We discuss power corrections associated with the infrared behavior of the
perturbative running coupling in the eikonal approximation to Drell-Yan and
other annihilation cross sections in hadron-hadron scattering. General
properties of the eikonal approximation imply that only even powers of the
energy scale are necessary.Comment: 4 pages, LaTeX. Contributed to 30th International Conference on
  High-Energy Physics (ICHEP 2000), Osaka, Japan, 27 Jul - 2 Aug 200

Laenen, Eric

Sterman, George

Vogelsang, Werner

English

arXiv

We discuss power corrections associated with the infrared behavior of the perturbative running coupling in the eikonal approximation to Drell-Yan and other annihilation cross sections in hadron-hadron scattering. General properties of the eikonal approximation imply that only even powers of the energy scale are necessary

Utrecht University Repository

arXiv:hep-ph/0010183v1  17 Oct 2000NIKHEF/00-31YITP-SB-00-60BNL-HET-00/38RBREC 144POWER CORRECTIONS IN EIKONAL CROSS SECTIONSERIC LAENENNIKHEF Theory Group, Kruislaan 4091098 SJ Amsterdam, The NetherlandsGEORGE STERMANC.N. Yang Institute for Theoretical Physics, SUNY Stony BrookStony Brook, New York 11794 – 3840, U.S.A.Physics Department, Brookhaven National Laboratory,Upton, NY 11973, USAWERNER VOGELSANGRIKEN-BNL Research Center, Brookhaven National Laboratory,Upton, NY 11973, U.S.A.We discuss power corrections associated with the infrared behavior of the perturbative running couplingin the eikonal approximation to Drell-Yan and other annihilation cross sections in hadron-hadronscattering. General properties of the eikonal approximation imply that only even powers of the energyscale are necessary.1 IntroductionPower corrections 1,2 are phenomenologicallysignificant in many QCD hard-scatteringcross sections for which the operator productexpansion is not directly available. Examplesthat have received considerable attention in-clude event shapes in electron-positron anni-hilation and transverse momentum distribu-tions in Drell-Yan cross sections. In each ofthese cases, a perturbative description of thecross section leads to integrals of the formIp ≡ Q−p∫ Q0 dµµp−1 αs(µ) with Q the hardscale and p ≥ 1. In perturbation theory witha fixed coupling, Ip is just a number, butwhen the coupling runs, the integral becomesill-defined at its lower limit. This observa-tion requires us to introduce a minimal setof power corrections of the form λp/Qp, onefor each ambiguous Ip that we encounter3,4.The perturbative expression is cut off, or oth-erwise regularized to make it finite withoutchanging the set of exponents p. The valuesof the coefficients λp are then to be found bycomparison with experiment; they depend onthe nature of the perturbative regularizationthat is employed. In any case, it is only thesum of regularized perturbation theory andpower corrections that has physical meaning.The first step in this process is to showthat in some self-consistent approximationthe cross section at hand may be written interms of integrals like the Ip above. In manycases, this step involves the resummation oflogarithms associated with soft gluon emis-sion, for which the eikonal approximation is1useful. In this talka, we discuss an expressionfor the eikonal approximation in hadronic col-lisions, where the analysis of power correc-tions through the running coupling is partic-ularly transparent.2 The Eikonal Cross SectionTo be specific, we discuss the eikonal approx-imation as it appears when partons a andb combine through an electroweak current,such as the Drell-Yan annihilation of quarkwith antiquark to a lepton pair or gluon fu-sion to a Higgs boson,σ(eik)ab (q) =∫d4x eiq·x〈0|W †ab(x)Wab(−0)|0〉.(1)The operators Wab are defined byWab(0) ≡ Φ†β′ (0)Φβ(0) , (2)in terms of nonabelian phase operators fora and b, Φβ(0) = P e−ig∫∞0dλβ·A(λβ), withlightlike velocities β and β′, β · β′ = 1.The eikonal cross sections reproducethe logarithms, as singular as (Q/q0)αnsln2n−1(q0/Q) and (Q/qT )αns ln2n−1(qT /Q),that characterize the edges of partonic phasespace at which the energy of radiation, q0, orits total transverse momentum, qT , vanish.The resummation of these logarithms is mostconvenient in terms of transforms,σ̃(eik)ab (N,b) =∫d4q e−Nq0−ib·q σ(eik)ab (q) .