1. Partonic Interpretation of GPDs Gary R. Goldstein Tufts University Simonetta Liuti University of Virginia Presentation for “QCD Evolution Workshop: from collinear to non-collinear case” Some of these ideas were developed in Trento ECT* & INT & JLab Titian (1543)- Pope Paul III presided over Council of Trent 1 1

2. Partonic Interpretation of GPDs Gary R. Goldstein Tufts University Simonetta Liuti University of Virginia Presentation for “QCD Evolution Workshop: from collinear to non-collinear case” Some of these ideas were developed in Trento ECT* & INT & JLab Thomas Jefferson 2 2

3. 3 04/08/2010QCD JLab Apr.2011 GR.Goldstein Outline of Discussion Dispersion Relations, DVCS & GPDs - DGLAP & ERBL – Thresholds & DRs  Unitarity & analyticity  Kinematic restrictions (Bjorken limit) Not target mass corrections.  Model calculations ERBL region & partons  Intermediate states  “semi-disconnected” diagrams  Abhor the vacuum!  Non-replacement for  >0. Beyond models. Spin dependence & spectator models: longitudinal & transverse quarks  Scalar diquark & chiral even  chiral odd relations  Axial diquark & different chiral even  chiral odd New parameterization  Reggeized spectator model  constrained by pdf’s, Form Factors, sum rules See: GG, Liuti, PRD80, 071501 (2009). GG, Liuti, ArXiv hep-ph/1006.0213

4. 4 t DVCS & DVMP γ*(Q 2 )+P→(γ or meson)+P’ partonic picture P' + =(1- ζ )P + P’ T =-Δ k + =XP + k' + =(X- ζ )P + q q+ Δ k' T =k T - Δ kTkT P+P+ } { ζ→0 Regge Quark-spectator quark+diquark Factorized “handbag” picture } X>ζ DGLAP X<ζ ERBL What is the meaning of k' + <0 or = 0? see GG&S.Liuti, arXiv:1006.0213 3/9/2016

5. 5 Momentum space nucleon matrix elements of quark correlators see, e.g. M. Diehl, Eur. Phys. J. C 19, 485 (2001).

6. 6 Chiral odd GPDs Eqns connecting GPD & helicity amps - M. Diehl, Eur.Phys.J.C19 (2001) 485; Boglione & Mulders, Phys.Rev.D 60 (1999) 054007. 3/9/2016

7. 7 04/08/2010QCD JLab Apr.2011 GR.Goldstein N NN  *  N NN  N Factorized “handbag” parton picture Convert kinematics s-channel unitarity hadron picture Imaginary part for on-shell intermediate states What about Analyticity? GPDs are real functions of x, ,t,Q 2. Functions of many variables - complex x not  & t? Connection of Dispersion Relations to GPDs? Consider Virtual Compton scattering from hadronic view. P+P+ XP +,k T (X-  )P +,k T -Δ T (1-  )P +,-Δ T q q( Q 2 ) qq qq GPD ∫ dk - d 2 k T   t

8. 8 04/08/2010QCD JLab Apr.2011 GR.Goldstein Dispersion relations: constraints on GPD modeling & experimental extraction Deeply Virtual Compton Scattering (DVCS) amplitudes satisfy unitarity, Lorentz convariance & analyticity (Long history - 1950’s, forward, fixed t, Lehmann ellipse, double DRs & Mandelstam Rep’n, Regge poles, duality,... ) (Generic) T( ,t,Q 2 ) (  =(s-u)/4M) has  or s & u analytic structure determined by nucleon pole & hadronic intermediate on-shell states - completeness d  /dt ∝ |T( ,t,Q 2 )| 2 s-channel unitarity hadron picture Imaginary part for on-shell intermediate states

