Lepton Flavor Violating Z′ Decays

• Lepton flavor violating decays of Z′ resonances are processes where a heavy Z′ boson decays into two charged leptons of different flavors, signaling physics beyond the Standard Model.
• Experimental searches set strict limits on LFV branching ratios, with collider and rare decay measurements constraining them to values between 10⁻⁷ and 1% depending on the model.
• Theoretical analyses explore tree- and loop-level mechanisms, including non-universal gauge symmetries and vector-like mixing, to explain potential LFV signals.

Lepton flavor violating (LFV) decays of Z′ resonances refer to processes in which a new heavy neutral gauge boson (denoted Z′) decays into a pair of charged leptons of different flavors, such as Z′ → eμ, Z′ → μτ, or Z′ → eτ. In contrast to the Standard Model Z boson, where such decays are forbidden at all orders apart from negligible effects due to neutrino masses, Z′ LFV decays can appear at tree or loop level in a wide variety of extensions to the Standard Model with new sources of flavor violation. These processes serve as sensitive probes of new physics, providing complementary constraints on the structure and couplings of extended gauge sectors, and are the subject of active experimental searches and theoretical analyses.

1. Theoretical Frameworks for Z′-Induced LFV

LFV in Z′ decays arises in theories where the new gauge boson has flavor-violating couplings to charged leptons. Such couplings may originate from:

• Non-universal gauge symmetries: Models with extra U(1)′ gauge groups in which the Z′ boson couples differently to each lepton generation. Family non-universal (FNU) charges lead, after rotation to the mass basis, to tree-level LFV couplings (Orduz-Ducuara, 2016, Espinosa-Gomez et al., 2023).
• Mixing with vector-like leptons: The addition of vector-like fermions, which mix with Standard Model leptons, can induce off-diagonal Z′ couplings via mass mixing matrices, resulting in LFV at tree level (Rocha-Moran et al., 2018, Chen et al., 2017).
• Supersymmetric, seesaw, or neutrino-mass–motivated extensions: Loop-induced LFV Z′ decays can occur due to slepton or heavy neutrino mixing, with Z′ couplings inherited from the structure of the neutrino or slepton mass matrices (Yang, 2010, Romeri et al., 2017).
• Generic effective field theory: The most general renormalizable Lagrangian allows for off-diagonal Z′–lepton-lepton couplings (Ω_L, Ω_R); these govern both tree– and loop–induced LFV decays (Aranda et al., 2012).

The generic Lagrangian for Z′–lepton interactions is

$$\mathcal{L}_\mathrm{NC} = \sum_{i \neq j} \bar{\ell}_i \gamma^\mu \left( \Omega_L^{ij} P_L + \Omega_R^{ij} P_R \right) \ell_j Z'_\mu + \text{h.c.}$$

where $P_{L,R} = (1 \mp \gamma^5)/2$.

2. Key Observables and Experimental Constraints

LFV Z′ decays are characterized by their branching ratios $\mathrm{Br}(Z' \to \ell_i \ell_j)$ and production cross sections $$\sigma(pp \to Z' + X) \times \mathrm{Br}(Z' \to \ell_i \ell_j)$$. Experimental limits depend on the Z′ mass, flavor combination, and collider.

Direct searches: The CMS Collaboration has set the most restrictive direct limits for the eμ channel in the mass range 110–500 GeV, with 95% CL upper limits on the product $\sigma \times \mathrm{Br}$ ranging from 7 fb (low mass) to 0.3 fb (500 GeV) (Collaboration, 11 Aug 2025). No excess with local $p$-value $<1\%$ has been observed.

Associated low-energy processes: The same LFV couplings $\Omega_{L,R}^{ij}$ contribute to rare decays and conversion:

• $\mu \to e\gamma$ (loop)
• $\mu \to 3e$ (tree)
• $\mu$–$e$ conversion in nuclei (tree)

These low-energy limits are typically much more restrictive than direct collider bounds, with upper limits on $\Omega_{L,R}^{ij}$ translating into $\mathrm{Br}(Z' \to e\mu) < 10^{-7}$–$10^{-4}$ for various benchmark Z′ models, depending on the process and chiral structure (Espinosa-Gomez et al., 2023).

Indirect limits from $\ell \to 3\ell$ or $\tau \to \ell\phi$ can dominate parameter space exclusions: e.g., in several $Z'$ models (e.g., LR model), the $\tau \to e\phi$ limit is more stringent than that from $\tau \to 3e$ (Yue et al., 2016).

3. Mechanisms for LFV Z′ Decays

The dominant mechanisms for LFV Z′ decays depend on the underlying theory:

Tree-Level Processes

• Occur in models where Z′ has direct off-diagonal couplings due to family non-universality or mixing with vector-like states (Espinosa-Gomez et al., 2023, Rocha-Moran et al., 2018, Orduz-Ducuara, 2016). In this case, the branching ratio is, schematically,

$$\mathrm{Br}(Z' \to \ell_i \ell_j) = \frac{1}{24\pi} \frac{m_{Z'}}{\Gamma_{Z'}} \{ [\ldots] (|\Omega_L|^2 + |\Omega_R|^2) + [\ldots] \mathrm{Re}(\Omega_L \Omega_R^*) \}$$

where $\Gamma_{Z'}$ is the total Z′ width and the coefficients are kinematic functions (Aranda et al., 2012).

