Algorithms and tools for iterated Eisenstein integrals

TL;DR

This paper introduces algorithms for iterated Eisenstein integrals, enabling their analytic continuation and efficient series expansion, with applications to hypergeometric functions and Feynman integrals like the sunrise graph.

Contribution

It provides novel algorithms for handling iterated Eisenstein integrals, facilitating their analytic continuation and series expansion in multi-loop Feynman integral computations.

Findings

Algorithms enable analytic continuation across all parameter regions.

Series representations converge rapidly in each region.

Applications demonstrated on hypergeometric functions and the sunrise graph.

Abstract

We present algorithms to work with iterated Eisenstein integrals that have recently appeared in the computation of multi-loop Feynman integrals. These algorithms allow one to analytically continue these integrals to all regions of the parameter space, and to obtain fast converging series representations in each region. We illustrate our approach on the examples of hypergeometric functions that evaluate to iterated Eisenstein integrals as well as the well-known sunrise graph.