  Tools from scattering amplitudes and effective field theory have recently
been repurposed to derive state-of-the-art results for the black hole binary
inspiral in the post-Minkowskian expansion. In the present work we extend this
approach to include the tidal effects of mass and current quadrupoles on the
conservative dynamics of non-spinning neutron star mergers. We compute the
leading and, for the first time, next-to-leading order post-Minkowskian finite
size corrections to the conservative Hamiltonian, together with their
associated scattering amplitudes and scattering angles. Our expressions are
gauge invariant and, in the extreme mass ratio limit, consistent with the
dynamics of a tidally deformed test body in a Schwarzschild background.
Furthermore, they agree completely with existing results at leading
post-Minkowskian and second post-Newtonian orders.
