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

1. 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

2. DeepInelastice-pScattering Incoming lepton outgoing lepton- registered K’ Deeply virtual photon k q Produced hadrons - not registered X p Incoming hadron

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 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

5. 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

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

7. 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

8. 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

9. 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 :

10. 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

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- 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

12. 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

13. 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

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-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:

15. 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

16. 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

17. 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

18. 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.

19. 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

20. 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

21. 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

22. 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

23. 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

24. 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