A dispersive approach to Sudakov resummation

TL;DR

This paper develops a comprehensive dispersive framework for Sudakov resummation in QCD, expressing the Sudakov exponent as a dispersion integral over spectral densities tied to process-dependent Sudakov effective charges. It unifies the non-Abelian dynamics with all-order running-coupling effects and establishes precise links to the scheme-invariant Borel formalism, including a universal infrared limit governed by the cusp anomalous dimension. The authors derive general representations for the jet and soft contributions, provide explicit characteristic functions and their moment-space expansions across multiple processes, and analyze power corrections and renormalons within both dispersive and Borel pictures. The approach supports Dressed Gluon Exponentiation and offers a practical pathway to incorporate infrared finite couplings and potential perturbative infrared fixed points, with broad applications to DIS, Drell–Yan, Higgs production, and heavy-quark decays.

Abstract

We present a general all-order formulation of Sudakov resummation in QCD in terms of dispersion integrals. We show that the Sudakov exponent can be written as a dispersion integral over spectral density functions, weighted by characteristic functions that encode information on power corrections. The characteristic functions are defined and computed analytically in the large-beta_0 limit. The spectral density functions encapsulate the non-Abelian nature of the interaction. They are defined by the time-like discontinuity of specific effective charges (couplings) that are directly related to the familiar Sudakov anomalous dimensions and can be computed order-by-order in perturbation theory. The dispersive approach provides a realization of Dressed Gluon Exponentiation, where Sudakov resummation is enhanced by an internal resummation of running-coupling corrections. We establish all-order relations between the scheme-invariant Borel formulation and the dispersive one, and address the difference in the treatment of power corrections. We find that in the context of Sudakov resummation the infrared-finite-coupling hypothesis is of special interest because the relevant coupling can be uniquely identified to any order, and may have an infrared fixed point already at the perturbative level. We prove that this infrared limit is universal: it is determined by the cusp anomalous dimension. To illustrate the formalism we discuss a few examples including B-meson decay spectra, deep inelastic structure functions and Drell-Yan or Higgs production.