Radiative lepton model in a non-invertible fusion rule

This paper proposes a radiative lepton mass model based on a non-invertible $Z_2$ gauging of a $Z_5$ fusion rule, where charged-lepton masses arise from dynamical symmetry breaking while neutrino masses are generated without breaking the rule, leading to testable predictions for lepton flavor violations, anomalous magnetic moments, electric dipole moments, and neutrinoless double beta decay that align with experimental data.

The Gist

Imagine the universe as a giant, complex kitchen where particles are the ingredients. For decades, physicists have been trying to figure out why some ingredients (like electrons and neutrinos) are so incredibly light, while others are heavy. This paper proposes a new recipe—a "Radiative Lepton Model"—that uses a very unusual set of cooking rules to explain how these light particles get their mass.

Here is the story of their discovery, broken down into simple concepts:

1. The Unusual Rulebook: The "Non-Invertible" Fusion

In most physics models, scientists use standard "symmetry rules" (like a recipe that says "if you mix A and B, you always get C"). These rules are like a strict librarian who never lets you change the order of books.

The authors of this paper introduce a new, stranger rulebook called a "Non-invertible Fusion Rule."

• The Analogy: Imagine a magic kitchen where mixing two ingredients doesn't just give you one result, but a mixture of possibilities. If you mix Ingredient A and Ingredient B, you might get a bowl containing both C and D.
• The Magic Trick: This rulebook has a special property: it can forbid certain dishes from being made at the start (the "tree level"), but it allows them to appear later if you cook them in a specific, roundabout way (the "loop level").

2. The Two Types of Particles: The "Forbidden" and the "Allowed"

The paper focuses on two types of particles: Charged Leptons (like electrons and muons) and Neutrinos (ghostly particles that barely interact with anything).

• The Charged Leptons (The "Dynamically Broken" Dish):
  The new rulebook says, "You cannot make an electron mass right now." It's forbidden. However, the authors show that if you cook the electron mass in a one-loop "fryer" (a complex quantum process involving other particles), the rulebook gets "broken" just enough to let the mass appear.

  • Analogy: It's like a security guard who won't let you enter the VIP room directly. But if you go through a back alley, knock on a specific door, and do a secret handshake, the guard lets you in. The door is only open because you took the long way around.

• The Neutrinos (The "Perfectly Protected" Dish):
  For neutrinos, the rulebook is stricter. Even after the long cooking process, the rulebook never breaks. The neutrino mass is generated in a way that respects the rule perfectly.

  • Analogy: Imagine a vault that is so secure, even if you try to pick the lock or blow it open, the vault remains sealed. Yet, somehow, the treasure inside (the neutrino mass) still gets created without ever opening the vault.

3. The "Spicy" Ingredients: CP Phases

The recipe includes some "spicy" ingredients called CP phases. In physics, these are like hidden flavors that can make matter behave differently from antimatter.

• The authors found that because their "back alley" cooking method (the charged lepton mass generation) is so complex, it creates these spicy flavors naturally.
• This is important because it predicts that these particles should have tiny "electric dipole moments" (EDMs). Think of an EDM as a tiny internal magnet or a slight wobble in the particle's shape. The paper predicts these wobbles are much larger than what simpler theories suggest, making them potentially detectable in future experiments.

4. The Taste Test: Numerical Results

The authors ran a massive computer simulation (a "taste test") to see if their recipe matches reality. They adjusted the amounts of ingredients (masses, angles, and phases) to see if they could reproduce what we see in the real world.

They tested two scenarios:

1. Normal Hierarchy (NH): Like a pyramid where the lightest particles are at the bottom.
2. Inverted Hierarchy (IH): Like an upside-down pyramid.

