The unequal mass sunrise integral expressed through iterated integrals on $\overline{\mathcal M}_{1,3}$
Christian Bogner, Stefan Müller-Stach, Stefan Weinzierl
TL;DR
The paper tackles the two-loop sunrise integral with unequal masses by achieving an $\varepsilon$-form for the differential equations of its master integrals and by mapping the kinematics to standard coordinates on the moduli space ${\overline{\mathcal M}}_{1,3}$. The solution is expressed as iterated integrals on ${\overline{\mathcal M}}_{1,3}$, reducing to elliptic polylogarithms for fixed modular parameter $\tau$ and to iterated modular forms in the equal-mass limit. Tadpole-like master integrals are computed to all orders in $\varepsilon$, while the remaining integrals are obtained order-by-order via Chen-style iterated integrals with kernels built from Kronecker-based functions $g^{(n)}(z,\tau)$ and related modular objects. The approach unifies modular-form and elliptic-polylogarithmic descriptions within a single geometric framework and provides a practical computational scheme for multi-scale elliptic Feynman integrals.
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 $τ=\mbox{const}$ our result reduces to elliptic polylogarithms. In the equal mass case our result reduces to iterated integrals of modular forms.