Canonical differential equations beyond polylogs

TL;DR

This paper introduces a systematic method to construct canonical differential equations for complex Feynman integrals involving elliptic curves and Calabi-Yau varieties, extending beyond multiple polylogarithms, with applications in physics calculations.

Contribution

It presents a new approach to derive canonical differential equations for integrals beyond polylogarithms, including elliptic and Calabi-Yau geometries.

Findings

Method applied to the two-loop sunrise integral

Simplifies the epsilon-expansion of complex integrals

Enhances analytic understanding of non-polylogarithmic integrals

Abstract

Feynman integrals whose associated geometries extend beyond the Riemann sphere, such as elliptic curves and Calabi-Yau varieties, are increasingly relevant in modern precision calculations. They arise not only in collider cross-section calculations, but also in the post-Minkowskian expansion of gravitational-wave scattering. A powerful approach to compute integrals of this type is via differential equations, particularly when cast in a canonical form, which simplifies their $\varepsilon$-expansion and makes analytic properties manifest. In these proceedings, we will present a method to systematically construct canonical differential equations even for integrals that evaluate beyond multiple polylogarithms. The discussion is kept as light as possible, focusing on the two-loop sunrise integral, deferring the technical details to the original publications.