Forbidden DM Annihilation: Dark Higgs Bosons

• Forbidden annihilation into dark Higgs bosons is a mechanism where kinetic thresholds enable DM to produce heavier dark sector states in high-temperature or high-velocity environments.
• The process relies on extended Higgs portal models with symmetry protections that allow only pair production of dark Higgs, ensuring valid relic density and consistent collider signatures.
• This phenomenon results in unique indirect detection signals, such as box-shaped gamma-ray spectra, while collider and direct detection experiments probe complementary aspects of the underlying model.

The concept of forbidden annihilation into dark Higgs bosons arises in many classes of dark matter (DM) models where the primary annihilation channel of DM is kinematically forbidden at zero velocity—i.e., the final state (often a dark Higgs or other heavier dark sector particle) is more massive than the DM itself—yet the process can proceed via the thermal tail in the early universe or be reactivated in astrophysical environments with significant DM acceleration. This mechanism plays a fundamental role in freeze-out relic abundance calculations, indirect detection prospects, and collider phenomenology, particularly when dark sector scalar mediators are present and mix (weakly or strongly) with the Standard Model Higgs boson.

1. Theoretical Foundation and Model Structures

In minimal Higgs portal frameworks, the Standard Model is extended by a dark sector containing one or more Higgs-like scalars responsible for dark symmetry breaking. The prototypical realization includes:

• An extra scalar singlet $S$ (giving rise to a physical dark Higgs $h_D$ after spontaneous symmetry breaking)
• A dark matter candidate (either scalar $\phi$, fermion $\chi$, or vector $X$), stabilized by a discrete symmetry (e.g., $Z_2$)
• A new $U(1)_D$ gauge sector (giving, e.g., a dark photon $A_D$), often broken by $S$ acquiring a vacuum expectation value.

Stability mechanisms such as H-parity (analogous to R-parity in supersymmetry) enforce the absence of dangerous triple couplings between dark Higgs and SM fields, with the Lagrangian containing only even powers of the dark Higgs field ($H$). For instance, in the 5D $SO(5)\otimes U(1)$ gauge-Higgs unification model (Alves, 2010), the effective 4D Lagrangian is

$$\mathcal{L}_\text{eff} = (g^2 v/4) \cos(2\theta_H) W_\mu W^\mu H H + (g^2/8c_W^2) \cos(2\theta_H) Z_\mu Z^\mu H H + \sum_f (m_f/2v^2)\cos(2\theta_H) (\bar{\psi}_f \psi_f) H H$$

evaluated at $\theta_H=\pi/2$ where all triple vertices vanish, ensuring absolute Higgs stability.

2. Kinematic Forbiddenness and Freeze-Out Physics

Forbidden annihilation refers to processes such as DM $+$ DM $\rightarrow$ heavier dark sector states (e.g., dark Higgs bosons), which are kinematically inaccessible for cold, stationary DM ($m_\text{DM} < m_{h_D}$). At finite temperature ($T$), the high-energy tail of the Boltzmann distribution enables these channels with rates suppressed by

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

as exemplified in vector-portal scalar DM models (Wojcik et al., 2021), where the forbidden $\phi^* \phi \to A_D A_D$ or $\phi^* \phi \to h_D h_D$ processes can be dominant for $1 < m_{A_D}/m_\phi < 2$, with annihilation amplitude resonantly enhanced when $m_{h_D} \approx 2 m_{A_D}$.

In composite dark matter scenarios with QCD-like SU($N_c$) gauge dynamics (Abe et al., 5 Apr 2024), chiral symmetry breaking yields nearly degenerate G-parity odd (DM) and even (heavier) dark pions, rendering $\chi\chi \to \pi\pi$ a forbidden channel that controls relic density for multi-TeV DM masses, elevating the preferred $m_\text{DM}$ far above that of conventional electroweakly interacting dark matter.

3. Role of Symmetry and Coupling Structure

Symmetry enforcement is crucial to forbidden annihilation. In Higgs unification models (Alves, 2010), H-parity forbids all odd powers of $H$ in the effective theory, excluding any process producing (or destroying) a single or odd-numbered dark Higgs boson. This restriction extends to DM annihilation and decay, guaranteeing dark Higgs stability and leading to the pair production topology in collider and indirect searches.

Higgs portal scenarios (Walker, 2013) impose unitarity constraints on the allowed dark Higgs mass: full relic abundance and perturbativity of the quartic couplings set $m_\rho \lesssim 8.5$ TeV (fermion exchange only) and $m_\rho \lesssim 45.5$ TeV (including Higgs exchange), with the mixing angle $\sin\theta$ scaling inversely with the new symmetry breaking scale $u$. Upper bounds on $u$ ($\lesssim 3$ TeV or $\lesssim 17$ TeV depending on the model) connect the viability of forbidden annihilation to collider and direct detection prospects.

