Calabi-Yau Feynman integrals in gravity: $\varepsilon$-factorized form for apparent singularities

TL;DR

This paper analyzes a complex four-loop Feynman integral with Calabi-Yau geometry relevant to black hole scattering in gravity, introducing an $ ext{ε}$-factorized form to handle apparent singularities in differential equations.

Contribution

It presents a novel method to achieve $ ext{ε}$-factorization for Feynman integrals with apparent singularities, demonstrated on a four-loop integral with Calabi-Yau geometry.

Findings

Successfully brought the integral into $ ext{ε}$-factorized form.

Handled $ ext{ε}$-dependent apparent singularities in differential equations.

Provided a proof of principle for the method.

Abstract

We study a recently identified four-loop Feynman integral that contains a three-dimensional Calabi-Yau geometry and contributes to the scattering of black holes in classical gravity at fifth post-Minkowskian and second self-force order (5PM 2SF) in the conservative sector. In contrast to previously studied Calabi-Yau Feynman integrals, the higher-order differential equation that this integral satisfies in dimensional regularization exhibits $\varepsilon$-dependent apparent singularities. We introduce an appropriate ansatz which allows us to bring such cases into an $\varepsilon$-factorized form. As a proof of principle, we apply it to the integral at hand.