Simple differential equations for Feynman integrals associated to elliptic curves

TL;DR

This paper explores the extension of the epsilon-form differential equation method to Feynman integrals associated with elliptic curves, demonstrating how to achieve this form for complex examples, including single-scale and multi-scale cases.

Contribution

It introduces methods to transform differential equations of elliptic Feynman integrals into epsilon-form, extending techniques beyond multiple polylogarithms.

Findings

Successfully brought elliptic Feynman integrals into epsilon-form

Applied methods to both single-scale and multi-scale cases

Demonstrated non-trivial examples of the approach

Abstract

The $\varepsilon$-form of a system of differential equations for Feynman integrals has led to tremendeous progress in our abilities to compute Feynman integrals, as long as they fall into the class of multiple polylogarithms. It is therefore of current interest, if these methods extend beyond the case of multiple polylogarithms. In this talk I discuss Feynman integrals, which are associated to elliptic curves and their differential equations. I show for non-trivial examples how the system of differential equations can be brought into an $\varepsilon$-form. Single-scale and multi-scale cases are discussed.