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

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

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

8. 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 functions depend on Q 2 and x = Q 2 /2pq, 0< x < 1. At small x: longitudinal spin-flip transverse spin -flip Theoretical study of g 1 and g 2 involves both Pert and Non-Pert QCD and therefore it is not model-independent

9. When the total energy and Q 2 are large compared to the mass scale, one can use factorization : k P q k P q k P q p p quarkgluon DIS off quark DIS off gluon Pert QCD Non-pert QCD = +

10. This allows to represent as a convolution of the partonic tensor and the probabilities to find a (polarized) parton (quark or gluon) in the hadron : W quark F quark W gluon F gluon qq pp DIS off quark DIS off gluon Probability to find a quark Probability to find a gluon

11. Analytically this convolution is written as follows : Perturbative QCD Non-pert QCD Pert QCD: analytical calculations of Feynman graphs Non-perturbative QCD: no regular methods

12. Then DIS off quarks and gluons can be studied with perturbative QCD, by calculating the Feynman graphs involved. The probabilities, F quark and F gluon involve non-perturbaive QCD. There is no regular analytic way to get them. Usually they are fitted from the experimental data at large x and Q 2, and 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

13. Some terminology Contributes to nonsinglet Contribute to singlet Initial quark Each structure function has both a non-singlet and a singlet components: g 1 = g 1 NS + g 1 S

14. The Standard Approach consists in using the perturbative Altarelli-Parisi or DGLAP Q 2 - Evolution Equations, together with fits for the initial parton densities. Evolution Equations: Altarelli-Parisi, Gribov-Lipatov, Dokshitzer Evolved quark distribution In particular, for the non-singlet g 1 : Coefficient function

15. The expression for the singlet g 1 is similar, though more involved. It includes coefficient functions and splitting functions in order to evolve the quark and gluon distributions Dq and Dg Splitting function where.

16. Coefficient function Anomalous dimension Initial quark density Using the Mellin transform, one obtains the expression for g 1 NS in a simpler form : Pert QCD Non-Pert QCD

17. 1/x 1 Q2Q2 m2m2 x-evolution of Dq with coefficient function Q 2 -evolution of dq with anomalous dimension evolved quark density Dq at x~1 and Q 2.>> m 2 g 1 at x<<1 and Q 2 >> m 2 dq at x ~1 and Q 2 ~ m 2 defined from fitting exp. data Starting point of Q2 -evolution Kinematics in the (1/x - Q 2 ) plane

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

19. LO splitting functions Ahmed-Ross, Altarelli-Parisi, Sasaki,… NLO splitting functions Floratos, Ross, Sachradja, Gonzale- Arroyo, Lopes, Yandurain, Kounnas, Lacaze, Curci, Furmanski, Petronzio, Zijlstra, Mertig, van Neerven, Vogelsang,… Coefficient functions C (1) k, C (2) k Bardeen, Buras, Muta, Duke, Altarelli, Kodaira, Efremov, Anselmino, Leader, Zijlstra, van Neerven,…

20. There are different fits for the initial parton densities. For example, Phenomenology of the fits for the parton densities Altarelli-Ball-Forte-Ridolfi, Blumlein-Botcher, Leader-Sidorov- Stamenov, Hirai et al.,… Altarelli-Ball- Forte-Ridolfi, The parameters should be fitted from experiment s. This combined phenomenology works well at large and small x, though strictly speaking, DGLAP is not supposed to work in the small- x region:

21. Large x 1/x 1 m2m2 Q2Q2 DGLAP –region ln(1/x) are small DGLAP accounts for logs(Q 2 ) to all orders in a s but neglects with k>2 ln(1/x) are large However, these contributions become leading at small x and should be accounted for to all orders in the QCD coupling. Small x

22. 1 Q2Q2 m2m2 Q 2 -evolution, total resummation of g 1 at small x and large Q 2 starting point x-evolution, total resummation of 1/x DGLAP yes no DGLAP cannot perform the resummation of logs of x because of the DGLAP-ordering, a keystone of DGLAP

23. 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 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 Obviously when x  0

