1. Michel Guidal (IPN Orsay)
NPQCD2015, 21/04/2015
GPD phenomenology
Michel Guidal (IPN Orsay)

2. y x z
ep a eX
(DIS)
(Parton Distribution
Functions: PDF)

3. ep a eX (DIS) ep a ep (elastic) y z (Parton Distribution x
Functions: PDF) y x z
ep a ep
(elastic)
(Form Factors: FFs)

4. ep a eX (DIS) ep a ep (elastic) ep a epg (DVCS) y z
(Parton Distribution
Functions: PDF) y x z
ep a ep
(elastic)
(Form Factors: FFs) x z
ep a epg
(DVCS)

5. ep a eX (DIS) ep a ep (elastic) ep a epg (DVCS) ep a epX (SIDIS) y z
(Parton Distribution
Functions: PDF) y x z
ep a ep
(elastic)
(Form Factors: FFs) x z
ep a epg
(DVCS)
(Generalized Parton
Distributions: GPDs)
ep a epX
(SIDIS) z
(Transverse Momentum
Dependent PDFs: TMDs) x

6. In the light-cone frame:
x+x : relative longitudinal momentum of initial quark
x-x : relative longitudinal momentum of final quark
t=D2 : total squared momentum transfer to the nucleon
(transverse component: )

7. p p’ p p’ p p’ GPDs probe the nucleon at amplitude level DIS : DVCS :x x
x+x
x-x p p’ p p’
q(x)~<p|F(x)OF(x)|p ’>
H(x,x)~<p|F(x-x)OF(x+x)|p ’> p p’
x+x
x-x z 1
x<x :
x>x :

8. (x,x ) dependence : Double Distributions
(Radyushkin)
Hq(x,x,0)~ db da d(x-b-ax)DDq(a,b)

9. (x,x ) dependence : Double Distributions
(Radyushkin)
22b+1G(2b+1)(1-|b|)2b+1
Hq(x,x,0)~ db da d(x-b-ax)DDq(a,b)
With :
and
DDq(a,b,t)=hb(a,b) q(b)
hb(a,b)=G(2b+2)[(1-|b|)2-a2]b

10. (x,x,t ) dependence : Reggeized Double Distributions
(Radyushkin, Muller, Polyakov,VdH, M.G.)
DDq(a,b,t)=q (b) hb(a,b)b-a’(1-b)t x b
(GeV-1)
Hu(x,b ) y z
Satisfies Form Factors sum rule and polynomiality

11. proton electromagnetic form factors
HqDD(x,0,t) q(x) x-a’1(1-x)t
EqDD(x,0,t) q(x)(1-x)hqx-a’2(1-x)t

13. DVCS data (until a few weeks ago)
JLab
Hall A
DVCS
B-pol. X-section
DVCS
unpol. X-section
JLab
CLAS
DVCS
BSA
DVCS
lTSA
HERMES
DVCS
BSA,lTSA,tTSA,BCA

14. In general, 8 GPD quantities accessible (Compton Form Factors)
with

15. Given the well-established LT-LO DVCS+BH amplitude
Bethe-Heitler
GPDs
Can one recover the 8 CFFs from the DVCS observables?
Obs= Amp(DVCS+BH) CFFs
Two (quasi-) model-independent approaches
to extract, at fixed xB, t and Q2 (« local » fitting),
the CFFs from the DVCS observables

16. 1/ «Brute force » fitting
c2 minimization (with MINUIT + MINOS) of the
available DVCS observables at a given xB, t and Q2 point
by varying the CFFs within a limited hyper-space (e.g. 5xVGG)
The problem can be (largely) underconstrained:
JLab Hall A: pol. and unpol. X-sections
JLab CLAS: BSA + TSA
2 constraints and 8 parameters !
However, as some observables are largely dominated by
a single or a few CFFs, there is a convergence (i.e. a well-defined
minimum c2) for these latter CFFs.
The contribution of the non-converging CFF enters in
the error bar of the converging ones. For instance (naive):
If -10<x<10:

