Astrid N. Hiller BlinAstrid N. Hiller Blin Institut fur¨ Kernphysik & PRISMA Cluster of Excellence, Johannes Gutenberg Universitat,¨ D-55099 Mainz, Germany Zhi-Feng Suny Research

Electromagnetic form factors of spin 1/2 doubly charmed baryons

Astrid N. Hiller Blin∗

Institut fur Kernphysik & PRISMA Cluster of Excellence,

Johannes Gutenberg Universitat, D-55099 Mainz, Germany

Zhi-Feng Sun†

Research Center for Hadron and CSR Physics,

Lanzhou University & Institute of Modern Physics of CAS, Lanzhou 730000, China and

School of Physical Science and Technology,

Lanzhou University, Lanzhou 730000, China

M. J. Vicente Vacas‡

Departamento de Fısica Teorica and Instituto de Fisica Corpuscular (IFIC),

Centro Mixto UVEG-CSIC, Valencia E-46071, Spain

(Dated: July 4, 2018)

Abstract

We study the electromagnetic form factors of the doubly charmed baryons, using covariant chiral per-

turbation theory within the extended on-mass-shell (EOMS) scheme. Vector-meson contributions are also

taken into account. We present results for the baryon magnetic moments, charge and magnetic radii. While

some of the chiral Lagrangian parameters could be set to values determined in previous works, the available

lattice results for Ξ+cc and Ω+

cc only allow for robust constraints on the low-energy constant (LEC) combi-

nation, c89(= − 13 c8 + 4c9). The couplings of the doubly charmed baryons to the vector mesons have been

estimated assuming the Okubo–Zweig–Iizuka (OZI) rule. We also give the expressions for the form factors

of the double beauty baryons considering the masses predicted in the framework of quark models. A com-

parison of our results with those obtained in heavy baryon chiral perturbation theory (HBChPT) at the same

chiral order is made.

∗ [email protected]† [email protected]‡ [email protected]

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I. INTRODUCTION

The recent announcement of the observation of the Ξ++cc particle by the LHCb collaboration [1]

has revived the interest on the physics of doubly heavy baryons. Up to now, the experimental

evidence for baryons with two heavy quarks was marginal. Only one cc baryon, Ξ+cc(3520), had

been included in the Review of Particle Physics by the PDG [2] and was labeled with one star.

The Ξ+cc(3520) baryon was first observed by the SELEX collaboration in the Λ+

c K−π+ channel [3]

and later corroborated in the pD+K− one [4]. However, neither BABAR [5], nor BELLE [6], nor

LHCb [7] could confirm the existence of this state.

On the other hand, the mass of Ξ++cc measured by LHCb is greater by more than 100 MeV than

that of the Ξ+cc(3520) particle. This large splitting, if confirmed, would suggest that the two states

are not isospin partners [8] and thus could belong to different multiplets.

The scarce and conflicting experimental information has not deterred the theoretical research on

the topic. Since the early works predicting the existence of doubly charmed baryons [9, 10], soon

after the discovery of hidden (cc) charm states, these particles have been studied in quark models,

QCD sum rules, lattice simulations, effective theories implementing heavy quark spin symmetry,

etc. See, e.g., Refs. [11, 12]. Most of those works address the spectroscopy of these baryons.

Besides, other matters such as their decays [13–16] and electromagnetic properties have been

abundantly studied. For instance, diverse quark models [17–24], the MIT bag model [25, 26], the

skyrmion model [27], HBChPT [28] and ChPT within the EOMS scheme [29] have been applied to

study the magnetic moments of doubly charmed baryons. Both, the magnetic and the electric form

factors (FF) have also been investigated in lattice QCD [30–32]. As expected, the lattice results

show that the doubly charmed baryons have smaller radii than the singly charmed ones and than

those composed of only light quarks such as the proton. However, there is some tension between

the lattice results for magnetic moments and those determined in other theoretical models [31].

Here, we focus on these FF, which are a quite interesting probe, as they offer an insight on the

hadron structure and how its constituents are distributed.

Our work is based on chiral perturbation theory (ChPT) [33–36], which provides a model in-

dependent and systematic framework to study the non-perturbative regime of the strong interac-

tion at low energies or for soft probes. Furthermore, it is well suited to analyse the lattice data

at quark(meson) masses different from the physical ones. The ChPT results are systematically

arranged as an expansion in powers of the Goldstone boson masses and the external (small) mo-

2

menta. The corresponding power counting involves some difficulties when baryon loops are in-

cluded in the calculation and different methods have been developed to overcome this issue such

as HBChPT [37], heavy hadron (HH)ChPT, using similar techniques [38, 39], and the covariant

approaches: infrared [40] and EOMS [41]. All these schemes have been widely and success-

fully used to investigate the electromagnetic structure of light baryons [36, 42–71]. There are

also some calculations of the electromagnetic properties of baryons with a single heavy quark

in HHChPT [39, 72–75]. Recently, the magnetic moments of doubly heavy baryons with spin12 and 3

2 have been studied in HBChPT [28, 76]. Here, we use the covariant EOMS framework

instead and we also calculate the electric and magnetic radii of the spin 12 triplet. The manifestly

Lorentz invariant EOMS scheme has been found to deliver a better chiral convergence than the

other schemes for most observables [77] and in particular for the magnetic moment of the light

baryons [63, 78]. Although the HB techniques are expected to work better the larger the baryon

mass is, the differences with the covariant calculation are not negligible. This point is also explored

both for di-charm and di-bottom baryons.

Our work is organized as follows. In Section II, the effective Lagrangian describing the inter-

action of doubly-heavy baryons and Goldstone bosons is given. The form factors of doubly heavy

baryons are introduced in Section III and the results are shown in Section IV. Finally, summary

and conclusions are given in Section V.

