A Feynman integral depending on two elliptic curves

TL;DR

This paper analyzes a two-loop four-point Feynman integral with one internal mass, revealing its dependence on two elliptic curves and transforming its differential equation into an epsilon form for better understanding.

Contribution

It introduces a method to transform the differential equation of a complex Feynman integral into an epsilon form, highlighting its dependence on two elliptic curves.

Findings

The integral depends on two elliptic curves.

The differential equation is successfully transformed into epsilon form.

Entries depending on both elliptic curves are studied in detail.

Abstract

We study a two-loop four-point function with one internal mass. This Feynman integral is one of the simplest Feynman integrals depending on two elliptic curves. We transform the associated differential equation into an $\varepsilon$-form. We study the entries of the differential equation, and in particular the entries which depend on both elliptic curves.