Regulating Sommerfeld resonances for multi-state systems and higher partial waves

Long-range attractive interactions between dark matter particles can significantly enhance their annihilation, particularly at low velocities. This ``Sommerfeld enhancement'' is typically computed by evaluating the deformation of the two-particle wavefunction due to the long-range potential, while ignoring the physics associated with the annihilation, and then scaling the appropriate annihilation matrix elements by factors that depend on the wavefunction in the limit where the particles approach zero relative separation. It has long been recognized that this approach is a valid approximation only in the limit where the annihilation rate is small, and breaks down in the regime where the enhanced annihilation rate approaches the unitarity bound, in which case ignoring the impact of the annihilation physics on the two-particle wavefunction cannot be justified and leads to apparent violations of unitarity. In the case where the physics relevant to annihilation occurs at a parametrically shorter distance scale (higher energy scale) compared with the long-range potential, we provide a simple prescription for correcting the Sommerfeld enhancement for the effects of the short-range physics, valid for all partial waves and for systems where multiple states are coupled by the long-range potential.