Dirac one-loop seesaw in a non-invertible fusion rule

This paper proposes a minimal radiative Dirac neutrino mass model stabilized by a non-invertible fusion rule from a $Z_3 \times Z_3'$ gauging, which successfully generates one-loop neutrino masses, provides a viable bosonic dark matter candidate, and remains consistent with current lepton flavor violation and magnetic moment constraints.

The Gist

Imagine the Standard Model of physics as a giant, incredibly complex Lego castle. It explains almost everything we see in the universe, from the atoms in your body to the stars in the sky. But there are two missing pieces that don't fit: Neutrinos (ghostly particles that have mass, but we don't know why) and Dark Matter (the invisible "glue" holding galaxies together).

This paper proposes a new, clever way to snap these two missing pieces into the castle using a special kind of "Lego rule" called a Non-Invertible Fusion Rule.

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

1. The Mystery of the Ghostly Neutrinos

Neutrinos are like ghosts. They pass through you by the billions every second without touching anything. For a long time, scientists thought they had no mass. But we now know they do have a tiny, tiny mass.

Usually, to give a particle mass, you need a "tree-level" connection (a direct link). But for neutrinos, that direct link is forbidden. It's like trying to build a bridge between two islands, but the blueprints say, "No bridges allowed!"

2. The "One-Loop" Detour

Since a direct bridge is forbidden, the authors propose a detour. Imagine you want to get from Island A to Island B, but you can't build a bridge. Instead, you take a boat to a small, hidden island (Island C), swap your boat for a different one, and then sail to Island B.

In physics terms:

• The Direct Path (Tree-level): Forbidden by the rules.
• The Detour (One-loop): Neutrinos gain mass by interacting with new, invisible particles (exotic fermions and inert scalars) in a loop. It's a roundabout way to get mass, which explains why the mass is so incredibly small.

3. The New "Lego Rule" (Non-Invertible Fusion)

This is the paper's biggest innovation. Usually, physicists use symmetries (like a mirror image) to forbid certain interactions. If you have a symmetry, you can usually "undo" it (invert it).

The authors use a Non-Invertible Fusion Rule. Think of this like a magical recipe:

• Normal Rule: If you mix Ingredient A and Ingredient B, you get C. If you mix C and B, you get A back. (Reversible).
• Non-Invertible Rule: If you mix Ingredient A and Ingredient B, you get C. But if you try to mix C and B, you don't get A back. The recipe changes!

Why is this cool?
This rule acts like a bouncer at a club.

• At the entrance (Tree-level): The bouncer says, "No direct mass for neutrinos!" (Because the rule forbids it).
• Inside the club (One-loop level): The bouncer relaxes the rules slightly. The neutrinos can sneak in through the back door (the loop) to get their mass.

This allows the model to be very "minimal" (using fewer new particles than other theories) because the rule does a lot of the heavy lifting automatically.

4. Solving Two Problems at Once: The Dark Matter Candidate

The model introduces new particles to create the neutrino mass loop. But here's the twist: one of these new particles is perfect for being Dark Matter.

• The Bosonic Candidate (The Invisible Shield): They found that a specific new particle (a boson) is stable, doesn't interact with light, and has the right amount of "stuff" in the universe to explain Dark Matter. It's like finding a spare tire that also happens to be a perfect shield for your car.
• The Fermionic Candidate (The Failed Attempt): They also tried using a different particle (a fermion) as Dark Matter. However, this particle is too "shy." It doesn't collide with other particles often enough to explain the amount of Dark Matter we see. It's like trying to fill a swimming pool with a single drop of water; it just doesn't work.

5. The Safety Check

The authors ran the numbers to make sure their new castle doesn't collapse.

• Lepton Flavor Violation: They checked if this model would cause electrons to randomly turn into muons (a type of particle). The answer? No. The model predicts these events are so rare that current experiments won't see them. This is good because we haven't seen them yet!
• Magnetic Moments: They checked if the model messes up the magnetic properties of electrons. Again, the numbers are tiny and safe.

