Forbidden Annihilation Channels in Dark Matter

• Forbidden annihilation channels are defined as processes where dark matter particles annihilate into heavier states that become accessible only at finite temperature due to thermal motion.
• The mechanism shows exponential sensitivity to mass splittings and kinetic states, fundamentally altering relic density calculations and constraining viable dark matter models.
• Distinct phenomenological signatures, including reactivation in high-velocity astrophysical environments, provide promising targets for upcoming terrestrial and astrophysical experiments.

A forbidden annihilation channel is a process wherein dark matter (DM) particles annihilate to heavier states that are kinematically inaccessible at zero temperature, but can proceed at finite temperature due to thermal motion. This mechanism is realized in multiple contexts: in the thermal freeze-out of relic DM, in the presence of strong background fields, and via environmental velocity boosts near compact objects. Its significance lies in the exponential sensitivity of the annihilation rate and relic abundance to particle mass splittings and the thermal/kinetic state of the universe, providing predictive constraints and distinctive phenomenological signatures.

1. Kinematical Structure and General Mechanism

A forbidden annihilation channel is defined by the condition that the annihilation of two DM particles, $\chi \chi \to XX$, is kinematically disallowed at zero velocity: $2 m_\chi < 2 m_X \qquad \text{(for identical %%%%1%%%%)}$ where $m_X > m_\chi$, and $X$ denotes SM or dark-sector states. However, in the early universe or in regions with high DM velocities, the nonzero kinetic energy of the DM makes the process possible for the fraction of the tail of the velocity distribution able to overcome the mass gap. The minimal velocity required for the channel to open is

$$v_{\text{min}} = \sqrt{2 (m_X - m_\chi)/m_\chi} = \sqrt{2 \delta}$$

where $\delta = (m_X - m_\chi)/m_\chi$ is the normalized splitting (Liu et al., 2023, D'Agnolo et al., 2020, Kopp et al., 2016).

The thermally averaged annihilation cross section then acquires an exponential Boltzmann suppression: $$\langle \sigma v \rangle \simeq \sigma_0 \, \exp(-2\delta x) \, v^n, \qquad x \equiv m_\chi/T$$ with $n=0$ for s-wave and $n=2$ for p-wave annihilation. This exponential dependence fundamentally alters the relic-density calculation and the phenomenology (Liu et al., 2023, Aboubrahim et al., 2023, D'Agnolo et al., 2020).

2. Model Implementations and Theoretical Realizations

Forbidden channels appear in diverse model frameworks:

• Leptophilic singlet-scalar mediation: Dirac fermion DM $\chi$ couples to a light real singlet scalar $\phi$ and SM leptons. Only one lepton flavor coupling may be significant (e.g., $\mu$-philic). The relevant interaction terms are:

$$-\mathcal{L} \supset g_{ij} \phi \bar{\ell}_i \ell_j + g^A_{ij} \phi \bar{\ell}_i \gamma_5 \ell_j + g_\chi \phi \bar{\chi} \chi + g^A_\chi \phi \bar{\chi} \gamma_5 \chi$$

Forbidden annihilation is realized for $m_\chi < m_\ell$ (Liu et al., 2023, Aboubrahim et al., 2023, D'Agnolo et al., 2020).

• Vector dark matter in a hidden sector: An Abelian $U(1)_X$ vector DM $X$ annihilates via $X X \to S S$, where $S$ is a heavier scalar mixed with the SM Higgs. Here, $m_S > m_X$ and the forbiddenness parameter $r \equiv m_S/m_X = 1 + \Delta$ is crucial (Yang, 2022).
• Composite dark matter: Chiral symmetry breaking in a QCD-like $SU(N_c)$ sector yields stable pions with near-degenerate G-parity odd (DM) and even (heavier) multiplets. Annihilation $\chi\chi \to \pi\pi$ becomes a forbidden channel as masses approach degeneracy (Abe et al., 2024).
• Portals to heavy SM species or mediators: Thermal freeze-out can be set by forbidden annihilation into top quarks via minimal flavor violation, or through a heavier mediator via a Breit–Wigner resonance above threshold (e.g., $m_\phi > 2 m_\chi$) (Delgado et al., 2016, Cheng et al., 2023).
• Strong field environments: In QED, processes such as $e^- e^+ \to \gamma$ are forbidden in vacuum but can be induced in strong background fields, as the field supplies the necessary 4-momentum (Blaschke et al., 2011).

3. Boltzmann Dynamics and Relic Density

The core dynamical equations differ from conventional freeze-out by (a) the exponential suppression in $\langle \sigma v \rangle$, and (b) the sensitivity to the DM temperature relative to the SM. The number density evolution is

$$\frac{dY}{dx} = -\frac{s}{H x} \langle \sigma v \rangle_{T_\chi} \big(Y^2 - Y_\text{eq}^2\big)$$

where $Y = n_\chi / s$. In forbidden scenarios, the kinetic decoupling of DM from the SM occurs earlier, at $x_\text{KD}\sim 20$, causing $T_\chi < T$. This self-cooling sharpens the exponential suppression: $$\langle \sigma v \rangle \sim \exp\left[-2\delta \frac{m_\chi}{T_\chi}\right]$$ and results in relic densities up to an order of magnitude higher than naive (single-equation, $T_\chi=T$) treatments (Liu et al., 2023, Aboubrahim et al., 2023).

