Strategies to extract GPDs from data
Simonetta Liuti
University of Virginia
&
Gary Goldstein
Tufts University
INT, 14-19 September 2009
Outline
Introduce 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
With 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)
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(P q)
y 1
(P1k1 )
  2xB
Amplitude
Angle between
transverse spin and
final state plane
M
Q2
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Azimuthal angle between planes
Comtpon Scattering and Bethe Heitler Processes
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Dynamics
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Look for instance at DVCS-BH Interference
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Off forward Parton Distributions (GPDs) are embedded in
soft matrix elements for deeply virtual Compton scattering (DVCS)
q
q’=q+
p+q
p+=XP+
p’+=(X-)P+
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P’+=(1- )P+
P+
1
1
2
2

PV

i


((
p

q)

m
)
2
2
2
2
( p  q)  m  i
( p  q)  m
1
1
1



Q 2  2( pq)  i
Q 2 / 2(Pq)  ( pq) / (Pq)   X
Amplitude
 1
1
dX Fq (X, ,t) 
   i  eq2 Fq ( , ,t)
1
  X X 
F q  P.V.
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”
+ Q2 Evolution
Quark-Diquark
Quark-Diquark model: two different time orderings/pole structure!
DGLAP: quark off shell, spectator on shell
X>
ERBL: quark on shell, spectator off-shell
X<
Quark anti-quark pair describes similar physics (dual to) Regge
t-channel exchange (JPC quantum numbers)
Vertex Structures

k’+=(X-)P+
k+=XP+

P+
PX+=(1-X)P+
S=0 or 1
PX+=(1-X)P+
P’+=(1- )P+
Focus e.g. on S=0
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   * (k ', P') (k, P)   * (k ', P') (k, P)
Vertex function 
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
2
O. Gonzalez Hernandez, S.L.
Fixed diquark mass formulation
DGLAP region
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ERBL region
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Reggeized diquark mass formulation
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
Diquark spectral function
(MX2-MX2)
 (MX2)
MX2
DIS  Brodsky, Close, Gunion ‘70s
Fitting Procedure


Fit at =0, t=0  Hq(x,0,0)=q(X)

3 parameters per quark flavor (MXq, q, q) + initial Qo2
Fit at =0, t0 

2 parameters per quark flavor (, p)
t
Regge
Quark-Diquark
Fit at 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,
Kumericki and D. Mueller, Guidal and Moutarde
 additional parameters (how many?)

=0,t=0
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
 = 0, t0
Nucleon Form Factors
S. Ahmad, H. Honkanen, S. L., S.K. Taneja, PRD75:094003,2007
Parameters from PDFs
Parameters from FFs
Some results…
Hu
Hd
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S. Ahmad, H. Honkanen, S. L., S.K. Taneja, (AHLT), PRD75:094003,2007
, 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
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 Extract dipole masses from
lattice data
 Relate dipole mass to “radius”
parameter
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 Chiral extrapolation of dip. mass
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Polynomiality from lattice results up to n=3
n=2
n=3
Results of Chiral Extrapolations
proton form factor
 Ashley et al. (2003)
 Ahmad et al. (2008)
-t (GeV2)
New Developments (H.Nguyen)
We repeated the calculation with improved lattice results
(Haegler et al., PRD 2007, arXiv:0705:4295)
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A20u-d
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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
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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.
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Dorati
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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)
Test with known, previously evaluated GPD, at 0
ERBL Region
Ahmad et al., EPJC (2009)
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Determined from lattice moments up to n=3
New Analysis
Results are more accurate one
can see trends
 both isovector and isoscalar
terms

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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)
Hall B (one binning, 11 more)
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Comparison with Jlab Hall A data (neutron)
Mazouz et al. (2007)
0
Fit to JLAB data: real part of CFF from d+ + dReal Part (work with S.Ahmad, H. Nguyen)
Schematically
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
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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
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to Mellin
moments
expansion
DVCS
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)
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Dispersion Relations (brief parenthesis…)
Dispersion
Direct
Difference
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Direct
Dispersion
G.Goldstein and S.L.,
Applications
Transversity
Simple Ansatz
h1(x,Q2) = q f1(x,Q2)

u
HT(x, , t,Q2) = q H(x,  , t,Q2)
d


ET(x,  , t,Q2) = Tq HT(x,  , t,Q2)
Related to Boer-Mulders function
Nuclei
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
quark's position
in nuclei
Burkardt-Soper
impact paramete
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
JPC=1--
,
,
, ..
JPC=1+-
b1, h1