The next-to-leading order (NLO) corrections to the BFKL equation in the BLM
optimal scale setting are briefly discussed. A striking feature of the BLM
approach is rather weak Q^2-dependence of the Pomeron intercept, which might
indicate an approximate conformal symmetry of the equation. An application of
the NLO BFKL resummation for the virtual gamma-gamma total cross section shows
a good agreement with recent L3 data at CERN LEP2 energies.Comment: 7 pages, 2 eps-figures, presented at International Symposium on
  Multiparticle Dynamics (ISMD99), August 9-13, 1999, Brown University,
  Providence, Rhode Islan

Kim, Victor T.

Lipatov, Lev N.

Pivovarov, Grigorii B.

English

arXiv

The next-to-leading order (NLO) corrections to the BFKL equation in the BLM optimal scale setting are briefly discussed. A striking feature of the BLM approach is rather weak Q^2-dependence of the Pomeron intercept, which might indicate an approximate conformal symmetry of the equation. An application of the NLO BFKL resummation for the virtual gamma-gamma total cross section shows a good agreement with recent L3 data at CERN LEP2 energies

Kim, V T

Lipatov, L N

Pivovarov, G B

CERN Document Server

IITAP-99-014hep-ph/9911242The Next-to-Leading Dynamics ofthe BFKL Pomeron aVictor T. Kimz&, Lev N. LipatovzandGrigorii B. Pivovarovx&z : St.Petersburg Nuclear Physics Institute, 188350 Gatchina, Russiax : Institute for Nuclear Research, 117312 Moscow, Russia& : International Institute of Theoretical and Applied Physics,Iowa State University, Ames, IA 50011, USAThe next-to-leading order (NLO) corrections to the BFKL equation in the BLMoptimal scale setting are briefly discussed. A striking feature of the BLM approachis rather weak Q2-dependence of the Pomeron intercept, which might indicatean approximate conformal symmetry of the equation. An application of the NLOBFKL resummation for the virtual gamma-gamma total cross section shows a goodagreement with recent L3 data at CERN LEP2 energies.The Balitsky-Fadin-Kuraev-Lipatov (BFKL) 1,2 resummation of energylogarithms is anticipated to be an important tool for exploring the high-energylimit of QCD. Namely, the highest eigenvalue, ωmax, of the BFKL equation 1 isrelated to the intercept of the Pomeron which in turn governs the high-energyasymptotics of the cross sections: σ  sαIP−1 = sωmax . The BFKL Pomeronintercept in the leading order (LO) turns out to be rather large: αIP − 1 =ωmaxLO = 12 ln 2 (αS/pi) ’ 0.55 for αS = 0.2; hence, it is very important toknow the next-to-leading order (NLO) corrections. In addition, the LO BFKLcalculations have restricted phenomenological applications because, e.g., therunning of the QCD coupling constant αS is not included, and the kinematicrange of validity of LO BFKL is not known.Recently the NLO corrections to the BFKL resummation of energy log-arithms were calculated; see Refs. 4,5 and references therein. The NLO cor-rections 4,5 to the highest eigenvalue of the BFKL equation turn out to beapresented by V. T. K. at the International Symposium on Multiparticle Dynamics (ISMD’99), August 9 - 13,1999, Brown University, Providence, Rhode Island, USA, to appear in the Proceedings1negative and even larger than the LO contribution for αS > 0.16.Eective eld theory approach for high-energy limit of QCD started in 6is an important issue since known LO 1 and NLO BFKL 4,5 contributions arelarge. It was observed that at large color number approximation the eectivehigh-energy QCD becomes an integrable 2D-model 7 and possesses interestingconformal and duality properties 8.We briefly mention here main available approaches: The gauge-invariantaction 9,3 is build, in particularly, to reproduce known LO and NLO BFKLcalculations. An approach 10 with eective right- and left- moving gluon eldswith emphasizing the renormalization group leads to an independent derivationof the Reggeization of gluon 1, a basic property for high-energy limit of QCD.An elegant nonlocal formulation 11 is based on evolution of Wilson lines. Largedensity of color charge at small-x allows to use quasiclassical methods 12,13 fortaking into account the non-perturbative contributions at the high-energy limitof QCD.While the above eective eld approaches are yet not accomplished, onecan get a closer look at the NLO BFKL calculations 4,5. It should be stressedthat the NLO calculations, as any nite-order perturbative results, containboth renormalization scheme and renormalization scale ambiguities. The NLOBFKL calculations 4,5 were performed by employing the modied minimalsubtraction scheme (MS) to regulate the ultraviolet divergences with arbitraryscale setting.In the recent work 14 it was found that the renormalization scheme depen-dence of the NLO BFKL resummation of energy logarithms 4,5 is not strong,i.e., value of the NLO BFKL term is practically the same in the known renor-malization schemes. To resolve the renormalization scale ambiguity due tothe large NLO BFKL term 4,5 the Brodsky-Lepage-Mackenzie (BLM) optimalscale setting 15 has been utilized. The BLM optimal scale setting eectivelyresums the conformal-violating