B meson mixing is a significant tool for probing new physics, providing stringent constraints on theoretical models. In the Standard Model (SM), the mixing originates from a box diagram involving a W boson and a top quark [27]. The dominant SM contribution to the mass difference between B s 0 and B ¯ s 0 mesons depends on QCD corrections, the top quark mass [28], and hadronic parameters like the decay constant and bag parameter [29]. Lattice QCD calculations have refined these parameters [30].

New Physics (NP) contributions, such as from a potential dark Z_D boson, modify the mass difference. These contributions depend on the Z_D mass and coupling constants [8]. The experimental mass difference aligns closely with SM predictions [24][31], placing tight constraints on NP parameters. Notably, lighter Z_D bosons require smaller couplings to remain consistent with observations, disallowing certain parameter combinations, particularly for Z_D masses below 60 MeV and couplings above 0.001. These findings underscore the sensitivity of B s 0 − B ¯ s 0 mixing to new physics.

We plot the sensitivity of B s 0 − B ¯ s 0 mixing to ϵ Z , Δ m i x , as a function of the dark Z mass for different values of ε Z in Figure 1. It is evident that lighter Z D require smaller values of ε Z for Δ m i x to lie within the 2 σ uncertainty of the SM prediction. We find that ε Z ≥ 0.001 is disallowed for M Z D ≤ 60 MeV.

Figure 1. Sensitivity of B s 0 − B ¯ s 0 mixing to ε Z as a function of M Z D . At leading order Δ m i x is independent of ε . The red band is the uncertainty in Δ m i x taken to be the 2 σ lower uncertainty in Δ M B s S M .

The rare decay B s → μ + μ − is a crucial probe for new physics. Its branching fraction depends on Standard Model (SM) parameters and possible contributions from new physics. In the Z D model, contributions from scalar and pseudoscalar Wilson coefficients are absent, leaving the dominant effect from C 10 [32].

The SM prediction for the branching fraction is

ℬ ( B s 0 → μ + μ − ) S M = ( 3.67 ± 0.15 ) × 10 − 9 [33][34], while the experimental measurement from LHCb is ℬ ( B s 0 → μ + μ − ) L H C b = ( 3.09 − 0.43 − 0.11 + 0.46 + 0.15 ) × 10 − 9 [35]. The close agreement between these values constrains potential new physics contributions, highlighting the importance of precision in both theoretical predictions and experimental measurements.

The Z D contribution to this rare decay is shown in Figure 2. Since the decay rate depends only on the new axial-vector interaction of the dark Z , it is independent of ε . It is evident that ε Z as large as 0.01 is allowed by the data at the 3 σ confidence level (CL).

The decays B → K ( ∗ ) ν ν ¯ are sensitive probes for new physics, particularly in the

Figure 2. Branching fraction for B s → μ + μ − for different values of ε Z . The horizontal red (light red) band denotes the 1 σ ( 3 σ ) allowed region from experiment [35]. The decay rate does not depend on ε .

context of the Z D model. These decays involve the on-shell production of Z D in B decay, followed by Z D → ν ν ¯ [36]. The effective interaction strengths and form factors governing these decays are derived from loop functions, and their detailed expressions are available in [37].

The Standard Model (SM) predicts branching fractions of ℬ ( B + → K + ν ν ¯ ) S M = ( 4.4 ± 0.7 ) × 10 − 6 and ℬ ( B 0 → K ∗ 0 ν ν ¯ ) S M = ( 11.6 ± 1.1 ) × 10 − 6 [38]. A recent search by Belle II for B + → K + ν ν ¯ sets a 90% confidence level (CL) upper bound of ℬ ( B + → K + ν ν ¯ ) < 4.1 × 10 − 5 [39], and the weighted average from existing data is ℬ ( B + → K + ν ν ¯ ) W A = ( 1.1 ± 0.4 ) × 10 − 5 [38], showing a slight enhancement over the SM prediction. However, this enhancement should not yet be interpreted as evidence for new physics.

For B → K ∗ ν ν , the most recent 90% CL upper bound from Belle is ℬ ( B 0 → K ∗ 0 ν ν ¯ ) < 1.8 × 10 − 5 [40]. These results impose constraints on new physics scenarios while leaving room for further exploration.

In Figure 3, we plot the branching fractions for some benchmark values of ε Z . We find that they are more than an order of magnitude smaller than the respective upper bounds even for ε Z as large as 0.1.

The flavor-changing decays K → π ν ν ¯ are governed by the s → d ν ν ¯ transition. The key decay modes are K + → π + ν ν ¯ and K L → π 0 ν ν ¯ . The most recent measurement from the NA62 experiment gives ℬ ( K + → π + ν ν ¯ ) = ( 10.6 − 3.4 + 4.0 ± 0.9 ) × 10 − 11 [41], while the KOTO experiment places a 90% confidence level (CL) upper bound of ℬ ( K L → π 0 ν ν ¯ ) < 4.9 × 10 − 9 [42]. These results align with the Standard Model (SM) predictions of ℬ ( K + → π + ν ν ¯ ) S M = ( 8.4 ± 1.0 ) × 10 − 11 and

Figure 3. Branching fraction of B → K ( ∗ ) ν ν ¯ as a function of the Z D mass for three values of ε Z and ε = 0.001 .

