Dark-Pion Dark Matter Models

• Dark-pion dark matter is defined as scenarios where dark matter consists of pseudo–Nambu–Goldstone bosons from a confining hidden-sector gauge theory.
• The models use chiral symmetry breaking and effective chiral Lagrangians to generate a multiplet of dark pions with calculable mass spectra and interactions.
• This framework yields rich phenomenology, featuring WIMP/SIMP-like relic mechanisms, forbidden channels, and diverse experimental signatures from direct detection to collider and gravitational wave observations.

Dark-pion dark matter refers to scenarios in which the observed dark matter is comprised of pseudo–Nambu–Goldstone bosons (pNGBs), analogous to pions in quantum chromodynamics (QCD), arising from a confining hidden-sector gauge theory. In these models, a spontaneously broken chiral or flavor symmetry in a new strongly-coupled sector yields a multiplet of light composite scalars—the "dark pions"—which can provide thermal relic dark matter with calculable, model-dependent properties. These frameworks have been extensively developed as candidates for both WIMP- and SIMP-like dark matter, with a rich phenomenology that connects ultraviolet model-building to measurable observables in direct, indirect, accelerator-based, and cosmological experiments.

1. Theoretical Framework: Dark QCD and Chiral Symmetry Breaking

Dark-pion dark matter models originate from strongly-interacting "dark QCD"-like sectors. The hidden sector possesses a gauge group (e.g., SU($N_d$), Sp($2N_c$), SO($N_c$)), and a set of "dark quarks" in representations allowing for a similar pattern of chiral symmetry breaking as in visible QCD. The typical symmetry-breaking pattern is $G_\text{flavor} \to H$ (e.g., SU($N_f)_L \times$ SU($N_f)_R \to$ SU($N_f)_V$), producing $N_f^2-1$ dark pions. The low-energy dynamics are encoded in an effective chiral Lagrangian: $$\mathcal{L}_\text{chiral} = \frac{f_\pi^2}{4} \mathrm{Tr}[\partial_\mu U^\dagger \partial^\mu U] + \frac{f_\pi^2 B_0}{2} \mathrm{Tr}[M(U + U^\dagger)]$$ where $U = \exp(2i \pi^a T^a/f_\pi)$ contains the dark pion fields $\pi^a$, $f_\pi$ is the decay constant, $M$ is the dark-quark mass matrix, and $B_0$ parametrizes the explicit breaking. Chiral anomaly and possible discrete symmetries ("dark G-parity") play a central role in determining stability and interactions (Bai et al., 2015).

2. Dark-Pion Masses, Spectra, and Stability

The mass spectrum is fixed by the Gell-Mann–Oakes–Renner relation: $m_\pi^2 \sim 2 B_0 m_q$ with model-dependent $B_0$ and dark-quark masses $m_q$. Dark-pion spectra can consist of both stable and unstable multiplets. Stability is typically ensured either by unbroken flavor symmetries (e.g., dark baryon number), accidental symmetries, or exact discrete symmetries such as dark G-parity. For example, in a 5 + 5̄ dark-QCD model, G-parity allows for a G-even pion (which can participate in collider anomalies such as a diphoton resonance) and a lightest G-odd pion that is absolutely stable and constitutes the dark matter (Bai et al., 2015, Abe et al., 5 Apr 2024, Beauchesne et al., 2019). The pattern of mass splittings, both radiative and explicit, determines which dark pions are stable and their possible decay channels.

Classification of dark-pion multiplets (Category I, II, III) by the kinematic accessibility of annihilation channels to unstable pions succinctly organizes the possible cosmological histories and indirect-detection phenomenology (Beauchesne et al., 2019).

3. Relic Abundance: Freeze-out, SIMP Mechanisms, and Forbidden Channels

Dark-pion models permit several qualitatively distinct mechanisms for setting the relic abundance:

WIMP-style freeze-out

In "WIMP-like" realizations, dark pions annihilate via $2\to2$ processes either into Standard Model (SM) states (if a portal exists, e.g., Higgs, dark photon) or into other dark-sector states (e.g., G-even pions or gluons via anomaly-induced operators). Cross sections are controlled by:

• Derivative interactions ($\sim m_\pi^2/f_\pi^4$ at threshold)
• Portal couplings (e.g., $U(1)_D$ kinetic mixing, Higgs-portal)
• Anomaly-induced couplings (for G-even/G-odd models, process-dependent) (Bai et al., 2015, Co et al., 2016, Bhattacharya et al., 2013)

The Boltzmann equation reduces to the standard form for a single species: $$\frac{dn}{dt} + 3Hn = -\langle \sigma v \rangle (n^2 - n_\text{eq}^2)$$ yielding the standard thermal relic prediction for $\Omega_\text{DM}$ as a function of $(m_\pi, f_\pi)$ and portal parameters.

