Master Final

S

Baryon spectrum and the proton spin from Lattice QCD

Christos Kallidonis

Supervisor Prof. Constantia Alexandrou

30 May 2013

o The Lattice QCD action for gluons and fermions o Simulation details o Baryon Spectrum

o Overview of calculations o Fixing the lattice spacing o Lattice artifacts Isospin symmetry breaking o Chiral extrapolations Comparison of lattice results

o Proton Spin o Generalized Parton Distributions and Generalized Form Factors o Extrapolations of spin observables for the proton

o Summary

OUTLINE

The Lattice QCD Action

The Lattice QCD Action Gluons Wilson plaquette action

Plaquette

Link variables

Lattice action

2. QCD on the lattice

The link variables U(x) and U(x+ae) are shown schematically in Fig. 2.1. In a typicalformulation of QCD on the lattice, link variables U(x) are chosen as the basic objects todescribe gluons instead of the usual gluon fields. There are four link variables for each lattice

site, which correspond to the four dimensions one can move on the lattice. Therefore, the

number of the degrees of freedom of the fields remain unchanged when switching to link

variables. In the following sections it will become clear how the concept of link variables

can be useful in formulating lattice QCD.

Figure 2.1: The link variables U(x) and U(x + ae)2.2 Gluons on the lattice

The gluon action in the continuum is given by

SG = 12g2

Tr d4xG(x)G(x) . (2.6)This action respects invariance under gauge transformations. Adopting this property as the

construction principle for the action, we will formulate the most simple non trivial gauge

invariant expression on the lattice using the link variables as introduced in Eq. 2.4. In order

to perform this task we first have to determine the gauge properties of the link variable itself.

If the lattice spacing is suciently small one can approximate a single link U(x), pointingin direction by

U(x) exp(igG(x)) 1 igG(x) . (2.7)On the other hand, even if a is not small enough for Eq. 2.7 to be a good approximation, one

can divide the integral over the path between x and x+ae into N small segments of length = aN and repeatedly apply Eq. 2.7, exploiting this way the path ordering property,

U(x) exp[igG(x)] exp[igG(x + e)] exp[igG(x + (a )e)] . (2.8)For each single factor in Eq. 2.8 we can use Eq. 2.7, therefore

U(x) expig aNG(x) 1 igG(x) . (2.9)

The gauge-transformed U (x) readsU (x) 1 igG(x) , (2.10)

16


plaquette and is symboled by P . It is defined as

P(x) = [U(x)U (x + ae)U (x + ae + ae)U (x + ae)] == U(x)U (x)U (x + ae)U (x + ae) . (2.16)It is easy to show that the plaquette transforms as

P(x) U P (x) = U(x)P(x)U (x) , (2.17)which immediately implies that

Tr P (x) = Tr [P(x)] . (2.18)Thus, the trace of the plaquette is a good candidate for describing the gluon action on the

lattice.

Figure 2.3: Plaquette P in the -plane.

The next step is to find the continuum analogue of the plaquette, i.e P(x) , a 0. Atfirst we consider the links in the plaquette separately. Assuming that a is suciently small

we approximate the line integral in the links as in Eq. 2.9. We take the value of G(x) inthe middle of the line and we keep O(a) terms in the expansion. One gets

U(x) exp iagG x + a2e exp iag G(x) + a

2@G(x) , (2.19)

and similarly for the other links

U (x) exp iag G(x) + a2@G(x) ,U (x + e) exp iagG x + ae + e2 exp iag G(x) + a@G(x) + a

2@G(x) ,

U (x + e) exp iag G(x) + a@G(x) + a2@G(x) . (2.20)

18






lattice.





2@G(x) , (2.19)



2@G(x) ,


18


The products of the above links yield

U(x)U (x)U (x + ae)U (x + ae) == expiga2 [@G(x) @G(x) + ig [G(x),G(x)]] +O(a4) . (2.21)Comparing this result with Eq. 1.3 one can see that the exponent is the gluon tensor, so

finally we have that

Pa0= exp iga2G(x) . (2.22)

Comparing this with the gluon action in the continuum, we can construct the corresponding

lattice version (SW , W stands for Wilson) that coincides with the continuum action in the

limit a 0SW = 6

g2 1 13R{Tr (P)} . (2.23)

The shortcut symbol denotes a single elementary plaquette on the lattice and symbol Rdenotes the real part. The sum extends over all possible plaquettes of the four dimensional

lattice. There are 6 planes in a four-dimensional lattice(x-y, x-z, x-t, y-z, y-t, z-t), thus

there are 6(Nx)3Nt elementary squares. One can replace the real part of the trace byR{Tr(P)} = 12Tr P + P . The Hermitian conjugate of the plaquette can be seen as theplaquette P with reversed orientation of the links (see Fig. 2.3). One can also generalizeEq. 2.23 to an arbitrary group SU(N), or equivalently to an arbitrary number of colour

flavours,

SSU(N)W = 1 1NR{Tr (P)} , (2.24)where in this case P contains appropriate unitary N N matrix-valued link variables and = 2Ng2. Many lattice QCD calculations use the SU(2) instead of the SU(3) groupbecause there are 3 gluons in SU(2), instead of 8 in SU(3) (see Appendix B). This reduces

the computational eorts significantly, while many features like asymptotic freedom and

confinement still exist for SU(2).

2.3 Fermions on the lattice

2.3.1 Discretization of free fermions

The discretization of the fermion part of the action is not as straight forward as in the

gluon part. Let us consider a naive discretization of the Dirac operator at first. The first

step has already been done in Sec. 2.1, where we introduced the four-dimensional hypercubic

lattice. We denote the set of all vectors of Eq. 2.1 by . Thus, now the spinors are placed

only on the lattice points, i.e. our fermion degrees of freedom are

(x) , (x) , x , (2.25)19

Continuum






lattice.





2@G(x) , (2.19)



2@G(x) ,


18






lattice.





2@G(x) , (2.19)



2@G(x) ,


18

Fermions Twisted mass formulation Nf = 2

3. Improved actions - Twisted mass fermions

and similarly, the rotated pseudoscalar and scalar densities are given by

Pa 5 a2 = P

a , a = 1,2cos(!)P 3 + i2 sin(!)S0 , a = 3 (3.19)

S0 = cos(!)S0 + 2i sin(!)P 3 . (3.20)It is easy to verify that the rotated currents and densities satisfy the PCAC and PCVC

relations of Eq. 3.14 with the transformed parameters mq and q of Eq. 3.10. Moreover,

the PCAC and PCVC relations, also referred to as Ward identities, are assumed their

standard form

@Aa = 2MPa ,@Va = 0 , (3.21)

if ! is related to the mass parameters as in Eq. 3.12. The PCAC and PCVC masses that

appear in Eq. 3.14 are the two components of the physical mass which is given by the polar

mass M (Eq. 3.7 and Eq. 3.21). A particular case is when one of the two masses vanishes;

then the physical masss is given by the non-vanishing mass. More specifically, it is of special

interest when one works at full twist, also called maximal twist. Full twist corresponds at

the classical level to mq = 0 or, equivalently, ! = 2. In that case the role of the physicalmass is fully played by the twisted mass q.

3.2.2 Wilson Twisted mass QCD on the lattice

As already stated, the lattice action is of the form

S[,, U] = SG[U] + SF [,, U] , (3.22)where we now work in the twisted basis and SG[U] is the gluon part of the action, Eq. 2.23.It is important that the formulation of twisted mass on the lattice should incorporate the

Wilson term as well since we still want the doublers to vanish on the lattice. Therefore, the

fermion action SF [,, U] will be of the same form as Eq. 2.54. In a notation where themass term m0 is written explicitly and including the twisted mass term, the fermion action

is given by

SF [,, U] = a4x(x) DW +m0 + iq5 3(x) , (3.23)

where the Wilson operator DW in correspondence with Eq. 2.55 can also be written as

DW = 12( + ) ar2 . (3.24)

33

twisted mass term


where D is the usual covariant derivative and 3 is the third Pauli matrix acting in flavour

space. The explicit form of the doublet is

a(x) = ua(x)da(x) , (3.4)where as always the Greek letters are Dirac indices and latin letters are colour indices. In

addition, writing explicitly all the indices in the action we have

SF [,,G] = d4xa,i(x) ()(D)abij + (mq)abij++ iq(5)( 3)ijabb,j(x) . (3.5)In this notation, i and j denote elements of the doublet , i.e. they are flavour indices. The

mass term of the twisted mass action can be written as

mq + iq5 3 =Mei53 , (3.6)where

M =m2q + 2q (3.7)is called the polar mass. Up to now the tmQCD action is just a rewriting of the standard

QCD action in a dierent basis. In fact, performing the following transformation

= ei!532 , = ei!532 , (3.8)

where ! is the so-called twist angle, the action form is left invariant, but the mass term

needs to be transformed into

Mei(!)53 . (3.9)Since the transformation of Eq. 3.8 acts as a two-dimensional rotation at an angle !, Eq.

