Diapositive 1

1
Progress on the A.Freund-LS code
Laurent Schoeffel (CEA Saclay)

• Different GPD NLO evolution codes avaible 1.
  Dieter 2. Freund
• Pb some caution with the code of Freund ?!
• gt Corrected and rebuit completly gt 2bis.
  FreundLS
• Some small checks are presented below to justify
  that
• everything is  ok  with the present version
• checks shown on H and G. The code is also
  correct for polarised
• case. Still some work to do on E

2
Evolution code Freund corrected and
rebuilt Test of the evolution code with a
skeewing -gt 0 gt standard PDF evol Evolution of u
distribution (singlet) CTEQ6m from
HEPDATA data web page _at_ 1.74 GeV²
(input cteq6m function) evolved at
7.14 GeV²
OK
Initial value for CTEQ6M
Q²1.69 GeV²
Q²7.14 GeV²
EVOL
3
Check of the evolution on the gluon distribution
Always in the limit skeewing -gt 0 gt standard
PDF evol CTEQ6m from HEPDATA data web
page _at_ 1.74 GeV² (input cteq6m
function) evolved at 7.14 GeV²
OK
Initial value for CTEQ6M
Q²1.69 GeV²
Q²7.14 GeV²
EVOL
4
In the limit of skeewing -gt 0 evolution code
is fine! Lets switch on the skeewing it means
several hard points 1. No  official 
comparison code to check with 2. Kernels of the
evolution eq. are more complicated (skeewing).
In practice, if we input a function with no
skeewing and evolve it, the skeewing effect
will be generated during the evolution and
the evolved function will verify the rules of
polynomiality! (non trivial) gt potential
sources of errors 3. Two sets of evolution eq.
DGLAP and ERBL domains gt continuity to
ensure between the two domains 4.  Ex post
facto  point large flexibility in the
definition of the input functions
5
Lets assume that everything is correct with the
present evol code as suggested by the good
results obtained in the zero skeewing limit We
can discuss the point 4. input functions! The
most simple hypothesis is no skeewing at the
input scale! (gt no need to define a profile
function which increases the flexibility of
the input param functions) Then
Hs(x,?,Q0²)Qs(x,Q0²) and G(x,?,Q0²)G(x,Q0²)
in Jis notations I have written the code in
Radyuskin variables gt H(X, ? ) Qs(X-?/2) /
(1-?/2) / (1-?/2) QsX-?/2 at low ?ltlt1 Note
the interest of this notation is that it is
more directly comparable to the PDF case as
0ltXlt1 Also, at zero skeewing, we immediatly find
the limit GPD PDF!
6
Reminder of one polynomiality rule ? dx x
Hq(x,?,t) Aq(t)4 ?²Cq(t) N1 rule, idem for
N2 etc. To build input functions that verifies
the rules DD functions (with profile functions
quite complicated procedure) OR our way
OR Guzey et al. (called dual param.) Some
technical points 1. D-term lives in the ERBL
domain no real constraint on this term at
the moment. Below, we work with D-term0 (but no
pb to include it) 2. Input function based on
D.A. continuity with the DGLAP param. sum
rules gt all this defines the ERBL part (Freund
03) there is still a non negligeable
flexibility in the ERBL param gt30
7
Input GPD function
Q²1.69 GeV²
?0.05
X.UsingletGPD _at_ input
in the DGLAP domain param in the
ERBL (see Freund 03)
Such that Us(X-?/2) / (1-?/2) /
(1-?/2) H(Us)(X,?) _at_ input scale
8
After Q² evolution
X(uubar)(X-?/2) / (1-?/2) / (1-?/2) with
(uubar)() from CTEQ6M Calculated at 7.14 GeV²
(again with HEPDATA web page)
Q²7.14 GeV²
X.UsingletGPD evolved
?0.05
? from gt skeewing generated
via the QCD evolution
zoom
skeewing effect from evol
9
Zoom at ?0.05
ratio GPD / PDF for Usinglet 2 (_at_1.69 GeV²)
and 2.7 (_at_7.14 GeV²)
Q²7.14 GeV²
Q²1.69 GeV²
Cteq6m PDF from hepdata
10
Q² evolution GPD/PDF

