Dark Hidden-Sector Model

• Dark Hidden-Sector Model is an extension of the Standard Model featuring a confining SU(2) gauge group and a Higgs portal that connects the hidden and SM sectors.
• It employs an accidental custodial SU(2) symmetry to naturally stabilize heavy vector bound states as dark matter candidates.
• Thermal relic density depends on the unique confinement scale, predicting dark matter masses in the 20–120 TeV range with nonperturbative strong dynamics.

A dark hidden-sector model describes an extension of the Standard Model (SM) in which one or more additional sectors are present that are neutral with respect to SM gauge charges but have their own nontrivial gauge dynamics, matter content, and possibly symmetry-breaking and confining phenomena. Communication between the dark hidden sector and the SM is mediated by "portal" couplings, typically a Higgs portal, kinetic mixing, or higher-dimensional operators. Dark matter (DM) candidates naturally arise as stable or long-lived states in the hidden sector, with stability often ensured by an accidental or emergent symmetry of the dark dynamics. These frameworks connect the properties of DM—relic density, self-interactions, direct and indirect detection prospects—to the gauge structure, matter content, and interaction strength scales of the hidden sector.

1. Model Architecture and Gauge Structure

A prototypical dark hidden-sector model employs a hidden SU(2) gauge group (SU(2)$_{HS}$) and a scalar field $\phi$ in the fundamental representation. The SM and the hidden sector are entirely decoupled at the renormalizable level except for a Higgs portal interaction: $$\mathcal{L} = \mathcal{L}_{SM} - \frac{1}{4} F_{HS}^{2} + (D_\mu \phi)^\dagger (D^\mu \phi) - m_\phi^2 \phi^\dagger \phi - \lambda_\phi (\phi^\dagger \phi)^2 - \lambda_m \phi^\dagger \phi H^\dagger H \tag{1}$$ where $$D_\mu \phi = \partial_\mu \phi - i g_{HS} A_{HS}^\mu \phi$$ and $H$ is the SM Higgs. The portal coupling $\lambda_m$ links the two sectors, but there is no kinetic mixing (forbidden by non-Abelian gauge invariance).

The hidden SU(2)$_{HS}$ sector is confining: below the confinement scale $\Lambda_{HS}$, the IR degrees of freedom are not elementary $\phi$ and $A_{HS}^\mu$, but composite bound states. Vector bound states arise naturally: these are triplets under a surviving global custodial SU(2) symmetry. The model features a minimal parameter set: the SU(2)$_{HS}$ coupling, the scalar mass, and quartic couplings, with all DM physics essentially controlled by the confinement scale.

2. Symmetry Structure and Stability: Custodial SU(2)

The long-term stability of the dark matter candidate is guaranteed not by an imposed global discrete symmetry, but by an exact custodial SU(2) symmetry in the hidden sector's IR. The custodial symmetry arises because the scalar in the fundamental representation can be forming real, SO(4)-like representations, allowing rearrangement into a 2×2 matrix transforming under a global SU(2). After confinement, the lowest-lying vector bound state transforms as a triplet of this custodial group, and decays into SM states are forbidden unless the custodial symmetry is broken.

This mechanism is structurally analogous to proton stability in QCD, which is a consequence of accidental baryon number conservation, itself rooted in gauge structure and matter content rather than an imposed symmetry.

3. Communication via the Higgs Portal

The only renormalizable connection to the SM is through the Higgs portal term: $-\lambda_m \phi^\dagger \phi H^\dagger H$ This coupling plays several critical roles:

• Thermalizes the hidden and SM sectors in the early Universe, enabling a freeze-out scenario for setting relic abundance.
• Allows hidden scalar singlets (composite in the strong phase) to mix with the SM Higgs, leading to interactions relevant for phenomenology such as direct detection.
• Can induce electroweak symmetry breaking in the SM sector via a negative induced mass term if $\lambda_m \langle \phi^\dagger \phi \rangle$ is sufficiently large and negative.

Effective interactions after confinement, such as $S h h$ and S–h mixing terms (S: singlet scalar bound state), emerge from this portal.