β0-terms into the running coupling in all ordersof the perturbation theory.It was shown 14 that the reliability of QCD predictions for the eectiveintercept of the BFKL Pomeron at NLO evaluated at the BLM scale set-ting within the non-Abelian physical schemes, such as the momentum spacesubtraction (MOM) scheme or the -scheme based on  ! ggg decay, issignicantly improved compared to the MS-scheme result 17,18.One of the striking features of the analysis 14 is that the NLO value for theintercept of the BFKL Pomeron, improved by the BLM procedure, has a veryweak dependence on the gluon virtuality Q2: αIP − 1 = ωmaxNLO =’ 0.13− 0.18at Q2 = 1−100 GeV2. It arises as a result of ne-tuned compensation betweenthe LO and NLO contributions. The minor Q2-dependence obtained leads to2Figure 1: Virtual gamma-gamma total cross section by the NLO BFKL Pomeron withinBLM approach vs L3 Collaboration data at energy 183 GeV of e+e− collisions. Solid curvescorrespond to NLO BFKL in BLM; dashed: LO BFKL and dotted: LO contribution. Twodierent curves are for two dierent choices of the Regge scale: s0 = Q2/2, s0 = 2Q2approximate conformal invariance.As a phenomenological application of the NLO BFKL improved by BLMprocedure one can consider the gamma-gamma scattering 19. This processis attractive because it is theoretically more under control than the hadron-hadron and lepton-hadron collisions, where non-perturbative hadronic struc-ture functions are involved. In addition, in the gamma-gamma scattering theunitarization (screening) corrections due to multiple Pomeron exchange wouldbe less important than in hadron collisions.The gamma-gamma cross sections with the BFKL resummation in theLO was considered in 2,20,21. In the NLO BFKL case one should obtain aformula analogous to LO BFKL 19. While exact NLO impact factor of gammais not known yet, one can use the LO impact factor of 2,20 assuming that themain energy-dependent NLO corrections come from the NLO BFKL subprocessrather than from the impact factors 19.3Figure 2: The same as Fig. 1 but for energy 189 GeV of e+e− collisions.In Figs. 1, 2 the comparison of BFKL predictions for LO and NLO BFKLimproved by the BLM procedure with L3 data 22,23 from CERN LEP is shown.The dierent curves reflect uncertainty with the choice of the Regge scaleparameter which denes the beginning of the asymptotic regime. For presentcalculations two variants have been choosen s0 = Q2/2 and s0 = 2Q2, whereQ2 is virtuality of photons. One can see from Figs. 1-2 that the agreement ofthe NLO BFKL improved by the BLM procedure is reasonably good at energiesof LEP2pse+e− = 183 - 189 GeV. One can notice also that sensitivity of theNLO BFKL results to the Regge parameter s0 is much smaller than in the caseof the LO BFKL.It was shown in Refs. 24,25 that the unitarization corrections in hadroncollisions can lead to higher value of the (bare) Pomeron intercept than theeective intercept value. Since the hadronic data t yields about 1.1 for theeective intercept value 25, then the bare Pomeron intercept value should beabove this value. Therefore, assuming small unitarization corrections in thegamma-gamma scattering at large Q2 one can accomodate the NLO BFKL4Pomeron intercept value 1.13 - 1.18 14 in the BLM optimal scale setting alongwith larger unitarization corrections in hadronic scattering 24, where they canlead to a smaller eective Pomeron intercept value about 1.1 for hadroniccollisions.Another possible application of the BFKL approach can be the collisionenergy dependence of the inclusive single jet production 26.There have been a number of recent papers which analyze the NLO BFKLpredictions in terms of rapidity correlations 27, angle-ordering 28, double trans-verse momentum logarithms 29,30,31, an additional log(1/x) enhancement 32and BLM scale setting for deep inelastic structure functions 33. Obviously, alot of work should be done to clarify the proper expansion parameter for BFKLregime and, also the relation between those papers and the result of the presentBLM approach. To conrm the result of 4,5 the independent NLO calculations(see 34 and references therein) for BFKL resummation are desirable.To conclude, we have shown that the NLO corrections to the BFKL equa-tion for the QCD Pomeron become controllable and meaningful provided oneuses physical renormalization scales and schemes relevant to non-Abelian gaugetheory. BLM optimal scale setting automatically sets the appropriate physi-cal renormalization scale by absorbing the non-conformal β-dependent coe-cients. The strong renormalization scheme and scale dependence of the NLOcorrections to BFKL resummation then largely disappears. A striking fea-ture of the NLO BFKL Pomeron intercept in the BLM approach is its veryweak Q2-dependence, which provides approximate conformal invariance. TheNLO BFKL application to the total gamma-gamma cross section shows a goodagreement with the L3 Collaboration data at CERN LEP2 energies. 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The Next-to-Leading Dynamics of the BFKL Pomeron

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