A walk on sunset boulevard

TL;DR

This paper explores advanced mathematical functions related to Feynman diagrams, including polylogarithms and elliptic curves, culminating in a comprehensive solution for the sunset integral in quantum field theory.

Contribution

It introduces the generalization of polylogarithms to elliptic functions and provides an all-order solution for the sunset integral with equal masses.

Findings

Elliptic generalization of polylogarithms is developed.

All-order solution for the sunset integral in the equal mass case is obtained.

Connections between Feynman diagrams and elliptic curves are established.

Abstract

A walk on sunset boulevard can teach us about transcendental functions associated to Feynman diagrams. On this guided tour we will see multiple polylogarithms, differential equations and elliptic curves. A highlight of the tour will be the generalisation of the polylogarithms to the elliptic setting and the all-order solution for the sunset integral in the equal mass case.