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Spin-05 Dubna Sept. 27- Oct. 1, 2005. S tructure of standard DGLAP inputs for initial parton densities and the role of the singular terms. B.I. Ermolaev, M. Greco, S.I. Troyan. Deep Inelastic e-p Scattering. Incoming lepton. outgoing lepton- registered. K’. Deeply virtual photon. k. q.

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s tructure of standard dglap inputs for initial parton densities and the role of the singular terms

Spin-05 Dubna Sept. 27- Oct. 1, 2005

Structure of standard DGLAP inputs for initial parton densities and the role of the singular terms

B.I. Ermolaev, M. Greco, S.I. Troyan

slide2

DeepInelastice-pScattering

Incoming lepton

outgoing lepton-

registered

K’

Deeply virtual photon

k

q

Produced hadrons

- not registered

X

p

Incoming hadron

slide3

Leptonic tensor

hadronic tensor W

hadronic tensor consists of two terms:

Does not depend on spin

Spin-dependent

slide4

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 > 0). Both of the functions depend on Q2 and x = Q2 /2pq, 0< x < 1.

At small x:

longitudinal spin-flip transverse spin -tlip

slide5

When the total energy and Q2are large compared to the mass scale, one

can use the factorization and representWas a convolution of the

the partonic tensor and probabilities to find a polarized parton

(quark or gluon) in the hadron :

q

q

Wquark

Wgluon

quark

gluon

p

p

slide6

In the analytic way this convolution is written as follows:

DIS off quark

DIS off gluon

Probability to find quark

Probability to find gluon

slide7

DIS off quark and gluon can be studied with perturbative QCD, with calculating involved Feynman graphs.

Probabilities,quark andgluoninvolve 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q andg .

The conventional form of the hadronic tensor is:

The standard instrument for theoretical investigation of the polarized DIS is DGLAP.

The DGLAP –expression for the non-singletg1 in the Mellin space is:

Dokshitzer-Gribov-

Lipatov-Altarelli-Parisi

slide8

Pert QCD

Coefficient function

Anomalous dimension

Evolved quark

distribution

Initial quark density

Non-Pert QCD

Coefficient function CDGLAP evolves the

initial quark densityq :

Anomalous dimension governs the Q2 -evolution ofq

Expression for the singletg1is similar, though more involved. It

includes more coefficient functions, the matrix of anomalous

dimensions and, in addition toq , the initial gluon densityg

slide9

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 :

slide10

SCIENCE

matrix of

Gribov, Lpatov, Ahmed, Ross, Altarelli, Parisi,

Dojshitzer

Floratos, Ross, Sachradja, Gonzale- Arroyo,

Lopes, Yandurain, Kounnas, Lacaze, Gurci,

Furmanski, Peronzio, Zijlstra, Merig, van Neervan,

Gluck, Reya, Vogelsang

matrix of 1

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q andg. For example,

Altarelli-Ball-

Forte-Ridolfi

Parameters N, , , ,  should be fixed from experiment

slide11

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- xregion:

1/x

ln(1/x) < ln(Q2)

ln(1/x)> ln(Q2)

DGLAP

1

2

Q2

DGLAP accounts for ln(Q2) to

all orders in sand neglects

with k>2

However, these contributions become leading at small x and should be

accounted for to all orders in the QCD coupling.

Total resummation of logs of x cannot be done because of the

DGLAP-ordering – the keystone of DGLAP

slide12

DGLAP –ordering:

q

K3

K2

K1

good approximation for large x whenlogs ofxcan

be neglected. At x << 1the ordering has to be lifted

p

DGLAP small-x asymptotics of g1 is well-known:

When the DGLAP –ordering is lifted, the asymptotics is different:

Bartels- Ermolaev-

Manaenkov-Ryskin

Non- singlet intercept

singlet intercept

The weakest point:sis fixed at unknown scale. DGLAP : runnings

Arguments in favor of the

DGLAP- parameterization

Bassetto-Ciafaloni-Marchesini

- Veneziano, Dokshitzer-Shirkov

slide13

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. Atx << 1 :

  • Obviously, this parameterization and the DGLAP one
  • converge when x is large but differ a lot at small x
  • So, in the small-x region, it is necessary:
  • Total resummation of logs of x
  • New parameterization ofs

valid when

k2>>2

The basic idea: the formula

it is necessary to introduce an infrared cut-offfork2

It is convenient to introduce in the transverse space:

k2 >> m2 Lipatov

slide14

As value of the cut-off is not fixed, one can evolve the structure functions

with respect tothe name of the method:

Infra-Red Evolution Equations (IREE)

IREE for the non-singletg1in the Mellin space looks similar to the DGLAP eq:

new anomalous dimension H()

accounts for the total resummation

of double- and single- logs of x

Contrary to DGLAP, H () and C () can be calculated with the same method.

