On Epsilon Factorized Differential Equations for Elliptic Feynman Integrals

TL;DR

This paper introduces a new method to derive epsilon factorized differential equations for elliptic Feynman integrals by selecting a basis that diagonalizes the period matrix, extending techniques from polylogarithmic cases.

Contribution

It generalizes existing methods to elliptic integrals, enabling systematic derivation of epsilon factorized differential equations for complex Feynman integrals.

Findings

Successfully applied to multiple elliptic Feynman integral families

Demonstrated the method's effectiveness in simplifying differential equations

Provides a framework for future elliptic integral computations

Abstract

In this paper we develop and demonstrate a method to obtain epsilon factorized differential equations for elliptic Feynman integrals. This method works by choosing an integral basis with the property that the period matrix obtained by integrating the basis over a complete set of integration cycles is diagonal. The method is a generalization of a similar method known to work for polylogarithmic Feynman integrals. We demonstrate the method explicitly for a number of Feynman integral families with an elliptic highest sector.