II. THE EFFECTIVE LAGRANGIAN

A. Interaction with light pseudoscalar mesons

The effective Lagrangian describing the interaction of double-charm baryons and the Goldstone

bosons up to second order was constructed in Refs. [79, 80]. It can be written as 1

L(1) = ψ(iD/ − m +gA

2γµγ5uµ)ψ, (1)

L(2) = c1ψ〈χ+〉ψ − c2

8m2 ψ〈uµuν〉Dµ,Dνψ + h.c.

−

c3

8m2 ψuµ, uνDµ,Dνψ + h.c.

+

c4

2ψ〈u2〉ψ

+c5

2ψu2ψ +

ic6

4ψσµν[uµ, uν]ψ + c7ψχ+ψ

1 In Refs. [79, 80], the c8 terms involves f +µν instead of f +

µν. However, both formulations are equivalent, and the

Lagrangian in Eq. (2) can be obtained by a redefinition of c9 in Refs. [79, 80]. In addition, note that the c6 term in

Eq. (14) of Ref. [79] is hermitian.

3

+c8

8mψσµν f +

µνψ +c9

8mψσµν〈 f +

µν〉ψ. (2)

The relevant pieces of the Lagrangian of order three can be obtained by considering chiral, parity

and charge conjugation symmetry. There are two terms contributing to the electromagnetic form

factors,

L(3) =

i2m

d1ψ[Dµ, f +µν]D

νψ + h.c.

+

2im

d2ψ[Dµ, 〈 f +µν〉]D

νψ + h.c.

+ ... . (3)

The Lagrangians for the double beauty baryons are analogous, only modifying m, their mass in

the chiral limit, and the coupling constants. In these equations, U = u2, which incorporates the

pseudoscalar meson field, is defined as

U = u2 = exp(iφ(x)

F

), (4)

where φ(x) is expressed as

φ(x) =

π0 + 1

√3η

√2π+

√2K+

√2π− −π0 + 1

√3η√

2K0

√2K−

√2K0 − 2

√3η

. (5)

The doubly-heavy baryon field ψ with spin 12 is a column vector in the flavor space, i.e.

ψ =

ΞuQQ

ΞdQQ

ΩsQQ

, (6)

where the subscript Q denotes the charm or beauty quark. In Eqs. (1) and (2), χ, χ±, fµν, f ±µν, uµ,

Γµ, Dµ have the following definitions

χ = diag(M2π,M

2π, 2M2

K − M2π), (7)

χ± = u†χu† ± uχ†u, (8)

fµν = −eQ∂µAν + eQ∂νAµ, (9)

f +µν = u† fµνu + u fµνu†, (10)

uµ = i[u†(∂µ + eQiAµ)u − u(∂µ + eQiAµ)u†], (11)

Γµ =12

[u†(∂µ + eQiAµ)u + u(∂µ + eQiAµ)u†], (12)

Dµ = ∂µ + Γµ, (13)

4

FIG. 1. Feynman diagram of the vector-meson contribution to the form factor.

with Aµ the photon field. For the double-charm baryons Q = diag(2, 1, 1), while for the double-

beauty baryons Q = diag(0,−1,−1). For a 3 × 3 matrix A in flavor space, we define A = A − 13〈A〉

with 〈A〉 the trace of A.

The interaction Lagrangian describing the Goldstone-boson interaction with a photon can be

extracted from the leading-order meson Lagrangian

L =F2

4Tr[DµU(DµU)†] (14)

as follows

Lφφγ =ie2

Tr[(φ∂µφ − ∂µφφ)Ql]Aµ. (15)

In Eq. (14), DµU = ∂µU + ieQlAµU − ieUQlAµ with Ql = diag(2/3,−1/3,−1/3).

B. Interaction with vector mesons

It is well known, in the case of light baryons, that the consideration of a pseudoscalar meson

cloud plus contact terms, even up to order O(q4), is not sufficient to provide a precise description

of the electromagnetic form factors in ChPT [52, 53, 71]. This is especially true for the Q2 depen-

dence and thus the charge and magnetic radii. The reason is the importance of the contribution of

vector meson mechanisms, see Fig. 1. We expect a similar situation for the case of heavy baryons.

Therefore, in order to model the behavior of the form factors at moderate momentum transfers,

the vector-meson contributions are also included. In the case of ideal mixing of the vector-meson

singlet and octet, the Lagrangian of the coupling of doubly-heavy baryons to the vector mesons

has the following structure

LVBB = (Ξ++QQ, Ξ+QQ

)(gΞQQ

v γµ + gΞQQt

σµν∂ν2mB

) 1√

2ρ0 + 1

√2ω ρ+

ρ− − 1√

2ρ0 + 1

√2ω

µ

Ξ++QQ

Ξ+QQ

+Ω+

QQ

(gΩQQ

v γµ + gΩQQt

σµν∂ν2mB

)φµΩ

+QQ. (16)

5

According to the OZI rule, ΞQQ/ΩQQ only couples to (ρ, ω)/φ. Furthermore, given the large

breaking of S U(3) symmetry, we take different values for the couplings of ΞQQ and ΩQQ.

The Lagrangian of the vector-meson coupling to the photon, needed to calculate the contribu-

tions of vector mesons to the form factors, is given by [81]

Lγ = −1

2√

2

FV

MV〈Vµν f +µν〉. (17)

Here, Vµν = ∂µVν − ∂νVµ with Vµ the 3×3 matrix

Vµ =

1√

2ρ0 + 1

√2ω ρ+ K∗+

ρ− − 1√

2ρ0 + 1

√2ω K∗0

K∗− K∗0 φ

. (18)

In Eq. (17), MV is the mass of the vector meson. FV can be obtained by calculating the decay

width V → e+e−

ΓV→e+e− = C2V

4πα2F2V

3MV(19)

with α = 1137 , and CV = 1, 1

3 ,−√

23 for ρ, ω and φ, respectively.

