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University of Virginia

&

Gary Goldstein

Tufts University

INT, 14-19 September 2009

APS DNP Meeting

Saturday, October 25th, 2008.

Oakland, USA

Strategies to extract GPDs from dataIntroduce a step by step analysis

Step 1, Step 2, Step 3, Step 4

Interesting applications: Access Chiral-Odd GPDs, Nuclei: DVCS and 0 electroproduction on 4He

Conclusions/Outlook

OutlineWith the new experimental analyses at HERMES, Jlab, Compass… we are entering a new, more advanced phase of extracting GPDs from data

Many concerns have been raised recently:

No longer simple parametrizations (K. Kumericki, D. Muller)

Q2 dependence (M. Diehl et al.)

What type of information and accuracy from simultaneous measurements of different observables? (M. Guidal, H. Moutarde)

How can one use Lattice + Chiral Extrapolations (P. Hägler, S.L.)

How can one connect various experiments, separate valence from sea, flavors separation (P. Kroll, T. Feldman)...

Use of dispersion relation: is it only necessary to measure imaginary part of DVCS, DVMP? (Anikin & Teryaev, Diehl & Ivanov, Vanderhaeghen, Goldstein & S.L.)

Global analysis exists for TMDs (simpler partonic interpretation than

GPDs) see e.g. M. Anselmino and collaborators

DVCS Cross Section (Belitsky, Kirchner, Muller, 2002)

Amplitude

Angle between transverse spin and final state plane

Azimuthal angle between planes

Off forward Parton Distributions (GPDs) are embedded in

soft matrix elements for deeply virtual Compton scattering (DVCS)

q

q’=q+

p+q

p’+=(X-)P+

p+=XP+

P’+=(1- )P+

P+

Amplitude

What goes into a theoretically motivated parametrization...?

The name of the game: Devise a form combining essential dynamical elements with a flexible model that allows for a fully quantitative analysis constrained by the data

Hq(X, , t)= R(X, , t) G(X, , t)

“Regge”

Quark-Diquark

+ Q2 Evolution

Quark-Diquark model: two different time orderings/pole structure!

DGLAP: quark off shell, spectator on shell

ERBL: quark on shell, spectator off-shell

X>

X<

Quark anti-quark pair describes similar physics (dual to) Regge

t-channel exchange (JPC quantum numbers)

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

Vertex function

2

2

O. Gonzalez Hernandez, S.L.

Reggeized diquark mass formulation

Diquark spectral function

(MX2-MX2)

(MX2)

MX2

DIS Brodsky, Close, Gunion ‘70s

- Fit at=0, t=0 Hq(x,0,0)=q(X)
- 3 parameters per quark flavor (MXq, q, q) + initial Qo2
- Fit at=0, t0
- 2 parameters per quark flavor (, p)
- Fit at 0, t0 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

- additional parameters (how many?)

t

Regge

Quark-Diquark

Parton Distribution Functions

Notice! GPD parametric

form is given at Q2=Qo2

and evolved to Q2 of data.

Notice! We provide a

parametrization for GPDs that simultaneously fits

the PDFs:

Hq(X,,t)= R(X,,t) G(X,,t)

Regge

Quark-Diquark

Parameters from FFs

, t S. Ahmad et al., EPJC (2009)

we were able to extend the parametrization to

taking into account lattice results on n=2,3 moments of GPDs

the new parametrization is valid for valence quarks only

(not expected to be extended sensibly, “as it is”, into

HERA/HERMES region: need sea quarks + gluons)

it works fine at Jefferson Lab kinematics

Use information from Lattice QCD:

(1) Assume lattice results follow dipole behavior for n=1,2,3

Extract dipole masses from

lattice data

Relate dipole mass to “radius”

parameter

Chiral extrapolation of dip. mass

Results of Chiral Extrapolations

proton form factor

Ashley et al. (2003)

Ahmad et al. (2008)

-t (GeV2)

We repeated the calculation with improved lattice results

(Haegler et al., PRD 2007, arXiv:0705:4295)

Results are comparable (up to n=2) to our

“phenomenological” extrapolation

We are investigating the impact of different chiral extrapolation

methods: “direct” extrapolation applicable up to n=2 only

M. Dorati, T. Gail and T. Hemmert (NPA 798, 2008)

(Also using P. Wang, A. Thomas et al. )

New Results are more precise and compatible with other chiral extrapolations

A20u-d vs. (-t)

Nguyen, S.L.

