Numerical integration of loop integrals through local cancellation of threshold singularities

TL;DR

This paper introduces a novel numerical method for calculating the real and imaginary parts of finite loop integrals separately, using local cancellation of threshold singularities to improve Monte Carlo integration efficiency.

Contribution

The authors develop a new momentum-space representation that separates real and imaginary parts of one-loop integrals, enhancing numerical stability and applicability for Monte Carlo methods.

Findings

Effective local cancellation of threshold singularities

Separate numerical calculation of real and imaginary parts

Promising results for extension to multi-loop integrals

Abstract

We propose a new approach that allows for the separate numerical calculation of the real and imaginary parts of finite loop integrals. We find that at one-loop the real part is given by the Loop-Tree Duality integral supplemented with suitable counterterms and the imaginary part is a sum of two-body phase space integrals, constituting a locally finite representation of the generalised optical theorem. These expressions are integrals in momentum space, whose integrands were specially designed to feature local cancellations of threshold singularities. Such a representation is well suited for Monte Carlo integration and avoids the drawbacks of a numerical contour deformation around remaining singularities. Our method is directly applicable to a range integrals with certain geometric properties but not yet fully generalised for arbitrary one-loop integrals. We demonstrate the computational performance with examples of one-loop integrals with various kinematic configurations, which gives promising prospects for an extension to multi-loop integrals.