Two-Loop Infrared Renormalization with On-shell Methods

TL;DR

This paper presents a method to compute two-loop infrared anomalous dimensions in massless theories using on-shell techniques, simplifying calculations and verifying results with traditional methods.

Contribution

It introduces a practical on-shell formula for two-loop infrared anomalous dimensions, extending previous one-loop methods and demonstrating its effectiveness through explicit examples.

Findings

Derived a two-loop formula relating anomalous dimensions to phase space integrals.

Validated the on-shell method by matching standard diagrammatic calculations.

Analyzed the cancellation of divergences in massless scalar and fermion theories.

Abstract

Within the framework proposed by Caron-Huot and Wilhelm, we give a recipe for computing infrared anomalous dimensions purely on-shell, efficiently up to two loops in any massless theory. After introducing the general formalism and reviewing the one-loop recipe, we extract a practical formula that relates two-loop infrared anomalous dimensions to certain two- and three-particle phase space integrals with tree-level form factors of conserved operators. We then provide several examples of the use of the two-loop formula and comment on some of its formal aspects, especially the cancellation of 'one-loop squared' spurious terms. The present version of the paper is augmented with a detailed treatment of the structure of infrared divergences in massless theories of scalars and fermions up to two loops. In the calculation we encounter divergent phase space integrals and show in detail how these cancel among each other as required by the finiteness of the anomalous dimension. As a non-trivial check of the method, we also perform the computation with a standard diagrammatic approach, finding perfect agreement.