DIS Polarized Structure Functions at small x

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DIS Polarized Structure Functions at small x

• B.I. Ermolaev, M. G., 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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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 of the functions depend on Q2
and x Q2 /2pq, 0lt x lt 1. At small x
longitudinal spin-flip transverse spin
-flip

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Leptonic tensor
hadronic tensor
Does not depend on spin
Spin-dependent
hadronic tensor consists of two terms
antisymmetric
symmetric
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When the total energy and Q2 are large compared
to the mass scale, one can use the factorization
and represent as a convolution of the
the partonic tensor and probabilities to find a
polarized parton (quark or gluon) in the hadron
q
Wquark
Wgluon
q
?quark
?gluon
p
p
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In the analytic way this convolution is written
as follows
DIS off quark
DIS off gluon
Probability to find quark
Probability to find gluon
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DIS off quark and gluon can be studied with
perturbative QCD, with calculating involved
Feynman graphs. Probabilities, ?quark and
?gluon involve non-perturbaive QCD. There is no a
regular analytic way to calculate them. Usually
they are defined from experimental data at large
x and small Q2 , 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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Terminology
Contribute to singlet
Contributes to nonsinglet
Initial quark
The standard instrument for theoretical
investigation of the polarized DIS is DGLAP.
Dokshitzer-Gribov- Lipatov-Altarelli-Parisi
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Pert QCD
Coefficient function
Anomalous dimension
Evolved quark distribution
Initial quark density
Non-Pert QCD
Coefficient function CDGLAP evolves the initial
quark density
Anomalous dimension governs the Q2 -evolution of
Expression for the singlet g1 is similar, though
more involved. It includes more coefficient
functions, the matrix of anomalous dimensions
and, in addition to ?q , the initial gluon
density ?g
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In DGLAP, coefficient functions and anomalous
dimensions are known with LO and NLO accuracy
LO
NLO
LO
NLO
One can say that DGLAP includes both Science and
Art
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SCIENCE
Gribov, Lipatov, Ahmed, Ross, Altarelli, Parisi,
Dokshitzer
matrix of LO anomalous dimensions
Floratos, Ross, Sachradja, Gonzales-Arroyo,
Lopes, Yndurain, Kounnas, Lacaze, Curci,
Furmanski, Petronzio, Zijlstra, Mertig, van
Neerven, Gluck, Reya, Vogelsang
matrix of NLO anomalous dimensions

Coefficient functions C(1)k , C(2)k
Bardeen, Buras, Duke, Altarelli, Kodaira,
Efremov, Anselmino, Leader, Zijlstra, van
Neerven
ART
There are different its for initial parton
densities. For example,
Altarelli-Ball- Forte-Ridolfi
Parameters should be
fixed from experiment
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This combination of science and art works well at
large and small x, though strictly speaking,
DGLAP is not supposed to work at the small- x
region
1/x
Small x
DGLAP region ln(1/x) are small
ln(1/x) are large
Large x
1
??
Q2
DGLAP accounts for ln(Q2) to all orders in ?s
and neglects
with kgt2
However, these contributions become leading at
small x and should be accounted for to all
orders in the QCD coupling.
DGLAP cannot do total resummation of logs of x
because of the DGLAP-ordering 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
p
DGLAP small-x asymptotics of g1 is well-known
when the initial parton densities
are not singular functions of x When the DGLAP
ordering is lifted and all double logarithms of
x are accounted for, the asymptotics is
different
Bartels- Ermolaev- Manaenkov-Ryskin
intercept
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Non- singlet intercept
singlet intercept
The weakest point of this approach the QCD
coupling is fixed at an unknown scale. On the
contrary, DGLAP equations always operate with
running ?s
DGLAP- parameterization
Bassetto-Ciafaloni-Marchesini - Veneziano,
Dokshitzer-Shirkov
Arguments in favor of the DGLAP-
parameterization
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Origin in each ladder rung
K K
K K
K K
DGLAP-parameterization
Ermolaev-Greco-Troyan
However, such a parameterization is good for
large x only. At x ltlt 1
When DGLAP- ordering and x 1
time-like argument Contributes to the Mellin
transform
space-like argument, no Mellin transform
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• Obviously, this parameterization and the DGLAP
  one
• coincide when x is large but differ a lot at
  small x
• So, for studying g1 in the small-x region, it is
  necessary to do
• Total resummation of logs of x
• New parameterization of the QCD coupling

