A Flexible Parameterization of GPDs & Their Role in DVCS & - PowerPoint PPT Presentation

A Flexible Parameterization of GPDs & Their Role in DVCS & Neutral Meson Leptoproduction Gary R. Goldstein Tufts University Simonetta Liuti, S. Ahmad, Osvaldo Gonzalez-Hernandez University of Virginia Presentation for PANIC MIT July 2011 These ideas were developed in Trento ECT*, INT, Jlab, DIS2011, Frascati INF, . . . 1 1

Outline New “ Flexible ” parameterization for Chiral Even GPDs Regge ✖ diquark spectator model Satisfies all constraints Results for DVCS (transverse γ *  transverse γ ) cross sections & asymmetries Extend to Chiral Odd GPDs via diquark spin relations Some simple relations between Chiral even & odd helicity amps π 0 , η , η ’ production data involve sizable γ * Transverse [Collins, Franfurt, Strikman] ) (factorization shown at leading twist for γ * Longitudinal γ * T requires requires chiral odd GPDs Q 2 dependence for π 0 depends on γ * +( ρ , b 1 )  π 0 π 0 cross sections & asymmetries 2 PANIC2011 GR.Goldstein

DVCS & DVMP γ *(Q 2 )+P → ( γ or meson)+P ’ 
 partonic picture q q+ Δ } { k T k' T =k T - Δ k + =XP + k' + =(X- ζ )P + GPD P + P' + =(1- ζ )P + } P ’ T =- Δ Factorized “ handbag ” ζ→ 0 t picture Regge Quark-spectator quark+diquark X> ζ DGLAP Δ T -> b T transverse spatial X< ζ ERBL x=(X- ζ /2)/(1- ζ /2); ξ = ζ /(2- ζ ) see Ahmad, GG, Liuti, PRD79, 054014, (2009) for first chiral odd GPD parameterization focused on pseudoscalar production 3 3 PANIC2011 GR.Goldstein 7/24/11

Momentum space nucleon matrix elements of quark field correlators see, e.g. M. Diehl, Eur. Phys. J. C 19, 485 (2001). Chiral even GPDs -> Ji sum rule x = 1 [ ] x � dx H ( x ,0,0) + E ( x ,0,0) J q 2 Chiral odd GPDs -> transversity 4 PANIC2011 GR.Goldstein 4 7/24/11

How to determine GPDs? Flexible Parameterization  Recursive Fit CHIRAL EVENS: O. Gonzalez Hernandez, G. G., S. Liuti, arXiV:1012.3776 PRD accepted 1 � dxH ( x , � , t ) = F 1 ( t ) Constraints from Form Factors Dirac 0 1 � dxE ( x , � , t ) = F 2 ( t ) Pauli, etc. including Axial 0 & Pseudoscalar How can these be independent of ξ ? Constraints from Polynomiality Result of Lorentz invariance & causality. Not necessarily built into models + 1 ( t ) � k + 1 � ( � 1) n � � n dx x n H ( x , � , t ) = C n ( t ) � n + 1 A n , k 2 k = 0, 2,.. � 1 1 n n - 1- (-1) � dx x n E ( x , � , t ) � = B n , k ( t ) � k C n ( t ) � n + 1 2 k = 0,2,... � 1 Constraints from pdf ’ s: H (x,0,0)= f 1 (x), H ~(x,0,0)= g 1 (x), H T (x,0,0)= h 1 (x) 7/24/11 5 5 PANIC2011 GR.Goldstein

Spectator inspired model of GPDs • 2 directions –  1. getting good parameterization of H, E & ~H, ~E satisfying many constraints ( see O. Gonzalez-Hernandez, GG, S. Liuti arXiv: 1012.3776)  2. getting 8 spin dependent GPDs • Chiral Odd GPDs π 0 production is testing ground (New results – preliminary) • Simplest Spectator -- scalar diquark – helicity amps for P  q ::::: q ’  P ’ Spin simplicity - P  q+diq and q ’ +diq  P ’ are spin disconnected => Chiral even related to Chiral odd GPDs Chiral even related to Chiral odd GPDs H, E, . .  helicity amp relations  H T , E T , . . • Axial diquark: more complex linear relations & distinction between u & d flavors 7/24/11 6 PANIC2011 GR.Goldstein 6

