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Partonic Interpretation of GPDs

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.

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Partonic Interpretation of GPDs

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  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

  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

  3. 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 04/08/2010 QCD JLab Apr.2011 GR.Goldstein 3

  4. DVCS & DVMP γ*(Q2)+P→(γ or meson)+P’partonic picture q q+Δ } { k'T=kT-Δ kT k+=XP+ k'+=(X-ζ)P+ P+ P'+=(1-ζ)P+ P’T=-Δ } Factorized “handbag” picture t ζ→0 Regge Quark-spectator quark+diquark X>ζ DGLAP X<ζ ERBL What is the meaning of k'+<0 or = 0? see GG&S.Liuti, arXiv:1006.0213

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

  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.

  7. What about Analyticity? GPDs are real functions of x,,t,Q2. Functions of many variables - complex x not  & t? Connection of Dispersion Relations to GPDs? Consider Virtual Compton scattering from hadronic view. q γ * q′ γ q(Q2) q′ γ * γ γ ∫dk -d2kT XP+,kT (X-ζ)P+,kT -ΔT N GPD P+ N′ N N′ N t (1-ζ)P+,-ΔT Factorized “handbag” parton picture Convert kinematics s-channel unitarity hadron picture Imaginary part for on-shell intermediate states 04/08/2010 QCD JLab Apr.2011 GR.Goldstein 7

  8. 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(ν,t,Q2) (ν=(s-u)/4M) has ν or s & u analytic structure determined by nucleon pole & hadronic intermediate on-shell states - completeness dσ/dt ∝ |T(ν,t,Q2)|2 s-channel unitarity hadron picture Imaginary part for on-shell intermediate states 04/08/2010 QCD JLab Apr.2011 GR.Goldstein 8

  9. Hadron to parton variables: s=(P+q)2=M2-Q2+2MνLab ν=(s-u)/4M=(2s+t+Q2-2M2)/4M = νLab+(t - Q2)/4M xBJ= Q2/2MνLab , X=k⋅P/q⋅P, ζ=q⋅P′/q⋅P ξ=ζ/(2-ζ), x=(2X-ζ)/(2-ζ) ξ=Q2/4Mν Integration variable x=Q2/4Mν′ • Teryaev; Anikin & Teryaev; Ivanov & Diehl; Vanderhaeghen, et al.; Müller, et al.; Brodsky, et al. Compton Form Factor 04/08/2010 QCD JLab Apr.2011 GR.Goldstein 9

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

  11. Dispersion Theory The amplitudes are analytic --in the chosen kinematical variables, , xBj, ,s -- except where the intermediate states are on shell k’+=(X-)P+ k+=XP+ P+ PX+=(1-X)P+ DIS cut epeX pole, =Q2/2M epe’p QCD JLab Apr.2011 GR.Goldstein

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

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

  14. Where is threshold? • Imaginary part from discontinuity across unitarity branch cut - from ν threshold for production of on-shell states - e.g. πN, ππN, ππΔ, etc. • Continuum starts at s =(M+mπ)2 ⇒ lowest hadronic threshold. • Physical region for non-zeroQ2 and t differs from this. • How to fill the gap? ν0 νphysical ν t s=(M+mπ)2 04/08/2010 QCD JLab Apr.2011 GR.Goldstein 14

  15.  (GeV) Physical region Q2=1.0 GeV2  -t(GeV2) s=(M+m

  16. (GeV) Lab Physical region Q2=1.0 GeV2  Physical region Q2=1.0 GeV2  -t(GeV2) s=(M+m

  17. (GeV) Lab (GeV) Physical region Q2=1.0 GeV2  Physical region Q2=1.0 GeV2  Physical region Q2=1.0 GeV2  -t(GeV2) Lab  s=(M+m 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  & Q2 ? -t=0.1

  18. Lab Physical region Q2=5.5 GeV2 Physical region Q2=5.5 GeV2 Lab  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. -t=0.1

  19. minLab Physical region Q2= 20 GeV2   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 Q2/2M.

