Gravitational Raman Scattering in Effective Field Theory: A Scalar Tidal Matching at $\mathcal{O}(G^3)$

We present a framework to compute amplitudes for the gravitational analog of the Raman process, a quasielastic scattering of waves off compact objects, in worldline effective field theory. As an example, we calculate third post-Minkowskian 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 effective field theory 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 post-Minkowskian orders.