On a procedure to derive $\epsilon$-factorised differential equations beyond polylogarithms

TL;DR

This paper presents a new method for deriving epsilon-factorized differential equations for complex Feynman integrals involving special functions beyond polylogarithms, applicable to multi-scale, multi-loop problems including elliptic and more complex geometries.

Contribution

The authors develop a procedure to obtain epsilon-factorized differential equations for Feynman integrals beyond multiple polylogarithms, extending applicability to elliptic and more complex geometries.

Findings

Successfully applied to equal-mass two-loop sunrise integral.

Extended method to elliptic and multi-scale Feynman integrals.

Demonstrated applicability to geometries beyond elliptic curves.

Abstract

In this manuscript, we elaborate on a procedure to derive $\epsilon$-factorised differential equations for multi-scale, multi-loop classes of Feynman integrals that evaluate to special functions beyond multiple polylogarithms. We demonstrate the applicability of our approach to diverse classes of problems, by working out $\epsilon$-factorised differential equations for single- and multi-scale problems of increasing complexity. To start we are reconsidering the well-studied equal-mass two-loop sunrise case, and move then to study other elliptic two-, three- and four-point problems depending on multiple different scales. Finally, we showcase how the same approach allows us to obtain $\epsilon$-factorised differential equations also for Feynman integrals that involve geometries beyond a single elliptic curve.