III. FORM FACTORS OF THE DOUBLY-HEAVY BARYONS

A. Definitions

Considering the baryon matrix elements of the electromagnetic vector current, the electromag-

netic form factors are defined as

〈B(p f )|Jµ(0)|B(pi)〉 = u(p f )[γµFB

1 (q2) +iσµνqν2mB

FB2 (q2)

]u(pi), (20)

where Jµ(x) =∑

q eqq(x)γµq(x) with q running over the quarks, and B denotes the baryon Ξ++cc ,

Ξ+cc, Ω+

cc or Ξ0bb, Ξ−bb, Ω−bb. The physical mass of the baryon B is given by mB, eq is the charge of

the quark q, and FB1 (q2) and FB

2 (q2) are the Dirac and Pauli form factors. The Dirac spinor of a

baryon with four-momentum pµ and mass m is denoted as u(p). The transferred four-momentum

qµ = pµf − pµi obeys q2 ≤ 0. The electric and magnetic form factors are defined as

GBE(q2) = FB

1 (q2) +q2

4m2B

FB2 (q2), (21)

GBM(q2) = FB

1 (q2) + FB2 (q2). (22)

6

FIG. 2. Feynman diagrams contributing to the electromagnetic form factors up to order O(q3). Wiggled,

dashed and solid lines correspond to photon, mesons and baryons respectively. The numbers in the circles

show the chiral order of the vertices.

Then, the magnetic moment is defined as

µB = GM(0)e

2mB, (23)

while the charge and magnetic radii of the baryons can be obtained from the slope of the electric

and magnetic form factors

〈r2E,M〉B =

6GB

E,M(0)

dGBE,M(q2)

dq2

∣∣∣∣∣∣∣q2=0

. (24)

For neutral baryons, an exception is made for the electric radius, which reads

〈r2E〉B = 6

dGBE(q2)

dq2

∣∣∣∣∣∣q2=0

. (25)

7

B. Calculation of the form factors

In Fig. 2, we show the diagrams derived from the Lagrangians of Eqs. (1), (2), (3) and (15)

which contribute to the electromagnetic current matrix element up to order O(q3) in the chiral

expansion. We use the standard ChPT definition for the order of a given diagram [33]. The

resulting lengthy expressions of the unrenormalized contributions to the Dirac and Pauli form

factors FB1 and FB

2 are given in the Appendix. We renormalize them following the EOMS scheme.

As customary, we perform a modified minimal subtraction (MS) 2. Later, the terms that still break

the nominal power counting, which come from the baryonic loops, are also subtracted. In fact, the

only subtraction terms required for the Pauli form factors read

∆F42 = C4

g2Am2

16π2F2 , ∆F82 = C8

g2Am2

32π2F2 , (26)

where C4 and C8 are shown in the Appendix. For the case of the Dirac form factor up to O(q3), the

subtraction vanishes exactly due to cancellations between diagrams.

To obtain the final expression, one needs to take into account the wave-function renormalization

(WFR) given by

Zλ,B = 1 −Cλ,Bg2

AM2λ

32π2F2λm2

√4m2 − M2

λ

√4m2 − M2

λ

[(−2 + 2 ln

mµ− 3 ln

Mλ

µ

)m2

+2 M2λ ln

Mλ

m

]−2Mλ(−3m2 + M2

λ)

arctanMλ√

4m2 − M2λ

+ arctan2m2 − M2

λ√4m2M2

λ − M4λ

. (27)

Here, the subscript λ denotes π±,0, K±,0, K0 or η, and Mλ is the mass of the pseudoscalar meson

λ. The Cλ,B are shown in Tab. I, and a sum over λ is inferred. The WFR constant only multiplies

the O(q) diagrams, since it provides a correction of O(q2). Its effect on other diagrams would only

start at O(q4), beyond the order of our calculation. Note that a proper inclusion of the WFR is

required to ensure that the total baryon charge F1(0) = GE(0) is conserved.

2 Namely, we subtract 1/(4 − n) + [ln(4π) + Γ′(1) + 1]/2, where n is the dimension of dimensional regularization.

8

Cλ,B π K η

Ξ++cc /Ξ

0bb 3 2 1

3

Ξ+cc/Ξ

−bb 3 2 1

3

Ω+cc/Ω

−bb 0 4 4

3

TABLE I. Value of the coefficients Cλ,B in Eq. (27).

C. Vector mesons

The contributions to the form factors originating from the coupling to the photon through vector

mesons, Fig. 1, are

FVB1 = −CVB

FV

MV

gBv q2

q2 − M2V + iε

,

FVB2 = CVB

FV

MV

gBt q2

q2 − M2V + iε

. (28)

In these equations, B denotes the doubly charmed (beauty) baryons Ξcc (Ξbb) and Ωcc (Ωbb). The

CVB values are in Tab. II. Obviously, the mechanism of Fig. 1 does not contribute to the Pauli

and Dirac form factors as q2 → 0. Therefore, the vector mesons do not affect the charge nor the

magnetic moment of the baryons.

TABLE II. Values of CVB in Eqs. (28).

Ξ++cc /Ξ

0bb Ξ+

cc/Ξ−bb Ω+

cc/Ω−bb

ρ 1√2

- 1√2

0

ω 13√

21

3√

20

φ 0 0 −√

23

IV. RESULTS

For the numerical results presented in this section, we take mΞcc = 3.621 GeV [1], mΩcc =

3.652 GeV [80], Fπ = 92 MeV, FK = 112 MeV and Fη = 110 MeV. Furthermore, we set the

renormalization scale in the loop diagrams to 1 GeV, and the chiral limit mass m to the physical

baryon mass mB. The coupling of the pseudoscalar mesons to the doubly charmed baryons is fixed

at gA = −0.2 [80]. For Mπ, MK , Mη, the nucleon mass mN , the vector meson masses and their

widths, we use the averaged PDG values [2].

