The three-loop equal-mass banana integral in $\epsilon$-factorised form with meromorphic modular forms

Abstract

We show that the differential equation for the three-loop equal-mass banana integral can be cast into an $\epsilon$-factorised form with entries constructed from (meromorphic) modular forms and one special function, which can be given as an iterated integral of meromorphic modular forms. The $\epsilon$-factorised form of the differential equation allows for a systematic solution to any order in the dimensional regularisation parameter $\epsilon$. The alphabet of the iterated integrals contains six letters.

30 pages, v2: version to be published