Dark-pion dark matter beyond leading order: unitarized chiral dynamics

TL;DR

The paper addresses the inadequacy of leading-order ChPT in predicting dark-pion DM phenomenology by applying the chiral unitary method, which enforces unitarity and correct analytic structure using only LO input and a single subtraction constant. By fixing this constant at its natural value, the unitarized amplitudes depend only on $m_\pi$ and $f_\pi$, and dynamically generate resonance poles that modify both SIMP self-scattering and WIMP annihilation through initial-state interactions. The analysis reveals substantial deviations from LO predictions, reshaping viable parameter regions, and links large deviations in the subtraction constant to the possible presence of additional resonances (compositeness) in the dark sector. Overall, the work provides a minimal, unitarity-respecting framework for dark-pion DM that highlights the necessity of resonances beyond LO ChPT in accurately predicting DM relic abundances and self-interactions, with implications for model-building and future studies of strongly coupled dark sectors.

Abstract

Dark pions are promising dark matter candidates, yet most analyses rely on leading-order (LO) chiral perturbation theory (ChPT). Motivated by the fact that, even for QCD pi-pi scattering, LO ChPT near threshold underestimates the isoscalar s-wave amplitude by an O(1) factor relative to high-precision dispersive results, we reassess the phenomenology of dark-pion dark matter in SIMP and WIMP scenarios using the chiral unitary method, a nonperturbative resummation that implements the correct analytic structure with minimal input, and quantify how unitarization modifies the standard LO ChPT picture. We fix the subtraction constant to its natural estimate, interpreted as an effective cutoff at the chiral scale, so that the unitarized amplitudes depend only on the dark-pion mass and decay constant. We show that, depending on the coupling, the unitarized amplitudes develop resonance poles absent at LO, leading to sizable departures in 2-to-2 self-scattering, relevant for SIMP scenarios, and in annihilation including initial-state interaction effects, relevant for WIMP scenarios. These modifications, in turn, affect the viable parameter space. Although the subtraction constant is, from a model-building perspective, merely a parameter, a substantial deviation from its natural value would point to additional elementary resonances with the same quantum numbers.

Figures (6)

• Figure 1: Analytic structure of the inverse partial-wave amplitude $T_\ell^{-1}(s)$. $s$-channel unitarity implies a right-hand or unitarity cut starting at threshold, while crossing symmetry implies a left-hand or dynamical cut. The red lines indicate the integration contours used in the dispersion relation.
• Figure 2: Phase shift $\delta_0^0$ (in degrees) up to $\sqrt{s}=1\,{\rm GeV}$. The dashed curve is the LO result computed with the $K$-matrix approximation Eq. (\ref{['eq:K-matrix approximation']}); the solid curve is the result from the chiral unitary method in Eq. (\ref{['eq:chiral unitary amplitude real pi']}); and the triangle and diamond markers indicate values extracted from experimental data Mennessier:1982fkOller:2000ma.
• Figure 3: Squared amplitudes $|T_0^0(s)|^2$ obtained from the LO calculation (dashed) and from the chiral unitary method (solid) for couplings $m_\pi/f_\pi = 1,2,3,4,5,6$ (brown, cyan, red, blue, magenta, green).
• Figure 4: Trajectory of the poles of the analytically continued $\ell=0$, $I=0$ scattering amplitude on the unphysical sheet as the coupling $m_\pi/f_\pi$ is varied from 1 to $4\pi$. For couplings $m_\pi/f_\pi \lesssim 3$, a pair of resonance and anti-resonance poles exists (blue lines). As the coupling increases, the poles merge below the threshold on the real axis to form a virtual state. One of the poles then moves onto the negative real axis as a virtual-state pole (green line), while the other moves in the positive direction and eventually reaches the physical sheet, becoming a bound-state pole (magenta line).
• Figure 5: Couplings $m_\pi/f_\pi$ (left axis) that reproduce $\Omega_{\rm DM}h^2\simeq0.12$ under Eq. (\ref{['eq:SIMP approximation']}) versus the dark-pion mass $m_\pi$, together with the corresponding self-scattering cross section $\sigma/m_\pi$ (right axis). Left panels: $SU(4)\times SU(4)/SU(4)$; right panels: $SU(4)/SO(4)$. Top: $x_f=15$; bottom: $x_f=20$. The phenomenologically preferred interval for $\sigma/m_\pi$ in Eq. (\ref{['eq:self-scattering condition']}), $0.1\,{\rm cm^2/g} \,\lesssim\, \sigma_{\rm DM}/m_{\rm DM} \,\lesssim\, 1.0\,{\rm cm^2/g}$, is shown as a shaded band. Blue solid (dashed): couplings from LO (tree-level $K$-matrix) relic-abundance matching. Green solid/dashed: $\sigma/m_\pi$ evaluated at LO $2\to2$ using those couplings without/with unitarization of the freeze-out $3\to2$ process. Red solid/dashed: $\sigma/m_\pi$ computed with the $2\to2$ amplitude improved by the chiral unitary method in Eq. (\ref{['eq:chiral unitary']}), based on couplings obtained without/with $3\to2$ unitarization.