LFV decays in a 3-4-1 model with minimal inverse seesaw neutrinos

This paper investigates an extended 3-4-1 model incorporating a new singly charged Higgs boson and the minimal inverse seesaw mechanism, demonstrating that it can simultaneously explain the electron and muon $(g-2)$ anomalies and lepton-flavor-violating decay rates while predicting strong, testable correlations between these observables that are consistent with current experimental bounds.

The Gist

Imagine the Standard Model of physics as a giant,精密ly tuned orchestra. For decades, every instrument has played its part perfectly, creating a beautiful symphony of how the universe works. But recently, the musicians playing the "electron" and "muon" (two types of tiny particles) have started playing slightly out of tune.

Scientists have measured how these particles spin and wobble in a magnetic field (a property called the magnetic moment, or $(g-2)$). The measurements don't quite match the sheet music (the Standard Model's predictions). It's like the conductor says, "You should be playing a C," but the instrument is clearly playing a C-sharp.

This paper is about a new theory proposed by a team of physicists from Vietnam to fix this "out-of-tune" orchestra. They suggest adding a few new instruments to the band to make the music sound right again, but they have to be careful not to create a new kind of noise (forbidden particle decays) that the audience would notice.

Here is the breakdown of their idea, using simple analogies:

1. The Problem: The "Out-of-Tune" Particles

The electron and the muon are like two siblings. The muon is the heavier, more energetic brother. Both are showing signs of "anomalies"—they are behaving in ways the current rules of physics can't explain.

• The Muon Anomaly: The muon's wobble is about 10 times bigger than expected.
• The Electron Anomaly: The electron's wobble is also off, but in a different direction.

2. The Solution: A New "3-4-1" Orchestra

The authors propose a new model called the 3-4-1 model with Minimal Inverse Seesaw (341mISS).

• The "3-4-1" Part: Imagine the orchestra is usually organized into groups of 3. This new model reorganizes the musicians into groups of 4, with one special "extra" member. This new structure allows for new interactions.
• The "Inverse Seesaw" Part: Think of neutrinos (ghostly particles that rarely interact) as a seesaw. Usually, to get a light neutrino, you need a super-heavy partner. But in this "Inverse" version, the seesaw is balanced by a tiny, almost invisible weight (a tiny violation of a conservation law). This allows the heavy partners to be light enough (at the "TeV scale") to be found by our current particle colliders, unlike the super-heavy ones from older theories.
• The "Singly Charged Higgs": They also add a new type of "Higgs boson" (the particle that gives others mass) that carries a single electric charge. Think of this as a new, special conductor's baton that can change the rhythm of the electron and muon, fixing their wobble.

3. The Catch: The "Forbidden Dance" (Lepton Flavor Violation)

In physics, there's a strict rule: A muon cannot just turn into an electron and a photon (a particle of light) on its own. It's like a dancer being forbidden to suddenly switch costumes and dance a different style in the middle of a performance. This is called Lepton Flavor Violation (LFV).

If the authors' new model is correct and fixes the "out-of-tune" wobble, it might accidentally allow these forbidden dances to happen.

• The Tightrope Walk: The paper investigates if they can tune the new model to fix the wobble without making the forbidden dances happen too often.
• The Result: They found a "sweet spot." They can adjust the parameters (like the size of the new particles and how strongly they interact) so that:
  1. The muon and electron wobbles are fixed to match the new experimental data.
  2. The rate of forbidden dances (like a muon turning into an electron) stays low enough to not have been caught by current experiments yet, but high enough that future, more sensitive experiments might see them soon.

4. The Prediction: What to Look For

The paper is essentially a "treasure map" for future experiments.

• The "Tau" Connection: They found a strong link between the muon's wobble and a specific forbidden dance: The Tau particle turning into a Muon and a Photon ($\tau \to \mu \gamma$).
• The Test: If the next generation of particle detectors (like the ones coming online in 2026 and beyond) sees this specific dance happening at a rate of about $5 \times 10^{-10}$ (a very tiny number, but detectable), it would be a massive "Aha!" moment. It would prove that this new 3-4-1 model is the correct way to fix the orchestra.
• The Warning: If the new detectors don't see this dance, or if they see it happening way more often than predicted, this specific model might be ruled out.

Summary

Think of this paper as a mechanic's blueprint.

