On a class of Feynman integrals evaluating to iterated integrals of modular forms

TL;DR

This paper explores a class of Feynman integrals that can be expressed as iterated integrals of modular forms, extending beyond multiple polylogarithms, and demonstrates methods to solve their differential equations efficiently.

Contribution

It introduces a framework for representing certain Feynman integrals as iterated integrals of modular forms and shows how to transform their differential equations into an $e$-form for solutions.

Findings

Feynman integrals can be expressed as iterated integrals of modular forms

Differential equations for these integrals can be transformed into an $e$-form

Solutions can be obtained in terms of iterated integrals

Abstract

In this talk we discuss a class of Feynman integrals, which can be expressed to all orders in the dimensional regularisation parameter as iterated integrals of modular forms. We review the mathematical prerequisites related to elliptic curves and modular forms. Feynman integrals, which evaluate to iterated integrals of modular forms go beyond the class of multiple polylogarithms. Nevertheless, we may bring for all examples considered the associated system of differential equations by a non-algebraic transformation to an $\varepsilon$-form, which makes a solution in terms of iterated integrals immediate.