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arxiv: 2604.11308 · v1 · submitted 2026-04-13 · ✦ hep-ph · hep-th

Recognition: unknown

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

Hiroshi Okada , Labh Singh
Authors on Pith no claims yet

Pith reviewed 2026-05-10 16:28 UTC · model grok-4.3

classification ✦ hep-ph hep-th
keywords radiative Dirac neutrino massnon-invertible fusion ruleZ3 x Z3 gaugingone-loop seesawdark matter relic densitylepton flavor violation
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The pith

A non-invertible fusion rule from Z3 x Z3' gauging forces Dirac neutrino masses to arise only at one loop while stabilizing a dark matter candidate.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs a minimal extension of the Standard Model in which a non-invertible fusion rule, obtained by gauging Z3 times Z3 prime, eliminates every tree-level Yukawa coupling. Neutrino masses are then generated radiatively through one-loop diagrams that exchange exotic fermions and inert scalars. The same symmetry stabilizes a bosonic singlet that can account for the observed dark matter relic density, while the fermionic alternative is ruled out by insufficient annihilation. Explicit calculations confirm that the resulting neutrino masses and mixings fit oscillation data, and that lepton-flavor-violating rates and magnetic-moment contributions remain far below current limits.

Core claim

We propose a radiative Dirac neutrino mass model stabilized by a non-invertible fusion rule originating from a Z3 × Z3' gauging. The imposed symmetry forbids tree-level Yukawa couplings and ensures that neutrino masses are generated only at the one-loop level through the exchange of exotic fermions and inert scalars. This minimal framework simultaneously accommodates neutrino masses and mixings consistent with current oscillation data, while providing a viable dark matter candidate. We analyze lepton flavor violating processes and lepton anomalous magnetic moments, finding that all contributions remain well below present experimental bounds. In the dark matter sector, the bosonic singlet is

What carries the argument

The non-invertible fusion rule arising from Z3 × Z3' gauging, which functions as a selection rule that eliminates all tree-level Yukawa operators while permitting specific one-loop diagrams involving exotic fermions and inert scalars.

If this is right

  • Neutrino oscillation parameters are reproduced by the one-loop mass matrix while satisfying all current bounds.
  • Lepton-flavor-violating branching ratios and anomalous magnetic moments lie well below experimental limits across the viable parameter space.
  • The bosonic singlet provides the correct thermal relic density, whereas the corresponding fermionic state is excluded by suppressed annihilation.
  • The exotic fermions and inert scalars remain light enough to be produced at future colliders.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar non-invertible rules could be applied to other sectors to suppress unwanted operators without introducing new mass scales.
  • Precision measurements of neutrino mixing angles at next-generation oscillation experiments could distinguish this one-loop pattern from tree-level alternatives.
  • Direct searches for the inert scalars or exotic fermions at the LHC or future colliders would provide an independent test independent of neutrino data.

Load-bearing premise

The non-invertible fusion rule must be realized exactly so that it removes every tree-level Yukawa coupling yet still allows the chosen one-loop diagrams to generate the observed neutrino masses without extra suppression.

What would settle it

A direct measurement showing that the bosonic singlet cannot achieve the observed relic density for any parameter choice consistent with the one-loop neutrino mass, or the discovery that tree-level Yukawa couplings are present despite the gauged symmetry.

Figures

Figures reproduced from arXiv: 2604.11308 by Hiroshi Okada, Labh Singh.

