We present the amplitude for classical scattering of gravitationally interacting massive scalars at third post-Minkowskian order. Our approach harnesses powerful tools from the modern amplitudes program such as generalized unitarity and the double-copy construction, which relates gravity integrands to simpler gauge-theory expressions. Adapting methods for integration and matching from effective field theory, we extract the conservative Hamiltonian for compact spinless binaries at third post-Minkowskian order. The resulting Hamiltonian is in complete agreement with corresponding terms in state-of-the-art expressions at fourth post-Newtonian order as well as the probe limit at all orders in velocity. We also derive the scattering angle at third post-Minkowskian order.Introduction. The recent discovery of gravitational waves at LIGO/Virgo [1] has launched an extraordinary new era in multi-messenger astronomy. Given expected improvements in detector sensitivity, high-precision theoretical predictions from general relativity will be crucial. Existing theory benchmarks come from a variety of approaches (see also Ref. [2] and references therein), including the effective one-body formalism [3], numerical relativity [4], the self-force formalism [5], and perturbative analysis using post-Newtonian (PN) [6-9], post-Minkowskian (PM) [10][11][12], and effective field theory (EFT) [13] methods.

We describe a systematic framework for computing the conservative potential of a compact binary system using modern tools from scattering amplitudes and effective field theory. Our approach combines methods for integration and matching adapted from effective field theory, generalized unitarity, and the double-copy construction, which relates gravity integrands to simpler gauge-theory expressions. With these methods we derive the third post-Minkowskian correction to the conservative two-body Hamiltonian for spinless black holes. We describe in some detail various checks of our integration methods and the resulting Hamiltonian.

We combine tools from effective field theory and generalized unitarity to construct a map between on-shell scattering amplitudes and the classical potential for interacting spinless particles. For general relativity, we obtain analytic expressions for the classical potential of a binary black hole system at second order in the gravitational constant and all orders in velocity. Our results exactly match all known results up to fourth post-Newtonian order, and offer a simple check of future higher order calculations. By design, these methods should extend to higher orders in perturbation theory.

Models of Weakly Interacting Massive Particles (WIMPs) specified at the electroweak scale are systematically matched to effective theories at hadronic scales where WIMP-nucleus scattering observables are evaluated. Anomalous dimensions and heavy quark threshold matching conditions are computed for the complete basis of lowest-dimension effective operators involving quarks and gluons. The resulting QCD renormalization group evolution equations are solved. The status of relevant hadronic matrix elements is reviewed and phenomenological illustrations are given, including details for the computation of the universal limit of nucleon scattering with heavy SU (2) W × U (1) Y charged WIMPs. Several cases of previously underestimated hadronic uncertainties are isolated. The results connect arbitrary models specified at the electroweak scale to a basis of n f = 3 flavor QCD operators. The complete basis of operators and Lorentz invariance constraints through order v 2 /c 2 in the nonrelativistic nucleon effective theory are derived.