The Bottom Line

This paper is like an architect presenting a new, elegant blueprint for the universe.

1. It explains Neutrino Mass: By using a "detour" (one-loop) instead of a direct path.
2. It explains Dark Matter: By identifying a specific new particle that fits the bill.
3. It uses a New Tool: The "Non-Invertible Fusion Rule" is a fresh mathematical concept that acts as a strict but clever gatekeeper, ensuring the model stays simple and consistent.

The Catch: Because the model is so "minimal" and the effects are so subtle, it's very hard to test in a lab right now. It's a beautiful theory that fits all the current data, but we might need bigger, more sensitive telescopes and particle colliders in the future to prove it's real.

In short: They found a clever, minimal way to explain two of the universe's biggest mysteries using a new set of mathematical "Lego rules."

Technical Summary

1. Problem Statement

The paper addresses two fundamental open questions in particle physics:

1. The Origin of Neutrino Masses: While the Standard Model (SM) predicts massless neutrinos, oscillation data confirms they have tiny masses. The nature of these masses (Dirac vs. Majorana) remains unresolved. The non-observation of neutrinoless double beta decay ($0\nu\beta\beta$) motivates the exploration of Dirac neutrino scenarios.
2. The Nature of Dark Matter (DM): The identity of the DM particle is unknown.
3. Symmetry Constraints: Conventional Dirac models often require continuous $U(1)$ or discrete symmetries to forbid Majorana mass terms. However, these symmetries are often stable at both tree and loop levels, requiring extensive particle content to generate neutrino masses radiatively. The authors aim to utilize non-invertible fusion rules (a class of generalized symmetries) to create a more minimal framework where tree-level couplings are forbidden, but loop-level generation is naturally induced.

2. Methodology and Model Setup

The authors propose a radiative Dirac neutrino mass model stabilized by a specific non-invertible symmetry.

• Symmetry Framework: The model employs a $Z_3$ gauging of the $Z_3 \times Z'_3$ fusion rule.

  • Unlike standard group symmetries, non-invertible fusion rules possess unique multiplication properties.
  • Crucially, this symmetry preserves invariance at the tree level but is broken at the one-loop level. This feature allows the model to forbid tree-level Yukawa couplings (preventing direct Dirac mass terms) while permitting them to be generated radiatively.
  • The specific generators $\{1, a, a^*, b, b^*\}$ are chosen such that Majorana mass terms (e.g., $\psi_R^C \psi_R$) are strictly forbidden, ensuring the Dirac nature of neutrinos.

• Particle Content:

  • SM Fields: Lepton doublets ($L_L$), charged lepton singlets ($\ell_R$), and the Higgs doublet ($H$).
  • New Fermions:
    • Vector-like neutral fermions: $\psi_{L,R}$ (assigned charge $b$).
    • Right-handed neutral fermions: $N_R$ (assigned charge $a$).
  • New Scalars (Inert Sector):
    • Singlet scalar: $S$ (assigned charge $b$).
    • Doublet scalar: $\eta = [\eta^+, \eta^0]^T$ (assigned charge $b$).

• Lagrangian and Mass Generation:

  • The Yukawa Lagrangian includes terms like $f \bar{L}_L \tilde{\eta} \psi_R$ and $g \psi_L N_R S$.
  • Tree-level terms like $\bar{L}_L \tilde{H} N_R$ are forbidden by the fusion rule.
  • One-Loop Mechanism: Neutrino masses are generated via a one-loop diagram involving the exchange of $\psi$, $\eta$, and $S$.
  • Mixing: A term $\mu(H^\dagger \eta S^*)$ in the Higgs potential induces mixing between the neutral component of $\eta$ ($\eta^0$) and the singlet $S$. This mixing is essential for closing the loop and generating the effective Dirac mass term.