Expressions for hem are:

B () is expressed through

conventional QCD parameters:

slide15

Expression for the non-singletg1 :

Expression for the singletg1is similar, though more involved.

Whenx  0,

intercepts

NS = 0.42S = 0.86.

Soffer-Teryaev, Kataev-

Sidorov-Parente, Kotikov-

Lipatov-Parente-Peshekhonov

-Krivokhijine-Zotov, Kochelev-

Lipka-Vento-Novak-Vinnikov

Thex-dependence perfectly agrees with

results of several groups who fitted

experimental data. The Q2 –dependence

has not been checked yet

slide16

Comparison between our and DGLAP results forg1depends on the

assumed shape of initial parton densities.

The simplest case: thebare quark input

in x- space

in Mellin space

Numerical comparison shows hat impact of the total resummation of logs

ofxbecomes quite sizable atx = 0.05approx. Hence, DGLAP should fail

atx < 0.05. However,it does not take place. In order to understand what

could be the reason to it, let us give more attention to structure of

Standard DGLAP fitsfor initial parton densities. For example,

Altarelli-Ball-Forte-

Ridolfi

normalization

singular

factor

slide17

In the Mellin space this fit is

Non-leading poles

 < 

Leading pole

  

the small-x DGLAPasymptotics of g1is (inessential factors dropped )

phenomenology

Comparison it to our asymptotics

calculations

shows that the singular factor x-in the DGLAP fit mimics the

total resummation of ln(1/x) . However, the value = 0.53

differs from our intercept

slide18

Comparison between our and DGLAP results forg1depends on the

assumed shape of initial parton densities.

The simplest case: thebare quark input

in x- space

in Mellin space

Numerical comparison shows hat impact of the total resummation of logs

ofxbecomes quite sizable atx = 0.05approx.

slide19

Hence, DGLAP should fail atx < 0.05. However,it does not take place. In order to understand what could be the reason to it, let us give more attention to structure of

Standard DGLAP fitsfor initial parton densities.

For example,

Altarelli-Ball-Forte-

Ridolfi

normalization

singular

factor

slide20

Although both our and DGLAP formulae lead to x- asymptotisc of

Regge type, they predict different Q2 -asymptotics: our prediction

Is the scaling

our calculations

x-asymptotics is checked

with extrapolating available

exp data to x 0.

Agrees with our values of 

Contradicts DGLAP

Q2–asymptotics has not

been checked yet.

whereas DGLAP predicts the steeper

x-behavior and

the flatter Q2 -behavior:

DGLAP fit

slide21

Common opinion:fits forq are singular butconvoluting them with

coefficient functions weakens the singularity

Obviously, it is not true,

q andq are equally

singular

Structure of DGLAP fit

x-dependence is weak at x<<1 and can be dropped

Can be dropped when ln(x) are resummed

slide22

Common opinion:DGLAP fits mimicstructure of hadrons, they describe effects of Non-Perturbative QCD, using many phenomenological parameters fixed from experiment.

Actually, singular factors in the fits mimic effects of Perturbative QCD and can be dropped when logarithms of x are resummed

Non-Perturbative QCD effects are accumulated in the regular parts of DGLAP fits. Obviously, impact of Non-Pert QCD is not strong in the region of small x. In this region, the fits approximately = overall factor N

slide23

DGLAP

our approach

Good at large x because

includes

exact two-loop calculations for

C and  but lacks the total

resummaion of ln(x)

Good at small x , includes

the total resummaionof ln(x) for C

and but bad at largexbecause

Neglects some contributions

essential in this region

WAY OUT – synthesis of our approach and DGLAP

  • Expand our formulae for coeff functions and anom dimensions into
  • series ins
  • Replace the first- and second- loop terms of the expansion by
  • corresponding DGLAP –expressions
  • New, “synthetic” formulae accumulate all advantages of the both
  • approaches and are equally good at large and small x
slide24

Conclusion

Total resummation of the double- and single- logarithmic contributions

New anomalous dimensions and coefficient functions

At x 0, asymptotics of g1is power-like in x and Q2

New scaling:

g1 ~ (Q2/x2)-

With fits regular in x, DGLAP would become unreliable at x=0.05 approx

Singular terms in the DGLAP fits ensure a steep rise of g1and mimic

the resummation of logs of x. With the resummation accounted for,

they can be dropped.

Regular factors can be dropped at x<<1, so the fits can be reduced

down to constants

DGLAP fits are expected to correspond to Non-Pert QCD. Instead,

they basically correspond to Pert QCDNon-Pert effects are

surprisingly small atx<<1