  For non-relativistic thermal dark matter, close-to-threshold effects largely
dominate the evolution of the number density for most of the times after
thermal freeze-out, and hence affect the cosmological relic density. A precise
evaluation of the relevant interaction rates in a thermal medium representing
the early universe includes accounting for the relative motion of the dark
matter particles and the thermal medium. We consider a model of dark fermions
interacting with a plasma of dark gauge bosons, which is equivalent to thermal
QED. The temperature is taken to be smaller than the dark fermion mass and the
inverse of the typical size of the dark fermion-antifermion bound states, which
allows for the use of non-relativistic effective field theories. For the
annihilation cross section, bound-state formation cross section, bound-state
dissociation width and bound-state transition width of dark matter
fermion-antifermion pairs, we compute the leading recoil effects in the
reference frame of both the plasma and the center-of-mass of the
fermion-antifermion pair. We explicitly verify the Lorentz transformations
among these quantities. We evaluate the impact of the recoil corrections on the
dark matter energy density. Our results can be directly applied to account for
the relative motion of quarkonia in the quark-gluon plasma formed in heavy-ion
collisions. They may be also used to precisely assess thermal effects in atomic
clocks based on atomic transitions; the present work provides a first field
theory derivation of time dilation for these processes in vacuum and in a
medium.
