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Three-loop induced neutrino mass model in a non-invertible symmetry

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper proposes that a non-invertible symmetry FR(6) forbids the tree-level Yukawa coupling that would give neutrinos mass, generating it at one loop through new fermions L' and a singlet S0, so neutrino masses are a three-loop effect.

desk verdict The central mechanism is invalid: the Eq. (18) interaction terms violate both U(1)_Y and the FR(6) charges listed in Table II, so the loop-induced Yukawa and three-loop neutrino mass are never defined. read the letter →

arxiv 2507.16198 v1 pith:X7MOEUJ3 submitted 2025-07-22 hep-ph

Hiroshi Okada , Yoshihiro Shigekami This is my paper
classification hep-ph
keywords radiativeneutrinomassthree-loopnon-invertiblesymmetryMamodelleptonflavorviolationmuong-2darkmatterinertdoublet
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proposes a way to make neutrino masses naturally tiny by generating them at three-loop order. Starting from the Ma radiative seesaw model, it adds three vector-like doublet fermions $L'$ and a neutral singlet scalar $S_0$ and imposes a non-invertible symmetry $F_R(6)$. Under this symmetry the direct Yukawa coupling $y_\eta$ that would give neutrinos mass at tree level is absent, but it is generated at one loop through $L'$ and $S_0$, so the resulting neutrino mass matrix is a three-loop effect. The authors show numerically that the model can accommodate neutrino oscillation data, lepton flavor violation bounds, the muon $g-2$ anomaly, and have either $S_0$ or the neutral component of $\eta$ as a dark matter candidate.

What carries the argument

The central object is the non-invertible symmetry $F_R(6)$, defined as a $\mathbb{Z}_2$ gauging of $\mathbb{Z}_6$, with fusion rules $\epsilon\otimes\epsilon=1\oplus\sigma$, $\sigma\otimes\sigma=1\oplus\sigma$, $\rho\otimes\rho=1$, $\epsilon\otimes\sigma=\epsilon\oplus\rho$, $\epsilon\otimes\rho=\sigma$, $\sigma\otimes\rho=\epsilon$. The charge assignments in Table II put the new fields ($L'_L$ and $S_0$) in nontrivial representations so that the tree-level operator that would give $y_\eta$ is not invariant. The one-loop diagram with $L'$ and $S_0$, controlled by the couplings $f$, $g$, and the $\mu$ term, then regenerates $y_\eta$, and this same $y_\eta$ feeds into the standard neutrino mass formula of the base model.

What would settle it

Multiply the $F_R(6)$ charges of the fields in each operator of Eqs. (18) and (19) using the fusion rules in Eq. (15) and check whether the identity representation $1$ appears in the product; if any operator's product contains only non-trivial representations (such as $\epsilon$ or $\rho$), then $y_\eta$ is not forbidden at tree level by this symmetry, and the three-loop mechanism is not realized.

Watch

Extended reading notes

Core claim

The central claim is that a non-invertible symmetry can do the work that a discrete parity does in the inert-doublet radiative seesaw model: it forbids the Yukawa coupling $y_\eta$ at tree level, but because the symmetry is non-invertible it is dynamically broken at one loop, so $y_\eta$ is generated radiatively. The new fields $L'$ and $S_0$, together with the scalar trilinear coupling $\mu|\eta|^2 S_0$, produce $y_\eta$ through a one-loop diagram, and since the neutrino mass formula of the base model is already one-loop in $y_\eta$, the physical neutrino mass matrix is a genuine three-loop quantity. The same scalar doublet $\eta$ participates both in generating $y_\eta$ and in the neutrino mass loop, so the model has a single inert doublet doing two jobs.

Load-bearing premise

The mechanism works only if the tree-level interaction terms in Eqs. (18) and (19) are genuinely invariant under the $F_R(6)$ charges assigned in Table II; if any of these operators transforms in a non-trivial representation rather than the identity, the symmetry would not forbid $y_\eta$ at tree level and the loop-generation chain would not start.

