Direct numerical integration for multi-loop integrals

TL;DR

This paper introduces a numerical method for multi-loop integrals that constructs contour deformations in loop momentum space, enabling direct numerical integration without Feynman parameters, applicable to both finite and divergent cases.

Contribution

The paper extends contour deformation techniques from one-loop to multi-loop integrals, allowing direct numerical evaluation in loop momentum space.

Findings

Applicable to two and three-loop integrals

Works for both finite and divergent integrals with subtraction

Avoids Feynman or Schwinger parameterization

Abstract

We present a method to construct a suitable contour deformation in loop momentum space for multi-loop integrals. This contour deformation can be used to perform the integration for multi-loop integrals numerically. The integration can be performed directly in loop momentum space without the introduction of Feynman or Schwinger parameters. The method can be applied to finite multi-loop integrals and to divergent multi-loop integrals with suitable subtraction terms. The algorithm extends techniques from the one-loop case to the multi-loop case. Examples at two and three loops are discussed explicitly.