The Results:

• Neutrinoless Double Beta Decay: This is a rare event where two neutrons turn into two protons without emitting neutrinos. The paper predicts that if the electron's "wobble" (EDM) is small enough to pass current safety checks, then this rare decay event has a very specific, limited range of probabilities. It's like saying, "If the cake isn't too sweet, it must be baked at exactly 350 degrees."
• The "Wobble" (EDMs): The paper predicts that the "wobble" for muons and taus (heavier cousins of the electron) is surprisingly large—thousands of times bigger than what older, simpler theories predicted. This is because the "spicy flavors" in their model come from a different source than in those older theories.
• Neutrino Mixing: The model successfully reproduces the known angles at which neutrinos "mix" (change from one type to another) as they travel through space.

5. What About Dark Matter?

The authors briefly mention that their model could have a candidate for Dark Matter (the invisible stuff holding galaxies together). However, after running their numbers, they found that in their specific setup, these candidates would decay (fall apart) too quickly to be the Dark Matter we see in the universe today. So, they decided to leave that part of the menu for another day and focus on the particles we can actually measure.

Summary

In short, this paper proposes a new way to cook up particle masses using a magical, non-standard rulebook.

• Electrons get their mass by sneaking through a loophole in the rules.
• Neutrinos get their mass while perfectly obeying the rules.
• This sneaky method creates unique "flavors" (CP phases) that predict measurable "wobbles" in particles, offering a new way to test if this recipe is the correct one for our universe.

The authors conclude that while they can't explain Dark Matter with this specific setup, their model offers a rich playground for testing new physics through precise measurements of particle behavior.

Technical Summary

Technical Summary: Radiative Lepton Model in a Noninvertible Fusion Rule

Problem and Motivation
The paper addresses the origin of minuscule lepton masses (active neutrinos and charged leptons) within the framework of physics beyond the Standard Model (BSM). While radiative seesaw models are attractive for generating small masses, the authors aim to utilize a specific theoretical structure: noninvertible fusion rules. Unlike conventional discrete symmetries (e.g., $Z_N$) or continuous global symmetries (e.g., $U(1)$), noninvertible fusion rules possess unique properties: they can forbid specific terms at the tree level while allowing them to be generated dynamically at the loop level, and they can provide exact symmetries (such as a $Z_2$) that remain unbroken to stabilize dark matter candidates. The authors note that standard symmetries often fail to realize scenarios where charged-lepton masses are radiatively generated via symmetry breaking while neutrino masses are generated without breaking the symmetry. This work proposes a model utilizing a $Z_2$ gauging of a $Z_5$ fusion rule, denoted as $Z_5^{NI}$, to achieve this specific separation of mechanisms.

Methodology and Model Setup
The authors construct a radiative lepton mass model extending the Standard Model particle content with:

1. Three families of left- and right-handed neutral Majorana fermions ($N_{L/R}$).
2. An isospin doublet inert boson $\eta = [\eta^+, (\eta_R + i\eta_I)/\sqrt{2}]^T$.
3. An isospin singlet singly charged boson $S^-$.

The model is governed by the $Z_5^{NI}$ fusion rule with elements $\{I, a, b\}$ and multiplication rules: $a \otimes a = I \oplus b$, $a \otimes b = a \oplus b$, and $b \otimes b = I \oplus a$. The charge assignments (Table I) are designed such that:

• Tree-level suppression: All renormalizable lepton mass terms are forbidden by the fusion rule.
• Charged-lepton generation: The charged-lepton mass matrix is generated at the one-loop level via the dynamical breaking of the fusion rule. This occurs through interactions involving the scalar potential term $\mu_0 (H^T i\tau_2 \eta S^-)$, which connects the inert doublet and the charged singlet. The resulting effective operator violates the fusion rule (specifically, the product $a \otimes b$ does not contain the identity $I$).
• Neutrino generation: The neutrino mass matrix is also generated at the one-loop level but without breaking the fusion rule. This is achieved via the interaction term $\lambda''_{H\eta} (H^\dagger \eta)^2$, where the fusion product $b \otimes b$ contains the identity $I$, preserving the symmetry. This mechanism is analogous to the Ma model, providing an accidental $Z_2$ symmetry that stabilizes potential dark matter candidates.