17. 1/ «Brute force » fitting
c2 minimization (with MINUIT + MINOS) of the
available DVCS observables at a given xB, t and Q2 point
by varying the CFFs within a limited hyper-space (e.g. 5xVGG)
The problem can be (largely) underconstrained:
JLab Hall A: pol. and unpol. X-sections
JLab CLAS: BSA + TSA
2 constraints and 8 parameters !
However, as some observables are largely dominated by
a single or a few CFFs, there is a convergence (i.e. a well-defined
minimum c2) for these latter CFFs.
The contribution of the non-converging CFF enters in
the error bar of the converging ones.
M.G. EPJA 37 (2008) 319
M.G. & H. Moutarde, EPJA 42 (2009) 71
M.G. PLB 689 (2010) 156
M.G. PLB 693 (2010) 17

18. 2/ Mapping and linearization
If enough observables measured, one has a system
of 8 equations with 8 unknowns
Given reasonable approximations (leading-twist dominance,
neglect of some 1/Q2 terms,...), the system can be linear
(practical for the error propagation) ~
DsLU ~ sinf Im{F1H + x(F1+F2)H -kF2E}df ~ ~
DsUL ~ sinfIm{F1H+x(F1+F2)(H + xB/2E) –xkF2 E+…}df
K. Kumericki, D. Mueller, M. Murray, arXiv: hep-ph, arXiv: hep-ph

19. Other approaches: Assume a functionnal shape and fit some parameters
*D. Mueller
& K. Kumericki
*H. Moutarde
*VGG by the
HERMES and
n-DVCS Hall A coll.
Les 2 precedentes methodes sont « quasi » independantes de modele, i.e. elles ne necessitent pas de supposer une forme fonctionelle avec des parametres a fitter. Les methodes qui sont sur ce slide supposent que les GPDs ont déjà une forme particuliere.

20. Hall A : s & DsLU , xB=0.36,Q2=2.3,t=.17,.23,.28,.33 c2=1.01 c2=0.92

22. unpol.sec.eff. + beam pol.sec.eff. c2 minimization
Avec « ma » methode, extraction obtenue pour Him en fittant les sec.eff. (pol. et non-pol.) du Hall A
c2 minimization

23. unpol.sec.eff. + beam pol.sec.eff. beam spin asym. +
long. pol. tar. asym
Avec « ma » methode, extraction obtenue pour Him en fittant les BSA et les TSA (Shifeng) de CLAS
c2 minimization

24. unpol.sec.eff. + beam pol.sec.eff. beam spin asym. +
long. pol. tar. asym
beam charge asym. +
beam spin asym …
Avec la methode de KM et la mienne, extraction obtenue pour Him en fittant les asymetries d’HERMES.
c2 minimization
linearization

25. unpol.sec.eff. + beam pol.sec.eff. beam spin asym. +
long. pol. tar. asym
beam charge asym. +
beam spin asym …
Compilation de toutes les methodes
c2 minimization
linearization
Moutarde 10 model/fit
VGG model
KM10 model/fit

26. the t-slope reflects the size of the probed object (Fourier transf.)
HIm: (~H(x,x,t) )
the t-slope reflects the size of the probed object (Fourier transf.)
Resultats pour HtildeIm
The sea quarks (low x) spread to the periphery of the nucleon while the valence quarks (large x) remain in the center

27. Resultats pour HtildeIm~
The axial charge (~Him) appears to be more « concentrated » than the electromagnetic charge (~Him)

28. From CFFs to spatial densities
How to go from momentum coordinates (t)
to space-time coordinates (b) ?
(with error propagation)
Burkardt (2000)
Applying a (model-dependent) “deskewing” factor:
and, in a first approach, neglecting the sea contribution

30. } 1/Smear the data according to their error bar 2/Fit by Aebt
3/Fourier transform (analytically)
4/Obtain a series of Fourier transform as a function of b
5/For each slice in b, obtain a (Gaussian) distribution which
is fitted so as to extract the mean and the standard deviation
~1000 times

31. CLAS data (xB=0.25) “skewed” HIm “deskewed” HIm
(fits applied to « deskewed » data)
“deskewed” HIm

32. HERMES data (xB=0.09) “skewed” HIm “deskewed” HIm
(fits applied to « deskewed » data)
“deskewed” HIm

34. xB=0.25
xB=0.09

35. GPDs contain a wealth of information on nucleon structure
and dynamics: space-momentum quark correlation, orbital
momentum, pion cloud, pressure forces within the nucleon,…