Editorial extensions

If this is right

  • If the mechanism is correct, neutrino masses are predicted to be three-loop suppressed, which naturally explains their smallness without invoking very heavy right-handed neutrinos.
  • The model predicts a positive contribution to the muon anomalous magnetic moment (because the photon attaches to charged fermions rather than charged bosons in the new loop), opposite in sign to the minimal inert-doublet version.
  • The fermion $N_{R1}$ cannot be the dominant dark matter because the radiatively generated $y_\eta$ is at most $10^{-4}$, far below the required order-one coupling; dark matter must instead be the scalar $S_0$ or the real inert doublet component $\eta_R$.
  • The allowed mixing angles $\theta_{12,23,13}$ show distinctive shapes, and the dark matter mass is bounded from above ($m_S \lesssim 800$ GeV for $S_0$ DM, $m_R \simeq 534 \pm 8.5$ GeV for $\eta_R$ DM), giving testable predictions for future direct-detection and collider searches.
  • All viable parameter points satisfy the $\Sigma m_\nu \leq 120$ meV bound from cosmology, and the model gives specific upper limits on the effective neutrinoless double-beta decay mass $m_{ee}$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A direct check of the $F_R(6)$ invariance of the interaction terms in Eqs. (18) and (19), using the fusion rules of Eq. (15), is a necessary consistency test; if any of these operators carries a nontrivial representation, the tree-level forbidding of $y_\eta$ would not hold and the mechanism would be ill-defined.
  • The same fusion-rule technique could be applied to other unwanted tree-level couplings (for instance, higher-dimensional lepton-number-violating operators) to push their appearance to even higher loops, generalizing the idea beyond the neutrino sector.
  • Because the non-invertible symmetry is dynamically broken at loop level, the model effectively links the smallness of neutrino mass to the loop order of symmetry breaking rather than to a hierarchy in Yukawa couplings, a pattern that could be probed by detecting the new fields $L'$ and $S_0$ at colliders.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper proposes a three-loop radiative neutrino mass model based on the Ma model, adding vector-like lepton doublets L' and a singlet scalar S0, and imposing a non-invertible symmetry FR(6) in the class of a Z2 gauging of Z6. The direct neutrino Yukawa coupling y^eta is forbidden at tree level by FR(6), and the authors claim that y^eta is generated at one loop through the couplings f, g, and the scalar cubic term mu, making the neutrino mass a three-loop effect. The rest of the paper performs numerical scans for neutrino oscillation data, lepton flavor violation, muon g-2, and two scalar dark matter candidates, S0 and eta_R, for normal and inverted hierarchies.

Significance. The general idea of using a non-invertible symmetry to forbid a coupling at tree level while allowing it radiatively is timely and potentially interesting. The paper gives explicit fusion rules, a full field-content table, a one-loop formula for the induced Yukawa coupling, and relic density calculations using micrOMEGAs, which is a strength. However, the central charge assignment is internally inconsistent: the very operators on which the loop mechanism depends are not invariant under the symmetry that defines the model. As written, the tree-level Lagrangian already violates FR(6), so the advertised mechanism cannot be constructed and the numerical results do not constrain a valid model.

