The iterated structure of the all-order result for the two-loop sunrise integral

TL;DR

The paper develops an all-order $\varepsilon$-expansion for the equal-mass two-loop sunrise integral in $D=2-2\varepsilon$ by introducing a generalized class of elliptic polylogarithms, called ELi, that combine genus-zero and genus-one structures via the elliptic nome $q$. Using a differential-equation framework, the authors transform the problem to a form where higher-order terms are generated iteratively from lower-order terms, ensuring all intermediate integrals stay within the ELi class. They provide explicit boundary values at a degenerate elliptic point and present detailed constructions for the first three orders of the expansion, including numerous ELi and polylogarithmic terms. The methodology generalizes the evaluation of certain elliptic Feynman integrals and offers an algorithm potentially applicable to other problems with elliptic geometry in perturbative quantum field theory.

Abstract

We present a method to compute the Laurent expansion of the two-loop sunrise integral with equal non-zero masses to arbitrary order in the dimensional regularisation $\varepsilon$. This is done by introducing a class of functions (generalisations of multiple polylogarithms to include the elliptic case) and by showing that all integrations can be carried out within this class of functions.