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

TL;DR

This work delivers a comprehensive geometry-based classification of Feynman integrals governing 2→2 black-hole scattering at 5PM (four loops) across all SF orders. By applying loop-by-loop Baikov leading singularity analysis and leveraging relations among topologies, the authors reduce the problem from 16,596 potentially contributing topologies to 70 essential ones, then identify the underlying non-polylogarithmic structures. They find two three-dimensional Calabi–Yau geometries and two K3 surfaces—manifesting in eight independent topologies (topologies 3,37 for CY3 and 4,5,6,38,39,40 for K3)—with parity determining whether a given topology contributes polylogarithmically or through these richer geometries. This classification informs the space of special functions needed for 5PM observables and provides a pathway to efficient, geometry-driven calculations for high-precision gravitational-wave predictions. The results align with recent 5PM1SF findings and set the stage for completing 5PM2SF computations using ε-factorized differential equations and geometry-based bootstrap methods.

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.

Figures (9)

• Figure 1: Independent four-loop PM Feynman integral topologies which contain non-trivial geometries. In the first row (1SF order), the first topology depends on a three-dimensional Calabi--Yau geometry, while the remaining ones depend on the same K3 surface that appeared at three loops Ruf:2021egkDlapa:2022wduFrellesvig:2024zph. In the second row (2SF order), the first topology depends on a different three-dimensional Calabi--Yau geometry, the next two involve the same K3 surface as in 1SF order, and the last one involves a different K3 surface.
• Figure 2: Kinematics for the scattering process, exemplified by the tree-level exchange. The massive scalars are depicted by thick lines, while the graviton propagator is denoted by a thin line.
• Figure 3: Subgraphs for which the corresponding Feynman integral is reducible to subsectors: (a) One-loop bubble with at least one vertex being cubic; (b) One-loop triangle with at least one cubic vertex at a matter line and cubic graviton self-interaction; (c) Two-loop dangling triangle with cubic matter vertices and a quartic graviton self-interaction; (d) Three-loop dangling triangle with cubic matter vertices and a quintic graviton self-interaction. These drawings should be conceived as being embedded in a bigger graph, which is indicated by the grey zone.
• Figure 4: Parametrization of the loop momenta for the 1SF integral topology $3$ of table \ref{['tab: 4-loop results 0SF and 1SF']}, which depends on a three-dimensional CY geometry in the odd-parity sector.
• Figure 5: Parametrization of the loop momenta for the 2SF integral topology $37$ of table \ref{['tab: 4-loop results 2SF']}, which depends on a three-dimensional CY geometry in the even-parity sector.