Conservative Black Hole Scattering at Fifth Post-Minkowskian and Second Self-Force Order

Abstract

Using the worldline quantum field theory formalism, we compute conservative contributions to the scattering angle and impulse for classical black hole scattering at fifth post-Minkowskian (5PM) and second self-force (2SF) order. This four-loop calculation involves non-planar Feynman integrals and requires advanced integration-by-parts reduction, novel differential-equation strategies, and efficient boundary-integral algorithms to solve a system of hundreds of master integrals in four integral families on high-performance computing systems. The resulting function space includes multiple polylogarithms as well as iterated integrals with a K3 period, which generate a spurious velocity divergence at $v/c=\sqrt{8}/3$, $\gamma =3$. This divergence is present in the potential region and must be canceled by contributions from the radiative memory region, while its dimensional-regularisation pole should cancel against the radiative tail region. As the standard use of Feynman propagators fails to ensure this cancellation, we instead propose a ($\gamma$-3) conservative prescription that realises both cancellations, leading to a physically sensible answer. All available low-velocity checks of our result against the post-Newtonian literature are satisfied.

Comments: 17 pages, 5 figures, 3 tables, accompanying Zenodo submission of ancillary files, see https://zenodo.org/records/18340334 ; v2: typos corrected & reference added; v3: minor correction & reference added; v4: version to be published in PRL