No abstract

We extend the resummation of dimensionally-regulated amplitudes to next-to-next-to-leading poles.This requires the calculation of two-loop anomalous dimension matrices for color mixing through soft gluon exchange. Remarkably, we find that they are proportional to the corresponding one-loop matrices.Using the color generator notation, we reproduce the two-loop single-pole quantities H (2) introduced by Catani for quark and gluon elastic scattering. Our results also make possible threshold and a variety of other resummations at next-to-next-to leading logarithm. All of these considerations apply to 2 → n processes with massless external lines.

The resummation of soft gluon exchange for QCD hard scattering requires a matrix of anomalous dimensions. We compute this matrix directly for arbitrary 2 → n massless processes for the first time at two loops. Using color generator notation, we show that it is proportional to the oneloop matrix. This result reproduces all pole terms in dimensional regularization of the explicit calculations of massless 2 → 2 amplitudes in the literature, and it predicts all poles at next-tonext-to-leading order in any 2 → n process that has been computed at next-to-leading order. The proportionality of the one-and two-loop matrices makes possible the resummation in closed form of the next-to-next-to-leading logarithms and poles in dimensional regularization for the 2 → n processes.The calculation of high-energy cross sections in perturbative quantum chromodynamics (QCD) for hadronic collisions involves the factorization of long-and shortdistance effects. Sensitivity to long-distance dynamics is enhanced by powers of logarithms whenever there is an incomplete cancellation of parton emission and virtual corrections. In such situations, it is useful to organize, or resum, these corrections to all orders in perturbation theory. Correspondingly, in partonic scattering or production amplitudes, it is necessary to organize poles in ε that arise in dimensional regularization (with D = 4 − 2ε). The resummation of these poles and related logarithmic enhancements is well-understood for inclusive reactions mediated by electroweak interactions, such as the Sudakov form factor [1, 2] and in Drell-Yan processes [3]. With recent advances in the computation of splitting functions [4], many such corrections can be resummed explicitly to next-to-next-to-leading level. Their structure at arbitrary level is known to be determined by a handful of anomalous dimensions.The situation for QCD hard scattering processes containing four or more partons -critical to understanding many types of backgrounds to new physics at hadron colliders [5] -is more complex. Resummation beyond leading logarithms or poles requires a matrix of additional anomalous dimensions [6,7,8,9]. These matrices are found in turn from the renormalization of the vacuum matrix elements of products of Wilson lines, one for each external parton in the underlying process [7]. In this paper, we investigate the structure of the two-loop anomalous dimension matrix. We will find that, remarkably, for every hard-scattering process involving only massless partons, this matrix is proportional to the one-loop matrix. We will concentrate below on the role that the matrix plays in partonic amplitudes. The full calculation of the two-loop matrix will be given elsewhere [10]. In this paper, we provide the simple calculation that is at the heart of the main result. We will show that certain color correlations due to two-loop diagrams that couple three Wilson lines vanish identically. We will also provide an explicit expression in terms of color generators [11,12] for all single-pole terms in massless 2 ...

We extend the Collins-Soper-Sterman (CSS) formalism to apply it to the spin-dependence governed by the Sivers function. We use it to give a correct numerical QCD evolution of existing fixedscale fits of the Sivers function. With the aid of approximations useful for the non-perturbative region, we present the results as parametrizations of a Gaussian form in transverse momentum space, rather than in the Fourier conjugate transverse coordinate space normally used in the CSS formalism. They are specifically valid at small transverse momentum. Since evolution has been applied, our results can be used to make predictions for Drell-Yan and semi-inclusive deep inelastic scattering at energies different from those where the original fits were made. Our evolved functions are of a form that they can be used in the same parton model factorization formulas as used in the original fits, but now with a predicted scale dependence in the fit parameters. We also present a method by which our evolved functions can be corrected to allow for twist-3 contributions at large parton transverse momentum.