due to the bad high-energy behavior of the π∆-production cross sections in BχPT
cf Fig 10 the dispersion relations in Eq (5) require further subtractions for a reconstruction of
the π∆-loop contribution to the spin-dependent VVCS amplitudes Therefore not all polarizabil-
ities could be verified but only those appearing as higher-order terms in the LEX of the VVCS
amplitudes such as γ0 [16]
29
proton neutron
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Neutron
02 04 06 08 100
50
100
150
200
250
[GeV]
T [b]
02 04 06 08 10
-50
0
50
100
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Neutron
02 04 06 08 100
50
100
150
200
250
[GeV]
T [b]
02 04 06 08 10
-50
0
50
100
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02040608100
50
100
150
200
250
300
[GeV]
T[b]
0204060810
-50
0
50
[GeV]
TT[b]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
0204060810
00
02
04
06
[GeV]
LTQ
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
0204060810
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
FIG 10 Photoabsorption cross sections for πN (red) and π∆ production (orange) with Q2 = 0
(solid) and Q2 = 01 GeV2 (dashed for πN and dotted for π∆ channel)
1 πN-production channel
In order to extract the response function σLT (νQ2) we have developed a method similar to
the one used to calculate σTT (νQ2) see for example Ref [86] For σLT (νQ2) however the
calculation is more complicated because one has to take into account that the associated Compton
process involves a spin-flip of the nucleon as illustrated in Fig 11 When calculating the cross
section the product of the incoming nucleon spinors has to reflect this flip
The forward VVCS amplitude related to σLT (νQ2) mdash and δLT (Q2) mdash is gLT (νQ2) It can
be extracted from Eq (A2) if one takes the modified polarization vector components in Eq (A3)
with ε = εL and εprimelowast = εlowastplusmn as input where εL = 1Q(|~q | 0 0 q0) and εplusmn = ∓ 1radic
2(0 1plusmni 0) are the
standard longitudinal and transverse polarization vectors respectively For εL and εlowastplusmn only the
choice of helicities hprime = plusmn12 and h = ∓12 gives a non-zero contribution and one obtains
χdaggerhprime T (νQ2)χh = χdaggerhprimeminusi~σ middot [(ε0~εprimelowast minus ~ε ε primelowast0 )times q] gLT (νQ2)χh =
radic2 gLT (νQ2) (B1)
where χh and χdaggerhprime are two-component Pauli spinors with opposite helicities or here spins
Let us now consider the related photoabsorption process and in particular the tree-level γlowastN rarr
30
s = 12
sprime = minus12
λ = 0 λprime = 1
rprime rprime
s=12sprime=minus1
2
λ=0λprime=1
rprimerprime
s=12
sprime=minus12
λ=0λprime=1
rprimerprime hacute=frac12h=-frac12
λacute=1λ=0
FIG 11 Relation between the forward Compton process and the photoabsorption process given
by the optical theorem In particular we show the longitudinal-transverse contribution The
double-line arrows represent the spin of the external particles while the dot represents the scalar
(longitudinal) polarization of the incoming photon Inside the blob the intermediate states are
represented eg nucleons with spins rprime (which are averaged in the calculation of the cross section)
and pions
πN channel see diagrams in Fig 8 of Ref [30] We define the πN -production amplitude as
T = uhB (PB)sumi
Ai(s t)Γi uhA(PA) (B2)
with the Dirac structures
Γ1 = γ5 (B3a)
Γ2 =1
2
[qA ε
]γ5 (B3b)
where uhA(PA) and udaggerhB (PB) are the Dirac spinors and PA and PB are the four-momenta of
the incoming and outgoing nucleons respectively When calculating the photoabsorption cross
section related to the VVCS amplitude in Eq (B1) the nucleon spin flip should be implemented
by uhprime(PA) in T dagger and uh(PA) in T together with the appropriate transverse and longitudinal
photon polarization vectors εlowastplusmn and εL
However if one wants to use the properties of the Dirac matrices it is more useful to construct
an operator to produce this spin flip in the external nucleons of Fig 11 This is accomplished by
introducing the projector ΓLT equiv 12radic
2(γ1 + iγ2)γ5 which also takes into account the extra factor
radic2 in Eq (B1) We checked that with this projector one correctly extracts δLT by comparing the
HB limit of our result to the HB result of Ref [50] where the authors calculate this polarizability
from the Compton amplitude directly With all those ingredients the longitudinal-transverse cross
section is calculated in the following way
σLT (νQ2) =1
64π2 s
|~pf |cm
|~pi|cm
int 1
minus1dcos θ
sumij
AiAdaggerjXij (B4)
31
with
Xij = Tr[(PB +MN )Γi(PA +MN )ΓLTγ0Γdaggerjγ
0] (B5)
where θ is the scattering angle in the center-of-mass (cm) frame and |~pi|cm (|~pf |cm) is the three-
momentum of an incoming (outgoing) particle in the cm frame An explicit calculation of the
matrix Xij leads to
X = MNQ
0 2(PB minus PA) middot εLminusradic
2|~qf |cm sin θ (sminus u)
(B6)
where |~qi|cm (|~qf |cm) is the relative three-momentum of the incoming (outgoing) particles in the
cm frame Here s t and u are the usual Mandelstam variables For the different γlowastN rarr πN
channels we obtain the following amplitudes Ai where we introduce qA as the four-momentum of
the incoming photon and qB as the four-momentum of the outgoing pion
bull γlowastprarr π0p
A1 =e gAMN
fπ
[2PA middot ε+ qA middot ε
sminusM2N
+2PB middot εminus qA middot ε
uminusM2N
] (B7a)
A2 =e gAMN
fπ
[1
sminusM2N
+1
uminusM2N
] (B7b)
bull γlowastprarr π+n
A1 =
radic2 e gAMN
fπ
[2PA middot ε+ qA middot ε
sminusM2N
+2(PA minus PB) middot ε+ qA middot ε
tminusm2π
] (B8a)
A2 =
radic2 e gAMN
fπ(sminusM2N )
(B8b)
bull γlowastnrarr π0n
A1 = 0 (B9a)
A2 = 0 (B9b)
bull γlowastnrarr πminusp
A1 =
radic2 e gAMN
fπ
[2PB middot εminus qA middot ε
uminusM2N
minus 2(PA minus PB) middot ε+ qA middot εtminusm2
π
] (B10a)
A2 =
radic2 e gAMN
fπ(uminusM2N )
(B10b)
The analytical expressions shown above were checked with the amplitudes given in Ref [87] An-
alytical expressions for the tree-level γlowastN rarr πN channel of the σLT (νQ2) and σTT (νQ2) cross
32
sections are given below (proton channels π+n and π0p neutron channel πminusp) We checked that
they reproduce the known results in the real-photon limit [45 86] To shorten the final expressions
for the cross sections which are considerably longer for finite Q2 than in the real-photon limit we
define the following dimensionless kinematic variables
αγ = (ENi )cmradics =
s+M2N +Q2
2s (B11)
απ = (ENf )cmradics =
s+M2N minusm2
π
2s (B12)
βγ = Eγcmradics =
sminusM2N minusQ2
2s (B13)
βπ = Eπcmradics =
sminusM2N +m2
π
2s (B14)
λγ = |~qi|cmradics =
radic(sminusM2
N minusQ2)2 + 4sQ2
2s (B15)
λπ = |~qf |cmradics =
radic(sminusM2
N +m2π)2 minus 4sm2
π
2s (B16)
Here (ENi )cm and (ENf )cm are the energies of the incoming and outgoing nucleon Eγcm is the energy
of the incoming photon Eπcm is the energy of the outgoing pion all in the cm frame
σ(π+n)TT = minus e2g2
AM2N
64πf2πs
3(sminusM2N )2λ4
γ
4sλπλγ
[(M2
N minus s)(M2N minusQ2 minus s)(Q2 + 2sβγβπ)
+ 2s(m2πQ
2 minus (M2N minus s)
(M2N + s(minus1 + 2βγβπ)
))λ2γ
]+ (M2
N minus s)(M2N minusQ2 minus s)
times (Q2 + 2sβγβπ minus 2sλπλγ)(Q2 + 2s(βγβπ + λπλγ)
)times log
(Q2 + 2sβγβπ minus 2sλγλπQ2 + 2sβγβπ + 2sλγλπ
) (B17)
33
σ(π0p)TT =
e2g2AM
2Nλπ
64πf2πs(sminusM2
N )2λγ
1
s2λπλ3γ
[(M2
N minus s)(M2N minusQ2 minus s)
(3M2
N + s(2βγβπ minus 3))(M2N
+ s(2βγβπ minus 1))
+ 2s
((2m2
πQ2 + (M2
N minus s)2)(M2
N minus s)minus 2(minusm2
πQ2 + (sminusM2
N )2)
times sβγβπ + 4s2(sminusM2N )β2
γβ2π
)λ2γ
]arctan
(2sλπλγ
M2N + s(2βγβπ minus 1)
)+ 2
[minus 2m2
πQ2
+ (M2N minus s)
(2(sminusM2
N ) + 4sβγβπ +(sminusM2
N +Q2)(3M2N minus 3s+ 2sβγβπ)
sλ2γ
+m2πQ
2(sminusM2N )
(M2N minus s+ 2sβγβπ)(M2
N + s(minus1 + 2βγβπ minus 2λπλγ))
+m2πQ
2(sminusM2N )
(M2N minus s+ 2sβγβπ)(M2
N + s(minus1 + 2βγβπ + 2λπλγ))
)
+1
λπλγ
(βγβπ
(minusm2
πQ2 + (M2
N minus s)(M2N minus s+ 2sβγβπ)
)minus (M2
N minus s)(M2N minusQ2 minus s)λ2
π
)log
(M2N + s(minus1 + 2βγβπ + 2λπλγ)
M2N + s(minus1 + 2βγβπ minus 2λπλγ)
)] (B18)
σ(πminusp)TT =
e2g2AM
2N
64πf2πs
3λ4γ(M2
N + s(2βπβγ minus 1))2 minus 4λ2πs
2λ2γ)
4sλπλγ
((minusM2
N +Q2 + s)(M2N
+ s(2βπβγ minus 1))
2 minus 2sλ2γ
(2λ2
πs(minusM2N +Q2 + s) +m2
πQ2))
+((M2N + s(2βπβγ minus 1)
)2
minus 4λ2πs
2λ2γ
)((M2N + s(2βπβγ minus 1)
)(M2N minus 2Q2 minus s(2βπβγ + 1)
)+ 4λ2
πs2λ2
γ
)times log
(minusM2
N minus 2βπsβγ minus 2λπsλγ + s)minus(M2N + s(2βπβγ minus 2λπλγ minus 1)
)(M2N + s(minus1
+ 2βπβγ + 2λπλγ))[((
M2N + s(2βπβγ minus 1)
)(M2N minus 2Q2 minus s(2βπβγ + 1)
)+ 4λ2
πs2λ2
γ
)times log
(minusM2
N minus 2βπsβγ + 2λπsλγ + s)
+((Q2 + 2βπsβγ)2 minus 4λ2
πs2λ2
γ
)times log
(Q2 + 2βπsβγ minus 2λπsλγQ2 + 2sβπβγ + 2λπsλγ
)]+ 4sλ2
γ
((M2
N + s(2βπβγ minus 1))2 minus 4λ2πs
2λ2γ
)times (M2
N minus s) arctanh
(2λπsλγ
M2N + 2βπsβγ minus s
) (B19)
34
σ(π+n)LT =
e2g2AM
3Nλπ
32πf2πQs
3(sminusM2N )2λ4
γ
2sλγ
[(M2
N minus s)(Q2 + 2sβ2γ)(Q2 + 2sβγβπ)
minus 4s
((M2
N minus s)(Q2 minus 2s(απ minus 1)βγ
)+Q2sβγβπ
)λ2γ + 8s3(minus1 + απ)λ4
γ
]minus M2
N minus sλπ
[(Q2 + 2sβ2
γ)(Q2 + 2sβγβπ)2 + 4s2(2(απ minus 1)βγ(Q2 + 2sβγβπ)
minusQ2λ2π
)λ2γ + 8s3(απ minus 1)2λ4
γ
]arctan
(2sλπλγ
Q2 + 2sβγβπ
) (B20)
σ(π0p)LT =
e2g2AM
3Nλπ
16πf2πQs(sminusM2
N )2λγ
1
minus2s(M2N + s(minus1 + 2βγβπ))2λ2
γ + 8s3λ2πλ
4γ
[minus 3M8
N (Q2
+ 2sβ2γ) + 2M4
Ns2
(minus (Q2 + 2sβ2
γ)(minus3 + 2βγβπ) +(Q2(minus3 + 2βγβπ + 6λ2
π) + 2sβγ
times (minus3 + 12απ + 2βγβπ minus 12απβγβπ + 4βγλ2π))λ2γ minus 4s(α2
π minus 1)λ4π
)+ 2M2
Ns3
[minus (Q2
+ 2sβ2γ)(minus1 + 2βγβπ)
(6 + βγβπ(minus9 + 2βγβπ)
)+
(Q2(3minus 12λ2
π + 4βγβπ(minus1 + 2βγβπ
+ λ2π))minus 2sβγ
(minus 3 + 4απ(3 + 2απ(3 + 2βγβπ(minus3 + βγβπ)) + 4βγ(βπ + (2minus βγβπ)
times λ2π)))λ2γ + 8s
((απ minus 1)(1 + απ minus 2βγβπ) + 2απβγλ
2π
)λ4γ
)+ s4
((Q2 + 2sβ2
γ)
times (1minus 2βγβπ)2(2βγβπ minus 3) + 2(Q2(minus1 + 6λ2
π + 2βγβπ((1minus 2βγβπ)2 minus 2λ2π)) + 2sβγ
times (minus1 + 2βγβπ + 4(minus1 + βγβπ)(minusαπ + 2απβγβπ minus βγλ2π)))λ2γ minus 8
(s(απ minus 1)(απ
+ (1minus 2βγβπ)2) + 2βγ(2sαπ +Q2βγ)λ2π
)λ4γ + 32s(απ minus 1)λ2
πλ6γ
)+ 2M6
Ns
(Q2(6
minus 7βγβπ + λ2γ) + 2sβγ(βγ(6minus 7βγβπ) + (1minus 4απ)λ2
γ)
)]+sminusM2
N
4s2λπλ3γ
[(Q2 + 2sβ2
γ)
times(3M2
N + s(2βγβπ minus 3))(M2N + s(2βγβπ minus 1)
)+ 2s
(minusM2
N (Q2 + 2sβγ minus 8sαπβγ)
+ s(2sβγ(1minus 4απ + 4απβγβπ) +Q2(1minus 2λ2π)))λ2γ + 8s3(απ minus 1)λ4
γ
]
times arctan
(2sλπλγ
M2N + s(2βγβπ minus 1)
) (B21)
35
σ(πminusp)LT =
e2g2AM
3N
64πf2πQs
3λ4γ(M2
N minusQ2 minus s)(M2N + s(2βπβγ + 2λπλγ minus 1))
times
1
M2N + s(2βπβγ minus 2λπλγ minus 1)
[4λπsλγ(M2
N minusQ2 minus s)(
2sλ2γ
(s((1minus 2λ2
π)Q2
+ 2(2απ minus 1)sβγ(2βπβγ minus 1))minusM2N (Q2 + 2(1minus 2απ)sβγ)
)+ (Q2 + 2sβ2
γ)(M2N + s(2βπβγ minus 1)
)2 + 8(απ minus 1)απs
3λ4γ
)]minus(M2N + s(2βπβγ + 2λπλγ minus 1)
)times[
log
(minusM2N minus 2βπsβγ minus 2λπsλγ + s
minusM2N minus 2βπsβγ + 2λπsλγ + s
)(minus (Q2 + 2sβ2
γ)(M2N + s(2βπβγ minus 1)
)(minusM2
N + 2Q2 + 2βπsβγ + s)minus 2sλ2γ
(minusM2
N (Q2 + 2sβγ) +Q4 +Q2s(4απβγ
minus 2βγ minus 2λ2π + 1) + 2s2βγ(4(απ minus 1)βπβγ + 1)
)minus 8(απ minus 1)2s3λ4
γ
)+ 2
(4s2λ2
γ
(2(απ minus 1)βγ(Q2 + 2βπsβγ)minus λ2
πQ2)
+ (Q2 + 2βπsβγ)2(Q2 + 2sβ2γ)
+ 8(απ minus 1)2s3λ4γ
)arctanh
(2λπsλγ
Q2 + 2βπsβγ
)] (B22)
2 ∆-production channel
The tree-level ∆-exchange diagram in Fig 2 of Ref [30] contributes to the non-Born part of
the VVCS amplitudes The contribution of the ∆ exchange to the VVCS amplitudes can be split
into [17]
S∆-exch1 (νQ2) = S∆-pole
1 (νQ2) + S∆-exch1 (νQ2) (B23a)
νS∆-exch2 (νQ2) = νS∆-pole
2 (νQ2) + νS2∆-exch
(νQ2) (B23b)
and similarly for the unpolarized VVCS amplitudes discussed in Ref [30] Here we introduced the
∆-pole contributions S∆-polei and the ∆-non-pole contributions S∆-exch
i The former amplitudes
feature a pole at the ∆(1232)-production threshold and thus are proportional to
1
[sminusM2∆][uminusM2
∆]=
1
4M2N
1
ν2∆ minus ν2
(B24)
36
They can be reconstructed from the dispersion relations in Eq (5) with the tree-level ∆-production
cross sections as input cf Fig 10 in Ref [30]
σTT (νQ2) =π2α
M2NM
2+|~q |
minus g2
MMN (M+ + ν)|~q |2 +g2E(∆minus ν)(Q2 minusMNν)2
MN(B25a)
+g2CQ
4s(∆minus ν)
MNM2∆
minus 4gMgE(Q2 minusMNν)|~q |2 minus 4gMgCQ2|~q |2
+2gEgCQ
2[minusMNM∆ |~q |2 + s(Q2 +∆ν)
]MNM∆
δ(ν minus ν∆)
σLT (νQ2) =Qπ2α
M2NM
2+|~q |
g2E(MNν minusQ2) [M∆(MN + ν)minus s]
MN(B25b)
+g2CQ
2[MNM∆|~q |2 minus s(Q2 +∆ν)
]MNM2
∆
+ gMgEM∆|~q |2 minusgMgC(Q2 minusMNν)|~q |2
M∆
+gEgC(ν minus∆)(M2
N |~q |2 minus 2Q2s)
MNM∆
δ(ν minus ν∆)
with ∆ = M∆ minusMN M+ = M∆ + MN and the Mandelstam variable s = M2N + 2MNν minus Q2
Analytical expressions for the spin structure functions g1(xQ2) and g2(xQ2) can be constructed
from Eq (3) with the flux factor K(νQ2) = |~q | =radicν2 +Q2
In the ∆-non-pole contributions to S1(νQ2) and νS2(νQ2) the pole in ν at the ∆(1232)-
production threshold has canceled out
S∆-exch1 (νQ2) =
πα
MNM2+
[g2MQ
2+ + g2
E
(∆2 minus 3Q2
)+
4g2CQ
4
M2∆
minus 8gMgEM∆ωminus (B26a)
minus2gMgCQ2(MN minus 4M∆)
M∆+
2gEgCQ2(3MN minus 2M∆)
M∆
]
νS2∆-exch
(νQ2) =2πα
MNM2+
[g2EM∆∆ωminus +
g2M MNQ
2+
2+g2C Q
2(Q2 minus∆2)
2M∆(B26b)
+gEgM M∆(M∆ω+ minus 4MNωminus)minus gEgC ∆(2Q2 +MNω+)
+gMgC Q2(4MN minus ω+)
]+S∆-exch
2 (νQ2)
ν
[M2
∆ ω2+
M2N
+ ν2
]
with Q+ =radic
(M∆ +MN )2 +Q2 and ωplusmn = (M2∆minusM2
N plusmnQ2)2M∆ and the non-pole contribution
to S2(νQ2)
S∆-exch2 (νQ2) = minus2παMNν
M∆M2+
[gM + gE
]gC (B27)
These amplitudes to the contrary are not described by the tree-level ∆-production cross sections
in the standard dispersive approach [17] This peculiarity has been previously missed e g in the
calculation of the ∆-exchange contribution to the hydrogen hyperfine splitting in Ref [88] The
importance of including the ∆-non-pole contribution is also evident when considering the BC sum
37
rule in Eq (14) The ∆-pole terms by themselves violate the BC sum rule but cancel exactly with
the ∆-non-pole terms
limνrarr0
νS∆-pole2 (νQ2) + lim
νrarr0νS2
∆-exch(νQ2) = 0 (B28)
Appendix C Polarizabilities at Q2 = 0
In this section we give analytical expressions for the static values and slopes at Q2 = 0 of
the spin polarizabilities and moments of polarized structure functions In particular we give the
HB expansion of the πN -loop contributions and the ∆-exchange contributions The complete
expressions also for the π∆-loop contributions can be found in the Supplemented material Recall
that IA(0) = I1(0) = d2(0) = 0 and dd2(Q2)dQ2
∣∣∣Q2=0
= 0
1 πN-loop contribution
Here we give analytical expressions for the πN -loop contributions to the proton and neutron
spin polarizabilities expanded in powers of micro = mπMN viz the HB expansion Note that we
choose to expand here to a high order in micro the strict HB expansion would only retain the leading
term in an analogous NLO calculation
bull Static polarizabilities (Q2 = 0)
γ0p =e2g2
A
96π3f2πm
2π
1minus 21πmicro
8minus(
59
2+ 26 logmicro
)micro2 +
1875πmicro3
64
+3
(3
2+ 26 logmicro
)micro4 +
(C1)
γ0n =e2g2
A
48π3f2πm
2π
1
2minus 9πmicro
16minus 2micro2 logmicro+
75πmicro3
128minus 3micro4
4+
(C2)
δLTp =e2g2
A
192π3f2πm
2π
1minus 9πmicro
8+
(13
2minus 2 logmicro
)micro2 minus 465πmicro3
64
minus(
47
2+ 42 logmicro
)micro4 +
(C3)
δLTn =e2g2
A
96π3f2πm
2π
1
2+
3πmicro
16+ (1 + 2 log micro)micro2 minus 105πmicro3
128+
5micro4
4+
(C4)
38
δLTn =e2g2
A
96π3f2πm
2π
1
2+
3πmicro
16+ (1 + 2 log micro)micro2 minus 105πmicro3
128+
5micro4
4+
(C5)
γ0p =e2g2
A
16π3f2πm
4π
4
45minus 3πmicro
16+
14micro2
5minus 1813πmicro3
384minus 192
5(1 + log micro)micro4
+80703πmicro5
2048+
(C6)
γ0n =e2g2
A
16π3f2πm
4π
4
45minus 5πmicro
48+
4micro2
5minus 245πmicro3
384minus 32micro4 logmicro
15
+1323πmicro5
2048+
(C7)
bull Slopes of polarizabilities at Q2 = 0
dγ0p(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
2minus 45πmicro
4+ 223micro2 minus 28515πmicro3
64
minus9
(1953
4+ 449 logmicro
)micro4 +
570255πmicro5
128+
(C8)
dγ0n(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
2minus 81πmicro
8+ 94micro2 minus 2535πmicro3
32
minus3 (1 + 90 logmicro)micro4 +84315πmicro5
1024+
(C9)
dδLTp(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
2880π3f2πm
4π
minus5
2minus 27πmicro
32+ 20micro2 minus 5865πmicro3
256
+3
(617
4+ 36 logmicro
)micro4 minus 2056845πmicro5
4096+
(C10)
dδLTn(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
minus5
4minus 81πmicro
64minus 11micro2 +
10005πmicro3
512
+15
8(11 + 48 logmicro)micro4 minus 267015πmicro5
8192+
(C11)
dIAp(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
96π2f2πmicro
2
1minus 15πmicro
4minus 1
2(115 + 88 logmicro)micro2 +
1839πmicro3
32
+5 (5 + 34 logmicro)micro4 +
(C12)
dIAn(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
48π2f2πmicro
2
1
2minus 11πmicro
8minus 1
4(1 + 20 logmicro)micro2 +
99πmicro3
64
minus25micro4
12+
(C13)
dI1p(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
96π2f2πmicro
3π
8+ 2 (4 + 3 logmicro)microminus 537πmicro2
64
minus1
2(15 + 56 logmicro)micro3 +
(C14)
39
dI1n(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
48π2f2πmicro
minus π
16+
1
4(3 + 4 logmicro)microminus 57πmicro2
128+
2micro3
3+
(C15)
γ0p(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2g2
A
16π3f2πm
6π
1
105minus 23πmicro
256+
377micro2
210minus 15551πmicro3
6144+
3371micro4
105
minus1640457πmicro5
32768+
(C16)
γ0n(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2g2
A
16π3f2πm
6π
1
105minus 153πmicro
1792+
69micro2
70minus 4615πmicro3
6144+
172micro4
35
minus120897πmicro5
32768+
(C17)
2 ∆-exchange contribution
Here we give analytical expressions for the tree-level ∆-exchange contributions to the nucleon
spin polarizabilities and their slopes at Q2 = 0 Note that the ∆-exchange contributes equally to
proton and neutron polarizabilities Recall that for the magnetic γlowastN∆ coupling we introduced a
dipole form factor to mimic vector-meson dominance gM rarr gM(1 +Q2Λ2)2
bull Static polarizabilities (Q2 = 0)
γ0 = minus e2
4πM2+
(g2M
∆2+
g2E
M2+
minus 4gMgEM+∆
) (C18)
δLT =e2M∆
4πM3+
(g2E
MNM++gMgE∆MN
minus gEgCM2
∆
) (C19)
γ0 =e2M2
N
π∆2M4+
(minusg
2M
∆2+
g2E
M2+
+4gMgE∆M+
) (C20)
bull Slopes of polarizabilities at Q2 = 0
dγ0(Q2)
dQ2
∣∣∣∣∣Q2=0
= minus e2
πM2+∆
(g2M
∆
[1
4∆2minus 1
∆M++
1
2M2+
]minus 1
Λ2
g2M
∆+
g2E
2M2+
[1
2∆minus 3
M+
]minusgMgE
M+
[1
∆2minus 5
∆M++
1
M2+
]+
1
Λ2
2gMgEM+
+2gMgC∆M2
+
minus gEgCM3
+
) (C21)
40
d δLT (Q2)
dQ2
∣∣∣∣∣Q2=0
=e2M∆∆
4πMNM2+
(g2E
∆2M2+
[1
∆minus 4
M+
]minus g2
C
∆M2∆M
2+
+gMgE∆2M+
[1
∆2
minus 3
∆M++
1
M2+
]minus 2
Λ2
gMgE∆2M+
+gMgC∆M2
∆
[1
2∆2minus 2
∆M++
1
2M2+
]minus gEgC
2M2∆M
2+
[7
∆+
1
M+
]) (C22)
dIA(Q2)
dQ2
∣∣∣∣∣Q2=0
= minusM2N
M2+
(g2M
2∆2+
g2E
MNM+minus 2gMgE
∆M+minus gEgCM∆M+
) (C23)
dI1(Q2)
dQ2
∣∣∣∣∣Q2=0
= minusM∆M2N
2M3+
(g2E
MNM∆minus gMgE∆MN
minus gEgCM2
∆
) (C24)
γ0(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2M2
N
π∆3M6+
(g2M
∆
[3M2
∆ + 2M∆MN minus 9M2N
∆2+
4M2+
Λ2
]
minus4gMgE
[5M2
∆ minus 9M2N
M+∆2+
2M+
Λ2
]+
g2E
∆M+[7M∆ minus 9MN ]
minus8gMgC∆
+4gEgCM+
) (C25)
[1] D Drechsel B Pasquini and M Vanderhaeghen Dispersion relations in real and virtual
Compton scattering Phys Rept 378 99 (2003) hep-ph0212124
[2] S E Kuhn J-P Chen and E Leader Spin structure of the nucleon mdash status and recent
results Prog Part Nucl Phys 63 1 (2009) arXiv08123535 [hep-ph]
[3] F Hagelstein R Miskimen and V Pascalutsa Nucleon polarizabilities from Compton scat-
tering to hydrogen atom Prog Part Nucl Phys 88 29 (2016) arXiv151203765 [nucl-th]
[4] B Pasquini and M Vanderhaeghen Dispersion theory in electromagnetic interactions Ann
Rev Nucl Part Sci 68 75 (2018) arXiv180510482 [hep-ph]
[5] S Gerasimov A Sum rule for magnetic moments and the damping of the nucleon magnetic
moment in nuclei Sov J Nucl Phys 2 430 (1966)
[6] S Drell and A C Hearn Exact sum rule for nucleon magnetic moments Phys Rev Lett
16 908 (1966)
[7] J S Schwinger Source theory viewpoints in deep inelastic scattering Proc Natl Acad Sci
USA 72 1 (1975)
[8] J S Schwinger Source theory viewpoints in deep inelastic scattering Electromagnetic In-
teractions and Field Theory Proceedings 14 Internationale Universitatswochen Schladming
Austria February 24-March 7 1975 Acta Phys Austriaca Suppl 14 471 (1975)
41
[9] J Schwinger Source theory discussion of deep inelastic scattering with polarized particles
Proc Natl Acad Sci USA 72 1559 (1975)
[10] A M Baldin Polarizability of nucleons Nucl Phys 18 310 (1960)
[11] M Gell-Mann M L Goldberger and W E Thirring Use of causality conditions in quantum
theory Phys Rev 95 1612 (1954)
[12] A Pineda Leading chiral logarithms to the hyperfine splitting of the hydrogen and muonic
hydrogen Phys Rev C 67 025201 (2003)
[13] C Peset and A Pineda The two-photon exchange contribution to muonic hydrogen from
chiral perturbation theory Nucl Phys B 887 69 (2014) arXiv14064524 [hep-ph]
[14] C Peset and A Pineda Model-independent determination of the two-photon exchange contri-
bution to hyperfine splitting in muonic hydrogen JHEP 04 060 arXiv161205206 [nucl-th]
[15] F Hagelstein and V Pascalutsa Proton structure in the hyperfine splitting of muonic hydro-
gen PoS CD15 077 (2016) arXiv151104301 [nucl-th]
[16] F Hagelstein Exciting Nucleons in Compton Scattering and Hydrogen-Like Atoms PhD
thesis Mainz U Inst Kernphys (2017) arXiv171000874 [nucl-th]
[17] F Hagelstein ∆(1232)-Resonance in the hydrogen spectrum Proceedings 11th International
Workshop on the Physics of Excited Nucleons (NSTAR 2017) Columbia SC USA August
20-23 2017 Few Body Syst 59 93 (2018) arXiv180109790 [nucl-th]
[18] Y Prok et al (CLAS) Moments of the spin structure functions gp1 and gd1 for 005 lt Q2 lt 30
GeV2 Phys Lett B 672 12 (2009) arXiv08022232 [nucl-ex]
[19] H Dutz et al (GDH) First measurement of the Gerasimov-Drell-Hearn sum rule for 1H from
07 GeV to 18 GeV at ELSA Phys Rev Lett 91 192001 (2003)
[20] M Amarian et al (Jefferson Lab E94010) Measurement of the generalized forward spin po-
larizabilities of the neutron Phys Rev Lett 93 152301 (2004) arXivnucl-ex0406005
[21] M Amarian et al The Q2 evolution of the generalized Gerasimov-Drell-Hearn integral for
the neutron using a 3He target Phys Rev Lett 89 242301 (2002) arXivnucl-ex0205020
[22] M Amarian et al (Jefferson Lab E94-010)Q2 evolution of the neutron spin structure moments
using a 3He target Phys Rev Lett 92 022301 (2004) arXivhep-ex0310003
[23] A Deur et al Experimental determination of the evolution of the Bjorken integral at low Q2
Phys Rev Lett 93 212001 (2004) arXivhep-ex0407007
[24] K Slifer Low Q2 measurement of gp2 and the δLT spin polarizability Spin structure at long dis-
tance Proceedings Workshop Newport News USA March 12ndash13 2009 nucl-ex09064775
(2009) AIP Conf Proc 1155 10106313203293 (2009) arXiv09064775 [nucl-ex]
[25] P Solvignon et al (E01-012) Moments of the neutron g2 structure function at intermediate
42
Q2 Phys Rev C 92 015208 (2015) arXiv13044497 [nucl-ex]
[26] A Deur Experimental studies at low Q2 of the spin structure of the nucleon at Jefferson Lab
in 9th International Workshop on Chiral Dynamics (CD18) Durham NC USA September
17-21 2018 (2019) arXiv190305661 [nucl-ex]
[27] V Sulkosky et al (Jefferson Lab E97-110) Measurement of the 3He spin-structure functions
and of neutron (3He) spin-dependent sum rules at 0035 le Q2 le 024 GeV2 Phys Lett B
805 135428 (2020) arXiv190805709 [nucl-ex]
[28] K Adhikari et al (CLAS) Measurement of the Q2 Dependence of the Deuteron Spin Structure
Function g1 and its Moments at Low Q2 with CLAS Phys Rev Lett 120 062501 (2018)
arXiv171101974 [nucl-ex]
[29] V Lensky F Hagelstein A Hiller Blin and V Pascalutsa Comment on rdquoMeasurement of
the Q2 Dependence of the Deuteron Spin Structure Function g1 and its Moments at Low Q2
with CLASrdquo (2018) arXiv180603219 [nucl-th]
[30] J M Alarcn F Hagelstein V Lensky and V Pascalutsa Forward doubly-virtual Compton
scattering off the nucleon in chiral perturbation theory at NLO the subtraction function and
moments of unpolarized structure functions (2020) arXiv200509518 [hep-ph]
[31] V Bernard E Epelbaum H Krebs and U-G Meiszligner New insights into the spin structure
of the nucleon Phys Rev D 87 054032 (2013) arXiv12092523 [hep-ph]
[32] V Lensky J M Alarcon and V Pascalutsa Moments of nucleon structure functions at
next-to-leading order in baryon chiral perturbation theory Phys Rev C 90 055202 (2014)
arXiv14072574 [hep-ph]
[33] V Pascalutsa and M Vanderhaeghen Chiral effective-field theory in the ∆(1232) region
I Pion electroproduction on the nucleon Phys Rev D 73 034003 (2006) arXivhep-
ph0512244
[34] V Pascalutsa and M Vanderhaeghen Electromagnetic nucleon-to-Delta transition in chiral
effective field theory Phys Rev Lett 95 232001 (2005) arXivhep-ph0508060
[35] T R Hemmert B R Holstein and J Kambor Systematic 1M expansion for spin 32
particles in baryon chiral perturbation theory Phys Lett B 395 89 (1997) arXivhep-
ph9606456
[36] V Pascalutsa and D R Phillips Effective theory of the ∆(1232) in Compton scattering off
the nucleon Phys Rev C 67 055202 (2003) arXivnucl-th0212024
[37] V Pascalutsa M Vanderhaeghen and S N Yang Electromagnetic excitation of the ∆(1232)-
resonance Phys Rept 437 125 (2007) arXivhep-ph0609004
[38] K A Olive et al (Particle Data Group) Review of Particle Physics Chin Phys C 38 090001
43
(2014)
[39] F E Low Scattering of light of very low frequency by systems of spin 12 Phys Rev 96
1428 (1954)
[40] M Gell-Mann and M L Goldberger Scattering of low-energy photons by particles of spin
12 Phys Rev 96 1433 (1954)
[41] A M Harun ar-Rashid A simple derivation of Schwingerrsquos sum rule for spin dependent
structure functions Nuovo Cim A 33 447 (1976)
[42] F Hagelstein and V Pascalutsa Dissecting the hadronic contributions to (g minus 2)micro by
Schwingerrsquos sum rule Phys Rev Lett 120 072002 (2018) arXiv171004571 [hep-ph]
[43] V Lensky V Pascalutsa M Vanderhaeghen and C Kao Spin-dependent sum rules con-
necting real and virtual Compton scattering verified Phys Rev D 95 074001 (2017)
arXiv170101947 [hep-ph]
[44] H Burkhardt and W N Cottingham Sum rules for forward virtual Compton scattering
Annals Phys 56 453 (1970)
[45] V Lensky and V Pascalutsa Predictive powers of chiral perturbation theory in Compton
scattering off protons Eur Phys J C 65 195 (2010) arXiv09070451 [hep-ph]
[46] H Grieszlighammer J McGovern D Phillips and G Feldman Using effective field theory to
analyse low-energy Compton scattering data from protons and light nuclei Prog Part Nucl
Phys 67 841 (2012) arXiv12036834 [nucl-th]
[47] H W Grieszlighammer J A McGovern and D R Phillips Nucleon polarisabilities at and
beyond physical pion masses Eur Phys J A 52 139 (2016) arXiv151101952 [nucl-th]
[48] E Epelbaum H Krebs and U-G Meiszligner Improved chiral nucleon-nucleon potential up to
next-to-next-to-next-to-leading order Eur Phys J A 51 53 (2015) arXiv14120142 [nucl-
th]
[49] E Epelbaum H Krebs and U-G Meiszligner Precision nucleon-nucleon potential at fifth order
in the chiral expansion Phys Rev Lett 115 122301 (2015) arXiv14124623 [nucl-th]
[50] C W Kao T Spitzenberg and M Vanderhaeghen Burkhardt-Cottingham sum rule and
forward spin polarizabilities in heavy baryon chiral perturbation theory Phys Rev D 67
016001 (2003) arXivhep-ph0209241
[51] C-W Kao D Drechsel S Kamalov and M Vanderhaeghen Higher moments of nucleon
spin structure functions in heavy baryon chiral perturbation theory and in a resonance model
Phys Rev D 69 056004 (2004) arXivhep-ph0312102
[52] T Becher and H Leutwyler Baryon chiral perturbation theory in manifestly Lorentz invariant
form Eur Phys J C 9 643 (1999) arXivhep-ph9901384
44
[53] J Gasser M E Sainio and A Svarc Nucleons with chiral loops Nucl Phys B 307 779
(1988)
[54] L S Geng J Martin Camalich L Alvarez-Ruso and M J Vicente Vacas Leading SU(3)-
breaking corrections to the baryon magnetic moments in chiral perturbation theory Phys
Rev Lett 101 222002 (2008) arXiv08051419 [hep-ph]
[55] D Drechsel S S Kamalov and L Tiator The GDH sum rule and related integrals Phys
Rev D 63 114010 (2001) arXivhep-ph0008306
[56] D Drechsel O Hanstein S S Kamalov and L Tiator A unitary isobar model for pion
photo- and electroproduction on the proton up to 1 GeV Nucl Phys A 645 145 (1999)
arXivnucl-th9807001
[57] L Tiator private communication (2020)
[58] V Bernard T R Hemmert and U-G Meiszligner Spin structure of the nucleon at low energies
Phys Rev D 67 076008 (2003) arXivhep-ph0212033
[59] O Gryniuk F Hagelstein and V Pascalutsa Evaluation of the forward Compton scattering
off protons II Spin-dependent amplitude and observables Phys Rev D 94 034043 (2016)
arXiv160400789 [nucl-th]
[60] R Zielinski The g2p Experiment A Measurement of the Protonrsquos Spin Structure Functions
PhD thesis New Hampshire U (2010) arXiv170808297 [nucl-ex]
[61] N Guler et al (CLAS) Precise determination of the deuteron spin structure at low to mod-
erate Q2 with CLAS and extraction of the neutron contribution Phys Rev C 92 055201
(2015) arXiv150507877 [nucl-ex]
[62] C E Carlson V Nazaryan and K Griffioen Proton structure corrections to electronic
and muonic hydrogen hyperfine splitting Phys Rev A 78 022517 (2008) arXiv08052603
[physicsatom-ph]
[63] V Bernard N Kaiser and U-G Meiszligner Chiral dynamics in nucleons and nuclei Int J
Mod Phys E 4 193 (1995) arXivhep-ph9501384
[64] T R Hemmert B R Holstein and J Kambor ∆(1232) and the polarizabilities of the nucleon
Phys Rev D 55 5598 (1997) arXivhep-ph9612374
[65] V Pascalutsa and R Timmermans Field theory of nucleon to higher-spin baryon transitions
Phys Rev C 60 042201 (1999) arXivnucl-th9905065
[66] V Pascalutsa Quantization of an interacting spin-32 field and the Delta isobar Phys Rev
D 58 096002 (1998) arXivhep-ph9802288
[67] H Krebs Double Virtual Compton Scattering and SpinStructure of the Nucleon PoS
CD2018 031 (2019)
45
[68] N Kochelev and Y Oh Axial anomaly and the δLT puzzle Phys Rev D 016012 (2012)
arXiv11034892 [hep-ph]
[69] D Drechsel S Kamalov and L Tiator Unitary isobar model ndash MAID2007 Eur Phys J A
69 (2007) available at httpsmaidkphuni-mainzde arXiv07100306 [nucl-th]
[70] P J Mohr B N Taylor and D B Newell CODATA recommended values of the fundamental
physical constants 2010 Rev Mod Phys 84 1527 (2012)
[71] J Ahrens et al (GDH A2) First measurement of the Gerasimov-Drell-Hearn integral for 1H
from 200 to 800 MeV Phys Rev Lett 87 022003 (2001) arXivhep-ex0105089 [hep-ex]
[72] K Helbing (GDH) Experimental verification of the GDH sum rule at ELSA and MAMI Nucl
Phys Proc Suppl 105 113 (2002)
[73] R Bradford A Bodek H S Budd and J Arrington A New parameterization of the nucleon
elastic form-factors NuInt05 proceedings of the 4th International Workshop on Neutrino-
Nucleus Interactions in the Few-GeV Region Okayama Japan 26-29 September 2005 Nucl
Phys Proc Suppl 159 127 (2006) arXivhep-ex0602017
[74] J D Bjorken Applications of the chiral U(6)timesU(6) algebra of current densities Phys Rev
148 1467 (1966)
[75] J D Bjorken Inelastic scattering of polarized leptons from polarized nucleons Phys Rev D
1 1376 (1970)
[76] V Bernard T R Hemmert and U-G Meiszligner Novel analysis of chiral loop effects in
the generalized Gerasimov-Drell-Hearn sum rule Phys Lett B 545 105 (2002) arXivhep-
ph0203167
[77] A Deur et al Experimental study of isovector spin sum rules Phys Rev D 78 032001
(2008) arXiv08023198 [nucl-ex]
[78] B Pasquini P Pedroni and D Drechsel Higher order forward spin polarizability Phys Lett
B 687 160 (2010) arXiv10014230 [hep-ph]
[79] R Jaffe g2ndashThe nucleonrsquos other spin-dependent structure function Comments Nucl Part
Phys 19 239 (1990)
[80] E V Shuryak and A Vainshtein Theory of power corrections to deep inelastic scattering in
quantum chromodynamics (II) Qminus4 effects polarized target Nucl Phys B 201 141 (1982)
[81] S Wandzura and F Wilczek Sum rules for spin dependent electroproduction Test of rela-
tivistic constituent quarks Phys Lett B 72 195 (1977)
[82] B W Filippone and X-D Ji The spin structure of the nucleon Adv Nucl Phys 26 1
(2001) arXivhep-ph0101224
[83] M Burkardt The g2 structure function Proceedings Workshop on Spin structure at long
46
distance Newport News USA March 12-13 2009 AIP Conf Proc 1155 26 (2009)
arXiv09054079 [hep-ph]
[84] A Deur S J Brodsky and G F de Teramond The spin structure of the nucleon Rept
Prog Phys 82 076201 (2019) arXiv180705250 [hep-ph]
[85] B R Holstein D Drechsel B Pasquini and M Vanderhaeghen Higher order polarizabilities
of the proton Phys Rev C 61 034316 (2000) arXivhep-ph9910427
[86] B R Holstein V Pascalutsa and M Vanderhaeghen Sum rules for magnetic moments and
polarizabilities in QED and chiral effective-field theory Phys Rev D 72 094014 (2005)
arXivhep-ph0507016
[87] B Pasquini D Drechsel and L Tiator Invariant amplitudes for pion electroproduction Eur
Phys J A 34 387 (2007) arXiv07122327 [hep-ph]
[88] A J Buchmann Non-spherical proton shape and hydrogen hyperfine splitting Proceedings
International Workshop on Precision Physics of Simple Atomic Systems (PSAS 2008) Wind-
sor Ontario Canada July 21-26 2008 Can J Phys 87 773 (2009) arXiv09104747
[physicsatom-ph]
47
Here we used the elastic Dirac and Pauli form factors F1(Q2) and F2(Q2) related to the
electric and magnetic Sachs form factors GE(Q2) and GM(Q2) through
F1(Q2) =GE(Q2) + τGM(Q2)
1 + τ (9a)
F2(Q2) =GM(Q2)minusGE(Q2)
1 + τ (9b)
where τ = Q24M2N
A low-energy expansion (LEX) of Eq (5) in combination with the unitarity relations
given in Eq (3) establishes various sum rules relating the nucleon properties (electromag-
netic moments polarizabilities) to experimentally observable response functions [1 3] The
leading terms in the LEX of the RCS amplitudes are determined uniquely by charge mass
and anomalous magnetic moment as the global properties of the nucleon These lowest-
order terms represent the celebrated low-energy theorem (LET) of Low Gell-Mann and
Goldberger [39 40] The polarizabilities related to the internal structure of the nucleon
enter the LEX at higher orders They make up the non-Born amplitudes and can be related
to moments of inelastic structure functions
The process of VVCS can be realized experimentally in electron-nucleon scattering where
a virtual photon is exchanged between the electron and the nucleon This virtual photon
acts as a probe whose resolution depends on its virtuality Q2 In this way one can access
the so-called generalized polarizabilities which extend the notion of polarizabilities to the
case of response to finite momentum transfer The generalized forward spin polarizability
γ0(Q2) and the longitudinal-transverse polarizability δLT (Q2) are most naturally defined via
the LEX of the non-Born part of the lab-frame VVCS amplitudes [1]
1
4πgnonpoleTT (νQ2) =
2α
M2N
IA(Q2) ν + γ0(Q2)ν3 + γ0(Q2)ν5 + (10a)
1
4πgnonpoleLT (νQ2) =
2α
M2N
I3(Q2)Q+ δLT (Q2)ν2Q+ (10b)
Their definitions in terms of integrals over structure functions are postponed to Eqs (19)
and (22) Here we only give the definition of the moment I3(Q2)
I3(Q2) =M2
N
4π2α
int infinν0
dνK(νQ2)
νQσLT (νQ2) =
2M2N
Q2
int x0
0
dx[g1(xQ2) + g2(xQ2)
] (11)
which is related to the Schwinger sum rule in the real photon limit [7ndash9 41 42] The LEX
of the non-pole part of the covariant VVCS amplitudes can be described entirely in terms
7
of moments of inelastic spin structure functions (up to O(ν4) [43])
1
4πSnonpole
1 (νQ2) =2α
MN
I1(Q2) +
2α
MNQ2
[IA(Q2)minus I1(Q2)
]+MNδLT (Q2)
ν2 (12a)
1
4πνSnonpole
2 (νQ2) = 2αI2(Q2) +2α
Q2
[I1(Q2)minus IA(Q2)
]ν2 (12b)
I1(Q2) and IA(Q2) are generalizations of the famous GerasimovndashDrellndashHearn (GDH) sum
rule [5 6] from RCS to the case of virtual photons [1] Their definitions are given in Eqs (26)
and (32) I2(Q2) is the well-known Burkhardt-Cottingham (BC) sum rule [44]
I2(Q2) equiv 2M2N
Q2
int x0
0
dx g2(x Q2) =1
4F2(Q2)GM(Q2) (13)
which can be written as a ldquosuperconvergence sum rulerdquo
Q2
16παM2N
limνrarr0
νS2(νQ2) =
int 1
0
dx g2 (x Q2) = 0 (14)
The latter is valid for any value of Q2 provided that the integral converges for x rarr 0
Combining Eq (5) with the above LEXs of the VVCS amplitudes we can relate IA(Q2)
I1(Q2) γ0(Q2) and δLT (Q2) to moments of inelastic structure functions see Sec III It is
important to note that only γ0(Q2) and δLT (Q2) are generalized polarizabilities The relation
of the inelastic moments IA(Q2) and I1(Q2) to polarizabilities will be discussed in details in
Secs III C and III D The difference between S1(νQ2) and Snonpole1 (νQ2) cf Eq (8a) will
be important in this context
C Details on χPT calculation and uncertainty estimate
In this work we calculated the NLO prediction of BχPT for the polarized non-Born
VVCS amplitudes This includes the leading pion-nucleon (πN) loops see Ref [32 Fig 1]
as well as the subleading tree-level Delta-exchange (∆-exchange) see Ref [30 Fig 2] and
the pion-Delta (π∆) loops see Ref [32 Fig 2] In the δ-power-counting scheme [36] the
LO and NLO non-Born VVCS amplitudes and polarizabilities are of O(p3) and O(p4∆)
respectively2 The LECs are listed in Table I sorted by the order at which they appear in
our calculation At the given orders there are no ldquonewrdquo LECs that would need to be fitted
from Compton processes For more details on the BχPT formalism we refer to Ref [30]
2 In the full Compton amplitude there is a lower order contribution coming from the Born terms leading
to a shift in nomenclature by one order the LO contribution referred to as the NLO contribution etc
see eg Ref [45]
8
where power counting predictive orders (Sec III A) and the renormalization procedure
(Sec III B) are discussed
A few remarks are in order for the inclusion of the ∆(1232) and the tree-level ∆-exchange
contribution In contrast to Ref [32] we include the Coulomb-quadrupole (C2) N rarr ∆
transition described by the LEC gC The relevant Lagrangian describing the non-minimal
γlowastN∆ coupling [33 34] (note that in these references the overall sign of gC is inconsistent
between the Lagrangian and Feynman rules) reads
L(2)∆ =
3e
2MNM+
N T3
igM F
microν partmicro∆ν minus gEγ5Fmicroν partmicro∆ν (15)
+igCM∆
γ5γα(partα∆ν minus partν∆α)partmicroF
microν
+ Hc
with M+ = MN + M∆ and the dual of the electromagnetic field strength tensor F microν =12εmicroνρλFρλ Even though the Coulomb coupling is subleading compared with the electric
and magnetic couplings (gE and gM) its relatively large magnitude cf Table I makes it
numerically important for instance in γ0(Q2) Furthermore we study the effect of modifying
the magnetic coupling using a dipole form factor
gM rarrgM[
1 + (QΛ)2 ]2 (16)
where Λ2 = 071 GeV2 The inclusion of this Q2 dependence mimics the form expected from
vector-meson dominance It is motivated by observing the importance of this form factor
for the correct description of the electroproduction data [33]
To estimate the uncertainties of our NLO predictions we define
δ(Q2) =
radic(∆
MN
)2
+
(Q2
2MN∆
)2
(17)
such that the neglected next-to-next-to-leading order terms are expected to be of relative
size δ2 [33] The uncertainties in the values of the parameters in Table I have a much
smaller impact compared to the truncation uncertainty and can be neglected Unfortunately
∆IA(Q2) γ0(Q2) and γ0(Q2) ie the sum rules involving the cross section σTT (νQ2)
as well as the polarizability ∆I1(Q2) turn out to be numerically small Their smallness
suggests a cancellation of leading orders (which can indeed be confirmed by looking at
separate contributions as shown below) Therefore an error of δ2(Q2)P (Q2) where P (Q2)
is a generalized polarizability might underestimate the theoretical uncertainty for some of
the NLO predictions To avoid this we estimate the uncertainty of our NLO polarizability
9
predictions by
σP (Q2) =
Max[δ4(0)P (0)2 δ4(0)P LO(0)2 δ2(0)PNLO(0)2
]+Max
[δ4(Q2)
[P (Q2)minus P (0)
]2 δ4(Q2)
[P LO(Q2)minus P LO(0)
]2
δ2(Q2)[PNLO(Q2)minus PNLO(0)
]2]12
(18)
where P LO(Q2) is the πN -loop contribution PNLO(Q2) are the ∆-exchange and π∆-loop
contributions and P (Q2) = P LO(Q2) + PNLO(Q2) This error prescription is similar to the
one used in eg Refs [46ndash49] Here since we are interested in the generalized polarizabil-
ities we added in quadrature the error due to the static piece P (0) and the Q2-dependent
remainder P (Q2) minus P (0) Note that the static values of IA(0) and I1(0) are given by the
elastic Pauli form factor which is not part of our BχPT prediction and is considered to be
exact
Note that our result for the spin polarizabilities (and the unpolarized moments [30]) are
NLO predictions only at low momentum transfers Q mπ At larger values of Q amp ∆ they
become incomplete LO predictions Indeed in this regime the ∆ propagators do not carry
additional suppression compared to the nucleon propagators and the π∆ loops are promoted
to LO In general we only expect a rather small contribution from omitted π∆ loops to the
Q2 dependence of the polarizabilities since π∆ loops show rather weak dependence on Q2
compared with the ∆ exchange or πN loops Nevertheless this issue has to be reflected in
the error estimate Since the static polarizabilities P (0) are not affected it is natural to
separate the error on the Q2-dependent remainder P (Q2) minus P (0) as done in Eq (18) To
accommodate for the potential loss of precision above Q amp ∆ we define the relative error
δ(Q2) as growing with increasing Q2 see Eq (17)
Upon expanding our results in powers of the inverse nucleon mass Mminus1N we are able to
reproduce existing results of heavy-baryon χPT (HBχPT) at LO We however do not see
a rationale to drop the higher-order Mminus1N terms when they are not negligible (ie when
their actual size exceeds by far the natural estimate for the size of higher-order terms)
Comparing our BχPT predictions to HBχPT we will also see a deficiency of HBχPT in
the description of the Q2 behaviour of the polarizabilities Note that the O(p4) HBχPT
results from Ref [50 51] which we use here for comparison do not include the ∆ These
references studied the leading effect of the latter in the HBχPT framework using the small-
scale expansion [35] observing no qualitative improvement in the HBχPT description of the
empirical data [50 51] when including it We therefore choose to use the O(p4) results as
the representative HBχPT curves
10
Another approach used in the literature to calculate the polarizabilities in χPT is the
infrared regularization (IR) scheme introduced in Ref [52] This covariant approach tries to
solve the power counting violation observed in Ref [53] by dropping the regular parts of the
loop integrals that contain the power-counting-breaking terms However this subtraction
scheme modifies the analytic structure of the loop contributions and may lead to unexpected
problems as was shown in Ref [54] As we will see in the next section the IR approach
also fails to describe the Q2 behaviour of the polarizabilities
III RESULTS AND DISCUSSION
We now present the NLO BχPT predictions for the nucleon polarizabilities and selected
moments of the nucleon spin structure functions Our results are obtained from the calcu-
lated non-Born VVCS amplitudes and the LEXs in Eqs (10) and (12) For a cross-check
we used the photoabsorption cross sections described in App B In addition to the full NLO
results we also analyse the individual contributions from the πN loops the ∆ exchange
and the π∆ loops
A γ0(Q2) mdash generalized forward spin polarizability
The forward spin polarizability
γ0(Q2) =1
2π2
int infinν0
dν
radic1 +
Q2
ν2
σTT (νQ2)
ν3(19)
=16αM2
N
Q6
int x0
0
dx x2
[g1(xQ2)minus 4M2
Nx2
Q2g2(xQ2)
]
provides information about the spin-dependent response of the nucleon to transversal pho-
ton probes The RCS analogue of the above generalized forward spin polarizability sum
rule is sometimes referred to as the Gell-Mann Goldberger and Thirring (GGT) sum rule
[11] At Q2 = 0 the forward spin polarizability is expressed through the lowest-order spin
polarizabilities of RCS as γ0 = minus(γE1E1 + γM1M1 + γE1M2 + γM1E2) The forward spin po-
larizability of the proton is relevant for an accurate knowledge of the (muonic-)hydrogen
hyperfine splitting as it controls the leading proton-polarizability correction [16 62]
The πN -loop ∆-exchange and π∆-loop contributions to the NLO BχPT prediction of
11
000 005 010 015 020 025 030
-4
-3
-2
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
4D
proton
000 005 010 015 020 025 030
-4
-3
-2
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
4D
neutron
000 005 010 015 020 025 030
0
1
2
Q2 GeV2D
∆LT
10-
4fm
4D
proton
000 005 010 015 020 025 030
0
1
2
3
Q2 GeV2D
∆LT
10-
4fm
4D
neutron
FIG 2 Upper panel Generalized forward spin polarizability for the proton (left) and neutron
(right) as function of Q2 The result of this work the NLO BχPT prediction is shown by the
blue solid line and the blue band The red line represents the LO BχPT result The purple short-
dashed line is the O(p4) HB result from Ref [50] note that the corresponding proton curve is
outside of the plotted range The black dotted line is the MAID model prediction [55ndash57] which
is taken from Ref [1] (proton) and Ref [20] (neutron) The pink band is the IR+∆ result from
Ref [58] and the gray band is the BχPT+∆ result from Ref [31] Empirical extractions for the
proton Ref [18] (blue dots) Ref [59] (purple square) and Ref [60] (orange triangle uncertainties
added in quadrature) and neutron Ref [20] (blue diamonds) and Ref [61] (green dots statistical
and systematic uncertainties added in quadrature) Lower Panel Longitudinal-transverse spin
polarizability for the proton (left) and neutron (right) The orange dot-dashed and purple short-
dashed lines are the O(p3) and O(p4) HB results from Ref [50] The pink band is the IR result
from Ref [58] and the gray band is the covariant BχPT+∆ result from Ref [31] The black dotted
line is the MAID model prediction [55ndash57] note that for the proton we use the updated estimate
from Ref [1] obtained using the π η ππ channels
12
the static forward spin polarizability amount to in units of 10minus4 fm4
γ0p = minus093(92) asymp 201minus 284minus 010 (20a)
γ0n = 003(92) asymp 298minus 284minus 010 (20b)
while the slope is composed as follows in units of 10minus4 fm6
dγ0p(Q2)
dQ2
∣∣∣∣Q2=0
= minus022(4) asymp minus033 + 011 + 001 (21a)
dγ0n(Q2)
dQ2
∣∣∣∣Q2=0
= minus061(7) asymp minus073 + 011 + 001 (21b)
Figure 2 upper panel shows our NLO prediction as well as the LO πN loops compared
to different experimental and theoretical results For the proton we have one determination
at the real-photon point by the GDH collaboration [19] γ0p = minus100(8)(12)times10minus4 fm4 and
further Jefferson Laboratory data [18 60] at very low Q2 For the neutron only data at finite
Q2 are available [20 61] The experimental data for the proton are fairly well reproduced
in the whole Q2 range considered here while for the neutron the agreement improves with
increasing Q2 The HB limit of our πN -loop contribution reproduces the results published
in Refs [50 63] for arbitrary Q2 In addition our prediction is compared to the MAID
model [1 20] the IR+∆ calculation of Ref [58] and the BχPT+∆ result of Ref [31]
The πN -production channel gives a positive contribution to the photoabsorption cross
section σTT (νQ2) at low Q2 cf Fig 10 Accordingly one observes that the πN loops give
a sizeable positive contribution to γ0(Q2) The Delta on the other hand has a very large
effect by cancelling the πN loops and bringing the result close to the empirical data From
Fig 3 upper panel one can see that it is the ∆ exchange which dominates while π∆
loops are negligible This was expected since the forward spin polarizability sum rule is an
integral over the helicity-difference cross section in which σ32 is governed by the Delta at
low energies (the relevant energy region for the sum rule)
To elucidate the difference between the present calculation and the one from Ref [31] we
note that the two calculations differ in the following important aspects Firstly Ref [31] uses
the small-scale counting [64] that considers ∆ and mπ as being of the same size ∆ sim mπ
In practice this results in a set of π∆-loop graphs which contains graphs with one or two
γ∆∆ couplings and hence two or three Delta propagators Such graphs are suppressed in
the δ-counting and thus omitted from our calculation while present in that of Ref [31]
Secondly the Lagrangians describing the interaction of the Delta are constructed differently
and assume slightly different values for the coupling constants In particular we employ
13
000 005 010 015 020 025 030
-3
-2
-1
0
1
2
Q2GeV
2D
Γ01
0-
4fm
4D
proton
000 005 010 015 020 025 030
-3
-2
-1
0
1
2
3
Q2GeV
2D
Γ01
0-
4fm
4D
neutron
000 005 010 015 020 025 030
00
05
10
15
Q2GeV
2D
∆LT1
0-
4fm
4D
proton
000 005 010 015 020 025 030
00
05
10
15
20
25
Q2GeV
2D
∆LT1
0-
4fm
4D
neutron
FIG 3 Contributions of the different orders to the chiral predictions of γ0(Q2) upper paneland δLT (Q2) lower panel for the proton (left) and neutron (right) Red solid line πN -loop
contribution green dot-dashed line ∆-exchange contribution orange dotted line π∆-loop con-
tribution blue long-dashed line total result purple dot-dot-dashed line total result without gCcontribution black short-dashed line total result without gM dipole
(where possible) the so-called ldquoconsistentrdquo couplings to the Delta field ie those couplings
that project out the spurious degree of freedom see Refs [37 65 66] The authors of
Ref [31] on the other hand use couplings where the consistency in this sense is not enforced
The effects of these differences are of higher order in the δ-counting expansion and their
contribution to the Q2 dependence of the considered polarizabilities is expected to be rather
small however the differences at Q2 = 0 could be noticeable [67]
Finally as mentioned in Sec II C the inclusion of the dipole form factor in the magnetic
coupling gM is expected to be important to generate the correct Q2 behaviour of the polar-
izabilities Comparing our predictions for the forward spin polarizability with and without
inclusion of the form factor see Fig 3 upper panel confirms this Without the dipole
our results for the proton and neutron are closer to the ones from Ref [31] where the form
factor is not included For the neutron our prediction without the dipole is able to describe
14
the experimental points at very low Q2 deviating from the data with increasing Q2 The
π∆-loop contribution does not modify the Q2 behavior of γ0(Q2) and only differs from
Ref [31] by a small global shift Note also the relatively large effect of gC which generates
a sign change for virtualities above sim 02 GeV2 see Fig 3 upper panel
B δLT (Q2) mdash longitudinal-transverse polarizability
The longitudinal-transverse spin polarizability
δLT (Q2) =1
2π2
int infinν0
dν
radic1 +
Q2
ν2
σLT (νQ2)
Qν2(22)
=16αM2
N
Q6
int x0
0
dx x2[g1(xQ2) + g2(xQ2)
]
contains information about the spin structure of the nucleon and is another important
input in the determination of the (muonic-)hydrogen hyperfine splitting [16 62] It is also
relevant in studies of higher-twist corrections to the structure function g2(xQ2) given by
the moment d2(Q2) [51] see Section III E The peculiarity of the response encoded in this
polarizability is that it involves a spin flip of the nucleon and a polarization change of the
photon see App B and Fig 11
It is expected that the Delta isobar gives only a small contribution to δLT (Q2) what makes
this polarizability a potentially clean test case for chiral calculations Consequently there are
relatively many different theoretical calculations of δLT (Q2) coming from different versions
of χPT with baryons (HB IR and covariant) Ref [50] found a systematic deviation of the
HB result for δLTn(Q2) from the MAID model prediction This disagreement was identified
by the authors of Ref [68] as a puzzle involving the neutron δLT polarizabilitymdashthe δLT
puzzle The IR calculation in Ref [58] also showed a deviation from the data and predicted
a rapid rise of δLT (Q2) with growing Q2 The problem is solved by keeping the relativistic
structure of the theory as the BχPT+∆ result of Ref [31] showed
As expected already the leading πN loops provide a reasonable agreement with the
experimental data cf Fig 2 lower panel Since the ∆-exchange contribution to δLT (Q2)
is small the effect of the gM form factor is negligible in this polarizability as is that of the gC
coupling cf Fig 3 lower panel In fact we predict both the ∆-exchange and the π∆-loop
contributions to be small and negative This is in agreement with the MAID model which
predicts a small and negative contribution of the P33 wave to δLT (Q2) However in the
calculation of Ref [31] which is different from the one presented here only in the way the
∆(1232) is included the contribution of this resonance to δLTp(Q2) is sizeable and positive
15
The authors of that work attributed this large contribution to diagrams where the photons
couple directly to the Delta inside a loop As mentioned in Sec III F the effect of such loop
diagrams does not change the Q2 behaviour of the polarizabilities On the other hand it
can produce a substantial shift of the δLT (Q2) as a whole A higher-order calculation should
resolve the discrepancy between the two covariant approaches however it will partially lose
the predictive power since the LECs appearing at higher orders will have to be fitted to
experimental data
The πN -loop ∆-exchange and π∆-loop contributions to the NLO BχPT prediction of
the static longitudinal-transverse polarizability are in units of 10minus4 fm4
δLTp = 132(15) asymp 150minus 016minus 002 (23a)
δLTn = 218(23) asymp 235minus 016minus 002 (23b)
while the slopes are in units of 10minus4 fm6
dδLTp(Q2)
dQ2
∣∣∣∣Q2=0
= minus085(8) asymp minus080minus 004minus 001 (24a)
dδLTn(Q2)
dQ2
∣∣∣∣Q2=0
= minus124(12) asymp minus119minus 004minus 001 (24b)
C IA(Q2) mdash a generalized GDH integral
The helicity-difference cross section σTT exhibits a faster fall-off in ν than its spin-averaged
counterpart σT This is due to a cancellation between the leading (constant) terms of σ12
and σ32 at large ν3 The resulting 1ν fall-off of the helicity-difference cross section allows
one to write an unsubtracted dispersion relation for the VVCS amplitude gnonpoleTT (νQ2) cf
Eq (10a) This is the origin of the GDH sum rule [5 6]
minus α
2M2N
κ2 =1
2π2
int infinν0
dνσTT (ν)
ν (25)
which establishes a relation to the anomalous magnetic moment κ It is experimentally
verified for the nucleon by MAMI (Mainz) and ELSA (Bonn) [71 72]
There are two extensions of the GDH sum rule to finite Q2 the generalized GDH integrals
3 Notice that a constant term in σTT at ν rarrinfin is forbidden by crossing symmetry
16
000 005 010 015 020 025 030
-30
-25
-20
-15
-10
-05
00
Q2 GeV2D
I Aproton
000 005 010 015 020 025 030
-25
-20
-15
-10
-05
00
Q2 GeV2D
I A
neutron
000 005 010 015 020 025 030
-05
00
05
Q2 GeV2D
I 1
proton
000 005 010 015 020 025 030
-10
-08
-06
-04
-02
00
02
Q2 GeV2D
I 1
neutron
FIG 4 Upper panel The generalized GDH integral IA(Q2) for the proton (left) and neutron
(right) as function of Q2 The result of this work the NLO BχPT prediction is shown by the blue
solid line and the blue band The red line represents the LO BχPT result The purple short-dashed
line is the O(p4) HB result from Ref [50 51] The gray band is the BχPT+∆ result from Ref [31]
The black dotted line is the MAID model prediction [69] Experimental extractions for the proton
Ref [60] (orange triangle uncertainties added in quadrature) and neutron Refs [21][27] where
magenta dotsorange diagonal crosses correspond to data and red squareslilac crosses correspond
to data plus extrapolation to unmeasured energy regions The green stars at the real-photon point
are derived from the anomalous magnetic moments κp asymp 1793 and κn asymp minus1913 [70] Lower
panel The generalized GDH integral I1(Q2) for the proton (left) and neutron (right) as function
of Q2 The purple short-dashed line is the HB result from Ref [51] Experimental extractions for
the proton Ref [18] (blue dots) and Ref [60] (orange triangle uncertainties added in quadrature)
and neutron Ref [61][27] (uncertainties added in quadrature) where green dotsorange diagonal
crosses correspond to data and gray squareslilac crosses correspond to data plus extrapolation to
unmeasured energy regions
17
IA(Q2) and I1(Q2) The latter will be discussed in Sec III D The former is defined as4
minus α
2M2N
IA(Q2) = minus 1
8π2
int infinν0
dν
radic1 +
Q2
ν2
σTT (νQ2)
ν(26)
=α
Q2
int x0
0
dx[4M2
Nx2
Q2g2(xQ2)minus g1(xQ2)
]
Due to its energy weighting the integral in Eq (26) converges slower than the one in the
generalized forward spin polarizability sum rule (19) Therefore knowledge of the cross
section at higher energies is required and the evaluation of the generalized GDH integral
IA(Q2) is not as simple as the evaluation of γ0(Q2)
The generalized GDH integral IA(Q2) is directly related to the non-pole amplitude
gnonpoleTT (νQ2) which differs from non-Born amplitude gTT (νQ2) by a term involving the
elastic Pauli form factor
gnonpoleTT (νQ2) = gTT (νQ2)minus 2παν
M2N
F 22 (Q2) (27)
cf Eqs (2a) and (8a) Consequently IA(Q2) is not a pure polarizability but also contains
an elastic contribution The ldquonon-polarizabilityrdquo or the Born part of IA(Q2) is given by
IBornA (Q2) = IA(Q2)minus∆IA(Q2) = minus1
4F 2
2 (Q2) (28)
where we refer to the polarizability part as ∆IA(Q2) The same is true for the generalized
GDH integral I1(Q2) which is directly related to Snonpole1 (νQ2)
IBorn1 (Q2) = I1(Q2)minus∆I1(Q2) = minus1
4F 2
2 (Q2) (29)
In the following we will add the Born parts to our LO and NLO BχPT predictions for
the polarizabilities ∆IA(Q2) and ∆I1(Q2) employing an empirical parametrization for the
elastic Pauli form factor [73] This allows us to compare to the experimental results for
IA(Q2) and I1(Q2) cf Fig 4 Note that the blue error bands only describe the uncertainties
of our BχPT predictions of the polarizabilities while the elastic contributions are considered
to be exact as explained in Sec II C The uncertainties of the polarizability predictions are
therefore better reflected in Fig 5 where we show the contributions of the different orders
to the BχPT predictions of ∆IA(Q2) and ∆I1(Q2) as well as the total results with error
bands
4 Note that IA(Q2) is sometimes called ITT (Q2)
18
000 005 010 015 020 025 030
-08
-06
-04
-02
00
02
Q2 GeV2D
DI A
proton
000 005 010 015 020 025 030
-10
-08
-06
-04
-02
00
02
Q2 GeV2D
DI A
neutron
000 005 010 015 020 025 030-005
000
005
010
015
020
025
Q2 GeV2D
DI 1
proton
000 005 010 015 020 025 030-015
-010
-005
000
005
010
015
Q2 GeV2D
DI 1
neutron
FIG 5 Contributions of the different orders to the chiral predictions of ∆IA(Q2) upper paneland ∆I1(Q2) lower panel for the proton (left) and neutron (right) Red solid line πN -loop
contribution green dot-dashed line ∆-exchange contribution orange dotted line π∆-loop contri-
bution blue solid line and blue band total result purple dot-dot-dashed line total result without
gC contribution black short-dashed line total result without gM dipole
The E97-110 experiment at Jefferson Lab has recently published their data for IAn(Q2) in
the region of 0035 GeV2 lt Q2 lt 024 GeV2 [27] In addition there are results for IAn(Q2)
from the earlier E94-010 experiment [21] and for IAp(Q2) from the E08-027 experiment [60]
The O(p4) HB calculation gives a large negative effect [51] which does not describe the data
The BχPT+∆ result from Ref [31] which mainly differs from our work by the absence of
the dipole form factor in gM looks similar to this HB result and only describes the data
points at lowest Q2 Our NLO prediction however follows closely the Q2 evolution of the
data In Fig 5 upper panel we show the polarizability ∆IA(Q2) whose Q2 evolution is
clearly dominated by the ∆ exchange Similar to the case of γ0p(Q2) inclusion of the dipole
in gM and the Coulomb coupling gC is very important in order to describe the experimental
data The LO prediction on the other hand slightly overestimates the data cf Fig 4
upper panel
19
At the real-photon point IA(0) = minusκ2
4and ∆IA(0) = 0 Therefore we give only the
slope of the polarizability ∆IA(Q2) [showing also the separate contributions from πN loops
∆ exchange and π∆ loops] in units of GeVminus2
d∆IAp(Q2)
dQ2
∣∣∣∣Q2=0
= minus858(343) asymp 238minus 1121 + 025 (30a)
d∆IAn(Q2)
dQ2
∣∣∣∣Q2=0
= minus955(343) asymp 141minus 1121 + 025 (30b)
Including the empirical Pauli form factor [73] we find in units of GeVminus2
dIAp(Q2)
dQ2
∣∣∣∣Q2=0
= minus318dIAn(Q2)
dQ2
∣∣∣∣Q2=0
= minus300 (31)
D Γ1(Q2) and I1(Q2) mdash the first moment of the structure function g1(xQ2)
The second variant for a generalization of the GDH sum rule to finite Q2 is defined as
minus α
2M2N
I1(Q2) = minus 1
8π2
int infinν0
dν1radic
ν2 +Q2
[σTT (νQ2) +
Q
νσLT (νQ2)
](32)
= minus α
Q2
int x0
0
dx g1(xQ2)
where I1(0) = minusκ2
4 This generalized GDH integral directly stems from the amplitude
Snonpole1 (νQ2) with the LEX from Eq (12a) It is given by the first moment of the struc-
ture function g1(xQ2) Γ1(Q2) =int x0
0dx g1(xQ2) as follows I1(Q2) =
2M2N
Q2 Γ1(Q2) The
isovector combination
Γ1(pminusn)(Q2) =
int x0
0
dx[g1p(xQ
2)minus g1n(xQ2)] (33)
is related to the axial coupling of the nucleon through the Bjorken sum rule [74 75]
limQ2rarrinfin
Γ1(pminusn)(Q2) =
gA6 (34)
As explained in Eq (28) the moment I1(Q2) splits into a polarizability part ∆I1(Q2) and
a Born part IBorn1 (Q2) Figure 4 lower panel shows the Q2 dependence of I1(Q2) which in
contrast to IA(Q2) shown in Figure 4 upper panel is clearly dominated by its Born part
and the elastic Pauli form factor The πN -loop ∆-exchange and π∆-loop contributions to
the polarizability ∆I1(Q2) are shown in Fig 5 lower panel Comparing to Fig 5 upper
20
000 005 010 015 020 025 030
-006
-004
-002
000
002
004
Q2 GeV2D
G1
proton
000 005 010 015 020 025 030
-006
-005
-004
-003
-002
-001
000
001
Q2 GeV2D
G1
neutron
FIG 6 First moment of the structure function g1(xQ2) for the proton (left) and neutron (right)
as function of Q2 The legend is the same as in Fig 4
panel one sees that ∆I1(Q2) is less sensitive to gC and the dipole form factor in gM than
∆IA(Q2)
For the proton our NLO BχPT prediction gives a very good description of the ex-
perimental data [18 60] and is in reasonable agreement with the MAID prediction [69]
For the neutron one observes good agreement with the empirical evaluations including ex-
trapolations to unmeasured energy regions starting from Q2 gt 01 GeV2 [27 61] In the
region of Q2 lt 005 GeV2 one observes an interesting tension between the recent E97-
110 experiment [27] and the data from CLAS [61] While the newest measurement finds
I1n(0035 GeV2) lt κ2n4 thus suggesting a negative slope at low Q the older measurement
found a rather large value for I1n(00496 GeV2) A similar but milder behaviour is seen in
the E97-110 [27] and E94-010 [21] data for IAn The MAID predictions do not agree with
the CODATA recommended values for the anomalous magnetic moments of the proton and
neutron [70] which in our work are imposed by using empirical parametrizations for the
elastic Pauli form factors [73] The slope of the HB result from Ref [51] is too large and
therefore only reproduces the data at very low Q2
Figure 6 shows the moment Γ1(Q2) for the proton and neutron while Fig 7 shows the
isovector combination Γ1 pminusn(Q2) The LO and NLO BχPT predictions are identical be-
cause our calculation produces the same Delta contributions for the proton and the neutron
For the isovector combination the MAID model only agrees with the data at very low
Q2 lt 010 GeV2 The same is true for the IR result [58 76] while all other chiral results
describe the data NLO BχPT (this work) BχPT+∆ [31] and HBχPT [51]
At the real-photon point I1(0) = minusκ2
4and ∆I1(0) = 0 Therefore we give only the
slope of the polarizability ∆I1(Q2) [showing also the separate contributions from πN loops
21
000 005 010 015 020 025 030-002
000
002
004
006
008
010
Q2 GeV2D
G1Hp-
nL
FIG 7 Isovector combination of Γ1(Q2) as function of Q2 The legend is the same as in Fig 6
The pink curve is the IR result from Ref [58 76] The experimental points are from Ref [77]
(brown dots) and Ref [23] (orange squares)
∆ exchange and π∆ loops] in units of GeVminus2
d∆I1p(Q2)
dQ2
∣∣∣∣Q2=0
= 039(4) asymp 034minus 053 + 058 (35a)
d∆I1n(Q2)
dQ2
∣∣∣∣Q2=0
= minus101(10) asymp minus107minus 053 + 058 (35b)
Including the empirical Pauli form factor [73] we find in units of GeVminus2
dI1p(Q2)
dQ2
∣∣∣∣Q2=0
= 580dI1n(Q2)
dQ2
∣∣∣∣Q2=0
= 553 (36)
E d2(Q2) mdash a measure of color polarizability
Another interesting moment to consider is d2(Q2) which is related to the twist-3 part of
the spin structure function g2(xQ2) [79 80]
d2(Q2) equiv 3
int 1
0
dx x2[g2(xQ2)minus gWW2 (xQ2)] (37)
where gWW2 (xQ2) is the twist-2 part of g2(xQ2) Using the Wandzura-Wilczek relation [81]
one can relate d2(Q2) to moments of the spin structure functions g1(xQ2) and g2(xQ2)
d2(Q2) =
int 1
0
dx x2 [3g2(xQ2) + 2g1(xQ2)] (38)
22
000 005 010 015 020 025 030
0000
0001
0002
0003
0004
0005
0006
0007
Q2 GeV2D
d2
proton
000 005 010 015 020 025 030
0000
0002
0004
0006
0008
0010
0012
Q2 GeV2D
d2
neutron
000 005 010 015 020 025 030
-1
0
1
2
Q2 GeV2D
Γ 010
-4
fm6
D
proton
000 005 010 015 020 025 030-2
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
6D
neutron
FIG 8 Upper panel The inelastic moment d2(Q2) for the proton (left) and neutron (right) as
function of Q2 The result of this work the NLO BχPT prediction is shown by the blue solid line
and the blue band The red line represents the LO BχPT result The purple short-dashed line is
the O(p4) HB result from Ref [50 51] The black dotted line is the MAID model prediction [69]
The experimental points for the neutron (cyan dots) are from Ref [22] Lower panel Fifth-order
generalized forward spin polarizability γ0(Q2) for the proton (left) and neutron (right) as function
of Q2 The experimental points for the proton are from Ref [59] (purple square) and Ref [78]
(orange dot)
This relation however only holds for asymptotically large Q2 It is also in the high-Q2
region where d2(Q2) is a measure of color polarizability [82 83] through its relation to the
gluon field strength tensor [80] We refer to Ref [84] for a recent review on the spin structure
of the nucleon including a discussion of sum rules for deep inelastic scattering and color
polarizabilities
What we consider in the following is the inelastic part of d2(Q2) defined as the moment
of g1(xQ2) and g2(xQ2) spin structure functions cf Eq (38)
d2(Q2) =
int x0
0
dx x2 [3g2(xQ2) + 2g1(xQ2)] (39)
23
This moment provides another testing ground for our BχPT predictions through comparison
with experiments on the neutron [22] Going towards the low-Q2 region the interpretation
of d2(Q2) in terms of color polarizabilities will fade out The above definition however
implies it is related to other VVCS polarizabilities
d2(Q2) =Q4
8M4N
[M2
NQ2
αδLT (Q2) + I1(Q2)minus IA(Q2)
] (40)
Note that d2(Q2) and its first two derivatives with respect to Q2 vanish at Q2 = 0 The
considerations in Eqs (28) and (29) have no effect on d2(Q2) since the Born contribution
from IA(Q2) and I1(Q2) cancel out Therefore d2(Q2) is a pure polarizability
In Fig 8 upper panel we show our NLO BχPT prediction and other results for d2(Q2)
While MAID [69] and BχPT describe the experimental data for the neutron [22] very well
the HB limit [50 51] is showing a fast growth with Q2 This illustrates the importance
of keeping the relativistic result Note also that even though the πN -loop contribution
is dominant both gC and the form factor in gM are essential to obtain a curvature that
reproduces the data cf Fig 9 upper panel For the proton there are to our knowledge
no experimental results to compare with However the agreement between the NLO BχPT
prediction and the MAID prediction at low energies is reasonable
F γ0(Q2) mdash fifth-order generalized forward spin polarizability
It is interesting to compare the generalized fifth-order forward spin polarizability sum
rule
γ0(Q2) =1
2π2
int infinν0
dν
radic1 +
Q2
ν2
σTT (νQ2)
ν5(41)
=64αM4
N
Q10
int x0
0
dx x4
[g1(xQ2)minus 4M2
Nx2
Q2g2(xQ2)
]
to the sum rule integrals for IA(Q2) and γ0(Q2) since they differ merely by their energy
weighting of σTT (νQ2) and a constant prefactor cf Eqs (19) (26) and (41) From IA(Q2)
to γ0(Q2) to γ0(Q2) the energy suppression is increasing by a factor of νminus2 respectively
Therefore the description of γ0(Q2) should be easiest in a low-energy effective-field theory
such as χPT whereas γ0(Q2) and IA(Q2) receive larger contributions from higher energies
In Fig 8 lower panel we show our LO and NLO BχPT predictions for γ0(Q2) One
can see that the πN -loop contribution is positive (in accordance to what we see for the cross
section σTT see Fig 10) The Delta shifts it substantially especially in the low Q2 region
24
000 005 010 015 020 025 030
-0004
-0002
0000
0002
0004
0006
0008
0010
Q2GeV
2D
d2
proton
000 005 010 015 020 025 030
-0005
0000
0005
0010
0015
Q2GeV
2D
d2
neutron
000 005 010 015 020 025 030
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
6D
proton
000 005 010 015 020 025 030
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
6D
neutron
FIG 9 Contributions of the different orders to the chiral predictions of d2(Q2) upper panel and
γ0(Q2) lower panel for the proton (left) and neutron (right) Red solid line πN -loop contribu-
tion green dot-dashed line ∆-exchange contribution orange dotted line π∆-loop contribution
blue long-dashed line total result purple dot-dot-dashed line total result without gC contribution
black short-dashed line total result without gM dipole
bringing it into a better agreement with data In general the BχPT curves start above the
empirical data points at the real-photon point and then decrease asymptotically to zero
above Q2 gt 01 GeV2 On the other hand the MAID prediction reproduces the empirical
data at the real-photon point then decreases to negative values until about Q2 gt 006 GeV2
from where it also starts to asymptotically approach zero Consequently our NLO BχPT
prediction of γ0(Q2) is consistently above the MAID prediction This is very different to
what we saw for IA(Q2) in Fig 4 upper panel where the MAID prediction at the real-
photon point is above the experimental value While the agreement of our predictions with
the empirical data is in general quite good for all moments of σTT (νQ2) one should point
out that both for γ0n(Q2) and γ0p(Q2) we overestimate the data at low Q2 For IA(Q2) such
observation cannot be made because ∆IA(0) = 0 and thus IA(0) is given by the empirical
Pauli form factor only From IA(Q2) γ0(Q2) and γ0(Q2) the latter has the smallest however
25
non-negligible dependence on gC and the dipole in gM cf Fig 9 lower panel
The πN -loop ∆-exchange and π∆-loop contributions to the NLO BχPT prediction of
the static fifth-order forward spin polarizability amount to in units of 10minus4 fm6
γ0p = 112(30) asymp 208minus 096minus 001 (42a)
γ0n = 195(30) asymp 292minus 096minus 001 (42b)
while the slope is composed as follows in units of 10minus4 fm8
dγ0p(Q2)
dQ2
∣∣∣∣Q2=0
= minus084(10) asymp minus100 + 016 + 000 (43a)
dγ0n(Q2)
dQ2
∣∣∣∣Q2=0
= minus142(15) asymp minus158 + 016 + 000 (43b)
Note that the HB prediction of the static polarizability γ0p (423 at O(p3) and 365 at O(ε3)
[78 85]) is almost one order of magnitude larger than the empirical value and therefore not
shown in Fig 8
G Summary
Our results are summarized in Table II where we give the contributions of the different
orders to the chiral predictions of the polarizabilities and their slopes at the real-photon
point A quantitative comparison of our predictions for the static spin polarizabilities to
the work of Bernard et al [31] and different empirical evaluations is shown in Table III We
can see that the inclusion of the Delta turns out to be very important for all moments of
the helicity-difference cross section To describe the Q2 behavior of the polarizabilities the
magnetic coupling of the N rarr ∆ transition should be modified by a dipole form factor as
has been observed previously in the description of electroproduction data [33] This dipole
form factor effectively takes account of vector-meson exchanges The Coulomb-quadrupole
N rarr ∆ transition despite its subleading order is important in the description of some
moments of spin structure functions This is contrary to what we saw for the moments of
unpolarized structure functions [30] where the Coulomb coupling had a negligible effect
The π∆ loops are mainly relevant for the generalized GDH integrals
26
TABLE II The NLO BχPT predictions for the forward VVCS polarizabilities and their slopes at
Q2 = 0 The contributions of the πN loops the ∆ exchange and the π∆ loops are shown together
with the combined total result Note that IA(0) = I1(0) = d2(0) = 0 and (d2)prime = 0
πN loops ∆ exchange π∆ loops Total
γ0 p 201 minus284 minus010minus093(92)
(10minus4 fm4) n 298 003(92)
δLT p 150 minus016 minus002132(15)
(10minus4 fm4) n 235 218(23)
γ0 p 208 minus096 minus001112(30)
(10minus4 fm6) n 292 195(30)
(γ0)prime p minus033011 001
minus022(4)
(10minus4 fm6) n minus073 minus061(7)
(δLT )prime p minus080 minus004 minus001minus085(8)
(10minus4 fm6) n minus119 minus124(12)
(γ0)prime p minus100016 000
minus084(10)
(10minus4 fm8) n minus158 minus142(15)
(∆IA)prime p 238 minus1121 025minus858(343)
(GeVminus2) n 141 minus955(343)
(∆I1)prime p 034 minus053 058039(4)
(GeVminus2) n minus107 minus101(10)
TABLE III Our NLO BχPT predictions for the spin polarizabilities at Q2 = 0 compared with
the BχPT+∆ predictions from Bernard et al [31] and the available empirical information Where
the reference is not given the empirical number is provided by the MAID analysis [55 69] with
unspecified uncertainty
Proton Neutron
This work BχPT+∆ Empirical This work BχPT+∆ Empirical
γ0 minus093(92) minus174(40) minus100(8)(12) [19] 003(92) minus077(40) minus0005
(10minus4 fm4) minus090(8)(11) [78] [MAID]
minus0929(105) [59]
δLT 132(15) 240(1) 134 218(23) 238(3) 203
(10minus4 fm4) [MAID] [MAID]
γ0 112(30) 060(7)(7) [78] 195(30) 123
(10minus4 fm6) 0484(82) [59] [MAID]
IV CONCLUSIONS
We have presented a complete NLO calculation of the polarized non-Born VVCS ampli-
tudes in covariant BχPT with pion nucleon and ∆(1232) fields The dispersion relations
between the VVCS amplitudes and the tree-level photoabsorption cross sections served as a
27
cross-check of these calculations
The obtained moments of the proton and neutron spin structure functions related to gen-
eralized polarizabilities and GDH-type integrals agree well with the available experimental
data The description of their Q2 evolution is improved compared to the previous χPT pre-
dictions In particular the NLO BχPT predictions obtained here give a better description
of the empirical data (eg from the Jefferson Laboratory ldquoSpin Physics Programrdquo) than
the HB [50 51] and IR [58] calculations
The demonstrated predictive power of the χPT framework amplitudes makes it well
suited for extending the χPT evaluation of the TPE effect in the hyperfine structure of
(muonic-)hydrogen [15ndash17] to next-to-leading order
ACKNOWLEDGEMENTS
We thank Lothar Tiator and Marc Vanderhaeghen for helpful discussions This work
is supported by the Deutsche Forschungsgemeinschaft (DFG) through the Collaborative
Research Center [The Low-Energy Frontier of the Standard Model (SFB 1044)] JMA ac-
knowledges support from the Community of Madrid through the ldquoPrograma de atraccion de
talento investigador 2017 (Modalidad 1)rdquo and the Spanish MECD grants FPA2016-77313-P
FH gratefully acknowledges financial support from the Swiss National Science Foundation
Appendix A Tensor decompositions of the VVCS amplitudes
In this appendix we review the decomposition of the forward VVCS process into tensor struc-
tures and scalar amplitudes In particular we consider the connection between the covariant and
the semi-relativistic decomposition in the lab frame that is defined in terms of the conventional
transverse longitudinal transverse-transverse and transverse-longitudinal amplitudes
As explained in Sec II A the process of forward VVCS off the nucleon can be described in
terms of four explicitly covariant amplitudes S1 2 and T1 2 [3]
T (νQ2) =
(minusgmicroν +
qmicroqν
q2
)T1(νQ2) +
1
M2N
(pmicro minus p middot q
q2qmicro)(
pν minus p middot qq2
qν)T2(νQ2) (A1)
minus 1
MNγmicroναqα S1(νQ2)minus 1
M2N
(γmicroνq2 + qmicroγναqα minus qνγmicroαqα
)S2(νQ2)
εprimelowastmicro εν
where εmicro (εprimelowastmicro ) are the incoming (outgoing) photon polarization vectors ν is the photon lab-frame
energy and Q2 is the photon virtuality Alternatively the decomposition in the laboratory frame
28
(which in the forward case coincides with the Breit frame) is parametrized in terms of the nucleon
Pauli matrices ~σ and the four scalar functions fL fT gTT and gLT
T (νQ2) = ε0 εprimelowast0 fL(νQ2) + (~ε primelowast middot ~ε ) fT (νQ2) + i~σ middot (~ε primelowast times ~ε ) gTT (νQ2) (A2)
minusi~σ middot [(ε0~εprimelowast minus ~ε ε primelowast0 )times q] gLT (νQ2)
Here ~q and q = ~q|~q | are the photon three-momentum in the lab system and its unit vector The
modified polarization vector components are given by
ε0 =
[ε0 minus
ν
|~q | (~ε middot q )
] |~q |Q ~ε = ~εminus q (~ε middot q ) (A3)
where ε = (ε0~ε ) is the usual incoming photon polarization vector and εprimelowast the outgoing polarization
vector The LEX of the lab frame amplitudes [Eq (10)] can serve in particular as the definition
of the generalized polarizabilities The lab frame amplitudes are also conveniently used for the
definition of the response functions see the example of the scalar amplitude gLT (νQ2) and the
corresponding response function σLT (νQ2) below in App B
Appendix B Photoabsorption cross sections
In the forward kinematics the spin-dependent VVCS amplitudes and the spin polarizabilities
can be described in terms of the polarized structure functions g1(xQ2) and g2(xQ2) or equiv-
alently the helicity-difference cross section σTT (νQ2) and the longitudinal-transverse response
function σLT (νQ2) with the help of dispersion relations (5) and the optical theorem (3) In this
way the photoabsorption cross sections measured in electroproduction processes form the basis
for most empirical evaluations shown throughout Sec III In the following we present the BχPT
predictions for the tree-level cross sections of πN - π∆- and ∆-production through photoabsorption
on the nucleon cf Figs 8 9 and 10 in Ref [30] In Secs B 1 and B 2 we will discuss the leading
πN -production channel and the ∆-production channel respectively We used these cross sections
to verify the polarizability predictions obtained otherwise from the calculated non-Born VVCS
amplitudes Due to the bad high-energy behavior of the π∆-production cross sections in BχPT
cf Fig 10 the dispersion relations in Eq (5) require further subtractions for a reconstruction of
the π∆-loop contribution to the spin-dependent VVCS amplitudes Therefore not all polarizabil-
ities could be verified but only those appearing as higher-order terms in the LEX of the VVCS
amplitudes such as γ0 [16]
29
proton neutron
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Neutron
02 04 06 08 100
50
100
150
200
250
[GeV]
T [b]
02 04 06 08 10
-50
0
50
100
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Neutron
02 04 06 08 100
50
100
150
200
250
[GeV]
T [b]
02 04 06 08 10
-50
0
50
100
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02040608100
50
100
150
200
250
300
[GeV]
T[b]
0204060810
-50
0
50
[GeV]
TT[b]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
0204060810
00
02
04
06
[GeV]
LTQ
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
0204060810
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
FIG 10 Photoabsorption cross sections for πN (red) and π∆ production (orange) with Q2 = 0
(solid) and Q2 = 01 GeV2 (dashed for πN and dotted for π∆ channel)
1 πN-production channel
In order to extract the response function σLT (νQ2) we have developed a method similar to
the one used to calculate σTT (νQ2) see for example Ref [86] For σLT (νQ2) however the
calculation is more complicated because one has to take into account that the associated Compton
process involves a spin-flip of the nucleon as illustrated in Fig 11 When calculating the cross
section the product of the incoming nucleon spinors has to reflect this flip
The forward VVCS amplitude related to σLT (νQ2) mdash and δLT (Q2) mdash is gLT (νQ2) It can
be extracted from Eq (A2) if one takes the modified polarization vector components in Eq (A3)
with ε = εL and εprimelowast = εlowastplusmn as input where εL = 1Q(|~q | 0 0 q0) and εplusmn = ∓ 1radic
2(0 1plusmni 0) are the
standard longitudinal and transverse polarization vectors respectively For εL and εlowastplusmn only the
choice of helicities hprime = plusmn12 and h = ∓12 gives a non-zero contribution and one obtains
χdaggerhprime T (νQ2)χh = χdaggerhprimeminusi~σ middot [(ε0~εprimelowast minus ~ε ε primelowast0 )times q] gLT (νQ2)χh =
radic2 gLT (νQ2) (B1)
where χh and χdaggerhprime are two-component Pauli spinors with opposite helicities or here spins
Let us now consider the related photoabsorption process and in particular the tree-level γlowastN rarr
30
s = 12
sprime = minus12
λ = 0 λprime = 1
rprime rprime
s=12sprime=minus1
2
λ=0λprime=1
rprimerprime
s=12
sprime=minus12
λ=0λprime=1
rprimerprime hacute=frac12h=-frac12
λacute=1λ=0
FIG 11 Relation between the forward Compton process and the photoabsorption process given
by the optical theorem In particular we show the longitudinal-transverse contribution The
double-line arrows represent the spin of the external particles while the dot represents the scalar
(longitudinal) polarization of the incoming photon Inside the blob the intermediate states are
represented eg nucleons with spins rprime (which are averaged in the calculation of the cross section)
and pions
πN channel see diagrams in Fig 8 of Ref [30] We define the πN -production amplitude as
T = uhB (PB)sumi
Ai(s t)Γi uhA(PA) (B2)
with the Dirac structures
Γ1 = γ5 (B3a)
Γ2 =1
2
[qA ε
]γ5 (B3b)
where uhA(PA) and udaggerhB (PB) are the Dirac spinors and PA and PB are the four-momenta of
the incoming and outgoing nucleons respectively When calculating the photoabsorption cross
section related to the VVCS amplitude in Eq (B1) the nucleon spin flip should be implemented
by uhprime(PA) in T dagger and uh(PA) in T together with the appropriate transverse and longitudinal
photon polarization vectors εlowastplusmn and εL
However if one wants to use the properties of the Dirac matrices it is more useful to construct
an operator to produce this spin flip in the external nucleons of Fig 11 This is accomplished by
introducing the projector ΓLT equiv 12radic
2(γ1 + iγ2)γ5 which also takes into account the extra factor
radic2 in Eq (B1) We checked that with this projector one correctly extracts δLT by comparing the
HB limit of our result to the HB result of Ref [50] where the authors calculate this polarizability
from the Compton amplitude directly With all those ingredients the longitudinal-transverse cross
section is calculated in the following way
σLT (νQ2) =1
64π2 s
|~pf |cm
|~pi|cm
int 1
minus1dcos θ
sumij
AiAdaggerjXij (B4)
31
with
Xij = Tr[(PB +MN )Γi(PA +MN )ΓLTγ0Γdaggerjγ
0] (B5)
where θ is the scattering angle in the center-of-mass (cm) frame and |~pi|cm (|~pf |cm) is the three-
momentum of an incoming (outgoing) particle in the cm frame An explicit calculation of the
matrix Xij leads to
X = MNQ
0 2(PB minus PA) middot εLminusradic
2|~qf |cm sin θ (sminus u)
(B6)
where |~qi|cm (|~qf |cm) is the relative three-momentum of the incoming (outgoing) particles in the
cm frame Here s t and u are the usual Mandelstam variables For the different γlowastN rarr πN
channels we obtain the following amplitudes Ai where we introduce qA as the four-momentum of
the incoming photon and qB as the four-momentum of the outgoing pion
bull γlowastprarr π0p
A1 =e gAMN
fπ
[2PA middot ε+ qA middot ε
sminusM2N
+2PB middot εminus qA middot ε
uminusM2N
] (B7a)
A2 =e gAMN
fπ
[1
sminusM2N
+1
uminusM2N
] (B7b)
bull γlowastprarr π+n
A1 =
radic2 e gAMN
fπ
[2PA middot ε+ qA middot ε
sminusM2N
+2(PA minus PB) middot ε+ qA middot ε
tminusm2π
] (B8a)
A2 =
radic2 e gAMN
fπ(sminusM2N )
(B8b)
bull γlowastnrarr π0n
A1 = 0 (B9a)
A2 = 0 (B9b)
bull γlowastnrarr πminusp
A1 =
radic2 e gAMN
fπ
[2PB middot εminus qA middot ε
uminusM2N
minus 2(PA minus PB) middot ε+ qA middot εtminusm2
π
] (B10a)
A2 =
radic2 e gAMN
fπ(uminusM2N )
(B10b)
The analytical expressions shown above were checked with the amplitudes given in Ref [87] An-
alytical expressions for the tree-level γlowastN rarr πN channel of the σLT (νQ2) and σTT (νQ2) cross
32
sections are given below (proton channels π+n and π0p neutron channel πminusp) We checked that
they reproduce the known results in the real-photon limit [45 86] To shorten the final expressions
for the cross sections which are considerably longer for finite Q2 than in the real-photon limit we
define the following dimensionless kinematic variables
αγ = (ENi )cmradics =
s+M2N +Q2
2s (B11)
απ = (ENf )cmradics =
s+M2N minusm2
π
2s (B12)
βγ = Eγcmradics =
sminusM2N minusQ2
2s (B13)
βπ = Eπcmradics =
sminusM2N +m2
π
2s (B14)
λγ = |~qi|cmradics =
radic(sminusM2
N minusQ2)2 + 4sQ2
2s (B15)
λπ = |~qf |cmradics =
radic(sminusM2
N +m2π)2 minus 4sm2
π
2s (B16)
Here (ENi )cm and (ENf )cm are the energies of the incoming and outgoing nucleon Eγcm is the energy
of the incoming photon Eπcm is the energy of the outgoing pion all in the cm frame
σ(π+n)TT = minus e2g2
AM2N
64πf2πs
3(sminusM2N )2λ4
γ
4sλπλγ
[(M2
N minus s)(M2N minusQ2 minus s)(Q2 + 2sβγβπ)
+ 2s(m2πQ
2 minus (M2N minus s)
(M2N + s(minus1 + 2βγβπ)
))λ2γ
]+ (M2
N minus s)(M2N minusQ2 minus s)
times (Q2 + 2sβγβπ minus 2sλπλγ)(Q2 + 2s(βγβπ + λπλγ)
)times log
(Q2 + 2sβγβπ minus 2sλγλπQ2 + 2sβγβπ + 2sλγλπ
) (B17)
33
σ(π0p)TT =
e2g2AM
2Nλπ
64πf2πs(sminusM2
N )2λγ
1
s2λπλ3γ
[(M2
N minus s)(M2N minusQ2 minus s)
(3M2
N + s(2βγβπ minus 3))(M2N
+ s(2βγβπ minus 1))
+ 2s
((2m2
πQ2 + (M2
N minus s)2)(M2
N minus s)minus 2(minusm2
πQ2 + (sminusM2
N )2)
times sβγβπ + 4s2(sminusM2N )β2
γβ2π
)λ2γ
]arctan
(2sλπλγ
M2N + s(2βγβπ minus 1)
)+ 2
[minus 2m2
πQ2
+ (M2N minus s)
(2(sminusM2
N ) + 4sβγβπ +(sminusM2
N +Q2)(3M2N minus 3s+ 2sβγβπ)
sλ2γ
+m2πQ
2(sminusM2N )
(M2N minus s+ 2sβγβπ)(M2
N + s(minus1 + 2βγβπ minus 2λπλγ))
+m2πQ
2(sminusM2N )
(M2N minus s+ 2sβγβπ)(M2
N + s(minus1 + 2βγβπ + 2λπλγ))
)
+1
λπλγ
(βγβπ
(minusm2
πQ2 + (M2
N minus s)(M2N minus s+ 2sβγβπ)
)minus (M2
N minus s)(M2N minusQ2 minus s)λ2
π
)log
(M2N + s(minus1 + 2βγβπ + 2λπλγ)
M2N + s(minus1 + 2βγβπ minus 2λπλγ)
)] (B18)
σ(πminusp)TT =
e2g2AM
2N
64πf2πs
3λ4γ(M2
N + s(2βπβγ minus 1))2 minus 4λ2πs
2λ2γ)
4sλπλγ
((minusM2
N +Q2 + s)(M2N
+ s(2βπβγ minus 1))
2 minus 2sλ2γ
(2λ2
πs(minusM2N +Q2 + s) +m2
πQ2))
+((M2N + s(2βπβγ minus 1)
)2
minus 4λ2πs
2λ2γ
)((M2N + s(2βπβγ minus 1)
)(M2N minus 2Q2 minus s(2βπβγ + 1)
)+ 4λ2
πs2λ2
γ
)times log
(minusM2
N minus 2βπsβγ minus 2λπsλγ + s)minus(M2N + s(2βπβγ minus 2λπλγ minus 1)
)(M2N + s(minus1
+ 2βπβγ + 2λπλγ))[((
M2N + s(2βπβγ minus 1)
)(M2N minus 2Q2 minus s(2βπβγ + 1)
)+ 4λ2
πs2λ2
γ
)times log
(minusM2
N minus 2βπsβγ + 2λπsλγ + s)
+((Q2 + 2βπsβγ)2 minus 4λ2
πs2λ2
γ
)times log
(Q2 + 2βπsβγ minus 2λπsλγQ2 + 2sβπβγ + 2λπsλγ
)]+ 4sλ2
γ
((M2
N + s(2βπβγ minus 1))2 minus 4λ2πs
2λ2γ
)times (M2
N minus s) arctanh
(2λπsλγ
M2N + 2βπsβγ minus s
) (B19)
34
σ(π+n)LT =
e2g2AM
3Nλπ
32πf2πQs
3(sminusM2N )2λ4
γ
2sλγ
[(M2
N minus s)(Q2 + 2sβ2γ)(Q2 + 2sβγβπ)
minus 4s
((M2
N minus s)(Q2 minus 2s(απ minus 1)βγ
)+Q2sβγβπ
)λ2γ + 8s3(minus1 + απ)λ4
γ
]minus M2
N minus sλπ
[(Q2 + 2sβ2
γ)(Q2 + 2sβγβπ)2 + 4s2(2(απ minus 1)βγ(Q2 + 2sβγβπ)
minusQ2λ2π
)λ2γ + 8s3(απ minus 1)2λ4
γ
]arctan
(2sλπλγ
Q2 + 2sβγβπ
) (B20)
σ(π0p)LT =
e2g2AM
3Nλπ
16πf2πQs(sminusM2
N )2λγ
1
minus2s(M2N + s(minus1 + 2βγβπ))2λ2
γ + 8s3λ2πλ
4γ
[minus 3M8
N (Q2
+ 2sβ2γ) + 2M4
Ns2
(minus (Q2 + 2sβ2
γ)(minus3 + 2βγβπ) +(Q2(minus3 + 2βγβπ + 6λ2
π) + 2sβγ
times (minus3 + 12απ + 2βγβπ minus 12απβγβπ + 4βγλ2π))λ2γ minus 4s(α2
π minus 1)λ4π
)+ 2M2
Ns3
[minus (Q2
+ 2sβ2γ)(minus1 + 2βγβπ)
(6 + βγβπ(minus9 + 2βγβπ)
)+
(Q2(3minus 12λ2
π + 4βγβπ(minus1 + 2βγβπ
+ λ2π))minus 2sβγ
(minus 3 + 4απ(3 + 2απ(3 + 2βγβπ(minus3 + βγβπ)) + 4βγ(βπ + (2minus βγβπ)
times λ2π)))λ2γ + 8s
((απ minus 1)(1 + απ minus 2βγβπ) + 2απβγλ
2π
)λ4γ
)+ s4
((Q2 + 2sβ2
γ)
times (1minus 2βγβπ)2(2βγβπ minus 3) + 2(Q2(minus1 + 6λ2
π + 2βγβπ((1minus 2βγβπ)2 minus 2λ2π)) + 2sβγ
times (minus1 + 2βγβπ + 4(minus1 + βγβπ)(minusαπ + 2απβγβπ minus βγλ2π)))λ2γ minus 8
(s(απ minus 1)(απ
+ (1minus 2βγβπ)2) + 2βγ(2sαπ +Q2βγ)λ2π
)λ4γ + 32s(απ minus 1)λ2
πλ6γ
)+ 2M6
Ns
(Q2(6
minus 7βγβπ + λ2γ) + 2sβγ(βγ(6minus 7βγβπ) + (1minus 4απ)λ2
γ)
)]+sminusM2
N
4s2λπλ3γ
[(Q2 + 2sβ2
γ)
times(3M2
N + s(2βγβπ minus 3))(M2N + s(2βγβπ minus 1)
)+ 2s
(minusM2
N (Q2 + 2sβγ minus 8sαπβγ)
+ s(2sβγ(1minus 4απ + 4απβγβπ) +Q2(1minus 2λ2π)))λ2γ + 8s3(απ minus 1)λ4
γ
]
times arctan
(2sλπλγ
M2N + s(2βγβπ minus 1)
) (B21)
35
σ(πminusp)LT =
e2g2AM
3N
64πf2πQs
3λ4γ(M2
N minusQ2 minus s)(M2N + s(2βπβγ + 2λπλγ minus 1))
times
1
M2N + s(2βπβγ minus 2λπλγ minus 1)
[4λπsλγ(M2
N minusQ2 minus s)(
2sλ2γ
(s((1minus 2λ2
π)Q2
+ 2(2απ minus 1)sβγ(2βπβγ minus 1))minusM2N (Q2 + 2(1minus 2απ)sβγ)
)+ (Q2 + 2sβ2
γ)(M2N + s(2βπβγ minus 1)
)2 + 8(απ minus 1)απs
3λ4γ
)]minus(M2N + s(2βπβγ + 2λπλγ minus 1)
)times[
log
(minusM2N minus 2βπsβγ minus 2λπsλγ + s
minusM2N minus 2βπsβγ + 2λπsλγ + s
)(minus (Q2 + 2sβ2
γ)(M2N + s(2βπβγ minus 1)
)(minusM2
N + 2Q2 + 2βπsβγ + s)minus 2sλ2γ
(minusM2
N (Q2 + 2sβγ) +Q4 +Q2s(4απβγ
minus 2βγ minus 2λ2π + 1) + 2s2βγ(4(απ minus 1)βπβγ + 1)
)minus 8(απ minus 1)2s3λ4
γ
)+ 2
(4s2λ2
γ
(2(απ minus 1)βγ(Q2 + 2βπsβγ)minus λ2
πQ2)
+ (Q2 + 2βπsβγ)2(Q2 + 2sβ2γ)
+ 8(απ minus 1)2s3λ4γ
)arctanh
(2λπsλγ
Q2 + 2βπsβγ
)] (B22)
2 ∆-production channel
The tree-level ∆-exchange diagram in Fig 2 of Ref [30] contributes to the non-Born part of
the VVCS amplitudes The contribution of the ∆ exchange to the VVCS amplitudes can be split
into [17]
S∆-exch1 (νQ2) = S∆-pole
1 (νQ2) + S∆-exch1 (νQ2) (B23a)
νS∆-exch2 (νQ2) = νS∆-pole
2 (νQ2) + νS2∆-exch
(νQ2) (B23b)
and similarly for the unpolarized VVCS amplitudes discussed in Ref [30] Here we introduced the
∆-pole contributions S∆-polei and the ∆-non-pole contributions S∆-exch
i The former amplitudes
feature a pole at the ∆(1232)-production threshold and thus are proportional to
1
[sminusM2∆][uminusM2
∆]=
1
4M2N
1
ν2∆ minus ν2
(B24)
36
They can be reconstructed from the dispersion relations in Eq (5) with the tree-level ∆-production
cross sections as input cf Fig 10 in Ref [30]
σTT (νQ2) =π2α
M2NM
2+|~q |
minus g2
MMN (M+ + ν)|~q |2 +g2E(∆minus ν)(Q2 minusMNν)2
MN(B25a)
+g2CQ
4s(∆minus ν)
MNM2∆
minus 4gMgE(Q2 minusMNν)|~q |2 minus 4gMgCQ2|~q |2
+2gEgCQ
2[minusMNM∆ |~q |2 + s(Q2 +∆ν)
]MNM∆
δ(ν minus ν∆)
σLT (νQ2) =Qπ2α
M2NM
2+|~q |
g2E(MNν minusQ2) [M∆(MN + ν)minus s]
MN(B25b)
+g2CQ
2[MNM∆|~q |2 minus s(Q2 +∆ν)
]MNM2
∆
+ gMgEM∆|~q |2 minusgMgC(Q2 minusMNν)|~q |2
M∆
+gEgC(ν minus∆)(M2
N |~q |2 minus 2Q2s)
MNM∆
δ(ν minus ν∆)
with ∆ = M∆ minusMN M+ = M∆ + MN and the Mandelstam variable s = M2N + 2MNν minus Q2
Analytical expressions for the spin structure functions g1(xQ2) and g2(xQ2) can be constructed
from Eq (3) with the flux factor K(νQ2) = |~q | =radicν2 +Q2
In the ∆-non-pole contributions to S1(νQ2) and νS2(νQ2) the pole in ν at the ∆(1232)-
production threshold has canceled out
S∆-exch1 (νQ2) =
πα
MNM2+
[g2MQ
2+ + g2
E
(∆2 minus 3Q2
)+
4g2CQ
4
M2∆
minus 8gMgEM∆ωminus (B26a)
minus2gMgCQ2(MN minus 4M∆)
M∆+
2gEgCQ2(3MN minus 2M∆)
M∆
]
νS2∆-exch
(νQ2) =2πα
MNM2+
[g2EM∆∆ωminus +
g2M MNQ
2+
2+g2C Q
2(Q2 minus∆2)
2M∆(B26b)
+gEgM M∆(M∆ω+ minus 4MNωminus)minus gEgC ∆(2Q2 +MNω+)
+gMgC Q2(4MN minus ω+)
]+S∆-exch
2 (νQ2)
ν
[M2
∆ ω2+
M2N
+ ν2
]
with Q+ =radic
(M∆ +MN )2 +Q2 and ωplusmn = (M2∆minusM2
N plusmnQ2)2M∆ and the non-pole contribution
to S2(νQ2)
S∆-exch2 (νQ2) = minus2παMNν
M∆M2+
[gM + gE
]gC (B27)
These amplitudes to the contrary are not described by the tree-level ∆-production cross sections
in the standard dispersive approach [17] This peculiarity has been previously missed e g in the
calculation of the ∆-exchange contribution to the hydrogen hyperfine splitting in Ref [88] The
importance of including the ∆-non-pole contribution is also evident when considering the BC sum
37
rule in Eq (14) The ∆-pole terms by themselves violate the BC sum rule but cancel exactly with
the ∆-non-pole terms
limνrarr0
νS∆-pole2 (νQ2) + lim
νrarr0νS2
∆-exch(νQ2) = 0 (B28)
Appendix C Polarizabilities at Q2 = 0
In this section we give analytical expressions for the static values and slopes at Q2 = 0 of
the spin polarizabilities and moments of polarized structure functions In particular we give the
HB expansion of the πN -loop contributions and the ∆-exchange contributions The complete
expressions also for the π∆-loop contributions can be found in the Supplemented material Recall
that IA(0) = I1(0) = d2(0) = 0 and dd2(Q2)dQ2
∣∣∣Q2=0
= 0
1 πN-loop contribution
Here we give analytical expressions for the πN -loop contributions to the proton and neutron
spin polarizabilities expanded in powers of micro = mπMN viz the HB expansion Note that we
choose to expand here to a high order in micro the strict HB expansion would only retain the leading
term in an analogous NLO calculation
bull Static polarizabilities (Q2 = 0)
γ0p =e2g2
A
96π3f2πm
2π
1minus 21πmicro
8minus(
59
2+ 26 logmicro
)micro2 +
1875πmicro3
64
+3
(3
2+ 26 logmicro
)micro4 +
(C1)
γ0n =e2g2
A
48π3f2πm
2π
1
2minus 9πmicro
16minus 2micro2 logmicro+
75πmicro3
128minus 3micro4
4+
(C2)
δLTp =e2g2
A
192π3f2πm
2π
1minus 9πmicro
8+
(13
2minus 2 logmicro
)micro2 minus 465πmicro3
64
minus(
47
2+ 42 logmicro
)micro4 +
(C3)
δLTn =e2g2
A
96π3f2πm
2π
1
2+
3πmicro
16+ (1 + 2 log micro)micro2 minus 105πmicro3
128+
5micro4
4+
(C4)
38
δLTn =e2g2
A
96π3f2πm
2π
1
2+
3πmicro
16+ (1 + 2 log micro)micro2 minus 105πmicro3
128+
5micro4
4+
(C5)
γ0p =e2g2
A
16π3f2πm
4π
4
45minus 3πmicro
16+
14micro2
5minus 1813πmicro3
384minus 192
5(1 + log micro)micro4
+80703πmicro5
2048+
(C6)
γ0n =e2g2
A
16π3f2πm
4π
4
45minus 5πmicro
48+
4micro2
5minus 245πmicro3
384minus 32micro4 logmicro
15
+1323πmicro5
2048+
(C7)
bull Slopes of polarizabilities at Q2 = 0
dγ0p(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
2minus 45πmicro
4+ 223micro2 minus 28515πmicro3
64
minus9
(1953
4+ 449 logmicro
)micro4 +
570255πmicro5
128+
(C8)
dγ0n(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
2minus 81πmicro
8+ 94micro2 minus 2535πmicro3
32
minus3 (1 + 90 logmicro)micro4 +84315πmicro5
1024+
(C9)
dδLTp(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
2880π3f2πm
4π
minus5
2minus 27πmicro
32+ 20micro2 minus 5865πmicro3
256
+3
(617
4+ 36 logmicro
)micro4 minus 2056845πmicro5
4096+
(C10)
dδLTn(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
minus5
4minus 81πmicro
64minus 11micro2 +
10005πmicro3
512
+15
8(11 + 48 logmicro)micro4 minus 267015πmicro5
8192+
(C11)
dIAp(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
96π2f2πmicro
2
1minus 15πmicro
4minus 1
2(115 + 88 logmicro)micro2 +
1839πmicro3
32
+5 (5 + 34 logmicro)micro4 +
(C12)
dIAn(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
48π2f2πmicro
2
1
2minus 11πmicro
8minus 1
4(1 + 20 logmicro)micro2 +
99πmicro3
64
minus25micro4
12+
(C13)
dI1p(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
96π2f2πmicro
3π
8+ 2 (4 + 3 logmicro)microminus 537πmicro2
64
minus1
2(15 + 56 logmicro)micro3 +
(C14)
39
dI1n(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
48π2f2πmicro
minus π
16+
1
4(3 + 4 logmicro)microminus 57πmicro2
128+
2micro3
3+
(C15)
γ0p(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2g2
A
16π3f2πm
6π
1
105minus 23πmicro
256+
377micro2
210minus 15551πmicro3
6144+
3371micro4
105
minus1640457πmicro5
32768+
(C16)
γ0n(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2g2
A
16π3f2πm
6π
1
105minus 153πmicro
1792+
69micro2
70minus 4615πmicro3
6144+
172micro4
35
minus120897πmicro5
32768+
(C17)
2 ∆-exchange contribution
Here we give analytical expressions for the tree-level ∆-exchange contributions to the nucleon
spin polarizabilities and their slopes at Q2 = 0 Note that the ∆-exchange contributes equally to
proton and neutron polarizabilities Recall that for the magnetic γlowastN∆ coupling we introduced a
dipole form factor to mimic vector-meson dominance gM rarr gM(1 +Q2Λ2)2
bull Static polarizabilities (Q2 = 0)
γ0 = minus e2
4πM2+
(g2M
∆2+
g2E
M2+
minus 4gMgEM+∆
) (C18)
δLT =e2M∆
4πM3+
(g2E
MNM++gMgE∆MN
minus gEgCM2
∆
) (C19)
γ0 =e2M2
N
π∆2M4+
(minusg
2M
∆2+
g2E
M2+
+4gMgE∆M+
) (C20)
bull Slopes of polarizabilities at Q2 = 0
dγ0(Q2)
dQ2
∣∣∣∣∣Q2=0
= minus e2
πM2+∆
(g2M
∆
[1
4∆2minus 1
∆M++
1
2M2+
]minus 1
Λ2
g2M
∆+
g2E
2M2+
[1
2∆minus 3
M+
]minusgMgE
M+
[1
∆2minus 5
∆M++
1
M2+
]+
1
Λ2
2gMgEM+
+2gMgC∆M2
+
minus gEgCM3
+
) (C21)
40
d δLT (Q2)
dQ2
∣∣∣∣∣Q2=0
=e2M∆∆
4πMNM2+
(g2E
∆2M2+
[1
∆minus 4
M+
]minus g2
C
∆M2∆M
2+
+gMgE∆2M+
[1
∆2
minus 3
∆M++
1
M2+
]minus 2
Λ2
gMgE∆2M+
+gMgC∆M2
∆
[1
2∆2minus 2
∆M++
1
2M2+
]minus gEgC
2M2∆M
2+
[7
∆+
1
M+
]) (C22)
dIA(Q2)
dQ2
∣∣∣∣∣Q2=0
= minusM2N
M2+
(g2M
2∆2+
g2E
MNM+minus 2gMgE
∆M+minus gEgCM∆M+
) (C23)
dI1(Q2)
dQ2
∣∣∣∣∣Q2=0
= minusM∆M2N
2M3+
(g2E
MNM∆minus gMgE∆MN
minus gEgCM2
∆
) (C24)
γ0(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2M2
N
π∆3M6+
(g2M
∆
[3M2
∆ + 2M∆MN minus 9M2N
∆2+
4M2+
Λ2
]
minus4gMgE
[5M2
∆ minus 9M2N
M+∆2+
2M+
Λ2
]+
g2E
∆M+[7M∆ minus 9MN ]
minus8gMgC∆
+4gEgCM+
) (C25)
[1] D Drechsel B Pasquini and M Vanderhaeghen Dispersion relations in real and virtual
Compton scattering Phys Rept 378 99 (2003) hep-ph0212124
[2] S E Kuhn J-P Chen and E Leader Spin structure of the nucleon mdash status and recent
results Prog Part Nucl Phys 63 1 (2009) arXiv08123535 [hep-ph]
[3] F Hagelstein R Miskimen and V Pascalutsa Nucleon polarizabilities from Compton scat-
tering to hydrogen atom Prog Part Nucl Phys 88 29 (2016) arXiv151203765 [nucl-th]
[4] B Pasquini and M Vanderhaeghen Dispersion theory in electromagnetic interactions Ann
Rev Nucl Part Sci 68 75 (2018) arXiv180510482 [hep-ph]
[5] S Gerasimov A Sum rule for magnetic moments and the damping of the nucleon magnetic
moment in nuclei Sov J Nucl Phys 2 430 (1966)
[6] S Drell and A C Hearn Exact sum rule for nucleon magnetic moments Phys Rev Lett
16 908 (1966)
[7] J S Schwinger Source theory viewpoints in deep inelastic scattering Proc Natl Acad Sci
USA 72 1 (1975)
[8] J S Schwinger Source theory viewpoints in deep inelastic scattering Electromagnetic In-
teractions and Field Theory Proceedings 14 Internationale Universitatswochen Schladming
Austria February 24-March 7 1975 Acta Phys Austriaca Suppl 14 471 (1975)
41
[9] J Schwinger Source theory discussion of deep inelastic scattering with polarized particles
Proc Natl Acad Sci USA 72 1559 (1975)
[10] A M Baldin Polarizability of nucleons Nucl Phys 18 310 (1960)
[11] M Gell-Mann M L Goldberger and W E Thirring Use of causality conditions in quantum
theory Phys Rev 95 1612 (1954)
[12] A Pineda Leading chiral logarithms to the hyperfine splitting of the hydrogen and muonic
hydrogen Phys Rev C 67 025201 (2003)
[13] C Peset and A Pineda The two-photon exchange contribution to muonic hydrogen from
chiral perturbation theory Nucl Phys B 887 69 (2014) arXiv14064524 [hep-ph]
[14] C Peset and A Pineda Model-independent determination of the two-photon exchange contri-
bution to hyperfine splitting in muonic hydrogen JHEP 04 060 arXiv161205206 [nucl-th]
[15] F Hagelstein and V Pascalutsa Proton structure in the hyperfine splitting of muonic hydro-
gen PoS CD15 077 (2016) arXiv151104301 [nucl-th]
[16] F Hagelstein Exciting Nucleons in Compton Scattering and Hydrogen-Like Atoms PhD
thesis Mainz U Inst Kernphys (2017) arXiv171000874 [nucl-th]
[17] F Hagelstein ∆(1232)-Resonance in the hydrogen spectrum Proceedings 11th International
Workshop on the Physics of Excited Nucleons (NSTAR 2017) Columbia SC USA August
20-23 2017 Few Body Syst 59 93 (2018) arXiv180109790 [nucl-th]
[18] Y Prok et al (CLAS) Moments of the spin structure functions gp1 and gd1 for 005 lt Q2 lt 30
GeV2 Phys Lett B 672 12 (2009) arXiv08022232 [nucl-ex]
[19] H Dutz et al (GDH) First measurement of the Gerasimov-Drell-Hearn sum rule for 1H from
07 GeV to 18 GeV at ELSA Phys Rev Lett 91 192001 (2003)
[20] M Amarian et al (Jefferson Lab E94010) Measurement of the generalized forward spin po-
larizabilities of the neutron Phys Rev Lett 93 152301 (2004) arXivnucl-ex0406005
[21] M Amarian et al The Q2 evolution of the generalized Gerasimov-Drell-Hearn integral for
the neutron using a 3He target Phys Rev Lett 89 242301 (2002) arXivnucl-ex0205020
[22] M Amarian et al (Jefferson Lab E94-010)Q2 evolution of the neutron spin structure moments
using a 3He target Phys Rev Lett 92 022301 (2004) arXivhep-ex0310003
[23] A Deur et al Experimental determination of the evolution of the Bjorken integral at low Q2
Phys Rev Lett 93 212001 (2004) arXivhep-ex0407007
[24] K Slifer Low Q2 measurement of gp2 and the δLT spin polarizability Spin structure at long dis-
tance Proceedings Workshop Newport News USA March 12ndash13 2009 nucl-ex09064775
(2009) AIP Conf Proc 1155 10106313203293 (2009) arXiv09064775 [nucl-ex]
[25] P Solvignon et al (E01-012) Moments of the neutron g2 structure function at intermediate
42
Q2 Phys Rev C 92 015208 (2015) arXiv13044497 [nucl-ex]
[26] A Deur Experimental studies at low Q2 of the spin structure of the nucleon at Jefferson Lab
in 9th International Workshop on Chiral Dynamics (CD18) Durham NC USA September
17-21 2018 (2019) arXiv190305661 [nucl-ex]
[27] V Sulkosky et al (Jefferson Lab E97-110) Measurement of the 3He spin-structure functions
and of neutron (3He) spin-dependent sum rules at 0035 le Q2 le 024 GeV2 Phys Lett B
805 135428 (2020) arXiv190805709 [nucl-ex]
[28] K Adhikari et al (CLAS) Measurement of the Q2 Dependence of the Deuteron Spin Structure
Function g1 and its Moments at Low Q2 with CLAS Phys Rev Lett 120 062501 (2018)
arXiv171101974 [nucl-ex]
[29] V Lensky F Hagelstein A Hiller Blin and V Pascalutsa Comment on rdquoMeasurement of
the Q2 Dependence of the Deuteron Spin Structure Function g1 and its Moments at Low Q2
with CLASrdquo (2018) arXiv180603219 [nucl-th]
[30] J M Alarcn F Hagelstein V Lensky and V Pascalutsa Forward doubly-virtual Compton
scattering off the nucleon in chiral perturbation theory at NLO the subtraction function and
moments of unpolarized structure functions (2020) arXiv200509518 [hep-ph]
[31] V Bernard E Epelbaum H Krebs and U-G Meiszligner New insights into the spin structure
of the nucleon Phys Rev D 87 054032 (2013) arXiv12092523 [hep-ph]
[32] V Lensky J M Alarcon and V Pascalutsa Moments of nucleon structure functions at
next-to-leading order in baryon chiral perturbation theory Phys Rev C 90 055202 (2014)
arXiv14072574 [hep-ph]
[33] V Pascalutsa and M Vanderhaeghen Chiral effective-field theory in the ∆(1232) region
I Pion electroproduction on the nucleon Phys Rev D 73 034003 (2006) arXivhep-
ph0512244
[34] V Pascalutsa and M Vanderhaeghen Electromagnetic nucleon-to-Delta transition in chiral
effective field theory Phys Rev Lett 95 232001 (2005) arXivhep-ph0508060
[35] T R Hemmert B R Holstein and J Kambor Systematic 1M expansion for spin 32
particles in baryon chiral perturbation theory Phys Lett B 395 89 (1997) arXivhep-
ph9606456
[36] V Pascalutsa and D R Phillips Effective theory of the ∆(1232) in Compton scattering off
the nucleon Phys Rev C 67 055202 (2003) arXivnucl-th0212024
[37] V Pascalutsa M Vanderhaeghen and S N Yang Electromagnetic excitation of the ∆(1232)-
resonance Phys Rept 437 125 (2007) arXivhep-ph0609004
[38] K A Olive et al (Particle Data Group) Review of Particle Physics Chin Phys C 38 090001
43
(2014)
[39] F E Low Scattering of light of very low frequency by systems of spin 12 Phys Rev 96
1428 (1954)
[40] M Gell-Mann and M L Goldberger Scattering of low-energy photons by particles of spin
12 Phys Rev 96 1433 (1954)
[41] A M Harun ar-Rashid A simple derivation of Schwingerrsquos sum rule for spin dependent
structure functions Nuovo Cim A 33 447 (1976)
[42] F Hagelstein and V Pascalutsa Dissecting the hadronic contributions to (g minus 2)micro by
Schwingerrsquos sum rule Phys Rev Lett 120 072002 (2018) arXiv171004571 [hep-ph]
[43] V Lensky V Pascalutsa M Vanderhaeghen and C Kao Spin-dependent sum rules con-
necting real and virtual Compton scattering verified Phys Rev D 95 074001 (2017)
arXiv170101947 [hep-ph]
[44] H Burkhardt and W N Cottingham Sum rules for forward virtual Compton scattering
Annals Phys 56 453 (1970)
[45] V Lensky and V Pascalutsa Predictive powers of chiral perturbation theory in Compton
scattering off protons Eur Phys J C 65 195 (2010) arXiv09070451 [hep-ph]
[46] H Grieszlighammer J McGovern D Phillips and G Feldman Using effective field theory to
analyse low-energy Compton scattering data from protons and light nuclei Prog Part Nucl
Phys 67 841 (2012) arXiv12036834 [nucl-th]
[47] H W Grieszlighammer J A McGovern and D R Phillips Nucleon polarisabilities at and
beyond physical pion masses Eur Phys J A 52 139 (2016) arXiv151101952 [nucl-th]
[48] E Epelbaum H Krebs and U-G Meiszligner Improved chiral nucleon-nucleon potential up to
next-to-next-to-next-to-leading order Eur Phys J A 51 53 (2015) arXiv14120142 [nucl-
th]
[49] E Epelbaum H Krebs and U-G Meiszligner Precision nucleon-nucleon potential at fifth order
in the chiral expansion Phys Rev Lett 115 122301 (2015) arXiv14124623 [nucl-th]
[50] C W Kao T Spitzenberg and M Vanderhaeghen Burkhardt-Cottingham sum rule and
forward spin polarizabilities in heavy baryon chiral perturbation theory Phys Rev D 67
016001 (2003) arXivhep-ph0209241
[51] C-W Kao D Drechsel S Kamalov and M Vanderhaeghen Higher moments of nucleon
spin structure functions in heavy baryon chiral perturbation theory and in a resonance model
Phys Rev D 69 056004 (2004) arXivhep-ph0312102
[52] T Becher and H Leutwyler Baryon chiral perturbation theory in manifestly Lorentz invariant
form Eur Phys J C 9 643 (1999) arXivhep-ph9901384
44
[53] J Gasser M E Sainio and A Svarc Nucleons with chiral loops Nucl Phys B 307 779
(1988)
[54] L S Geng J Martin Camalich L Alvarez-Ruso and M J Vicente Vacas Leading SU(3)-
breaking corrections to the baryon magnetic moments in chiral perturbation theory Phys
Rev Lett 101 222002 (2008) arXiv08051419 [hep-ph]
[55] D Drechsel S S Kamalov and L Tiator The GDH sum rule and related integrals Phys
Rev D 63 114010 (2001) arXivhep-ph0008306
[56] D Drechsel O Hanstein S S Kamalov and L Tiator A unitary isobar model for pion
photo- and electroproduction on the proton up to 1 GeV Nucl Phys A 645 145 (1999)
arXivnucl-th9807001
[57] L Tiator private communication (2020)
[58] V Bernard T R Hemmert and U-G Meiszligner Spin structure of the nucleon at low energies
Phys Rev D 67 076008 (2003) arXivhep-ph0212033
[59] O Gryniuk F Hagelstein and V Pascalutsa Evaluation of the forward Compton scattering
off protons II Spin-dependent amplitude and observables Phys Rev D 94 034043 (2016)
arXiv160400789 [nucl-th]
[60] R Zielinski The g2p Experiment A Measurement of the Protonrsquos Spin Structure Functions
PhD thesis New Hampshire U (2010) arXiv170808297 [nucl-ex]
[61] N Guler et al (CLAS) Precise determination of the deuteron spin structure at low to mod-
erate Q2 with CLAS and extraction of the neutron contribution Phys Rev C 92 055201
(2015) arXiv150507877 [nucl-ex]
[62] C E Carlson V Nazaryan and K Griffioen Proton structure corrections to electronic
and muonic hydrogen hyperfine splitting Phys Rev A 78 022517 (2008) arXiv08052603
[physicsatom-ph]
[63] V Bernard N Kaiser and U-G Meiszligner Chiral dynamics in nucleons and nuclei Int J
Mod Phys E 4 193 (1995) arXivhep-ph9501384
[64] T R Hemmert B R Holstein and J Kambor ∆(1232) and the polarizabilities of the nucleon
Phys Rev D 55 5598 (1997) arXivhep-ph9612374
[65] V Pascalutsa and R Timmermans Field theory of nucleon to higher-spin baryon transitions
Phys Rev C 60 042201 (1999) arXivnucl-th9905065
[66] V Pascalutsa Quantization of an interacting spin-32 field and the Delta isobar Phys Rev
D 58 096002 (1998) arXivhep-ph9802288
[67] H Krebs Double Virtual Compton Scattering and SpinStructure of the Nucleon PoS
CD2018 031 (2019)
45
[68] N Kochelev and Y Oh Axial anomaly and the δLT puzzle Phys Rev D 016012 (2012)
arXiv11034892 [hep-ph]
[69] D Drechsel S Kamalov and L Tiator Unitary isobar model ndash MAID2007 Eur Phys J A
69 (2007) available at httpsmaidkphuni-mainzde arXiv07100306 [nucl-th]
[70] P J Mohr B N Taylor and D B Newell CODATA recommended values of the fundamental
physical constants 2010 Rev Mod Phys 84 1527 (2012)
[71] J Ahrens et al (GDH A2) First measurement of the Gerasimov-Drell-Hearn integral for 1H
from 200 to 800 MeV Phys Rev Lett 87 022003 (2001) arXivhep-ex0105089 [hep-ex]
[72] K Helbing (GDH) Experimental verification of the GDH sum rule at ELSA and MAMI Nucl
Phys Proc Suppl 105 113 (2002)
[73] R Bradford A Bodek H S Budd and J Arrington A New parameterization of the nucleon
elastic form-factors NuInt05 proceedings of the 4th International Workshop on Neutrino-
Nucleus Interactions in the Few-GeV Region Okayama Japan 26-29 September 2005 Nucl
Phys Proc Suppl 159 127 (2006) arXivhep-ex0602017
[74] J D Bjorken Applications of the chiral U(6)timesU(6) algebra of current densities Phys Rev
148 1467 (1966)
[75] J D Bjorken Inelastic scattering of polarized leptons from polarized nucleons Phys Rev D
1 1376 (1970)
[76] V Bernard T R Hemmert and U-G Meiszligner Novel analysis of chiral loop effects in
the generalized Gerasimov-Drell-Hearn sum rule Phys Lett B 545 105 (2002) arXivhep-
ph0203167
[77] A Deur et al Experimental study of isovector spin sum rules Phys Rev D 78 032001
(2008) arXiv08023198 [nucl-ex]
[78] B Pasquini P Pedroni and D Drechsel Higher order forward spin polarizability Phys Lett
B 687 160 (2010) arXiv10014230 [hep-ph]
[79] R Jaffe g2ndashThe nucleonrsquos other spin-dependent structure function Comments Nucl Part
Phys 19 239 (1990)
[80] E V Shuryak and A Vainshtein Theory of power corrections to deep inelastic scattering in
quantum chromodynamics (II) Qminus4 effects polarized target Nucl Phys B 201 141 (1982)
[81] S Wandzura and F Wilczek Sum rules for spin dependent electroproduction Test of rela-
tivistic constituent quarks Phys Lett B 72 195 (1977)
[82] B W Filippone and X-D Ji The spin structure of the nucleon Adv Nucl Phys 26 1
(2001) arXivhep-ph0101224
[83] M Burkardt The g2 structure function Proceedings Workshop on Spin structure at long
46
distance Newport News USA March 12-13 2009 AIP Conf Proc 1155 26 (2009)
arXiv09054079 [hep-ph]
[84] A Deur S J Brodsky and G F de Teramond The spin structure of the nucleon Rept
Prog Phys 82 076201 (2019) arXiv180705250 [hep-ph]
[85] B R Holstein D Drechsel B Pasquini and M Vanderhaeghen Higher order polarizabilities
of the proton Phys Rev C 61 034316 (2000) arXivhep-ph9910427
[86] B R Holstein V Pascalutsa and M Vanderhaeghen Sum rules for magnetic moments and
polarizabilities in QED and chiral effective-field theory Phys Rev D 72 094014 (2005)
arXivhep-ph0507016
[87] B Pasquini D Drechsel and L Tiator Invariant amplitudes for pion electroproduction Eur
Phys J A 34 387 (2007) arXiv07122327 [hep-ph]
[88] A J Buchmann Non-spherical proton shape and hydrogen hyperfine splitting Proceedings
International Workshop on Precision Physics of Simple Atomic Systems (PSAS 2008) Wind-
sor Ontario Canada July 21-26 2008 Can J Phys 87 773 (2009) arXiv09104747
[physicsatom-ph]
47
of moments of inelastic spin structure functions (up to O(ν4) [43])
1
4πSnonpole
1 (νQ2) =2α
MN
I1(Q2) +
2α
MNQ2
[IA(Q2)minus I1(Q2)
]+MNδLT (Q2)
ν2 (12a)
1
4πνSnonpole
2 (νQ2) = 2αI2(Q2) +2α
Q2
[I1(Q2)minus IA(Q2)
]ν2 (12b)
I1(Q2) and IA(Q2) are generalizations of the famous GerasimovndashDrellndashHearn (GDH) sum
rule [5 6] from RCS to the case of virtual photons [1] Their definitions are given in Eqs (26)
and (32) I2(Q2) is the well-known Burkhardt-Cottingham (BC) sum rule [44]
I2(Q2) equiv 2M2N
Q2
int x0
0
dx g2(x Q2) =1
4F2(Q2)GM(Q2) (13)
which can be written as a ldquosuperconvergence sum rulerdquo
Q2
16παM2N
limνrarr0
νS2(νQ2) =
int 1
0
dx g2 (x Q2) = 0 (14)
The latter is valid for any value of Q2 provided that the integral converges for x rarr 0
Combining Eq (5) with the above LEXs of the VVCS amplitudes we can relate IA(Q2)
I1(Q2) γ0(Q2) and δLT (Q2) to moments of inelastic structure functions see Sec III It is
important to note that only γ0(Q2) and δLT (Q2) are generalized polarizabilities The relation
of the inelastic moments IA(Q2) and I1(Q2) to polarizabilities will be discussed in details in
Secs III C and III D The difference between S1(νQ2) and Snonpole1 (νQ2) cf Eq (8a) will
be important in this context
C Details on χPT calculation and uncertainty estimate
In this work we calculated the NLO prediction of BχPT for the polarized non-Born
VVCS amplitudes This includes the leading pion-nucleon (πN) loops see Ref [32 Fig 1]
as well as the subleading tree-level Delta-exchange (∆-exchange) see Ref [30 Fig 2] and
the pion-Delta (π∆) loops see Ref [32 Fig 2] In the δ-power-counting scheme [36] the
LO and NLO non-Born VVCS amplitudes and polarizabilities are of O(p3) and O(p4∆)
respectively2 The LECs are listed in Table I sorted by the order at which they appear in
our calculation At the given orders there are no ldquonewrdquo LECs that would need to be fitted
from Compton processes For more details on the BχPT formalism we refer to Ref [30]
2 In the full Compton amplitude there is a lower order contribution coming from the Born terms leading
to a shift in nomenclature by one order the LO contribution referred to as the NLO contribution etc
see eg Ref [45]
8
where power counting predictive orders (Sec III A) and the renormalization procedure
(Sec III B) are discussed
A few remarks are in order for the inclusion of the ∆(1232) and the tree-level ∆-exchange
contribution In contrast to Ref [32] we include the Coulomb-quadrupole (C2) N rarr ∆
transition described by the LEC gC The relevant Lagrangian describing the non-minimal
γlowastN∆ coupling [33 34] (note that in these references the overall sign of gC is inconsistent
between the Lagrangian and Feynman rules) reads
L(2)∆ =
3e
2MNM+
N T3
igM F
microν partmicro∆ν minus gEγ5Fmicroν partmicro∆ν (15)
+igCM∆
γ5γα(partα∆ν minus partν∆α)partmicroF
microν
+ Hc
with M+ = MN + M∆ and the dual of the electromagnetic field strength tensor F microν =12εmicroνρλFρλ Even though the Coulomb coupling is subleading compared with the electric
and magnetic couplings (gE and gM) its relatively large magnitude cf Table I makes it
numerically important for instance in γ0(Q2) Furthermore we study the effect of modifying
the magnetic coupling using a dipole form factor
gM rarrgM[
1 + (QΛ)2 ]2 (16)
where Λ2 = 071 GeV2 The inclusion of this Q2 dependence mimics the form expected from
vector-meson dominance It is motivated by observing the importance of this form factor
for the correct description of the electroproduction data [33]
To estimate the uncertainties of our NLO predictions we define
δ(Q2) =
radic(∆
MN
)2
+
(Q2
2MN∆
)2
(17)
such that the neglected next-to-next-to-leading order terms are expected to be of relative
size δ2 [33] The uncertainties in the values of the parameters in Table I have a much
smaller impact compared to the truncation uncertainty and can be neglected Unfortunately
∆IA(Q2) γ0(Q2) and γ0(Q2) ie the sum rules involving the cross section σTT (νQ2)
as well as the polarizability ∆I1(Q2) turn out to be numerically small Their smallness
suggests a cancellation of leading orders (which can indeed be confirmed by looking at
separate contributions as shown below) Therefore an error of δ2(Q2)P (Q2) where P (Q2)
is a generalized polarizability might underestimate the theoretical uncertainty for some of
the NLO predictions To avoid this we estimate the uncertainty of our NLO polarizability
9
predictions by
σP (Q2) =
Max[δ4(0)P (0)2 δ4(0)P LO(0)2 δ2(0)PNLO(0)2
]+Max
[δ4(Q2)
[P (Q2)minus P (0)
]2 δ4(Q2)
[P LO(Q2)minus P LO(0)
]2
δ2(Q2)[PNLO(Q2)minus PNLO(0)
]2]12
(18)
where P LO(Q2) is the πN -loop contribution PNLO(Q2) are the ∆-exchange and π∆-loop
contributions and P (Q2) = P LO(Q2) + PNLO(Q2) This error prescription is similar to the
one used in eg Refs [46ndash49] Here since we are interested in the generalized polarizabil-
ities we added in quadrature the error due to the static piece P (0) and the Q2-dependent
remainder P (Q2) minus P (0) Note that the static values of IA(0) and I1(0) are given by the
elastic Pauli form factor which is not part of our BχPT prediction and is considered to be
exact
Note that our result for the spin polarizabilities (and the unpolarized moments [30]) are
NLO predictions only at low momentum transfers Q mπ At larger values of Q amp ∆ they
become incomplete LO predictions Indeed in this regime the ∆ propagators do not carry
additional suppression compared to the nucleon propagators and the π∆ loops are promoted
to LO In general we only expect a rather small contribution from omitted π∆ loops to the
Q2 dependence of the polarizabilities since π∆ loops show rather weak dependence on Q2
compared with the ∆ exchange or πN loops Nevertheless this issue has to be reflected in
the error estimate Since the static polarizabilities P (0) are not affected it is natural to
separate the error on the Q2-dependent remainder P (Q2) minus P (0) as done in Eq (18) To
accommodate for the potential loss of precision above Q amp ∆ we define the relative error
δ(Q2) as growing with increasing Q2 see Eq (17)
Upon expanding our results in powers of the inverse nucleon mass Mminus1N we are able to
reproduce existing results of heavy-baryon χPT (HBχPT) at LO We however do not see
a rationale to drop the higher-order Mminus1N terms when they are not negligible (ie when
their actual size exceeds by far the natural estimate for the size of higher-order terms)
Comparing our BχPT predictions to HBχPT we will also see a deficiency of HBχPT in
the description of the Q2 behaviour of the polarizabilities Note that the O(p4) HBχPT
results from Ref [50 51] which we use here for comparison do not include the ∆ These
references studied the leading effect of the latter in the HBχPT framework using the small-
scale expansion [35] observing no qualitative improvement in the HBχPT description of the
empirical data [50 51] when including it We therefore choose to use the O(p4) results as
the representative HBχPT curves
10
Another approach used in the literature to calculate the polarizabilities in χPT is the
infrared regularization (IR) scheme introduced in Ref [52] This covariant approach tries to
solve the power counting violation observed in Ref [53] by dropping the regular parts of the
loop integrals that contain the power-counting-breaking terms However this subtraction
scheme modifies the analytic structure of the loop contributions and may lead to unexpected
problems as was shown in Ref [54] As we will see in the next section the IR approach
also fails to describe the Q2 behaviour of the polarizabilities
III RESULTS AND DISCUSSION
We now present the NLO BχPT predictions for the nucleon polarizabilities and selected
moments of the nucleon spin structure functions Our results are obtained from the calcu-
lated non-Born VVCS amplitudes and the LEXs in Eqs (10) and (12) For a cross-check
we used the photoabsorption cross sections described in App B In addition to the full NLO
results we also analyse the individual contributions from the πN loops the ∆ exchange
and the π∆ loops
A γ0(Q2) mdash generalized forward spin polarizability
The forward spin polarizability
γ0(Q2) =1
2π2
int infinν0
dν
radic1 +
Q2
ν2
σTT (νQ2)
ν3(19)
=16αM2
N
Q6
int x0
0
dx x2
[g1(xQ2)minus 4M2
Nx2
Q2g2(xQ2)
]
provides information about the spin-dependent response of the nucleon to transversal pho-
ton probes The RCS analogue of the above generalized forward spin polarizability sum
rule is sometimes referred to as the Gell-Mann Goldberger and Thirring (GGT) sum rule
[11] At Q2 = 0 the forward spin polarizability is expressed through the lowest-order spin
polarizabilities of RCS as γ0 = minus(γE1E1 + γM1M1 + γE1M2 + γM1E2) The forward spin po-
larizability of the proton is relevant for an accurate knowledge of the (muonic-)hydrogen
hyperfine splitting as it controls the leading proton-polarizability correction [16 62]
The πN -loop ∆-exchange and π∆-loop contributions to the NLO BχPT prediction of
11
000 005 010 015 020 025 030
-4
-3
-2
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
4D
proton
000 005 010 015 020 025 030
-4
-3
-2
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
4D
neutron
000 005 010 015 020 025 030
0
1
2
Q2 GeV2D
∆LT
10-
4fm
4D
proton
000 005 010 015 020 025 030
0
1
2
3
Q2 GeV2D
∆LT
10-
4fm
4D
neutron
FIG 2 Upper panel Generalized forward spin polarizability for the proton (left) and neutron
(right) as function of Q2 The result of this work the NLO BχPT prediction is shown by the
blue solid line and the blue band The red line represents the LO BχPT result The purple short-
dashed line is the O(p4) HB result from Ref [50] note that the corresponding proton curve is
outside of the plotted range The black dotted line is the MAID model prediction [55ndash57] which
is taken from Ref [1] (proton) and Ref [20] (neutron) The pink band is the IR+∆ result from
Ref [58] and the gray band is the BχPT+∆ result from Ref [31] Empirical extractions for the
proton Ref [18] (blue dots) Ref [59] (purple square) and Ref [60] (orange triangle uncertainties
added in quadrature) and neutron Ref [20] (blue diamonds) and Ref [61] (green dots statistical
and systematic uncertainties added in quadrature) Lower Panel Longitudinal-transverse spin
polarizability for the proton (left) and neutron (right) The orange dot-dashed and purple short-
dashed lines are the O(p3) and O(p4) HB results from Ref [50] The pink band is the IR result
from Ref [58] and the gray band is the covariant BχPT+∆ result from Ref [31] The black dotted
line is the MAID model prediction [55ndash57] note that for the proton we use the updated estimate
from Ref [1] obtained using the π η ππ channels
12
the static forward spin polarizability amount to in units of 10minus4 fm4
γ0p = minus093(92) asymp 201minus 284minus 010 (20a)
γ0n = 003(92) asymp 298minus 284minus 010 (20b)
while the slope is composed as follows in units of 10minus4 fm6
dγ0p(Q2)
dQ2
∣∣∣∣Q2=0
= minus022(4) asymp minus033 + 011 + 001 (21a)
dγ0n(Q2)
dQ2
∣∣∣∣Q2=0
= minus061(7) asymp minus073 + 011 + 001 (21b)
Figure 2 upper panel shows our NLO prediction as well as the LO πN loops compared
to different experimental and theoretical results For the proton we have one determination
at the real-photon point by the GDH collaboration [19] γ0p = minus100(8)(12)times10minus4 fm4 and
further Jefferson Laboratory data [18 60] at very low Q2 For the neutron only data at finite
Q2 are available [20 61] The experimental data for the proton are fairly well reproduced
in the whole Q2 range considered here while for the neutron the agreement improves with
increasing Q2 The HB limit of our πN -loop contribution reproduces the results published
in Refs [50 63] for arbitrary Q2 In addition our prediction is compared to the MAID
model [1 20] the IR+∆ calculation of Ref [58] and the BχPT+∆ result of Ref [31]
The πN -production channel gives a positive contribution to the photoabsorption cross
section σTT (νQ2) at low Q2 cf Fig 10 Accordingly one observes that the πN loops give
a sizeable positive contribution to γ0(Q2) The Delta on the other hand has a very large
effect by cancelling the πN loops and bringing the result close to the empirical data From
Fig 3 upper panel one can see that it is the ∆ exchange which dominates while π∆
loops are negligible This was expected since the forward spin polarizability sum rule is an
integral over the helicity-difference cross section in which σ32 is governed by the Delta at
low energies (the relevant energy region for the sum rule)
To elucidate the difference between the present calculation and the one from Ref [31] we
note that the two calculations differ in the following important aspects Firstly Ref [31] uses
the small-scale counting [64] that considers ∆ and mπ as being of the same size ∆ sim mπ
In practice this results in a set of π∆-loop graphs which contains graphs with one or two
γ∆∆ couplings and hence two or three Delta propagators Such graphs are suppressed in
the δ-counting and thus omitted from our calculation while present in that of Ref [31]
Secondly the Lagrangians describing the interaction of the Delta are constructed differently
and assume slightly different values for the coupling constants In particular we employ
13
000 005 010 015 020 025 030
-3
-2
-1
0
1
2
Q2GeV
2D
Γ01
0-
4fm
4D
proton
000 005 010 015 020 025 030
-3
-2
-1
0
1
2
3
Q2GeV
2D
Γ01
0-
4fm
4D
neutron
000 005 010 015 020 025 030
00
05
10
15
Q2GeV
2D
∆LT1
0-
4fm
4D
proton
000 005 010 015 020 025 030
00
05
10
15
20
25
Q2GeV
2D
∆LT1
0-
4fm
4D
neutron
FIG 3 Contributions of the different orders to the chiral predictions of γ0(Q2) upper paneland δLT (Q2) lower panel for the proton (left) and neutron (right) Red solid line πN -loop
contribution green dot-dashed line ∆-exchange contribution orange dotted line π∆-loop con-
tribution blue long-dashed line total result purple dot-dot-dashed line total result without gCcontribution black short-dashed line total result without gM dipole
(where possible) the so-called ldquoconsistentrdquo couplings to the Delta field ie those couplings
that project out the spurious degree of freedom see Refs [37 65 66] The authors of
Ref [31] on the other hand use couplings where the consistency in this sense is not enforced
The effects of these differences are of higher order in the δ-counting expansion and their
contribution to the Q2 dependence of the considered polarizabilities is expected to be rather
small however the differences at Q2 = 0 could be noticeable [67]
Finally as mentioned in Sec II C the inclusion of the dipole form factor in the magnetic
coupling gM is expected to be important to generate the correct Q2 behaviour of the polar-
izabilities Comparing our predictions for the forward spin polarizability with and without
inclusion of the form factor see Fig 3 upper panel confirms this Without the dipole
our results for the proton and neutron are closer to the ones from Ref [31] where the form
factor is not included For the neutron our prediction without the dipole is able to describe
14
the experimental points at very low Q2 deviating from the data with increasing Q2 The
π∆-loop contribution does not modify the Q2 behavior of γ0(Q2) and only differs from
Ref [31] by a small global shift Note also the relatively large effect of gC which generates
a sign change for virtualities above sim 02 GeV2 see Fig 3 upper panel
B δLT (Q2) mdash longitudinal-transverse polarizability
The longitudinal-transverse spin polarizability
δLT (Q2) =1
2π2
int infinν0
dν
radic1 +
Q2
ν2
σLT (νQ2)
Qν2(22)
=16αM2
N
Q6
int x0
0
dx x2[g1(xQ2) + g2(xQ2)
]
contains information about the spin structure of the nucleon and is another important
input in the determination of the (muonic-)hydrogen hyperfine splitting [16 62] It is also
relevant in studies of higher-twist corrections to the structure function g2(xQ2) given by
the moment d2(Q2) [51] see Section III E The peculiarity of the response encoded in this
polarizability is that it involves a spin flip of the nucleon and a polarization change of the
photon see App B and Fig 11
It is expected that the Delta isobar gives only a small contribution to δLT (Q2) what makes
this polarizability a potentially clean test case for chiral calculations Consequently there are
relatively many different theoretical calculations of δLT (Q2) coming from different versions
of χPT with baryons (HB IR and covariant) Ref [50] found a systematic deviation of the
HB result for δLTn(Q2) from the MAID model prediction This disagreement was identified
by the authors of Ref [68] as a puzzle involving the neutron δLT polarizabilitymdashthe δLT
puzzle The IR calculation in Ref [58] also showed a deviation from the data and predicted
a rapid rise of δLT (Q2) with growing Q2 The problem is solved by keeping the relativistic
structure of the theory as the BχPT+∆ result of Ref [31] showed
As expected already the leading πN loops provide a reasonable agreement with the
experimental data cf Fig 2 lower panel Since the ∆-exchange contribution to δLT (Q2)
is small the effect of the gM form factor is negligible in this polarizability as is that of the gC
coupling cf Fig 3 lower panel In fact we predict both the ∆-exchange and the π∆-loop
contributions to be small and negative This is in agreement with the MAID model which
predicts a small and negative contribution of the P33 wave to δLT (Q2) However in the
calculation of Ref [31] which is different from the one presented here only in the way the
∆(1232) is included the contribution of this resonance to δLTp(Q2) is sizeable and positive
15
The authors of that work attributed this large contribution to diagrams where the photons
couple directly to the Delta inside a loop As mentioned in Sec III F the effect of such loop
diagrams does not change the Q2 behaviour of the polarizabilities On the other hand it
can produce a substantial shift of the δLT (Q2) as a whole A higher-order calculation should
resolve the discrepancy between the two covariant approaches however it will partially lose
the predictive power since the LECs appearing at higher orders will have to be fitted to
experimental data
The πN -loop ∆-exchange and π∆-loop contributions to the NLO BχPT prediction of
the static longitudinal-transverse polarizability are in units of 10minus4 fm4
δLTp = 132(15) asymp 150minus 016minus 002 (23a)
δLTn = 218(23) asymp 235minus 016minus 002 (23b)
while the slopes are in units of 10minus4 fm6
dδLTp(Q2)
dQ2
∣∣∣∣Q2=0
= minus085(8) asymp minus080minus 004minus 001 (24a)
dδLTn(Q2)
dQ2
∣∣∣∣Q2=0
= minus124(12) asymp minus119minus 004minus 001 (24b)
C IA(Q2) mdash a generalized GDH integral
The helicity-difference cross section σTT exhibits a faster fall-off in ν than its spin-averaged
counterpart σT This is due to a cancellation between the leading (constant) terms of σ12
and σ32 at large ν3 The resulting 1ν fall-off of the helicity-difference cross section allows
one to write an unsubtracted dispersion relation for the VVCS amplitude gnonpoleTT (νQ2) cf
Eq (10a) This is the origin of the GDH sum rule [5 6]
minus α
2M2N
κ2 =1
2π2
int infinν0
dνσTT (ν)
ν (25)
which establishes a relation to the anomalous magnetic moment κ It is experimentally
verified for the nucleon by MAMI (Mainz) and ELSA (Bonn) [71 72]
There are two extensions of the GDH sum rule to finite Q2 the generalized GDH integrals
3 Notice that a constant term in σTT at ν rarrinfin is forbidden by crossing symmetry
16
000 005 010 015 020 025 030
-30
-25
-20
-15
-10
-05
00
Q2 GeV2D
I Aproton
000 005 010 015 020 025 030
-25
-20
-15
-10
-05
00
Q2 GeV2D
I A
neutron
000 005 010 015 020 025 030
-05
00
05
Q2 GeV2D
I 1
proton
000 005 010 015 020 025 030
-10
-08
-06
-04
-02
00
02
Q2 GeV2D
I 1
neutron
FIG 4 Upper panel The generalized GDH integral IA(Q2) for the proton (left) and neutron
(right) as function of Q2 The result of this work the NLO BχPT prediction is shown by the blue
solid line and the blue band The red line represents the LO BχPT result The purple short-dashed
line is the O(p4) HB result from Ref [50 51] The gray band is the BχPT+∆ result from Ref [31]
The black dotted line is the MAID model prediction [69] Experimental extractions for the proton
Ref [60] (orange triangle uncertainties added in quadrature) and neutron Refs [21][27] where
magenta dotsorange diagonal crosses correspond to data and red squareslilac crosses correspond
to data plus extrapolation to unmeasured energy regions The green stars at the real-photon point
are derived from the anomalous magnetic moments κp asymp 1793 and κn asymp minus1913 [70] Lower
panel The generalized GDH integral I1(Q2) for the proton (left) and neutron (right) as function
of Q2 The purple short-dashed line is the HB result from Ref [51] Experimental extractions for
the proton Ref [18] (blue dots) and Ref [60] (orange triangle uncertainties added in quadrature)
and neutron Ref [61][27] (uncertainties added in quadrature) where green dotsorange diagonal
crosses correspond to data and gray squareslilac crosses correspond to data plus extrapolation to
unmeasured energy regions
17
IA(Q2) and I1(Q2) The latter will be discussed in Sec III D The former is defined as4
minus α
2M2N
IA(Q2) = minus 1
8π2
int infinν0
dν
radic1 +
Q2
ν2
σTT (νQ2)
ν(26)
=α
Q2
int x0
0
dx[4M2
Nx2
Q2g2(xQ2)minus g1(xQ2)
]
Due to its energy weighting the integral in Eq (26) converges slower than the one in the
generalized forward spin polarizability sum rule (19) Therefore knowledge of the cross
section at higher energies is required and the evaluation of the generalized GDH integral
IA(Q2) is not as simple as the evaluation of γ0(Q2)
The generalized GDH integral IA(Q2) is directly related to the non-pole amplitude
gnonpoleTT (νQ2) which differs from non-Born amplitude gTT (νQ2) by a term involving the
elastic Pauli form factor
gnonpoleTT (νQ2) = gTT (νQ2)minus 2παν
M2N
F 22 (Q2) (27)
cf Eqs (2a) and (8a) Consequently IA(Q2) is not a pure polarizability but also contains
an elastic contribution The ldquonon-polarizabilityrdquo or the Born part of IA(Q2) is given by
IBornA (Q2) = IA(Q2)minus∆IA(Q2) = minus1
4F 2
2 (Q2) (28)
where we refer to the polarizability part as ∆IA(Q2) The same is true for the generalized
GDH integral I1(Q2) which is directly related to Snonpole1 (νQ2)
IBorn1 (Q2) = I1(Q2)minus∆I1(Q2) = minus1
4F 2
2 (Q2) (29)
In the following we will add the Born parts to our LO and NLO BχPT predictions for
the polarizabilities ∆IA(Q2) and ∆I1(Q2) employing an empirical parametrization for the
elastic Pauli form factor [73] This allows us to compare to the experimental results for
IA(Q2) and I1(Q2) cf Fig 4 Note that the blue error bands only describe the uncertainties
of our BχPT predictions of the polarizabilities while the elastic contributions are considered
to be exact as explained in Sec II C The uncertainties of the polarizability predictions are
therefore better reflected in Fig 5 where we show the contributions of the different orders
to the BχPT predictions of ∆IA(Q2) and ∆I1(Q2) as well as the total results with error
bands
4 Note that IA(Q2) is sometimes called ITT (Q2)
18
000 005 010 015 020 025 030
-08
-06
-04
-02
00
02
Q2 GeV2D
DI A
proton
000 005 010 015 020 025 030
-10
-08
-06
-04
-02
00
02
Q2 GeV2D
DI A
neutron
000 005 010 015 020 025 030-005
000
005
010
015
020
025
Q2 GeV2D
DI 1
proton
000 005 010 015 020 025 030-015
-010
-005
000
005
010
015
Q2 GeV2D
DI 1
neutron
FIG 5 Contributions of the different orders to the chiral predictions of ∆IA(Q2) upper paneland ∆I1(Q2) lower panel for the proton (left) and neutron (right) Red solid line πN -loop
contribution green dot-dashed line ∆-exchange contribution orange dotted line π∆-loop contri-
bution blue solid line and blue band total result purple dot-dot-dashed line total result without
gC contribution black short-dashed line total result without gM dipole
The E97-110 experiment at Jefferson Lab has recently published their data for IAn(Q2) in
the region of 0035 GeV2 lt Q2 lt 024 GeV2 [27] In addition there are results for IAn(Q2)
from the earlier E94-010 experiment [21] and for IAp(Q2) from the E08-027 experiment [60]
The O(p4) HB calculation gives a large negative effect [51] which does not describe the data
The BχPT+∆ result from Ref [31] which mainly differs from our work by the absence of
the dipole form factor in gM looks similar to this HB result and only describes the data
points at lowest Q2 Our NLO prediction however follows closely the Q2 evolution of the
data In Fig 5 upper panel we show the polarizability ∆IA(Q2) whose Q2 evolution is
clearly dominated by the ∆ exchange Similar to the case of γ0p(Q2) inclusion of the dipole
in gM and the Coulomb coupling gC is very important in order to describe the experimental
data The LO prediction on the other hand slightly overestimates the data cf Fig 4
upper panel
19
At the real-photon point IA(0) = minusκ2
4and ∆IA(0) = 0 Therefore we give only the
slope of the polarizability ∆IA(Q2) [showing also the separate contributions from πN loops
∆ exchange and π∆ loops] in units of GeVminus2
d∆IAp(Q2)
dQ2
∣∣∣∣Q2=0
= minus858(343) asymp 238minus 1121 + 025 (30a)
d∆IAn(Q2)
dQ2
∣∣∣∣Q2=0
= minus955(343) asymp 141minus 1121 + 025 (30b)
Including the empirical Pauli form factor [73] we find in units of GeVminus2
dIAp(Q2)
dQ2
∣∣∣∣Q2=0
= minus318dIAn(Q2)
dQ2
∣∣∣∣Q2=0
= minus300 (31)
D Γ1(Q2) and I1(Q2) mdash the first moment of the structure function g1(xQ2)
The second variant for a generalization of the GDH sum rule to finite Q2 is defined as
minus α
2M2N
I1(Q2) = minus 1
8π2
int infinν0
dν1radic
ν2 +Q2
[σTT (νQ2) +
Q
νσLT (νQ2)
](32)
= minus α
Q2
int x0
0
dx g1(xQ2)
where I1(0) = minusκ2
4 This generalized GDH integral directly stems from the amplitude
Snonpole1 (νQ2) with the LEX from Eq (12a) It is given by the first moment of the struc-
ture function g1(xQ2) Γ1(Q2) =int x0
0dx g1(xQ2) as follows I1(Q2) =
2M2N
Q2 Γ1(Q2) The
isovector combination
Γ1(pminusn)(Q2) =
int x0
0
dx[g1p(xQ
2)minus g1n(xQ2)] (33)
is related to the axial coupling of the nucleon through the Bjorken sum rule [74 75]
limQ2rarrinfin
Γ1(pminusn)(Q2) =
gA6 (34)
As explained in Eq (28) the moment I1(Q2) splits into a polarizability part ∆I1(Q2) and
a Born part IBorn1 (Q2) Figure 4 lower panel shows the Q2 dependence of I1(Q2) which in
contrast to IA(Q2) shown in Figure 4 upper panel is clearly dominated by its Born part
and the elastic Pauli form factor The πN -loop ∆-exchange and π∆-loop contributions to
the polarizability ∆I1(Q2) are shown in Fig 5 lower panel Comparing to Fig 5 upper
20
000 005 010 015 020 025 030
-006
-004
-002
000
002
004
Q2 GeV2D
G1
proton
000 005 010 015 020 025 030
-006
-005
-004
-003
-002
-001
000
001
Q2 GeV2D
G1
neutron
FIG 6 First moment of the structure function g1(xQ2) for the proton (left) and neutron (right)
as function of Q2 The legend is the same as in Fig 4
panel one sees that ∆I1(Q2) is less sensitive to gC and the dipole form factor in gM than
∆IA(Q2)
For the proton our NLO BχPT prediction gives a very good description of the ex-
perimental data [18 60] and is in reasonable agreement with the MAID prediction [69]
For the neutron one observes good agreement with the empirical evaluations including ex-
trapolations to unmeasured energy regions starting from Q2 gt 01 GeV2 [27 61] In the
region of Q2 lt 005 GeV2 one observes an interesting tension between the recent E97-
110 experiment [27] and the data from CLAS [61] While the newest measurement finds
I1n(0035 GeV2) lt κ2n4 thus suggesting a negative slope at low Q the older measurement
found a rather large value for I1n(00496 GeV2) A similar but milder behaviour is seen in
the E97-110 [27] and E94-010 [21] data for IAn The MAID predictions do not agree with
the CODATA recommended values for the anomalous magnetic moments of the proton and
neutron [70] which in our work are imposed by using empirical parametrizations for the
elastic Pauli form factors [73] The slope of the HB result from Ref [51] is too large and
therefore only reproduces the data at very low Q2
Figure 6 shows the moment Γ1(Q2) for the proton and neutron while Fig 7 shows the
isovector combination Γ1 pminusn(Q2) The LO and NLO BχPT predictions are identical be-
cause our calculation produces the same Delta contributions for the proton and the neutron
For the isovector combination the MAID model only agrees with the data at very low
Q2 lt 010 GeV2 The same is true for the IR result [58 76] while all other chiral results
describe the data NLO BχPT (this work) BχPT+∆ [31] and HBχPT [51]
At the real-photon point I1(0) = minusκ2
4and ∆I1(0) = 0 Therefore we give only the
slope of the polarizability ∆I1(Q2) [showing also the separate contributions from πN loops
21
000 005 010 015 020 025 030-002
000
002
004
006
008
010
Q2 GeV2D
G1Hp-
nL
FIG 7 Isovector combination of Γ1(Q2) as function of Q2 The legend is the same as in Fig 6
The pink curve is the IR result from Ref [58 76] The experimental points are from Ref [77]
(brown dots) and Ref [23] (orange squares)
∆ exchange and π∆ loops] in units of GeVminus2
d∆I1p(Q2)
dQ2
∣∣∣∣Q2=0
= 039(4) asymp 034minus 053 + 058 (35a)
d∆I1n(Q2)
dQ2
∣∣∣∣Q2=0
= minus101(10) asymp minus107minus 053 + 058 (35b)
Including the empirical Pauli form factor [73] we find in units of GeVminus2
dI1p(Q2)
dQ2
∣∣∣∣Q2=0
= 580dI1n(Q2)
dQ2
∣∣∣∣Q2=0
= 553 (36)
E d2(Q2) mdash a measure of color polarizability
Another interesting moment to consider is d2(Q2) which is related to the twist-3 part of
the spin structure function g2(xQ2) [79 80]
d2(Q2) equiv 3
int 1
0
dx x2[g2(xQ2)minus gWW2 (xQ2)] (37)
where gWW2 (xQ2) is the twist-2 part of g2(xQ2) Using the Wandzura-Wilczek relation [81]
one can relate d2(Q2) to moments of the spin structure functions g1(xQ2) and g2(xQ2)
d2(Q2) =
int 1
0
dx x2 [3g2(xQ2) + 2g1(xQ2)] (38)
22
000 005 010 015 020 025 030
0000
0001
0002
0003
0004
0005
0006
0007
Q2 GeV2D
d2
proton
000 005 010 015 020 025 030
0000
0002
0004
0006
0008
0010
0012
Q2 GeV2D
d2
neutron
000 005 010 015 020 025 030
-1
0
1
2
Q2 GeV2D
Γ 010
-4
fm6
D
proton
000 005 010 015 020 025 030-2
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
6D
neutron
FIG 8 Upper panel The inelastic moment d2(Q2) for the proton (left) and neutron (right) as
function of Q2 The result of this work the NLO BχPT prediction is shown by the blue solid line
and the blue band The red line represents the LO BχPT result The purple short-dashed line is
the O(p4) HB result from Ref [50 51] The black dotted line is the MAID model prediction [69]
The experimental points for the neutron (cyan dots) are from Ref [22] Lower panel Fifth-order
generalized forward spin polarizability γ0(Q2) for the proton (left) and neutron (right) as function
of Q2 The experimental points for the proton are from Ref [59] (purple square) and Ref [78]
(orange dot)
This relation however only holds for asymptotically large Q2 It is also in the high-Q2
region where d2(Q2) is a measure of color polarizability [82 83] through its relation to the
gluon field strength tensor [80] We refer to Ref [84] for a recent review on the spin structure
of the nucleon including a discussion of sum rules for deep inelastic scattering and color
polarizabilities
What we consider in the following is the inelastic part of d2(Q2) defined as the moment
of g1(xQ2) and g2(xQ2) spin structure functions cf Eq (38)
d2(Q2) =
int x0
0
dx x2 [3g2(xQ2) + 2g1(xQ2)] (39)
23
This moment provides another testing ground for our BχPT predictions through comparison
with experiments on the neutron [22] Going towards the low-Q2 region the interpretation
of d2(Q2) in terms of color polarizabilities will fade out The above definition however
implies it is related to other VVCS polarizabilities
d2(Q2) =Q4
8M4N
[M2
NQ2
αδLT (Q2) + I1(Q2)minus IA(Q2)
] (40)
Note that d2(Q2) and its first two derivatives with respect to Q2 vanish at Q2 = 0 The
considerations in Eqs (28) and (29) have no effect on d2(Q2) since the Born contribution
from IA(Q2) and I1(Q2) cancel out Therefore d2(Q2) is a pure polarizability
In Fig 8 upper panel we show our NLO BχPT prediction and other results for d2(Q2)
While MAID [69] and BχPT describe the experimental data for the neutron [22] very well
the HB limit [50 51] is showing a fast growth with Q2 This illustrates the importance
of keeping the relativistic result Note also that even though the πN -loop contribution
is dominant both gC and the form factor in gM are essential to obtain a curvature that
reproduces the data cf Fig 9 upper panel For the proton there are to our knowledge
no experimental results to compare with However the agreement between the NLO BχPT
prediction and the MAID prediction at low energies is reasonable
F γ0(Q2) mdash fifth-order generalized forward spin polarizability
It is interesting to compare the generalized fifth-order forward spin polarizability sum
rule
γ0(Q2) =1
2π2
int infinν0
dν
radic1 +
Q2
ν2
σTT (νQ2)
ν5(41)
=64αM4
N
Q10
int x0
0
dx x4
[g1(xQ2)minus 4M2
Nx2
Q2g2(xQ2)
]
to the sum rule integrals for IA(Q2) and γ0(Q2) since they differ merely by their energy
weighting of σTT (νQ2) and a constant prefactor cf Eqs (19) (26) and (41) From IA(Q2)
to γ0(Q2) to γ0(Q2) the energy suppression is increasing by a factor of νminus2 respectively
Therefore the description of γ0(Q2) should be easiest in a low-energy effective-field theory
such as χPT whereas γ0(Q2) and IA(Q2) receive larger contributions from higher energies
In Fig 8 lower panel we show our LO and NLO BχPT predictions for γ0(Q2) One
can see that the πN -loop contribution is positive (in accordance to what we see for the cross
section σTT see Fig 10) The Delta shifts it substantially especially in the low Q2 region
24
000 005 010 015 020 025 030
-0004
-0002
0000
0002
0004
0006
0008
0010
Q2GeV
2D
d2
proton
000 005 010 015 020 025 030
-0005
0000
0005
0010
0015
Q2GeV
2D
d2
neutron
000 005 010 015 020 025 030
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
6D
proton
000 005 010 015 020 025 030
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
6D
neutron
FIG 9 Contributions of the different orders to the chiral predictions of d2(Q2) upper panel and
γ0(Q2) lower panel for the proton (left) and neutron (right) Red solid line πN -loop contribu-
tion green dot-dashed line ∆-exchange contribution orange dotted line π∆-loop contribution
blue long-dashed line total result purple dot-dot-dashed line total result without gC contribution
black short-dashed line total result without gM dipole
bringing it into a better agreement with data In general the BχPT curves start above the
empirical data points at the real-photon point and then decrease asymptotically to zero
above Q2 gt 01 GeV2 On the other hand the MAID prediction reproduces the empirical
data at the real-photon point then decreases to negative values until about Q2 gt 006 GeV2
from where it also starts to asymptotically approach zero Consequently our NLO BχPT
prediction of γ0(Q2) is consistently above the MAID prediction This is very different to
what we saw for IA(Q2) in Fig 4 upper panel where the MAID prediction at the real-
photon point is above the experimental value While the agreement of our predictions with
the empirical data is in general quite good for all moments of σTT (νQ2) one should point
out that both for γ0n(Q2) and γ0p(Q2) we overestimate the data at low Q2 For IA(Q2) such
observation cannot be made because ∆IA(0) = 0 and thus IA(0) is given by the empirical
Pauli form factor only From IA(Q2) γ0(Q2) and γ0(Q2) the latter has the smallest however
25
non-negligible dependence on gC and the dipole in gM cf Fig 9 lower panel
The πN -loop ∆-exchange and π∆-loop contributions to the NLO BχPT prediction of
the static fifth-order forward spin polarizability amount to in units of 10minus4 fm6
γ0p = 112(30) asymp 208minus 096minus 001 (42a)
γ0n = 195(30) asymp 292minus 096minus 001 (42b)
while the slope is composed as follows in units of 10minus4 fm8
dγ0p(Q2)
dQ2
∣∣∣∣Q2=0
= minus084(10) asymp minus100 + 016 + 000 (43a)
dγ0n(Q2)
dQ2
∣∣∣∣Q2=0
= minus142(15) asymp minus158 + 016 + 000 (43b)
Note that the HB prediction of the static polarizability γ0p (423 at O(p3) and 365 at O(ε3)
[78 85]) is almost one order of magnitude larger than the empirical value and therefore not
shown in Fig 8
G Summary
Our results are summarized in Table II where we give the contributions of the different
orders to the chiral predictions of the polarizabilities and their slopes at the real-photon
point A quantitative comparison of our predictions for the static spin polarizabilities to
the work of Bernard et al [31] and different empirical evaluations is shown in Table III We
can see that the inclusion of the Delta turns out to be very important for all moments of
the helicity-difference cross section To describe the Q2 behavior of the polarizabilities the
magnetic coupling of the N rarr ∆ transition should be modified by a dipole form factor as
has been observed previously in the description of electroproduction data [33] This dipole
form factor effectively takes account of vector-meson exchanges The Coulomb-quadrupole
N rarr ∆ transition despite its subleading order is important in the description of some
moments of spin structure functions This is contrary to what we saw for the moments of
unpolarized structure functions [30] where the Coulomb coupling had a negligible effect
The π∆ loops are mainly relevant for the generalized GDH integrals
26
TABLE II The NLO BχPT predictions for the forward VVCS polarizabilities and their slopes at
Q2 = 0 The contributions of the πN loops the ∆ exchange and the π∆ loops are shown together
with the combined total result Note that IA(0) = I1(0) = d2(0) = 0 and (d2)prime = 0
πN loops ∆ exchange π∆ loops Total
γ0 p 201 minus284 minus010minus093(92)
(10minus4 fm4) n 298 003(92)
δLT p 150 minus016 minus002132(15)
(10minus4 fm4) n 235 218(23)
γ0 p 208 minus096 minus001112(30)
(10minus4 fm6) n 292 195(30)
(γ0)prime p minus033011 001
minus022(4)
(10minus4 fm6) n minus073 minus061(7)
(δLT )prime p minus080 minus004 minus001minus085(8)
(10minus4 fm6) n minus119 minus124(12)
(γ0)prime p minus100016 000
minus084(10)
(10minus4 fm8) n minus158 minus142(15)
(∆IA)prime p 238 minus1121 025minus858(343)
(GeVminus2) n 141 minus955(343)
(∆I1)prime p 034 minus053 058039(4)
(GeVminus2) n minus107 minus101(10)
TABLE III Our NLO BχPT predictions for the spin polarizabilities at Q2 = 0 compared with
the BχPT+∆ predictions from Bernard et al [31] and the available empirical information Where
the reference is not given the empirical number is provided by the MAID analysis [55 69] with
unspecified uncertainty
Proton Neutron
This work BχPT+∆ Empirical This work BχPT+∆ Empirical
γ0 minus093(92) minus174(40) minus100(8)(12) [19] 003(92) minus077(40) minus0005
(10minus4 fm4) minus090(8)(11) [78] [MAID]
minus0929(105) [59]
δLT 132(15) 240(1) 134 218(23) 238(3) 203
(10minus4 fm4) [MAID] [MAID]
γ0 112(30) 060(7)(7) [78] 195(30) 123
(10minus4 fm6) 0484(82) [59] [MAID]
IV CONCLUSIONS
We have presented a complete NLO calculation of the polarized non-Born VVCS ampli-
tudes in covariant BχPT with pion nucleon and ∆(1232) fields The dispersion relations
between the VVCS amplitudes and the tree-level photoabsorption cross sections served as a
27
cross-check of these calculations
The obtained moments of the proton and neutron spin structure functions related to gen-
eralized polarizabilities and GDH-type integrals agree well with the available experimental
data The description of their Q2 evolution is improved compared to the previous χPT pre-
dictions In particular the NLO BχPT predictions obtained here give a better description
of the empirical data (eg from the Jefferson Laboratory ldquoSpin Physics Programrdquo) than
the HB [50 51] and IR [58] calculations
The demonstrated predictive power of the χPT framework amplitudes makes it well
suited for extending the χPT evaluation of the TPE effect in the hyperfine structure of
(muonic-)hydrogen [15ndash17] to next-to-leading order
ACKNOWLEDGEMENTS
We thank Lothar Tiator and Marc Vanderhaeghen for helpful discussions This work
is supported by the Deutsche Forschungsgemeinschaft (DFG) through the Collaborative
Research Center [The Low-Energy Frontier of the Standard Model (SFB 1044)] JMA ac-
knowledges support from the Community of Madrid through the ldquoPrograma de atraccion de
talento investigador 2017 (Modalidad 1)rdquo and the Spanish MECD grants FPA2016-77313-P
FH gratefully acknowledges financial support from the Swiss National Science Foundation
Appendix A Tensor decompositions of the VVCS amplitudes
In this appendix we review the decomposition of the forward VVCS process into tensor struc-
tures and scalar amplitudes In particular we consider the connection between the covariant and
the semi-relativistic decomposition in the lab frame that is defined in terms of the conventional
transverse longitudinal transverse-transverse and transverse-longitudinal amplitudes
As explained in Sec II A the process of forward VVCS off the nucleon can be described in
terms of four explicitly covariant amplitudes S1 2 and T1 2 [3]
T (νQ2) =
(minusgmicroν +
qmicroqν
q2
)T1(νQ2) +
1
M2N
(pmicro minus p middot q
q2qmicro)(
pν minus p middot qq2
qν)T2(νQ2) (A1)
minus 1
MNγmicroναqα S1(νQ2)minus 1
M2N
(γmicroνq2 + qmicroγναqα minus qνγmicroαqα
)S2(νQ2)
εprimelowastmicro εν
where εmicro (εprimelowastmicro ) are the incoming (outgoing) photon polarization vectors ν is the photon lab-frame
energy and Q2 is the photon virtuality Alternatively the decomposition in the laboratory frame
28
(which in the forward case coincides with the Breit frame) is parametrized in terms of the nucleon
Pauli matrices ~σ and the four scalar functions fL fT gTT and gLT
T (νQ2) = ε0 εprimelowast0 fL(νQ2) + (~ε primelowast middot ~ε ) fT (νQ2) + i~σ middot (~ε primelowast times ~ε ) gTT (νQ2) (A2)
minusi~σ middot [(ε0~εprimelowast minus ~ε ε primelowast0 )times q] gLT (νQ2)
Here ~q and q = ~q|~q | are the photon three-momentum in the lab system and its unit vector The
modified polarization vector components are given by
ε0 =
[ε0 minus
ν
|~q | (~ε middot q )
] |~q |Q ~ε = ~εminus q (~ε middot q ) (A3)
where ε = (ε0~ε ) is the usual incoming photon polarization vector and εprimelowast the outgoing polarization
vector The LEX of the lab frame amplitudes [Eq (10)] can serve in particular as the definition
of the generalized polarizabilities The lab frame amplitudes are also conveniently used for the
definition of the response functions see the example of the scalar amplitude gLT (νQ2) and the
corresponding response function σLT (νQ2) below in App B
Appendix B Photoabsorption cross sections
In the forward kinematics the spin-dependent VVCS amplitudes and the spin polarizabilities
can be described in terms of the polarized structure functions g1(xQ2) and g2(xQ2) or equiv-
alently the helicity-difference cross section σTT (νQ2) and the longitudinal-transverse response
function σLT (νQ2) with the help of dispersion relations (5) and the optical theorem (3) In this
way the photoabsorption cross sections measured in electroproduction processes form the basis
for most empirical evaluations shown throughout Sec III In the following we present the BχPT
predictions for the tree-level cross sections of πN - π∆- and ∆-production through photoabsorption
on the nucleon cf Figs 8 9 and 10 in Ref [30] In Secs B 1 and B 2 we will discuss the leading
πN -production channel and the ∆-production channel respectively We used these cross sections
to verify the polarizability predictions obtained otherwise from the calculated non-Born VVCS
amplitudes Due to the bad high-energy behavior of the π∆-production cross sections in BχPT
cf Fig 10 the dispersion relations in Eq (5) require further subtractions for a reconstruction of
the π∆-loop contribution to the spin-dependent VVCS amplitudes Therefore not all polarizabil-
ities could be verified but only those appearing as higher-order terms in the LEX of the VVCS
amplitudes such as γ0 [16]
29
proton neutron
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Neutron
02 04 06 08 100
50
100
150
200
250
[GeV]
T [b]
02 04 06 08 10
-50
0
50
100
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Neutron
02 04 06 08 100
50
100
150
200
250
[GeV]
T [b]
02 04 06 08 10
-50
0
50
100
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02040608100
50
100
150
200
250
300
[GeV]
T[b]
0204060810
-50
0
50
[GeV]
TT[b]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
0204060810
00
02
04
06
[GeV]
LTQ
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
0204060810
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
FIG 10 Photoabsorption cross sections for πN (red) and π∆ production (orange) with Q2 = 0
(solid) and Q2 = 01 GeV2 (dashed for πN and dotted for π∆ channel)
1 πN-production channel
In order to extract the response function σLT (νQ2) we have developed a method similar to
the one used to calculate σTT (νQ2) see for example Ref [86] For σLT (νQ2) however the
calculation is more complicated because one has to take into account that the associated Compton
process involves a spin-flip of the nucleon as illustrated in Fig 11 When calculating the cross
section the product of the incoming nucleon spinors has to reflect this flip
The forward VVCS amplitude related to σLT (νQ2) mdash and δLT (Q2) mdash is gLT (νQ2) It can
be extracted from Eq (A2) if one takes the modified polarization vector components in Eq (A3)
with ε = εL and εprimelowast = εlowastplusmn as input where εL = 1Q(|~q | 0 0 q0) and εplusmn = ∓ 1radic
2(0 1plusmni 0) are the
standard longitudinal and transverse polarization vectors respectively For εL and εlowastplusmn only the
choice of helicities hprime = plusmn12 and h = ∓12 gives a non-zero contribution and one obtains
χdaggerhprime T (νQ2)χh = χdaggerhprimeminusi~σ middot [(ε0~εprimelowast minus ~ε ε primelowast0 )times q] gLT (νQ2)χh =
radic2 gLT (νQ2) (B1)
where χh and χdaggerhprime are two-component Pauli spinors with opposite helicities or here spins
Let us now consider the related photoabsorption process and in particular the tree-level γlowastN rarr
30
s = 12
sprime = minus12
λ = 0 λprime = 1
rprime rprime
s=12sprime=minus1
2
λ=0λprime=1
rprimerprime
s=12
sprime=minus12
λ=0λprime=1
rprimerprime hacute=frac12h=-frac12
λacute=1λ=0
FIG 11 Relation between the forward Compton process and the photoabsorption process given
by the optical theorem In particular we show the longitudinal-transverse contribution The
double-line arrows represent the spin of the external particles while the dot represents the scalar
(longitudinal) polarization of the incoming photon Inside the blob the intermediate states are
represented eg nucleons with spins rprime (which are averaged in the calculation of the cross section)
and pions
πN channel see diagrams in Fig 8 of Ref [30] We define the πN -production amplitude as
T = uhB (PB)sumi
Ai(s t)Γi uhA(PA) (B2)
with the Dirac structures
Γ1 = γ5 (B3a)
Γ2 =1
2
[qA ε
]γ5 (B3b)
where uhA(PA) and udaggerhB (PB) are the Dirac spinors and PA and PB are the four-momenta of
the incoming and outgoing nucleons respectively When calculating the photoabsorption cross
section related to the VVCS amplitude in Eq (B1) the nucleon spin flip should be implemented
by uhprime(PA) in T dagger and uh(PA) in T together with the appropriate transverse and longitudinal
photon polarization vectors εlowastplusmn and εL
However if one wants to use the properties of the Dirac matrices it is more useful to construct
an operator to produce this spin flip in the external nucleons of Fig 11 This is accomplished by
introducing the projector ΓLT equiv 12radic
2(γ1 + iγ2)γ5 which also takes into account the extra factor
radic2 in Eq (B1) We checked that with this projector one correctly extracts δLT by comparing the
HB limit of our result to the HB result of Ref [50] where the authors calculate this polarizability
from the Compton amplitude directly With all those ingredients the longitudinal-transverse cross
section is calculated in the following way
σLT (νQ2) =1
64π2 s
|~pf |cm
|~pi|cm
int 1
minus1dcos θ
sumij
AiAdaggerjXij (B4)
31
with
Xij = Tr[(PB +MN )Γi(PA +MN )ΓLTγ0Γdaggerjγ
0] (B5)
where θ is the scattering angle in the center-of-mass (cm) frame and |~pi|cm (|~pf |cm) is the three-
momentum of an incoming (outgoing) particle in the cm frame An explicit calculation of the
matrix Xij leads to
X = MNQ
0 2(PB minus PA) middot εLminusradic
2|~qf |cm sin θ (sminus u)
(B6)
where |~qi|cm (|~qf |cm) is the relative three-momentum of the incoming (outgoing) particles in the
cm frame Here s t and u are the usual Mandelstam variables For the different γlowastN rarr πN
channels we obtain the following amplitudes Ai where we introduce qA as the four-momentum of
the incoming photon and qB as the four-momentum of the outgoing pion
bull γlowastprarr π0p
A1 =e gAMN
fπ
[2PA middot ε+ qA middot ε
sminusM2N
+2PB middot εminus qA middot ε
uminusM2N
] (B7a)
A2 =e gAMN
fπ
[1
sminusM2N
+1
uminusM2N
] (B7b)
bull γlowastprarr π+n
A1 =
radic2 e gAMN
fπ
[2PA middot ε+ qA middot ε
sminusM2N
+2(PA minus PB) middot ε+ qA middot ε
tminusm2π
] (B8a)
A2 =
radic2 e gAMN
fπ(sminusM2N )
(B8b)
bull γlowastnrarr π0n
A1 = 0 (B9a)
A2 = 0 (B9b)
bull γlowastnrarr πminusp
A1 =
radic2 e gAMN
fπ
[2PB middot εminus qA middot ε
uminusM2N
minus 2(PA minus PB) middot ε+ qA middot εtminusm2
π
] (B10a)
A2 =
radic2 e gAMN
fπ(uminusM2N )
(B10b)
The analytical expressions shown above were checked with the amplitudes given in Ref [87] An-
alytical expressions for the tree-level γlowastN rarr πN channel of the σLT (νQ2) and σTT (νQ2) cross
32
sections are given below (proton channels π+n and π0p neutron channel πminusp) We checked that
they reproduce the known results in the real-photon limit [45 86] To shorten the final expressions
for the cross sections which are considerably longer for finite Q2 than in the real-photon limit we
define the following dimensionless kinematic variables
αγ = (ENi )cmradics =
s+M2N +Q2
2s (B11)
απ = (ENf )cmradics =
s+M2N minusm2
π
2s (B12)
βγ = Eγcmradics =
sminusM2N minusQ2
2s (B13)
βπ = Eπcmradics =
sminusM2N +m2
π
2s (B14)
λγ = |~qi|cmradics =
radic(sminusM2
N minusQ2)2 + 4sQ2
2s (B15)
λπ = |~qf |cmradics =
radic(sminusM2
N +m2π)2 minus 4sm2
π
2s (B16)
Here (ENi )cm and (ENf )cm are the energies of the incoming and outgoing nucleon Eγcm is the energy
of the incoming photon Eπcm is the energy of the outgoing pion all in the cm frame
σ(π+n)TT = minus e2g2
AM2N
64πf2πs
3(sminusM2N )2λ4
γ
4sλπλγ
[(M2
N minus s)(M2N minusQ2 minus s)(Q2 + 2sβγβπ)
+ 2s(m2πQ
2 minus (M2N minus s)
(M2N + s(minus1 + 2βγβπ)
))λ2γ
]+ (M2
N minus s)(M2N minusQ2 minus s)
times (Q2 + 2sβγβπ minus 2sλπλγ)(Q2 + 2s(βγβπ + λπλγ)
)times log
(Q2 + 2sβγβπ minus 2sλγλπQ2 + 2sβγβπ + 2sλγλπ
) (B17)
33
σ(π0p)TT =
e2g2AM
2Nλπ
64πf2πs(sminusM2
N )2λγ
1
s2λπλ3γ
[(M2
N minus s)(M2N minusQ2 minus s)
(3M2
N + s(2βγβπ minus 3))(M2N
+ s(2βγβπ minus 1))
+ 2s
((2m2
πQ2 + (M2
N minus s)2)(M2
N minus s)minus 2(minusm2
πQ2 + (sminusM2
N )2)
times sβγβπ + 4s2(sminusM2N )β2
γβ2π
)λ2γ
]arctan
(2sλπλγ
M2N + s(2βγβπ minus 1)
)+ 2
[minus 2m2
πQ2
+ (M2N minus s)
(2(sminusM2
N ) + 4sβγβπ +(sminusM2
N +Q2)(3M2N minus 3s+ 2sβγβπ)
sλ2γ
+m2πQ
2(sminusM2N )
(M2N minus s+ 2sβγβπ)(M2
N + s(minus1 + 2βγβπ minus 2λπλγ))
+m2πQ
2(sminusM2N )
(M2N minus s+ 2sβγβπ)(M2
N + s(minus1 + 2βγβπ + 2λπλγ))
)
+1
λπλγ
(βγβπ
(minusm2
πQ2 + (M2
N minus s)(M2N minus s+ 2sβγβπ)
)minus (M2
N minus s)(M2N minusQ2 minus s)λ2
π
)log
(M2N + s(minus1 + 2βγβπ + 2λπλγ)
M2N + s(minus1 + 2βγβπ minus 2λπλγ)
)] (B18)
σ(πminusp)TT =
e2g2AM
2N
64πf2πs
3λ4γ(M2
N + s(2βπβγ minus 1))2 minus 4λ2πs
2λ2γ)
4sλπλγ
((minusM2
N +Q2 + s)(M2N
+ s(2βπβγ minus 1))
2 minus 2sλ2γ
(2λ2
πs(minusM2N +Q2 + s) +m2
πQ2))
+((M2N + s(2βπβγ minus 1)
)2
minus 4λ2πs
2λ2γ
)((M2N + s(2βπβγ minus 1)
)(M2N minus 2Q2 minus s(2βπβγ + 1)
)+ 4λ2
πs2λ2
γ
)times log
(minusM2
N minus 2βπsβγ minus 2λπsλγ + s)minus(M2N + s(2βπβγ minus 2λπλγ minus 1)
)(M2N + s(minus1
+ 2βπβγ + 2λπλγ))[((
M2N + s(2βπβγ minus 1)
)(M2N minus 2Q2 minus s(2βπβγ + 1)
)+ 4λ2
πs2λ2
γ
)times log
(minusM2
N minus 2βπsβγ + 2λπsλγ + s)
+((Q2 + 2βπsβγ)2 minus 4λ2
πs2λ2
γ
)times log
(Q2 + 2βπsβγ minus 2λπsλγQ2 + 2sβπβγ + 2λπsλγ
)]+ 4sλ2
γ
((M2
N + s(2βπβγ minus 1))2 minus 4λ2πs
2λ2γ
)times (M2
N minus s) arctanh
(2λπsλγ
M2N + 2βπsβγ minus s
) (B19)
34
σ(π+n)LT =
e2g2AM
3Nλπ
32πf2πQs
3(sminusM2N )2λ4
γ
2sλγ
[(M2
N minus s)(Q2 + 2sβ2γ)(Q2 + 2sβγβπ)
minus 4s
((M2
N minus s)(Q2 minus 2s(απ minus 1)βγ
)+Q2sβγβπ
)λ2γ + 8s3(minus1 + απ)λ4
γ
]minus M2
N minus sλπ
[(Q2 + 2sβ2
γ)(Q2 + 2sβγβπ)2 + 4s2(2(απ minus 1)βγ(Q2 + 2sβγβπ)
minusQ2λ2π
)λ2γ + 8s3(απ minus 1)2λ4
γ
]arctan
(2sλπλγ
Q2 + 2sβγβπ
) (B20)
σ(π0p)LT =
e2g2AM
3Nλπ
16πf2πQs(sminusM2
N )2λγ
1
minus2s(M2N + s(minus1 + 2βγβπ))2λ2
γ + 8s3λ2πλ
4γ
[minus 3M8
N (Q2
+ 2sβ2γ) + 2M4
Ns2
(minus (Q2 + 2sβ2
γ)(minus3 + 2βγβπ) +(Q2(minus3 + 2βγβπ + 6λ2
π) + 2sβγ
times (minus3 + 12απ + 2βγβπ minus 12απβγβπ + 4βγλ2π))λ2γ minus 4s(α2
π minus 1)λ4π
)+ 2M2
Ns3
[minus (Q2
+ 2sβ2γ)(minus1 + 2βγβπ)
(6 + βγβπ(minus9 + 2βγβπ)
)+
(Q2(3minus 12λ2
π + 4βγβπ(minus1 + 2βγβπ
+ λ2π))minus 2sβγ
(minus 3 + 4απ(3 + 2απ(3 + 2βγβπ(minus3 + βγβπ)) + 4βγ(βπ + (2minus βγβπ)
times λ2π)))λ2γ + 8s
((απ minus 1)(1 + απ minus 2βγβπ) + 2απβγλ
2π
)λ4γ
)+ s4
((Q2 + 2sβ2
γ)
times (1minus 2βγβπ)2(2βγβπ minus 3) + 2(Q2(minus1 + 6λ2
π + 2βγβπ((1minus 2βγβπ)2 minus 2λ2π)) + 2sβγ
times (minus1 + 2βγβπ + 4(minus1 + βγβπ)(minusαπ + 2απβγβπ minus βγλ2π)))λ2γ minus 8
(s(απ minus 1)(απ
+ (1minus 2βγβπ)2) + 2βγ(2sαπ +Q2βγ)λ2π
)λ4γ + 32s(απ minus 1)λ2
πλ6γ
)+ 2M6
Ns
(Q2(6
minus 7βγβπ + λ2γ) + 2sβγ(βγ(6minus 7βγβπ) + (1minus 4απ)λ2
γ)
)]+sminusM2
N
4s2λπλ3γ
[(Q2 + 2sβ2
γ)
times(3M2
N + s(2βγβπ minus 3))(M2N + s(2βγβπ minus 1)
)+ 2s
(minusM2
N (Q2 + 2sβγ minus 8sαπβγ)
+ s(2sβγ(1minus 4απ + 4απβγβπ) +Q2(1minus 2λ2π)))λ2γ + 8s3(απ minus 1)λ4
γ
]
times arctan
(2sλπλγ
M2N + s(2βγβπ minus 1)
) (B21)
35
σ(πminusp)LT =
e2g2AM
3N
64πf2πQs
3λ4γ(M2
N minusQ2 minus s)(M2N + s(2βπβγ + 2λπλγ minus 1))
times
1
M2N + s(2βπβγ minus 2λπλγ minus 1)
[4λπsλγ(M2
N minusQ2 minus s)(
2sλ2γ
(s((1minus 2λ2
π)Q2
+ 2(2απ minus 1)sβγ(2βπβγ minus 1))minusM2N (Q2 + 2(1minus 2απ)sβγ)
)+ (Q2 + 2sβ2
γ)(M2N + s(2βπβγ minus 1)
)2 + 8(απ minus 1)απs
3λ4γ
)]minus(M2N + s(2βπβγ + 2λπλγ minus 1)
)times[
log
(minusM2N minus 2βπsβγ minus 2λπsλγ + s
minusM2N minus 2βπsβγ + 2λπsλγ + s
)(minus (Q2 + 2sβ2
γ)(M2N + s(2βπβγ minus 1)
)(minusM2
N + 2Q2 + 2βπsβγ + s)minus 2sλ2γ
(minusM2
N (Q2 + 2sβγ) +Q4 +Q2s(4απβγ
minus 2βγ minus 2λ2π + 1) + 2s2βγ(4(απ minus 1)βπβγ + 1)
)minus 8(απ minus 1)2s3λ4
γ
)+ 2
(4s2λ2
γ
(2(απ minus 1)βγ(Q2 + 2βπsβγ)minus λ2
πQ2)
+ (Q2 + 2βπsβγ)2(Q2 + 2sβ2γ)
+ 8(απ minus 1)2s3λ4γ
)arctanh
(2λπsλγ
Q2 + 2βπsβγ
)] (B22)
2 ∆-production channel
The tree-level ∆-exchange diagram in Fig 2 of Ref [30] contributes to the non-Born part of
the VVCS amplitudes The contribution of the ∆ exchange to the VVCS amplitudes can be split
into [17]
S∆-exch1 (νQ2) = S∆-pole
1 (νQ2) + S∆-exch1 (νQ2) (B23a)
νS∆-exch2 (νQ2) = νS∆-pole
2 (νQ2) + νS2∆-exch
(νQ2) (B23b)
and similarly for the unpolarized VVCS amplitudes discussed in Ref [30] Here we introduced the
∆-pole contributions S∆-polei and the ∆-non-pole contributions S∆-exch
i The former amplitudes
feature a pole at the ∆(1232)-production threshold and thus are proportional to
1
[sminusM2∆][uminusM2
∆]=
1
4M2N
1
ν2∆ minus ν2
(B24)
36
They can be reconstructed from the dispersion relations in Eq (5) with the tree-level ∆-production
cross sections as input cf Fig 10 in Ref [30]
σTT (νQ2) =π2α
M2NM
2+|~q |
minus g2
MMN (M+ + ν)|~q |2 +g2E(∆minus ν)(Q2 minusMNν)2
MN(B25a)
+g2CQ
4s(∆minus ν)
MNM2∆
minus 4gMgE(Q2 minusMNν)|~q |2 minus 4gMgCQ2|~q |2
+2gEgCQ
2[minusMNM∆ |~q |2 + s(Q2 +∆ν)
]MNM∆
δ(ν minus ν∆)
σLT (νQ2) =Qπ2α
M2NM
2+|~q |
g2E(MNν minusQ2) [M∆(MN + ν)minus s]
MN(B25b)
+g2CQ
2[MNM∆|~q |2 minus s(Q2 +∆ν)
]MNM2
∆
+ gMgEM∆|~q |2 minusgMgC(Q2 minusMNν)|~q |2
M∆
+gEgC(ν minus∆)(M2
N |~q |2 minus 2Q2s)
MNM∆
δ(ν minus ν∆)
with ∆ = M∆ minusMN M+ = M∆ + MN and the Mandelstam variable s = M2N + 2MNν minus Q2
Analytical expressions for the spin structure functions g1(xQ2) and g2(xQ2) can be constructed
from Eq (3) with the flux factor K(νQ2) = |~q | =radicν2 +Q2
In the ∆-non-pole contributions to S1(νQ2) and νS2(νQ2) the pole in ν at the ∆(1232)-
production threshold has canceled out
S∆-exch1 (νQ2) =
πα
MNM2+
[g2MQ
2+ + g2
E
(∆2 minus 3Q2
)+
4g2CQ
4
M2∆
minus 8gMgEM∆ωminus (B26a)
minus2gMgCQ2(MN minus 4M∆)
M∆+
2gEgCQ2(3MN minus 2M∆)
M∆
]
νS2∆-exch
(νQ2) =2πα
MNM2+
[g2EM∆∆ωminus +
g2M MNQ
2+
2+g2C Q
2(Q2 minus∆2)
2M∆(B26b)
+gEgM M∆(M∆ω+ minus 4MNωminus)minus gEgC ∆(2Q2 +MNω+)
+gMgC Q2(4MN minus ω+)
]+S∆-exch
2 (νQ2)
ν
[M2
∆ ω2+
M2N
+ ν2
]
with Q+ =radic
(M∆ +MN )2 +Q2 and ωplusmn = (M2∆minusM2
N plusmnQ2)2M∆ and the non-pole contribution
to S2(νQ2)
S∆-exch2 (νQ2) = minus2παMNν
M∆M2+
[gM + gE
]gC (B27)
These amplitudes to the contrary are not described by the tree-level ∆-production cross sections
in the standard dispersive approach [17] This peculiarity has been previously missed e g in the
calculation of the ∆-exchange contribution to the hydrogen hyperfine splitting in Ref [88] The
importance of including the ∆-non-pole contribution is also evident when considering the BC sum
37
rule in Eq (14) The ∆-pole terms by themselves violate the BC sum rule but cancel exactly with
the ∆-non-pole terms
limνrarr0
νS∆-pole2 (νQ2) + lim
νrarr0νS2
∆-exch(νQ2) = 0 (B28)
Appendix C Polarizabilities at Q2 = 0
In this section we give analytical expressions for the static values and slopes at Q2 = 0 of
the spin polarizabilities and moments of polarized structure functions In particular we give the
HB expansion of the πN -loop contributions and the ∆-exchange contributions The complete
expressions also for the π∆-loop contributions can be found in the Supplemented material Recall
that IA(0) = I1(0) = d2(0) = 0 and dd2(Q2)dQ2
∣∣∣Q2=0
= 0
1 πN-loop contribution
Here we give analytical expressions for the πN -loop contributions to the proton and neutron
spin polarizabilities expanded in powers of micro = mπMN viz the HB expansion Note that we
choose to expand here to a high order in micro the strict HB expansion would only retain the leading
term in an analogous NLO calculation
bull Static polarizabilities (Q2 = 0)
γ0p =e2g2
A
96π3f2πm
2π
1minus 21πmicro
8minus(
59
2+ 26 logmicro
)micro2 +
1875πmicro3
64
+3
(3
2+ 26 logmicro
)micro4 +
(C1)
γ0n =e2g2
A
48π3f2πm
2π
1
2minus 9πmicro
16minus 2micro2 logmicro+
75πmicro3
128minus 3micro4
4+
(C2)
δLTp =e2g2
A
192π3f2πm
2π
1minus 9πmicro
8+
(13
2minus 2 logmicro
)micro2 minus 465πmicro3
64
minus(
47
2+ 42 logmicro
)micro4 +
(C3)
δLTn =e2g2
A
96π3f2πm
2π
1
2+
3πmicro
16+ (1 + 2 log micro)micro2 minus 105πmicro3
128+
5micro4
4+
(C4)
38
δLTn =e2g2
A
96π3f2πm
2π
1
2+
3πmicro
16+ (1 + 2 log micro)micro2 minus 105πmicro3
128+
5micro4
4+
(C5)
γ0p =e2g2
A
16π3f2πm
4π
4
45minus 3πmicro
16+
14micro2
5minus 1813πmicro3
384minus 192
5(1 + log micro)micro4
+80703πmicro5
2048+
(C6)
γ0n =e2g2
A
16π3f2πm
4π
4
45minus 5πmicro
48+
4micro2
5minus 245πmicro3
384minus 32micro4 logmicro
15
+1323πmicro5
2048+
(C7)
bull Slopes of polarizabilities at Q2 = 0
dγ0p(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
2minus 45πmicro
4+ 223micro2 minus 28515πmicro3
64
minus9
(1953
4+ 449 logmicro
)micro4 +
570255πmicro5
128+
(C8)
dγ0n(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
2minus 81πmicro
8+ 94micro2 minus 2535πmicro3
32
minus3 (1 + 90 logmicro)micro4 +84315πmicro5
1024+
(C9)
dδLTp(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
2880π3f2πm
4π
minus5
2minus 27πmicro
32+ 20micro2 minus 5865πmicro3
256
+3
(617
4+ 36 logmicro
)micro4 minus 2056845πmicro5
4096+
(C10)
dδLTn(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
minus5
4minus 81πmicro
64minus 11micro2 +
10005πmicro3
512
+15
8(11 + 48 logmicro)micro4 minus 267015πmicro5
8192+
(C11)
dIAp(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
96π2f2πmicro
2
1minus 15πmicro
4minus 1
2(115 + 88 logmicro)micro2 +
1839πmicro3
32
+5 (5 + 34 logmicro)micro4 +
(C12)
dIAn(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
48π2f2πmicro
2
1
2minus 11πmicro
8minus 1
4(1 + 20 logmicro)micro2 +
99πmicro3
64
minus25micro4
12+
(C13)
dI1p(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
96π2f2πmicro
3π
8+ 2 (4 + 3 logmicro)microminus 537πmicro2
64
minus1
2(15 + 56 logmicro)micro3 +
(C14)
39
dI1n(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
48π2f2πmicro
minus π
16+
1
4(3 + 4 logmicro)microminus 57πmicro2
128+
2micro3
3+
(C15)
γ0p(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2g2
A
16π3f2πm
6π
1
105minus 23πmicro
256+
377micro2
210minus 15551πmicro3
6144+
3371micro4
105
minus1640457πmicro5
32768+
(C16)
γ0n(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2g2
A
16π3f2πm
6π
1
105minus 153πmicro
1792+
69micro2
70minus 4615πmicro3
6144+
172micro4
35
minus120897πmicro5
32768+
(C17)
2 ∆-exchange contribution
Here we give analytical expressions for the tree-level ∆-exchange contributions to the nucleon
spin polarizabilities and their slopes at Q2 = 0 Note that the ∆-exchange contributes equally to
proton and neutron polarizabilities Recall that for the magnetic γlowastN∆ coupling we introduced a
dipole form factor to mimic vector-meson dominance gM rarr gM(1 +Q2Λ2)2
bull Static polarizabilities (Q2 = 0)
γ0 = minus e2
4πM2+
(g2M
∆2+
g2E
M2+
minus 4gMgEM+∆
) (C18)
δLT =e2M∆
4πM3+
(g2E
MNM++gMgE∆MN
minus gEgCM2
∆
) (C19)
γ0 =e2M2
N
π∆2M4+
(minusg
2M
∆2+
g2E
M2+
+4gMgE∆M+
) (C20)
bull Slopes of polarizabilities at Q2 = 0
dγ0(Q2)
dQ2
∣∣∣∣∣Q2=0
= minus e2
πM2+∆
(g2M
∆
[1
4∆2minus 1
∆M++
1
2M2+
]minus 1
Λ2
g2M
∆+
g2E
2M2+
[1
2∆minus 3
M+
]minusgMgE
M+
[1
∆2minus 5
∆M++
1
M2+
]+
1
Λ2
2gMgEM+
+2gMgC∆M2
+
minus gEgCM3
+
) (C21)
40
d δLT (Q2)
dQ2
∣∣∣∣∣Q2=0
=e2M∆∆
4πMNM2+
(g2E
∆2M2+
[1
∆minus 4
M+
]minus g2
C
∆M2∆M
2+
+gMgE∆2M+
[1
∆2
minus 3
∆M++
1
M2+
]minus 2
Λ2
gMgE∆2M+
+gMgC∆M2
∆
[1
2∆2minus 2
∆M++
1
2M2+
]minus gEgC
2M2∆M
2+
[7
∆+
1
M+
]) (C22)
dIA(Q2)
dQ2
∣∣∣∣∣Q2=0
= minusM2N
M2+
(g2M
2∆2+
g2E
MNM+minus 2gMgE
∆M+minus gEgCM∆M+
) (C23)
dI1(Q2)
dQ2
∣∣∣∣∣Q2=0
= minusM∆M2N
2M3+
(g2E
MNM∆minus gMgE∆MN
minus gEgCM2
∆
) (C24)
γ0(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2M2
N
π∆3M6+
(g2M
∆
[3M2
∆ + 2M∆MN minus 9M2N
∆2+
4M2+
Λ2
]
minus4gMgE
[5M2
∆ minus 9M2N
M+∆2+
2M+
Λ2
]+
g2E
∆M+[7M∆ minus 9MN ]
minus8gMgC∆
+4gEgCM+
) (C25)
[1] D Drechsel B Pasquini and M Vanderhaeghen Dispersion relations in real and virtual
Compton scattering Phys Rept 378 99 (2003) hep-ph0212124
[2] S E Kuhn J-P Chen and E Leader Spin structure of the nucleon mdash status and recent
results Prog Part Nucl Phys 63 1 (2009) arXiv08123535 [hep-ph]
[3] F Hagelstein R Miskimen and V Pascalutsa Nucleon polarizabilities from Compton scat-
tering to hydrogen atom Prog Part Nucl Phys 88 29 (2016) arXiv151203765 [nucl-th]
[4] B Pasquini and M Vanderhaeghen Dispersion theory in electromagnetic interactions Ann
Rev Nucl Part Sci 68 75 (2018) arXiv180510482 [hep-ph]
[5] S Gerasimov A Sum rule for magnetic moments and the damping of the nucleon magnetic
moment in nuclei Sov J Nucl Phys 2 430 (1966)
[6] S Drell and A C Hearn Exact sum rule for nucleon magnetic moments Phys Rev Lett
16 908 (1966)
[7] J S Schwinger Source theory viewpoints in deep inelastic scattering Proc Natl Acad Sci
USA 72 1 (1975)
[8] J S Schwinger Source theory viewpoints in deep inelastic scattering Electromagnetic In-
teractions and Field Theory Proceedings 14 Internationale Universitatswochen Schladming
Austria February 24-March 7 1975 Acta Phys Austriaca Suppl 14 471 (1975)
41
[9] J Schwinger Source theory discussion of deep inelastic scattering with polarized particles
Proc Natl Acad Sci USA 72 1559 (1975)
[10] A M Baldin Polarizability of nucleons Nucl Phys 18 310 (1960)
[11] M Gell-Mann M L Goldberger and W E Thirring Use of causality conditions in quantum
theory Phys Rev 95 1612 (1954)
[12] A Pineda Leading chiral logarithms to the hyperfine splitting of the hydrogen and muonic
hydrogen Phys Rev C 67 025201 (2003)
[13] C Peset and A Pineda The two-photon exchange contribution to muonic hydrogen from
chiral perturbation theory Nucl Phys B 887 69 (2014) arXiv14064524 [hep-ph]
[14] C Peset and A Pineda Model-independent determination of the two-photon exchange contri-
bution to hyperfine splitting in muonic hydrogen JHEP 04 060 arXiv161205206 [nucl-th]
[15] F Hagelstein and V Pascalutsa Proton structure in the hyperfine splitting of muonic hydro-
gen PoS CD15 077 (2016) arXiv151104301 [nucl-th]
[16] F Hagelstein Exciting Nucleons in Compton Scattering and Hydrogen-Like Atoms PhD
thesis Mainz U Inst Kernphys (2017) arXiv171000874 [nucl-th]
[17] F Hagelstein ∆(1232)-Resonance in the hydrogen spectrum Proceedings 11th International
Workshop on the Physics of Excited Nucleons (NSTAR 2017) Columbia SC USA August
20-23 2017 Few Body Syst 59 93 (2018) arXiv180109790 [nucl-th]
[18] Y Prok et al (CLAS) Moments of the spin structure functions gp1 and gd1 for 005 lt Q2 lt 30
GeV2 Phys Lett B 672 12 (2009) arXiv08022232 [nucl-ex]
[19] H Dutz et al (GDH) First measurement of the Gerasimov-Drell-Hearn sum rule for 1H from
07 GeV to 18 GeV at ELSA Phys Rev Lett 91 192001 (2003)
[20] M Amarian et al (Jefferson Lab E94010) Measurement of the generalized forward spin po-
larizabilities of the neutron Phys Rev Lett 93 152301 (2004) arXivnucl-ex0406005
[21] M Amarian et al The Q2 evolution of the generalized Gerasimov-Drell-Hearn integral for
the neutron using a 3He target Phys Rev Lett 89 242301 (2002) arXivnucl-ex0205020
[22] M Amarian et al (Jefferson Lab E94-010)Q2 evolution of the neutron spin structure moments
using a 3He target Phys Rev Lett 92 022301 (2004) arXivhep-ex0310003
[23] A Deur et al Experimental determination of the evolution of the Bjorken integral at low Q2
Phys Rev Lett 93 212001 (2004) arXivhep-ex0407007
[24] K Slifer Low Q2 measurement of gp2 and the δLT spin polarizability Spin structure at long dis-
tance Proceedings Workshop Newport News USA March 12ndash13 2009 nucl-ex09064775
(2009) AIP Conf Proc 1155 10106313203293 (2009) arXiv09064775 [nucl-ex]
[25] P Solvignon et al (E01-012) Moments of the neutron g2 structure function at intermediate
42
Q2 Phys Rev C 92 015208 (2015) arXiv13044497 [nucl-ex]
[26] A Deur Experimental studies at low Q2 of the spin structure of the nucleon at Jefferson Lab
in 9th International Workshop on Chiral Dynamics (CD18) Durham NC USA September
17-21 2018 (2019) arXiv190305661 [nucl-ex]
[27] V Sulkosky et al (Jefferson Lab E97-110) Measurement of the 3He spin-structure functions
and of neutron (3He) spin-dependent sum rules at 0035 le Q2 le 024 GeV2 Phys Lett B
805 135428 (2020) arXiv190805709 [nucl-ex]
[28] K Adhikari et al (CLAS) Measurement of the Q2 Dependence of the Deuteron Spin Structure
Function g1 and its Moments at Low Q2 with CLAS Phys Rev Lett 120 062501 (2018)
arXiv171101974 [nucl-ex]
[29] V Lensky F Hagelstein A Hiller Blin and V Pascalutsa Comment on rdquoMeasurement of
the Q2 Dependence of the Deuteron Spin Structure Function g1 and its Moments at Low Q2
with CLASrdquo (2018) arXiv180603219 [nucl-th]
[30] J M Alarcn F Hagelstein V Lensky and V Pascalutsa Forward doubly-virtual Compton
scattering off the nucleon in chiral perturbation theory at NLO the subtraction function and
moments of unpolarized structure functions (2020) arXiv200509518 [hep-ph]
[31] V Bernard E Epelbaum H Krebs and U-G Meiszligner New insights into the spin structure
of the nucleon Phys Rev D 87 054032 (2013) arXiv12092523 [hep-ph]
[32] V Lensky J M Alarcon and V Pascalutsa Moments of nucleon structure functions at
next-to-leading order in baryon chiral perturbation theory Phys Rev C 90 055202 (2014)
arXiv14072574 [hep-ph]
[33] V Pascalutsa and M Vanderhaeghen Chiral effective-field theory in the ∆(1232) region
I Pion electroproduction on the nucleon Phys Rev D 73 034003 (2006) arXivhep-
ph0512244
[34] V Pascalutsa and M Vanderhaeghen Electromagnetic nucleon-to-Delta transition in chiral
effective field theory Phys Rev Lett 95 232001 (2005) arXivhep-ph0508060
[35] T R Hemmert B R Holstein and J Kambor Systematic 1M expansion for spin 32
particles in baryon chiral perturbation theory Phys Lett B 395 89 (1997) arXivhep-
ph9606456
[36] V Pascalutsa and D R Phillips Effective theory of the ∆(1232) in Compton scattering off
the nucleon Phys Rev C 67 055202 (2003) arXivnucl-th0212024
[37] V Pascalutsa M Vanderhaeghen and S N Yang Electromagnetic excitation of the ∆(1232)-
resonance Phys Rept 437 125 (2007) arXivhep-ph0609004
[38] K A Olive et al (Particle Data Group) Review of Particle Physics Chin Phys C 38 090001
43
(2014)
[39] F E Low Scattering of light of very low frequency by systems of spin 12 Phys Rev 96
1428 (1954)
[40] M Gell-Mann and M L Goldberger Scattering of low-energy photons by particles of spin
12 Phys Rev 96 1433 (1954)
[41] A M Harun ar-Rashid A simple derivation of Schwingerrsquos sum rule for spin dependent
structure functions Nuovo Cim A 33 447 (1976)
[42] F Hagelstein and V Pascalutsa Dissecting the hadronic contributions to (g minus 2)micro by
Schwingerrsquos sum rule Phys Rev Lett 120 072002 (2018) arXiv171004571 [hep-ph]
[43] V Lensky V Pascalutsa M Vanderhaeghen and C Kao Spin-dependent sum rules con-
necting real and virtual Compton scattering verified Phys Rev D 95 074001 (2017)
arXiv170101947 [hep-ph]
[44] H Burkhardt and W N Cottingham Sum rules for forward virtual Compton scattering
Annals Phys 56 453 (1970)
[45] V Lensky and V Pascalutsa Predictive powers of chiral perturbation theory in Compton
scattering off protons Eur Phys J C 65 195 (2010) arXiv09070451 [hep-ph]
[46] H Grieszlighammer J McGovern D Phillips and G Feldman Using effective field theory to
analyse low-energy Compton scattering data from protons and light nuclei Prog Part Nucl
Phys 67 841 (2012) arXiv12036834 [nucl-th]
[47] H W Grieszlighammer J A McGovern and D R Phillips Nucleon polarisabilities at and
beyond physical pion masses Eur Phys J A 52 139 (2016) arXiv151101952 [nucl-th]
[48] E Epelbaum H Krebs and U-G Meiszligner Improved chiral nucleon-nucleon potential up to
next-to-next-to-next-to-leading order Eur Phys J A 51 53 (2015) arXiv14120142 [nucl-
th]
[49] E Epelbaum H Krebs and U-G Meiszligner Precision nucleon-nucleon potential at fifth order
in the chiral expansion Phys Rev Lett 115 122301 (2015) arXiv14124623 [nucl-th]
[50] C W Kao T Spitzenberg and M Vanderhaeghen Burkhardt-Cottingham sum rule and
forward spin polarizabilities in heavy baryon chiral perturbation theory Phys Rev D 67
016001 (2003) arXivhep-ph0209241
[51] C-W Kao D Drechsel S Kamalov and M Vanderhaeghen Higher moments of nucleon
spin structure functions in heavy baryon chiral perturbation theory and in a resonance model
Phys Rev D 69 056004 (2004) arXivhep-ph0312102
[52] T Becher and H Leutwyler Baryon chiral perturbation theory in manifestly Lorentz invariant
form Eur Phys J C 9 643 (1999) arXivhep-ph9901384
44
[53] J Gasser M E Sainio and A Svarc Nucleons with chiral loops Nucl Phys B 307 779
(1988)
[54] L S Geng J Martin Camalich L Alvarez-Ruso and M J Vicente Vacas Leading SU(3)-
breaking corrections to the baryon magnetic moments in chiral perturbation theory Phys
Rev Lett 101 222002 (2008) arXiv08051419 [hep-ph]
[55] D Drechsel S S Kamalov and L Tiator The GDH sum rule and related integrals Phys
Rev D 63 114010 (2001) arXivhep-ph0008306
[56] D Drechsel O Hanstein S S Kamalov and L Tiator A unitary isobar model for pion
photo- and electroproduction on the proton up to 1 GeV Nucl Phys A 645 145 (1999)
arXivnucl-th9807001
[57] L Tiator private communication (2020)
[58] V Bernard T R Hemmert and U-G Meiszligner Spin structure of the nucleon at low energies
Phys Rev D 67 076008 (2003) arXivhep-ph0212033
[59] O Gryniuk F Hagelstein and V Pascalutsa Evaluation of the forward Compton scattering
off protons II Spin-dependent amplitude and observables Phys Rev D 94 034043 (2016)
arXiv160400789 [nucl-th]
[60] R Zielinski The g2p Experiment A Measurement of the Protonrsquos Spin Structure Functions
PhD thesis New Hampshire U (2010) arXiv170808297 [nucl-ex]
[61] N Guler et al (CLAS) Precise determination of the deuteron spin structure at low to mod-
erate Q2 with CLAS and extraction of the neutron contribution Phys Rev C 92 055201
(2015) arXiv150507877 [nucl-ex]
[62] C E Carlson V Nazaryan and K Griffioen Proton structure corrections to electronic
and muonic hydrogen hyperfine splitting Phys Rev A 78 022517 (2008) arXiv08052603
[physicsatom-ph]
[63] V Bernard N Kaiser and U-G Meiszligner Chiral dynamics in nucleons and nuclei Int J
Mod Phys E 4 193 (1995) arXivhep-ph9501384
[64] T R Hemmert B R Holstein and J Kambor ∆(1232) and the polarizabilities of the nucleon
Phys Rev D 55 5598 (1997) arXivhep-ph9612374
[65] V Pascalutsa and R Timmermans Field theory of nucleon to higher-spin baryon transitions
Phys Rev C 60 042201 (1999) arXivnucl-th9905065
[66] V Pascalutsa Quantization of an interacting spin-32 field and the Delta isobar Phys Rev
D 58 096002 (1998) arXivhep-ph9802288
[67] H Krebs Double Virtual Compton Scattering and SpinStructure of the Nucleon PoS
CD2018 031 (2019)
45
[68] N Kochelev and Y Oh Axial anomaly and the δLT puzzle Phys Rev D 016012 (2012)
arXiv11034892 [hep-ph]
[69] D Drechsel S Kamalov and L Tiator Unitary isobar model ndash MAID2007 Eur Phys J A
69 (2007) available at httpsmaidkphuni-mainzde arXiv07100306 [nucl-th]
[70] P J Mohr B N Taylor and D B Newell CODATA recommended values of the fundamental
physical constants 2010 Rev Mod Phys 84 1527 (2012)
[71] J Ahrens et al (GDH A2) First measurement of the Gerasimov-Drell-Hearn integral for 1H
from 200 to 800 MeV Phys Rev Lett 87 022003 (2001) arXivhep-ex0105089 [hep-ex]
[72] K Helbing (GDH) Experimental verification of the GDH sum rule at ELSA and MAMI Nucl
Phys Proc Suppl 105 113 (2002)
[73] R Bradford A Bodek H S Budd and J Arrington A New parameterization of the nucleon
elastic form-factors NuInt05 proceedings of the 4th International Workshop on Neutrino-
Nucleus Interactions in the Few-GeV Region Okayama Japan 26-29 September 2005 Nucl
Phys Proc Suppl 159 127 (2006) arXivhep-ex0602017
[74] J D Bjorken Applications of the chiral U(6)timesU(6) algebra of current densities Phys Rev
148 1467 (1966)
[75] J D Bjorken Inelastic scattering of polarized leptons from polarized nucleons Phys Rev D
1 1376 (1970)
[76] V Bernard T R Hemmert and U-G Meiszligner Novel analysis of chiral loop effects in
the generalized Gerasimov-Drell-Hearn sum rule Phys Lett B 545 105 (2002) arXivhep-
ph0203167
[77] A Deur et al Experimental study of isovector spin sum rules Phys Rev D 78 032001
(2008) arXiv08023198 [nucl-ex]
[78] B Pasquini P Pedroni and D Drechsel Higher order forward spin polarizability Phys Lett
B 687 160 (2010) arXiv10014230 [hep-ph]
[79] R Jaffe g2ndashThe nucleonrsquos other spin-dependent structure function Comments Nucl Part
Phys 19 239 (1990)
[80] E V Shuryak and A Vainshtein Theory of power corrections to deep inelastic scattering in
quantum chromodynamics (II) Qminus4 effects polarized target Nucl Phys B 201 141 (1982)
[81] S Wandzura and F Wilczek Sum rules for spin dependent electroproduction Test of rela-
tivistic constituent quarks Phys Lett B 72 195 (1977)
[82] B W Filippone and X-D Ji The spin structure of the nucleon Adv Nucl Phys 26 1
(2001) arXivhep-ph0101224
[83] M Burkardt The g2 structure function Proceedings Workshop on Spin structure at long
46
distance Newport News USA March 12-13 2009 AIP Conf Proc 1155 26 (2009)
arXiv09054079 [hep-ph]
[84] A Deur S J Brodsky and G F de Teramond The spin structure of the nucleon Rept
Prog Phys 82 076201 (2019) arXiv180705250 [hep-ph]
[85] B R Holstein D Drechsel B Pasquini and M Vanderhaeghen Higher order polarizabilities
of the proton Phys Rev C 61 034316 (2000) arXivhep-ph9910427
[86] B R Holstein V Pascalutsa and M Vanderhaeghen Sum rules for magnetic moments and
polarizabilities in QED and chiral effective-field theory Phys Rev D 72 094014 (2005)
arXivhep-ph0507016
[87] B Pasquini D Drechsel and L Tiator Invariant amplitudes for pion electroproduction Eur
Phys J A 34 387 (2007) arXiv07122327 [hep-ph]
[88] A J Buchmann Non-spherical proton shape and hydrogen hyperfine splitting Proceedings
International Workshop on Precision Physics of Simple Atomic Systems (PSAS 2008) Wind-
sor Ontario Canada July 21-26 2008 Can J Phys 87 773 (2009) arXiv09104747
[physicsatom-ph]
47
where power counting predictive orders (Sec III A) and the renormalization procedure
(Sec III B) are discussed
A few remarks are in order for the inclusion of the ∆(1232) and the tree-level ∆-exchange
contribution In contrast to Ref [32] we include the Coulomb-quadrupole (C2) N rarr ∆
transition described by the LEC gC The relevant Lagrangian describing the non-minimal
γlowastN∆ coupling [33 34] (note that in these references the overall sign of gC is inconsistent
between the Lagrangian and Feynman rules) reads
L(2)∆ =
3e
2MNM+
N T3
igM F
microν partmicro∆ν minus gEγ5Fmicroν partmicro∆ν (15)
+igCM∆
γ5γα(partα∆ν minus partν∆α)partmicroF
microν
+ Hc
with M+ = MN + M∆ and the dual of the electromagnetic field strength tensor F microν =12εmicroνρλFρλ Even though the Coulomb coupling is subleading compared with the electric
and magnetic couplings (gE and gM) its relatively large magnitude cf Table I makes it
numerically important for instance in γ0(Q2) Furthermore we study the effect of modifying
the magnetic coupling using a dipole form factor
gM rarrgM[
1 + (QΛ)2 ]2 (16)
where Λ2 = 071 GeV2 The inclusion of this Q2 dependence mimics the form expected from
vector-meson dominance It is motivated by observing the importance of this form factor
for the correct description of the electroproduction data [33]
To estimate the uncertainties of our NLO predictions we define
δ(Q2) =
radic(∆
MN
)2
+
(Q2
2MN∆
)2
(17)
such that the neglected next-to-next-to-leading order terms are expected to be of relative
size δ2 [33] The uncertainties in the values of the parameters in Table I have a much
smaller impact compared to the truncation uncertainty and can be neglected Unfortunately
∆IA(Q2) γ0(Q2) and γ0(Q2) ie the sum rules involving the cross section σTT (νQ2)
as well as the polarizability ∆I1(Q2) turn out to be numerically small Their smallness
suggests a cancellation of leading orders (which can indeed be confirmed by looking at
separate contributions as shown below) Therefore an error of δ2(Q2)P (Q2) where P (Q2)
is a generalized polarizability might underestimate the theoretical uncertainty for some of
the NLO predictions To avoid this we estimate the uncertainty of our NLO polarizability
9
predictions by
σP (Q2) =
Max[δ4(0)P (0)2 δ4(0)P LO(0)2 δ2(0)PNLO(0)2
]+Max
[δ4(Q2)
[P (Q2)minus P (0)
]2 δ4(Q2)
[P LO(Q2)minus P LO(0)
]2
δ2(Q2)[PNLO(Q2)minus PNLO(0)
]2]12
(18)
where P LO(Q2) is the πN -loop contribution PNLO(Q2) are the ∆-exchange and π∆-loop
contributions and P (Q2) = P LO(Q2) + PNLO(Q2) This error prescription is similar to the
one used in eg Refs [46ndash49] Here since we are interested in the generalized polarizabil-
ities we added in quadrature the error due to the static piece P (0) and the Q2-dependent
remainder P (Q2) minus P (0) Note that the static values of IA(0) and I1(0) are given by the
elastic Pauli form factor which is not part of our BχPT prediction and is considered to be
exact
Note that our result for the spin polarizabilities (and the unpolarized moments [30]) are
NLO predictions only at low momentum transfers Q mπ At larger values of Q amp ∆ they
become incomplete LO predictions Indeed in this regime the ∆ propagators do not carry
additional suppression compared to the nucleon propagators and the π∆ loops are promoted
to LO In general we only expect a rather small contribution from omitted π∆ loops to the
Q2 dependence of the polarizabilities since π∆ loops show rather weak dependence on Q2
compared with the ∆ exchange or πN loops Nevertheless this issue has to be reflected in
the error estimate Since the static polarizabilities P (0) are not affected it is natural to
separate the error on the Q2-dependent remainder P (Q2) minus P (0) as done in Eq (18) To
accommodate for the potential loss of precision above Q amp ∆ we define the relative error
δ(Q2) as growing with increasing Q2 see Eq (17)
Upon expanding our results in powers of the inverse nucleon mass Mminus1N we are able to
reproduce existing results of heavy-baryon χPT (HBχPT) at LO We however do not see
a rationale to drop the higher-order Mminus1N terms when they are not negligible (ie when
their actual size exceeds by far the natural estimate for the size of higher-order terms)
Comparing our BχPT predictions to HBχPT we will also see a deficiency of HBχPT in
the description of the Q2 behaviour of the polarizabilities Note that the O(p4) HBχPT
results from Ref [50 51] which we use here for comparison do not include the ∆ These
references studied the leading effect of the latter in the HBχPT framework using the small-
scale expansion [35] observing no qualitative improvement in the HBχPT description of the
empirical data [50 51] when including it We therefore choose to use the O(p4) results as
the representative HBχPT curves
10
Another approach used in the literature to calculate the polarizabilities in χPT is the
infrared regularization (IR) scheme introduced in Ref [52] This covariant approach tries to
solve the power counting violation observed in Ref [53] by dropping the regular parts of the
loop integrals that contain the power-counting-breaking terms However this subtraction
scheme modifies the analytic structure of the loop contributions and may lead to unexpected
problems as was shown in Ref [54] As we will see in the next section the IR approach
also fails to describe the Q2 behaviour of the polarizabilities
III RESULTS AND DISCUSSION
We now present the NLO BχPT predictions for the nucleon polarizabilities and selected
moments of the nucleon spin structure functions Our results are obtained from the calcu-
lated non-Born VVCS amplitudes and the LEXs in Eqs (10) and (12) For a cross-check
we used the photoabsorption cross sections described in App B In addition to the full NLO
results we also analyse the individual contributions from the πN loops the ∆ exchange
and the π∆ loops
A γ0(Q2) mdash generalized forward spin polarizability
The forward spin polarizability
γ0(Q2) =1
2π2
int infinν0
dν
radic1 +
Q2
ν2
σTT (νQ2)
ν3(19)
=16αM2
N
Q6
int x0
0
dx x2
[g1(xQ2)minus 4M2
Nx2
Q2g2(xQ2)
]
provides information about the spin-dependent response of the nucleon to transversal pho-
ton probes The RCS analogue of the above generalized forward spin polarizability sum
rule is sometimes referred to as the Gell-Mann Goldberger and Thirring (GGT) sum rule
[11] At Q2 = 0 the forward spin polarizability is expressed through the lowest-order spin
polarizabilities of RCS as γ0 = minus(γE1E1 + γM1M1 + γE1M2 + γM1E2) The forward spin po-
larizability of the proton is relevant for an accurate knowledge of the (muonic-)hydrogen
hyperfine splitting as it controls the leading proton-polarizability correction [16 62]
The πN -loop ∆-exchange and π∆-loop contributions to the NLO BχPT prediction of
11
000 005 010 015 020 025 030
-4
-3
-2
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
4D
proton
000 005 010 015 020 025 030
-4
-3
-2
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
4D
neutron
000 005 010 015 020 025 030
0
1
2
Q2 GeV2D
∆LT
10-
4fm
4D
proton
000 005 010 015 020 025 030
0
1
2
3
Q2 GeV2D
∆LT
10-
4fm
4D
neutron
FIG 2 Upper panel Generalized forward spin polarizability for the proton (left) and neutron
(right) as function of Q2 The result of this work the NLO BχPT prediction is shown by the
blue solid line and the blue band The red line represents the LO BχPT result The purple short-
dashed line is the O(p4) HB result from Ref [50] note that the corresponding proton curve is
outside of the plotted range The black dotted line is the MAID model prediction [55ndash57] which
is taken from Ref [1] (proton) and Ref [20] (neutron) The pink band is the IR+∆ result from
Ref [58] and the gray band is the BχPT+∆ result from Ref [31] Empirical extractions for the
proton Ref [18] (blue dots) Ref [59] (purple square) and Ref [60] (orange triangle uncertainties
added in quadrature) and neutron Ref [20] (blue diamonds) and Ref [61] (green dots statistical
and systematic uncertainties added in quadrature) Lower Panel Longitudinal-transverse spin
polarizability for the proton (left) and neutron (right) The orange dot-dashed and purple short-
dashed lines are the O(p3) and O(p4) HB results from Ref [50] The pink band is the IR result
from Ref [58] and the gray band is the covariant BχPT+∆ result from Ref [31] The black dotted
line is the MAID model prediction [55ndash57] note that for the proton we use the updated estimate
from Ref [1] obtained using the π η ππ channels
12
the static forward spin polarizability amount to in units of 10minus4 fm4
γ0p = minus093(92) asymp 201minus 284minus 010 (20a)
γ0n = 003(92) asymp 298minus 284minus 010 (20b)
while the slope is composed as follows in units of 10minus4 fm6
dγ0p(Q2)
dQ2
∣∣∣∣Q2=0
= minus022(4) asymp minus033 + 011 + 001 (21a)
dγ0n(Q2)
dQ2
∣∣∣∣Q2=0
= minus061(7) asymp minus073 + 011 + 001 (21b)
Figure 2 upper panel shows our NLO prediction as well as the LO πN loops compared
to different experimental and theoretical results For the proton we have one determination
at the real-photon point by the GDH collaboration [19] γ0p = minus100(8)(12)times10minus4 fm4 and
further Jefferson Laboratory data [18 60] at very low Q2 For the neutron only data at finite
Q2 are available [20 61] The experimental data for the proton are fairly well reproduced
in the whole Q2 range considered here while for the neutron the agreement improves with
increasing Q2 The HB limit of our πN -loop contribution reproduces the results published
in Refs [50 63] for arbitrary Q2 In addition our prediction is compared to the MAID
model [1 20] the IR+∆ calculation of Ref [58] and the BχPT+∆ result of Ref [31]
The πN -production channel gives a positive contribution to the photoabsorption cross
section σTT (νQ2) at low Q2 cf Fig 10 Accordingly one observes that the πN loops give
a sizeable positive contribution to γ0(Q2) The Delta on the other hand has a very large
effect by cancelling the πN loops and bringing the result close to the empirical data From
Fig 3 upper panel one can see that it is the ∆ exchange which dominates while π∆
loops are negligible This was expected since the forward spin polarizability sum rule is an
integral over the helicity-difference cross section in which σ32 is governed by the Delta at
low energies (the relevant energy region for the sum rule)
To elucidate the difference between the present calculation and the one from Ref [31] we
note that the two calculations differ in the following important aspects Firstly Ref [31] uses
the small-scale counting [64] that considers ∆ and mπ as being of the same size ∆ sim mπ
In practice this results in a set of π∆-loop graphs which contains graphs with one or two
γ∆∆ couplings and hence two or three Delta propagators Such graphs are suppressed in
the δ-counting and thus omitted from our calculation while present in that of Ref [31]
Secondly the Lagrangians describing the interaction of the Delta are constructed differently
and assume slightly different values for the coupling constants In particular we employ
13
000 005 010 015 020 025 030
-3
-2
-1
0
1
2
Q2GeV
2D
Γ01
0-
4fm
4D
proton
000 005 010 015 020 025 030
-3
-2
-1
0
1
2
3
Q2GeV
2D
Γ01
0-
4fm
4D
neutron
000 005 010 015 020 025 030
00
05
10
15
Q2GeV
2D
∆LT1
0-
4fm
4D
proton
000 005 010 015 020 025 030
00
05
10
15
20
25
Q2GeV
2D
∆LT1
0-
4fm
4D
neutron
FIG 3 Contributions of the different orders to the chiral predictions of γ0(Q2) upper paneland δLT (Q2) lower panel for the proton (left) and neutron (right) Red solid line πN -loop
contribution green dot-dashed line ∆-exchange contribution orange dotted line π∆-loop con-
tribution blue long-dashed line total result purple dot-dot-dashed line total result without gCcontribution black short-dashed line total result without gM dipole
(where possible) the so-called ldquoconsistentrdquo couplings to the Delta field ie those couplings
that project out the spurious degree of freedom see Refs [37 65 66] The authors of
Ref [31] on the other hand use couplings where the consistency in this sense is not enforced
The effects of these differences are of higher order in the δ-counting expansion and their
contribution to the Q2 dependence of the considered polarizabilities is expected to be rather
small however the differences at Q2 = 0 could be noticeable [67]
Finally as mentioned in Sec II C the inclusion of the dipole form factor in the magnetic
coupling gM is expected to be important to generate the correct Q2 behaviour of the polar-
izabilities Comparing our predictions for the forward spin polarizability with and without
inclusion of the form factor see Fig 3 upper panel confirms this Without the dipole
our results for the proton and neutron are closer to the ones from Ref [31] where the form
factor is not included For the neutron our prediction without the dipole is able to describe
14
the experimental points at very low Q2 deviating from the data with increasing Q2 The
π∆-loop contribution does not modify the Q2 behavior of γ0(Q2) and only differs from
Ref [31] by a small global shift Note also the relatively large effect of gC which generates
a sign change for virtualities above sim 02 GeV2 see Fig 3 upper panel
B δLT (Q2) mdash longitudinal-transverse polarizability
The longitudinal-transverse spin polarizability
δLT (Q2) =1
2π2
int infinν0
dν
radic1 +
Q2
ν2
σLT (νQ2)
Qν2(22)
=16αM2
N
Q6
int x0
0
dx x2[g1(xQ2) + g2(xQ2)
]
contains information about the spin structure of the nucleon and is another important
input in the determination of the (muonic-)hydrogen hyperfine splitting [16 62] It is also
relevant in studies of higher-twist corrections to the structure function g2(xQ2) given by
the moment d2(Q2) [51] see Section III E The peculiarity of the response encoded in this
polarizability is that it involves a spin flip of the nucleon and a polarization change of the
photon see App B and Fig 11
It is expected that the Delta isobar gives only a small contribution to δLT (Q2) what makes
this polarizability a potentially clean test case for chiral calculations Consequently there are
relatively many different theoretical calculations of δLT (Q2) coming from different versions
of χPT with baryons (HB IR and covariant) Ref [50] found a systematic deviation of the
HB result for δLTn(Q2) from the MAID model prediction This disagreement was identified
by the authors of Ref [68] as a puzzle involving the neutron δLT polarizabilitymdashthe δLT
puzzle The IR calculation in Ref [58] also showed a deviation from the data and predicted
a rapid rise of δLT (Q2) with growing Q2 The problem is solved by keeping the relativistic
structure of the theory as the BχPT+∆ result of Ref [31] showed
As expected already the leading πN loops provide a reasonable agreement with the
experimental data cf Fig 2 lower panel Since the ∆-exchange contribution to δLT (Q2)
is small the effect of the gM form factor is negligible in this polarizability as is that of the gC
coupling cf Fig 3 lower panel In fact we predict both the ∆-exchange and the π∆-loop
contributions to be small and negative This is in agreement with the MAID model which
predicts a small and negative contribution of the P33 wave to δLT (Q2) However in the
calculation of Ref [31] which is different from the one presented here only in the way the
∆(1232) is included the contribution of this resonance to δLTp(Q2) is sizeable and positive
15
The authors of that work attributed this large contribution to diagrams where the photons
couple directly to the Delta inside a loop As mentioned in Sec III F the effect of such loop
diagrams does not change the Q2 behaviour of the polarizabilities On the other hand it
can produce a substantial shift of the δLT (Q2) as a whole A higher-order calculation should
resolve the discrepancy between the two covariant approaches however it will partially lose
the predictive power since the LECs appearing at higher orders will have to be fitted to
experimental data
The πN -loop ∆-exchange and π∆-loop contributions to the NLO BχPT prediction of
the static longitudinal-transverse polarizability are in units of 10minus4 fm4
δLTp = 132(15) asymp 150minus 016minus 002 (23a)
δLTn = 218(23) asymp 235minus 016minus 002 (23b)
while the slopes are in units of 10minus4 fm6
dδLTp(Q2)
dQ2
∣∣∣∣Q2=0
= minus085(8) asymp minus080minus 004minus 001 (24a)
dδLTn(Q2)
dQ2
∣∣∣∣Q2=0
= minus124(12) asymp minus119minus 004minus 001 (24b)
C IA(Q2) mdash a generalized GDH integral
The helicity-difference cross section σTT exhibits a faster fall-off in ν than its spin-averaged
counterpart σT This is due to a cancellation between the leading (constant) terms of σ12
and σ32 at large ν3 The resulting 1ν fall-off of the helicity-difference cross section allows
one to write an unsubtracted dispersion relation for the VVCS amplitude gnonpoleTT (νQ2) cf
Eq (10a) This is the origin of the GDH sum rule [5 6]
minus α
2M2N
κ2 =1
2π2
int infinν0
dνσTT (ν)
ν (25)
which establishes a relation to the anomalous magnetic moment κ It is experimentally
verified for the nucleon by MAMI (Mainz) and ELSA (Bonn) [71 72]
There are two extensions of the GDH sum rule to finite Q2 the generalized GDH integrals
3 Notice that a constant term in σTT at ν rarrinfin is forbidden by crossing symmetry
16
000 005 010 015 020 025 030
-30
