Landau singularity of the K3 maximal cut in black-hole scattering at fifth post-Minkowskian order

Abstract

Modeling the gravitational waves from merging black holes requires precise calculations of the two-body problem. The post-Minkowskian expansion of black-hole scattering, in powers of Newton's constant, now stands at fifth order, four loops. At this order one part of the deflection diverges when the black holes pass at about 94 percent of the speed of light. The divergence is known to come from periods of a K3 surface, which has a node at this speed. We show that the node lies on the integration contour of the K3 maximal cut, which makes this speed a Landau singularity of the cut. Requiring the divergence to cancel, without introducing new infinities, has been shown to fix the memory contribution, named for the permanent displacement that the waves leave in test masses, up to one constant. We derive the leading coefficients of its boundary integrals in closed form. We obtain both results from maximal cuts and boundary limits, without an integration-by-parts reduction of the full four-loop integrand. We also continue the K3 and Calabi--Yau periods of these integrals to the energies of bound orbits, as in the inspiral before a merger.