3.9 is equivalent with the transformation

mqq = cos(!) sin(!) sin(!) cos(!) mqq (3.10)of the mass parameters mq and q.

Particularly, the standard QCD action for Nf = 2 degenerate quarks (M is the polarmass)

SF [ , ,G] = d4x (D +M) (3.11)is obtained from the twisted basis (Eq. 3.3) if the twist angle ! appearing in Eq. 3.8 satisfies

31


where D is the usual covariant derivative and 3 is the third Pauli matrix acting in flavour

space. The explicit form of the doublet is

a(x) = ua(x)da(x) , (3.4)where as always the Greek letters are Dirac indices and latin letters are colour indices. In

addition, writing explicitly all the indices in the action we have

SF [,,G] = d4xa,i(x) ()(D)abij + (mq)abij++ iq(5)( 3)ijabb,j(x) . (3.5)In this notation, i and j denote elements of the doublet , i.e. they are flavour indices. The

mass term of the twisted mass action can be written as

mq + iq5 3 =Mei53 , (3.6)where

M =m2q + 2q (3.7)is called the polar mass. Up to now the tmQCD action is just a rewriting of the standard

QCD action in a dierent basis. In fact, performing the following transformation

= ei!532 , = ei!532 , (3.8)

where ! is the so-called twist angle, the action form is left invariant, but the mass term

needs to be transformed into

Mei(!)53 . (3.9)Since the transformation of Eq. 3.8 acts as a two-dimensional rotation at an angle !, Eq.

3.9 is equivalent with the transformation

mqq = cos(!) sin(!) sin(!) cos(!) mqq (3.10)of the mass parameters mq and q.

Particularly, the standard QCD action for Nf = 2 degenerate quarks (M is the polarmass)

SF [ , ,G] = d4x (D +M) (3.11)is obtained from the twisted basis (Eq. 3.3) if the twist angle ! appearing in Eq. 3.8 satisfies

31

Physical basis: tan! = qm0

(1)

1

, Maximal twist at ! = 2m0 = 0 (1)

1

Nf = 2+1+1

6. Baryon spectrum

strange and charm quarks is introduced, described by the action [35, 46]

S(h)F (h),(h), U = a4x(h)(x)DW [U] +m0,h + i5 1 + 3(h)(x) (6.6)

where m0,h is the bare untwisted quark mass for the heavy doublet, is the bare twisted

mass along the 1 direction and is the mass splitting in the 3 direction. The quark fields

for the heavy quarks in the physical basis are obtained from the twisted basis through the

transformation

(h)(x) = 1211 + i 15(h)(x), (h)(x) = (h)(x) 1

211 + i 15 . (6.7)

Throughout, unless otherwise stated, the quark fields will be understood as physical fields,

, in particular when we define the baryonic interpolating fields.

The form of the fermion action in Eq. 6.2 breaks parity and isospin at non-vanishing

lattice spacing at O(a2) [31], as also discussed in Chapter 3. Maximally twisted Wilsonquarks are obtained by setting the untwisted quark mass m0 to its critical value mcr, while

the twisted quark mass parameter is kept non-vanishing in order to work away from the

chiral limit. We exploit the crucial advantage of the twisted mass formulation, namely we

tune the bare untwisted quark mass m0 to its critical value mcr, thus gaining automaticO(a) improvement in all physical observables [31, 35]. In practice, we implement maximaltwist of Wilson quarks by tuning to zero the bare untwisted current quark mass, commonly

called PCAC mass, mPCAC [58, 59], which is proportional tom0mcr up to O(a) corrections.A convenient way to evaluate mPCAC is through

mPCAC = limta1 x@4Ab4(x, t)P b(0)xP b(x, t)P b(0) b = 1,2 , (6.8)

where Ab = 5 b2 is the axial vector current and P b = 5 b2 is the pseudoscalar densityin the twisted basis. The large ta limit is required in order to isolate the contribution ofthe lowest-lying charged pseudoscalar meson state in the correlators of Eq. 6.8. This way of

determining mPCAC is equivalent to imposing on the lattice the validity of the axial Ward

identity (first expression of Eq. 3.14), between the vacuum and the charged zero three-

momentum one-pion state. When m0 is taken such that mPCAC vanishes, this Ward identity

expresses isospin conservation, as it becomes clear by rewriting it in the physical quark

basis. The value of mcr is determined at each value at the lowest twisted mass used in our

simulations, a procedure that preserves O(a) improvement and keeps O(a2) small [58, 59].A tuning of the bare twisted strange and charm quark masses is performed and these

two heavy quarks are added as Osterwalder-Seiler valence quarks. In the twisted mass

approach this is realized by introducing two additional doublets of strange and charm quarks,

(s) = (s+, s) and (c) = (c+, c), whose action is the same as for the doublet of light quarks,Eq. 6.2, taking m0 to be equal to the critical mass determined in the light sector. With

61

+ _ Automatic improvement

cut-o eects no operator improvement needed

Parity and Isospin symmetry breaking up to (restored in the continuum)

! = 2m0 = 0 (1)O(a)O(a2)

1

! = 2m0 = 0 (1)O(a)O(a2)

1

! = 2m0 = 0 (1)O(a)O(a2)

1

The Lattice QCD Action

R. Frezzotti and G.C. Rossi. Twisted mass lattice QCD with mass nondegenerate quarks. Nucl.Phys.Proc.Suppl., 128:193202, 2004. doi: 10.1016/S0920-5632(03)02477-0.

R. Frezzotti and G.C. Rossi. Chirally improving Wilson fermions. 1. O(a) improvement. JHEP, 0408: 007, 2004. doi: 10.1088/1126-6708/2004/08/007.

Simulation details

Simulation Details Baryon Spectrum

17 simulations with Nf = 2+1+1 dynamical twisted mass fermions 3 values of the lattice spacing 4 lattice volumes Pion mass range 210 470 MeV

Proton spin 11 simulations with Nf = 2 and 2 Nf = 2+1+1 dynamical twisted mass

fermions Results for pion mass as low as 210 MeV

Computer Resources

Our computations demand sophisticated programming codes which run on state-of-the-art highly parallel computer clusters

need access, granted through competitive proposals update of existing codes local workshops organized for user training

Non-negligible human and computer time devoted in porting, tuning AND testing of lattice QCD applications on such machines

Simulation details

JUQUEEN, IBM Blue Gene/Q, Juelich Supercomputing Center, Germany ranked 5th worldwide as of November 2012, according to www.top500.org theoretical peak performance 5 PFlop/s ~ 1015 arithmetic operations per second (!!!) 393216 cores and 393216 GB of memory

(conventional computers: 40 90 Gop/s, 4 cores and 4GB of memory)

CyTera, IBM iDataPlex M3, CaSToRC, The Cyprus Institute 116 twelve-core compute nodes, 18 of which with dual NVidia M2070 GPUs theoretical peak performance 30.5 Top/s 4700 GB of memory, 300 TB of storage

Simulation details

Baryon Spectrum

does Lattice QCD reproduce already known results for the masses of baryons? Can it safely predict masses of baryons still to be observed?

are the lattice artifacts small and under control?

Overview of calculations

Interpolating elds quantum numbers of the baryon in interest Overview of calculations

Baryon Spectrum

Outline Lattice QCD Charmed mesons Charmed baryons Summary

SU(4) representations+c

+c0c

udc

uds

usc

dsc

Flavour symmetry is not respected.

Simplest way to see which baryons should exist.

SU(4): 4 4 4 = 20 20 20 4!!! = !!! !!

!!!!

!!!

Gunnar Bali (Uni Regensburg) Charmed hadrons 22 / 30

Outline Lattice QCD Charmed mesons Charmed baryons Summary

SU(4) representations+c

+c0c

udc

uds

usc

dsc

Flavour symmetry is not respected.

Simplest way to see which baryons should exist.

SU(4): 4 4 4 = 20 20 20 4!!! = !!! !!

!!!!

!!!

