Deep Inelastic Scattering and the limit of small parton energy fraction.

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Deep Inelastic Scattering and the limit of small
parton energy fraction.
Frascati, 14 May 2007

• M. Greco
• talk based on results obtained in collaboration
  with B.I. Ermolaev and S.I. Troyan

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Deep Inelastic e-p Scattering
Incoming lepton
outgoing lepton- detected
k
Deeply virtual photon
k
q
Produced hadrons - not detected

X
p
Incoming hadron
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Leptonic tensor
q
q
hadronic tensor
p
p
Does not depend on spin
Spin-dependent
Hadronic tensor consists of two terms
antisymmetric
symmetric
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The spin-dependent part of Wmn is parameterized
by two structure functions
Structure functions
where m, p and S are the hadron mass, momentum
and spin q is the virtual photon momentum (Q2
- q2 gt 0). Both functions depend on Q2 and x
Q2 /2pq, 0lt x lt 1. At small x
longitudinal spin-flip transverse spin
-flip
Theoretical study of g1 and g2 involves both
Pert and Non-Pert QCD and therefore it is not
model-independent

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When the total energy and Q2 are large compared
to the mass scale, one can use factorization
Pert QCD
DIS off gluon
DIS off quark
k
k
k
q
q
q


p
p
quark
gluon
P
P
P
Non-pert QCD
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This allows to represent as a
convolution of the partonic tensor and the
probabilities to find a (polarized) parton
(quark or gluon) in the hadron
DIS off gluon
q
q
DIS off quark
Wquark
Wgluon
Fquark
Fgluon
p
p
Probability to find a quark
Probability to find a gluon
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Analytically this convolution is written as
follows
Perturbative QCD
Perturbative QCD
Non-pert QCD
Non-pert QCD
Pert QCD analytical calculations of Feynman
graphs
Non-perturbative QCD no regular methods
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Then DIS off quarks and gluons can be studied
with perturbative QCD, by calculating the Feynman
graphs involved. The probabilities, Fquark and
Fgluon involve non-perturbaive QCD. There is no
regular analytic way to get them. Usually they
are fitted from the experimental data at large x
and Q2 , and they are called the initial quark
and gluon densities and are denoted dq and dg .
So, the conventional form of the hadronic tensor
is
Initial quark distribution
Initial gluon distribution
DIS off the quark
DIS off the gluon
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Some terminology
Contribute to singlet
Contributes to nonsinglet
Initial quark
Each structure function has both a non-singlet
and a singlet components g1 g1NS g1S
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The Standard Approach consists in using the
perturbative Altarelli-Parisi or DGLAP Q2-
Evolution Equations, together with fits for the
initial parton densities. Evolution Equations
Altarelli-Parisi, Gribov-Lipatov, Dokshitzer
In particular, for the non-singlet g1
Evolved quark distribution
Coefficient function
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where
Splitting function
The expression for the singlet g1 is similar,
though more involved. It includes coefficient
functions and splitting
functions
.
in order to evolve the quark and gluon
distributions Dq and Dg
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Using the Mellin transform, one obtains the
expression for g1NS in a simpler form
Initial quark density
Non-Pert QCD
Anomalous dimension
Coefficient function
Pert QCD
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Kinematics in the (1/x - Q2) plane

1/x
g1 at xltlt1 and Q2 gtgt m2
x-evolution of Dq with coefficient function
Q2 -evolution of dq with anomalous dimension
Dq at x1 and Q2 .gtgt m2
1
Q2
m2
dq at x 1 and Q2 m2 defined from fitting exp.
data
evolved quark density
Starting point of Q2 -evolution
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In DGLAP, the coefficient functions and anomalous
dimensions are known with LO and NLO accuracy.
LO
NLO
LO
NLO
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LO splitting functions
Ahmed-Ross, Altarelli-Parisi, Sasaki,

