Elliptic polylogarithms and iterated integrals on elliptic curves II: an application to the sunrise integral

TL;DR

This paper introduces elliptic multiple polylogarithms as a new class of iterated integrals on elliptic curves and applies them to compute sunrise Feynman integrals with arbitrary masses.

Contribution

It develops a formalism using elliptic polylogarithms and integral kernels to evaluate sunrise integrals, extending the mathematical toolkit for Feynman integral calculations.

Findings

Successfully expressed sunrise integrals in terms of elliptic polylogarithms

Provided a systematic method using integral kernels based on branch points

Expected broad applicability to other high-energy physics integrals

Abstract

We introduce a class of iterated integrals that generalize multiple polylogarithms to elliptic curves. These elliptic multiple polylogarithms are closely related to similar functions defined in pure math- ematics and string theory. We then focus on the equal-mass and non-equal-mass sunrise integrals, and we develop a formalism that enables us to compute these Feynman integrals in terms of our iterated integrals on elliptic curves. The key idea is to use integration-by-parts identities to identify a set of integral kernels, whose precise form is determined by the branch points of the integral in question. These kernels allow us to express all iterated integrals on an elliptic curve in terms of them. The flexibility of our approach leads us to expect that it will be applicable to a large variety of integrals in high-energy physics.