Canonical Differential Equations Beyond Genus One

TL;DR

This paper extends the theory of canonical differential equations to hyperelliptic Feynman integrals of genus two, introducing new functions, methods, and connections to Siegel modular forms.

Contribution

It constructs canonical differential equations for hyperelliptic Lauricella functions of genus two, advancing the understanding of higher-genus Feynman integrals.

Findings

Application of existing methods to higher-genus cases

Introduction of new ideas on twisted cohomology intersection matrices

Identification of Siegel modular forms in the differential equations

Abstract

We discuss for the first time canonical differential equations for hyperelliptic Feynman integrals. We study hyperelliptic Lauricella functions that include in particular the maximal cut of the two-loop non-planar double box, which is known to involve a hyperlliptic curve of genus two. We consider specifically three- and four-parameter Lauricella functions, each associated to a hyperelliptic curve of genus two, and construct their canonical differential equations. Whilst core steps of this construction rely on existing methods $\unicode{x2014}$ that we show to be applicable in the higher-genus case $\unicode{x2014}$ we use new ideas on the structure of the twisted cohomology intersection matrix associated to the integral family in canonical form to obtain a better understanding of the appearing new functions. We further observe the appearance of Siegel modular forms in the $\varepsilon$-factorized differential equation matrix, nicely generalizing similar observations from the elliptic case.