Post-Newtonian Multipoles from the Next-to-Leading Post-Minkowskian Gravitational Waveform

We consider the frequency-domain LO and NLO post-Minkowskian (PM) waveforms obtained from the tree-level and one-loop amplitudes describing the scattering of two massive scalar objects and the emission of one graviton. We explicitly calculate their post-Newtonian (PN) limit obtaining an expansion up to the third subleading PN order in all ingredients: the tree-level amplitude, the odd and even parts of the real one-loop kernel, and the Compton or "rescattering" cuts, thus reaching 3PN precision for the latter. We provide explicit expressions for the multipole decomposition of these results in the center-of-mass frame and compare them with the results obtained from the classical Multipolar post-Minkowskian (MPM) method. We find perfect agreement between the two, once the BMS supertranslation frame is properly adjusted and the infrared divergences due to rescattering are suitably subtracted in dimensional regularization. This shows that the approach proposed in arXiv:2312.07452 can be applied beyond the soft-regime ensuring the agreement between amplitude-based and MPM results for generic frequencies.