Vector–Scalar Portal Models in Dark Matter

• Vector–Scalar Portal Models are frameworks that connect hidden gauge sectors with the Standard Model through renormalizable scalar-vector interactions, enabling studies of dark matter and collider phenomena.
• They incorporate hidden U(1), SU(2), or SU(3) gauge symmetries with scalar-induced mass generation and discrete symmetries to stabilize potential dark matter candidates.
• Experimental implications include precise relic density predictions, interference effects in mediator interactions, and distinctive collider and direct detection signatures that constrain model parameters.

Vector–Scalar Portal Models are frameworks that connect hidden sector degrees of freedom—vector bosons and scalars—to the Standard Model (SM) through renormalizable interactions. They are central in dark matter phenomenology, collider signatures, model building for extended gauge sectors, and in the interpretation of rare processes with nucleon triggers. Typical realizations involve extensions of the SM by hidden U(1), SU(2), or SU(3) gauge symmetries whose associated vector bosons acquire mass via scalar sector dynamics, rendering them stable dark matter candidates. The portal arises through scalar–vector interactions that mediate couplings, mixing effects, or loop–induced transitions between SM and hidden states.

1. Model Structure and Portal Mechanisms

Vector–scalar portal constructions generically extend the SM with a hidden gauge sector (commonly abelian U(1)ₓ, but also non-abelian SU(2)ₓ or SU(3)ₓ) and an associated scalar (either a singlet or multiplet charged under the hidden gauge group). The general Lagrangian adopts the form: $$\mathcal{L} = \mathcal{L}_{\text{SM}} - \frac{1}{4} X_{\mu\nu} X^{\mu\nu} + |D_\mu S|^2 - V(S, H) + \text{portal},$$ where $X_\mu$ denotes the hidden vector, $S$ the hidden scalar, $V(S,H)$ the scalar potential involving both the SM Higgs doublet $H$ and $S$, and “portal” covers renormalizable terms like $(\lambda_{hv}/4) H^\dagger H X_\mu X^\mu$ or $(\lambda_{hs}/4) H^\dagger H |S|^2$ (Lebedev et al., 2011). The vector acquires mass through either the Stueckelberg mechanism (with an axionlike field) or conventional hidden-sector Higgs mechanism (spontaneous breaking via $⟨S⟩$). Discrete symmetries, typically $Z_2: X_\mu \to -X_\mu$, are automatic in minimal setups, ensuring cosmological stability of the vector state when no additional hidden sector matter is present.

In non-abelian realizations ($SU(2)_D$), the vector bosons (e.g. $V^0_{D\pm}$) emerge as DM candidates with mass $M_V = g_D v_S$, where $v_S$ arises from the dark Higgs VEV (Belyaev et al., 2022). Mixing with the SM Higgs, through scalar portal interactions, is often suppressed, but higher-dimensional portals may be induced through loop effects or specific mediator structures.

2. Mass Generation, Stability, and Symmetry Protection

Vector masses originate from spontaneous symmetry breaking in the hidden sector. In the Stueckelberg mechanism,

$$\mathcal{L}_{\text{St}} = -\frac{1}{4} F_{\mu\nu} F^{\mu\nu} + \frac{1}{2} m^2 X_\mu X^\mu,$$

with $X_\mu \equiv X'_\mu + (1/\mu)\partial_\mu \phi$ and $\phi$ the axionic field (Lebedev et al., 2011). Gauge invariance is maintained by appropriate transformations of $X'_\mu$ and $\phi$. Field-dependent masses such as $m^2 = \mu^2 (1 + \zeta H^\dagger H / M^2)$ introduce portal couplings $\lambda_{hv} = 2\zeta (\mu^2/M^2)$.

Alternatively, hidden–sector Higgs mechanisms employ a complex scalar $S$, leading to mass terms $g_X X_\mu X^\mu$, post symmetry breaking. The $Z_2$ symmetry, often realized as charge conjugation $S \to S^*, X_\mu \to -X_\mu$, ensures stability of the lightest (vector) hidden particle. Minimal models (no extra charged matter apart from longitudinal modes) guarantee exact parity and thus viable DM.

