Gravitational waveforms: A tale of two formalisms

We revisit the quantum-amplitude-based derivation of the gravitational waveform emitted by the scattering of two spinless massive bodies at the third order in Newton's constant, $h\sim G+G^2+G^3$ (one-loop level), and correspondingly update its comparison with its classically derived multipolar-post-Minkowskian counterpart. A spurious-pole-free reorganization of the one-loop five-point amplitude substantially simplifies the post-Newtonian expansion. We find complete agreement between the two results up to the fifth order in the small velocity expansion after taking into account three subtle aspects of the amplitude derivation: (1) in agreement with [A. Georgoudis et al., J. High Energy Phys. 03 (2024) 089], the term quadratic in the amplitude in the observable-based formalism [D. A. Kosower et al., J. High Energy Phys. 02 (2019) 137] generates a frame rotation by half the classical scattering angle; (2) the dimensional regularization of the infrared divergences of the amplitude introduces an additional $(d-4)/(d-4)$ finite term; and (3) zero-frequency gravitons are found to contribute additional terms both at order $h\sim G^1$ and at order $h\sim G^3$ when including disconnected diagrams in the observable-based formalism.