Generalized Parton Distributions Recent Progress

Generalized Parton Distributions Recent Progress (Mostly a summary of various talks at SIR2005@Jlab in May 2005 Pervez Hoodbhoy Quaid-e-Azam University Islamabad

γ* γ,π,ρ hard soft x+ξ x-ξ GPDs P P t Factorisation: Q 2 large, -t<1 GeV 2

What is a GPD? It is a proton matrix element which is a hybrid of elastic form factors and Feynman distributions GPDs depend upon: x : fraction of the longitudinal momentum carried by struck parton t: t-channel momentum transfer squared ξ: skewness parameter (a( a new variable coming from selection of a light-cone direction) Q 2 : probing scale

DVCS DVMP hard gluon longitudinal only hard vertices DVCS cannot separate u/d quark contributions. M = ρ/ω select H, E, for u/d flavors M = π, η, K select H, E

Formal definition of GPDs: d i x 1 1 ( 2 ) ( 2 ) (,, ) (,, ) i + ν λ λ e p q n q n p H x t u u E x t u q + + σ ν λ γ λ = ξ γ + ξ u 2π 2M x and x are the momentum fractions of the struck i f quark, and x = ( x + x ). 1 2 ξ =( x - x )/2 is skewness. Depends on lightcone direction. f dxh ( x, ξ, t)=f (t) dxe( x, ξ, t)=f (t) i 1 2 i f

Relation of GPDs to Angular Momentum Generalized form factor and quark angular momentum: Total quark angular momentum: Quark angular momentum (Ji s sum rule) 1 q = 1 J G = 1 J xdx H 2 2 1 [ ( x, ξ,0) + E ( x, ξ,0) ] q q X. Ji, Phy.Rev.Lett.78,610(1997)

GPDs And Orbital Angular Momentum Distribution: 1 2 n ( β µ 1 µ 2 O ψγ id id id = βµ µ µ µ ) Define generalized angular momentum tensor: α β M = ξ O ξ O αβµ µ µ βµ µ µ αµ µ µ 1 2 n 1 2 n 1 2 n ξ ( ξ) = tensor structures ( 2 π) δ (0) αβµ 1µ 2 µ n d p M p J n ψ (minus traces) 4 4 4 n reduced matrix element % % 3 12 +++ + + + + + + d ξ p M ( ξ ) p = S + L + L 1 L( x) = xq( x) + xe( x) q( x) 2 [ ] Ji+Hoodbhoy

TMD Parton Distributions These appear in the processes in which hadron transverse-momentum is measured, often together with TMD fragmentation functions. The leading-twist ones are classified by Boer, Mulders,, and Tangerman (1996,1998) There are 8 of them q(x,, k ), q T (x,, k ), q L (x, k ), q T (x, k ), δq(x, k ), δ L q(x, k ), δ T q(x, k ), δ T q(x, k )

Wigner parton distributions (WPD) O( x, p) = dxdpo( x, p) W ( x, p) When integrated over p, one gets the coordinate space density ρ(x)= (x)= ψ(x) (x) 2 When integrated over x, one gets the coordinate space density n(p)= )= ψ(p) 2 X. Ji

Wigner parton distributions & offsprings (Ji) Mother Dis. W(r,p) q(x,, r, k ) Red. Wig. q(x,r) X. Ji TMDPD q (x, k ) PDF q(x) Density ρ(r)

Wigner distributions for quarks in proton Wigner operator (X.( Ji,PRL91:062001,2003) Wigner distribution: density for quarks having position r and 4-momentum 4 k µ (off-shell) X. Ji

Reduced Wigner Distributions and GPDs The 4D reduced Wigner distribution f(r,x,x)) is related to Generalized parton distributions (GPD) H and E through simple FT, t= q 2 ξ ~ q z X. Ji H,E depend only on 3 variables. There is a rotational symmetry in the transverse plane..

TMD W pu (x,k,r) Parent Wigner distributions d 3 r TMD PDFs: f pu (x,k T ),g 1,f 1T, h 1,h 1L (FT) d 2 k T GPD Probability to find a quark u in a nucleon P with a certain polarization in a position r and momentum k ~ ~ GPDs: H pu (x,ξ,t), E pu (x,ξ,t), H,E, Measure momentum transfer to quark k T distributions also important for exclusive studies d 2 k T ξ=0,t=0 dx Measure momentum transfer to target Exclusive meson data important in understanding of SIDIS measurements PDFs f pu (x,k T ), g 1, h 1 FFs F 1pu (t),f 2pu (t).. Some PDFs same in exclusive and semi-inclusive analysis Analysis of SIDIS and DVMP are complementary

Holography is "lensless photography" in which an image is captured not as an image focused on film, but as an interference pattern at the film. Typically, coherent light from a laser is reflected from an object and combined at the film with light from a reference beam. This recorded interference pattern actually contains much more information that a focused image, and enables the viewer to view a true three-dimensional image which exhibits parallax.

