The sunrise integral around two and four space-time dimensions in terms of elliptic polylogarithms

TL;DR

This paper presents solutions for the sunrise integral in two and four space-time dimensions using elliptic polylogarithms, revealing geometric interpretations and expressing four-dimensional results via lower-dimensional solutions and derivatives.

Contribution

It introduces a novel elliptic polylogarithm framework for the sunrise integral in different dimensions, connecting geometric insights with integral solutions.

Findings

In two dimensions, the sunrise integral is expressed as a sum of three elliptic dilogarithms.

In four dimensions, the integral is related to two-dimensional solutions and their derivatives.

The arguments of elliptic dilogarithms have a geometric interpretation.

Abstract

In this talk we discuss the solution for the sunrise integral around two and four space-time dimensions in terms of a generalised elliptic version of the multiple polylogarithms. In two space-time dimensions we obtain a sum of three elliptic dilogarithms. The arguments of the elliptic dilogarithms have a nice geometric interpretation. In four space-time dimensions the sunrise integral can be expressed with the $\epsilon^0$- and $\epsilon^1$-solution around two dimensions, mass derivatives thereof and simpler terms.