• Ratio GPD / PDF for Usinglet
• obtained with the evolution code
• for GPD and PDF (id from HEPDATA
• webpage or the evolution code)
• gt Increase with Q²
• Do we have a potential comparison
• with avaible data ?
• If YES, it would be a good check
• of the code input functions!

11
The good point is that we have measured F2 and
DVCS xs! gt We can extract the ratio of DVCS and
DIS imaginary part using the formulae below
Im(A(?p??p)) 4?²?/Q² F2 FT
4?²?/W² ltegt² PDFS(xBj)
(only at LO) Im(A(?Tp??p)) 16?b
?DVCS /(1?²)1/2
4?²?/W² ltegt² GPDS(?,?) with ?xBj/2
And this ratio will give directly the
ratio of the GPD / PDF BUT, be careful the
relations above between Im() and GPD or PDF is
only valid at LO (we come back later on the NLO
case) What do we get from the data ?
12
R Im(A(?p??p))/ Im(A(?p??p))
R 1.8 (Q²lt12 GeV²) almost contant with a
small increase at larger Q² Is it compatible
with what we have obtained previously ?
13
The answer is YES due to NLO corrections!
There are 2 main graphs to compute for DVCS
process

• DVCS Amplitude ? dx 1/ (x- ? i?) - 1/(x?
  i?) ?eq²Hq(x,?,t) axial O(m/Q)
• gt Im(A(?p)) 4?²?/W² ?eq²HqS(?,?,t)
  with ?xBj/(2-xBj)
• Re(A (?p)) P.V.

• At NLO, the gg graph contributes with
  Im(ANLO) Im(ALO) 0.7
• (see talk of 09/2006 _at_ compass GPD meeting)
• The reduction factor of 0.7 depends (of course)
  of the size of
• the Gluon component, which is increasing with Q²
  (_at_ low x)

14
The Gluon distribution _at_ input scale and evolved
Q²1.69 GeV²
Q²7.14 GeV²
Cteq6m PDF from hepdata
15
Effect of the gg graph illustration on xs
Modeling GPD at low scale HS,V,g(x,?) ?
QS,V,g(x) (Ji notations) at an intital scale 1
GeV² ERBL param. (symetries) and ? dependence
generated by the GPD evolution (DGLAPERBL)
LO gt NLO AG lt0 (-30
tot. Amplitude)
?(?p??p) ImA(?p??p)² (1?²) / (16? b)
16
Q² evolution GPD/PDF _at_ NLO
with NLO corrections gt OK
compatible with the experimental
result
R(Q²) GPD(Singlet)(xbj/2, xbj/2,Q²)f(Q²) /
PDF(Singlet)(xbj,Q²)
17
Lastest Results vs our GPD model
ImA
GPD model
? R(Q²)
18

• In conclusion the evolution code seems to be
  correct
• gt Possible to start with an input GPD and evolve
  in Q²
• Fast evolution for the Singlet distribution
  compensated
• for the Im() by  gg  graph which makes the Q²
  behaviour
• of Im(DVCS) and Im(DIS) compatible!
• And our result is in agreement with present
  measurement!
• Can we move to the BCA ?
• For BCA, we integrate over the ERBL and DGLAP
  domain
• gt No reason not to beleive our evolution in the
  ERBL domain
• even if we have no way to cross-check
  directly that everything
• is technically fine (indirectly, by showing
  that all is OK in
• the DGLAP domain, we have a hint that it is
  the case in the ERBL
• region as both evolutions are linked by
  continuity relations and
• sum rules)

19
To calculate BCA, there is another technical
step from GPDs Re(A (?p)) P.V.Coef?GPDs _at_
NLO This part of the code still under
cross-checks Then BCA proportional to F1 Re(H
(?p)) cos(?) In the low x part, Re() and Im()
are linked via dispersion relations gt then, we
can evaluate Re() directly from Im() and check if
we get the same answer from P.V. calculus
correct at 10 level
20
Using our GPD model, we get some predictions for
all other published measurements

• In the future, it will be possible to perform a
  global NLO analysis
• gt constraints on the DGLAP ERBL domains in a
  wide range of x
• VERY interesting perspective!

