Integration by parts identities in integer numbers of dimensions. A criterion for decoupling systems of differential equations

TL;DR

This paper introduces a method to analyze IBP identities at fixed integer dimensions to identify a basis of master integrals with decoupled differential equations, simplifying their solution.

Contribution

It presents a novel approach to decouple systems of differential equations for Feynman integrals by studying IBPs at fixed integer dimensions.

Findings

Decoupling of differential equations as dimension approaches an integer value

Simplification of solving Feynman integrals via Laurent expansion

New basis of master integrals with easier-to-solve differential equations

Abstract

Integration by parts identities (IBPs) can be used to express large numbers of apparently different d-dimensional Feynman Integrals in terms of a small subset of so-called master integrals (MIs). Using the IBPs one can moreover show that the MIs fulfil linear systems of coupled differential equations in the external invariants. With the increase in number of loops and external legs, one is left in general with an increasing number of MIs and consequently also with an increasing number of coupled differential equations, which can turn out to be very difficult to solve. In this paper we show how studying the IBPs in fixed integer numbers of dimension d=n with $n \in \mathbb{N}$ one can extract the information useful to determine a new basis of MIs, whose differential equations decouple as $d \to n$ and can therefore be more easily solved as Laurent expansion in (d-n).