⚛️ phenomenology

Lepton flavor violating decays $l_j \rightarrow l_i\gamma$ in the $U(1)_X$ VLFM

This paper investigates lepton flavor violating decays $l_j \to l_i \gamma$ within the $U(1)_X$ vector-like fermion model, demonstrating that specific parameter regions allow branching ratios large enough to be detectable by future experiments.

Original authors: Shuang Di, Wei-Hang Zhang, Zi-Xuan Su, Guo-Zhu Ning, Xing-Xing Dong, Shu-Min Zhao

Published 2026-08-19

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CC BY 4.0

Original authors: Shuang Di, Wei-Hang Zhang, Zi-Xuan Su, Guo-Zhu Ning, Xing-Xing Dong, Shu-Min Zhao

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 world of particle physics, there is a fundamental rule that has held up for decades: the universe treats different types of electrons, muons, and taus as distinct families that do not mix. These particles, known as leptons, are like three different species of birds that never interbreed. In the standard model of physics, which describes how the basic building blocks of matter interact, a heavy muon should never spontaneously transform into a lighter electron while emitting a flash of light. The theory predicts this event is so incredibly rare that it would effectively never happen in the lifetime of the universe. Yet, we know from experiments with neutrinos that nature does allow for mixing in other contexts, suggesting that this strict separation of lepton families might be an illusion rather than a law. If scientists could catch a muon turning into an electron, it would be a smoking gun for new, undiscovered physics that lies beyond our current understanding.

A team of researchers has taken a close look at this possibility using a specific theoretical framework that extends the standard model. They explored a scenario where the universe contains not just the known particles, but also a hidden layer of "vector-like" fermions—particles that behave differently than the ones we see in everyday matter—and an extra force carrier similar to the photon but with a different charge. In this expanded universe, the strict rules preventing lepton families from mixing are loosened. The researchers calculated how these new particles and forces would influence the decay of heavy leptons into lighter ones, specifically looking at the process where a muon or a tau particle decays into an electron or a muon while releasing a photon. Their work involves mapping out the vast landscape of possible values for the masses and interaction strengths of these new particles to see if any combination could make these forbidden decays common enough to be detected by future experiments.

The study reveals that while the standard model predicts these decays are impossible to observe, this new framework allows them to happen at rates that are within reach of current and upcoming detectors. The researchers found that the likelihood of these events depends heavily on the strength of the connections between the known particles and the new, hidden ones. When these connections are strong, the probability of a muon turning into an electron increases significantly. However, this probability is also sensitive to the mass of the new particles involved; heavier particles suppress the effect, making the decay rarer. By running extensive simulations, the team identified specific regions where the decay rates could be large enough to challenge the current experimental limits, which stand at a branching ratio of less than 4.2 times 10 to the power of negative 13 for the muon-to-electron transition. In simpler terms, if you were to watch a trillion trillion muons decay, the current rules say you would see zero instances of this transformation, but this new model suggests that under the right conditions, you might see a few.

The analysis also examined how the different generations of particles contribute to these decays. The researchers discovered that the first two generations of particles play a dominant role in the muon-to-electron transition, while the third generation becomes crucial when a tau particle is involved. This selective behavior helps explain why certain decays might be more visible than others. Furthermore, the team checked if these new particles would cause other observable effects, such as the Higgs boson decaying into different lepton flavors or the existence of a new heavy particle called a Z-prime. They found that the model remains consistent with all existing data: the predicted rates for Higgs decays are far too low to be seen, and the mass of the new Z-prime particle is heavy enough to have escaped detection so far, sitting just above the current experimental threshold of 5.1 tera-electronvolts.

Ultimately, this work provides a clear roadmap for experimentalists. It defines exactly which parameters of this new theoretical model are allowed by current data and which ones are ruled out. The researchers show that if the new particles exist with specific masses and interaction strengths, the next generation of experiments could finally catch a glimpse of lepton flavor violation. This would not only confirm the existence of these hidden particles but also open a window into the deeper structure of the universe, proving that the strict separation of lepton families is indeed a feature of our low-energy world rather than a fundamental law of nature. The study serves as a vital guide, telling us where to look and what to expect if the universe is indeed hiding a richer, more complex reality just beyond our current reach.

Technical Summary: Lepton Flavor Violating Decays $l_j \to l_i \gamma$ in the U(1) $_X$ Vector-Like Fermion Model

Problem Statement
The Standard Model (SM) predicts Lepton Flavor Violating (LFV) decays, such as $\mu \to e\gamma$ , to be extremely suppressed due to the Glashow-Iliopoulos-Maiani (GIM) mechanism, yielding branching ratios on the order of $10^{-55}$ . This is far below current experimental sensitivities (e.g., $Br(\mu \to e\gamma) < 4.2 \times 10^{-13}$ ). However, neutrino oscillation experiments have confirmed non-zero neutrino masses and significant lepton flavor mixing, challenging the SM assumption of lepton flavor conservation. The observation of LFV decays in future experiments would constitute conclusive evidence for physics beyond the SM (BSM). This paper investigates these processes within the framework of the U(1) $_X$ Vector-Like Fermion Model (U(1) $_X$ VLFM), a model designed to address SM shortcomings such as neutrino mass generation and the fermion mass hierarchy while introducing new sources of flavor violation.

