Three-loop banana integrals with three equal masses

TL;DR

This work resolves the canonical differential equations for the three-loop banana integrals with three equal masses in dimensional regularisation, exploiting a $K3$ surface that factorises into two elliptic curves. This factorisation enables expressing the differential forms in terms of meromorphic modular forms and solving via a path-ordered exponential, yielding explicit master integral expressions as iterated integrals of modular forms (and their integrals). The authors rigorously prove that the differential forms have only simple poles and define independent cohomology classes, and they provide complete analytic results for all master integrals, including boundary expansions and checks against numerical results. This represents a first full analytic solution for a Feynman integral tied to a CY geometry depending on two dimensionless ratios, using a two-variable modular framework with potential application to other CY-related integrals.

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.

Figures (1)

• Figure 1: The three-loop banana graph with three equal masses.