1. overview GPD and nucleon structure Experiments – Deeply Virtual Compton Scattering –

Theory – Chiral Quark Soliton Model –

2. lattice QCD Axial FFs and quark spin Moments of GPD(vector) and total angular momentumSummary

格子格子 QCDQCD による核子の一般化パートン分布とによる核子の一般化パートン分布とクォーク角運動量の解析クォーク角運動量の解析

10 Jun. 2009@ Komaba

大谷 宗久 ( 杏林大学 )

IntroductionIntroduction

H, E, …P P'

x+ x －

momentum transfer squared:

t = (P' － P)2

longitudinal mt. transfer:

－ n ·

Generalized Parton Distributions of Nucleon

GPDs encode important information on Nucleon structure !

q q'

Intrinsic quark momentum: k Impact parameter: b Momentum transfer: q

Wigner distribution in phase spaceWigner distribution in phase space

dkW(k,b)

GPD H(x,b), E(x,b),…F.T

.

GPD H(x,, t), E(x,, t),…

Wigner fn.

k dep. dist. (TMD)Sivers fn.: f1T

(x, k),Collins fn.: H1

(x, k), …

db W(k,b)

X.Ji, PRL91(2003) A.Belitsky et.al., PRD69(2004)

W(k,b) ∝ dr dq eik·r-iq·bP+q/2|†r/2r/2|Pq/2

Generalized Parton DistributionsGeneralized Parton Distributions

PDF q(x) q(x) q(x)

Fourier transf.

Quark density in b plane

q(x, b) = d2 e–i b H (x, , )

= H (x,0,0) = H (x,0,0) = HT(x,0,0) forward

limit

Form Factors

F1(t) = dx H (x, , t)

GA(t) = dx H (x, , t)

GT(t) = dx HT(x, , t)

local limit

as moments in the forward limit

J q = 1/2 dx x (H(x, , 0) + E(x, , 0)) u+d 1/2 ( A20 + B20 )

s q = 1/2 dx H(x, , 0)u+d 1/2 A10

Angular momentum

X.Ji, PRL78(1997)

GPDs

DVCS Bethe-Heitler

Deeply Virtual Compton ScatteringDeeply Virtual Compton Scattering

e －

P

FiH,E,..

F1×H, F2×E, …

Beam Charge AsymmetryBeam Charge Asymmetry HERMES coll. PRD 75 (2007) 011103

: angle btw e scat. plane & prod. plane

Longitudinal Target Spin AsymmetryLongitudinal Target Spin Asymmetry CLAS coll. PRL 97 (2006) 072002

: angle btw e － scat. plane & prod. plane

GPDs in Chiral quark soliton modelGPDs in Chiral quark soliton model M.Wakamatsu & H.Tsujimoto, PRD71(2005)

H(x, , 0) + E(x, , 0) 1/2 x (H(x, , 0) + E(x, , 0) )

K.Goeke et.al., PRC75(2007)

x dependence ca ｌ culable no dynamical gluon

low energy scale

1. overview GPD and nucleon structure Experiments – Deeply Virtual Compton Scattering –

Theory – Chiral Quark Soliton Model –

2. lattice QCD Axial FFs and quark spin Moments of GPD and total angular momentumSummary

for UKQCD-QCDSF collab.

Moments of GPDMoments of GPD: : Generalized Form Factors Generalized Form Factors

An,2k , Bn,2k , Cn are related to P | DDn} |P'

X.Ji, J.Phys.G24(1998)1181 Polynomiality

LHPC, PRD68(2003)034505

Calculate ratio of 2pt & 3pt correlation functions on lattice

Extract GFF

Simulation parametersSimulation parameters

Nf =2 Wilson fermions w/ clover improvement

# of config: 400-2200 for each(,)

Physical unit translated by r0

c / a

O(a) improved operators

non-perturbative renormalization into MS @ = 2 GeV

1.3470.9560.670

0.0856163 32〃 〃〃〃

0.134200.135000.13550

5.20mGeVa [fm]volume

5.25

5.29

5.40

0.134600.135200.135750.136000.134000.135000.135500.135900.136200.136320.135000.135600.136100.136250.136400.13660

163 32〃

243 48〃

243 48〃〃〃

323 64

0.0794

0.0753

0.0672

1.2250.9490.6350.4571.5111.1020.8570.6290.4140.2791.1830.9170.6480.5590.451

243 48〃〃〃〃〃

323 64 0.255

V.Bernard et.al. J.Phys.G 28(2002)R1

cf. A10u-d

from expt: p + e e' + + n

t dependence of Axial Form Factort dependence of Axial Form Factor

t [GeV2]

Dipole form:

A10 22 )/1(

#

Amt

A10u+d (t) with = 5.29,= 1.3632

GFF and physical quantitiesGFF and physical quantities

A10 B10u-d

@ t = 0

Slope, mpole

Q

r 2 EM mV2

Vector Meson Dominance ?