Methodology
The authors perform a detailed theoretical and numerical analysis of the LFV radiative decays $l_j \to l_i \gamma$ (where $j = \tau, \mu$ and $i = \mu, e$ ) within the U(1) $_X$ VLFM.

Model Framework: The model extends the SM gauge group to $SU(3)_C \otimes SU(2)_L \otimes U(1)_Y \otimes U(1)_X$ . It introduces:
- Three generations of right-handed neutrinos ( $\nu_R$ ).
- Two scalar singlets ( $\phi$ and $S$ ) in addition to the SM Higgs doublet ( $H$ ).
- One generation of vector-like fermions (quarks, leptons, and neutrinos) to ensure anomaly cancellation.
- A new gauge boson $Z'$ arising from the $U(1)_X$ symmetry, which mixes kinetically with the hypercharge boson.
Analytical Calculation: The authors derive the analytical expressions for the branching ratios of the LFV processes. The calculation involves:
- Constructing the mass matrices for leptons, neutrinos, and neutral scalars, which are diagonalized by unitary mixing matrices.
- Calculating one-loop contributions to the decay amplitude. The relevant Feynman diagrams include loops involving virtual $W$ bosons and neutrinos, $Z$ bosons and leptons, charged Goldstone bosons ( $G^\pm$ ) and neutrinos, and neutral scalars ( $h$ ) and leptons.
- Deriving the Wilson coefficients ( $C_{1,2}^{L,R}$ ) for the effective dipole operators, summing contributions from all relevant loops.
- Expressing the decay width $\Gamma(l_j \to l_i \gamma)$ in terms of these coefficients.
Numerical Analysis: The authors conduct parameter scans to evaluate the branching ratios under current experimental constraints.
- Fixed Parameters: Neutrino Yukawa couplings are fixed to reproduce the normal mass ordering (NO) of neutrino oscillation data. Other parameters include specific values for scalar potential couplings and gauge mixing parameters.
- Variable Parameters: The study varies Yukawa couplings involving vector-like leptons ( $Y_{Xe}$ , $Y_{Pe}$ ), gauge mixing parameters ( $g_{YX}$ , $g_X$ ), and vacuum expectation values (VEVs) of the scalar fields ( $v_S$ , $v_P$ ).
- Constraints: The analysis is constrained by the current 90% confidence level upper limits from the MEG experiment for $\mu \to e\gamma$ , and limits for $\tau \to e\gamma$ and $\tau \to \mu\gamma$ . Constraints from $Z'$ searches (dilepton resonances) and Higgs LFV decays ( $h \to l_i l_j$ ) are also applied.

Key Contributions and Results

Analytical Formulation: The paper provides explicit analytical formulas for the Wilson coefficients and branching ratios in the U(1) $_X$ VLFM, detailing the contributions from $W$ , $Z$ , Goldstone, and scalar loops, including the specific mixing matrices and couplings involved.
Parameter Sensitivity:
- $\mu \to e\gamma$ : The branching ratio is highly sensitive to the first- and second-generation Yukawa couplings ( $Y_{Xe11}$ , $Y_{Xe22}$ ). The study finds that $Y_{Xe11}$ and $Y_{Xe22}$ dominate this process, while the third-generation coupling ( $Y_{Xe33}$ ) has a negligible impact. The branching ratio increases with these couplings and decreases as the scalar VEVs ( $v_S, v_P$ ) increase (due to the mass suppression of new particles).
- $\tau \to e\gamma$ : This process shows a significant dependence on both the first-generation ( $Y_{Xe11}$ ) and third-generation ( $Y_{Xe33}$ ) couplings. The branching ratio increases synergistically with these parameters.
- $\tau \to \mu\gamma$ : The decay rate is strongly dependent on the second-generation coupling ( $Y_{Xe22}$ ) and the new coupling $Y_{Pe}$ . It increases with $Y_{Xe22}$ and decreases with $Y_{Pe}$ .
Experimental Constraints:
- The parameter space is constrained such that for large Yukawa couplings or low scalar VEVs, the predicted branching ratios can exceed current experimental limits (e.g., $Br(\mu \to e\gamma) > 4.2 \times 10^{-13}$ ).
- The study identifies "safe" regions in the parameter space where the model remains consistent with current data but predicts branching ratios potentially within the reach of future experiments (e.g., MEG II).
- Tree-level Higgs LFV decays ( $h \to l_i l_j$ ) are calculated and found to be far below current experimental limits ( $Br \sim 10^{-16} - 10^{-21}$ ), posing no additional constraints.
- The $Z'$ boson mass is constrained to be $m_{Z'} > 5.1$ TeV based on LHC dilepton resonance searches, which is satisfied in the model's viable parameter space ( $m_{Z'} \approx 5.17$ TeV).

Significance
The paper claims that the U(1) $_X$ VLFM provides a rich source of lepton flavor violation through the mixing of vector-like leptons with SM leptons, as well as through extended scalar and gauge sectors. The study demonstrates that the model can generate branching ratios for $l_j \to l_i \gamma$ that are large enough to be probed by current or upcoming experiments, unlike the SM prediction. By mapping the allowed parameter space and identifying the specific sensitivity of different decay channels to various generations of Yukawa couplings, the work offers theoretical guidance and constraints for exploring new physics. The authors conclude that the model has the potential to be tested or excluded in the next generation of LFV experiments, providing a clear theoretical basis for interpreting future observations of lepton flavor violation.

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