Simplifying differential equations for multi-scale Feynman integrals beyond multiple polylogarithms
Luise Adams, Ekta Chaubey, Stefan Weinzierl
TL;DR
The paper addresses the challenge of solving differential equations for multi-scale Feynman integrals that do not generically admit an $\varepsilon$-form, especially when elliptic sectors appear. It introduces a systematic method that reduces to a single scale, constructs a Picard-Fuchs operator for a master integral, and factorises this operator to decouple the ε^0 system into smaller blocks corresponding to irreducible factors. In the linear-factor special case, it provides an explicit transformation that yields an $\varepsilon$-form, while in general the method decomposes multi-scale systems into more tractable sub-systems, as illustrated by two-loop examples. The approach offers a practical route to precise NNLO calculations by simplifying complex coupled differential equations and clarifying when an $\varepsilon$-form is attainable.
Abstract
In this paper we exploit factorisation properties of Picard-Fuchs operators to decouple differential equations for multi-scale Feynman integrals. The algorithm reduces the differential equations to blocks of the size of the order of the irreducible factors of the Picard-Fuchs operator. As a side product, our method can be used to easily convert the differential equations for Feynman integrals which evaluate to multiple polylogarithms to $\varepsilon$-form.