Breakdown of Dimensional Regularization in the Sudakov Problem

TL;DR

The paper demonstrates that dimensional regularization can fail to regulate infrared singularities that appear in Minkowski-space Sudakov expansions. Using a concrete one-loop triangle graph, it applies the As-operation to construct a distributional expansion in the small parameter and identifies necessary counterterms localized on pinch manifolds, including delta-function terms with non-analytic, mass-dependent coefficients. It shows that a non-Euclidean singularity at k = 0 cannot be tamed by dimensional regularization, signaling a fundamental limitation in this regime. The work highlights general features of non-Euclidean asymptotics, with implications for factorization at higher twists and for the development of robust asymptotic methods beyond Euclidean settings.

Abstract

An explicit example is presented (a one-loop triangle graph) where dimensional regularization fails to regulate the infra-red singularities that emerge at intermediate steps of studying large-$Q^2$ Sudakov factorization. The mathematical nature of the phenomenon is explained within the framework of the theory of the $As$-operation.