1. quark angular momentum in lattice QCD
Munehisa Ohtani ( Kyorin Univ., Regensburg Univ.) for UKQCD-QCDSF collab.
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
Introduction
Form Factors and Observables
Numerical results of lattice simulation
- Axial FF and quark spin
- Moments of GPD(vector) and total angular momentum
Summary
8 Apr. RIKEN

2. Introduction Non perturbative study on Nucleon structure
 Generalized Parton Distributions of Nucleon q q'
momentum transfer squared:
t = (D  P'－P)2
longitudinal mt. transfer:
 = －n ·D / 2
x+
x－
H, E, … P P'
Non perturbative study on Nucleon structure

3. Generalized Parton Distributions
PDF
Form Factors
F1(t) =  dx H (x, x, t)
GA(t) =  dx H (x, x, t)
GT(t) =  dx HT(x, x, t) 
local limit
q(x)
Dq(x)
dq(x) 
= H (x,0,0)
= HT(x,0,0)
forward limit
GPDs
J q = 1/2  dx x (H(x, x, 0) + E(x, x, 0)) u+d  1/2 ( A20 + B20 )
s q = 1/2  dx H(x, x, 0)u+d  1/2 A10 
Angular momentum
as moments in the
forward limit
Fourier transf.
Quark density in b plane
q(x, b2) =  d2 D e–i Db H (x, x=0, D2)

4. Moments of GPD: Generalized Form Factors
 Polynomiality
 X.-D.Ji, J.Phys.G24(1998)1181
An,2k , Bn,2k , Cn are related to  P |q g {m 1Dm 2  Dm n}q |P' 
 LHPC, PRD68(2003)034505
Calculate ratio of 2pt & 3pt
correlation functions on lattice
Extract GFF
Vector current, EM tensor

5. Simulation parametersb k
volume
a [fm]
mp[GeV]
 Nf =2 Wilson fermions w/ clover improvement
 # of config:
for each(b,k)
 Physical unit translated by r0c / a
O(a) improved operators
 non-perturbative
renormalization into
m = 2 GeV
5.20
0.0856
163  32 〃
1.347
0.956
0.670
5.25
0.0794
1.225
0.949
0.635
0.457
243  48 〃
163  32 〃
5.29
0.0753
1.511
1.102
0.857
0.629
0.414
0.345
243  48 〃
323  64
5.40
0.0672
243  48 〃
1.183
0.917
0.648
0.559
0.450

6. t dependence of Axial Form Factor
 V.Bernard et.al. J.Phys.G 28(2002)R1
cf. A10u-d from expt: p + e  e' + p + + n 
A10u+d (t) with b = 5.29,k =
Dipole form:
A10  
-t [GeV2]

7. Chiral extrapolation and quark spin
A10u+d (t=0) : 2 s u+d = DSu+d
 0.024 mp = .14GeV)
Strong mp dependence
by “chiral log” term
HERMES,
PRD75(2007) 
Heavy Baryon Chiral Perturbation Theory
 M.Diehl, A.Manashov, A.Schäfer, EPJ.A 29(2006)
mp 2[GeV2]

8. t dependence of the 2nd Moments
A20u-d (t), B20u-d (t) and C20u-d (t) with b = 5.29,k =
-t [GeV2]

9. Dipole mass of A20 and tensor meson
mD [GeV]
Dipole form:
A20 
x
(1 - t / mD2)2
in CQSM for comparison
 K. Goeke et. al., PRC75(2007) 
Observed
mass of f2

10. Chiral extrapolation of A20u+d(t =0)
 M.Dorati et.al. nucl-th/
A20u+d (t=0)
 x u+d  0.012
mp = .14GeV)
 CTEQ6
@m 2 = 4GeV2
mp 2[GeV2]

11. Chiral extrapolation of A20u-d(t =0)
 x u-d  0.008
mp = .14GeV)
A20u-d (t=0)
 CTEQ6
@m 2 = 4GeV2
Strong mp dependence
by “chiral log” term
mp 2[GeV2]

12. Generalized Form Factors in Chiral Perturbation
 M.Dorati, T.A.Gail and T.R.Hemmert, nucl-th/
 3 param. in each GFFs
 t dependence via

13. Dipole fit and forward limit of B20u,d(t )
 2 J q -  x q
t  0
B20q (t)
in CQSM
 J u+d = 1/2
 x u+d = 1
) no dynamical gluon 
 K.Goeke et. al., PRC75(2007)
in CQSM for comparison
-t [GeV2]

14. B20u+d( t ) and covariantized Baryon ChPT

15. Chiral extrapolation of B20u+d(0)
mp = .14GeV)
B20u+d (0)
Strong mp dependence
by “chiral log” term
mp 2[GeV2]

16. B20u-d(t ) and covariantized Baryon ChPT

17. Chiral extrapolation of B20u-d(0)
mp = .14GeV)
B20u-d (0)
mp 2[GeV2]

18. Chiral extrapolation of Ju , Jd
Ji’s sum rule :  J q = 1/2 [ A20q (0) +B20q(0) ]
 J u  0.008
 J d  0.008
mp = .14GeV)
mp 2[GeV2]

19. decomposition of quark angular momentum
J u+d = 1/2 [ A20u+d (0) +B20u+d(0) ]
 s u+d = 1/2 A10u+d (0)
; Lu+d =  J u+d -  s u+d 
 J u+d  0.013
 s u+d  0.024
 L u+d  0.027
mp = .14GeV)
mp 2[GeV2]

20. Summary 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 B20u+d have strong “chiral log” corrections.
Chiral extrapolation of A20 (0) & B20 (0) via BChPT nucl-th/ leads
 J u  0.008
 J d  0.008
lighter mp , larger volume (for t 0), Finite size corrections, Continuum limit, disconnected diagram, Chirally improved fermions, higher twist, angular momentum of gluon, …
 J u+d  0.013
 s u+d  0.024 mp = .14GeV)
 L u+d  0.027