9. 9 04/08/2010QCD JLab Apr.2011 GR.Goldstein Hadron to parton variables: s=(P+q) 2 =M 2 -Q 2 +2M  Lab  =(s-u)/4M=(2s+t+Q 2 -2M 2 )/4M =  Lab +(t - Q 2 )/4M x BJ = Q 2 /2M  Lab, X=k  P/q  P,  =q  P  /q  P  =  /(2-  ), x=(2X-  )/(2-  )  =Q 2 /4M  Integration variable x=Q 2 /4M    Teryaev; Anikin & Teryaev; Ivanov & Diehl; Vanderhaeghen, et al.; M ü ller, et al.; Brodsky, et al. Compton Form Factor

10. Dispersion Relations (Anikin, Teryaev, Diehl, Ivanov, Vanderhaeghen…)  H ( x,x,t ) + C(t) G.Goldstein and S.L.,PRD’09 (graph: D. Müller) All information contained In the “ridge” x=  ? Branch cut -1<x<+1.  QCD JLab Apr.2011 GR.Goldstein

11. The amplitudes are analytic --in the chosen kinematical variables,, x Bj,,s -- except where the intermediate states are on shell Dispersion Theory k + =XP + P+P+ k’ + =(X-  )P + P X + =(1-X)P + cut ep  eX DIS pole, =Q 2 /2M ep  e’p QCD JLab Apr.2011 GR.Goldstein

12. From DR + Optical Theorem to Mellin moments expansion OPE is seeded in DRs (see e.g. Jaffe’s SPIN Lectures)  =1/x QCD JLab Apr.2011 GR.Goldstein

13. Because t  0, the quark + spectator’s kinematical “physical threshold” does not match the one required for the dispersion relations to be valid DVCS: Where is the threshold? G.G and S.Liuti, PRD’09  Continuum starts at s =(M+m  ) 2  lowest hadronic threshold. Continuum threshold QCD JLab Apr.2011 GR.Goldstein

14. 14 04/08/2010QCD JLab Apr.2011 GR.Goldstein Where is threshold?  Imaginary part from discontinuity across unitarity branch cut - from  threshold for production of on-shell states - e.g.  N,  N,  Δ, etc.  Continuum starts at s =(M+m  ) 2  lowest hadronic threshold.  Physical region for non-zero Q 2 and t differs from this.  How to fill the gap? s=(M+m  ) 2  t 00  physical

15. (GeV)   Physical region Q 2 =1.0 GeV 2 -t (GeV 2 ) s=(M+m   

16. (GeV)   Physical region Q 2 =1.0 GeV 2 -t (GeV 2 ) s=(M+m    Physical region Q 2 =1.0 GeV 2 Lab

17. (GeV)   Physical region Q 2 =1.0 GeV 2 -t (GeV 2 ) s=(M+m    Physical region Q 2 =1.0 GeV 2 Lab (GeV) Lab  -t=0.1 Physical region Q 2 =1.0 GeV 2 Blue arrow shows gap between continuum threshold for t=-0.1 and physical threshold where measurements begin. Gap of 0.9 GeV. For -t>0.6 no small |t| gap. What about higher & Q 2 ? 

18. Lab Physical region Q 2 =5.5 GeV 2  -t=0.1 Blue arrow shows gap between continuum threshold for t=-0.1 and physical threshold. Gap of 6.9 GeV. For -t > 4.0 no |t| gap until backward at 5.7. Physical region Q 2 =5.5 GeV 2

19. Physical region Q 2 = 20 GeV 2 min Lab   Blue arrow shows gap between continuum threshold for t=-0.1 and physical threshold. Gap of 30 GeV. For -t > 10.0 no |t| gap until backward at 20+. min Lab at -t from 10 to 20 is nearly flat & has value approximately Q 2 /2M.

20. physical  How does one fill the gap? Analytic continuation?…problem for experimental extraction.  Range of x limited experimentally.  What is reasonable scheme for continuation? Model dependence. Physical threshold obtained by imposing  T 2 > 0 (same as t min = -M 2  2 /(1-  ) 2 ) Q 2 =2.0 GeV 2  -t  Physical s=(M+m  ) 2 continuum QCD JLab Apr.2011 GR.Goldstein

21. 21 04/08/2010QCD JLab Apr.2011 GR.Goldstein Examples of DVCS dispersion integrals & threshold dependences Regge form for H(X, ,t) or H R ( ,Q 2,t) =  (t,Q 2 )(1-e -i  (t) )(  /  0 )  (t) So ReH R ( ,Q 2,t) = tan(  (t)/2) ImH( ,Q 2,t) Dispersion: This is exact for  Threshold =0. For Q 2,  Threshold = -t/4M this is asymptotic.

