The observation of replica bands by angle-resolved photoemission spectroscopy has ignited interest in the study of electron-phonon coupling at low carrier densities, particularly in monolayer FeSe/SrTiO$_3$, where the appearance of replica bands has motivated theoretical work suggesting that the interfacial coupling of electrons in the FeSe layer to optical phonons in the SrTiO$_3$ substrate might contribute to the enhanced superconducting pairing temperature. Alternatively, it has also been recently proposed that such replica bands might instead originate from extrinsic final state losses associated with the photoemission process. Here, we perform a quantitative examination of replica bands in monolayer FeSe/SrTiO$_3$, where we are able to conclusively demonstrate that the replica bands are indeed signatures of intrinsic electron-boson coupling, and not associated with final state effects. A detailed analysis of the energy splittings between the higher-order replicas, as well as other self-energy effects, allow us to determine that the interfacial electron-phonon coupling in the system corresponds to a value of $lambda = 0.19 pm 0.02$.
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