Computed Tomography Computed Tomography (CT) is a powerful nondestructive evaluation (NDE) technique for producing 2-D and 3-D cross-sectional images of an object from flat X-ray images. Characteristics of the internal structure of an object such as dimensions, shape, internal defects, and density are readily available from CT images.

From Holography to Tomography mirror A. Belitsky, B. Mueller, NPA711 (2002) 118 An Apple mirror mirror A Proton mirror detector By varying the energy and momentum transfer to the proton we probe its interior and generate tomographic images of the proton ( femto tomography ). Burkert

Burkardt

Imaging quarks at fixed Feynman-x For every choice of x, one can use the Wigner distributions to picture the nucleon in 3-space; 3 quantum phase-space tomography! z b x b y X. Ji

Zanotti GPDs ON A LATTICE

1 2 q 2 = q 2 10 1 A ( Q ) F ( Q ) q 2 = q 2 10 2 B ( Q ) F ( Q ) A% ( Q ) G ( Q ) q 10 q 10 2 = q 2 A B% ( Q ) G ( Q ) 2 = q 2 P q = 1 q + q 2 20 20 J ( A (0) B (0)) q Σ = A% (0) q 10

Zanotti

Fleming

hoodbhoy

GPDs - Experimental Aspects DVCS measured at HERA (at H1 and Zeus) DVCS measured at JLab (fixed target,clas) DVCS planned at COMPASS, CERN DVMP measured at HERA DVMP measured at JLab DVMP measured (old data, 2002) at COMPASS DDVCS planned at JLab

Some Generalities 1 1 = P iπδ ( x ξ ) x ξ + iε x ξ { = π ξ ξ ξ ξ 2 q 2 q 2 Im{ F} eq F (,, t, Q ) mf (,, t, Q ) + 1 2 q 2 1 1 Re{ F} = eq P dxf ( x, ξ, t, Q ) ± x ξ x + ξ 1 } γγ*p plane e - e - γ* ee γ* plane Θ γγ γ p φ

DVCS small signal BH big noise GPDs FF A A LU C LU r s dσ ( φ) dσ ( φ) ( φ) = r s (Beam Spin Asymmetry, BSA) dσ ( φ) + dσ ( φ) + dσ ( φ) dσ ( φ) ( φ) = (Beam Charge Asymmetry, BCA) + dσ ( φ) + dσ ( φ) find that: A ( φ) Im( M% )sin φ and A ( φ) Re( M% )cosφ where: % 0 M = C t t F1 Η + ξ ( F1 + F2 ) Η% t 2 2m Ε 4m

Kinematical domain E=190, 100GeV Nx2 Collider : H1 & ZEUS 0.0001<x<0.01 Fixed target : JLAB 6-11GeV SSA,BCA? HERMES 27 GeV SSA,BCA COMPASS could provide data on : Cross section (190 GeV) BCA (100 GeV) Wide Q 2 and x bj ranges Limitation due to luminosity Burtin

Nowak

e e p GPD p Helicity-flip GPDs P. Hoodbhoy and X. Ji, PR D 58 (1998) 054006

Nowak

Ellinghaus

GPD Reaction Obs. Expt Status H ( ±ξ, ξ, t) ~ H ( ±ξ, ξ, t) E( ±ξ, ξ, t) ( u + d) H ( x < ξ, ξ, t) H, E ( u + d) x x H, E (2u d) ep epγ (DVCS) BSA CLAS 4.2 GeV Published PRL CLAS 4.8 GeV Preliminary CLAS 5.75 GeV Preliminary (+ σ) Hall A 5.75 GeV Fall 04 CLAS 5.75 GeV Spring 05 ep epγ (DVCS) TSA CLAS 5.65 GeV Preliminary e(n) enγ (DVCS) BSA Hall A 5.75 GeV Fall 04 ed edγ (DVCS) BSA CLAS 5.4 GeV under analysis ep epe + e - (DDVCS) BSA CLAS 5.75 GeV under analysis ep epρ σ L CLAS 4.2 GeV Published PLB CLAS 5.75 GeV under analysis ep epω (σ L ) CLAS 5.75 GeV Accepted EPJA + other meson production channels π, η, Φ under analyses in the three Halls. From ep epx Dedicated set-up M.Garcon