• The Car: The Standard Model of physics.
• The Problem: The engine (muon/electron behavior) is making a weird noise.
• The Fix: A new part (the 3-4-1 model with new Higgs and neutrinos) that can silence the noise.
• The Risk: Installing this new part might cause a leak (forbidden particle decays).
• The Verdict: The mechanic calculated that if you install the part just right, the noise stops, and the leak is small enough to be invisible now, but visible soon.

The authors are telling the experimentalists: "Keep your eyes on the Tau-to-Muon decay channel. If you see it there, you've found the new physics we predicted!"

Technical Summary

1. Problem Statement

The paper addresses two major anomalies in particle physics that remain unexplained by the Standard Model (SM):

1. The $(g-2)_{e,\mu}$ Anomalies: Significant deviations between experimental measurements and SM predictions for the anomalous magnetic moments of the electron ($a_e$) and muon ($a_\mu$). Specifically, the muon anomaly shows a $1\sigma$ deviation of $\Delta a_\mu \approx (3.8 \pm 6.3) \times 10^{-10}$, while electron data suggests potential deviations of order $O(10^{-13})$.
2. Lepton Flavor Violation (LFV): The SM predicts negligible rates for charged lepton flavor-violating decays (e.g., $\mu \to e\gamma$, $\tau \to \mu\gamma$, $h \to \mu e$, $Z \to \mu e$). Current and future experiments (MEG II, Belle II, HL-LHC, FCC-ee) are pushing sensitivity limits, creating a tension between models that explain $(g-2)$ anomalies and those that satisfy stringent LFV bounds.

Previous versions of the 3-4-1 model (based on $SU(4)_L \times U(1)_X$ gauge symmetry) struggled to explain large $(g-2)$ deviations because they typically predict very heavy gauge bosons (TeV scale), leading to suppressed one-loop contributions ($\Delta a_\mu \le O(10^{-11})$).

2. Methodology

The authors propose and analyze an extended 3-4-1 model with Minimal Inverse Seesaw (mISS) neutrinos and a new singly charged Higgs boson ($s^\pm$).

• Model Structure:

  • Gauge Group: $SU(4)_L \times U(1)_X$.
  • Lepton Sector: Three left-handed lepton quadruplets ($L_a$) containing standard leptons, new heavy charged leptons ($E'$), and new neutral fermions. Right-handed neutrinos are assigned to these quadruplets.
  • Neutrino Mass Mechanism: Implements the Minimal Inverse Seesaw (mISS) with only two pairs of SM gauge-singlet neutrinos ($X_R$). This reduces the particle content compared to full ISS models. The mass matrix is $7 \times 7$, predicting one massless light neutrino at tree level (a distinctive feature of mISS).
  • Higgs Sector: Includes four Higgs quadruplets ($\chi, \phi, \rho, \eta$) and a new singly charged Higgs boson ($s^\pm$). The model utilizes a Two-Higgs-Doublet-like structure ($\rho, \eta$) at the electroweak scale after symmetry breaking.
  • Symmetry Breaking: The symmetry breaks in steps: $SU(4)_L \times U(1)_X \to SU(3)_L \times U(1)_N \to SU(2)_L \times U(1)_Y \to U(1)_{EM}$.

• Analytical Framework:

  • Yukawa Interactions: Derived couplings for charged leptons, neutrinos, and the new charged Higgs.
  • Mixing Matrices: Defined the unitary mixing matrix $U^\nu$ relating flavor states to mass eigenstates (3 light + 4 heavy neutrinos).
  • One-Loop Calculations: Calculated contributions to:
    • Anomalous magnetic moments ($\Delta a_e, \Delta a_\mu$).
    • Charged Lepton Flavor Violation (cLFV): $l_b \to l_a \gamma$.
    • LFV Higgs decays ($h \to l_a l_b$).
    • LFV Z boson decays ($Z \to l_a l_b$).
  • Dominant Contributions: The authors determined that contributions from $SU(4)_L$ gauge bosons ($W_{24}, W_{23}$) are suppressed due to their heavy masses. The dominant contributions to $\Delta a_{e,\mu}$ and LFV processes arise from the exchange of heavy neutrinos and the singly charged Higgs bosons ($h^\pm_{1,2}$).

• Numerical Analysis:

  • Scanned the parameter space using neutrino oscillation data (Normal Ordering assumed).
  • Free Parameters: Heavy neutrino masses ($0.1 - 5$ TeV), charged Higgs masses ($0.5 - 5$ TeV), $\tan\beta$ ($5-25$), mixing angles, and Yukawa couplings.
  • Constraints: Enforced consistency with current experimental upper bounds on LFV branching ratios (Table I in the paper) and the $1\sigma$ range of $(g-2)_\mu$.

