Vector-Portal Kinetic-Mixing Model

• The Vector-Portal Kinetic-Mixing Model is a theoretical framework that introduces a dark photon, interacting with the Standard Model through a renormalizable kinetic mixing operator.
• It employs a minimal extension to the SM Lagrangian to bridge visible matter and a hidden dark sector, with key predictions confirmed by precision electroweak measurements.
• Experimental analyses, from collider data to differential scattering measurements, robustly constrain the kinetic mixing parameter ε to values typically below 0.03 over a wide mass range.

The Vector-Portal Kinetic-Mixing Model is a theoretical framework that posits the existence of a new abelian vector boson ("dark photon" or $A'$), associated with a hidden $U(1)_D$ gauge symmetry, which interacts with the Standard Model (SM) primarily by kinetic mixing with the hypercharge gauge boson. The model provides a minimal and robust portal between visible matter and a dark sector via a renormalizable operator, enabling experimental access through precision electroweak measurements, collider experiments, and cosmological observables.

1. Theoretical Foundation and Lagrangian Structure

The vector-portal kinetic-mixing mechanism is defined at the renormalizable level by extending the SM Lagrangian with an additional $U(1)$ field strength and a kinetic mixing operator: $$\mathcal{L} = \mathcal{L}_{\mathrm{SM}} - \frac{1}{4} F'_{\mu\nu} F'^{\mu\nu} - \frac{\sin\epsilon}{2} F'_{\mu\nu} B^{\mu\nu} + \frac{1}{2} m_{A'}^2 A'_\mu A'^\mu + \tilde{g} J_{A'}^\mu A'_\mu \tag{1}$$ where $F'_{\mu\nu}$ is the field strength of the new abelian vector boson $A'$, $B^{\mu\nu}$ is the field strength of the SM hypercharge, $\epsilon$ is the kinetic mixing angle, $m_{A'}$ is the mass of the new state, and $\tilde{g} J_{A'}^\mu$ represents additional couplings to a hidden/dark sector.

Upon diagonalization of both the kinetic and mass terms, the physical neutral current eigenstates ($A$, $Z$, $Z'$) are rotated from the gauge basis $(A, Z^0, A')$, with kinetic mixing modifying the canonical couplings and shifting the mass matrix. The physical $Z$ mass is altered by the presence of the $Z'$: $$m_Z^2 = \frac{Z_0^2 - m_{Z'}^2 \sin^2 \xi}{\cos^2\xi}$$ where $\xi$ is the $Z^0$–$Z'$ mixing angle, determined by

$$\tan(2\xi) = \frac{2\Delta \left(m_{Z'}^2 - Z_0^2 \right)} {\left(m_{Z'}^2 - Z_0^2\right)^2 - \Delta^2}, \quad \Delta = -Z_0^2 \sin\theta_w \tan\epsilon$$

Compared to tree-level SM predictions, the precision measurements constrain the allowed value of $\epsilon$ since the $Z$ mass is measured to a level $\leq 2.5 \times 10^{-5}$ relative to its tree-level expectation.

2. Experimental Constraints and Data Analysis Methodology

A defining feature of the vector-portal model is the model-independent bound on the kinetic mixing parameter, $\epsilon$. The analysis employs a comprehensive suite of high-precision $e^+e^-$ collider data spanning $1$ GeV to $1$ TeV, focusing on observables sensitive to new virtual neutral-current contributions:

• Differential Bhabha Scattering: Sensitive to modifications in the running of $\alpha_{\mathrm{EM}}(q^2)$ at various $q^2$, primarily constraining $\epsilon$ near and above the $Z^0$ pole.
• Forward–Backward Asymmetries and Hadronic Cross Sections: Precision measurements of $e^+e^-\to f\bar{f}$ rates and asymmetries at the $Z$ pole, primarily for $b$, $c$, $\mu$, and $\tau$ final states.
• Resonant Production at LEP2: Enhancement of cross-sections when $\sqrt{s}\sim m_{Z'}$. Even if $Z'$ decays dominantly to the dark sector, interference effects proportional to $\epsilon^2$ affect visible cross-sections.