Dorati

Lattice results are used to model/fit the ERBL Region…

We know the area from n=1 moment + constrained DGLAP

Reconstruction of GPDs from Bernstein moments

Weighted Average Value

Location of X-bin

Dispersion (error in X)

*

* Algebra a bit more complicated for to transformation, details in EPJC(2009)

- Results are more accurate one can see trends
- both isovector and isoscalar terms

Summary of first three steps towards parametrization

0

7 + 1 (Qo) parameters

v1

10 + 1 (Qo) parameters

v2

use v1 for DGLAP region(X > )

0

use lattice calculations for ERBL region(X < )

BSA data are predicted at this stage

Munoz Camacho et al., PRL(2006)

Comparison with Jlab Hall A data (neutron)

Mazouz et al. (2007)

Fitted directly at Q2 of data

Cusp from reggeized ERBL (-X)

either from phenom.“DA type” shape, or diquark model

Behavior determined by Jlab data on Real Part and Q2 dependence

- Consistent with lattice determination!

Dispersion Relations (brief parenthesis…)

Dispersion

Direct

Difference

Direct

Dispersion

G.Goldstein and S.L.,arXiv:0905.4753 [hep-ph]

Dispersion relations cannot be directly applied to DVCS because one misses a

fundamental hypothesis: “all intermediate states need to be summed over”

This happens because “t” is not zero t-dependent threshold cuts out

physical states

It is not an issue in DIS (see your favorite textbook, LeBellac, Muta,

Jaffe’s lectures…) because of optical theorem

From DR

to Mellin moments expansion

One proceeds backwards, from polynomiality analytic properties (Teryaev)

But here one is forced to look into the nature of intermediate states because there is no optical theorem

t-dependent thresholds are important: counter-intuitively as Q2 increases the DRs start failing because the physical threshold is farther away from the continuum one (from factorization)

Is the mismatch between the limits obtained from factorization and the physical limits from DRs a signature of the “limits of standard kinematical approximations”? (Collins, Rogers, Stasto and Accardi, Qiu)

Dispersion Relations (brief parenthesis…)

Dispersion

Direct

Difference

Direct

Dispersion

G.Goldstein and S.L.,

Simple Ansatz

h1(x,Q2) = q f1(x,Q2)

u

HT(x, , t,Q2) = q H(x, , t,Q2)

d

ET(x, , t,Q2) = TqHT(x, , t,Q2)

Related to Boer-Mulders function

GPDs & hadron tensor for Spin 0 nuclear target(Liuti and Taneja, PRC 2005)Exclusive o production from 4He (with G. Goldstein)

OAM sum rule in deuterium (with S.K. Taneja)

Jefferson Lab approved experiment, H. Egiyan, F.X Girod, K. Hafidi,

S.L. and E. Voutier

Spatial structure of quarks and gluons in nuclei

Burkardt-Soper

impact parameter

quark\'s position

in nuclei

New! Test OAM SR in Spin 1 system: Deuteron

(S.L. and S. Taneja)

Conclusions and Outlook

- Approaching “Global Analysis” for GPDs is a more complex problem than for PDFs and TMDs:
- combinations of GPDs enter simultaneously the physical observables
- dependence on several kinematical variables: X,,t,Q2 of which…
- …X always appears integrated over
- Strategies to extract GPDs from data are based on multistep analyses: we propose one of such analyses using a physically motivated parametrization + lattice results
- Focus of the present work was on H and E in “valence” region
- Several applications and extensions: extraction of tensor charge and transverse anomalous moment from neutral pion production data, studies of spatial structure of nuclei…
- …but analysis is underway that takes into account all GPDs
- This analysis is possible thanks to the flexibility offered by our parametrization/model

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