Amplitudes, Supersymmetric Black Hole Scattering at $\mathcal{O}(G^5)$, and Loop Integration

TL;DR

This paper calculates the classical scattering amplitude and related quantities for extremal black holes in N=8 supergravity at the fifth post-Minkowskian order, revealing surprisingly simple results despite complex integrals involved.

Contribution

It provides a detailed computation of the five-loop order scattering amplitude in supergravity, including master integrals and differential equations, with results simplified to polylogarithmic functions.

Findings

Final supergravity amplitude expressed with polylogarithms up to weight 2.

Systematic analysis of elliptic integrals applicable beyond supersymmetry.

Methodology for reducing complex integrals to manageable forms in gravitational theories.

Abstract

We compute the potential-graviton contribution to the scattering amplitude, the radial action, and the scattering angle of two extremal black holes in N = 8 supergravity at the fifth post-Minkowskian order to next-to-leading order in a large mass expansion (first self-force order). Properties of classical unitarity cuts allow us to focus on the integration-by-parts reduction of planar integrals, while nonplanar integrals at this order are obtained from the planar ones by straightforward manipulations. We present all master integrals and solve their associated differential equations necessary to evaluate the classical scattering amplitudes of massive scalar particles at this order in all gravitational theories, in particular in N = 8 supergravity, and in general relativity. Despite the appearance of higher-weight generalized polylogarithms and elliptic functions in the solution to the differential equation for master integrals, the final supergravity answer is remarkably simple and contains only (harmonic) polylogarithmic functions up to weight 2. The systematic analysis of elliptic integrals discussed here, as well as the particular organization of boundary integrals in N = 8 observables are independent of supersymmetry and may have wider applications, including to aspects of collider physics.