Scattering Amplitudes and Conservative Binary Dynamics at ${\cal O}(G^4)$

TL;DR

Using scattering amplitudes, the potential contributions to conservative binary dynamics in general relativity at fourth post-Minkowskian order O(G^{4}) are obtained and the radial action directly from the amplitude is derived, and the corresponding Hamiltonian in isotropic gauge is determined.

Abstract

Using scattering amplitudes, we obtain the potential contributions to conservative binary dynamics in general relativity at fourth post-Minkowskian order, ${\cal O}(G^4)$. As in previous lower-order calculations, we harness powerful tools from the modern scattering amplitudes program including generalized unitarity, the double copy, and advanced multiloop integration methods, in combination with effective field theory. The classical amplitude involves polylogarithms with up to transcendental weight two and elliptic integrals. We derive the radial action directly from the amplitude, and determine the corresponding Hamiltonian in isotropic gauge. Our results are in agreement with known overlapping terms up to sixth post-Newtonian order, and with the probe limit. We also determine the post-Minkowskian energy loss from radiation emission at ${\cal O}(G^3)$ via its relation to the tail effect.

Figures (2)

• Figure 1: Generalized unitarity cuts encoding potential-region contributions to binary dynamics. Ovals represent tree amplitudes while exposed lines depict on-shell states. Thin and thick lines denote gravitons and massive scalars, respectively.
• Figure 2: Sample diagrams at ${\cal O}(G^4)$. From left to right: a contribution in the probe limit, a nonplanar diagram that contains iteration terms, and a diagram that contains contributions related to the tail effect.