Gunnar Bali (Uni Regensburg) Charmed hadrons 22 / 30

4 quark avours

Baryons (qqq)

SU(3) sub-groups of SU(4)

20plet of spin-1/2 baryons 20plet of spin-3/2 baryons

20 = 8 6 3 320 =10 6 31

Calculation of two-point correlation functions

Major part of the work was to write programming code for all baryons

Baryon Spectrum Overview of calculations

4. Lattice Techniques

general structure of an operator for a baryon consisting of quark flavours q1, q2 and q3 is

J X = abc (qa1)TAqb2Bqc3 , (4.15)or even a sum of such terms after permuting q1, q2 and q3. Three possible combinations of

A and B which give rise to quantum numbers JP = 12+ are (A,B) = (C5,1), (C,5) and(i4C5,1). Baryons with J = 32 can be represented with combinations (A,B) = (Cj,1).The reason for this is that j matrices give diquarks with J = 1, and together with the thirdquark there are contributions to spin which give states with J = 32 and J = 12 [38].

In the same sense as in the meson case, we now perform the contractions to compute the

two-point function for the proton in the propagator level. We use the interpolating operator

of Eq. 4.11 and we note that there are two ways to contract the up quark fields, hence the

proton two-point function turns out to be a sum of two terms

J N (x)J N(0) == abcua(x)(C5)db(x)uc(x)abcuc(0)db(0)(C5)ua(0) == abcabc C5G(x0)C5bb Gcc(x0)Gaa(x0) Gac(x0)Gca(x0) . (4.16)

In the above expression the sums over and have already been performed. The two-pointfunction still carries two open indices ( and ). These will be summed over once the two-point function is multiplied with the parity operator Eq. 4.13. A Feynman diagram which

corresponds to the two-point function of the proton is shown in Fig. 4.3.

Figure 4.3: Diagram of a two-point function for proton.

4.2 The Eective mass

The goal here is to show that the mass of hadrons can be extracted from the two-point

correlator and to demonstrate this with an example computation on the lattice. We start

o by writing the two-point correlator at the hadron level

C(x, t) = J (x, t)J (0,0) , (4.17)where J (x, t) is an interpolating operator that acts on the vacuum creating a hadron withthe same quantum numbers as those of the hadron in interest. In the Heisenberg picture, one

43

sink source

J ++ = abc uTaCubucJ N = abc(ua(C5)db)uc

J0 =23abc uTaCdb sc + dTaCsbuc + sTaCubdc

1

Baryon Spectrum

Eective mass

9

In addition, the source location is chosen randomly on the whole lattice for each configuration, in order to decreasecorrelation among measurements. Then, masses can be extracted from eective mass calculations. The eective massis defined by

mXe(t) = log

CX(t)

CX(t+ 1)

= mX + log

1 +

P1i=1 cie

it

1 +P1

i=1 ciei(t+1)

!t!1mX (18)

where i = mi mX is the mass dierence of the excited state i with respect to the ground mass mX .All results in this work have been extracted from correlators where Gaussian smearing is applied both at the source

and sink. In general, eective masses from local correlators are expected to have the same value in the large timelimit, but smearing suppresses excited state contaminations, therefore yielding a plateau region at earlier source-sinktime separations and better accuracy in the extraction of the mass. Our fitting procedure to extract mX is as follows:The sum over excited states in the eective mass given in Eq. (18) is truncated, keeping only the first excited state.The upper time slice boundary is kept fixed, and, allowing a separation of a couple of time slices the eective massis fitted to the form given in Eq. (19). This yields an estimation for the parameters c1 and 1 = m1 mX . Thena constant fit is carried out, increasing the lower time slice boundary (hence decreasing the fitting range) until thecontribution to mX due to the first excited state is less than 50% of the statistical error of mX from the constant fit.This criterion is in most cases in agreement with 2/d.o.f. < 1. In the cases in which this criterion is not satisfied acareful examination of the eective mass is made to ensure that the fit range is in the plateau region.

mXe(t) mX + log

1 + c1e1t

1 + c1e1(t+1)

!t!1mX (19)

We show representative results of the eective masses of baryons considered in this work in Fig. 4. The error bandson the constant fits are obtained using jackknife analysis.

FIG. 4. Representative eective mass plots for = 1.95,al = 0.0055 where both the constant and the exponential fits aredisplayed.

9


mXe(t) = log

CX(t)

CX(t+ 1)

= mX + log

1 +

P1i=1 cie

it

1 +P1

i=1 ciei(t+1)

!t!1mX (18)



mXe(t) mX + log

1 + c1e1t

1 + c1e1(t+1)

!t!1mX (19)



9


mXe(t) = log

CX(t)

CX(t+ 1)

= mX + log

1 +

P1i=1 cie

it

1 +P1

i=1 ciei(t+1)

!t!1mX (18)



mXe(t) mX + log

1 + c1e1t

1 + c1e1(t+1)

!t!1mX (19)



9


mXe(t) = log

CX(t)

CX(t+ 1)

= mX + log

1 +

P1i=1 cie

it

1 +P1

i=1 ciei(t+1)

!t!1mX (18)



mXe(t) mX + log

1 + c1e1t

1 + c1e1(t+1)

!t!1mX (19)



6

Charm Strange BaryonQuark

Interpolating field I Izcontent

c = 0 s = 2?0 uss 1p

3abc

2sTaCub

sc +

sTaCsb

uc

1/2 +1/2

? dss 1p3abc

2sTaCdb

sc +

sTaCsb

dc

1/2 -1/2

c = 1 s = 1?+c usc

q23 abc

uTaCsb

cc +

sTaCcb

uc +

cTaCub

sc

1/2 +1/2

?0c dscq

23 abc

dTaCsb

cc +

sTaCcb

dc +

cTaCdb

sc

1/2 -1/2

s = 2 ?0c ssc1p3abc

2sTaCcb

sc +

sTaCsb

cc

0 0

c = 2 s = 0?++cc ucc

1p3abc

2cTaCub

cc +

cTaCcb

uc

1/2 +1/2

?+cc dcc1p3abc

2cTaCdb

cc +

cTaCcb

dc

1/2 -1/2

s = 1 ?+cc scc1p3abc

2cTaCsb

cc +

cTaCcb

sc

0 0

TABLE VI. Alternative interpolating fields for the spin-3/2 baryons.

= 1.95 = 2.1

APEn 20 50

0.5 0.5

Gaussiann 50 110

4 4

TABLE VII. Smearing parameters for the ensembles at = 1.95 and = 2.1.

mXe(t) = log

CX(t)

CX(t+ 1)

= mX + log

1 +

P1i=1 cie

it

1 +P1

i=1 ciei(t+1)

!t!1mX (17)



mXe(t) mX + log

1 + c1e1t

1 + c1e1(t+1)

!t!1mX (18)


mcX mX12 (m

cX +mX)

12mcXCriterion:

Overview of calculations

Isospin symmetry breaking

Isospin symmetry breaking

Twisted mass action breaks isospin symmetry explicitly to Size of terms determine how large the breaking will be It is expected to be zero in the continuum limit Manifests itself as mass splitting between baryons belonging to the

same isospin multiplets due to lattice artifacts

Masses of light baryons calculated on the lattice are not expected to exert any degree of isospin symmetry breaking

Baryon Spectrum

! = 2m0 = 0 (1)O(a)O(a2)

1

! = 2m0 = 0 (1)O(a)O(a2)

1

baryons

No isospin splitting for light baryons (u and d quarks are degenerate)

B55.32

D15.48

Baryon Spectrum Isospin symmetry breaking

baryons Strange spin-1/2 states

splitting between baryons belonging

to same isospin multiplets

Strange spin-3/2 states

No isospin splitting

Circles: B55.32 Squares: D15.48

Circles: B55.32 Squares: D15.48


Charm cc baryons

No isospin symmetry breaking for charm spin-1/2 AND spin-3/2 baryons

Circles: B55.32 Triangles: B35.32 Squares: D15.48

Circles: B55.32 Triangles: B35.32 Squares: D15.48


Isospin splitting as a function of lattice spacing

Isospin splitting eects are small and reduce for smaller values of the lattice spacing for spin-1/2 baryons

consistent with zero for spin-3/2 baryons (as for the baryons)


Fixing the lattice spacing

Fixing the lattice spacing Nucleon

HBPT (leading order)

HBPT (SSE Scheme)

