An algorithmic approach to finding canonical differential equations for elliptic Feynman integrals

TL;DR

This paper introduces an algorithmic method to find canonical differential equations for elliptic Feynman integrals, simplifying their computation and expanding the set of known canonical forms.

Contribution

It presents a new algorithmic approach to derive canonical forms for elliptic Feynman integrals, including first-time results and an updated software tool.

Findings

Successfully reproduces known canonical forms

Derives new canonical forms for elliptic integrals

Provides a publicly available implementation of the algorithm

Abstract

In recent years, differential equations have become the method of choice to compute multi-loop Feynman integrals. Whenever they can be cast into canonical form, their solution in terms of special functions is straightforward. Recently, progress has been made in understanding the precise canonical form for Feynman integrals involving elliptic polylogarithms. In this article, we make use of an algorithmic approach that proves powerful to find canonical forms for these cases. To illustrate the method, we reproduce several known canonical forms from the literature and present examples where a canonical form is deduced for the first time. Together with this article, we also release an update for INITIAL, a publicly available Mathematica implementation of the algorithm.