-25
-20
-15
-10
-05
00
Q2 GeV2D
I Aproton
000 005 010 015 020 025 030
-25
-20
-15
-10
-05
00
Q2 GeV2D
I A
neutron
000 005 010 015 020 025 030
-05
00
05
Q2 GeV2D
I 1
proton
000 005 010 015 020 025 030
-10
-08
-06
-04
-02
00
02
Q2 GeV2D
I 1
neutron
FIG 4 Upper panel The generalized GDH integral IA(Q2) for the proton (left) and neutron
(right) as function of Q2 The result of this work the NLO BχPT prediction is shown by the blue
solid line and the blue band The red line represents the LO BχPT result The purple short-dashed
line is the O(p4) HB result from Ref [50 51] The gray band is the BχPT+∆ result from Ref [31]
The black dotted line is the MAID model prediction [69] Experimental extractions for the proton
Ref [60] (orange triangle uncertainties added in quadrature) and neutron Refs [21][27] where
magenta dotsorange diagonal crosses correspond to data and red squareslilac crosses correspond
to data plus extrapolation to unmeasured energy regions The green stars at the real-photon point
are derived from the anomalous magnetic moments κp asymp 1793 and κn asymp minus1913 [70] Lower
panel The generalized GDH integral I1(Q2) for the proton (left) and neutron (right) as function
of Q2 The purple short-dashed line is the HB result from Ref [51] Experimental extractions for
the proton Ref [18] (blue dots) and Ref [60] (orange triangle uncertainties added in quadrature)
and neutron Ref [61][27] (uncertainties added in quadrature) where green dotsorange diagonal
crosses correspond to data and gray squareslilac crosses correspond to data plus extrapolation to
unmeasured energy regions
17
IA(Q2) and I1(Q2) The latter will be discussed in Sec III D The former is defined as4
minus α
2M2N
IA(Q2) = minus 1
8π2
int infinν0
dν
radic1 +
Q2
ν2
σTT (νQ2)
ν(26)
=α
Q2
int x0
0
dx[4M2
Nx2
Q2g2(xQ2)minus g1(xQ2)
]
Due to its energy weighting the integral in Eq (26) converges slower than the one in the
generalized forward spin polarizability sum rule (19) Therefore knowledge of the cross
section at higher energies is required and the evaluation of the generalized GDH integral
IA(Q2) is not as simple as the evaluation of γ0(Q2)
The generalized GDH integral IA(Q2) is directly related to the non-pole amplitude
gnonpoleTT (νQ2) which differs from non-Born amplitude gTT (νQ2) by a term involving the
elastic Pauli form factor
gnonpoleTT (νQ2) = gTT (νQ2)minus 2παν
M2N
F 22 (Q2) (27)
cf Eqs (2a) and (8a) Consequently IA(Q2) is not a pure polarizability but also contains
an elastic contribution The ldquonon-polarizabilityrdquo or the Born part of IA(Q2) is given by
IBornA (Q2) = IA(Q2)minus∆IA(Q2) = minus1
4F 2
2 (Q2) (28)
where we refer to the polarizability part as ∆IA(Q2) The same is true for the generalized
GDH integral I1(Q2) which is directly related to Snonpole1 (νQ2)
IBorn1 (Q2) = I1(Q2)minus∆I1(Q2) = minus1
4F 2
2 (Q2) (29)
In the following we will add the Born parts to our LO and NLO BχPT predictions for
the polarizabilities ∆IA(Q2) and ∆I1(Q2) employing an empirical parametrization for the
elastic Pauli form factor [73] This allows us to compare to the experimental results for
IA(Q2) and I1(Q2) cf Fig 4 Note that the blue error bands only describe the uncertainties
of our BχPT predictions of the polarizabilities while the elastic contributions are considered
to be exact as explained in Sec II C The uncertainties of the polarizability predictions are
therefore better reflected in Fig 5 where we show the contributions of the different orders
to the BχPT predictions of ∆IA(Q2) and ∆I1(Q2) as well as the total results with error
bands
4 Note that IA(Q2) is sometimes called ITT (Q2)
18
000 005 010 015 020 025 030
-08
-06
-04
-02
00
02
Q2 GeV2D
DI A
proton
000 005 010 015 020 025 030
-10
-08
-06
-04
-02
00
02
Q2 GeV2D
DI A
neutron
000 005 010 015 020 025 030-005
000
005
010
015
020
025
Q2 GeV2D
DI 1
proton
000 005 010 015 020 025 030-015
-010
-005
000
005
010
015
Q2 GeV2D
DI 1
neutron
FIG 5 Contributions of the different orders to the chiral predictions of ∆IA(Q2) upper paneland ∆I1(Q2) lower panel for the proton (left) and neutron (right) Red solid line πN -loop
contribution green dot-dashed line ∆-exchange contribution orange dotted line π∆-loop contri-
bution blue solid line and blue band total result purple dot-dot-dashed line total result without
gC contribution black short-dashed line total result without gM dipole
The E97-110 experiment at Jefferson Lab has recently published their data for IAn(Q2) in
the region of 0035 GeV2 lt Q2 lt 024 GeV2 [27] In addition there are results for IAn(Q2)
from the earlier E94-010 experiment [21] and for IAp(Q2) from the E08-027 experiment [60]
The O(p4) HB calculation gives a large negative effect [51] which does not describe the data
The BχPT+∆ result from Ref [31] which mainly differs from our work by the absence of
the dipole form factor in gM looks similar to this HB result and only describes the data
points at lowest Q2 Our NLO prediction however follows closely the Q2 evolution of the
data In Fig 5 upper panel we show the polarizability ∆IA(Q2) whose Q2 evolution is
clearly dominated by the ∆ exchange Similar to the case of γ0p(Q2) inclusion of the dipole
in gM and the Coulomb coupling gC is very important in order to describe the experimental
data The LO prediction on the other hand slightly overestimates the data cf Fig 4
upper panel
19
At the real-photon point IA(0) = minusκ2
4and ∆IA(0) = 0 Therefore we give only the
slope of the polarizability ∆IA(Q2) [showing also the separate contributions from πN loops
∆ exchange and π∆ loops] in units of GeVminus2
d∆IAp(Q2)
dQ2
∣∣∣∣Q2=0
= minus858(343) asymp 238minus 1121 + 025 (30a)
d∆IAn(Q2)
dQ2
∣∣∣∣Q2=0
= minus955(343) asymp 141minus 1121 + 025 (30b)
Including the empirical Pauli form factor [73] we find in units of GeVminus2
dIAp(Q2)
dQ2
∣∣∣∣Q2=0
= minus318dIAn(Q2)
dQ2
∣∣∣∣Q2=0
= minus300 (31)
D Γ1(Q2) and I1(Q2) mdash the first moment of the structure function g1(xQ2)
The second variant for a generalization of the GDH sum rule to finite Q2 is defined as
minus α
2M2N
I1(Q2) = minus 1
8π2
int infinν0
dν1radic
ν2 +Q2
[σTT (νQ2) +
Q
νσLT (νQ2)
](32)
= minus α
Q2
int x0
0
dx g1(xQ2)
where I1(0) = minusκ2
4 This generalized GDH integral directly stems from the amplitude
Snonpole1 (νQ2) with the LEX from Eq (12a) It is given by the first moment of the struc-
ture function g1(xQ2) Γ1(Q2) =int x0
0dx g1(xQ2) as follows I1(Q2) =
2M2N
Q2 Γ1(Q2) The
isovector combination
Γ1(pminusn)(Q2) =
int x0
0
dx[g1p(xQ
2)minus g1n(xQ2)] (33)
is related to the axial coupling of the nucleon through the Bjorken sum rule [74 75]
limQ2rarrinfin
Γ1(pminusn)(Q2) =
gA6 (34)
As explained in Eq (28) the moment I1(Q2) splits into a polarizability part ∆I1(Q2) and
a Born part IBorn1 (Q2) Figure 4 lower panel shows the Q2 dependence of I1(Q2) which in
contrast to IA(Q2) shown in Figure 4 upper panel is clearly dominated by its Born part
and the elastic Pauli form factor The πN -loop ∆-exchange and π∆-loop contributions to
the polarizability ∆I1(Q2) are shown in Fig 5 lower panel Comparing to Fig 5 upper
20
000 005 010 015 020 025 030
-006
-004
-002
000
002
004
Q2 GeV2D
G1
proton
000 005 010 015 020 025 030
-006
-005
-004
-003
-002
-001
000
001
Q2 GeV2D
G1
neutron
FIG 6 First moment of the structure function g1(xQ2) for the proton (left) and neutron (right)
as function of Q2 The legend is the same as in Fig 4
panel one sees that ∆I1(Q2) is less sensitive to gC and the dipole form factor in gM than
∆IA(Q2)
For the proton our NLO BχPT prediction gives a very good description of the ex-
perimental data [18 60] and is in reasonable agreement with the MAID prediction [69]
For the neutron one observes good agreement with the empirical evaluations including ex-
trapolations to unmeasured energy regions starting from Q2 gt 01 GeV2 [27 61] In the
region of Q2 lt 005 GeV2 one observes an interesting tension between the recent E97-
110 experiment [27] and the data from CLAS [61] While the newest measurement finds
I1n(0035 GeV2) lt κ2n4 thus suggesting a negative slope at low Q the older measurement
found a rather large value for I1n(00496 GeV2) A similar but milder behaviour is seen in
the E97-110 [27] and E94-010 [21] data for IAn The MAID predictions do not agree with
the CODATA recommended values for the anomalous magnetic moments of the proton and
neutron [70] which in our work are imposed by using empirical parametrizations for the
elastic Pauli form factors [73] The slope of the HB result from Ref [51] is too large and
therefore only reproduces the data at very low Q2
Figure 6 shows the moment Γ1(Q2) for the proton and neutron while Fig 7 shows the
isovector combination Γ1 pminusn(Q2) The LO and NLO BχPT predictions are identical be-
cause our calculation produces the same Delta contributions for the proton and the neutron
For the isovector combination the MAID model only agrees with the data at very low
Q2 lt 010 GeV2 The same is true for the IR result [58 76] while all other chiral results
describe the data NLO BχPT (this work) BχPT+∆ [31] and HBχPT [51]
At the real-photon point I1(0) = minusκ2
4and ∆I1(0) = 0 Therefore we give only the
slope of the polarizability ∆I1(Q2) [showing also the separate contributions from πN loops
21
000 005 010 015 020 025 030-002
000
002
004
006
008
010
Q2 GeV2D
G1Hp-
nL
FIG 7 Isovector combination of Γ1(Q2) as function of Q2 The legend is the same as in Fig 6
The pink curve is the IR result from Ref [58 76] The experimental points are from Ref [77]
(brown dots) and Ref [23] (orange squares)
∆ exchange and π∆ loops] in units of GeVminus2
d∆I1p(Q2)
dQ2
∣∣∣∣Q2=0
= 039(4) asymp 034minus 053 + 058 (35a)
d∆I1n(Q2)
dQ2
∣∣∣∣Q2=0
= minus101(10) asymp minus107minus 053 + 058 (35b)
Including the empirical Pauli form factor [73] we find in units of GeVminus2
dI1p(Q2)
dQ2
∣∣∣∣Q2=0
= 580dI1n(Q2)
dQ2
∣∣∣∣Q2=0
= 553 (36)
E d2(Q2) mdash a measure of color polarizability
Another interesting moment to consider is d2(Q2) which is related to the twist-3 part of
the spin structure function g2(xQ2) [79 80]
d2(Q2) equiv 3
int 1
0
dx x2[g2(xQ2)minus gWW2 (xQ2)] (37)
where gWW2 (xQ2) is the twist-2 part of g2(xQ2) Using the Wandzura-Wilczek relation [81]
one can relate d2(Q2) to moments of the spin structure functions g1(xQ2) and g2(xQ2)
d2(Q2) =
int 1
0
dx x2 [3g2(xQ2) + 2g1(xQ2)] (38)
22
000 005 010 015 020 025 030
0000
0001
0002
0003
0004
0005
0006
0007
Q2 GeV2D
d2
proton
000 005 010 015 020 025 030
0000
0002
0004
0006
0008
0010
0012
Q2 GeV2D
d2
neutron
000 005 010 015 020 025 030
-1
0
1
2
Q2 GeV2D
Γ 010
-4
fm6
D
proton
000 005 010 015 020 025 030-2
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
6D
neutron
FIG 8 Upper panel The inelastic moment d2(Q2) for the proton (left) and neutron (right) as
function of Q2 The result of this work the NLO BχPT prediction is shown by the blue solid line
and the blue band The red line represents the LO BχPT result The purple short-dashed line is
the O(p4) HB result from Ref [50 51] The black dotted line is the MAID model prediction [69]
The experimental points for the neutron (cyan dots) are from Ref [22] Lower panel Fifth-order
generalized forward spin polarizability γ0(Q2) for the proton (left) and neutron (right) as function
of Q2 The experimental points for the proton are from Ref [59] (purple square) and Ref [78]
(orange dot)
This relation however only holds for asymptotically large Q2 It is also in the high-Q2
region where d2(Q2) is a measure of color polarizability [82 83] through its relation to the
gluon field strength tensor [80] We refer to Ref [84] for a recent review on the spin structure
of the nucleon including a discussion of sum rules for deep inelastic scattering and color
polarizabilities
What we consider in the following is the inelastic part of d2(Q2) defined as the moment
of g1(xQ2) and g2(xQ2) spin structure functions cf Eq (38)
d2(Q2) =
int x0
0
dx x2 [3g2(xQ2) + 2g1(xQ2)] (39)
23
This moment provides another testing ground for our BχPT predictions through comparison
with experiments on the neutron [22] Going towards the low-Q2 region the interpretation
of d2(Q2) in terms of color polarizabilities will fade out The above definition however
implies it is related to other VVCS polarizabilities
d2(Q2) =Q4
8M4N
[M2
NQ2
αδLT (Q2) + I1(Q2)minus IA(Q2)
] (40)
Note that d2(Q2) and its first two derivatives with respect to Q2 vanish at Q2 = 0 The
considerations in Eqs (28) and (29) have no effect on d2(Q2) since the Born contribution
from IA(Q2) and I1(Q2) cancel out Therefore d2(Q2) is a pure polarizability
In Fig 8 upper panel we show our NLO BχPT prediction and other results for d2(Q2)
While MAID [69] and BχPT describe the experimental data for the neutron [22] very well
the HB limit [50 51] is showing a fast growth with Q2 This illustrates the importance
of keeping the relativistic result Note also that even though the πN -loop contribution
is dominant both gC and the form factor in gM are essential to obtain a curvature that
reproduces the data cf Fig 9 upper panel For the proton there are to our knowledge
no experimental results to compare with However the agreement between the NLO BχPT
prediction and the MAID prediction at low energies is reasonable
F γ0(Q2) mdash fifth-order generalized forward spin polarizability
It is interesting to compare the generalized fifth-order forward spin polarizability sum
rule
γ0(Q2) =1
2π2
int infinν0
dν
radic1 +
Q2
ν2
σTT (νQ2)
ν5(41)
=64αM4
N
Q10
int x0
0
dx x4
[g1(xQ2)minus 4M2
Nx2
Q2g2(xQ2)
]
to the sum rule integrals for IA(Q2) and γ0(Q2) since they differ merely by their energy
weighting of σTT (νQ2) and a constant prefactor cf Eqs (19) (26) and (41) From IA(Q2)
to γ0(Q2) to γ0(Q2) the energy suppression is increasing by a factor of νminus2 respectively
Therefore the description of γ0(Q2) should be easiest in a low-energy effective-field theory
such as χPT whereas γ0(Q2) and IA(Q2) receive larger contributions from higher energies
In Fig 8 lower panel we show our LO and NLO BχPT predictions for γ0(Q2) One
can see that the πN -loop contribution is positive (in accordance to what we see for the cross
section σTT see Fig 10) The Delta shifts it substantially especially in the low Q2 region
24
000 005 010 015 020 025 030
-0004
-0002
0000
0002
0004
0006
0008
0010
Q2GeV
2D
d2
proton
000 005 010 015 020 025 030
-0005
0000
0005
0010
0015
Q2GeV
2D
d2
neutron
000 005 010 015 020 025 030
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
6D
proton
000 005 010 015 020 025 030
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
6D
neutron
FIG 9 Contributions of the different orders to the chiral predictions of d2(Q2) upper panel and
γ0(Q2) lower panel for the proton (left) and neutron (right) Red solid line πN -loop contribu-
tion green dot-dashed line ∆-exchange contribution orange dotted line π∆-loop contribution
blue long-dashed line total result purple dot-dot-dashed line total result without gC contribution
black short-dashed line total result without gM dipole
bringing it into a better agreement with data In general the BχPT curves start above the
empirical data points at the real-photon point and then decrease asymptotically to zero
above Q2 gt 01 GeV2 On the other hand the MAID prediction reproduces the empirical
data at the real-photon point then decreases to negative values until about Q2 gt 006 GeV2
from where it also starts to asymptotically approach zero Consequently our NLO BχPT
prediction of γ0(Q2) is consistently above the MAID prediction This is very different to
what we saw for IA(Q2) in Fig 4 upper panel where the MAID prediction at the real-
photon point is above the experimental value While the agreement of our predictions with
the empirical data is in general quite good for all moments of σTT (νQ2) one should point
out that both for γ0n(Q2) and γ0p(Q2) we overestimate the data at low Q2 For IA(Q2) such
observation cannot be made because ∆IA(0) = 0 and thus IA(0) is given by the empirical
Pauli form factor only From IA(Q2) γ0(Q2) and γ0(Q2) the latter has the smallest however
25
non-negligible dependence on gC and the dipole in gM cf Fig 9 lower panel
The πN -loop ∆-exchange and π∆-loop contributions to the NLO BχPT prediction of
the static fifth-order forward spin polarizability amount to in units of 10minus4 fm6
γ0p = 112(30) asymp 208minus 096minus 001 (42a)
γ0n = 195(30) asymp 292minus 096minus 001 (42b)
while the slope is composed as follows in units of 10minus4 fm8
dγ0p(Q2)
dQ2
∣∣∣∣Q2=0
= minus084(10) asymp minus100 + 016 + 000 (43a)
dγ0n(Q2)
dQ2
∣∣∣∣Q2=0
= minus142(15) asymp minus158 + 016 + 000 (43b)
Note that the HB prediction of the static polarizability γ0p (423 at O(p3) and 365 at O(ε3)
[78 85]) is almost one order of magnitude larger than the empirical value and therefore not
shown in Fig 8
G Summary
Our results are summarized in Table II where we give the contributions of the different
orders to the chiral predictions of the polarizabilities and their slopes at the real-photon
point A quantitative comparison of our predictions for the static spin polarizabilities to
the work of Bernard et al [31] and different empirical evaluations is shown in Table III We
can see that the inclusion of the Delta turns out to be very important for all moments of
the helicity-difference cross section To describe the Q2 behavior of the polarizabilities the
magnetic coupling of the N rarr ∆ transition should be modified by a dipole form factor as
has been observed previously in the description of electroproduction data [33] This dipole
form factor effectively takes account of vector-meson exchanges The Coulomb-quadrupole
N rarr ∆ transition despite its subleading order is important in the description of some
moments of spin structure functions This is contrary to what we saw for the moments of
unpolarized structure functions [30] where the Coulomb coupling had a negligible effect
The π∆ loops are mainly relevant for the generalized GDH integrals
26
TABLE II The NLO BχPT predictions for the forward VVCS polarizabilities and their slopes at
Q2 = 0 The contributions of the πN loops the ∆ exchange and the π∆ loops are shown together
with the combined total result Note that IA(0) = I1(0) = d2(0) = 0 and (d2)prime = 0
πN loops ∆ exchange π∆ loops Total
γ0 p 201 minus284 minus010minus093(92)
(10minus4 fm4) n 298 003(92)
δLT p 150 minus016 minus002132(15)
(10minus4 fm4) n 235 218(23)
γ0 p 208 minus096 minus001112(30)
(10minus4 fm6) n 292 195(30)
(γ0)prime p minus033011 001
minus022(4)
(10minus4 fm6) n minus073 minus061(7)
(δLT )prime p minus080 minus004 minus001minus085(8)
(10minus4 fm6) n minus119 minus124(12)
(γ0)prime p minus100016 000
minus084(10)
(10minus4 fm8) n minus158 minus142(15)
(∆IA)prime p 238 minus1121 025minus858(343)
(GeVminus2) n 141 minus955(343)
(∆I1)prime p 034 minus053 058039(4)
(GeVminus2) n minus107 minus101(10)
TABLE III Our NLO BχPT predictions for the spin polarizabilities at Q2 = 0 compared with
the BχPT+∆ predictions from Bernard et al [31] and the available empirical information Where
the reference is not given the empirical number is provided by the MAID analysis [55 69] with
unspecified uncertainty
Proton Neutron
This work BχPT+∆ Empirical This work BχPT+∆ Empirical
γ0 minus093(92) minus174(40) minus100(8)(12) [19] 003(92) minus077(40) minus0005
(10minus4 fm4) minus090(8)(11) [78] [MAID]
minus0929(105) [59]
δLT 132(15) 240(1) 134 218(23) 238(3) 203
(10minus4 fm4) [MAID] [MAID]
γ0 112(30) 060(7)(7) [78] 195(30) 123
(10minus4 fm6) 0484(82) [59] [MAID]
IV CONCLUSIONS
We have presented a complete NLO calculation of the polarized non-Born VVCS ampli-
tudes in covariant BχPT with pion nucleon and ∆(1232) fields The dispersion relations
between the VVCS amplitudes and the tree-level photoabsorption cross sections served as a
27
cross-check of these calculations
The obtained moments of the proton and neutron spin structure functions related to gen-
eralized polarizabilities and GDH-type integrals agree well with the available experimental
data The description of their Q2 evolution is improved compared to the previous χPT pre-
dictions In particular the NLO BχPT predictions obtained here give a better description
of the empirical data (eg from the Jefferson Laboratory ldquoSpin Physics Programrdquo) than
the HB [50 51] and IR [58] calculations
The demonstrated predictive power of the χPT framework amplitudes makes it well
suited for extending the χPT evaluation of the TPE effect in the hyperfine structure of
(muonic-)hydrogen [15ndash17] to next-to-leading order
ACKNOWLEDGEMENTS
We thank Lothar Tiator and Marc Vanderhaeghen for helpful discussions This work
is supported by the Deutsche Forschungsgemeinschaft (DFG) through the Collaborative
Research Center [The Low-Energy Frontier of the Standard Model (SFB 1044)] JMA ac-
knowledges support from the Community of Madrid through the ldquoPrograma de atraccion de
talento investigador 2017 (Modalidad 1)rdquo and the Spanish MECD grants FPA2016-77313-P
FH gratefully acknowledges financial support from the Swiss National Science Foundation
Appendix A Tensor decompositions of the VVCS amplitudes
In this appendix we review the decomposition of the forward VVCS process into tensor struc-
tures and scalar amplitudes In particular we consider the connection between the covariant and
the semi-relativistic decomposition in the lab frame that is defined in terms of the conventional
transverse longitudinal transverse-transverse and transverse-longitudinal amplitudes
As explained in Sec II A the process of forward VVCS off the nucleon can be described in
terms of four explicitly covariant amplitudes S1 2 and T1 2 [3]
T (νQ2) =
(minusgmicroν +
qmicroqν
q2
)T1(νQ2) +
1
M2N
(pmicro minus p middot q
q2qmicro)(
pν minus p middot qq2
qν)T2(νQ2) (A1)
minus 1
MNγmicroναqα S1(νQ2)minus 1
M2N
(γmicroνq2 + qmicroγναqα minus qνγmicroαqα
)S2(νQ2)
εprimelowastmicro εν
where εmicro (εprimelowastmicro ) are the incoming (outgoing) photon polarization vectors ν is the photon lab-frame
energy and Q2 is the photon virtuality Alternatively the decomposition in the laboratory frame
28
(which in the forward case coincides with the Breit frame) is parametrized in terms of the nucleon
Pauli matrices ~σ and the four scalar functions fL fT gTT and gLT
T (νQ2) = ε0 εprimelowast0 fL(νQ2) + (~ε primelowast middot ~ε ) fT (νQ2) + i~σ middot (~ε primelowast times ~ε ) gTT (νQ2) (A2)
minusi~σ middot [(ε0~εprimelowast minus ~ε ε primelowast0 )times q] gLT (νQ2)
Here ~q and q = ~q|~q | are the photon three-momentum in the lab system and its unit vector The
modified polarization vector components are given by
ε0 =
[ε0 minus
ν
|~q | (~ε middot q )
] |~q |Q ~ε = ~εminus q (~ε middot q ) (A3)
where ε = (ε0~ε ) is the usual incoming photon polarization vector and εprimelowast the outgoing polarization
vector The LEX of the lab frame amplitudes [Eq (10)] can serve in particular as the definition
of the generalized polarizabilities The lab frame amplitudes are also conveniently used for the
definition of the response functions see the example of the scalar amplitude gLT (νQ2) and the
corresponding response function σLT (νQ2) below in App B
Appendix B Photoabsorption cross sections
In the forward kinematics the spin-dependent VVCS amplitudes and the spin polarizabilities
can be described in terms of the polarized structure functions g1(xQ2) and g2(xQ2) or equiv-
alently the helicity-difference cross section σTT (νQ2) and the longitudinal-transverse response
function σLT (νQ2) with the help of dispersion relations (5) and the optical theorem (3) In this
way the photoabsorption cross sections measured in electroproduction processes form the basis
for most empirical evaluations shown throughout Sec III In the following we present the BχPT
predictions for the tree-level cross sections of πN - π∆- and ∆-production through photoabsorption
on the nucleon cf Figs 8 9 and 10 in Ref [30] In Secs B 1 and B 2 we will discuss the leading
πN -production channel and the ∆-production channel respectively We used these cross sections
to verify the polarizability predictions obtained otherwise from the calculated non-Born VVCS
amplitudes Due to the bad high-energy behavior of the π∆-production cross sections in BχPT
cf Fig 10 the dispersion relations in Eq (5) require further subtractions for a reconstruction of
the π∆-loop contribution to the spin-dependent VVCS amplitudes Therefore not all polarizabil-
ities could be verified but only those appearing as higher-order terms in the LEX of the VVCS
amplitudes such as γ0 [16]
29
proton neutron
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Neutron
02 04 06 08 100
50
100
150
200
250
[GeV]
T [b]
02 04 06 08 10
-50
0
50
100
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Neutron
02 04 06 08 100
50
100
150
200
250
[GeV]
T [b]
02 04 06 08 10
-50
0
50
100
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02040608100
50
100
150
200
250
300
[GeV]
T[b]
0204060810
-50
0
50
[GeV]
TT[b]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
0204060810
00
02
04
06
[GeV]
LTQ
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
0204060810
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
FIG 10 Photoabsorption cross sections for πN (red) and π∆ production (orange) with Q2 = 0
(solid) and Q2 = 01 GeV2 (dashed for πN and dotted for π∆ channel)
1 πN-production channel
In order to extract the response function σLT (νQ2) we have developed a method similar to
the one used to calculate σTT (νQ2) see for example Ref [86] For σLT (νQ2) however the
calculation is more complicated because one has to take into account that the associated Compton
process involves a spin-flip of the nucleon as illustrated in Fig 11 When calculating the cross
section the product of the incoming nucleon spinors has to reflect this flip
The forward VVCS amplitude related to σLT (νQ2) mdash and δLT (Q2) mdash is gLT (νQ2) It can
be extracted from Eq (A2) if one takes the modified polarization vector components in Eq (A3)
with ε = εL and εprimelowast = εlowastplusmn as input where εL = 1Q(|~q | 0 0 q0) and εplusmn = ∓ 1radic
2(0 1plusmni 0) are the
standard longitudinal and transverse polarization vectors respectively For εL and εlowastplusmn only the
choice of helicities hprime = plusmn12 and h = ∓12 gives a non-zero contribution and one obtains
χdaggerhprime T (νQ2)χh = χdaggerhprimeminusi~σ middot [(ε0~εprimelowast minus ~ε ε primelowast0 )times q] gLT (νQ2)χh =
radic2 gLT (νQ2) (B1)
where χh and χdaggerhprime are two-component Pauli spinors with opposite helicities or here spins
Let us now consider the related photoabsorption process and in particular the tree-level γlowastN rarr
30
s = 12
sprime = minus12
λ = 0 λprime = 1
rprime rprime
s=12sprime=minus1
2
λ=0λprime=1
rprimerprime
s=12
sprime=minus12
λ=0λprime=1
rprimerprime hacute=frac12h=-frac12
λacute=1λ=0
FIG 11 Relation between the forward Compton process and the photoabsorption process given
by the optical theorem In particular we show the longitudinal-transverse contribution The
double-line arrows represent the spin of the external particles while the dot represents the scalar
(longitudinal) polarization of the incoming photon Inside the blob the intermediate states are
represented eg nucleons with spins rprime (which are averaged in the calculation of the cross section)
and pions
πN channel see diagrams in Fig 8 of Ref [30] We define the πN -production amplitude as
T = uhB (PB)sumi
Ai(s t)Γi uhA(PA) (B2)
with the Dirac structures
Γ1 = γ5 (B3a)
Γ2 =1
2
[qA ε
]γ5 (B3b)
where uhA(PA) and udaggerhB (PB) are the Dirac spinors and PA and PB are the four-momenta of
the incoming and outgoing nucleons respectively When calculating the photoabsorption cross
section related to the VVCS amplitude in Eq (B1) the nucleon spin flip should be implemented
by uhprime(PA) in T dagger and uh(PA) in T together with the appropriate transverse and longitudinal
photon polarization vectors εlowastplusmn and εL
However if one wants to use the properties of the Dirac matrices it is more useful to construct
an operator to produce this spin flip in the external nucleons of Fig 11 This is accomplished by
introducing the projector ΓLT equiv 12radic
2(γ1 + iγ2)γ5 which also takes into account the extra factor
radic2 in Eq (B1) We checked that with this projector one correctly extracts δLT by comparing the
HB limit of our result to the HB result of Ref [50] where the authors calculate this polarizability
from the Compton amplitude directly With all those ingredients the longitudinal-transverse cross
section is calculated in the following way
σLT (νQ2) =1
64π2 s
|~pf |cm
|~pi|cm
int 1
minus1dcos θ
sumij
AiAdaggerjXij (B4)
31
with
Xij = Tr[(PB +MN )Γi(PA +MN )ΓLTγ0Γdaggerjγ
0] (B5)
where θ is the scattering angle in the center-of-mass (cm) frame and |~pi|cm (|~pf |cm) is the three-
momentum of an incoming (outgoing) particle in the cm frame An explicit calculation of the
matrix Xij leads to
X = MNQ
0 2(PB minus PA) middot εLminusradic
2|~qf |cm sin θ (sminus u)
(B6)
where |~qi|cm (|~qf |cm) is the relative three-momentum of the incoming (outgoing) particles in the
cm frame Here s t and u are the usual Mandelstam variables For the different γlowastN rarr πN
channels we obtain the following amplitudes Ai where we introduce qA as the four-momentum of
the incoming photon and qB as the four-momentum of the outgoing pion
bull γlowastprarr π0p
A1 =e gAMN
fπ
[2PA middot ε+ qA middot ε
sminusM2N
+2PB middot εminus qA middot ε
uminusM2N
] (B7a)
A2 =e gAMN
fπ
[1
sminusM2N
+1
uminusM2N
] (B7b)
bull γlowastprarr π+n
A1 =
radic2 e gAMN
fπ
[2PA middot ε+ qA middot ε
sminusM2N
+2(PA minus PB) middot ε+ qA middot ε
tminusm2π
] (B8a)
A2 =
radic2 e gAMN
fπ(sminusM2N )
(B8b)
bull γlowastnrarr π0n
A1 = 0 (B9a)
A2 = 0 (B9b)
bull γlowastnrarr πminusp
A1 =
radic2 e gAMN
fπ
[2PB middot εminus qA middot ε
uminusM2N
minus 2(PA minus PB) middot ε+ qA middot εtminusm2
π
] (B10a)
A2 =
radic2 e gAMN
fπ(uminusM2N )
(B10b)
The analytical expressions shown above were checked with the amplitudes given in Ref [87] An-
alytical expressions for the tree-level γlowastN rarr πN channel of the σLT (νQ2) and σTT (νQ2) cross
32
sections are given below (proton channels π+n and π0p neutron channel πminusp) We checked that
they reproduce the known results in the real-photon limit [45 86] To shorten the final expressions
for the cross sections which are considerably longer for finite Q2 than in the real-photon limit we
define the following dimensionless kinematic variables
αγ = (ENi )cmradics =
s+M2N +Q2
2s (B11)
απ = (ENf )cmradics =
s+M2N minusm2
π
2s (B12)
βγ = Eγcmradics =
sminusM2N minusQ2
2s (B13)
βπ = Eπcmradics =
sminusM2N +m2
π
2s (B14)
λγ = |~qi|cmradics =
radic(sminusM2
N minusQ2)2 + 4sQ2
2s (B15)
λπ = |~qf |cmradics =
radic(sminusM2
N +m2π)2 minus 4sm2
π
2s (B16)
Here (ENi )cm and (ENf )cm are the energies of the incoming and outgoing nucleon Eγcm is the energy
of the incoming photon Eπcm is the energy of the outgoing pion all in the cm frame
σ(π+n)TT = minus e2g2
AM2N
64πf2πs
3(sminusM2N )2λ4
γ
4sλπλγ
[(M2
N minus s)(M2N minusQ2 minus s)(Q2 + 2sβγβπ)
+ 2s(m2πQ
2 minus (M2N minus s)
(M2N + s(minus1 + 2βγβπ)
))λ2γ
]+ (M2
N minus s)(M2N minusQ2 minus s)
times (Q2 + 2sβγβπ minus 2sλπλγ)(Q2 + 2s(βγβπ + λπλγ)
)times log
(Q2 + 2sβγβπ minus 2sλγλπQ2 + 2sβγβπ + 2sλγλπ
) (B17)
33
σ(π0p)TT =
e2g2AM
2Nλπ
64πf2πs(sminusM2
N )2λγ
1
s2λπλ3γ
[(M2
N minus s)(M2N minusQ2 minus s)
(3M2
N + s(2βγβπ minus 3))(M2N
+ s(2βγβπ minus 1))
+ 2s
((2m2
πQ2 + (M2
N minus s)2)(M2
N minus s)minus 2(minusm2
πQ2 + (sminusM2
N )2)
times sβγβπ + 4s2(sminusM2N )β2
γβ2π
)λ2γ
]arctan
(2sλπλγ
M2N + s(2βγβπ minus 1)
)+ 2
[minus 2m2
πQ2
+ (M2N minus s)
(2(sminusM2
N ) + 4sβγβπ +(sminusM2
N +Q2)(3M2N minus 3s+ 2sβγβπ)
sλ2γ
+m2πQ
2(sminusM2N )
(M2N minus s+ 2sβγβπ)(M2
N + s(minus1 + 2βγβπ minus 2λπλγ))
+m2πQ
2(sminusM2N )
(M2N minus s+ 2sβγβπ)(M2
N + s(minus1 + 2βγβπ + 2λπλγ))
)
+1
λπλγ
(βγβπ
(minusm2
πQ2 + (M2
N minus s)(M2N minus s+ 2sβγβπ)
)minus (M2
N minus s)(M2N minusQ2 minus s)λ2
π
)log
(M2N + s(minus1 + 2βγβπ + 2λπλγ)
M2N + s(minus1 + 2βγβπ minus 2λπλγ)
)] (B18)
σ(πminusp)TT =
e2g2AM
2N
64πf2πs
3λ4γ(M2
N + s(2βπβγ minus 1))2 minus 4λ2πs
2λ2γ)
4sλπλγ
((minusM2
N +Q2 + s)(M2N
+ s(2βπβγ minus 1))
2 minus 2sλ2γ
(2λ2
πs(minusM2N +Q2 + s) +m2
πQ2))
+((M2N + s(2βπβγ minus 1)
)2
minus 4λ2πs
2λ2γ
)((M2N + s(2βπβγ minus 1)
)(M2N minus 2Q2 minus s(2βπβγ + 1)
)+ 4λ2
πs2λ2
γ
)times log
(minusM2
N minus 2βπsβγ minus 2λπsλγ + s)minus(M2N + s(2βπβγ minus 2λπλγ minus 1)
)(M2N + s(minus1
+ 2βπβγ + 2λπλγ))[((
M2N + s(2βπβγ minus 1)
)(M2N minus 2Q2 minus s(2βπβγ + 1)
)+ 4λ2
πs2λ2
γ
)times log
(minusM2
N minus 2βπsβγ + 2λπsλγ + s)
+((Q2 + 2βπsβγ)2 minus 4λ2
πs2λ2
γ
)times log
(Q2 + 2βπsβγ minus 2λπsλγQ2 + 2sβπβγ + 2λπsλγ
)]+ 4sλ2
γ
((M2
N + s(2βπβγ minus 1))2 minus 4λ2πs
2λ2γ
)times (M2
N minus s) arctanh
(2λπsλγ
M2N + 2βπsβγ minus s
) (B19)
34
σ(π+n)LT =
e2g2AM
3Nλπ
32πf2πQs
3(sminusM2N )2λ4
γ
2sλγ
[(M2
N minus s)(Q2 + 2sβ2γ)(Q2 + 2sβγβπ)
minus 4s
((M2
N minus s)(Q2 minus 2s(απ minus 1)βγ
)+Q2sβγβπ
)λ2γ + 8s3(minus1 + απ)λ4
γ
]minus M2
N minus sλπ
[(Q2 + 2sβ2
γ)(Q2 + 2sβγβπ)2 + 4s2(2(απ minus 1)βγ(Q2 + 2sβγβπ)
minusQ2λ2π
)λ2γ + 8s3(απ minus 1)2λ4
γ
]arctan
(2sλπλγ
Q2 + 2sβγβπ
) (B20)
σ(π0p)LT =
e2g2AM
3Nλπ
16πf2πQs(sminusM2
N )2λγ
1
minus2s(M2N + s(minus1 + 2βγβπ))2λ2
γ + 8s3λ2πλ
4γ
[minus 3M8
N (Q2
+ 2sβ2γ) + 2M4
Ns2
(minus (Q2 + 2sβ2
γ)(minus3 + 2βγβπ) +(Q2(minus3 + 2βγβπ + 6λ2
π) + 2sβγ
times (minus3 + 12απ + 2βγβπ minus 12απβγβπ + 4βγλ2π))λ2γ minus 4s(α2
π minus 1)λ4π
)+ 2M2
Ns3
[minus (Q2
+ 2sβ2γ)(minus1 + 2βγβπ)
(6 + βγβπ(minus9 + 2βγβπ)
)+
(Q2(3minus 12λ2
π + 4βγβπ(minus1 + 2βγβπ
+ λ2π))minus 2sβγ
(minus 3 + 4απ(3 + 2απ(3 + 2βγβπ(minus3 + βγβπ)) + 4βγ(βπ + (2minus βγβπ)
times λ2π)))λ2γ + 8s
((απ minus 1)(1 + απ minus 2βγβπ) + 2απβγλ
2π
)λ4γ
)+ s4
((Q2 + 2sβ2
γ)
times (1minus 2βγβπ)2(2βγβπ minus 3) + 2(Q2(minus1 + 6λ2
π + 2βγβπ((1minus 2βγβπ)2 minus 2λ2π)) + 2sβγ
times (minus1 + 2βγβπ + 4(minus1 + βγβπ)(minusαπ + 2απβγβπ minus βγλ2π)))λ2γ minus 8
(s(απ minus 1)(απ
+ (1minus 2βγβπ)2) + 2βγ(2sαπ +Q2βγ)λ2π
)λ4γ + 32s(απ minus 1)λ2
πλ6γ
)+ 2M6
Ns
(Q2(6
minus 7βγβπ + λ2γ) + 2sβγ(βγ(6minus 7βγβπ) + (1minus 4απ)λ2
γ)
)]+sminusM2
N
4s2λπλ3γ
[(Q2 + 2sβ2
γ)
times(3M2
N + s(2βγβπ minus 3))(M2N + s(2βγβπ minus 1)
)+ 2s
(minusM2
N (Q2 + 2sβγ minus 8sαπβγ)
+ s(2sβγ(1minus 4απ + 4απβγβπ) +Q2(1minus 2λ2π)))λ2γ + 8s3(απ minus 1)λ4
γ
]
times arctan
(2sλπλγ
M2N + s(2βγβπ minus 1)
) (B21)
35
σ(πminusp)LT =
e2g2AM
3N
64πf2πQs
3λ4γ(M2
N minusQ2 minus s)(M2N + s(2βπβγ + 2λπλγ minus 1))
times
1
M2N + s(2βπβγ minus 2λπλγ minus 1)
[4λπsλγ(M2
N minusQ2 minus s)(
2sλ2γ
(s((1minus 2λ2
π)Q2
+ 2(2απ minus 1)sβγ(2βπβγ minus 1))minusM2N (Q2 + 2(1minus 2απ)sβγ)
)+ (Q2 + 2sβ2
γ)(M2N + s(2βπβγ minus 1)
)2 + 8(απ minus 1)απs
3λ4γ
)]minus(M2N + s(2βπβγ + 2λπλγ minus 1)
)times[
log
(minusM2N minus 2βπsβγ minus 2λπsλγ + s
minusM2N minus 2βπsβγ + 2λπsλγ + s
)(minus (Q2 + 2sβ2
γ)(M2N + s(2βπβγ minus 1)
)(minusM2
N + 2Q2 + 2βπsβγ + s)minus 2sλ2γ
(minusM2
N (Q2 + 2sβγ) +Q4 +Q2s(4απβγ
minus 2βγ minus 2λ2π + 1) + 2s2βγ(4(απ minus 1)βπβγ + 1)
)minus 8(απ minus 1)2s3λ4
γ
)+ 2
(4s2λ2
γ
(2(απ minus 1)βγ(Q2 + 2βπsβγ)minus λ2
πQ2)
+ (Q2 + 2βπsβγ)2(Q2 + 2sβ2γ)
+ 8(απ minus 1)2s3λ4γ
)arctanh
(2λπsλγ
Q2 + 2βπsβγ
)] (B22)
2 ∆-production channel
The tree-level ∆-exchange diagram in Fig 2 of Ref [30] contributes to the non-Born part of
the VVCS amplitudes The contribution of the ∆ exchange to the VVCS amplitudes can be split
into [17]
S∆-exch1 (νQ2) = S∆-pole
1 (νQ2) + S∆-exch1 (νQ2) (B23a)
νS∆-exch2 (νQ2) = νS∆-pole
2 (νQ2) + νS2∆-exch
(νQ2) (B23b)
and similarly for the unpolarized VVCS amplitudes discussed in Ref [30] Here we introduced the
∆-pole contributions S∆-polei and the ∆-non-pole contributions S∆-exch
i The former amplitudes
feature a pole at the ∆(1232)-production threshold and thus are proportional to
1
[sminusM2∆][uminusM2
∆]=
1
4M2N
1
ν2∆ minus ν2
(B24)
36
They can be reconstructed from the dispersion relations in Eq (5) with the tree-level ∆-production
cross sections as input cf Fig 10 in Ref [30]
σTT (νQ2) =π2α
M2NM
2+|~q |
minus g2
MMN (M+ + ν)|~q |2 +g2E(∆minus ν)(Q2 minusMNν)2
MN(B25a)
+g2CQ
4s(∆minus ν)
MNM2∆
minus 4gMgE(Q2 minusMNν)|~q |2 minus 4gMgCQ2|~q |2
+2gEgCQ
2[minusMNM∆ |~q |2 + s(Q2 +∆ν)
]MNM∆
δ(ν minus ν∆)
σLT (νQ2) =Qπ2α
M2NM
2+|~q |
g2E(MNν minusQ2) [M∆(MN + ν)minus s]
MN(B25b)
+g2CQ
2[MNM∆|~q |2 minus s(Q2 +∆ν)
]MNM2
∆
+ gMgEM∆|~q |2 minusgMgC(Q2 minusMNν)|~q |2
M∆
+gEgC(ν minus∆)(M2
N |~q |2 minus 2Q2s)
MNM∆
δ(ν minus ν∆)
with ∆ = M∆ minusMN M+ = M∆ + MN and the Mandelstam variable s = M2N + 2MNν minus Q2
Analytical expressions for the spin structure functions g1(xQ2) and g2(xQ2) can be constructed
from Eq (3) with the flux factor K(νQ2) = |~q | =radicν2 +Q2
In the ∆-non-pole contributions to S1(νQ2) and νS2(νQ2) the pole in ν at the ∆(1232)-
production threshold has canceled out
S∆-exch1 (νQ2) =
πα
MNM2+
[g2MQ
2+ + g2
E
(∆2 minus 3Q2
)+
4g2CQ
4
M2∆
minus 8gMgEM∆ωminus (B26a)
minus2gMgCQ2(MN minus 4M∆)
M∆+
2gEgCQ2(3MN minus 2M∆)
M∆
]
νS2∆-exch
(νQ2) =2πα
MNM2+
[g2EM∆∆ωminus +
g2M MNQ
2+
2+g2C Q
2(Q2 minus∆2)
2M∆(B26b)
+gEgM M∆(M∆ω+ minus 4MNωminus)minus gEgC ∆(2Q2 +MNω+)
+gMgC Q2(4MN minus ω+)
]+S∆-exch
2 (νQ2)
ν
[M2
∆ ω2+
M2N
+ ν2
]
with Q+ =radic
(M∆ +MN )2 +Q2 and ωplusmn = (M2∆minusM2
N plusmnQ2)2M∆ and the non-pole contribution
to S2(νQ2)
S∆-exch2 (νQ2) = minus2παMNν
M∆M2+
[gM + gE
]gC (B27)
These amplitudes to the contrary are not described by the tree-level ∆-production cross sections
in the standard dispersive approach [17] This peculiarity has been previously missed e g in the
calculation of the ∆-exchange contribution to the hydrogen hyperfine splitting in Ref [88] The
importance of including the ∆-non-pole contribution is also evident when considering the BC sum
37
rule in Eq (14) The ∆-pole terms by themselves violate the BC sum rule but cancel exactly with
the ∆-non-pole terms
limνrarr0
νS∆-pole2 (νQ2) + lim
νrarr0νS2
∆-exch(νQ2) = 0 (B28)
Appendix C Polarizabilities at Q2 = 0
In this section we give analytical expressions for the static values and slopes at Q2 = 0 of
the spin polarizabilities and moments of polarized structure functions In particular we give the
HB expansion of the πN -loop contributions and the ∆-exchange contributions The complete
expressions also for the π∆-loop contributions can be found in the Supplemented material Recall
that IA(0) = I1(0) = d2(0) = 0 and dd2(Q2)dQ2
∣∣∣Q2=0
= 0
1 πN-loop contribution
Here we give analytical expressions for the πN -loop contributions to the proton and neutron
spin polarizabilities expanded in powers of micro = mπMN viz the HB expansion Note that we
choose to expand here to a high order in micro the strict HB expansion would only retain the leading
term in an analogous NLO calculation
bull Static polarizabilities (Q2 = 0)
γ0p =e2g2
A
96π3f2πm
2π
1minus 21πmicro
8minus(
59
2+ 26 logmicro
)micro2 +
1875πmicro3
64
+3
(3
2+ 26 logmicro
)micro4 +
(C1)
γ0n =e2g2
A
48π3f2πm
2π
1
2minus 9πmicro
16minus 2micro2 logmicro+
75πmicro3
128minus 3micro4
4+
(C2)
δLTp =e2g2
A
192π3f2πm
2π
1minus 9πmicro
8+
(13
2minus 2 logmicro
)micro2 minus 465πmicro3
64
minus(
47
2+ 42 logmicro
)micro4 +
(C3)
δLTn =e2g2
A
96π3f2πm
2π
1
2+
3πmicro
16+ (1 + 2 log micro)micro2 minus 105πmicro3
128+
5micro4
4+
(C4)
38
δLTn =e2g2
A
96π3f2πm
2π
1
2+
3πmicro
16+ (1 + 2 log micro)micro2 minus 105πmicro3
128+
5micro4
4+
(C5)
γ0p =e2g2
A
16π3f2πm
4π
4
45minus 3πmicro
16+
14micro2
5minus 1813πmicro3
384minus 192
5(1 + log micro)micro4
+80703πmicro5
2048+
(C6)
γ0n =e2g2
A
16π3f2πm
4π
4
45minus 5πmicro
48+
4micro2
5minus 245πmicro3
384minus 32micro4 logmicro
15
+1323πmicro5
2048+
(C7)
bull Slopes of polarizabilities at Q2 = 0
dγ0p(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
2minus 45πmicro
4+ 223micro2 minus 28515πmicro3
64
minus9
(1953
4+ 449 logmicro
)micro4 +
570255πmicro5
128+
(C8)
dγ0n(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
2minus 81πmicro
8+ 94micro2 minus 2535πmicro3
32
minus3 (1 + 90 logmicro)micro4 +84315πmicro5
1024+
(C9)
dδLTp(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
2880π3f2πm
4π
minus5
2minus 27πmicro
32+ 20micro2 minus 5865πmicro3
256
+3
(617
4+ 36 logmicro
)micro4 minus 2056845πmicro5
4096+
(C10)
dδLTn(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
minus5
4minus 81πmicro
64minus 11micro2 +
10005πmicro3
512
+15
8(11 + 48 logmicro)micro4 minus 267015πmicro5
8192+
(C11)
dIAp(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
96π2f2πmicro
2
1minus 15πmicro
4minus 1
2(115 + 88 logmicro)micro2 +
1839πmicro3
32
+5 (5 + 34 logmicro)micro4 +
(C12)
dIAn(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
48π2f2πmicro
2
1
2minus 11πmicro
8minus 1
4(1 + 20 logmicro)micro2 +
99πmicro3
64
minus25micro4
12+
(C13)
dI1p(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
96π2f2πmicro
3π
8+ 2 (4 + 3 logmicro)microminus 537πmicro2
64
minus1
2(15 + 56 logmicro)micro3 +
(C14)
39
dI1n(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
48π2f2πmicro
minus π
16+
1
4(3 + 4 logmicro)microminus 57πmicro2
128+
2micro3
3+
(C15)
γ0p(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2g2
A
16π3f2πm
6π
1
105minus 23πmicro
256+
377micro2
210minus 15551πmicro3
6144+
3371micro4
105
minus1640457πmicro5
32768+
(C16)
γ0n(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2g2
A
16π3f2πm
6π
1
105minus 153πmicro
1792+
69micro2
70minus 4615πmicro3
6144+
172micro4
35
minus120897πmicro5
32768+
(C17)
2 ∆-exchange contribution
Here we give analytical expressions for the tree-level ∆-exchange contributions to the nucleon
spin polarizabilities and their slopes at Q2 = 0 Note that the ∆-exchange contributes equally to
proton and neutron polarizabilities Recall that for the magnetic γlowastN∆ coupling we introduced a
dipole form factor to mimic vector-meson dominance gM rarr gM(1 +Q2Λ2)2
bull Static polarizabilities (Q2 = 0)
γ0 = minus e2
4πM2+
(g2M
∆2+
g2E
M2+
minus 4gMgEM+∆
) (C18)
δLT =e2M∆
4πM3+
(g2E
MNM++gMgE∆MN
minus gEgCM2
∆
) (C19)
γ0 =e2M2
N
π∆2M4+
(minusg
2M
∆2+
g2E
M2+
+4gMgE∆M+
) (C20)
bull Slopes of polarizabilities at Q2 = 0
dγ0(Q2)
dQ2
∣∣∣∣∣Q2=0
= minus e2
πM2+∆
(g2M
∆
[1
4∆2minus 1
∆M++
1
2M2+
]minus 1
Λ2
g2M
∆+
g2E
2M2+
[1
2∆minus 3
M+
]minusgMgE
M+
[1
∆2minus 5
∆M++
1
M2+
]+
1
Λ2
2gMgEM+
+2gMgC∆M2
+
minus gEgCM3
+
) (C21)
40
d δLT (Q2)
dQ2
∣∣∣∣∣Q2=0
=e2M∆∆
4πMNM2+
(g2E
∆2M2+
[1
∆minus 4
M+
]minus g2
C
∆M2∆M
2+
+gMgE∆2M+
[1
∆2
minus 3
∆M++
1
M2+
]minus 2
Λ2
gMgE∆2M+
+gMgC∆M2
∆
[1
2∆2minus 2
∆M++
1
2M2+
]minus gEgC
2M2∆M
2+
[7
∆+
1
M+
]) (C22)
dIA(Q2)
dQ2
∣∣∣∣∣Q2=0
= minusM2N
M2+
(g2M
2∆2+
g2E
MNM+minus 2gMgE
∆M+minus gEgCM∆M+
) (C23)
dI1(Q2)
dQ2
∣∣∣∣∣Q2=0
= minusM∆M2N
2M3+
(g2E
MNM∆minus gMgE∆MN
minus gEgCM2
∆
) (C24)
γ0(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2M2
N
π∆3M6+
(g2M
∆
[3M2
∆ + 2M∆MN minus 9M2N
∆2+
4M2+
Λ2
]
minus4gMgE
[5M2
∆ minus 9M2N
M+∆2+
2M+
Λ2
]+
g2E
∆M+[7M∆ minus 9MN ]
minus8gMgC∆
+4gEgCM+
) (C25)
[1] D Drechsel B Pasquini and M Vanderhaeghen Dispersion relations in real and virtual
Compton scattering Phys Rept 378 99 (2003) hep-ph0212124
[2] S E Kuhn J-P Chen and E Leader Spin structure of the nucleon mdash status and recent
results Prog Part Nucl Phys 63 1 (2009) arXiv08123535 [hep-ph]
[3] F Hagelstein R Miskimen and V Pascalutsa Nucleon polarizabilities from Compton scat-
tering to hydrogen atom Prog Part Nucl Phys 88 29 (2016) arXiv151203765 [nucl-th]
[4] B Pasquini and M Vanderhaeghen Dispersion theory in electromagnetic interactions Ann
Rev Nucl Part Sci 68 75 (2018) arXiv180510482 [hep-ph]
[5] S Gerasimov A Sum rule for magnetic moments and the damping of the nucleon magnetic
moment in nuclei Sov J Nucl Phys 2 430 (1966)
[6] S Drell and A C Hearn Exact sum rule for nucleon magnetic moments Phys Rev Lett
16 908 (1966)
[7] J S Schwinger Source theory viewpoints in deep inelastic scattering Proc Natl Acad Sci
USA 72 1 (1975)
[8] J S Schwinger Source theory viewpoints in deep inelastic scattering Electromagnetic In-
teractions and Field Theory Proceedings 14 Internationale Universitatswochen Schladming
Austria February 24-March 7 1975 Acta Phys Austriaca Suppl 14 471 (1975)
41
[9] J Schwinger Source theory discussion of deep inelastic scattering with polarized particles
Proc Natl Acad Sci USA 72 1559 (1975)
[10] A M Baldin Polarizability of nucleons Nucl Phys 18 310 (1960)
[11] M Gell-Mann M L Goldberger and W E Thirring Use of causality conditions in quantum
theory Phys Rev 95 1612 (1954)
[12] A Pineda Leading chiral logarithms to the hyperfine splitting of the hydrogen and muonic
hydrogen Phys Rev C 67 025201 (2003)
[13] C Peset and A Pineda The two-photon exchange contribution to muonic hydrogen from
chiral perturbation theory Nucl Phys B 887 69 (2014) arXiv14064524 [hep-ph]
[14] C Peset and A Pineda Model-independent determination of the two-photon exchange contri-
bution to hyperfine splitting in muonic hydrogen JHEP 04 060 arXiv161205206 [nucl-th]
[15] F Hagelstein and V Pascalutsa Proton structure in the hyperfine splitting of muonic hydro-
gen PoS CD15 077 (2016) arXiv151104301 [nucl-th]
[16] F Hagelstein Exciting Nucleons in Compton Scattering and Hydrogen-Like Atoms PhD
thesis Mainz U Inst Kernphys (2017) arXiv171000874 [nucl-th]
[17] F Hagelstein ∆(1232)-Resonance in the hydrogen spectrum Proceedings 11th International
Workshop on the Physics of Excited Nucleons (NSTAR 2017) Columbia SC USA August
20-23 2017 Few Body Syst 59 93 (2018) arXiv180109790 [nucl-th]
[18] Y Prok et al (CLAS) Moments of the spin structure functions gp1 and gd1 for 005 lt Q2 lt 30
GeV2 Phys Lett B 672 12 (2009) arXiv08022232 [nucl-ex]
[19] H Dutz et al (GDH) First measurement of the Gerasimov-Drell-Hearn sum rule for 1H from
07 GeV to 18 GeV at ELSA Phys Rev Lett 91 192001 (2003)
[20] M Amarian et al (Jefferson Lab E94010) Measurement of the generalized forward spin po-
larizabilities of the neutron Phys Rev Lett 93 152301 (2004) arXivnucl-ex0406005
[21] M Amarian et al The Q2 evolution of the generalized Gerasimov-Drell-Hearn integral for
the neutron using a 3He target Phys Rev Lett 89 242301 (2002) arXivnucl-ex0205020
[22] M Amarian et al (Jefferson Lab E94-010)Q2 evolution of the neutron spin structure moments
using a 3He target Phys Rev Lett 92 022301 (2004) arXivhep-ex0310003
[23] A Deur et al Experimental determination of the evolution of the Bjorken integral at low Q2
Phys Rev Lett 93 212001 (2004) arXivhep-ex0407007
[24] K Slifer Low Q2 measurement of gp2 and the δLT spin polarizability Spin structure at long dis-
tance Proceedings Workshop Newport News USA March 12ndash13 2009 nucl-ex09064775
(2009) AIP Conf Proc 1155 10106313203293 (2009) arXiv09064775 [nucl-ex]
[25] P Solvignon et al (E01-012) Moments of the neutron g2 structure function at intermediate
42
Q2 Phys Rev C 92 015208 (2015) arXiv13044497 [nucl-ex]
[26] A Deur Experimental studies at low Q2 of the spin structure of the nucleon at Jefferson Lab
in 9th International Workshop on Chiral Dynamics (CD18) Durham NC USA September
17-21 2018 (2019) arXiv190305661 [nucl-ex]
[27] V Sulkosky et al (Jefferson Lab E97-110) Measurement of the 3He spin-structure functions
and of neutron (3He) spin-dependent sum rules at 0035 le Q2 le 024 GeV2 Phys Lett B
805 135428 (2020) arXiv190805709 [nucl-ex]
[28] K Adhikari et al (CLAS) Measurement of the Q2 Dependence of the Deuteron Spin Structure
Function g1 and its Moments at Low Q2 with CLAS Phys Rev Lett 120 062501 (2018)
arXiv171101974 [nucl-ex]
[29] V Lensky F Hagelstein A Hiller Blin and V Pascalutsa Comment on rdquoMeasurement of
the Q2 Dependence of the Deuteron Spin Structure Function g1 and its Moments at Low Q2
with CLASrdquo (2018) arXiv180603219 [nucl-th]
[30] J M Alarcn F Hagelstein V Lensky and V Pascalutsa Forward doubly-virtual Compton
scattering off the nucleon in chiral perturbation theory at NLO the subtraction function and
moments of unpolarized structure functions (2020) arXiv200509518 [hep-ph]
[31] V Bernard E Epelbaum H Krebs and U-G Meiszligner New insights into the spin structure
of the nucleon Phys Rev D 87 054032 (2013) arXiv12092523 [hep-ph]
[32] V Lensky J M Alarcon and V Pascalutsa Moments of nucleon structure functions at
next-to-leading order in baryon chiral perturbation theory Phys Rev C 90 055202 (2014)
arXiv14072574 [hep-ph]
[33] V Pascalutsa and M Vanderhaeghen Chiral effective-field theory in the ∆(1232) region
I Pion electroproduction on the nucleon Phys Rev D 73 034003 (2006) arXivhep-
ph0512244
[34] V Pascalutsa and M Vanderhaeghen Electromagnetic nucleon-to-Delta transition in chiral
effective field theory Phys Rev Lett 95 232001 (2005) arXivhep-ph0508060
[35] T R Hemmert B R Holstein and J Kambor Systematic 1M expansion for spin 32
particles in baryon chiral perturbation theory Phys Lett B 395 89 (1997) arXivhep-
ph9606456
[36] V Pascalutsa and D R Phillips Effective theory of the ∆(1232) in Compton scattering off
the nucleon Phys Rev C 67 055202 (2003) arXivnucl-th0212024
[37] V Pascalutsa M Vanderhaeghen and S N Yang Electromagnetic excitation of the ∆(1232)-
resonance Phys Rept 437 125 (2007) arXivhep-ph0609004
[38] K A Olive et al (Particle Data Group) Review of Particle Physics Chin Phys C 38 090001
43
(2014)
[39] F E Low Scattering of light of very low frequency by systems of spin 12 Phys Rev 96
1428 (1954)
[40] M Gell-Mann and M L Goldberger Scattering of low-energy photons by particles of spin
12 Phys Rev 96 1433 (1954)
[41] A M Harun ar-Rashid A simple derivation of Schwingerrsquos sum rule for spin dependent
structure functions Nuovo Cim A 33 447 (1976)
[42] F Hagelstein and V Pascalutsa Dissecting the hadronic contributions to (g minus 2)micro by
Schwingerrsquos sum rule Phys Rev Lett 120 072002 (2018) arXiv171004571 [hep-ph]
[43] V Lensky V Pascalutsa M Vanderhaeghen and C Kao Spin-dependent sum rules con-
necting real and virtual Compton scattering verified Phys Rev D 95 074001 (2017)
arXiv170101947 [hep-ph]
[44] H Burkhardt and W N Cottingham Sum rules for forward virtual Compton scattering
Annals Phys 56 453 (1970)
[45] V Lensky and V Pascalutsa Predictive powers of chiral perturbation theory in Compton
scattering off protons Eur Phys J C 65 195 (2010) arXiv09070451 [hep-ph]
[46] H Grieszlighammer J McGovern D Phillips and G Feldman Using effective field theory to
analyse low-energy Compton scattering data from protons and light nuclei Prog Part Nucl
Phys 67 841 (2012) arXiv12036834 [nucl-th]
[47] H W Grieszlighammer J A McGovern and D R Phillips Nucleon polarisabilities at and
beyond physical pion masses Eur Phys J A 52 139 (2016) arXiv151101952 [nucl-th]
[48] E Epelbaum H Krebs and U-G Meiszligner Improved chiral nucleon-nucleon potential up to
next-to-next-to-next-to-leading order Eur Phys J A 51 53 (2015) arXiv14120142 [nucl-
th]
[49] E Epelbaum H Krebs and U-G Meiszligner Precision nucleon-nucleon potential at fifth order
in the chiral expansion Phys Rev Lett 115 122301 (2015) arXiv14124623 [nucl-th]
[50] C W Kao T Spitzenberg and M Vanderhaeghen Burkhardt-Cottingham sum rule and
forward spin polarizabilities in heavy baryon chiral perturbation theory Phys Rev D 67
016001 (2003) arXivhep-ph0209241
[51] C-W Kao D Drechsel S Kamalov and M Vanderhaeghen Higher moments of nucleon
spin structure functions in heavy baryon chiral perturbation theory and in a resonance model
Phys Rev D 69 056004 (2004) arXivhep-ph0312102
[52] T Becher and H Leutwyler Baryon chiral perturbation theory in manifestly Lorentz invariant
form Eur Phys J C 9 643 (1999) arXivhep-ph9901384
44
[53] J Gasser M E Sainio and A Svarc Nucleons with chiral loops Nucl Phys B 307 779
(1988)
[54] L S Geng J Martin Camalich L Alvarez-Ruso and M J Vicente Vacas Leading SU(3)-
breaking corrections to the baryon magnetic moments in chiral perturbation theory Phys
Rev Lett 101 222002 (2008) arXiv08051419 [hep-ph]
[55] D Drechsel S S Kamalov and L Tiator The GDH sum rule and related integrals Phys
Rev D 63 114010 (2001) arXivhep-ph0008306
[56] D Drechsel O Hanstein S S Kamalov and L Tiator A unitary isobar model for pion
photo- and electroproduction on the proton up to 1 GeV Nucl Phys A 645 145 (1999)
arXivnucl-th9807001
[57] L Tiator private communication (2020)
[58] V Bernard T R Hemmert and U-G Meiszligner Spin structure of the nucleon at low energies
Phys Rev D 67 076008 (2003) arXivhep-ph0212033
[59] O Gryniuk F Hagelstein and V Pascalutsa Evaluation of the forward Compton scattering
off protons II Spin-dependent amplitude and observables Phys Rev D 94 034043 (2016)
arXiv160400789 [nucl-th]
[60] R Zielinski The g2p Experiment A Measurement of the Protonrsquos Spin Structure Functions
PhD thesis New Hampshire U (2010) arXiv170808297 [nucl-ex]
[61] N Guler et al (CLAS) Precise determination of the deuteron spin structure at low to mod-
erate Q2 with CLAS and extraction of the neutron contribution Phys Rev C 92 055201
(2015) arXiv150507877 [nucl-ex]
[62] C E Carlson V Nazaryan and K Griffioen Proton structure corrections to electronic
and muonic hydrogen hyperfine splitting Phys Rev A 78 022517 (2008) arXiv08052603
[physicsatom-ph]
[63] V Bernard N Kaiser and U-G Meiszligner Chiral dynamics in nucleons and nuclei Int J
Mod Phys E 4 193 (1995) arXivhep-ph9501384
[64] T R Hemmert B R Holstein and J Kambor ∆(1232) and the polarizabilities of the nucleon
Phys Rev D 55 5598 (1997) arXivhep-ph9612374
[65] V Pascalutsa and R Timmermans Field theory of nucleon to higher-spin baryon transitions
Phys Rev C 60 042201 (1999) arXivnucl-th9905065
[66] V Pascalutsa Quantization of an interacting spin-32 field and the Delta isobar Phys Rev
D 58 096002 (1998) arXivhep-ph9802288
[67] H Krebs Double Virtual Compton Scattering and SpinStructure of the Nucleon PoS
CD2018 031 (2019)
45
[68] N Kochelev and Y Oh Axial anomaly and the δLT puzzle Phys Rev D 016012 (2012)
arXiv11034892 [hep-ph]
[69] D Drechsel S Kamalov and L Tiator Unitary isobar model ndash MAID2007 Eur Phys J A
69 (2007) available at httpsmaidkphuni-mainzde arXiv07100306 [nucl-th]
[70] P J Mohr B N Taylor and D B Newell CODATA recommended values of the fundamental
physical constants 2010 Rev Mod Phys 84 1527 (2012)
[71] J Ahrens et al (GDH A2) First measurement of the Gerasimov-Drell-Hearn integral for 1H
from 200 to 800 MeV Phys Rev Lett 87 022003 (2001) arXivhep-ex0105089 [hep-ex]
[72] K Helbing (GDH) Experimental verification of the GDH sum rule at ELSA and MAMI Nucl
Phys Proc Suppl 105 113 (2002)
[73] R Bradford A Bodek H S Budd and J Arrington A New parameterization of the nucleon
elastic form-factors NuInt05 proceedings of the 4th International Workshop on Neutrino-
Nucleus Interactions in the Few-GeV Region Okayama Japan 26-29 September 2005 Nucl
Phys Proc Suppl 159 127 (2006) arXivhep-ex0602017
[74] J D Bjorken Applications of the chiral U(6)timesU(6) algebra of current densities Phys Rev
148 1467 (1966)
[75] J D Bjorken Inelastic scattering of polarized leptons from polarized nucleons Phys Rev D
1 1376 (1970)
[76] V Bernard T R Hemmert and U-G Meiszligner Novel analysis of chiral loop effects in
the generalized Gerasimov-Drell-Hearn sum rule Phys Lett B 545 105 (2002) arXivhep-
ph0203167
[77] A Deur et al Experimental study of isovector spin sum rules Phys Rev D 78 032001
(2008) arXiv08023198 [nucl-ex]
[78] B Pasquini P Pedroni and D Drechsel Higher order forward spin polarizability Phys Lett
B 687 160 (2010) arXiv10014230 [hep-ph]
[79] R Jaffe g2ndashThe nucleonrsquos other spin-dependent structure function Comments Nucl Part
Phys 19 239 (1990)
[80] E V Shuryak and A Vainshtein Theory of power corrections to deep inelastic scattering in
quantum chromodynamics (II) Qminus4 effects polarized target Nucl Phys B 201 141 (1982)
[81] S Wandzura and F Wilczek Sum rules for spin dependent electroproduction Test of rela-
tivistic constituent quarks Phys Lett B 72 195 (1977)
[82] B W Filippone and X-D Ji The spin structure of the nucleon Adv Nucl Phys 26 1
(2001) arXivhep-ph0101224
[83] M Burkardt The g2 structure function Proceedings Workshop on Spin structure at long
46
distance Newport News USA March 12-13 2009 AIP Conf Proc 1155 26 (2009)
arXiv09054079 [hep-ph]
[84] A Deur S J Brodsky and G F de Teramond The spin structure of the nucleon Rept
Prog Phys 82 076201 (2019) arXiv180705250 [hep-ph]
[85] B R Holstein D Drechsel B Pasquini and M Vanderhaeghen Higher order polarizabilities
of the proton Phys Rev C 61 034316 (2000) arXivhep-ph9910427
[86] B R Holstein V Pascalutsa and M Vanderhaeghen Sum rules for magnetic moments and
polarizabilities in QED and chiral effective-field theory Phys Rev D 72 094014 (2005)
arXivhep-ph0507016
[87] B Pasquini D Drechsel and L Tiator Invariant amplitudes for pion electroproduction Eur
Phys J A 34 387 (2007) arXiv07122327 [hep-ph]
[88] A J Buchmann Non-spherical proton shape and hydrogen hyperfine splitting Proceedings
International Workshop on Precision Physics of Simple Atomic Systems (PSAS 2008) Wind-
sor Ontario Canada July 21-26 2008 Can J Phys 87 773 (2009) arXiv09104747
[physicsatom-ph]
47
predictions by
σP (Q2) =
Max[δ4(0)P (0)2 δ4(0)P LO(0)2 δ2(0)PNLO(0)2
]+Max
[δ4(Q2)
[P (Q2)minus P (0)
]2 δ4(Q2)
[P LO(Q2)minus P LO(0)
]2
δ2(Q2)[PNLO(Q2)minus PNLO(0)
]2]12
(18)
where P LO(Q2) is the πN -loop contribution PNLO(Q2) are the ∆-exchange and π∆-loop
contributions and P (Q2) = P LO(Q2) + PNLO(Q2) This error prescription is similar to the
one used in eg Refs [46ndash49] Here since we are interested in the generalized polarizabil-
ities we added in quadrature the error due to the static piece P (0) and the Q2-dependent
remainder P (Q2) minus P (0) Note that the static values of IA(0) and I1(0) are given by the
elastic Pauli form factor which is not part of our BχPT prediction and is considered to be
exact
Note that our result for the spin polarizabilities (and the unpolarized moments [30]) are
NLO predictions only at low momentum transfers Q mπ At larger values of Q amp ∆ they
become incomplete LO predictions Indeed in this regime the ∆ propagators do not carry
additional suppression compared to the nucleon propagators and the π∆ loops are promoted
to LO In general we only expect a rather small contribution from omitted π∆ loops to the
Q2 dependence of the polarizabilities since π∆ loops show rather weak dependence on Q2
compared with the ∆ exchange or πN loops Nevertheless this issue has to be reflected in
the error estimate Since the static polarizabilities P (0) are not affected it is natural to
separate the error on the Q2-dependent remainder P (Q2) minus P (0) as done in Eq (18) To
accommodate for the potential loss of precision above Q amp ∆ we define the relative error
δ(Q2) as growing with increasing Q2 see Eq (17)
Upon expanding our results in powers of the inverse nucleon mass Mminus1N we are able to
reproduce existing results of heavy-baryon χPT (HBχPT) at LO We however do not see
a rationale to drop the higher-order Mminus1N terms when they are not negligible (ie when
their actual size exceeds by far the natural estimate for the size of higher-order terms)
Comparing our BχPT predictions to HBχPT we will also see a deficiency of HBχPT in
the description of the Q2 behaviour of the polarizabilities Note that the O(p4) HBχPT
results from Ref [50 51] which we use here for comparison do not include the ∆ These
references studied the leading effect of the latter in the HBχPT framework using the small-
scale expansion [35] observing no qualitative improvement in the HBχPT description of the
empirical data [50 51] when including it We therefore choose to use the O(p4) results as
the representative HBχPT curves
10
Another approach used in the literature to calculate the polarizabilities in χPT is the
infrared regularization (IR) scheme introduced in Ref [52] This covariant approach tries to
solve the power counting violation observed in Ref [53] by dropping the regular parts of the
loop integrals that contain the power-counting-breaking terms However this subtraction
scheme modifies the analytic structure of the loop contributions and may lead to unexpected
problems as was shown in Ref [54] As we will see in the next section the IR approach
also fails to describe the Q2 behaviour of the polarizabilities
III RESULTS AND DISCUSSION
We now present the NLO BχPT predictions for the nucleon polarizabilities and selected
moments of the nucleon spin structure functions Our results are obtained from the calcu-
lated non-Born VVCS amplitudes and the LEXs in Eqs (10) and (12) For a cross-check
we used the photoabsorption cross sections described in App B In addition to the full NLO
results we also analyse the individual contributions from the πN loops the ∆ exchange
and the π∆ loops
A γ0(Q2) mdash generalized forward spin polarizability
The forward spin polarizability
γ0(Q2) =1
2π2
int infinν0
dν
radic1 +
Q2
ν2
σTT (νQ2)
ν3(19)
=16αM2
N
Q6
int x0
0
dx x2
[g1(xQ2)minus 4M2
Nx2
Q2g2(xQ2)
]
provides information about the spin-dependent response of the nucleon to transversal pho-
ton probes The RCS analogue of the above generalized forward spin polarizability sum
rule is sometimes referred to as the Gell-Mann Goldberger and Thirring (GGT) sum rule
[11] At Q2 = 0 the forward spin polarizability is expressed through the lowest-order spin
polarizabilities of RCS as γ0 = minus(γE1E1 + γM1M1 + γE1M2 + γM1E2) The forward spin po-
larizability of the proton is relevant for an accurate knowledge of the (muonic-)hydrogen
hyperfine splitting as it controls the leading proton-polarizability correction [16 62]
The πN -loop ∆-exchange and π∆-loop contributions to the NLO BχPT prediction of
11
000 005 010 015 020 025 030
-4
-3
-2
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
4D
proton
000 005 010 015 020 025 030
-4
-3
-2
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
4D
neutron
000 005 010 015 020 025 030
0
1
2
Q2 GeV2D
∆LT
10-
4fm
4D
proton
000 005 010 015 020 025 030
0
1
2
3
Q2 GeV2D
∆LT
10-
4fm
4D
neutron
FIG 2 Upper panel Generalized forward spin polarizability for the proton (left) and neutron
(right) as function of Q2 The result of this work the NLO BχPT prediction is shown by the
blue solid line and the blue band The red line represents the LO BχPT result The purple short-
dashed line is the O(p4) HB result from Ref [50] note that the corresponding proton curve is
outside of the plotted range The black dotted line is the MAID model prediction [55ndash57] which
is taken from Ref [1] (proton) and Ref [20] (neutron) The pink band is the IR+∆ result from
Ref [58] and the gray band is the BχPT+∆ result from Ref [31] Empirical extractions for the
proton Ref [18] (blue dots) Ref [59] (purple square) and Ref [60] (orange triangle uncertainties
added in quadrature) and neutron Ref [20] (blue diamonds) and Ref [61] (green dots statistical
and systematic uncertainties added in quadrature) Lower Panel Longitudinal-transverse spin
polarizability for the proton (left) and neutron (right) The orange dot-dashed and purple short-
dashed lines are the O(p3) and O(p4) HB results from Ref [50] The pink band is the IR result
from Ref [58] and the gray band is the covariant BχPT+∆ result from Ref [31] The black dotted
line is the MAID model prediction [55ndash57] note that for the proton we use the updated estimate
from Ref [1] obtained using the π η ππ channels
12
the static forward spin polarizability amount to in units of 10minus4 fm4
γ0p = minus093(92) asymp 201minus 284minus 010 (20a)
γ0n = 003(92) asymp 298minus 284minus 010 (20b)
while the slope is composed as follows in units of 10minus4 fm6
dγ0p(Q2)
dQ2
∣∣∣∣Q2=0
= minus022(4) asymp minus033 + 011 + 001 (21a)
dγ0n(Q2)
dQ2
∣∣∣∣Q2=0
= minus061(7) asymp minus073 + 011 + 001 (21b)
Figure 2 upper panel shows our NLO prediction as well as the LO πN loops compared
to different experimental and theoretical results For the proton we have one determination
at the real-photon point by the GDH collaboration [19] γ0p = minus100(8)(12)times10minus4 fm4 and
further Jefferson Laboratory data [18 60] at very low Q2 For the neutron only data at finite
Q2 are available [20 61] The experimental data for the proton are fairly well reproduced
in the whole Q2 range considered here while for the neutron the agreement improves with
increasing Q2 The HB limit of our πN -loop contribution reproduces the results published
in Refs [50 63] for arbitrary Q2 In addition our prediction is compared to the MAID
model [1 20] the IR+∆ calculation of Ref [58] and the BχPT+∆ result of Ref [31]
The πN -production channel gives a positive contribution to the photoabsorption cross
section σTT (νQ2) at low Q2 cf Fig 10 Accordingly one observes that the πN loops give
a sizeable positive contribution to γ0(Q2) The Delta on the other hand has a very large
effect by cancelling the πN loops and bringing the result close to the empirical data From
Fig 3 upper panel one can see that it is the ∆ exchange which dominates while π∆
loops are negligible This was expected since the forward spin polarizability sum rule is an
integral over the helicity-difference cross section in which σ32 is governed by the Delta at
low energies (the relevant energy region for the sum rule)
To elucidate the difference between the present calculation and the one from Ref [31] we
note that the two calculations differ in the following important aspects Firstly Ref [31] uses
the small-scale counting [64] that considers ∆ and mπ as being of the same size ∆ sim mπ
In practice this results in a set of π∆-loop graphs which contains graphs with one or two
γ∆∆ couplings and hence two or three Delta propagators Such graphs are suppressed in
the δ-counting and thus omitted from our calculation while present in that of Ref [31]
Secondly the Lagrangians describing the interaction of the Delta are constructed differently
and assume slightly different values for the coupling constants In particular we employ
13
000 005 010 015 020 025 030
-3
-2
-1
0
1
2
Q2GeV
2D
Γ01
0-
4fm
4D
proton
000 005 010 015 020 025 030
-3
-2
-1
0
1
2
3
Q2GeV
2D
Γ01
0-
4fm
4D
neutron
000 005 010 015 020 025 030
00
05
10
15
Q2GeV
2D
∆LT1
0-
4fm
4D
proton
000 005 010 015 020 025 030
00
05
10
15
20
25
Q2GeV
2D
∆LT1
0-
4fm
4D
neutron
FIG 3 Contributions of the different orders to the chiral predictions of γ0(Q2) upper paneland δLT (Q2) lower panel for the proton (left) and neutron (right) Red solid line πN -loop
contribution green dot-dashed line ∆-exchange contribution orange dotted line π∆-loop con-
tribution blue long-dashed line total result purple dot-dot-dashed line total result without gCcontribution black short-dashed line total result without gM dipole
(where possible) the so-called ldquoconsistentrdquo couplings to the Delta field ie those couplings
that project out the spurious degree of freedom see Refs [37 65 66] The authors of
Ref [31] on the other hand use couplings where the consistency in this sense is not enforced
The effects of these differences are of higher order in the δ-counting expansion and their
contribution to the Q2 dependence of the considered polarizabilities is expected to be rather
small however the differences at Q2 = 0 could be noticeable [67]
Finally as mentioned in Sec II C the inclusion of the dipole form factor in the magnetic
coupling gM is expected to be important to generate the correct Q2 behaviour of the polar-
izabilities Comparing our predictions for the forward spin polarizability with and without
inclusion of the form factor see Fig 3 upper panel confirms this Without the dipole
our results for the proton and neutron are closer to the ones from Ref [31] where the form
factor is not included For the neutron our prediction without the dipole is able to describe
14
the experimental points at very low Q2 deviating from the data with increasing Q2 The
π∆-loop contribution does not modify the Q2 behavior of γ0(Q2) and only differs from
Ref [31] by a small global shift Note also the relatively large effect of gC which generates
a sign change for virtualities above sim 02 GeV2 see Fig 3 upper panel
B δLT (Q2) mdash longitudinal-transverse polarizability
The longitudinal-transverse spin polarizability
δLT (Q2) =1
2π2
int infinν0
dν
radic1 +
Q2
ν2
σLT (νQ2)
Qν2(22)
=16αM2
N
Q6
int x0
0
dx x2[g1(xQ2) + g2(xQ2)
]
contains information about the spin structure of the nucleon and is another important
input in the determination of the (muonic-)hydrogen hyperfine splitting [16 62] It is also
relevant in studies of higher-twist corrections to the structure function g2(xQ2) given by
the moment d2(Q2) [51] see Section III E The peculiarity of the response encoded in this
polarizability is that it involves a spin flip of the nucleon and a polarization change of the
photon see App B and Fig 11
It is expected that the Delta isobar gives only a small contribution to δLT (Q2) what makes
this polarizability a potentially clean test case for chiral calculations Consequently there are
relatively many different theoretical calculations of δLT (Q2) coming from different versions
of χPT with baryons (HB IR and covariant) Ref [50] found a systematic deviation of the
HB result for δLTn(Q2) from the MAID model prediction This disagreement was identified
by the authors of Ref [68] as a puzzle involving the neutron δLT polarizabilitymdashthe δLT
puzzle The IR calculation in Ref [58] also showed a deviation from the data and predicted
a rapid rise of δLT (Q2) with growing Q2 The problem is solved by keeping the relativistic
structure of the theory as the BχPT+∆ result of Ref [31] showed
As expected already the leading πN loops provide a reasonable agreement with the
experimental data cf Fig 2 lower panel Since the ∆-exchange contribution to δLT (Q2)
is small the effect of the gM form factor is negligible in this polarizability as is that of the gC
coupling cf Fig 3 lower panel In fact we predict both the ∆-exchange and the π∆-loop
contributions to be small and negative This is in agreement with the MAID model which
predicts a small and negative contribution of the P33 wave to δLT (Q2) However in the
calculation of Ref [31] which is different from the one presented here only in the way the
∆(1232) is included the contribution of this resonance to δLTp(Q2) is sizeable and positive
15
The authors of that work attributed this large contribution to diagrams where the photons
couple directly to the Delta inside a loop As mentioned in Sec III F the effect of such loop
diagrams does not change the Q2 behaviour of the polarizabilities On the other hand it
can produce a substantial shift of the δLT (Q2) as a whole A higher-order calculation should
resolve the discrepancy between the two covariant approaches however it will partially lose
the predictive power since the LECs appearing at higher orders will have to be fitted to
experimental data
The πN -loop ∆-exchange and π∆-loop contributions to the NLO BχPT prediction of
the static longitudinal-transverse polarizability are in units of 10minus4 fm4
δLTp = 132(15) asymp 150minus 016minus 002 (23a)
δLTn = 218(23) asymp 235minus 016minus 002 (23b)
while the slopes are in units of 10minus4 fm6
dδLTp(Q2)
dQ2
∣∣∣∣Q2=0
= minus085(8) asymp minus080minus 004minus 001 (24a)
dδLTn(Q2)
dQ2
∣∣∣∣Q2=0
= minus124(12) asymp minus119minus 004minus 001 (24b)
C IA(Q2) mdash a generalized GDH integral
The helicity-difference cross section σTT exhibits a faster fall-off in ν than its spin-averaged
counterpart σT This is due to a cancellation between the leading (constant) terms of σ12
and σ32 at large ν3 The resulting 1ν fall-off of the helicity-difference cross section allows
one to write an unsubtracted dispersion relation for the VVCS amplitude gnonpoleTT (νQ2) cf
Eq (10a) This is the origin of the GDH sum rule [5 6]
minus α
2M2N
κ2 =1
2π2
int infinν0
dνσTT (ν)
ν (25)
which establishes a relation to the anomalous magnetic moment κ It is experimentally
verified for the nucleon by MAMI (Mainz) and ELSA (Bonn) [71 72]
There are two extensions of the GDH sum rule to finite Q2 the generalized GDH integrals
3 Notice that a constant term in σTT at ν rarrinfin is forbidden by crossing symmetry
16
000 005 010 015 020 025 030
-30
-25
-20
-15
-10
-05
00
Q2 GeV2D
I Aproton
000 005 010 015 020 025 030
-25
-20
-15
-10
-05
00
Q2 GeV2D
I A
neutron
000 005 010 015 020 025 030
-05
00
05
Q2 GeV2D
I 1
proton
000 005 010 015 020 025 030
-10
-08
-06
-04
-02
00
02
Q2 GeV2D
I 1
neutron
FIG 4 Upper panel The generalized GDH integral IA(Q2) for the proton (left) and neutron
(right) as function of Q2 The result of this work the NLO BχPT prediction is shown by the blue
solid line and the blue band The red line represents the LO BχPT result The purple short-dashed
line is the O(p4) HB result from Ref [50 51] The gray band is the BχPT+∆ result from Ref [31]
The black dotted line is the MAID model prediction [69] Experimental extractions for the proton
Ref [60] (orange triangle uncertainties added in quadrature) and neutron Refs [21][27] where
magenta dotsorange diagonal crosses correspond to data and red squareslilac crosses correspond
to data plus extrapolation to unmeasured energy regions The green stars at the real-photon point
are derived from the anomalous magnetic moments κp asymp 1793 and κn asymp minus1913 [70] Lower
panel The generalized GDH integral I1(Q2) for the proton (left) and neutron (right) as function
of Q2 The purple short-dashed line is the HB result from Ref [51] Experimental extractions for
the proton Ref [18] (blue dots) and Ref [60] (orange triangle uncertainties added in quadrature)
and neutron Ref [61][27] (uncertainties added in quadrature) where green dotsorange diagonal
crosses correspond to data and gray squareslilac crosses correspond to data plus extrapolation to
unmeasured energy regions
17
IA(Q2) and I1(Q2) The latter will be discussed in Sec III D The former is defined as4
minus α
2M2N
IA(Q2) = minus 1
8π2
int infinν0
dν
radic1 +
Q2
ν2
σTT (νQ2)
ν(26)
=α
Q2
int x0
0
dx[4M2
Nx2
Q2g2(xQ2)minus g1(xQ2)
]
Due to its energy weighting the integral in Eq (26) converges slower than the one in the
generalized forward spin polarizability sum rule (19) Therefore knowledge of the cross
section at higher energies is required and the evaluation of the generalized GDH integral
IA(Q2) is not as simple as the evaluation of γ0(Q2)
The generalized GDH integral IA(Q2) is directly related to the non-pole amplitude
gnonpoleTT (νQ2) which differs from non-Born amplitude gTT (νQ2) by a term involving the
elastic Pauli form factor
gnonpoleTT (νQ2) = gTT (νQ2)minus 2παν
M2N
F 22 (Q2) (27)
cf Eqs (2a) and (8a) Consequently IA(Q2) is not a pure polarizability but also contains
an elastic contribution The ldquonon-polarizabilityrdquo or the Born part of IA(Q2) is given by
IBornA (Q2) = IA(Q2)minus∆IA(Q2) = minus1
4F 2
2 (Q2) (28)
where we refer to the polarizability part as ∆IA(Q2) The same is true for the generalized
GDH integral I1(Q2) which is directly related to Snonpole1 (νQ2)
IBorn1 (Q2) = I1(Q2)minus∆I1(Q2) = minus1
4F 2
2 (Q2) (29)
In the following we will add the Born parts to our LO and NLO BχPT predictions for
the polarizabilities ∆IA(Q2) and ∆I1(Q2) employing an empirical parametrization for the
elastic Pauli form factor [73] This allows us to compare to the experimental results for
IA(Q2) and I1(Q2) cf Fig 4 Note that the blue error bands only describe the uncertainties
of our BχPT predictions of the polarizabilities while the elastic contributions are considered
to be exact as explained in Sec II C The uncertainties of the polarizability predictions are
therefore better reflected in Fig 5 where we show the contributions of the different orders
to the BχPT predictions of ∆IA(Q2) and ∆I1(Q2) as well as the total results with error
bands
4 Note that IA(Q2) is sometimes called ITT (Q2)
18
000 005 010 015 020 025 030
-08
-06
-04
-02
00
02
Q2 GeV2D
DI A
proton
000 005 010 015 020 025 030
-10
-08
-06
-04
-02
00
02
Q2 GeV2D
DI A
neutron
000 005 010 015 020 025 030-005
000
005
010
015
020
025
Q2 GeV2D
DI 1
proton
000 005 010 015 020 025 030-015
-010
-005
000
005
010
015
Q2 GeV2D
DI 1
neutron
FIG 5 Contributions of the different orders to the chiral predictions of ∆IA(Q2) upper paneland ∆I1(Q2) lower panel for the proton (left) and neutron (right) Red solid line πN -loop
contribution green dot-dashed line ∆-exchange contribution orange dotted line π∆-loop contri-
bution blue solid line and blue band total result purple dot-dot-dashed line total result without
gC contribution black short-dashed line total result without gM dipole
The E97-110 experiment at Jefferson Lab has recently published their data for IAn(Q2) in
the region of 0035 GeV2 lt Q2 lt 024 GeV2 [27] In addition there are results for IAn(Q2)
from the earlier E94-010 experiment [21] and for IAp(Q2) from the E08-027 experiment [60]
The O(p4) HB calculation gives a large negative effect [51] which does not describe the data
The BχPT+∆ result from Ref [31] which mainly differs from our work by the absence of
the dipole form factor in gM looks similar to this HB result and only describes the data
points at lowest Q2 Our NLO prediction however follows closely the Q2 evolution of the
data In Fig 5 upper panel we show the polarizability ∆IA(Q2) whose Q2 evolution is
clearly dominated by the ∆ exchange Similar to the case of γ0p(Q2) inclusion of the dipole
in gM and the Coulomb coupling gC is very important in order to describe the experimental
data The LO prediction on the other hand slightly overestimates the data cf Fig 4
upper panel
19
At the real-photon point IA(0) = minusκ2
4and ∆IA(0) = 0 Therefore we give only the
slope of the polarizability ∆IA(Q2) [showing also the separate contributions from πN loops
∆ exchange and π∆ loops] in units of GeVminus2
d∆IAp(Q2)
dQ2
∣∣∣∣Q2=0
= minus858(343) asymp 238minus 1121 + 025 (30a)
d∆IAn(Q2)
dQ2
∣∣∣∣Q2=0
= minus955(343) asymp 141minus 1121 + 025 (30b)
Including the empirical Pauli form factor [73] we find in units of GeVminus2
dIAp(Q2)
dQ2
∣∣∣∣Q2=0
= minus318dIAn(Q2)
dQ2
∣∣∣∣Q2=0
= minus300 (31)
D Γ1(Q2) and I1(Q2) mdash the first moment of the structure function g1(xQ2)
The second variant for a generalization of the GDH sum rule to finite Q2 is defined as
minus α
2M2N
I1(Q2) = minus 1
8π2
int infinν0
dν1radic
ν2 +Q2
[σTT (νQ2) +
Q
νσLT (νQ2)
](32)
= minus α
Q2
int x0
0
dx g1(xQ2)
where I1(0) = minusκ2
4 This generalized GDH integral directly stems from the amplitude
Snonpole1 (νQ2) with the LEX from Eq (12a) It is given by the first moment of the struc-
ture function g1(xQ2) Γ1(Q2) =int x0
0dx g1(xQ2) as follows I1(Q2) =
2M2N
Q2 Γ1(Q2) The
isovector combination
Γ1(pminusn)(Q2) =
int x0
0
dx[g1p(xQ
2)minus g1n(xQ2)] (33)
is related to the axial coupling of the nucleon through the Bjorken sum rule [74 75]
limQ2rarrinfin
Γ1(pminusn)(Q2) =
gA6 (34)
As explained in Eq (28) the moment I1(Q2) splits into a polarizability part ∆I1(Q2) and
a Born part IBorn1 (Q2) Figure 4 lower panel shows the Q2 dependence of I1(Q2) which in
contrast to IA(Q2) shown in Figure 4 upper panel is clearly dominated by its Born part
and the elastic Pauli form factor The πN -loop ∆-exchange and π∆-loop contributions to
the polarizability ∆I1(Q2) are shown in Fig 5 lower panel Comparing to Fig 5 upper
20
000 005 010 015 020 025 030
-006
-004
-002
000
002
004
Q2 GeV2D
G1
proton
000 005 010 015 020 025 030
-006
-005
-004
-003
-002
-001
000
001
Q2 GeV2D
G1
neutron
FIG 6 First moment of the structure function g1(xQ2) for the proton (left) and neutron (right)
as function of Q2 The legend is the same as in Fig 4
panel one sees that ∆I1(Q2) is less sensitive to gC and the dipole form factor in gM than
∆IA(Q2)
For the proton our NLO BχPT prediction gives a very good description of the ex-
perimental data [18 60] and is in reasonable agreement with the MAID prediction [69]
For the neutron one observes good agreement with the empirical evaluations including ex-
trapolations to unmeasured energy regions starting from Q2 gt 01 GeV2 [27 61] In the
region of Q2 lt 005 GeV2 one observes an interesting tension between the recent E97-
110 experiment [27] and the data from CLAS [61] While the newest measurement finds
I1n(0035 GeV2) lt κ2n4 thus suggesting a negative slope at low Q the older measurement
found a rather large value for I1n(00496 GeV2) A similar but milder behaviour is seen in
the E97-110 [27] and E94-010 [21] data for IAn The MAID predictions do not agree with
the CODATA recommended values for the anomalous magnetic moments of the proton and
neutron [70] which in our work are imposed by using empirical parametrizations for the
elastic Pauli form factors [73] The slope of the HB result from Ref [51] is too large and
therefore only reproduces the data at very low Q2
Figure 6 shows the moment Γ1(Q2) for the proton and neutron while Fig 7 shows the
isovector combination Γ1 pminusn(Q2) The LO and NLO BχPT predictions are identical be-
cause our calculation produces the same Delta contributions for the proton and the neutron
For the isovector combination the MAID model only agrees with the data at very low
Q2 lt 010 GeV2 The same is true for the IR result [58 76] while all other chiral results
describe the data NLO BχPT (this work) BχPT+∆ [31] and HBχPT [51]
At the real-photon point I1(0) = minusκ2
4and ∆I1(0) = 0 Therefore we give only the
slope of the polarizability ∆I1(Q2) [showing also the separate contributions from πN loops
21
000 005 010 015 020 025 030-002
000
002
004
006
008
010
Q2 GeV2D
G1Hp-
nL
FIG 7 Isovector combination of Γ1(Q2) as function of Q2 The legend is the same as in Fig 6
The pink curve is the IR result from Ref [58 76] The experimental points are from Ref [77]
(brown dots) and Ref [23] (orange squares)
∆ exchange and π∆ loops] in units of GeVminus2
d∆I1p(Q2)
dQ2
∣∣∣∣Q2=0
= 039(4) asymp 034minus 053 + 058 (35a)
d∆I1n(Q2)
dQ2
∣∣∣∣Q2=0
= minus101(10) asymp minus107minus 053 + 058 (35b)
Including the empirical Pauli form factor [73] we find in units of GeVminus2
dI1p(Q2)
dQ2
∣∣∣∣Q2=0
= 580dI1n(Q2)
dQ2
∣∣∣∣Q2=0
= 553 (36)
E d2(Q2) mdash a measure of color polarizability
Another interesting moment to consider is d2(Q2) which is related to the twist-3 part of
the spin structure function g2(xQ2) [79 80]
d2(Q2) equiv 3
int 1
0
dx x2[g2(xQ2)minus gWW2 (xQ2)] (37)
where gWW2 (xQ2) is the twist-2 part of g2(xQ2) Using the Wandzura-Wilczek relation [81]
one can relate d2(Q2) to moments of the spin structure functions g1(xQ2) and g2(xQ2)
d2(Q2) =
int 1
0
dx x2 [3g2(xQ2) + 2g1(xQ2)] (38)
22
000 005 010 015 020 025 030
0000
0001
0002
0003
0004
0005
0006
0007
Q2 GeV2D
d2
proton
000 005 010 015 020 025 030
0000
0002
0004
0006
0008
0010
0012
Q2 GeV2D
d2
neutron
000 005 010 015 020 025 030
-1
0
1
2
Q2 GeV2D
Γ 010
-4
fm6
D
proton
000 005 010 015 020 025 030-2
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
6D
neutron
FIG 8 Upper panel The inelastic moment d2(Q2) for the proton (left) and neutron (right) as
function of Q2 The result of this work the NLO BχPT prediction is shown by the blue solid line
and the blue band The red line represents the LO BχPT result The purple short-dashed line is
the O(p4) HB result from Ref [50 51] The black dotted line is the MAID model prediction [69]
The experimental points for the neutron (cyan dots) are from Ref [22] Lower panel Fifth-order
generalized forward spin polarizability γ0(Q2) for the proton (left) and neutron (right) as function
of Q2 The experimental points for the proton are from Ref [59] (purple square) and Ref [78]
(orange dot)
This relation however only holds for asymptotically large Q2 It is also in the high-Q2
region where d2(Q2) is a measure of color polarizability [82 83] through its relation to the
gluon field strength tensor [80] We refer to Ref [84] for a recent review on the spin structure
of the nucleon including a discussion of sum rules for deep inelastic scattering and color
polarizabilities
What we consider in the following is the inelastic part of d2(Q2) defined as the moment
of g1(xQ2) and g2(xQ2) spin structure functions cf Eq (38)
d2(Q2) =
int x0
0
dx x2 [3g2(xQ2) + 2g1(xQ2)] (39)
23
This moment provides another testing ground for our BχPT predictions through comparison
with experiments on the neutron [22] Going towards the low-Q2 region the interpretation
of d2(Q2) in terms of color polarizabilities will fade out The above definition however
implies it is related to other VVCS polarizabilities
d2(Q2) =Q4
8M4N
[M2
NQ2
αδLT (Q2) + I1(Q2)minus IA(Q2)
] (40)
Note that d2(Q2) and its first two derivatives with respect to Q2 vanish at Q2 = 0 The
considerations in Eqs (28) and (29) have no effect on d2(Q2) since the Born contribution
from IA(Q2) and I1(Q2) cancel out Therefore d2(Q2) is a pure polarizability
In Fig 8 upper panel we show our NLO BχPT prediction and other results for d2(Q2)
While MAID [69] and BχPT describe the experimental data for the neutron [22] very well
the HB limit [50 51] is showing a fast growth with Q2 This illustrates the importance
of keeping the relativistic result Note also that even though the πN -loop contribution
is dominant both gC and the form factor in gM are essential to obtain a curvature that
reproduces the data cf Fig 9 upper panel For the proton there are to our knowledge
no experimental results to compare with However the agreement between the NLO BχPT
prediction and the MAID prediction at low energies is reasonable
F γ0(Q2) mdash fifth-order generalized forward spin polarizability
It is interesting to compare the generalized fifth-order forward spin polarizability sum
rule
γ0(Q2) =1
2π2
int infinν0
dν
radic1 +
Q2
ν2
σTT (νQ2)
ν5(41)
=64αM4
N
Q10
int x0
0
dx x4
[g1(xQ2)minus 4M2
Nx2
Q2g2(xQ2)
]
to the sum rule integrals for IA(Q2) and γ0(Q2) since they differ merely by their energy
weighting of σTT (νQ2) and a constant prefactor cf Eqs (19) (26) and (41) From IA(Q2)
to γ0(Q2) to γ0(Q2) the energy suppression is increasing by a factor of νminus2 respectively
Therefore the description of γ0(Q2) should be easiest in a low-energy effective-field theory
such as χPT whereas γ0(Q2) and IA(Q2) receive larger contributions from higher energies
In Fig 8 lower panel we show our LO and NLO BχPT predictions for γ0(Q2) One
can see that the πN -loop contribution is positive (in accordance to what we see for the cross
section σTT see Fig 10) The Delta shifts it substantially especially in the low Q2 region
24
000 005 010 015 020 025 030
-0004
-0002
0000
0002
0004
0006
0008
0010
Q2GeV
2D
d2
proton
000 005 010 015 020 025 030
-0005
0000
0005
0010
0015
Q2GeV
2D
d2
neutron
000 005 010 015 020 025 030
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
6D
proton
000 005 010 015 020 025 030
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
6D
neutron
FIG 9 Contributions of the different orders to the chiral predictions of d2(Q2) upper panel and
γ0(Q2) lower panel for the proton (left) and neutron (right) Red solid line πN -loop contribu-
tion green dot-dashed line ∆-exchange contribution orange dotted line π∆-loop contribution
blue long-dashed line total result purple dot-dot-dashed line total result without gC contribution
black short-dashed line total result without gM dipole
bringing it into a better agreement with data In general the BχPT curves start above the
empirical data points at the real-photon point and then decrease asymptotically to zero
above Q2 gt 01 GeV2 On the other hand the MAID prediction reproduces the empirical
data at the real-photon point then decreases to negative values until about Q2 gt 006 GeV2
from where it also starts to asymptotically approach zero Consequently our NLO BχPT
prediction of γ0(Q2) is consistently above the MAID prediction This is very different to
what we saw for IA(Q2) in Fig 4 upper panel where the MAID prediction at the real-
photon point is above the experimental value While the agreement of our predictions with
the empirical data is in general quite good for all moments of σTT (νQ2) one should point
out that both for γ0n(Q2) and γ0p(Q2) we overestimate the data at low Q2 For IA(Q2) such
observation cannot be made because ∆IA(0) = 0 and thus IA(0) is given by the empirical
Pauli form factor only From IA(Q2) γ0(Q2) and γ0(Q2) the latter has the smallest however
25
non-negligible dependence on gC and the dipole in gM cf Fig 9 lower panel
The πN -loop ∆-exchange and π∆-loop contributions to the NLO BχPT prediction of
the static fifth-order forward spin polarizability amount to in units of 10minus4 fm6
γ0p = 112(30) asymp 208minus 096minus 001 (42a)
γ0n = 195(30) asymp 292minus 096minus 001 (42b)
while the slope is composed as follows in units of 10minus4 fm8
dγ0p(Q2)
dQ2
∣∣∣∣Q2=0
= minus084(10) asymp minus100 + 016 + 000 (43a)
dγ0n(Q2)
dQ2
∣∣∣∣Q2=0
= minus142(15) asymp minus158 + 016 + 000 (43b)
Note that the HB prediction of the static polarizability γ0p (423 at O(p3) and 365 at O(ε3)
[78 85]) is almost one order of magnitude larger than the empirical value and therefore not
shown in Fig 8
G Summary
Our results are summarized in Table II where we give the contributions of the different
orders to the chiral predictions of the polarizabilities and their slopes at the real-photon
point A quantitative comparison of our predictions for the static spin polarizabilities to
the work of Bernard et al [31] and different empirical evaluations is shown in Table III We
can see that the inclusion of the Delta turns out to be very important for all moments of
the helicity-difference cross section To describe the Q2 behavior of the polarizabilities the
magnetic coupling of the N rarr ∆ transition should be modified by a dipole form factor as
has been observed previously in the description of electroproduction data [33] This dipole
form factor effectively takes account of vector-meson exchanges The Coulomb-quadrupole
N rarr ∆ transition despite its subleading order is important in the description of some
moments of spin structure functions This is contrary to what we saw for the moments of
unpolarized structure functions [30] where the Coulomb coupling had a negligible effect
The π∆ loops are mainly relevant for the generalized GDH integrals
26
TABLE II The NLO BχPT predictions for the forward VVCS polarizabilities and their slopes at
Q2 = 0 The contributions of the πN loops the ∆ exchange and the π∆ loops are shown together
with the combined total result Note that IA(0) = I1(0) = d2(0) = 0 and (d2)prime = 0
πN loops ∆ exchange π∆ loops Total
γ0 p 201 minus284 minus010minus093(92)
(10minus4 fm4) n 298 003(92)
δLT p 150 minus016 minus002132(15)
(10minus4 fm4) n 235 218(23)
γ0 p 208 minus096 minus001112(30)
(10minus4 fm6) n 292 195(30)
(γ0)prime p minus033011 001
minus022(4)
(10minus4 fm6) n minus073 minus061(7)
(δLT )prime p minus080 minus004 minus001minus085(8)
(10minus4 fm6) n minus119 minus124(12)
(γ0)prime p minus100016 000
minus084(10)
(10minus4 fm8) n minus158 minus142(15)
(∆IA)prime p 238 minus1121 025minus858(343)
(GeVminus2) n 141 minus955(343)
(∆I1)prime p 034 minus053 058039(4)
(GeVminus2) n minus107 minus101(10)
TABLE III Our NLO BχPT predictions for the spin polarizabilities at Q2 = 0 compared with
the BχPT+∆ predictions from Bernard et al [31] and the available empirical information Where
the reference is not given the empirical number is provided by the MAID analysis [55 69] with
unspecified uncertainty
Proton Neutron
This work BχPT+∆ Empirical This work BχPT+∆ Empirical
γ0 minus093(92) minus174(40) minus100(8)(12) [19] 003(92) minus077(40) minus0005
(10minus4 fm4) minus090(8)(11) [78] [MAID]
minus0929(105) [59]
δLT 132(15) 240(1) 134 218(23) 238(3) 203
(10minus4 fm4) [MAID] [MAID]
γ0 112(30) 060(7)(7) [78] 195(30) 123
(10minus4 fm6) 0484(82) [59] [MAID]
IV CONCLUSIONS
We have presented a complete NLO calculation of the polarized non-Born VVCS ampli-
tudes in covariant BχPT with pion nucleon and ∆(1232) fields The dispersion relations
between the VVCS amplitudes and the tree-level photoabsorption cross sections served as a
27
cross-check of these calculations
The obtained moments of the proton and neutron spin structure functions related to gen-
eralized polarizabilities and GDH-type integrals agree well with the available experimental
data The description of their Q2 evolution is improved compared to the previous χPT pre-
dictions In particular the NLO BχPT predictions obtained here give a better description
of the empirical data (eg from the Jefferson Laboratory ldquoSpin Physics Programrdquo) than
the HB [50 51] and IR [58] calculations
The demonstrated predictive power of the χPT framework amplitudes makes it well
suited for extending the χPT evaluation of the TPE effect in the hyperfine structure of
(muonic-)hydrogen [15ndash17] to next-to-leading order
ACKNOWLEDGEMENTS
We thank Lothar Tiator and Marc Vanderhaeghen for helpful discussions This work
is supported by the Deutsche Forschungsgemeinschaft (DFG) through the Collaborative
Research Center [The Low-Energy Frontier of the Standard Model (SFB 1044)] JMA ac-
knowledges support from the Community of Madrid through the ldquoPrograma de atraccion de
talento investigador 2017 (Modalidad 1)rdquo and the Spanish MECD grants FPA2016-77313-P
FH gratefully acknowledges financial support from the Swiss National Science Foundation
Appendix A Tensor decompositions of the VVCS amplitudes
In this appendix we review the decomposition of the forward VVCS process into tensor struc-
tures and scalar amplitudes In particular we consider the connection between the covariant and
the semi-relativistic decomposition in the lab frame that is defined in terms of the conventional
transverse longitudinal transverse-transverse and transverse-longitudinal amplitudes
As explained in Sec II A the process of forward VVCS off the nucleon can be described in
terms of four explicitly covariant amplitudes S1 2 and T1 2 [3]
T (νQ2) =
(minusgmicroν +
qmicroqν
q2
)T1(νQ2) +
1
M2N
(pmicro minus p middot q
q2qmicro)(
pν minus p middot qq2
qν)T2(νQ2) (A1)
minus 1
MNγmicroναqα S1(νQ2)minus 1
M2N
(γmicroνq2 + qmicroγναqα minus qνγmicroαqα
)S2(νQ2)
εprimelowastmicro εν
where εmicro (εprimelowastmicro ) are the incoming (outgoing) photon polarization vectors ν is the photon lab-frame
energy and Q2 is the photon virtuality Alternatively the decomposition in the laboratory frame
28
(which in the forward case coincides with the Breit frame) is parametrized in terms of the nucleon
Pauli matrices ~σ and the four scalar functions fL fT gTT and gLT
T (νQ2) = ε0 εprimelowast0 fL(νQ2) + (~ε primelowast middot ~ε ) fT (νQ2) + i~σ middot (~ε primelowast times ~ε ) gTT (νQ2) (A2)
minusi~σ middot [(ε0~εprimelowast minus ~ε ε primelowast0 )times q] gLT (νQ2)
Here ~q and q = ~q|~q | are the photon three-momentum in the lab system and its unit vector The
modified polarization vector components are given by
ε0 =
[ε0 minus
ν
|~q | (~ε middot q )
] |~q |Q ~ε = ~εminus q (~ε middot q ) (A3)
where ε = (ε0~ε ) is the usual incoming photon polarization vector and εprimelowast the outgoing polarization
vector The LEX of the lab frame amplitudes [Eq (10)] can serve in particular as the definition
of the generalized polarizabilities The lab frame amplitudes are also conveniently used for the
definition of the response functions see the example of the scalar amplitude gLT (νQ2) and the
corresponding response function σLT (νQ2) below in App B
Appendix B Photoabsorption cross sections
In the forward kinematics the spin-dependent VVCS amplitudes and the spin polarizabilities
can be described in terms of the polarized structure functions g1(xQ2) and g2(xQ2) or equiv-
alently the helicity-difference cross section σTT (νQ2) and the longitudinal-transverse response
function σLT (νQ2) with the help of dispersion relations (5) and the optical theorem (3) In this
way the photoabsorption cross sections measured in electroproduction processes form the basis
for most empirical evaluations shown throughout Sec III In the following we present the BχPT
predictions for the tree-level cross sections of πN - π∆- and ∆-production through photoabsorption
on the nucleon cf Figs 8 9 and 10 in Ref [30] In Secs B 1 and B 2 we will discuss the leading
πN -production channel and the ∆-production channel respectively We used these cross sections
to verify the polarizability predictions obtained otherwise from the calculated non-Born VVCS
amplitudes Due to the bad high-energy behavior of the π∆-production cross sections in BχPT
cf Fig 10 the dispersion relations in Eq (5) require further subtractions for a reconstruction of
the π∆-loop contribution to the spin-dependent VVCS amplitudes Therefore not all polarizabil-
ities could be verified but only those appearing as higher-order terms in the LEX of the VVCS
amplitudes such as γ0 [16]
29
proton neutron
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Neutron
02 04 06 08 100
50
100
150
200
250
[GeV]
T [b]
02 04 06 08 10
-50
0
50
100
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Neutron
02 04 06 08 100
50
100
150
200
250
[GeV]
T [b]
02 04 06 08 10
-50
0
50
100
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02040608100
50
100
150
200
250
300
[GeV]
T[b]
0204060810
-50
0
50
[GeV]
TT[b]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
0204060810
00
02
04
06
[GeV]
LTQ
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
0204060810
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
FIG 10 Photoabsorption cross sections for πN (red) and π∆ production (orange) with Q2 = 0
(solid) and Q2 = 01 GeV2 (dashed for πN and dotted for π∆ channel)
1 πN-production channel
In order to extract the response function σLT (νQ2) we have developed a method similar to
the one used to calculate σTT (νQ2) see for example Ref [86] For σLT (νQ2) however the
calculation is more complicated because one has to take into account that the associated Compton
process involves a spin-flip of the nucleon as illustrated in Fig 11 When calculating the cross
section the product of the incoming nucleon spinors has to reflect this flip
The forward VVCS amplitude related to σLT (νQ2) mdash and δLT (Q2) mdash is gLT (νQ2) It can
be extracted from Eq (A2) if one takes the modified polarization vector components in Eq (A3)
with ε = εL and εprimelowast = εlowastplusmn as input where εL = 1Q(|~q | 0 0 q0) and εplusmn = ∓ 1radic
2(0 1plusmni 0) are the
standard longitudinal and transverse polarization vectors respectively For εL and εlowastplusmn only the
choice of helicities hprime = plusmn12 and h = ∓12 gives a non-zero contribution and one obtains
χdaggerhprime T (νQ2)χh = χdaggerhprimeminusi~σ middot [(ε0~εprimelowast minus ~ε ε primelowast0 )times q] gLT (νQ2)χh =
radic2 gLT (νQ2) (B1)
where χh and χdaggerhprime are two-component Pauli spinors with opposite helicities or here spins
Let us now consider the related photoabsorption process and in particular the tree-level γlowastN rarr
30
s = 12
sprime = minus12
λ = 0 λprime = 1
rprime rprime
s=12sprime=minus1
2
λ=0λprime=1
rprimerprime
s=12
sprime=minus12
λ=0λprime=1
rprimerprime hacute=frac12h=-frac12
λacute=1λ=0
FIG 11 Relation between the forward Compton process and the photoabsorption process given
by the optical theorem In particular we show the longitudinal-transverse contribution The
double-line arrows represent the spin of the external particles while the dot represents the scalar
(longitudinal) polarization of the incoming photon Inside the blob the intermediate states are
represented eg nucleons with spins rprime (which are averaged in the calculation of the cross section)
and pions
πN channel see diagrams in Fig 8 of Ref [30] We define the πN -production amplitude as
T = uhB (PB)sumi
Ai(s t)Γi uhA(PA) (B2)
with the Dirac structures
Γ1 = γ5 (B3a)
Γ2 =1
2
[qA ε
]γ5 (B3b)
where uhA(PA) and udaggerhB (PB) are the Dirac spinors and PA and PB are the four-momenta of
the incoming and outgoing nucleons respectively When calculating the photoabsorption cross
section related to the VVCS amplitude in Eq (B1) the nucleon spin flip should be implemented
by uhprime(PA) in T dagger and uh(PA) in T together with the appropriate transverse and longitudinal
photon polarization vectors εlowastplusmn and εL
However if one wants to use the properties of the Dirac matrices it is more useful to construct
an operator to produce this spin flip in the external nucleons of Fig 11 This is accomplished by
introducing the projector ΓLT equiv 12radic
2(γ1 + iγ2)γ5 which also takes into account the extra factor
radic2 in Eq (B1) We checked that with this projector one correctly extracts δLT by comparing the
HB limit of our result to the HB result of Ref [50] where the authors calculate this polarizability
from the Compton amplitude directly With all those ingredients the longitudinal-transverse cross
section is calculated in the following way
σLT (νQ2) =1
64π2 s
|~pf |cm
|~pi|cm
int 1
minus1dcos θ
sumij
AiAdaggerjXij (B4)
31
with
Xij = Tr[(PB +MN )Γi(PA +MN )ΓLTγ0Γdaggerjγ
0] (B5)
where θ is the scattering angle in the center-of-mass (cm) frame and |~pi|cm (|~pf |cm) is the three-
momentum of an incoming (outgoing) particle in the cm frame An explicit calculation of the
matrix Xij leads to
X = MNQ
0 2(PB minus PA) middot εLminusradic
2|~qf |cm sin θ (sminus u)
(B6)
where |~qi|cm (|~qf |cm) is the relative three-momentum of the incoming (outgoing) particles in the
cm frame Here s t and u are the usual Mandelstam variables For the different γlowastN rarr πN
channels we obtain the following amplitudes Ai where we introduce qA as the four-momentum of
the incoming photon and qB as the four-momentum of the outgoing pion
bull γlowastprarr π0p
A1 =e gAMN
fπ
[2PA middot ε+ qA middot ε
sminusM2N
+2PB middot εminus qA middot ε
uminusM2N
] (B7a)
A2 =e gAMN
fπ
[1
sminusM2N
+1
uminusM2N
] (B7b)
bull γlowastprarr π+n
A1 =
radic2 e gAMN
fπ
[2PA middot ε+ qA middot ε
sminusM2N
+2(PA minus PB) middot ε+ qA middot ε
tminusm2π
] (B8a)
A2 =
radic2 e gAMN
fπ(sminusM2N )
(B8b)
bull γlowastnrarr π0n
A1 = 0 (B9a)
A2 = 0 (B9b)
bull γlowastnrarr πminusp
A1 =
radic2 e gAMN
fπ
[2PB middot εminus qA middot ε
uminusM2N
minus 2(PA minus PB) middot ε+ qA middot εtminusm2
π
] (B10a)
A2 =
radic2 e gAMN
fπ(uminusM2N )
(B10b)
The analytical expressions shown above were checked with the amplitudes given in Ref [87] An-
alytical expressions for the tree-level γlowastN rarr πN channel of the σLT (νQ2) and σTT (νQ2) cross
32
sections are given below (proton channels π+n and π0p neutron channel πminusp) We checked that
they reproduce the known results in the real-photon limit [45 86] To shorten the final expressions
for the cross sections which are considerably longer for finite Q2 than in the real-photon limit we
define the following dimensionless kinematic variables
αγ = (ENi )cmradics =
s+M2N +Q2
2s (B11)
απ = (ENf )cmradics =
s+M2N minusm2
π
2s (B12)
βγ = Eγcmradics =
sminusM2N minusQ2
2s (B13)
βπ = Eπcmradics =
sminusM2N +m2
π
2s (B14)
λγ = |~qi|cmradics =
radic(sminusM2
N minusQ2)2 + 4sQ2
2s (B15)
λπ = |~qf |cmradics =
radic(sminusM2
N +m2π)2 minus 4sm2
π
2s (B16)
Here (ENi )cm and (ENf )cm are the energies of the incoming and outgoing nucleon Eγcm is the energy
of the incoming photon Eπcm is the energy of the outgoing pion all in the cm frame
σ(π+n)TT = minus e2g2
AM2N
64πf2πs
3(sminusM2N )2λ4
γ
4sλπλγ
[(M2
N minus s)(M2N minusQ2 minus s)(Q2 + 2sβγβπ)
+ 2s(m2πQ
2 minus (M2N minus s)
(M2N + s(minus1 + 2βγβπ)
))λ2γ
]+ (M2
N minus s)(M2N minusQ2 minus s)
times (Q2 + 2sβγβπ minus 2sλπλγ)(Q2 + 2s(βγβπ + λπλγ)
)times log
(Q2 + 2sβγβπ minus 2sλγλπQ2 + 2sβγβπ + 2sλγλπ
) (B17)
33
σ(π0p)TT =
e2g2AM
2Nλπ
64πf2πs(sminusM2
N )2λγ
1
s2λπλ3γ
[(M2
N minus s)(M2N minusQ2 minus s)
(3M2
N + s(2βγβπ minus 3))(M2N
+ s(2βγβπ minus 1))
+ 2s
((2m2
πQ2 + (M2
N minus s)2)(M2
N minus s)minus 2(minusm2
πQ2 + (sminusM2
N )2)
times sβγβπ + 4s2(sminusM2N )β2
γβ2π
)λ2γ
]arctan
(2sλπλγ
M2N + s(2βγβπ minus 1)
)+ 2
[minus 2m2
πQ2
+ (M2N minus s)
(2(sminusM2
N ) + 4sβγβπ +(sminusM2
N +Q2)(3M2N minus 3s+ 2sβγβπ)
sλ2γ
+m2πQ
2(sminusM2N )
(M2N minus s+ 2sβγβπ)(M2
N + s(minus1 + 2βγβπ minus 2λπλγ))
+m2πQ
2(sminusM2N )
(M2N minus s+ 2sβγβπ)(M2
N + s(minus1 + 2βγβπ + 2λπλγ))
)
+1
λπλγ
(βγβπ
(minusm2
πQ2 + (M2
N minus s)(M2N minus s+ 2sβγβπ)
)minus (M2
N minus s)(M2N minusQ2 minus s)λ2
π
)log
(M2N + s(minus1 + 2βγβπ + 2λπλγ)
M2N + s(minus1 + 2βγβπ minus 2λπλγ)
)] (B18)
σ(πminusp)TT =
e2g2AM
2N
64πf2πs
3λ4γ(M2
N + s(2βπβγ minus 1))2 minus 4λ2πs
2λ2γ)
4sλπλγ
((minusM2
N +Q2 + s)(M2N
+ s(2βπβγ minus 1))
2 minus 2sλ2γ
(2λ2
πs(minusM2N +Q2 + s) +m2
πQ2))
+((M2N + s(2βπβγ minus 1)
)2
minus 4λ2πs
2λ2γ
)((M2N + s(2βπβγ minus 1)
)(M2N minus 2Q2 minus s(2βπβγ + 1)
)+ 4λ2
πs2λ2
γ
)times log
(minusM2
N minus 2βπsβγ minus 2λπsλγ + s)minus(M2N + s(2βπβγ minus 2λπλγ minus 1)
)(M2N + s(minus1
+ 2βπβγ + 2λπλγ))[((
M2N + s(2βπβγ minus 1)
)(M2N minus 2Q2 minus s(2βπβγ + 1)
)+ 4λ2
πs2λ2
γ
)times log
(minusM2
N minus 2βπsβγ + 2λπsλγ + s)
+((Q2 + 2βπsβγ)2 minus 4λ2
πs2λ2
γ
)times log
(Q2 + 2βπsβγ minus 2λπsλγQ2 + 2sβπβγ + 2λπsλγ
)]+ 4sλ2
γ
((M2
N + s(2βπβγ minus 1))2 minus 4λ2πs
2λ2γ
)times (M2
N minus s) arctanh
(2λπsλγ
M2N + 2βπsβγ minus s
) (B19)
34
σ(π+n)LT =
e2g2AM
3Nλπ
32πf2πQs
3(sminusM2N )2λ4
γ
2sλγ
[(M2
N minus s)(Q2 + 2sβ2γ)(Q2 + 2sβγβπ)
minus 4s
((M2
N minus s)(Q2 minus 2s(απ minus 1)βγ
)+Q2sβγβπ
)λ2γ + 8s3(minus1 + απ)λ4
γ
]minus M2
N minus sλπ
[(Q2 + 2sβ2
γ)(Q2 + 2sβγβπ)2 + 4s2(2(απ minus 1)βγ(Q2 + 2sβγβπ)
minusQ2λ2π
)λ2γ + 8s3(απ minus 1)2λ4
γ
]arctan
(2sλπλγ
Q2 + 2sβγβπ
) (B20)
σ(π0p)LT =
e2g2AM
3Nλπ
16πf2πQs(sminusM2
N )2λγ
1
minus2s(M2N + s(minus1 + 2βγβπ))2λ2
γ + 8s3λ2πλ
4γ
[minus 3M8
N (Q2
+ 2sβ2γ) + 2M4
Ns2
(minus (Q2 + 2sβ2
γ)(minus3 + 2βγβπ) +(Q2(minus3 + 2βγβπ + 6λ2
π) + 2sβγ
times (minus3 + 12απ + 2βγβπ minus 12απβγβπ + 4βγλ2π))λ2γ minus 4s(α2
π minus 1)λ4π
)+ 2M2
Ns3
[minus (Q2
+ 2sβ2γ)(minus1 + 2βγβπ)
(6 + βγβπ(minus9 + 2βγβπ)
)+
(Q2(3minus 12λ2
π + 4βγβπ(minus1 + 2βγβπ
+ λ2π))minus 2sβγ
(minus 3 + 4απ(3 + 2απ(3 + 2βγβπ(minus3 + βγβπ)) + 4βγ(βπ + (2minus βγβπ)
times λ2π)))λ2γ + 8s
((απ minus 1)(1 + απ minus 2βγβπ) + 2απβγλ
2π
)λ4γ
)+ s4
((Q2 + 2sβ2
γ)
times (1minus 2βγβπ)2(2βγβπ minus 3) + 2(Q2(minus1 + 6λ2
π + 2βγβπ((1minus 2βγβπ)2 minus 2λ2π)) + 2sβγ
times (minus1 + 2βγβπ + 4(minus1 + βγβπ)(minusαπ + 2απβγβπ minus βγλ2π)))λ2γ minus 8
(s(απ minus 1)(απ
+ (1minus 2βγβπ)2) + 2βγ(2sαπ +Q2βγ)λ2π
)λ4γ + 32s(απ minus 1)λ2
πλ6γ
)+ 2M6
Ns
(Q2(6
minus 7βγβπ + λ2γ) + 2sβγ(βγ(6minus 7βγβπ) + (1minus 4απ)λ2
γ)
)]+sminusM2
N
4s2λπλ3γ
[(Q2 + 2sβ2
γ)
times(3M2
N + s(2βγβπ minus 3))(M2N + s(2βγβπ minus 1)
)+ 2s
(minusM2
N (Q2 + 2sβγ minus 8sαπβγ)
+ s(2sβγ(1minus 4απ + 4απβγβπ) +Q2(1minus 2λ2π)))λ2γ + 8s3(απ minus 1)λ4
γ
]
times arctan
(2sλπλγ
M2N + s(2βγβπ minus 1)
) (B21)
35
σ(πminusp)LT =
e2g2AM
3N
64πf2πQs
3λ4γ(M2
N minusQ2 minus s)(M2N + s(2βπβγ + 2λπλγ minus 1))
times
1
M2N + s(2βπβγ minus 2λπλγ minus 1)
[4λπsλγ(M2
N minusQ2 minus s)(
2sλ2γ
(s((1minus 2λ2
π)Q2
+ 2(2απ minus 1)sβγ(2βπβγ minus 1))minusM2N (Q2 + 2(1minus 2απ)sβγ)
)+ (Q2 + 2sβ2
γ)(M2N + s(2βπβγ minus 1)
)2 + 8(απ minus 1)απs
3λ4γ
)]minus(M2N + s(2βπβγ + 2λπλγ minus 1)
)times[
log
(minusM2N minus 2βπsβγ minus 2λπsλγ + s
minusM2N minus 2βπsβγ + 2λπsλγ + s
)(minus (Q2 + 2sβ2
γ)(M2N + s(2βπβγ minus 1)
)(minusM2
N + 2Q2 + 2βπsβγ + s)minus 2sλ2γ
(minusM2
N (Q2 + 2sβγ) +Q4 +Q2s(4απβγ
minus 2βγ minus 2λ2π + 1) + 2s2βγ(4(απ minus 1)βπβγ + 1)
)minus 8(απ minus 1)2s3λ4
γ
)+ 2
(4s2λ2
γ
(2(απ minus 1)βγ(Q2 + 2βπsβγ)minus λ2
πQ2)
+ (Q2 + 2βπsβγ)2(Q2 + 2sβ2γ)
+ 8(απ minus 1)2s3λ4γ
)arctanh
(2λπsλγ
Q2 + 2βπsβγ
)] (B22)
2 ∆-production channel
The tree-level ∆-exchange diagram in Fig 2 of Ref [30] contributes to the non-Born part of
the VVCS amplitudes The contribution of the ∆ exchange to the VVCS amplitudes can be split
into [17]
S∆-exch1 (νQ2) = S∆-pole
1 (νQ2) + S∆-exch1 (νQ2) (B23a)
νS∆-exch2 (νQ2) = νS∆-pole
2 (νQ2) + νS2∆-exch
(νQ2) (B23b)
and similarly for the unpolarized VVCS amplitudes discussed in Ref [30] Here we introduced the
∆-pole contributions S∆-polei and the ∆-non-pole contributions S∆-exch
i The former amplitudes
feature a pole at the ∆(1232)-production threshold and thus are proportional to
1
[sminusM2∆][uminusM2
∆]=
1
4M2N
1
ν2∆ minus ν2
(B24)
36
They can be reconstructed from the dispersion relations in Eq (5) with the tree-level ∆-production
cross sections as input cf Fig 10 in Ref [30]
σTT (νQ2) =π2α
M2NM
2+|~q |
minus g2
MMN (M+ + ν)|~q |2 +g2E(∆minus ν)(Q2 minusMNν)2
MN(B25a)
+g2CQ
4s(∆minus ν)
MNM2∆
minus 4gMgE(Q2 minusMNν)|~q |2 minus 4gMgCQ2|~q |2
+2gEgCQ
2[minusMNM∆ |~q |2 + s(Q2 +∆ν)
]MNM∆
δ(ν minus ν∆)
σLT (νQ2) =Qπ2α
M2NM
2+|~q |
g2E(MNν minusQ2) [M∆(MN + ν)minus s]
MN(B25b)
+g2CQ
2[MNM∆|~q |2 minus s(Q2 +∆ν)
]MNM2
∆
+ gMgEM∆|~q |2 minusgMgC(Q2 minusMNν)|~q |2
M∆
+gEgC(ν minus∆)(M2
N |~q |2 minus 2Q2s)
MNM∆
δ(ν minus ν∆)
with ∆ = M∆ minusMN M+ = M∆ + MN and the Mandelstam variable s = M2N + 2MNν minus Q2
Analytical expressions for the spin structure functions g1(xQ2) and g2(xQ2) can be constructed
from Eq (3) with the flux factor K(νQ2) = |~q | =radicν2 +Q2
In the ∆-non-pole contributions to S1(νQ2) and νS2(νQ2) the pole in ν at the ∆(1232)-
production threshold has canceled out
S∆-exch1 (νQ2) =
πα
MNM2+
[g2MQ
2+ + g2
E
(∆2 minus 3Q2
)+
4g2CQ
4
M2∆
minus 8gMgEM∆ωminus (B26a)
minus2gMgCQ2(MN minus 4M∆)
M∆+
2gEgCQ2(3MN minus 2M∆)
M∆
]
νS2∆-exch
(νQ2) =2πα
MNM2+
[g2EM∆∆ωminus +
g2M MNQ
2+
2+g2C Q
2(Q2 minus∆2)
2M∆(B26b)
+gEgM M∆(M∆ω+ minus 4MNωminus)minus gEgC ∆(2Q2 +MNω+)
+gMgC Q2(4MN minus ω+)
]+S∆-exch
2 (νQ2)
ν
[M2
∆ ω2+
M2N
+ ν2
]
with Q+ =radic
(M∆ +MN )2 +Q2 and ωplusmn = (M2∆minusM2
N plusmnQ2)2M∆ and the non-pole contribution
to S2(νQ2)
S∆-exch2 (νQ2) = minus2παMNν
M∆M2+
[gM + gE
]gC (B27)
These amplitudes to the contrary are not described by the tree-level ∆-production cross sections
in the standard dispersive approach [17] This peculiarity has been previously missed e g in the
calculation of the ∆-exchange contribution to the hydrogen hyperfine splitting in Ref [88] The
importance of including the ∆-non-pole contribution is also evident when considering the BC sum
37
rule in Eq (14) The ∆-pole terms by themselves violate the BC sum rule but cancel exactly with
the ∆-non-pole terms
limνrarr0
νS∆-pole2 (νQ2) + lim
νrarr0νS2
∆-exch(νQ2) = 0 (B28)
Appendix C Polarizabilities at Q2 = 0
In this section we give analytical expressions for the static values and slopes at Q2 = 0 of
the spin polarizabilities and moments of polarized structure functions In particular we give the
HB expansion of the πN -loop contributions and the ∆-exchange contributions The complete
expressions also for the π∆-loop contributions can be found in the Supplemented material Recall
that IA(0) = I1(0) = d2(0) = 0 and dd2(Q2)dQ2
∣∣∣Q2=0
= 0
1 πN-loop contribution
Here we give analytical expressions for the πN -loop contributions to the proton and neutron
spin polarizabilities expanded in powers of micro = mπMN viz the HB expansion Note that we
choose to expand here to a high order in micro the strict HB expansion would only retain the leading
term in an analogous NLO calculation
bull Static polarizabilities (Q2 = 0)
γ0p =e2g2
A
96π3f2πm
2π
1minus 21πmicro
8minus(
59
2+ 26 logmicro
)micro2 +
1875πmicro3
64
+3
(3
2+ 26 logmicro
)micro4 +
(C1)
γ0n =e2g2
A
48π3f2πm
2π
1
2minus 9πmicro
16minus 2micro2 logmicro+
75πmicro3
128minus 3micro4
4+
(C2)
δLTp =e2g2
A
192π3f2πm
2π
1minus 9πmicro
8+
(13
2minus 2 logmicro
)micro2 minus 465πmicro3
64
minus(
47
2+ 42 logmicro
)micro4 +
(C3)
δLTn =e2g2
A
96π3f2πm
2π
1
2+
3πmicro
16+ (1 + 2 log micro)micro2 minus 105πmicro3
128+
5micro4
4+
(C4)
38
δLTn =e2g2
A
96π3f2πm
2π
1
2+
3πmicro
16+ (1 + 2 log micro)micro2 minus 105πmicro3
128+
5micro4
4+
(C5)
γ0p =e2g2
A
16π3f2πm
4π
4
45minus 3πmicro
16+
14micro2
5minus 1813πmicro3
384minus 192
5(1 + log micro)micro4
+80703πmicro5
2048+
(C6)
γ0n =e2g2
A
16π3f2πm
4π
4
45minus 5πmicro
48+
4micro2
5minus 245πmicro3
384minus 32micro4 logmicro
15
+1323πmicro5
2048+
(C7)
bull Slopes of polarizabilities at Q2 = 0
dγ0p(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
2minus 45πmicro
4+ 223micro2 minus 28515πmicro3
64
minus9
(1953
4+ 449 logmicro
)micro4 +
570255πmicro5
128+
(C8)
dγ0n(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
2minus 81πmicro
8+ 94micro2 minus 2535πmicro3
32
minus3 (1 + 90 logmicro)micro4 +84315πmicro5
1024+
(C9)
dδLTp(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
2880π3f2πm
4π
minus5
2minus 27πmicro
32+ 20micro2 minus 5865πmicro3
256
+3
(617
4+ 36 logmicro
)micro4 minus 2056845πmicro5
4096+
(C10)
dδLTn(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
minus5
4minus 81πmicro
64minus 11micro2 +
10005πmicro3
512
+15
8(11 + 48 logmicro)micro4 minus 267015πmicro5
8192+
(C11)
dIAp(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
96π2f2πmicro
2
1minus 15πmicro
4minus 1
2(115 + 88 logmicro)micro2 +
1839πmicro3
32
+5 (5 + 34 logmicro)micro4 +
(C12)
dIAn(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
48π2f2πmicro
2
1
2minus 11πmicro
8minus 1
4(1 + 20 logmicro)micro2 +
99πmicro3
64
minus25micro4
12+
(C13)
dI1p(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
96π2f2πmicro
3π
8+ 2 (4 + 3 logmicro)microminus 537πmicro2
64
minus1
2(15 + 56 logmicro)micro3 +
(C14)
39
dI1n(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
48π2f2πmicro
minus π
16+
1
4(3 + 4 logmicro)microminus 57πmicro2
128+
2micro3
3+
(C15)
γ0p(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2g2
A
16π3f2πm
6π
1
105minus 23πmicro
256+
377micro2
210minus 15551πmicro3
6144+
3371micro4
105
minus1640457πmicro5
32768+
(C16)
γ0n(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2g2
A
16π3f2πm
6π
1
105minus 153πmicro
1792+
69micro2
70minus 4615πmicro3
6144+
172micro4
35
minus120897πmicro5
32768+
(C17)
2 ∆-exchange contribution
Here we give analytical expressions for the tree-level ∆-exchange contributions to the nucleon
spin polarizabilities and their slopes at Q2 = 0 Note that the ∆-exchange contributes equally to
proton and neutron polarizabilities Recall that for the magnetic γlowastN∆ coupling we introduced a
dipole form factor to mimic vector-meson dominance gM rarr gM(1 +Q2Λ2)2
bull Static polarizabilities (Q2 = 0)
γ0 = minus e2
4πM2+
(g2M
∆2+
g2E
M2+
minus 4gMgEM+∆
) (C18)
δLT =e2M∆
4πM3+
(g2E
MNM++gMgE∆MN
minus gEgCM2
∆
) (C19)
γ0 =e2M2
N
π∆2M4+
(minusg
2M
∆2+
g2E
M2+
+4gMgE∆M+
) (C20)
bull Slopes of polarizabilities at Q2 = 0
dγ0(Q2)
dQ2
∣∣∣∣∣Q2=0
= minus e2
πM2+∆
(g2M
∆
[1
4∆2minus 1
∆M++
1
2M2+
]minus 1
Λ2
g2M
∆+
g2E
2M2+
[1
2∆minus 3
M+
]minusgMgE
M+
[1
∆2minus 5
∆M++
1
M2+
]+
1
Λ2
2gMgEM+
+2gMgC∆M2
+
minus gEgCM3
+
) (C21)
40
d δLT (Q2)
dQ2
∣∣∣∣∣Q2=0
=e2M∆∆
4πMNM2+
(g2E
∆2M2+
[1
∆minus 4
M+
]minus g2
C
∆M2∆M
2+
+gMgE∆2M+
[1
∆2
minus 3
∆M++
1
M2+
]minus 2
Λ2
gMgE∆2M+
+gMgC∆M2
∆
[1
2∆2minus 2
∆M++
1
2M2+
]minus gEgC
2M2∆M
2+
[7
∆+
1
M+
]) (C22)
dIA(Q2)
dQ2
∣∣∣∣∣Q2=0
= minusM2N
M2+
(g2M
2∆2+
g2E
MNM+minus 2gMgE
∆M+minus gEgCM∆M+
) (C23)
dI1(Q2)
dQ2
∣∣∣∣∣Q2=0
= minusM∆M2N
2M3+
(g2E
MNM∆minus gMgE∆MN
minus gEgCM2
∆
) (C24)
γ0(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2M2
N
π∆3M6+
(g2M
∆
[3M2
∆ + 2M∆MN minus 9M2N
∆2+
4M2+
Λ2
]
minus4gMgE
[5M2
∆ minus 9M2N
M+∆2+
2M+
Λ2
]+
g2E
∆M+[7M∆ minus 9MN ]
minus8gMgC∆
+4gEgCM+
) (C25)
[1] D Drechsel B Pasquini and M Vanderhaeghen Dispersion relations in real and virtual
Compton scattering Phys Rept 378 99 (2003) hep-ph0212124
[2] S E Kuhn J-P Chen and E Leader Spin structure of the nucleon mdash status and recent
results Prog Part Nucl Phys 63 1 (2009) arXiv08123535 [hep-ph]
[3] F Hagelstein R Miskimen and V Pascalutsa Nucleon polarizabilities from Compton scat-
tering to hydrogen atom Prog Part Nucl Phys 88 29 (2016) arXiv151203765 [nucl-th]
[4] B Pasquini and M Vanderhaeghen Dispersion theory in electromagnetic interactions Ann
Rev Nucl Part Sci 68 75 (2018) arXiv180510482 [hep-ph]
[5] S Gerasimov A Sum rule for magnetic moments and the damping of the nucleon magnetic
moment in nuclei Sov J Nucl Phys 2 430 (1966)
[6] S Drell and A C Hearn Exact sum rule for nucleon magnetic moments Phys Rev Lett
16 908 (1966)
[7] J S Schwinger Source theory viewpoints in deep inelastic scattering Proc Natl Acad Sci
USA 72 1 (1975)
[8] J S Schwinger Source theory viewpoints in deep inelastic scattering Electromagnetic In-
teractions and Field Theory Proceedings 14 Internationale Universitatswochen Schladming
Austria February 24-March 7 1975 Acta Phys Austriaca Suppl 14 471 (1975)
41
[9] J Schwinger Source theory discussion of deep inelastic scattering with polarized particles
Proc Natl Acad Sci USA 72 1559 (1975)
[10] A M Baldin Polarizability of nucleons Nucl Phys 18 310 (1960)
[11] M Gell-Mann M L Goldberger and W E Thirring Use of causality conditions in quantum
theory Phys Rev 95 1612 (1954)
[12] A Pineda Leading chiral logarithms to the hyperfine splitting of the hydrogen and muonic
hydrogen Phys Rev C 67 025201 (2003)
[13] C Peset and A Pineda The two-photon exchange contribution to muonic hydrogen from
chiral perturbation theory Nucl Phys B 887 69 (2014) arXiv14064524 [hep-ph]
[14] C Peset and A Pineda Model-independent determination of the two-photon exchange contri-
bution to hyperfine splitting in muonic hydrogen JHEP 04 060 arXiv161205206 [nucl-th]
[15] F Hagelstein and V Pascalutsa Proton structure in the hyperfine splitting of muonic hydro-
gen PoS CD15 077 (2016) arXiv151104301 [nucl-th]
[16] F Hagelstein Exciting Nucleons in Compton Scattering and Hydrogen-Like Atoms PhD
thesis Mainz U Inst Kernphys (2017) arXiv171000874 [nucl-th]
[17] F Hagelstein ∆(1232)-Resonance in the hydrogen spectrum Proceedings 11th International
Workshop on the Physics of Excited Nucleons (NSTAR 2017) Columbia SC USA August
20-23 2017 Few Body Syst 59 93 (2018) arXiv180109790 [nucl-th]
[18] Y Prok et al (CLAS) Moments of the spin structure functions gp1 and gd1 for 005 lt Q2 lt 30
GeV2 Phys Lett B 672 12 (2009) arXiv08022232 [nucl-ex]
[19] H Dutz et al (GDH) First measurement of the Gerasimov-Drell-Hearn sum rule for 1H from
07 GeV to 18 GeV at ELSA Phys Rev Lett 91 192001 (2003)
[20] M Amarian et al (Jefferson Lab E94010) Measurement of the generalized forward spin po-
larizabilities of the neutron Phys Rev Lett 93 152301 (2004) arXivnucl-ex0406005
[21] M Amarian et al The Q2 evolution of the generalized Gerasimov-Drell-Hearn integral for
the neutron using a 3He target Phys Rev Lett 89 242301 (2002) arXivnucl-ex0205020
[22] M Amarian et al (Jefferson Lab E94-010)Q2 evolution of the neutron spin structure moments
using a 3He target Phys Rev Lett 92 022301 (2004) arXivhep-ex0310003
[23] A Deur et al Experimental determination of the evolution of the Bjorken integral at low Q2
Phys Rev Lett 93 212001 (2004) arXivhep-ex0407007
[24] K Slifer Low Q2 measurement of gp2 and the δLT spin polarizability Spin structure at long dis-
tance Proceedings Workshop Newport News USA March 12ndash13 2009 nucl-ex09064775
(2009) AIP Conf Proc 1155 10106313203293 (2009) arXiv09064775 [nucl-ex]
[25] P Solvignon et al (E01-012) Moments of the neutron g2 structure function at intermediate
42
Q2 Phys Rev C 92 015208 (2015) arXiv13044497 [nucl-ex]
[26] A Deur Experimental studies at low Q2 of the spin structure of the nucleon at Jefferson Lab
in 9th International Workshop on Chiral Dynamics (CD18) Durham NC USA September
17-21 2018 (2019) arXiv190305661 [nucl-ex]
[27] V Sulkosky et al (Jefferson Lab E97-110) Measurement of the 3He spin-structure functions
and of neutron (3He) spin-dependent sum rules at 0035 le Q2 le 024 GeV2 Phys Lett B
805 135428 (2020) arXiv190805709 [nucl-ex]
[28] K Adhikari et al (CLAS) Measurement of the Q2 Dependence of the Deuteron Spin Structure
Function g1 and its Moments at Low Q2 with CLAS Phys Rev Lett 120 062501 (2018)
arXiv171101974 [nucl-ex]
[29] V Lensky F Hagelstein A Hiller Blin and V Pascalutsa Comment on rdquoMeasurement of
the Q2 Dependence of the Deuteron Spin Structure Function g1 and its Moments at Low Q2
with CLASrdquo (2018) arXiv180603219 [nucl-th]
[30] J M Alarcn F Hagelstein V Lensky and V Pascalutsa Forward doubly-virtual Compton
scattering off the nucleon in chiral perturbation theory at NLO the subtraction function and
moments of unpolarized structure functions (2020) arXiv200509518 [hep-ph]
[31] V Bernard E Epelbaum H Krebs and U-G Meiszligner New insights into the spin structure
of the nucleon Phys Rev D 87 054032 (2013) arXiv12092523 [hep-ph]
[32] V Lensky J M Alarcon and V Pascalutsa Moments of nucleon structure functions at
next-to-leading order in baryon chiral perturbation theory Phys Rev C 90 055202 (2014)
arXiv14072574 [hep-ph]
[33] V Pascalutsa and M Vanderhaeghen Chiral effective-field theory in the ∆(1232) region
I Pion electroproduction on the nucleon Phys Rev D 73 034003 (2006) arXivhep-
ph0512244
[34] V Pascalutsa and M Vanderhaeghen Electromagnetic nucleon-to-Delta transition in chiral
effective field theory Phys Rev Lett 95 232001 (2005) arXivhep-ph0508060
[35] T R Hemmert B R Holstein and J Kambor Systematic 1M expansion for spin 32
particles in baryon chiral perturbation theory Phys Lett B 395 89 (1997) arXivhep-
ph9606456
[36] V Pascalutsa and D R Phillips Effective theory of the ∆(1232) in Compton scattering off
the nucleon Phys Rev C 67 055202 (2003) arXivnucl-th0212024
[37] V Pascalutsa M Vanderhaeghen and S N Yang Electromagnetic excitation of the ∆(1232)-
resonance Phys Rept 437 125 (2007) arXivhep-ph0609004
[38] K A Olive et al (Particle Data Group) Review of Particle Physics Chin Phys C 38 090001
43
(2014)
[39] F E Low Scattering of light of very low frequency by systems of spin 12 Phys Rev 96
1428 (1954)
[40] M Gell-Mann and M L Goldberger Scattering of low-energy photons by particles of spin
12 Phys Rev 96 1433 (1954)
[41] A M Harun ar-Rashid A simple derivation of Schwingerrsquos sum rule for spin dependent
structure functions Nuovo Cim A 33 447 (1976)
[42] F Hagelstein and V Pascalutsa Dissecting the hadronic contributions to (g minus 2)micro by
Schwingerrsquos sum rule Phys Rev Lett 120 072002 (2018) arXiv171004571 [hep-ph]
[43] V Lensky V Pascalutsa M Vanderhaeghen and C Kao Spin-dependent sum rules con-
necting real and virtual Compton scattering verified Phys Rev D 95 074001 (2017)
arXiv170101947 [hep-ph]
[44] H Burkhardt and W N Cottingham Sum rules for forward virtual Compton scattering
Annals Phys 56 453 (1970)
[45] V Lensky and V Pascalutsa Predictive powers of chiral perturbation theory in Compton
scattering off protons Eur Phys J C 65 195 (2010) arXiv09070451 [hep-ph]
[46] H Grieszlighammer J McGovern D Phillips and G Feldman Using effective field theory to
analyse low-energy Compton scattering data from protons and light nuclei Prog Part Nucl
Phys 67 841 (2012) arXiv12036834 [nucl-th]
[47] H W Grieszlighammer J A McGovern and D R Phillips Nucleon polarisabilities at and
beyond physical pion masses Eur Phys J A 52 139 (2016) arXiv151101952 [nucl-th]
[48] E Epelbaum H Krebs and U-G Meiszligner Improved chiral nucleon-nucleon potential up to
next-to-next-to-next-to-leading order Eur Phys J A 51 53 (2015) arXiv14120142 [nucl-
th]
[49] E Epelbaum H Krebs and U-G Meiszligner Precision nucleon-nucleon potential at fifth order
in the chiral expansion Phys Rev Lett 115 122301 (2015) arXiv14124623 [nucl-th]
[50] C W Kao T Spitzenberg and M Vanderhaeghen Burkhardt-Cottingham sum rule and
forward spin polarizabilities in heavy baryon chiral perturbation theory Phys Rev D 67
016001 (2003) arXivhep-ph0209241
[51] C-W Kao D Drechsel S Kamalov and M Vanderhaeghen Higher moments of nucleon
spin structure functions in heavy baryon chiral perturbation theory and in a resonance model
Phys Rev D 69 056004 (2004) arXivhep-ph0312102
[52] T Becher and H Leutwyler Baryon chiral perturbation theory in manifestly Lorentz invariant
form Eur Phys J C 9 643 (1999) arXivhep-ph9901384
44
[53] J Gasser M E Sainio and A Svarc Nucleons with chiral loops Nucl Phys B 307 779
(1988)
[54] L S Geng J Martin Camalich L Alvarez-Ruso and M J Vicente Vacas Leading SU(3)-
breaking corrections to the baryon magnetic moments in chiral perturbation theory Phys
Rev Lett 101 222002 (2008) arXiv08051419 [hep-ph]
[55] D Drechsel S S Kamalov and L Tiator The GDH sum rule and related integrals Phys
Rev D 63 114010 (2001) arXivhep-ph0008306
[56] D Drechsel O Hanstein S S Kamalov and L Tiator A unitary isobar model for pion
photo- and electroproduction on the proton up to 1 GeV Nucl Phys A 645 145 (1999)
arXivnucl-th9807001
[57] L Tiator private communication (2020)
[58] V Bernard T R Hemmert and U-G Meiszligner Spin structure of the nucleon at low energies
Phys Rev D 67 076008 (2003) arXivhep-ph0212033
[59] O Gryniuk F Hagelstein and V Pascalutsa Evaluation of the forward Compton scattering
off protons II Spin-dependent amplitude and observables Phys Rev D 94 034043 (2016)
arXiv160400789 [nucl-th]
[60] R Zielinski The g2p Experiment A Measurement of the Protonrsquos Spin Structure Functions
PhD thesis New Hampshire U (2010) arXiv170808297 [nucl-ex]
[61] N Guler et al (CLAS) Precise determination of the deuteron spin structure at low to mod-
erate Q2 with CLAS and extraction of the neutron contribution Phys Rev C 92 055201
(2015) arXiv150507877 [nucl-ex]
[62] C E Carlson V Nazaryan and K Griffioen Proton structure corrections to electronic
and muonic hydrogen hyperfine splitting Phys Rev A 78 022517 (2008) arXiv08052603
[physicsatom-ph]
[63] V Bernard N Kaiser and U-G Meiszligner Chiral dynamics in nucleons and nuclei Int J
Mod Phys E 4 193 (1995) arXivhep-ph9501384
[64] T R Hemmert B R Holstein and J Kambor ∆(1232) and the polarizabilities of the nucleon
Phys Rev D 55 5598 (1997) arXivhep-ph9612374
[65] V Pascalutsa and R Timmermans Field theory of nucleon to higher-spin baryon transitions
Phys Rev C 60 042201 (1999) arXivnucl-th9905065
[66] V Pascalutsa Quantization of an interacting spin-32 field and the Delta isobar Phys Rev
D 58 096002 (1998) arXivhep-ph9802288
[67] H Krebs Double Virtual Compton Scattering and SpinStructure of the Nucleon PoS
CD2018 031 (2019)
45
[68] N Kochelev and Y Oh Axial anomaly and the δLT puzzle Phys Rev D 016012 (2012)
arXiv11034892 [hep-ph]
[69] D Drechsel S Kamalov and L Tiator Unitary isobar model ndash MAID2007 Eur Phys J A
69 (2007) available at httpsmaidkphuni-mainzde arXiv07100306 [nucl-th]
[70] P J Mohr B N Taylor and D B Newell CODATA recommended values of the fundamental
physical constants 2010 Rev Mod Phys 84 1527 (2012)
[71] J Ahrens et al (GDH A2) First measurement of the Gerasimov-Drell-Hearn integral for 1H
from 200 to 800 MeV Phys Rev Lett 87 022003 (2001) arXivhep-ex0105089 [hep-ex]
[72] K Helbing (GDH) Experimental verification of the GDH sum rule at ELSA and MAMI Nucl
Phys Proc Suppl 105 113 (2002)
[73] R Bradford A Bodek H S Budd and J Arrington A New parameterization of the nucleon
elastic form-factors NuInt05 proceedings of the 4th International Workshop on Neutrino-
Nucleus Interactions in the Few-GeV Region Okayama Japan 26-29 September 2005 Nucl
Phys Proc Suppl 159 127 (2006) arXivhep-ex0602017
[74] J D Bjorken Applications of the chiral U(6)timesU(6) algebra of current densities Phys Rev
148 1467 (1966)
[75] J D Bjorken Inelastic scattering of polarized leptons from polarized nucleons Phys Rev D
1 1376 (1970)
[76] V Bernard T R Hemmert and U-G Meiszligner Novel analysis of chiral loop effects in
the generalized Gerasimov-Drell-Hearn sum rule Phys Lett B 545 105 (2002) arXivhep-
ph0203167
[77] A Deur et al Experimental study of isovector spin sum rules Phys Rev D 78 032001
(2008) arXiv08023198 [nucl-ex]
[78] B Pasquini P Pedroni and D Drechsel Higher order forward spin polarizability Phys Lett
B 687 160 (2010) arXiv10014230 [hep-ph]
[79] R Jaffe g2ndashThe nucleonrsquos other spin-dependent structure function Comments Nucl Part
Phys 19 239 (1990)
[80] E V Shuryak and A Vainshtein Theory of power corrections to deep inelastic scattering in
quantum chromodynamics (II) Qminus4 effects polarized target Nucl Phys B 201 141 (1982)
[81] S Wandzura and F Wilczek Sum rules for spin dependent electroproduction Test of rela-
tivistic constituent quarks Phys Lett B 72 195 (1977)
[82] B W Filippone and X-D Ji The spin structure of the nucleon Adv Nucl Phys 26 1
(2001) arXivhep-ph0101224
[83] M Burkardt The g2 structure function Proceedings Workshop on Spin structure at long
46
distance Newport News USA March 12-13 2009 AIP Conf Proc 1155 26 (2009)
arXiv09054079 [hep-ph]
[84] A Deur S J Brodsky and G F de Teramond The spin structure of the nucleon Rept
Prog Phys 82 076201 (2019) arXiv180705250 [hep-ph]
[85] B R Holstein D Drechsel B Pasquini and M Vanderhaeghen Higher order polarizabilities
of the proton Phys Rev C 61 034316 (2000) arXivhep-ph9910427
[86] B R Holstein V Pascalutsa and M Vanderhaeghen Sum rules for magnetic moments and
polarizabilities in QED and chiral effective-field theory Phys Rev D 72 094014 (2005)
arXivhep-ph0507016
[87] B Pasquini D Drechsel and L Tiator Invariant amplitudes for pion electroproduction Eur
Phys J A 34 387 (2007) arXiv07122327 [hep-ph]
[88] A J Buchmann Non-spherical proton shape and hydrogen hyperfine splitting Proceedings
International Workshop on Precision Physics of Simple Atomic Systems (PSAS 2008) Wind-
sor Ontario Canada July 21-26 2008 Can J Phys 87 773 (2009) arXiv09104747
[physicsatom-ph]
47
Another approach used in the literature to calculate the polarizabilities in χPT is the
infrared regularization (IR) scheme introduced in Ref [52] This covariant approach tries to
solve the power counting violation observed in Ref [53] by dropping the regular parts of the
loop integrals that contain the power-counting-breaking terms However this subtraction
scheme modifies the analytic structure of the loop contributions and may lead to unexpected
problems as was shown in Ref [54] As we will see in the next section the IR approach
also fails to describe the Q2 behaviour of the polarizabilities
III RESULTS AND DISCUSSION
We now present the NLO BχPT predictions for the nucleon polarizabilities and selected
moments of the nucleon spin structure functions Our results are obtained from the calcu-
lated non-Born VVCS amplitudes and the LEXs in Eqs (10) and (12) For a cross-check
we used the photoabsorption cross sections described in App B In addition to the full NLO
results we also analyse the individual contributions from the πN loops the ∆ exchange
and the π∆ loops
A γ0(Q2) mdash generalized forward spin polarizability
The forward spin polarizability
γ0(Q2) =1
2π2
int infinν0
dν
radic1 +
Q2
ν2
σTT (νQ2)
ν3(19)
=16αM2
N
Q6
int x0
0
dx x2
[g1(xQ2)minus 4M2
Nx2
Q2g2(xQ2)
]
provides information about the spin-dependent response of the nucleon to transversal pho-
ton probes The RCS analogue of the above generalized forward spin polarizability sum
rule is sometimes referred to as the Gell-Mann Goldberger and Thirring (GGT) sum rule
[11] At Q2 = 0 the forward spin polarizability is expressed through the lowest-order spin
polarizabilities of RCS as γ0 = minus(γE1E1 + γM1M1 + γE1M2 + γM1E2) The forward spin po-
larizability of the proton is relevant for an accurate knowledge of the (muonic-)hydrogen
hyperfine splitting as it controls the leading proton-polarizability correction [16 62]
The πN -loop ∆-exchange and π∆-loop contributions to the NLO BχPT prediction of
11
000 005 010 015 020 025 030
-4
-3
-2
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
4D
proton
000 005 010 015 020 025 030
-4
-3
-2
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
4D
neutron
000 005 010 015 020 025 030
0
1
2
Q2 GeV2D
∆LT
10-
4fm
4D
proton
000 005 010 015 020 025 030
0
1
2
3
Q2 GeV2D
∆LT
10-
4fm
4D
neutron
FIG 2 Upper panel Generalized forward spin polarizability for the proton (left) and neutron
(right) as function of Q2 The result of this work the NLO BχPT prediction is shown by the
blue solid line and the blue band The red line represents the LO BχPT result The purple short-
dashed line is the O(p4) HB result from Ref [50] note that the corresponding proton curve is
outside of the plotted range The black dotted line is the MAID model prediction [55ndash57] which
is taken from Ref [1] (proton) and Ref [20] (neutron) The pink band is the IR+∆ result from
Ref [58] and the gray band is the BχPT+∆ result from Ref [31] Empirical extractions for the
proton Ref [18] (blue dots) Ref [59] (purple square) and Ref [60] (orange triangle uncertainties
added in quadrature) and neutron Ref [20] (blue diamonds) and Ref [61] (green dots statistical
and systematic uncertainties added in quadrature) Lower Panel Longitudinal-transverse spin
polarizability for the proton (left) and neutron (right) The orange dot-dashed and purple short-
dashed lines are the O(p3) and O(p4) HB results from Ref [50] The pink band is the IR result
from Ref [58] and the gray band is the covariant BχPT+∆ result from Ref [31] The black dotted
line is the MAID model prediction [55ndash57] note that for the proton we use the updated estimate
from Ref [1] obtained using the π η ππ channels
12
the static forward spin polarizability amount to in units of 10minus4 fm4
γ0p = minus093(92) asymp 201minus 284minus 010 (20a)
γ0n = 003(92) asymp 298minus 284minus 010 (20b)
while the slope is composed as follows in units of 10minus4 fm6
dγ0p(Q2)
dQ2
∣∣∣∣Q2=0
= minus022(4) asymp minus033 + 011 + 001 (21a)
dγ0n(Q2)
dQ2
∣∣∣∣Q2=0
= minus061(7) asymp minus073 + 011 + 001 (21b)
Figure 2 upper panel shows our NLO prediction as well as the LO πN loops compared
to different experimental and theoretical results For the proton we have one determination
at the real-photon point by the GDH collaboration [19] γ0p = minus100(8)(12)times10minus4 fm4 and
further Jefferson Laboratory data [18 60] at very low Q2 For the neutron only data at finite
Q2 are available [20 61] The experimental data for the proton are fairly well reproduced
in the whole Q2 range considered here while for the neutron the agreement improves with
increasing Q2 The HB limit of our πN -loop contribution reproduces the results published
in Refs [50 63] for arbitrary Q2 In addition our prediction is compared to the MAID
model [1 20] the IR+∆ calculation of Ref [58] and the BχPT+∆ result of Ref [31]
The πN -production channel gives a positive contribution to the photoabsorption cross
section σTT (νQ2) at low Q2 cf Fig 10 Accordingly one observes that the πN loops give
a sizeable positive contribution to γ0(Q2) The Delta on the other hand has a very large
effect by cancelling the πN loops and bringing the result close to the empirical data From
Fig 3 upper panel one can see that it is the ∆ exchange which dominates while π∆
loops are negligible This was expected since the forward spin polarizability sum rule is an
integral over the helicity-difference cross section in which σ32 is governed by the Delta at
low energies (the relevant energy region for the sum rule)
To elucidate the difference between the present calculation and the one from Ref [31] we
note that the two calculations differ in the following important aspects Firstly Ref [31] uses
the small-scale counting [64] that considers ∆ and mπ as being of the same size ∆ sim mπ
In practice this results in a set of π∆-loop graphs which contains graphs with one or two
γ∆∆ couplings and hence two or three Delta propagators Such graphs are suppressed in
the δ-counting and thus omitted from our calculation while present in that of Ref [31]
Secondly the Lagrangians describing the interaction of the Delta are constructed differently
and assume slightly different values for the coupling constants In particular we employ
13
000 005 010 015 020 025 030
-3
-2
-1
0
1
2
Q2GeV
2D
Γ01
0-
4fm
4D
proton
000 005 010 015 020 025 030
-3
-2
-1
0
1
2
3
Q2GeV
2D
Γ01
0-
4fm
4D
neutron
000 005 010 015 020 025 030
00
05
10
15
Q2GeV
2D
∆LT1
0-
4fm
4D
proton
000 005 010 015 020 025 030
00
05
10
15
20
25
Q2GeV
2D
∆LT1
0-
4fm
4D
neutron
FIG 3 Contributions of the different orders to the chiral predictions of γ0(Q2) upper paneland δLT (Q2) lower panel for the proton (left) and neutron (right) Red solid line πN -loop
contribution green dot-dashed line ∆-exchange contribution orange dotted line π∆-loop con-
tribution blue long-dashed line total result purple dot-dot-dashed line total result without gCcontribution black short-dashed line total result without gM dipole
(where possible) the so-called ldquoconsistentrdquo couplings to the Delta field ie those couplings
that project out the spurious degree of freedom see Refs [37 65 66] The authors of
Ref [31] on the other hand use couplings where the consistency in this sense is not enforced
The effects of these differences are of higher order in the δ-counting expansion and their
contribution to the Q2 dependence of the considered polarizabilities is expected to be rather
small however the differences at Q2 = 0 could be noticeable [67]
Finally as mentioned in Sec II C the inclusion of the dipole form factor in the magnetic
coupling gM is expected to be important to generate the correct Q2 behaviour of the polar-
izabilities Comparing our predictions for the forward spin polarizability with and without
inclusion of the form factor see Fig 3 upper panel confirms this Without the dipole
our results for the proton and neutron are closer to the ones from Ref [31] where the form
factor is not included For the neutron our prediction without the dipole is able to describe
14
the experimental points at very low Q2 deviating from the data with increasing Q2 The
π∆-loop contribution does not modify the Q2 behavior of γ0(Q2) and only differs from
Ref [31] by a small global shift Note also the relatively large effect of gC which generates
a sign change for virtualities above sim 02 GeV2 see Fig 3 upper panel
B δLT (Q2) mdash longitudinal-transverse polarizability
The longitudinal-transverse spin polarizability
δLT (Q2) =1
2π2
int infinν0
dν
radic1 +
Q2
ν2
σLT (νQ2)
Qν2(22)
=16αM2
N
Q6
int x0
0
dx x2[g1(xQ2) + g2(xQ2)
]
contains information about the spin structure of the nucleon and is another important
input in the determination of the (muonic-)hydrogen hyperfine splitting [16 62] It is also
relevant in studies of higher-twist corrections to the structure function g2(xQ2) given by
the moment d2(Q2) [51] see Section III E The peculiarity of the response encoded in this
polarizability is that it involves a spin flip of the nucleon and a polarization change of the
photon see App B and Fig 11
It is expected that the Delta isobar gives only a small contribution to δLT (Q2) what makes
this polarizability a potentially clean test case for chiral calculations Consequently there are
relatively many different theoretical calculations of δLT (Q2) coming from different versions
of χPT with baryons (HB IR and covariant) Ref [50] found a systematic deviation of the
HB result for δLTn(Q2) from the MAID model prediction This disagreement was identified
by the authors of Ref [68] as a puzzle involving the neutron δLT polarizabilitymdashthe δLT
puzzle The IR calculation in Ref [58] also showed a deviation from the data and predicted
a rapid rise of δLT (Q2) with growing Q2 The problem is solved by keeping the relativistic
structure of the theory as the BχPT+∆ result of Ref [31] showed
As expected already the leading πN loops provide a reasonable agreement with the
experimental data cf Fig 2 lower panel Since the ∆-exchange contribution to δLT (Q2)
is small the effect of the gM form factor is negligible in this polarizability as is that of the gC
coupling cf Fig 3 lower panel In fact we predict both the ∆-exchange and the π∆-loop
contributions to be small and negative This is in agreement with the MAID model which
predicts a small and negative contribution of the P33 wave to δLT (Q2) However in the
calculation of Ref [31] which is different from the one presented here only in the way the
∆(1232) is included the contribution of this resonance to δLTp(Q2) is sizeable and positive
15
The authors of that work attributed this large contribution to diagrams where the photons
couple directly to the Delta inside a loop As mentioned in Sec III F the effect of such loop
diagrams does not change the Q2 behaviour of the polarizabilities On the other hand it
can produce a substantial shift of the δLT (Q2) as a whole A higher-order calculation should
resolve the discrepancy between the two covariant approaches however it will partially lose
the predictive power since the LECs appearing at higher orders will have to be fitted to
experimental data
The πN -loop ∆-exchange and π∆-loop contributions to the NLO BχPT prediction of
the static longitudinal-transverse polarizability are in units of 10minus4 fm4
δLTp = 132(15) asymp 150minus 016minus 002 (23a)
δLTn = 218(23) asymp 235minus 016minus 002 (23b)
while the slopes are in units of 10minus4 fm6
dδLTp(Q2)
dQ2
∣∣∣∣Q2=0
= minus085(8) asymp minus080minus 004minus 001 (24a)
dδLTn(Q2)
dQ2
∣∣∣∣Q2=0
= minus124(12) asymp minus119minus 004minus 001 (24b)
C IA(Q2) mdash a generalized GDH integral
The helicity-difference cross section σTT exhibits a faster fall-off in ν than its spin-averaged
counterpart σT This is due to a cancellation between the leading (constant) terms of σ12
and σ32 at large ν3 The resulting 1ν fall-off of the helicity-difference cross section allows
one to write an unsubtracted dispersion relation for the VVCS amplitude gnonpoleTT (νQ2) cf
Eq (10a) This is the origin of the GDH sum rule [5 6]
minus α
2M2N
κ2 =1
2π2
int infinν0
dνσTT (ν)
ν (25)
which establishes a relation to the anomalous magnetic moment κ It is experimentally
verified for the nucleon by MAMI (Mainz) and ELSA (Bonn) [71 72]
There are two extensions of the GDH sum rule to finite Q2 the generalized GDH integrals
3 Notice that a constant term in σTT at ν rarrinfin is forbidden by crossing symmetry
16
000 005 010 015 020 025 030
-30
-25
-20
-15
-10
-05
00
Q2 GeV2D
I Aproton
000 005 010 015 020 025 030
-25
-20
-15
-10
-05
00
Q2 GeV2D
I A
neutron
000 005 010 015 020 025 030
-05
00
05
Q2 GeV2D
I 1
proton
000 005 010 015 020 025 030
-10
-08
-06
-04
-02
00
02
Q2 GeV2D
I 1
neutron
FIG 4 Upper panel The generalized GDH integral IA(Q2) for the proton (left) and neutron
(right) as function of Q2 The result of this work the NLO BχPT prediction is shown by the blue
solid line and the blue band The red line represents the LO BχPT result The purple short-dashed
line is the O(p4) HB result from Ref [50 51] The gray band is the BχPT+∆ result from Ref [31]
The black dotted line is the MAID model prediction [69] Experimental extractions for the proton
Ref [60] (orange triangle uncertainties added in quadrature) and neutron Refs [21][27] where
magenta dotsorange diagonal crosses correspond to data and red squareslilac crosses correspond
to data plus extrapolation to unmeasured energy regions The green stars at the real-photon point
are derived from the anomalous magnetic moments κp asymp 1793 and κn asymp minus1913 [70] Lower
panel The generalized GDH integral I1(Q2) for the proton (left) and neutron (right) as function
of Q2 The purple short-dashed line is the HB result from Ref [51] Experimental extractions for
the proton Ref [18] (blue dots) and Ref [60] (orange triangle uncertainties added in quadrature)
and neutron Ref [61][27] (uncertainties added in quadrature) where green dotsorange diagonal
crosses correspond to data and gray squareslilac crosses correspond to data plus extrapolation to
unmeasured energy regions
17
IA(Q2) and I1(Q2) The latter will be discussed in Sec III D The former is defined as4
minus α
2M2N
IA(Q2) = minus 1
8π2
int infinν0
dν
radic1 +
Q2
ν2
σTT (νQ2)
ν(26)
=α
Q2
int x0
0
dx[4M2
Nx2
Q2g2(xQ2)minus g1(xQ2)
]
Due to its energy weighting the integral in Eq (26) converges slower than the one in the
generalized forward spin polarizability sum rule (19) Therefore knowledge of the cross
section at higher energies is required and the evaluation of the generalized GDH integral
IA(Q2) is not as simple as the evaluation of γ0(Q2)
The generalized GDH integral IA(Q2) is directly related to the non-pole amplitude
gnonpoleTT (νQ2) which differs from non-Born amplitude gTT (νQ2) by a term involving the
elastic Pauli form factor
gnonpoleTT (νQ2) = gTT (νQ2)minus 2παν
M2N
F 22 (Q2) (27)
cf Eqs (2a) and (8a) Consequently IA(Q2) is not a pure polarizability but also contains
an elastic contribution The ldquonon-polarizabilityrdquo or the Born part of IA(Q2) is given by
IBornA (Q2) = IA(Q2)minus∆IA(Q2) = minus1
4F 2
2 (Q2) (28)
where we refer to the polarizability part as ∆IA(Q2) The same is true for the generalized
GDH integral I1(Q2) which is directly related to Snonpole1 (νQ2)
IBorn1 (Q2) = I1(Q2)minus∆I1(Q2) = minus1
4F 2
2 (Q2) (29)
In the following we will add the Born parts to our LO and NLO BχPT predictions for
the polarizabilities ∆IA(Q2) and ∆I1(Q2) employing an empirical parametrization for the
elastic Pauli form factor [73] This allows us to compare to the experimental results for
IA(Q2) and I1(Q2) cf Fig 4 Note that the blue error bands only describe the uncertainties
of our BχPT predictions of the polarizabilities while the elastic contributions are considered
to be exact as explained in Sec II C The uncertainties of the polarizability predictions are
therefore better reflected in Fig 5 where we show the contributions of the different orders
to the BχPT predictions of ∆IA(Q2) and ∆I1(Q2) as well as the total results with error
bands
4 Note that IA(Q2) is sometimes called ITT (Q2)
18
000 005 010 015 020 025 030
-08
-06
-04
-02
00
02
Q2 GeV2D
DI A
proton
000 005 010 015 020 025 030
-10
-08
-06
-04
-02
00
02
Q2 GeV2D
DI A
neutron
000 005 010 015 020 025 030-005
000
005
010
015
020
025
Q2 GeV2D
DI 1
proton
000 005 010 015 020 025 030-015
-010
-005
000
005
010
015
Q2 GeV2D
DI 1
neutron
FIG 5 Contributions of the different orders to the chiral predictions of ∆IA(Q2) upper paneland ∆I1(Q2) lower panel for the proton (left) and neutron (right) Red solid line πN -loop
contribution green dot-dashed line ∆-exchange contribution orange dotted line π∆-loop contri-
bution blue solid line and blue band total result purple dot-dot-dashed line total result without
gC contribution black short-dashed line total result without gM dipole
The E97-110 experiment at Jefferson Lab has recently published their data for IAn(Q2) in
the region of 0035 GeV2 lt Q2 lt 024 GeV2 [27] In addition there are results for IAn(Q2)
from the earlier E94-010 experiment [21] and for IAp(Q2) from the E08-027 experiment [60]
The O(p4) HB calculation gives a large negative effect [51] which does not describe the data
The BχPT+∆ result from Ref [31] which mainly differs from our work by the absence of
the dipole form factor in gM looks similar to this HB result and only describes the data
points at lowest Q2 Our NLO prediction however follows closely the Q2 evolution of the
data In Fig 5 upper panel we show the polarizability ∆IA(Q2) whose Q2 evolution is
clearly dominated by the ∆ exchange Similar to the case of γ0p(Q2) inclusion of the dipole
in gM and the Coulomb coupling gC is very important in order to describe the experimental
data The LO prediction on the other hand slightly overestimates the data cf Fig 4
upper panel
19
At the real-photon point IA(0) = minusκ2
4and ∆IA(0) = 0 Therefore we give only the
slope of the polarizability ∆IA(Q2) [showing also the separate contributions from πN loops
∆ exchange and π∆ loops] in units of GeVminus2
d∆IAp(Q2)
dQ2
∣∣∣∣Q2=0
= minus858(343) asymp 238minus 1121 + 025 (30a)
d∆IAn(Q2)
dQ2
∣∣∣∣Q2=0
= minus955(343) asymp 141minus 1121 + 025 (30b)
Including the empirical Pauli form factor [73] we find in units of GeVminus2
dIAp(Q2)
dQ2
∣∣∣∣Q2=0
= minus318dIAn(Q2)
dQ2
∣∣∣∣Q2=0
= minus300 (31)
D Γ1(Q2) and I1(Q2) mdash the first moment of the structure function g1(xQ2)
The second variant for a generalization of the GDH sum rule to finite Q2 is defined as
minus α
2M2N
I1(Q2) = minus 1
8π2
int infinν0
dν1radic
ν2 +Q2
[σTT (νQ2) +
Q
νσLT (νQ2)
](32)
= minus α
Q2
int x0
0
dx g1(xQ2)
where I1(0) = minusκ2
4 This generalized GDH integral directly stems from the amplitude
Snonpole1 (νQ2) with the LEX from Eq (12a) It is given by the first moment of the struc-
ture function g1(xQ2) Γ1(Q2) =int x0
0dx g1(xQ2) as follows I1(Q2) =
2M2N
Q2 Γ1(Q2) The
isovector combination
Γ1(pminusn)(Q2) =
int x0
0
dx[g1p(xQ
2)minus g1n(xQ2)] (33)
is related to the axial coupling of the nucleon through the Bjorken sum rule [74 75]
limQ2rarrinfin
Γ1(pminusn)(Q2) =
gA6 (34)
As explained in Eq (28) the moment I1(Q2) splits into a polarizability part ∆I1(Q2) and
a Born part IBorn1 (Q2) Figure 4 lower panel shows the Q2 dependence of I1(Q2) which in
contrast to IA(Q2) shown in Figure 4 upper panel is clearly dominated by its Born part
and the elastic Pauli form factor The πN -loop ∆-exchange and π∆-loop contributions to
the polarizability ∆I1(Q2) are shown in Fig 5 lower panel Comparing to Fig 5 upper
20
000 005 010 015 020 025 030
-006
-004
-002
000
002
004
Q2 GeV2D
G1
proton
000 005 010 015 020 025 030
-006
-005
-004
-003
-002
-001
000
001
Q2 GeV2D
G1
neutron
FIG 6 First moment of the structure function g1(xQ2) for the proton (left) and neutron (right)
as function of Q2 The legend is the same as in Fig 4
panel one sees that ∆I1(Q2) is less sensitive to gC and the dipole form factor in gM than
∆IA(Q2)
For the proton our NLO BχPT prediction gives a very good description of the ex-
perimental data [18 60] and is in reasonable agreement with the MAID prediction [69]
For the neutron one observes good agreement with the empirical evaluations including ex-
trapolations to unmeasured energy regions starting from Q2 gt 01 GeV2 [27 61] In the
region of Q2 lt 005 GeV2 one observes an interesting tension between the recent E97-
110 experiment [27] and the data from CLAS [61] While the newest measurement finds
I1n(0035 GeV2) lt κ2n4 thus suggesting a negative slope at low Q the older measurement
found a rather large value for I1n(00496 GeV2) A similar but milder behaviour is seen in
the E97-110 [27] and E94-010 [21] data for IAn The MAID predictions do not agree with
the CODATA recommended values for the anomalous magnetic moments of the proton and
neutron [70] which in our work are imposed by using empirical parametrizations for the
elastic Pauli form factors [73] The slope of the HB result from Ref [51] is too large and
therefore only reproduces the data at very low Q2
Figure 6 shows the moment Γ1(Q2) for the proton and neutron while Fig 7 shows the
isovector combination Γ1 pminusn(Q2) The LO and NLO BχPT predictions are identical be-
cause our calculation produces the same Delta contributions for the proton and the neutron
For the isovector combination the MAID model only agrees with the data at very low
Q2 lt 010 GeV2 The same is true for the IR result [58 76] while all other chiral results
describe the data NLO BχPT (this work) BχPT+∆ [31] and HBχPT [51]
At the real-photon point I1(0) = minusκ2
4and ∆I1(0) = 0 Therefore we give only the
slope of the polarizability ∆I1(Q2) [showing also the separate contributions from πN loops
21
000 005 010 015 020 025 030-002
000
002
004
006
008
010
Q2 GeV2D
G1Hp-
nL
FIG 7 Isovector combination of Γ1(Q2) as function of Q2 The legend is the same as in Fig 6
The pink curve is the IR result from Ref [58 76] The experimental points are from Ref [77]
(brown dots) and Ref [23] (orange squares)
∆ exchange and π∆ loops] in units of GeVminus2
d∆I1p(Q2)
dQ2
∣∣∣∣Q2=0
= 039(4) asymp 034minus 053 + 058 (35a)
d∆I1n(Q2)
dQ2
∣∣∣∣Q2=0
= minus101(10) asymp minus107minus 053 + 058 (35b)
Including the empirical Pauli form factor [73] we find in units of GeVminus2
dI1p(Q2)
dQ2
∣∣∣∣Q2=0
= 580dI1n(Q2)
dQ2
∣∣∣∣Q2=0
= 553 (36)
E d2(Q2) mdash a measure of color polarizability
Another interesting moment to consider is d2(Q2) which is related to the twist-3 part of
the spin structure function g2(xQ2) [79 80]
d2(Q2) equiv 3
int 1
0
dx x2[g2(xQ2)minus gWW2 (xQ2)] (37)
where gWW2 (xQ2) is the twist-2 part of g2(xQ2) Using the Wandzura-Wilczek relation [81]
one can relate d2(Q2) to moments of the spin structure functions g1(xQ2) and g2(xQ2)
d2(Q2) =
int 1
0
dx x2 [3g2(xQ2) + 2g1(xQ2)] (38)
22
000 005 010 015 020 025 030
0000
0001
0002
0003
0004
0005
0006
0007
Q2 GeV2D
d2
proton
000 005 010 015 020 025 030
0000
0002
0004
0006
0008
0010
0012
Q2 GeV2D
d2
neutron
000 005 010 015 020 025 030
-1
0
1
2
Q2 GeV2D
Γ 010
-4
fm6
D
proton
000 005 010 015 020 025 030-2
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
6D
neutron
FIG 8 Upper panel The inelastic moment d2(Q2) for the proton (left) and neutron (right) as
function of Q2 The result of this work the NLO BχPT prediction is shown by the blue solid line
and the blue band The red line represents the LO BχPT result The purple short-dashed line is
the O(p4) HB result from Ref [50 51] The black dotted line is the MAID model prediction [69]
The experimental points for the neutron (cyan dots) are from Ref [22] Lower panel Fifth-order
generalized forward spin polarizability γ0(Q2) for the proton (left) and neutron (right) as function
of Q2 The experimental points for the proton are from Ref [59] (purple square) and Ref [78]
(orange dot)
This relation however only holds for asymptotically large Q2 It is also in the high-Q2
region where d2(Q2) is a measure of color polarizability [82 83] through its relation to the
gluon field strength tensor [80] We refer to Ref [84] for a recent review on the spin structure
of the nucleon including a discussion of sum rules for deep inelastic scattering and color
polarizabilities
What we consider in the following is the inelastic part of d2(Q2) defined as the moment
of g1(xQ2) and g2(xQ2) spin structure functions cf Eq (38)
d2(Q2) =
int x0
0
dx x2 [3g2(xQ2) + 2g1(xQ2)] (39)
23
This moment provides another testing ground for our BχPT predictions through comparison
with experiments on the neutron [22] Going towards the low-Q2 region the interpretation
of d2(Q2) in terms of color polarizabilities will fade out The above definition however
implies it is related to other VVCS polarizabilities
d2(Q2) =Q4
8M4N
[M2
NQ2
αδLT (Q2) + I1(Q2)minus IA(Q2)
] (40)
Note that d2(Q2) and its first two derivatives with respect to Q2 vanish at Q2 = 0 The
considerations in Eqs (28) and (29) have no effect on d2(Q2) since the Born contribution
from IA(Q2) and I1(Q2) cancel out Therefore d2(Q2) is a pure polarizability
In Fig 8 upper panel we show our NLO BχPT prediction and other results for d2(Q2)
While MAID [69] and BχPT describe the experimental data for the neutron [22] very well
the HB limit [50 51] is showing a fast growth with Q2 This illustrates the importance
of keeping the relativistic result Note also that even though the πN -loop contribution
is dominant both gC and the form factor in gM are essential to obtain a curvature that
reproduces the data cf Fig 9 upper panel For the proton there are to our knowledge
no experimental results to compare with However the agreement between the NLO BχPT
prediction and the MAID prediction at low energies is reasonable
F γ0(Q2) mdash fifth-order generalized forward spin polarizability
It is interesting to compare the generalized fifth-order forward spin polarizability sum
rule
γ0(Q2) =1
2π2
int infinν0
dν
radic1 +
Q2
ν2
σTT (νQ2)
ν5(41)
=64αM4
N
Q10
int x0
0
dx x4
[g1(xQ2)minus 4M2
Nx2
Q2g2(xQ2)
]
to the sum rule integrals for IA(Q2) and γ0(Q2) since they differ merely by their energy
weighting of σTT (νQ2) and a constant prefactor cf Eqs (19) (26) and (41) From IA(Q2)
to γ0(Q2) to γ0(Q2) the energy suppression is increasing by a factor of νminus2 respectively
Therefore the description of γ0(Q2) should be easiest in a low-energy effective-field theory
such as χPT whereas γ0(Q2) and IA(Q2) receive larger contributions from higher energies
In Fig 8 lower panel we show our LO and NLO BχPT predictions for γ0(Q2) One
can see that the πN -loop contribution is positive (in accordance to what we see for the cross
section σTT see Fig 10) The Delta shifts it substantially especially in the low Q2 region
24
000 005 010 015 020 025 030
-0004
-0002
0000
0002
0004
0006
0008
0010
Q2GeV
2D
d2
proton
000 005 010 015 020 025 030
-0005
0000
0005
0010
0015
Q2GeV
2D
d2
neutron
000 005 010 015 020 025 030
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
6D
proton
000 005 010 015 020 025 030
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
6D
neutron
FIG 9 Contributions of the different orders to the chiral predictions of d2(Q2) upper panel and
γ0(Q2) lower panel for the proton (left) and neutron (right) Red solid line πN -loop contribu-
tion green dot-dashed line ∆-exchange contribution orange dotted line π∆-loop contribution
blue long-dashed line total result purple dot-dot-dashed line total result without gC contribution
black short-dashed line total result without gM dipole
bringing it into a better agreement with data In general the BχPT curves start above the
empirical data points at the real-photon point and then decrease asymptotically to zero
above Q2 gt 01 GeV2 On the other hand the MAID prediction reproduces the empirical
data at the real-photon point then decreases to negative values until about Q2 gt 006 GeV2
from where it also starts to asymptotically approach zero Consequently our NLO BχPT
prediction of γ0(Q2) is consistently above the MAID prediction This is very different to
what we saw for IA(Q2) in Fig 4 upper panel where the MAID prediction at the real-
photon point is above the experimental value While the agreement of our predictions with
the empirical data is in general quite good for all moments of σTT (νQ2) one should point
out that both for γ0n(Q2) and γ0p(Q2) we overestimate the data at low Q2 For IA(Q2) such
observation cannot be made because ∆IA(0) = 0 and thus IA(0) is given by the empirical
Pauli form factor only From IA(Q2) γ0(Q2) and γ0(Q2) the latter has the smallest however
25
non-negligible dependence on gC and the dipole in gM cf Fig 9 lower panel
The πN -loop ∆-exchange and π∆-loop contributions to the NLO BχPT prediction of
the static fifth-order forward spin polarizability amount to in units of 10minus4 fm6
γ0p = 112(30) asymp 208minus 096minus 001 (42a)
γ0n = 195(30) asymp 292minus 096minus 001 (42b)
while the slope is composed as follows in units of 10minus4 fm8
dγ0p(Q2)
dQ2
∣∣∣∣Q2=0
= minus084(10) asymp minus100 + 016 + 000 (43a)
dγ0n(Q2)
dQ2
∣∣∣∣Q2=0
= minus142(15) asymp minus158 + 016 + 000 (43b)
Note that the HB prediction of the static polarizability γ0p (423 at O(p3) and 365 at O(ε3)
[78 85]) is almost one order of magnitude larger than the empirical value and therefore not
shown in Fig 8
G Summary
Our results are summarized in Table II where we give the contributions of the different
orders to the chiral predictions of the polarizabilities and their slopes at the real-photon
point A quantitative comparison of our predictions for the static spin polarizabilities to
the work of Bernard et al [31] and different empirical evaluations is shown in Table III We
can see that the inclusion of the Delta turns out to be very important for all moments of
the helicity-difference cross section To describe the Q2 behavior of the polarizabilities the
magnetic coupling of the N rarr ∆ transition should be modified by a dipole form factor as
has been observed previously in the description of electroproduction data [33] This dipole
form factor effectively takes account of vector-meson exchanges The Coulomb-quadrupole
N rarr ∆ transition despite its subleading order is important in the description of some
moments of spin structure functions This is contrary to what we saw for the moments of
unpolarized structure functions [30] where the Coulomb coupling had a negligible effect
The π∆ loops are mainly relevant for the generalized GDH integrals
26
TABLE II The NLO BχPT predictions for the forward VVCS polarizabilities and their slopes at
Q2 = 0 The contributions of the πN loops the ∆ exchange and the π∆ loops are shown together
with the combined total result Note that IA(0) = I1(0) = d2(0) = 0 and (d2)prime = 0
πN loops ∆ exchange π∆ loops Total
γ0 p 201 minus284 minus010minus093(92)
(10minus4 fm4) n 298 003(92)
δLT p 150 minus016 minus002132(15)
(10minus4 fm4) n 235 218(23)
γ0 p 208 minus096 minus001112(30)
(10minus4 fm6) n 292 195(30)
(γ0)prime p minus033011 001
minus022(4)
(10minus4 fm6) n minus073 minus061(7)
(δLT )prime p minus080 minus004 minus001minus085(8)
(10minus4 fm6) n minus119 minus124(12)
(γ0)prime p minus100016 000
minus084(10)
(10minus4 fm8) n minus158 minus142(15)
(∆IA)prime p 238 minus1121 025minus858(343)
(GeVminus2) n 141 minus955(343)
(∆I1)prime p 034 minus053 058039(4)
(GeVminus2) n minus107 minus101(10)
TABLE III Our NLO BχPT predictions for the spin polarizabilities at Q2 = 0 compared with
the BχPT+∆ predictions from Bernard et al [31] and the available empirical information Where
the reference is not given the empirical number is provided by the MAID analysis [55 69] with
unspecified uncertainty
Proton Neutron
This work BχPT+∆ Empirical This work BχPT+∆ Empirical
γ0 minus093(92) minus174(40) minus100(8)(12) [19] 003(92) minus077(40) minus0005
(10minus4 fm4) minus090(8)(11) [78] [MAID]
minus0929(105) [59]
δLT 132(15) 240(1) 134 218(23) 238(3) 203
(10minus4 fm4) [MAID] [MAID]
γ0 112(30) 060(7)(7) [78] 195(30) 123
(10minus4 fm6) 0484(82) [59] [MAID]
IV CONCLUSIONS
We have presented a complete NLO calculation of the polarized non-Born VVCS ampli-
tudes in covariant BχPT with pion nucleon and ∆(1232) fields The dispersion relations
between the VVCS amplitudes and the tree-level photoabsorption cross sections served as a
27
cross-check of these calculations
The obtained moments of the proton and neutron spin structure functions related to gen-
eralized polarizabilities and GDH-type integrals agree well with the available experimental
data The description of their Q2 evolution is improved compared to the previous χPT pre-
dictions In particular the NLO BχPT predictions obtained here give a better description
of the empirical data (eg from the Jefferson Laboratory ldquoSpin Physics Programrdquo) than
the HB [50 51] and IR [58] calculations
The demonstrated predictive power of the χPT framework amplitudes makes it well
suited for extending the χPT evaluation of the TPE effect in the hyperfine structure of
(muonic-)hydrogen [15ndash17] to next-to-leading order
ACKNOWLEDGEMENTS
We thank Lothar Tiator and Marc Vanderhaeghen for helpful discussions This work
is supported by the Deutsche Forschungsgemeinschaft (DFG) through the Collaborative
Research Center [The Low-Energy Frontier of the Standard Model (SFB 1044)] JMA ac-
knowledges support from the Community of Madrid through the ldquoPrograma de atraccion de
talento investigador 2017 (Modalidad 1)rdquo and the Spanish MECD grants FPA2016-77313-P
FH gratefully acknowledges financial support from the Swiss National Science Foundation
Appendix A Tensor decompositions of the VVCS amplitudes
In this appendix we review the decomposition of the forward VVCS process into tensor struc-
tures and scalar amplitudes In particular we consider the connection between the covariant and
the semi-relativistic decomposition in the lab frame that is defined in terms of the conventional
transverse longitudinal transverse-transverse and transverse-longitudinal amplitudes
As explained in Sec II A the process of forward VVCS off the nucleon can be described in
terms of four explicitly covariant amplitudes S1 2 and T1 2 [3]
T (νQ2) =
(minusgmicroν +
qmicroqν
q2
)T1(νQ2) +
1
M2N
(pmicro minus p middot q
q2qmicro)(
pν minus p middot qq2
qν)T2(νQ2) (A1)
minus 1
MNγmicroναqα S1(νQ2)minus 1
M2N
(γmicroνq2 + qmicroγναqα minus qνγmicroαqα
)S2(νQ2)
εprimelowastmicro εν
where εmicro (εprimelowastmicro ) are the incoming (outgoing) photon polarization vectors ν is the photon lab-frame
energy and Q2 is the photon virtuality Alternatively the decomposition in the laboratory frame
28
(which in the forward case coincides with the Breit frame) is parametrized in terms of the nucleon
Pauli matrices ~σ and the four scalar functions fL fT gTT and gLT
T (νQ2) = ε0 εprimelowast0 fL(νQ2) + (~ε primelowast middot ~ε ) fT (νQ2) + i~σ middot (~ε primelowast times ~ε ) gTT (νQ2) (A2)
minusi~σ middot [(ε0~εprimelowast minus ~ε ε primelowast0 )times q] gLT (νQ2)
Here ~q and q = ~q|~q | are the photon three-momentum in the lab system and its unit vector The
modified polarization vector components are given by
ε0 =
[ε0 minus
ν
|~q | (~ε middot q )
] |~q |Q ~ε = ~εminus q (~ε middot q ) (A3)
where ε = (ε0~ε ) is the usual incoming photon polarization vector and εprimelowast the outgoing polarization
vector The LEX of the lab frame amplitudes [Eq (10)] can serve in particular as the definition
of the generalized polarizabilities The lab frame amplitudes are also conveniently used for the
definition of the response functions see the example of the scalar amplitude gLT (νQ2) and the
corresponding response function σLT (νQ2) below in App B
Appendix B Photoabsorption cross sections
In the forward kinematics the spin-dependent VVCS amplitudes and the spin polarizabilities
can be described in terms of the polarized structure functions g1(xQ2) and g2(xQ2) or equiv-
alently the helicity-difference cross section σTT (νQ2) and the longitudinal-transverse response
function σLT (νQ2) with the help of dispersion relations (5) and the optical theorem (3) In this
way the photoabsorption cross sections measured in electroproduction processes form the basis
for most empirical evaluations shown throughout Sec III In the following we present the BχPT
predictions for the tree-level cross sections of πN - π∆- and ∆-production through photoabsorption
on the nucleon cf Figs 8 9 and 10 in Ref [30] In Secs B 1 and B 2 we will discuss the leading
πN -production channel and the ∆-production channel respectively We used these cross sections
to verify the polarizability predictions obtained otherwise from the calculated non-Born VVCS
amplitudes Due to the bad high-energy behavior of the π∆-production cross sections in BχPT
cf Fig 10 the dispersion relations in Eq (5) require further subtractions for a reconstruction of
the π∆-loop contribution to the spin-dependent VVCS amplitudes Therefore not all polarizabil-
ities could be verified but only those appearing as higher-order terms in the LEX of the VVCS
amplitudes such as γ0 [16]
29
proton neutron
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Neutron
02 04 06 08 100
50
100
150
200
250
[GeV]
T [b]
02 04 06 08 10
-50
0
50
100
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Neutron
02 04 06 08 100
50
100
150
200
250
[GeV]
T [b]
02 04 06 08 10
-50
0
50
100
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02040608100
50
100
150
200
250
300
[GeV]
T[b]
0204060810
-50
0
50
[GeV]
TT[b]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
0204060810
00
02
04
06
[GeV]
LTQ
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
0204060810
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
FIG 10 Photoabsorption cross sections for πN (red) and π∆ production (orange) with Q2 = 0
(solid) and Q2 = 01 GeV2 (dashed for πN and dotted for π∆ channel)
1 πN-production channel
In order to extract the response function σLT (νQ2) we have developed a method similar to
the one used to calculate σTT (νQ2) see for example Ref [86] For σLT (νQ2) however the
calculation is more complicated because one has to take into account that the associated Compton
process involves a spin-flip of the nucleon as illustrated in Fig 11 When calculating the cross
section the product of the incoming nucleon spinors has to reflect this flip
The forward VVCS amplitude related to σLT (νQ2) mdash and δLT (Q2) mdash is gLT (νQ2) It can
be extracted from Eq (A2) if one takes the modified polarization vector components in Eq (A3)
with ε = εL and εprimelowast = εlowastplusmn as input where εL = 1Q(|~q | 0 0 q0) and εplusmn = ∓ 1radic
2(0 1plusmni 0) are the
standard longitudinal and transverse polarization vectors respectively For εL and εlowastplusmn only the
choice of helicities hprime = plusmn12 and h = ∓12 gives a non-zero contribution and one obtains
χdaggerhprime T (νQ2)χh = χdaggerhprimeminusi~σ middot [(ε0~εprimelowast minus ~ε ε primelowast0 )times q] gLT (νQ2)χh =
radic2 gLT (νQ2) (B1)
where χh and χdaggerhprime are two-component Pauli spinors with opposite helicities or here spins
Let us now consider the related photoabsorption process and in particular the tree-level γlowastN rarr
30
s = 12
sprime = minus12
λ = 0 λprime = 1
rprime rprime
s=12sprime=minus1
2
λ=0λprime=1
rprimerprime
s=12
sprime=minus12
λ=0λprime=1
rprimerprime hacute=frac12h=-frac12
λacute=1λ=0
FIG 11 Relation between the forward Compton process and the photoabsorption process given
by the optical theorem In particular we show the longitudinal-transverse contribution The
double-line arrows represent the spin of the external particles while the dot represents the scalar
(longitudinal) polarization of the incoming photon Inside the blob the intermediate states are
represented eg nucleons with spins rprime (which are averaged in the calculation of the cross section)
and pions
πN channel see diagrams in Fig 8 of Ref [30] We define the πN -production amplitude as
T = uhB (PB)sumi
Ai(s t)Γi uhA(PA) (B2)
with the Dirac structures
Γ1 = γ5 (B3a)
Γ2 =1
2
[qA ε
]γ5 (B3b)
where uhA(PA) and udaggerhB (PB) are the Dirac spinors and PA and PB are the four-momenta of
the incoming and outgoing nucleons respectively When calculating the photoabsorption cross
section related to the VVCS amplitude in Eq (B1) the nucleon spin flip should be implemented
by uhprime(PA) in T dagger and uh(PA) in T together with the appropriate transverse and longitudinal
photon polarization vectors εlowastplusmn and εL
However if one wants to use the properties of the Dirac matrices it is more useful to construct
an operator to produce this spin flip in the external nucleons of Fig 11 This is accomplished by
introducing the projector ΓLT equiv 12radic
2(γ1 + iγ2)γ5 which also takes into account the extra factor
radic2 in Eq (B1) We checked that with this projector one correctly extracts δLT by comparing the
HB limit of our result to the HB result of Ref [50] where the authors calculate this polarizability
from the Compton amplitude directly With all those ingredients the longitudinal-transverse cross
section is calculated in the following way
σLT (νQ2) =1
64π2 s
|~pf |cm
|~pi|cm
int 1
minus1dcos θ
sumij
AiAdaggerjXij (B4)
31
with
Xij = Tr[(PB +MN )Γi(PA +MN )ΓLTγ0Γdaggerjγ
0] (B5)
where θ is the scattering angle in the center-of-mass (cm) frame and |~pi|cm (|~pf |cm) is the three-
momentum of an incoming (outgoing) particle in the cm frame An explicit calculation of the
matrix Xij leads to
X = MNQ
0 2(PB minus PA) middot εLminusradic
2|~qf |cm sin θ (sminus u)
(B6)
where |~qi|cm (|~qf |cm) is the relative three-momentum of the incoming (outgoing) particles in the
cm frame Here s t and u are the usual Mandelstam variables For the different γlowastN rarr πN
channels we obtain the following amplitudes Ai where we introduce qA as the four-momentum of
the incoming photon and qB as the four-momentum of the outgoing pion
bull γlowastprarr π0p
A1 =e gAMN
fπ
[2PA middot ε+ qA middot ε
sminusM2N
+2PB middot εminus qA middot ε
uminusM2N
] (B7a)
A2 =e gAMN
fπ
[1
sminusM2N
+1
uminusM2N
] (B7b)
bull γlowastprarr π+n
A1 =
radic2 e gAMN
fπ
[2PA middot ε+ qA middot ε
sminusM2N
+2(PA minus PB) middot ε+ qA middot ε
tminusm2π
] (B8a)
A2 =
radic2 e gAMN
fπ(sminusM2N )
(B8b)
bull γlowastnrarr π0n
A1 = 0 (B9a)
A2 = 0 (B9b)
bull γlowastnrarr πminusp
A1 =
radic2 e gAMN
fπ
[2PB middot εminus qA middot ε
uminusM2N
minus 2(PA minus PB) middot ε+ qA middot εtminusm2
π
] (B10a)
A2 =
radic2 e gAMN
fπ(uminusM2N )
(B10b)
The analytical expressions shown above were checked with the amplitudes given in Ref [87] An-
alytical expressions for the tree-level γlowastN rarr πN channel of the σLT (νQ2) and σTT (νQ2) cross
32
sections are given below (proton channels π+n and π0p neutron channel πminusp) We checked that
they reproduce the known results in the real-photon limit [45 86] To shorten the final expressions
for the cross sections which are considerably longer for finite Q2 than in the real-photon limit we
define the following dimensionless kinematic variables
αγ = (ENi )cmradics =
s+M2N +Q2
2s (B11)
απ = (ENf )cmradics =
s+M2N minusm2
π
2s (B12)
βγ = Eγcmradics =
sminusM2N minusQ2
2s (B13)
βπ = Eπcmradics =
sminusM2N +m2
π
2s (B14)
λγ = |~qi|cmradics =
radic(sminusM2
N minusQ2)2 + 4sQ2
2s (B15)
λπ = |~qf |cmradics =
radic(sminusM2
N +m2π)2 minus 4sm2
π
2s (B16)
Here (ENi )cm and (ENf )cm are the energies of the incoming and outgoing nucleon Eγcm is the energy
of the incoming photon Eπcm is the energy of the outgoing pion all in the cm frame
σ(π+n)TT = minus e2g2
AM2N
64πf2πs
3(sminusM2N )2λ4
γ
4sλπλγ
[(M2
N minus s)(M2N minusQ2 minus s)(Q2 + 2sβγβπ)
+ 2s(m2πQ
2 minus (M2N minus s)
(M2N + s(minus1 + 2βγβπ)
))λ2γ
]+ (M2
N minus s)(M2N minusQ2 minus s)
times (Q2 + 2sβγβπ minus 2sλπλγ)(Q2 + 2s(βγβπ + λπλγ)
)times log
(Q2 + 2sβγβπ minus 2sλγλπQ2 + 2sβγβπ + 2sλγλπ
) (B17)
33
σ(π0p)TT =
e2g2AM
2Nλπ
64πf2πs(sminusM2
N )2λγ
1
s2λπλ3γ
[(M2
N minus s)(M2N minusQ2 minus s)
(3M2
N + s(2βγβπ minus 3))(M2N
+ s(2βγβπ minus 1))
+ 2s
((2m2
πQ2 + (M2
N minus s)2)(M2
N minus s)minus 2(minusm2
πQ2 + (sminusM2
N )2)
times sβγβπ + 4s2(sminusM2N )β2
γβ2π
)λ2γ
]arctan
(2sλπλγ
M2N + s(2βγβπ minus 1)
)+ 2
[minus 2m2
πQ2
+ (M2N minus s)
(2(sminusM2
N ) + 4sβγβπ +(sminusM2
N +Q2)(3M2N minus 3s+ 2sβγβπ)
sλ2γ
+m2πQ
2(sminusM2N )
(M2N minus s+ 2sβγβπ)(M2
N + s(minus1 + 2βγβπ minus 2λπλγ))
+m2πQ
2(sminusM2N )
(M2N minus s+ 2sβγβπ)(M2
N + s(minus1 + 2βγβπ + 2λπλγ))
)
+1
λπλγ
(βγβπ
(minusm2
πQ2 + (M2
N minus s)(M2N minus s+ 2sβγβπ)
)minus (M2
N minus s)(M2N minusQ2 minus s)λ2
π
)log
(M2N + s(minus1 + 2βγβπ + 2λπλγ)
M2N + s(minus1 + 2βγβπ minus 2λπλγ)
)] (B18)
σ(πminusp)TT =
e2g2AM
2N
64πf2πs
3λ4γ(M2
N + s(2βπβγ minus 1))2 minus 4λ2πs
2λ2γ)
4sλπλγ
((minusM2
N +Q2 + s)(M2N
+ s(2βπβγ minus 1))
2 minus 2sλ2γ
(2λ2
πs(minusM2N +Q2 + s) +m2
πQ2))
+((M2N + s(2βπβγ minus 1)
)2
minus 4λ2πs
2λ2γ
)((M2N + s(2βπβγ minus 1)
)(M2N minus 2Q2 minus s(2βπβγ + 1)
)+ 4λ2
πs2λ2
γ
)times log
(minusM2
N minus 2βπsβγ minus 2λπsλγ + s)minus(M2N + s(2βπβγ minus 2λπλγ minus 1)
)(M2N + s(minus1
+ 2βπβγ + 2λπλγ))[((
M2N + s(2βπβγ minus 1)
)(M2N minus 2Q2 minus s(2βπβγ + 1)
)+ 4λ2
πs2λ2
γ
)times log
(minusM2
N minus 2βπsβγ + 2λπsλγ + s)
+((Q2 + 2βπsβγ)2 minus 4λ2
πs2λ2
γ
)times log
(Q2 + 2βπsβγ minus 2λπsλγQ2 + 2sβπβγ + 2λπsλγ
)]+ 4sλ2
γ
((M2
N + s(2βπβγ minus 1))2 minus 4λ2πs
2λ2γ
)times (M2
N minus s) arctanh
(2λπsλγ
M2N + 2βπsβγ minus s
) (B19)
34
σ(π+n)LT =
e2g2AM
3Nλπ
32πf2πQs
3(sminusM2N )2λ4
γ
2sλγ
[(M2
N minus s)(Q2 + 2sβ2γ)(Q2 + 2sβγβπ)
minus 4s
((M2
N minus s)(Q2 minus 2s(απ minus 1)βγ
)+Q2sβγβπ
)λ2γ + 8s3(minus1 + απ)λ4
γ
]minus M2
N minus sλπ
[(Q2 + 2sβ2
γ)(Q2 + 2sβγβπ)2 + 4s2(2(απ minus 1)βγ(Q2 + 2sβγβπ)
minusQ2λ2π
)λ2γ + 8s3(απ minus 1)2λ4
γ
]arctan
(2sλπλγ
Q2 + 2sβγβπ
) (B20)
σ(π0p)LT =
e2g2AM
3Nλπ
16πf2πQs(sminusM2
N )2λγ
1
minus2s(M2N + s(minus1 + 2βγβπ))2λ2
γ + 8s3λ2πλ
4γ
[minus 3M8
N (Q2
+ 2sβ2γ) + 2M4
Ns2
(minus (Q2 + 2sβ2
γ)(minus3 + 2βγβπ) +(Q2(minus3 + 2βγβπ + 6λ2
π) + 2sβγ
times (minus3 + 12απ + 2βγβπ minus 12απβγβπ + 4βγλ2π))λ2γ minus 4s(α2
π minus 1)λ4π
)+ 2M2
Ns3
[minus (Q2
+ 2sβ2γ)(minus1 + 2βγβπ)
(6 + βγβπ(minus9 + 2βγβπ)
)+
(Q2(3minus 12λ2
π + 4βγβπ(minus1 + 2βγβπ
+ λ2π))minus 2sβγ
(minus 3 + 4απ(3 + 2απ(3 + 2βγβπ(minus3 + βγβπ)) + 4βγ(βπ + (2minus βγβπ)
times λ2π)))λ2γ + 8s
((απ minus 1)(1 + απ minus 2βγβπ) + 2απβγλ
2π
)λ4γ
)+ s4
((Q2 + 2sβ2
γ)
times (1minus 2βγβπ)2(2βγβπ minus 3) + 2(Q2(minus1 + 6λ2
π + 2βγβπ((1minus 2βγβπ)2 minus 2λ2π)) + 2sβγ
times (minus1 + 2βγβπ + 4(minus1 + βγβπ)(minusαπ + 2απβγβπ minus βγλ2π)))λ2γ minus 8
(s(απ minus 1)(απ
+ (1minus 2βγβπ)2) + 2βγ(2sαπ +Q2βγ)λ2π
)λ4γ + 32s(απ minus 1)λ2
πλ6γ
)+ 2M6
Ns
(Q2(6
minus 7βγβπ + λ2γ) + 2sβγ(βγ(6minus 7βγβπ) + (1minus 4απ)λ2
γ)
)]+sminusM2
N
4s2λπλ3γ
[(Q2 + 2sβ2
γ)
times(3M2
N + s(2βγβπ minus 3))(M2N + s(2βγβπ minus 1)
)+ 2s
(minusM2
N (Q2 + 2sβγ minus 8sαπβγ)
+ s(2sβγ(1minus 4απ + 4απβγβπ) +Q2(1minus 2λ2π)))λ2γ + 8s3(απ minus 1)λ4
γ
]
times arctan
(2sλπλγ
M2N + s(2βγβπ minus 1)
) (B21)
35
σ(πminusp)LT =
e2g2AM
3N
64πf2πQs
3λ4γ(M2
N minusQ2 minus s)(M2N + s(2βπβγ + 2λπλγ minus 1))
times
1
M2N + s(2βπβγ minus 2λπλγ minus 1)
[4λπsλγ(M2
N minusQ2 minus s)(
2sλ2γ
(s((1minus 2λ2
π)Q2
+ 2(2απ minus 1)sβγ(2βπβγ minus 1))minusM2N (Q2 + 2(1minus 2απ)sβγ)
)+ (Q2 + 2sβ2
γ)(M2N + s(2βπβγ minus 1)
)2 + 8(απ minus 1)απs
3λ4γ
)]minus(M2N + s(2βπβγ + 2λπλγ minus 1)
)times[
log
(minusM2N minus 2βπsβγ minus 2λπsλγ + s
minusM2N minus 2βπsβγ + 2λπsλγ + s
)(minus (Q2 + 2sβ2
γ)(M2N + s(2βπβγ minus 1)
)(minusM2
N + 2Q2 + 2βπsβγ + s)minus 2sλ2γ
(minusM2
N (Q2 + 2sβγ) +Q4 +Q2s(4απβγ
minus 2βγ minus 2λ2π + 1) + 2s2βγ(4(απ minus 1)βπβγ + 1)
)minus 8(απ minus 1)2s3λ4
γ
)+ 2
(4s2λ2
γ
(2(απ minus 1)βγ(Q2 + 2βπsβγ)minus λ2
πQ2)
+ (Q2 + 2βπsβγ)2(Q2 + 2sβ2γ)
+ 8(απ minus 1)2s3λ4γ
)arctanh
(2λπsλγ
Q2 + 2βπsβγ
)] (B22)
2 ∆-production channel
The tree-level ∆-exchange diagram in Fig 2 of Ref [30] contributes to the non-Born part of
the VVCS amplitudes The contribution of the ∆ exchange to the VVCS amplitudes can be split
into [17]
S∆-exch1 (νQ2) = S∆-pole
1 (νQ2) + S∆-exch1 (νQ2) (B23a)
νS∆-exch2 (νQ2) = νS∆-pole
2 (νQ2) + νS2∆-exch
(νQ2) (B23b)
and similarly for the unpolarized VVCS amplitudes discussed in Ref [30] Here we introduced the
∆-pole contributions S∆-polei and the ∆-non-pole contributions S∆-exch
i The former amplitudes
feature a pole at the ∆(1232)-production threshold and thus are proportional to
1
[sminusM2∆][uminusM2
∆]=
1
4M2N
1
ν2∆ minus ν2
(B24)
36
They can be reconstructed from the dispersion relations in Eq (5) with the tree-level ∆-production
cross sections as input cf Fig 10 in Ref [30]
σTT (νQ2) =π2α
M2NM
2+|~q |
minus g2
MMN (M+ + ν)|~q |2 +g2E(∆minus ν)(Q2 minusMNν)2
MN(B25a)
+g2CQ
4s(∆minus ν)
MNM2∆
minus 4gMgE(Q2 minusMNν)|~q |2 minus 4gMgCQ2|~q |2
+2gEgCQ
2[minusMNM∆ |~q |2 + s(Q2 +∆ν)
]MNM∆
δ(ν minus ν∆)
σLT (νQ2) =Qπ2α
M2NM
2+|~q |
g2E(MNν minusQ2) [M∆(MN + ν)minus s]
MN(B25b)
+g2CQ
2[MNM∆|~q |2 minus s(Q2 +∆ν)
]MNM2
∆
+ gMgEM∆|~q |2 minusgMgC(Q2 minusMNν)|~q |2
M∆
+gEgC(ν minus∆)(M2
N |~q |2 minus 2Q2s)
MNM∆
δ(ν minus ν∆)
with ∆ = M∆ minusMN M+ = M∆ + MN and the Mandelstam variable s = M2N + 2MNν minus Q2
Analytical expressions for the spin structure functions g1(xQ2) and g2(xQ2) can be constructed
from Eq (3) with the flux factor K(νQ2) = |~q | =radicν2 +Q2
In the ∆-non-pole contributions to S1(νQ2) and νS2(νQ2) the pole in ν at the ∆(1232)-
production threshold has canceled out
S∆-exch1 (νQ2) =
πα
MNM2+
[g2MQ
2+ + g2
E
(∆2 minus 3Q2
)+
4g2CQ
4
M2∆
minus 8gMgEM∆ωminus (B26a)
minus2gMgCQ2(MN minus 4M∆)
M∆+
2gEgCQ2(3MN minus 2M∆)
M∆
]
νS2∆-exch
(νQ2) =2πα
MNM2+
[g2EM∆∆ωminus +
g2M MNQ
2+
2+g2C Q
2(Q2 minus∆2)
2M∆(B26b)
+gEgM M∆(M∆ω+ minus 4MNωminus)minus gEgC ∆(2Q2 +MNω+)
+gMgC Q2(4MN minus ω+)
]+S∆-exch
2 (νQ2)
ν
[M2
∆ ω2+
M2N
+ ν2
]
with Q+ =radic
(M∆ +MN )2 +Q2 and ωplusmn = (M2∆minusM2
N plusmnQ2)2M∆ and the non-pole contribution
to S2(νQ2)
S∆-exch2 (νQ2) = minus2παMNν
M∆M2+
[gM + gE
]gC (B27)
These amplitudes to the contrary are not described by the tree-level ∆-production cross sections
in the standard dispersive approach [17] This peculiarity has been previously missed e g in the
calculation of the ∆-exchange contribution to the hydrogen hyperfine splitting in Ref [88] The
importance of including the ∆-non-pole contribution is also evident when considering the BC sum
37
rule in Eq (14) The ∆-pole terms by themselves violate the BC sum rule but cancel exactly with
the ∆-non-pole terms
limνrarr0
νS∆-pole2 (νQ2) + lim
νrarr0νS2
∆-exch(νQ2) = 0 (B28)
Appendix C Polarizabilities at Q2 = 0
In this section we give analytical expressions for the static values and slopes at Q2 = 0 of
the spin polarizabilities and moments of polarized structure functions In particular we give the
HB expansion of the πN -loop contributions and the ∆-exchange contributions The complete
expressions also for the π∆-loop contributions can be found in the Supplemented material Recall
that IA(0) = I1(0) = d2(0) = 0 and dd2(Q2)dQ2
∣∣∣Q2=0
= 0
1 πN-loop contribution
Here we give analytical expressions for the πN -loop contributions to the proton and neutron
spin polarizabilities expanded in powers of micro = mπMN viz the HB expansion Note that we
choose to expand here to a high order in micro the strict HB expansion would only retain the leading
term in an analogous NLO calculation
bull Static polarizabilities (Q2 = 0)
γ0p =e2g2
A
96π3f2πm
2π
1minus 21πmicro
8minus(
59
2+ 26 logmicro
)micro2 +
1875πmicro3
64
+3
(3
2+ 26 logmicro
)micro4 +
(C1)
γ0n =e2g2
A
48π3f2πm
2π
1
2minus 9πmicro
16minus 2micro2 logmicro+
75πmicro3
128minus 3micro4
4+
(C2)
δLTp =e2g2
A
192π3f2πm
2π
1minus 9πmicro
8+
(13
2minus 2 logmicro
)micro2 minus 465πmicro3
64
minus(
47
2+ 42 logmicro
)micro4 +
(C3)
δLTn =e2g2
A
96π3f2πm
2π
1
2+
3πmicro
16+ (1 + 2 log micro)micro2 minus 105πmicro3
128+
5micro4
4+
(C4)
38
δLTn =e2g2
A
96π3f2πm
2π
1
2+
3πmicro
16+ (1 + 2 log micro)micro2 minus 105πmicro3
128+
5micro4
4+
(C5)
γ0p =e2g2
A
16π3f2πm
4π
4
45minus 3πmicro
16+
14micro2
5minus 1813πmicro3
384minus 192
5(1 + log micro)micro4
+80703πmicro5
2048+
(C6)
γ0n =e2g2
A
16π3f2πm
4π
4
45minus 5πmicro
48+
4micro2
5minus 245πmicro3
384minus 32micro4 logmicro
15
+1323πmicro5
2048+
(C7)
bull Slopes of polarizabilities at Q2 = 0
dγ0p(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
2minus 45πmicro
4+ 223micro2 minus 28515πmicro3
64
minus9
(1953
4+ 449 logmicro
)micro4 +
570255πmicro5
128+
(C8)
dγ0n(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
2minus 81πmicro
8+ 94micro2 minus 2535πmicro3
32
minus3 (1 + 90 logmicro)micro4 +84315πmicro5
1024+
(C9)
dδLTp(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
2880π3f2πm
4π
minus5
2minus 27πmicro
32+ 20micro2 minus 5865πmicro3
256
+3
(617
4+ 36 logmicro
)micro4 minus 2056845πmicro5
4096+
(C10)
dδLTn(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
minus5
4minus 81πmicro
64minus 11micro2 +
10005πmicro3
512
+15
8(11 + 48 logmicro)micro4 minus 267015πmicro5
8192+
(C11)
dIAp(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
96π2f2πmicro
2
1minus 15πmicro
4minus 1
2(115 + 88 logmicro)micro2 +
1839πmicro3
32
+5 (5 + 34 logmicro)micro4 +
(C12)
dIAn(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
48π2f2πmicro
2
1
2minus 11πmicro
8minus 1
4(1 + 20 logmicro)micro2 +
99πmicro3
64
minus25micro4
12+
(C13)
dI1p(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
96π2f2πmicro
3π
8+ 2 (4 + 3 logmicro)microminus 537πmicro2
64
minus1
2(15 + 56 logmicro)micro3 +
(C14)
39
dI1n(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
48π2f2πmicro
minus π
16+
1
4(3 + 4 logmicro)microminus 57πmicro2
128+
2micro3
3+
(C15)
γ0p(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2g2
A
16π3f2πm
6π
1
105minus 23πmicro
256+
377micro2
210minus 15551πmicro3
6144+
3371micro4
105
minus1640457πmicro5
32768+
(C16)
γ0n(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2g2
A
16π3f2πm
6π
1
105minus 153πmicro
1792+
69micro2
70minus 4615πmicro3
6144+
172micro4
35
minus120897πmicro5
32768+
(C17)
2 ∆-exchange contribution
Here we give analytical expressions for the tree-level ∆-exchange contributions to the nucleon
spin polarizabilities and their slopes at Q2 = 0 Note that the ∆-exchange contributes equally to
proton and neutron polarizabilities Recall that for the magnetic γlowastN∆ coupling we introduced a
dipole form factor to mimic vector-meson dominance gM rarr gM(1 +Q2Λ2)2
bull Static polarizabilities (Q2 = 0)
γ0 = minus e2
4πM2+
(g2M
∆2+
g2E
M2+
minus 4gMgEM+∆
) (C18)
δLT =e2M∆
4πM3+
(g2E
MNM++gMgE∆MN
minus gEgCM2
∆
) (C19)
γ0 =e2M2
N
π∆2M4+
(minusg
2M
∆2+
g2E
M2+
+4gMgE∆M+
) (C20)
bull Slopes of polarizabilities at Q2 = 0
dγ0(Q2)
dQ2
∣∣∣∣∣Q2=0
= minus e2
πM2+∆
(g2M
∆
[1
4∆2minus 1
∆M++
1
2M2+
]minus 1
Λ2
g2M
∆+
g2E
2M2+
[1
2∆minus 3
M+
]minusgMgE
M+
[1
∆2minus 5
∆M++
1
M2+
]+
1
Λ2
2gMgEM+
+2gMgC∆M2
+
minus gEgCM3
+
) (C21)
40
d δLT (Q2)
dQ2
∣∣∣∣∣Q2=0
=e2M∆∆
4πMNM2+
(g2E
∆2M2+
[1
∆minus 4
M+
]minus g2
C
∆M2∆M
2+
+gMgE∆2M+
[1
∆2
minus 3
∆M++
1
M2+
]minus 2
Λ2
gMgE∆2M+
+gMgC∆M2
∆
[1
2∆2minus 2
∆M++
1
2M2+
]minus gEgC
2M2∆M
2+
[7
∆+
1
M+
]) (C22)
dIA(Q2)
dQ2
∣∣∣∣∣Q2=0
= minusM2N
M2+
(g2M
2∆2+
g2E
MNM+minus 2gMgE
∆M+minus gEgCM∆M+
) (C23)
dI1(Q2)
dQ2
∣∣∣∣∣Q2=0
= minusM∆M2N
2M3+
(g2E
MNM∆minus gMgE∆MN
minus gEgCM2
∆
) (C24)
γ0(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2M2
N
π∆3M6+
(g2M
∆
[3M2
∆ + 2M∆MN minus 9M2N
∆2+
4M2+
Λ2
]
minus4gMgE
[5M2
∆ minus 9M2N
M+∆2+
2M+
Λ2
]+
g2E
∆M+[7M∆ minus 9MN ]
minus8gMgC∆
+4gEgCM+
) (C25)
[1] D Drechsel B Pasquini and M Vanderhaeghen Dispersion relations in real and virtual
Compton scattering Phys Rept 378 99 (2003) hep-ph0212124
[2] S E Kuhn J-P Chen and E Leader Spin structure of the nucleon mdash status and recent
results Prog Part Nucl Phys 63 1 (2009) arXiv08123535 [hep-ph]
[3] F Hagelstein R Miskimen and V Pascalutsa Nucleon polarizabilities from Compton scat-
tering to hydrogen atom Prog Part Nucl Phys 88 29 (2016) arXiv151203765 [nucl-th]
[4] B Pasquini and M Vanderhaeghen Dispersion theory in electromagnetic interactions Ann
Rev Nucl Part Sci 68 75 (2018) arXiv180510482 [hep-ph]
[5] S Gerasimov A Sum rule for magnetic moments and the damping of the nucleon magnetic
moment in nuclei Sov J Nucl Phys 2 430 (1966)
[6] S Drell and A C Hearn Exact sum rule for nucleon magnetic moments Phys Rev Lett
16 908 (1966)
[7] J S Schwinger Source theory viewpoints in deep inelastic scattering Proc Natl Acad Sci
USA 72 1 (1975)
[8] J S Schwinger Source theory viewpoints in deep inelastic scattering Electromagnetic In-
teractions and Field Theory Proceedings 14 Internationale Universitatswochen Schladming
Austria February 24-March 7 1975 Acta Phys Austriaca Suppl 14 471 (1975)
41
[9] J Schwinger Source theory discussion of deep inelastic scattering with polarized particles
Proc Natl Acad Sci USA 72 1559 (1975)
[10] A M Baldin Polarizability of nucleons Nucl Phys 18 310 (1960)
[11] M Gell-Mann M L Goldberger and W E Thirring Use of causality conditions in quantum
theory Phys Rev 95 1612 (1954)
[12] A Pineda Leading chiral logarithms to the hyperfine splitting of the hydrogen and muonic
hydrogen Phys Rev C 67 025201 (2003)
[13] C Peset and A Pineda The two-photon exchange contribution to muonic hydrogen from
chiral perturbation theory Nucl Phys B 887 69 (2014) arXiv14064524 [hep-ph]
[14] C Peset and A Pineda Model-independent determination of the two-photon exchange contri-
bution to hyperfine splitting in muonic hydrogen JHEP 04 060 arXiv161205206 [nucl-th]
[15] F Hagelstein and V Pascalutsa Proton structure in the hyperfine splitting of muonic hydro-
gen PoS CD15 077 (2016) arXiv151104301 [nucl-th]
[16] F Hagelstein Exciting Nucleons in Compton Scattering and Hydrogen-Like Atoms PhD
thesis Mainz U Inst Kernphys (2017) arXiv171000874 [nucl-th]
[17] F Hagelstein ∆(1232)-Resonance in the hydrogen spectrum Proceedings 11th International
Workshop on the Physics of Excited Nucleons (NSTAR 2017) Columbia SC USA August
20-23 2017 Few Body Syst 59 93 (2018) arXiv180109790 [nucl-th]
[18] Y Prok et al (CLAS) Moments of the spin structure functions gp1 and gd1 for 005 lt Q2 lt 30
GeV2 Phys Lett B 672 12 (2009) arXiv08022232 [nucl-ex]
[19] H Dutz et al (GDH) First measurement of the Gerasimov-Drell-Hearn sum rule for 1H from
07 GeV to 18 GeV at ELSA Phys Rev Lett 91 192001 (2003)
[20] M Amarian et al (Jefferson Lab E94010) Measurement of the generalized forward spin po-
larizabilities of the neutron Phys Rev Lett 93 152301 (2004) arXivnucl-ex0406005
[21] M Amarian et al The Q2 evolution of the generalized Gerasimov-Drell-Hearn integral for
the neutron using a 3He target Phys Rev Lett 89 242301 (2002) arXivnucl-ex0205020
[22] M Amarian et al (Jefferson Lab E94-010)Q2 evolution of the neutron spin structure moments
using a 3He target Phys Rev Lett 92 022301 (2004) arXivhep-ex0310003
[23] A Deur et al Experimental determination of the evolution of the Bjorken integral at low Q2
Phys Rev Lett 93 212001 (2004) arXivhep-ex0407007
[24] K Slifer Low Q2 measurement of gp2 and the δLT spin polarizability Spin structure at long dis-
tance Proceedings Workshop Newport News USA March 12ndash13 2009 nucl-ex09064775
(2009) AIP Conf Proc 1155 10106313203293 (2009) arXiv09064775 [nucl-ex]
[25] P Solvignon et al (E01-012) Moments of the neutron g2 structure function at intermediate
42
Q2 Phys Rev C 92 015208 (2015) arXiv13044497 [nucl-ex]
[26] A Deur Experimental studies at low Q2 of the spin structure of the nucleon at Jefferson Lab
in 9th International Workshop on Chiral Dynamics (CD18) Durham NC USA September
17-21 2018 (2019) arXiv190305661 [nucl-ex]
[27] V Sulkosky et al (Jefferson Lab E97-110) Measurement of the 3He spin-structure functions
and of neutron (3He) spin-dependent sum rules at 0035 le Q2 le 024 GeV2 Phys Lett B
805 135428 (2020) arXiv190805709 [nucl-ex]
[28] K Adhikari et al (CLAS) Measurement of the Q2 Dependence of the Deuteron Spin Structure
Function g1 and its Moments at Low Q2 with CLAS Phys Rev Lett 120 062501 (2018)
arXiv171101974 [nucl-ex]
[29] V Lensky F Hagelstein A Hiller Blin and V Pascalutsa Comment on rdquoMeasurement of
the Q2 Dependence of the Deuteron Spin Structure Function g1 and its Moments at Low Q2
with CLASrdquo (2018) arXiv180603219 [nucl-th]
[30] J M Alarcn F Hagelstein V Lensky and V Pascalutsa Forward doubly-virtual Compton
scattering off the nucleon in chiral perturbation theory at NLO the subtraction function and
moments of unpolarized structure functions (2020) arXiv200509518 [hep-ph]
[31] V Bernard E Epelbaum H Krebs and U-G Meiszligner New insights into the spin structure
of the nucleon Phys Rev D 87 054032 (2013) arXiv12092523 [hep-ph]
[32] V Lensky J M Alarcon and V Pascalutsa Moments of nucleon structure functions at
next-to-leading order in baryon chiral perturbation theory Phys Rev C 90 055202 (2014)
arXiv14072574 [hep-ph]
[33] V Pascalutsa and M Vanderhaeghen Chiral effective-field theory in the ∆(1232) region
I Pion electroproduction on the nucleon Phys Rev D 73 034003 (2006) arXivhep-
ph0512244
[34] V Pascalutsa and M Vanderhaeghen Electromagnetic nucleon-to-Delta transition in chiral
effective field theory Phys Rev Lett 95 232001 (2005) arXivhep-ph0508060
[35] T R Hemmert B R Holstein and J Kambor Systematic 1M expansion for spin 32
particles in baryon chiral perturbation theory Phys Lett B 395 89 (1997) arXivhep-
ph9606456
[36] V Pascalutsa and D R Phillips Effective theory of the ∆(1232) in Compton scattering off
the nucleon Phys Rev C 67 055202 (2003) arXivnucl-th0212024
[37] V Pascalutsa M Vanderhaeghen and S N Yang Electromagnetic excitation of the ∆(1232)-
resonance Phys Rept 437 125 (2007) arXivhep-ph0609004
[38] K A Olive et al (Particle Data Group) Review of Particle Physics Chin Phys C 38 090001
43
(2014)
[39] F E Low Scattering of light of very low frequency by systems of spin 12 Phys Rev 96
1428 (1954)
[40] M Gell-Mann and M L Goldberger Scattering of low-energy photons by particles of spin
12 Phys Rev 96 1433 (1954)
[41] A M Harun ar-Rashid A simple derivation of Schwingerrsquos sum rule for spin dependent
structure functions Nuovo Cim A 33 447 (1976)
[42] F Hagelstein and V Pascalutsa Dissecting the hadronic contributions to (g minus 2)micro by
Schwingerrsquos sum rule Phys Rev Lett 120 072002 (2018) arXiv171004571 [hep-ph]
[43] V Lensky V Pascalutsa M Vanderhaeghen and C Kao Spin-dependent sum rules con-
necting real and virtual Compton scattering verified Phys Rev D 95 074001 (2017)
arXiv170101947 [hep-ph]
[44] H Burkhardt and W N Cottingham Sum rules for forward virtual Compton scattering
Annals Phys 56 453 (1970)
[45] V Lensky and V Pascalutsa Predictive powers of chiral perturbation theory in Compton
scattering off protons Eur Phys J C 65 195 (2010) arXiv09070451 [hep-ph]
[46] H Grieszlighammer J McGovern D Phillips and G Feldman Using effective field theory to
analyse low-energy Compton scattering data from protons and light nuclei Prog Part Nucl
Phys 67 841 (2012) arXiv12036834 [nucl-th]
[47] H W Grieszlighammer J A McGovern and D R Phillips Nucleon polarisabilities at and
beyond physical pion masses Eur Phys J A 52 139 (2016) arXiv151101952 [nucl-th]
[48] E Epelbaum H Krebs and U-G Meiszligner Improved chiral nucleon-nucleon potential up to
next-to-next-to-next-to-leading order Eur Phys J A 51 53 (2015) arXiv14120142 [nucl-
th]
[49] E Epelbaum H Krebs and U-G Meiszligner Precision nucleon-nucleon potential at fifth order
in the chiral expansion Phys Rev Lett 115 122301 (2015) arXiv14124623 [nucl-th]
[50] C W Kao T Spitzenberg and M Vanderhaeghen Burkhardt-Cottingham sum rule and
forward spin polarizabilities in heavy baryon chiral perturbation theory Phys Rev D 67
016001 (2003) arXivhep-ph0209241
[51] C-W Kao D Drechsel S Kamalov and M Vanderhaeghen Higher moments of nucleon
spin structure functions in heavy baryon chiral perturbation theory and in a resonance model
Phys Rev D 69 056004 (2004) arXivhep-ph0312102
[52] T Becher and H Leutwyler Baryon chiral perturbation theory in manifestly Lorentz invariant
form Eur Phys J C 9 643 (1999) arXivhep-ph9901384
44
[53] J Gasser M E Sainio and A Svarc Nucleons with chiral loops Nucl Phys B 307 779
(1988)
[54] L S Geng J Martin Camalich L Alvarez-Ruso and M J Vicente Vacas Leading SU(3)-
breaking corrections to the baryon magnetic moments in chiral perturbation theory Phys
Rev Lett 101 222002 (2008) arXiv08051419 [hep-ph]
[55] D Drechsel S S Kamalov and L Tiator The GDH sum rule and related integrals Phys
Rev D 63 114010 (2001) arXivhep-ph0008306
[56] D Drechsel O Hanstein S S Kamalov and L Tiator A unitary isobar model for pion
photo- and electroproduction on the proton up to 1 GeV Nucl Phys A 645 145 (1999)
arXivnucl-th9807001
[57] L Tiator private communication (2020)
[58] V Bernard T R Hemmert and U-G Meiszligner Spin structure of the nucleon at low energies
Phys Rev D 67 076008 (2003) arXivhep-ph0212033
[59] O Gryniuk F Hagelstein and V Pascalutsa Evaluation of the forward Compton scattering
off protons II Spin-dependent amplitude and observables Phys Rev D 94 034043 (2016)
arXiv160400789 [nucl-th]
[60] R Zielinski The g2p Experiment A Measurement of the Protonrsquos Spin Structure Functions
PhD thesis New Hampshire U (2010) arXiv170808297 [nucl-ex]
[61] N Guler et al (CLAS) Precise determination of the deuteron spin structure at low to mod-
erate Q2 with CLAS and extraction of the neutron contribution Phys Rev C 92 055201
(2015) arXiv150507877 [nucl-ex]
[62] C E Carlson V Nazaryan and K Griffioen Proton structure corrections to electronic
and muonic hydrogen hyperfine splitting Phys Rev A 78 022517 (2008) arXiv08052603
[physicsatom-ph]
[63] V Bernard N Kaiser and U-G Meiszligner Chiral dynamics in nucleons and nuclei Int J
Mod Phys E 4 193 (1995) arXivhep-ph9501384
[64] T R Hemmert B R Holstein and J Kambor ∆(1232) and the polarizabilities of the nucleon
Phys Rev D 55 5598 (1997) arXivhep-ph9612374
[65] V Pascalutsa and R Timmermans Field theory of nucleon to higher-spin baryon transitions
Phys Rev C 60 042201 (1999) arXivnucl-th9905065
[66] V Pascalutsa Quantization of an interacting spin-32 field and the Delta isobar Phys Rev
D 58 096002 (1998) arXivhep-ph9802288
[67] H Krebs Double Virtual Compton Scattering and SpinStructure of the Nucleon PoS
CD2018 031 (2019)
45
[68] N Kochelev and Y Oh Axial anomaly and the δLT puzzle Phys Rev D 016012 (2012)
arXiv11034892 [hep-ph]
[69] D Drechsel S Kamalov and L Tiator Unitary isobar model ndash MAID2007 Eur Phys J A
69 (2007) available at httpsmaidkphuni-mainzde arXiv07100306 [nucl-th]
[70] P J Mohr B N Taylor and D B Newell CODATA recommended values of the fundamental
physical constants 2010 Rev Mod Phys 84 1527 (2012)
[71] J Ahrens et al (GDH A2) First measurement of the Gerasimov-Drell-Hearn integral for 1H
from 200 to 800 MeV Phys Rev Lett 87 022003 (2001) arXivhep-ex0105089 [hep-ex]
[72] K Helbing (GDH) Experimental verification of the GDH sum rule at ELSA and MAMI Nucl
Phys Proc Suppl 105 113 (2002)
[73] R Bradford A Bodek H S Budd and J Arrington A New parameterization of the nucleon
elastic form-factors NuInt05 proceedings of the 4th International Workshop on Neutrino-
Nucleus Interactions in the Few-GeV Region Okayama Japan 26-29 September 2005 Nucl
Phys Proc Suppl 159 127 (2006) arXivhep-ex0602017
[74] J D Bjorken Applications of the chiral U(6)timesU(6) algebra of current densities Phys Rev
148 1467 (1966)
[75] J D Bjorken Inelastic scattering of polarized leptons from polarized nucleons Phys Rev D
1 1376 (1970)
[76] V Bernard T R Hemmert and U-G Meiszligner Novel analysis of chiral loop effects in
the generalized Gerasimov-Drell-Hearn sum rule Phys Lett B 545 105 (2002) arXivhep-
ph0203167
[77] A Deur et al Experimental study of isovector spin sum rules Phys Rev D 78 032001
(2008) arXiv08023198 [nucl-ex]
[78] B Pasquini P Pedroni and D Drechsel Higher order forward spin polarizability Phys Lett
B 687 160 (2010) arXiv10014230 [hep-ph]
[79] R Jaffe g2ndashThe nucleonrsquos other spin-dependent structure function Comments Nucl Part
Phys 19 239 (1990)
[80] E V Shuryak and A Vainshtein Theory of power corrections to deep inelastic scattering in
quantum chromodynamics (II) Qminus4 effects polarized target Nucl Phys B 201 141 (1982)
[81] S Wandzura and F Wilczek Sum rules for spin dependent electroproduction Test of rela-
tivistic constituent quarks Phys Lett B 72 195 (1977)
[82] B W Filippone and X-D Ji The spin structure of the nucleon Adv Nucl Phys 26 1
(2001) arXivhep-ph0101224
[83] M Burkardt The g2 structure function Proceedings Workshop on Spin structure at long
46
distance Newport News USA March 12-13 2009 AIP Conf Proc 1155 26 (2009)
arXiv09054079 [hep-ph]
[84] A Deur S J Brodsky and G F de Teramond The spin structure of the nucleon Rept
Prog Phys 82 076201 (2019) arXiv180705250 [hep-ph]
[85] B R Holstein D Drechsel B Pasquini and M Vanderhaeghen Higher order polarizabilities
of the proton Phys Rev C 61 034316 (2000) arXivhep-ph9910427
[86] B R Holstein V Pascalutsa and M Vanderhaeghen Sum rules for magnetic moments and
polarizabilities in QED and chiral effective-field theory Phys Rev D 72 094014 (2005)
arXivhep-ph0507016
[87] B Pasquini D Drechsel and L Tiator Invariant amplitudes for pion electroproduction Eur
Phys J A 34 387 (2007) arXiv07122327 [hep-ph]
[88] A J Buchmann Non-spherical proton shape and hydrogen hyperfine splitting Proceedings
International Workshop on Precision Physics of Simple Atomic Systems (PSAS 2008) Wind-
sor Ontario Canada July 21-26 2008 Can J Phys 87 773 (2009) arXiv09104747
[physicsatom-ph]
47
000 005 010 015 020 025 030
-4
-3
-2
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
4D
proton
000 005 010 015 020 025 030
-4
-3
-2
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
4D
neutron
000 005 010 015 020 025 030
0
1
2
Q2 GeV2D
∆LT
10-
4fm
4D
proton
000 005 010 015 020 025 030
0
1
2
3
Q2 GeV2D
∆LT
10-
4fm
4D
neutron
FIG 2 Upper panel Generalized forward spin polarizability for the proton (left) and neutron
(right) as function of Q2 The result of this work the NLO BχPT prediction is shown by the
blue solid line and the blue band The red line represents the LO BχPT result The purple short-
dashed line is the O(p4) HB result from Ref [50] note that the corresponding proton curve is
outside of the plotted range The black dotted line is the MAID model prediction [55ndash57] which
is taken from Ref [1] (proton) and Ref [20] (neutron) The pink band is the IR+∆ result from
Ref [58] and the gray band is the BχPT+∆ result from Ref [31] Empirical extractions for the
proton Ref [18] (blue dots) Ref [59] (purple square) and Ref [60] (orange triangle uncertainties
added in quadrature) and neutron Ref [20] (blue diamonds) and Ref [61] (green dots statistical
and systematic uncertainties added in quadrature) Lower Panel Longitudinal-transverse spin
polarizability for the proton (left) and neutron (right) The orange dot-dashed and purple short-
dashed lines are the O(p3) and O(p4) HB results from Ref [50] The pink band is the IR result
from Ref [58] and the gray band is the covariant BχPT+∆ result from Ref [31] The black dotted
line is the MAID model prediction [55ndash57] note that for the proton we use the updated estimate
from Ref [1] obtained using the π η ππ channels
12
the static forward spin polarizability amount to in units of 10minus4 fm4
γ0p = minus093(92) asymp 201minus 284minus 010 (20a)
γ0n = 003(92) asymp 298minus 284minus 010 (20b)
while the slope is composed as follows in units of 10minus4 fm6
dγ0p(Q2)
dQ2
∣∣∣∣Q2=0
= minus022(4) asymp minus033 + 011 + 001 (21a)
dγ0n(Q2)
dQ2
∣∣∣∣Q2=0
= minus061(7) asymp minus073 + 011 + 001 (21b)
Figure 2 upper panel shows our NLO prediction as well as the LO πN loops compared
to different experimental and theoretical results For the proton we have one determination
at the real-photon point by the GDH collaboration [19] γ0p = minus100(8)(12)times10minus4 fm4 and
further Jefferson Laboratory data [18 60] at very low Q2 For the neutron only data at finite
Q2 are available [20 61] The experimental data for the proton are fairly well reproduced
in the whole Q2 range considered here while for the neutron the agreement improves with
increasing Q2 The HB limit of our πN -loop contribution reproduces the results published
in Refs [50 63] for arbitrary Q2 In addition our prediction is compared to the MAID
model [1 20] the IR+∆ calculation of Ref [58] and the BχPT+∆ result of Ref [31]
The πN -production channel gives a positive contribution to the photoabsorption cross
section σTT (νQ2) at low Q2 cf Fig 10 Accordingly one observes that the πN loops give
a sizeable positive contribution to γ0(Q2) The Delta on the other hand has a very large
effect by cancelling the πN loops and bringing the result close to the empirical data From
Fig 3 upper panel one can see that it is the ∆ exchange which dominates while π∆
loops are negligible This was expected since the forward spin polarizability sum rule is an
integral over the helicity-difference cross section in which σ32 is governed by the Delta at
low energies (the relevant energy region for the sum rule)
To elucidate the difference between the present calculation and the one from Ref [31] we
note that the two calculations differ in the following important aspects Firstly Ref [31] uses
the small-scale counting [64] that considers ∆ and mπ as being of the same size ∆ sim mπ
In practice this results in a set of π∆-loop graphs which contains graphs with one or two
γ∆∆ couplings and hence two or three Delta propagators Such graphs are suppressed in
the δ-counting and thus omitted from our calculation while present in that of Ref [31]
Secondly the Lagrangians describing the interaction of the Delta are constructed differently
and assume slightly different values for the coupling constants In particular we employ
13
000 005 010 015 020 025 030
-3
-2
-1
0
1
2
Q2GeV
2D
Γ01
0-
4fm
4D
proton
000 005 010 015 020 025 030
-3
-2
-1
0
1
2
3
Q2GeV
2D
Γ01
0-
4fm
4D
neutron
000 005 010 015 020 025 030
00
05
10
15
Q2GeV
2D
∆LT1
0-
4fm
4D
proton
000 005 010 015 020 025 030
00
05
10
15
20
25
Q2GeV
2D
∆LT1
0-
4fm
4D
neutron
FIG 3 Contributions of the different orders to the chiral predictions of γ0(Q2) upper paneland δLT (Q2) lower panel for the proton (left) and neutron (right) Red solid line πN -loop
contribution green dot-dashed line ∆-exchange contribution orange dotted line π∆-loop con-
tribution blue long-dashed line total result purple dot-dot-dashed line total result without gCcontribution black short-dashed line total result without gM dipole
(where possible) the so-called ldquoconsistentrdquo couplings to the Delta field ie those couplings
that project out the spurious degree of freedom see Refs [37 65 66] The authors of
Ref [31] on the other hand use couplings where the consistency in this sense is not enforced
The effects of these differences are of higher order in the δ-counting expansion and their
contribution to the Q2 dependence of the considered polarizabilities is expected to be rather
small however the differences at Q2 = 0 could be noticeable [67]
Finally as mentioned in Sec II C the inclusion of the dipole form factor in the magnetic
coupling gM is expected to be important to generate the correct Q2 behaviour of the polar-
izabilities Comparing our predictions for the forward spin polarizability with and without
inclusion of the form factor see Fig 3 upper panel confirms this Without the dipole
our results for the proton and neutron are closer to the ones from Ref [31] where the form
factor is not included For the neutron our prediction without the dipole is able to describe
14
the experimental points at very low Q2 deviating from the data with increasing Q2 The
π∆-loop contribution does not modify the Q2 behavior of γ0(Q2) and only differs from
Ref [31] by a small global shift Note also the relatively large effect of gC which generates
a sign change for virtualities above sim 02 GeV2 see Fig 3 upper panel
B δLT (Q2) mdash longitudinal-transverse polarizability
The longitudinal-transverse spin polarizability
δLT (Q2) =1
2π2
int infinν0
dν
radic1 +
Q2
ν2
σLT (νQ2)
Qν2(22)
=16αM2
N
Q6
int x0
0
dx x2[g1(xQ2) + g2(xQ2)
]
contains information about the spin structure of the nucleon and is another important
