A natural realization of inverse seesaw model in a non-invertible selection rule

Hiroshi Okada; Shilpa Jangid

REVIEW 3 major objections 2 minor 2 cited by

A natural realization of inverse seesaw model in a non-invertible selection rule

T0 review · 3 major / 2 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read This paper proposes that a Z3 Tambara-Yamagami fusion rule, applied to neutral fermions, naturally generates inverse-seesaw neutrino masses at one loop and supplies a dark matter candidate.

desk verdict Interesting idea, unreadable text — I can judge the abstract but not the math. read the letter →

arxiv 2508.16174 v1 pith:HIDXKFCP submitted 2025-08-22 hep-ph

Shilpa Jangid , Hiroshi Okada This is my paper

classification hep-ph

keywords inverseseesawneutrinomassnon-invertiblesymmetryTambara-YamagamifusionruleMajoranadarkmatterloop-generatedhierarchyZ3

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 new way to get the small neutrino masses of the inverse seesaw mechanism without tuning: impose a Z3 Tambara-Yamagami fusion rule on the neutral fields. The rule forbids tree-level Majorana mass terms for both right- and left-handed neutrinos, so the usual tiny lepton-number-breaking parameter is absent by symmetry. Instead, those masses are generated at one-loop order when the symmetry is dynamically broken, naturally producing the hierarchy between heavy and light neutral fermions. The same neutral sector supplies a dark matter candidate, and the model fits current neutrino-oscillation data for both normal and inverted mass ordering.

What carries the argument

The Z3 Tambara-Yamagami fusion rule is a non-invertible algebraic selection rule on field charges: it includes a single non-invertible object whose fusion with the Z3 group objects follows the Tambara-Yamagami category rule rather than ordinary group multiplication, so only certain operator products are allowed. The paper uses this rule to forbid tree-level Majorana mass operators while leaving the one-loop diagrams open, and its dynamical breaking is what converts a formally forbidden mass into a small, calculable generated mass.

What would settle it

Compute the one-loop effective potential of the model: if its minimum leaves the Z3 Tambara-Yamagami fusion symmetry unbroken, or if the would-be Majorana mass integral vanishes or requires a counterterm that reintroduces a tree-level mass, then the claimed natural hierarchy is not realized. Observing a tree-level N_R Majorana mass comparable in size to the one-loop term would also falsify the selection rule.

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Extended reading notes

Core claim

The central claim is that the inverse seesaw hierarchy can be realized naturally by treating the Z3 Tambara-Yamagami fusion rule as a selection rule for the Lagrangian. Under this non-invertible symmetry, Majorana mass terms for N_R and N_L are forbidden at tree level and only appear through one-loop diagrams after the symmetry is dynamically broken. This loop suppression replaces the ad hoc small lepton-number-violating parameter of the usual inverse seesaw. The paper further shows that the neutral boson or fermion added to mediate the loop can be a dark matter candidate, and that the resulting neutrino mass matrix can reproduce the best-fit values for both normal and inverted hierarchies w

Load-bearing premise

The argument holds only if the Z3 Tambara-Yamagami fusion rule can be consistently imposed on the full Standard-Model-extended Lagrangian, with anomalies cancelled and a vacuum that breaks the symmetry dynamically in exactly the way the one-loop calculation assumes.

Editorial extensions

If this is right

Tree-level Majorana masses are naturally absent, so no tiny lepton-number-breaking coupling needs to be inserted by hand.
The light neutrino mass scale is set by a one-loop factor, offering a calculable explanation for the smallness of neutrino masses.
The same neutral sector contains a boson or fermion stable enough to serve as a dark matter candidate.
The neutrino mass matrix can fit both normal and inverted mass ordering under current oscillation data.

Reading between the lines

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

If this mechanism generalizes, other non-invertible fusion rules could yield a family of models where mass hierarchies are determined purely by the loop order at which symmetry breaking first appears.
A decisive internal test is whether the effective potential has a minimum that actually breaks the fusion symmetry in the required way; if the minimum preserves it, the one-loop generation would not occur.
The dark matter stability may be tied to the same non-invertible selection rule that forbids the tree-level Majorana mass, making the candidate a structural consequence rather than an add-on.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Zack,

Honestly, I can't evaluate the physics because the full text is a mangled stream of replacement characters — no equations, no Lagrangian, no field content beyond the abstract. So this is a letter about the abstract and whether the program is plausible.

What's new: applying a Z3 Tambara-Yamagami fusion rule to an inverse seesaw, using the non-invertible symmetry to kill tree-level Majorana masses for N_R and N_L and generating them only radiatively. That is a genuinely fresh move. The idea that a non-invertible selection rule can play a structural role in the neutrino mass mechanism is worth taking seriously. The connection to a dark matter candidate is reasonable. If the derivation is correct, this is a solid model-building contribution, not a revolution, but a real addition to the toolbox.