SIMP (Strongly Interacting Massive Particle) and $3\to2$ Processes

When $2\to2$ channels are suppressed or forbidden, $3\to2$ number-changing processes via the Wess–Zumino–Witten (WZW) term can dominate freeze-out: $$\langle \sigma v^2 \rangle_{3\to2} \sim \frac{N_c^2 m_\pi^5}{f_\pi^{10}} x^2$$ With $x=m_\pi/T$ near freeze-out, this mechanism prefers sub-GeV pions and large $m_\pi/f_\pi$ ratios (in tension with lattice/holography unless additional resonances contribute, see (Alfano et al., 5 Sep 2025, Tsumura et al., 2017, Braat et al., 2023)).

Forbidden and Semi-forbidden Annihilations

If the stable dark pion has mass below that of heavier (unstable) pions but close to degeneracy, "forbidden" or "semi-forbidden" channels ($\pi_\text{DM}\pi_\text{DM} \to$ heavier pions, suppressed at low $T$) can dominate, allowing for heavier dark matter ($\sim$ TeV – 100 TeV) while maintaining the correct relic (Abe et al., 5 Apr 2024). The relic yield exhibits an exponential Boltzmann suppression controlled by the mass difference,

$$\langle \sigma v \rangle \propto \exp[-2 (m_\text{heavy} - m_\pi)/T]$$

leading to a sharply different cosmology from standard WIMP/SIMP models.

4. Dark-Pion Interactions: Unitarity, Resonances, and Higher-Order Effects

Leading-order chiral perturbation theory (ChPT) systematically underestimates dark-pion self-scattering and annihilation amplitudes near threshold in strongly-coupled regimes. Higher-order, unitarized, and resonance contributions can dramatically impact phenomenology:

• Unitarization via the inverse amplitude or N/D method resums higher-order corrections, dynamically generating resonance poles ($\sigma$, $\rho$, etc.) that enhance or suppress cross sections (Watanabe, 8 Dec 2025).
• NLO chiral Lagrangian (low-energy constants extracted from lattice simulations) is crucial in accurately delimiting allowed $(m_\pi, f_\pi)$ windows and imposing astrophysical bounds (e.g., Bullet Cluster) (Kolešová et al., 8 Sep 2025).
• Resonant and semi-annihilation channels (e.g., $3\pi\to\pi\rho$, $2\pi\to\rho\rho$) can dominate, widen the viable parameter space, and decouple the relic abundance from pure pion dynamics (Bernreuther et al., 2023, Alfano et al., 5 Sep 2025).

The inclusion of vector ($\rho$), scalar ($\sigma$), and glueball states at low energy is both motivated by lattice/holographic modeling and required where large $m_\pi/f_\pi$ is unattainable in pure-pion scenarios.

5. Portal Interactions and Experimental Signals

Portal couplings connecting the dark sector to the SM dictate testability and consistency with cosmological constraints:

• Dark photon portals (kinetic mixing with $U(1)_Y$ or $U(1)_\text{em}$) allow for direct and indirect detection, visible/invisible decay searches, and thermalization between dark and visible sectors (Co et al., 2016, Harigaya et al., 2016, Kondo et al., 2022, Braat et al., 2023).
• Higgs portals permit $2\to2$ annihilation through the SM Higgs, contributing to direct-detection signatures (Bhattacharya et al., 2013).
• Heavy mediator exchange (e.g., in “sneaky” dark matter) controls both relic abundance (via co-annihilation or co-scattering) and collider signals (emerging/semi-visible jets) (Carmona et al., 22 Nov 2024).

The presence and properties of such portals (mass, kinetic mixing $\varepsilon$, gauge coupling $e_D$) are highly model-dependent and determine the viability of parameter space in light of direct detection (XENONnT, LZ), indirect detection (Fermi, Planck), and collider (LHC, Belle II, SHiP) bounds.

