Spin- and velocity-dependent non-relativistic potentials in modified electrodynamics

We investigate the interparticle potential between spin-0, -1/2 and -1 sources interacting in modified electrodynamics in the non-relativistic regime. By keeping terms of $\mathcal{O}( |{\bf p}|^2/m^2 )$ in the amplitudes, we obtain spin- and velocity-dependent interaction energies. We find well-known effects such as spin-orbit couplings, as well as spin-spin (dipole-dipole) interactions. For concreteness, we consider the cases of electrodynamics with higher derivatives (Podolsky-Lee-Wick) and hidden photons.