An analytic solution for the equal-mass banana graph

TL;DR

The paper solves for all master integrals of the equal-mass three-loop banana graph in d = 2 using a differential-equation approach. It shows the homogeneous solution space is governed by the same modular forms that appear in the two-loop equal-mass sunrise, enabling a representation in terms of iterated integrals of modular forms for Gamma1(6) and, equivalently, elliptic polylogarithms at rational points. A semi-simple/unipotent decomposition yields uniformly-weight expressions, and Mellin-Barnes boundary analysis provides leading asymptotics. The results establish a concrete analytic link between banana and sunrise topologies, with implications for higher-loop computations and Higgs phenomenology.

Abstract

We present fully analytic results for all master integrals for the three-loop banana graph with four equal and non-zero masses. The results are remarkably simple and all integrals are expressed as linear combinations of iterated integrals of modular forms of uniform weight for the same congruence subgroup as for the two-loop equal-mass sunrise graph. We also show how to write the results in terms of elliptic polylogarithms evaluated at rational points.

Figures (1)

• Figure 1: The three-loop banana graph.