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II. Generalized Parton Distributions (GPDs)

MarcVanderhaeghen

Johannes Gutenberg Universität, Mainz

HANUC, Jyväskylä, August 25-29, 2008

Outline : GPDs

1) link of FF to Generalized Parton Distributions :

nucleon “tomography”

2) GPDs and spin of nucleon

3) Hard exclusive processes : DVCS, …

Generalized Parton Distributions : yield 3-dim quark structure of nucleon

Burkardt (2000,2003)

Belitsky,Ji,Yuan (2004)

Elastic Scattering

transverse quark

distribution in

coordinate space

DIS

longitudinal

quark distribution

in momentum space

DES (GPDs)

fully-correlated

quark distribution in

both coordinate and

momentum space

Q2 >>

t = Δ2

x + ξ

x - ξ

P - Δ/2

P + Δ/2

GPD (x, ξ ,t)

Generalized Parton Distributions

low –t process : -t << Q2

Δ+ = - (2 ξ) P+

light-cone dominance, nμ(1, 0, 0, -1) / (2 P+)

P - Δ/2

P + Δ/2

known information on GPDs

forward limit : ordinaryparton distributions

unpolarized quark distr

polarized quark distr

: do NOT appear in DIS new information

first moments : nucleonelectroweak form factors

Dirac

Pauli

axial

ξ independence : Lorentz invariance

pseudo-scalar

+1

-1

-ξ

0

ξ

GPDs : x and ξdependence

forward parton distr

Hu (x,ξ,0)

x + ξ

x - ξ

quark distribution

anti-quark distribution

q q distribution amplitude

Unique tool to explore the internal landscape of the nucleon :

3D quark/gluon imaging of nucleon

Access to static properties :

constrained (sum rules) by precision measurements of charge/magnetization

orbital angular momentum carried by quarks

GPDs : 3D quark/gluon imaging of nucleon

Fourier transform of GPDs :

simultaneous distributions of quarks w.r.t. longitudinal momentum x P and transverse positionb

theoretical parametrization needed

modified Regge parametrization : Guidal, Polyakov, Radyushkin, Vdh (2004)

Input : forward parton distributions atm2 = 1 GeV2 (MRST2002 NNLO)

Drell-Yan-West relation : exp(- α΄ t ) -> exp(- α΄ (1 – x) t) : Burkardt (2001)

parameters :

regge slopes : α’1 = α’2 determined from rms radii

determined from F2 / F1 at large -t

future constraints : moments from lattice QCD

Regge model

modified Regge model

experiment

consistent with slopes of meson Regge trajectories ( ~ 0.9 GeV-2 )

Regge slope

connection large Q2 of FF <-> large x of GPD

at large Q2 : integral dominated by maximum of f(x,Q2), remainder region is exp. suppressed (method of steepest descent)

f(x,Q2) reaches maximum for :

“Drell-Yan-West” relation for PDF/GPD :

at large Q2 : I is dominated by its behavior around x -> 1

PROTON

NEUTRON

world data (2006)

modified Regge GPD parameterization

1 : Regge slope -> protonDirac (Pauli) radius

2, 3 :large x behavior of GPD Eu, Ed ->large Q2 behavior of F2p, F2n

3-parameter fit

ηu = 1.713,ηd = 0.566

Guidal, Polyakov, Radyushkin, Vdh (2005)

also Diehl, Feldmann, Jakob, Kroll (2005)

scaling behavior of N and N -> Δ F.F.

PQCD

N

N, Δ

+ collinear quarks

F1p ~ 1/Q4

F2p / F1p ~ 1/Q2

GM* ~ 1/Q4

GPD

modified Regge model

Guidal, Polyakov, Radyushkin, Vdh (2005)

GPDs : 3D quark/gluon imaging of nucleon

Fourier transform of GPDs :

simultaneous distributions of quarks w.r.t. longitudinal momentum x P and transverse positionb

GPDs : transverse image of the nucleon (tomography)

Hu(x, b? )

x

Guidal, Polyakov, Radyushkin, Vdh,

PRD 72, 054013 (2005)

b?(fm)

Moments of flavor-NS GPDs

Lattice results : m = 740 MeV

x

b (fm)

LHPC/SESAM

Decrease slope : decreasing transverse size as x -> 1Burkardt

-T2

lattice QCD for moment of PDFs : present state-of-the-art

full lattice QCD : domain wall valence quarks,

2+1 flavor staggered sea quarks

LHPC (2007)

Ratio:

EXPERIMENT / Lattice

Lattice (normalized to one)

isovector quantities

u - d

(no disconnected contributions)

Handbag (bilocal) operator : new way to probe the nucleon

Y-

y0

y

0

generalized probe

( Y ≈ 0 )

( W±, Z0 ) probe

spin 2 (graviton) probe

electroweak form factors

energy-momentum form factors

P - Δ/2

P + Δ/2

Energy momentum form factors

G

nucleon in external classical gravitational field

G couples toenergy-momentum tensor

Gordon

identity

Total system : energy-momentum conservation

A(0) = 1

Physical interpretation in terms of :

quarks : Aq(0)

Aq(0) + Ag(0) = 1

&

gluons : Ag(0)

consider N in rest frame : Pμ (M,0,0,0) Sμ (0,0,0,1)

Angular momentum sum rule (cont.)

