New algorithms for Feynman integral reduction and $\varepsilon$-factorised differential equations

TL;DR

This paper presents a new algorithm for Feynman integral reduction and solving differential equations that depend on the regularization parameter, using geometric order relations and $ ext{ε}$-factorization techniques.

Contribution

It introduces a geometric order relation for integral reduction and a method for $ ext{ε}$-factorising differential equations, applicable to complex Feynman integrals.

Findings

The algorithm produces a basis of master integrals with Laurent polynomial differential equations in ε.

The method successfully applies to various complex Feynman integral examples.

The approach combines rational function techniques with algebraic and transcendental function analysis.

Abstract

In this paper, we give a detailed account of the algorithm outlined in [1] for Feynman integral reduction and $\varepsilon$-factorised differential equations. The algorithm consists of two steps. In the first step, we use a new geometric order relation in the integration-by-parts reduction to obtain a basis of master integrals, whose differential equations on the maximal cut are of a Laurent polynomial form in the regularisation parameter $\varepsilon$ and compatible with a filtration. This step works entirely with rational functions. In a second step, we provide a method to $\varepsilon$-factorise the aforementioned Laurent differential equations. The second step may introduce algebraic and transcendental functions. We illustrate the versatility of the algorithm by applying it to different examples with a wide range of complexity.