24. singlet intercept The weakest point of this approach: the QCD coupling a s is fixed at an unknown scale. On the contrary, DGLAP equations have always a running a s Bassetto-Ciafaloni-Marchesini - Veneziano, Dokshitzer-Shirkov Arguments in favor of the Q 2 - parameterization: non- singlet intercept DGLAP- parameterization Intercepts of g 1 in Double-Logarithmic Approximation:

25. K K’ K K’ K K’ Origin: in each ladder rung 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

26. 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 g 1 in the small -x region, it is necessary: 1.Total resummation of logs of x 2.New parametrization of the QCD coupling How: the formula is valid when Then it is necessary to introduce an infrared cut-off for k 2 in the transverse space: Lipatov

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

28. Essence of the method s t Introduce IR cut-off: for all virtual particle momenta, 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 come from the integration region where both in the longitudinal and in the transverse space. Typical Feynman graph DL and SL contributions of softest particles can be factorized g g qq

29. Case A: the softest particle is a non-ladder gluon. It can be factorized: = + + + symmetric contributions k k k is the lowest limit of integration over Factorized softest gluons of the softest gluon only. It does not involve other momenta

30. = + + + symmetric contributions k k k New IR cut-offs for integrations over transverse momenta of other virtual particles Is replaced by In the blobs with factorized gluons

31. Case B. The softest particle is a ladder quark or gluon. It can also be factorized: = + k k DL contributions come from the region When This case does not contribute when s ~ -t i.e. in the case of hard kinematics such contributions disappear after applying Combining case A and case B and adding Born terms leads to the IREE quark pair gluon pair

32. 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 Inverse transform : and difference with the standard Mellin transform

33. For singlet g 1 Structure function Forward Compton amplitude with negative signature Compton off quark Compton off gluon System of IREE for the Compton amplitudes : Anomalous dimension matrix. Sums up DL and part of SL contributions and where

34. IREE for the non-singlet g 1 is simpler: Single Logarithms Double Logarithms New anomalous dimension H NS (w) accounts for the total resummation of DL and a part of SL of x new non-singlet anomalous dimension, sums up DLs and SLs

35. Expression for the non-singlet g 1 at large Q 2 : Q 2 >> 1 GeV 2 Coefficient function Anomalous dimension Initial quark density

36. Expression for the singlet g 1 at large Q 2 : here Large Q 2 means

37. Small –x asymptotics of g 1 : when x  0, the saddle-point method leads to intercept At large x, g 1 NS and g 1 S are positive large Q 2 small Q 2 As x 0 > x, dependence on x is weak In the whole range of x at any Q 2 COMPASS: Q 2 << 1 GeV 2

38. where intercept Asymptotics of the singlet g 1 are more involved (we assume that dq and dg are just constants)

39. Case A Case C Case B Sign of singlet g 1 : g 1 is positive at large x and goes to zero at small x g 1 is positive at large x and negative at small x g 1 is positive at large and small x Negative and large Negative but not large or positive strong correlation = fine tuning

40. Note on the singlet intercept: A.Graphs with gluons only: violates unitarity B. All graphs No violation of unitarity 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 similar to LO BFKL non-singlet intercept singlet intercept

41. Comparison of our results to DGLAP at finite x (no asymptotic formulae used) Comparison depends on the assumed shape of initial parton densities. The simplest option: use the bare quark input in x- space in Mellin 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 < 0.05. However, it does not. Why?

42. Altarelli-Ball-Forte- Ridolfi singular factor normalization 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. regular factors The parameters are fixed from fitting experimental data at large x

43. In the Mellin space this fit is Leading pole a =0.58 >0 Non-leading poles -k +a <0 The small- x DGLAP asymptotics of g 1 is (inessential factors dropped): Comparison with our asymptotics shows that the singular factor in the DGLAP fit mimics the total resummation of ln(1/x). However, the value a = 0.58 differs from our non-singlet intercept =0.4 phenomenology calculations

44. Although our and DGLAP asymptotics lead to an x- behavior of Regge type, they predict different intercepts for the x- dependence and different Q 2 - dependence: whereas DGLAP predicts the steeper x-behavior and a flatter Q 2 -behavior: x -asymptotics was checked by extrapolating the available exp. data to x  0. It agrees with our values of D Contradicts DGLAP our and DGLAP Q 2 –asymptotics have not been checked yet. our calculations 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.

