Vector Dark Matter Model

• Vector dark matter (VDM) models are frameworks where a stable, electrically neutral vector boson arises from a new U(1) gauge symmetry with an exact Z₂ symmetry ensuring its stability.
• They extend the Standard Model by introducing complex scalars and spontaneous symmetry breaking, providing mass to the VDM through a Higgs–scalar portal, with distinct minimal and extended formulations.
• Phenomenologically, these models predict observable signals in collider experiments, direct detection setups, and indirect searches, constrained by relic density and invisible Higgs decay measurements.

Vector dark matter (VDM) models are theoretical frameworks in which the dark matter candidate is a stable, electrically neutral vector boson—typically the gauge boson of a new dark U(1) gauge symmetry beyond the Standard Model (SM). The archetype of such models is constructed by augmenting the SM gauge group with an additional Abelian U(1)X symmetry and introducing new complex scalars charged under U(1)_X but not under the SM gauge group. The spontaneous breaking of U(1)_X, while maintaining an exact discrete Z₂ symmetry under which the vector boson is odd, yields a massive vector state (Vμ) that is stable and thus a viable dark matter constituent (Farzan et al., 2012).

1. Theoretical Construction of the VDM Framework

The canonical vector dark matter model extends the SM gauge structure to $G_{\rm SM} \times U(1)_X$. The new gauge field $V_{\mu}$ is associated with the U(1)_X symmetry. A critical requirement for VDM stability is the existence of a discrete Z₂ symmetry under which $V_{\mu}$ is odd and all SM fields are even. This forbids both kinetic mixing between $V_{\mu}$ and the SM hypercharge boson and any VDM decays to SM states or other dark states of lower mass. The scalar content consists of one or more complex scalars $\Phi$, which are SM singlets but carry nonzero U(1)_X charge. The breaking of U(1)_X via a vacuum expectation value (vev) $\langle \Phi \rangle = v_\phi / \sqrt{2}$ imparts a mass to $V_{\mu}$ given by

$m_V = g_V v_\phi,$

where $g_V$ is the U(1)_X gauge coupling. The Lagrangian contains a dark scalar potential of the form: $$V = -\mu^2_\phi |\Phi|^2 - \mu^2|H|^2 + \lambda_\phi |\Phi|^4 + \lambda |H|^4 + \lambda_{H\phi} |\Phi|^2|H|^2,$$ and all interactions between the dark and visible sectors proceed through the Higgs–scalar portal couplings $\lambda_{H\phi}$ (Farzan et al., 2012).

2. Scalar Sector: Minimal and Extended Models

In the minimal scenario, a single complex scalar $\Phi$ is sufficient to break U(1)_X. Its decomposition,

$\Phi = \frac{1}{\sqrt{2}} (\phi_r + i\phi_i),$

yields a massive real scalar $\phi_r$ and an eaten Goldstone mode ($\phi_i$) in the unitary gauge; the latter becomes the longitudinal mode of $V_\mu$. Diagonalization of the CP-even scalar mass matrix in the $(\phi_r, h)$ basis, with $h$ the SM Higgs field, produces a mixing—parametrized by $\lambda_{H\phi}$—between the two sectors: $$\begin{pmatrix} 2\lambda_\phi v_r^2 & \lambda_{H\phi} v v_r \ \lambda_{H\phi} v v_r & 2\lambda v^2 \end{pmatrix}.$$ In the extended model, two complex scalars $\Phi^T = (\phi_1, \phi_2)$ are introduced, forming a doublet that allows for richer symmetry-breaking patterns and multiple physical phases:

• Phase I: Both $\operatorname{Re}(\Phi)$ and $\operatorname{Im}(\Phi)$ obtain vevs, yielding spontaneous CP violation and a 3 × 3 mixing between $\phi_r, \phi_i, h$.
• Phase II/III: Only one scalar component acquires a vev; an accidental Z₂ symmetry stabilizes an extra scalar, realizing two-component dark matter. CP is preserved in these phases.

The phenomenological consequences—CP violation, scalar stability, and multiplicity of dark matter candidates—are tightly linked to the pattern of vevs in the extended scalar sector (Farzan et al., 2012).

