The unequal-mass three-loop banana integral

TL;DR

This paper computes a complex three-loop Feynman integral with four different masses, revealing its connection to K3 surfaces and employing an algorithmic method to derive an epsilon-factorised differential equation without prior geometric assumptions.

Contribution

It introduces an algorithmic approach to derive epsilon-factorised differential equations for Feynman integrals associated with K3 surface geometries, extending methods beyond elliptic cases.

Findings

Successfully computed the three-loop banana integral with four unequal masses.

Established a connection between the integral and K3 surface geometry.

Developed an algorithmic procedure to achieve epsilon-factorisation without prior geometric knowledge.

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.