22. Difference Direct Dispersion Regge Model QCD JLab Apr.2011 GR.Goldstein Diquark Model

23. SUMMARY OF PART 1 Dispersion relations cannot be directly applied to DVCS because one misses a fundamental hypothesis: “all intermediate states need to be summed over” For DVCS one is forced to look into the nature of intermediate states because there is no optical theorem This happens because “t” is not zero and there is a mismatch between the photons initial and final Q 2  t- dependent threshold cuts out physical states “counter-intuitively as Q 2 increases the DRs start failing because the physical threshold is farther away from the continuum” QCD JLab Apr.2011 GR.Goldstein

24. q k' + =(X-  )P +, k T -  T q'=q+  k + =XP +, k T P' + =(1-  )P +, -  T P+P+ DVCS Kinematics What about X=  -- returning quark with no + momentum (stopped on light cone) & X<  “central” or ERBL region How to interpret? ERBL similar to contributions other than the parton model mentioned by Jaffe (1983) QCD JLab Apr.2011 GR.Goldstein

25. Interpreting off-forward GPDs see Jaffe NPB229, 205(1983) For pdf’s - forward elastic Relate T-product to cut diagrams partons emerge + Completeness QCD JLab Apr.2011 GR.Goldstein

26. Re k - Im k - 3'1' 2' Re k - Im k - 2'3' 1' X>  X<  3 1 2 Diquark spectator on-shell struck quark on-shell Returning quark =outgoing anti-q QCD JLab Apr.2011 GR.Goldstein Singularities in covariant picture along with s and u cuts

27. P+P+ k’ + =(  -X)P + P’ + =(1-  )P + In ERBL region struck quark, k, is on-shell k + =XP + P+P+ k’ + =(X-  )P + P’ + =(1-  )P + P X + =(1-X)P + In DGLAP region spectator with diquark q. numbers is on-shell Analysis done for DIS/forward case by Jaffe NPB(1983) In forward kinematics can use alternative set of connected diagrams that have partonic interpretation. QCD JLab Apr.2011 GR.Goldstein

28. P+P+ k’ + =(  -X)P + P’ + =(1-  )P + ERBL region corresponds to semi-disconnected diagrams: no partonic interpretation What is meaning of diagram? note: Diehl & Gousset (‘98) - for off-forward generalization of Jaffe’s argument, still can replace T-ordered  † (z)  (0) with cut diagrams & interpret ERBL region as q+anti-q QCD JLab Apr.2011 GR.Goldstein

29. Is there a way to restore a partonic interpretation? We consider multiparton configurations  FSI k + =XP + P+P+ P X + =(1-X)P + k + =XP + P' + =(1-  )P + l + =(X-X')P + = y P + P' X + =(1-X')P + P X + =(1-X)P + P+P+ k' + =(X'-  )P + BA 2'  3 2 1 3' 1' Planar QCD JLab Apr.2011 GR.Goldstein

30. k + =XP + P+P+ P X + =(1-X)P + k + =XP + P' + =(1-  )P + l + =(X-X')P + = y P + P' X + =(1-X')P + P X + =(1-X)P + P+P+ k' + =(X'-  )P + BA 2'  3 2 1 3' 1' k + =XP + P+P+ P X + =(1-X)P + k + =XP + P' + =(1-  )P + l + =(X-X')P + = y P + P' X + =(1-X')P + P X + =(1-X)P + P+P+ k' + =(X'-  )P + BA 2'  3 2 1 3' 1' Non-Planar FSI“promotion” QCD JLab Apr.2011 GR.Goldstein

31. Summary of part 2: GPDs in ERBL region can be described within QCD, consistently with factorization theorems, only by multiparton configurations, possibly higher twist. QCD JLab Apr.2011 GR.Goldstein

