This talk summarises the current status of the NLL corrections to BFKL
physics and discusses the question of small-x factorisation.Comment: 4 pages, 2 figures. Uses moriond.sty (included). Talk presented at
  the XXXVth Rencontres de Moriond, QCD and High Energy Hadronic Interactions,
  Les Arcs, France, March 18-25, 200

Salam, G. P.

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This talk summarises the current status of the NLL corrections to BFKL physics and discusses the question of small-x factorisation

Salam, G.P.

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CERN–TH/2000–150hep-ph/0005304May 2000The status of NLL BFKL aG.P. SalamTH Division, CERN, CH-1211 Gene`ve 23.This talk summarises the current status of the NLL corrections to BFKL physics and discussesthe question of small-x factorisation.1 Introduction and total cross sectionsUnderstanding perturbative high-energy hadronic scattering is one of the fundamental problemsof QCD, and one which has seen considerable progress in the past couple of years.The reaction under consideration is that of the scattering of objects with some transversescale Q2 at a centre-of-mass energyps, in the limit s  Q2  Λ2QCD. This was first examinedin the mid 1970’s with the resummation of the leading logarithmic (LL) terms (s ln s)n to givethe result that the cross section   s4 ln 2αsNC/pi. This is known as the BFKL pomeron.1 Thisresult however seems to be in contradiction with numerous experimental results, which invariablyindicate that while there is a rise of the cross section at large s, the power is somewhat lower,of the order of 0:3 as opposed to the 0:5 expected from the LL calculation (for s ’ 0:2).It was expected that this discrepancy would be resolved by the inclusion of the next-to-leading (NLL) corrections s(s ln s)n, but it turns out that they modify the power ! as follows:2! = 2:65s − 16:32s ’ −0:12 (for s = 0:2) ; (1)which is just as incompatible with the data as the LL result. Closer inspection reveals thatthe NLL corrections, taken literally, also imply negative cross sections for the scattering oftwo objects of different transverse sizes3 (essentially, in such a collinear limit the convergencebecomes even worse).There have been several significantly different attempts to explain the large size of thecorrections and to thus estimate yet higher-order corrections with the aim of obtaining stableaTalk presented at the XXXVth Rencontres de Moriond, QCD and High Energy Hadronic Interactions, LesArcs, France, March 18-25, 2000.00.050.10.150.20.250.30.350 0.05 0.1 0.15 0.2 0.25 0.3503 100 40 10.1 4.3ωαs (MSbar)Q2 [GeV2]LLωsωcNLLFigure 1: The prediction for the BFKL power as a function of αs at LL, NLL and after the all-order inclusionof collinearly enhanced terms (ωs). The uncertainty on ωs is of the order of 10%. The curve labelled ωc is theBFKL power for splitting functions. The results, shown for nf = 0, depend only weakly on nf .predictions. One involves the idea of a minimum rapidity gap between emissions,4 while anotherargues that a meaningful answer can be obtained using a more ‘natural’ renormalisation schemecoupled with BLM resummation.5 Both suffer from instabilities (with respect to the choice ofthe rapidity gap and the renormalisation scheme, respectively) and have difficulty solving theproblem of the negative cross sections, essentially because they do not take into account theworsening of the convergence in the collinear limit.An alternative approach indeed starts from a consideration of the collinear limit, and observesthat a large part of the NLL corrections (even for non-collinear scattering) come precisely fromterms that are enhanced in the collinear limit. These can be resummed by requiring that thecross section satisfy the renormalisation group properties in both collinear limits (there are twocollinear limits corresponding to which of the scattering objects is the smaller).6 One finds thatthe problem of negative cross sections goes away completely and that the power of the crosssection growth is relatively stable with respect changes of scheme. The result for the power asa function of s is shown in fig. 1.The cleanest (most inclusive) measurement of the BFKL power comes from γ∗γ∗ scattering,by the L3 collaboration7 at scales Q2 = 3:5 and 14:5 GeV2. They quote a value for the power of! = 0:37  0:04. The quoted error is probably an underestimate: in the fit procedure the nor-malisation is fixed to be the LL value, whereas one should fit also for the normalisation becauseit too is liable to be subject to significant (but as yet uncalculated) higher-order corrections. Amore realistic error is about three times larger. There are also systematic errors associated withthe functional form used to fit for the power. So the measurement is roughly in accord with thetheoretical expectation which goes from 0:29 to 0:33 for this range of scales (for comparison theLL power is in the range 0:6 to 0:8), but a satisfactory comparison requires, on the experimentalfront, higher precision data and larger energies (e.g. at the NLC), and on the theoretical fronta calculation of the expected normalisation (i.e. the impact factors to NLL order).2 Scaling violationsSo far we have discussed the total cross section in a context where both scattering objects areperturbative. An important situation is that in which only one of the objects is perturbative,0.010.11101001000100001000001e-30 1e-25 1e-20 1e-15 1e-10 1e-05 1x|P(x)|xabc0.110.120.130.140.151e-06 0.001 1Figure 2: The splitting function in LL BFKL with running coupling for three different sets (a,b,c) of infra-redregularisation and initial transverse scale. (Shown for αs = 0.126.) The inset shows the