Three-loop banana integrals with three equal masses

TL;DR

This paper derives and solves differential equations for three-loop banana Feynman integrals with three equal masses, expressing results in terms of modular forms related to elliptic curves and K3 surfaces.

Contribution

It provides the first explicit expression of a Feynman integral with K3 geometry in terms of known modular functions, advancing analytical techniques in multi-loop calculations.

Findings

Differential equations are solved using elliptic and modular forms.

Explicit expressions for all master integrals are obtained.

The associated K3 surface factorizes into elliptic curves, enabling modular form representation.

Abstract

We obtain and solve the canonical differential equations for the three-loop banana integrals in dimensional regularisation when three of the four masses are equal. The K3 surface associated with the maximal cuts factorises into a product of two elliptic curves. This allows us to express the differential forms in the canonical differential equations in terms of meromorphic modular forms. We present a rigorous proof that these differential forms only have simple poles and that they define independent cohomology classes. We also present explicit results for all master integrals in terms of iterated integrals of meromorphic modular forms (and integrals thereof). This is the first time that it was possible to express the results of a Feynman integral associated with a K3 geometry and depending on two dimensionless ratios in terms of functions that have previously been studied in the literature.