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Dispersion Relations in Hard Exclusive Processes: Theory and Applications

Explore the utilization of dispersion relations in hard exclusive processes, focusing on deep virtual Compton scattering (DVCS) amplitudes, constraints on GPD modeling, and the connection between dispersion relations and Generalized Parton Distributions (GPDs). Learn about the significance of unitarity, analyticity, and kinematic restrictions in modeling and experimental extractions.

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Dispersion Relations in Hard Exclusive Processes: Theory and Applications

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  1. The Use of Dispersion Relations in Hard Exclusive Processes Gary R. Goldstein Tufts University Simonetta Liuti University of Virginia Presentation for Hard Meson and Photon Production ECT*Oct, 2010 Titian (1543)- Pope Paul III presided over Council of Trent 10/14/2010 ECT* Oct.2010 G.R. Goldstein 1

  2. Outline of Discussion Dispersion Relations, DVCS & GPDs - DGLAP & ERBL • Thresholds & DRs • Unitarity & analyticity • Threshold limits • Kinematic restrictions • Model calculations • ERBL region & partons • Intermediate states • “semi-disconnected” diagrams • Non-replacement for >0 See: GG, Liuti, PRD80, 071501 (2009). GG, Liuti, ArXiv hep-ph/1006.0213 10/14/2010 ECT* Oct.2010 GR.Goldstein 2

  3. Are GPDsHeretical? Pasquale Cati (1588)- Council of Trent with Church Triumphant

  4. Phenomenological Constraints on GPDsGPD challenge - How unique?Models; Parameterizations; Theoretical constraints Constraints from PDFs Form factors: • Further Constraints: • Polynomiality Sensible hadron shape at x~1 • & x→0 Regge behavior Sensible prediction for t dependence - positivity • see Ahmad, Liuti et al. (AHLT) (2008,09) • & ongoing work (S.Liuti, GG, O. Gonzalez Hernandez in this session) • (Connection with TMDs? Wigner distributions? Beyond models?) ECT* Oct.2010 GR.Goldstein 4

  5. 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 10/14/2010 ECT* Oct.2010 GR.Goldstein 5

  6. 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 10/14/2010 ECT* Oct.2010 GR.Goldstein 6

  7. 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 10/14/2010 ECT* Oct.2010 GR.Goldstein 7

  8. 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) ECT* Oct.2010 GR.Goldstein

  9. 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 ECT* Oct.2010 GR.Goldstein

  10. OPE is seeded in DRs (see e.g. Jaffe’s SPIN Lectures) From DR + Optical Theorem =1/x to Mellin moments expansion ECT* Oct.2010 GR.Goldstein

  11. DVCS: Where is the threshold?G.Goldstein and S.L., 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. ECT* Oct.2010 GR.Goldstein

  12. 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 10/14/2010 ECT* Oct.2010 GR.Goldstein 12

  13. ν Q2=1.0 GeV2 νPhysical -t s=(M+mπ)2 B.Pasquini, et al.,Eur.Phys.J.A11,185 (2001) 10/14/2010 ECT* Oct.2010 GR.Goldstein 13

  14. Gaps in dispersion integrals ν ν Q2=2.0 GeV2 Q2=1.0 GeV2 -1.1>t>-2.7 GeV2 Physical region has no gap for Q2=2.0 GeV2 νPhysical νPhysical s=(M+mπ)2 -t -t s=(M+mπ)2 -0.60>t>-1.34 GeV2 Physical region has no gap for Q2=1.0 GeV2 ν Q2=5.5 GeV2 -2.4>t>-7.4 GeV2 Physical region has no gap for Q2=5.5 GeV2 νPhysical M+2mπ next threshold . . . M+nmπ, etc. s=(M+mπ)2 -t 10/14/2010 ECT* Oct.2010 GR.Goldstein 14

  15. 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. ECT* Oct.2010 GR.Goldstein