6. Astrophysical, Cosmological, and Collider Phenomenology

Dark-pion dark matter models yield a rich phenomenology:

• Self-scattering: Cross sections in the range $0.1 \text{–} 10$ cm$^2$/g (velocity-dependent or resonant) can address small-scale structure problems (cusp-core, diversity) but are strongly constrained by cluster mergers (Kondo et al., 2022, García-Cely et al., 28 Aug 2025).
• Indirect detection: Stable pions annihilating to unstable ones (which decay to SM) produce observable photon or lepton spectra if cross sections are unsuppressed at $T=0$. However, in forbidden or "sneaky" scenarios, velocity suppression allows such models to evade indirect and CMB constraints (Beauchesne et al., 2019, Carmona et al., 22 Nov 2024).
• Direct detection: Portal couplings lead to nuclear or electron recoils, with spin-independent cross sections calculable in chiral effective theory, providing clear exclusion regions or targets for next-generation detectors (Bai et al., 2015, Co et al., 2016, Bhattacharya et al., 2013).
• Collider signatures: Emerging jets, semi-visible jets, and invisible decays of exotic resonances are predicted depending on the spectrum and portal structure. Macroscopic decay lengths for unstable pions offer distinctive displaced-vertex signatures (Carmona et al., 22 Nov 2024, Beauchesne et al., 2019).
• Gravitational waves: A first-order chiral symmetry-breaking transition in the dark sector can generate a stochastic gravitational-wave background in models with weak explicit breaking and suitable phase structure (Tsumura et al., 2017, Abe et al., 5 Apr 2024).

7. Model-building Variations and Current Constraints

Various extensions and limits of dark-pion models exist:

• Millicharged ultra-light dark pions can act as fuzzy dark matter, with mixed pionic/baryonic compositions and rich solitonic structure in halos (Maleknejad et al., 2022, Kouvaris, 2013).
• Theta-vacuum dynamics and CP violation can lead to distinctive kinetic and static observables, such as electron EDMs (García-Cely et al., 28 Aug 2025, Abe et al., 5 Apr 2024).
• Nonthermal production (e.g., via freeze-in, misalignment, or mini-inflation) expands the accessible parameter space and allows scenario reconcilability with entropy and baryon-asymmetry requirements (Yamanaka et al., 2014, Maleknejad et al., 2022).
• Chiral dark-sector models with accidental symmetries can yield mixed dark-pion/dark-baryon dark matter and potentially constitute dark radiation (Co et al., 2016).

Table: Summary of Relic Mechanisms and Typical Parameter Ranges

Mechanism	$m_\pi$ (GeV)	$f_\pi$ (GeV)	Portal	Notes
WIMP annihilation	$0.1$–$10^3$	$10^2$–$10^3$	Higgs, $U(1)$	Benchmark classic scenario
SIMP $3\to2$ freeze-out	$0.1$–$3$	$0.1$–$1$	$U(1)$	Large $m_\pi/f_\pi$ needed
Forbidden/semi-forbidden	$1$–$100$	$1$–$10$	SU(2)	Relic via suppressed $2\to2$
Nonthermal (freeze-in)	$10^{-6}$–$10^3$	Model-dependent	$U(1)$	Millicharged or misalignment
“Sneaky”/impeded freeze-out	$1$–$10$	$1$–$10$	$t$-channel	Co-annihilation, collider signatures

• Dark G-parity WIMP and 750 GeV resonance: (Bai et al., 2015)
• Composite forbidden DM and relics: (Abe et al., 5 Apr 2024)
• Chiral DM, accidental symmetries: (Co et al., 2016)
• Holographic dark-pion landscapes: (Alfano et al., 5 Sep 2025)
• Unitarity, chiral N/D, and resonances: (Watanabe, 8 Dec 2025, Kolešová et al., 8 Sep 2025)
• SIMP, linear-sigma, resonance self-interactions: (Kondo et al., 2022, Braat et al., 2023)
• Dark-pion classification, collider: (Beauchesne et al., 2019, Carmona et al., 22 Nov 2024)
• Light vector mesons, enhanced $3\to2$: (Bernreuther et al., 2023)
• $\theta$ vacuum, velocity-dependent SIDM: (García-Cely et al., 28 Aug 2025)
• Mini-inflation resolution of overabundance: (Yamanaka et al., 2014)
• Ultra-light millicharged pions and WIMPzillas: (Maleknejad et al., 2022)

Dark-pion dark matter thus realizes a broad category of models, tightly connecting high-energy theory, chiral dynamics, resonance physics, cosmological relic mechanisms, and experimental searches at multiple frontiers. The allowed parameter spaces are sharply delineated by theoretical (unitarity, nonperturbativity) and experimental (direct/indirect, collider, cosmological) constraints, with ongoing and planned experiments expected to probe or constrain significant portions of the viable regimes in the near future.