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

TL;DR

The paper introduces a criterion to simplify systems of differential equations for multi-loop master integrals by studying IBPs at fixed integer dimensions $d=2n$, exposing degeneracies that decouple subsets of integrals as $d\to 2n$. Through a sequence of explicit two- and three-loop examples, it demonstrates how to construct rotated bases $\mathcal{J}$ that reveal decoupled blocks, often with an explicit $(d-2n)$ factor, enabling iterative quadrature. The approach clarifies when solutions can be expressed in terms of multiple polylogarithms and when elliptic generalizations become necessary, and it connects dimension-shifting identities to practical basis changes. The results offer a practical path toward classifying diagrams by the expected function class and toward automating part of the IBP/DE reduction workflow via dimension-specific degeneracies.

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).