II. Generalized Parton Distributions (GPDs)

1. II. Generalized Parton Distributions (GPDs) MarcVanderhaeghen Johannes Gutenberg Universität, Mainz HANUC, Jyväskylä, August 25-29, 2008

2. Outline : GPDs 1) link of FF to Generalized Parton Distributions : nucleon “tomography” 2) GPDs and spin of nucleon 3) Hard exclusive processes : DVCS, …

3. Generalized Parton Distributions : yield 3-dim quark structure of nucleon Burkardt (2000,2003) Belitsky,Ji,Yuan (2004) Elastic Scattering transverse quark distribution in coordinate space DIS longitudinal quark distribution in momentum space DES (GPDs) fully-correlated quark distribution in both coordinate and momentum space

4. * Q2 >> t = Δ2 x + ξ x - ξ P - Δ/2 P + Δ/2 GPD (x, ξ ,t) Generalized Parton Distributions low –t process : -t << Q2 Δ+ = - (2 ξ) P+ light-cone dominance, nμ(1, 0, 0, -1) / (2 P+)

5. Δ P - Δ/2 P + Δ/2 known information on GPDs forward limit : ordinaryparton distributions unpolarized quark distr polarized quark distr : do NOT appear in DIS new information first moments : nucleonelectroweak form factors Dirac Pauli axial ξ independence : Lorentz invariance pseudo-scalar

6. x +1 -1 -ξ 0 ξ GPDs : x and ξdependence forward parton distr Hu (x,ξ,0) x + ξ x - ξ quark distribution anti-quark distribution q q distribution amplitude

7. Why GPDs are interesting Unique tool to explore the internal landscape of the nucleon : 3D quark/gluon imaging of nucleon Access to static properties : constrained (sum rules) by precision measurements of charge/magnetization orbital angular momentum carried by quarks

8. GPDs : 3D quark/gluon imaging of nucleon Fourier transform of GPDs : simultaneous distributions of quarks w.r.t. longitudinal momentum x P and transverse positionb theoretical parametrization needed

9. GPDs : t dependence modified Regge parametrization : Guidal, Polyakov, Radyushkin, Vdh (2004) Input : forward parton distributions atm2 = 1 GeV2 (MRST2002 NNLO) Drell-Yan-West relation : exp(- α΄ t ) -> exp(- α΄ (1 – x) t) : Burkardt (2001) parameters : regge slopes : α’1 = α’2 determined from rms radii determined from F2 / F1 at large -t future constraints : moments from lattice QCD

10. proton & neutron charge radii Regge model modified Regge model experiment consistent with slopes of meson Regge trajectories ( ~ 0.9 GeV-2 ) Regge slope

11. connection large Q2 of FF <-> large x of GPD at large Q2 : integral dominated by maximum of f(x,Q2), remainder region is exp. suppressed (method of steepest descent) f(x,Q2) reaches maximum for : “Drell-Yan-West” relation for PDF/GPD : at large Q2 : I is dominated by its behavior around x -> 1

12. electromagnetic form factors PROTON NEUTRON world data (2006) modified Regge GPD parameterization 1 : Regge slope -> protonDirac (Pauli) radius 2, 3 :large x behavior of GPD Eu, Ed ->large Q2 behavior of F2p, F2n 3-parameter fit ηu = 1.713,ηd = 0.566 Guidal, Polyakov, Radyushkin, Vdh (2005) also Diehl, Feldmann, Jakob, Kroll (2005)

13. scaling behavior of N and N -> Δ F.F. PQCD N N, Δ + collinear quarks F1p ~ 1/Q4 F2p / F1p ~ 1/Q2 GM* ~ 1/Q4 GPD modified Regge model Guidal, Polyakov, Radyushkin, Vdh (2005)

14. GPDs : 3D quark/gluon imaging of nucleon Fourier transform of GPDs : simultaneous distributions of quarks w.r.t. longitudinal momentum x P and transverse positionb

15. GPDs : transverse image of the nucleon (tomography) Hu(x, b? ) x Guidal, Polyakov, Radyushkin, Vdh, PRD 72, 054013 (2005) b?(fm)

16. Moments of flavor-NS GPDs Lattice results : m = 740 MeV x b (fm) LHPC/SESAM Decrease slope : decreasing transverse size as x -> 1Burkardt -T2

17. lattice QCD for moment of PDFs : present state-of-the-art full lattice QCD : domain wall valence quarks, 2+1 flavor staggered sea quarks LHPC (2007) Ratio: EXPERIMENT / Lattice Lattice (normalized to one) isovector quantities u - d (no disconnected contributions)

18. y3 Handbag (bilocal) operator : new way to probe the nucleon Y- y0 y 0 generalized probe ( Y ≈ 0 ) ( W±, Z0 ) probe spin 2 (graviton) probe electroweak form factors energy-momentum form factors