input in the determination of the (muonic-)hydrogen hyperfine splitting [16 62] It is also
relevant in studies of higher-twist corrections to the structure function g2(xQ2) given by
the moment d2(Q2) [51] see Section III E The peculiarity of the response encoded in this
polarizability is that it involves a spin flip of the nucleon and a polarization change of the
photon see App B and Fig 11
It is expected that the Delta isobar gives only a small contribution to δLT (Q2) what makes
this polarizability a potentially clean test case for chiral calculations Consequently there are
relatively many different theoretical calculations of δLT (Q2) coming from different versions
of χPT with baryons (HB IR and covariant) Ref [50] found a systematic deviation of the
HB result for δLTn(Q2) from the MAID model prediction This disagreement was identified
by the authors of Ref [68] as a puzzle involving the neutron δLT polarizabilitymdashthe δLT
puzzle The IR calculation in Ref [58] also showed a deviation from the data and predicted
a rapid rise of δLT (Q2) with growing Q2 The problem is solved by keeping the relativistic
structure of the theory as the BχPT+∆ result of Ref [31] showed
As expected already the leading πN loops provide a reasonable agreement with the
experimental data cf Fig 2 lower panel Since the ∆-exchange contribution to δLT (Q2)
is small the effect of the gM form factor is negligible in this polarizability as is that of the gC
coupling cf Fig 3 lower panel In fact we predict both the ∆-exchange and the π∆-loop
contributions to be small and negative This is in agreement with the MAID model which
predicts a small and negative contribution of the P33 wave to δLT (Q2) However in the
calculation of Ref [31] which is different from the one presented here only in the way the
∆(1232) is included the contribution of this resonance to δLTp(Q2) is sizeable and positive
15
The authors of that work attributed this large contribution to diagrams where the photons
couple directly to the Delta inside a loop As mentioned in Sec III F the effect of such loop
diagrams does not change the Q2 behaviour of the polarizabilities On the other hand it
can produce a substantial shift of the δLT (Q2) as a whole A higher-order calculation should
resolve the discrepancy between the two covariant approaches however it will partially lose
the predictive power since the LECs appearing at higher orders will have to be fitted to
experimental data
The πN -loop ∆-exchange and π∆-loop contributions to the NLO BχPT prediction of
the static longitudinal-transverse polarizability are in units of 10minus4 fm4
δLTp = 132(15) asymp 150minus 016minus 002 (23a)
δLTn = 218(23) asymp 235minus 016minus 002 (23b)
while the slopes are in units of 10minus4 fm6
dδLTp(Q2)
dQ2
∣∣∣∣Q2=0
= minus085(8) asymp minus080minus 004minus 001 (24a)
dδLTn(Q2)
dQ2
∣∣∣∣Q2=0
= minus124(12) asymp minus119minus 004minus 001 (24b)
C IA(Q2) mdash a generalized GDH integral
The helicity-difference cross section σTT exhibits a faster fall-off in ν than its spin-averaged
counterpart σT This is due to a cancellation between the leading (constant) terms of σ12
and σ32 at large ν3 The resulting 1ν fall-off of the helicity-difference cross section allows
one to write an unsubtracted dispersion relation for the VVCS amplitude gnonpoleTT (νQ2) cf
Eq (10a) This is the origin of the GDH sum rule [5 6]
minus α
2M2N
κ2 =1
2π2
int infinν0
dνσTT (ν)
ν (25)
which establishes a relation to the anomalous magnetic moment κ It is experimentally
verified for the nucleon by MAMI (Mainz) and ELSA (Bonn) [71 72]
There are two extensions of the GDH sum rule to finite Q2 the generalized GDH integrals
3 Notice that a constant term in σTT at ν rarrinfin is forbidden by crossing symmetry
16
000 005 010 015 020 025 030
-30
-25
-20
-15
-10
-05
00
Q2 GeV2D
I Aproton
000 005 010 015 020 025 030
-25
-20
-15
-10
-05
00
Q2 GeV2D
I A
neutron
000 005 010 015 020 025 030
-05
00
05
Q2 GeV2D
I 1
proton
000 005 010 015 020 025 030
-10
-08
-06
-04
-02
00
02
Q2 GeV2D
I 1
neutron
FIG 4 Upper panel The generalized GDH integral IA(Q2) for the proton (left) and neutron
(right) as function of Q2 The result of this work the NLO BχPT prediction is shown by the blue
solid line and the blue band The red line represents the LO BχPT result The purple short-dashed
line is the O(p4) HB result from Ref [50 51] The gray band is the BχPT+∆ result from Ref [31]
The black dotted line is the MAID model prediction [69] Experimental extractions for the proton
Ref [60] (orange triangle uncertainties added in quadrature) and neutron Refs [21][27] where
magenta dotsorange diagonal crosses correspond to data and red squareslilac crosses correspond
to data plus extrapolation to unmeasured energy regions The green stars at the real-photon point
are derived from the anomalous magnetic moments κp asymp 1793 and κn asymp minus1913 [70] Lower
panel The generalized GDH integral I1(Q2) for the proton (left) and neutron (right) as function
of Q2 The purple short-dashed line is the HB result from Ref [51] Experimental extractions for
the proton Ref [18] (blue dots) and Ref [60] (orange triangle uncertainties added in quadrature)
and neutron Ref [61][27] (uncertainties added in quadrature) where green dotsorange diagonal
crosses correspond to data and gray squareslilac crosses correspond to data plus extrapolation to
unmeasured energy regions
17
IA(Q2) and I1(Q2) The latter will be discussed in Sec III D The former is defined as4
minus α
2M2N
IA(Q2) = minus 1
8π2
int infinν0
dν
radic1 +
Q2
ν2
σTT (νQ2)
ν(26)
=α
Q2
int x0
0
dx[4M2
Nx2
Q2g2(xQ2)minus g1(xQ2)
]
Due to its energy weighting the integral in Eq (26) converges slower than the one in the
generalized forward spin polarizability sum rule (19) Therefore knowledge of the cross
section at higher energies is required and the evaluation of the generalized GDH integral
IA(Q2) is not as simple as the evaluation of γ0(Q2)
The generalized GDH integral IA(Q2) is directly related to the non-pole amplitude
gnonpoleTT (νQ2) which differs from non-Born amplitude gTT (νQ2) by a term involving the
elastic Pauli form factor
gnonpoleTT (νQ2) = gTT (νQ2)minus 2παν
M2N
F 22 (Q2) (27)
cf Eqs (2a) and (8a) Consequently IA(Q2) is not a pure polarizability but also contains
an elastic contribution The ldquonon-polarizabilityrdquo or the Born part of IA(Q2) is given by
IBornA (Q2) = IA(Q2)minus∆IA(Q2) = minus1
4F 2
2 (Q2) (28)
where we refer to the polarizability part as ∆IA(Q2) The same is true for the generalized
GDH integral I1(Q2) which is directly related to Snonpole1 (νQ2)
IBorn1 (Q2) = I1(Q2)minus∆I1(Q2) = minus1
4F 2
2 (Q2) (29)
In the following we will add the Born parts to our LO and NLO BχPT predictions for
the polarizabilities ∆IA(Q2) and ∆I1(Q2) employing an empirical parametrization for the
elastic Pauli form factor [73] This allows us to compare to the experimental results for
IA(Q2) and I1(Q2) cf Fig 4 Note that the blue error bands only describe the uncertainties
of our BχPT predictions of the polarizabilities while the elastic contributions are considered
to be exact as explained in Sec II C The uncertainties of the polarizability predictions are
therefore better reflected in Fig 5 where we show the contributions of the different orders
to the BχPT predictions of ∆IA(Q2) and ∆I1(Q2) as well as the total results with error
bands
4 Note that IA(Q2) is sometimes called ITT (Q2)
18
000 005 010 015 020 025 030
-08
-06
-04
-02
00
02
Q2 GeV2D
DI A
proton
000 005 010 015 020 025 030
-10
-08
-06
-04
-02
00
02
Q2 GeV2D
DI A
neutron
000 005 010 015 020 025 030-005
000
005
010
015
020
025
Q2 GeV2D
DI 1
proton
000 005 010 015 020 025 030-015
-010
-005
000
005
010
015
Q2 GeV2D
DI 1
neutron
FIG 5 Contributions of the different orders to the chiral predictions of ∆IA(Q2) upper paneland ∆I1(Q2) lower panel for the proton (left) and neutron (right) Red solid line πN -loop
contribution green dot-dashed line ∆-exchange contribution orange dotted line π∆-loop contri-
bution blue solid line and blue band total result purple dot-dot-dashed line total result without
gC contribution black short-dashed line total result without gM dipole
The E97-110 experiment at Jefferson Lab has recently published their data for IAn(Q2) in
the region of 0035 GeV2 lt Q2 lt 024 GeV2 [27] In addition there are results for IAn(Q2)
from the earlier E94-010 experiment [21] and for IAp(Q2) from the E08-027 experiment [60]
The O(p4) HB calculation gives a large negative effect [51] which does not describe the data
The BχPT+∆ result from Ref [31] which mainly differs from our work by the absence of
the dipole form factor in gM looks similar to this HB result and only describes the data
points at lowest Q2 Our NLO prediction however follows closely the Q2 evolution of the
data In Fig 5 upper panel we show the polarizability ∆IA(Q2) whose Q2 evolution is
clearly dominated by the ∆ exchange Similar to the case of γ0p(Q2) inclusion of the dipole
in gM and the Coulomb coupling gC is very important in order to describe the experimental
data The LO prediction on the other hand slightly overestimates the data cf Fig 4
upper panel
19
At the real-photon point IA(0) = minusκ2
4and ∆IA(0) = 0 Therefore we give only the
slope of the polarizability ∆IA(Q2) [showing also the separate contributions from πN loops
∆ exchange and π∆ loops] in units of GeVminus2
d∆IAp(Q2)
dQ2
∣∣∣∣Q2=0
= minus858(343) asymp 238minus 1121 + 025 (30a)
d∆IAn(Q2)
dQ2
∣∣∣∣Q2=0
= minus955(343) asymp 141minus 1121 + 025 (30b)
Including the empirical Pauli form factor [73] we find in units of GeVminus2
dIAp(Q2)
dQ2
∣∣∣∣Q2=0
= minus318dIAn(Q2)
dQ2
∣∣∣∣Q2=0
= minus300 (31)
D Γ1(Q2) and I1(Q2) mdash the first moment of the structure function g1(xQ2)
The second variant for a generalization of the GDH sum rule to finite Q2 is defined as
minus α
2M2N
I1(Q2) = minus 1
8π2
int infinν0
dν1radic
ν2 +Q2
[σTT (νQ2) +
Q
νσLT (νQ2)
](32)
= minus α
Q2
int x0
0
dx g1(xQ2)
where I1(0) = minusκ2
4 This generalized GDH integral directly stems from the amplitude
Snonpole1 (νQ2) with the LEX from Eq (12a) It is given by the first moment of the struc-
ture function g1(xQ2) Γ1(Q2) =int x0
0dx g1(xQ2) as follows I1(Q2) =
2M2N
Q2 Γ1(Q2) The
isovector combination
Γ1(pminusn)(Q2) =
int x0
0
dx[g1p(xQ
2)minus g1n(xQ2)] (33)
is related to the axial coupling of the nucleon through the Bjorken sum rule [74 75]
limQ2rarrinfin
Γ1(pminusn)(Q2) =
gA6 (34)
As explained in Eq (28) the moment I1(Q2) splits into a polarizability part ∆I1(Q2) and
a Born part IBorn1 (Q2) Figure 4 lower panel shows the Q2 dependence of I1(Q2) which in
contrast to IA(Q2) shown in Figure 4 upper panel is clearly dominated by its Born part
and the elastic Pauli form factor The πN -loop ∆-exchange and π∆-loop contributions to
the polarizability ∆I1(Q2) are shown in Fig 5 lower panel Comparing to Fig 5 upper
20
000 005 010 015 020 025 030
-006
-004
-002
000
002
004
Q2 GeV2D
G1
proton
000 005 010 015 020 025 030
-006
-005
-004
-003
-002
-001
000
001
Q2 GeV2D
G1
neutron
FIG 6 First moment of the structure function g1(xQ2) for the proton (left) and neutron (right)
as function of Q2 The legend is the same as in Fig 4
panel one sees that ∆I1(Q2) is less sensitive to gC and the dipole form factor in gM than
∆IA(Q2)
For the proton our NLO BχPT prediction gives a very good description of the ex-
perimental data [18 60] and is in reasonable agreement with the MAID prediction [69]
For the neutron one observes good agreement with the empirical evaluations including ex-
trapolations to unmeasured energy regions starting from Q2 gt 01 GeV2 [27 61] In the
region of Q2 lt 005 GeV2 one observes an interesting tension between the recent E97-
110 experiment [27] and the data from CLAS [61] While the newest measurement finds
I1n(0035 GeV2) lt κ2n4 thus suggesting a negative slope at low Q the older measurement
found a rather large value for I1n(00496 GeV2) A similar but milder behaviour is seen in
the E97-110 [27] and E94-010 [21] data for IAn The MAID predictions do not agree with
the CODATA recommended values for the anomalous magnetic moments of the proton and
neutron [70] which in our work are imposed by using empirical parametrizations for the
elastic Pauli form factors [73] The slope of the HB result from Ref [51] is too large and
therefore only reproduces the data at very low Q2
Figure 6 shows the moment Γ1(Q2) for the proton and neutron while Fig 7 shows the
isovector combination Γ1 pminusn(Q2) The LO and NLO BχPT predictions are identical be-
cause our calculation produces the same Delta contributions for the proton and the neutron
For the isovector combination the MAID model only agrees with the data at very low
Q2 lt 010 GeV2 The same is true for the IR result [58 76] while all other chiral results
describe the data NLO BχPT (this work) BχPT+∆ [31] and HBχPT [51]
At the real-photon point I1(0) = minusκ2
4and ∆I1(0) = 0 Therefore we give only the
slope of the polarizability ∆I1(Q2) [showing also the separate contributions from πN loops
21
000 005 010 015 020 025 030-002
000
002
004
006
008
010
Q2 GeV2D
G1Hp-
nL
FIG 7 Isovector combination of Γ1(Q2) as function of Q2 The legend is the same as in Fig 6
The pink curve is the IR result from Ref [58 76] The experimental points are from Ref [77]
(brown dots) and Ref [23] (orange squares)
∆ exchange and π∆ loops] in units of GeVminus2
d∆I1p(Q2)
dQ2
∣∣∣∣Q2=0
= 039(4) asymp 034minus 053 + 058 (35a)
d∆I1n(Q2)
dQ2
∣∣∣∣Q2=0
= minus101(10) asymp minus107minus 053 + 058 (35b)
Including the empirical Pauli form factor [73] we find in units of GeVminus2
dI1p(Q2)
dQ2
∣∣∣∣Q2=0
= 580dI1n(Q2)
dQ2
∣∣∣∣Q2=0
= 553 (36)
E d2(Q2) mdash a measure of color polarizability
Another interesting moment to consider is d2(Q2) which is related to the twist-3 part of
the spin structure function g2(xQ2) [79 80]
d2(Q2) equiv 3
int 1
0
dx x2[g2(xQ2)minus gWW2 (xQ2)] (37)
where gWW2 (xQ2) is the twist-2 part of g2(xQ2) Using the Wandzura-Wilczek relation [81]
one can relate d2(Q2) to moments of the spin structure functions g1(xQ2) and g2(xQ2)
d2(Q2) =
int 1
0
dx x2 [3g2(xQ2) + 2g1(xQ2)] (38)
22
000 005 010 015 020 025 030
0000
0001
0002
0003
0004
0005
0006
0007
Q2 GeV2D
d2
proton
000 005 010 015 020 025 030
0000
0002
0004
0006
0008
0010
0012
Q2 GeV2D
d2
neutron
000 005 010 015 020 025 030
-1
0
1
2
Q2 GeV2D
Γ 010
-4
fm6
D
proton
000 005 010 015 020 025 030-2
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
6D
neutron
FIG 8 Upper panel The inelastic moment d2(Q2) for the proton (left) and neutron (right) as
function of Q2 The result of this work the NLO BχPT prediction is shown by the blue solid line
and the blue band The red line represents the LO BχPT result The purple short-dashed line is
the O(p4) HB result from Ref [50 51] The black dotted line is the MAID model prediction [69]
The experimental points for the neutron (cyan dots) are from Ref [22] Lower panel Fifth-order
generalized forward spin polarizability γ0(Q2) for the proton (left) and neutron (right) as function
of Q2 The experimental points for the proton are from Ref [59] (purple square) and Ref [78]
(orange dot)
This relation however only holds for asymptotically large Q2 It is also in the high-Q2
region where d2(Q2) is a measure of color polarizability [82 83] through its relation to the
gluon field strength tensor [80] We refer to Ref [84] for a recent review on the spin structure
of the nucleon including a discussion of sum rules for deep inelastic scattering and color
polarizabilities
What we consider in the following is the inelastic part of d2(Q2) defined as the moment
of g1(xQ2) and g2(xQ2) spin structure functions cf Eq (38)
d2(Q2) =
int x0
0
dx x2 [3g2(xQ2) + 2g1(xQ2)] (39)
23
This moment provides another testing ground for our BχPT predictions through comparison
with experiments on the neutron [22] Going towards the low-Q2 region the interpretation
of d2(Q2) in terms of color polarizabilities will fade out The above definition however
implies it is related to other VVCS polarizabilities
d2(Q2) =Q4
8M4N
[M2
NQ2
αδLT (Q2) + I1(Q2)minus IA(Q2)
] (40)
Note that d2(Q2) and its first two derivatives with respect to Q2 vanish at Q2 = 0 The
considerations in Eqs (28) and (29) have no effect on d2(Q2) since the Born contribution
from IA(Q2) and I1(Q2) cancel out Therefore d2(Q2) is a pure polarizability
In Fig 8 upper panel we show our NLO BχPT prediction and other results for d2(Q2)
While MAID [69] and BχPT describe the experimental data for the neutron [22] very well
the HB limit [50 51] is showing a fast growth with Q2 This illustrates the importance
of keeping the relativistic result Note also that even though the πN -loop contribution
is dominant both gC and the form factor in gM are essential to obtain a curvature that
reproduces the data cf Fig 9 upper panel For the proton there are to our knowledge
no experimental results to compare with However the agreement between the NLO BχPT
prediction and the MAID prediction at low energies is reasonable
F γ0(Q2) mdash fifth-order generalized forward spin polarizability
It is interesting to compare the generalized fifth-order forward spin polarizability sum
rule
γ0(Q2) =1
2π2
int infinν0
dν
radic1 +
Q2
ν2
σTT (νQ2)
ν5(41)
=64αM4
N
Q10
int x0
0
dx x4
[g1(xQ2)minus 4M2
Nx2
Q2g2(xQ2)
]
to the sum rule integrals for IA(Q2) and γ0(Q2) since they differ merely by their energy
weighting of σTT (νQ2) and a constant prefactor cf Eqs (19) (26) and (41) From IA(Q2)
to γ0(Q2) to γ0(Q2) the energy suppression is increasing by a factor of νminus2 respectively
Therefore the description of γ0(Q2) should be easiest in a low-energy effective-field theory
such as χPT whereas γ0(Q2) and IA(Q2) receive larger contributions from higher energies
In Fig 8 lower panel we show our LO and NLO BχPT predictions for γ0(Q2) One
can see that the πN -loop contribution is positive (in accordance to what we see for the cross
section σTT see Fig 10) The Delta shifts it substantially especially in the low Q2 region
24
000 005 010 015 020 025 030
-0004
-0002
0000
0002
0004
0006
0008
0010
Q2GeV
2D
d2
proton
000 005 010 015 020 025 030
-0005
0000
0005
0010
0015
Q2GeV
2D
d2
neutron
000 005 010 015 020 025 030
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
6D
proton
000 005 010 015 020 025 030
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
6D
neutron
FIG 9 Contributions of the different orders to the chiral predictions of d2(Q2) upper panel and
γ0(Q2) lower panel for the proton (left) and neutron (right) Red solid line πN -loop contribu-
tion green dot-dashed line ∆-exchange contribution orange dotted line π∆-loop contribution
blue long-dashed line total result purple dot-dot-dashed line total result without gC contribution
black short-dashed line total result without gM dipole
bringing it into a better agreement with data In general the BχPT curves start above the
empirical data points at the real-photon point and then decrease asymptotically to zero
above Q2 gt 01 GeV2 On the other hand the MAID prediction reproduces the empirical
data at the real-photon point then decreases to negative values until about Q2 gt 006 GeV2
from where it also starts to asymptotically approach zero Consequently our NLO BχPT
prediction of γ0(Q2) is consistently above the MAID prediction This is very different to
what we saw for IA(Q2) in Fig 4 upper panel where the MAID prediction at the real-
photon point is above the experimental value While the agreement of our predictions with
the empirical data is in general quite good for all moments of σTT (νQ2) one should point
out that both for γ0n(Q2) and γ0p(Q2) we overestimate the data at low Q2 For IA(Q2) such
observation cannot be made because ∆IA(0) = 0 and thus IA(0) is given by the empirical
Pauli form factor only From IA(Q2) γ0(Q2) and γ0(Q2) the latter has the smallest however
25
non-negligible dependence on gC and the dipole in gM cf Fig 9 lower panel
The πN -loop ∆-exchange and π∆-loop contributions to the NLO BχPT prediction of
the static fifth-order forward spin polarizability amount to in units of 10minus4 fm6
γ0p = 112(30) asymp 208minus 096minus 001 (42a)
γ0n = 195(30) asymp 292minus 096minus 001 (42b)
while the slope is composed as follows in units of 10minus4 fm8
dγ0p(Q2)
dQ2
∣∣∣∣Q2=0
= minus084(10) asymp minus100 + 016 + 000 (43a)
dγ0n(Q2)
dQ2
∣∣∣∣Q2=0
= minus142(15) asymp minus158 + 016 + 000 (43b)
Note that the HB prediction of the static polarizability γ0p (423 at O(p3) and 365 at O(ε3)
[78 85]) is almost one order of magnitude larger than the empirical value and therefore not
shown in Fig 8
G Summary
Our results are summarized in Table II where we give the contributions of the different
orders to the chiral predictions of the polarizabilities and their slopes at the real-photon
point A quantitative comparison of our predictions for the static spin polarizabilities to
the work of Bernard et al [31] and different empirical evaluations is shown in Table III We
can see that the inclusion of the Delta turns out to be very important for all moments of
the helicity-difference cross section To describe the Q2 behavior of the polarizabilities the
magnetic coupling of the N rarr ∆ transition should be modified by a dipole form factor as
has been observed previously in the description of electroproduction data [33] This dipole
form factor effectively takes account of vector-meson exchanges The Coulomb-quadrupole
N rarr ∆ transition despite its subleading order is important in the description of some
moments of spin structure functions This is contrary to what we saw for the moments of
unpolarized structure functions [30] where the Coulomb coupling had a negligible effect
The π∆ loops are mainly relevant for the generalized GDH integrals
26
TABLE II The NLO BχPT predictions for the forward VVCS polarizabilities and their slopes at
Q2 = 0 The contributions of the πN loops the ∆ exchange and the π∆ loops are shown together
with the combined total result Note that IA(0) = I1(0) = d2(0) = 0 and (d2)prime = 0
πN loops ∆ exchange π∆ loops Total
γ0 p 201 minus284 minus010minus093(92)
(10minus4 fm4) n 298 003(92)
δLT p 150 minus016 minus002132(15)
(10minus4 fm4) n 235 218(23)
γ0 p 208 minus096 minus001112(30)
(10minus4 fm6) n 292 195(30)
(γ0)prime p minus033011 001
minus022(4)
(10minus4 fm6) n minus073 minus061(7)
(δLT )prime p minus080 minus004 minus001minus085(8)
(10minus4 fm6) n minus119 minus124(12)
(γ0)prime p minus100016 000
minus084(10)
(10minus4 fm8) n minus158 minus142(15)
(∆IA)prime p 238 minus1121 025minus858(343)
(GeVminus2) n 141 minus955(343)
(∆I1)prime p 034 minus053 058039(4)
(GeVminus2) n minus107 minus101(10)
TABLE III Our NLO BχPT predictions for the spin polarizabilities at Q2 = 0 compared with
the BχPT+∆ predictions from Bernard et al [31] and the available empirical information Where
the reference is not given the empirical number is provided by the MAID analysis [55 69] with
unspecified uncertainty
Proton Neutron
This work BχPT+∆ Empirical This work BχPT+∆ Empirical
γ0 minus093(92) minus174(40) minus100(8)(12) [19] 003(92) minus077(40) minus0005
(10minus4 fm4) minus090(8)(11) [78] [MAID]
minus0929(105) [59]
δLT 132(15) 240(1) 134 218(23) 238(3) 203
(10minus4 fm4) [MAID] [MAID]
γ0 112(30) 060(7)(7) [78] 195(30) 123
(10minus4 fm6) 0484(82) [59] [MAID]
IV CONCLUSIONS
We have presented a complete NLO calculation of the polarized non-Born VVCS ampli-
tudes in covariant BχPT with pion nucleon and ∆(1232) fields The dispersion relations
between the VVCS amplitudes and the tree-level photoabsorption cross sections served as a
27
cross-check of these calculations
The obtained moments of the proton and neutron spin structure functions related to gen-
eralized polarizabilities and GDH-type integrals agree well with the available experimental
data The description of their Q2 evolution is improved compared to the previous χPT pre-
dictions In particular the NLO BχPT predictions obtained here give a better description
of the empirical data (eg from the Jefferson Laboratory ldquoSpin Physics Programrdquo) than
the HB [50 51] and IR [58] calculations
The demonstrated predictive power of the χPT framework amplitudes makes it well
suited for extending the χPT evaluation of the TPE effect in the hyperfine structure of
(muonic-)hydrogen [15ndash17] to next-to-leading order
ACKNOWLEDGEMENTS
We thank Lothar Tiator and Marc Vanderhaeghen for helpful discussions This work
is supported by the Deutsche Forschungsgemeinschaft (DFG) through the Collaborative
Research Center [The Low-Energy Frontier of the Standard Model (SFB 1044)] JMA ac-
knowledges support from the Community of Madrid through the ldquoPrograma de atraccion de
talento investigador 2017 (Modalidad 1)rdquo and the Spanish MECD grants FPA2016-77313-P
FH gratefully acknowledges financial support from the Swiss National Science Foundation
Appendix A Tensor decompositions of the VVCS amplitudes
In this appendix we review the decomposition of the forward VVCS process into tensor struc-
tures and scalar amplitudes In particular we consider the connection between the covariant and
the semi-relativistic decomposition in the lab frame that is defined in terms of the conventional
transverse longitudinal transverse-transverse and transverse-longitudinal amplitudes
As explained in Sec II A the process of forward VVCS off the nucleon can be described in
terms of four explicitly covariant amplitudes S1 2 and T1 2 [3]
T (νQ2) =
(minusgmicroν +
qmicroqν
q2
)T1(νQ2) +
1
M2N
(pmicro minus p middot q
q2qmicro)(
pν minus p middot qq2
qν)T2(νQ2) (A1)
minus 1
MNγmicroναqα S1(νQ2)minus 1
M2N
(γmicroνq2 + qmicroγναqα minus qνγmicroαqα
)S2(νQ2)
εprimelowastmicro εν
where εmicro (εprimelowastmicro ) are the incoming (outgoing) photon polarization vectors ν is the photon lab-frame
energy and Q2 is the photon virtuality Alternatively the decomposition in the laboratory frame
28
(which in the forward case coincides with the Breit frame) is parametrized in terms of the nucleon
Pauli matrices ~σ and the four scalar functions fL fT gTT and gLT
T (νQ2) = ε0 εprimelowast0 fL(νQ2) + (~ε primelowast middot ~ε ) fT (νQ2) + i~σ middot (~ε primelowast times ~ε ) gTT (νQ2) (A2)
minusi~σ middot [(ε0~εprimelowast minus ~ε ε primelowast0 )times q] gLT (νQ2)
Here ~q and q = ~q|~q | are the photon three-momentum in the lab system and its unit vector The
modified polarization vector components are given by
ε0 =
[ε0 minus
ν
|~q | (~ε middot q )
] |~q |Q ~ε = ~εminus q (~ε middot q ) (A3)
where ε = (ε0~ε ) is the usual incoming photon polarization vector and εprimelowast the outgoing polarization
vector The LEX of the lab frame amplitudes [Eq (10)] can serve in particular as the definition
of the generalized polarizabilities The lab frame amplitudes are also conveniently used for the
definition of the response functions see the example of the scalar amplitude gLT (νQ2) and the
corresponding response function σLT (νQ2) below in App B
Appendix B Photoabsorption cross sections
In the forward kinematics the spin-dependent VVCS amplitudes and the spin polarizabilities
can be described in terms of the polarized structure functions g1(xQ2) and g2(xQ2) or equiv-
alently the helicity-difference cross section σTT (νQ2) and the longitudinal-transverse response
function σLT (νQ2) with the help of dispersion relations (5) and the optical theorem (3) In this
way the photoabsorption cross sections measured in electroproduction processes form the basis
for most empirical evaluations shown throughout Sec III In the following we present the BχPT
predictions for the tree-level cross sections of πN - π∆- and ∆-production through photoabsorption
on the nucleon cf Figs 8 9 and 10 in Ref [30] In Secs B 1 and B 2 we will discuss the leading
πN -production channel and the ∆-production channel respectively We used these cross sections
to verify the polarizability predictions obtained otherwise from the calculated non-Born VVCS
amplitudes Due to the bad high-energy behavior of the π∆-production cross sections in BχPT
cf Fig 10 the dispersion relations in Eq (5) require further subtractions for a reconstruction of
the π∆-loop contribution to the spin-dependent VVCS amplitudes Therefore not all polarizabil-
ities could be verified but only those appearing as higher-order terms in the LEX of the VVCS
amplitudes such as γ0 [16]
29
proton neutron
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Neutron
02 04 06 08 100
50
100
150
200
250
[GeV]
T [b]
02 04 06 08 10
-50
0
50
100
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Neutron
02 04 06 08 100
50
100
150
200
250
[GeV]
T [b]
02 04 06 08 10
-50
0
50
100
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02040608100
50
100
150
200
250
300
[GeV]
T[b]
0204060810
-50
0
50
[GeV]
TT[b]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
0204060810
00
02
04
06
[GeV]
LTQ
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
0204060810
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
FIG 10 Photoabsorption cross sections for πN (red) and π∆ production (orange) with Q2 = 0
(solid) and Q2 = 01 GeV2 (dashed for πN and dotted for π∆ channel)
1 πN-production channel
In order to extract the response function σLT (νQ2) we have developed a method similar to
the one used to calculate σTT (νQ2) see for example Ref [86] For σLT (νQ2) however the
calculation is more complicated because one has to take into account that the associated Compton
process involves a spin-flip of the nucleon as illustrated in Fig 11 When calculating the cross
section the product of the incoming nucleon spinors has to reflect this flip
The forward VVCS amplitude related to σLT (νQ2) mdash and δLT (Q2) mdash is gLT (νQ2) It can
be extracted from Eq (A2) if one takes the modified polarization vector components in Eq (A3)
with ε = εL and εprimelowast = εlowastplusmn as input where εL = 1Q(|~q | 0 0 q0) and εplusmn = ∓ 1radic
2(0 1plusmni 0) are the
standard longitudinal and transverse polarization vectors respectively For εL and εlowastplusmn only the
choice of helicities hprime = plusmn12 and h = ∓12 gives a non-zero contribution and one obtains
χdaggerhprime T (νQ2)χh = χdaggerhprimeminusi~σ middot [(ε0~εprimelowast minus ~ε ε primelowast0 )times q] gLT (νQ2)χh =
radic2 gLT (νQ2) (B1)
where χh and χdaggerhprime are two-component Pauli spinors with opposite helicities or here spins
Let us now consider the related photoabsorption process and in particular the tree-level γlowastN rarr
30
s = 12
sprime = minus12
λ = 0 λprime = 1
rprime rprime
s=12sprime=minus1
2
λ=0λprime=1
rprimerprime
s=12
sprime=minus12
λ=0λprime=1
rprimerprime hacute=frac12h=-frac12
λacute=1λ=0
FIG 11 Relation between the forward Compton process and the photoabsorption process given
by the optical theorem In particular we show the longitudinal-transverse contribution The
double-line arrows represent the spin of the external particles while the dot represents the scalar
(longitudinal) polarization of the incoming photon Inside the blob the intermediate states are
represented eg nucleons with spins rprime (which are averaged in the calculation of the cross section)
and pions
πN channel see diagrams in Fig 8 of Ref [30] We define the πN -production amplitude as
T = uhB (PB)sumi
Ai(s t)Γi uhA(PA) (B2)
with the Dirac structures
Γ1 = γ5 (B3a)
Γ2 =1
2
[qA ε
]γ5 (B3b)
where uhA(PA) and udaggerhB (PB) are the Dirac spinors and PA and PB are the four-momenta of
the incoming and outgoing nucleons respectively When calculating the photoabsorption cross
section related to the VVCS amplitude in Eq (B1) the nucleon spin flip should be implemented
by uhprime(PA) in T dagger and uh(PA) in T together with the appropriate transverse and longitudinal
photon polarization vectors εlowastplusmn and εL
However if one wants to use the properties of the Dirac matrices it is more useful to construct
an operator to produce this spin flip in the external nucleons of Fig 11 This is accomplished by
introducing the projector ΓLT equiv 12radic
2(γ1 + iγ2)γ5 which also takes into account the extra factor
radic2 in Eq (B1) We checked that with this projector one correctly extracts δLT by comparing the
HB limit of our result to the HB result of Ref [50] where the authors calculate this polarizability
from the Compton amplitude directly With all those ingredients the longitudinal-transverse cross
section is calculated in the following way
σLT (νQ2) =1
64π2 s
|~pf |cm
|~pi|cm
int 1
minus1dcos θ
sumij
AiAdaggerjXij (B4)
31
with
Xij = Tr[(PB +MN )Γi(PA +MN )ΓLTγ0Γdaggerjγ
0] (B5)
where θ is the scattering angle in the center-of-mass (cm) frame and |~pi|cm (|~pf |cm) is the three-
momentum of an incoming (outgoing) particle in the cm frame An explicit calculation of the
matrix Xij leads to
X = MNQ
0 2(PB minus PA) middot εLminusradic
2|~qf |cm sin θ (sminus u)
(B6)
where |~qi|cm (|~qf |cm) is the relative three-momentum of the incoming (outgoing) particles in the
cm frame Here s t and u are the usual Mandelstam variables For the different γlowastN rarr πN
channels we obtain the following amplitudes Ai where we introduce qA as the four-momentum of
the incoming photon and qB as the four-momentum of the outgoing pion
bull γlowastprarr π0p
A1 =e gAMN
fπ
[2PA middot ε+ qA middot ε
sminusM2N
+2PB middot εminus qA middot ε
uminusM2N
] (B7a)
A2 =e gAMN
fπ
[1
sminusM2N
+1
uminusM2N
] (B7b)
bull γlowastprarr π+n
A1 =
radic2 e gAMN
fπ
[2PA middot ε+ qA middot ε
sminusM2N
+2(PA minus PB) middot ε+ qA middot ε
tminusm2π
] (B8a)
A2 =
radic2 e gAMN
fπ(sminusM2N )
(B8b)
bull γlowastnrarr π0n
A1 = 0 (B9a)
A2 = 0 (B9b)
bull γlowastnrarr πminusp
A1 =
radic2 e gAMN
fπ
[2PB middot εminus qA middot ε
uminusM2N
minus 2(PA minus PB) middot ε+ qA middot εtminusm2
π
] (B10a)
A2 =
radic2 e gAMN
fπ(uminusM2N )
(B10b)
The analytical expressions shown above were checked with the amplitudes given in Ref [87] An-
alytical expressions for the tree-level γlowastN rarr πN channel of the σLT (νQ2) and σTT (νQ2) cross
32
sections are given below (proton channels π+n and π0p neutron channel πminusp) We checked that
they reproduce the known results in the real-photon limit [45 86] To shorten the final expressions
for the cross sections which are considerably longer for finite Q2 than in the real-photon limit we
define the following dimensionless kinematic variables
αγ = (ENi )cmradics =
s+M2N +Q2
2s (B11)
απ = (ENf )cmradics =
s+M2N minusm2
π
2s (B12)
βγ = Eγcmradics =
sminusM2N minusQ2
2s (B13)
βπ = Eπcmradics =
sminusM2N +m2
π
2s (B14)
λγ = |~qi|cmradics =
radic(sminusM2
N minusQ2)2 + 4sQ2
2s (B15)
λπ = |~qf |cmradics =
radic(sminusM2
N +m2π)2 minus 4sm2
π
2s (B16)
Here (ENi )cm and (ENf )cm are the energies of the incoming and outgoing nucleon Eγcm is the energy
of the incoming photon Eπcm is the energy of the outgoing pion all in the cm frame
σ(π+n)TT = minus e2g2
AM2N
64πf2πs
3(sminusM2N )2λ4
γ
4sλπλγ
[(M2
N minus s)(M2N minusQ2 minus s)(Q2 + 2sβγβπ)
+ 2s(m2πQ
2 minus (M2N minus s)
(M2N + s(minus1 + 2βγβπ)
))λ2γ
]+ (M2
N minus s)(M2N minusQ2 minus s)
times (Q2 + 2sβγβπ minus 2sλπλγ)(Q2 + 2s(βγβπ + λπλγ)
)times log
(Q2 + 2sβγβπ minus 2sλγλπQ2 + 2sβγβπ + 2sλγλπ
) (B17)
33
σ(π0p)TT =
e2g2AM
2Nλπ
64πf2πs(sminusM2
N )2λγ
1
s2λπλ3γ
[(M2
N minus s)(M2N minusQ2 minus s)
(3M2
N + s(2βγβπ minus 3))(M2N
+ s(2βγβπ minus 1))
+ 2s
((2m2
πQ2 + (M2
N minus s)2)(M2
N minus s)minus 2(minusm2
πQ2 + (sminusM2
N )2)
times sβγβπ + 4s2(sminusM2N )β2
γβ2π
)λ2γ
]arctan
(2sλπλγ
M2N + s(2βγβπ minus 1)
)+ 2
[minus 2m2
πQ2
+ (M2N minus s)
(2(sminusM2
N ) + 4sβγβπ +(sminusM2
N +Q2)(3M2N minus 3s+ 2sβγβπ)
sλ2γ
+m2πQ
2(sminusM2N )
(M2N minus s+ 2sβγβπ)(M2
N + s(minus1 + 2βγβπ minus 2λπλγ))
+m2πQ
2(sminusM2N )
(M2N minus s+ 2sβγβπ)(M2
N + s(minus1 + 2βγβπ + 2λπλγ))
)
+1
λπλγ
(βγβπ
(minusm2
πQ2 + (M2
N minus s)(M2N minus s+ 2sβγβπ)
)minus (M2
N minus s)(M2N minusQ2 minus s)λ2
π
)log
(M2N + s(minus1 + 2βγβπ + 2λπλγ)
M2N + s(minus1 + 2βγβπ minus 2λπλγ)
)] (B18)
σ(πminusp)TT =
e2g2AM
2N
64πf2πs
3λ4γ(M2
N + s(2βπβγ minus 1))2 minus 4λ2πs
2λ2γ)
4sλπλγ
((minusM2
N +Q2 + s)(M2N
+ s(2βπβγ minus 1))
2 minus 2sλ2γ
(2λ2
πs(minusM2N +Q2 + s) +m2
πQ2))
+((M2N + s(2βπβγ minus 1)
)2
minus 4λ2πs
2λ2γ
)((M2N + s(2βπβγ minus 1)
)(M2N minus 2Q2 minus s(2βπβγ + 1)
)+ 4λ2
πs2λ2
γ
)times log
(minusM2
N minus 2βπsβγ minus 2λπsλγ + s)minus(M2N + s(2βπβγ minus 2λπλγ minus 1)
)(M2N + s(minus1
+ 2βπβγ + 2λπλγ))[((
M2N + s(2βπβγ minus 1)
)(M2N minus 2Q2 minus s(2βπβγ + 1)
)+ 4λ2
πs2λ2
γ
)times log
(minusM2
N minus 2βπsβγ + 2λπsλγ + s)
+((Q2 + 2βπsβγ)2 minus 4λ2
πs2λ2
γ
)times log
(Q2 + 2βπsβγ minus 2λπsλγQ2 + 2sβπβγ + 2λπsλγ
)]+ 4sλ2
γ
((M2
N + s(2βπβγ minus 1))2 minus 4λ2πs
2λ2γ
)times (M2
N minus s) arctanh
(2λπsλγ
M2N + 2βπsβγ minus s
) (B19)
34
σ(π+n)LT =
e2g2AM
3Nλπ
32πf2πQs
3(sminusM2N )2λ4
γ
2sλγ
[(M2
N minus s)(Q2 + 2sβ2γ)(Q2 + 2sβγβπ)
minus 4s
((M2
N minus s)(Q2 minus 2s(απ minus 1)βγ
)+Q2sβγβπ
)λ2γ + 8s3(minus1 + απ)λ4
γ
]minus M2
N minus sλπ
[(Q2 + 2sβ2
γ)(Q2 + 2sβγβπ)2 + 4s2(2(απ minus 1)βγ(Q2 + 2sβγβπ)
minusQ2λ2π
)λ2γ + 8s3(απ minus 1)2λ4
γ
]arctan
(2sλπλγ
Q2 + 2sβγβπ
) (B20)
σ(π0p)LT =
e2g2AM
3Nλπ
16πf2πQs(sminusM2
N )2λγ
1
minus2s(M2N + s(minus1 + 2βγβπ))2λ2
γ + 8s3λ2πλ
4γ
[minus 3M8
N (Q2
+ 2sβ2γ) + 2M4
Ns2
(minus (Q2 + 2sβ2
γ)(minus3 + 2βγβπ) +(Q2(minus3 + 2βγβπ + 6λ2
π) + 2sβγ
times (minus3 + 12απ + 2βγβπ minus 12απβγβπ + 4βγλ2π))λ2γ minus 4s(α2
π minus 1)λ4π
)+ 2M2
Ns3
[minus (Q2
+ 2sβ2γ)(minus1 + 2βγβπ)
(6 + βγβπ(minus9 + 2βγβπ)
)+
(Q2(3minus 12λ2
π + 4βγβπ(minus1 + 2βγβπ
+ λ2π))minus 2sβγ
(minus 3 + 4απ(3 + 2απ(3 + 2βγβπ(minus3 + βγβπ)) + 4βγ(βπ + (2minus βγβπ)
times λ2π)))λ2γ + 8s
((απ minus 1)(1 + απ minus 2βγβπ) + 2απβγλ
2π
)λ4γ
)+ s4
((Q2 + 2sβ2
γ)
times (1minus 2βγβπ)2(2βγβπ minus 3) + 2(Q2(minus1 + 6λ2
π + 2βγβπ((1minus 2βγβπ)2 minus 2λ2π)) + 2sβγ
times (minus1 + 2βγβπ + 4(minus1 + βγβπ)(minusαπ + 2απβγβπ minus βγλ2π)))λ2γ minus 8
(s(απ minus 1)(απ
+ (1minus 2βγβπ)2) + 2βγ(2sαπ +Q2βγ)λ2π
)λ4γ + 32s(απ minus 1)λ2
πλ6γ
)+ 2M6
Ns
(Q2(6
minus 7βγβπ + λ2γ) + 2sβγ(βγ(6minus 7βγβπ) + (1minus 4απ)λ2
γ)
)]+sminusM2
N
4s2λπλ3γ
[(Q2 + 2sβ2
γ)
times(3M2
N + s(2βγβπ minus 3))(M2N + s(2βγβπ minus 1)
)+ 2s
(minusM2
N (Q2 + 2sβγ minus 8sαπβγ)
+ s(2sβγ(1minus 4απ + 4απβγβπ) +Q2(1minus 2λ2π)))λ2γ + 8s3(απ minus 1)λ4
γ
]
times arctan
(2sλπλγ
M2N + s(2βγβπ minus 1)
) (B21)
35
σ(πminusp)LT =
e2g2AM
3N
64πf2πQs
3λ4γ(M2
N minusQ2 minus s)(M2N + s(2βπβγ + 2λπλγ minus 1))
times
1
M2N + s(2βπβγ minus 2λπλγ minus 1)
[4λπsλγ(M2
N minusQ2 minus s)(
2sλ2γ
(s((1minus 2λ2
π)Q2
+ 2(2απ minus 1)sβγ(2βπβγ minus 1))minusM2N (Q2 + 2(1minus 2απ)sβγ)
)+ (Q2 + 2sβ2
γ)(M2N + s(2βπβγ minus 1)
)2 + 8(απ minus 1)απs
3λ4γ
)]minus(M2N + s(2βπβγ + 2λπλγ minus 1)
)times[
log
(minusM2N minus 2βπsβγ minus 2λπsλγ + s
minusM2N minus 2βπsβγ + 2λπsλγ + s
)(minus (Q2 + 2sβ2
γ)(M2N + s(2βπβγ minus 1)
)(minusM2
N + 2Q2 + 2βπsβγ + s)minus 2sλ2γ
(minusM2
N (Q2 + 2sβγ) +Q4 +Q2s(4απβγ
minus 2βγ minus 2λ2π + 1) + 2s2βγ(4(απ minus 1)βπβγ + 1)
)minus 8(απ minus 1)2s3λ4
γ
)+ 2
(4s2λ2
γ
(2(απ minus 1)βγ(Q2 + 2βπsβγ)minus λ2
πQ2)
+ (Q2 + 2βπsβγ)2(Q2 + 2sβ2γ)
+ 8(απ minus 1)2s3λ4γ
)arctanh
(2λπsλγ
Q2 + 2βπsβγ
)] (B22)
2 ∆-production channel
The tree-level ∆-exchange diagram in Fig 2 of Ref [30] contributes to the non-Born part of
the VVCS amplitudes The contribution of the ∆ exchange to the VVCS amplitudes can be split
into [17]
S∆-exch1 (νQ2) = S∆-pole
1 (νQ2) + S∆-exch1 (νQ2) (B23a)
νS∆-exch2 (νQ2) = νS∆-pole
2 (νQ2) + νS2∆-exch
(νQ2) (B23b)
and similarly for the unpolarized VVCS amplitudes discussed in Ref [30] Here we introduced the
∆-pole contributions S∆-polei and the ∆-non-pole contributions S∆-exch
i The former amplitudes
feature a pole at the ∆(1232)-production threshold and thus are proportional to
1
[sminusM2∆][uminusM2
∆]=
1
4M2N
1
ν2∆ minus ν2
(B24)
36
They can be reconstructed from the dispersion relations in Eq (5) with the tree-level ∆-production
cross sections as input cf Fig 10 in Ref [30]
σTT (νQ2) =π2α
M2NM
2+|~q |
minus g2
MMN (M+ + ν)|~q |2 +g2E(∆minus ν)(Q2 minusMNν)2
MN(B25a)
+g2CQ
4s(∆minus ν)
MNM2∆
minus 4gMgE(Q2 minusMNν)|~q |2 minus 4gMgCQ2|~q |2
+2gEgCQ
2[minusMNM∆ |~q |2 + s(Q2 +∆ν)
]MNM∆
δ(ν minus ν∆)
σLT (νQ2) =Qπ2α
M2NM
2+|~q |
g2E(MNν minusQ2) [M∆(MN + ν)minus s]
MN(B25b)
+g2CQ
2[MNM∆|~q |2 minus s(Q2 +∆ν)
]MNM2
∆
+ gMgEM∆|~q |2 minusgMgC(Q2 minusMNν)|~q |2
M∆
+gEgC(ν minus∆)(M2
N |~q |2 minus 2Q2s)
MNM∆
δ(ν minus ν∆)
with ∆ = M∆ minusMN M+ = M∆ + MN and the Mandelstam variable s = M2N + 2MNν minus Q2
Analytical expressions for the spin structure functions g1(xQ2) and g2(xQ2) can be constructed
from Eq (3) with the flux factor K(νQ2) = |~q | =radicν2 +Q2
In the ∆-non-pole contributions to S1(νQ2) and νS2(νQ2) the pole in ν at the ∆(1232)-
production threshold has canceled out
S∆-exch1 (νQ2) =
πα
MNM2+
[g2MQ
2+ + g2
E
(∆2 minus 3Q2
)+
4g2CQ
4
M2∆
minus 8gMgEM∆ωminus (B26a)
minus2gMgCQ2(MN minus 4M∆)
M∆+
2gEgCQ2(3MN minus 2M∆)
M∆
]
νS2∆-exch
(νQ2) =2πα
MNM2+
[g2EM∆∆ωminus +
g2M MNQ
2+
2+g2C Q
2(Q2 minus∆2)
2M∆(B26b)
+gEgM M∆(M∆ω+ minus 4MNωminus)minus gEgC ∆(2Q2 +MNω+)
+gMgC Q2(4MN minus ω+)
]+S∆-exch
2 (νQ2)
ν
[M2
∆ ω2+
M2N
+ ν2
]
with Q+ =radic
(M∆ +MN )2 +Q2 and ωplusmn = (M2∆minusM2
N plusmnQ2)2M∆ and the non-pole contribution
to S2(νQ2)
S∆-exch2 (νQ2) = minus2παMNν
M∆M2+
[gM + gE
]gC (B27)
These amplitudes to the contrary are not described by the tree-level ∆-production cross sections
in the standard dispersive approach [17] This peculiarity has been previously missed e g in the
calculation of the ∆-exchange contribution to the hydrogen hyperfine splitting in Ref [88] The
importance of including the ∆-non-pole contribution is also evident when considering the BC sum
37
rule in Eq (14) The ∆-pole terms by themselves violate the BC sum rule but cancel exactly with
the ∆-non-pole terms
limνrarr0
νS∆-pole2 (νQ2) + lim
νrarr0νS2
∆-exch(νQ2) = 0 (B28)
Appendix C Polarizabilities at Q2 = 0
In this section we give analytical expressions for the static values and slopes at Q2 = 0 of
the spin polarizabilities and moments of polarized structure functions In particular we give the
HB expansion of the πN -loop contributions and the ∆-exchange contributions The complete
expressions also for the π∆-loop contributions can be found in the Supplemented material Recall
that IA(0) = I1(0) = d2(0) = 0 and dd2(Q2)dQ2
∣∣∣Q2=0
= 0
1 πN-loop contribution
Here we give analytical expressions for the πN -loop contributions to the proton and neutron
spin polarizabilities expanded in powers of micro = mπMN viz the HB expansion Note that we
choose to expand here to a high order in micro the strict HB expansion would only retain the leading
term in an analogous NLO calculation
bull Static polarizabilities (Q2 = 0)
γ0p =e2g2
A
96π3f2πm
2π
1minus 21πmicro
8minus(
59
2+ 26 logmicro
)micro2 +
1875πmicro3
64
+3
(3
2+ 26 logmicro
)micro4 +
(C1)
γ0n =e2g2
A
48π3f2πm
2π
1
2minus 9πmicro
16minus 2micro2 logmicro+
75πmicro3
128minus 3micro4
4+
(C2)
δLTp =e2g2
A
192π3f2πm
2π
1minus 9πmicro
8+
(13
2minus 2 logmicro
)micro2 minus 465πmicro3
64
minus(
47
2+ 42 logmicro
)micro4 +
(C3)
δLTn =e2g2
A
96π3f2πm
2π
1
2+
3πmicro
16+ (1 + 2 log micro)micro2 minus 105πmicro3
128+
5micro4
4+
(C4)
38
δLTn =e2g2
A
96π3f2πm
2π
1
2+
3πmicro
16+ (1 + 2 log micro)micro2 minus 105πmicro3
128+
5micro4
4+
(C5)
γ0p =e2g2
A
16π3f2πm
4π
4
45minus 3πmicro
16+
14micro2
5minus 1813πmicro3
384minus 192
5(1 + log micro)micro4
+80703πmicro5
2048+
(C6)
γ0n =e2g2
A
16π3f2πm
4π
4
45minus 5πmicro
48+
4micro2
5minus 245πmicro3
384minus 32micro4 logmicro
15
+1323πmicro5
2048+
(C7)
bull Slopes of polarizabilities at Q2 = 0
dγ0p(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
2minus 45πmicro
4+ 223micro2 minus 28515πmicro3
64
minus9
(1953
4+ 449 logmicro
)micro4 +
570255πmicro5
128+
(C8)
dγ0n(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
2minus 81πmicro
8+ 94micro2 minus 2535πmicro3
32
minus3 (1 + 90 logmicro)micro4 +84315πmicro5
1024+
(C9)
dδLTp(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
2880π3f2πm
4π
minus5
2minus 27πmicro
32+ 20micro2 minus 5865πmicro3
256
+3
(617
4+ 36 logmicro
)micro4 minus 2056845πmicro5
4096+
(C10)
dδLTn(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
minus5
4minus 81πmicro
64minus 11micro2 +
10005πmicro3
512
+15
8(11 + 48 logmicro)micro4 minus 267015πmicro5
8192+
(C11)
dIAp(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
96π2f2πmicro
2
1minus 15πmicro
4minus 1
2(115 + 88 logmicro)micro2 +
1839πmicro3
32
+5 (5 + 34 logmicro)micro4 +
(C12)
dIAn(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
48π2f2πmicro
2
1
2minus 11πmicro
8minus 1
4(1 + 20 logmicro)micro2 +
99πmicro3
64
minus25micro4
12+
(C13)
dI1p(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
96π2f2πmicro
3π
8+ 2 (4 + 3 logmicro)microminus 537πmicro2
64
minus1
2(15 + 56 logmicro)micro3 +
(C14)
39
dI1n(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
48π2f2πmicro
minus π
16+
1
4(3 + 4 logmicro)microminus 57πmicro2
128+
2micro3
3+
(C15)
γ0p(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2g2
A
16π3f2πm
6π
1
105minus 23πmicro
256+
377micro2
210minus 15551πmicro3
6144+
3371micro4
105
minus1640457πmicro5
32768+
(C16)
γ0n(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2g2
A
16π3f2πm
6π
1
105minus 153πmicro
1792+
69micro2
70minus 4615πmicro3
6144+
172micro4
35
minus120897πmicro5
32768+
(C17)
2 ∆-exchange contribution
Here we give analytical expressions for the tree-level ∆-exchange contributions to the nucleon
spin polarizabilities and their slopes at Q2 = 0 Note that the ∆-exchange contributes equally to
proton and neutron polarizabilities Recall that for the magnetic γlowastN∆ coupling we introduced a
dipole form factor to mimic vector-meson dominance gM rarr gM(1 +Q2Λ2)2
bull Static polarizabilities (Q2 = 0)
γ0 = minus e2
4πM2+
(g2M
∆2+
g2E
M2+
minus 4gMgEM+∆
) (C18)
δLT =e2M∆
4πM3+
(g2E
MNM++gMgE∆MN
minus gEgCM2
∆
) (C19)
γ0 =e2M2
N
π∆2M4+
(minusg
2M
∆2+
g2E
M2+
+4gMgE∆M+
) (C20)
bull Slopes of polarizabilities at Q2 = 0
dγ0(Q2)
dQ2
∣∣∣∣∣Q2=0
= minus e2
πM2+∆
(g2M
∆
[1
4∆2minus 1
∆M++
1
2M2+
]minus 1
Λ2
g2M
∆+
g2E
2M2+
[1
2∆minus 3
M+
]minusgMgE
M+
[1
∆2minus 5
∆M++
1
M2+
]+
1
Λ2
2gMgEM+
+2gMgC∆M2
+
minus gEgCM3
+
) (C21)
40
d δLT (Q2)
dQ2
∣∣∣∣∣Q2=0
=e2M∆∆
4πMNM2+
(g2E
∆2M2+
[1
∆minus 4
M+
]minus g2
C
∆M2∆M
2+
+gMgE∆2M+
[1
∆2
minus 3
∆M++
1
M2+
]minus 2
Λ2
gMgE∆2M+
+gMgC∆M2
∆
[1
2∆2minus 2
∆M++
1
2M2+
]minus gEgC
2M2∆M
2+
[7
∆+
1
M+
]) (C22)
dIA(Q2)
dQ2
∣∣∣∣∣Q2=0
= minusM2N
M2+
(g2M
2∆2+
g2E
MNM+minus 2gMgE
∆M+minus gEgCM∆M+
) (C23)
dI1(Q2)
dQ2
∣∣∣∣∣Q2=0
= minusM∆M2N
2M3+
(g2E
MNM∆minus gMgE∆MN
minus gEgCM2
∆
) (C24)
γ0(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2M2
N
π∆3M6+
(g2M
∆
[3M2
∆ + 2M∆MN minus 9M2N
∆2+
4M2+
Λ2
]
minus4gMgE
[5M2
∆ minus 9M2N
M+∆2+
2M+
Λ2
]+
g2E
∆M+[7M∆ minus 9MN ]
minus8gMgC∆
+4gEgCM+
) (C25)
[1] D Drechsel B Pasquini and M Vanderhaeghen Dispersion relations in real and virtual
Compton scattering Phys Rept 378 99 (2003) hep-ph0212124
[2] S E Kuhn J-P Chen and E Leader Spin structure of the nucleon mdash status and recent
results Prog Part Nucl Phys 63 1 (2009) arXiv08123535 [hep-ph]
[3] F Hagelstein R Miskimen and V Pascalutsa Nucleon polarizabilities from Compton scat-
tering to hydrogen atom Prog Part Nucl Phys 88 29 (2016) arXiv151203765 [nucl-th]
[4] B Pasquini and M Vanderhaeghen Dispersion theory in electromagnetic interactions Ann
Rev Nucl Part Sci 68 75 (2018) arXiv180510482 [hep-ph]
[5] S Gerasimov A Sum rule for magnetic moments and the damping of the nucleon magnetic
moment in nuclei Sov J Nucl Phys 2 430 (1966)
[6] S Drell and A C Hearn Exact sum rule for nucleon magnetic moments Phys Rev Lett
16 908 (1966)
[7] J S Schwinger Source theory viewpoints in deep inelastic scattering Proc Natl Acad Sci
USA 72 1 (1975)
[8] J S Schwinger Source theory viewpoints in deep inelastic scattering Electromagnetic In-
teractions and Field Theory Proceedings 14 Internationale Universitatswochen Schladming
Austria February 24-March 7 1975 Acta Phys Austriaca Suppl 14 471 (1975)
41
[9] J Schwinger Source theory discussion of deep inelastic scattering with polarized particles
Proc Natl Acad Sci USA 72 1559 (1975)
[10] A M Baldin Polarizability of nucleons Nucl Phys 18 310 (1960)
[11] M Gell-Mann M L Goldberger and W E Thirring Use of causality conditions in quantum
theory Phys Rev 95 1612 (1954)
[12] A Pineda Leading chiral logarithms to the hyperfine splitting of the hydrogen and muonic
hydrogen Phys Rev C 67 025201 (2003)
[13] C Peset and A Pineda The two-photon exchange contribution to muonic hydrogen from
chiral perturbation theory Nucl Phys B 887 69 (2014) arXiv14064524 [hep-ph]
[14] C Peset and A Pineda Model-independent determination of the two-photon exchange contri-
bution to hyperfine splitting in muonic hydrogen JHEP 04 060 arXiv161205206 [nucl-th]
[15] F Hagelstein and V Pascalutsa Proton structure in the hyperfine splitting of muonic hydro-
gen PoS CD15 077 (2016) arXiv151104301 [nucl-th]
[16] F Hagelstein Exciting Nucleons in Compton Scattering and Hydrogen-Like Atoms PhD
thesis Mainz U Inst Kernphys (2017) arXiv171000874 [nucl-th]
[17] F Hagelstein ∆(1232)-Resonance in the hydrogen spectrum Proceedings 11th International
Workshop on the Physics of Excited Nucleons (NSTAR 2017) Columbia SC USA August
20-23 2017 Few Body Syst 59 93 (2018) arXiv180109790 [nucl-th]
[18] Y Prok et al (CLAS) Moments of the spin structure functions gp1 and gd1 for 005 lt Q2 lt 30
GeV2 Phys Lett B 672 12 (2009) arXiv08022232 [nucl-ex]
[19] H Dutz et al (GDH) First measurement of the Gerasimov-Drell-Hearn sum rule for 1H from
07 GeV to 18 GeV at ELSA Phys Rev Lett 91 192001 (2003)
[20] M Amarian et al (Jefferson Lab E94010) Measurement of the generalized forward spin po-
larizabilities of the neutron Phys Rev Lett 93 152301 (2004) arXivnucl-ex0406005
[21] M Amarian et al The Q2 evolution of the generalized Gerasimov-Drell-Hearn integral for
the neutron using a 3He target Phys Rev Lett 89 242301 (2002) arXivnucl-ex0205020
[22] M Amarian et al (Jefferson Lab E94-010)Q2 evolution of the neutron spin structure moments
using a 3He target Phys Rev Lett 92 022301 (2004) arXivhep-ex0310003
[23] A Deur et al Experimental determination of the evolution of the Bjorken integral at low Q2
Phys Rev Lett 93 212001 (2004) arXivhep-ex0407007
[24] K Slifer Low Q2 measurement of gp2 and the δLT spin polarizability Spin structure at long dis-
tance Proceedings Workshop Newport News USA March 12ndash13 2009 nucl-ex09064775
(2009) AIP Conf Proc 1155 10106313203293 (2009) arXiv09064775 [nucl-ex]
[25] P Solvignon et al (E01-012) Moments of the neutron g2 structure function at intermediate
42
Q2 Phys Rev C 92 015208 (2015) arXiv13044497 [nucl-ex]
[26] A Deur Experimental studies at low Q2 of the spin structure of the nucleon at Jefferson Lab
in 9th International Workshop on Chiral Dynamics (CD18) Durham NC USA September
17-21 2018 (2019) arXiv190305661 [nucl-ex]
[27] V Sulkosky et al (Jefferson Lab E97-110) Measurement of the 3He spin-structure functions
and of neutron (3He) spin-dependent sum rules at 0035 le Q2 le 024 GeV2 Phys Lett B
805 135428 (2020) arXiv190805709 [nucl-ex]
[28] K Adhikari et al (CLAS) Measurement of the Q2 Dependence of the Deuteron Spin Structure
Function g1 and its Moments at Low Q2 with CLAS Phys Rev Lett 120 062501 (2018)
arXiv171101974 [nucl-ex]
[29] V Lensky F Hagelstein A Hiller Blin and V Pascalutsa Comment on rdquoMeasurement of
the Q2 Dependence of the Deuteron Spin Structure Function g1 and its Moments at Low Q2
with CLASrdquo (2018) arXiv180603219 [nucl-th]
[30] J M Alarcn F Hagelstein V Lensky and V Pascalutsa Forward doubly-virtual Compton
scattering off the nucleon in chiral perturbation theory at NLO the subtraction function and
moments of unpolarized structure functions (2020) arXiv200509518 [hep-ph]
[31] V Bernard E Epelbaum H Krebs and U-G Meiszligner New insights into the spin structure
of the nucleon Phys Rev D 87 054032 (2013) arXiv12092523 [hep-ph]
[32] V Lensky J M Alarcon and V Pascalutsa Moments of nucleon structure functions at
next-to-leading order in baryon chiral perturbation theory Phys Rev C 90 055202 (2014)
arXiv14072574 [hep-ph]
[33] V Pascalutsa and M Vanderhaeghen Chiral effective-field theory in the ∆(1232) region
I Pion electroproduction on the nucleon Phys Rev D 73 034003 (2006) arXivhep-
ph0512244
[34] V Pascalutsa and M Vanderhaeghen Electromagnetic nucleon-to-Delta transition in chiral
effective field theory Phys Rev Lett 95 232001 (2005) arXivhep-ph0508060
[35] T R Hemmert B R Holstein and J Kambor Systematic 1M expansion for spin 32
particles in baryon chiral perturbation theory Phys Lett B 395 89 (1997) arXivhep-
ph9606456
[36] V Pascalutsa and D R Phillips Effective theory of the ∆(1232) in Compton scattering off
the nucleon Phys Rev C 67 055202 (2003) arXivnucl-th0212024
[37] V Pascalutsa M Vanderhaeghen and S N Yang Electromagnetic excitation of the ∆(1232)-
resonance Phys Rept 437 125 (2007) arXivhep-ph0609004
[38] K A Olive et al (Particle Data Group) Review of Particle Physics Chin Phys C 38 090001
43
(2014)
[39] F E Low Scattering of light of very low frequency by systems of spin 12 Phys Rev 96
1428 (1954)
[40] M Gell-Mann and M L Goldberger Scattering of low-energy photons by particles of spin
12 Phys Rev 96 1433 (1954)
[41] A M Harun ar-Rashid A simple derivation of Schwingerrsquos sum rule for spin dependent
structure functions Nuovo Cim A 33 447 (1976)
[42] F Hagelstein and V Pascalutsa Dissecting the hadronic contributions to (g minus 2)micro by
Schwingerrsquos sum rule Phys Rev Lett 120 072002 (2018) arXiv171004571 [hep-ph]
[43] V Lensky V Pascalutsa M Vanderhaeghen and C Kao Spin-dependent sum rules con-
necting real and virtual Compton scattering verified Phys Rev D 95 074001 (2017)
arXiv170101947 [hep-ph]
[44] H Burkhardt and W N Cottingham Sum rules for forward virtual Compton scattering
Annals Phys 56 453 (1970)
[45] V Lensky and V Pascalutsa Predictive powers of chiral perturbation theory in Compton
scattering off protons Eur Phys J C 65 195 (2010) arXiv09070451 [hep-ph]
[46] H Grieszlighammer J McGovern D Phillips and G Feldman Using effective field theory to
analyse low-energy Compton scattering data from protons and light nuclei Prog Part Nucl
Phys 67 841 (2012) arXiv12036834 [nucl-th]
[47] H W Grieszlighammer J A McGovern and D R Phillips Nucleon polarisabilities at and
beyond physical pion masses Eur Phys J A 52 139 (2016) arXiv151101952 [nucl-th]
[48] E Epelbaum H Krebs and U-G Meiszligner Improved chiral nucleon-nucleon potential up to
next-to-next-to-next-to-leading order Eur Phys J A 51 53 (2015) arXiv14120142 [nucl-
th]
[49] E Epelbaum H Krebs and U-G Meiszligner Precision nucleon-nucleon potential at fifth order
in the chiral expansion Phys Rev Lett 115 122301 (2015) arXiv14124623 [nucl-th]
[50] C W Kao T Spitzenberg and M Vanderhaeghen Burkhardt-Cottingham sum rule and
forward spin polarizabilities in heavy baryon chiral perturbation theory Phys Rev D 67
016001 (2003) arXivhep-ph0209241
[51] C-W Kao D Drechsel S Kamalov and M Vanderhaeghen Higher moments of nucleon
spin structure functions in heavy baryon chiral perturbation theory and in a resonance model
Phys Rev D 69 056004 (2004) arXivhep-ph0312102
[52] T Becher and H Leutwyler Baryon chiral perturbation theory in manifestly Lorentz invariant
form Eur Phys J C 9 643 (1999) arXivhep-ph9901384
44
[53] J Gasser M E Sainio and A Svarc Nucleons with chiral loops Nucl Phys B 307 779
(1988)
[54] L S Geng J Martin Camalich L Alvarez-Ruso and M J Vicente Vacas Leading SU(3)-
breaking corrections to the baryon magnetic moments in chiral perturbation theory Phys
Rev Lett 101 222002 (2008) arXiv08051419 [hep-ph]
[55] D Drechsel S S Kamalov and L Tiator The GDH sum rule and related integrals Phys
Rev D 63 114010 (2001) arXivhep-ph0008306
[56] D Drechsel O Hanstein S S Kamalov and L Tiator A unitary isobar model for pion
photo- and electroproduction on the proton up to 1 GeV Nucl Phys A 645 145 (1999)
arXivnucl-th9807001
[57] L Tiator private communication (2020)
[58] V Bernard T R Hemmert and U-G Meiszligner Spin structure of the nucleon at low energies
Phys Rev D 67 076008 (2003) arXivhep-ph0212033
[59] O Gryniuk F Hagelstein and V Pascalutsa Evaluation of the forward Compton scattering
off protons II Spin-dependent amplitude and observables Phys Rev D 94 034043 (2016)
arXiv160400789 [nucl-th]
[60] R Zielinski The g2p Experiment A Measurement of the Protonrsquos Spin Structure Functions
PhD thesis New Hampshire U (2010) arXiv170808297 [nucl-ex]
[61] N Guler et al (CLAS) Precise determination of the deuteron spin structure at low to mod-
erate Q2 with CLAS and extraction of the neutron contribution Phys Rev C 92 055201
(2015) arXiv150507877 [nucl-ex]
[62] C E Carlson V Nazaryan and K Griffioen Proton structure corrections to electronic
and muonic hydrogen hyperfine splitting Phys Rev A 78 022517 (2008) arXiv08052603
[physicsatom-ph]
[63] V Bernard N Kaiser and U-G Meiszligner Chiral dynamics in nucleons and nuclei Int J
Mod Phys E 4 193 (1995) arXivhep-ph9501384
[64] T R Hemmert B R Holstein and J Kambor ∆(1232) and the polarizabilities of the nucleon
Phys Rev D 55 5598 (1997) arXivhep-ph9612374
[65] V Pascalutsa and R Timmermans Field theory of nucleon to higher-spin baryon transitions
Phys Rev C 60 042201 (1999) arXivnucl-th9905065
[66] V Pascalutsa Quantization of an interacting spin-32 field and the Delta isobar Phys Rev
D 58 096002 (1998) arXivhep-ph9802288
[67] H Krebs Double Virtual Compton Scattering and SpinStructure of the Nucleon PoS
CD2018 031 (2019)
45
[68] N Kochelev and Y Oh Axial anomaly and the δLT puzzle Phys Rev D 016012 (2012)
arXiv11034892 [hep-ph]
[69] D Drechsel S Kamalov and L Tiator Unitary isobar model ndash MAID2007 Eur Phys J A
69 (2007) available at httpsmaidkphuni-mainzde arXiv07100306 [nucl-th]
[70] P J Mohr B N Taylor and D B Newell CODATA recommended values of the fundamental
physical constants 2010 Rev Mod Phys 84 1527 (2012)
[71] J Ahrens et al (GDH A2) First measurement of the Gerasimov-Drell-Hearn integral for 1H
from 200 to 800 MeV Phys Rev Lett 87 022003 (2001) arXivhep-ex0105089 [hep-ex]
[72] K Helbing (GDH) Experimental verification of the GDH sum rule at ELSA and MAMI Nucl
Phys Proc Suppl 105 113 (2002)
[73] R Bradford A Bodek H S Budd and J Arrington A New parameterization of the nucleon
elastic form-factors NuInt05 proceedings of the 4th International Workshop on Neutrino-
Nucleus Interactions in the Few-GeV Region Okayama Japan 26-29 September 2005 Nucl
Phys Proc Suppl 159 127 (2006) arXivhep-ex0602017
[74] J D Bjorken Applications of the chiral U(6)timesU(6) algebra of current densities Phys Rev
148 1467 (1966)
[75] J D Bjorken Inelastic scattering of polarized leptons from polarized nucleons Phys Rev D
1 1376 (1970)
[76] V Bernard T R Hemmert and U-G Meiszligner Novel analysis of chiral loop effects in
the generalized Gerasimov-Drell-Hearn sum rule Phys Lett B 545 105 (2002) arXivhep-
ph0203167
[77] A Deur et al Experimental study of isovector spin sum rules Phys Rev D 78 032001
(2008) arXiv08023198 [nucl-ex]
[78] B Pasquini P Pedroni and D Drechsel Higher order forward spin polarizability Phys Lett
B 687 160 (2010) arXiv10014230 [hep-ph]
[79] R Jaffe g2ndashThe nucleonrsquos other spin-dependent structure function Comments Nucl Part
Phys 19 239 (1990)
[80] E V Shuryak and A Vainshtein Theory of power corrections to deep inelastic scattering in
quantum chromodynamics (II) Qminus4 effects polarized target Nucl Phys B 201 141 (1982)
[81] S Wandzura and F Wilczek Sum rules for spin dependent electroproduction Test of rela-
tivistic constituent quarks Phys Lett B 72 195 (1977)
[82] B W Filippone and X-D Ji The spin structure of the nucleon Adv Nucl Phys 26 1
(2001) arXivhep-ph0101224
[83] M Burkardt The g2 structure function Proceedings Workshop on Spin structure at long
46
distance Newport News USA March 12-13 2009 AIP Conf Proc 1155 26 (2009)
arXiv09054079 [hep-ph]
[84] A Deur S J Brodsky and G F de Teramond The spin structure of the nucleon Rept
Prog Phys 82 076201 (2019) arXiv180705250 [hep-ph]
[85] B R Holstein D Drechsel B Pasquini and M Vanderhaeghen Higher order polarizabilities
of the proton Phys Rev C 61 034316 (2000) arXivhep-ph9910427
[86] B R Holstein V Pascalutsa and M Vanderhaeghen Sum rules for magnetic moments and
polarizabilities in QED and chiral effective-field theory Phys Rev D 72 094014 (2005)
arXivhep-ph0507016
[87] B Pasquini D Drechsel and L Tiator Invariant amplitudes for pion electroproduction Eur
Phys J A 34 387 (2007) arXiv07122327 [hep-ph]
[88] A J Buchmann Non-spherical proton shape and hydrogen hyperfine splitting Proceedings
International Workshop on Precision Physics of Simple Atomic Systems (PSAS 2008) Wind-
sor Ontario Canada July 21-26 2008 Can J Phys 87 773 (2009) arXiv09104747
[physicsatom-ph]
47
the static forward spin polarizability amount to in units of 10minus4 fm4
γ0p = minus093(92) asymp 201minus 284minus 010 (20a)
γ0n = 003(92) asymp 298minus 284minus 010 (20b)
while the slope is composed as follows in units of 10minus4 fm6
dγ0p(Q2)
dQ2
∣∣∣∣Q2=0
= minus022(4) asymp minus033 + 011 + 001 (21a)
dγ0n(Q2)
dQ2
∣∣∣∣Q2=0
= minus061(7) asymp minus073 + 011 + 001 (21b)
Figure 2 upper panel shows our NLO prediction as well as the LO πN loops compared
to different experimental and theoretical results For the proton we have one determination
at the real-photon point by the GDH collaboration [19] γ0p = minus100(8)(12)times10minus4 fm4 and
further Jefferson Laboratory data [18 60] at very low Q2 For the neutron only data at finite
Q2 are available [20 61] The experimental data for the proton are fairly well reproduced
in the whole Q2 range considered here while for the neutron the agreement improves with
increasing Q2 The HB limit of our πN -loop contribution reproduces the results published
in Refs [50 63] for arbitrary Q2 In addition our prediction is compared to the MAID
model [1 20] the IR+∆ calculation of Ref [58] and the BχPT+∆ result of Ref [31]
The πN -production channel gives a positive contribution to the photoabsorption cross
section σTT (νQ2) at low Q2 cf Fig 10 Accordingly one observes that the πN loops give
a sizeable positive contribution to γ0(Q2) The Delta on the other hand has a very large
effect by cancelling the πN loops and bringing the result close to the empirical data From
Fig 3 upper panel one can see that it is the ∆ exchange which dominates while π∆
loops are negligible This was expected since the forward spin polarizability sum rule is an
integral over the helicity-difference cross section in which σ32 is governed by the Delta at
low energies (the relevant energy region for the sum rule)
To elucidate the difference between the present calculation and the one from Ref [31] we
note that the two calculations differ in the following important aspects Firstly Ref [31] uses
the small-scale counting [64] that considers ∆ and mπ as being of the same size ∆ sim mπ
In practice this results in a set of π∆-loop graphs which contains graphs with one or two
γ∆∆ couplings and hence two or three Delta propagators Such graphs are suppressed in
the δ-counting and thus omitted from our calculation while present in that of Ref [31]
Secondly the Lagrangians describing the interaction of the Delta are constructed differently
and assume slightly different values for the coupling constants In particular we employ
13
000 005 010 015 020 025 030
-3
-2
-1
0
1
2
Q2GeV
2D
Γ01
0-
4fm
4D
proton
000 005 010 015 020 025 030
-3
-2
-1
0
1
2
3
Q2GeV
2D
Γ01
0-
4fm
4D
neutron
000 005 010 015 020 025 030
00
05
10
15
Q2GeV
2D
∆LT1
0-
4fm
4D
proton
000 005 010 015 020 025 030
00
05
10
15
20
25
Q2GeV
2D
∆LT1
0-
4fm
4D
neutron
FIG 3 Contributions of the different orders to the chiral predictions of γ0(Q2) upper paneland δLT (Q2) lower panel for the proton (left) and neutron (right) Red solid line πN -loop
contribution green dot-dashed line ∆-exchange contribution orange dotted line π∆-loop con-
tribution blue long-dashed line total result purple dot-dot-dashed line total result without gCcontribution black short-dashed line total result without gM dipole
(where possible) the so-called ldquoconsistentrdquo couplings to the Delta field ie those couplings
that project out the spurious degree of freedom see Refs [37 65 66] The authors of
Ref [31] on the other hand use couplings where the consistency in this sense is not enforced
The effects of these differences are of higher order in the δ-counting expansion and their
contribution to the Q2 dependence of the considered polarizabilities is expected to be rather
small however the differences at Q2 = 0 could be noticeable [67]
Finally as mentioned in Sec II C the inclusion of the dipole form factor in the magnetic
coupling gM is expected to be important to generate the correct Q2 behaviour of the polar-
izabilities Comparing our predictions for the forward spin polarizability with and without
inclusion of the form factor see Fig 3 upper panel confirms this Without the dipole
our results for the proton and neutron are closer to the ones from Ref [31] where the form
factor is not included For the neutron our prediction without the dipole is able to describe
14
the experimental points at very low Q2 deviating from the data with increasing Q2 The
π∆-loop contribution does not modify the Q2 behavior of γ0(Q2) and only differs from
Ref [31] by a small global shift Note also the relatively large effect of gC which generates
a sign change for virtualities above sim 02 GeV2 see Fig 3 upper panel
B δLT (Q2) mdash longitudinal-transverse polarizability
The longitudinal-transverse spin polarizability
δLT (Q2) =1
2π2
int infinν0
dν
radic1 +
Q2
ν2
σLT (νQ2)
Qν2(22)
=16αM2
N
Q6
int x0
0
dx x2[g1(xQ2) + g2(xQ2)
]
contains information about the spin structure of the nucleon and is another important
input in the determination of the (muonic-)hydrogen hyperfine splitting [16 62] It is also
relevant in studies of higher-twist corrections to the structure function g2(xQ2) given by
the moment d2(Q2) [51] see Section III E The peculiarity of the response encoded in this
polarizability is that it involves a spin flip of the nucleon and a polarization change of the
photon see App B and Fig 11
It is expected that the Delta isobar gives only a small contribution to δLT (Q2) what makes
this polarizability a potentially clean test case for chiral calculations Consequently there are
relatively many different theoretical calculations of δLT (Q2) coming from different versions
of χPT with baryons (HB IR and covariant) Ref [50] found a systematic deviation of the
HB result for δLTn(Q2) from the MAID model prediction This disagreement was identified
by the authors of Ref [68] as a puzzle involving the neutron δLT polarizabilitymdashthe δLT
puzzle The IR calculation in Ref [58] also showed a deviation from the data and predicted
a rapid rise of δLT (Q2) with growing Q2 The problem is solved by keeping the relativistic
structure of the theory as the BχPT+∆ result of Ref [31] showed
As expected already the leading πN loops provide a reasonable agreement with the
experimental data cf Fig 2 lower panel Since the ∆-exchange contribution to δLT (Q2)
is small the effect of the gM form factor is negligible in this polarizability as is that of the gC
coupling cf Fig 3 lower panel In fact we predict both the ∆-exchange and the π∆-loop
contributions to be small and negative This is in agreement with the MAID model which
predicts a small and negative contribution of the P33 wave to δLT (Q2) However in the
calculation of Ref [31] which is different from the one presented here only in the way the
∆(1232) is included the contribution of this resonance to δLTp(Q2) is sizeable and positive
15
The authors of that work attributed this large contribution to diagrams where the photons
couple directly to the Delta inside a loop As mentioned in Sec III F the effect of such loop
diagrams does not change the Q2 behaviour of the polarizabilities On the other hand it
can produce a substantial shift of the δLT (Q2) as a whole A higher-order calculation should
resolve the discrepancy between the two covariant approaches however it will partially lose
the predictive power since the LECs appearing at higher orders will have to be fitted to
experimental data
The πN -loop ∆-exchange and π∆-loop contributions to the NLO BχPT prediction of
the static longitudinal-transverse polarizability are in units of 10minus4 fm4
δLTp = 132(15) asymp 150minus 016minus 002 (23a)
δLTn = 218(23) asymp 235minus 016minus 002 (23b)
while the slopes are in units of 10minus4 fm6
dδLTp(Q2)
dQ2
∣∣∣∣Q2=0
= minus085(8) asymp minus080minus 004minus 001 (24a)
dδLTn(Q2)
dQ2
∣∣∣∣Q2=0
= minus124(12) asymp minus119minus 004minus 001 (24b)
C IA(Q2) mdash a generalized GDH integral
The helicity-difference cross section σTT exhibits a faster fall-off in ν than its spin-averaged
counterpart σT This is due to a cancellation between the leading (constant) terms of σ12
and σ32 at large ν3 The resulting 1ν fall-off of the helicity-difference cross section allows
one to write an unsubtracted dispersion relation for the VVCS amplitude gnonpoleTT (νQ2) cf
Eq (10a) This is the origin of the GDH sum rule [5 6]
minus α
2M2N
κ2 =1
2π2
int infinν0
dνσTT (ν)
ν (25)
which establishes a relation to the anomalous magnetic moment κ It is experimentally
verified for the nucleon by MAMI (Mainz) and ELSA (Bonn) [71 72]
There are two extensions of the GDH sum rule to finite Q2 the generalized GDH integrals
3 Notice that a constant term in σTT at ν rarrinfin is forbidden by crossing symmetry
16
000 005 010 015 020 025 030
-30
-25
-20
-15
-10
-05
00
Q2 GeV2D
I Aproton
000 005 010 015 020 025 030
-25
-20
-15
-10
-05
00
Q2 GeV2D
I A
neutron
000 005 010 015 020 025 030
-05
00
05
Q2 GeV2D
I 1
proton
000 005 010 015 020 025 030
-10
-08
-06
-04
-02
00
02
Q2 GeV2D
I 1
neutron
FIG 4 Upper panel The generalized GDH integral IA(Q2) for the proton (left) and neutron
(right) as function of Q2 The result of this work the NLO BχPT prediction is shown by the blue
solid line and the blue band The red line represents the LO BχPT result The purple short-dashed
line is the O(p4) HB result from Ref [50 51] The gray band is the BχPT+∆ result from Ref [31]
The black dotted line is the MAID model prediction [69] Experimental extractions for the proton
Ref [60] (orange triangle uncertainties added in quadrature) and neutron Refs [21][27] where
magenta dotsorange diagonal crosses correspond to data and red squareslilac crosses correspond
to data plus extrapolation to unmeasured energy regions The green stars at the real-photon point
are derived from the anomalous magnetic moments κp asymp 1793 and κn asymp minus1913 [70] Lower
panel The generalized GDH integral I1(Q2) for the proton (left) and neutron (right) as function
of Q2 The purple short-dashed line is the HB result from Ref [51] Experimental extractions for
the proton Ref [18] (blue dots) and Ref [60] (orange triangle uncertainties added in quadrature)
and neutron Ref [61][27] (uncertainties added in quadrature) where green dotsorange diagonal
crosses correspond to data and gray squareslilac crosses correspond to data plus extrapolation to
unmeasured energy regions
17
IA(Q2) and I1(Q2) The latter will be discussed in Sec III D The former is defined as4
minus α
2M2N
IA(Q2) = minus 1
8π2
int infinν0
dν
radic1 +
Q2
ν2
σTT (νQ2)
ν(26)
=α
Q2
int x0
0
dx[4M2
Nx2
Q2g2(xQ2)minus g1(xQ2)
]
Due to its energy weighting the integral in Eq (26) converges slower than the one in the
generalized forward spin polarizability sum rule (19) Therefore knowledge of the cross
section at higher energies is required and the evaluation of the generalized GDH integral
IA(Q2) is not as simple as the evaluation of γ0(Q2)
The generalized GDH integral IA(Q2) is directly related to the non-pole amplitude
gnonpoleTT (νQ2) which differs from non-Born amplitude gTT (νQ2) by a term involving the
elastic Pauli form factor
gnonpoleTT (νQ2) = gTT (νQ2)minus 2παν
M2N
F 22 (Q2) (27)
cf Eqs (2a) and (8a) Consequently IA(Q2) is not a pure polarizability but also contains
an elastic contribution The ldquonon-polarizabilityrdquo or the Born part of IA(Q2) is given by
IBornA (Q2) = IA(Q2)minus∆IA(Q2) = minus1
4F 2
2 (Q2) (28)
where we refer to the polarizability part as ∆IA(Q2) The same is true for the generalized
GDH integral I1(Q2) which is directly related to Snonpole1 (νQ2)
IBorn1 (Q2) = I1(Q2)minus∆I1(Q2) = minus1
4F 2
2 (Q2) (29)
In the following we will add the Born parts to our LO and NLO BχPT predictions for
the polarizabilities ∆IA(Q2) and ∆I1(Q2) employing an empirical parametrization for the
elastic Pauli form factor [73] This allows us to compare to the experimental results for
IA(Q2) and I1(Q2) cf Fig 4 Note that the blue error bands only describe the uncertainties
of our BχPT predictions of the polarizabilities while the elastic contributions are considered
to be exact as explained in Sec II C The uncertainties of the polarizability predictions are
therefore better reflected in Fig 5 where we show the contributions of the different orders
to the BχPT predictions of ∆IA(Q2) and ∆I1(Q2) as well as the total results with error
bands
4 Note that IA(Q2) is sometimes called ITT (Q2)
18
000 005 010 015 020 025 030
-08
-06
-04
-02
00
02
Q2 GeV2D
DI A
proton
000 005 010 015 020 025 030
-10
-08
-06
-04
-02
00
02
Q2 GeV2D
DI A
neutron
000 005 010 015 020 025 030-005
000
005
010
015
020
025
Q2 GeV2D
DI 1
proton
000 005 010 015 020 025 030-015
-010
-005
000
005
010
015
Q2 GeV2D
DI 1
neutron
FIG 5 Contributions of the different orders to the chiral predictions of ∆IA(Q2) upper paneland ∆I1(Q2) lower panel for the proton (left) and neutron (right) Red solid line πN -loop
contribution green dot-dashed line ∆-exchange contribution orange dotted line π∆-loop contri-
bution blue solid line and blue band total result purple dot-dot-dashed line total result without
gC contribution black short-dashed line total result without gM dipole
The E97-110 experiment at Jefferson Lab has recently published their data for IAn(Q2) in
the region of 0035 GeV2 lt Q2 lt 024 GeV2 [27] In addition there are results for IAn(Q2)
from the earlier E94-010 experiment [21] and for IAp(Q2) from the E08-027 experiment [60]
The O(p4) HB calculation gives a large negative effect [51] which does not describe the data
The BχPT+∆ result from Ref [31] which mainly differs from our work by the absence of
the dipole form factor in gM looks similar to this HB result and only describes the data
points at lowest Q2 Our NLO prediction however follows closely the Q2 evolution of the
data In Fig 5 upper panel we show the polarizability ∆IA(Q2) whose Q2 evolution is
clearly dominated by the ∆ exchange Similar to the case of γ0p(Q2) inclusion of the dipole
in gM and the Coulomb coupling gC is very important in order to describe the experimental
data The LO prediction on the other hand slightly overestimates the data cf Fig 4
upper panel
19
At the real-photon point IA(0) = minusκ2
4and ∆IA(0) = 0 Therefore we give only the
slope of the polarizability ∆IA(Q2) [showing also the separate contributions from πN loops
∆ exchange and π∆ loops] in units of GeVminus2
d∆IAp(Q2)
dQ2
∣∣∣∣Q2=0
= minus858(343) asymp 238minus 1121 + 025 (30a)
d∆IAn(Q2)
dQ2
∣∣∣∣Q2=0
= minus955(343) asymp 141minus 1121 + 025 (30b)
Including the empirical Pauli form factor [73] we find in units of GeVminus2
dIAp(Q2)
dQ2
∣∣∣∣Q2=0
= minus318dIAn(Q2)
dQ2
∣∣∣∣Q2=0
= minus300 (31)
D Γ1(Q2) and I1(Q2) mdash the first moment of the structure function g1(xQ2)
The second variant for a generalization of the GDH sum rule to finite Q2 is defined as
minus α
2M2N
I1(Q2) = minus 1
8π2
int infinν0
dν1radic
ν2 +Q2
[σTT (νQ2) +
Q
νσLT (νQ2)
](32)
= minus α
Q2
int x0
0
dx g1(xQ2)
where I1(0) = minusκ2
4 This generalized GDH integral directly stems from the amplitude
Snonpole1 (νQ2) with the LEX from Eq (12a) It is given by the first moment of the struc-
ture function g1(xQ2) Γ1(Q2) =int x0
0dx g1(xQ2) as follows I1(Q2) =
2M2N
Q2 Γ1(Q2) The
isovector combination
Γ1(pminusn)(Q2) =
int x0
0
dx[g1p(xQ
2)minus g1n(xQ2)] (33)
is related to the axial coupling of the nucleon through the Bjorken sum rule [74 75]
limQ2rarrinfin
Γ1(pminusn)(Q2) =
gA6 (34)
As explained in Eq (28) the moment I1(Q2) splits into a polarizability part ∆I1(Q2) and
a Born part IBorn1 (Q2) Figure 4 lower panel shows the Q2 dependence of I1(Q2) which in
contrast to IA(Q2) shown in Figure 4 upper panel is clearly dominated by its Born part
and the elastic Pauli form factor The πN -loop ∆-exchange and π∆-loop contributions to
the polarizability ∆I1(Q2) are shown in Fig 5 lower panel Comparing to Fig 5 upper
20
000 005 010 015 020 025 030
-006
-004
-002
000
002
004
Q2 GeV2D
G1
proton
000 005 010 015 020 025 030
-006
-005
-004
-003
-002
-001
000
001
Q2 GeV2D
G1
neutron
FIG 6 First moment of the structure function g1(xQ2) for the proton (left) and neutron (right)
as function of Q2 The legend is the same as in Fig 4
panel one sees that ∆I1(Q2) is less sensitive to gC and the dipole form factor in gM than
∆IA(Q2)
For the proton our NLO BχPT prediction gives a very good description of the ex-
perimental data [18 60] and is in reasonable agreement with the MAID prediction [69]
For the neutron one observes good agreement with the empirical evaluations including ex-
trapolations to unmeasured energy regions starting from Q2 gt 01 GeV2 [27 61] In the
region of Q2 lt 005 GeV2 one observes an interesting tension between the recent E97-
110 experiment [27] and the data from CLAS [61] While the newest measurement finds
I1n(0035 GeV2) lt κ2n4 thus suggesting a negative slope at low Q the older measurement
found a rather large value for I1n(00496 GeV2) A similar but milder behaviour is seen in
the E97-110 [27] and E94-010 [21] data for IAn The MAID predictions do not agree with
the CODATA recommended values for the anomalous magnetic moments of the proton and
neutron [70] which in our work are imposed by using empirical parametrizations for the
elastic Pauli form factors [73] The slope of the HB result from Ref [51] is too large and
therefore only reproduces the data at very low Q2
Figure 6 shows the moment Γ1(Q2) for the proton and neutron while Fig 7 shows the
isovector combination Γ1 pminusn(Q2) The LO and NLO BχPT predictions are identical be-
cause our calculation produces the same Delta contributions for the proton and the neutron
For the isovector combination the MAID model only agrees with the data at very low
Q2 lt 010 GeV2 The same is true for the IR result [58 76] while all other chiral results
describe the data NLO BχPT (this work) BχPT+∆ [31] and HBχPT [51]
At the real-photon point I1(0) = minusκ2
4and ∆I1(0) = 0 Therefore we give only the
slope of the polarizability ∆I1(Q2) [showing also the separate contributions from πN loops
21
000 005 010 015 020 025 030-002
000
002
004
006
008
010
Q2 GeV2D
G1Hp-
nL
FIG 7 Isovector combination of Γ1(Q2) as function of Q2 The legend is the same as in Fig 6
The pink curve is the IR result from Ref [58 76] The experimental points are from Ref [77]
(brown dots) and Ref [23] (orange squares)
∆ exchange and π∆ loops] in units of GeVminus2
d∆I1p(Q2)
dQ2
∣∣∣∣Q2=0
= 039(4) asymp 034minus 053 + 058 (35a)
d∆I1n(Q2)
dQ2
∣∣∣∣Q2=0
= minus101(10) asymp minus107minus 053 + 058 (35b)
Including the empirical Pauli form factor [73] we find in units of GeVminus2
dI1p(Q2)
dQ2
∣∣∣∣Q2=0
= 580dI1n(Q2)
dQ2
∣∣∣∣Q2=0
= 553 (36)
E d2(Q2) mdash a measure of color polarizability
Another interesting moment to consider is d2(Q2) which is related to the twist-3 part of
the spin structure function g2(xQ2) [79 80]
d2(Q2) equiv 3
int 1
0
dx x2[g2(xQ2)minus gWW2 (xQ2)] (37)
where gWW2 (xQ2) is the twist-2 part of g2(xQ2) Using the Wandzura-Wilczek relation [81]
one can relate d2(Q2) to moments of the spin structure functions g1(xQ2) and g2(xQ2)
d2(Q2) =
int 1
0
dx x2 [3g2(xQ2) + 2g1(xQ2)] (38)
22
000 005 010 015 020 025 030
0000
0001
0002
0003
0004
0005
0006
0007
Q2 GeV2D
d2
proton
000 005 010 015 020 025 030
0000
0002
0004
0006
0008
0010
0012
Q2 GeV2D
d2
neutron
000 005 010 015 020 025 030
-1
0
1
2
Q2 GeV2D
Γ 010
-4
fm6
D
proton
000 005 010 015 020 025 030-2
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
6D
neutron
FIG 8 Upper panel The inelastic moment d2(Q2) for the proton (left) and neutron (right) as
function of Q2 The result of this work the NLO BχPT prediction is shown by the blue solid line
and the blue band The red line represents the LO BχPT result The purple short-dashed line is
the O(p4) HB result from Ref [50 51] The black dotted line is the MAID model prediction [69]
The experimental points for the neutron (cyan dots) are from Ref [22] Lower panel Fifth-order
generalized forward spin polarizability γ0(Q2) for the proton (left) and neutron (right) as function
of Q2 The experimental points for the proton are from Ref [59] (purple square) and Ref [78]
(orange dot)
This relation however only holds for asymptotically large Q2 It is also in the high-Q2
region where d2(Q2) is a measure of color polarizability [82 83] through its relation to the
gluon field strength tensor [80] We refer to Ref [84] for a recent review on the spin structure
of the nucleon including a discussion of sum rules for deep inelastic scattering and color
polarizabilities
What we consider in the following is the inelastic part of d2(Q2) defined as the moment
of g1(xQ2) and g2(xQ2) spin structure functions cf Eq (38)
d2(Q2) =
int x0
0
dx x2 [3g2(xQ2) + 2g1(xQ2)] (39)
23
This moment provides another testing ground for our BχPT predictions through comparison
with experiments on the neutron [22] Going towards the low-Q2 region the interpretation
of d2(Q2) in terms of color polarizabilities will fade out The above definition however
implies it is related to other VVCS polarizabilities
d2(Q2) =Q4
8M4N
[M2
NQ2
αδLT (Q2) + I1(Q2)minus IA(Q2)
] (40)
Note that d2(Q2) and its first two derivatives with respect to Q2 vanish at Q2 = 0 The
considerations in Eqs (28) and (29) have no effect on d2(Q2) since the Born contribution
from IA(Q2) and I1(Q2) cancel out Therefore d2(Q2) is a pure polarizability
In Fig 8 upper panel we show our NLO BχPT prediction and other results for d2(Q2)
While MAID [69] and BχPT describe the experimental data for the neutron [22] very well
the HB limit [50 51] is showing a fast growth with Q2 This illustrates the importance
of keeping the relativistic result Note also that even though the πN -loop contribution
is dominant both gC and the form factor in gM are essential to obtain a curvature that
reproduces the data cf Fig 9 upper panel For the proton there are to our knowledge
no experimental results to compare with However the agreement between the NLO BχPT
prediction and the MAID prediction at low energies is reasonable
F γ0(Q2) mdash fifth-order generalized forward spin polarizability
It is interesting to compare the generalized fifth-order forward spin polarizability sum
rule
γ0(Q2) =1
2π2
int infinν0
dν
radic1 +
Q2
ν2
σTT (νQ2)
ν5(41)
=64αM4
N
Q10
int x0
0
dx x4
[g1(xQ2)minus 4M2
Nx2
Q2g2(xQ2)
]
to the sum rule integrals for IA(Q2) and γ0(Q2) since they differ merely by their energy
weighting of σTT (νQ2) and a constant prefactor cf Eqs (19) (26) and (41) From IA(Q2)
to γ0(Q2) to γ0(Q2) the energy suppression is increasing by a factor of νminus2 respectively
Therefore the description of γ0(Q2) should be easiest in a low-energy effective-field theory
such as χPT whereas γ0(Q2) and IA(Q2) receive larger contributions from higher energies
In Fig 8 lower panel we show our LO and NLO BχPT predictions for γ0(Q2) One
can see that the πN -loop contribution is positive (in accordance to what we see for the cross
section σTT see Fig 10) The Delta shifts it substantially especially in the low Q2 region
24
000 005 010 015 020 025 030
-0004
-0002
0000
0002
0004
0006
0008
0010
Q2GeV
2D
d2
proton
000 005 010 015 020 025 030
-0005
0000
0005
0010
0015
Q2GeV
2D
d2
neutron
000 005 010 015 020 025 030
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
6D
proton
000 005 010 015 020 025 030
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
6D
neutron
FIG 9 Contributions of the different orders to the chiral predictions of d2(Q2) upper panel and
γ0(Q2) lower panel for the proton (left) and neutron (right) Red solid line πN -loop contribu-
tion green dot-dashed line ∆-exchange contribution orange dotted line π∆-loop contribution
blue long-dashed line total result purple dot-dot-dashed line total result without gC contribution
black short-dashed line total result without gM dipole
bringing it into a better agreement with data In general the BχPT curves start above the
empirical data points at the real-photon point and then decrease asymptotically to zero
above Q2 gt 01 GeV2 On the other hand the MAID prediction reproduces the empirical
data at the real-photon point then decreases to negative values until about Q2 gt 006 GeV2
from where it also starts to asymptotically approach zero Consequently our NLO BχPT
prediction of γ0(Q2) is consistently above the MAID prediction This is very different to
what we saw for IA(Q2) in Fig 4 upper panel where the MAID prediction at the real-
photon point is above the experimental value While the agreement of our predictions with
the empirical data is in general quite good for all moments of σTT (νQ2) one should point
out that both for γ0n(Q2) and γ0p(Q2) we overestimate the data at low Q2 For IA(Q2) such
observation cannot be made because ∆IA(0) = 0 and thus IA(0) is given by the empirical
Pauli form factor only From IA(Q2) γ0(Q2) and γ0(Q2) the latter has the smallest however
25
non-negligible dependence on gC and the dipole in gM cf Fig 9 lower panel
The πN -loop ∆-exchange and π∆-loop contributions to the NLO BχPT prediction of
the static fifth-order forward spin polarizability amount to in units of 10minus4 fm6
γ0p = 112(30) asymp 208minus 096minus 001 (42a)
γ0n = 195(30) asymp 292minus 096minus 001 (42b)
while the slope is composed as follows in units of 10minus4 fm8
dγ0p(Q2)
dQ2
∣∣∣∣Q2=0
= minus084(10) asymp minus100 + 016 + 000 (43a)
dγ0n(Q2)
dQ2
∣∣∣∣Q2=0
= minus142(15) asymp minus158 + 016 + 000 (43b)
Note that the HB prediction of the static polarizability γ0p (423 at O(p3) and 365 at O(ε3)
[78 85]) is almost one order of magnitude larger than the empirical value and therefore not
shown in Fig 8
G Summary
Our results are summarized in Table II where we give the contributions of the different
orders to the chiral predictions of the polarizabilities and their slopes at the real-photon
point A quantitative comparison of our predictions for the static spin polarizabilities to
the work of Bernard et al [31] and different empirical evaluations is shown in Table III We
can see that the inclusion of the Delta turns out to be very important for all moments of
the helicity-difference cross section To describe the Q2 behavior of the polarizabilities the
magnetic coupling of the N rarr ∆ transition should be modified by a dipole form factor as
has been observed previously in the description of electroproduction data [33] This dipole
form factor effectively takes account of vector-meson exchanges The Coulomb-quadrupole
N rarr ∆ transition despite its subleading order is important in the description of some
moments of spin structure functions This is contrary to what we saw for the moments of
unpolarized structure functions [30] where the Coulomb coupling had a negligible effect
The π∆ loops are mainly relevant for the generalized GDH integrals
26
TABLE II The NLO BχPT predictions for the forward VVCS polarizabilities and their slopes at
Q2 = 0 The contributions of the πN loops the ∆ exchange and the π∆ loops are shown together
with the combined total result Note that IA(0) = I1(0) = d2(0) = 0 and (d2)prime = 0
πN loops ∆ exchange π∆ loops Total
γ0 p 201 minus284 minus010minus093(92)
(10minus4 fm4) n 298 003(92)
δLT p 150 minus016 minus002132(15)
(10minus4 fm4) n 235 218(23)
γ0 p 208 minus096 minus001112(30)
(10minus4 fm6) n 292 195(30)
(γ0)prime p minus033011 001
minus022(4)
(10minus4 fm6) n minus073 minus061(7)
(δLT )prime p minus080 minus004 minus001minus085(8)
(10minus4 fm6) n minus119 minus124(12)
(γ0)prime p minus100016 000
minus084(10)
(10minus4 fm8) n minus158 minus142(15)
(∆IA)prime p 238 minus1121 025minus858(343)
(GeVminus2) n 141 minus955(343)
(∆I1)prime p 034 minus053 058039(4)
(GeVminus2) n minus107 minus101(10)
TABLE III Our NLO BχPT predictions for the spin polarizabilities at Q2 = 0 compared with
the BχPT+∆ predictions from Bernard et al [31] and the available empirical information Where
the reference is not given the empirical number is provided by the MAID analysis [55 69] with
unspecified uncertainty
Proton Neutron
This work BχPT+∆ Empirical This work BχPT+∆ Empirical
γ0 minus093(92) minus174(40) minus100(8)(12) [19] 003(92) minus077(40) minus0005
(10minus4 fm4) minus090(8)(11) [78] [MAID]
minus0929(105) [59]
δLT 132(15) 240(1) 134 218(23) 238(3) 203
(10minus4 fm4) [MAID] [MAID]
γ0 112(30) 060(7)(7) [78] 195(30) 123
(10minus4 fm6) 0484(82) [59] [MAID]
IV CONCLUSIONS
We have presented a complete NLO calculation of the polarized non-Born VVCS ampli-
tudes in covariant BχPT with pion nucleon and ∆(1232) fields The dispersion relations
between the VVCS amplitudes and the tree-level photoabsorption cross sections served as a
27
cross-check of these calculations
The obtained moments of the proton and neutron spin structure functions related to gen-
eralized polarizabilities and GDH-type integrals agree well with the available experimental
data The description of their Q2 evolution is improved compared to the previous χPT pre-
dictions In particular the NLO BχPT predictions obtained here give a better description
of the empirical data (eg from the Jefferson Laboratory ldquoSpin Physics Programrdquo) than
the HB [50 51] and IR [58] calculations
The demonstrated predictive power of the χPT framework amplitudes makes it well
suited for extending the χPT evaluation of the TPE effect in the hyperfine structure of
(muonic-)hydrogen [15ndash17] to next-to-leading order
ACKNOWLEDGEMENTS
We thank Lothar Tiator and Marc Vanderhaeghen for helpful discussions This work
is supported by the Deutsche Forschungsgemeinschaft (DFG) through the Collaborative
Research Center [The Low-Energy Frontier of the Standard Model (SFB 1044)] JMA ac-
knowledges support from the Community of Madrid through the ldquoPrograma de atraccion de
talento investigador 2017 (Modalidad 1)rdquo and the Spanish MECD grants FPA2016-77313-P
FH gratefully acknowledges financial support from the Swiss National Science Foundation
Appendix A Tensor decompositions of the VVCS amplitudes
In this appendix we review the decomposition of the forward VVCS process into tensor struc-
tures and scalar amplitudes In particular we consider the connection between the covariant and
the semi-relativistic decomposition in the lab frame that is defined in terms of the conventional
transverse longitudinal transverse-transverse and transverse-longitudinal amplitudes
As explained in Sec II A the process of forward VVCS off the nucleon can be described in
terms of four explicitly covariant amplitudes S1 2 and T1 2 [3]
T (νQ2) =
(minusgmicroν +
qmicroqν
q2
)T1(νQ2) +
1
M2N
(pmicro minus p middot q
q2qmicro)(
pν minus p middot qq2
qν)T2(νQ2) (A1)
minus 1
MNγmicroναqα S1(νQ2)minus 1
M2N
(γmicroνq2 + qmicroγναqα minus qνγmicroαqα
)S2(νQ2)
εprimelowastmicro εν
where εmicro (εprimelowastmicro ) are the incoming (outgoing) photon polarization vectors ν is the photon lab-frame
energy and Q2 is the photon virtuality Alternatively the decomposition in the laboratory frame
28
(which in the forward case coincides with the Breit frame) is parametrized in terms of the nucleon
Pauli matrices ~σ and the four scalar functions fL fT gTT and gLT
T (νQ2) = ε0 εprimelowast0 fL(νQ2) + (~ε primelowast middot ~ε ) fT (νQ2) + i~σ middot (~ε primelowast times ~ε ) gTT (νQ2) (A2)
minusi~σ middot [(ε0~εprimelowast minus ~ε ε primelowast0 )times q] gLT (νQ2)
Here ~q and q = ~q|~q | are the photon three-momentum in the lab system and its unit vector The
modified polarization vector components are given by
ε0 =
[ε0 minus
ν
|~q | (~ε middot q )
] |~q |Q ~ε = ~εminus q (~ε middot q ) (A3)
where ε = (ε0~ε ) is the usual incoming photon polarization vector and εprimelowast the outgoing polarization
vector The LEX of the lab frame amplitudes [Eq (10)] can serve in particular as the definition
of the generalized polarizabilities The lab frame amplitudes are also conveniently used for the
definition of the response functions see the example of the scalar amplitude gLT (νQ2) and the
corresponding response function σLT (νQ2) below in App B
Appendix B Photoabsorption cross sections
In the forward kinematics the spin-dependent VVCS amplitudes and the spin polarizabilities
can be described in terms of the polarized structure functions g1(xQ2) and g2(xQ2) or equiv-
alently the helicity-difference cross section σTT (νQ2) and the longitudinal-transverse response
function σLT (νQ2) with the help of dispersion relations (5) and the optical theorem (3) In this
way the photoabsorption cross sections measured in electroproduction processes form the basis
for most empirical evaluations shown throughout Sec III In the following we present the BχPT
predictions for the tree-level cross sections of πN - π∆- and ∆-production through photoabsorption
on the nucleon cf Figs 8 9 and 10 in Ref [30] In Secs B 1 and B 2 we will discuss the leading
πN -production channel and the ∆-production channel respectively We used these cross sections
to verify the polarizability predictions obtained otherwise from the calculated non-Born VVCS
amplitudes Due to the bad high-energy behavior of the π∆-production cross sections in BχPT
cf Fig 10 the dispersion relations in Eq (5) require further subtractions for a reconstruction of
the π∆-loop contribution to the spin-dependent VVCS amplitudes Therefore not all polarizabil-
ities could be verified but only those appearing as higher-order terms in the LEX of the VVCS
amplitudes such as γ0 [16]
29
proton neutron
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Neutron
02 04 06 08 100
50
100
150
200
250
[GeV]
T [b]
02 04 06 08 10
-50
0
50
100
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Neutron
02 04 06 08 100
50
100
150
200
250
[GeV]
T [b]
02 04 06 08 10
-50
0
50
100
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02040608100
50
100
150
200
250
300
[GeV]
T[b]
0204060810
-50
0
50
[GeV]
TT[b]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
0204060810
00
02
04
06
[GeV]
LTQ
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
0204060810
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
FIG 10 Photoabsorption cross sections for πN (red) and π∆ production (orange) with Q2 = 0
(solid) and Q2 = 01 GeV2 (dashed for πN and dotted for π∆ channel)
1 πN-production channel
In order to extract the response function σLT (νQ2) we have developed a method similar to
the one used to calculate σTT (νQ2) see for example Ref [86] For σLT (νQ2) however the
calculation is more complicated because one has to take into account that the associated Compton
process involves a spin-flip of the nucleon as illustrated in Fig 11 When calculating the cross
section the product of the incoming nucleon spinors has to reflect this flip
The forward VVCS amplitude related to σLT (νQ2) mdash and δLT (Q2) mdash is gLT (νQ2) It can
be extracted from Eq (A2) if one takes the modified polarization vector components in Eq (A3)
with ε = εL and εprimelowast = εlowastplusmn as input where εL = 1Q(|~q | 0 0 q0) and εplusmn = ∓ 1radic
2(0 1plusmni 0) are the
standard longitudinal and transverse polarization vectors respectively For εL and εlowastplusmn only the
choice of helicities hprime = plusmn12 and h = ∓12 gives a non-zero contribution and one obtains
χdaggerhprime T (νQ2)χh = χdaggerhprimeminusi~σ middot [(ε0~εprimelowast minus ~ε ε primelowast0 )times q] gLT (νQ2)χh =
radic2 gLT (νQ2) (B1)
where χh and χdaggerhprime are two-component Pauli spinors with opposite helicities or here spins
Let us now consider the related photoabsorption process and in particular the tree-level γlowastN rarr
30
s = 12
sprime = minus12
λ = 0 λprime = 1
rprime rprime
s=12sprime=minus1
2
λ=0λprime=1
rprimerprime
s=12
sprime=minus12
λ=0λprime=1
rprimerprime hacute=frac12h=-frac12
λacute=1λ=0
FIG 11 Relation between the forward Compton process and the photoabsorption process given
by the optical theorem In particular we show the longitudinal-transverse contribution The
double-line arrows represent the spin of the external particles while the dot represents the scalar
(longitudinal) polarization of the incoming photon Inside the blob the intermediate states are
represented eg nucleons with spins rprime (which are averaged in the calculation of the cross section)
and pions
πN channel see diagrams in Fig 8 of Ref [30] We define the πN -production amplitude as
T = uhB (PB)sumi
Ai(s t)Γi uhA(PA) (B2)
with the Dirac structures
Γ1 = γ5 (B3a)
Γ2 =1
2
[qA ε
]γ5 (B3b)
where uhA(PA) and udaggerhB (PB) are the Dirac spinors and PA and PB are the four-momenta of
the incoming and outgoing nucleons respectively When calculating the photoabsorption cross
section related to the VVCS amplitude in Eq (B1) the nucleon spin flip should be implemented
by uhprime(PA) in T dagger and uh(PA) in T together with the appropriate transverse and longitudinal
photon polarization vectors εlowastplusmn and εL
However if one wants to use the properties of the Dirac matrices it is more useful to construct
an operator to produce this spin flip in the external nucleons of Fig 11 This is accomplished by
introducing the projector ΓLT equiv 12radic
2(γ1 + iγ2)γ5 which also takes into account the extra factor
radic2 in Eq (B1) We checked that with this projector one correctly extracts δLT by comparing the
HB limit of our result to the HB result of Ref [50] where the authors calculate this polarizability
from the Compton amplitude directly With all those ingredients the longitudinal-transverse cross
section is calculated in the following way
σLT (νQ2) =1
64π2 s
|~pf |cm
|~pi|cm
int 1
minus1dcos θ
sumij
AiAdaggerjXij (B4)
31
with
Xij = Tr[(PB +MN )Γi(PA +MN )ΓLTγ0Γdaggerjγ
0] (B5)
where θ is the scattering angle in the center-of-mass (cm) frame and |~pi|cm (|~pf |cm) is the three-
momentum of an incoming (outgoing) particle in the cm frame An explicit calculation of the
matrix Xij leads to
X = MNQ
0 2(PB minus PA) middot εLminusradic
2|~qf |cm sin θ (sminus u)
(B6)
where |~qi|cm (|~qf |cm) is the relative three-momentum of the incoming (outgoing) particles in the
cm frame Here s t and u are the usual Mandelstam variables For the different γlowastN rarr πN
channels we obtain the following amplitudes Ai where we introduce qA as the four-momentum of
the incoming photon and qB as the four-momentum of the outgoing pion
bull γlowastprarr π0p
A1 =e gAMN
fπ
[2PA middot ε+ qA middot ε
sminusM2N
+2PB middot εminus qA middot ε
uminusM2N
] (B7a)
A2 =e gAMN
fπ
[1
sminusM2N
+1
uminusM2N
] (B7b)
bull γlowastprarr π+n
A1 =
radic2 e gAMN
fπ
[2PA middot ε+ qA middot ε
sminusM2N
+2(PA minus PB) middot ε+ qA middot ε
tminusm2π
] (B8a)
A2 =
radic2 e gAMN
fπ(sminusM2N )
(B8b)
bull γlowastnrarr π0n
A1 = 0 (B9a)
A2 = 0 (B9b)
bull γlowastnrarr πminusp
A1 =
radic2 e gAMN
fπ
[2PB middot εminus qA middot ε
uminusM2N
minus 2(PA minus PB) middot ε+ qA middot εtminusm2
π
] (B10a)
A2 =
radic2 e gAMN
fπ(uminusM2N )
(B10b)
The analytical expressions shown above were checked with the amplitudes given in Ref [87] An-
alytical expressions for the tree-level γlowastN rarr πN channel of the σLT (νQ2) and σTT (νQ2) cross
32
sections are given below (proton channels π+n and π0p neutron channel πminusp) We checked that
they reproduce the known results in the real-photon limit [45 86] To shorten the final expressions
for the cross sections which are considerably longer for finite Q2 than in the real-photon limit we
define the following dimensionless kinematic variables
αγ = (ENi )cmradics =
s+M2N +Q2
2s (B11)
απ = (ENf )cmradics =
s+M2N minusm2
π
2s (B12)
βγ = Eγcmradics =
sminusM2N minusQ2
2s (B13)
βπ = Eπcmradics =
sminusM2N +m2
π
2s (B14)
λγ = |~qi|cmradics =
radic(sminusM2
N minusQ2)2 + 4sQ2
2s (B15)
λπ = |~qf |cmradics =
radic(sminusM2
N +m2π)2 minus 4sm2
π
2s (B16)
Here (ENi )cm and (ENf )cm are the energies of the incoming and outgoing nucleon Eγcm is the energy
of the incoming photon Eπcm is the energy of the outgoing pion all in the cm frame
σ(π+n)TT = minus e2g2
AM2N
64πf2πs
3(sminusM2N )2λ4
γ
4sλπλγ
[(M2
N minus s)(M2N minusQ2 minus s)(Q2 + 2sβγβπ)
+ 2s(m2πQ
2 minus (M2N minus s)
(M2N + s(minus1 + 2βγβπ)
))λ2γ
]+ (M2
N minus s)(M2N minusQ2 minus s)
times (Q2 + 2sβγβπ minus 2sλπλγ)(Q2 + 2s(βγβπ + λπλγ)
)times log
(Q2 + 2sβγβπ minus 2sλγλπQ2 + 2sβγβπ + 2sλγλπ
) (B17)
33
σ(π0p)TT =
e2g2AM
2Nλπ
64πf2πs(sminusM2
N )2λγ
1
s2λπλ3γ
[(M2
N minus s)(M2N minusQ2 minus s)
(3M2
N + s(2βγβπ minus 3))(M2N
+ s(2βγβπ minus 1))
+ 2s
((2m2
πQ2 + (M2
N minus s)2)(M2
N minus s)minus 2(minusm2
πQ2 + (sminusM2
N )2)
times sβγβπ + 4s2(sminusM2N )β2
γβ2π
)λ2γ
]arctan
(2sλπλγ
M2N + s(2βγβπ minus 1)
)+ 2
[minus 2m2
πQ2
+ (M2N minus s)
(2(sminusM2
N ) + 4sβγβπ +(sminusM2
N +Q2)(3M2N minus 3s+ 2sβγβπ)
sλ2γ
+m2πQ
2(sminusM2N )
(M2N minus s+ 2sβγβπ)(M2
N + s(minus1 + 2βγβπ minus 2λπλγ))
+m2πQ
2(sminusM2N )
(M2N minus s+ 2sβγβπ)(M2
N + s(minus1 + 2βγβπ + 2λπλγ))
)
+1
λπλγ
(βγβπ
(minusm2
πQ2 + (M2
N minus s)(M2N minus s+ 2sβγβπ)
)minus (M2
N minus s)(M2N minusQ2 minus s)λ2
π
)log
(M2N + s(minus1 + 2βγβπ + 2λπλγ)
M2N + s(minus1 + 2βγβπ minus 2λπλγ)
)] (B18)
σ(πminusp)TT =
e2g2AM
2N
64πf2πs
3λ4γ(M2
N + s(2βπβγ minus 1))2 minus 4λ2πs
2λ2γ)
4sλπλγ
((minusM2
N +Q2 + s)(M2N
+ s(2βπβγ minus 1))
2 minus 2sλ2γ
(2λ2
πs(minusM2N +Q2 + s) +m2
πQ2))
+((M2N + s(2βπβγ minus 1)
)2
minus 4λ2πs
2λ2γ
)((M2N + s(2βπβγ minus 1)
)(M2N minus 2Q2 minus s(2βπβγ + 1)
)+ 4λ2
πs2λ2
γ
)times log
(minusM2
N minus 2βπsβγ minus 2λπsλγ + s)minus(M2N + s(2βπβγ minus 2λπλγ minus 1)
)(M2N + s(minus1
+ 2βπβγ + 2λπλγ))[((
M2N + s(2βπβγ minus 1)
)(M2N minus 2Q2 minus s(2βπβγ + 1)
)+ 4λ2
πs2λ2
γ
)times log
(minusM2
N minus 2βπsβγ + 2λπsλγ + s)
+((Q2 + 2βπsβγ)2 minus 4λ2
πs2λ2
γ
)times log
(Q2 + 2βπsβγ minus 2λπsλγQ2 + 2sβπβγ + 2λπsλγ
)]+ 4sλ2
γ
((M2
N + s(2βπβγ minus 1))2 minus 4λ2πs
2λ2γ
)times (M2
N minus s) arctanh
(2λπsλγ
M2N + 2βπsβγ minus s
) (B19)
34
σ(π+n)LT =
e2g2AM
3Nλπ
32πf2πQs
3(sminusM2N )2λ4
γ
2sλγ
[(M2
N minus s)(Q2 + 2sβ2γ)(Q2 + 2sβγβπ)
minus 4s
((M2
N minus s)(Q2 minus 2s(απ minus 1)βγ
)+Q2sβγβπ
)λ2γ + 8s3(minus1 + απ)λ4
γ
]minus M2
N minus sλπ
[(Q2 + 2sβ2
γ)(Q2 + 2sβγβπ)2 + 4s2(2(απ minus 1)βγ(Q2 + 2sβγβπ)
minusQ2λ2π
)λ2γ + 8s3(απ minus 1)2λ4
γ
]arctan
(2sλπλγ
Q2 + 2sβγβπ
) (B20)
σ(π0p)LT =
e2g2AM
3Nλπ
16πf2πQs(sminusM2
N )2λγ
1
minus2s(M2N + s(minus1 + 2βγβπ))2λ2
γ + 8s3λ2πλ
4γ
[minus 3M8
N (Q2
+ 2sβ2γ) + 2M4
Ns2
(minus (Q2 + 2sβ2
γ)(minus3 + 2βγβπ) +(Q2(minus3 + 2βγβπ + 6λ2
π) + 2sβγ
times (minus3 + 12απ + 2βγβπ minus 12απβγβπ + 4βγλ2π))λ2γ minus 4s(α2
π minus 1)λ4π
)+ 2M2
Ns3
[minus (Q2
+ 2sβ2γ)(minus1 + 2βγβπ)
(6 + βγβπ(minus9 + 2βγβπ)
)+
(Q2(3minus 12λ2
π + 4βγβπ(minus1 + 2βγβπ
+ λ2π))minus 2sβγ
(minus 3 + 4απ(3 + 2απ(3 + 2βγβπ(minus3 + βγβπ)) + 4βγ(βπ + (2minus βγβπ)
times λ2π)))λ2γ + 8s
((απ minus 1)(1 + απ minus 2βγβπ) + 2απβγλ
2π
)λ4γ
)+ s4
((Q2 + 2sβ2
γ)
times (1minus 2βγβπ)2(2βγβπ minus 3) + 2(Q2(minus1 + 6λ2
π + 2βγβπ((1minus 2βγβπ)2 minus 2λ2π)) + 2sβγ
times (minus1 + 2βγβπ + 4(minus1 + βγβπ)(minusαπ + 2απβγβπ minus βγλ2π)))λ2γ minus 8
(s(απ minus 1)(απ
+ (1minus 2βγβπ)2) + 2βγ(2sαπ +Q2βγ)λ2π
)λ4γ + 32s(απ minus 1)λ2
πλ6γ
)+ 2M6
Ns
(Q2(6
minus 7βγβπ + λ2γ) + 2sβγ(βγ(6minus 7βγβπ) + (1minus 4απ)λ2
γ)
)]+sminusM2
N
4s2λπλ3γ
[(Q2 + 2sβ2
γ)
times(3M2
N + s(2βγβπ minus 3))(M2N + s(2βγβπ minus 1)
)+ 2s
(minusM2
N (Q2 + 2sβγ minus 8sαπβγ)
+ s(2sβγ(1minus 4απ + 4απβγβπ) +Q2(1minus 2λ2π)))λ2γ + 8s3(απ minus 1)λ4
γ
]
times arctan
(2sλπλγ
M2N + s(2βγβπ minus 1)
) (B21)
35
σ(πminusp)LT =
e2g2AM
3N
64πf2πQs
3λ4γ(M2
N minusQ2 minus s)(M2N + s(2βπβγ + 2λπλγ minus 1))
times
1
M2N + s(2βπβγ minus 2λπλγ minus 1)
[4λπsλγ(M2
N minusQ2 minus s)(
2sλ2γ
(s((1minus 2λ2
π)Q2
+ 2(2απ minus 1)sβγ(2βπβγ minus 1))minusM2N (Q2 + 2(1minus 2απ)sβγ)
)+ (Q2 + 2sβ2
γ)(M2N + s(2βπβγ minus 1)
)2 + 8(απ minus 1)απs
3λ4γ
)]minus(M2N + s(2βπβγ + 2λπλγ minus 1)
)times[
log
(minusM2N minus 2βπsβγ minus 2λπsλγ + s
minusM2N minus 2βπsβγ + 2λπsλγ + s
)(minus (Q2 + 2sβ2
γ)(M2N + s(2βπβγ minus 1)
)(minusM2
N + 2Q2 + 2βπsβγ + s)minus 2sλ2γ
(minusM2
N (Q2 + 2sβγ) +Q4 +Q2s(4απβγ
minus 2βγ minus 2λ2π + 1) + 2s2βγ(4(απ minus 1)βπβγ + 1)
)minus 8(απ minus 1)2s3λ4
γ
)+ 2
(4s2λ2
γ
(2(απ minus 1)βγ(Q2 + 2βπsβγ)minus λ2
πQ2)
+ (Q2 + 2βπsβγ)2(Q2 + 2sβ2γ)
+ 8(απ minus 1)2s3λ4γ
)arctanh
(2λπsλγ
Q2 + 2βπsβγ
)] (B22)
2 ∆-production channel
The tree-level ∆-exchange diagram in Fig 2 of Ref [30] contributes to the non-Born part of
the VVCS amplitudes The contribution of the ∆ exchange to the VVCS amplitudes can be split
into [17]
S∆-exch1 (νQ2) = S∆-pole
1 (νQ2) + S∆-exch1 (νQ2) (B23a)
νS∆-exch2 (νQ2) = νS∆-pole
2 (νQ2) + νS2∆-exch
(νQ2) (B23b)
and similarly for the unpolarized VVCS amplitudes discussed in Ref [30] Here we introduced the
∆-pole contributions S∆-polei and the ∆-non-pole contributions S∆-exch
i The former amplitudes
feature a pole at the ∆(1232)-production threshold and thus are proportional to
1
[sminusM2∆][uminusM2
∆]=
1
4M2N
1
ν2∆ minus ν2
(B24)
36
They can be reconstructed from the dispersion relations in Eq (5) with the tree-level ∆-production
cross sections as input cf Fig 10 in Ref [30]
σTT (νQ2) =π2α
M2NM
2+|~q |
minus g2
MMN (M+ + ν)|~q |2 +g2E(∆minus ν)(Q2 minusMNν)2
MN(B25a)
+g2CQ
4s(∆minus ν)
MNM2∆
minus 4gMgE(Q2 minusMNν)|~q |2 minus 4gMgCQ2|~q |2
+2gEgCQ
2[minusMNM∆ |~q |2 + s(Q2 +∆ν)
]MNM∆
δ(ν minus ν∆)
σLT (νQ2) =Qπ2α
M2NM
2+|~q |
g2E(MNν minusQ2) [M∆(MN + ν)minus s]
MN(B25b)
+g2CQ
2[MNM∆|~q |2 minus s(Q2 +∆ν)
]MNM2
∆
+ gMgEM∆|~q |2 minusgMgC(Q2 minusMNν)|~q |2
M∆
+gEgC(ν minus∆)(M2
N |~q |2 minus 2Q2s)
MNM∆
δ(ν minus ν∆)
with ∆ = M∆ minusMN M+ = M∆ + MN and the Mandelstam variable s = M2N + 2MNν minus Q2
Analytical expressions for the spin structure functions g1(xQ2) and g2(xQ2) can be constructed
from Eq (3) with the flux factor K(νQ2) = |~q | =radicν2 +Q2
In the ∆-non-pole contributions to S1(νQ2) and νS2(νQ2) the pole in ν at the ∆(1232)-
production threshold has canceled out
S∆-exch1 (νQ2) =
πα
MNM2+
[g2MQ
2+ + g2
E
(∆2 minus 3Q2
)+
4g2CQ
4
M2∆
minus 8gMgEM∆ωminus (B26a)
minus2gMgCQ2(MN minus 4M∆)
M∆+
2gEgCQ2(3MN minus 2M∆)
M∆
]
νS2∆-exch
(νQ2) =2πα
MNM2+
[g2EM∆∆ωminus +
g2M MNQ
2+
2+g2C Q
2(Q2 minus∆2)
2M∆(B26b)
+gEgM M∆(M∆ω+ minus 4MNωminus)minus gEgC ∆(2Q2 +MNω+)
+gMgC Q2(4MN minus ω+)
]+S∆-exch
2 (νQ2)
ν
[M2
∆ ω2+
M2N
+ ν2
]
with Q+ =radic
(M∆ +MN )2 +Q2 and ωplusmn = (M2∆minusM2
N plusmnQ2)2M∆ and the non-pole contribution
to S2(νQ2)
S∆-exch2 (νQ2) = minus2παMNν
M∆M2+
[gM + gE
]gC (B27)
These amplitudes to the contrary are not described by the tree-level ∆-production cross sections
in the standard dispersive approach [17] This peculiarity has been previously missed e g in the
calculation of the ∆-exchange contribution to the hydrogen hyperfine splitting in Ref [88] The
importance of including the ∆-non-pole contribution is also evident when considering the BC sum
37
rule in Eq (14) The ∆-pole terms by themselves violate the BC sum rule but cancel exactly with
the ∆-non-pole terms
limνrarr0
νS∆-pole2 (νQ2) + lim
νrarr0νS2
∆-exch(νQ2) = 0 (B28)
Appendix C Polarizabilities at Q2 = 0
In this section we give analytical expressions for the static values and slopes at Q2 = 0 of
the spin polarizabilities and moments of polarized structure functions In particular we give the
HB expansion of the πN -loop contributions and the ∆-exchange contributions The complete
expressions also for the π∆-loop contributions can be found in the Supplemented material Recall
that IA(0) = I1(0) = d2(0) = 0 and dd2(Q2)dQ2
∣∣∣Q2=0
= 0
1 πN-loop contribution
Here we give analytical expressions for the πN -loop contributions to the proton and neutron
spin polarizabilities expanded in powers of micro = mπMN viz the HB expansion Note that we
choose to expand here to a high order in micro the strict HB expansion would only retain the leading
term in an analogous NLO calculation
bull Static polarizabilities (Q2 = 0)
γ0p =e2g2
A
96π3f2πm
2π
1minus 21πmicro
8minus(
59
2+ 26 logmicro
)micro2 +
1875πmicro3
64
+3
(3
2+ 26 logmicro
)micro4 +
(C1)
γ0n =e2g2
A
48π3f2πm
2π
1
2minus 9πmicro
16minus 2micro2 logmicro+
75πmicro3
128minus 3micro4
4+
(C2)
δLTp =e2g2
A
192π3f2πm
2π
1minus 9πmicro
8+
(13
2minus 2 logmicro
)micro2 minus 465πmicro3
64
minus(
47
2+ 42 logmicro
)micro4 +
(C3)
δLTn =e2g2
A
96π3f2πm
2π
1
2+
3πmicro
16+ (1 + 2 log micro)micro2 minus 105πmicro3
128+
5micro4
4+
(C4)
38
δLTn =e2g2
A
96π3f2πm
2π
1
2+
3πmicro
16+ (1 + 2 log micro)micro2 minus 105πmicro3
128+
5micro4
4+
(C5)
γ0p =e2g2
A
16π3f2πm
4π
4
45minus 3πmicro
16+
14micro2
5minus 1813πmicro3
384minus 192
5(1 + log micro)micro4
+80703πmicro5
2048+
(C6)
γ0n =e2g2
A
16π3f2πm
4π
4
45minus 5πmicro
48+
4micro2
5minus 245πmicro3
384minus 32micro4 logmicro
15
+1323πmicro5
2048+
(C7)
bull Slopes of polarizabilities at Q2 = 0
dγ0p(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
2minus 45πmicro
4+ 223micro2 minus 28515πmicro3
64
minus9
(1953
4+ 449 logmicro
)micro4 +
570255πmicro5
128+
(C8)
dγ0n(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
2minus 81πmicro
8+ 94micro2 minus 2535πmicro3
32
minus3 (1 + 90 logmicro)micro4 +84315πmicro5
1024+
(C9)
dδLTp(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
2880π3f2πm
4π
minus5
2minus 27πmicro
32+ 20micro2 minus 5865πmicro3
256
+3
(617
4+ 36 logmicro
)micro4 minus 2056845πmicro5
4096+
(C10)
dδLTn(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
minus5
4minus 81πmicro
64minus 11micro2 +
10005πmicro3
512
+15
8(11 + 48 logmicro)micro4 minus 267015πmicro5
8192+
(C11)
dIAp(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
96π2f2πmicro
2
1minus 15πmicro
4minus 1
2(115 + 88 logmicro)micro2 +
1839πmicro3
32
+5 (5 + 34 logmicro)micro4 +
(C12)
dIAn(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
48π2f2πmicro
2
1
2minus 11πmicro
8minus 1
4(1 + 20 logmicro)micro2 +
99πmicro3
64
minus25micro4
12+
(C13)
dI1p(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
96π2f2πmicro
3π
8+ 2 (4 + 3 logmicro)microminus 537πmicro2
64
minus1
2(15 + 56 logmicro)micro3 +
(C14)
39
dI1n(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
48π2f2πmicro
minus π
16+
1
4(3 + 4 logmicro)microminus 57πmicro2
128+
2micro3
3+
(C15)
γ0p(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2g2
A
16π3f2πm
6π
1
105minus 23πmicro
256+
377micro2
210minus 15551πmicro3
6144+
3371micro4
105
minus1640457πmicro5
32768+
(C16)
γ0n(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2g2
A
16π3f2πm
6π
1
105minus 153πmicro
1792+
69micro2
70minus 4615πmicro3
6144+
172micro4
35
minus120897πmicro5
32768+
(C17)
2 ∆-exchange contribution
Here we give analytical expressions for the tree-level ∆-exchange contributions to the nucleon
spin polarizabilities and their slopes at Q2 = 0 Note that the ∆-exchange contributes equally to
proton and neutron polarizabilities Recall that for the magnetic γlowastN∆ coupling we introduced a
dipole form factor to mimic vector-meson dominance gM rarr gM(1 +Q2Λ2)2
bull Static polarizabilities (Q2 = 0)
γ0 = minus e2
4πM2+
(g2M
∆2+
g2E
M2+
minus 4gMgEM+∆
) (C18)
δLT =e2M∆
4πM3+
(g2E
MNM++gMgE∆MN
minus gEgCM2
∆
) (C19)
γ0 =e2M2
N
π∆2M4+
(minusg
2M
∆2+
g2E
M2+
+4gMgE∆M+
) (C20)
bull Slopes of polarizabilities at Q2 = 0
dγ0(Q2)
dQ2
∣∣∣∣∣Q2=0
= minus e2
πM2+∆
(g2M
∆
[1
4∆2minus 1
∆M++
1
2M2+
]minus 1
Λ2
g2M
∆+
g2E
2M2+
[1
2∆minus 3
M+
]minusgMgE
M+
[1
∆2minus 5
∆M++
1
M2+
]+
1
Λ2
2gMgEM+
+2gMgC∆M2
+
minus gEgCM3
+
) (C21)
40
d δLT (Q2)
dQ2
∣∣∣∣∣Q2=0
=e2M∆∆
4πMNM2+
(g2E
∆2M2+
[1
∆minus 4
M+
]minus g2
C
∆M2∆M
2+
+gMgE∆2M+
[1
∆2
minus 3
∆M++
1
M2+
]minus 2
Λ2
gMgE∆2M+
+gMgC∆M2
∆
[1
2∆2minus 2
∆M++
1
2M2+
]minus gEgC
2M2∆M
2+
[7
∆+
1
M+
]) (C22)
dIA(Q2)
dQ2
∣∣∣∣∣Q2=0
= minusM2N
M2+
(g2M
2∆2+
g2E
MNM+minus 2gMgE
∆M+minus gEgCM∆M+
) (C23)
dI1(Q2)
dQ2
∣∣∣∣∣Q2=0
= minusM∆M2N
2M3+
(g2E
MNM∆minus gMgE∆MN
minus gEgCM2
∆
) (C24)
γ0(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2M2
N
π∆3M6+
(g2M
∆
[3M2
∆ + 2M∆MN minus 9M2N
∆2+
4M2+
Λ2
]
minus4gMgE
[5M2
∆ minus 9M2N
M+∆2+
2M+
Λ2
]+
g2E
∆M+[7M∆ minus 9MN ]
minus8gMgC∆
+4gEgCM+
) (C25)
[1] D Drechsel B Pasquini and M Vanderhaeghen Dispersion relations in real and virtual
Compton scattering Phys Rept 378 99 (2003) hep-ph0212124
[2] S E Kuhn J-P Chen and E Leader Spin structure of the nucleon mdash status and recent
results Prog Part Nucl Phys 63 1 (2009) arXiv08123535 [hep-ph]
[3] F Hagelstein R Miskimen and V Pascalutsa Nucleon polarizabilities from Compton scat-
tering to hydrogen atom Prog Part Nucl Phys 88 29 (2016) arXiv151203765 [nucl-th]
[4] B Pasquini and M Vanderhaeghen Dispersion theory in electromagnetic interactions Ann
Rev Nucl Part Sci 68 75 (2018) arXiv180510482 [hep-ph]
[5] S Gerasimov A Sum rule for magnetic moments and the damping of the nucleon magnetic
moment in nuclei Sov J Nucl Phys 2 430 (1966)
[6] S Drell and A C Hearn Exact sum rule for nucleon magnetic moments Phys Rev Lett
16 908 (1966)
[7] J S Schwinger Source theory viewpoints in deep inelastic scattering Proc Natl Acad Sci
USA 72 1 (1975)
[8] J S Schwinger Source theory viewpoints in deep inelastic scattering Electromagnetic In-
teractions and Field Theory Proceedings 14 Internationale Universitatswochen Schladming
Austria February 24-March 7 1975 Acta Phys Austriaca Suppl 14 471 (1975)
41
[9] J Schwinger Source theory discussion of deep inelastic scattering with polarized particles
Proc Natl Acad Sci USA 72 1559 (1975)
[10] A M Baldin Polarizability of nucleons Nucl Phys 18 310 (1960)
[11] M Gell-Mann M L Goldberger and W E Thirring Use of causality conditions in quantum
theory Phys Rev 95 1612 (1954)
[12] A Pineda Leading chiral logarithms to the hyperfine splitting of the hydrogen and muonic
hydrogen Phys Rev C 67 025201 (2003)
[13] C Peset and A Pineda The two-photon exchange contribution to muonic hydrogen from
chiral perturbation theory Nucl Phys B 887 69 (2014) arXiv14064524 [hep-ph]
[14] C Peset and A Pineda Model-independent determination of the two-photon exchange contri-
bution to hyperfine splitting in muonic hydrogen JHEP 04 060 arXiv161205206 [nucl-th]
[15] F Hagelstein and V Pascalutsa Proton structure in the hyperfine splitting of muonic hydro-
gen PoS CD15 077 (2016) arXiv151104301 [nucl-th]
[16] F Hagelstein Exciting Nucleons in Compton Scattering and Hydrogen-Like Atoms PhD
thesis Mainz U Inst Kernphys (2017) arXiv171000874 [nucl-th]
[17] F Hagelstein ∆(1232)-Resonance in the hydrogen spectrum Proceedings 11th International
Workshop on the Physics of Excited Nucleons (NSTAR 2017) Columbia SC USA August
20-23 2017 Few Body Syst 59 93 (2018) arXiv180109790 [nucl-th]
[18] Y Prok et al (CLAS) Moments of the spin structure functions gp1 and gd1 for 005 lt Q2 lt 30
GeV2 Phys Lett B 672 12 (2009) arXiv08022232 [nucl-ex]
[19] H Dutz et al (GDH) First measurement of the Gerasimov-Drell-Hearn sum rule for 1H from
07 GeV to 18 GeV at ELSA Phys Rev Lett 91 192001 (2003)
[20] M Amarian et al (Jefferson Lab E94010) Measurement of the generalized forward spin po-
larizabilities of the neutron Phys Rev Lett 93 152301 (2004) arXivnucl-ex0406005
[21] M Amarian et al The Q2 evolution of the generalized Gerasimov-Drell-Hearn integral for
the neutron using a 3He target Phys Rev Lett 89 242301 (2002) arXivnucl-ex0205020
[22] M Amarian et al (Jefferson Lab E94-010)Q2 evolution of the neutron spin structure moments
using a 3He target Phys Rev Lett 92 022301 (2004) arXivhep-ex0310003
[23] A Deur et al Experimental determination of the evolution of the Bjorken integral at low Q2
Phys Rev Lett 93 212001 (2004) arXivhep-ex0407007
[24] K Slifer Low Q2 measurement of gp2 and the δLT spin polarizability Spin structure at long dis-
tance Proceedings Workshop Newport News USA March 12ndash13 2009 nucl-ex09064775
(2009) AIP Conf Proc 1155 10106313203293 (2009) arXiv09064775 [nucl-ex]
[25] P Solvignon et al (E01-012) Moments of the neutron g2 structure function at intermediate
42
Q2 Phys Rev C 92 015208 (2015) arXiv13044497 [nucl-ex]
[26] A Deur Experimental studies at low Q2 of the spin structure of the nucleon at Jefferson Lab
in 9th International Workshop on Chiral Dynamics (CD18) Durham NC USA September
17-21 2018 (2019) arXiv190305661 [nucl-ex]
[27] V Sulkosky et al (Jefferson Lab E97-110) Measurement of the 3He spin-structure functions
and of neutron (3He) spin-dependent sum rules at 0035 le Q2 le 024 GeV2 Phys Lett B
805 135428 (2020) arXiv190805709 [nucl-ex]
[28] K Adhikari et al (CLAS) Measurement of the Q2 Dependence of the Deuteron Spin Structure
Function g1 and its Moments at Low Q2 with CLAS Phys Rev Lett 120 062501 (2018)
arXiv171101974 [nucl-ex]
[29] V Lensky F Hagelstein A Hiller Blin and V Pascalutsa Comment on rdquoMeasurement of
the Q2 Dependence of the Deuteron Spin Structure Function g1 and its Moments at Low Q2
with CLASrdquo (2018) arXiv180603219 [nucl-th]
[30] J M Alarcn F Hagelstein V Lensky and V Pascalutsa Forward doubly-virtual Compton
scattering off the nucleon in chiral perturbation theory at NLO the subtraction function and
moments of unpolarized structure functions (2020) arXiv200509518 [hep-ph]
[31] V Bernard E Epelbaum H Krebs and U-G Meiszligner New insights into the spin structure
of the nucleon Phys Rev D 87 054032 (2013) arXiv12092523 [hep-ph]
[32] V Lensky J M Alarcon and V Pascalutsa Moments of nucleon structure functions at
next-to-leading order in baryon chiral perturbation theory Phys Rev C 90 055202 (2014)
arXiv14072574 [hep-ph]
[33] V Pascalutsa and M Vanderhaeghen Chiral effective-field theory in the ∆(1232) region
I Pion electroproduction on the nucleon Phys Rev D 73 034003 (2006) arXivhep-
ph0512244
[34] V Pascalutsa and M Vanderhaeghen Electromagnetic nucleon-to-Delta transition in chiral
effective field theory Phys Rev Lett 95 232001 (2005) arXivhep-ph0508060
[35] T R Hemmert B R Holstein and J Kambor Systematic 1M expansion for spin 32
particles in baryon chiral perturbation theory Phys Lett B 395 89 (1997) arXivhep-
ph9606456
[36] V Pascalutsa and D R Phillips Effective theory of the ∆(1232) in Compton scattering off
the nucleon Phys Rev C 67 055202 (2003) arXivnucl-th0212024
[37] V Pascalutsa M Vanderhaeghen and S N Yang Electromagnetic excitation of the ∆(1232)-
resonance Phys Rept 437 125 (2007) arXivhep-ph0609004
[38] K A Olive et al (Particle Data Group) Review of Particle Physics Chin Phys C 38 090001
43
(2014)
[39] F E Low Scattering of light of very low frequency by systems of spin 12 Phys Rev 96
1428 (1954)
[40] M Gell-Mann and M L Goldberger Scattering of low-energy photons by particles of spin
12 Phys Rev 96 1433 (1954)
[41] A M Harun ar-Rashid A simple derivation of Schwingerrsquos sum rule for spin dependent
structure functions Nuovo Cim A 33 447 (1976)
[42] F Hagelstein and V Pascalutsa Dissecting the hadronic contributions to (g minus 2)micro by
Schwingerrsquos sum rule Phys Rev Lett 120 072002 (2018) arXiv171004571 [hep-ph]
[43] V Lensky V Pascalutsa M Vanderhaeghen and C Kao Spin-dependent sum rules con-
necting real and virtual Compton scattering verified Phys Rev D 95 074001 (2017)
arXiv170101947 [hep-ph]
[44] H Burkhardt and W N Cottingham Sum rules for forward virtual Compton scattering
Annals Phys 56 453 (1970)
[45] V Lensky and V Pascalutsa Predictive powers of chiral perturbation theory in Compton
scattering off protons Eur Phys J C 65 195 (2010) arXiv09070451 [hep-ph]
[46] H Grieszlighammer J McGovern D Phillips and G Feldman Using effective field theory to
analyse low-energy Compton scattering data from protons and light nuclei Prog Part Nucl
Phys 67 841 (2012) arXiv12036834 [nucl-th]
[47] H W Grieszlighammer J A McGovern and D R Phillips Nucleon polarisabilities at and
beyond physical pion masses Eur Phys J A 52 139 (2016) arXiv151101952 [nucl-th]
[48] E Epelbaum H Krebs and U-G Meiszligner Improved chiral nucleon-nucleon potential up to
next-to-next-to-next-to-leading order Eur Phys J A 51 53 (2015) arXiv14120142 [nucl-
th]
[49] E Epelbaum H Krebs and U-G Meiszligner Precision nucleon-nucleon potential at fifth order
in the chiral expansion Phys Rev Lett 115 122301 (2015) arXiv14124623 [nucl-th]
[50] C W Kao T Spitzenberg and M Vanderhaeghen Burkhardt-Cottingham sum rule and
forward spin polarizabilities in heavy baryon chiral perturbation theory Phys Rev D 67
016001 (2003) arXivhep-ph0209241
[51] C-W Kao D Drechsel S Kamalov and M Vanderhaeghen Higher moments of nucleon
spin structure functions in heavy baryon chiral perturbation theory and in a resonance model
Phys Rev D 69 056004 (2004) arXivhep-ph0312102
[52] T Becher and H Leutwyler Baryon chiral perturbation theory in manifestly Lorentz invariant
form Eur Phys J C 9 643 (1999) arXivhep-ph9901384
44
[53] J Gasser M E Sainio and A Svarc Nucleons with chiral loops Nucl Phys B 307 779
(1988)
[54] L S Geng J Martin Camalich L Alvarez-Ruso and M J Vicente Vacas Leading SU(3)-
breaking corrections to the baryon magnetic moments in chiral perturbation theory Phys
Rev Lett 101 222002 (2008) arXiv08051419 [hep-ph]
[55] D Drechsel S S Kamalov and L Tiator The GDH sum rule and related integrals Phys
Rev D 63 114010 (2001) arXivhep-ph0008306
[56] D Drechsel O Hanstein S S Kamalov and L Tiator A unitary isobar model for pion
photo- and electroproduction on the proton up to 1 GeV Nucl Phys A 645 145 (1999)
arXivnucl-th9807001
[57] L Tiator private communication (2020)
[58] V Bernard T R Hemmert and U-G Meiszligner Spin structure of the nucleon at low energies
Phys Rev D 67 076008 (2003) arXivhep-ph0212033
[59] O Gryniuk F Hagelstein and V Pascalutsa Evaluation of the forward Compton scattering
off protons II Spin-dependent amplitude and observables Phys Rev D 94 034043 (2016)
arXiv160400789 [nucl-th]
[60] R Zielinski The g2p Experiment A Measurement of the Protonrsquos Spin Structure Functions
PhD thesis New Hampshire U (2010) arXiv170808297 [nucl-ex]
[61] N Guler et al (CLAS) Precise determination of the deuteron spin structure at low to mod-
erate Q2 with CLAS and extraction of the neutron contribution Phys Rev C 92 055201
(2015) arXiv150507877 [nucl-ex]
[62] C E Carlson V Nazaryan and K Griffioen Proton structure corrections to electronic
and muonic hydrogen hyperfine splitting Phys Rev A 78 022517 (2008) arXiv08052603
[physicsatom-ph]
[63] V Bernard N Kaiser and U-G Meiszligner Chiral dynamics in nucleons and nuclei Int J
Mod Phys E 4 193 (1995) arXivhep-ph9501384
[64] T R Hemmert B R Holstein and J Kambor ∆(1232) and the polarizabilities of the nucleon
Phys Rev D 55 5598 (1997) arXivhep-ph9612374
[65] V Pascalutsa and R Timmermans Field theory of nucleon to higher-spin baryon transitions
Phys Rev C 60 042201 (1999) arXivnucl-th9905065
[66] V Pascalutsa Quantization of an interacting spin-32 field and the Delta isobar Phys Rev
D 58 096002 (1998) arXivhep-ph9802288
[67] H Krebs Double Virtual Compton Scattering and SpinStructure of the Nucleon PoS
CD2018 031 (2019)
45
[68] N Kochelev and Y Oh Axial anomaly and the δLT puzzle Phys Rev D 016012 (2012)
arXiv11034892 [hep-ph]
[69] D Drechsel S Kamalov and L Tiator Unitary isobar model ndash MAID2007 Eur Phys J A
69 (2007) available at httpsmaidkphuni-mainzde arXiv07100306 [nucl-th]
[70] P J Mohr B N Taylor and D B Newell CODATA recommended values of the fundamental
physical constants 2010 Rev Mod Phys 84 1527 (2012)
[71] J Ahrens et al (GDH A2) First measurement of the Gerasimov-Drell-Hearn integral for 1H
from 200 to 800 MeV Phys Rev Lett 87 022003 (2001) arXivhep-ex0105089 [hep-ex]
[72] K Helbing (GDH) Experimental verification of the GDH sum rule at ELSA and MAMI Nucl
Phys Proc Suppl 105 113 (2002)
[73] R Bradford A Bodek H S Budd and J Arrington A New parameterization of the nucleon
elastic form-factors NuInt05 proceedings of the 4th International Workshop on Neutrino-
Nucleus Interactions in the Few-GeV Region Okayama Japan 26-29 September 2005 Nucl
Phys Proc Suppl 159 127 (2006) arXivhep-ex0602017
[74] J D Bjorken Applications of the chiral U(6)timesU(6) algebra of current densities Phys Rev
148 1467 (1966)
[75] J D Bjorken Inelastic scattering of polarized leptons from polarized nucleons Phys Rev D
1 1376 (1970)
[76] V Bernard T R Hemmert and U-G Meiszligner Novel analysis of chiral loop effects in
the generalized Gerasimov-Drell-Hearn sum rule Phys Lett B 545 105 (2002) arXivhep-
ph0203167
[77] A Deur et al Experimental study of isovector spin sum rules Phys Rev D 78 032001
(2008) arXiv08023198 [nucl-ex]
[78] B Pasquini P Pedroni and D Drechsel Higher order forward spin polarizability Phys Lett
B 687 160 (2010) arXiv10014230 [hep-ph]
[79] R Jaffe g2ndashThe nucleonrsquos other spin-dependent structure function Comments Nucl Part
Phys 19 239 (1990)
[80] E V Shuryak and A Vainshtein Theory of power corrections to deep inelastic scattering in
quantum chromodynamics (II) Qminus4 effects polarized target Nucl Phys B 201 141 (1982)
[81] S Wandzura and F Wilczek Sum rules for spin dependent electroproduction Test of rela-
tivistic constituent quarks Phys Lett B 72 195 (1977)
[82] B W Filippone and X-D Ji The spin structure of the nucleon Adv Nucl Phys 26 1
(2001) arXivhep-ph0101224
[83] M Burkardt The g2 structure function Proceedings Workshop on Spin structure at long
46
distance Newport News USA March 12-13 2009 AIP Conf Proc 1155 26 (2009)
arXiv09054079 [hep-ph]
[84] A Deur S J Brodsky and G F de Teramond The spin structure of the nucleon Rept
Prog Phys 82 076201 (2019) arXiv180705250 [hep-ph]
[85] B R Holstein D Drechsel B Pasquini and M Vanderhaeghen Higher order polarizabilities
of the proton Phys Rev C 61 034316 (2000) arXivhep-ph9910427
[86] B R Holstein V Pascalutsa and M Vanderhaeghen Sum rules for magnetic moments and
polarizabilities in QED and chiral effective-field theory Phys Rev D 72 094014 (2005)
arXivhep-ph0507016
[87] B Pasquini D Drechsel and L Tiator Invariant amplitudes for pion electroproduction Eur
Phys J A 34 387 (2007) arXiv07122327 [hep-ph]
[88] A J Buchmann Non-spherical proton shape and hydrogen hyperfine splitting Proceedings
International Workshop on Precision Physics of Simple Atomic Systems (PSAS 2008) Wind-
sor Ontario Canada July 21-26 2008 Can J Phys 87 773 (2009) arXiv09104747
[physicsatom-ph]
47
000 005 010 015 020 025 030
-3
-2
-1
0
1
2
Q2GeV
2D
Γ01
0-
4fm
4D
proton
000 005 010 015 020 025 030
-3
-2
-1
0
1
2
3
Q2GeV
2D
Γ01
0-
4fm
4D
neutron
000 005 010 015 020 025 030
00
05
10
15
Q2GeV
2D
∆LT1
0-
4fm
4D
proton
000 005 010 015 020 025 030
00
05
10
15
20
25
Q2GeV
2D
∆LT1
0-
4fm
4D
neutron
FIG 3 Contributions of the different orders to the chiral predictions of γ0(Q2) upper paneland δLT (Q2) lower panel for the proton (left) and neutron (right) Red solid line πN -loop
contribution green dot-dashed line ∆-exchange contribution orange dotted line π∆-loop con-
tribution blue long-dashed line total result purple dot-dot-dashed line total result without gCcontribution black short-dashed line total result without gM dipole
(where possible) the so-called ldquoconsistentrdquo couplings to the Delta field ie those couplings
that project out the spurious degree of freedom see Refs [37 65 66] The authors of
Ref [31] on the other hand use couplings where the consistency in this sense is not enforced
The effects of these differences are of higher order in the δ-counting expansion and their
contribution to the Q2 dependence of the considered polarizabilities is expected to be rather
small however the differences at Q2 = 0 could be noticeable [67]
Finally as mentioned in Sec II C the inclusion of the dipole form factor in the magnetic
coupling gM is expected to be important to generate the correct Q2 behaviour of the polar-
izabilities Comparing our predictions for the forward spin polarizability with and without
inclusion of the form factor see Fig 3 upper panel confirms this Without the dipole
our results for the proton and neutron are closer to the ones from Ref [31] where the form
factor is not included For the neutron our prediction without the dipole is able to describe
14
the experimental points at very low Q2 deviating from the data with increasing Q2 The
π∆-loop contribution does not modify the Q2 behavior of γ0(Q2) and only differs from
Ref [31] by a small global shift Note also the relatively large effect of gC which generates
a sign change for virtualities above sim 02 GeV2 see Fig 3 upper panel
B δLT (Q2) mdash longitudinal-transverse polarizability
The longitudinal-transverse spin polarizability
δLT (Q2) =1
2π2
int infinν0
dν
radic1 +
Q2
ν2
σLT (νQ2)
Qν2(22)
=16αM2
N
Q6
int x0
0
dx x2[g1(xQ2) + g2(xQ2)
]
contains information about the spin structure of the nucleon and is another important
input in the determination of the (muonic-)hydrogen hyperfine splitting [16 62] It is also
relevant in studies of higher-twist corrections to the structure function g2(xQ2) given by
the moment d2(Q2) [51] see Section III E The peculiarity of the response encoded in this
polarizability is that it involves a spin flip of the nucleon and a polarization change of the
photon see App B and Fig 11
It is expected that the Delta isobar gives only a small contribution to δLT (Q2) what makes
this polarizability a potentially clean test case for chiral calculations Consequently there are
relatively many different theoretical calculations of δLT (Q2) coming from different versions
of χPT with baryons (HB IR and covariant) Ref [50] found a systematic deviation of the
HB result for δLTn(Q2) from the MAID model prediction This disagreement was identified
by the authors of Ref [68] as a puzzle involving the neutron δLT polarizabilitymdashthe δLT
puzzle The IR calculation in Ref [58] also showed a deviation from the data and predicted
a rapid rise of δLT (Q2) with growing Q2 The problem is solved by keeping the relativistic
structure of the theory as the BχPT+∆ result of Ref [31] showed
As expected already the leading πN loops provide a reasonable agreement with the
experimental data cf Fig 2 lower panel Since the ∆-exchange contribution to δLT (Q2)
is small the effect of the gM form factor is negligible in this polarizability as is that of the gC
coupling cf Fig 3 lower panel In fact we predict both the ∆-exchange and the π∆-loop
contributions to be small and negative This is in agreement with the MAID model which
predicts a small and negative contribution of the P33 wave to δLT (Q2) However in the
calculation of Ref [31] which is different from the one presented here only in the way the
∆(1232) is included the contribution of this resonance to δLTp(Q2) is sizeable and positive
15
The authors of that work attributed this large contribution to diagrams where the photons
couple directly to the Delta inside a loop As mentioned in Sec III F the effect of such loop
diagrams does not change the Q2 behaviour of the polarizabilities On the other hand it
can produce a substantial shift of the δLT (Q2) as a whole A higher-order calculation should
resolve the discrepancy between the two covariant approaches however it will partially lose
the predictive power since the LECs appearing at higher orders will have to be fitted to
experimental data
The πN -loop ∆-exchange and π∆-loop contributions to the NLO BχPT prediction of
the static longitudinal-transverse polarizability are in units of 10minus4 fm4
δLTp = 132(15) asymp 150minus 016minus 002 (23a)
δLTn = 218(23) asymp 235minus 016minus 002 (23b)
while the slopes are in units of 10minus4 fm6
dδLTp(Q2)
dQ2
∣∣∣∣Q2=0
= minus085(8) asymp minus080minus 004minus 001 (24a)
dδLTn(Q2)
dQ2
∣∣∣∣Q2=0
= minus124(12) asymp minus119minus 004minus 001 (24b)
C IA(Q2) mdash a generalized GDH integral
The helicity-difference cross section σTT exhibits a faster fall-off in ν than its spin-averaged
counterpart σT This is due to a cancellation between the leading (constant) terms of σ12
and σ32 at large ν3 The resulting 1ν fall-off of the helicity-difference cross section allows
one to write an unsubtracted dispersion relation for the VVCS amplitude gnonpoleTT (νQ2) cf
Eq (10a) This is the origin of the GDH sum rule [5 6]
minus α
2M2N
κ2 =1
2π2
int infinν0
dνσTT (ν)
ν (25)
which establishes a relation to the anomalous magnetic moment κ It is experimentally
verified for the nucleon by MAMI (Mainz) and ELSA (Bonn) [71 72]
There are two extensions of the GDH sum rule to finite Q2 the generalized GDH integrals
3 Notice that a constant term in σTT at ν rarrinfin is forbidden by crossing symmetry
16
000 005 010 015 020 025 030
-30
-25
-20
-15
-10
-05
00
Q2 GeV2D
I Aproton
000 005 010 015 020 025 030
-25
-20
-15
-10
-05
00
Q2 GeV2D
I A
neutron
000 005 010 015 020 025 030
-05
00
05
Q2 GeV2D
I 1
proton
000 005 010 015 020 025 030
-10
-08
-06
-04
-02
00
02
Q2 GeV2D
I 1
neutron
FIG 4 Upper panel The generalized GDH integral IA(Q2) for the proton (left) and neutron
(right) as function of Q2 The result of this work the NLO BχPT prediction is shown by the blue
solid line and the blue band The red line represents the LO BχPT result The purple short-dashed
line is the O(p4) HB result from Ref [50 51] The gray band is the BχPT+∆ result from Ref [31]
The black dotted line is the MAID model prediction [69] Experimental extractions for the proton
Ref [60] (orange triangle uncertainties added in quadrature) and neutron Refs [21][27] where
magenta dotsorange diagonal crosses correspond to data and red squareslilac crosses correspond
to data plus extrapolation to unmeasured energy regions The green stars at the real-photon point
are derived from the anomalous magnetic moments κp asymp 1793 and κn asymp minus1913 [70] Lower
panel The generalized GDH integral I1(Q2) for the proton (left) and neutron (right) as function
of Q2 The purple short-dashed line is the HB result from Ref [51] Experimental extractions for
the proton Ref [18] (blue dots) and Ref [60] (orange triangle uncertainties added in quadrature)
and neutron Ref [61][27] (uncertainties added in quadrature) where green dotsorange diagonal
crosses correspond to data and gray squareslilac crosses correspond to data plus extrapolation to
unmeasured energy regions
17
IA(Q2) and I1(Q2) The latter will be discussed in Sec III D The former is defined as4
minus α
2M2N
IA(Q2) = minus 1
8π2
int infinν0
dν
radic1 +
Q2
ν2
σTT (νQ2)
ν(26)
=α
Q2
int x0
0
dx[4M2
Nx2
Q2g2(xQ2)minus g1(xQ2)
]
Due to its energy weighting the integral in Eq (26) converges slower than the one in the
generalized forward spin polarizability sum rule (19) Therefore knowledge of the cross
section at higher energies is required and the evaluation of the generalized GDH integral
IA(Q2) is not as simple as the evaluation of γ0(Q2)
The generalized GDH integral IA(Q2) is directly related to the non-pole amplitude
gnonpoleTT (νQ2) which differs from non-Born amplitude gTT (νQ2) by a term involving the
elastic Pauli form factor
gnonpoleTT (νQ2) = gTT (νQ2)minus 2παν
M2N
F 22 (Q2) (27)
cf Eqs (2a) and (8a) Consequently IA(Q2) is not a pure polarizability but also contains
an elastic contribution The ldquonon-polarizabilityrdquo or the Born part of IA(Q2) is given by
IBornA (Q2) = IA(Q2)minus∆IA(Q2) = minus1
4F 2
2 (Q2) (28)
where we refer to the polarizability part as ∆IA(Q2) The same is true for the generalized
GDH integral I1(Q2) which is directly related to Snonpole1 (νQ2)
IBorn1 (Q2) = I1(Q2)minus∆I1(Q2) = minus1
4F 2
2 (Q2) (29)
In the following we will add the Born parts to our LO and NLO BχPT predictions for
the polarizabilities ∆IA(Q2) and ∆I1(Q2) employing an empirical parametrization for the
elastic Pauli form factor [73] This allows us to compare to the experimental results for
IA(Q2) and I1(Q2) cf Fig 4 Note that the blue error bands only describe the uncertainties
of our BχPT predictions of the polarizabilities while the elastic contributions are considered
to be exact as explained in Sec II C The uncertainties of the polarizability predictions are
therefore better reflected in Fig 5 where we show the contributions of the different orders
to the BχPT predictions of ∆IA(Q2) and ∆I1(Q2) as well as the total results with error
bands
4 Note that IA(Q2) is sometimes called ITT (Q2)
18
000 005 010 015 020 025 030
-08
-06
-04
-02
00
02
Q2 GeV2D
DI A
proton
000 005 010 015 020 025 030
-10
-08
-06
-04
-02
00
02
Q2 GeV2D
DI A
neutron
000 005 010 015 020 025 030-005
000
005
010
015
020
025
Q2 GeV2D
DI 1
proton
000 005 010 015 020 025 030-015
-010
-005
000
005
010
015
Q2 GeV2D
DI 1
neutron
FIG 5 Contributions of the different orders to the chiral predictions of ∆IA(Q2) upper paneland ∆I1(Q2) lower panel for the proton (left) and neutron (right) Red solid line πN -loop
contribution green dot-dashed line ∆-exchange contribution orange dotted line π∆-loop contri-
bution blue solid line and blue band total result purple dot-dot-dashed line total result without
gC contribution black short-dashed line total result without gM dipole
The E97-110 experiment at Jefferson Lab has recently published their data for IAn(Q2) in
the region of 0035 GeV2 lt Q2 lt 024 GeV2 [27] In addition there are results for IAn(Q2)
from the earlier E94-010 experiment [21] and for IAp(Q2) from the E08-027 experiment [60]
The O(p4) HB calculation gives a large negative effect [51] which does not describe the data
The BχPT+∆ result from Ref [31] which mainly differs from our work by the absence of
the dipole form factor in gM looks similar to this HB result and only describes the data
points at lowest Q2 Our NLO prediction however follows closely the Q2 evolution of the
data In Fig 5 upper panel we show the polarizability ∆IA(Q2) whose Q2 evolution is
clearly dominated by the ∆ exchange Similar to the case of γ0p(Q2) inclusion of the dipole
in gM and the Coulomb coupling gC is very important in order to describe the experimental
data The LO prediction on the other hand slightly overestimates the data cf Fig 4
upper panel
19
At the real-photon point IA(0) = minusκ2
4and ∆IA(0) = 0 Therefore we give only the
slope of the polarizability ∆IA(Q2) [showing also the separate contributions from πN loops
∆ exchange and π∆ loops] in units of GeVminus2
d∆IAp(Q2)
dQ2
∣∣∣∣Q2=0
= minus858(343) asymp 238minus 1121 + 025 (30a)
d∆IAn(Q2)
dQ2
∣∣∣∣Q2=0
= minus955(343) asymp 141minus 1121 + 025 (30b)
Including the empirical Pauli form factor [73] we find in units of GeVminus2
dIAp(Q2)
dQ2
∣∣∣∣Q2=0
= minus318dIAn(Q2)
dQ2
∣∣∣∣Q2=0
= minus300 (31)
D Γ1(Q2) and I1(Q2) mdash the first moment of the structure function g1(xQ2)
The second variant for a generalization of the GDH sum rule to finite Q2 is defined as
minus α
2M2N
I1(Q2) = minus 1
8π2
int infinν0
dν1radic
ν2 +Q2
[σTT (νQ2) +
Q
νσLT (νQ2)
](32)
= minus α
Q2
int x0
0
dx g1(xQ2)
where I1(0) = minusκ2
4 This generalized GDH integral directly stems from the amplitude
Snonpole1 (νQ2) with the LEX from Eq (12a) It is given by the first moment of the struc-
ture function g1(xQ2) Γ1(Q2) =int x0
0dx g1(xQ2) as follows I1(Q2) =
2M2N
Q2 Γ1(Q2) The
isovector combination
Γ1(pminusn)(Q2) =
int x0
0
dx[g1p(xQ
2)minus g1n(xQ2)] (33)
is related to the axial coupling of the nucleon through the Bjorken sum rule [74 75]
limQ2rarrinfin
Γ1(pminusn)(Q2) =
gA6 (34)
As explained in Eq (28) the moment I1(Q2) splits into a polarizability part ∆I1(Q2) and
a Born part IBorn1 (Q2) Figure 4 lower panel shows the Q2 dependence of I1(Q2) which in
contrast to IA(Q2) shown in Figure 4 upper panel is clearly dominated by its Born part
and the elastic Pauli form factor The πN -loop ∆-exchange and π∆-loop contributions to
the polarizability ∆I1(Q2) are shown in Fig 5 lower panel Comparing to Fig 5 upper
20
000 005 010 015 020 025 030
-006
-004
-002
000
002
004
Q2 GeV2D
G1
proton
000 005 010 015 020 025 030
-006
-005
-004
-003
-002
-001
000
001
Q2 GeV2D
G1
neutron
FIG 6 First moment of the structure function g1(xQ2) for the proton (left) and neutron (right)
as function of Q2 The legend is the same as in Fig 4
panel one sees that ∆I1(Q2) is less sensitive to gC and the dipole form factor in gM than
∆IA(Q2)
For the proton our NLO BχPT prediction gives a very good description of the ex-
perimental data [18 60] and is in reasonable agreement with the MAID prediction [69]
For the neutron one observes good agreement with the empirical evaluations including ex-
trapolations to unmeasured energy regions starting from Q2 gt 01 GeV2 [27 61] In the
region of Q2 lt 005 GeV2 one observes an interesting tension between the recent E97-
110 experiment [27] and the data from CLAS [61] While the newest measurement finds
I1n(0035 GeV2) lt κ2n4 thus suggesting a negative slope at low Q the older measurement
found a rather large value for I1n(00496 GeV2) A similar but milder behaviour is seen in
the E97-110 [27] and E94-010 [21] data for IAn The MAID predictions do not agree with
the CODATA recommended values for the anomalous magnetic moments of the proton and
neutron [70] which in our work are imposed by using empirical parametrizations for the
elastic Pauli form factors [73] The slope of the HB result from Ref [51] is too large and
therefore only reproduces the data at very low Q2
Figure 6 shows the moment Γ1(Q2) for the proton and neutron while Fig 7 shows the
isovector combination Γ1 pminusn(Q2) The LO and NLO BχPT predictions are identical be-
cause our calculation produces the same Delta contributions for the proton and the neutron
For the isovector combination the MAID model only agrees with the data at very low
Q2 lt 010 GeV2 The same is true for the IR result [58 76] while all other chiral results
describe the data NLO BχPT (this work) BχPT+∆ [31] and HBχPT [51]
At the real-photon point I1(0) = minusκ2
4and ∆I1(0) = 0 Therefore we give only the
slope of the polarizability ∆I1(Q2) [showing also the separate contributions from πN loops
21
000 005 010 015 020 025 030-002
000
002
004
006
008
010
Q2 GeV2D
G1Hp-
nL
FIG 7 Isovector combination of Γ1(Q2) as function of Q2 The legend is the same as in Fig 6
The pink curve is the IR result from Ref [58 76] The experimental points are from Ref [77]
(brown dots) and Ref [23] (orange squares)
∆ exchange and π∆ loops] in units of GeVminus2
d∆I1p(Q2)
dQ2
∣∣∣∣Q2=0
= 039(4) asymp 034minus 053 + 058 (35a)
d∆I1n(Q2)
dQ2
∣∣∣∣Q2=0
= minus101(10) asymp minus107minus 053 + 058 (35b)
Including the empirical Pauli form factor [73] we find in units of GeVminus2
dI1p(Q2)
dQ2
∣∣∣∣Q2=0
= 580dI1n(Q2)
dQ2
∣∣∣∣Q2=0
= 553 (36)
E d2(Q2) mdash a measure of color polarizability
Another interesting moment to consider is d2(Q2) which is related to the twist-3 part of
the spin structure function g2(xQ2) [79 80]
d2(Q2) equiv 3
int 1
0
dx x2[g2(xQ2)minus gWW2 (xQ2)] (37)
where gWW2 (xQ2) is the twist-2 part of g2(xQ2) Using the Wandzura-Wilczek relation [81]
one can relate d2(Q2) to moments of the spin structure functions g1(xQ2) and g2(xQ2)
d2(Q2) =
int 1
0
dx x2 [3g2(xQ2) + 2g1(xQ2)] (38)
22
000 005 010 015 020 025 030
0000
0001
0002
0003
0004
0005
0006
0007
Q2 GeV2D
d2
proton
000 005 010 015 020 025 030
0000
0002
0004
0006
0008
0010
0012
Q2 GeV2D
d2
neutron
000 005 010 015 020 025 030
-1
0
1
2
Q2 GeV2D
Γ 010
-4
fm6
D
proton
000 005 010 015 020 025 030-2
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
6D
neutron
FIG 8 Upper panel The inelastic moment d2(Q2) for the proton (left) and neutron (right) as
function of Q2 The result of this work the NLO BχPT prediction is shown by the blue solid line
and the blue band The red line represents the LO BχPT result The purple short-dashed line is
the O(p4) HB result from Ref [50 51] The black dotted line is the MAID model prediction [69]
The experimental points for the neutron (cyan dots) are from Ref [22] Lower panel Fifth-order
generalized forward spin polarizability γ0(Q2) for the proton (left) and neutron (right) as function
of Q2 The experimental points for the proton are from Ref [59] (purple square) and Ref [78]
(orange dot)
This relation however only holds for asymptotically large Q2 It is also in the high-Q2
region where d2(Q2) is a measure of color polarizability [82 83] through its relation to the
gluon field strength tensor [80] We refer to Ref [84] for a recent review on the spin structure
of the nucleon including a discussion of sum rules for deep inelastic scattering and color
polarizabilities
What we consider in the following is the inelastic part of d2(Q2) defined as the moment
of g1(xQ2) and g2(xQ2) spin structure functions cf Eq (38)
d2(Q2) =
int x0
0
dx x2 [3g2(xQ2) + 2g1(xQ2)] (39)
23
This moment provides another testing ground for our BχPT predictions through comparison
with experiments on the neutron [22] Going towards the low-Q2 region the interpretation
of d2(Q2) in terms of color polarizabilities will fade out The above definition however
implies it is related to other VVCS polarizabilities
d2(Q2) =Q4
8M4N
[M2
NQ2
αδLT (Q2) + I1(Q2)minus IA(Q2)
] (40)
Note that d2(Q2) and its first two derivatives with respect to Q2 vanish at Q2 = 0 The
considerations in Eqs (28) and (29) have no effect on d2(Q2) since the Born contribution
from IA(Q2) and I1(Q2) cancel out Therefore d2(Q2) is a pure polarizability
In Fig 8 upper panel we show our NLO BχPT prediction and other results for d2(Q2)
While MAID [69] and BχPT describe the experimental data for the neutron [22] very well
the HB limit [50 51] is showing a fast growth with Q2 This illustrates the importance
of keeping the relativistic result Note also that even though the πN -loop contribution
is dominant both gC and the form factor in gM are essential to obtain a curvature that
reproduces the data cf Fig 9 upper panel For the proton there are to our knowledge
no experimental results to compare with However the agreement between the NLO BχPT
prediction and the MAID prediction at low energies is reasonable
F γ0(Q2) mdash fifth-order generalized forward spin polarizability
It is interesting to compare the generalized fifth-order forward spin polarizability sum
rule
γ0(Q2) =1
2π2
int infinν0
dν
radic1 +
Q2
ν2
σTT (νQ2)
ν5(41)
=64αM4
N
Q10
int x0
0
dx x4
[g1(xQ2)minus 4M2
Nx2
Q2g2(xQ2)
]
to the sum rule integrals for IA(Q2) and γ0(Q2) since they differ merely by their energy
weighting of σTT (νQ2) and a constant prefactor cf Eqs (19) (26) and (41) From IA(Q2)
to γ0(Q2) to γ0(Q2) the energy suppression is increasing by a factor of νminus2 respectively
Therefore the description of γ0(Q2) should be easiest in a low-energy effective-field theory
such as χPT whereas γ0(Q2) and IA(Q2) receive larger contributions from higher energies
In Fig 8 lower panel we show our LO and NLO BχPT predictions for γ0(Q2) One
can see that the πN -loop contribution is positive (in accordance to what we see for the cross
section σTT see Fig 10) The Delta shifts it substantially especially in the low Q2 region
24
000 005 010 015 020 025 030
-0004
-0002
0000
0002
0004
0006
0008
0010
Q2GeV
2D
d2
proton
000 005 010 015 020 025 030
-0005
0000
0005
0010
0015
Q2GeV
2D
d2
neutron
000 005 010 015 020 025 030
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
6D
proton
000 005 010 015 020 025 030
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
6D
neutron
FIG 9 Contributions of the different orders to the chiral predictions of d2(Q2) upper panel and
γ0(Q2) lower panel for the proton (left) and neutron (right) Red solid line πN -loop contribu-
tion green dot-dashed line ∆-exchange contribution orange dotted line π∆-loop contribution
blue long-dashed line total result purple dot-dot-dashed line total result without gC contribution
black short-dashed line total result without gM dipole
bringing it into a better agreement with data In general the BχPT curves start above the
empirical data points at the real-photon point and then decrease asymptotically to zero
above Q2 gt 01 GeV2 On the other hand the MAID prediction reproduces the empirical
data at the real-photon point then decreases to negative values until about Q2 gt 006 GeV2
from where it also starts to asymptotically approach zero Consequently our NLO BχPT
prediction of γ0(Q2) is consistently above the MAID prediction This is very different to
what we saw for IA(Q2) in Fig 4 upper panel where the MAID prediction at the real-
photon point is above the experimental value While the agreement of our predictions with
the empirical data is in general quite good for all moments of σTT (νQ2) one should point
out that both for γ0n(Q2) and γ0p(Q2) we overestimate the data at low Q2 For IA(Q2) such
observation cannot be made because ∆IA(0) = 0 and thus IA(0) is given by the empirical
Pauli form factor only From IA(Q2) γ0(Q2) and γ0(Q2) the latter has the smallest however
25
non-negligible dependence on gC and the dipole in gM cf Fig 9 lower panel
The πN -loop ∆-exchange and π∆-loop contributions to the NLO BχPT prediction of
the static fifth-order forward spin polarizability amount to in units of 10minus4 fm6
γ0p = 112(30) asymp 208minus 096minus 001 (42a)
γ0n = 195(30) asymp 292minus 096minus 001 (42b)
while the slope is composed as follows in units of 10minus4 fm8
dγ0p(Q2)
dQ2
∣∣∣∣Q2=0
= minus084(10) asymp minus100 + 016 + 000 (43a)
dγ0n(Q2)
dQ2
∣∣∣∣Q2=0
= minus142(15) asymp minus158 + 016 + 000 (43b)
Note that the HB prediction of the static polarizability γ0p (423 at O(p3) and 365 at O(ε3)
[78 85]) is almost one order of magnitude larger than the empirical value and therefore not
shown in Fig 8
G Summary
Our results are summarized in Table II where we give the contributions of the different
orders to the chiral predictions of the polarizabilities and their slopes at the real-photon
point A quantitative comparison of our predictions for the static spin polarizabilities to
the work of Bernard et al [31] and different empirical evaluations is shown in Table III We
can see that the inclusion of the Delta turns out to be very important for all moments of
the helicity-difference cross section To describe the Q2 behavior of the polarizabilities the
magnetic coupling of the N rarr ∆ transition should be modified by a dipole form factor as
has been observed previously in the description of electroproduction data [33] This dipole
form factor effectively takes account of vector-meson exchanges The Coulomb-quadrupole
N rarr ∆ transition despite its subleading order is important in the description of some
moments of spin structure functions This is contrary to what we saw for the moments of
unpolarized structure functions [30] where the Coulomb coupling had a negligible effect
The π∆ loops are mainly relevant for the generalized GDH integrals
26
TABLE II The NLO BχPT predictions for the forward VVCS polarizabilities and their slopes at
Q2 = 0 The contributions of the πN loops the ∆ exchange and the π∆ loops are shown together
with the combined total result Note that IA(0) = I1(0) = d2(0) = 0 and (d2)prime = 0
πN loops ∆ exchange π∆ loops Total
γ0 p 201 minus284 minus010minus093(92)
(10minus4 fm4) n 298 003(92)
δLT p 150 minus016 minus002132(15)
(10minus4 fm4) n 235 218(23)
γ0 p 208 minus096 minus001112(30)
(10minus4 fm6) n 292 195(30)
(γ0)prime p minus033011 001
minus022(4)
(10minus4 fm6) n minus073 minus061(7)
(δLT )prime p minus080 minus004 minus001minus085(8)
(10minus4 fm6) n minus119 minus124(12)
(γ0)prime p minus100016 000
minus084(10)
(10minus4 fm8) n minus158 minus142(15)
(∆IA)prime p 238 minus1121 025minus858(343)
(GeVminus2) n 141 minus955(343)
(∆I1)prime p 034 minus053 058039(4)
(GeVminus2) n minus107 minus101(10)
TABLE III Our NLO BχPT predictions for the spin polarizabilities at Q2 = 0 compared with
the BχPT+∆ predictions from Bernard et al [31] and the available empirical information Where
the reference is not given the empirical number is provided by the MAID analysis [55 69] with
unspecified uncertainty
Proton Neutron
This work BχPT+∆ Empirical This work BχPT+∆ Empirical
γ0 minus093(92) minus174(40) minus100(8)(12) [19] 003(92) minus077(40) minus0005
(10minus4 fm4) minus090(8)(11) [78] [MAID]
minus0929(105) [59]
δLT 132(15) 240(1) 134 218(23) 238(3) 203
(10minus4 fm4) [MAID] [MAID]
γ0 112(30) 060(7)(7) [78] 195(30) 123
(10minus4 fm6) 0484(82) [59] [MAID]
IV CONCLUSIONS
We have presented a complete NLO calculation of the polarized non-Born VVCS ampli-
tudes in covariant BχPT with pion nucleon and ∆(1232) fields The dispersion relations
between the VVCS amplitudes and the tree-level photoabsorption cross sections served as a
27
cross-check of these calculations
The obtained moments of the proton and neutron spin structure functions related to gen-
eralized polarizabilities and GDH-type integrals agree well with the available experimental
data The description of their Q2 evolution is improved compared to the previous χPT pre-
dictions In particular the NLO BχPT predictions obtained here give a better description
of the empirical data (eg from the Jefferson Laboratory ldquoSpin Physics Programrdquo) than
the HB [50 51] and IR [58] calculations
The demonstrated predictive power of the χPT framework amplitudes makes it well
suited for extending the χPT evaluation of the TPE effect in the hyperfine structure of
(muonic-)hydrogen [15ndash17] to next-to-leading order
ACKNOWLEDGEMENTS
We thank Lothar Tiator and Marc Vanderhaeghen for helpful discussions This work
is supported by the Deutsche Forschungsgemeinschaft (DFG) through the Collaborative
Research Center [The Low-Energy Frontier of the Standard Model (SFB 1044)] JMA ac-
knowledges support from the Community of Madrid through the ldquoPrograma de atraccion de
talento investigador 2017 (Modalidad 1)rdquo and the Spanish MECD grants FPA2016-77313-P
FH gratefully acknowledges financial support from the Swiss National Science Foundation
Appendix A Tensor decompositions of the VVCS amplitudes
In this appendix we review the decomposition of the forward VVCS process into tensor struc-
tures and scalar amplitudes In particular we consider the connection between the covariant and
the semi-relativistic decomposition in the lab frame that is defined in terms of the conventional
transverse longitudinal transverse-transverse and transverse-longitudinal amplitudes
As explained in Sec II A the process of forward VVCS off the nucleon can be described in
terms of four explicitly covariant amplitudes S1 2 and T1 2 [3]
T (νQ2) =
(minusgmicroν +
qmicroqν
q2
)T1(νQ2) +
1
M2N
(pmicro minus p middot q
q2qmicro)(
pν minus p middot qq2
qν)T2(νQ2) (A1)
minus 1
MNγmicroναqα S1(νQ2)minus 1
M2N
(γmicroνq2 + qmicroγναqα minus qνγmicroαqα
)S2(νQ2)
εprimelowastmicro εν
where εmicro (εprimelowastmicro ) are the incoming (outgoing) photon polarization vectors ν is the photon lab-frame
energy and Q2 is the photon virtuality Alternatively the decomposition in the laboratory frame
28
(which in the forward case coincides with the Breit frame) is parametrized in terms of the nucleon
Pauli matrices ~σ and the four scalar functions fL fT gTT and gLT
T (νQ2) = ε0 εprimelowast0 fL(νQ2) + (~ε primelowast middot ~ε ) fT (νQ2) + i~σ middot (~ε primelowast times ~ε ) gTT (νQ2) (A2)
minusi~σ middot [(ε0~εprimelowast minus ~ε ε primelowast0 )times q] gLT (νQ2)
Here ~q and q = ~q|~q | are the photon three-momentum in the lab system and its unit vector The
modified polarization vector components are given by
ε0 =
[ε0 minus
ν
|~q | (~ε middot q )
] |~q |Q ~ε = ~εminus q (~ε middot q ) (A3)
where ε = (ε0~ε ) is the usual incoming photon polarization vector and εprimelowast the outgoing polarization
vector The LEX of the lab frame amplitudes [Eq (10)] can serve in particular as the definition
of the generalized polarizabilities The lab frame amplitudes are also conveniently used for the
definition of the response functions see the example of the scalar amplitude gLT (νQ2) and the
corresponding response function σLT (νQ2) below in App B
Appendix B Photoabsorption cross sections
In the forward kinematics the spin-dependent VVCS amplitudes and the spin polarizabilities
can be described in terms of the polarized structure functions g1(xQ2) and g2(xQ2) or equiv-
alently the helicity-difference cross section σTT (νQ2) and the longitudinal-transverse response
function σLT (νQ2) with the help of dispersion relations (5) and the optical theorem (3) In this
way the photoabsorption cross sections measured in electroproduction processes form the basis
for most empirical evaluations shown throughout Sec III In the following we present the BχPT
predictions for the tree-level cross sections of πN - π∆- and ∆-production through photoabsorption
on the nucleon cf Figs 8 9 and 10 in Ref [30] In Secs B 1 and B 2 we will discuss the leading
πN -production channel and the ∆-production channel respectively We used these cross sections
to verify the polarizability predictions obtained otherwise from the calculated non-Born VVCS
amplitudes Due to the bad high-energy behavior of the π∆-production cross sections in BχPT
cf Fig 10 the dispersion relations in Eq (5) require further subtractions for a reconstruction of
the π∆-loop contribution to the spin-dependent VVCS amplitudes Therefore not all polarizabil-
ities could be verified but only those appearing as higher-order terms in the LEX of the VVCS
amplitudes such as γ0 [16]
29
proton neutron
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Neutron
02 04 06 08 100
50
100
150
200
250
[GeV]
T [b]
02 04 06 08 10
-50
0
50
100
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Neutron
02 04 06 08 100
50
100
150
200
250
[GeV]
T [b]
02 04 06 08 10
-50
0
50
100
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02040608100
50
100
150
200
250
300
[GeV]
T[b]
0204060810
-50
0
50
[GeV]
TT[b]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
0204060810
00
02
04
06
[GeV]
LTQ
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
0204060810
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
FIG 10 Photoabsorption cross sections for πN (red) and π∆ production (orange) with Q2 = 0
(solid) and Q2 = 01 GeV2 (dashed for πN and dotted for π∆ channel)
1 πN-production channel
In order to extract the response function σLT (νQ2) we have developed a method similar to
the one used to calculate σTT (νQ2) see for example Ref [86] For σLT (νQ2) however the
calculation is more complicated because one has to take into account that the associated Compton
process involves a spin-flip of the nucleon as illustrated in Fig 11 When calculating the cross
section the product of the incoming nucleon spinors has to reflect this flip
The forward VVCS amplitude related to σLT (νQ2) mdash and δLT (Q2) mdash is gLT (νQ2) It can
be extracted from Eq (A2) if one takes the modified polarization vector components in Eq (A3)
with ε = εL and εprimelowast = εlowastplusmn as input where εL = 1Q(|~q | 0 0 q0) and εplusmn = ∓ 1radic
2(0 1plusmni 0) are the
standard longitudinal and transverse polarization vectors respectively For εL and εlowastplusmn only the
choice of helicities hprime = plusmn12 and h = ∓12 gives a non-zero contribution and one obtains
χdaggerhprime T (νQ2)χh = χdaggerhprimeminusi~σ middot [(ε0~εprimelowast minus ~ε ε primelowast0 )times q] gLT (νQ2)χh =
radic2 gLT (νQ2) (B1)
where χh and χdaggerhprime are two-component Pauli spinors with opposite helicities or here spins
Let us now consider the related photoabsorption process and in particular the tree-level γlowastN rarr
30
s = 12
sprime = minus12
λ = 0 λprime = 1
rprime rprime
s=12sprime=minus1
2
λ=0λprime=1
rprimerprime
s=12
sprime=minus12
λ=0λprime=1
rprimerprime hacute=frac12h=-frac12
λacute=1λ=0
FIG 11 Relation between the forward Compton process and the photoabsorption process given
by the optical theorem In particular we show the longitudinal-transverse contribution The
double-line arrows represent the spin of the external particles while the dot represents the scalar
(longitudinal) polarization of the incoming photon Inside the blob the intermediate states are
represented eg nucleons with spins rprime (which are averaged in the calculation of the cross section)
and pions
πN channel see diagrams in Fig 8 of Ref [30] We define the πN -production amplitude as
T = uhB (PB)sumi
Ai(s t)Γi uhA(PA) (B2)
with the Dirac structures
Γ1 = γ5 (B3a)
Γ2 =1
2
[qA ε
]γ5 (B3b)
where uhA(PA) and udaggerhB (PB) are the Dirac spinors and PA and PB are the four-momenta of
the incoming and outgoing nucleons respectively When calculating the photoabsorption cross
section related to the VVCS amplitude in Eq (B1) the nucleon spin flip should be implemented
by uhprime(PA) in T dagger and uh(PA) in T together with the appropriate transverse and longitudinal
photon polarization vectors εlowastplusmn and εL
However if one wants to use the properties of the Dirac matrices it is more useful to construct
an operator to produce this spin flip in the external nucleons of Fig 11 This is accomplished by
introducing the projector ΓLT equiv 12radic
2(γ1 + iγ2)γ5 which also takes into account the extra factor
radic2 in Eq (B1) We checked that with this projector one correctly extracts δLT by comparing the
HB limit of our result to the HB result of Ref [50] where the authors calculate this polarizability
from the Compton amplitude directly With all those ingredients the longitudinal-transverse cross
section is calculated in the following way
σLT (νQ2) =1
64π2 s
|~pf |cm
|~pi|cm
int 1
minus1dcos θ
sumij
AiAdaggerjXij (B4)
31
with
Xij = Tr[(PB +MN )Γi(PA +MN )ΓLTγ0Γdaggerjγ
0] (B5)
where θ is the scattering angle in the center-of-mass (cm) frame and |~pi|cm (|~pf |cm) is the three-
momentum of an incoming (outgoing) particle in the cm frame An explicit calculation of the
matrix Xij leads to
X = MNQ
0 2(PB minus PA) middot εLminusradic
2|~qf |cm sin θ (sminus u)
(B6)
where |~qi|cm (|~qf |cm) is the relative three-momentum of the incoming (outgoing) particles in the
cm frame Here s t and u are the usual Mandelstam variables For the different γlowastN rarr πN
channels we obtain the following amplitudes Ai where we introduce qA as the four-momentum of
the incoming photon and qB as the four-momentum of the outgoing pion
bull γlowastprarr π0p
A1 =e gAMN
fπ
[2PA middot ε+ qA middot ε
sminusM2N
+2PB middot εminus qA middot ε
uminusM2N
] (B7a)
A2 =e gAMN
fπ
[1
sminusM2N
+1
uminusM2N
] (B7b)
bull γlowastprarr π+n
A1 =
radic2 e gAMN
fπ
[2PA middot ε+ qA middot ε
sminusM2N
+2(PA minus PB) middot ε+ qA middot ε
tminusm2π
] (B8a)
A2 =
radic2 e gAMN
fπ(sminusM2N )
(B8b)
bull γlowastnrarr π0n
A1 = 0 (B9a)
A2 = 0 (B9b)
bull γlowastnrarr πminusp
A1 =
radic2 e gAMN
fπ
[2PB middot εminus qA middot ε
uminusM2N
minus 2(PA minus PB) middot ε+ qA middot εtminusm2
π
] (B10a)
A2 =
radic2 e gAMN
fπ(uminusM2N )
(B10b)
The analytical expressions shown above were checked with the amplitudes given in Ref [87] An-
alytical expressions for the tree-level γlowastN rarr πN channel of the σLT (νQ2) and σTT (νQ2) cross
32
sections are given below (proton channels π+n and π0p neutron channel πminusp) We checked that
they reproduce the known results in the real-photon limit [45 86] To shorten the final expressions
for the cross sections which are considerably longer for finite Q2 than in the real-photon limit we
define the following dimensionless kinematic variables
αγ = (ENi )cmradics =
s+M2N +Q2
2s (B11)
απ = (ENf )cmradics =
s+M2N minusm2
π
2s (B12)
βγ = Eγcmradics =
sminusM2N minusQ2
2s (B13)
βπ = Eπcmradics =
sminusM2N +m2
π
2s (B14)
λγ = |~qi|cmradics =
radic(sminusM2
N minusQ2)2 + 4sQ2
2s (B15)
λπ = |~qf |cmradics =
radic(sminusM2
N +m2π)2 minus 4sm2
π
2s (B16)
Here (ENi )cm and (ENf )cm are the energies of the incoming and outgoing nucleon Eγcm is the energy
of the incoming photon Eπcm is the energy of the outgoing pion all in the cm frame
σ(π+n)TT = minus e2g2
AM2N
64πf2πs
3(sminusM2N )2λ4
γ
4sλπλγ
[(M2
N minus s)(M2N minusQ2 minus s)(Q2 + 2sβγβπ)
+ 2s(m2πQ
2 minus (M2N minus s)
(M2N + s(minus1 + 2βγβπ)
))λ2γ
]+ (M2
N minus s)(M2N minusQ2 minus s)
times (Q2 + 2sβγβπ minus 2sλπλγ)(Q2 + 2s(βγβπ + λπλγ)
)times log
(Q2 + 2sβγβπ minus 2sλγλπQ2 + 2sβγβπ + 2sλγλπ
) (B17)
33
σ(π0p)TT =
e2g2AM
2Nλπ
64πf2πs(sminusM2
N )2λγ
1
s2λπλ3γ
[(M2
N minus s)(M2N minusQ2 minus s)
(3M2
N + s(2βγβπ minus 3))(M2N
+ s(2βγβπ minus 1))
+ 2s
((2m2
πQ2 + (M2
N minus s)2)(M2
N minus s)minus 2(minusm2
πQ2 + (sminusM2
N )2)
times sβγβπ + 4s2(sminusM2N )β2
γβ2π
)λ2γ
]arctan
(2sλπλγ
M2N + s(2βγβπ minus 1)
)+ 2
[minus 2m2
πQ2
+ (M2N minus s)
(2(sminusM2
N ) + 4sβγβπ +(sminusM2
N +Q2)(3M2N minus 3s+ 2sβγβπ)
sλ2γ
+m2πQ
2(sminusM2N )
(M2N minus s+ 2sβγβπ)(M2
N + s(minus1 + 2βγβπ minus 2λπλγ))
+m2πQ
2(sminusM2N )
(M2N minus s+ 2sβγβπ)(M2
N + s(minus1 + 2βγβπ + 2λπλγ))
)
+1
λπλγ
(βγβπ
(minusm2
πQ2 + (M2
N minus s)(M2N minus s+ 2sβγβπ)
)minus (M2
N minus s)(M2N minusQ2 minus s)λ2
π
)log
(M2N + s(minus1 + 2βγβπ + 2λπλγ)
M2N + s(minus1 + 2βγβπ minus 2λπλγ)
)] (B18)
σ(πminusp)TT =
e2g2AM
2N
64πf2πs
3λ4γ(M2
N + s(2βπβγ minus 1))2 minus 4λ2πs
2λ2γ)
4sλπλγ
((minusM2
N +Q2 + s)(M2N
+ s(2βπβγ minus 1))
2 minus 2sλ2γ
(2λ2
πs(minusM2N +Q2 + s) +m2
πQ2))
+((M2N + s(2βπβγ minus 1)
)2
minus 4λ2πs
2λ2γ
)((M2N + s(2βπβγ minus 1)
)(M2N minus 2Q2 minus s(2βπβγ + 1)
)+ 4λ2
πs2λ2
γ
)times log
(minusM2
N minus 2βπsβγ minus 2λπsλγ + s)minus(M2N + s(2βπβγ minus 2λπλγ minus 1)
)(M2N + s(minus1
+ 2βπβγ + 2λπλγ))[((
M2N + s(2βπβγ minus 1)
)(M2N minus 2Q2 minus s(2βπβγ + 1)
)+ 4λ2
πs2λ2
γ
)times log
(minusM2
N minus 2βπsβγ + 2λπsλγ + s)
+((Q2 + 2βπsβγ)2 minus 4λ2
πs2λ2
γ
)times log
(Q2 + 2βπsβγ minus 2λπsλγQ2 + 2sβπβγ + 2λπsλγ
)]+ 4sλ2
γ
((M2
N + s(2βπβγ minus 1))2 minus 4λ2πs
2λ2γ
)times (M2
N minus s) arctanh
(2λπsλγ
M2N + 2βπsβγ minus s
) (B19)
34
σ(π+n)LT =
e2g2AM
3Nλπ
32πf2πQs
3(sminusM2N )2λ4
γ
2sλγ
[(M2
N minus s)(Q2 + 2sβ2γ)(Q2 + 2sβγβπ)
minus 4s
((M2
N minus s)(Q2 minus 2s(απ minus 1)βγ
)+Q2sβγβπ
)λ2γ + 8s3(minus1 + απ)λ4
γ
]minus M2
N minus sλπ
[(Q2 + 2sβ2
γ)(Q2 + 2sβγβπ)2 + 4s2(2(απ minus 1)βγ(Q2 + 2sβγβπ)
minusQ2λ2π
)λ2γ + 8s3(απ minus 1)2λ4
γ
]arctan
(2sλπλγ
Q2 + 2sβγβπ
) (B20)
σ(π0p)LT =
e2g2AM
3Nλπ
16πf2πQs(sminusM2
N )2λγ
1
minus2s(M2N + s(minus1 + 2βγβπ))2λ2
γ + 8s3λ2πλ
4γ
[minus 3M8
N (Q2
+ 2sβ2γ) + 2M4
Ns2
(minus (Q2 + 2sβ2
γ)(minus3 + 2βγβπ) +(Q2(minus3 + 2βγβπ + 6λ2
π) + 2sβγ
times (minus3 + 12απ + 2βγβπ minus 12απβγβπ + 4βγλ2π))λ2γ minus 4s(α2
π minus 1)λ4π
)+ 2M2
Ns3
[minus (Q2
+ 2sβ2γ)(minus1 + 2βγβπ)
(6 + βγβπ(minus9 + 2βγβπ)
)+
(Q2(3minus 12λ2
π + 4βγβπ(minus1 + 2βγβπ
+ λ2π))minus 2sβγ
(minus 3 + 4απ(3 + 2απ(3 + 2βγβπ(minus3 + βγβπ)) + 4βγ(βπ + (2minus βγβπ)
times λ2π)))λ2γ + 8s
((απ minus 1)(1 + απ minus 2βγβπ) + 2απβγλ
2π
)λ4γ
)+ s4
((Q2 + 2sβ2
γ)
times (1minus 2βγβπ)2(2βγβπ minus 3) + 2(Q2(minus1 + 6λ2
π + 2βγβπ((1minus 2βγβπ)2 minus 2λ2π)) + 2sβγ
times (minus1 + 2βγβπ + 4(minus1 + βγβπ)(minusαπ + 2απβγβπ minus βγλ2π)))λ2γ minus 8
(s(απ minus 1)(απ
+ (1minus 2βγβπ)2) + 2βγ(2sαπ +Q2βγ)λ2π
)λ4γ + 32s(απ minus 1)λ2
πλ6γ
)+ 2M6
Ns
(Q2(6
minus 7βγβπ + λ2γ) + 2sβγ(βγ(6minus 7βγβπ) + (1minus 4απ)λ2
γ)
)]+sminusM2
N
4s2λπλ3γ
[(Q2 + 2sβ2
γ)
times(3M2
N + s(2βγβπ minus 3))(M2N + s(2βγβπ minus 1)
)+ 2s
(minusM2
N (Q2 + 2sβγ minus 8sαπβγ)
+ s(2sβγ(1minus 4απ + 4απβγβπ) +Q2(1minus 2λ2π)))λ2γ + 8s3(απ minus 1)λ4
γ
]
times arctan
(2sλπλγ
M2N + s(2βγβπ minus 1)
) (B21)
35
σ(πminusp)LT =
e2g2AM
3N
64πf2πQs
3λ4γ(M2
N minusQ2 minus s)(M2N + s(2βπβγ + 2λπλγ minus 1))
times
1
M2N + s(2βπβγ minus 2λπλγ minus 1)
[4λπsλγ(M2
N minusQ2 minus s)(
2sλ2γ
(s((1minus 2λ2
π)Q2
+ 2(2απ minus 1)sβγ(2βπβγ minus 1))minusM2N (Q2 + 2(1minus 2απ)sβγ)
)+ (Q2 + 2sβ2
γ)(M2N + s(2βπβγ minus 1)
)2 + 8(απ minus 1)απs
3λ4γ
)]minus(M2N + s(2βπβγ + 2λπλγ minus 1)
)times[
log
(minusM2N minus 2βπsβγ minus 2λπsλγ + s
minusM2N minus 2βπsβγ + 2λπsλγ + s
)(minus (Q2 + 2sβ2
γ)(M2N + s(2βπβγ minus 1)
)(minusM2
N + 2Q2 + 2βπsβγ + s)minus 2sλ2γ
(minusM2
N (Q2 + 2sβγ) +Q4 +Q2s(4απβγ
minus 2βγ minus 2λ2π + 1) + 2s2βγ(4(απ minus 1)βπβγ + 1)
)minus 8(απ minus 1)2s3λ4
γ
)+ 2
(4s2λ2
γ
(2(απ minus 1)βγ(Q2 + 2βπsβγ)minus λ2
πQ2)
+ (Q2 + 2βπsβγ)2(Q2 + 2sβ2γ)
+ 8(απ minus 1)2s3λ4γ
)arctanh
(2λπsλγ
Q2 + 2βπsβγ
)] (B22)
2 ∆-production channel
The tree-level ∆-exchange diagram in Fig 2 of Ref [30] contributes to the non-Born part of
the VVCS amplitudes The contribution of the ∆ exchange to the VVCS amplitudes can be split
into [17]
S∆-exch1 (νQ2) = S∆-pole
1 (νQ2) + S∆-exch1 (νQ2) (B23a)
νS∆-exch2 (νQ2) = νS∆-pole
2 (νQ2) + νS2∆-exch
(νQ2) (B23b)
and similarly for the unpolarized VVCS amplitudes discussed in Ref [30] Here we introduced the
∆-pole contributions S∆-polei and the ∆-non-pole contributions S∆-exch
i The former amplitudes
feature a pole at the ∆(1232)-production threshold and thus are proportional to
1
[sminusM2∆][uminusM2
∆]=
1
4M2N
1
ν2∆ minus ν2
(B24)
36
They can be reconstructed from the dispersion relations in Eq (5) with the tree-level ∆-production
cross sections as input cf Fig 10 in Ref [30]
σTT (νQ2) =π2α
M2NM
2+|~q |
minus g2
MMN (M+ + ν)|~q |2 +g2E(∆minus ν)(Q2 minusMNν)2
MN(B25a)
+g2CQ
4s(∆minus ν)
MNM2∆
minus 4gMgE(Q2 minusMNν)|~q |2 minus 4gMgCQ2|~q |2
+2gEgCQ
2[minusMNM∆ |~q |2 + s(Q2 +∆ν)
]MNM∆
δ(ν minus ν∆)
σLT (νQ2) =Qπ2α
M2NM
2+|~q |
g2E(MNν minusQ2) [M∆(MN + ν)minus s]
MN(B25b)
+g2CQ
2[MNM∆|~q |2 minus s(Q2 +∆ν)
]MNM2
∆
+ gMgEM∆|~q |2 minusgMgC(Q2 minusMNν)|~q |2
M∆
+gEgC(ν minus∆)(M2
N |~q |2 minus 2Q2s)
MNM∆
δ(ν minus ν∆)
with ∆ = M∆ minusMN M+ = M∆ + MN and the Mandelstam variable s = M2N + 2MNν minus Q2
Analytical expressions for the spin structure functions g1(xQ2) and g2(xQ2) can be constructed
from Eq (3) with the flux factor K(νQ2) = |~q | =radicν2 +Q2
In the ∆-non-pole contributions to S1(νQ2) and νS2(νQ2) the pole in ν at the ∆(1232)-
production threshold has canceled out
S∆-exch1 (νQ2) =
πα
MNM2+
[g2MQ
2+ + g2
E
(∆2 minus 3Q2
)+
4g2CQ
4
M2∆
minus 8gMgEM∆ωminus (B26a)
minus2gMgCQ2(MN minus 4M∆)
M∆+
2gEgCQ2(3MN minus 2M∆)
M∆
]
νS2∆-exch
(νQ2) =2πα
MNM2+
[g2EM∆∆ωminus +
g2M MNQ
2+
2+g2C Q
2(Q2 minus∆2)
2M∆(B26b)
+gEgM M∆(M∆ω+ minus 4MNωminus)minus gEgC ∆(2Q2 +MNω+)
+gMgC Q2(4MN minus ω+)
]+S∆-exch
2 (νQ2)
ν
[M2
∆ ω2+
M2N
+ ν2
]
with Q+ =radic
(M∆ +MN )2 +Q2 and ωplusmn = (M2∆minusM2
N plusmnQ2)2M∆ and the non-pole contribution
to S2(νQ2)
S∆-exch2 (νQ2) = minus2παMNν
M∆M2+
[gM + gE
]gC (B27)
These amplitudes to the contrary are not described by the tree-level ∆-production cross sections
in the standard dispersive approach [17] This peculiarity has been previously missed e g in the
calculation of the ∆-exchange contribution to the hydrogen hyperfine splitting in Ref [88] The
importance of including the ∆-non-pole contribution is also evident when considering the BC sum
37
rule in Eq (14) The ∆-pole terms by themselves violate the BC sum rule but cancel exactly with
the ∆-non-pole terms
limνrarr0
νS∆-pole2 (νQ2) + lim
νrarr0νS2
∆-exch(νQ2) = 0 (B28)
Appendix C Polarizabilities at Q2 = 0
In this section we give analytical expressions for the static values and slopes at Q2 = 0 of
the spin polarizabilities and moments of polarized structure functions In particular we give the
HB expansion of the πN -loop contributions and the ∆-exchange contributions The complete
expressions also for the π∆-loop contributions can be found in the Supplemented material Recall
that IA(0) = I1(0) = d2(0) = 0 and dd2(Q2)dQ2
∣∣∣Q2=0
= 0
1 πN-loop contribution
Here we give analytical expressions for the πN -loop contributions to the proton and neutron
spin polarizabilities expanded in powers of micro = mπMN viz the HB expansion Note that we
choose to expand here to a high order in micro the strict HB expansion would only retain the leading
term in an analogous NLO calculation
bull Static polarizabilities (Q2 = 0)
γ0p =e2g2
A
96π3f2πm
2π
1minus 21πmicro
8minus(
59
2+ 26 logmicro
)micro2 +
1875πmicro3
64
+3
(3
2+ 26 logmicro
)micro4 +
(C1)
γ0n =e2g2
A
48π3f2πm
2π
1
2minus 9πmicro
16minus 2micro2 logmicro+
75πmicro3
128minus 3micro4
4+
(C2)
δLTp =e2g2
A
192π3f2πm
2π
1minus 9πmicro
8+
(13
2minus 2 logmicro
)micro2 minus 465πmicro3
64
minus(
47
2+ 42 logmicro
)micro4 +
(C3)
δLTn =e2g2
A
96π3f2πm
2π
1
2+
3πmicro
16+ (1 + 2 log micro)micro2 minus 105πmicro3
128+
5micro4
4+
(C4)
38
δLTn =e2g2
A
96π3f2πm
2π
1
2+
3πmicro
16+ (1 + 2 log micro)micro2 minus 105πmicro3
128+
5micro4
4+
(C5)
γ0p =e2g2
A
16π3f2πm
4π
4
45minus 3πmicro
16+
14micro2
5minus 1813πmicro3
384minus 192
5(1 + log micro)micro4
+80703πmicro5
2048+
(C6)
γ0n =e2g2
A
16π3f2πm
4π
4
45minus 5πmicro
48+
4micro2
5minus 245πmicro3
384minus 32micro4 logmicro
15
+1323πmicro5
2048+
(C7)
bull Slopes of polarizabilities at Q2 = 0
dγ0p(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
2minus 45πmicro
4+ 223micro2 minus 28515πmicro3
64
minus9
(1953
4+ 449 logmicro
)micro4 +
570255πmicro5
128+
(C8)
dγ0n(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
2minus 81πmicro
8+ 94micro2 minus 2535πmicro3
32
minus3 (1 + 90 logmicro)micro4 +84315πmicro5
1024+
(C9)
dδLTp(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
2880π3f2πm
4π
minus5
2minus 27πmicro
32+ 20micro2 minus 5865πmicro3
256
+3
(617
4+ 36 logmicro
)micro4 minus 2056845πmicro5
4096+
(C10)
dδLTn(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
minus5
4minus 81πmicro
64minus 11micro2 +
10005πmicro3
512
+15
8(11 + 48 logmicro)micro4 minus 267015πmicro5
8192+
(C11)
dIAp(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
96π2f2πmicro
2
1minus 15πmicro
4minus 1
2(115 + 88 logmicro)micro2 +
1839πmicro3
32
+5 (5 + 34 logmicro)micro4 +
(C12)
dIAn(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
48π2f2πmicro
2
1
2minus 11πmicro
8minus 1
4(1 + 20 logmicro)micro2 +
99πmicro3
64
minus25micro4
12+
(C13)
dI1p(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
96π2f2πmicro
3π
8+ 2 (4 + 3 logmicro)microminus 537πmicro2
64
minus1
2(15 + 56 logmicro)micro3 +
(C14)
39
dI1n(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
48π2f2πmicro
minus π
16+
1
4(3 + 4 logmicro)microminus 57πmicro2
128+
2micro3
3+
(C15)
γ0p(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2g2
A
16π3f2πm
6π
1
105minus 23πmicro
256+
377micro2
210minus 15551πmicro3
6144+
3371micro4
105
minus1640457πmicro5
32768+
(C16)
γ0n(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2g2
A
16π3f2πm
6π
1
105minus 153πmicro
1792+
69micro2
70minus 4615πmicro3
6144+
172micro4
35
minus120897πmicro5
32768+
(C17)
2 ∆-exchange contribution
Here we give analytical expressions for the tree-level ∆-exchange contributions to the nucleon
spin polarizabilities and their slopes at Q2 = 0 Note that the ∆-exchange contributes equally to
proton and neutron polarizabilities Recall that for the magnetic γlowastN∆ coupling we introduced a
dipole form factor to mimic vector-meson dominance gM rarr gM(1 +Q2Λ2)2
bull Static polarizabilities (Q2 = 0)
γ0 = minus e2
4πM2+
(g2M
∆2+
g2E
M2+
minus 4gMgEM+∆
) (C18)
δLT =e2M∆
4πM3+
(g2E
MNM++gMgE∆MN
minus gEgCM2
∆
) (C19)
γ0 =e2M2
N
π∆2M4+
(minusg
2M
∆2+
g2E
M2+
+4gMgE∆M+
) (C20)
bull Slopes of polarizabilities at Q2 = 0
dγ0(Q2)
dQ2
∣∣∣∣∣Q2=0
= minus e2
πM2+∆
(g2M
∆
[1
4∆2minus 1
∆M++
1
2M2+
]minus 1
Λ2
g2M
∆+
g2E
2M2+
[1
2∆minus 3
M+
]minusgMgE
M+
[1
∆2minus 5
∆M++
1
M2+
]+
1
Λ2
2gMgEM+
+2gMgC∆M2
+
minus gEgCM3
+
) (C21)
40
d δLT (Q2)
dQ2
∣∣∣∣∣Q2=0
=e2M∆∆
4πMNM2+
(g2E
∆2M2+
[1
∆minus 4
M+
]minus g2
C
∆M2∆M
2+
+gMgE∆2M+
[1
∆2
minus 3
∆M++
1
M2+
]minus 2
Λ2
gMgE∆2M+
+gMgC∆M2
∆
[1
2∆2minus 2
∆M++
1
2M2+
]minus gEgC
2M2∆M
2+
[7
∆+
1
M+
]) (C22)
dIA(Q2)
dQ2
∣∣∣∣∣Q2=0
= minusM2N
M2+
(g2M
2∆2+
g2E
MNM+minus 2gMgE
∆M+minus gEgCM∆M+
) (C23)
dI1(Q2)
dQ2
∣∣∣∣∣Q2=0
= minusM∆M2N
2M3+
(g2E
MNM∆minus gMgE∆MN
minus gEgCM2
∆
) (C24)
γ0(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2M2
N
π∆3M6+
(g2M
∆
[3M2
∆ + 2M∆MN minus 9M2N
∆2+
4M2+
Λ2
]
minus4gMgE
[5M2
∆ minus 9M2N
M+∆2+
2M+
Λ2
]+
g2E
∆M+[7M∆ minus 9MN ]
minus8gMgC∆
+4gEgCM+
) (C25)
[1] D Drechsel B Pasquini and M Vanderhaeghen Dispersion relations in real and virtual
Compton scattering Phys Rept 378 99 (2003) hep-ph0212124
[2] S E Kuhn J-P Chen and E Leader Spin structure of the nucleon mdash status and recent
results Prog Part Nucl Phys 63 1 (2009) arXiv08123535 [hep-ph]
[3] F Hagelstein R Miskimen and V Pascalutsa Nucleon polarizabilities from Compton scat-
tering to hydrogen atom Prog Part Nucl Phys 88 29 (2016) arXiv151203765 [nucl-th]
[4] B Pasquini and M Vanderhaeghen Dispersion theory in electromagnetic interactions Ann
Rev Nucl Part Sci 68 75 (2018) arXiv180510482 [hep-ph]
[5] S Gerasimov A Sum rule for magnetic moments and the damping of the nucleon magnetic
moment in nuclei Sov J Nucl Phys 2 430 (1966)
[6] S Drell and A C Hearn Exact sum rule for nucleon magnetic moments Phys Rev Lett
16 908 (1966)
[7] J S Schwinger Source theory viewpoints in deep inelastic scattering Proc Natl Acad Sci
USA 72 1 (1975)
[8] J S Schwinger Source theory viewpoints in deep inelastic scattering Electromagnetic In-
teractions and Field Theory Proceedings 14 Internationale Universitatswochen Schladming
Austria February 24-March 7 1975 Acta Phys Austriaca Suppl 14 471 (1975)
41
[9] J Schwinger Source theory discussion of deep inelastic scattering with polarized particles
Proc Natl Acad Sci USA 72 1559 (1975)
[10] A M Baldin Polarizability of nucleons Nucl Phys 18 310 (1960)
[11] M Gell-Mann M L Goldberger and W E Thirring Use of causality conditions in quantum
theory Phys Rev 95 1612 (1954)
[12] A Pineda Leading chiral logarithms to the hyperfine splitting of the hydrogen and muonic
hydrogen Phys Rev C 67 025201 (2003)
[13] C Peset and A Pineda The two-photon exchange contribution to muonic hydrogen from
chiral perturbation theory Nucl Phys B 887 69 (2014) arXiv14064524 [hep-ph]
[14] C Peset and A Pineda Model-independent determination of the two-photon exchange contri-
bution to hyperfine splitting in muonic hydrogen JHEP 04 060 arXiv161205206 [nucl-th]
[15] F Hagelstein and V Pascalutsa Proton structure in the hyperfine splitting of muonic hydro-
gen PoS CD15 077 (2016) arXiv151104301 [nucl-th]
[16] F Hagelstein Exciting Nucleons in Compton Scattering and Hydrogen-Like Atoms PhD
thesis Mainz U Inst Kernphys (2017) arXiv171000874 [nucl-th]
[17] F Hagelstein ∆(1232)-Resonance in the hydrogen spectrum Proceedings 11th International
Workshop on the Physics of Excited Nucleons (NSTAR 2017) Columbia SC USA August
20-23 2017 Few Body Syst 59 93 (2018) arXiv180109790 [nucl-th]
[18] Y Prok et al (CLAS) Moments of the spin structure functions gp1 and gd1 for 005 lt Q2 lt 30
GeV2 Phys Lett B 672 12 (2009) arXiv08022232 [nucl-ex]
[19] H Dutz et al (GDH) First measurement of the Gerasimov-Drell-Hearn sum rule for 1H from
07 GeV to 18 GeV at ELSA Phys Rev Lett 91 192001 (2003)
[20] M Amarian et al (Jefferson Lab E94010) Measurement of the generalized forward spin po-
larizabilities of the neutron Phys Rev Lett 93 152301 (2004) arXivnucl-ex0406005
[21] M Amarian et al The Q2 evolution of the generalized Gerasimov-Drell-Hearn integral for
the neutron using a 3He target Phys Rev Lett 89 242301 (2002) arXivnucl-ex0205020
[22] M Amarian et al (Jefferson Lab E94-010)Q2 evolution of the neutron spin structure moments
using a 3He target Phys Rev Lett 92 022301 (2004) arXivhep-ex0310003
[23] A Deur et al Experimental determination of the evolution of the Bjorken integral at low Q2
Phys Rev Lett 93 212001 (2004) arXivhep-ex0407007
[24] K Slifer Low Q2 measurement of gp2 and the δLT spin polarizability Spin structure at long dis-
tance Proceedings Workshop Newport News USA March 12ndash13 2009 nucl-ex09064775
(2009) AIP Conf Proc 1155 10106313203293 (2009) arXiv09064775 [nucl-ex]
[25] P Solvignon et al (E01-012) Moments of the neutron g2 structure function at intermediate
42
Q2 Phys Rev C 92 015208 (2015) arXiv13044497 [nucl-ex]
[26] A Deur Experimental studies at low Q2 of the spin structure of the nucleon at Jefferson Lab
in 9th International Workshop on Chiral Dynamics (CD18) Durham NC USA September
17-21 2018 (2019) arXiv190305661 [nucl-ex]
[27] V Sulkosky et al (Jefferson Lab E97-110) Measurement of the 3He spin-structure functions
and of neutron (3He) spin-dependent sum rules at 0035 le Q2 le 024 GeV2 Phys Lett B
805 135428 (2020) arXiv190805709 [nucl-ex]
[28] K Adhikari et al (CLAS) Measurement of the Q2 Dependence of the Deuteron Spin Structure
Function g1 and its Moments at Low Q2 with CLAS Phys Rev Lett 120 062501 (2018)
arXiv171101974 [nucl-ex]
[29] V Lensky F Hagelstein A Hiller Blin and V Pascalutsa Comment on rdquoMeasurement of
the Q2 Dependence of the Deuteron Spin Structure Function g1 and its Moments at Low Q2
with CLASrdquo (2018) arXiv180603219 [nucl-th]
[30] J M Alarcn F Hagelstein V Lensky and V Pascalutsa Forward doubly-virtual Compton
scattering off the nucleon in chiral perturbation theory at NLO the subtraction function and
moments of unpolarized structure functions (2020) arXiv200509518 [hep-ph]
[31] V Bernard E Epelbaum H Krebs and U-G Meiszligner New insights into the spin structure
of the nucleon Phys Rev D 87 054032 (2013) arXiv12092523 [hep-ph]
[32] V Lensky J M Alarcon and V Pascalutsa Moments of nucleon structure functions at
next-to-leading order in baryon chiral perturbation theory Phys Rev C 90 055202 (2014)
arXiv14072574 [hep-ph]
[33] V Pascalutsa and M Vanderhaeghen Chiral effective-field theory in the ∆(1232) region
I Pion electroproduction on the nucleon Phys Rev D 73 034003 (2006) arXivhep-
ph0512244
[34] V Pascalutsa and M Vanderhaeghen Electromagnetic nucleon-to-Delta transition in chiral
effective field theory Phys Rev Lett 95 232001 (2005) arXivhep-ph0508060
[35] T R Hemmert B R Holstein and J Kambor Systematic 1M expansion for spin 32
particles in baryon chiral perturbation theory Phys Lett B 395 89 (1997) arXivhep-
ph9606456
[36] V Pascalutsa and D R Phillips Effective theory of the ∆(1232) in Compton scattering off
the nucleon Phys Rev C 67 055202 (2003) arXivnucl-th0212024
[37] V Pascalutsa M Vanderhaeghen and S N Yang Electromagnetic excitation of the ∆(1232)-
resonance Phys Rept 437 125 (2007) arXivhep-ph0609004
[38] K A Olive et al (Particle Data Group) Review of Particle Physics Chin Phys C 38 090001
43
(2014)
[39] F E Low Scattering of light of very low frequency by systems of spin 12 Phys Rev 96
1428 (1954)
[40] M Gell-Mann and M L Goldberger Scattering of low-energy photons by particles of spin
12 Phys Rev 96 1433 (1954)
[41] A M Harun ar-Rashid A simple derivation of Schwingerrsquos sum rule for spin dependent
structure functions Nuovo Cim A 33 447 (1976)
[42] F Hagelstein and V Pascalutsa Dissecting the hadronic contributions to (g minus 2)micro by
Schwingerrsquos sum rule Phys Rev Lett 120 072002 (2018) arXiv171004571 [hep-ph]
[43] V Lensky V Pascalutsa M Vanderhaeghen and C Kao Spin-dependent sum rules con-
necting real and virtual Compton scattering verified Phys Rev D 95 074001 (2017)
arXiv170101947 [hep-ph]
[44] H Burkhardt and W N Cottingham Sum rules for forward virtual Compton scattering
Annals Phys 56 453 (1970)
[45] V Lensky and V Pascalutsa Predictive powers of chiral perturbation theory in Compton
scattering off protons Eur Phys J C 65 195 (2010) arXiv09070451 [hep-ph]
[46] H Grieszlighammer J McGovern D Phillips and G Feldman Using effective field theory to
analyse low-energy Compton scattering data from protons and light nuclei Prog Part Nucl
Phys 67 841 (2012) arXiv12036834 [nucl-th]
[47] H W Grieszlighammer J A McGovern and D R Phillips Nucleon polarisabilities at and
beyond physical pion masses Eur Phys J A 52 139 (2016) arXiv151101952 [nucl-th]
[48] E Epelbaum H Krebs and U-G Meiszligner Improved chiral nucleon-nucleon potential up to
next-to-next-to-next-to-leading order Eur Phys J A 51 53 (2015) arXiv14120142 [nucl-
th]
[49] E Epelbaum H Krebs and U-G Meiszligner Precision nucleon-nucleon potential at fifth order
in the chiral expansion Phys Rev Lett 115 122301 (2015) arXiv14124623 [nucl-th]
[50] C W Kao T Spitzenberg and M Vanderhaeghen Burkhardt-Cottingham sum rule and
forward spin polarizabilities in heavy baryon chiral perturbation theory Phys Rev D 67
016001 (2003) arXivhep-ph0209241
[51] C-W Kao D Drechsel S Kamalov and M Vanderhaeghen Higher moments of nucleon
spin structure functions in heavy baryon chiral perturbation theory and in a resonance model
Phys Rev D 69 056004 (2004) arXivhep-ph0312102
[52] T Becher and H Leutwyler Baryon chiral perturbation theory in manifestly Lorentz invariant
form Eur Phys J C 9 643 (1999) arXivhep-ph9901384
44
[53] J Gasser M E Sainio and A Svarc Nucleons with chiral loops Nucl Phys B 307 779
(1988)
[54] L S Geng J Martin Camalich L Alvarez-Ruso and M J Vicente Vacas Leading SU(3)-
breaking corrections to the baryon magnetic moments in chiral perturbation theory Phys
Rev Lett 101 222002 (2008) arXiv08051419 [hep-ph]
[55] D Drechsel S S Kamalov and L Tiator The GDH sum rule and related integrals Phys
Rev D 63 114010 (2001) arXivhep-ph0008306
[56] D Drechsel O Hanstein S S Kamalov and L Tiator A unitary isobar model for pion
photo- and electroproduction on the proton up to 1 GeV Nucl Phys A 645 145 (1999)
arXivnucl-th9807001
[57] L Tiator private communication (2020)
[58] V Bernard T R Hemmert and U-G Meiszligner Spin structure of the nucleon at low energies
Phys Rev D 67 076008 (2003) arXivhep-ph0212033
[59] O Gryniuk F Hagelstein and V Pascalutsa Evaluation of the forward Compton scattering
off protons II Spin-dependent amplitude and observables Phys Rev D 94 034043 (2016)
arXiv160400789 [nucl-th]
[60] R Zielinski The g2p Experiment A Measurement of the Protonrsquos Spin Structure Functions
PhD thesis New Hampshire U (2010) arXiv170808297 [nucl-ex]
[61] N Guler et al (CLAS) Precise determination of the deuteron spin structure at low to mod-
erate Q2 with CLAS and extraction of the neutron contribution Phys Rev C 92 055201
(2015) arXiv150507877 [nucl-ex]
[62] C E Carlson V Nazaryan and K Griffioen Proton structure corrections to electronic
and muonic hydrogen hyperfine splitting Phys Rev A 78 022517 (2008) arXiv08052603
[physicsatom-ph]
[63] V Bernard N Kaiser and U-G Meiszligner Chiral dynamics in nucleons and nuclei Int J
Mod Phys E 4 193 (1995) arXivhep-ph9501384
[64] T R Hemmert B R Holstein and J Kambor ∆(1232) and the polarizabilities of the nucleon
Phys Rev D 55 5598 (1997) arXivhep-ph9612374
[65] V Pascalutsa and R Timmermans Field theory of nucleon to higher-spin baryon transitions
Phys Rev C 60 042201 (1999) arXivnucl-th9905065
[66] V Pascalutsa Quantization of an interacting spin-32 field and the Delta isobar Phys Rev
D 58 096002 (1998) arXivhep-ph9802288
[67] H Krebs Double Virtual Compton Scattering and SpinStructure of the Nucleon PoS
CD2018 031 (2019)
45
[68] N Kochelev and Y Oh Axial anomaly and the δLT puzzle Phys Rev D 016012 (2012)
arXiv11034892 [hep-ph]
[69] D Drechsel S Kamalov and L Tiator Unitary isobar model ndash MAID2007 Eur Phys J A
69 (2007) available at httpsmaidkphuni-mainzde arXiv07100306 [nucl-th]
[70] P J Mohr B N Taylor and D B Newell CODATA recommended values of the fundamental
physical constants 2010 Rev Mod Phys 84 1527 (2012)
[71] J Ahrens et al (GDH A2) First measurement of the Gerasimov-Drell-Hearn integral for 1H
from 200 to 800 MeV Phys Rev Lett 87 022003 (2001) arXivhep-ex0105089 [hep-ex]
[72] K Helbing (GDH) Experimental verification of the GDH sum rule at ELSA and MAMI Nucl
Phys Proc Suppl 105 113 (2002)
[73] R Bradford A Bodek H S Budd and J Arrington A New parameterization of the nucleon
elastic form-factors NuInt05 proceedings of the 4th International Workshop on Neutrino-
Nucleus Interactions in the Few-GeV Region Okayama Japan 26-29 September 2005 Nucl
Phys Proc Suppl 159 127 (2006) arXivhep-ex0602017
[74] J D Bjorken Applications of the chiral U(6)timesU(6) algebra of current densities Phys Rev
148 1467 (1966)
[75] J D Bjorken Inelastic scattering of polarized leptons from polarized nucleons Phys Rev D
1 1376 (1970)
[76] V Bernard T R Hemmert and U-G Meiszligner Novel analysis of chiral loop effects in
the generalized Gerasimov-Drell-Hearn sum rule Phys Lett B 545 105 (2002) arXivhep-
ph0203167
[77] A Deur et al Experimental study of isovector spin sum rules Phys Rev D 78 032001
(2008) arXiv08023198 [nucl-ex]
[78] B Pasquini P Pedroni and D Drechsel Higher order forward spin polarizability Phys Lett
B 687 160 (2010) arXiv10014230 [hep-ph]
[79] R Jaffe g2ndashThe nucleonrsquos other spin-dependent structure function Comments Nucl Part
Phys 19 239 (1990)
[80] E V Shuryak and A Vainshtein Theory of power corrections to deep inelastic scattering in
quantum chromodynamics (II) Qminus4 effects polarized target Nucl Phys B 201 141 (1982)
[81] S Wandzura and F Wilczek Sum rules for spin dependent electroproduction Test of rela-
tivistic constituent quarks Phys Lett B 72 195 (1977)
[82] B W Filippone and X-D Ji The spin structure of the nucleon Adv Nucl Phys 26 1
(2001) arXivhep-ph0101224
[83] M Burkardt The g2 structure function Proceedings Workshop on Spin structure at long
46
distance Newport News USA March 12-13 2009 AIP Conf Proc 1155 26 (2009)
arXiv09054079 [hep-ph]
[84] A Deur S J Brodsky and G F de Teramond The spin structure of the nucleon Rept
Prog Phys 82 076201 (2019) arXiv180705250 [hep-ph]
[85] B R Holstein D Drechsel B Pasquini and M Vanderhaeghen Higher order polarizabilities
of the proton Phys Rev C 61 034316 (2000) arXivhep-ph9910427
[86] B R Holstein V Pascalutsa and M Vanderhaeghen Sum rules for magnetic moments and
polarizabilities in QED and chiral effective-field theory Phys Rev D 72 094014 (2005)
arXivhep-ph0507016
[87] B Pasquini D Drechsel and L Tiator Invariant amplitudes for pion electroproduction Eur
Phys J A 34 387 (2007) arXiv07122327 [hep-ph]
[88] A J Buchmann Non-spherical proton shape and hydrogen hyperfine splitting Proceedings
International Workshop on Precision Physics of Simple Atomic Systems (PSAS 2008) Wind-
sor Ontario Canada July 21-26 2008 Can J Phys 87 773 (2009) arXiv09104747
[physicsatom-ph]
47
the experimental points at very low Q2 deviating from the data with increasing Q2 The
π∆-loop contribution does not modify the Q2 behavior of γ0(Q2) and only differs from
Ref [31] by a small global shift Note also the relatively large effect of gC which generates
a sign change for virtualities above sim 02 GeV2 see Fig 3 upper panel
B δLT (Q2) mdash longitudinal-transverse polarizability
The longitudinal-transverse spin polarizability
δLT (Q2) =1
2π2
int infinν0
dν
radic1 +
Q2
ν2
σLT (νQ2)
Qν2(22)
=16αM2
N
Q6
int x0
0
dx x2[g1(xQ2) + g2(xQ2)
]
contains information about the spin structure of the nucleon and is another important
input in the determination of the (muonic-)hydrogen hyperfine splitting [16 62] It is also
relevant in studies of higher-twist corrections to the structure function g2(xQ2) given by
the moment d2(Q2) [51] see Section III E The peculiarity of the response encoded in this
polarizability is that it involves a spin flip of the nucleon and a polarization change of the
photon see App B and Fig 11
It is expected that the Delta isobar gives only a small contribution to δLT (Q2) what makes
this polarizability a potentially clean test case for chiral calculations Consequently there are
relatively many different theoretical calculations of δLT (Q2) coming from different versions
of χPT with baryons (HB IR and covariant) Ref [50] found a systematic deviation of the
HB result for δLTn(Q2) from the MAID model prediction This disagreement was identified
by the authors of Ref [68] as a puzzle involving the neutron δLT polarizabilitymdashthe δLT
puzzle The IR calculation in Ref [58] also showed a deviation from the data and predicted
a rapid rise of δLT (Q2) with growing Q2 The problem is solved by keeping the relativistic
structure of the theory as the BχPT+∆ result of Ref [31] showed
As expected already the leading πN loops provide a reasonable agreement with the
experimental data cf Fig 2 lower panel Since the ∆-exchange contribution to δLT (Q2)
is small the effect of the gM form factor is negligible in this polarizability as is that of the gC
coupling cf Fig 3 lower panel In fact we predict both the ∆-exchange and the π∆-loop
contributions to be small and negative This is in agreement with the MAID model which
predicts a small and negative contribution of the P33 wave to δLT (Q2) However in the
calculation of Ref [31] which is different from the one presented here only in the way the
∆(1232) is included the contribution of this resonance to δLTp(Q2) is sizeable and positive
15
The authors of that work attributed this large contribution to diagrams where the photons
couple directly to the Delta inside a loop As mentioned in Sec III F the effect of such loop
diagrams does not change the Q2 behaviour of the polarizabilities On the other hand it
can produce a substantial shift of the δLT (Q2) as a whole A higher-order calculation should
resolve the discrepancy between the two covariant approaches however it will partially lose
the predictive power since the LECs appearing at higher orders will have to be fitted to
experimental data
The πN -loop ∆-exchange and π∆-loop contributions to the NLO BχPT prediction of
the static longitudinal-transverse polarizability are in units of 10minus4 fm4
δLTp = 132(15) asymp 150minus 016minus 002 (23a)
δLTn = 218(23) asymp 235minus 016minus 002 (23b)
while the slopes are in units of 10minus4 fm6
dδLTp(Q2)
dQ2
∣∣∣∣Q2=0
= minus085(8) asymp minus080minus 004minus 001 (24a)
dδLTn(Q2)
dQ2
∣∣∣∣Q2=0
= minus124(12) asymp minus119minus 004minus 001 (24b)
C IA(Q2) mdash a generalized GDH integral
The helicity-difference cross section σTT exhibits a faster fall-off in ν than its spin-averaged
counterpart σT This is due to a cancellation between the leading (constant) terms of σ12
and σ32 at large ν3 The resulting 1ν fall-off of the helicity-difference cross section allows
one to write an unsubtracted dispersion relation for the VVCS amplitude gnonpoleTT (νQ2) cf
Eq (10a) This is the origin of the GDH sum rule [5 6]
minus α
2M2N
κ2 =1
2π2
int infinν0
dνσTT (ν)
ν (25)
which establishes a relation to the anomalous magnetic moment κ It is experimentally
verified for the nucleon by MAMI (Mainz) and ELSA (Bonn) [71 72]
There are two extensions of the GDH sum rule to finite Q2 the generalized GDH integrals
3 Notice that a constant term in σTT at ν rarrinfin is forbidden by crossing symmetry
16
000 005 010 015 020 025 030
-30
-25
-20
-15
-10
-05
00
Q2 GeV2D
I Aproton
000 005 010 015 020 025 030
-25
-20
-15
-10
-05
00
Q2 GeV2D
I A
neutron
000 005 010 015 020 025 030
-05
00
05
Q2 GeV2D
I 1
proton
000 005 010 015 020 025 030
-10
-08
-06
-04
-02
00
02
Q2 GeV2D
I 1
neutron
FIG 4 Upper panel The generalized GDH integral IA(Q2) for the proton (left) and neutron
(right) as function of Q2 The result of this work the NLO BχPT prediction is shown by the blue
solid line and the blue band The red line represents the LO BχPT result The purple short-dashed
line is the O(p4) HB result from Ref [50 51] The gray band is the BχPT+∆ result from Ref [31]
The black dotted line is the MAID model prediction [69] Experimental extractions for the proton
Ref [60] (orange triangle uncertainties added in quadrature) and neutron Refs [21][27] where
magenta dotsorange diagonal crosses correspond to data and red squareslilac crosses correspond
to data plus extrapolation to unmeasured energy regions The green stars at the real-photon point
are derived from the anomalous magnetic moments κp asymp 1793 and κn asymp minus1913 [70] Lower
panel The generalized GDH integral I1(Q2) for the proton (left) and neutron (right) as function
of Q2 The purple short-dashed line is the HB result from Ref [51] Experimental extractions for
the proton Ref [18] (blue dots) and Ref [60] (orange triangle uncertainties added in quadrature)
and neutron Ref [61][27] (uncertainties added in quadrature) where green dotsorange diagonal
crosses correspond to data and gray squareslilac crosses correspond to data plus extrapolation to
unmeasured energy regions
17
IA(Q2) and I1(Q2) The latter will be discussed in Sec III D The former is defined as4
minus α
2M2N
IA(Q2) = minus 1
8π2
int infinν0
dν
radic1 +
Q2
ν2
σTT (νQ2)
ν(26)
=α
Q2
int x0
0
dx[4M2
Nx2
Q2g2(xQ2)minus g1(xQ2)
]
Due to its energy weighting the integral in Eq (26) converges slower than the one in the
generalized forward spin polarizability sum rule (19) Therefore knowledge of the cross
section at higher energies is required and the evaluation of the generalized GDH integral
IA(Q2) is not as simple as the evaluation of γ0(Q2)
The generalized GDH integral IA(Q2) is directly related to the non-pole amplitude
gnonpoleTT (νQ2) which differs from non-Born amplitude gTT (νQ2) by a term involving the
elastic Pauli form factor
gnonpoleTT (νQ2) = gTT (νQ2)minus 2παν
M2N
F 22 (Q2) (27)
cf Eqs (2a) and (8a) Consequently IA(Q2) is not a pure polarizability but also contains
an elastic contribution The ldquonon-polarizabilityrdquo or the Born part of IA(Q2) is given by
IBornA (Q2) = IA(Q2)minus∆IA(Q2) = minus1
4F 2
2 (Q2) (28)
where we refer to the polarizability part as ∆IA(Q2) The same is true for the generalized
GDH integral I1(Q2) which is directly related to Snonpole1 (νQ2)
IBorn1 (Q2) = I1(Q2)minus∆I1(Q2) = minus1
4F 2
2 (Q2) (29)
In the following we will add the Born parts to our LO and NLO BχPT predictions for
the polarizabilities ∆IA(Q2) and ∆I1(Q2) employing an empirical parametrization for the
elastic Pauli form factor [73] This allows us to compare to the experimental results for
IA(Q2) and I1(Q2) cf Fig 4 Note that the blue error bands only describe the uncertainties
of our BχPT predictions of the polarizabilities while the elastic contributions are considered
to be exact as explained in Sec II C The uncertainties of the polarizability predictions are
therefore better reflected in Fig 5 where we show the contributions of the different orders
to the BχPT predictions of ∆IA(Q2) and ∆I1(Q2) as well as the total results with error
bands
4 Note that IA(Q2) is sometimes called ITT (Q2)
18
000 005 010 015 020 025 030
-08
-06
-04
-02
00
02
Q2 GeV2D
DI A
proton
000 005 010 015 020 025 030
-10
-08
-06
-04
-02
00
02
Q2 GeV2D
DI A
neutron
000 005 010 015 020 025 030-005
000
005
010
015
020
025
Q2 GeV2D
DI 1
proton
000 005 010 015 020 025 030-015
-010
-005
000
005
010
015
Q2 GeV2D
DI 1
neutron
FIG 5 Contributions of the different orders to the chiral predictions of ∆IA(Q2) upper paneland ∆I1(Q2) lower panel for the proton (left) and neutron (right) Red solid line πN -loop
contribution green dot-dashed line ∆-exchange contribution orange dotted line π∆-loop contri-
bution blue solid line and blue band total result purple dot-dot-dashed line total result without
gC contribution black short-dashed line total result without gM dipole
The E97-110 experiment at Jefferson Lab has recently published their data for IAn(Q2) in
the region of 0035 GeV2 lt Q2 lt 024 GeV2 [27] In addition there are results for IAn(Q2)
from the earlier E94-010 experiment [21] and for IAp(Q2) from the E08-027 experiment [60]
The O(p4) HB calculation gives a large negative effect [51] which does not describe the data
The BχPT+∆ result from Ref [31] which mainly differs from our work by the absence of
the dipole form factor in gM looks similar to this HB result and only describes the data
points at lowest Q2 Our NLO prediction however follows closely the Q2 evolution of the
data In Fig 5 upper panel we show the polarizability ∆IA(Q2) whose Q2 evolution is
clearly dominated by the ∆ exchange Similar to the case of γ0p(Q2) inclusion of the dipole
in gM and the Coulomb coupling gC is very important in order to describe the experimental
data The LO prediction on the other hand slightly overestimates the data cf Fig 4
upper panel
19
At the real-photon point IA(0) = minusκ2
4and ∆IA(0) = 0 Therefore we give only the
slope of the polarizability ∆IA(Q2) [showing also the separate contributions from πN loops
∆ exchange and π∆ loops] in units of GeVminus2
d∆IAp(Q2)
dQ2
∣∣∣∣Q2=0
= minus858(343) asymp 238minus 1121 + 025 (30a)
d∆IAn(Q2)
dQ2
∣∣∣∣Q2=0
= minus955(343) asymp 141minus 1121 + 025 (30b)
Including the empirical Pauli form factor [73] we find in units of GeVminus2
dIAp(Q2)
dQ2
∣∣∣∣Q2=0
= minus318dIAn(Q2)
dQ2
∣∣∣∣Q2=0
= minus300 (31)
D Γ1(Q2) and I1(Q2) mdash the first moment of the structure function g1(xQ2)
The second variant for a generalization of the GDH sum rule to finite Q2 is defined as
minus α
2M2N
I1(Q2) = minus 1
8π2
int infinν0
dν1radic
ν2 +Q2
[σTT (νQ2) +
Q
νσLT (νQ2)
](32)
= minus α
Q2
int x0
0
dx g1(xQ2)
where I1(0) = minusκ2
4 This generalized GDH integral directly stems from the amplitude
Snonpole1 (νQ2) with the LEX from Eq (12a) It is given by the first moment of the struc-
ture function g1(xQ2) Γ1(Q2) =int x0
0dx g1(xQ2) as follows I1(Q2) =
2M2N
Q2 Γ1(Q2) The
isovector combination
Γ1(pminusn)(Q2) =
int x0
0
dx[g1p(xQ
2)minus g1n(xQ2)] (33)
is related to the axial coupling of the nucleon through the Bjorken sum rule [74 75]
limQ2rarrinfin
Γ1(pminusn)(Q2) =
gA6 (34)
As explained in Eq (28) the moment I1(Q2) splits into a polarizability part ∆I1(Q2) and
a Born part IBorn1 (Q2) Figure 4 lower panel shows the Q2 dependence of I1(Q2) which in
contrast to IA(Q2) shown in Figure 4 upper panel is clearly dominated by its Born part
and the elastic Pauli form factor The πN -loop ∆-exchange and π∆-loop contributions to
the polarizability ∆I1(Q2) are shown in Fig 5 lower panel Comparing to Fig 5 upper
20
000 005 010 015 020 025 030
-006
-004
-002
000
002
004
Q2 GeV2D
G1
proton
000 005 010 015 020 025 030
-006
-005
-004
-003
-002
-001
000
001
Q2 GeV2D
G1
neutron
FIG 6 First moment of the structure function g1(xQ2) for the proton (left) and neutron (right)
as function of Q2 The legend is the same as in Fig 4
panel one sees that ∆I1(Q2) is less sensitive to gC and the dipole form factor in gM than
∆IA(Q2)
For the proton our NLO BχPT prediction gives a very good description of the ex-
perimental data [18 60] and is in reasonable agreement with the MAID prediction [69]
For the neutron one observes good agreement with the empirical evaluations including ex-
trapolations to unmeasured energy regions starting from Q2 gt 01 GeV2 [27 61] In the
region of Q2 lt 005 GeV2 one observes an interesting tension between the recent E97-
110 experiment [27] and the data from CLAS [61] While the newest measurement finds
I1n(0035 GeV2) lt κ2n4 thus suggesting a negative slope at low Q the older measurement
found a rather large value for I1n(00496 GeV2) A similar but milder behaviour is seen in
the E97-110 [27] and E94-010 [21] data for IAn The MAID predictions do not agree with
the CODATA recommended values for the anomalous magnetic moments of the proton and
neutron [70] which in our work are imposed by using empirical parametrizations for the
elastic Pauli form factors [73] The slope of the HB result from Ref [51] is too large and
therefore only reproduces the data at very low Q2
Figure 6 shows the moment Γ1(Q2) for the proton and neutron while Fig 7 shows the
isovector combination Γ1 pminusn(Q2) The LO and NLO BχPT predictions are identical be-
cause our calculation produces the same Delta contributions for the proton and the neutron
For the isovector combination the MAID model only agrees with the data at very low
Q2 lt 010 GeV2 The same is true for the IR result [58 76] while all other chiral results
describe the data NLO BχPT (this work) BχPT+∆ [31] and HBχPT [51]
At the real-photon point I1(0) = minusκ2
4and ∆I1(0) = 0 Therefore we give only the
slope of the polarizability ∆I1(Q2) [showing also the separate contributions from πN loops
21
000 005 010 015 020 025 030-002
000
002
004
006
008
010
Q2 GeV2D
G1Hp-
nL
FIG 7 Isovector combination of Γ1(Q2) as function of Q2 The legend is the same as in Fig 6
The pink curve is the IR result from Ref [58 76] The experimental points are from Ref [77]
(brown dots) and Ref [23] (orange squares)
∆ exchange and π∆ loops] in units of GeVminus2
d∆I1p(Q2)
dQ2
∣∣∣∣Q2=0
= 039(4) asymp 034minus 053 + 058 (35a)
d∆I1n(Q2)
dQ2
∣∣∣∣Q2=0
= minus101(10) asymp minus107minus 053 + 058 (35b)
Including the empirical Pauli form factor [73] we find in units of GeVminus2
dI1p(Q2)
dQ2
∣∣∣∣Q2=0
= 580dI1n(Q2)
dQ2
∣∣∣∣Q2=0
= 553 (36)
E d2(Q2) mdash a measure of color polarizability
Another interesting moment to consider is d2(Q2) which is related to the twist-3 part of
the spin structure function g2(xQ2) [79 80]
d2(Q2) equiv 3
int 1
0
dx x2[g2(xQ2)minus gWW2 (xQ2)] (37)
where gWW2 (xQ2) is the twist-2 part of g2(xQ2) Using the Wandzura-Wilczek relation [81]
one can relate d2(Q2) to moments of the spin structure functions g1(xQ2) and g2(xQ2)
d2(Q2) =
int 1
0
dx x2 [3g2(xQ2) + 2g1(xQ2)] (38)
22
000 005 010 015 020 025 030
0000
0001
0002
0003
0004
0005
0006
0007
Q2 GeV2D
d2
proton
000 005 010 015 020 025 030
0000
0002
0004
0006
0008
0010
0012
Q2 GeV2D
d2
neutron
000 005 010 015 020 025 030
-1
0
1
2
Q2 GeV2D
Γ 010
-4
fm6
D
proton
000 005 010 015 020 025 030-2
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
6D
neutron
FIG 8 Upper panel The inelastic moment d2(Q2) for the proton (left) and neutron (right) as
function of Q2 The result of this work the NLO BχPT prediction is shown by the blue solid line
and the blue band The red line represents the LO BχPT result The purple short-dashed line is
the O(p4) HB result from Ref [50 51] The black dotted line is the MAID model prediction [69]
