# An analytic solution for the equal-mass banana graph

## TL;DR

This paper provides a fully analytic solution for all master integrals of the three-loop banana graph with equal masses, expressing them in terms of modular forms and elliptic polylogarithms, advancing the understanding of complex Feynman integrals.

## Contribution

It introduces a novel analytic approach to evaluate all master integrals of the three-loop banana graph with equal masses using modular forms and elliptic polylogarithms.

## Key findings

- All integrals expressed as linear combinations of iterated integrals of modular forms.
- Results are remarkably simple and involve elliptic polylogarithms at rational points.
- The approach generalizes previous methods used for the two-loop sunrise graph.

## 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.