From elliptic curves to Feynman integrals

TL;DR

This paper explores Feynman integrals associated with elliptic curves, demonstrating multiple elliptic curves can appear and showing how to simplify their differential equations for easier solutions.

Contribution

It introduces a method to identify multiple elliptic curves in Feynman integrals and transforms the differential equations into a linear form in epsilon for straightforward integration.

Findings

Multiple elliptic curves can occur in Feynman integrals

Maximal cuts help identify elliptic curves

Transformed differential equations are linear in epsilon

Abstract

In this talk we discuss Feynman integrals which are related to elliptic curves. We show with the help of an explicit example that in the set of master integrals more than one elliptic curve may occur. The technique of maximal cuts is a useful tool to identify the elliptic curves. By a suitable transformation of the master integrals the system of differential equations for our example can be brought into a form linear in $\varepsilon$, where the $\varepsilon^0$-term is strictly lower-triangular. This system is easily solved in terms of iterated integrals.