In SU(2)$_D$ or SU(3)$_D$ generalizations, stability is achieved through residual discrete symmetries, often linked to global U(1)$_D$ charge assignments. The mass splittings between vector eigenstates (e.g., $V_{D\,\pm}$ and $V'$) follow from one-loop corrections and play a role in coannihilation and direct/indirect detection cross sections (Belyaev et al., 2022).

3. Relic Density, Detection, and Collider Constraints

The dark matter relic abundance is typically set by s-channel Higgs exchange, with the annihilation rate into SM fermions mediated by portal couplings such as (Lebedev et al., 2011): $$\langle \sigma_{f\bar{f}} v \rangle = \frac{\lambda_{hv}^2 m_f^2}{48\pi} \frac{(1 - m_f^2/m_X^2)^{3/2}}{(4 m_X^2 - m_h^2)^2}.$$ Direct detection constraints arise from elastic nuclear scattering via Higgs exchange: $$\sigma_\text{SI} \propto \frac{\lambda_{hv}^2}{16\pi m_h^4} \frac{m_N^4 f_N^2}{(m_X + m_N)^2}.$$ The strongest limits to date, for appropriate DM mass and coupling, push $\lambda_{hv} \lesssim 0.1$ for $m_X \sim 100$ GeV (Lebedev et al., 2011, Djouadi et al., 2011).

For invisible Higgs decays ($m_X < m_h/2$), the partial width into vectors is enhanced (notably by a $m_h^3$ factor, reflecting longitudinal mode coupling): $$\Gamma^\text{inv}_{h\to XX} = \frac{\lambda_{hv}^2 v^2 m_h^3}{256\pi m_X^4}(1 - 4 m_X^2/m_h^2 + 12 m_X^4/m_h^4) \sqrt{1 - 4 m_X^2/m_h^2}.$$ Combined ATLAS/CMS constraints on invisible Higgs width—buttressed by XENON100/XENON1T data—now constrain vector and scalar DM masses below 60 GeV, with fermion DM Higgs portals essentially excluded under standard cosmological assumptions (Djouadi et al., 2011, Collaboration et al., 2018).

Collider searches at the LHC (Higgs properties, missing energy, mono-$Z$ recoil distributions) and planned lepton colliders (e.g., ILC) provide sensitivity to portal-induced DM production channels, threshold and interference shapes, and rare widths (Ko et al., 2016).

4. Interference, Unitarity, and Ultraviolet Completions

Vector–scalar portal models must respect unitarity constraints arising from the longitudinal vector mode, requiring careful handling of parameter space (e.g. $\lambda_{hv} < 2 m_X^2 / v^2$) (Collaboration et al., 2018). Ultraviolet-complete versions, with extra gauge symmetry (e.g. $U(1)_D$, $SU(2)_D$), recast the portal in terms of scalar mixing between the SM Higgs and new scalar eigenstates, giving rise to physical Higgses $h_1$, $h_2$ (Duch et al., 2015, Arcadi et al., 2021).

Interference between multiple scalar mediators (SM-like and dark Higgs) can both suppress and enhance direct detection signals, invisible widths, and collider cross sections. In processes such as $e^+e^- \to Z h \to Z$ + DM, the recoil-mass spectrum shape (threshold onset, high-$t$ tail) allow one to distinguish scalar and vector/fermion DM via interference patterns between mediators (Ko et al., 2016).

Effective field theory treatments are valid only in the decoupling limit of extra scalars; regions with light mediators require the full propagator structure for correct interpretation of direct detection, decay widths, and relic densities (Arcadi et al., 2021). In extended non-Abelian dark sectors ($SU(3)_D$), additional vector degrees of freedom and scalar mixing further enrich phenomenology, potentially enhancing invisible signals or modifying relic density predictions.

