The Sommerfeld enhancement for dark matter with an excited state
  We present an analysis of the Sommerfeld enhancement to dark matter
annihilation in the presence of an excited state, where the interaction
inducing the enhancement is purely off-diagonal, such as in models of exciting
or inelastic dark matter. We derive a simple and accurate semi-analytic
approximation for the s-wave enhancement, which is valid provided the mass
splitting between the ground and excited states is not too large, and discuss
the cutoff of the enhancement for large mass splittings. We reproduce
previously derived results in the appropriate limits, and demonstrate excellent
agreement with numerical calculations of the enhancement. We show that the
presence of an excited state leads to generically larger values of the
Sommerfeld enhancement, larger resonances, and shifting of the resonances to
lower mediator masses. Furthermore, in the presence of a mass splitting the
enhancement is no longer a monotonic function of velocity: the enhancement
where the kinetic energy is close to that required to excite the higher state
can be up to twice as large as the enhancement at zero velocity.
