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

格子 QCD 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. 大谷 宗久 ( 杏林大学 ). 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 momentum

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

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  1. 格子QCD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 大谷 宗久 (杏林大学 ) 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 momentum Summary 10 Jun. 2009@ Komaba

  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' • GPDs encode important information on Nucleon structure !

  3.  db W(k,b)  dkW(k,b) k dep. dist. (TMD) GPDH(x,b), E(x,b),… Sivers fn.: f1T(x, k), Collins fn.: H1(x, k), … F.T. GPDH(x,x, t), E(x,x, t),… Wigner distribution in phase space  X.Ji, PRL91(2003) A.Belitsky et.al., PRD69(2004) Wigner fn. W(k,b)∝ dr dq eik·r-iq·bP+q/2|y†(r/2) Gy(-r/2)|P- q/2 Intrinsic quark momentum:k Impact parameter: b Momentum transfer: q

  4. 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. as moments in the forward limit Quark densityinb plane q(x, b) = d2 D e–i Db H (x, x=0, D2) Angular momentum  X.Ji, PRL78(1997) 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  1/2A10 Generalized Parton Distributions PDF q(x) Dq(x) dq(x) GPDs

  5. Deeply Virtual Compton Scattering Bethe-Heitler DVCS g e－ Fi H,E,.. P F1×H, F2×E, …

  6. Beam Charge Asymmetry HERMES coll. PRD 75 (2007)011103 f : angle btw e scat. plane & prod. plane

  7. Longitudinal Target Spin Asymmetry CLAS coll. PRL 97 (2006)072002 f : angle btw e－scat. plane & prod. plane

  8.  K.Goeke et.al., PRC75(2007) GPDs in Chiral quark soliton model  M.Wakamatsu & H.Tsujimoto, PRD71(2005) 1/2x (H(x, 0, 0) + E(x, 0, 0) ) H(x, 0, 0) + E(x, 0, 0) x dependence caｌculable- no dynamical gluon- low energy scale

  9. forUKQCD-QCDSF collab. 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 momentum Summary

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

  11. 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 163 32 〃 〃 〃 〃 0.13420 0.13500 0.13550 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 0.279 243 48 〃 〃 〃 323 64 5.40 0.0672 0.13500 0.13560 0.13610 0.13625 0.136400.13660 1.183 0.917 0.648 0.559 0.451 243 48 〃 〃 〃 〃 〃323 64 0.255

  12. 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]

  13.   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 ?

  14.  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]

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

  16. M.Dorati et.al. nucl-th/0703073 CTEQ6 @m 2 = 4GeV2  xu+d 0.563 0.014 (@ mp = .14GeV) Chiral extrapolation of A20u+d(t =0) A20u+d (t=0) mp 2[GeV2]

  17.  xu-d 0.193 0.004 (@ 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]

  18.  3param. in each GFFs tdependence via Generalized Form Factors in Chiral Perturbation M.Dorati, T.A.Gail and T.R.Hemmert, nucl-th/0703073

  19. 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]

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

  21. 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]

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

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

  24.  J u 0.226 0.009  J d -0.005 0.009 (@ mp = .14GeV) Chiral extrapolation of Ju ,Jd Ji’s sum rule :  J q = 1/2 [ A20q(0) +B20q(0) ] mp 2[GeV2]

  25.  J u+d 0.222 0.014  s u+d 0.201 0.024  L u+d 0.021 0.028 (@ mp = .14GeV) 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  mp 2[GeV2]

  26. -t = (DE)2- [(pf+qf/ L) - (pi+qi/ L)]2  2p/L  Twisted boundary condition q(xk+L) = e iq kq(xk)

  27. Summary and outlook • Generaized Parton Distribution • spin content, transverse quark distribution, Form factors,… - accessible experimentaly viaDVCS - theoretical calculations in CQSM, Skyrme model,… • moments of GPD in lattice QCD -A20u-d and B20u+d have strong “chiral log” corrections. - Chiral extrapolation of A20(0) & B20(0)via BChPTnucl-th/0703073 leads •  J u 0.226 0.009 •  J d -0.005 0.009 • - lighter mp , 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 (@ mp = .14GeV)  L u+d 0.021 0.028

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