• Introduction • Form Factors and Observables • Numerical results of lattice simulation - Axial FFs and quark spin - 2nd Moments of GPD(vector) and total angular momentum• Summary

格子格子 QCDQCD による 核子の一般化形状因子とによる 核子の一般化形状因子とクォーク角運動量の解析クォーク角運動量の解析

20 Nov. 横断研究会 @ KEK

Munehisa Ohtani (Univ. Regensburg ) for

QCDSF

with D. Brömmel, M. Göckeler, Ph. Hägler, R. Horsley, Y. Nakamura, D. Pleiter, P.E.L. Rakow, A. Schäfer, G. Schierholz, W. Schroers, H. Stüben, J.M. Zanotti

IntroductionIntroduction

H, E, …P P'

x+ x －

momentum transfer squared:

t = (P' － P)2

longitudinal mt. transfer:

－ n ·

Generalized Parton Distributions of Nucleon

Non perturbative study on Nucleon structure

q q'

Generalized Parton DistributionsGeneralized Parton Distributions

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

Fourier transf.

Quark density in b plane

q(x, b2) = d2 e–i b H (x, , )

GPDs

= 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

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 A20

Angular momentum

as moments in the forward limit

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

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

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

LHPC, PRD68(2003)034505

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

Extract GFF

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 q2 J q x q

mf2 , ma2 ?

· DVCS - spin asymmetry - charge asymmetry

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

0.134600.135200.135750.136000.134000.135000.135500.135900.136200.13632

0.135000.135600.136100.136250.13640

163 32〃

243 48〃

243 48〃〃〃

323 64

0.0794

0.0753

0.0672

1.2250.9490.6350.457

1.5111.1020.8570.6290.4140.345

1.1830.9170.6480.5590.450

243 48〃〃〃〃〃

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

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

Moments of GPD: AMoments of GPD: A1010 vs. A vs. A2020 An0

u+d (t)/ An0u+d (0) with = 5.29,= 1.3632

t [GeV2]

x ~ 1 contribution in A20

broader tail of A20 in t

Dipole mass of ADipole mass of A20 20 and tensor mesonand tensor meson

mD[GeV]

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

Observedmass of f2

Dipole form:

A20 x .

(1 t / mD2)2

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

AA2020u+du+d(t ) and covariantized Baryon ChPT(t ) and covariantized Baryon ChPT

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

A20u+d (t=0)

x u+d 0.572 0.012

(@ m= .14GeV)

CTEQ6 @2 = 4GeV2

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

m[GeV]

AA2020u-du-d(t ) and covariantized Baryon ChPT(t ) and covariantized Baryon ChPT

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

A20u-d (t=0)

x u-d 0.198 0.008 (@ m= .14GeV)

m[GeV]

Strong m dependenceby “chiral log” term

CTEQ6 @2 = 4GeV2

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 0.230 0.008

J d 0.004 0.008

(@ 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 A20u+d (0)

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

J u+d 0.226 0.013

s u+d 0.201 0.024

L u+d 0.025 0.027

(@ m= .14GeV)

Summary and outlookSummary and outlook

lattice simulation of moments of Generaized Parton Distribution spin content, transverse quark distribution, DVCS,…

dipole mass of A20 is comparable with tensor meson mass.

A20u-d and B20

u+d have strong “chiral log” corrections.

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

0.230 0.008 J d

0.004 0.008

as a preliminary.

lighter m, larger volume (for t 0), Finite size corrections, Continuum limit, disconnected diagram, Chirally improved fermions, higher twist, angular momentum of gluon, …

J u+d 0.226 0.013

s u+d 0.201 0.024 (@

m= .14GeV)

L u+d 0.025 0.027