Unitarization of the Sommerfeld enhancement through the renormalization group

TL;DR

This paper develops a renormalization group method to improve the Sommerfeld enhancement calculation, ensuring unitarity and accurately accounting for bound state decay widths in dark matter annihilation.

Contribution

It introduces a renormalization group approach to address unitarity violations in Sommerfeld enhancement calculations, incorporating decay widths of bound states.

Findings

The improved amplitude maintains unitarity across resonances.

Poles in the enhancement acquire an imaginary part indicating decay widths.

The method can be extended to higher-order processes like bound-state formation.

Abstract

When a pair of dark matter particles interacts via a long-range force mediated by a light particle, their nonrelativistic annihilation cross section can be significantly enhanced - a phenomenon known as the Sommerfeld enhancement. This enhancement exhibits resonant behavior if the long-range potential supports shallow bound states or narrow resonances, which can lead to violations of the partial-wave unitarity bound. We identify the origin of this pathological behavior as the emergence of secular terms in perturbative expansions associated with low-energy composite states of the long-range potential. To address this issue, we propose a renormalization group improvement of the perturbative series. The resulting improved amplitude provides a unitarity-consistent form of the Sommerfeld enhancement, with its poles acquiring an imaginary part that reflects the decay width of the annihilating bound states. We also briefly discuss the implications of our approach from the perspective of Wilsonian renormalization group, and comment on its potential application to higher-order annihilation processes such as bound-state formation.