Figure 1
Figure 1. Figure 1: FIG. 1: One loop-induced Dirac neutrino masses. All vertices are invariant under FR. But one [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: The allowed regions of ˜m [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Allowed regions of branching ratios of [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: The allowed regions of ˜m [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Allowed regions of branching ratios of [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Allowed region of the DM cross section to explain the relic density in terms of DM mass. [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗
read the original abstract

We propose a radiative Dirac neutrino mass model stabilized by a non-invertible fusion rule originating from a $Z_3 \times Z_3'$ gauging. The imposed symmetry forbids tree-level Yukawa couplings and ensures that neutrino masses are generated only at the one-loop level through the exchange of exotic fermions and inert scalars. This minimal framework simultaneously accommodates neutrino masses and mixings consistent with current oscillation data, while providing a viable dark matter candidate. We analyze lepton flavor violating processes and lepton anomalous magnetic moments, finding that all contributions remain well below present experimental bounds. In the dark matter sector, the bosonic singlet emerges as a promising candidate with relic density compatible with cosmological observations, whereas the fermionic option is strongly disfavored due to suppressed annihilation cross sections. Our study demonstrates that non-invertible fusion rules can serve as a powerful organizing principle for constructing minimal and phenomenologically consistent extensions of the Standard Model, linking neutrino physics and dark matter within a unified radiative framework.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript proposes a radiative Dirac neutrino mass model stabilized by a non-invertible fusion rule from Z_3 × Z_3' gauging. The symmetry is claimed to forbid all tree-level Yukawa couplings, generating neutrino masses at one loop via exotic fermions and inert scalars. The framework is asserted to accommodate neutrino oscillation data, keep lepton flavor violation and g-2 contributions below experimental bounds, and yield a viable bosonic singlet dark matter candidate with the correct relic density (while disfavoring the fermionic option).

Significance. If the fusion rules rigorously enforce the absence of tree-level operators while permitting the specified one-loop topology, the work illustrates a potentially useful organizing principle for minimal SM extensions that simultaneously address neutrino masses and dark matter. The approach is novel in its application of non-invertible symmetries, but its significance hinges on explicit verification of the symmetry constraints and the robustness of the phenomenological fits.

major comments (2)
  1. [Abstract] Abstract: the central claim that the Z_3 × Z_3' non-invertible fusion rule forbids every tree-level Dirac Yukawa coupling (while allowing the one-loop diagram) is load-bearing for the entire radiative construction. The manuscript must explicitly list the charge assignments for L, H, N, the exotic fermions, and inert scalars, then compute the fusion products to demonstrate that no dimension-4 operator fuses to the identity.
  2. [Model construction] Model construction (likely §2–3): without the explicit fusion-rule verification above, the assertions that neutrino masses arise only radiatively, that DM stability is protected, and that LFV is suppressed all rest on an unconfirmed symmetry condition. If even one Yukawa term is allowed, the one-loop seesaw claim collapses.
minor comments (1)
  1. The abstract states consistency with oscillation data and relic density but provides no numerical values, loop integrals, or parameter ranges; including a brief table of benchmark points would improve clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting the need for explicit verification of the symmetry constraints. We address the major comments point by point below and will incorporate the requested details in a revised version.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that the Z_3 × Z_3' non-invertible fusion rule forbids every tree-level Dirac Yukawa coupling (while allowing the one-loop diagram) is load-bearing for the entire radiative construction. The manuscript must explicitly list the charge assignments for L, H, N, the exotic fermions, and inert scalars, then compute the fusion products to demonstrate that no dimension-4 operator fuses to the identity.

    Authors: We agree that an explicit demonstration is required to substantiate the central claim. In the revised manuscript we will add a new subsection (in §2) that lists the Z_3 × Z_3' charge assignments for all relevant fields—lepton doublets L, the Higgs H, right-handed neutrinos N, the exotic fermions, and the inert scalars. We will then compute the fusion products of every possible dimension-4 operator and show that none fuse to the identity, while the operators appearing in the one-loop diagram are permitted. This addition will make the symmetry argument fully transparent and directly support the statements in the abstract. revision: yes

  2. Referee: [Model construction] Model construction (likely §2–3): without the explicit fusion-rule verification above, the assertions that neutrino masses arise only radiatively, that DM stability is protected, and that LFV is suppressed all rest on an unconfirmed symmetry condition. If even one Yukawa term is allowed, the one-loop seesaw claim collapses.

