Radiative classical gravitational observables at $$ \mathcal{O} $$(G3) from scattering amplitudes

Abstract

We compute classical gravitational observables for the scattering of two spinless black holes in general relativity and $$ \mathcal{N} $$ N =8 supergravity in the formalism of Kosower, Maybee, and O’Connell (KMOC). We focus on the gravitational impulse with radiation reaction and the radiated momentum in black hole scattering at $$ \mathcal{O} $$ O (G3) to all orders in the velocity. These classical observables require the construction and evaluation of certain loop-level quantities which are greatly simplified by harnessing recent advances from scattering amplitudes and collider physics. In particular, we make use of generalized unitarity to construct the relevant loop integrands, employ reverse unitarity, the method of regions, integration-by-parts (IBP), and (canonical) differential equations to simplify and evaluate all loop and phase-space integrals to obtain the classical gravitational observables of interest to two-loop order. The KMOC formalism naturally incorporates radiation effects which enables us to explore these classical quantities beyond the conservative two-body dynamics. From the impulse and the radiated momentum, we extract the scattering angle and the radiated energy. Finally, we discuss universality of the impulse in the high-energy limit and the relation to the eikonal phase.

Radiated momentum in the post-Minkowskian worldline approach via reverse unitarity

“…This result agrees with the one recently derived via other methods [75,76,80], and complete the leading-order radiated sector derived with an EFT worldline approch [1]. From eq.…”

“…The advantage of this procedure is that we can now solve the 2-loop integral all at once, making use of the powerful computational tools routinely employed in high-energy physics -IBP reduction into master integrals [82][83][84] and differential equation methods [85][86][87][88] to solve the latter -without the need of deriving the Fourier-space gravitational waveform. This is analogous to the calculations recently performed in [75,76,80]. However, here we do not have to consider any intermediate quantum or super-classical contributions in our integrals: the amplitude A λ is a classical observable from the start.…”

“…Finally, the same radiated energy also appears as a tail effect in the 4PM Hamiltonian computed in [45]. We refer to [75,76,104] for a more thorough discussion.…”

“…Scattering bodies are accompanied by emission of gravitational Bremsstrahlung radiation [17,[59][60][61][62][63][64], which is the unbound analog of the gravitational waves emitted by inspiral binaries and is suppressed by three powers of G. Although radiation reaction effects were thoroughly investigated in the past in the Regge limit (i.e., when the centerof-mass energy is much larger than the momentum transfer) [65][66][67][68] or in association to the loss of angular momentum in the collision [69][70][71][72][73][74], the full leading-order emitted momentum has been obtained only very recently in [75,76] via the formalism of [77], which derives classical observables from quantum scattering (see also [78,79] for extensions of the formalism of [77] to spin and classical observables in Yang-Mills theories), and in [80] using the eikonal approach to classical gravitational scattering. These calculations require evaluating the classical limit of relevant two-loop Feynman integrals, that can be solved by combining different techniques borrowed from particle physics, as shown in [81], namely reduction to master integrals by Integration-by-Parts (IBP) identities [82][83][84] and differential equations [85][86][87][88] to solve the latter, using the near-static regime as initial conditions.…”

Extracting Einstein from the loop-level double-copy

“…There are an increasing number of classical (and semi-classical) field theory solutions, both perturbative and non-perturbative, guided by quantum color-dual double-copy, 2 including quantum double-copies against a classically double-copied background [41,42]. Critically, amplitude calculation of novel precision observables relevant to gravitational-wave physics have been achieved using both EFT and direct approaches [43][44][45][46][47][48][49][50]. Indeed, the highest precision post-Minkowskian O(G 4 ) correction to the scattering of classical non-rotating black holes has involved a synthesis of effective field theory techniques, advanced multiloop integration innovation, and double-copy applied to tree-level scattering amplitudes [47].…”