Spectator inspired model (cont ’ d) • u + scalar (ud+du) -> u struck quark • axial (uu, ud-du, dd) -> d and u struck : 3 – relations among helicity amps => at ξ & t not 0 • E=- ξ E T + E T ~ , ξ E ~ = - 2H T ~ - E T + ξ E T ~ (multiplied by √ (t 0 -t)) H T +(t 0 -t)H T ~ /4M 2 = (H+H ~ )/2– ξ [(1+ ξ /2)E+ ξ E ~ ]/(1- ξ 2 ) and H T ~ in terms of chiral evens Could apply to u-quark GPDs – part that has scalar if same Regge factors • Regge-like behavior 1/x α (t) could differentiate among GPDs depending on t -channel quantum numbers 7 PANIC2011 GR.Goldstein

Vertex Structures λ λ ’ k ’ + =(X- ζ )P + k + =XP + Λ P ’ + =(1- ζ )P + P + P X + =(1-X)P + P X + =(1-X)P + Λ ’ S=0 or 1 First focus e.g. on S=0 pure spectator * ( k ', P ') � ++ ( k , P ) + � * ( k ', P ') � � + ( k , P ) H � � + + � + * ( k ', P ') � + � ( k , P ) + � * ( k ', P ') � + + ( k , P ) E � � ++ + � * ( k ', P ') � ++ ( k , P ) � � * ( k ', P ') � � + ( k , P ) � H � � ++ � + * ( k ', P ') � + � ( k , P ) � � * ( k ', P ') � ++ ( k , P ) � E � � ++ + � Vertex functions Φ 8 PANIC2011 GR.Goldstein

Vertex Structures λ λ ’ k ’ + =(X- ζ )P + k + =XP + Λ P ’ + =(1- ζ )P + P + P X + =(1-X)P + P X + =(1-X)P + Λ ’ S=0 or 1 First focus e.g. on S=0 pure spectator Note that by switching * ( k ', P ') � ++ ( k , P ) + � * ( k ', P ') � � + ( k , P ) H � � λ  - λ & Λ  - Λ (Parity) + + � + will have chiral evens * ( k ', P ') � + � ( k , P ) + � * ( k ', P ') � + + ( k , P ) E � � go to ± chiral odds ++ + � giving relations – * ( k ', P ') � ++ ( k , P ) � � * ( k ', P ') � � + ( k , P ) � H � � before k integrations ++ � + * ( k ', P ') � + � ( k , P ) � � * ( k ', P ') � ++ ( k , P ) � A( Λ ’ λ ’ ; Λλ )  E � � ± A( Λ ’ , λ ’ ;- Λ ,- λ ) ++ + � Vertex function Φ but then ( Λ ’ - λ ’ )-( Λ - λ ) ≠ ( Λ ’ - λ ’ )+( Λ - λ ) unless Λ = λ 9 PANIC2011 GR.Goldstein

Helicity amps (q’+N->q+N’) are linear combinations of GPDs In diquark spectator models A ++;++ , etc. are calculated directly. Inverted -> GPDs 7/24/11 10 PANIC2011 GR.Goldstein 10

Invert to obtain model for GPDs A ++,-+ = - A ++,+- A -+,++ = - A +-,++ A ++,++ = A ++,-- double flip 7/24/11 11 PANIC2011 GR.Goldstein 11

Fitting Procedure e.g. for H and E Fit at ζ =0, t=0 ⇒ H q (x,0,0)=q(X) ✔ 3 parameters per quark flavor (M X q , Λ q , α q ) + initial Q o 2 ✔ Fit at ζ =0, t ≠ 0 ⇒ ✔ 2 parameters per quark flavor ( β , p) ✔ R = X � [ � + � '(1 � X ) p t + � ( � ) t ] Regge factor Quark-Diquark Fit at ζ≠ 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, ✔ 12 Kumericki and D. Mueller, Guidal and Moutarde

