# The Sommerfeld enhancement for dark matter with an excited state

## TL;DR

This paper analyzes the Sommerfeld enhancement in dark matter annihilation involving an excited state, providing a semi-analytic approximation and exploring how excited states affect the enhancement's magnitude and resonance behavior.

## Contribution

It introduces a simple semi-analytic approximation for the s-wave Sommerfeld enhancement in models with excited dark matter states, extending previous results and analyzing the effects of mass splitting.

## Key findings

- Excited states generally increase the Sommerfeld enhancement.
- Resonances are larger and shifted to lower mediator masses.
- Enhancement can be up to twice as large near excitation energy velocities.

## Abstract

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.