Total system : angular momentum conservation

A(0) + B(0) = 1

B(0) = 0

Physical interpretation in terms of quarks & gluons

½ = Jq + Jg = ½ ΔΣ + Lq + Jg

½ΔΣ

Lq

Tμν form factors in terms of GPDs

of both lhs and rhs

P - Δ/2

P + Δ/2

Energy momentum form factors /spinof nucleon

G

nucleon in external classical gravitational field

G couples toenergy-momentum tensor

link to GPDs :

X. Ji (1997)

SPIN

sum rule

quark contribution to proton spin

X. Ji

(1997)

with

parametrizations for E q :

GPD : based on MRST2002

μ2 = 2 GeV2

lattice : full QCD,

no disconnected diagrams so far

orbital angular momentumcarried by quarks : solving thespin puzzle

with

Ji (1997)

evaluated at μ2 = 2.5 GeV2

Deeply Virtual Compton Scattering

*

Q2 large

t = Δ2

low –t process : -t << Q2

+ diagram with photons crossed

x + ξ

x - ξ

P - Δ/2

P + Δ/2

GPD (x, ξ ,t)

twist-2 DVCS amplitude independent of Q SCALING !

DVCS amplitude is complex : imaginary part -> x = ξ

Q2 , ξ, and t are kinematic variables

with

variable x is integrated over ( x ≠ xB ! )

DVCS amplitude is sensitive to GPDs weighted with coefficient functions

ALU =

(BH) * Im(DVCS) * sin Φ

Bethe-Heitler

DVCS

Q2 = 1 – 1.5 GeV2 , xB = 0.15 – 0.25, -t = 0.1 - 0.25 GeV2

Q2 = 2.6 GeV2 , xB = 0.11, -t = 0.27 GeV2

JLab/CLAS (2001)

HERMES (2001)

twist-2 + twist-3 : Kivel, Polyakov, Vdh (2000)

DVCS on proton

DVCS

JLab/Hall A @ 6 GeV(2006)

GPDs

Difference of polarized cross sections

Q2 ≈ 2 GeV2 xB = 0.36

Unpolarized cross sections

also JLab/CLAS, HERMES, H1 / ZEUS

0 because F1(t) is small

0 because of cancelation of u and d quarks

n-DVCSgives access to the least known and constrained GPD,E

JLab /Hall A (E03-106) : preliminary data

proton

JLab/CLAS

mainly sensitive to

GPD

calculation :

Vdh, Guichon, Guidal

(1999)

Hard electroproduction of mesons (ρ0,± , ω , φ, π, …)

M

*

TH

GPDs

hard scattering amplitude

Factorization theorem shown for longitudinal photon

Collins, Frankfurt, Strikman (1997)

longitudinally pol. Vector meson

PseudoScalar meson

M

M

±

TH

TH

Vector meson : accesses unpolarized GPDs H and E

~

~

PseudoScalar meson : accesses polarized GPDs H and E

Hard electroproduction of vector mesons (ρ0,± , ω , φ)

amplitude for longitudinally polarized vector meson

leading (1 gluon exchange) amplitude depends on αsgoes as 1/Q

dependence on meson distribution amplitudeΦV

Hard electroproduction of mesons : target normal spin asymmetry

lepton plane

hadron plane

in leading order (in Q)2 observables

Asymmetry :

Target polarization normal to hadron plane

Hard electroprod. of vector mesons : target normal spin asymmetry

A GPD H

B GPD E

unpolarized cross section

linear dependence on GPD E

ratio : less sensitive to NLO and higher twist effects

sensitivity to Ju and Jd

measure of TOTAL angular momentum contribution to proton spin

2000hrs

sL dominance

DQ2 =1

Dt = 0.2

-t = 0.5GeV2

exclusive 0 prod. with transverse target

2D (Im(AB*))/p

T

A~ (2Hu +Hd)

AUT = -

r0

|A|2(1-x2) - |B|2(x2+t/4m2) - Re(AB*)2x2

B~ (2Eu + Ed)

AUT

Q2 = 5GeV2

r0

Asymmetry depends

linearly on the GPD E,

which enters

Ji’s sum rule.

K. Goeke, M.V. Polyakov,

M. Vdh (2001)

xB

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