major comments (3)
  1. [Sec. II, Table II and Eq. (18)] The tree-level Lagrangian is not invariant under FR(6) with the charges given in Table II. Using the fusion rules in Eq. (15), the coupling g_ab \bar{L}'_{La} N_{Rb} \tilde{\eta} transforms as \sigma\otimes\rho\otimes\sigma = (\sigma\otimes\rho)\otimes\sigma = \epsilon\otimes\sigma = \epsilon\oplus\rho, which does not contain the identity. Similarly, h_ab \bar{L}'^c_{Ra} N_{Rb} \eta transforms as 1\otimes\rho\otimes\sigma = \epsilon, and the vector-like mass term M_{L'} \bar{L}'_L L'_R transforms as \sigma\otimes 1 = \sigma. None of these operators is invariant. In particular, the g term is non-invariant for any choice of the L'_R and S0 charges, because the charges of L'_L = \sigma, N_R = \rho, and \eta = \sigma are fixed by the table. The one-loop formula in Eq. (20) therefore has no basis in the symmetric Lagrangian, and the model's central mechanism cannot be defined.
  2. [Sec. II, Table II; Eq. (18)] The hypercharge assignments in Table II are inconsistent with the 4-component notation used in Eq. (1). With Y(L'_L) = -1/2 and Y(L'_R) = +1/2, the Dirac mass term M_{L'} \bar{L}'_L L'_R carries hypercharge (+1/2) + (+1/2) = +1, and the coupling f \bar{L}_L L'_R S0 also carries net hypercharge +1; the analogous f' term is likewise not U(1)_Y invariant. If the authors intended two-component Weyl notation, then the table's hypercharges and the explicit bar and charge-conjugation symbols in Eq. (18) must be rewritten consistently. As it stands, the loop diagram behind Eq. (20) is not gauge-invariant.
  3. [Sec. IV, Eqs. (22)-(23)] The numerical analysis treats f and g as free input couplings and uses the Casas-Ibarra parametrization to reproduce neutrino oscillation data. But because these couplings are not part of the FR(6)-invariant Lagrangian, the scans in Sec. IV describe an effective theory in which the non-invertible symmetry has been removed by hand. The claim that the observed neutrino parameters follow from a three-loop mechanism generated by the symmetric model is therefore not supported by the presented calculation.
minor comments (5)
  1. [Eq. (20)] The second logarithm in Eq. (20) is written with M_{L'a} inside the argument, while the prefactor sums over the index b; this should be M_{L'b} for consistency.
  2. [Eq. (13)] The quoted value \Delta a_\mu \simeq (39 \pm 64) \times 10^{-1} has an implausible normalization; a factor of 10^{-11} or 10^{-10} is presumably intended.
  3. [After Eq. (8)] The sentence defining C_{21}, C_{31}, and C_{21} repeats C_{21} twice and uses 0.1784 versus 0.173648; one of these should presumably be C_{32}.
  4. [Table II] The FR(6) row in Table II appears to contain only seven entries for eight fields, leaving the charges of S0 and possibly eta ambiguous. Since these charges are the basis of the whole construction, the table must be corrected to assign a charge to every field explicitly.
  5. [Sec. II, Eq. (18)] The notation for the L' fields should be clarified: if L'_L and L'_R are 4-component chiral projections, the vector-like mass term requires equal hypercharges; if they are two-component Weyl spinors, the appearance of \bar{L}'_L and L'^c_R in the same equation is a mixed notation that needs to be defined.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the neutrino-sector fit is an input-constrained parameter scan, not a prediction, and no load-bearing claim reduces to its own input.

full rationale

The derivation chain is not circular. The FR(6) charges, fusion rules, and the one-loop formula for y_eta in Eq. (20) are assumed or computed ingredients; they are not derived from the numerical outputs. The closest candidate for circularity is Eqs. (21)-(23), where f and g are re-expressed via Casas-Ibarra from the observed neutrino masses and PMNS matrix, but this is an input-constrained parametrization, not a prediction of D_nu or U. The paper explicitly summarizes its numerical work as demonstrating 'allowed space for our input parameters' rather than as a parameter-free prediction. The LFV, muon g-2, and dark-matter observables are computed after imposing the neutrino fit and can exclude points, so no fitted quantity is renamed as a prediction and no result is forced by construction. The self-citations to non-invertible-symmetry papers supply background formalism, but the central loop calculation is self-contained once the charge assignments are accepted. A separate consistency issue exists: the terms in Eq. (18) may violate the U(1)_Y and FR(6) assignments in Table II, which would be an internal-consistency flaw rather than a circularity, and is not counted in this score.

Assumptions & free parameters 7 free parameters · 3 assumptions · 3 invented entities

The model rests on many scanned masses and couplings, the assumed fusion rules of FR(6), and the incorrect assumption that the added interaction terms respect the symmetry. These constitute the bulk of the model's input, and the central claim depends on them.

