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We complete the calculation of conservative two-body scattering dynamics atfourth post-Minkowskian order, i.e. $O(G^4)$ and all orders in velocity,including radiative contributions corresponding to the tail effect in generalrelativity. As in previous calculations, we harness powerful tools from themodern scattering amplitudes program including generalized unitarity, thedouble copy, and advanced multiloop integration methods, in combination witheffective field theory. The classical amplitude involves complete ellipticintegrals, and polylogarithms with up to transcendental weight two. Using theamplitude-action relation, we obtain the radial action directly from theamplitude, and match the known overlapping terms in the post-Newtonianexpansion.

Scattering Amplitudes, the Tail Effect, and Conservative Binary Dynamics at $O(G^4)$