Three-loop banana integrals with four unequal masses

TL;DR

This work constructs canonical differential equations for the three-loop banana integral family with four distinct non-zero masses in $D=2-2\varepsilon$, enabling analytic results in terms of iterated integrals once initial data at small masses are supplied. The authors integrate a sequence of basis rotations inspired by $\varepsilon$-factorization with the geometric structure of a K3 surface, and use twisted cohomology to relate and reduce the number of new functions appearing in the differential equation matrix. They show that 23 $\varepsilon$-functions can be constrained down to 13, and, further exploiting permutation symmetries, effectively to two core functions, yielding a compact, implementable canonical form. The final system, together with boundary conditions in the small-mass limit, provides a practical route to high-order analytic results for these multiloop integrals, with potential applicability to other Calabi–Yau- or nontrivial-geometry-based Feynman integrals.

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.

Figures (1)

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