  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
$\lambda = 0.19 \pm 0.02$.