3. Key Contributions

1. Novel Symmetry Application: The paper demonstrates the first application of a non-invertible fusion rule ($Z_3$ gauging of $Z_3 \times Z'_3$) to construct a minimal Dirac seesaw model. It shows how these rules can naturally suppress tree-level interactions while allowing loop-level generation.
2. Unified Framework: The model simultaneously explains the smallness of neutrino masses (via the loop suppression factor $(4\pi)^{-2}$) and provides a Dark Matter candidate within the same particle content.
3. Phenomenological Analysis: The authors perform a comprehensive numerical analysis covering:
   • Neutrino oscillation parameters (mixing angles and mass squared differences).
   • Lepton Flavor Violation (LFV) processes ($\ell_i \to \ell_j \gamma$).
   • Lepton anomalous magnetic moments ($g-2$).
   • Dark Matter relic density and direct detection constraints.

4. Results

A. Neutrino Masses and Oscillations

• The model successfully reproduces the observed neutrino mass spectrum and mixing patterns (PMNS matrix) consistent with NuFit 6.1 and recent JUNO data.
• The sum of neutrino masses ($\sum m_\nu$) is constrained to be $\lesssim 72$ meV (consistent with DESI/CMB data), with the model predicting values localized around 59–64 meV for the Normal Hierarchy (NH) case.

B. Lepton Flavor Violation (LFV) and $g-2$

• LFV: The branching ratios for processes like $\mu \to e\gamma$, $\tau \to e\gamma$, and $\tau \to \mu\gamma$ are calculated. The results are far below current experimental bounds (e.g., $BR(\mu \to e\gamma) < 1.5 \times 10^{-13}$).
• $g-2$: Contributions to the electron and muon anomalous magnetic moments are negligible and well within experimental uncertainties.
• Implication: While the model is consistent, the predicted signals for LFV and $g-2$ are too small to be tested by current or near-future experiments.

C. Dark Matter Candidates

The model offers two potential DM candidates, but their viability differs significantly:

1. Bosonic Candidate ($S$):

   • The singlet scalar $S$ is a viable DM candidate.
   • It interacts with the SM only via the Higgs portal.
   • Relic Density: The correct relic density ($\Omega h^2 \approx 0.12$) is achieved via resonant annihilation when the mass of $S$ is approximately half the Higgs mass ($m_S \approx m_h/2 \approx 63$ GeV).
   • This scenario is consistent with direct detection limits.

2. Fermionic Candidate ($\psi_1$):

   • The lightest vector-like fermion $\psi_1$ is stabilized by the symmetry.
   • Relic Density Issue: The annihilation cross-section $\langle \sigma v \rangle$ for $\psi_1$ is strongly suppressed (orders of magnitude lower than the required $\sim 10^{-9} \text{ GeV}^{-2}$).
   • Conclusion: The fermionic option is strongly disfavored as the dominant DM component because it cannot account for the observed relic density.

5. Significance and Conclusion

• Theoretical Innovation: The work establishes non-invertible fusion rules as a powerful organizing principle for constructing minimal extensions of the Standard Model. It provides a natural mechanism to forbid Majorana masses while allowing Dirac masses to arise radiatively.
• Phenomenological Viability: The model is fully consistent with all current experimental data, including neutrino oscillations, LFV limits, and cosmological constraints.
• Predictions:
  • The bosonic singlet $S$ with a mass near 63 GeV is the primary viable DM candidate in this framework.
  • The model predicts extremely small rates for LFV and $g-2$ deviations, making direct detection of the new physics challenging in the near term.
  • Future tests may rely on precision cosmology (refining the sum of neutrino masses) or collider searches for the inert scalars and vector-like fermions.

In summary, the paper presents a robust, minimal theoretical framework linking neutrino physics and dark matter through the novel application of non-invertible symmetries, with the bosonic singlet emerging as the most promising phenomenological outcome.