The Infrared Behavior of QCD Cross Sections at Next-to-Next-to-Leading Order

In this talk we examine how one-loop soft and collinear splitting functions occur in the calculation of next-to-next-to-leading order (NNLO) corrections to production rates, and we present the one-loop gluon soft and splitting functions, computed to all orders in the dimensional regularization parameter ǫ. We apply the one-loop gluon soft function to the calculation of the next-to-leading logarithmic corrections to the Lipatov vertex to all orders in ǫ. On leave of absence from I.N.F.N., Sezione di Torino, Italy. Rapporteur at the Corfu Summer Institute on Elementary Particle Physics, 1998. The single most important parameter of perturbative QCD is the strong coupling constant, αs, which has been determined in several ways [1]. Some of the most promising ones are due to hadron production in e+e− collisions; e.g., the hadronic branching ratio of the Z0 or global event shape variables in e+e− → 3 jets. The hadronic branching ratio RZ is known in perturbative QCD to three loops; however, the usefulness of this observable in the determination of αs is limited by the sensitivity of RZ to other Standard Model parameters [2] (for an overview, see ref. [3]). On the contrary, e+e− → 3 jets, which is known only to next-to-leading order (NLO) [4, 5], does not suffer from the above limitations. Thus a next-to-next-to-leading order (NNLO) calculation of this process could yield a significant reduction of the theoretical uncertainty in the determination of αs. In order to understand the general features of a calculation at NNLO, we begin by outlining how a higher-order calculation of a scattering process is performed. At leading order (LO) in αs the cross section is obtained by squaring the tree amplitudes. If n particles are produced in the scattering, each of them will be resolved in the final state. Thus no singularities appear in the LO cross section. At LO the coupling αs is evaluated with one-loop running, so that there is an implicit dependence on an arbitrary renormalization scale μR. In addition, if one or both of the scattering particles are strongly interacting, the cross section will factorize into the convolution of parton density functions (to be determined experimentally) and a hard partonic cross section, which is computed as an expansion in αs. This procedure introduces into both the parton densities and the partonic cross section a dependence on a second arbitrary parameter, the factorization scale μF [6]. Typically, the dependence on μR and μF is maximal at LO. The calculation of the cross section at next-to-leading order (NLO) in αs is less straightforward. Two series of amplitudes are required in the squared matrix elements: a) tree and one-loop amplitudes for the production of n particles; b) tree amplitudes for the production of n + 1 particles. The one-loop amplitudes typically have virtual ultraviolet and infrared singularities, which may be regularized using dimensional regularization. This involves analytically continuing the loop momenta into D = 4−2ǫ dimensions, so that the one-loop amplitude is now a function of ǫ. If this is expanded in ǫ, the ultraviolet singularities appear as single poles in ǫ, which can be removed by renormalizing the amplitude. This introduces an explicit dependence on the renormalization scale μR. At NLO the structure of the infrared singularities has been extensively studied. Virtual infrared singularities appear as double poles in ǫ, when they are both soft and collinear, and single poles in ǫ, when they are either soft or collinear. Real infrared singularities occur in the phase-space integral over the n + 1 final-state particles of the squared tree amplitudes, either when any gluon becomes soft or when any two massless particles become collinear, thus yielding single poles in ǫ. If one of the two collinear particles is soft, a double pole in ǫ arises. The singularities occur in a universal way, i.e. independent of the particular amplitude considered. Accordingly, soft singularities can be accounted for by universal tree soft functions [7, 8], and collinear singularities by universal tree splitting functions [9]. These have also been combined into a single function [10]. A detailed discussion of the infrared singularities at NLO for e+e− → jets may be found, for example, in ref. [11]. For processes with no strongly interacting scattering particles, all infrared divergences cancel