Reducing differential equations for multiloop master integrals

TL;DR

This paper introduces an algorithm to reduce differential equations for multiloop master integrals into a Fuchsian form with epsilon-dependent matrices, facilitating calculations in quantum field theory.

Contribution

The paper presents a novel algorithm for transforming differential equations of master integrals into a simplified Fuchsian form using rational linear transformations.

Findings

Algorithm successfully reduces differential equations to Fuchsian form

Handles epsilon-dependent matrices in the transformation process

Identifies cases where reduction is not possible (degenerate cases)

Abstract

We present an algorithm of the reduction of the differential equations for master integrals the Fuchsian form with the right-hand side matrix linearly depending on dimensional regularization parameter $\epsilon$. We consider linear transformations of the functions column which are rational in the variable and in $\epsilon$. Apart from some degenerate cases described below, the algorithm allows one to obtain the required transformation or to ascertain irreducibility to the form required. Degenerate cases are quite anticipated and likely to correspond to irreducible systems.