Finite Feynman Integrals

TL;DR

This paper introduces an algorithm based on Landau singularities to classify and construct finite bases of Feynman integrals, simplifying two-loop gluon scattering amplitude calculations.

Contribution

It presents a novel method to organize Feynman integrals by their infrared properties and construct finite bases that simplify complex amplitude computations.

Findings

Classified all configurations leading to infrared divergences.

Constructed bases of numerators that cancel singularities.

Simplified the two-loop four-gluon scattering amplitude.

Abstract

We describe an algorithm to organize Feynman integrals in terms of their infrared properties. Our approach builds upon the theory of Landau singularities, which we use to classify all configurations of loop momenta that can give rise to infrared divergences. We then construct bases of numerators for arbitrary Feynman integrals, which cancel all singularities and render the integrals finite. Through the same analysis, one can also classify so-called evanescent and evanescently finite Feynman integrals. These are integrals whose vanishing or finiteness relies on properties of dimensional regularization. To illustrate the use of these integrals, we display how to obtain a simpler form for the leading-color two-loop four-gluon scattering amplitude through the choice of a suitable basis of finite integrals. In particular, when all gluon helicities are equal, we show that with our basis the most complicated double-box integrals do not contribute to the finite remainder of the scattering amplitude.