Feynman parameter integration through differential equations

TL;DR

This paper introduces a new algorithmic method for numerically evaluating multi-loop Feynman integrals by solving differential equations with series expansions, simplifying the process without manual boundary conditions.

Contribution

The authors develop a fully automated approach to compute Feynman integrals using differential equations and series expansions, reducing manual effort and complexity.

Findings

Efficient numerical computation of multi-loop Feynman integrals.

Reduction in the number of master integrals for complex topologies.

Successful application to a five-point two-loop integral family.

Abstract

We present a new method for numerically computing generic multi-loop Feynman integrals. The method relies on an iterative application of Feynman's trick for combining two propagators. Each application of Feynman's trick introduces a simplified Feynman integral topology which depends on a Feynman parameter that should be integrated over. For each integral family, we set up a system of differential equations which we solve in terms of a piecewise collection of generalized series expansions in the Feynman parameter. These generalized series expansions can be efficiently integrated term by term, and segment by segment. This approach leads to a fully algorithmic method for computing Feynman integrals from differential equations, which does not require the manual determination of boundary conditions. Furthermore, the most complicated topology that appears in the method often has less master integrals than the original one. We illustrate the strength of our method with a five-point two-loop integral family.