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Introduction Form Factors and Observables Numerical results of lattice simulation

格子 QCD 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. Munehisa Ohtani ( Univ. Regensburg ) for QCDSF with D. Br ö mmel, M. G ö ckeler, Ph. H ä gler, R. Horsley, Y. Nakamura, D. Pleiter, P.E.L. Rakow, A. Sch ä fer, G. Schierholz,

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Introduction Form Factors and Observables Numerical results of lattice simulation

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  1. 格子QCD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 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 • 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 20 Nov. 横断研究会@ KEK

  2. Introduction  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. Form Factors = H (x,0,0) = H (x,0,0) = HT(x,0,0)  F1(t) =dx H (x, x, t) GA(t)= dx H (x, x, t) GT(t)=dxHT(x, x, t)  local limit forward limit Fourier transf. Quark densityinb plane q(x, b2) = d2 D e–i Db H (x, x=0, D2) Generalized Parton Distributions PDF q(x) Dq(x) dq(x) GPDs as moments in the forward limit Angular momentum 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  1/2A20

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

  5.   A10u-d A10u+d B10u-d A20B20 gA 2 sq gP  xq 2 Jq-  xq  r 2A mp  mf2 ,ma2 ? ·DVCS - spin asymmetry - charge asymmetry ·nNscattering ·pion electroproduction ·muon capture GFF and physical quantities A10 B10u-d @ t = 0 Q Dk Slope, mpole  r 2EM  mV-2 ·eNscattering experiments Vector Meson Dominance ? PCAC & G-T relation Tensor Meson Dominance ?

  6. Simulation parameters b k volume a [fm] mp[GeV] • Nf =2Wilson fermionsw/ clover improvement • # of config: • 400-2200for each(b,k) • Physical unit translated byr0c/a • O(a) improved operators • non-perturbative • renormalization into • MS @ m = 2 GeV 5.20 0.0856 0.13420 0.13500 0.13550 1.347 0.956 0.670 163 32 〃 〃 〃 〃 5.25 0.0794 1.225 0.949 0.635 0.457 0.13460 0.13520 0.13575 0.13600 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 243 48 〃 〃 〃 〃 〃 1.183 0.917 0.648 0.559 0.450 0.13500 0.13560 0.13610 0.13625 0.13640

  7. cf. A10u-d from expt: p + e e' + p + + n V.Bernard et.al. J.Phys.G 28(2002)R1 Dipole form: A10  t dependence of Axial Form Factor  A10u+d (t) withb = 5.29,k = 1.3632 -t [GeV2]

  8.  0.402 0.024 (@ mp = .14GeV) Heavy Baryon Chiral Perturbation Theory M.Diehl, A.Manashov, A.Schäfer, EPJ.A 29(2006) Chiral extrapolation and quark spin  A10u+d (t=0) : 2 su+d= DSu+d Strongmp dependence by “chiral log” term • HERMES, PRD75(2007)012007  mp 2[GeV2]

  9. t dependence of the 2nd Moments A20u-d (t), B20u-d (t) andC20u-d (t) withb = 5.29,k = 1.3632 -t [GeV2]

  10. x ~1 contribution inA20 broader tail ofA20int Moments of GPD: A10 vs. A20 An0u+d (t)/ An0u+d (0) withb = 5.29,k = 1.3632 -t [GeV2]

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

  12.  3param. in each GFFs tdependence via Generalized Form Factors in Chiral Perturbation M.Dorati, T.A.Gail and T.R.Hemmert, nucl-th/0703073

  13. A20u+d(t ) and covariantized Baryon ChPT

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

  15. A20u-d(t ) and covariantized Baryon ChPT

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

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

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

  19. B20u+d (0) -0.120 0.023 (@ mp = .14GeV) Chiral extrapolation of B20u+d(0) B20u+d (0) Strongmp dependence by “chiral log” term mp 2[GeV2]

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

  21. B20u-d (0)0.269 0.020 (@ mp = .14GeV) Chiral extrapolation of B20u-d(0) B20u-d (0) mp 2[GeV2]

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

  23.  J u+d 0.226 0.013  s u+d 0.201 0.024  L u+d 0.025 0.027 (@ mp = .14GeV) decomposition of quark angular momentum • Ju+d = 1/2 [ A20u+d (0) +B20u+d(0) ]  su+d = 1/2 A20u+d (0) ; Lu+d = Ju+d- su+d  mp 2[GeV2]

  24. 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 BChPTnucl-th/0703073 leads •  J u 0.230 0.008 •  J d -0.004 0.008 • as a preliminary. • 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

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