The unequal-mass three-loop banana integral

Abstract

We compute the three-loop banana integral with four unequal masses in dimensional regularisation. This integral is associated to a family of K3 surfaces, thus representing an example for Feynman integrals with geometries beyond elliptic curves. We evaluate the integral by deriving an $\varepsilon$-factorised differential equation, for which we rely on the algorithm presented in a recent publication. Equipping the space of differential forms in Baikov representation by a set of filtrations inspired by Hodge theory, we first obtain a differential equation with entries as Laurent polynomials in $\varepsilon$. Via a sequence of basis rotations we then remove any non-$\varepsilon$-factorising terms. This procedure is algorithmic and at no point relies on prior knowledge of the underlying geometry.