17

one-loop order of the pion mass dependence for the nucleon mass in HBPT is given by

mN = m0N 4c1m2

3g2A32f2

m3 (20)

where m0N is the nucleon mass at the chiral limit and together with c1 are treated as fit parameters. This lowest orderresult for the nucleon in HBPT was first derived in Ref. [22] and successfully fitted lattice data are also discussed inprevious studies [23, 24]. This result is the same in HBPT with dimensional and infra-red regularization as well aswhen the degree of freedom is explicitly included. It is also the same in manifestly Lorentz-invariant formulationwith infra-red regularization. Therefore we will use this result to fix the lattice spacing using the nucleon mass asinput. This can also provide a non-trivial check of our lattice formulation. The lattice spacings a=1.90, a=1.95 anda=2.10 are considered as additional independent fit parameters in a combined fit of our data at = 1.90, = 1.95and = 2.10. We constrain our fit so that the fitted curve passes through the physical point by fixing the value of c1,hence the lattice spacings are determined using the nucleon mass at the physical point as the only input. The physicalvalues of f and gA are used in the fits, namely f = 0.092419(7)(25) and gA = 1.2695(29), which is common practicein chiral fits to lattice data on the nucleon mass [2527]. The left plot of Fig. 18 shows the fit to the O(p3) result ofEq. (20) on the nucleon mass. The error band and the errors on the fit parameters are obtained from super-jackknifeanalysis. As can be seen, the O(p3) result provides a very good fit to our lattice data, which in fact confirms thatcut-o and finite volume eects are small for the -values used. The results on the fit parameter m0N and the latticespacings a=1.90, a=1.95 and a=2.10 are collected in Table XI. The physical spatial lengths of the lattices used inthis calculation are obtained from the resulting lattice spacings of the fit and their values are labeled in Fig. ??.We perform the same analysis with higher order chiral corrections to Eq. (20). These O(p4) corrections are known

within several expansion schemes. In the so called small scale expansion (SSE) scheme [27], the -degress of freedomare explicitly included in covariant baryon PT by introducing as an additional counting parameter the -nucleonmass splitting, m mN , taking O(/mN ) O(m/mN ). In SSE the nucleon mass is given by

mN = m0N 4c1m2

3g2A32f2

m3 4E1()m4 3g2A + 3c

2A

642f2m

0N

m4 3g2A + 10c

2A

322f2m

0N

m4 logm

c

2A

32f2

1 +

2m0N

4m2 +

3 3

2m2

log

m2

+2 m2

R (m)

(21)

where R (m) = pm2 2 cos1

m

for m > and R (m) =

p2 m2 log

m

+q2

m2 1

for m < .

We take the cut-o scale = 1 GeV, c1 = 1.127 [27] and treat the counter-term E1 as an additional fit parameter.As in the O(p3) case we use the physical values of gA and f. The corresponding plot is shown on the right panel ofFig. 18. The error band as well as the errors on the fit parameters are obtained using super-jackknife analysis. Onecan see that this formulation provides a good description of the lattice data as well and yields values of the latticespacings and m0N which are consistent with those obtained in O(p3) of HBPT. The resulting parameters of this fitare given in Table XI.

FIG. 18. The nucleon mass as a function of m2 for = 1.90, = 1.95 and = 2.10 in O(p3) HBPT (left) and in O(p4) ofSSE scheme (right). The physical nucleon mass is shown with the black asterisk.

Baryon Spectrum

! = 2m0 = 0 (1)O(p3)O(p4)

1

! = 2m0 = 0 (1)O(p3)O(p4)

1

Results

Baryon Spectrum Fixing the lattice spacing

a=1.90 (fm) a=1.95 (fm) a=2.10 (fm) m0N (GeV) E1() GeV3 2d.o.fO(p3) HBPT 0.0929(14) 0.0815(11) 0.0641(8) 0.8681(15) 1.4314O(p4) SSE 0.0967(19) 0.0854(17) 0.0668(12) 0.8826(48) -2.5849(2530) 0.9303Table 1: Fit parameters a=1.90, a=1.95, a=2.10 in fm, m0N in GeV and E1() in GeV3 fromO(p3) HBPT and O(p4) SSE using the nucleon mass.

a=1.90 (fm) a=1.95 (fm) a=2.10 (fm) m0 GeV 2d.o.fSU(2) HBPT 0.1018(5) 0.0857(4) 0.0660(4) 1.6673(16) 2.0925

Table 2: Fit parameters a=1.90, a=1.95, a=2.10 in fm and m0 in GeV from SU(2) HBPTusing the mass.

1

Other methods for detemining the lattice spacing Pion decay constant Sommer scale r0

Systematics arise and are under investigation

Chiral Extrapolations - Comparison

Chiral Extrapolations Comparison Nucleon

S. Aoki et. al. (PACS-CS), Phys. Rev. D 79, 034503 (2009), 0807.1661 S. Durr et. al., Science 322, 1224 (2008) A. Walker-Loud, H.-W.Lin, D. Richards R. Edwards, M. Engelhardt, et. al., Phys.Rev. D 79, 054502 (2009), 0806.4549

Baryon Spectrum

baryon

8

JX1/2 = P1/2JX .

P1/2 = P3/2 =

1

3

Cij(t) =

ij 1

3ijC 3

2(t) +

1

3ijC 1

2(t)

C 32(t) =

1

3Tr[C] +

1

6

3Xi 6=j

ijCij

Cw/p(t) =1

3Tr[C]

m = m0 4cm2

g216f2

m3

m = m0 4cm2

25

27

g216f2

m3

m = m0 4cm2

10

9

g216f2

m3

II. LATTICE RESULTS

A. Isospin breaking

The twisted mass action breaks isospin explicitly to O(a2) and the size of O(a2) terms determines how large thisbreaking would be. It is expected that the splitting is zero in the continuum limit. In general, isospin symmetrybreaking manifests itself as a mass splitting between baryons belonging to the same multiplets. Therefore, it would beinteresting to examine at first the degree of isospin splitting between baryons belonging to the same isospin multipletsdue to lattice artifacts. We begin this analysis by presenting the eective masses of the light baryons in the decupletrepresentation of SU(3), namely the ++, , + and 0 shown in Fig. 4.As shown in Fig. 4, the masses of the baryons show consistency within errors, as one would expect, since the

light quarks u and d are in fact degenerate. Therefore, no isospin splitting eects are observed for the light baryons.However, the strange baryons of the octet representation of SU(3) is a case where some degree of isospin splitting

is clearly visible for = 1.95, whereas for = 2.1 the splitting is decreased as expected. To demonstrate this, weshow results on the masses for the strange particles and in Fig. 5. A more quantitative determination of thedegree of isospin breaking will follow.It is of interest to notice that the same behaviour is not observed for the corresponding spin-3/2 strange baryon

states and . Here, the splitting eect is not apparent within errors, neither for = 1.95, nor for = 2.1. Thisis shown in Fig. 6, where we plot the eective masses of the strange spin-3/2 baryons of the decuplet. NOTE: Ineed some help for commenting on thisWe continue our analysis by studying the isospin breaking on the charm baryons. To this end we concentrate on

the results obtained for the charm c baryons, containing light quarks and a charm quark as well as the charm cbaryons, which contain a light, a strange and a charm quark. As before, we plot results for = 1.95 and = 2.1 tocompare between dierent lattice spacings.

baryons (averaged)

8

JX1/2 = P1/2JX .

P1/2 = P3/2 =

1

3

Cij(t) =

ij 1

3ijC 3

2(t) +

1

3ijC 1

2(t)

C 32(t) =

1

3Tr[C] +

1

6

3Xi 6=j

ijCij

Cw/p(t) =1

3Tr[C]

m = m0 4cm2

g216f2

m3

m = m0 4cm2

25

27

g216f2

m3

m = m0 4cm2

10

9

g216f2

m3

II. LATTICE RESULTS

A. Isospin breaking

The twisted mass action breaks isospin explicitly to O(a2) and the size of O(a2) terms determines how large thisbreaking would be. It is expected that the splitting is zero in the continuum limit. In general, isospin symmetrybreaking manifests itself as a mass splitting between baryons belonging to the same multiplets. Therefore, it would beinteresting to examine at first the degree of isospin splitting between baryons belonging to the same isospin multipletsdue to lattice artifacts. We begin this analysis by presenting the eective masses of the light baryons in the decupletrepresentation of SU(3), namely the ++, , + and 0 shown in Fig. 4.As shown in Fig. 4, the masses of the baryons show consistency within errors, as one would expect, since the

light quarks u and d are in fact degenerate. Therefore, no isospin splitting eects are observed for the light baryons.However, the strange baryons of the octet representation of SU(3) is a case where some degree of isospin splitting

is clearly visible for = 1.95, whereas for = 2.1 the splitting is decreased as expected. To demonstrate this, weshow results on the masses for the strange particles and in Fig. 5. A more quantitative determination of thedegree of isospin breaking will follow.It is of interest to notice that the same behaviour is not observed for the corresponding spin-3/2 strange baryon

states and . Here, the splitting eect is not apparent within errors, neither for = 1.95, nor for = 2.1. Thisis shown in Fig. 6, where we plot the eective masses of the strange spin-3/2 baryons of the decuplet. NOTE: Ineed some help for commenting on thisWe continue our analysis by studying the isospin breaking on the charm baryons. To this end we concentrate on

the results obtained for the charm c baryons, containing light quarks and a charm quark as well as the charm cbaryons, which contain a light, a strange and a charm quark. As before, we plot results for = 1.95 and = 2.1 tocompare between dierent lattice spacings.

