Numerical evaluation of multi-loop integrals using subtraction terms

TL;DR

This paper introduces a numerical method for evaluating multi-loop integrals by subtracting divergences and using complex contour deformation, demonstrated with examples and implemented in the NICODEMOS software.

Contribution

It presents a novel formalism for numerical multi-loop integral evaluation using subtraction terms and contour deformation, with an available implementation.

Findings

Effective removal of divergences in numerical integrals

Robust convergence achieved with complex contour deformation

Demonstrated success on several one- and two-loop examples

Abstract

A formalism for the numerical integration of one- and two-loop integrals is presented. It is based on subtraction terms which remove the soft, collinear and some of the ultraviolet divergences from the integrand. The numerical integral is performed in the Feynman parameter space, using a complex contour deformation to ensure robust convergence even in the presence of physical thresholds. The application of the proposed procedure is demonstrated with several one- and two-loop examples. An implementation in the program NICODEMOS is publicly available, which currently incorporates only one-loop functionality, but an extension to two-loop cases is planned for future versions.