5. Phenomenological Diversity and Astrophysical Probes

Vector–scalar portal frameworks accommodate a range of derivate models, such as loop-induced scalar portals (e.g. via heavy vector-like quark mediation (Dutta et al., 2017)) and nucleon-rescaled portals for forbidden decays (e.g. $\eta\to\pi^0\gamma\gamma$ via scalar-induced nucleon current triggers (Balytskyi, 13 Jul 2025)). Such "nucleon-triggered" interactions manifest contact operators of the form $$(\bar{N}N)\, \widetilde{V}_{\mathcal{B}}^{\mu\nu} F_{\mu\nu} P$$, modifying branching ratios in nucleon environments (photoproduction, charge-exchange) while leaving purely leptonic channels SM-like. This mechanism predicts A$^2$ scaling on heavy nuclei and distinctive shifts in related branching ratios, with experimental validation achievable by cross-comparison between nucleon- and lepton-induced reactions and searches for narrow, nucleon-philic vectors in the 2–5 GeV range (Balytskyi, 13 Jul 2025).

Other exotic realizations include supercooled Higgs portal scenarios where radiative symmetry breaking induces both vector and electroweak mass scales, linked through small scalar mixing angles. These frameworks feature strong first-order phase transitions, generating a stochastic gravitational wave background potentially observable at LISA or TianQin, complementing limits from direct detection experiments (Frandsen et al., 2022). Interference between SM Higgs and "scalon" propagators can create detection blind spots while GW observatories probe the underlying dark sector phase transition dynamics.

6. Cancellation Mechanisms and Suppression of Direct Detection Signals

Discussions of pseudo-Nambu–Goldstone boson (pNGB) dark matter models highlight a robust cancellation mechanism for DM–nucleon scattering amplitudes. When the trilinear coupling between a scalar mediator and on-shell DM is momentum-suppressed (e.g., arising as $(s/v_S)\partial^2 \chi^2$), direct detection cross-sections vanish in the zero-momentum transfer limit, evading XENON/LUX bounds (Cai et al., 2021). Extensions to SO(N) symmetry structures and vector–portal models (SU(2)$_L\times$U(1)$_Y\times$U(1)$_X$, or SU(2)$_L\times$U(1)$_Y\times$U(1)$_{B-L}\times$U(1)$_X$) preserve cancellation in the absence of unsuppressed kinetic mixing terms, or require fine-tuning when such terms arise.

7. Collider and Indirect-Signature Opportunities

Precision collider probes constrain new vector bosons via direct couplings (singleton portals) or kinetic mixing (twin Higgs sectors, singleton portals) (Bishara et al., 2018). Dijet, dilepton, or ditau resonance searches, electroweak precision observables (e.g. $T$ parameter, $Z$ branching ratios), and displaced vertex studies for nearly degenerate vector-like fermion intermediaries offer experimental access to extended vector–scalar sectors (Belyaev et al., 2022, Bishara et al., 2018).

Indirect signals, such as gamma-ray lines from vector DM annihilation in resonance-enhanced models (Choi et al., 2013), or spectral features from radiative corrections in top-philic vector-like portals (Colucci et al., 2018), provide complementary avenues for constraint and characterization.

Concluding Remarks

Vector–Scalar Portal Models epitomize the synergy between symmetry-based stability, renormalizable interactions, and portal–induced phenomenology in dark matter and hidden-sector physics. Their minimal realization connects deep theoretical issues (vacuum stability, mass generation, unitarity) with an array of experimental signals—invisible Higgs decays, direct detection, gravitational waves, rare meson decays, and collider signatures. Cancellation mechanisms, interference patterns, and rich scalar–vector sector dynamics continue to reveal model discriminants and experimental strategies across multiple frontiers. Future measurements—at colliders, underground and astrophysical observatories—will critically test the viable parameter space and possibly uncover novel vector–scalar portal connections linking dark and visible matter.