  We present a framework to compute amplitudes for the gravitational analog of
the Raman process, a quasi-elastic scattering of waves off compact objects, in
worldline effective field theory (EFT). As an example, we calculate third
post-Minkowskian (PM) order ($\mathcal{O}(G^3)$), or two-loop, phase shifts for
the scattering of a massless scalar field including all tidal effects and
dissipation. Our calculation unveils two sources of the classical
renormalization-group flow of dynamical Love numbers: a universal running
independent of the nature of the compact object, and a running self-induced by
tides. Restricting to the black hole case, we find that our EFT phase shifts
agree exactly with those from general relativity, provided that the relevant
static Love numbers are set to zero. In addition, we carry out a complete
matching of the leading scalar dynamical Love number required to renormalize a
universal short scale divergence in the S-wave. Our results pave the way for
systematic calculations of gravitational Raman scattering at higher PM orders.
