  If the annihilation products of dark matter (DM) are non-relativistic and if
there is some long-range force between them, there can be Sommerfeld effect for
the final state particles. We study this effect on DM relic abundance in the
thermal freeze-out scenario. As a proof of concept, we consider the case of a
DM pair annihilation into a final state pair, assuming that the mutual
interactions between the two final state particles give rise to a Coulomb-like
potential, and that the masses of the initial and final state particles are
similar, so that both the initial and final state particles are
non-relativistic. The size of the final state Sommerfeld (FSS) effect depends
on the strength of the potential, as well as on the mass ratio of the final and
initial state particles. We find that the impact of the FSS effect on DM relic
abundance can be significant, and an electroweak sized long-range interaction
is large enough to make a correction well beyond the observational accuracy.
Another feature of the FSS effect is that it could be suppressed when its time
scale is longer than the lifetime of the final state particles. As a corollary,
we also study in the DM coannihilation scenario where the initial state
Sommerfeld effect between two coannihilators could be reduced due to their
instability, which may need to be taken into account for an accurate
calculation of the DM relic abundance.
