⚛️ phenomenology

Lepton-flavor violation and muon $(g-2)$ in the flavor-dependent $U(1)_F$ model

This paper investigates how a flavor-dependent $U(1)_F$ extension of the Standard Model can simultaneously address the muon anomalous magnetic moment and predict lepton-flavor violation processes with branching ratios potentially within the reach of future experiments.

Original authors: Yu-Ju Peng, Feng-Yan Niu, Qi-Zhen Qin, Zhan Cao, Jin-Lei Yang

Published 2026-09-03

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CC BY 4.0

Original authors: Yu-Ju Peng, Feng-Yan Niu, Qi-Zhen Qin, Zhan Cao, Jin-Lei Yang

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the universe of particle physics, there is a set of rules known as the Standard Model that describes how the smallest building blocks of matter behave and interact. For decades, this framework has been remarkably successful, yet it leaves a few mysteries unsolved. One of the most persistent puzzles involves the "flavor" of particles. Just as humans have different names and characteristics, fundamental particles like electrons and muons come in different types, or flavors. In the Standard Model, these flavors are strictly conserved; a muon, for instance, should never spontaneously turn into an electron. If such a transformation were ever observed, it would be a clear signal that the Standard Model is incomplete and that new, hidden forces are at work. Alongside this, physicists have been trying to reconcile a tiny discrepancy between the predicted and measured behavior of the muon's magnetic field, a property that acts like a tiny internal compass. The tension between what theory predicts and what experiments measure in these two areas—flavor changes and magnetic behavior—has driven researchers to look for new theories that could explain both at once.

A team of researchers from Hebei University has taken a fresh look at this problem by exploring a specific theoretical extension called the flavor-dependent U(1)F model. This model proposes the existence of a new force carrier, a particle similar to the photon but associated with a new symmetry that distinguishes between the different flavors of matter. In this framework, the new force does not treat all particles equally; instead, its strength depends on which "flavor" of particle it is interacting with. The researchers built a detailed mathematical structure for this model, introducing new particles and forces to see if they could simultaneously explain the muon's magnetic anomaly and predict how often leptons might change their flavor. They focused on two main areas: the magnetic moment of the muon, which is one of the most precisely measured quantities in physics, and rare decay processes where a heavier lepton, like a muon or a tau, transforms into lighter ones, such as an electron, while emitting a photon or other particles.

The team performed extensive calculations to determine how the new particles in their model would influence these physical phenomena. They found that the new interactions could indeed account for the observed difference in the muon's magnetic moment without contradicting other known experimental data. Furthermore, the study indicates that the new interactions in this model can make significant contributions to lepton flavor violation processes. By running simulations across a wide range of possible values for the model's parameters, the researchers discovered that the predicted branching ratios for these rare decays can well reach the sensitivity of future experiments. They determined that the model's predictions are consistent with current experimental limits, provided that certain parameters, such as the vacuum expectation value $v_1$ , fall within specific ranges. For instance, the mass of a new scalar particle and the strength of the new force must be such that the predicted rates remain below current detection limits but are high enough to be observable by next-generation detectors.

The study reveals that while the model can successfully accommodate the latest measurements of the muon's magnetic moment, it places specific constraints on variables like $v_1$ and the elements of the Yukawa coupling matrix. The researchers mapped out how the likelihood of these rare decays changes as they vary the mass of the new scalar particles and the energy scale of the new force. Their results show that the predicted rates for processes like a muon turning into an electron and a photon are currently just below the detection limits of existing experiments, yet they are well within the reach of future sensitivity. This suggests that the next generation of particle detectors, which are designed to be significantly more sensitive, will be able to test this model directly. If these future experiments observe these rare decays at the levels predicted by the model, it would provide strong evidence for the existence of this new flavor-dependent force. Conversely, if the experiments find nothing, the specific version of the model explored here would be ruled out.

Ultimately, the work serves as a bridge between theoretical speculation and experimental reality. The researchers did not claim to have discovered new particles, but rather provided a clear roadmap for how to find them. They demonstrated that a universe with this specific type of new force is mathematically consistent with current data, provided that the new particles are heavy enough to hide from our current instruments but light enough to leave a trace in the muon's magnetic behavior. The paper concludes that the predicted branching ratios for these flavor-changing processes are within the reach of future experiments, offering a tangible target for physicists to aim for. The findings suggest that the answer to the muon's magnetic mystery and the question of lepton flavor violation may lie in the same new physics, waiting to be uncovered by more precise measurements in the coming years.

Technical Summary: Lepton-Flavor Violation and Muon (g −2) in the Flavor-Dependent U(1)F Model

Problem Statement
Lepton flavor violation (LFV) processes, such as $l^-_j \to l^-_i \gamma$ and $l^-_j \to l^-_i l^-_i l^+_i$ , are strictly forbidden in the Standard Model (SM) due to the conservation of lepton flavor numbers. Consequently, their observation would constitute unambiguous evidence of New Physics (NP). While the SM successfully predicts many observables, the muon anomalous magnetic dipole moment ( $a_\mu$ ) has historically shown tension with theoretical predictions, though recent Fermilab results indicate a reduced discrepancy ( $0.5\sigma$ ). The authors investigate whether a specific extension of the SM, the Flavor-Dependent $U(1)_F$ model (FDM), can simultaneously accommodate the latest experimental constraints on the muon $(g-2)$ and generate significant, potentially observable contributions to LFV processes.