9

In order to estimate the relevant low-energy constants c8, c9, d1 and d2, we use the lattice results

from Refs. [30–32]. There, the magnetic moments and electromagnetic form factors of Ω+cc and Ξ+

cc

at different values of Q2 were obtained for different lattice configurations, and therefore different

meson and baryon masses. Since the scale of ChPT is approximately Λ ∼ 1 GeV, we take into

account for the fit the lattice results up to Q2 < 0.4 GeV2 and M2π < 0.4 GeV2, meaning a total of 34

data points. For the fit, we set the pion and baryon masses in the ChPT calculations to those given

by the lattice collaboration. For the kaon and the η meson masses, not explicit in Refs. [30–32],

we use the Gell-Mann, Oakes, and Renner relations [33, 82] taking into account that the strange

quark mass is fixed to its physical value.

A. Magnetic Moments

In Table III, we show our results for the double-charm baryon magnetic moments µB. The

vector mesons do not contribute to this observable. The tree diagram contributions are the same for

Ξ+cc and Ω+

cc. For the loop terms, we show the analytic expression of the leading-order heavy-baryon

expansion, which reproduces the findings of Ref. [28]3, and compare the numerical results in HB

with the covariant EOMS scheme. We find appreciable differences between the two schemes,

especially for Ξ++cc .

TABLE III. Contributions to µB for the double-charm baryons, split into tree-level and loop terms. The last

four columns are in units of µN . Best-fit results for the magnetic moments are shown in boldface. Lattice

estimations from the quadratic fit of Ref. [31] are in the last column.

Tree Loops HB Loop HB [µN] Loop EOMS [µN] µ [µN] Ref. [31]

Ξ++cc 2 + 2

3 c8 + 4c9 −g2

A8π

[MπmΞcc

F2π

+MKmΩcc

F2K

]-2.09g2

A −1.21g2A — —

Ξ+cc 1 − 1

3 c8 + 4c9g2

AmΞcc8π

Mπ

F2π

0.60g2A 0.80g2

A 0.37(2) 0.425(29)

Ω+cc 1 − 1

3 c8 + 4c9g2

AmΞcc8π

MKF2

K1.46g2

A 1.59g2A 0.40(3) 0.413(24)

Up to O(q3), the magnetic moments depend only on c8 and c9 and several known parame-

ters. Furthermore, the available lattice data for magnetic moments correspond to the Ω+cc and Ξ+

cc

baryons. For these two particles, c8 and c9 appear just in the combination c89 = −13c8 + 4c9. Thus,

3 There is a factor 2 discrepancy between our work and Ref. [28] in the vertex definitions involving gA, leading to an

overall factor 4 difference in the analytic heavy-baryon results. This simply translates into different values for gA

when fitting to data. Indeed, while we use the value gA = −0.2 [80], their estimate is of gA = −0.5.

10

making a fit to the 13 lattice data for magnetic moments, the only free parameter is c89, and we

can obtain an estimate for this constant: c89 = 0.32(2).4 The agreement of our µΞ+cc and µΩ+

cc results

with the simple extrapolations of the lattice data to the physical point done in Refs. [30–32] is

good considering the uncertainties.

As can be seen, the loop corrections obtained from the relativistic EOMS renormalization are

larger than in the HB approach for Ξ+cc and Ω+

cc. The main reason for this is that most of the loop

diagrams in Fig. 2 enter only at O(q4) in HB (only diagram (8) contributes at O(q3)), while in the

EOMS scheme they are all non-vanishing already at O(q3).

In these results, the LEC uncertainties are purely statistical. However, the chiral error esti-

mates, δµ, of the magnetic moments are performed as in Refs. [87, 88] and try to account for the

systematic error due to the truncation of the chiral series. For our particular case we have

δµ = max[µ(1)

(Mπ

Λ

)3

,

|µ(k) − µ( j)|

(Mπ

Λ

)3− j], 1 ≤ j ≤ k ≤ 3, (29)

where µ(i) are the magnetic moments obtained with our best-fit parameters, up to the order O(pi).

TABLE IV. Contributions to µB for the double-beauty baryons, split into tree-level and loop terms. For the

latter, we show the analytic expression for the leading-order heavy-baryon (HB) expansion, and compare

the numerical results in HB with the covariant EOMS scheme, in units of µN .

Tree Loops HB Loops HB [µN] Loops EOMS [µN]

Ξ0bb

23 c8 − 2c9 −

g2A

8π

[MπmΞbb

F2π

+MKmΩbb

F2K

]-2.11g2

A −1.92g2A

Ξ−bb −1 − 13 c8 − 2c9

g2AmΞbb

8πMπ

F2π

0.60g2A 0.44g2

A

Ω−bb −1 − 13 c8 − 2c9

g2AmΞbb

8πMKF2

K1.46g2

A 1.02g2A

We show the analogous results for the double-beauty magnetic moments in Table IV. Again,

due to the symmetry, the tree-level expressions are the same for the two baryons with the same

charge, Ξ−bb and Ω−bb. The only difference between the HB expansions of the charm triplet and the

beauty triplet are the baryon masses. We take mΞbb = 10.314 GeV and mΩbb = 10.476 GeV [83]

for the numerical calculations. Since the double-beauty baryons are substantially heavier than the

double-charm ones, a HB approach is expected to give a better approximation of the full relativistic4 In Ref. [30], only the Ξcc was studied, and there is no information on the Ωcc lattice mass needed for the loop

contributions. Thus, for these data we use the values of mΩcc given for the same lattice configurations in Ref. [31].

