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 $\epsilon$-factorized differential equation matrix, nicely generalizing similar observations from the elliptic case.

52 pages

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• Classification of Feynman integral geometries for black-hole scattering at 5PM order
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• Electroweak double-box integrals for Moller scattering
• $\text{history}$: A tool for fully-differential cross sections at next-to-next-to-leading order
• The spectrum of Feynman-integral geometries at two loops
• A construction of single-valued elliptic polylogarithms