On factorization hierarchy of equations for banana Feynman amplitudes

TL;DR

This paper reviews the differential equations governing banana Feynman amplitudes, exploring their factorization properties and relations across different representations, aiming to uncover their integrable structure.

Contribution

It provides a detailed analysis of the factorization and interrelation of equations for banana Feynman diagrams in various forms, advancing understanding of their mathematical structure.

Findings

Factorization properties at equal mass locus

Relations between Fourier and Picard-Fuchs equations

Identification of equations as counterparts to Virasoro constraints

Abstract

We present a review of the relations between various equations for maximal cut banana Feynman diagrams, i.e. amplitudes with propagators substituted with $\delta$-functions. We consider both equal and generic masses. There are three types of equation to consider: those in coordinate space, their Fourier transform and Picard-Fuchs equations originating from the parametric representation. First, we review the properties of the corresponding differential operators themselves, mainly their factorization properties at the equal mass locus and their form at special values of the dimension. Then we study the relation between the Fourier transform of the coordinate space equations and the Picard-Fuchs equations and show that they are related by factorization as well. The equations in question are the counterparts of the Virasoro constraints in the much-better studied theory of eigenvalue matrix models and are the first step towards building a full-fledged theory of Feynman integrals, which will reveal their hidden integrable structure.