32. 32 How to determine GPDs? Flexible Parameterization  Recursive Fit O. Gonzalez Hernandez, G. G., S. Liuti, arXiV:1012.3776 Constraints from Form Factors Dirac Pauli, etc. How can these be independent of ξ? Constraints from Polynomiality & pdf’s Result of Lorentz invariance & causality. Not necessarily built in to models 3/9/2016 QCD JLab Apr.2011 GR.Goldstein

33. Spectator inspired model of GPDs 3/9/2016 QCD JLab Apr.2011 GR.Goldstein 33

34. k + =XP + k’ + =(X-  )P + P’ + =(1-  )P + P X + =(1-X)P + Vertex Structures P+P+ P X + =(1-X)P +  S=0 or 1 Focus e.g. on S=0 Vertex function  note that by switching λ  - λ & Λ  -Λ (Parity) will have chiral evens go to ±chiral odds giving relations shown Before k integration A(Λ’λ’;Λλ)   ±A(Λ’,λ’;-Λ,-λ) but then (Λ’-λ’)-(Λ-λ) ≠(Λ’-λ’)+(Λ-λ) unless Λ=λ

35. recall that helicity amps are linear combinations of GPDs 3/9/2016 QCD JLab Apr.2011 GR.Goldstein 35 In diquark spectator models A ++;++, etc. are calculated directly. Can be inverted -> GPDs

36. scalar diquark model (with vertex Γ(k) for N  quark+diquark) 3/9/2016 QCD JLab Apr.2011 GR.Goldstein 36

37. the k + integration sets k + = XP + k - integration is determined by propagator poles in k - complex plane  in DGLAP region (x>ζ) diquark is on-shell (pole stands alone) k azimuthal dependence can be integrated analytically (only enters in k T  T terms) Leaves k T integration 2 shown here: get relations: A ++,-+ = - A ++,+-, A -+,++ = - A +-,++, A ++,++ = A ++,-- 3/9/2016 QCD JLab Apr.2011 GR.Goldstein 37

38. Invert to obtain model for GPDs 3/9/2016 QCD JLab Apr.2011 GR.Goldstein 38 A ++,-+ = - A ++,+-, A -+,++ = - A +-,++, A ++,++ = A ++,--

39. Fitting Procedure e.g. for H and E  Fit at  =0, t=0  H q (x,0,0)=q(X)  3 parameters per quark flavor (M X q,  q,  q ) + initial Q o 2  Fit at  =0, t  0   2 parameters per quark flavor ( , p)  Fit at  0, t  0  DVCS, DVMP,… data (convolutions of GPDs with Wilson coefficient functions) + lattice results (Mellin Moments of GPDs)  Note! This is a multivariable analysis  see e.g. Moutarde, Kumericki and D. Mueller, Guidal and Moutarde Regge Quark-Diquark

40. Reggeized diquark mass formulation Following DIS work by Brodsky, Close, Gunion (1973) Diquark spectral function  MX2MX2  (M X 2 )   (M X 2 -M X 2 )

41. Flexible Parametrization of Generalized Parton Distributions from Deeply Virtual Compton Scattering Observables Gary R. Goldstein, J. Osvaldo Gonzalez Hernandez, † and Simonetta Liuti arXiv:1012.3776

43. Hall B dataHall B data

44. Hermes data

46. 46 04/08/2010QCD JLab Apr.2011 GR.Goldstein Lessons from examples Moderate reach of E  (Jlab<12 GeV) or s<25 GeV 2 =(5 GeV) 2 not far from t-dependent thresholds Non-forward dispersion relations require some model-dependent analytic continuation Difference between direct & dispersion Real H( ,t) depends on thresholds DVCS & Bethe-Heitler interference via cross sections & asymmetries measures complex H( ,t) directly Interpreting ERBL “partonic” picture unclear - higher twist? FSI? N  q+anti-q +N’ ? N  Meson + N’ ? ERBL region exists. Significance? Model using (anti)symmetric form &/or fsi.