Baryon Spectrum Chiral Extrapolations - Comparison

Octet and Decuplet at the physical point

Baryon Spectrum Chiral Extrapolations - Comparison

Only statistical errors are shown Systematic errors are important, their detection and estimation is not a

trivial task

they are currently under investigation

The Proton Spin

discovery by EMC in 1987 for the small contribution of the quarks to the proton spin led to the proton spin crisis

how much of the proton spin is carried by the quarks after all?

Generalized Parton Distributions

Generalized Parton Distributions and Generalized Form Factors

The GPDs encode important information related to baryon structure

related to baryon form factors (e.g. Dirac and Pauli nucleon form factors)

dened as matrix elements of local operators and parameterized in terms of the generalized form factors (GFFs)

Proton Spin

7. GPDs and the spin of the proton

Depending on the structure ( = or = 5), the V and A subscriptions denoteVector and Axial-vector operators respectively. Curly braces around indices represent a

symmetrization and the subtraction of traces of the indices, e.g.

O{} 12(O +O) 1

4

O . (7.8)

In addition, the derivativeD is defined via

D 1

4 + + , (7.9)

where and are the forward and backward derivatives already defined in Eq. 3.25 andEq. 3.26. In this work we limit the values of n to n = 1,2, namely we will calculate matrixelements of local operators and operators with a single derivative. Moreover, we consider

a = 3 which corresponds to the isovector combination. Considering specifically the nucleonstate, N(p, s), the matrix elements of the one derivative operators are parametrized interms of the generalized form factors (GFFs), A20(q2), B20(q2), C20(q2) for the vectoroperators and A20(q2), B20(q2) for the axial-vector operators, according toN(p, s)OV3 N(p, s) = uN(p, s) A20(q2){P } +B20(q2)i{qP }2m ++C20(q2) 1

mq{q} 1

2uN(p, s) ,

N(p, s)OA3 N(p, s) = uN(p, s) A20(q2){P }5 + B20(q2)q{P }2m 5 12uN(p, s)(7.10)

We note here that the GFFs depend only on the momentum transfer squared, q2 = (pp)2,with p and p being the final and initial momentum respectively. This implies, together withEq. 7.3 and Eq. 7.4, that the moments of the GPDs are polynomial in and the coecients

are the GFFs,

Hn=1(, q2) = A10(q2) ,Hn=2(, q2) = A20(q2) + (2)2C20(q2) ,En=1(, q2) = B10(q2) ,En=2(, q2) = B20(q2) (2)2C20(q2) , (7.11)

92

Evaluation of GFFs

Evaluation of the GFFs Proton Spin


and

Hn=1(, q2) = A10(q2) ,Hn=2(, q2) = A20(q2) ,En=1(, q2) = B10(q2) ,En=2(, q2) = B20(q2) , (7.12)

Taking together Eqs. 7.4, 7.5, 7.11 and 7.12 in the forward limit one gets the moments

xn1q 11 dxxn1q(x) =Hn(0,0) = An0(0) ,xn1q 11 dxxn1q(x) = H(0,0) = An0(0) . (7.13)These are of particular interest, especially for n = 1 and n = 2. For n = 2 in the forward limitwe have A20(0) = xq and A20(0) = xq, which are the first moments of the unpolarizedand polarized quark distributions respectively. By obtaining the GFFs one can evaluate

the quark distribution to the nucleon spin using Jis sum rule Jq = 12[Aq20(0) +Bq20(0)], aswe will discuss in the following sections. In addition, it holds that Aq10 q = gqA, whereq is the quark helicity. Thus, one can calculate on the lattice and compare with the

measured value from experiment. Moreover, the decomposition Jq = 12q +Lq allows for astudy of the role of the quark orbital angular momentum Lq.

7.3 Lattice evaluation

The GFFs are extracted from dimensionless ratios involving two-point and three-point

functions which in the sense of Eqns. 4.17 and 4.29 can be written as

C(q, tf ti) =xf 0J(tf , xf)J(ti, xi)ei(xfxi)q (7.14)and

G1n( , q, t) = x,xf J(tf , xf)O1n (t, x)J(ti, xi)ei(xxi)q (7.15)respectively. We use the operating field operator for the proton, J(t, x) as defined in Table.C.1. The projection matrices are given by

0 = 14(1 + 0) and k = 0i5k . (7.16)

The kinematics are such that the final momentum p = 0. As with the baryon spectrum, weuse Gaussian smeared quark fields at the source and sink in order to improve overlap with the

93


and

Hn=1(, q2) = A10(q2) ,Hn=2(, q2) = A20(q2) ,En=1(, q2) = B10(q2) ,En=2(, q2) = B20(q2) , (7.12)

Taking together Eqs. 7.4, 7.5, 7.11 and 7.12 in the forward limit one gets the moments

xn1q 11 dxxn1q(x) =Hn(0,0) = An0(0) ,xn1q 11 dxxn1q(x) = H(0,0) = An0(0) . (7.13)These are of particular interest, especially for n = 1 and n = 2. For n = 2 in the forward limitwe have A20(0) = xq and A20(0) = xq, which are the first moments of the unpolarizedand polarized quark distributions respectively. By obtaining the GFFs one can evaluate

the quark distribution to the nucleon spin using Jis sum rule Jq = 12[Aq20(0) +Bq20(0)], aswe will discuss in the following sections. In addition, it holds that Aq10 q = gqA, whereq is the quark helicity. Thus, one can calculate on the lattice and compare with the

measured value from experiment. Moreover, the decomposition Jq = 12q +Lq allows for astudy of the role of the quark orbital angular momentum Lq.

7.3 Lattice evaluation

The GFFs are extracted from dimensionless ratios involving two-point and three-point

functions which in the sense of Eqns. 4.17 and 4.29 can be written as

C(q, tf ti) =xf 0J(tf , xf)J(ti, xi)ei(xfxi)q (7.14)and

G1n( , q, t) = x,xf J(tf , xf)O1n (t, x)J(ti, xi)ei(xxi)q (7.15)respectively. We use the operating field operator for the proton, J(t, x) as defined in Table.C.1. The projection matrices are given by

0 = 14(1 + 0) and k = 0i5k . (7.16)

The kinematics are such that the final momentum p = 0. As with the baryon spectrum, weuse Gaussian smeared quark fields at the source and sink in order to improve overlap with the

93


proton state and decrease excited state contaminations. For the insertion, O1n , the onederivative vector ( {1D2} ) and axial-vector ( 5{1D2} ) operators are employed.

Figure 7.2: Connected piece of a nucleon three-point function.

At this stage, we point out that in three-point correlators of the isovector combination

of operators, OIV Ou Od, the disconnected diagrams are zero up to lattice artifacts andcan be safely neglected as we approach the continuum limit. Thus, the connected diagram

of Fig. 7.2 is an exact evaluation in the isovector case. The sequential inversion through the

sink [105] is employed. This means that the time separation between the source and sink,

tf ti, needs to be kept fixed but three-point correlators for various operators and for allpossible momentum transfers and insertion times can be obtained with a single sequential

inversion per choice of sink time. The standard procedure is to take the creation operator at

a fixed position, xi = 0 (source). The annihilation operator at a later time (sink) tf carriesmomentum p = 0. The current couples to a quark at an intermediate time t, as in Fig. 7.2and carries momentum q. Translation invariance demands q = p. An alternative approachis the stochastic computation of the spatial all-to-all propagator, which recent studies, e.g.

Ref. [106], have shown to be suitable for the evaluation of nucleon three-point functions.