Floratos, Ross, Sachradja, Gonzale- Arroyo,
Lopes, Yandurain, Kounnas, Lacaze, Curci,
Furmanski, Petronzio, Zijlstra, Mertig, van
Neerven, Vogelsang,
NLO splitting functions
Coefficient functions C(1)k , C(2)k
Bardeen, Buras, Muta, Duke, Altarelli, Kodaira,
Efremov, Anselmino, Leader, Zijlstra, van
Neerven,
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Phenomenology of the fits for the parton
densities
Altarelli-Ball-Forte-Ridolfi, Blumlein-Botcher,
Leader-Sidorov- Stamenov, Hirai et al.,
There are different fits for the initial parton
densities. For example,
Altarelli-Ball- Forte-Ridolfi,
The parameters
should be fitted from experiments.
This combined phenomenology works well at large
and small x, though strictly speaking, DGLAP is
not supposed to work in the small- x region
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1/x
Small x
DGLAP region ln(1/x) are small
ln(1/x) are large
Large x
1
m2
Q2
DGLAP accounts for logs(Q2) to all orders in as
but neglects
with kgt2
However, these contributions become leading at
small x and should be accounted for to all
orders in the QCD coupling.
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1/x
g1 at small x and large Q2
x-evolution, total resummation of
starting point
1
no
Q2
m2
DGLAP
Q2 -evolution , total resummation of
yes
DGLAP cannot perform the resummation of logs of x
because of the DGLAP-ordering, a keystone of
DGLAP
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DGLAP ordering
q
K3 K2 K1
good approximation for large x when logs of x
can be neglected. At x ltlt 1 the ordering has
to be lifted
DGLAP small-x asymptotics of g1 is well-known
p
when the initial parton densities
are not singular functions of x When the DGLAP
ordering is lifted all double logarithms of x
can be accounted for, and the asymptotics is
different
Bartels- Ermolaev- Manaenkov-Ryskin
intercept
when x? 0
Obviously
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Intercepts of g1 in Double-Logarithmic
Approximation
non- singlet intercept
singlet intercept
The weakest point of this approach the QCD
coupling as is fixed at an unknown scale. On
the contrary, DGLAP equations have always a
running as
DGLAP- parameterization
Bassetto-Ciafaloni-Marchesini - Veneziano,
Dokshitzer-Shirkov
Arguments in favor of the Q2- parameterization
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Origin in each ladder rung
K K
K K
K K
DGLAP-parameterization
However, such a parameterization is good for
large x only. At small x
Ermolaev-Greco-Troyan
When DGLAP- ordering is used and x 1
time-like argument Contributes in the Mellin
transform
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• Obviously, this new parameterization and the
  DGLAP one
• converge when x is large but they differ a lot at
  small x.
• In this new approach for studying g1 in the
  small-x region, it is necessary
• Total resummation of logs of x
• New parametrization of the QCD coupling

How the formula
is valid when
Then it is necessary to introduce an infrared
cut-off for k2
in the transverse space
Lipatov





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As the value of the cut-off is not fixed, one
can evolve the structure functions with respect
to m Infra-Red Evolution Equations
(IREE)

(name of the method)
Method prevoiusly used by Gribov, Lipatov,
Kirschner, Bartels-Ermolaev-Manaenkov-Ryskin,
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Essence of the method
t
g
Typical Feynman graph
g
s
q
q
Introduce IR cut-off
for all virtual particle momenta,
both in the longitudinal and in the transverse
space.
DL and SL contributions come from the integration
region where
DL contributions, in particular, come from the
region where all transverse momenta are widely
different so, one can factorize the
phase space into a set of separable sub-regions,
in each region some virtual particle has a
minimal . Let us call such a particle
the softest one.
DL and SL contributions of softest particles can
be factorized
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Case A the softest particle is a non-ladder
gluon. It can be factorized
k



k
symmetric contributions
k
Factorized softest gluons
is the lowest limit of integration over
of the softest gluon only. It does not involve
other momenta
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k



k
symmetric contributions
k
New IR cut-offs for integrations over transverse
momenta of other virtual particles
Is replaced by In the blobs with
factorized gluons
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Case B. The softest particle is a ladder quark
or gluon. It can also be factorized


k
k
gluon pair
quark pair
DL contributions come from the region
This case does not contribute when s -t i.e.
in the case of hard kinematics

