  We present the two-body Hamiltonian and associated eikonal phase, to leading
post-Minkowskian order, for infinitely many tidal deformations described by
operators with arbitrary powers of the curvature tensor. Scattering amplitudes
in momentum and position space provide systematic complementary approaches. For
the tidal operators quadratic in curvature, which describe the linear response
to an external gravitational field, we work out the leading post-Minkowskian
contributions using a basis of operators with arbitrary numbers of derivatives
which are in one-to-one correspondence with the worldline multipole operators.
Explicit examples are used to show that the same techniques apply to both
bodies interacting tidally with a spinning particle, for which we find the
leading contributions from quadratic in curvature tidal operators with an
arbitrary number of derivatives, and to effective field theory extensions of
general relativity. We also note that the leading post-Minkowskian order
contributions from higher-dimension operators manifest double-copy relations.
Finally, we comment on the structure of higher-order corrections.