Having calculated the two- and three-point functions defined in Eqns. 7.14 and 7.15 and

considering operators with up to one derivative we form the ratio [97] (in correspondence

with Eq. 4.37)

R(, q, t) = G(, q, t)C(0, tf ti)

C(p, tf t)C(0, t ti)C(0, tf ti)C(0, tf t)C(p, t ti)C(p, tf ti) . (7.17)

This ratio is optimized because it does not contain potentially noisy two-point functions at

large separations and because correlations between its dierent factors reduce the statistical

noise. For suciently large separations tf t and t ti this ratio becomes time-independentand a plateau region can be identified

limtft limttiR(, q, t) = (, q) . (7.18)

From the plateau values of the renormalized asymptotic ratio R(, q) = Z(, q) thenucleon matrix elements of all the operators can be extracted. The expressions relating

94

-0.15-0.1

-0.05 0

R00 ( K

0 , p= (

0 ,0 ,

0 ))

0.03

0.04

0.05

Rk0 ( K

0 , p= (

0 ,0 ,

0 ))

-0.15-0.1

-0.05

R00 ( K

0 , p= (

1 ,0 ,

0 ))

0.01 0.02 0.03 0.04

1 2 3 4 5 6 7 8 9 10 11 12Rij ( K

k , p= (

1 ,0 ,

0 ))

t/a


Depending on the structure ( = or = 5), the V and A subscriptions denoteVector and Axial-vector operators respectively. Curly braces around indices represent a

symmetrization and the subtraction of traces of the indices, e.g.

O{} 12(O +O) 1

4

O . (7.8)

In addition, the derivativeD is defined via

D 1

4 + + , (7.9)

where and are the forward and backward derivatives already defined in Eq. 3.25 andEq. 3.26. In this work we limit the values of n to n = 1,2, namely we will calculate matrixelements of local operators and operators with a single derivative. Moreover, we consider

a = 3 which corresponds to the isovector combination. Considering specifically the nucleonstate, N(p, s), the matrix elements of the one derivative operators are parametrized interms of the generalized form factors (GFFs), A20(q2), B20(q2), C20(q2) for the vectoroperators and A20(q2), B20(q2) for the axial-vector operators, according toN(p, s)OV3 N(p, s) = uN(p, s) A20(q2){P } +B20(q2)i{qP }2m ++C20(q2) 1

mq{q} 1

2uN(p, s) ,

N(p, s)OA3 N(p, s) = uN(p, s) A20(q2){P }5 + B20(q2)q{P }2m 5 12uN(p, s)(7.10)

We note here that the GFFs depend only on the momentum transfer squared, q2 = (pp)2,with p and p being the final and initial momentum respectively. This implies, together withEq. 7.3 and Eq. 7.4, that the moments of the GPDs are polynomial in and the coecients

are the GFFs,

Hn=1(, q2) = A10(q2) ,Hn=2(, q2) = A20(q2) + (2)2C20(q2) ,En=1(, q2) = B10(q2) ,En=2(, q2) = B20(q2) (2)2C20(q2) , (7.11)

92


proton state and decrease excited state contaminations. For the insertion, O1n , the onederivative vector ( {1D2} ) and axial-vector ( 5{1D2} ) operators are employed.

Figure 7.2: Connected piece of a nucleon three-point function.

At this stage, we point out that in three-point correlators of the isovector combination

of operators, OIV Ou Od, the disconnected diagrams are zero up to lattice artifacts andcan be safely neglected as we approach the continuum limit. Thus, the connected diagram

of Fig. 7.2 is an exact evaluation in the isovector case. The sequential inversion through the

sink [105] is employed. This means that the time separation between the source and sink,

tf ti, needs to be kept fixed but three-point correlators for various operators and for allpossible momentum transfers and insertion times can be obtained with a single sequential

inversion per choice of sink time. The standard procedure is to take the creation operator at

a fixed position, xi = 0 (source). The annihilation operator at a later time (sink) tf carriesmomentum p = 0. The current couples to a quark at an intermediate time t, as in Fig. 7.2and carries momentum q. Translation invariance demands q = p. An alternative approachis the stochastic computation of the spatial all-to-all propagator, which recent studies, e.g.

Ref. [106], have shown to be suitable for the evaluation of nucleon three-point functions.

Having calculated the two- and three-point functions defined in Eqns. 7.14 and 7.15 and

considering operators with up to one derivative we form the ratio [97] (in correspondence

with Eq. 4.37)

R(, q, t) = G(, q, t)C(0, tf ti)

C(p, tf t)C(0, t ti)C(0, tf ti)C(0, tf t)C(p, t ti)C(p, tf ti) . (7.17)

This ratio is optimized because it does not contain potentially noisy two-point functions at

large separations and because correlations between its dierent factors reduce the statistical

noise. For suciently large separations tf t and t ti this ratio becomes time-independentand a plateau region can be identified

limtft limttiR(, q, t) = (, q) . (7.18)

From the plateau values of the renormalized asymptotic ratio R(, q) = Z(, q) thenucleon matrix elements of all the operators can be extracted. The expressions relating

94

Results on the GFFs

Results on the GFFs

Good description of lattice data Consistency between Nf = 2 and

Nf = 2+1+1 results

Larger value at the physical point

Proton Spin

0.1

0.2

0.3

0.4

0 0.05 0.1 0.15 0.2 0.25

A20u-

d

0.2

0.4

0.6

0 0.05 0.1 0.15 0.2 0.25

B20u-

d

m2 (GeV2)

Appendix E

Expressions for chiral extrapolation of

GFFs and quark spin components

In this Appendix we collect the expressions used to extrapolate our lattice data for the

quark spin to the physical point. Throughout, we use 2 = 1 GeV2, f = 0.0924 GeV andgA = 1.267.

In HBPT the expressions for A20(0) and B20(0) for the isovector combination are givenby [89]

AI=120 (0) = AI=1(0)20 1 m2(4f)2 (3g2A + 1) lnm22 + 2g2A +AI=1(2,m)20 m2 ,BI=120 (0) = BI=1(0)20 1 m2(4f)2 (2g2A + 1) lnm22 + 2g2A ++ AI=1(0)20 m2g2A(4f)2 lnm22 +BI=1(2,m)20 m2 (E.1)

and for the isoscalar by

AI=020 (0) = AI=0(0)20 +AI=0(2,m)20 m2 ,BI=020 (0) = BI=0(0)20 1 3g2Am2(4f)2 lnm22 AI=0(0)20 3g2Am2(4f)2 lnm22+ BI=0(2,m)20 m2 +BI=0(2,)20 . (E.2)

The spin carried by the quarks is given by the axial coupling gA or A10(0) asud = Aud10 = AI=110 (0)u+d = Au+d10 = AI=010 (0) . (E.3)

119

Appendix E

Expressions for chiral extrapolation of

GFFs and quark spin components

In this Appendix we collect the expressions used to extrapolate our lattice data for the

quark spin to the physical point. Throughout, we use 2 = 1 GeV2, f = 0.0924 GeV andgA = 1.267.

In HBPT the expressions for A20(0) and B20(0) for the isovector combination are givenby [89]

AI=120 (0) = AI=1(0)20 1 m2(4f)2 (3g2A + 1) lnm22 + 2g2A +AI=1(2,m)20 m2 ,BI=120 (0) = BI=1(0)20 1 m2(4f)2 (2g2A + 1) lnm22 + 2g2A ++ AI=1(0)20 m2g2A(4f)2 lnm22 +BI=1(2,m)20 m2 (E.1)

and for the isoscalar by

AI=020 (0) = AI=0(0)20 +AI=0(2,m)20 m2 ,BI=020 (0) = BI=0(0)20 1 3g2Am2(4f)2 lnm22 AI=0(0)20 3g2Am2(4f)2 lnm22+ BI=0(2,m)20 m2 +BI=0(2,)20 . (E.2)

The spin carried by the quarks is given by the axial coupling gA or A10(0) asud = Aud10 = AI=110 (0)u+d = Au+d10 = AI=010 (0) . (E.3)

119

C. Alexandrou, J. Carbonell, M. Constantinou, P.A. Harraud, P. Guichon, K. Jansen, C. Kallidonis, T. Korzec, and M Papinutto. Moments of nucleon generalized parton distributions from lattice QCD. Phys.Rev., D83:114513, 2011. doi: 10.1103/PhysRevD.83.114513., 1104.1600

C. Alexandrou, M. Constantinou, S. Dinter, V. Drach, K. Jansen, C. Kallidonis, and G. Koutsou. Nucleon form factors and moments of generalized parton distributions using Nf = 2 + 1 + 1 twisted mass fermions. 2013. , 1303.5979

Spin Observables

Spin observables Jis sum rule

Total angular momentum decomposition

Proton Spin

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0 0.05 0.1 0.15 0.2 0.25

Co n

t r ib u

t i on s

t on u

c le o

ns p

i n

m2pi (GeV2)