free parameters (7)
  • m_S = [0.1-1000] GeV (scanned)
    Mass of the singlet scalar S0; controls DM relic density and enters the y_eta loop.
  • m_eta = [m_S-1000] GeV (scanned)
    Mass of the inert doublet; enters the Ma loop and the y_eta loop via m_0.
  • M_L' = [m_S-1000] GeV (scanned)
    Masses of the three vector-like fermions in the y_eta loop.
  • M_N = [m_S-1000] GeV (scanned)
    Masses of the right-handed neutrinos in the Ma loop.
  • mu = [0-1000] GeV (scanned)
    Scalar trilinear coupling mu|eta|^2 S0; directly multiplies the y_eta loop function.
  • O_N angles (3 complex) = |theta| in [0,pi], arguments in [0,2pi]
    Casas-Ibarra orthogonal matrix parameters scanned to reproduce neutrino oscillation data.
  • lightest neutrino mass and Majorana phases = D_nu1(nu3) <= 1 eV, phases in (-pi,pi)
    Scanned to satisfy sum of masses and neutrinoless double beta decay bounds.
assumptions (3)
  • domain assumption The FR(6) fusion rules in Eq. (15) are taken as the correct non-invertible symmetry algebra.
    The paper imports these rules from the non-invertible symmetry literature without derivation; the model's charge assignments depend on them.
  • domain assumption The non-invertible symmetry is dynamically broken at loop level, so operators that are not invariant can be generated radiatively.
    The paper relies on this property to justify generating y_eta at one loop; it is stated in the introduction and Sec. II.
  • ad hoc to paper The Yukawa and scalar terms in Eq. (18) and Eq. (19) are invariant under the charge assignments in Table II.
    This is asserted by 'According to this charge assignments, we add the following terms', but direct computation with Eq. (15) shows the products do not contain the identity, so the assumption is false.
invented entities (3)
  • Non-invertible symmetry FR(6)
    purpose: Forbids the tree-level neutrino Yukawa coupling y_eta, forcing loop generation.
    No independent experimental handle; it is an abstract symmetry imposed on the model.
  • Vector-like lepton doublets L'
    purpose: Mediate the one-loop generation of y_eta.
    No predicted mass or signature that would show up in existing data; masses are scanned parameters.
  • Singlet scalar S0
    purpose: Mediates the y_eta loop and serves as a dark matter candidate.
    DM relic density is computed from its annihilations, but the mass and couplings are free parameters.

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Cite this review

Pith. "Pith review of Three-loop induced neutrino mass model in a non-invertible symmetry." pith.science (2026). https://pith.science/paper/X7MOEUJ3

@misc{pith2026250716198,
  author       = {Pith},
  title        = {Pith review of: Three-loop induced neutrino mass model in a non-invertible symmetry},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/X7MOEUJ3}},
  note         = {Machine review of arXiv:2507.16198}
}
abstract

We propose a new type of radiatively induced neutrino masses at three-loop level based on the Ma model, introducing a non-invertible symmetry in the class under a ${\mathbb Z_2}$ gauging of ${\mathbb Z_6}$ symmetry and adding three isospin doublet vector-like fermions $L'$ and singlet boson $S_0$. Under this symmetry, the Yukawa interactions directly related to the neutrino masses are not allowed at tree-level. However it is allowed at one-loop level due to $L'$ and $S_0$ as well as $\eta$, which is no longer invariant under this symmetry. Therefore, the symmetry is dynamically broken. Intriguingly, $\eta$ plays important roles in contributing to both the radiative matrices $y^\eta$ and $m_\nu$. After constructing our model, we show some numerical analyses to satisfy the lepton flavor violations, muon anomalous magnetic dipole moment, and a boson dark matter candidate $S_0$ or $\eta_R$ for the cases of normal hierarchy and inverted hierarchy. Then, we demonstrate allowed space for our input parameters.

Figures

Figures reproduced from arXiv: 2507.16198 by the authors.

Figure 1
Figure 1. shows allowed regions for θ12 (left), θ23 (center), θ13 (right) for the NH case. Blue points are allowed only when PDν ≤ 120 meV is satisfied, otherwise points are colored by red. Hereafter, we use this color legend for similar plots, without mentioning. The vertical axis represents argu￾ments, while the horizontal axis does the absolute values for these mixing angles. These figures imply that they are unique shapes… view at source ↗
Figure 2
Figure 2. shows allowed regions for muon g − 2 in terms of the mass of DM. As can be seen the figure, the predicted muon g − 2, ∆aµ ≲ 3.2 × 10−17, has almost no discrepancy from the SM prediction, and therefore, the result is consistent with the current discrepancy between theoretical [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Same plots as Fig. 1, for the IH case with [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Same plot as Fig. 2, for the IH case with [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: Plots for the relic density in the case of the [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]
Figure 6
Figure 6. Figure 6: shows allowed regions for θ12 (left), θ23 (center), θ13 (right). The vertical axis represents arguments, while the horizontal axis does the absolute values for these mixing angles. These figures imply that allowed regions are similar to the S0 DM case but there are two…
Figure 7
Figure 7. Figure 7: FIG. 7: Allowed regions for muon [PITH_FULL_IMAGE:figures/full_fig_p014_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8: Same plot as Fig. 6, for the IH case with [PITH_FULL_IMAGE:figures/full_fig_p015_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9: Same plot as Fig. 7, for the IH case with [PITH_FULL_IMAGE:figures/full_fig_p015_9.png]

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Forward citations

Cited by 3 Pith papers

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