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

TL;DR

This work addresses the breakdown of naive Sommerfeld enhancement near the unitarity bound by introducing a scale-separated, unitarity-preserving prescription that treats short-range annihilation physics as a boundary condition at a matching radius $a$ external to the long-range potential. The authors formulate a general framework for both single- and multi-state two-body systems across arbitrary partial waves, deriving a compact S-matrix expression and cross sections that factor in short-distance amplitudes and long-range dynamics through matrix quantities like $\Sigma_{0,\ell}$, $\kappa_\ell$, and $\bar{f}_{s,\ell}$. They show how to extract these quantities by solving the long-range Schrödinger equation (via the variable phase method) and performing perturbative matching to perturbative QFT input for annihilation, ensuring unitarity and applicability to realistic DM scenarios such as wino dark matter. The paper provides explicit examples (spherical well, Coulomb, finite-range Coulomb) and a detailed wino DM case study with LO/NLO potentials, demonstrating resonance regulation, $a$-independence of final results, and compatibility with existing unitarized approaches. Overall, the framework offers a practical, general method to compute regulated Sommerfeld-enhanced cross sections for complex DM sectors, with clear prescriptions for incorporating short-range physics and multi-state couplings into the S-matrix formalism.

Abstract

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.

Figures (19)

• Figure 1:
• Figure 2: Left:$\tilde{z}_\ell(p;0) R^{2\ell+1}$ for the finite square-well potential for $\ell=0$ ( upper), $\ell = 1$ ( middle), and $\ell = 2$ ( lower). Right:$\tilde{z}_\ell(p;0) / p^{2\ell+1} C_\ell^2$ for the finite square-well potential for $\ell=0$ ( upper), $\ell = 1$ ( middle), and $\ell = 2$ ( lower). Solid (dashed) curves indicate positive (negative) $\tilde{z}_\ell(p;0)$.
• Figure 3: $|\tilde{z}_\ell(0;0) R^{2\ell+1}|$ for the finite square-well potential with $\ell = 0$ ( left), $\ell = 1$ ( middle), and $\ell = 2$ ( right). Solid (dashed) curves indicate positive (negative) $\tilde{z}_\ell(0;0)$.
• Figure 4: The ratio between the annihilation cross sections with and without the Sommerfeld effect (i.e. the Sommerfeld enhancement factor) for the finite square-well potential, for the $\ell=0$ ( left), $\ell=1$ ( middle) and $\ell=2$ ( right) cases. The black solid curves shows the results computed using the conventional formula in Eq. \ref{['eq:conventional SE formula']}, the red solid curves correspond to the full corrected result given in Eq. \ref{['eq:onestateannfinal']}, and the blue dashed curves correspond to the simplified formula of Eqs. \ref{['eq:sigma simplified swave']}, \ref{['eq:sigma simplified higherell']}. We take $k(p_0) = (10^3 + 10^3 i)R^{-1}$ for $s$-wave, $(10^6 + 10^6 i)R^{-3}$ for $p$-wave, and $(10^9 + 10^9 i)R^{-5}$ for $d$-wave, and choose $p = 10^{-2} R^{-1}$ in all cases.
• Figure 5: $\tilde{z}_\ell a_B^{2\ell+1}$ for the Coulomb potential with $\ell = 0$ ( left), $\ell = 1$ ( middle), and $\ell = 2$ ( right). $a$ is taken to be $10^{-5} a_B$ (solid curves), $10^{-10} a_B$ (dashed curves), and $10^{-20} a_B$ (dot-dashed curves).