12

u+d

Lu+d

-0.1

0

0.1

0.2

0.3

0.4

0 0.05 0.1 0.15 0.2 0.25

Co n

t r ib u

t i on s

t on u

c le o

ns p

i n

m2pi (GeV2)

Ju

Jd


As already mentioned, the isovector Aud20 , Bud20 and isoscalar Au+d20 , Bu+d20 moments areneeded in order to obtain information on the spin content of the nucleon since, according

to Jis sum rule it holds that

Jq = 12(Aq20 +Bq20) , (7.20)

where q can be any of u, d, u d and u + d. In addition, the total angular momentum Jqcan be further decomposed into its orbital angular momentum Lq and its spin component

q as

Jq = 12q +Lq . (7.21)

In addition to the moments A20 and B20, we need results for the n = 1 moments at Q2 = 0,since the spin carried by the u and d quarks is determined by u+d = Au+d10 and ud =Aud10 . The expressions for A10 are found in Appendix E (Eq. E.4). The isoscalar quantitiesreceive contributions from disconnected diagrams, which are notoriously dicult to calculate

and are neglected in most current evaluations of GFFs. Under the assumption that they

are small, we also neglect these contributions and extract information on the fraction of the

nucleon spin carried by the quarks. Having both isoscalar and isovector quantities we can

extract the angular momentum Ju and Jd carried by the u and d quarks, by noting that

Ju = 12(Ju+d + Jud) ,

Jd = 12(Ju+d Jud) (7.22)

The above expression also holds for u,d and Lu,d. The latter can be extracted from

Lq = Jq q. Thus, using Jis sum rule of Eq. 7.20 and the above expressions one cancalculate apart from the total angular momentum J and the spin component which are

found from the GFFs, also the orbital angular momentum L.

We begin by comparing at first our results for the quark angular momentum Jq, spin

q and orbital angular momentum Lq, q = u, d to those obtained using the hybrid actionof Ref. [43]. The comparison is shown in Fig. 7.6. The experimental values for are

shown by the black asterisks and are taken from the HERMES 2007 analysis [112]. As we

can see, the lattice data are in agreement within our statistical errors indicating that lattice

artifacts are smaller that the current statistical errors for these quantities.

In order to get an indication for the value of Jq at the physical point we perform a chiral

extrapolation using HBPT. Combining the expressions for A20 and B20 of Eqns. E.1 and

E.2 as well as Eq. 7.22 we obtain an Ansatz for Jq

Jq = aq0 m2(4f)2 lnm22 + aq1m2 + aq2 (7.23)where we take 2 = 1 GeV2. We also carry out a similar procedure using the expressions fromCBPT. All relevant expressions are collected in Appendix E. In Fig. 7.7 we show the chiral

fits on Jq using HBPT and CBPT. The error bands are constructed from super-jackknife

98




Jq = 12(Aq20 +Bq20) , (7.20)


q as

Jq = 12q +Lq . (7.21)






Ju = 12(Ju+d + Jud) ,

Jd = 12(Ju+d Jud) (7.22)














98




Jq = 12(Aq20 +Bq20) , (7.20)


q as

Jq = 12q +Lq . (7.21)






Ju = 12(Ju+d + Jud) ,

Jd = 12(Ju+d Jud) (7.22)














98

A. Airapetian et al. Precise determination of the spin structure function g(1) of the proton, deuteron and neutron. Phys.Rev., D75:012007, 2007. doi: 10.1103/PhysRevD.75.012007.

Proton Spin

-0.1

0

0.1

0.2

0.3

0.4

0 0.1 0.2 0.3 0.4 0.5 0.6

Co n

t r ib u

t i on s

t on u

c le o

ns p

i n

m2pi (GeV2)

Ju

Jd-0.2

0

0.2

0.4

0 0.1 0.2 0.3 0.4 0.5 0.6

Co n

t r ib u

t i on s

t on u

c le o

ns p

i n

m2pi (GeV2)

12

u

12

d

Lu

Ld

Comparison with results from other lattice formulations

Very good agreement with other lattice formulations at our pion mass range

Nf = 2+1+1 results show consistency

Spin Observables

SUMMARY

o Isospin breaking is present only for strange spin-1/2 baryons. In general it is under control and vanishes in the continuum limit

o TM results have overall agreement with results from other collaborations and with experiment, except for baryons with heavy strange content (e.g. , -)

o The GFFs provide a way of getting an insight into observables regarding the proton spin

o Quark contribution to the total angular momentum of the proton is small (less than 50%)

o Our results for the proton spin agree in general with similar calculations carried out using other lattice simulations and any discrepancies at the physical point

could be solved with simulations closer to or even at the physical pion mass

Acknowledgements

o Prof. Constantia Alexandrou

o Dr. Giannis Koutsou

o Dr. Martha Constantinou

o Kyriakos Hadjiyiannakou

o ETMC members

S

Thank you

3 = 1.90, a = 0.0929(14) fm r0/a = 5.231(38)

203 48, L = 1.9 fma 0.0040No. of Confs 308m (GeV) 0.3166mL 2.99

243 48, L = 2.2 fma 0.0040 0.0060 0.0080 0.0100No. of Confs 523 479 449 501m (GeV) 0.3074 0.3664 0.4216 0.4729mL 3.48 4.15 4.77 5.35

323 64, L = 3.0 fma 0.0030 0.0040 0.0050No. of Confs 185 389 193m (GeV) 0.2629 0.3000 0.3351mL 3.97 4.53 5.05

= 1.95, a = 0.0815(11) fm, r0/a = 5.710(41)

243 48, L = 2.1 fma 0.0085No. of Confs 0.4689m (GeV) 0.4428mL 4.66

323 64, L = 2.6 fma 0.0025 0.0035 0.0055 0.0075No. of Confs 723 546 4644 521m (GeV) 0.2582 0.3047 0.3752 0.4357mL 3.42 4.03 4.97 5.77

= 2.10, a = 0.0641(8) fm r0/a = 7.538(58)

323 64, L = 2.1 fma 0.0045No. of Confs 952m (GeV) 0.3712mL 3.87

483 96, L = 3.1 fma 0.0015 0.002 0.003No. of Confs 606 186 113m (GeV) 0.2142 0.2471 0.3004mL 3.35 3.86 4.69

TABLE I. Input parameters (, L, ) of our lattice simulations and corresponding lattice spacing (a) and pion mass (m).Row Stat. refers to the number of gauge configurations used in the calculations. In row Abbr. we use an abbreviation for theensembles of the form Xl.L with X referring to the value used, in order to quickly refer to the ensembles later on. Thelattice spacing was determined using the nucleon mass in this work. NEEDS CORRECTION!!!

For consistency and cross check of the tuned masses, we have also performed two linear fits in constrained ranges ofah, one in the sector of the kaon mass and one in the sector of the D-meson mass, according to the expressions

a2M2PS(ah) = c1 + c2ah for the strange quark,

aMPS(ah) = d1 + d2ah for the charm quark. (11)

The linear fits are shown on the right panel of Fig. 1. We list the resulting polynomial fit parameters ai, the linearfit parameters ci and di as well as the tuned values of the strange and charm quark masses in Table II. As one cansee, the tuned values of the strange and charm quark masses are compatible among the polynomial and linear fits andthey are also in agreement with other similar calculations CITATION. Such agreements are satisfactory and showthat both fit procedures lead to the same tuned values within one standard deviation.

D. Interpolating fields

In order to extract the masses of the baryons in lattice QCD, we evaluate two-point functions. The baryon states arecreated from the vacuum with the use of interpolating fields which are constructed such that they have the quantumnumbers of the baryon in interest and reduce to the quark model wave functions in the non-relativistic limit. Thelow-lying baryons belonging to the octet and decuplet representation of SU(3) are given in Figs. 2 and 3 respectively.They are classified through the isospin, I, the third component of the isospin, Iz, the strangeness (s) and the electricalcharge (q). The corresponding interpolating fields for these baryons are collected in Tables III [15, 16] and IV [15, 17].

Simulation details 3 / 30 Baryon Spectrum

Calculation of two-point correlation functions

Major part of the work was to write programming code for all baryons

Average to gain statistics

8

JX1/2 = P1/2JX .