PCAC &G-T relation

Tensor Meson Dominance ?

experiments · eN scattering

A10

u-d

gA gP

r 2 A m

A10u+d

2 s q

B10u-d

· N scattering· pion electroproduction· muon capture

A20 B20

x q 2 J q x q

mf2 , ma2 ?

· DVCS - spin asymmetry - charge asymmetry

0.402 0.024 (@ m= .14GeV)

Chiral extrapolation and quark spinChiral extrapolation and quark spin

A10u+d (t=0) 2 s u+d

= u+d

HERMES,

PRD75(2007)012007

Heavy Baryon Chiral Perturbation Theory

M.Diehl, A.Manashov, A.Schäfer, EPJ.A 29(2006)

m[GeV]

Strong m dependenceby “chiral log” term

t dependence of the 2t dependence of the 2ndnd Moments Moments

t [GeV2]

A20u-d (t), B20

u-d (t) and C20u-d (t) with = 5.29,= 1.3632

Chiral extrapolation of AChiral extrapolation of A2020u+du+d(t =0)(t =0)

A20u+d (t=0)

x u+d 0.563 0.014

(@ m= .14GeV)

M.Dorati et.al. nucl-th/0703073

m[GeV]

CTEQ6@2 = 4GeV2

Chiral extrapolation of AChiral extrapolation of A2020u-du-d(t =0)(t =0)

A20u-d (t=0)

x u-d 0.193 0.004 (@ m= .14GeV)

m[GeV]

CTEQ6 @2 = 4GeV2

Strong m dependenceby “chiral log” term

Generalized Form Factors in Chiral PerturbationGeneralized Form Factors in Chiral Perturbation M.Dorati, T.A.Gail and T.R.Hemmert, nucl-th/0703073

3 param. in each GFFs t dependence via

Dipole fit and forward limit of BDipole fit and forward limit of B2020u,du,d(t )(t )

B20q (t)

t [GeV2]

in CQSM

J u+d = 1/2

x u+d = 1 ) no dynamical gluon

2 J q x qt 0

K.Goeke et. al., PRC75(2007)in CQSM for comparison

BB2020u+du+d(( tt ) and covariantized Baryon ChPT) and covariantized Baryon ChPT

Chiral extrapolation of BChiral extrapolation of B2020u+du+d(0)(0)

B20u+d (0)

B20u+d (0) 0.120 0.023 (@ m= .14GeV)

m[GeV]

Strong m dependenceby “chiral log” term

BB2020u-du-d(t ) and covariantized Baryon ChPT(t ) and covariantized Baryon ChPT

Chiral extrapolation of BChiral extrapolation of B2020u-du-d(0)(0)

B20u-d (0)

B20u-d (0) 0.269 0.020 (@ m= .14GeV)

m[GeV]

m[GeV]

Chiral extrapolation of JChiral extrapolation of Ju u ,, JJdd

Ji’s sum rule : J q = 1/2 [ A20q (0) +B20

q(0) ]

J u 0.226 0.009 J d 0.005 0.009

(@ m= .14GeV)

decomposition of quark angular momentumdecomposition of quark angular momentum

m[GeV]

J u+d = 1/2 [ A20u+d (0) +B20

u+d(0) ]

s u+d = 1/2 A10u+d (0)

; Lu+d = J u+d s u+d

J u+d 0.222 0.014

s u+d 0.201 0.024

L u+d 0.021 0.028

(@ m= .14GeV)

Twisted boundary conditionTwisted boundary condition

q(xk +L) = e ik q(xk) t = (EpffLpiiL

2/L

Summary and outlookSummary and outlook

Generaized Parton Distribution spin content, transverse quark distribution, Form factors,…

- accessible experimentaly via DVCS - theoretical calculations in CQSM, Skyrme model,…

moments of GPD in lattice QCD - A20

u-d and B20u+d have strong “chiral log” corrections.

- Chiral extrapolation of A20 (0) & B20 (0) via BChPT nucl-th/0703073 leads J u 0.226 0.009 J d 0.005 0.009

- lighter m, larger volume (for t 0), Finite size corrections, Continuum limit, disconnected diagram, …

J u+d 0.222 0.014

s u+d 0.201 0.024 (@ m= .14GeV)

L u+d 0.021 0.028