  We apply the Hilbert series to extend the gravitational action for a scalar
field to a complete, non-redundant basis of higher-dimensional operators that
is quadratic in the scalars and the Weyl tensor. Such an extension of the
action fully describes tidal effects arising from operators involving two
powers of the curvature. As an application of this new action, we compute all
spinless tidal effects at the leading post-Minkowskian order. This computation
is greatly simplified by appealing to the heavy limit, where only a severely
constrained set of operators can contribute classically at the one-loop level.
Finally, we use this amplitude to derive the $\mathcal{O}(G^{2})$ tidal
corrections to the Hamiltonian and the scattering angle.