The constraints are obtained through a global $\chi^2$ minimization comparing theoretical predictions including kinetic mixing (and treating the $Z'$ width as either "narrow" or "wide" to encompass both visible and invisible decays) with experimental residuals: $$\delta_{\mathrm{th}}(i, \epsilon, m_{Z'}) = \frac{X_{\mathrm{th}}(i, \epsilon, m_{Z'}) - X_{\mathrm{th}}^{\mathrm{SM}}(i)}{X_{\mathrm{th}}^{\mathrm{SM}}(i)}, \quad \delta_{\exp}(i) = \frac{X_{\exp}(i) - X_{\exp}^{\mathrm{SM}}(i)}{X_{\exp}^{\mathrm{SM}}(i)}$$

Only measurements with significance $E_{\min}=0.5$ or greater are included to avoid dilution of sensitivity.

3. Robustness and Assumptions of the Model-Independent Bound

Several characteristics ensure the broad validity of these constraints:

• Model Independence: The analysis is independent of $Z'$ decay specifics, robust against additional dark sector interactions, since the bounds are extracted from interference and virtual effects rather than missing energy signatures.
• Tree-Level Calculations: The theory predictions are made at tree level, while SM radiative corrections are incorporated via comparison with corrected experimental values.
• Dominant Coupling via Kinetic Mixing: The only assumed non-negligible coupling of $A'$ to the SM is through the kinetic mixing parameter $\epsilon$. Any new physics that would mimic or cancel the kinetic mixing effect could invalidate the bound, but such fine-tuned scenarios are not generic.

As a result, for $$1~\mathrm{GeV} \lesssim m_{Z'} \lesssim 1~\mathrm{TeV}$$, the upper bound $\epsilon \lesssim 0.03$ is robust for most of the mass range, with relaxation only in the narrow region $\vert m_{Z'} - m_Z \vert \lesssim 0.1~\mathrm{GeV}$ where the $Z$ mass shift vanishes due to the crossing of sign in the mixing angle $\xi$.

4. Implications for Model Building and Parameter Space

This model-independent constraint has several key consequences for both theoretical model building and phenomenological exploration:

• Limits on Visible-Dark Sector Coupling: Even if the $Z'$ has large couplings to additional dark sector states, its interaction with the SM is completely set by $\epsilon$, and is stringently constrained.
• On-Shell and Off-Shell Effects: The analysis accounts for both off-shell (virtual) effects on neutral current processes and on-shell $Z'$ production, providing comprehensive coverage.
• Dark Sector Decays: The bound on $\epsilon$ is unaffected by whether $Z'$ decays primarily to the SM or to invisible (dark sector) final states, making the result powerful for a broad class of hidden sector models.
• Precision Electroweak Constraints: High-precision Z pole measurements (e.g., at LEP and SLD) are particularly sensitive, making the bounds essentially "hard" unless new physics conspires to cancel the shift.

5. Summary Table: Principal Constraints and Features

Quantity	Formula	Typical Bound
$Z$ mass shift	$m_Z^2 = (Z_0^2 - m_{Z'}^2 \sin^2 \xi)/\cos^2\xi$	$\leq 2.5\times 10^{-5}$
Mixing angle	$$\tan(2\xi) = \frac{2\Delta (m_{Z'}^2 - Z_0^2)}{(m_{Z'}^2 - Z_0^2)^2 - \Delta^2}$$, $\Delta = -Z_0^2\sin\theta_w\tan\epsilon$	—
Global bound	$\epsilon \lesssim 0.03$	$1$ GeV $\leq m_{Z'} \leq 1$ TeV
Inclusion of dark decays	—	Constraints unchanged
Narrow window near $m_{Z'}\approx m_Z$	Relaxed, controlled by other observables	—

The measurement-based bound $\epsilon \lesssim 0.03$ is largely insensitive to the $Z'$ mass for $m_{Z'}$ far from $m_Z$, with a narrow weak region close to the $Z$ pole.

6. Prospects for Future Searches and Model Extensions

Precision improvements at future lepton colliders (such as FCC-ee) or new $e^+e^-$ facilities with increased statistical power could refine the bound on $\epsilon$ for $m_{A'}$ above the GeV scale, provided systematic and SM uncertainties improve correspondingly. Extensions to models with non-minimal dark sectors, with multiple $U(1)$'s or nonabelian gauge groups, or those with additional new physics at the electroweak scale must ensure that observable shifts in the neutral current sector from kinetic mixing remain subdominant to maintain compatibility with the robust constraints derived.

The model-independent analysis establishes concrete boundaries in parameter space for vector-portal models, ensuring that any viable dark sector scenario employing abelian kinetic mixing adheres to the severe restriction $\epsilon \lesssim 0.03$, as dictated by precision data in the $1~\mathrm{GeV} - 1~\mathrm{TeV}$ regime (Hook et al., 2010).