Similarly, in Ref. [32] only the Ωcc is studied, and we use the linear fit from Ref. [31] for the Ξcc mass, mΞcc =

3.660 GeV. The sensitivity of the results to these choices is negligible.

11

result. Indeed, for Ξ0bb the differences between HB and EOMS results become smaller. However,

for the other two baryons the differences are still large, as was the case in the double-charm sector.

In the double-beauty sector, all the magnetic moments are systematically of a smaller magnitude

when calculated in EOMS than when determined in a HB approach.5

B. Electric and magnetic radii

The electric and magnetic radii, rBE,M, measure the derivative with respect to q2 of the GB

E,M form

factors. The tree-level results for F1 and F2 are shown in Tab. V. Since the LECs c8 and c9 appear

as the combination c89, and d1 and d2 as d12 ≡ −d13 + 4d2 for both the Ω+

cc and the Ξ+cc, if we analyze

data for only these latter particles, the number of degrees of freedom from the chiral Lagrangian

is reduced to just two.

As it happens for light baryons, we expect the vector mesons to be relevant for these observ-

ables. As simplifying conjectures, we assume the OZI rule and ideal mixing, which implies that

the vector-meson contributions to the Ξcc form factors come from ρ and ω and φ is the only vector-

meson contributing in the Ωcc case. Still, this amounts to four unknown parameters: gΞccv , gΞcc

t , gΩccv

and gΩcct . In the magnetic form factors, gΞcc

v and gΞcct appear as the combination gΞcc

v − gΞcct ≡ gΞcc

vt .

Therefore, in order to separate them, one needs additionally information on the electric form fac-

tors. The same is true for gΩccv and gΩcc

t . Since we have lattice results on GΩ+cc

E , but not on GΞ+cc

E , we

can obtain the values of gΩccv and gΩcc

t separately, but only the combination gΞccvt can be determined.

Fitting the full set of lattice GE,M(Q2) results with Q2 < 0.4 GeV2 and M2π < 0.4 GeV2 from

[30–32], we obtain c89 = 0.32(2), d12 = (−0.12 ± 0.11), gΞccvt = −10.4(7), gΩcc

v = (−3.7 ± 3.9) and

gΩcct = (−18.9 ± 4.2), with a reduced χ2 ≈ 2.1. The value for the parameter c89 coincides with the

determination obtained using only the magnetic moments. In Fig. 3, we show our results for the

central values of the LECs, compared with the corresponding lattice data. The agreement is fair

in the range of Q2 considered. Notice, however, the large uncertainties, not reflected in the figure,

in some of the LECs. The quality of this Q2 description depends heavily on the inclusion of the

vector-meson effects. Indeed, a fit which only includes chiral Lagrangian terms leads to a much

higher reduced χ2 ≈ 9.9.6 The alternative to the explicit vector-meson consideration would be a5 It should be mentioned that the EOMS results are not simply obtained by the substitution of the double-charm

baryon masses in the loops by their double-beauty partners. Since the beauty triplet contains a baryon of neutral

charge, some of the contributions from diagram (4) in Fig. 2 to the charm triplet now vanish.6 Furthermore, the numerical values of c89 and d12 would change drastically to 0.09(3) and −0.02(2), respectively.

12

TABLE V. Tree-level contributions to the double charm F1 and F2 from the chiral Lagrangian (χPT) and

vector-meson diagrams (VM).

χPT F1 VM F1 χPT F2 VM F2

Ξ++cc 2 − 4d1

3 t − 8d2t −∑

V=ρ,ω,φ

CVBFV tMV

gΞccv

t − M2V

23 c8 + 4c9 +

4d13 t + 8d2t

∑V=ρ,ω,φ

CVBFV tMV

gΞcct

t − M2V

Ξ+cc 1 +

2d13 t − 8d2t −

∑V=ρ,ω,φ

CVBFV tMV

gΞccv

t − M2V

− 13 c8 + 4c9 −

2d13 t + 8d2t

∑V=ρ,ω,φ

CVBFV tMV

gΞcct

t − M2V

Ω+cc 1 +

2d13 t − 8d2t −

∑V=ρ,ω,φ

CVBFV tMV

gΩccv

t − M2V

− 13 c8 + 4c9 −

2d13 t + 8d2t

∑V=ρ,ω,φ

CVBFV tMV

gΩcct

t − M2V

TABLE VI. Tree-level contributions to the double-beauty F1 and F2, from the chiral Lagrangian (χPT) and

vector-meson diagrams (VM).

χPT F1 VM F1 χPT F2 VM F2

Ξ0bb −

4d13 t + 4d2t −

∑V=ρ,ω,φ

CVBFV tMV

gΞbbv

t − M2V

23 c8 − 2c9 +

4d13 t − 4d2t

∑V=ρ,ω,φ

CVBFV tMV

gΞbbt

t − M2V

Ξ−bb −1 +2d13 t + 4d2t −

∑V=ρ,ω,φ

CVBFV tMV

gΞbbv

t − M2V

−13 c8 − 2c9 −

2d13 t − 4d2t

∑V=ρ,ω,φ

CVBFV tMV

gΞbbt

t − M2V

Ω−bb −1 +2d13 t + 4d2t −

∑V=ρ,ω,φ

CVBFV tMV

gΩbbv

t − M2V

−13 c8 − 2c9 −

2d13 t − 4d2t

∑V=ρ,ω,φ

CVBFV tMV

gΩbbt

t − M2V

calculation at a higher chiral order.