ℬ ( K L → π 0 ν ν ¯ ) S M = ( 3.4 ± 0.6 ) × 10 − 11 [43][44]. The two modes are related through the Grossman-Nir bound, which limits ℬ ( K L → π 0 ν ν ¯ ) based on ℬ ( K + → π + ν ν ¯ ) [45].

In the context of new physics, including the Z D model, contributions to these decays are resonant but highly suppressed due to the weak s → d Z D transition. Consequently, the branching fractions remain small even for relatively large Z D coupling parameters ( ϵ Z ~ 0.01 ).

Neutral kaon oscillations ( K 0 ↔ K ¯ 0 ) are also relevant, with a mass difference Δ M K that matches the SM prediction of 3.484 ( 6 ) × 10 − 12   MeV [32]. The dominant SM contribution arises from loop-level processes with QCD corrections and hadronic parameters, while new physics contributions from the Z D model, for example, are found to be negligible under typical parameter values, such as M Z D = 10   MeV and ϵ Z = 0.001 . Thus, Δ M K does not strongly constrain this model.

The three-body decay K + → μ + ν μ Z D serves as a radiative correction to the standard K + → μ + ν μ decay, where the dark boson Z D is emitted from the muon leg and subsequently decays invisibly. This decay is relevant for scenarios where the dark boson mass is below 2 m μ , allowing it to be produced on-shell. Experimental constraints on this process have been established by the NA62 experiment, which sets a 90% confidence level upper limit on the branching fraction for K + → μ + ν ν ν ¯ at 1.0 × 10⁻⁶ [46]. This bound imposes restrictions on the coupling parameters of the dark boson in various model scenarios.

In one scenario, referred to as Case A, where the emission is suppressed by mixing parameters, the branching fraction remains below the experimental limit even for relatively large couplings, as illustrated in the left panel of Figure 4. Conversely, in Case B, where the direct coupling g D μ dominates, the branching fraction increases with g D μ , necessitating g D μ < 0.01 to remain consistent with the data. This behavior is shown in the right panel of Figure 4. The interplay between mixing and direct coupling highlights distinct constraints depending on the underlying assumptions of the model.

Figure 4. Dependence of the K + → μ + + invisible branching fraction on the mixing parameters in Case A (left) and on the direct coupling g D μ in Case B (right). The red shaded region shows the 90% CL upper limit on the branching fraction.

In addition to invisible decays, the dark boson Z D can also decay into visible final states, such as e + e − . This contributes to the decay K + → μ + ν e + e − , for which the experimentally measured branching fraction is ( 7.06 ± 0.31 ) × 10 − 8 [24][47] for invariant electron-positron masses above 145 MeV. Within the context of Case B, the coupling g D μ can enhance the branching fraction significantly for 145 < M Z D < 200   MeV when ϵ Z < ϵ and Z D predominantly decays to e + e − . This dependence is illustrated in Figure 5, where the constraints from data are clearly visible. Notably, these constraints do not apply to scenarios like Case C, where Z D lacks coupling to electrons.

Overall, the analysis of both invisible and visible decay modes of the dark boson provides stringent experimental constraints on its properties, demonstrating how rare kaon decays can serve as sensitive probes of new physics.

Figure 5. Dependence of the K + → μ + ν e + e − branching fraction on g D μ in Case B. The red shaded region shows the 3 σ interval of the branching fraction.

The decay process π + → μ + ν μ Z D , analogous to radiative kaon decay, can be enhanced when the dark boson Z D mass satisfies 0 < M Z D < m π − m μ . The PIENU experiment provides constraints on this process by setting an upper bound on the ratio R π μ ν X = Γ ( π + → μ + ν μ X ) / Γ ( π + → μ + ν μ ) , where X is an invisible decay product [48], in the mass range 0 < M X < 33.9   MeV perimental bound is used to constrain the parameters of the dark boson.

The decay rate depends on the coupling structure. In Case A, it is suppressed by mixing parameters, while in Cases B and C, it scales with ( g D μ ) 2 . The amplitude squared for the process is derived from the general interaction and is adjusted for pion decay by substituting the relevant constants (e.g., f K → f π , V u s → V u d , and m K → m π ).

Figure 6 illustrates the relationship between R π μ ν X and g D μ , with experimental limits from PIENU shown for two ranges of muon kinetic energy: T μ > 1.2   MeV (solid black curve) and T μ < 1.2   MeV (solid red curve). The results indicate that g D μ < 0.1 is consistent with the data for M Z D < 34   MeV if the dark Z does not predominantly decay to neutrinos (i.e., ϵ Z ≪ ϵ ). However, for ϵ Z ≥ ϵ , where B ( Z D → ν ν ¯ ) is significant, g D μ ≥ 0.1 is excluded for M Z D < 15   MeV . This highlights the sensitivity of π + → μ + ν μ Z D decays to the coupling parameters and branching ratios of the dark boson.

