Meromorphic modular forms and the three-loop equal-mass banana integral

TL;DR

This paper develops a framework using meromorphic modular forms to solve multi-loop Feynman integrals, providing explicit decompositions and applying these methods to obtain new analytic results for the three-loop banana integral.

Contribution

It introduces a novel approach linking modular forms to multi-loop Feynman integrals and decomposes modular form spaces for arbitrary finite-index subgroups, enabling new analytic solutions.

Findings

Explicit solutions for the three-loop banana integral in dimensional regularisation

Decomposition of modular form spaces into derivatives and basis forms

Connection between modular forms and monodromy groups in differential equations

Abstract

We consider a class of differential equations for multi-loop Feynman integrals which can be solved to all orders in dimensional regularisation in terms of iterated integrals of meromorphic modular forms. We show that the subgroup under which the modular forms transform can naturally be identified with the monodromy group of a certain second-order differential operator. We provide an explicit decomposition of the spaces of modular forms into a direct sum of total derivatives and a basis of modular forms that cannot be written as derivatives of other functions, thereby generalising a result by one of the authors form the full modular group to arbitrary finite-index subgroups of genus zero. Finally, we apply our results to the two- and three-loop equal-mass banana integrals, and we obtain in particular for the first time complete analytic results for the higher orders in dimensional regularisation for the three-loop case, which involves iterated integrals of meromorphic modular forms.