The experimental points for the neutron (cyan dots) are from Ref [22] Lower panel Fifth-order
generalized forward spin polarizability γ0(Q2) for the proton (left) and neutron (right) as function
of Q2 The experimental points for the proton are from Ref [59] (purple square) and Ref [78]
(orange dot)
This relation however only holds for asymptotically large Q2 It is also in the high-Q2
region where d2(Q2) is a measure of color polarizability [82 83] through its relation to the
gluon field strength tensor [80] We refer to Ref [84] for a recent review on the spin structure
of the nucleon including a discussion of sum rules for deep inelastic scattering and color
polarizabilities
What we consider in the following is the inelastic part of d2(Q2) defined as the moment
of g1(xQ2) and g2(xQ2) spin structure functions cf Eq (38)
d2(Q2) =
int x0
0
dx x2 [3g2(xQ2) + 2g1(xQ2)] (39)
23
This moment provides another testing ground for our BχPT predictions through comparison
with experiments on the neutron [22] Going towards the low-Q2 region the interpretation
of d2(Q2) in terms of color polarizabilities will fade out The above definition however
implies it is related to other VVCS polarizabilities
d2(Q2) =Q4
8M4N
[M2
NQ2
αδLT (Q2) + I1(Q2)minus IA(Q2)
] (40)
Note that d2(Q2) and its first two derivatives with respect to Q2 vanish at Q2 = 0 The
considerations in Eqs (28) and (29) have no effect on d2(Q2) since the Born contribution
from IA(Q2) and I1(Q2) cancel out Therefore d2(Q2) is a pure polarizability
In Fig 8 upper panel we show our NLO BχPT prediction and other results for d2(Q2)
While MAID [69] and BχPT describe the experimental data for the neutron [22] very well
the HB limit [50 51] is showing a fast growth with Q2 This illustrates the importance
of keeping the relativistic result Note also that even though the πN -loop contribution
is dominant both gC and the form factor in gM are essential to obtain a curvature that
reproduces the data cf Fig 9 upper panel For the proton there are to our knowledge
no experimental results to compare with However the agreement between the NLO BχPT
prediction and the MAID prediction at low energies is reasonable
F γ0(Q2) mdash fifth-order generalized forward spin polarizability
It is interesting to compare the generalized fifth-order forward spin polarizability sum
rule
γ0(Q2) =1
2π2
int infinν0
dν
radic1 +
Q2
ν2
σTT (νQ2)
ν5(41)
=64αM4
N
Q10
int x0
0
dx x4
[g1(xQ2)minus 4M2
Nx2
Q2g2(xQ2)
]
to the sum rule integrals for IA(Q2) and γ0(Q2) since they differ merely by their energy
weighting of σTT (νQ2) and a constant prefactor cf Eqs (19) (26) and (41) From IA(Q2)
to γ0(Q2) to γ0(Q2) the energy suppression is increasing by a factor of νminus2 respectively
Therefore the description of γ0(Q2) should be easiest in a low-energy effective-field theory
such as χPT whereas γ0(Q2) and IA(Q2) receive larger contributions from higher energies
In Fig 8 lower panel we show our LO and NLO BχPT predictions for γ0(Q2) One
can see that the πN -loop contribution is positive (in accordance to what we see for the cross
section σTT see Fig 10) The Delta shifts it substantially especially in the low Q2 region
24
000 005 010 015 020 025 030
-0004
-0002
0000
0002
0004
0006
0008
0010
Q2GeV
2D
d2
proton
000 005 010 015 020 025 030
-0005
0000
0005
0010
0015
Q2GeV
2D
d2
neutron
000 005 010 015 020 025 030
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
6D
proton
000 005 010 015 020 025 030
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
6D
neutron
FIG 9 Contributions of the different orders to the chiral predictions of d2(Q2) upper panel and
γ0(Q2) lower panel for the proton (left) and neutron (right) Red solid line πN -loop contribu-
tion green dot-dashed line ∆-exchange contribution orange dotted line π∆-loop contribution
blue long-dashed line total result purple dot-dot-dashed line total result without gC contribution
black short-dashed line total result without gM dipole
bringing it into a better agreement with data In general the BχPT curves start above the
empirical data points at the real-photon point and then decrease asymptotically to zero
above Q2 gt 01 GeV2 On the other hand the MAID prediction reproduces the empirical
data at the real-photon point then decreases to negative values until about Q2 gt 006 GeV2
from where it also starts to asymptotically approach zero Consequently our NLO BχPT
prediction of γ0(Q2) is consistently above the MAID prediction This is very different to
what we saw for IA(Q2) in Fig 4 upper panel where the MAID prediction at the real-
photon point is above the experimental value While the agreement of our predictions with
the empirical data is in general quite good for all moments of σTT (νQ2) one should point
out that both for γ0n(Q2) and γ0p(Q2) we overestimate the data at low Q2 For IA(Q2) such
observation cannot be made because ∆IA(0) = 0 and thus IA(0) is given by the empirical
Pauli form factor only From IA(Q2) γ0(Q2) and γ0(Q2) the latter has the smallest however
25
non-negligible dependence on gC and the dipole in gM cf Fig 9 lower panel
The πN -loop ∆-exchange and π∆-loop contributions to the NLO BχPT prediction of
the static fifth-order forward spin polarizability amount to in units of 10minus4 fm6
γ0p = 112(30) asymp 208minus 096minus 001 (42a)
γ0n = 195(30) asymp 292minus 096minus 001 (42b)
while the slope is composed as follows in units of 10minus4 fm8
dγ0p(Q2)
dQ2
∣∣∣∣Q2=0
= minus084(10) asymp minus100 + 016 + 000 (43a)
dγ0n(Q2)
dQ2
∣∣∣∣Q2=0
= minus142(15) asymp minus158 + 016 + 000 (43b)
Note that the HB prediction of the static polarizability γ0p (423 at O(p3) and 365 at O(ε3)
[78 85]) is almost one order of magnitude larger than the empirical value and therefore not
shown in Fig 8
G Summary
Our results are summarized in Table II where we give the contributions of the different
orders to the chiral predictions of the polarizabilities and their slopes at the real-photon
point A quantitative comparison of our predictions for the static spin polarizabilities to
the work of Bernard et al [31] and different empirical evaluations is shown in Table III We
can see that the inclusion of the Delta turns out to be very important for all moments of
the helicity-difference cross section To describe the Q2 behavior of the polarizabilities the
magnetic coupling of the N rarr ∆ transition should be modified by a dipole form factor as
has been observed previously in the description of electroproduction data [33] This dipole
form factor effectively takes account of vector-meson exchanges The Coulomb-quadrupole
N rarr ∆ transition despite its subleading order is important in the description of some
moments of spin structure functions This is contrary to what we saw for the moments of
unpolarized structure functions [30] where the Coulomb coupling had a negligible effect
The π∆ loops are mainly relevant for the generalized GDH integrals
26
TABLE II The NLO BχPT predictions for the forward VVCS polarizabilities and their slopes at
Q2 = 0 The contributions of the πN loops the ∆ exchange and the π∆ loops are shown together
with the combined total result Note that IA(0) = I1(0) = d2(0) = 0 and (d2)prime = 0
πN loops ∆ exchange π∆ loops Total
γ0 p 201 minus284 minus010minus093(92)
(10minus4 fm4) n 298 003(92)
δLT p 150 minus016 minus002132(15)
(10minus4 fm4) n 235 218(23)
γ0 p 208 minus096 minus001112(30)
(10minus4 fm6) n 292 195(30)
(γ0)prime p minus033011 001
minus022(4)
(10minus4 fm6) n minus073 minus061(7)
(δLT )prime p minus080 minus004 minus001minus085(8)
(10minus4 fm6) n minus119 minus124(12)
(γ0)prime p minus100016 000
minus084(10)
(10minus4 fm8) n minus158 minus142(15)
(∆IA)prime p 238 minus1121 025minus858(343)
(GeVminus2) n 141 minus955(343)
(∆I1)prime p 034 minus053 058039(4)
(GeVminus2) n minus107 minus101(10)
TABLE III Our NLO BχPT predictions for the spin polarizabilities at Q2 = 0 compared with
the BχPT+∆ predictions from Bernard et al [31] and the available empirical information Where
the reference is not given the empirical number is provided by the MAID analysis [55 69] with
unspecified uncertainty
Proton Neutron
This work BχPT+∆ Empirical This work BχPT+∆ Empirical
γ0 minus093(92) minus174(40) minus100(8)(12) [19] 003(92) minus077(40) minus0005
(10minus4 fm4) minus090(8)(11) [78] [MAID]
minus0929(105) [59]
δLT 132(15) 240(1) 134 218(23) 238(3) 203
(10minus4 fm4) [MAID] [MAID]
γ0 112(30) 060(7)(7) [78] 195(30) 123
(10minus4 fm6) 0484(82) [59] [MAID]
IV CONCLUSIONS
We have presented a complete NLO calculation of the polarized non-Born VVCS ampli-
tudes in covariant BχPT with pion nucleon and ∆(1232) fields The dispersion relations
between the VVCS amplitudes and the tree-level photoabsorption cross sections served as a
27
cross-check of these calculations
The obtained moments of the proton and neutron spin structure functions related to gen-
eralized polarizabilities and GDH-type integrals agree well with the available experimental
data The description of their Q2 evolution is improved compared to the previous χPT pre-
dictions In particular the NLO BχPT predictions obtained here give a better description
of the empirical data (eg from the Jefferson Laboratory ldquoSpin Physics Programrdquo) than
the HB [50 51] and IR [58] calculations
The demonstrated predictive power of the χPT framework amplitudes makes it well
suited for extending the χPT evaluation of the TPE effect in the hyperfine structure of
(muonic-)hydrogen [15ndash17] to next-to-leading order
ACKNOWLEDGEMENTS
We thank Lothar Tiator and Marc Vanderhaeghen for helpful discussions This work
is supported by the Deutsche Forschungsgemeinschaft (DFG) through the Collaborative
Research Center [The Low-Energy Frontier of the Standard Model (SFB 1044)] JMA ac-
knowledges support from the Community of Madrid through the ldquoPrograma de atraccion de
talento investigador 2017 (Modalidad 1)rdquo and the Spanish MECD grants FPA2016-77313-P
FH gratefully acknowledges financial support from the Swiss National Science Foundation
Appendix A Tensor decompositions of the VVCS amplitudes
In this appendix we review the decomposition of the forward VVCS process into tensor struc-
tures and scalar amplitudes In particular we consider the connection between the covariant and
the semi-relativistic decomposition in the lab frame that is defined in terms of the conventional
transverse longitudinal transverse-transverse and transverse-longitudinal amplitudes
As explained in Sec II A the process of forward VVCS off the nucleon can be described in
terms of four explicitly covariant amplitudes S1 2 and T1 2 [3]
T (νQ2) =
(minusgmicroν +
qmicroqν
q2
)T1(νQ2) +
1
M2N
(pmicro minus p middot q
q2qmicro)(
pν minus p middot qq2
qν)T2(νQ2) (A1)
minus 1
MNγmicroναqα S1(νQ2)minus 1
M2N
(γmicroνq2 + qmicroγναqα minus qνγmicroαqα
)S2(νQ2)
εprimelowastmicro εν
where εmicro (εprimelowastmicro ) are the incoming (outgoing) photon polarization vectors ν is the photon lab-frame
energy and Q2 is the photon virtuality Alternatively the decomposition in the laboratory frame
28
(which in the forward case coincides with the Breit frame) is parametrized in terms of the nucleon
Pauli matrices ~σ and the four scalar functions fL fT gTT and gLT
T (νQ2) = ε0 εprimelowast0 fL(νQ2) + (~ε primelowast middot ~ε ) fT (νQ2) + i~σ middot (~ε primelowast times ~ε ) gTT (νQ2) (A2)
minusi~σ middot [(ε0~εprimelowast minus ~ε ε primelowast0 )times q] gLT (νQ2)
Here ~q and q = ~q|~q | are the photon three-momentum in the lab system and its unit vector The
modified polarization vector components are given by
ε0 =
[ε0 minus
ν
|~q | (~ε middot q )
] |~q |Q ~ε = ~εminus q (~ε middot q ) (A3)
where ε = (ε0~ε ) is the usual incoming photon polarization vector and εprimelowast the outgoing polarization
vector The LEX of the lab frame amplitudes [Eq (10)] can serve in particular as the definition
of the generalized polarizabilities The lab frame amplitudes are also conveniently used for the
definition of the response functions see the example of the scalar amplitude gLT (νQ2) and the
corresponding response function σLT (νQ2) below in App B
Appendix B Photoabsorption cross sections
In the forward kinematics the spin-dependent VVCS amplitudes and the spin polarizabilities
can be described in terms of the polarized structure functions g1(xQ2) and g2(xQ2) or equiv-
alently the helicity-difference cross section σTT (νQ2) and the longitudinal-transverse response
function σLT (νQ2) with the help of dispersion relations (5) and the optical theorem (3) In this
way the photoabsorption cross sections measured in electroproduction processes form the basis
for most empirical evaluations shown throughout Sec III In the following we present the BχPT
predictions for the tree-level cross sections of πN - π∆- and ∆-production through photoabsorption
on the nucleon cf Figs 8 9 and 10 in Ref [30] In Secs B 1 and B 2 we will discuss the leading
πN -production channel and the ∆-production channel respectively We used these cross sections
to verify the polarizability predictions obtained otherwise from the calculated non-Born VVCS
amplitudes Due to the bad high-energy behavior of the π∆-production cross sections in BχPT
cf Fig 10 the dispersion relations in Eq (5) require further subtractions for a reconstruction of
the π∆-loop contribution to the spin-dependent VVCS amplitudes Therefore not all polarizabil-
ities could be verified but only those appearing as higher-order terms in the LEX of the VVCS
amplitudes such as γ0 [16]
29
proton neutron
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Neutron
02 04 06 08 100
50
100
150
200
250
[GeV]
T [b]
02 04 06 08 10
-50
0
50
100
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Neutron
02 04 06 08 100
50
100
150
200
250
[GeV]
T [b]
02 04 06 08 10
-50
0
50
100
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02040608100
50
100
150
200
250
300
[GeV]
T[b]
0204060810
-50
0
50
[GeV]
TT[b]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
0204060810
00
02
04
06
[GeV]
LTQ
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
0204060810
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
FIG 10 Photoabsorption cross sections for πN (red) and π∆ production (orange) with Q2 = 0
(solid) and Q2 = 01 GeV2 (dashed for πN and dotted for π∆ channel)
1 πN-production channel
In order to extract the response function σLT (νQ2) we have developed a method similar to
the one used to calculate σTT (νQ2) see for example Ref [86] For σLT (νQ2) however the
calculation is more complicated because one has to take into account that the associated Compton
process involves a spin-flip of the nucleon as illustrated in Fig 11 When calculating the cross
section the product of the incoming nucleon spinors has to reflect this flip
The forward VVCS amplitude related to σLT (νQ2) mdash and δLT (Q2) mdash is gLT (νQ2) It can
be extracted from Eq (A2) if one takes the modified polarization vector components in Eq (A3)
with ε = εL and εprimelowast = εlowastplusmn as input where εL = 1Q(|~q | 0 0 q0) and εplusmn = ∓ 1radic
2(0 1plusmni 0) are the
standard longitudinal and transverse polarization vectors respectively For εL and εlowastplusmn only the
choice of helicities hprime = plusmn12 and h = ∓12 gives a non-zero contribution and one obtains
χdaggerhprime T (νQ2)χh = χdaggerhprimeminusi~σ middot [(ε0~εprimelowast minus ~ε ε primelowast0 )times q] gLT (νQ2)χh =
radic2 gLT (νQ2) (B1)
where χh and χdaggerhprime are two-component Pauli spinors with opposite helicities or here spins
Let us now consider the related photoabsorption process and in particular the tree-level γlowastN rarr
30
s = 12
sprime = minus12
λ = 0 λprime = 1
rprime rprime
s=12sprime=minus1
2
λ=0λprime=1
rprimerprime
s=12
sprime=minus12
λ=0λprime=1
rprimerprime hacute=frac12h=-frac12
λacute=1λ=0
FIG 11 Relation between the forward Compton process and the photoabsorption process given
by the optical theorem In particular we show the longitudinal-transverse contribution The
double-line arrows represent the spin of the external particles while the dot represents the scalar
(longitudinal) polarization of the incoming photon Inside the blob the intermediate states are
represented eg nucleons with spins rprime (which are averaged in the calculation of the cross section)
and pions
πN channel see diagrams in Fig 8 of Ref [30] We define the πN -production amplitude as
T = uhB (PB)sumi
Ai(s t)Γi uhA(PA) (B2)
with the Dirac structures
Γ1 = γ5 (B3a)
Γ2 =1
2
[qA ε
]γ5 (B3b)
where uhA(PA) and udaggerhB (PB) are the Dirac spinors and PA and PB are the four-momenta of
the incoming and outgoing nucleons respectively When calculating the photoabsorption cross
section related to the VVCS amplitude in Eq (B1) the nucleon spin flip should be implemented
by uhprime(PA) in T dagger and uh(PA) in T together with the appropriate transverse and longitudinal
photon polarization vectors εlowastplusmn and εL
However if one wants to use the properties of the Dirac matrices it is more useful to construct
an operator to produce this spin flip in the external nucleons of Fig 11 This is accomplished by
introducing the projector ΓLT equiv 12radic
2(γ1 + iγ2)γ5 which also takes into account the extra factor
radic2 in Eq (B1) We checked that with this projector one correctly extracts δLT by comparing the
HB limit of our result to the HB result of Ref [50] where the authors calculate this polarizability
from the Compton amplitude directly With all those ingredients the longitudinal-transverse cross
section is calculated in the following way
σLT (νQ2) =1
64π2 s
|~pf |cm
|~pi|cm
int 1
minus1dcos θ
sumij
AiAdaggerjXij (B4)
31
with
Xij = Tr[(PB +MN )Γi(PA +MN )ΓLTγ0Γdaggerjγ
0] (B5)
where θ is the scattering angle in the center-of-mass (cm) frame and |~pi|cm (|~pf |cm) is the three-
momentum of an incoming (outgoing) particle in the cm frame An explicit calculation of the
matrix Xij leads to
X = MNQ
0 2(PB minus PA) middot εLminusradic
2|~qf |cm sin θ (sminus u)
(B6)
where |~qi|cm (|~qf |cm) is the relative three-momentum of the incoming (outgoing) particles in the
cm frame Here s t and u are the usual Mandelstam variables For the different γlowastN rarr πN
channels we obtain the following amplitudes Ai where we introduce qA as the four-momentum of
the incoming photon and qB as the four-momentum of the outgoing pion
bull γlowastprarr π0p
A1 =e gAMN
fπ
[2PA middot ε+ qA middot ε
sminusM2N
+2PB middot εminus qA middot ε
uminusM2N
] (B7a)
A2 =e gAMN
fπ
[1
sminusM2N
+1
uminusM2N
] (B7b)
bull γlowastprarr π+n
A1 =
radic2 e gAMN
fπ
[2PA middot ε+ qA middot ε
sminusM2N
+2(PA minus PB) middot ε+ qA middot ε
tminusm2π
] (B8a)
A2 =
radic2 e gAMN
fπ(sminusM2N )
(B8b)
bull γlowastnrarr π0n
A1 = 0 (B9a)
A2 = 0 (B9b)
bull γlowastnrarr πminusp
A1 =
radic2 e gAMN
fπ
[2PB middot εminus qA middot ε
uminusM2N
minus 2(PA minus PB) middot ε+ qA middot εtminusm2
π
] (B10a)
A2 =
radic2 e gAMN
fπ(uminusM2N )
(B10b)
The analytical expressions shown above were checked with the amplitudes given in Ref [87] An-
alytical expressions for the tree-level γlowastN rarr πN channel of the σLT (νQ2) and σTT (νQ2) cross
32
sections are given below (proton channels π+n and π0p neutron channel πminusp) We checked that
they reproduce the known results in the real-photon limit [45 86] To shorten the final expressions
for the cross sections which are considerably longer for finite Q2 than in the real-photon limit we
define the following dimensionless kinematic variables
αγ = (ENi )cmradics =
s+M2N +Q2
2s (B11)
απ = (ENf )cmradics =
s+M2N minusm2
π
2s (B12)
βγ = Eγcmradics =
sminusM2N minusQ2
2s (B13)
βπ = Eπcmradics =
sminusM2N +m2
π
2s (B14)
λγ = |~qi|cmradics =
radic(sminusM2
N minusQ2)2 + 4sQ2
2s (B15)
λπ = |~qf |cmradics =
radic(sminusM2
N +m2π)2 minus 4sm2
π
2s (B16)
Here (ENi )cm and (ENf )cm are the energies of the incoming and outgoing nucleon Eγcm is the energy
of the incoming photon Eπcm is the energy of the outgoing pion all in the cm frame
σ(π+n)TT = minus e2g2
AM2N
64πf2πs
3(sminusM2N )2λ4
γ
4sλπλγ
[(M2
N minus s)(M2N minusQ2 minus s)(Q2 + 2sβγβπ)
+ 2s(m2πQ
2 minus (M2N minus s)
(M2N + s(minus1 + 2βγβπ)
))λ2γ
]+ (M2
N minus s)(M2N minusQ2 minus s)
times (Q2 + 2sβγβπ minus 2sλπλγ)(Q2 + 2s(βγβπ + λπλγ)
)times log
(Q2 + 2sβγβπ minus 2sλγλπQ2 + 2sβγβπ + 2sλγλπ
) (B17)
33
σ(π0p)TT =
e2g2AM
2Nλπ
64πf2πs(sminusM2
N )2λγ
1
s2λπλ3γ
[(M2
N minus s)(M2N minusQ2 minus s)
(3M2
N + s(2βγβπ minus 3))(M2N
+ s(2βγβπ minus 1))
+ 2s
((2m2
πQ2 + (M2
N minus s)2)(M2
N minus s)minus 2(minusm2
πQ2 + (sminusM2
N )2)
times sβγβπ + 4s2(sminusM2N )β2
γβ2π
)λ2γ
]arctan
(2sλπλγ
M2N + s(2βγβπ minus 1)
)+ 2
[minus 2m2
πQ2
+ (M2N minus s)
(2(sminusM2
N ) + 4sβγβπ +(sminusM2
N +Q2)(3M2N minus 3s+ 2sβγβπ)
sλ2γ
+m2πQ
2(sminusM2N )
(M2N minus s+ 2sβγβπ)(M2
N + s(minus1 + 2βγβπ minus 2λπλγ))
+m2πQ
2(sminusM2N )
(M2N minus s+ 2sβγβπ)(M2
N + s(minus1 + 2βγβπ + 2λπλγ))
)
+1
λπλγ
(βγβπ
(minusm2
πQ2 + (M2
N minus s)(M2N minus s+ 2sβγβπ)
)minus (M2
N minus s)(M2N minusQ2 minus s)λ2
π
)log
(M2N + s(minus1 + 2βγβπ + 2λπλγ)
M2N + s(minus1 + 2βγβπ minus 2λπλγ)
)] (B18)
σ(πminusp)TT =
e2g2AM
2N
64πf2πs
3λ4γ(M2
N + s(2βπβγ minus 1))2 minus 4λ2πs
2λ2γ)
4sλπλγ
((minusM2
N +Q2 + s)(M2N
+ s(2βπβγ minus 1))
2 minus 2sλ2γ
(2λ2
πs(minusM2N +Q2 + s) +m2
πQ2))
+((M2N + s(2βπβγ minus 1)
)2
minus 4λ2πs
2λ2γ
)((M2N + s(2βπβγ minus 1)
)(M2N minus 2Q2 minus s(2βπβγ + 1)
)+ 4λ2
πs2λ2
γ
)times log
(minusM2
N minus 2βπsβγ minus 2λπsλγ + s)minus(M2N + s(2βπβγ minus 2λπλγ minus 1)
)(M2N + s(minus1
+ 2βπβγ + 2λπλγ))[((
M2N + s(2βπβγ minus 1)
)(M2N minus 2Q2 minus s(2βπβγ + 1)
)+ 4λ2
πs2λ2
γ
)times log
(minusM2
N minus 2βπsβγ + 2λπsλγ + s)
+((Q2 + 2βπsβγ)2 minus 4λ2
πs2λ2
γ
)times log
(Q2 + 2βπsβγ minus 2λπsλγQ2 + 2sβπβγ + 2λπsλγ
)]+ 4sλ2
γ
((M2
N + s(2βπβγ minus 1))2 minus 4λ2πs
2λ2γ
)times (M2
N minus s) arctanh
(2λπsλγ
M2N + 2βπsβγ minus s
) (B19)
34
σ(π+n)LT =
e2g2AM
3Nλπ
32πf2πQs
3(sminusM2N )2λ4
γ
2sλγ
[(M2
N minus s)(Q2 + 2sβ2γ)(Q2 + 2sβγβπ)
minus 4s
((M2
N minus s)(Q2 minus 2s(απ minus 1)βγ
)+Q2sβγβπ
)λ2γ + 8s3(minus1 + απ)λ4
γ
]minus M2
N minus sλπ
[(Q2 + 2sβ2
γ)(Q2 + 2sβγβπ)2 + 4s2(2(απ minus 1)βγ(Q2 + 2sβγβπ)
minusQ2λ2π
)λ2γ + 8s3(απ minus 1)2λ4
γ
]arctan
(2sλπλγ
Q2 + 2sβγβπ
) (B20)
σ(π0p)LT =
e2g2AM
3Nλπ
16πf2πQs(sminusM2
N )2λγ
1
minus2s(M2N + s(minus1 + 2βγβπ))2λ2
γ + 8s3λ2πλ
4γ
[minus 3M8
N (Q2
+ 2sβ2γ) + 2M4
Ns2
(minus (Q2 + 2sβ2
γ)(minus3 + 2βγβπ) +(Q2(minus3 + 2βγβπ + 6λ2
π) + 2sβγ
times (minus3 + 12απ + 2βγβπ minus 12απβγβπ + 4βγλ2π))λ2γ minus 4s(α2
π minus 1)λ4π
)+ 2M2
Ns3
[minus (Q2
+ 2sβ2γ)(minus1 + 2βγβπ)
(6 + βγβπ(minus9 + 2βγβπ)
)+
(Q2(3minus 12λ2
π + 4βγβπ(minus1 + 2βγβπ
+ λ2π))minus 2sβγ
(minus 3 + 4απ(3 + 2απ(3 + 2βγβπ(minus3 + βγβπ)) + 4βγ(βπ + (2minus βγβπ)
times λ2π)))λ2γ + 8s
((απ minus 1)(1 + απ minus 2βγβπ) + 2απβγλ
2π
)λ4γ
)+ s4
((Q2 + 2sβ2
γ)
times (1minus 2βγβπ)2(2βγβπ minus 3) + 2(Q2(minus1 + 6λ2
π + 2βγβπ((1minus 2βγβπ)2 minus 2λ2π)) + 2sβγ
times (minus1 + 2βγβπ + 4(minus1 + βγβπ)(minusαπ + 2απβγβπ minus βγλ2π)))λ2γ minus 8
(s(απ minus 1)(απ
+ (1minus 2βγβπ)2) + 2βγ(2sαπ +Q2βγ)λ2π
)λ4γ + 32s(απ minus 1)λ2
πλ6γ
)+ 2M6
Ns
(Q2(6
minus 7βγβπ + λ2γ) + 2sβγ(βγ(6minus 7βγβπ) + (1minus 4απ)λ2
γ)
)]+sminusM2
N
4s2λπλ3γ
[(Q2 + 2sβ2
γ)
times(3M2
N + s(2βγβπ minus 3))(M2N + s(2βγβπ minus 1)
)+ 2s
(minusM2
N (Q2 + 2sβγ minus 8sαπβγ)
+ s(2sβγ(1minus 4απ + 4απβγβπ) +Q2(1minus 2λ2π)))λ2γ + 8s3(απ minus 1)λ4
γ
]
times arctan
(2sλπλγ
M2N + s(2βγβπ minus 1)
) (B21)
35
σ(πminusp)LT =
e2g2AM
3N
64πf2πQs
3λ4γ(M2
N minusQ2 minus s)(M2N + s(2βπβγ + 2λπλγ minus 1))
times
1
M2N + s(2βπβγ minus 2λπλγ minus 1)
[4λπsλγ(M2
N minusQ2 minus s)(
2sλ2γ
(s((1minus 2λ2
π)Q2
+ 2(2απ minus 1)sβγ(2βπβγ minus 1))minusM2N (Q2 + 2(1minus 2απ)sβγ)
)+ (Q2 + 2sβ2
γ)(M2N + s(2βπβγ minus 1)
)2 + 8(απ minus 1)απs
3λ4γ
)]minus(M2N + s(2βπβγ + 2λπλγ minus 1)
)times[
log
(minusM2N minus 2βπsβγ minus 2λπsλγ + s
minusM2N minus 2βπsβγ + 2λπsλγ + s
)(minus (Q2 + 2sβ2
γ)(M2N + s(2βπβγ minus 1)
)(minusM2
N + 2Q2 + 2βπsβγ + s)minus 2sλ2γ
(minusM2
N (Q2 + 2sβγ) +Q4 +Q2s(4απβγ
minus 2βγ minus 2λ2π + 1) + 2s2βγ(4(απ minus 1)βπβγ + 1)
)minus 8(απ minus 1)2s3λ4
γ
)+ 2
(4s2λ2
γ
(2(απ minus 1)βγ(Q2 + 2βπsβγ)minus λ2
πQ2)
+ (Q2 + 2βπsβγ)2(Q2 + 2sβ2γ)
+ 8(απ minus 1)2s3λ4γ
)arctanh
(2λπsλγ
Q2 + 2βπsβγ
)] (B22)
2 ∆-production channel
The tree-level ∆-exchange diagram in Fig 2 of Ref [30] contributes to the non-Born part of
the VVCS amplitudes The contribution of the ∆ exchange to the VVCS amplitudes can be split
into [17]
S∆-exch1 (νQ2) = S∆-pole
1 (νQ2) + S∆-exch1 (νQ2) (B23a)
νS∆-exch2 (νQ2) = νS∆-pole
2 (νQ2) + νS2∆-exch
(νQ2) (B23b)
and similarly for the unpolarized VVCS amplitudes discussed in Ref [30] Here we introduced the
∆-pole contributions S∆-polei and the ∆-non-pole contributions S∆-exch
i The former amplitudes
feature a pole at the ∆(1232)-production threshold and thus are proportional to
1
[sminusM2∆][uminusM2
∆]=
1
4M2N
1
ν2∆ minus ν2
(B24)
36
They can be reconstructed from the dispersion relations in Eq (5) with the tree-level ∆-production
cross sections as input cf Fig 10 in Ref [30]
σTT (νQ2) =π2α
M2NM
2+|~q |
minus g2
MMN (M+ + ν)|~q |2 +g2E(∆minus ν)(Q2 minusMNν)2
MN(B25a)
+g2CQ
4s(∆minus ν)
MNM2∆
minus 4gMgE(Q2 minusMNν)|~q |2 minus 4gMgCQ2|~q |2
+2gEgCQ
2[minusMNM∆ |~q |2 + s(Q2 +∆ν)
]MNM∆
δ(ν minus ν∆)
σLT (νQ2) =Qπ2α
M2NM
2+|~q |
g2E(MNν minusQ2) [M∆(MN + ν)minus s]
MN(B25b)
+g2CQ
2[MNM∆|~q |2 minus s(Q2 +∆ν)
]MNM2
∆
+ gMgEM∆|~q |2 minusgMgC(Q2 minusMNν)|~q |2
M∆
+gEgC(ν minus∆)(M2
N |~q |2 minus 2Q2s)
MNM∆
δ(ν minus ν∆)
with ∆ = M∆ minusMN M+ = M∆ + MN and the Mandelstam variable s = M2N + 2MNν minus Q2
Analytical expressions for the spin structure functions g1(xQ2) and g2(xQ2) can be constructed
from Eq (3) with the flux factor K(νQ2) = |~q | =radicν2 +Q2
In the ∆-non-pole contributions to S1(νQ2) and νS2(νQ2) the pole in ν at the ∆(1232)-
production threshold has canceled out
S∆-exch1 (νQ2) =
πα
MNM2+
[g2MQ
2+ + g2
E
(∆2 minus 3Q2
)+
4g2CQ
4
M2∆
minus 8gMgEM∆ωminus (B26a)
minus2gMgCQ2(MN minus 4M∆)
M∆+
2gEgCQ2(3MN minus 2M∆)
M∆
]
νS2∆-exch
(νQ2) =2πα
MNM2+
[g2EM∆∆ωminus +
g2M MNQ
2+
2+g2C Q
2(Q2 minus∆2)
2M∆(B26b)
+gEgM M∆(M∆ω+ minus 4MNωminus)minus gEgC ∆(2Q2 +MNω+)
+gMgC Q2(4MN minus ω+)
]+S∆-exch
2 (νQ2)
ν
[M2
∆ ω2+
M2N
+ ν2
]
with Q+ =radic
(M∆ +MN )2 +Q2 and ωplusmn = (M2∆minusM2
N plusmnQ2)2M∆ and the non-pole contribution
to S2(νQ2)
S∆-exch2 (νQ2) = minus2παMNν
M∆M2+
[gM + gE
]gC (B27)
These amplitudes to the contrary are not described by the tree-level ∆-production cross sections
in the standard dispersive approach [17] This peculiarity has been previously missed e g in the
calculation of the ∆-exchange contribution to the hydrogen hyperfine splitting in Ref [88] The
importance of including the ∆-non-pole contribution is also evident when considering the BC sum
37
rule in Eq (14) The ∆-pole terms by themselves violate the BC sum rule but cancel exactly with
the ∆-non-pole terms
limνrarr0
νS∆-pole2 (νQ2) + lim
νrarr0νS2
∆-exch(νQ2) = 0 (B28)
Appendix C Polarizabilities at Q2 = 0
In this section we give analytical expressions for the static values and slopes at Q2 = 0 of
the spin polarizabilities and moments of polarized structure functions In particular we give the
HB expansion of the πN -loop contributions and the ∆-exchange contributions The complete
expressions also for the π∆-loop contributions can be found in the Supplemented material Recall
that IA(0) = I1(0) = d2(0) = 0 and dd2(Q2)dQ2
∣∣∣Q2=0
= 0
1 πN-loop contribution
Here we give analytical expressions for the πN -loop contributions to the proton and neutron
spin polarizabilities expanded in powers of micro = mπMN viz the HB expansion Note that we
choose to expand here to a high order in micro the strict HB expansion would only retain the leading
term in an analogous NLO calculation
bull Static polarizabilities (Q2 = 0)
γ0p =e2g2
A
96π3f2πm
2π
1minus 21πmicro
8minus(
59
2+ 26 logmicro
)micro2 +
1875πmicro3
64
+3
(3
2+ 26 logmicro
)micro4 +
(C1)
γ0n =e2g2
A
48π3f2πm
2π
1
2minus 9πmicro
16minus 2micro2 logmicro+
75πmicro3
128minus 3micro4
4+
(C2)
δLTp =e2g2
A
192π3f2πm
2π
1minus 9πmicro
8+
(13
2minus 2 logmicro
)micro2 minus 465πmicro3
64
minus(
47
2+ 42 logmicro
)micro4 +
(C3)
δLTn =e2g2
A
96π3f2πm
2π
1
2+
3πmicro
16+ (1 + 2 log micro)micro2 minus 105πmicro3
128+
5micro4
4+
(C4)
38
δLTn =e2g2
A
96π3f2πm
2π
1
2+
3πmicro
16+ (1 + 2 log micro)micro2 minus 105πmicro3
128+
5micro4
4+
(C5)
γ0p =e2g2
A
16π3f2πm
4π
4
45minus 3πmicro
16+
14micro2
5minus 1813πmicro3
384minus 192
5(1 + log micro)micro4
+80703πmicro5
2048+
(C6)
γ0n =e2g2
A
16π3f2πm
4π
4
45minus 5πmicro
48+
4micro2
5minus 245πmicro3
384minus 32micro4 logmicro
15
+1323πmicro5
2048+
(C7)
bull Slopes of polarizabilities at Q2 = 0
dγ0p(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
2minus 45πmicro
4+ 223micro2 minus 28515πmicro3
64
minus9
(1953
4+ 449 logmicro
)micro4 +
570255πmicro5
128+
(C8)
dγ0n(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
2minus 81πmicro
8+ 94micro2 minus 2535πmicro3
32
minus3 (1 + 90 logmicro)micro4 +84315πmicro5
1024+
(C9)
dδLTp(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
2880π3f2πm
4π
minus5
2minus 27πmicro
32+ 20micro2 minus 5865πmicro3
256
+3
(617
4+ 36 logmicro
)micro4 minus 2056845πmicro5
4096+
(C10)
dδLTn(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
minus5
4minus 81πmicro
64minus 11micro2 +
10005πmicro3
512
+15
8(11 + 48 logmicro)micro4 minus 267015πmicro5
8192+
(C11)
dIAp(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
96π2f2πmicro
2
1minus 15πmicro
4minus 1
2(115 + 88 logmicro)micro2 +
1839πmicro3
32
+5 (5 + 34 logmicro)micro4 +
(C12)
dIAn(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
48π2f2πmicro
2
1
2minus 11πmicro
8minus 1
4(1 + 20 logmicro)micro2 +
99πmicro3
64
minus25micro4
12+
(C13)
dI1p(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
96π2f2πmicro
3π
8+ 2 (4 + 3 logmicro)microminus 537πmicro2
64
minus1
2(15 + 56 logmicro)micro3 +
(C14)
39
dI1n(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
48π2f2πmicro
minus π
16+
1
4(3 + 4 logmicro)microminus 57πmicro2
128+
2micro3
3+
(C15)
γ0p(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2g2
A
16π3f2πm
6π
1
105minus 23πmicro
256+
377micro2
210minus 15551πmicro3
6144+
3371micro4
105
minus1640457πmicro5
32768+
(C16)
γ0n(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2g2
A
16π3f2πm
6π
1
105minus 153πmicro
1792+
69micro2
70minus 4615πmicro3
6144+
172micro4
35
minus120897πmicro5
32768+
(C17)
2 ∆-exchange contribution
Here we give analytical expressions for the tree-level ∆-exchange contributions to the nucleon
spin polarizabilities and their slopes at Q2 = 0 Note that the ∆-exchange contributes equally to
proton and neutron polarizabilities Recall that for the magnetic γlowastN∆ coupling we introduced a
dipole form factor to mimic vector-meson dominance gM rarr gM(1 +Q2Λ2)2
bull Static polarizabilities (Q2 = 0)
γ0 = minus e2
4πM2+
(g2M
∆2+
g2E
M2+
minus 4gMgEM+∆
) (C18)
δLT =e2M∆
4πM3+
(g2E
MNM++gMgE∆MN
minus gEgCM2
∆
) (C19)
γ0 =e2M2
N
π∆2M4+
(minusg
2M
∆2+
g2E
M2+
+4gMgE∆M+
) (C20)
bull Slopes of polarizabilities at Q2 = 0
dγ0(Q2)
dQ2
∣∣∣∣∣Q2=0
= minus e2
πM2+∆
(g2M
∆
[1
4∆2minus 1
∆M++
1
2M2+
]minus 1
Λ2
g2M
∆+
g2E
2M2+
[1
2∆minus 3
M+
]minusgMgE
M+
[1
∆2minus 5
∆M++
1
M2+
]+
1
Λ2
2gMgEM+
+2gMgC∆M2
+
minus gEgCM3
+
) (C21)
40
d δLT (Q2)
dQ2
∣∣∣∣∣Q2=0
=e2M∆∆
4πMNM2+
(g2E
∆2M2+
[1
∆minus 4
M+
]minus g2
C
∆M2∆M
2+
+gMgE∆2M+
[1
∆2
minus 3
∆M++
1
M2+
]minus 2
Λ2
gMgE∆2M+
+gMgC∆M2
∆
[1
2∆2minus 2
∆M++
1
2M2+
]minus gEgC
2M2∆M
2+
[7
∆+
1
M+
]) (C22)
dIA(Q2)
dQ2
∣∣∣∣∣Q2=0
= minusM2N
M2+
(g2M
2∆2+
g2E
MNM+minus 2gMgE
∆M+minus gEgCM∆M+
) (C23)
dI1(Q2)
dQ2
∣∣∣∣∣Q2=0
= minusM∆M2N
2M3+
(g2E
MNM∆minus gMgE∆MN
minus gEgCM2
∆
) (C24)
γ0(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2M2
N
π∆3M6+
(g2M
∆
[3M2
∆ + 2M∆MN minus 9M2N
∆2+
4M2+
Λ2
]
minus4gMgE
[5M2
∆ minus 9M2N
M+∆2+
2M+
Λ2
]+
g2E
∆M+[7M∆ minus 9MN ]
minus8gMgC∆
+4gEgCM+
) (C25)
[1] D Drechsel B Pasquini and M Vanderhaeghen Dispersion relations in real and virtual
Compton scattering Phys Rept 378 99 (2003) hep-ph0212124
[2] S E Kuhn J-P Chen and E Leader Spin structure of the nucleon mdash status and recent
results Prog Part Nucl Phys 63 1 (2009) arXiv08123535 [hep-ph]
[3] F Hagelstein R Miskimen and V Pascalutsa Nucleon polarizabilities from Compton scat-
tering to hydrogen atom Prog Part Nucl Phys 88 29 (2016) arXiv151203765 [nucl-th]
[4] B Pasquini and M Vanderhaeghen Dispersion theory in electromagnetic interactions Ann
Rev Nucl Part Sci 68 75 (2018) arXiv180510482 [hep-ph]
[5] S Gerasimov A Sum rule for magnetic moments and the damping of the nucleon magnetic
moment in nuclei Sov J Nucl Phys 2 430 (1966)
[6] S Drell and A C Hearn Exact sum rule for nucleon magnetic moments Phys Rev Lett
16 908 (1966)
[7] J S Schwinger Source theory viewpoints in deep inelastic scattering Proc Natl Acad Sci
USA 72 1 (1975)
[8] J S Schwinger Source theory viewpoints in deep inelastic scattering Electromagnetic In-
teractions and Field Theory Proceedings 14 Internationale Universitatswochen Schladming
Austria February 24-March 7 1975 Acta Phys Austriaca Suppl 14 471 (1975)
41
[9] J Schwinger Source theory discussion of deep inelastic scattering with polarized particles
Proc Natl Acad Sci USA 72 1559 (1975)
[10] A M Baldin Polarizability of nucleons Nucl Phys 18 310 (1960)
[11] M Gell-Mann M L Goldberger and W E Thirring Use of causality conditions in quantum
theory Phys Rev 95 1612 (1954)
[12] A Pineda Leading chiral logarithms to the hyperfine splitting of the hydrogen and muonic
hydrogen Phys Rev C 67 025201 (2003)
[13] C Peset and A Pineda The two-photon exchange contribution to muonic hydrogen from
chiral perturbation theory Nucl Phys B 887 69 (2014) arXiv14064524 [hep-ph]
[14] C Peset and A Pineda Model-independent determination of the two-photon exchange contri-
bution to hyperfine splitting in muonic hydrogen JHEP 04 060 arXiv161205206 [nucl-th]
[15] F Hagelstein and V Pascalutsa Proton structure in the hyperfine splitting of muonic hydro-
gen PoS CD15 077 (2016) arXiv151104301 [nucl-th]
[16] F Hagelstein Exciting Nucleons in Compton Scattering and Hydrogen-Like Atoms PhD
thesis Mainz U Inst Kernphys (2017) arXiv171000874 [nucl-th]
[17] F Hagelstein ∆(1232)-Resonance in the hydrogen spectrum Proceedings 11th International
Workshop on the Physics of Excited Nucleons (NSTAR 2017) Columbia SC USA August
20-23 2017 Few Body Syst 59 93 (2018) arXiv180109790 [nucl-th]
[18] Y Prok et al (CLAS) Moments of the spin structure functions gp1 and gd1 for 005 lt Q2 lt 30
GeV2 Phys Lett B 672 12 (2009) arXiv08022232 [nucl-ex]
[19] H Dutz et al (GDH) First measurement of the Gerasimov-Drell-Hearn sum rule for 1H from
07 GeV to 18 GeV at ELSA Phys Rev Lett 91 192001 (2003)
[20] M Amarian et al (Jefferson Lab E94010) Measurement of the generalized forward spin po-
larizabilities of the neutron Phys Rev Lett 93 152301 (2004) arXivnucl-ex0406005
[21] M Amarian et al The Q2 evolution of the generalized Gerasimov-Drell-Hearn integral for
the neutron using a 3He target Phys Rev Lett 89 242301 (2002) arXivnucl-ex0205020
[22] M Amarian et al (Jefferson Lab E94-010)Q2 evolution of the neutron spin structure moments
using a 3He target Phys Rev Lett 92 022301 (2004) arXivhep-ex0310003
[23] A Deur et al Experimental determination of the evolution of the Bjorken integral at low Q2
Phys Rev Lett 93 212001 (2004) arXivhep-ex0407007
[24] K Slifer Low Q2 measurement of gp2 and the δLT spin polarizability Spin structure at long dis-
tance Proceedings Workshop Newport News USA March 12ndash13 2009 nucl-ex09064775
(2009) AIP Conf Proc 1155 10106313203293 (2009) arXiv09064775 [nucl-ex]
[25] P Solvignon et al (E01-012) Moments of the neutron g2 structure function at intermediate
42
Q2 Phys Rev C 92 015208 (2015) arXiv13044497 [nucl-ex]
[26] A Deur Experimental studies at low Q2 of the spin structure of the nucleon at Jefferson Lab
in 9th International Workshop on Chiral Dynamics (CD18) Durham NC USA September
17-21 2018 (2019) arXiv190305661 [nucl-ex]
[27] V Sulkosky et al (Jefferson Lab E97-110) Measurement of the 3He spin-structure functions
and of neutron (3He) spin-dependent sum rules at 0035 le Q2 le 024 GeV2 Phys Lett B
805 135428 (2020) arXiv190805709 [nucl-ex]
[28] K Adhikari et al (CLAS) Measurement of the Q2 Dependence of the Deuteron Spin Structure
Function g1 and its Moments at Low Q2 with CLAS Phys Rev Lett 120 062501 (2018)
arXiv171101974 [nucl-ex]
[29] V Lensky F Hagelstein A Hiller Blin and V Pascalutsa Comment on rdquoMeasurement of
the Q2 Dependence of the Deuteron Spin Structure Function g1 and its Moments at Low Q2
with CLASrdquo (2018) arXiv180603219 [nucl-th]
[30] J M Alarcn F Hagelstein V Lensky and V Pascalutsa Forward doubly-virtual Compton
scattering off the nucleon in chiral perturbation theory at NLO the subtraction function and
moments of unpolarized structure functions (2020) arXiv200509518 [hep-ph]
[31] V Bernard E Epelbaum H Krebs and U-G Meiszligner New insights into the spin structure
of the nucleon Phys Rev D 87 054032 (2013) arXiv12092523 [hep-ph]
[32] V Lensky J M Alarcon and V Pascalutsa Moments of nucleon structure functions at
next-to-leading order in baryon chiral perturbation theory Phys Rev C 90 055202 (2014)
arXiv14072574 [hep-ph]
[33] V Pascalutsa and M Vanderhaeghen Chiral effective-field theory in the ∆(1232) region
I Pion electroproduction on the nucleon Phys Rev D 73 034003 (2006) arXivhep-
ph0512244
[34] V Pascalutsa and M Vanderhaeghen Electromagnetic nucleon-to-Delta transition in chiral
effective field theory Phys Rev Lett 95 232001 (2005) arXivhep-ph0508060
[35] T R Hemmert B R Holstein and J Kambor Systematic 1M expansion for spin 32
particles in baryon chiral perturbation theory Phys Lett B 395 89 (1997) arXivhep-
ph9606456
[36] V Pascalutsa and D R Phillips Effective theory of the ∆(1232) in Compton scattering off
the nucleon Phys Rev C 67 055202 (2003) arXivnucl-th0212024
[37] V Pascalutsa M Vanderhaeghen and S N Yang Electromagnetic excitation of the ∆(1232)-
resonance Phys Rept 437 125 (2007) arXivhep-ph0609004
[38] K A Olive et al (Particle Data Group) Review of Particle Physics Chin Phys C 38 090001
43
(2014)
[39] F E Low Scattering of light of very low frequency by systems of spin 12 Phys Rev 96
1428 (1954)
[40] M Gell-Mann and M L Goldberger Scattering of low-energy photons by particles of spin
12 Phys Rev 96 1433 (1954)
[41] A M Harun ar-Rashid A simple derivation of Schwingerrsquos sum rule for spin dependent
structure functions Nuovo Cim A 33 447 (1976)
[42] F Hagelstein and V Pascalutsa Dissecting the hadronic contributions to (g minus 2)micro by
Schwingerrsquos sum rule Phys Rev Lett 120 072002 (2018) arXiv171004571 [hep-ph]
[43] V Lensky V Pascalutsa M Vanderhaeghen and C Kao Spin-dependent sum rules con-
necting real and virtual Compton scattering verified Phys Rev D 95 074001 (2017)
arXiv170101947 [hep-ph]
[44] H Burkhardt and W N Cottingham Sum rules for forward virtual Compton scattering
Annals Phys 56 453 (1970)
[45] V Lensky and V Pascalutsa Predictive powers of chiral perturbation theory in Compton
scattering off protons Eur Phys J C 65 195 (2010) arXiv09070451 [hep-ph]
[46] H Grieszlighammer J McGovern D Phillips and G Feldman Using effective field theory to
analyse low-energy Compton scattering data from protons and light nuclei Prog Part Nucl
Phys 67 841 (2012) arXiv12036834 [nucl-th]
[47] H W Grieszlighammer J A McGovern and D R Phillips Nucleon polarisabilities at and
beyond physical pion masses Eur Phys J A 52 139 (2016) arXiv151101952 [nucl-th]
[48] E Epelbaum H Krebs and U-G Meiszligner Improved chiral nucleon-nucleon potential up to
next-to-next-to-next-to-leading order Eur Phys J A 51 53 (2015) arXiv14120142 [nucl-
th]
[49] E Epelbaum H Krebs and U-G Meiszligner Precision nucleon-nucleon potential at fifth order
in the chiral expansion Phys Rev Lett 115 122301 (2015) arXiv14124623 [nucl-th]
[50] C W Kao T Spitzenberg and M Vanderhaeghen Burkhardt-Cottingham sum rule and
forward spin polarizabilities in heavy baryon chiral perturbation theory Phys Rev D 67
016001 (2003) arXivhep-ph0209241
[51] C-W Kao D Drechsel S Kamalov and M Vanderhaeghen Higher moments of nucleon
spin structure functions in heavy baryon chiral perturbation theory and in a resonance model
Phys Rev D 69 056004 (2004) arXivhep-ph0312102
[52] T Becher and H Leutwyler Baryon chiral perturbation theory in manifestly Lorentz invariant
form Eur Phys J C 9 643 (1999) arXivhep-ph9901384
44
[53] J Gasser M E Sainio and A Svarc Nucleons with chiral loops Nucl Phys B 307 779
(1988)
[54] L S Geng J Martin Camalich L Alvarez-Ruso and M J Vicente Vacas Leading SU(3)-
breaking corrections to the baryon magnetic moments in chiral perturbation theory Phys
Rev Lett 101 222002 (2008) arXiv08051419 [hep-ph]
[55] D Drechsel S S Kamalov and L Tiator The GDH sum rule and related integrals Phys
Rev D 63 114010 (2001) arXivhep-ph0008306
[56] D Drechsel O Hanstein S S Kamalov and L Tiator A unitary isobar model for pion
photo- and electroproduction on the proton up to 1 GeV Nucl Phys A 645 145 (1999)
arXivnucl-th9807001
[57] L Tiator private communication (2020)
[58] V Bernard T R Hemmert and U-G Meiszligner Spin structure of the nucleon at low energies
Phys Rev D 67 076008 (2003) arXivhep-ph0212033
[59] O Gryniuk F Hagelstein and V Pascalutsa Evaluation of the forward Compton scattering
off protons II Spin-dependent amplitude and observables Phys Rev D 94 034043 (2016)
arXiv160400789 [nucl-th]
[60] R Zielinski The g2p Experiment A Measurement of the Protonrsquos Spin Structure Functions
PhD thesis New Hampshire U (2010) arXiv170808297 [nucl-ex]
[61] N Guler et al (CLAS) Precise determination of the deuteron spin structure at low to mod-
erate Q2 with CLAS and extraction of the neutron contribution Phys Rev C 92 055201
(2015) arXiv150507877 [nucl-ex]
[62] C E Carlson V Nazaryan and K Griffioen Proton structure corrections to electronic
and muonic hydrogen hyperfine splitting Phys Rev A 78 022517 (2008) arXiv08052603
[physicsatom-ph]
[63] V Bernard N Kaiser and U-G Meiszligner Chiral dynamics in nucleons and nuclei Int J
Mod Phys E 4 193 (1995) arXivhep-ph9501384
[64] T R Hemmert B R Holstein and J Kambor ∆(1232) and the polarizabilities of the nucleon
Phys Rev D 55 5598 (1997) arXivhep-ph9612374
[65] V Pascalutsa and R Timmermans Field theory of nucleon to higher-spin baryon transitions
Phys Rev C 60 042201 (1999) arXivnucl-th9905065
[66] V Pascalutsa Quantization of an interacting spin-32 field and the Delta isobar Phys Rev
D 58 096002 (1998) arXivhep-ph9802288
[67] H Krebs Double Virtual Compton Scattering and SpinStructure of the Nucleon PoS
CD2018 031 (2019)
45
[68] N Kochelev and Y Oh Axial anomaly and the δLT puzzle Phys Rev D 016012 (2012)
arXiv11034892 [hep-ph]
[69] D Drechsel S Kamalov and L Tiator Unitary isobar model ndash MAID2007 Eur Phys J A
69 (2007) available at httpsmaidkphuni-mainzde arXiv07100306 [nucl-th]
[70] P J Mohr B N Taylor and D B Newell CODATA recommended values of the fundamental
physical constants 2010 Rev Mod Phys 84 1527 (2012)
[71] J Ahrens et al (GDH A2) First measurement of the Gerasimov-Drell-Hearn integral for 1H
from 200 to 800 MeV Phys Rev Lett 87 022003 (2001) arXivhep-ex0105089 [hep-ex]
[72] K Helbing (GDH) Experimental verification of the GDH sum rule at ELSA and MAMI Nucl
Phys Proc Suppl 105 113 (2002)
[73] R Bradford A Bodek H S Budd and J Arrington A New parameterization of the nucleon
elastic form-factors NuInt05 proceedings of the 4th International Workshop on Neutrino-
Nucleus Interactions in the Few-GeV Region Okayama Japan 26-29 September 2005 Nucl
Phys Proc Suppl 159 127 (2006) arXivhep-ex0602017
[74] J D Bjorken Applications of the chiral U(6)timesU(6) algebra of current densities Phys Rev
148 1467 (1966)
[75] J D Bjorken Inelastic scattering of polarized leptons from polarized nucleons Phys Rev D
1 1376 (1970)
[76] V Bernard T R Hemmert and U-G Meiszligner Novel analysis of chiral loop effects in
the generalized Gerasimov-Drell-Hearn sum rule Phys Lett B 545 105 (2002) arXivhep-
ph0203167
[77] A Deur et al Experimental study of isovector spin sum rules Phys Rev D 78 032001
(2008) arXiv08023198 [nucl-ex]
[78] B Pasquini P Pedroni and D Drechsel Higher order forward spin polarizability Phys Lett
B 687 160 (2010) arXiv10014230 [hep-ph]
[79] R Jaffe g2ndashThe nucleonrsquos other spin-dependent structure function Comments Nucl Part
Phys 19 239 (1990)
[80] E V Shuryak and A Vainshtein Theory of power corrections to deep inelastic scattering in
quantum chromodynamics (II) Qminus4 effects polarized target Nucl Phys B 201 141 (1982)
[81] S Wandzura and F Wilczek Sum rules for spin dependent electroproduction Test of rela-
tivistic constituent quarks Phys Lett B 72 195 (1977)
[82] B W Filippone and X-D Ji The spin structure of the nucleon Adv Nucl Phys 26 1
(2001) arXivhep-ph0101224
[83] M Burkardt The g2 structure function Proceedings Workshop on Spin structure at long
46
distance Newport News USA March 12-13 2009 AIP Conf Proc 1155 26 (2009)
arXiv09054079 [hep-ph]
[84] A Deur S J Brodsky and G F de Teramond The spin structure of the nucleon Rept
Prog Phys 82 076201 (2019) arXiv180705250 [hep-ph]
[85] B R Holstein D Drechsel B Pasquini and M Vanderhaeghen Higher order polarizabilities
of the proton Phys Rev C 61 034316 (2000) arXivhep-ph9910427
[86] B R Holstein V Pascalutsa and M Vanderhaeghen Sum rules for magnetic moments and
polarizabilities in QED and chiral effective-field theory Phys Rev D 72 094014 (2005)
arXivhep-ph0507016
[87] B Pasquini D Drechsel and L Tiator Invariant amplitudes for pion electroproduction Eur
Phys J A 34 387 (2007) arXiv07122327 [hep-ph]
[88] A J Buchmann Non-spherical proton shape and hydrogen hyperfine splitting Proceedings
International Workshop on Precision Physics of Simple Atomic Systems (PSAS 2008) Wind-
sor Ontario Canada July 21-26 2008 Can J Phys 87 773 (2009) arXiv09104747
[physicsatom-ph]
47
The authors of that work attributed this large contribution to diagrams where the photons
couple directly to the Delta inside a loop As mentioned in Sec III F the effect of such loop
diagrams does not change the Q2 behaviour of the polarizabilities On the other hand it
can produce a substantial shift of the δLT (Q2) as a whole A higher-order calculation should
resolve the discrepancy between the two covariant approaches however it will partially lose
the predictive power since the LECs appearing at higher orders will have to be fitted to
experimental data
The πN -loop ∆-exchange and π∆-loop contributions to the NLO BχPT prediction of
the static longitudinal-transverse polarizability are in units of 10minus4 fm4
δLTp = 132(15) asymp 150minus 016minus 002 (23a)
δLTn = 218(23) asymp 235minus 016minus 002 (23b)
while the slopes are in units of 10minus4 fm6
dδLTp(Q2)
dQ2
∣∣∣∣Q2=0
= minus085(8) asymp minus080minus 004minus 001 (24a)
dδLTn(Q2)
dQ2
∣∣∣∣Q2=0
= minus124(12) asymp minus119minus 004minus 001 (24b)
C IA(Q2) mdash a generalized GDH integral
The helicity-difference cross section σTT exhibits a faster fall-off in ν than its spin-averaged
counterpart σT This is due to a cancellation between the leading (constant) terms of σ12
and σ32 at large ν3 The resulting 1ν fall-off of the helicity-difference cross section allows
one to write an unsubtracted dispersion relation for the VVCS amplitude gnonpoleTT (νQ2) cf
Eq (10a) This is the origin of the GDH sum rule [5 6]
minus α
2M2N
κ2 =1
2π2
int infinν0
dνσTT (ν)
ν (25)
which establishes a relation to the anomalous magnetic moment κ It is experimentally
verified for the nucleon by MAMI (Mainz) and ELSA (Bonn) [71 72]
There are two extensions of the GDH sum rule to finite Q2 the generalized GDH integrals
3 Notice that a constant term in σTT at ν rarrinfin is forbidden by crossing symmetry
16
000 005 010 015 020 025 030
-30
-25
-20
-15
-10
-05
00
Q2 GeV2D
I Aproton
000 005 010 015 020 025 030
-25
-20
-15
-10
-05
00
Q2 GeV2D
I A
neutron
000 005 010 015 020 025 030
-05
00
05
Q2 GeV2D
I 1
proton
000 005 010 015 020 025 030
-10
-08
-06
-04
-02
00
02
Q2 GeV2D
I 1
neutron
FIG 4 Upper panel The generalized GDH integral IA(Q2) for the proton (left) and neutron
(right) as function of Q2 The result of this work the NLO BχPT prediction is shown by the blue
solid line and the blue band The red line represents the LO BχPT result The purple short-dashed
line is the O(p4) HB result from Ref [50 51] The gray band is the BχPT+∆ result from Ref [31]
The black dotted line is the MAID model prediction [69] Experimental extractions for the proton
Ref [60] (orange triangle uncertainties added in quadrature) and neutron Refs [21][27] where
magenta dotsorange diagonal crosses correspond to data and red squareslilac crosses correspond
to data plus extrapolation to unmeasured energy regions The green stars at the real-photon point
are derived from the anomalous magnetic moments κp asymp 1793 and κn asymp minus1913 [70] Lower
panel The generalized GDH integral I1(Q2) for the proton (left) and neutron (right) as function
of Q2 The purple short-dashed line is the HB result from Ref [51] Experimental extractions for
the proton Ref [18] (blue dots) and Ref [60] (orange triangle uncertainties added in quadrature)
and neutron Ref [61][27] (uncertainties added in quadrature) where green dotsorange diagonal
crosses correspond to data and gray squareslilac crosses correspond to data plus extrapolation to
unmeasured energy regions
17
IA(Q2) and I1(Q2) The latter will be discussed in Sec III D The former is defined as4
minus α
2M2N
IA(Q2) = minus 1
8π2
int infinν0
dν
radic1 +
Q2
ν2
σTT (νQ2)
ν(26)
=α
Q2
int x0
0
dx[4M2
Nx2
Q2g2(xQ2)minus g1(xQ2)
]
Due to its energy weighting the integral in Eq (26) converges slower than the one in the
generalized forward spin polarizability sum rule (19) Therefore knowledge of the cross
section at higher energies is required and the evaluation of the generalized GDH integral
IA(Q2) is not as simple as the evaluation of γ0(Q2)
The generalized GDH integral IA(Q2) is directly related to the non-pole amplitude
gnonpoleTT (νQ2) which differs from non-Born amplitude gTT (νQ2) by a term involving the
elastic Pauli form factor
gnonpoleTT (νQ2) = gTT (νQ2)minus 2παν
M2N
F 22 (Q2) (27)
cf Eqs (2a) and (8a) Consequently IA(Q2) is not a pure polarizability but also contains
an elastic contribution The ldquonon-polarizabilityrdquo or the Born part of IA(Q2) is given by
IBornA (Q2) = IA(Q2)minus∆IA(Q2) = minus1
4F 2
2 (Q2) (28)
where we refer to the polarizability part as ∆IA(Q2) The same is true for the generalized
GDH integral I1(Q2) which is directly related to Snonpole1 (νQ2)
IBorn1 (Q2) = I1(Q2)minus∆I1(Q2) = minus1
4F 2
2 (Q2) (29)
In the following we will add the Born parts to our LO and NLO BχPT predictions for
the polarizabilities ∆IA(Q2) and ∆I1(Q2) employing an empirical parametrization for the
elastic Pauli form factor [73] This allows us to compare to the experimental results for
IA(Q2) and I1(Q2) cf Fig 4 Note that the blue error bands only describe the uncertainties
of our BχPT predictions of the polarizabilities while the elastic contributions are considered
to be exact as explained in Sec II C The uncertainties of the polarizability predictions are
therefore better reflected in Fig 5 where we show the contributions of the different orders
to the BχPT predictions of ∆IA(Q2) and ∆I1(Q2) as well as the total results with error
bands
4 Note that IA(Q2) is sometimes called ITT (Q2)
18
000 005 010 015 020 025 030
-08
-06
-04
-02
00
02
Q2 GeV2D
DI A
proton
000 005 010 015 020 025 030
-10
-08
-06
-04
-02
00
02
Q2 GeV2D
DI A
neutron
000 005 010 015 020 025 030-005
000
005
010
015
020
025
Q2 GeV2D
DI 1
proton
000 005 010 015 020 025 030-015
-010
-005
000
005
010
015
Q2 GeV2D
DI 1
neutron
FIG 5 Contributions of the different orders to the chiral predictions of ∆IA(Q2) upper paneland ∆I1(Q2) lower panel for the proton (left) and neutron (right) Red solid line πN -loop
contribution green dot-dashed line ∆-exchange contribution orange dotted line π∆-loop contri-
bution blue solid line and blue band total result purple dot-dot-dashed line total result without
gC contribution black short-dashed line total result without gM dipole
The E97-110 experiment at Jefferson Lab has recently published their data for IAn(Q2) in
the region of 0035 GeV2 lt Q2 lt 024 GeV2 [27] In addition there are results for IAn(Q2)
from the earlier E94-010 experiment [21] and for IAp(Q2) from the E08-027 experiment [60]
The O(p4) HB calculation gives a large negative effect [51] which does not describe the data
The BχPT+∆ result from Ref [31] which mainly differs from our work by the absence of
the dipole form factor in gM looks similar to this HB result and only describes the data
points at lowest Q2 Our NLO prediction however follows closely the Q2 evolution of the
data In Fig 5 upper panel we show the polarizability ∆IA(Q2) whose Q2 evolution is
clearly dominated by the ∆ exchange Similar to the case of γ0p(Q2) inclusion of the dipole
in gM and the Coulomb coupling gC is very important in order to describe the experimental
data The LO prediction on the other hand slightly overestimates the data cf Fig 4
upper panel
19
At the real-photon point IA(0) = minusκ2
4and ∆IA(0) = 0 Therefore we give only the
slope of the polarizability ∆IA(Q2) [showing also the separate contributions from πN loops
∆ exchange and π∆ loops] in units of GeVminus2
d∆IAp(Q2)
dQ2
∣∣∣∣Q2=0
= minus858(343) asymp 238minus 1121 + 025 (30a)
d∆IAn(Q2)
dQ2
∣∣∣∣Q2=0
= minus955(343) asymp 141minus 1121 + 025 (30b)
Including the empirical Pauli form factor [73] we find in units of GeVminus2
dIAp(Q2)
dQ2
∣∣∣∣Q2=0
= minus318dIAn(Q2)
dQ2
∣∣∣∣Q2=0
= minus300 (31)
D Γ1(Q2) and I1(Q2) mdash the first moment of the structure function g1(xQ2)
The second variant for a generalization of the GDH sum rule to finite Q2 is defined as
minus α
2M2N
I1(Q2) = minus 1
8π2
int infinν0
dν1radic
ν2 +Q2
[σTT (νQ2) +
Q
νσLT (νQ2)
](32)
= minus α
Q2
int x0
0
dx g1(xQ2)
where I1(0) = minusκ2
4 This generalized GDH integral directly stems from the amplitude
Snonpole1 (νQ2) with the LEX from Eq (12a) It is given by the first moment of the struc-
ture function g1(xQ2) Γ1(Q2) =int x0
0dx g1(xQ2) as follows I1(Q2) =
2M2N
Q2 Γ1(Q2) The
isovector combination
Γ1(pminusn)(Q2) =
int x0
0
dx[g1p(xQ
2)minus g1n(xQ2)] (33)
is related to the axial coupling of the nucleon through the Bjorken sum rule [74 75]
limQ2rarrinfin
Γ1(pminusn)(Q2) =
gA6 (34)
As explained in Eq (28) the moment I1(Q2) splits into a polarizability part ∆I1(Q2) and
a Born part IBorn1 (Q2) Figure 4 lower panel shows the Q2 dependence of I1(Q2) which in
contrast to IA(Q2) shown in Figure 4 upper panel is clearly dominated by its Born part
and the elastic Pauli form factor The πN -loop ∆-exchange and π∆-loop contributions to
the polarizability ∆I1(Q2) are shown in Fig 5 lower panel Comparing to Fig 5 upper
20
000 005 010 015 020 025 030
-006
-004
-002
000
002
004
Q2 GeV2D
G1
proton
000 005 010 015 020 025 030
-006
-005
-004
-003
-002
-001
000
001
Q2 GeV2D
G1
neutron
FIG 6 First moment of the structure function g1(xQ2) for the proton (left) and neutron (right)
as function of Q2 The legend is the same as in Fig 4
panel one sees that ∆I1(Q2) is less sensitive to gC and the dipole form factor in gM than
∆IA(Q2)
For the proton our NLO BχPT prediction gives a very good description of the ex-
perimental data [18 60] and is in reasonable agreement with the MAID prediction [69]
For the neutron one observes good agreement with the empirical evaluations including ex-
trapolations to unmeasured energy regions starting from Q2 gt 01 GeV2 [27 61] In the
region of Q2 lt 005 GeV2 one observes an interesting tension between the recent E97-
110 experiment [27] and the data from CLAS [61] While the newest measurement finds
I1n(0035 GeV2) lt κ2n4 thus suggesting a negative slope at low Q the older measurement
found a rather large value for I1n(00496 GeV2) A similar but milder behaviour is seen in
the E97-110 [27] and E94-010 [21] data for IAn The MAID predictions do not agree with
the CODATA recommended values for the anomalous magnetic moments of the proton and
neutron [70] which in our work are imposed by using empirical parametrizations for the
elastic Pauli form factors [73] The slope of the HB result from Ref [51] is too large and
therefore only reproduces the data at very low Q2
Figure 6 shows the moment Γ1(Q2) for the proton and neutron while Fig 7 shows the
isovector combination Γ1 pminusn(Q2) The LO and NLO BχPT predictions are identical be-
cause our calculation produces the same Delta contributions for the proton and the neutron
For the isovector combination the MAID model only agrees with the data at very low
Q2 lt 010 GeV2 The same is true for the IR result [58 76] while all other chiral results
describe the data NLO BχPT (this work) BχPT+∆ [31] and HBχPT [51]
At the real-photon point I1(0) = minusκ2
4and ∆I1(0) = 0 Therefore we give only the
slope of the polarizability ∆I1(Q2) [showing also the separate contributions from πN loops
21
000 005 010 015 020 025 030-002
000
002
004
006
008
010
Q2 GeV2D
G1Hp-
nL
FIG 7 Isovector combination of Γ1(Q2) as function of Q2 The legend is the same as in Fig 6
The pink curve is the IR result from Ref [58 76] The experimental points are from Ref [77]
(brown dots) and Ref [23] (orange squares)
∆ exchange and π∆ loops] in units of GeVminus2
d∆I1p(Q2)
dQ2
∣∣∣∣Q2=0
= 039(4) asymp 034minus 053 + 058 (35a)
d∆I1n(Q2)
dQ2
∣∣∣∣Q2=0
= minus101(10) asymp minus107minus 053 + 058 (35b)
Including the empirical Pauli form factor [73] we find in units of GeVminus2
dI1p(Q2)
dQ2
∣∣∣∣Q2=0
= 580dI1n(Q2)
dQ2
∣∣∣∣Q2=0
= 553 (36)
E d2(Q2) mdash a measure of color polarizability
Another interesting moment to consider is d2(Q2) which is related to the twist-3 part of
the spin structure function g2(xQ2) [79 80]
d2(Q2) equiv 3
int 1
0
dx x2[g2(xQ2)minus gWW2 (xQ2)] (37)
where gWW2 (xQ2) is the twist-2 part of g2(xQ2) Using the Wandzura-Wilczek relation [81]
one can relate d2(Q2) to moments of the spin structure functions g1(xQ2) and g2(xQ2)
d2(Q2) =
int 1
0
dx x2 [3g2(xQ2) + 2g1(xQ2)] (38)
22
000 005 010 015 020 025 030
0000
0001
0002
0003
0004
0005
0006
0007
Q2 GeV2D
d2
proton
000 005 010 015 020 025 030
0000
0002
0004
0006
0008
0010
0012
Q2 GeV2D
d2
neutron
000 005 010 015 020 025 030
-1
0
1
2
Q2 GeV2D
Γ 010
-4
fm6
D
proton
000 005 010 015 020 025 030-2
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
6D
neutron
FIG 8 Upper panel The inelastic moment d2(Q2) for the proton (left) and neutron (right) as
function of Q2 The result of this work the NLO BχPT prediction is shown by the blue solid line
and the blue band The red line represents the LO BχPT result The purple short-dashed line is
the O(p4) HB result from Ref [50 51] The black dotted line is the MAID model prediction [69]
The experimental points for the neutron (cyan dots) are from Ref [22] Lower panel Fifth-order
generalized forward spin polarizability γ0(Q2) for the proton (left) and neutron (right) as function
of Q2 The experimental points for the proton are from Ref [59] (purple square) and Ref [78]
(orange dot)
This relation however only holds for asymptotically large Q2 It is also in the high-Q2
region where d2(Q2) is a measure of color polarizability [82 83] through its relation to the
gluon field strength tensor [80] We refer to Ref [84] for a recent review on the spin structure
of the nucleon including a discussion of sum rules for deep inelastic scattering and color
polarizabilities
What we consider in the following is the inelastic part of d2(Q2) defined as the moment
of g1(xQ2) and g2(xQ2) spin structure functions cf Eq (38)
d2(Q2) =
int x0
0
dx x2 [3g2(xQ2) + 2g1(xQ2)] (39)
23
This moment provides another testing ground for our BχPT predictions through comparison
with experiments on the neutron [22] Going towards the low-Q2 region the interpretation
of d2(Q2) in terms of color polarizabilities will fade out The above definition however
implies it is related to other VVCS polarizabilities
d2(Q2) =Q4
8M4N
[M2
NQ2
αδLT (Q2) + I1(Q2)minus IA(Q2)
] (40)
Note that d2(Q2) and its first two derivatives with respect to Q2 vanish at Q2 = 0 The
considerations in Eqs (28) and (29) have no effect on d2(Q2) since the Born contribution
from IA(Q2) and I1(Q2) cancel out Therefore d2(Q2) is a pure polarizability
In Fig 8 upper panel we show our NLO BχPT prediction and other results for d2(Q2)
While MAID [69] and BχPT describe the experimental data for the neutron [22] very well
the HB limit [50 51] is showing a fast growth with Q2 This illustrates the importance
of keeping the relativistic result Note also that even though the πN -loop contribution
is dominant both gC and the form factor in gM are essential to obtain a curvature that
reproduces the data cf Fig 9 upper panel For the proton there are to our knowledge
no experimental results to compare with However the agreement between the NLO BχPT
prediction and the MAID prediction at low energies is reasonable
F γ0(Q2) mdash fifth-order generalized forward spin polarizability
It is interesting to compare the generalized fifth-order forward spin polarizability sum
rule
γ0(Q2) =1
2π2
int infinν0
dν
radic1 +
Q2
ν2
σTT (νQ2)
ν5(41)
=64αM4
N
Q10
int x0
0
dx x4
[g1(xQ2)minus 4M2
Nx2
Q2g2(xQ2)
]
to the sum rule integrals for IA(Q2) and γ0(Q2) since they differ merely by their energy
weighting of σTT (νQ2) and a constant prefactor cf Eqs (19) (26) and (41) From IA(Q2)
to γ0(Q2) to γ0(Q2) the energy suppression is increasing by a factor of νminus2 respectively
Therefore the description of γ0(Q2) should be easiest in a low-energy effective-field theory
such as χPT whereas γ0(Q2) and IA(Q2) receive larger contributions from higher energies
In Fig 8 lower panel we show our LO and NLO BχPT predictions for γ0(Q2) One
can see that the πN -loop contribution is positive (in accordance to what we see for the cross
section σTT see Fig 10) The Delta shifts it substantially especially in the low Q2 region
24
000 005 010 015 020 025 030
-0004
-0002
0000
0002
0004
0006
0008
0010
Q2GeV
2D
d2
proton
000 005 010 015 020 025 030
-0005
0000
0005
0010
0015
Q2GeV
2D
d2
neutron
000 005 010 015 020 025 030
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
6D
proton
000 005 010 015 020 025 030
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
6D
neutron
FIG 9 Contributions of the different orders to the chiral predictions of d2(Q2) upper panel and
γ0(Q2) lower panel for the proton (left) and neutron (right) Red solid line πN -loop contribu-
tion green dot-dashed line ∆-exchange contribution orange dotted line π∆-loop contribution
blue long-dashed line total result purple dot-dot-dashed line total result without gC contribution
black short-dashed line total result without gM dipole
bringing it into a better agreement with data In general the BχPT curves start above the
empirical data points at the real-photon point and then decrease asymptotically to zero
above Q2 gt 01 GeV2 On the other hand the MAID prediction reproduces the empirical
data at the real-photon point then decreases to negative values until about Q2 gt 006 GeV2
from where it also starts to asymptotically approach zero Consequently our NLO BχPT
prediction of γ0(Q2) is consistently above the MAID prediction This is very different to
what we saw for IA(Q2) in Fig 4 upper panel where the MAID prediction at the real-
photon point is above the experimental value While the agreement of our predictions with
the empirical data is in general quite good for all moments of σTT (νQ2) one should point
out that both for γ0n(Q2) and γ0p(Q2) we overestimate the data at low Q2 For IA(Q2) such
observation cannot be made because ∆IA(0) = 0 and thus IA(0) is given by the empirical
Pauli form factor only From IA(Q2) γ0(Q2) and γ0(Q2) the latter has the smallest however
25
non-negligible dependence on gC and the dipole in gM cf Fig 9 lower panel
The πN -loop ∆-exchange and π∆-loop contributions to the NLO BχPT prediction of
the static fifth-order forward spin polarizability amount to in units of 10minus4 fm6
γ0p = 112(30) asymp 208minus 096minus 001 (42a)
γ0n = 195(30) asymp 292minus 096minus 001 (42b)
while the slope is composed as follows in units of 10minus4 fm8
dγ0p(Q2)
dQ2
∣∣∣∣Q2=0
= minus084(10) asymp minus100 + 016 + 000 (43a)
dγ0n(Q2)
dQ2
∣∣∣∣Q2=0
= minus142(15) asymp minus158 + 016 + 000 (43b)
Note that the HB prediction of the static polarizability γ0p (423 at O(p3) and 365 at O(ε3)
[78 85]) is almost one order of magnitude larger than the empirical value and therefore not
shown in Fig 8
G Summary
Our results are summarized in Table II where we give the contributions of the different
orders to the chiral predictions of the polarizabilities and their slopes at the real-photon
point A quantitative comparison of our predictions for the static spin polarizabilities to
the work of Bernard et al [31] and different empirical evaluations is shown in Table III We
can see that the inclusion of the Delta turns out to be very important for all moments of
the helicity-difference cross section To describe the Q2 behavior of the polarizabilities the
magnetic coupling of the N rarr ∆ transition should be modified by a dipole form factor as
has been observed previously in the description of electroproduction data [33] This dipole
form factor effectively takes account of vector-meson exchanges The Coulomb-quadrupole
N rarr ∆ transition despite its subleading order is important in the description of some
moments of spin structure functions This is contrary to what we saw for the moments of
unpolarized structure functions [30] where the Coulomb coupling had a negligible effect
The π∆ loops are mainly relevant for the generalized GDH integrals
26
TABLE II The NLO BχPT predictions for the forward VVCS polarizabilities and their slopes at
Q2 = 0 The contributions of the πN loops the ∆ exchange and the π∆ loops are shown together
with the combined total result Note that IA(0) = I1(0) = d2(0) = 0 and (d2)prime = 0
πN loops ∆ exchange π∆ loops Total
γ0 p 201 minus284 minus010minus093(92)
(10minus4 fm4) n 298 003(92)
δLT p 150 minus016 minus002132(15)
(10minus4 fm4) n 235 218(23)
γ0 p 208 minus096 minus001112(30)
(10minus4 fm6) n 292 195(30)
(γ0)prime p minus033011 001
minus022(4)
(10minus4 fm6) n minus073 minus061(7)
(δLT )prime p minus080 minus004 minus001minus085(8)
(10minus4 fm6) n minus119 minus124(12)
(γ0)prime p minus100016 000
minus084(10)
(10minus4 fm8) n minus158 minus142(15)
(∆IA)prime p 238 minus1121 025minus858(343)
(GeVminus2) n 141 minus955(343)
(∆I1)prime p 034 minus053 058039(4)
(GeVminus2) n minus107 minus101(10)
TABLE III Our NLO BχPT predictions for the spin polarizabilities at Q2 = 0 compared with
the BχPT+∆ predictions from Bernard et al [31] and the available empirical information Where
the reference is not given the empirical number is provided by the MAID analysis [55 69] with
unspecified uncertainty
Proton Neutron
This work BχPT+∆ Empirical This work BχPT+∆ Empirical
γ0 minus093(92) minus174(40) minus100(8)(12) [19] 003(92) minus077(40) minus0005
(10minus4 fm4) minus090(8)(11) [78] [MAID]
minus0929(105) [59]
δLT 132(15) 240(1) 134 218(23) 238(3) 203
(10minus4 fm4) [MAID] [MAID]
γ0 112(30) 060(7)(7) [78] 195(30) 123
(10minus4 fm6) 0484(82) [59] [MAID]
IV CONCLUSIONS
We have presented a complete NLO calculation of the polarized non-Born VVCS ampli-
tudes in covariant BχPT with pion nucleon and ∆(1232) fields The dispersion relations
between the VVCS amplitudes and the tree-level photoabsorption cross sections served as a
27
cross-check of these calculations
The obtained moments of the proton and neutron spin structure functions related to gen-
eralized polarizabilities and GDH-type integrals agree well with the available experimental
data The description of their Q2 evolution is improved compared to the previous χPT pre-
dictions In particular the NLO BχPT predictions obtained here give a better description
of the empirical data (eg from the Jefferson Laboratory ldquoSpin Physics Programrdquo) than
the HB [50 51] and IR [58] calculations
The demonstrated predictive power of the χPT framework amplitudes makes it well
suited for extending the χPT evaluation of the TPE effect in the hyperfine structure of
(muonic-)hydrogen [15ndash17] to next-to-leading order
ACKNOWLEDGEMENTS
We thank Lothar Tiator and Marc Vanderhaeghen for helpful discussions This work
is supported by the Deutsche Forschungsgemeinschaft (DFG) through the Collaborative
Research Center [The Low-Energy Frontier of the Standard Model (SFB 1044)] JMA ac-
knowledges support from the Community of Madrid through the ldquoPrograma de atraccion de
talento investigador 2017 (Modalidad 1)rdquo and the Spanish MECD grants FPA2016-77313-P
FH gratefully acknowledges financial support from the Swiss National Science Foundation
Appendix A Tensor decompositions of the VVCS amplitudes
In this appendix we review the decomposition of the forward VVCS process into tensor struc-
tures and scalar amplitudes In particular we consider the connection between the covariant and
the semi-relativistic decomposition in the lab frame that is defined in terms of the conventional
transverse longitudinal transverse-transverse and transverse-longitudinal amplitudes
As explained in Sec II A the process of forward VVCS off the nucleon can be described in
terms of four explicitly covariant amplitudes S1 2 and T1 2 [3]
T (νQ2) =
(minusgmicroν +
qmicroqν
q2
)T1(νQ2) +
1
M2N
(pmicro minus p middot q
q2qmicro)(
pν minus p middot qq2
qν)T2(νQ2) (A1)
minus 1
MNγmicroναqα S1(νQ2)minus 1
M2N
(γmicroνq2 + qmicroγναqα minus qνγmicroαqα
)S2(νQ2)
εprimelowastmicro εν
where εmicro (εprimelowastmicro ) are the incoming (outgoing) photon polarization vectors ν is the photon lab-frame
energy and Q2 is the photon virtuality Alternatively the decomposition in the laboratory frame
28
(which in the forward case coincides with the Breit frame) is parametrized in terms of the nucleon
Pauli matrices ~σ and the four scalar functions fL fT gTT and gLT
T (νQ2) = ε0 εprimelowast0 fL(νQ2) + (~ε primelowast middot ~ε ) fT (νQ2) + i~σ middot (~ε primelowast times ~ε ) gTT (νQ2) (A2)
minusi~σ middot [(ε0~εprimelowast minus ~ε ε primelowast0 )times q] gLT (νQ2)
Here ~q and q = ~q|~q | are the photon three-momentum in the lab system and its unit vector The
modified polarization vector components are given by
ε0 =
[ε0 minus
ν
|~q | (~ε middot q )
] |~q |Q ~ε = ~εminus q (~ε middot q ) (A3)
where ε = (ε0~ε ) is the usual incoming photon polarization vector and εprimelowast the outgoing polarization
vector The LEX of the lab frame amplitudes [Eq (10)] can serve in particular as the definition
of the generalized polarizabilities The lab frame amplitudes are also conveniently used for the
definition of the response functions see the example of the scalar amplitude gLT (νQ2) and the
corresponding response function σLT (νQ2) below in App B
Appendix B Photoabsorption cross sections
In the forward kinematics the spin-dependent VVCS amplitudes and the spin polarizabilities
can be described in terms of the polarized structure functions g1(xQ2) and g2(xQ2) or equiv-
alently the helicity-difference cross section σTT (νQ2) and the longitudinal-transverse response
function σLT (νQ2) with the help of dispersion relations (5) and the optical theorem (3) In this
way the photoabsorption cross sections measured in electroproduction processes form the basis
for most empirical evaluations shown throughout Sec III In the following we present the BχPT
predictions for the tree-level cross sections of πN - π∆- and ∆-production through photoabsorption
on the nucleon cf Figs 8 9 and 10 in Ref [30] In Secs B 1 and B 2 we will discuss the leading
πN -production channel and the ∆-production channel respectively We used these cross sections
to verify the polarizability predictions obtained otherwise from the calculated non-Born VVCS
amplitudes Due to the bad high-energy behavior of the π∆-production cross sections in BχPT
cf Fig 10 the dispersion relations in Eq (5) require further subtractions for a reconstruction of
the π∆-loop contribution to the spin-dependent VVCS amplitudes Therefore not all polarizabil-
ities could be verified but only those appearing as higher-order terms in the LEX of the VVCS
amplitudes such as γ0 [16]
29
proton neutron
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Neutron
02 04 06 08 100
50
100
150
200
250
[GeV]
T [b]
02 04 06 08 10
-50
0
50
100
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Neutron
02 04 06 08 100
50
100
150
200
250
[GeV]
T [b]
02 04 06 08 10
-50
0
50
100
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02040608100
50
100
150
200
250
300
[GeV]
T[b]
0204060810
-50
0
50
[GeV]
TT[b]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
0204060810
00
02
04
06
[GeV]
LTQ
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
0204060810
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
FIG 10 Photoabsorption cross sections for πN (red) and π∆ production (orange) with Q2 = 0
(solid) and Q2 = 01 GeV2 (dashed for πN and dotted for π∆ channel)
1 πN-production channel
In order to extract the response function σLT (νQ2) we have developed a method similar to
the one used to calculate σTT (νQ2) see for example Ref [86] For σLT (νQ2) however the
calculation is more complicated because one has to take into account that the associated Compton
process involves a spin-flip of the nucleon as illustrated in Fig 11 When calculating the cross
section the product of the incoming nucleon spinors has to reflect this flip
The forward VVCS amplitude related to σLT (νQ2) mdash and δLT (Q2) mdash is gLT (νQ2) It can
be extracted from Eq (A2) if one takes the modified polarization vector components in Eq (A3)
with ε = εL and εprimelowast = εlowastplusmn as input where εL = 1Q(|~q | 0 0 q0) and εplusmn = ∓ 1radic
2(0 1plusmni 0) are the
standard longitudinal and transverse polarization vectors respectively For εL and εlowastplusmn only the
choice of helicities hprime = plusmn12 and h = ∓12 gives a non-zero contribution and one obtains
χdaggerhprime T (νQ2)χh = χdaggerhprimeminusi~σ middot [(ε0~εprimelowast minus ~ε ε primelowast0 )times q] gLT (νQ2)χh =
radic2 gLT (νQ2) (B1)
where χh and χdaggerhprime are two-component Pauli spinors with opposite helicities or here spins
Let us now consider the related photoabsorption process and in particular the tree-level γlowastN rarr
30
s = 12
sprime = minus12
λ = 0 λprime = 1
rprime rprime
s=12sprime=minus1
2
λ=0λprime=1
rprimerprime
s=12
sprime=minus12
λ=0λprime=1
rprimerprime hacute=frac12h=-frac12
λacute=1λ=0
FIG 11 Relation between the forward Compton process and the photoabsorption process given
by the optical theorem In particular we show the longitudinal-transverse contribution The
double-line arrows represent the spin of the external particles while the dot represents the scalar
(longitudinal) polarization of the incoming photon Inside the blob the intermediate states are
represented eg nucleons with spins rprime (which are averaged in the calculation of the cross section)
and pions
πN channel see diagrams in Fig 8 of Ref [30] We define the πN -production amplitude as
T = uhB (PB)sumi
Ai(s t)Γi uhA(PA) (B2)
with the Dirac structures
Γ1 = γ5 (B3a)
Γ2 =1
2
[qA ε
]γ5 (B3b)
where uhA(PA) and udaggerhB (PB) are the Dirac spinors and PA and PB are the four-momenta of
the incoming and outgoing nucleons respectively When calculating the photoabsorption cross
section related to the VVCS amplitude in Eq (B1) the nucleon spin flip should be implemented
by uhprime(PA) in T dagger and uh(PA) in T together with the appropriate transverse and longitudinal
photon polarization vectors εlowastplusmn and εL
However if one wants to use the properties of the Dirac matrices it is more useful to construct
an operator to produce this spin flip in the external nucleons of Fig 11 This is accomplished by
introducing the projector ΓLT equiv 12radic
2(γ1 + iγ2)γ5 which also takes into account the extra factor
radic2 in Eq (B1) We checked that with this projector one correctly extracts δLT by comparing the
HB limit of our result to the HB result of Ref [50] where the authors calculate this polarizability
from the Compton amplitude directly With all those ingredients the longitudinal-transverse cross
section is calculated in the following way
σLT (νQ2) =1
64π2 s
|~pf |cm
|~pi|cm
int 1
minus1dcos θ
sumij
AiAdaggerjXij (B4)
31
with
Xij = Tr[(PB +MN )Γi(PA +MN )ΓLTγ0Γdaggerjγ
0] (B5)
where θ is the scattering angle in the center-of-mass (cm) frame and |~pi|cm (|~pf |cm) is the three-
momentum of an incoming (outgoing) particle in the cm frame An explicit calculation of the
matrix Xij leads to
X = MNQ
0 2(PB minus PA) middot εLminusradic
2|~qf |cm sin θ (sminus u)
(B6)
where |~qi|cm (|~qf |cm) is the relative three-momentum of the incoming (outgoing) particles in the
cm frame Here s t and u are the usual Mandelstam variables For the different γlowastN rarr πN
channels we obtain the following amplitudes Ai where we introduce qA as the four-momentum of
the incoming photon and qB as the four-momentum of the outgoing pion
bull γlowastprarr π0p
A1 =e gAMN
fπ
[2PA middot ε+ qA middot ε
sminusM2N
+2PB middot εminus qA middot ε
uminusM2N
] (B7a)
A2 =e gAMN
fπ
[1
sminusM2N
+1
uminusM2N
] (B7b)
bull γlowastprarr π+n
A1 =
radic2 e gAMN
fπ
[2PA middot ε+ qA middot ε
sminusM2N
+2(PA minus PB) middot ε+ qA middot ε
tminusm2π
] (B8a)
A2 =
radic2 e gAMN
fπ(sminusM2N )
(B8b)
bull γlowastnrarr π0n
A1 = 0 (B9a)
A2 = 0 (B9b)
bull γlowastnrarr πminusp
A1 =
radic2 e gAMN
fπ
[2PB middot εminus qA middot ε
uminusM2N
minus 2(PA minus PB) middot ε+ qA middot εtminusm2
π
] (B10a)
A2 =
radic2 e gAMN
fπ(uminusM2N )
(B10b)
The analytical expressions shown above were checked with the amplitudes given in Ref [87] An-
alytical expressions for the tree-level γlowastN rarr πN channel of the σLT (νQ2) and σTT (νQ2) cross
32
sections are given below (proton channels π+n and π0p neutron channel πminusp) We checked that
they reproduce the known results in the real-photon limit [45 86] To shorten the final expressions
for the cross sections which are considerably longer for finite Q2 than in the real-photon limit we
define the following dimensionless kinematic variables
αγ = (ENi )cmradics =
s+M2N +Q2
2s (B11)
απ = (ENf )cmradics =
s+M2N minusm2
π
2s (B12)
βγ = Eγcmradics =
sminusM2N minusQ2
2s (B13)
βπ = Eπcmradics =
sminusM2N +m2
π
2s (B14)
λγ = |~qi|cmradics =
radic(sminusM2
N minusQ2)2 + 4sQ2
2s (B15)
λπ = |~qf |cmradics =
radic(sminusM2
N +m2π)2 minus 4sm2
π
2s (B16)
Here (ENi )cm and (ENf )cm are the energies of the incoming and outgoing nucleon Eγcm is the energy
of the incoming photon Eπcm is the energy of the outgoing pion all in the cm frame
σ(π+n)TT = minus e2g2
AM2N
64πf2πs
3(sminusM2N )2λ4
γ
4sλπλγ
[(M2
N minus s)(M2N minusQ2 minus s)(Q2 + 2sβγβπ)
+ 2s(m2πQ
2 minus (M2N minus s)
(M2N + s(minus1 + 2βγβπ)
))λ2γ
]+ (M2
N minus s)(M2N minusQ2 minus s)
times (Q2 + 2sβγβπ minus 2sλπλγ)(Q2 + 2s(βγβπ + λπλγ)
)times log
(Q2 + 2sβγβπ minus 2sλγλπQ2 + 2sβγβπ + 2sλγλπ
) (B17)
33
σ(π0p)TT =
e2g2AM
2Nλπ
64πf2πs(sminusM2
N )2λγ
1
s2λπλ3γ
[(M2
N minus s)(M2N minusQ2 minus s)
(3M2
N + s(2βγβπ minus 3))(M2N
+ s(2βγβπ minus 1))
+ 2s
((2m2
πQ2 + (M2
N minus s)2)(M2
N minus s)minus 2(minusm2
πQ2 + (sminusM2
N )2)
times sβγβπ + 4s2(sminusM2N )β2
γβ2π
)λ2γ
]arctan
(2sλπλγ
M2N + s(2βγβπ minus 1)
)+ 2
[minus 2m2
πQ2
+ (M2N minus s)
(2(sminusM2
N ) + 4sβγβπ +(sminusM2
N +Q2)(3M2N minus 3s+ 2sβγβπ)
sλ2γ
+m2πQ
2(sminusM2N )
(M2N minus s+ 2sβγβπ)(M2
N + s(minus1 + 2βγβπ minus 2λπλγ))
+m2πQ
2(sminusM2N )
(M2N minus s+ 2sβγβπ)(M2
N + s(minus1 + 2βγβπ + 2λπλγ))
)
+1
λπλγ
(βγβπ
(minusm2
πQ2 + (M2
N minus s)(M2N minus s+ 2sβγβπ)
)minus (M2
N minus s)(M2N minusQ2 minus s)λ2
π
)log
(M2N + s(minus1 + 2βγβπ + 2λπλγ)
M2N + s(minus1 + 2βγβπ minus 2λπλγ)
)] (B18)
σ(πminusp)TT =
e2g2AM
2N
64πf2πs
3λ4γ(M2
N + s(2βπβγ minus 1))2 minus 4λ2πs
2λ2γ)
4sλπλγ
((minusM2
N +Q2 + s)(M2N
+ s(2βπβγ minus 1))
2 minus 2sλ2γ
(2λ2
πs(minusM2N +Q2 + s) +m2
πQ2))
+((M2N + s(2βπβγ minus 1)
)2
minus 4λ2πs
2λ2γ
)((M2N + s(2βπβγ minus 1)
)(M2N minus 2Q2 minus s(2βπβγ + 1)
)+ 4λ2
πs2λ2
γ
)times log
(minusM2
N minus 2βπsβγ minus 2λπsλγ + s)minus(M2N + s(2βπβγ minus 2λπλγ minus 1)
)(M2N + s(minus1
+ 2βπβγ + 2λπλγ))[((
M2N + s(2βπβγ minus 1)
)(M2N minus 2Q2 minus s(2βπβγ + 1)
)+ 4λ2
πs2λ2
γ
)times log
(minusM2
N minus 2βπsβγ + 2λπsλγ + s)
+((Q2 + 2βπsβγ)2 minus 4λ2
πs2λ2
γ
)times log
(Q2 + 2βπsβγ minus 2λπsλγQ2 + 2sβπβγ + 2λπsλγ
)]+ 4sλ2
γ
((M2
N + s(2βπβγ minus 1))2 minus 4λ2πs
2λ2γ
)times (M2
N minus s) arctanh
(2λπsλγ
M2N + 2βπsβγ minus s
) (B19)
34
σ(π+n)LT =
e2g2AM
3Nλπ
32πf2πQs
3(sminusM2N )2λ4
γ
2sλγ
[(M2
N minus s)(Q2 + 2sβ2γ)(Q2 + 2sβγβπ)
minus 4s
((M2
N minus s)(Q2 minus 2s(απ minus 1)βγ
)+Q2sβγβπ
)λ2γ + 8s3(minus1 + απ)λ4
γ
]minus M2
N minus sλπ
[(Q2 + 2sβ2
γ)(Q2 + 2sβγβπ)2 + 4s2(2(απ minus 1)βγ(Q2 + 2sβγβπ)
minusQ2λ2π
)λ2γ + 8s3(απ minus 1)2λ4
γ
]arctan
(2sλπλγ
Q2 + 2sβγβπ
) (B20)
σ(π0p)LT =
e2g2AM
3Nλπ
16πf2πQs(sminusM2
N )2λγ
1
minus2s(M2N + s(minus1 + 2βγβπ))2λ2
γ + 8s3λ2πλ
4γ
[minus 3M8
N (Q2
+ 2sβ2γ) + 2M4
Ns2
(minus (Q2 + 2sβ2
γ)(minus3 + 2βγβπ) +(Q2(minus3 + 2βγβπ + 6λ2
π) + 2sβγ
times (minus3 + 12απ + 2βγβπ minus 12απβγβπ + 4βγλ2π))λ2γ minus 4s(α2
π minus 1)λ4π
)+ 2M2
Ns3
[minus (Q2
+ 2sβ2γ)(minus1 + 2βγβπ)
(6 + βγβπ(minus9 + 2βγβπ)
)+
(Q2(3minus 12λ2
π + 4βγβπ(minus1 + 2βγβπ
+ λ2π))minus 2sβγ
(minus 3 + 4απ(3 + 2απ(3 + 2βγβπ(minus3 + βγβπ)) + 4βγ(βπ + (2minus βγβπ)
times λ2π)))λ2γ + 8s
((απ minus 1)(1 + απ minus 2βγβπ) + 2απβγλ
2π
)λ4γ
)+ s4
((Q2 + 2sβ2
γ)
times (1minus 2βγβπ)2(2βγβπ minus 3) + 2(Q2(minus1 + 6λ2
π + 2βγβπ((1minus 2βγβπ)2 minus 2λ2π)) + 2sβγ
times (minus1 + 2βγβπ + 4(minus1 + βγβπ)(minusαπ + 2απβγβπ minus βγλ2π)))λ2γ minus 8
(s(απ minus 1)(απ
+ (1minus 2βγβπ)2) + 2βγ(2sαπ +Q2βγ)λ2π
)λ4γ + 32s(απ minus 1)λ2
πλ6γ
)+ 2M6
Ns
(Q2(6
minus 7βγβπ + λ2γ) + 2sβγ(βγ(6minus 7βγβπ) + (1minus 4απ)λ2
γ)
)]+sminusM2
N
4s2λπλ3γ
[(Q2 + 2sβ2
γ)
times(3M2
N + s(2βγβπ minus 3))(M2N + s(2βγβπ minus 1)
)+ 2s
(minusM2
N (Q2 + 2sβγ minus 8sαπβγ)
+ s(2sβγ(1minus 4απ + 4απβγβπ) +Q2(1minus 2λ2π)))λ2γ + 8s3(απ minus 1)λ4
γ
]
times arctan
(2sλπλγ
M2N + s(2βγβπ minus 1)
) (B21)
35
σ(πminusp)LT =
e2g2AM
3N
64πf2πQs
3λ4γ(M2
N minusQ2 minus s)(M2N + s(2βπβγ + 2λπλγ minus 1))
times
1
M2N + s(2βπβγ minus 2λπλγ minus 1)
[4λπsλγ(M2
N minusQ2 minus s)(
2sλ2γ
(s((1minus 2λ2
π)Q2
+ 2(2απ minus 1)sβγ(2βπβγ minus 1))minusM2N (Q2 + 2(1minus 2απ)sβγ)
)+ (Q2 + 2sβ2
γ)(M2N + s(2βπβγ minus 1)
)2 + 8(απ minus 1)απs
3λ4γ
)]minus(M2N + s(2βπβγ + 2λπλγ minus 1)
)times[
log
(minusM2N minus 2βπsβγ minus 2λπsλγ + s
minusM2N minus 2βπsβγ + 2λπsλγ + s
)(minus (Q2 + 2sβ2
γ)(M2N + s(2βπβγ minus 1)
)(minusM2
N + 2Q2 + 2βπsβγ + s)minus 2sλ2γ
(minusM2
N (Q2 + 2sβγ) +Q4 +Q2s(4απβγ
minus 2βγ minus 2λ2π + 1) + 2s2βγ(4(απ minus 1)βπβγ + 1)
)minus 8(απ minus 1)2s3λ4
γ
)+ 2
(4s2λ2
γ
(2(απ minus 1)βγ(Q2 + 2βπsβγ)minus λ2
πQ2)
+ (Q2 + 2βπsβγ)2(Q2 + 2sβ2γ)
+ 8(απ minus 1)2s3λ4γ
)arctanh
(2λπsλγ
Q2 + 2βπsβγ
)] (B22)
2 ∆-production channel
The tree-level ∆-exchange diagram in Fig 2 of Ref [30] contributes to the non-Born part of
the VVCS amplitudes The contribution of the ∆ exchange to the VVCS amplitudes can be split
into [17]
S∆-exch1 (νQ2) = S∆-pole
1 (νQ2) + S∆-exch1 (νQ2) (B23a)
νS∆-exch2 (νQ2) = νS∆-pole
2 (νQ2) + νS2∆-exch
(νQ2) (B23b)
and similarly for the unpolarized VVCS amplitudes discussed in Ref [30] Here we introduced the
∆-pole contributions S∆-polei and the ∆-non-pole contributions S∆-exch
i The former amplitudes
feature a pole at the ∆(1232)-production threshold and thus are proportional to
1
[sminusM2∆][uminusM2
∆]=
1
4M2N
1
ν2∆ minus ν2
(B24)
36
They can be reconstructed from the dispersion relations in Eq (5) with the tree-level ∆-production
cross sections as input cf Fig 10 in Ref [30]
σTT (νQ2) =π2α
M2NM
2+|~q |
minus g2
MMN (M+ + ν)|~q |2 +g2E(∆minus ν)(Q2 minusMNν)2
MN(B25a)
+g2CQ
4s(∆minus ν)
MNM2∆
minus 4gMgE(Q2 minusMNν)|~q |2 minus 4gMgCQ2|~q |2
+2gEgCQ
2[minusMNM∆ |~q |2 + s(Q2 +∆ν)
]MNM∆
δ(ν minus ν∆)
σLT (νQ2) =Qπ2α
M2NM
2+|~q |
g2E(MNν minusQ2) [M∆(MN + ν)minus s]
MN(B25b)
+g2CQ
2[MNM∆|~q |2 minus s(Q2 +∆ν)
]MNM2
∆
+ gMgEM∆|~q |2 minusgMgC(Q2 minusMNν)|~q |2
M∆
+gEgC(ν minus∆)(M2
N |~q |2 minus 2Q2s)
MNM∆
δ(ν minus ν∆)
with ∆ = M∆ minusMN M+ = M∆ + MN and the Mandelstam variable s = M2N + 2MNν minus Q2
Analytical expressions for the spin structure functions g1(xQ2) and g2(xQ2) can be constructed
from Eq (3) with the flux factor K(νQ2) = |~q | =radicν2 +Q2
In the ∆-non-pole contributions to S1(νQ2) and νS2(νQ2) the pole in ν at the ∆(1232)-
production threshold has canceled out
S∆-exch1 (νQ2) =
πα
MNM2+
[g2MQ
2+ + g2
E
(∆2 minus 3Q2
)+
4g2CQ
4
M2∆
minus 8gMgEM∆ωminus (B26a)
minus2gMgCQ2(MN minus 4M∆)
M∆+
2gEgCQ2(3MN minus 2M∆)
M∆
]
νS2∆-exch
(νQ2) =2πα
MNM2+
[g2EM∆∆ωminus +
g2M MNQ
2+
2+g2C Q
2(Q2 minus∆2)
2M∆(B26b)
+gEgM M∆(M∆ω+ minus 4MNωminus)minus gEgC ∆(2Q2 +MNω+)
+gMgC Q2(4MN minus ω+)
]+S∆-exch
2 (νQ2)
ν
[M2
∆ ω2+
M2N
+ ν2
]
with Q+ =radic
(M∆ +MN )2 +Q2 and ωplusmn = (M2∆minusM2
N plusmnQ2)2M∆ and the non-pole contribution
to S2(νQ2)
S∆-exch2 (νQ2) = minus2παMNν
M∆M2+
[gM + gE
]gC (B27)
These amplitudes to the contrary are not described by the tree-level ∆-production cross sections
in the standard dispersive approach [17] This peculiarity has been previously missed e g in the
calculation of the ∆-exchange contribution to the hydrogen hyperfine splitting in Ref [88] The
importance of including the ∆-non-pole contribution is also evident when considering the BC sum
37
rule in Eq (14) The ∆-pole terms by themselves violate the BC sum rule but cancel exactly with
the ∆-non-pole terms
limνrarr0
νS∆-pole2 (νQ2) + lim
νrarr0νS2
∆-exch(νQ2) = 0 (B28)
Appendix C Polarizabilities at Q2 = 0
In this section we give analytical expressions for the static values and slopes at Q2 = 0 of
the spin polarizabilities and moments of polarized structure functions In particular we give the
HB expansion of the πN -loop contributions and the ∆-exchange contributions The complete
expressions also for the π∆-loop contributions can be found in the Supplemented material Recall
that IA(0) = I1(0) = d2(0) = 0 and dd2(Q2)dQ2
∣∣∣Q2=0
= 0
1 πN-loop contribution
Here we give analytical expressions for the πN -loop contributions to the proton and neutron
spin polarizabilities expanded in powers of micro = mπMN viz the HB expansion Note that we
choose to expand here to a high order in micro the strict HB expansion would only retain the leading
term in an analogous NLO calculation
bull Static polarizabilities (Q2 = 0)
γ0p =e2g2
A
96π3f2πm
2π
1minus 21πmicro
8minus(
59
2+ 26 logmicro
)micro2 +
1875πmicro3
64
+3
(3
2+ 26 logmicro
)micro4 +
(C1)
γ0n =e2g2
A
48π3f2πm
2π
1
2minus 9πmicro
16minus 2micro2 logmicro+
75πmicro3
128minus 3micro4
4+
(C2)
δLTp =e2g2
A
192π3f2πm
2π
1minus 9πmicro
8+
(13
2minus 2 logmicro
)micro2 minus 465πmicro3
64
minus(
47
2+ 42 logmicro
)micro4 +
(C3)
δLTn =e2g2
A
96π3f2πm
2π
1
2+
3πmicro
16+ (1 + 2 log micro)micro2 minus 105πmicro3
128+
5micro4
4+
(C4)
38
δLTn =e2g2
A
96π3f2πm
2π
1
2+
3πmicro
16+ (1 + 2 log micro)micro2 minus 105πmicro3
128+
5micro4
4+
(C5)
γ0p =e2g2
A
16π3f2πm
4π
4
45minus 3πmicro
16+
14micro2
5minus 1813πmicro3
384minus 192
5(1 + log micro)micro4
+80703πmicro5
2048+
(C6)
γ0n =e2g2
A
16π3f2πm
4π
4
45minus 5πmicro
48+
4micro2
5minus 245πmicro3
384minus 32micro4 logmicro
15
+1323πmicro5
2048+
(C7)
bull Slopes of polarizabilities at Q2 = 0
dγ0p(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
2minus 45πmicro
4+ 223micro2 minus 28515πmicro3
64
minus9
(1953
4+ 449 logmicro
)micro4 +
570255πmicro5
128+
(C8)
dγ0n(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
2minus 81πmicro
8+ 94micro2 minus 2535πmicro3
32
minus3 (1 + 90 logmicro)micro4 +84315πmicro5
1024+
(C9)
dδLTp(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
2880π3f2πm
4π
minus5
2minus 27πmicro
32+ 20micro2 minus 5865πmicro3
256
+3
(617
4+ 36 logmicro
)micro4 minus 2056845πmicro5
4096+
(C10)
dδLTn(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
minus5
4minus 81πmicro
64minus 11micro2 +
10005πmicro3
512
+15
8(11 + 48 logmicro)micro4 minus 267015πmicro5
8192+
(C11)
dIAp(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
96π2f2πmicro
2
1minus 15πmicro
4minus 1
2(115 + 88 logmicro)micro2 +
1839πmicro3
32
+5 (5 + 34 logmicro)micro4 +
(C12)
dIAn(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
48π2f2πmicro
2
1
2minus 11πmicro
8minus 1
4(1 + 20 logmicro)micro2 +
99πmicro3
64
minus25micro4
12+
(C13)
dI1p(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
96π2f2πmicro
3π
8+ 2 (4 + 3 logmicro)microminus 537πmicro2
64
minus1
2(15 + 56 logmicro)micro3 +
(C14)
39
dI1n(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
48π2f2πmicro
minus π
16+
1
4(3 + 4 logmicro)microminus 57πmicro2
128+
2micro3
3+
(C15)
γ0p(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2g2
A
16π3f2πm
6π
1
105minus 23πmicro
256+
377micro2
210minus 15551πmicro3
6144+
3371micro4
105
minus1640457πmicro5
32768+
(C16)
γ0n(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2g2
A
16π3f2πm
6π
1
105minus 153πmicro
1792+
69micro2
70minus 4615πmicro3
6144+
172micro4
35
minus120897πmicro5
32768+
(C17)
2 ∆-exchange contribution
Here we give analytical expressions for the tree-level ∆-exchange contributions to the nucleon
spin polarizabilities and their slopes at Q2 = 0 Note that the ∆-exchange contributes equally to
proton and neutron polarizabilities Recall that for the magnetic γlowastN∆ coupling we introduced a
dipole form factor to mimic vector-meson dominance gM rarr gM(1 +Q2Λ2)2
bull Static polarizabilities (Q2 = 0)
γ0 = minus e2
4πM2+
(g2M
∆2+
g2E
M2+
minus 4gMgEM+∆
) (C18)
δLT =e2M∆
4πM3+
(g2E
MNM++gMgE∆MN
minus gEgCM2
∆
) (C19)
γ0 =e2M2
N
π∆2M4+
(minusg
2M
∆2+
g2E
M2+
+4gMgE∆M+
) (C20)
bull Slopes of polarizabilities at Q2 = 0
dγ0(Q2)
dQ2
∣∣∣∣∣Q2=0
= minus e2
πM2+∆
(g2M
∆
[1
4∆2minus 1
∆M++
1
2M2+
]minus 1
Λ2
g2M
∆+
g2E
2M2+
[1
2∆minus 3
M+
]minusgMgE
M+
[1
∆2minus 5
∆M++
1
M2+
]+
1
Λ2
2gMgEM+
+2gMgC∆M2
+
minus gEgCM3
+
) (C21)
40
d δLT (Q2)
dQ2
∣∣∣∣∣Q2=0
=e2M∆∆
4πMNM2+
(g2E
∆2M2+
[1
∆minus 4
M+
]minus g2
C
∆M2∆M
2+
+gMgE∆2M+
[1
∆2
minus 3
∆M++
1
M2+
]minus 2
Λ2
gMgE∆2M+
+gMgC∆M2
∆
[1
2∆2minus 2
∆M++
1
2M2+
]minus gEgC
2M2∆M
2+
[7
∆+
1
M+
]) (C22)
dIA(Q2)
dQ2
∣∣∣∣∣Q2=0
= minusM2N
M2+
(g2M
2∆2+
g2E
MNM+minus 2gMgE
∆M+minus gEgCM∆M+
) (C23)
dI1(Q2)
dQ2
∣∣∣∣∣Q2=0
= minusM∆M2N
2M3+
(g2E
MNM∆minus gMgE∆MN
minus gEgCM2
∆
) (C24)
γ0(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2M2
N
π∆3M6+
(g2M
∆
[3M2
∆ + 2M∆MN minus 9M2N
∆2+
4M2+
Λ2
]
minus4gMgE
[5M2
∆ minus 9M2N
M+∆2+
2M+
Λ2
]+
g2E
∆M+[7M∆ minus 9MN ]
minus8gMgC∆
+4gEgCM+
) (C25)
[1] D Drechsel B Pasquini and M Vanderhaeghen Dispersion relations in real and virtual
Compton scattering Phys Rept 378 99 (2003) hep-ph0212124
[2] S E Kuhn J-P Chen and E Leader Spin structure of the nucleon mdash status and recent
results Prog Part Nucl Phys 63 1 (2009) arXiv08123535 [hep-ph]
[3] F Hagelstein R Miskimen and V Pascalutsa Nucleon polarizabilities from Compton scat-
tering to hydrogen atom Prog Part Nucl Phys 88 29 (2016) arXiv151203765 [nucl-th]
[4] B Pasquini and M Vanderhaeghen Dispersion theory in electromagnetic interactions Ann
Rev Nucl Part Sci 68 75 (2018) arXiv180510482 [hep-ph]
[5] S Gerasimov A Sum rule for magnetic moments and the damping of the nucleon magnetic
moment in nuclei Sov J Nucl Phys 2 430 (1966)
[6] S Drell and A C Hearn Exact sum rule for nucleon magnetic moments Phys Rev Lett
16 908 (1966)
[7] J S Schwinger Source theory viewpoints in deep inelastic scattering Proc Natl Acad Sci
USA 72 1 (1975)
[8] J S Schwinger Source theory viewpoints in deep inelastic scattering Electromagnetic In-
teractions and Field Theory Proceedings 14 Internationale Universitatswochen Schladming
Austria February 24-March 7 1975 Acta Phys Austriaca Suppl 14 471 (1975)
41
[9] J Schwinger Source theory discussion of deep inelastic scattering with polarized particles
Proc Natl Acad Sci USA 72 1559 (1975)
[10] A M Baldin Polarizability of nucleons Nucl Phys 18 310 (1960)
[11] M Gell-Mann M L Goldberger and W E Thirring Use of causality conditions in quantum
theory Phys Rev 95 1612 (1954)
[12] A Pineda Leading chiral logarithms to the hyperfine splitting of the hydrogen and muonic
hydrogen Phys Rev C 67 025201 (2003)
[13] C Peset and A Pineda The two-photon exchange contribution to muonic hydrogen from
chiral perturbation theory Nucl Phys B 887 69 (2014) arXiv14064524 [hep-ph]
[14] C Peset and A Pineda Model-independent determination of the two-photon exchange contri-
bution to hyperfine splitting in muonic hydrogen JHEP 04 060 arXiv161205206 [nucl-th]
[15] F Hagelstein and V Pascalutsa Proton structure in the hyperfine splitting of muonic hydro-
gen PoS CD15 077 (2016) arXiv151104301 [nucl-th]
[16] F Hagelstein Exciting Nucleons in Compton Scattering and Hydrogen-Like Atoms PhD
thesis Mainz U Inst Kernphys (2017) arXiv171000874 [nucl-th]
[17] F Hagelstein ∆(1232)-Resonance in the hydrogen spectrum Proceedings 11th International
Workshop on the Physics of Excited Nucleons (NSTAR 2017) Columbia SC USA August
20-23 2017 Few Body Syst 59 93 (2018) arXiv180109790 [nucl-th]
[18] Y Prok et al (CLAS) Moments of the spin structure functions gp1 and gd1 for 005 lt Q2 lt 30
GeV2 Phys Lett B 672 12 (2009) arXiv08022232 [nucl-ex]
[19] H Dutz et al (GDH) First measurement of the Gerasimov-Drell-Hearn sum rule for 1H from
07 GeV to 18 GeV at ELSA Phys Rev Lett 91 192001 (2003)
[20] M Amarian et al (Jefferson Lab E94010) Measurement of the generalized forward spin po-
larizabilities of the neutron Phys Rev Lett 93 152301 (2004) arXivnucl-ex0406005
[21] M Amarian et al The Q2 evolution of the generalized Gerasimov-Drell-Hearn integral for
the neutron using a 3He target Phys Rev Lett 89 242301 (2002) arXivnucl-ex0205020
[22] M Amarian et al (Jefferson Lab E94-010)Q2 evolution of the neutron spin structure moments
using a 3He target Phys Rev Lett 92 022301 (2004) arXivhep-ex0310003
[23] A Deur et al Experimental determination of the evolution of the Bjorken integral at low Q2
Phys Rev Lett 93 212001 (2004) arXivhep-ex0407007
[24] K Slifer Low Q2 measurement of gp2 and the δLT spin polarizability Spin structure at long dis-
tance Proceedings Workshop Newport News USA March 12ndash13 2009 nucl-ex09064775
(2009) AIP Conf Proc 1155 10106313203293 (2009) arXiv09064775 [nucl-ex]
[25] P Solvignon et al (E01-012) Moments of the neutron g2 structure function at intermediate
42
Q2 Phys Rev C 92 015208 (2015) arXiv13044497 [nucl-ex]
[26] A Deur Experimental studies at low Q2 of the spin structure of the nucleon at Jefferson Lab
in 9th International Workshop on Chiral Dynamics (CD18) Durham NC USA September
17-21 2018 (2019) arXiv190305661 [nucl-ex]
[27] V Sulkosky et al (Jefferson Lab E97-110) Measurement of the 3He spin-structure functions
and of neutron (3He) spin-dependent sum rules at 0035 le Q2 le 024 GeV2 Phys Lett B
805 135428 (2020) arXiv190805709 [nucl-ex]
[28] K Adhikari et al (CLAS) Measurement of the Q2 Dependence of the Deuteron Spin Structure
Function g1 and its Moments at Low Q2 with CLAS Phys Rev Lett 120 062501 (2018)
arXiv171101974 [nucl-ex]
[29] V Lensky F Hagelstein A Hiller Blin and V Pascalutsa Comment on rdquoMeasurement of
the Q2 Dependence of the Deuteron Spin Structure Function g1 and its Moments at Low Q2
with CLASrdquo (2018) arXiv180603219 [nucl-th]
[30] J M Alarcn F Hagelstein V Lensky and V Pascalutsa Forward doubly-virtual Compton
scattering off the nucleon in chiral perturbation theory at NLO the subtraction function and
moments of unpolarized structure functions (2020) arXiv200509518 [hep-ph]
[31] V Bernard E Epelbaum H Krebs and U-G Meiszligner New insights into the spin structure
of the nucleon Phys Rev D 87 054032 (2013) arXiv12092523 [hep-ph]
[32] V Lensky J M Alarcon and V Pascalutsa Moments of nucleon structure functions at
next-to-leading order in baryon chiral perturbation theory Phys Rev C 90 055202 (2014)
arXiv14072574 [hep-ph]
[33] V Pascalutsa and M Vanderhaeghen Chiral effective-field theory in the ∆(1232) region
I Pion electroproduction on the nucleon Phys Rev D 73 034003 (2006) arXivhep-
ph0512244
[34] V Pascalutsa and M Vanderhaeghen Electromagnetic nucleon-to-Delta transition in chiral
effective field theory Phys Rev Lett 95 232001 (2005) arXivhep-ph0508060
[35] T R Hemmert B R Holstein and J Kambor Systematic 1M expansion for spin 32
particles in baryon chiral perturbation theory Phys Lett B 395 89 (1997) arXivhep-
ph9606456
[36] V Pascalutsa and D R Phillips Effective theory of the ∆(1232) in Compton scattering off
the nucleon Phys Rev C 67 055202 (2003) arXivnucl-th0212024
[37] V Pascalutsa M Vanderhaeghen and S N Yang Electromagnetic excitation of the ∆(1232)-
resonance Phys Rept 437 125 (2007) arXivhep-ph0609004
[38] K A Olive et al (Particle Data Group) Review of Particle Physics Chin Phys C 38 090001
43
(2014)
[39] F E Low Scattering of light of very low frequency by systems of spin 12 Phys Rev 96
1428 (1954)
[40] M Gell-Mann and M L Goldberger Scattering of low-energy photons by particles of spin
12 Phys Rev 96 1433 (1954)
[41] A M Harun ar-Rashid A simple derivation of Schwingerrsquos sum rule for spin dependent
structure functions Nuovo Cim A 33 447 (1976)
[42] F Hagelstein and V Pascalutsa Dissecting the hadronic contributions to (g minus 2)micro by
Schwingerrsquos sum rule Phys Rev Lett 120 072002 (2018) arXiv171004571 [hep-ph]
[43] V Lensky V Pascalutsa M Vanderhaeghen and C Kao Spin-dependent sum rules con-
necting real and virtual Compton scattering verified Phys Rev D 95 074001 (2017)
arXiv170101947 [hep-ph]
[44] H Burkhardt and W N Cottingham Sum rules for forward virtual Compton scattering
Annals Phys 56 453 (1970)
[45] V Lensky and V Pascalutsa Predictive powers of chiral perturbation theory in Compton
scattering off protons Eur Phys J C 65 195 (2010) arXiv09070451 [hep-ph]
[46] H Grieszlighammer J McGovern D Phillips and G Feldman Using effective field theory to
analyse low-energy Compton scattering data from protons and light nuclei Prog Part Nucl
Phys 67 841 (2012) arXiv12036834 [nucl-th]
[47] H W Grieszlighammer J A McGovern and D R Phillips Nucleon polarisabilities at and
beyond physical pion masses Eur Phys J A 52 139 (2016) arXiv151101952 [nucl-th]
[48] E Epelbaum H Krebs and U-G Meiszligner Improved chiral nucleon-nucleon potential up to
next-to-next-to-next-to-leading order Eur Phys J A 51 53 (2015) arXiv14120142 [nucl-
th]
[49] E Epelbaum H Krebs and U-G Meiszligner Precision nucleon-nucleon potential at fifth order
in the chiral expansion Phys Rev Lett 115 122301 (2015) arXiv14124623 [nucl-th]
[50] C W Kao T Spitzenberg and M Vanderhaeghen Burkhardt-Cottingham sum rule and
forward spin polarizabilities in heavy baryon chiral perturbation theory Phys Rev D 67
016001 (2003) arXivhep-ph0209241
[51] C-W Kao D Drechsel S Kamalov and M Vanderhaeghen Higher moments of nucleon
spin structure functions in heavy baryon chiral perturbation theory and in a resonance model
Phys Rev D 69 056004 (2004) arXivhep-ph0312102
[52] T Becher and H Leutwyler Baryon chiral perturbation theory in manifestly Lorentz invariant
form Eur Phys J C 9 643 (1999) arXivhep-ph9901384
44
[53] J Gasser M E Sainio and A Svarc Nucleons with chiral loops Nucl Phys B 307 779
(1988)
[54] L S Geng J Martin Camalich L Alvarez-Ruso and M J Vicente Vacas Leading SU(3)-
breaking corrections to the baryon magnetic moments in chiral perturbation theory Phys
Rev Lett 101 222002 (2008) arXiv08051419 [hep-ph]
[55] D Drechsel S S Kamalov and L Tiator The GDH sum rule and related integrals Phys
Rev D 63 114010 (2001) arXivhep-ph0008306
[56] D Drechsel O Hanstein S S Kamalov and L Tiator A unitary isobar model for pion
photo- and electroproduction on the proton up to 1 GeV Nucl Phys A 645 145 (1999)
arXivnucl-th9807001
[57] L Tiator private communication (2020)
[58] V Bernard T R Hemmert and U-G Meiszligner Spin structure of the nucleon at low energies
Phys Rev D 67 076008 (2003) arXivhep-ph0212033
[59] O Gryniuk F Hagelstein and V Pascalutsa Evaluation of the forward Compton scattering
off protons II Spin-dependent amplitude and observables Phys Rev D 94 034043 (2016)
arXiv160400789 [nucl-th]
[60] R Zielinski The g2p Experiment A Measurement of the Protonrsquos Spin Structure Functions
PhD thesis New Hampshire U (2010) arXiv170808297 [nucl-ex]
[61] N Guler et al (CLAS) Precise determination of the deuteron spin structure at low to mod-
erate Q2 with CLAS and extraction of the neutron contribution Phys Rev C 92 055201
(2015) arXiv150507877 [nucl-ex]
[62] C E Carlson V Nazaryan and K Griffioen Proton structure corrections to electronic
and muonic hydrogen hyperfine splitting Phys Rev A 78 022517 (2008) arXiv08052603
[physicsatom-ph]
[63] V Bernard N Kaiser and U-G Meiszligner Chiral dynamics in nucleons and nuclei Int J
Mod Phys E 4 193 (1995) arXivhep-ph9501384
[64] T R Hemmert B R Holstein and J Kambor ∆(1232) and the polarizabilities of the nucleon
Phys Rev D 55 5598 (1997) arXivhep-ph9612374
[65] V Pascalutsa and R Timmermans Field theory of nucleon to higher-spin baryon transitions
Phys Rev C 60 042201 (1999) arXivnucl-th9905065
[66] V Pascalutsa Quantization of an interacting spin-32 field and the Delta isobar Phys Rev
D 58 096002 (1998) arXivhep-ph9802288
[67] H Krebs Double Virtual Compton Scattering and SpinStructure of the Nucleon PoS
CD2018 031 (2019)
45
[68] N Kochelev and Y Oh Axial anomaly and the δLT puzzle Phys Rev D 016012 (2012)
arXiv11034892 [hep-ph]
[69] D Drechsel S Kamalov and L Tiator Unitary isobar model ndash MAID2007 Eur Phys J A
69 (2007) available at httpsmaidkphuni-mainzde arXiv07100306 [nucl-th]
[70] P J Mohr B N Taylor and D B Newell CODATA recommended values of the fundamental
physical constants 2010 Rev Mod Phys 84 1527 (2012)
[71] J Ahrens et al (GDH A2) First measurement of the Gerasimov-Drell-Hearn integral for 1H
from 200 to 800 MeV Phys Rev Lett 87 022003 (2001) arXivhep-ex0105089 [hep-ex]
[72] K Helbing (GDH) Experimental verification of the GDH sum rule at ELSA and MAMI Nucl
Phys Proc Suppl 105 113 (2002)
[73] R Bradford A Bodek H S Budd and J Arrington A New parameterization of the nucleon
elastic form-factors NuInt05 proceedings of the 4th International Workshop on Neutrino-
Nucleus Interactions in the Few-GeV Region Okayama Japan 26-29 September 2005 Nucl
Phys Proc Suppl 159 127 (2006) arXivhep-ex0602017
[74] J D Bjorken Applications of the chiral U(6)timesU(6) algebra of current densities Phys Rev
148 1467 (1966)
[75] J D Bjorken Inelastic scattering of polarized leptons from polarized nucleons Phys Rev D
1 1376 (1970)
[76] V Bernard T R Hemmert and U-G Meiszligner Novel analysis of chiral loop effects in
the generalized Gerasimov-Drell-Hearn sum rule Phys Lett B 545 105 (2002) arXivhep-
ph0203167
[77] A Deur et al Experimental study of isovector spin sum rules Phys Rev D 78 032001
(2008) arXiv08023198 [nucl-ex]
[78] B Pasquini P Pedroni and D Drechsel Higher order forward spin polarizability Phys Lett
B 687 160 (2010) arXiv10014230 [hep-ph]
[79] R Jaffe g2ndashThe nucleonrsquos other spin-dependent structure function Comments Nucl Part
Phys 19 239 (1990)
[80] E V Shuryak and A Vainshtein Theory of power corrections to deep inelastic scattering in
quantum chromodynamics (II) Qminus4 effects polarized target Nucl Phys B 201 141 (1982)
[81] S Wandzura and F Wilczek Sum rules for spin dependent electroproduction Test of rela-
tivistic constituent quarks Phys Lett B 72 195 (1977)
[82] B W Filippone and X-D Ji The spin structure of the nucleon Adv Nucl Phys 26 1
(2001) arXivhep-ph0101224
[83] M Burkardt The g2 structure function Proceedings Workshop on Spin structure at long
46
distance Newport News USA March 12-13 2009 AIP Conf Proc 1155 26 (2009)
arXiv09054079 [hep-ph]
[84] A Deur S J Brodsky and G F de Teramond The spin structure of the nucleon Rept
Prog Phys 82 076201 (2019) arXiv180705250 [hep-ph]
[85] B R Holstein D Drechsel B Pasquini and M Vanderhaeghen Higher order polarizabilities
of the proton Phys Rev C 61 034316 (2000) arXivhep-ph9910427
[86] B R Holstein V Pascalutsa and M Vanderhaeghen Sum rules for magnetic moments and
polarizabilities in QED and chiral effective-field theory Phys Rev D 72 094014 (2005)
arXivhep-ph0507016
[87] B Pasquini D Drechsel and L Tiator Invariant amplitudes for pion electroproduction Eur
Phys J A 34 387 (2007) arXiv07122327 [hep-ph]
[88] A J Buchmann Non-spherical proton shape and hydrogen hyperfine splitting Proceedings
International Workshop on Precision Physics of Simple Atomic Systems (PSAS 2008) Wind-
sor Ontario Canada July 21-26 2008 Can J Phys 87 773 (2009) arXiv09104747
[physicsatom-ph]
47
000 005 010 015 020 025 030
-30
-25
-20
-15
-10
-05
00
Q2 GeV2D
I Aproton
000 005 010 015 020 025 030
-25
-20
-15
-10
-05
00
Q2 GeV2D
I A
neutron
000 005 010 015 020 025 030
-05
00
05
Q2 GeV2D
I 1
proton
000 005 010 015 020 025 030
-10
-08
-06
-04
-02
00
02
Q2 GeV2D
I 1
neutron
FIG 4 Upper panel The generalized GDH integral IA(Q2) for the proton (left) and neutron
(right) as function of Q2 The result of this work the NLO BχPT prediction is shown by the blue
solid line and the blue band The red line represents the LO BχPT result The purple short-dashed
line is the O(p4) HB result from Ref [50 51] The gray band is the BχPT+∆ result from Ref [31]
The black dotted line is the MAID model prediction [69] Experimental extractions for the proton
Ref [60] (orange triangle uncertainties added in quadrature) and neutron Refs [21][27] where
magenta dotsorange diagonal crosses correspond to data and red squareslilac crosses correspond
to data plus extrapolation to unmeasured energy regions The green stars at the real-photon point
are derived from the anomalous magnetic moments κp asymp 1793 and κn asymp minus1913 [70] Lower
panel The generalized GDH integral I1(Q2) for the proton (left) and neutron (right) as function
of Q2 The purple short-dashed line is the HB result from Ref [51] Experimental extractions for
the proton Ref [18] (blue dots) and Ref [60] (orange triangle uncertainties added in quadrature)
and neutron Ref [61][27] (uncertainties added in quadrature) where green dotsorange diagonal
crosses correspond to data and gray squareslilac crosses correspond to data plus extrapolation to
unmeasured energy regions
17
IA(Q2) and I1(Q2) The latter will be discussed in Sec III D The former is defined as4
minus α
2M2N
IA(Q2) = minus 1
8π2
int infinν0
dν
radic1 +
Q2
ν2
σTT (νQ2)
ν(26)
=α
Q2
int x0
0
dx[4M2
Nx2
Q2g2(xQ2)minus g1(xQ2)
]
Due to its energy weighting the integral in Eq (26) converges slower than the one in the
generalized forward spin polarizability sum rule (19) Therefore knowledge of the cross
section at higher energies is required and the evaluation of the generalized GDH integral
IA(Q2) is not as simple as the evaluation of γ0(Q2)
The generalized GDH integral IA(Q2) is directly related to the non-pole amplitude
gnonpoleTT (νQ2) which differs from non-Born amplitude gTT (νQ2) by a term involving the
elastic Pauli form factor
gnonpoleTT (νQ2) = gTT (νQ2)minus 2παν
M2N
F 22 (Q2) (27)
cf Eqs (2a) and (8a) Consequently IA(Q2) is not a pure polarizability but also contains
an elastic contribution The ldquonon-polarizabilityrdquo or the Born part of IA(Q2) is given by
IBornA (Q2) = IA(Q2)minus∆IA(Q2) = minus1
4F 2
2 (Q2) (28)
where we refer to the polarizability part as ∆IA(Q2) The same is true for the generalized
GDH integral I1(Q2) which is directly related to Snonpole1 (νQ2)
IBorn1 (Q2) = I1(Q2)minus∆I1(Q2) = minus1
4F 2
2 (Q2) (29)
In the following we will add the Born parts to our LO and NLO BχPT predictions for
the polarizabilities ∆IA(Q2) and ∆I1(Q2) employing an empirical parametrization for the
elastic Pauli form factor [73] This allows us to compare to the experimental results for
IA(Q2) and I1(Q2) cf Fig 4 Note that the blue error bands only describe the uncertainties
of our BχPT predictions of the polarizabilities while the elastic contributions are considered
to be exact as explained in Sec II C The uncertainties of the polarizability predictions are
therefore better reflected in Fig 5 where we show the contributions of the different orders
to the BχPT predictions of ∆IA(Q2) and ∆I1(Q2) as well as the total results with error
bands
4 Note that IA(Q2) is sometimes called ITT (Q2)
18
000 005 010 015 020 025 030
-08
-06
-04
-02
00
02
Q2 GeV2D
DI A
proton
000 005 010 015 020 025 030
-10
-08
-06
-04
-02
00
02
Q2 GeV2D
DI A
neutron
000 005 010 015 020 025 030-005
000
005
010
015
020
025
Q2 GeV2D
DI 1
proton
000 005 010 015 020 025 030-015
-010
-005
000
005
010
015
Q2 GeV2D
DI 1
neutron
FIG 5 Contributions of the different orders to the chiral predictions of ∆IA(Q2) upper paneland ∆I1(Q2) lower panel for the proton (left) and neutron (right) Red solid line πN -loop
contribution green dot-dashed line ∆-exchange contribution orange dotted line π∆-loop contri-
bution blue solid line and blue band total result purple dot-dot-dashed line total result without
gC contribution black short-dashed line total result without gM dipole
The E97-110 experiment at Jefferson Lab has recently published their data for IAn(Q2) in
the region of 0035 GeV2 lt Q2 lt 024 GeV2 [27] In addition there are results for IAn(Q2)
from the earlier E94-010 experiment [21] and for IAp(Q2) from the E08-027 experiment [60]
The O(p4) HB calculation gives a large negative effect [51] which does not describe the data
The BχPT+∆ result from Ref [31] which mainly differs from our work by the absence of
the dipole form factor in gM looks similar to this HB result and only describes the data
points at lowest Q2 Our NLO prediction however follows closely the Q2 evolution of the
data In Fig 5 upper panel we show the polarizability ∆IA(Q2) whose Q2 evolution is
clearly dominated by the ∆ exchange Similar to the case of γ0p(Q2) inclusion of the dipole
in gM and the Coulomb coupling gC is very important in order to describe the experimental
data The LO prediction on the other hand slightly overestimates the data cf Fig 4
upper panel
19
At the real-photon point IA(0) = minusκ2
4and ∆IA(0) = 0 Therefore we give only the
slope of the polarizability ∆IA(Q2) [showing also the separate contributions from πN loops
∆ exchange and π∆ loops] in units of GeVminus2
d∆IAp(Q2)
dQ2
∣∣∣∣Q2=0
= minus858(343) asymp 238minus 1121 + 025 (30a)
d∆IAn(Q2)
dQ2
∣∣∣∣Q2=0
= minus955(343) asymp 141minus 1121 + 025 (30b)
Including the empirical Pauli form factor [73] we find in units of GeVminus2
dIAp(Q2)
dQ2
∣∣∣∣Q2=0
= minus318dIAn(Q2)
dQ2
∣∣∣∣Q2=0
= minus300 (31)
D Γ1(Q2) and I1(Q2) mdash the first moment of the structure function g1(xQ2)
The second variant for a generalization of the GDH sum rule to finite Q2 is defined as
minus α
2M2N
I1(Q2) = minus 1
8π2
int infinν0
dν1radic
ν2 +Q2
[σTT (νQ2) +
Q
νσLT (νQ2)
](32)
= minus α
Q2
int x0
0
dx g1(xQ2)
where I1(0) = minusκ2
4 This generalized GDH integral directly stems from the amplitude
Snonpole1 (νQ2) with the LEX from Eq (12a) It is given by the first moment of the struc-
ture function g1(xQ2) Γ1(Q2) =int x0
0dx g1(xQ2) as follows I1(Q2) =
2M2N
Q2 Γ1(Q2) The
isovector combination
Γ1(pminusn)(Q2) =
int x0
0
dx[g1p(xQ
2)minus g1n(xQ2)] (33)
is related to the axial coupling of the nucleon through the Bjorken sum rule [74 75]
limQ2rarrinfin
Γ1(pminusn)(Q2) =
gA6 (34)
As explained in Eq (28) the moment I1(Q2) splits into a polarizability part ∆I1(Q2) and
a Born part IBorn1 (Q2) Figure 4 lower panel shows the Q2 dependence of I1(Q2) which in
contrast to IA(Q2) shown in Figure 4 upper panel is clearly dominated by its Born part
and the elastic Pauli form factor The πN -loop ∆-exchange and π∆-loop contributions to
the polarizability ∆I1(Q2) are shown in Fig 5 lower panel Comparing to Fig 5 upper
20
000 005 010 015 020 025 030
-006
-004
-002
000
002
004
Q2 GeV2D
G1
proton
000 005 010 015 020 025 030
-006
-005
-004
-003
-002
-001
000
001
Q2 GeV2D
G1
neutron
FIG 6 First moment of the structure function g1(xQ2) for the proton (left) and neutron (right)
as function of Q2 The legend is the same as in Fig 4
panel one sees that ∆I1(Q2) is less sensitive to gC and the dipole form factor in gM than
∆IA(Q2)
For the proton our NLO BχPT prediction gives a very good description of the ex-
perimental data [18 60] and is in reasonable agreement with the MAID prediction [69]
For the neutron one observes good agreement with the empirical evaluations including ex-
trapolations to unmeasured energy regions starting from Q2 gt 01 GeV2 [27 61] In the
region of Q2 lt 005 GeV2 one observes an interesting tension between the recent E97-
110 experiment [27] and the data from CLAS [61] While the newest measurement finds
I1n(0035 GeV2) lt κ2n4 thus suggesting a negative slope at low Q the older measurement
found a rather large value for I1n(00496 GeV2) A similar but milder behaviour is seen in
the E97-110 [27] and E94-010 [21] data for IAn The MAID predictions do not agree with
the CODATA recommended values for the anomalous magnetic moments of the proton and
neutron [70] which in our work are imposed by using empirical parametrizations for the
elastic Pauli form factors [73] The slope of the HB result from Ref [51] is too large and
therefore only reproduces the data at very low Q2
Figure 6 shows the moment Γ1(Q2) for the proton and neutron while Fig 7 shows the
isovector combination Γ1 pminusn(Q2) The LO and NLO BχPT predictions are identical be-
cause our calculation produces the same Delta contributions for the proton and the neutron
For the isovector combination the MAID model only agrees with the data at very low
Q2 lt 010 GeV2 The same is true for the IR result [58 76] while all other chiral results
describe the data NLO BχPT (this work) BχPT+∆ [31] and HBχPT [51]
At the real-photon point I1(0) = minusκ2
4and ∆I1(0) = 0 Therefore we give only the
slope of the polarizability ∆I1(Q2) [showing also the separate contributions from πN loops
21
000 005 010 015 020 025 030-002
000
002
004
006
008
010
Q2 GeV2D
G1Hp-
nL
FIG 7 Isovector combination of Γ1(Q2) as function of Q2 The legend is the same as in Fig 6
The pink curve is the IR result from Ref [58 76] The experimental points are from Ref [77]
(brown dots) and Ref [23] (orange squares)
∆ exchange and π∆ loops] in units of GeVminus2
d∆I1p(Q2)
dQ2
∣∣∣∣Q2=0
= 039(4) asymp 034minus 053 + 058 (35a)
d∆I1n(Q2)
dQ2
∣∣∣∣Q2=0
= minus101(10) asymp minus107minus 053 + 058 (35b)
Including the empirical Pauli form factor [73] we find in units of GeVminus2
dI1p(Q2)
dQ2
∣∣∣∣Q2=0
= 580dI1n(Q2)
dQ2
∣∣∣∣Q2=0
= 553 (36)
E d2(Q2) mdash a measure of color polarizability
Another interesting moment to consider is d2(Q2) which is related to the twist-3 part of
the spin structure function g2(xQ2) [79 80]
d2(Q2) equiv 3
int 1
0
dx x2[g2(xQ2)minus gWW2 (xQ2)] (37)
where gWW2 (xQ2) is the twist-2 part of g2(xQ2) Using the Wandzura-Wilczek relation [81]
one can relate d2(Q2) to moments of the spin structure functions g1(xQ2) and g2(xQ2)
d2(Q2) =
int 1
0
dx x2 [3g2(xQ2) + 2g1(xQ2)] (38)
22
000 005 010 015 020 025 030
0000
0001
0002
0003
0004
0005
0006
0007
Q2 GeV2D
d2
proton
000 005 010 015 020 025 030
0000
0002
0004
0006
0008
0010
0012
Q2 GeV2D
d2
neutron
000 005 010 015 020 025 030
-1
0
1
2
Q2 GeV2D
Γ 010
-4
fm6
D
proton
000 005 010 015 020 025 030-2
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
6D
neutron
FIG 8 Upper panel The inelastic moment d2(Q2) for the proton (left) and neutron (right) as
function of Q2 The result of this work the NLO BχPT prediction is shown by the blue solid line
and the blue band The red line represents the LO BχPT result The purple short-dashed line is
the O(p4) HB result from Ref [50 51] The black dotted line is the MAID model prediction [69]
The experimental points for the neutron (cyan dots) are from Ref [22] Lower panel Fifth-order
generalized forward spin polarizability γ0(Q2) for the proton (left) and neutron (right) as function
of Q2 The experimental points for the proton are from Ref [59] (purple square) and Ref [78]
(orange dot)
This relation however only holds for asymptotically large Q2 It is also in the high-Q2
region where d2(Q2) is a measure of color polarizability [82 83] through its relation to the
gluon field strength tensor [80] We refer to Ref [84] for a recent review on the spin structure
of the nucleon including a discussion of sum rules for deep inelastic scattering and color
polarizabilities
What we consider in the following is the inelastic part of d2(Q2) defined as the moment
of g1(xQ2) and g2(xQ2) spin structure functions cf Eq (38)
d2(Q2) =
int x0
0
dx x2 [3g2(xQ2) + 2g1(xQ2)] (39)
23
This moment provides another testing ground for our BχPT predictions through comparison
with experiments on the neutron [22] Going towards the low-Q2 region the interpretation
of d2(Q2) in terms of color polarizabilities will fade out The above definition however
implies it is related to other VVCS polarizabilities
d2(Q2) =Q4
8M4N
[M2
NQ2
αδLT (Q2) + I1(Q2)minus IA(Q2)
] (40)
Note that d2(Q2) and its first two derivatives with respect to Q2 vanish at Q2 = 0 The
considerations in Eqs (28) and (29) have no effect on d2(Q2) since the Born contribution
from IA(Q2) and I1(Q2) cancel out Therefore d2(Q2) is a pure polarizability
In Fig 8 upper panel we show our NLO BχPT prediction and other results for d2(Q2)
While MAID [69] and BχPT describe the experimental data for the neutron [22] very well
the HB limit [50 51] is showing a fast growth with Q2 This illustrates the importance
of keeping the relativistic result Note also that even though the πN -loop contribution
is dominant both gC and the form factor in gM are essential to obtain a curvature that
reproduces the data cf Fig 9 upper panel For the proton there are to our knowledge
no experimental results to compare with However the agreement between the NLO BχPT
prediction and the MAID prediction at low energies is reasonable
F γ0(Q2) mdash fifth-order generalized forward spin polarizability
It is interesting to compare the generalized fifth-order forward spin polarizability sum
rule
γ0(Q2) =1
2π2
int infinν0
dν
radic1 +
Q2
ν2
σTT (νQ2)
ν5(41)
=64αM4
N
Q10
int x0
0
dx x4
[g1(xQ2)minus 4M2
Nx2
Q2g2(xQ2)
]
to the sum rule integrals for IA(Q2) and γ0(Q2) since they differ merely by their energy
weighting of σTT (νQ2) and a constant prefactor cf Eqs (19) (26) and (41) From IA(Q2)
to γ0(Q2) to γ0(Q2) the energy suppression is increasing by a factor of νminus2 respectively
Therefore the description of γ0(Q2) should be easiest in a low-energy effective-field theory
such as χPT whereas γ0(Q2) and IA(Q2) receive larger contributions from higher energies
In Fig 8 lower panel we show our LO and NLO BχPT predictions for γ0(Q2) One
can see that the πN -loop contribution is positive (in accordance to what we see for the cross
section σTT see Fig 10) The Delta shifts it substantially especially in the low Q2 region
24
000 005 010 015 020 025 030
-0004
-0002
0000
0002
0004
0006
0008
0010
Q2GeV
2D
d2
proton
000 005 010 015 020 025 030
-0005
0000
0005
0010
0015
Q2GeV
2D
d2
neutron
000 005 010 015 020 025 030
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
6D
proton
000 005 010 015 020 025 030
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
6D
neutron
FIG 9 Contributions of the different orders to the chiral predictions of d2(Q2) upper panel and
γ0(Q2) lower panel for the proton (left) and neutron (right) Red solid line πN -loop contribu-
tion green dot-dashed line ∆-exchange contribution orange dotted line π∆-loop contribution
blue long-dashed line total result purple dot-dot-dashed line total result without gC contribution
black short-dashed line total result without gM dipole
bringing it into a better agreement with data In general the BχPT curves start above the
empirical data points at the real-photon point and then decrease asymptotically to zero
above Q2 gt 01 GeV2 On the other hand the MAID prediction reproduces the empirical
data at the real-photon point then decreases to negative values until about Q2 gt 006 GeV2
from where it also starts to asymptotically approach zero Consequently our NLO BχPT
prediction of γ0(Q2) is consistently above the MAID prediction This is very different to
what we saw for IA(Q2) in Fig 4 upper panel where the MAID prediction at the real-
photon point is above the experimental value While the agreement of our predictions with
the empirical data is in general quite good for all moments of σTT (νQ2) one should point
out that both for γ0n(Q2) and γ0p(Q2) we overestimate the data at low Q2 For IA(Q2) such
observation cannot be made because ∆IA(0) = 0 and thus IA(0) is given by the empirical
Pauli form factor only From IA(Q2) γ0(Q2) and γ0(Q2) the latter has the smallest however
25
non-negligible dependence on gC and the dipole in gM cf Fig 9 lower panel
The πN -loop ∆-exchange and π∆-loop contributions to the NLO BχPT prediction of
the static fifth-order forward spin polarizability amount to in units of 10minus4 fm6
γ0p = 112(30) asymp 208minus 096minus 001 (42a)
γ0n = 195(30) asymp 292minus 096minus 001 (42b)
while the slope is composed as follows in units of 10minus4 fm8
dγ0p(Q2)
dQ2
∣∣∣∣Q2=0
= minus084(10) asymp minus100 + 016 + 000 (43a)
dγ0n(Q2)
dQ2
∣∣∣∣Q2=0
= minus142(15) asymp minus158 + 016 + 000 (43b)
Note that the HB prediction of the static polarizability γ0p (423 at O(p3) and 365 at O(ε3)
[78 85]) is almost one order of magnitude larger than the empirical value and therefore not
shown in Fig 8
G Summary
Our results are summarized in Table II where we give the contributions of the different
orders to the chiral predictions of the polarizabilities and their slopes at the real-photon
point A quantitative comparison of our predictions for the static spin polarizabilities to
the work of Bernard et al [31] and different empirical evaluations is shown in Table III We
can see that the inclusion of the Delta turns out to be very important for all moments of
the helicity-difference cross section To describe the Q2 behavior of the polarizabilities the
magnetic coupling of the N rarr ∆ transition should be modified by a dipole form factor as
has been observed previously in the description of electroproduction data [33] This dipole
form factor effectively takes account of vector-meson exchanges The Coulomb-quadrupole
N rarr ∆ transition despite its subleading order is important in the description of some
moments of spin structure functions This is contrary to what we saw for the moments of
unpolarized structure functions [30] where the Coulomb coupling had a negligible effect
The π∆ loops are mainly relevant for the generalized GDH integrals
26
TABLE II The NLO BχPT predictions for the forward VVCS polarizabilities and their slopes at
Q2 = 0 The contributions of the πN loops the ∆ exchange and the π∆ loops are shown together
with the combined total result Note that IA(0) = I1(0) = d2(0) = 0 and (d2)prime = 0
πN loops ∆ exchange π∆ loops Total
γ0 p 201 minus284 minus010minus093(92)
(10minus4 fm4) n 298 003(92)
δLT p 150 minus016 minus002132(15)
(10minus4 fm4) n 235 218(23)
γ0 p 208 minus096 minus001112(30)
(10minus4 fm6) n 292 195(30)
(γ0)prime p minus033011 001
minus022(4)
(10minus4 fm6) n minus073 minus061(7)
(δLT )prime p minus080 minus004 minus001minus085(8)
(10minus4 fm6) n minus119 minus124(12)
(γ0)prime p minus100016 000
minus084(10)
(10minus4 fm8) n minus158 minus142(15)
(∆IA)prime p 238 minus1121 025minus858(343)
(GeVminus2) n 141 minus955(343)
(∆I1)prime p 034 minus053 058039(4)
(GeVminus2) n minus107 minus101(10)
TABLE III Our NLO BχPT predictions for the spin polarizabilities at Q2 = 0 compared with
the BχPT+∆ predictions from Bernard et al [31] and the available empirical information Where
the reference is not given the empirical number is provided by the MAID analysis [55 69] with
unspecified uncertainty
Proton Neutron
This work BχPT+∆ Empirical This work BχPT+∆ Empirical
γ0 minus093(92) minus174(40) minus100(8)(12) [19] 003(92) minus077(40) minus0005
(10minus4 fm4) minus090(8)(11) [78] [MAID]
minus0929(105) [59]
δLT 132(15) 240(1) 134 218(23) 238(3) 203
(10minus4 fm4) [MAID] [MAID]
γ0 112(30) 060(7)(7) [78] 195(30) 123
(10minus4 fm6) 0484(82) [59] [MAID]
IV CONCLUSIONS
We have presented a complete NLO calculation of the polarized non-Born VVCS ampli-
tudes in covariant BχPT with pion nucleon and ∆(1232) fields The dispersion relations
between the VVCS amplitudes and the tree-level photoabsorption cross sections served as a
27
cross-check of these calculations
The obtained moments of the proton and neutron spin structure functions related to gen-
eralized polarizabilities and GDH-type integrals agree well with the available experimental
data The description of their Q2 evolution is improved compared to the previous χPT pre-
dictions In particular the NLO BχPT predictions obtained here give a better description
of the empirical data (eg from the Jefferson Laboratory ldquoSpin Physics Programrdquo) than
the HB [50 51] and IR [58] calculations
The demonstrated predictive power of the χPT framework amplitudes makes it well
suited for extending the χPT evaluation of the TPE effect in the hyperfine structure of
(muonic-)hydrogen [15ndash17] to next-to-leading order
ACKNOWLEDGEMENTS
We thank Lothar Tiator and Marc Vanderhaeghen for helpful discussions This work
is supported by the Deutsche Forschungsgemeinschaft (DFG) through the Collaborative
Research Center [The Low-Energy Frontier of the Standard Model (SFB 1044)] JMA ac-
knowledges support from the Community of Madrid through the ldquoPrograma de atraccion de
talento investigador 2017 (Modalidad 1)rdquo and the Spanish MECD grants FPA2016-77313-P
FH gratefully acknowledges financial support from the Swiss National Science Foundation
Appendix A Tensor decompositions of the VVCS amplitudes
In this appendix we review the decomposition of the forward VVCS process into tensor struc-
tures and scalar amplitudes In particular we consider the connection between the covariant and
the semi-relativistic decomposition in the lab frame that is defined in terms of the conventional
transverse longitudinal transverse-transverse and transverse-longitudinal amplitudes
As explained in Sec II A the process of forward VVCS off the nucleon can be described in
terms of four explicitly covariant amplitudes S1 2 and T1 2 [3]
T (νQ2) =
(minusgmicroν +
qmicroqν
q2
)T1(νQ2) +
1
M2N
(pmicro minus p middot q
q2qmicro)(
pν minus p middot qq2
qν)T2(νQ2) (A1)
minus 1
MNγmicroναqα S1(νQ2)minus 1
M2N
(γmicroνq2 + qmicroγναqα minus qνγmicroαqα
)S2(νQ2)
εprimelowastmicro εν
where εmicro (εprimelowastmicro ) are the incoming (outgoing) photon polarization vectors ν is the photon lab-frame
energy and Q2 is the photon virtuality Alternatively the decomposition in the laboratory frame
28
(which in the forward case coincides with the Breit frame) is parametrized in terms of the nucleon
Pauli matrices ~σ and the four scalar functions fL fT gTT and gLT
T (νQ2) = ε0 εprimelowast0 fL(νQ2) + (~ε primelowast middot ~ε ) fT (νQ2) + i~σ middot (~ε primelowast times ~ε ) gTT (νQ2) (A2)
minusi~σ middot [(ε0~εprimelowast minus ~ε ε primelowast0 )times q] gLT (νQ2)
Here ~q and q = ~q|~q | are the photon three-momentum in the lab system and its unit vector The
modified polarization vector components are given by
ε0 =
[ε0 minus
ν
|~q | (~ε middot q )
] |~q |Q ~ε = ~εminus q (~ε middot q ) (A3)
where ε = (ε0~ε ) is the usual incoming photon polarization vector and εprimelowast the outgoing polarization
vector The LEX of the lab frame amplitudes [Eq (10)] can serve in particular as the definition
of the generalized polarizabilities The lab frame amplitudes are also conveniently used for the
definition of the response functions see the example of the scalar amplitude gLT (νQ2) and the
corresponding response function σLT (νQ2) below in App B
Appendix B Photoabsorption cross sections
In the forward kinematics the spin-dependent VVCS amplitudes and the spin polarizabilities
can be described in terms of the polarized structure functions g1(xQ2) and g2(xQ2) or equiv-
alently the helicity-difference cross section σTT (νQ2) and the longitudinal-transverse response
function σLT (νQ2) with the help of dispersion relations (5) and the optical theorem (3) In this
way the photoabsorption cross sections measured in electroproduction processes form the basis
for most empirical evaluations shown throughout Sec III In the following we present the BχPT
predictions for the tree-level cross sections of πN - π∆- and ∆-production through photoabsorption
on the nucleon cf Figs 8 9 and 10 in Ref [30] In Secs B 1 and B 2 we will discuss the leading
πN -production channel and the ∆-production channel respectively We used these cross sections
to verify the polarizability predictions obtained otherwise from the calculated non-Born VVCS
amplitudes Due to the bad high-energy behavior of the π∆-production cross sections in BχPT
cf Fig 10 the dispersion relations in Eq (5) require further subtractions for a reconstruction of
the π∆-loop contribution to the spin-dependent VVCS amplitudes Therefore not all polarizabil-
ities could be verified but only those appearing as higher-order terms in the LEX of the VVCS
amplitudes such as γ0 [16]
29
proton neutron
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Neutron
02 04 06 08 100
50
100
150
200
250
[GeV]
T [b]
02 04 06 08 10
-50
0
50
100
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Neutron
02 04 06 08 100
50
100
150
200
250
[GeV]
T [b]
02 04 06 08 10
-50
0
50
100
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02040608100
50
100
150
200
250
300
[GeV]
T[b]
0204060810
-50
0
50
[GeV]
TT[b]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
0204060810
00
02
04
06
[GeV]
LTQ
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
0204060810
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
FIG 10 Photoabsorption cross sections for πN (red) and π∆ production (orange) with Q2 = 0
(solid) and Q2 = 01 GeV2 (dashed for πN and dotted for π∆ channel)
1 πN-production channel
In order to extract the response function σLT (νQ2) we have developed a method similar to
the one used to calculate σTT (νQ2) see for example Ref [86] For σLT (νQ2) however the
calculation is more complicated because one has to take into account that the associated Compton
process involves a spin-flip of the nucleon as illustrated in Fig 11 When calculating the cross
section the product of the incoming nucleon spinors has to reflect this flip
The forward VVCS amplitude related to σLT (νQ2) mdash and δLT (Q2) mdash is gLT (νQ2) It can
be extracted from Eq (A2) if one takes the modified polarization vector components in Eq (A3)
with ε = εL and εprimelowast = εlowastplusmn as input where εL = 1Q(|~q | 0 0 q0) and εplusmn = ∓ 1radic
2(0 1plusmni 0) are the
standard longitudinal and transverse polarization vectors respectively For εL and εlowastplusmn only the
choice of helicities hprime = plusmn12 and h = ∓12 gives a non-zero contribution and one obtains
χdaggerhprime T (νQ2)χh = χdaggerhprimeminusi~σ middot [(ε0~εprimelowast minus ~ε ε primelowast0 )times q] gLT (νQ2)χh =
radic2 gLT (νQ2) (B1)
where χh and χdaggerhprime are two-component Pauli spinors with opposite helicities or here spins
Let us now consider the related photoabsorption process and in particular the tree-level γlowastN rarr
30
s = 12
sprime = minus12
λ = 0 λprime = 1
rprime rprime
s=12sprime=minus1
2
λ=0λprime=1
rprimerprime
s=12
sprime=minus12
λ=0λprime=1
rprimerprime hacute=frac12h=-frac12
λacute=1λ=0
FIG 11 Relation between the forward Compton process and the photoabsorption process given
by the optical theorem In particular we show the longitudinal-transverse contribution The
double-line arrows represent the spin of the external particles while the dot represents the scalar
(longitudinal) polarization of the incoming photon Inside the blob the intermediate states are
represented eg nucleons with spins rprime (which are averaged in the calculation of the cross section)
and pions
πN channel see diagrams in Fig 8 of Ref [30] We define the πN -production amplitude as
T = uhB (PB)sumi
Ai(s t)Γi uhA(PA) (B2)
with the Dirac structures
Γ1 = γ5 (B3a)
Γ2 =1
2
[qA ε
]γ5 (B3b)
where uhA(PA) and udaggerhB (PB) are the Dirac spinors and PA and PB are the four-momenta of
the incoming and outgoing nucleons respectively When calculating the photoabsorption cross
section related to the VVCS amplitude in Eq (B1) the nucleon spin flip should be implemented
by uhprime(PA) in T dagger and uh(PA) in T together with the appropriate transverse and longitudinal
photon polarization vectors εlowastplusmn and εL
However if one wants to use the properties of the Dirac matrices it is more useful to construct
an operator to produce this spin flip in the external nucleons of Fig 11 This is accomplished by
introducing the projector ΓLT equiv 12radic
2(γ1 + iγ2)γ5 which also takes into account the extra factor
radic2 in Eq (B1) We checked that with this projector one correctly extracts δLT by comparing the
HB limit of our result to the HB result of Ref [50] where the authors calculate this polarizability
from the Compton amplitude directly With all those ingredients the longitudinal-transverse cross
section is calculated in the following way
σLT (νQ2) =1
64π2 s
|~pf |cm
|~pi|cm
int 1
minus1dcos θ
sumij
AiAdaggerjXij (B4)
31
with
Xij = Tr[(PB +MN )Γi(PA +MN )ΓLTγ0Γdaggerjγ
0] (B5)
where θ is the scattering angle in the center-of-mass (cm) frame and |~pi|cm (|~pf |cm) is the three-
momentum of an incoming (outgoing) particle in the cm frame An explicit calculation of the
matrix Xij leads to
X = MNQ
0 2(PB minus PA) middot εLminusradic
2|~qf |cm sin θ (sminus u)
(B6)
where |~qi|cm (|~qf |cm) is the relative three-momentum of the incoming (outgoing) particles in the
cm frame Here s t and u are the usual Mandelstam variables For the different γlowastN rarr πN
channels we obtain the following amplitudes Ai where we introduce qA as the four-momentum of
the incoming photon and qB as the four-momentum of the outgoing pion
bull γlowastprarr π0p
A1 =e gAMN
fπ
[2PA middot ε+ qA middot ε
sminusM2N
+2PB middot εminus qA middot ε
uminusM2N
] (B7a)
A2 =e gAMN
fπ
[1
sminusM2N
+1
uminusM2N
] (B7b)
bull γlowastprarr π+n
A1 =
radic2 e gAMN
fπ
[2PA middot ε+ qA middot ε
sminusM2N
+2(PA minus PB) middot ε+ qA middot ε
tminusm2π
] (B8a)
A2 =
radic2 e gAMN
fπ(sminusM2N )
(B8b)
bull γlowastnrarr π0n
A1 = 0 (B9a)
A2 = 0 (B9b)
bull γlowastnrarr πminusp
A1 =
radic2 e gAMN
fπ
[2PB middot εminus qA middot ε
uminusM2N
minus 2(PA minus PB) middot ε+ qA middot εtminusm2
π
] (B10a)
A2 =
radic2 e gAMN
fπ(uminusM2N )
(B10b)
The analytical expressions shown above were checked with the amplitudes given in Ref [87] An-
alytical expressions for the tree-level γlowastN rarr πN channel of the σLT (νQ2) and σTT (νQ2) cross
32
sections are given below (proton channels π+n and π0p neutron channel πminusp) We checked that
they reproduce the known results in the real-photon limit [45 86] To shorten the final expressions
for the cross sections which are considerably longer for finite Q2 than in the real-photon limit we
define the following dimensionless kinematic variables
αγ = (ENi )cmradics =
s+M2N +Q2
2s (B11)
απ = (ENf )cmradics =
s+M2N minusm2
π
2s (B12)
βγ = Eγcmradics =
sminusM2N minusQ2
2s (B13)
βπ = Eπcmradics =
sminusM2N +m2
π
2s (B14)
λγ = |~qi|cmradics =
radic(sminusM2
N minusQ2)2 + 4sQ2
2s (B15)
λπ = |~qf |cmradics =
radic(sminusM2
N +m2π)2 minus 4sm2
π
2s (B16)
Here (ENi )cm and (ENf )cm are the energies of the incoming and outgoing nucleon Eγcm is the energy
of the incoming photon Eπcm is the energy of the outgoing pion all in the cm frame
σ(π+n)TT = minus e2g2
AM2N
64πf2πs
3(sminusM2N )2λ4
γ
4sλπλγ
[(M2
N minus s)(M2N minusQ2 minus s)(Q2 + 2sβγβπ)
+ 2s(m2πQ
2 minus (M2N minus s)
(M2N + s(minus1 + 2βγβπ)
))λ2γ
]+ (M2
N minus s)(M2N minusQ2 minus s)
times (Q2 + 2sβγβπ minus 2sλπλγ)(Q2 + 2s(βγβπ + λπλγ)
)times log
(Q2 + 2sβγβπ minus 2sλγλπQ2 + 2sβγβπ + 2sλγλπ
) (B17)
33
σ(π0p)TT =
e2g2AM
2Nλπ
64πf2πs(sminusM2
N )2λγ
1
s2λπλ3γ
[(M2
N minus s)(M2N minusQ2 minus s)
(3M2
N + s(2βγβπ minus 3))(M2N
+ s(2βγβπ minus 1))
+ 2s
((2m2
πQ2 + (M2
N minus s)2)(M2
N minus s)minus 2(minusm2
πQ2 + (sminusM2
N )2)
times sβγβπ + 4s2(sminusM2N )β2
γβ2π
)λ2γ
]arctan
(2sλπλγ
M2N + s(2βγβπ minus 1)
)+ 2
[minus 2m2
πQ2
+ (M2N minus s)
(2(sminusM2
N ) + 4sβγβπ +(sminusM2
N +Q2)(3M2N minus 3s+ 2sβγβπ)
sλ2γ
+m2πQ
2(sminusM2N )
(M2N minus s+ 2sβγβπ)(M2
N + s(minus1 + 2βγβπ minus 2λπλγ))
+m2πQ
2(sminusM2N )
(M2N minus s+ 2sβγβπ)(M2
N + s(minus1 + 2βγβπ + 2λπλγ))
)
+1
λπλγ
(βγβπ
(minusm2
πQ2 + (M2
N minus s)(M2N minus s+ 2sβγβπ)
)minus (M2
N minus s)(M2N minusQ2 minus s)λ2
π
)log
(M2N + s(minus1 + 2βγβπ + 2λπλγ)
M2N + s(minus1 + 2βγβπ minus 2λπλγ)
)] (B18)
σ(πminusp)TT =
e2g2AM
2N
64πf2πs
3λ4γ(M2
N + s(2βπβγ minus 1))2 minus 4λ2πs
2λ2γ)
4sλπλγ
((minusM2
N +Q2 + s)(M2N
+ s(2βπβγ minus 1))
2 minus 2sλ2γ
(2λ2
πs(minusM2N +Q2 + s) +m2
πQ2))
+((M2N + s(2βπβγ minus 1)
)2
minus 4λ2πs
2λ2γ
)((M2N + s(2βπβγ minus 1)
)(M2N minus 2Q2 minus s(2βπβγ + 1)
)+ 4λ2
πs2λ2
γ
)times log
(minusM2
N minus 2βπsβγ minus 2λπsλγ + s)minus(M2N + s(2βπβγ minus 2λπλγ minus 1)
)(M2N + s(minus1
+ 2βπβγ + 2λπλγ))[((
M2N + s(2βπβγ minus 1)
)(M2N minus 2Q2 minus s(2βπβγ + 1)
)+ 4λ2
πs2λ2
γ
)times log
(minusM2
N minus 2βπsβγ + 2λπsλγ + s)
+((Q2 + 2βπsβγ)2 minus 4λ2
πs2λ2
γ
)times log
(Q2 + 2βπsβγ minus 2λπsλγQ2 + 2sβπβγ + 2λπsλγ
)]+ 4sλ2
γ
((M2
N + s(2βπβγ minus 1))2 minus 4λ2πs
2λ2γ
)times (M2
N minus s) arctanh
(2λπsλγ
M2N + 2βπsβγ minus s
) (B19)
34
σ(π+n)LT =
e2g2AM
3Nλπ
32πf2πQs
3(sminusM2N )2λ4
γ
2sλγ
[(M2
N minus s)(Q2 + 2sβ2γ)(Q2 + 2sβγβπ)
minus 4s
((M2
N minus s)(Q2 minus 2s(απ minus 1)βγ
)+Q2sβγβπ
)λ2γ + 8s3(minus1 + απ)λ4
γ
]minus M2
N minus sλπ
[(Q2 + 2sβ2
γ)(Q2 + 2sβγβπ)2 + 4s2(2(απ minus 1)βγ(Q2 + 2sβγβπ)
minusQ2λ2π
)λ2γ + 8s3(απ minus 1)2λ4
γ
]arctan
(2sλπλγ
Q2 + 2sβγβπ
) (B20)
σ(π0p)LT =
e2g2AM
3Nλπ
16πf2πQs(sminusM2
N )2λγ
1
minus2s(M2N + s(minus1 + 2βγβπ))2λ2
γ + 8s3λ2πλ
4γ
[minus 3M8
N (Q2
+ 2sβ2γ) + 2M4
Ns2
(minus (Q2 + 2sβ2
γ)(minus3 + 2βγβπ) +(Q2(minus3 + 2βγβπ + 6λ2
π) + 2sβγ
times (minus3 + 12απ + 2βγβπ minus 12απβγβπ + 4βγλ2π))λ2γ minus 4s(α2
π minus 1)λ4π
)+ 2M2
Ns3
[minus (Q2
+ 2sβ2γ)(minus1 + 2βγβπ)
(6 + βγβπ(minus9 + 2βγβπ)
)+
(Q2(3minus 12λ2
π + 4βγβπ(minus1 + 2βγβπ
+ λ2π))minus 2sβγ
(minus 3 + 4απ(3 + 2απ(3 + 2βγβπ(minus3 + βγβπ)) + 4βγ(βπ + (2minus βγβπ)
times λ2π)))λ2γ + 8s
((απ minus 1)(1 + απ minus 2βγβπ) + 2απβγλ
2π
)λ4γ
)+ s4
((Q2 + 2sβ2
γ)
times (1minus 2βγβπ)2(2βγβπ minus 3) + 2(Q2(minus1 + 6λ2
π + 2βγβπ((1minus 2βγβπ)2 minus 2λ2π)) + 2sβγ
times (minus1 + 2βγβπ + 4(minus1 + βγβπ)(minusαπ + 2απβγβπ minus βγλ2π)))λ2γ minus 8
(s(απ minus 1)(απ
+ (1minus 2βγβπ)2) + 2βγ(2sαπ +Q2βγ)λ2π
)λ4γ + 32s(απ minus 1)λ2
πλ6γ
)+ 2M6
Ns
(Q2(6
minus 7βγβπ + λ2γ) + 2sβγ(βγ(6minus 7βγβπ) + (1minus 4απ)λ2
γ)
)]+sminusM2
N
4s2λπλ3γ
[(Q2 + 2sβ2
γ)
times(3M2
N + s(2βγβπ minus 3))(M2N + s(2βγβπ minus 1)
)+ 2s
(minusM2
N (Q2 + 2sβγ minus 8sαπβγ)
+ s(2sβγ(1minus 4απ + 4απβγβπ) +Q2(1minus 2λ2π)))λ2γ + 8s3(απ minus 1)λ4
γ
]
times arctan
(2sλπλγ
M2N + s(2βγβπ minus 1)
) (B21)
35
σ(πminusp)LT =
e2g2AM
3N
64πf2πQs
3λ4γ(M2
N minusQ2 minus s)(M2N + s(2βπβγ + 2λπλγ minus 1))
times
1
M2N + s(2βπβγ minus 2λπλγ minus 1)
[4λπsλγ(M2
N minusQ2 minus s)(
2sλ2γ
(s((1minus 2λ2
π)Q2
+ 2(2απ minus 1)sβγ(2βπβγ minus 1))minusM2N (Q2 + 2(1minus 2απ)sβγ)
)+ (Q2 + 2sβ2
γ)(M2N + s(2βπβγ minus 1)
)2 + 8(απ minus 1)απs
3λ4γ
)]minus(M2N + s(2βπβγ + 2λπλγ minus 1)
)times[
log
(minusM2N minus 2βπsβγ minus 2λπsλγ + s
minusM2N minus 2βπsβγ + 2λπsλγ + s
)(minus (Q2 + 2sβ2
γ)(M2N + s(2βπβγ minus 1)
)(minusM2
N + 2Q2 + 2βπsβγ + s)minus 2sλ2γ
(minusM2
N (Q2 + 2sβγ) +Q4 +Q2s(4απβγ
minus 2βγ minus 2λ2π + 1) + 2s2βγ(4(απ minus 1)βπβγ + 1)
)minus 8(απ minus 1)2s3λ4
γ
)+ 2
(4s2λ2
γ
(2(απ minus 1)βγ(Q2 + 2βπsβγ)minus λ2
πQ2)
+ (Q2 + 2βπsβγ)2(Q2 + 2sβ2γ)
+ 8(απ minus 1)2s3λ4γ
)arctanh
(2λπsλγ
Q2 + 2βπsβγ
)] (B22)
2 ∆-production channel
The tree-level ∆-exchange diagram in Fig 2 of Ref [30] contributes to the non-Born part of
the VVCS amplitudes The contribution of the ∆ exchange to the VVCS amplitudes can be split
into [17]
S∆-exch1 (νQ2) = S∆-pole
1 (νQ2) + S∆-exch1 (νQ2) (B23a)
νS∆-exch2 (νQ2) = νS∆-pole
2 (νQ2) + νS2∆-exch
(νQ2) (B23b)
and similarly for the unpolarized VVCS amplitudes discussed in Ref [30] Here we introduced the
∆-pole contributions S∆-polei and the ∆-non-pole contributions S∆-exch
i The former amplitudes
feature a pole at the ∆(1232)-production threshold and thus are proportional to
1
[sminusM2∆][uminusM2
∆]=
1
4M2N
1
ν2∆ minus ν2
(B24)
36
They can be reconstructed from the dispersion relations in Eq (5) with the tree-level ∆-production
cross sections as input cf Fig 10 in Ref [30]
σTT (νQ2) =π2α
M2NM
2+|~q |
minus g2
MMN (M+ + ν)|~q |2 +g2E(∆minus ν)(Q2 minusMNν)2
MN(B25a)
+g2CQ
4s(∆minus ν)
MNM2∆
minus 4gMgE(Q2 minusMNν)|~q |2 minus 4gMgCQ2|~q |2
+2gEgCQ
2[minusMNM∆ |~q |2 + s(Q2 +∆ν)
]MNM∆
δ(ν minus ν∆)
σLT (νQ2) =Qπ2α
M2NM
2+|~q |
g2E(MNν minusQ2) [M∆(MN + ν)minus s]
MN(B25b)
+g2CQ
2[MNM∆|~q |2 minus s(Q2 +∆ν)
]MNM2
∆
+ gMgEM∆|~q |2 minusgMgC(Q2 minusMNν)|~q |2
M∆
+gEgC(ν minus∆)(M2
N |~q |2 minus 2Q2s)
MNM∆
δ(ν minus ν∆)
with ∆ = M∆ minusMN M+ = M∆ + MN and the Mandelstam variable s = M2N + 2MNν minus Q2
Analytical expressions for the spin structure functions g1(xQ2) and g2(xQ2) can be constructed
from Eq (3) with the flux factor K(νQ2) = |~q | =radicν2 +Q2
In the ∆-non-pole contributions to S1(νQ2) and νS2(νQ2) the pole in ν at the ∆(1232)-
production threshold has canceled out
S∆-exch1 (νQ2) =
πα
MNM2+
[g2MQ
2+ + g2
E
(∆2 minus 3Q2
)+
4g2CQ
4
M2∆
minus 8gMgEM∆ωminus (B26a)
minus2gMgCQ2(MN minus 4M∆)
M∆+
2gEgCQ2(3MN minus 2M∆)
M∆
]
νS2∆-exch
(νQ2) =2πα
MNM2+
[g2EM∆∆ωminus +
g2M MNQ
2+
2+g2C Q
2(Q2 minus∆2)
2M∆(B26b)
+gEgM M∆(M∆ω+ minus 4MNωminus)minus gEgC ∆(2Q2 +MNω+)
+gMgC Q2(4MN minus ω+)
]+S∆-exch
2 (νQ2)
ν
[M2
∆ ω2+
M2N
+ ν2
]
with Q+ =radic
(M∆ +MN )2 +Q2 and ωplusmn = (M2∆minusM2
N plusmnQ2)2M∆ and the non-pole contribution
to S2(νQ2)
S∆-exch2 (νQ2) = minus2παMNν
M∆M2+
[gM + gE
]gC (B27)
These amplitudes to the contrary are not described by the tree-level ∆-production cross sections
in the standard dispersive approach [17] This peculiarity has been previously missed e g in the
calculation of the ∆-exchange contribution to the hydrogen hyperfine splitting in Ref [88] The
importance of including the ∆-non-pole contribution is also evident when considering the BC sum
37
rule in Eq (14) The ∆-pole terms by themselves violate the BC sum rule but cancel exactly with
the ∆-non-pole terms
limνrarr0
νS∆-pole2 (νQ2) + lim
νrarr0νS2
∆-exch(νQ2) = 0 (B28)
Appendix C Polarizabilities at Q2 = 0
In this section we give analytical expressions for the static values and slopes at Q2 = 0 of
the spin polarizabilities and moments of polarized structure functions In particular we give the
HB expansion of the πN -loop contributions and the ∆-exchange contributions The complete
expressions also for the π∆-loop contributions can be found in the Supplemented material Recall
that IA(0) = I1(0) = d2(0) = 0 and dd2(Q2)dQ2
∣∣∣Q2=0
= 0
1 πN-loop contribution
Here we give analytical expressions for the πN -loop contributions to the proton and neutron
spin polarizabilities expanded in powers of micro = mπMN viz the HB expansion Note that we
choose to expand here to a high order in micro the strict HB expansion would only retain the leading
term in an analogous NLO calculation
bull Static polarizabilities (Q2 = 0)
γ0p =e2g2
A
96π3f2πm
2π
1minus 21πmicro
8minus(
59
2+ 26 logmicro
)micro2 +
1875πmicro3
64
+3
(3
2+ 26 logmicro
)micro4 +
(C1)
γ0n =e2g2
A
48π3f2πm
2π
1
2minus 9πmicro
16minus 2micro2 logmicro+
75πmicro3
128minus 3micro4
4+
(C2)
δLTp =e2g2
A
192π3f2πm
2π
1minus 9πmicro
8+
(13
2minus 2 logmicro
)micro2 minus 465πmicro3
64
minus(
47
2+ 42 logmicro
)micro4 +
(C3)
δLTn =e2g2
A
96π3f2πm
2π
1
2+
3πmicro
16+ (1 + 2 log micro)micro2 minus 105πmicro3
128+
5micro4
4+
(C4)
38
δLTn =e2g2
A
96π3f2πm
2π
1
2+
3πmicro
16+ (1 + 2 log micro)micro2 minus 105πmicro3
128+
5micro4
4+
(C5)
γ0p =e2g2
A
16π3f2πm
4π
4
45minus 3πmicro
16+
14micro2
5minus 1813πmicro3
384minus 192
5(1 + log micro)micro4
+80703πmicro5
2048+
(C6)
γ0n =e2g2
A
16π3f2πm
4π
4
45minus 5πmicro
48+
4micro2
5minus 245πmicro3
384minus 32micro4 logmicro
15
+1323πmicro5
2048+
(C7)
bull Slopes of polarizabilities at Q2 = 0
dγ0p(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
2minus 45πmicro
4+ 223micro2 minus 28515πmicro3
64
minus9
(1953
4+ 449 logmicro
)micro4 +
570255πmicro5
128+
(C8)
dγ0n(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
2minus 81πmicro
8+ 94micro2 minus 2535πmicro3
32
minus3 (1 + 90 logmicro)micro4 +84315πmicro5
1024+
(C9)
dδLTp(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
2880π3f2πm
4π
minus5
2minus 27πmicro
32+ 20micro2 minus 5865πmicro3
256
+3
(617
4+ 36 logmicro
)micro4 minus 2056845πmicro5
4096+
(C10)
dδLTn(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
minus5
4minus 81πmicro
64minus 11micro2 +
10005πmicro3
512
+15
8(11 + 48 logmicro)micro4 minus 267015πmicro5
8192+
(C11)
dIAp(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
96π2f2πmicro
2
1minus 15πmicro
4minus 1
2(115 + 88 logmicro)micro2 +
1839πmicro3
32
+5 (5 + 34 logmicro)micro4 +
(C12)
dIAn(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
48π2f2πmicro
2
1
2minus 11πmicro
8minus 1
4(1 + 20 logmicro)micro2 +
99πmicro3
64
minus25micro4
12+
(C13)
dI1p(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
96π2f2πmicro
3π
8+ 2 (4 + 3 logmicro)microminus 537πmicro2
64
minus1
2(15 + 56 logmicro)micro3 +
(C14)
39
dI1n(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
48π2f2πmicro
minus π
16+
1
4(3 + 4 logmicro)microminus 57πmicro2
128+
2micro3
3+
(C15)
γ0p(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2g2
A
16π3f2πm
6π
1
105minus 23πmicro
256+
377micro2
210minus 15551πmicro3
6144+
3371micro4
105
minus1640457πmicro5
32768+
(C16)
γ0n(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2g2
A
16π3f2πm
6π
1
105minus 153πmicro
1792+
69micro2
70minus 4615πmicro3
6144+
172micro4
35
minus120897πmicro5
32768+
(C17)
2 ∆-exchange contribution
Here we give analytical expressions for the tree-level ∆-exchange contributions to the nucleon
spin polarizabilities and their slopes at Q2 = 0 Note that the ∆-exchange contributes equally to
proton and neutron polarizabilities Recall that for the magnetic γlowastN∆ coupling we introduced a
dipole form factor to mimic vector-meson dominance gM rarr gM(1 +Q2Λ2)2
bull Static polarizabilities (Q2 = 0)
γ0 = minus e2
4πM2+
(g2M
∆2+
g2E
M2+
minus 4gMgEM+∆
) (C18)
δLT =e2M∆
4πM3+
(g2E
MNM++gMgE∆MN
minus gEgCM2
∆
) (C19)
γ0 =e2M2
N
π∆2M4+
(minusg
2M
∆2+
g2E
M2+
+4gMgE∆M+
) (C20)
bull Slopes of polarizabilities at Q2 = 0
dγ0(Q2)
dQ2
∣∣∣∣∣Q2=0
= minus e2
πM2+∆
(g2M
∆
[1
4∆2minus 1
∆M++
1
2M2+
]minus 1
Λ2
g2M
∆+
g2E
2M2+
[1
2∆minus 3
M+
]minusgMgE
M+
[1
∆2minus 5
∆M++
1
M2+
]+
1
Λ2
2gMgEM+
+2gMgC∆M2
+
minus gEgCM3
+
) (C21)
40
d δLT (Q2)
dQ2
∣∣∣∣∣Q2=0
=e2M∆∆
4πMNM2+
(g2E
∆2M2+
[1
∆minus 4
M+
]minus g2
C
∆M2∆M
2+
+gMgE∆2M+
[1
∆2
minus 3
∆M++
1
M2+
]minus 2
Λ2
gMgE∆2M+
+gMgC∆M2
∆
[1
2∆2minus 2
∆M++
1
2M2+
]minus gEgC
2M2∆M
2+
[7
∆+
1
M+
]) (C22)
dIA(Q2)
dQ2
∣∣∣∣∣Q2=0
= minusM2N
M2+
(g2M
2∆2+
g2E
MNM+minus 2gMgE
∆M+minus gEgCM∆M+
) (C23)
dI1(Q2)
dQ2
∣∣∣∣∣Q2=0
= minusM∆M2N
2M3+
(g2E
MNM∆minus gMgE∆MN
minus gEgCM2
∆
) (C24)
γ0(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2M2
N
π∆3M6+
(g2M
∆
[3M2
∆ + 2M∆MN minus 9M2N
∆2+
4M2+
Λ2
]
minus4gMgE
[5M2
∆ minus 9M2N
M+∆2+
2M+
Λ2
]+
g2E
∆M+[7M∆ minus 9MN ]
minus8gMgC∆
+4gEgCM+
) (C25)
[1] D Drechsel B Pasquini and M Vanderhaeghen Dispersion relations in real and virtual
Compton scattering Phys Rept 378 99 (2003) hep-ph0212124
[2] S E Kuhn J-P Chen and E Leader Spin structure of the nucleon mdash status and recent
results Prog Part Nucl Phys 63 1 (2009) arXiv08123535 [hep-ph]
[3] F Hagelstein R Miskimen and V Pascalutsa Nucleon polarizabilities from Compton scat-
tering to hydrogen atom Prog Part Nucl Phys 88 29 (2016) arXiv151203765 [nucl-th]
[4] B Pasquini and M Vanderhaeghen Dispersion theory in electromagnetic interactions Ann
Rev Nucl Part Sci 68 75 (2018) arXiv180510482 [hep-ph]
[5] S Gerasimov A Sum rule for magnetic moments and the damping of the nucleon magnetic
moment in nuclei Sov J Nucl Phys 2 430 (1966)
[6] S Drell and A C Hearn Exact sum rule for nucleon magnetic moments Phys Rev Lett
16 908 (1966)
[7] J S Schwinger Source theory viewpoints in deep inelastic scattering Proc Natl Acad Sci
USA 72 1 (1975)
[8] J S Schwinger Source theory viewpoints in deep inelastic scattering Electromagnetic In-
teractions and Field Theory Proceedings 14 Internationale Universitatswochen Schladming
Austria February 24-March 7 1975 Acta Phys Austriaca Suppl 14 471 (1975)
41
[9] J Schwinger Source theory discussion of deep inelastic scattering with polarized particles
Proc Natl Acad Sci USA 72 1559 (1975)
[10] A M Baldin Polarizability of nucleons Nucl Phys 18 310 (1960)
[11] M Gell-Mann M L Goldberger and W E Thirring Use of causality conditions in quantum
theory Phys Rev 95 1612 (1954)
[12] A Pineda Leading chiral logarithms to the hyperfine splitting of the hydrogen and muonic
hydrogen Phys Rev C 67 025201 (2003)
[13] C Peset and A Pineda The two-photon exchange contribution to muonic hydrogen from
chiral perturbation theory Nucl Phys B 887 69 (2014) arXiv14064524 [hep-ph]
[14] C Peset and A Pineda Model-independent determination of the two-photon exchange contri-
bution to hyperfine splitting in muonic hydrogen JHEP 04 060 arXiv161205206 [nucl-th]
[15] F Hagelstein and V Pascalutsa Proton structure in the hyperfine splitting of muonic hydro-
gen PoS CD15 077 (2016) arXiv151104301 [nucl-th]
[16] F Hagelstein Exciting Nucleons in Compton Scattering and Hydrogen-Like Atoms PhD
thesis Mainz U Inst Kernphys (2017) arXiv171000874 [nucl-th]
[17] F Hagelstein ∆(1232)-Resonance in the hydrogen spectrum Proceedings 11th International
Workshop on the Physics of Excited Nucleons (NSTAR 2017) Columbia SC USA August
20-23 2017 Few Body Syst 59 93 (2018) arXiv180109790 [nucl-th]
[18] Y Prok et al (CLAS) Moments of the spin structure functions gp1 and gd1 for 005 lt Q2 lt 30
GeV2 Phys Lett B 672 12 (2009) arXiv08022232 [nucl-ex]
[19] H Dutz et al (GDH) First measurement of the Gerasimov-Drell-Hearn sum rule for 1H from
07 GeV to 18 GeV at ELSA Phys Rev Lett 91 192001 (2003)
[20] M Amarian et al (Jefferson Lab E94010) Measurement of the generalized forward spin po-
larizabilities of the neutron Phys Rev Lett 93 152301 (2004) arXivnucl-ex0406005
[21] M Amarian et al The Q2 evolution of the generalized Gerasimov-Drell-Hearn integral for
the neutron using a 3He target Phys Rev Lett 89 242301 (2002) arXivnucl-ex0205020
[22] M Amarian et al (Jefferson Lab E94-010)Q2 evolution of the neutron spin structure moments
using a 3He target Phys Rev Lett 92 022301 (2004) arXivhep-ex0310003
[23] A Deur et al Experimental determination of the evolution of the Bjorken integral at low Q2
Phys Rev Lett 93 212001 (2004) arXivhep-ex0407007
[24] K Slifer Low Q2 measurement of gp2 and the δLT spin polarizability Spin structure at long dis-
tance Proceedings Workshop Newport News USA March 12ndash13 2009 nucl-ex09064775
(2009) AIP Conf Proc 1155 10106313203293 (2009) arXiv09064775 [nucl-ex]
[25] P Solvignon et al (E01-012) Moments of the neutron g2 structure function at intermediate
42
Q2 Phys Rev C 92 015208 (2015) arXiv13044497 [nucl-ex]
[26] A Deur Experimental studies at low Q2 of the spin structure of the nucleon at Jefferson Lab
in 9th International Workshop on Chiral Dynamics (CD18) Durham NC USA September
17-21 2018 (2019) arXiv190305661 [nucl-ex]
[27] V Sulkosky et al (Jefferson Lab E97-110) Measurement of the 3He spin-structure functions
and of neutron (3He) spin-dependent sum rules at 0035 le Q2 le 024 GeV2 Phys Lett B
805 135428 (2020) arXiv190805709 [nucl-ex]
[28] K Adhikari et al (CLAS) Measurement of the Q2 Dependence of the Deuteron Spin Structure
Function g1 and its Moments at Low Q2 with CLAS Phys Rev Lett 120 062501 (2018)
arXiv171101974 [nucl-ex]
[29] V Lensky F Hagelstein A Hiller Blin and V Pascalutsa Comment on rdquoMeasurement of
the Q2 Dependence of the Deuteron Spin Structure Function g1 and its Moments at Low Q2
with CLASrdquo (2018) arXiv180603219 [nucl-th]
[30] J M Alarcn F Hagelstein V Lensky and V Pascalutsa Forward doubly-virtual Compton
scattering off the nucleon in chiral perturbation theory at NLO the subtraction function and
moments of unpolarized structure functions (2020) arXiv200509518 [hep-ph]
[31] V Bernard E Epelbaum H Krebs and U-G Meiszligner New insights into the spin structure
of the nucleon Phys Rev D 87 054032 (2013) arXiv12092523 [hep-ph]
[32] V Lensky J M Alarcon and V Pascalutsa Moments of nucleon structure functions at
next-to-leading order in baryon chiral perturbation theory Phys Rev C 90 055202 (2014)
arXiv14072574 [hep-ph]
[33] V Pascalutsa and M Vanderhaeghen Chiral effective-field theory in the ∆(1232) region
I Pion electroproduction on the nucleon Phys Rev D 73 034003 (2006) arXivhep-
ph0512244
[34] V Pascalutsa and M Vanderhaeghen Electromagnetic nucleon-to-Delta transition in chiral
effective field theory Phys Rev Lett 95 232001 (2005) arXivhep-ph0508060
[35] T R Hemmert B R Holstein and J Kambor Systematic 1M expansion for spin 32
particles in baryon chiral perturbation theory Phys Lett B 395 89 (1997) arXivhep-
ph9606456
[36] V Pascalutsa and D R Phillips Effective theory of the ∆(1232) in Compton scattering off
the nucleon Phys Rev C 67 055202 (2003) arXivnucl-th0212024
[37] V Pascalutsa M Vanderhaeghen and S N Yang Electromagnetic excitation of the ∆(1232)-
resonance Phys Rept 437 125 (2007) arXivhep-ph0609004
[38] K A Olive et al (Particle Data Group) Review of Particle Physics Chin Phys C 38 090001
43
(2014)
[39] F E Low Scattering of light of very low frequency by systems of spin 12 Phys Rev 96
1428 (1954)
[40] M Gell-Mann and M L Goldberger Scattering of low-energy photons by particles of spin
12 Phys Rev 96 1433 (1954)
[41] A M Harun ar-Rashid A simple derivation of Schwingerrsquos sum rule for spin dependent
structure functions Nuovo Cim A 33 447 (1976)
[42] F Hagelstein and V Pascalutsa Dissecting the hadronic contributions to (g minus 2)micro by
Schwingerrsquos sum rule Phys Rev Lett 120 072002 (2018) arXiv171004571 [hep-ph]
[43] V Lensky V Pascalutsa M Vanderhaeghen and C Kao Spin-dependent sum rules con-
necting real and virtual Compton scattering verified Phys Rev D 95 074001 (2017)
arXiv170101947 [hep-ph]
[44] H Burkhardt and W N Cottingham Sum rules for forward virtual Compton scattering
Annals Phys 56 453 (1970)
[45] V Lensky and V Pascalutsa Predictive powers of chiral perturbation theory in Compton
scattering off protons Eur Phys J C 65 195 (2010) arXiv09070451 [hep-ph]
[46] H Grieszlighammer J McGovern D Phillips and G Feldman Using effective field theory to
analyse low-energy Compton scattering data from protons and light nuclei Prog Part Nucl
Phys 67 841 (2012) arXiv12036834 [nucl-th]
[47] H W Grieszlighammer J A McGovern and D R Phillips Nucleon polarisabilities at and
beyond physical pion masses Eur Phys J A 52 139 (2016) arXiv151101952 [nucl-th]
[48] E Epelbaum H Krebs and U-G Meiszligner Improved chiral nucleon-nucleon potential up to
next-to-next-to-next-to-leading order Eur Phys J A 51 53 (2015) arXiv14120142 [nucl-
th]
[49] E Epelbaum H Krebs and U-G Meiszligner Precision nucleon-nucleon potential at fifth order
in the chiral expansion Phys Rev Lett 115 122301 (2015) arXiv14124623 [nucl-th]
[50] C W Kao T Spitzenberg and M Vanderhaeghen Burkhardt-Cottingham sum rule and
forward spin polarizabilities in heavy baryon chiral perturbation theory Phys Rev D 67
016001 (2003) arXivhep-ph0209241
[51] C-W Kao D Drechsel S Kamalov and M Vanderhaeghen Higher moments of nucleon
spin structure functions in heavy baryon chiral perturbation theory and in a resonance model
Phys Rev D 69 056004 (2004) arXivhep-ph0312102
[52] T Becher and H Leutwyler Baryon chiral perturbation theory in manifestly Lorentz invariant
form Eur Phys J C 9 643 (1999) arXivhep-ph9901384
44
[53] J Gasser M E Sainio and A Svarc Nucleons with chiral loops Nucl Phys B 307 779
(1988)
[54] L S Geng J Martin Camalich L Alvarez-Ruso and M J Vicente Vacas Leading SU(3)-
breaking corrections to the baryon magnetic moments in chiral perturbation theory Phys
Rev Lett 101 222002 (2008) arXiv08051419 [hep-ph]
[55] D Drechsel S S Kamalov and L Tiator The GDH sum rule and related integrals Phys
Rev D 63 114010 (2001) arXivhep-ph0008306
[56] D Drechsel O Hanstein S S Kamalov and L Tiator A unitary isobar model for pion
photo- and electroproduction on the proton up to 1 GeV Nucl Phys A 645 145 (1999)
arXivnucl-th9807001
[57] L Tiator private communication (2020)
[58] V Bernard T R Hemmert and U-G Meiszligner Spin structure of the nucleon at low energies
Phys Rev D 67 076008 (2003) arXivhep-ph0212033
[59] O Gryniuk F Hagelstein and V Pascalutsa Evaluation of the forward Compton scattering
off protons II Spin-dependent amplitude and observables Phys Rev D 94 034043 (2016)
arXiv160400789 [nucl-th]
[60] R Zielinski The g2p Experiment A Measurement of the Protonrsquos Spin Structure Functions
PhD thesis New Hampshire U (2010) arXiv170808297 [nucl-ex]
[61] N Guler et al (CLAS) Precise determination of the deuteron spin structure at low to mod-
erate Q2 with CLAS and extraction of the neutron contribution Phys Rev C 92 055201
(2015) arXiv150507877 [nucl-ex]
[62] C E Carlson V Nazaryan and K Griffioen Proton structure corrections to electronic
and muonic hydrogen hyperfine splitting Phys Rev A 78 022517 (2008) arXiv08052603
[physicsatom-ph]
[63] V Bernard N Kaiser and U-G Meiszligner Chiral dynamics in nucleons and nuclei Int J
Mod Phys E 4 193 (1995) arXivhep-ph9501384
[64] T R Hemmert B R Holstein and J Kambor ∆(1232) and the polarizabilities of the nucleon
Phys Rev D 55 5598 (1997) arXivhep-ph9612374
[65] V Pascalutsa and R Timmermans Field theory of nucleon to higher-spin baryon transitions
Phys Rev C 60 042201 (1999) arXivnucl-th9905065
[66] V Pascalutsa Quantization of an interacting spin-32 field and the Delta isobar Phys Rev
D 58 096002 (1998) arXivhep-ph9802288
[67] H Krebs Double Virtual Compton Scattering and SpinStructure of the Nucleon PoS
CD2018 031 (2019)
45
[68] N Kochelev and Y Oh Axial anomaly and the δLT puzzle Phys Rev D 016012 (2012)
arXiv11034892 [hep-ph]
[69] D Drechsel S Kamalov and L Tiator Unitary isobar model ndash MAID2007 Eur Phys J A
69 (2007) available at httpsmaidkphuni-mainzde arXiv07100306 [nucl-th]
[70] P J Mohr B N Taylor and D B Newell CODATA recommended values of the fundamental
physical constants 2010 Rev Mod Phys 84 1527 (2012)
[71] J Ahrens et al (GDH A2) First measurement of the Gerasimov-Drell-Hearn integral for 1H
from 200 to 800 MeV Phys Rev Lett 87 022003 (2001) arXivhep-ex0105089 [hep-ex]
[72] K Helbing (GDH) Experimental verification of the GDH sum rule at ELSA and MAMI Nucl
Phys Proc Suppl 105 113 (2002)
[73] R Bradford A Bodek H S Budd and J Arrington A New parameterization of the nucleon
elastic form-factors NuInt05 proceedings of the 4th International Workshop on Neutrino-
Nucleus Interactions in the Few-GeV Region Okayama Japan 26-29 September 2005 Nucl
Phys Proc Suppl 159 127 (2006) arXivhep-ex0602017
[74] J D Bjorken Applications of the chiral U(6)timesU(6) algebra of current densities Phys Rev
148 1467 (1966)
[75] J D Bjorken Inelastic scattering of polarized leptons from polarized nucleons Phys Rev D
1 1376 (1970)
[76] V Bernard T R Hemmert and U-G Meiszligner Novel analysis of chiral loop effects in
the generalized Gerasimov-Drell-Hearn sum rule Phys Lett B 545 105 (2002) arXivhep-
ph0203167
[77] A Deur et al Experimental study of isovector spin sum rules Phys Rev D 78 032001
(2008) arXiv08023198 [nucl-ex]
[78] B Pasquini P Pedroni and D Drechsel Higher order forward spin polarizability Phys Lett
B 687 160 (2010) arXiv10014230 [hep-ph]
[79] R Jaffe g2ndashThe nucleonrsquos other spin-dependent structure function Comments Nucl Part
Phys 19 239 (1990)
[80] E V Shuryak and A Vainshtein Theory of power corrections to deep inelastic scattering in
quantum chromodynamics (II) Qminus4 effects polarized target Nucl Phys B 201 141 (1982)
[81] S Wandzura and F Wilczek Sum rules for spin dependent electroproduction Test of rela-
tivistic constituent quarks Phys Lett B 72 195 (1977)
[82] B W Filippone and X-D Ji The spin structure of the nucleon Adv Nucl Phys 26 1
(2001) arXivhep-ph0101224
[83] M Burkardt The g2 structure function Proceedings Workshop on Spin structure at long
46
distance Newport News USA March 12-13 2009 AIP Conf Proc 1155 26 (2009)
arXiv09054079 [hep-ph]
[84] A Deur S J Brodsky and G F de Teramond The spin structure of the nucleon Rept
Prog Phys 82 076201 (2019) arXiv180705250 [hep-ph]
[85] B R Holstein D Drechsel B Pasquini and M Vanderhaeghen Higher order polarizabilities
of the proton Phys Rev C 61 034316 (2000) arXivhep-ph9910427
[86] B R Holstein V Pascalutsa and M Vanderhaeghen Sum rules for magnetic moments and
polarizabilities in QED and chiral effective-field theory Phys Rev D 72 094014 (2005)
arXivhep-ph0507016
[87] B Pasquini D Drechsel and L Tiator Invariant amplitudes for pion electroproduction Eur
Phys J A 34 387 (2007) arXiv07122327 [hep-ph]
[88] A J Buchmann Non-spherical proton shape and hydrogen hyperfine splitting Proceedings
International Workshop on Precision Physics of Simple Atomic Systems (PSAS 2008) Wind-
sor Ontario Canada July 21-26 2008 Can J Phys 87 773 (2009) arXiv09104747
[physicsatom-ph]
47
IA(Q2) and I1(Q2) The latter will be discussed in Sec III D The former is defined as4
minus α
2M2N
IA(Q2) = minus 1
8π2
int infinν0
dν
radic1 +
Q2
ν2
σTT (νQ2)
ν(26)
=α
Q2
int x0
0
dx[4M2
Nx2
Q2g2(xQ2)minus g1(xQ2)
]
Due to its energy weighting the integral in Eq (26) converges slower than the one in the
generalized forward spin polarizability sum rule (19) Therefore knowledge of the cross
section at higher energies is required and the evaluation of the generalized GDH integral
IA(Q2) is not as simple as the evaluation of γ0(Q2)
The generalized GDH integral IA(Q2) is directly related to the non-pole amplitude
gnonpoleTT (νQ2) which differs from non-Born amplitude gTT (νQ2) by a term involving the
elastic Pauli form factor
gnonpoleTT (νQ2) = gTT (νQ2)minus 2παν
M2N
F 22 (Q2) (27)
cf Eqs (2a) and (8a) Consequently IA(Q2) is not a pure polarizability but also contains
an elastic contribution The ldquonon-polarizabilityrdquo or the Born part of IA(Q2) is given by
IBornA (Q2) = IA(Q2)minus∆IA(Q2) = minus1
4F 2
2 (Q2) (28)
where we refer to the polarizability part as ∆IA(Q2) The same is true for the generalized
GDH integral I1(Q2) which is directly related to Snonpole1 (νQ2)
IBorn1 (Q2) = I1(Q2)minus∆I1(Q2) = minus1
4F 2
2 (Q2) (29)
In the following we will add the Born parts to our LO and NLO BχPT predictions for
the polarizabilities ∆IA(Q2) and ∆I1(Q2) employing an empirical parametrization for the
elastic Pauli form factor [73] This allows us to compare to the experimental results for
IA(Q2) and I1(Q2) cf Fig 4 Note that the blue error bands only describe the uncertainties
of our BχPT predictions of the polarizabilities while the elastic contributions are considered
to be exact as explained in Sec II C The uncertainties of the polarizability predictions are
therefore better reflected in Fig 5 where we show the contributions of the different orders
to the BχPT predictions of ∆IA(Q2) and ∆I1(Q2) as well as the total results with error
bands
4 Note that IA(Q2) is sometimes called ITT (Q2)
18
000 005 010 015 020 025 030
-08
-06
-04
-02
00
02
Q2 GeV2D
DI A
proton
000 005 010 015 020 025 030
-10
-08
-06
-04
-02
00
02
Q2 GeV2D
DI A
neutron
000 005 010 015 020 025 030-005
000
005
010
015
020
025
Q2 GeV2D
DI 1
proton
000 005 010 015 020 025 030-015
-010
-005
000
005
010
015
Q2 GeV2D
DI 1
neutron
FIG 5 Contributions of the different orders to the chiral predictions of ∆IA(Q2) upper paneland ∆I1(Q2) lower panel for the proton (left) and neutron (right) Red solid line πN -loop
contribution green dot-dashed line ∆-exchange contribution orange dotted line π∆-loop contri-
bution blue solid line and blue band total result purple dot-dot-dashed line total result without
gC contribution black short-dashed line total result without gM dipole
The E97-110 experiment at Jefferson Lab has recently published their data for IAn(Q2) in
the region of 0035 GeV2 lt Q2 lt 024 GeV2 [27] In addition there are results for IAn(Q2)
from the earlier E94-010 experiment [21] and for IAp(Q2) from the E08-027 experiment [60]
The O(p4) HB calculation gives a large negative effect [51] which does not describe the data
The BχPT+∆ result from Ref [31] which mainly differs from our work by the absence of
the dipole form factor in gM looks similar to this HB result and only describes the data
points at lowest Q2 Our NLO prediction however follows closely the Q2 evolution of the
data In Fig 5 upper panel we show the polarizability ∆IA(Q2) whose Q2 evolution is
clearly dominated by the ∆ exchange Similar to the case of γ0p(Q2) inclusion of the dipole
in gM and the Coulomb coupling gC is very important in order to describe the experimental
data The LO prediction on the other hand slightly overestimates the data cf Fig 4
upper panel
19
At the real-photon point IA(0) = minusκ2
4and ∆IA(0) = 0 Therefore we give only the
slope of the polarizability ∆IA(Q2) [showing also the separate contributions from πN loops
∆ exchange and π∆ loops] in units of GeVminus2
d∆IAp(Q2)
dQ2
∣∣∣∣Q2=0
= minus858(343) asymp 238minus 1121 + 025 (30a)
d∆IAn(Q2)
dQ2
∣∣∣∣Q2=0
= minus955(343) asymp 141minus 1121 + 025 (30b)
Including the empirical Pauli form factor [73] we find in units of GeVminus2
dIAp(Q2)
dQ2
∣∣∣∣Q2=0
= minus318dIAn(Q2)
dQ2
∣∣∣∣Q2=0
= minus300 (31)
D Γ1(Q2) and I1(Q2) mdash the first moment of the structure function g1(xQ2)
The second variant for a generalization of the GDH sum rule to finite Q2 is defined as
minus α
2M2N
I1(Q2) = minus 1
8π2
int infinν0
dν1radic
ν2 +Q2
[σTT (νQ2) +
Q
νσLT (νQ2)
](32)
= minus α
Q2
int x0
0
dx g1(xQ2)
where I1(0) = minusκ2
4 This generalized GDH integral directly stems from the amplitude
Snonpole1 (νQ2) with the LEX from Eq (12a) It is given by the first moment of the struc-
ture function g1(xQ2) Γ1(Q2) =int x0
0dx g1(xQ2) as follows I1(Q2) =
2M2N
Q2 Γ1(Q2) The
isovector combination
Γ1(pminusn)(Q2) =
int x0
0
dx[g1p(xQ
2)minus g1n(xQ2)] (33)
is related to the axial coupling of the nucleon through the Bjorken sum rule [74 75]
limQ2rarrinfin
Γ1(pminusn)(Q2) =
gA6 (34)
As explained in Eq (28) the moment I1(Q2) splits into a polarizability part ∆I1(Q2) and
a Born part IBorn1 (Q2) Figure 4 lower panel shows the Q2 dependence of I1(Q2) which in
contrast to IA(Q2) shown in Figure 4 upper panel is clearly dominated by its Born part
and the elastic Pauli form factor The πN -loop ∆-exchange and π∆-loop contributions to
the polarizability ∆I1(Q2) are shown in Fig 5 lower panel Comparing to Fig 5 upper
20
000 005 010 015 020 025 030
-006
-004
-002
000
002
004
Q2 GeV2D
G1
proton
000 005 010 015 020 025 030
-006
-005
-004
-003
-002
-001
000
001
Q2 GeV2D
G1
neutron
FIG 6 First moment of the structure function g1(xQ2) for the proton (left) and neutron (right)
as function of Q2 The legend is the same as in Fig 4
panel one sees that ∆I1(Q2) is less sensitive to gC and the dipole form factor in gM than
∆IA(Q2)
For the proton our NLO BχPT prediction gives a very good description of the ex-
perimental data [18 60] and is in reasonable agreement with the MAID prediction [69]
For the neutron one observes good agreement with the empirical evaluations including ex-
trapolations to unmeasured energy regions starting from Q2 gt 01 GeV2 [27 61] In the
region of Q2 lt 005 GeV2 one observes an interesting tension between the recent E97-
110 experiment [27] and the data from CLAS [61] While the newest measurement finds
I1n(0035 GeV2) lt κ2n4 thus suggesting a negative slope at low Q the older measurement
found a rather large value for I1n(00496 GeV2) A similar but milder behaviour is seen in
the E97-110 [27] and E94-010 [21] data for IAn The MAID predictions do not agree with
the CODATA recommended values for the anomalous magnetic moments of the proton and
neutron [70] which in our work are imposed by using empirical parametrizations for the
elastic Pauli form factors [73] The slope of the HB result from Ref [51] is too large and
therefore only reproduces the data at very low Q2
Figure 6 shows the moment Γ1(Q2) for the proton and neutron while Fig 7 shows the
isovector combination Γ1 pminusn(Q2) The LO and NLO BχPT predictions are identical be-
cause our calculation produces the same Delta contributions for the proton and the neutron
For the isovector combination the MAID model only agrees with the data at very low
Q2 lt 010 GeV2 The same is true for the IR result [58 76] while all other chiral results
describe the data NLO BχPT (this work) BχPT+∆ [31] and HBχPT [51]
At the real-photon point I1(0) = minusκ2
4and ∆I1(0) = 0 Therefore we give only the
slope of the polarizability ∆I1(Q2) [showing also the separate contributions from πN loops
21
000 005 010 015 020 025 030-002
000
002
004
006
008
010
Q2 GeV2D
G1Hp-
nL
FIG 7 Isovector combination of Γ1(Q2) as function of Q2 The legend is the same as in Fig 6
The pink curve is the IR result from Ref [58 76] The experimental points are from Ref [77]
(brown dots) and Ref [23] (orange squares)
∆ exchange and π∆ loops] in units of GeVminus2
d∆I1p(Q2)
dQ2
∣∣∣∣Q2=0
= 039(4) asymp 034minus 053 + 058 (35a)
d∆I1n(Q2)
dQ2
∣∣∣∣Q2=0
= minus101(10) asymp minus107minus 053 + 058 (35b)
Including the empirical Pauli form factor [73] we find in units of GeVminus2
dI1p(Q2)
dQ2
∣∣∣∣Q2=0
= 580dI1n(Q2)
dQ2
∣∣∣∣Q2=0
= 553 (36)
E d2(Q2) mdash a measure of color polarizability
Another interesting moment to consider is d2(Q2) which is related to the twist-3 part of
the spin structure function g2(xQ2) [79 80]
d2(Q2) equiv 3
int 1
0
dx x2[g2(xQ2)minus gWW2 (xQ2)] (37)
where gWW2 (xQ2) is the twist-2 part of g2(xQ2) Using the Wandzura-Wilczek relation [81]
one can relate d2(Q2) to moments of the spin structure functions g1(xQ2) and g2(xQ2)
d2(Q2) =
int 1
0
dx x2 [3g2(xQ2) + 2g1(xQ2)] (38)
22
000 005 010 015 020 025 030
0000
0001
0002
0003
0004
0005
0006
0007
Q2 GeV2D
d2
proton
000 005 010 015 020 025 030
0000
0002
0004
0006
0008
0010
0012
Q2 GeV2D
d2
neutron
000 005 010 015 020 025 030
-1
0
1
2
Q2 GeV2D
Γ 010
-4
fm6
D
proton
000 005 010 015 020 025 030-2
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
6D
neutron
FIG 8 Upper panel The inelastic moment d2(Q2) for the proton (left) and neutron (right) as
function of Q2 The result of this work the NLO BχPT prediction is shown by the blue solid line
and the blue band The red line represents the LO BχPT result The purple short-dashed line is
the O(p4) HB result from Ref [50 51] The black dotted line is the MAID model prediction [69]
The experimental points for the neutron (cyan dots) are from Ref [22] Lower panel Fifth-order
generalized forward spin polarizability γ0(Q2) for the proton (left) and neutron (right) as function
of Q2 The experimental points for the proton are from Ref [59] (purple square) and Ref [78]
(orange dot)
This relation however only holds for asymptotically large Q2 It is also in the high-Q2
region where d2(Q2) is a measure of color polarizability [82 83] through its relation to the
gluon field strength tensor [80] We refer to Ref [84] for a recent review on the spin structure
of the nucleon including a discussion of sum rules for deep inelastic scattering and color
polarizabilities
What we consider in the following is the inelastic part of d2(Q2) defined as the moment
of g1(xQ2) and g2(xQ2) spin structure functions cf Eq (38)
d2(Q2) =
int x0
0
dx x2 [3g2(xQ2) + 2g1(xQ2)] (39)
23
This moment provides another testing ground for our BχPT predictions through comparison
with experiments on the neutron [22] Going towards the low-Q2 region the interpretation
of d2(Q2) in terms of color polarizabilities will fade out The above definition however
implies it is related to other VVCS polarizabilities
d2(Q2) =Q4
8M4N
[M2
NQ2
αδLT (Q2) + I1(Q2)minus IA(Q2)
] (40)
Note that d2(Q2) and its first two derivatives with respect to Q2 vanish at Q2 = 0 The
considerations in Eqs (28) and (29) have no effect on d2(Q2) since the Born contribution
from IA(Q2) and I1(Q2) cancel out Therefore d2(Q2) is a pure polarizability
In Fig 8 upper panel we show our NLO BχPT prediction and other results for d2(Q2)
While MAID [69] and BχPT describe the experimental data for the neutron [22] very well
the HB limit [50 51] is showing a fast growth with Q2 This illustrates the importance
of keeping the relativistic result Note also that even though the πN -loop contribution
is dominant both gC and the form factor in gM are essential to obtain a curvature that
reproduces the data cf Fig 9 upper panel For the proton there are to our knowledge
no experimental results to compare with However the agreement between the NLO BχPT
prediction and the MAID prediction at low energies is reasonable
F γ0(Q2) mdash fifth-order generalized forward spin polarizability
It is interesting to compare the generalized fifth-order forward spin polarizability sum
rule
γ0(Q2) =1
2π2
int infinν0
dν
radic1 +
Q2
ν2
σTT (νQ2)
ν5(41)
=64αM4
N
Q10
int x0
0
dx x4
[g1(xQ2)minus 4M2
Nx2
Q2g2(xQ2)
]
to the sum rule integrals for IA(Q2) and γ0(Q2) since they differ merely by their energy
weighting of σTT (νQ2) and a constant prefactor cf Eqs (19) (26) and (41) From IA(Q2)
to γ0(Q2) to γ0(Q2) the energy suppression is increasing by a factor of νminus2 respectively
Therefore the description of γ0(Q2) should be easiest in a low-energy effective-field theory
such as χPT whereas γ0(Q2) and IA(Q2) receive larger contributions from higher energies
In Fig 8 lower panel we show our LO and NLO BχPT predictions for γ0(Q2) One
can see that the πN -loop contribution is positive (in accordance to what we see for the cross
section σTT see Fig 10) The Delta shifts it substantially especially in the low Q2 region
24
000 005 010 015 020 025 030
-0004
-0002
0000
0002
0004
0006
0008
0010
Q2GeV
2D
d2
proton
000 005 010 015 020 025 030
-0005
0000
0005
0010
0015
Q2GeV
2D
d2
neutron
000 005 010 015 020 025 030
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
6D
proton
000 005 010 015 020 025 030
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
6D
neutron
FIG 9 Contributions of the different orders to the chiral predictions of d2(Q2) upper panel and
γ0(Q2) lower panel for the proton (left) and neutron (right) Red solid line πN -loop contribu-
tion green dot-dashed line ∆-exchange contribution orange dotted line π∆-loop contribution
blue long-dashed line total result purple dot-dot-dashed line total result without gC contribution
black short-dashed line total result without gM dipole
bringing it into a better agreement with data In general the BχPT curves start above the
empirical data points at the real-photon point and then decrease asymptotically to zero
above Q2 gt 01 GeV2 On the other hand the MAID prediction reproduces the empirical
data at the real-photon point then decreases to negative values until about Q2 gt 006 GeV2
from where it also starts to asymptotically approach zero Consequently our NLO BχPT
prediction of γ0(Q2) is consistently above the MAID prediction This is very different to
what we saw for IA(Q2) in Fig 4 upper panel where the MAID prediction at the real-
photon point is above the experimental value While the agreement of our predictions with
the empirical data is in general quite good for all moments of σTT (νQ2) one should point
out that both for γ0n(Q2) and γ0p(Q2) we overestimate the data at low Q2 For IA(Q2) such
observation cannot be made because ∆IA(0) = 0 and thus IA(0) is given by the empirical
Pauli form factor only From IA(Q2) γ0(Q2) and γ0(Q2) the latter has the smallest however
25
non-negligible dependence on gC and the dipole in gM cf Fig 9 lower panel
The πN -loop ∆-exchange and π∆-loop contributions to the NLO BχPT prediction of
the static fifth-order forward spin polarizability amount to in units of 10minus4 fm6
γ0p = 112(30) asymp 208minus 096minus 001 (42a)
γ0n = 195(30) asymp 292minus 096minus 001 (42b)
while the slope is composed as follows in units of 10minus4 fm8
dγ0p(Q2)
dQ2
∣∣∣∣Q2=0
= minus084(10) asymp minus100 + 016 + 000 (43a)
dγ0n(Q2)
dQ2
∣∣∣∣Q2=0
= minus142(15) asymp minus158 + 016 + 000 (43b)
Note that the HB prediction of the static polarizability γ0p (423 at O(p3) and 365 at O(ε3)
[78 85]) is almost one order of magnitude larger than the empirical value and therefore not
shown in Fig 8
G Summary
Our results are summarized in Table II where we give the contributions of the different
orders to the chiral predictions of the polarizabilities and their slopes at the real-photon
point A quantitative comparison of our predictions for the static spin polarizabilities to
the work of Bernard et al [31] and different empirical evaluations is shown in Table III We
can see that the inclusion of the Delta turns out to be very important for all moments of
the helicity-difference cross section To describe the Q2 behavior of the polarizabilities the
magnetic coupling of the N rarr ∆ transition should be modified by a dipole form factor as
has been observed previously in the description of electroproduction data [33] This dipole
form factor effectively takes account of vector-meson exchanges The Coulomb-quadrupole
N rarr ∆ transition despite its subleading order is important in the description of some
moments of spin structure functions This is contrary to what we saw for the moments of
unpolarized structure functions [30] where the Coulomb coupling had a negligible effect
The π∆ loops are mainly relevant for the generalized GDH integrals
26
TABLE II The NLO BχPT predictions for the forward VVCS polarizabilities and their slopes at
Q2 = 0 The contributions of the πN loops the ∆ exchange and the π∆ loops are shown together
with the combined total result Note that IA(0) = I1(0) = d2(0) = 0 and (d2)prime = 0
πN loops ∆ exchange π∆ loops Total
γ0 p 201 minus284 minus010minus093(92)
(10minus4 fm4) n 298 003(92)
δLT p 150 minus016 minus002132(15)
(10minus4 fm4) n 235 218(23)
γ0 p 208 minus096 minus001112(30)
(10minus4 fm6) n 292 195(30)
(γ0)prime p minus033011 001
minus022(4)
(10minus4 fm6) n minus073 minus061(7)
(δLT )prime p minus080 minus004 minus001minus085(8)
(10minus4 fm6) n minus119 minus124(12)
(γ0)prime p minus100016 000
minus084(10)
(10minus4 fm8) n minus158 minus142(15)
(∆IA)prime p 238 minus1121 025minus858(343)
(GeVminus2) n 141 minus955(343)
(∆I1)prime p 034 minus053 058039(4)
(GeVminus2) n minus107 minus101(10)
TABLE III Our NLO BχPT predictions for the spin polarizabilities at Q2 = 0 compared with
the BχPT+∆ predictions from Bernard et al [31] and the available empirical information Where
the reference is not given the empirical number is provided by the MAID analysis [55 69] with
unspecified uncertainty
Proton Neutron
This work BχPT+∆ Empirical This work BχPT+∆ Empirical
γ0 minus093(92) minus174(40) minus100(8)(12) [19] 003(92) minus077(40) minus0005
(10minus4 fm4) minus090(8)(11) [78] [MAID]
minus0929(105) [59]
δLT 132(15) 240(1) 134 218(23) 238(3) 203
(10minus4 fm4) [MAID] [MAID]
γ0 112(30) 060(7)(7) [78] 195(30) 123
(10minus4 fm6) 0484(82) [59] [MAID]
IV CONCLUSIONS
We have presented a complete NLO calculation of the polarized non-Born VVCS ampli-
tudes in covariant BχPT with pion nucleon and ∆(1232) fields The dispersion relations
between the VVCS amplitudes and the tree-level photoabsorption cross sections served as a
27
cross-check of these calculations
The obtained moments of the proton and neutron spin structure functions related to gen-
eralized polarizabilities and GDH-type integrals agree well with the available experimental
data The description of their Q2 evolution is improved compared to the previous χPT pre-
dictions In particular the NLO BχPT predictions obtained here give a better description
of the empirical data (eg from the Jefferson Laboratory ldquoSpin Physics Programrdquo) than
the HB [50 51] and IR [58] calculations
The demonstrated predictive power of the χPT framework amplitudes makes it well
suited for extending the χPT evaluation of the TPE effect in the hyperfine structure of
(muonic-)hydrogen [15ndash17] to next-to-leading order
ACKNOWLEDGEMENTS
We thank Lothar Tiator and Marc Vanderhaeghen for helpful discussions This work
is supported by the Deutsche Forschungsgemeinschaft (DFG) through the Collaborative
Research Center [The Low-Energy Frontier of the Standard Model (SFB 1044)] JMA ac-
knowledges support from the Community of Madrid through the ldquoPrograma de atraccion de
talento investigador 2017 (Modalidad 1)rdquo and the Spanish MECD grants FPA2016-77313-P
FH gratefully acknowledges financial support from the Swiss National Science Foundation
Appendix A Tensor decompositions of the VVCS amplitudes
In this appendix we review the decomposition of the forward VVCS process into tensor struc-
tures and scalar amplitudes In particular we consider the connection between the covariant and
the semi-relativistic decomposition in the lab frame that is defined in terms of the conventional
transverse longitudinal transverse-transverse and transverse-longitudinal amplitudes
As explained in Sec II A the process of forward VVCS off the nucleon can be described in
terms of four explicitly covariant amplitudes S1 2 and T1 2 [3]
T (νQ2) =
(minusgmicroν +
qmicroqν
q2
)T1(νQ2) +
1
M2N
(pmicro minus p middot q
q2qmicro)(
pν minus p middot qq2
qν)T2(νQ2) (A1)
minus 1
MNγmicroναqα S1(νQ2)minus 1
M2N
(γmicroνq2 + qmicroγναqα minus qνγmicroαqα
)S2(νQ2)
εprimelowastmicro εν
where εmicro (εprimelowastmicro ) are the incoming (outgoing) photon polarization vectors ν is the photon lab-frame
energy and Q2 is the photon virtuality Alternatively the decomposition in the laboratory frame
28
(which in the forward case coincides with the Breit frame) is parametrized in terms of the nucleon
Pauli matrices ~σ and the four scalar functions fL fT gTT and gLT
T (νQ2) = ε0 εprimelowast0 fL(νQ2) + (~ε primelowast middot ~ε ) fT (νQ2) + i~σ middot (~ε primelowast times ~ε ) gTT (νQ2) (A2)
minusi~σ middot [(ε0~εprimelowast minus ~ε ε primelowast0 )times q] gLT (νQ2)
Here ~q and q = ~q|~q | are the photon three-momentum in the lab system and its unit vector The
modified polarization vector components are given by
ε0 =
[ε0 minus
ν
|~q | (~ε middot q )
] |~q |Q ~ε = ~εminus q (~ε middot q ) (A3)
where ε = (ε0~ε ) is the usual incoming photon polarization vector and εprimelowast the outgoing polarization
vector The LEX of the lab frame amplitudes [Eq (10)] can serve in particular as the definition
of the generalized polarizabilities The lab frame amplitudes are also conveniently used for the
definition of the response functions see the example of the scalar amplitude gLT (νQ2) and the
corresponding response function σLT (νQ2) below in App B
Appendix B Photoabsorption cross sections
In the forward kinematics the spin-dependent VVCS amplitudes and the spin polarizabilities
can be described in terms of the polarized structure functions g1(xQ2) and g2(xQ2) or equiv-
alently the helicity-difference cross section σTT (νQ2) and the longitudinal-transverse response
function σLT (νQ2) with the help of dispersion relations (5) and the optical theorem (3) In this
way the photoabsorption cross sections measured in electroproduction processes form the basis
for most empirical evaluations shown throughout Sec III In the following we present the BχPT
predictions for the tree-level cross sections of πN - π∆- and ∆-production through photoabsorption
on the nucleon cf Figs 8 9 and 10 in Ref [30] In Secs B 1 and B 2 we will discuss the leading
πN -production channel and the ∆-production channel respectively We used these cross sections
to verify the polarizability predictions obtained otherwise from the calculated non-Born VVCS
amplitudes Due to the bad high-energy behavior of the π∆-production cross sections in BχPT
cf Fig 10 the dispersion relations in Eq (5) require further subtractions for a reconstruction of
the π∆-loop contribution to the spin-dependent VVCS amplitudes Therefore not all polarizabil-
ities could be verified but only those appearing as higher-order terms in the LEX of the VVCS
amplitudes such as γ0 [16]
29
proton neutron
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Neutron
02 04 06 08 100
50
100
150
200
250
[GeV]
T [b]
02 04 06 08 10
-50
0
50
100
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Neutron
02 04 06 08 100
50
100
150
200
250
[GeV]
T [b]
02 04 06 08 10
-50
0
50
100
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02040608100
50
100
150
200
250
300
[GeV]
T[b]
0204060810
-50
0
50
[GeV]
TT[b]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
0204060810
00
02
04
06
[GeV]
LTQ
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
0204060810
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
FIG 10 Photoabsorption cross sections for πN (red) and π∆ production (orange) with Q2 = 0
(solid) and Q2 = 01 GeV2 (dashed for πN and dotted for π∆ channel)
1 πN-production channel
In order to extract the response function σLT (νQ2) we have developed a method similar to
the one used to calculate σTT (νQ2) see for example Ref [86] For σLT (νQ2) however the
calculation is more complicated because one has to take into account that the associated Compton
process involves a spin-flip of the nucleon as illustrated in Fig 11 When calculating the cross
section the product of the incoming nucleon spinors has to reflect this flip
The forward VVCS amplitude related to σLT (νQ2) mdash and δLT (Q2) mdash is gLT (νQ2) It can
be extracted from Eq (A2) if one takes the modified polarization vector components in Eq (A3)
with ε = εL and εprimelowast = εlowastplusmn as input where εL = 1Q(|~q | 0 0 q0) and εplusmn = ∓ 1radic
2(0 1plusmni 0) are the
standard longitudinal and transverse polarization vectors respectively For εL and εlowastplusmn only the
choice of helicities hprime = plusmn12 and h = ∓12 gives a non-zero contribution and one obtains
χdaggerhprime T (νQ2)χh = χdaggerhprimeminusi~σ middot [(ε0~εprimelowast minus ~ε ε primelowast0 )times q] gLT (νQ2)χh =
radic2 gLT (νQ2) (B1)
where χh and χdaggerhprime are two-component Pauli spinors with opposite helicities or here spins
Let us now consider the related photoabsorption process and in particular the tree-level γlowastN rarr
30
s = 12
sprime = minus12
λ = 0 λprime = 1
rprime rprime
s=12sprime=minus1
2
λ=0λprime=1
rprimerprime
s=12
sprime=minus12
λ=0λprime=1
rprimerprime hacute=frac12h=-frac12
λacute=1λ=0
FIG 11 Relation between the forward Compton process and the photoabsorption process given
by the optical theorem In particular we show the longitudinal-transverse contribution The
double-line arrows represent the spin of the external particles while the dot represents the scalar
(longitudinal) polarization of the incoming photon Inside the blob the intermediate states are
represented eg nucleons with spins rprime (which are averaged in the calculation of the cross section)
and pions
πN channel see diagrams in Fig 8 of Ref [30] We define the πN -production amplitude as
T = uhB (PB)sumi
Ai(s t)Γi uhA(PA) (B2)
with the Dirac structures
Γ1 = γ5 (B3a)
Γ2 =1
2
[qA ε
]γ5 (B3b)
where uhA(PA) and udaggerhB (PB) are the Dirac spinors and PA and PB are the four-momenta of
the incoming and outgoing nucleons respectively When calculating the photoabsorption cross
section related to the VVCS amplitude in Eq (B1) the nucleon spin flip should be implemented
by uhprime(PA) in T dagger and uh(PA) in T together with the appropriate transverse and longitudinal
photon polarization vectors εlowastplusmn and εL
However if one wants to use the properties of the Dirac matrices it is more useful to construct
an operator to produce this spin flip in the external nucleons of Fig 11 This is accomplished by
introducing the projector ΓLT equiv 12radic
2(γ1 + iγ2)γ5 which also takes into account the extra factor
radic2 in Eq (B1) We checked that with this projector one correctly extracts δLT by comparing the
HB limit of our result to the HB result of Ref [50] where the authors calculate this polarizability
from the Compton amplitude directly With all those ingredients the longitudinal-transverse cross
section is calculated in the following way
σLT (νQ2) =1
64π2 s
|~pf |cm
|~pi|cm
int 1
minus1dcos θ
sumij
AiAdaggerjXij (B4)
31
with
Xij = Tr[(PB +MN )Γi(PA +MN )ΓLTγ0Γdaggerjγ
0] (B5)
where θ is the scattering angle in the center-of-mass (cm) frame and |~pi|cm (|~pf |cm) is the three-
momentum of an incoming (outgoing) particle in the cm frame An explicit calculation of the
matrix Xij leads to
X = MNQ
0 2(PB minus PA) middot εLminusradic
2|~qf |cm sin θ (sminus u)
(B6)
where |~qi|cm (|~qf |cm) is the relative three-momentum of the incoming (outgoing) particles in the
cm frame Here s t and u are the usual Mandelstam variables For the different γlowastN rarr πN
channels we obtain the following amplitudes Ai where we introduce qA as the four-momentum of
the incoming photon and qB as the four-momentum of the outgoing pion
bull γlowastprarr π0p
A1 =e gAMN
fπ
[2PA middot ε+ qA middot ε
sminusM2N
+2PB middot εminus qA middot ε
uminusM2N
] (B7a)
A2 =e gAMN
fπ
[1
sminusM2N
+1
uminusM2N
] (B7b)
bull γlowastprarr π+n
A1 =
radic2 e gAMN
fπ
[2PA middot ε+ qA middot ε
sminusM2N
+2(PA minus PB) middot ε+ qA middot ε
tminusm2π
] (B8a)
A2 =
radic2 e gAMN
fπ(sminusM2N )
(B8b)
bull γlowastnrarr π0n
A1 = 0 (B9a)
A2 = 0 (B9b)
bull γlowastnrarr πminusp
A1 =
radic2 e gAMN
fπ
[2PB middot εminus qA middot ε
uminusM2N
minus 2(PA minus PB) middot ε+ qA middot εtminusm2
π
] (B10a)
A2 =
radic2 e gAMN
fπ(uminusM2N )
(B10b)
The analytical expressions shown above were checked with the amplitudes given in Ref [87] An-
alytical expressions for the tree-level γlowastN rarr πN channel of the σLT (νQ2) and σTT (νQ2) cross
32
sections are given below (proton channels π+n and π0p neutron channel πminusp) We checked that
they reproduce the known results in the real-photon limit [45 86] To shorten the final expressions
for the cross sections which are considerably longer for finite Q2 than in the real-photon limit we
define the following dimensionless kinematic variables
αγ = (ENi )cmradics =
s+M2N +Q2
2s (B11)
απ = (ENf )cmradics =
s+M2N minusm2
π
2s (B12)
βγ = Eγcmradics =
sminusM2N minusQ2
2s (B13)
βπ = Eπcmradics =
sminusM2N +m2
π
2s (B14)
λγ = |~qi|cmradics =
radic(sminusM2
N minusQ2)2 + 4sQ2
2s (B15)
λπ = |~qf |cmradics =
radic(sminusM2
N +m2π)2 minus 4sm2
π
2s (B16)
Here (ENi )cm and (ENf )cm are the energies of the incoming and outgoing nucleon Eγcm is the energy
of the incoming photon Eπcm is the energy of the outgoing pion all in the cm frame
σ(π+n)TT = minus e2g2
AM2N
64πf2πs
3(sminusM2N )2λ4
γ
4sλπλγ
[(M2
N minus s)(M2N minusQ2 minus s)(Q2 + 2sβγβπ)
+ 2s(m2πQ
2 minus (M2N minus s)
(M2N + s(minus1 + 2βγβπ)
))λ2γ
]+ (M2
N minus s)(M2N minusQ2 minus s)
times (Q2 + 2sβγβπ minus 2sλπλγ)(Q2 + 2s(βγβπ + λπλγ)
)times log
(Q2 + 2sβγβπ minus 2sλγλπQ2 + 2sβγβπ + 2sλγλπ
) (B17)
33
σ(π0p)TT =
e2g2AM
2Nλπ
64πf2πs(sminusM2
N )2λγ
1
s2λπλ3γ
[(M2
N minus s)(M2N minusQ2 minus s)
(3M2
N + s(2βγβπ minus 3))(M2N
+ s(2βγβπ minus 1))
+ 2s
((2m2
πQ2 + (M2
N minus s)2)(M2
N minus s)minus 2(minusm2
πQ2 + (sminusM2
N )2)
times sβγβπ + 4s2(sminusM2N )β2
γβ2π
)λ2γ
]arctan
(2sλπλγ
M2N + s(2βγβπ minus 1)
)+ 2
[minus 2m2
πQ2
+ (M2N minus s)
(2(sminusM2
N ) + 4sβγβπ +(sminusM2
N +Q2)(3M2N minus 3s+ 2sβγβπ)
sλ2γ
+m2πQ
2(sminusM2N )
(M2N minus s+ 2sβγβπ)(M2
N + s(minus1 + 2βγβπ minus 2λπλγ))
+m2πQ
2(sminusM2N )
(M2N minus s+ 2sβγβπ)(M2
N + s(minus1 + 2βγβπ + 2λπλγ))
)
+1
λπλγ
(βγβπ
(minusm2
πQ2 + (M2
N minus s)(M2N minus s+ 2sβγβπ)
)minus (M2
N minus s)(M2N minusQ2 minus s)λ2
π
)log
(M2N + s(minus1 + 2βγβπ + 2λπλγ)
M2N + s(minus1 + 2βγβπ minus 2λπλγ)
)] (B18)
σ(πminusp)TT =
e2g2AM
2N
64πf2πs
3λ4γ(M2
N + s(2βπβγ minus 1))2 minus 4λ2πs
2λ2γ)
4sλπλγ
((minusM2
N +Q2 + s)(M2N
+ s(2βπβγ minus 1))
2 minus 2sλ2γ
(2λ2
πs(minusM2N +Q2 + s) +m2
πQ2))
+((M2N + s(2βπβγ minus 1)
)2
minus 4λ2πs
2λ2γ
)((M2N + s(2βπβγ minus 1)
)(M2N minus 2Q2 minus s(2βπβγ + 1)
)+ 4λ2
πs2λ2
γ
)times log
(minusM2
N minus 2βπsβγ minus 2λπsλγ + s)minus(M2N + s(2βπβγ minus 2λπλγ minus 1)
)(M2N + s(minus1
+ 2βπβγ + 2λπλγ))[((
M2N + s(2βπβγ minus 1)
)(M2N minus 2Q2 minus s(2βπβγ + 1)
)+ 4λ2
πs2λ2
γ
)times log
(minusM2
N minus 2βπsβγ + 2λπsλγ + s)
+((Q2 + 2βπsβγ)2 minus 4λ2
πs2λ2
γ
)times log
(Q2 + 2βπsβγ minus 2λπsλγQ2 + 2sβπβγ + 2λπsλγ
)]+ 4sλ2
γ
((M2
N + s(2βπβγ minus 1))2 minus 4λ2πs
2λ2γ
)times (M2
N minus s) arctanh
(2λπsλγ
M2N + 2βπsβγ minus s
) (B19)
34
σ(π+n)LT =
e2g2AM
3Nλπ
32πf2πQs
3(sminusM2N )2λ4
γ
2sλγ
[(M2
N minus s)(Q2 + 2sβ2γ)(Q2 + 2sβγβπ)
minus 4s
((M2
N minus s)(Q2 minus 2s(απ minus 1)βγ
)+Q2sβγβπ
)λ2γ + 8s3(minus1 + απ)λ4
γ
]minus M2
N minus sλπ
[(Q2 + 2sβ2
γ)(Q2 + 2sβγβπ)2 + 4s2(2(απ minus 1)βγ(Q2 + 2sβγβπ)
minusQ2λ2π
)λ2γ + 8s3(απ minus 1)2λ4
γ
]arctan
(2sλπλγ
Q2 + 2sβγβπ
) (B20)
σ(π0p)LT =
e2g2AM
3Nλπ
16πf2πQs(sminusM2
N )2λγ
1
minus2s(M2N + s(minus1 + 2βγβπ))2λ2
γ + 8s3λ2πλ
4γ
[minus 3M8
N (Q2
+ 2sβ2γ) + 2M4
Ns2
(minus (Q2 + 2sβ2
γ)(minus3 + 2βγβπ) +(Q2(minus3 + 2βγβπ + 6λ2
π) + 2sβγ
times (minus3 + 12απ + 2βγβπ minus 12απβγβπ + 4βγλ2π))λ2γ minus 4s(α2
π minus 1)λ4π
)+ 2M2
Ns3
[minus (Q2
+ 2sβ2γ)(minus1 + 2βγβπ)
(6 + βγβπ(minus9 + 2βγβπ)
)+
(Q2(3minus 12λ2
π + 4βγβπ(minus1 + 2βγβπ
+ λ2π))minus 2sβγ
(minus 3 + 4απ(3 + 2απ(3 + 2βγβπ(minus3 + βγβπ)) + 4βγ(βπ + (2minus βγβπ)
times λ2π)))λ2γ + 8s
((απ minus 1)(1 + απ minus 2βγβπ) + 2απβγλ
2π
)λ4γ
)+ s4
((Q2 + 2sβ2
γ)
times (1minus 2βγβπ)2(2βγβπ minus 3) + 2(Q2(minus1 + 6λ2
π + 2βγβπ((1minus 2βγβπ)2 minus 2λ2π)) + 2sβγ
times (minus1 + 2βγβπ + 4(minus1 + βγβπ)(minusαπ + 2απβγβπ minus βγλ2π)))λ2γ minus 8
(s(απ minus 1)(απ
+ (1minus 2βγβπ)2) + 2βγ(2sαπ +Q2βγ)λ2π
)λ4γ + 32s(απ minus 1)λ2
πλ6γ
)+ 2M6
Ns
(Q2(6
minus 7βγβπ + λ2γ) + 2sβγ(βγ(6minus 7βγβπ) + (1minus 4απ)λ2
γ)
)]+sminusM2
N
4s2λπλ3γ
[(Q2 + 2sβ2
γ)
times(3M2
N + s(2βγβπ minus 3))(M2N + s(2βγβπ minus 1)
)+ 2s
(minusM2
N (Q2 + 2sβγ minus 8sαπβγ)
+ s(2sβγ(1minus 4απ + 4απβγβπ) +Q2(1minus 2λ2π)))λ2γ + 8s3(απ minus 1)λ4
γ
]
times arctan
(2sλπλγ
M2N + s(2βγβπ minus 1)
) (B21)
35
σ(πminusp)LT =
e2g2AM
3N
64πf2πQs
3λ4γ(M2
N minusQ2 minus s)(M2N + s(2βπβγ + 2λπλγ minus 1))
times
1
M2N + s(2βπβγ minus 2λπλγ minus 1)
[4λπsλγ(M2
N minusQ2 minus s)(
2sλ2γ
(s((1minus 2λ2
π)Q2
+ 2(2απ minus 1)sβγ(2βπβγ minus 1))minusM2N (Q2 + 2(1minus 2απ)sβγ)
)+ (Q2 + 2sβ2
γ)(M2N + s(2βπβγ minus 1)
)2 + 8(απ minus 1)απs
3λ4γ
)]minus(M2N + s(2βπβγ + 2λπλγ minus 1)
)times[
log
(minusM2N minus 2βπsβγ minus 2λπsλγ + s
minusM2N minus 2βπsβγ + 2λπsλγ + s
)(minus (Q2 + 2sβ2
γ)(M2N + s(2βπβγ minus 1)
)(minusM2
N + 2Q2 + 2βπsβγ + s)minus 2sλ2γ
(minusM2
N (Q2 + 2sβγ) +Q4 +Q2s(4απβγ
minus 2βγ minus 2λ2π + 1) + 2s2βγ(4(απ minus 1)βπβγ + 1)
)minus 8(απ minus 1)2s3λ4
γ
)+ 2
(4s2λ2
γ
(2(απ minus 1)βγ(Q2 + 2βπsβγ)minus λ2
πQ2)
+ (Q2 + 2βπsβγ)2(Q2 + 2sβ2γ)
+ 8(απ minus 1)2s3λ4γ
)arctanh
(2λπsλγ
Q2 + 2βπsβγ
)] (B22)
2 ∆-production channel
The tree-level ∆-exchange diagram in Fig 2 of Ref [30] contributes to the non-Born part of
the VVCS amplitudes The contribution of the ∆ exchange to the VVCS amplitudes can be split
into [17]
S∆-exch1 (νQ2) = S∆-pole
1 (νQ2) + S∆-exch1 (νQ2) (B23a)
νS∆-exch2 (νQ2) = νS∆-pole
2 (νQ2) + νS2∆-exch
(νQ2) (B23b)
and similarly for the unpolarized VVCS amplitudes discussed in Ref [30] Here we introduced the
∆-pole contributions S∆-polei and the ∆-non-pole contributions S∆-exch
i The former amplitudes
feature a pole at the ∆(1232)-production threshold and thus are proportional to
1
[sminusM2∆][uminusM2
∆]=
1
4M2N
1
ν2∆ minus ν2
(B24)
36
They can be reconstructed from the dispersion relations in Eq (5) with the tree-level ∆-production
cross sections as input cf Fig 10 in Ref [30]
σTT (νQ2) =π2α
M2NM
2+|~q |
minus g2
MMN (M+ + ν)|~q |2 +g2E(∆minus ν)(Q2 minusMNν)2
MN(B25a)
+g2CQ
4s(∆minus ν)
MNM2∆
minus 4gMgE(Q2 minusMNν)|~q |2 minus 4gMgCQ2|~q |2
+2gEgCQ
2[minusMNM∆ |~q |2 + s(Q2 +∆ν)
]MNM∆
δ(ν minus ν∆)
σLT (νQ2) =Qπ2α
M2NM
2+|~q |
g2E(MNν minusQ2) [M∆(MN + ν)minus s]
MN(B25b)
+g2CQ
2[MNM∆|~q |2 minus s(Q2 +∆ν)
]MNM2
∆
+ gMgEM∆|~q |2 minusgMgC(Q2 minusMNν)|~q |2
M∆
+gEgC(ν minus∆)(M2
N |~q |2 minus 2Q2s)
MNM∆
δ(ν minus ν∆)
with ∆ = M∆ minusMN M+ = M∆ + MN and the Mandelstam variable s = M2N + 2MNν minus Q2
Analytical expressions for the spin structure functions g1(xQ2) and g2(xQ2) can be constructed
from Eq (3) with the flux factor K(νQ2) = |~q | =radicν2 +Q2
In the ∆-non-pole contributions to S1(νQ2) and νS2(νQ2) the pole in ν at the ∆(1232)-
production threshold has canceled out
S∆-exch1 (νQ2) =
πα
MNM2+
[g2MQ
2+ + g2
E
(∆2 minus 3Q2
)+
4g2CQ
4
M2∆
minus 8gMgEM∆ωminus (B26a)
minus2gMgCQ2(MN minus 4M∆)
M∆+
2gEgCQ2(3MN minus 2M∆)
M∆
]
νS2∆-exch
(νQ2) =2πα
MNM2+
[g2EM∆∆ωminus +
g2M MNQ
2+
2+g2C Q
2(Q2 minus∆2)
2M∆(B26b)
+gEgM M∆(M∆ω+ minus 4MNωminus)minus gEgC ∆(2Q2 +MNω+)
+gMgC Q2(4MN minus ω+)
]+S∆-exch
2 (νQ2)
ν
[M2
∆ ω2+
M2N
+ ν2
]
with Q+ =radic
(M∆ +MN )2 +Q2 and ωplusmn = (M2∆minusM2
N plusmnQ2)2M∆ and the non-pole contribution
to S2(νQ2)
S∆-exch2 (νQ2) = minus2παMNν
M∆M2+
[gM + gE
]gC (B27)
These amplitudes to the contrary are not described by the tree-level ∆-production cross sections
in the standard dispersive approach [17] This peculiarity has been previously missed e g in the
calculation of the ∆-exchange contribution to the hydrogen hyperfine splitting in Ref [88] The
importance of including the ∆-non-pole contribution is also evident when considering the BC sum
37
rule in Eq (14) The ∆-pole terms by themselves violate the BC sum rule but cancel exactly with
the ∆-non-pole terms
limνrarr0
νS∆-pole2 (νQ2) + lim
νrarr0νS2
∆-exch(νQ2) = 0 (B28)
Appendix C Polarizabilities at Q2 = 0
In this section we give analytical expressions for the static values and slopes at Q2 = 0 of
the spin polarizabilities and moments of polarized structure functions In particular we give the
HB expansion of the πN -loop contributions and the ∆-exchange contributions The complete
expressions also for the π∆-loop contributions can be found in the Supplemented material Recall
that IA(0) = I1(0) = d2(0) = 0 and dd2(Q2)dQ2
∣∣∣Q2=0
= 0
1 πN-loop contribution
Here we give analytical expressions for the πN -loop contributions to the proton and neutron
spin polarizabilities expanded in powers of micro = mπMN viz the HB expansion Note that we
choose to expand here to a high order in micro the strict HB expansion would only retain the leading
term in an analogous NLO calculation
bull Static polarizabilities (Q2 = 0)
γ0p =e2g2
A
96π3f2πm
2π
1minus 21πmicro
8minus(
59
2+ 26 logmicro
)micro2 +
1875πmicro3
64
+3
(3
2+ 26 logmicro
)micro4 +
(C1)
γ0n =e2g2
A
48π3f2πm
2π
1
2minus 9πmicro
16minus 2micro2 logmicro+
75πmicro3
128minus 3micro4
4+
(C2)
δLTp =e2g2
A
192π3f2πm
2π
1minus 9πmicro
8+
(13
2minus 2 logmicro
)micro2 minus 465πmicro3
64
minus(
47
2+ 42 logmicro
)micro4 +
(C3)
δLTn =e2g2
A
96π3f2πm
2π
1
2+
3πmicro
16+ (1 + 2 log micro)micro2 minus 105πmicro3
128+
5micro4
4+
(C4)
38
δLTn =e2g2
A
96π3f2πm
2π
1
2+
3πmicro
16+ (1 + 2 log micro)micro2 minus 105πmicro3
128+
5micro4
4+
(C5)
γ0p =e2g2
A
16π3f2πm
4π
4
45minus 3πmicro
16+
14micro2
5minus 1813πmicro3
384minus 192
5(1 + log micro)micro4
+80703πmicro5
2048+
(C6)
γ0n =e2g2
A
16π3f2πm
4π
4
45minus 5πmicro
48+
4micro2
5minus 245πmicro3
384minus 32micro4 logmicro
15
+1323πmicro5
2048+
(C7)
bull Slopes of polarizabilities at Q2 = 0
dγ0p(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
2minus 45πmicro
4+ 223micro2 minus 28515πmicro3
64
minus9
(1953
4+ 449 logmicro
)micro4 +
570255πmicro5
128+
(C8)
dγ0n(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
2minus 81πmicro
8+ 94micro2 minus 2535πmicro3
32
minus3 (1 + 90 logmicro)micro4 +84315πmicro5
1024+
(C9)
dδLTp(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
2880π3f2πm
4π
minus5
2minus 27πmicro
32+ 20micro2 minus 5865πmicro3
256
+3
(617
4+ 36 logmicro
)micro4 minus 2056845πmicro5
4096+
(C10)
dδLTn(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
minus5
4minus 81πmicro
64minus 11micro2 +
10005πmicro3
512
+15
8(11 + 48 logmicro)micro4 minus 267015πmicro5
8192+
(C11)
dIAp(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
96π2f2πmicro
2
1minus 15πmicro
4minus 1
2(115 + 88 logmicro)micro2 +
1839πmicro3
32
+5 (5 + 34 logmicro)micro4 +
(C12)
dIAn(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
48π2f2πmicro
2
1
2minus 11πmicro
8minus 1
4(1 + 20 logmicro)micro2 +
99πmicro3
64
minus25micro4
12+
(C13)
dI1p(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
96π2f2πmicro
3π
8+ 2 (4 + 3 logmicro)microminus 537πmicro2
64
minus1
2(15 + 56 logmicro)micro3 +
(C14)
39
dI1n(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
48π2f2πmicro
minus π
16+
1
4(3 + 4 logmicro)microminus 57πmicro2
128+
2micro3
3+
(C15)
γ0p(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2g2
A
16π3f2πm
6π
1
105minus 23πmicro
256+
377micro2
210minus 15551πmicro3
6144+
3371micro4
105
minus1640457πmicro5
32768+
(C16)
γ0n(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2g2
A
16π3f2πm
6π
1
105minus 153πmicro
1792+
69micro2
70minus 4615πmicro3
6144+
172micro4
35
minus120897πmicro5
32768+
(C17)
2 ∆-exchange contribution
Here we give analytical expressions for the tree-level ∆-exchange contributions to the nucleon
spin polarizabilities and their slopes at Q2 = 0 Note that the ∆-exchange contributes equally to
proton and neutron polarizabilities Recall that for the magnetic γlowastN∆ coupling we introduced a
dipole form factor to mimic vector-meson dominance gM rarr gM(1 +Q2Λ2)2
bull Static polarizabilities (Q2 = 0)
γ0 = minus e2
4πM2+
(g2M
∆2+
g2E
M2+
minus 4gMgEM+∆
) (C18)
δLT =e2M∆
4πM3+
(g2E
MNM++gMgE∆MN
minus gEgCM2
∆
) (C19)
γ0 =e2M2
N
π∆2M4+
(minusg
2M
∆2+
g2E
M2+
+4gMgE∆M+
) (C20)
bull Slopes of polarizabilities at Q2 = 0
dγ0(Q2)
dQ2
∣∣∣∣∣Q2=0
= minus e2
πM2+∆
(g2M
∆
[1
4∆2minus 1
∆M++
1
2M2+
]minus 1
Λ2
g2M
∆+
g2E
2M2+
[1
2∆minus 3
M+
]minusgMgE
M+
[1
∆2minus 5
∆M++
1
M2+
]+
1
Λ2
2gMgEM+
+2gMgC∆M2
+
minus gEgCM3
+
) (C21)
40
d δLT (Q2)
dQ2
∣∣∣∣∣Q2=0
=e2M∆∆
4πMNM2+
(g2E
∆2M2+
[1
∆minus 4
M+
]minus g2
C
∆M2∆M
2+
+gMgE∆2M+
[1
∆2
minus 3
∆M++
1
M2+
]minus 2
Λ2
gMgE∆2M+
+gMgC∆M2
∆
[1
2∆2minus 2
∆M++
1
2M2+
]minus gEgC
2M2∆M
2+
[7
∆+
1
M+
]) (C22)
dIA(Q2)
dQ2
∣∣∣∣∣Q2=0
= minusM2N
M2+
(g2M
2∆2+
g2E
MNM+minus 2gMgE
∆M+minus gEgCM∆M+
) (C23)
dI1(Q2)
dQ2
∣∣∣∣∣Q2=0
= minusM∆M2N
2M3+
(g2E
MNM∆minus gMgE∆MN
minus gEgCM2
∆
) (C24)
γ0(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2M2
N
π∆3M6+
(g2M
∆
[3M2
∆ + 2M∆MN minus 9M2N
∆2+
4M2+
Λ2
]
minus4gMgE
[5M2
∆ minus 9M2N
M+∆2+
2M+
Λ2
]+
g2E
∆M+[7M∆ minus 9MN ]
minus8gMgC∆
+4gEgCM+
) (C25)
[1] D Drechsel B Pasquini and M Vanderhaeghen Dispersion relations in real and virtual
Compton scattering Phys Rept 378 99 (2003) hep-ph0212124
[2] S E Kuhn J-P Chen and E Leader Spin structure of the nucleon mdash status and recent
results Prog Part Nucl Phys 63 1 (2009) arXiv08123535 [hep-ph]
[3] F Hagelstein R Miskimen and V Pascalutsa Nucleon polarizabilities from Compton scat-
tering to hydrogen atom Prog Part Nucl Phys 88 29 (2016) arXiv151203765 [nucl-th]
[4] B Pasquini and M Vanderhaeghen Dispersion theory in electromagnetic interactions Ann
Rev Nucl Part Sci 68 75 (2018) arXiv180510482 [hep-ph]
[5] S Gerasimov A Sum rule for magnetic moments and the damping of the nucleon magnetic
moment in nuclei Sov J Nucl Phys 2 430 (1966)
[6] S Drell and A C Hearn Exact sum rule for nucleon magnetic moments Phys Rev Lett
16 908 (1966)
[7] J S Schwinger Source theory viewpoints in deep inelastic scattering Proc Natl Acad Sci
USA 72 1 (1975)
[8] J S Schwinger Source theory viewpoints in deep inelastic scattering Electromagnetic In-
teractions and Field Theory Proceedings 14 Internationale Universitatswochen Schladming
Austria February 24-March 7 1975 Acta Phys Austriaca Suppl 14 471 (1975)
41
[9] J Schwinger Source theory discussion of deep inelastic scattering with polarized particles
Proc Natl Acad Sci USA 72 1559 (1975)
[10] A M Baldin Polarizability of nucleons Nucl Phys 18 310 (1960)
[11] M Gell-Mann M L Goldberger and W E Thirring Use of causality conditions in quantum
theory Phys Rev 95 1612 (1954)
[12] A Pineda Leading chiral logarithms to the hyperfine splitting of the hydrogen and muonic
hydrogen Phys Rev C 67 025201 (2003)
[13] C Peset and A Pineda The two-photon exchange contribution to muonic hydrogen from
chiral perturbation theory Nucl Phys B 887 69 (2014) arXiv14064524 [hep-ph]
[14] C Peset and A Pineda Model-independent determination of the two-photon exchange contri-
bution to hyperfine splitting in muonic hydrogen JHEP 04 060 arXiv161205206 [nucl-th]
