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. Spin-Hadron@ RIKEN

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

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)

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

Simulation parameters b k volume a [fm] mp[GeV]  Nf =2 Wilson fermions w/ clover improvement  # of config: 400-2200 for each(b,k)  Physical unit translated by r0c / a O(a) improved operators  non-perturbative renormalization into MS @ m = 2 GeV 5.20 0.0856 0.13420 0.13500 0.13550 163  32 〃 1.347 0.956 0.670 5.25 0.0794 0.13460 0.13520 0.13575 0.13600 1.225 0.949 0.635 0.457 243  48 〃 163  32 〃 5.29 0.0753 0.13400 0.13500 0.13550 0.13590 0.13620 0.13632 1.511 1.102 0.857 0.629 0.414 0.345 243  48 〃 323  64 5.40 0.0672 0.13500 0.13560 0.13610 0.13625 0.13640 243  48 〃 1.183 0.917 0.648 0.559 0.450

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 = 1.3632 Dipole form: A10   -t [GeV2]

Chiral extrapolation and quark spin  A10u+d (t=0) : 2 s u+d = DSu+d  0.402 0.024 (@ mp = .14GeV) Strong mp dependence by “chiral log” term HERMES, PRD75(2007)012007  Heavy Baryon Chiral Perturbation Theory  M.Diehl, A.Manashov, A.Schäfer, EPJ.A 29(2006) mp 2[GeV2]

t dependence of the 2nd Moments A20u-d (t), B20u-d (t) and C20u-d (t) with b = 5.29,k = 1.3632 -t [GeV2]

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

Chiral extrapolation of A20u+d(t =0)  M.Dorati et.al. nucl-th/0703073 A20u+d (t=0)  x u+d  0.572 0.012 (@ mp = .14GeV)  CTEQ6 @m 2 = 4GeV2 mp 2[GeV2]

Chiral extrapolation of A20u-d(t =0)  x u-d  0.198 0.008 (@ mp = .14GeV) A20u-d (t=0)  CTEQ6 @m 2 = 4GeV2 Strong mp dependence by “chiral log” term mp 2[GeV2]

Generalized 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 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]

B20u+d( t ) and covariantized Baryon ChPT

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

B20u-d(t ) and covariantized Baryon ChPT

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

Chiral extrapolation of Ju , Jd Ji’s sum rule :  J q = 1/2 [ A20q (0) +B20q(0) ]  J u  0.230 0.008  J d  -0.004 0.008 (@ mp = .14GeV) mp 2[GeV2]

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  0.226 0.013  s u+d  0.201 0.024  L u+d  0.025 0.027 (@ mp = .14GeV) mp 2[GeV2]

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/0703073 leads  J u  0.230 0.008  J d  -0.004 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  0.226 0.013  s u+d  0.201 0.024 (@ mp = .14GeV)  L u+d  0.025 0.027