Reggeized diquark mass formulation Diquark spectral function “ Regge ” ρ ∝δ (M X 2 -M X 2 ) ∝ (M X 2 ) α + Q 2 Evolution M X 2 Following DIS work by Brodsky, Close, Gunion (1973) 13 PANIC2011 GR.Goldstein

Flexible Parametrization of Generalized Parton Distributions from Deeply Virtual Compton Scattering Observables Gary R. Goldstein, J. Osvaldo Gonzalez Hernandez , Simonetta Liuti arXiv:1012.3776 14 PANIC2011 GR.Goldstein

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Compton Form Factors  Real & Imaginary Parts 16 PANIC2011 GR.Goldstein

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Observables Note GPD & Bethe-Heitler separate amplitudes -> interference linear in GPD 18 PANIC2011 GR.Goldstein

Hall B data A LU (90 o ) 19

Hermes data 20 PANIC2011 GR.Goldstein

A LU (90 o ) -t=0.118 GeV 2 -t=0.118 GeV 2 x=0.097 0.1 Q 2 =2.5 GeV 2 Q 2 =2.5 GeV 2 x=0.097 0 -0.1 -0.2 -0.3 -0.4 -0.5 Hermes kinematics -0.6 0.1 0 -0.1 -0.2 -0.3 -0.4 no DVCS -0.5 Hermes kinematics -0.6 -2 -1 -1 10 10 1 1 10 -t (GeV 2 ) Q 2 (GeV 2 ) x Bj 21 PANIC2011 GR.Goldstein

Interference contribution to cos( φ ) term in DVCS cross section 22 PANIC2011 GR.Goldstein

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Exclusive Lepto-production of π o or η , η ’ to measure chiral odd GPDs π o +1,0 + 1/2 q nos of C-odd +1,0 1 - - exchange - 1/2 + 1/2 - 1/2 1 +- exchange + 1/2 N b 1 & h 1 N - 1/2 + 1/2 e.g. f +1+,0- (s,t,Q 2 ) π o g +1+,0- A ++,- - What about coupling of π to q → q ′ ? Assumed γ 5 vertex H T +1,0 Then for m quark =0 has to flip helicity b 1 & h 1 for q →π +q ′ and q ⋅ q ′ ≠ 0. Naïve twist 3 ψ bar γ 5 ψ - 1/2 + 1/2 N Rather than γ µ γ 5 – does not flip twist 2. But q ’ γ µ γ 5 q N will not contribute to transverse γ . Differs from t- channel approach to Regge factorization 24 24 PANIC2011 GR.Goldstein 7/24/11

Q 2 dependent form factors t-channel view see Belitsky, Ji, Yuan (2007) & Ng (2007) 7/24/11 25 25 PANIC2011 G.R.Goldstein

New Chiral Odd GPDs & exclusive π 0 electroproduction 50 x Bj =0.41 40 Q 2 =3.2 GeV 2 30 Cross Section σ T + εσ L 20 10 σ LT 0 -10 σ TT -20 0 0.2 0.4 0.6 0.8 1 1.2 1.4 -t (GeV 2 ) 26 PANIC2011 GR.Goldstein

New Chiral Odd GPDs & exclusive π 0 electroproduction 50 x Bj =0.41 40 Q 2 =3.2 GeV 2 30 Cross Section 20 10 0 -10 -20 0 0.2 0.4 0.6 0.8 1 1.2 1.4 -t (GeV 2 ) σ T + εσ L data:Q 2 ≈ 3.2 GeV 2 , x ≈ 0.41 σ LT σ TT preliminary data courtesy V. Kubarovsky, HallB 27 PANIC2011 GR.Goldstein

New Chiral Odd GPDs & exclusive π 0 electroproduction 100 x Bj =0.19 80 Q 2 =2.3 GeV 2 60 Cross Section 40 σ T + εσ L 20 σ LT 0 -20 σ TT -40 0 0.2 0.4 0.6 0.8 1 1.2 1.4 -t (GeV 2 ) 28 PANIC2011 GR.Goldstein