45. Numerical comparison of DGLAP with our approach at small but finite x, using the same DGLAP fit for initial quark density. R = g 1 our / g 1 DGLAP Regular term in g 1 our vs regular + singular in g 1 DGLAP Only regular factors in g 1 our and g 1 DGLAP x

46. R = g 1 our / g 1 DGLAP as function of Q 2 at different x X = 10 -4 X = 10 -3 X = 10 -2 Q 2 - dependence of R is flatter than the x -dependence Q2Q2 Q2Q2 R = g 1 our / g 1 DGLAP

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

48. Common opinion : in DGLAP analyses the fits to the initial parton densities are related to the structure 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 and a linear term in x

49. Comparison between DGLAP and our approach at small x DGLAPour approach Coeff. functions and anom. dimensions are calculated with two-loop accuracy Coeff. functions and anom. dimensions sum up DL and SL terms to all orders To ensure the Regge behaviour, singular terms in x are used in initial partonic densities. Regge behavior is achieved automatically, even when the initial densities are regular in x This could be equivalent phenomenologically, but could be Asymptotic formulae of g 1 are never used in expressions for g 1 at finite x unreliable for g 1 at very small x.

50. SUGGESTION: merging of our approach and DGLAP 1.Expand our formulae for coefficient functions and anomalous dimensions into series in the QCD coupling 2.Replace the first- and second- loop terms of the expansion by corresponding DGLAP –expressions DGLAP Good at large x because includes exact two-loop calculations but bad at small x as lacks the total resummaion of ln(x) our approach Good at small x, includes the total resummaion of ln(x) but bad at large x because neglects some contributions essential in this region Comparison between DGLAP and our approach at any x

51. New “synthetic” formulae include all advantages of both approaches and should be equally good at large and small x. New fits for the part. densities should not involve singular factors. Our expressions anomalous dimension coefficient function First tems of the perturbative expansions in series New “synthetic” formulae:

52. 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. CERN 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

53. 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

54. COMPASS runs at small Q 2 and small x In order to generalize our results to the region of small Q 2, one should recall that is the result of the integration However this is valid for large Q 2 only. For arbitrary Q 2 : Introduce a “mass” of virtual quarks and gluons to regulate infrared singularities This is suggested by the relevant Feynman diagrams involved.

55. Coefficient function Anomalous dimension Initial quark density weak x -dependence weak Q 2 -dependence It leads to new expressions: Small Q 2 non-singlet g 1

56. again weak Q 2 - dependence Small Q 2 Singlet g 1 At Q 2 < 1 GeV 2, the x- dependence is almost flat even at x<<1 PREDICTION:

57. C O M P A S S (Phys.Lett.B647, 330, 2007)

58. Power Q 2 -corrections In practice, the power corrections are obtained phenomenologically through a discrepancy between DGLAP –predictions and experimental data. However, There exist contributions ~1/(Q 2 ) k, with k = 1,2,.. They can be of perturbative (renormalons) and non-perturbative (higher twists) origin. when and when New power terms. Taken Into account, they can change impact of higher twists

59. Conclusions Standard Approach is based on the DGLAP evolution equations and phenomenological fits for the initial parton densities. DGLAP was originally developed, and is very successful, in the region where both x and Q 2 are large, but neglects logs of (1/x), so it cannot applied safely at very small x. In order to extend SA to the region of small x, phenomenological singular factors x -a have to be incorporated into the parton densities. These factors mimic the total resummation of leading logs of x that we perform in our approach, which leads naturally to the Regge behaviour of g 1 at small x with predicted intercepts, in agreement with exps. SUGGESTION: merging of our approach and DGLAP The region of small Q 2 is also beyond the reach of SA. In our model of extension at small Q 2 we predict that g 1 is almost independent of x, even at x<< 1. Instead, it depends on 2pq only. In particular g1 can be pretty close to zero in the range of 2pq investigated experimentally by COMPASS. This agrees with the data.