[15] F Hagelstein and V Pascalutsa Proton structure in the hyperfine splitting of muonic hydro-
gen PoS CD15 077 (2016) arXiv151104301 [nucl-th]
[16] F Hagelstein Exciting Nucleons in Compton Scattering and Hydrogen-Like Atoms PhD
thesis Mainz U Inst Kernphys (2017) arXiv171000874 [nucl-th]
[17] F Hagelstein ∆(1232)-Resonance in the hydrogen spectrum Proceedings 11th International
Workshop on the Physics of Excited Nucleons (NSTAR 2017) Columbia SC USA August
20-23 2017 Few Body Syst 59 93 (2018) arXiv180109790 [nucl-th]
[18] Y Prok et al (CLAS) Moments of the spin structure functions gp1 and gd1 for 005 lt Q2 lt 30
GeV2 Phys Lett B 672 12 (2009) arXiv08022232 [nucl-ex]
[19] H Dutz et al (GDH) First measurement of the Gerasimov-Drell-Hearn sum rule for 1H from
07 GeV to 18 GeV at ELSA Phys Rev Lett 91 192001 (2003)
[20] M Amarian et al (Jefferson Lab E94010) Measurement of the generalized forward spin po-
larizabilities of the neutron Phys Rev Lett 93 152301 (2004) arXivnucl-ex0406005
[21] M Amarian et al The Q2 evolution of the generalized Gerasimov-Drell-Hearn integral for
the neutron using a 3He target Phys Rev Lett 89 242301 (2002) arXivnucl-ex0205020
[22] M Amarian et al (Jefferson Lab E94-010)Q2 evolution of the neutron spin structure moments
using a 3He target Phys Rev Lett 92 022301 (2004) arXivhep-ex0310003
[23] A Deur et al Experimental determination of the evolution of the Bjorken integral at low Q2
Phys Rev Lett 93 212001 (2004) arXivhep-ex0407007
[24] K Slifer Low Q2 measurement of gp2 and the δLT spin polarizability Spin structure at long dis-
tance Proceedings Workshop Newport News USA March 12ndash13 2009 nucl-ex09064775
(2009) AIP Conf Proc 1155 10106313203293 (2009) arXiv09064775 [nucl-ex]
[25] P Solvignon et al (E01-012) Moments of the neutron g2 structure function at intermediate
42
Q2 Phys Rev C 92 015208 (2015) arXiv13044497 [nucl-ex]
[26] A Deur Experimental studies at low Q2 of the spin structure of the nucleon at Jefferson Lab
in 9th International Workshop on Chiral Dynamics (CD18) Durham NC USA September
17-21 2018 (2019) arXiv190305661 [nucl-ex]
[27] V Sulkosky et al (Jefferson Lab E97-110) Measurement of the 3He spin-structure functions
and of neutron (3He) spin-dependent sum rules at 0035 le Q2 le 024 GeV2 Phys Lett B
805 135428 (2020) arXiv190805709 [nucl-ex]
[28] K Adhikari et al (CLAS) Measurement of the Q2 Dependence of the Deuteron Spin Structure
Function g1 and its Moments at Low Q2 with CLAS Phys Rev Lett 120 062501 (2018)
arXiv171101974 [nucl-ex]
[29] V Lensky F Hagelstein A Hiller Blin and V Pascalutsa Comment on rdquoMeasurement of
the Q2 Dependence of the Deuteron Spin Structure Function g1 and its Moments at Low Q2
with CLASrdquo (2018) arXiv180603219 [nucl-th]
[30] J M Alarcn F Hagelstein V Lensky and V Pascalutsa Forward doubly-virtual Compton
scattering off the nucleon in chiral perturbation theory at NLO the subtraction function and
moments of unpolarized structure functions (2020) arXiv200509518 [hep-ph]
[31] V Bernard E Epelbaum H Krebs and U-G Meiszligner New insights into the spin structure
of the nucleon Phys Rev D 87 054032 (2013) arXiv12092523 [hep-ph]
[32] V Lensky J M Alarcon and V Pascalutsa Moments of nucleon structure functions at
next-to-leading order in baryon chiral perturbation theory Phys Rev C 90 055202 (2014)
arXiv14072574 [hep-ph]
[33] V Pascalutsa and M Vanderhaeghen Chiral effective-field theory in the ∆(1232) region
I Pion electroproduction on the nucleon Phys Rev D 73 034003 (2006) arXivhep-
ph0512244
[34] V Pascalutsa and M Vanderhaeghen Electromagnetic nucleon-to-Delta transition in chiral
effective field theory Phys Rev Lett 95 232001 (2005) arXivhep-ph0508060
[35] T R Hemmert B R Holstein and J Kambor Systematic 1M expansion for spin 32
particles in baryon chiral perturbation theory Phys Lett B 395 89 (1997) arXivhep-
ph9606456
[36] V Pascalutsa and D R Phillips Effective theory of the ∆(1232) in Compton scattering off
the nucleon Phys Rev C 67 055202 (2003) arXivnucl-th0212024
[37] V Pascalutsa M Vanderhaeghen and S N Yang Electromagnetic excitation of the ∆(1232)-
resonance Phys Rept 437 125 (2007) arXivhep-ph0609004
[38] K A Olive et al (Particle Data Group) Review of Particle Physics Chin Phys C 38 090001
43
(2014)
[39] F E Low Scattering of light of very low frequency by systems of spin 12 Phys Rev 96
1428 (1954)
[40] M Gell-Mann and M L Goldberger Scattering of low-energy photons by particles of spin
12 Phys Rev 96 1433 (1954)
[41] A M Harun ar-Rashid A simple derivation of Schwingerrsquos sum rule for spin dependent
structure functions Nuovo Cim A 33 447 (1976)
[42] F Hagelstein and V Pascalutsa Dissecting the hadronic contributions to (g minus 2)micro by
Schwingerrsquos sum rule Phys Rev Lett 120 072002 (2018) arXiv171004571 [hep-ph]
[43] V Lensky V Pascalutsa M Vanderhaeghen and C Kao Spin-dependent sum rules con-
necting real and virtual Compton scattering verified Phys Rev D 95 074001 (2017)
arXiv170101947 [hep-ph]
[44] H Burkhardt and W N Cottingham Sum rules for forward virtual Compton scattering
Annals Phys 56 453 (1970)
[45] V Lensky and V Pascalutsa Predictive powers of chiral perturbation theory in Compton
scattering off protons Eur Phys J C 65 195 (2010) arXiv09070451 [hep-ph]
[46] H Grieszlighammer J McGovern D Phillips and G Feldman Using effective field theory to
analyse low-energy Compton scattering data from protons and light nuclei Prog Part Nucl
Phys 67 841 (2012) arXiv12036834 [nucl-th]
[47] H W Grieszlighammer J A McGovern and D R Phillips Nucleon polarisabilities at and
beyond physical pion masses Eur Phys J A 52 139 (2016) arXiv151101952 [nucl-th]
[48] E Epelbaum H Krebs and U-G Meiszligner Improved chiral nucleon-nucleon potential up to
next-to-next-to-next-to-leading order Eur Phys J A 51 53 (2015) arXiv14120142 [nucl-
th]
[49] E Epelbaum H Krebs and U-G Meiszligner Precision nucleon-nucleon potential at fifth order
in the chiral expansion Phys Rev Lett 115 122301 (2015) arXiv14124623 [nucl-th]
[50] C W Kao T Spitzenberg and M Vanderhaeghen Burkhardt-Cottingham sum rule and
forward spin polarizabilities in heavy baryon chiral perturbation theory Phys Rev D 67
016001 (2003) arXivhep-ph0209241
[51] C-W Kao D Drechsel S Kamalov and M Vanderhaeghen Higher moments of nucleon
spin structure functions in heavy baryon chiral perturbation theory and in a resonance model
Phys Rev D 69 056004 (2004) arXivhep-ph0312102
[52] T Becher and H Leutwyler Baryon chiral perturbation theory in manifestly Lorentz invariant
form Eur Phys J C 9 643 (1999) arXivhep-ph9901384
44
[53] J Gasser M E Sainio and A Svarc Nucleons with chiral loops Nucl Phys B 307 779
(1988)
[54] L S Geng J Martin Camalich L Alvarez-Ruso and M J Vicente Vacas Leading SU(3)-
breaking corrections to the baryon magnetic moments in chiral perturbation theory Phys
Rev Lett 101 222002 (2008) arXiv08051419 [hep-ph]
[55] D Drechsel S S Kamalov and L Tiator The GDH sum rule and related integrals Phys
Rev D 63 114010 (2001) arXivhep-ph0008306
[56] D Drechsel O Hanstein S S Kamalov and L Tiator A unitary isobar model for pion
photo- and electroproduction on the proton up to 1 GeV Nucl Phys A 645 145 (1999)
arXivnucl-th9807001
[57] L Tiator private communication (2020)
[58] V Bernard T R Hemmert and U-G Meiszligner Spin structure of the nucleon at low energies
Phys Rev D 67 076008 (2003) arXivhep-ph0212033
[59] O Gryniuk F Hagelstein and V Pascalutsa Evaluation of the forward Compton scattering
off protons II Spin-dependent amplitude and observables Phys Rev D 94 034043 (2016)
arXiv160400789 [nucl-th]
[60] R Zielinski The g2p Experiment A Measurement of the Protonrsquos Spin Structure Functions
PhD thesis New Hampshire U (2010) arXiv170808297 [nucl-ex]
[61] N Guler et al (CLAS) Precise determination of the deuteron spin structure at low to mod-
erate Q2 with CLAS and extraction of the neutron contribution Phys Rev C 92 055201
(2015) arXiv150507877 [nucl-ex]
[62] C E Carlson V Nazaryan and K Griffioen Proton structure corrections to electronic
and muonic hydrogen hyperfine splitting Phys Rev A 78 022517 (2008) arXiv08052603
[physicsatom-ph]
[63] V Bernard N Kaiser and U-G Meiszligner Chiral dynamics in nucleons and nuclei Int J
Mod Phys E 4 193 (1995) arXivhep-ph9501384
[64] T R Hemmert B R Holstein and J Kambor ∆(1232) and the polarizabilities of the nucleon
Phys Rev D 55 5598 (1997) arXivhep-ph9612374
[65] V Pascalutsa and R Timmermans Field theory of nucleon to higher-spin baryon transitions
Phys Rev C 60 042201 (1999) arXivnucl-th9905065
[66] V Pascalutsa Quantization of an interacting spin-32 field and the Delta isobar Phys Rev
D 58 096002 (1998) arXivhep-ph9802288
[67] H Krebs Double Virtual Compton Scattering and SpinStructure of the Nucleon PoS
CD2018 031 (2019)
45
[68] N Kochelev and Y Oh Axial anomaly and the δLT puzzle Phys Rev D 016012 (2012)
arXiv11034892 [hep-ph]
[69] D Drechsel S Kamalov and L Tiator Unitary isobar model ndash MAID2007 Eur Phys J A
69 (2007) available at httpsmaidkphuni-mainzde arXiv07100306 [nucl-th]
[70] P J Mohr B N Taylor and D B Newell CODATA recommended values of the fundamental
physical constants 2010 Rev Mod Phys 84 1527 (2012)
[71] J Ahrens et al (GDH A2) First measurement of the Gerasimov-Drell-Hearn integral for 1H
from 200 to 800 MeV Phys Rev Lett 87 022003 (2001) arXivhep-ex0105089 [hep-ex]
[72] K Helbing (GDH) Experimental verification of the GDH sum rule at ELSA and MAMI Nucl
Phys Proc Suppl 105 113 (2002)
[73] R Bradford A Bodek H S Budd and J Arrington A New parameterization of the nucleon
elastic form-factors NuInt05 proceedings of the 4th International Workshop on Neutrino-
Nucleus Interactions in the Few-GeV Region Okayama Japan 26-29 September 2005 Nucl
Phys Proc Suppl 159 127 (2006) arXivhep-ex0602017
[74] J D Bjorken Applications of the chiral U(6)timesU(6) algebra of current densities Phys Rev
148 1467 (1966)
[75] J D Bjorken Inelastic scattering of polarized leptons from polarized nucleons Phys Rev D
1 1376 (1970)
[76] V Bernard T R Hemmert and U-G Meiszligner Novel analysis of chiral loop effects in
the generalized Gerasimov-Drell-Hearn sum rule Phys Lett B 545 105 (2002) arXivhep-
ph0203167
[77] A Deur et al Experimental study of isovector spin sum rules Phys Rev D 78 032001
(2008) arXiv08023198 [nucl-ex]
[78] B Pasquini P Pedroni and D Drechsel Higher order forward spin polarizability Phys Lett
B 687 160 (2010) arXiv10014230 [hep-ph]
[79] R Jaffe g2ndashThe nucleonrsquos other spin-dependent structure function Comments Nucl Part
Phys 19 239 (1990)
[80] E V Shuryak and A Vainshtein Theory of power corrections to deep inelastic scattering in
quantum chromodynamics (II) Qminus4 effects polarized target Nucl Phys B 201 141 (1982)
[81] S Wandzura and F Wilczek Sum rules for spin dependent electroproduction Test of rela-
tivistic constituent quarks Phys Lett B 72 195 (1977)
[82] B W Filippone and X-D Ji The spin structure of the nucleon Adv Nucl Phys 26 1
(2001) arXivhep-ph0101224
[83] M Burkardt The g2 structure function Proceedings Workshop on Spin structure at long
46
distance Newport News USA March 12-13 2009 AIP Conf Proc 1155 26 (2009)
arXiv09054079 [hep-ph]
[84] A Deur S J Brodsky and G F de Teramond The spin structure of the nucleon Rept
Prog Phys 82 076201 (2019) arXiv180705250 [hep-ph]
[85] B R Holstein D Drechsel B Pasquini and M Vanderhaeghen Higher order polarizabilities
of the proton Phys Rev C 61 034316 (2000) arXivhep-ph9910427
[86] B R Holstein V Pascalutsa and M Vanderhaeghen Sum rules for magnetic moments and
polarizabilities in QED and chiral effective-field theory Phys Rev D 72 094014 (2005)
arXivhep-ph0507016
[87] B Pasquini D Drechsel and L Tiator Invariant amplitudes for pion electroproduction Eur
Phys J A 34 387 (2007) arXiv07122327 [hep-ph]
[88] A J Buchmann Non-spherical proton shape and hydrogen hyperfine splitting Proceedings
International Workshop on Precision Physics of Simple Atomic Systems (PSAS 2008) Wind-
sor Ontario Canada July 21-26 2008 Can J Phys 87 773 (2009) arXiv09104747
[physicsatom-ph]
47
000 005 010 015 020 025 030
-08
-06
-04
-02
00
02
Q2 GeV2D
DI A
proton
000 005 010 015 020 025 030
-10
-08
-06
-04
-02
00
02
Q2 GeV2D
DI A
neutron
000 005 010 015 020 025 030-005
000
005
010
015
020
025
Q2 GeV2D
DI 1
proton
000 005 010 015 020 025 030-015
-010
-005
000
005
010
015
Q2 GeV2D
DI 1
neutron
FIG 5 Contributions of the different orders to the chiral predictions of ∆IA(Q2) upper paneland ∆I1(Q2) lower panel for the proton (left) and neutron (right) Red solid line πN -loop
contribution green dot-dashed line ∆-exchange contribution orange dotted line π∆-loop contri-
bution blue solid line and blue band total result purple dot-dot-dashed line total result without
gC contribution black short-dashed line total result without gM dipole
The E97-110 experiment at Jefferson Lab has recently published their data for IAn(Q2) in
the region of 0035 GeV2 lt Q2 lt 024 GeV2 [27] In addition there are results for IAn(Q2)
from the earlier E94-010 experiment [21] and for IAp(Q2) from the E08-027 experiment [60]
The O(p4) HB calculation gives a large negative effect [51] which does not describe the data
The BχPT+∆ result from Ref [31] which mainly differs from our work by the absence of
the dipole form factor in gM looks similar to this HB result and only describes the data
points at lowest Q2 Our NLO prediction however follows closely the Q2 evolution of the
data In Fig 5 upper panel we show the polarizability ∆IA(Q2) whose Q2 evolution is
clearly dominated by the ∆ exchange Similar to the case of γ0p(Q2) inclusion of the dipole
in gM and the Coulomb coupling gC is very important in order to describe the experimental
data The LO prediction on the other hand slightly overestimates the data cf Fig 4
upper panel
19
At the real-photon point IA(0) = minusκ2
4and ∆IA(0) = 0 Therefore we give only the
slope of the polarizability ∆IA(Q2) [showing also the separate contributions from πN loops
∆ exchange and π∆ loops] in units of GeVminus2
d∆IAp(Q2)
dQ2
∣∣∣∣Q2=0
= minus858(343) asymp 238minus 1121 + 025 (30a)
d∆IAn(Q2)
dQ2
∣∣∣∣Q2=0
= minus955(343) asymp 141minus 1121 + 025 (30b)
Including the empirical Pauli form factor [73] we find in units of GeVminus2
dIAp(Q2)
dQ2
∣∣∣∣Q2=0
= minus318dIAn(Q2)
dQ2
∣∣∣∣Q2=0
= minus300 (31)
D Γ1(Q2) and I1(Q2) mdash the first moment of the structure function g1(xQ2)
The second variant for a generalization of the GDH sum rule to finite Q2 is defined as
minus α
2M2N
I1(Q2) = minus 1
8π2
int infinν0
dν1radic
ν2 +Q2
[σTT (νQ2) +
Q
νσLT (νQ2)
](32)
= minus α
Q2
int x0
0
dx g1(xQ2)
where I1(0) = minusκ2
4 This generalized GDH integral directly stems from the amplitude
Snonpole1 (νQ2) with the LEX from Eq (12a) It is given by the first moment of the struc-
ture function g1(xQ2) Γ1(Q2) =int x0
0dx g1(xQ2) as follows I1(Q2) =
2M2N
Q2 Γ1(Q2) The
isovector combination
Γ1(pminusn)(Q2) =
int x0
0
dx[g1p(xQ
2)minus g1n(xQ2)] (33)
is related to the axial coupling of the nucleon through the Bjorken sum rule [74 75]
limQ2rarrinfin
Γ1(pminusn)(Q2) =
gA6 (34)
As explained in Eq (28) the moment I1(Q2) splits into a polarizability part ∆I1(Q2) and
a Born part IBorn1 (Q2) Figure 4 lower panel shows the Q2 dependence of I1(Q2) which in
contrast to IA(Q2) shown in Figure 4 upper panel is clearly dominated by its Born part
and the elastic Pauli form factor The πN -loop ∆-exchange and π∆-loop contributions to
the polarizability ∆I1(Q2) are shown in Fig 5 lower panel Comparing to Fig 5 upper
20
000 005 010 015 020 025 030
-006
-004
-002
000
002
004
Q2 GeV2D
G1
proton
000 005 010 015 020 025 030
-006
-005
-004
-003
-002
-001
000
001
Q2 GeV2D
G1
neutron
FIG 6 First moment of the structure function g1(xQ2) for the proton (left) and neutron (right)
as function of Q2 The legend is the same as in Fig 4
panel one sees that ∆I1(Q2) is less sensitive to gC and the dipole form factor in gM than
∆IA(Q2)
For the proton our NLO BχPT prediction gives a very good description of the ex-
perimental data [18 60] and is in reasonable agreement with the MAID prediction [69]
For the neutron one observes good agreement with the empirical evaluations including ex-
trapolations to unmeasured energy regions starting from Q2 gt 01 GeV2 [27 61] In the
region of Q2 lt 005 GeV2 one observes an interesting tension between the recent E97-
110 experiment [27] and the data from CLAS [61] While the newest measurement finds
I1n(0035 GeV2) lt κ2n4 thus suggesting a negative slope at low Q the older measurement
found a rather large value for I1n(00496 GeV2) A similar but milder behaviour is seen in
the E97-110 [27] and E94-010 [21] data for IAn The MAID predictions do not agree with
the CODATA recommended values for the anomalous magnetic moments of the proton and
neutron [70] which in our work are imposed by using empirical parametrizations for the
elastic Pauli form factors [73] The slope of the HB result from Ref [51] is too large and
therefore only reproduces the data at very low Q2
Figure 6 shows the moment Γ1(Q2) for the proton and neutron while Fig 7 shows the
isovector combination Γ1 pminusn(Q2) The LO and NLO BχPT predictions are identical be-
cause our calculation produces the same Delta contributions for the proton and the neutron
For the isovector combination the MAID model only agrees with the data at very low
Q2 lt 010 GeV2 The same is true for the IR result [58 76] while all other chiral results
describe the data NLO BχPT (this work) BχPT+∆ [31] and HBχPT [51]
At the real-photon point I1(0) = minusκ2
4and ∆I1(0) = 0 Therefore we give only the
slope of the polarizability ∆I1(Q2) [showing also the separate contributions from πN loops
21
000 005 010 015 020 025 030-002
000
002
004
006
008
010
Q2 GeV2D
G1Hp-
nL
FIG 7 Isovector combination of Γ1(Q2) as function of Q2 The legend is the same as in Fig 6
The pink curve is the IR result from Ref [58 76] The experimental points are from Ref [77]
(brown dots) and Ref [23] (orange squares)
∆ exchange and π∆ loops] in units of GeVminus2
d∆I1p(Q2)
dQ2
∣∣∣∣Q2=0
= 039(4) asymp 034minus 053 + 058 (35a)
d∆I1n(Q2)
dQ2
∣∣∣∣Q2=0
= minus101(10) asymp minus107minus 053 + 058 (35b)
Including the empirical Pauli form factor [73] we find in units of GeVminus2
dI1p(Q2)
dQ2
∣∣∣∣Q2=0
= 580dI1n(Q2)
dQ2
∣∣∣∣Q2=0
= 553 (36)
E d2(Q2) mdash a measure of color polarizability
Another interesting moment to consider is d2(Q2) which is related to the twist-3 part of
the spin structure function g2(xQ2) [79 80]
d2(Q2) equiv 3
int 1
0
dx x2[g2(xQ2)minus gWW2 (xQ2)] (37)
where gWW2 (xQ2) is the twist-2 part of g2(xQ2) Using the Wandzura-Wilczek relation [81]
one can relate d2(Q2) to moments of the spin structure functions g1(xQ2) and g2(xQ2)
d2(Q2) =
int 1
0
dx x2 [3g2(xQ2) + 2g1(xQ2)] (38)
22
000 005 010 015 020 025 030
0000
0001
0002
0003
0004
0005
0006
0007
Q2 GeV2D
d2
proton
000 005 010 015 020 025 030
0000
0002
0004
0006
0008
0010
0012
Q2 GeV2D
d2
neutron
000 005 010 015 020 025 030
-1
0
1
2
Q2 GeV2D
Γ 010
-4
fm6
D
proton
000 005 010 015 020 025 030-2
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
6D
neutron
FIG 8 Upper panel The inelastic moment d2(Q2) for the proton (left) and neutron (right) as
function of Q2 The result of this work the NLO BχPT prediction is shown by the blue solid line
and the blue band The red line represents the LO BχPT result The purple short-dashed line is
the O(p4) HB result from Ref [50 51] The black dotted line is the MAID model prediction [69]
The experimental points for the neutron (cyan dots) are from Ref [22] Lower panel Fifth-order
generalized forward spin polarizability γ0(Q2) for the proton (left) and neutron (right) as function
of Q2 The experimental points for the proton are from Ref [59] (purple square) and Ref [78]
(orange dot)
This relation however only holds for asymptotically large Q2 It is also in the high-Q2
region where d2(Q2) is a measure of color polarizability [82 83] through its relation to the
gluon field strength tensor [80] We refer to Ref [84] for a recent review on the spin structure
of the nucleon including a discussion of sum rules for deep inelastic scattering and color
polarizabilities
What we consider in the following is the inelastic part of d2(Q2) defined as the moment
of g1(xQ2) and g2(xQ2) spin structure functions cf Eq (38)
d2(Q2) =
int x0
0
dx x2 [3g2(xQ2) + 2g1(xQ2)] (39)
23
This moment provides another testing ground for our BχPT predictions through comparison
with experiments on the neutron [22] Going towards the low-Q2 region the interpretation
of d2(Q2) in terms of color polarizabilities will fade out The above definition however
implies it is related to other VVCS polarizabilities
d2(Q2) =Q4
8M4N
[M2
NQ2
αδLT (Q2) + I1(Q2)minus IA(Q2)
] (40)
Note that d2(Q2) and its first two derivatives with respect to Q2 vanish at Q2 = 0 The
considerations in Eqs (28) and (29) have no effect on d2(Q2) since the Born contribution
from IA(Q2) and I1(Q2) cancel out Therefore d2(Q2) is a pure polarizability
In Fig 8 upper panel we show our NLO BχPT prediction and other results for d2(Q2)
While MAID [69] and BχPT describe the experimental data for the neutron [22] very well
the HB limit [50 51] is showing a fast growth with Q2 This illustrates the importance
of keeping the relativistic result Note also that even though the πN -loop contribution
is dominant both gC and the form factor in gM are essential to obtain a curvature that
reproduces the data cf Fig 9 upper panel For the proton there are to our knowledge
no experimental results to compare with However the agreement between the NLO BχPT
prediction and the MAID prediction at low energies is reasonable
F γ0(Q2) mdash fifth-order generalized forward spin polarizability
It is interesting to compare the generalized fifth-order forward spin polarizability sum
rule
γ0(Q2) =1
2π2
int infinν0
dν
radic1 +
Q2
ν2
σTT (νQ2)
ν5(41)
=64αM4
N
Q10
int x0
0
dx x4
[g1(xQ2)minus 4M2
Nx2
Q2g2(xQ2)
]
to the sum rule integrals for IA(Q2) and γ0(Q2) since they differ merely by their energy
weighting of σTT (νQ2) and a constant prefactor cf Eqs (19) (26) and (41) From IA(Q2)
to γ0(Q2) to γ0(Q2) the energy suppression is increasing by a factor of νminus2 respectively
Therefore the description of γ0(Q2) should be easiest in a low-energy effective-field theory
such as χPT whereas γ0(Q2) and IA(Q2) receive larger contributions from higher energies
In Fig 8 lower panel we show our LO and NLO BχPT predictions for γ0(Q2) One
can see that the πN -loop contribution is positive (in accordance to what we see for the cross
section σTT see Fig 10) The Delta shifts it substantially especially in the low Q2 region
24
000 005 010 015 020 025 030
-0004
-0002
0000
0002
0004
0006
0008
0010
Q2GeV
2D
d2
proton
000 005 010 015 020 025 030
-0005
0000
0005
0010
0015
Q2GeV
2D
d2
neutron
000 005 010 015 020 025 030
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
6D
proton
000 005 010 015 020 025 030
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
6D
neutron
FIG 9 Contributions of the different orders to the chiral predictions of d2(Q2) upper panel and
γ0(Q2) lower panel for the proton (left) and neutron (right) Red solid line πN -loop contribu-
tion green dot-dashed line ∆-exchange contribution orange dotted line π∆-loop contribution
blue long-dashed line total result purple dot-dot-dashed line total result without gC contribution
black short-dashed line total result without gM dipole
bringing it into a better agreement with data In general the BχPT curves start above the
empirical data points at the real-photon point and then decrease asymptotically to zero
above Q2 gt 01 GeV2 On the other hand the MAID prediction reproduces the empirical
data at the real-photon point then decreases to negative values until about Q2 gt 006 GeV2
from where it also starts to asymptotically approach zero Consequently our NLO BχPT
prediction of γ0(Q2) is consistently above the MAID prediction This is very different to
what we saw for IA(Q2) in Fig 4 upper panel where the MAID prediction at the real-
photon point is above the experimental value While the agreement of our predictions with
the empirical data is in general quite good for all moments of σTT (νQ2) one should point
out that both for γ0n(Q2) and γ0p(Q2) we overestimate the data at low Q2 For IA(Q2) such
observation cannot be made because ∆IA(0) = 0 and thus IA(0) is given by the empirical
Pauli form factor only From IA(Q2) γ0(Q2) and γ0(Q2) the latter has the smallest however
25
non-negligible dependence on gC and the dipole in gM cf Fig 9 lower panel
The πN -loop ∆-exchange and π∆-loop contributions to the NLO BχPT prediction of
the static fifth-order forward spin polarizability amount to in units of 10minus4 fm6
γ0p = 112(30) asymp 208minus 096minus 001 (42a)
γ0n = 195(30) asymp 292minus 096minus 001 (42b)
while the slope is composed as follows in units of 10minus4 fm8
dγ0p(Q2)
dQ2
∣∣∣∣Q2=0
= minus084(10) asymp minus100 + 016 + 000 (43a)
dγ0n(Q2)
dQ2
∣∣∣∣Q2=0
= minus142(15) asymp minus158 + 016 + 000 (43b)
Note that the HB prediction of the static polarizability γ0p (423 at O(p3) and 365 at O(ε3)
[78 85]) is almost one order of magnitude larger than the empirical value and therefore not
shown in Fig 8
G Summary
Our results are summarized in Table II where we give the contributions of the different
orders to the chiral predictions of the polarizabilities and their slopes at the real-photon
point A quantitative comparison of our predictions for the static spin polarizabilities to
the work of Bernard et al [31] and different empirical evaluations is shown in Table III We
can see that the inclusion of the Delta turns out to be very important for all moments of
the helicity-difference cross section To describe the Q2 behavior of the polarizabilities the
magnetic coupling of the N rarr ∆ transition should be modified by a dipole form factor as
has been observed previously in the description of electroproduction data [33] This dipole
form factor effectively takes account of vector-meson exchanges The Coulomb-quadrupole
N rarr ∆ transition despite its subleading order is important in the description of some
moments of spin structure functions This is contrary to what we saw for the moments of
unpolarized structure functions [30] where the Coulomb coupling had a negligible effect
The π∆ loops are mainly relevant for the generalized GDH integrals
26
TABLE II The NLO BχPT predictions for the forward VVCS polarizabilities and their slopes at
Q2 = 0 The contributions of the πN loops the ∆ exchange and the π∆ loops are shown together
with the combined total result Note that IA(0) = I1(0) = d2(0) = 0 and (d2)prime = 0
πN loops ∆ exchange π∆ loops Total
γ0 p 201 minus284 minus010minus093(92)
(10minus4 fm4) n 298 003(92)
δLT p 150 minus016 minus002132(15)
(10minus4 fm4) n 235 218(23)
γ0 p 208 minus096 minus001112(30)
(10minus4 fm6) n 292 195(30)
(γ0)prime p minus033011 001
minus022(4)
(10minus4 fm6) n minus073 minus061(7)
(δLT )prime p minus080 minus004 minus001minus085(8)
(10minus4 fm6) n minus119 minus124(12)
(γ0)prime p minus100016 000
minus084(10)
(10minus4 fm8) n minus158 minus142(15)
(∆IA)prime p 238 minus1121 025minus858(343)
(GeVminus2) n 141 minus955(343)
(∆I1)prime p 034 minus053 058039(4)
(GeVminus2) n minus107 minus101(10)
TABLE III Our NLO BχPT predictions for the spin polarizabilities at Q2 = 0 compared with
the BχPT+∆ predictions from Bernard et al [31] and the available empirical information Where
the reference is not given the empirical number is provided by the MAID analysis [55 69] with
unspecified uncertainty
Proton Neutron
This work BχPT+∆ Empirical This work BχPT+∆ Empirical
γ0 minus093(92) minus174(40) minus100(8)(12) [19] 003(92) minus077(40) minus0005
(10minus4 fm4) minus090(8)(11) [78] [MAID]
minus0929(105) [59]
δLT 132(15) 240(1) 134 218(23) 238(3) 203
(10minus4 fm4) [MAID] [MAID]
γ0 112(30) 060(7)(7) [78] 195(30) 123
(10minus4 fm6) 0484(82) [59] [MAID]
IV CONCLUSIONS
We have presented a complete NLO calculation of the polarized non-Born VVCS ampli-
tudes in covariant BχPT with pion nucleon and ∆(1232) fields The dispersion relations
between the VVCS amplitudes and the tree-level photoabsorption cross sections served as a
27
cross-check of these calculations
The obtained moments of the proton and neutron spin structure functions related to gen-
eralized polarizabilities and GDH-type integrals agree well with the available experimental
data The description of their Q2 evolution is improved compared to the previous χPT pre-
dictions In particular the NLO BχPT predictions obtained here give a better description
of the empirical data (eg from the Jefferson Laboratory ldquoSpin Physics Programrdquo) than
the HB [50 51] and IR [58] calculations
The demonstrated predictive power of the χPT framework amplitudes makes it well
suited for extending the χPT evaluation of the TPE effect in the hyperfine structure of
(muonic-)hydrogen [15ndash17] to next-to-leading order
ACKNOWLEDGEMENTS
We thank Lothar Tiator and Marc Vanderhaeghen for helpful discussions This work
is supported by the Deutsche Forschungsgemeinschaft (DFG) through the Collaborative
Research Center [The Low-Energy Frontier of the Standard Model (SFB 1044)] JMA ac-
knowledges support from the Community of Madrid through the ldquoPrograma de atraccion de
talento investigador 2017 (Modalidad 1)rdquo and the Spanish MECD grants FPA2016-77313-P
FH gratefully acknowledges financial support from the Swiss National Science Foundation
Appendix A Tensor decompositions of the VVCS amplitudes
In this appendix we review the decomposition of the forward VVCS process into tensor struc-
tures and scalar amplitudes In particular we consider the connection between the covariant and
the semi-relativistic decomposition in the lab frame that is defined in terms of the conventional
transverse longitudinal transverse-transverse and transverse-longitudinal amplitudes
As explained in Sec II A the process of forward VVCS off the nucleon can be described in
terms of four explicitly covariant amplitudes S1 2 and T1 2 [3]
T (νQ2) =
(minusgmicroν +
qmicroqν
q2
)T1(νQ2) +
1
M2N
(pmicro minus p middot q
q2qmicro)(
pν minus p middot qq2
qν)T2(νQ2) (A1)
minus 1
MNγmicroναqα S1(νQ2)minus 1
M2N
(γmicroνq2 + qmicroγναqα minus qνγmicroαqα
)S2(νQ2)
εprimelowastmicro εν
where εmicro (εprimelowastmicro ) are the incoming (outgoing) photon polarization vectors ν is the photon lab-frame
energy and Q2 is the photon virtuality Alternatively the decomposition in the laboratory frame
28
(which in the forward case coincides with the Breit frame) is parametrized in terms of the nucleon
Pauli matrices ~σ and the four scalar functions fL fT gTT and gLT
T (νQ2) = ε0 εprimelowast0 fL(νQ2) + (~ε primelowast middot ~ε ) fT (νQ2) + i~σ middot (~ε primelowast times ~ε ) gTT (νQ2) (A2)
minusi~σ middot [(ε0~εprimelowast minus ~ε ε primelowast0 )times q] gLT (νQ2)
Here ~q and q = ~q|~q | are the photon three-momentum in the lab system and its unit vector The
modified polarization vector components are given by
ε0 =
[ε0 minus
ν
|~q | (~ε middot q )
] |~q |Q ~ε = ~εminus q (~ε middot q ) (A3)
where ε = (ε0~ε ) is the usual incoming photon polarization vector and εprimelowast the outgoing polarization
vector The LEX of the lab frame amplitudes [Eq (10)] can serve in particular as the definition
of the generalized polarizabilities The lab frame amplitudes are also conveniently used for the
definition of the response functions see the example of the scalar amplitude gLT (νQ2) and the
corresponding response function σLT (νQ2) below in App B
Appendix B Photoabsorption cross sections
In the forward kinematics the spin-dependent VVCS amplitudes and the spin polarizabilities
can be described in terms of the polarized structure functions g1(xQ2) and g2(xQ2) or equiv-
alently the helicity-difference cross section σTT (νQ2) and the longitudinal-transverse response
function σLT (νQ2) with the help of dispersion relations (5) and the optical theorem (3) In this
way the photoabsorption cross sections measured in electroproduction processes form the basis
for most empirical evaluations shown throughout Sec III In the following we present the BχPT
predictions for the tree-level cross sections of πN - π∆- and ∆-production through photoabsorption
on the nucleon cf Figs 8 9 and 10 in Ref [30] In Secs B 1 and B 2 we will discuss the leading
πN -production channel and the ∆-production channel respectively We used these cross sections
to verify the polarizability predictions obtained otherwise from the calculated non-Born VVCS
amplitudes Due to the bad high-energy behavior of the π∆-production cross sections in BχPT
cf Fig 10 the dispersion relations in Eq (5) require further subtractions for a reconstruction of
the π∆-loop contribution to the spin-dependent VVCS amplitudes Therefore not all polarizabil-
ities could be verified but only those appearing as higher-order terms in the LEX of the VVCS
amplitudes such as γ0 [16]
29
proton neutron
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Neutron
02 04 06 08 100
50
100
150
200
250
[GeV]
T [b]
02 04 06 08 10
-50
0
50
100
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Neutron
02 04 06 08 100
50
100
150
200
250
[GeV]
T [b]
02 04 06 08 10
-50
0
50
100
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02040608100
50
100
150
200
250
300
[GeV]
T[b]
0204060810
-50
0
50
[GeV]
TT[b]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
0204060810
00
02
04
06
[GeV]
LTQ
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
0204060810
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
FIG 10 Photoabsorption cross sections for πN (red) and π∆ production (orange) with Q2 = 0
(solid) and Q2 = 01 GeV2 (dashed for πN and dotted for π∆ channel)
1 πN-production channel
In order to extract the response function σLT (νQ2) we have developed a method similar to
the one used to calculate σTT (νQ2) see for example Ref [86] For σLT (νQ2) however the
calculation is more complicated because one has to take into account that the associated Compton
process involves a spin-flip of the nucleon as illustrated in Fig 11 When calculating the cross
section the product of the incoming nucleon spinors has to reflect this flip
The forward VVCS amplitude related to σLT (νQ2) mdash and δLT (Q2) mdash is gLT (νQ2) It can
be extracted from Eq (A2) if one takes the modified polarization vector components in Eq (A3)
with ε = εL and εprimelowast = εlowastplusmn as input where εL = 1Q(|~q | 0 0 q0) and εplusmn = ∓ 1radic
2(0 1plusmni 0) are the
standard longitudinal and transverse polarization vectors respectively For εL and εlowastplusmn only the
choice of helicities hprime = plusmn12 and h = ∓12 gives a non-zero contribution and one obtains
χdaggerhprime T (νQ2)χh = χdaggerhprimeminusi~σ middot [(ε0~εprimelowast minus ~ε ε primelowast0 )times q] gLT (νQ2)χh =
radic2 gLT (νQ2) (B1)
where χh and χdaggerhprime are two-component Pauli spinors with opposite helicities or here spins
Let us now consider the related photoabsorption process and in particular the tree-level γlowastN rarr
30
s = 12
sprime = minus12
λ = 0 λprime = 1
rprime rprime
s=12sprime=minus1
2
λ=0λprime=1
rprimerprime
s=12
sprime=minus12
λ=0λprime=1
rprimerprime hacute=frac12h=-frac12
λacute=1λ=0
FIG 11 Relation between the forward Compton process and the photoabsorption process given
by the optical theorem In particular we show the longitudinal-transverse contribution The
double-line arrows represent the spin of the external particles while the dot represents the scalar
(longitudinal) polarization of the incoming photon Inside the blob the intermediate states are
represented eg nucleons with spins rprime (which are averaged in the calculation of the cross section)
and pions
πN channel see diagrams in Fig 8 of Ref [30] We define the πN -production amplitude as
T = uhB (PB)sumi
Ai(s t)Γi uhA(PA) (B2)
with the Dirac structures
Γ1 = γ5 (B3a)
Γ2 =1
2
[qA ε
]γ5 (B3b)
where uhA(PA) and udaggerhB (PB) are the Dirac spinors and PA and PB are the four-momenta of
the incoming and outgoing nucleons respectively When calculating the photoabsorption cross
section related to the VVCS amplitude in Eq (B1) the nucleon spin flip should be implemented
by uhprime(PA) in T dagger and uh(PA) in T together with the appropriate transverse and longitudinal
photon polarization vectors εlowastplusmn and εL
However if one wants to use the properties of the Dirac matrices it is more useful to construct
an operator to produce this spin flip in the external nucleons of Fig 11 This is accomplished by
introducing the projector ΓLT equiv 12radic
2(γ1 + iγ2)γ5 which also takes into account the extra factor
radic2 in Eq (B1) We checked that with this projector one correctly extracts δLT by comparing the
HB limit of our result to the HB result of Ref [50] where the authors calculate this polarizability
from the Compton amplitude directly With all those ingredients the longitudinal-transverse cross
section is calculated in the following way
σLT (νQ2) =1
64π2 s
|~pf |cm
|~pi|cm
int 1
minus1dcos θ
sumij
AiAdaggerjXij (B4)
31
with
Xij = Tr[(PB +MN )Γi(PA +MN )ΓLTγ0Γdaggerjγ
0] (B5)
where θ is the scattering angle in the center-of-mass (cm) frame and |~pi|cm (|~pf |cm) is the three-
momentum of an incoming (outgoing) particle in the cm frame An explicit calculation of the
matrix Xij leads to
X = MNQ
0 2(PB minus PA) middot εLminusradic
2|~qf |cm sin θ (sminus u)
(B6)
where |~qi|cm (|~qf |cm) is the relative three-momentum of the incoming (outgoing) particles in the
cm frame Here s t and u are the usual Mandelstam variables For the different γlowastN rarr πN
channels we obtain the following amplitudes Ai where we introduce qA as the four-momentum of
the incoming photon and qB as the four-momentum of the outgoing pion
bull γlowastprarr π0p
A1 =e gAMN
fπ
[2PA middot ε+ qA middot ε
sminusM2N
+2PB middot εminus qA middot ε
uminusM2N
] (B7a)
A2 =e gAMN
fπ
[1
sminusM2N
+1
uminusM2N
] (B7b)
bull γlowastprarr π+n
A1 =
radic2 e gAMN
fπ
[2PA middot ε+ qA middot ε
sminusM2N
+2(PA minus PB) middot ε+ qA middot ε
tminusm2π
] (B8a)
A2 =
radic2 e gAMN
fπ(sminusM2N )
(B8b)
bull γlowastnrarr π0n
A1 = 0 (B9a)
A2 = 0 (B9b)
bull γlowastnrarr πminusp
A1 =
radic2 e gAMN
fπ
[2PB middot εminus qA middot ε
uminusM2N
minus 2(PA minus PB) middot ε+ qA middot εtminusm2
π
] (B10a)
A2 =
radic2 e gAMN
fπ(uminusM2N )
(B10b)
The analytical expressions shown above were checked with the amplitudes given in Ref [87] An-
alytical expressions for the tree-level γlowastN rarr πN channel of the σLT (νQ2) and σTT (νQ2) cross
32
sections are given below (proton channels π+n and π0p neutron channel πminusp) We checked that
they reproduce the known results in the real-photon limit [45 86] To shorten the final expressions
for the cross sections which are considerably longer for finite Q2 than in the real-photon limit we
define the following dimensionless kinematic variables
αγ = (ENi )cmradics =
s+M2N +Q2
2s (B11)
απ = (ENf )cmradics =
s+M2N minusm2
π
2s (B12)
βγ = Eγcmradics =
sminusM2N minusQ2
2s (B13)
βπ = Eπcmradics =
sminusM2N +m2
π
2s (B14)
λγ = |~qi|cmradics =
radic(sminusM2
N minusQ2)2 + 4sQ2
2s (B15)
λπ = |~qf |cmradics =
radic(sminusM2
N +m2π)2 minus 4sm2
π
2s (B16)
Here (ENi )cm and (ENf )cm are the energies of the incoming and outgoing nucleon Eγcm is the energy
of the incoming photon Eπcm is the energy of the outgoing pion all in the cm frame
σ(π+n)TT = minus e2g2
AM2N
64πf2πs
3(sminusM2N )2λ4
γ
4sλπλγ
[(M2
N minus s)(M2N minusQ2 minus s)(Q2 + 2sβγβπ)
+ 2s(m2πQ
2 minus (M2N minus s)
(M2N + s(minus1 + 2βγβπ)
))λ2γ
]+ (M2
N minus s)(M2N minusQ2 minus s)
times (Q2 + 2sβγβπ minus 2sλπλγ)(Q2 + 2s(βγβπ + λπλγ)
)times log
(Q2 + 2sβγβπ minus 2sλγλπQ2 + 2sβγβπ + 2sλγλπ
) (B17)
33
σ(π0p)TT =
e2g2AM
2Nλπ
64πf2πs(sminusM2
N )2λγ
1
s2λπλ3γ
[(M2
N minus s)(M2N minusQ2 minus s)
(3M2
N + s(2βγβπ minus 3))(M2N
+ s(2βγβπ minus 1))
+ 2s
((2m2
πQ2 + (M2
N minus s)2)(M2
N minus s)minus 2(minusm2
πQ2 + (sminusM2
N )2)
times sβγβπ + 4s2(sminusM2N )β2
γβ2π
)λ2γ
]arctan
(2sλπλγ
M2N + s(2βγβπ minus 1)
)+ 2
[minus 2m2
πQ2
+ (M2N minus s)
(2(sminusM2
N ) + 4sβγβπ +(sminusM2
N +Q2)(3M2N minus 3s+ 2sβγβπ)
sλ2γ
+m2πQ
2(sminusM2N )
(M2N minus s+ 2sβγβπ)(M2
N + s(minus1 + 2βγβπ minus 2λπλγ))
+m2πQ
2(sminusM2N )
(M2N minus s+ 2sβγβπ)(M2
N + s(minus1 + 2βγβπ + 2λπλγ))
)
+1
λπλγ
(βγβπ
(minusm2
πQ2 + (M2
N minus s)(M2N minus s+ 2sβγβπ)
)minus (M2
N minus s)(M2N minusQ2 minus s)λ2
π
)log
(M2N + s(minus1 + 2βγβπ + 2λπλγ)
M2N + s(minus1 + 2βγβπ minus 2λπλγ)
)] (B18)
σ(πminusp)TT =
e2g2AM
2N
64πf2πs
3λ4γ(M2
N + s(2βπβγ minus 1))2 minus 4λ2πs
2λ2γ)
4sλπλγ
((minusM2
N +Q2 + s)(M2N
+ s(2βπβγ minus 1))
2 minus 2sλ2γ
(2λ2
πs(minusM2N +Q2 + s) +m2
πQ2))
+((M2N + s(2βπβγ minus 1)
)2
minus 4λ2πs
2λ2γ
)((M2N + s(2βπβγ minus 1)
)(M2N minus 2Q2 minus s(2βπβγ + 1)
)+ 4λ2
πs2λ2
γ
)times log
(minusM2
N minus 2βπsβγ minus 2λπsλγ + s)minus(M2N + s(2βπβγ minus 2λπλγ minus 1)
)(M2N + s(minus1
+ 2βπβγ + 2λπλγ))[((
M2N + s(2βπβγ minus 1)
)(M2N minus 2Q2 minus s(2βπβγ + 1)
)+ 4λ2
πs2λ2
γ
)times log
(minusM2
N minus 2βπsβγ + 2λπsλγ + s)
+((Q2 + 2βπsβγ)2 minus 4λ2
πs2λ2
γ
)times log
(Q2 + 2βπsβγ minus 2λπsλγQ2 + 2sβπβγ + 2λπsλγ
)]+ 4sλ2
γ
((M2
N + s(2βπβγ minus 1))2 minus 4λ2πs
2λ2γ
)times (M2
N minus s) arctanh
(2λπsλγ
M2N + 2βπsβγ minus s
) (B19)
34
σ(π+n)LT =
e2g2AM
3Nλπ
32πf2πQs
3(sminusM2N )2λ4
γ
2sλγ
[(M2
N minus s)(Q2 + 2sβ2γ)(Q2 + 2sβγβπ)
minus 4s
((M2
N minus s)(Q2 minus 2s(απ minus 1)βγ
)+Q2sβγβπ
)λ2γ + 8s3(minus1 + απ)λ4
γ
]minus M2
N minus sλπ
[(Q2 + 2sβ2
γ)(Q2 + 2sβγβπ)2 + 4s2(2(απ minus 1)βγ(Q2 + 2sβγβπ)
minusQ2λ2π
)λ2γ + 8s3(απ minus 1)2λ4
γ
]arctan
(2sλπλγ
Q2 + 2sβγβπ
) (B20)
σ(π0p)LT =
e2g2AM
3Nλπ
16πf2πQs(sminusM2
N )2λγ
1
minus2s(M2N + s(minus1 + 2βγβπ))2λ2
γ + 8s3λ2πλ
4γ
[minus 3M8
N (Q2
+ 2sβ2γ) + 2M4
Ns2
(minus (Q2 + 2sβ2
γ)(minus3 + 2βγβπ) +(Q2(minus3 + 2βγβπ + 6λ2
π) + 2sβγ
times (minus3 + 12απ + 2βγβπ minus 12απβγβπ + 4βγλ2π))λ2γ minus 4s(α2
π minus 1)λ4π
)+ 2M2
Ns3
[minus (Q2
+ 2sβ2γ)(minus1 + 2βγβπ)
(6 + βγβπ(minus9 + 2βγβπ)
)+
(Q2(3minus 12λ2
π + 4βγβπ(minus1 + 2βγβπ
+ λ2π))minus 2sβγ
(minus 3 + 4απ(3 + 2απ(3 + 2βγβπ(minus3 + βγβπ)) + 4βγ(βπ + (2minus βγβπ)
times λ2π)))λ2γ + 8s
((απ minus 1)(1 + απ minus 2βγβπ) + 2απβγλ
2π
)λ4γ
)+ s4
((Q2 + 2sβ2
γ)
times (1minus 2βγβπ)2(2βγβπ minus 3) + 2(Q2(minus1 + 6λ2
π + 2βγβπ((1minus 2βγβπ)2 minus 2λ2π)) + 2sβγ
times (minus1 + 2βγβπ + 4(minus1 + βγβπ)(minusαπ + 2απβγβπ minus βγλ2π)))λ2γ minus 8
(s(απ minus 1)(απ
+ (1minus 2βγβπ)2) + 2βγ(2sαπ +Q2βγ)λ2π
)λ4γ + 32s(απ minus 1)λ2
πλ6γ
)+ 2M6
Ns
(Q2(6
minus 7βγβπ + λ2γ) + 2sβγ(βγ(6minus 7βγβπ) + (1minus 4απ)λ2
γ)
)]+sminusM2
N
4s2λπλ3γ
[(Q2 + 2sβ2
γ)
times(3M2
N + s(2βγβπ minus 3))(M2N + s(2βγβπ minus 1)
)+ 2s
(minusM2
N (Q2 + 2sβγ minus 8sαπβγ)
+ s(2sβγ(1minus 4απ + 4απβγβπ) +Q2(1minus 2λ2π)))λ2γ + 8s3(απ minus 1)λ4
γ
]
times arctan
(2sλπλγ
M2N + s(2βγβπ minus 1)
) (B21)
35
σ(πminusp)LT =
e2g2AM
3N
64πf2πQs
3λ4γ(M2
N minusQ2 minus s)(M2N + s(2βπβγ + 2λπλγ minus 1))
times
1
M2N + s(2βπβγ minus 2λπλγ minus 1)
[4λπsλγ(M2
N minusQ2 minus s)(
2sλ2γ
(s((1minus 2λ2
π)Q2
+ 2(2απ minus 1)sβγ(2βπβγ minus 1))minusM2N (Q2 + 2(1minus 2απ)sβγ)
)+ (Q2 + 2sβ2
γ)(M2N + s(2βπβγ minus 1)
)2 + 8(απ minus 1)απs
3λ4γ
)]minus(M2N + s(2βπβγ + 2λπλγ minus 1)
)times[
log
(minusM2N minus 2βπsβγ minus 2λπsλγ + s
minusM2N minus 2βπsβγ + 2λπsλγ + s
)(minus (Q2 + 2sβ2
γ)(M2N + s(2βπβγ minus 1)
)(minusM2
N + 2Q2 + 2βπsβγ + s)minus 2sλ2γ
(minusM2
N (Q2 + 2sβγ) +Q4 +Q2s(4απβγ
minus 2βγ minus 2λ2π + 1) + 2s2βγ(4(απ minus 1)βπβγ + 1)
)minus 8(απ minus 1)2s3λ4
γ
)+ 2
(4s2λ2
γ
(2(απ minus 1)βγ(Q2 + 2βπsβγ)minus λ2
πQ2)
+ (Q2 + 2βπsβγ)2(Q2 + 2sβ2γ)
+ 8(απ minus 1)2s3λ4γ
)arctanh
(2λπsλγ
Q2 + 2βπsβγ
)] (B22)
2 ∆-production channel
The tree-level ∆-exchange diagram in Fig 2 of Ref [30] contributes to the non-Born part of
the VVCS amplitudes The contribution of the ∆ exchange to the VVCS amplitudes can be split
into [17]
S∆-exch1 (νQ2) = S∆-pole
1 (νQ2) + S∆-exch1 (νQ2) (B23a)
νS∆-exch2 (νQ2) = νS∆-pole
2 (νQ2) + νS2∆-exch
(νQ2) (B23b)
and similarly for the unpolarized VVCS amplitudes discussed in Ref [30] Here we introduced the
∆-pole contributions S∆-polei and the ∆-non-pole contributions S∆-exch
i The former amplitudes
feature a pole at the ∆(1232)-production threshold and thus are proportional to
1
[sminusM2∆][uminusM2
∆]=
1
4M2N
1
ν2∆ minus ν2
(B24)
36
They can be reconstructed from the dispersion relations in Eq (5) with the tree-level ∆-production
cross sections as input cf Fig 10 in Ref [30]
σTT (νQ2) =π2α
M2NM
2+|~q |
minus g2
MMN (M+ + ν)|~q |2 +g2E(∆minus ν)(Q2 minusMNν)2
MN(B25a)
+g2CQ
4s(∆minus ν)
MNM2∆
minus 4gMgE(Q2 minusMNν)|~q |2 minus 4gMgCQ2|~q |2
+2gEgCQ
2[minusMNM∆ |~q |2 + s(Q2 +∆ν)
]MNM∆
δ(ν minus ν∆)
σLT (νQ2) =Qπ2α
M2NM
2+|~q |
g2E(MNν minusQ2) [M∆(MN + ν)minus s]
MN(B25b)
+g2CQ
2[MNM∆|~q |2 minus s(Q2 +∆ν)
]MNM2
∆
+ gMgEM∆|~q |2 minusgMgC(Q2 minusMNν)|~q |2
M∆
+gEgC(ν minus∆)(M2
N |~q |2 minus 2Q2s)
MNM∆
δ(ν minus ν∆)
with ∆ = M∆ minusMN M+ = M∆ + MN and the Mandelstam variable s = M2N + 2MNν minus Q2
Analytical expressions for the spin structure functions g1(xQ2) and g2(xQ2) can be constructed
from Eq (3) with the flux factor K(νQ2) = |~q | =radicν2 +Q2
In the ∆-non-pole contributions to S1(νQ2) and νS2(νQ2) the pole in ν at the ∆(1232)-
production threshold has canceled out
S∆-exch1 (νQ2) =
πα
MNM2+
[g2MQ
2+ + g2
E
(∆2 minus 3Q2
)+
4g2CQ
4
M2∆
minus 8gMgEM∆ωminus (B26a)
minus2gMgCQ2(MN minus 4M∆)
M∆+
2gEgCQ2(3MN minus 2M∆)
M∆
]
νS2∆-exch
(νQ2) =2πα
MNM2+
[g2EM∆∆ωminus +
g2M MNQ
2+
2+g2C Q
2(Q2 minus∆2)
2M∆(B26b)
+gEgM M∆(M∆ω+ minus 4MNωminus)minus gEgC ∆(2Q2 +MNω+)
+gMgC Q2(4MN minus ω+)
]+S∆-exch
2 (νQ2)
ν
[M2
∆ ω2+
M2N
+ ν2
]
with Q+ =radic
(M∆ +MN )2 +Q2 and ωplusmn = (M2∆minusM2
N plusmnQ2)2M∆ and the non-pole contribution
to S2(νQ2)
S∆-exch2 (νQ2) = minus2παMNν
M∆M2+
[gM + gE
]gC (B27)
These amplitudes to the contrary are not described by the tree-level ∆-production cross sections
in the standard dispersive approach [17] This peculiarity has been previously missed e g in the
calculation of the ∆-exchange contribution to the hydrogen hyperfine splitting in Ref [88] The
importance of including the ∆-non-pole contribution is also evident when considering the BC sum
37
rule in Eq (14) The ∆-pole terms by themselves violate the BC sum rule but cancel exactly with
the ∆-non-pole terms
limνrarr0
νS∆-pole2 (νQ2) + lim
νrarr0νS2
∆-exch(νQ2) = 0 (B28)
Appendix C Polarizabilities at Q2 = 0
In this section we give analytical expressions for the static values and slopes at Q2 = 0 of
the spin polarizabilities and moments of polarized structure functions In particular we give the
HB expansion of the πN -loop contributions and the ∆-exchange contributions The complete
expressions also for the π∆-loop contributions can be found in the Supplemented material Recall
that IA(0) = I1(0) = d2(0) = 0 and dd2(Q2)dQ2
∣∣∣Q2=0
= 0
1 πN-loop contribution
Here we give analytical expressions for the πN -loop contributions to the proton and neutron
spin polarizabilities expanded in powers of micro = mπMN viz the HB expansion Note that we
choose to expand here to a high order in micro the strict HB expansion would only retain the leading
term in an analogous NLO calculation
bull Static polarizabilities (Q2 = 0)
γ0p =e2g2
A
96π3f2πm
2π
1minus 21πmicro
8minus(
59
2+ 26 logmicro
)micro2 +
1875πmicro3
64
+3
(3
2+ 26 logmicro
)micro4 +
(C1)
γ0n =e2g2
A
48π3f2πm
2π
1
2minus 9πmicro
16minus 2micro2 logmicro+
75πmicro3
128minus 3micro4
4+
(C2)
δLTp =e2g2
A
192π3f2πm
2π
1minus 9πmicro
8+
(13
2minus 2 logmicro
)micro2 minus 465πmicro3
64
minus(
47
2+ 42 logmicro
)micro4 +
(C3)
δLTn =e2g2
A
96π3f2πm
2π
1
2+
3πmicro
16+ (1 + 2 log micro)micro2 minus 105πmicro3
128+
5micro4
4+
(C4)
38
δLTn =e2g2
A
96π3f2πm
2π
1
2+
3πmicro
16+ (1 + 2 log micro)micro2 minus 105πmicro3
128+
5micro4
4+
(C5)
γ0p =e2g2
A
16π3f2πm
4π
4
45minus 3πmicro
16+
14micro2
5minus 1813πmicro3
384minus 192
5(1 + log micro)micro4
+80703πmicro5
2048+
(C6)
γ0n =e2g2
A
16π3f2πm
4π
4
45minus 5πmicro
48+
4micro2
5minus 245πmicro3
384minus 32micro4 logmicro
15
+1323πmicro5
2048+
(C7)
bull Slopes of polarizabilities at Q2 = 0
dγ0p(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
2minus 45πmicro
4+ 223micro2 minus 28515πmicro3
64
minus9
(1953
4+ 449 logmicro
)micro4 +
570255πmicro5
128+
(C8)
dγ0n(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
2minus 81πmicro
8+ 94micro2 minus 2535πmicro3
32
minus3 (1 + 90 logmicro)micro4 +84315πmicro5
1024+
(C9)
dδLTp(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
2880π3f2πm
4π
minus5
2minus 27πmicro
32+ 20micro2 minus 5865πmicro3
256
+3
(617
4+ 36 logmicro
)micro4 minus 2056845πmicro5
4096+
(C10)
dδLTn(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
minus5
4minus 81πmicro
64minus 11micro2 +
10005πmicro3
512
+15
8(11 + 48 logmicro)micro4 minus 267015πmicro5
8192+
(C11)
dIAp(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
96π2f2πmicro
2
1minus 15πmicro
4minus 1
2(115 + 88 logmicro)micro2 +
1839πmicro3
32
+5 (5 + 34 logmicro)micro4 +
(C12)
dIAn(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
48π2f2πmicro
2
1
2minus 11πmicro
8minus 1
4(1 + 20 logmicro)micro2 +
99πmicro3
64
minus25micro4
12+
(C13)
dI1p(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
96π2f2πmicro
3π
8+ 2 (4 + 3 logmicro)microminus 537πmicro2
64
minus1
2(15 + 56 logmicro)micro3 +
(C14)
39
dI1n(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
48π2f2πmicro
minus π
16+
1
4(3 + 4 logmicro)microminus 57πmicro2
128+
2micro3
3+
(C15)
γ0p(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2g2
A
16π3f2πm
6π
1
105minus 23πmicro
256+
377micro2
210minus 15551πmicro3
6144+
3371micro4
105
minus1640457πmicro5
32768+
(C16)
γ0n(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2g2
A
16π3f2πm
6π
1
105minus 153πmicro
1792+
69micro2
70minus 4615πmicro3
6144+
172micro4
35
minus120897πmicro5
32768+
(C17)
2 ∆-exchange contribution
Here we give analytical expressions for the tree-level ∆-exchange contributions to the nucleon
spin polarizabilities and their slopes at Q2 = 0 Note that the ∆-exchange contributes equally to
proton and neutron polarizabilities Recall that for the magnetic γlowastN∆ coupling we introduced a
dipole form factor to mimic vector-meson dominance gM rarr gM(1 +Q2Λ2)2
bull Static polarizabilities (Q2 = 0)
γ0 = minus e2
4πM2+
(g2M
∆2+
g2E
M2+
minus 4gMgEM+∆
) (C18)
δLT =e2M∆
4πM3+
(g2E
MNM++gMgE∆MN
minus gEgCM2
∆
) (C19)
γ0 =e2M2
N
π∆2M4+
(minusg
2M
∆2+
g2E
M2+
+4gMgE∆M+
) (C20)
bull Slopes of polarizabilities at Q2 = 0
dγ0(Q2)
dQ2
∣∣∣∣∣Q2=0
= minus e2
πM2+∆
(g2M
∆
[1
4∆2minus 1
∆M++
1
2M2+
]minus 1
Λ2
g2M
∆+
g2E
2M2+
[1
2∆minus 3
M+
]minusgMgE
M+
[1
∆2minus 5
∆M++
1
M2+
]+
1
Λ2
2gMgEM+
+2gMgC∆M2
+
minus gEgCM3
+
) (C21)
40
d δLT (Q2)
dQ2
∣∣∣∣∣Q2=0
=e2M∆∆
4πMNM2+
(g2E
∆2M2+
[1
∆minus 4
M+
]minus g2
C
∆M2∆M
2+
+gMgE∆2M+
[1
∆2
minus 3
∆M++
1
M2+
]minus 2
Λ2
gMgE∆2M+
+gMgC∆M2
∆
[1
2∆2minus 2
∆M++
1
2M2+
]minus gEgC
2M2∆M
2+
[7
∆+
1
M+
]) (C22)
dIA(Q2)
dQ2
∣∣∣∣∣Q2=0
= minusM2N
M2+
(g2M
2∆2+
g2E
MNM+minus 2gMgE
∆M+minus gEgCM∆M+
) (C23)
dI1(Q2)
dQ2
∣∣∣∣∣Q2=0
= minusM∆M2N
2M3+
(g2E
MNM∆minus gMgE∆MN
minus gEgCM2
∆
) (C24)
γ0(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2M2
N
π∆3M6+
(g2M
∆
[3M2
∆ + 2M∆MN minus 9M2N
∆2+
4M2+
Λ2
]
minus4gMgE
[5M2
∆ minus 9M2N
M+∆2+
2M+
Λ2
]+
g2E
∆M+[7M∆ minus 9MN ]
minus8gMgC∆
+4gEgCM+
) (C25)
[1] D Drechsel B Pasquini and M Vanderhaeghen Dispersion relations in real and virtual
Compton scattering Phys Rept 378 99 (2003) hep-ph0212124
[2] S E Kuhn J-P Chen and E Leader Spin structure of the nucleon mdash status and recent
results Prog Part Nucl Phys 63 1 (2009) arXiv08123535 [hep-ph]
[3] F Hagelstein R Miskimen and V Pascalutsa Nucleon polarizabilities from Compton scat-
tering to hydrogen atom Prog Part Nucl Phys 88 29 (2016) arXiv151203765 [nucl-th]
[4] B Pasquini and M Vanderhaeghen Dispersion theory in electromagnetic interactions Ann
Rev Nucl Part Sci 68 75 (2018) arXiv180510482 [hep-ph]
[5] S Gerasimov A Sum rule for magnetic moments and the damping of the nucleon magnetic
moment in nuclei Sov J Nucl Phys 2 430 (1966)
[6] S Drell and A C Hearn Exact sum rule for nucleon magnetic moments Phys Rev Lett
16 908 (1966)
[7] J S Schwinger Source theory viewpoints in deep inelastic scattering Proc Natl Acad Sci
USA 72 1 (1975)
[8] J S Schwinger Source theory viewpoints in deep inelastic scattering Electromagnetic In-
teractions and Field Theory Proceedings 14 Internationale Universitatswochen Schladming
Austria February 24-March 7 1975 Acta Phys Austriaca Suppl 14 471 (1975)
41
[9] J Schwinger Source theory discussion of deep inelastic scattering with polarized particles
Proc Natl Acad Sci USA 72 1559 (1975)
[10] A M Baldin Polarizability of nucleons Nucl Phys 18 310 (1960)
[11] M Gell-Mann M L Goldberger and W E Thirring Use of causality conditions in quantum
theory Phys Rev 95 1612 (1954)
[12] A Pineda Leading chiral logarithms to the hyperfine splitting of the hydrogen and muonic
hydrogen Phys Rev C 67 025201 (2003)
[13] C Peset and A Pineda The two-photon exchange contribution to muonic hydrogen from
chiral perturbation theory Nucl Phys B 887 69 (2014) arXiv14064524 [hep-ph]
[14] C Peset and A Pineda Model-independent determination of the two-photon exchange contri-
bution to hyperfine splitting in muonic hydrogen JHEP 04 060 arXiv161205206 [nucl-th]
[15] F Hagelstein and V Pascalutsa Proton structure in the hyperfine splitting of muonic hydro-
gen PoS CD15 077 (2016) arXiv151104301 [nucl-th]
[16] F Hagelstein Exciting Nucleons in Compton Scattering and Hydrogen-Like Atoms PhD
thesis Mainz U Inst Kernphys (2017) arXiv171000874 [nucl-th]
[17] F Hagelstein ∆(1232)-Resonance in the hydrogen spectrum Proceedings 11th International
Workshop on the Physics of Excited Nucleons (NSTAR 2017) Columbia SC USA August
20-23 2017 Few Body Syst 59 93 (2018) arXiv180109790 [nucl-th]
[18] Y Prok et al (CLAS) Moments of the spin structure functions gp1 and gd1 for 005 lt Q2 lt 30
GeV2 Phys Lett B 672 12 (2009) arXiv08022232 [nucl-ex]
[19] H Dutz et al (GDH) First measurement of the Gerasimov-Drell-Hearn sum rule for 1H from
07 GeV to 18 GeV at ELSA Phys Rev Lett 91 192001 (2003)
[20] M Amarian et al (Jefferson Lab E94010) Measurement of the generalized forward spin po-
larizabilities of the neutron Phys Rev Lett 93 152301 (2004) arXivnucl-ex0406005
[21] M Amarian et al The Q2 evolution of the generalized Gerasimov-Drell-Hearn integral for
the neutron using a 3He target Phys Rev Lett 89 242301 (2002) arXivnucl-ex0205020
[22] M Amarian et al (Jefferson Lab E94-010)Q2 evolution of the neutron spin structure moments
using a 3He target Phys Rev Lett 92 022301 (2004) arXivhep-ex0310003
[23] A Deur et al Experimental determination of the evolution of the Bjorken integral at low Q2
Phys Rev Lett 93 212001 (2004) arXivhep-ex0407007
[24] K Slifer Low Q2 measurement of gp2 and the δLT spin polarizability Spin structure at long dis-
tance Proceedings Workshop Newport News USA March 12ndash13 2009 nucl-ex09064775
(2009) AIP Conf Proc 1155 10106313203293 (2009) arXiv09064775 [nucl-ex]
[25] P Solvignon et al (E01-012) Moments of the neutron g2 structure function at intermediate
42
Q2 Phys Rev C 92 015208 (2015) arXiv13044497 [nucl-ex]
[26] A Deur Experimental studies at low Q2 of the spin structure of the nucleon at Jefferson Lab
in 9th International Workshop on Chiral Dynamics (CD18) Durham NC USA September
17-21 2018 (2019) arXiv190305661 [nucl-ex]
[27] V Sulkosky et al (Jefferson Lab E97-110) Measurement of the 3He spin-structure functions
and of neutron (3He) spin-dependent sum rules at 0035 le Q2 le 024 GeV2 Phys Lett B
805 135428 (2020) arXiv190805709 [nucl-ex]
[28] K Adhikari et al (CLAS) Measurement of the Q2 Dependence of the Deuteron Spin Structure
Function g1 and its Moments at Low Q2 with CLAS Phys Rev Lett 120 062501 (2018)
arXiv171101974 [nucl-ex]
[29] V Lensky F Hagelstein A Hiller Blin and V Pascalutsa Comment on rdquoMeasurement of
the Q2 Dependence of the Deuteron Spin Structure Function g1 and its Moments at Low Q2
with CLASrdquo (2018) arXiv180603219 [nucl-th]
[30] J M Alarcn F Hagelstein V Lensky and V Pascalutsa Forward doubly-virtual Compton
scattering off the nucleon in chiral perturbation theory at NLO the subtraction function and
moments of unpolarized structure functions (2020) arXiv200509518 [hep-ph]
[31] V Bernard E Epelbaum H Krebs and U-G Meiszligner New insights into the spin structure
of the nucleon Phys Rev D 87 054032 (2013) arXiv12092523 [hep-ph]
[32] V Lensky J M Alarcon and V Pascalutsa Moments of nucleon structure functions at
next-to-leading order in baryon chiral perturbation theory Phys Rev C 90 055202 (2014)
arXiv14072574 [hep-ph]
[33] V Pascalutsa and M Vanderhaeghen Chiral effective-field theory in the ∆(1232) region
I Pion electroproduction on the nucleon Phys Rev D 73 034003 (2006) arXivhep-
ph0512244
[34] V Pascalutsa and M Vanderhaeghen Electromagnetic nucleon-to-Delta transition in chiral
effective field theory Phys Rev Lett 95 232001 (2005) arXivhep-ph0508060
[35] T R Hemmert B R Holstein and J Kambor Systematic 1M expansion for spin 32
particles in baryon chiral perturbation theory Phys Lett B 395 89 (1997) arXivhep-
ph9606456
[36] V Pascalutsa and D R Phillips Effective theory of the ∆(1232) in Compton scattering off
the nucleon Phys Rev C 67 055202 (2003) arXivnucl-th0212024
[37] V Pascalutsa M Vanderhaeghen and S N Yang Electromagnetic excitation of the ∆(1232)-
resonance Phys Rept 437 125 (2007) arXivhep-ph0609004
[38] K A Olive et al (Particle Data Group) Review of Particle Physics Chin Phys C 38 090001
43
(2014)
[39] F E Low Scattering of light of very low frequency by systems of spin 12 Phys Rev 96
1428 (1954)
[40] M Gell-Mann and M L Goldberger Scattering of low-energy photons by particles of spin
12 Phys Rev 96 1433 (1954)
[41] A M Harun ar-Rashid A simple derivation of Schwingerrsquos sum rule for spin dependent
structure functions Nuovo Cim A 33 447 (1976)
[42] F Hagelstein and V Pascalutsa Dissecting the hadronic contributions to (g minus 2)micro by
Schwingerrsquos sum rule Phys Rev Lett 120 072002 (2018) arXiv171004571 [hep-ph]
[43] V Lensky V Pascalutsa M Vanderhaeghen and C Kao Spin-dependent sum rules con-
necting real and virtual Compton scattering verified Phys Rev D 95 074001 (2017)
arXiv170101947 [hep-ph]
[44] H Burkhardt and W N Cottingham Sum rules for forward virtual Compton scattering
Annals Phys 56 453 (1970)
[45] V Lensky and V Pascalutsa Predictive powers of chiral perturbation theory in Compton
scattering off protons Eur Phys J C 65 195 (2010) arXiv09070451 [hep-ph]
[46] H Grieszlighammer J McGovern D Phillips and G Feldman Using effective field theory to
analyse low-energy Compton scattering data from protons and light nuclei Prog Part Nucl
Phys 67 841 (2012) arXiv12036834 [nucl-th]
[47] H W Grieszlighammer J A McGovern and D R Phillips Nucleon polarisabilities at and
beyond physical pion masses Eur Phys J A 52 139 (2016) arXiv151101952 [nucl-th]
[48] E Epelbaum H Krebs and U-G Meiszligner Improved chiral nucleon-nucleon potential up to
next-to-next-to-next-to-leading order Eur Phys J A 51 53 (2015) arXiv14120142 [nucl-
th]
[49] E Epelbaum H Krebs and U-G Meiszligner Precision nucleon-nucleon potential at fifth order
in the chiral expansion Phys Rev Lett 115 122301 (2015) arXiv14124623 [nucl-th]
[50] C W Kao T Spitzenberg and M Vanderhaeghen Burkhardt-Cottingham sum rule and
forward spin polarizabilities in heavy baryon chiral perturbation theory Phys Rev D 67
016001 (2003) arXivhep-ph0209241
[51] C-W Kao D Drechsel S Kamalov and M Vanderhaeghen Higher moments of nucleon
spin structure functions in heavy baryon chiral perturbation theory and in a resonance model
Phys Rev D 69 056004 (2004) arXivhep-ph0312102
[52] T Becher and H Leutwyler Baryon chiral perturbation theory in manifestly Lorentz invariant
form Eur Phys J C 9 643 (1999) arXivhep-ph9901384
44
[53] J Gasser M E Sainio and A Svarc Nucleons with chiral loops Nucl Phys B 307 779
(1988)
[54] L S Geng J Martin Camalich L Alvarez-Ruso and M J Vicente Vacas Leading SU(3)-
breaking corrections to the baryon magnetic moments in chiral perturbation theory Phys
Rev Lett 101 222002 (2008) arXiv08051419 [hep-ph]
[55] D Drechsel S S Kamalov and L Tiator The GDH sum rule and related integrals Phys
Rev D 63 114010 (2001) arXivhep-ph0008306
[56] D Drechsel O Hanstein S S Kamalov and L Tiator A unitary isobar model for pion
photo- and electroproduction on the proton up to 1 GeV Nucl Phys A 645 145 (1999)
arXivnucl-th9807001
[57] L Tiator private communication (2020)
[58] V Bernard T R Hemmert and U-G Meiszligner Spin structure of the nucleon at low energies
Phys Rev D 67 076008 (2003) arXivhep-ph0212033
[59] O Gryniuk F Hagelstein and V Pascalutsa Evaluation of the forward Compton scattering
off protons II Spin-dependent amplitude and observables Phys Rev D 94 034043 (2016)
arXiv160400789 [nucl-th]
[60] R Zielinski The g2p Experiment A Measurement of the Protonrsquos Spin Structure Functions
PhD thesis New Hampshire U (2010) arXiv170808297 [nucl-ex]
[61] N Guler et al (CLAS) Precise determination of the deuteron spin structure at low to mod-
erate Q2 with CLAS and extraction of the neutron contribution Phys Rev C 92 055201
(2015) arXiv150507877 [nucl-ex]
[62] C E Carlson V Nazaryan and K Griffioen Proton structure corrections to electronic
and muonic hydrogen hyperfine splitting Phys Rev A 78 022517 (2008) arXiv08052603
[physicsatom-ph]
[63] V Bernard N Kaiser and U-G Meiszligner Chiral dynamics in nucleons and nuclei Int J
Mod Phys E 4 193 (1995) arXivhep-ph9501384
[64] T R Hemmert B R Holstein and J Kambor ∆(1232) and the polarizabilities of the nucleon
Phys Rev D 55 5598 (1997) arXivhep-ph9612374
[65] V Pascalutsa and R Timmermans Field theory of nucleon to higher-spin baryon transitions
Phys Rev C 60 042201 (1999) arXivnucl-th9905065
[66] V Pascalutsa Quantization of an interacting spin-32 field and the Delta isobar Phys Rev
D 58 096002 (1998) arXivhep-ph9802288
[67] H Krebs Double Virtual Compton Scattering and SpinStructure of the Nucleon PoS
CD2018 031 (2019)
45
[68] N Kochelev and Y Oh Axial anomaly and the δLT puzzle Phys Rev D 016012 (2012)
arXiv11034892 [hep-ph]
[69] D Drechsel S Kamalov and L Tiator Unitary isobar model ndash MAID2007 Eur Phys J A
69 (2007) available at httpsmaidkphuni-mainzde arXiv07100306 [nucl-th]
[70] P J Mohr B N Taylor and D B Newell CODATA recommended values of the fundamental
physical constants 2010 Rev Mod Phys 84 1527 (2012)
[71] J Ahrens et al (GDH A2) First measurement of the Gerasimov-Drell-Hearn integral for 1H
from 200 to 800 MeV Phys Rev Lett 87 022003 (2001) arXivhep-ex0105089 [hep-ex]
[72] K Helbing (GDH) Experimental verification of the GDH sum rule at ELSA and MAMI Nucl
Phys Proc Suppl 105 113 (2002)
[73] R Bradford A Bodek H S Budd and J Arrington A New parameterization of the nucleon
elastic form-factors NuInt05 proceedings of the 4th International Workshop on Neutrino-
Nucleus Interactions in the Few-GeV Region Okayama Japan 26-29 September 2005 Nucl
Phys Proc Suppl 159 127 (2006) arXivhep-ex0602017
[74] J D Bjorken Applications of the chiral U(6)timesU(6) algebra of current densities Phys Rev
148 1467 (1966)
[75] J D Bjorken Inelastic scattering of polarized leptons from polarized nucleons Phys Rev D
1 1376 (1970)
[76] V Bernard T R Hemmert and U-G Meiszligner Novel analysis of chiral loop effects in
the generalized Gerasimov-Drell-Hearn sum rule Phys Lett B 545 105 (2002) arXivhep-
ph0203167
[77] A Deur et al Experimental study of isovector spin sum rules Phys Rev D 78 032001
(2008) arXiv08023198 [nucl-ex]
[78] B Pasquini P Pedroni and D Drechsel Higher order forward spin polarizability Phys Lett
B 687 160 (2010) arXiv10014230 [hep-ph]
[79] R Jaffe g2ndashThe nucleonrsquos other spin-dependent structure function Comments Nucl Part
Phys 19 239 (1990)
[80] E V Shuryak and A Vainshtein Theory of power corrections to deep inelastic scattering in
quantum chromodynamics (II) Qminus4 effects polarized target Nucl Phys B 201 141 (1982)
[81] S Wandzura and F Wilczek Sum rules for spin dependent electroproduction Test of rela-
tivistic constituent quarks Phys Lett B 72 195 (1977)
[82] B W Filippone and X-D Ji The spin structure of the nucleon Adv Nucl Phys 26 1
(2001) arXivhep-ph0101224
[83] M Burkardt The g2 structure function Proceedings Workshop on Spin structure at long
46
distance Newport News USA March 12-13 2009 AIP Conf Proc 1155 26 (2009)
arXiv09054079 [hep-ph]
[84] A Deur S J Brodsky and G F de Teramond The spin structure of the nucleon Rept
Prog Phys 82 076201 (2019) arXiv180705250 [hep-ph]
[85] B R Holstein D Drechsel B Pasquini and M Vanderhaeghen Higher order polarizabilities
of the proton Phys Rev C 61 034316 (2000) arXivhep-ph9910427
[86] B R Holstein V Pascalutsa and M Vanderhaeghen Sum rules for magnetic moments and
polarizabilities in QED and chiral effective-field theory Phys Rev D 72 094014 (2005)
arXivhep-ph0507016
[87] B Pasquini D Drechsel and L Tiator Invariant amplitudes for pion electroproduction Eur
Phys J A 34 387 (2007) arXiv07122327 [hep-ph]
[88] A J Buchmann Non-spherical proton shape and hydrogen hyperfine splitting Proceedings
International Workshop on Precision Physics of Simple Atomic Systems (PSAS 2008) Wind-
sor Ontario Canada July 21-26 2008 Can J Phys 87 773 (2009) arXiv09104747
[physicsatom-ph]
47
At the real-photon point IA(0) = minusκ2
4and ∆IA(0) = 0 Therefore we give only the
slope of the polarizability ∆IA(Q2) [showing also the separate contributions from πN loops
∆ exchange and π∆ loops] in units of GeVminus2
d∆IAp(Q2)
dQ2
∣∣∣∣Q2=0
= minus858(343) asymp 238minus 1121 + 025 (30a)
d∆IAn(Q2)
dQ2
∣∣∣∣Q2=0
= minus955(343) asymp 141minus 1121 + 025 (30b)
Including the empirical Pauli form factor [73] we find in units of GeVminus2
dIAp(Q2)
dQ2
∣∣∣∣Q2=0
= minus318dIAn(Q2)
dQ2
∣∣∣∣Q2=0
= minus300 (31)
D Γ1(Q2) and I1(Q2) mdash the first moment of the structure function g1(xQ2)
The second variant for a generalization of the GDH sum rule to finite Q2 is defined as
minus α
2M2N
I1(Q2) = minus 1
8π2
int infinν0
dν1radic
ν2 +Q2
[σTT (νQ2) +
Q
νσLT (νQ2)
](32)
= minus α
Q2
int x0
0
dx g1(xQ2)
where I1(0) = minusκ2
4 This generalized GDH integral directly stems from the amplitude
Snonpole1 (νQ2) with the LEX from Eq (12a) It is given by the first moment of the struc-
ture function g1(xQ2) Γ1(Q2) =int x0
0dx g1(xQ2) as follows I1(Q2) =
2M2N
Q2 Γ1(Q2) The
isovector combination
Γ1(pminusn)(Q2) =
int x0
0
dx[g1p(xQ
2)minus g1n(xQ2)] (33)
is related to the axial coupling of the nucleon through the Bjorken sum rule [74 75]
limQ2rarrinfin
Γ1(pminusn)(Q2) =
gA6 (34)
As explained in Eq (28) the moment I1(Q2) splits into a polarizability part ∆I1(Q2) and
a Born part IBorn1 (Q2) Figure 4 lower panel shows the Q2 dependence of I1(Q2) which in
contrast to IA(Q2) shown in Figure 4 upper panel is clearly dominated by its Born part
and the elastic Pauli form factor The πN -loop ∆-exchange and π∆-loop contributions to
the polarizability ∆I1(Q2) are shown in Fig 5 lower panel Comparing to Fig 5 upper
20
000 005 010 015 020 025 030
-006
-004
-002
000
002
004
Q2 GeV2D
G1
proton
000 005 010 015 020 025 030
-006
-005
-004
-003
-002
-001
000
001
Q2 GeV2D
G1
neutron
FIG 6 First moment of the structure function g1(xQ2) for the proton (left) and neutron (right)
as function of Q2 The legend is the same as in Fig 4
panel one sees that ∆I1(Q2) is less sensitive to gC and the dipole form factor in gM than
∆IA(Q2)
For the proton our NLO BχPT prediction gives a very good description of the ex-
perimental data [18 60] and is in reasonable agreement with the MAID prediction [69]
For the neutron one observes good agreement with the empirical evaluations including ex-
trapolations to unmeasured energy regions starting from Q2 gt 01 GeV2 [27 61] In the
region of Q2 lt 005 GeV2 one observes an interesting tension between the recent E97-
110 experiment [27] and the data from CLAS [61] While the newest measurement finds
I1n(0035 GeV2) lt κ2n4 thus suggesting a negative slope at low Q the older measurement
found a rather large value for I1n(00496 GeV2) A similar but milder behaviour is seen in
the E97-110 [27] and E94-010 [21] data for IAn The MAID predictions do not agree with
the CODATA recommended values for the anomalous magnetic moments of the proton and
neutron [70] which in our work are imposed by using empirical parametrizations for the
elastic Pauli form factors [73] The slope of the HB result from Ref [51] is too large and
therefore only reproduces the data at very low Q2
Figure 6 shows the moment Γ1(Q2) for the proton and neutron while Fig 7 shows the
isovector combination Γ1 pminusn(Q2) The LO and NLO BχPT predictions are identical be-
cause our calculation produces the same Delta contributions for the proton and the neutron
For the isovector combination the MAID model only agrees with the data at very low
Q2 lt 010 GeV2 The same is true for the IR result [58 76] while all other chiral results
describe the data NLO BχPT (this work) BχPT+∆ [31] and HBχPT [51]
At the real-photon point I1(0) = minusκ2
4and ∆I1(0) = 0 Therefore we give only the
slope of the polarizability ∆I1(Q2) [showing also the separate contributions from πN loops
21
000 005 010 015 020 025 030-002
000
002
004
006
008
010
Q2 GeV2D
G1Hp-
nL
FIG 7 Isovector combination of Γ1(Q2) as function of Q2 The legend is the same as in Fig 6
The pink curve is the IR result from Ref [58 76] The experimental points are from Ref [77]
(brown dots) and Ref [23] (orange squares)
∆ exchange and π∆ loops] in units of GeVminus2
d∆I1p(Q2)
dQ2
∣∣∣∣Q2=0
= 039(4) asymp 034minus 053 + 058 (35a)
d∆I1n(Q2)
dQ2
∣∣∣∣Q2=0
= minus101(10) asymp minus107minus 053 + 058 (35b)
Including the empirical Pauli form factor [73] we find in units of GeVminus2
dI1p(Q2)
dQ2
∣∣∣∣Q2=0
= 580dI1n(Q2)
dQ2
∣∣∣∣Q2=0
= 553 (36)
E d2(Q2) mdash a measure of color polarizability
Another interesting moment to consider is d2(Q2) which is related to the twist-3 part of
the spin structure function g2(xQ2) [79 80]
d2(Q2) equiv 3
int 1
0
dx x2[g2(xQ2)minus gWW2 (xQ2)] (37)
where gWW2 (xQ2) is the twist-2 part of g2(xQ2) Using the Wandzura-Wilczek relation [81]
one can relate d2(Q2) to moments of the spin structure functions g1(xQ2) and g2(xQ2)
d2(Q2) =
int 1
0
dx x2 [3g2(xQ2) + 2g1(xQ2)] (38)
22
000 005 010 015 020 025 030
0000
0001
0002
0003
0004
0005
0006
0007
Q2 GeV2D
d2
proton
000 005 010 015 020 025 030
0000
0002
0004
0006
0008
0010
0012
Q2 GeV2D
d2
neutron
000 005 010 015 020 025 030
-1
0
1
2
Q2 GeV2D
Γ 010
-4
fm6
D
proton
000 005 010 015 020 025 030-2
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
6D
neutron
FIG 8 Upper panel The inelastic moment d2(Q2) for the proton (left) and neutron (right) as
function of Q2 The result of this work the NLO BχPT prediction is shown by the blue solid line
and the blue band The red line represents the LO BχPT result The purple short-dashed line is
the O(p4) HB result from Ref [50 51] The black dotted line is the MAID model prediction [69]
The experimental points for the neutron (cyan dots) are from Ref [22] Lower panel Fifth-order
generalized forward spin polarizability γ0(Q2) for the proton (left) and neutron (right) as function
of Q2 The experimental points for the proton are from Ref [59] (purple square) and Ref [78]
(orange dot)
This relation however only holds for asymptotically large Q2 It is also in the high-Q2
region where d2(Q2) is a measure of color polarizability [82 83] through its relation to the
gluon field strength tensor [80] We refer to Ref [84] for a recent review on the spin structure
of the nucleon including a discussion of sum rules for deep inelastic scattering and color
polarizabilities
What we consider in the following is the inelastic part of d2(Q2) defined as the moment
of g1(xQ2) and g2(xQ2) spin structure functions cf Eq (38)
d2(Q2) =
int x0
0
dx x2 [3g2(xQ2) + 2g1(xQ2)] (39)
23
This moment provides another testing ground for our BχPT predictions through comparison
with experiments on the neutron [22] Going towards the low-Q2 region the interpretation
of d2(Q2) in terms of color polarizabilities will fade out The above definition however
implies it is related to other VVCS polarizabilities
d2(Q2) =Q4
8M4N
[M2
NQ2
αδLT (Q2) + I1(Q2)minus IA(Q2)
] (40)
Note that d2(Q2) and its first two derivatives with respect to Q2 vanish at Q2 = 0 The
considerations in Eqs (28) and (29) have no effect on d2(Q2) since the Born contribution
from IA(Q2) and I1(Q2) cancel out Therefore d2(Q2) is a pure polarizability
In Fig 8 upper panel we show our NLO BχPT prediction and other results for d2(Q2)
While MAID [69] and BχPT describe the experimental data for the neutron [22] very well
the HB limit [50 51] is showing a fast growth with Q2 This illustrates the importance
of keeping the relativistic result Note also that even though the πN -loop contribution
is dominant both gC and the form factor in gM are essential to obtain a curvature that
reproduces the data cf Fig 9 upper panel For the proton there are to our knowledge
no experimental results to compare with However the agreement between the NLO BχPT
prediction and the MAID prediction at low energies is reasonable
F γ0(Q2) mdash fifth-order generalized forward spin polarizability
It is interesting to compare the generalized fifth-order forward spin polarizability sum
rule
γ0(Q2) =1
2π2
int infinν0
dν
radic1 +
Q2
ν2
σTT (νQ2)
ν5(41)
=64αM4
N
Q10
int x0
0
dx x4
[g1(xQ2)minus 4M2
Nx2
Q2g2(xQ2)
]
to the sum rule integrals for IA(Q2) and γ0(Q2) since they differ merely by their energy
weighting of σTT (νQ2) and a constant prefactor cf Eqs (19) (26) and (41) From IA(Q2)
to γ0(Q2) to γ0(Q2) the energy suppression is increasing by a factor of νminus2 respectively
Therefore the description of γ0(Q2) should be easiest in a low-energy effective-field theory
such as χPT whereas γ0(Q2) and IA(Q2) receive larger contributions from higher energies
In Fig 8 lower panel we show our LO and NLO BχPT predictions for γ0(Q2) One
can see that the πN -loop contribution is positive (in accordance to what we see for the cross
section σTT see Fig 10) The Delta shifts it substantially especially in the low Q2 region
24
000 005 010 015 020 025 030
-0004
-0002
0000
0002
0004
0006
0008
0010
Q2GeV
2D
d2
proton
000 005 010 015 020 025 030
-0005
0000
0005
0010
0015
Q2GeV
2D
d2
neutron
000 005 010 015 020 025 030
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
6D
proton
000 005 010 015 020 025 030
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
6D
neutron
FIG 9 Contributions of the different orders to the chiral predictions of d2(Q2) upper panel and
γ0(Q2) lower panel for the proton (left) and neutron (right) Red solid line πN -loop contribu-
tion green dot-dashed line ∆-exchange contribution orange dotted line π∆-loop contribution
blue long-dashed line total result purple dot-dot-dashed line total result without gC contribution
black short-dashed line total result without gM dipole
bringing it into a better agreement with data In general the BχPT curves start above the
empirical data points at the real-photon point and then decrease asymptotically to zero
above Q2 gt 01 GeV2 On the other hand the MAID prediction reproduces the empirical
data at the real-photon point then decreases to negative values until about Q2 gt 006 GeV2
from where it also starts to asymptotically approach zero Consequently our NLO BχPT
prediction of γ0(Q2) is consistently above the MAID prediction This is very different to
what we saw for IA(Q2) in Fig 4 upper panel where the MAID prediction at the real-
photon point is above the experimental value While the agreement of our predictions with
the empirical data is in general quite good for all moments of σTT (νQ2) one should point
out that both for γ0n(Q2) and γ0p(Q2) we overestimate the data at low Q2 For IA(Q2) such
observation cannot be made because ∆IA(0) = 0 and thus IA(0) is given by the empirical
Pauli form factor only From IA(Q2) γ0(Q2) and γ0(Q2) the latter has the smallest however
25
non-negligible dependence on gC and the dipole in gM cf Fig 9 lower panel
The πN -loop ∆-exchange and π∆-loop contributions to the NLO BχPT prediction of
the static fifth-order forward spin polarizability amount to in units of 10minus4 fm6
γ0p = 112(30) asymp 208minus 096minus 001 (42a)
γ0n = 195(30) asymp 292minus 096minus 001 (42b)
while the slope is composed as follows in units of 10minus4 fm8
dγ0p(Q2)
dQ2
∣∣∣∣Q2=0
= minus084(10) asymp minus100 + 016 + 000 (43a)
dγ0n(Q2)
dQ2
∣∣∣∣Q2=0
= minus142(15) asymp minus158 + 016 + 000 (43b)
Note that the HB prediction of the static polarizability γ0p (423 at O(p3) and 365 at O(ε3)
[78 85]) is almost one order of magnitude larger than the empirical value and therefore not
shown in Fig 8
G Summary
Our results are summarized in Table II where we give the contributions of the different
orders to the chiral predictions of the polarizabilities and their slopes at the real-photon
point A quantitative comparison of our predictions for the static spin polarizabilities to
the work of Bernard et al [31] and different empirical evaluations is shown in Table III We
can see that the inclusion of the Delta turns out to be very important for all moments of
the helicity-difference cross section To describe the Q2 behavior of the polarizabilities the
magnetic coupling of the N rarr ∆ transition should be modified by a dipole form factor as
has been observed previously in the description of electroproduction data [33] This dipole
form factor effectively takes account of vector-meson exchanges The Coulomb-quadrupole
N rarr ∆ transition despite its subleading order is important in the description of some
moments of spin structure functions This is contrary to what we saw for the moments of
unpolarized structure functions [30] where the Coulomb coupling had a negligible effect
The π∆ loops are mainly relevant for the generalized GDH integrals
26
TABLE II The NLO BχPT predictions for the forward VVCS polarizabilities and their slopes at
Q2 = 0 The contributions of the πN loops the ∆ exchange and the π∆ loops are shown together
with the combined total result Note that IA(0) = I1(0) = d2(0) = 0 and (d2)prime = 0
πN loops ∆ exchange π∆ loops Total
γ0 p 201 minus284 minus010minus093(92)
(10minus4 fm4) n 298 003(92)
δLT p 150 minus016 minus002132(15)
(10minus4 fm4) n 235 218(23)
γ0 p 208 minus096 minus001112(30)
(10minus4 fm6) n 292 195(30)
(γ0)prime p minus033011 001
minus022(4)
(10minus4 fm6) n minus073 minus061(7)
(δLT )prime p minus080 minus004 minus001minus085(8)
(10minus4 fm6) n minus119 minus124(12)
(γ0)prime p minus100016 000
minus084(10)
(10minus4 fm8) n minus158 minus142(15)
(∆IA)prime p 238 minus1121 025minus858(343)
(GeVminus2) n 141 minus955(343)
(∆I1)prime p 034 minus053 058039(4)
(GeVminus2) n minus107 minus101(10)
TABLE III Our NLO BχPT predictions for the spin polarizabilities at Q2 = 0 compared with
the BχPT+∆ predictions from Bernard et al [31] and the available empirical information Where
the reference is not given the empirical number is provided by the MAID analysis [55 69] with
unspecified uncertainty
Proton Neutron
This work BχPT+∆ Empirical This work BχPT+∆ Empirical
γ0 minus093(92) minus174(40) minus100(8)(12) [19] 003(92) minus077(40) minus0005
(10minus4 fm4) minus090(8)(11) [78] [MAID]
minus0929(105) [59]
δLT 132(15) 240(1) 134 218(23) 238(3) 203
(10minus4 fm4) [MAID] [MAID]
γ0 112(30) 060(7)(7) [78] 195(30) 123
(10minus4 fm6) 0484(82) [59] [MAID]
IV CONCLUSIONS
We have presented a complete NLO calculation of the polarized non-Born VVCS ampli-
tudes in covariant BχPT with pion nucleon and ∆(1232) fields The dispersion relations
between the VVCS amplitudes and the tree-level photoabsorption cross sections served as a
27
cross-check of these calculations
The obtained moments of the proton and neutron spin structure functions related to gen-
eralized polarizabilities and GDH-type integrals agree well with the available experimental
data The description of their Q2 evolution is improved compared to the previous χPT pre-
dictions In particular the NLO BχPT predictions obtained here give a better description
of the empirical data (eg from the Jefferson Laboratory ldquoSpin Physics Programrdquo) than
the HB [50 51] and IR [58] calculations
The demonstrated predictive power of the χPT framework amplitudes makes it well
suited for extending the χPT evaluation of the TPE effect in the hyperfine structure of
(muonic-)hydrogen [15ndash17] to next-to-leading order
ACKNOWLEDGEMENTS
We thank Lothar Tiator and Marc Vanderhaeghen for helpful discussions This work
is supported by the Deutsche Forschungsgemeinschaft (DFG) through the Collaborative
Research Center [The Low-Energy Frontier of the Standard Model (SFB 1044)] JMA ac-
knowledges support from the Community of Madrid through the ldquoPrograma de atraccion de
talento investigador 2017 (Modalidad 1)rdquo and the Spanish MECD grants FPA2016-77313-P
FH gratefully acknowledges financial support from the Swiss National Science Foundation
Appendix A Tensor decompositions of the VVCS amplitudes
In this appendix we review the decomposition of the forward VVCS process into tensor struc-
tures and scalar amplitudes In particular we consider the connection between the covariant and
the semi-relativistic decomposition in the lab frame that is defined in terms of the conventional
transverse longitudinal transverse-transverse and transverse-longitudinal amplitudes
As explained in Sec II A the process of forward VVCS off the nucleon can be described in
terms of four explicitly covariant amplitudes S1 2 and T1 2 [3]
T (νQ2) =
(minusgmicroν +
qmicroqν
q2
)T1(νQ2) +
1
M2N
(pmicro minus p middot q
q2qmicro)(
pν minus p middot qq2
qν)T2(νQ2) (A1)
minus 1
MNγmicroναqα S1(νQ2)minus 1
M2N
(γmicroνq2 + qmicroγναqα minus qνγmicroαqα
)S2(νQ2)
εprimelowastmicro εν
where εmicro (εprimelowastmicro ) are the incoming (outgoing) photon polarization vectors ν is the photon lab-frame
energy and Q2 is the photon virtuality Alternatively the decomposition in the laboratory frame
28
(which in the forward case coincides with the Breit frame) is parametrized in terms of the nucleon
Pauli matrices ~σ and the four scalar functions fL fT gTT and gLT
T (νQ2) = ε0 εprimelowast0 fL(νQ2) + (~ε primelowast middot ~ε ) fT (νQ2) + i~σ middot (~ε primelowast times ~ε ) gTT (νQ2) (A2)
minusi~σ middot [(ε0~εprimelowast minus ~ε ε primelowast0 )times q] gLT (νQ2)
Here ~q and q = ~q|~q | are the photon three-momentum in the lab system and its unit vector The
modified polarization vector components are given by
ε0 =
[ε0 minus
ν
|~q | (~ε middot q )
] |~q |Q ~ε = ~εminus q (~ε middot q ) (A3)
where ε = (ε0~ε ) is the usual incoming photon polarization vector and εprimelowast the outgoing polarization
vector The LEX of the lab frame amplitudes [Eq (10)] can serve in particular as the definition
of the generalized polarizabilities The lab frame amplitudes are also conveniently used for the
definition of the response functions see the example of the scalar amplitude gLT (νQ2) and the
corresponding response function σLT (νQ2) below in App B
Appendix B Photoabsorption cross sections
In the forward kinematics the spin-dependent VVCS amplitudes and the spin polarizabilities
can be described in terms of the polarized structure functions g1(xQ2) and g2(xQ2) or equiv-
alently the helicity-difference cross section σTT (νQ2) and the longitudinal-transverse response
function σLT (νQ2) with the help of dispersion relations (5) and the optical theorem (3) In this
way the photoabsorption cross sections measured in electroproduction processes form the basis
for most empirical evaluations shown throughout Sec III In the following we present the BχPT
predictions for the tree-level cross sections of πN - π∆- and ∆-production through photoabsorption
on the nucleon cf Figs 8 9 and 10 in Ref [30] In Secs B 1 and B 2 we will discuss the leading
πN -production channel and the ∆-production channel respectively We used these cross sections
to verify the polarizability predictions obtained otherwise from the calculated non-Born VVCS
amplitudes Due to the bad high-energy behavior of the π∆-production cross sections in BχPT
cf Fig 10 the dispersion relations in Eq (5) require further subtractions for a reconstruction of
the π∆-loop contribution to the spin-dependent VVCS amplitudes Therefore not all polarizabil-
ities could be verified but only those appearing as higher-order terms in the LEX of the VVCS
amplitudes such as γ0 [16]
29
proton neutron
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Neutron
02 04 06 08 100
50
100
150
200
250
[GeV]
T [b]
02 04 06 08 10
-50
0
50
100
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Neutron
02 04 06 08 100
50
100
150
200
250
[GeV]
T [b]
02 04 06 08 10
-50
0
50
100
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02040608100
50
100
150
200
250
300
[GeV]
T[b]
0204060810
-50
0
50
[GeV]
TT[b]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
0204060810
00
02
04
06
[GeV]
LTQ
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
0204060810
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
FIG 10 Photoabsorption cross sections for πN (red) and π∆ production (orange) with Q2 = 0
(solid) and Q2 = 01 GeV2 (dashed for πN and dotted for π∆ channel)
1 πN-production channel
In order to extract the response function σLT (νQ2) we have developed a method similar to
the one used to calculate σTT (νQ2) see for example Ref [86] For σLT (νQ2) however the
calculation is more complicated because one has to take into account that the associated Compton
process involves a spin-flip of the nucleon as illustrated in Fig 11 When calculating the cross
section the product of the incoming nucleon spinors has to reflect this flip
The forward VVCS amplitude related to σLT (νQ2) mdash and δLT (Q2) mdash is gLT (νQ2) It can
be extracted from Eq (A2) if one takes the modified polarization vector components in Eq (A3)
with ε = εL and εprimelowast = εlowastplusmn as input where εL = 1Q(|~q | 0 0 q0) and εplusmn = ∓ 1radic
2(0 1plusmni 0) are the
standard longitudinal and transverse polarization vectors respectively For εL and εlowastplusmn only the
choice of helicities hprime = plusmn12 and h = ∓12 gives a non-zero contribution and one obtains
χdaggerhprime T (νQ2)χh = χdaggerhprimeminusi~σ middot [(ε0~εprimelowast minus ~ε ε primelowast0 )times q] gLT (νQ2)χh =
radic2 gLT (νQ2) (B1)
where χh and χdaggerhprime are two-component Pauli spinors with opposite helicities or here spins
Let us now consider the related photoabsorption process and in particular the tree-level γlowastN rarr
30
s = 12
sprime = minus12
λ = 0 λprime = 1
rprime rprime
s=12sprime=minus1
2
λ=0λprime=1
rprimerprime
s=12
sprime=minus12
λ=0λprime=1
rprimerprime hacute=frac12h=-frac12
λacute=1λ=0
FIG 11 Relation between the forward Compton process and the photoabsorption process given
by the optical theorem In particular we show the longitudinal-transverse contribution The
double-line arrows represent the spin of the external particles while the dot represents the scalar
(longitudinal) polarization of the incoming photon Inside the blob the intermediate states are
represented eg nucleons with spins rprime (which are averaged in the calculation of the cross section)
and pions
πN channel see diagrams in Fig 8 of Ref [30] We define the πN -production amplitude as
T = uhB (PB)sumi
Ai(s t)Γi uhA(PA) (B2)
with the Dirac structures
Γ1 = γ5 (B3a)
Γ2 =1
2
[qA ε
]γ5 (B3b)
where uhA(PA) and udaggerhB (PB) are the Dirac spinors and PA and PB are the four-momenta of
the incoming and outgoing nucleons respectively When calculating the photoabsorption cross
section related to the VVCS amplitude in Eq (B1) the nucleon spin flip should be implemented
by uhprime(PA) in T dagger and uh(PA) in T together with the appropriate transverse and longitudinal
photon polarization vectors εlowastplusmn and εL
However if one wants to use the properties of the Dirac matrices it is more useful to construct
an operator to produce this spin flip in the external nucleons of Fig 11 This is accomplished by
introducing the projector ΓLT equiv 12radic
2(γ1 + iγ2)γ5 which also takes into account the extra factor
radic2 in Eq (B1) We checked that with this projector one correctly extracts δLT by comparing the
HB limit of our result to the HB result of Ref [50] where the authors calculate this polarizability
from the Compton amplitude directly With all those ingredients the longitudinal-transverse cross
section is calculated in the following way
σLT (νQ2) =1
64π2 s
|~pf |cm
|~pi|cm
int 1
minus1dcos θ
sumij
AiAdaggerjXij (B4)
31
with
Xij = Tr[(PB +MN )Γi(PA +MN )ΓLTγ0Γdaggerjγ
0] (B5)
where θ is the scattering angle in the center-of-mass (cm) frame and |~pi|cm (|~pf |cm) is the three-
momentum of an incoming (outgoing) particle in the cm frame An explicit calculation of the
matrix Xij leads to
X = MNQ
0 2(PB minus PA) middot εLminusradic
2|~qf |cm sin θ (sminus u)
(B6)
where |~qi|cm (|~qf |cm) is the relative three-momentum of the incoming (outgoing) particles in the
cm frame Here s t and u are the usual Mandelstam variables For the different γlowastN rarr πN
channels we obtain the following amplitudes Ai where we introduce qA as the four-momentum of
the incoming photon and qB as the four-momentum of the outgoing pion
bull γlowastprarr π0p
A1 =e gAMN
fπ
[2PA middot ε+ qA middot ε
sminusM2N
+2PB middot εminus qA middot ε
uminusM2N
] (B7a)
A2 =e gAMN
fπ
[1
sminusM2N
+1
uminusM2N
] (B7b)
bull γlowastprarr π+n
A1 =
radic2 e gAMN
fπ
[2PA middot ε+ qA middot ε
sminusM2N
+2(PA minus PB) middot ε+ qA middot ε
tminusm2π
] (B8a)
A2 =
radic2 e gAMN
fπ(sminusM2N )
(B8b)
bull γlowastnrarr π0n
A1 = 0 (B9a)
A2 = 0 (B9b)
bull γlowastnrarr πminusp
A1 =
radic2 e gAMN
fπ
[2PB middot εminus qA middot ε
uminusM2N
minus 2(PA minus PB) middot ε+ qA middot εtminusm2
π
] (B10a)
A2 =
radic2 e gAMN
fπ(uminusM2N )
(B10b)
The analytical expressions shown above were checked with the amplitudes given in Ref [87] An-
alytical expressions for the tree-level γlowastN rarr πN channel of the σLT (νQ2) and σTT (νQ2) cross
32
sections are given below (proton channels π+n and π0p neutron channel πminusp) We checked that
they reproduce the known results in the real-photon limit [45 86] To shorten the final expressions
for the cross sections which are considerably longer for finite Q2 than in the real-photon limit we
define the following dimensionless kinematic variables
αγ = (ENi )cmradics =
s+M2N +Q2
2s (B11)
απ = (ENf )cmradics =
s+M2N minusm2
π
2s (B12)
βγ = Eγcmradics =
sminusM2N minusQ2
2s (B13)
βπ = Eπcmradics =
sminusM2N +m2
π
2s (B14)
λγ = |~qi|cmradics =
radic(sminusM2
N minusQ2)2 + 4sQ2
2s (B15)
λπ = |~qf |cmradics =
radic(sminusM2
N +m2π)2 minus 4sm2
π
2s (B16)
Here (ENi )cm and (ENf )cm are the energies of the incoming and outgoing nucleon Eγcm is the energy
of the incoming photon Eπcm is the energy of the outgoing pion all in the cm frame
σ(π+n)TT = minus e2g2
AM2N
64πf2πs
3(sminusM2N )2λ4
γ
4sλπλγ
[(M2
N minus s)(M2N minusQ2 minus s)(Q2 + 2sβγβπ)
+ 2s(m2πQ
2 minus (M2N minus s)
(M2N + s(minus1 + 2βγβπ)
))λ2γ
]+ (M2
N minus s)(M2N minusQ2 minus s)
times (Q2 + 2sβγβπ minus 2sλπλγ)(Q2 + 2s(βγβπ + λπλγ)
)times log
(Q2 + 2sβγβπ minus 2sλγλπQ2 + 2sβγβπ + 2sλγλπ
) (B17)
33
σ(π0p)TT =
e2g2AM
2Nλπ
64πf2πs(sminusM2
N )2λγ
1
s2λπλ3γ
[(M2
N minus s)(M2N minusQ2 minus s)
(3M2
N + s(2βγβπ minus 3))(M2N
+ s(2βγβπ minus 1))
+ 2s
((2m2
πQ2 + (M2
N minus s)2)(M2
N minus s)minus 2(minusm2
πQ2 + (sminusM2
N )2)
times sβγβπ + 4s2(sminusM2N )β2
γβ2π
)λ2γ
]arctan
(2sλπλγ
M2N + s(2βγβπ minus 1)
)+ 2
[minus 2m2
πQ2
+ (M2N minus s)
(2(sminusM2
N ) + 4sβγβπ +(sminusM2
N +Q2)(3M2N minus 3s+ 2sβγβπ)
sλ2γ
+m2πQ
2(sminusM2N )
(M2N minus s+ 2sβγβπ)(M2
N + s(minus1 + 2βγβπ minus 2λπλγ))
+m2πQ
2(sminusM2N )
(M2N minus s+ 2sβγβπ)(M2
N + s(minus1 + 2βγβπ + 2λπλγ))
)
+1
λπλγ
(βγβπ
(minusm2
πQ2 + (M2
N minus s)(M2N minus s+ 2sβγβπ)
)minus (M2
N minus s)(M2N minusQ2 minus s)λ2
π
)log
(M2N + s(minus1 + 2βγβπ + 2λπλγ)
M2N + s(minus1 + 2βγβπ minus 2λπλγ)
)] (B18)
σ(πminusp)TT =
e2g2AM
2N
64πf2πs
3λ4γ(M2
N + s(2βπβγ minus 1))2 minus 4λ2πs
2λ2γ)
4sλπλγ
((minusM2
N +Q2 + s)(M2N
+ s(2βπβγ minus 1))
2 minus 2sλ2γ
(2λ2
πs(minusM2N +Q2 + s) +m2
πQ2))
+((M2N + s(2βπβγ minus 1)
)2
minus 4λ2πs
2λ2γ
)((M2N + s(2βπβγ minus 1)
)(M2N minus 2Q2 minus s(2βπβγ + 1)
)+ 4λ2
πs2λ2
γ
)times log
(minusM2
N minus 2βπsβγ minus 2λπsλγ + s)minus(M2N + s(2βπβγ minus 2λπλγ minus 1)
)(M2N + s(minus1
+ 2βπβγ + 2λπλγ))[((
M2N + s(2βπβγ minus 1)
)(M2N minus 2Q2 minus s(2βπβγ + 1)
)+ 4λ2
πs2λ2
γ
)times log
(minusM2
N minus 2βπsβγ + 2λπsλγ + s)
+((Q2 + 2βπsβγ)2 minus 4λ2
πs2λ2
γ
)times log
(Q2 + 2βπsβγ minus 2λπsλγQ2 + 2sβπβγ + 2λπsλγ
)]+ 4sλ2
γ
((M2
N + s(2βπβγ minus 1))2 minus 4λ2πs
2λ2γ
)times (M2
N minus s) arctanh
(2λπsλγ
M2N + 2βπsβγ minus s
) (B19)
34
σ(π+n)LT =
e2g2AM
3Nλπ
32πf2πQs
3(sminusM2N )2λ4
γ
2sλγ
[(M2
N minus s)(Q2 + 2sβ2γ)(Q2 + 2sβγβπ)
minus 4s
((M2
N minus s)(Q2 minus 2s(απ minus 1)βγ
)+Q2sβγβπ
)λ2γ + 8s3(minus1 + απ)λ4
γ
]minus M2
N minus sλπ
[(Q2 + 2sβ2
γ)(Q2 + 2sβγβπ)2 + 4s2(2(απ minus 1)βγ(Q2 + 2sβγβπ)
minusQ2λ2π
)λ2γ + 8s3(απ minus 1)2λ4
γ
]arctan
(2sλπλγ
Q2 + 2sβγβπ
) (B20)
σ(π0p)LT =
e2g2AM
3Nλπ
16πf2πQs(sminusM2
N )2λγ
1
minus2s(M2N + s(minus1 + 2βγβπ))2λ2
γ + 8s3λ2πλ
4γ
[minus 3M8
N (Q2
+ 2sβ2γ) + 2M4
Ns2
(minus (Q2 + 2sβ2
γ)(minus3 + 2βγβπ) +(Q2(minus3 + 2βγβπ + 6λ2
π) + 2sβγ
times (minus3 + 12απ + 2βγβπ minus 12απβγβπ + 4βγλ2π))λ2γ minus 4s(α2
π minus 1)λ4π
)+ 2M2
Ns3
[minus (Q2
+ 2sβ2γ)(minus1 + 2βγβπ)
(6 + βγβπ(minus9 + 2βγβπ)
)+
(Q2(3minus 12λ2
π + 4βγβπ(minus1 + 2βγβπ
+ λ2π))minus 2sβγ
(minus 3 + 4απ(3 + 2απ(3 + 2βγβπ(minus3 + βγβπ)) + 4βγ(βπ + (2minus βγβπ)
times λ2π)))λ2γ + 8s
((απ minus 1)(1 + απ minus 2βγβπ) + 2απβγλ
2π
)λ4γ
)+ s4
((Q2 + 2sβ2
γ)
times (1minus 2βγβπ)2(2βγβπ minus 3) + 2(Q2(minus1 + 6λ2
π + 2βγβπ((1minus 2βγβπ)2 minus 2λ2π)) + 2sβγ
times (minus1 + 2βγβπ + 4(minus1 + βγβπ)(minusαπ + 2απβγβπ minus βγλ2π)))λ2γ minus 8
(s(απ minus 1)(απ
+ (1minus 2βγβπ)2) + 2βγ(2sαπ +Q2βγ)λ2π
)λ4γ + 32s(απ minus 1)λ2
πλ6γ
)+ 2M6
Ns
(Q2(6
minus 7βγβπ + λ2γ) + 2sβγ(βγ(6minus 7βγβπ) + (1minus 4απ)λ2
γ)
)]+sminusM2
N
4s2λπλ3γ
[(Q2 + 2sβ2
γ)
times(3M2
N + s(2βγβπ minus 3))(M2N + s(2βγβπ minus 1)
)+ 2s
(minusM2
N (Q2 + 2sβγ minus 8sαπβγ)
+ s(2sβγ(1minus 4απ + 4απβγβπ) +Q2(1minus 2λ2π)))λ2γ + 8s3(απ minus 1)λ4
γ
]
times arctan
(2sλπλγ
M2N + s(2βγβπ minus 1)
) (B21)
35
σ(πminusp)LT =
e2g2AM
3N
64πf2πQs
3λ4γ(M2
N minusQ2 minus s)(M2N + s(2βπβγ + 2λπλγ minus 1))
times
1
M2N + s(2βπβγ minus 2λπλγ minus 1)
[4λπsλγ(M2
N minusQ2 minus s)(
2sλ2γ
(s((1minus 2λ2
π)Q2
+ 2(2απ minus 1)sβγ(2βπβγ minus 1))minusM2N (Q2 + 2(1minus 2απ)sβγ)
)+ (Q2 + 2sβ2
γ)(M2N + s(2βπβγ minus 1)
)2 + 8(απ minus 1)απs
3λ4γ
)]minus(M2N + s(2βπβγ + 2λπλγ minus 1)
)times[
log
(minusM2N minus 2βπsβγ minus 2λπsλγ + s
minusM2N minus 2βπsβγ + 2λπsλγ + s
)(minus (Q2 + 2sβ2
γ)(M2N + s(2βπβγ minus 1)
)(minusM2
N + 2Q2 + 2βπsβγ + s)minus 2sλ2γ
(minusM2
N (Q2 + 2sβγ) +Q4 +Q2s(4απβγ
minus 2βγ minus 2λ2π + 1) + 2s2βγ(4(απ minus 1)βπβγ + 1)
)minus 8(απ minus 1)2s3λ4
γ
)+ 2
(4s2λ2
γ
(2(απ minus 1)βγ(Q2 + 2βπsβγ)minus λ2
πQ2)
+ (Q2 + 2βπsβγ)2(Q2 + 2sβ2γ)
+ 8(απ minus 1)2s3λ4γ
)arctanh
(2λπsλγ
Q2 + 2βπsβγ
)] (B22)
2 ∆-production channel
The tree-level ∆-exchange diagram in Fig 2 of Ref [30] contributes to the non-Born part of
the VVCS amplitudes The contribution of the ∆ exchange to the VVCS amplitudes can be split
into [17]
S∆-exch1 (νQ2) = S∆-pole
1 (νQ2) + S∆-exch1 (νQ2) (B23a)
νS∆-exch2 (νQ2) = νS∆-pole
2 (νQ2) + νS2∆-exch
(νQ2) (B23b)
and similarly for the unpolarized VVCS amplitudes discussed in Ref [30] Here we introduced the
∆-pole contributions S∆-polei and the ∆-non-pole contributions S∆-exch
i The former amplitudes
feature a pole at the ∆(1232)-production threshold and thus are proportional to
1
[sminusM2∆][uminusM2
∆]=
1
4M2N
1
ν2∆ minus ν2
(B24)
36
They can be reconstructed from the dispersion relations in Eq (5) with the tree-level ∆-production
cross sections as input cf Fig 10 in Ref [30]
σTT (νQ2) =π2α
M2NM
2+|~q |
minus g2
MMN (M+ + ν)|~q |2 +g2E(∆minus ν)(Q2 minusMNν)2
MN(B25a)
+g2CQ
4s(∆minus ν)
MNM2∆
minus 4gMgE(Q2 minusMNν)|~q |2 minus 4gMgCQ2|~q |2
+2gEgCQ
2[minusMNM∆ |~q |2 + s(Q2 +∆ν)
]MNM∆
δ(ν minus ν∆)
σLT (νQ2) =Qπ2α
M2NM
2+|~q |
g2E(MNν minusQ2) [M∆(MN + ν)minus s]
MN(B25b)
+g2CQ
2[MNM∆|~q |2 minus s(Q2 +∆ν)
]MNM2
∆
+ gMgEM∆|~q |2 minusgMgC(Q2 minusMNν)|~q |2
M∆
+gEgC(ν minus∆)(M2
N |~q |2 minus 2Q2s)
MNM∆
δ(ν minus ν∆)
with ∆ = M∆ minusMN M+ = M∆ + MN and the Mandelstam variable s = M2N + 2MNν minus Q2
Analytical expressions for the spin structure functions g1(xQ2) and g2(xQ2) can be constructed
from Eq (3) with the flux factor K(νQ2) = |~q | =radicν2 +Q2
In the ∆-non-pole contributions to S1(νQ2) and νS2(νQ2) the pole in ν at the ∆(1232)-
production threshold has canceled out
S∆-exch1 (νQ2) =
πα
MNM2+
[g2MQ
2+ + g2
E
(∆2 minus 3Q2
)+
4g2CQ
4
M2∆
minus 8gMgEM∆ωminus (B26a)
minus2gMgCQ2(MN minus 4M∆)
M∆+
2gEgCQ2(3MN minus 2M∆)
M∆
]
νS2∆-exch
(νQ2) =2πα
MNM2+
[g2EM∆∆ωminus +
g2M MNQ
2+
2+g2C Q
2(Q2 minus∆2)
2M∆(B26b)
+gEgM M∆(M∆ω+ minus 4MNωminus)minus gEgC ∆(2Q2 +MNω+)
+gMgC Q2(4MN minus ω+)
]+S∆-exch
2 (νQ2)
ν
[M2
∆ ω2+
M2N
+ ν2
]
with Q+ =radic
(M∆ +MN )2 +Q2 and ωplusmn = (M2∆minusM2
N plusmnQ2)2M∆ and the non-pole contribution
to S2(νQ2)
S∆-exch2 (νQ2) = minus2παMNν
M∆M2+
[gM + gE
]gC (B27)
These amplitudes to the contrary are not described by the tree-level ∆-production cross sections
in the standard dispersive approach [17] This peculiarity has been previously missed e g in the
calculation of the ∆-exchange contribution to the hydrogen hyperfine splitting in Ref [88] The
importance of including the ∆-non-pole contribution is also evident when considering the BC sum
37
rule in Eq (14) The ∆-pole terms by themselves violate the BC sum rule but cancel exactly with
the ∆-non-pole terms
limνrarr0
νS∆-pole2 (νQ2) + lim
νrarr0νS2
∆-exch(νQ2) = 0 (B28)
Appendix C Polarizabilities at Q2 = 0
In this section we give analytical expressions for the static values and slopes at Q2 = 0 of
the spin polarizabilities and moments of polarized structure functions In particular we give the
HB expansion of the πN -loop contributions and the ∆-exchange contributions The complete
expressions also for the π∆-loop contributions can be found in the Supplemented material Recall
that IA(0) = I1(0) = d2(0) = 0 and dd2(Q2)dQ2
∣∣∣Q2=0
= 0
1 πN-loop contribution
Here we give analytical expressions for the πN -loop contributions to the proton and neutron
spin polarizabilities expanded in powers of micro = mπMN viz the HB expansion Note that we
choose to expand here to a high order in micro the strict HB expansion would only retain the leading
term in an analogous NLO calculation
bull Static polarizabilities (Q2 = 0)
γ0p =e2g2
A
96π3f2πm
2π
1minus 21πmicro
8minus(
59
2+ 26 logmicro
)micro2 +
1875πmicro3
64
+3
(3
2+ 26 logmicro
)micro4 +
(C1)
γ0n =e2g2
A
48π3f2πm
2π
1
2minus 9πmicro
16minus 2micro2 logmicro+
75πmicro3
128minus 3micro4
4+
(C2)
δLTp =e2g2
A
192π3f2πm
2π
1minus 9πmicro
8+
(13
2minus 2 logmicro
)micro2 minus 465πmicro3
64
minus(
47
2+ 42 logmicro
)micro4 +
(C3)
δLTn =e2g2
A
96π3f2πm
2π
1
2+
3πmicro
16+ (1 + 2 log micro)micro2 minus 105πmicro3
128+
5micro4
4+
(C4)
38
δLTn =e2g2
A
96π3f2πm
2π
1
2+
3πmicro
16+ (1 + 2 log micro)micro2 minus 105πmicro3
128+
5micro4
4+
(C5)
γ0p =e2g2
A
16π3f2πm
4π
4
45minus 3πmicro
16+
14micro2
5minus 1813πmicro3
384minus 192
5(1 + log micro)micro4
+80703πmicro5
2048+
(C6)
γ0n =e2g2
A
16π3f2πm
4π
4
45minus 5πmicro
48+
4micro2
5minus 245πmicro3
384minus 32micro4 logmicro
15
+1323πmicro5
2048+
(C7)
bull Slopes of polarizabilities at Q2 = 0
dγ0p(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
2minus 45πmicro
4+ 223micro2 minus 28515πmicro3
64
minus9
(1953
4+ 449 logmicro
)micro4 +
570255πmicro5
128+
(C8)
dγ0n(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
2minus 81πmicro
8+ 94micro2 minus 2535πmicro3
32
minus3 (1 + 90 logmicro)micro4 +84315πmicro5
1024+
(C9)
dδLTp(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
2880π3f2πm
4π
minus5
2minus 27πmicro
32+ 20micro2 minus 5865πmicro3
256
+3
(617
4+ 36 logmicro
)micro4 minus 2056845πmicro5
4096+
(C10)
dδLTn(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
minus5
4minus 81πmicro
64minus 11micro2 +
10005πmicro3
512
+15
8(11 + 48 logmicro)micro4 minus 267015πmicro5
8192+
(C11)
dIAp(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
96π2f2πmicro
2
1minus 15πmicro
4minus 1
2(115 + 88 logmicro)micro2 +
1839πmicro3
32
+5 (5 + 34 logmicro)micro4 +
(C12)
dIAn(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
48π2f2πmicro
2
1
2minus 11πmicro
8minus 1
4(1 + 20 logmicro)micro2 +
99πmicro3
64
minus25micro4
12+
(C13)
dI1p(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
96π2f2πmicro
3π
8+ 2 (4 + 3 logmicro)microminus 537πmicro2
64
minus1
2(15 + 56 logmicro)micro3 +
(C14)
39
dI1n(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
48π2f2πmicro
minus π
16+
1
4(3 + 4 logmicro)microminus 57πmicro2
128+
2micro3
3+
(C15)
γ0p(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2g2
A
16π3f2πm
6π
1
105minus 23πmicro
256+
377micro2
210minus 15551πmicro3
6144+
3371micro4
105
minus1640457πmicro5
32768+
(C16)
γ0n(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2g2
A
16π3f2πm
6π
1
105minus 153πmicro
1792+
69micro2
70minus 4615πmicro3
6144+
172micro4
35
minus120897πmicro5
32768+
(C17)
2 ∆-exchange contribution
Here we give analytical expressions for the tree-level ∆-exchange contributions to the nucleon
spin polarizabilities and their slopes at Q2 = 0 Note that the ∆-exchange contributes equally to
proton and neutron polarizabilities Recall that for the magnetic γlowastN∆ coupling we introduced a
dipole form factor to mimic vector-meson dominance gM rarr gM(1 +Q2Λ2)2
bull Static polarizabilities (Q2 = 0)
γ0 = minus e2
4πM2+
(g2M
∆2+
g2E
M2+
minus 4gMgEM+∆
) (C18)
δLT =e2M∆
4πM3+
(g2E
MNM++gMgE∆MN
minus gEgCM2
∆
) (C19)
γ0 =e2M2
N
π∆2M4+
(minusg
2M
∆2+
g2E
M2+
+4gMgE∆M+
) (C20)
bull Slopes of polarizabilities at Q2 = 0
dγ0(Q2)
dQ2
∣∣∣∣∣Q2=0
= minus e2
πM2+∆
(g2M
∆
[1
4∆2minus 1
∆M++
1
2M2+
]minus 1
Λ2
g2M
∆+
g2E
2M2+
[1
2∆minus 3
M+
]minusgMgE
M+
[1
∆2minus 5
∆M++
1
M2+
]+
1
Λ2
2gMgEM+
+2gMgC∆M2
+
minus gEgCM3
+
) (C21)
40
d δLT (Q2)
dQ2
∣∣∣∣∣Q2=0
=e2M∆∆
4πMNM2+
(g2E
∆2M2+
[1
∆minus 4
M+
]minus g2
C
∆M2∆M
2+
+gMgE∆2M+
[1
∆2
minus 3
∆M++
1
M2+
]minus 2
Λ2
gMgE∆2M+
+gMgC∆M2
∆
[1
2∆2minus 2
∆M++
1
2M2+
]minus gEgC
2M2∆M
2+
[7
∆+
1
M+
]) (C22)
dIA(Q2)
dQ2
∣∣∣∣∣Q2=0
= minusM2N
M2+
(g2M
2∆2+
g2E
MNM+minus 2gMgE
∆M+minus gEgCM∆M+
) (C23)
dI1(Q2)
dQ2
∣∣∣∣∣Q2=0
= minusM∆M2N
2M3+
(g2E
MNM∆minus gMgE∆MN
minus gEgCM2
∆
) (C24)
γ0(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2M2
N
π∆3M6+
(g2M
∆
[3M2
∆ + 2M∆MN minus 9M2N
∆2+
4M2+
Λ2
]
minus4gMgE
[5M2
∆ minus 9M2N
M+∆2+
2M+
Λ2
]+
g2E
∆M+[7M∆ minus 9MN ]
minus8gMgC∆
+4gEgCM+
) (C25)
[1] D Drechsel B Pasquini and M Vanderhaeghen Dispersion relations in real and virtual
Compton scattering Phys Rept 378 99 (2003) hep-ph0212124
[2] S E Kuhn J-P Chen and E Leader Spin structure of the nucleon mdash status and recent
results Prog Part Nucl Phys 63 1 (2009) arXiv08123535 [hep-ph]
[3] F Hagelstein R Miskimen and V Pascalutsa Nucleon polarizabilities from Compton scat-
tering to hydrogen atom Prog Part Nucl Phys 88 29 (2016) arXiv151203765 [nucl-th]
[4] B Pasquini and M Vanderhaeghen Dispersion theory in electromagnetic interactions Ann
Rev Nucl Part Sci 68 75 (2018) arXiv180510482 [hep-ph]
[5] S Gerasimov A Sum rule for magnetic moments and the damping of the nucleon magnetic
moment in nuclei Sov J Nucl Phys 2 430 (1966)
[6] S Drell and A C Hearn Exact sum rule for nucleon magnetic moments Phys Rev Lett
16 908 (1966)
[7] J S Schwinger Source theory viewpoints in deep inelastic scattering Proc Natl Acad Sci
USA 72 1 (1975)
[8] J S Schwinger Source theory viewpoints in deep inelastic scattering Electromagnetic In-
teractions and Field Theory Proceedings 14 Internationale Universitatswochen Schladming
Austria February 24-March 7 1975 Acta Phys Austriaca Suppl 14 471 (1975)
41
[9] J Schwinger Source theory discussion of deep inelastic scattering with polarized particles
Proc Natl Acad Sci USA 72 1559 (1975)
[10] A M Baldin Polarizability of nucleons Nucl Phys 18 310 (1960)
[11] M Gell-Mann M L Goldberger and W E Thirring Use of causality conditions in quantum
theory Phys Rev 95 1612 (1954)
[12] A Pineda Leading chiral logarithms to the hyperfine splitting of the hydrogen and muonic
hydrogen Phys Rev C 67 025201 (2003)
[13] C Peset and A Pineda The two-photon exchange contribution to muonic hydrogen from
chiral perturbation theory Nucl Phys B 887 69 (2014) arXiv14064524 [hep-ph]
[14] C Peset and A Pineda Model-independent determination of the two-photon exchange contri-
bution to hyperfine splitting in muonic hydrogen JHEP 04 060 arXiv161205206 [nucl-th]
[15] F Hagelstein and V Pascalutsa Proton structure in the hyperfine splitting of muonic hydro-
gen PoS CD15 077 (2016) arXiv151104301 [nucl-th]
[16] F Hagelstein Exciting Nucleons in Compton Scattering and Hydrogen-Like Atoms PhD
thesis Mainz U Inst Kernphys (2017) arXiv171000874 [nucl-th]
[17] F Hagelstein ∆(1232)-Resonance in the hydrogen spectrum Proceedings 11th International
Workshop on the Physics of Excited Nucleons (NSTAR 2017) Columbia SC USA August
20-23 2017 Few Body Syst 59 93 (2018) arXiv180109790 [nucl-th]
[18] Y Prok et al (CLAS) Moments of the spin structure functions gp1 and gd1 for 005 lt Q2 lt 30
GeV2 Phys Lett B 672 12 (2009) arXiv08022232 [nucl-ex]
[19] H Dutz et al (GDH) First measurement of the Gerasimov-Drell-Hearn sum rule for 1H from
07 GeV to 18 GeV at ELSA Phys Rev Lett 91 192001 (2003)
[20] M Amarian et al (Jefferson Lab E94010) Measurement of the generalized forward spin po-
larizabilities of the neutron Phys Rev Lett 93 152301 (2004) arXivnucl-ex0406005
[21] M Amarian et al The Q2 evolution of the generalized Gerasimov-Drell-Hearn integral for
the neutron using a 3He target Phys Rev Lett 89 242301 (2002) arXivnucl-ex0205020
[22] M Amarian et al (Jefferson Lab E94-010)Q2 evolution of the neutron spin structure moments
using a 3He target Phys Rev Lett 92 022301 (2004) arXivhep-ex0310003
[23] A Deur et al Experimental determination of the evolution of the Bjorken integral at low Q2
Phys Rev Lett 93 212001 (2004) arXivhep-ex0407007
[24] K Slifer Low Q2 measurement of gp2 and the δLT spin polarizability Spin structure at long dis-
tance Proceedings Workshop Newport News USA March 12ndash13 2009 nucl-ex09064775
(2009) AIP Conf Proc 1155 10106313203293 (2009) arXiv09064775 [nucl-ex]
[25] P Solvignon et al (E01-012) Moments of the neutron g2 structure function at intermediate
42
Q2 Phys Rev C 92 015208 (2015) arXiv13044497 [nucl-ex]
[26] A Deur Experimental studies at low Q2 of the spin structure of the nucleon at Jefferson Lab
in 9th International Workshop on Chiral Dynamics (CD18) Durham NC USA September
17-21 2018 (2019) arXiv190305661 [nucl-ex]
[27] V Sulkosky et al (Jefferson Lab E97-110) Measurement of the 3He spin-structure functions
and of neutron (3He) spin-dependent sum rules at 0035 le Q2 le 024 GeV2 Phys Lett B
805 135428 (2020) arXiv190805709 [nucl-ex]
[28] K Adhikari et al (CLAS) Measurement of the Q2 Dependence of the Deuteron Spin Structure
Function g1 and its Moments at Low Q2 with CLAS Phys Rev Lett 120 062501 (2018)
arXiv171101974 [nucl-ex]
[29] V Lensky F Hagelstein A Hiller Blin and V Pascalutsa Comment on rdquoMeasurement of
the Q2 Dependence of the Deuteron Spin Structure Function g1 and its Moments at Low Q2
with CLASrdquo (2018) arXiv180603219 [nucl-th]
[30] J M Alarcn F Hagelstein V Lensky and V Pascalutsa Forward doubly-virtual Compton
scattering off the nucleon in chiral perturbation theory at NLO the subtraction function and
moments of unpolarized structure functions (2020) arXiv200509518 [hep-ph]
[31] V Bernard E Epelbaum H Krebs and U-G Meiszligner New insights into the spin structure
of the nucleon Phys Rev D 87 054032 (2013) arXiv12092523 [hep-ph]
[32] V Lensky J M Alarcon and V Pascalutsa Moments of nucleon structure functions at
next-to-leading order in baryon chiral perturbation theory Phys Rev C 90 055202 (2014)
arXiv14072574 [hep-ph]
[33] V Pascalutsa and M Vanderhaeghen Chiral effective-field theory in the ∆(1232) region
I Pion electroproduction on the nucleon Phys Rev D 73 034003 (2006) arXivhep-
ph0512244
[34] V Pascalutsa and M Vanderhaeghen Electromagnetic nucleon-to-Delta transition in chiral
effective field theory Phys Rev Lett 95 232001 (2005) arXivhep-ph0508060
[35] T R Hemmert B R Holstein and J Kambor Systematic 1M expansion for spin 32
particles in baryon chiral perturbation theory Phys Lett B 395 89 (1997) arXivhep-
ph9606456
[36] V Pascalutsa and D R Phillips Effective theory of the ∆(1232) in Compton scattering off
the nucleon Phys Rev C 67 055202 (2003) arXivnucl-th0212024
[37] V Pascalutsa M Vanderhaeghen and S N Yang Electromagnetic excitation of the ∆(1232)-
resonance Phys Rept 437 125 (2007) arXivhep-ph0609004
[38] K A Olive et al (Particle Data Group) Review of Particle Physics Chin Phys C 38 090001
43
(2014)
[39] F E Low Scattering of light of very low frequency by systems of spin 12 Phys Rev 96
1428 (1954)
[40] M Gell-Mann and M L Goldberger Scattering of low-energy photons by particles of spin
12 Phys Rev 96 1433 (1954)
[41] A M Harun ar-Rashid A simple derivation of Schwingerrsquos sum rule for spin dependent
structure functions Nuovo Cim A 33 447 (1976)
[42] F Hagelstein and V Pascalutsa Dissecting the hadronic contributions to (g minus 2)micro by
Schwingerrsquos sum rule Phys Rev Lett 120 072002 (2018) arXiv171004571 [hep-ph]
[43] V Lensky V Pascalutsa M Vanderhaeghen and C Kao Spin-dependent sum rules con-
necting real and virtual Compton scattering verified Phys Rev D 95 074001 (2017)
arXiv170101947 [hep-ph]
[44] H Burkhardt and W N Cottingham Sum rules for forward virtual Compton scattering
Annals Phys 56 453 (1970)
[45] V Lensky and V Pascalutsa Predictive powers of chiral perturbation theory in Compton
scattering off protons Eur Phys J C 65 195 (2010) arXiv09070451 [hep-ph]
[46] H Grieszlighammer J McGovern D Phillips and G Feldman Using effective field theory to
analyse low-energy Compton scattering data from protons and light nuclei Prog Part Nucl
Phys 67 841 (2012) arXiv12036834 [nucl-th]
[47] H W Grieszlighammer J A McGovern and D R Phillips Nucleon polarisabilities at and
beyond physical pion masses Eur Phys J A 52 139 (2016) arXiv151101952 [nucl-th]
[48] E Epelbaum H Krebs and U-G Meiszligner Improved chiral nucleon-nucleon potential up to
next-to-next-to-next-to-leading order Eur Phys J A 51 53 (2015) arXiv14120142 [nucl-
th]
[49] E Epelbaum H Krebs and U-G Meiszligner Precision nucleon-nucleon potential at fifth order
in the chiral expansion Phys Rev Lett 115 122301 (2015) arXiv14124623 [nucl-th]
[50] C W Kao T Spitzenberg and M Vanderhaeghen Burkhardt-Cottingham sum rule and
forward spin polarizabilities in heavy baryon chiral perturbation theory Phys Rev D 67
016001 (2003) arXivhep-ph0209241
[51] C-W Kao D Drechsel S Kamalov and M Vanderhaeghen Higher moments of nucleon
spin structure functions in heavy baryon chiral perturbation theory and in a resonance model
Phys Rev D 69 056004 (2004) arXivhep-ph0312102
[52] T Becher and H Leutwyler Baryon chiral perturbation theory in manifestly Lorentz invariant
form Eur Phys J C 9 643 (1999) arXivhep-ph9901384
44
[53] J Gasser M E Sainio and A Svarc Nucleons with chiral loops Nucl Phys B 307 779
(1988)
[54] L S Geng J Martin Camalich L Alvarez-Ruso and M J Vicente Vacas Leading SU(3)-
breaking corrections to the baryon magnetic moments in chiral perturbation theory Phys
Rev Lett 101 222002 (2008) arXiv08051419 [hep-ph]
[55] D Drechsel S S Kamalov and L Tiator The GDH sum rule and related integrals Phys
Rev D 63 114010 (2001) arXivhep-ph0008306
[56] D Drechsel O Hanstein S S Kamalov and L Tiator A unitary isobar model for pion
photo- and electroproduction on the proton up to 1 GeV Nucl Phys A 645 145 (1999)
arXivnucl-th9807001
[57] L Tiator private communication (2020)
[58] V Bernard T R Hemmert and U-G Meiszligner Spin structure of the nucleon at low energies
Phys Rev D 67 076008 (2003) arXivhep-ph0212033
[59] O Gryniuk F Hagelstein and V Pascalutsa Evaluation of the forward Compton scattering
off protons II Spin-dependent amplitude and observables Phys Rev D 94 034043 (2016)
arXiv160400789 [nucl-th]
[60] R Zielinski The g2p Experiment A Measurement of the Protonrsquos Spin Structure Functions
PhD thesis New Hampshire U (2010) arXiv170808297 [nucl-ex]
[61] N Guler et al (CLAS) Precise determination of the deuteron spin structure at low to mod-
erate Q2 with CLAS and extraction of the neutron contribution Phys Rev C 92 055201
(2015) arXiv150507877 [nucl-ex]
[62] C E Carlson V Nazaryan and K Griffioen Proton structure corrections to electronic
and muonic hydrogen hyperfine splitting Phys Rev A 78 022517 (2008) arXiv08052603
[physicsatom-ph]
[63] V Bernard N Kaiser and U-G Meiszligner Chiral dynamics in nucleons and nuclei Int J
Mod Phys E 4 193 (1995) arXivhep-ph9501384
[64] T R Hemmert B R Holstein and J Kambor ∆(1232) and the polarizabilities of the nucleon
Phys Rev D 55 5598 (1997) arXivhep-ph9612374
[65] V Pascalutsa and R Timmermans Field theory of nucleon to higher-spin baryon transitions
Phys Rev C 60 042201 (1999) arXivnucl-th9905065
[66] V Pascalutsa Quantization of an interacting spin-32 field and the Delta isobar Phys Rev
D 58 096002 (1998) arXivhep-ph9802288
[67] H Krebs Double Virtual Compton Scattering and SpinStructure of the Nucleon PoS
CD2018 031 (2019)
45
[68] N Kochelev and Y Oh Axial anomaly and the δLT puzzle Phys Rev D 016012 (2012)
arXiv11034892 [hep-ph]
[69] D Drechsel S Kamalov and L Tiator Unitary isobar model ndash MAID2007 Eur Phys J A
69 (2007) available at httpsmaidkphuni-mainzde arXiv07100306 [nucl-th]
[70] P J Mohr B N Taylor and D B Newell CODATA recommended values of the fundamental
physical constants 2010 Rev Mod Phys 84 1527 (2012)
[71] J Ahrens et al (GDH A2) First measurement of the Gerasimov-Drell-Hearn integral for 1H
from 200 to 800 MeV Phys Rev Lett 87 022003 (2001) arXivhep-ex0105089 [hep-ex]
[72] K Helbing (GDH) Experimental verification of the GDH sum rule at ELSA and MAMI Nucl
Phys Proc Suppl 105 113 (2002)
[73] R Bradford A Bodek H S Budd and J Arrington A New parameterization of the nucleon
elastic form-factors NuInt05 proceedings of the 4th International Workshop on Neutrino-
Nucleus Interactions in the Few-GeV Region Okayama Japan 26-29 September 2005 Nucl
Phys Proc Suppl 159 127 (2006) arXivhep-ex0602017
[74] J D Bjorken Applications of the chiral U(6)timesU(6) algebra of current densities Phys Rev
148 1467 (1966)
[75] J D Bjorken Inelastic scattering of polarized leptons from polarized nucleons Phys Rev D
1 1376 (1970)
[76] V Bernard T R Hemmert and U-G Meiszligner Novel analysis of chiral loop effects in
the generalized Gerasimov-Drell-Hearn sum rule Phys Lett B 545 105 (2002) arXivhep-
ph0203167
[77] A Deur et al Experimental study of isovector spin sum rules Phys Rev D 78 032001
(2008) arXiv08023198 [nucl-ex]
[78] B Pasquini P Pedroni and D Drechsel Higher order forward spin polarizability Phys Lett
B 687 160 (2010) arXiv10014230 [hep-ph]
[79] R Jaffe g2ndashThe nucleonrsquos other spin-dependent structure function Comments Nucl Part
Phys 19 239 (1990)
[80] E V Shuryak and A Vainshtein Theory of power corrections to deep inelastic scattering in
quantum chromodynamics (II) Qminus4 effects polarized target Nucl Phys B 201 141 (1982)
[81] S Wandzura and F Wilczek Sum rules for spin dependent electroproduction Test of rela-
tivistic constituent quarks Phys Lett B 72 195 (1977)
[82] B W Filippone and X-D Ji The spin structure of the nucleon Adv Nucl Phys 26 1
(2001) arXivhep-ph0101224
[83] M Burkardt The g2 structure function Proceedings Workshop on Spin structure at long
46
distance Newport News USA March 12-13 2009 AIP Conf Proc 1155 26 (2009)
arXiv09054079 [hep-ph]
[84] A Deur S J Brodsky and G F de Teramond The spin structure of the nucleon Rept
Prog Phys 82 076201 (2019) arXiv180705250 [hep-ph]
[85] B R Holstein D Drechsel B Pasquini and M Vanderhaeghen Higher order polarizabilities
of the proton Phys Rev C 61 034316 (2000) arXivhep-ph9910427
[86] B R Holstein V Pascalutsa and M Vanderhaeghen Sum rules for magnetic moments and
polarizabilities in QED and chiral effective-field theory Phys Rev D 72 094014 (2005)
arXivhep-ph0507016
[87] B Pasquini D Drechsel and L Tiator Invariant amplitudes for pion electroproduction Eur
Phys J A 34 387 (2007) arXiv07122327 [hep-ph]
[88] A J Buchmann Non-spherical proton shape and hydrogen hyperfine splitting Proceedings
International Workshop on Precision Physics of Simple Atomic Systems (PSAS 2008) Wind-
sor Ontario Canada July 21-26 2008 Can J Phys 87 773 (2009) arXiv09104747
[physicsatom-ph]
47
000 005 010 015 020 025 030
-006
-004
-002
000
002
004
Q2 GeV2D
G1
proton
000 005 010 015 020 025 030
-006
-005
-004
-003
-002
-001
000
001
Q2 GeV2D
G1
neutron
FIG 6 First moment of the structure function g1(xQ2) for the proton (left) and neutron (right)
as function of Q2 The legend is the same as in Fig 4
panel one sees that ∆I1(Q2) is less sensitive to gC and the dipole form factor in gM than
∆IA(Q2)
For the proton our NLO BχPT prediction gives a very good description of the ex-
perimental data [18 60] and is in reasonable agreement with the MAID prediction [69]
For the neutron one observes good agreement with the empirical evaluations including ex-
trapolations to unmeasured energy regions starting from Q2 gt 01 GeV2 [27 61] In the
region of Q2 lt 005 GeV2 one observes an interesting tension between the recent E97-
110 experiment [27] and the data from CLAS [61] While the newest measurement finds
I1n(0035 GeV2) lt κ2n4 thus suggesting a negative slope at low Q the older measurement
found a rather large value for I1n(00496 GeV2) A similar but milder behaviour is seen in
the E97-110 [27] and E94-010 [21] data for IAn The MAID predictions do not agree with
the CODATA recommended values for the anomalous magnetic moments of the proton and
neutron [70] which in our work are imposed by using empirical parametrizations for the
elastic Pauli form factors [73] The slope of the HB result from Ref [51] is too large and
therefore only reproduces the data at very low Q2
Figure 6 shows the moment Γ1(Q2) for the proton and neutron while Fig 7 shows the
isovector combination Γ1 pminusn(Q2) The LO and NLO BχPT predictions are identical be-
cause our calculation produces the same Delta contributions for the proton and the neutron
For the isovector combination the MAID model only agrees with the data at very low
Q2 lt 010 GeV2 The same is true for the IR result [58 76] while all other chiral results
describe the data NLO BχPT (this work) BχPT+∆ [31] and HBχPT [51]
At the real-photon point I1(0) = minusκ2
4and ∆I1(0) = 0 Therefore we give only the
slope of the polarizability ∆I1(Q2) [showing also the separate contributions from πN loops
21
000 005 010 015 020 025 030-002
000
002
004
006
008
010
Q2 GeV2D
G1Hp-
nL
FIG 7 Isovector combination of Γ1(Q2) as function of Q2 The legend is the same as in Fig 6
The pink curve is the IR result from Ref [58 76] The experimental points are from Ref [77]
(brown dots) and Ref [23] (orange squares)
∆ exchange and π∆ loops] in units of GeVminus2
d∆I1p(Q2)
dQ2
∣∣∣∣Q2=0
= 039(4) asymp 034minus 053 + 058 (35a)
d∆I1n(Q2)
dQ2
∣∣∣∣Q2=0
= minus101(10) asymp minus107minus 053 + 058 (35b)
Including the empirical Pauli form factor [73] we find in units of GeVminus2
dI1p(Q2)
dQ2
∣∣∣∣Q2=0
= 580dI1n(Q2)
dQ2
∣∣∣∣Q2=0
= 553 (36)
E d2(Q2) mdash a measure of color polarizability
Another interesting moment to consider is d2(Q2) which is related to the twist-3 part of
the spin structure function g2(xQ2) [79 80]
d2(Q2) equiv 3
int 1
0
dx x2[g2(xQ2)minus gWW2 (xQ2)] (37)
where gWW2 (xQ2) is the twist-2 part of g2(xQ2) Using the Wandzura-Wilczek relation [81]
one can relate d2(Q2) to moments of the spin structure functions g1(xQ2) and g2(xQ2)
d2(Q2) =
int 1
0
dx x2 [3g2(xQ2) + 2g1(xQ2)] (38)
22
000 005 010 015 020 025 030
0000
0001
0002
0003
0004
0005
0006
0007
Q2 GeV2D
d2
proton
000 005 010 015 020 025 030
0000
0002
0004
0006
0008
0010
0012
Q2 GeV2D
d2
neutron
000 005 010 015 020 025 030
-1
0
1
2
Q2 GeV2D
Γ 010
-4
fm6
D
proton
000 005 010 015 020 025 030-2
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
6D
neutron
FIG 8 Upper panel The inelastic moment d2(Q2) for the proton (left) and neutron (right) as
function of Q2 The result of this work the NLO BχPT prediction is shown by the blue solid line
and the blue band The red line represents the LO BχPT result The purple short-dashed line is
the O(p4) HB result from Ref [50 51] The black dotted line is the MAID model prediction [69]
The experimental points for the neutron (cyan dots) are from Ref [22] Lower panel Fifth-order
generalized forward spin polarizability γ0(Q2) for the proton (left) and neutron (right) as function
of Q2 The experimental points for the proton are from Ref [59] (purple square) and Ref [78]
(orange dot)
This relation however only holds for asymptotically large Q2 It is also in the high-Q2
region where d2(Q2) is a measure of color polarizability [82 83] through its relation to the
gluon field strength tensor [80] We refer to Ref [84] for a recent review on the spin structure
of the nucleon including a discussion of sum rules for deep inelastic scattering and color
polarizabilities
What we consider in the following is the inelastic part of d2(Q2) defined as the moment
of g1(xQ2) and g2(xQ2) spin structure functions cf Eq (38)
d2(Q2) =
int x0
0
dx x2 [3g2(xQ2) + 2g1(xQ2)] (39)
23
This moment provides another testing ground for our BχPT predictions through comparison
with experiments on the neutron [22] Going towards the low-Q2 region the interpretation
of d2(Q2) in terms of color polarizabilities will fade out The above definition however
implies it is related to other VVCS polarizabilities
d2(Q2) =Q4
8M4N
[M2
NQ2
αδLT (Q2) + I1(Q2)minus IA(Q2)
] (40)
Note that d2(Q2) and its first two derivatives with respect to Q2 vanish at Q2 = 0 The
considerations in Eqs (28) and (29) have no effect on d2(Q2) since the Born contribution
from IA(Q2) and I1(Q2) cancel out Therefore d2(Q2) is a pure polarizability
In Fig 8 upper panel we show our NLO BχPT prediction and other results for d2(Q2)
While MAID [69] and BχPT describe the experimental data for the neutron [22] very well
the HB limit [50 51] is showing a fast growth with Q2 This illustrates the importance
of keeping the relativistic result Note also that even though the πN -loop contribution
is dominant both gC and the form factor in gM are essential to obtain a curvature that
reproduces the data cf Fig 9 upper panel For the proton there are to our knowledge
no experimental results to compare with However the agreement between the NLO BχPT
prediction and the MAID prediction at low energies is reasonable
F γ0(Q2) mdash fifth-order generalized forward spin polarizability
It is interesting to compare the generalized fifth-order forward spin polarizability sum
rule
γ0(Q2) =1
2π2
int infinν0
dν
radic1 +
Q2
ν2
σTT (νQ2)
ν5(41)
=64αM4
N
Q10
int x0
0
dx x4
[g1(xQ2)minus 4M2
Nx2
Q2g2(xQ2)
]
to the sum rule integrals for IA(Q2) and γ0(Q2) since they differ merely by their energy
weighting of σTT (νQ2) and a constant prefactor cf Eqs (19) (26) and (41) From IA(Q2)
to γ0(Q2) to γ0(Q2) the energy suppression is increasing by a factor of νminus2 respectively
Therefore the description of γ0(Q2) should be easiest in a low-energy effective-field theory
such as χPT whereas γ0(Q2) and IA(Q2) receive larger contributions from higher energies
In Fig 8 lower panel we show our LO and NLO BχPT predictions for γ0(Q2) One
can see that the πN -loop contribution is positive (in accordance to what we see for the cross
section σTT see Fig 10) The Delta shifts it substantially especially in the low Q2 region
24
000 005 010 015 020 025 030
-0004
-0002
0000
0002
0004
0006
0008
0010
Q2GeV
2D
d2
proton
000 005 010 015 020 025 030
-0005
0000
0005
0010
0015
Q2GeV
2D
d2
neutron
000 005 010 015 020 025 030
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
6D
proton
000 005 010 015 020 025 030
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
6D
neutron
FIG 9 Contributions of the different orders to the chiral predictions of d2(Q2) upper panel and
γ0(Q2) lower panel for the proton (left) and neutron (right) Red solid line πN -loop contribu-
tion green dot-dashed line ∆-exchange contribution orange dotted line π∆-loop contribution
blue long-dashed line total result purple dot-dot-dashed line total result without gC contribution
black short-dashed line total result without gM dipole
bringing it into a better agreement with data In general the BχPT curves start above the
empirical data points at the real-photon point and then decrease asymptotically to zero
above Q2 gt 01 GeV2 On the other hand the MAID prediction reproduces the empirical
data at the real-photon point then decreases to negative values until about Q2 gt 006 GeV2
from where it also starts to asymptotically approach zero Consequently our NLO BχPT
prediction of γ0(Q2) is consistently above the MAID prediction This is very different to
what we saw for IA(Q2) in Fig 4 upper panel where the MAID prediction at the real-
photon point is above the experimental value While the agreement of our predictions with
the empirical data is in general quite good for all moments of σTT (νQ2) one should point
out that both for γ0n(Q2) and γ0p(Q2) we overestimate the data at low Q2 For IA(Q2) such
observation cannot be made because ∆IA(0) = 0 and thus IA(0) is given by the empirical
Pauli form factor only From IA(Q2) γ0(Q2) and γ0(Q2) the latter has the smallest however
25
non-negligible dependence on gC and the dipole in gM cf Fig 9 lower panel
The πN -loop ∆-exchange and π∆-loop contributions to the NLO BχPT prediction of
the static fifth-order forward spin polarizability amount to in units of 10minus4 fm6
γ0p = 112(30) asymp 208minus 096minus 001 (42a)
γ0n = 195(30) asymp 292minus 096minus 001 (42b)
while the slope is composed as follows in units of 10minus4 fm8
dγ0p(Q2)
dQ2
∣∣∣∣Q2=0
= minus084(10) asymp minus100 + 016 + 000 (43a)
dγ0n(Q2)
dQ2
∣∣∣∣Q2=0
= minus142(15) asymp minus158 + 016 + 000 (43b)
Note that the HB prediction of the static polarizability γ0p (423 at O(p3) and 365 at O(ε3)
[78 85]) is almost one order of magnitude larger than the empirical value and therefore not
shown in Fig 8
G Summary
Our results are summarized in Table II where we give the contributions of the different
orders to the chiral predictions of the polarizabilities and their slopes at the real-photon
point A quantitative comparison of our predictions for the static spin polarizabilities to
the work of Bernard et al [31] and different empirical evaluations is shown in Table III We
can see that the inclusion of the Delta turns out to be very important for all moments of
the helicity-difference cross section To describe the Q2 behavior of the polarizabilities the
magnetic coupling of the N rarr ∆ transition should be modified by a dipole form factor as
has been observed previously in the description of electroproduction data [33] This dipole
form factor effectively takes account of vector-meson exchanges The Coulomb-quadrupole
N rarr ∆ transition despite its subleading order is important in the description of some
moments of spin structure functions This is contrary to what we saw for the moments of
unpolarized structure functions [30] where the Coulomb coupling had a negligible effect
The π∆ loops are mainly relevant for the generalized GDH integrals
26
TABLE II The NLO BχPT predictions for the forward VVCS polarizabilities and their slopes at
Q2 = 0 The contributions of the πN loops the ∆ exchange and the π∆ loops are shown together
with the combined total result Note that IA(0) = I1(0) = d2(0) = 0 and (d2)prime = 0
πN loops ∆ exchange π∆ loops Total
γ0 p 201 minus284 minus010minus093(92)
(10minus4 fm4) n 298 003(92)
δLT p 150 minus016 minus002132(15)
(10minus4 fm4) n 235 218(23)
γ0 p 208 minus096 minus001112(30)
(10minus4 fm6) n 292 195(30)
(γ0)prime p minus033011 001
minus022(4)
(10minus4 fm6) n minus073 minus061(7)
(δLT )prime p minus080 minus004 minus001minus085(8)
(10minus4 fm6) n minus119 minus124(12)
(γ0)prime p minus100016 000
minus084(10)
(10minus4 fm8) n minus158 minus142(15)
(∆IA)prime p 238 minus1121 025minus858(343)
(GeVminus2) n 141 minus955(343)
(∆I1)prime p 034 minus053 058039(4)
(GeVminus2) n minus107 minus101(10)
TABLE III Our NLO BχPT predictions for the spin polarizabilities at Q2 = 0 compared with
the BχPT+∆ predictions from Bernard et al [31] and the available empirical information Where
the reference is not given the empirical number is provided by the MAID analysis [55 69] with
unspecified uncertainty
Proton Neutron
This work BχPT+∆ Empirical This work BχPT+∆ Empirical
γ0 minus093(92) minus174(40) minus100(8)(12) [19] 003(92) minus077(40) minus0005
(10minus4 fm4) minus090(8)(11) [78] [MAID]
minus0929(105) [59]
δLT 132(15) 240(1) 134 218(23) 238(3) 203
(10minus4 fm4) [MAID] [MAID]
γ0 112(30) 060(7)(7) [78] 195(30) 123
(10minus4 fm6) 0484(82) [59] [MAID]
IV CONCLUSIONS
We have presented a complete NLO calculation of the polarized non-Born VVCS ampli-
tudes in covariant BχPT with pion nucleon and ∆(1232) fields The dispersion relations
between the VVCS amplitudes and the tree-level photoabsorption cross sections served as a
27
cross-check of these calculations
The obtained moments of the proton and neutron spin structure functions related to gen-
eralized polarizabilities and GDH-type integrals agree well with the available experimental
data The description of their Q2 evolution is improved compared to the previous χPT pre-
dictions In particular the NLO BχPT predictions obtained here give a better description
of the empirical data (eg from the Jefferson Laboratory ldquoSpin Physics Programrdquo) than
the HB [50 51] and IR [58] calculations
The demonstrated predictive power of the χPT framework amplitudes makes it well
suited for extending the χPT evaluation of the TPE effect in the hyperfine structure of
(muonic-)hydrogen [15ndash17] to next-to-leading order
ACKNOWLEDGEMENTS
We thank Lothar Tiator and Marc Vanderhaeghen for helpful discussions This work
is supported by the Deutsche Forschungsgemeinschaft (DFG) through the Collaborative
Research Center [The Low-Energy Frontier of the Standard Model (SFB 1044)] JMA ac-
knowledges support from the Community of Madrid through the ldquoPrograma de atraccion de
talento investigador 2017 (Modalidad 1)rdquo and the Spanish MECD grants FPA2016-77313-P
FH gratefully acknowledges financial support from the Swiss National Science Foundation
Appendix A Tensor decompositions of the VVCS amplitudes
In this appendix we review the decomposition of the forward VVCS process into tensor struc-
tures and scalar amplitudes In particular we consider the connection between the covariant and
the semi-relativistic decomposition in the lab frame that is defined in terms of the conventional
transverse longitudinal transverse-transverse and transverse-longitudinal amplitudes
As explained in Sec II A the process of forward VVCS off the nucleon can be described in
terms of four explicitly covariant amplitudes S1 2 and T1 2 [3]
T (νQ2) =
(minusgmicroν +
qmicroqν
q2
)T1(νQ2) +
1
M2N
(pmicro minus p middot q
q2qmicro)(
pν minus p middot qq2
qν)T2(νQ2) (A1)
minus 1
MNγmicroναqα S1(νQ2)minus 1
M2N
(γmicroνq2 + qmicroγναqα minus qνγmicroαqα
)S2(νQ2)
εprimelowastmicro εν
where εmicro (εprimelowastmicro ) are the incoming (outgoing) photon polarization vectors ν is the photon lab-frame
energy and Q2 is the photon virtuality Alternatively the decomposition in the laboratory frame
28
(which in the forward case coincides with the Breit frame) is parametrized in terms of the nucleon
Pauli matrices ~σ and the four scalar functions fL fT gTT and gLT
T (νQ2) = ε0 εprimelowast0 fL(νQ2) + (~ε primelowast middot ~ε ) fT (νQ2) + i~σ middot (~ε primelowast times ~ε ) gTT (νQ2) (A2)
minusi~σ middot [(ε0~εprimelowast minus ~ε ε primelowast0 )times q] gLT (νQ2)
Here ~q and q = ~q|~q | are the photon three-momentum in the lab system and its unit vector The
modified polarization vector components are given by
ε0 =
[ε0 minus
ν
|~q | (~ε middot q )
] |~q |Q ~ε = ~εminus q (~ε middot q ) (A3)
where ε = (ε0~ε ) is the usual incoming photon polarization vector and εprimelowast the outgoing polarization
vector The LEX of the lab frame amplitudes [Eq (10)] can serve in particular as the definition
of the generalized polarizabilities The lab frame amplitudes are also conveniently used for the
definition of the response functions see the example of the scalar amplitude gLT (νQ2) and the
corresponding response function σLT (νQ2) below in App B
Appendix B Photoabsorption cross sections
In the forward kinematics the spin-dependent VVCS amplitudes and the spin polarizabilities
can be described in terms of the polarized structure functions g1(xQ2) and g2(xQ2) or equiv-
alently the helicity-difference cross section σTT (νQ2) and the longitudinal-transverse response
function σLT (νQ2) with the help of dispersion relations (5) and the optical theorem (3) In this
way the photoabsorption cross sections measured in electroproduction processes form the basis
for most empirical evaluations shown throughout Sec III In the following we present the BχPT
predictions for the tree-level cross sections of πN - π∆- and ∆-production through photoabsorption
on the nucleon cf Figs 8 9 and 10 in Ref [30] In Secs B 1 and B 2 we will discuss the leading
πN -production channel and the ∆-production channel respectively We used these cross sections
to verify the polarizability predictions obtained otherwise from the calculated non-Born VVCS
amplitudes Due to the bad high-energy behavior of the π∆-production cross sections in BχPT
cf Fig 10 the dispersion relations in Eq (5) require further subtractions for a reconstruction of
the π∆-loop contribution to the spin-dependent VVCS amplitudes Therefore not all polarizabil-
ities could be verified but only those appearing as higher-order terms in the LEX of the VVCS
amplitudes such as γ0 [16]
29
proton neutron
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Neutron
02 04 06 08 100
50
100
150
200
250
[GeV]
T [b]
02 04 06 08 10
-50
0
50
100
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Neutron
02 04 06 08 100
50
100
150
200
250
[GeV]
T [b]
02 04 06 08 10
-50
0
50
100
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02040608100
50
100
150
200
250
300
[GeV]
T[b]
0204060810
-50
0
50
[GeV]
TT[b]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
0204060810
00
02
04
06
[GeV]
LTQ
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
0204060810
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
FIG 10 Photoabsorption cross sections for πN (red) and π∆ production (orange) with Q2 = 0
(solid) and Q2 = 01 GeV2 (dashed for πN and dotted for π∆ channel)
1 πN-production channel
In order to extract the response function σLT (νQ2) we have developed a method similar to
the one used to calculate σTT (νQ2) see for example Ref [86] For σLT (νQ2) however the
calculation is more complicated because one has to take into account that the associated Compton
process involves a spin-flip of the nucleon as illustrated in Fig 11 When calculating the cross
section the product of the incoming nucleon spinors has to reflect this flip
The forward VVCS amplitude related to σLT (νQ2) mdash and δLT (Q2) mdash is gLT (νQ2) It can
be extracted from Eq (A2) if one takes the modified polarization vector components in Eq (A3)
with ε = εL and εprimelowast = εlowastplusmn as input where εL = 1Q(|~q | 0 0 q0) and εplusmn = ∓ 1radic
2(0 1plusmni 0) are the
standard longitudinal and transverse polarization vectors respectively For εL and εlowastplusmn only the
choice of helicities hprime = plusmn12 and h = ∓12 gives a non-zero contribution and one obtains
χdaggerhprime T (νQ2)χh = χdaggerhprimeminusi~σ middot [(ε0~εprimelowast minus ~ε ε primelowast0 )times q] gLT (νQ2)χh =
radic2 gLT (νQ2) (B1)
where χh and χdaggerhprime are two-component Pauli spinors with opposite helicities or here spins
Let us now consider the related photoabsorption process and in particular the tree-level γlowastN rarr
30
s = 12
sprime = minus12
λ = 0 λprime = 1
rprime rprime
s=12sprime=minus1
2
λ=0λprime=1
rprimerprime
s=12
sprime=minus12
λ=0λprime=1
rprimerprime hacute=frac12h=-frac12
λacute=1λ=0
FIG 11 Relation between the forward Compton process and the photoabsorption process given
by the optical theorem In particular we show the longitudinal-transverse contribution The
double-line arrows represent the spin of the external particles while the dot represents the scalar
(longitudinal) polarization of the incoming photon Inside the blob the intermediate states are
represented eg nucleons with spins rprime (which are averaged in the calculation of the cross section)
and pions
πN channel see diagrams in Fig 8 of Ref [30] We define the πN -production amplitude as
T = uhB (PB)sumi
Ai(s t)Γi uhA(PA) (B2)
with the Dirac structures
Γ1 = γ5 (B3a)
Γ2 =1
2
[qA ε
]γ5 (B3b)
where uhA(PA) and udaggerhB (PB) are the Dirac spinors and PA and PB are the four-momenta of
the incoming and outgoing nucleons respectively When calculating the photoabsorption cross
section related to the VVCS amplitude in Eq (B1) the nucleon spin flip should be implemented
by uhprime(PA) in T dagger and uh(PA) in T together with the appropriate transverse and longitudinal
photon polarization vectors εlowastplusmn and εL
However if one wants to use the properties of the Dirac matrices it is more useful to construct
an operator to produce this spin flip in the external nucleons of Fig 11 This is accomplished by
introducing the projector ΓLT equiv 12radic
2(γ1 + iγ2)γ5 which also takes into account the extra factor
radic2 in Eq (B1) We checked that with this projector one correctly extracts δLT by comparing the
HB limit of our result to the HB result of Ref [50] where the authors calculate this polarizability
from the Compton amplitude directly With all those ingredients the longitudinal-transverse cross
section is calculated in the following way
σLT (νQ2) =1
64π2 s
|~pf |cm
|~pi|cm
int 1
minus1dcos θ
sumij
AiAdaggerjXij (B4)
31
with
Xij = Tr[(PB +MN )Γi(PA +MN )ΓLTγ0Γdaggerjγ
0] (B5)
where θ is the scattering angle in the center-of-mass (cm) frame and |~pi|cm (|~pf |cm) is the three-
momentum of an incoming (outgoing) particle in the cm frame An explicit calculation of the
matrix Xij leads to
X = MNQ
0 2(PB minus PA) middot εLminusradic
2|~qf |cm sin θ (sminus u)
(B6)
where |~qi|cm (|~qf |cm) is the relative three-momentum of the incoming (outgoing) particles in the
cm frame Here s t and u are the usual Mandelstam variables For the different γlowastN rarr πN
channels we obtain the following amplitudes Ai where we introduce qA as the four-momentum of
the incoming photon and qB as the four-momentum of the outgoing pion
bull γlowastprarr π0p
A1 =e gAMN
fπ
[2PA middot ε+ qA middot ε
sminusM2N
+2PB middot εminus qA middot ε
uminusM2N
] (B7a)
A2 =e gAMN
fπ
[1
sminusM2N
+1
uminusM2N
] (B7b)
bull γlowastprarr π+n
A1 =
radic2 e gAMN
fπ
[2PA middot ε+ qA middot ε
sminusM2N
+2(PA minus PB) middot ε+ qA middot ε
tminusm2π
] (B8a)
A2 =
radic2 e gAMN
fπ(sminusM2N )
(B8b)
bull γlowastnrarr π0n
A1 = 0 (B9a)
A2 = 0 (B9b)
bull γlowastnrarr πminusp
A1 =
radic2 e gAMN
fπ
[2PB middot εminus qA middot ε
uminusM2N
minus 2(PA minus PB) middot ε+ qA middot εtminusm2
π
] (B10a)
A2 =
radic2 e gAMN
fπ(uminusM2N )
(B10b)
The analytical expressions shown above were checked with the amplitudes given in Ref [87] An-
alytical expressions for the tree-level γlowastN rarr πN channel of the σLT (νQ2) and σTT (νQ2) cross
32
sections are given below (proton channels π+n and π0p neutron channel πminusp) We checked that
they reproduce the known results in the real-photon limit [45 86] To shorten the final expressions
for the cross sections which are considerably longer for finite Q2 than in the real-photon limit we
define the following dimensionless kinematic variables
αγ = (ENi )cmradics =
s+M2N +Q2
2s (B11)
απ = (ENf )cmradics =
s+M2N minusm2
π
2s (B12)
βγ = Eγcmradics =
sminusM2N minusQ2
2s (B13)
βπ = Eπcmradics =
sminusM2N +m2
π
2s (B14)
λγ = |~qi|cmradics =
radic(sminusM2
N minusQ2)2 + 4sQ2
2s (B15)
λπ = |~qf |cmradics =
radic(sminusM2
N +m2π)2 minus 4sm2
π
2s (B16)
Here (ENi )cm and (ENf )cm are the energies of the incoming and outgoing nucleon Eγcm is the energy
of the incoming photon Eπcm is the energy of the outgoing pion all in the cm frame
σ(π+n)TT = minus e2g2
AM2N
64πf2πs
3(sminusM2N )2λ4
γ
4sλπλγ
[(M2
N minus s)(M2N minusQ2 minus s)(Q2 + 2sβγβπ)
+ 2s(m2πQ
2 minus (M2N minus s)
(M2N + s(minus1 + 2βγβπ)
))λ2γ
]+ (M2
N minus s)(M2N minusQ2 minus s)
times (Q2 + 2sβγβπ minus 2sλπλγ)(Q2 + 2s(βγβπ + λπλγ)
)times log
(Q2 + 2sβγβπ minus 2sλγλπQ2 + 2sβγβπ + 2sλγλπ
) (B17)
33
σ(π0p)TT =
e2g2AM
2Nλπ
64πf2πs(sminusM2
N )2λγ
1
s2λπλ3γ
[(M2
N minus s)(M2N minusQ2 minus s)
(3M2
N + s(2βγβπ minus 3))(M2N
+ s(2βγβπ minus 1))
+ 2s
((2m2
πQ2 + (M2
N minus s)2)(M2
N minus s)minus 2(minusm2
πQ2 + (sminusM2
N )2)
times sβγβπ + 4s2(sminusM2N )β2
γβ2π
)λ2γ
]arctan
(2sλπλγ
M2N + s(2βγβπ minus 1)
)+ 2
[minus 2m2
πQ2
+ (M2N minus s)
(2(sminusM2
N ) + 4sβγβπ +(sminusM2
N +Q2)(3M2N minus 3s+ 2sβγβπ)
sλ2γ
+m2πQ
2(sminusM2N )
(M2N minus s+ 2sβγβπ)(M2
N + s(minus1 + 2βγβπ minus 2λπλγ))
+m2πQ
2(sminusM2N )
(M2N minus s+ 2sβγβπ)(M2
N + s(minus1 + 2βγβπ + 2λπλγ))
)
+1
λπλγ
(βγβπ
(minusm2
πQ2 + (M2
N minus s)(M2N minus s+ 2sβγβπ)
)minus (M2
N minus s)(M2N minusQ2 minus s)λ2
π
)log
(M2N + s(minus1 + 2βγβπ + 2λπλγ)
M2N + s(minus1 + 2βγβπ minus 2λπλγ)
)] (B18)
σ(πminusp)TT =
e2g2AM
2N
64πf2πs
3λ4γ(M2
N + s(2βπβγ minus 1))2 minus 4λ2πs
2λ2γ)
4sλπλγ
((minusM2
N +Q2 + s)(M2N
+ s(2βπβγ minus 1))
2 minus 2sλ2γ
(2λ2
πs(minusM2N +Q2 + s) +m2
πQ2))
+((M2N + s(2βπβγ minus 1)
)2
minus 4λ2πs
2λ2γ
)((M2N + s(2βπβγ minus 1)
)(M2N minus 2Q2 minus s(2βπβγ + 1)
)+ 4λ2
πs2λ2
γ
)times log
(minusM2
N minus 2βπsβγ minus 2λπsλγ + s)minus(M2N + s(2βπβγ minus 2λπλγ minus 1)
)(M2N + s(minus1
+ 2βπβγ + 2λπλγ))[((
M2N + s(2βπβγ minus 1)
)(M2N minus 2Q2 minus s(2βπβγ + 1)
)+ 4λ2
πs2λ2
γ
)times log
(minusM2
N minus 2βπsβγ + 2λπsλγ + s)
+((Q2 + 2βπsβγ)2 minus 4λ2
πs2λ2
γ
)times log
(Q2 + 2βπsβγ minus 2λπsλγQ2 + 2sβπβγ + 2λπsλγ
)]+ 4sλ2
γ
((M2
N + s(2βπβγ minus 1))2 minus 4λ2πs
2λ2γ
)times (M2
N minus s) arctanh
(2λπsλγ
M2N + 2βπsβγ minus s
) (B19)
34
σ(π+n)LT =
e2g2AM
3Nλπ
32πf2πQs
3(sminusM2N )2λ4
γ
2sλγ
[(M2
N minus s)(Q2 + 2sβ2γ)(Q2 + 2sβγβπ)
minus 4s
((M2
N minus s)(Q2 minus 2s(απ minus 1)βγ
)+Q2sβγβπ
)λ2γ + 8s3(minus1 + απ)λ4
γ
]minus M2
N minus sλπ
[(Q2 + 2sβ2
γ)(Q2 + 2sβγβπ)2 + 4s2(2(απ minus 1)βγ(Q2 + 2sβγβπ)
minusQ2λ2π
)λ2γ + 8s3(απ minus 1)2λ4
γ
]arctan
(2sλπλγ
Q2 + 2sβγβπ
) (B20)
σ(π0p)LT =
e2g2AM
3Nλπ
16πf2πQs(sminusM2
N )2λγ
1
minus2s(M2N + s(minus1 + 2βγβπ))2λ2
γ + 8s3λ2πλ
4γ
[minus 3M8
N (Q2
+ 2sβ2γ) + 2M4
Ns2
(minus (Q2 + 2sβ2
γ)(minus3 + 2βγβπ) +(Q2(minus3 + 2βγβπ + 6λ2
π) + 2sβγ
times (minus3 + 12απ + 2βγβπ minus 12απβγβπ + 4βγλ2π))λ2γ minus 4s(α2
π minus 1)λ4π
)+ 2M2
Ns3
[minus (Q2
+ 2sβ2γ)(minus1 + 2βγβπ)
(6 + βγβπ(minus9 + 2βγβπ)
)+
(Q2(3minus 12λ2
π + 4βγβπ(minus1 + 2βγβπ
+ λ2π))minus 2sβγ
(minus 3 + 4απ(3 + 2απ(3 + 2βγβπ(minus3 + βγβπ)) + 4βγ(βπ + (2minus βγβπ)
times λ2π)))λ2γ + 8s
((απ minus 1)(1 + απ minus 2βγβπ) + 2απβγλ
2π
)λ4γ
)+ s4
((Q2 + 2sβ2
γ)
times (1minus 2βγβπ)2(2βγβπ minus 3) + 2(Q2(minus1 + 6λ2
π + 2βγβπ((1minus 2βγβπ)2 minus 2λ2π)) + 2sβγ
times (minus1 + 2βγβπ + 4(minus1 + βγβπ)(minusαπ + 2απβγβπ minus βγλ2π)))λ2γ minus 8
(s(απ minus 1)(απ
+ (1minus 2βγβπ)2) + 2βγ(2sαπ +Q2βγ)λ2π
)λ4γ + 32s(απ minus 1)λ2
πλ6γ
)+ 2M6
Ns
(Q2(6
minus 7βγβπ + λ2γ) + 2sβγ(βγ(6minus 7βγβπ) + (1minus 4απ)λ2
γ)
)]+sminusM2
N
4s2λπλ3γ
[(Q2 + 2sβ2
γ)
times(3M2
N + s(2βγβπ minus 3))(M2N + s(2βγβπ minus 1)
)+ 2s
(minusM2
N (Q2 + 2sβγ minus 8sαπβγ)
+ s(2sβγ(1minus 4απ + 4απβγβπ) +Q2(1minus 2λ2π)))λ2γ + 8s3(απ minus 1)λ4
γ
]
times arctan
(2sλπλγ
M2N + s(2βγβπ minus 1)
) (B21)
35
σ(πminusp)LT =
e2g2AM
3N
64πf2πQs
3λ4γ(M2
N minusQ2 minus s)(M2N + s(2βπβγ + 2λπλγ minus 1))
times
1
M2N + s(2βπβγ minus 2λπλγ minus 1)
[4λπsλγ(M2
N minusQ2 minus s)(
2sλ2γ
(s((1minus 2λ2
π)Q2
+ 2(2απ minus 1)sβγ(2βπβγ minus 1))minusM2N (Q2 + 2(1minus 2απ)sβγ)
)+ (Q2 + 2sβ2
γ)(M2N + s(2βπβγ minus 1)
)2 + 8(απ minus 1)απs
3λ4γ
)]minus(M2N + s(2βπβγ + 2λπλγ minus 1)
)times[
log
(minusM2N minus 2βπsβγ minus 2λπsλγ + s
minusM2N minus 2βπsβγ + 2λπsλγ + s
)(minus (Q2 + 2sβ2
γ)(M2N + s(2βπβγ minus 1)
)(minusM2
N + 2Q2 + 2βπsβγ + s)minus 2sλ2γ
(minusM2
N (Q2 + 2sβγ) +Q4 +Q2s(4απβγ
minus 2βγ minus 2λ2π + 1) + 2s2βγ(4(απ minus 1)βπβγ + 1)
)minus 8(απ minus 1)2s3λ4
γ
)+ 2
(4s2λ2
γ
(2(απ minus 1)βγ(Q2 + 2βπsβγ)minus λ2
πQ2)
+ (Q2 + 2βπsβγ)2(Q2 + 2sβ2γ)
+ 8(απ minus 1)2s3λ4γ
)arctanh
(2λπsλγ
Q2 + 2βπsβγ
)] (B22)
2 ∆-production channel
The tree-level ∆-exchange diagram in Fig 2 of Ref [30] contributes to the non-Born part of
the VVCS amplitudes The contribution of the ∆ exchange to the VVCS amplitudes can be split
into [17]
S∆-exch1 (νQ2) = S∆-pole
1 (νQ2) + S∆-exch1 (νQ2) (B23a)
νS∆-exch2 (νQ2) = νS∆-pole
2 (νQ2) + νS2∆-exch
(νQ2) (B23b)
and similarly for the unpolarized VVCS amplitudes discussed in Ref [30] Here we introduced the
∆-pole contributions S∆-polei and the ∆-non-pole contributions S∆-exch
i The former amplitudes
feature a pole at the ∆(1232)-production threshold and thus are proportional to
1
[sminusM2∆][uminusM2
∆]=
1
4M2N
1
ν2∆ minus ν2
(B24)
36
They can be reconstructed from the dispersion relations in Eq (5) with the tree-level ∆-production
cross sections as input cf Fig 10 in Ref [30]
σTT (νQ2) =π2α
M2NM
2+|~q |
minus g2
MMN (M+ + ν)|~q |2 +g2E(∆minus ν)(Q2 minusMNν)2
MN(B25a)
+g2CQ
4s(∆minus ν)
MNM2∆
minus 4gMgE(Q2 minusMNν)|~q |2 minus 4gMgCQ2|~q |2
+2gEgCQ
2[minusMNM∆ |~q |2 + s(Q2 +∆ν)
]MNM∆
δ(ν minus ν∆)
σLT (νQ2) =Qπ2α
M2NM
2+|~q |
g2E(MNν minusQ2) [M∆(MN + ν)minus s]
MN(B25b)
+g2CQ
2[MNM∆|~q |2 minus s(Q2 +∆ν)
]MNM2
∆
+ gMgEM∆|~q |2 minusgMgC(Q2 minusMNν)|~q |2
M∆
+gEgC(ν minus∆)(M2
N |~q |2 minus 2Q2s)
MNM∆
δ(ν minus ν∆)
with ∆ = M∆ minusMN M+ = M∆ + MN and the Mandelstam variable s = M2N + 2MNν minus Q2
Analytical expressions for the spin structure functions g1(xQ2) and g2(xQ2) can be constructed
from Eq (3) with the flux factor K(νQ2) = |~q | =radicν2 +Q2
In the ∆-non-pole contributions to S1(νQ2) and νS2(νQ2) the pole in ν at the ∆(1232)-
production threshold has canceled out
S∆-exch1 (νQ2) =
πα
MNM2+
[g2MQ
2+ + g2
E
(∆2 minus 3Q2
)+
4g2CQ
4
M2∆
minus 8gMgEM∆ωminus (B26a)
minus2gMgCQ2(MN minus 4M∆)
M∆+
2gEgCQ2(3MN minus 2M∆)
M∆
]
νS2∆-exch
(νQ2) =2πα
MNM2+
[g2EM∆∆ωminus +
g2M MNQ
2+
2+g2C Q
2(Q2 minus∆2)
2M∆(B26b)
+gEgM M∆(M∆ω+ minus 4MNωminus)minus gEgC ∆(2Q2 +MNω+)
+gMgC Q2(4MN minus ω+)
]+S∆-exch
2 (νQ2)
ν
[M2
∆ ω2+
M2N
+ ν2
]
with Q+ =radic
(M∆ +MN )2 +Q2 and ωplusmn = (M2∆minusM2
N plusmnQ2)2M∆ and the non-pole contribution
to S2(νQ2)
S∆-exch2 (νQ2) = minus2παMNν
M∆M2+
[gM + gE
]gC (B27)
These amplitudes to the contrary are not described by the tree-level ∆-production cross sections
in the standard dispersive approach [17] This peculiarity has been previously missed e g in the
calculation of the ∆-exchange contribution to the hydrogen hyperfine splitting in Ref [88] The
importance of including the ∆-non-pole contribution is also evident when considering the BC sum
37
rule in Eq (14) The ∆-pole terms by themselves violate the BC sum rule but cancel exactly with
the ∆-non-pole terms
limνrarr0
νS∆-pole2 (νQ2) + lim
νrarr0νS2
∆-exch(νQ2) = 0 (B28)
Appendix C Polarizabilities at Q2 = 0
In this section we give analytical expressions for the static values and slopes at Q2 = 0 of
the spin polarizabilities and moments of polarized structure functions In particular we give the
HB expansion of the πN -loop contributions and the ∆-exchange contributions The complete
expressions also for the π∆-loop contributions can be found in the Supplemented material Recall
that IA(0) = I1(0) = d2(0) = 0 and dd2(Q2)dQ2
∣∣∣Q2=0
= 0
1 πN-loop contribution
Here we give analytical expressions for the πN -loop contributions to the proton and neutron
spin polarizabilities expanded in powers of micro = mπMN viz the HB expansion Note that we
choose to expand here to a high order in micro the strict HB expansion would only retain the leading
term in an analogous NLO calculation
bull Static polarizabilities (Q2 = 0)
γ0p =e2g2
A
96π3f2πm
2π
1minus 21πmicro
8minus(
59
2+ 26 logmicro
)micro2 +
1875πmicro3
64
+3
(3
2+ 26 logmicro
)micro4 +
(C1)
γ0n =e2g2
A
48π3f2πm
2π
1
2minus 9πmicro
16minus 2micro2 logmicro+
75πmicro3
128minus 3micro4
4+
(C2)
δLTp =e2g2
A
192π3f2πm
2π
1minus 9πmicro
8+
(13
2minus 2 logmicro
)micro2 minus 465πmicro3
64
minus(
47
2+ 42 logmicro
)micro4 +
(C3)
δLTn =e2g2
A
96π3f2πm
2π
1
2+
3πmicro
16+ (1 + 2 log micro)micro2 minus 105πmicro3
128+
5micro4
4+
(C4)
38
δLTn =e2g2
A
96π3f2πm
2π
1
2+
3πmicro
16+ (1 + 2 log micro)micro2 minus 105πmicro3
128+
5micro4
4+
(C5)
γ0p =e2g2
A
16π3f2πm
4π
4
45minus 3πmicro
16+
14micro2
5minus 1813πmicro3
384minus 192
5(1 + log micro)micro4
+80703πmicro5
2048+
(C6)
γ0n =e2g2
A
16π3f2πm
4π
4
45minus 5πmicro
48+
4micro2
5minus 245πmicro3
384minus 32micro4 logmicro
15
+1323πmicro5
2048+
(C7)
bull Slopes of polarizabilities at Q2 = 0
dγ0p(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
2minus 45πmicro
4+ 223micro2 minus 28515πmicro3
64
minus9
(1953
4+ 449 logmicro
)micro4 +
570255πmicro5
128+
(C8)
dγ0n(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
2minus 81πmicro
8+ 94micro2 minus 2535πmicro3
32
minus3 (1 + 90 logmicro)micro4 +84315πmicro5
1024+
(C9)
dδLTp(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
2880π3f2πm
4π
minus5
2minus 27πmicro
32+ 20micro2 minus 5865πmicro3
256
+3
(617
4+ 36 logmicro
)micro4 minus 2056845πmicro5
4096+
(C10)
dδLTn(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
minus5
4minus 81πmicro
64minus 11micro2 +
10005πmicro3
512
+15
8(11 + 48 logmicro)micro4 minus 267015πmicro5
8192+
(C11)
dIAp(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
96π2f2πmicro
2
1minus 15πmicro
4minus 1
2(115 + 88 logmicro)micro2 +
1839πmicro3
32
+5 (5 + 34 logmicro)micro4 +
(C12)
dIAn(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
48π2f2πmicro
2
1
2minus 11πmicro
8minus 1
4(1 + 20 logmicro)micro2 +
99πmicro3
64
minus25micro4
12+
(C13)
dI1p(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
96π2f2πmicro
3π
8+ 2 (4 + 3 logmicro)microminus 537πmicro2
64
minus1
2(15 + 56 logmicro)micro3 +
(C14)
39
dI1n(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
48π2f2πmicro
minus π
16+
1
4(3 + 4 logmicro)microminus 57πmicro2
128+
2micro3
3+
(C15)
γ0p(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2g2
A
16π3f2πm
6π
1
105minus 23πmicro
256+
377micro2
210minus 15551πmicro3
6144+
3371micro4
105
minus1640457πmicro5
32768+
(C16)
γ0n(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2g2
A
16π3f2πm
6π
1
105minus 153πmicro
1792+
69micro2
70minus 4615πmicro3
6144+
172micro4
35
minus120897πmicro5
32768+
(C17)
2 ∆-exchange contribution
Here we give analytical expressions for the tree-level ∆-exchange contributions to the nucleon
spin polarizabilities and their slopes at Q2 = 0 Note that the ∆-exchange contributes equally to
proton and neutron polarizabilities Recall that for the magnetic γlowastN∆ coupling we introduced a
dipole form factor to mimic vector-meson dominance gM rarr gM(1 +Q2Λ2)2
bull Static polarizabilities (Q2 = 0)
γ0 = minus e2
4πM2+
(g2M
∆2+
g2E
M2+
minus 4gMgEM+∆
) (C18)
δLT =e2M∆
4πM3+
(g2E
MNM++gMgE∆MN
minus gEgCM2
∆
) (C19)
γ0 =e2M2
N
π∆2M4+
(minusg
2M
∆2+
g2E
M2+
+4gMgE∆M+
) (C20)
bull Slopes of polarizabilities at Q2 = 0
dγ0(Q2)
dQ2
∣∣∣∣∣Q2=0
= minus e2
πM2+∆
(g2M
∆
[1
4∆2minus 1
∆M++
1
2M2+
]minus 1
Λ2
g2M
∆+
g2E
2M2+
[1
2∆minus 3
M+
]minusgMgE
M+
[1
∆2minus 5
∆M++
1
M2+
]+
1
Λ2
2gMgEM+
+2gMgC∆M2
+
minus gEgCM3
+
) (C21)
40
d δLT (Q2)
dQ2
∣∣∣∣∣Q2=0
=e2M∆∆
4πMNM2+
(g2E
∆2M2+
[1
∆minus 4
M+
]minus g2
C
∆M2∆M
2+
+gMgE∆2M+
[1
∆2
minus 3
∆M++
1
M2+
]minus 2
Λ2
gMgE∆2M+
+gMgC∆M2
∆
[1
2∆2minus 2
∆M++
1
2M2+
]minus gEgC
2M2∆M
2+
[7
∆+
1
M+
]) (C22)
dIA(Q2)
dQ2
∣∣∣∣∣Q2=0
= minusM2N
M2+
(g2M
2∆2+
g2E
MNM+minus 2gMgE
∆M+minus gEgCM∆M+
) (C23)
dI1(Q2)
dQ2
∣∣∣∣∣Q2=0
= minusM∆M2N
2M3+
(g2E
MNM∆minus gMgE∆MN
minus gEgCM2
∆
) (C24)
γ0(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2M2
N
π∆3M6+
(g2M
∆
[3M2
∆ + 2M∆MN minus 9M2N
∆2+
4M2+
Λ2
]
minus4gMgE
[5M2
∆ minus 9M2N
M+∆2+
2M+
Λ2
]+
g2E
∆M+[7M∆ minus 9MN ]
minus8gMgC∆
+4gEgCM+
) (C25)
[1] D Drechsel B Pasquini and M Vanderhaeghen Dispersion relations in real and virtual
Compton scattering Phys Rept 378 99 (2003) hep-ph0212124
[2] S E Kuhn J-P Chen and E Leader Spin structure of the nucleon mdash status and recent
results Prog Part Nucl Phys 63 1 (2009) arXiv08123535 [hep-ph]
[3] F Hagelstein R Miskimen and V Pascalutsa Nucleon polarizabilities from Compton scat-
tering to hydrogen atom Prog Part Nucl Phys 88 29 (2016) arXiv151203765 [nucl-th]
[4] B Pasquini and M Vanderhaeghen Dispersion theory in electromagnetic interactions Ann
Rev Nucl Part Sci 68 75 (2018) arXiv180510482 [hep-ph]
[5] S Gerasimov A Sum rule for magnetic moments and the damping of the nucleon magnetic
moment in nuclei Sov J Nucl Phys 2 430 (1966)
[6] S Drell and A C Hearn Exact sum rule for nucleon magnetic moments Phys Rev Lett
16 908 (1966)
[7] J S Schwinger Source theory viewpoints in deep inelastic scattering Proc Natl Acad Sci
USA 72 1 (1975)
[8] J S Schwinger Source theory viewpoints in deep inelastic scattering Electromagnetic In-
teractions and Field Theory Proceedings 14 Internationale Universitatswochen Schladming
Austria February 24-March 7 1975 Acta Phys Austriaca Suppl 14 471 (1975)
41
[9] J Schwinger Source theory discussion of deep inelastic scattering with polarized particles
Proc Natl Acad Sci USA 72 1559 (1975)
[10] A M Baldin Polarizability of nucleons Nucl Phys 18 310 (1960)
[11] M Gell-Mann M L Goldberger and W E Thirring Use of causality conditions in quantum
theory Phys Rev 95 1612 (1954)
[12] A Pineda Leading chiral logarithms to the hyperfine splitting of the hydrogen and muonic
hydrogen Phys Rev C 67 025201 (2003)
[13] C Peset and A Pineda The two-photon exchange contribution to muonic hydrogen from
chiral perturbation theory Nucl Phys B 887 69 (2014) arXiv14064524 [hep-ph]
[14] C Peset and A Pineda Model-independent determination of the two-photon exchange contri-
bution to hyperfine splitting in muonic hydrogen JHEP 04 060 arXiv161205206 [nucl-th]
[15] F Hagelstein and V Pascalutsa Proton structure in the hyperfine splitting of muonic hydro-
gen PoS CD15 077 (2016) arXiv151104301 [nucl-th]
[16] F Hagelstein Exciting Nucleons in Compton Scattering and Hydrogen-Like Atoms PhD
thesis Mainz U Inst Kernphys (2017) arXiv171000874 [nucl-th]
[17] F Hagelstein ∆(1232)-Resonance in the hydrogen spectrum Proceedings 11th International
Workshop on the Physics of Excited Nucleons (NSTAR 2017) Columbia SC USA August
20-23 2017 Few Body Syst 59 93 (2018) arXiv180109790 [nucl-th]
[18] Y Prok et al (CLAS) Moments of the spin structure functions gp1 and gd1 for 005 lt Q2 lt 30
GeV2 Phys Lett B 672 12 (2009) arXiv08022232 [nucl-ex]
[19] H Dutz et al (GDH) First measurement of the Gerasimov-Drell-Hearn sum rule for 1H from
07 GeV to 18 GeV at ELSA Phys Rev Lett 91 192001 (2003)
[20] M Amarian et al (Jefferson Lab E94010) Measurement of the generalized forward spin po-
larizabilities of the neutron Phys Rev Lett 93 152301 (2004) arXivnucl-ex0406005
[21] M Amarian et al The Q2 evolution of the generalized Gerasimov-Drell-Hearn integral for
the neutron using a 3He target Phys Rev Lett 89 242301 (2002) arXivnucl-ex0205020
[22] M Amarian et al (Jefferson Lab E94-010)Q2 evolution of the neutron spin structure moments
using a 3He target Phys Rev Lett 92 022301 (2004) arXivhep-ex0310003
[23] A Deur et al Experimental determination of the evolution of the Bjorken integral at low Q2
Phys Rev Lett 93 212001 (2004) arXivhep-ex0407007
[24] K Slifer Low Q2 measurement of gp2 and the δLT spin polarizability Spin structure at long dis-
tance Proceedings Workshop Newport News USA March 12ndash13 2009 nucl-ex09064775
(2009) AIP Conf Proc 1155 10106313203293 (2009) arXiv09064775 [nucl-ex]
[25] P Solvignon et al (E01-012) Moments of the neutron g2 structure function at intermediate
42
Q2 Phys Rev C 92 015208 (2015) arXiv13044497 [nucl-ex]
[26] A Deur Experimental studies at low Q2 of the spin structure of the nucleon at Jefferson Lab
in 9th International Workshop on Chiral Dynamics (CD18) Durham NC USA September
17-21 2018 (2019) arXiv190305661 [nucl-ex]
[27] V Sulkosky et al (Jefferson Lab E97-110) Measurement of the 3He spin-structure functions
and of neutron (3He) spin-dependent sum rules at 0035 le Q2 le 024 GeV2 Phys Lett B
805 135428 (2020) arXiv190805709 [nucl-ex]
[28] K Adhikari et al (CLAS) Measurement of the Q2 Dependence of the Deuteron Spin Structure
Function g1 and its Moments at Low Q2 with CLAS Phys Rev Lett 120 062501 (2018)
arXiv171101974 [nucl-ex]
[29] V Lensky F Hagelstein A Hiller Blin and V Pascalutsa Comment on rdquoMeasurement of
the Q2 Dependence of the Deuteron Spin Structure Function g1 and its Moments at Low Q2
with CLASrdquo (2018) arXiv180603219 [nucl-th]
[30] J M Alarcn F Hagelstein V Lensky and V Pascalutsa Forward doubly-virtual Compton
scattering off the nucleon in chiral perturbation theory at NLO the subtraction function and
moments of unpolarized structure functions (2020) arXiv200509518 [hep-ph]
[31] V Bernard E Epelbaum H Krebs and U-G Meiszligner New insights into the spin structure
of the nucleon Phys Rev D 87 054032 (2013) arXiv12092523 [hep-ph]
[32] V Lensky J M Alarcon and V Pascalutsa Moments of nucleon structure functions at
next-to-leading order in baryon chiral perturbation theory Phys Rev C 90 055202 (2014)
arXiv14072574 [hep-ph]
[33] V Pascalutsa and M Vanderhaeghen Chiral effective-field theory in the ∆(1232) region
I Pion electroproduction on the nucleon Phys Rev D 73 034003 (2006) arXivhep-
ph0512244
[34] V Pascalutsa and M Vanderhaeghen Electromagnetic nucleon-to-Delta transition in chiral
effective field theory Phys Rev Lett 95 232001 (2005) arXivhep-ph0508060
[35] T R Hemmert B R Holstein and J Kambor Systematic 1M expansion for spin 32
particles in baryon chiral perturbation theory Phys Lett B 395 89 (1997) arXivhep-
ph9606456
[36] V Pascalutsa and D R Phillips Effective theory of the ∆(1232) in Compton scattering off
the nucleon Phys Rev C 67 055202 (2003) arXivnucl-th0212024
[37] V Pascalutsa M Vanderhaeghen and S N Yang Electromagnetic excitation of the ∆(1232)-
resonance Phys Rept 437 125 (2007) arXivhep-ph0609004
[38] K A Olive et al (Particle Data Group) Review of Particle Physics Chin Phys C 38 090001
43
(2014)
[39] F E Low Scattering of light of very low frequency by systems of spin 12 Phys Rev 96
1428 (1954)
[40] M Gell-Mann and M L Goldberger Scattering of low-energy photons by particles of spin
12 Phys Rev 96 1433 (1954)
[41] A M Harun ar-Rashid A simple derivation of Schwingerrsquos sum rule for spin dependent
structure functions Nuovo Cim A 33 447 (1976)
[42] F Hagelstein and V Pascalutsa Dissecting the hadronic contributions to (g minus 2)micro by
Schwingerrsquos sum rule Phys Rev Lett 120 072002 (2018) arXiv171004571 [hep-ph]
[43] V Lensky V Pascalutsa M Vanderhaeghen and C Kao Spin-dependent sum rules con-
necting real and virtual Compton scattering verified Phys Rev D 95 074001 (2017)
arXiv170101947 [hep-ph]
[44] H Burkhardt and W N Cottingham Sum rules for forward virtual Compton scattering
Annals Phys 56 453 (1970)
[45] V Lensky and V Pascalutsa Predictive powers of chiral perturbation theory in Compton
scattering off protons Eur Phys J C 65 195 (2010) arXiv09070451 [hep-ph]
[46] H Grieszlighammer J McGovern D Phillips and G Feldman Using effective field theory to
analyse low-energy Compton scattering data from protons and light nuclei Prog Part Nucl
Phys 67 841 (2012) arXiv12036834 [nucl-th]
[47] H W Grieszlighammer J A McGovern and D R Phillips Nucleon polarisabilities at and
beyond physical pion masses Eur Phys J A 52 139 (2016) arXiv151101952 [nucl-th]
[48] E Epelbaum H Krebs and U-G Meiszligner Improved chiral nucleon-nucleon potential up to
next-to-next-to-next-to-leading order Eur Phys J A 51 53 (2015) arXiv14120142 [nucl-
th]
[49] E Epelbaum H Krebs and U-G Meiszligner Precision nucleon-nucleon potential at fifth order
in the chiral expansion Phys Rev Lett 115 122301 (2015) arXiv14124623 [nucl-th]
[50] C W Kao T Spitzenberg and M Vanderhaeghen Burkhardt-Cottingham sum rule and
forward spin polarizabilities in heavy baryon chiral perturbation theory Phys Rev D 67
016001 (2003) arXivhep-ph0209241
[51] C-W Kao D Drechsel S Kamalov and M Vanderhaeghen Higher moments of nucleon
spin structure functions in heavy baryon chiral perturbation theory and in a resonance model
Phys Rev D 69 056004 (2004) arXivhep-ph0312102
[52] T Becher and H Leutwyler Baryon chiral perturbation theory in manifestly Lorentz invariant
form Eur Phys J C 9 643 (1999) arXivhep-ph9901384
44
[53] J Gasser M E Sainio and A Svarc Nucleons with chiral loops Nucl Phys B 307 779
(1988)
[54] L S Geng J Martin Camalich L Alvarez-Ruso and M J Vicente Vacas Leading SU(3)-
breaking corrections to the baryon magnetic moments in chiral perturbation theory Phys
Rev Lett 101 222002 (2008) arXiv08051419 [hep-ph]
[55] D Drechsel S S Kamalov and L Tiator The GDH sum rule and related integrals Phys
Rev D 63 114010 (2001) arXivhep-ph0008306
[56] D Drechsel O Hanstein S S Kamalov and L Tiator A unitary isobar model for pion
photo- and electroproduction on the proton up to 1 GeV Nucl Phys A 645 145 (1999)
arXivnucl-th9807001
[57] L Tiator private communication (2020)
[58] V Bernard T R Hemmert and U-G Meiszligner Spin structure of the nucleon at low energies
Phys Rev D 67 076008 (2003) arXivhep-ph0212033
[59] O Gryniuk F Hagelstein and V Pascalutsa Evaluation of the forward Compton scattering
off protons II Spin-dependent amplitude and observables Phys Rev D 94 034043 (2016)
arXiv160400789 [nucl-th]
[60] R Zielinski The g2p Experiment A Measurement of the Protonrsquos Spin Structure Functions
PhD thesis New Hampshire U (2010) arXiv170808297 [nucl-ex]
[61] N Guler et al (CLAS) Precise determination of the deuteron spin structure at low to mod-
erate Q2 with CLAS and extraction of the neutron contribution Phys Rev C 92 055201
(2015) arXiv150507877 [nucl-ex]
[62] C E Carlson V Nazaryan and K Griffioen Proton structure corrections to electronic
and muonic hydrogen hyperfine splitting Phys Rev A 78 022517 (2008) arXiv08052603
[physicsatom-ph]
[63] V Bernard N Kaiser and U-G Meiszligner Chiral dynamics in nucleons and nuclei Int J
Mod Phys E 4 193 (1995) arXivhep-ph9501384
[64] T R Hemmert B R Holstein and J Kambor ∆(1232) and the polarizabilities of the nucleon
Phys Rev D 55 5598 (1997) arXivhep-ph9612374
[65] V Pascalutsa and R Timmermans Field theory of nucleon to higher-spin baryon transitions
Phys Rev C 60 042201 (1999) arXivnucl-th9905065
[66] V Pascalutsa Quantization of an interacting spin-32 field and the Delta isobar Phys Rev
D 58 096002 (1998) arXivhep-ph9802288
[67] H Krebs Double Virtual Compton Scattering and SpinStructure of the Nucleon PoS
CD2018 031 (2019)
45
[68] N Kochelev and Y Oh Axial anomaly and the δLT puzzle Phys Rev D 016012 (2012)
arXiv11034892 [hep-ph]
[69] D Drechsel S Kamalov and L Tiator Unitary isobar model ndash MAID2007 Eur Phys J A
69 (2007) available at httpsmaidkphuni-mainzde arXiv07100306 [nucl-th]
[70] P J Mohr B N Taylor and D B Newell CODATA recommended values of the fundamental
physical constants 2010 Rev Mod Phys 84 1527 (2012)
[71] J Ahrens et al (GDH A2) First measurement of the Gerasimov-Drell-Hearn integral for 1H
from 200 to 800 MeV Phys Rev Lett 87 022003 (2001) arXivhep-ex0105089 [hep-ex]
[72] K Helbing (GDH) Experimental verification of the GDH sum rule at ELSA and MAMI Nucl
Phys Proc Suppl 105 113 (2002)
[73] R Bradford A Bodek H S Budd and J Arrington A New parameterization of the nucleon
elastic form-factors NuInt05 proceedings of the 4th International Workshop on Neutrino-
Nucleus Interactions in the Few-GeV Region Okayama Japan 26-29 September 2005 Nucl
Phys Proc Suppl 159 127 (2006) arXivhep-ex0602017
[74] J D Bjorken Applications of the chiral U(6)timesU(6) algebra of current densities Phys Rev
148 1467 (1966)
[75] J D Bjorken Inelastic scattering of polarized leptons from polarized nucleons Phys Rev D
1 1376 (1970)
[76] V Bernard T R Hemmert and U-G Meiszligner Novel analysis of chiral loop effects in
the generalized Gerasimov-Drell-Hearn sum rule Phys Lett B 545 105 (2002) arXivhep-
ph0203167
[77] A Deur et al Experimental study of isovector spin sum rules Phys Rev D 78 032001
(2008) arXiv08023198 [nucl-ex]
[78] B Pasquini P Pedroni and D Drechsel Higher order forward spin polarizability Phys Lett
B 687 160 (2010) arXiv10014230 [hep-ph]
[79] R Jaffe g2ndashThe nucleonrsquos other spin-dependent structure function Comments Nucl Part
Phys 19 239 (1990)
[80] E V Shuryak and A Vainshtein Theory of power corrections to deep inelastic scattering in
quantum chromodynamics (II) Qminus4 effects polarized target Nucl Phys B 201 141 (1982)
[81] S Wandzura and F Wilczek Sum rules for spin dependent electroproduction Test of rela-
tivistic constituent quarks Phys Lett B 72 195 (1977)
[82] B W Filippone and X-D Ji The spin structure of the nucleon Adv Nucl Phys 26 1
(2001) arXivhep-ph0101224
[83] M Burkardt The g2 structure function Proceedings Workshop on Spin structure at long
46
distance Newport News USA March 12-13 2009 AIP Conf Proc 1155 26 (2009)
arXiv09054079 [hep-ph]
[84] A Deur S J Brodsky and G F de Teramond The spin structure of the nucleon Rept
Prog Phys 82 076201 (2019) arXiv180705250 [hep-ph]
[85] B R Holstein D Drechsel B Pasquini and M Vanderhaeghen Higher order polarizabilities
of the proton Phys Rev C 61 034316 (2000) arXivhep-ph9910427
[86] B R Holstein V Pascalutsa and M Vanderhaeghen Sum rules for magnetic moments and
polarizabilities in QED and chiral effective-field theory Phys Rev D 72 094014 (2005)
arXivhep-ph0507016
[87] B Pasquini D Drechsel and L Tiator Invariant amplitudes for pion electroproduction Eur
Phys J A 34 387 (2007) arXiv07122327 [hep-ph]
[88] A J Buchmann Non-spherical proton shape and hydrogen hyperfine splitting Proceedings
International Workshop on Precision Physics of Simple Atomic Systems (PSAS 2008) Wind-
sor Ontario Canada July 21-26 2008 Can J Phys 87 773 (2009) arXiv09104747
[physicsatom-ph]
47
000 005 010 015 020 025 030-002
000
002
004
006
008
010
Q2 GeV2D
G1Hp-
nL
FIG 7 Isovector combination of Γ1(Q2) as function of Q2 The legend is the same as in Fig 6
The pink curve is the IR result from Ref [58 76] The experimental points are from Ref [77]
(brown dots) and Ref [23] (orange squares)
∆ exchange and π∆ loops] in units of GeVminus2
d∆I1p(Q2)
dQ2
∣∣∣∣Q2=0
= 039(4) asymp 034minus 053 + 058 (35a)
d∆I1n(Q2)
dQ2
∣∣∣∣Q2=0
= minus101(10) asymp minus107minus 053 + 058 (35b)
Including the empirical Pauli form factor [73] we find in units of GeVminus2
dI1p(Q2)
dQ2
∣∣∣∣Q2=0
= 580dI1n(Q2)
dQ2
∣∣∣∣Q2=0
= 553 (36)
E d2(Q2) mdash a measure of color polarizability
Another interesting moment to consider is d2(Q2) which is related to the twist-3 part of
the spin structure function g2(xQ2) [79 80]
d2(Q2) equiv 3
int 1
0
dx x2[g2(xQ2)minus gWW2 (xQ2)] (37)
where gWW2 (xQ2) is the twist-2 part of g2(xQ2) Using the Wandzura-Wilczek relation [81]
one can relate d2(Q2) to moments of the spin structure functions g1(xQ2) and g2(xQ2)
d2(Q2) =
int 1
0
dx x2 [3g2(xQ2) + 2g1(xQ2)] (38)
22
000 005 010 015 020 025 030
0000
0001
0002
0003
0004
0005
0006
0007
Q2 GeV2D
d2
proton
000 005 010 015 020 025 030
0000
0002
0004
0006
0008
0010
0012
Q2 GeV2D
d2
neutron
000 005 010 015 020 025 030
-1
0
1
2
Q2 GeV2D
Γ 010
-4
fm6
D
proton
000 005 010 015 020 025 030-2
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
6D
neutron
FIG 8 Upper panel The inelastic moment d2(Q2) for the proton (left) and neutron (right) as
function of Q2 The result of this work the NLO BχPT prediction is shown by the blue solid line
and the blue band The red line represents the LO BχPT result The purple short-dashed line is
the O(p4) HB result from Ref [50 51] The black dotted line is the MAID model prediction [69]
The experimental points for the neutron (cyan dots) are from Ref [22] Lower panel Fifth-order
generalized forward spin polarizability γ0(Q2) for the proton (left) and neutron (right) as function
of Q2 The experimental points for the proton are from Ref [59] (purple square) and Ref [78]
(orange dot)
This relation however only holds for asymptotically large Q2 It is also in the high-Q2
region where d2(Q2) is a measure of color polarizability [82 83] through its relation to the
gluon field strength tensor [80] We refer to Ref [84] for a recent review on the spin structure
of the nucleon including a discussion of sum rules for deep inelastic scattering and color
polarizabilities
What we consider in the following is the inelastic part of d2(Q2) defined as the moment
of g1(xQ2) and g2(xQ2) spin structure functions cf Eq (38)
d2(Q2) =
int x0
0
dx x2 [3g2(xQ2) + 2g1(xQ2)] (39)
23
This moment provides another testing ground for our BχPT predictions through comparison
with experiments on the neutron [22] Going towards the low-Q2 region the interpretation
of d2(Q2) in terms of color polarizabilities will fade out The above definition however
implies it is related to other VVCS polarizabilities
d2(Q2) =Q4
8M4N
[M2
NQ2
αδLT (Q2) + I1(Q2)minus IA(Q2)
] (40)
Note that d2(Q2) and its first two derivatives with respect to Q2 vanish at Q2 = 0 The
considerations in Eqs (28) and (29) have no effect on d2(Q2) since the Born contribution
from IA(Q2) and I1(Q2) cancel out Therefore d2(Q2) is a pure polarizability
In Fig 8 upper panel we show our NLO BχPT prediction and other results for d2(Q2)
While MAID [69] and BχPT describe the experimental data for the neutron [22] very well
the HB limit [50 51] is showing a fast growth with Q2 This illustrates the importance
of keeping the relativistic result Note also that even though the πN -loop contribution
is dominant both gC and the form factor in gM are essential to obtain a curvature that
reproduces the data cf Fig 9 upper panel For the proton there are to our knowledge
no experimental results to compare with However the agreement between the NLO BχPT
prediction and the MAID prediction at low energies is reasonable
F γ0(Q2) mdash fifth-order generalized forward spin polarizability
It is interesting to compare the generalized fifth-order forward spin polarizability sum
rule
γ0(Q2) =1
2π2
int infinν0
dν
radic1 +
Q2
ν2
σTT (νQ2)
ν5(41)
=64αM4
N
Q10
int x0
0
dx x4
[g1(xQ2)minus 4M2
Nx2
Q2g2(xQ2)
]
to the sum rule integrals for IA(Q2) and γ0(Q2) since they differ merely by their energy
weighting of σTT (νQ2) and a constant prefactor cf Eqs (19) (26) and (41) From IA(Q2)
to γ0(Q2) to γ0(Q2) the energy suppression is increasing by a factor of νminus2 respectively
Therefore the description of γ0(Q2) should be easiest in a low-energy effective-field theory
such as χPT whereas γ0(Q2) and IA(Q2) receive larger contributions from higher energies
In Fig 8 lower panel we show our LO and NLO BχPT predictions for γ0(Q2) One
can see that the πN -loop contribution is positive (in accordance to what we see for the cross
section σTT see Fig 10) The Delta shifts it substantially especially in the low Q2 region
24
000 005 010 015 020 025 030
-0004
-0002
0000
0002
0004
0006
0008
0010
Q2GeV
2D
d2
proton
000 005 010 015 020 025 030
-0005
0000
0005
0010
0015
Q2GeV
2D
d2
neutron
000 005 010 015 020 025 030
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
6D
proton
000 005 010 015 020 025 030
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
6D
neutron
FIG 9 Contributions of the different orders to the chiral predictions of d2(Q2) upper panel and
γ0(Q2) lower panel for the proton (left) and neutron (right) Red solid line πN -loop contribu-
tion green dot-dashed line ∆-exchange contribution orange dotted line π∆-loop contribution
blue long-dashed line total result purple dot-dot-dashed line total result without gC contribution
black short-dashed line total result without gM dipole
bringing it into a better agreement with data In general the BχPT curves start above the
empirical data points at the real-photon point and then decrease asymptotically to zero
above Q2 gt 01 GeV2 On the other hand the MAID prediction reproduces the empirical
data at the real-photon point then decreases to negative values until about Q2 gt 006 GeV2
from where it also starts to asymptotically approach zero Consequently our NLO BχPT
prediction of γ0(Q2) is consistently above the MAID prediction This is very different to
what we saw for IA(Q2) in Fig 4 upper panel where the MAID prediction at the real-
photon point is above the experimental value While the agreement of our predictions with
the empirical data is in general quite good for all moments of σTT (νQ2) one should point
out that both for γ0n(Q2) and γ0p(Q2) we overestimate the data at low Q2 For IA(Q2) such
observation cannot be made because ∆IA(0) = 0 and thus IA(0) is given by the empirical
Pauli form factor only From IA(Q2) γ0(Q2) and γ0(Q2) the latter has the smallest however
25
non-negligible dependence on gC and the dipole in gM cf Fig 9 lower panel
The πN -loop ∆-exchange and π∆-loop contributions to the NLO BχPT prediction of
the static fifth-order forward spin polarizability amount to in units of 10minus4 fm6
γ0p = 112(30) asymp 208minus 096minus 001 (42a)
γ0n = 195(30) asymp 292minus 096minus 001 (42b)
while the slope is composed as follows in units of 10minus4 fm8
dγ0p(Q2)
dQ2
∣∣∣∣Q2=0
= minus084(10) asymp minus100 + 016 + 000 (43a)
dγ0n(Q2)
dQ2
∣∣∣∣Q2=0
= minus142(15) asymp minus158 + 016 + 000 (43b)
Note that the HB prediction of the static polarizability γ0p (423 at O(p3) and 365 at O(ε3)
[78 85]) is almost one order of magnitude larger than the empirical value and therefore not
shown in Fig 8
G Summary
Our results are summarized in Table II where we give the contributions of the different
orders to the chiral predictions of the polarizabilities and their slopes at the real-photon
point A quantitative comparison of our predictions for the static spin polarizabilities to
the work of Bernard et al [31] and different empirical evaluations is shown in Table III We
can see that the inclusion of the Delta turns out to be very important for all moments of
the helicity-difference cross section To describe the Q2 behavior of the polarizabilities the
magnetic coupling of the N rarr ∆ transition should be modified by a dipole form factor as
has been observed previously in the description of electroproduction data [33] This dipole
form factor effectively takes account of vector-meson exchanges The Coulomb-quadrupole
N rarr ∆ transition despite its subleading order is important in the description of some
moments of spin structure functions This is contrary to what we saw for the moments of
unpolarized structure functions [30] where the Coulomb coupling had a negligible effect
The π∆ loops are mainly relevant for the generalized GDH integrals
26
TABLE II The NLO BχPT predictions for the forward VVCS polarizabilities and their slopes at
Q2 = 0 The contributions of the πN loops the ∆ exchange and the π∆ loops are shown together
with the combined total result Note that IA(0) = I1(0) = d2(0) = 0 and (d2)prime = 0
πN loops ∆ exchange π∆ loops Total
γ0 p 201 minus284 minus010minus093(92)
(10minus4 fm4) n 298 003(92)
δLT p 150 minus016 minus002132(15)
(10minus4 fm4) n 235 218(23)
γ0 p 208 minus096 minus001112(30)
(10minus4 fm6) n 292 195(30)
(γ0)prime p minus033011 001
minus022(4)
(10minus4 fm6) n minus073 minus061(7)
(δLT )prime p minus080 minus004 minus001minus085(8)
(10minus4 fm6) n minus119 minus124(12)
(γ0)prime p minus100016 000
minus084(10)
(10minus4 fm8) n minus158 minus142(15)
(∆IA)prime p 238 minus1121 025minus858(343)
(GeVminus2) n 141 minus955(343)
(∆I1)prime p 034 minus053 058039(4)
(GeVminus2) n minus107 minus101(10)
TABLE III Our NLO BχPT predictions for the spin polarizabilities at Q2 = 0 compared with
the BχPT+∆ predictions from Bernard et al [31] and the available empirical information Where
the reference is not given the empirical number is provided by the MAID analysis [55 69] with
unspecified uncertainty
Proton Neutron
This work BχPT+∆ Empirical This work BχPT+∆ Empirical
γ0 minus093(92) minus174(40) minus100(8)(12) [19] 003(92) minus077(40) minus0005
(10minus4 fm4) minus090(8)(11) [78] [MAID]
minus0929(105) [59]
δLT 132(15) 240(1) 134 218(23) 238(3) 203
(10minus4 fm4) [MAID] [MAID]
γ0 112(30) 060(7)(7) [78] 195(30) 123
(10minus4 fm6) 0484(82) [59] [MAID]
IV CONCLUSIONS
We have presented a complete NLO calculation of the polarized non-Born VVCS ampli-
tudes in covariant BχPT with pion nucleon and ∆(1232) fields The dispersion relations
between the VVCS amplitudes and the tree-level photoabsorption cross sections served as a
27
cross-check of these calculations
The obtained moments of the proton and neutron spin structure functions related to gen-
eralized polarizabilities and GDH-type integrals agree well with the available experimental
data The description of their Q2 evolution is improved compared to the previous χPT pre-
dictions In particular the NLO BχPT predictions obtained here give a better description
of the empirical data (eg from the Jefferson Laboratory ldquoSpin Physics Programrdquo) than
the HB [50 51] and IR [58] calculations
The demonstrated predictive power of the χPT framework amplitudes makes it well
suited for extending the χPT evaluation of the TPE effect in the hyperfine structure of
(muonic-)hydrogen [15ndash17] to next-to-leading order
ACKNOWLEDGEMENTS
We thank Lothar Tiator and Marc Vanderhaeghen for helpful discussions This work
is supported by the Deutsche Forschungsgemeinschaft (DFG) through the Collaborative
Research Center [The Low-Energy Frontier of the Standard Model (SFB 1044)] JMA ac-
knowledges support from the Community of Madrid through the ldquoPrograma de atraccion de
talento investigador 2017 (Modalidad 1)rdquo and the Spanish MECD grants FPA2016-77313-P
FH gratefully acknowledges financial support from the Swiss National Science Foundation
Appendix A Tensor decompositions of the VVCS amplitudes
In this appendix we review the decomposition of the forward VVCS process into tensor struc-
tures and scalar amplitudes In particular we consider the connection between the covariant and
the semi-relativistic decomposition in the lab frame that is defined in terms of the conventional
transverse longitudinal transverse-transverse and transverse-longitudinal amplitudes
As explained in Sec II A the process of forward VVCS off the nucleon can be described in
terms of four explicitly covariant amplitudes S1 2 and T1 2 [3]
T (νQ2) =
(minusgmicroν +
qmicroqν
q2
)T1(νQ2) +
1
M2N
(pmicro minus p middot q
q2qmicro)(
pν minus p middot qq2
qν)T2(νQ2) (A1)
minus 1
MNγmicroναqα S1(νQ2)minus 1
M2N
(γmicroνq2 + qmicroγναqα minus qνγmicroαqα
)S2(νQ2)
εprimelowastmicro εν
where εmicro (εprimelowastmicro ) are the incoming (outgoing) photon polarization vectors ν is the photon lab-frame
energy and Q2 is the photon virtuality Alternatively the decomposition in the laboratory frame
28
(which in the forward case coincides with the Breit frame) is parametrized in terms of the nucleon
Pauli matrices ~σ and the four scalar functions fL fT gTT and gLT
T (νQ2) = ε0 εprimelowast0 fL(νQ2) + (~ε primelowast middot ~ε ) fT (νQ2) + i~σ middot (~ε primelowast times ~ε ) gTT (νQ2) (A2)
minusi~σ middot [(ε0~εprimelowast minus ~ε ε primelowast0 )times q] gLT (νQ2)
Here ~q and q = ~q|~q | are the photon three-momentum in the lab system and its unit vector The
modified polarization vector components are given by
ε0 =
[ε0 minus
ν
|~q | (~ε middot q )
] |~q |Q ~ε = ~εminus q (~ε middot q ) (A3)
where ε = (ε0~ε ) is the usual incoming photon polarization vector and εprimelowast the outgoing polarization
vector The LEX of the lab frame amplitudes [Eq (10)] can serve in particular as the definition
of the generalized polarizabilities The lab frame amplitudes are also conveniently used for the
definition of the response functions see the example of the scalar amplitude gLT (νQ2) and the
corresponding response function σLT (νQ2) below in App B
Appendix B Photoabsorption cross sections
In the forward kinematics the spin-dependent VVCS amplitudes and the spin polarizabilities
can be described in terms of the polarized structure functions g1(xQ2) and g2(xQ2) or equiv-
alently the helicity-difference cross section σTT (νQ2) and the longitudinal-transverse response
function σLT (νQ2) with the help of dispersion relations (5) and the optical theorem (3) In this
way the photoabsorption cross sections measured in electroproduction processes form the basis
for most empirical evaluations shown throughout Sec III In the following we present the BχPT
predictions for the tree-level cross sections of πN - π∆- and ∆-production through photoabsorption
on the nucleon cf Figs 8 9 and 10 in Ref [30] In Secs B 1 and B 2 we will discuss the leading
πN -production channel and the ∆-production channel respectively We used these cross sections
to verify the polarizability predictions obtained otherwise from the calculated non-Born VVCS
amplitudes Due to the bad high-energy behavior of the π∆-production cross sections in BχPT
cf Fig 10 the dispersion relations in Eq (5) require further subtractions for a reconstruction of
the π∆-loop contribution to the spin-dependent VVCS amplitudes Therefore not all polarizabil-
ities could be verified but only those appearing as higher-order terms in the LEX of the VVCS
amplitudes such as γ0 [16]
29
proton neutron
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Neutron
02 04 06 08 100
50
100
150
200
250
[GeV]
T [b]
02 04 06 08 10
-50
0
50
100
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Neutron
02 04 06 08 100
50
100
150
200
250
[GeV]
T [b]
02 04 06 08 10
-50
0
50
100
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02040608100
50
100
150
200
250
300
[GeV]
T[b]
0204060810
-50
0
50
[GeV]
TT[b]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
0204060810
00
02
04
06
[GeV]
LTQ
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
0204060810
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
FIG 10 Photoabsorption cross sections for πN (red) and π∆ production (orange) with Q2 = 0
(solid) and Q2 = 01 GeV2 (dashed for πN and dotted for π∆ channel)
1 πN-production channel
In order to extract the response function σLT (νQ2) we have developed a method similar to
the one used to calculate σTT (νQ2) see for example Ref [86] For σLT (νQ2) however the
calculation is more complicated because one has to take into account that the associated Compton
process involves a spin-flip of the nucleon as illustrated in Fig 11 When calculating the cross
section the product of the incoming nucleon spinors has to reflect this flip
The forward VVCS amplitude related to σLT (νQ2) mdash and δLT (Q2) mdash is gLT (νQ2) It can
be extracted from Eq (A2) if one takes the modified polarization vector components in Eq (A3)
with ε = εL and εprimelowast = εlowastplusmn as input where εL = 1Q(|~q | 0 0 q0) and εplusmn = ∓ 1radic
2(0 1plusmni 0) are the
standard longitudinal and transverse polarization vectors respectively For εL and εlowastplusmn only the
choice of helicities hprime = plusmn12 and h = ∓12 gives a non-zero contribution and one obtains
χdaggerhprime T (νQ2)χh = χdaggerhprimeminusi~σ middot [(ε0~εprimelowast minus ~ε ε primelowast0 )times q] gLT (νQ2)χh =
radic2 gLT (νQ2) (B1)
where χh and χdaggerhprime are two-component Pauli spinors with opposite helicities or here spins
Let us now consider the related photoabsorption process and in particular the tree-level γlowastN rarr
30
s = 12
sprime = minus12
λ = 0 λprime = 1
rprime rprime
s=12sprime=minus1
2
λ=0λprime=1
rprimerprime
s=12
sprime=minus12
λ=0λprime=1
rprimerprime hacute=frac12h=-frac12
λacute=1λ=0
FIG 11 Relation between the forward Compton process and the photoabsorption process given
by the optical theorem In particular we show the longitudinal-transverse contribution The
double-line arrows represent the spin of the external particles while the dot represents the scalar
(longitudinal) polarization of the incoming photon Inside the blob the intermediate states are
represented eg nucleons with spins rprime (which are averaged in the calculation of the cross section)
and pions
πN channel see diagrams in Fig 8 of Ref [30] We define the πN -production amplitude as
T = uhB (PB)sumi
Ai(s t)Γi uhA(PA) (B2)
with the Dirac structures
Γ1 = γ5 (B3a)
Γ2 =1
2
[qA ε
]γ5 (B3b)
where uhA(PA) and udaggerhB (PB) are the Dirac spinors and PA and PB are the four-momenta of
the incoming and outgoing nucleons respectively When calculating the photoabsorption cross
section related to the VVCS amplitude in Eq (B1) the nucleon spin flip should be implemented
by uhprime(PA) in T dagger and uh(PA) in T together with the appropriate transverse and longitudinal
photon polarization vectors εlowastplusmn and εL
However if one wants to use the properties of the Dirac matrices it is more useful to construct
an operator to produce this spin flip in the external nucleons of Fig 11 This is accomplished by
introducing the projector ΓLT equiv 12radic
2(γ1 + iγ2)γ5 which also takes into account the extra factor
radic2 in Eq (B1) We checked that with this projector one correctly extracts δLT by comparing the
HB limit of our result to the HB result of Ref [50] where the authors calculate this polarizability
from the Compton amplitude directly With all those ingredients the longitudinal-transverse cross
section is calculated in the following way
σLT (νQ2) =1
64π2 s
|~pf |cm
|~pi|cm
int 1
minus1dcos θ
sumij
AiAdaggerjXij (B4)
31
with
Xij = Tr[(PB +MN )Γi(PA +MN )ΓLTγ0Γdaggerjγ
0] (B5)
where θ is the scattering angle in the center-of-mass (cm) frame and |~pi|cm (|~pf |cm) is the three-
momentum of an incoming (outgoing) particle in the cm frame An explicit calculation of the
matrix Xij leads to
X = MNQ
0 2(PB minus PA) middot εLminusradic
2|~qf |cm sin θ (sminus u)
(B6)
where |~qi|cm (|~qf |cm) is the relative three-momentum of the incoming (outgoing) particles in the
cm frame Here s t and u are the usual Mandelstam variables For the different γlowastN rarr πN
channels we obtain the following amplitudes Ai where we introduce qA as the four-momentum of
the incoming photon and qB as the four-momentum of the outgoing pion
bull γlowastprarr π0p
A1 =e gAMN
fπ
[2PA middot ε+ qA middot ε
sminusM2N
+2PB middot εminus qA middot ε
uminusM2N
] (B7a)
A2 =e gAMN
fπ
[1
sminusM2N
+1
uminusM2N
] (B7b)
bull γlowastprarr π+n
A1 =
radic2 e gAMN
fπ
[2PA middot ε+ qA middot ε
sminusM2N
+2(PA minus PB) middot ε+ qA middot ε
tminusm2π
] (B8a)
A2 =
radic2 e gAMN
fπ(sminusM2N )
(B8b)
bull γlowastnrarr π0n
A1 = 0 (B9a)
A2 = 0 (B9b)
bull γlowastnrarr πminusp
A1 =
radic2 e gAMN
fπ
[2PB middot εminus qA middot ε
uminusM2N
minus 2(PA minus PB) middot ε+ qA middot εtminusm2
π
] (B10a)
A2 =
radic2 e gAMN
fπ(uminusM2N )
(B10b)
The analytical expressions shown above were checked with the amplitudes given in Ref [87] An-
alytical expressions for the tree-level γlowastN rarr πN channel of the σLT (νQ2) and σTT (νQ2) cross
32
sections are given below (proton channels π+n and π0p neutron channel πminusp) We checked that
they reproduce the known results in the real-photon limit [45 86] To shorten the final expressions
for the cross sections which are considerably longer for finite Q2 than in the real-photon limit we
define the following dimensionless kinematic variables
αγ = (ENi )cmradics =
s+M2N +Q2
2s (B11)
απ = (ENf )cmradics =
s+M2N minusm2
π
2s (B12)
βγ = Eγcmradics =
sminusM2N minusQ2
2s (B13)
βπ = Eπcmradics =
sminusM2N +m2
π
2s (B14)
λγ = |~qi|cmradics =
radic(sminusM2
N minusQ2)2 + 4sQ2
2s (B15)
λπ = |~qf |cmradics =
radic(sminusM2
N +m2π)2 minus 4sm2
π
2s (B16)
Here (ENi )cm and (ENf )cm are the energies of the incoming and outgoing nucleon Eγcm is the energy
of the incoming photon Eπcm is the energy of the outgoing pion all in the cm frame
σ(π+n)TT = minus e2g2
AM2N
64πf2πs
3(sminusM2N )2λ4
γ
4sλπλγ
[(M2
N minus s)(M2N minusQ2 minus s)(Q2 + 2sβγβπ)
+ 2s(m2πQ
2 minus (M2N minus s)
(M2N + s(minus1 + 2βγβπ)
))λ2γ
]+ (M2
N minus s)(M2N minusQ2 minus s)
times (Q2 + 2sβγβπ minus 2sλπλγ)(Q2 + 2s(βγβπ + λπλγ)
)times log
(Q2 + 2sβγβπ minus 2sλγλπQ2 + 2sβγβπ + 2sλγλπ
) (B17)
33
σ(π0p)TT =
e2g2AM
2Nλπ
64πf2πs(sminusM2
N )2λγ
1
s2λπλ3γ
[(M2
N minus s)(M2N minusQ2 minus s)
(3M2
N + s(2βγβπ minus 3))(M2N
+ s(2βγβπ minus 1))
+ 2s
((2m2
πQ2 + (M2
N minus s)2)(M2
N minus s)minus 2(minusm2
πQ2 + (sminusM2
N )2)
times sβγβπ + 4s2(sminusM2N )β2
γβ2π
)λ2γ
]arctan
(2sλπλγ
M2N + s(2βγβπ minus 1)
)+ 2
[minus 2m2
πQ2
+ (M2N minus s)
(2(sminusM2
N ) + 4sβγβπ +(sminusM2
N +Q2)(3M2N minus 3s+ 2sβγβπ)
sλ2γ
+m2πQ
2(sminusM2N )
(M2N minus s+ 2sβγβπ)(M2
N + s(minus1 + 2βγβπ minus 2λπλγ))
+m2πQ
2(sminusM2N )
(M2N minus s+ 2sβγβπ)(M2
N + s(minus1 + 2βγβπ + 2λπλγ))
)
+1
λπλγ
(βγβπ
(minusm2
πQ2 + (M2
N minus s)(M2N minus s+ 2sβγβπ)
)minus (M2
N minus s)(M2N minusQ2 minus s)λ2
π
)log
(M2N + s(minus1 + 2βγβπ + 2λπλγ)
M2N + s(minus1 + 2βγβπ minus 2λπλγ)
)] (B18)
σ(πminusp)TT =
e2g2AM
2N
64πf2πs
3λ4γ(M2
N + s(2βπβγ minus 1))2 minus 4λ2πs
2λ2γ)
4sλπλγ
((minusM2
N +Q2 + s)(M2N
+ s(2βπβγ minus 1))
2 minus 2sλ2γ
(2λ2
πs(minusM2N +Q2 + s) +m2
πQ2))
+((M2N + s(2βπβγ minus 1)
)2
minus 4λ2πs
2λ2γ
)((M2N + s(2βπβγ minus 1)
)(M2N minus 2Q2 minus s(2βπβγ + 1)
)+ 4λ2
πs2λ2
γ
)times log
(minusM2
N minus 2βπsβγ minus 2λπsλγ + s)minus(M2N + s(2βπβγ minus 2λπλγ minus 1)
)(M2N + s(minus1
+ 2βπβγ + 2λπλγ))[((
M2N + s(2βπβγ minus 1)
)(M2N minus 2Q2 minus s(2βπβγ + 1)
)+ 4λ2
πs2λ2
γ
)times log
(minusM2
N minus 2βπsβγ + 2λπsλγ + s)
+((Q2 + 2βπsβγ)2 minus 4λ2
πs2λ2
γ
)times log
(Q2 + 2βπsβγ minus 2λπsλγQ2 + 2sβπβγ + 2λπsλγ
)]+ 4sλ2
γ
((M2
N + s(2βπβγ minus 1))2 minus 4λ2πs
2λ2γ
)times (M2
N minus s) arctanh
(2λπsλγ
M2N + 2βπsβγ minus s
) (B19)
34
σ(π+n)LT =
e2g2AM
3Nλπ
32πf2πQs
3(sminusM2N )2λ4
γ
2sλγ
[(M2
N minus s)(Q2 + 2sβ2γ)(Q2 + 2sβγβπ)
minus 4s
((M2
N minus s)(Q2 minus 2s(απ minus 1)βγ
)+Q2sβγβπ
)λ2γ + 8s3(minus1 + απ)λ4
γ
]minus M2
N minus sλπ
[(Q2 + 2sβ2
γ)(Q2 + 2sβγβπ)2 + 4s2(2(απ minus 1)βγ(Q2 + 2sβγβπ)
minusQ2λ2π
)λ2γ + 8s3(απ minus 1)2λ4
γ
]arctan
(2sλπλγ
Q2 + 2sβγβπ
) (B20)
σ(π0p)LT =
e2g2AM
3Nλπ
16πf2πQs(sminusM2
N )2λγ
1
minus2s(M2N + s(minus1 + 2βγβπ))2λ2
γ + 8s3λ2πλ
4γ
[minus 3M8
N (Q2
+ 2sβ2γ) + 2M4
Ns2
(minus (Q2 + 2sβ2
γ)(minus3 + 2βγβπ) +(Q2(minus3 + 2βγβπ + 6λ2
π) + 2sβγ
times (minus3 + 12απ + 2βγβπ minus 12απβγβπ + 4βγλ2π))λ2γ minus 4s(α2
π minus 1)λ4π
)+ 2M2
Ns3
[minus (Q2
+ 2sβ2γ)(minus1 + 2βγβπ)
(6 + βγβπ(minus9 + 2βγβπ)
)+
(Q2(3minus 12λ2
π + 4βγβπ(minus1 + 2βγβπ
+ λ2π))minus 2sβγ
(minus 3 + 4απ(3 + 2απ(3 + 2βγβπ(minus3 + βγβπ)) + 4βγ(βπ + (2minus βγβπ)
times λ2π)))λ2γ + 8s
((απ minus 1)(1 + απ minus 2βγβπ) + 2απβγλ
2π
)λ4γ
)+ s4
((Q2 + 2sβ2
γ)
times (1minus 2βγβπ)2(2βγβπ minus 3) + 2(Q2(minus1 + 6λ2
π + 2βγβπ((1minus 2βγβπ)2 minus 2λ2π)) + 2sβγ
times (minus1 + 2βγβπ + 4(minus1 + βγβπ)(minusαπ + 2απβγβπ minus βγλ2π)))λ2γ minus 8
(s(απ minus 1)(απ
+ (1minus 2βγβπ)2) + 2βγ(2sαπ +Q2βγ)λ2π
)λ4γ + 32s(απ minus 1)λ2
πλ6γ
)+ 2M6
Ns
(Q2(6
minus 7βγβπ + λ2γ) + 2sβγ(βγ(6minus 7βγβπ) + (1minus 4απ)λ2
γ)
)]+sminusM2
N
4s2λπλ3γ
[(Q2 + 2sβ2
γ)
times(3M2
N + s(2βγβπ minus 3))(M2N + s(2βγβπ minus 1)
)+ 2s
(minusM2
N (Q2 + 2sβγ minus 8sαπβγ)
+ s(2sβγ(1minus 4απ + 4απβγβπ) +Q2(1minus 2λ2π)))λ2γ + 8s3(απ minus 1)λ4
γ
]
times arctan
(2sλπλγ
M2N + s(2βγβπ minus 1)
) (B21)
35
σ(πminusp)LT =
e2g2AM
3N
64πf2πQs
3λ4γ(M2
N minusQ2 minus s)(M2N + s(2βπβγ + 2λπλγ minus 1))
times
1
M2N + s(2βπβγ minus 2λπλγ minus 1)
[4λπsλγ(M2
N minusQ2 minus s)(
2sλ2γ
(s((1minus 2λ2
π)Q2
+ 2(2απ minus 1)sβγ(2βπβγ minus 1))minusM2N (Q2 + 2(1minus 2απ)sβγ)
)+ (Q2 + 2sβ2
γ)(M2N + s(2βπβγ minus 1)
)2 + 8(απ minus 1)απs
3λ4γ
)]minus(M2N + s(2βπβγ + 2λπλγ minus 1)
)times[
log
(minusM2N minus 2βπsβγ minus 2λπsλγ + s
minusM2N minus 2βπsβγ + 2λπsλγ + s
)(minus (Q2 + 2sβ2
γ)(M2N + s(2βπβγ minus 1)
)(minusM2
N + 2Q2 + 2βπsβγ + s)minus 2sλ2γ
(minusM2
N (Q2 + 2sβγ) +Q4 +Q2s(4απβγ
minus 2βγ minus 2λ2π + 1) + 2s2βγ(4(απ minus 1)βπβγ + 1)
)minus 8(απ minus 1)2s3λ4
γ
)+ 2
(4s2λ2
γ
(2(απ minus 1)βγ(Q2 + 2βπsβγ)minus λ2
πQ2)
+ (Q2 + 2βπsβγ)2(Q2 + 2sβ2γ)
+ 8(απ minus 1)2s3λ4γ
)arctanh
(2λπsλγ
Q2 + 2βπsβγ
)] (B22)
2 ∆-production channel
The tree-level ∆-exchange diagram in Fig 2 of Ref [30] contributes to the non-Born part of
the VVCS amplitudes The contribution of the ∆ exchange to the VVCS amplitudes can be split
into [17]
S∆-exch1 (νQ2) = S∆-pole
1 (νQ2) + S∆-exch1 (νQ2) (B23a)
νS∆-exch2 (νQ2) = νS∆-pole
2 (νQ2) + νS2∆-exch
(νQ2) (B23b)
and similarly for the unpolarized VVCS amplitudes discussed in Ref [30] Here we introduced the
∆-pole contributions S∆-polei and the ∆-non-pole contributions S∆-exch
i The former amplitudes
feature a pole at the ∆(1232)-production threshold and thus are proportional to
1
[sminusM2∆][uminusM2
∆]=
1
4M2N
1
ν2∆ minus ν2
(B24)
36
They can be reconstructed from the dispersion relations in Eq (5) with the tree-level ∆-production
cross sections as input cf Fig 10 in Ref [30]
σTT (νQ2) =π2α
M2NM
2+|~q |
minus g2
MMN (M+ + ν)|~q |2 +g2E(∆minus ν)(Q2 minusMNν)2
MN(B25a)
+g2CQ
4s(∆minus ν)
MNM2∆
minus 4gMgE(Q2 minusMNν)|~q |2 minus 4gMgCQ2|~q |2
+2gEgCQ
2[minusMNM∆ |~q |2 + s(Q2 +∆ν)
]MNM∆
δ(ν minus ν∆)
σLT (νQ2) =Qπ2α
M2NM
2+|~q |
g2E(MNν minusQ2) [M∆(MN + ν)minus s]
MN(B25b)
+g2CQ
2[MNM∆|~q |2 minus s(Q2 +∆ν)
]MNM2
∆
+ gMgEM∆|~q |2 minusgMgC(Q2 minusMNν)|~q |2
M∆
+gEgC(ν minus∆)(M2
N |~q |2 minus 2Q2s)
MNM∆
δ(ν minus ν∆)
with ∆ = M∆ minusMN M+ = M∆ + MN and the Mandelstam variable s = M2N + 2MNν minus Q2
Analytical expressions for the spin structure functions g1(xQ2) and g2(xQ2) can be constructed
from Eq (3) with the flux factor K(νQ2) = |~q | =radicν2 +Q2
In the ∆-non-pole contributions to S1(νQ2) and νS2(νQ2) the pole in ν at the ∆(1232)-
production threshold has canceled out
S∆-exch1 (νQ2) =
πα
MNM2+
[g2MQ
2+ + g2
E
(∆2 minus 3Q2
)+
4g2CQ
4
M2∆
minus 8gMgEM∆ωminus (B26a)
minus2gMgCQ2(MN minus 4M∆)
M∆+
2gEgCQ2(3MN minus 2M∆)
M∆
]
νS2∆-exch
(νQ2) =2πα
MNM2+
[g2EM∆∆ωminus +
g2M MNQ
2+
2+g2C Q
2(Q2 minus∆2)
2M∆(B26b)
+gEgM M∆(M∆ω+ minus 4MNωminus)minus gEgC ∆(2Q2 +MNω+)
+gMgC Q2(4MN minus ω+)
]+S∆-exch
2 (νQ2)
ν
[M2
∆ ω2+
M2N
+ ν2
]
with Q+ =radic
(M∆ +MN )2 +Q2 and ωplusmn = (M2∆minusM2
N plusmnQ2)2M∆ and the non-pole contribution
to S2(νQ2)
S∆-exch2 (νQ2) = minus2παMNν
M∆M2+
[gM + gE
]gC (B27)
These amplitudes to the contrary are not described by the tree-level ∆-production cross sections
in the standard dispersive approach [17] This peculiarity has been previously missed e g in the
calculation of the ∆-exchange contribution to the hydrogen hyperfine splitting in Ref [88] The
importance of including the ∆-non-pole contribution is also evident when considering the BC sum
37
rule in Eq (14) The ∆-pole terms by themselves violate the BC sum rule but cancel exactly with
the ∆-non-pole terms
limνrarr0
νS∆-pole2 (νQ2) + lim
νrarr0νS2
∆-exch(νQ2) = 0 (B28)
Appendix C Polarizabilities at Q2 = 0
In this section we give analytical expressions for the static values and slopes at Q2 = 0 of
the spin polarizabilities and moments of polarized structure functions In particular we give the
HB expansion of the πN -loop contributions and the ∆-exchange contributions The complete
expressions also for the π∆-loop contributions can be found in the Supplemented material Recall
that IA(0) = I1(0) = d2(0) = 0 and dd2(Q2)dQ2
∣∣∣Q2=0
= 0
1 πN-loop contribution
Here we give analytical expressions for the πN -loop contributions to the proton and neutron
spin polarizabilities expanded in powers of micro = mπMN viz the HB expansion Note that we
choose to expand here to a high order in micro the strict HB expansion would only retain the leading
term in an analogous NLO calculation
bull Static polarizabilities (Q2 = 0)
γ0p =e2g2
A
96π3f2πm
2π
1minus 21πmicro
8minus(
59
2+ 26 logmicro
)micro2 +
1875πmicro3
64
+3
(3
2+ 26 logmicro
)micro4 +
(C1)
γ0n =e2g2
A
48π3f2πm
2π
1
2minus 9πmicro
16minus 2micro2 logmicro+
75πmicro3
128minus 3micro4
4+
(C2)
δLTp =e2g2
A
192π3f2πm
2π
1minus 9πmicro
8+
(13
2minus 2 logmicro
)micro2 minus 465πmicro3
64
minus(
47
2+ 42 logmicro
)micro4 +
(C3)
δLTn =e2g2
A
96π3f2πm
2π
1
2+
3πmicro
16+ (1 + 2 log micro)micro2 minus 105πmicro3
128+
5micro4
4+
(C4)
38
δLTn =e2g2
A
96π3f2πm
2π
1
2+
3πmicro
16+ (1 + 2 log micro)micro2 minus 105πmicro3
128+
5micro4
4+
(C5)
γ0p =e2g2
A
16π3f2πm
4π
4
45minus 3πmicro
16+
14micro2
5minus 1813πmicro3
384minus 192
5(1 + log micro)micro4
+80703πmicro5
2048+
(C6)
γ0n =e2g2
A
16π3f2πm
4π
4
45minus 5πmicro
48+
4micro2
5minus 245πmicro3
384minus 32micro4 logmicro
15
+1323πmicro5
2048+
(C7)
bull Slopes of polarizabilities at Q2 = 0
dγ0p(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
2minus 45πmicro
4+ 223micro2 minus 28515πmicro3
64
minus9
(1953
4+ 449 logmicro
)micro4 +
570255πmicro5
128+
(C8)
dγ0n(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
2minus 81πmicro
8+ 94micro2 minus 2535πmicro3
32
minus3 (1 + 90 logmicro)micro4 +84315πmicro5
1024+
(C9)
dδLTp(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
2880π3f2πm
4π
minus5
2minus 27πmicro
32+ 20micro2 minus 5865πmicro3
256
+3
(617
4+ 36 logmicro
)micro4 minus 2056845πmicro5
4096+
(C10)
dδLTn(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
minus5
4minus 81πmicro
64minus 11micro2 +
10005πmicro3
512
+15
8(11 + 48 logmicro)micro4 minus 267015πmicro5
8192+
(C11)
dIAp(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
96π2f2πmicro
2
1minus 15πmicro
4minus 1
2(115 + 88 logmicro)micro2 +
1839πmicro3
32
+5 (5 + 34 logmicro)micro4 +
(C12)
dIAn(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
48π2f2πmicro
2
1
2minus 11πmicro
8minus 1
4(1 + 20 logmicro)micro2 +
99πmicro3
64
minus25micro4
12+
(C13)
dI1p(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
96π2f2πmicro
3π
8+ 2 (4 + 3 logmicro)microminus 537πmicro2
64
minus1
2(15 + 56 logmicro)micro3 +
(C14)
39
dI1n(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
48π2f2πmicro
minus π
16+
1
4(3 + 4 logmicro)microminus 57πmicro2
128+
2micro3
3+
(C15)
γ0p(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2g2
A
16π3f2πm
6π
1
105minus 23πmicro
256+
377micro2
210minus 15551πmicro3
6144+
3371micro4
105
minus1640457πmicro5
32768+
(C16)
γ0n(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2g2
A
16π3f2πm
6π
1
105minus 153πmicro
1792+
69micro2
70minus 4615πmicro3
6144+
172micro4
35
minus120897πmicro5
32768+
(C17)
2 ∆-exchange contribution
Here we give analytical expressions for the tree-level ∆-exchange contributions to the nucleon
spin polarizabilities and their slopes at Q2 = 0 Note that the ∆-exchange contributes equally to
proton and neutron polarizabilities Recall that for the magnetic γlowastN∆ coupling we introduced a
dipole form factor to mimic vector-meson dominance gM rarr gM(1 +Q2Λ2)2
bull Static polarizabilities (Q2 = 0)
γ0 = minus e2
4πM2+
(g2M
∆2+
g2E
M2+
minus 4gMgEM+∆
) (C18)
δLT =e2M∆
4πM3+
(g2E
MNM++gMgE∆MN
minus gEgCM2
∆
) (C19)
γ0 =e2M2
N
π∆2M4+
(minusg
2M
∆2+
g2E
M2+
+4gMgE∆M+
) (C20)
bull Slopes of polarizabilities at Q2 = 0
dγ0(Q2)
dQ2
∣∣∣∣∣Q2=0
= minus e2
πM2+∆
(g2M
∆
[1
4∆2minus 1
∆M++
1
2M2+
]minus 1
Λ2
g2M
∆+
g2E
2M2+
[1
2∆minus 3
M+
]minusgMgE
M+
[1
∆2minus 5
∆M++
1
M2+
]+
1
Λ2
2gMgEM+
+2gMgC∆M2
+
minus gEgCM3
+
) (C21)
40
d δLT (Q2)
dQ2
∣∣∣∣∣Q2=0
=e2M∆∆
4πMNM2+
(g2E
∆2M2+
[1
∆minus 4
M+
]minus g2
C
∆M2∆M
2+
+gMgE∆2M+
[1
∆2
minus 3
∆M++
1
M2+
]minus 2
Λ2
gMgE∆2M+
+gMgC∆M2
∆
[1
2∆2minus 2
∆M++
1
2M2+
]minus gEgC
2M2∆M
2+
[7
∆+
1
M+
]) (C22)
dIA(Q2)
dQ2
∣∣∣∣∣Q2=0
= minusM2N
M2+
(g2M
2∆2+
g2E
MNM+minus 2gMgE
∆M+minus gEgCM∆M+
) (C23)
dI1(Q2)
dQ2
∣∣∣∣∣Q2=0
= minusM∆M2N
2M3+
(g2E
MNM∆minus gMgE∆MN
minus gEgCM2
∆
) (C24)
γ0(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2M2
N
π∆3M6+
(g2M
∆
[3M2
∆ + 2M∆MN minus 9M2N
∆2+
4M2+
Λ2
]
minus4gMgE
[5M2
∆ minus 9M2N
M+∆2+
2M+
Λ2
]+
g2E
∆M+[7M∆ minus 9MN ]
minus8gMgC∆
+4gEgCM+
) (C25)
[1] D Drechsel B Pasquini and M Vanderhaeghen Dispersion relations in real and virtual
Compton scattering Phys Rept 378 99 (2003) hep-ph0212124
[2] S E Kuhn J-P Chen and E Leader Spin structure of the nucleon mdash status and recent
results Prog Part Nucl Phys 63 1 (2009) arXiv08123535 [hep-ph]
[3] F Hagelstein R Miskimen and V Pascalutsa Nucleon polarizabilities from Compton scat-
tering to hydrogen atom Prog Part Nucl Phys 88 29 (2016) arXiv151203765 [nucl-th]
[4] B Pasquini and M Vanderhaeghen Dispersion theory in electromagnetic interactions Ann
Rev Nucl Part Sci 68 75 (2018) arXiv180510482 [hep-ph]
[5] S Gerasimov A Sum rule for magnetic moments and the damping of the nucleon magnetic
moment in nuclei Sov J Nucl Phys 2 430 (1966)
[6] S Drell and A C Hearn Exact sum rule for nucleon magnetic moments Phys Rev Lett
16 908 (1966)
[7] J S Schwinger Source theory viewpoints in deep inelastic scattering Proc Natl Acad Sci
USA 72 1 (1975)
[8] J S Schwinger Source theory viewpoints in deep inelastic scattering Electromagnetic In-
teractions and Field Theory Proceedings 14 Internationale Universitatswochen Schladming
Austria February 24-March 7 1975 Acta Phys Austriaca Suppl 14 471 (1975)
41
[9] J Schwinger Source theory discussion of deep inelastic scattering with polarized particles
Proc Natl Acad Sci USA 72 1559 (1975)
[10] A M Baldin Polarizability of nucleons Nucl Phys 18 310 (1960)
[11] M Gell-Mann M L Goldberger and W E Thirring Use of causality conditions in quantum
theory Phys Rev 95 1612 (1954)
[12] A Pineda Leading chiral logarithms to the hyperfine splitting of the hydrogen and muonic
hydrogen Phys Rev C 67 025201 (2003)
[13] C Peset and A Pineda The two-photon exchange contribution to muonic hydrogen from
chiral perturbation theory Nucl Phys B 887 69 (2014) arXiv14064524 [hep-ph]
[14] C Peset and A Pineda Model-independent determination of the two-photon exchange contri-
bution to hyperfine splitting in muonic hydrogen JHEP 04 060 arXiv161205206 [nucl-th]
[15] F Hagelstein and V Pascalutsa Proton structure in the hyperfine splitting of muonic hydro-
gen PoS CD15 077 (2016) arXiv151104301 [nucl-th]
[16] F Hagelstein Exciting Nucleons in Compton Scattering and Hydrogen-Like Atoms PhD
thesis Mainz U Inst Kernphys (2017) arXiv171000874 [nucl-th]
[17] F Hagelstein ∆(1232)-Resonance in the hydrogen spectrum Proceedings 11th International
Workshop on the Physics of Excited Nucleons (NSTAR 2017) Columbia SC USA August
20-23 2017 Few Body Syst 59 93 (2018) arXiv180109790 [nucl-th]
[18] Y Prok et al (CLAS) Moments of the spin structure functions gp1 and gd1 for 005 lt Q2 lt 30
GeV2 Phys Lett B 672 12 (2009) arXiv08022232 [nucl-ex]
[19] H Dutz et al (GDH) First measurement of the Gerasimov-Drell-Hearn sum rule for 1H from
07 GeV to 18 GeV at ELSA Phys Rev Lett 91 192001 (2003)
[20] M Amarian et al (Jefferson Lab E94010) Measurement of the generalized forward spin po-
larizabilities of the neutron Phys Rev Lett 93 152301 (2004) arXivnucl-ex0406005
[21] M Amarian et al The Q2 evolution of the generalized Gerasimov-Drell-Hearn integral for
the neutron using a 3He target Phys Rev Lett 89 242301 (2002) arXivnucl-ex0205020
[22] M Amarian et al (Jefferson Lab E94-010)Q2 evolution of the neutron spin structure moments
using a 3He target Phys Rev Lett 92 022301 (2004) arXivhep-ex0310003
[23] A Deur et al Experimental determination of the evolution of the Bjorken integral at low Q2
Phys Rev Lett 93 212001 (2004) arXivhep-ex0407007
[24] K Slifer Low Q2 measurement of gp2 and the δLT spin polarizability Spin structure at long dis-
tance Proceedings Workshop Newport News USA March 12ndash13 2009 nucl-ex09064775
(2009) AIP Conf Proc 1155 10106313203293 (2009) arXiv09064775 [nucl-ex]
[25] P Solvignon et al (E01-012) Moments of the neutron g2 structure function at intermediate
42
Q2 Phys Rev C 92 015208 (2015) arXiv13044497 [nucl-ex]
[26] A Deur Experimental studies at low Q2 of the spin structure of the nucleon at Jefferson Lab
in 9th International Workshop on Chiral Dynamics (CD18) Durham NC USA September
17-21 2018 (2019) arXiv190305661 [nucl-ex]
[27] V Sulkosky et al (Jefferson Lab E97-110) Measurement of the 3He spin-structure functions
and of neutron (3He) spin-dependent sum rules at 0035 le Q2 le 024 GeV2 Phys Lett B
805 135428 (2020) arXiv190805709 [nucl-ex]
[28] K Adhikari et al (CLAS) Measurement of the Q2 Dependence of the Deuteron Spin Structure
Function g1 and its Moments at Low Q2 with CLAS Phys Rev Lett 120 062501 (2018)
arXiv171101974 [nucl-ex]
[29] V Lensky F Hagelstein A Hiller Blin and V Pascalutsa Comment on rdquoMeasurement of
