On the Structure of Infrared Singularities of Gauge-Theory Amplitudes

TL;DR

This paper derives a comprehensive formula for the infrared singularities in massless gauge-theory scattering amplitudes, revealing constraints on their structure and extending previous two-loop results to three loops and beyond.

Contribution

It provides an all-order conjecture for the anomalous-dimension matrix governing infrared singularities, incorporating new constraints and extending known results to three-loop order.

Findings

Infrared singularities involve only two-parton correlations at three loops.

Casimir scaling of the cusp anomalous dimension holds to all orders.

Higher Casimir invariants do not appear at four loops.

Abstract

A closed formula is obtained for the infrared singularities of dimensionally regularized, massless gauge-theory scattering amplitudes with an arbitrary number of legs and loops. It follows from an all-order conjecture for the anomalous-dimension matrix of n-jet operators in soft-collinear effective theory. We show that the form of this anomalous dimension is severely constrained by soft-collinear factorization, non-abelian exponentiation, and the behavior of amplitudes in collinear limits. Using a diagrammatic analysis, we demonstrate that these constraints imply that to three-loop order the anomalous dimension involves only two-parton correlations, with the possible exception of a single color structure multiplying a function of conformal cross ratios depending on the momenta of four external partons, which would have to vanish in all two-particle collinear limits. We argue that such a function does not appear at three-loop order, and that the same is true in higher orders. Our formula predicts Casimir scaling of the cusp anomalous dimension to all orders in perturbation theory, and we explicitly check that the constraints exclude the appearance of higher Casimir invariants at four loops. Using known results for the quark and gluon form factors, we derive the three-loop coefficients of the 1/epsilon^n pole terms (with n=1,...,6) for an arbitrary n-parton scattering amplitude in massless QCD. This generalizes Catani's two-loop formula proposed in 1998.