From these electric and magnetic form factors we can extract the corresponding radii. We ob-

tain < r2E >Ω+

cc= 0.00(10)(1) fm2, < r2M >Ξ+

cc= 0.18(2)(1) fm2 and < r2M >Ω+

cc= 0.147(92)(1) fm2.

In this case, the error estimates arise mostly due to the large uncertainties of the fitted parameters.

The second number in parentheses corresponds to the uncertainty coming from the chiral trunca-

tion calculated as in Eq. (29). These values for the radii support the expectations of the double

heavy baryons being substantially smaller than the single heavy ones or the light baryons.

Within their large uncertainties, these radii are compatible with the lattice results from Refs. [30–

32]. It is worth mentioning, though, that the extrapolation to the physical point with a linear or

quadratic fit, as done in Refs. [30–32], is not expected to give the correct results at the physical

point, since the non-trivial behavior due to chiral loops cannot be taken into account. Specifi-

cally, the pion cloud effects, very relevant at low Q2 values, lead to an unavoidable logarithmic

dependence on the pion mass and, therefore, to a rapid curvature of the radii when approaching

the physical mass, absent in the extrapolations of Refs. [30–32].

13

1

1.2

1.4

1.6

1.8

2

2.2

-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6

GM

Q2 [GeV2]

Ξcc+

ChPT, Mπ=0.57 GeV

ChPT, Mπ=0.41 GeV

ChPT, Mπ=0.3 GeV

ChPT, physical pointL 1, Mπ=0.57 GeV

L 1, Mπ=0.41 GeV

L 1, Mπ=0.3 GeV

L 2, Mπ=0.57 GeV

L 2, Mπ=0.41 GeV

L 2, Mπ=0.3 GeV

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6

GM

Q2 [GeV2]

Ωcc+

ChPT, Mπ=0.57 GeV

ChPT, Mπ=0.41 GeV

ChPT, Mπ=0.3 GeV


L 1, Mπ=0.41 GeV

L 1, Mπ=0.3 GeV

L 3, Mπ=0.156 GeV

0.9

0.92

0.94

0.96

0.98

1

0 0.1 0.2 0.3 0.4 0.5 0.6

GE

Q2 [GeV2]

Ωcc+

ChPT, Mπ=0.57 GeV

ChPT, Mπ=0.41 GeV

ChPT, Mπ=0.3 GeV


L 1, Mπ=0.41 GeV

L 1, Mπ=0.3 GeV

FIG. 3. Magnetic and electric form factor fit results compared to the lattice data from Refs. [31], [30] and

[32], denoted as L1, L2 and L3, respectively. We show the results for four different pion mass configura-

tions: at the physical point (dotted line), at Mπ = 0.3 GeV (blue), Mπ = 0.41 GeV (red), Mπ = 0.57 GeV

(green). The data points for Mπ = 0.3 GeV(Mπ = 0.57 GeV) were shifted to the left(right) for better

visibility, but they all correspond to the same Q2 value as those for Mπ = 0.41 GeV.

For completeness, we also show the analytical expressions of the tree-level contributions to the

form factors for the double beauty baryons in Tab. VI.

V. SUMMARY

In this work, we have studied the electromagnetic form factors of the doubly heavy baryons

within the framework of covariant ChPT up to the chiral order O(q3). We have applied the co-

variant extended on-mass-shell renormalization scheme to generate a systematic power counting.

The vector meson contributions have also been included to describe the behavior of form factors

at momentum transfer different from zero.

14

From the Lagrangian constructed in Eqs. (1), (2), (3) and (16), we have obtained the Dirac

and Pauli form factors, from which we have extracted magnetic moments, charge and magnetic

radii. We have also compared our results with those extracted in HBChPT. We have found that

the differences between the loop term contributions to µB in HBChPT and EOMS approaches are

around 10% ∼ 70% for the double charm sector and 10% ∼ 40% for the double beauty sector.

In order to obtain first estimates for the LECs, we have fitted our model to the available lattice

results. We have found that the vector meson contribution is necessary for a good description.

However, the lattice data are not sufficient to fix all the parameters. Instead, since for Ω+cc and Ξ+

cc

the LECs c8 and c9 appear as c89 = −13c8 + 4c9, and d1 and d2 as d12 = −1

3d1 +4d2, we fit only these

combinations, thus reducing the amount of fitting parameters. We also assume the validity of the

OZI rule and ideal mixing to further constrain the vector meson contributions. With these caveats,

we obtain c89 = 0.32(2), d12 = −0.12(11), gΞccvt = −10.4(7), gΩcc

v = −3.7±3.9 and gΩcct = −18.9±4.2

with a small χ2. As a consequence, we deduce the values µΞ+cc = 0.37(2)µN and µΩ+

cc = 0.40(3)µN

for the magnetic moments.

Once more lattice data are available, a determination of the separate LECs d1, d2, c8 and c9 and

of the vector-meson parameters will be possible, together with a better extraction of charge and

magnetic radii of the doubly-heavy baryons.

ACKNOWLEDGMENTS

This research is supported by MINECO (Spain) and the ERDF (European Commission) grants

No. FIS2014-51948-C2-2-P, FIS2017-84038-C2-2-P, and SEV-2014-0398, and by the General-

itat Valenciana under Contract PROMETEOII/2014/0068. It is also supported by the Deutsche

Forschungsgemeinschaft DFG and by the National Science Foundation for Young Scientists of

China under Grants NO. 11705069 and the Fundamental Research Funds for the Central Univer-

sities.

