From modular forms to differential equations for Feynman integrals

TL;DR

This paper introduces a new representation for modular forms using elliptic integrals, facilitating the calculation of Feynman integrals via differential equations and iterated integrals.

Contribution

It provides a novel representation of modular forms in terms of elliptic integrals, aiding the computation of Feynman integrals in quantum field theory.

Findings

Rewritten elliptic multiple zeta values as iterated integrals over elliptic integrals.

Applied the representation to differential equations for sunrise and kite integrals.

Demonstrated the effectiveness of the approach on several examples.

Abstract

In these proceedings we discuss a representation for modular forms that is more suitable for their application to the calculation of Feynman integrals in the context of iterated integrals and the differential equation method. In particular, we show that for every modular form we can find a representation in terms of powers of complete elliptic integrals of the first kind multiplied by algebraic functions. We illustrate this result on several examples. In particular, we show how to explicitly rewrite elliptic multiple zeta values as iterated integrals over powers of complete elliptic integrals and rational functions, and we discuss how to use our results in the context of the system of differential equations satisfied by the sunrise and kite integrals.