On canonical differential equations for Calabi-Yau multi-scale Feynman integrals

TL;DR

This work extends the canonical differential equations framework to multi-scale Feynman integrals with Calabi–Yau geometry. By leveraging Picard–Fuchs operators, mixed Hodge structures, and mirror-map data, the authors construct canonical bases that yield $ε$-factorised differential equations with connections having at most simple poles, validated on the unequal-mass sunrise and new multi-loop banana integrals (three- and four-loop) with two unequal masses. They present two complementary elliptic/CY-based strategies (democratic and decoupling) and show gauge-equivalence between approaches and with prior elliptic-form results, enabling systematic treatment of higher-dimensional Calabi–Yau sectors. The results provide practical tools for evaluating complex multiscale FIs and deepen the link between Feynman integrals and Calabi–Yau geometry, with potential extensions to higher-genus and more intricate graphs.

Abstract

We generalise a method recently introduced in the literature, that derives canonical differential equations, to multi-scale Feynman integrals with an underlying Calabi-Yau geometry. We start by recomputing a canonical form for the sunrise integral with all unequal masses. Additionally, we compute for the first time a canonical form for the three-loop banana integral with two unequal masses and for a four-loop banana integral with two unequal masses. For the integrals we compute, we find an $ε$-form whose connection has at most simple poles. We motivate our construction by studying the Picard-Fuchs operators acting on the integrals considered. In the appendices, we give a constructive explanation for why our generalisation works.

Figures (1)

• Figure 1: The $l$-loop banana graph.