  20. Physical threshold obtained by imposing T2 > 0 (same as tmin = -M2 2/(1-)2) ν physical Q2=2.0 GeV2  νPhysical s=(M+mπ)2 -t continuum • 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. QCD JLab Apr.2011 GR.Goldstein

  21. Examples of DVCS dispersion integrals & threshold dependences Regge form for H(X,ζ,t) or HR(ν,Q2,t) = β(t,Q2)(1-e-iπα(t))(ν/ν0)α(t) So ReHR(ν,Q2,t) = tan(πα(t)/2) ImH(ν,Q2,t) Dispersion: This is exact for νThreshold=0. For Q2 , νThreshold = -t/4M this is asymptotic. 04/08/2010 QCD JLab Apr.2011 GR.Goldstein 21

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

  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 Q2 t-dependent threshold cuts out physical states “counter-intuitively as Q2 increases the DRs start failing because the physical threshold is farther away from the continuum” QCD JLab Apr.2011 GR.Goldstein

  24. DVCS Kinematics q q'=q+ k+=XP+, kT k'+=(X- )P+, kT- T P'+=(1-  )P+, - T P+ 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. Singularities in covariant picture along with s and u cuts X> Im k- X< Im k- 1' 3' 1' 2' 3' 2' Re k- Re k- Diquark spectator on-shell struck quark on-shell Returning quark =outgoing anti-q 2 3 1 QCD JLab Apr.2011 GR.Goldstein

  27. In DGLAP region spectator with diquark q. numbers is on-shell k’+=(X-)P+ k+=XP+ P’+=(1- )P+ P+ PX+=(1-X)P+ In ERBL region struck quark, k, is on-shell k’+=(-X)P+ P’+=(1- )P+ P+ 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. 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 k’+=(-X)P+ P’+=(1- )P+ P+ QCD JLab Apr.2011 GR.Goldstein

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

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

  33. QCD JLab Apr.2011 GR.Goldstein Spectator inspired model of GPDs • 2 directions – • 1. getting good parameterization of H, E • 2. getting 8 spin dependent GPDs • Simplest Spectator: scalar diquark: Spin simplicity - Pq+diq and q’+diqP’ are spin disconnected => Chiral even related to Chiral odd GPDs H relationsHT and same for other 3 pairs • Axial diquark: more complex linear relations & distinction between u & d flavors • scalar (ud+du) -> u struck quark • axial (uu, ud-du, dd) -> d and u struck • Regge-like behavior 1/xα(t) could differentiate among GPDs • pure scalar diquark: 3 – relations among helicity amps => at ξ & t not 0 • E=-ξET + ET~ , ξE~ = - 2HT~ - ET + ξET~ (multiplied by √(t0-t) ) HT = (H+H~)/2 – (1+ξ2)ξE/(1-ξ2)+. . . and HT~ in terms of chiral evens Could apply to u-quark GPDs – part that has scalar if same Regge factors

  34. Vertex Structures  k’+=(X-)P+ k+=XP+  P’+=(1- )P+ P+ PX+=(1-X)P+ PX+=(1-X)P+ S=0 or 1 Focus e.g. on S=0 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 Vertex function 

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

  36. QCD JLab Apr.2011 GR.Goldstein scalar diquark model (with vertex Γ(k) for Nquark+diquark) there are 2 independent Φ (k,p)’s & 2 Φ (k’,p’)’s => A’s not all independent

  37. QCD JLab Apr.2011 GR.Goldstein obtaining A’s • 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 kTΔT terms) • Leaves kT integration • 2 shown here: • get relations: • A++,-+= - A++,+- , A-+,++ = - A+-,++ , • A++,++ = A++,--

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

  39. Fitting Procedure e.g. for H and E • Fit at=0, t=0  Hq(x,0,0)=q(X) • 3 parameters per quark flavor (MXq, q, q) + initial Qo2 • 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 Diquark spectral function  (MX2-MX2)  (MX2) MX2 Following DIS work byBrodsky, Close, Gunion (1973)

  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

  42. Hall B data

  43. Hermes data

  44. Lessons from examples • Moderate reach of Eγ (Jlab<12 GeV) or s<25 GeV2=(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(ζ,t) depends on thresholds • DVCS & Bethe-Heitler interference via cross sections & asymmetries measures complex H(ζ,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. 04/08/2010 QCD JLab Apr.2011 GR.Goldstein 46

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