P1/2 = P3/2 =

1

3

Cij(t) =

ij 1

3ij

C 3

2(t) +

1

3ij

C 1

2(t)

C 32(t) =

1

3Tr[C] +

1

6

3Xi 6=j

ijCij

Cw/p(t) =1

3Tr[C]

m = m0 4cm2

g216f2

m3

m = m0 4cm2

25

27

g216f2

m3

m = m0 4cm2

10

9

g216f2

m3

J++ = abcuTa ub

uc

J 0 = abcsTaC5ub

sc

J 0 = abcsTa ub

sc

J 0 =1p3abc

2sTaCub

sc +

sTaCsb

uc

C+X(t) = CX(T t)

8

As local interpolating fields are not optimal for suppressing excited state contributions, we apply Gaussian smearingto each quark field q(x, t) [19, 20]. The smeared quark field is given by qsmear(x, t) =

Py F (x,y;U(t))q(y, t), where

we have used the gauge invariant smearing function

F (x,y;U(t)) = (1 + H)n (x,y;U(t)), (12)

constructed from the hopping matrix understood as a matrix in coordinate, color and spin space,

H(x,y;U(t)) =3X

i=1

Ui(x, t)x,yai + U

i (x ai, t)x,y+ai

. (13)

In addition, we apply APE smearing to the spatial links that enter the hopping matrix. The parameters and nof the Gaussian and APE smearing at each value of are collected in Table IX.We need to explain why use these parameters?

= 1.95 = 2.1

APEn 20 50

0.5 0.5

Gaussiann 50 110

4 4

TABLE IX. Smearing parameters for the ensembles at = 1.95 and = 2.1.

E. Two-point correlators

The interpolating fields for the spin-3/2 baryons as defined in Tables IV,VI and VIII have overlap with their heavierspin-1/2 excitations as well. These overlaps can be removed with the incorporation of a spin-3/2 projector in thedefinitions of the interpolating fields

JX3/2 = P3/2JX . (14)

For non-zero momentum, P3/2 is defined by

P3/2 = 1

3 1

3p2( 6 pp + p 6 p) . (15)

In correspondance, the spin-1/2 interpolating field JX1/2 that has overlap only with the excited spin-1/2 state can

be obtained by acting with the spin-1/2 projector P1/2 = g P3/2 on JX . In this work we study the mass spectrum

of the baryons in the rest frame (~p = ~0) and therefore the last term of Eq. (15) vanishes. In order to extract masseswe consider two-point correlation functions defined by

CX(t, ~p = ~0) =1

4Tr (1 0)

Xxsink

hJX (xsink, tsink) JX (xsource, tsource)i, t = tsink tsource (16)

Space-time reflection symmetries of the action and the anti-periodic boundary conditions in the temporal directionfor the quark fields imply, for zero three-momentum correlators, that C+X(t) = CX(T t). Therefore in order todecrease errors we average correlators in the forward and backward direction and define

CX(t) = C+X(t) CX(T t) . (17)

Nucleon case average proton and neutron correlation functions

Anti-periodic boundary conditions in t-direction

Baryon Spectrum Overview of calculations


Fixing the lattice spacing HBPT (SSE Scheme)

17

one-loop order of the pion mass dependence for the nucleon mass in HBPT is given by

mN = m0N 4c1m2

3g2A32f2

m3 (20)

where m0N is the nucleon mass at the chiral limit and together with c1 are treated as fit parameters. This lowest orderresult for the nucleon in HBPT was first derived in Ref. [22] and successfully fitted lattice data are also discussed inprevious studies [23, 24]. This result is the same in HBPT with dimensional and infra-red regularization as well aswhen the degree of freedom is explicitly included. It is also the same in manifestly Lorentz-invariant formulationwith infra-red regularization. Therefore we will use this result to fix the lattice spacing using the nucleon mass asinput. This can also provide a non-trivial check of our lattice formulation. The lattice spacings a=1.90, a=1.95 anda=2.10 are considered as additional independent fit parameters in a combined fit of our data at = 1.90, = 1.95and = 2.10. We constrain our fit so that the fitted curve passes through the physical point by fixing the value of c1,hence the lattice spacings are determined using the nucleon mass at the physical point as the only input. The physicalvalues of f and gA are used in the fits, namely f = 0.092419(7)(25) and gA = 1.2695(29), which is common practicein chiral fits to lattice data on the nucleon mass [2527]. The left plot of Fig. 18 shows the fit to the O(p3) result ofEq. (20) on the nucleon mass. The error band and the errors on the fit parameters are obtained from super-jackknifeanalysis. As can be seen, the O(p3) result provides a very good fit to our lattice data, which in fact confirms thatcut-o and finite volume eects are small for the -values used. The results on the fit parameter m0N and the latticespacings a=1.90, a=1.95 and a=2.10 are collected in Table XI. The physical spatial lengths of the lattices used inthis calculation are obtained from the resulting lattice spacings of the fit and their values are labeled in Fig. ??.We perform the same analysis with higher order chiral corrections to Eq. (20). These O(p4) corrections are known

within several expansion schemes. In the so called small scale expansion (SSE) scheme [27], the -degress of freedomare explicitly included in covariant baryon PT by introducing as an additional counting parameter the -nucleonmass splitting, m mN , taking O(/mN ) O(m/mN ). In SSE the nucleon mass is given by

mN = m0N 4c1m2

3g2A32f2

m3 4E1()m4 3g2A + 3c

2A

642f2m

0N

m4 3g2A + 10c

2A

322f2m

0N

m4 logm

c

2A

32f2

1 +

2m0N

4m2 +

3 3

2m2

log

m2

+2 m2

R (m)

(21)

where R (m) = pm2 2 cos1

m

for m > and R (m) =

p2 m2 log

m

+q2

m2 1

for m < .

We take the cut-o scale = 1 GeV, c1 = 1.127 [27] and treat the counter-term E1 as an additional fit parameter.As in the O(p3) case we use the physical values of gA and f. The corresponding plot is shown on the right panel ofFig. 18. The error band as well as the errors on the fit parameters are obtained using super-jackknife analysis. Onecan see that this formulation provides a good description of the lattice data as well and yields values of the latticespacings and m0N which are consistent with those obtained in O(p3) of HBPT. The resulting parameters of this fitare given in Table XI.

FIG. 18. The nucleon mass as a function of m2 for = 1.90, = 1.95 and = 2.10 in O(p3) HBPT (left) and in O(p4) ofSSE scheme (right). The physical nucleon mass is shown with the black asterisk.

Baryon Spectrum

! = 2m0 = 0 (1)O(p3)O(p4)

1

Omega HBPT (leading order)

18

a=1.90 a=1.95 a=2.10 m0N E1()

2/d.o.f

O(p3) HBPT 0.0934(14) 0.0819(11) 0.0645(8) 0.8681(15) 1.4314O(p4) SSE 0.0972(19) 0.0858(17) 0.0672(12) 0.8826(48) -2.5849(2530) 0.9303

TABLE XI. Fit parameters a=1.90, a=1.95, a=2.10 in fm, m0N in GeV and E1() in GeV

3 from O(p3) HBPT and O(p4)SSE using the nucleon mass.

B. mass

Having determined that the tuning of the bare strange quark mass in the mixed action approach reproduces themass of the in the unitary setup, we perform a similar analysis as for the nucleon mass to fix the lattice spacingusing the mass as input. This could provide a cross-check of the results when using the nucleon mass and alsoconfirm that the tuning of the strange quark mass is optimal. In the case of the , the leading one-loop order HBPTresult reduces to the simple form

m = m0 4c1m2 (22)

where m0 is the mass at the chiral limit and as in the nucleon mass case, the value of c1 is kept fixed such as thefitted curve passes through the physical point. In Fig. 19 we show the fit on the result of Eq. (22) as a function of m2.In addition, we provide the resulting parameters of this fit in Table XII. The physical spatial lengths of the latticesused are obtained from the resulting lattice spacings of the fit and their values are shown in Fig. 19. As one can seewhen comparing the results from the nucleon and the , the lattice spacings a=1.95 and a=2.10 show consistency,while for = 1.90 the value is larger in the case and its outside error bars, when compared to the value extractedfrom the nucleon. To this end, we conclude that fixing the lattice spacing using the mass is not as reliable as usingthe nucleon, since the case encompasses assumptions which in principle may not be accurate. Such assumptionsinclude deviations of the physical kaon mass from the simulated kaon mass, which is used as input to perform thestrange quark tuning.

FIG. 19. The mass as a function of m2 for = 1.90, = 1.95 and = 2.10 in SU(2) HBPT. The physical mass is shownwith the black asterisk.

1.62

1.64

1.66

1.68

1.7

1.72

1.74

1.76

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

m

- (G

eV)

m2 (GeV2)

=1.90, L/a=32, L=3.3fm=1.90, L/a=24, L=2.5fm=1.95, L/a=32, L=2.8fm=2.10, L/a=48, L=3.2fm

Baryon Spectrum ! = 2m0 = 0 (1)O(p3)O(p4)

1


Documents

Master Final