Amplitudes, supersymmetric black hole scattering at O(G5), and loop integration

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 and 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 the solution to the
differential equations for all master integrals 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.