(3)In the transformed functions we find loga-rithms at each order up to αns ln2n N andαns ln2n(bQ), which exponentiate. The ex-ponentiation of energy logarithms is knownas threshold resummation 5,6,7, of transversemomentum logarithms as kT resummation7,8.aContributed to the 30th International Conference onHigh-Energy Physics (ICHEP 2000), Osaka, Japan,27 Jul - 2 Aug 20003 ExponentiationTransforms of the eikonal cross section maybe written in exponential form on the ba-sis of algebraic considerations that have beenknown for a long time,σ̃(eik)ab (N,b) = exp[E(eik)ab (N,bQ, ǫ)], (4)where the exponent is an integral over func-tions wab, sometimes called “webs”9, whichare defined by a modified set of diagrammaticrules,E(eik)ab = 2∫ Q d4−2ǫkΩ1−2ǫ× wab(k2,k · βk · β′β · β′, µ2, αs(µ2), ǫ)×(e−N(k0/Q)+ik·b − 1). (5)The variable k in this expression may bethought of as the momentum contributed bythe web to the final state. The webs factorfrom each other under the transforms, andindeed in any symmetric integral over phasespace. 11Webs have a number of restrictive prop-erties. At fixed k, they are invariant un-der rescalings of the velocities in the eikonalphases, which corresponds to boost invari-ance under the axis defined by the two. Inaddition, at any fixed order, the web func-tion has only one overall collinear and IRdivergence, from kT → 0 and k0 → 0, re-spectively. Finally, the web functions haveno overall renormalization:µddµwab(k2,k · βk · β′β · β′, µ2, αs(µ2), ǫ)= 0 .(6)Using boost invariance in the large-Nlimit, we find that the exponent takes theformE(eik)ab = 2∫d2−2ǫkTΩ1−2ǫ(7)×∫ Q2−k2T0dk2 wab(k2, k2T + k2)2×[e−ib·kT K02N√k2T + k2Q2− ln√Q2k2T + k2]+O(e−N).This expression, which requires dimensionalregularization for its collinear divergences, iscompletely general for the eikonal cross sec-tion.4 FactorizationThe factorized hard-scattering function,σ̂(eik)ab , for the eikonal cross section may beconstructed in moment space to O(1/N) bydividing by moments of eikonal distributions,10φ̃(eik)f (N,µ, ǫ) (8)= exp[− ln (NeγE )∫ µ20dµ′2µ′2Af(αs(µ′2))],where Aa, with A(1)a = Ca, is the coefficientof lnN in the Nth moment of the a → a split-ting function. Factorization theorems ensurethe cancellation of collinear divergences inthe resulting hard-scattering functions. In-voking this requirement, we find an explicitrelation between the webs and the anomalousdimensions,∫ Q2−k2T0dk2 wab(k2, k2T + k2)(9)=Aa(αs(k2T ))+Ab(αs(k2T ))(k2T )1−2ǫ+ . . . .In this fashion, we derive a general formfor the eikonal approximation to the hard-scattering functions σ̂ab of electroweak anni-hilation,σ̂(eik)ab (N,b, Q, µ) =σ(eik)ab (N,b, Q)φ̃a(N,µ)φb(N,µ)= exp[Ê(eik)ab (N, b,Q)], (10)where the collinear-finite exponent (hereshown in a simplified form, accurate to NLL)is:Ê(eik)ab =∫ Q20dk2Tk2T∑i=a,bAi(αs(k2T ))×[J0 (bkT ) K0(2NkTQ)+ ln(NeγEkTQ)].(11)This result is the basis of the joint threshold-kT resummation12 developed in Ref. 13.As described above, the resummation oflogarithms as in (11) requires the inclusionof power corrections in both Q−1 as well asb, to compensate for the ill-defined behaviorof the strong coupling at low scales. In 3 itwas shown that for the Drell-Yan cross sec-tion only integer powers ofQ−1 are necessary;in 14 models of the running coupling were in-voked to suggest that power corrections beginat order Q−2. Eq. (7) implies that only evenpowers of Q are present in all generality forthe eikonal approximation. This is because,up to a single log, the expansion of the Besselfunction K0(z) at small z involves only evenpowers of z. This conclusion includes, andrequires, an expression which, like Eq. (7),is accurate to the level of “constant terms”,(lnN)0. Other consequences of this approachhave been discussed in Ref. 13.AcknowledgementsThe work of E.L. is part of the research pro-gram of the Foundation for Fundamental Re-search of Matter (FOM) and the NationalOrganization for Scientific Research (NWO).The work of G.S. was supported in partby the National Science Foundation, grantPHY9722101. G.S. thanks Brookhaven Na-tional Laboratory for its hospitality. 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Power corrections in eikonal cross sections

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