Sommerfeld enhancement from unstable final-state particles in dark matter annihilation

We study the Sommerfeld enhancement of the annihilation cross section of dark matter into heavier unstable particles. In this process, the annihilation products become non-relativistic near the kinematical threshold. If they experience long-range interactions with each other, their wave function is distorted from a plane wave, and the annihilation cross section can be significantly enhanced. When evaluating the Sommerfeld enhancement from the long-range interactions between the annihilation products, the decay of the products needs to be taken into account. We treat this issue by including the decay width in the Schr\"odinger equations of the two-body wave function of the annihilation products. We find that bound states of the annihilation products with a narrow decay width enhance the annihilation cross section through a resonant effect. At the same time, this formulation automatically includes the annihilation process with off-shell final state particles, which is relevant for a wide decay width. We show that the resonant effect significantly affects the prediction of the dark matter relic abundance.