Exclusive ρ production on transverse target A UT ~ Im(AB*) ρ 0 A ~ 2H u + H d B ~ 2E u + E d A UT ρ + A ~ H u - H d B ~ E u - E d Asymmetry depends linearly on the GPD E in Ji s sum rule. ρ 0 ρ 0 and ρ + measurements allow separation of E u, E d CLAS12 projected K. Goeke, M.V. Polyakov, M. Vanderhaeghen, 2001 x B Burkert

Exclusive ρ 0 production on transverse target Α UT = 2 (Im(AB*))/π T Α 2 (1 ξ 2 ) Β 2 (ξ 2 +t/4m 2 ) - Re(ΑΒ )2ξ 2 Q 2 =5 GeV 2 ρ 0 A ~ 2H u + H d B ~ 2E u + E d ρ 0 E u, E d needed for angular momentum sum rule. Burkert B K. Goeke, M.V. Polyakov, M. Vanderhaeghen, 2001

Exclusive ep epρ 0 production L CLAS (4.3 GeV) x B =0.38 HERMES (27GeV) W=5.4 GeV Q 2 (GeV 2 ) GPD formalism approximately describes CLAS and HERMES data Q 2 > 2 GeV 2 Burkert

Deeply virtual meson production Deeply virtual meson production Meson and Pomeron (or two-gluon) exchange (Photoproduction) ω π, f 2, P ρ 0 ω Φ (σ), f 2, P π, f 2, P P or scattering at the quark level? Flavor sensitivity of DVMP on the proton: ρ 0 2u+d, 9g/4 γ* L ω L ω 2u-d, 3g/4 Φ ρ + s, g u-d dσ L dt 1 α ψ ( z) 1 ξ S M 4 Q Q z x ± µ ( ah iε + be)( x, ξ, t) dxdz 2 f ( ξ, t) 6 Q M.Garcon

Exclusive ρ meson production: ep epρ Exclusive ρ meson production: ep epρ CLAS (4.2 GeV) CLAS (5.75 GeV) Regge (JML) C. Hadjidakis et al., PLB 605 GPD (MG-MVdh) Analysis in progress GPD formalism (beyond leading order) describes approximately data for x B <0.4, Q 2 >1.5 GeV 2 Two-pion invariant mass spectra M.Garcon

DDVCS (Double (Double Deeply Deeply Virtual Virtual Compton Compton Scattering) Scattering) DDVCS-BH interference generates a beam spin asymmetry sensitive to e - e - e + e - Im T DDVCS ~ H ( ± x( ξ, q'), ξ, t) + Κ γ* T γ* T p p The (continuously varying) virtuality of the outgoing photon allows to tune the kinematical point (x,ξ,t) at which the GPDs are sampled (with x < ξ). M. Guidal & M. Vanderhaeghen, PRL 90 A. V. Belitsky & D. Müller, PRL 90 M.Garcon

DDVCS: first observation of ep epe + e - DDVCS: first observation of ep epe + e - * Positrons identified among large background of positive pions * ep epe + e - cleanly selected (mostly) through missing mass ep epe + X * Φ distribution of outgoing γ* and beam spin asymmetry extracted (integrated over γ* virtuality) A problem for both experiment and theory: but * 2 electrons in the final state antisymmetrisation was not included in calculations, define domain of validity for exchange diagram. * data analysis was performed assuming two different hypotheses Lepton pair squared invariant mass either detected electron = scattered electron or detected electron belongs to lepton pair from γ* Hyp. 2 seems the most valid quasi-real photoproduction of vector mesons M.Garcon

GPD CHALLENGES Goal: map out the full dependence on x, ξ, t, Q Develop models consistent with known forward distributions, form factors, polynomiality constraints, positivity, More lattice moments, smaller pion masses, towards unquenched QCD, Launch a world-wide wide program for analyzing GPDs perhaps along the lines of CTEQ for PDFs. High energy, high luminosity is needed to map out GPDs in deeply virtual exclusive processes such as DDVCS (JLab( with 12GeV unique). 2