The Lagrangian includes Yukawa couplings ($y_\ell, f_{R/L}, g_{R/L}$) and a scalar potential. The physical CP phases arise from complex parameters in the Yukawa sector and the Majorana mass terms, which cannot be fully rephased away.

Key Contributions and Numerical Analysis
The authors perform a numerical $\chi^2$ analysis to constrain the model's parameter space, fixing the absolute values of free parameters to satisfy experimental data for lepton masses, mixing angles, and mass-squared differences (using NuFit 6.0). They analyze two scenarios: Normal Hierarchy (NH) and Inverted Hierarchy (IH).

1. Phenomenological Constraints: The model is tested against:

   • Neutrino oscillation data ($\sin^2 \theta_{12,13,23}$, $\Delta m^2_{sol,atm}$).
   • Lepton Flavor Violations (LFVs): $BR(\mu \to e\gamma)$, $BR(\tau \to e\gamma)$, $BR(\tau \to \mu\gamma)$.
   • Anomalous magnetic moments: $\Delta a_e$ and $\Delta a_\mu$.
   • Charged-lepton Electric Dipole Moments (EDMs): $d_e, d_\mu, d_\tau$.
   • Neutrinoless double beta decay ($\langle m_{ee} \rangle$).

2. Results:

   • NH Case: The best-fit point yields $\Delta \chi^2_{min} \approx 1.06$. The model predicts allowed regions for the Dirac CP phase ($30^\circ \lesssim \delta_{CP} \lesssim 60^\circ$ and $200^\circ \lesssim \delta_{CP} \lesssim 240^\circ$) and Majorana phases. The predicted electron EDM ($|d_e|$) reaches the current experimental upper bound ($4.1 \times 10^{-30}$ e cm), which in turn constrains the neutrinoless double beta decay effective mass to $\langle m_{ee} \rangle \lesssim 28$ meV.
   • IH Case: The best-fit point yields $\Delta \chi^2_{min} \approx 0.539$. The model predicts $\delta_{CP} \sim 0^\circ, 180^\circ$ and specific Majorana phase ranges. The constraint on $|d_e|$ leads to a maximum $\langle m_{ee} \rangle \approx 33.5$ meV.
   • EDM Magnitudes: A notable finding is that the predicted muon and tau EDMs ($|d_\mu|, |d_\tau|$) are several orders of magnitude larger than those predicted by minimal flavor violation relations (e.g., $|d_\mu| \sim 10^{-25}$ e cm). This is attributed to the distinct origins of the physical CP phases for different lepton flavors in this model.
   • Dark Matter: While the model contains potential dark matter candidates (the lightest neutral fermion or the lightest neutral component of $\eta$), the numerical analysis suggests their masses would be at the PeV scale for $m_S \sim 100$ GeV, leading to rapid decay. Thus, the authors conclude these candidates are not viable dark matter in the specific parameter space explored, shifting focus to the charged-lepton phenomenology.

Significance and Claims
The paper claims to demonstrate a viable scenario where a noninvertible fusion rule successfully differentiates the generation mechanisms of charged-lepton and neutrino masses. Specifically, it shows that:

• Noninvertible symmetries can forbid tree-level charged-lepton masses while allowing them to be generated radiatively through symmetry breaking, a feature not achievable with standard Abelian or non-Abelian discrete symmetries.
• The same framework can generate neutrino masses without breaking the symmetry, preserving a stabilizing $Z_2$ symmetry.
• The model provides testable predictions for lepton flavor violations, anomalous magnetic moments, and EDMs.
• The constraint on the electron EDM serves as a powerful indirect probe, limiting the neutrinoless double beta decay effective mass to values near the current experimental sensitivity limits.

The authors modestly state that while they have constructed a model where all lepton masses are radiative, the primary goal was to demonstrate the feasibility of this mechanism rather than to derive unique, specific predictions for all observables due to the large number of free parameters. They also suggest that the mechanism could be extended to the quark sector (specifically down-type quarks) but leave that analysis for future work.