APPENDIX: ANALYTICAL RESULTS AND LOOP INTEGRALS

The unrenormalized Dirac form factors corresponding to the diagrams in Fig. 2 read

F41 = C4

g2A

4F2

(pi · p f + m2)IBB(q2) + IB + M2

[4I00

λBB(q2) + P2IPPλBB(q2)

15

+q2IqqλBB(q2) +

132π2

]+ 2(m + mB)2I00

λBB(q2) − (m + mB)2[4I00λBB(q2)

+P2IPPλBB(q2) + q2Iqq

λBB(q2) +1

32π2 ] − P2[IBB(q2) + 2M2IPλBB(q2)]

+4[pi · p f I00λBB(q2) + pi · Pp f · PIPP

λBB(q2) + pi · qp f · qIqqλBB(q2)]

+4mB(m + mB)[I00λBB(q2) + q2Iqq

λBB(q2)] + 8m2B(m + mB)2IPP

λBB(q2)

+2(m + mB)mB[IBB(q2) + 2M2IPλBB(q2)] − 8(m + mB)mB

×[I00λBB(q2) + pi · PIPP

λBB(q2) − pi · qIqqλBB(q2)]

,

F51 = C5

18F2 Iλ,

F61 = F7

1 = C67g2

A

4F2

−IB − M2IλB(m2

B) − (m − mB)mBI pλB(m2

B),

F81 = C8

g2A

8F2

−4mB(m + mB)I00

λλB(q2) − 2q2I00λλ(q

2) + 2(m2B − m2)I00

λλB(q2)− F8

2 ,

F91 = C9

14F2 q2I00

λλ(q2), (30)

F42 = −C4

g2A

4F2

4(m + mB)2m2

BIPPλBB(q2) + (m + mB)mB[IBB(q2) + 2M2IP

λBB(q2)]

−4(m + mB)mB [I00λBB(q2) + pi · PIPP

λBB(q2) − pi · qIqqλBB(q2)]

,

F52 = F6

2 = F72 = 0,

F82 = C8

g2A

8F2

8m2

B(m + mB)[2mBIPPλλB(q2) + mBIP

λλB(q2)] − 8m3B(m + mB)IP

λλB(q2)

−2mB(m + mB)[4I00λλB(q2) + (4m2

B − q2)IPPλλB(q2) + q2Iqq

λλB(q2) +1

32π2 ]

−8(m + mB)mB[I00λλB(q2) + (2m2

B − q2/2)IPPλλB(q2) +

q2

2IqqλλB(q2) + m2

BIPλλB(q2)]

+4(m + mB)mB[(2m2B −

q2

2)IPλλB(q2) +

q2

4IλλB(q2)]

+4(m + mB)mB[IλB(m2B) + M2(IP

λλB(q2) +12

IλλB(q2))]

−2mB[mBI pλB(m2

B) + 2M2mBIPλλB(q2)] − 4mB(m2

B − m2)[2mBIPPλλB(q2) + mBIP

λλB(q2)]

+4mB[mBI00λλB(q2) + 2mB(2m2

B − q2/2)IPPλλB(q2) +

mBq2

2IPλλB(q2)]

,

F92 = 0.

In these equations, the lower index of the F’s is 1(2) for the Dirac(Pauli) form factor. The upper

index corresponds to the Fig. 2 label. The values of the constants C4, C5, C67, C8 and C9 are

listed in Tab. VII for the different baryons and mesons. A sum over the mesons (λ-subindex) is

understood. Also, m is the doubly heavy baryon mass in the chiral limit, mB is the corresponding

16

physical mass, qµ is the four momentum of the photon, piµ and p fµ are the four momenta of the

initial and final doubly heavy baryons, and Pµ is defined as Pµ = piµ + p fµ.

TABLE VII. Values of the parameters C4, C5, C67, C8, C9 of Eq. (30).

C4 C5 C67 C8 C9

π K η π K η π K η π K η π K η

Ξ++cc 4 2 2

3 -4 -4 0 -2 -2 0 4 4 0 4 4 0

Ξ+cc 5 2 1

3 4 0 0 2 0 0 -4 0 0 -4 0 0

Ω+cc 0 6 4

3 0 4 0 0 2 0 0 -4 0 0 -4 0

Ξ0bb -2 -2 0 -4 -4 0 -2 -2 0 4 4 0 4 4 0

Ξ−bb -1 -2 − 13 4 0 0 2 0 0 -4 0 0 -4 0 0

Ω−bb 0 -2 − 43 0 4 0 0 2 0 0 -4 0 0 -4 0

The loop integrals of Eq. (30) are given below.

IB = iµ4−n∫

dnk(2π)n

1k2 − m2 + iε

= 2m2

R +1

32π2 lnm2

µ2

,

Iλ = iµ4−n∫

dnk(2π)n

1k2 − M2

λ + iε= 2M2

λR +M2

λ

(4π)2 lnM2

λ

µ2 ,

IλB(p2) = iµ4−n∫

dnk(2π)n

1(p − k)2 − m2 + iε

1k2 − M2

λ + iε

= 2

R +1

32π2 lnm2

µ2

+

1(4π)2

(−1 +

p2 − m2 + M2λ

2p2 lnM2

λ

m2 + f0

),

Iλλ(q2) = iµ4−n∫

dnk(2π)n

1(q + k)2 − M2

λ + iε1

k2 − M2λ + iε

= 2R +1

16π2

[1 + 2 ln

Mµ

+ f ′0

(q2

M2

)],

IBB(q2) = iµ4−n∫

dnk(2π)n

1(q + k)2 − m2 + iε

1k2 − m2 + iε

= 2

R +1

32π2 lnm2

µ2

+

116π2

[1 + f ′0

(q2

m2

)],

Iµλλ(q) = iµ4−n∫

dnk(2π)n

kµ

[(q + k)2 − M2λ + iε](k2 − M2

λ + iε)= qµIq

λλ(q2),

Iµνλλ(q) = iµ4−n∫

dnk(2π)n

kµkν

[(q + k)2 − M2λ + iε](k2 − M2

λ + iε)= gµνq2I00

λλ(q2) + qµqνIqq

λλ(q2),

IµλB(p) = iµ4−n∫

dnk(2π)n

kµ

(p − k)2 − m2 + iε1

k2 − M2λ + iε

= pµI pλB(p2), (31)