19. Δ P - Δ/2 P + Δ/2 Energy momentum form factors G nucleon in external classical gravitational field G couples toenergy-momentum tensor Gordon identity

20. Momentum sum rule

21. Momentum sum rule (cont.) Total system : energy-momentum conservation A(0) = 1 Physical interpretation in terms of : quarks : Aq(0) Aq(0) + Ag(0) = 1 & gluons : Ag(0)

22. Angular momentum sum rule consider N in rest frame : Pμ (M,0,0,0) Sμ (0,0,0,1)

23. Angular momentum sum rule (cont.) M 0 in rest frame 2 M

24. Angular momentum sum rule (cont.) Total system : angular momentum conservation A(0) + B(0) = 1 B(0) = 0 Physical interpretation in terms of quarks & gluons ½ = Jq + Jg = ½ ΔΣ + Lq + Jg ½ΔΣ Lq

25. Tμν form factors in terms of GPDs of both lhs and rhs

26. Δ P - Δ/2 P + Δ/2 Energy momentum form factors /spinof nucleon G nucleon in external classical gravitational field G couples toenergy-momentum tensor link to GPDs : X. Ji (1997) SPIN sum rule

27. quark contribution to proton spin X. Ji (1997) with parametrizations for E q : GPD : based on MRST2002 μ2 = 2 GeV2 lattice : full QCD, no disconnected diagrams so far

28. orbital angular momentumcarried by quarks : solving thespin puzzle with Ji (1997) evaluated at μ2 = 2.5 GeV2

29. Deeply Virtual Compton Scattering * Q2 large t = Δ2 low –t process : -t << Q2 + diagram with photons crossed x + ξ x - ξ P - Δ/2 P + Δ/2 GPD (x, ξ ,t)

30. DVCS (cont.) twist-2 DVCS amplitude independent of Q SCALING ! DVCS amplitude is complex : imaginary part -> x = ξ Q2 , ξ, and t are kinematic variables with variable x is integrated over ( x ≠ xB ! ) DVCS amplitude is sensitive to GPDs weighted with coefficient functions

31. DVCS : beam spin asymmetry ALU = (BH) * Im(DVCS) * sin Φ Bethe-Heitler DVCS Q2 = 1 – 1.5 GeV2 , xB = 0.15 – 0.25, -t = 0.1 - 0.25 GeV2 Q2 = 2.6 GeV2 , xB = 0.11, -t = 0.27 GeV2 JLab/CLAS (2001) HERMES (2001) twist-2 + twist-3 : Kivel, Polyakov, Vdh (2000)

32. Bethe-Heitler DVCS on proton DVCS JLab/Hall A @ 6 GeV(2006) GPDs Difference of polarized cross sections Q2 ≈ 2 GeV2 xB = 0.36 Unpolarized cross sections also JLab/CLAS, HERMES, H1 / ZEUS

33. DVCS on neutron 0 because F1(t) is small 0 because of cancelation of u and d quarks n-DVCSgives access to the least known and constrained GPD,E JLab /Hall A (E03-106) : preliminary data

34. DVCS : target spin asymmetry proton JLab/CLAS mainly sensitive to GPD calculation : Vdh, Guichon, Guidal (1999)

35. Hard electroproduction of mesons (ρ0,± , ω , φ, π, …) M * TH GPDs hard scattering amplitude Factorization theorem shown for longitudinal photon Collins, Frankfurt, Strikman (1997)

36. Meson acts as helicity filter longitudinally pol. Vector meson PseudoScalar meson M M ± TH TH Vector meson : accesses unpolarized GPDs H and E ~ ~ PseudoScalar meson : accesses polarized GPDs H and E

37. Hard electroproduction of vector mesons (ρ0,± , ω , φ) amplitude for longitudinally polarized vector meson leading (1 gluon exchange) amplitude depends on αsgoes as 1/Q dependence on meson distribution amplitudeΦV

38. Flavor decomposition of GPDs H and E ρ0 ρ± ω

39. Hard electroproduction of mesons : target normal spin asymmetry lepton plane hadron plane in leading order (in Q)2 observables Asymmetry : Target polarization normal to hadron plane

40. Hard electroprod. of vector mesons : target normal spin asymmetry A GPD H B GPD E unpolarized cross section linear dependence on GPD E ratio : less sensitive to NLO and higher twist effects sensitivity to Ju and Jd measure of TOTAL angular momentum contribution to proton spin

41. L=1035cm-2s-1 2000hrs sL dominance DQ2 =1 Dt = 0.2 -t = 0.5GeV2 exclusive 0 prod. with transverse target 2D (Im(AB*))/p T A~ (2Hu +Hd) AUT = - r0 |A|2(1-x2) - |B|2(x2+t/4m2) - Re(AB*)2x2 B~ (2Eu + Ed) AUT Q2 = 5GeV2 r0 Asymmetry depends linearly on the GPD E, which enters Ji’s sum rule. K. Goeke, M.V. Polyakov, M. Vdh (2001) xB