Modular transformations of elliptic Feynman integrals

TL;DR

This paper studies how elliptic Feynman integrals behave under modular transformations, enabling faster numerical evaluations by transforming the nome squared to a small value, and introduces a combined approach of coordinate and basis transformations.

Contribution

It introduces a combined modular and basis transformation method for elliptic Feynman integrals, extending the understanding of their behavior under modular transformations.

Findings

Modular transformations can optimize numerical evaluation of elliptic integrals.

A combined coordinate and basis transformation preserves the class of functions.

The approach improves computational efficiency for elliptic Feynman integrals.

Abstract

We investigate the behaviour of elliptic Feynman integrals under modular transformations. This has a practical motivation: Through a suitable modular transformation we can achieve that the nome squared is a small quantity, leading to fast numerical evaluations. Contrary to the case of multiple polylogarithms, where it is sufficient to consider just variable transformations for the numerical evaluations of multiple polylogarithms, it is more natural in the elliptic case to consider a combination of a variable transformation (i.e. a modular transformation) together with a redefinition of the master integrals. Thus we combine a coordinate transformation on the base manifold with a basis transformation in the fibre. Only in the combination of the two transformations we stay within the same class of functions.