the Q2 Dependence of the Deuteron Spin Structure Function g1 and its Moments at Low Q2
with CLASrdquo (2018) arXiv180603219 [nucl-th]
[30] J M Alarcn F Hagelstein V Lensky and V Pascalutsa Forward doubly-virtual Compton
scattering off the nucleon in chiral perturbation theory at NLO the subtraction function and
moments of unpolarized structure functions (2020) arXiv200509518 [hep-ph]
[31] V Bernard E Epelbaum H Krebs and U-G Meiszligner New insights into the spin structure
of the nucleon Phys Rev D 87 054032 (2013) arXiv12092523 [hep-ph]
[32] V Lensky J M Alarcon and V Pascalutsa Moments of nucleon structure functions at
next-to-leading order in baryon chiral perturbation theory Phys Rev C 90 055202 (2014)
arXiv14072574 [hep-ph]
[33] V Pascalutsa and M Vanderhaeghen Chiral effective-field theory in the ∆(1232) region
I Pion electroproduction on the nucleon Phys Rev D 73 034003 (2006) arXivhep-
ph0512244
[34] V Pascalutsa and M Vanderhaeghen Electromagnetic nucleon-to-Delta transition in chiral
effective field theory Phys Rev Lett 95 232001 (2005) arXivhep-ph0508060
[35] T R Hemmert B R Holstein and J Kambor Systematic 1M expansion for spin 32
particles in baryon chiral perturbation theory Phys Lett B 395 89 (1997) arXivhep-
ph9606456
[36] V Pascalutsa and D R Phillips Effective theory of the ∆(1232) in Compton scattering off
the nucleon Phys Rev C 67 055202 (2003) arXivnucl-th0212024
[37] V Pascalutsa M Vanderhaeghen and S N Yang Electromagnetic excitation of the ∆(1232)-
resonance Phys Rept 437 125 (2007) arXivhep-ph0609004
[38] K A Olive et al (Particle Data Group) Review of Particle Physics Chin Phys C 38 090001
43
(2014)
[39] F E Low Scattering of light of very low frequency by systems of spin 12 Phys Rev 96
1428 (1954)
[40] M Gell-Mann and M L Goldberger Scattering of low-energy photons by particles of spin
12 Phys Rev 96 1433 (1954)
[41] A M Harun ar-Rashid A simple derivation of Schwingerrsquos sum rule for spin dependent
structure functions Nuovo Cim A 33 447 (1976)
[42] F Hagelstein and V Pascalutsa Dissecting the hadronic contributions to (g minus 2)micro by
Schwingerrsquos sum rule Phys Rev Lett 120 072002 (2018) arXiv171004571 [hep-ph]
[43] V Lensky V Pascalutsa M Vanderhaeghen and C Kao Spin-dependent sum rules con-
necting real and virtual Compton scattering verified Phys Rev D 95 074001 (2017)
arXiv170101947 [hep-ph]
[44] H Burkhardt and W N Cottingham Sum rules for forward virtual Compton scattering
Annals Phys 56 453 (1970)
[45] V Lensky and V Pascalutsa Predictive powers of chiral perturbation theory in Compton
scattering off protons Eur Phys J C 65 195 (2010) arXiv09070451 [hep-ph]
[46] H Grieszlighammer J McGovern D Phillips and G Feldman Using effective field theory to
analyse low-energy Compton scattering data from protons and light nuclei Prog Part Nucl
Phys 67 841 (2012) arXiv12036834 [nucl-th]
[47] H W Grieszlighammer J A McGovern and D R Phillips Nucleon polarisabilities at and
beyond physical pion masses Eur Phys J A 52 139 (2016) arXiv151101952 [nucl-th]
[48] E Epelbaum H Krebs and U-G Meiszligner Improved chiral nucleon-nucleon potential up to
next-to-next-to-next-to-leading order Eur Phys J A 51 53 (2015) arXiv14120142 [nucl-
th]
[49] E Epelbaum H Krebs and U-G Meiszligner Precision nucleon-nucleon potential at fifth order
in the chiral expansion Phys Rev Lett 115 122301 (2015) arXiv14124623 [nucl-th]
[50] C W Kao T Spitzenberg and M Vanderhaeghen Burkhardt-Cottingham sum rule and
forward spin polarizabilities in heavy baryon chiral perturbation theory Phys Rev D 67
016001 (2003) arXivhep-ph0209241
[51] C-W Kao D Drechsel S Kamalov and M Vanderhaeghen Higher moments of nucleon
spin structure functions in heavy baryon chiral perturbation theory and in a resonance model
Phys Rev D 69 056004 (2004) arXivhep-ph0312102
[52] T Becher and H Leutwyler Baryon chiral perturbation theory in manifestly Lorentz invariant
form Eur Phys J C 9 643 (1999) arXivhep-ph9901384
44
[53] J Gasser M E Sainio and A Svarc Nucleons with chiral loops Nucl Phys B 307 779
(1988)
[54] L S Geng J Martin Camalich L Alvarez-Ruso and M J Vicente Vacas Leading SU(3)-
breaking corrections to the baryon magnetic moments in chiral perturbation theory Phys
Rev Lett 101 222002 (2008) arXiv08051419 [hep-ph]
[55] D Drechsel S S Kamalov and L Tiator The GDH sum rule and related integrals Phys
Rev D 63 114010 (2001) arXivhep-ph0008306
[56] D Drechsel O Hanstein S S Kamalov and L Tiator A unitary isobar model for pion
photo- and electroproduction on the proton up to 1 GeV Nucl Phys A 645 145 (1999)
arXivnucl-th9807001
[57] L Tiator private communication (2020)
[58] V Bernard T R Hemmert and U-G Meiszligner Spin structure of the nucleon at low energies
Phys Rev D 67 076008 (2003) arXivhep-ph0212033
[59] O Gryniuk F Hagelstein and V Pascalutsa Evaluation of the forward Compton scattering
off protons II Spin-dependent amplitude and observables Phys Rev D 94 034043 (2016)
arXiv160400789 [nucl-th]
[60] R Zielinski The g2p Experiment A Measurement of the Protonrsquos Spin Structure Functions
PhD thesis New Hampshire U (2010) arXiv170808297 [nucl-ex]
[61] N Guler et al (CLAS) Precise determination of the deuteron spin structure at low to mod-
erate Q2 with CLAS and extraction of the neutron contribution Phys Rev C 92 055201
(2015) arXiv150507877 [nucl-ex]
[62] C E Carlson V Nazaryan and K Griffioen Proton structure corrections to electronic
and muonic hydrogen hyperfine splitting Phys Rev A 78 022517 (2008) arXiv08052603
[physicsatom-ph]
[63] V Bernard N Kaiser and U-G Meiszligner Chiral dynamics in nucleons and nuclei Int J
Mod Phys E 4 193 (1995) arXivhep-ph9501384
[64] T R Hemmert B R Holstein and J Kambor ∆(1232) and the polarizabilities of the nucleon
Phys Rev D 55 5598 (1997) arXivhep-ph9612374
[65] V Pascalutsa and R Timmermans Field theory of nucleon to higher-spin baryon transitions
Phys Rev C 60 042201 (1999) arXivnucl-th9905065
[66] V Pascalutsa Quantization of an interacting spin-32 field and the Delta isobar Phys Rev
D 58 096002 (1998) arXivhep-ph9802288
[67] H Krebs Double Virtual Compton Scattering and SpinStructure of the Nucleon PoS
CD2018 031 (2019)
45
[68] N Kochelev and Y Oh Axial anomaly and the δLT puzzle Phys Rev D 016012 (2012)
arXiv11034892 [hep-ph]
[69] D Drechsel S Kamalov and L Tiator Unitary isobar model ndash MAID2007 Eur Phys J A
69 (2007) available at httpsmaidkphuni-mainzde arXiv07100306 [nucl-th]
[70] P J Mohr B N Taylor and D B Newell CODATA recommended values of the fundamental
physical constants 2010 Rev Mod Phys 84 1527 (2012)
[71] J Ahrens et al (GDH A2) First measurement of the Gerasimov-Drell-Hearn integral for 1H
from 200 to 800 MeV Phys Rev Lett 87 022003 (2001) arXivhep-ex0105089 [hep-ex]
[72] K Helbing (GDH) Experimental verification of the GDH sum rule at ELSA and MAMI Nucl
Phys Proc Suppl 105 113 (2002)
[73] R Bradford A Bodek H S Budd and J Arrington A New parameterization of the nucleon
elastic form-factors NuInt05 proceedings of the 4th International Workshop on Neutrino-
Nucleus Interactions in the Few-GeV Region Okayama Japan 26-29 September 2005 Nucl
Phys Proc Suppl 159 127 (2006) arXivhep-ex0602017
[74] J D Bjorken Applications of the chiral U(6)timesU(6) algebra of current densities Phys Rev
148 1467 (1966)
[75] J D Bjorken Inelastic scattering of polarized leptons from polarized nucleons Phys Rev D
1 1376 (1970)
[76] V Bernard T R Hemmert and U-G Meiszligner Novel analysis of chiral loop effects in
the generalized Gerasimov-Drell-Hearn sum rule Phys Lett B 545 105 (2002) arXivhep-
ph0203167
[77] A Deur et al Experimental study of isovector spin sum rules Phys Rev D 78 032001
(2008) arXiv08023198 [nucl-ex]
[78] B Pasquini P Pedroni and D Drechsel Higher order forward spin polarizability Phys Lett
B 687 160 (2010) arXiv10014230 [hep-ph]
[79] R Jaffe g2ndashThe nucleonrsquos other spin-dependent structure function Comments Nucl Part
Phys 19 239 (1990)
[80] E V Shuryak and A Vainshtein Theory of power corrections to deep inelastic scattering in
quantum chromodynamics (II) Qminus4 effects polarized target Nucl Phys B 201 141 (1982)
[81] S Wandzura and F Wilczek Sum rules for spin dependent electroproduction Test of rela-
tivistic constituent quarks Phys Lett B 72 195 (1977)
[82] B W Filippone and X-D Ji The spin structure of the nucleon Adv Nucl Phys 26 1
(2001) arXivhep-ph0101224
[83] M Burkardt The g2 structure function Proceedings Workshop on Spin structure at long
46
distance Newport News USA March 12-13 2009 AIP Conf Proc 1155 26 (2009)
arXiv09054079 [hep-ph]
[84] A Deur S J Brodsky and G F de Teramond The spin structure of the nucleon Rept
Prog Phys 82 076201 (2019) arXiv180705250 [hep-ph]
[85] B R Holstein D Drechsel B Pasquini and M Vanderhaeghen Higher order polarizabilities
of the proton Phys Rev C 61 034316 (2000) arXivhep-ph9910427
[86] B R Holstein V Pascalutsa and M Vanderhaeghen Sum rules for magnetic moments and
polarizabilities in QED and chiral effective-field theory Phys Rev D 72 094014 (2005)
arXivhep-ph0507016
[87] B Pasquini D Drechsel and L Tiator Invariant amplitudes for pion electroproduction Eur
Phys J A 34 387 (2007) arXiv07122327 [hep-ph]
[88] A J Buchmann Non-spherical proton shape and hydrogen hyperfine splitting Proceedings
International Workshop on Precision Physics of Simple Atomic Systems (PSAS 2008) Wind-
sor Ontario Canada July 21-26 2008 Can J Phys 87 773 (2009) arXiv09104747
[physicsatom-ph]
47
000 005 010 015 020 025 030
0000
0001
0002
0003
0004
0005
0006
0007
Q2 GeV2D
d2
proton
000 005 010 015 020 025 030
0000
0002
0004
0006
0008
0010
0012
Q2 GeV2D
d2
neutron
000 005 010 015 020 025 030
-1
0
1
2
Q2 GeV2D
Γ 010
-4
fm6
D
proton
000 005 010 015 020 025 030-2
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
6D
neutron
FIG 8 Upper panel The inelastic moment d2(Q2) for the proton (left) and neutron (right) as
function of Q2 The result of this work the NLO BχPT prediction is shown by the blue solid line
and the blue band The red line represents the LO BχPT result The purple short-dashed line is
the O(p4) HB result from Ref [50 51] The black dotted line is the MAID model prediction [69]
The experimental points for the neutron (cyan dots) are from Ref [22] Lower panel Fifth-order
generalized forward spin polarizability γ0(Q2) for the proton (left) and neutron (right) as function
of Q2 The experimental points for the proton are from Ref [59] (purple square) and Ref [78]
(orange dot)
This relation however only holds for asymptotically large Q2 It is also in the high-Q2
region where d2(Q2) is a measure of color polarizability [82 83] through its relation to the
gluon field strength tensor [80] We refer to Ref [84] for a recent review on the spin structure
of the nucleon including a discussion of sum rules for deep inelastic scattering and color
polarizabilities
What we consider in the following is the inelastic part of d2(Q2) defined as the moment
of g1(xQ2) and g2(xQ2) spin structure functions cf Eq (38)
d2(Q2) =
int x0
0
dx x2 [3g2(xQ2) + 2g1(xQ2)] (39)
23
This moment provides another testing ground for our BχPT predictions through comparison
with experiments on the neutron [22] Going towards the low-Q2 region the interpretation
of d2(Q2) in terms of color polarizabilities will fade out The above definition however
implies it is related to other VVCS polarizabilities
d2(Q2) =Q4
8M4N
[M2
NQ2
αδLT (Q2) + I1(Q2)minus IA(Q2)
] (40)
Note that d2(Q2) and its first two derivatives with respect to Q2 vanish at Q2 = 0 The
considerations in Eqs (28) and (29) have no effect on d2(Q2) since the Born contribution
from IA(Q2) and I1(Q2) cancel out Therefore d2(Q2) is a pure polarizability
In Fig 8 upper panel we show our NLO BχPT prediction and other results for d2(Q2)
While MAID [69] and BχPT describe the experimental data for the neutron [22] very well
the HB limit [50 51] is showing a fast growth with Q2 This illustrates the importance
of keeping the relativistic result Note also that even though the πN -loop contribution
is dominant both gC and the form factor in gM are essential to obtain a curvature that
reproduces the data cf Fig 9 upper panel For the proton there are to our knowledge
no experimental results to compare with However the agreement between the NLO BχPT
prediction and the MAID prediction at low energies is reasonable
F γ0(Q2) mdash fifth-order generalized forward spin polarizability
It is interesting to compare the generalized fifth-order forward spin polarizability sum
rule
γ0(Q2) =1
2π2
int infinν0
dν
radic1 +
Q2
ν2
σTT (νQ2)
ν5(41)
=64αM4
N
Q10
int x0
0
dx x4
[g1(xQ2)minus 4M2
Nx2
Q2g2(xQ2)
]
to the sum rule integrals for IA(Q2) and γ0(Q2) since they differ merely by their energy
weighting of σTT (νQ2) and a constant prefactor cf Eqs (19) (26) and (41) From IA(Q2)
to γ0(Q2) to γ0(Q2) the energy suppression is increasing by a factor of νminus2 respectively
Therefore the description of γ0(Q2) should be easiest in a low-energy effective-field theory
such as χPT whereas γ0(Q2) and IA(Q2) receive larger contributions from higher energies
In Fig 8 lower panel we show our LO and NLO BχPT predictions for γ0(Q2) One
can see that the πN -loop contribution is positive (in accordance to what we see for the cross
section σTT see Fig 10) The Delta shifts it substantially especially in the low Q2 region
24
000 005 010 015 020 025 030
-0004
-0002
0000
0002
0004
0006
0008
0010
Q2GeV
2D
d2
proton
000 005 010 015 020 025 030
-0005
0000
0005
0010
0015
Q2GeV
2D
d2
neutron
000 005 010 015 020 025 030
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
6D
proton
000 005 010 015 020 025 030
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
6D
neutron
FIG 9 Contributions of the different orders to the chiral predictions of d2(Q2) upper panel and
γ0(Q2) lower panel for the proton (left) and neutron (right) Red solid line πN -loop contribu-
tion green dot-dashed line ∆-exchange contribution orange dotted line π∆-loop contribution
blue long-dashed line total result purple dot-dot-dashed line total result without gC contribution
black short-dashed line total result without gM dipole
bringing it into a better agreement with data In general the BχPT curves start above the
empirical data points at the real-photon point and then decrease asymptotically to zero
above Q2 gt 01 GeV2 On the other hand the MAID prediction reproduces the empirical
data at the real-photon point then decreases to negative values until about Q2 gt 006 GeV2
from where it also starts to asymptotically approach zero Consequently our NLO BχPT
prediction of γ0(Q2) is consistently above the MAID prediction This is very different to
what we saw for IA(Q2) in Fig 4 upper panel where the MAID prediction at the real-
photon point is above the experimental value While the agreement of our predictions with
the empirical data is in general quite good for all moments of σTT (νQ2) one should point
out that both for γ0n(Q2) and γ0p(Q2) we overestimate the data at low Q2 For IA(Q2) such
observation cannot be made because ∆IA(0) = 0 and thus IA(0) is given by the empirical
Pauli form factor only From IA(Q2) γ0(Q2) and γ0(Q2) the latter has the smallest however
25
non-negligible dependence on gC and the dipole in gM cf Fig 9 lower panel
The πN -loop ∆-exchange and π∆-loop contributions to the NLO BχPT prediction of
the static fifth-order forward spin polarizability amount to in units of 10minus4 fm6
γ0p = 112(30) asymp 208minus 096minus 001 (42a)
γ0n = 195(30) asymp 292minus 096minus 001 (42b)
while the slope is composed as follows in units of 10minus4 fm8
dγ0p(Q2)
dQ2
∣∣∣∣Q2=0
= minus084(10) asymp minus100 + 016 + 000 (43a)
dγ0n(Q2)
dQ2
∣∣∣∣Q2=0
= minus142(15) asymp minus158 + 016 + 000 (43b)
Note that the HB prediction of the static polarizability γ0p (423 at O(p3) and 365 at O(ε3)
[78 85]) is almost one order of magnitude larger than the empirical value and therefore not
shown in Fig 8
G Summary
Our results are summarized in Table II where we give the contributions of the different
orders to the chiral predictions of the polarizabilities and their slopes at the real-photon
point A quantitative comparison of our predictions for the static spin polarizabilities to
the work of Bernard et al [31] and different empirical evaluations is shown in Table III We
can see that the inclusion of the Delta turns out to be very important for all moments of
the helicity-difference cross section To describe the Q2 behavior of the polarizabilities the
magnetic coupling of the N rarr ∆ transition should be modified by a dipole form factor as
has been observed previously in the description of electroproduction data [33] This dipole
form factor effectively takes account of vector-meson exchanges The Coulomb-quadrupole
N rarr ∆ transition despite its subleading order is important in the description of some
moments of spin structure functions This is contrary to what we saw for the moments of
unpolarized structure functions [30] where the Coulomb coupling had a negligible effect
The π∆ loops are mainly relevant for the generalized GDH integrals
26
TABLE II The NLO BχPT predictions for the forward VVCS polarizabilities and their slopes at
Q2 = 0 The contributions of the πN loops the ∆ exchange and the π∆ loops are shown together
with the combined total result Note that IA(0) = I1(0) = d2(0) = 0 and (d2)prime = 0
πN loops ∆ exchange π∆ loops Total
γ0 p 201 minus284 minus010minus093(92)
(10minus4 fm4) n 298 003(92)
δLT p 150 minus016 minus002132(15)
(10minus4 fm4) n 235 218(23)
γ0 p 208 minus096 minus001112(30)
(10minus4 fm6) n 292 195(30)
(γ0)prime p minus033011 001
minus022(4)
(10minus4 fm6) n minus073 minus061(7)
(δLT )prime p minus080 minus004 minus001minus085(8)
(10minus4 fm6) n minus119 minus124(12)
(γ0)prime p minus100016 000
minus084(10)
(10minus4 fm8) n minus158 minus142(15)
(∆IA)prime p 238 minus1121 025minus858(343)
(GeVminus2) n 141 minus955(343)
(∆I1)prime p 034 minus053 058039(4)
(GeVminus2) n minus107 minus101(10)
TABLE III Our NLO BχPT predictions for the spin polarizabilities at Q2 = 0 compared with
the BχPT+∆ predictions from Bernard et al [31] and the available empirical information Where
the reference is not given the empirical number is provided by the MAID analysis [55 69] with
unspecified uncertainty
Proton Neutron
This work BχPT+∆ Empirical This work BχPT+∆ Empirical
γ0 minus093(92) minus174(40) minus100(8)(12) [19] 003(92) minus077(40) minus0005
(10minus4 fm4) minus090(8)(11) [78] [MAID]
minus0929(105) [59]
δLT 132(15) 240(1) 134 218(23) 238(3) 203
(10minus4 fm4) [MAID] [MAID]
γ0 112(30) 060(7)(7) [78] 195(30) 123
(10minus4 fm6) 0484(82) [59] [MAID]
IV CONCLUSIONS
We have presented a complete NLO calculation of the polarized non-Born VVCS ampli-
tudes in covariant BχPT with pion nucleon and ∆(1232) fields The dispersion relations
between the VVCS amplitudes and the tree-level photoabsorption cross sections served as a
27
cross-check of these calculations
The obtained moments of the proton and neutron spin structure functions related to gen-
eralized polarizabilities and GDH-type integrals agree well with the available experimental
data The description of their Q2 evolution is improved compared to the previous χPT pre-
dictions In particular the NLO BχPT predictions obtained here give a better description
of the empirical data (eg from the Jefferson Laboratory ldquoSpin Physics Programrdquo) than
the HB [50 51] and IR [58] calculations
The demonstrated predictive power of the χPT framework amplitudes makes it well
suited for extending the χPT evaluation of the TPE effect in the hyperfine structure of
(muonic-)hydrogen [15ndash17] to next-to-leading order
ACKNOWLEDGEMENTS
We thank Lothar Tiator and Marc Vanderhaeghen for helpful discussions This work
is supported by the Deutsche Forschungsgemeinschaft (DFG) through the Collaborative
Research Center [The Low-Energy Frontier of the Standard Model (SFB 1044)] JMA ac-
knowledges support from the Community of Madrid through the ldquoPrograma de atraccion de
talento investigador 2017 (Modalidad 1)rdquo and the Spanish MECD grants FPA2016-77313-P
FH gratefully acknowledges financial support from the Swiss National Science Foundation
Appendix A Tensor decompositions of the VVCS amplitudes
In this appendix we review the decomposition of the forward VVCS process into tensor struc-
tures and scalar amplitudes In particular we consider the connection between the covariant and
the semi-relativistic decomposition in the lab frame that is defined in terms of the conventional
transverse longitudinal transverse-transverse and transverse-longitudinal amplitudes
As explained in Sec II A the process of forward VVCS off the nucleon can be described in
terms of four explicitly covariant amplitudes S1 2 and T1 2 [3]
T (νQ2) =
(minusgmicroν +
qmicroqν
q2
)T1(νQ2) +
1
M2N
(pmicro minus p middot q
q2qmicro)(
pν minus p middot qq2
qν)T2(νQ2) (A1)
minus 1
MNγmicroναqα S1(νQ2)minus 1
M2N
(γmicroνq2 + qmicroγναqα minus qνγmicroαqα
)S2(νQ2)
εprimelowastmicro εν
where εmicro (εprimelowastmicro ) are the incoming (outgoing) photon polarization vectors ν is the photon lab-frame
energy and Q2 is the photon virtuality Alternatively the decomposition in the laboratory frame
28
(which in the forward case coincides with the Breit frame) is parametrized in terms of the nucleon
Pauli matrices ~σ and the four scalar functions fL fT gTT and gLT
T (νQ2) = ε0 εprimelowast0 fL(νQ2) + (~ε primelowast middot ~ε ) fT (νQ2) + i~σ middot (~ε primelowast times ~ε ) gTT (νQ2) (A2)
minusi~σ middot [(ε0~εprimelowast minus ~ε ε primelowast0 )times q] gLT (νQ2)
Here ~q and q = ~q|~q | are the photon three-momentum in the lab system and its unit vector The
modified polarization vector components are given by
ε0 =
[ε0 minus
ν
|~q | (~ε middot q )
] |~q |Q ~ε = ~εminus q (~ε middot q ) (A3)
where ε = (ε0~ε ) is the usual incoming photon polarization vector and εprimelowast the outgoing polarization
vector The LEX of the lab frame amplitudes [Eq (10)] can serve in particular as the definition
of the generalized polarizabilities The lab frame amplitudes are also conveniently used for the
definition of the response functions see the example of the scalar amplitude gLT (νQ2) and the
corresponding response function σLT (νQ2) below in App B
Appendix B Photoabsorption cross sections
In the forward kinematics the spin-dependent VVCS amplitudes and the spin polarizabilities
can be described in terms of the polarized structure functions g1(xQ2) and g2(xQ2) or equiv-
alently the helicity-difference cross section σTT (νQ2) and the longitudinal-transverse response
function σLT (νQ2) with the help of dispersion relations (5) and the optical theorem (3) In this
way the photoabsorption cross sections measured in electroproduction processes form the basis
for most empirical evaluations shown throughout Sec III In the following we present the BχPT
predictions for the tree-level cross sections of πN - π∆- and ∆-production through photoabsorption
on the nucleon cf Figs 8 9 and 10 in Ref [30] In Secs B 1 and B 2 we will discuss the leading
πN -production channel and the ∆-production channel respectively We used these cross sections
to verify the polarizability predictions obtained otherwise from the calculated non-Born VVCS
amplitudes Due to the bad high-energy behavior of the π∆-production cross sections in BχPT
cf Fig 10 the dispersion relations in Eq (5) require further subtractions for a reconstruction of
the π∆-loop contribution to the spin-dependent VVCS amplitudes Therefore not all polarizabil-
ities could be verified but only those appearing as higher-order terms in the LEX of the VVCS
amplitudes such as γ0 [16]
29
proton neutron
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Neutron
02 04 06 08 100
50
100
150
200
250
[GeV]
T [b]
02 04 06 08 10
-50
0
50
100
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Neutron
02 04 06 08 100
50
100
150
200
250
[GeV]
T [b]
02 04 06 08 10
-50
0
50
100
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02040608100
50
100
150
200
250
300
[GeV]
T[b]
0204060810
-50
0
50
[GeV]
TT[b]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
0204060810
00
02
04
06
[GeV]
LTQ
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
0204060810
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
FIG 10 Photoabsorption cross sections for πN (red) and π∆ production (orange) with Q2 = 0
(solid) and Q2 = 01 GeV2 (dashed for πN and dotted for π∆ channel)
1 πN-production channel
In order to extract the response function σLT (νQ2) we have developed a method similar to
the one used to calculate σTT (νQ2) see for example Ref [86] For σLT (νQ2) however the
calculation is more complicated because one has to take into account that the associated Compton
process involves a spin-flip of the nucleon as illustrated in Fig 11 When calculating the cross
section the product of the incoming nucleon spinors has to reflect this flip
The forward VVCS amplitude related to σLT (νQ2) mdash and δLT (Q2) mdash is gLT (νQ2) It can
be extracted from Eq (A2) if one takes the modified polarization vector components in Eq (A3)
with ε = εL and εprimelowast = εlowastplusmn as input where εL = 1Q(|~q | 0 0 q0) and εplusmn = ∓ 1radic
2(0 1plusmni 0) are the
standard longitudinal and transverse polarization vectors respectively For εL and εlowastplusmn only the
choice of helicities hprime = plusmn12 and h = ∓12 gives a non-zero contribution and one obtains
χdaggerhprime T (νQ2)χh = χdaggerhprimeminusi~σ middot [(ε0~εprimelowast minus ~ε ε primelowast0 )times q] gLT (νQ2)χh =
radic2 gLT (νQ2) (B1)
where χh and χdaggerhprime are two-component Pauli spinors with opposite helicities or here spins
Let us now consider the related photoabsorption process and in particular the tree-level γlowastN rarr
30
s = 12
sprime = minus12
λ = 0 λprime = 1
rprime rprime
s=12sprime=minus1
2
λ=0λprime=1
rprimerprime
s=12
sprime=minus12
λ=0λprime=1
rprimerprime hacute=frac12h=-frac12
λacute=1λ=0
FIG 11 Relation between the forward Compton process and the photoabsorption process given
by the optical theorem In particular we show the longitudinal-transverse contribution The
double-line arrows represent the spin of the external particles while the dot represents the scalar
(longitudinal) polarization of the incoming photon Inside the blob the intermediate states are
represented eg nucleons with spins rprime (which are averaged in the calculation of the cross section)
and pions
πN channel see diagrams in Fig 8 of Ref [30] We define the πN -production amplitude as
T = uhB (PB)sumi
Ai(s t)Γi uhA(PA) (B2)
with the Dirac structures
Γ1 = γ5 (B3a)
Γ2 =1
2
[qA ε
]γ5 (B3b)
where uhA(PA) and udaggerhB (PB) are the Dirac spinors and PA and PB are the four-momenta of
the incoming and outgoing nucleons respectively When calculating the photoabsorption cross
section related to the VVCS amplitude in Eq (B1) the nucleon spin flip should be implemented
by uhprime(PA) in T dagger and uh(PA) in T together with the appropriate transverse and longitudinal
photon polarization vectors εlowastplusmn and εL
However if one wants to use the properties of the Dirac matrices it is more useful to construct
an operator to produce this spin flip in the external nucleons of Fig 11 This is accomplished by
introducing the projector ΓLT equiv 12radic
2(γ1 + iγ2)γ5 which also takes into account the extra factor
radic2 in Eq (B1) We checked that with this projector one correctly extracts δLT by comparing the
HB limit of our result to the HB result of Ref [50] where the authors calculate this polarizability
from the Compton amplitude directly With all those ingredients the longitudinal-transverse cross
section is calculated in the following way
σLT (νQ2) =1
64π2 s
|~pf |cm
|~pi|cm
int 1
minus1dcos θ
sumij
AiAdaggerjXij (B4)
31
with
Xij = Tr[(PB +MN )Γi(PA +MN )ΓLTγ0Γdaggerjγ
0] (B5)
where θ is the scattering angle in the center-of-mass (cm) frame and |~pi|cm (|~pf |cm) is the three-
momentum of an incoming (outgoing) particle in the cm frame An explicit calculation of the
matrix Xij leads to
X = MNQ
0 2(PB minus PA) middot εLminusradic
2|~qf |cm sin θ (sminus u)
(B6)
where |~qi|cm (|~qf |cm) is the relative three-momentum of the incoming (outgoing) particles in the
cm frame Here s t and u are the usual Mandelstam variables For the different γlowastN rarr πN
channels we obtain the following amplitudes Ai where we introduce qA as the four-momentum of
the incoming photon and qB as the four-momentum of the outgoing pion
bull γlowastprarr π0p
A1 =e gAMN
fπ
[2PA middot ε+ qA middot ε
sminusM2N
+2PB middot εminus qA middot ε
uminusM2N
] (B7a)
A2 =e gAMN
fπ
[1
sminusM2N
+1
uminusM2N
] (B7b)
bull γlowastprarr π+n
A1 =
radic2 e gAMN
fπ
[2PA middot ε+ qA middot ε
sminusM2N
+2(PA minus PB) middot ε+ qA middot ε
tminusm2π
] (B8a)
A2 =
radic2 e gAMN
fπ(sminusM2N )
(B8b)
bull γlowastnrarr π0n
A1 = 0 (B9a)
A2 = 0 (B9b)
bull γlowastnrarr πminusp
A1 =
radic2 e gAMN
fπ
[2PB middot εminus qA middot ε
uminusM2N
minus 2(PA minus PB) middot ε+ qA middot εtminusm2
π
] (B10a)
A2 =
radic2 e gAMN
fπ(uminusM2N )
(B10b)
The analytical expressions shown above were checked with the amplitudes given in Ref [87] An-
alytical expressions for the tree-level γlowastN rarr πN channel of the σLT (νQ2) and σTT (νQ2) cross
32
sections are given below (proton channels π+n and π0p neutron channel πminusp) We checked that
they reproduce the known results in the real-photon limit [45 86] To shorten the final expressions
for the cross sections which are considerably longer for finite Q2 than in the real-photon limit we
define the following dimensionless kinematic variables
αγ = (ENi )cmradics =
s+M2N +Q2
2s (B11)
απ = (ENf )cmradics =
s+M2N minusm2
π
2s (B12)
βγ = Eγcmradics =
sminusM2N minusQ2
2s (B13)
βπ = Eπcmradics =
sminusM2N +m2
π
2s (B14)
λγ = |~qi|cmradics =
radic(sminusM2
N minusQ2)2 + 4sQ2
2s (B15)
λπ = |~qf |cmradics =
radic(sminusM2
N +m2π)2 minus 4sm2
π
2s (B16)
Here (ENi )cm and (ENf )cm are the energies of the incoming and outgoing nucleon Eγcm is the energy
of the incoming photon Eπcm is the energy of the outgoing pion all in the cm frame
σ(π+n)TT = minus e2g2
AM2N
64πf2πs
3(sminusM2N )2λ4
γ
4sλπλγ
[(M2
N minus s)(M2N minusQ2 minus s)(Q2 + 2sβγβπ)
+ 2s(m2πQ
2 minus (M2N minus s)
(M2N + s(minus1 + 2βγβπ)
))λ2γ
]+ (M2
N minus s)(M2N minusQ2 minus s)
times (Q2 + 2sβγβπ minus 2sλπλγ)(Q2 + 2s(βγβπ + λπλγ)
)times log
(Q2 + 2sβγβπ minus 2sλγλπQ2 + 2sβγβπ + 2sλγλπ
) (B17)
33
σ(π0p)TT =
e2g2AM
2Nλπ
64πf2πs(sminusM2
N )2λγ
1
s2λπλ3γ
[(M2
N minus s)(M2N minusQ2 minus s)
(3M2
N + s(2βγβπ minus 3))(M2N
+ s(2βγβπ minus 1))
+ 2s
((2m2
πQ2 + (M2
N minus s)2)(M2
N minus s)minus 2(minusm2
πQ2 + (sminusM2
N )2)
times sβγβπ + 4s2(sminusM2N )β2
γβ2π
)λ2γ
]arctan
(2sλπλγ
M2N + s(2βγβπ minus 1)
)+ 2
[minus 2m2
πQ2
+ (M2N minus s)
(2(sminusM2
N ) + 4sβγβπ +(sminusM2
N +Q2)(3M2N minus 3s+ 2sβγβπ)
sλ2γ
+m2πQ
2(sminusM2N )
(M2N minus s+ 2sβγβπ)(M2
N + s(minus1 + 2βγβπ minus 2λπλγ))
+m2πQ
2(sminusM2N )
(M2N minus s+ 2sβγβπ)(M2
N + s(minus1 + 2βγβπ + 2λπλγ))
)
+1
λπλγ
(βγβπ
(minusm2
πQ2 + (M2
N minus s)(M2N minus s+ 2sβγβπ)
)minus (M2
N minus s)(M2N minusQ2 minus s)λ2
π
)log
(M2N + s(minus1 + 2βγβπ + 2λπλγ)
M2N + s(minus1 + 2βγβπ minus 2λπλγ)
)] (B18)
σ(πminusp)TT =
e2g2AM
2N
64πf2πs
3λ4γ(M2
N + s(2βπβγ minus 1))2 minus 4λ2πs
2λ2γ)
4sλπλγ
((minusM2
N +Q2 + s)(M2N
+ s(2βπβγ minus 1))
2 minus 2sλ2γ
(2λ2
πs(minusM2N +Q2 + s) +m2
πQ2))
+((M2N + s(2βπβγ minus 1)
)2
minus 4λ2πs
2λ2γ
)((M2N + s(2βπβγ minus 1)
)(M2N minus 2Q2 minus s(2βπβγ + 1)
)+ 4λ2
πs2λ2
γ
)times log
(minusM2
N minus 2βπsβγ minus 2λπsλγ + s)minus(M2N + s(2βπβγ minus 2λπλγ minus 1)
)(M2N + s(minus1
+ 2βπβγ + 2λπλγ))[((
M2N + s(2βπβγ minus 1)
)(M2N minus 2Q2 minus s(2βπβγ + 1)
)+ 4λ2
πs2λ2
γ
)times log
(minusM2
N minus 2βπsβγ + 2λπsλγ + s)
+((Q2 + 2βπsβγ)2 minus 4λ2
πs2λ2
γ
)times log
(Q2 + 2βπsβγ minus 2λπsλγQ2 + 2sβπβγ + 2λπsλγ
)]+ 4sλ2
γ
((M2
N + s(2βπβγ minus 1))2 minus 4λ2πs
2λ2γ
)times (M2
N minus s) arctanh
(2λπsλγ
M2N + 2βπsβγ minus s
) (B19)
34
σ(π+n)LT =
e2g2AM
3Nλπ
32πf2πQs
3(sminusM2N )2λ4
γ
2sλγ
[(M2
N minus s)(Q2 + 2sβ2γ)(Q2 + 2sβγβπ)
minus 4s
((M2
N minus s)(Q2 minus 2s(απ minus 1)βγ
)+Q2sβγβπ
)λ2γ + 8s3(minus1 + απ)λ4
γ
]minus M2
N minus sλπ
[(Q2 + 2sβ2
γ)(Q2 + 2sβγβπ)2 + 4s2(2(απ minus 1)βγ(Q2 + 2sβγβπ)
minusQ2λ2π
)λ2γ + 8s3(απ minus 1)2λ4
γ
]arctan
(2sλπλγ
Q2 + 2sβγβπ
) (B20)
σ(π0p)LT =
e2g2AM
3Nλπ
16πf2πQs(sminusM2
N )2λγ
1
minus2s(M2N + s(minus1 + 2βγβπ))2λ2
γ + 8s3λ2πλ
4γ
[minus 3M8
N (Q2
+ 2sβ2γ) + 2M4
Ns2
(minus (Q2 + 2sβ2
γ)(minus3 + 2βγβπ) +(Q2(minus3 + 2βγβπ + 6λ2
π) + 2sβγ
times (minus3 + 12απ + 2βγβπ minus 12απβγβπ + 4βγλ2π))λ2γ minus 4s(α2
π minus 1)λ4π
)+ 2M2
Ns3
[minus (Q2
+ 2sβ2γ)(minus1 + 2βγβπ)
(6 + βγβπ(minus9 + 2βγβπ)
)+
(Q2(3minus 12λ2
π + 4βγβπ(minus1 + 2βγβπ
+ λ2π))minus 2sβγ
(minus 3 + 4απ(3 + 2απ(3 + 2βγβπ(minus3 + βγβπ)) + 4βγ(βπ + (2minus βγβπ)
times λ2π)))λ2γ + 8s
((απ minus 1)(1 + απ minus 2βγβπ) + 2απβγλ
2π
)λ4γ
)+ s4
((Q2 + 2sβ2
γ)
times (1minus 2βγβπ)2(2βγβπ minus 3) + 2(Q2(minus1 + 6λ2
π + 2βγβπ((1minus 2βγβπ)2 minus 2λ2π)) + 2sβγ
times (minus1 + 2βγβπ + 4(minus1 + βγβπ)(minusαπ + 2απβγβπ minus βγλ2π)))λ2γ minus 8
(s(απ minus 1)(απ
+ (1minus 2βγβπ)2) + 2βγ(2sαπ +Q2βγ)λ2π
)λ4γ + 32s(απ minus 1)λ2
πλ6γ
)+ 2M6
Ns
(Q2(6
minus 7βγβπ + λ2γ) + 2sβγ(βγ(6minus 7βγβπ) + (1minus 4απ)λ2
γ)
)]+sminusM2
N
4s2λπλ3γ
[(Q2 + 2sβ2
γ)
times(3M2
N + s(2βγβπ minus 3))(M2N + s(2βγβπ minus 1)
)+ 2s
(minusM2
N (Q2 + 2sβγ minus 8sαπβγ)
+ s(2sβγ(1minus 4απ + 4απβγβπ) +Q2(1minus 2λ2π)))λ2γ + 8s3(απ minus 1)λ4
γ
]
times arctan
(2sλπλγ
M2N + s(2βγβπ minus 1)
) (B21)
35
σ(πminusp)LT =
e2g2AM
3N
64πf2πQs
3λ4γ(M2
N minusQ2 minus s)(M2N + s(2βπβγ + 2λπλγ minus 1))
times
1
M2N + s(2βπβγ minus 2λπλγ minus 1)
[4λπsλγ(M2
N minusQ2 minus s)(
2sλ2γ
(s((1minus 2λ2
π)Q2
+ 2(2απ minus 1)sβγ(2βπβγ minus 1))minusM2N (Q2 + 2(1minus 2απ)sβγ)
)+ (Q2 + 2sβ2
γ)(M2N + s(2βπβγ minus 1)
)2 + 8(απ minus 1)απs
3λ4γ
)]minus(M2N + s(2βπβγ + 2λπλγ minus 1)
)times[
log
(minusM2N minus 2βπsβγ minus 2λπsλγ + s
minusM2N minus 2βπsβγ + 2λπsλγ + s
)(minus (Q2 + 2sβ2
γ)(M2N + s(2βπβγ minus 1)
)(minusM2
N + 2Q2 + 2βπsβγ + s)minus 2sλ2γ
(minusM2
N (Q2 + 2sβγ) +Q4 +Q2s(4απβγ
minus 2βγ minus 2λ2π + 1) + 2s2βγ(4(απ minus 1)βπβγ + 1)
)minus 8(απ minus 1)2s3λ4
γ
)+ 2
(4s2λ2
γ
(2(απ minus 1)βγ(Q2 + 2βπsβγ)minus λ2
πQ2)
+ (Q2 + 2βπsβγ)2(Q2 + 2sβ2γ)
+ 8(απ minus 1)2s3λ4γ
)arctanh
(2λπsλγ
Q2 + 2βπsβγ
)] (B22)
2 ∆-production channel
The tree-level ∆-exchange diagram in Fig 2 of Ref [30] contributes to the non-Born part of
the VVCS amplitudes The contribution of the ∆ exchange to the VVCS amplitudes can be split
into [17]
S∆-exch1 (νQ2) = S∆-pole
1 (νQ2) + S∆-exch1 (νQ2) (B23a)
νS∆-exch2 (νQ2) = νS∆-pole
2 (νQ2) + νS2∆-exch
(νQ2) (B23b)
and similarly for the unpolarized VVCS amplitudes discussed in Ref [30] Here we introduced the
∆-pole contributions S∆-polei and the ∆-non-pole contributions S∆-exch
i The former amplitudes
feature a pole at the ∆(1232)-production threshold and thus are proportional to
1
[sminusM2∆][uminusM2
∆]=
1
4M2N
1
ν2∆ minus ν2
(B24)
36
They can be reconstructed from the dispersion relations in Eq (5) with the tree-level ∆-production
cross sections as input cf Fig 10 in Ref [30]
σTT (νQ2) =π2α
M2NM
2+|~q |
minus g2
MMN (M+ + ν)|~q |2 +g2E(∆minus ν)(Q2 minusMNν)2
MN(B25a)
+g2CQ
4s(∆minus ν)
MNM2∆
minus 4gMgE(Q2 minusMNν)|~q |2 minus 4gMgCQ2|~q |2
+2gEgCQ
2[minusMNM∆ |~q |2 + s(Q2 +∆ν)
]MNM∆
δ(ν minus ν∆)
σLT (νQ2) =Qπ2α
M2NM
2+|~q |
g2E(MNν minusQ2) [M∆(MN + ν)minus s]
MN(B25b)
+g2CQ
2[MNM∆|~q |2 minus s(Q2 +∆ν)
]MNM2
∆
+ gMgEM∆|~q |2 minusgMgC(Q2 minusMNν)|~q |2
M∆
+gEgC(ν minus∆)(M2
N |~q |2 minus 2Q2s)
MNM∆
δ(ν minus ν∆)
with ∆ = M∆ minusMN M+ = M∆ + MN and the Mandelstam variable s = M2N + 2MNν minus Q2
Analytical expressions for the spin structure functions g1(xQ2) and g2(xQ2) can be constructed
from Eq (3) with the flux factor K(νQ2) = |~q | =radicν2 +Q2
In the ∆-non-pole contributions to S1(νQ2) and νS2(νQ2) the pole in ν at the ∆(1232)-
production threshold has canceled out
S∆-exch1 (νQ2) =
πα
MNM2+
[g2MQ
2+ + g2
E
(∆2 minus 3Q2
)+
4g2CQ
4
M2∆
minus 8gMgEM∆ωminus (B26a)
minus2gMgCQ2(MN minus 4M∆)
M∆+
2gEgCQ2(3MN minus 2M∆)
M∆
]
νS2∆-exch
(νQ2) =2πα
MNM2+
[g2EM∆∆ωminus +
g2M MNQ
2+
2+g2C Q
2(Q2 minus∆2)
2M∆(B26b)
+gEgM M∆(M∆ω+ minus 4MNωminus)minus gEgC ∆(2Q2 +MNω+)
+gMgC Q2(4MN minus ω+)
]+S∆-exch
2 (νQ2)
ν
[M2
∆ ω2+
M2N
+ ν2
]
with Q+ =radic
(M∆ +MN )2 +Q2 and ωplusmn = (M2∆minusM2
N plusmnQ2)2M∆ and the non-pole contribution
to S2(νQ2)
S∆-exch2 (νQ2) = minus2παMNν
M∆M2+
[gM + gE
]gC (B27)
These amplitudes to the contrary are not described by the tree-level ∆-production cross sections
in the standard dispersive approach [17] This peculiarity has been previously missed e g in the
calculation of the ∆-exchange contribution to the hydrogen hyperfine splitting in Ref [88] The
importance of including the ∆-non-pole contribution is also evident when considering the BC sum
37
rule in Eq (14) The ∆-pole terms by themselves violate the BC sum rule but cancel exactly with
the ∆-non-pole terms
limνrarr0
νS∆-pole2 (νQ2) + lim
νrarr0νS2
∆-exch(νQ2) = 0 (B28)
Appendix C Polarizabilities at Q2 = 0
In this section we give analytical expressions for the static values and slopes at Q2 = 0 of
the spin polarizabilities and moments of polarized structure functions In particular we give the
HB expansion of the πN -loop contributions and the ∆-exchange contributions The complete
expressions also for the π∆-loop contributions can be found in the Supplemented material Recall
that IA(0) = I1(0) = d2(0) = 0 and dd2(Q2)dQ2
∣∣∣Q2=0
= 0
1 πN-loop contribution
Here we give analytical expressions for the πN -loop contributions to the proton and neutron
spin polarizabilities expanded in powers of micro = mπMN viz the HB expansion Note that we
choose to expand here to a high order in micro the strict HB expansion would only retain the leading
term in an analogous NLO calculation
bull Static polarizabilities (Q2 = 0)
γ0p =e2g2
A
96π3f2πm
2π
1minus 21πmicro
8minus(
59
2+ 26 logmicro
)micro2 +
1875πmicro3
64
+3
(3
2+ 26 logmicro
)micro4 +
(C1)
γ0n =e2g2
A
48π3f2πm
2π
1
2minus 9πmicro
16minus 2micro2 logmicro+
75πmicro3
128minus 3micro4
4+
(C2)
δLTp =e2g2
A
192π3f2πm
2π
1minus 9πmicro
8+
(13
2minus 2 logmicro
)micro2 minus 465πmicro3
64
minus(
47
2+ 42 logmicro
)micro4 +
(C3)
δLTn =e2g2
A
96π3f2πm
2π
1
2+
3πmicro
16+ (1 + 2 log micro)micro2 minus 105πmicro3
128+
5micro4
4+
(C4)
38
δLTn =e2g2
A
96π3f2πm
2π
1
2+
3πmicro
16+ (1 + 2 log micro)micro2 minus 105πmicro3
128+
5micro4
4+
(C5)
γ0p =e2g2
A
16π3f2πm
4π
4
45minus 3πmicro
16+
14micro2
5minus 1813πmicro3
384minus 192
5(1 + log micro)micro4
+80703πmicro5
2048+
(C6)
γ0n =e2g2
A
16π3f2πm
4π
4
45minus 5πmicro
48+
4micro2
5minus 245πmicro3
384minus 32micro4 logmicro
15
+1323πmicro5
2048+
(C7)
bull Slopes of polarizabilities at Q2 = 0
dγ0p(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
2minus 45πmicro
4+ 223micro2 minus 28515πmicro3
64
minus9
(1953
4+ 449 logmicro
)micro4 +
570255πmicro5
128+
(C8)
dγ0n(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
2minus 81πmicro
8+ 94micro2 minus 2535πmicro3
32
minus3 (1 + 90 logmicro)micro4 +84315πmicro5
1024+
(C9)
dδLTp(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
2880π3f2πm
4π
minus5
2minus 27πmicro
32+ 20micro2 minus 5865πmicro3
256
+3
(617
4+ 36 logmicro
)micro4 minus 2056845πmicro5
4096+
(C10)
dδLTn(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
minus5
4minus 81πmicro
64minus 11micro2 +
10005πmicro3
512
+15
8(11 + 48 logmicro)micro4 minus 267015πmicro5
8192+
(C11)
dIAp(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
96π2f2πmicro
2
1minus 15πmicro
4minus 1
2(115 + 88 logmicro)micro2 +
1839πmicro3
32
+5 (5 + 34 logmicro)micro4 +
(C12)
dIAn(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
48π2f2πmicro
2
1
2minus 11πmicro
8minus 1
4(1 + 20 logmicro)micro2 +
99πmicro3
64
minus25micro4
12+
(C13)
dI1p(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
96π2f2πmicro
3π
8+ 2 (4 + 3 logmicro)microminus 537πmicro2
64
minus1
2(15 + 56 logmicro)micro3 +
(C14)
39
dI1n(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
48π2f2πmicro
minus π
16+
1
4(3 + 4 logmicro)microminus 57πmicro2
128+
2micro3
3+
(C15)
γ0p(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2g2
A
16π3f2πm
6π
1
105minus 23πmicro
256+
377micro2
210minus 15551πmicro3
6144+
3371micro4
105
minus1640457πmicro5
32768+
(C16)
γ0n(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2g2
A
16π3f2πm
6π
1
105minus 153πmicro
1792+
69micro2
70minus 4615πmicro3
6144+
172micro4
35
minus120897πmicro5
32768+
(C17)
2 ∆-exchange contribution
Here we give analytical expressions for the tree-level ∆-exchange contributions to the nucleon
spin polarizabilities and their slopes at Q2 = 0 Note that the ∆-exchange contributes equally to
proton and neutron polarizabilities Recall that for the magnetic γlowastN∆ coupling we introduced a
dipole form factor to mimic vector-meson dominance gM rarr gM(1 +Q2Λ2)2
bull Static polarizabilities (Q2 = 0)
γ0 = minus e2
4πM2+
(g2M
∆2+
g2E
M2+
minus 4gMgEM+∆
) (C18)
δLT =e2M∆
4πM3+
(g2E
MNM++gMgE∆MN
minus gEgCM2
∆
) (C19)
γ0 =e2M2
N
π∆2M4+
(minusg
2M
∆2+
g2E
M2+
+4gMgE∆M+
) (C20)
bull Slopes of polarizabilities at Q2 = 0
dγ0(Q2)
dQ2
∣∣∣∣∣Q2=0
= minus e2
πM2+∆
(g2M
∆
[1
4∆2minus 1
∆M++
1
2M2+
]minus 1
Λ2
g2M
∆+
g2E
2M2+
[1
2∆minus 3
M+
]minusgMgE
M+
[1
∆2minus 5
∆M++
1
M2+
]+
1
Λ2
2gMgEM+
+2gMgC∆M2
+
minus gEgCM3
+
) (C21)
40
d δLT (Q2)
dQ2
∣∣∣∣∣Q2=0
=e2M∆∆
4πMNM2+
(g2E
∆2M2+
[1
∆minus 4
M+
]minus g2
C
∆M2∆M
2+
+gMgE∆2M+
[1
∆2
minus 3
∆M++
1
M2+
]minus 2
Λ2
gMgE∆2M+
+gMgC∆M2
∆
[1
2∆2minus 2
∆M++
1
2M2+
]minus gEgC
2M2∆M
2+
[7
∆+
1
M+
]) (C22)
dIA(Q2)
dQ2
∣∣∣∣∣Q2=0
= minusM2N
M2+
(g2M
2∆2+
g2E
MNM+minus 2gMgE
∆M+minus gEgCM∆M+
) (C23)
dI1(Q2)
dQ2
∣∣∣∣∣Q2=0
= minusM∆M2N
2M3+
(g2E
MNM∆minus gMgE∆MN
minus gEgCM2
∆
) (C24)
γ0(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2M2
N
π∆3M6+
(g2M
∆
[3M2
∆ + 2M∆MN minus 9M2N
∆2+
4M2+
Λ2
]
minus4gMgE
[5M2
∆ minus 9M2N
M+∆2+
2M+
Λ2
]+
g2E
∆M+[7M∆ minus 9MN ]
minus8gMgC∆
+4gEgCM+
) (C25)
[1] D Drechsel B Pasquini and M Vanderhaeghen Dispersion relations in real and virtual
Compton scattering Phys Rept 378 99 (2003) hep-ph0212124
[2] S E Kuhn J-P Chen and E Leader Spin structure of the nucleon mdash status and recent
results Prog Part Nucl Phys 63 1 (2009) arXiv08123535 [hep-ph]
[3] F Hagelstein R Miskimen and V Pascalutsa Nucleon polarizabilities from Compton scat-
tering to hydrogen atom Prog Part Nucl Phys 88 29 (2016) arXiv151203765 [nucl-th]
[4] B Pasquini and M Vanderhaeghen Dispersion theory in electromagnetic interactions Ann
Rev Nucl Part Sci 68 75 (2018) arXiv180510482 [hep-ph]
[5] S Gerasimov A Sum rule for magnetic moments and the damping of the nucleon magnetic
moment in nuclei Sov J Nucl Phys 2 430 (1966)
[6] S Drell and A C Hearn Exact sum rule for nucleon magnetic moments Phys Rev Lett
16 908 (1966)
[7] J S Schwinger Source theory viewpoints in deep inelastic scattering Proc Natl Acad Sci
USA 72 1 (1975)
[8] J S Schwinger Source theory viewpoints in deep inelastic scattering Electromagnetic In-
teractions and Field Theory Proceedings 14 Internationale Universitatswochen Schladming
Austria February 24-March 7 1975 Acta Phys Austriaca Suppl 14 471 (1975)
41
[9] J Schwinger Source theory discussion of deep inelastic scattering with polarized particles
Proc Natl Acad Sci USA 72 1559 (1975)
[10] A M Baldin Polarizability of nucleons Nucl Phys 18 310 (1960)
[11] M Gell-Mann M L Goldberger and W E Thirring Use of causality conditions in quantum
theory Phys Rev 95 1612 (1954)
[12] A Pineda Leading chiral logarithms to the hyperfine splitting of the hydrogen and muonic
hydrogen Phys Rev C 67 025201 (2003)
[13] C Peset and A Pineda The two-photon exchange contribution to muonic hydrogen from
chiral perturbation theory Nucl Phys B 887 69 (2014) arXiv14064524 [hep-ph]
[14] C Peset and A Pineda Model-independent determination of the two-photon exchange contri-
bution to hyperfine splitting in muonic hydrogen JHEP 04 060 arXiv161205206 [nucl-th]
[15] F Hagelstein and V Pascalutsa Proton structure in the hyperfine splitting of muonic hydro-
gen PoS CD15 077 (2016) arXiv151104301 [nucl-th]
[16] F Hagelstein Exciting Nucleons in Compton Scattering and Hydrogen-Like Atoms PhD
thesis Mainz U Inst Kernphys (2017) arXiv171000874 [nucl-th]
[17] F Hagelstein ∆(1232)-Resonance in the hydrogen spectrum Proceedings 11th International
Workshop on the Physics of Excited Nucleons (NSTAR 2017) Columbia SC USA August
20-23 2017 Few Body Syst 59 93 (2018) arXiv180109790 [nucl-th]
[18] Y Prok et al (CLAS) Moments of the spin structure functions gp1 and gd1 for 005 lt Q2 lt 30
GeV2 Phys Lett B 672 12 (2009) arXiv08022232 [nucl-ex]
[19] H Dutz et al (GDH) First measurement of the Gerasimov-Drell-Hearn sum rule for 1H from
07 GeV to 18 GeV at ELSA Phys Rev Lett 91 192001 (2003)
[20] M Amarian et al (Jefferson Lab E94010) Measurement of the generalized forward spin po-
larizabilities of the neutron Phys Rev Lett 93 152301 (2004) arXivnucl-ex0406005
[21] M Amarian et al The Q2 evolution of the generalized Gerasimov-Drell-Hearn integral for
the neutron using a 3He target Phys Rev Lett 89 242301 (2002) arXivnucl-ex0205020
[22] M Amarian et al (Jefferson Lab E94-010)Q2 evolution of the neutron spin structure moments
using a 3He target Phys Rev Lett 92 022301 (2004) arXivhep-ex0310003
[23] A Deur et al Experimental determination of the evolution of the Bjorken integral at low Q2
Phys Rev Lett 93 212001 (2004) arXivhep-ex0407007
[24] K Slifer Low Q2 measurement of gp2 and the δLT spin polarizability Spin structure at long dis-
tance Proceedings Workshop Newport News USA March 12ndash13 2009 nucl-ex09064775
(2009) AIP Conf Proc 1155 10106313203293 (2009) arXiv09064775 [nucl-ex]
[25] P Solvignon et al (E01-012) Moments of the neutron g2 structure function at intermediate
42
Q2 Phys Rev C 92 015208 (2015) arXiv13044497 [nucl-ex]
[26] A Deur Experimental studies at low Q2 of the spin structure of the nucleon at Jefferson Lab
in 9th International Workshop on Chiral Dynamics (CD18) Durham NC USA September
17-21 2018 (2019) arXiv190305661 [nucl-ex]
[27] V Sulkosky et al (Jefferson Lab E97-110) Measurement of the 3He spin-structure functions
and of neutron (3He) spin-dependent sum rules at 0035 le Q2 le 024 GeV2 Phys Lett B
805 135428 (2020) arXiv190805709 [nucl-ex]
[28] K Adhikari et al (CLAS) Measurement of the Q2 Dependence of the Deuteron Spin Structure
Function g1 and its Moments at Low Q2 with CLAS Phys Rev Lett 120 062501 (2018)
arXiv171101974 [nucl-ex]
[29] V Lensky F Hagelstein A Hiller Blin and V Pascalutsa Comment on rdquoMeasurement of
the Q2 Dependence of the Deuteron Spin Structure Function g1 and its Moments at Low Q2
with CLASrdquo (2018) arXiv180603219 [nucl-th]
[30] J M Alarcn F Hagelstein V Lensky and V Pascalutsa Forward doubly-virtual Compton
scattering off the nucleon in chiral perturbation theory at NLO the subtraction function and
moments of unpolarized structure functions (2020) arXiv200509518 [hep-ph]
[31] V Bernard E Epelbaum H Krebs and U-G Meiszligner New insights into the spin structure
of the nucleon Phys Rev D 87 054032 (2013) arXiv12092523 [hep-ph]
[32] V Lensky J M Alarcon and V Pascalutsa Moments of nucleon structure functions at
next-to-leading order in baryon chiral perturbation theory Phys Rev C 90 055202 (2014)
arXiv14072574 [hep-ph]
[33] V Pascalutsa and M Vanderhaeghen Chiral effective-field theory in the ∆(1232) region
I Pion electroproduction on the nucleon Phys Rev D 73 034003 (2006) arXivhep-
ph0512244
[34] V Pascalutsa and M Vanderhaeghen Electromagnetic nucleon-to-Delta transition in chiral
effective field theory Phys Rev Lett 95 232001 (2005) arXivhep-ph0508060
[35] T R Hemmert B R Holstein and J Kambor Systematic 1M expansion for spin 32
particles in baryon chiral perturbation theory Phys Lett B 395 89 (1997) arXivhep-
ph9606456
[36] V Pascalutsa and D R Phillips Effective theory of the ∆(1232) in Compton scattering off
the nucleon Phys Rev C 67 055202 (2003) arXivnucl-th0212024
[37] V Pascalutsa M Vanderhaeghen and S N Yang Electromagnetic excitation of the ∆(1232)-
resonance Phys Rept 437 125 (2007) arXivhep-ph0609004
[38] K A Olive et al (Particle Data Group) Review of Particle Physics Chin Phys C 38 090001
43
(2014)
[39] F E Low Scattering of light of very low frequency by systems of spin 12 Phys Rev 96
1428 (1954)
[40] M Gell-Mann and M L Goldberger Scattering of low-energy photons by particles of spin
12 Phys Rev 96 1433 (1954)
[41] A M Harun ar-Rashid A simple derivation of Schwingerrsquos sum rule for spin dependent
structure functions Nuovo Cim A 33 447 (1976)
[42] F Hagelstein and V Pascalutsa Dissecting the hadronic contributions to (g minus 2)micro by
Schwingerrsquos sum rule Phys Rev Lett 120 072002 (2018) arXiv171004571 [hep-ph]
[43] V Lensky V Pascalutsa M Vanderhaeghen and C Kao Spin-dependent sum rules con-
necting real and virtual Compton scattering verified Phys Rev D 95 074001 (2017)
arXiv170101947 [hep-ph]
[44] H Burkhardt and W N Cottingham Sum rules for forward virtual Compton scattering
Annals Phys 56 453 (1970)
[45] V Lensky and V Pascalutsa Predictive powers of chiral perturbation theory in Compton
scattering off protons Eur Phys J C 65 195 (2010) arXiv09070451 [hep-ph]
[46] H Grieszlighammer J McGovern D Phillips and G Feldman Using effective field theory to
analyse low-energy Compton scattering data from protons and light nuclei Prog Part Nucl
Phys 67 841 (2012) arXiv12036834 [nucl-th]
[47] H W Grieszlighammer J A McGovern and D R Phillips Nucleon polarisabilities at and
beyond physical pion masses Eur Phys J A 52 139 (2016) arXiv151101952 [nucl-th]
[48] E Epelbaum H Krebs and U-G Meiszligner Improved chiral nucleon-nucleon potential up to
next-to-next-to-next-to-leading order Eur Phys J A 51 53 (2015) arXiv14120142 [nucl-
th]
[49] E Epelbaum H Krebs and U-G Meiszligner Precision nucleon-nucleon potential at fifth order
in the chiral expansion Phys Rev Lett 115 122301 (2015) arXiv14124623 [nucl-th]
[50] C W Kao T Spitzenberg and M Vanderhaeghen Burkhardt-Cottingham sum rule and
forward spin polarizabilities in heavy baryon chiral perturbation theory Phys Rev D 67
016001 (2003) arXivhep-ph0209241
[51] C-W Kao D Drechsel S Kamalov and M Vanderhaeghen Higher moments of nucleon
spin structure functions in heavy baryon chiral perturbation theory and in a resonance model
Phys Rev D 69 056004 (2004) arXivhep-ph0312102
[52] T Becher and H Leutwyler Baryon chiral perturbation theory in manifestly Lorentz invariant
form Eur Phys J C 9 643 (1999) arXivhep-ph9901384
44
[53] J Gasser M E Sainio and A Svarc Nucleons with chiral loops Nucl Phys B 307 779
(1988)
[54] L S Geng J Martin Camalich L Alvarez-Ruso and M J Vicente Vacas Leading SU(3)-
breaking corrections to the baryon magnetic moments in chiral perturbation theory Phys
Rev Lett 101 222002 (2008) arXiv08051419 [hep-ph]
[55] D Drechsel S S Kamalov and L Tiator The GDH sum rule and related integrals Phys
Rev D 63 114010 (2001) arXivhep-ph0008306
[56] D Drechsel O Hanstein S S Kamalov and L Tiator A unitary isobar model for pion
photo- and electroproduction on the proton up to 1 GeV Nucl Phys A 645 145 (1999)
arXivnucl-th9807001
[57] L Tiator private communication (2020)
[58] V Bernard T R Hemmert and U-G Meiszligner Spin structure of the nucleon at low energies
Phys Rev D 67 076008 (2003) arXivhep-ph0212033
[59] O Gryniuk F Hagelstein and V Pascalutsa Evaluation of the forward Compton scattering
off protons II Spin-dependent amplitude and observables Phys Rev D 94 034043 (2016)
arXiv160400789 [nucl-th]
[60] R Zielinski The g2p Experiment A Measurement of the Protonrsquos Spin Structure Functions
PhD thesis New Hampshire U (2010) arXiv170808297 [nucl-ex]
[61] N Guler et al (CLAS) Precise determination of the deuteron spin structure at low to mod-
erate Q2 with CLAS and extraction of the neutron contribution Phys Rev C 92 055201
(2015) arXiv150507877 [nucl-ex]
[62] C E Carlson V Nazaryan and K Griffioen Proton structure corrections to electronic
and muonic hydrogen hyperfine splitting Phys Rev A 78 022517 (2008) arXiv08052603
[physicsatom-ph]
[63] V Bernard N Kaiser and U-G Meiszligner Chiral dynamics in nucleons and nuclei Int J
Mod Phys E 4 193 (1995) arXivhep-ph9501384
[64] T R Hemmert B R Holstein and J Kambor ∆(1232) and the polarizabilities of the nucleon
Phys Rev D 55 5598 (1997) arXivhep-ph9612374
[65] V Pascalutsa and R Timmermans Field theory of nucleon to higher-spin baryon transitions
Phys Rev C 60 042201 (1999) arXivnucl-th9905065
[66] V Pascalutsa Quantization of an interacting spin-32 field and the Delta isobar Phys Rev
D 58 096002 (1998) arXivhep-ph9802288
[67] H Krebs Double Virtual Compton Scattering and SpinStructure of the Nucleon PoS
CD2018 031 (2019)
45
[68] N Kochelev and Y Oh Axial anomaly and the δLT puzzle Phys Rev D 016012 (2012)
arXiv11034892 [hep-ph]
[69] D Drechsel S Kamalov and L Tiator Unitary isobar model ndash MAID2007 Eur Phys J A
69 (2007) available at httpsmaidkphuni-mainzde arXiv07100306 [nucl-th]
[70] P J Mohr B N Taylor and D B Newell CODATA recommended values of the fundamental
physical constants 2010 Rev Mod Phys 84 1527 (2012)
[71] J Ahrens et al (GDH A2) First measurement of the Gerasimov-Drell-Hearn integral for 1H
from 200 to 800 MeV Phys Rev Lett 87 022003 (2001) arXivhep-ex0105089 [hep-ex]
[72] K Helbing (GDH) Experimental verification of the GDH sum rule at ELSA and MAMI Nucl
Phys Proc Suppl 105 113 (2002)
[73] R Bradford A Bodek H S Budd and J Arrington A New parameterization of the nucleon
elastic form-factors NuInt05 proceedings of the 4th International Workshop on Neutrino-
Nucleus Interactions in the Few-GeV Region Okayama Japan 26-29 September 2005 Nucl
Phys Proc Suppl 159 127 (2006) arXivhep-ex0602017
[74] J D Bjorken Applications of the chiral U(6)timesU(6) algebra of current densities Phys Rev
148 1467 (1966)
[75] J D Bjorken Inelastic scattering of polarized leptons from polarized nucleons Phys Rev D
1 1376 (1970)
[76] V Bernard T R Hemmert and U-G Meiszligner Novel analysis of chiral loop effects in
the generalized Gerasimov-Drell-Hearn sum rule Phys Lett B 545 105 (2002) arXivhep-
ph0203167
[77] A Deur et al Experimental study of isovector spin sum rules Phys Rev D 78 032001
(2008) arXiv08023198 [nucl-ex]
[78] B Pasquini P Pedroni and D Drechsel Higher order forward spin polarizability Phys Lett
B 687 160 (2010) arXiv10014230 [hep-ph]
[79] R Jaffe g2ndashThe nucleonrsquos other spin-dependent structure function Comments Nucl Part
Phys 19 239 (1990)
[80] E V Shuryak and A Vainshtein Theory of power corrections to deep inelastic scattering in
quantum chromodynamics (II) Qminus4 effects polarized target Nucl Phys B 201 141 (1982)
[81] S Wandzura and F Wilczek Sum rules for spin dependent electroproduction Test of rela-
tivistic constituent quarks Phys Lett B 72 195 (1977)
[82] B W Filippone and X-D Ji The spin structure of the nucleon Adv Nucl Phys 26 1
(2001) arXivhep-ph0101224
[83] M Burkardt The g2 structure function Proceedings Workshop on Spin structure at long
46
distance Newport News USA March 12-13 2009 AIP Conf Proc 1155 26 (2009)
arXiv09054079 [hep-ph]
[84] A Deur S J Brodsky and G F de Teramond The spin structure of the nucleon Rept
Prog Phys 82 076201 (2019) arXiv180705250 [hep-ph]
[85] B R Holstein D Drechsel B Pasquini and M Vanderhaeghen Higher order polarizabilities
of the proton Phys Rev C 61 034316 (2000) arXivhep-ph9910427
[86] B R Holstein V Pascalutsa and M Vanderhaeghen Sum rules for magnetic moments and
polarizabilities in QED and chiral effective-field theory Phys Rev D 72 094014 (2005)
arXivhep-ph0507016
[87] B Pasquini D Drechsel and L Tiator Invariant amplitudes for pion electroproduction Eur
Phys J A 34 387 (2007) arXiv07122327 [hep-ph]
[88] A J Buchmann Non-spherical proton shape and hydrogen hyperfine splitting Proceedings
International Workshop on Precision Physics of Simple Atomic Systems (PSAS 2008) Wind-
sor Ontario Canada July 21-26 2008 Can J Phys 87 773 (2009) arXiv09104747
[physicsatom-ph]
47
This moment provides another testing ground for our BχPT predictions through comparison
with experiments on the neutron [22] Going towards the low-Q2 region the interpretation
of d2(Q2) in terms of color polarizabilities will fade out The above definition however
implies it is related to other VVCS polarizabilities
d2(Q2) =Q4
8M4N
[M2
NQ2
αδLT (Q2) + I1(Q2)minus IA(Q2)
] (40)
Note that d2(Q2) and its first two derivatives with respect to Q2 vanish at Q2 = 0 The
considerations in Eqs (28) and (29) have no effect on d2(Q2) since the Born contribution
from IA(Q2) and I1(Q2) cancel out Therefore d2(Q2) is a pure polarizability
In Fig 8 upper panel we show our NLO BχPT prediction and other results for d2(Q2)
While MAID [69] and BχPT describe the experimental data for the neutron [22] very well
the HB limit [50 51] is showing a fast growth with Q2 This illustrates the importance
of keeping the relativistic result Note also that even though the πN -loop contribution
is dominant both gC and the form factor in gM are essential to obtain a curvature that
reproduces the data cf Fig 9 upper panel For the proton there are to our knowledge
no experimental results to compare with However the agreement between the NLO BχPT
prediction and the MAID prediction at low energies is reasonable
F γ0(Q2) mdash fifth-order generalized forward spin polarizability
It is interesting to compare the generalized fifth-order forward spin polarizability sum
rule
γ0(Q2) =1
2π2
int infinν0
dν
radic1 +
Q2
ν2
σTT (νQ2)
ν5(41)
=64αM4
N
Q10
int x0
0
dx x4
[g1(xQ2)minus 4M2
Nx2
Q2g2(xQ2)
]
to the sum rule integrals for IA(Q2) and γ0(Q2) since they differ merely by their energy
weighting of σTT (νQ2) and a constant prefactor cf Eqs (19) (26) and (41) From IA(Q2)
to γ0(Q2) to γ0(Q2) the energy suppression is increasing by a factor of νminus2 respectively
Therefore the description of γ0(Q2) should be easiest in a low-energy effective-field theory
such as χPT whereas γ0(Q2) and IA(Q2) receive larger contributions from higher energies
In Fig 8 lower panel we show our LO and NLO BχPT predictions for γ0(Q2) One
can see that the πN -loop contribution is positive (in accordance to what we see for the cross
section σTT see Fig 10) The Delta shifts it substantially especially in the low Q2 region
24
000 005 010 015 020 025 030
-0004
-0002
0000
0002
0004
0006
0008
0010
Q2GeV
2D
d2
proton
000 005 010 015 020 025 030
-0005
0000
0005
0010
0015
Q2GeV
2D
d2
neutron
000 005 010 015 020 025 030
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
6D
proton
000 005 010 015 020 025 030
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
6D
neutron
FIG 9 Contributions of the different orders to the chiral predictions of d2(Q2) upper panel and
γ0(Q2) lower panel for the proton (left) and neutron (right) Red solid line πN -loop contribu-
tion green dot-dashed line ∆-exchange contribution orange dotted line π∆-loop contribution
blue long-dashed line total result purple dot-dot-dashed line total result without gC contribution
black short-dashed line total result without gM dipole
bringing it into a better agreement with data In general the BχPT curves start above the
empirical data points at the real-photon point and then decrease asymptotically to zero
above Q2 gt 01 GeV2 On the other hand the MAID prediction reproduces the empirical
data at the real-photon point then decreases to negative values until about Q2 gt 006 GeV2
from where it also starts to asymptotically approach zero Consequently our NLO BχPT
prediction of γ0(Q2) is consistently above the MAID prediction This is very different to
what we saw for IA(Q2) in Fig 4 upper panel where the MAID prediction at the real-
photon point is above the experimental value While the agreement of our predictions with
the empirical data is in general quite good for all moments of σTT (νQ2) one should point
out that both for γ0n(Q2) and γ0p(Q2) we overestimate the data at low Q2 For IA(Q2) such
observation cannot be made because ∆IA(0) = 0 and thus IA(0) is given by the empirical
Pauli form factor only From IA(Q2) γ0(Q2) and γ0(Q2) the latter has the smallest however
25
non-negligible dependence on gC and the dipole in gM cf Fig 9 lower panel
The πN -loop ∆-exchange and π∆-loop contributions to the NLO BχPT prediction of
the static fifth-order forward spin polarizability amount to in units of 10minus4 fm6
γ0p = 112(30) asymp 208minus 096minus 001 (42a)
γ0n = 195(30) asymp 292minus 096minus 001 (42b)
while the slope is composed as follows in units of 10minus4 fm8
dγ0p(Q2)
dQ2
∣∣∣∣Q2=0
= minus084(10) asymp minus100 + 016 + 000 (43a)
dγ0n(Q2)
dQ2
∣∣∣∣Q2=0
= minus142(15) asymp minus158 + 016 + 000 (43b)
Note that the HB prediction of the static polarizability γ0p (423 at O(p3) and 365 at O(ε3)
[78 85]) is almost one order of magnitude larger than the empirical value and therefore not
shown in Fig 8
G Summary
Our results are summarized in Table II where we give the contributions of the different
orders to the chiral predictions of the polarizabilities and their slopes at the real-photon
point A quantitative comparison of our predictions for the static spin polarizabilities to
the work of Bernard et al [31] and different empirical evaluations is shown in Table III We
can see that the inclusion of the Delta turns out to be very important for all moments of
the helicity-difference cross section To describe the Q2 behavior of the polarizabilities the
magnetic coupling of the N rarr ∆ transition should be modified by a dipole form factor as
has been observed previously in the description of electroproduction data [33] This dipole
form factor effectively takes account of vector-meson exchanges The Coulomb-quadrupole
N rarr ∆ transition despite its subleading order is important in the description of some
moments of spin structure functions This is contrary to what we saw for the moments of
unpolarized structure functions [30] where the Coulomb coupling had a negligible effect
The π∆ loops are mainly relevant for the generalized GDH integrals
26
TABLE II The NLO BχPT predictions for the forward VVCS polarizabilities and their slopes at
Q2 = 0 The contributions of the πN loops the ∆ exchange and the π∆ loops are shown together
with the combined total result Note that IA(0) = I1(0) = d2(0) = 0 and (d2)prime = 0
πN loops ∆ exchange π∆ loops Total
γ0 p 201 minus284 minus010minus093(92)
(10minus4 fm4) n 298 003(92)
δLT p 150 minus016 minus002132(15)
(10minus4 fm4) n 235 218(23)
γ0 p 208 minus096 minus001112(30)
(10minus4 fm6) n 292 195(30)
(γ0)prime p minus033011 001
minus022(4)
(10minus4 fm6) n minus073 minus061(7)
(δLT )prime p minus080 minus004 minus001minus085(8)
(10minus4 fm6) n minus119 minus124(12)
(γ0)prime p minus100016 000
minus084(10)
(10minus4 fm8) n minus158 minus142(15)
(∆IA)prime p 238 minus1121 025minus858(343)
(GeVminus2) n 141 minus955(343)
(∆I1)prime p 034 minus053 058039(4)
(GeVminus2) n minus107 minus101(10)
TABLE III Our NLO BχPT predictions for the spin polarizabilities at Q2 = 0 compared with
the BχPT+∆ predictions from Bernard et al [31] and the available empirical information Where
the reference is not given the empirical number is provided by the MAID analysis [55 69] with
unspecified uncertainty
Proton Neutron
This work BχPT+∆ Empirical This work BχPT+∆ Empirical
γ0 minus093(92) minus174(40) minus100(8)(12) [19] 003(92) minus077(40) minus0005
(10minus4 fm4) minus090(8)(11) [78] [MAID]
minus0929(105) [59]
δLT 132(15) 240(1) 134 218(23) 238(3) 203
(10minus4 fm4) [MAID] [MAID]
γ0 112(30) 060(7)(7) [78] 195(30) 123
(10minus4 fm6) 0484(82) [59] [MAID]
IV CONCLUSIONS
We have presented a complete NLO calculation of the polarized non-Born VVCS ampli-
tudes in covariant BχPT with pion nucleon and ∆(1232) fields The dispersion relations
between the VVCS amplitudes and the tree-level photoabsorption cross sections served as a
27
cross-check of these calculations
The obtained moments of the proton and neutron spin structure functions related to gen-
eralized polarizabilities and GDH-type integrals agree well with the available experimental
data The description of their Q2 evolution is improved compared to the previous χPT pre-
dictions In particular the NLO BχPT predictions obtained here give a better description
of the empirical data (eg from the Jefferson Laboratory ldquoSpin Physics Programrdquo) than
the HB [50 51] and IR [58] calculations
The demonstrated predictive power of the χPT framework amplitudes makes it well
suited for extending the χPT evaluation of the TPE effect in the hyperfine structure of
(muonic-)hydrogen [15ndash17] to next-to-leading order
ACKNOWLEDGEMENTS
We thank Lothar Tiator and Marc Vanderhaeghen for helpful discussions This work
is supported by the Deutsche Forschungsgemeinschaft (DFG) through the Collaborative
Research Center [The Low-Energy Frontier of the Standard Model (SFB 1044)] JMA ac-
knowledges support from the Community of Madrid through the ldquoPrograma de atraccion de
talento investigador 2017 (Modalidad 1)rdquo and the Spanish MECD grants FPA2016-77313-P
FH gratefully acknowledges financial support from the Swiss National Science Foundation
Appendix A Tensor decompositions of the VVCS amplitudes
In this appendix we review the decomposition of the forward VVCS process into tensor struc-
tures and scalar amplitudes In particular we consider the connection between the covariant and
the semi-relativistic decomposition in the lab frame that is defined in terms of the conventional
transverse longitudinal transverse-transverse and transverse-longitudinal amplitudes
As explained in Sec II A the process of forward VVCS off the nucleon can be described in
terms of four explicitly covariant amplitudes S1 2 and T1 2 [3]
T (νQ2) =
(minusgmicroν +
qmicroqν
q2
)T1(νQ2) +
1
M2N
(pmicro minus p middot q
q2qmicro)(
pν minus p middot qq2
qν)T2(νQ2) (A1)
minus 1
MNγmicroναqα S1(νQ2)minus 1
M2N
(γmicroνq2 + qmicroγναqα minus qνγmicroαqα
)S2(νQ2)
εprimelowastmicro εν
where εmicro (εprimelowastmicro ) are the incoming (outgoing) photon polarization vectors ν is the photon lab-frame
energy and Q2 is the photon virtuality Alternatively the decomposition in the laboratory frame
28
(which in the forward case coincides with the Breit frame) is parametrized in terms of the nucleon
Pauli matrices ~σ and the four scalar functions fL fT gTT and gLT
T (νQ2) = ε0 εprimelowast0 fL(νQ2) + (~ε primelowast middot ~ε ) fT (νQ2) + i~σ middot (~ε primelowast times ~ε ) gTT (νQ2) (A2)
minusi~σ middot [(ε0~εprimelowast minus ~ε ε primelowast0 )times q] gLT (νQ2)
Here ~q and q = ~q|~q | are the photon three-momentum in the lab system and its unit vector The
modified polarization vector components are given by
ε0 =
[ε0 minus
ν
|~q | (~ε middot q )
] |~q |Q ~ε = ~εminus q (~ε middot q ) (A3)
where ε = (ε0~ε ) is the usual incoming photon polarization vector and εprimelowast the outgoing polarization
vector The LEX of the lab frame amplitudes [Eq (10)] can serve in particular as the definition
of the generalized polarizabilities The lab frame amplitudes are also conveniently used for the
definition of the response functions see the example of the scalar amplitude gLT (νQ2) and the
corresponding response function σLT (νQ2) below in App B
Appendix B Photoabsorption cross sections
In the forward kinematics the spin-dependent VVCS amplitudes and the spin polarizabilities
can be described in terms of the polarized structure functions g1(xQ2) and g2(xQ2) or equiv-
alently the helicity-difference cross section σTT (νQ2) and the longitudinal-transverse response
function σLT (νQ2) with the help of dispersion relations (5) and the optical theorem (3) In this
way the photoabsorption cross sections measured in electroproduction processes form the basis
for most empirical evaluations shown throughout Sec III In the following we present the BχPT
predictions for the tree-level cross sections of πN - π∆- and ∆-production through photoabsorption
on the nucleon cf Figs 8 9 and 10 in Ref [30] In Secs B 1 and B 2 we will discuss the leading
πN -production channel and the ∆-production channel respectively We used these cross sections
to verify the polarizability predictions obtained otherwise from the calculated non-Born VVCS
amplitudes Due to the bad high-energy behavior of the π∆-production cross sections in BχPT
cf Fig 10 the dispersion relations in Eq (5) require further subtractions for a reconstruction of
the π∆-loop contribution to the spin-dependent VVCS amplitudes Therefore not all polarizabil-
ities could be verified but only those appearing as higher-order terms in the LEX of the VVCS
amplitudes such as γ0 [16]
29
proton neutron
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Neutron
02 04 06 08 100
50
100
150
200
250
[GeV]
T [b]
02 04 06 08 10
-50
0
50
100
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Neutron
02 04 06 08 100
50
100
150
200
250
[GeV]
T [b]
02 04 06 08 10
-50
0
50
100
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02040608100
50
100
150
200
250
300
[GeV]
T[b]
0204060810
-50
0
50
[GeV]
TT[b]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
0204060810
00
02
04
06
[GeV]
LTQ
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
0204060810
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
FIG 10 Photoabsorption cross sections for πN (red) and π∆ production (orange) with Q2 = 0
(solid) and Q2 = 01 GeV2 (dashed for πN and dotted for π∆ channel)
1 πN-production channel
In order to extract the response function σLT (νQ2) we have developed a method similar to
the one used to calculate σTT (νQ2) see for example Ref [86] For σLT (νQ2) however the
calculation is more complicated because one has to take into account that the associated Compton
process involves a spin-flip of the nucleon as illustrated in Fig 11 When calculating the cross
section the product of the incoming nucleon spinors has to reflect this flip
The forward VVCS amplitude related to σLT (νQ2) mdash and δLT (Q2) mdash is gLT (νQ2) It can
be extracted from Eq (A2) if one takes the modified polarization vector components in Eq (A3)
with ε = εL and εprimelowast = εlowastplusmn as input where εL = 1Q(|~q | 0 0 q0) and εplusmn = ∓ 1radic
2(0 1plusmni 0) are the
standard longitudinal and transverse polarization vectors respectively For εL and εlowastplusmn only the
choice of helicities hprime = plusmn12 and h = ∓12 gives a non-zero contribution and one obtains
χdaggerhprime T (νQ2)χh = χdaggerhprimeminusi~σ middot [(ε0~εprimelowast minus ~ε ε primelowast0 )times q] gLT (νQ2)χh =
radic2 gLT (νQ2) (B1)
where χh and χdaggerhprime are two-component Pauli spinors with opposite helicities or here spins
Let us now consider the related photoabsorption process and in particular the tree-level γlowastN rarr
30
s = 12
sprime = minus12
λ = 0 λprime = 1
rprime rprime
s=12sprime=minus1
2
λ=0λprime=1
rprimerprime
s=12
sprime=minus12
λ=0λprime=1
rprimerprime hacute=frac12h=-frac12
λacute=1λ=0
FIG 11 Relation between the forward Compton process and the photoabsorption process given
by the optical theorem In particular we show the longitudinal-transverse contribution The
double-line arrows represent the spin of the external particles while the dot represents the scalar
(longitudinal) polarization of the incoming photon Inside the blob the intermediate states are
represented eg nucleons with spins rprime (which are averaged in the calculation of the cross section)
and pions
πN channel see diagrams in Fig 8 of Ref [30] We define the πN -production amplitude as
T = uhB (PB)sumi
Ai(s t)Γi uhA(PA) (B2)
with the Dirac structures
Γ1 = γ5 (B3a)
Γ2 =1
2
[qA ε
]γ5 (B3b)
where uhA(PA) and udaggerhB (PB) are the Dirac spinors and PA and PB are the four-momenta of
the incoming and outgoing nucleons respectively When calculating the photoabsorption cross
section related to the VVCS amplitude in Eq (B1) the nucleon spin flip should be implemented
by uhprime(PA) in T dagger and uh(PA) in T together with the appropriate transverse and longitudinal
photon polarization vectors εlowastplusmn and εL
However if one wants to use the properties of the Dirac matrices it is more useful to construct
an operator to produce this spin flip in the external nucleons of Fig 11 This is accomplished by
introducing the projector ΓLT equiv 12radic
2(γ1 + iγ2)γ5 which also takes into account the extra factor
radic2 in Eq (B1) We checked that with this projector one correctly extracts δLT by comparing the
HB limit of our result to the HB result of Ref [50] where the authors calculate this polarizability
from the Compton amplitude directly With all those ingredients the longitudinal-transverse cross
section is calculated in the following way
σLT (νQ2) =1
64π2 s
|~pf |cm
|~pi|cm
int 1
minus1dcos θ
sumij
AiAdaggerjXij (B4)
31
with
Xij = Tr[(PB +MN )Γi(PA +MN )ΓLTγ0Γdaggerjγ
0] (B5)
where θ is the scattering angle in the center-of-mass (cm) frame and |~pi|cm (|~pf |cm) is the three-
momentum of an incoming (outgoing) particle in the cm frame An explicit calculation of the
matrix Xij leads to
X = MNQ
0 2(PB minus PA) middot εLminusradic
2|~qf |cm sin θ (sminus u)
(B6)
where |~qi|cm (|~qf |cm) is the relative three-momentum of the incoming (outgoing) particles in the
cm frame Here s t and u are the usual Mandelstam variables For the different γlowastN rarr πN
channels we obtain the following amplitudes Ai where we introduce qA as the four-momentum of
the incoming photon and qB as the four-momentum of the outgoing pion
bull γlowastprarr π0p
A1 =e gAMN
fπ
[2PA middot ε+ qA middot ε
sminusM2N
+2PB middot εminus qA middot ε
uminusM2N
] (B7a)
A2 =e gAMN
fπ
[1
sminusM2N
+1
uminusM2N
] (B7b)
bull γlowastprarr π+n
A1 =
radic2 e gAMN
fπ
[2PA middot ε+ qA middot ε
sminusM2N
+2(PA minus PB) middot ε+ qA middot ε
tminusm2π
] (B8a)
A2 =
radic2 e gAMN
fπ(sminusM2N )
(B8b)
bull γlowastnrarr π0n
A1 = 0 (B9a)
A2 = 0 (B9b)
bull γlowastnrarr πminusp
A1 =
radic2 e gAMN
fπ
[2PB middot εminus qA middot ε
uminusM2N
minus 2(PA minus PB) middot ε+ qA middot εtminusm2
π
] (B10a)
A2 =
radic2 e gAMN
fπ(uminusM2N )
(B10b)
The analytical expressions shown above were checked with the amplitudes given in Ref [87] An-
alytical expressions for the tree-level γlowastN rarr πN channel of the σLT (νQ2) and σTT (νQ2) cross
32
sections are given below (proton channels π+n and π0p neutron channel πminusp) We checked that
they reproduce the known results in the real-photon limit [45 86] To shorten the final expressions
for the cross sections which are considerably longer for finite Q2 than in the real-photon limit we
define the following dimensionless kinematic variables
αγ = (ENi )cmradics =
s+M2N +Q2
2s (B11)
απ = (ENf )cmradics =
s+M2N minusm2
π
2s (B12)
βγ = Eγcmradics =
sminusM2N minusQ2
2s (B13)
βπ = Eπcmradics =
sminusM2N +m2
π
2s (B14)
λγ = |~qi|cmradics =
radic(sminusM2
N minusQ2)2 + 4sQ2
2s (B15)
λπ = |~qf |cmradics =
radic(sminusM2
N +m2π)2 minus 4sm2
π
2s (B16)
Here (ENi )cm and (ENf )cm are the energies of the incoming and outgoing nucleon Eγcm is the energy
of the incoming photon Eπcm is the energy of the outgoing pion all in the cm frame
σ(π+n)TT = minus e2g2
AM2N
64πf2πs
3(sminusM2N )2λ4
γ
4sλπλγ
[(M2
N minus s)(M2N minusQ2 minus s)(Q2 + 2sβγβπ)
+ 2s(m2πQ
2 minus (M2N minus s)
(M2N + s(minus1 + 2βγβπ)
))λ2γ
]+ (M2
N minus s)(M2N minusQ2 minus s)
times (Q2 + 2sβγβπ minus 2sλπλγ)(Q2 + 2s(βγβπ + λπλγ)
)times log
(Q2 + 2sβγβπ minus 2sλγλπQ2 + 2sβγβπ + 2sλγλπ
) (B17)
33
σ(π0p)TT =
e2g2AM
2Nλπ
64πf2πs(sminusM2
N )2λγ
1
s2λπλ3γ
[(M2
N minus s)(M2N minusQ2 minus s)
(3M2
N + s(2βγβπ minus 3))(M2N
+ s(2βγβπ minus 1))
+ 2s
((2m2
πQ2 + (M2
N minus s)2)(M2
N minus s)minus 2(minusm2
πQ2 + (sminusM2
N )2)
times sβγβπ + 4s2(sminusM2N )β2
γβ2π
)λ2γ
]arctan
(2sλπλγ
M2N + s(2βγβπ minus 1)
)+ 2
[minus 2m2
πQ2
+ (M2N minus s)
(2(sminusM2
N ) + 4sβγβπ +(sminusM2
N +Q2)(3M2N minus 3s+ 2sβγβπ)
sλ2γ
+m2πQ
2(sminusM2N )
(M2N minus s+ 2sβγβπ)(M2
N + s(minus1 + 2βγβπ minus 2λπλγ))
+m2πQ
2(sminusM2N )
(M2N minus s+ 2sβγβπ)(M2
N + s(minus1 + 2βγβπ + 2λπλγ))
)
+1
λπλγ
(βγβπ
(minusm2
πQ2 + (M2
N minus s)(M2N minus s+ 2sβγβπ)
)minus (M2
N minus s)(M2N minusQ2 minus s)λ2
π
)log
(M2N + s(minus1 + 2βγβπ + 2λπλγ)
M2N + s(minus1 + 2βγβπ minus 2λπλγ)
)] (B18)
σ(πminusp)TT =
e2g2AM
2N
64πf2πs
3λ4γ(M2
N + s(2βπβγ minus 1))2 minus 4λ2πs
2λ2γ)
4sλπλγ
((minusM2
N +Q2 + s)(M2N
+ s(2βπβγ minus 1))
2 minus 2sλ2γ
(2λ2
πs(minusM2N +Q2 + s) +m2
πQ2))
+((M2N + s(2βπβγ minus 1)
)2
minus 4λ2πs
2λ2γ
)((M2N + s(2βπβγ minus 1)
)(M2N minus 2Q2 minus s(2βπβγ + 1)
)+ 4λ2
πs2λ2
γ
)times log
(minusM2
N minus 2βπsβγ minus 2λπsλγ + s)minus(M2N + s(2βπβγ minus 2λπλγ minus 1)
)(M2N + s(minus1
+ 2βπβγ + 2λπλγ))[((
M2N + s(2βπβγ minus 1)
)(M2N minus 2Q2 minus s(2βπβγ + 1)
)+ 4λ2
πs2λ2
γ
)times log
(minusM2
N minus 2βπsβγ + 2λπsλγ + s)
+((Q2 + 2βπsβγ)2 minus 4λ2
πs2λ2
γ
)times log
(Q2 + 2βπsβγ minus 2λπsλγQ2 + 2sβπβγ + 2λπsλγ
)]+ 4sλ2
γ
((M2
N + s(2βπβγ minus 1))2 minus 4λ2πs
2λ2γ
)times (M2
N minus s) arctanh
(2λπsλγ
M2N + 2βπsβγ minus s
) (B19)
34
σ(π+n)LT =
e2g2AM
3Nλπ
32πf2πQs
3(sminusM2N )2λ4
γ
2sλγ
[(M2
N minus s)(Q2 + 2sβ2γ)(Q2 + 2sβγβπ)
minus 4s
((M2
N minus s)(Q2 minus 2s(απ minus 1)βγ
)+Q2sβγβπ
)λ2γ + 8s3(minus1 + απ)λ4
γ
]minus M2
N minus sλπ
[(Q2 + 2sβ2
γ)(Q2 + 2sβγβπ)2 + 4s2(2(απ minus 1)βγ(Q2 + 2sβγβπ)
minusQ2λ2π
)λ2γ + 8s3(απ minus 1)2λ4
γ
]arctan
(2sλπλγ
Q2 + 2sβγβπ
) (B20)
σ(π0p)LT =
e2g2AM
3Nλπ
16πf2πQs(sminusM2
N )2λγ
1
minus2s(M2N + s(minus1 + 2βγβπ))2λ2
γ + 8s3λ2πλ
4γ
[minus 3M8
N (Q2
+ 2sβ2γ) + 2M4
Ns2
(minus (Q2 + 2sβ2
γ)(minus3 + 2βγβπ) +(Q2(minus3 + 2βγβπ + 6λ2
π) + 2sβγ
times (minus3 + 12απ + 2βγβπ minus 12απβγβπ + 4βγλ2π))λ2γ minus 4s(α2
π minus 1)λ4π
)+ 2M2
Ns3
[minus (Q2
+ 2sβ2γ)(minus1 + 2βγβπ)
(6 + βγβπ(minus9 + 2βγβπ)
)+
(Q2(3minus 12λ2
π + 4βγβπ(minus1 + 2βγβπ
+ λ2π))minus 2sβγ
(minus 3 + 4απ(3 + 2απ(3 + 2βγβπ(minus3 + βγβπ)) + 4βγ(βπ + (2minus βγβπ)
times λ2π)))λ2γ + 8s
((απ minus 1)(1 + απ minus 2βγβπ) + 2απβγλ
2π
)λ4γ
)+ s4
((Q2 + 2sβ2
γ)
times (1minus 2βγβπ)2(2βγβπ minus 3) + 2(Q2(minus1 + 6λ2
π + 2βγβπ((1minus 2βγβπ)2 minus 2λ2π)) + 2sβγ
times (minus1 + 2βγβπ + 4(minus1 + βγβπ)(minusαπ + 2απβγβπ minus βγλ2π)))λ2γ minus 8
(s(απ minus 1)(απ
+ (1minus 2βγβπ)2) + 2βγ(2sαπ +Q2βγ)λ2π
)λ4γ + 32s(απ minus 1)λ2
πλ6γ
)+ 2M6
Ns
(Q2(6
minus 7βγβπ + λ2γ) + 2sβγ(βγ(6minus 7βγβπ) + (1minus 4απ)λ2
γ)
)]+sminusM2
N
4s2λπλ3γ
[(Q2 + 2sβ2
γ)
times(3M2
N + s(2βγβπ minus 3))(M2N + s(2βγβπ minus 1)
)+ 2s
(minusM2
N (Q2 + 2sβγ minus 8sαπβγ)
+ s(2sβγ(1minus 4απ + 4απβγβπ) +Q2(1minus 2λ2π)))λ2γ + 8s3(απ minus 1)λ4
γ
]
times arctan
(2sλπλγ
M2N + s(2βγβπ minus 1)
) (B21)
35
σ(πminusp)LT =
e2g2AM
3N
64πf2πQs
3λ4γ(M2
N minusQ2 minus s)(M2N + s(2βπβγ + 2λπλγ minus 1))
times
1
M2N + s(2βπβγ minus 2λπλγ minus 1)
[4λπsλγ(M2
N minusQ2 minus s)(
2sλ2γ
(s((1minus 2λ2
π)Q2
+ 2(2απ minus 1)sβγ(2βπβγ minus 1))minusM2N (Q2 + 2(1minus 2απ)sβγ)
)+ (Q2 + 2sβ2
γ)(M2N + s(2βπβγ minus 1)
)2 + 8(απ minus 1)απs
3λ4γ
)]minus(M2N + s(2βπβγ + 2λπλγ minus 1)
)times[
log
(minusM2N minus 2βπsβγ minus 2λπsλγ + s
minusM2N minus 2βπsβγ + 2λπsλγ + s
)(minus (Q2 + 2sβ2
γ)(M2N + s(2βπβγ minus 1)
)(minusM2
N + 2Q2 + 2βπsβγ + s)minus 2sλ2γ
(minusM2
N (Q2 + 2sβγ) +Q4 +Q2s(4απβγ
minus 2βγ minus 2λ2π + 1) + 2s2βγ(4(απ minus 1)βπβγ + 1)
)minus 8(απ minus 1)2s3λ4
γ
)+ 2
(4s2λ2
γ
(2(απ minus 1)βγ(Q2 + 2βπsβγ)minus λ2
πQ2)
+ (Q2 + 2βπsβγ)2(Q2 + 2sβ2γ)
+ 8(απ minus 1)2s3λ4γ
)arctanh
(2λπsλγ
Q2 + 2βπsβγ
)] (B22)
2 ∆-production channel
The tree-level ∆-exchange diagram in Fig 2 of Ref [30] contributes to the non-Born part of
the VVCS amplitudes The contribution of the ∆ exchange to the VVCS amplitudes can be split
into [17]
S∆-exch1 (νQ2) = S∆-pole
1 (νQ2) + S∆-exch1 (νQ2) (B23a)
νS∆-exch2 (νQ2) = νS∆-pole
2 (νQ2) + νS2∆-exch
(νQ2) (B23b)
and similarly for the unpolarized VVCS amplitudes discussed in Ref [30] Here we introduced the
∆-pole contributions S∆-polei and the ∆-non-pole contributions S∆-exch
i The former amplitudes
feature a pole at the ∆(1232)-production threshold and thus are proportional to
1
[sminusM2∆][uminusM2
∆]=
1
4M2N
1
ν2∆ minus ν2
(B24)
36
They can be reconstructed from the dispersion relations in Eq (5) with the tree-level ∆-production
cross sections as input cf Fig 10 in Ref [30]
σTT (νQ2) =π2α
M2NM
2+|~q |
minus g2
MMN (M+ + ν)|~q |2 +g2E(∆minus ν)(Q2 minusMNν)2
MN(B25a)
+g2CQ
4s(∆minus ν)
MNM2∆
minus 4gMgE(Q2 minusMNν)|~q |2 minus 4gMgCQ2|~q |2
+2gEgCQ
2[minusMNM∆ |~q |2 + s(Q2 +∆ν)
]MNM∆
δ(ν minus ν∆)
σLT (νQ2) =Qπ2α
M2NM
2+|~q |
g2E(MNν minusQ2) [M∆(MN + ν)minus s]
MN(B25b)
+g2CQ
2[MNM∆|~q |2 minus s(Q2 +∆ν)
]MNM2
∆
+ gMgEM∆|~q |2 minusgMgC(Q2 minusMNν)|~q |2
M∆
+gEgC(ν minus∆)(M2
N |~q |2 minus 2Q2s)
MNM∆
δ(ν minus ν∆)
with ∆ = M∆ minusMN M+ = M∆ + MN and the Mandelstam variable s = M2N + 2MNν minus Q2
Analytical expressions for the spin structure functions g1(xQ2) and g2(xQ2) can be constructed
from Eq (3) with the flux factor K(νQ2) = |~q | =radicν2 +Q2
In the ∆-non-pole contributions to S1(νQ2) and νS2(νQ2) the pole in ν at the ∆(1232)-
production threshold has canceled out
S∆-exch1 (νQ2) =
πα
MNM2+
[g2MQ
2+ + g2
E
(∆2 minus 3Q2
)+
4g2CQ
4
M2∆
minus 8gMgEM∆ωminus (B26a)
minus2gMgCQ2(MN minus 4M∆)
M∆+
2gEgCQ2(3MN minus 2M∆)
M∆
]
νS2∆-exch
(νQ2) =2πα
MNM2+
[g2EM∆∆ωminus +
g2M MNQ
2+
2+g2C Q
2(Q2 minus∆2)
2M∆(B26b)
+gEgM M∆(M∆ω+ minus 4MNωminus)minus gEgC ∆(2Q2 +MNω+)
+gMgC Q2(4MN minus ω+)
]+S∆-exch
2 (νQ2)
ν
[M2
∆ ω2+
M2N
+ ν2
]
with Q+ =radic
(M∆ +MN )2 +Q2 and ωplusmn = (M2∆minusM2
N plusmnQ2)2M∆ and the non-pole contribution
to S2(νQ2)
S∆-exch2 (νQ2) = minus2παMNν
M∆M2+
[gM + gE
]gC (B27)
These amplitudes to the contrary are not described by the tree-level ∆-production cross sections
in the standard dispersive approach [17] This peculiarity has been previously missed e g in the
calculation of the ∆-exchange contribution to the hydrogen hyperfine splitting in Ref [88] The
importance of including the ∆-non-pole contribution is also evident when considering the BC sum
37
rule in Eq (14) The ∆-pole terms by themselves violate the BC sum rule but cancel exactly with
the ∆-non-pole terms
limνrarr0
νS∆-pole2 (νQ2) + lim
νrarr0νS2
∆-exch(νQ2) = 0 (B28)
Appendix C Polarizabilities at Q2 = 0
In this section we give analytical expressions for the static values and slopes at Q2 = 0 of
the spin polarizabilities and moments of polarized structure functions In particular we give the
HB expansion of the πN -loop contributions and the ∆-exchange contributions The complete
expressions also for the π∆-loop contributions can be found in the Supplemented material Recall
that IA(0) = I1(0) = d2(0) = 0 and dd2(Q2)dQ2
∣∣∣Q2=0
= 0
1 πN-loop contribution
Here we give analytical expressions for the πN -loop contributions to the proton and neutron
spin polarizabilities expanded in powers of micro = mπMN viz the HB expansion Note that we
choose to expand here to a high order in micro the strict HB expansion would only retain the leading
term in an analogous NLO calculation
bull Static polarizabilities (Q2 = 0)
γ0p =e2g2
A
96π3f2πm
2π
1minus 21πmicro
8minus(
59
2+ 26 logmicro
)micro2 +
1875πmicro3
64
+3
(3
2+ 26 logmicro
)micro4 +
(C1)
γ0n =e2g2
A
48π3f2πm
2π
1
2minus 9πmicro
16minus 2micro2 logmicro+
75πmicro3
128minus 3micro4
4+
(C2)
δLTp =e2g2
A
192π3f2πm
2π
1minus 9πmicro
8+
(13
2minus 2 logmicro
)micro2 minus 465πmicro3
64
minus(
47
2+ 42 logmicro
)micro4 +
(C3)
δLTn =e2g2
A
96π3f2πm
2π
1
2+
3πmicro
16+ (1 + 2 log micro)micro2 minus 105πmicro3
128+
5micro4
4+
(C4)
38
δLTn =e2g2
A
96π3f2πm
2π
1
2+
3πmicro
16+ (1 + 2 log micro)micro2 minus 105πmicro3
128+
5micro4
4+
(C5)
γ0p =e2g2
A
16π3f2πm
4π
4
45minus 3πmicro
16+
14micro2
5minus 1813πmicro3
384minus 192
5(1 + log micro)micro4
+80703πmicro5
2048+
(C6)
γ0n =e2g2
A
16π3f2πm
4π
4
45minus 5πmicro
48+
4micro2
5minus 245πmicro3
384minus 32micro4 logmicro
15
+1323πmicro5
2048+
(C7)
bull Slopes of polarizabilities at Q2 = 0
dγ0p(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
2minus 45πmicro
4+ 223micro2 minus 28515πmicro3
64
minus9
(1953
4+ 449 logmicro
)micro4 +
570255πmicro5
128+
(C8)
dγ0n(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
2minus 81πmicro
8+ 94micro2 minus 2535πmicro3
32
minus3 (1 + 90 logmicro)micro4 +84315πmicro5
1024+
(C9)
dδLTp(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
2880π3f2πm
4π
minus5
2minus 27πmicro
32+ 20micro2 minus 5865πmicro3
256
+3
(617
4+ 36 logmicro
)micro4 minus 2056845πmicro5
4096+
(C10)
dδLTn(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
minus5
4minus 81πmicro
64minus 11micro2 +
10005πmicro3
512
+15
8(11 + 48 logmicro)micro4 minus 267015πmicro5
8192+
(C11)
dIAp(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
96π2f2πmicro
2
1minus 15πmicro
4minus 1
2(115 + 88 logmicro)micro2 +
1839πmicro3
32
+5 (5 + 34 logmicro)micro4 +
(C12)
dIAn(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
48π2f2πmicro
2
1
2minus 11πmicro
8minus 1
4(1 + 20 logmicro)micro2 +
99πmicro3
64
minus25micro4
12+
(C13)
dI1p(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
96π2f2πmicro
3π
8+ 2 (4 + 3 logmicro)microminus 537πmicro2
64
minus1
2(15 + 56 logmicro)micro3 +
(C14)
39
dI1n(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
48π2f2πmicro
minus π
16+
1
4(3 + 4 logmicro)microminus 57πmicro2
128+
2micro3
3+
(C15)
γ0p(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2g2
A
16π3f2πm
6π
1
105minus 23πmicro
256+
377micro2
210minus 15551πmicro3
6144+
3371micro4
105
minus1640457πmicro5
32768+
(C16)
γ0n(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2g2
A
16π3f2πm
6π
1
105minus 153πmicro
1792+
69micro2
70minus 4615πmicro3
6144+
172micro4
35
minus120897πmicro5
32768+
(C17)
2 ∆-exchange contribution
Here we give analytical expressions for the tree-level ∆-exchange contributions to the nucleon
spin polarizabilities and their slopes at Q2 = 0 Note that the ∆-exchange contributes equally to
proton and neutron polarizabilities Recall that for the magnetic γlowastN∆ coupling we introduced a
dipole form factor to mimic vector-meson dominance gM rarr gM(1 +Q2Λ2)2
bull Static polarizabilities (Q2 = 0)
γ0 = minus e2
4πM2+
(g2M
∆2+
g2E
M2+
minus 4gMgEM+∆
) (C18)
δLT =e2M∆
4πM3+
(g2E
MNM++gMgE∆MN
minus gEgCM2
∆
) (C19)
γ0 =e2M2
N
π∆2M4+
(minusg
2M
∆2+
g2E
M2+
+4gMgE∆M+
) (C20)
bull Slopes of polarizabilities at Q2 = 0
dγ0(Q2)
dQ2
∣∣∣∣∣Q2=0
= minus e2
πM2+∆
(g2M
∆
[1
4∆2minus 1
∆M++
1
2M2+
]minus 1
Λ2
g2M
∆+
g2E
2M2+
[1
2∆minus 3
M+
]minusgMgE
M+
[1
∆2minus 5
∆M++
1
M2+
]+
1
Λ2
2gMgEM+
+2gMgC∆M2
+
minus gEgCM3
+
) (C21)
40
d δLT (Q2)
dQ2
∣∣∣∣∣Q2=0
=e2M∆∆
4πMNM2+
(g2E
∆2M2+
[1
∆minus 4
M+
]minus g2
C
∆M2∆M
2+
+gMgE∆2M+
[1
∆2
minus 3
∆M++
1
M2+
]minus 2
Λ2
gMgE∆2M+
+gMgC∆M2
∆
[1
2∆2minus 2
∆M++
1
2M2+
]minus gEgC
2M2∆M
2+
[7
∆+
1
M+
]) (C22)
dIA(Q2)
dQ2
∣∣∣∣∣Q2=0
= minusM2N
M2+
(g2M
2∆2+
g2E
MNM+minus 2gMgE
∆M+minus gEgCM∆M+
) (C23)
dI1(Q2)
dQ2
∣∣∣∣∣Q2=0
= minusM∆M2N
2M3+
(g2E
MNM∆minus gMgE∆MN
minus gEgCM2
∆
) (C24)
γ0(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2M2
N
π∆3M6+
(g2M
∆
[3M2
∆ + 2M∆MN minus 9M2N
∆2+
4M2+
Λ2
]
minus4gMgE
[5M2
∆ minus 9M2N
M+∆2+
2M+
Λ2
]+
g2E
∆M+[7M∆ minus 9MN ]
minus8gMgC∆
+4gEgCM+
) (C25)
[1] D Drechsel B Pasquini and M Vanderhaeghen Dispersion relations in real and virtual
Compton scattering Phys Rept 378 99 (2003) hep-ph0212124
[2] S E Kuhn J-P Chen and E Leader Spin structure of the nucleon mdash status and recent
results Prog Part Nucl Phys 63 1 (2009) arXiv08123535 [hep-ph]
[3] F Hagelstein R Miskimen and V Pascalutsa Nucleon polarizabilities from Compton scat-
tering to hydrogen atom Prog Part Nucl Phys 88 29 (2016) arXiv151203765 [nucl-th]
[4] B Pasquini and M Vanderhaeghen Dispersion theory in electromagnetic interactions Ann
Rev Nucl Part Sci 68 75 (2018) arXiv180510482 [hep-ph]
[5] S Gerasimov A Sum rule for magnetic moments and the damping of the nucleon magnetic
moment in nuclei Sov J Nucl Phys 2 430 (1966)
[6] S Drell and A C Hearn Exact sum rule for nucleon magnetic moments Phys Rev Lett
16 908 (1966)
[7] J S Schwinger Source theory viewpoints in deep inelastic scattering Proc Natl Acad Sci
USA 72 1 (1975)
[8] J S Schwinger Source theory viewpoints in deep inelastic scattering Electromagnetic In-
teractions and Field Theory Proceedings 14 Internationale Universitatswochen Schladming
Austria February 24-March 7 1975 Acta Phys Austriaca Suppl 14 471 (1975)
41
[9] J Schwinger Source theory discussion of deep inelastic scattering with polarized particles
Proc Natl Acad Sci USA 72 1559 (1975)
[10] A M Baldin Polarizability of nucleons Nucl Phys 18 310 (1960)
[11] M Gell-Mann M L Goldberger and W E Thirring Use of causality conditions in quantum
theory Phys Rev 95 1612 (1954)
[12] A Pineda Leading chiral logarithms to the hyperfine splitting of the hydrogen and muonic
hydrogen Phys Rev C 67 025201 (2003)
[13] C Peset and A Pineda The two-photon exchange contribution to muonic hydrogen from
chiral perturbation theory Nucl Phys B 887 69 (2014) arXiv14064524 [hep-ph]
[14] C Peset and A Pineda Model-independent determination of the two-photon exchange contri-
bution to hyperfine splitting in muonic hydrogen JHEP 04 060 arXiv161205206 [nucl-th]
[15] F Hagelstein and V Pascalutsa Proton structure in the hyperfine splitting of muonic hydro-
gen PoS CD15 077 (2016) arXiv151104301 [nucl-th]
[16] F Hagelstein Exciting Nucleons in Compton Scattering and Hydrogen-Like Atoms PhD
thesis Mainz U Inst Kernphys (2017) arXiv171000874 [nucl-th]
[17] F Hagelstein ∆(1232)-Resonance in the hydrogen spectrum Proceedings 11th International
Workshop on the Physics of Excited Nucleons (NSTAR 2017) Columbia SC USA August
20-23 2017 Few Body Syst 59 93 (2018) arXiv180109790 [nucl-th]
[18] Y Prok et al (CLAS) Moments of the spin structure functions gp1 and gd1 for 005 lt Q2 lt 30
GeV2 Phys Lett B 672 12 (2009) arXiv08022232 [nucl-ex]
[19] H Dutz et al (GDH) First measurement of the Gerasimov-Drell-Hearn sum rule for 1H from
07 GeV to 18 GeV at ELSA Phys Rev Lett 91 192001 (2003)
[20] M Amarian et al (Jefferson Lab E94010) Measurement of the generalized forward spin po-
larizabilities of the neutron Phys Rev Lett 93 152301 (2004) arXivnucl-ex0406005
[21] M Amarian et al The Q2 evolution of the generalized Gerasimov-Drell-Hearn integral for
the neutron using a 3He target Phys Rev Lett 89 242301 (2002) arXivnucl-ex0205020
[22] M Amarian et al (Jefferson Lab E94-010)Q2 evolution of the neutron spin structure moments
using a 3He target Phys Rev Lett 92 022301 (2004) arXivhep-ex0310003
[23] A Deur et al Experimental determination of the evolution of the Bjorken integral at low Q2
Phys Rev Lett 93 212001 (2004) arXivhep-ex0407007
[24] K Slifer Low Q2 measurement of gp2 and the δLT spin polarizability Spin structure at long dis-
tance Proceedings Workshop Newport News USA March 12ndash13 2009 nucl-ex09064775
(2009) AIP Conf Proc 1155 10106313203293 (2009) arXiv09064775 [nucl-ex]
[25] P Solvignon et al (E01-012) Moments of the neutron g2 structure function at intermediate
42
Q2 Phys Rev C 92 015208 (2015) arXiv13044497 [nucl-ex]
[26] A Deur Experimental studies at low Q2 of the spin structure of the nucleon at Jefferson Lab
in 9th International Workshop on Chiral Dynamics (CD18) Durham NC USA September
17-21 2018 (2019) arXiv190305661 [nucl-ex]
[27] V Sulkosky et al (Jefferson Lab E97-110) Measurement of the 3He spin-structure functions
and of neutron (3He) spin-dependent sum rules at 0035 le Q2 le 024 GeV2 Phys Lett B
805 135428 (2020) arXiv190805709 [nucl-ex]
[28] K Adhikari et al (CLAS) Measurement of the Q2 Dependence of the Deuteron Spin Structure
Function g1 and its Moments at Low Q2 with CLAS Phys Rev Lett 120 062501 (2018)
arXiv171101974 [nucl-ex]
[29] V Lensky F Hagelstein A Hiller Blin and V Pascalutsa Comment on rdquoMeasurement of
the Q2 Dependence of the Deuteron Spin Structure Function g1 and its Moments at Low Q2
with CLASrdquo (2018) arXiv180603219 [nucl-th]
[30] J M Alarcn F Hagelstein V Lensky and V Pascalutsa Forward doubly-virtual Compton
scattering off the nucleon in chiral perturbation theory at NLO the subtraction function and
moments of unpolarized structure functions (2020) arXiv200509518 [hep-ph]
[31] V Bernard E Epelbaum H Krebs and U-G Meiszligner New insights into the spin structure
of the nucleon Phys Rev D 87 054032 (2013) arXiv12092523 [hep-ph]
[32] V Lensky J M Alarcon and V Pascalutsa Moments of nucleon structure functions at
next-to-leading order in baryon chiral perturbation theory Phys Rev C 90 055202 (2014)
arXiv14072574 [hep-ph]
[33] V Pascalutsa and M Vanderhaeghen Chiral effective-field theory in the ∆(1232) region
I Pion electroproduction on the nucleon Phys Rev D 73 034003 (2006) arXivhep-
ph0512244
[34] V Pascalutsa and M Vanderhaeghen Electromagnetic nucleon-to-Delta transition in chiral
effective field theory Phys Rev Lett 95 232001 (2005) arXivhep-ph0508060
[35] T R Hemmert B R Holstein and J Kambor Systematic 1M expansion for spin 32
particles in baryon chiral perturbation theory Phys Lett B 395 89 (1997) arXivhep-
ph9606456
[36] V Pascalutsa and D R Phillips Effective theory of the ∆(1232) in Compton scattering off
the nucleon Phys Rev C 67 055202 (2003) arXivnucl-th0212024
[37] V Pascalutsa M Vanderhaeghen and S N Yang Electromagnetic excitation of the ∆(1232)-
resonance Phys Rept 437 125 (2007) arXivhep-ph0609004
[38] K A Olive et al (Particle Data Group) Review of Particle Physics Chin Phys C 38 090001
43
(2014)
[39] F E Low Scattering of light of very low frequency by systems of spin 12 Phys Rev 96
1428 (1954)
[40] M Gell-Mann and M L Goldberger Scattering of low-energy photons by particles of spin
12 Phys Rev 96 1433 (1954)
[41] A M Harun ar-Rashid A simple derivation of Schwingerrsquos sum rule for spin dependent
structure functions Nuovo Cim A 33 447 (1976)
[42] F Hagelstein and V Pascalutsa Dissecting the hadronic contributions to (g minus 2)micro by
Schwingerrsquos sum rule Phys Rev Lett 120 072002 (2018) arXiv171004571 [hep-ph]
[43] V Lensky V Pascalutsa M Vanderhaeghen and C Kao Spin-dependent sum rules con-
necting real and virtual Compton scattering verified Phys Rev D 95 074001 (2017)
arXiv170101947 [hep-ph]
[44] H Burkhardt and W N Cottingham Sum rules for forward virtual Compton scattering
Annals Phys 56 453 (1970)
[45] V Lensky and V Pascalutsa Predictive powers of chiral perturbation theory in Compton
scattering off protons Eur Phys J C 65 195 (2010) arXiv09070451 [hep-ph]
[46] H Grieszlighammer J McGovern D Phillips and G Feldman Using effective field theory to
analyse low-energy Compton scattering data from protons and light nuclei Prog Part Nucl
Phys 67 841 (2012) arXiv12036834 [nucl-th]
[47] H W Grieszlighammer J A McGovern and D R Phillips Nucleon polarisabilities at and
beyond physical pion masses Eur Phys J A 52 139 (2016) arXiv151101952 [nucl-th]
[48] E Epelbaum H Krebs and U-G Meiszligner Improved chiral nucleon-nucleon potential up to
next-to-next-to-next-to-leading order Eur Phys J A 51 53 (2015) arXiv14120142 [nucl-
th]
[49] E Epelbaum H Krebs and U-G Meiszligner Precision nucleon-nucleon potential at fifth order
in the chiral expansion Phys Rev Lett 115 122301 (2015) arXiv14124623 [nucl-th]
[50] C W Kao T Spitzenberg and M Vanderhaeghen Burkhardt-Cottingham sum rule and
forward spin polarizabilities in heavy baryon chiral perturbation theory Phys Rev D 67
016001 (2003) arXivhep-ph0209241
[51] C-W Kao D Drechsel S Kamalov and M Vanderhaeghen Higher moments of nucleon
spin structure functions in heavy baryon chiral perturbation theory and in a resonance model
Phys Rev D 69 056004 (2004) arXivhep-ph0312102
[52] T Becher and H Leutwyler Baryon chiral perturbation theory in manifestly Lorentz invariant
form Eur Phys J C 9 643 (1999) arXivhep-ph9901384
44
[53] J Gasser M E Sainio and A Svarc Nucleons with chiral loops Nucl Phys B 307 779
(1988)
[54] L S Geng J Martin Camalich L Alvarez-Ruso and M J Vicente Vacas Leading SU(3)-
breaking corrections to the baryon magnetic moments in chiral perturbation theory Phys
Rev Lett 101 222002 (2008) arXiv08051419 [hep-ph]
[55] D Drechsel S S Kamalov and L Tiator The GDH sum rule and related integrals Phys
Rev D 63 114010 (2001) arXivhep-ph0008306
[56] D Drechsel O Hanstein S S Kamalov and L Tiator A unitary isobar model for pion
photo- and electroproduction on the proton up to 1 GeV Nucl Phys A 645 145 (1999)
arXivnucl-th9807001
[57] L Tiator private communication (2020)
[58] V Bernard T R Hemmert and U-G Meiszligner Spin structure of the nucleon at low energies
Phys Rev D 67 076008 (2003) arXivhep-ph0212033
[59] O Gryniuk F Hagelstein and V Pascalutsa Evaluation of the forward Compton scattering
off protons II Spin-dependent amplitude and observables Phys Rev D 94 034043 (2016)
arXiv160400789 [nucl-th]
[60] R Zielinski The g2p Experiment A Measurement of the Protonrsquos Spin Structure Functions
PhD thesis New Hampshire U (2010) arXiv170808297 [nucl-ex]
[61] N Guler et al (CLAS) Precise determination of the deuteron spin structure at low to mod-
erate Q2 with CLAS and extraction of the neutron contribution Phys Rev C 92 055201
(2015) arXiv150507877 [nucl-ex]
[62] C E Carlson V Nazaryan and K Griffioen Proton structure corrections to electronic
and muonic hydrogen hyperfine splitting Phys Rev A 78 022517 (2008) arXiv08052603
[physicsatom-ph]
[63] V Bernard N Kaiser and U-G Meiszligner Chiral dynamics in nucleons and nuclei Int J
Mod Phys E 4 193 (1995) arXivhep-ph9501384
[64] T R Hemmert B R Holstein and J Kambor ∆(1232) and the polarizabilities of the nucleon
Phys Rev D 55 5598 (1997) arXivhep-ph9612374
[65] V Pascalutsa and R Timmermans Field theory of nucleon to higher-spin baryon transitions
Phys Rev C 60 042201 (1999) arXivnucl-th9905065
[66] V Pascalutsa Quantization of an interacting spin-32 field and the Delta isobar Phys Rev
D 58 096002 (1998) arXivhep-ph9802288
[67] H Krebs Double Virtual Compton Scattering and SpinStructure of the Nucleon PoS
CD2018 031 (2019)
45
[68] N Kochelev and Y Oh Axial anomaly and the δLT puzzle Phys Rev D 016012 (2012)
arXiv11034892 [hep-ph]
[69] D Drechsel S Kamalov and L Tiator Unitary isobar model ndash MAID2007 Eur Phys J A
69 (2007) available at httpsmaidkphuni-mainzde arXiv07100306 [nucl-th]
[70] P J Mohr B N Taylor and D B Newell CODATA recommended values of the fundamental
physical constants 2010 Rev Mod Phys 84 1527 (2012)
[71] J Ahrens et al (GDH A2) First measurement of the Gerasimov-Drell-Hearn integral for 1H
from 200 to 800 MeV Phys Rev Lett 87 022003 (2001) arXivhep-ex0105089 [hep-ex]
[72] K Helbing (GDH) Experimental verification of the GDH sum rule at ELSA and MAMI Nucl
Phys Proc Suppl 105 113 (2002)
[73] R Bradford A Bodek H S Budd and J Arrington A New parameterization of the nucleon
elastic form-factors NuInt05 proceedings of the 4th International Workshop on Neutrino-
Nucleus Interactions in the Few-GeV Region Okayama Japan 26-29 September 2005 Nucl
Phys Proc Suppl 159 127 (2006) arXivhep-ex0602017
[74] J D Bjorken Applications of the chiral U(6)timesU(6) algebra of current densities Phys Rev
148 1467 (1966)
[75] J D Bjorken Inelastic scattering of polarized leptons from polarized nucleons Phys Rev D
1 1376 (1970)
[76] V Bernard T R Hemmert and U-G Meiszligner Novel analysis of chiral loop effects in
the generalized Gerasimov-Drell-Hearn sum rule Phys Lett B 545 105 (2002) arXivhep-
ph0203167
[77] A Deur et al Experimental study of isovector spin sum rules Phys Rev D 78 032001
(2008) arXiv08023198 [nucl-ex]
[78] B Pasquini P Pedroni and D Drechsel Higher order forward spin polarizability Phys Lett
B 687 160 (2010) arXiv10014230 [hep-ph]
[79] R Jaffe g2ndashThe nucleonrsquos other spin-dependent structure function Comments Nucl Part
Phys 19 239 (1990)
[80] E V Shuryak and A Vainshtein Theory of power corrections to deep inelastic scattering in
quantum chromodynamics (II) Qminus4 effects polarized target Nucl Phys B 201 141 (1982)
[81] S Wandzura and F Wilczek Sum rules for spin dependent electroproduction Test of rela-
tivistic constituent quarks Phys Lett B 72 195 (1977)
[82] B W Filippone and X-D Ji The spin structure of the nucleon Adv Nucl Phys 26 1
(2001) arXivhep-ph0101224
[83] M Burkardt The g2 structure function Proceedings Workshop on Spin structure at long
46
distance Newport News USA March 12-13 2009 AIP Conf Proc 1155 26 (2009)
arXiv09054079 [hep-ph]
[84] A Deur S J Brodsky and G F de Teramond The spin structure of the nucleon Rept
Prog Phys 82 076201 (2019) arXiv180705250 [hep-ph]
[85] B R Holstein D Drechsel B Pasquini and M Vanderhaeghen Higher order polarizabilities
of the proton Phys Rev C 61 034316 (2000) arXivhep-ph9910427
[86] B R Holstein V Pascalutsa and M Vanderhaeghen Sum rules for magnetic moments and
polarizabilities in QED and chiral effective-field theory Phys Rev D 72 094014 (2005)
arXivhep-ph0507016
[87] B Pasquini D Drechsel and L Tiator Invariant amplitudes for pion electroproduction Eur
Phys J A 34 387 (2007) arXiv07122327 [hep-ph]
[88] A J Buchmann Non-spherical proton shape and hydrogen hyperfine splitting Proceedings
International Workshop on Precision Physics of Simple Atomic Systems (PSAS 2008) Wind-
sor Ontario Canada July 21-26 2008 Can J Phys 87 773 (2009) arXiv09104747
[physicsatom-ph]
47
000 005 010 015 020 025 030
-0004
-0002
0000
0002
0004
0006
0008
0010
Q2GeV
2D
d2
proton
000 005 010 015 020 025 030
-0005
0000
0005
0010
0015
Q2GeV
2D
d2
neutron
000 005 010 015 020 025 030
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
6D
proton
000 005 010 015 020 025 030
-1
0
1
2
3
Q2 GeV2D
Γ0
10-
4fm
6D
neutron
FIG 9 Contributions of the different orders to the chiral predictions of d2(Q2) upper panel and
γ0(Q2) lower panel for the proton (left) and neutron (right) Red solid line πN -loop contribu-
tion green dot-dashed line ∆-exchange contribution orange dotted line π∆-loop contribution
blue long-dashed line total result purple dot-dot-dashed line total result without gC contribution
black short-dashed line total result without gM dipole
bringing it into a better agreement with data In general the BχPT curves start above the
empirical data points at the real-photon point and then decrease asymptotically to zero
above Q2 gt 01 GeV2 On the other hand the MAID prediction reproduces the empirical
data at the real-photon point then decreases to negative values until about Q2 gt 006 GeV2
from where it also starts to asymptotically approach zero Consequently our NLO BχPT
prediction of γ0(Q2) is consistently above the MAID prediction This is very different to
what we saw for IA(Q2) in Fig 4 upper panel where the MAID prediction at the real-
photon point is above the experimental value While the agreement of our predictions with
the empirical data is in general quite good for all moments of σTT (νQ2) one should point
out that both for γ0n(Q2) and γ0p(Q2) we overestimate the data at low Q2 For IA(Q2) such
observation cannot be made because ∆IA(0) = 0 and thus IA(0) is given by the empirical
Pauli form factor only From IA(Q2) γ0(Q2) and γ0(Q2) the latter has the smallest however
25
non-negligible dependence on gC and the dipole in gM cf Fig 9 lower panel
The πN -loop ∆-exchange and π∆-loop contributions to the NLO BχPT prediction of
the static fifth-order forward spin polarizability amount to in units of 10minus4 fm6
γ0p = 112(30) asymp 208minus 096minus 001 (42a)
γ0n = 195(30) asymp 292minus 096minus 001 (42b)
while the slope is composed as follows in units of 10minus4 fm8
dγ0p(Q2)
dQ2
∣∣∣∣Q2=0
= minus084(10) asymp minus100 + 016 + 000 (43a)
dγ0n(Q2)
dQ2
∣∣∣∣Q2=0
= minus142(15) asymp minus158 + 016 + 000 (43b)
Note that the HB prediction of the static polarizability γ0p (423 at O(p3) and 365 at O(ε3)
[78 85]) is almost one order of magnitude larger than the empirical value and therefore not
shown in Fig 8
G Summary
Our results are summarized in Table II where we give the contributions of the different
orders to the chiral predictions of the polarizabilities and their slopes at the real-photon
point A quantitative comparison of our predictions for the static spin polarizabilities to
the work of Bernard et al [31] and different empirical evaluations is shown in Table III We
can see that the inclusion of the Delta turns out to be very important for all moments of
the helicity-difference cross section To describe the Q2 behavior of the polarizabilities the
magnetic coupling of the N rarr ∆ transition should be modified by a dipole form factor as
has been observed previously in the description of electroproduction data [33] This dipole
form factor effectively takes account of vector-meson exchanges The Coulomb-quadrupole
N rarr ∆ transition despite its subleading order is important in the description of some
moments of spin structure functions This is contrary to what we saw for the moments of
unpolarized structure functions [30] where the Coulomb coupling had a negligible effect
The π∆ loops are mainly relevant for the generalized GDH integrals
26
TABLE II The NLO BχPT predictions for the forward VVCS polarizabilities and their slopes at
Q2 = 0 The contributions of the πN loops the ∆ exchange and the π∆ loops are shown together
with the combined total result Note that IA(0) = I1(0) = d2(0) = 0 and (d2)prime = 0
πN loops ∆ exchange π∆ loops Total
γ0 p 201 minus284 minus010minus093(92)
(10minus4 fm4) n 298 003(92)
δLT p 150 minus016 minus002132(15)
(10minus4 fm4) n 235 218(23)
γ0 p 208 minus096 minus001112(30)
(10minus4 fm6) n 292 195(30)
(γ0)prime p minus033011 001
minus022(4)
(10minus4 fm6) n minus073 minus061(7)
(δLT )prime p minus080 minus004 minus001minus085(8)
(10minus4 fm6) n minus119 minus124(12)
(γ0)prime p minus100016 000
minus084(10)
(10minus4 fm8) n minus158 minus142(15)
(∆IA)prime p 238 minus1121 025minus858(343)
(GeVminus2) n 141 minus955(343)
(∆I1)prime p 034 minus053 058039(4)
(GeVminus2) n minus107 minus101(10)
TABLE III Our NLO BχPT predictions for the spin polarizabilities at Q2 = 0 compared with
the BχPT+∆ predictions from Bernard et al [31] and the available empirical information Where
the reference is not given the empirical number is provided by the MAID analysis [55 69] with
unspecified uncertainty
Proton Neutron
This work BχPT+∆ Empirical This work BχPT+∆ Empirical
γ0 minus093(92) minus174(40) minus100(8)(12) [19] 003(92) minus077(40) minus0005
(10minus4 fm4) minus090(8)(11) [78] [MAID]
minus0929(105) [59]
δLT 132(15) 240(1) 134 218(23) 238(3) 203
(10minus4 fm4) [MAID] [MAID]
γ0 112(30) 060(7)(7) [78] 195(30) 123
(10minus4 fm6) 0484(82) [59] [MAID]
IV CONCLUSIONS
We have presented a complete NLO calculation of the polarized non-Born VVCS ampli-
tudes in covariant BχPT with pion nucleon and ∆(1232) fields The dispersion relations
between the VVCS amplitudes and the tree-level photoabsorption cross sections served as a
27
cross-check of these calculations
The obtained moments of the proton and neutron spin structure functions related to gen-
eralized polarizabilities and GDH-type integrals agree well with the available experimental
data The description of their Q2 evolution is improved compared to the previous χPT pre-
dictions In particular the NLO BχPT predictions obtained here give a better description
of the empirical data (eg from the Jefferson Laboratory ldquoSpin Physics Programrdquo) than
the HB [50 51] and IR [58] calculations
The demonstrated predictive power of the χPT framework amplitudes makes it well
suited for extending the χPT evaluation of the TPE effect in the hyperfine structure of
(muonic-)hydrogen [15ndash17] to next-to-leading order
ACKNOWLEDGEMENTS
We thank Lothar Tiator and Marc Vanderhaeghen for helpful discussions This work
is supported by the Deutsche Forschungsgemeinschaft (DFG) through the Collaborative
Research Center [The Low-Energy Frontier of the Standard Model (SFB 1044)] JMA ac-
knowledges support from the Community of Madrid through the ldquoPrograma de atraccion de
talento investigador 2017 (Modalidad 1)rdquo and the Spanish MECD grants FPA2016-77313-P
FH gratefully acknowledges financial support from the Swiss National Science Foundation
Appendix A Tensor decompositions of the VVCS amplitudes
In this appendix we review the decomposition of the forward VVCS process into tensor struc-
tures and scalar amplitudes In particular we consider the connection between the covariant and
the semi-relativistic decomposition in the lab frame that is defined in terms of the conventional
transverse longitudinal transverse-transverse and transverse-longitudinal amplitudes
As explained in Sec II A the process of forward VVCS off the nucleon can be described in
terms of four explicitly covariant amplitudes S1 2 and T1 2 [3]
T (νQ2) =
(minusgmicroν +
qmicroqν
q2
)T1(νQ2) +
1
M2N
(pmicro minus p middot q
q2qmicro)(
pν minus p middot qq2
qν)T2(νQ2) (A1)
minus 1
MNγmicroναqα S1(νQ2)minus 1
M2N
(γmicroνq2 + qmicroγναqα minus qνγmicroαqα
)S2(νQ2)
εprimelowastmicro εν
where εmicro (εprimelowastmicro ) are the incoming (outgoing) photon polarization vectors ν is the photon lab-frame
energy and Q2 is the photon virtuality Alternatively the decomposition in the laboratory frame
28
(which in the forward case coincides with the Breit frame) is parametrized in terms of the nucleon
Pauli matrices ~σ and the four scalar functions fL fT gTT and gLT
T (νQ2) = ε0 εprimelowast0 fL(νQ2) + (~ε primelowast middot ~ε ) fT (νQ2) + i~σ middot (~ε primelowast times ~ε ) gTT (νQ2) (A2)
minusi~σ middot [(ε0~εprimelowast minus ~ε ε primelowast0 )times q] gLT (νQ2)
Here ~q and q = ~q|~q | are the photon three-momentum in the lab system and its unit vector The
modified polarization vector components are given by
ε0 =
[ε0 minus
ν
|~q | (~ε middot q )
] |~q |Q ~ε = ~εminus q (~ε middot q ) (A3)
where ε = (ε0~ε ) is the usual incoming photon polarization vector and εprimelowast the outgoing polarization
vector The LEX of the lab frame amplitudes [Eq (10)] can serve in particular as the definition
of the generalized polarizabilities The lab frame amplitudes are also conveniently used for the
definition of the response functions see the example of the scalar amplitude gLT (νQ2) and the
corresponding response function σLT (νQ2) below in App B
Appendix B Photoabsorption cross sections
In the forward kinematics the spin-dependent VVCS amplitudes and the spin polarizabilities
can be described in terms of the polarized structure functions g1(xQ2) and g2(xQ2) or equiv-
alently the helicity-difference cross section σTT (νQ2) and the longitudinal-transverse response
function σLT (νQ2) with the help of dispersion relations (5) and the optical theorem (3) In this
way the photoabsorption cross sections measured in electroproduction processes form the basis
for most empirical evaluations shown throughout Sec III In the following we present the BχPT
predictions for the tree-level cross sections of πN - π∆- and ∆-production through photoabsorption
on the nucleon cf Figs 8 9 and 10 in Ref [30] In Secs B 1 and B 2 we will discuss the leading
πN -production channel and the ∆-production channel respectively We used these cross sections
to verify the polarizability predictions obtained otherwise from the calculated non-Born VVCS
amplitudes Due to the bad high-energy behavior of the π∆-production cross sections in BχPT
cf Fig 10 the dispersion relations in Eq (5) require further subtractions for a reconstruction of
the π∆-loop contribution to the spin-dependent VVCS amplitudes Therefore not all polarizabil-
ities could be verified but only those appearing as higher-order terms in the LEX of the VVCS
amplitudes such as γ0 [16]
29
proton neutron
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Neutron
02 04 06 08 100
50
100
150
200
250
[GeV]
T [b]
02 04 06 08 10
-50
0
50
100
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Neutron
02 04 06 08 100
50
100
150
200
250
[GeV]
T [b]
02 04 06 08 10
-50
0
50
100
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02040608100
50
100
150
200
250
300
[GeV]
T[b]
0204060810
-50
0
50
[GeV]
TT[b]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
0204060810
00
02
04
06
[GeV]
LTQ
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
0204060810
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
FIG 10 Photoabsorption cross sections for πN (red) and π∆ production (orange) with Q2 = 0
(solid) and Q2 = 01 GeV2 (dashed for πN and dotted for π∆ channel)
1 πN-production channel
In order to extract the response function σLT (νQ2) we have developed a method similar to
the one used to calculate σTT (νQ2) see for example Ref [86] For σLT (νQ2) however the
calculation is more complicated because one has to take into account that the associated Compton
process involves a spin-flip of the nucleon as illustrated in Fig 11 When calculating the cross
section the product of the incoming nucleon spinors has to reflect this flip
The forward VVCS amplitude related to σLT (νQ2) mdash and δLT (Q2) mdash is gLT (νQ2) It can
be extracted from Eq (A2) if one takes the modified polarization vector components in Eq (A3)
with ε = εL and εprimelowast = εlowastplusmn as input where εL = 1Q(|~q | 0 0 q0) and εplusmn = ∓ 1radic
2(0 1plusmni 0) are the
standard longitudinal and transverse polarization vectors respectively For εL and εlowastplusmn only the
choice of helicities hprime = plusmn12 and h = ∓12 gives a non-zero contribution and one obtains
χdaggerhprime T (νQ2)χh = χdaggerhprimeminusi~σ middot [(ε0~εprimelowast minus ~ε ε primelowast0 )times q] gLT (νQ2)χh =
radic2 gLT (νQ2) (B1)
where χh and χdaggerhprime are two-component Pauli spinors with opposite helicities or here spins
Let us now consider the related photoabsorption process and in particular the tree-level γlowastN rarr
30
s = 12
sprime = minus12
λ = 0 λprime = 1
rprime rprime
s=12sprime=minus1
2
λ=0λprime=1
rprimerprime
s=12
sprime=minus12
λ=0λprime=1
rprimerprime hacute=frac12h=-frac12
λacute=1λ=0
FIG 11 Relation between the forward Compton process and the photoabsorption process given
by the optical theorem In particular we show the longitudinal-transverse contribution The
double-line arrows represent the spin of the external particles while the dot represents the scalar
(longitudinal) polarization of the incoming photon Inside the blob the intermediate states are
represented eg nucleons with spins rprime (which are averaged in the calculation of the cross section)
and pions
πN channel see diagrams in Fig 8 of Ref [30] We define the πN -production amplitude as
T = uhB (PB)sumi
Ai(s t)Γi uhA(PA) (B2)
with the Dirac structures
Γ1 = γ5 (B3a)
Γ2 =1
2
[qA ε
]γ5 (B3b)
where uhA(PA) and udaggerhB (PB) are the Dirac spinors and PA and PB are the four-momenta of
the incoming and outgoing nucleons respectively When calculating the photoabsorption cross
section related to the VVCS amplitude in Eq (B1) the nucleon spin flip should be implemented
by uhprime(PA) in T dagger and uh(PA) in T together with the appropriate transverse and longitudinal
photon polarization vectors εlowastplusmn and εL
However if one wants to use the properties of the Dirac matrices it is more useful to construct
an operator to produce this spin flip in the external nucleons of Fig 11 This is accomplished by
introducing the projector ΓLT equiv 12radic
2(γ1 + iγ2)γ5 which also takes into account the extra factor
radic2 in Eq (B1) We checked that with this projector one correctly extracts δLT by comparing the
HB limit of our result to the HB result of Ref [50] where the authors calculate this polarizability
from the Compton amplitude directly With all those ingredients the longitudinal-transverse cross
section is calculated in the following way
σLT (νQ2) =1
64π2 s
|~pf |cm
|~pi|cm
int 1
minus1dcos θ
sumij
AiAdaggerjXij (B4)
31
with
Xij = Tr[(PB +MN )Γi(PA +MN )ΓLTγ0Γdaggerjγ
0] (B5)
where θ is the scattering angle in the center-of-mass (cm) frame and |~pi|cm (|~pf |cm) is the three-
momentum of an incoming (outgoing) particle in the cm frame An explicit calculation of the
matrix Xij leads to
X = MNQ
0 2(PB minus PA) middot εLminusradic
2|~qf |cm sin θ (sminus u)
(B6)
where |~qi|cm (|~qf |cm) is the relative three-momentum of the incoming (outgoing) particles in the
cm frame Here s t and u are the usual Mandelstam variables For the different γlowastN rarr πN
channels we obtain the following amplitudes Ai where we introduce qA as the four-momentum of
the incoming photon and qB as the four-momentum of the outgoing pion
bull γlowastprarr π0p
A1 =e gAMN
fπ
[2PA middot ε+ qA middot ε
sminusM2N
+2PB middot εminus qA middot ε
uminusM2N
] (B7a)
A2 =e gAMN
fπ
[1
sminusM2N
+1
uminusM2N
] (B7b)
bull γlowastprarr π+n
A1 =
radic2 e gAMN
fπ
[2PA middot ε+ qA middot ε
sminusM2N
+2(PA minus PB) middot ε+ qA middot ε
tminusm2π
] (B8a)
A2 =
radic2 e gAMN
fπ(sminusM2N )
(B8b)
bull γlowastnrarr π0n
A1 = 0 (B9a)
A2 = 0 (B9b)
bull γlowastnrarr πminusp
A1 =
radic2 e gAMN
fπ
[2PB middot εminus qA middot ε
uminusM2N
minus 2(PA minus PB) middot ε+ qA middot εtminusm2
π
] (B10a)
A2 =
radic2 e gAMN
fπ(uminusM2N )
(B10b)
The analytical expressions shown above were checked with the amplitudes given in Ref [87] An-
alytical expressions for the tree-level γlowastN rarr πN channel of the σLT (νQ2) and σTT (νQ2) cross
32
sections are given below (proton channels π+n and π0p neutron channel πminusp) We checked that
they reproduce the known results in the real-photon limit [45 86] To shorten the final expressions
for the cross sections which are considerably longer for finite Q2 than in the real-photon limit we
define the following dimensionless kinematic variables
αγ = (ENi )cmradics =
s+M2N +Q2
2s (B11)
απ = (ENf )cmradics =
s+M2N minusm2
π
2s (B12)
βγ = Eγcmradics =
sminusM2N minusQ2
2s (B13)
βπ = Eπcmradics =
sminusM2N +m2
π
2s (B14)
λγ = |~qi|cmradics =
radic(sminusM2
N minusQ2)2 + 4sQ2
2s (B15)
λπ = |~qf |cmradics =
radic(sminusM2
N +m2π)2 minus 4sm2
π
2s (B16)
Here (ENi )cm and (ENf )cm are the energies of the incoming and outgoing nucleon Eγcm is the energy
of the incoming photon Eπcm is the energy of the outgoing pion all in the cm frame
σ(π+n)TT = minus e2g2
AM2N
64πf2πs
3(sminusM2N )2λ4
γ
4sλπλγ
[(M2
N minus s)(M2N minusQ2 minus s)(Q2 + 2sβγβπ)
+ 2s(m2πQ
2 minus (M2N minus s)
(M2N + s(minus1 + 2βγβπ)
))λ2γ
]+ (M2
N minus s)(M2N minusQ2 minus s)
times (Q2 + 2sβγβπ minus 2sλπλγ)(Q2 + 2s(βγβπ + λπλγ)
)times log
(Q2 + 2sβγβπ minus 2sλγλπQ2 + 2sβγβπ + 2sλγλπ
) (B17)
33
σ(π0p)TT =
e2g2AM
2Nλπ
64πf2πs(sminusM2
N )2λγ
1
s2λπλ3γ
[(M2
N minus s)(M2N minusQ2 minus s)
(3M2
N + s(2βγβπ minus 3))(M2N
+ s(2βγβπ minus 1))
+ 2s
((2m2
πQ2 + (M2
N minus s)2)(M2
N minus s)minus 2(minusm2
πQ2 + (sminusM2
N )2)
times sβγβπ + 4s2(sminusM2N )β2
γβ2π
)λ2γ
]arctan
(2sλπλγ
M2N + s(2βγβπ minus 1)
)+ 2
[minus 2m2
πQ2
+ (M2N minus s)
(2(sminusM2
N ) + 4sβγβπ +(sminusM2
N +Q2)(3M2N minus 3s+ 2sβγβπ)
sλ2γ
+m2πQ
2(sminusM2N )
(M2N minus s+ 2sβγβπ)(M2
N + s(minus1 + 2βγβπ minus 2λπλγ))
+m2πQ
2(sminusM2N )
(M2N minus s+ 2sβγβπ)(M2
N + s(minus1 + 2βγβπ + 2λπλγ))
)
+1
λπλγ
(βγβπ
(minusm2
πQ2 + (M2
N minus s)(M2N minus s+ 2sβγβπ)
)minus (M2
N minus s)(M2N minusQ2 minus s)λ2
π
)log
(M2N + s(minus1 + 2βγβπ + 2λπλγ)
M2N + s(minus1 + 2βγβπ minus 2λπλγ)
)] (B18)
σ(πminusp)TT =
e2g2AM
2N
64πf2πs
3λ4γ(M2
N + s(2βπβγ minus 1))2 minus 4λ2πs
2λ2γ)
4sλπλγ
((minusM2
N +Q2 + s)(M2N
+ s(2βπβγ minus 1))
2 minus 2sλ2γ
(2λ2
πs(minusM2N +Q2 + s) +m2
πQ2))
+((M2N + s(2βπβγ minus 1)
)2
minus 4λ2πs
2λ2γ
)((M2N + s(2βπβγ minus 1)
)(M2N minus 2Q2 minus s(2βπβγ + 1)
)+ 4λ2
πs2λ2
γ
)times log
(minusM2
N minus 2βπsβγ minus 2λπsλγ + s)minus(M2N + s(2βπβγ minus 2λπλγ minus 1)
)(M2N + s(minus1
+ 2βπβγ + 2λπλγ))[((
M2N + s(2βπβγ minus 1)
)(M2N minus 2Q2 minus s(2βπβγ + 1)
)+ 4λ2
πs2λ2
γ
)times log
(minusM2
N minus 2βπsβγ + 2λπsλγ + s)
+((Q2 + 2βπsβγ)2 minus 4λ2
πs2λ2
γ
)times log
(Q2 + 2βπsβγ minus 2λπsλγQ2 + 2sβπβγ + 2λπsλγ
)]+ 4sλ2
γ
((M2
N + s(2βπβγ minus 1))2 minus 4λ2πs
2λ2γ
)times (M2
N minus s) arctanh
(2λπsλγ
M2N + 2βπsβγ minus s
) (B19)
34
σ(π+n)LT =
e2g2AM
3Nλπ
32πf2πQs
3(sminusM2N )2λ4
γ
2sλγ
[(M2
N minus s)(Q2 + 2sβ2γ)(Q2 + 2sβγβπ)
minus 4s
((M2
N minus s)(Q2 minus 2s(απ minus 1)βγ
)+Q2sβγβπ
)λ2γ + 8s3(minus1 + απ)λ4
γ
]minus M2
N minus sλπ
[(Q2 + 2sβ2
γ)(Q2 + 2sβγβπ)2 + 4s2(2(απ minus 1)βγ(Q2 + 2sβγβπ)
minusQ2λ2π
)λ2γ + 8s3(απ minus 1)2λ4
γ
]arctan
(2sλπλγ
Q2 + 2sβγβπ
) (B20)
σ(π0p)LT =
e2g2AM
3Nλπ
16πf2πQs(sminusM2
N )2λγ
1
minus2s(M2N + s(minus1 + 2βγβπ))2λ2
γ + 8s3λ2πλ
4γ
[minus 3M8
N (Q2
+ 2sβ2γ) + 2M4
Ns2
(minus (Q2 + 2sβ2
γ)(minus3 + 2βγβπ) +(Q2(minus3 + 2βγβπ + 6λ2
π) + 2sβγ
times (minus3 + 12απ + 2βγβπ minus 12απβγβπ + 4βγλ2π))λ2γ minus 4s(α2
π minus 1)λ4π
)+ 2M2
Ns3
[minus (Q2
+ 2sβ2γ)(minus1 + 2βγβπ)
(6 + βγβπ(minus9 + 2βγβπ)
)+
(Q2(3minus 12λ2
π + 4βγβπ(minus1 + 2βγβπ
+ λ2π))minus 2sβγ
(minus 3 + 4απ(3 + 2απ(3 + 2βγβπ(minus3 + βγβπ)) + 4βγ(βπ + (2minus βγβπ)
times λ2π)))λ2γ + 8s
((απ minus 1)(1 + απ minus 2βγβπ) + 2απβγλ
2π
)λ4γ
)+ s4
((Q2 + 2sβ2
γ)
times (1minus 2βγβπ)2(2βγβπ minus 3) + 2(Q2(minus1 + 6λ2
π + 2βγβπ((1minus 2βγβπ)2 minus 2λ2π)) + 2sβγ
times (minus1 + 2βγβπ + 4(minus1 + βγβπ)(minusαπ + 2απβγβπ minus βγλ2π)))λ2γ minus 8
(s(απ minus 1)(απ
+ (1minus 2βγβπ)2) + 2βγ(2sαπ +Q2βγ)λ2π
)λ4γ + 32s(απ minus 1)λ2
πλ6γ
)+ 2M6
Ns
(Q2(6
minus 7βγβπ + λ2γ) + 2sβγ(βγ(6minus 7βγβπ) + (1minus 4απ)λ2
γ)
)]+sminusM2
N
4s2λπλ3γ
[(Q2 + 2sβ2
γ)
times(3M2
N + s(2βγβπ minus 3))(M2N + s(2βγβπ minus 1)
)+ 2s
(minusM2
N (Q2 + 2sβγ minus 8sαπβγ)
+ s(2sβγ(1minus 4απ + 4απβγβπ) +Q2(1minus 2λ2π)))λ2γ + 8s3(απ minus 1)λ4
γ
]
times arctan
(2sλπλγ
M2N + s(2βγβπ minus 1)
) (B21)
35
σ(πminusp)LT =
e2g2AM
3N
64πf2πQs
3λ4γ(M2
N minusQ2 minus s)(M2N + s(2βπβγ + 2λπλγ minus 1))
times
1
M2N + s(2βπβγ minus 2λπλγ minus 1)
[4λπsλγ(M2
N minusQ2 minus s)(
2sλ2γ
(s((1minus 2λ2
π)Q2
+ 2(2απ minus 1)sβγ(2βπβγ minus 1))minusM2N (Q2 + 2(1minus 2απ)sβγ)
)+ (Q2 + 2sβ2
γ)(M2N + s(2βπβγ minus 1)
)2 + 8(απ minus 1)απs
3λ4γ
)]minus(M2N + s(2βπβγ + 2λπλγ minus 1)
)times[
log
(minusM2N minus 2βπsβγ minus 2λπsλγ + s
minusM2N minus 2βπsβγ + 2λπsλγ + s
)(minus (Q2 + 2sβ2
γ)(M2N + s(2βπβγ minus 1)
)(minusM2
N + 2Q2 + 2βπsβγ + s)minus 2sλ2γ
(minusM2
N (Q2 + 2sβγ) +Q4 +Q2s(4απβγ
minus 2βγ minus 2λ2π + 1) + 2s2βγ(4(απ minus 1)βπβγ + 1)
)minus 8(απ minus 1)2s3λ4
γ
)+ 2
(4s2λ2
γ
(2(απ minus 1)βγ(Q2 + 2βπsβγ)minus λ2
πQ2)
+ (Q2 + 2βπsβγ)2(Q2 + 2sβ2γ)
+ 8(απ minus 1)2s3λ4γ
)arctanh
(2λπsλγ
Q2 + 2βπsβγ
)] (B22)
2 ∆-production channel
The tree-level ∆-exchange diagram in Fig 2 of Ref [30] contributes to the non-Born part of
the VVCS amplitudes The contribution of the ∆ exchange to the VVCS amplitudes can be split
into [17]
S∆-exch1 (νQ2) = S∆-pole
1 (νQ2) + S∆-exch1 (νQ2) (B23a)
νS∆-exch2 (νQ2) = νS∆-pole
2 (νQ2) + νS2∆-exch
(νQ2) (B23b)
and similarly for the unpolarized VVCS amplitudes discussed in Ref [30] Here we introduced the
∆-pole contributions S∆-polei and the ∆-non-pole contributions S∆-exch
i The former amplitudes
feature a pole at the ∆(1232)-production threshold and thus are proportional to
1
[sminusM2∆][uminusM2
∆]=
1
4M2N
1
ν2∆ minus ν2
(B24)
36
They can be reconstructed from the dispersion relations in Eq (5) with the tree-level ∆-production
cross sections as input cf Fig 10 in Ref [30]
σTT (νQ2) =π2α
M2NM
2+|~q |
minus g2
MMN (M+ + ν)|~q |2 +g2E(∆minus ν)(Q2 minusMNν)2
MN(B25a)
+g2CQ
4s(∆minus ν)
MNM2∆
minus 4gMgE(Q2 minusMNν)|~q |2 minus 4gMgCQ2|~q |2
+2gEgCQ
2[minusMNM∆ |~q |2 + s(Q2 +∆ν)
]MNM∆
δ(ν minus ν∆)
σLT (νQ2) =Qπ2α
M2NM
2+|~q |
g2E(MNν minusQ2) [M∆(MN + ν)minus s]
MN(B25b)
+g2CQ
2[MNM∆|~q |2 minus s(Q2 +∆ν)
]MNM2
∆
+ gMgEM∆|~q |2 minusgMgC(Q2 minusMNν)|~q |2
M∆
+gEgC(ν minus∆)(M2
N |~q |2 minus 2Q2s)
MNM∆
δ(ν minus ν∆)
with ∆ = M∆ minusMN M+ = M∆ + MN and the Mandelstam variable s = M2N + 2MNν minus Q2
Analytical expressions for the spin structure functions g1(xQ2) and g2(xQ2) can be constructed
from Eq (3) with the flux factor K(νQ2) = |~q | =radicν2 +Q2
In the ∆-non-pole contributions to S1(νQ2) and νS2(νQ2) the pole in ν at the ∆(1232)-
production threshold has canceled out
S∆-exch1 (νQ2) =
πα
MNM2+
[g2MQ
2+ + g2
E
(∆2 minus 3Q2
)+
4g2CQ
4
M2∆
minus 8gMgEM∆ωminus (B26a)
minus2gMgCQ2(MN minus 4M∆)
M∆+
2gEgCQ2(3MN minus 2M∆)
M∆
]
νS2∆-exch
(νQ2) =2πα
MNM2+
[g2EM∆∆ωminus +
g2M MNQ
2+
2+g2C Q
2(Q2 minus∆2)
2M∆(B26b)
+gEgM M∆(M∆ω+ minus 4MNωminus)minus gEgC ∆(2Q2 +MNω+)
+gMgC Q2(4MN minus ω+)
]+S∆-exch
2 (νQ2)
ν
[M2
∆ ω2+
M2N
+ ν2
]
with Q+ =radic
(M∆ +MN )2 +Q2 and ωplusmn = (M2∆minusM2
N plusmnQ2)2M∆ and the non-pole contribution
to S2(νQ2)
S∆-exch2 (νQ2) = minus2παMNν
M∆M2+
[gM + gE
]gC (B27)
These amplitudes to the contrary are not described by the tree-level ∆-production cross sections
in the standard dispersive approach [17] This peculiarity has been previously missed e g in the
calculation of the ∆-exchange contribution to the hydrogen hyperfine splitting in Ref [88] The
importance of including the ∆-non-pole contribution is also evident when considering the BC sum
37
rule in Eq (14) The ∆-pole terms by themselves violate the BC sum rule but cancel exactly with
the ∆-non-pole terms
limνrarr0
νS∆-pole2 (νQ2) + lim
νrarr0νS2
∆-exch(νQ2) = 0 (B28)
Appendix C Polarizabilities at Q2 = 0
In this section we give analytical expressions for the static values and slopes at Q2 = 0 of
the spin polarizabilities and moments of polarized structure functions In particular we give the
HB expansion of the πN -loop contributions and the ∆-exchange contributions The complete
expressions also for the π∆-loop contributions can be found in the Supplemented material Recall
that IA(0) = I1(0) = d2(0) = 0 and dd2(Q2)dQ2
∣∣∣Q2=0
= 0
1 πN-loop contribution
Here we give analytical expressions for the πN -loop contributions to the proton and neutron
spin polarizabilities expanded in powers of micro = mπMN viz the HB expansion Note that we
choose to expand here to a high order in micro the strict HB expansion would only retain the leading
term in an analogous NLO calculation
bull Static polarizabilities (Q2 = 0)
γ0p =e2g2
A
96π3f2πm
2π
1minus 21πmicro
8minus(
59
2+ 26 logmicro
)micro2 +
1875πmicro3
64
+3
(3
2+ 26 logmicro
)micro4 +
(C1)
γ0n =e2g2
A
48π3f2πm
2π
1
2minus 9πmicro
16minus 2micro2 logmicro+
75πmicro3
128minus 3micro4
4+
(C2)
δLTp =e2g2
A
192π3f2πm
2π
1minus 9πmicro
8+
(13
2minus 2 logmicro
)micro2 minus 465πmicro3
64
minus(
47
2+ 42 logmicro
)micro4 +
(C3)
δLTn =e2g2
A
96π3f2πm
2π
1
2+
3πmicro
16+ (1 + 2 log micro)micro2 minus 105πmicro3
128+
5micro4
4+
(C4)
38
δLTn =e2g2
A
96π3f2πm
2π
1
2+
3πmicro
16+ (1 + 2 log micro)micro2 minus 105πmicro3
128+
5micro4
4+
(C5)
γ0p =e2g2
A
16π3f2πm
4π
4
45minus 3πmicro
16+
14micro2
5minus 1813πmicro3
384minus 192
5(1 + log micro)micro4
+80703πmicro5
2048+
(C6)
γ0n =e2g2
A
16π3f2πm
4π
4
45minus 5πmicro
48+
4micro2
5minus 245πmicro3
384minus 32micro4 logmicro
15
+1323πmicro5
2048+
(C7)
bull Slopes of polarizabilities at Q2 = 0
dγ0p(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
2minus 45πmicro
4+ 223micro2 minus 28515πmicro3
64
minus9
(1953
4+ 449 logmicro
)micro4 +
570255πmicro5
128+
(C8)
dγ0n(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
2minus 81πmicro
8+ 94micro2 minus 2535πmicro3
32
minus3 (1 + 90 logmicro)micro4 +84315πmicro5
1024+
(C9)
dδLTp(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
2880π3f2πm
4π
minus5
2minus 27πmicro
32+ 20micro2 minus 5865πmicro3
256
+3
(617
4+ 36 logmicro
)micro4 minus 2056845πmicro5
4096+
(C10)
dδLTn(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
minus5
4minus 81πmicro
64minus 11micro2 +
10005πmicro3
512
+15
8(11 + 48 logmicro)micro4 minus 267015πmicro5
8192+
(C11)
dIAp(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
96π2f2πmicro
2
1minus 15πmicro
4minus 1
2(115 + 88 logmicro)micro2 +
1839πmicro3
32
+5 (5 + 34 logmicro)micro4 +
(C12)
dIAn(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
48π2f2πmicro
2
1
2minus 11πmicro
8minus 1
4(1 + 20 logmicro)micro2 +
99πmicro3
64
minus25micro4
12+
(C13)
dI1p(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
96π2f2πmicro
3π
8+ 2 (4 + 3 logmicro)microminus 537πmicro2
64
minus1
2(15 + 56 logmicro)micro3 +
(C14)
39
dI1n(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
48π2f2πmicro
minus π
16+
1
4(3 + 4 logmicro)microminus 57πmicro2
128+
2micro3
3+
(C15)
γ0p(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2g2
A
16π3f2πm
6π
1
105minus 23πmicro
256+
377micro2
210minus 15551πmicro3
6144+
3371micro4
105
minus1640457πmicro5
32768+
(C16)
γ0n(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2g2
A
16π3f2πm
6π
1
105minus 153πmicro
1792+
69micro2
70minus 4615πmicro3
6144+
172micro4
35
minus120897πmicro5
32768+
(C17)
2 ∆-exchange contribution
Here we give analytical expressions for the tree-level ∆-exchange contributions to the nucleon
spin polarizabilities and their slopes at Q2 = 0 Note that the ∆-exchange contributes equally to
proton and neutron polarizabilities Recall that for the magnetic γlowastN∆ coupling we introduced a
dipole form factor to mimic vector-meson dominance gM rarr gM(1 +Q2Λ2)2
bull Static polarizabilities (Q2 = 0)
γ0 = minus e2
4πM2+
(g2M
∆2+
g2E
M2+
minus 4gMgEM+∆
) (C18)
δLT =e2M∆
4πM3+
(g2E
MNM++gMgE∆MN
minus gEgCM2
∆
) (C19)
γ0 =e2M2
N
π∆2M4+
(minusg
2M
∆2+
g2E
M2+
+4gMgE∆M+
) (C20)
bull Slopes of polarizabilities at Q2 = 0
dγ0(Q2)
dQ2
∣∣∣∣∣Q2=0
= minus e2
πM2+∆
(g2M
∆
[1
4∆2minus 1
∆M++
1
2M2+
]minus 1
Λ2
g2M
∆+
g2E
2M2+
[1
2∆minus 3
M+
]minusgMgE
M+
[1
∆2minus 5
∆M++
1
M2+
]+
1
Λ2
2gMgEM+
+2gMgC∆M2
+
minus gEgCM3
+
) (C21)
40
d δLT (Q2)
dQ2
∣∣∣∣∣Q2=0
=e2M∆∆
4πMNM2+
(g2E
∆2M2+
[1
∆minus 4
M+
]minus g2
C
∆M2∆M
2+
+gMgE∆2M+
[1
∆2
minus 3
∆M++
1
M2+
]minus 2
Λ2
gMgE∆2M+
+gMgC∆M2
∆
[1
2∆2minus 2
∆M++
1
2M2+
]minus gEgC
2M2∆M
2+
[7
∆+
1
M+
]) (C22)
dIA(Q2)
dQ2
∣∣∣∣∣Q2=0
= minusM2N
M2+
(g2M
2∆2+
g2E
MNM+minus 2gMgE
∆M+minus gEgCM∆M+
) (C23)
dI1(Q2)
dQ2
∣∣∣∣∣Q2=0
= minusM∆M2N
2M3+
(g2E
MNM∆minus gMgE∆MN
minus gEgCM2
∆
) (C24)
γ0(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2M2
N
π∆3M6+
(g2M
∆
[3M2
∆ + 2M∆MN minus 9M2N
∆2+
4M2+
Λ2
]
minus4gMgE
[5M2
∆ minus 9M2N
M+∆2+
2M+
Λ2
]+
g2E
∆M+[7M∆ minus 9MN ]
minus8gMgC∆
+4gEgCM+
) (C25)
[1] D Drechsel B Pasquini and M Vanderhaeghen Dispersion relations in real and virtual
Compton scattering Phys Rept 378 99 (2003) hep-ph0212124
[2] S E Kuhn J-P Chen and E Leader Spin structure of the nucleon mdash status and recent
results Prog Part Nucl Phys 63 1 (2009) arXiv08123535 [hep-ph]
[3] F Hagelstein R Miskimen and V Pascalutsa Nucleon polarizabilities from Compton scat-
tering to hydrogen atom Prog Part Nucl Phys 88 29 (2016) arXiv151203765 [nucl-th]
[4] B Pasquini and M Vanderhaeghen Dispersion theory in electromagnetic interactions Ann
Rev Nucl Part Sci 68 75 (2018) arXiv180510482 [hep-ph]
[5] S Gerasimov A Sum rule for magnetic moments and the damping of the nucleon magnetic
moment in nuclei Sov J Nucl Phys 2 430 (1966)
[6] S Drell and A C Hearn Exact sum rule for nucleon magnetic moments Phys Rev Lett
16 908 (1966)
[7] J S Schwinger Source theory viewpoints in deep inelastic scattering Proc Natl Acad Sci
USA 72 1 (1975)
[8] J S Schwinger Source theory viewpoints in deep inelastic scattering Electromagnetic In-
teractions and Field Theory Proceedings 14 Internationale Universitatswochen Schladming
Austria February 24-March 7 1975 Acta Phys Austriaca Suppl 14 471 (1975)
41
[9] J Schwinger Source theory discussion of deep inelastic scattering with polarized particles
Proc Natl Acad Sci USA 72 1559 (1975)
[10] A M Baldin Polarizability of nucleons Nucl Phys 18 310 (1960)
[11] M Gell-Mann M L Goldberger and W E Thirring Use of causality conditions in quantum
theory Phys Rev 95 1612 (1954)
[12] A Pineda Leading chiral logarithms to the hyperfine splitting of the hydrogen and muonic
hydrogen Phys Rev C 67 025201 (2003)
[13] C Peset and A Pineda The two-photon exchange contribution to muonic hydrogen from
chiral perturbation theory Nucl Phys B 887 69 (2014) arXiv14064524 [hep-ph]
[14] C Peset and A Pineda Model-independent determination of the two-photon exchange contri-
bution to hyperfine splitting in muonic hydrogen JHEP 04 060 arXiv161205206 [nucl-th]
[15] F Hagelstein and V Pascalutsa Proton structure in the hyperfine splitting of muonic hydro-
gen PoS CD15 077 (2016) arXiv151104301 [nucl-th]
[16] F Hagelstein Exciting Nucleons in Compton Scattering and Hydrogen-Like Atoms PhD
thesis Mainz U Inst Kernphys (2017) arXiv171000874 [nucl-th]
[17] F Hagelstein ∆(1232)-Resonance in the hydrogen spectrum Proceedings 11th International
Workshop on the Physics of Excited Nucleons (NSTAR 2017) Columbia SC USA August
20-23 2017 Few Body Syst 59 93 (2018) arXiv180109790 [nucl-th]
[18] Y Prok et al (CLAS) Moments of the spin structure functions gp1 and gd1 for 005 lt Q2 lt 30
GeV2 Phys Lett B 672 12 (2009) arXiv08022232 [nucl-ex]
[19] H Dutz et al (GDH) First measurement of the Gerasimov-Drell-Hearn sum rule for 1H from
07 GeV to 18 GeV at ELSA Phys Rev Lett 91 192001 (2003)
[20] M Amarian et al (Jefferson Lab E94010) Measurement of the generalized forward spin po-
larizabilities of the neutron Phys Rev Lett 93 152301 (2004) arXivnucl-ex0406005
[21] M Amarian et al The Q2 evolution of the generalized Gerasimov-Drell-Hearn integral for
the neutron using a 3He target Phys Rev Lett 89 242301 (2002) arXivnucl-ex0205020
[22] M Amarian et al (Jefferson Lab E94-010)Q2 evolution of the neutron spin structure moments
using a 3He target Phys Rev Lett 92 022301 (2004) arXivhep-ex0310003
[23] A Deur et al Experimental determination of the evolution of the Bjorken integral at low Q2
Phys Rev Lett 93 212001 (2004) arXivhep-ex0407007
[24] K Slifer Low Q2 measurement of gp2 and the δLT spin polarizability Spin structure at long dis-
tance Proceedings Workshop Newport News USA March 12ndash13 2009 nucl-ex09064775
(2009) AIP Conf Proc 1155 10106313203293 (2009) arXiv09064775 [nucl-ex]
[25] P Solvignon et al (E01-012) Moments of the neutron g2 structure function at intermediate
42
Q2 Phys Rev C 92 015208 (2015) arXiv13044497 [nucl-ex]
[26] A Deur Experimental studies at low Q2 of the spin structure of the nucleon at Jefferson Lab
in 9th International Workshop on Chiral Dynamics (CD18) Durham NC USA September
17-21 2018 (2019) arXiv190305661 [nucl-ex]
[27] V Sulkosky et al (Jefferson Lab E97-110) Measurement of the 3He spin-structure functions
and of neutron (3He) spin-dependent sum rules at 0035 le Q2 le 024 GeV2 Phys Lett B
805 135428 (2020) arXiv190805709 [nucl-ex]
[28] K Adhikari et al (CLAS) Measurement of the Q2 Dependence of the Deuteron Spin Structure
Function g1 and its Moments at Low Q2 with CLAS Phys Rev Lett 120 062501 (2018)
arXiv171101974 [nucl-ex]
[29] V Lensky F Hagelstein A Hiller Blin and V Pascalutsa Comment on rdquoMeasurement of
the Q2 Dependence of the Deuteron Spin Structure Function g1 and its Moments at Low Q2
with CLASrdquo (2018) arXiv180603219 [nucl-th]
[30] J M Alarcn F Hagelstein V Lensky and V Pascalutsa Forward doubly-virtual Compton
scattering off the nucleon in chiral perturbation theory at NLO the subtraction function and
moments of unpolarized structure functions (2020) arXiv200509518 [hep-ph]
[31] V Bernard E Epelbaum H Krebs and U-G Meiszligner New insights into the spin structure
of the nucleon Phys Rev D 87 054032 (2013) arXiv12092523 [hep-ph]
[32] V Lensky J M Alarcon and V Pascalutsa Moments of nucleon structure functions at
next-to-leading order in baryon chiral perturbation theory Phys Rev C 90 055202 (2014)
arXiv14072574 [hep-ph]
[33] V Pascalutsa and M Vanderhaeghen Chiral effective-field theory in the ∆(1232) region
I Pion electroproduction on the nucleon Phys Rev D 73 034003 (2006) arXivhep-
ph0512244
[34] V Pascalutsa and M Vanderhaeghen Electromagnetic nucleon-to-Delta transition in chiral
effective field theory Phys Rev Lett 95 232001 (2005) arXivhep-ph0508060
[35] T R Hemmert B R Holstein and J Kambor Systematic 1M expansion for spin 32
particles in baryon chiral perturbation theory Phys Lett B 395 89 (1997) arXivhep-
ph9606456
[36] V Pascalutsa and D R Phillips Effective theory of the ∆(1232) in Compton scattering off
the nucleon Phys Rev C 67 055202 (2003) arXivnucl-th0212024
[37] V Pascalutsa M Vanderhaeghen and S N Yang Electromagnetic excitation of the ∆(1232)-
resonance Phys Rept 437 125 (2007) arXivhep-ph0609004
[38] K A Olive et al (Particle Data Group) Review of Particle Physics Chin Phys C 38 090001
43
(2014)
[39] F E Low Scattering of light of very low frequency by systems of spin 12 Phys Rev 96
1428 (1954)
[40] M Gell-Mann and M L Goldberger Scattering of low-energy photons by particles of spin
12 Phys Rev 96 1433 (1954)
[41] A M Harun ar-Rashid A simple derivation of Schwingerrsquos sum rule for spin dependent
structure functions Nuovo Cim A 33 447 (1976)
[42] F Hagelstein and V Pascalutsa Dissecting the hadronic contributions to (g minus 2)micro by
Schwingerrsquos sum rule Phys Rev Lett 120 072002 (2018) arXiv171004571 [hep-ph]
[43] V Lensky V Pascalutsa M Vanderhaeghen and C Kao Spin-dependent sum rules con-
necting real and virtual Compton scattering verified Phys Rev D 95 074001 (2017)
arXiv170101947 [hep-ph]
[44] H Burkhardt and W N Cottingham Sum rules for forward virtual Compton scattering
Annals Phys 56 453 (1970)
[45] V Lensky and V Pascalutsa Predictive powers of chiral perturbation theory in Compton
scattering off protons Eur Phys J C 65 195 (2010) arXiv09070451 [hep-ph]
[46] H Grieszlighammer J McGovern D Phillips and G Feldman Using effective field theory to
analyse low-energy Compton scattering data from protons and light nuclei Prog Part Nucl
Phys 67 841 (2012) arXiv12036834 [nucl-th]
[47] H W Grieszlighammer J A McGovern and D R Phillips Nucleon polarisabilities at and
beyond physical pion masses Eur Phys J A 52 139 (2016) arXiv151101952 [nucl-th]
[48] E Epelbaum H Krebs and U-G Meiszligner Improved chiral nucleon-nucleon potential up to
next-to-next-to-next-to-leading order Eur Phys J A 51 53 (2015) arXiv14120142 [nucl-
th]
[49] E Epelbaum H Krebs and U-G Meiszligner Precision nucleon-nucleon potential at fifth order
in the chiral expansion Phys Rev Lett 115 122301 (2015) arXiv14124623 [nucl-th]
[50] C W Kao T Spitzenberg and M Vanderhaeghen Burkhardt-Cottingham sum rule and
forward spin polarizabilities in heavy baryon chiral perturbation theory Phys Rev D 67
016001 (2003) arXivhep-ph0209241
[51] C-W Kao D Drechsel S Kamalov and M Vanderhaeghen Higher moments of nucleon
spin structure functions in heavy baryon chiral perturbation theory and in a resonance model
Phys Rev D 69 056004 (2004) arXivhep-ph0312102
[52] T Becher and H Leutwyler Baryon chiral perturbation theory in manifestly Lorentz invariant
form Eur Phys J C 9 643 (1999) arXivhep-ph9901384
44
[53] J Gasser M E Sainio and A Svarc Nucleons with chiral loops Nucl Phys B 307 779
(1988)
[54] L S Geng J Martin Camalich L Alvarez-Ruso and M J Vicente Vacas Leading SU(3)-
breaking corrections to the baryon magnetic moments in chiral perturbation theory Phys
Rev Lett 101 222002 (2008) arXiv08051419 [hep-ph]
[55] D Drechsel S S Kamalov and L Tiator The GDH sum rule and related integrals Phys
Rev D 63 114010 (2001) arXivhep-ph0008306
[56] D Drechsel O Hanstein S S Kamalov and L Tiator A unitary isobar model for pion
photo- and electroproduction on the proton up to 1 GeV Nucl Phys A 645 145 (1999)
arXivnucl-th9807001
[57] L Tiator private communication (2020)
[58] V Bernard T R Hemmert and U-G Meiszligner Spin structure of the nucleon at low energies
Phys Rev D 67 076008 (2003) arXivhep-ph0212033
[59] O Gryniuk F Hagelstein and V Pascalutsa Evaluation of the forward Compton scattering
off protons II Spin-dependent amplitude and observables Phys Rev D 94 034043 (2016)
arXiv160400789 [nucl-th]
[60] R Zielinski The g2p Experiment A Measurement of the Protonrsquos Spin Structure Functions
PhD thesis New Hampshire U (2010) arXiv170808297 [nucl-ex]
[61] N Guler et al (CLAS) Precise determination of the deuteron spin structure at low to mod-
erate Q2 with CLAS and extraction of the neutron contribution Phys Rev C 92 055201
(2015) arXiv150507877 [nucl-ex]
[62] C E Carlson V Nazaryan and K Griffioen Proton structure corrections to electronic
and muonic hydrogen hyperfine splitting Phys Rev A 78 022517 (2008) arXiv08052603
[physicsatom-ph]
[63] V Bernard N Kaiser and U-G Meiszligner Chiral dynamics in nucleons and nuclei Int J
Mod Phys E 4 193 (1995) arXivhep-ph9501384
[64] T R Hemmert B R Holstein and J Kambor ∆(1232) and the polarizabilities of the nucleon
Phys Rev D 55 5598 (1997) arXivhep-ph9612374
[65] V Pascalutsa and R Timmermans Field theory of nucleon to higher-spin baryon transitions
Phys Rev C 60 042201 (1999) arXivnucl-th9905065
[66] V Pascalutsa Quantization of an interacting spin-32 field and the Delta isobar Phys Rev
D 58 096002 (1998) arXivhep-ph9802288
[67] H Krebs Double Virtual Compton Scattering and SpinStructure of the Nucleon PoS
CD2018 031 (2019)
45
[68] N Kochelev and Y Oh Axial anomaly and the δLT puzzle Phys Rev D 016012 (2012)
arXiv11034892 [hep-ph]
[69] D Drechsel S Kamalov and L Tiator Unitary isobar model ndash MAID2007 Eur Phys J A
69 (2007) available at httpsmaidkphuni-mainzde arXiv07100306 [nucl-th]
[70] P J Mohr B N Taylor and D B Newell CODATA recommended values of the fundamental
physical constants 2010 Rev Mod Phys 84 1527 (2012)
[71] J Ahrens et al (GDH A2) First measurement of the Gerasimov-Drell-Hearn integral for 1H
from 200 to 800 MeV Phys Rev Lett 87 022003 (2001) arXivhep-ex0105089 [hep-ex]
[72] K Helbing (GDH) Experimental verification of the GDH sum rule at ELSA and MAMI Nucl
Phys Proc Suppl 105 113 (2002)
[73] R Bradford A Bodek H S Budd and J Arrington A New parameterization of the nucleon
elastic form-factors NuInt05 proceedings of the 4th International Workshop on Neutrino-
Nucleus Interactions in the Few-GeV Region Okayama Japan 26-29 September 2005 Nucl
Phys Proc Suppl 159 127 (2006) arXivhep-ex0602017
[74] J D Bjorken Applications of the chiral U(6)timesU(6) algebra of current densities Phys Rev
148 1467 (1966)
[75] J D Bjorken Inelastic scattering of polarized leptons from polarized nucleons Phys Rev D
1 1376 (1970)
[76] V Bernard T R Hemmert and U-G Meiszligner Novel analysis of chiral loop effects in
the generalized Gerasimov-Drell-Hearn sum rule Phys Lett B 545 105 (2002) arXivhep-
ph0203167
[77] A Deur et al Experimental study of isovector spin sum rules Phys Rev D 78 032001
(2008) arXiv08023198 [nucl-ex]
[78] B Pasquini P Pedroni and D Drechsel Higher order forward spin polarizability Phys Lett
B 687 160 (2010) arXiv10014230 [hep-ph]
[79] R Jaffe g2ndashThe nucleonrsquos other spin-dependent structure function Comments Nucl Part
Phys 19 239 (1990)
[80] E V Shuryak and A Vainshtein Theory of power corrections to deep inelastic scattering in
quantum chromodynamics (II) Qminus4 effects polarized target Nucl Phys B 201 141 (1982)
[81] S Wandzura and F Wilczek Sum rules for spin dependent electroproduction Test of rela-
tivistic constituent quarks Phys Lett B 72 195 (1977)
[82] B W Filippone and X-D Ji The spin structure of the nucleon Adv Nucl Phys 26 1
(2001) arXivhep-ph0101224
[83] M Burkardt The g2 structure function Proceedings Workshop on Spin structure at long
46
distance Newport News USA March 12-13 2009 AIP Conf Proc 1155 26 (2009)
arXiv09054079 [hep-ph]
[84] A Deur S J Brodsky and G F de Teramond The spin structure of the nucleon Rept
Prog Phys 82 076201 (2019) arXiv180705250 [hep-ph]
[85] B R Holstein D Drechsel B Pasquini and M Vanderhaeghen Higher order polarizabilities
of the proton Phys Rev C 61 034316 (2000) arXivhep-ph9910427
[86] B R Holstein V Pascalutsa and M Vanderhaeghen Sum rules for magnetic moments and
polarizabilities in QED and chiral effective-field theory Phys Rev D 72 094014 (2005)
arXivhep-ph0507016
[87] B Pasquini D Drechsel and L Tiator Invariant amplitudes for pion electroproduction Eur
Phys J A 34 387 (2007) arXiv07122327 [hep-ph]
[88] A J Buchmann Non-spherical proton shape and hydrogen hyperfine splitting Proceedings
International Workshop on Precision Physics of Simple Atomic Systems (PSAS 2008) Wind-
sor Ontario Canada July 21-26 2008 Can J Phys 87 773 (2009) arXiv09104747
[physicsatom-ph]
47
non-negligible dependence on gC and the dipole in gM cf Fig 9 lower panel
The πN -loop ∆-exchange and π∆-loop contributions to the NLO BχPT prediction of
the static fifth-order forward spin polarizability amount to in units of 10minus4 fm6
γ0p = 112(30) asymp 208minus 096minus 001 (42a)
γ0n = 195(30) asymp 292minus 096minus 001 (42b)
while the slope is composed as follows in units of 10minus4 fm8
dγ0p(Q2)
dQ2
∣∣∣∣Q2=0
= minus084(10) asymp minus100 + 016 + 000 (43a)
dγ0n(Q2)
dQ2
∣∣∣∣Q2=0
= minus142(15) asymp minus158 + 016 + 000 (43b)
Note that the HB prediction of the static polarizability γ0p (423 at O(p3) and 365 at O(ε3)
[78 85]) is almost one order of magnitude larger than the empirical value and therefore not
shown in Fig 8
G Summary
Our results are summarized in Table II where we give the contributions of the different
orders to the chiral predictions of the polarizabilities and their slopes at the real-photon
point A quantitative comparison of our predictions for the static spin polarizabilities to
the work of Bernard et al [31] and different empirical evaluations is shown in Table III We
can see that the inclusion of the Delta turns out to be very important for all moments of
the helicity-difference cross section To describe the Q2 behavior of the polarizabilities the
magnetic coupling of the N rarr ∆ transition should be modified by a dipole form factor as
has been observed previously in the description of electroproduction data [33] This dipole
form factor effectively takes account of vector-meson exchanges The Coulomb-quadrupole
N rarr ∆ transition despite its subleading order is important in the description of some
moments of spin structure functions This is contrary to what we saw for the moments of
unpolarized structure functions [30] where the Coulomb coupling had a negligible effect
The π∆ loops are mainly relevant for the generalized GDH integrals
26
TABLE II The NLO BχPT predictions for the forward VVCS polarizabilities and their slopes at
Q2 = 0 The contributions of the πN loops the ∆ exchange and the π∆ loops are shown together
with the combined total result Note that IA(0) = I1(0) = d2(0) = 0 and (d2)prime = 0
πN loops ∆ exchange π∆ loops Total
γ0 p 201 minus284 minus010minus093(92)
(10minus4 fm4) n 298 003(92)
δLT p 150 minus016 minus002132(15)
(10minus4 fm4) n 235 218(23)
γ0 p 208 minus096 minus001112(30)
(10minus4 fm6) n 292 195(30)
(γ0)prime p minus033011 001
minus022(4)
(10minus4 fm6) n minus073 minus061(7)
(δLT )prime p minus080 minus004 minus001minus085(8)
(10minus4 fm6) n minus119 minus124(12)
(γ0)prime p minus100016 000
minus084(10)
(10minus4 fm8) n minus158 minus142(15)
(∆IA)prime p 238 minus1121 025minus858(343)
(GeVminus2) n 141 minus955(343)
(∆I1)prime p 034 minus053 058039(4)
(GeVminus2) n minus107 minus101(10)
TABLE III Our NLO BχPT predictions for the spin polarizabilities at Q2 = 0 compared with
the BχPT+∆ predictions from Bernard et al [31] and the available empirical information Where
the reference is not given the empirical number is provided by the MAID analysis [55 69] with
unspecified uncertainty
Proton Neutron
This work BχPT+∆ Empirical This work BχPT+∆ Empirical
γ0 minus093(92) minus174(40) minus100(8)(12) [19] 003(92) minus077(40) minus0005
(10minus4 fm4) minus090(8)(11) [78] [MAID]
minus0929(105) [59]
δLT 132(15) 240(1) 134 218(23) 238(3) 203
(10minus4 fm4) [MAID] [MAID]
γ0 112(30) 060(7)(7) [78] 195(30) 123
(10minus4 fm6) 0484(82) [59] [MAID]
IV CONCLUSIONS
We have presented a complete NLO calculation of the polarized non-Born VVCS ampli-
tudes in covariant BχPT with pion nucleon and ∆(1232) fields The dispersion relations
between the VVCS amplitudes and the tree-level photoabsorption cross sections served as a
27
cross-check of these calculations
The obtained moments of the proton and neutron spin structure functions related to gen-
eralized polarizabilities and GDH-type integrals agree well with the available experimental
data The description of their Q2 evolution is improved compared to the previous χPT pre-
dictions In particular the NLO BχPT predictions obtained here give a better description
of the empirical data (eg from the Jefferson Laboratory ldquoSpin Physics Programrdquo) than
the HB [50 51] and IR [58] calculations
The demonstrated predictive power of the χPT framework amplitudes makes it well
suited for extending the χPT evaluation of the TPE effect in the hyperfine structure of
(muonic-)hydrogen [15ndash17] to next-to-leading order
ACKNOWLEDGEMENTS
We thank Lothar Tiator and Marc Vanderhaeghen for helpful discussions This work
is supported by the Deutsche Forschungsgemeinschaft (DFG) through the Collaborative
Research Center [The Low-Energy Frontier of the Standard Model (SFB 1044)] JMA ac-
knowledges support from the Community of Madrid through the ldquoPrograma de atraccion de
talento investigador 2017 (Modalidad 1)rdquo and the Spanish MECD grants FPA2016-77313-P
FH gratefully acknowledges financial support from the Swiss National Science Foundation
Appendix A Tensor decompositions of the VVCS amplitudes
In this appendix we review the decomposition of the forward VVCS process into tensor struc-
tures and scalar amplitudes In particular we consider the connection between the covariant and
the semi-relativistic decomposition in the lab frame that is defined in terms of the conventional
transverse longitudinal transverse-transverse and transverse-longitudinal amplitudes
As explained in Sec II A the process of forward VVCS off the nucleon can be described in
terms of four explicitly covariant amplitudes S1 2 and T1 2 [3]
T (νQ2) =
(minusgmicroν +
qmicroqν
q2
)T1(νQ2) +
1
M2N
(pmicro minus p middot q
q2qmicro)(
pν minus p middot qq2
qν)T2(νQ2) (A1)
minus 1
MNγmicroναqα S1(νQ2)minus 1
M2N
(γmicroνq2 + qmicroγναqα minus qνγmicroαqα
)S2(νQ2)
εprimelowastmicro εν
where εmicro (εprimelowastmicro ) are the incoming (outgoing) photon polarization vectors ν is the photon lab-frame
energy and Q2 is the photon virtuality Alternatively the decomposition in the laboratory frame
28
(which in the forward case coincides with the Breit frame) is parametrized in terms of the nucleon
Pauli matrices ~σ and the four scalar functions fL fT gTT and gLT
T (νQ2) = ε0 εprimelowast0 fL(νQ2) + (~ε primelowast middot ~ε ) fT (νQ2) + i~σ middot (~ε primelowast times ~ε ) gTT (νQ2) (A2)
minusi~σ middot [(ε0~εprimelowast minus ~ε ε primelowast0 )times q] gLT (νQ2)
Here ~q and q = ~q|~q | are the photon three-momentum in the lab system and its unit vector The
modified polarization vector components are given by
ε0 =
[ε0 minus
ν
|~q | (~ε middot q )
] |~q |Q ~ε = ~εminus q (~ε middot q ) (A3)
where ε = (ε0~ε ) is the usual incoming photon polarization vector and εprimelowast the outgoing polarization
vector The LEX of the lab frame amplitudes [Eq (10)] can serve in particular as the definition
of the generalized polarizabilities The lab frame amplitudes are also conveniently used for the
definition of the response functions see the example of the scalar amplitude gLT (νQ2) and the
corresponding response function σLT (νQ2) below in App B
Appendix B Photoabsorption cross sections
In the forward kinematics the spin-dependent VVCS amplitudes and the spin polarizabilities
can be described in terms of the polarized structure functions g1(xQ2) and g2(xQ2) or equiv-
alently the helicity-difference cross section σTT (νQ2) and the longitudinal-transverse response
function σLT (νQ2) with the help of dispersion relations (5) and the optical theorem (3) In this
way the photoabsorption cross sections measured in electroproduction processes form the basis
for most empirical evaluations shown throughout Sec III In the following we present the BχPT
predictions for the tree-level cross sections of πN - π∆- and ∆-production through photoabsorption
on the nucleon cf Figs 8 9 and 10 in Ref [30] In Secs B 1 and B 2 we will discuss the leading
πN -production channel and the ∆-production channel respectively We used these cross sections
to verify the polarizability predictions obtained otherwise from the calculated non-Born VVCS
amplitudes Due to the bad high-energy behavior of the π∆-production cross sections in BχPT
cf Fig 10 the dispersion relations in Eq (5) require further subtractions for a reconstruction of
the π∆-loop contribution to the spin-dependent VVCS amplitudes Therefore not all polarizabil-
ities could be verified but only those appearing as higher-order terms in the LEX of the VVCS
amplitudes such as γ0 [16]
29
proton neutron
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Neutron
02 04 06 08 100
50
100
150
200
250
[GeV]
T [b]
02 04 06 08 10
-50
0
50
100
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Neutron
02 04 06 08 100
50
100
150
200
250
[GeV]
T [b]
02 04 06 08 10
-50
0
50
100
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02040608100
50
100
150
200
250
300
[GeV]
T[b]
0204060810
-50
0
50
[GeV]
TT[b]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
0204060810
00
02
04
06
[GeV]
LTQ
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
0204060810
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
FIG 10 Photoabsorption cross sections for πN (red) and π∆ production (orange) with Q2 = 0
(solid) and Q2 = 01 GeV2 (dashed for πN and dotted for π∆ channel)
1 πN-production channel
In order to extract the response function σLT (νQ2) we have developed a method similar to
the one used to calculate σTT (νQ2) see for example Ref [86] For σLT (νQ2) however the
calculation is more complicated because one has to take into account that the associated Compton
process involves a spin-flip of the nucleon as illustrated in Fig 11 When calculating the cross
section the product of the incoming nucleon spinors has to reflect this flip
The forward VVCS amplitude related to σLT (νQ2) mdash and δLT (Q2) mdash is gLT (νQ2) It can
be extracted from Eq (A2) if one takes the modified polarization vector components in Eq (A3)
with ε = εL and εprimelowast = εlowastplusmn as input where εL = 1Q(|~q | 0 0 q0) and εplusmn = ∓ 1radic
2(0 1plusmni 0) are the
standard longitudinal and transverse polarization vectors respectively For εL and εlowastplusmn only the
choice of helicities hprime = plusmn12 and h = ∓12 gives a non-zero contribution and one obtains
χdaggerhprime T (νQ2)χh = χdaggerhprimeminusi~σ middot [(ε0~εprimelowast minus ~ε ε primelowast0 )times q] gLT (νQ2)χh =
radic2 gLT (νQ2) (B1)
where χh and χdaggerhprime are two-component Pauli spinors with opposite helicities or here spins
Let us now consider the related photoabsorption process and in particular the tree-level γlowastN rarr
30
s = 12
sprime = minus12
λ = 0 λprime = 1
rprime rprime
s=12sprime=minus1
2
λ=0λprime=1
rprimerprime
s=12
sprime=minus12
λ=0λprime=1
rprimerprime hacute=frac12h=-frac12
λacute=1λ=0
FIG 11 Relation between the forward Compton process and the photoabsorption process given
by the optical theorem In particular we show the longitudinal-transverse contribution The
double-line arrows represent the spin of the external particles while the dot represents the scalar
(longitudinal) polarization of the incoming photon Inside the blob the intermediate states are
represented eg nucleons with spins rprime (which are averaged in the calculation of the cross section)
and pions
πN channel see diagrams in Fig 8 of Ref [30] We define the πN -production amplitude as
T = uhB (PB)sumi
Ai(s t)Γi uhA(PA) (B2)
with the Dirac structures
Γ1 = γ5 (B3a)
Γ2 =1
2
[qA ε
]γ5 (B3b)
where uhA(PA) and udaggerhB (PB) are the Dirac spinors and PA and PB are the four-momenta of
the incoming and outgoing nucleons respectively When calculating the photoabsorption cross
section related to the VVCS amplitude in Eq (B1) the nucleon spin flip should be implemented
by uhprime(PA) in T dagger and uh(PA) in T together with the appropriate transverse and longitudinal
photon polarization vectors εlowastplusmn and εL
However if one wants to use the properties of the Dirac matrices it is more useful to construct
an operator to produce this spin flip in the external nucleons of Fig 11 This is accomplished by
introducing the projector ΓLT equiv 12radic
2(γ1 + iγ2)γ5 which also takes into account the extra factor
radic2 in Eq (B1) We checked that with this projector one correctly extracts δLT by comparing the
HB limit of our result to the HB result of Ref [50] where the authors calculate this polarizability
from the Compton amplitude directly With all those ingredients the longitudinal-transverse cross
section is calculated in the following way
σLT (νQ2) =1
64π2 s
|~pf |cm
|~pi|cm
int 1
minus1dcos θ
sumij
AiAdaggerjXij (B4)
31
with
Xij = Tr[(PB +MN )Γi(PA +MN )ΓLTγ0Γdaggerjγ
0] (B5)
where θ is the scattering angle in the center-of-mass (cm) frame and |~pi|cm (|~pf |cm) is the three-
momentum of an incoming (outgoing) particle in the cm frame An explicit calculation of the
matrix Xij leads to
X = MNQ
0 2(PB minus PA) middot εLminusradic
2|~qf |cm sin θ (sminus u)
(B6)
where |~qi|cm (|~qf |cm) is the relative three-momentum of the incoming (outgoing) particles in the
cm frame Here s t and u are the usual Mandelstam variables For the different γlowastN rarr πN
channels we obtain the following amplitudes Ai where we introduce qA as the four-momentum of
the incoming photon and qB as the four-momentum of the outgoing pion
bull γlowastprarr π0p
A1 =e gAMN
fπ
[2PA middot ε+ qA middot ε
sminusM2N
+2PB middot εminus qA middot ε
uminusM2N
] (B7a)
A2 =e gAMN
fπ
[1
sminusM2N
+1
uminusM2N
] (B7b)
bull γlowastprarr π+n
A1 =
radic2 e gAMN
fπ
[2PA middot ε+ qA middot ε
sminusM2N
+2(PA minus PB) middot ε+ qA middot ε
tminusm2π
] (B8a)
A2 =
radic2 e gAMN
fπ(sminusM2N )
(B8b)
bull γlowastnrarr π0n
A1 = 0 (B9a)
A2 = 0 (B9b)
bull γlowastnrarr πminusp
A1 =
radic2 e gAMN
fπ
[2PB middot εminus qA middot ε
uminusM2N
minus 2(PA minus PB) middot ε+ qA middot εtminusm2
π
] (B10a)
A2 =
radic2 e gAMN
fπ(uminusM2N )
(B10b)
The analytical expressions shown above were checked with the amplitudes given in Ref [87] An-
alytical expressions for the tree-level γlowastN rarr πN channel of the σLT (νQ2) and σTT (νQ2) cross
32
sections are given below (proton channels π+n and π0p neutron channel πminusp) We checked that
they reproduce the known results in the real-photon limit [45 86] To shorten the final expressions
for the cross sections which are considerably longer for finite Q2 than in the real-photon limit we
define the following dimensionless kinematic variables
αγ = (ENi )cmradics =
s+M2N +Q2
2s (B11)
απ = (ENf )cmradics =
s+M2N minusm2
π
2s (B12)
βγ = Eγcmradics =
sminusM2N minusQ2
2s (B13)
βπ = Eπcmradics =
sminusM2N +m2
π
2s (B14)
λγ = |~qi|cmradics =
radic(sminusM2
N minusQ2)2 + 4sQ2
2s (B15)
λπ = |~qf |cmradics =
radic(sminusM2
N +m2π)2 minus 4sm2
π
2s (B16)
Here (ENi )cm and (ENf )cm are the energies of the incoming and outgoing nucleon Eγcm is the energy
of the incoming photon Eπcm is the energy of the outgoing pion all in the cm frame
σ(π+n)TT = minus e2g2
AM2N
64πf2πs
3(sminusM2N )2λ4
γ
4sλπλγ
[(M2
N minus s)(M2N minusQ2 minus s)(Q2 + 2sβγβπ)
+ 2s(m2πQ
2 minus (M2N minus s)
(M2N + s(minus1 + 2βγβπ)
))λ2γ
]+ (M2
N minus s)(M2N minusQ2 minus s)
times (Q2 + 2sβγβπ minus 2sλπλγ)(Q2 + 2s(βγβπ + λπλγ)
)times log
(Q2 + 2sβγβπ minus 2sλγλπQ2 + 2sβγβπ + 2sλγλπ
) (B17)
33
σ(π0p)TT =
e2g2AM
2Nλπ
64πf2πs(sminusM2
N )2λγ
1
s2λπλ3γ
[(M2
N minus s)(M2N minusQ2 minus s)
(3M2
N + s(2βγβπ minus 3))(M2N
+ s(2βγβπ minus 1))
+ 2s
((2m2
πQ2 + (M2
N minus s)2)(M2
N minus s)minus 2(minusm2
πQ2 + (sminusM2
N )2)
times sβγβπ + 4s2(sminusM2N )β2
γβ2π
)λ2γ
]arctan
(2sλπλγ
M2N + s(2βγβπ minus 1)
)+ 2
[minus 2m2
πQ2
+ (M2N minus s)
(2(sminusM2
N ) + 4sβγβπ +(sminusM2
N +Q2)(3M2N minus 3s+ 2sβγβπ)
sλ2γ
+m2πQ
2(sminusM2N )
(M2N minus s+ 2sβγβπ)(M2
N + s(minus1 + 2βγβπ minus 2λπλγ))
+m2πQ
2(sminusM2N )
(M2N minus s+ 2sβγβπ)(M2
N + s(minus1 + 2βγβπ + 2λπλγ))
)
+1
λπλγ
(βγβπ
(minusm2
πQ2 + (M2
N minus s)(M2N minus s+ 2sβγβπ)
)minus (M2
N minus s)(M2N minusQ2 minus s)λ2
π
)log
(M2N + s(minus1 + 2βγβπ + 2λπλγ)
M2N + s(minus1 + 2βγβπ minus 2λπλγ)
)] (B18)
σ(πminusp)TT =
e2g2AM
2N
64πf2πs
3λ4γ(M2
N + s(2βπβγ minus 1))2 minus 4λ2πs
2λ2γ)
4sλπλγ
((minusM2
N +Q2 + s)(M2N
+ s(2βπβγ minus 1))
2 minus 2sλ2γ
(2λ2
πs(minusM2N +Q2 + s) +m2
πQ2))
+((M2N + s(2βπβγ minus 1)
)2
minus 4λ2πs
2λ2γ
)((M2N + s(2βπβγ minus 1)
)(M2N minus 2Q2 minus s(2βπβγ + 1)
)+ 4λ2
πs2λ2
γ
)times log
(minusM2
N minus 2βπsβγ minus 2λπsλγ + s)minus(M2N + s(2βπβγ minus 2λπλγ minus 1)
)(M2N + s(minus1
+ 2βπβγ + 2λπλγ))[((
M2N + s(2βπβγ minus 1)
)(M2N minus 2Q2 minus s(2βπβγ + 1)
)+ 4λ2
πs2λ2
γ
)times log
(minusM2
N minus 2βπsβγ + 2λπsλγ + s)
+((Q2 + 2βπsβγ)2 minus 4λ2
πs2λ2
γ
)times log
(Q2 + 2βπsβγ minus 2λπsλγQ2 + 2sβπβγ + 2λπsλγ
)]+ 4sλ2
γ
((M2
N + s(2βπβγ minus 1))2 minus 4λ2πs
2λ2γ
)times (M2
N minus s) arctanh
(2λπsλγ
M2N + 2βπsβγ minus s
) (B19)
34
σ(π+n)LT =
e2g2AM
3Nλπ
32πf2πQs
3(sminusM2N )2λ4
γ
2sλγ
[(M2
N minus s)(Q2 + 2sβ2γ)(Q2 + 2sβγβπ)
minus 4s
((M2
N minus s)(Q2 minus 2s(απ minus 1)βγ
)+Q2sβγβπ
)λ2γ + 8s3(minus1 + απ)λ4
γ
]minus M2
N minus sλπ
[(Q2 + 2sβ2
γ)(Q2 + 2sβγβπ)2 + 4s2(2(απ minus 1)βγ(Q2 + 2sβγβπ)
minusQ2λ2π
)λ2γ + 8s3(απ minus 1)2λ4
γ
]arctan
(2sλπλγ
Q2 + 2sβγβπ
) (B20)
σ(π0p)LT =
e2g2AM
3Nλπ
16πf2πQs(sminusM2
N )2λγ
1
minus2s(M2N + s(minus1 + 2βγβπ))2λ2
γ + 8s3λ2πλ
4γ
[minus 3M8
N (Q2
+ 2sβ2γ) + 2M4
Ns2
(minus (Q2 + 2sβ2
γ)(minus3 + 2βγβπ) +(Q2(minus3 + 2βγβπ + 6λ2
π) + 2sβγ
times (minus3 + 12απ + 2βγβπ minus 12απβγβπ + 4βγλ2π))λ2γ minus 4s(α2
π minus 1)λ4π
)+ 2M2
Ns3
[minus (Q2
+ 2sβ2γ)(minus1 + 2βγβπ)
(6 + βγβπ(minus9 + 2βγβπ)
)+
(Q2(3minus 12λ2
π + 4βγβπ(minus1 + 2βγβπ
+ λ2π))minus 2sβγ
(minus 3 + 4απ(3 + 2απ(3 + 2βγβπ(minus3 + βγβπ)) + 4βγ(βπ + (2minus βγβπ)
times λ2π)))λ2γ + 8s
((απ minus 1)(1 + απ minus 2βγβπ) + 2απβγλ
2π
)λ4γ
)+ s4
((Q2 + 2sβ2
γ)
times (1minus 2βγβπ)2(2βγβπ minus 3) + 2(Q2(minus1 + 6λ2
π + 2βγβπ((1minus 2βγβπ)2 minus 2λ2π)) + 2sβγ
times (minus1 + 2βγβπ + 4(minus1 + βγβπ)(minusαπ + 2απβγβπ minus βγλ2π)))λ2γ minus 8
(s(απ minus 1)(απ
+ (1minus 2βγβπ)2) + 2βγ(2sαπ +Q2βγ)λ2π
)λ4γ + 32s(απ minus 1)λ2
πλ6γ
)+ 2M6
Ns
(Q2(6
minus 7βγβπ + λ2γ) + 2sβγ(βγ(6minus 7βγβπ) + (1minus 4απ)λ2
γ)
)]+sminusM2
N
4s2λπλ3γ
[(Q2 + 2sβ2
γ)
times(3M2
N + s(2βγβπ minus 3))(M2N + s(2βγβπ minus 1)
)+ 2s
(minusM2
N (Q2 + 2sβγ minus 8sαπβγ)
+ s(2sβγ(1minus 4απ + 4απβγβπ) +Q2(1minus 2λ2π)))λ2γ + 8s3(απ minus 1)λ4
γ
]
times arctan
(2sλπλγ
M2N + s(2βγβπ minus 1)
) (B21)
35
σ(πminusp)LT =
e2g2AM
3N
64πf2πQs
3λ4γ(M2
N minusQ2 minus s)(M2N + s(2βπβγ + 2λπλγ minus 1))
times
1
M2N + s(2βπβγ minus 2λπλγ minus 1)
[4λπsλγ(M2
N minusQ2 minus s)(
2sλ2γ
(s((1minus 2λ2
π)Q2
+ 2(2απ minus 1)sβγ(2βπβγ minus 1))minusM2N (Q2 + 2(1minus 2απ)sβγ)
)+ (Q2 + 2sβ2
γ)(M2N + s(2βπβγ minus 1)
)2 + 8(απ minus 1)απs
3λ4γ
)]minus(M2N + s(2βπβγ + 2λπλγ minus 1)
)times[
log
(minusM2N minus 2βπsβγ minus 2λπsλγ + s
minusM2N minus 2βπsβγ + 2λπsλγ + s
)(minus (Q2 + 2sβ2
γ)(M2N + s(2βπβγ minus 1)
)(minusM2
N + 2Q2 + 2βπsβγ + s)minus 2sλ2γ
(minusM2
N (Q2 + 2sβγ) +Q4 +Q2s(4απβγ
minus 2βγ minus 2λ2π + 1) + 2s2βγ(4(απ minus 1)βπβγ + 1)
)minus 8(απ minus 1)2s3λ4
γ
)+ 2
(4s2λ2
γ
(2(απ minus 1)βγ(Q2 + 2βπsβγ)minus λ2
πQ2)
+ (Q2 + 2βπsβγ)2(Q2 + 2sβ2γ)
+ 8(απ minus 1)2s3λ4γ
)arctanh
(2λπsλγ
Q2 + 2βπsβγ
)] (B22)
2 ∆-production channel
The tree-level ∆-exchange diagram in Fig 2 of Ref [30] contributes to the non-Born part of
the VVCS amplitudes The contribution of the ∆ exchange to the VVCS amplitudes can be split
into [17]
S∆-exch1 (νQ2) = S∆-pole
1 (νQ2) + S∆-exch1 (νQ2) (B23a)
νS∆-exch2 (νQ2) = νS∆-pole
2 (νQ2) + νS2∆-exch
(νQ2) (B23b)
and similarly for the unpolarized VVCS amplitudes discussed in Ref [30] Here we introduced the
∆-pole contributions S∆-polei and the ∆-non-pole contributions S∆-exch
i The former amplitudes
feature a pole at the ∆(1232)-production threshold and thus are proportional to
1
[sminusM2∆][uminusM2
∆]=
1
4M2N
1
ν2∆ minus ν2
(B24)
36
They can be reconstructed from the dispersion relations in Eq (5) with the tree-level ∆-production
cross sections as input cf Fig 10 in Ref [30]
σTT (νQ2) =π2α
M2NM
2+|~q |
minus g2
MMN (M+ + ν)|~q |2 +g2E(∆minus ν)(Q2 minusMNν)2
MN(B25a)
+g2CQ
4s(∆minus ν)
MNM2∆
minus 4gMgE(Q2 minusMNν)|~q |2 minus 4gMgCQ2|~q |2
+2gEgCQ
2[minusMNM∆ |~q |2 + s(Q2 +∆ν)
]MNM∆
δ(ν minus ν∆)
σLT (νQ2) =Qπ2α
M2NM
2+|~q |
g2E(MNν minusQ2) [M∆(MN + ν)minus s]
MN(B25b)
+g2CQ
2[MNM∆|~q |2 minus s(Q2 +∆ν)
]MNM2
∆
+ gMgEM∆|~q |2 minusgMgC(Q2 minusMNν)|~q |2
M∆
+gEgC(ν minus∆)(M2
N |~q |2 minus 2Q2s)
MNM∆
δ(ν minus ν∆)
with ∆ = M∆ minusMN M+ = M∆ + MN and the Mandelstam variable s = M2N + 2MNν minus Q2
Analytical expressions for the spin structure functions g1(xQ2) and g2(xQ2) can be constructed
from Eq (3) with the flux factor K(νQ2) = |~q | =radicν2 +Q2
In the ∆-non-pole contributions to S1(νQ2) and νS2(νQ2) the pole in ν at the ∆(1232)-
production threshold has canceled out
S∆-exch1 (νQ2) =
πα
MNM2+
[g2MQ
2+ + g2
E
(∆2 minus 3Q2
)+
4g2CQ
4
M2∆
minus 8gMgEM∆ωminus (B26a)
minus2gMgCQ2(MN minus 4M∆)
M∆+
2gEgCQ2(3MN minus 2M∆)
M∆
]
νS2∆-exch
(νQ2) =2πα
MNM2+
[g2EM∆∆ωminus +
g2M MNQ
2+
2+g2C Q
2(Q2 minus∆2)
2M∆(B26b)
+gEgM M∆(M∆ω+ minus 4MNωminus)minus gEgC ∆(2Q2 +MNω+)
+gMgC Q2(4MN minus ω+)
]+S∆-exch
2 (νQ2)
ν
[M2
∆ ω2+
M2N
+ ν2
]
with Q+ =radic
(M∆ +MN )2 +Q2 and ωplusmn = (M2∆minusM2
N plusmnQ2)2M∆ and the non-pole contribution
to S2(νQ2)
S∆-exch2 (νQ2) = minus2παMNν
M∆M2+
[gM + gE
]gC (B27)
These amplitudes to the contrary are not described by the tree-level ∆-production cross sections
in the standard dispersive approach [17] This peculiarity has been previously missed e g in the
calculation of the ∆-exchange contribution to the hydrogen hyperfine splitting in Ref [88] The
importance of including the ∆-non-pole contribution is also evident when considering the BC sum
37
rule in Eq (14) The ∆-pole terms by themselves violate the BC sum rule but cancel exactly with
the ∆-non-pole terms
limνrarr0
νS∆-pole2 (νQ2) + lim
νrarr0νS2
∆-exch(νQ2) = 0 (B28)
Appendix C Polarizabilities at Q2 = 0
In this section we give analytical expressions for the static values and slopes at Q2 = 0 of
the spin polarizabilities and moments of polarized structure functions In particular we give the
HB expansion of the πN -loop contributions and the ∆-exchange contributions The complete
expressions also for the π∆-loop contributions can be found in the Supplemented material Recall
that IA(0) = I1(0) = d2(0) = 0 and dd2(Q2)dQ2
∣∣∣Q2=0
= 0
1 πN-loop contribution
Here we give analytical expressions for the πN -loop contributions to the proton and neutron
spin polarizabilities expanded in powers of micro = mπMN viz the HB expansion Note that we
choose to expand here to a high order in micro the strict HB expansion would only retain the leading
term in an analogous NLO calculation
bull Static polarizabilities (Q2 = 0)
γ0p =e2g2
A
96π3f2πm
2π
1minus 21πmicro
8minus(
59
2+ 26 logmicro
)micro2 +
1875πmicro3
64
+3
(3
2+ 26 logmicro
)micro4 +
(C1)
γ0n =e2g2
A
48π3f2πm
2π
1
2minus 9πmicro
16minus 2micro2 logmicro+
75πmicro3
128minus 3micro4
4+
(C2)
δLTp =e2g2
A
192π3f2πm
2π
1minus 9πmicro
8+
(13
2minus 2 logmicro
)micro2 minus 465πmicro3
64
minus(
47
2+ 42 logmicro
)micro4 +
(C3)
δLTn =e2g2
A
96π3f2πm
2π
1
2+
3πmicro
16+ (1 + 2 log micro)micro2 minus 105πmicro3
128+
5micro4
4+
(C4)
38
δLTn =e2g2
A
96π3f2πm
2π
1
2+
3πmicro
16+ (1 + 2 log micro)micro2 minus 105πmicro3
128+
5micro4
4+
(C5)
γ0p =e2g2
A
16π3f2πm
4π
4
45minus 3πmicro
16+
14micro2
5minus 1813πmicro3
384minus 192
5(1 + log micro)micro4
+80703πmicro5
2048+
(C6)
γ0n =e2g2
A
16π3f2πm
4π
4
45minus 5πmicro
48+
4micro2
5minus 245πmicro3
384minus 32micro4 logmicro
15
+1323πmicro5
2048+
(C7)
bull Slopes of polarizabilities at Q2 = 0
dγ0p(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
2minus 45πmicro
4+ 223micro2 minus 28515πmicro3
64
minus9
(1953
4+ 449 logmicro
)micro4 +
570255πmicro5
128+
(C8)
dγ0n(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
2minus 81πmicro
8+ 94micro2 minus 2535πmicro3
32
minus3 (1 + 90 logmicro)micro4 +84315πmicro5
1024+
(C9)
dδLTp(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
2880π3f2πm
4π
minus5
2minus 27πmicro
32+ 20micro2 minus 5865πmicro3
256
+3
(617
4+ 36 logmicro
)micro4 minus 2056845πmicro5
4096+
(C10)
dδLTn(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
minus5
4minus 81πmicro
64minus 11micro2 +
10005πmicro3
512
+15
8(11 + 48 logmicro)micro4 minus 267015πmicro5
8192+
(C11)
dIAp(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
96π2f2πmicro
2
1minus 15πmicro
4minus 1
2(115 + 88 logmicro)micro2 +
1839πmicro3
32
+5 (5 + 34 logmicro)micro4 +
(C12)
dIAn(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
48π2f2πmicro
2
1
2minus 11πmicro
8minus 1
4(1 + 20 logmicro)micro2 +
99πmicro3
64
minus25micro4
12+
(C13)
dI1p(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
96π2f2πmicro
3π
8+ 2 (4 + 3 logmicro)microminus 537πmicro2
64
minus1
2(15 + 56 logmicro)micro3 +
(C14)
39
dI1n(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
48π2f2πmicro
minus π
16+
1
4(3 + 4 logmicro)microminus 57πmicro2
128+
2micro3
3+
(C15)
γ0p(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2g2
A
16π3f2πm
6π
1
105minus 23πmicro
256+
377micro2
210minus 15551πmicro3
6144+
3371micro4
105
minus1640457πmicro5
32768+
(C16)
γ0n(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2g2
A
16π3f2πm
6π
1
105minus 153πmicro
1792+
69micro2
70minus 4615πmicro3
6144+
172micro4
35
minus120897πmicro5
32768+
(C17)
2 ∆-exchange contribution
Here we give analytical expressions for the tree-level ∆-exchange contributions to the nucleon
spin polarizabilities and their slopes at Q2 = 0 Note that the ∆-exchange contributes equally to
proton and neutron polarizabilities Recall that for the magnetic γlowastN∆ coupling we introduced a
dipole form factor to mimic vector-meson dominance gM rarr gM(1 +Q2Λ2)2
bull Static polarizabilities (Q2 = 0)
γ0 = minus e2
4πM2+
(g2M
∆2+
g2E
M2+
minus 4gMgEM+∆
) (C18)
δLT =e2M∆
4πM3+
(g2E
MNM++gMgE∆MN
minus gEgCM2
∆
) (C19)
γ0 =e2M2
N
π∆2M4+
(minusg
2M
∆2+
g2E
M2+
+4gMgE∆M+
) (C20)
bull Slopes of polarizabilities at Q2 = 0
dγ0(Q2)
dQ2
∣∣∣∣∣Q2=0
= minus e2
πM2+∆
(g2M
∆
[1
4∆2minus 1
∆M++
1
2M2+
]minus 1
Λ2
g2M
∆+
g2E
2M2+
[1
2∆minus 3
M+
]minusgMgE
M+
[1
∆2minus 5
∆M++
1
M2+
]+
1
Λ2
2gMgEM+
+2gMgC∆M2
+
minus gEgCM3
+
) (C21)
40
d δLT (Q2)
dQ2
∣∣∣∣∣Q2=0
=e2M∆∆
4πMNM2+
(g2E
∆2M2+
[1
∆minus 4
M+
]minus g2
C
∆M2∆M
2+
+gMgE∆2M+
[1
∆2
minus 3
∆M++
1
M2+
]minus 2
Λ2
gMgE∆2M+
+gMgC∆M2
∆
[1
2∆2minus 2
∆M++
1
2M2+
]minus gEgC
2M2∆M
2+
[7
∆+
1
M+
]) (C22)
dIA(Q2)
dQ2
∣∣∣∣∣Q2=0
= minusM2N
M2+
(g2M
2∆2+
g2E
MNM+minus 2gMgE
∆M+minus gEgCM∆M+
) (C23)
dI1(Q2)
dQ2
∣∣∣∣∣Q2=0
= minusM∆M2N
2M3+
(g2E
MNM∆minus gMgE∆MN
minus gEgCM2
∆
) (C24)
γ0(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2M2
N
π∆3M6+
(g2M
∆
[3M2
∆ + 2M∆MN minus 9M2N
∆2+
4M2+
Λ2
]
minus4gMgE
[5M2
∆ minus 9M2N
M+∆2+
2M+
Λ2
]+
g2E
∆M+[7M∆ minus 9MN ]
minus8gMgC∆
+4gEgCM+
) (C25)
[1] D Drechsel B Pasquini and M Vanderhaeghen Dispersion relations in real and virtual
Compton scattering Phys Rept 378 99 (2003) hep-ph0212124
[2] S E Kuhn J-P Chen and E Leader Spin structure of the nucleon mdash status and recent
results Prog Part Nucl Phys 63 1 (2009) arXiv08123535 [hep-ph]
[3] F Hagelstein R Miskimen and V Pascalutsa Nucleon polarizabilities from Compton scat-
tering to hydrogen atom Prog Part Nucl Phys 88 29 (2016) arXiv151203765 [nucl-th]
[4] B Pasquini and M Vanderhaeghen Dispersion theory in electromagnetic interactions Ann
Rev Nucl Part Sci 68 75 (2018) arXiv180510482 [hep-ph]
[5] S Gerasimov A Sum rule for magnetic moments and the damping of the nucleon magnetic
moment in nuclei Sov J Nucl Phys 2 430 (1966)
[6] S Drell and A C Hearn Exact sum rule for nucleon magnetic moments Phys Rev Lett
16 908 (1966)
[7] J S Schwinger Source theory viewpoints in deep inelastic scattering Proc Natl Acad Sci
USA 72 1 (1975)
[8] J S Schwinger Source theory viewpoints in deep inelastic scattering Electromagnetic In-
teractions and Field Theory Proceedings 14 Internationale Universitatswochen Schladming
Austria February 24-March 7 1975 Acta Phys Austriaca Suppl 14 471 (1975)
41
[9] J Schwinger Source theory discussion of deep inelastic scattering with polarized particles
Proc Natl Acad Sci USA 72 1559 (1975)
[10] A M Baldin Polarizability of nucleons Nucl Phys 18 310 (1960)
[11] M Gell-Mann M L Goldberger and W E Thirring Use of causality conditions in quantum
theory Phys Rev 95 1612 (1954)
[12] A Pineda Leading chiral logarithms to the hyperfine splitting of the hydrogen and muonic
hydrogen Phys Rev C 67 025201 (2003)
[13] C Peset and A Pineda The two-photon exchange contribution to muonic hydrogen from
chiral perturbation theory Nucl Phys B 887 69 (2014) arXiv14064524 [hep-ph]
[14] C Peset and A Pineda Model-independent determination of the two-photon exchange contri-
bution to hyperfine splitting in muonic hydrogen JHEP 04 060 arXiv161205206 [nucl-th]
[15] F Hagelstein and V Pascalutsa Proton structure in the hyperfine splitting of muonic hydro-
gen PoS CD15 077 (2016) arXiv151104301 [nucl-th]
[16] F Hagelstein Exciting Nucleons in Compton Scattering and Hydrogen-Like Atoms PhD
thesis Mainz U Inst Kernphys (2017) arXiv171000874 [nucl-th]
[17] F Hagelstein ∆(1232)-Resonance in the hydrogen spectrum Proceedings 11th International
Workshop on the Physics of Excited Nucleons (NSTAR 2017) Columbia SC USA August
20-23 2017 Few Body Syst 59 93 (2018) arXiv180109790 [nucl-th]
[18] Y Prok et al (CLAS) Moments of the spin structure functions gp1 and gd1 for 005 lt Q2 lt 30
GeV2 Phys Lett B 672 12 (2009) arXiv08022232 [nucl-ex]
[19] H Dutz et al (GDH) First measurement of the Gerasimov-Drell-Hearn sum rule for 1H from
07 GeV to 18 GeV at ELSA Phys Rev Lett 91 192001 (2003)
[20] M Amarian et al (Jefferson Lab E94010) Measurement of the generalized forward spin po-
larizabilities of the neutron Phys Rev Lett 93 152301 (2004) arXivnucl-ex0406005
[21] M Amarian et al The Q2 evolution of the generalized Gerasimov-Drell-Hearn integral for
the neutron using a 3He target Phys Rev Lett 89 242301 (2002) arXivnucl-ex0205020
[22] M Amarian et al (Jefferson Lab E94-010)Q2 evolution of the neutron spin structure moments
using a 3He target Phys Rev Lett 92 022301 (2004) arXivhep-ex0310003
[23] A Deur et al Experimental determination of the evolution of the Bjorken integral at low Q2
Phys Rev Lett 93 212001 (2004) arXivhep-ex0407007
[24] K Slifer Low Q2 measurement of gp2 and the δLT spin polarizability Spin structure at long dis-
tance Proceedings Workshop Newport News USA March 12ndash13 2009 nucl-ex09064775
(2009) AIP Conf Proc 1155 10106313203293 (2009) arXiv09064775 [nucl-ex]
[25] P Solvignon et al (E01-012) Moments of the neutron g2 structure function at intermediate
42
Q2 Phys Rev C 92 015208 (2015) arXiv13044497 [nucl-ex]
[26] A Deur Experimental studies at low Q2 of the spin structure of the nucleon at Jefferson Lab
in 9th International Workshop on Chiral Dynamics (CD18) Durham NC USA September
17-21 2018 (2019) arXiv190305661 [nucl-ex]
[27] V Sulkosky et al (Jefferson Lab E97-110) Measurement of the 3He spin-structure functions
and of neutron (3He) spin-dependent sum rules at 0035 le Q2 le 024 GeV2 Phys Lett B
805 135428 (2020) arXiv190805709 [nucl-ex]
[28] K Adhikari et al (CLAS) Measurement of the Q2 Dependence of the Deuteron Spin Structure
Function g1 and its Moments at Low Q2 with CLAS Phys Rev Lett 120 062501 (2018)
arXiv171101974 [nucl-ex]
[29] V Lensky F Hagelstein A Hiller Blin and V Pascalutsa Comment on rdquoMeasurement of
the Q2 Dependence of the Deuteron Spin Structure Function g1 and its Moments at Low Q2
with CLASrdquo (2018) arXiv180603219 [nucl-th]
[30] J M Alarcn F Hagelstein V Lensky and V Pascalutsa Forward doubly-virtual Compton
scattering off the nucleon in chiral perturbation theory at NLO the subtraction function and
moments of unpolarized structure functions (2020) arXiv200509518 [hep-ph]
[31] V Bernard E Epelbaum H Krebs and U-G Meiszligner New insights into the spin structure
of the nucleon Phys Rev D 87 054032 (2013) arXiv12092523 [hep-ph]
[32] V Lensky J M Alarcon and V Pascalutsa Moments of nucleon structure functions at
next-to-leading order in baryon chiral perturbation theory Phys Rev C 90 055202 (2014)
arXiv14072574 [hep-ph]
[33] V Pascalutsa and M Vanderhaeghen Chiral effective-field theory in the ∆(1232) region
I Pion electroproduction on the nucleon Phys Rev D 73 034003 (2006) arXivhep-
ph0512244
[34] V Pascalutsa and M Vanderhaeghen Electromagnetic nucleon-to-Delta transition in chiral
effective field theory Phys Rev Lett 95 232001 (2005) arXivhep-ph0508060
[35] T R Hemmert B R Holstein and J Kambor Systematic 1M expansion for spin 32
particles in baryon chiral perturbation theory Phys Lett B 395 89 (1997) arXivhep-
ph9606456
[36] V Pascalutsa and D R Phillips Effective theory of the ∆(1232) in Compton scattering off
the nucleon Phys Rev C 67 055202 (2003) arXivnucl-th0212024
[37] V Pascalutsa M Vanderhaeghen and S N Yang Electromagnetic excitation of the ∆(1232)-
resonance Phys Rept 437 125 (2007) arXivhep-ph0609004
[38] K A Olive et al (Particle Data Group) Review of Particle Physics Chin Phys C 38 090001
43
(2014)
[39] F E Low Scattering of light of very low frequency by systems of spin 12 Phys Rev 96
1428 (1954)
[40] M Gell-Mann and M L Goldberger Scattering of low-energy photons by particles of spin
12 Phys Rev 96 1433 (1954)
[41] A M Harun ar-Rashid A simple derivation of Schwingerrsquos sum rule for spin dependent
structure functions Nuovo Cim A 33 447 (1976)
[42] F Hagelstein and V Pascalutsa Dissecting the hadronic contributions to (g minus 2)micro by
Schwingerrsquos sum rule Phys Rev Lett 120 072002 (2018) arXiv171004571 [hep-ph]
[43] V Lensky V Pascalutsa M Vanderhaeghen and C Kao Spin-dependent sum rules con-
necting real and virtual Compton scattering verified Phys Rev D 95 074001 (2017)
arXiv170101947 [hep-ph]
[44] H Burkhardt and W N Cottingham Sum rules for forward virtual Compton scattering
Annals Phys 56 453 (1970)
[45] V Lensky and V Pascalutsa Predictive powers of chiral perturbation theory in Compton
scattering off protons Eur Phys J C 65 195 (2010) arXiv09070451 [hep-ph]
[46] H Grieszlighammer J McGovern D Phillips and G Feldman Using effective field theory to
analyse low-energy Compton scattering data from protons and light nuclei Prog Part Nucl
Phys 67 841 (2012) arXiv12036834 [nucl-th]
[47] H W Grieszlighammer J A McGovern and D R Phillips Nucleon polarisabilities at and
beyond physical pion masses Eur Phys J A 52 139 (2016) arXiv151101952 [nucl-th]
[48] E Epelbaum H Krebs and U-G Meiszligner Improved chiral nucleon-nucleon potential up to
next-to-next-to-next-to-leading order Eur Phys J A 51 53 (2015) arXiv14120142 [nucl-
th]
[49] E Epelbaum H Krebs and U-G Meiszligner Precision nucleon-nucleon potential at fifth order
in the chiral expansion Phys Rev Lett 115 122301 (2015) arXiv14124623 [nucl-th]
[50] C W Kao T Spitzenberg and M Vanderhaeghen Burkhardt-Cottingham sum rule and
forward spin polarizabilities in heavy baryon chiral perturbation theory Phys Rev D 67
016001 (2003) arXivhep-ph0209241
[51] C-W Kao D Drechsel S Kamalov and M Vanderhaeghen Higher moments of nucleon
spin structure functions in heavy baryon chiral perturbation theory and in a resonance model
Phys Rev D 69 056004 (2004) arXivhep-ph0312102
[52] T Becher and H Leutwyler Baryon chiral perturbation theory in manifestly Lorentz invariant
form Eur Phys J C 9 643 (1999) arXivhep-ph9901384
44
[53] J Gasser M E Sainio and A Svarc Nucleons with chiral loops Nucl Phys B 307 779
(1988)
[54] L S Geng J Martin Camalich L Alvarez-Ruso and M J Vicente Vacas Leading SU(3)-
breaking corrections to the baryon magnetic moments in chiral perturbation theory Phys
Rev Lett 101 222002 (2008) arXiv08051419 [hep-ph]
[55] D Drechsel S S Kamalov and L Tiator The GDH sum rule and related integrals Phys
Rev D 63 114010 (2001) arXivhep-ph0008306
[56] D Drechsel O Hanstein S S Kamalov and L Tiator A unitary isobar model for pion
photo- and electroproduction on the proton up to 1 GeV Nucl Phys A 645 145 (1999)
arXivnucl-th9807001
[57] L Tiator private communication (2020)
[58] V Bernard T R Hemmert and U-G Meiszligner Spin structure of the nucleon at low energies
Phys Rev D 67 076008 (2003) arXivhep-ph0212033
[59] O Gryniuk F Hagelstein and V Pascalutsa Evaluation of the forward Compton scattering
off protons II Spin-dependent amplitude and observables Phys Rev D 94 034043 (2016)
arXiv160400789 [nucl-th]
[60] R Zielinski The g2p Experiment A Measurement of the Protonrsquos Spin Structure Functions
PhD thesis New Hampshire U (2010) arXiv170808297 [nucl-ex]
[61] N Guler et al (CLAS) Precise determination of the deuteron spin structure at low to mod-
erate Q2 with CLAS and extraction of the neutron contribution Phys Rev C 92 055201
(2015) arXiv150507877 [nucl-ex]
[62] C E Carlson V Nazaryan and K Griffioen Proton structure corrections to electronic
and muonic hydrogen hyperfine splitting Phys Rev A 78 022517 (2008) arXiv08052603
[physicsatom-ph]
[63] V Bernard N Kaiser and U-G Meiszligner Chiral dynamics in nucleons and nuclei Int J
Mod Phys E 4 193 (1995) arXivhep-ph9501384
[64] T R Hemmert B R Holstein and J Kambor ∆(1232) and the polarizabilities of the nucleon
Phys Rev D 55 5598 (1997) arXivhep-ph9612374
[65] V Pascalutsa and R Timmermans Field theory of nucleon to higher-spin baryon transitions
Phys Rev C 60 042201 (1999) arXivnucl-th9905065
[66] V Pascalutsa Quantization of an interacting spin-32 field and the Delta isobar Phys Rev
D 58 096002 (1998) arXivhep-ph9802288
[67] H Krebs Double Virtual Compton Scattering and SpinStructure of the Nucleon PoS
CD2018 031 (2019)
45
[68] N Kochelev and Y Oh Axial anomaly and the δLT puzzle Phys Rev D 016012 (2012)
arXiv11034892 [hep-ph]
[69] D Drechsel S Kamalov and L Tiator Unitary isobar model ndash MAID2007 Eur Phys J A
69 (2007) available at httpsmaidkphuni-mainzde arXiv07100306 [nucl-th]
[70] P J Mohr B N Taylor and D B Newell CODATA recommended values of the fundamental
physical constants 2010 Rev Mod Phys 84 1527 (2012)
[71] J Ahrens et al (GDH A2) First measurement of the Gerasimov-Drell-Hearn integral for 1H
from 200 to 800 MeV Phys Rev Lett 87 022003 (2001) arXivhep-ex0105089 [hep-ex]
[72] K Helbing (GDH) Experimental verification of the GDH sum rule at ELSA and MAMI Nucl
Phys Proc Suppl 105 113 (2002)
[73] R Bradford A Bodek H S Budd and J Arrington A New parameterization of the nucleon
elastic form-factors NuInt05 proceedings of the 4th International Workshop on Neutrino-
Nucleus Interactions in the Few-GeV Region Okayama Japan 26-29 September 2005 Nucl
Phys Proc Suppl 159 127 (2006) arXivhep-ex0602017
[74] J D Bjorken Applications of the chiral U(6)timesU(6) algebra of current densities Phys Rev
148 1467 (1966)
[75] J D Bjorken Inelastic scattering of polarized leptons from polarized nucleons Phys Rev D
1 1376 (1970)
[76] V Bernard T R Hemmert and U-G Meiszligner Novel analysis of chiral loop effects in
the generalized Gerasimov-Drell-Hearn sum rule Phys Lett B 545 105 (2002) arXivhep-
ph0203167
[77] A Deur et al Experimental study of isovector spin sum rules Phys Rev D 78 032001
(2008) arXiv08023198 [nucl-ex]
[78] B Pasquini P Pedroni and D Drechsel Higher order forward spin polarizability Phys Lett
B 687 160 (2010) arXiv10014230 [hep-ph]
[79] R Jaffe g2ndashThe nucleonrsquos other spin-dependent structure function Comments Nucl Part
Phys 19 239 (1990)
[80] E V Shuryak and A Vainshtein Theory of power corrections to deep inelastic scattering in
quantum chromodynamics (II) Qminus4 effects polarized target Nucl Phys B 201 141 (1982)
[81] S Wandzura and F Wilczek Sum rules for spin dependent electroproduction Test of rela-
tivistic constituent quarks Phys Lett B 72 195 (1977)
[82] B W Filippone and X-D Ji The spin structure of the nucleon Adv Nucl Phys 26 1
(2001) arXivhep-ph0101224
[83] M Burkardt The g2 structure function Proceedings Workshop on Spin structure at long
46
distance Newport News USA March 12-13 2009 AIP Conf Proc 1155 26 (2009)
arXiv09054079 [hep-ph]
[84] A Deur S J Brodsky and G F de Teramond The spin structure of the nucleon Rept
Prog Phys 82 076201 (2019) arXiv180705250 [hep-ph]
[85] B R Holstein D Drechsel B Pasquini and M Vanderhaeghen Higher order polarizabilities
of the proton Phys Rev C 61 034316 (2000) arXivhep-ph9910427
[86] B R Holstein V Pascalutsa and M Vanderhaeghen Sum rules for magnetic moments and
polarizabilities in QED and chiral effective-field theory Phys Rev D 72 094014 (2005)
arXivhep-ph0507016
[87] B Pasquini D Drechsel and L Tiator Invariant amplitudes for pion electroproduction Eur
Phys J A 34 387 (2007) arXiv07122327 [hep-ph]
[88] A J Buchmann Non-spherical proton shape and hydrogen hyperfine splitting Proceedings
International Workshop on Precision Physics of Simple Atomic Systems (PSAS 2008) Wind-
sor Ontario Canada July 21-26 2008 Can J Phys 87 773 (2009) arXiv09104747
[physicsatom-ph]
47
TABLE II The NLO BχPT predictions for the forward VVCS polarizabilities and their slopes at
Q2 = 0 The contributions of the πN loops the ∆ exchange and the π∆ loops are shown together
with the combined total result Note that IA(0) = I1(0) = d2(0) = 0 and (d2)prime = 0
πN loops ∆ exchange π∆ loops Total
γ0 p 201 minus284 minus010minus093(92)
(10minus4 fm4) n 298 003(92)
δLT p 150 minus016 minus002132(15)
(10minus4 fm4) n 235 218(23)
γ0 p 208 minus096 minus001112(30)
(10minus4 fm6) n 292 195(30)
(γ0)prime p minus033011 001
minus022(4)
(10minus4 fm6) n minus073 minus061(7)
(δLT )prime p minus080 minus004 minus001minus085(8)
(10minus4 fm6) n minus119 minus124(12)
(γ0)prime p minus100016 000
minus084(10)
(10minus4 fm8) n minus158 minus142(15)
(∆IA)prime p 238 minus1121 025minus858(343)
(GeVminus2) n 141 minus955(343)
(∆I1)prime p 034 minus053 058039(4)
(GeVminus2) n minus107 minus101(10)
TABLE III Our NLO BχPT predictions for the spin polarizabilities at Q2 = 0 compared with
the BχPT+∆ predictions from Bernard et al [31] and the available empirical information Where
the reference is not given the empirical number is provided by the MAID analysis [55 69] with
unspecified uncertainty
Proton Neutron
This work BχPT+∆ Empirical This work BχPT+∆ Empirical
γ0 minus093(92) minus174(40) minus100(8)(12) [19] 003(92) minus077(40) minus0005
(10minus4 fm4) minus090(8)(11) [78] [MAID]
minus0929(105) [59]
δLT 132(15) 240(1) 134 218(23) 238(3) 203
(10minus4 fm4) [MAID] [MAID]
γ0 112(30) 060(7)(7) [78] 195(30) 123
(10minus4 fm6) 0484(82) [59] [MAID]
IV CONCLUSIONS
We have presented a complete NLO calculation of the polarized non-Born VVCS ampli-
tudes in covariant BχPT with pion nucleon and ∆(1232) fields The dispersion relations
between the VVCS amplitudes and the tree-level photoabsorption cross sections served as a
27
cross-check of these calculations
The obtained moments of the proton and neutron spin structure functions related to gen-
eralized polarizabilities and GDH-type integrals agree well with the available experimental
data The description of their Q2 evolution is improved compared to the previous χPT pre-
dictions In particular the NLO BχPT predictions obtained here give a better description
of the empirical data (eg from the Jefferson Laboratory ldquoSpin Physics Programrdquo) than
the HB [50 51] and IR [58] calculations
The demonstrated predictive power of the χPT framework amplitudes makes it well
suited for extending the χPT evaluation of the TPE effect in the hyperfine structure of
(muonic-)hydrogen [15ndash17] to next-to-leading order
ACKNOWLEDGEMENTS
We thank Lothar Tiator and Marc Vanderhaeghen for helpful discussions This work
is supported by the Deutsche Forschungsgemeinschaft (DFG) through the Collaborative
Research Center [The Low-Energy Frontier of the Standard Model (SFB 1044)] JMA ac-
knowledges support from the Community of Madrid through the ldquoPrograma de atraccion de
talento investigador 2017 (Modalidad 1)rdquo and the Spanish MECD grants FPA2016-77313-P
FH gratefully acknowledges financial support from the Swiss National Science Foundation
Appendix A Tensor decompositions of the VVCS amplitudes
In this appendix we review the decomposition of the forward VVCS process into tensor struc-
tures and scalar amplitudes In particular we consider the connection between the covariant and
the semi-relativistic decomposition in the lab frame that is defined in terms of the conventional
transverse longitudinal transverse-transverse and transverse-longitudinal amplitudes
As explained in Sec II A the process of forward VVCS off the nucleon can be described in
terms of four explicitly covariant amplitudes S1 2 and T1 2 [3]
T (νQ2) =
(minusgmicroν +
qmicroqν
q2
)T1(νQ2) +
1
M2N
(pmicro minus p middot q
q2qmicro)(
pν minus p middot qq2
qν)T2(νQ2) (A1)
minus 1
MNγmicroναqα S1(νQ2)minus 1
M2N
(γmicroνq2 + qmicroγναqα minus qνγmicroαqα
)S2(νQ2)
εprimelowastmicro εν
where εmicro (εprimelowastmicro ) are the incoming (outgoing) photon polarization vectors ν is the photon lab-frame
energy and Q2 is the photon virtuality Alternatively the decomposition in the laboratory frame
28
(which in the forward case coincides with the Breit frame) is parametrized in terms of the nucleon
Pauli matrices ~σ and the four scalar functions fL fT gTT and gLT
T (νQ2) = ε0 εprimelowast0 fL(νQ2) + (~ε primelowast middot ~ε ) fT (νQ2) + i~σ middot (~ε primelowast times ~ε ) gTT (νQ2) (A2)
minusi~σ middot [(ε0~εprimelowast minus ~ε ε primelowast0 )times q] gLT (νQ2)
Here ~q and q = ~q|~q | are the photon three-momentum in the lab system and its unit vector The
modified polarization vector components are given by
ε0 =
[ε0 minus
ν
|~q | (~ε middot q )
] |~q |Q ~ε = ~εminus q (~ε middot q ) (A3)
where ε = (ε0~ε ) is the usual incoming photon polarization vector and εprimelowast the outgoing polarization
vector The LEX of the lab frame amplitudes [Eq (10)] can serve in particular as the definition
of the generalized polarizabilities The lab frame amplitudes are also conveniently used for the
definition of the response functions see the example of the scalar amplitude gLT (νQ2) and the
corresponding response function σLT (νQ2) below in App B
Appendix B Photoabsorption cross sections
In the forward kinematics the spin-dependent VVCS amplitudes and the spin polarizabilities
can be described in terms of the polarized structure functions g1(xQ2) and g2(xQ2) or equiv-
alently the helicity-difference cross section σTT (νQ2) and the longitudinal-transverse response
function σLT (νQ2) with the help of dispersion relations (5) and the optical theorem (3) In this
way the photoabsorption cross sections measured in electroproduction processes form the basis
for most empirical evaluations shown throughout Sec III In the following we present the BχPT
predictions for the tree-level cross sections of πN - π∆- and ∆-production through photoabsorption
on the nucleon cf Figs 8 9 and 10 in Ref [30] In Secs B 1 and B 2 we will discuss the leading
πN -production channel and the ∆-production channel respectively We used these cross sections
to verify the polarizability predictions obtained otherwise from the calculated non-Born VVCS
amplitudes Due to the bad high-energy behavior of the π∆-production cross sections in BχPT
cf Fig 10 the dispersion relations in Eq (5) require further subtractions for a reconstruction of
the π∆-loop contribution to the spin-dependent VVCS amplitudes Therefore not all polarizabil-
ities could be verified but only those appearing as higher-order terms in the LEX of the VVCS
amplitudes such as γ0 [16]
29
proton neutron
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Neutron
02 04 06 08 100
50
100
150
200
250
[GeV]
T [b]
02 04 06 08 10
-50
0
50
100
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Neutron
02 04 06 08 100
50
100
150
200
250
[GeV]
T [b]
02 04 06 08 10
-50
0
50
100
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02040608100
50
100
150
200
250
300
[GeV]
T[b]
0204060810
-50
0
50
[GeV]
TT[b]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
0204060810
00
02
04
06
[GeV]
LTQ
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
0204060810
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
FIG 10 Photoabsorption cross sections for πN (red) and π∆ production (orange) with Q2 = 0
(solid) and Q2 = 01 GeV2 (dashed for πN and dotted for π∆ channel)
1 πN-production channel
In order to extract the response function σLT (νQ2) we have developed a method similar to
the one used to calculate σTT (νQ2) see for example Ref [86] For σLT (νQ2) however the
calculation is more complicated because one has to take into account that the associated Compton
process involves a spin-flip of the nucleon as illustrated in Fig 11 When calculating the cross
section the product of the incoming nucleon spinors has to reflect this flip
The forward VVCS amplitude related to σLT (νQ2) mdash and δLT (Q2) mdash is gLT (νQ2) It can
be extracted from Eq (A2) if one takes the modified polarization vector components in Eq (A3)
with ε = εL and εprimelowast = εlowastplusmn as input where εL = 1Q(|~q | 0 0 q0) and εplusmn = ∓ 1radic
2(0 1plusmni 0) are the
standard longitudinal and transverse polarization vectors respectively For εL and εlowastplusmn only the
choice of helicities hprime = plusmn12 and h = ∓12 gives a non-zero contribution and one obtains
χdaggerhprime T (νQ2)χh = χdaggerhprimeminusi~σ middot [(ε0~εprimelowast minus ~ε ε primelowast0 )times q] gLT (νQ2)χh =
radic2 gLT (νQ2) (B1)
where χh and χdaggerhprime are two-component Pauli spinors with opposite helicities or here spins
Let us now consider the related photoabsorption process and in particular the tree-level γlowastN rarr
30
s = 12
sprime = minus12
λ = 0 λprime = 1
rprime rprime
s=12sprime=minus1
2
λ=0λprime=1
rprimerprime
s=12
sprime=minus12
λ=0λprime=1
rprimerprime hacute=frac12h=-frac12
λacute=1λ=0
FIG 11 Relation between the forward Compton process and the photoabsorption process given
by the optical theorem In particular we show the longitudinal-transverse contribution The
double-line arrows represent the spin of the external particles while the dot represents the scalar
(longitudinal) polarization of the incoming photon Inside the blob the intermediate states are
represented eg nucleons with spins rprime (which are averaged in the calculation of the cross section)
and pions
πN channel see diagrams in Fig 8 of Ref [30] We define the πN -production amplitude as
T = uhB (PB)sumi
Ai(s t)Γi uhA(PA) (B2)
with the Dirac structures
Γ1 = γ5 (B3a)
Γ2 =1
2
[qA ε
]γ5 (B3b)
where uhA(PA) and udaggerhB (PB) are the Dirac spinors and PA and PB are the four-momenta of
the incoming and outgoing nucleons respectively When calculating the photoabsorption cross
section related to the VVCS amplitude in Eq (B1) the nucleon spin flip should be implemented
by uhprime(PA) in T dagger and uh(PA) in T together with the appropriate transverse and longitudinal
photon polarization vectors εlowastplusmn and εL
However if one wants to use the properties of the Dirac matrices it is more useful to construct
an operator to produce this spin flip in the external nucleons of Fig 11 This is accomplished by
introducing the projector ΓLT equiv 12radic
2(γ1 + iγ2)γ5 which also takes into account the extra factor
radic2 in Eq (B1) We checked that with this projector one correctly extracts δLT by comparing the
HB limit of our result to the HB result of Ref [50] where the authors calculate this polarizability
from the Compton amplitude directly With all those ingredients the longitudinal-transverse cross
section is calculated in the following way
σLT (νQ2) =1
64π2 s
|~pf |cm
|~pi|cm
int 1
minus1dcos θ
sumij
AiAdaggerjXij (B4)
31
with
Xij = Tr[(PB +MN )Γi(PA +MN )ΓLTγ0Γdaggerjγ
0] (B5)
where θ is the scattering angle in the center-of-mass (cm) frame and |~pi|cm (|~pf |cm) is the three-
momentum of an incoming (outgoing) particle in the cm frame An explicit calculation of the
matrix Xij leads to
X = MNQ
0 2(PB minus PA) middot εLminusradic
2|~qf |cm sin θ (sminus u)
(B6)
where |~qi|cm (|~qf |cm) is the relative three-momentum of the incoming (outgoing) particles in the
cm frame Here s t and u are the usual Mandelstam variables For the different γlowastN rarr πN
channels we obtain the following amplitudes Ai where we introduce qA as the four-momentum of
the incoming photon and qB as the four-momentum of the outgoing pion
bull γlowastprarr π0p
A1 =e gAMN
fπ
[2PA middot ε+ qA middot ε
sminusM2N
+2PB middot εminus qA middot ε
uminusM2N
] (B7a)
A2 =e gAMN
fπ
[1
sminusM2N
+1
uminusM2N
] (B7b)
bull γlowastprarr π+n
A1 =
radic2 e gAMN
fπ
[2PA middot ε+ qA middot ε
sminusM2N
+2(PA minus PB) middot ε+ qA middot ε
tminusm2π
] (B8a)
A2 =
radic2 e gAMN
fπ(sminusM2N )
(B8b)
bull γlowastnrarr π0n
A1 = 0 (B9a)
A2 = 0 (B9b)
bull γlowastnrarr πminusp
A1 =
radic2 e gAMN
fπ
[2PB middot εminus qA middot ε
uminusM2N
minus 2(PA minus PB) middot ε+ qA middot εtminusm2
π
] (B10a)
A2 =
radic2 e gAMN
fπ(uminusM2N )
(B10b)
The analytical expressions shown above were checked with the amplitudes given in Ref [87] An-
alytical expressions for the tree-level γlowastN rarr πN channel of the σLT (νQ2) and σTT (νQ2) cross
32
sections are given below (proton channels π+n and π0p neutron channel πminusp) We checked that
they reproduce the known results in the real-photon limit [45 86] To shorten the final expressions
for the cross sections which are considerably longer for finite Q2 than in the real-photon limit we
define the following dimensionless kinematic variables
αγ = (ENi )cmradics =
s+M2N +Q2
2s (B11)
απ = (ENf )cmradics =
s+M2N minusm2
π
2s (B12)
βγ = Eγcmradics =
sminusM2N minusQ2
2s (B13)
βπ = Eπcmradics =
sminusM2N +m2
π
2s (B14)
λγ = |~qi|cmradics =
radic(sminusM2
N minusQ2)2 + 4sQ2
2s (B15)
λπ = |~qf |cmradics =
radic(sminusM2
N +m2π)2 minus 4sm2
π
2s (B16)
Here (ENi )cm and (ENf )cm are the energies of the incoming and outgoing nucleon Eγcm is the energy
of the incoming photon Eπcm is the energy of the outgoing pion all in the cm frame
σ(π+n)TT = minus e2g2
AM2N
64πf2πs
3(sminusM2N )2λ4
γ
4sλπλγ
[(M2
N minus s)(M2N minusQ2 minus s)(Q2 + 2sβγβπ)
+ 2s(m2πQ
2 minus (M2N minus s)
(M2N + s(minus1 + 2βγβπ)
))λ2γ
]+ (M2
N minus s)(M2N minusQ2 minus s)
times (Q2 + 2sβγβπ minus 2sλπλγ)(Q2 + 2s(βγβπ + λπλγ)
)times log
(Q2 + 2sβγβπ minus 2sλγλπQ2 + 2sβγβπ + 2sλγλπ
) (B17)
33
σ(π0p)TT =
e2g2AM
2Nλπ
64πf2πs(sminusM2
N )2λγ
1
s2λπλ3γ
[(M2
N minus s)(M2N minusQ2 minus s)
(3M2
N + s(2βγβπ minus 3))(M2N
+ s(2βγβπ minus 1))
+ 2s
((2m2
πQ2 + (M2
N minus s)2)(M2
N minus s)minus 2(minusm2
πQ2 + (sminusM2
N )2)
times sβγβπ + 4s2(sminusM2N )β2
γβ2π
)λ2γ
]arctan
(2sλπλγ
M2N + s(2βγβπ minus 1)
)+ 2
[minus 2m2
πQ2
+ (M2N minus s)
(2(sminusM2
N ) + 4sβγβπ +(sminusM2
N +Q2)(3M2N minus 3s+ 2sβγβπ)
sλ2γ
+m2πQ
2(sminusM2N )
(M2N minus s+ 2sβγβπ)(M2
N + s(minus1 + 2βγβπ minus 2λπλγ))
+m2πQ
2(sminusM2N )
(M2N minus s+ 2sβγβπ)(M2
N + s(minus1 + 2βγβπ + 2λπλγ))
)
+1
λπλγ
(βγβπ
(minusm2
πQ2 + (M2
N minus s)(M2N minus s+ 2sβγβπ)
)minus (M2
N minus s)(M2N minusQ2 minus s)λ2
π
)log
(M2N + s(minus1 + 2βγβπ + 2λπλγ)
M2N + s(minus1 + 2βγβπ minus 2λπλγ)
)] (B18)
σ(πminusp)TT =
e2g2AM
2N
64πf2πs
3λ4γ(M2
N + s(2βπβγ minus 1))2 minus 4λ2πs
2λ2γ)
4sλπλγ
((minusM2
N +Q2 + s)(M2N
+ s(2βπβγ minus 1))
2 minus 2sλ2γ
(2λ2
πs(minusM2N +Q2 + s) +m2
πQ2))
+((M2N + s(2βπβγ minus 1)
)2
minus 4λ2πs
2λ2γ
)((M2N + s(2βπβγ minus 1)
)(M2N minus 2Q2 minus s(2βπβγ + 1)
)+ 4λ2
πs2λ2
γ
)times log
(minusM2
N minus 2βπsβγ minus 2λπsλγ + s)minus(M2N + s(2βπβγ minus 2λπλγ minus 1)
)(M2N + s(minus1
+ 2βπβγ + 2λπλγ))[((
M2N + s(2βπβγ minus 1)
)(M2N minus 2Q2 minus s(2βπβγ + 1)
)+ 4λ2
πs2λ2
γ
)times log
(minusM2
N minus 2βπsβγ + 2λπsλγ + s)
+((Q2 + 2βπsβγ)2 minus 4λ2
πs2λ2
γ
)times log
(Q2 + 2βπsβγ minus 2λπsλγQ2 + 2sβπβγ + 2λπsλγ
)]+ 4sλ2
γ
((M2
N + s(2βπβγ minus 1))2 minus 4λ2πs
2λ2γ
)times (M2
N minus s) arctanh
(2λπsλγ
M2N + 2βπsβγ minus s
) (B19)
34
σ(π+n)LT =
e2g2AM
3Nλπ
32πf2πQs
3(sminusM2N )2λ4
γ
2sλγ
[(M2
N minus s)(Q2 + 2sβ2γ)(Q2 + 2sβγβπ)
minus 4s
((M2
N minus s)(Q2 minus 2s(απ minus 1)βγ
)+Q2sβγβπ
)λ2γ + 8s3(minus1 + απ)λ4
γ
]minus M2
N minus sλπ
[(Q2 + 2sβ2
γ)(Q2 + 2sβγβπ)2 + 4s2(2(απ minus 1)βγ(Q2 + 2sβγβπ)
minusQ2λ2π
)λ2γ + 8s3(απ minus 1)2λ4
γ
]arctan
(2sλπλγ
Q2 + 2sβγβπ
) (B20)
σ(π0p)LT =
e2g2AM
3Nλπ
16πf2πQs(sminusM2
N )2λγ
1
minus2s(M2N + s(minus1 + 2βγβπ))2λ2
γ + 8s3λ2πλ
4γ
[minus 3M8
N (Q2
+ 2sβ2γ) + 2M4
Ns2
(minus (Q2 + 2sβ2
γ)(minus3 + 2βγβπ) +(Q2(minus3 + 2βγβπ + 6λ2
π) + 2sβγ
times (minus3 + 12απ + 2βγβπ minus 12απβγβπ + 4βγλ2π))λ2γ minus 4s(α2
π minus 1)λ4π
)+ 2M2
Ns3
[minus (Q2
+ 2sβ2γ)(minus1 + 2βγβπ)
(6 + βγβπ(minus9 + 2βγβπ)
)+
(Q2(3minus 12λ2
π + 4βγβπ(minus1 + 2βγβπ
+ λ2π))minus 2sβγ
(minus 3 + 4απ(3 + 2απ(3 + 2βγβπ(minus3 + βγβπ)) + 4βγ(βπ + (2minus βγβπ)
times λ2π)))λ2γ + 8s
((απ minus 1)(1 + απ minus 2βγβπ) + 2απβγλ
2π
)λ4γ
)+ s4
((Q2 + 2sβ2
γ)
times (1minus 2βγβπ)2(2βγβπ minus 3) + 2(Q2(minus1 + 6λ2
π + 2βγβπ((1minus 2βγβπ)2 minus 2λ2π)) + 2sβγ
times (minus1 + 2βγβπ + 4(minus1 + βγβπ)(minusαπ + 2απβγβπ minus βγλ2π)))λ2γ minus 8
(s(απ minus 1)(απ
+ (1minus 2βγβπ)2) + 2βγ(2sαπ +Q2βγ)λ2π
)λ4γ + 32s(απ minus 1)λ2
πλ6γ
)+ 2M6
Ns
(Q2(6
minus 7βγβπ + λ2γ) + 2sβγ(βγ(6minus 7βγβπ) + (1minus 4απ)λ2
γ)
)]+sminusM2
N
4s2λπλ3γ
[(Q2 + 2sβ2
γ)
times(3M2
N + s(2βγβπ minus 3))(M2N + s(2βγβπ minus 1)
)+ 2s
(minusM2
N (Q2 + 2sβγ minus 8sαπβγ)
+ s(2sβγ(1minus 4απ + 4απβγβπ) +Q2(1minus 2λ2π)))λ2γ + 8s3(απ minus 1)λ4
γ
]
times arctan
(2sλπλγ
M2N + s(2βγβπ minus 1)
) (B21)
35
σ(πminusp)LT =
e2g2AM
3N
64πf2πQs
3λ4γ(M2
N minusQ2 minus s)(M2N + s(2βπβγ + 2λπλγ minus 1))
times
1
M2N + s(2βπβγ minus 2λπλγ minus 1)
[4λπsλγ(M2
N minusQ2 minus s)(
2sλ2γ
(s((1minus 2λ2
π)Q2
+ 2(2απ minus 1)sβγ(2βπβγ minus 1))minusM2N (Q2 + 2(1minus 2απ)sβγ)
)+ (Q2 + 2sβ2
γ)(M2N + s(2βπβγ minus 1)
)2 + 8(απ minus 1)απs
3λ4γ
)]minus(M2N + s(2βπβγ + 2λπλγ minus 1)
)times[
log
(minusM2N minus 2βπsβγ minus 2λπsλγ + s
minusM2N minus 2βπsβγ + 2λπsλγ + s
)(minus (Q2 + 2sβ2
γ)(M2N + s(2βπβγ minus 1)
)(minusM2
N + 2Q2 + 2βπsβγ + s)minus 2sλ2γ
(minusM2
N (Q2 + 2sβγ) +Q4 +Q2s(4απβγ
minus 2βγ minus 2λ2π + 1) + 2s2βγ(4(απ minus 1)βπβγ + 1)
)minus 8(απ minus 1)2s3λ4
γ
)+ 2
(4s2λ2
γ
(2(απ minus 1)βγ(Q2 + 2βπsβγ)minus λ2
πQ2)
+ (Q2 + 2βπsβγ)2(Q2 + 2sβ2γ)
+ 8(απ minus 1)2s3λ4γ
)arctanh
(2λπsλγ
Q2 + 2βπsβγ
)] (B22)
2 ∆-production channel
The tree-level ∆-exchange diagram in Fig 2 of Ref [30] contributes to the non-Born part of
the VVCS amplitudes The contribution of the ∆ exchange to the VVCS amplitudes can be split
into [17]
S∆-exch1 (νQ2) = S∆-pole
1 (νQ2) + S∆-exch1 (νQ2) (B23a)
νS∆-exch2 (νQ2) = νS∆-pole
2 (νQ2) + νS2∆-exch
(νQ2) (B23b)
and similarly for the unpolarized VVCS amplitudes discussed in Ref [30] Here we introduced the
∆-pole contributions S∆-polei and the ∆-non-pole contributions S∆-exch
i The former amplitudes
feature a pole at the ∆(1232)-production threshold and thus are proportional to
1
[sminusM2∆][uminusM2
∆]=
1
4M2N
1
ν2∆ minus ν2
(B24)
36
They can be reconstructed from the dispersion relations in Eq (5) with the tree-level ∆-production
cross sections as input cf Fig 10 in Ref [30]
σTT (νQ2) =π2α
M2NM
2+|~q |
minus g2
MMN (M+ + ν)|~q |2 +g2E(∆minus ν)(Q2 minusMNν)2
MN(B25a)
+g2CQ
4s(∆minus ν)
MNM2∆
minus 4gMgE(Q2 minusMNν)|~q |2 minus 4gMgCQ2|~q |2
+2gEgCQ
2[minusMNM∆ |~q |2 + s(Q2 +∆ν)
]MNM∆
δ(ν minus ν∆)
σLT (νQ2) =Qπ2α
M2NM
2+|~q |
g2E(MNν minusQ2) [M∆(MN + ν)minus s]
MN(B25b)
+g2CQ
2[MNM∆|~q |2 minus s(Q2 +∆ν)
]MNM2
∆
+ gMgEM∆|~q |2 minusgMgC(Q2 minusMNν)|~q |2
M∆
+gEgC(ν minus∆)(M2
N |~q |2 minus 2Q2s)
MNM∆
δ(ν minus ν∆)
with ∆ = M∆ minusMN M+ = M∆ + MN and the Mandelstam variable s = M2N + 2MNν minus Q2
Analytical expressions for the spin structure functions g1(xQ2) and g2(xQ2) can be constructed
from Eq (3) with the flux factor K(νQ2) = |~q | =radicν2 +Q2
In the ∆-non-pole contributions to S1(νQ2) and νS2(νQ2) the pole in ν at the ∆(1232)-
production threshold has canceled out
S∆-exch1 (νQ2) =
πα
MNM2+
[g2MQ
2+ + g2
E
(∆2 minus 3Q2
)+
4g2CQ
4
M2∆
minus 8gMgEM∆ωminus (B26a)
minus2gMgCQ2(MN minus 4M∆)
M∆+
2gEgCQ2(3MN minus 2M∆)
M∆
]
νS2∆-exch
(νQ2) =2πα
MNM2+
[g2EM∆∆ωminus +
g2M MNQ
2+
2+g2C Q
2(Q2 minus∆2)
2M∆(B26b)
+gEgM M∆(M∆ω+ minus 4MNωminus)minus gEgC ∆(2Q2 +MNω+)
+gMgC Q2(4MN minus ω+)
]+S∆-exch
2 (νQ2)
ν
[M2
∆ ω2+
M2N
+ ν2
]
with Q+ =radic
(M∆ +MN )2 +Q2 and ωplusmn = (M2∆minusM2
N plusmnQ2)2M∆ and the non-pole contribution
to S2(νQ2)
S∆-exch2 (νQ2) = minus2παMNν
M∆M2+
[gM + gE
]gC (B27)
These amplitudes to the contrary are not described by the tree-level ∆-production cross sections
in the standard dispersive approach [17] This peculiarity has been previously missed e g in the
calculation of the ∆-exchange contribution to the hydrogen hyperfine splitting in Ref [88] The
importance of including the ∆-non-pole contribution is also evident when considering the BC sum
37
rule in Eq (14) The ∆-pole terms by themselves violate the BC sum rule but cancel exactly with
the ∆-non-pole terms
limνrarr0
νS∆-pole2 (νQ2) + lim
νrarr0νS2
∆-exch(νQ2) = 0 (B28)
Appendix C Polarizabilities at Q2 = 0
In this section we give analytical expressions for the static values and slopes at Q2 = 0 of
the spin polarizabilities and moments of polarized structure functions In particular we give the
HB expansion of the πN -loop contributions and the ∆-exchange contributions The complete
expressions also for the π∆-loop contributions can be found in the Supplemented material Recall
that IA(0) = I1(0) = d2(0) = 0 and dd2(Q2)dQ2
∣∣∣Q2=0
= 0
1 πN-loop contribution
Here we give analytical expressions for the πN -loop contributions to the proton and neutron
spin polarizabilities expanded in powers of micro = mπMN viz the HB expansion Note that we
choose to expand here to a high order in micro the strict HB expansion would only retain the leading
term in an analogous NLO calculation
bull Static polarizabilities (Q2 = 0)
γ0p =e2g2
A
96π3f2πm
2π
1minus 21πmicro
8minus(
59
2+ 26 logmicro
)micro2 +
1875πmicro3
64
+3
(3
2+ 26 logmicro
)micro4 +
(C1)
γ0n =e2g2
A
48π3f2πm
2π
1
2minus 9πmicro
16minus 2micro2 logmicro+
75πmicro3
128minus 3micro4
4+
(C2)
δLTp =e2g2
A
192π3f2πm
2π
1minus 9πmicro
8+
(13
2minus 2 logmicro
)micro2 minus 465πmicro3
64
minus(
47
2+ 42 logmicro
)micro4 +
(C3)
δLTn =e2g2
A
96π3f2πm
2π
1
2+
3πmicro
16+ (1 + 2 log micro)micro2 minus 105πmicro3
128+
5micro4
4+
(C4)
38
δLTn =e2g2
A
96π3f2πm
2π
1
2+
3πmicro
16+ (1 + 2 log micro)micro2 minus 105πmicro3
128+
5micro4
4+
(C5)
γ0p =e2g2
A
16π3f2πm
4π
4
45minus 3πmicro
16+
14micro2
5minus 1813πmicro3
384minus 192
5(1 + log micro)micro4
+80703πmicro5
2048+
(C6)
γ0n =e2g2
A
16π3f2πm
4π
4
45minus 5πmicro
48+
4micro2
5minus 245πmicro3
384minus 32micro4 logmicro
15
+1323πmicro5
2048+
(C7)
bull Slopes of polarizabilities at Q2 = 0
dγ0p(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
2minus 45πmicro
4+ 223micro2 minus 28515πmicro3
64
minus9
(1953
4+ 449 logmicro
)micro4 +
570255πmicro5
128+
(C8)
dγ0n(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
2minus 81πmicro
8+ 94micro2 minus 2535πmicro3
32
minus3 (1 + 90 logmicro)micro4 +84315πmicro5
1024+
(C9)
dδLTp(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
2880π3f2πm
4π
minus5
2minus 27πmicro
32+ 20micro2 minus 5865πmicro3
256
+3
(617
4+ 36 logmicro
)micro4 minus 2056845πmicro5
4096+
(C10)
dδLTn(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
minus5
4minus 81πmicro
64minus 11micro2 +
10005πmicro3
512
+15
8(11 + 48 logmicro)micro4 minus 267015πmicro5
8192+
(C11)
dIAp(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
96π2f2πmicro
2
1minus 15πmicro
4minus 1
2(115 + 88 logmicro)micro2 +
1839πmicro3
32
+5 (5 + 34 logmicro)micro4 +
(C12)
dIAn(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
48π2f2πmicro
2
1
2minus 11πmicro
8minus 1
4(1 + 20 logmicro)micro2 +
99πmicro3
64
minus25micro4
12+
(C13)
dI1p(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
96π2f2πmicro
3π
8+ 2 (4 + 3 logmicro)microminus 537πmicro2
64
minus1
2(15 + 56 logmicro)micro3 +
(C14)
39
dI1n(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
48π2f2πmicro
minus π
16+
1
4(3 + 4 logmicro)microminus 57πmicro2
128+
2micro3
3+
(C15)
γ0p(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2g2
A
16π3f2πm
6π
1
105minus 23πmicro
256+
377micro2
210minus 15551πmicro3
6144+
3371micro4
105
minus1640457πmicro5
32768+
(C16)
γ0n(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2g2
A
16π3f2πm
6π
1
105minus 153πmicro
1792+
69micro2
70minus 4615πmicro3
6144+
172micro4
35
minus120897πmicro5
32768+
(C17)
2 ∆-exchange contribution
Here we give analytical expressions for the tree-level ∆-exchange contributions to the nucleon
spin polarizabilities and their slopes at Q2 = 0 Note that the ∆-exchange contributes equally to
proton and neutron polarizabilities Recall that for the magnetic γlowastN∆ coupling we introduced a
dipole form factor to mimic vector-meson dominance gM rarr gM(1 +Q2Λ2)2
bull Static polarizabilities (Q2 = 0)
γ0 = minus e2
4πM2+
(g2M
∆2+
g2E
M2+
minus 4gMgEM+∆
) (C18)
δLT =e2M∆
4πM3+
(g2E
MNM++gMgE∆MN
minus gEgCM2
∆
) (C19)
γ0 =e2M2
N
π∆2M4+
(minusg
2M
∆2+
g2E
M2+
+4gMgE∆M+
) (C20)
bull Slopes of polarizabilities at Q2 = 0
dγ0(Q2)
dQ2
∣∣∣∣∣Q2=0
= minus e2
πM2+∆
(g2M
∆
[1
4∆2minus 1
∆M++
1
2M2+
]minus 1
Λ2
g2M
∆+
g2E
2M2+
[1
2∆minus 3
M+
]minusgMgE
M+
[1
∆2minus 5
∆M++
1
M2+
]+
1
Λ2
2gMgEM+
+2gMgC∆M2
+
minus gEgCM3
+
) (C21)
40
d δLT (Q2)
dQ2
∣∣∣∣∣Q2=0
=e2M∆∆
4πMNM2+
(g2E
∆2M2+
[1
∆minus 4
M+
]minus g2
C
∆M2∆M
2+
+gMgE∆2M+
[1
∆2
minus 3
∆M++
1
M2+
]minus 2
Λ2
gMgE∆2M+
+gMgC∆M2
∆
[1
2∆2minus 2
∆M++
1
2M2+
]minus gEgC
2M2∆M
2+
[7
∆+
1
M+
]) (C22)
dIA(Q2)
dQ2
∣∣∣∣∣Q2=0
= minusM2N
M2+
(g2M
2∆2+
g2E
MNM+minus 2gMgE
∆M+minus gEgCM∆M+
) (C23)
dI1(Q2)
dQ2
∣∣∣∣∣Q2=0
= minusM∆M2N
2M3+
(g2E
MNM∆minus gMgE∆MN
minus gEgCM2
∆
) (C24)
γ0(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2M2
N
π∆3M6+
(g2M
∆
[3M2
∆ + 2M∆MN minus 9M2N
∆2+
4M2+
Λ2
]
minus4gMgE
[5M2
∆ minus 9M2N
M+∆2+
2M+
Λ2
]+
g2E
∆M+[7M∆ minus 9MN ]
minus8gMgC∆
+4gEgCM+
) (C25)
[1] D Drechsel B Pasquini and M Vanderhaeghen Dispersion relations in real and virtual
Compton scattering Phys Rept 378 99 (2003) hep-ph0212124
[2] S E Kuhn J-P Chen and E Leader Spin structure of the nucleon mdash status and recent
results Prog Part Nucl Phys 63 1 (2009) arXiv08123535 [hep-ph]
[3] F Hagelstein R Miskimen and V Pascalutsa Nucleon polarizabilities from Compton scat-
tering to hydrogen atom Prog Part Nucl Phys 88 29 (2016) arXiv151203765 [nucl-th]
[4] B Pasquini and M Vanderhaeghen Dispersion theory in electromagnetic interactions Ann
Rev Nucl Part Sci 68 75 (2018) arXiv180510482 [hep-ph]
[5] S Gerasimov A Sum rule for magnetic moments and the damping of the nucleon magnetic
moment in nuclei Sov J Nucl Phys 2 430 (1966)
[6] S Drell and A C Hearn Exact sum rule for nucleon magnetic moments Phys Rev Lett
16 908 (1966)
[7] J S Schwinger Source theory viewpoints in deep inelastic scattering Proc Natl Acad Sci
USA 72 1 (1975)
[8] J S Schwinger Source theory viewpoints in deep inelastic scattering Electromagnetic In-
teractions and Field Theory Proceedings 14 Internationale Universitatswochen Schladming
Austria February 24-March 7 1975 Acta Phys Austriaca Suppl 14 471 (1975)
41
[9] J Schwinger Source theory discussion of deep inelastic scattering with polarized particles
Proc Natl Acad Sci USA 72 1559 (1975)
[10] A M Baldin Polarizability of nucleons Nucl Phys 18 310 (1960)
[11] M Gell-Mann M L Goldberger and W E Thirring Use of causality conditions in quantum
theory Phys Rev 95 1612 (1954)
[12] A Pineda Leading chiral logarithms to the hyperfine splitting of the hydrogen and muonic
hydrogen Phys Rev C 67 025201 (2003)
[13] C Peset and A Pineda The two-photon exchange contribution to muonic hydrogen from
chiral perturbation theory Nucl Phys B 887 69 (2014) arXiv14064524 [hep-ph]
[14] C Peset and A Pineda Model-independent determination of the two-photon exchange contri-
bution to hyperfine splitting in muonic hydrogen JHEP 04 060 arXiv161205206 [nucl-th]
[15] F Hagelstein and V Pascalutsa Proton structure in the hyperfine splitting of muonic hydro-
gen PoS CD15 077 (2016) arXiv151104301 [nucl-th]
[16] F Hagelstein Exciting Nucleons in Compton Scattering and Hydrogen-Like Atoms PhD
thesis Mainz U Inst Kernphys (2017) arXiv171000874 [nucl-th]
[17] F Hagelstein ∆(1232)-Resonance in the hydrogen spectrum Proceedings 11th International
Workshop on the Physics of Excited Nucleons (NSTAR 2017) Columbia SC USA August
20-23 2017 Few Body Syst 59 93 (2018) arXiv180109790 [nucl-th]
[18] Y Prok et al (CLAS) Moments of the spin structure functions gp1 and gd1 for 005 lt Q2 lt 30
GeV2 Phys Lett B 672 12 (2009) arXiv08022232 [nucl-ex]
[19] H Dutz et al (GDH) First measurement of the Gerasimov-Drell-Hearn sum rule for 1H from
07 GeV to 18 GeV at ELSA Phys Rev Lett 91 192001 (2003)
[20] M Amarian et al (Jefferson Lab E94010) Measurement of the generalized forward spin po-
larizabilities of the neutron Phys Rev Lett 93 152301 (2004) arXivnucl-ex0406005
[21] M Amarian et al The Q2 evolution of the generalized Gerasimov-Drell-Hearn integral for
the neutron using a 3He target Phys Rev Lett 89 242301 (2002) arXivnucl-ex0205020
[22] M Amarian et al (Jefferson Lab E94-010)Q2 evolution of the neutron spin structure moments
using a 3He target Phys Rev Lett 92 022301 (2004) arXivhep-ex0310003
[23] A Deur et al Experimental determination of the evolution of the Bjorken integral at low Q2
Phys Rev Lett 93 212001 (2004) arXivhep-ex0407007
[24] K Slifer Low Q2 measurement of gp2 and the δLT spin polarizability Spin structure at long dis-
tance Proceedings Workshop Newport News USA March 12ndash13 2009 nucl-ex09064775
(2009) AIP Conf Proc 1155 10106313203293 (2009) arXiv09064775 [nucl-ex]
[25] P Solvignon et al (E01-012) Moments of the neutron g2 structure function at intermediate
42
Q2 Phys Rev C 92 015208 (2015) arXiv13044497 [nucl-ex]
[26] A Deur Experimental studies at low Q2 of the spin structure of the nucleon at Jefferson Lab
in 9th International Workshop on Chiral Dynamics (CD18) Durham NC USA September
17-21 2018 (2019) arXiv190305661 [nucl-ex]
[27] V Sulkosky et al (Jefferson Lab E97-110) Measurement of the 3He spin-structure functions
and of neutron (3He) spin-dependent sum rules at 0035 le Q2 le 024 GeV2 Phys Lett B
805 135428 (2020) arXiv190805709 [nucl-ex]
[28] K Adhikari et al (CLAS) Measurement of the Q2 Dependence of the Deuteron Spin Structure
Function g1 and its Moments at Low Q2 with CLAS Phys Rev Lett 120 062501 (2018)
arXiv171101974 [nucl-ex]
[29] V Lensky F Hagelstein A Hiller Blin and V Pascalutsa Comment on rdquoMeasurement of
the Q2 Dependence of the Deuteron Spin Structure Function g1 and its Moments at Low Q2
with CLASrdquo (2018) arXiv180603219 [nucl-th]
[30] J M Alarcn F Hagelstein V Lensky and V Pascalutsa Forward doubly-virtual Compton
scattering off the nucleon in chiral perturbation theory at NLO the subtraction function and
moments of unpolarized structure functions (2020) arXiv200509518 [hep-ph]
[31] V Bernard E Epelbaum H Krebs and U-G Meiszligner New insights into the spin structure
of the nucleon Phys Rev D 87 054032 (2013) arXiv12092523 [hep-ph]
[32] V Lensky J M Alarcon and V Pascalutsa Moments of nucleon structure functions at
next-to-leading order in baryon chiral perturbation theory Phys Rev C 90 055202 (2014)
arXiv14072574 [hep-ph]
[33] V Pascalutsa and M Vanderhaeghen Chiral effective-field theory in the ∆(1232) region
I Pion electroproduction on the nucleon Phys Rev D 73 034003 (2006) arXivhep-
ph0512244
[34] V Pascalutsa and M Vanderhaeghen Electromagnetic nucleon-to-Delta transition in chiral
effective field theory Phys Rev Lett 95 232001 (2005) arXivhep-ph0508060
[35] T R Hemmert B R Holstein and J Kambor Systematic 1M expansion for spin 32
particles in baryon chiral perturbation theory Phys Lett B 395 89 (1997) arXivhep-
ph9606456
[36] V Pascalutsa and D R Phillips Effective theory of the ∆(1232) in Compton scattering off
the nucleon Phys Rev C 67 055202 (2003) arXivnucl-th0212024
[37] V Pascalutsa M Vanderhaeghen and S N Yang Electromagnetic excitation of the ∆(1232)-
resonance Phys Rept 437 125 (2007) arXivhep-ph0609004
[38] K A Olive et al (Particle Data Group) Review of Particle Physics Chin Phys C 38 090001
43
(2014)
[39] F E Low Scattering of light of very low frequency by systems of spin 12 Phys Rev 96
1428 (1954)
[40] M Gell-Mann and M L Goldberger Scattering of low-energy photons by particles of spin
12 Phys Rev 96 1433 (1954)
[41] A M Harun ar-Rashid A simple derivation of Schwingerrsquos sum rule for spin dependent
structure functions Nuovo Cim A 33 447 (1976)
[42] F Hagelstein and V Pascalutsa Dissecting the hadronic contributions to (g minus 2)micro by
Schwingerrsquos sum rule Phys Rev Lett 120 072002 (2018) arXiv171004571 [hep-ph]
[43] V Lensky V Pascalutsa M Vanderhaeghen and C Kao Spin-dependent sum rules con-
necting real and virtual Compton scattering verified Phys Rev D 95 074001 (2017)
arXiv170101947 [hep-ph]
[44] H Burkhardt and W N Cottingham Sum rules for forward virtual Compton scattering
Annals Phys 56 453 (1970)
[45] V Lensky and V Pascalutsa Predictive powers of chiral perturbation theory in Compton
scattering off protons Eur Phys J C 65 195 (2010) arXiv09070451 [hep-ph]
[46] H Grieszlighammer J McGovern D Phillips and G Feldman Using effective field theory to
analyse low-energy Compton scattering data from protons and light nuclei Prog Part Nucl
Phys 67 841 (2012) arXiv12036834 [nucl-th]
[47] H W Grieszlighammer J A McGovern and D R Phillips Nucleon polarisabilities at and
beyond physical pion masses Eur Phys J A 52 139 (2016) arXiv151101952 [nucl-th]
[48] E Epelbaum H Krebs and U-G Meiszligner Improved chiral nucleon-nucleon potential up to
next-to-next-to-next-to-leading order Eur Phys J A 51 53 (2015) arXiv14120142 [nucl-
th]
[49] E Epelbaum H Krebs and U-G Meiszligner Precision nucleon-nucleon potential at fifth order
in the chiral expansion Phys Rev Lett 115 122301 (2015) arXiv14124623 [nucl-th]
[50] C W Kao T Spitzenberg and M Vanderhaeghen Burkhardt-Cottingham sum rule and
forward spin polarizabilities in heavy baryon chiral perturbation theory Phys Rev D 67
016001 (2003) arXivhep-ph0209241
[51] C-W Kao D Drechsel S Kamalov and M Vanderhaeghen Higher moments of nucleon
spin structure functions in heavy baryon chiral perturbation theory and in a resonance model
Phys Rev D 69 056004 (2004) arXivhep-ph0312102
[52] T Becher and H Leutwyler Baryon chiral perturbation theory in manifestly Lorentz invariant
form Eur Phys J C 9 643 (1999) arXivhep-ph9901384
44
[53] J Gasser M E Sainio and A Svarc Nucleons with chiral loops Nucl Phys B 307 779
(1988)
[54] L S Geng J Martin Camalich L Alvarez-Ruso and M J Vicente Vacas Leading SU(3)-
breaking corrections to the baryon magnetic moments in chiral perturbation theory Phys
Rev Lett 101 222002 (2008) arXiv08051419 [hep-ph]
[55] D Drechsel S S Kamalov and L Tiator The GDH sum rule and related integrals Phys
Rev D 63 114010 (2001) arXivhep-ph0008306
[56] D Drechsel O Hanstein S S Kamalov and L Tiator A unitary isobar model for pion
photo- and electroproduction on the proton up to 1 GeV Nucl Phys A 645 145 (1999)
arXivnucl-th9807001
[57] L Tiator private communication (2020)
[58] V Bernard T R Hemmert and U-G Meiszligner Spin structure of the nucleon at low energies
Phys Rev D 67 076008 (2003) arXivhep-ph0212033
[59] O Gryniuk F Hagelstein and V Pascalutsa Evaluation of the forward Compton scattering
off protons II Spin-dependent amplitude and observables Phys Rev D 94 034043 (2016)
arXiv160400789 [nucl-th]
[60] R Zielinski The g2p Experiment A Measurement of the Protonrsquos Spin Structure Functions
PhD thesis New Hampshire U (2010) arXiv170808297 [nucl-ex]
[61] N Guler et al (CLAS) Precise determination of the deuteron spin structure at low to mod-
erate Q2 with CLAS and extraction of the neutron contribution Phys Rev C 92 055201
(2015) arXiv150507877 [nucl-ex]
[62] C E Carlson V Nazaryan and K Griffioen Proton structure corrections to electronic
and muonic hydrogen hyperfine splitting Phys Rev A 78 022517 (2008) arXiv08052603
[physicsatom-ph]
[63] V Bernard N Kaiser and U-G Meiszligner Chiral dynamics in nucleons and nuclei Int J
Mod Phys E 4 193 (1995) arXivhep-ph9501384
[64] T R Hemmert B R Holstein and J Kambor ∆(1232) and the polarizabilities of the nucleon
Phys Rev D 55 5598 (1997) arXivhep-ph9612374
[65] V Pascalutsa and R Timmermans Field theory of nucleon to higher-spin baryon transitions
Phys Rev C 60 042201 (1999) arXivnucl-th9905065
[66] V Pascalutsa Quantization of an interacting spin-32 field and the Delta isobar Phys Rev
D 58 096002 (1998) arXivhep-ph9802288
[67] H Krebs Double Virtual Compton Scattering and SpinStructure of the Nucleon PoS
CD2018 031 (2019)
45
[68] N Kochelev and Y Oh Axial anomaly and the δLT puzzle Phys Rev D 016012 (2012)
arXiv11034892 [hep-ph]
[69] D Drechsel S Kamalov and L Tiator Unitary isobar model ndash MAID2007 Eur Phys J A
69 (2007) available at httpsmaidkphuni-mainzde arXiv07100306 [nucl-th]
[70] P J Mohr B N Taylor and D B Newell CODATA recommended values of the fundamental
physical constants 2010 Rev Mod Phys 84 1527 (2012)
[71] J Ahrens et al (GDH A2) First measurement of the Gerasimov-Drell-Hearn integral for 1H
from 200 to 800 MeV Phys Rev Lett 87 022003 (2001) arXivhep-ex0105089 [hep-ex]
[72] K Helbing (GDH) Experimental verification of the GDH sum rule at ELSA and MAMI Nucl
Phys Proc Suppl 105 113 (2002)
[73] R Bradford A Bodek H S Budd and J Arrington A New parameterization of the nucleon
elastic form-factors NuInt05 proceedings of the 4th International Workshop on Neutrino-
Nucleus Interactions in the Few-GeV Region Okayama Japan 26-29 September 2005 Nucl
Phys Proc Suppl 159 127 (2006) arXivhep-ex0602017
[74] J D Bjorken Applications of the chiral U(6)timesU(6) algebra of current densities Phys Rev
148 1467 (1966)
[75] J D Bjorken Inelastic scattering of polarized leptons from polarized nucleons Phys Rev D
1 1376 (1970)
[76] V Bernard T R Hemmert and U-G Meiszligner Novel analysis of chiral loop effects in
the generalized Gerasimov-Drell-Hearn sum rule Phys Lett B 545 105 (2002) arXivhep-
ph0203167
[77] A Deur et al Experimental study of isovector spin sum rules Phys Rev D 78 032001
(2008) arXiv08023198 [nucl-ex]
[78] B Pasquini P Pedroni and D Drechsel Higher order forward spin polarizability Phys Lett
B 687 160 (2010) arXiv10014230 [hep-ph]
[79] R Jaffe g2ndashThe nucleonrsquos other spin-dependent structure function Comments Nucl Part
Phys 19 239 (1990)
[80] E V Shuryak and A Vainshtein Theory of power corrections to deep inelastic scattering in
quantum chromodynamics (II) Qminus4 effects polarized target Nucl Phys B 201 141 (1982)
[81] S Wandzura and F Wilczek Sum rules for spin dependent electroproduction Test of rela-
tivistic constituent quarks Phys Lett B 72 195 (1977)
[82] B W Filippone and X-D Ji The spin structure of the nucleon Adv Nucl Phys 26 1
(2001) arXivhep-ph0101224
[83] M Burkardt The g2 structure function Proceedings Workshop on Spin structure at long
46
distance Newport News USA March 12-13 2009 AIP Conf Proc 1155 26 (2009)
arXiv09054079 [hep-ph]
[84] A Deur S J Brodsky and G F de Teramond The spin structure of the nucleon Rept
Prog Phys 82 076201 (2019) arXiv180705250 [hep-ph]
[85] B R Holstein D Drechsel B Pasquini and M Vanderhaeghen Higher order polarizabilities
of the proton Phys Rev C 61 034316 (2000) arXivhep-ph9910427
[86] B R Holstein V Pascalutsa and M Vanderhaeghen Sum rules for magnetic moments and
polarizabilities in QED and chiral effective-field theory Phys Rev D 72 094014 (2005)
arXivhep-ph0507016
[87] B Pasquini D Drechsel and L Tiator Invariant amplitudes for pion electroproduction Eur
Phys J A 34 387 (2007) arXiv07122327 [hep-ph]
[88] A J Buchmann Non-spherical proton shape and hydrogen hyperfine splitting Proceedings
International Workshop on Precision Physics of Simple Atomic Systems (PSAS 2008) Wind-
sor Ontario Canada July 21-26 2008 Can J Phys 87 773 (2009) arXiv09104747
[physicsatom-ph]
47
cross-check of these calculations
The obtained moments of the proton and neutron spin structure functions related to gen-
eralized polarizabilities and GDH-type integrals agree well with the available experimental
data The description of their Q2 evolution is improved compared to the previous χPT pre-
dictions In particular the NLO BχPT predictions obtained here give a better description
of the empirical data (eg from the Jefferson Laboratory ldquoSpin Physics Programrdquo) than
the HB [50 51] and IR [58] calculations
The demonstrated predictive power of the χPT framework amplitudes makes it well
suited for extending the χPT evaluation of the TPE effect in the hyperfine structure of
(muonic-)hydrogen [15ndash17] to next-to-leading order
ACKNOWLEDGEMENTS
We thank Lothar Tiator and Marc Vanderhaeghen for helpful discussions This work
is supported by the Deutsche Forschungsgemeinschaft (DFG) through the Collaborative
Research Center [The Low-Energy Frontier of the Standard Model (SFB 1044)] JMA ac-
knowledges support from the Community of Madrid through the ldquoPrograma de atraccion de
talento investigador 2017 (Modalidad 1)rdquo and the Spanish MECD grants FPA2016-77313-P
FH gratefully acknowledges financial support from the Swiss National Science Foundation
Appendix A Tensor decompositions of the VVCS amplitudes
In this appendix we review the decomposition of the forward VVCS process into tensor struc-
tures and scalar amplitudes In particular we consider the connection between the covariant and
the semi-relativistic decomposition in the lab frame that is defined in terms of the conventional
transverse longitudinal transverse-transverse and transverse-longitudinal amplitudes
As explained in Sec II A the process of forward VVCS off the nucleon can be described in
terms of four explicitly covariant amplitudes S1 2 and T1 2 [3]
T (νQ2) =
(minusgmicroν +
qmicroqν
q2
)T1(νQ2) +
1
M2N
(pmicro minus p middot q
q2qmicro)(
pν minus p middot qq2
qν)T2(νQ2) (A1)
minus 1
MNγmicroναqα S1(νQ2)minus 1
M2N
(γmicroνq2 + qmicroγναqα minus qνγmicroαqα
)S2(νQ2)
εprimelowastmicro εν
where εmicro (εprimelowastmicro ) are the incoming (outgoing) photon polarization vectors ν is the photon lab-frame
energy and Q2 is the photon virtuality Alternatively the decomposition in the laboratory frame
28
(which in the forward case coincides with the Breit frame) is parametrized in terms of the nucleon
Pauli matrices ~σ and the four scalar functions fL fT gTT and gLT
T (νQ2) = ε0 εprimelowast0 fL(νQ2) + (~ε primelowast middot ~ε ) fT (νQ2) + i~σ middot (~ε primelowast times ~ε ) gTT (νQ2) (A2)
minusi~σ middot [(ε0~εprimelowast minus ~ε ε primelowast0 )times q] gLT (νQ2)
Here ~q and q = ~q|~q | are the photon three-momentum in the lab system and its unit vector The
modified polarization vector components are given by
ε0 =
[ε0 minus
ν
|~q | (~ε middot q )
] |~q |Q ~ε = ~εminus q (~ε middot q ) (A3)
where ε = (ε0~ε ) is the usual incoming photon polarization vector and εprimelowast the outgoing polarization
vector The LEX of the lab frame amplitudes [Eq (10)] can serve in particular as the definition
of the generalized polarizabilities The lab frame amplitudes are also conveniently used for the
definition of the response functions see the example of the scalar amplitude gLT (νQ2) and the
corresponding response function σLT (νQ2) below in App B
Appendix B Photoabsorption cross sections
In the forward kinematics the spin-dependent VVCS amplitudes and the spin polarizabilities
can be described in terms of the polarized structure functions g1(xQ2) and g2(xQ2) or equiv-
alently the helicity-difference cross section σTT (νQ2) and the longitudinal-transverse response
function σLT (νQ2) with the help of dispersion relations (5) and the optical theorem (3) In this
way the photoabsorption cross sections measured in electroproduction processes form the basis
for most empirical evaluations shown throughout Sec III In the following we present the BχPT
predictions for the tree-level cross sections of πN - π∆- and ∆-production through photoabsorption
on the nucleon cf Figs 8 9 and 10 in Ref [30] In Secs B 1 and B 2 we will discuss the leading
πN -production channel and the ∆-production channel respectively We used these cross sections
to verify the polarizability predictions obtained otherwise from the calculated non-Born VVCS
amplitudes Due to the bad high-energy behavior of the π∆-production cross sections in BχPT
cf Fig 10 the dispersion relations in Eq (5) require further subtractions for a reconstruction of
the π∆-loop contribution to the spin-dependent VVCS amplitudes Therefore not all polarizabil-
ities could be verified but only those appearing as higher-order terms in the LEX of the VVCS
amplitudes such as γ0 [16]
29
proton neutron
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Neutron
02 04 06 08 100
50
100
150
200
250
[GeV]
T [b]
02 04 06 08 10
-50
0
50
100
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Neutron
02 04 06 08 100
50
100
150
200
250
[GeV]
T [b]
02 04 06 08 10
-50
0
50
100
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02040608100
50
100
150
200
250
300
[GeV]
T[b]
0204060810
-50
0
50
[GeV]
TT[b]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
0204060810
00
02
04
06
[GeV]
LTQ
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
0204060810
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
FIG 10 Photoabsorption cross sections for πN (red) and π∆ production (orange) with Q2 = 0
(solid) and Q2 = 01 GeV2 (dashed for πN and dotted for π∆ channel)
1 πN-production channel
In order to extract the response function σLT (νQ2) we have developed a method similar to
the one used to calculate σTT (νQ2) see for example Ref [86] For σLT (νQ2) however the
calculation is more complicated because one has to take into account that the associated Compton
process involves a spin-flip of the nucleon as illustrated in Fig 11 When calculating the cross
section the product of the incoming nucleon spinors has to reflect this flip
The forward VVCS amplitude related to σLT (νQ2) mdash and δLT (Q2) mdash is gLT (νQ2) It can
be extracted from Eq (A2) if one takes the modified polarization vector components in Eq (A3)
with ε = εL and εprimelowast = εlowastplusmn as input where εL = 1Q(|~q | 0 0 q0) and εplusmn = ∓ 1radic
2(0 1plusmni 0) are the
standard longitudinal and transverse polarization vectors respectively For εL and εlowastplusmn only the
choice of helicities hprime = plusmn12 and h = ∓12 gives a non-zero contribution and one obtains
χdaggerhprime T (νQ2)χh = χdaggerhprimeminusi~σ middot [(ε0~εprimelowast minus ~ε ε primelowast0 )times q] gLT (νQ2)χh =
radic2 gLT (νQ2) (B1)
where χh and χdaggerhprime are two-component Pauli spinors with opposite helicities or here spins
Let us now consider the related photoabsorption process and in particular the tree-level γlowastN rarr
30
s = 12
sprime = minus12
λ = 0 λprime = 1
rprime rprime
s=12sprime=minus1
2
λ=0λprime=1
rprimerprime
s=12
sprime=minus12
λ=0λprime=1
rprimerprime hacute=frac12h=-frac12
λacute=1λ=0
FIG 11 Relation between the forward Compton process and the photoabsorption process given
by the optical theorem In particular we show the longitudinal-transverse contribution The
double-line arrows represent the spin of the external particles while the dot represents the scalar
(longitudinal) polarization of the incoming photon Inside the blob the intermediate states are
represented eg nucleons with spins rprime (which are averaged in the calculation of the cross section)
and pions
πN channel see diagrams in Fig 8 of Ref [30] We define the πN -production amplitude as
T = uhB (PB)sumi
Ai(s t)Γi uhA(PA) (B2)
with the Dirac structures
Γ1 = γ5 (B3a)
Γ2 =1
2
[qA ε
]γ5 (B3b)
where uhA(PA) and udaggerhB (PB) are the Dirac spinors and PA and PB are the four-momenta of
the incoming and outgoing nucleons respectively When calculating the photoabsorption cross
section related to the VVCS amplitude in Eq (B1) the nucleon spin flip should be implemented
by uhprime(PA) in T dagger and uh(PA) in T together with the appropriate transverse and longitudinal
photon polarization vectors εlowastplusmn and εL
However if one wants to use the properties of the Dirac matrices it is more useful to construct
an operator to produce this spin flip in the external nucleons of Fig 11 This is accomplished by
introducing the projector ΓLT equiv 12radic
2(γ1 + iγ2)γ5 which also takes into account the extra factor
radic2 in Eq (B1) We checked that with this projector one correctly extracts δLT by comparing the
HB limit of our result to the HB result of Ref [50] where the authors calculate this polarizability
from the Compton amplitude directly With all those ingredients the longitudinal-transverse cross
section is calculated in the following way
σLT (νQ2) =1
64π2 s
|~pf |cm
|~pi|cm
int 1
minus1dcos θ
sumij
AiAdaggerjXij (B4)
31
with
Xij = Tr[(PB +MN )Γi(PA +MN )ΓLTγ0Γdaggerjγ
0] (B5)
where θ is the scattering angle in the center-of-mass (cm) frame and |~pi|cm (|~pf |cm) is the three-
momentum of an incoming (outgoing) particle in the cm frame An explicit calculation of the
matrix Xij leads to
X = MNQ
0 2(PB minus PA) middot εLminusradic
2|~qf |cm sin θ (sminus u)
(B6)
where |~qi|cm (|~qf |cm) is the relative three-momentum of the incoming (outgoing) particles in the
cm frame Here s t and u are the usual Mandelstam variables For the different γlowastN rarr πN
channels we obtain the following amplitudes Ai where we introduce qA as the four-momentum of
the incoming photon and qB as the four-momentum of the outgoing pion
bull γlowastprarr π0p
A1 =e gAMN
fπ
[2PA middot ε+ qA middot ε
sminusM2N
+2PB middot εminus qA middot ε
uminusM2N
] (B7a)
A2 =e gAMN
fπ
[1
sminusM2N
+1
uminusM2N
] (B7b)
bull γlowastprarr π+n
A1 =
radic2 e gAMN
fπ
[2PA middot ε+ qA middot ε
sminusM2N
+2(PA minus PB) middot ε+ qA middot ε
tminusm2π
] (B8a)
A2 =
radic2 e gAMN
fπ(sminusM2N )
(B8b)
bull γlowastnrarr π0n
A1 = 0 (B9a)
A2 = 0 (B9b)
bull γlowastnrarr πminusp
A1 =
radic2 e gAMN
fπ
[2PB middot εminus qA middot ε
uminusM2N
minus 2(PA minus PB) middot ε+ qA middot εtminusm2
π
] (B10a)
A2 =
radic2 e gAMN
fπ(uminusM2N )
(B10b)
The analytical expressions shown above were checked with the amplitudes given in Ref [87] An-
alytical expressions for the tree-level γlowastN rarr πN channel of the σLT (νQ2) and σTT (νQ2) cross
32
sections are given below (proton channels π+n and π0p neutron channel πminusp) We checked that
they reproduce the known results in the real-photon limit [45 86] To shorten the final expressions
for the cross sections which are considerably longer for finite Q2 than in the real-photon limit we
define the following dimensionless kinematic variables
αγ = (ENi )cmradics =
s+M2N +Q2
2s (B11)
απ = (ENf )cmradics =
s+M2N minusm2
π
2s (B12)
βγ = Eγcmradics =
sminusM2N minusQ2
2s (B13)
βπ = Eπcmradics =
sminusM2N +m2
π
2s (B14)
λγ = |~qi|cmradics =
radic(sminusM2
N minusQ2)2 + 4sQ2
2s (B15)
λπ = |~qf |cmradics =
radic(sminusM2
N +m2π)2 minus 4sm2
π
2s (B16)
Here (ENi )cm and (ENf )cm are the energies of the incoming and outgoing nucleon Eγcm is the energy
of the incoming photon Eπcm is the energy of the outgoing pion all in the cm frame
σ(π+n)TT = minus e2g2
AM2N
64πf2πs
3(sminusM2N )2λ4
γ
4sλπλγ
[(M2
N minus s)(M2N minusQ2 minus s)(Q2 + 2sβγβπ)
+ 2s(m2πQ
2 minus (M2N minus s)
(M2N + s(minus1 + 2βγβπ)
))λ2γ
]+ (M2
N minus s)(M2N minusQ2 minus s)
times (Q2 + 2sβγβπ minus 2sλπλγ)(Q2 + 2s(βγβπ + λπλγ)
)times log
(Q2 + 2sβγβπ minus 2sλγλπQ2 + 2sβγβπ + 2sλγλπ
) (B17)
33
σ(π0p)TT =
e2g2AM
2Nλπ
64πf2πs(sminusM2
N )2λγ
1
s2λπλ3γ
[(M2
N minus s)(M2N minusQ2 minus s)
(3M2
N + s(2βγβπ minus 3))(M2N
+ s(2βγβπ minus 1))
+ 2s
((2m2
πQ2 + (M2
N minus s)2)(M2
N minus s)minus 2(minusm2
πQ2 + (sminusM2
N )2)
times sβγβπ + 4s2(sminusM2N )β2
γβ2π
)λ2γ
]arctan
(2sλπλγ
M2N + s(2βγβπ minus 1)
)+ 2
[minus 2m2
πQ2
+ (M2N minus s)
(2(sminusM2
N ) + 4sβγβπ +(sminusM2
N +Q2)(3M2N minus 3s+ 2sβγβπ)
sλ2γ
+m2πQ
2(sminusM2N )
(M2N minus s+ 2sβγβπ)(M2
N + s(minus1 + 2βγβπ minus 2λπλγ))
+m2πQ
2(sminusM2N )
(M2N minus s+ 2sβγβπ)(M2
N + s(minus1 + 2βγβπ + 2λπλγ))
)
+1
λπλγ
(βγβπ
(minusm2
πQ2 + (M2
N minus s)(M2N minus s+ 2sβγβπ)
)minus (M2
N minus s)(M2N minusQ2 minus s)λ2
π
)log
(M2N + s(minus1 + 2βγβπ + 2λπλγ)
M2N + s(minus1 + 2βγβπ minus 2λπλγ)
)] (B18)
σ(πminusp)TT =
e2g2AM
2N
64πf2πs
3λ4γ(M2
N + s(2βπβγ minus 1))2 minus 4λ2πs
2λ2γ)
4sλπλγ
((minusM2
N +Q2 + s)(M2N
+ s(2βπβγ minus 1))
2 minus 2sλ2γ
(2λ2
πs(minusM2N +Q2 + s) +m2
πQ2))
+((M2N + s(2βπβγ minus 1)
)2
minus 4λ2πs
2λ2γ
)((M2N + s(2βπβγ minus 1)
)(M2N minus 2Q2 minus s(2βπβγ + 1)
)+ 4λ2
πs2λ2
γ
)times log
(minusM2
N minus 2βπsβγ minus 2λπsλγ + s)minus(M2N + s(2βπβγ minus 2λπλγ minus 1)
)(M2N + s(minus1
+ 2βπβγ + 2λπλγ))[((
M2N + s(2βπβγ minus 1)
)(M2N minus 2Q2 minus s(2βπβγ + 1)
)+ 4λ2
πs2λ2
γ
)times log
(minusM2
N minus 2βπsβγ + 2λπsλγ + s)
+((Q2 + 2βπsβγ)2 minus 4λ2
πs2λ2
γ
)times log
(Q2 + 2βπsβγ minus 2λπsλγQ2 + 2sβπβγ + 2λπsλγ
)]+ 4sλ2
γ
((M2
N + s(2βπβγ minus 1))2 minus 4λ2πs
2λ2γ
)times (M2
N minus s) arctanh
(2λπsλγ
M2N + 2βπsβγ minus s
) (B19)
34
σ(π+n)LT =
e2g2AM
3Nλπ
32πf2πQs
3(sminusM2N )2λ4
γ
2sλγ
[(M2
N minus s)(Q2 + 2sβ2γ)(Q2 + 2sβγβπ)
minus 4s
((M2
N minus s)(Q2 minus 2s(απ minus 1)βγ
)+Q2sβγβπ
)λ2γ + 8s3(minus1 + απ)λ4
γ
]minus M2
N minus sλπ
[(Q2 + 2sβ2
γ)(Q2 + 2sβγβπ)2 + 4s2(2(απ minus 1)βγ(Q2 + 2sβγβπ)
minusQ2λ2π
)λ2γ + 8s3(απ minus 1)2λ4
γ
]arctan
(2sλπλγ
Q2 + 2sβγβπ
) (B20)
σ(π0p)LT =
e2g2AM
3Nλπ
16πf2πQs(sminusM2
N )2λγ
1
minus2s(M2N + s(minus1 + 2βγβπ))2λ2
γ + 8s3λ2πλ
4γ
[minus 3M8
N (Q2
+ 2sβ2γ) + 2M4
Ns2
(minus (Q2 + 2sβ2
γ)(minus3 + 2βγβπ) +(Q2(minus3 + 2βγβπ + 6λ2
π) + 2sβγ
times (minus3 + 12απ + 2βγβπ minus 12απβγβπ + 4βγλ2π))λ2γ minus 4s(α2
π minus 1)λ4π
)+ 2M2
Ns3
[minus (Q2
+ 2sβ2γ)(minus1 + 2βγβπ)
(6 + βγβπ(minus9 + 2βγβπ)
)+
(Q2(3minus 12λ2
π + 4βγβπ(minus1 + 2βγβπ
+ λ2π))minus 2sβγ
(minus 3 + 4απ(3 + 2απ(3 + 2βγβπ(minus3 + βγβπ)) + 4βγ(βπ + (2minus βγβπ)
times λ2π)))λ2γ + 8s
((απ minus 1)(1 + απ minus 2βγβπ) + 2απβγλ
2π
)λ4γ
)+ s4
((Q2 + 2sβ2
γ)
times (1minus 2βγβπ)2(2βγβπ minus 3) + 2(Q2(minus1 + 6λ2
π + 2βγβπ((1minus 2βγβπ)2 minus 2λ2π)) + 2sβγ
times (minus1 + 2βγβπ + 4(minus1 + βγβπ)(minusαπ + 2απβγβπ minus βγλ2π)))λ2γ minus 8
(s(απ minus 1)(απ
+ (1minus 2βγβπ)2) + 2βγ(2sαπ +Q2βγ)λ2π
)λ4γ + 32s(απ minus 1)λ2
πλ6γ
)+ 2M6
Ns
(Q2(6
minus 7βγβπ + λ2γ) + 2sβγ(βγ(6minus 7βγβπ) + (1minus 4απ)λ2
γ)
)]+sminusM2
N
4s2λπλ3γ
[(Q2 + 2sβ2
γ)
times(3M2
N + s(2βγβπ minus 3))(M2N + s(2βγβπ minus 1)
)+ 2s
(minusM2
N (Q2 + 2sβγ minus 8sαπβγ)
+ s(2sβγ(1minus 4απ + 4απβγβπ) +Q2(1minus 2λ2π)))λ2γ + 8s3(απ minus 1)λ4
γ
]
times arctan
(2sλπλγ
M2N + s(2βγβπ minus 1)
) (B21)
35
σ(πminusp)LT =
e2g2AM
3N
64πf2πQs
3λ4γ(M2
N minusQ2 minus s)(M2N + s(2βπβγ + 2λπλγ minus 1))
times
1
M2N + s(2βπβγ minus 2λπλγ minus 1)
[4λπsλγ(M2
N minusQ2 minus s)(
2sλ2γ
(s((1minus 2λ2
π)Q2
+ 2(2απ minus 1)sβγ(2βπβγ minus 1))minusM2N (Q2 + 2(1minus 2απ)sβγ)
)+ (Q2 + 2sβ2
γ)(M2N + s(2βπβγ minus 1)
)2 + 8(απ minus 1)απs
3λ4γ
)]minus(M2N + s(2βπβγ + 2λπλγ minus 1)
)times[
log
(minusM2N minus 2βπsβγ minus 2λπsλγ + s
minusM2N minus 2βπsβγ + 2λπsλγ + s
)(minus (Q2 + 2sβ2
γ)(M2N + s(2βπβγ minus 1)
)(minusM2
N + 2Q2 + 2βπsβγ + s)minus 2sλ2γ
(minusM2
N (Q2 + 2sβγ) +Q4 +Q2s(4απβγ
minus 2βγ minus 2λ2π + 1) + 2s2βγ(4(απ minus 1)βπβγ + 1)
)minus 8(απ minus 1)2s3λ4
γ
)+ 2
(4s2λ2
γ
(2(απ minus 1)βγ(Q2 + 2βπsβγ)minus λ2
πQ2)
+ (Q2 + 2βπsβγ)2(Q2 + 2sβ2γ)
+ 8(απ minus 1)2s3λ4γ
)arctanh
(2λπsλγ
Q2 + 2βπsβγ
)] (B22)
2 ∆-production channel
The tree-level ∆-exchange diagram in Fig 2 of Ref [30] contributes to the non-Born part of
the VVCS amplitudes The contribution of the ∆ exchange to the VVCS amplitudes can be split
into [17]
S∆-exch1 (νQ2) = S∆-pole
1 (νQ2) + S∆-exch1 (νQ2) (B23a)
νS∆-exch2 (νQ2) = νS∆-pole
2 (νQ2) + νS2∆-exch
(νQ2) (B23b)
and similarly for the unpolarized VVCS amplitudes discussed in Ref [30] Here we introduced the
∆-pole contributions S∆-polei and the ∆-non-pole contributions S∆-exch
i The former amplitudes
feature a pole at the ∆(1232)-production threshold and thus are proportional to
1
[sminusM2∆][uminusM2
∆]=
1
4M2N
1
ν2∆ minus ν2
(B24)
36
They can be reconstructed from the dispersion relations in Eq (5) with the tree-level ∆-production
cross sections as input cf Fig 10 in Ref [30]
σTT (νQ2) =π2α
M2NM
2+|~q |
minus g2
MMN (M+ + ν)|~q |2 +g2E(∆minus ν)(Q2 minusMNν)2
MN(B25a)
+g2CQ
4s(∆minus ν)
MNM2∆
minus 4gMgE(Q2 minusMNν)|~q |2 minus 4gMgCQ2|~q |2
+2gEgCQ
2[minusMNM∆ |~q |2 + s(Q2 +∆ν)
]MNM∆
δ(ν minus ν∆)
σLT (νQ2) =Qπ2α
M2NM
2+|~q |
g2E(MNν minusQ2) [M∆(MN + ν)minus s]
MN(B25b)
+g2CQ
2[MNM∆|~q |2 minus s(Q2 +∆ν)
]MNM2
∆
+ gMgEM∆|~q |2 minusgMgC(Q2 minusMNν)|~q |2
M∆
+gEgC(ν minus∆)(M2
N |~q |2 minus 2Q2s)
MNM∆
δ(ν minus ν∆)
with ∆ = M∆ minusMN M+ = M∆ + MN and the Mandelstam variable s = M2N + 2MNν minus Q2
Analytical expressions for the spin structure functions g1(xQ2) and g2(xQ2) can be constructed
from Eq (3) with the flux factor K(νQ2) = |~q | =radicν2 +Q2
In the ∆-non-pole contributions to S1(νQ2) and νS2(νQ2) the pole in ν at the ∆(1232)-
production threshold has canceled out
S∆-exch1 (νQ2) =
πα
MNM2+
[g2MQ
2+ + g2
E
(∆2 minus 3Q2
)+
4g2CQ
4
M2∆
minus 8gMgEM∆ωminus (B26a)
minus2gMgCQ2(MN minus 4M∆)
M∆+
2gEgCQ2(3MN minus 2M∆)
M∆
]
νS2∆-exch
(νQ2) =2πα
MNM2+
[g2EM∆∆ωminus +
g2M MNQ
2+
2+g2C Q
2(Q2 minus∆2)
2M∆(B26b)
+gEgM M∆(M∆ω+ minus 4MNωminus)minus gEgC ∆(2Q2 +MNω+)
+gMgC Q2(4MN minus ω+)
]+S∆-exch
2 (νQ2)
ν
[M2
∆ ω2+
M2N
+ ν2
]
with Q+ =radic
(M∆ +MN )2 +Q2 and ωplusmn = (M2∆minusM2
N plusmnQ2)2M∆ and the non-pole contribution
to S2(νQ2)
S∆-exch2 (νQ2) = minus2παMNν
M∆M2+
[gM + gE
]gC (B27)
These amplitudes to the contrary are not described by the tree-level ∆-production cross sections
in the standard dispersive approach [17] This peculiarity has been previously missed e g in the
calculation of the ∆-exchange contribution to the hydrogen hyperfine splitting in Ref [88] The
importance of including the ∆-non-pole contribution is also evident when considering the BC sum
37
rule in Eq (14) The ∆-pole terms by themselves violate the BC sum rule but cancel exactly with
the ∆-non-pole terms
limνrarr0
νS∆-pole2 (νQ2) + lim
νrarr0νS2
∆-exch(νQ2) = 0 (B28)
Appendix C Polarizabilities at Q2 = 0
In this section we give analytical expressions for the static values and slopes at Q2 = 0 of
the spin polarizabilities and moments of polarized structure functions In particular we give the
HB expansion of the πN -loop contributions and the ∆-exchange contributions The complete
expressions also for the π∆-loop contributions can be found in the Supplemented material Recall
that IA(0) = I1(0) = d2(0) = 0 and dd2(Q2)dQ2
∣∣∣Q2=0
= 0
1 πN-loop contribution
Here we give analytical expressions for the πN -loop contributions to the proton and neutron
spin polarizabilities expanded in powers of micro = mπMN viz the HB expansion Note that we
choose to expand here to a high order in micro the strict HB expansion would only retain the leading
term in an analogous NLO calculation
bull Static polarizabilities (Q2 = 0)
γ0p =e2g2
A
96π3f2πm
2π
1minus 21πmicro
8minus(
59
2+ 26 logmicro
)micro2 +
1875πmicro3
64
+3
(3
2+ 26 logmicro
)micro4 +
(C1)
γ0n =e2g2
A
48π3f2πm
2π
1
2minus 9πmicro
16minus 2micro2 logmicro+
75πmicro3
128minus 3micro4
4+
(C2)
δLTp =e2g2
A
192π3f2πm
2π
1minus 9πmicro
8+
(13
2minus 2 logmicro
)micro2 minus 465πmicro3
64
minus(
47
2+ 42 logmicro
)micro4 +
(C3)
δLTn =e2g2
A
96π3f2πm
2π
1
2+
3πmicro
16+ (1 + 2 log micro)micro2 minus 105πmicro3
128+
5micro4
4+
(C4)
38
δLTn =e2g2
A
96π3f2πm
2π
1
2+
3πmicro
16+ (1 + 2 log micro)micro2 minus 105πmicro3
128+
5micro4
4+
(C5)
γ0p =e2g2
A
16π3f2πm
4π
4
45minus 3πmicro
16+
14micro2
5minus 1813πmicro3
384minus 192
5(1 + log micro)micro4
+80703πmicro5
2048+
(C6)
γ0n =e2g2
A
16π3f2πm
4π
4
45minus 5πmicro
48+
4micro2
5minus 245πmicro3
384minus 32micro4 logmicro
15
+1323πmicro5
2048+
(C7)
bull Slopes of polarizabilities at Q2 = 0
dγ0p(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
2minus 45πmicro
4+ 223micro2 minus 28515πmicro3
64
minus9
(1953
4+ 449 logmicro
)micro4 +
570255πmicro5
128+
(C8)
dγ0n(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
2minus 81πmicro
8+ 94micro2 minus 2535πmicro3
32
minus3 (1 + 90 logmicro)micro4 +84315πmicro5
1024+
(C9)
dδLTp(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
2880π3f2πm
4π
minus5
2minus 27πmicro
32+ 20micro2 minus 5865πmicro3
256
+3
(617
4+ 36 logmicro
)micro4 minus 2056845πmicro5
4096+
(C10)
dδLTn(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
minus5
4minus 81πmicro
64minus 11micro2 +
10005πmicro3
512
+15
8(11 + 48 logmicro)micro4 minus 267015πmicro5
8192+
(C11)
dIAp(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
96π2f2πmicro
2
1minus 15πmicro
4minus 1
2(115 + 88 logmicro)micro2 +
1839πmicro3
32
+5 (5 + 34 logmicro)micro4 +
(C12)
dIAn(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
48π2f2πmicro
2
1
2minus 11πmicro
8minus 1
4(1 + 20 logmicro)micro2 +
99πmicro3
64
minus25micro4
12+
(C13)
dI1p(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
96π2f2πmicro
3π
8+ 2 (4 + 3 logmicro)microminus 537πmicro2
64
minus1
2(15 + 56 logmicro)micro3 +
(C14)
39
dI1n(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
48π2f2πmicro
minus π
16+
1
4(3 + 4 logmicro)microminus 57πmicro2
128+
2micro3
3+
(C15)
γ0p(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2g2
A
16π3f2πm
6π
1
105minus 23πmicro
256+
377micro2
210minus 15551πmicro3
6144+
3371micro4
105
minus1640457πmicro5
32768+
(C16)
γ0n(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2g2
A
16π3f2πm
6π
1
105minus 153πmicro
1792+
69micro2
70minus 4615πmicro3
6144+
172micro4
35
minus120897πmicro5
32768+
(C17)
2 ∆-exchange contribution
Here we give analytical expressions for the tree-level ∆-exchange contributions to the nucleon
spin polarizabilities and their slopes at Q2 = 0 Note that the ∆-exchange contributes equally to
proton and neutron polarizabilities Recall that for the magnetic γlowastN∆ coupling we introduced a
dipole form factor to mimic vector-meson dominance gM rarr gM(1 +Q2Λ2)2
bull Static polarizabilities (Q2 = 0)
γ0 = minus e2
4πM2+
(g2M
∆2+
g2E
M2+
minus 4gMgEM+∆
) (C18)
δLT =e2M∆
4πM3+
(g2E
MNM++gMgE∆MN
minus gEgCM2
∆
) (C19)
γ0 =e2M2
N
π∆2M4+
(minusg
2M
∆2+
g2E
M2+
+4gMgE∆M+
) (C20)
bull Slopes of polarizabilities at Q2 = 0
dγ0(Q2)
dQ2
∣∣∣∣∣Q2=0
= minus e2
πM2+∆
(g2M
∆
[1
4∆2minus 1
∆M++
1
2M2+
]minus 1
Λ2
g2M
∆+
g2E
2M2+
[1
2∆minus 3
M+
]minusgMgE
M+
[1
∆2minus 5
∆M++
1
M2+
]+
1
Λ2
2gMgEM+
+2gMgC∆M2
+
minus gEgCM3
+
) (C21)
40
d δLT (Q2)
dQ2
∣∣∣∣∣Q2=0
=e2M∆∆
4πMNM2+
(g2E
∆2M2+
[1
∆minus 4
M+
]minus g2
C
∆M2∆M
2+
+gMgE∆2M+
[1
∆2
minus 3
∆M++
1
M2+
]minus 2
Λ2
gMgE∆2M+
+gMgC∆M2
∆
[1
2∆2minus 2
∆M++
1
2M2+
]minus gEgC
2M2∆M
2+
[7
∆+
1
M+
]) (C22)
dIA(Q2)
dQ2
∣∣∣∣∣Q2=0
= minusM2N
M2+
(g2M
2∆2+
g2E
MNM+minus 2gMgE
∆M+minus gEgCM∆M+
) (C23)
dI1(Q2)
dQ2
∣∣∣∣∣Q2=0
= minusM∆M2N
2M3+
(g2E
MNM∆minus gMgE∆MN
minus gEgCM2
∆
) (C24)
γ0(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2M2
N
π∆3M6+
(g2M
∆
[3M2
∆ + 2M∆MN minus 9M2N
∆2+
4M2+
Λ2
]
minus4gMgE
[5M2
∆ minus 9M2N
M+∆2+
2M+
Λ2
]+
g2E
∆M+[7M∆ minus 9MN ]
minus8gMgC∆
+4gEgCM+
) (C25)
[1] D Drechsel B Pasquini and M Vanderhaeghen Dispersion relations in real and virtual
Compton scattering Phys Rept 378 99 (2003) hep-ph0212124
[2] S E Kuhn J-P Chen and E Leader Spin structure of the nucleon mdash status and recent
results Prog Part Nucl Phys 63 1 (2009) arXiv08123535 [hep-ph]
[3] F Hagelstein R Miskimen and V Pascalutsa Nucleon polarizabilities from Compton scat-
tering to hydrogen atom Prog Part Nucl Phys 88 29 (2016) arXiv151203765 [nucl-th]
[4] B Pasquini and M Vanderhaeghen Dispersion theory in electromagnetic interactions Ann
Rev Nucl Part Sci 68 75 (2018) arXiv180510482 [hep-ph]
[5] S Gerasimov A Sum rule for magnetic moments and the damping of the nucleon magnetic
moment in nuclei Sov J Nucl Phys 2 430 (1966)
[6] S Drell and A C Hearn Exact sum rule for nucleon magnetic moments Phys Rev Lett
16 908 (1966)
[7] J S Schwinger Source theory viewpoints in deep inelastic scattering Proc Natl Acad Sci
USA 72 1 (1975)
[8] J S Schwinger Source theory viewpoints in deep inelastic scattering Electromagnetic In-
teractions and Field Theory Proceedings 14 Internationale Universitatswochen Schladming
Austria February 24-March 7 1975 Acta Phys Austriaca Suppl 14 471 (1975)
41
[9] J Schwinger Source theory discussion of deep inelastic scattering with polarized particles
Proc Natl Acad Sci USA 72 1559 (1975)
[10] A M Baldin Polarizability of nucleons Nucl Phys 18 310 (1960)
[11] M Gell-Mann M L Goldberger and W E Thirring Use of causality conditions in quantum
theory Phys Rev 95 1612 (1954)
[12] A Pineda Leading chiral logarithms to the hyperfine splitting of the hydrogen and muonic
hydrogen Phys Rev C 67 025201 (2003)
[13] C Peset and A Pineda The two-photon exchange contribution to muonic hydrogen from
chiral perturbation theory Nucl Phys B 887 69 (2014) arXiv14064524 [hep-ph]
[14] C Peset and A Pineda Model-independent determination of the two-photon exchange contri-
bution to hyperfine splitting in muonic hydrogen JHEP 04 060 arXiv161205206 [nucl-th]
[15] F Hagelstein and V Pascalutsa Proton structure in the hyperfine splitting of muonic hydro-
gen PoS CD15 077 (2016) arXiv151104301 [nucl-th]
[16] F Hagelstein Exciting Nucleons in Compton Scattering and Hydrogen-Like Atoms PhD
thesis Mainz U Inst Kernphys (2017) arXiv171000874 [nucl-th]
[17] F Hagelstein ∆(1232)-Resonance in the hydrogen spectrum Proceedings 11th International
Workshop on the Physics of Excited Nucleons (NSTAR 2017) Columbia SC USA August
20-23 2017 Few Body Syst 59 93 (2018) arXiv180109790 [nucl-th]
[18] Y Prok et al (CLAS) Moments of the spin structure functions gp1 and gd1 for 005 lt Q2 lt 30
GeV2 Phys Lett B 672 12 (2009) arXiv08022232 [nucl-ex]
[19] H Dutz et al (GDH) First measurement of the Gerasimov-Drell-Hearn sum rule for 1H from
07 GeV to 18 GeV at ELSA Phys Rev Lett 91 192001 (2003)
[20] M Amarian et al (Jefferson Lab E94010) Measurement of the generalized forward spin po-
larizabilities of the neutron Phys Rev Lett 93 152301 (2004) arXivnucl-ex0406005
[21] M Amarian et al The Q2 evolution of the generalized Gerasimov-Drell-Hearn integral for
the neutron using a 3He target Phys Rev Lett 89 242301 (2002) arXivnucl-ex0205020
[22] M Amarian et al (Jefferson Lab E94-010)Q2 evolution of the neutron spin structure moments
using a 3He target Phys Rev Lett 92 022301 (2004) arXivhep-ex0310003
[23] A Deur et al Experimental determination of the evolution of the Bjorken integral at low Q2
Phys Rev Lett 93 212001 (2004) arXivhep-ex0407007
[24] K Slifer Low Q2 measurement of gp2 and the δLT spin polarizability Spin structure at long dis-
tance Proceedings Workshop Newport News USA March 12ndash13 2009 nucl-ex09064775
(2009) AIP Conf Proc 1155 10106313203293 (2009) arXiv09064775 [nucl-ex]
[25] P Solvignon et al (E01-012) Moments of the neutron g2 structure function at intermediate
42
Q2 Phys Rev C 92 015208 (2015) arXiv13044497 [nucl-ex]
[26] A Deur Experimental studies at low Q2 of the spin structure of the nucleon at Jefferson Lab
in 9th International Workshop on Chiral Dynamics (CD18) Durham NC USA September
17-21 2018 (2019) arXiv190305661 [nucl-ex]
[27] V Sulkosky et al (Jefferson Lab E97-110) Measurement of the 3He spin-structure functions
and of neutron (3He) spin-dependent sum rules at 0035 le Q2 le 024 GeV2 Phys Lett B
805 135428 (2020) arXiv190805709 [nucl-ex]
[28] K Adhikari et al (CLAS) Measurement of the Q2 Dependence of the Deuteron Spin Structure
Function g1 and its Moments at Low Q2 with CLAS Phys Rev Lett 120 062501 (2018)
arXiv171101974 [nucl-ex]
[29] V Lensky F Hagelstein A Hiller Blin and V Pascalutsa Comment on rdquoMeasurement of
the Q2 Dependence of the Deuteron Spin Structure Function g1 and its Moments at Low Q2
with CLASrdquo (2018) arXiv180603219 [nucl-th]
[30] J M Alarcn F Hagelstein V Lensky and V Pascalutsa Forward doubly-virtual Compton
scattering off the nucleon in chiral perturbation theory at NLO the subtraction function and
moments of unpolarized structure functions (2020) arXiv200509518 [hep-ph]
[31] V Bernard E Epelbaum H Krebs and U-G Meiszligner New insights into the spin structure
of the nucleon Phys Rev D 87 054032 (2013) arXiv12092523 [hep-ph]
[32] V Lensky J M Alarcon and V Pascalutsa Moments of nucleon structure functions at
next-to-leading order in baryon chiral perturbation theory Phys Rev C 90 055202 (2014)
arXiv14072574 [hep-ph]
[33] V Pascalutsa and M Vanderhaeghen Chiral effective-field theory in the ∆(1232) region
I Pion electroproduction on the nucleon Phys Rev D 73 034003 (2006) arXivhep-
ph0512244
[34] V Pascalutsa and M Vanderhaeghen Electromagnetic nucleon-to-Delta transition in chiral
effective field theory Phys Rev Lett 95 232001 (2005) arXivhep-ph0508060
[35] T R Hemmert B R Holstein and J Kambor Systematic 1M expansion for spin 32
particles in baryon chiral perturbation theory Phys Lett B 395 89 (1997) arXivhep-
ph9606456
[36] V Pascalutsa and D R Phillips Effective theory of the ∆(1232) in Compton scattering off
the nucleon Phys Rev C 67 055202 (2003) arXivnucl-th0212024
[37] V Pascalutsa M Vanderhaeghen and S N Yang Electromagnetic excitation of the ∆(1232)-
resonance Phys Rept 437 125 (2007) arXivhep-ph0609004
[38] K A Olive et al (Particle Data Group) Review of Particle Physics Chin Phys C 38 090001
43
(2014)
[39] F E Low Scattering of light of very low frequency by systems of spin 12 Phys Rev 96
1428 (1954)
[40] M Gell-Mann and M L Goldberger Scattering of low-energy photons by particles of spin
12 Phys Rev 96 1433 (1954)
[41] A M Harun ar-Rashid A simple derivation of Schwingerrsquos sum rule for spin dependent
structure functions Nuovo Cim A 33 447 (1976)
[42] F Hagelstein and V Pascalutsa Dissecting the hadronic contributions to (g minus 2)micro by
Schwingerrsquos sum rule Phys Rev Lett 120 072002 (2018) arXiv171004571 [hep-ph]
[43] V Lensky V Pascalutsa M Vanderhaeghen and C Kao Spin-dependent sum rules con-
necting real and virtual Compton scattering verified Phys Rev D 95 074001 (2017)
arXiv170101947 [hep-ph]
[44] H Burkhardt and W N Cottingham Sum rules for forward virtual Compton scattering
Annals Phys 56 453 (1970)
[45] V Lensky and V Pascalutsa Predictive powers of chiral perturbation theory in Compton
scattering off protons Eur Phys J C 65 195 (2010) arXiv09070451 [hep-ph]
[46] H Grieszlighammer J McGovern D Phillips and G Feldman Using effective field theory to
analyse low-energy Compton scattering data from protons and light nuclei Prog Part Nucl
Phys 67 841 (2012) arXiv12036834 [nucl-th]
[47] H W Grieszlighammer J A McGovern and D R Phillips Nucleon polarisabilities at and
beyond physical pion masses Eur Phys J A 52 139 (2016) arXiv151101952 [nucl-th]
[48] E Epelbaum H Krebs and U-G Meiszligner Improved chiral nucleon-nucleon potential up to
next-to-next-to-next-to-leading order Eur Phys J A 51 53 (2015) arXiv14120142 [nucl-
th]
[49] E Epelbaum H Krebs and U-G Meiszligner Precision nucleon-nucleon potential at fifth order
in the chiral expansion Phys Rev Lett 115 122301 (2015) arXiv14124623 [nucl-th]
[50] C W Kao T Spitzenberg and M Vanderhaeghen Burkhardt-Cottingham sum rule and
forward spin polarizabilities in heavy baryon chiral perturbation theory Phys Rev D 67
016001 (2003) arXivhep-ph0209241
[51] C-W Kao D Drechsel S Kamalov and M Vanderhaeghen Higher moments of nucleon
spin structure functions in heavy baryon chiral perturbation theory and in a resonance model
Phys Rev D 69 056004 (2004) arXivhep-ph0312102
[52] T Becher and H Leutwyler Baryon chiral perturbation theory in manifestly Lorentz invariant
form Eur Phys J C 9 643 (1999) arXivhep-ph9901384
44
[53] J Gasser M E Sainio and A Svarc Nucleons with chiral loops Nucl Phys B 307 779
(1988)
[54] L S Geng J Martin Camalich L Alvarez-Ruso and M J Vicente Vacas Leading SU(3)-
breaking corrections to the baryon magnetic moments in chiral perturbation theory Phys
Rev Lett 101 222002 (2008) arXiv08051419 [hep-ph]
[55] D Drechsel S S Kamalov and L Tiator The GDH sum rule and related integrals Phys
Rev D 63 114010 (2001) arXivhep-ph0008306
[56] D Drechsel O Hanstein S S Kamalov and L Tiator A unitary isobar model for pion
photo- and electroproduction on the proton up to 1 GeV Nucl Phys A 645 145 (1999)
arXivnucl-th9807001
[57] L Tiator private communication (2020)
[58] V Bernard T R Hemmert and U-G Meiszligner Spin structure of the nucleon at low energies
Phys Rev D 67 076008 (2003) arXivhep-ph0212033
[59] O Gryniuk F Hagelstein and V Pascalutsa Evaluation of the forward Compton scattering
off protons II Spin-dependent amplitude and observables Phys Rev D 94 034043 (2016)
arXiv160400789 [nucl-th]
[60] R Zielinski The g2p Experiment A Measurement of the Protonrsquos Spin Structure Functions
PhD thesis New Hampshire U (2010) arXiv170808297 [nucl-ex]
[61] N Guler et al (CLAS) Precise determination of the deuteron spin structure at low to mod-
erate Q2 with CLAS and extraction of the neutron contribution Phys Rev C 92 055201
(2015) arXiv150507877 [nucl-ex]
[62] C E Carlson V Nazaryan and K Griffioen Proton structure corrections to electronic
and muonic hydrogen hyperfine splitting Phys Rev A 78 022517 (2008) arXiv08052603
[physicsatom-ph]
[63] V Bernard N Kaiser and U-G Meiszligner Chiral dynamics in nucleons and nuclei Int J
Mod Phys E 4 193 (1995) arXivhep-ph9501384
[64] T R Hemmert B R Holstein and J Kambor ∆(1232) and the polarizabilities of the nucleon
Phys Rev D 55 5598 (1997) arXivhep-ph9612374
[65] V Pascalutsa and R Timmermans Field theory of nucleon to higher-spin baryon transitions
Phys Rev C 60 042201 (1999) arXivnucl-th9905065
[66] V Pascalutsa Quantization of an interacting spin-32 field and the Delta isobar Phys Rev
D 58 096002 (1998) arXivhep-ph9802288
[67] H Krebs Double Virtual Compton Scattering and SpinStructure of the Nucleon PoS
CD2018 031 (2019)
45
[68] N Kochelev and Y Oh Axial anomaly and the δLT puzzle Phys Rev D 016012 (2012)
arXiv11034892 [hep-ph]
[69] D Drechsel S Kamalov and L Tiator Unitary isobar model ndash MAID2007 Eur Phys J A
69 (2007) available at httpsmaidkphuni-mainzde arXiv07100306 [nucl-th]
[70] P J Mohr B N Taylor and D B Newell CODATA recommended values of the fundamental
physical constants 2010 Rev Mod Phys 84 1527 (2012)
[71] J Ahrens et al (GDH A2) First measurement of the Gerasimov-Drell-Hearn integral for 1H
from 200 to 800 MeV Phys Rev Lett 87 022003 (2001) arXivhep-ex0105089 [hep-ex]
[72] K Helbing (GDH) Experimental verification of the GDH sum rule at ELSA and MAMI Nucl
Phys Proc Suppl 105 113 (2002)
[73] R Bradford A Bodek H S Budd and J Arrington A New parameterization of the nucleon
elastic form-factors NuInt05 proceedings of the 4th International Workshop on Neutrino-
Nucleus Interactions in the Few-GeV Region Okayama Japan 26-29 September 2005 Nucl
Phys Proc Suppl 159 127 (2006) arXivhep-ex0602017
[74] J D Bjorken Applications of the chiral U(6)timesU(6) algebra of current densities Phys Rev
148 1467 (1966)
[75] J D Bjorken Inelastic scattering of polarized leptons from polarized nucleons Phys Rev D
1 1376 (1970)
[76] V Bernard T R Hemmert and U-G Meiszligner Novel analysis of chiral loop effects in
the generalized Gerasimov-Drell-Hearn sum rule Phys Lett B 545 105 (2002) arXivhep-
ph0203167
[77] A Deur et al Experimental study of isovector spin sum rules Phys Rev D 78 032001
(2008) arXiv08023198 [nucl-ex]
[78] B Pasquini P Pedroni and D Drechsel Higher order forward spin polarizability Phys Lett
B 687 160 (2010) arXiv10014230 [hep-ph]
[79] R Jaffe g2ndashThe nucleonrsquos other spin-dependent structure function Comments Nucl Part
Phys 19 239 (1990)
[80] E V Shuryak and A Vainshtein Theory of power corrections to deep inelastic scattering in
quantum chromodynamics (II) Qminus4 effects polarized target Nucl Phys B 201 141 (1982)
[81] S Wandzura and F Wilczek Sum rules for spin dependent electroproduction Test of rela-
tivistic constituent quarks Phys Lett B 72 195 (1977)
[82] B W Filippone and X-D Ji The spin structure of the nucleon Adv Nucl Phys 26 1
(2001) arXivhep-ph0101224
[83] M Burkardt The g2 structure function Proceedings Workshop on Spin structure at long
46
distance Newport News USA March 12-13 2009 AIP Conf Proc 1155 26 (2009)
arXiv09054079 [hep-ph]
[84] A Deur S J Brodsky and G F de Teramond The spin structure of the nucleon Rept
Prog Phys 82 076201 (2019) arXiv180705250 [hep-ph]
[85] B R Holstein D Drechsel B Pasquini and M Vanderhaeghen Higher order polarizabilities
of the proton Phys Rev C 61 034316 (2000) arXivhep-ph9910427
[86] B R Holstein V Pascalutsa and M Vanderhaeghen Sum rules for magnetic moments and
polarizabilities in QED and chiral effective-field theory Phys Rev D 72 094014 (2005)
arXivhep-ph0507016
[87] B Pasquini D Drechsel and L Tiator Invariant amplitudes for pion electroproduction Eur
Phys J A 34 387 (2007) arXiv07122327 [hep-ph]
[88] A J Buchmann Non-spherical proton shape and hydrogen hyperfine splitting Proceedings
International Workshop on Precision Physics of Simple Atomic Systems (PSAS 2008) Wind-
sor Ontario Canada July 21-26 2008 Can J Phys 87 773 (2009) arXiv09104747
[physicsatom-ph]
47
(which in the forward case coincides with the Breit frame) is parametrized in terms of the nucleon
Pauli matrices ~σ and the four scalar functions fL fT gTT and gLT
T (νQ2) = ε0 εprimelowast0 fL(νQ2) + (~ε primelowast middot ~ε ) fT (νQ2) + i~σ middot (~ε primelowast times ~ε ) gTT (νQ2) (A2)
minusi~σ middot [(ε0~εprimelowast minus ~ε ε primelowast0 )times q] gLT (νQ2)
Here ~q and q = ~q|~q | are the photon three-momentum in the lab system and its unit vector The
modified polarization vector components are given by
ε0 =
[ε0 minus
ν
|~q | (~ε middot q )
] |~q |Q ~ε = ~εminus q (~ε middot q ) (A3)
where ε = (ε0~ε ) is the usual incoming photon polarization vector and εprimelowast the outgoing polarization
vector The LEX of the lab frame amplitudes [Eq (10)] can serve in particular as the definition
of the generalized polarizabilities The lab frame amplitudes are also conveniently used for the
definition of the response functions see the example of the scalar amplitude gLT (νQ2) and the
corresponding response function σLT (νQ2) below in App B
Appendix B Photoabsorption cross sections
In the forward kinematics the spin-dependent VVCS amplitudes and the spin polarizabilities
can be described in terms of the polarized structure functions g1(xQ2) and g2(xQ2) or equiv-
alently the helicity-difference cross section σTT (νQ2) and the longitudinal-transverse response
function σLT (νQ2) with the help of dispersion relations (5) and the optical theorem (3) In this
way the photoabsorption cross sections measured in electroproduction processes form the basis
for most empirical evaluations shown throughout Sec III In the following we present the BχPT
predictions for the tree-level cross sections of πN - π∆- and ∆-production through photoabsorption
on the nucleon cf Figs 8 9 and 10 in Ref [30] In Secs B 1 and B 2 we will discuss the leading
πN -production channel and the ∆-production channel respectively We used these cross sections
to verify the polarizability predictions obtained otherwise from the calculated non-Born VVCS
amplitudes Due to the bad high-energy behavior of the π∆-production cross sections in BχPT
cf Fig 10 the dispersion relations in Eq (5) require further subtractions for a reconstruction of
the π∆-loop contribution to the spin-dependent VVCS amplitudes Therefore not all polarizabil-
ities could be verified but only those appearing as higher-order terms in the LEX of the VVCS
amplitudes such as γ0 [16]
29
proton neutron
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Neutron
02 04 06 08 100
50
100
150
200
250
[GeV]
T [b]
02 04 06 08 10
-50
0
50
100
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Neutron
02 04 06 08 100
50
100
150
200
250
[GeV]
T [b]
02 04 06 08 10
-50
0
50
100
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02040608100
50
100
150
200
250
300
[GeV]
T[b]
0204060810
-50
0
50
[GeV]
TT[b]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
0204060810
00
02
04
06
[GeV]
LTQ
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
0204060810
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
FIG 10 Photoabsorption cross sections for πN (red) and π∆ production (orange) with Q2 = 0
(solid) and Q2 = 01 GeV2 (dashed for πN and dotted for π∆ channel)
1 πN-production channel
In order to extract the response function σLT (νQ2) we have developed a method similar to
the one used to calculate σTT (νQ2) see for example Ref [86] For σLT (νQ2) however the
calculation is more complicated because one has to take into account that the associated Compton
process involves a spin-flip of the nucleon as illustrated in Fig 11 When calculating the cross
section the product of the incoming nucleon spinors has to reflect this flip
The forward VVCS amplitude related to σLT (νQ2) mdash and δLT (Q2) mdash is gLT (νQ2) It can
be extracted from Eq (A2) if one takes the modified polarization vector components in Eq (A3)
with ε = εL and εprimelowast = εlowastplusmn as input where εL = 1Q(|~q | 0 0 q0) and εplusmn = ∓ 1radic
2(0 1plusmni 0) are the
standard longitudinal and transverse polarization vectors respectively For εL and εlowastplusmn only the
choice of helicities hprime = plusmn12 and h = ∓12 gives a non-zero contribution and one obtains
χdaggerhprime T (νQ2)χh = χdaggerhprimeminusi~σ middot [(ε0~εprimelowast minus ~ε ε primelowast0 )times q] gLT (νQ2)χh =
radic2 gLT (νQ2) (B1)
where χh and χdaggerhprime are two-component Pauli spinors with opposite helicities or here spins
Let us now consider the related photoabsorption process and in particular the tree-level γlowastN rarr
30
s = 12
sprime = minus12
λ = 0 λprime = 1
rprime rprime
s=12sprime=minus1
2
λ=0λprime=1
rprimerprime
s=12
sprime=minus12
λ=0λprime=1
rprimerprime hacute=frac12h=-frac12
λacute=1λ=0
FIG 11 Relation between the forward Compton process and the photoabsorption process given
by the optical theorem In particular we show the longitudinal-transverse contribution The
double-line arrows represent the spin of the external particles while the dot represents the scalar
(longitudinal) polarization of the incoming photon Inside the blob the intermediate states are
represented eg nucleons with spins rprime (which are averaged in the calculation of the cross section)
and pions
πN channel see diagrams in Fig 8 of Ref [30] We define the πN -production amplitude as
T = uhB (PB)sumi
Ai(s t)Γi uhA(PA) (B2)
with the Dirac structures
Γ1 = γ5 (B3a)
Γ2 =1
2
[qA ε
]γ5 (B3b)
where uhA(PA) and udaggerhB (PB) are the Dirac spinors and PA and PB are the four-momenta of
the incoming and outgoing nucleons respectively When calculating the photoabsorption cross
section related to the VVCS amplitude in Eq (B1) the nucleon spin flip should be implemented
by uhprime(PA) in T dagger and uh(PA) in T together with the appropriate transverse and longitudinal
photon polarization vectors εlowastplusmn and εL
However if one wants to use the properties of the Dirac matrices it is more useful to construct
an operator to produce this spin flip in the external nucleons of Fig 11 This is accomplished by
introducing the projector ΓLT equiv 12radic
2(γ1 + iγ2)γ5 which also takes into account the extra factor
radic2 in Eq (B1) We checked that with this projector one correctly extracts δLT by comparing the
HB limit of our result to the HB result of Ref [50] where the authors calculate this polarizability
from the Compton amplitude directly With all those ingredients the longitudinal-transverse cross
section is calculated in the following way
σLT (νQ2) =1
64π2 s
|~pf |cm
|~pi|cm
int 1
minus1dcos θ
sumij
AiAdaggerjXij (B4)
31
with
Xij = Tr[(PB +MN )Γi(PA +MN )ΓLTγ0Γdaggerjγ
0] (B5)
where θ is the scattering angle in the center-of-mass (cm) frame and |~pi|cm (|~pf |cm) is the three-
momentum of an incoming (outgoing) particle in the cm frame An explicit calculation of the
matrix Xij leads to
X = MNQ
0 2(PB minus PA) middot εLminusradic
2|~qf |cm sin θ (sminus u)
(B6)
where |~qi|cm (|~qf |cm) is the relative three-momentum of the incoming (outgoing) particles in the
cm frame Here s t and u are the usual Mandelstam variables For the different γlowastN rarr πN
channels we obtain the following amplitudes Ai where we introduce qA as the four-momentum of
the incoming photon and qB as the four-momentum of the outgoing pion
bull γlowastprarr π0p
A1 =e gAMN
fπ
[2PA middot ε+ qA middot ε
sminusM2N
+2PB middot εminus qA middot ε
uminusM2N
] (B7a)
A2 =e gAMN
fπ
[1
sminusM2N
+1
uminusM2N
] (B7b)
bull γlowastprarr π+n
A1 =
radic2 e gAMN
fπ
[2PA middot ε+ qA middot ε
sminusM2N
+2(PA minus PB) middot ε+ qA middot ε
tminusm2π
] (B8a)
A2 =
radic2 e gAMN
fπ(sminusM2N )
(B8b)
bull γlowastnrarr π0n
A1 = 0 (B9a)
A2 = 0 (B9b)
bull γlowastnrarr πminusp
A1 =
radic2 e gAMN
fπ
[2PB middot εminus qA middot ε
uminusM2N
minus 2(PA minus PB) middot ε+ qA middot εtminusm2
π
] (B10a)
A2 =
radic2 e gAMN
fπ(uminusM2N )
(B10b)
The analytical expressions shown above were checked with the amplitudes given in Ref [87] An-
alytical expressions for the tree-level γlowastN rarr πN channel of the σLT (νQ2) and σTT (νQ2) cross
32
sections are given below (proton channels π+n and π0p neutron channel πminusp) We checked that
they reproduce the known results in the real-photon limit [45 86] To shorten the final expressions
for the cross sections which are considerably longer for finite Q2 than in the real-photon limit we
define the following dimensionless kinematic variables
αγ = (ENi )cmradics =
s+M2N +Q2
2s (B11)
απ = (ENf )cmradics =
s+M2N minusm2
π
2s (B12)
βγ = Eγcmradics =
sminusM2N minusQ2
2s (B13)
βπ = Eπcmradics =
sminusM2N +m2
π
2s (B14)
λγ = |~qi|cmradics =
radic(sminusM2
N minusQ2)2 + 4sQ2
2s (B15)
λπ = |~qf |cmradics =
radic(sminusM2
N +m2π)2 minus 4sm2
π
2s (B16)
Here (ENi )cm and (ENf )cm are the energies of the incoming and outgoing nucleon Eγcm is the energy
of the incoming photon Eπcm is the energy of the outgoing pion all in the cm frame
σ(π+n)TT = minus e2g2
AM2N
64πf2πs
3(sminusM2N )2λ4
γ
4sλπλγ
[(M2
N minus s)(M2N minusQ2 minus s)(Q2 + 2sβγβπ)
+ 2s(m2πQ
2 minus (M2N minus s)
(M2N + s(minus1 + 2βγβπ)
))λ2γ
]+ (M2
N minus s)(M2N minusQ2 minus s)
times (Q2 + 2sβγβπ minus 2sλπλγ)(Q2 + 2s(βγβπ + λπλγ)
)times log
(Q2 + 2sβγβπ minus 2sλγλπQ2 + 2sβγβπ + 2sλγλπ
) (B17)
33
σ(π0p)TT =
e2g2AM
2Nλπ
64πf2πs(sminusM2
N )2λγ
1
s2λπλ3γ
[(M2
N minus s)(M2N minusQ2 minus s)
(3M2
N + s(2βγβπ minus 3))(M2N
+ s(2βγβπ minus 1))
+ 2s
((2m2
πQ2 + (M2
N minus s)2)(M2
N minus s)minus 2(minusm2
πQ2 + (sminusM2
N )2)
times sβγβπ + 4s2(sminusM2N )β2
γβ2π
)λ2γ
]arctan
(2sλπλγ
M2N + s(2βγβπ minus 1)
)+ 2
[minus 2m2
πQ2
+ (M2N minus s)
(2(sminusM2
N ) + 4sβγβπ +(sminusM2
N +Q2)(3M2N minus 3s+ 2sβγβπ)
sλ2γ
+m2πQ
2(sminusM2N )
(M2N minus s+ 2sβγβπ)(M2
N + s(minus1 + 2βγβπ minus 2λπλγ))
+m2πQ
2(sminusM2N )
(M2N minus s+ 2sβγβπ)(M2
N + s(minus1 + 2βγβπ + 2λπλγ))
)
+1
λπλγ
(βγβπ
(minusm2
πQ2 + (M2
N minus s)(M2N minus s+ 2sβγβπ)
)minus (M2
N minus s)(M2N minusQ2 minus s)λ2
π
)log
(M2N + s(minus1 + 2βγβπ + 2λπλγ)
M2N + s(minus1 + 2βγβπ minus 2λπλγ)
)] (B18)
σ(πminusp)TT =
e2g2AM
2N
64πf2πs
3λ4γ(M2
N + s(2βπβγ minus 1))2 minus 4λ2πs
2λ2γ)
4sλπλγ
((minusM2
N +Q2 + s)(M2N
+ s(2βπβγ minus 1))
2 minus 2sλ2γ
(2λ2
πs(minusM2N +Q2 + s) +m2
πQ2))
+((M2N + s(2βπβγ minus 1)
)2
minus 4λ2πs
2λ2γ
)((M2N + s(2βπβγ minus 1)
)(M2N minus 2Q2 minus s(2βπβγ + 1)
)+ 4λ2
πs2λ2
γ
)times log
(minusM2
N minus 2βπsβγ minus 2λπsλγ + s)minus(M2N + s(2βπβγ minus 2λπλγ minus 1)
)(M2N + s(minus1
+ 2βπβγ + 2λπλγ))[((
M2N + s(2βπβγ minus 1)
)(M2N minus 2Q2 minus s(2βπβγ + 1)
)+ 4λ2
πs2λ2
γ
)times log
(minusM2
N minus 2βπsβγ + 2λπsλγ + s)
+((Q2 + 2βπsβγ)2 minus 4λ2
πs2λ2
γ
)times log
(Q2 + 2βπsβγ minus 2λπsλγQ2 + 2sβπβγ + 2λπsλγ
)]+ 4sλ2
γ
((M2
N + s(2βπβγ minus 1))2 minus 4λ2πs
2λ2γ
)times (M2
N minus s) arctanh
(2λπsλγ
M2N + 2βπsβγ minus s
) (B19)
34
σ(π+n)LT =
e2g2AM
3Nλπ
32πf2πQs
3(sminusM2N )2λ4
γ
2sλγ
[(M2
N minus s)(Q2 + 2sβ2γ)(Q2 + 2sβγβπ)
minus 4s
((M2
N minus s)(Q2 minus 2s(απ minus 1)βγ
)+Q2sβγβπ
)λ2γ + 8s3(minus1 + απ)λ4
γ
]minus M2
N minus sλπ
[(Q2 + 2sβ2
γ)(Q2 + 2sβγβπ)2 + 4s2(2(απ minus 1)βγ(Q2 + 2sβγβπ)
minusQ2λ2π
)λ2γ + 8s3(απ minus 1)2λ4
γ
]arctan
(2sλπλγ
Q2 + 2sβγβπ
) (B20)
σ(π0p)LT =
e2g2AM
3Nλπ
16πf2πQs(sminusM2
N )2λγ
1
minus2s(M2N + s(minus1 + 2βγβπ))2λ2
γ + 8s3λ2πλ
4γ
[minus 3M8
N (Q2
+ 2sβ2γ) + 2M4
Ns2
(minus (Q2 + 2sβ2
γ)(minus3 + 2βγβπ) +(Q2(minus3 + 2βγβπ + 6λ2
π) + 2sβγ
times (minus3 + 12απ + 2βγβπ minus 12απβγβπ + 4βγλ2π))λ2γ minus 4s(α2
π minus 1)λ4π
)+ 2M2
Ns3
[minus (Q2
+ 2sβ2γ)(minus1 + 2βγβπ)
(6 + βγβπ(minus9 + 2βγβπ)
)+
(Q2(3minus 12λ2
π + 4βγβπ(minus1 + 2βγβπ
+ λ2π))minus 2sβγ
(minus 3 + 4απ(3 + 2απ(3 + 2βγβπ(minus3 + βγβπ)) + 4βγ(βπ + (2minus βγβπ)
times λ2π)))λ2γ + 8s
((απ minus 1)(1 + απ minus 2βγβπ) + 2απβγλ
2π
)λ4γ
)+ s4
((Q2 + 2sβ2
γ)
times (1minus 2βγβπ)2(2βγβπ minus 3) + 2(Q2(minus1 + 6λ2
π + 2βγβπ((1minus 2βγβπ)2 minus 2λ2π)) + 2sβγ
times (minus1 + 2βγβπ + 4(minus1 + βγβπ)(minusαπ + 2απβγβπ minus βγλ2π)))λ2γ minus 8
(s(απ minus 1)(απ
+ (1minus 2βγβπ)2) + 2βγ(2sαπ +Q2βγ)λ2π
)λ4γ + 32s(απ minus 1)λ2
πλ6γ
)+ 2M6
Ns
(Q2(6
minus 7βγβπ + λ2γ) + 2sβγ(βγ(6minus 7βγβπ) + (1minus 4απ)λ2
γ)
)]+sminusM2
N
4s2λπλ3γ
[(Q2 + 2sβ2
γ)
times(3M2
N + s(2βγβπ minus 3))(M2N + s(2βγβπ minus 1)
)+ 2s
(minusM2
N (Q2 + 2sβγ minus 8sαπβγ)
+ s(2sβγ(1minus 4απ + 4απβγβπ) +Q2(1minus 2λ2π)))λ2γ + 8s3(απ minus 1)λ4
γ
]
times arctan
(2sλπλγ
M2N + s(2βγβπ minus 1)
) (B21)
35
σ(πminusp)LT =
e2g2AM
3N
64πf2πQs
3λ4γ(M2
N minusQ2 minus s)(M2N + s(2βπβγ + 2λπλγ minus 1))
times
1
M2N + s(2βπβγ minus 2λπλγ minus 1)
[4λπsλγ(M2
N minusQ2 minus s)(
2sλ2γ
(s((1minus 2λ2
π)Q2
+ 2(2απ minus 1)sβγ(2βπβγ minus 1))minusM2N (Q2 + 2(1minus 2απ)sβγ)
)+ (Q2 + 2sβ2
γ)(M2N + s(2βπβγ minus 1)
)2 + 8(απ minus 1)απs
3λ4γ
)]minus(M2N + s(2βπβγ + 2λπλγ minus 1)
)times[
log
(minusM2N minus 2βπsβγ minus 2λπsλγ + s
minusM2N minus 2βπsβγ + 2λπsλγ + s
)(minus (Q2 + 2sβ2
γ)(M2N + s(2βπβγ minus 1)
)(minusM2
N + 2Q2 + 2βπsβγ + s)minus 2sλ2γ
(minusM2
N (Q2 + 2sβγ) +Q4 +Q2s(4απβγ
minus 2βγ minus 2λ2π + 1) + 2s2βγ(4(απ minus 1)βπβγ + 1)
)minus 8(απ minus 1)2s3λ4
γ
)+ 2
(4s2λ2
γ
(2(απ minus 1)βγ(Q2 + 2βπsβγ)minus λ2
πQ2)
+ (Q2 + 2βπsβγ)2(Q2 + 2sβ2γ)
+ 8(απ minus 1)2s3λ4γ
)arctanh
(2λπsλγ
Q2 + 2βπsβγ
)] (B22)
2 ∆-production channel
The tree-level ∆-exchange diagram in Fig 2 of Ref [30] contributes to the non-Born part of
the VVCS amplitudes The contribution of the ∆ exchange to the VVCS amplitudes can be split
into [17]
S∆-exch1 (νQ2) = S∆-pole
1 (νQ2) + S∆-exch1 (νQ2) (B23a)
νS∆-exch2 (νQ2) = νS∆-pole
2 (νQ2) + νS2∆-exch
(νQ2) (B23b)
and similarly for the unpolarized VVCS amplitudes discussed in Ref [30] Here we introduced the
∆-pole contributions S∆-polei and the ∆-non-pole contributions S∆-exch
i The former amplitudes
feature a pole at the ∆(1232)-production threshold and thus are proportional to
1
[sminusM2∆][uminusM2
∆]=
1
4M2N
1
ν2∆ minus ν2
(B24)
36
They can be reconstructed from the dispersion relations in Eq (5) with the tree-level ∆-production
cross sections as input cf Fig 10 in Ref [30]
σTT (νQ2) =π2α
M2NM
2+|~q |
minus g2
MMN (M+ + ν)|~q |2 +g2E(∆minus ν)(Q2 minusMNν)2
MN(B25a)
+g2CQ
4s(∆minus ν)
MNM2∆
minus 4gMgE(Q2 minusMNν)|~q |2 minus 4gMgCQ2|~q |2
+2gEgCQ
2[minusMNM∆ |~q |2 + s(Q2 +∆ν)
]MNM∆
δ(ν minus ν∆)
σLT (νQ2) =Qπ2α
M2NM
2+|~q |
g2E(MNν minusQ2) [M∆(MN + ν)minus s]
MN(B25b)
+g2CQ
2[MNM∆|~q |2 minus s(Q2 +∆ν)
]MNM2
∆
+ gMgEM∆|~q |2 minusgMgC(Q2 minusMNν)|~q |2
M∆
+gEgC(ν minus∆)(M2
N |~q |2 minus 2Q2s)
MNM∆
δ(ν minus ν∆)
with ∆ = M∆ minusMN M+ = M∆ + MN and the Mandelstam variable s = M2N + 2MNν minus Q2
Analytical expressions for the spin structure functions g1(xQ2) and g2(xQ2) can be constructed
from Eq (3) with the flux factor K(νQ2) = |~q | =radicν2 +Q2
In the ∆-non-pole contributions to S1(νQ2) and νS2(νQ2) the pole in ν at the ∆(1232)-
production threshold has canceled out
S∆-exch1 (νQ2) =
πα
MNM2+
[g2MQ
2+ + g2
E
(∆2 minus 3Q2
)+
4g2CQ
4
M2∆
minus 8gMgEM∆ωminus (B26a)
minus2gMgCQ2(MN minus 4M∆)
M∆+
2gEgCQ2(3MN minus 2M∆)
M∆
]
νS2∆-exch
(νQ2) =2πα
MNM2+
[g2EM∆∆ωminus +
g2M MNQ
2+
2+g2C Q
2(Q2 minus∆2)
2M∆(B26b)
+gEgM M∆(M∆ω+ minus 4MNωminus)minus gEgC ∆(2Q2 +MNω+)
+gMgC Q2(4MN minus ω+)
]+S∆-exch
2 (νQ2)
ν
[M2
∆ ω2+
M2N
+ ν2
]
with Q+ =radic
(M∆ +MN )2 +Q2 and ωplusmn = (M2∆minusM2
N plusmnQ2)2M∆ and the non-pole contribution
to S2(νQ2)
S∆-exch2 (νQ2) = minus2παMNν
M∆M2+
[gM + gE
]gC (B27)
These amplitudes to the contrary are not described by the tree-level ∆-production cross sections
in the standard dispersive approach [17] This peculiarity has been previously missed e g in the
calculation of the ∆-exchange contribution to the hydrogen hyperfine splitting in Ref [88] The
importance of including the ∆-non-pole contribution is also evident when considering the BC sum
37
rule in Eq (14) The ∆-pole terms by themselves violate the BC sum rule but cancel exactly with
the ∆-non-pole terms
limνrarr0
νS∆-pole2 (νQ2) + lim
νrarr0νS2
∆-exch(νQ2) = 0 (B28)
Appendix C Polarizabilities at Q2 = 0
In this section we give analytical expressions for the static values and slopes at Q2 = 0 of
the spin polarizabilities and moments of polarized structure functions In particular we give the
HB expansion of the πN -loop contributions and the ∆-exchange contributions The complete
expressions also for the π∆-loop contributions can be found in the Supplemented material Recall
that IA(0) = I1(0) = d2(0) = 0 and dd2(Q2)dQ2
∣∣∣Q2=0
= 0
1 πN-loop contribution
Here we give analytical expressions for the πN -loop contributions to the proton and neutron
spin polarizabilities expanded in powers of micro = mπMN viz the HB expansion Note that we
choose to expand here to a high order in micro the strict HB expansion would only retain the leading
term in an analogous NLO calculation
bull Static polarizabilities (Q2 = 0)
γ0p =e2g2
A
96π3f2πm
2π
1minus 21πmicro
8minus(
59
2+ 26 logmicro
)micro2 +
1875πmicro3
64
+3
(3
2+ 26 logmicro
)micro4 +
(C1)
γ0n =e2g2
A
48π3f2πm
2π
1
2minus 9πmicro
16minus 2micro2 logmicro+
75πmicro3
128minus 3micro4
4+
(C2)
δLTp =e2g2
A
192π3f2πm
2π
1minus 9πmicro
8+
(13
2minus 2 logmicro
)micro2 minus 465πmicro3
64
minus(
47
2+ 42 logmicro
)micro4 +
(C3)
δLTn =e2g2
A
96π3f2πm
2π
1
2+
3πmicro
16+ (1 + 2 log micro)micro2 minus 105πmicro3
128+
5micro4
4+
(C4)
38
δLTn =e2g2
A
96π3f2πm
2π
1
2+
3πmicro
16+ (1 + 2 log micro)micro2 minus 105πmicro3
128+
5micro4
4+
(C5)
γ0p =e2g2
A
16π3f2πm
4π
4
45minus 3πmicro
16+
14micro2
5minus 1813πmicro3
384minus 192
5(1 + log micro)micro4
+80703πmicro5
2048+
(C6)
γ0n =e2g2
A
16π3f2πm
4π
4
45minus 5πmicro
48+
4micro2
5minus 245πmicro3
384minus 32micro4 logmicro
15
+1323πmicro5
2048+
(C7)
bull Slopes of polarizabilities at Q2 = 0
dγ0p(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
2minus 45πmicro
4+ 223micro2 minus 28515πmicro3
64
minus9
(1953
4+ 449 logmicro
)micro4 +
570255πmicro5
128+
(C8)
dγ0n(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
2minus 81πmicro
8+ 94micro2 minus 2535πmicro3
32
minus3 (1 + 90 logmicro)micro4 +84315πmicro5
1024+
(C9)
dδLTp(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
2880π3f2πm
4π
minus5
2minus 27πmicro
32+ 20micro2 minus 5865πmicro3
256
+3
(617
4+ 36 logmicro
)micro4 minus 2056845πmicro5
4096+
(C10)
dδLTn(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
minus5
4minus 81πmicro
64minus 11micro2 +
10005πmicro3
512
+15
8(11 + 48 logmicro)micro4 minus 267015πmicro5
8192+
(C11)
dIAp(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
96π2f2πmicro
2
1minus 15πmicro
4minus 1
2(115 + 88 logmicro)micro2 +
1839πmicro3
32
+5 (5 + 34 logmicro)micro4 +
(C12)
dIAn(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
48π2f2πmicro
2
1
2minus 11πmicro
8minus 1
4(1 + 20 logmicro)micro2 +
99πmicro3
64
minus25micro4
12+
(C13)
dI1p(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
96π2f2πmicro
3π
8+ 2 (4 + 3 logmicro)microminus 537πmicro2
64
minus1
2(15 + 56 logmicro)micro3 +
(C14)
39
dI1n(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
48π2f2πmicro
minus π
16+
1
4(3 + 4 logmicro)microminus 57πmicro2
128+
2micro3
3+
(C15)
γ0p(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2g2
A
16π3f2πm
6π
1
105minus 23πmicro
256+
377micro2
210minus 15551πmicro3
6144+
3371micro4
105
minus1640457πmicro5
32768+
(C16)
γ0n(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2g2
A
16π3f2πm
6π
1
105minus 153πmicro
1792+
69micro2
70minus 4615πmicro3
6144+
172micro4
35
minus120897πmicro5
32768+
(C17)
2 ∆-exchange contribution
Here we give analytical expressions for the tree-level ∆-exchange contributions to the nucleon
spin polarizabilities and their slopes at Q2 = 0 Note that the ∆-exchange contributes equally to
proton and neutron polarizabilities Recall that for the magnetic γlowastN∆ coupling we introduced a
dipole form factor to mimic vector-meson dominance gM rarr gM(1 +Q2Λ2)2
bull Static polarizabilities (Q2 = 0)
γ0 = minus e2
4πM2+
(g2M
∆2+
g2E
M2+
minus 4gMgEM+∆
) (C18)
δLT =e2M∆
4πM3+
(g2E
MNM++gMgE∆MN
minus gEgCM2
∆
) (C19)
γ0 =e2M2
N
π∆2M4+
(minusg
2M
∆2+
g2E
M2+
+4gMgE∆M+
) (C20)
bull Slopes of polarizabilities at Q2 = 0
dγ0(Q2)
dQ2
∣∣∣∣∣Q2=0
= minus e2
πM2+∆
(g2M
∆
[1
4∆2minus 1
∆M++
1
2M2+
]minus 1
Λ2
g2M
∆+
g2E
2M2+
[1
2∆minus 3
M+
]minusgMgE
M+
[1
∆2minus 5
∆M++
1
M2+
]+
1
Λ2
2gMgEM+
+2gMgC∆M2
+
minus gEgCM3
+
) (C21)
40
d δLT (Q2)
dQ2
∣∣∣∣∣Q2=0
=e2M∆∆
4πMNM2+
(g2E
∆2M2+
[1
∆minus 4
M+
]minus g2
C
∆M2∆M
2+
+gMgE∆2M+
[1
∆2
minus 3
∆M++
1
M2+
]minus 2
Λ2
gMgE∆2M+
+gMgC∆M2
∆
[1
2∆2minus 2
∆M++
1
2M2+
]minus gEgC
2M2∆M
2+
[7
∆+
1
M+
]) (C22)
dIA(Q2)
dQ2
∣∣∣∣∣Q2=0
= minusM2N
M2+
(g2M
2∆2+
g2E
MNM+minus 2gMgE
∆M+minus gEgCM∆M+
) (C23)
dI1(Q2)
dQ2
∣∣∣∣∣Q2=0
= minusM∆M2N
2M3+
(g2E
MNM∆minus gMgE∆MN
minus gEgCM2
∆
) (C24)
γ0(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2M2
N
π∆3M6+
(g2M
∆
[3M2
∆ + 2M∆MN minus 9M2N
∆2+
4M2+
Λ2
]
minus4gMgE
[5M2
∆ minus 9M2N
M+∆2+
2M+
Λ2
]+
g2E
∆M+[7M∆ minus 9MN ]
minus8gMgC∆
+4gEgCM+
) (C25)
[1] D Drechsel B Pasquini and M Vanderhaeghen Dispersion relations in real and virtual
Compton scattering Phys Rept 378 99 (2003) hep-ph0212124
[2] S E Kuhn J-P Chen and E Leader Spin structure of the nucleon mdash status and recent
results Prog Part Nucl Phys 63 1 (2009) arXiv08123535 [hep-ph]
[3] F Hagelstein R Miskimen and V Pascalutsa Nucleon polarizabilities from Compton scat-
tering to hydrogen atom Prog Part Nucl Phys 88 29 (2016) arXiv151203765 [nucl-th]
[4] B Pasquini and M Vanderhaeghen Dispersion theory in electromagnetic interactions Ann
Rev Nucl Part Sci 68 75 (2018) arXiv180510482 [hep-ph]
[5] S Gerasimov A Sum rule for magnetic moments and the damping of the nucleon magnetic
moment in nuclei Sov J Nucl Phys 2 430 (1966)
[6] S Drell and A C Hearn Exact sum rule for nucleon magnetic moments Phys Rev Lett
16 908 (1966)
[7] J S Schwinger Source theory viewpoints in deep inelastic scattering Proc Natl Acad Sci
USA 72 1 (1975)
[8] J S Schwinger Source theory viewpoints in deep inelastic scattering Electromagnetic In-
teractions and Field Theory Proceedings 14 Internationale Universitatswochen Schladming
Austria February 24-March 7 1975 Acta Phys Austriaca Suppl 14 471 (1975)
41
[9] J Schwinger Source theory discussion of deep inelastic scattering with polarized particles
Proc Natl Acad Sci USA 72 1559 (1975)
[10] A M Baldin Polarizability of nucleons Nucl Phys 18 310 (1960)
[11] M Gell-Mann M L Goldberger and W E Thirring Use of causality conditions in quantum
theory Phys Rev 95 1612 (1954)
[12] A Pineda Leading chiral logarithms to the hyperfine splitting of the hydrogen and muonic
hydrogen Phys Rev C 67 025201 (2003)
[13] C Peset and A Pineda The two-photon exchange contribution to muonic hydrogen from
chiral perturbation theory Nucl Phys B 887 69 (2014) arXiv14064524 [hep-ph]
[14] C Peset and A Pineda Model-independent determination of the two-photon exchange contri-
bution to hyperfine splitting in muonic hydrogen JHEP 04 060 arXiv161205206 [nucl-th]
[15] F Hagelstein and V Pascalutsa Proton structure in the hyperfine splitting of muonic hydro-
gen PoS CD15 077 (2016) arXiv151104301 [nucl-th]
[16] F Hagelstein Exciting Nucleons in Compton Scattering and Hydrogen-Like Atoms PhD
thesis Mainz U Inst Kernphys (2017) arXiv171000874 [nucl-th]
[17] F Hagelstein ∆(1232)-Resonance in the hydrogen spectrum Proceedings 11th International
Workshop on the Physics of Excited Nucleons (NSTAR 2017) Columbia SC USA August
20-23 2017 Few Body Syst 59 93 (2018) arXiv180109790 [nucl-th]
[18] Y Prok et al (CLAS) Moments of the spin structure functions gp1 and gd1 for 005 lt Q2 lt 30
GeV2 Phys Lett B 672 12 (2009) arXiv08022232 [nucl-ex]
[19] H Dutz et al (GDH) First measurement of the Gerasimov-Drell-Hearn sum rule for 1H from
07 GeV to 18 GeV at ELSA Phys Rev Lett 91 192001 (2003)
[20] M Amarian et al (Jefferson Lab E94010) Measurement of the generalized forward spin po-
larizabilities of the neutron Phys Rev Lett 93 152301 (2004) arXivnucl-ex0406005
[21] M Amarian et al The Q2 evolution of the generalized Gerasimov-Drell-Hearn integral for
the neutron using a 3He target Phys Rev Lett 89 242301 (2002) arXivnucl-ex0205020
[22] M Amarian et al (Jefferson Lab E94-010)Q2 evolution of the neutron spin structure moments
using a 3He target Phys Rev Lett 92 022301 (2004) arXivhep-ex0310003
[23] A Deur et al Experimental determination of the evolution of the Bjorken integral at low Q2
Phys Rev Lett 93 212001 (2004) arXivhep-ex0407007
[24] K Slifer Low Q2 measurement of gp2 and the δLT spin polarizability Spin structure at long dis-
tance Proceedings Workshop Newport News USA March 12ndash13 2009 nucl-ex09064775
(2009) AIP Conf Proc 1155 10106313203293 (2009) arXiv09064775 [nucl-ex]
[25] P Solvignon et al (E01-012) Moments of the neutron g2 structure function at intermediate
42
Q2 Phys Rev C 92 015208 (2015) arXiv13044497 [nucl-ex]
[26] A Deur Experimental studies at low Q2 of the spin structure of the nucleon at Jefferson Lab
in 9th International Workshop on Chiral Dynamics (CD18) Durham NC USA September
17-21 2018 (2019) arXiv190305661 [nucl-ex]
[27] V Sulkosky et al (Jefferson Lab E97-110) Measurement of the 3He spin-structure functions
and of neutron (3He) spin-dependent sum rules at 0035 le Q2 le 024 GeV2 Phys Lett B
805 135428 (2020) arXiv190805709 [nucl-ex]
[28] K Adhikari et al (CLAS) Measurement of the Q2 Dependence of the Deuteron Spin Structure
Function g1 and its Moments at Low Q2 with CLAS Phys Rev Lett 120 062501 (2018)
arXiv171101974 [nucl-ex]
[29] V Lensky F Hagelstein A Hiller Blin and V Pascalutsa Comment on rdquoMeasurement of
the Q2 Dependence of the Deuteron Spin Structure Function g1 and its Moments at Low Q2
with CLASrdquo (2018) arXiv180603219 [nucl-th]
[30] J M Alarcn F Hagelstein V Lensky and V Pascalutsa Forward doubly-virtual Compton
scattering off the nucleon in chiral perturbation theory at NLO the subtraction function and
moments of unpolarized structure functions (2020) arXiv200509518 [hep-ph]
[31] V Bernard E Epelbaum H Krebs and U-G Meiszligner New insights into the spin structure
of the nucleon Phys Rev D 87 054032 (2013) arXiv12092523 [hep-ph]
[32] V Lensky J M Alarcon and V Pascalutsa Moments of nucleon structure functions at
next-to-leading order in baryon chiral perturbation theory Phys Rev C 90 055202 (2014)
arXiv14072574 [hep-ph]
[33] V Pascalutsa and M Vanderhaeghen Chiral effective-field theory in the ∆(1232) region
I Pion electroproduction on the nucleon Phys Rev D 73 034003 (2006) arXivhep-
ph0512244
[34] V Pascalutsa and M Vanderhaeghen Electromagnetic nucleon-to-Delta transition in chiral
effective field theory Phys Rev Lett 95 232001 (2005) arXivhep-ph0508060
[35] T R Hemmert B R Holstein and J Kambor Systematic 1M expansion for spin 32
particles in baryon chiral perturbation theory Phys Lett B 395 89 (1997) arXivhep-
ph9606456
[36] V Pascalutsa and D R Phillips Effective theory of the ∆(1232) in Compton scattering off
the nucleon Phys Rev C 67 055202 (2003) arXivnucl-th0212024
[37] V Pascalutsa M Vanderhaeghen and S N Yang Electromagnetic excitation of the ∆(1232)-
resonance Phys Rept 437 125 (2007) arXivhep-ph0609004
[38] K A Olive et al (Particle Data Group) Review of Particle Physics Chin Phys C 38 090001
43
(2014)
[39] F E Low Scattering of light of very low frequency by systems of spin 12 Phys Rev 96
1428 (1954)
[40] M Gell-Mann and M L Goldberger Scattering of low-energy photons by particles of spin
12 Phys Rev 96 1433 (1954)
[41] A M Harun ar-Rashid A simple derivation of Schwingerrsquos sum rule for spin dependent
structure functions Nuovo Cim A 33 447 (1976)
[42] F Hagelstein and V Pascalutsa Dissecting the hadronic contributions to (g minus 2)micro by
Schwingerrsquos sum rule Phys Rev Lett 120 072002 (2018) arXiv171004571 [hep-ph]
[43] V Lensky V Pascalutsa M Vanderhaeghen and C Kao Spin-dependent sum rules con-
necting real and virtual Compton scattering verified Phys Rev D 95 074001 (2017)
arXiv170101947 [hep-ph]
[44] H Burkhardt and W N Cottingham Sum rules for forward virtual Compton scattering
Annals Phys 56 453 (1970)
[45] V Lensky and V Pascalutsa Predictive powers of chiral perturbation theory in Compton
scattering off protons Eur Phys J C 65 195 (2010) arXiv09070451 [hep-ph]
[46] H Grieszlighammer J McGovern D Phillips and G Feldman Using effective field theory to
analyse low-energy Compton scattering data from protons and light nuclei Prog Part Nucl
Phys 67 841 (2012) arXiv12036834 [nucl-th]
[47] H W Grieszlighammer J A McGovern and D R Phillips Nucleon polarisabilities at and
beyond physical pion masses Eur Phys J A 52 139 (2016) arXiv151101952 [nucl-th]
[48] E Epelbaum H Krebs and U-G Meiszligner Improved chiral nucleon-nucleon potential up to
next-to-next-to-next-to-leading order Eur Phys J A 51 53 (2015) arXiv14120142 [nucl-
th]
[49] E Epelbaum H Krebs and U-G Meiszligner Precision nucleon-nucleon potential at fifth order
in the chiral expansion Phys Rev Lett 115 122301 (2015) arXiv14124623 [nucl-th]
[50] C W Kao T Spitzenberg and M Vanderhaeghen Burkhardt-Cottingham sum rule and
forward spin polarizabilities in heavy baryon chiral perturbation theory Phys Rev D 67
016001 (2003) arXivhep-ph0209241
[51] C-W Kao D Drechsel S Kamalov and M Vanderhaeghen Higher moments of nucleon
spin structure functions in heavy baryon chiral perturbation theory and in a resonance model
Phys Rev D 69 056004 (2004) arXivhep-ph0312102
[52] T Becher and H Leutwyler Baryon chiral perturbation theory in manifestly Lorentz invariant
form Eur Phys J C 9 643 (1999) arXivhep-ph9901384
44
[53] J Gasser M E Sainio and A Svarc Nucleons with chiral loops Nucl Phys B 307 779
(1988)
[54] L S Geng J Martin Camalich L Alvarez-Ruso and M J Vicente Vacas Leading SU(3)-
breaking corrections to the baryon magnetic moments in chiral perturbation theory Phys
Rev Lett 101 222002 (2008) arXiv08051419 [hep-ph]
[55] D Drechsel S S Kamalov and L Tiator The GDH sum rule and related integrals Phys
Rev D 63 114010 (2001) arXivhep-ph0008306
[56] D Drechsel O Hanstein S S Kamalov and L Tiator A unitary isobar model for pion
photo- and electroproduction on the proton up to 1 GeV Nucl Phys A 645 145 (1999)
arXivnucl-th9807001
[57] L Tiator private communication (2020)
[58] V Bernard T R Hemmert and U-G Meiszligner Spin structure of the nucleon at low energies
Phys Rev D 67 076008 (2003) arXivhep-ph0212033
[59] O Gryniuk F Hagelstein and V Pascalutsa Evaluation of the forward Compton scattering
off protons II Spin-dependent amplitude and observables Phys Rev D 94 034043 (2016)
arXiv160400789 [nucl-th]
[60] R Zielinski The g2p Experiment A Measurement of the Protonrsquos Spin Structure Functions
PhD thesis New Hampshire U (2010) arXiv170808297 [nucl-ex]
[61] N Guler et al (CLAS) Precise determination of the deuteron spin structure at low to mod-
erate Q2 with CLAS and extraction of the neutron contribution Phys Rev C 92 055201
(2015) arXiv150507877 [nucl-ex]
[62] C E Carlson V Nazaryan and K Griffioen Proton structure corrections to electronic
and muonic hydrogen hyperfine splitting Phys Rev A 78 022517 (2008) arXiv08052603
[physicsatom-ph]
[63] V Bernard N Kaiser and U-G Meiszligner Chiral dynamics in nucleons and nuclei Int J
Mod Phys E 4 193 (1995) arXivhep-ph9501384
[64] T R Hemmert B R Holstein and J Kambor ∆(1232) and the polarizabilities of the nucleon
Phys Rev D 55 5598 (1997) arXivhep-ph9612374
[65] V Pascalutsa and R Timmermans Field theory of nucleon to higher-spin baryon transitions
Phys Rev C 60 042201 (1999) arXivnucl-th9905065
[66] V Pascalutsa Quantization of an interacting spin-32 field and the Delta isobar Phys Rev
D 58 096002 (1998) arXivhep-ph9802288
[67] H Krebs Double Virtual Compton Scattering and SpinStructure of the Nucleon PoS
CD2018 031 (2019)
45
[68] N Kochelev and Y Oh Axial anomaly and the δLT puzzle Phys Rev D 016012 (2012)
arXiv11034892 [hep-ph]
[69] D Drechsel S Kamalov and L Tiator Unitary isobar model ndash MAID2007 Eur Phys J A
69 (2007) available at httpsmaidkphuni-mainzde arXiv07100306 [nucl-th]
[70] P J Mohr B N Taylor and D B Newell CODATA recommended values of the fundamental
physical constants 2010 Rev Mod Phys 84 1527 (2012)
[71] J Ahrens et al (GDH A2) First measurement of the Gerasimov-Drell-Hearn integral for 1H
from 200 to 800 MeV Phys Rev Lett 87 022003 (2001) arXivhep-ex0105089 [hep-ex]
[72] K Helbing (GDH) Experimental verification of the GDH sum rule at ELSA and MAMI Nucl
Phys Proc Suppl 105 113 (2002)
[73] R Bradford A Bodek H S Budd and J Arrington A New parameterization of the nucleon
elastic form-factors NuInt05 proceedings of the 4th International Workshop on Neutrino-
Nucleus Interactions in the Few-GeV Region Okayama Japan 26-29 September 2005 Nucl
Phys Proc Suppl 159 127 (2006) arXivhep-ex0602017
[74] J D Bjorken Applications of the chiral U(6)timesU(6) algebra of current densities Phys Rev
148 1467 (1966)
[75] J D Bjorken Inelastic scattering of polarized leptons from polarized nucleons Phys Rev D
1 1376 (1970)
[76] V Bernard T R Hemmert and U-G Meiszligner Novel analysis of chiral loop effects in
the generalized Gerasimov-Drell-Hearn sum rule Phys Lett B 545 105 (2002) arXivhep-
ph0203167
[77] A Deur et al Experimental study of isovector spin sum rules Phys Rev D 78 032001
(2008) arXiv08023198 [nucl-ex]
[78] B Pasquini P Pedroni and D Drechsel Higher order forward spin polarizability Phys Lett
B 687 160 (2010) arXiv10014230 [hep-ph]
[79] R Jaffe g2ndashThe nucleonrsquos other spin-dependent structure function Comments Nucl Part
Phys 19 239 (1990)
[80] E V Shuryak and A Vainshtein Theory of power corrections to deep inelastic scattering in
quantum chromodynamics (II) Qminus4 effects polarized target Nucl Phys B 201 141 (1982)
[81] S Wandzura and F Wilczek Sum rules for spin dependent electroproduction Test of rela-
tivistic constituent quarks Phys Lett B 72 195 (1977)
[82] B W Filippone and X-D Ji The spin structure of the nucleon Adv Nucl Phys 26 1
(2001) arXivhep-ph0101224
[83] M Burkardt The g2 structure function Proceedings Workshop on Spin structure at long
46
distance Newport News USA March 12-13 2009 AIP Conf Proc 1155 26 (2009)
arXiv09054079 [hep-ph]
[84] A Deur S J Brodsky and G F de Teramond The spin structure of the nucleon Rept
Prog Phys 82 076201 (2019) arXiv180705250 [hep-ph]
[85] B R Holstein D Drechsel B Pasquini and M Vanderhaeghen Higher order polarizabilities
of the proton Phys Rev C 61 034316 (2000) arXivhep-ph9910427
[86] B R Holstein V Pascalutsa and M Vanderhaeghen Sum rules for magnetic moments and
polarizabilities in QED and chiral effective-field theory Phys Rev D 72 094014 (2005)
arXivhep-ph0507016
[87] B Pasquini D Drechsel and L Tiator Invariant amplitudes for pion electroproduction Eur
Phys J A 34 387 (2007) arXiv07122327 [hep-ph]
[88] A J Buchmann Non-spherical proton shape and hydrogen hyperfine splitting Proceedings
International Workshop on Precision Physics of Simple Atomic Systems (PSAS 2008) Wind-
sor Ontario Canada July 21-26 2008 Can J Phys 87 773 (2009) arXiv09104747
[physicsatom-ph]
47
proton neutron
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Neutron
02 04 06 08 100
50
100
150
200
250
[GeV]
T [b]
02 04 06 08 10
-50
0
50
100
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Neutron
02 04 06 08 100
50
100
150
200
250
[GeV]
T [b]
02 04 06 08 10
-50
0
50
100
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
5
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
08
[GeV]
LTQ
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02040608100
50
100
150
200
250
300
[GeV]
T[b]
0204060810
-50
0
50
[GeV]
TT[b]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
0204060810
00
02
04
06
[GeV]
LTQ
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02040608100
1
2
3
4
[GeV]
LQ2
[GeV-4]
0204060810
00
02
04
06
[GeV]
LTQ
[GeV-3]
Proton
02 04 06 08 100
50
100
150
200
250
300
[GeV]
T [b]
02 04 06 08 10
-50
0
50
[GeV]
TT [b]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 100
1
2
3
4
[GeV]
LQ2
[GeV-4]
02 04 06 08 10
00
02
04
06
[GeV]
LTQ
[GeV-3]
FIG 10 Photoabsorption cross sections for πN (red) and π∆ production (orange) with Q2 = 0
(solid) and Q2 = 01 GeV2 (dashed for πN and dotted for π∆ channel)
1 πN-production channel
In order to extract the response function σLT (νQ2) we have developed a method similar to
the one used to calculate σTT (νQ2) see for example Ref [86] For σLT (νQ2) however the
calculation is more complicated because one has to take into account that the associated Compton
process involves a spin-flip of the nucleon as illustrated in Fig 11 When calculating the cross
section the product of the incoming nucleon spinors has to reflect this flip
The forward VVCS amplitude related to σLT (νQ2) mdash and δLT (Q2) mdash is gLT (νQ2) It can
be extracted from Eq (A2) if one takes the modified polarization vector components in Eq (A3)
with ε = εL and εprimelowast = εlowastplusmn as input where εL = 1Q(|~q | 0 0 q0) and εplusmn = ∓ 1radic
2(0 1plusmni 0) are the
standard longitudinal and transverse polarization vectors respectively For εL and εlowastplusmn only the
choice of helicities hprime = plusmn12 and h = ∓12 gives a non-zero contribution and one obtains
χdaggerhprime T (νQ2)χh = χdaggerhprimeminusi~σ middot [(ε0~εprimelowast minus ~ε ε primelowast0 )times q] gLT (νQ2)χh =
radic2 gLT (νQ2) (B1)
where χh and χdaggerhprime are two-component Pauli spinors with opposite helicities or here spins
Let us now consider the related photoabsorption process and in particular the tree-level γlowastN rarr
30
s = 12
sprime = minus12
λ = 0 λprime = 1
rprime rprime
s=12sprime=minus1
2
λ=0λprime=1
rprimerprime
s=12
sprime=minus12
λ=0λprime=1
rprimerprime hacute=frac12h=-frac12
λacute=1λ=0
FIG 11 Relation between the forward Compton process and the photoabsorption process given
by the optical theorem In particular we show the longitudinal-transverse contribution The
double-line arrows represent the spin of the external particles while the dot represents the scalar
(longitudinal) polarization of the incoming photon Inside the blob the intermediate states are
represented eg nucleons with spins rprime (which are averaged in the calculation of the cross section)
and pions
πN channel see diagrams in Fig 8 of Ref [30] We define the πN -production amplitude as
T = uhB (PB)sumi
Ai(s t)Γi uhA(PA) (B2)
with the Dirac structures
Γ1 = γ5 (B3a)
Γ2 =1
2
[qA ε
]γ5 (B3b)
where uhA(PA) and udaggerhB (PB) are the Dirac spinors and PA and PB are the four-momenta of
the incoming and outgoing nucleons respectively When calculating the photoabsorption cross
section related to the VVCS amplitude in Eq (B1) the nucleon spin flip should be implemented
by uhprime(PA) in T dagger and uh(PA) in T together with the appropriate transverse and longitudinal
photon polarization vectors εlowastplusmn and εL
However if one wants to use the properties of the Dirac matrices it is more useful to construct
an operator to produce this spin flip in the external nucleons of Fig 11 This is accomplished by
introducing the projector ΓLT equiv 12radic
2(γ1 + iγ2)γ5 which also takes into account the extra factor
radic2 in Eq (B1) We checked that with this projector one correctly extracts δLT by comparing the
HB limit of our result to the HB result of Ref [50] where the authors calculate this polarizability
from the Compton amplitude directly With all those ingredients the longitudinal-transverse cross
section is calculated in the following way
σLT (νQ2) =1
64π2 s
|~pf |cm
|~pi|cm
int 1
minus1dcos θ
sumij
AiAdaggerjXij (B4)
31
with
Xij = Tr[(PB +MN )Γi(PA +MN )ΓLTγ0Γdaggerjγ
0] (B5)
where θ is the scattering angle in the center-of-mass (cm) frame and |~pi|cm (|~pf |cm) is the three-
momentum of an incoming (outgoing) particle in the cm frame An explicit calculation of the
matrix Xij leads to
X = MNQ
0 2(PB minus PA) middot εLminusradic
2|~qf |cm sin θ (sminus u)
(B6)
where |~qi|cm (|~qf |cm) is the relative three-momentum of the incoming (outgoing) particles in the
cm frame Here s t and u are the usual Mandelstam variables For the different γlowastN rarr πN
channels we obtain the following amplitudes Ai where we introduce qA as the four-momentum of
the incoming photon and qB as the four-momentum of the outgoing pion
bull γlowastprarr π0p
A1 =e gAMN
fπ
[2PA middot ε+ qA middot ε
sminusM2N
+2PB middot εminus qA middot ε
uminusM2N
] (B7a)
A2 =e gAMN
fπ
[1
sminusM2N
+1
uminusM2N
] (B7b)
bull γlowastprarr π+n
A1 =
radic2 e gAMN
fπ
[2PA middot ε+ qA middot ε
sminusM2N
+2(PA minus PB) middot ε+ qA middot ε
tminusm2π
] (B8a)
A2 =
radic2 e gAMN
fπ(sminusM2N )
(B8b)
bull γlowastnrarr π0n
A1 = 0 (B9a)
A2 = 0 (B9b)
bull γlowastnrarr πminusp
A1 =
radic2 e gAMN
fπ
[2PB middot εminus qA middot ε
uminusM2N
minus 2(PA minus PB) middot ε+ qA middot εtminusm2
π
] (B10a)
A2 =
radic2 e gAMN
fπ(uminusM2N )
(B10b)
The analytical expressions shown above were checked with the amplitudes given in Ref [87] An-
alytical expressions for the tree-level γlowastN rarr πN channel of the σLT (νQ2) and σTT (νQ2) cross
32
sections are given below (proton channels π+n and π0p neutron channel πminusp) We checked that
they reproduce the known results in the real-photon limit [45 86] To shorten the final expressions
for the cross sections which are considerably longer for finite Q2 than in the real-photon limit we
define the following dimensionless kinematic variables
αγ = (ENi )cmradics =
s+M2N +Q2
2s (B11)
απ = (ENf )cmradics =
s+M2N minusm2
π
2s (B12)
βγ = Eγcmradics =
sminusM2N minusQ2
2s (B13)
βπ = Eπcmradics =
sminusM2N +m2
π
2s (B14)
λγ = |~qi|cmradics =
radic(sminusM2
N minusQ2)2 + 4sQ2
2s (B15)
λπ = |~qf |cmradics =
radic(sminusM2
N +m2π)2 minus 4sm2
π
2s (B16)
Here (ENi )cm and (ENf )cm are the energies of the incoming and outgoing nucleon Eγcm is the energy
of the incoming photon Eπcm is the energy of the outgoing pion all in the cm frame
σ(π+n)TT = minus e2g2
AM2N
64πf2πs
3(sminusM2N )2λ4
γ
4sλπλγ
[(M2
N minus s)(M2N minusQ2 minus s)(Q2 + 2sβγβπ)
+ 2s(m2πQ
2 minus (M2N minus s)
(M2N + s(minus1 + 2βγβπ)
))λ2γ
]+ (M2
N minus s)(M2N minusQ2 minus s)
times (Q2 + 2sβγβπ minus 2sλπλγ)(Q2 + 2s(βγβπ + λπλγ)
)times log
(Q2 + 2sβγβπ minus 2sλγλπQ2 + 2sβγβπ + 2sλγλπ
) (B17)
33
σ(π0p)TT =
e2g2AM
2Nλπ
64πf2πs(sminusM2
N )2λγ
1
s2λπλ3γ
[(M2
N minus s)(M2N minusQ2 minus s)
(3M2
N + s(2βγβπ minus 3))(M2N
+ s(2βγβπ minus 1))
+ 2s
((2m2
πQ2 + (M2
N minus s)2)(M2
N minus s)minus 2(minusm2
πQ2 + (sminusM2
N )2)
times sβγβπ + 4s2(sminusM2N )β2
γβ2π
)λ2γ
]arctan
(2sλπλγ
M2N + s(2βγβπ minus 1)
)+ 2
[minus 2m2
πQ2
+ (M2N minus s)
(2(sminusM2
N ) + 4sβγβπ +(sminusM2
N +Q2)(3M2N minus 3s+ 2sβγβπ)
sλ2γ
+m2πQ
2(sminusM2N )
(M2N minus s+ 2sβγβπ)(M2
N + s(minus1 + 2βγβπ minus 2λπλγ))
+m2πQ
2(sminusM2N )
(M2N minus s+ 2sβγβπ)(M2
N + s(minus1 + 2βγβπ + 2λπλγ))
)
+1
λπλγ
(βγβπ
(minusm2
πQ2 + (M2
N minus s)(M2N minus s+ 2sβγβπ)
)minus (M2
N minus s)(M2N minusQ2 minus s)λ2
π
)log
(M2N + s(minus1 + 2βγβπ + 2λπλγ)
M2N + s(minus1 + 2βγβπ minus 2λπλγ)
)] (B18)
σ(πminusp)TT =
e2g2AM
2N
64πf2πs
3λ4γ(M2
N + s(2βπβγ minus 1))2 minus 4λ2πs
2λ2γ)
4sλπλγ
((minusM2
N +Q2 + s)(M2N
+ s(2βπβγ minus 1))
2 minus 2sλ2γ
(2λ2
πs(minusM2N +Q2 + s) +m2
πQ2))
+((M2N + s(2βπβγ minus 1)
)2
minus 4λ2πs
2λ2γ
)((M2N + s(2βπβγ minus 1)
)(M2N minus 2Q2 minus s(2βπβγ + 1)
)+ 4λ2
πs2λ2
γ
)times log
(minusM2
N minus 2βπsβγ minus 2λπsλγ + s)minus(M2N + s(2βπβγ minus 2λπλγ minus 1)
)(M2N + s(minus1
+ 2βπβγ + 2λπλγ))[((
M2N + s(2βπβγ minus 1)
)(M2N minus 2Q2 minus s(2βπβγ + 1)
)+ 4λ2
πs2λ2
γ
)times log
(minusM2
N minus 2βπsβγ + 2λπsλγ + s)
+((Q2 + 2βπsβγ)2 minus 4λ2
πs2λ2
γ
)times log
(Q2 + 2βπsβγ minus 2λπsλγQ2 + 2sβπβγ + 2λπsλγ
)]+ 4sλ2
γ
((M2
N + s(2βπβγ minus 1))2 minus 4λ2πs
2λ2γ
)times (M2
N minus s) arctanh
(2λπsλγ
M2N + 2βπsβγ minus s
) (B19)
34
σ(π+n)LT =
e2g2AM
3Nλπ
32πf2πQs
3(sminusM2N )2λ4
γ
2sλγ
[(M2
N minus s)(Q2 + 2sβ2γ)(Q2 + 2sβγβπ)
minus 4s
((M2
N minus s)(Q2 minus 2s(απ minus 1)βγ
)+Q2sβγβπ
)λ2γ + 8s3(minus1 + απ)λ4
γ
]minus M2
N minus sλπ
[(Q2 + 2sβ2
γ)(Q2 + 2sβγβπ)2 + 4s2(2(απ minus 1)βγ(Q2 + 2sβγβπ)
minusQ2λ2π
)λ2γ + 8s3(απ minus 1)2λ4
γ
]arctan
(2sλπλγ
Q2 + 2sβγβπ
) (B20)
σ(π0p)LT =
e2g2AM
3Nλπ
16πf2πQs(sminusM2
N )2λγ
1
minus2s(M2N + s(minus1 + 2βγβπ))2λ2
γ + 8s3λ2πλ
4γ
[minus 3M8
N (Q2
+ 2sβ2γ) + 2M4
Ns2
(minus (Q2 + 2sβ2
γ)(minus3 + 2βγβπ) +(Q2(minus3 + 2βγβπ + 6λ2
π) + 2sβγ
times (minus3 + 12απ + 2βγβπ minus 12απβγβπ + 4βγλ2π))λ2γ minus 4s(α2
π minus 1)λ4π
)+ 2M2
Ns3
[minus (Q2
+ 2sβ2γ)(minus1 + 2βγβπ)
(6 + βγβπ(minus9 + 2βγβπ)
)+
(Q2(3minus 12λ2
π + 4βγβπ(minus1 + 2βγβπ
+ λ2π))minus 2sβγ
(minus 3 + 4απ(3 + 2απ(3 + 2βγβπ(minus3 + βγβπ)) + 4βγ(βπ + (2minus βγβπ)
times λ2π)))λ2γ + 8s
((απ minus 1)(1 + απ minus 2βγβπ) + 2απβγλ
2π
)λ4γ
)+ s4
((Q2 + 2sβ2
γ)
times (1minus 2βγβπ)2(2βγβπ minus 3) + 2(Q2(minus1 + 6λ2
π + 2βγβπ((1minus 2βγβπ)2 minus 2λ2π)) + 2sβγ
times (minus1 + 2βγβπ + 4(minus1 + βγβπ)(minusαπ + 2απβγβπ minus βγλ2π)))λ2γ minus 8
(s(απ minus 1)(απ
+ (1minus 2βγβπ)2) + 2βγ(2sαπ +Q2βγ)λ2π
)λ4γ + 32s(απ minus 1)λ2
πλ6γ
)+ 2M6
Ns
(Q2(6
minus 7βγβπ + λ2γ) + 2sβγ(βγ(6minus 7βγβπ) + (1minus 4απ)λ2
γ)
)]+sminusM2
N
4s2λπλ3γ
[(Q2 + 2sβ2
γ)
times(3M2
N + s(2βγβπ minus 3))(M2N + s(2βγβπ minus 1)
)+ 2s
(minusM2
N (Q2 + 2sβγ minus 8sαπβγ)
+ s(2sβγ(1minus 4απ + 4απβγβπ) +Q2(1minus 2λ2π)))λ2γ + 8s3(απ minus 1)λ4
γ
]
times arctan
(2sλπλγ
M2N + s(2βγβπ minus 1)
) (B21)
35
σ(πminusp)LT =
e2g2AM
3N
64πf2πQs
3λ4γ(M2
N minusQ2 minus s)(M2N + s(2βπβγ + 2λπλγ minus 1))
times
1
M2N + s(2βπβγ minus 2λπλγ minus 1)
[4λπsλγ(M2
N minusQ2 minus s)(
2sλ2γ
(s((1minus 2λ2
π)Q2
+ 2(2απ minus 1)sβγ(2βπβγ minus 1))minusM2N (Q2 + 2(1minus 2απ)sβγ)
)+ (Q2 + 2sβ2
γ)(M2N + s(2βπβγ minus 1)
)2 + 8(απ minus 1)απs
3λ4γ
)]minus(M2N + s(2βπβγ + 2λπλγ minus 1)
)times[
log
(minusM2N minus 2βπsβγ minus 2λπsλγ + s
minusM2N minus 2βπsβγ + 2λπsλγ + s
)(minus (Q2 + 2sβ2
γ)(M2N + s(2βπβγ minus 1)
)(minusM2
N + 2Q2 + 2βπsβγ + s)minus 2sλ2γ
(minusM2
N (Q2 + 2sβγ) +Q4 +Q2s(4απβγ
minus 2βγ minus 2λ2π + 1) + 2s2βγ(4(απ minus 1)βπβγ + 1)
)minus 8(απ minus 1)2s3λ4
γ
)+ 2
(4s2λ2
γ
(2(απ minus 1)βγ(Q2 + 2βπsβγ)minus λ2
πQ2)
+ (Q2 + 2βπsβγ)2(Q2 + 2sβ2γ)
+ 8(απ minus 1)2s3λ4γ
)arctanh
(2λπsλγ
Q2 + 2βπsβγ
)] (B22)
2 ∆-production channel
The tree-level ∆-exchange diagram in Fig 2 of Ref [30] contributes to the non-Born part of
the VVCS amplitudes The contribution of the ∆ exchange to the VVCS amplitudes can be split
into [17]
S∆-exch1 (νQ2) = S∆-pole
1 (νQ2) + S∆-exch1 (νQ2) (B23a)
νS∆-exch2 (νQ2) = νS∆-pole
2 (νQ2) + νS2∆-exch
(νQ2) (B23b)
and similarly for the unpolarized VVCS amplitudes discussed in Ref [30] Here we introduced the
∆-pole contributions S∆-polei and the ∆-non-pole contributions S∆-exch
i The former amplitudes
feature a pole at the ∆(1232)-production threshold and thus are proportional to
1
[sminusM2∆][uminusM2
∆]=
1
4M2N
1
ν2∆ minus ν2
(B24)
36
They can be reconstructed from the dispersion relations in Eq (5) with the tree-level ∆-production
cross sections as input cf Fig 10 in Ref [30]
σTT (νQ2) =π2α
M2NM
2+|~q |
minus g2
MMN (M+ + ν)|~q |2 +g2E(∆minus ν)(Q2 minusMNν)2
MN(B25a)
+g2CQ
4s(∆minus ν)
MNM2∆
minus 4gMgE(Q2 minusMNν)|~q |2 minus 4gMgCQ2|~q |2
+2gEgCQ
2[minusMNM∆ |~q |2 + s(Q2 +∆ν)
]MNM∆
δ(ν minus ν∆)
σLT (νQ2) =Qπ2α
M2NM
2+|~q |
g2E(MNν minusQ2) [M∆(MN + ν)minus s]
MN(B25b)
+g2CQ
2[MNM∆|~q |2 minus s(Q2 +∆ν)
]MNM2
∆
+ gMgEM∆|~q |2 minusgMgC(Q2 minusMNν)|~q |2
M∆
+gEgC(ν minus∆)(M2
N |~q |2 minus 2Q2s)
MNM∆
δ(ν minus ν∆)
with ∆ = M∆ minusMN M+ = M∆ + MN and the Mandelstam variable s = M2N + 2MNν minus Q2
Analytical expressions for the spin structure functions g1(xQ2) and g2(xQ2) can be constructed
from Eq (3) with the flux factor K(νQ2) = |~q | =radicν2 +Q2
In the ∆-non-pole contributions to S1(νQ2) and νS2(νQ2) the pole in ν at the ∆(1232)-
production threshold has canceled out
S∆-exch1 (νQ2) =
πα
MNM2+
[g2MQ
2+ + g2
E
(∆2 minus 3Q2
)+
4g2CQ
4
M2∆
minus 8gMgEM∆ωminus (B26a)
minus2gMgCQ2(MN minus 4M∆)
M∆+
2gEgCQ2(3MN minus 2M∆)
M∆
]
νS2∆-exch
(νQ2) =2πα
MNM2+
[g2EM∆∆ωminus +
g2M MNQ
2+
2+g2C Q
2(Q2 minus∆2)
2M∆(B26b)
+gEgM M∆(M∆ω+ minus 4MNωminus)minus gEgC ∆(2Q2 +MNω+)
+gMgC Q2(4MN minus ω+)
]+S∆-exch
2 (νQ2)
ν
[M2
∆ ω2+
M2N
+ ν2
]
with Q+ =radic
(M∆ +MN )2 +Q2 and ωplusmn = (M2∆minusM2
N plusmnQ2)2M∆ and the non-pole contribution
to S2(νQ2)
S∆-exch2 (νQ2) = minus2παMNν
M∆M2+
[gM + gE
]gC (B27)
These amplitudes to the contrary are not described by the tree-level ∆-production cross sections
in the standard dispersive approach [17] This peculiarity has been previously missed e g in the
calculation of the ∆-exchange contribution to the hydrogen hyperfine splitting in Ref [88] The
importance of including the ∆-non-pole contribution is also evident when considering the BC sum
37
rule in Eq (14) The ∆-pole terms by themselves violate the BC sum rule but cancel exactly with
the ∆-non-pole terms
limνrarr0
νS∆-pole2 (νQ2) + lim
νrarr0νS2
∆-exch(νQ2) = 0 (B28)
Appendix C Polarizabilities at Q2 = 0
In this section we give analytical expressions for the static values and slopes at Q2 = 0 of
the spin polarizabilities and moments of polarized structure functions In particular we give the
HB expansion of the πN -loop contributions and the ∆-exchange contributions The complete
expressions also for the π∆-loop contributions can be found in the Supplemented material Recall
that IA(0) = I1(0) = d2(0) = 0 and dd2(Q2)dQ2
∣∣∣Q2=0
= 0
1 πN-loop contribution
Here we give analytical expressions for the πN -loop contributions to the proton and neutron
spin polarizabilities expanded in powers of micro = mπMN viz the HB expansion Note that we
choose to expand here to a high order in micro the strict HB expansion would only retain the leading
term in an analogous NLO calculation
bull Static polarizabilities (Q2 = 0)
γ0p =e2g2
A
96π3f2πm
2π
1minus 21πmicro
8minus(
59
2+ 26 logmicro
)micro2 +
1875πmicro3
64
+3
(3
2+ 26 logmicro
)micro4 +
(C1)
γ0n =e2g2
A
48π3f2πm
2π
1
2minus 9πmicro
16minus 2micro2 logmicro+
75πmicro3
128minus 3micro4
4+
(C2)
δLTp =e2g2
A
192π3f2πm
2π
1minus 9πmicro
8+
(13
2minus 2 logmicro
)micro2 minus 465πmicro3
64
minus(
47
2+ 42 logmicro
)micro4 +
(C3)
δLTn =e2g2
A
96π3f2πm
2π
1
2+
3πmicro
16+ (1 + 2 log micro)micro2 minus 105πmicro3
128+
5micro4
4+
(C4)
38
δLTn =e2g2
A
96π3f2πm
2π
1
2+
3πmicro
16+ (1 + 2 log micro)micro2 minus 105πmicro3
128+
5micro4
4+
(C5)
γ0p =e2g2
A
16π3f2πm
4π
4
45minus 3πmicro
16+
14micro2
5minus 1813πmicro3
384minus 192
5(1 + log micro)micro4
+80703πmicro5
2048+
(C6)
γ0n =e2g2
A
16π3f2πm
4π
4
45minus 5πmicro
48+
4micro2
5minus 245πmicro3
384minus 32micro4 logmicro
15
+1323πmicro5
2048+
(C7)
bull Slopes of polarizabilities at Q2 = 0
dγ0p(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
2minus 45πmicro
4+ 223micro2 minus 28515πmicro3
64
minus9
(1953
4+ 449 logmicro
)micro4 +
570255πmicro5
128+
(C8)
dγ0n(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
2minus 81πmicro
8+ 94micro2 minus 2535πmicro3
32
minus3 (1 + 90 logmicro)micro4 +84315πmicro5
1024+
(C9)
dδLTp(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
2880π3f2πm
4π
minus5
2minus 27πmicro
32+ 20micro2 minus 5865πmicro3
256
+3
(617
4+ 36 logmicro
)micro4 minus 2056845πmicro5
4096+
(C10)
dδLTn(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
minus5
4minus 81πmicro
64minus 11micro2 +
10005πmicro3
512
+15
8(11 + 48 logmicro)micro4 minus 267015πmicro5
8192+
(C11)
dIAp(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
96π2f2πmicro
2
1minus 15πmicro
4minus 1
2(115 + 88 logmicro)micro2 +
1839πmicro3
32
+5 (5 + 34 logmicro)micro4 +
(C12)
dIAn(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
48π2f2πmicro
2
1
2minus 11πmicro
8minus 1
4(1 + 20 logmicro)micro2 +
99πmicro3
64
minus25micro4
12+
(C13)
dI1p(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
96π2f2πmicro
3π
8+ 2 (4 + 3 logmicro)microminus 537πmicro2
64
minus1
2(15 + 56 logmicro)micro3 +
(C14)
39
dI1n(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
48π2f2πmicro
minus π
16+
1
4(3 + 4 logmicro)microminus 57πmicro2
128+
2micro3
3+
(C15)
γ0p(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2g2
A
16π3f2πm
6π
1
105minus 23πmicro
256+
377micro2
210minus 15551πmicro3
6144+
3371micro4
105
minus1640457πmicro5
32768+
(C16)
γ0n(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2g2
A
16π3f2πm
6π
1
105minus 153πmicro
1792+
69micro2
70minus 4615πmicro3
6144+
172micro4
35
minus120897πmicro5
32768+
(C17)
2 ∆-exchange contribution
Here we give analytical expressions for the tree-level ∆-exchange contributions to the nucleon
spin polarizabilities and their slopes at Q2 = 0 Note that the ∆-exchange contributes equally to
proton and neutron polarizabilities Recall that for the magnetic γlowastN∆ coupling we introduced a
dipole form factor to mimic vector-meson dominance gM rarr gM(1 +Q2Λ2)2
bull Static polarizabilities (Q2 = 0)
γ0 = minus e2
4πM2+
(g2M
∆2+
g2E
M2+
minus 4gMgEM+∆
) (C18)
δLT =e2M∆
4πM3+
(g2E
MNM++gMgE∆MN
minus gEgCM2
∆
) (C19)
γ0 =e2M2
N
π∆2M4+
(minusg
2M
∆2+
g2E
M2+
+4gMgE∆M+
) (C20)
bull Slopes of polarizabilities at Q2 = 0
dγ0(Q2)
dQ2
∣∣∣∣∣Q2=0
= minus e2
πM2+∆
(g2M
∆
[1
4∆2minus 1
∆M++
1
2M2+
]minus 1
Λ2
g2M
∆+
g2E
2M2+
[1
2∆minus 3
M+
]minusgMgE
M+
[1
∆2minus 5
∆M++
1
M2+
]+
1
Λ2
2gMgEM+
+2gMgC∆M2
+
minus gEgCM3
+
) (C21)
40
d δLT (Q2)
dQ2
∣∣∣∣∣Q2=0
=e2M∆∆
4πMNM2+
(g2E
∆2M2+
[1
∆minus 4
M+
]minus g2
C
∆M2∆M
2+
+gMgE∆2M+
[1
∆2
minus 3
∆M++
1
M2+
]minus 2
Λ2
gMgE∆2M+
+gMgC∆M2
∆
[1
2∆2minus 2
∆M++
1
2M2+
]minus gEgC
2M2∆M
2+
[7
∆+
1
M+
]) (C22)
dIA(Q2)
dQ2
∣∣∣∣∣Q2=0
= minusM2N
M2+
(g2M
2∆2+
g2E
MNM+minus 2gMgE
∆M+minus gEgCM∆M+
) (C23)
dI1(Q2)
dQ2
∣∣∣∣∣Q2=0
= minusM∆M2N
2M3+
(g2E
MNM∆minus gMgE∆MN
minus gEgCM2
∆
) (C24)
γ0(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2M2
N
π∆3M6+
(g2M
∆
[3M2
∆ + 2M∆MN minus 9M2N
∆2+
4M2+
Λ2
]
minus4gMgE
[5M2
∆ minus 9M2N
M+∆2+
2M+
Λ2
]+
g2E
∆M+[7M∆ minus 9MN ]
minus8gMgC∆
+4gEgCM+
) (C25)
[1] D Drechsel B Pasquini and M Vanderhaeghen Dispersion relations in real and virtual
Compton scattering Phys Rept 378 99 (2003) hep-ph0212124
[2] S E Kuhn J-P Chen and E Leader Spin structure of the nucleon mdash status and recent
results Prog Part Nucl Phys 63 1 (2009) arXiv08123535 [hep-ph]
[3] F Hagelstein R Miskimen and V Pascalutsa Nucleon polarizabilities from Compton scat-
tering to hydrogen atom Prog Part Nucl Phys 88 29 (2016) arXiv151203765 [nucl-th]
[4] B Pasquini and M Vanderhaeghen Dispersion theory in electromagnetic interactions Ann
Rev Nucl Part Sci 68 75 (2018) arXiv180510482 [hep-ph]
[5] S Gerasimov A Sum rule for magnetic moments and the damping of the nucleon magnetic
moment in nuclei Sov J Nucl Phys 2 430 (1966)
[6] S Drell and A C Hearn Exact sum rule for nucleon magnetic moments Phys Rev Lett
16 908 (1966)
[7] J S Schwinger Source theory viewpoints in deep inelastic scattering Proc Natl Acad Sci
USA 72 1 (1975)
[8] J S Schwinger Source theory viewpoints in deep inelastic scattering Electromagnetic In-
teractions and Field Theory Proceedings 14 Internationale Universitatswochen Schladming
Austria February 24-March 7 1975 Acta Phys Austriaca Suppl 14 471 (1975)
41
[9] J Schwinger Source theory discussion of deep inelastic scattering with polarized particles
Proc Natl Acad Sci USA 72 1559 (1975)
[10] A M Baldin Polarizability of nucleons Nucl Phys 18 310 (1960)
[11] M Gell-Mann M L Goldberger and W E Thirring Use of causality conditions in quantum
theory Phys Rev 95 1612 (1954)
[12] A Pineda Leading chiral logarithms to the hyperfine splitting of the hydrogen and muonic
hydrogen Phys Rev C 67 025201 (2003)
[13] C Peset and A Pineda The two-photon exchange contribution to muonic hydrogen from
chiral perturbation theory Nucl Phys B 887 69 (2014) arXiv14064524 [hep-ph]
[14] C Peset and A Pineda Model-independent determination of the two-photon exchange contri-
bution to hyperfine splitting in muonic hydrogen JHEP 04 060 arXiv161205206 [nucl-th]
[15] F Hagelstein and V Pascalutsa Proton structure in the hyperfine splitting of muonic hydro-
gen PoS CD15 077 (2016) arXiv151104301 [nucl-th]
[16] F Hagelstein Exciting Nucleons in Compton Scattering and Hydrogen-Like Atoms PhD
thesis Mainz U Inst Kernphys (2017) arXiv171000874 [nucl-th]
[17] F Hagelstein ∆(1232)-Resonance in the hydrogen spectrum Proceedings 11th International
Workshop on the Physics of Excited Nucleons (NSTAR 2017) Columbia SC USA August
20-23 2017 Few Body Syst 59 93 (2018) arXiv180109790 [nucl-th]
[18] Y Prok et al (CLAS) Moments of the spin structure functions gp1 and gd1 for 005 lt Q2 lt 30
GeV2 Phys Lett B 672 12 (2009) arXiv08022232 [nucl-ex]
[19] H Dutz et al (GDH) First measurement of the Gerasimov-Drell-Hearn sum rule for 1H from
07 GeV to 18 GeV at ELSA Phys Rev Lett 91 192001 (2003)
[20] M Amarian et al (Jefferson Lab E94010) Measurement of the generalized forward spin po-
larizabilities of the neutron Phys Rev Lett 93 152301 (2004) arXivnucl-ex0406005
[21] M Amarian et al The Q2 evolution of the generalized Gerasimov-Drell-Hearn integral for
the neutron using a 3He target Phys Rev Lett 89 242301 (2002) arXivnucl-ex0205020
[22] M Amarian et al (Jefferson Lab E94-010)Q2 evolution of the neutron spin structure moments
using a 3He target Phys Rev Lett 92 022301 (2004) arXivhep-ex0310003
[23] A Deur et al Experimental determination of the evolution of the Bjorken integral at low Q2
Phys Rev Lett 93 212001 (2004) arXivhep-ex0407007
[24] K Slifer Low Q2 measurement of gp2 and the δLT spin polarizability Spin structure at long dis-
tance Proceedings Workshop Newport News USA March 12ndash13 2009 nucl-ex09064775
(2009) AIP Conf Proc 1155 10106313203293 (2009) arXiv09064775 [nucl-ex]
[25] P Solvignon et al (E01-012) Moments of the neutron g2 structure function at intermediate
42
Q2 Phys Rev C 92 015208 (2015) arXiv13044497 [nucl-ex]
[26] A Deur Experimental studies at low Q2 of the spin structure of the nucleon at Jefferson Lab
in 9th International Workshop on Chiral Dynamics (CD18) Durham NC USA September
17-21 2018 (2019) arXiv190305661 [nucl-ex]
[27] V Sulkosky et al (Jefferson Lab E97-110) Measurement of the 3He spin-structure functions
and of neutron (3He) spin-dependent sum rules at 0035 le Q2 le 024 GeV2 Phys Lett B
805 135428 (2020) arXiv190805709 [nucl-ex]
[28] K Adhikari et al (CLAS) Measurement of the Q2 Dependence of the Deuteron Spin Structure
Function g1 and its Moments at Low Q2 with CLAS Phys Rev Lett 120 062501 (2018)
arXiv171101974 [nucl-ex]
[29] V Lensky F Hagelstein A Hiller Blin and V Pascalutsa Comment on rdquoMeasurement of
the Q2 Dependence of the Deuteron Spin Structure Function g1 and its Moments at Low Q2
with CLASrdquo (2018) arXiv180603219 [nucl-th]
[30] J M Alarcn F Hagelstein V Lensky and V Pascalutsa Forward doubly-virtual Compton
scattering off the nucleon in chiral perturbation theory at NLO the subtraction function and
moments of unpolarized structure functions (2020) arXiv200509518 [hep-ph]
[31] V Bernard E Epelbaum H Krebs and U-G Meiszligner New insights into the spin structure
of the nucleon Phys Rev D 87 054032 (2013) arXiv12092523 [hep-ph]
[32] V Lensky J M Alarcon and V Pascalutsa Moments of nucleon structure functions at
next-to-leading order in baryon chiral perturbation theory Phys Rev C 90 055202 (2014)
arXiv14072574 [hep-ph]
[33] V Pascalutsa and M Vanderhaeghen Chiral effective-field theory in the ∆(1232) region
I Pion electroproduction on the nucleon Phys Rev D 73 034003 (2006) arXivhep-
ph0512244
[34] V Pascalutsa and M Vanderhaeghen Electromagnetic nucleon-to-Delta transition in chiral
effective field theory Phys Rev Lett 95 232001 (2005) arXivhep-ph0508060
[35] T R Hemmert B R Holstein and J Kambor Systematic 1M expansion for spin 32
particles in baryon chiral perturbation theory Phys Lett B 395 89 (1997) arXivhep-
ph9606456
[36] V Pascalutsa and D R Phillips Effective theory of the ∆(1232) in Compton scattering off
the nucleon Phys Rev C 67 055202 (2003) arXivnucl-th0212024
[37] V Pascalutsa M Vanderhaeghen and S N Yang Electromagnetic excitation of the ∆(1232)-
resonance Phys Rept 437 125 (2007) arXivhep-ph0609004
[38] K A Olive et al (Particle Data Group) Review of Particle Physics Chin Phys C 38 090001
43
(2014)
[39] F E Low Scattering of light of very low frequency by systems of spin 12 Phys Rev 96
1428 (1954)
[40] M Gell-Mann and M L Goldberger Scattering of low-energy photons by particles of spin
12 Phys Rev 96 1433 (1954)
[41] A M Harun ar-Rashid A simple derivation of Schwingerrsquos sum rule for spin dependent
structure functions Nuovo Cim A 33 447 (1976)
[42] F Hagelstein and V Pascalutsa Dissecting the hadronic contributions to (g minus 2)micro by
Schwingerrsquos sum rule Phys Rev Lett 120 072002 (2018) arXiv171004571 [hep-ph]
[43] V Lensky V Pascalutsa M Vanderhaeghen and C Kao Spin-dependent sum rules con-
necting real and virtual Compton scattering verified Phys Rev D 95 074001 (2017)
arXiv170101947 [hep-ph]
[44] H Burkhardt and W N Cottingham Sum rules for forward virtual Compton scattering
Annals Phys 56 453 (1970)
[45] V Lensky and V Pascalutsa Predictive powers of chiral perturbation theory in Compton
scattering off protons Eur Phys J C 65 195 (2010) arXiv09070451 [hep-ph]
[46] H Grieszlighammer J McGovern D Phillips and G Feldman Using effective field theory to
analyse low-energy Compton scattering data from protons and light nuclei Prog Part Nucl
Phys 67 841 (2012) arXiv12036834 [nucl-th]
[47] H W Grieszlighammer J A McGovern and D R Phillips Nucleon polarisabilities at and
beyond physical pion masses Eur Phys J A 52 139 (2016) arXiv151101952 [nucl-th]
[48] E Epelbaum H Krebs and U-G Meiszligner Improved chiral nucleon-nucleon potential up to
next-to-next-to-next-to-leading order Eur Phys J A 51 53 (2015) arXiv14120142 [nucl-
th]
[49] E Epelbaum H Krebs and U-G Meiszligner Precision nucleon-nucleon potential at fifth order
in the chiral expansion Phys Rev Lett 115 122301 (2015) arXiv14124623 [nucl-th]
[50] C W Kao T Spitzenberg and M Vanderhaeghen Burkhardt-Cottingham sum rule and
forward spin polarizabilities in heavy baryon chiral perturbation theory Phys Rev D 67
016001 (2003) arXivhep-ph0209241
[51] C-W Kao D Drechsel S Kamalov and M Vanderhaeghen Higher moments of nucleon
spin structure functions in heavy baryon chiral perturbation theory and in a resonance model
Phys Rev D 69 056004 (2004) arXivhep-ph0312102
[52] T Becher and H Leutwyler Baryon chiral perturbation theory in manifestly Lorentz invariant
form Eur Phys J C 9 643 (1999) arXivhep-ph9901384
44
[53] J Gasser M E Sainio and A Svarc Nucleons with chiral loops Nucl Phys B 307 779
(1988)
[54] L S Geng J Martin Camalich L Alvarez-Ruso and M J Vicente Vacas Leading SU(3)-
breaking corrections to the baryon magnetic moments in chiral perturbation theory Phys
Rev Lett 101 222002 (2008) arXiv08051419 [hep-ph]
[55] D Drechsel S S Kamalov and L Tiator The GDH sum rule and related integrals Phys
Rev D 63 114010 (2001) arXivhep-ph0008306
[56] D Drechsel O Hanstein S S Kamalov and L Tiator A unitary isobar model for pion
photo- and electroproduction on the proton up to 1 GeV Nucl Phys A 645 145 (1999)
arXivnucl-th9807001
[57] L Tiator private communication (2020)
[58] V Bernard T R Hemmert and U-G Meiszligner Spin structure of the nucleon at low energies
Phys Rev D 67 076008 (2003) arXivhep-ph0212033
[59] O Gryniuk F Hagelstein and V Pascalutsa Evaluation of the forward Compton scattering
off protons II Spin-dependent amplitude and observables Phys Rev D 94 034043 (2016)
arXiv160400789 [nucl-th]
[60] R Zielinski The g2p Experiment A Measurement of the Protonrsquos Spin Structure Functions
PhD thesis New Hampshire U (2010) arXiv170808297 [nucl-ex]
[61] N Guler et al (CLAS) Precise determination of the deuteron spin structure at low to mod-
erate Q2 with CLAS and extraction of the neutron contribution Phys Rev C 92 055201
(2015) arXiv150507877 [nucl-ex]
[62] C E Carlson V Nazaryan and K Griffioen Proton structure corrections to electronic
and muonic hydrogen hyperfine splitting Phys Rev A 78 022517 (2008) arXiv08052603
[physicsatom-ph]
[63] V Bernard N Kaiser and U-G Meiszligner Chiral dynamics in nucleons and nuclei Int J
Mod Phys E 4 193 (1995) arXivhep-ph9501384
[64] T R Hemmert B R Holstein and J Kambor ∆(1232) and the polarizabilities of the nucleon
Phys Rev D 55 5598 (1997) arXivhep-ph9612374
[65] V Pascalutsa and R Timmermans Field theory of nucleon to higher-spin baryon transitions
Phys Rev C 60 042201 (1999) arXivnucl-th9905065
[66] V Pascalutsa Quantization of an interacting spin-32 field and the Delta isobar Phys Rev
D 58 096002 (1998) arXivhep-ph9802288
[67] H Krebs Double Virtual Compton Scattering and SpinStructure of the Nucleon PoS
CD2018 031 (2019)
45
[68] N Kochelev and Y Oh Axial anomaly and the δLT puzzle Phys Rev D 016012 (2012)
arXiv11034892 [hep-ph]
[69] D Drechsel S Kamalov and L Tiator Unitary isobar model ndash MAID2007 Eur Phys J A
69 (2007) available at httpsmaidkphuni-mainzde arXiv07100306 [nucl-th]
[70] P J Mohr B N Taylor and D B Newell CODATA recommended values of the fundamental
physical constants 2010 Rev Mod Phys 84 1527 (2012)
[71] J Ahrens et al (GDH A2) First measurement of the Gerasimov-Drell-Hearn integral for 1H
from 200 to 800 MeV Phys Rev Lett 87 022003 (2001) arXivhep-ex0105089 [hep-ex]
[72] K Helbing (GDH) Experimental verification of the GDH sum rule at ELSA and MAMI Nucl
Phys Proc Suppl 105 113 (2002)
[73] R Bradford A Bodek H S Budd and J Arrington A New parameterization of the nucleon
elastic form-factors NuInt05 proceedings of the 4th International Workshop on Neutrino-
Nucleus Interactions in the Few-GeV Region Okayama Japan 26-29 September 2005 Nucl
Phys Proc Suppl 159 127 (2006) arXivhep-ex0602017
[74] J D Bjorken Applications of the chiral U(6)timesU(6) algebra of current densities Phys Rev
148 1467 (1966)
[75] J D Bjorken Inelastic scattering of polarized leptons from polarized nucleons Phys Rev D
1 1376 (1970)
[76] V Bernard T R Hemmert and U-G Meiszligner Novel analysis of chiral loop effects in
the generalized Gerasimov-Drell-Hearn sum rule Phys Lett B 545 105 (2002) arXivhep-
ph0203167
[77] A Deur et al Experimental study of isovector spin sum rules Phys Rev D 78 032001
(2008) arXiv08023198 [nucl-ex]
[78] B Pasquini P Pedroni and D Drechsel Higher order forward spin polarizability Phys Lett
B 687 160 (2010) arXiv10014230 [hep-ph]
[79] R Jaffe g2ndashThe nucleonrsquos other spin-dependent structure function Comments Nucl Part
Phys 19 239 (1990)
[80] E V Shuryak and A Vainshtein Theory of power corrections to deep inelastic scattering in
quantum chromodynamics (II) Qminus4 effects polarized target Nucl Phys B 201 141 (1982)
[81] S Wandzura and F Wilczek Sum rules for spin dependent electroproduction Test of rela-
tivistic constituent quarks Phys Lett B 72 195 (1977)
[82] B W Filippone and X-D Ji The spin structure of the nucleon Adv Nucl Phys 26 1
(2001) arXivhep-ph0101224
[83] M Burkardt The g2 structure function Proceedings Workshop on Spin structure at long
46
distance Newport News USA March 12-13 2009 AIP Conf Proc 1155 26 (2009)
arXiv09054079 [hep-ph]
[84] A Deur S J Brodsky and G F de Teramond The spin structure of the nucleon Rept
Prog Phys 82 076201 (2019) arXiv180705250 [hep-ph]
[85] B R Holstein D Drechsel B Pasquini and M Vanderhaeghen Higher order polarizabilities
of the proton Phys Rev C 61 034316 (2000) arXivhep-ph9910427
[86] B R Holstein V Pascalutsa and M Vanderhaeghen Sum rules for magnetic moments and
polarizabilities in QED and chiral effective-field theory Phys Rev D 72 094014 (2005)
arXivhep-ph0507016
[87] B Pasquini D Drechsel and L Tiator Invariant amplitudes for pion electroproduction Eur
Phys J A 34 387 (2007) arXiv07122327 [hep-ph]
[88] A J Buchmann Non-spherical proton shape and hydrogen hyperfine splitting Proceedings
International Workshop on Precision Physics of Simple Atomic Systems (PSAS 2008) Wind-
sor Ontario Canada July 21-26 2008 Can J Phys 87 773 (2009) arXiv09104747
[physicsatom-ph]
47
s = 12
sprime = minus12
λ = 0 λprime = 1
rprime rprime
s=12sprime=minus1
2
λ=0λprime=1
rprimerprime
s=12
sprime=minus12
λ=0λprime=1
rprimerprime hacute=frac12h=-frac12
λacute=1λ=0
FIG 11 Relation between the forward Compton process and the photoabsorption process given
by the optical theorem In particular we show the longitudinal-transverse contribution The
double-line arrows represent the spin of the external particles while the dot represents the scalar
(longitudinal) polarization of the incoming photon Inside the blob the intermediate states are
represented eg nucleons with spins rprime (which are averaged in the calculation of the cross section)
and pions
πN channel see diagrams in Fig 8 of Ref [30] We define the πN -production amplitude as
T = uhB (PB)sumi
Ai(s t)Γi uhA(PA) (B2)
with the Dirac structures
Γ1 = γ5 (B3a)
Γ2 =1
2
[qA ε
]γ5 (B3b)
where uhA(PA) and udaggerhB (PB) are the Dirac spinors and PA and PB are the four-momenta of
the incoming and outgoing nucleons respectively When calculating the photoabsorption cross
section related to the VVCS amplitude in Eq (B1) the nucleon spin flip should be implemented
by uhprime(PA) in T dagger and uh(PA) in T together with the appropriate transverse and longitudinal
photon polarization vectors εlowastplusmn and εL
However if one wants to use the properties of the Dirac matrices it is more useful to construct
an operator to produce this spin flip in the external nucleons of Fig 11 This is accomplished by
introducing the projector ΓLT equiv 12radic
2(γ1 + iγ2)γ5 which also takes into account the extra factor
radic2 in Eq (B1) We checked that with this projector one correctly extracts δLT by comparing the
HB limit of our result to the HB result of Ref [50] where the authors calculate this polarizability
from the Compton amplitude directly With all those ingredients the longitudinal-transverse cross
section is calculated in the following way
σLT (νQ2) =1
64π2 s
|~pf |cm
|~pi|cm
int 1
minus1dcos θ
sumij
AiAdaggerjXij (B4)
31
with
Xij = Tr[(PB +MN )Γi(PA +MN )ΓLTγ0Γdaggerjγ
0] (B5)
where θ is the scattering angle in the center-of-mass (cm) frame and |~pi|cm (|~pf |cm) is the three-
momentum of an incoming (outgoing) particle in the cm frame An explicit calculation of the
matrix Xij leads to
X = MNQ
0 2(PB minus PA) middot εLminusradic
2|~qf |cm sin θ (sminus u)
(B6)
where |~qi|cm (|~qf |cm) is the relative three-momentum of the incoming (outgoing) particles in the
cm frame Here s t and u are the usual Mandelstam variables For the different γlowastN rarr πN
channels we obtain the following amplitudes Ai where we introduce qA as the four-momentum of
the incoming photon and qB as the four-momentum of the outgoing pion
bull γlowastprarr π0p
A1 =e gAMN
fπ
[2PA middot ε+ qA middot ε
sminusM2N
+2PB middot εminus qA middot ε
uminusM2N
] (B7a)
A2 =e gAMN
fπ
[1
sminusM2N
+1
uminusM2N
] (B7b)
bull γlowastprarr π+n
A1 =
radic2 e gAMN
fπ
[2PA middot ε+ qA middot ε
sminusM2N
+2(PA minus PB) middot ε+ qA middot ε
tminusm2π
] (B8a)
A2 =
radic2 e gAMN
fπ(sminusM2N )
(B8b)
bull γlowastnrarr π0n
A1 = 0 (B9a)
A2 = 0 (B9b)
bull γlowastnrarr πminusp
A1 =
radic2 e gAMN
fπ
[2PB middot εminus qA middot ε
uminusM2N
minus 2(PA minus PB) middot ε+ qA middot εtminusm2
π
] (B10a)
A2 =
radic2 e gAMN
fπ(uminusM2N )
(B10b)
The analytical expressions shown above were checked with the amplitudes given in Ref [87] An-
alytical expressions for the tree-level γlowastN rarr πN channel of the σLT (νQ2) and σTT (νQ2) cross
32
sections are given below (proton channels π+n and π0p neutron channel πminusp) We checked that
they reproduce the known results in the real-photon limit [45 86] To shorten the final expressions
for the cross sections which are considerably longer for finite Q2 than in the real-photon limit we
define the following dimensionless kinematic variables
αγ = (ENi )cmradics =
s+M2N +Q2
2s (B11)
απ = (ENf )cmradics =
s+M2N minusm2
π
2s (B12)
βγ = Eγcmradics =
sminusM2N minusQ2
2s (B13)
βπ = Eπcmradics =
sminusM2N +m2
π
2s (B14)
λγ = |~qi|cmradics =
radic(sminusM2
N minusQ2)2 + 4sQ2
2s (B15)
λπ = |~qf |cmradics =
radic(sminusM2
N +m2π)2 minus 4sm2
π
2s (B16)
Here (ENi )cm and (ENf )cm are the energies of the incoming and outgoing nucleon Eγcm is the energy
of the incoming photon Eπcm is the energy of the outgoing pion all in the cm frame
σ(π+n)TT = minus e2g2
AM2N
64πf2πs
3(sminusM2N )2λ4
γ
4sλπλγ
[(M2
N minus s)(M2N minusQ2 minus s)(Q2 + 2sβγβπ)
+ 2s(m2πQ
2 minus (M2N minus s)
(M2N + s(minus1 + 2βγβπ)
))λ2γ
]+ (M2
N minus s)(M2N minusQ2 minus s)
times (Q2 + 2sβγβπ minus 2sλπλγ)(Q2 + 2s(βγβπ + λπλγ)
)times log
(Q2 + 2sβγβπ minus 2sλγλπQ2 + 2sβγβπ + 2sλγλπ
) (B17)
33
σ(π0p)TT =
e2g2AM
2Nλπ
64πf2πs(sminusM2
N )2λγ
1
s2λπλ3γ
[(M2
N minus s)(M2N minusQ2 minus s)
(3M2
N + s(2βγβπ minus 3))(M2N
+ s(2βγβπ minus 1))
+ 2s
((2m2
πQ2 + (M2
N minus s)2)(M2
N minus s)minus 2(minusm2
πQ2 + (sminusM2
N )2)
times sβγβπ + 4s2(sminusM2N )β2
γβ2π
)λ2γ
]arctan
(2sλπλγ
M2N + s(2βγβπ minus 1)
)+ 2
[minus 2m2
πQ2
+ (M2N minus s)
(2(sminusM2
N ) + 4sβγβπ +(sminusM2
N +Q2)(3M2N minus 3s+ 2sβγβπ)
sλ2γ
+m2πQ
2(sminusM2N )
(M2N minus s+ 2sβγβπ)(M2
N + s(minus1 + 2βγβπ minus 2λπλγ))
+m2πQ
2(sminusM2N )
(M2N minus s+ 2sβγβπ)(M2
N + s(minus1 + 2βγβπ + 2λπλγ))
)
+1
λπλγ
(βγβπ
(minusm2
πQ2 + (M2
N minus s)(M2N minus s+ 2sβγβπ)
)minus (M2
N minus s)(M2N minusQ2 minus s)λ2
π
)log
(M2N + s(minus1 + 2βγβπ + 2λπλγ)
M2N + s(minus1 + 2βγβπ minus 2λπλγ)
)] (B18)
σ(πminusp)TT =
e2g2AM
2N
64πf2πs
3λ4γ(M2
N + s(2βπβγ minus 1))2 minus 4λ2πs
2λ2γ)
4sλπλγ
((minusM2
N +Q2 + s)(M2N
+ s(2βπβγ minus 1))
2 minus 2sλ2γ
(2λ2
πs(minusM2N +Q2 + s) +m2
πQ2))
+((M2N + s(2βπβγ minus 1)
)2
minus 4λ2πs
2λ2γ
)((M2N + s(2βπβγ minus 1)
)(M2N minus 2Q2 minus s(2βπβγ + 1)
)+ 4λ2
πs2λ2
γ
)times log
(minusM2
N minus 2βπsβγ minus 2λπsλγ + s)minus(M2N + s(2βπβγ minus 2λπλγ minus 1)
)(M2N + s(minus1
+ 2βπβγ + 2λπλγ))[((
M2N + s(2βπβγ minus 1)
)(M2N minus 2Q2 minus s(2βπβγ + 1)
)+ 4λ2
πs2λ2
γ
)times log
(minusM2
N minus 2βπsβγ + 2λπsλγ + s)
+((Q2 + 2βπsβγ)2 minus 4λ2
πs2λ2
γ
)times log
(Q2 + 2βπsβγ minus 2λπsλγQ2 + 2sβπβγ + 2λπsλγ
)]+ 4sλ2
γ
((M2
N + s(2βπβγ minus 1))2 minus 4λ2πs
2λ2γ
)times (M2
N minus s) arctanh
(2λπsλγ
M2N + 2βπsβγ minus s
) (B19)
34
σ(π+n)LT =
e2g2AM
3Nλπ
32πf2πQs
3(sminusM2N )2λ4
γ
2sλγ
[(M2
N minus s)(Q2 + 2sβ2γ)(Q2 + 2sβγβπ)
minus 4s
((M2
N minus s)(Q2 minus 2s(απ minus 1)βγ
)+Q2sβγβπ
)λ2γ + 8s3(minus1 + απ)λ4
γ
]minus M2
N minus sλπ
[(Q2 + 2sβ2
γ)(Q2 + 2sβγβπ)2 + 4s2(2(απ minus 1)βγ(Q2 + 2sβγβπ)
minusQ2λ2π
)λ2γ + 8s3(απ minus 1)2λ4
γ
]arctan
(2sλπλγ
Q2 + 2sβγβπ
) (B20)
σ(π0p)LT =
e2g2AM
3Nλπ
16πf2πQs(sminusM2
N )2λγ
1
minus2s(M2N + s(minus1 + 2βγβπ))2λ2
γ + 8s3λ2πλ
4γ
[minus 3M8
N (Q2
+ 2sβ2γ) + 2M4
Ns2
(minus (Q2 + 2sβ2
γ)(minus3 + 2βγβπ) +(Q2(minus3 + 2βγβπ + 6λ2
π) + 2sβγ
times (minus3 + 12απ + 2βγβπ minus 12απβγβπ + 4βγλ2π))λ2γ minus 4s(α2
π minus 1)λ4π
)+ 2M2
Ns3
[minus (Q2
+ 2sβ2γ)(minus1 + 2βγβπ)
(6 + βγβπ(minus9 + 2βγβπ)
)+
(Q2(3minus 12λ2
π + 4βγβπ(minus1 + 2βγβπ
+ λ2π))minus 2sβγ
(minus 3 + 4απ(3 + 2απ(3 + 2βγβπ(minus3 + βγβπ)) + 4βγ(βπ + (2minus βγβπ)
times λ2π)))λ2γ + 8s
((απ minus 1)(1 + απ minus 2βγβπ) + 2απβγλ
2π
)λ4γ
)+ s4
((Q2 + 2sβ2
γ)
times (1minus 2βγβπ)2(2βγβπ minus 3) + 2(Q2(minus1 + 6λ2
π + 2βγβπ((1minus 2βγβπ)2 minus 2λ2π)) + 2sβγ
times (minus1 + 2βγβπ + 4(minus1 + βγβπ)(minusαπ + 2απβγβπ minus βγλ2π)))λ2γ minus 8
(s(απ minus 1)(απ
+ (1minus 2βγβπ)2) + 2βγ(2sαπ +Q2βγ)λ2π
)λ4γ + 32s(απ minus 1)λ2
πλ6γ
)+ 2M6
Ns
(Q2(6
minus 7βγβπ + λ2γ) + 2sβγ(βγ(6minus 7βγβπ) + (1minus 4απ)λ2
γ)
)]+sminusM2
N
4s2λπλ3γ
[(Q2 + 2sβ2
γ)
times(3M2
N + s(2βγβπ minus 3))(M2N + s(2βγβπ minus 1)
)+ 2s
(minusM2
N (Q2 + 2sβγ minus 8sαπβγ)
+ s(2sβγ(1minus 4απ + 4απβγβπ) +Q2(1minus 2λ2π)))λ2γ + 8s3(απ minus 1)λ4
γ
]
times arctan
(2sλπλγ
M2N + s(2βγβπ minus 1)
) (B21)
35
σ(πminusp)LT =
e2g2AM
3N
64πf2πQs
3λ4γ(M2
N minusQ2 minus s)(M2N + s(2βπβγ + 2λπλγ minus 1))
times
1
M2N + s(2βπβγ minus 2λπλγ minus 1)
[4λπsλγ(M2
N minusQ2 minus s)(
2sλ2γ
(s((1minus 2λ2
π)Q2
+ 2(2απ minus 1)sβγ(2βπβγ minus 1))minusM2N (Q2 + 2(1minus 2απ)sβγ)
)+ (Q2 + 2sβ2
γ)(M2N + s(2βπβγ minus 1)
)2 + 8(απ minus 1)απs
3λ4γ
)]minus(M2N + s(2βπβγ + 2λπλγ minus 1)
)times[
log
(minusM2N minus 2βπsβγ minus 2λπsλγ + s
minusM2N minus 2βπsβγ + 2λπsλγ + s
)(minus (Q2 + 2sβ2
γ)(M2N + s(2βπβγ minus 1)
)(minusM2
N + 2Q2 + 2βπsβγ + s)minus 2sλ2γ
(minusM2
N (Q2 + 2sβγ) +Q4 +Q2s(4απβγ
minus 2βγ minus 2λ2π + 1) + 2s2βγ(4(απ minus 1)βπβγ + 1)
)minus 8(απ minus 1)2s3λ4
γ
)+ 2
(4s2λ2
γ
(2(απ minus 1)βγ(Q2 + 2βπsβγ)minus λ2
πQ2)
+ (Q2 + 2βπsβγ)2(Q2 + 2sβ2γ)
+ 8(απ minus 1)2s3λ4γ
)arctanh
(2λπsλγ
Q2 + 2βπsβγ
)] (B22)
2 ∆-production channel
The tree-level ∆-exchange diagram in Fig 2 of Ref [30] contributes to the non-Born part of
the VVCS amplitudes The contribution of the ∆ exchange to the VVCS amplitudes can be split
into [17]
S∆-exch1 (νQ2) = S∆-pole
1 (νQ2) + S∆-exch1 (νQ2) (B23a)
νS∆-exch2 (νQ2) = νS∆-pole
2 (νQ2) + νS2∆-exch
(νQ2) (B23b)
and similarly for the unpolarized VVCS amplitudes discussed in Ref [30] Here we introduced the
∆-pole contributions S∆-polei and the ∆-non-pole contributions S∆-exch
i The former amplitudes
feature a pole at the ∆(1232)-production threshold and thus are proportional to
1
[sminusM2∆][uminusM2
∆]=
1
4M2N
1
ν2∆ minus ν2
(B24)
36
They can be reconstructed from the dispersion relations in Eq (5) with the tree-level ∆-production
cross sections as input cf Fig 10 in Ref [30]
σTT (νQ2) =π2α
M2NM
2+|~q |
minus g2
MMN (M+ + ν)|~q |2 +g2E(∆minus ν)(Q2 minusMNν)2
MN(B25a)
+g2CQ
4s(∆minus ν)
MNM2∆
minus 4gMgE(Q2 minusMNν)|~q |2 minus 4gMgCQ2|~q |2
+2gEgCQ
2[minusMNM∆ |~q |2 + s(Q2 +∆ν)
]MNM∆
δ(ν minus ν∆)
σLT (νQ2) =Qπ2α
M2NM
2+|~q |
g2E(MNν minusQ2) [M∆(MN + ν)minus s]
MN(B25b)
+g2CQ
2[MNM∆|~q |2 minus s(Q2 +∆ν)
]MNM2
∆
+ gMgEM∆|~q |2 minusgMgC(Q2 minusMNν)|~q |2
M∆
+gEgC(ν minus∆)(M2
N |~q |2 minus 2Q2s)
MNM∆
δ(ν minus ν∆)
with ∆ = M∆ minusMN M+ = M∆ + MN and the Mandelstam variable s = M2N + 2MNν minus Q2
Analytical expressions for the spin structure functions g1(xQ2) and g2(xQ2) can be constructed
from Eq (3) with the flux factor K(νQ2) = |~q | =radicν2 +Q2
In the ∆-non-pole contributions to S1(νQ2) and νS2(νQ2) the pole in ν at the ∆(1232)-
production threshold has canceled out
S∆-exch1 (νQ2) =
πα
MNM2+
[g2MQ
2+ + g2
E
(∆2 minus 3Q2
)+
4g2CQ
4
M2∆
minus 8gMgEM∆ωminus (B26a)
minus2gMgCQ2(MN minus 4M∆)
M∆+
2gEgCQ2(3MN minus 2M∆)
M∆
]
νS2∆-exch
(νQ2) =2πα
MNM2+
[g2EM∆∆ωminus +
g2M MNQ
2+
2+g2C Q
2(Q2 minus∆2)
2M∆(B26b)
+gEgM M∆(M∆ω+ minus 4MNωminus)minus gEgC ∆(2Q2 +MNω+)
+gMgC Q2(4MN minus ω+)
]+S∆-exch
2 (νQ2)
ν
[M2
∆ ω2+
M2N
+ ν2
]
with Q+ =radic
(M∆ +MN )2 +Q2 and ωplusmn = (M2∆minusM2
N plusmnQ2)2M∆ and the non-pole contribution
to S2(νQ2)
S∆-exch2 (νQ2) = minus2παMNν
M∆M2+
[gM + gE
]gC (B27)
These amplitudes to the contrary are not described by the tree-level ∆-production cross sections
in the standard dispersive approach [17] This peculiarity has been previously missed e g in the
calculation of the ∆-exchange contribution to the hydrogen hyperfine splitting in Ref [88] The
importance of including the ∆-non-pole contribution is also evident when considering the BC sum
37
rule in Eq (14) The ∆-pole terms by themselves violate the BC sum rule but cancel exactly with
the ∆-non-pole terms
limνrarr0
νS∆-pole2 (νQ2) + lim
νrarr0νS2
∆-exch(νQ2) = 0 (B28)
Appendix C Polarizabilities at Q2 = 0
In this section we give analytical expressions for the static values and slopes at Q2 = 0 of
the spin polarizabilities and moments of polarized structure functions In particular we give the
HB expansion of the πN -loop contributions and the ∆-exchange contributions The complete
expressions also for the π∆-loop contributions can be found in the Supplemented material Recall
that IA(0) = I1(0) = d2(0) = 0 and dd2(Q2)dQ2
∣∣∣Q2=0
= 0
1 πN-loop contribution
Here we give analytical expressions for the πN -loop contributions to the proton and neutron
spin polarizabilities expanded in powers of micro = mπMN viz the HB expansion Note that we
choose to expand here to a high order in micro the strict HB expansion would only retain the leading
term in an analogous NLO calculation
bull Static polarizabilities (Q2 = 0)
γ0p =e2g2
A
96π3f2πm
2π
1minus 21πmicro
8minus(
59
2+ 26 logmicro
)micro2 +
1875πmicro3
64
+3
(3
2+ 26 logmicro
)micro4 +
(C1)
γ0n =e2g2
A
48π3f2πm
2π
1
2minus 9πmicro
16minus 2micro2 logmicro+
75πmicro3
128minus 3micro4
4+
(C2)
δLTp =e2g2
A
192π3f2πm
2π
1minus 9πmicro
8+
(13
2minus 2 logmicro
)micro2 minus 465πmicro3
64
minus(
47
2+ 42 logmicro
)micro4 +
(C3)
δLTn =e2g2
A
96π3f2πm
2π
1
2+
3πmicro
16+ (1 + 2 log micro)micro2 minus 105πmicro3
128+
5micro4
4+
(C4)
38
δLTn =e2g2
A
96π3f2πm
2π
1
2+
3πmicro
16+ (1 + 2 log micro)micro2 minus 105πmicro3
128+
5micro4
4+
(C5)
γ0p =e2g2
A
16π3f2πm
4π
4
45minus 3πmicro
16+
14micro2
5minus 1813πmicro3
384minus 192
5(1 + log micro)micro4
+80703πmicro5
2048+
(C6)
γ0n =e2g2
A
16π3f2πm
4π
4
45minus 5πmicro
48+
4micro2
5minus 245πmicro3
384minus 32micro4 logmicro
15
+1323πmicro5
2048+
(C7)
bull Slopes of polarizabilities at Q2 = 0
dγ0p(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
2minus 45πmicro
4+ 223micro2 minus 28515πmicro3
64
minus9
(1953
4+ 449 logmicro
)micro4 +
570255πmicro5
128+
(C8)
dγ0n(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
2minus 81πmicro
8+ 94micro2 minus 2535πmicro3
32
minus3 (1 + 90 logmicro)micro4 +84315πmicro5
1024+
(C9)
dδLTp(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
2880π3f2πm
4π
minus5
2minus 27πmicro
32+ 20micro2 minus 5865πmicro3
256
+3
(617
4+ 36 logmicro
)micro4 minus 2056845πmicro5
4096+
(C10)
dδLTn(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
minus5
4minus 81πmicro
64minus 11micro2 +
10005πmicro3
512
+15
8(11 + 48 logmicro)micro4 minus 267015πmicro5
8192+
(C11)
dIAp(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
96π2f2πmicro
2
1minus 15πmicro
4minus 1
2(115 + 88 logmicro)micro2 +
1839πmicro3
32
+5 (5 + 34 logmicro)micro4 +
(C12)
dIAn(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
48π2f2πmicro
2
1
2minus 11πmicro
8minus 1
4(1 + 20 logmicro)micro2 +
99πmicro3
64
minus25micro4
12+
(C13)
dI1p(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
96π2f2πmicro
3π
8+ 2 (4 + 3 logmicro)microminus 537πmicro2
64
minus1
2(15 + 56 logmicro)micro3 +
(C14)
39
dI1n(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
48π2f2πmicro
minus π
16+
1
4(3 + 4 logmicro)microminus 57πmicro2
128+
2micro3
3+
(C15)
γ0p(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2g2
A
16π3f2πm
6π
1
105minus 23πmicro
256+
377micro2
210minus 15551πmicro3
6144+
3371micro4
105
minus1640457πmicro5
32768+
(C16)
γ0n(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2g2
A
16π3f2πm
6π
1
105minus 153πmicro
1792+
69micro2
70minus 4615πmicro3
6144+
172micro4
35
minus120897πmicro5
32768+
(C17)
2 ∆-exchange contribution
Here we give analytical expressions for the tree-level ∆-exchange contributions to the nucleon
spin polarizabilities and their slopes at Q2 = 0 Note that the ∆-exchange contributes equally to
proton and neutron polarizabilities Recall that for the magnetic γlowastN∆ coupling we introduced a
dipole form factor to mimic vector-meson dominance gM rarr gM(1 +Q2Λ2)2
bull Static polarizabilities (Q2 = 0)
γ0 = minus e2
4πM2+
(g2M
∆2+
g2E
M2+
minus 4gMgEM+∆
) (C18)
δLT =e2M∆
4πM3+
(g2E
MNM++gMgE∆MN
minus gEgCM2
∆
) (C19)
γ0 =e2M2
N
π∆2M4+
(minusg
2M
∆2+
g2E
M2+
+4gMgE∆M+
) (C20)
bull Slopes of polarizabilities at Q2 = 0
dγ0(Q2)
dQ2
∣∣∣∣∣Q2=0
= minus e2
πM2+∆
(g2M
∆
[1
4∆2minus 1
∆M++
1
2M2+
]minus 1
Λ2
g2M
∆+
g2E
2M2+
[1
2∆minus 3
M+
]minusgMgE
M+
[1
∆2minus 5
∆M++
1
M2+
]+
1
Λ2
2gMgEM+
+2gMgC∆M2
+
minus gEgCM3
+
) (C21)
40
d δLT (Q2)
dQ2
∣∣∣∣∣Q2=0
=e2M∆∆
4πMNM2+
(g2E
∆2M2+
[1
∆minus 4
M+
]minus g2
C
∆M2∆M
2+
+gMgE∆2M+
[1
∆2
minus 3
∆M++
1
M2+
]minus 2
Λ2
gMgE∆2M+
+gMgC∆M2
∆
[1
2∆2minus 2
∆M++
1
2M2+
]minus gEgC
2M2∆M
2+
[7
∆+
1
M+
]) (C22)
dIA(Q2)
dQ2
∣∣∣∣∣Q2=0
= minusM2N
M2+
(g2M
2∆2+
g2E
MNM+minus 2gMgE
∆M+minus gEgCM∆M+
) (C23)
dI1(Q2)
dQ2
∣∣∣∣∣Q2=0
= minusM∆M2N
2M3+
(g2E
MNM∆minus gMgE∆MN
minus gEgCM2
∆
) (C24)
γ0(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2M2
N
π∆3M6+
(g2M
∆
[3M2
∆ + 2M∆MN minus 9M2N
∆2+
4M2+
Λ2
]
minus4gMgE
[5M2
∆ minus 9M2N
M+∆2+
2M+
Λ2
]+
g2E
∆M+[7M∆ minus 9MN ]
minus8gMgC∆
+4gEgCM+
) (C25)
[1] D Drechsel B Pasquini and M Vanderhaeghen Dispersion relations in real and virtual
Compton scattering Phys Rept 378 99 (2003) hep-ph0212124
[2] S E Kuhn J-P Chen and E Leader Spin structure of the nucleon mdash status and recent
results Prog Part Nucl Phys 63 1 (2009) arXiv08123535 [hep-ph]
[3] F Hagelstein R Miskimen and V Pascalutsa Nucleon polarizabilities from Compton scat-
tering to hydrogen atom Prog Part Nucl Phys 88 29 (2016) arXiv151203765 [nucl-th]
[4] B Pasquini and M Vanderhaeghen Dispersion theory in electromagnetic interactions Ann
Rev Nucl Part Sci 68 75 (2018) arXiv180510482 [hep-ph]
[5] S Gerasimov A Sum rule for magnetic moments and the damping of the nucleon magnetic
moment in nuclei Sov J Nucl Phys 2 430 (1966)
[6] S Drell and A C Hearn Exact sum rule for nucleon magnetic moments Phys Rev Lett
16 908 (1966)
[7] J S Schwinger Source theory viewpoints in deep inelastic scattering Proc Natl Acad Sci
USA 72 1 (1975)
[8] J S Schwinger Source theory viewpoints in deep inelastic scattering Electromagnetic In-
teractions and Field Theory Proceedings 14 Internationale Universitatswochen Schladming
Austria February 24-March 7 1975 Acta Phys Austriaca Suppl 14 471 (1975)
41
[9] J Schwinger Source theory discussion of deep inelastic scattering with polarized particles
Proc Natl Acad Sci USA 72 1559 (1975)
[10] A M Baldin Polarizability of nucleons Nucl Phys 18 310 (1960)
[11] M Gell-Mann M L Goldberger and W E Thirring Use of causality conditions in quantum
theory Phys Rev 95 1612 (1954)
[12] A Pineda Leading chiral logarithms to the hyperfine splitting of the hydrogen and muonic
hydrogen Phys Rev C 67 025201 (2003)
[13] C Peset and A Pineda The two-photon exchange contribution to muonic hydrogen from
chiral perturbation theory Nucl Phys B 887 69 (2014) arXiv14064524 [hep-ph]
[14] C Peset and A Pineda Model-independent determination of the two-photon exchange contri-
bution to hyperfine splitting in muonic hydrogen JHEP 04 060 arXiv161205206 [nucl-th]
[15] F Hagelstein and V Pascalutsa Proton structure in the hyperfine splitting of muonic hydro-
gen PoS CD15 077 (2016) arXiv151104301 [nucl-th]
[16] F Hagelstein Exciting Nucleons in Compton Scattering and Hydrogen-Like Atoms PhD
thesis Mainz U Inst Kernphys (2017) arXiv171000874 [nucl-th]
[17] F Hagelstein ∆(1232)-Resonance in the hydrogen spectrum Proceedings 11th International
Workshop on the Physics of Excited Nucleons (NSTAR 2017) Columbia SC USA August
20-23 2017 Few Body Syst 59 93 (2018) arXiv180109790 [nucl-th]
[18] Y Prok et al (CLAS) Moments of the spin structure functions gp1 and gd1 for 005 lt Q2 lt 30
GeV2 Phys Lett B 672 12 (2009) arXiv08022232 [nucl-ex]
[19] H Dutz et al (GDH) First measurement of the Gerasimov-Drell-Hearn sum rule for 1H from
07 GeV to 18 GeV at ELSA Phys Rev Lett 91 192001 (2003)
[20] M Amarian et al (Jefferson Lab E94010) Measurement of the generalized forward spin po-
larizabilities of the neutron Phys Rev Lett 93 152301 (2004) arXivnucl-ex0406005
[21] M Amarian et al The Q2 evolution of the generalized Gerasimov-Drell-Hearn integral for
the neutron using a 3He target Phys Rev Lett 89 242301 (2002) arXivnucl-ex0205020
[22] M Amarian et al (Jefferson Lab E94-010)Q2 evolution of the neutron spin structure moments
using a 3He target Phys Rev Lett 92 022301 (2004) arXivhep-ex0310003
[23] A Deur et al Experimental determination of the evolution of the Bjorken integral at low Q2
Phys Rev Lett 93 212001 (2004) arXivhep-ex0407007
[24] K Slifer Low Q2 measurement of gp2 and the δLT spin polarizability Spin structure at long dis-
tance Proceedings Workshop Newport News USA March 12ndash13 2009 nucl-ex09064775
(2009) AIP Conf Proc 1155 10106313203293 (2009) arXiv09064775 [nucl-ex]
[25] P Solvignon et al (E01-012) Moments of the neutron g2 structure function at intermediate
42
Q2 Phys Rev C 92 015208 (2015) arXiv13044497 [nucl-ex]
[26] A Deur Experimental studies at low Q2 of the spin structure of the nucleon at Jefferson Lab
in 9th International Workshop on Chiral Dynamics (CD18) Durham NC USA September
17-21 2018 (2019) arXiv190305661 [nucl-ex]
[27] V Sulkosky et al (Jefferson Lab E97-110) Measurement of the 3He spin-structure functions
and of neutron (3He) spin-dependent sum rules at 0035 le Q2 le 024 GeV2 Phys Lett B
805 135428 (2020) arXiv190805709 [nucl-ex]
[28] K Adhikari et al (CLAS) Measurement of the Q2 Dependence of the Deuteron Spin Structure
Function g1 and its Moments at Low Q2 with CLAS Phys Rev Lett 120 062501 (2018)
arXiv171101974 [nucl-ex]
[29] V Lensky F Hagelstein A Hiller Blin and V Pascalutsa Comment on rdquoMeasurement of
the Q2 Dependence of the Deuteron Spin Structure Function g1 and its Moments at Low Q2
with CLASrdquo (2018) arXiv180603219 [nucl-th]
[30] J M Alarcn F Hagelstein V Lensky and V Pascalutsa Forward doubly-virtual Compton
scattering off the nucleon in chiral perturbation theory at NLO the subtraction function and
moments of unpolarized structure functions (2020) arXiv200509518 [hep-ph]
[31] V Bernard E Epelbaum H Krebs and U-G Meiszligner New insights into the spin structure
of the nucleon Phys Rev D 87 054032 (2013) arXiv12092523 [hep-ph]
[32] V Lensky J M Alarcon and V Pascalutsa Moments of nucleon structure functions at
next-to-leading order in baryon chiral perturbation theory Phys Rev C 90 055202 (2014)
arXiv14072574 [hep-ph]
[33] V Pascalutsa and M Vanderhaeghen Chiral effective-field theory in the ∆(1232) region
I Pion electroproduction on the nucleon Phys Rev D 73 034003 (2006) arXivhep-
ph0512244
[34] V Pascalutsa and M Vanderhaeghen Electromagnetic nucleon-to-Delta transition in chiral
effective field theory Phys Rev Lett 95 232001 (2005) arXivhep-ph0508060
[35] T R Hemmert B R Holstein and J Kambor Systematic 1M expansion for spin 32
particles in baryon chiral perturbation theory Phys Lett B 395 89 (1997) arXivhep-
ph9606456
[36] V Pascalutsa and D R Phillips Effective theory of the ∆(1232) in Compton scattering off
the nucleon Phys Rev C 67 055202 (2003) arXivnucl-th0212024
[37] V Pascalutsa M Vanderhaeghen and S N Yang Electromagnetic excitation of the ∆(1232)-
resonance Phys Rept 437 125 (2007) arXivhep-ph0609004
[38] K A Olive et al (Particle Data Group) Review of Particle Physics Chin Phys C 38 090001
43
(2014)
[39] F E Low Scattering of light of very low frequency by systems of spin 12 Phys Rev 96
1428 (1954)
[40] M Gell-Mann and M L Goldberger Scattering of low-energy photons by particles of spin
12 Phys Rev 96 1433 (1954)
[41] A M Harun ar-Rashid A simple derivation of Schwingerrsquos sum rule for spin dependent
structure functions Nuovo Cim A 33 447 (1976)
[42] F Hagelstein and V Pascalutsa Dissecting the hadronic contributions to (g minus 2)micro by
Schwingerrsquos sum rule Phys Rev Lett 120 072002 (2018) arXiv171004571 [hep-ph]
[43] V Lensky V Pascalutsa M Vanderhaeghen and C Kao Spin-dependent sum rules con-
necting real and virtual Compton scattering verified Phys Rev D 95 074001 (2017)
arXiv170101947 [hep-ph]
[44] H Burkhardt and W N Cottingham Sum rules for forward virtual Compton scattering
Annals Phys 56 453 (1970)
[45] V Lensky and V Pascalutsa Predictive powers of chiral perturbation theory in Compton
scattering off protons Eur Phys J C 65 195 (2010) arXiv09070451 [hep-ph]
[46] H Grieszlighammer J McGovern D Phillips and G Feldman Using effective field theory to
analyse low-energy Compton scattering data from protons and light nuclei Prog Part Nucl
Phys 67 841 (2012) arXiv12036834 [nucl-th]
[47] H W Grieszlighammer J A McGovern and D R Phillips Nucleon polarisabilities at and
beyond physical pion masses Eur Phys J A 52 139 (2016) arXiv151101952 [nucl-th]
[48] E Epelbaum H Krebs and U-G Meiszligner Improved chiral nucleon-nucleon potential up to
next-to-next-to-next-to-leading order Eur Phys J A 51 53 (2015) arXiv14120142 [nucl-
th]
[49] E Epelbaum H Krebs and U-G Meiszligner Precision nucleon-nucleon potential at fifth order
in the chiral expansion Phys Rev Lett 115 122301 (2015) arXiv14124623 [nucl-th]
[50] C W Kao T Spitzenberg and M Vanderhaeghen Burkhardt-Cottingham sum rule and
forward spin polarizabilities in heavy baryon chiral perturbation theory Phys Rev D 67
016001 (2003) arXivhep-ph0209241
[51] C-W Kao D Drechsel S Kamalov and M Vanderhaeghen Higher moments of nucleon
spin structure functions in heavy baryon chiral perturbation theory and in a resonance model
Phys Rev D 69 056004 (2004) arXivhep-ph0312102
[52] T Becher and H Leutwyler Baryon chiral perturbation theory in manifestly Lorentz invariant
form Eur Phys J C 9 643 (1999) arXivhep-ph9901384
44
[53] J Gasser M E Sainio and A Svarc Nucleons with chiral loops Nucl Phys B 307 779
(1988)
[54] L S Geng J Martin Camalich L Alvarez-Ruso and M J Vicente Vacas Leading SU(3)-
breaking corrections to the baryon magnetic moments in chiral perturbation theory Phys
Rev Lett 101 222002 (2008) arXiv08051419 [hep-ph]
[55] D Drechsel S S Kamalov and L Tiator The GDH sum rule and related integrals Phys
Rev D 63 114010 (2001) arXivhep-ph0008306
[56] D Drechsel O Hanstein S S Kamalov and L Tiator A unitary isobar model for pion
photo- and electroproduction on the proton up to 1 GeV Nucl Phys A 645 145 (1999)
arXivnucl-th9807001
[57] L Tiator private communication (2020)
[58] V Bernard T R Hemmert and U-G Meiszligner Spin structure of the nucleon at low energies
Phys Rev D 67 076008 (2003) arXivhep-ph0212033
[59] O Gryniuk F Hagelstein and V Pascalutsa Evaluation of the forward Compton scattering
off protons II Spin-dependent amplitude and observables Phys Rev D 94 034043 (2016)
arXiv160400789 [nucl-th]
[60] R Zielinski The g2p Experiment A Measurement of the Protonrsquos Spin Structure Functions
PhD thesis New Hampshire U (2010) arXiv170808297 [nucl-ex]
[61] N Guler et al (CLAS) Precise determination of the deuteron spin structure at low to mod-
erate Q2 with CLAS and extraction of the neutron contribution Phys Rev C 92 055201
(2015) arXiv150507877 [nucl-ex]
[62] C E Carlson V Nazaryan and K Griffioen Proton structure corrections to electronic
and muonic hydrogen hyperfine splitting Phys Rev A 78 022517 (2008) arXiv08052603
[physicsatom-ph]
[63] V Bernard N Kaiser and U-G Meiszligner Chiral dynamics in nucleons and nuclei Int J
Mod Phys E 4 193 (1995) arXivhep-ph9501384
[64] T R Hemmert B R Holstein and J Kambor ∆(1232) and the polarizabilities of the nucleon
Phys Rev D 55 5598 (1997) arXivhep-ph9612374
[65] V Pascalutsa and R Timmermans Field theory of nucleon to higher-spin baryon transitions
Phys Rev C 60 042201 (1999) arXivnucl-th9905065
[66] V Pascalutsa Quantization of an interacting spin-32 field and the Delta isobar Phys Rev
D 58 096002 (1998) arXivhep-ph9802288
[67] H Krebs Double Virtual Compton Scattering and SpinStructure of the Nucleon PoS
CD2018 031 (2019)
45
[68] N Kochelev and Y Oh Axial anomaly and the δLT puzzle Phys Rev D 016012 (2012)
arXiv11034892 [hep-ph]
[69] D Drechsel S Kamalov and L Tiator Unitary isobar model ndash MAID2007 Eur Phys J A
69 (2007) available at httpsmaidkphuni-mainzde arXiv07100306 [nucl-th]
[70] P J Mohr B N Taylor and D B Newell CODATA recommended values of the fundamental
physical constants 2010 Rev Mod Phys 84 1527 (2012)
[71] J Ahrens et al (GDH A2) First measurement of the Gerasimov-Drell-Hearn integral for 1H
from 200 to 800 MeV Phys Rev Lett 87 022003 (2001) arXivhep-ex0105089 [hep-ex]
[72] K Helbing (GDH) Experimental verification of the GDH sum rule at ELSA and MAMI Nucl
Phys Proc Suppl 105 113 (2002)
[73] R Bradford A Bodek H S Budd and J Arrington A New parameterization of the nucleon
elastic form-factors NuInt05 proceedings of the 4th International Workshop on Neutrino-
Nucleus Interactions in the Few-GeV Region Okayama Japan 26-29 September 2005 Nucl
Phys Proc Suppl 159 127 (2006) arXivhep-ex0602017
[74] J D Bjorken Applications of the chiral U(6)timesU(6) algebra of current densities Phys Rev
148 1467 (1966)
[75] J D Bjorken Inelastic scattering of polarized leptons from polarized nucleons Phys Rev D
1 1376 (1970)
[76] V Bernard T R Hemmert and U-G Meiszligner Novel analysis of chiral loop effects in
the generalized Gerasimov-Drell-Hearn sum rule Phys Lett B 545 105 (2002) arXivhep-
ph0203167
[77] A Deur et al Experimental study of isovector spin sum rules Phys Rev D 78 032001
(2008) arXiv08023198 [nucl-ex]
[78] B Pasquini P Pedroni and D Drechsel Higher order forward spin polarizability Phys Lett
B 687 160 (2010) arXiv10014230 [hep-ph]
[79] R Jaffe g2ndashThe nucleonrsquos other spin-dependent structure function Comments Nucl Part
Phys 19 239 (1990)
[80] E V Shuryak and A Vainshtein Theory of power corrections to deep inelastic scattering in
quantum chromodynamics (II) Qminus4 effects polarized target Nucl Phys B 201 141 (1982)
[81] S Wandzura and F Wilczek Sum rules for spin dependent electroproduction Test of rela-
tivistic constituent quarks Phys Lett B 72 195 (1977)
[82] B W Filippone and X-D Ji The spin structure of the nucleon Adv Nucl Phys 26 1
(2001) arXivhep-ph0101224
[83] M Burkardt The g2 structure function Proceedings Workshop on Spin structure at long
46
distance Newport News USA March 12-13 2009 AIP Conf Proc 1155 26 (2009)
arXiv09054079 [hep-ph]
[84] A Deur S J Brodsky and G F de Teramond The spin structure of the nucleon Rept
Prog Phys 82 076201 (2019) arXiv180705250 [hep-ph]
[85] B R Holstein D Drechsel B Pasquini and M Vanderhaeghen Higher order polarizabilities
of the proton Phys Rev C 61 034316 (2000) arXivhep-ph9910427
[86] B R Holstein V Pascalutsa and M Vanderhaeghen Sum rules for magnetic moments and
polarizabilities in QED and chiral effective-field theory Phys Rev D 72 094014 (2005)
arXivhep-ph0507016
[87] B Pasquini D Drechsel and L Tiator Invariant amplitudes for pion electroproduction Eur
Phys J A 34 387 (2007) arXiv07122327 [hep-ph]
[88] A J Buchmann Non-spherical proton shape and hydrogen hyperfine splitting Proceedings
International Workshop on Precision Physics of Simple Atomic Systems (PSAS 2008) Wind-
sor Ontario Canada July 21-26 2008 Can J Phys 87 773 (2009) arXiv09104747
[physicsatom-ph]
47
with
Xij = Tr[(PB +MN )Γi(PA +MN )ΓLTγ0Γdaggerjγ
0] (B5)
where θ is the scattering angle in the center-of-mass (cm) frame and |~pi|cm (|~pf |cm) is the three-
momentum of an incoming (outgoing) particle in the cm frame An explicit calculation of the
matrix Xij leads to
X = MNQ
0 2(PB minus PA) middot εLminusradic
2|~qf |cm sin θ (sminus u)
(B6)
where |~qi|cm (|~qf |cm) is the relative three-momentum of the incoming (outgoing) particles in the
cm frame Here s t and u are the usual Mandelstam variables For the different γlowastN rarr πN
channels we obtain the following amplitudes Ai where we introduce qA as the four-momentum of
the incoming photon and qB as the four-momentum of the outgoing pion
bull γlowastprarr π0p
A1 =e gAMN
fπ
[2PA middot ε+ qA middot ε
sminusM2N
+2PB middot εminus qA middot ε
uminusM2N
] (B7a)
A2 =e gAMN
fπ
[1
sminusM2N
+1
uminusM2N
] (B7b)
bull γlowastprarr π+n
A1 =
radic2 e gAMN
fπ
[2PA middot ε+ qA middot ε
sminusM2N
+2(PA minus PB) middot ε+ qA middot ε
tminusm2π
] (B8a)
A2 =
radic2 e gAMN
fπ(sminusM2N )
(B8b)
bull γlowastnrarr π0n
A1 = 0 (B9a)
A2 = 0 (B9b)
bull γlowastnrarr πminusp
A1 =
radic2 e gAMN
fπ
[2PB middot εminus qA middot ε
uminusM2N
minus 2(PA minus PB) middot ε+ qA middot εtminusm2
π
] (B10a)
A2 =
radic2 e gAMN
fπ(uminusM2N )
(B10b)
The analytical expressions shown above were checked with the amplitudes given in Ref [87] An-
alytical expressions for the tree-level γlowastN rarr πN channel of the σLT (νQ2) and σTT (νQ2) cross
32
sections are given below (proton channels π+n and π0p neutron channel πminusp) We checked that
they reproduce the known results in the real-photon limit [45 86] To shorten the final expressions
for the cross sections which are considerably longer for finite Q2 than in the real-photon limit we
define the following dimensionless kinematic variables
αγ = (ENi )cmradics =
s+M2N +Q2
2s (B11)
απ = (ENf )cmradics =
s+M2N minusm2
π
2s (B12)
βγ = Eγcmradics =
sminusM2N minusQ2
2s (B13)
βπ = Eπcmradics =
sminusM2N +m2
π
2s (B14)
λγ = |~qi|cmradics =
radic(sminusM2
N minusQ2)2 + 4sQ2
2s (B15)
λπ = |~qf |cmradics =
radic(sminusM2
N +m2π)2 minus 4sm2
π
2s (B16)
Here (ENi )cm and (ENf )cm are the energies of the incoming and outgoing nucleon Eγcm is the energy
of the incoming photon Eπcm is the energy of the outgoing pion all in the cm frame
σ(π+n)TT = minus e2g2
AM2N
64πf2πs
3(sminusM2N )2λ4
γ
4sλπλγ
[(M2
N minus s)(M2N minusQ2 minus s)(Q2 + 2sβγβπ)
+ 2s(m2πQ
2 minus (M2N minus s)
(M2N + s(minus1 + 2βγβπ)
))λ2γ
]+ (M2
N minus s)(M2N minusQ2 minus s)
times (Q2 + 2sβγβπ minus 2sλπλγ)(Q2 + 2s(βγβπ + λπλγ)
)times log
(Q2 + 2sβγβπ minus 2sλγλπQ2 + 2sβγβπ + 2sλγλπ
) (B17)
33
σ(π0p)TT =
e2g2AM
2Nλπ
64πf2πs(sminusM2
N )2λγ
1
s2λπλ3γ
[(M2
N minus s)(M2N minusQ2 minus s)
(3M2
N + s(2βγβπ minus 3))(M2N
+ s(2βγβπ minus 1))
+ 2s
((2m2
πQ2 + (M2
N minus s)2)(M2
N minus s)minus 2(minusm2
πQ2 + (sminusM2
N )2)
times sβγβπ + 4s2(sminusM2N )β2
γβ2π
)λ2γ
]arctan
(2sλπλγ
M2N + s(2βγβπ minus 1)
)+ 2
[minus 2m2
πQ2
+ (M2N minus s)
(2(sminusM2
N ) + 4sβγβπ +(sminusM2
N +Q2)(3M2N minus 3s+ 2sβγβπ)
sλ2γ
+m2πQ
2(sminusM2N )
(M2N minus s+ 2sβγβπ)(M2
N + s(minus1 + 2βγβπ minus 2λπλγ))
+m2πQ
2(sminusM2N )
(M2N minus s+ 2sβγβπ)(M2
N + s(minus1 + 2βγβπ + 2λπλγ))
)
+1
λπλγ
(βγβπ
(minusm2
πQ2 + (M2
N minus s)(M2N minus s+ 2sβγβπ)
)minus (M2
N minus s)(M2N minusQ2 minus s)λ2
π
)log
(M2N + s(minus1 + 2βγβπ + 2λπλγ)
M2N + s(minus1 + 2βγβπ minus 2λπλγ)
)] (B18)
σ(πminusp)TT =
e2g2AM
2N
64πf2πs
3λ4γ(M2
N + s(2βπβγ minus 1))2 minus 4λ2πs
2λ2γ)
4sλπλγ
((minusM2
N +Q2 + s)(M2N
+ s(2βπβγ minus 1))
2 minus 2sλ2γ
(2λ2
πs(minusM2N +Q2 + s) +m2
πQ2))
+((M2N + s(2βπβγ minus 1)
)2
minus 4λ2πs
2λ2γ
)((M2N + s(2βπβγ minus 1)
)(M2N minus 2Q2 minus s(2βπβγ + 1)
)+ 4λ2
πs2λ2
γ
)times log
(minusM2
N minus 2βπsβγ minus 2λπsλγ + s)minus(M2N + s(2βπβγ minus 2λπλγ minus 1)
)(M2N + s(minus1
+ 2βπβγ + 2λπλγ))[((
M2N + s(2βπβγ minus 1)
)(M2N minus 2Q2 minus s(2βπβγ + 1)
)+ 4λ2
πs2λ2
γ
)times log
(minusM2
N minus 2βπsβγ + 2λπsλγ + s)
+((Q2 + 2βπsβγ)2 minus 4λ2
πs2λ2
γ
)times log
(Q2 + 2βπsβγ minus 2λπsλγQ2 + 2sβπβγ + 2λπsλγ
)]+ 4sλ2
γ
((M2
N + s(2βπβγ minus 1))2 minus 4λ2πs
2λ2γ
)times (M2
N minus s) arctanh
(2λπsλγ
M2N + 2βπsβγ minus s
) (B19)
34
σ(π+n)LT =
e2g2AM
3Nλπ
32πf2πQs
3(sminusM2N )2λ4
γ
2sλγ
[(M2
N minus s)(Q2 + 2sβ2γ)(Q2 + 2sβγβπ)
minus 4s
((M2
N minus s)(Q2 minus 2s(απ minus 1)βγ
)+Q2sβγβπ
)λ2γ + 8s3(minus1 + απ)λ4
γ
]minus M2
N minus sλπ
[(Q2 + 2sβ2
γ)(Q2 + 2sβγβπ)2 + 4s2(2(απ minus 1)βγ(Q2 + 2sβγβπ)
minusQ2λ2π
)λ2γ + 8s3(απ minus 1)2λ4
γ
]arctan
(2sλπλγ
Q2 + 2sβγβπ
) (B20)
σ(π0p)LT =
e2g2AM
3Nλπ
16πf2πQs(sminusM2
N )2λγ
1
minus2s(M2N + s(minus1 + 2βγβπ))2λ2
γ + 8s3λ2πλ
4γ
[minus 3M8
N (Q2
+ 2sβ2γ) + 2M4
Ns2
(minus (Q2 + 2sβ2
γ)(minus3 + 2βγβπ) +(Q2(minus3 + 2βγβπ + 6λ2
π) + 2sβγ
times (minus3 + 12απ + 2βγβπ minus 12απβγβπ + 4βγλ2π))λ2γ minus 4s(α2
π minus 1)λ4π
)+ 2M2
Ns3
[minus (Q2
+ 2sβ2γ)(minus1 + 2βγβπ)
(6 + βγβπ(minus9 + 2βγβπ)
)+
(Q2(3minus 12λ2
π + 4βγβπ(minus1 + 2βγβπ
+ λ2π))minus 2sβγ
(minus 3 + 4απ(3 + 2απ(3 + 2βγβπ(minus3 + βγβπ)) + 4βγ(βπ + (2minus βγβπ)
times λ2π)))λ2γ + 8s
((απ minus 1)(1 + απ minus 2βγβπ) + 2απβγλ
2π
)λ4γ
)+ s4
((Q2 + 2sβ2
γ)
times (1minus 2βγβπ)2(2βγβπ minus 3) + 2(Q2(minus1 + 6λ2
π + 2βγβπ((1minus 2βγβπ)2 minus 2λ2π)) + 2sβγ
times (minus1 + 2βγβπ + 4(minus1 + βγβπ)(minusαπ + 2απβγβπ minus βγλ2π)))λ2γ minus 8
(s(απ minus 1)(απ
+ (1minus 2βγβπ)2) + 2βγ(2sαπ +Q2βγ)λ2π
)λ4γ + 32s(απ minus 1)λ2
πλ6γ
)+ 2M6
Ns
(Q2(6
minus 7βγβπ + λ2γ) + 2sβγ(βγ(6minus 7βγβπ) + (1minus 4απ)λ2
γ)
)]+sminusM2
N
4s2λπλ3γ
[(Q2 + 2sβ2
γ)
times(3M2
N + s(2βγβπ minus 3))(M2N + s(2βγβπ minus 1)
)+ 2s
(minusM2
N (Q2 + 2sβγ minus 8sαπβγ)
+ s(2sβγ(1minus 4απ + 4απβγβπ) +Q2(1minus 2λ2π)))λ2γ + 8s3(απ minus 1)λ4
γ
]
times arctan
(2sλπλγ
M2N + s(2βγβπ minus 1)
) (B21)
35
σ(πminusp)LT =
e2g2AM
3N
64πf2πQs
3λ4γ(M2
N minusQ2 minus s)(M2N + s(2βπβγ + 2λπλγ minus 1))
times
1
M2N + s(2βπβγ minus 2λπλγ minus 1)
[4λπsλγ(M2
N minusQ2 minus s)(
2sλ2γ
(s((1minus 2λ2
π)Q2
+ 2(2απ minus 1)sβγ(2βπβγ minus 1))minusM2N (Q2 + 2(1minus 2απ)sβγ)
)+ (Q2 + 2sβ2
γ)(M2N + s(2βπβγ minus 1)
)2 + 8(απ minus 1)απs
3λ4γ
)]minus(M2N + s(2βπβγ + 2λπλγ minus 1)
)times[
log
(minusM2N minus 2βπsβγ minus 2λπsλγ + s
minusM2N minus 2βπsβγ + 2λπsλγ + s
)(minus (Q2 + 2sβ2
γ)(M2N + s(2βπβγ minus 1)
)(minusM2
N + 2Q2 + 2βπsβγ + s)minus 2sλ2γ
(minusM2
N (Q2 + 2sβγ) +Q4 +Q2s(4απβγ
minus 2βγ minus 2λ2π + 1) + 2s2βγ(4(απ minus 1)βπβγ + 1)
)minus 8(απ minus 1)2s3λ4
γ
)+ 2
(4s2λ2
γ
(2(απ minus 1)βγ(Q2 + 2βπsβγ)minus λ2
πQ2)
+ (Q2 + 2βπsβγ)2(Q2 + 2sβ2γ)
+ 8(απ minus 1)2s3λ4γ
)arctanh
(2λπsλγ
Q2 + 2βπsβγ
)] (B22)
2 ∆-production channel
The tree-level ∆-exchange diagram in Fig 2 of Ref [30] contributes to the non-Born part of
the VVCS amplitudes The contribution of the ∆ exchange to the VVCS amplitudes can be split
into [17]
S∆-exch1 (νQ2) = S∆-pole
1 (νQ2) + S∆-exch1 (νQ2) (B23a)
νS∆-exch2 (νQ2) = νS∆-pole
2 (νQ2) + νS2∆-exch
(νQ2) (B23b)
and similarly for the unpolarized VVCS amplitudes discussed in Ref [30] Here we introduced the
∆-pole contributions S∆-polei and the ∆-non-pole contributions S∆-exch
i The former amplitudes
feature a pole at the ∆(1232)-production threshold and thus are proportional to
1
[sminusM2∆][uminusM2
∆]=
1
4M2N
1
ν2∆ minus ν2
(B24)
36
They can be reconstructed from the dispersion relations in Eq (5) with the tree-level ∆-production
cross sections as input cf Fig 10 in Ref [30]
σTT (νQ2) =π2α
M2NM
2+|~q |
minus g2
MMN (M+ + ν)|~q |2 +g2E(∆minus ν)(Q2 minusMNν)2
MN(B25a)
+g2CQ
4s(∆minus ν)
MNM2∆
minus 4gMgE(Q2 minusMNν)|~q |2 minus 4gMgCQ2|~q |2
+2gEgCQ
2[minusMNM∆ |~q |2 + s(Q2 +∆ν)
]MNM∆
δ(ν minus ν∆)
σLT (νQ2) =Qπ2α
M2NM
2+|~q |
g2E(MNν minusQ2) [M∆(MN + ν)minus s]
MN(B25b)
+g2CQ
2[MNM∆|~q |2 minus s(Q2 +∆ν)
]MNM2
∆
+ gMgEM∆|~q |2 minusgMgC(Q2 minusMNν)|~q |2
M∆
+gEgC(ν minus∆)(M2
N |~q |2 minus 2Q2s)
MNM∆
δ(ν minus ν∆)
with ∆ = M∆ minusMN M+ = M∆ + MN and the Mandelstam variable s = M2N + 2MNν minus Q2
Analytical expressions for the spin structure functions g1(xQ2) and g2(xQ2) can be constructed
from Eq (3) with the flux factor K(νQ2) = |~q | =radicν2 +Q2
In the ∆-non-pole contributions to S1(νQ2) and νS2(νQ2) the pole in ν at the ∆(1232)-
production threshold has canceled out
S∆-exch1 (νQ2) =
πα
MNM2+
[g2MQ
2+ + g2
E
(∆2 minus 3Q2
)+
4g2CQ
4
M2∆
minus 8gMgEM∆ωminus (B26a)
minus2gMgCQ2(MN minus 4M∆)
M∆+
2gEgCQ2(3MN minus 2M∆)
M∆
]
νS2∆-exch
(νQ2) =2πα
MNM2+
[g2EM∆∆ωminus +
g2M MNQ
2+
2+g2C Q
2(Q2 minus∆2)
2M∆(B26b)
+gEgM M∆(M∆ω+ minus 4MNωminus)minus gEgC ∆(2Q2 +MNω+)
+gMgC Q2(4MN minus ω+)
]+S∆-exch
2 (νQ2)
ν
[M2
∆ ω2+
M2N
+ ν2
]
with Q+ =radic
(M∆ +MN )2 +Q2 and ωplusmn = (M2∆minusM2
N plusmnQ2)2M∆ and the non-pole contribution
to S2(νQ2)
S∆-exch2 (νQ2) = minus2παMNν
M∆M2+
[gM + gE
]gC (B27)
These amplitudes to the contrary are not described by the tree-level ∆-production cross sections
in the standard dispersive approach [17] This peculiarity has been previously missed e g in the
calculation of the ∆-exchange contribution to the hydrogen hyperfine splitting in Ref [88] The
importance of including the ∆-non-pole contribution is also evident when considering the BC sum
37
rule in Eq (14) The ∆-pole terms by themselves violate the BC sum rule but cancel exactly with
the ∆-non-pole terms
limνrarr0
νS∆-pole2 (νQ2) + lim
νrarr0νS2
∆-exch(νQ2) = 0 (B28)
Appendix C Polarizabilities at Q2 = 0
In this section we give analytical expressions for the static values and slopes at Q2 = 0 of
the spin polarizabilities and moments of polarized structure functions In particular we give the
HB expansion of the πN -loop contributions and the ∆-exchange contributions The complete
expressions also for the π∆-loop contributions can be found in the Supplemented material Recall
that IA(0) = I1(0) = d2(0) = 0 and dd2(Q2)dQ2
∣∣∣Q2=0
= 0
1 πN-loop contribution
Here we give analytical expressions for the πN -loop contributions to the proton and neutron
spin polarizabilities expanded in powers of micro = mπMN viz the HB expansion Note that we
choose to expand here to a high order in micro the strict HB expansion would only retain the leading
term in an analogous NLO calculation
bull Static polarizabilities (Q2 = 0)
γ0p =e2g2
A
96π3f2πm
2π
1minus 21πmicro
8minus(
59
2+ 26 logmicro
)micro2 +
1875πmicro3
64
+3
(3
2+ 26 logmicro
)micro4 +
(C1)
γ0n =e2g2
A
48π3f2πm
2π
1
2minus 9πmicro
16minus 2micro2 logmicro+
75πmicro3
128minus 3micro4
4+
(C2)
δLTp =e2g2
A
192π3f2πm
2π
1minus 9πmicro
8+
(13
2minus 2 logmicro
)micro2 minus 465πmicro3
64
minus(
47
2+ 42 logmicro
)micro4 +
(C3)
δLTn =e2g2
A
96π3f2πm
2π
1
2+
3πmicro
16+ (1 + 2 log micro)micro2 minus 105πmicro3
128+
5micro4
4+
(C4)
38
δLTn =e2g2
A
96π3f2πm
2π
1
2+
3πmicro
16+ (1 + 2 log micro)micro2 minus 105πmicro3
128+
5micro4
4+
(C5)
γ0p =e2g2
A
16π3f2πm
4π
4
45minus 3πmicro
16+
14micro2
5minus 1813πmicro3
384minus 192
5(1 + log micro)micro4
+80703πmicro5
2048+
(C6)
γ0n =e2g2
A
16π3f2πm
4π
4
45minus 5πmicro
48+
4micro2
5minus 245πmicro3
384minus 32micro4 logmicro
15
+1323πmicro5
2048+
(C7)
bull Slopes of polarizabilities at Q2 = 0
dγ0p(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
2minus 45πmicro
4+ 223micro2 minus 28515πmicro3
64
minus9
(1953
4+ 449 logmicro
)micro4 +
570255πmicro5
128+
(C8)
dγ0n(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
2minus 81πmicro
8+ 94micro2 minus 2535πmicro3
32
minus3 (1 + 90 logmicro)micro4 +84315πmicro5
1024+
(C9)
dδLTp(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
2880π3f2πm
4π
minus5
2minus 27πmicro
32+ 20micro2 minus 5865πmicro3
256
+3
(617
4+ 36 logmicro
)micro4 minus 2056845πmicro5
4096+
(C10)
dδLTn(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
minus5
4minus 81πmicro
64minus 11micro2 +
10005πmicro3
512
+15
8(11 + 48 logmicro)micro4 minus 267015πmicro5
8192+
(C11)
dIAp(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
96π2f2πmicro
2
1minus 15πmicro
4minus 1
2(115 + 88 logmicro)micro2 +
1839πmicro3
32
+5 (5 + 34 logmicro)micro4 +
(C12)
dIAn(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
48π2f2πmicro
2
1
2minus 11πmicro
8minus 1
4(1 + 20 logmicro)micro2 +
99πmicro3
64
minus25micro4
12+
(C13)
dI1p(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
96π2f2πmicro
3π
8+ 2 (4 + 3 logmicro)microminus 537πmicro2
64
minus1
2(15 + 56 logmicro)micro3 +
(C14)
39
dI1n(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
48π2f2πmicro
minus π
16+
1
4(3 + 4 logmicro)microminus 57πmicro2
128+
2micro3
3+
(C15)
γ0p(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2g2
A
16π3f2πm
6π
1
105minus 23πmicro
256+
377micro2
210minus 15551πmicro3
6144+
3371micro4
105
minus1640457πmicro5
32768+
(C16)
γ0n(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2g2
A
16π3f2πm
6π
1
105minus 153πmicro
1792+
69micro2
70minus 4615πmicro3
6144+
172micro4
35
minus120897πmicro5
32768+
(C17)
2 ∆-exchange contribution
Here we give analytical expressions for the tree-level ∆-exchange contributions to the nucleon
spin polarizabilities and their slopes at Q2 = 0 Note that the ∆-exchange contributes equally to
proton and neutron polarizabilities Recall that for the magnetic γlowastN∆ coupling we introduced a
dipole form factor to mimic vector-meson dominance gM rarr gM(1 +Q2Λ2)2
bull Static polarizabilities (Q2 = 0)
γ0 = minus e2
4πM2+
(g2M
∆2+
g2E
M2+
minus 4gMgEM+∆
) (C18)
δLT =e2M∆
4πM3+
(g2E
MNM++gMgE∆MN
minus gEgCM2
∆
) (C19)
γ0 =e2M2
N
π∆2M4+
(minusg
2M
∆2+
g2E
M2+
+4gMgE∆M+
) (C20)
bull Slopes of polarizabilities at Q2 = 0
dγ0(Q2)
dQ2
∣∣∣∣∣Q2=0
= minus e2
πM2+∆
(g2M
∆
[1
4∆2minus 1
∆M++
1
2M2+
]minus 1
Λ2
g2M
∆+
g2E
2M2+
[1
2∆minus 3
M+
]minusgMgE
M+
[1
∆2minus 5
∆M++
1
M2+
]+
1
Λ2
2gMgEM+
+2gMgC∆M2
+
minus gEgCM3
+
) (C21)
40
d δLT (Q2)
dQ2
∣∣∣∣∣Q2=0
=e2M∆∆
4πMNM2+
(g2E
∆2M2+
[1
∆minus 4
M+
]minus g2
C
∆M2∆M
2+
+gMgE∆2M+
[1
∆2
minus 3
∆M++
1
M2+
]minus 2
Λ2
gMgE∆2M+
+gMgC∆M2
∆
[1
2∆2minus 2
∆M++
1
2M2+
]minus gEgC
2M2∆M
2+
[7
∆+
1
M+
]) (C22)
dIA(Q2)
dQ2
∣∣∣∣∣Q2=0
= minusM2N
M2+
(g2M
2∆2+
g2E
MNM+minus 2gMgE
∆M+minus gEgCM∆M+
) (C23)
dI1(Q2)
dQ2
∣∣∣∣∣Q2=0
= minusM∆M2N
2M3+
(g2E
MNM∆minus gMgE∆MN
minus gEgCM2
∆
) (C24)
γ0(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2M2
N
π∆3M6+
(g2M
∆
[3M2
∆ + 2M∆MN minus 9M2N
∆2+
4M2+
Λ2
]
minus4gMgE
[5M2
∆ minus 9M2N
M+∆2+
2M+
Λ2
]+
g2E
∆M+[7M∆ minus 9MN ]
minus8gMgC∆
+4gEgCM+
) (C25)
[1] D Drechsel B Pasquini and M Vanderhaeghen Dispersion relations in real and virtual
Compton scattering Phys Rept 378 99 (2003) hep-ph0212124
[2] S E Kuhn J-P Chen and E Leader Spin structure of the nucleon mdash status and recent
results Prog Part Nucl Phys 63 1 (2009) arXiv08123535 [hep-ph]
[3] F Hagelstein R Miskimen and V Pascalutsa Nucleon polarizabilities from Compton scat-
tering to hydrogen atom Prog Part Nucl Phys 88 29 (2016) arXiv151203765 [nucl-th]
[4] B Pasquini and M Vanderhaeghen Dispersion theory in electromagnetic interactions Ann
Rev Nucl Part Sci 68 75 (2018) arXiv180510482 [hep-ph]
[5] S Gerasimov A Sum rule for magnetic moments and the damping of the nucleon magnetic
moment in nuclei Sov J Nucl Phys 2 430 (1966)
[6] S Drell and A C Hearn Exact sum rule for nucleon magnetic moments Phys Rev Lett
16 908 (1966)
[7] J S Schwinger Source theory viewpoints in deep inelastic scattering Proc Natl Acad Sci
USA 72 1 (1975)
[8] J S Schwinger Source theory viewpoints in deep inelastic scattering Electromagnetic In-
teractions and Field Theory Proceedings 14 Internationale Universitatswochen Schladming
Austria February 24-March 7 1975 Acta Phys Austriaca Suppl 14 471 (1975)
41
[9] J Schwinger Source theory discussion of deep inelastic scattering with polarized particles
Proc Natl Acad Sci USA 72 1559 (1975)
[10] A M Baldin Polarizability of nucleons Nucl Phys 18 310 (1960)
[11] M Gell-Mann M L Goldberger and W E Thirring Use of causality conditions in quantum
theory Phys Rev 95 1612 (1954)
[12] A Pineda Leading chiral logarithms to the hyperfine splitting of the hydrogen and muonic
hydrogen Phys Rev C 67 025201 (2003)
[13] C Peset and A Pineda The two-photon exchange contribution to muonic hydrogen from
chiral perturbation theory Nucl Phys B 887 69 (2014) arXiv14064524 [hep-ph]
[14] C Peset and A Pineda Model-independent determination of the two-photon exchange contri-
bution to hyperfine splitting in muonic hydrogen JHEP 04 060 arXiv161205206 [nucl-th]
[15] F Hagelstein and V Pascalutsa Proton structure in the hyperfine splitting of muonic hydro-
gen PoS CD15 077 (2016) arXiv151104301 [nucl-th]
[16] F Hagelstein Exciting Nucleons in Compton Scattering and Hydrogen-Like Atoms PhD
thesis Mainz U Inst Kernphys (2017) arXiv171000874 [nucl-th]
[17] F Hagelstein ∆(1232)-Resonance in the hydrogen spectrum Proceedings 11th International
Workshop on the Physics of Excited Nucleons (NSTAR 2017) Columbia SC USA August
20-23 2017 Few Body Syst 59 93 (2018) arXiv180109790 [nucl-th]
[18] Y Prok et al (CLAS) Moments of the spin structure functions gp1 and gd1 for 005 lt Q2 lt 30
GeV2 Phys Lett B 672 12 (2009) arXiv08022232 [nucl-ex]
[19] H Dutz et al (GDH) First measurement of the Gerasimov-Drell-Hearn sum rule for 1H from
07 GeV to 18 GeV at ELSA Phys Rev Lett 91 192001 (2003)
[20] M Amarian et al (Jefferson Lab E94010) Measurement of the generalized forward spin po-
larizabilities of the neutron Phys Rev Lett 93 152301 (2004) arXivnucl-ex0406005
[21] M Amarian et al The Q2 evolution of the generalized Gerasimov-Drell-Hearn integral for
the neutron using a 3He target Phys Rev Lett 89 242301 (2002) arXivnucl-ex0205020
[22] M Amarian et al (Jefferson Lab E94-010)Q2 evolution of the neutron spin structure moments
using a 3He target Phys Rev Lett 92 022301 (2004) arXivhep-ex0310003
[23] A Deur et al Experimental determination of the evolution of the Bjorken integral at low Q2
Phys Rev Lett 93 212001 (2004) arXivhep-ex0407007
[24] K Slifer Low Q2 measurement of gp2 and the δLT spin polarizability Spin structure at long dis-
tance Proceedings Workshop Newport News USA March 12ndash13 2009 nucl-ex09064775
(2009) AIP Conf Proc 1155 10106313203293 (2009) arXiv09064775 [nucl-ex]
[25] P Solvignon et al (E01-012) Moments of the neutron g2 structure function at intermediate
42
Q2 Phys Rev C 92 015208 (2015) arXiv13044497 [nucl-ex]
[26] A Deur Experimental studies at low Q2 of the spin structure of the nucleon at Jefferson Lab
in 9th International Workshop on Chiral Dynamics (CD18) Durham NC USA September
17-21 2018 (2019) arXiv190305661 [nucl-ex]
[27] V Sulkosky et al (Jefferson Lab E97-110) Measurement of the 3He spin-structure functions
and of neutron (3He) spin-dependent sum rules at 0035 le Q2 le 024 GeV2 Phys Lett B
805 135428 (2020) arXiv190805709 [nucl-ex]
[28] K Adhikari et al (CLAS) Measurement of the Q2 Dependence of the Deuteron Spin Structure
Function g1 and its Moments at Low Q2 with CLAS Phys Rev Lett 120 062501 (2018)
arXiv171101974 [nucl-ex]
[29] V Lensky F Hagelstein A Hiller Blin and V Pascalutsa Comment on rdquoMeasurement of
the Q2 Dependence of the Deuteron Spin Structure Function g1 and its Moments at Low Q2
with CLASrdquo (2018) arXiv180603219 [nucl-th]
[30] J M Alarcn F Hagelstein V Lensky and V Pascalutsa Forward doubly-virtual Compton
scattering off the nucleon in chiral perturbation theory at NLO the subtraction function and
moments of unpolarized structure functions (2020) arXiv200509518 [hep-ph]
[31] V Bernard E Epelbaum H Krebs and U-G Meiszligner New insights into the spin structure
of the nucleon Phys Rev D 87 054032 (2013) arXiv12092523 [hep-ph]
[32] V Lensky J M Alarcon and V Pascalutsa Moments of nucleon structure functions at
next-to-leading order in baryon chiral perturbation theory Phys Rev C 90 055202 (2014)
arXiv14072574 [hep-ph]
[33] V Pascalutsa and M Vanderhaeghen Chiral effective-field theory in the ∆(1232) region
I Pion electroproduction on the nucleon Phys Rev D 73 034003 (2006) arXivhep-
ph0512244
[34] V Pascalutsa and M Vanderhaeghen Electromagnetic nucleon-to-Delta transition in chiral
effective field theory Phys Rev Lett 95 232001 (2005) arXivhep-ph0508060
[35] T R Hemmert B R Holstein and J Kambor Systematic 1M expansion for spin 32
particles in baryon chiral perturbation theory Phys Lett B 395 89 (1997) arXivhep-
ph9606456
[36] V Pascalutsa and D R Phillips Effective theory of the ∆(1232) in Compton scattering off
the nucleon Phys Rev C 67 055202 (2003) arXivnucl-th0212024
[37] V Pascalutsa M Vanderhaeghen and S N Yang Electromagnetic excitation of the ∆(1232)-
resonance Phys Rept 437 125 (2007) arXivhep-ph0609004
[38] K A Olive et al (Particle Data Group) Review of Particle Physics Chin Phys C 38 090001
43
(2014)
[39] F E Low Scattering of light of very low frequency by systems of spin 12 Phys Rev 96
1428 (1954)
[40] M Gell-Mann and M L Goldberger Scattering of low-energy photons by particles of spin
12 Phys Rev 96 1433 (1954)
[41] A M Harun ar-Rashid A simple derivation of Schwingerrsquos sum rule for spin dependent
structure functions Nuovo Cim A 33 447 (1976)
[42] F Hagelstein and V Pascalutsa Dissecting the hadronic contributions to (g minus 2)micro by
Schwingerrsquos sum rule Phys Rev Lett 120 072002 (2018) arXiv171004571 [hep-ph]
[43] V Lensky V Pascalutsa M Vanderhaeghen and C Kao Spin-dependent sum rules con-
necting real and virtual Compton scattering verified Phys Rev D 95 074001 (2017)
arXiv170101947 [hep-ph]
[44] H Burkhardt and W N Cottingham Sum rules for forward virtual Compton scattering
Annals Phys 56 453 (1970)
[45] V Lensky and V Pascalutsa Predictive powers of chiral perturbation theory in Compton
scattering off protons Eur Phys J C 65 195 (2010) arXiv09070451 [hep-ph]
[46] H Grieszlighammer J McGovern D Phillips and G Feldman Using effective field theory to
analyse low-energy Compton scattering data from protons and light nuclei Prog Part Nucl
Phys 67 841 (2012) arXiv12036834 [nucl-th]
[47] H W Grieszlighammer J A McGovern and D R Phillips Nucleon polarisabilities at and
beyond physical pion masses Eur Phys J A 52 139 (2016) arXiv151101952 [nucl-th]
[48] E Epelbaum H Krebs and U-G Meiszligner Improved chiral nucleon-nucleon potential up to
next-to-next-to-next-to-leading order Eur Phys J A 51 53 (2015) arXiv14120142 [nucl-
th]
[49] E Epelbaum H Krebs and U-G Meiszligner Precision nucleon-nucleon potential at fifth order
in the chiral expansion Phys Rev Lett 115 122301 (2015) arXiv14124623 [nucl-th]
[50] C W Kao T Spitzenberg and M Vanderhaeghen Burkhardt-Cottingham sum rule and
forward spin polarizabilities in heavy baryon chiral perturbation theory Phys Rev D 67
016001 (2003) arXivhep-ph0209241
[51] C-W Kao D Drechsel S Kamalov and M Vanderhaeghen Higher moments of nucleon
spin structure functions in heavy baryon chiral perturbation theory and in a resonance model
Phys Rev D 69 056004 (2004) arXivhep-ph0312102
[52] T Becher and H Leutwyler Baryon chiral perturbation theory in manifestly Lorentz invariant
form Eur Phys J C 9 643 (1999) arXivhep-ph9901384
44
[53] J Gasser M E Sainio and A Svarc Nucleons with chiral loops Nucl Phys B 307 779
(1988)
[54] L S Geng J Martin Camalich L Alvarez-Ruso and M J Vicente Vacas Leading SU(3)-
breaking corrections to the baryon magnetic moments in chiral perturbation theory Phys
Rev Lett 101 222002 (2008) arXiv08051419 [hep-ph]
[55] D Drechsel S S Kamalov and L Tiator The GDH sum rule and related integrals Phys
Rev D 63 114010 (2001) arXivhep-ph0008306
[56] D Drechsel O Hanstein S S Kamalov and L Tiator A unitary isobar model for pion
photo- and electroproduction on the proton up to 1 GeV Nucl Phys A 645 145 (1999)
arXivnucl-th9807001
[57] L Tiator private communication (2020)
[58] V Bernard T R Hemmert and U-G Meiszligner Spin structure of the nucleon at low energies
Phys Rev D 67 076008 (2003) arXivhep-ph0212033
[59] O Gryniuk F Hagelstein and V Pascalutsa Evaluation of the forward Compton scattering
off protons II Spin-dependent amplitude and observables Phys Rev D 94 034043 (2016)
arXiv160400789 [nucl-th]
[60] R Zielinski The g2p Experiment A Measurement of the Protonrsquos Spin Structure Functions
PhD thesis New Hampshire U (2010) arXiv170808297 [nucl-ex]
[61] N Guler et al (CLAS) Precise determination of the deuteron spin structure at low to mod-
erate Q2 with CLAS and extraction of the neutron contribution Phys Rev C 92 055201
(2015) arXiv150507877 [nucl-ex]
[62] C E Carlson V Nazaryan and K Griffioen Proton structure corrections to electronic
and muonic hydrogen hyperfine splitting Phys Rev A 78 022517 (2008) arXiv08052603
[physicsatom-ph]
[63] V Bernard N Kaiser and U-G Meiszligner Chiral dynamics in nucleons and nuclei Int J
Mod Phys E 4 193 (1995) arXivhep-ph9501384
[64] T R Hemmert B R Holstein and J Kambor ∆(1232) and the polarizabilities of the nucleon
Phys Rev D 55 5598 (1997) arXivhep-ph9612374
[65] V Pascalutsa and R Timmermans Field theory of nucleon to higher-spin baryon transitions
Phys Rev C 60 042201 (1999) arXivnucl-th9905065
[66] V Pascalutsa Quantization of an interacting spin-32 field and the Delta isobar Phys Rev
D 58 096002 (1998) arXivhep-ph9802288
[67] H Krebs Double Virtual Compton Scattering and SpinStructure of the Nucleon PoS
CD2018 031 (2019)
45
[68] N Kochelev and Y Oh Axial anomaly and the δLT puzzle Phys Rev D 016012 (2012)
arXiv11034892 [hep-ph]
[69] D Drechsel S Kamalov and L Tiator Unitary isobar model ndash MAID2007 Eur Phys J A
69 (2007) available at httpsmaidkphuni-mainzde arXiv07100306 [nucl-th]
[70] P J Mohr B N Taylor and D B Newell CODATA recommended values of the fundamental
physical constants 2010 Rev Mod Phys 84 1527 (2012)
[71] J Ahrens et al (GDH A2) First measurement of the Gerasimov-Drell-Hearn integral for 1H
from 200 to 800 MeV Phys Rev Lett 87 022003 (2001) arXivhep-ex0105089 [hep-ex]
[72] K Helbing (GDH) Experimental verification of the GDH sum rule at ELSA and MAMI Nucl
Phys Proc Suppl 105 113 (2002)
[73] R Bradford A Bodek H S Budd and J Arrington A New parameterization of the nucleon
elastic form-factors NuInt05 proceedings of the 4th International Workshop on Neutrino-
Nucleus Interactions in the Few-GeV Region Okayama Japan 26-29 September 2005 Nucl
Phys Proc Suppl 159 127 (2006) arXivhep-ex0602017
[74] J D Bjorken Applications of the chiral U(6)timesU(6) algebra of current densities Phys Rev
148 1467 (1966)
[75] J D Bjorken Inelastic scattering of polarized leptons from polarized nucleons Phys Rev D
1 1376 (1970)
[76] V Bernard T R Hemmert and U-G Meiszligner Novel analysis of chiral loop effects in
the generalized Gerasimov-Drell-Hearn sum rule Phys Lett B 545 105 (2002) arXivhep-
ph0203167
[77] A Deur et al Experimental study of isovector spin sum rules Phys Rev D 78 032001
(2008) arXiv08023198 [nucl-ex]
[78] B Pasquini P Pedroni and D Drechsel Higher order forward spin polarizability Phys Lett
B 687 160 (2010) arXiv10014230 [hep-ph]
[79] R Jaffe g2ndashThe nucleonrsquos other spin-dependent structure function Comments Nucl Part
Phys 19 239 (1990)
[80] E V Shuryak and A Vainshtein Theory of power corrections to deep inelastic scattering in
quantum chromodynamics (II) Qminus4 effects polarized target Nucl Phys B 201 141 (1982)
[81] S Wandzura and F Wilczek Sum rules for spin dependent electroproduction Test of rela-
tivistic constituent quarks Phys Lett B 72 195 (1977)
[82] B W Filippone and X-D Ji The spin structure of the nucleon Adv Nucl Phys 26 1
(2001) arXivhep-ph0101224
[83] M Burkardt The g2 structure function Proceedings Workshop on Spin structure at long
46
distance Newport News USA March 12-13 2009 AIP Conf Proc 1155 26 (2009)
arXiv09054079 [hep-ph]
[84] A Deur S J Brodsky and G F de Teramond The spin structure of the nucleon Rept
Prog Phys 82 076201 (2019) arXiv180705250 [hep-ph]
[85] B R Holstein D Drechsel B Pasquini and M Vanderhaeghen Higher order polarizabilities
of the proton Phys Rev C 61 034316 (2000) arXivhep-ph9910427
[86] B R Holstein V Pascalutsa and M Vanderhaeghen Sum rules for magnetic moments and
polarizabilities in QED and chiral effective-field theory Phys Rev D 72 094014 (2005)
arXivhep-ph0507016
[87] B Pasquini D Drechsel and L Tiator Invariant amplitudes for pion electroproduction Eur
Phys J A 34 387 (2007) arXiv07122327 [hep-ph]
[88] A J Buchmann Non-spherical proton shape and hydrogen hyperfine splitting Proceedings
International Workshop on Precision Physics of Simple Atomic Systems (PSAS 2008) Wind-
sor Ontario Canada July 21-26 2008 Can J Phys 87 773 (2009) arXiv09104747
[physicsatom-ph]
47
sections are given below (proton channels π+n and π0p neutron channel πminusp) We checked that
they reproduce the known results in the real-photon limit [45 86] To shorten the final expressions
for the cross sections which are considerably longer for finite Q2 than in the real-photon limit we
define the following dimensionless kinematic variables
αγ = (ENi )cmradics =
s+M2N +Q2
2s (B11)
απ = (ENf )cmradics =
s+M2N minusm2
π
2s (B12)
βγ = Eγcmradics =
sminusM2N minusQ2
2s (B13)
βπ = Eπcmradics =
sminusM2N +m2
π
2s (B14)
λγ = |~qi|cmradics =
radic(sminusM2
N minusQ2)2 + 4sQ2
2s (B15)
λπ = |~qf |cmradics =
radic(sminusM2
N +m2π)2 minus 4sm2
π
2s (B16)
Here (ENi )cm and (ENf )cm are the energies of the incoming and outgoing nucleon Eγcm is the energy
of the incoming photon Eπcm is the energy of the outgoing pion all in the cm frame
σ(π+n)TT = minus e2g2
AM2N
64πf2πs
3(sminusM2N )2λ4
γ
4sλπλγ
[(M2
N minus s)(M2N minusQ2 minus s)(Q2 + 2sβγβπ)
+ 2s(m2πQ
2 minus (M2N minus s)
(M2N + s(minus1 + 2βγβπ)
))λ2γ
]+ (M2
N minus s)(M2N minusQ2 minus s)
times (Q2 + 2sβγβπ minus 2sλπλγ)(Q2 + 2s(βγβπ + λπλγ)
)times log
(Q2 + 2sβγβπ minus 2sλγλπQ2 + 2sβγβπ + 2sλγλπ
) (B17)
33
σ(π0p)TT =
e2g2AM
2Nλπ
64πf2πs(sminusM2
N )2λγ
1
s2λπλ3γ
[(M2
N minus s)(M2N minusQ2 minus s)
(3M2
N + s(2βγβπ minus 3))(M2N
+ s(2βγβπ minus 1))
+ 2s
((2m2
πQ2 + (M2
N minus s)2)(M2
N minus s)minus 2(minusm2
πQ2 + (sminusM2
N )2)
times sβγβπ + 4s2(sminusM2N )β2
γβ2π
)λ2γ
]arctan
(2sλπλγ
M2N + s(2βγβπ minus 1)
)+ 2
[minus 2m2
πQ2
+ (M2N minus s)
(2(sminusM2
N ) + 4sβγβπ +(sminusM2
N +Q2)(3M2N minus 3s+ 2sβγβπ)
sλ2γ
+m2πQ
2(sminusM2N )
(M2N minus s+ 2sβγβπ)(M2
N + s(minus1 + 2βγβπ minus 2λπλγ))
+m2πQ
2(sminusM2N )
(M2N minus s+ 2sβγβπ)(M2
N + s(minus1 + 2βγβπ + 2λπλγ))
)
+1
λπλγ
(βγβπ
(minusm2
πQ2 + (M2
N minus s)(M2N minus s+ 2sβγβπ)
)minus (M2
N minus s)(M2N minusQ2 minus s)λ2
π
)log
(M2N + s(minus1 + 2βγβπ + 2λπλγ)
M2N + s(minus1 + 2βγβπ minus 2λπλγ)
)] (B18)
σ(πminusp)TT =
e2g2AM
2N
64πf2πs
3λ4γ(M2
N + s(2βπβγ minus 1))2 minus 4λ2πs
2λ2γ)
4sλπλγ
((minusM2
N +Q2 + s)(M2N
+ s(2βπβγ minus 1))
2 minus 2sλ2γ
(2λ2
πs(minusM2N +Q2 + s) +m2
πQ2))
+((M2N + s(2βπβγ minus 1)
)2
minus 4λ2πs
2λ2γ
)((M2N + s(2βπβγ minus 1)
)(M2N minus 2Q2 minus s(2βπβγ + 1)
)+ 4λ2
πs2λ2
γ
)times log
(minusM2
N minus 2βπsβγ minus 2λπsλγ + s)minus(M2N + s(2βπβγ minus 2λπλγ minus 1)
)(M2N + s(minus1
+ 2βπβγ + 2λπλγ))[((
M2N + s(2βπβγ minus 1)
)(M2N minus 2Q2 minus s(2βπβγ + 1)
)+ 4λ2
πs2λ2
γ
)times log
(minusM2
N minus 2βπsβγ + 2λπsλγ + s)
+((Q2 + 2βπsβγ)2 minus 4λ2
πs2λ2
γ
)times log
(Q2 + 2βπsβγ minus 2λπsλγQ2 + 2sβπβγ + 2λπsλγ
)]+ 4sλ2
γ
((M2
N + s(2βπβγ minus 1))2 minus 4λ2πs
2λ2γ
)times (M2
N minus s) arctanh
(2λπsλγ
M2N + 2βπsβγ minus s
) (B19)
34
σ(π+n)LT =
e2g2AM
3Nλπ
32πf2πQs
3(sminusM2N )2λ4
γ
2sλγ
[(M2
N minus s)(Q2 + 2sβ2γ)(Q2 + 2sβγβπ)
minus 4s
((M2
N minus s)(Q2 minus 2s(απ minus 1)βγ
)+Q2sβγβπ
)λ2γ + 8s3(minus1 + απ)λ4
γ
]minus M2
N minus sλπ
[(Q2 + 2sβ2
γ)(Q2 + 2sβγβπ)2 + 4s2(2(απ minus 1)βγ(Q2 + 2sβγβπ)
minusQ2λ2π
)λ2γ + 8s3(απ minus 1)2λ4
γ
]arctan
(2sλπλγ
Q2 + 2sβγβπ
) (B20)
σ(π0p)LT =
e2g2AM
3Nλπ
16πf2πQs(sminusM2
N )2λγ
1
minus2s(M2N + s(minus1 + 2βγβπ))2λ2
γ + 8s3λ2πλ
4γ
[minus 3M8
N (Q2
+ 2sβ2γ) + 2M4
Ns2
(minus (Q2 + 2sβ2
γ)(minus3 + 2βγβπ) +(Q2(minus3 + 2βγβπ + 6λ2
π) + 2sβγ
times (minus3 + 12απ + 2βγβπ minus 12απβγβπ + 4βγλ2π))λ2γ minus 4s(α2
π minus 1)λ4π
)+ 2M2
Ns3
[minus (Q2
+ 2sβ2γ)(minus1 + 2βγβπ)
(6 + βγβπ(minus9 + 2βγβπ)
)+
(Q2(3minus 12λ2
π + 4βγβπ(minus1 + 2βγβπ
+ λ2π))minus 2sβγ
(minus 3 + 4απ(3 + 2απ(3 + 2βγβπ(minus3 + βγβπ)) + 4βγ(βπ + (2minus βγβπ)
times λ2π)))λ2γ + 8s
((απ minus 1)(1 + απ minus 2βγβπ) + 2απβγλ
2π
)λ4γ
)+ s4
((Q2 + 2sβ2
γ)
times (1minus 2βγβπ)2(2βγβπ minus 3) + 2(Q2(minus1 + 6λ2
π + 2βγβπ((1minus 2βγβπ)2 minus 2λ2π)) + 2sβγ
times (minus1 + 2βγβπ + 4(minus1 + βγβπ)(minusαπ + 2απβγβπ minus βγλ2π)))λ2γ minus 8
(s(απ minus 1)(απ
+ (1minus 2βγβπ)2) + 2βγ(2sαπ +Q2βγ)λ2π
)λ4γ + 32s(απ minus 1)λ2
πλ6γ
)+ 2M6
Ns
(Q2(6
minus 7βγβπ + λ2γ) + 2sβγ(βγ(6minus 7βγβπ) + (1minus 4απ)λ2
γ)
)]+sminusM2
N
4s2λπλ3γ
[(Q2 + 2sβ2
γ)
times(3M2
N + s(2βγβπ minus 3))(M2N + s(2βγβπ minus 1)
)+ 2s
(minusM2
N (Q2 + 2sβγ minus 8sαπβγ)
+ s(2sβγ(1minus 4απ + 4απβγβπ) +Q2(1minus 2λ2π)))λ2γ + 8s3(απ minus 1)λ4
γ
]
times arctan
(2sλπλγ
M2N + s(2βγβπ minus 1)
) (B21)
35
σ(πminusp)LT =
e2g2AM
3N
64πf2πQs
3λ4γ(M2
N minusQ2 minus s)(M2N + s(2βπβγ + 2λπλγ minus 1))
times
1
M2N + s(2βπβγ minus 2λπλγ minus 1)
[4λπsλγ(M2
N minusQ2 minus s)(
2sλ2γ
(s((1minus 2λ2
π)Q2
+ 2(2απ minus 1)sβγ(2βπβγ minus 1))minusM2N (Q2 + 2(1minus 2απ)sβγ)
)+ (Q2 + 2sβ2
γ)(M2N + s(2βπβγ minus 1)
)2 + 8(απ minus 1)απs
3λ4γ
)]minus(M2N + s(2βπβγ + 2λπλγ minus 1)
)times[
log
(minusM2N minus 2βπsβγ minus 2λπsλγ + s
minusM2N minus 2βπsβγ + 2λπsλγ + s
)(minus (Q2 + 2sβ2
γ)(M2N + s(2βπβγ minus 1)
)(minusM2
N + 2Q2 + 2βπsβγ + s)minus 2sλ2γ
(minusM2
N (Q2 + 2sβγ) +Q4 +Q2s(4απβγ
minus 2βγ minus 2λ2π + 1) + 2s2βγ(4(απ minus 1)βπβγ + 1)
)minus 8(απ minus 1)2s3λ4
γ
)+ 2
(4s2λ2
γ
(2(απ minus 1)βγ(Q2 + 2βπsβγ)minus λ2
πQ2)
+ (Q2 + 2βπsβγ)2(Q2 + 2sβ2γ)
+ 8(απ minus 1)2s3λ4γ
)arctanh
(2λπsλγ
Q2 + 2βπsβγ
)] (B22)
2 ∆-production channel
The tree-level ∆-exchange diagram in Fig 2 of Ref [30] contributes to the non-Born part of
the VVCS amplitudes The contribution of the ∆ exchange to the VVCS amplitudes can be split
into [17]
S∆-exch1 (νQ2) = S∆-pole
1 (νQ2) + S∆-exch1 (νQ2) (B23a)
νS∆-exch2 (νQ2) = νS∆-pole
2 (νQ2) + νS2∆-exch
(νQ2) (B23b)
and similarly for the unpolarized VVCS amplitudes discussed in Ref [30] Here we introduced the
∆-pole contributions S∆-polei and the ∆-non-pole contributions S∆-exch
i The former amplitudes
feature a pole at the ∆(1232)-production threshold and thus are proportional to
1
[sminusM2∆][uminusM2
∆]=
1
4M2N
1
ν2∆ minus ν2
(B24)
36
They can be reconstructed from the dispersion relations in Eq (5) with the tree-level ∆-production
cross sections as input cf Fig 10 in Ref [30]
σTT (νQ2) =π2α
M2NM
2+|~q |
minus g2
MMN (M+ + ν)|~q |2 +g2E(∆minus ν)(Q2 minusMNν)2
MN(B25a)
+g2CQ
4s(∆minus ν)
MNM2∆
minus 4gMgE(Q2 minusMNν)|~q |2 minus 4gMgCQ2|~q |2
+2gEgCQ
2[minusMNM∆ |~q |2 + s(Q2 +∆ν)
]MNM∆
δ(ν minus ν∆)
σLT (νQ2) =Qπ2α
M2NM
2+|~q |
g2E(MNν minusQ2) [M∆(MN + ν)minus s]
MN(B25b)
+g2CQ
2[MNM∆|~q |2 minus s(Q2 +∆ν)
]MNM2
∆
+ gMgEM∆|~q |2 minusgMgC(Q2 minusMNν)|~q |2
M∆
+gEgC(ν minus∆)(M2
N |~q |2 minus 2Q2s)
MNM∆
δ(ν minus ν∆)
with ∆ = M∆ minusMN M+ = M∆ + MN and the Mandelstam variable s = M2N + 2MNν minus Q2
Analytical expressions for the spin structure functions g1(xQ2) and g2(xQ2) can be constructed
from Eq (3) with the flux factor K(νQ2) = |~q | =radicν2 +Q2
In the ∆-non-pole contributions to S1(νQ2) and νS2(νQ2) the pole in ν at the ∆(1232)-
production threshold has canceled out
S∆-exch1 (νQ2) =
πα
MNM2+
[g2MQ
2+ + g2
E
(∆2 minus 3Q2
)+
4g2CQ
4
M2∆
minus 8gMgEM∆ωminus (B26a)
minus2gMgCQ2(MN minus 4M∆)
M∆+
2gEgCQ2(3MN minus 2M∆)
M∆
]
νS2∆-exch
(νQ2) =2πα
MNM2+
[g2EM∆∆ωminus +
g2M MNQ
2+
2+g2C Q
2(Q2 minus∆2)
2M∆(B26b)
+gEgM M∆(M∆ω+ minus 4MNωminus)minus gEgC ∆(2Q2 +MNω+)
+gMgC Q2(4MN minus ω+)
]+S∆-exch
2 (νQ2)
ν
[M2
∆ ω2+
M2N
+ ν2
]
with Q+ =radic
(M∆ +MN )2 +Q2 and ωplusmn = (M2∆minusM2
N plusmnQ2)2M∆ and the non-pole contribution
to S2(νQ2)
S∆-exch2 (νQ2) = minus2παMNν
M∆M2+
[gM + gE
]gC (B27)
These amplitudes to the contrary are not described by the tree-level ∆-production cross sections
in the standard dispersive approach [17] This peculiarity has been previously missed e g in the
calculation of the ∆-exchange contribution to the hydrogen hyperfine splitting in Ref [88] The
importance of including the ∆-non-pole contribution is also evident when considering the BC sum
37
rule in Eq (14) The ∆-pole terms by themselves violate the BC sum rule but cancel exactly with
the ∆-non-pole terms
limνrarr0
νS∆-pole2 (νQ2) + lim
νrarr0νS2
∆-exch(νQ2) = 0 (B28)
Appendix C Polarizabilities at Q2 = 0
In this section we give analytical expressions for the static values and slopes at Q2 = 0 of
the spin polarizabilities and moments of polarized structure functions In particular we give the
HB expansion of the πN -loop contributions and the ∆-exchange contributions The complete
expressions also for the π∆-loop contributions can be found in the Supplemented material Recall
that IA(0) = I1(0) = d2(0) = 0 and dd2(Q2)dQ2
∣∣∣Q2=0
= 0
1 πN-loop contribution
Here we give analytical expressions for the πN -loop contributions to the proton and neutron
spin polarizabilities expanded in powers of micro = mπMN viz the HB expansion Note that we
choose to expand here to a high order in micro the strict HB expansion would only retain the leading
term in an analogous NLO calculation
bull Static polarizabilities (Q2 = 0)
γ0p =e2g2
A
96π3f2πm
2π
1minus 21πmicro
8minus(
59
2+ 26 logmicro
)micro2 +
1875πmicro3
64
+3
(3
2+ 26 logmicro
)micro4 +
(C1)
γ0n =e2g2
A
48π3f2πm
2π
1
2minus 9πmicro
16minus 2micro2 logmicro+
75πmicro3
128minus 3micro4
4+
(C2)
δLTp =e2g2
A
192π3f2πm
2π
1minus 9πmicro
8+
(13
2minus 2 logmicro
)micro2 minus 465πmicro3
64
minus(
47
2+ 42 logmicro
)micro4 +
(C3)
δLTn =e2g2
A
96π3f2πm
2π
1
2+
3πmicro
16+ (1 + 2 log micro)micro2 minus 105πmicro3
128+
5micro4
4+
(C4)
38
δLTn =e2g2
A
96π3f2πm
2π
1
2+
3πmicro
16+ (1 + 2 log micro)micro2 minus 105πmicro3
128+
5micro4
4+
(C5)
γ0p =e2g2
A
16π3f2πm
4π
4
45minus 3πmicro
16+
14micro2
5minus 1813πmicro3
384minus 192
5(1 + log micro)micro4
+80703πmicro5
2048+
(C6)
γ0n =e2g2
A
16π3f2πm
4π
4
45minus 5πmicro
48+
4micro2
5minus 245πmicro3
384minus 32micro4 logmicro
15
+1323πmicro5
2048+
(C7)
bull Slopes of polarizabilities at Q2 = 0
dγ0p(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
2minus 45πmicro
4+ 223micro2 minus 28515πmicro3
64
minus9
(1953
4+ 449 logmicro
)micro4 +
570255πmicro5
128+
(C8)
dγ0n(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
2minus 81πmicro
8+ 94micro2 minus 2535πmicro3
32
minus3 (1 + 90 logmicro)micro4 +84315πmicro5
1024+
(C9)
dδLTp(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
2880π3f2πm
4π
minus5
2minus 27πmicro
32+ 20micro2 minus 5865πmicro3
256
+3
(617
4+ 36 logmicro
)micro4 minus 2056845πmicro5
4096+
(C10)
dδLTn(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
minus5
4minus 81πmicro
64minus 11micro2 +
10005πmicro3
512
+15
8(11 + 48 logmicro)micro4 minus 267015πmicro5
8192+
(C11)
dIAp(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
96π2f2πmicro
2
1minus 15πmicro
4minus 1
2(115 + 88 logmicro)micro2 +
1839πmicro3
32
+5 (5 + 34 logmicro)micro4 +
(C12)
dIAn(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
48π2f2πmicro
2
1
2minus 11πmicro
8minus 1
4(1 + 20 logmicro)micro2 +
99πmicro3
64
minus25micro4
12+
(C13)
dI1p(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
96π2f2πmicro
3π
8+ 2 (4 + 3 logmicro)microminus 537πmicro2
64
minus1
2(15 + 56 logmicro)micro3 +
(C14)
39
dI1n(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
48π2f2πmicro
minus π
16+
1
4(3 + 4 logmicro)microminus 57πmicro2
128+
2micro3
3+
(C15)
γ0p(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2g2
A
16π3f2πm
6π
1
105minus 23πmicro
256+
377micro2
210minus 15551πmicro3
6144+
3371micro4
105
minus1640457πmicro5
32768+
(C16)
γ0n(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2g2
A
16π3f2πm
6π
1
105minus 153πmicro
1792+
69micro2
70minus 4615πmicro3
6144+
172micro4
35
minus120897πmicro5
32768+
(C17)
2 ∆-exchange contribution
Here we give analytical expressions for the tree-level ∆-exchange contributions to the nucleon
spin polarizabilities and their slopes at Q2 = 0 Note that the ∆-exchange contributes equally to
proton and neutron polarizabilities Recall that for the magnetic γlowastN∆ coupling we introduced a
dipole form factor to mimic vector-meson dominance gM rarr gM(1 +Q2Λ2)2
bull Static polarizabilities (Q2 = 0)
γ0 = minus e2
4πM2+
(g2M
∆2+
g2E
M2+
minus 4gMgEM+∆
) (C18)
δLT =e2M∆
4πM3+
(g2E
MNM++gMgE∆MN
minus gEgCM2
∆
) (C19)
γ0 =e2M2
N
π∆2M4+
(minusg
2M
∆2+
g2E
M2+
+4gMgE∆M+
) (C20)
bull Slopes of polarizabilities at Q2 = 0
dγ0(Q2)
dQ2
∣∣∣∣∣Q2=0
= minus e2
πM2+∆
(g2M
∆
[1
4∆2minus 1
∆M++
1
2M2+
]minus 1
Λ2
g2M
∆+
g2E
2M2+
[1
2∆minus 3
M+
]minusgMgE
M+
[1
∆2minus 5
∆M++
1
M2+
]+
1
Λ2
2gMgEM+
+2gMgC∆M2
+
minus gEgCM3
+
) (C21)
40
d δLT (Q2)
dQ2
∣∣∣∣∣Q2=0
=e2M∆∆
4πMNM2+
(g2E
∆2M2+
[1
∆minus 4
M+
]minus g2
C
∆M2∆M
2+
+gMgE∆2M+
[1
∆2
minus 3
∆M++
1
M2+
]minus 2
Λ2
gMgE∆2M+
+gMgC∆M2
∆
[1
2∆2minus 2
∆M++
1
2M2+
]minus gEgC
2M2∆M
2+
[7
∆+
1
M+
]) (C22)
dIA(Q2)
dQ2
∣∣∣∣∣Q2=0
= minusM2N
M2+
(g2M
2∆2+
g2E
MNM+minus 2gMgE
∆M+minus gEgCM∆M+
) (C23)
dI1(Q2)
dQ2
∣∣∣∣∣Q2=0
= minusM∆M2N
2M3+
(g2E
MNM∆minus gMgE∆MN
minus gEgCM2
∆
) (C24)
γ0(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2M2
N
π∆3M6+
(g2M
∆
[3M2
∆ + 2M∆MN minus 9M2N
∆2+
4M2+
Λ2
]
minus4gMgE
[5M2
∆ minus 9M2N
M+∆2+
2M+
Λ2
]+
g2E
∆M+[7M∆ minus 9MN ]
minus8gMgC∆
+4gEgCM+
) (C25)
[1] D Drechsel B Pasquini and M Vanderhaeghen Dispersion relations in real and virtual
Compton scattering Phys Rept 378 99 (2003) hep-ph0212124
[2] S E Kuhn J-P Chen and E Leader Spin structure of the nucleon mdash status and recent
results Prog Part Nucl Phys 63 1 (2009) arXiv08123535 [hep-ph]
[3] F Hagelstein R Miskimen and V Pascalutsa Nucleon polarizabilities from Compton scat-
tering to hydrogen atom Prog Part Nucl Phys 88 29 (2016) arXiv151203765 [nucl-th]
[4] B Pasquini and M Vanderhaeghen Dispersion theory in electromagnetic interactions Ann
Rev Nucl Part Sci 68 75 (2018) arXiv180510482 [hep-ph]
[5] S Gerasimov A Sum rule for magnetic moments and the damping of the nucleon magnetic
moment in nuclei Sov J Nucl Phys 2 430 (1966)
[6] S Drell and A C Hearn Exact sum rule for nucleon magnetic moments Phys Rev Lett
16 908 (1966)
[7] J S Schwinger Source theory viewpoints in deep inelastic scattering Proc Natl Acad Sci
USA 72 1 (1975)
[8] J S Schwinger Source theory viewpoints in deep inelastic scattering Electromagnetic In-
teractions and Field Theory Proceedings 14 Internationale Universitatswochen Schladming
Austria February 24-March 7 1975 Acta Phys Austriaca Suppl 14 471 (1975)
41
[9] J Schwinger Source theory discussion of deep inelastic scattering with polarized particles
Proc Natl Acad Sci USA 72 1559 (1975)
[10] A M Baldin Polarizability of nucleons Nucl Phys 18 310 (1960)
[11] M Gell-Mann M L Goldberger and W E Thirring Use of causality conditions in quantum
theory Phys Rev 95 1612 (1954)
[12] A Pineda Leading chiral logarithms to the hyperfine splitting of the hydrogen and muonic
hydrogen Phys Rev C 67 025201 (2003)
[13] C Peset and A Pineda The two-photon exchange contribution to muonic hydrogen from
chiral perturbation theory Nucl Phys B 887 69 (2014) arXiv14064524 [hep-ph]
[14] C Peset and A Pineda Model-independent determination of the two-photon exchange contri-
bution to hyperfine splitting in muonic hydrogen JHEP 04 060 arXiv161205206 [nucl-th]
[15] F Hagelstein and V Pascalutsa Proton structure in the hyperfine splitting of muonic hydro-
gen PoS CD15 077 (2016) arXiv151104301 [nucl-th]
[16] F Hagelstein Exciting Nucleons in Compton Scattering and Hydrogen-Like Atoms PhD
thesis Mainz U Inst Kernphys (2017) arXiv171000874 [nucl-th]
[17] F Hagelstein ∆(1232)-Resonance in the hydrogen spectrum Proceedings 11th International
Workshop on the Physics of Excited Nucleons (NSTAR 2017) Columbia SC USA August
20-23 2017 Few Body Syst 59 93 (2018) arXiv180109790 [nucl-th]
[18] Y Prok et al (CLAS) Moments of the spin structure functions gp1 and gd1 for 005 lt Q2 lt 30
GeV2 Phys Lett B 672 12 (2009) arXiv08022232 [nucl-ex]
[19] H Dutz et al (GDH) First measurement of the Gerasimov-Drell-Hearn sum rule for 1H from
07 GeV to 18 GeV at ELSA Phys Rev Lett 91 192001 (2003)
[20] M Amarian et al (Jefferson Lab E94010) Measurement of the generalized forward spin po-
larizabilities of the neutron Phys Rev Lett 93 152301 (2004) arXivnucl-ex0406005
[21] M Amarian et al The Q2 evolution of the generalized Gerasimov-Drell-Hearn integral for
the neutron using a 3He target Phys Rev Lett 89 242301 (2002) arXivnucl-ex0205020
[22] M Amarian et al (Jefferson Lab E94-010)Q2 evolution of the neutron spin structure moments
using a 3He target Phys Rev Lett 92 022301 (2004) arXivhep-ex0310003
[23] A Deur et al Experimental determination of the evolution of the Bjorken integral at low Q2
Phys Rev Lett 93 212001 (2004) arXivhep-ex0407007
[24] K Slifer Low Q2 measurement of gp2 and the δLT spin polarizability Spin structure at long dis-
tance Proceedings Workshop Newport News USA March 12ndash13 2009 nucl-ex09064775
(2009) AIP Conf Proc 1155 10106313203293 (2009) arXiv09064775 [nucl-ex]
[25] P Solvignon et al (E01-012) Moments of the neutron g2 structure function at intermediate
42
Q2 Phys Rev C 92 015208 (2015) arXiv13044497 [nucl-ex]
[26] A Deur Experimental studies at low Q2 of the spin structure of the nucleon at Jefferson Lab
in 9th International Workshop on Chiral Dynamics (CD18) Durham NC USA September
17-21 2018 (2019) arXiv190305661 [nucl-ex]
[27] V Sulkosky et al (Jefferson Lab E97-110) Measurement of the 3He spin-structure functions
and of neutron (3He) spin-dependent sum rules at 0035 le Q2 le 024 GeV2 Phys Lett B
805 135428 (2020) arXiv190805709 [nucl-ex]
[28] K Adhikari et al (CLAS) Measurement of the Q2 Dependence of the Deuteron Spin Structure
Function g1 and its Moments at Low Q2 with CLAS Phys Rev Lett 120 062501 (2018)
arXiv171101974 [nucl-ex]
[29] V Lensky F Hagelstein A Hiller Blin and V Pascalutsa Comment on rdquoMeasurement of
the Q2 Dependence of the Deuteron Spin Structure Function g1 and its Moments at Low Q2
with CLASrdquo (2018) arXiv180603219 [nucl-th]
[30] J M Alarcn F Hagelstein V Lensky and V Pascalutsa Forward doubly-virtual Compton
scattering off the nucleon in chiral perturbation theory at NLO the subtraction function and
moments of unpolarized structure functions (2020) arXiv200509518 [hep-ph]
[31] V Bernard E Epelbaum H Krebs and U-G Meiszligner New insights into the spin structure
of the nucleon Phys Rev D 87 054032 (2013) arXiv12092523 [hep-ph]
[32] V Lensky J M Alarcon and V Pascalutsa Moments of nucleon structure functions at
next-to-leading order in baryon chiral perturbation theory Phys Rev C 90 055202 (2014)
arXiv14072574 [hep-ph]
[33] V Pascalutsa and M Vanderhaeghen Chiral effective-field theory in the ∆(1232) region
I Pion electroproduction on the nucleon Phys Rev D 73 034003 (2006) arXivhep-
ph0512244
[34] V Pascalutsa and M Vanderhaeghen Electromagnetic nucleon-to-Delta transition in chiral
effective field theory Phys Rev Lett 95 232001 (2005) arXivhep-ph0508060
[35] T R Hemmert B R Holstein and J Kambor Systematic 1M expansion for spin 32
particles in baryon chiral perturbation theory Phys Lett B 395 89 (1997) arXivhep-
ph9606456
[36] V Pascalutsa and D R Phillips Effective theory of the ∆(1232) in Compton scattering off
the nucleon Phys Rev C 67 055202 (2003) arXivnucl-th0212024
[37] V Pascalutsa M Vanderhaeghen and S N Yang Electromagnetic excitation of the ∆(1232)-
resonance Phys Rept 437 125 (2007) arXivhep-ph0609004
[38] K A Olive et al (Particle Data Group) Review of Particle Physics Chin Phys C 38 090001
43
(2014)
[39] F E Low Scattering of light of very low frequency by systems of spin 12 Phys Rev 96
1428 (1954)
[40] M Gell-Mann and M L Goldberger Scattering of low-energy photons by particles of spin
12 Phys Rev 96 1433 (1954)
[41] A M Harun ar-Rashid A simple derivation of Schwingerrsquos sum rule for spin dependent
structure functions Nuovo Cim A 33 447 (1976)
[42] F Hagelstein and V Pascalutsa Dissecting the hadronic contributions to (g minus 2)micro by
Schwingerrsquos sum rule Phys Rev Lett 120 072002 (2018) arXiv171004571 [hep-ph]
[43] V Lensky V Pascalutsa M Vanderhaeghen and C Kao Spin-dependent sum rules con-
necting real and virtual Compton scattering verified Phys Rev D 95 074001 (2017)
arXiv170101947 [hep-ph]
[44] H Burkhardt and W N Cottingham Sum rules for forward virtual Compton scattering
Annals Phys 56 453 (1970)
[45] V Lensky and V Pascalutsa Predictive powers of chiral perturbation theory in Compton
scattering off protons Eur Phys J C 65 195 (2010) arXiv09070451 [hep-ph]
[46] H Grieszlighammer J McGovern D Phillips and G Feldman Using effective field theory to
analyse low-energy Compton scattering data from protons and light nuclei Prog Part Nucl
Phys 67 841 (2012) arXiv12036834 [nucl-th]
[47] H W Grieszlighammer J A McGovern and D R Phillips Nucleon polarisabilities at and
beyond physical pion masses Eur Phys J A 52 139 (2016) arXiv151101952 [nucl-th]
[48] E Epelbaum H Krebs and U-G Meiszligner Improved chiral nucleon-nucleon potential up to
next-to-next-to-next-to-leading order Eur Phys J A 51 53 (2015) arXiv14120142 [nucl-
th]
[49] E Epelbaum H Krebs and U-G Meiszligner Precision nucleon-nucleon potential at fifth order
in the chiral expansion Phys Rev Lett 115 122301 (2015) arXiv14124623 [nucl-th]
[50] C W Kao T Spitzenberg and M Vanderhaeghen Burkhardt-Cottingham sum rule and
forward spin polarizabilities in heavy baryon chiral perturbation theory Phys Rev D 67
016001 (2003) arXivhep-ph0209241
[51] C-W Kao D Drechsel S Kamalov and M Vanderhaeghen Higher moments of nucleon
spin structure functions in heavy baryon chiral perturbation theory and in a resonance model
Phys Rev D 69 056004 (2004) arXivhep-ph0312102
[52] T Becher and H Leutwyler Baryon chiral perturbation theory in manifestly Lorentz invariant
form Eur Phys J C 9 643 (1999) arXivhep-ph9901384
44
[53] J Gasser M E Sainio and A Svarc Nucleons with chiral loops Nucl Phys B 307 779
(1988)
[54] L S Geng J Martin Camalich L Alvarez-Ruso and M J Vicente Vacas Leading SU(3)-
breaking corrections to the baryon magnetic moments in chiral perturbation theory Phys
Rev Lett 101 222002 (2008) arXiv08051419 [hep-ph]
[55] D Drechsel S S Kamalov and L Tiator The GDH sum rule and related integrals Phys
Rev D 63 114010 (2001) arXivhep-ph0008306
[56] D Drechsel O Hanstein S S Kamalov and L Tiator A unitary isobar model for pion
photo- and electroproduction on the proton up to 1 GeV Nucl Phys A 645 145 (1999)
arXivnucl-th9807001
[57] L Tiator private communication (2020)
[58] V Bernard T R Hemmert and U-G Meiszligner Spin structure of the nucleon at low energies
Phys Rev D 67 076008 (2003) arXivhep-ph0212033
[59] O Gryniuk F Hagelstein and V Pascalutsa Evaluation of the forward Compton scattering
off protons II Spin-dependent amplitude and observables Phys Rev D 94 034043 (2016)
arXiv160400789 [nucl-th]
[60] R Zielinski The g2p Experiment A Measurement of the Protonrsquos Spin Structure Functions
PhD thesis New Hampshire U (2010) arXiv170808297 [nucl-ex]
[61] N Guler et al (CLAS) Precise determination of the deuteron spin structure at low to mod-
erate Q2 with CLAS and extraction of the neutron contribution Phys Rev C 92 055201
(2015) arXiv150507877 [nucl-ex]
[62] C E Carlson V Nazaryan and K Griffioen Proton structure corrections to electronic
and muonic hydrogen hyperfine splitting Phys Rev A 78 022517 (2008) arXiv08052603
[physicsatom-ph]
[63] V Bernard N Kaiser and U-G Meiszligner Chiral dynamics in nucleons and nuclei Int J
Mod Phys E 4 193 (1995) arXivhep-ph9501384
