Positive Integrands from Feynman Integrals in the Minkowski Regime

TL;DR

This paper introduces a method to rewrite Feynman integrals in the Minkowski regime as sums of positive integrands, enabling more efficient numerical and analytical evaluations without contour deformation.

Contribution

The authors develop an algorithm to construct positive integrand representations for a wide class of Feynman integrals in Minkowski space, including complex multi-loop cases.

Findings

Performance gains of up to four orders of magnitude in evaluation speed.

Applicability to integrals with internal masses and off-shell external legs.

Compatibility with existing techniques like sector decomposition.

Abstract

We present a method for rewriting dimensionally regulated Feynman parameter integrals in the Minkowski regime as a sum of real, positive integrands multiplied by complex prefactors. This representation eliminates the need for contour deformation, allowing for direct numerical or analytic evaluation of the integrals. We develop an algorithm to construct such representations for a broad class of integrals and demonstrate its generalisation through selected examples. Our approach is applied to integrals up to three loops, including cases with internal masses and off-shell external legs. The resulting expressions are suitable for evaluation using existing techniques, such as sector decomposition, where we observe performance gains of up to four orders of magnitude in certain cases.