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

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 $\epsilon$-form, which makes a solution in terms of iterated integrals immediate.

Comments: 21 pages, talk given at the KMPB conference "Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory"