Canonical Differential Equations Beyond Genus One

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.