Zee models with a non-invertible $Z_M$ symmetry

This paper systematically classifies Zee models incorporating non-invertible $Z_M$ symmetries to identify viable candidates consistent with experimental data, with a detailed numerical analysis of a representative $Z_7$ benchmark model yielding specific predictions for neutrino observables and charged lepton flavor violation.

The Gist

Imagine the universe as a giant, complex puzzle. For a long time, physicists have been trying to figure out why certain puzzle pieces called neutrinos (tiny, ghost-like particles) have mass, while the standard rules of physics suggested they shouldn't.

This paper is like a team of detectives (the authors) trying to solve that puzzle by building a new, more specific version of an old theory called the Zee model. Instead of using standard "rules of symmetry" (like how a snowflake looks the same when you rotate it), they decided to use a very strange, new kind of rule called non-invertible symmetry.

Here is a simple breakdown of what they did and found:

1. The New Rulebook: "Non-Invertible Symmetry"

Think of traditional symmetry like a dance where if you do a move forward, you can always do the exact same move backward to return to the start.
Non-invertible symmetry is like a dance where some moves cannot be undone. If you step forward, you might end up in a place where you can't just step backward to get back to where you started.

The authors used this "un-undoable" rule (specifically a version called $Z_M^{NI}$) to dictate how particles interact. It acts like a strict bouncer at a club:

• It decides which particles are allowed to talk to each other.
• It forbids certain interactions that would normally be allowed.
• This creates a very specific "menu" of allowed interactions, which helps the scientists predict what the universe should look like.

2. The Setup: The Zee Model Kitchen

The Zee model is like a kitchen where neutrino mass is "cooked up" not instantly, but through a slow, one-step cooking process (a "one-loop" mechanism).

• The Ingredients: They added extra "chefs" (new particles like extra Higgs bosons and charged scalars) to the kitchen.
• The Recipe: The new "non-invertible bouncer" rules how these chefs mix their ingredients.
• The Goal: To create a recipe that produces the exact amount of neutrino mass we see in experiments, without adding too many random ingredients (free parameters) that make the theory messy.

3. The Investigation: Sorting the Candidates

The authors went through a massive sorting process:

• They tried assigning different "bouncer codes" (symmetry classes) to the three generations of particles (electrons, muons, and taus).
• They checked which assignments resulted in a "neutrino mass matrix" (a blueprint for how heavy the neutrinos are) that actually matches real-world data.
• The Result: They found that many combinations were "bad recipes" (they didn't fit the data). However, they identified a few "viable candidates" that worked.

4. The Star Player: The $Z_7$ Model

To prove their idea works, they picked one specific, promising recipe based on a $Z_7$ symmetry (think of it as a 7-step dance rule) and ran detailed computer simulations on it.

What they found in this specific model:

• The Texture of Mass: Depending on a specific setting in their model (called $\tan \beta$, which is like a "flavor intensity" knob), the blueprint for neutrino mass changes shape.
  • At some settings, the blueprint has one zero (one missing piece).
  • At other settings, it has two zeros (two missing pieces).
  • This is a unique fingerprint that distinguishes their model from others.
• Predictions:
  • Neutrino Mass: They predict the total mass of neutrinos is quite light (around 60–70 "milli-electron-volts"), which fits within current cosmic limits.
  • Rare Events: They predict that certain extremely rare particle decay events (like a tau particle turning into three muons) should happen at very specific, tiny rates. Currently, these events are too rare to be seen, but their model gives a target for future experiments to look for.
  • CP Violation: They predict specific values for how these particles behave differently from their mirror images (CP phases), which could be tested by future neutrino experiments.

5. The Conclusion

The paper concludes that using these strange, "non-invertible" rules is a powerful new way to build theories about the universe. It naturally filters out bad ideas and leaves behind a few very specific, testable models.

In short: The authors built a new theory using a "un-undoable" rule to explain why neutrinos have mass. They tested one specific version of this theory and found it fits the data well, predicting specific, tiny signals that future experiments might be able to catch. If those signals are found, it would be a huge win for this new way of thinking about particle physics.

Technical Summary

Technical Summary: Zee models with a non-invertible ZM symmetry

Problem Statement
The origin of neutrino masses and mixings remains a primary open question in particle physics, necessitating physics beyond the Standard Model (BSM). While the Zee model is an attractive radiative mechanism generating neutrino masses at the one-loop level via an extended scalar sector, its predictive power is often limited by a large number of free parameters in the Yukawa sector. Traditional symmetry approaches (e.g., $Z_2$, global $U(1)$, or discrete flavor symmetries) have been employed to constrain these parameters, but some specific realizations (like the original $Z_2$ Zee model) have been excluded by experimental data. Furthermore, standard group-based symmetries impose selection rules that may not capture all possible interaction structures. The authors aim to address these limitations by investigating the application of non-invertible symmetries, specifically the non-invertible $\mathbb{Z}_M$ symmetry ($\mathbb{Z}_M^{NI}$), within the Zee model framework to systematically classify viable models and enhance predictability.

Methodology
The authors construct Zee models extended by a second Higgs doublet ($\Phi$) and a charged scalar singlet ($S^+$). They introduce a softly-broken non-invertible $\mathbb{Z}_M^{NI}$ symmetry, where fields are assigned to classes $[g_k]$ derived from the generator $g$ of an underlying $\mathbb{Z}_M$ discrete symmetry.