Figure 6. Dependence of R π μ ν X = Γ ( π + → μ + ν μ Z D ( → ν ν ¯ ) ) / Γ ( π + → μ + ν μ ) on g D μ for ε Z = 10 − 5 (left) and 10^-4 (right). The black and red solid curves punctuated with points show the 90% CL upper limit from PIENU for the muon kinetic energy ranges, T μ > 1.2 MeV and T μ < 1.2 MeV, respectively.

The dark boson Z D can couple to first-generation Standard Model (SM) fermions via mixing, leading to stringent constraints from atomic parity-violating (APV) observables. These interactions affect the weak charge Q W of the proton and certain nuclei, such as cesium (133Cs), whose measured values are Q W p , e x p = 0.0719 ( 45 ) and Q W 133 Cs , e x p = − 72.82 ( 42 ) [44][49][50], respectivelynfluenced by modifications to the Fermi constant G F and the weak mixing angle θ W , both of which are altered by the dark boson interaction. The parameters ρ d and κ d encode these modifications, depending on the mixing parameters ( ϵ , ϵ Z ), the mass of the dark boson ( M Z D ), and the momentum transfer ( Q 2 ). For example, ρ d and κ d involve terms proportional to mixing coefficients and the ratio of M Z D to the SM Z -boson mass.

The function f ( Q 2 / M Z D 2 ) , which characterizes the momentum dependence, varies for different systems. For protons, f decreases with increasing Q 2 , while for cesium, it is approximately constant. For instance, f ≈ 0.5 at M Z D ≈ 2.4   MeV and f ≈ 1 for M Z D ≈ 100   MeV . Using these dependencies, the model’s consistency with experimental data from APV measurements provides constraints on the mixing parameters and the mass of Z D .

In Figure 7, we plot the 3 σ CL upper bound from APV on ε Z for different values of ε . By and large, for larger values of ε , ε Z is more constrained. Among the constraints discussed so far, APV places the strongest constraint on ε Z in the few MeV-GeV mass range. The coupling g D μ which appears in Case B is unconstrained by APV. Again, because of our fine-tuned choice of g D e to cancel the Z D coupling to electrons, Case C is also unconstrained by APV.

Muon neutrinos can scatter off a nucleus and produce a pair of muons via a weak

Figure 7. The 3 σ CL upper bound on ε Z from measurements of the proton and cesium weak charges in atomic parity violation experiments.

interaction known as neutrino trident production, which occurs through Z D exchange. This process has been measured in neutrino beam experiments, such as CHARM-II [51] and CCFR [52], with results indicating a ratio of the experimental to Standard Model cross sections close to 1 for both experiments. The absence of any excess in these measurements places strong upper limits on the coupling of Z D with muons. In particular, bounds derived for a Z ′ based on the U ( 1 ) L μ − L τ symmetry [53] are adapted to constrain the model’s parameters in the M Z D and ϵ Z plane, with the bounds being less stringent for Case A, where both vector and axial-vector interactions are involved.

Additionally, the COHERENT experiment, which observed coherent elastic neutrino-nucleus scattering (CEνNS) [54], provides further constraints on the model. The data from COHERENT set limits on the ϵ Z parameter for a given mass of Z D . These bounds, derived by rescaling earlier results from U ( 1 ) X models [55]-[57], turn out to be much weaker compared to direct gauge coupling constraints. For instance, for M Z D ~ 10   MeV , the upper bound on ϵ Z is found to be approximately 0.0005.

The dark boson Z D can be produced through both on-shell and off-shell decays of the Z boson, such as in the process Z → ℓ ℓ Z D , which could lead to final states like Z → 4 ℓ . Searches for such events at ATLAS [58][59] and CMS [60][61] have shown results consistent with the Standard Model, placing a lower bound on M Z D of about 5 GeV. However, due to suppression of the Z D -lepton coupling in the model, the decay rate for Z D → 4 ℓ is small, and the resulting bounds are not impactful for this model.

Belle II has also conducted a search for invisibly decaying Z ′ bosons in the e + e − → μ + μ − plus missing energy channel [62], placing an upper limit on the coupling of Z ′ to muons for M Z ′ < 6   GeV . However, this bound is weaker than those from low-energy experiments and does not significantly constrain the model.

The dark boson could contribute to the leptonic decay width of the W boson, particularly in processes where the Z D decays invisibly. The contribution to the decay width is found to be small enough to be consistent with the measured W decay width. For Case A, assuming M Z D is below the dimuon threshold, the contribution is constrained to be within the uncertainty of the measured W width, setting an upper limit on ϵ Z for different values of M Z D . This limit, however, is not as stringent as the other constraints discussed.

Additionally, LHCb searches for dark photons in dimuon samples have set strong bounds on the mixing parameter ϵ . For M Z D ~ 200   MeV , the bound on ϵ is ~10⁻⁴, and for M Z D ~ 2   GeV , it is ~0.0005, which directly applies to Case A. For Cases B and C, this bound is recast in terms of the g D μ coupling, which rules out g D μ > 10 − 3 for M Z D > 210   MeV .