same curves in a morelimited range of x values.namely deep inelastic scattering. Since the other object is non-perturbative the calculation ofthe total cross section is beyond perturbation theory. When the s is of the same order as Q2 (i.e.x not too small) we know however that the cross section can be factorised into non-perturbativeparton distributions convoluted with perturbative coefficient functions, and the scaling violationsof the parton distributions can be calculated perturbatively through the DGLAP equations.8 Theintuitive justification for this picture is that emissions are ordered in transverse scale, so thatthe non-perturbative parton distribution contains emissions only below a given scale Q20, whilethe coefficient function (and parton distribution evolution) involves only subsequent emissions,which are all above Q20.However in high-energy scattering (i.e. at small x) it has long been known that emissions arenot ordered in transverse momentum. This led to the suggestion9 that the factorisation whichholds at moderate x might break down at small x. (Recently, on this basis, it has been arguedthat small-x splitting functions are beyond calculation and can at best be fitted.10)Last summer an explicit counterexample was presented,11 namely a toy model which re-tained the fundamental property of BFKL evolution, namely emissions which are not orderedin transverse scale, but in which the property of factorisation could be demonstrated for all x.Recently, numerical methods have been developed12 which allow the study of factorisation inthe full BFKL equation (for the time being at LL, with running coupling). In this approachone solves the BFKL equation to obtain a gluon distribution as a function of x and Q2. Thisdistribution depends intrinsically on the regularisation of s in the infra-red (IR) and the initialtransverse scale. One then performs a deconvolution of the scaling violations from the gluondistribution, to obtain an effective splitting function.The results of this procedure are shown in fig. 2, which shows the effective splitting functionfor three different sets of infra-red regularisations. One sees that the splitting function is inde-pendent of the IR regularisation, except at very small x, where a non-perturbative componenttakes over. More detailed investigation reveals that this second component is higher-twist, anddespite its much stronger x dependence, the scaling violations themselves can be predicted towithin a relative higher-twist term which is not x enhanced, thus confirming that factorisationholds even at small x.A rough explanation of why factorisation holds is the following: the fact that the evolutionis not ordered in transverse momentum means that the separation into a parton distributionand coefficient or splitting functions is less trivial. If one chooses to separate the two at somescale Q0, then the parton distribution now depends on emissions both below and above Q20. Butsignificantly, the splitting and coefficient functions turn out to still only depend on emissionsabove Q20.This brings us to the question of what exactly the small-x splitting functions look like. Afundamental property is that the power-growth only sets in at relatively small x. It is alsosignificantly weaker than what would be observed in total cross sections at the same scale. Thisis the reason for the presence of two powers in fig. 1: !s is that which applies to total crosssections, while !c is that which is relevant for splitting functions. These two points (which havebeen noted also by Thorne13) mean that for the x values that are relevant at HERA, the small-x resummed splitting functions are probably quite similar to the DGLAP splitting functions,explaining the unexpected success of the latter in reproducing the observed scaling violations.AcknowledgementsI wish to thank Marcello Ciafaloni and Dimitri Colferai (in collaboration with whom many ofthe results presented here have been obtained) for discussions. This work was supported in partby the E.U. QCDNET contract FMRX-CT98-0194.References1. L.N. Lipatov, Sov. J. Nucl. Phys. 23 (1976) 338; E.A. Kuraev, L.N. Lipatov and V.S.Fadin, Sov. Phys. JETP 44 (1976) 443; E.A. Kuraev, L.N. Lipatov and V.S. Fadin, Sov.Phys. JETP 45 (1977) 199; Ya. Balitskii and L.N. Lipatov, Sov. J. Nucl. Phys. 28(1978) 822.2. V.S. Fadin and L.N. Lipatov, Phys. Lett. B 429 (1998) 127 [hep-ph/9802290]; M. Ciafaloniand G. Camici, Phys. Lett. B 430 (1998) 349 [hep-ph/9803389]; references therein.3. D.A. Ross, Phys. Lett. B 431 (1998) 161 [hep-ph/9804332].4. C. Schmidt, Phys. Rev. D 60 (1999) 074003 [hep-ph/9901397].5. S.J. Brodsky, V.S. Fadin, V.T. Kim, L.N. Lipatov and G.B. Pivovarov, JETP Lett. 70(1999) 155 [hep-ph/9901229].6. G.P. Salam, JHEP 07 (1998) 19 [hep-ph/9806482]; M. Ciafaloni and D. Colferai, Phys.Lett. B 452 (1999) 372 [hep-ph/9812366]; M. Ciafaloni, D. Colferai and G.P. Salam, Phys.Rev. D 60 (1999) 114036 [hep-ph/9905566].7. L3 Collaboration (M. Acciarri et al.), Phys. Lett. B 453 (1999) 333; C.H. LIN, theseproceedings.8. V.N. Gribov and L.N. Lipatov, Sov. J. Nucl. Phys. 15 (1972) 438;G. Altarelli and G. Parisi, Nucl. Phys. B 126 (1977) 298;Yu.L. Dokshitzer, Sov. Phys. JETP 46 (1977) 641.9. A.H. Mueller, Phys. Lett. B 396 (1997) 251 [hep-ph/9612251].10. G. Altarelli, R.D. Ball and S. Forte, hep-ph/9911273.11. M. Ciafaloni, D. Colferai and G.P. Salam, JHEP 10 (1999) 017 [hep-ph/9907409].12. M. Ciafaloni, D. Colferai, G.P. Salam and M. Taiuti, in preparation.13. R.S. Thorne, Phys. Lett. B 474 (2000) 372 [hep-ph/9912284].

The status of NLL BFKL

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The status of NLL BFKL

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