Accurate computation requires solving coupled equations for $Y$ and $T_\chi$, accounting for the full (non-Fokker–Planck) elastic collision operator. The freeze-out temperature $x_f$ is shifted higher, and the viable parameter space is substantially reduced (Aboubrahim et al., 2023). Analytical estimates confirm that only small splittings $\delta \lesssim 0.1$ are compatible with the observed relic abundance for perturbative couplings (D'Agnolo et al., 2020, Liu et al., 2023, Yang, 2022).

4. Phenomenology, Astrophysical Reactivation, and Experimental Probes

Late-time annihilation through forbidden channels is suppressed (e.g., $$\langle \sigma v \rangle(T) \propto e^{-2\delta m_\chi/T}$$ with $T \ll m_\chi$), rendering CMB and diffuse $\gamma$-ray constraints ineffectual in typical galactic or CMB environments (Liu et al., 2023, D'Agnolo et al., 2020, Aboubrahim et al., 2023, Kopp et al., 2016).

However, in environments where DM is gravitationally accelerated to high velocities—most notably in DM spikes near supermassive black holes—these forbidden channels can "reactivate." The local DM velocity $v_\text{d}(r) \simeq \sqrt{G M/r}$ becomes large enough that $v_\text{rel} > v_\text{th}$ for a substantial fraction of the DM, allowing annihilation into heavier products (e.g., $\chi\chi \to FF$). The photon spectra from such processes are sharply localized (box- or line-shaped) and serve as unique signatures in $\gamma$-ray observations toward Sgr A* (Cheng et al., 2022, Cheng et al., 2023, Yang et al., 2024, Lu et al., 2024, Cheng et al., 2023).

Key terrestrial experimental constraints include:

• Beam-dump and fixed-target searches: SLAC E137, BDX, NA64–μ, and M³ (for muon-philic mediators) (Liu et al., 2023, D'Agnolo et al., 2020, Aboubrahim et al., 2023).
• $e^+e^-$ colliders: BaBar, Belle II, and future Z-factories (FCC-ee, CEPC) (D'Agnolo et al., 2020, Aboubrahim et al., 2023).
• Kaon factories: NA62 bounds (Liu et al., 2023).
• CMB: Planck, with CMB-S4 poised to probe even mild splittings for s-wave forbidden annihilation (Aboubrahim et al., 2023).
• Indirect detection: Fermi-LAT, DAMPE, and future MeV-$\gamma$-ray telescopes have set or can set limits on DM models with reactivated forbidden channels near SMBHs (Cheng et al., 2022, Yang et al., 2024, Lu et al., 2024, Cheng et al., 2023).

5. Theoretical Variants and Related Mechanisms

Variants include:

• Not-Forbidden Dark Matter: If $2\to2$ forbidden annihilation is closed but $3\to2$ processes such as $\chi\chi\chi \to \chi X$ are open, the DM relic density can be set by $3\to2$ annihilations. This hybrid scenario (NFDM) produces relics for MeV–GeV DM with large self-interactions and is robust to late-universe bounds (Cline et al., 2017).
• Impeded Annihilation: When mass splittings are extremely small ($\Delta \to 0^\pm$), the annihilation cross section may depend linearly on velocity or remain suppressed, leading to highly environment-dependent indirect signals and possible evasion of CMB and dwarf-galaxy constraints (Kopp et al., 2016).
• Assisted Annihilation: When forbidden $2\to2$ channels are closed, processes with one or more "assisters" in the initial state (e.g., $\chi\chi A \to f\bar f$) can set the relic density, with characteristic parametric and experimental signatures for keV–MeV dark matter (Dey et al., 2016).
• Forbidden Channel with Phase Transition: A first-order dark-sector phase transition can dynamically induce the forbidden mass gap, tightly correlating DM mass, freeze-out temperature, and potentially observable gravitational-wave backgrounds (Mahapatra et al., 18 Jan 2026).

6. Implications, Controlling Parameters, and Future Prospects

The forbidden annihilation scenario has highly predictive features:

• Relic density sets DM mass: The observed relic density sharply fixes $m_\chi$ relative to the heavier final state—often to within a few percent—given the exponential sensitivity to $\delta$ (D'Agnolo et al., 2020, Yang, 2022, Delgado et al., 2016).
• Shrinking allowed parameter space: Inclusion of kinetic decoupling and the full Boltzmann equation shrinks viable couplings and mass splittings by factors of ten or more compared to prior equilibrium-based analyses (Liu et al., 2023, Aboubrahim et al., 2023).
• Proximity to discovery: The required large couplings to overcome Boltzmann suppression place most viable models within reach of next-generation fixed-target, collider, and CMB experiments, save for narrow resonant regions (Aboubrahim et al., 2023, D'Agnolo et al., 2020).
• Environmental modulation of annihilation: The annihilation rate varies exponentially among environments (CMB, dwarf galaxies, BH spikes), suppressing indirect signals in most, but allowing unique signals in high-velocity regimes (Cheng et al., 2022, Cheng et al., 2023).
• Multi-messenger signatures: In first-order phase transition models, the dark-sector phase boundary yielding forbidden annihilation also sources a stochastic gravitational-wave background correlated with DM parameters (Mahapatra et al., 18 Jan 2026).

The forbidden channel mechanism provides a highly constrained and testable alternative to the conventional WIMP paradigm and forms a theoretically robust target for a suite of upcoming terrestrial and astrophysical probes (Liu et al., 2023, D'Agnolo et al., 2020, Aboubrahim et al., 2023, Cheng et al., 2022, Mahapatra et al., 18 Jan 2026).