1. Structure of standard DGLAP inputs for initial parton densities and the role of the singular terms B.I. Ermolaev, M. Greco, S.I. Troyan Spin-05 Dubna Sept. 27- Oct. 1, 2005

2. Deep Inelastic e-p Scattering Incoming lepton outgoing lepton- registered Incoming hadron Produced hadrons - not registered k p K’ X Deeply virtual photon q

3. Leptonic tensor hadronic tensor W  hadronic tensor consists of two terms: Does not depend on spin Spin-dependent

4. The spin-dependent part of W mn 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 ( Q 2 = - q 2 > 0 ). Both of the functions depend on Q 2 and x = Q 2 /2pq, 0< x < 1. At small x: longitudinal spin-flip transverse spin -tlip

5. When the total energy and Q 2 are large compared to the mass scale, one can use the factorization and represent W  as a convolution of the the partonic tensor and probabilities to find a polarized parton (quark or gluon) in the hadron : W quark  quark W gluon  gluon q q p p

6. Probability to find quark Probability to find gluon DIS off quark DIS off gluon In the analytic way this convolution is written as follows :

7. 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 Q 2, 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-singlet g 1 in the Mellin space is : Dokshitzer-Gribov- Lipatov-Altarelli-Parisi

8. Coefficient function Anomalous dimension Initial quark density Expression for the singlet g 1 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 Coefficient function C DGLAP evolves the initial quark density  q : Anomalous dimension governs the Q 2 - evolution of  q Evolved quark distribution Pert QCD Non-Pert QCD

9. LO NLO In DGLAP, coefficient functions and anomalous dimensions are known with LO and NLO accuracy One can say that DGLAP includes both Science and Art :

10. matrix of   Gribov, Lpatov, Ahmed, Ross, Altarelli, Parisi, Dojshitzer matrix of   1  Floratos, Ross, Sachradja, Gonzale- Arroyo, Lopes, Yandurain, Kounnas, Lacaze, Gurci, Furmanski, Peronzio, Zijlstra, Merig, van Neervan, Gluck, Reya, Vogelsang Coefficient functions C (1) k, C (2) k Bardeen, Buras, Duke, Altarelli, Kodaira, Efremov, Anselmino, Leader, Zijlstra, van Neerven SCIENCE ART There are different its for  q and  g. For example, Altarelli-Ball- Forte-Ridolfi Parameters N, , , ,  should be fixed from experiment

11. 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: DGLAP 1/x 1 Q2Q2 ln(1/x) < ln(Q 2 ) DGLAP accounts for ln(Q 2 ) to all orders in  s and neglects with k>2 ln(1/x)> ln(Q 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 22

12. K3K2K1K3K2K1 DGLAP –ordering: good approximation for large x when logs of x can be neglected. At x << 1 the ordering has to be lifted q p DGLAP small -x asymptotics of g 1 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:  s is fixed at unknown scale. DGLAP : running  s Arguments in favor of the DGLAP- parameterization Bassetto-Ciafaloni-Marchesini - Veneziano, Dokshitzer-Shirkov

13. K K’ K K’ K K’ Origin: in each ladder rung DGLAP-parameterization However, such a parameterization is good for large x only. At x << 1 : Ermolaev- Greco- Troyan 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: 1.Total resummation of logs of x 2.New parameterization of  s The basic idea: the formula valid when k 2 >>  2 it is necessary to introduce an infrared cut-off  for k 2 It is convenient to introduce  in the transverse space: k 2 >>  m 2 Lipatov

14. 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-singlet g 1 in 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:

15. Expression for the non-singlet g 1 : Expression for the singlet g 1 is similar, though more involved. When x  0, The x -dependence perfectly agrees with results of several groups who fitted experimental data. The Q 2 –dependence has not been checked yet Soffer-Teryaev, Kataev- Sidorov-Parente, Kotikov- Lipatov-Parente-Peshekhonov -Krivokhijine-Zotov, Kochelev- Lipka-Vento-Novak-Vinnikov intercepts  NS = 0.42  S = 0.86.

16. Comparison between our and DGLAP results for g 1 depends on the assumed shape of initial parton densities. The simplest case: the bare quark input in x- space in Mellin space Numerical comparison shows hat impact of the total resummation of logs of x becomes quite sizable at x = 0.05 approx. Hence, DGLAP should fail at x < 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 fits for initial parton densities. For example, Altarelli-Ball-Forte- Ridolfi normalization singular factor

17. In the Mellin space this fit is Leading pole      Non-leading poles  <  the small- x DGLAP asymptotics of g 1 is (inessential factors dropped ) Comparison it to our asymptotics 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 phenomenology calculations

18. Comparison between our and DGLAP results for g 1 depends on the assumed shape of initial parton densities. The simplest case: the bare quark input Numerical comparison shows hat impact of the total resummation of logs of x becomes quite sizable at x = 0.05 approx. in x- space in Mellin space

19. Hence, DGLAP should fail at x < 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 fits for initial parton densities. For example, Altarelli-Ball-Forte- Ridolfi normalization singular factor

20. Although both our and DGLAP formulae lead to x- asymptotisc of Regge type, they predict different Q 2 - asymptotics: our prediction Is the scaling whereas DGLAP predicts the steeper x-behavior and the flatter Q 2 -behavior: x-asymptotics is checked with extrapolating available exp data to x  0. Agrees with our values of  Contradicts DGLAP Q 2 –asymptotics has not been checked yet. our calculations DGLAP fit

21. Structure of DGLAP fit Can be dropped when ln(x) are resummed x-dependence is weak at x<<1 and can be dropped Common opinion: fits for  q are singular but convoluting them with coefficient functions weakens the singularity Obviously, it is not true,  q and  q are equally singular

22. Common opinion : DGLAP fits mimic structure 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

23. WAY OUT – synthesis of our approach and DGLAP 1.Expand our formulae for coeff functions and anom dimensions into series in  s 2.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 DGLAP Good at large x because includes exact two-loop calculations for C and  but lacks the total resummaion of ln(x) our approach Good at small x, includes the total resummaion of ln(x) for C and  but bad at large x because Neglects some contributions essential in this region

24. Conclusion Total resummation of the double- and single- logarithmic contributions New anomalous dimensions and coefficient functions At x  0, asymptotics of g 1 is power-like in x and Q 2 New scaling: g1 ~ (Q 2 /x 2 ) -  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 g 1 and 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 at x<<1