Loop-Level Processes

• In cases with only flavor-conserving couplings at tree level, loop-induced LFV arises via heavy lepton or sneutrino exchange, often correlated with the mechanism for neutrino masses (Yang, 2010, Romeri et al., 2017). These contributions are typically suppressed by the scale of new physics and mixing angles.

Constraints from Radiative and Three-Body Decays

• Diagrams contributing to $\ell_i \to \ell_j\gamma$ or $\ell_i \to 3\ell_j$ (mediated by Z′ exchange) set tight constraints on the products of couplings. For Z′ bosons accessible at TeV scales, these processes force LFV couplings below $\sim 10^{-2}$, resulting in suppressed $\mathrm{Br}(Z' \to \ell_i \ell_j)$ in most models (Aranda et al., 2012, Espinosa-Gomez et al., 2023).

4. Numerical Results and Model Benchmarks

The range of possible branching ratios in representative models is as follows:

Model/Scenario	$\mathrm{Br}(Z'\to \ell_i \ell_j)$ upper bound	Dominant constraint
E₆-derived Z′ (e.g. $Z_\eta$)	$< 10^{-4}$	$\mu \to e\gamma$ (Espinosa-Gomez et al., 2023)
Sequential/LR Z′	$< 2\times 10^{-7}$	$\mu$–$e$ conversion
General vector-like	$\sim\mathcal{O}(10^{-2})$ (maximal)	$\tau\to 3\mu$
Tree-level flavor violation	up to $\sim1\%$ in less constrained $Z_\chi$ (1-2 TeV)	rare $\tau$ decays (Aranda et al., 2012)
Loop-induced (SUSY/see-saw)	typically $\ll 10^{-9}$	low-energy LFV decays

These bounds can be further tightened by improved experimental sensitivities, especially for low energy rare processes.

Direct collider searches are currently not sensitive to branching fractions below $10^{-7}$ for $Z'\to e\mu$ in the $\sim100$–$500$ GeV range (Collaboration, 11 Aug 2025). Rare decay and conversion processes continue to provide the strongest constraints for most flavor combinations and model parameters.

5. Experimental Techniques and Searches

The search strategy for LFV Z′ decays at colliders such as the LHC employs

• Final state selection: Events with one electron and one muon (or other lepton flavors as appropriate), requiring opposite charges and high-quality reconstruction.
• Invariant mass spectrum analysis: A "sliding window" approach is taken to scan the reconstructed $\ell_i \ell_j$ invariant mass ($m_{e\mu}$, etc.) across a wide mass range, optimizing for possible narrow resonant structures consistent with Z′ production (Collaboration, 11 Aug 2025).
• Multivariate classifiers: To improve background rejection, boosted decision tree (BDT) algorithms are used with variables sensitive to SM background versus LFV signal.
• Background modeling: Data-driven methods, including fits to sideband regions and composite models (Gaussian plus polynomial components), are used. The "discrete profiling" technique allows for systematic uncertainties in the background choice to be accounted for in limit setting.
• Limit extraction: The modified frequentist (CL$_s$) method is used with the combined signal and background models to set upper limits, assuming the narrow width approximation for the signal.

Current analyses are statistically limited, with no significant excess observed above SM backgrounds. All local p-values for excesses exceed 1%, with global p-values well above discovery thresholds (Collaboration, 11 Aug 2025).

6. Implications for Model Building and Future Directions

The results of the most sensitive searches, together with low-energy constraints, significantly constrain the parameter space of new physics models:

• Non-universal Z′ models must arrange for flavor-violating couplings below the limits set by both flavor physics and collider searches (Espinosa-Gomez et al., 2023, Orduz-Ducuara, 2016).
• Heavy Z′ bosons ($\gtrsim$ few TeV) with LFV couplings are tightly constrained, but can still induce detectable LFV in high-luminosity scenarios or if couplings/masses allow (Aranda et al., 2012).
• Models with TeV-scale heavy neutrinos can induce LFV Z′ decays in scenarios with large mixings, though indirect constraints from radiative and three-body decays are often dominant (Romeri et al., 2017, Yang, 2010).
• Future experiments, with increased integrated luminosity or running at high-luminosity/dedicated $e^+e^-$ colliders, have the potential to probe LFV branching ratios below current direct search limits, possibly reaching the regime predicted by certain classes of models.
• Discovery of an LFV Z′ decay would point unequivocally to new physics and would provide a direct handle on the flavor structure and coupling constants of extended gauge sectors, requiring a reassessment of existing flavor symmetries and BSM scenarios.

7. Summary Table: Experimental and Theoretical Bounds

Mass Range	Experimental CLFV Bound (eμ)	Theoretical Max $\mathrm{Br}(Z'\to e\mu)$	Typical Models	Notes
110–500 GeV	$\sigma \times \mathrm{Br} < 7\ \mbox{fb} \rightarrow 0.3$ fb (Collaboration, 11 Aug 2025)	$10^{-7}$–$10^{-4}$ (Espinosa-Gomez et al., 2023)	Non-universal $Z'$, vector-like mixing	Limited by low-energy rare decay constraints
TeV-scale	(no direct limit; rare decays dominate)	$\lesssim 1\%$ (least restricted); $\ll 10^{-6}$ (most restricted)	$Z_\chi$, $Z_{LR}$, $Z_\eta$ (Aranda et al., 2012, Espinosa-Gomez et al., 2023)	Model-dependent; rare processes exclude most of parameter space

A robust experimental program combining high-statistics collider searches and rare decay measurements is essential for advancing tests of LFV Z′ decays and discriminating among new physics scenarios.