Classification of Feynman integral geometries for black-hole scattering at 5PM order

TL;DR

This paper classifies the geometries of Feynman integrals relevant to black-hole scattering at 5PM order, revealing a limited set of integral topologies and associated Calabi-Yau and K3 geometries that extend beyond polylogarithmic functions.

Contribution

It provides a complete classification of integral geometries at 5PM order, identifying only 70 key topologies and their underlying Calabi-Yau and K3 geometries, advancing understanding of complex Feynman integrals.

Findings

Only 70 integral topologies out of 16,596 are relevant.

Two Calabi-Yau geometries and two K3 surfaces characterize the integrals.

Integrals evaluate to functions beyond polylogarithms.

Abstract

We provide a complete classification of the Feynman integral geometries relevant to the scattering of two black holes at fifth order in the post-Minkowskian (PM) expansion, i.e. at four loops. The analysis includes integrals relevant to both the conservative and dissipative dynamics, as well as to all orders in the self-force (SF) expansion, i.e. the 0SF, 1SF and 2SF orders. By relating the geometries of integrals across different loop orders and integral families, we find that out of the 16,596 potentially contributing integral topologies, only 70 need to be analyzed in detail. By further computing their leading singularities using the loop-by-loop Baikov representation, we show that there only appear two different three-dimensional Calabi-Yau geometries and two different K3 surfaces at this loop order, which together characterize the space of functions beyond polylogarithms to which the 5PM integrals evaluate.