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

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  1. Frascati, 14 May 2007 Deep Inelastic Scattering and the limit of small parton energy fraction. M. Greco talk based on results obtained in collaboration withB.I. Ermolaev and S.I. Troyan

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

  3. Leptonic tensor q q hadronic tensor p p Does not depend on spin Spin-dependent Hadronic tensor consists of two terms: antisymmetric symmetric

  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 functions depend on Q2 and x = Q2 /2pq, 0< x < 1. At small x: longitudinal spin-flip transverse spin -flip Theoretical study of g1and g2involves both Pert and Non-Pert QCDand therefore it is not model-independent

  5. When the total energy and Q2are 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

  6. This allows to representas a convolution of the partonic tensor and the probabilities to find a (polarized) parton (quark or gluon) in the hadron : DIS off gluon DIS off quark q q Wquark Wgluon Fquark Fgluon p p Probability to find a quark Probability to find a gluon

  7. 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

  8. ThenDIS off quarks and gluons can be studied with perturbative QCD, by calculating the Feynman graphs involved. The probabilities,Fquark and Fgluoninvolve 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 denoteddq 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

  9. 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

  10. 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: Coefficient function Evolved quark distribution

  11. where Splitting function The expression for the singletg1is similar, though more involved. It includes coefficient functions and splitting functions . in order to evolve the quarkand gluon distributions Dq andDg

  12. Using the Mellin transform, one obtains the expression for g1NSina simpler form : Initial quark density Non-Pert QCD Anomalous dimension Coefficient function Pert QCD

  13. Kinematics in the (1/x - Q2) plane 1/x g1 at x<<1 and Q2 >> m2 x-evolution of Dq with coefficient function Q2 -evolution of dq with anomalous dimension Dq at x~1 and Q2 .>> m2 1 Q2 m2 dq at x ~1 and Q2 ~ m2 defined from fitting exp. data evolved quark density Starting point of Q2 -evolution

  14. In DGLAP, the coefficient functions and anomalous dimensions are known with LO and NLO accuracy. LO NLO LO NLO

  15. 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,…

  16. Phenomenology of the fits for the parton densities Altarelli-Ball-Forte-Ridolfi, Blumlein-Botcher, Leader-Sidorov- Stamenov, Hirai et al.,… There are different fits forthe initial parton densities.For example, Altarelli-Ball- Forte-Ridolfi, The parametersshould 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- xregion:

  17. 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 k>2 However, these contributions become leading at small x and should be accounted for to all orders in the QCD coupling.

  18. 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

  19. DGLAP –ordering: q K3 K2 K1 good approximation for large x whenlogs ofxcan be neglected. At x << 1the 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

  20. 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 runningas DGLAP- parameterization Arguments in favor of the Q2- parameterization: Bassetto-Ciafaloni-Marchesini - Veneziano, Dokshitzer-Shirkov

  21. 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

  22. 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-xregion, it is necessary: • Total resummation of logs ofx • New parametrization of the QCD coupling How: the formula is valid when Then it is necessary to introduce an infrared cut-offfork2 in the transverse space:Lipatov

  23. As the value of the cut-off is not fixed, one can evolve the structure functions with respect tom Infra-Red Evolution Equations (IREE) (name of the method) Method prevoiusly used by Gribov, Lipatov, Kirschner, Bartels-Ermolaev-Manaenkov-Ryskin, …

  24. Essence of the method t g g Typical Feynman graph 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

  25. 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

  26. 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

  27. 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

  28. IREE look simpler when an integral transform (Mellin) is applied. For the Regge kinematics s >> -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

  29. 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

  30. IREEfor the non-singlet g1is 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

  31. Expression for the non-singlet g1at large Q2: Q2 >> 1 GeV2 Initial quark density Coefficient function Anomalous dimension

  32. Expression for the singletg1at largeQ2: Large Q2means here

  33. Small –x asymptotics of g1: whenx  0, the saddle-point method leads to large Q2 small Q2 COMPASS: Q2 << 1 GeV2 As x0 > x, dependence on x is weak intercept At large x,g1NS and g1Sare positive In the whole range of x at any Q2

  34. Asymptotics of the singlet g1are more involved intercept where (we assume that dq and dg are just constants)

  35. Sign of singlet g1: Case A g1is positive at large and small x Negative and large Case B g1is positive at large x and negative at small x Negative but not large or positive g1is positive at large xand goes to zero at small x Case C strong correlation = fine tuning

  36. 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-Zotov, 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

  37. Comparison of our results to DGLAP at finitex (no asymptotic formulae used) Comparisondepends on the assumed shape of initial parton densities. The simplest option: usethebare quark input in x- space in Mellin space Numerical comparison shows that the impact of the total resummation of logs ofxbecomes quite sizable atx = 0.05approx. Hence, DGLAP should have failed atx < 0.05. However,it does not. Why?

  38. 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 normalization singular factor regular factors The parameters are fixed from fitting experimental data at largex

  39. Non-leading poles -k +a<0 In the Mellin space this fit is Leading polea=0.58 >0 The small-x DGLAPasymptotics of g1is (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 valuea = 0.58 differs from our non-singlet intercept =0.4

  40. 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 ofD 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.

  41. Numerical comparison of DGLAP with our approach at small but finitex, using the same DGLAP fitfor initial quark density. R = g1our/g1DGLAP Only regular factors in g1our and g1DGLAP Regular term in g1our vs regular + singular in g1DGLAP x

  42. R = g1our/g1DGLAP as function ofQ2at differentx X = 10-4 R = g1our/g1DGLAP X = 10-3 Q2 X = 10-2 Q2 Q2 -dependence of R is flatter than the x-dependence

  43. Structure of DGLAP fit x-dependence is weak at x<<1 and can be dropped Can be dropped when ln(x) are resummed Therefore at x << 1

  44. Common opinion:in DGLAP analyses the fits to the initial parton densities are related to thestructure 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 anda linear term in x

  45. 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 usedin 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 forg1at very small x.

  46. 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 resummaionof ln(x) but bad at largexbecause 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 inthe QCD coupling • Replace the first- and second- loop terms of the expansion by • corresponding DGLAP –expressions