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Radiative Classical Gravitational Observables at $\mathcal O(G^3)$ from Scattering Amplitudes

Enrico Herrmann, Julio Parra-Martinez, Michael S. Ruf, Mao Zeng

Abstract

We compute classical gravitational observables for the scattering of two spinless black holes in general relativity and $\mathcal N {=} 8$ 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 $\mathcal O(G^3)$ 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.

Radiative Classical Gravitational Observables at $\mathcal O(G^3)$ from Scattering Amplitudes

Abstract

We compute classical gravitational observables for the scattering of two spinless black holes in general relativity and 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 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.

Paper Structure

This paper contains 43 sections, 181 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Gravitational observables via the KMOC formalism
  3. Gravitational Impulse
  4. Radiated momentum
  5. Setup and review
  6. The method of regions and the classical limit
  7. Loop integrands and generalized unitarity
  8. Soft expansion, integration-by-parts and differential equations
  9. Full soft integrands and reverse unitarity
  10. New contributions from the soft region
  11. Soft expansion and partial fractioning
  12. Reverse unitarity
  13. Evaluation of soft master integrals
  14. Soft one-loop integrals: Euclidean region and analytic continuation
  15. Virtual two-loop integrals
  16. ...and 28 more sections

Figures (11)

  • Figure 1: Cubic diagrams relevant for the potential-region amplitude in classical GR.
  • Figure 2: Spanning set of unitarity cuts relevant for the potential region $\mathcal{O}(G^3)$.
  • Figure 3: Depiction of the parameterization of external momenta useful for the soft expansion.
  • Figure 4: Sample diagrams that do not contribute in the classical limit. (a) and (b) are purely quantum from simple $|q|$ counting arguments, whereas (c) is naively classical but scaleless in the soft region.
  • Figure 5: Cubic diagrams relevant for the classical $\mathcal{O}(G^3)$ impulse and radiated momentum in general relativity (including radiative contributions).
  • ...and 6 more figures