The basic idea
the formula
is valid when
it is necessary to introduce an infrared cut-off
for k2
It is convenient to introduce the cut-off 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
?. This method is known as Infra-Red
Evolution Equations (IREE)
IREE for the non-singlet g1 in the Mellin space
can be written similarly to the DGLAP equation
Double logarithms
Single logarithms
new anomalous dimension H accounts for the total
resummation of double and single logarithms of x
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In contrast to DGLAP, H and C can be
calculated with the same method. Expressions for
them are
A is the QCD coupling in the Mellin space
B,A,D are expressed through conventional QCD
parameters
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Expression for the non-singlet g1
Expression for the singlet g1 is similar, though
more involved. When x ? 0,
intercepts ? NS 0.42 ? S 0.86.
Soffer-Teryaev, Kataev- Sidorov-Parente,
Kotikov- Lipatov-Parente-Peshekhonov -Krivokhijine
-Zotov, Kochelev- Lipka-Vento-Novak-Vinnikov
The x-dependence perfectly agrees with results
of several groups who have analyzed experimental
data. The Q2 dependence has not been checked yet
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Comparison between our and DGLAP results for g1
depends on the assumed shape of initial parton
densities. The simplest case in DGLAP and
our expression for non-singlet lets choose the
bare quark input
in Mellin space
in x- 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.
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In order to understand what could be the reason
for this unexpected success of DGLAP at small x,
let us consider in more detail the standard
DGLAP fits for initial parton densities. For
example
Altarelli-Ball-Forte- Ridolfi
singular factor
normalization
regular factors
The parameters
are fixed from fitting exp data at large x
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In the Mellin space this fit gives
Non-leading poles ?lt?
Leading pole ?????
At small-x DGLAP asymptotics of g1 is then
(inessential factors dropped) )
phenomenology
Comparison to our asymptotics
calculations
shows that the singular factor in the DGLAP fit
mimics the total resummation of ln(1/x) .
However, the value ? 0.58 sizably differs
from our non-singlet intercept 0.4
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Although our and DGLAP phenom. asymptotics lead
to the x- behaviour of Regge type, they predict
different intercepts and different Q2
-dependence our prediction is the scaling
our calculation
x-asymptotics was checked with available exp
data extrapolating to x? 0. It agrees with our
values of ? Contradicts DGLAP our and the
DGLAP Q2 asymptotics have not been checked yet.

whereas DGLAP predicts a steeper x-behaviour
and a flatter Q2 - behaviour
DGLAP fit
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Alternatively DGLAP fails at x lt 0.05 when the
simplest, bare quark fit is used. Let us
numerically compare DGLAP with our approach at
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
g1 our and g1 DGLAP regular singular
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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Usual comment DGLAP fits for ?q are singular,
but being defined at large x, convoluting them
with coefficient functions makes the singularity
weaker
Obviously, this is not true They are both
equally singular
initial
evolved
Structure of DGLAP fit once again
Can be dropped if ln(x) are resummed
x-dependence is weak at xltlt1 and can be dropped
Therefore at x ltlt 1
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Common opinion DGLAP initial parton densities
are related to the structure of hadrons, so they
mimic the effects of Non-Perturbative QCD, and
using phenomenological parameters they are fixed
from experiments. Actually, singular factors in
the fits mimic the effects of Perturbative QCD
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 so
strong in the region of small x. In this region,
the fits approximately give an overall factor
N
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Comparison between DGLAP and our approach at
small x
DGLAP
our approach
Coeff. functions and anom. dimensions sum DL and
SL terms to all orders
Coeff. functions and anom. dimensions are
calculated with two-loop accuracy
Regge behaviour is achieved automatically, even
when the initial densities are regular in x
To ensure Regge behaviour, singular terms in x
are used in the initial partonic densities
Equivalent to inserting a phenomenological
asymptotic factor into expressions for g1
Asymptotics of g1 are never used in
expressions for g1 at finite x
Warning asymptotic formulae for g1 are
unreliable at x gt 10-5
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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 resummation of ln(x)
Good at small x , includes the total resummation
of ln(x) but bad at large x because neglects
some contributions essential in this region
WAY OUT merging of our approach and DGLAP

• Expand our formulae for coeff. functions and
  anom. dimensions into a series in the QCD
  coupling
• Replace the first- and second- loop terms of the
  expansion by
• corresponding DGLAP expressions
• New, synthetic formulae have the advantage of
  the both
• approaches and are equally good at large and
  small x.
• Only regular terms in x for the init. part.
  densities are required

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Both for small x
Comparison between our results and BFKL
BFKL
Our approach
describes unpolarized processes, does not
contributes to spin-flips and asymmetries No
model-independent BFKL expressions for F1, F2
Violates unitarity saturation required
Special arrangements for QCD coupling are
needed
describes polarized processes, contributes to
spin-flips and asymmetries Explicit expressions
for g1 Obtained with Pert QCD No unitarity
violation QCD coupling is running in
each Feynman graph
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t
pb
p1
Cross section (1/s) Im
s
pn
pa
Regge kinematics sgtgt t
BFKL
universal ingredient for all processes with
unpolarized particles
Impact factors They depend on the process
For polarized particles, with S S
our reggeon
BFKL
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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
33
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 operates with small Q2 and small x
In order to generalize our results to the region
of small Q2 , one should remember that
is the result of the integration
However it is valid for large Q2 only. For
arbitrary Q2
Introduced as a mass of virtual quarks and
gluons to regulate infrared singularities
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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
Where again
weak Q2-dependence
At Q2 lt 1 GeV2, the x-dependence is almost flat
even at xltlt1
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Power Q2 -corrections
There are known 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
through 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 I
Total resummation of the double- and single-
logarithmic contributions
New anomalous dimensions and coefficient
functions.
New scaling
At x? 0, asymptotics of g1 is power-like in x and
Q2
With init. densities regular in x, DGLAP becomes
unreliable at x0.05 approximately.
Singular terms in the DGLAP initial parton
densities ensure a steep rise of g1 at small x
and mimic the resummation of logs of x. With the
resummation accounted for, they can be
dropped. x-dependent terms in the regular
factors can also be dropped at xltlt1, so the fits
can be reduced down to a constant DGLAP init.
dens. are expected to describe non-Pert QCD .
Instead, they basically correspond to Pert QCD.
? Non-Pert effects are surprisingly small at xltlt1
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Conclusions
II The region of small Q2 is also beyond the
reach of SA. We predict that g1 at small Q2 is
almost independent of x, even at xltlt 1. Instead,
it depends on 2pq only. At a certain relation
between the initial quark and gluon densities,
g1 can be pretty close to zero in the range of
2pq investigated now experimentally by COMPASS.