3. Key Contributions

1. Resolution of $(g-2)$ Anomalies in 3-4-1: Demonstrated that the 3-4-1 model, when supplemented with mISS neutrinos and a light singly charged Higgs, can naturally accommodate large deviations in both $(g-2)_\mu$ (up to $10^{-9}$) and $(g-2)_e$ (up to $O(10^{-13})$), overcoming the suppression issues of previous 3-4-1 realizations.
2. Minimal Neutrino Content: Showed that only four heavy right-handed neutrinos (two pairs) are sufficient to generate the required phenomenology, making the model more predictive and testable than models with three pairs.
3. Strong Correlations: Identified a critical, strong correlation between the muon anomalous magnetic moment ($\Delta a_\mu$) and the branching ratio of the tau decay $Br(\tau \to \mu\gamma)$.
4. Predictive Power: Established that the model is more predictive than previous versions because the parameter space allowing large $\Delta a_\mu$ is tightly constrained by upcoming LFV sensitivities.

4. Results

• $(g-2)_\mu$ and LFV Constraints:

  • The model can explain the current $1\sigma$ deviation of $\Delta a_\mu \approx 10^{-9}$.
  • However, achieving this large deviation requires specific parameter regions (typically small $\tan\beta$).
  • Crucial Finding: The current upper bound on $Br(\tau \to \mu\gamma)$ imposes a stringent constraint. If $\Delta a_\mu$ is near the upper limit of the $1\sigma$ range ($\sim 10^{-9}$), the predicted $Br(\tau \to \mu\gamma)$ is close to the current experimental limit ($4.2 \times 10^{-8}$).
  • Future Sensitivity: The projected sensitivity of future experiments ($Br(\tau \to \mu\gamma) < 6.9 \times 10^{-9}$) will reduce the allowed maximum deviation for $\Delta a_\mu$ to approximately $5 \times 10^{-10}$. This implies that future non-observation of $\tau \to \mu\gamma$ will severely restrict the model's ability to explain the full $(g-2)_\mu$ anomaly.

• Electron Anomaly ($\Delta a_e$):

  • The model can accommodate $\Delta a_e$ in the range $O(10^{-13})$.
  • Unlike the muon case, $\Delta a_e$ shows a weaker correlation with $Br(\mu \to e\gamma)$, though $Br(\mu \to e\gamma)$ remains the most stringent bound overall.

• Other LFV Channels:

  • Higgs Decays ($h \to \mu e, \tau \mu, \tau e$): The model predicts rates that can reach current experimental limits (e.g., $Br(h \to \mu e)$ can approach $O(10^{-5})$) but generally remain below the most stringent bounds.
  • Z Boson Decays ($Z \to \mu e, \tau \mu, \tau e$): Predicted rates are generally below current limits but could be probed by future colliders (HL-LHC, FCC-ee).
  • Tau Electron Decay ($\tau \to e\gamma$): This channel shows a strong correlation with $\Delta a_e$ but remains below current bounds, potentially becoming observable with improved sensitivity.

5. Significance

• Testability: The paper transforms the 3-4-1 model from a theoretical construct into a highly testable framework. The strong correlation between $\Delta a_\mu$ and $Br(\tau \to \mu\gamma)$ means that the next generation of experiments (MEG II, Belle II) can effectively validate or rule out this specific realization of the 3-4-1 model.
• Model Discrimination: The results distinguish the 341mISS model from other BSM scenarios. The requirement of a light charged Higgs and specific heavy neutrino masses to satisfy both $(g-2)$ and LFV constraints provides a unique "fingerprint" for this model.
• Future Outlook: The study concludes that if future experiments confirm the $(g-2)_\mu$ anomaly at the $10^{-9}$ level, the model predicts a near-future discovery of $\tau \to \mu\gamma$. Conversely, if $\tau \to \mu\gamma$ is not observed at the projected sensitivity, the model will be forced to predict a smaller $\Delta a_\mu$ (closer to $5 \times 10^{-10}$), potentially requiring a re-evaluation of the model's parameters or the nature of the anomaly itself.

In summary, this work provides a rigorous phenomenological analysis of a minimal extension of the 3-4-1 model, demonstrating its viability in explaining lepton magnetic moment anomalies while highlighting the critical role of future LFV searches in constraining the model's parameter space.