Methodology
The study employs the FDM, an extension of the SM gauge group $SU(3)_C \otimes SU(2)_L \otimes U(1)_Y$ by an additional local gauge symmetry $U(1)_F$ . The model is characterized by the following features:

Scalar Sector: To generate fermion masses and mixings, the model introduces two additional Higgs doublets ( $\Phi_1, \Phi_2$ ) and one scalar singlet ( $\chi$ ), alongside the SM-like doublet ( $\Phi_3$ ). The third generation of fermions acquires mass via tree-level couplings to $\Phi_3$ , while the first two generations obtain mass through tree-level mixings with the third generation.
Fermion Sector: The model utilizes the Type-I seesaw mechanism to generate neutrino masses. Due to the specific $U(1)_F$ charge assignments (where the third-generation right-handed neutrino is a trivial singlet), only two right-handed neutrinos with non-zero charges are introduced, leaving one active neutrino massless at the tree level.
Gauge Sector: The model includes a new gauge boson $Z'$ associated with the $U(1)_F$ symmetry. The gauge couplings are flavor-dependent, directly linking the new interactions to the flavor structure of fermions.

The authors perform analytical calculations at the one-loop level to derive expressions for:

The muon anomalous magnetic moment ( $\Delta a_\mu$ ).
The decay widths and branching ratios for LFV processes: $l^-_j \to l^-_i \gamma$ and $l^-_j \to l^-_i l^-_i l^+_i$ .

These calculations involve contributions from charged Higgs bosons ( $H^\pm$ ), neutral scalars/pseudoscalars ( $h_\alpha, A_\alpha$ ), neutrinos, and the $Z'$ boson. The numerical analysis scans the model's parameter space (including vacuum stability and perturbativity constraints, collider limits on $Z'$ mass, and Higgs signal strengths) to identify regions consistent with current data.

Key Contributions and Results

Muon $(g-2)$ Constraints: The numerical analysis demonstrates that the new contributions from the FDM to $\Delta a_\mu$ can satisfy the latest experimental results. The study finds that $\Delta a_\mu$ exhibits a monotonically decreasing trend with increasing charged Higgs mass ( $M_{H^\pm}$ ). For relatively light charged Higgs bosons ( $0 < M_{H^\pm} < 100$ GeV), the model predicts a wider dispersion of values, but the dominant contributions are suppressed by heavy scalar masses.
LFV Branching Ratios: The paper presents numerical predictions for the branching ratios of various LFV transitions ( $\mu \to e\gamma$ $μ \to e γ$ , $\mu \to 3e$ $μ \to 3 e$ , $\tau \to e\gamma$ $τ \to e γ$ , $\tau \to 3e$ $τ \to 3 e$ , $\tau \to \mu\gamma$ $τ \to μ γ$ , $\tau \to 3\mu$ $τ \to 3 μ$ ).
- Dependence on $v_1$ : The branching ratios are found to decrease as the vacuum expectation value (VEV) $v_1$ increases. For small values of $v_1$ , the predicted rates for $\mu \to e\gamma$ and $\mu \to 3e$ exceed current experimental upper bounds, thereby imposing strict lower limits on $v_1$ .
- Dependence on Yukawa Couplings: The parameter $|m_{e,13}|$ (related to the electron mass matrix element) shows a strong correlation with $\mu \to e\gamma$ and $\mu \to 3e$ rates; increasing $|m_{e,13}|$ suppresses these branching ratios. This suppressive effect is less pronounced for $\tau$ -lepton transitions.
- CP Violation Phases: The CP-violating phases ( $\theta_{e,ij}$ ) significantly influence the mixing strength between flavors, affecting the predictions for $\mu \to e\gamma$ , $\mu \to 3e$ , $\tau \to e\gamma$ , and $\tau \to 3e$ , while having negligible effects on $\tau \to \mu\gamma$ and $\tau \to 3\mu$ .
Future Sensitivity: The authors note that while current experimental limits constrain the parameter space, the predicted branching ratios for several LFV processes are within the reach of future experimental sensitivities (e.g., $\mu \to e\gamma$ at $\sim 6 \times 10^{-14}$ and $\mu \to 3e$ at $\sim 10^{-16}$ ).

Significance and Claims
The paper claims that the Flavor-Dependent $U(1)_F$ model offers a simultaneous resolution to the puzzles of flavor mixings and fermion mass hierarchies while naturally generating neutrino masses. The primary significance of this work lies in demonstrating that the model's new interactions, constrained by the latest precise measurements of the muon anomalous magnetic moment, can still yield significant contributions to LFV processes.

The authors conclude that the predicted branching ratios for LFV transitions can reach the sensitivity levels of future experiments. Specifically, the model suggests that processes like $\mu \to e\gamma$ and $\mu \to 3e$ are promising candidates for observation in next-generation experiments, provided the model parameters (such as $v_1$ and Yukawa couplings) fall within the allowed regions defined by current $(g-2)$ and collider constraints. The work underscores the complementary nature of precision $(g-2)$ measurements and LFV searches in probing flavor-dependent new physics.

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