When
such contributions disappear after applying
Combining case A and case B and adding Born terms
leads to the IREE
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IREE look simpler when an integral transform
(Mellin) is applied. For the Regge kinematics s
gtgt -t, one should use the Sommerfeld-Watson
transform or its simplified, Mellin-like,
version
where the signature factor
and
Inverse transform
difference with the standard Mellin transform
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Structure function
Forward Compton amplitude with negative signature
For singlet g1
Compton off gluon
Compton off quark
System of IREE for the Compton amplitudes
where
and
Anomalous dimension matrix. Sums up DL and part
of SL contributions
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IREE for the non-singlet g1 is simpler
new non-singlet anomalous dimension, sums up DLs
and SLs
Double Logarithms
Single Logarithms
New anomalous dimension HNS(w) accounts for the
total resummation of DL and a part of SL of x
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Expression for the non-singlet g1 at large Q2 Q2
gtgt 1 GeV2
Initial quark density
Coefficient function
Anomalous dimension
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Expression for the singlet g1 at large Q2
Large Q2 means
here
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Small x asymptotics of g1 when x ? 0, the
saddle-point method leads to
large Q2
small Q2
COMPASS Q2 ltlt 1 GeV2
As x0 gt x, dependence on x is weak
intercept
At large x, g1NS and g1S are positive
In the whole range of x at any Q2
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Asymptotics of the singlet g1 are more involved
intercept
where
(we assume that dq and dg are just constants)
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Sign of singlet g1
Case A
g1 is positive at large and small x
Negative and large
Case B
g1 is positive at large x and negative at small x
Negative but not large or positive
g1 is positive at large x and goes to zero at
small x
Case C
strong correlation fine tuning
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Values of the predicted intercepts
perfectly agree with results of several groups
who have fitted the experimental data.
Soffer-Teryaev, Kataev-Sidorov-Parente,
Kotikov-Lipatov-Parente-Peshekhonov-Krivokhijine-Z
otov, Kochelev-Lipka-Vento-Novak-Vinnikov
non-singlet intercept
singlet intercept
Note on the singlet intercept
violates unitarity

• Graphs with
• gluons only

similar to LO BFKL
B. All graphs
No violation of unitarity
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Comparison of our results to DGLAP at finite x
(no asymptotic formulae used) Comparison
depends on the assumed shape of initial parton
densities. The simplest option use the bare
quark input
in x- space
in Mellin space
Numerical comparison shows that the impact of the
total resummation of logs of x becomes quite
sizable at x 0.05 approx. Hence, DGLAP should
have failed at x lt 0.05. However, it does not.
Why?
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In order to understand what could be the reason
for the success of DGLAP at small x, let us
consider in more detail the standard fits for
initial parton densities.
Altarelli-Ball-Forte- Ridolfi
singular factor
normalization
regular factors
The parameters

are fixed from fitting experimental data at large
x
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Non-leading poles -k alt0
In the Mellin space this fit is
Leading pole a0.58 gt0
The small-x DGLAP asymptotics of g1 is
(inessential factors dropped)
phenomenology
Comparison with our asymptotics
calculations
shows that the singular factor in the DGLAP fit
mimics the total resummation of ln(1/x) .
However, the value a 0.58 differs from our
non-singlet intercept 0.4
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Although our and DGLAP asymptotics lead to an x-
behavior of Regge type, they predict different
intercepts for the x- dependence and
different Q2 -dependence
our calculations
x-asymptotics was checked by extrapolating the
available exp. data to x? 0. It agrees with our
values of D Contradicts DGLAP our and DGLAP
Q2 asymptotics have not been checked yet.
whereas DGLAP predicts the steeper x-behavior
and a flatter Q2 -behavior
DGLAP
Common opinion total resummation is not
relevant at available values of x. Actually the
resummation has been accounted for through the
fits to the parton densities, however without
realizing it.
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Numerical comparison of DGLAP with our approach
at small but finite x, using the same DGLAP fit
for initial quark density.
R g1 our/g1 DGLAP
Only regular factors in g1 our and g1 DGLAP
Regular term in g1 our vs regular singular in
g1 DGLAP
x
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R g1 our/g1 DGLAP as function of Q2 at
different x
X 10-4
R g1 our/g1 DGLAP
X 10-3
Q2
X 10-2
Q2
Q2 -dependence of R is flatter than the
x-dependence
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Structure of DGLAP fit
x-dependence is weak at xltlt1 and can be dropped
Can be dropped when ln(x) are resummed
Therefore at x ltlt 1
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Common opinion in DGLAP analyses the fits to the
initial parton densities are related to the
structure of hadrons, so they mimic effects of
Non-Perturbative QCD, using the phenomenological
parameters fixed from experiments . Actually,
the singular factors introduced in the fits mimic
the effects of Perturbative QCD at small x and
can be dropped when logarithms of x are resummed
Non-Perturbative QCD effects are included in
the regular parts of DGLAP fits. Obviously, the
impact of Non-Pert QCD is not strong in the
region of small x. In this region, the p.
densities can be approximated by an overall
factors N and a linear term in x
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Comparison between DGLAP and our approach at
small x
DGLAP
our approach
Coeff. functions and anom. dimensions sum up DL
and SL terms to all orders
Coeff. functions and anom. dimensions are
calculated with two-loop accuracy
Regge behavior is achieved automatically, even
when the initial densities are regular in x
To ensure the Regge behaviour, singular terms in
x are used in initial partonic densities.
This could be equivalent phenomenologically,
but could be
Asymptotic formulae of g1 are never used in
expressions for g1 at finite x
unreliable for g1 at very small x.
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Comparison between DGLAP and our approach at any x
DGLAP
our approach
Good at large x because includes exact two-loop
calculations but bad at small x as lacks the
total resummaion of ln(x)
Good at small x , includes the total resummaion
of ln(x) but bad at large x because neglects
some contributions essential in this region
SUGGESTION merging of our approach and DGLAP