    Authors: We acknowledge that the radiative nature of the neutrino masses, the stability of the dark-matter candidate, and the suppression of lepton-flavor violation all rely on the fusion rules being correctly enforced. The explicit calculations described in the response to the first comment will be placed in §2 and cross-referenced in §3. With these verifications in place, the revised text will confirm that no tree-level Yukawa operator is allowed, that the bosonic singlet remains stable, and that tree-level LFV contributions are absent. The one-loop topology remains the leading contribution, preserving the overall phenomenological conclusions. revision: yes

Circularity Check

0 steps flagged

No significant circularity; explicit symmetry construction checked against data

full rationale

The paper constructs a model by imposing a non-invertible fusion rule from Z3 × Z3' gauging that is stated to forbid all tree-level Yukawa operators while permitting a specific one-loop diagram involving exotic fermions and inert scalars. Neutrino masses/mixings, LFV rates, g-2 contributions, and DM relic density are then shown to be compatible with data via suitable parameter choices in the allowed Lagrangian. This is standard model-building: the symmetry assignments and fusion products are presented as the defining input, the one-loop topology follows directly from the charge rules, and numerical scans confirm viability without claiming parameter-free predictions or deriving the symmetry from the observables themselves. No load-bearing step reduces by construction to a fit, self-citation chain, or renamed ansatz; the central result (viability of the radiative framework) remains independent of the fitted values.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 2 invented entities

The model rests on the assumption that the chosen symmetry forbids tree-level terms, introduces several new particles whose masses and couplings are adjusted to data, and postulates a stable bosonic singlet as DM without external falsifiable evidence beyond internal consistency.

free parameters (1)
  • masses and couplings of exotic fermions and inert scalars
    Adjusted to reproduce neutrino oscillation parameters and DM relic density.
axioms (1)
  • domain assumption Z3 x Z3' gauging produces a non-invertible fusion rule that forbids tree-level Yukawa couplings.
    Central premise stated in the abstract for generating radiative masses only.
invented entities (2)
  • exotic fermions and inert scalars no independent evidence
    purpose: mediate the one-loop Dirac neutrino mass
    New particles required to realize the radiative mechanism.
  • bosonic singlet no independent evidence
    purpose: stable dark matter candidate
    Emerges from the symmetry and claimed to match relic density.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. A theoretical account of tiny multi-Higgs vacuum expectation values from non-invertible symmetry

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    Non-invertible symmetry from the minimal Fibonacci fusion rule forbids tree-level VEVs for multi-Higgs fields H4 and H5, generating them radiatively at one-loop with naturally small values of 10^{-3}-10^{-2} GeV that ...

Reference graph

Works this paper leans on

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    Bosonic DM case:Sstate SinceSdoes not couple to Standard Model fields through Yukawa interactions, its only portal to the visible sector arisesviascalar interactions in the Higgs potential. In such a 8 case, detailed analysis is already explored, and the correct relic density of DM is obtained near the half of Higgs mass by the resonant enhancement [89] w...

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    Bosonic DM:Sstate Table II lists the BP that minimizes ∆χ 2, where our DM is fixed to be half of the SM Higgs mass∼63 GeV and the resulting minimum value is approximately 0.200. The corresponding 14 FIG. 6: Allowed region of the DM cross section to explain the relic density in terms of DM mass. All the color legends and the line are the same as the ones i...

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    Fermionic DM:ψ 1 state Table III lists the BP that minimizes ∆χ 2. The resulting minimum value is approximately 0.148. The corresponding values for ˜gand ˜fare also provided as follows: ˜g×103 ≈   75.3899−79.9135i−1.10263 + 0.0792865i0.0115587−0.0801839i −7.77716 + 26.6837i−459.569−3588.96i−2.9311 + 11.6813i 0.0249052 + 0.108523i8.89349 + 16.6716i10 ...

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