1. Symmetry Rules: The product of two classes follows the rule $[g_k][g_{k'}] = [g_{k+k'}] + [g_{M-k+k'}]$. Interaction terms are allowed only if the product of the classes of the participating fields contains the trivial class $[g_0]$.
2. Model Classification: The authors systematically assign these classes to the three generations of Standard Model leptons ($L_L, \ell_R$) and the new scalar fields ($\Phi_1, S^+$). They impose constraints to ensure the charged lepton Yukawa matrix ($y_\ell$) remains diagonal and that quark interactions are not restricted.
3. Mass Matrix Derivation: The neutrino mass matrix ($m_\nu$) is generated via one-loop diagrams involving charged scalars and charged leptons. The structure of $m_\nu$ is determined by the product $F m_\ell Y^\dagger$, where $F$ and $Y$ are Yukawa coupling matrices constrained by the $\mathbb{Z}_M^{NI}$ assignments.
4. Numerical Analysis: A representative benchmark model based on $\mathbb{Z}_7^{NI}$ (Model 1 in their classification) is selected for detailed numerical analysis. The authors scan the parameter space of the remaining free Yukawa couplings and the $\tan\beta$ parameter. They perform a $\chi^2$ analysis against global neutrino oscillation data (NuFit 6.1) and impose constraints from charged lepton flavor violation (CLFV) processes (e.g., $\mu \to e\gamma$, $\tau \to \mu\gamma$, $\mu \to eee$, $\tau \to \mu\mu\mu$) and cosmological bounds on the sum of neutrino masses.

Key Contributions

• Systematic Classification: The paper provides a comprehensive classification of Zee models under non-invertible $\mathbb{Z}_M^{NI}$ symmetries for $M=4$ to $M=7$. It identifies viable model candidates that yield neutrino mass structures compatible with current oscillation data.
• Novel Selection Rules: The work demonstrates that non-invertible symmetries generate specific Yukawa coupling structures (zero textures) that differ from those achievable by traditional invertible group symmetries.
• Benchmark Model Analysis: A detailed phenomenological study of a specific $\mathbb{Z}_7^{NI}$ benchmark model is presented. The authors show that the neutrino mass matrix texture depends critically on the value of $\tan\beta$:
  • For finite $\tan\beta$, the model exhibits a one-zero texture.
  • In the limit $\tan\beta \to \infty$, the model approaches a two-zero texture.
• Phenomenological Predictions: The study yields specific predictions for neutrino observables (mixing angles, CP phases, mass sum) and CLFV branching ratios within the allowed parameter space.

Results

• Viability: Models with $\mathbb{Z}_4^{NI}$ through $\mathbb{Z}_7^{NI}$ were examined. The authors found that many assignments lead to neutrino mass structures incompatible with data (e.g., excessive zero textures in the $\tan\beta \to \infty$ limit). The viable models are listed in the appendices.
• Benchmark Model ($\mathbb{Z}_7^{NI}$):
  • Neutrino Observables: For the benchmark model, the sum of neutrino masses ($\sum m_\nu$) is predicted to be in the range of approximately 60–70 meV, consistent with current cosmological limits (Planck+DESI).
  • CP Phases: The Dirac CP phase ($\delta_{CP}$) and Majorana phases ($\alpha_{21}, \alpha_{31}$) show specific correlations depending on $\tan\beta$. For $\tan\beta=1$, $\delta_{CP}$ is preferred in the ranges $[0, 50^\circ]$ and $[300, 360^\circ]$. For higher $\tan\beta$, constraints on phases relax or change character.
  • CLFV: The model predicts highly suppressed CLFV rates. Specifically, $BR(\tau \to 3\mu)$ is concentrated around $1.5 \times 10^{-14}$ for $\tan\beta=1$, and $BR(\tau \to \mu\gamma)$ ranges from $10^{-18}$ to $10^{-16}$. These values are well below current experimental upper bounds but could be probed by future experiments.
  • $\tan\beta$ Dependence: The analysis highlights a strong dependence of predictions on $\tan\beta$. Low $\tan\beta$ imposes tighter constraints on neutrino observables due to the equal contribution of $y_\Phi$ and $y_\ell$ to the charged lepton mass matrix, whereas high $\tan\beta$ leads to a more diagonal charged lepton mass matrix and different correlation patterns.

Significance and Claims
The authors claim that the introduction of non-invertible $\mathbb{Z}_M^{NI}$ symmetry provides a "novel conceptual framework" for constructing BSM models, offering selection rules unattainable by traditional group symmetries. The paper asserts that this framework leads to "unique predictions" for neutrino observables and lepton flavor violations. By identifying viable models and performing a detailed numerical analysis of a benchmark case, the work illustrates the "unique phenomenological features" of the Zee model under non-invertible symmetries. The authors conclude that the distinctive signatures identified, particularly the specific correlations between CP phases and mass sums, and the predicted CLFV rates, suggest these models can be "rigorously tested" by upcoming high-precision neutrino experiments and searches for lepton flavor violations. They also note that the model could be tested via heavy scalar production signals at colliders (LHC, FCC, CEPC, muon colliders), though a detailed study of these signals is left for future work.