# Removing infrared divergences from two-loop integrals

## TL;DR

This paper introduces a systematic method to remove infrared divergences from two-loop Feynman integrals by adding counterterms, simplifying the evaluation of scattering amplitudes in massless particle processes.

## Contribution

The authors develop a novel subtraction technique that cancels infrared divergences at the integrand level, enabling finite evaluations in four dimensions and simplifying the extraction of singular behavior.

## Key findings

- Counterterms effectively cancel divergences in two-loop integrals.
- Remaining finite parts can be computed directly after subtraction.
- Method aids in analyzing asymptotic behavior of amplitudes with small mass parameters.

## Abstract

Feynman amplitudes at higher orders in perturbation theory generically have complex singular structures. Notwithstanding the emergence of many powerful new methods, the presence of infrared divergences poses significant challenges for their evaluation. In this article, we develop a systematic method for the removal of the infrared singularities, by adding appropriate counterterms that approximate and cancel divergent limits point-by-point at the level of the integrand. We provide a proof of concept for our method by applying it to master-integrals that are found in scattering amplitudes for representative two-to-two scattering processes of massless particles. We demonstrate that, after the introduction of counterterms, the remainder is finite in four dimensions. In addition, we find in these cases that the complete singular dependence of the integrals can be obtained simply by analytically integrating the counterterms. Finally, we observe that our subtraction method can be also useful in order to extract in a simple way the asymptotic behavior of Feynman amplitudes in the limit of small mass parameters.