• Expand our formulae for coefficient functions and
  anomalous dimensions into series in the QCD
  coupling
• Replace the first- and second- loop terms of the
  expansion by
• corresponding DGLAP expressions

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Our expressions
anomalous dimension
coefficient function
First tems of the perturbative expansions in
series
New synthetic formulae
New synthetic formulae include all advantages
of both approaches and should be equally good
at large and small x. New fits for the part.
densities should not involve singular factors.
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COMPASS is a high-energy physics experiment at
the Super Proton Synchrotron (SPS) at CERN in
Geneva, Switzerland. The purpose of this
experiment is the study of hadron structure and
hadron spectroscopy with high intensity muon and
hadron beams.  On February 1997 the experiment
was approved conditionally by  CERN and the final
Memorandum of Understanding was signed in
September 1998. The spectrometer was installed in
1999 - 2000 and was commissioned during a
technical run in 2001. Data taking started in
summer 2002 and continued until fall 2004. After
one year shutdown in 2005, COMPASS will resume
data taking in 2006.   Nearly 240 physicists
from 11 countries and 28 institutions work in
COMPASS
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COMPASS COmmon Muon Proton Apparatus for
Structure and Spectroscopy
Artistic view of the 60 m long COMPASS two-stage
spectrometer. The two dipole magnets are
indicated in red
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COMPASS runs at small Q2 and small
x
In order to generalize our results to the region
of small Q2 , one should recall that
is the result of the integration
However this is valid for large Q2 only. For
arbitrary Q2
Introduce a mass of virtual quarks and gluons
to regulate infrared singularities
This is suggested by the relevant Feynman
diagrams involved.
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It leads to new expressions
Small Q2 non-singlet g1
weak x -dependence
Anomalous dimension
weak Q2 -dependence
Coefficient function
Initial quark density
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Small Q2 Singlet g1
again
weak Q2-dependence
PREDICTION
At Q2 lt 1 GeV2, the x-dependence is almost flat
even at xltlt1
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(Phys.Lett.B647, 330, 2007)
C O M P A S S
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Power Q2 -corrections
There exist contributions 1/(Q2)k , with k
1,2,.. They can be of perturbative (renormalons)
and non-perturbative (higher twists) origin.
In practice, the power corrections are obtained
phenomenologically through a discrepancy between
DGLAP predictions and experimental data.
However,
New power terms. Taken Into account, they can
change impact of higher twists
when
and
when
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Conclusions
Standard Approach is based on the DGLAP
evolution equations and phenomenological fits
for the initial parton densities. DGLAP was
originally developed, and is very successful, in
the region where both x and Q2 are large, but
neglects logs of (1/x), so it cannot applied
safely at very small x. In order to extend SA
to the region of small x, phenomenological
singular factors x-a have to be incorporated
into the parton densities. These factors mimic
the total resummation of leading logs of x that
we perform in our approach, which leads naturally
to the Regge behaviour of g1 at small x with
predicted intercepts, in agreement with exps.
SUGGESTION merging of
our approach and DGLAP The region of small Q2
is also beyond the reach of SA. In our model of
extension at small Q2 we predict that g1 is
almost independent of x, even at xltlt 1. Instead,
it depends on 2pq only. In particular g1 can be
pretty close to zero in the range of 2pq
investigated experimentally by COMPASS. This
agrees with the data.