The Sommerfeld enhancement at NLO and the dark matter unitarity bound

We reexamine the consequences of perturbative unitarity on dark matter freeze-out when both Sommerfeld enhancement and bound state formation affect dark matter annihilations. At leading order (LO) the annihilation cross-section is infrared dominated and the connection between the unitarity bound and the upper bound on the dark matter mass depends only on how the different partial waves are populated. We compute how this picture is modified at next-to-leading order (NLO) with the goal of assigning a reliable theory uncertainty to the freeze-out predictions. We explicitly compute NLO corrections in a simple model with abelian gauge interactions and provide an estimate of the theoretical uncertainty for the thermal masses of heavy electroweak $n$-plets. Along the way, we clarify the regularization and matching procedure necessary to deal with singular potentials in quantum mechanics with a calculable, relativistic UV completion.