[64] T R Hemmert B R Holstein and J Kambor ∆(1232) and the polarizabilities of the nucleon
Phys Rev D 55 5598 (1997) arXivhep-ph9612374
[65] V Pascalutsa and R Timmermans Field theory of nucleon to higher-spin baryon transitions
Phys Rev C 60 042201 (1999) arXivnucl-th9905065
[66] V Pascalutsa Quantization of an interacting spin-32 field and the Delta isobar Phys Rev
D 58 096002 (1998) arXivhep-ph9802288
[67] H Krebs Double Virtual Compton Scattering and SpinStructure of the Nucleon PoS
CD2018 031 (2019)
45
[68] N Kochelev and Y Oh Axial anomaly and the δLT puzzle Phys Rev D 016012 (2012)
arXiv11034892 [hep-ph]
[69] D Drechsel S Kamalov and L Tiator Unitary isobar model ndash MAID2007 Eur Phys J A
69 (2007) available at httpsmaidkphuni-mainzde arXiv07100306 [nucl-th]
[70] P J Mohr B N Taylor and D B Newell CODATA recommended values of the fundamental
physical constants 2010 Rev Mod Phys 84 1527 (2012)
[71] J Ahrens et al (GDH A2) First measurement of the Gerasimov-Drell-Hearn integral for 1H
from 200 to 800 MeV Phys Rev Lett 87 022003 (2001) arXivhep-ex0105089 [hep-ex]
[72] K Helbing (GDH) Experimental verification of the GDH sum rule at ELSA and MAMI Nucl
Phys Proc Suppl 105 113 (2002)
[73] R Bradford A Bodek H S Budd and J Arrington A New parameterization of the nucleon
elastic form-factors NuInt05 proceedings of the 4th International Workshop on Neutrino-
Nucleus Interactions in the Few-GeV Region Okayama Japan 26-29 September 2005 Nucl
Phys Proc Suppl 159 127 (2006) arXivhep-ex0602017
[74] J D Bjorken Applications of the chiral U(6)timesU(6) algebra of current densities Phys Rev
148 1467 (1966)
[75] J D Bjorken Inelastic scattering of polarized leptons from polarized nucleons Phys Rev D
1 1376 (1970)
[76] V Bernard T R Hemmert and U-G Meiszligner Novel analysis of chiral loop effects in
the generalized Gerasimov-Drell-Hearn sum rule Phys Lett B 545 105 (2002) arXivhep-
ph0203167
[77] A Deur et al Experimental study of isovector spin sum rules Phys Rev D 78 032001
(2008) arXiv08023198 [nucl-ex]
[78] B Pasquini P Pedroni and D Drechsel Higher order forward spin polarizability Phys Lett
B 687 160 (2010) arXiv10014230 [hep-ph]
[79] R Jaffe g2ndashThe nucleonrsquos other spin-dependent structure function Comments Nucl Part
Phys 19 239 (1990)
[80] E V Shuryak and A Vainshtein Theory of power corrections to deep inelastic scattering in
quantum chromodynamics (II) Qminus4 effects polarized target Nucl Phys B 201 141 (1982)
[81] S Wandzura and F Wilczek Sum rules for spin dependent electroproduction Test of rela-
tivistic constituent quarks Phys Lett B 72 195 (1977)
[82] B W Filippone and X-D Ji The spin structure of the nucleon Adv Nucl Phys 26 1
(2001) arXivhep-ph0101224
[83] M Burkardt The g2 structure function Proceedings Workshop on Spin structure at long
46
distance Newport News USA March 12-13 2009 AIP Conf Proc 1155 26 (2009)
arXiv09054079 [hep-ph]
[84] A Deur S J Brodsky and G F de Teramond The spin structure of the nucleon Rept
Prog Phys 82 076201 (2019) arXiv180705250 [hep-ph]
[85] B R Holstein D Drechsel B Pasquini and M Vanderhaeghen Higher order polarizabilities
of the proton Phys Rev C 61 034316 (2000) arXivhep-ph9910427
[86] B R Holstein V Pascalutsa and M Vanderhaeghen Sum rules for magnetic moments and
polarizabilities in QED and chiral effective-field theory Phys Rev D 72 094014 (2005)
arXivhep-ph0507016
[87] B Pasquini D Drechsel and L Tiator Invariant amplitudes for pion electroproduction Eur
Phys J A 34 387 (2007) arXiv07122327 [hep-ph]
[88] A J Buchmann Non-spherical proton shape and hydrogen hyperfine splitting Proceedings
International Workshop on Precision Physics of Simple Atomic Systems (PSAS 2008) Wind-
sor Ontario Canada July 21-26 2008 Can J Phys 87 773 (2009) arXiv09104747
[physicsatom-ph]
47
σ(π0p)TT =
e2g2AM
2Nλπ
64πf2πs(sminusM2
N )2λγ
1
s2λπλ3γ
[(M2
N minus s)(M2N minusQ2 minus s)
(3M2
N + s(2βγβπ minus 3))(M2N
+ s(2βγβπ minus 1))
+ 2s
((2m2
πQ2 + (M2
N minus s)2)(M2
N minus s)minus 2(minusm2
πQ2 + (sminusM2
N )2)
times sβγβπ + 4s2(sminusM2N )β2
γβ2π
)λ2γ
]arctan
(2sλπλγ
M2N + s(2βγβπ minus 1)
)+ 2
[minus 2m2
πQ2
+ (M2N minus s)
(2(sminusM2
N ) + 4sβγβπ +(sminusM2
N +Q2)(3M2N minus 3s+ 2sβγβπ)
sλ2γ
+m2πQ
2(sminusM2N )
(M2N minus s+ 2sβγβπ)(M2
N + s(minus1 + 2βγβπ minus 2λπλγ))
+m2πQ
2(sminusM2N )
(M2N minus s+ 2sβγβπ)(M2
N + s(minus1 + 2βγβπ + 2λπλγ))
)
+1
λπλγ
(βγβπ
(minusm2
πQ2 + (M2
N minus s)(M2N minus s+ 2sβγβπ)
)minus (M2
N minus s)(M2N minusQ2 minus s)λ2
π
)log
(M2N + s(minus1 + 2βγβπ + 2λπλγ)
M2N + s(minus1 + 2βγβπ minus 2λπλγ)
)] (B18)
σ(πminusp)TT =
e2g2AM
2N
64πf2πs
3λ4γ(M2
N + s(2βπβγ minus 1))2 minus 4λ2πs
2λ2γ)
4sλπλγ
((minusM2
N +Q2 + s)(M2N
+ s(2βπβγ minus 1))
2 minus 2sλ2γ
(2λ2
πs(minusM2N +Q2 + s) +m2
πQ2))
+((M2N + s(2βπβγ minus 1)
)2
minus 4λ2πs
2λ2γ
)((M2N + s(2βπβγ minus 1)
)(M2N minus 2Q2 minus s(2βπβγ + 1)
)+ 4λ2
πs2λ2
γ
)times log
(minusM2
N minus 2βπsβγ minus 2λπsλγ + s)minus(M2N + s(2βπβγ minus 2λπλγ minus 1)
)(M2N + s(minus1
+ 2βπβγ + 2λπλγ))[((
M2N + s(2βπβγ minus 1)
)(M2N minus 2Q2 minus s(2βπβγ + 1)
)+ 4λ2
πs2λ2
γ
)times log
(minusM2
N minus 2βπsβγ + 2λπsλγ + s)
+((Q2 + 2βπsβγ)2 minus 4λ2
πs2λ2
γ
)times log
(Q2 + 2βπsβγ minus 2λπsλγQ2 + 2sβπβγ + 2λπsλγ
)]+ 4sλ2
γ
((M2
N + s(2βπβγ minus 1))2 minus 4λ2πs
2λ2γ
)times (M2
N minus s) arctanh
(2λπsλγ
M2N + 2βπsβγ minus s
) (B19)
34
σ(π+n)LT =
e2g2AM
3Nλπ
32πf2πQs
3(sminusM2N )2λ4
γ
2sλγ
[(M2
N minus s)(Q2 + 2sβ2γ)(Q2 + 2sβγβπ)
minus 4s
((M2
N minus s)(Q2 minus 2s(απ minus 1)βγ
)+Q2sβγβπ
)λ2γ + 8s3(minus1 + απ)λ4
γ
]minus M2
N minus sλπ
[(Q2 + 2sβ2
γ)(Q2 + 2sβγβπ)2 + 4s2(2(απ minus 1)βγ(Q2 + 2sβγβπ)
minusQ2λ2π
)λ2γ + 8s3(απ minus 1)2λ4
γ
]arctan
(2sλπλγ
Q2 + 2sβγβπ
) (B20)
σ(π0p)LT =
e2g2AM
3Nλπ
16πf2πQs(sminusM2
N )2λγ
1
minus2s(M2N + s(minus1 + 2βγβπ))2λ2
γ + 8s3λ2πλ
4γ
[minus 3M8
N (Q2
+ 2sβ2γ) + 2M4
Ns2
(minus (Q2 + 2sβ2
γ)(minus3 + 2βγβπ) +(Q2(minus3 + 2βγβπ + 6λ2
π) + 2sβγ
times (minus3 + 12απ + 2βγβπ minus 12απβγβπ + 4βγλ2π))λ2γ minus 4s(α2
π minus 1)λ4π
)+ 2M2
Ns3
[minus (Q2
+ 2sβ2γ)(minus1 + 2βγβπ)
(6 + βγβπ(minus9 + 2βγβπ)
)+
(Q2(3minus 12λ2
π + 4βγβπ(minus1 + 2βγβπ
+ λ2π))minus 2sβγ
(minus 3 + 4απ(3 + 2απ(3 + 2βγβπ(minus3 + βγβπ)) + 4βγ(βπ + (2minus βγβπ)
times λ2π)))λ2γ + 8s
((απ minus 1)(1 + απ minus 2βγβπ) + 2απβγλ
2π
)λ4γ
)+ s4
((Q2 + 2sβ2
γ)
times (1minus 2βγβπ)2(2βγβπ minus 3) + 2(Q2(minus1 + 6λ2
π + 2βγβπ((1minus 2βγβπ)2 minus 2λ2π)) + 2sβγ
times (minus1 + 2βγβπ + 4(minus1 + βγβπ)(minusαπ + 2απβγβπ minus βγλ2π)))λ2γ minus 8
(s(απ minus 1)(απ
+ (1minus 2βγβπ)2) + 2βγ(2sαπ +Q2βγ)λ2π
)λ4γ + 32s(απ minus 1)λ2
πλ6γ
)+ 2M6
Ns
(Q2(6
minus 7βγβπ + λ2γ) + 2sβγ(βγ(6minus 7βγβπ) + (1minus 4απ)λ2
γ)
)]+sminusM2
N
4s2λπλ3γ
[(Q2 + 2sβ2
γ)
times(3M2
N + s(2βγβπ minus 3))(M2N + s(2βγβπ minus 1)
)+ 2s
(minusM2
N (Q2 + 2sβγ minus 8sαπβγ)
+ s(2sβγ(1minus 4απ + 4απβγβπ) +Q2(1minus 2λ2π)))λ2γ + 8s3(απ minus 1)λ4
γ
]
times arctan
(2sλπλγ
M2N + s(2βγβπ minus 1)
) (B21)
35
σ(πminusp)LT =
e2g2AM
3N
64πf2πQs
3λ4γ(M2
N minusQ2 minus s)(M2N + s(2βπβγ + 2λπλγ minus 1))
times
1
M2N + s(2βπβγ minus 2λπλγ minus 1)
[4λπsλγ(M2
N minusQ2 minus s)(
2sλ2γ
(s((1minus 2λ2
π)Q2
+ 2(2απ minus 1)sβγ(2βπβγ minus 1))minusM2N (Q2 + 2(1minus 2απ)sβγ)
)+ (Q2 + 2sβ2
γ)(M2N + s(2βπβγ minus 1)
)2 + 8(απ minus 1)απs
3λ4γ
)]minus(M2N + s(2βπβγ + 2λπλγ minus 1)
)times[
log
(minusM2N minus 2βπsβγ minus 2λπsλγ + s
minusM2N minus 2βπsβγ + 2λπsλγ + s
)(minus (Q2 + 2sβ2
γ)(M2N + s(2βπβγ minus 1)
)(minusM2
N + 2Q2 + 2βπsβγ + s)minus 2sλ2γ
(minusM2
N (Q2 + 2sβγ) +Q4 +Q2s(4απβγ
minus 2βγ minus 2λ2π + 1) + 2s2βγ(4(απ minus 1)βπβγ + 1)
)minus 8(απ minus 1)2s3λ4
γ
)+ 2
(4s2λ2
γ
(2(απ minus 1)βγ(Q2 + 2βπsβγ)minus λ2
πQ2)
+ (Q2 + 2βπsβγ)2(Q2 + 2sβ2γ)
+ 8(απ minus 1)2s3λ4γ
)arctanh
(2λπsλγ
Q2 + 2βπsβγ
)] (B22)
2 ∆-production channel
The tree-level ∆-exchange diagram in Fig 2 of Ref [30] contributes to the non-Born part of
the VVCS amplitudes The contribution of the ∆ exchange to the VVCS amplitudes can be split
into [17]
S∆-exch1 (νQ2) = S∆-pole
1 (νQ2) + S∆-exch1 (νQ2) (B23a)
νS∆-exch2 (νQ2) = νS∆-pole
2 (νQ2) + νS2∆-exch
(νQ2) (B23b)
and similarly for the unpolarized VVCS amplitudes discussed in Ref [30] Here we introduced the
∆-pole contributions S∆-polei and the ∆-non-pole contributions S∆-exch
i The former amplitudes
feature a pole at the ∆(1232)-production threshold and thus are proportional to
1
[sminusM2∆][uminusM2
∆]=
1
4M2N
1
ν2∆ minus ν2
(B24)
36
They can be reconstructed from the dispersion relations in Eq (5) with the tree-level ∆-production
cross sections as input cf Fig 10 in Ref [30]
σTT (νQ2) =π2α
M2NM
2+|~q |
minus g2
MMN (M+ + ν)|~q |2 +g2E(∆minus ν)(Q2 minusMNν)2
MN(B25a)
+g2CQ
4s(∆minus ν)
MNM2∆
minus 4gMgE(Q2 minusMNν)|~q |2 minus 4gMgCQ2|~q |2
+2gEgCQ
2[minusMNM∆ |~q |2 + s(Q2 +∆ν)
]MNM∆
δ(ν minus ν∆)
σLT (νQ2) =Qπ2α
M2NM
2+|~q |
g2E(MNν minusQ2) [M∆(MN + ν)minus s]
MN(B25b)
+g2CQ
2[MNM∆|~q |2 minus s(Q2 +∆ν)
]MNM2
∆
+ gMgEM∆|~q |2 minusgMgC(Q2 minusMNν)|~q |2
M∆
+gEgC(ν minus∆)(M2
N |~q |2 minus 2Q2s)
MNM∆
δ(ν minus ν∆)
with ∆ = M∆ minusMN M+ = M∆ + MN and the Mandelstam variable s = M2N + 2MNν minus Q2
Analytical expressions for the spin structure functions g1(xQ2) and g2(xQ2) can be constructed
from Eq (3) with the flux factor K(νQ2) = |~q | =radicν2 +Q2
In the ∆-non-pole contributions to S1(νQ2) and νS2(νQ2) the pole in ν at the ∆(1232)-
production threshold has canceled out
S∆-exch1 (νQ2) =
πα
MNM2+
[g2MQ
2+ + g2
E
(∆2 minus 3Q2
)+
4g2CQ
4
M2∆
minus 8gMgEM∆ωminus (B26a)
minus2gMgCQ2(MN minus 4M∆)
M∆+
2gEgCQ2(3MN minus 2M∆)
M∆
]
νS2∆-exch
(νQ2) =2πα
MNM2+
[g2EM∆∆ωminus +
g2M MNQ
2+
2+g2C Q
2(Q2 minus∆2)
2M∆(B26b)
+gEgM M∆(M∆ω+ minus 4MNωminus)minus gEgC ∆(2Q2 +MNω+)
+gMgC Q2(4MN minus ω+)
]+S∆-exch
2 (νQ2)
ν
[M2
∆ ω2+
M2N
+ ν2
]
with Q+ =radic
(M∆ +MN )2 +Q2 and ωplusmn = (M2∆minusM2
N plusmnQ2)2M∆ and the non-pole contribution
to S2(νQ2)
S∆-exch2 (νQ2) = minus2παMNν
M∆M2+
[gM + gE
]gC (B27)
These amplitudes to the contrary are not described by the tree-level ∆-production cross sections
in the standard dispersive approach [17] This peculiarity has been previously missed e g in the
calculation of the ∆-exchange contribution to the hydrogen hyperfine splitting in Ref [88] The
importance of including the ∆-non-pole contribution is also evident when considering the BC sum
37
rule in Eq (14) The ∆-pole terms by themselves violate the BC sum rule but cancel exactly with
the ∆-non-pole terms
limνrarr0
νS∆-pole2 (νQ2) + lim
νrarr0νS2
∆-exch(νQ2) = 0 (B28)
Appendix C Polarizabilities at Q2 = 0
In this section we give analytical expressions for the static values and slopes at Q2 = 0 of
the spin polarizabilities and moments of polarized structure functions In particular we give the
HB expansion of the πN -loop contributions and the ∆-exchange contributions The complete
expressions also for the π∆-loop contributions can be found in the Supplemented material Recall
that IA(0) = I1(0) = d2(0) = 0 and dd2(Q2)dQ2
∣∣∣Q2=0
= 0
1 πN-loop contribution
Here we give analytical expressions for the πN -loop contributions to the proton and neutron
spin polarizabilities expanded in powers of micro = mπMN viz the HB expansion Note that we
choose to expand here to a high order in micro the strict HB expansion would only retain the leading
term in an analogous NLO calculation
bull Static polarizabilities (Q2 = 0)
γ0p =e2g2
A
96π3f2πm
2π
1minus 21πmicro
8minus(
59
2+ 26 logmicro
)micro2 +
1875πmicro3
64
+3
(3
2+ 26 logmicro
)micro4 +
(C1)
γ0n =e2g2
A
48π3f2πm
2π
1
2minus 9πmicro
16minus 2micro2 logmicro+
75πmicro3
128minus 3micro4
4+
(C2)
δLTp =e2g2
A
192π3f2πm
2π
1minus 9πmicro
8+
(13
2minus 2 logmicro
)micro2 minus 465πmicro3
64
minus(
47
2+ 42 logmicro
)micro4 +
(C3)
δLTn =e2g2
A
96π3f2πm
2π
1
2+
3πmicro
16+ (1 + 2 log micro)micro2 minus 105πmicro3
128+
5micro4
4+
(C4)
38
δLTn =e2g2
A
96π3f2πm
2π
1
2+
3πmicro
16+ (1 + 2 log micro)micro2 minus 105πmicro3
128+
5micro4
4+
(C5)
γ0p =e2g2
A
16π3f2πm
4π
4
45minus 3πmicro
16+
14micro2
5minus 1813πmicro3
384minus 192
5(1 + log micro)micro4
+80703πmicro5
2048+
(C6)
γ0n =e2g2
A
16π3f2πm
4π
4
45minus 5πmicro
48+
4micro2
5minus 245πmicro3
384minus 32micro4 logmicro
15
+1323πmicro5
2048+
(C7)
bull Slopes of polarizabilities at Q2 = 0
dγ0p(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
2minus 45πmicro
4+ 223micro2 minus 28515πmicro3
64
minus9
(1953
4+ 449 logmicro
)micro4 +
570255πmicro5
128+
(C8)
dγ0n(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
2minus 81πmicro
8+ 94micro2 minus 2535πmicro3
32
minus3 (1 + 90 logmicro)micro4 +84315πmicro5
1024+
(C9)
dδLTp(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
2880π3f2πm
4π
minus5
2minus 27πmicro
32+ 20micro2 minus 5865πmicro3
256
+3
(617
4+ 36 logmicro
)micro4 minus 2056845πmicro5
4096+
(C10)
dδLTn(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
minus5
4minus 81πmicro
64minus 11micro2 +
10005πmicro3
512
+15
8(11 + 48 logmicro)micro4 minus 267015πmicro5
8192+
(C11)
dIAp(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
96π2f2πmicro
2
1minus 15πmicro
4minus 1
2(115 + 88 logmicro)micro2 +
1839πmicro3
32
+5 (5 + 34 logmicro)micro4 +
(C12)
dIAn(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
48π2f2πmicro
2
1
2minus 11πmicro
8minus 1
4(1 + 20 logmicro)micro2 +
99πmicro3
64
minus25micro4
12+
(C13)
dI1p(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
96π2f2πmicro
3π
8+ 2 (4 + 3 logmicro)microminus 537πmicro2
64
minus1
2(15 + 56 logmicro)micro3 +
(C14)
39
dI1n(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
48π2f2πmicro
minus π
16+
1
4(3 + 4 logmicro)microminus 57πmicro2
128+
2micro3
3+
(C15)
γ0p(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2g2
A
16π3f2πm
6π
1
105minus 23πmicro
256+
377micro2
210minus 15551πmicro3
6144+
3371micro4
105
minus1640457πmicro5
32768+
(C16)
γ0n(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2g2
A
16π3f2πm
6π
1
105minus 153πmicro
1792+
69micro2
70minus 4615πmicro3
6144+
172micro4
35
minus120897πmicro5
32768+
(C17)
2 ∆-exchange contribution
Here we give analytical expressions for the tree-level ∆-exchange contributions to the nucleon
spin polarizabilities and their slopes at Q2 = 0 Note that the ∆-exchange contributes equally to
proton and neutron polarizabilities Recall that for the magnetic γlowastN∆ coupling we introduced a
dipole form factor to mimic vector-meson dominance gM rarr gM(1 +Q2Λ2)2
bull Static polarizabilities (Q2 = 0)
γ0 = minus e2
4πM2+
(g2M
∆2+
g2E
M2+
minus 4gMgEM+∆
) (C18)
δLT =e2M∆
4πM3+
(g2E
MNM++gMgE∆MN
minus gEgCM2
∆
) (C19)
γ0 =e2M2
N
π∆2M4+
(minusg
2M
∆2+
g2E
M2+
+4gMgE∆M+
) (C20)
bull Slopes of polarizabilities at Q2 = 0
dγ0(Q2)
dQ2
∣∣∣∣∣Q2=0
= minus e2
πM2+∆
(g2M
∆
[1
4∆2minus 1
∆M++
1
2M2+
]minus 1
Λ2
g2M
∆+
g2E
2M2+
[1
2∆minus 3
M+
]minusgMgE
M+
[1
∆2minus 5
∆M++
1
M2+
]+
1
Λ2
2gMgEM+
+2gMgC∆M2
+
minus gEgCM3
+
) (C21)
40
d δLT (Q2)
dQ2
∣∣∣∣∣Q2=0
=e2M∆∆
4πMNM2+
(g2E
∆2M2+
[1
∆minus 4
M+
]minus g2
C
∆M2∆M
2+
+gMgE∆2M+
[1
∆2
minus 3
∆M++
1
M2+
]minus 2
Λ2
gMgE∆2M+
+gMgC∆M2
∆
[1
2∆2minus 2
∆M++
1
2M2+
]minus gEgC
2M2∆M
2+
[7
∆+
1
M+
]) (C22)
dIA(Q2)
dQ2
∣∣∣∣∣Q2=0
= minusM2N
M2+
(g2M
2∆2+
g2E
MNM+minus 2gMgE
∆M+minus gEgCM∆M+
) (C23)
dI1(Q2)
dQ2
∣∣∣∣∣Q2=0
= minusM∆M2N
2M3+
(g2E
MNM∆minus gMgE∆MN
minus gEgCM2
∆
) (C24)
γ0(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2M2
N
π∆3M6+
(g2M
∆
[3M2
∆ + 2M∆MN minus 9M2N
∆2+
4M2+
Λ2
]
minus4gMgE
[5M2
∆ minus 9M2N
M+∆2+
2M+
Λ2
]+
g2E
∆M+[7M∆ minus 9MN ]
minus8gMgC∆
+4gEgCM+
) (C25)
[1] D Drechsel B Pasquini and M Vanderhaeghen Dispersion relations in real and virtual
Compton scattering Phys Rept 378 99 (2003) hep-ph0212124
[2] S E Kuhn J-P Chen and E Leader Spin structure of the nucleon mdash status and recent
results Prog Part Nucl Phys 63 1 (2009) arXiv08123535 [hep-ph]
[3] F Hagelstein R Miskimen and V Pascalutsa Nucleon polarizabilities from Compton scat-
tering to hydrogen atom Prog Part Nucl Phys 88 29 (2016) arXiv151203765 [nucl-th]
[4] B Pasquini and M Vanderhaeghen Dispersion theory in electromagnetic interactions Ann
Rev Nucl Part Sci 68 75 (2018) arXiv180510482 [hep-ph]
[5] S Gerasimov A Sum rule for magnetic moments and the damping of the nucleon magnetic
moment in nuclei Sov J Nucl Phys 2 430 (1966)
[6] S Drell and A C Hearn Exact sum rule for nucleon magnetic moments Phys Rev Lett
16 908 (1966)
[7] J S Schwinger Source theory viewpoints in deep inelastic scattering Proc Natl Acad Sci
USA 72 1 (1975)
[8] J S Schwinger Source theory viewpoints in deep inelastic scattering Electromagnetic In-
teractions and Field Theory Proceedings 14 Internationale Universitatswochen Schladming
Austria February 24-March 7 1975 Acta Phys Austriaca Suppl 14 471 (1975)
41
[9] J Schwinger Source theory discussion of deep inelastic scattering with polarized particles
Proc Natl Acad Sci USA 72 1559 (1975)
[10] A M Baldin Polarizability of nucleons Nucl Phys 18 310 (1960)
[11] M Gell-Mann M L Goldberger and W E Thirring Use of causality conditions in quantum
theory Phys Rev 95 1612 (1954)
[12] A Pineda Leading chiral logarithms to the hyperfine splitting of the hydrogen and muonic
hydrogen Phys Rev C 67 025201 (2003)
[13] C Peset and A Pineda The two-photon exchange contribution to muonic hydrogen from
chiral perturbation theory Nucl Phys B 887 69 (2014) arXiv14064524 [hep-ph]
[14] C Peset and A Pineda Model-independent determination of the two-photon exchange contri-
bution to hyperfine splitting in muonic hydrogen JHEP 04 060 arXiv161205206 [nucl-th]
[15] F Hagelstein and V Pascalutsa Proton structure in the hyperfine splitting of muonic hydro-
gen PoS CD15 077 (2016) arXiv151104301 [nucl-th]
[16] F Hagelstein Exciting Nucleons in Compton Scattering and Hydrogen-Like Atoms PhD
thesis Mainz U Inst Kernphys (2017) arXiv171000874 [nucl-th]
[17] F Hagelstein ∆(1232)-Resonance in the hydrogen spectrum Proceedings 11th International
Workshop on the Physics of Excited Nucleons (NSTAR 2017) Columbia SC USA August
20-23 2017 Few Body Syst 59 93 (2018) arXiv180109790 [nucl-th]
[18] Y Prok et al (CLAS) Moments of the spin structure functions gp1 and gd1 for 005 lt Q2 lt 30
GeV2 Phys Lett B 672 12 (2009) arXiv08022232 [nucl-ex]
[19] H Dutz et al (GDH) First measurement of the Gerasimov-Drell-Hearn sum rule for 1H from
07 GeV to 18 GeV at ELSA Phys Rev Lett 91 192001 (2003)
[20] M Amarian et al (Jefferson Lab E94010) Measurement of the generalized forward spin po-
larizabilities of the neutron Phys Rev Lett 93 152301 (2004) arXivnucl-ex0406005
[21] M Amarian et al The Q2 evolution of the generalized Gerasimov-Drell-Hearn integral for
the neutron using a 3He target Phys Rev Lett 89 242301 (2002) arXivnucl-ex0205020
[22] M Amarian et al (Jefferson Lab E94-010)Q2 evolution of the neutron spin structure moments
using a 3He target Phys Rev Lett 92 022301 (2004) arXivhep-ex0310003
[23] A Deur et al Experimental determination of the evolution of the Bjorken integral at low Q2
Phys Rev Lett 93 212001 (2004) arXivhep-ex0407007
[24] K Slifer Low Q2 measurement of gp2 and the δLT spin polarizability Spin structure at long dis-
tance Proceedings Workshop Newport News USA March 12ndash13 2009 nucl-ex09064775
(2009) AIP Conf Proc 1155 10106313203293 (2009) arXiv09064775 [nucl-ex]
[25] P Solvignon et al (E01-012) Moments of the neutron g2 structure function at intermediate
42
Q2 Phys Rev C 92 015208 (2015) arXiv13044497 [nucl-ex]
[26] A Deur Experimental studies at low Q2 of the spin structure of the nucleon at Jefferson Lab
in 9th International Workshop on Chiral Dynamics (CD18) Durham NC USA September
17-21 2018 (2019) arXiv190305661 [nucl-ex]
[27] V Sulkosky et al (Jefferson Lab E97-110) Measurement of the 3He spin-structure functions
and of neutron (3He) spin-dependent sum rules at 0035 le Q2 le 024 GeV2 Phys Lett B
805 135428 (2020) arXiv190805709 [nucl-ex]
[28] K Adhikari et al (CLAS) Measurement of the Q2 Dependence of the Deuteron Spin Structure
Function g1 and its Moments at Low Q2 with CLAS Phys Rev Lett 120 062501 (2018)
arXiv171101974 [nucl-ex]
[29] V Lensky F Hagelstein A Hiller Blin and V Pascalutsa Comment on rdquoMeasurement of
the Q2 Dependence of the Deuteron Spin Structure Function g1 and its Moments at Low Q2
with CLASrdquo (2018) arXiv180603219 [nucl-th]
[30] J M Alarcn F Hagelstein V Lensky and V Pascalutsa Forward doubly-virtual Compton
scattering off the nucleon in chiral perturbation theory at NLO the subtraction function and
moments of unpolarized structure functions (2020) arXiv200509518 [hep-ph]
[31] V Bernard E Epelbaum H Krebs and U-G Meiszligner New insights into the spin structure
of the nucleon Phys Rev D 87 054032 (2013) arXiv12092523 [hep-ph]
[32] V Lensky J M Alarcon and V Pascalutsa Moments of nucleon structure functions at
next-to-leading order in baryon chiral perturbation theory Phys Rev C 90 055202 (2014)
arXiv14072574 [hep-ph]
[33] V Pascalutsa and M Vanderhaeghen Chiral effective-field theory in the ∆(1232) region
I Pion electroproduction on the nucleon Phys Rev D 73 034003 (2006) arXivhep-
ph0512244
[34] V Pascalutsa and M Vanderhaeghen Electromagnetic nucleon-to-Delta transition in chiral
effective field theory Phys Rev Lett 95 232001 (2005) arXivhep-ph0508060
[35] T R Hemmert B R Holstein and J Kambor Systematic 1M expansion for spin 32
particles in baryon chiral perturbation theory Phys Lett B 395 89 (1997) arXivhep-
ph9606456
[36] V Pascalutsa and D R Phillips Effective theory of the ∆(1232) in Compton scattering off
the nucleon Phys Rev C 67 055202 (2003) arXivnucl-th0212024
[37] V Pascalutsa M Vanderhaeghen and S N Yang Electromagnetic excitation of the ∆(1232)-
resonance Phys Rept 437 125 (2007) arXivhep-ph0609004
[38] K A Olive et al (Particle Data Group) Review of Particle Physics Chin Phys C 38 090001
43
(2014)
[39] F E Low Scattering of light of very low frequency by systems of spin 12 Phys Rev 96
1428 (1954)
[40] M Gell-Mann and M L Goldberger Scattering of low-energy photons by particles of spin
12 Phys Rev 96 1433 (1954)
[41] A M Harun ar-Rashid A simple derivation of Schwingerrsquos sum rule for spin dependent
structure functions Nuovo Cim A 33 447 (1976)
[42] F Hagelstein and V Pascalutsa Dissecting the hadronic contributions to (g minus 2)micro by
Schwingerrsquos sum rule Phys Rev Lett 120 072002 (2018) arXiv171004571 [hep-ph]
[43] V Lensky V Pascalutsa M Vanderhaeghen and C Kao Spin-dependent sum rules con-
necting real and virtual Compton scattering verified Phys Rev D 95 074001 (2017)
arXiv170101947 [hep-ph]
[44] H Burkhardt and W N Cottingham Sum rules for forward virtual Compton scattering
Annals Phys 56 453 (1970)
[45] V Lensky and V Pascalutsa Predictive powers of chiral perturbation theory in Compton
scattering off protons Eur Phys J C 65 195 (2010) arXiv09070451 [hep-ph]
[46] H Grieszlighammer J McGovern D Phillips and G Feldman Using effective field theory to
analyse low-energy Compton scattering data from protons and light nuclei Prog Part Nucl
Phys 67 841 (2012) arXiv12036834 [nucl-th]
[47] H W Grieszlighammer J A McGovern and D R Phillips Nucleon polarisabilities at and
beyond physical pion masses Eur Phys J A 52 139 (2016) arXiv151101952 [nucl-th]
[48] E Epelbaum H Krebs and U-G Meiszligner Improved chiral nucleon-nucleon potential up to
next-to-next-to-next-to-leading order Eur Phys J A 51 53 (2015) arXiv14120142 [nucl-
th]
[49] E Epelbaum H Krebs and U-G Meiszligner Precision nucleon-nucleon potential at fifth order
in the chiral expansion Phys Rev Lett 115 122301 (2015) arXiv14124623 [nucl-th]
[50] C W Kao T Spitzenberg and M Vanderhaeghen Burkhardt-Cottingham sum rule and
forward spin polarizabilities in heavy baryon chiral perturbation theory Phys Rev D 67
016001 (2003) arXivhep-ph0209241
[51] C-W Kao D Drechsel S Kamalov and M Vanderhaeghen Higher moments of nucleon
spin structure functions in heavy baryon chiral perturbation theory and in a resonance model
Phys Rev D 69 056004 (2004) arXivhep-ph0312102
[52] T Becher and H Leutwyler Baryon chiral perturbation theory in manifestly Lorentz invariant
form Eur Phys J C 9 643 (1999) arXivhep-ph9901384
44
[53] J Gasser M E Sainio and A Svarc Nucleons with chiral loops Nucl Phys B 307 779
(1988)
[54] L S Geng J Martin Camalich L Alvarez-Ruso and M J Vicente Vacas Leading SU(3)-
breaking corrections to the baryon magnetic moments in chiral perturbation theory Phys
Rev Lett 101 222002 (2008) arXiv08051419 [hep-ph]
[55] D Drechsel S S Kamalov and L Tiator The GDH sum rule and related integrals Phys
Rev D 63 114010 (2001) arXivhep-ph0008306
[56] D Drechsel O Hanstein S S Kamalov and L Tiator A unitary isobar model for pion
photo- and electroproduction on the proton up to 1 GeV Nucl Phys A 645 145 (1999)
arXivnucl-th9807001
[57] L Tiator private communication (2020)
[58] V Bernard T R Hemmert and U-G Meiszligner Spin structure of the nucleon at low energies
Phys Rev D 67 076008 (2003) arXivhep-ph0212033
[59] O Gryniuk F Hagelstein and V Pascalutsa Evaluation of the forward Compton scattering
off protons II Spin-dependent amplitude and observables Phys Rev D 94 034043 (2016)
arXiv160400789 [nucl-th]
[60] R Zielinski The g2p Experiment A Measurement of the Protonrsquos Spin Structure Functions
PhD thesis New Hampshire U (2010) arXiv170808297 [nucl-ex]
[61] N Guler et al (CLAS) Precise determination of the deuteron spin structure at low to mod-
erate Q2 with CLAS and extraction of the neutron contribution Phys Rev C 92 055201
(2015) arXiv150507877 [nucl-ex]
[62] C E Carlson V Nazaryan and K Griffioen Proton structure corrections to electronic
and muonic hydrogen hyperfine splitting Phys Rev A 78 022517 (2008) arXiv08052603
[physicsatom-ph]
[63] V Bernard N Kaiser and U-G Meiszligner Chiral dynamics in nucleons and nuclei Int J
Mod Phys E 4 193 (1995) arXivhep-ph9501384
[64] T R Hemmert B R Holstein and J Kambor ∆(1232) and the polarizabilities of the nucleon
Phys Rev D 55 5598 (1997) arXivhep-ph9612374
[65] V Pascalutsa and R Timmermans Field theory of nucleon to higher-spin baryon transitions
Phys Rev C 60 042201 (1999) arXivnucl-th9905065
[66] V Pascalutsa Quantization of an interacting spin-32 field and the Delta isobar Phys Rev
D 58 096002 (1998) arXivhep-ph9802288
[67] H Krebs Double Virtual Compton Scattering and SpinStructure of the Nucleon PoS
CD2018 031 (2019)
45
[68] N Kochelev and Y Oh Axial anomaly and the δLT puzzle Phys Rev D 016012 (2012)
arXiv11034892 [hep-ph]
[69] D Drechsel S Kamalov and L Tiator Unitary isobar model ndash MAID2007 Eur Phys J A
69 (2007) available at httpsmaidkphuni-mainzde arXiv07100306 [nucl-th]
[70] P J Mohr B N Taylor and D B Newell CODATA recommended values of the fundamental
physical constants 2010 Rev Mod Phys 84 1527 (2012)
[71] J Ahrens et al (GDH A2) First measurement of the Gerasimov-Drell-Hearn integral for 1H
from 200 to 800 MeV Phys Rev Lett 87 022003 (2001) arXivhep-ex0105089 [hep-ex]
[72] K Helbing (GDH) Experimental verification of the GDH sum rule at ELSA and MAMI Nucl
Phys Proc Suppl 105 113 (2002)
[73] R Bradford A Bodek H S Budd and J Arrington A New parameterization of the nucleon
elastic form-factors NuInt05 proceedings of the 4th International Workshop on Neutrino-
Nucleus Interactions in the Few-GeV Region Okayama Japan 26-29 September 2005 Nucl
Phys Proc Suppl 159 127 (2006) arXivhep-ex0602017
[74] J D Bjorken Applications of the chiral U(6)timesU(6) algebra of current densities Phys Rev
148 1467 (1966)
[75] J D Bjorken Inelastic scattering of polarized leptons from polarized nucleons Phys Rev D
1 1376 (1970)
[76] V Bernard T R Hemmert and U-G Meiszligner Novel analysis of chiral loop effects in
the generalized Gerasimov-Drell-Hearn sum rule Phys Lett B 545 105 (2002) arXivhep-
ph0203167
[77] A Deur et al Experimental study of isovector spin sum rules Phys Rev D 78 032001
(2008) arXiv08023198 [nucl-ex]
[78] B Pasquini P Pedroni and D Drechsel Higher order forward spin polarizability Phys Lett
B 687 160 (2010) arXiv10014230 [hep-ph]
[79] R Jaffe g2ndashThe nucleonrsquos other spin-dependent structure function Comments Nucl Part
Phys 19 239 (1990)
[80] E V Shuryak and A Vainshtein Theory of power corrections to deep inelastic scattering in
quantum chromodynamics (II) Qminus4 effects polarized target Nucl Phys B 201 141 (1982)
[81] S Wandzura and F Wilczek Sum rules for spin dependent electroproduction Test of rela-
tivistic constituent quarks Phys Lett B 72 195 (1977)
[82] B W Filippone and X-D Ji The spin structure of the nucleon Adv Nucl Phys 26 1
(2001) arXivhep-ph0101224
[83] M Burkardt The g2 structure function Proceedings Workshop on Spin structure at long
46
distance Newport News USA March 12-13 2009 AIP Conf Proc 1155 26 (2009)
arXiv09054079 [hep-ph]
[84] A Deur S J Brodsky and G F de Teramond The spin structure of the nucleon Rept
Prog Phys 82 076201 (2019) arXiv180705250 [hep-ph]
[85] B R Holstein D Drechsel B Pasquini and M Vanderhaeghen Higher order polarizabilities
of the proton Phys Rev C 61 034316 (2000) arXivhep-ph9910427
[86] B R Holstein V Pascalutsa and M Vanderhaeghen Sum rules for magnetic moments and
polarizabilities in QED and chiral effective-field theory Phys Rev D 72 094014 (2005)
arXivhep-ph0507016
[87] B Pasquini D Drechsel and L Tiator Invariant amplitudes for pion electroproduction Eur
Phys J A 34 387 (2007) arXiv07122327 [hep-ph]
[88] A J Buchmann Non-spherical proton shape and hydrogen hyperfine splitting Proceedings
International Workshop on Precision Physics of Simple Atomic Systems (PSAS 2008) Wind-
sor Ontario Canada July 21-26 2008 Can J Phys 87 773 (2009) arXiv09104747
[physicsatom-ph]
47
σ(π+n)LT =
e2g2AM
3Nλπ
32πf2πQs
3(sminusM2N )2λ4
γ
2sλγ
[(M2
N minus s)(Q2 + 2sβ2γ)(Q2 + 2sβγβπ)
minus 4s
((M2
N minus s)(Q2 minus 2s(απ minus 1)βγ
)+Q2sβγβπ
)λ2γ + 8s3(minus1 + απ)λ4
γ
]minus M2
N minus sλπ
[(Q2 + 2sβ2
γ)(Q2 + 2sβγβπ)2 + 4s2(2(απ minus 1)βγ(Q2 + 2sβγβπ)
minusQ2λ2π
)λ2γ + 8s3(απ minus 1)2λ4
γ
]arctan
(2sλπλγ
Q2 + 2sβγβπ
) (B20)
σ(π0p)LT =
e2g2AM
3Nλπ
16πf2πQs(sminusM2
N )2λγ
1
minus2s(M2N + s(minus1 + 2βγβπ))2λ2
γ + 8s3λ2πλ
4γ
[minus 3M8
N (Q2
+ 2sβ2γ) + 2M4
Ns2
(minus (Q2 + 2sβ2
γ)(minus3 + 2βγβπ) +(Q2(minus3 + 2βγβπ + 6λ2
π) + 2sβγ
times (minus3 + 12απ + 2βγβπ minus 12απβγβπ + 4βγλ2π))λ2γ minus 4s(α2
π minus 1)λ4π
)+ 2M2
Ns3
[minus (Q2
+ 2sβ2γ)(minus1 + 2βγβπ)
(6 + βγβπ(minus9 + 2βγβπ)
)+
(Q2(3minus 12λ2
π + 4βγβπ(minus1 + 2βγβπ
+ λ2π))minus 2sβγ
(minus 3 + 4απ(3 + 2απ(3 + 2βγβπ(minus3 + βγβπ)) + 4βγ(βπ + (2minus βγβπ)
times λ2π)))λ2γ + 8s
((απ minus 1)(1 + απ minus 2βγβπ) + 2απβγλ
2π
)λ4γ
)+ s4
((Q2 + 2sβ2
γ)
times (1minus 2βγβπ)2(2βγβπ minus 3) + 2(Q2(minus1 + 6λ2
π + 2βγβπ((1minus 2βγβπ)2 minus 2λ2π)) + 2sβγ
times (minus1 + 2βγβπ + 4(minus1 + βγβπ)(minusαπ + 2απβγβπ minus βγλ2π)))λ2γ minus 8
(s(απ minus 1)(απ
+ (1minus 2βγβπ)2) + 2βγ(2sαπ +Q2βγ)λ2π
)λ4γ + 32s(απ minus 1)λ2
πλ6γ
)+ 2M6
Ns
(Q2(6
minus 7βγβπ + λ2γ) + 2sβγ(βγ(6minus 7βγβπ) + (1minus 4απ)λ2
γ)
)]+sminusM2
N
4s2λπλ3γ
[(Q2 + 2sβ2
γ)
times(3M2
N + s(2βγβπ minus 3))(M2N + s(2βγβπ minus 1)
)+ 2s
(minusM2
N (Q2 + 2sβγ minus 8sαπβγ)
+ s(2sβγ(1minus 4απ + 4απβγβπ) +Q2(1minus 2λ2π)))λ2γ + 8s3(απ minus 1)λ4
γ
]
times arctan
(2sλπλγ
M2N + s(2βγβπ minus 1)
) (B21)
35
σ(πminusp)LT =
e2g2AM
3N
64πf2πQs
3λ4γ(M2
N minusQ2 minus s)(M2N + s(2βπβγ + 2λπλγ minus 1))
times
1
M2N + s(2βπβγ minus 2λπλγ minus 1)
[4λπsλγ(M2
N minusQ2 minus s)(
2sλ2γ
(s((1minus 2λ2
π)Q2
+ 2(2απ minus 1)sβγ(2βπβγ minus 1))minusM2N (Q2 + 2(1minus 2απ)sβγ)
)+ (Q2 + 2sβ2
γ)(M2N + s(2βπβγ minus 1)
)2 + 8(απ minus 1)απs
3λ4γ
)]minus(M2N + s(2βπβγ + 2λπλγ minus 1)
)times[
log
(minusM2N minus 2βπsβγ minus 2λπsλγ + s
minusM2N minus 2βπsβγ + 2λπsλγ + s
)(minus (Q2 + 2sβ2
γ)(M2N + s(2βπβγ minus 1)
)(minusM2
N + 2Q2 + 2βπsβγ + s)minus 2sλ2γ
(minusM2
N (Q2 + 2sβγ) +Q4 +Q2s(4απβγ
minus 2βγ minus 2λ2π + 1) + 2s2βγ(4(απ minus 1)βπβγ + 1)
)minus 8(απ minus 1)2s3λ4
γ
)+ 2
(4s2λ2
γ
(2(απ minus 1)βγ(Q2 + 2βπsβγ)minus λ2
πQ2)
+ (Q2 + 2βπsβγ)2(Q2 + 2sβ2γ)
+ 8(απ minus 1)2s3λ4γ
)arctanh
(2λπsλγ
Q2 + 2βπsβγ
)] (B22)
2 ∆-production channel
The tree-level ∆-exchange diagram in Fig 2 of Ref [30] contributes to the non-Born part of
the VVCS amplitudes The contribution of the ∆ exchange to the VVCS amplitudes can be split
into [17]
S∆-exch1 (νQ2) = S∆-pole
1 (νQ2) + S∆-exch1 (νQ2) (B23a)
νS∆-exch2 (νQ2) = νS∆-pole
2 (νQ2) + νS2∆-exch
(νQ2) (B23b)
and similarly for the unpolarized VVCS amplitudes discussed in Ref [30] Here we introduced the
∆-pole contributions S∆-polei and the ∆-non-pole contributions S∆-exch
i The former amplitudes
feature a pole at the ∆(1232)-production threshold and thus are proportional to
1
[sminusM2∆][uminusM2
∆]=
1
4M2N
1
ν2∆ minus ν2
(B24)
36
They can be reconstructed from the dispersion relations in Eq (5) with the tree-level ∆-production
cross sections as input cf Fig 10 in Ref [30]
σTT (νQ2) =π2α
M2NM
2+|~q |
minus g2
MMN (M+ + ν)|~q |2 +g2E(∆minus ν)(Q2 minusMNν)2
MN(B25a)
+g2CQ
4s(∆minus ν)
MNM2∆
minus 4gMgE(Q2 minusMNν)|~q |2 minus 4gMgCQ2|~q |2
+2gEgCQ
2[minusMNM∆ |~q |2 + s(Q2 +∆ν)
]MNM∆
δ(ν minus ν∆)
σLT (νQ2) =Qπ2α
M2NM
2+|~q |
g2E(MNν minusQ2) [M∆(MN + ν)minus s]
MN(B25b)
+g2CQ
2[MNM∆|~q |2 minus s(Q2 +∆ν)
]MNM2
∆
+ gMgEM∆|~q |2 minusgMgC(Q2 minusMNν)|~q |2
M∆
+gEgC(ν minus∆)(M2
N |~q |2 minus 2Q2s)
MNM∆
δ(ν minus ν∆)
with ∆ = M∆ minusMN M+ = M∆ + MN and the Mandelstam variable s = M2N + 2MNν minus Q2
Analytical expressions for the spin structure functions g1(xQ2) and g2(xQ2) can be constructed
from Eq (3) with the flux factor K(νQ2) = |~q | =radicν2 +Q2
In the ∆-non-pole contributions to S1(νQ2) and νS2(νQ2) the pole in ν at the ∆(1232)-
production threshold has canceled out
S∆-exch1 (νQ2) =
πα
MNM2+
[g2MQ
2+ + g2
E
(∆2 minus 3Q2
)+
4g2CQ
4
M2∆
minus 8gMgEM∆ωminus (B26a)
minus2gMgCQ2(MN minus 4M∆)
M∆+
2gEgCQ2(3MN minus 2M∆)
M∆
]
νS2∆-exch
(νQ2) =2πα
MNM2+
[g2EM∆∆ωminus +
g2M MNQ
2+
2+g2C Q
2(Q2 minus∆2)
2M∆(B26b)
+gEgM M∆(M∆ω+ minus 4MNωminus)minus gEgC ∆(2Q2 +MNω+)
+gMgC Q2(4MN minus ω+)
]+S∆-exch
2 (νQ2)
ν
[M2
∆ ω2+
M2N
+ ν2
]
with Q+ =radic
(M∆ +MN )2 +Q2 and ωplusmn = (M2∆minusM2
N plusmnQ2)2M∆ and the non-pole contribution
to S2(νQ2)
S∆-exch2 (νQ2) = minus2παMNν
M∆M2+
[gM + gE
]gC (B27)
These amplitudes to the contrary are not described by the tree-level ∆-production cross sections
in the standard dispersive approach [17] This peculiarity has been previously missed e g in the
calculation of the ∆-exchange contribution to the hydrogen hyperfine splitting in Ref [88] The
importance of including the ∆-non-pole contribution is also evident when considering the BC sum
37
rule in Eq (14) The ∆-pole terms by themselves violate the BC sum rule but cancel exactly with
the ∆-non-pole terms
limνrarr0
νS∆-pole2 (νQ2) + lim
νrarr0νS2
∆-exch(νQ2) = 0 (B28)
Appendix C Polarizabilities at Q2 = 0
In this section we give analytical expressions for the static values and slopes at Q2 = 0 of
the spin polarizabilities and moments of polarized structure functions In particular we give the
HB expansion of the πN -loop contributions and the ∆-exchange contributions The complete
expressions also for the π∆-loop contributions can be found in the Supplemented material Recall
that IA(0) = I1(0) = d2(0) = 0 and dd2(Q2)dQ2
∣∣∣Q2=0
= 0
1 πN-loop contribution
Here we give analytical expressions for the πN -loop contributions to the proton and neutron
spin polarizabilities expanded in powers of micro = mπMN viz the HB expansion Note that we
choose to expand here to a high order in micro the strict HB expansion would only retain the leading
term in an analogous NLO calculation
bull Static polarizabilities (Q2 = 0)
γ0p =e2g2
A
96π3f2πm
2π
1minus 21πmicro
8minus(
59
2+ 26 logmicro
)micro2 +
1875πmicro3
64
+3
(3
2+ 26 logmicro
)micro4 +
(C1)
γ0n =e2g2
A
48π3f2πm
2π
1
2minus 9πmicro
16minus 2micro2 logmicro+
75πmicro3
128minus 3micro4
4+
(C2)
δLTp =e2g2
A
192π3f2πm
2π
1minus 9πmicro
8+
(13
2minus 2 logmicro
)micro2 minus 465πmicro3
64
minus(
47
2+ 42 logmicro
)micro4 +
(C3)
δLTn =e2g2
A
96π3f2πm
2π
1
2+
3πmicro
16+ (1 + 2 log micro)micro2 minus 105πmicro3
128+
5micro4
4+
(C4)
38
δLTn =e2g2
A
96π3f2πm
2π
1
2+
3πmicro
16+ (1 + 2 log micro)micro2 minus 105πmicro3
128+
5micro4
4+
(C5)
γ0p =e2g2
A
16π3f2πm
4π
4
45minus 3πmicro
16+
14micro2
5minus 1813πmicro3
384minus 192
5(1 + log micro)micro4
+80703πmicro5
2048+
(C6)
γ0n =e2g2
A
16π3f2πm
4π
4
45minus 5πmicro
48+
4micro2
5minus 245πmicro3
384minus 32micro4 logmicro
15
+1323πmicro5
2048+
(C7)
bull Slopes of polarizabilities at Q2 = 0
dγ0p(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
2minus 45πmicro
4+ 223micro2 minus 28515πmicro3
64
minus9
(1953
4+ 449 logmicro
)micro4 +
570255πmicro5
128+
(C8)
dγ0n(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
2minus 81πmicro
8+ 94micro2 minus 2535πmicro3
32
minus3 (1 + 90 logmicro)micro4 +84315πmicro5
1024+
(C9)
dδLTp(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
2880π3f2πm
4π
minus5
2minus 27πmicro
32+ 20micro2 minus 5865πmicro3
256
+3
(617
4+ 36 logmicro
)micro4 minus 2056845πmicro5
4096+
(C10)
dδLTn(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
minus5
4minus 81πmicro
64minus 11micro2 +
10005πmicro3
512
+15
8(11 + 48 logmicro)micro4 minus 267015πmicro5
8192+
(C11)
dIAp(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
96π2f2πmicro
2
1minus 15πmicro
4minus 1
2(115 + 88 logmicro)micro2 +
1839πmicro3
32
+5 (5 + 34 logmicro)micro4 +
(C12)
dIAn(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
48π2f2πmicro
2
1
2minus 11πmicro
8minus 1
4(1 + 20 logmicro)micro2 +
99πmicro3
64
minus25micro4
12+
(C13)
dI1p(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
96π2f2πmicro
3π
8+ 2 (4 + 3 logmicro)microminus 537πmicro2
64
minus1
2(15 + 56 logmicro)micro3 +
(C14)
39
dI1n(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
48π2f2πmicro
minus π
16+
1
4(3 + 4 logmicro)microminus 57πmicro2
128+
2micro3
3+
(C15)
γ0p(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2g2
A
16π3f2πm
6π
1
105minus 23πmicro
256+
377micro2
210minus 15551πmicro3
6144+
3371micro4
105
minus1640457πmicro5
32768+
(C16)
γ0n(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2g2
A
16π3f2πm
6π
1
105minus 153πmicro
1792+
69micro2
70minus 4615πmicro3
6144+
172micro4
35
minus120897πmicro5
32768+
(C17)
2 ∆-exchange contribution
Here we give analytical expressions for the tree-level ∆-exchange contributions to the nucleon
spin polarizabilities and their slopes at Q2 = 0 Note that the ∆-exchange contributes equally to
proton and neutron polarizabilities Recall that for the magnetic γlowastN∆ coupling we introduced a
dipole form factor to mimic vector-meson dominance gM rarr gM(1 +Q2Λ2)2
bull Static polarizabilities (Q2 = 0)
γ0 = minus e2
4πM2+
(g2M
∆2+
g2E
M2+
minus 4gMgEM+∆
) (C18)
δLT =e2M∆
4πM3+
(g2E
MNM++gMgE∆MN
minus gEgCM2
∆
) (C19)
γ0 =e2M2
N
π∆2M4+
(minusg
2M
∆2+
g2E
M2+
+4gMgE∆M+
) (C20)
bull Slopes of polarizabilities at Q2 = 0
dγ0(Q2)
dQ2
∣∣∣∣∣Q2=0
= minus e2
πM2+∆
(g2M
∆
[1
4∆2minus 1
∆M++
1
2M2+
]minus 1
Λ2
g2M
∆+
g2E
2M2+
[1
2∆minus 3
M+
]minusgMgE
M+
[1
∆2minus 5
∆M++
1
M2+
]+
1
Λ2
2gMgEM+
+2gMgC∆M2
+
minus gEgCM3
+
) (C21)
40
d δLT (Q2)
dQ2
∣∣∣∣∣Q2=0
=e2M∆∆
4πMNM2+
(g2E
∆2M2+
[1
∆minus 4
M+
]minus g2
C
∆M2∆M
2+
+gMgE∆2M+
[1
∆2
minus 3
∆M++
1
M2+
]minus 2
Λ2
gMgE∆2M+
+gMgC∆M2
∆
[1
2∆2minus 2
∆M++
1
2M2+
]minus gEgC
2M2∆M
2+
[7
∆+
1
M+
]) (C22)
dIA(Q2)
dQ2
∣∣∣∣∣Q2=0
= minusM2N
M2+
(g2M
2∆2+
g2E
MNM+minus 2gMgE
∆M+minus gEgCM∆M+
) (C23)
dI1(Q2)
dQ2
∣∣∣∣∣Q2=0
= minusM∆M2N
2M3+
(g2E
MNM∆minus gMgE∆MN
minus gEgCM2
∆
) (C24)
γ0(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2M2
N
π∆3M6+
(g2M
∆
[3M2
∆ + 2M∆MN minus 9M2N
∆2+
4M2+
Λ2
]
minus4gMgE
[5M2
∆ minus 9M2N
M+∆2+
2M+
Λ2
]+
g2E
∆M+[7M∆ minus 9MN ]
minus8gMgC∆
+4gEgCM+
) (C25)
[1] D Drechsel B Pasquini and M Vanderhaeghen Dispersion relations in real and virtual
Compton scattering Phys Rept 378 99 (2003) hep-ph0212124
[2] S E Kuhn J-P Chen and E Leader Spin structure of the nucleon mdash status and recent
results Prog Part Nucl Phys 63 1 (2009) arXiv08123535 [hep-ph]
[3] F Hagelstein R Miskimen and V Pascalutsa Nucleon polarizabilities from Compton scat-
tering to hydrogen atom Prog Part Nucl Phys 88 29 (2016) arXiv151203765 [nucl-th]
[4] B Pasquini and M Vanderhaeghen Dispersion theory in electromagnetic interactions Ann
Rev Nucl Part Sci 68 75 (2018) arXiv180510482 [hep-ph]
[5] S Gerasimov A Sum rule for magnetic moments and the damping of the nucleon magnetic
moment in nuclei Sov J Nucl Phys 2 430 (1966)
[6] S Drell and A C Hearn Exact sum rule for nucleon magnetic moments Phys Rev Lett
16 908 (1966)
[7] J S Schwinger Source theory viewpoints in deep inelastic scattering Proc Natl Acad Sci
USA 72 1 (1975)
[8] J S Schwinger Source theory viewpoints in deep inelastic scattering Electromagnetic In-
teractions and Field Theory Proceedings 14 Internationale Universitatswochen Schladming
Austria February 24-March 7 1975 Acta Phys Austriaca Suppl 14 471 (1975)
41
[9] J Schwinger Source theory discussion of deep inelastic scattering with polarized particles
Proc Natl Acad Sci USA 72 1559 (1975)
[10] A M Baldin Polarizability of nucleons Nucl Phys 18 310 (1960)
[11] M Gell-Mann M L Goldberger and W E Thirring Use of causality conditions in quantum
theory Phys Rev 95 1612 (1954)
[12] A Pineda Leading chiral logarithms to the hyperfine splitting of the hydrogen and muonic
hydrogen Phys Rev C 67 025201 (2003)
[13] C Peset and A Pineda The two-photon exchange contribution to muonic hydrogen from
chiral perturbation theory Nucl Phys B 887 69 (2014) arXiv14064524 [hep-ph]
[14] C Peset and A Pineda Model-independent determination of the two-photon exchange contri-
bution to hyperfine splitting in muonic hydrogen JHEP 04 060 arXiv161205206 [nucl-th]
[15] F Hagelstein and V Pascalutsa Proton structure in the hyperfine splitting of muonic hydro-
gen PoS CD15 077 (2016) arXiv151104301 [nucl-th]
[16] F Hagelstein Exciting Nucleons in Compton Scattering and Hydrogen-Like Atoms PhD
thesis Mainz U Inst Kernphys (2017) arXiv171000874 [nucl-th]
[17] F Hagelstein ∆(1232)-Resonance in the hydrogen spectrum Proceedings 11th International
Workshop on the Physics of Excited Nucleons (NSTAR 2017) Columbia SC USA August
20-23 2017 Few Body Syst 59 93 (2018) arXiv180109790 [nucl-th]
[18] Y Prok et al (CLAS) Moments of the spin structure functions gp1 and gd1 for 005 lt Q2 lt 30
GeV2 Phys Lett B 672 12 (2009) arXiv08022232 [nucl-ex]
[19] H Dutz et al (GDH) First measurement of the Gerasimov-Drell-Hearn sum rule for 1H from
07 GeV to 18 GeV at ELSA Phys Rev Lett 91 192001 (2003)
[20] M Amarian et al (Jefferson Lab E94010) Measurement of the generalized forward spin po-
larizabilities of the neutron Phys Rev Lett 93 152301 (2004) arXivnucl-ex0406005
[21] M Amarian et al The Q2 evolution of the generalized Gerasimov-Drell-Hearn integral for
the neutron using a 3He target Phys Rev Lett 89 242301 (2002) arXivnucl-ex0205020
[22] M Amarian et al (Jefferson Lab E94-010)Q2 evolution of the neutron spin structure moments
using a 3He target Phys Rev Lett 92 022301 (2004) arXivhep-ex0310003
[23] A Deur et al Experimental determination of the evolution of the Bjorken integral at low Q2
Phys Rev Lett 93 212001 (2004) arXivhep-ex0407007
[24] K Slifer Low Q2 measurement of gp2 and the δLT spin polarizability Spin structure at long dis-
tance Proceedings Workshop Newport News USA March 12ndash13 2009 nucl-ex09064775
(2009) AIP Conf Proc 1155 10106313203293 (2009) arXiv09064775 [nucl-ex]
[25] P Solvignon et al (E01-012) Moments of the neutron g2 structure function at intermediate
42
Q2 Phys Rev C 92 015208 (2015) arXiv13044497 [nucl-ex]
[26] A Deur Experimental studies at low Q2 of the spin structure of the nucleon at Jefferson Lab
in 9th International Workshop on Chiral Dynamics (CD18) Durham NC USA September
17-21 2018 (2019) arXiv190305661 [nucl-ex]
[27] V Sulkosky et al (Jefferson Lab E97-110) Measurement of the 3He spin-structure functions
and of neutron (3He) spin-dependent sum rules at 0035 le Q2 le 024 GeV2 Phys Lett B
805 135428 (2020) arXiv190805709 [nucl-ex]
[28] K Adhikari et al (CLAS) Measurement of the Q2 Dependence of the Deuteron Spin Structure
Function g1 and its Moments at Low Q2 with CLAS Phys Rev Lett 120 062501 (2018)
arXiv171101974 [nucl-ex]
[29] V Lensky F Hagelstein A Hiller Blin and V Pascalutsa Comment on rdquoMeasurement of
the Q2 Dependence of the Deuteron Spin Structure Function g1 and its Moments at Low Q2
with CLASrdquo (2018) arXiv180603219 [nucl-th]
[30] J M Alarcn F Hagelstein V Lensky and V Pascalutsa Forward doubly-virtual Compton
scattering off the nucleon in chiral perturbation theory at NLO the subtraction function and
moments of unpolarized structure functions (2020) arXiv200509518 [hep-ph]
[31] V Bernard E Epelbaum H Krebs and U-G Meiszligner New insights into the spin structure
of the nucleon Phys Rev D 87 054032 (2013) arXiv12092523 [hep-ph]
[32] V Lensky J M Alarcon and V Pascalutsa Moments of nucleon structure functions at
next-to-leading order in baryon chiral perturbation theory Phys Rev C 90 055202 (2014)
arXiv14072574 [hep-ph]
[33] V Pascalutsa and M Vanderhaeghen Chiral effective-field theory in the ∆(1232) region
I Pion electroproduction on the nucleon Phys Rev D 73 034003 (2006) arXivhep-
ph0512244
[34] V Pascalutsa and M Vanderhaeghen Electromagnetic nucleon-to-Delta transition in chiral
effective field theory Phys Rev Lett 95 232001 (2005) arXivhep-ph0508060
[35] T R Hemmert B R Holstein and J Kambor Systematic 1M expansion for spin 32
particles in baryon chiral perturbation theory Phys Lett B 395 89 (1997) arXivhep-
ph9606456
[36] V Pascalutsa and D R Phillips Effective theory of the ∆(1232) in Compton scattering off
the nucleon Phys Rev C 67 055202 (2003) arXivnucl-th0212024
[37] V Pascalutsa M Vanderhaeghen and S N Yang Electromagnetic excitation of the ∆(1232)-
resonance Phys Rept 437 125 (2007) arXivhep-ph0609004
[38] K A Olive et al (Particle Data Group) Review of Particle Physics Chin Phys C 38 090001
43
(2014)
[39] F E Low Scattering of light of very low frequency by systems of spin 12 Phys Rev 96
1428 (1954)
[40] M Gell-Mann and M L Goldberger Scattering of low-energy photons by particles of spin
12 Phys Rev 96 1433 (1954)
[41] A M Harun ar-Rashid A simple derivation of Schwingerrsquos sum rule for spin dependent
structure functions Nuovo Cim A 33 447 (1976)
[42] F Hagelstein and V Pascalutsa Dissecting the hadronic contributions to (g minus 2)micro by
Schwingerrsquos sum rule Phys Rev Lett 120 072002 (2018) arXiv171004571 [hep-ph]
[43] V Lensky V Pascalutsa M Vanderhaeghen and C Kao Spin-dependent sum rules con-
necting real and virtual Compton scattering verified Phys Rev D 95 074001 (2017)
arXiv170101947 [hep-ph]
[44] H Burkhardt and W N Cottingham Sum rules for forward virtual Compton scattering
Annals Phys 56 453 (1970)
[45] V Lensky and V Pascalutsa Predictive powers of chiral perturbation theory in Compton
scattering off protons Eur Phys J C 65 195 (2010) arXiv09070451 [hep-ph]
[46] H Grieszlighammer J McGovern D Phillips and G Feldman Using effective field theory to
analyse low-energy Compton scattering data from protons and light nuclei Prog Part Nucl
Phys 67 841 (2012) arXiv12036834 [nucl-th]
[47] H W Grieszlighammer J A McGovern and D R Phillips Nucleon polarisabilities at and
beyond physical pion masses Eur Phys J A 52 139 (2016) arXiv151101952 [nucl-th]
[48] E Epelbaum H Krebs and U-G Meiszligner Improved chiral nucleon-nucleon potential up to
next-to-next-to-next-to-leading order Eur Phys J A 51 53 (2015) arXiv14120142 [nucl-
th]
[49] E Epelbaum H Krebs and U-G Meiszligner Precision nucleon-nucleon potential at fifth order
in the chiral expansion Phys Rev Lett 115 122301 (2015) arXiv14124623 [nucl-th]
[50] C W Kao T Spitzenberg and M Vanderhaeghen Burkhardt-Cottingham sum rule and
forward spin polarizabilities in heavy baryon chiral perturbation theory Phys Rev D 67
016001 (2003) arXivhep-ph0209241
[51] C-W Kao D Drechsel S Kamalov and M Vanderhaeghen Higher moments of nucleon
spin structure functions in heavy baryon chiral perturbation theory and in a resonance model
Phys Rev D 69 056004 (2004) arXivhep-ph0312102
[52] T Becher and H Leutwyler Baryon chiral perturbation theory in manifestly Lorentz invariant
form Eur Phys J C 9 643 (1999) arXivhep-ph9901384
44
[53] J Gasser M E Sainio and A Svarc Nucleons with chiral loops Nucl Phys B 307 779
(1988)
[54] L S Geng J Martin Camalich L Alvarez-Ruso and M J Vicente Vacas Leading SU(3)-
breaking corrections to the baryon magnetic moments in chiral perturbation theory Phys
Rev Lett 101 222002 (2008) arXiv08051419 [hep-ph]
[55] D Drechsel S S Kamalov and L Tiator The GDH sum rule and related integrals Phys
Rev D 63 114010 (2001) arXivhep-ph0008306
[56] D Drechsel O Hanstein S S Kamalov and L Tiator A unitary isobar model for pion
photo- and electroproduction on the proton up to 1 GeV Nucl Phys A 645 145 (1999)
arXivnucl-th9807001
[57] L Tiator private communication (2020)
[58] V Bernard T R Hemmert and U-G Meiszligner Spin structure of the nucleon at low energies
Phys Rev D 67 076008 (2003) arXivhep-ph0212033
[59] O Gryniuk F Hagelstein and V Pascalutsa Evaluation of the forward Compton scattering
off protons II Spin-dependent amplitude and observables Phys Rev D 94 034043 (2016)
arXiv160400789 [nucl-th]
[60] R Zielinski The g2p Experiment A Measurement of the Protonrsquos Spin Structure Functions
PhD thesis New Hampshire U (2010) arXiv170808297 [nucl-ex]
[61] N Guler et al (CLAS) Precise determination of the deuteron spin structure at low to mod-
erate Q2 with CLAS and extraction of the neutron contribution Phys Rev C 92 055201
(2015) arXiv150507877 [nucl-ex]
[62] C E Carlson V Nazaryan and K Griffioen Proton structure corrections to electronic
and muonic hydrogen hyperfine splitting Phys Rev A 78 022517 (2008) arXiv08052603
[physicsatom-ph]
[63] V Bernard N Kaiser and U-G Meiszligner Chiral dynamics in nucleons and nuclei Int J
Mod Phys E 4 193 (1995) arXivhep-ph9501384
[64] T R Hemmert B R Holstein and J Kambor ∆(1232) and the polarizabilities of the nucleon
Phys Rev D 55 5598 (1997) arXivhep-ph9612374
[65] V Pascalutsa and R Timmermans Field theory of nucleon to higher-spin baryon transitions
Phys Rev C 60 042201 (1999) arXivnucl-th9905065
[66] V Pascalutsa Quantization of an interacting spin-32 field and the Delta isobar Phys Rev
D 58 096002 (1998) arXivhep-ph9802288
[67] H Krebs Double Virtual Compton Scattering and SpinStructure of the Nucleon PoS
CD2018 031 (2019)
45
[68] N Kochelev and Y Oh Axial anomaly and the δLT puzzle Phys Rev D 016012 (2012)
arXiv11034892 [hep-ph]
[69] D Drechsel S Kamalov and L Tiator Unitary isobar model ndash MAID2007 Eur Phys J A
69 (2007) available at httpsmaidkphuni-mainzde arXiv07100306 [nucl-th]
[70] P J Mohr B N Taylor and D B Newell CODATA recommended values of the fundamental
physical constants 2010 Rev Mod Phys 84 1527 (2012)
[71] J Ahrens et al (GDH A2) First measurement of the Gerasimov-Drell-Hearn integral for 1H
from 200 to 800 MeV Phys Rev Lett 87 022003 (2001) arXivhep-ex0105089 [hep-ex]
[72] K Helbing (GDH) Experimental verification of the GDH sum rule at ELSA and MAMI Nucl
Phys Proc Suppl 105 113 (2002)
[73] R Bradford A Bodek H S Budd and J Arrington A New parameterization of the nucleon
elastic form-factors NuInt05 proceedings of the 4th International Workshop on Neutrino-
Nucleus Interactions in the Few-GeV Region Okayama Japan 26-29 September 2005 Nucl
Phys Proc Suppl 159 127 (2006) arXivhep-ex0602017
[74] J D Bjorken Applications of the chiral U(6)timesU(6) algebra of current densities Phys Rev
148 1467 (1966)
[75] J D Bjorken Inelastic scattering of polarized leptons from polarized nucleons Phys Rev D
1 1376 (1970)
[76] V Bernard T R Hemmert and U-G Meiszligner Novel analysis of chiral loop effects in
the generalized Gerasimov-Drell-Hearn sum rule Phys Lett B 545 105 (2002) arXivhep-
ph0203167
[77] A Deur et al Experimental study of isovector spin sum rules Phys Rev D 78 032001
(2008) arXiv08023198 [nucl-ex]
[78] B Pasquini P Pedroni and D Drechsel Higher order forward spin polarizability Phys Lett
B 687 160 (2010) arXiv10014230 [hep-ph]
[79] R Jaffe g2ndashThe nucleonrsquos other spin-dependent structure function Comments Nucl Part
Phys 19 239 (1990)
[80] E V Shuryak and A Vainshtein Theory of power corrections to deep inelastic scattering in
quantum chromodynamics (II) Qminus4 effects polarized target Nucl Phys B 201 141 (1982)
[81] S Wandzura and F Wilczek Sum rules for spin dependent electroproduction Test of rela-
tivistic constituent quarks Phys Lett B 72 195 (1977)
[82] B W Filippone and X-D Ji The spin structure of the nucleon Adv Nucl Phys 26 1
(2001) arXivhep-ph0101224
[83] M Burkardt The g2 structure function Proceedings Workshop on Spin structure at long
46
distance Newport News USA March 12-13 2009 AIP Conf Proc 1155 26 (2009)
arXiv09054079 [hep-ph]
[84] A Deur S J Brodsky and G F de Teramond The spin structure of the nucleon Rept
Prog Phys 82 076201 (2019) arXiv180705250 [hep-ph]
[85] B R Holstein D Drechsel B Pasquini and M Vanderhaeghen Higher order polarizabilities
of the proton Phys Rev C 61 034316 (2000) arXivhep-ph9910427
[86] B R Holstein V Pascalutsa and M Vanderhaeghen Sum rules for magnetic moments and
polarizabilities in QED and chiral effective-field theory Phys Rev D 72 094014 (2005)
arXivhep-ph0507016
[87] B Pasquini D Drechsel and L Tiator Invariant amplitudes for pion electroproduction Eur
Phys J A 34 387 (2007) arXiv07122327 [hep-ph]
[88] A J Buchmann Non-spherical proton shape and hydrogen hyperfine splitting Proceedings
International Workshop on Precision Physics of Simple Atomic Systems (PSAS 2008) Wind-
sor Ontario Canada July 21-26 2008 Can J Phys 87 773 (2009) arXiv09104747
[physicsatom-ph]
47
σ(πminusp)LT =
e2g2AM
3N
64πf2πQs
3λ4γ(M2
N minusQ2 minus s)(M2N + s(2βπβγ + 2λπλγ minus 1))
times
1
M2N + s(2βπβγ minus 2λπλγ minus 1)
[4λπsλγ(M2
N minusQ2 minus s)(
2sλ2γ
(s((1minus 2λ2
π)Q2
+ 2(2απ minus 1)sβγ(2βπβγ minus 1))minusM2N (Q2 + 2(1minus 2απ)sβγ)
)+ (Q2 + 2sβ2
γ)(M2N + s(2βπβγ minus 1)
)2 + 8(απ minus 1)απs
3λ4γ
)]minus(M2N + s(2βπβγ + 2λπλγ minus 1)
)times[
log
(minusM2N minus 2βπsβγ minus 2λπsλγ + s
minusM2N minus 2βπsβγ + 2λπsλγ + s
)(minus (Q2 + 2sβ2
γ)(M2N + s(2βπβγ minus 1)
)(minusM2
N + 2Q2 + 2βπsβγ + s)minus 2sλ2γ
(minusM2
N (Q2 + 2sβγ) +Q4 +Q2s(4απβγ
minus 2βγ minus 2λ2π + 1) + 2s2βγ(4(απ minus 1)βπβγ + 1)
)minus 8(απ minus 1)2s3λ4
γ
)+ 2
(4s2λ2
γ
(2(απ minus 1)βγ(Q2 + 2βπsβγ)minus λ2
πQ2)
+ (Q2 + 2βπsβγ)2(Q2 + 2sβ2γ)
+ 8(απ minus 1)2s3λ4γ
)arctanh
(2λπsλγ
Q2 + 2βπsβγ
)] (B22)
2 ∆-production channel
The tree-level ∆-exchange diagram in Fig 2 of Ref [30] contributes to the non-Born part of
the VVCS amplitudes The contribution of the ∆ exchange to the VVCS amplitudes can be split
into [17]
S∆-exch1 (νQ2) = S∆-pole
1 (νQ2) + S∆-exch1 (νQ2) (B23a)
νS∆-exch2 (νQ2) = νS∆-pole
2 (νQ2) + νS2∆-exch
(νQ2) (B23b)
and similarly for the unpolarized VVCS amplitudes discussed in Ref [30] Here we introduced the
∆-pole contributions S∆-polei and the ∆-non-pole contributions S∆-exch
i The former amplitudes
feature a pole at the ∆(1232)-production threshold and thus are proportional to
1
[sminusM2∆][uminusM2
∆]=
1
4M2N
1
ν2∆ minus ν2
(B24)
36
They can be reconstructed from the dispersion relations in Eq (5) with the tree-level ∆-production
cross sections as input cf Fig 10 in Ref [30]
σTT (νQ2) =π2α
M2NM
2+|~q |
minus g2
MMN (M+ + ν)|~q |2 +g2E(∆minus ν)(Q2 minusMNν)2
MN(B25a)
+g2CQ
4s(∆minus ν)
MNM2∆
minus 4gMgE(Q2 minusMNν)|~q |2 minus 4gMgCQ2|~q |2
+2gEgCQ
2[minusMNM∆ |~q |2 + s(Q2 +∆ν)
]MNM∆
δ(ν minus ν∆)
σLT (νQ2) =Qπ2α
M2NM
2+|~q |
g2E(MNν minusQ2) [M∆(MN + ν)minus s]
MN(B25b)
+g2CQ
2[MNM∆|~q |2 minus s(Q2 +∆ν)
]MNM2
∆
+ gMgEM∆|~q |2 minusgMgC(Q2 minusMNν)|~q |2
M∆
+gEgC(ν minus∆)(M2
N |~q |2 minus 2Q2s)
MNM∆
δ(ν minus ν∆)
with ∆ = M∆ minusMN M+ = M∆ + MN and the Mandelstam variable s = M2N + 2MNν minus Q2
Analytical expressions for the spin structure functions g1(xQ2) and g2(xQ2) can be constructed
from Eq (3) with the flux factor K(νQ2) = |~q | =radicν2 +Q2
In the ∆-non-pole contributions to S1(νQ2) and νS2(νQ2) the pole in ν at the ∆(1232)-
production threshold has canceled out
S∆-exch1 (νQ2) =
πα
MNM2+
[g2MQ
2+ + g2
E
(∆2 minus 3Q2
)+
4g2CQ
4
M2∆
minus 8gMgEM∆ωminus (B26a)
minus2gMgCQ2(MN minus 4M∆)
M∆+
2gEgCQ2(3MN minus 2M∆)
M∆
]
νS2∆-exch
(νQ2) =2πα
MNM2+
[g2EM∆∆ωminus +
g2M MNQ
2+
2+g2C Q
2(Q2 minus∆2)
2M∆(B26b)
+gEgM M∆(M∆ω+ minus 4MNωminus)minus gEgC ∆(2Q2 +MNω+)
+gMgC Q2(4MN minus ω+)
]+S∆-exch
2 (νQ2)
ν
[M2
∆ ω2+
M2N
+ ν2
]
with Q+ =radic
(M∆ +MN )2 +Q2 and ωplusmn = (M2∆minusM2
N plusmnQ2)2M∆ and the non-pole contribution
to S2(νQ2)
S∆-exch2 (νQ2) = minus2παMNν
M∆M2+
[gM + gE
]gC (B27)
These amplitudes to the contrary are not described by the tree-level ∆-production cross sections
in the standard dispersive approach [17] This peculiarity has been previously missed e g in the
calculation of the ∆-exchange contribution to the hydrogen hyperfine splitting in Ref [88] The
importance of including the ∆-non-pole contribution is also evident when considering the BC sum
37
rule in Eq (14) The ∆-pole terms by themselves violate the BC sum rule but cancel exactly with
the ∆-non-pole terms
limνrarr0
νS∆-pole2 (νQ2) + lim
νrarr0νS2
∆-exch(νQ2) = 0 (B28)
Appendix C Polarizabilities at Q2 = 0
In this section we give analytical expressions for the static values and slopes at Q2 = 0 of
the spin polarizabilities and moments of polarized structure functions In particular we give the
HB expansion of the πN -loop contributions and the ∆-exchange contributions The complete
expressions also for the π∆-loop contributions can be found in the Supplemented material Recall
that IA(0) = I1(0) = d2(0) = 0 and dd2(Q2)dQ2
∣∣∣Q2=0
= 0
1 πN-loop contribution
Here we give analytical expressions for the πN -loop contributions to the proton and neutron
spin polarizabilities expanded in powers of micro = mπMN viz the HB expansion Note that we
choose to expand here to a high order in micro the strict HB expansion would only retain the leading
term in an analogous NLO calculation
bull Static polarizabilities (Q2 = 0)
γ0p =e2g2
A
96π3f2πm
2π
1minus 21πmicro
8minus(
59
2+ 26 logmicro
)micro2 +
1875πmicro3
64
+3
(3
2+ 26 logmicro
)micro4 +
(C1)
γ0n =e2g2
A
48π3f2πm
2π
1
2minus 9πmicro
16minus 2micro2 logmicro+
75πmicro3
128minus 3micro4
4+
(C2)
δLTp =e2g2
A
192π3f2πm
2π
1minus 9πmicro
8+
(13
2minus 2 logmicro
)micro2 minus 465πmicro3
64
minus(
47
2+ 42 logmicro
)micro4 +
(C3)
δLTn =e2g2
A
96π3f2πm
2π
1
2+
3πmicro
16+ (1 + 2 log micro)micro2 minus 105πmicro3
128+
5micro4
4+
(C4)
38
δLTn =e2g2
A
96π3f2πm
2π
1
2+
3πmicro
16+ (1 + 2 log micro)micro2 minus 105πmicro3
128+
5micro4
4+
(C5)
γ0p =e2g2
A
16π3f2πm
4π
4
45minus 3πmicro
16+
14micro2
5minus 1813πmicro3
384minus 192
5(1 + log micro)micro4
+80703πmicro5
2048+
(C6)
γ0n =e2g2
A
16π3f2πm
4π
4
45minus 5πmicro
48+
4micro2
5minus 245πmicro3
384minus 32micro4 logmicro
15
+1323πmicro5
2048+
(C7)
bull Slopes of polarizabilities at Q2 = 0
dγ0p(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
2minus 45πmicro
4+ 223micro2 minus 28515πmicro3
64
minus9
(1953
4+ 449 logmicro
)micro4 +
570255πmicro5
128+
(C8)
dγ0n(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
2minus 81πmicro
8+ 94micro2 minus 2535πmicro3
32
minus3 (1 + 90 logmicro)micro4 +84315πmicro5
1024+
(C9)
dδLTp(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
2880π3f2πm
4π
minus5
2minus 27πmicro
32+ 20micro2 minus 5865πmicro3
256
+3
(617
4+ 36 logmicro
)micro4 minus 2056845πmicro5
4096+
(C10)
dδLTn(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
minus5
4minus 81πmicro
64minus 11micro2 +
10005πmicro3
512
+15
8(11 + 48 logmicro)micro4 minus 267015πmicro5
8192+
(C11)
dIAp(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
96π2f2πmicro
2
1minus 15πmicro
4minus 1
2(115 + 88 logmicro)micro2 +
1839πmicro3
32
+5 (5 + 34 logmicro)micro4 +
(C12)
dIAn(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
48π2f2πmicro
2
1
2minus 11πmicro
8minus 1
4(1 + 20 logmicro)micro2 +
99πmicro3
64
minus25micro4
12+
(C13)
dI1p(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
96π2f2πmicro
3π
8+ 2 (4 + 3 logmicro)microminus 537πmicro2
64
minus1
2(15 + 56 logmicro)micro3 +
(C14)
39
dI1n(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
48π2f2πmicro
minus π
16+
1
4(3 + 4 logmicro)microminus 57πmicro2
128+
2micro3
3+
(C15)
γ0p(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2g2
A
16π3f2πm
6π
1
105minus 23πmicro
256+
377micro2
210minus 15551πmicro3
6144+
3371micro4
105
minus1640457πmicro5
32768+
(C16)
γ0n(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2g2
A
16π3f2πm
6π
1
105minus 153πmicro
1792+
69micro2
70minus 4615πmicro3
6144+
172micro4
35
minus120897πmicro5
32768+
(C17)
2 ∆-exchange contribution
Here we give analytical expressions for the tree-level ∆-exchange contributions to the nucleon
spin polarizabilities and their slopes at Q2 = 0 Note that the ∆-exchange contributes equally to
proton and neutron polarizabilities Recall that for the magnetic γlowastN∆ coupling we introduced a
dipole form factor to mimic vector-meson dominance gM rarr gM(1 +Q2Λ2)2
bull Static polarizabilities (Q2 = 0)
γ0 = minus e2
4πM2+
(g2M
∆2+
g2E
M2+
minus 4gMgEM+∆
) (C18)
δLT =e2M∆
4πM3+
(g2E
MNM++gMgE∆MN
minus gEgCM2
∆
) (C19)
γ0 =e2M2
N
π∆2M4+
(minusg
2M
∆2+
g2E
M2+
+4gMgE∆M+
) (C20)
bull Slopes of polarizabilities at Q2 = 0
dγ0(Q2)
dQ2
∣∣∣∣∣Q2=0
= minus e2
πM2+∆
(g2M
∆
[1
4∆2minus 1
∆M++
1
2M2+
]minus 1
Λ2
g2M
∆+
g2E
2M2+
[1
2∆minus 3
M+
]minusgMgE
M+
[1
∆2minus 5
∆M++
1
M2+
]+
1
Λ2
2gMgEM+
+2gMgC∆M2
+
minus gEgCM3
+
) (C21)
40
d δLT (Q2)
dQ2
∣∣∣∣∣Q2=0
=e2M∆∆
4πMNM2+
(g2E
∆2M2+
[1
∆minus 4
M+
]minus g2
C
∆M2∆M
2+
+gMgE∆2M+
[1
∆2
minus 3
∆M++
1
M2+
]minus 2
Λ2
gMgE∆2M+
+gMgC∆M2
∆
[1
2∆2minus 2
∆M++
1
2M2+
]minus gEgC
2M2∆M
2+
[7
∆+
1
M+
]) (C22)
dIA(Q2)
dQ2
∣∣∣∣∣Q2=0
= minusM2N
M2+
(g2M
2∆2+
g2E
MNM+minus 2gMgE
∆M+minus gEgCM∆M+
) (C23)
dI1(Q2)
dQ2
∣∣∣∣∣Q2=0
= minusM∆M2N
2M3+
(g2E
MNM∆minus gMgE∆MN
minus gEgCM2
∆
) (C24)
γ0(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2M2
N
π∆3M6+
(g2M
∆
[3M2
∆ + 2M∆MN minus 9M2N
∆2+
4M2+
Λ2
]
minus4gMgE
[5M2
∆ minus 9M2N
M+∆2+
2M+
Λ2
]+
g2E
∆M+[7M∆ minus 9MN ]
minus8gMgC∆
+4gEgCM+
) (C25)
[1] D Drechsel B Pasquini and M Vanderhaeghen Dispersion relations in real and virtual
Compton scattering Phys Rept 378 99 (2003) hep-ph0212124
[2] S E Kuhn J-P Chen and E Leader Spin structure of the nucleon mdash status and recent
results Prog Part Nucl Phys 63 1 (2009) arXiv08123535 [hep-ph]
[3] F Hagelstein R Miskimen and V Pascalutsa Nucleon polarizabilities from Compton scat-
tering to hydrogen atom Prog Part Nucl Phys 88 29 (2016) arXiv151203765 [nucl-th]
[4] B Pasquini and M Vanderhaeghen Dispersion theory in electromagnetic interactions Ann
Rev Nucl Part Sci 68 75 (2018) arXiv180510482 [hep-ph]
[5] S Gerasimov A Sum rule for magnetic moments and the damping of the nucleon magnetic
moment in nuclei Sov J Nucl Phys 2 430 (1966)
[6] S Drell and A C Hearn Exact sum rule for nucleon magnetic moments Phys Rev Lett
16 908 (1966)
[7] J S Schwinger Source theory viewpoints in deep inelastic scattering Proc Natl Acad Sci
USA 72 1 (1975)
[8] J S Schwinger Source theory viewpoints in deep inelastic scattering Electromagnetic In-
teractions and Field Theory Proceedings 14 Internationale Universitatswochen Schladming
Austria February 24-March 7 1975 Acta Phys Austriaca Suppl 14 471 (1975)
41
[9] J Schwinger Source theory discussion of deep inelastic scattering with polarized particles
Proc Natl Acad Sci USA 72 1559 (1975)
[10] A M Baldin Polarizability of nucleons Nucl Phys 18 310 (1960)
[11] M Gell-Mann M L Goldberger and W E Thirring Use of causality conditions in quantum
theory Phys Rev 95 1612 (1954)
[12] A Pineda Leading chiral logarithms to the hyperfine splitting of the hydrogen and muonic
hydrogen Phys Rev C 67 025201 (2003)
[13] C Peset and A Pineda The two-photon exchange contribution to muonic hydrogen from
chiral perturbation theory Nucl Phys B 887 69 (2014) arXiv14064524 [hep-ph]
[14] C Peset and A Pineda Model-independent determination of the two-photon exchange contri-
bution to hyperfine splitting in muonic hydrogen JHEP 04 060 arXiv161205206 [nucl-th]
[15] F Hagelstein and V Pascalutsa Proton structure in the hyperfine splitting of muonic hydro-
gen PoS CD15 077 (2016) arXiv151104301 [nucl-th]
[16] F Hagelstein Exciting Nucleons in Compton Scattering and Hydrogen-Like Atoms PhD
thesis Mainz U Inst Kernphys (2017) arXiv171000874 [nucl-th]
[17] F Hagelstein ∆(1232)-Resonance in the hydrogen spectrum Proceedings 11th International
Workshop on the Physics of Excited Nucleons (NSTAR 2017) Columbia SC USA August
20-23 2017 Few Body Syst 59 93 (2018) arXiv180109790 [nucl-th]
[18] Y Prok et al (CLAS) Moments of the spin structure functions gp1 and gd1 for 005 lt Q2 lt 30
GeV2 Phys Lett B 672 12 (2009) arXiv08022232 [nucl-ex]
[19] H Dutz et al (GDH) First measurement of the Gerasimov-Drell-Hearn sum rule for 1H from
07 GeV to 18 GeV at ELSA Phys Rev Lett 91 192001 (2003)
[20] M Amarian et al (Jefferson Lab E94010) Measurement of the generalized forward spin po-
larizabilities of the neutron Phys Rev Lett 93 152301 (2004) arXivnucl-ex0406005
[21] M Amarian et al The Q2 evolution of the generalized Gerasimov-Drell-Hearn integral for
the neutron using a 3He target Phys Rev Lett 89 242301 (2002) arXivnucl-ex0205020
[22] M Amarian et al (Jefferson Lab E94-010)Q2 evolution of the neutron spin structure moments
using a 3He target Phys Rev Lett 92 022301 (2004) arXivhep-ex0310003
[23] A Deur et al Experimental determination of the evolution of the Bjorken integral at low Q2
Phys Rev Lett 93 212001 (2004) arXivhep-ex0407007
[24] K Slifer Low Q2 measurement of gp2 and the δLT spin polarizability Spin structure at long dis-
tance Proceedings Workshop Newport News USA March 12ndash13 2009 nucl-ex09064775
(2009) AIP Conf Proc 1155 10106313203293 (2009) arXiv09064775 [nucl-ex]
[25] P Solvignon et al (E01-012) Moments of the neutron g2 structure function at intermediate
42
Q2 Phys Rev C 92 015208 (2015) arXiv13044497 [nucl-ex]
[26] A Deur Experimental studies at low Q2 of the spin structure of the nucleon at Jefferson Lab
in 9th International Workshop on Chiral Dynamics (CD18) Durham NC USA September
17-21 2018 (2019) arXiv190305661 [nucl-ex]
[27] V Sulkosky et al (Jefferson Lab E97-110) Measurement of the 3He spin-structure functions
and of neutron (3He) spin-dependent sum rules at 0035 le Q2 le 024 GeV2 Phys Lett B
805 135428 (2020) arXiv190805709 [nucl-ex]
[28] K Adhikari et al (CLAS) Measurement of the Q2 Dependence of the Deuteron Spin Structure
Function g1 and its Moments at Low Q2 with CLAS Phys Rev Lett 120 062501 (2018)
arXiv171101974 [nucl-ex]
[29] V Lensky F Hagelstein A Hiller Blin and V Pascalutsa Comment on rdquoMeasurement of
the Q2 Dependence of the Deuteron Spin Structure Function g1 and its Moments at Low Q2
with CLASrdquo (2018) arXiv180603219 [nucl-th]
[30] J M Alarcn F Hagelstein V Lensky and V Pascalutsa Forward doubly-virtual Compton
scattering off the nucleon in chiral perturbation theory at NLO the subtraction function and
moments of unpolarized structure functions (2020) arXiv200509518 [hep-ph]
[31] V Bernard E Epelbaum H Krebs and U-G Meiszligner New insights into the spin structure
of the nucleon Phys Rev D 87 054032 (2013) arXiv12092523 [hep-ph]
[32] V Lensky J M Alarcon and V Pascalutsa Moments of nucleon structure functions at
next-to-leading order in baryon chiral perturbation theory Phys Rev C 90 055202 (2014)
arXiv14072574 [hep-ph]
[33] V Pascalutsa and M Vanderhaeghen Chiral effective-field theory in the ∆(1232) region
I Pion electroproduction on the nucleon Phys Rev D 73 034003 (2006) arXivhep-
ph0512244
[34] V Pascalutsa and M Vanderhaeghen Electromagnetic nucleon-to-Delta transition in chiral
effective field theory Phys Rev Lett 95 232001 (2005) arXivhep-ph0508060
[35] T R Hemmert B R Holstein and J Kambor Systematic 1M expansion for spin 32
particles in baryon chiral perturbation theory Phys Lett B 395 89 (1997) arXivhep-
ph9606456
[36] V Pascalutsa and D R Phillips Effective theory of the ∆(1232) in Compton scattering off
the nucleon Phys Rev C 67 055202 (2003) arXivnucl-th0212024
[37] V Pascalutsa M Vanderhaeghen and S N Yang Electromagnetic excitation of the ∆(1232)-
resonance Phys Rept 437 125 (2007) arXivhep-ph0609004
[38] K A Olive et al (Particle Data Group) Review of Particle Physics Chin Phys C 38 090001
43
(2014)
[39] F E Low Scattering of light of very low frequency by systems of spin 12 Phys Rev 96
1428 (1954)
[40] M Gell-Mann and M L Goldberger Scattering of low-energy photons by particles of spin
12 Phys Rev 96 1433 (1954)
[41] A M Harun ar-Rashid A simple derivation of Schwingerrsquos sum rule for spin dependent
structure functions Nuovo Cim A 33 447 (1976)
[42] F Hagelstein and V Pascalutsa Dissecting the hadronic contributions to (g minus 2)micro by
Schwingerrsquos sum rule Phys Rev Lett 120 072002 (2018) arXiv171004571 [hep-ph]
[43] V Lensky V Pascalutsa M Vanderhaeghen and C Kao Spin-dependent sum rules con-
necting real and virtual Compton scattering verified Phys Rev D 95 074001 (2017)
arXiv170101947 [hep-ph]
[44] H Burkhardt and W N Cottingham Sum rules for forward virtual Compton scattering
Annals Phys 56 453 (1970)
[45] V Lensky and V Pascalutsa Predictive powers of chiral perturbation theory in Compton
scattering off protons Eur Phys J C 65 195 (2010) arXiv09070451 [hep-ph]
[46] H Grieszlighammer J McGovern D Phillips and G Feldman Using effective field theory to
analyse low-energy Compton scattering data from protons and light nuclei Prog Part Nucl
Phys 67 841 (2012) arXiv12036834 [nucl-th]
[47] H W Grieszlighammer J A McGovern and D R Phillips Nucleon polarisabilities at and
beyond physical pion masses Eur Phys J A 52 139 (2016) arXiv151101952 [nucl-th]
[48] E Epelbaum H Krebs and U-G Meiszligner Improved chiral nucleon-nucleon potential up to
next-to-next-to-next-to-leading order Eur Phys J A 51 53 (2015) arXiv14120142 [nucl-
th]
[49] E Epelbaum H Krebs and U-G Meiszligner Precision nucleon-nucleon potential at fifth order
in the chiral expansion Phys Rev Lett 115 122301 (2015) arXiv14124623 [nucl-th]
[50] C W Kao T Spitzenberg and M Vanderhaeghen Burkhardt-Cottingham sum rule and
forward spin polarizabilities in heavy baryon chiral perturbation theory Phys Rev D 67
016001 (2003) arXivhep-ph0209241
[51] C-W Kao D Drechsel S Kamalov and M Vanderhaeghen Higher moments of nucleon
spin structure functions in heavy baryon chiral perturbation theory and in a resonance model
Phys Rev D 69 056004 (2004) arXivhep-ph0312102
[52] T Becher and H Leutwyler Baryon chiral perturbation theory in manifestly Lorentz invariant
form Eur Phys J C 9 643 (1999) arXivhep-ph9901384
44
[53] J Gasser M E Sainio and A Svarc Nucleons with chiral loops Nucl Phys B 307 779
(1988)
[54] L S Geng J Martin Camalich L Alvarez-Ruso and M J Vicente Vacas Leading SU(3)-
breaking corrections to the baryon magnetic moments in chiral perturbation theory Phys
Rev Lett 101 222002 (2008) arXiv08051419 [hep-ph]
[55] D Drechsel S S Kamalov and L Tiator The GDH sum rule and related integrals Phys
Rev D 63 114010 (2001) arXivhep-ph0008306
[56] D Drechsel O Hanstein S S Kamalov and L Tiator A unitary isobar model for pion
photo- and electroproduction on the proton up to 1 GeV Nucl Phys A 645 145 (1999)
arXivnucl-th9807001
[57] L Tiator private communication (2020)
[58] V Bernard T R Hemmert and U-G Meiszligner Spin structure of the nucleon at low energies
Phys Rev D 67 076008 (2003) arXivhep-ph0212033
[59] O Gryniuk F Hagelstein and V Pascalutsa Evaluation of the forward Compton scattering
off protons II Spin-dependent amplitude and observables Phys Rev D 94 034043 (2016)
arXiv160400789 [nucl-th]
[60] R Zielinski The g2p Experiment A Measurement of the Protonrsquos Spin Structure Functions
PhD thesis New Hampshire U (2010) arXiv170808297 [nucl-ex]
[61] N Guler et al (CLAS) Precise determination of the deuteron spin structure at low to mod-
erate Q2 with CLAS and extraction of the neutron contribution Phys Rev C 92 055201
(2015) arXiv150507877 [nucl-ex]
[62] C E Carlson V Nazaryan and K Griffioen Proton structure corrections to electronic
and muonic hydrogen hyperfine splitting Phys Rev A 78 022517 (2008) arXiv08052603
[physicsatom-ph]
[63] V Bernard N Kaiser and U-G Meiszligner Chiral dynamics in nucleons and nuclei Int J
Mod Phys E 4 193 (1995) arXivhep-ph9501384
[64] T R Hemmert B R Holstein and J Kambor ∆(1232) and the polarizabilities of the nucleon
Phys Rev D 55 5598 (1997) arXivhep-ph9612374
[65] V Pascalutsa and R Timmermans Field theory of nucleon to higher-spin baryon transitions
Phys Rev C 60 042201 (1999) arXivnucl-th9905065
[66] V Pascalutsa Quantization of an interacting spin-32 field and the Delta isobar Phys Rev
D 58 096002 (1998) arXivhep-ph9802288
[67] H Krebs Double Virtual Compton Scattering and SpinStructure of the Nucleon PoS
CD2018 031 (2019)
45
[68] N Kochelev and Y Oh Axial anomaly and the δLT puzzle Phys Rev D 016012 (2012)
arXiv11034892 [hep-ph]
[69] D Drechsel S Kamalov and L Tiator Unitary isobar model ndash MAID2007 Eur Phys J A
69 (2007) available at httpsmaidkphuni-mainzde arXiv07100306 [nucl-th]
[70] P J Mohr B N Taylor and D B Newell CODATA recommended values of the fundamental
physical constants 2010 Rev Mod Phys 84 1527 (2012)
[71] J Ahrens et al (GDH A2) First measurement of the Gerasimov-Drell-Hearn integral for 1H
from 200 to 800 MeV Phys Rev Lett 87 022003 (2001) arXivhep-ex0105089 [hep-ex]
[72] K Helbing (GDH) Experimental verification of the GDH sum rule at ELSA and MAMI Nucl
Phys Proc Suppl 105 113 (2002)
[73] R Bradford A Bodek H S Budd and J Arrington A New parameterization of the nucleon
elastic form-factors NuInt05 proceedings of the 4th International Workshop on Neutrino-
Nucleus Interactions in the Few-GeV Region Okayama Japan 26-29 September 2005 Nucl
Phys Proc Suppl 159 127 (2006) arXivhep-ex0602017
[74] J D Bjorken Applications of the chiral U(6)timesU(6) algebra of current densities Phys Rev
148 1467 (1966)
[75] J D Bjorken Inelastic scattering of polarized leptons from polarized nucleons Phys Rev D
1 1376 (1970)
[76] V Bernard T R Hemmert and U-G Meiszligner Novel analysis of chiral loop effects in
the generalized Gerasimov-Drell-Hearn sum rule Phys Lett B 545 105 (2002) arXivhep-
ph0203167
[77] A Deur et al Experimental study of isovector spin sum rules Phys Rev D 78 032001
(2008) arXiv08023198 [nucl-ex]
[78] B Pasquini P Pedroni and D Drechsel Higher order forward spin polarizability Phys Lett
B 687 160 (2010) arXiv10014230 [hep-ph]
[79] R Jaffe g2ndashThe nucleonrsquos other spin-dependent structure function Comments Nucl Part
Phys 19 239 (1990)
[80] E V Shuryak and A Vainshtein Theory of power corrections to deep inelastic scattering in
quantum chromodynamics (II) Qminus4 effects polarized target Nucl Phys B 201 141 (1982)
[81] S Wandzura and F Wilczek Sum rules for spin dependent electroproduction Test of rela-
tivistic constituent quarks Phys Lett B 72 195 (1977)
[82] B W Filippone and X-D Ji The spin structure of the nucleon Adv Nucl Phys 26 1
(2001) arXivhep-ph0101224
[83] M Burkardt The g2 structure function Proceedings Workshop on Spin structure at long
46
distance Newport News USA March 12-13 2009 AIP Conf Proc 1155 26 (2009)
arXiv09054079 [hep-ph]
[84] A Deur S J Brodsky and G F de Teramond The spin structure of the nucleon Rept
Prog Phys 82 076201 (2019) arXiv180705250 [hep-ph]
[85] B R Holstein D Drechsel B Pasquini and M Vanderhaeghen Higher order polarizabilities
of the proton Phys Rev C 61 034316 (2000) arXivhep-ph9910427
[86] B R Holstein V Pascalutsa and M Vanderhaeghen Sum rules for magnetic moments and
polarizabilities in QED and chiral effective-field theory Phys Rev D 72 094014 (2005)
arXivhep-ph0507016
[87] B Pasquini D Drechsel and L Tiator Invariant amplitudes for pion electroproduction Eur
Phys J A 34 387 (2007) arXiv07122327 [hep-ph]
[88] A J Buchmann Non-spherical proton shape and hydrogen hyperfine splitting Proceedings
International Workshop on Precision Physics of Simple Atomic Systems (PSAS 2008) Wind-
sor Ontario Canada July 21-26 2008 Can J Phys 87 773 (2009) arXiv09104747
[physicsatom-ph]
47
They can be reconstructed from the dispersion relations in Eq (5) with the tree-level ∆-production
cross sections as input cf Fig 10 in Ref [30]
σTT (νQ2) =π2α
M2NM
2+|~q |
minus g2
MMN (M+ + ν)|~q |2 +g2E(∆minus ν)(Q2 minusMNν)2
MN(B25a)
+g2CQ
4s(∆minus ν)
MNM2∆
minus 4gMgE(Q2 minusMNν)|~q |2 minus 4gMgCQ2|~q |2
+2gEgCQ
2[minusMNM∆ |~q |2 + s(Q2 +∆ν)
]MNM∆
δ(ν minus ν∆)
σLT (νQ2) =Qπ2α
M2NM
2+|~q |
g2E(MNν minusQ2) [M∆(MN + ν)minus s]
MN(B25b)
+g2CQ
2[MNM∆|~q |2 minus s(Q2 +∆ν)
]MNM2
∆
+ gMgEM∆|~q |2 minusgMgC(Q2 minusMNν)|~q |2
M∆
+gEgC(ν minus∆)(M2
N |~q |2 minus 2Q2s)
MNM∆
δ(ν minus ν∆)
with ∆ = M∆ minusMN M+ = M∆ + MN and the Mandelstam variable s = M2N + 2MNν minus Q2
Analytical expressions for the spin structure functions g1(xQ2) and g2(xQ2) can be constructed
from Eq (3) with the flux factor K(νQ2) = |~q | =radicν2 +Q2
In the ∆-non-pole contributions to S1(νQ2) and νS2(νQ2) the pole in ν at the ∆(1232)-
production threshold has canceled out
S∆-exch1 (νQ2) =
πα
MNM2+
[g2MQ
2+ + g2
E
(∆2 minus 3Q2
)+
4g2CQ
4
M2∆
minus 8gMgEM∆ωminus (B26a)
minus2gMgCQ2(MN minus 4M∆)
M∆+
2gEgCQ2(3MN minus 2M∆)
M∆
]
νS2∆-exch
(νQ2) =2πα
MNM2+
[g2EM∆∆ωminus +
g2M MNQ
2+
2+g2C Q
2(Q2 minus∆2)
2M∆(B26b)
+gEgM M∆(M∆ω+ minus 4MNωminus)minus gEgC ∆(2Q2 +MNω+)
+gMgC Q2(4MN minus ω+)
]+S∆-exch
2 (νQ2)
ν
[M2
∆ ω2+
M2N
+ ν2
]
with Q+ =radic
(M∆ +MN )2 +Q2 and ωplusmn = (M2∆minusM2
N plusmnQ2)2M∆ and the non-pole contribution
to S2(νQ2)
S∆-exch2 (νQ2) = minus2παMNν
M∆M2+
[gM + gE
]gC (B27)
These amplitudes to the contrary are not described by the tree-level ∆-production cross sections
in the standard dispersive approach [17] This peculiarity has been previously missed e g in the
calculation of the ∆-exchange contribution to the hydrogen hyperfine splitting in Ref [88] The
importance of including the ∆-non-pole contribution is also evident when considering the BC sum
37
rule in Eq (14) The ∆-pole terms by themselves violate the BC sum rule but cancel exactly with
the ∆-non-pole terms
limνrarr0
νS∆-pole2 (νQ2) + lim
νrarr0νS2
∆-exch(νQ2) = 0 (B28)
Appendix C Polarizabilities at Q2 = 0
In this section we give analytical expressions for the static values and slopes at Q2 = 0 of
the spin polarizabilities and moments of polarized structure functions In particular we give the
HB expansion of the πN -loop contributions and the ∆-exchange contributions The complete
expressions also for the π∆-loop contributions can be found in the Supplemented material Recall
that IA(0) = I1(0) = d2(0) = 0 and dd2(Q2)dQ2
∣∣∣Q2=0
= 0
1 πN-loop contribution
Here we give analytical expressions for the πN -loop contributions to the proton and neutron
spin polarizabilities expanded in powers of micro = mπMN viz the HB expansion Note that we
choose to expand here to a high order in micro the strict HB expansion would only retain the leading
term in an analogous NLO calculation
bull Static polarizabilities (Q2 = 0)
γ0p =e2g2
A
96π3f2πm
2π
1minus 21πmicro
8minus(
59
2+ 26 logmicro
)micro2 +
1875πmicro3
64
+3
(3
2+ 26 logmicro
)micro4 +
(C1)
γ0n =e2g2
A
48π3f2πm
2π
1
2minus 9πmicro
16minus 2micro2 logmicro+
75πmicro3
128minus 3micro4
4+
(C2)
δLTp =e2g2
A
192π3f2πm
2π
1minus 9πmicro
8+
(13
2minus 2 logmicro
)micro2 minus 465πmicro3
64
minus(
47
2+ 42 logmicro
)micro4 +
(C3)
δLTn =e2g2
A
96π3f2πm
2π
1
2+
3πmicro
16+ (1 + 2 log micro)micro2 minus 105πmicro3
128+
5micro4
4+
(C4)
38
δLTn =e2g2
A
96π3f2πm
2π
1
2+
3πmicro
16+ (1 + 2 log micro)micro2 minus 105πmicro3
128+
5micro4
4+
(C5)
γ0p =e2g2
A
16π3f2πm
4π
4
45minus 3πmicro
16+
14micro2
5minus 1813πmicro3
384minus 192
5(1 + log micro)micro4
+80703πmicro5
2048+
(C6)
γ0n =e2g2
A
16π3f2πm
4π
4
45minus 5πmicro
48+
4micro2
5minus 245πmicro3
384minus 32micro4 logmicro
15
+1323πmicro5
2048+
(C7)
bull Slopes of polarizabilities at Q2 = 0
dγ0p(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
2minus 45πmicro
4+ 223micro2 minus 28515πmicro3
64
minus9
(1953
4+ 449 logmicro
)micro4 +
570255πmicro5
128+
(C8)
dγ0n(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
2minus 81πmicro
8+ 94micro2 minus 2535πmicro3
32
minus3 (1 + 90 logmicro)micro4 +84315πmicro5
1024+
(C9)
dδLTp(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
2880π3f2πm
4π
minus5
2minus 27πmicro
32+ 20micro2 minus 5865πmicro3
256
+3
(617
4+ 36 logmicro
)micro4 minus 2056845πmicro5
4096+
(C10)
dδLTn(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
minus5
4minus 81πmicro
64minus 11micro2 +
10005πmicro3
512
+15
8(11 + 48 logmicro)micro4 minus 267015πmicro5
8192+
(C11)
dIAp(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
96π2f2πmicro
2
1minus 15πmicro
4minus 1
2(115 + 88 logmicro)micro2 +
1839πmicro3
32
+5 (5 + 34 logmicro)micro4 +
(C12)
dIAn(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
48π2f2πmicro
2
1
2minus 11πmicro
8minus 1
4(1 + 20 logmicro)micro2 +
99πmicro3
64
minus25micro4
12+
(C13)
dI1p(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
96π2f2πmicro
3π
8+ 2 (4 + 3 logmicro)microminus 537πmicro2
64
minus1
2(15 + 56 logmicro)micro3 +
(C14)
39
dI1n(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
48π2f2πmicro
minus π
16+
1
4(3 + 4 logmicro)microminus 57πmicro2
128+
2micro3
3+
(C15)
γ0p(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2g2
A
16π3f2πm
6π
1
105minus 23πmicro
256+
377micro2
210minus 15551πmicro3
6144+
3371micro4
105
minus1640457πmicro5
32768+
(C16)
γ0n(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2g2
A
16π3f2πm
6π
1
105minus 153πmicro
1792+
69micro2
70minus 4615πmicro3
6144+
172micro4
35
minus120897πmicro5
32768+
(C17)
2 ∆-exchange contribution
Here we give analytical expressions for the tree-level ∆-exchange contributions to the nucleon
spin polarizabilities and their slopes at Q2 = 0 Note that the ∆-exchange contributes equally to
proton and neutron polarizabilities Recall that for the magnetic γlowastN∆ coupling we introduced a
dipole form factor to mimic vector-meson dominance gM rarr gM(1 +Q2Λ2)2
bull Static polarizabilities (Q2 = 0)
γ0 = minus e2
4πM2+
(g2M
∆2+
g2E
M2+
minus 4gMgEM+∆
) (C18)
δLT =e2M∆
4πM3+
(g2E
MNM++gMgE∆MN
minus gEgCM2
∆
) (C19)
γ0 =e2M2
N
π∆2M4+
(minusg
2M
∆2+
g2E
M2+
+4gMgE∆M+
) (C20)
bull Slopes of polarizabilities at Q2 = 0
dγ0(Q2)
dQ2
∣∣∣∣∣Q2=0
= minus e2
πM2+∆
(g2M
∆
[1
4∆2minus 1
∆M++
1
2M2+
]minus 1
Λ2
g2M
∆+
g2E
2M2+
[1
2∆minus 3
M+
]minusgMgE
M+
[1
∆2minus 5
∆M++
1
M2+
]+
1
Λ2
2gMgEM+
+2gMgC∆M2
+
minus gEgCM3
+
) (C21)
40
d δLT (Q2)
dQ2
∣∣∣∣∣Q2=0
=e2M∆∆
4πMNM2+
(g2E
∆2M2+
[1
∆minus 4
M+
]minus g2
C
∆M2∆M
2+
+gMgE∆2M+
[1
∆2
minus 3
∆M++
1
M2+
]minus 2
Λ2
gMgE∆2M+
+gMgC∆M2
∆
[1
2∆2minus 2
∆M++
1
2M2+
]minus gEgC
2M2∆M
2+
[7
∆+
1
M+
]) (C22)
dIA(Q2)
dQ2
∣∣∣∣∣Q2=0
= minusM2N
M2+
(g2M
2∆2+
g2E
MNM+minus 2gMgE
∆M+minus gEgCM∆M+
) (C23)
dI1(Q2)
dQ2
∣∣∣∣∣Q2=0
= minusM∆M2N
2M3+
(g2E
MNM∆minus gMgE∆MN
minus gEgCM2
∆
) (C24)
γ0(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2M2
N
π∆3M6+
(g2M
∆
[3M2
∆ + 2M∆MN minus 9M2N
∆2+
4M2+
Λ2
]
minus4gMgE
[5M2
∆ minus 9M2N
M+∆2+
2M+
Λ2
]+
g2E
∆M+[7M∆ minus 9MN ]
minus8gMgC∆
+4gEgCM+
) (C25)
[1] D Drechsel B Pasquini and M Vanderhaeghen Dispersion relations in real and virtual
Compton scattering Phys Rept 378 99 (2003) hep-ph0212124
[2] S E Kuhn J-P Chen and E Leader Spin structure of the nucleon mdash status and recent
results Prog Part Nucl Phys 63 1 (2009) arXiv08123535 [hep-ph]
[3] F Hagelstein R Miskimen and V Pascalutsa Nucleon polarizabilities from Compton scat-
tering to hydrogen atom Prog Part Nucl Phys 88 29 (2016) arXiv151203765 [nucl-th]
[4] B Pasquini and M Vanderhaeghen Dispersion theory in electromagnetic interactions Ann
Rev Nucl Part Sci 68 75 (2018) arXiv180510482 [hep-ph]
[5] S Gerasimov A Sum rule for magnetic moments and the damping of the nucleon magnetic
moment in nuclei Sov J Nucl Phys 2 430 (1966)
[6] S Drell and A C Hearn Exact sum rule for nucleon magnetic moments Phys Rev Lett
16 908 (1966)
[7] J S Schwinger Source theory viewpoints in deep inelastic scattering Proc Natl Acad Sci
USA 72 1 (1975)
[8] J S Schwinger Source theory viewpoints in deep inelastic scattering Electromagnetic In-
teractions and Field Theory Proceedings 14 Internationale Universitatswochen Schladming
Austria February 24-March 7 1975 Acta Phys Austriaca Suppl 14 471 (1975)
41
[9] J Schwinger Source theory discussion of deep inelastic scattering with polarized particles
Proc Natl Acad Sci USA 72 1559 (1975)
[10] A M Baldin Polarizability of nucleons Nucl Phys 18 310 (1960)
[11] M Gell-Mann M L Goldberger and W E Thirring Use of causality conditions in quantum
theory Phys Rev 95 1612 (1954)
[12] A Pineda Leading chiral logarithms to the hyperfine splitting of the hydrogen and muonic
hydrogen Phys Rev C 67 025201 (2003)
[13] C Peset and A Pineda The two-photon exchange contribution to muonic hydrogen from
chiral perturbation theory Nucl Phys B 887 69 (2014) arXiv14064524 [hep-ph]
[14] C Peset and A Pineda Model-independent determination of the two-photon exchange contri-
bution to hyperfine splitting in muonic hydrogen JHEP 04 060 arXiv161205206 [nucl-th]
[15] F Hagelstein and V Pascalutsa Proton structure in the hyperfine splitting of muonic hydro-
gen PoS CD15 077 (2016) arXiv151104301 [nucl-th]
[16] F Hagelstein Exciting Nucleons in Compton Scattering and Hydrogen-Like Atoms PhD
thesis Mainz U Inst Kernphys (2017) arXiv171000874 [nucl-th]
[17] F Hagelstein ∆(1232)-Resonance in the hydrogen spectrum Proceedings 11th International
Workshop on the Physics of Excited Nucleons (NSTAR 2017) Columbia SC USA August
20-23 2017 Few Body Syst 59 93 (2018) arXiv180109790 [nucl-th]
[18] Y Prok et al (CLAS) Moments of the spin structure functions gp1 and gd1 for 005 lt Q2 lt 30
GeV2 Phys Lett B 672 12 (2009) arXiv08022232 [nucl-ex]
[19] H Dutz et al (GDH) First measurement of the Gerasimov-Drell-Hearn sum rule for 1H from
07 GeV to 18 GeV at ELSA Phys Rev Lett 91 192001 (2003)
[20] M Amarian et al (Jefferson Lab E94010) Measurement of the generalized forward spin po-
larizabilities of the neutron Phys Rev Lett 93 152301 (2004) arXivnucl-ex0406005
[21] M Amarian et al The Q2 evolution of the generalized Gerasimov-Drell-Hearn integral for
the neutron using a 3He target Phys Rev Lett 89 242301 (2002) arXivnucl-ex0205020
[22] M Amarian et al (Jefferson Lab E94-010)Q2 evolution of the neutron spin structure moments
using a 3He target Phys Rev Lett 92 022301 (2004) arXivhep-ex0310003
[23] A Deur et al Experimental determination of the evolution of the Bjorken integral at low Q2
Phys Rev Lett 93 212001 (2004) arXivhep-ex0407007
[24] K Slifer Low Q2 measurement of gp2 and the δLT spin polarizability Spin structure at long dis-
tance Proceedings Workshop Newport News USA March 12ndash13 2009 nucl-ex09064775
(2009) AIP Conf Proc 1155 10106313203293 (2009) arXiv09064775 [nucl-ex]
[25] P Solvignon et al (E01-012) Moments of the neutron g2 structure function at intermediate
42
Q2 Phys Rev C 92 015208 (2015) arXiv13044497 [nucl-ex]
[26] A Deur Experimental studies at low Q2 of the spin structure of the nucleon at Jefferson Lab
in 9th International Workshop on Chiral Dynamics (CD18) Durham NC USA September
17-21 2018 (2019) arXiv190305661 [nucl-ex]
[27] V Sulkosky et al (Jefferson Lab E97-110) Measurement of the 3He spin-structure functions
and of neutron (3He) spin-dependent sum rules at 0035 le Q2 le 024 GeV2 Phys Lett B
805 135428 (2020) arXiv190805709 [nucl-ex]
[28] K Adhikari et al (CLAS) Measurement of the Q2 Dependence of the Deuteron Spin Structure
Function g1 and its Moments at Low Q2 with CLAS Phys Rev Lett 120 062501 (2018)
arXiv171101974 [nucl-ex]
[29] V Lensky F Hagelstein A Hiller Blin and V Pascalutsa Comment on rdquoMeasurement of
the Q2 Dependence of the Deuteron Spin Structure Function g1 and its Moments at Low Q2
with CLASrdquo (2018) arXiv180603219 [nucl-th]
[30] J M Alarcn F Hagelstein V Lensky and V Pascalutsa Forward doubly-virtual Compton
scattering off the nucleon in chiral perturbation theory at NLO the subtraction function and
moments of unpolarized structure functions (2020) arXiv200509518 [hep-ph]
[31] V Bernard E Epelbaum H Krebs and U-G Meiszligner New insights into the spin structure
of the nucleon Phys Rev D 87 054032 (2013) arXiv12092523 [hep-ph]
[32] V Lensky J M Alarcon and V Pascalutsa Moments of nucleon structure functions at
next-to-leading order in baryon chiral perturbation theory Phys Rev C 90 055202 (2014)
arXiv14072574 [hep-ph]
[33] V Pascalutsa and M Vanderhaeghen Chiral effective-field theory in the ∆(1232) region
I Pion electroproduction on the nucleon Phys Rev D 73 034003 (2006) arXivhep-
ph0512244
[34] V Pascalutsa and M Vanderhaeghen Electromagnetic nucleon-to-Delta transition in chiral
effective field theory Phys Rev Lett 95 232001 (2005) arXivhep-ph0508060
[35] T R Hemmert B R Holstein and J Kambor Systematic 1M expansion for spin 32
particles in baryon chiral perturbation theory Phys Lett B 395 89 (1997) arXivhep-
ph9606456
[36] V Pascalutsa and D R Phillips Effective theory of the ∆(1232) in Compton scattering off
the nucleon Phys Rev C 67 055202 (2003) arXivnucl-th0212024
[37] V Pascalutsa M Vanderhaeghen and S N Yang Electromagnetic excitation of the ∆(1232)-
resonance Phys Rept 437 125 (2007) arXivhep-ph0609004
[38] K A Olive et al (Particle Data Group) Review of Particle Physics Chin Phys C 38 090001
43
(2014)
[39] F E Low Scattering of light of very low frequency by systems of spin 12 Phys Rev 96
1428 (1954)
[40] M Gell-Mann and M L Goldberger Scattering of low-energy photons by particles of spin
12 Phys Rev 96 1433 (1954)
[41] A M Harun ar-Rashid A simple derivation of Schwingerrsquos sum rule for spin dependent
structure functions Nuovo Cim A 33 447 (1976)
[42] F Hagelstein and V Pascalutsa Dissecting the hadronic contributions to (g minus 2)micro by
Schwingerrsquos sum rule Phys Rev Lett 120 072002 (2018) arXiv171004571 [hep-ph]
[43] V Lensky V Pascalutsa M Vanderhaeghen and C Kao Spin-dependent sum rules con-
necting real and virtual Compton scattering verified Phys Rev D 95 074001 (2017)
arXiv170101947 [hep-ph]
[44] H Burkhardt and W N Cottingham Sum rules for forward virtual Compton scattering
Annals Phys 56 453 (1970)
[45] V Lensky and V Pascalutsa Predictive powers of chiral perturbation theory in Compton
scattering off protons Eur Phys J C 65 195 (2010) arXiv09070451 [hep-ph]
[46] H Grieszlighammer J McGovern D Phillips and G Feldman Using effective field theory to
analyse low-energy Compton scattering data from protons and light nuclei Prog Part Nucl
Phys 67 841 (2012) arXiv12036834 [nucl-th]
[47] H W Grieszlighammer J A McGovern and D R Phillips Nucleon polarisabilities at and
beyond physical pion masses Eur Phys J A 52 139 (2016) arXiv151101952 [nucl-th]
[48] E Epelbaum H Krebs and U-G Meiszligner Improved chiral nucleon-nucleon potential up to
next-to-next-to-next-to-leading order Eur Phys J A 51 53 (2015) arXiv14120142 [nucl-
th]
[49] E Epelbaum H Krebs and U-G Meiszligner Precision nucleon-nucleon potential at fifth order
in the chiral expansion Phys Rev Lett 115 122301 (2015) arXiv14124623 [nucl-th]
[50] C W Kao T Spitzenberg and M Vanderhaeghen Burkhardt-Cottingham sum rule and
forward spin polarizabilities in heavy baryon chiral perturbation theory Phys Rev D 67
016001 (2003) arXivhep-ph0209241
[51] C-W Kao D Drechsel S Kamalov and M Vanderhaeghen Higher moments of nucleon
spin structure functions in heavy baryon chiral perturbation theory and in a resonance model
Phys Rev D 69 056004 (2004) arXivhep-ph0312102
[52] T Becher and H Leutwyler Baryon chiral perturbation theory in manifestly Lorentz invariant
form Eur Phys J C 9 643 (1999) arXivhep-ph9901384
44
[53] J Gasser M E Sainio and A Svarc Nucleons with chiral loops Nucl Phys B 307 779
(1988)
[54] L S Geng J Martin Camalich L Alvarez-Ruso and M J Vicente Vacas Leading SU(3)-
breaking corrections to the baryon magnetic moments in chiral perturbation theory Phys
Rev Lett 101 222002 (2008) arXiv08051419 [hep-ph]
[55] D Drechsel S S Kamalov and L Tiator The GDH sum rule and related integrals Phys
Rev D 63 114010 (2001) arXivhep-ph0008306
[56] D Drechsel O Hanstein S S Kamalov and L Tiator A unitary isobar model for pion
photo- and electroproduction on the proton up to 1 GeV Nucl Phys A 645 145 (1999)
arXivnucl-th9807001
[57] L Tiator private communication (2020)
[58] V Bernard T R Hemmert and U-G Meiszligner Spin structure of the nucleon at low energies
Phys Rev D 67 076008 (2003) arXivhep-ph0212033
[59] O Gryniuk F Hagelstein and V Pascalutsa Evaluation of the forward Compton scattering
off protons II Spin-dependent amplitude and observables Phys Rev D 94 034043 (2016)
arXiv160400789 [nucl-th]
[60] R Zielinski The g2p Experiment A Measurement of the Protonrsquos Spin Structure Functions
PhD thesis New Hampshire U (2010) arXiv170808297 [nucl-ex]
[61] N Guler et al (CLAS) Precise determination of the deuteron spin structure at low to mod-
erate Q2 with CLAS and extraction of the neutron contribution Phys Rev C 92 055201
(2015) arXiv150507877 [nucl-ex]
[62] C E Carlson V Nazaryan and K Griffioen Proton structure corrections to electronic
and muonic hydrogen hyperfine splitting Phys Rev A 78 022517 (2008) arXiv08052603
[physicsatom-ph]
[63] V Bernard N Kaiser and U-G Meiszligner Chiral dynamics in nucleons and nuclei Int J
Mod Phys E 4 193 (1995) arXivhep-ph9501384
[64] T R Hemmert B R Holstein and J Kambor ∆(1232) and the polarizabilities of the nucleon
Phys Rev D 55 5598 (1997) arXivhep-ph9612374
[65] V Pascalutsa and R Timmermans Field theory of nucleon to higher-spin baryon transitions
Phys Rev C 60 042201 (1999) arXivnucl-th9905065
[66] V Pascalutsa Quantization of an interacting spin-32 field and the Delta isobar Phys Rev
D 58 096002 (1998) arXivhep-ph9802288
[67] H Krebs Double Virtual Compton Scattering and SpinStructure of the Nucleon PoS
CD2018 031 (2019)
45
[68] N Kochelev and Y Oh Axial anomaly and the δLT puzzle Phys Rev D 016012 (2012)
arXiv11034892 [hep-ph]
[69] D Drechsel S Kamalov and L Tiator Unitary isobar model ndash MAID2007 Eur Phys J A
69 (2007) available at httpsmaidkphuni-mainzde arXiv07100306 [nucl-th]
[70] P J Mohr B N Taylor and D B Newell CODATA recommended values of the fundamental
physical constants 2010 Rev Mod Phys 84 1527 (2012)
[71] J Ahrens et al (GDH A2) First measurement of the Gerasimov-Drell-Hearn integral for 1H
from 200 to 800 MeV Phys Rev Lett 87 022003 (2001) arXivhep-ex0105089 [hep-ex]
[72] K Helbing (GDH) Experimental verification of the GDH sum rule at ELSA and MAMI Nucl
Phys Proc Suppl 105 113 (2002)
[73] R Bradford A Bodek H S Budd and J Arrington A New parameterization of the nucleon
elastic form-factors NuInt05 proceedings of the 4th International Workshop on Neutrino-
Nucleus Interactions in the Few-GeV Region Okayama Japan 26-29 September 2005 Nucl
Phys Proc Suppl 159 127 (2006) arXivhep-ex0602017
[74] J D Bjorken Applications of the chiral U(6)timesU(6) algebra of current densities Phys Rev
148 1467 (1966)
[75] J D Bjorken Inelastic scattering of polarized leptons from polarized nucleons Phys Rev D
1 1376 (1970)
[76] V Bernard T R Hemmert and U-G Meiszligner Novel analysis of chiral loop effects in
the generalized Gerasimov-Drell-Hearn sum rule Phys Lett B 545 105 (2002) arXivhep-
ph0203167
[77] A Deur et al Experimental study of isovector spin sum rules Phys Rev D 78 032001
(2008) arXiv08023198 [nucl-ex]
[78] B Pasquini P Pedroni and D Drechsel Higher order forward spin polarizability Phys Lett
B 687 160 (2010) arXiv10014230 [hep-ph]
[79] R Jaffe g2ndashThe nucleonrsquos other spin-dependent structure function Comments Nucl Part
Phys 19 239 (1990)
[80] E V Shuryak and A Vainshtein Theory of power corrections to deep inelastic scattering in
quantum chromodynamics (II) Qminus4 effects polarized target Nucl Phys B 201 141 (1982)
[81] S Wandzura and F Wilczek Sum rules for spin dependent electroproduction Test of rela-
tivistic constituent quarks Phys Lett B 72 195 (1977)
[82] B W Filippone and X-D Ji The spin structure of the nucleon Adv Nucl Phys 26 1
(2001) arXivhep-ph0101224
[83] M Burkardt The g2 structure function Proceedings Workshop on Spin structure at long
46
distance Newport News USA March 12-13 2009 AIP Conf Proc 1155 26 (2009)
arXiv09054079 [hep-ph]
[84] A Deur S J Brodsky and G F de Teramond The spin structure of the nucleon Rept
Prog Phys 82 076201 (2019) arXiv180705250 [hep-ph]
[85] B R Holstein D Drechsel B Pasquini and M Vanderhaeghen Higher order polarizabilities
of the proton Phys Rev C 61 034316 (2000) arXivhep-ph9910427
[86] B R Holstein V Pascalutsa and M Vanderhaeghen Sum rules for magnetic moments and
polarizabilities in QED and chiral effective-field theory Phys Rev D 72 094014 (2005)
arXivhep-ph0507016
[87] B Pasquini D Drechsel and L Tiator Invariant amplitudes for pion electroproduction Eur
Phys J A 34 387 (2007) arXiv07122327 [hep-ph]
[88] A J Buchmann Non-spherical proton shape and hydrogen hyperfine splitting Proceedings
International Workshop on Precision Physics of Simple Atomic Systems (PSAS 2008) Wind-
sor Ontario Canada July 21-26 2008 Can J Phys 87 773 (2009) arXiv09104747
[physicsatom-ph]
47
rule in Eq (14) The ∆-pole terms by themselves violate the BC sum rule but cancel exactly with
the ∆-non-pole terms
limνrarr0
νS∆-pole2 (νQ2) + lim
νrarr0νS2
∆-exch(νQ2) = 0 (B28)
Appendix C Polarizabilities at Q2 = 0
In this section we give analytical expressions for the static values and slopes at Q2 = 0 of
the spin polarizabilities and moments of polarized structure functions In particular we give the
HB expansion of the πN -loop contributions and the ∆-exchange contributions The complete
expressions also for the π∆-loop contributions can be found in the Supplemented material Recall
that IA(0) = I1(0) = d2(0) = 0 and dd2(Q2)dQ2
∣∣∣Q2=0
= 0
1 πN-loop contribution
Here we give analytical expressions for the πN -loop contributions to the proton and neutron
spin polarizabilities expanded in powers of micro = mπMN viz the HB expansion Note that we
choose to expand here to a high order in micro the strict HB expansion would only retain the leading
term in an analogous NLO calculation
bull Static polarizabilities (Q2 = 0)
γ0p =e2g2
A
96π3f2πm
2π
1minus 21πmicro
8minus(
59
2+ 26 logmicro
)micro2 +
1875πmicro3
64
+3
(3
2+ 26 logmicro
)micro4 +
(C1)
γ0n =e2g2
A
48π3f2πm
2π
1
2minus 9πmicro
16minus 2micro2 logmicro+
75πmicro3
128minus 3micro4
4+
(C2)
δLTp =e2g2
A
192π3f2πm
2π
1minus 9πmicro
8+
(13
2minus 2 logmicro
)micro2 minus 465πmicro3
64
minus(
47
2+ 42 logmicro
)micro4 +
(C3)
δLTn =e2g2
A
96π3f2πm
2π
1
2+
3πmicro
16+ (1 + 2 log micro)micro2 minus 105πmicro3
128+
5micro4
4+
(C4)
38
δLTn =e2g2
A
96π3f2πm
2π
1
2+
3πmicro
16+ (1 + 2 log micro)micro2 minus 105πmicro3
128+
5micro4
4+
(C5)
γ0p =e2g2
A
16π3f2πm
4π
4
45minus 3πmicro
16+
14micro2
5minus 1813πmicro3
384minus 192
5(1 + log micro)micro4
+80703πmicro5
2048+
(C6)
γ0n =e2g2
A
16π3f2πm
4π
4
45minus 5πmicro
48+
4micro2
5minus 245πmicro3
384minus 32micro4 logmicro
15
+1323πmicro5
2048+
(C7)
bull Slopes of polarizabilities at Q2 = 0
dγ0p(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
2minus 45πmicro
4+ 223micro2 minus 28515πmicro3
64
minus9
(1953
4+ 449 logmicro
)micro4 +
570255πmicro5
128+
(C8)
dγ0n(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
2minus 81πmicro
8+ 94micro2 minus 2535πmicro3
32
minus3 (1 + 90 logmicro)micro4 +84315πmicro5
1024+
(C9)
dδLTp(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
2880π3f2πm
4π
minus5
2minus 27πmicro
32+ 20micro2 minus 5865πmicro3
256
+3
(617
4+ 36 logmicro
)micro4 minus 2056845πmicro5
4096+
(C10)
dδLTn(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
minus5
4minus 81πmicro
64minus 11micro2 +
10005πmicro3
512
+15
8(11 + 48 logmicro)micro4 minus 267015πmicro5
8192+
(C11)
dIAp(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
96π2f2πmicro
2
1minus 15πmicro
4minus 1
2(115 + 88 logmicro)micro2 +
1839πmicro3
32
+5 (5 + 34 logmicro)micro4 +
(C12)
dIAn(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
48π2f2πmicro
2
1
2minus 11πmicro
8minus 1
4(1 + 20 logmicro)micro2 +
99πmicro3
64
minus25micro4
12+
(C13)
dI1p(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
96π2f2πmicro
3π
8+ 2 (4 + 3 logmicro)microminus 537πmicro2
64
minus1
2(15 + 56 logmicro)micro3 +
(C14)
39
dI1n(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
48π2f2πmicro
minus π
16+
1
4(3 + 4 logmicro)microminus 57πmicro2
128+
2micro3
3+
(C15)
γ0p(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2g2
A
16π3f2πm
6π
1
105minus 23πmicro
256+
377micro2
210minus 15551πmicro3
6144+
3371micro4
105
minus1640457πmicro5
32768+
(C16)
γ0n(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2g2
A
16π3f2πm
6π
1
105minus 153πmicro
1792+
69micro2
70minus 4615πmicro3
6144+
172micro4
35
minus120897πmicro5
32768+
(C17)
2 ∆-exchange contribution
Here we give analytical expressions for the tree-level ∆-exchange contributions to the nucleon
spin polarizabilities and their slopes at Q2 = 0 Note that the ∆-exchange contributes equally to
proton and neutron polarizabilities Recall that for the magnetic γlowastN∆ coupling we introduced a
dipole form factor to mimic vector-meson dominance gM rarr gM(1 +Q2Λ2)2
bull Static polarizabilities (Q2 = 0)
γ0 = minus e2
4πM2+
(g2M
∆2+
g2E
M2+
minus 4gMgEM+∆
) (C18)
δLT =e2M∆
4πM3+
(g2E
MNM++gMgE∆MN
minus gEgCM2
∆
) (C19)
γ0 =e2M2
N
π∆2M4+
(minusg
2M
∆2+
g2E
M2+
+4gMgE∆M+
) (C20)
bull Slopes of polarizabilities at Q2 = 0
dγ0(Q2)
dQ2
∣∣∣∣∣Q2=0
= minus e2
πM2+∆
(g2M
∆
[1
4∆2minus 1
∆M++
1
2M2+
]minus 1
Λ2
g2M
∆+
g2E
2M2+
[1
2∆minus 3
M+
]minusgMgE
M+
[1
∆2minus 5
∆M++
1
M2+
]+
1
Λ2
2gMgEM+
+2gMgC∆M2
+
minus gEgCM3
+
) (C21)
40
d δLT (Q2)
dQ2
∣∣∣∣∣Q2=0
=e2M∆∆
4πMNM2+
(g2E
∆2M2+
[1
∆minus 4
M+
]minus g2
C
∆M2∆M
2+
+gMgE∆2M+
[1
∆2
minus 3
∆M++
1
M2+
]minus 2
Λ2
gMgE∆2M+
+gMgC∆M2
∆
[1
2∆2minus 2
∆M++
1
2M2+
]minus gEgC
2M2∆M
2+
[7
∆+
1
M+
]) (C22)
dIA(Q2)
dQ2
∣∣∣∣∣Q2=0
= minusM2N
M2+
(g2M
2∆2+
g2E
MNM+minus 2gMgE
∆M+minus gEgCM∆M+
) (C23)
dI1(Q2)
dQ2
∣∣∣∣∣Q2=0
= minusM∆M2N
2M3+
(g2E
MNM∆minus gMgE∆MN
minus gEgCM2
∆
) (C24)
γ0(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2M2
N
π∆3M6+
(g2M
∆
[3M2
∆ + 2M∆MN minus 9M2N
∆2+
4M2+
Λ2
]
minus4gMgE
[5M2
∆ minus 9M2N
M+∆2+
2M+
Λ2
]+
g2E
∆M+[7M∆ minus 9MN ]
minus8gMgC∆
+4gEgCM+
) (C25)
[1] D Drechsel B Pasquini and M Vanderhaeghen Dispersion relations in real and virtual
Compton scattering Phys Rept 378 99 (2003) hep-ph0212124
[2] S E Kuhn J-P Chen and E Leader Spin structure of the nucleon mdash status and recent
results Prog Part Nucl Phys 63 1 (2009) arXiv08123535 [hep-ph]
[3] F Hagelstein R Miskimen and V Pascalutsa Nucleon polarizabilities from Compton scat-
tering to hydrogen atom Prog Part Nucl Phys 88 29 (2016) arXiv151203765 [nucl-th]
[4] B Pasquini and M Vanderhaeghen Dispersion theory in electromagnetic interactions Ann
Rev Nucl Part Sci 68 75 (2018) arXiv180510482 [hep-ph]
[5] S Gerasimov A Sum rule for magnetic moments and the damping of the nucleon magnetic
moment in nuclei Sov J Nucl Phys 2 430 (1966)
[6] S Drell and A C Hearn Exact sum rule for nucleon magnetic moments Phys Rev Lett
16 908 (1966)
[7] J S Schwinger Source theory viewpoints in deep inelastic scattering Proc Natl Acad Sci
USA 72 1 (1975)
[8] J S Schwinger Source theory viewpoints in deep inelastic scattering Electromagnetic In-
teractions and Field Theory Proceedings 14 Internationale Universitatswochen Schladming
Austria February 24-March 7 1975 Acta Phys Austriaca Suppl 14 471 (1975)
41
[9] J Schwinger Source theory discussion of deep inelastic scattering with polarized particles
Proc Natl Acad Sci USA 72 1559 (1975)
[10] A M Baldin Polarizability of nucleons Nucl Phys 18 310 (1960)
[11] M Gell-Mann M L Goldberger and W E Thirring Use of causality conditions in quantum
theory Phys Rev 95 1612 (1954)
[12] A Pineda Leading chiral logarithms to the hyperfine splitting of the hydrogen and muonic
hydrogen Phys Rev C 67 025201 (2003)
[13] C Peset and A Pineda The two-photon exchange contribution to muonic hydrogen from
chiral perturbation theory Nucl Phys B 887 69 (2014) arXiv14064524 [hep-ph]
[14] C Peset and A Pineda Model-independent determination of the two-photon exchange contri-
bution to hyperfine splitting in muonic hydrogen JHEP 04 060 arXiv161205206 [nucl-th]
[15] F Hagelstein and V Pascalutsa Proton structure in the hyperfine splitting of muonic hydro-
gen PoS CD15 077 (2016) arXiv151104301 [nucl-th]
[16] F Hagelstein Exciting Nucleons in Compton Scattering and Hydrogen-Like Atoms PhD
thesis Mainz U Inst Kernphys (2017) arXiv171000874 [nucl-th]
[17] F Hagelstein ∆(1232)-Resonance in the hydrogen spectrum Proceedings 11th International
Workshop on the Physics of Excited Nucleons (NSTAR 2017) Columbia SC USA August
20-23 2017 Few Body Syst 59 93 (2018) arXiv180109790 [nucl-th]
[18] Y Prok et al (CLAS) Moments of the spin structure functions gp1 and gd1 for 005 lt Q2 lt 30
GeV2 Phys Lett B 672 12 (2009) arXiv08022232 [nucl-ex]
[19] H Dutz et al (GDH) First measurement of the Gerasimov-Drell-Hearn sum rule for 1H from
07 GeV to 18 GeV at ELSA Phys Rev Lett 91 192001 (2003)
[20] M Amarian et al (Jefferson Lab E94010) Measurement of the generalized forward spin po-
larizabilities of the neutron Phys Rev Lett 93 152301 (2004) arXivnucl-ex0406005
[21] M Amarian et al The Q2 evolution of the generalized Gerasimov-Drell-Hearn integral for
the neutron using a 3He target Phys Rev Lett 89 242301 (2002) arXivnucl-ex0205020
[22] M Amarian et al (Jefferson Lab E94-010)Q2 evolution of the neutron spin structure moments
using a 3He target Phys Rev Lett 92 022301 (2004) arXivhep-ex0310003
[23] A Deur et al Experimental determination of the evolution of the Bjorken integral at low Q2
Phys Rev Lett 93 212001 (2004) arXivhep-ex0407007
[24] K Slifer Low Q2 measurement of gp2 and the δLT spin polarizability Spin structure at long dis-
tance Proceedings Workshop Newport News USA March 12ndash13 2009 nucl-ex09064775
(2009) AIP Conf Proc 1155 10106313203293 (2009) arXiv09064775 [nucl-ex]
[25] P Solvignon et al (E01-012) Moments of the neutron g2 structure function at intermediate
42
Q2 Phys Rev C 92 015208 (2015) arXiv13044497 [nucl-ex]
[26] A Deur Experimental studies at low Q2 of the spin structure of the nucleon at Jefferson Lab
in 9th International Workshop on Chiral Dynamics (CD18) Durham NC USA September
17-21 2018 (2019) arXiv190305661 [nucl-ex]
[27] V Sulkosky et al (Jefferson Lab E97-110) Measurement of the 3He spin-structure functions
and of neutron (3He) spin-dependent sum rules at 0035 le Q2 le 024 GeV2 Phys Lett B
805 135428 (2020) arXiv190805709 [nucl-ex]
[28] K Adhikari et al (CLAS) Measurement of the Q2 Dependence of the Deuteron Spin Structure
Function g1 and its Moments at Low Q2 with CLAS Phys Rev Lett 120 062501 (2018)
arXiv171101974 [nucl-ex]
[29] V Lensky F Hagelstein A Hiller Blin and V Pascalutsa Comment on rdquoMeasurement of
the Q2 Dependence of the Deuteron Spin Structure Function g1 and its Moments at Low Q2
with CLASrdquo (2018) arXiv180603219 [nucl-th]
[30] J M Alarcn F Hagelstein V Lensky and V Pascalutsa Forward doubly-virtual Compton
scattering off the nucleon in chiral perturbation theory at NLO the subtraction function and
moments of unpolarized structure functions (2020) arXiv200509518 [hep-ph]
[31] V Bernard E Epelbaum H Krebs and U-G Meiszligner New insights into the spin structure
of the nucleon Phys Rev D 87 054032 (2013) arXiv12092523 [hep-ph]
[32] V Lensky J M Alarcon and V Pascalutsa Moments of nucleon structure functions at
next-to-leading order in baryon chiral perturbation theory Phys Rev C 90 055202 (2014)
arXiv14072574 [hep-ph]
[33] V Pascalutsa and M Vanderhaeghen Chiral effective-field theory in the ∆(1232) region
I Pion electroproduction on the nucleon Phys Rev D 73 034003 (2006) arXivhep-
ph0512244
[34] V Pascalutsa and M Vanderhaeghen Electromagnetic nucleon-to-Delta transition in chiral
effective field theory Phys Rev Lett 95 232001 (2005) arXivhep-ph0508060
[35] T R Hemmert B R Holstein and J Kambor Systematic 1M expansion for spin 32
particles in baryon chiral perturbation theory Phys Lett B 395 89 (1997) arXivhep-
ph9606456
[36] V Pascalutsa and D R Phillips Effective theory of the ∆(1232) in Compton scattering off
the nucleon Phys Rev C 67 055202 (2003) arXivnucl-th0212024
[37] V Pascalutsa M Vanderhaeghen and S N Yang Electromagnetic excitation of the ∆(1232)-
resonance Phys Rept 437 125 (2007) arXivhep-ph0609004
[38] K A Olive et al (Particle Data Group) Review of Particle Physics Chin Phys C 38 090001
43
(2014)
[39] F E Low Scattering of light of very low frequency by systems of spin 12 Phys Rev 96
1428 (1954)
[40] M Gell-Mann and M L Goldberger Scattering of low-energy photons by particles of spin
12 Phys Rev 96 1433 (1954)
[41] A M Harun ar-Rashid A simple derivation of Schwingerrsquos sum rule for spin dependent
structure functions Nuovo Cim A 33 447 (1976)
[42] F Hagelstein and V Pascalutsa Dissecting the hadronic contributions to (g minus 2)micro by
Schwingerrsquos sum rule Phys Rev Lett 120 072002 (2018) arXiv171004571 [hep-ph]
[43] V Lensky V Pascalutsa M Vanderhaeghen and C Kao Spin-dependent sum rules con-
necting real and virtual Compton scattering verified Phys Rev D 95 074001 (2017)
arXiv170101947 [hep-ph]
[44] H Burkhardt and W N Cottingham Sum rules for forward virtual Compton scattering
Annals Phys 56 453 (1970)
[45] V Lensky and V Pascalutsa Predictive powers of chiral perturbation theory in Compton
scattering off protons Eur Phys J C 65 195 (2010) arXiv09070451 [hep-ph]
[46] H Grieszlighammer J McGovern D Phillips and G Feldman Using effective field theory to
analyse low-energy Compton scattering data from protons and light nuclei Prog Part Nucl
Phys 67 841 (2012) arXiv12036834 [nucl-th]
[47] H W Grieszlighammer J A McGovern and D R Phillips Nucleon polarisabilities at and
beyond physical pion masses Eur Phys J A 52 139 (2016) arXiv151101952 [nucl-th]
[48] E Epelbaum H Krebs and U-G Meiszligner Improved chiral nucleon-nucleon potential up to
next-to-next-to-next-to-leading order Eur Phys J A 51 53 (2015) arXiv14120142 [nucl-
th]
[49] E Epelbaum H Krebs and U-G Meiszligner Precision nucleon-nucleon potential at fifth order
in the chiral expansion Phys Rev Lett 115 122301 (2015) arXiv14124623 [nucl-th]
[50] C W Kao T Spitzenberg and M Vanderhaeghen Burkhardt-Cottingham sum rule and
forward spin polarizabilities in heavy baryon chiral perturbation theory Phys Rev D 67
016001 (2003) arXivhep-ph0209241
[51] C-W Kao D Drechsel S Kamalov and M Vanderhaeghen Higher moments of nucleon
spin structure functions in heavy baryon chiral perturbation theory and in a resonance model
Phys Rev D 69 056004 (2004) arXivhep-ph0312102
[52] T Becher and H Leutwyler Baryon chiral perturbation theory in manifestly Lorentz invariant
form Eur Phys J C 9 643 (1999) arXivhep-ph9901384
44
[53] J Gasser M E Sainio and A Svarc Nucleons with chiral loops Nucl Phys B 307 779
(1988)
[54] L S Geng J Martin Camalich L Alvarez-Ruso and M J Vicente Vacas Leading SU(3)-
breaking corrections to the baryon magnetic moments in chiral perturbation theory Phys
Rev Lett 101 222002 (2008) arXiv08051419 [hep-ph]
[55] D Drechsel S S Kamalov and L Tiator The GDH sum rule and related integrals Phys
Rev D 63 114010 (2001) arXivhep-ph0008306
[56] D Drechsel O Hanstein S S Kamalov and L Tiator A unitary isobar model for pion
photo- and electroproduction on the proton up to 1 GeV Nucl Phys A 645 145 (1999)
arXivnucl-th9807001
[57] L Tiator private communication (2020)
[58] V Bernard T R Hemmert and U-G Meiszligner Spin structure of the nucleon at low energies
Phys Rev D 67 076008 (2003) arXivhep-ph0212033
[59] O Gryniuk F Hagelstein and V Pascalutsa Evaluation of the forward Compton scattering
off protons II Spin-dependent amplitude and observables Phys Rev D 94 034043 (2016)
arXiv160400789 [nucl-th]
[60] R Zielinski The g2p Experiment A Measurement of the Protonrsquos Spin Structure Functions
PhD thesis New Hampshire U (2010) arXiv170808297 [nucl-ex]
[61] N Guler et al (CLAS) Precise determination of the deuteron spin structure at low to mod-
erate Q2 with CLAS and extraction of the neutron contribution Phys Rev C 92 055201
(2015) arXiv150507877 [nucl-ex]
[62] C E Carlson V Nazaryan and K Griffioen Proton structure corrections to electronic
and muonic hydrogen hyperfine splitting Phys Rev A 78 022517 (2008) arXiv08052603
[physicsatom-ph]
[63] V Bernard N Kaiser and U-G Meiszligner Chiral dynamics in nucleons and nuclei Int J
Mod Phys E 4 193 (1995) arXivhep-ph9501384
[64] T R Hemmert B R Holstein and J Kambor ∆(1232) and the polarizabilities of the nucleon
Phys Rev D 55 5598 (1997) arXivhep-ph9612374
[65] V Pascalutsa and R Timmermans Field theory of nucleon to higher-spin baryon transitions
Phys Rev C 60 042201 (1999) arXivnucl-th9905065
[66] V Pascalutsa Quantization of an interacting spin-32 field and the Delta isobar Phys Rev
D 58 096002 (1998) arXivhep-ph9802288
[67] H Krebs Double Virtual Compton Scattering and SpinStructure of the Nucleon PoS
CD2018 031 (2019)
45
[68] N Kochelev and Y Oh Axial anomaly and the δLT puzzle Phys Rev D 016012 (2012)
arXiv11034892 [hep-ph]
[69] D Drechsel S Kamalov and L Tiator Unitary isobar model ndash MAID2007 Eur Phys J A
69 (2007) available at httpsmaidkphuni-mainzde arXiv07100306 [nucl-th]
[70] P J Mohr B N Taylor and D B Newell CODATA recommended values of the fundamental
physical constants 2010 Rev Mod Phys 84 1527 (2012)
[71] J Ahrens et al (GDH A2) First measurement of the Gerasimov-Drell-Hearn integral for 1H
from 200 to 800 MeV Phys Rev Lett 87 022003 (2001) arXivhep-ex0105089 [hep-ex]
[72] K Helbing (GDH) Experimental verification of the GDH sum rule at ELSA and MAMI Nucl
Phys Proc Suppl 105 113 (2002)
[73] R Bradford A Bodek H S Budd and J Arrington A New parameterization of the nucleon
elastic form-factors NuInt05 proceedings of the 4th International Workshop on Neutrino-
Nucleus Interactions in the Few-GeV Region Okayama Japan 26-29 September 2005 Nucl
Phys Proc Suppl 159 127 (2006) arXivhep-ex0602017
[74] J D Bjorken Applications of the chiral U(6)timesU(6) algebra of current densities Phys Rev
148 1467 (1966)
[75] J D Bjorken Inelastic scattering of polarized leptons from polarized nucleons Phys Rev D
1 1376 (1970)
[76] V Bernard T R Hemmert and U-G Meiszligner Novel analysis of chiral loop effects in
the generalized Gerasimov-Drell-Hearn sum rule Phys Lett B 545 105 (2002) arXivhep-
ph0203167
[77] A Deur et al Experimental study of isovector spin sum rules Phys Rev D 78 032001
(2008) arXiv08023198 [nucl-ex]
[78] B Pasquini P Pedroni and D Drechsel Higher order forward spin polarizability Phys Lett
B 687 160 (2010) arXiv10014230 [hep-ph]
[79] R Jaffe g2ndashThe nucleonrsquos other spin-dependent structure function Comments Nucl Part
Phys 19 239 (1990)
[80] E V Shuryak and A Vainshtein Theory of power corrections to deep inelastic scattering in
quantum chromodynamics (II) Qminus4 effects polarized target Nucl Phys B 201 141 (1982)
[81] S Wandzura and F Wilczek Sum rules for spin dependent electroproduction Test of rela-
tivistic constituent quarks Phys Lett B 72 195 (1977)
[82] B W Filippone and X-D Ji The spin structure of the nucleon Adv Nucl Phys 26 1
(2001) arXivhep-ph0101224
[83] M Burkardt The g2 structure function Proceedings Workshop on Spin structure at long
46
distance Newport News USA March 12-13 2009 AIP Conf Proc 1155 26 (2009)
arXiv09054079 [hep-ph]
[84] A Deur S J Brodsky and G F de Teramond The spin structure of the nucleon Rept
Prog Phys 82 076201 (2019) arXiv180705250 [hep-ph]
[85] B R Holstein D Drechsel B Pasquini and M Vanderhaeghen Higher order polarizabilities
of the proton Phys Rev C 61 034316 (2000) arXivhep-ph9910427
[86] B R Holstein V Pascalutsa and M Vanderhaeghen Sum rules for magnetic moments and
polarizabilities in QED and chiral effective-field theory Phys Rev D 72 094014 (2005)
arXivhep-ph0507016
[87] B Pasquini D Drechsel and L Tiator Invariant amplitudes for pion electroproduction Eur
Phys J A 34 387 (2007) arXiv07122327 [hep-ph]
[88] A J Buchmann Non-spherical proton shape and hydrogen hyperfine splitting Proceedings
International Workshop on Precision Physics of Simple Atomic Systems (PSAS 2008) Wind-
sor Ontario Canada July 21-26 2008 Can J Phys 87 773 (2009) arXiv09104747
[physicsatom-ph]
47
δLTn =e2g2
A
96π3f2πm
2π
1
2+
3πmicro
16+ (1 + 2 log micro)micro2 minus 105πmicro3
128+
5micro4
4+
(C5)
γ0p =e2g2
A
16π3f2πm
4π
4
45minus 3πmicro
16+
14micro2
5minus 1813πmicro3
384minus 192
5(1 + log micro)micro4
+80703πmicro5
2048+
(C6)
γ0n =e2g2
A
16π3f2πm
4π
4
45minus 5πmicro
48+
4micro2
5minus 245πmicro3
384minus 32micro4 logmicro
15
+1323πmicro5
2048+
(C7)
bull Slopes of polarizabilities at Q2 = 0
dγ0p(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
2minus 45πmicro
4+ 223micro2 minus 28515πmicro3
64
minus9
(1953
4+ 449 logmicro
)micro4 +
570255πmicro5
128+
(C8)
dγ0n(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
2minus 81πmicro
8+ 94micro2 minus 2535πmicro3
32
minus3 (1 + 90 logmicro)micro4 +84315πmicro5
1024+
(C9)
dδLTp(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
2880π3f2πm
4π
minus5
2minus 27πmicro
32+ 20micro2 minus 5865πmicro3
256
+3
(617
4+ 36 logmicro
)micro4 minus 2056845πmicro5
4096+
(C10)
dδLTn(Q2)
dQ2
∣∣∣∣Q2=0
=e2g2
A
1440π3f2πm
4π
minus5
4minus 81πmicro
64minus 11micro2 +
10005πmicro3
512
+15
8(11 + 48 logmicro)micro4 minus 267015πmicro5
8192+
(C11)
dIAp(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
96π2f2πmicro
2
1minus 15πmicro
4minus 1
2(115 + 88 logmicro)micro2 +
1839πmicro3
32
+5 (5 + 34 logmicro)micro4 +
(C12)
dIAn(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
48π2f2πmicro
2
1
2minus 11πmicro
8minus 1
4(1 + 20 logmicro)micro2 +
99πmicro3
64
minus25micro4
12+
(C13)
dI1p(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
96π2f2πmicro
3π
8+ 2 (4 + 3 logmicro)microminus 537πmicro2
64
minus1
2(15 + 56 logmicro)micro3 +
(C14)
39
dI1n(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
48π2f2πmicro
minus π
16+
1
4(3 + 4 logmicro)microminus 57πmicro2
128+
2micro3
3+
(C15)
γ0p(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2g2
A
16π3f2πm
6π
1
105minus 23πmicro
256+
377micro2
210minus 15551πmicro3
6144+
3371micro4
105
minus1640457πmicro5
32768+
(C16)
γ0n(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2g2
A
16π3f2πm
6π
1
105minus 153πmicro
1792+
69micro2
70minus 4615πmicro3
6144+
172micro4
35
minus120897πmicro5
32768+
(C17)
2 ∆-exchange contribution
Here we give analytical expressions for the tree-level ∆-exchange contributions to the nucleon
spin polarizabilities and their slopes at Q2 = 0 Note that the ∆-exchange contributes equally to
proton and neutron polarizabilities Recall that for the magnetic γlowastN∆ coupling we introduced a
dipole form factor to mimic vector-meson dominance gM rarr gM(1 +Q2Λ2)2
bull Static polarizabilities (Q2 = 0)
γ0 = minus e2
4πM2+
(g2M
∆2+
g2E
M2+
minus 4gMgEM+∆
) (C18)
δLT =e2M∆
4πM3+
(g2E
MNM++gMgE∆MN
minus gEgCM2
∆
) (C19)
γ0 =e2M2
N
π∆2M4+
(minusg
2M
∆2+
g2E
M2+
+4gMgE∆M+
) (C20)
bull Slopes of polarizabilities at Q2 = 0
dγ0(Q2)
dQ2
∣∣∣∣∣Q2=0
= minus e2
πM2+∆
(g2M
∆
[1
4∆2minus 1
∆M++
1
2M2+
]minus 1
Λ2
g2M
∆+
g2E
2M2+
[1
2∆minus 3
M+
]minusgMgE
M+
[1
∆2minus 5
∆M++
1
M2+
]+
1
Λ2
2gMgEM+
+2gMgC∆M2
+
minus gEgCM3
+
) (C21)
40
d δLT (Q2)
dQ2
∣∣∣∣∣Q2=0
=e2M∆∆
4πMNM2+
(g2E
∆2M2+
[1
∆minus 4
M+
]minus g2
C
∆M2∆M
2+
+gMgE∆2M+
[1
∆2
minus 3
∆M++
1
M2+
]minus 2
Λ2
gMgE∆2M+
+gMgC∆M2
∆
[1
2∆2minus 2
∆M++
1
2M2+
]minus gEgC
2M2∆M
2+
[7
∆+
1
M+
]) (C22)
dIA(Q2)
dQ2
∣∣∣∣∣Q2=0
= minusM2N
M2+
(g2M
2∆2+
g2E
MNM+minus 2gMgE
∆M+minus gEgCM∆M+
) (C23)
dI1(Q2)
dQ2
∣∣∣∣∣Q2=0
= minusM∆M2N
2M3+
(g2E
MNM∆minus gMgE∆MN
minus gEgCM2
∆
) (C24)
γ0(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2M2
N
π∆3M6+
(g2M
∆
[3M2
∆ + 2M∆MN minus 9M2N
∆2+
4M2+
Λ2
]
minus4gMgE
[5M2
∆ minus 9M2N
M+∆2+
2M+
Λ2
]+
g2E
∆M+[7M∆ minus 9MN ]
minus8gMgC∆
+4gEgCM+
) (C25)
[1] D Drechsel B Pasquini and M Vanderhaeghen Dispersion relations in real and virtual
Compton scattering Phys Rept 378 99 (2003) hep-ph0212124
[2] S E Kuhn J-P Chen and E Leader Spin structure of the nucleon mdash status and recent
results Prog Part Nucl Phys 63 1 (2009) arXiv08123535 [hep-ph]
[3] F Hagelstein R Miskimen and V Pascalutsa Nucleon polarizabilities from Compton scat-
tering to hydrogen atom Prog Part Nucl Phys 88 29 (2016) arXiv151203765 [nucl-th]
[4] B Pasquini and M Vanderhaeghen Dispersion theory in electromagnetic interactions Ann
Rev Nucl Part Sci 68 75 (2018) arXiv180510482 [hep-ph]
[5] S Gerasimov A Sum rule for magnetic moments and the damping of the nucleon magnetic
moment in nuclei Sov J Nucl Phys 2 430 (1966)
[6] S Drell and A C Hearn Exact sum rule for nucleon magnetic moments Phys Rev Lett
16 908 (1966)
[7] J S Schwinger Source theory viewpoints in deep inelastic scattering Proc Natl Acad Sci
USA 72 1 (1975)
[8] J S Schwinger Source theory viewpoints in deep inelastic scattering Electromagnetic In-
teractions and Field Theory Proceedings 14 Internationale Universitatswochen Schladming
Austria February 24-March 7 1975 Acta Phys Austriaca Suppl 14 471 (1975)
41
[9] J Schwinger Source theory discussion of deep inelastic scattering with polarized particles
Proc Natl Acad Sci USA 72 1559 (1975)
[10] A M Baldin Polarizability of nucleons Nucl Phys 18 310 (1960)
[11] M Gell-Mann M L Goldberger and W E Thirring Use of causality conditions in quantum
theory Phys Rev 95 1612 (1954)
[12] A Pineda Leading chiral logarithms to the hyperfine splitting of the hydrogen and muonic
hydrogen Phys Rev C 67 025201 (2003)
[13] C Peset and A Pineda The two-photon exchange contribution to muonic hydrogen from
chiral perturbation theory Nucl Phys B 887 69 (2014) arXiv14064524 [hep-ph]
[14] C Peset and A Pineda Model-independent determination of the two-photon exchange contri-
bution to hyperfine splitting in muonic hydrogen JHEP 04 060 arXiv161205206 [nucl-th]
[15] F Hagelstein and V Pascalutsa Proton structure in the hyperfine splitting of muonic hydro-
gen PoS CD15 077 (2016) arXiv151104301 [nucl-th]
[16] F Hagelstein Exciting Nucleons in Compton Scattering and Hydrogen-Like Atoms PhD
thesis Mainz U Inst Kernphys (2017) arXiv171000874 [nucl-th]
[17] F Hagelstein ∆(1232)-Resonance in the hydrogen spectrum Proceedings 11th International
Workshop on the Physics of Excited Nucleons (NSTAR 2017) Columbia SC USA August
20-23 2017 Few Body Syst 59 93 (2018) arXiv180109790 [nucl-th]
[18] Y Prok et al (CLAS) Moments of the spin structure functions gp1 and gd1 for 005 lt Q2 lt 30
GeV2 Phys Lett B 672 12 (2009) arXiv08022232 [nucl-ex]
[19] H Dutz et al (GDH) First measurement of the Gerasimov-Drell-Hearn sum rule for 1H from
07 GeV to 18 GeV at ELSA Phys Rev Lett 91 192001 (2003)
[20] M Amarian et al (Jefferson Lab E94010) Measurement of the generalized forward spin po-
larizabilities of the neutron Phys Rev Lett 93 152301 (2004) arXivnucl-ex0406005
[21] M Amarian et al The Q2 evolution of the generalized Gerasimov-Drell-Hearn integral for
the neutron using a 3He target Phys Rev Lett 89 242301 (2002) arXivnucl-ex0205020
[22] M Amarian et al (Jefferson Lab E94-010)Q2 evolution of the neutron spin structure moments
using a 3He target Phys Rev Lett 92 022301 (2004) arXivhep-ex0310003
[23] A Deur et al Experimental determination of the evolution of the Bjorken integral at low Q2
Phys Rev Lett 93 212001 (2004) arXivhep-ex0407007
[24] K Slifer Low Q2 measurement of gp2 and the δLT spin polarizability Spin structure at long dis-
tance Proceedings Workshop Newport News USA March 12ndash13 2009 nucl-ex09064775
(2009) AIP Conf Proc 1155 10106313203293 (2009) arXiv09064775 [nucl-ex]
[25] P Solvignon et al (E01-012) Moments of the neutron g2 structure function at intermediate
42
Q2 Phys Rev C 92 015208 (2015) arXiv13044497 [nucl-ex]
[26] A Deur Experimental studies at low Q2 of the spin structure of the nucleon at Jefferson Lab
in 9th International Workshop on Chiral Dynamics (CD18) Durham NC USA September
17-21 2018 (2019) arXiv190305661 [nucl-ex]
[27] V Sulkosky et al (Jefferson Lab E97-110) Measurement of the 3He spin-structure functions
and of neutron (3He) spin-dependent sum rules at 0035 le Q2 le 024 GeV2 Phys Lett B
805 135428 (2020) arXiv190805709 [nucl-ex]
[28] K Adhikari et al (CLAS) Measurement of the Q2 Dependence of the Deuteron Spin Structure
Function g1 and its Moments at Low Q2 with CLAS Phys Rev Lett 120 062501 (2018)
arXiv171101974 [nucl-ex]
[29] V Lensky F Hagelstein A Hiller Blin and V Pascalutsa Comment on rdquoMeasurement of
the Q2 Dependence of the Deuteron Spin Structure Function g1 and its Moments at Low Q2
with CLASrdquo (2018) arXiv180603219 [nucl-th]
[30] J M Alarcn F Hagelstein V Lensky and V Pascalutsa Forward doubly-virtual Compton
scattering off the nucleon in chiral perturbation theory at NLO the subtraction function and
moments of unpolarized structure functions (2020) arXiv200509518 [hep-ph]
[31] V Bernard E Epelbaum H Krebs and U-G Meiszligner New insights into the spin structure
of the nucleon Phys Rev D 87 054032 (2013) arXiv12092523 [hep-ph]
[32] V Lensky J M Alarcon and V Pascalutsa Moments of nucleon structure functions at
next-to-leading order in baryon chiral perturbation theory Phys Rev C 90 055202 (2014)
arXiv14072574 [hep-ph]
[33] V Pascalutsa and M Vanderhaeghen Chiral effective-field theory in the ∆(1232) region
I Pion electroproduction on the nucleon Phys Rev D 73 034003 (2006) arXivhep-
ph0512244
[34] V Pascalutsa and M Vanderhaeghen Electromagnetic nucleon-to-Delta transition in chiral
effective field theory Phys Rev Lett 95 232001 (2005) arXivhep-ph0508060
[35] T R Hemmert B R Holstein and J Kambor Systematic 1M expansion for spin 32
particles in baryon chiral perturbation theory Phys Lett B 395 89 (1997) arXivhep-
ph9606456
[36] V Pascalutsa and D R Phillips Effective theory of the ∆(1232) in Compton scattering off
the nucleon Phys Rev C 67 055202 (2003) arXivnucl-th0212024
[37] V Pascalutsa M Vanderhaeghen and S N Yang Electromagnetic excitation of the ∆(1232)-
resonance Phys Rept 437 125 (2007) arXivhep-ph0609004
[38] K A Olive et al (Particle Data Group) Review of Particle Physics Chin Phys C 38 090001
43
(2014)
[39] F E Low Scattering of light of very low frequency by systems of spin 12 Phys Rev 96
1428 (1954)
[40] M Gell-Mann and M L Goldberger Scattering of low-energy photons by particles of spin
12 Phys Rev 96 1433 (1954)
[41] A M Harun ar-Rashid A simple derivation of Schwingerrsquos sum rule for spin dependent
structure functions Nuovo Cim A 33 447 (1976)
[42] F Hagelstein and V Pascalutsa Dissecting the hadronic contributions to (g minus 2)micro by
Schwingerrsquos sum rule Phys Rev Lett 120 072002 (2018) arXiv171004571 [hep-ph]
[43] V Lensky V Pascalutsa M Vanderhaeghen and C Kao Spin-dependent sum rules con-
necting real and virtual Compton scattering verified Phys Rev D 95 074001 (2017)
arXiv170101947 [hep-ph]
[44] H Burkhardt and W N Cottingham Sum rules for forward virtual Compton scattering
Annals Phys 56 453 (1970)
[45] V Lensky and V Pascalutsa Predictive powers of chiral perturbation theory in Compton
scattering off protons Eur Phys J C 65 195 (2010) arXiv09070451 [hep-ph]
[46] H Grieszlighammer J McGovern D Phillips and G Feldman Using effective field theory to
analyse low-energy Compton scattering data from protons and light nuclei Prog Part Nucl
Phys 67 841 (2012) arXiv12036834 [nucl-th]
[47] H W Grieszlighammer J A McGovern and D R Phillips Nucleon polarisabilities at and
beyond physical pion masses Eur Phys J A 52 139 (2016) arXiv151101952 [nucl-th]
[48] E Epelbaum H Krebs and U-G Meiszligner Improved chiral nucleon-nucleon potential up to
next-to-next-to-next-to-leading order Eur Phys J A 51 53 (2015) arXiv14120142 [nucl-
th]
[49] E Epelbaum H Krebs and U-G Meiszligner Precision nucleon-nucleon potential at fifth order
in the chiral expansion Phys Rev Lett 115 122301 (2015) arXiv14124623 [nucl-th]
[50] C W Kao T Spitzenberg and M Vanderhaeghen Burkhardt-Cottingham sum rule and
forward spin polarizabilities in heavy baryon chiral perturbation theory Phys Rev D 67
016001 (2003) arXivhep-ph0209241
[51] C-W Kao D Drechsel S Kamalov and M Vanderhaeghen Higher moments of nucleon
spin structure functions in heavy baryon chiral perturbation theory and in a resonance model
Phys Rev D 69 056004 (2004) arXivhep-ph0312102
[52] T Becher and H Leutwyler Baryon chiral perturbation theory in manifestly Lorentz invariant
form Eur Phys J C 9 643 (1999) arXivhep-ph9901384
44
[53] J Gasser M E Sainio and A Svarc Nucleons with chiral loops Nucl Phys B 307 779
(1988)
[54] L S Geng J Martin Camalich L Alvarez-Ruso and M J Vicente Vacas Leading SU(3)-
breaking corrections to the baryon magnetic moments in chiral perturbation theory Phys
Rev Lett 101 222002 (2008) arXiv08051419 [hep-ph]
[55] D Drechsel S S Kamalov and L Tiator The GDH sum rule and related integrals Phys
Rev D 63 114010 (2001) arXivhep-ph0008306
[56] D Drechsel O Hanstein S S Kamalov and L Tiator A unitary isobar model for pion
photo- and electroproduction on the proton up to 1 GeV Nucl Phys A 645 145 (1999)
arXivnucl-th9807001
[57] L Tiator private communication (2020)
[58] V Bernard T R Hemmert and U-G Meiszligner Spin structure of the nucleon at low energies
Phys Rev D 67 076008 (2003) arXivhep-ph0212033
[59] O Gryniuk F Hagelstein and V Pascalutsa Evaluation of the forward Compton scattering
off protons II Spin-dependent amplitude and observables Phys Rev D 94 034043 (2016)
arXiv160400789 [nucl-th]
[60] R Zielinski The g2p Experiment A Measurement of the Protonrsquos Spin Structure Functions
PhD thesis New Hampshire U (2010) arXiv170808297 [nucl-ex]
[61] N Guler et al (CLAS) Precise determination of the deuteron spin structure at low to mod-
erate Q2 with CLAS and extraction of the neutron contribution Phys Rev C 92 055201
(2015) arXiv150507877 [nucl-ex]
[62] C E Carlson V Nazaryan and K Griffioen Proton structure corrections to electronic
and muonic hydrogen hyperfine splitting Phys Rev A 78 022517 (2008) arXiv08052603
[physicsatom-ph]
[63] V Bernard N Kaiser and U-G Meiszligner Chiral dynamics in nucleons and nuclei Int J
Mod Phys E 4 193 (1995) arXivhep-ph9501384
[64] T R Hemmert B R Holstein and J Kambor ∆(1232) and the polarizabilities of the nucleon
Phys Rev D 55 5598 (1997) arXivhep-ph9612374
[65] V Pascalutsa and R Timmermans Field theory of nucleon to higher-spin baryon transitions
Phys Rev C 60 042201 (1999) arXivnucl-th9905065
[66] V Pascalutsa Quantization of an interacting spin-32 field and the Delta isobar Phys Rev
D 58 096002 (1998) arXivhep-ph9802288
[67] H Krebs Double Virtual Compton Scattering and SpinStructure of the Nucleon PoS
CD2018 031 (2019)
45
[68] N Kochelev and Y Oh Axial anomaly and the δLT puzzle Phys Rev D 016012 (2012)
arXiv11034892 [hep-ph]
[69] D Drechsel S Kamalov and L Tiator Unitary isobar model ndash MAID2007 Eur Phys J A
69 (2007) available at httpsmaidkphuni-mainzde arXiv07100306 [nucl-th]
[70] P J Mohr B N Taylor and D B Newell CODATA recommended values of the fundamental
physical constants 2010 Rev Mod Phys 84 1527 (2012)
[71] J Ahrens et al (GDH A2) First measurement of the Gerasimov-Drell-Hearn integral for 1H
from 200 to 800 MeV Phys Rev Lett 87 022003 (2001) arXivhep-ex0105089 [hep-ex]
[72] K Helbing (GDH) Experimental verification of the GDH sum rule at ELSA and MAMI Nucl
Phys Proc Suppl 105 113 (2002)
[73] R Bradford A Bodek H S Budd and J Arrington A New parameterization of the nucleon
elastic form-factors NuInt05 proceedings of the 4th International Workshop on Neutrino-
Nucleus Interactions in the Few-GeV Region Okayama Japan 26-29 September 2005 Nucl
Phys Proc Suppl 159 127 (2006) arXivhep-ex0602017
[74] J D Bjorken Applications of the chiral U(6)timesU(6) algebra of current densities Phys Rev
148 1467 (1966)
[75] J D Bjorken Inelastic scattering of polarized leptons from polarized nucleons Phys Rev D
1 1376 (1970)
[76] V Bernard T R Hemmert and U-G Meiszligner Novel analysis of chiral loop effects in
the generalized Gerasimov-Drell-Hearn sum rule Phys Lett B 545 105 (2002) arXivhep-
ph0203167
[77] A Deur et al Experimental study of isovector spin sum rules Phys Rev D 78 032001
(2008) arXiv08023198 [nucl-ex]
[78] B Pasquini P Pedroni and D Drechsel Higher order forward spin polarizability Phys Lett
B 687 160 (2010) arXiv10014230 [hep-ph]
[79] R Jaffe g2ndashThe nucleonrsquos other spin-dependent structure function Comments Nucl Part
Phys 19 239 (1990)
[80] E V Shuryak and A Vainshtein Theory of power corrections to deep inelastic scattering in
quantum chromodynamics (II) Qminus4 effects polarized target Nucl Phys B 201 141 (1982)
[81] S Wandzura and F Wilczek Sum rules for spin dependent electroproduction Test of rela-
tivistic constituent quarks Phys Lett B 72 195 (1977)
[82] B W Filippone and X-D Ji The spin structure of the nucleon Adv Nucl Phys 26 1
(2001) arXivhep-ph0101224
[83] M Burkardt The g2 structure function Proceedings Workshop on Spin structure at long
46
distance Newport News USA March 12-13 2009 AIP Conf Proc 1155 26 (2009)
arXiv09054079 [hep-ph]
[84] A Deur S J Brodsky and G F de Teramond The spin structure of the nucleon Rept
Prog Phys 82 076201 (2019) arXiv180705250 [hep-ph]
[85] B R Holstein D Drechsel B Pasquini and M Vanderhaeghen Higher order polarizabilities
of the proton Phys Rev C 61 034316 (2000) arXivhep-ph9910427
[86] B R Holstein V Pascalutsa and M Vanderhaeghen Sum rules for magnetic moments and
polarizabilities in QED and chiral effective-field theory Phys Rev D 72 094014 (2005)
arXivhep-ph0507016
[87] B Pasquini D Drechsel and L Tiator Invariant amplitudes for pion electroproduction Eur
Phys J A 34 387 (2007) arXiv07122327 [hep-ph]
[88] A J Buchmann Non-spherical proton shape and hydrogen hyperfine splitting Proceedings
International Workshop on Precision Physics of Simple Atomic Systems (PSAS 2008) Wind-
sor Ontario Canada July 21-26 2008 Can J Phys 87 773 (2009) arXiv09104747
[physicsatom-ph]
47
dI1n(Q2)
dQ2
∣∣∣∣Q2=0
=g2A
48π2f2πmicro
minus π
16+
1
4(3 + 4 logmicro)microminus 57πmicro2
128+
2micro3
3+
(C15)
γ0p(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2g2
A
16π3f2πm
6π
1
105minus 23πmicro
256+
377micro2
210minus 15551πmicro3
6144+
3371micro4
105
minus1640457πmicro5
32768+
(C16)
γ0n(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2g2
A
16π3f2πm
6π
1
105minus 153πmicro
1792+
69micro2
70minus 4615πmicro3
6144+
172micro4
35
minus120897πmicro5
32768+
(C17)
2 ∆-exchange contribution
Here we give analytical expressions for the tree-level ∆-exchange contributions to the nucleon
spin polarizabilities and their slopes at Q2 = 0 Note that the ∆-exchange contributes equally to
proton and neutron polarizabilities Recall that for the magnetic γlowastN∆ coupling we introduced a
dipole form factor to mimic vector-meson dominance gM rarr gM(1 +Q2Λ2)2
bull Static polarizabilities (Q2 = 0)
γ0 = minus e2
4πM2+
(g2M
∆2+
g2E
M2+
minus 4gMgEM+∆
) (C18)
δLT =e2M∆
4πM3+
(g2E
MNM++gMgE∆MN
minus gEgCM2
∆
) (C19)
γ0 =e2M2
N
π∆2M4+
(minusg
2M
∆2+
g2E
M2+
+4gMgE∆M+
) (C20)
bull Slopes of polarizabilities at Q2 = 0
dγ0(Q2)
dQ2
∣∣∣∣∣Q2=0
= minus e2
πM2+∆
(g2M
∆
[1
4∆2minus 1
∆M++
1
2M2+
]minus 1
Λ2
g2M
∆+
g2E
2M2+
[1
2∆minus 3
M+
]minusgMgE
M+
[1
∆2minus 5
∆M++
1
M2+
]+
1
Λ2
2gMgEM+
+2gMgC∆M2
+
minus gEgCM3
+
) (C21)
40
d δLT (Q2)
dQ2
∣∣∣∣∣Q2=0
=e2M∆∆
4πMNM2+
(g2E
∆2M2+
[1
∆minus 4
M+
]minus g2
C
∆M2∆M
2+
+gMgE∆2M+
[1
∆2
minus 3
∆M++
1
M2+
]minus 2
Λ2
gMgE∆2M+
+gMgC∆M2
∆
[1
2∆2minus 2
∆M++
1
2M2+
]minus gEgC
2M2∆M
2+
[7
∆+
1
M+
]) (C22)
dIA(Q2)
dQ2
∣∣∣∣∣Q2=0
= minusM2N
M2+
(g2M
2∆2+
g2E
MNM+minus 2gMgE
∆M+minus gEgCM∆M+
) (C23)
dI1(Q2)
dQ2
∣∣∣∣∣Q2=0
= minusM∆M2N
2M3+
(g2E
MNM∆minus gMgE∆MN
minus gEgCM2
∆
) (C24)
γ0(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2M2
N
π∆3M6+
(g2M
∆
[3M2
∆ + 2M∆MN minus 9M2N
∆2+
4M2+
Λ2
]
minus4gMgE
[5M2
∆ minus 9M2N
M+∆2+
2M+
Λ2
]+
g2E
∆M+[7M∆ minus 9MN ]
minus8gMgC∆
+4gEgCM+
) (C25)
[1] D Drechsel B Pasquini and M Vanderhaeghen Dispersion relations in real and virtual
Compton scattering Phys Rept 378 99 (2003) hep-ph0212124
[2] S E Kuhn J-P Chen and E Leader Spin structure of the nucleon mdash status and recent
results Prog Part Nucl Phys 63 1 (2009) arXiv08123535 [hep-ph]
[3] F Hagelstein R Miskimen and V Pascalutsa Nucleon polarizabilities from Compton scat-
tering to hydrogen atom Prog Part Nucl Phys 88 29 (2016) arXiv151203765 [nucl-th]
[4] B Pasquini and M Vanderhaeghen Dispersion theory in electromagnetic interactions Ann
Rev Nucl Part Sci 68 75 (2018) arXiv180510482 [hep-ph]
[5] S Gerasimov A Sum rule for magnetic moments and the damping of the nucleon magnetic
moment in nuclei Sov J Nucl Phys 2 430 (1966)
[6] S Drell and A C Hearn Exact sum rule for nucleon magnetic moments Phys Rev Lett
16 908 (1966)
[7] J S Schwinger Source theory viewpoints in deep inelastic scattering Proc Natl Acad Sci
USA 72 1 (1975)
[8] J S Schwinger Source theory viewpoints in deep inelastic scattering Electromagnetic In-
teractions and Field Theory Proceedings 14 Internationale Universitatswochen Schladming
Austria February 24-March 7 1975 Acta Phys Austriaca Suppl 14 471 (1975)
41
[9] J Schwinger Source theory discussion of deep inelastic scattering with polarized particles
Proc Natl Acad Sci USA 72 1559 (1975)
[10] A M Baldin Polarizability of nucleons Nucl Phys 18 310 (1960)
[11] M Gell-Mann M L Goldberger and W E Thirring Use of causality conditions in quantum
theory Phys Rev 95 1612 (1954)
[12] A Pineda Leading chiral logarithms to the hyperfine splitting of the hydrogen and muonic
hydrogen Phys Rev C 67 025201 (2003)
[13] C Peset and A Pineda The two-photon exchange contribution to muonic hydrogen from
chiral perturbation theory Nucl Phys B 887 69 (2014) arXiv14064524 [hep-ph]
[14] C Peset and A Pineda Model-independent determination of the two-photon exchange contri-
bution to hyperfine splitting in muonic hydrogen JHEP 04 060 arXiv161205206 [nucl-th]
[15] F Hagelstein and V Pascalutsa Proton structure in the hyperfine splitting of muonic hydro-
gen PoS CD15 077 (2016) arXiv151104301 [nucl-th]
[16] F Hagelstein Exciting Nucleons in Compton Scattering and Hydrogen-Like Atoms PhD
thesis Mainz U Inst Kernphys (2017) arXiv171000874 [nucl-th]
[17] F Hagelstein ∆(1232)-Resonance in the hydrogen spectrum Proceedings 11th International
Workshop on the Physics of Excited Nucleons (NSTAR 2017) Columbia SC USA August
20-23 2017 Few Body Syst 59 93 (2018) arXiv180109790 [nucl-th]
[18] Y Prok et al (CLAS) Moments of the spin structure functions gp1 and gd1 for 005 lt Q2 lt 30
GeV2 Phys Lett B 672 12 (2009) arXiv08022232 [nucl-ex]
[19] H Dutz et al (GDH) First measurement of the Gerasimov-Drell-Hearn sum rule for 1H from
07 GeV to 18 GeV at ELSA Phys Rev Lett 91 192001 (2003)
[20] M Amarian et al (Jefferson Lab E94010) Measurement of the generalized forward spin po-
larizabilities of the neutron Phys Rev Lett 93 152301 (2004) arXivnucl-ex0406005
[21] M Amarian et al The Q2 evolution of the generalized Gerasimov-Drell-Hearn integral for
the neutron using a 3He target Phys Rev Lett 89 242301 (2002) arXivnucl-ex0205020
[22] M Amarian et al (Jefferson Lab E94-010)Q2 evolution of the neutron spin structure moments
using a 3He target Phys Rev Lett 92 022301 (2004) arXivhep-ex0310003
[23] A Deur et al Experimental determination of the evolution of the Bjorken integral at low Q2
Phys Rev Lett 93 212001 (2004) arXivhep-ex0407007
[24] K Slifer Low Q2 measurement of gp2 and the δLT spin polarizability Spin structure at long dis-
tance Proceedings Workshop Newport News USA March 12ndash13 2009 nucl-ex09064775
(2009) AIP Conf Proc 1155 10106313203293 (2009) arXiv09064775 [nucl-ex]
[25] P Solvignon et al (E01-012) Moments of the neutron g2 structure function at intermediate
42
Q2 Phys Rev C 92 015208 (2015) arXiv13044497 [nucl-ex]
[26] A Deur Experimental studies at low Q2 of the spin structure of the nucleon at Jefferson Lab
in 9th International Workshop on Chiral Dynamics (CD18) Durham NC USA September
17-21 2018 (2019) arXiv190305661 [nucl-ex]
[27] V Sulkosky et al (Jefferson Lab E97-110) Measurement of the 3He spin-structure functions
and of neutron (3He) spin-dependent sum rules at 0035 le Q2 le 024 GeV2 Phys Lett B
805 135428 (2020) arXiv190805709 [nucl-ex]
[28] K Adhikari et al (CLAS) Measurement of the Q2 Dependence of the Deuteron Spin Structure
Function g1 and its Moments at Low Q2 with CLAS Phys Rev Lett 120 062501 (2018)
arXiv171101974 [nucl-ex]
[29] V Lensky F Hagelstein A Hiller Blin and V Pascalutsa Comment on rdquoMeasurement of
the Q2 Dependence of the Deuteron Spin Structure Function g1 and its Moments at Low Q2
with CLASrdquo (2018) arXiv180603219 [nucl-th]
[30] J M Alarcn F Hagelstein V Lensky and V Pascalutsa Forward doubly-virtual Compton
scattering off the nucleon in chiral perturbation theory at NLO the subtraction function and
moments of unpolarized structure functions (2020) arXiv200509518 [hep-ph]
[31] V Bernard E Epelbaum H Krebs and U-G Meiszligner New insights into the spin structure
of the nucleon Phys Rev D 87 054032 (2013) arXiv12092523 [hep-ph]
[32] V Lensky J M Alarcon and V Pascalutsa Moments of nucleon structure functions at
next-to-leading order in baryon chiral perturbation theory Phys Rev C 90 055202 (2014)
arXiv14072574 [hep-ph]
[33] V Pascalutsa and M Vanderhaeghen Chiral effective-field theory in the ∆(1232) region
I Pion electroproduction on the nucleon Phys Rev D 73 034003 (2006) arXivhep-
ph0512244
[34] V Pascalutsa and M Vanderhaeghen Electromagnetic nucleon-to-Delta transition in chiral
effective field theory Phys Rev Lett 95 232001 (2005) arXivhep-ph0508060
[35] T R Hemmert B R Holstein and J Kambor Systematic 1M expansion for spin 32
particles in baryon chiral perturbation theory Phys Lett B 395 89 (1997) arXivhep-
ph9606456
[36] V Pascalutsa and D R Phillips Effective theory of the ∆(1232) in Compton scattering off
the nucleon Phys Rev C 67 055202 (2003) arXivnucl-th0212024
[37] V Pascalutsa M Vanderhaeghen and S N Yang Electromagnetic excitation of the ∆(1232)-
resonance Phys Rept 437 125 (2007) arXivhep-ph0609004
[38] K A Olive et al (Particle Data Group) Review of Particle Physics Chin Phys C 38 090001
43
(2014)
[39] F E Low Scattering of light of very low frequency by systems of spin 12 Phys Rev 96
1428 (1954)
[40] M Gell-Mann and M L Goldberger Scattering of low-energy photons by particles of spin
12 Phys Rev 96 1433 (1954)
[41] A M Harun ar-Rashid A simple derivation of Schwingerrsquos sum rule for spin dependent
structure functions Nuovo Cim A 33 447 (1976)
[42] F Hagelstein and V Pascalutsa Dissecting the hadronic contributions to (g minus 2)micro by
Schwingerrsquos sum rule Phys Rev Lett 120 072002 (2018) arXiv171004571 [hep-ph]
[43] V Lensky V Pascalutsa M Vanderhaeghen and C Kao Spin-dependent sum rules con-
necting real and virtual Compton scattering verified Phys Rev D 95 074001 (2017)
arXiv170101947 [hep-ph]
[44] H Burkhardt and W N Cottingham Sum rules for forward virtual Compton scattering
Annals Phys 56 453 (1970)
[45] V Lensky and V Pascalutsa Predictive powers of chiral perturbation theory in Compton
scattering off protons Eur Phys J C 65 195 (2010) arXiv09070451 [hep-ph]
[46] H Grieszlighammer J McGovern D Phillips and G Feldman Using effective field theory to
analyse low-energy Compton scattering data from protons and light nuclei Prog Part Nucl
Phys 67 841 (2012) arXiv12036834 [nucl-th]
[47] H W Grieszlighammer J A McGovern and D R Phillips Nucleon polarisabilities at and
beyond physical pion masses Eur Phys J A 52 139 (2016) arXiv151101952 [nucl-th]
[48] E Epelbaum H Krebs and U-G Meiszligner Improved chiral nucleon-nucleon potential up to
next-to-next-to-next-to-leading order Eur Phys J A 51 53 (2015) arXiv14120142 [nucl-
th]
[49] E Epelbaum H Krebs and U-G Meiszligner Precision nucleon-nucleon potential at fifth order
in the chiral expansion Phys Rev Lett 115 122301 (2015) arXiv14124623 [nucl-th]
[50] C W Kao T Spitzenberg and M Vanderhaeghen Burkhardt-Cottingham sum rule and
forward spin polarizabilities in heavy baryon chiral perturbation theory Phys Rev D 67
016001 (2003) arXivhep-ph0209241
[51] C-W Kao D Drechsel S Kamalov and M Vanderhaeghen Higher moments of nucleon
spin structure functions in heavy baryon chiral perturbation theory and in a resonance model
Phys Rev D 69 056004 (2004) arXivhep-ph0312102
[52] T Becher and H Leutwyler Baryon chiral perturbation theory in manifestly Lorentz invariant
form Eur Phys J C 9 643 (1999) arXivhep-ph9901384
44
[53] J Gasser M E Sainio and A Svarc Nucleons with chiral loops Nucl Phys B 307 779
(1988)
[54] L S Geng J Martin Camalich L Alvarez-Ruso and M J Vicente Vacas Leading SU(3)-
breaking corrections to the baryon magnetic moments in chiral perturbation theory Phys
Rev Lett 101 222002 (2008) arXiv08051419 [hep-ph]
[55] D Drechsel S S Kamalov and L Tiator The GDH sum rule and related integrals Phys
Rev D 63 114010 (2001) arXivhep-ph0008306
[56] D Drechsel O Hanstein S S Kamalov and L Tiator A unitary isobar model for pion
photo- and electroproduction on the proton up to 1 GeV Nucl Phys A 645 145 (1999)
arXivnucl-th9807001
[57] L Tiator private communication (2020)
[58] V Bernard T R Hemmert and U-G Meiszligner Spin structure of the nucleon at low energies
Phys Rev D 67 076008 (2003) arXivhep-ph0212033
[59] O Gryniuk F Hagelstein and V Pascalutsa Evaluation of the forward Compton scattering
off protons II Spin-dependent amplitude and observables Phys Rev D 94 034043 (2016)
arXiv160400789 [nucl-th]
[60] R Zielinski The g2p Experiment A Measurement of the Protonrsquos Spin Structure Functions
PhD thesis New Hampshire U (2010) arXiv170808297 [nucl-ex]
[61] N Guler et al (CLAS) Precise determination of the deuteron spin structure at low to mod-
erate Q2 with CLAS and extraction of the neutron contribution Phys Rev C 92 055201
(2015) arXiv150507877 [nucl-ex]
[62] C E Carlson V Nazaryan and K Griffioen Proton structure corrections to electronic
and muonic hydrogen hyperfine splitting Phys Rev A 78 022517 (2008) arXiv08052603
[physicsatom-ph]
[63] V Bernard N Kaiser and U-G Meiszligner Chiral dynamics in nucleons and nuclei Int J
Mod Phys E 4 193 (1995) arXivhep-ph9501384
[64] T R Hemmert B R Holstein and J Kambor ∆(1232) and the polarizabilities of the nucleon
Phys Rev D 55 5598 (1997) arXivhep-ph9612374
[65] V Pascalutsa and R Timmermans Field theory of nucleon to higher-spin baryon transitions
Phys Rev C 60 042201 (1999) arXivnucl-th9905065
[66] V Pascalutsa Quantization of an interacting spin-32 field and the Delta isobar Phys Rev
D 58 096002 (1998) arXivhep-ph9802288
[67] H Krebs Double Virtual Compton Scattering and SpinStructure of the Nucleon PoS
CD2018 031 (2019)
45
[68] N Kochelev and Y Oh Axial anomaly and the δLT puzzle Phys Rev D 016012 (2012)
arXiv11034892 [hep-ph]
[69] D Drechsel S Kamalov and L Tiator Unitary isobar model ndash MAID2007 Eur Phys J A
69 (2007) available at httpsmaidkphuni-mainzde arXiv07100306 [nucl-th]
[70] P J Mohr B N Taylor and D B Newell CODATA recommended values of the fundamental
physical constants 2010 Rev Mod Phys 84 1527 (2012)
[71] J Ahrens et al (GDH A2) First measurement of the Gerasimov-Drell-Hearn integral for 1H
from 200 to 800 MeV Phys Rev Lett 87 022003 (2001) arXivhep-ex0105089 [hep-ex]
[72] K Helbing (GDH) Experimental verification of the GDH sum rule at ELSA and MAMI Nucl
Phys Proc Suppl 105 113 (2002)
[73] R Bradford A Bodek H S Budd and J Arrington A New parameterization of the nucleon
elastic form-factors NuInt05 proceedings of the 4th International Workshop on Neutrino-
Nucleus Interactions in the Few-GeV Region Okayama Japan 26-29 September 2005 Nucl
Phys Proc Suppl 159 127 (2006) arXivhep-ex0602017
[74] J D Bjorken Applications of the chiral U(6)timesU(6) algebra of current densities Phys Rev
148 1467 (1966)
[75] J D Bjorken Inelastic scattering of polarized leptons from polarized nucleons Phys Rev D
1 1376 (1970)
[76] V Bernard T R Hemmert and U-G Meiszligner Novel analysis of chiral loop effects in
the generalized Gerasimov-Drell-Hearn sum rule Phys Lett B 545 105 (2002) arXivhep-
ph0203167
[77] A Deur et al Experimental study of isovector spin sum rules Phys Rev D 78 032001
(2008) arXiv08023198 [nucl-ex]
[78] B Pasquini P Pedroni and D Drechsel Higher order forward spin polarizability Phys Lett
B 687 160 (2010) arXiv10014230 [hep-ph]
[79] R Jaffe g2ndashThe nucleonrsquos other spin-dependent structure function Comments Nucl Part
Phys 19 239 (1990)
[80] E V Shuryak and A Vainshtein Theory of power corrections to deep inelastic scattering in
quantum chromodynamics (II) Qminus4 effects polarized target Nucl Phys B 201 141 (1982)
[81] S Wandzura and F Wilczek Sum rules for spin dependent electroproduction Test of rela-
tivistic constituent quarks Phys Lett B 72 195 (1977)
[82] B W Filippone and X-D Ji The spin structure of the nucleon Adv Nucl Phys 26 1
(2001) arXivhep-ph0101224
[83] M Burkardt The g2 structure function Proceedings Workshop on Spin structure at long
46
distance Newport News USA March 12-13 2009 AIP Conf Proc 1155 26 (2009)
arXiv09054079 [hep-ph]
[84] A Deur S J Brodsky and G F de Teramond The spin structure of the nucleon Rept
Prog Phys 82 076201 (2019) arXiv180705250 [hep-ph]
[85] B R Holstein D Drechsel B Pasquini and M Vanderhaeghen Higher order polarizabilities
of the proton Phys Rev C 61 034316 (2000) arXivhep-ph9910427
[86] B R Holstein V Pascalutsa and M Vanderhaeghen Sum rules for magnetic moments and
polarizabilities in QED and chiral effective-field theory Phys Rev D 72 094014 (2005)
arXivhep-ph0507016
[87] B Pasquini D Drechsel and L Tiator Invariant amplitudes for pion electroproduction Eur
Phys J A 34 387 (2007) arXiv07122327 [hep-ph]
[88] A J Buchmann Non-spherical proton shape and hydrogen hyperfine splitting Proceedings
International Workshop on Precision Physics of Simple Atomic Systems (PSAS 2008) Wind-
sor Ontario Canada July 21-26 2008 Can J Phys 87 773 (2009) arXiv09104747
[physicsatom-ph]
47
d δLT (Q2)
dQ2
∣∣∣∣∣Q2=0
=e2M∆∆
4πMNM2+
(g2E
∆2M2+
[1
∆minus 4
M+
]minus g2
C
∆M2∆M
2+
+gMgE∆2M+
[1
∆2
minus 3
∆M++
1
M2+
]minus 2
Λ2
gMgE∆2M+
+gMgC∆M2
∆
[1
2∆2minus 2
∆M++
1
2M2+
]minus gEgC
2M2∆M
2+
[7
∆+
1
M+
]) (C22)
dIA(Q2)
dQ2
∣∣∣∣∣Q2=0
= minusM2N
M2+
(g2M
2∆2+
g2E
MNM+minus 2gMgE
∆M+minus gEgCM∆M+
) (C23)
dI1(Q2)
dQ2
∣∣∣∣∣Q2=0
= minusM∆M2N
2M3+
(g2E
MNM∆minus gMgE∆MN
minus gEgCM2
∆
) (C24)
γ0(Q2)
dQ2
∣∣∣∣∣Q2=0
=e2M2
N
π∆3M6+
(g2M
∆
[3M2
∆ + 2M∆MN minus 9M2N
∆2+
4M2+
Λ2
]
minus4gMgE
[5M2
∆ minus 9M2N
M+∆2+
2M+
Λ2
]+
g2E
∆M+[7M∆ minus 9MN ]
minus8gMgC∆
+4gEgCM+
) (C25)
[1] D Drechsel B Pasquini and M Vanderhaeghen Dispersion relations in real and virtual
Compton scattering Phys Rept 378 99 (2003) hep-ph0212124
[2] S E Kuhn J-P Chen and E Leader Spin structure of the nucleon mdash status and recent
results Prog Part Nucl Phys 63 1 (2009) arXiv08123535 [hep-ph]
[3] F Hagelstein R Miskimen and V Pascalutsa Nucleon polarizabilities from Compton scat-
tering to hydrogen atom Prog Part Nucl Phys 88 29 (2016) arXiv151203765 [nucl-th]
[4] B Pasquini and M Vanderhaeghen Dispersion theory in electromagnetic interactions Ann
Rev Nucl Part Sci 68 75 (2018) arXiv180510482 [hep-ph]
[5] S Gerasimov A Sum rule for magnetic moments and the damping of the nucleon magnetic
moment in nuclei Sov J Nucl Phys 2 430 (1966)
[6] S Drell and A C Hearn Exact sum rule for nucleon magnetic moments Phys Rev Lett
16 908 (1966)
[7] J S Schwinger Source theory viewpoints in deep inelastic scattering Proc Natl Acad Sci
USA 72 1 (1975)
[8] J S Schwinger Source theory viewpoints in deep inelastic scattering Electromagnetic In-
teractions and Field Theory Proceedings 14 Internationale Universitatswochen Schladming
Austria February 24-March 7 1975 Acta Phys Austriaca Suppl 14 471 (1975)
41
[9] J Schwinger Source theory discussion of deep inelastic scattering with polarized particles
Proc Natl Acad Sci USA 72 1559 (1975)
[10] A M Baldin Polarizability of nucleons Nucl Phys 18 310 (1960)
[11] M Gell-Mann M L Goldberger and W E Thirring Use of causality conditions in quantum
theory Phys Rev 95 1612 (1954)
[12] A Pineda Leading chiral logarithms to the hyperfine splitting of the hydrogen and muonic
hydrogen Phys Rev C 67 025201 (2003)
[13] C Peset and A Pineda The two-photon exchange contribution to muonic hydrogen from
chiral perturbation theory Nucl Phys B 887 69 (2014) arXiv14064524 [hep-ph]
[14] C Peset and A Pineda Model-independent determination of the two-photon exchange contri-
bution to hyperfine splitting in muonic hydrogen JHEP 04 060 arXiv161205206 [nucl-th]
[15] F Hagelstein and V Pascalutsa Proton structure in the hyperfine splitting of muonic hydro-
gen PoS CD15 077 (2016) arXiv151104301 [nucl-th]
[16] F Hagelstein Exciting Nucleons in Compton Scattering and Hydrogen-Like Atoms PhD
thesis Mainz U Inst Kernphys (2017) arXiv171000874 [nucl-th]
[17] F Hagelstein ∆(1232)-Resonance in the hydrogen spectrum Proceedings 11th International
Workshop on the Physics of Excited Nucleons (NSTAR 2017) Columbia SC USA August
20-23 2017 Few Body Syst 59 93 (2018) arXiv180109790 [nucl-th]
[18] Y Prok et al (CLAS) Moments of the spin structure functions gp1 and gd1 for 005 lt Q2 lt 30
GeV2 Phys Lett B 672 12 (2009) arXiv08022232 [nucl-ex]
[19] H Dutz et al (GDH) First measurement of the Gerasimov-Drell-Hearn sum rule for 1H from
07 GeV to 18 GeV at ELSA Phys Rev Lett 91 192001 (2003)
[20] M Amarian et al (Jefferson Lab E94010) Measurement of the generalized forward spin po-
larizabilities of the neutron Phys Rev Lett 93 152301 (2004) arXivnucl-ex0406005
[21] M Amarian et al The Q2 evolution of the generalized Gerasimov-Drell-Hearn integral for
the neutron using a 3He target Phys Rev Lett 89 242301 (2002) arXivnucl-ex0205020
[22] M Amarian et al (Jefferson Lab E94-010)Q2 evolution of the neutron spin structure moments
using a 3He target Phys Rev Lett 92 022301 (2004) arXivhep-ex0310003
[23] A Deur et al Experimental determination of the evolution of the Bjorken integral at low Q2
Phys Rev Lett 93 212001 (2004) arXivhep-ex0407007
[24] K Slifer Low Q2 measurement of gp2 and the δLT spin polarizability Spin structure at long dis-
tance Proceedings Workshop Newport News USA March 12ndash13 2009 nucl-ex09064775
(2009) AIP Conf Proc 1155 10106313203293 (2009) arXiv09064775 [nucl-ex]
[25] P Solvignon et al (E01-012) Moments of the neutron g2 structure function at intermediate
42
Q2 Phys Rev C 92 015208 (2015) arXiv13044497 [nucl-ex]
[26] A Deur Experimental studies at low Q2 of the spin structure of the nucleon at Jefferson Lab
in 9th International Workshop on Chiral Dynamics (CD18) Durham NC USA September
17-21 2018 (2019) arXiv190305661 [nucl-ex]
[27] V Sulkosky et al (Jefferson Lab E97-110) Measurement of the 3He spin-structure functions
and of neutron (3He) spin-dependent sum rules at 0035 le Q2 le 024 GeV2 Phys Lett B
805 135428 (2020) arXiv190805709 [nucl-ex]
[28] K Adhikari et al (CLAS) Measurement of the Q2 Dependence of the Deuteron Spin Structure
Function g1 and its Moments at Low Q2 with CLAS Phys Rev Lett 120 062501 (2018)
arXiv171101974 [nucl-ex]
[29] V Lensky F Hagelstein A Hiller Blin and V Pascalutsa Comment on rdquoMeasurement of
the Q2 Dependence of the Deuteron Spin Structure Function g1 and its Moments at Low Q2
with CLASrdquo (2018) arXiv180603219 [nucl-th]
[30] J M Alarcn F Hagelstein V Lensky and V Pascalutsa Forward doubly-virtual Compton
scattering off the nucleon in chiral perturbation theory at NLO the subtraction function and
moments of unpolarized structure functions (2020) arXiv200509518 [hep-ph]
[31] V Bernard E Epelbaum H Krebs and U-G Meiszligner New insights into the spin structure
of the nucleon Phys Rev D 87 054032 (2013) arXiv12092523 [hep-ph]
[32] V Lensky J M Alarcon and V Pascalutsa Moments of nucleon structure functions at
next-to-leading order in baryon chiral perturbation theory Phys Rev C 90 055202 (2014)
arXiv14072574 [hep-ph]
[33] V Pascalutsa and M Vanderhaeghen Chiral effective-field theory in the ∆(1232) region
I Pion electroproduction on the nucleon Phys Rev D 73 034003 (2006) arXivhep-
ph0512244
[34] V Pascalutsa and M Vanderhaeghen Electromagnetic nucleon-to-Delta transition in chiral
effective field theory Phys Rev Lett 95 232001 (2005) arXivhep-ph0508060
[35] T R Hemmert B R Holstein and J Kambor Systematic 1M expansion for spin 32
particles in baryon chiral perturbation theory Phys Lett B 395 89 (1997) arXivhep-
ph9606456
[36] V Pascalutsa and D R Phillips Effective theory of the ∆(1232) in Compton scattering off
the nucleon Phys Rev C 67 055202 (2003) arXivnucl-th0212024
[37] V Pascalutsa M Vanderhaeghen and S N Yang Electromagnetic excitation of the ∆(1232)-
resonance Phys Rept 437 125 (2007) arXivhep-ph0609004
[38] K A Olive et al (Particle Data Group) Review of Particle Physics Chin Phys C 38 090001
43
(2014)
[39] F E Low Scattering of light of very low frequency by systems of spin 12 Phys Rev 96
1428 (1954)
[40] M Gell-Mann and M L Goldberger Scattering of low-energy photons by particles of spin
12 Phys Rev 96 1433 (1954)
[41] A M Harun ar-Rashid A simple derivation of Schwingerrsquos sum rule for spin dependent
structure functions Nuovo Cim A 33 447 (1976)
[42] F Hagelstein and V Pascalutsa Dissecting the hadronic contributions to (g minus 2)micro by
Schwingerrsquos sum rule Phys Rev Lett 120 072002 (2018) arXiv171004571 [hep-ph]
[43] V Lensky V Pascalutsa M Vanderhaeghen and C Kao Spin-dependent sum rules con-
necting real and virtual Compton scattering verified Phys Rev D 95 074001 (2017)
arXiv170101947 [hep-ph]
[44] H Burkhardt and W N Cottingham Sum rules for forward virtual Compton scattering
Annals Phys 56 453 (1970)
[45] V Lensky and V Pascalutsa Predictive powers of chiral perturbation theory in Compton
scattering off protons Eur Phys J C 65 195 (2010) arXiv09070451 [hep-ph]
[46] H Grieszlighammer J McGovern D Phillips and G Feldman Using effective field theory to
analyse low-energy Compton scattering data from protons and light nuclei Prog Part Nucl
Phys 67 841 (2012) arXiv12036834 [nucl-th]
[47] H W Grieszlighammer J A McGovern and D R Phillips Nucleon polarisabilities at and
beyond physical pion masses Eur Phys J A 52 139 (2016) arXiv151101952 [nucl-th]
[48] E Epelbaum H Krebs and U-G Meiszligner Improved chiral nucleon-nucleon potential up to
next-to-next-to-next-to-leading order Eur Phys J A 51 53 (2015) arXiv14120142 [nucl-
th]
[49] E Epelbaum H Krebs and U-G Meiszligner Precision nucleon-nucleon potential at fifth order
in the chiral expansion Phys Rev Lett 115 122301 (2015) arXiv14124623 [nucl-th]
[50] C W Kao T Spitzenberg and M Vanderhaeghen Burkhardt-Cottingham sum rule and
forward spin polarizabilities in heavy baryon chiral perturbation theory Phys Rev D 67
016001 (2003) arXivhep-ph0209241
[51] C-W Kao D Drechsel S Kamalov and M Vanderhaeghen Higher moments of nucleon
spin structure functions in heavy baryon chiral perturbation theory and in a resonance model
Phys Rev D 69 056004 (2004) arXivhep-ph0312102
[52] T Becher and H Leutwyler Baryon chiral perturbation theory in manifestly Lorentz invariant
form Eur Phys J C 9 643 (1999) arXivhep-ph9901384
44
[53] J Gasser M E Sainio and A Svarc Nucleons with chiral loops Nucl Phys B 307 779
(1988)
[54] L S Geng J Martin Camalich L Alvarez-Ruso and M J Vicente Vacas Leading SU(3)-
breaking corrections to the baryon magnetic moments in chiral perturbation theory Phys
Rev Lett 101 222002 (2008) arXiv08051419 [hep-ph]
[55] D Drechsel S S Kamalov and L Tiator The GDH sum rule and related integrals Phys
Rev D 63 114010 (2001) arXivhep-ph0008306
[56] D Drechsel O Hanstein S S Kamalov and L Tiator A unitary isobar model for pion
photo- and electroproduction on the proton up to 1 GeV Nucl Phys A 645 145 (1999)
arXivnucl-th9807001
[57] L Tiator private communication (2020)
[58] V Bernard T R Hemmert and U-G Meiszligner Spin structure of the nucleon at low energies
Phys Rev D 67 076008 (2003) arXivhep-ph0212033
[59] O Gryniuk F Hagelstein and V Pascalutsa Evaluation of the forward Compton scattering
off protons II Spin-dependent amplitude and observables Phys Rev D 94 034043 (2016)
arXiv160400789 [nucl-th]
[60] R Zielinski The g2p Experiment A Measurement of the Protonrsquos Spin Structure Functions
PhD thesis New Hampshire U (2010) arXiv170808297 [nucl-ex]
[61] N Guler et al (CLAS) Precise determination of the deuteron spin structure at low to mod-
erate Q2 with CLAS and extraction of the neutron contribution Phys Rev C 92 055201
(2015) arXiv150507877 [nucl-ex]
[62] C E Carlson V Nazaryan and K Griffioen Proton structure corrections to electronic
and muonic hydrogen hyperfine splitting Phys Rev A 78 022517 (2008) arXiv08052603
[physicsatom-ph]
[63] V Bernard N Kaiser and U-G Meiszligner Chiral dynamics in nucleons and nuclei Int J
Mod Phys E 4 193 (1995) arXivhep-ph9501384
[64] T R Hemmert B R Holstein and J Kambor ∆(1232) and the polarizabilities of the nucleon
Phys Rev D 55 5598 (1997) arXivhep-ph9612374
[65] V Pascalutsa and R Timmermans Field theory of nucleon to higher-spin baryon transitions
Phys Rev C 60 042201 (1999) arXivnucl-th9905065
[66] V Pascalutsa Quantization of an interacting spin-32 field and the Delta isobar Phys Rev
D 58 096002 (1998) arXivhep-ph9802288
[67] H Krebs Double Virtual Compton Scattering and SpinStructure of the Nucleon PoS
CD2018 031 (2019)
45
[68] N Kochelev and Y Oh Axial anomaly and the δLT puzzle Phys Rev D 016012 (2012)
arXiv11034892 [hep-ph]
[69] D Drechsel S Kamalov and L Tiator Unitary isobar model ndash MAID2007 Eur Phys J A
69 (2007) available at httpsmaidkphuni-mainzde arXiv07100306 [nucl-th]
[70] P J Mohr B N Taylor and D B Newell CODATA recommended values of the fundamental
physical constants 2010 Rev Mod Phys 84 1527 (2012)
[71] J Ahrens et al (GDH A2) First measurement of the Gerasimov-Drell-Hearn integral for 1H
from 200 to 800 MeV Phys Rev Lett 87 022003 (2001) arXivhep-ex0105089 [hep-ex]
[72] K Helbing (GDH) Experimental verification of the GDH sum rule at ELSA and MAMI Nucl
Phys Proc Suppl 105 113 (2002)
[73] R Bradford A Bodek H S Budd and J Arrington A New parameterization of the nucleon
elastic form-factors NuInt05 proceedings of the 4th International Workshop on Neutrino-
Nucleus Interactions in the Few-GeV Region Okayama Japan 26-29 September 2005 Nucl
Phys Proc Suppl 159 127 (2006) arXivhep-ex0602017
[74] J D Bjorken Applications of the chiral U(6)timesU(6) algebra of current densities Phys Rev
148 1467 (1966)
[75] J D Bjorken Inelastic scattering of polarized leptons from polarized nucleons Phys Rev D
1 1376 (1970)
[76] V Bernard T R Hemmert and U-G Meiszligner Novel analysis of chiral loop effects in
the generalized Gerasimov-Drell-Hearn sum rule Phys Lett B 545 105 (2002) arXivhep-
ph0203167
[77] A Deur et al Experimental study of isovector spin sum rules Phys Rev D 78 032001
(2008) arXiv08023198 [nucl-ex]
[78] B Pasquini P Pedroni and D Drechsel Higher order forward spin polarizability Phys Lett
B 687 160 (2010) arXiv10014230 [hep-ph]
[79] R Jaffe g2ndashThe nucleonrsquos other spin-dependent structure function Comments Nucl Part
Phys 19 239 (1990)
[80] E V Shuryak and A Vainshtein Theory of power corrections to deep inelastic scattering in
quantum chromodynamics (II) Qminus4 effects polarized target Nucl Phys B 201 141 (1982)
[81] S Wandzura and F Wilczek Sum rules for spin dependent electroproduction Test of rela-
tivistic constituent quarks Phys Lett B 72 195 (1977)
[82] B W Filippone and X-D Ji The spin structure of the nucleon Adv Nucl Phys 26 1
(2001) arXivhep-ph0101224
[83] M Burkardt The g2 structure function Proceedings Workshop on Spin structure at long
46
distance Newport News USA March 12-13 2009 AIP Conf Proc 1155 26 (2009)
arXiv09054079 [hep-ph]
[84] A Deur S J Brodsky and G F de Teramond The spin structure of the nucleon Rept
Prog Phys 82 076201 (2019) arXiv180705250 [hep-ph]
[85] B R Holstein D Drechsel B Pasquini and M Vanderhaeghen Higher order polarizabilities
of the proton Phys Rev C 61 034316 (2000) arXivhep-ph9910427
[86] B R Holstein V Pascalutsa and M Vanderhaeghen Sum rules for magnetic moments and
polarizabilities in QED and chiral effective-field theory Phys Rev D 72 094014 (2005)
arXivhep-ph0507016
[87] B Pasquini D Drechsel and L Tiator Invariant amplitudes for pion electroproduction Eur
Phys J A 34 387 (2007) arXiv07122327 [hep-ph]
[88] A J Buchmann Non-spherical proton shape and hydrogen hyperfine splitting Proceedings
International Workshop on Precision Physics of Simple Atomic Systems (PSAS 2008) Wind-
sor Ontario Canada July 21-26 2008 Can J Phys 87 773 (2009) arXiv09104747
[physicsatom-ph]
47
[9] J Schwinger Source theory discussion of deep inelastic scattering with polarized particles
Proc Natl Acad Sci USA 72 1559 (1975)
[10] A M Baldin Polarizability of nucleons Nucl Phys 18 310 (1960)
[11] M Gell-Mann M L Goldberger and W E Thirring Use of causality conditions in quantum
theory Phys Rev 95 1612 (1954)
[12] A Pineda Leading chiral logarithms to the hyperfine splitting of the hydrogen and muonic
hydrogen Phys Rev C 67 025201 (2003)
[13] C Peset and A Pineda The two-photon exchange contribution to muonic hydrogen from
chiral perturbation theory Nucl Phys B 887 69 (2014) arXiv14064524 [hep-ph]
[14] C Peset and A Pineda Model-independent determination of the two-photon exchange contri-
bution to hyperfine splitting in muonic hydrogen JHEP 04 060 arXiv161205206 [nucl-th]
[15] F Hagelstein and V Pascalutsa Proton structure in the hyperfine splitting of muonic hydro-
gen PoS CD15 077 (2016) arXiv151104301 [nucl-th]
[16] F Hagelstein Exciting Nucleons in Compton Scattering and Hydrogen-Like Atoms PhD
thesis Mainz U Inst Kernphys (2017) arXiv171000874 [nucl-th]
[17] F Hagelstein ∆(1232)-Resonance in the hydrogen spectrum Proceedings 11th International
Workshop on the Physics of Excited Nucleons (NSTAR 2017) Columbia SC USA August
20-23 2017 Few Body Syst 59 93 (2018) arXiv180109790 [nucl-th]
[18] Y Prok et al (CLAS) Moments of the spin structure functions gp1 and gd1 for 005 lt Q2 lt 30
GeV2 Phys Lett B 672 12 (2009) arXiv08022232 [nucl-ex]
[19] H Dutz et al (GDH) First measurement of the Gerasimov-Drell-Hearn sum rule for 1H from
07 GeV to 18 GeV at ELSA Phys Rev Lett 91 192001 (2003)
[20] M Amarian et al (Jefferson Lab E94010) Measurement of the generalized forward spin po-
larizabilities of the neutron Phys Rev Lett 93 152301 (2004) arXivnucl-ex0406005
[21] M Amarian et al The Q2 evolution of the generalized Gerasimov-Drell-Hearn integral for
the neutron using a 3He target Phys Rev Lett 89 242301 (2002) arXivnucl-ex0205020
[22] M Amarian et al (Jefferson Lab E94-010)Q2 evolution of the neutron spin structure moments
using a 3He target Phys Rev Lett 92 022301 (2004) arXivhep-ex0310003
[23] A Deur et al Experimental determination of the evolution of the Bjorken integral at low Q2
Phys Rev Lett 93 212001 (2004) arXivhep-ex0407007
[24] K Slifer Low Q2 measurement of gp2 and the δLT spin polarizability Spin structure at long dis-
tance Proceedings Workshop Newport News USA March 12ndash13 2009 nucl-ex09064775
(2009) AIP Conf Proc 1155 10106313203293 (2009) arXiv09064775 [nucl-ex]
[25] P Solvignon et al (E01-012) Moments of the neutron g2 structure function at intermediate
42
Q2 Phys Rev C 92 015208 (2015) arXiv13044497 [nucl-ex]
[26] A Deur Experimental studies at low Q2 of the spin structure of the nucleon at Jefferson Lab
in 9th International Workshop on Chiral Dynamics (CD18) Durham NC USA September
17-21 2018 (2019) arXiv190305661 [nucl-ex]
[27] V Sulkosky et al (Jefferson Lab E97-110) Measurement of the 3He spin-structure functions
and of neutron (3He) spin-dependent sum rules at 0035 le Q2 le 024 GeV2 Phys Lett B
805 135428 (2020) arXiv190805709 [nucl-ex]
[28] K Adhikari et al (CLAS) Measurement of the Q2 Dependence of the Deuteron Spin Structure
Function g1 and its Moments at Low Q2 with CLAS Phys Rev Lett 120 062501 (2018)
arXiv171101974 [nucl-ex]
[29] V Lensky F Hagelstein A Hiller Blin and V Pascalutsa Comment on rdquoMeasurement of
the Q2 Dependence of the Deuteron Spin Structure Function g1 and its Moments at Low Q2
with CLASrdquo (2018) arXiv180603219 [nucl-th]
[30] J M Alarcn F Hagelstein V Lensky and V Pascalutsa Forward doubly-virtual Compton
scattering off the nucleon in chiral perturbation theory at NLO the subtraction function and
moments of unpolarized structure functions (2020) arXiv200509518 [hep-ph]
[31] V Bernard E Epelbaum H Krebs and U-G Meiszligner New insights into the spin structure
of the nucleon Phys Rev D 87 054032 (2013) arXiv12092523 [hep-ph]
[32] V Lensky J M Alarcon and V Pascalutsa Moments of nucleon structure functions at
next-to-leading order in baryon chiral perturbation theory Phys Rev C 90 055202 (2014)
arXiv14072574 [hep-ph]
[33] V Pascalutsa and M Vanderhaeghen Chiral effective-field theory in the ∆(1232) region
I Pion electroproduction on the nucleon Phys Rev D 73 034003 (2006) arXivhep-
ph0512244
[34] V Pascalutsa and M Vanderhaeghen Electromagnetic nucleon-to-Delta transition in chiral
effective field theory Phys Rev Lett 95 232001 (2005) arXivhep-ph0508060
[35] T R Hemmert B R Holstein and J Kambor Systematic 1M expansion for spin 32
particles in baryon chiral perturbation theory Phys Lett B 395 89 (1997) arXivhep-
ph9606456
[36] V Pascalutsa and D R Phillips Effective theory of the ∆(1232) in Compton scattering off
the nucleon Phys Rev C 67 055202 (2003) arXivnucl-th0212024
[37] V Pascalutsa M Vanderhaeghen and S N Yang Electromagnetic excitation of the ∆(1232)-
resonance Phys Rept 437 125 (2007) arXivhep-ph0609004
[38] K A Olive et al (Particle Data Group) Review of Particle Physics Chin Phys C 38 090001
43
(2014)
[39] F E Low Scattering of light of very low frequency by systems of spin 12 Phys Rev 96
1428 (1954)
[40] M Gell-Mann and M L Goldberger Scattering of low-energy photons by particles of spin
12 Phys Rev 96 1433 (1954)
[41] A M Harun ar-Rashid A simple derivation of Schwingerrsquos sum rule for spin dependent
structure functions Nuovo Cim A 33 447 (1976)
[42] F Hagelstein and V Pascalutsa Dissecting the hadronic contributions to (g minus 2)micro by
Schwingerrsquos sum rule Phys Rev Lett 120 072002 (2018) arXiv171004571 [hep-ph]
[43] V Lensky V Pascalutsa M Vanderhaeghen and C Kao Spin-dependent sum rules con-
necting real and virtual Compton scattering verified Phys Rev D 95 074001 (2017)
arXiv170101947 [hep-ph]
[44] H Burkhardt and W N Cottingham Sum rules for forward virtual Compton scattering
Annals Phys 56 453 (1970)
[45] V Lensky and V Pascalutsa Predictive powers of chiral perturbation theory in Compton
scattering off protons Eur Phys J C 65 195 (2010) arXiv09070451 [hep-ph]
[46] H Grieszlighammer J McGovern D Phillips and G Feldman Using effective field theory to
analyse low-energy Compton scattering data from protons and light nuclei Prog Part Nucl
Phys 67 841 (2012) arXiv12036834 [nucl-th]
[47] H W Grieszlighammer J A McGovern and D R Phillips Nucleon polarisabilities at and
beyond physical pion masses Eur Phys J A 52 139 (2016) arXiv151101952 [nucl-th]
[48] E Epelbaum H Krebs and U-G Meiszligner Improved chiral nucleon-nucleon potential up to
next-to-next-to-next-to-leading order Eur Phys J A 51 53 (2015) arXiv14120142 [nucl-
th]
[49] E Epelbaum H Krebs and U-G Meiszligner Precision nucleon-nucleon potential at fifth order
in the chiral expansion Phys Rev Lett 115 122301 (2015) arXiv14124623 [nucl-th]
[50] C W Kao T Spitzenberg and M Vanderhaeghen Burkhardt-Cottingham sum rule and
forward spin polarizabilities in heavy baryon chiral perturbation theory Phys Rev D 67
016001 (2003) arXivhep-ph0209241
[51] C-W Kao D Drechsel S Kamalov and M Vanderhaeghen Higher moments of nucleon
spin structure functions in heavy baryon chiral perturbation theory and in a resonance model
Phys Rev D 69 056004 (2004) arXivhep-ph0312102
[52] T Becher and H Leutwyler Baryon chiral perturbation theory in manifestly Lorentz invariant
form Eur Phys J C 9 643 (1999) arXivhep-ph9901384
44
[53] J Gasser M E Sainio and A Svarc Nucleons with chiral loops Nucl Phys B 307 779
(1988)
[54] L S Geng J Martin Camalich L Alvarez-Ruso and M J Vicente Vacas Leading SU(3)-
breaking corrections to the baryon magnetic moments in chiral perturbation theory Phys
Rev Lett 101 222002 (2008) arXiv08051419 [hep-ph]
[55] D Drechsel S S Kamalov and L Tiator The GDH sum rule and related integrals Phys
Rev D 63 114010 (2001) arXivhep-ph0008306
[56] D Drechsel O Hanstein S S Kamalov and L Tiator A unitary isobar model for pion
photo- and electroproduction on the proton up to 1 GeV Nucl Phys A 645 145 (1999)
arXivnucl-th9807001
[57] L Tiator private communication (2020)
[58] V Bernard T R Hemmert and U-G Meiszligner Spin structure of the nucleon at low energies
Phys Rev D 67 076008 (2003) arXivhep-ph0212033
[59] O Gryniuk F Hagelstein and V Pascalutsa Evaluation of the forward Compton scattering
off protons II Spin-dependent amplitude and observables Phys Rev D 94 034043 (2016)
arXiv160400789 [nucl-th]
[60] R Zielinski The g2p Experiment A Measurement of the Protonrsquos Spin Structure Functions
PhD thesis New Hampshire U (2010) arXiv170808297 [nucl-ex]
[61] N Guler et al (CLAS) Precise determination of the deuteron spin structure at low to mod-
erate Q2 with CLAS and extraction of the neutron contribution Phys Rev C 92 055201
(2015) arXiv150507877 [nucl-ex]
[62] C E Carlson V Nazaryan and K Griffioen Proton structure corrections to electronic
and muonic hydrogen hyperfine splitting Phys Rev A 78 022517 (2008) arXiv08052603
[physicsatom-ph]
[63] V Bernard N Kaiser and U-G Meiszligner Chiral dynamics in nucleons and nuclei Int J
Mod Phys E 4 193 (1995) arXivhep-ph9501384
[64] T R Hemmert B R Holstein and J Kambor ∆(1232) and the polarizabilities of the nucleon
Phys Rev D 55 5598 (1997) arXivhep-ph9612374
[65] V Pascalutsa and R Timmermans Field theory of nucleon to higher-spin baryon transitions
Phys Rev C 60 042201 (1999) arXivnucl-th9905065
[66] V Pascalutsa Quantization of an interacting spin-32 field and the Delta isobar Phys Rev
D 58 096002 (1998) arXivhep-ph9802288
[67] H Krebs Double Virtual Compton Scattering and SpinStructure of the Nucleon PoS
CD2018 031 (2019)
45
[68] N Kochelev and Y Oh Axial anomaly and the δLT puzzle Phys Rev D 016012 (2012)
arXiv11034892 [hep-ph]
[69] D Drechsel S Kamalov and L Tiator Unitary isobar model ndash MAID2007 Eur Phys J A
69 (2007) available at httpsmaidkphuni-mainzde arXiv07100306 [nucl-th]
[70] P J Mohr B N Taylor and D B Newell CODATA recommended values of the fundamental
physical constants 2010 Rev Mod Phys 84 1527 (2012)
[71] J Ahrens et al (GDH A2) First measurement of the Gerasimov-Drell-Hearn integral for 1H
from 200 to 800 MeV Phys Rev Lett 87 022003 (2001) arXivhep-ex0105089 [hep-ex]
[72] K Helbing (GDH) Experimental verification of the GDH sum rule at ELSA and MAMI Nucl
Phys Proc Suppl 105 113 (2002)
[73] R Bradford A Bodek H S Budd and J Arrington A New parameterization of the nucleon
elastic form-factors NuInt05 proceedings of the 4th International Workshop on Neutrino-
Nucleus Interactions in the Few-GeV Region Okayama Japan 26-29 September 2005 Nucl
Phys Proc Suppl 159 127 (2006) arXivhep-ex0602017
[74] J D Bjorken Applications of the chiral U(6)timesU(6) algebra of current densities Phys Rev
148 1467 (1966)
[75] J D Bjorken Inelastic scattering of polarized leptons from polarized nucleons Phys Rev D
1 1376 (1970)
[76] V Bernard T R Hemmert and U-G Meiszligner Novel analysis of chiral loop effects in
the generalized Gerasimov-Drell-Hearn sum rule Phys Lett B 545 105 (2002) arXivhep-
ph0203167
[77] A Deur et al Experimental study of isovector spin sum rules Phys Rev D 78 032001
(2008) arXiv08023198 [nucl-ex]
[78] B Pasquini P Pedroni and D Drechsel Higher order forward spin polarizability Phys Lett
B 687 160 (2010) arXiv10014230 [hep-ph]
[79] R Jaffe g2ndashThe nucleonrsquos other spin-dependent structure function Comments Nucl Part
Phys 19 239 (1990)
[80] E V Shuryak and A Vainshtein Theory of power corrections to deep inelastic scattering in
quantum chromodynamics (II) Qminus4 effects polarized target Nucl Phys B 201 141 (1982)
[81] S Wandzura and F Wilczek Sum rules for spin dependent electroproduction Test of rela-
tivistic constituent quarks Phys Lett B 72 195 (1977)
[82] B W Filippone and X-D Ji The spin structure of the nucleon Adv Nucl Phys 26 1
(2001) arXivhep-ph0101224
[83] M Burkardt The g2 structure function Proceedings Workshop on Spin structure at long
46
distance Newport News USA March 12-13 2009 AIP Conf Proc 1155 26 (2009)
arXiv09054079 [hep-ph]
[84] A Deur S J Brodsky and G F de Teramond The spin structure of the nucleon Rept
Prog Phys 82 076201 (2019) arXiv180705250 [hep-ph]
[85] B R Holstein D Drechsel B Pasquini and M Vanderhaeghen Higher order polarizabilities
of the proton Phys Rev C 61 034316 (2000) arXivhep-ph9910427
[86] B R Holstein V Pascalutsa and M Vanderhaeghen Sum rules for magnetic moments and
polarizabilities in QED and chiral effective-field theory Phys Rev D 72 094014 (2005)
arXivhep-ph0507016
[87] B Pasquini D Drechsel and L Tiator Invariant amplitudes for pion electroproduction Eur
Phys J A 34 387 (2007) arXiv07122327 [hep-ph]
[88] A J Buchmann Non-spherical proton shape and hydrogen hyperfine splitting Proceedings
International Workshop on Precision Physics of Simple Atomic Systems (PSAS 2008) Wind-
sor Ontario Canada July 21-26 2008 Can J Phys 87 773 (2009) arXiv09104747
[physicsatom-ph]
47
Q2 Phys Rev C 92 015208 (2015) arXiv13044497 [nucl-ex]
[26] A Deur Experimental studies at low Q2 of the spin structure of the nucleon at Jefferson Lab
in 9th International Workshop on Chiral Dynamics (CD18) Durham NC USA September
17-21 2018 (2019) arXiv190305661 [nucl-ex]
[27] V Sulkosky et al (Jefferson Lab E97-110) Measurement of the 3He spin-structure functions
and of neutron (3He) spin-dependent sum rules at 0035 le Q2 le 024 GeV2 Phys Lett B
805 135428 (2020) arXiv190805709 [nucl-ex]
[28] K Adhikari et al (CLAS) Measurement of the Q2 Dependence of the Deuteron Spin Structure
Function g1 and its Moments at Low Q2 with CLAS Phys Rev Lett 120 062501 (2018)
arXiv171101974 [nucl-ex]
[29] V Lensky F Hagelstein A Hiller Blin and V Pascalutsa Comment on rdquoMeasurement of
the Q2 Dependence of the Deuteron Spin Structure Function g1 and its Moments at Low Q2
with CLASrdquo (2018) arXiv180603219 [nucl-th]
[30] J M Alarcn F Hagelstein V Lensky and V Pascalutsa Forward doubly-virtual Compton
scattering off the nucleon in chiral perturbation theory at NLO the subtraction function and
moments of unpolarized structure functions (2020) arXiv200509518 [hep-ph]
[31] V Bernard E Epelbaum H Krebs and U-G Meiszligner New insights into the spin structure
of the nucleon Phys Rev D 87 054032 (2013) arXiv12092523 [hep-ph]
[32] V Lensky J M Alarcon and V Pascalutsa Moments of nucleon structure functions at
next-to-leading order in baryon chiral perturbation theory Phys Rev C 90 055202 (2014)
arXiv14072574 [hep-ph]
[33] V Pascalutsa and M Vanderhaeghen Chiral effective-field theory in the ∆(1232) region
I Pion electroproduction on the nucleon Phys Rev D 73 034003 (2006) arXivhep-
ph0512244
[34] V Pascalutsa and M Vanderhaeghen Electromagnetic nucleon-to-Delta transition in chiral
effective field theory Phys Rev Lett 95 232001 (2005) arXivhep-ph0508060
[35] T R Hemmert B R Holstein and J Kambor Systematic 1M expansion for spin 32
particles in baryon chiral perturbation theory Phys Lett B 395 89 (1997) arXivhep-
ph9606456
[36] V Pascalutsa and D R Phillips Effective theory of the ∆(1232) in Compton scattering off
the nucleon Phys Rev C 67 055202 (2003) arXivnucl-th0212024
[37] V Pascalutsa M Vanderhaeghen and S N Yang Electromagnetic excitation of the ∆(1232)-
resonance Phys Rept 437 125 (2007) arXivhep-ph0609004
[38] K A Olive et al (Particle Data Group) Review of Particle Physics Chin Phys C 38 090001
43
(2014)
[39] F E Low Scattering of light of very low frequency by systems of spin 12 Phys Rev 96
1428 (1954)
[40] M Gell-Mann and M L Goldberger Scattering of low-energy photons by particles of spin
12 Phys Rev 96 1433 (1954)
[41] A M Harun ar-Rashid A simple derivation of Schwingerrsquos sum rule for spin dependent
structure functions Nuovo Cim A 33 447 (1976)
[42] F Hagelstein and V Pascalutsa Dissecting the hadronic contributions to (g minus 2)micro by
Schwingerrsquos sum rule Phys Rev Lett 120 072002 (2018) arXiv171004571 [hep-ph]
[43] V Lensky V Pascalutsa M Vanderhaeghen and C Kao Spin-dependent sum rules con-
necting real and virtual Compton scattering verified Phys Rev D 95 074001 (2017)
arXiv170101947 [hep-ph]
[44] H Burkhardt and W N Cottingham Sum rules for forward virtual Compton scattering
Annals Phys 56 453 (1970)
[45] V Lensky and V Pascalutsa Predictive powers of chiral perturbation theory in Compton
scattering off protons Eur Phys J C 65 195 (2010) arXiv09070451 [hep-ph]
[46] H Grieszlighammer J McGovern D Phillips and G Feldman Using effective field theory to
analyse low-energy Compton scattering data from protons and light nuclei Prog Part Nucl
Phys 67 841 (2012) arXiv12036834 [nucl-th]
[47] H W Grieszlighammer J A McGovern and D R Phillips Nucleon polarisabilities at and
beyond physical pion masses Eur Phys J A 52 139 (2016) arXiv151101952 [nucl-th]
[48] E Epelbaum H Krebs and U-G Meiszligner Improved chiral nucleon-nucleon potential up to
next-to-next-to-next-to-leading order Eur Phys J A 51 53 (2015) arXiv14120142 [nucl-
th]
[49] E Epelbaum H Krebs and U-G Meiszligner Precision nucleon-nucleon potential at fifth order
in the chiral expansion Phys Rev Lett 115 122301 (2015) arXiv14124623 [nucl-th]
[50] C W Kao T Spitzenberg and M Vanderhaeghen Burkhardt-Cottingham sum rule and
forward spin polarizabilities in heavy baryon chiral perturbation theory Phys Rev D 67
016001 (2003) arXivhep-ph0209241
[51] C-W Kao D Drechsel S Kamalov and M Vanderhaeghen Higher moments of nucleon
spin structure functions in heavy baryon chiral perturbation theory and in a resonance model
Phys Rev D 69 056004 (2004) arXivhep-ph0312102
[52] T Becher and H Leutwyler Baryon chiral perturbation theory in manifestly Lorentz invariant
form Eur Phys J C 9 643 (1999) arXivhep-ph9901384
44
[53] J Gasser M E Sainio and A Svarc Nucleons with chiral loops Nucl Phys B 307 779
(1988)
[54] L S Geng J Martin Camalich L Alvarez-Ruso and M J Vicente Vacas Leading SU(3)-
breaking corrections to the baryon magnetic moments in chiral perturbation theory Phys
Rev Lett 101 222002 (2008) arXiv08051419 [hep-ph]
[55] D Drechsel S S Kamalov and L Tiator The GDH sum rule and related integrals Phys
Rev D 63 114010 (2001) arXivhep-ph0008306
[56] D Drechsel O Hanstein S S Kamalov and L Tiator A unitary isobar model for pion
photo- and electroproduction on the proton up to 1 GeV Nucl Phys A 645 145 (1999)
arXivnucl-th9807001
[57] L Tiator private communication (2020)
[58] V Bernard T R Hemmert and U-G Meiszligner Spin structure of the nucleon at low energies
Phys Rev D 67 076008 (2003) arXivhep-ph0212033
[59] O Gryniuk F Hagelstein and V Pascalutsa Evaluation of the forward Compton scattering
off protons II Spin-dependent amplitude and observables Phys Rev D 94 034043 (2016)
arXiv160400789 [nucl-th]
[60] R Zielinski The g2p Experiment A Measurement of the Protonrsquos Spin Structure Functions
PhD thesis New Hampshire U (2010) arXiv170808297 [nucl-ex]
[61] N Guler et al (CLAS) Precise determination of the deuteron spin structure at low to mod-
erate Q2 with CLAS and extraction of the neutron contribution Phys Rev C 92 055201
(2015) arXiv150507877 [nucl-ex]
[62] C E Carlson V Nazaryan and K Griffioen Proton structure corrections to electronic
and muonic hydrogen hyperfine splitting Phys Rev A 78 022517 (2008) arXiv08052603
[physicsatom-ph]
[63] V Bernard N Kaiser and U-G Meiszligner Chiral dynamics in nucleons and nuclei Int J
Mod Phys E 4 193 (1995) arXivhep-ph9501384
[64] T R Hemmert B R Holstein and J Kambor ∆(1232) and the polarizabilities of the nucleon
Phys Rev D 55 5598 (1997) arXivhep-ph9612374
[65] V Pascalutsa and R Timmermans Field theory of nucleon to higher-spin baryon transitions
Phys Rev C 60 042201 (1999) arXivnucl-th9905065
[66] V Pascalutsa Quantization of an interacting spin-32 field and the Delta isobar Phys Rev
D 58 096002 (1998) arXivhep-ph9802288
[67] H Krebs Double Virtual Compton Scattering and SpinStructure of the Nucleon PoS
CD2018 031 (2019)
45
[68] N Kochelev and Y Oh Axial anomaly and the δLT puzzle Phys Rev D 016012 (2012)
arXiv11034892 [hep-ph]
[69] D Drechsel S Kamalov and L Tiator Unitary isobar model ndash MAID2007 Eur Phys J A
69 (2007) available at httpsmaidkphuni-mainzde arXiv07100306 [nucl-th]
[70] P J Mohr B N Taylor and D B Newell CODATA recommended values of the fundamental
physical constants 2010 Rev Mod Phys 84 1527 (2012)
[71] J Ahrens et al (GDH A2) First measurement of the Gerasimov-Drell-Hearn integral for 1H
from 200 to 800 MeV Phys Rev Lett 87 022003 (2001) arXivhep-ex0105089 [hep-ex]
[72] K Helbing (GDH) Experimental verification of the GDH sum rule at ELSA and MAMI Nucl
Phys Proc Suppl 105 113 (2002)
[73] R Bradford A Bodek H S Budd and J Arrington A New parameterization of the nucleon
elastic form-factors NuInt05 proceedings of the 4th International Workshop on Neutrino-
Nucleus Interactions in the Few-GeV Region Okayama Japan 26-29 September 2005 Nucl
Phys Proc Suppl 159 127 (2006) arXivhep-ex0602017
[74] J D Bjorken Applications of the chiral U(6)timesU(6) algebra of current densities Phys Rev
148 1467 (1966)
[75] J D Bjorken Inelastic scattering of polarized leptons from polarized nucleons Phys Rev D
1 1376 (1970)
[76] V Bernard T R Hemmert and U-G Meiszligner Novel analysis of chiral loop effects in
the generalized Gerasimov-Drell-Hearn sum rule Phys Lett B 545 105 (2002) arXivhep-
ph0203167
[77] A Deur et al Experimental study of isovector spin sum rules Phys Rev D 78 032001
(2008) arXiv08023198 [nucl-ex]
[78] B Pasquini P Pedroni and D Drechsel Higher order forward spin polarizability Phys Lett
B 687 160 (2010) arXiv10014230 [hep-ph]
[79] R Jaffe g2ndashThe nucleonrsquos other spin-dependent structure function Comments Nucl Part
Phys 19 239 (1990)
[80] E V Shuryak and A Vainshtein Theory of power corrections to deep inelastic scattering in
quantum chromodynamics (II) Qminus4 effects polarized target Nucl Phys B 201 141 (1982)
[81] S Wandzura and F Wilczek Sum rules for spin dependent electroproduction Test of rela-
tivistic constituent quarks Phys Lett B 72 195 (1977)
[82] B W Filippone and X-D Ji The spin structure of the nucleon Adv Nucl Phys 26 1
(2001) arXivhep-ph0101224
[83] M Burkardt The g2 structure function Proceedings Workshop on Spin structure at long
46
distance Newport News USA March 12-13 2009 AIP Conf Proc 1155 26 (2009)
arXiv09054079 [hep-ph]
[84] A Deur S J Brodsky and G F de Teramond The spin structure of the nucleon Rept
Prog Phys 82 076201 (2019) arXiv180705250 [hep-ph]
[85] B R Holstein D Drechsel B Pasquini and M Vanderhaeghen Higher order polarizabilities
of the proton Phys Rev C 61 034316 (2000) arXivhep-ph9910427
[86] B R Holstein V Pascalutsa and M Vanderhaeghen Sum rules for magnetic moments and
polarizabilities in QED and chiral effective-field theory Phys Rev D 72 094014 (2005)
arXivhep-ph0507016
[87] B Pasquini D Drechsel and L Tiator Invariant amplitudes for pion electroproduction Eur
Phys J A 34 387 (2007) arXiv07122327 [hep-ph]
[88] A J Buchmann Non-spherical proton shape and hydrogen hyperfine splitting Proceedings
International Workshop on Precision Physics of Simple Atomic Systems (PSAS 2008) Wind-
sor Ontario Canada July 21-26 2008 Can J Phys 87 773 (2009) arXiv09104747
[physicsatom-ph]
47
(2014)
[39] F E Low Scattering of light of very low frequency by systems of spin 12 Phys Rev 96
1428 (1954)
[40] M Gell-Mann and M L Goldberger Scattering of low-energy photons by particles of spin
12 Phys Rev 96 1433 (1954)
[41] A M Harun ar-Rashid A simple derivation of Schwingerrsquos sum rule for spin dependent
structure functions Nuovo Cim A 33 447 (1976)
[42] F Hagelstein and V Pascalutsa Dissecting the hadronic contributions to (g minus 2)micro by
Schwingerrsquos sum rule Phys Rev Lett 120 072002 (2018) arXiv171004571 [hep-ph]
[43] V Lensky V Pascalutsa M Vanderhaeghen and C Kao Spin-dependent sum rules con-
necting real and virtual Compton scattering verified Phys Rev D 95 074001 (2017)
arXiv170101947 [hep-ph]
[44] H Burkhardt and W N Cottingham Sum rules for forward virtual Compton scattering
Annals Phys 56 453 (1970)
[45] V Lensky and V Pascalutsa Predictive powers of chiral perturbation theory in Compton
scattering off protons Eur Phys J C 65 195 (2010) arXiv09070451 [hep-ph]
[46] H Grieszlighammer J McGovern D Phillips and G Feldman Using effective field theory to
analyse low-energy Compton scattering data from protons and light nuclei Prog Part Nucl
Phys 67 841 (2012) arXiv12036834 [nucl-th]
[47] H W Grieszlighammer J A McGovern and D R Phillips Nucleon polarisabilities at and
beyond physical pion masses Eur Phys J A 52 139 (2016) arXiv151101952 [nucl-th]
[48] E Epelbaum H Krebs and U-G Meiszligner Improved chiral nucleon-nucleon potential up to
next-to-next-to-next-to-leading order Eur Phys J A 51 53 (2015) arXiv14120142 [nucl-
th]
[49] E Epelbaum H Krebs and U-G Meiszligner Precision nucleon-nucleon potential at fifth order
in the chiral expansion Phys Rev Lett 115 122301 (2015) arXiv14124623 [nucl-th]
[50] C W Kao T Spitzenberg and M Vanderhaeghen Burkhardt-Cottingham sum rule and
forward spin polarizabilities in heavy baryon chiral perturbation theory Phys Rev D 67
016001 (2003) arXivhep-ph0209241
[51] C-W Kao D Drechsel S Kamalov and M Vanderhaeghen Higher moments of nucleon
spin structure functions in heavy baryon chiral perturbation theory and in a resonance model
Phys Rev D 69 056004 (2004) arXivhep-ph0312102
[52] T Becher and H Leutwyler Baryon chiral perturbation theory in manifestly Lorentz invariant
form Eur Phys J C 9 643 (1999) arXivhep-ph9901384
44
[53] J Gasser M E Sainio and A Svarc Nucleons with chiral loops Nucl Phys B 307 779
(1988)
[54] L S Geng J Martin Camalich L Alvarez-Ruso and M J Vicente Vacas Leading SU(3)-
breaking corrections to the baryon magnetic moments in chiral perturbation theory Phys
Rev Lett 101 222002 (2008) arXiv08051419 [hep-ph]
[55] D Drechsel S S Kamalov and L Tiator The GDH sum rule and related integrals Phys
Rev D 63 114010 (2001) arXivhep-ph0008306
[56] D Drechsel O Hanstein S S Kamalov and L Tiator A unitary isobar model for pion
photo- and electroproduction on the proton up to 1 GeV Nucl Phys A 645 145 (1999)
arXivnucl-th9807001
[57] L Tiator private communication (2020)
[58] V Bernard T R Hemmert and U-G Meiszligner Spin structure of the nucleon at low energies
Phys Rev D 67 076008 (2003) arXivhep-ph0212033
[59] O Gryniuk F Hagelstein and V Pascalutsa Evaluation of the forward Compton scattering
off protons II Spin-dependent amplitude and observables Phys Rev D 94 034043 (2016)
arXiv160400789 [nucl-th]
[60] R Zielinski The g2p Experiment A Measurement of the Protonrsquos Spin Structure Functions
PhD thesis New Hampshire U (2010) arXiv170808297 [nucl-ex]
[61] N Guler et al (CLAS) Precise determination of the deuteron spin structure at low to mod-
erate Q2 with CLAS and extraction of the neutron contribution Phys Rev C 92 055201
(2015) arXiv150507877 [nucl-ex]
[62] C E Carlson V Nazaryan and K Griffioen Proton structure corrections to electronic
and muonic hydrogen hyperfine splitting Phys Rev A 78 022517 (2008) arXiv08052603
[physicsatom-ph]
[63] V Bernard N Kaiser and U-G Meiszligner Chiral dynamics in nucleons and nuclei Int J
Mod Phys E 4 193 (1995) arXivhep-ph9501384
[64] T R Hemmert B R Holstein and J Kambor ∆(1232) and the polarizabilities of the nucleon
Phys Rev D 55 5598 (1997) arXivhep-ph9612374
[65] V Pascalutsa and R Timmermans Field theory of nucleon to higher-spin baryon transitions
Phys Rev C 60 042201 (1999) arXivnucl-th9905065
[66] V Pascalutsa Quantization of an interacting spin-32 field and the Delta isobar Phys Rev
D 58 096002 (1998) arXivhep-ph9802288
[67] H Krebs Double Virtual Compton Scattering and SpinStructure of the Nucleon PoS
CD2018 031 (2019)
45
[68] N Kochelev and Y Oh Axial anomaly and the δLT puzzle Phys Rev D 016012 (2012)
arXiv11034892 [hep-ph]
[69] D Drechsel S Kamalov and L Tiator Unitary isobar model ndash MAID2007 Eur Phys J A
69 (2007) available at httpsmaidkphuni-mainzde arXiv07100306 [nucl-th]
[70] P J Mohr B N Taylor and D B Newell CODATA recommended values of the fundamental
physical constants 2010 Rev Mod Phys 84 1527 (2012)
[71] J Ahrens et al (GDH A2) First measurement of the Gerasimov-Drell-Hearn integral for 1H
from 200 to 800 MeV Phys Rev Lett 87 022003 (2001) arXivhep-ex0105089 [hep-ex]
[72] K Helbing (GDH) Experimental verification of the GDH sum rule at ELSA and MAMI Nucl
Phys Proc Suppl 105 113 (2002)
[73] R Bradford A Bodek H S Budd and J Arrington A New parameterization of the nucleon
elastic form-factors NuInt05 proceedings of the 4th International Workshop on Neutrino-
Nucleus Interactions in the Few-GeV Region Okayama Japan 26-29 September 2005 Nucl
Phys Proc Suppl 159 127 (2006) arXivhep-ex0602017
[74] J D Bjorken Applications of the chiral U(6)timesU(6) algebra of current densities Phys Rev
148 1467 (1966)
[75] J D Bjorken Inelastic scattering of polarized leptons from polarized nucleons Phys Rev D
1 1376 (1970)
[76] V Bernard T R Hemmert and U-G Meiszligner Novel analysis of chiral loop effects in
the generalized Gerasimov-Drell-Hearn sum rule Phys Lett B 545 105 (2002) arXivhep-
ph0203167
[77] A Deur et al Experimental study of isovector spin sum rules Phys Rev D 78 032001
(2008) arXiv08023198 [nucl-ex]
[78] B Pasquini P Pedroni and D Drechsel Higher order forward spin polarizability Phys Lett
B 687 160 (2010) arXiv10014230 [hep-ph]
[79] R Jaffe g2ndashThe nucleonrsquos other spin-dependent structure function Comments Nucl Part
Phys 19 239 (1990)
[80] E V Shuryak and A Vainshtein Theory of power corrections to deep inelastic scattering in
quantum chromodynamics (II) Qminus4 effects polarized target Nucl Phys B 201 141 (1982)
[81] S Wandzura and F Wilczek Sum rules for spin dependent electroproduction Test of rela-
tivistic constituent quarks Phys Lett B 72 195 (1977)
[82] B W Filippone and X-D Ji The spin structure of the nucleon Adv Nucl Phys 26 1
(2001) arXivhep-ph0101224
[83] M Burkardt The g2 structure function Proceedings Workshop on Spin structure at long
46
distance Newport News USA March 12-13 2009 AIP Conf Proc 1155 26 (2009)
arXiv09054079 [hep-ph]
[84] A Deur S J Brodsky and G F de Teramond The spin structure of the nucleon Rept
Prog Phys 82 076201 (2019) arXiv180705250 [hep-ph]
[85] B R Holstein D Drechsel B Pasquini and M Vanderhaeghen Higher order polarizabilities
of the proton Phys Rev C 61 034316 (2000) arXivhep-ph9910427
[86] B R Holstein V Pascalutsa and M Vanderhaeghen Sum rules for magnetic moments and
polarizabilities in QED and chiral effective-field theory Phys Rev D 72 094014 (2005)
arXivhep-ph0507016
[87] B Pasquini D Drechsel and L Tiator Invariant amplitudes for pion electroproduction Eur
Phys J A 34 387 (2007) arXiv07122327 [hep-ph]
[88] A J Buchmann Non-spherical proton shape and hydrogen hyperfine splitting Proceedings
International Workshop on Precision Physics of Simple Atomic Systems (PSAS 2008) Wind-
sor Ontario Canada July 21-26 2008 Can J Phys 87 773 (2009) arXiv09104747
[physicsatom-ph]
47
[53] J Gasser M E Sainio and A Svarc Nucleons with chiral loops Nucl Phys B 307 779
(1988)
[54] L S Geng J Martin Camalich L Alvarez-Ruso and M J Vicente Vacas Leading SU(3)-
breaking corrections to the baryon magnetic moments in chiral perturbation theory Phys
Rev Lett 101 222002 (2008) arXiv08051419 [hep-ph]
[55] D Drechsel S S Kamalov and L Tiator The GDH sum rule and related integrals Phys
Rev D 63 114010 (2001) arXivhep-ph0008306
[56] D Drechsel O Hanstein S S Kamalov and L Tiator A unitary isobar model for pion
photo- and electroproduction on the proton up to 1 GeV Nucl Phys A 645 145 (1999)
arXivnucl-th9807001
[57] L Tiator private communication (2020)
[58] V Bernard T R Hemmert and U-G Meiszligner Spin structure of the nucleon at low energies
Phys Rev D 67 076008 (2003) arXivhep-ph0212033
[59] O Gryniuk F Hagelstein and V Pascalutsa Evaluation of the forward Compton scattering
off protons II Spin-dependent amplitude and observables Phys Rev D 94 034043 (2016)
arXiv160400789 [nucl-th]
[60] R Zielinski The g2p Experiment A Measurement of the Protonrsquos Spin Structure Functions
PhD thesis New Hampshire U (2010) arXiv170808297 [nucl-ex]
[61] N Guler et al (CLAS) Precise determination of the deuteron spin structure at low to mod-
erate Q2 with CLAS and extraction of the neutron contribution Phys Rev C 92 055201
(2015) arXiv150507877 [nucl-ex]
[62] C E Carlson V Nazaryan and K Griffioen Proton structure corrections to electronic
and muonic hydrogen hyperfine splitting Phys Rev A 78 022517 (2008) arXiv08052603
[physicsatom-ph]
[63] V Bernard N Kaiser and U-G Meiszligner Chiral dynamics in nucleons and nuclei Int J
Mod Phys E 4 193 (1995) arXivhep-ph9501384
[64] T R Hemmert B R Holstein and J Kambor ∆(1232) and the polarizabilities of the nucleon
Phys Rev D 55 5598 (1997) arXivhep-ph9612374
[65] V Pascalutsa and R Timmermans Field theory of nucleon to higher-spin baryon transitions
Phys Rev C 60 042201 (1999) arXivnucl-th9905065
[66] V Pascalutsa Quantization of an interacting spin-32 field and the Delta isobar Phys Rev
D 58 096002 (1998) arXivhep-ph9802288
[67] H Krebs Double Virtual Compton Scattering and SpinStructure of the Nucleon PoS
CD2018 031 (2019)
45
[68] N Kochelev and Y Oh Axial anomaly and the δLT puzzle Phys Rev D 016012 (2012)
arXiv11034892 [hep-ph]
[69] D Drechsel S Kamalov and L Tiator Unitary isobar model ndash MAID2007 Eur Phys J A
69 (2007) available at httpsmaidkphuni-mainzde arXiv07100306 [nucl-th]
[70] P J Mohr B N Taylor and D B Newell CODATA recommended values of the fundamental
physical constants 2010 Rev Mod Phys 84 1527 (2012)
[71] J Ahrens et al (GDH A2) First measurement of the Gerasimov-Drell-Hearn integral for 1H
from 200 to 800 MeV Phys Rev Lett 87 022003 (2001) arXivhep-ex0105089 [hep-ex]
[72] K Helbing (GDH) Experimental verification of the GDH sum rule at ELSA and MAMI Nucl
Phys Proc Suppl 105 113 (2002)
[73] R Bradford A Bodek H S Budd and J Arrington A New parameterization of the nucleon
elastic form-factors NuInt05 proceedings of the 4th International Workshop on Neutrino-
Nucleus Interactions in the Few-GeV Region Okayama Japan 26-29 September 2005 Nucl
Phys Proc Suppl 159 127 (2006) arXivhep-ex0602017
[74] J D Bjorken Applications of the chiral U(6)timesU(6) algebra of current densities Phys Rev
148 1467 (1966)
[75] J D Bjorken Inelastic scattering of polarized leptons from polarized nucleons Phys Rev D
1 1376 (1970)
[76] V Bernard T R Hemmert and U-G Meiszligner Novel analysis of chiral loop effects in
the generalized Gerasimov-Drell-Hearn sum rule Phys Lett B 545 105 (2002) arXivhep-
ph0203167
[77] A Deur et al Experimental study of isovector spin sum rules Phys Rev D 78 032001
(2008) arXiv08023198 [nucl-ex]
[78] B Pasquini P Pedroni and D Drechsel Higher order forward spin polarizability Phys Lett
B 687 160 (2010) arXiv10014230 [hep-ph]
[79] R Jaffe g2ndashThe nucleonrsquos other spin-dependent structure function Comments Nucl Part
Phys 19 239 (1990)
[80] E V Shuryak and A Vainshtein Theory of power corrections to deep inelastic scattering in
quantum chromodynamics (II) Qminus4 effects polarized target Nucl Phys B 201 141 (1982)
[81] S Wandzura and F Wilczek Sum rules for spin dependent electroproduction Test of rela-
tivistic constituent quarks Phys Lett B 72 195 (1977)
[82] B W Filippone and X-D Ji The spin structure of the nucleon Adv Nucl Phys 26 1
(2001) arXivhep-ph0101224
[83] M Burkardt The g2 structure function Proceedings Workshop on Spin structure at long
46
distance Newport News USA March 12-13 2009 AIP Conf Proc 1155 26 (2009)
arXiv09054079 [hep-ph]
[84] A Deur S J Brodsky and G F de Teramond The spin structure of the nucleon Rept
Prog Phys 82 076201 (2019) arXiv180705250 [hep-ph]
[85] B R Holstein D Drechsel B Pasquini and M Vanderhaeghen Higher order polarizabilities
of the proton Phys Rev C 61 034316 (2000) arXivhep-ph9910427
[86] B R Holstein V Pascalutsa and M Vanderhaeghen Sum rules for magnetic moments and
polarizabilities in QED and chiral effective-field theory Phys Rev D 72 094014 (2005)
arXivhep-ph0507016
[87] B Pasquini D Drechsel and L Tiator Invariant amplitudes for pion electroproduction Eur
Phys J A 34 387 (2007) arXiv07122327 [hep-ph]
[88] A J Buchmann Non-spherical proton shape and hydrogen hyperfine splitting Proceedings
International Workshop on Precision Physics of Simple Atomic Systems (PSAS 2008) Wind-
sor Ontario Canada July 21-26 2008 Can J Phys 87 773 (2009) arXiv09104747
[physicsatom-ph]
47
[68] N Kochelev and Y Oh Axial anomaly and the δLT puzzle Phys Rev D 016012 (2012)
arXiv11034892 [hep-ph]
[69] D Drechsel S Kamalov and L Tiator Unitary isobar model ndash MAID2007 Eur Phys J A
69 (2007) available at httpsmaidkphuni-mainzde arXiv07100306 [nucl-th]
[70] P J Mohr B N Taylor and D B Newell CODATA recommended values of the fundamental
physical constants 2010 Rev Mod Phys 84 1527 (2012)
[71] J Ahrens et al (GDH A2) First measurement of the Gerasimov-Drell-Hearn integral for 1H
from 200 to 800 MeV Phys Rev Lett 87 022003 (2001) arXivhep-ex0105089 [hep-ex]
[72] K Helbing (GDH) Experimental verification of the GDH sum rule at ELSA and MAMI Nucl
Phys Proc Suppl 105 113 (2002)
[73] R Bradford A Bodek H S Budd and J Arrington A New parameterization of the nucleon
elastic form-factors NuInt05 proceedings of the 4th International Workshop on Neutrino-
Nucleus Interactions in the Few-GeV Region Okayama Japan 26-29 September 2005 Nucl
Phys Proc Suppl 159 127 (2006) arXivhep-ex0602017
[74] J D Bjorken Applications of the chiral U(6)timesU(6) algebra of current densities Phys Rev
148 1467 (1966)
[75] J D Bjorken Inelastic scattering of polarized leptons from polarized nucleons Phys Rev D
1 1376 (1970)
[76] V Bernard T R Hemmert and U-G Meiszligner Novel analysis of chiral loop effects in
the generalized Gerasimov-Drell-Hearn sum rule Phys Lett B 545 105 (2002) arXivhep-
ph0203167
[77] A Deur et al Experimental study of isovector spin sum rules Phys Rev D 78 032001
(2008) arXiv08023198 [nucl-ex]
[78] B Pasquini P Pedroni and D Drechsel Higher order forward spin polarizability Phys Lett
B 687 160 (2010) arXiv10014230 [hep-ph]
[79] R Jaffe g2ndashThe nucleonrsquos other spin-dependent structure function Comments Nucl Part
Phys 19 239 (1990)
[80] E V Shuryak and A Vainshtein Theory of power corrections to deep inelastic scattering in
quantum chromodynamics (II) Qminus4 effects polarized target Nucl Phys B 201 141 (1982)
[81] S Wandzura and F Wilczek Sum rules for spin dependent electroproduction Test of rela-
tivistic constituent quarks Phys Lett B 72 195 (1977)
[82] B W Filippone and X-D Ji The spin structure of the nucleon Adv Nucl Phys 26 1
(2001) arXivhep-ph0101224
[83] M Burkardt The g2 structure function Proceedings Workshop on Spin structure at long
46
distance Newport News USA March 12-13 2009 AIP Conf Proc 1155 26 (2009)
arXiv09054079 [hep-ph]
[84] A Deur S J Brodsky and G F de Teramond The spin structure of the nucleon Rept
Prog Phys 82 076201 (2019) arXiv180705250 [hep-ph]
[85] B R Holstein D Drechsel B Pasquini and M Vanderhaeghen Higher order polarizabilities
of the proton Phys Rev C 61 034316 (2000) arXivhep-ph9910427
[86] B R Holstein V Pascalutsa and M Vanderhaeghen Sum rules for magnetic moments and
polarizabilities in QED and chiral effective-field theory Phys Rev D 72 094014 (2005)
arXivhep-ph0507016
[87] B Pasquini D Drechsel and L Tiator Invariant amplitudes for pion electroproduction Eur
Phys J A 34 387 (2007) arXiv07122327 [hep-ph]
[88] A J Buchmann Non-spherical proton shape and hydrogen hyperfine splitting Proceedings
International Workshop on Precision Physics of Simple Atomic Systems (PSAS 2008) Wind-
sor Ontario Canada July 21-26 2008 Can J Phys 87 773 (2009) arXiv09104747
[physicsatom-ph]
47
distance Newport News USA March 12-13 2009 AIP Conf Proc 1155 26 (2009)
arXiv09054079 [hep-ph]
[84] A Deur S J Brodsky and G F de Teramond The spin structure of the nucleon Rept
Prog Phys 82 076201 (2019) arXiv180705250 [hep-ph]
[85] B R Holstein D Drechsel B Pasquini and M Vanderhaeghen Higher order polarizabilities
of the proton Phys Rev C 61 034316 (2000) arXivhep-ph9910427
[86] B R Holstein V Pascalutsa and M Vanderhaeghen Sum rules for magnetic moments and
polarizabilities in QED and chiral effective-field theory Phys Rev D 72 094014 (2005)
arXivhep-ph0507016
[87] B Pasquini D Drechsel and L Tiator Invariant amplitudes for pion electroproduction Eur
Phys J A 34 387 (2007) arXiv07122327 [hep-ph]
[88] A J Buchmann Non-spherical proton shape and hydrogen hyperfine splitting Proceedings
International Workshop on Precision Physics of Simple Atomic Systems (PSAS 2008) Wind-
sor Ontario Canada July 21-26 2008 Can J Phys 87 773 (2009) arXiv09104747
[physicsatom-ph]
47