where

R =1

(4π)2

[−

1ε−

12

(ln

4πm2 + 1 − γ

)],

17

f0 =

√Θ2−∆2

p2 arccos −∆Θ, −1 < ∆

Θ< 1,

√∆2−Θ2

2p2 ln ∆+√

∆2−Θ2

∆−√

∆2−Θ2, ∆

Θ< −1,

√∆2−Θ2

2p2 ln ∆+√

∆2−Θ2

∆−√

∆2−Θ2− iπ

√∆2−Θ2

p2 , ∆Θ> 1,

0, ∆Θ

= ±1,

f ′0(x) =

−2 − σ ln

(σ−1σ+1

), x < 0,

−2 + 2√

4x − 1 arccot

(√4x − 1

), 0 ≤ x ≤ 4,

−2 − σ ln(σ−1σ+1

)− iπσ, x > 4,

Iqλλ(q

2) = −12

Iλλ(q2),

Iqqλλ(q

2) =1t

[13

Iλ +13

(t − M2λ)Iλλ(q

2) +1

144π2

(3M2

λ −t2

)],

I00λλ(q

2) =1t

[16

Iλ +1

12(4M2

λ − t)Iλλ(q2) −1

8π2

(M2

λ

6−

t36

)],

I pλB(p2) =

12p2 [IN − Iλ + (p2 − m2 + M2

λ)IλB(p2)]. (32)

Here ∆ = p2 − m2 − M2λ, Θ = 2mMλ, σ =

√1 − 4

x with x < [0, 4], and pµ is an arbitrary four

momentum (p2 , 0), which, in our case, corresponds to the four momentum of the initial or final

doubly heavy baryon.

Furthermore, in the following loop integrals corresponding to three internal lines, the on-shell

condition (i.e. p2i = p2

f = m2B) is assumed.

IλλB(q2) = iµ4−n∫

dnk(2π)n

1(pi − k)2 − m2 + iε

1k2 − M2

λ + iε1

(k + q)2 − M2λ + iε

,

IµλλB(q, P) = iµ4−n∫

dnk(2π)n

kµ

(pi − k)2 − m2 + iε1

k2 − M2λ + iε

1(k + q)2 − M2

λ + iε,

IµνλλB(q, P) = iµ4−n∫

dnk(2π)n

kµkν

(pi − k)2 − m2 + iε1

k2 − M2λ + iε

1(k + q)2 − M2

λ + iε,

IλBB(q2) = iµ4−n∫

dnk(2π)n

1(k − pi)2 − m2 + iε

1k2 − M2

λ + iε1

(k − p f )2 − m2 + iε,

IµλBB(q, P) = iµ4−n∫

dnk(2π)n

kµ

(k − pi)2 − m2 + iε1

k2 − M2λ + iε

1(k − p f )2 − m2 + iε

,

IµνλBB(q, P) = iµ4−n∫

dnk(2π)n

kµkν

(k − pi)2 − m2 + iε1

k2 − M2λ + iε

1(k − p f )2 − m2 + iε

. (33)

The tensor integrals defined above can be reduced to the scalar ones as follows

IµλBB(q, P) = PµIPλBB(q2),

IµνλBB(q, P) = gµνI00λBB(q2) + PµPνIPP

λBB(q2) + qµqνIqqλBB(q2),

18

IµλλB(q, P) = PµIPλλB(q2) −

12

qµIλλB(q2),

IµνλλB(q, P) = gµνI00λλB(q2) + PµPνIPP

λλB(q2) + qµqνIqqλλB(q2) −

qµPν + Pµqν

2IPλλB(q2), (34)

where

IPλλB(q2) =

12(4m2

B − q2)[(2M2

λ − q2)IλλB(q2) + 2IλB(m2B) − 2Iλλ(q2)],

I00λλB(q2) =

18

[2IλB(m2

B) + (4M2λ − q2)IλλB(q2) − 2(2M2

λ − q2)IPλλB(q2) −

18π2

],

IPPλλB(q2) =

18(4m2

B − q2)

[−2IλB(m2

B) + 4I pλB(m2

B) − (4M2λ − q2)IλλB(q2) + 6(2M2

λ − q2)IPλλB(q2) +

18π2

],

IqqλλB(q2) =

18q2

[2IλB(m2

B) − 4I pλB(m2

B) − (4M2λ − 3q2)IλλB(q2) + 2(2M2

λ − q2)IPλλB(q2) +

18π2

],

IPλBB(q2) =

14m2 − q2 [M2

λIλBB(q2) − IλB(m2B) + IBB(q2)],

I00λBB(q2) =

12

[M2

λ(IλBB(q2) − IPλBB(q2)) +

12

IBB(q2) −1

32π2

],

IPPλBB(q2) =

12(4m2

B − q2)

[3IP

λBB(q2) − IλBB(q2)]M2λ − I p

λB(m2B) +

12

IBB(q2) +1

32π2

,

IqqλBB(q2) =

12q2

[IPλBB(q2) − IλBB(q2)]M2

λ + I pλB(m2

B) −12

IBB(q2) +1

32π2

. (35)

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Astrid N. Hiller BlinAstrid N. Hiller Blin Institut fur¨ Kernphysik & PRISMA Cluster of Excellence, Johannes Gutenberg Universitat,¨ D-55099 Mainz, Germany Zhi-Feng Suny Research