36. GPDs contain a wealth of information on nucleon structure
and dynamics: space-momentum quark correlation, orbital
momentum, pion cloud, pressure forces within the nucleon,…
Many more DVCS data coming with JLab 6 GeV:
Cross sections on p, BSA for DVCS on He4,…

37. To be published in 2015, on ArXiv
CLAS
(courtesy H.S. Jo)
To be published in 2015, on ArXiv

39. GPDs contain a wealth of information on nucleon structure
and dynamics: space-momentum quark correlation, orbital
momentum, pion cloud, pressure forces within the nucleon,…
Many more DVCS data coming with JLab 6 GeV:
Cross sections on p, BSA for DVCS on He4,…
DVMP data also provide access to GPDs:
PS mesons transversity GPDs,…
First new insights on nucleon structure already emerging
from current data with new fitting algorithms
Large flow of new observables and data expected soon
(JLab6,JLab12,COMPASS) will allow a precise nucleon
tomography in the valence region
(Other DVCS-related processes planned for JLab 12 GeV
(TCS, DDVCS)
Theory developments: higher twists, target mass corrections,
NLO corrections, …

41. Double beam-target spin asym.
Long. target spin asym.
Double beam-target spin asym.
CLAS
(courtesy S. Niccolai)
To be published in 2014

42. Mass corrections in propagators–
In lightcone frame, q and P collinear along z-axis
using lightcone vectors pµ, nµ
along + and – z direction
and ξ and ξ' = + component
of ∆ and q ~ ~ ~ –
Kinematical variables: experimentaly accessible
Light-cone momentum fraction,
include mass terms
At asymptotic limit:

43. Gauge invariance correction
We have: qµHµνLO = ½ (∆⊥)κHκνLO and q'νHµνLO = – ½ (∆⊥)λHκλLO
Restore gauge invariance: qµHµν = qνHµν ≡ 0
twist 2
vector part

44. Hard scattering amplitude of TCS- - - - - -

45. 1) The HERMES Recoil detector. A. Airapetian et al, e-Print: arXiv:1302.6092
2) Beam-helicity asymmetry arising from deeply virtual Compton scattering measured with kinematically complete event reconstruction A. Airapetian et al, JHEP10(2012)042
3) Beam-helicity and beam-charge asymmetries associated with deeply virtual Compton scattering on the unpolarised proton A. Airapetian et al, JHEP 07 (2012) 032
Many analysis at JLab in progress (CLAS X-sections and lTSA, new Hall A X-sections at different beam
energies, proton and neutron channels,… with publications planned for 2013/2014

46. HIm:the t-slope reflects the size of the probed object (Fourier transf
The sea quarks (low x) spread to the periphery of the nucleon while the valence quarks (large x) remain in the center

47. ~
The axial charge (~Him) appears to be more « concentrated » than the electromagnetic charge (~Him)

48. ~ New A_UL and A_LL CLAS data Confirms the apparent
Phys.Rev. D91 (2015) 5,
Les nouvelles donnees de CLAS pour A_UL et A_LL.
Confirms the apparent
« xB-independence » and
flatter « t-dependence » of HIm ~

49. ~ Im{Hp, Hp, Ep} ~ ~ Im{Hp, Hp} Re{Hp, Hp} Im{Hp, Ep} f g e’ e N’
leptonic plane
hadronic
plane N’ e’ e
x= xB/(2-xB) k=-t/4M2
DsLU ~ sinf Im{F1H + x(F1+F2)H -kF2E}df ~
Polarized beam, unpolarized target (BSA) :
Im{Hp, Hp, Ep}
Unpolarized beam, longitudinal target (lTSA) :
DsUL ~ sinfIm{F1H+x(F1+F2)(H + xB/2E) –xkF2 E+…}df ~
Im{Hp, Hp}
nDVCS complements pDVCS for the different sensitivity to GPDs of the various observables
Polarized beam, longitudinal target (BlTSA) :
DsLL ~ (A+Bcosf)Re{F1H+x(F1+F2)(H + xB/2E)…}df ~
Re{Hp, Hp}
Unpolarized beam, transverse target (tTSA) :
DsUT ~ cosfIm{k(F2H – F1E) + ….. }df
Im{Hp, Ep}