21
Example of prediction for the COMPASS
kinematics with E100 GeV of the BCA (cos? term
only) Some sources of theory uncertainties 1.
Note that the absolute scale of the prediction
depends a lot on the t dependence a
measurement of t xs is necessary in the
COMPASS regime! 10 uncertainty (see appendix
A) 2. Also, the BCA depends on the D-term gt
large theory uncertainty gt10 only from
?(D-term) 3. Uncertainty due to the ERBL param
gt30 error
Q² 4 GeV²
2 hypothesis for F(t)
All this errors can be reduced with a BCA
measurement at COMPASS
22
Conclusions and Outlook

• From the TOOL box no major pb from the
  evolution code
• BCA, with the present formulae from Belitsky et
  al. is OK also
• All this at NLO! The code of D.Mueller is also
  avaible soon
• Comparison with VGG to be discussed as VGG is a
  LO param ?! ()
• and we have seen the large influence of NLO
  corrections
• A few notes
• BCA essential to constraint the ERBL part
• Measurement of the t dependence necessary
• gt spatial transverse distributions
• With the other experiments global view of GPDs
• We should not forget that COMPASS kinematic
  domain  may 
• be in the BFKL/dipole regime gt very interesting
  as many issues
• are associated as saturation, geometric scaling
  etc. (no GPDs in
• this approach) (see appendix B)

23
Appendix A Some more words on t xs measurement
From F.T. lt?²gt 4 d/dtH(x,t)/H(x,0) Ex b
6 GeV-2 gt lt?²gt2b0.46 fm²
24
Question for the future do we verify this
shape(x) for the impact parameter ?
A future global analysis could Verify this
intuitive view ?
Sea quarks and Gluon Domain gt  large  b
Core of the proton (valence domain)  low  b
HERA
COMPASS
From M. Diehl
CLAS
dependence on the resolution scale
25
end of appendix A
scale dependence of b() (Diehl et al.)
Similar for singlet and gluon gt b decreases
with ?²
26
Appendix B Geometric Scaling for
DVCS C.Marquet, LS (06)
?dip

• for DVCS xxIP only  T  contributes with
  the hypothesis T()T(r.Qs)
• At low x (large W), this model gt
  ?DVCS(x,Q²)?DVCS(?Q²/Qs²)
• 
  with QsQ0 (x/x0)?/2

27
Geometric scaling some results
From hep-ph/0606079 PLB 2006
gt Scaling verified
28
Geometric Scaling for DVCS (comments)

• T()T(r.Qs) appears to be a genuine property
• of the dipole-proton elastic scattering
  amplitude,
• with T() verifying the non-linear BK equation.
• QsQ0 ?s with ?s a dimensionless saturation
• scale that can be obtained from the linear part
• of the equation (BFKL term)
• In practice T()T(r,b,x)S(b)T(r.Qs) and b
  is integrated over
• with an approx. (gaussian) for this dependence
• Latest results work with the FT of T(), namely
  Tc(r,t,x) (conjugate
• variable t appears instead of b) gt need a
  reformulation of the
• equations in momentum space etc. (see Peschanski,
  Iancu et al.)
• gt Possible saturation scale which depends also
  on  t 
• if the xs is given in bins of this variable

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Geometric Scaling for DVCS (latest
results) Property verified (global and t)!
end of appendix B