Direct numerical evaluation of multi-loop integrals without contour deformation

TL;DR

This paper introduces a numerical method for evaluating multi-loop integrals directly in Minkowski space without contour deformation, enabling precise Monte Carlo estimates for complex quantum field theory calculations.

Contribution

The authors develop a novel numerical approach that avoids explicit contour deformation for multi-loop integrals, improving accuracy and applicability in Minkowski space.

Findings

Validated method with examples from one to three loops

Achieved precise Monte Carlo estimates for multi-scale integrals

Extended approach to divergent integrals with regularization techniques

Abstract

We propose a method for computing numerically integrals defined via $i \epsilon$ deformations acting on single-pole singularities. We achieve this without an explicit analytic contour deformation. Our solution is then used to produce precise Monte Carlo estimates of multi-scale multi-loop integrals directly in Minkowski space. We corroborate the validity of our strategy by presenting several examples ranging from one to three loops. When used in connection with four-dimensional regularization techniques, our treatment can be extended to ultraviolet and infrared divergent integrals.