Numerical Loop-Tree Duality: contour deformation and subtraction

TL;DR

This paper presents a new, systematic contour deformation method within Loop-Tree Duality for numerically evaluating complex loop integrals, effectively handling threshold singularities and advancing towards fully numerical higher-order scattering calculations.

Contribution

A novel, automatic contour deformation technique for Loop-Tree Duality that simplifies numerical loop integral computations without fine-tuning.

Findings

Successfully applied to over 100 finite scalar integrals up to six loops.

Extended to one-loop infrared divergent integrals and photon production amplitudes.

Demonstrated potential for fully numerical higher-order scattering cross-section calculations.

Abstract

We introduce a novel construction of a contour deformation within the framework of Loop-Tree Duality for the numerical computation of loop integrals featuring threshold singularities in momentum space. The functional form of our contour deformation automatically satisfies all constraints without the need for fine-tuning. We demonstrate that our construction is systematic and efficient by applying it to more than 100 examples of finite scalar integrals featuring up to six loops. We also showcase a first step towards handling non-integrable singularities by applying our work to one-loop infrared divergent scalar integrals and to the one-loop amplitude for the ordered production of two and three photons. This requires the combination of our contour deformation with local counterterms that regulate soft, collinear and ultraviolet divergences. This work is an important step towards computing higher-order corrections to relevant scattering cross-sections in a fully numerical fashion.