Resummation for QCD Hard Scattering

TL;DR

The paper develops a general method to resumm threshold logarithms in QCD hard-scattering cross sections by working in moment space and treating color structure with a matrix in color space. It builds a factorization framework into hard, jet, and soft components and derives a renormalization-group equation for the soft-function matrix, enabling NLL resummation through diagonalization and requiring ordered exponentials beyond that. Explicit one-loop soft anomalous-dimension matrices are calculated for heavy-quark production in both $q\bar{q}$ and $gg$ channels, with threshold deltas and mass effects analyzed and consistency checks performed in various limits. The approach promises improved perturbative predictions for heavy-quark production and related high-energy processes, with clear paths to extensions to dijet and transverse-momentum observables.

Abstract

We resum distributions that are singular at partonic threshold (the elastic limit) in heavy quark production, in terms of logarithmic behavior in moment space. The method may be applied to a variety of cross sections sensitive to the edge of phase space, including transverse momentum distributions. Beyond leading logarithm, dependence on the moment variable is controlled by a matrix renormalization group equation, reflecting the evolution of composite operators that represent the color structure of the underlying hard scattering. At next-to-leading logarithmic accuracy, these evolution equations may be diagonalized, and moment dependence in the cross section is a sum of exponentials. Beyond next-to-leading logarithm, resummation involves matrix-ordering. We give a detailed analysis for the case of heavy quark production by light quark annihilation and gluon fusion.