The unequal mass sunrise integral expressed through iterated integrals on M‾1,3

Abstract

We solve the two-loop sunrise integral with unequal masses systematically to all orders in the dimensional regularisation parameter ε. In order to do so, we transform the system of differential equations for the master integrals to an ε-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 M1,3 of a genus one Riemann surface with three marked points. This gives us the solution as iterated integrals on M‾1,3. On the hypersurface τ=const our result reduces to elliptic polylogarithms. In the equal mass case our result reduces to iterated integrals of modular forms.