On the equivalence of unitarization prescriptions for the Sommerfeld enhancement

TL;DR

This paper compares different methods of regulating the Sommerfeld enhancement in dark matter annihilation, demonstrating their consistency and proposing a regulator-independent unitarization approach for multi-state systems.

Contribution

It provides a detailed comparison of unitarization prescriptions, showing their equivalence and introducing a regulator-independent method for multi-state systems.

Findings

Regulated Sommerfeld cross sections are largely independent of UV regulators.

Different regulation schemes agree at leading order when unitarity corrections are large.

A new, regulator-independent unitarization prescription is proposed for multi-state systems.

Abstract

The annihilation of self-interacting dark matter with long-range interactions can be significantly enhanced at low velocities through the Sommerfeld effect. At special points in parameter space, where near-zero-energy resonances exist in the spectrum of the theory, the standard calculation of this enhancement appears to violate unitarity. Recently, several approaches have been proposed to regulate this behavior, some introducing explicit ultraviolet (UV) scales and others not, raising the question of whether these prescriptions are consistent. In this paper, we compare these approaches and show that even in nominally cutoff-dependent methods, the regulated Sommerfeld-enhanced cross sections are independent of the UV regulator to a good approximation, and that when the unitarity-preserving corrections are large, the regulation schemes coincide to leading order. We use these insights to write down a regulator-independent prescription for unitarization applicable to multi-state systems, where the modified enhancement can be written solely in terms of the standard enhancement factor, the hard annihilation amplitude, and the $S$-matrix for scattering in the long-range potential.