Automated computation of one-loop integrals in massless theories

TL;DR

We address the problem of efficiently evaluating one-loop tensor and scalar integrals in massless quantum field theories for high-multiplicity external states. The authors implement a spinor-based tensor-reduction approach that uses light-like momenta $l_1$ and $l_2$ to decompose loop momentum, along with dimension-shifting techniques to handle higher-dimensional contributions, reducing any $n$-point integral to the basic scalar $I_2$, $I_3$, and $I_4$ functions. The paper provides detailed reduction schemes for $n=2$ through $12$, including pentagon and hexagon relations and a Gram-SVD based method for $n\ge7$, together with a full numerical implementation in Fortran and C++ and extensive cross-checks. This work enables automated next-to-leading order calculations for $2\rightarrow n$ processes in massless QCD and offers scalable techniques for high-multiplicity jet analyses, constrained mainly by computational resources.

Abstract

We consider one-loop tensor and scalar integrals, which occur in a massless quantum field theory and we report on the implementation into a numerical program of an algorithm for the automated computation of these one-loop integrals. The number of external legs of the loop integrals is not restricted. All calculations are done within dimensional regularization.

Figures (1)

• Figure 1: The labelling for a generic one-loop integral. The arrows denote the momentum flow.