Elliptic Feynman integrals and pure functions

TL;DR

This paper introduces a new class of elliptic multiple polylogarithms with specific singularity properties, demonstrating their effectiveness in expressing complex two-loop Feynman integrals as pure, uniform-weight functions.

Contribution

It proposes a novel variant of elliptic multiple polylogarithms with desirable mathematical properties and applies them to express and analyze elliptic Feynman integrals, revealing uniform weight structures.

Findings

Elliptic Feynman integrals can be expressed in terms of the new functions.

The functions exhibit at most logarithmic singularities and satisfy specific differential equations.

First observation of uniform weight in elliptic polylogarithm evaluations of Feynman integrals.

Abstract

We propose a variant of elliptic multiple polylogarithms that have at most logarithmic singularities in all variables and satisfy a differential equation without homogeneous term. We investigate several non-trivial elliptic two-loop Feynman integrals with up to three external legs and express them in terms of our functions. We observe that in all cases they evaluate to pure combinations of elliptic multiple polylogarithms of uniform weight. This is the first time that a notion of uniform weight is observed in the context of Feynman integrals that evaluate to elliptic polylogarithms.