Sudakov Factorization and Resummation

TL;DR

This paper presents a unified, factorization-based framework for Sudakov resummation, decomposing cross sections into hard, jet, and soft components that capture the elastic-limit logarithmic enhancements. By enforcing renormalization-group consistency and introducing a momentum-transfer evolution, it derives exponentiated Sudakov structures that apply to electroweak-induced processes and extend to QCD hard scattering with color coherence. The formalism yields explicit resummed expressions for DIS and Drell–Yan in both DIS and MS-bar schemes, including two-loop coefficients, and demonstrates agreement with known results while outlining pathways to EFT treatments and broader QCD applications. The approach clarifies how large logarithms organize into universal functions and their evolution, providing practical tools for precise predictions near phase-space edges.

Abstract

We present a unified derivation of the resummation of Sudakov logarithms, directly from the factorization properties of cross sections in which they occur. We rederive in this manner the well-known exponentiation of leading and nonleading logarithmic enhancements near the edge of phase space for cross sections such as deeply inelastic scattering, which are induced by an electroweak hard scattering. For QCD hard-scattering processes, such as heavy-quark production, we show that the resummation of nonleading logarithms requires in general mixing in the space of the color tensors of the hard scattering. The exponentiation of Sudakov logarithms implies that many weighted cross sections obey particular evolution equations in momentum transfer, which streamline the computation of their Sudakov exponents. We illustrate this method with the resummation of soft-gluon enhancements of the inclusive Drell-Yan cross section, in both DIS and $\overline{\rm MS}$ factorization schemes.