Three-loop banana integrals with four unequal masses

TL;DR

This paper derives a system of differential equations for three-loop banana Feynman integrals with four different masses, enabling analytic solutions in terms of iterated integrals using recent geometric and cohomological methods.

Contribution

It introduces a canonical differential equation framework for these complex integrals, utilizing K3 geometry and twisted cohomology to simplify and solve them analytically.

Findings

Derived differential equations for three-loop banana integrals with four masses.

Established a method to reduce the complexity of related geometric functions.

Provided a pathway for computing multiloop integrals with non-trivial geometries.

Abstract

We present a system of canonical differential equations satisfied by the three-loop banana integrals with four distinct non-zero masses in $D = 2-2\eps$ dimensions. Together with the initial condition in the small-mass limit, this provides all the ingredients to find analytic results for three-loop banana integrals in terms of iterated integrals to any desired order in the dimensional regulator. To obtain this result, we rely on recent advances in understanding the K3 geometry underlying these integrals and in how to construct rotations to an $\eps$-factorized basis. This rotation typically involves the introduction of objects defined as integrals of (derivatives of) K3 periods and rational functions. We apply and extend a method based on results from twisted cohomology to identify relations among these functions, which allows us to reduce their number considerably. We expect that the methods that we have applied here will prove useful to compute further multiloop multiscale Feynman integrals attached to non-trivial geometries.