  16. 10/14/2010 ECT* Oct.2010 GR.Goldstein 16

  17. Crossing symmetry Analyticity in the energy variable ν=(s-u)/4M ∝ 1/x requires crossing (anti)symmetric amplitudes. At GPD level need: H (+) (x,ξ,t ) = H (x,ξ,t) − H (−x,ξ,t) H (−) (x, ξ ) = H (x,ξ,t) + H (−x,ξ,t) singlet non-singlet Each is multiplied by hard part from γ* q→γ q’ C ±(x/ξ) = 1 /(−X + ζ − iϵ) ∓ 1/ (X − iϵ) or 1/(ξ – x − iϵ) ∓ 1/(ξ + x − iϵ) H(+)(ν,t)∝ναP(t)-1whereαPomeron(0)=1+δ H(-)(ν,t)∝ναR(t)-1where αRegge(0)≈1/2 DR needs subtraction Unsubtracted DR ECT* Oct.2010 GR.Goldstein 17

  18. 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. 10/14/2010 ECT* Oct.2010 GR.Goldstein 18

  19. 10/14/2010 ECT* Oct.2010 GR.Goldstein 19

  20. Diquark spectator model - simple, no spin Valence quark model Based on Brodsky & Llanes-Estrada model q′ q γ * γ ζ=0.12 t = -0.5,-1.0,-2.0 GeV2 k k′ X ERBL region DGLAP region P′ P N′ N No symmetry around x=0 or X=ς/2 scalar diquark - subprocess û-channel pole with nucleon→quark+diquark form factors 10/14/2010 ECT* Oct.2010 GR.Goldstein 20

  21. Scalar diquark, polynomiality & DR H(x,x,t=-0.5 GeV2) x2 moment of H(x,ξ,t) vs. ξfor 3 t values ξ 10/14/2010 ECT* Oct.2010 GR.Goldstein 21

  22. 10/14/2010 ECT* Oct.2010 GR.Goldstein 22

  23. Dispersion relations for AHLT GPDs Real part H(ζ,t) for Q2=5.5 GeV2 Evolved H(X,ζ,t) from low Q2 scale, then get Im H (ζ,t) or H(X,X,t) & direct integration for Re H (ζ,t) 10/14/2010 ECT* Oct.2010 GR.Goldstein 23

  24. 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” ECT* Oct.2010 GR.Goldstein

  25. Dispersion Direct Regge Model Difference Direct Dispersion ECT* Oct.2010 GR.Goldstein

  26. When deeply virtual processes involve directly final states - as in exclusive or semi-inclusive processes - “standard kinematic approximations should be questioned” (Collins, Rogers, Stasto, 2007, Accardi, Qiu, 2008) H is calculated off the ridge ECT* Oct.2010 GR.Goldstein

  27. ECT* Oct.2010 GR.Goldstein

  28. Summary of part 1: dispersion relations cannot be applied straighforwardly to DVCS Need model for continuation into unphysical regions.. The “ridge” does not seem to contain all the information. ECT* Oct.2010 GR.Goldstein

  29. 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) ECT* Oct.2010 GR.Goldstein

  30. Interpreting off-forward GPDs see Jaffe NPB229, 205(1983) For pdf’s - forward elastic Relate T-product to cut diagrams partons emerge + Completeness ECT* Oct.2010 GR.Goldstein

  31. 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 ECT* Oct.2010 GR.Goldstein

  32. 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. ECT* Oct.2010 GR.Goldstein

  33. 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+ ECT* Oct.2010 GR.Goldstein

  34. 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+ ECT* Oct.2010 GR.Goldstein

  35. 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+ ECT* Oct.2010 GR.Goldstein

  36. Summary of part 2: GPDs in ERBL region can be described within QCD, consistently with factorization theorems, only by multiparton configurations, possibly higher twist. ECT* Oct.2010 GR.Goldstein

  37. 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. 10/14/2010 ECT* Oct.2010 GR.Goldstein 37

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