3. Phenomenology: Dark Matter Interactions and Phases

After symmetry breaking, VDM models generically communicate with the SM only through Higgs–scalar mixing. Key phenomenological aspects include:

• Annihilation: Dark matter pairs annihilate to SM final states primarily via s-channel scalar exchange. The general annihilation cross section (in the minimal model, for $m_{\phi_r} > m_V$) is

$$\langle \sigma (VV \rightarrow {\rm SM}) v_{rel} \rangle = \frac{64}{3} g_V^4 \left[\frac{\lambda_{H\phi} v v_r}{(m_h^2 - 4m_V^2)(m_{\phi_r}^2 - 4m_V^2)}\right]^2 F,$$

where $F$ is a sum over SM final-state phase-space factors.

• Direct Detection: Elastic scattering is spin-independent and mediated by the Higgs portal. A representative cross section is

$$\sigma_N = \frac{g_V^4 M_r^2 m_N^2}{\pi m_V^2 v_H^2} \left[\frac{\lambda_{H\phi} v v_r}{m_h^2 m_{\phi_r}^2}\right]^2 f^2,$$

with $M_r$ the reduced DM–nucleon mass and $f$ the nucleon matrix element (typically $0.14 < f < 0.66$).

• Invisible Higgs Decay: If extra scalars are lighter than $m_h/2$, the SM Higgs develops an invisible width; current LHC bounds restrict $\lambda_{H\phi}$ accordingly (Farzan et al., 2012).
• Multiple DM Components: In extended models, accidental symmetries in certain phases can render both a scalar and the vector boson stable, producing multi-component dark matter.

Mixings, masses, and couplings are constrained by collider searches (modified Higgs signals, invisible decays), relic density measurements, and underground direct detection limits (e.g., XENON100).

4. Experimental Tests and Discovery Prospects

The only Standard Model portal to the dark sector in this framework is via Higgs–scalar mixing:

• Collider Signatures: Modifications in Higgs couplings, exotic Higgs decays, and possible production of additional scalar states, e.g., $h \rightarrow VV$ or $h \rightarrow$ new scalars, are the main signatures. LHC limits on invisible Higgs decays already exclude part of the parameter space.
• Direct Detection: Predicted scattering rates are below current bounds for reasonable parameters but accessible with improved sensitivity in future experiments.
• Indirect Detection: DM annihilation produces secondary SM particles (antiprotons, gamma rays) that may be detectable in astronomical searches. Multi-scalar scenarios permit distinctive indirect signatures if both vector and scalar DM components are sufficiently stable.
• Parameter Constraints: The allowed masses and couplings for $V_\mu$ and the scalar sector are determined by annihilation cross-section requirements for relic abundance and by non-observation in direct and collider searches (Farzan et al., 2012).

5. Mathematical Structure and Key Formulas

The essential dynamical features of the model are encapsulated by:

• Vector Mass: $m_V = g_V v_r$ (or $g_V \sqrt{v_r^2+v_i^2+v^{\prime 2}}$ in the extended case).
• Scalar Potential (minimal): $$V = -\mu_\phi^2 |\Phi|^2 - \mu^2|H|^2 + \lambda_\phi |\Phi|^4 + \lambda|H|^4 + \lambda_{H\phi}|\Phi|^2|H|^2$$.
• Mixing Matrix (minimal): see above.
• Annihilation Cross Section: as detailed for $VV \to$ SM final states.
• Direct Detection Cross Section: as detailed above.

In the extended scalar sector, mixing matrices expand to 3 × 3 and dark matter–nucleon cross sections generalize accordingly, with mixing elements $a_{ij}$ replacing simple couplings.

6. Model Generalizations and Outlook

The vector dark matter framework based on U(1)_X × Z₂ symmetry is generalizable in several directions:

• Adding more scalars introduces richer CP-violation and accidental symmetries, resulting in multi-component DM scenarios.
• Non-Abelian extensions (not covered in this particular model but widespread in the literature) provide alternative stabilizing symmetries and alter the spectrum and phenomenology.
• Phenomenological connection to other portals: The exclusive reliance on the Higgs–scalar portal for SM–dark sector communication distinguishes this framework from those with kinetic mixing or higher-dimension operators.

This class of models offers a well-defined, minimal and testable realization of stable vector dark matter. The simplicity ensures predictive power, with key signals lying in scalar mass spectra, portal couplings, and associated signatures in dark matter detection experiments and collider measurements (Farzan et al., 2012).