# The unequal mass sunrise integral expressed through iterated integrals on $\overline{\mathcal M}_{1,3}$

## TL;DR

This paper provides a systematic all-orders solution for the two-loop sunrise integral with unequal masses using iterated integrals on a moduli space, unifying elliptic polylogarithms and modular forms.

## Contribution

It introduces a change of variables to express the sunrise integral as iterated integrals on ar al M_{1,3}, enabling solutions in terms of elliptic polylogarithms and modular forms.

## Key findings

- Solution expressed as iterated integrals on ar al M_{1,3}
- Reduces to elliptic polylogarithms at constant au
- Special case reduces to iterated integrals of modular forms

## Abstract

We solve the two-loop sunrise integral with unequal masses systematically to all orders in the dimensional regularisation parameter $\varepsilon$. In order to do so, we transform the system of differential equations for the master integrals to an $\varepsilon$-form. The sunrise integral with unequal masses depends on three kinematical variables. We perform a change of variables to standard coordinates on the moduli space ${\mathcal M}_{1,3}$ of a genus one Riemann surface with three marked points. This gives us the solution as iterated integrals on $\overline{\mathcal M}_{1,3}$. On the hypersurface $\tau=\mbox{const}$ our result reduces to elliptic polylogarithms. In the equal mass case our result reduces to iterated integrals of modular forms.