4. Indirect Detection and Astrophysical Reactivation

Forbidden channels are typically suppressed at present-day galactic velocities ($v \sim 10^{-3}c$), but can be “reactivated” in high-energy environments such as near supermassive black holes (SMBHs) (Cheng et al., 2022, Lu et al., 26 Dec 2024). There, DM is accelerated to $v \sim 0.5c$, and the local density spike boosts annihilation rates: $s = 4 m_\chi^2 / \left(1 - v_\text{rel}^2/4\right)$ enabling $m_\chi < m_{h_D}$ final states, previously forbidden, to be produced on-shell. This mechanism naturally produces unique gamma-ray spectral features (e.g., box-shaped spectra from mediator decays) observable in Fermi-LAT data near Sgr A*, with preliminary constraints on the thermally averaged cross-section matching relic density requirements.

5. Collider and Direct Detection Constraints

At colliders, forbidden annihilation restricts possible final states. In Higgs unification models (Alves, 2010), the only observable production channel is pair production via weak boson fusion ($pp\to jjHH\to jj + E_T^\text{miss}$), with key selection variables including far-forward jets and large missing transverse energy. Achieving a $5\sigma$ observation requires integrated luminosities in the regime of $240-260\,\mathrm{fb}^{-1}$.

Direct detection signals are set by DM–electron or DM–nucleon cross-sections mediated through the Higgs portal (with scaling $\sigma_{e\phi}\propto g_D^2 \epsilon^2 / m_{A_D}^4$ (Wojcik et al., 2021)). Near forbidden channel resonance (e.g., $m_{h_D}\approx 2 m_{A_D}$), the required coupling $g_D$ to reproduce relic abundance is reduced, suppressing direct detection rates, yet forthcoming experiments (such as SENSEI, SuperCDMS) can probe most of the viable parameter space.

6. Phenomenological Consequences and Model Comparisons

Forbidden annihilation modifies the expected cosmic ray and gamma-ray spectra detectable in indirect searches. Internal scalar bremsstrahlung processes (e.g., Higgs-strahlung) lift s-wave helicity suppression in Majorana DM models (Luo et al., 2013, Bringmann et al., 2017), with the resulting $3$-body final states sharply enhancing the annihilation rates compared to suppressed $2$-body channels, especially for sub-TeV DM.

In multi-Higgs sector models, forbidden annihilation into heavier dark Higgs (mediator) states can be suppressed by appropriate mass hierarchy tuning, as in scenarios fitting the Galactic Center gamma-ray excess (Ipek, 2015, Yang, 2018). This tuning ensures indirect detection (visible final states such as $b\bar{b}$ or $\tau^+\tau^-$) dominates over invisible channels, and allows connections with other observables such as the muon $g-2$ anomaly.

7. Outlook and Future Directions

The realization of forbidden annihilation into dark Higgs bosons has profound implications for particle cosmology. It establishes strong connections between:

• High-scale symmetry breaking physics and experimental observables (dark Higgs mass and mixing bounds via unitarity and relic density calculations)
• Galactic and SMBH environments as laboratories to probe otherwise inaccessible DM interactions
• Collider strategies focusing on pair production topologies and large missing energy signals
• Indirect detection via gamma-ray spectral features distinct from background astrophysical processes.

Forthcoming experiments (VLAST, high-luminosity LHC upgrades, lepton colliders) possess the potential to further constrain or detect signals consistent with forbidden dark Higgs annihilation scenarios, especially in model parameter regimes where direct detection continues to lack sensitivity yet astrophysical or collider searches offer complementary reach.

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Table: Model Features and Forbidden Annihilation Properties

Model Class	Forbidden Channel	Symmetry Mechanism	Experimental Consequence
Gauge-Higgs Unification	Single/odd Higgs final	H-parity	Only HH pair production at LHC
Scalar DM + dark Higgs	$\phi^*\phi \to h_D h_D$	$Z_2$ or generalized parity	Relic density controlled by resonance
Composite DM (CQD)	$\chi\chi \to \pi\pi$	G-parity under SU($N_c$)	Multi-TeV DM, changes relic abundance
Portal + SMBH acceleration	$\chi\chi \to h_{s,p}h_{s,p}$	Velocity-induced kinematics	Gamma-ray spikes in Galactic Center

Forbidden annihilation into dark Higgs bosons thus represents a crucial intersection of dark sector model building, symmetry protection, kinematic thresholds, and observational prospects, anchoring both current experimental programs and future theoretical explorations.