Unitarization of the Sommerfeld enhancement through the renormalization group

TL;DR

The paper analyzes how Sommerfeld enhancement can violate partial-wave unitarity when a long-range force generates shallow bound states or near-threshold resonances. It identifies secular terms in the perturbative expansion as the origin and implements a renormalization-group improvement to resum these terms, yielding a unitarity-consistent amplitude with bound-state poles acquiring decay widths. The main result is an RG-improved amplitude with a shifted Jost function, $\mathscr{J}^{\rm imp}_\ell(p) = \mathscr{J}_\ell(p) - \frac{2ip f^S_\ell(p)}{\mathscr{J}_\ell(-p)-\mathscr{J}_\ell(p)}$, and a corresponding unitarized Sommerfeld factor $S^{\rm imp}_\ell(p)=1/|\mathscr{J}^{\rm imp}_\ell(p)|^2$, demonstrated in a spherical-well test case. The work bridges nonperturbative resummation with Wilsonian RG intuition, clarifies the role of bound-state widths in restoring unitarity, and sets the stage for applying the framework to bound-state formation and other higher-order annihilation processes in dark-matter phenomenology.

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.

Figures (6)

• Figure 1: Analytic structure of the scattering amplitude on the complex $p$ and $E = p^2/2\mu$ plane. The left panel shows the complex $p$-plane, while the middle and right panels represent the physical and unphysical sheets of the energy Riemann surface, respectively. The two sheets are connected across the branch cut, depicted by the wavy lines along the positive real axis. Sample paths illustrating analytic continuation around the branch point, as well as the corresponding trajectories in the $p$-plane and possible pole singularities, are also indicated. See the main text for details.
• Figure 2: Sommerfeld factors as functions of momentum for the $s$-wave (left panel) and $p$-wave (right panel) with the spherical well potential. The dashed lines correspond to the conventional calculation (Con), while the solid lines represent the RG-improved results (RG). The red line indicates the upper bound imposed by unitarity. The mass parameter and the coupling constant are take to be $\mu R = 10$ and $\alpha_D = 10^{-2}$, respectively. See the main text for details.
• Figure 3: Self-scattering cross sections as functions of momentum for the $s$-wave (left panel) and $p$-wave (right panel) with the spherical well potential. The dashed lines correspond to the conventional calculation (Con), while the solid lines represent the RG-improved results (RG). The red line shows the unitarity bound for the self-scattering cross section. The mass parameter and the coupling constant are take to be $\mu R = 10$ and $\alpha_D = 10^{-2}$, respectively. See the main text for details.
• Figure 4: Diagrams for scattering processes contributing to self-scattering, induced from the UV theory. The left diagram shows the contribution from a direct decay process, while the right diagram depicts a contribution from a vertex mediating a transition to a bound state via the emission of a light boson. Solid lines denote the scattering particles, dashed lines their daughter particles, and double lines the bound states formed by the scattering particles. The dotted vertical lines represent Cutkosky cuts, indicating that these diagrams have nonzero imaginary parts.
• Figure 5: Comparison of the unitarized s-wave Sommerfeld factor computed with the RG method (RG; solid) and by solving the Schrödinger equation (Scr; dashed). The left panel uses well depths $p_VR = 1,\,10^{-1},\,10^{-2}$; the right panel uses $p_VR = \pi/2-10^{-3},\,\pi/2-10^{-5},\,3\pi/2$. The mass parameter and coupling are fixed to $\mu R = 10$ and $\alpha_D = 10^{-2}$, respectively.