4. Relic Density Determination and Confinement Scale

The observed relic density of dark matter imposes a precise requirement on the hidden sector's confinement scale since the annihilation cross section is determined by the mass of the vector bound state ($m_{DM} \sim \Lambda_{HS}$): $$\langle \sigma v_{rel} \rangle \simeq \frac{A}{m_{DM}^2} \tag{2}$$ with A an order-one (to $4\pi$) number set by strong dynamics. For thermal relics consistent with WMAP/Planck ($\langle \sigma v_{rel} \rangle \sim 1$ pb), the required mass range is: $m_{DM} \approx 20\,\text{–}\,120\,\text{TeV}$ This scale is fixed by the requirement that freeze-out in the early Universe leaves the correct relic abundance. Since the hidden sector has a single dynamically generated scale, both the DM mass and all relevant cross sections are set by $\Lambda_{HS}$.

5. Mathematical Formulation

Key equations and relationships include:

• The Lagrangian (Eq.(1)) and definition of the Higgs portal term.
• Annihilation cross section (Eq.(2)): $\langle \sigma v_{rel} \rangle = A / m_{DM}^2$, reflecting the dominance of annihilation to lighter hidden sector bound states.
• The effective mass mixing in the Higgs sector: $$\Delta \mathcal{L} \sim \lambda_m \langle \phi^\dagger \phi \rangle H^\dagger H \equiv m_{eff}^2 H^\dagger H$$
• Spin-independent cross section for direct detection (on a nucleus of mass $m_N$): $$\sigma_{SI}^{(N)} \propto \frac{\lambda_m^2 f^2 m_N^4}{m_{DM}^2 m_h^4}$$ where $f \simeq 0.3$ parameterizes the Higgs–nucleon coupling, and $m_h$ is the SM Higgs mass.

6. Comparison to Other Hidden-Sector and WIMP Models

The confined hidden vector dark matter model (based on (0907.1007)) exhibits several features distinguishing it from other hidden-sector and weakly interacting massive particle (WIMP) constructions:

• Emergent Stability: Rather than imposing stability by hand, as with R-parity in SUSY models or discrete $Z_2$ symmetry in scalar singlet models, stability is a natural consequence of the gauge and matter content via custodial symmetry.
• Scale Minimalism: Only a single confinement scale controls all dark sector properties, with no need for additional free parameters to set the DM mass or couplings.
• Minimal Portal: Communication with the SM is restricted to the Higgs portal, excluding models with kinetic mixing or additional mediators. All SM–dark matter interactions are sensitive to $\lambda_m$ and the nonperturbative dynamics.
• Strong Dynamics Regime: Predictive power is limited by nonperturbative uncertainty in strong-coupling computations, but the model covers both SIMP-like (strongly interacting) and WIMP-like regimes, depending on the portal coupling and confinement scale.
• Relic Mass Scale: The natural DM mass range is in the tens to hundreds of TeV, which is considerably above electroweak-scale WIMP expectations.

Unlike "hidden photon" or dark force models involving light mediators and (typically) keV–GeV-scale DM, the present construction does not rely on kinetic mixing or cascade decays and does not accommodate indirect detection signals from low-mass dark matter but generates heavy, stable bound states largely inaccessible to traditional direct and indirect probes.

7. Model Limitations and Phenomenological Challenges

The main theoretical challenge is the difficulty of precise quantitative predictions in the nonperturbative strong coupling regime ($\Lambda_{HS}$), making cross section and spectrum computations indirect. Phenomenologically:

• Direct detection cross sections are typically suppressed due to the heavy scale and small Higgs-portal mixing.
• Constraints from Higgs-singlet scalar mixing must be checked carefully, e.g., from LEP precision data.
• Self-interaction and indirect detection constraints, as from bounds on DM–DM scatterings or cosmic-ray channels, require detailed model-dependent investigations.

This model serves as a benchmark for dark-sector theories emphasizing emergent stability, strong dynamics, and single-scale simplicity, and exemplifies how hidden gauge dynamics can naturally generate all salient features of a viable and minimal dark matter candidate.