The soft spots are ones I can't check, not ones I've seen. The abstract gives no anomaly check, no statement that the one-loop mass is finite, no vacuum for the dynamical breaking, no fit uncertainties. The 'best fit value' is exactly that — a fit — so it doesn't serve as a prediction of the symmetry. That doesn't make the paper circular; it just means the numerical part is standard phenomenology. The stress-test's worry about whether the non-invertible rule is a genuine symmetry or bookkeeping is the right question, but it's a question for the full text, not a strike against what I can see.

Bottom line: the paper is unreadable in this version, so I can't pass judgment on the math. But the abstract is not a red flag — it reads like a serious attempt. If the editor can get a clean PDF, I'd send it to a referee with a solid background in non-invertible symmetries and neutrino model building. The referee should focus on whether the selection rule is consistent with gauge invariance and whether the one-loop mass is finite and parametrically controlled. I wouldn't cite it until I can read the derivation; I'd bring the abstract to the reading group as an example of a non-standard application, but the actual paper needs a readable copy.

Referee Report

3 major / 2 minor

Summary. The manuscript proposes an inverse seesaw model in which a Z3 Tambara-Yamagami fusion rule is imposed as a non-invertible selection rule on the theory. It claims that this rule forbids the tree-level Majorana mass terms for the right- and left-handed neutral fermions N_R and N_L, while one-loop effects, after dynamical breaking of the symmetry, generate these masses naturally. The paper also states that a neutral boson or fermion introduced for the loop mechanism can serve as a dark matter candidate, and that a best-fit value for the neutrino sector is found for normal and inverted hierarchies. The full text, as supplied, is unreadable: it consists of block-replacement characters with no recoverable equations, tables, or derivations. The only legible content is the abstract.

Significance. If the claimed mechanism works, the paper would offer an interesting and novel connection between non-invertible symmetries and the naturalness of small Majorana masses in an inverse seesaw, potentially with a dark matter candidate. However, as presented, the central results cannot be verified or even inspected: the Lagrangian, charge assignments, anomaly cancellation, one-loop calculation, vacuum stability, and the neutrino fit are all absent from the readable portion. No reproducible code, machine-checked proofs, or parameter-free derivations are available. The significance therefore cannot be assessed beyond the abstract level; the idea is intriguing but entirely unsupported in the provided manuscript.

major comments (3)

[Full text (entire)] The body of the manuscript is unreadable: it contains only replacement characters (��) with no recoverable equations, tables, or prose. None of the central technical claims can be checked: there is no Lagrangian, no explicit operator analysis, no charge assignments, no anomaly cancellation argument, and no derivation of the one-loop Majorana mass. This is a load-bearing deficiency: the abstract alone cannot establish that the Z3 Tambara-Yamagami rule is consistently imposed on the full Standard-Model extension.
[Abstract, first sentence] The paper asserts that the Z3 Tambara-Yamagami fusion rule forbids the tree-level Majorana masses for N_R and N_L. For this to be meaningful, the selection rule must be shown to act on gauge-invariant composite operators, not just on fields. The abstract provides no evidence that the non-invertible symmetry survives gauging or that it is not anomalous. Without the full operator analysis, the central claim that the rule is a genuine symmetry of the theory is unverifiable.
[Abstract, final sentence] The claim of a 'best fit value of the neutrino sector for normal and inverted hierarchies' is unsupported: no chi-square, pulls, number of experimental data points, or confidence intervals are reported. As written, this is a bare assertion. Even if the full text were readable, the abstract's lack of quantitative assessment means we cannot judge whether the fit is meaningful or merely a post-hoc adjustment.

minor comments (2)

[Abstract] The term 'Z3 Tambara-Yamagami fusion rule' is used without definition or reference. Since this is not a standard particle-physics symmetry for most readers, a brief explanation of the fusion rule and its realization (e.g., as a non-invertible symmetry of the Lagrangian) is needed.
[Abstract] The phrase 'either of which can be an appropriate dark matter candidate' is vague: no mass range, stability mechanism, or relic density argument is given even in outline.

Circularity Check

0 steps flagged · score 0.0 of 10

No detectable circularity in the readable abstract: the fusion-rule mechanism is a structural model-building claim, and the neutrino-sector numbers are explicitly presented as a best fit, not as a symmetry-only prediction.

full rationale

The manuscript body is almost entirely corrupted/unreadable in the provided version; only the abstract is usable as evidence. On that basis, no load-bearing step can be shown to reduce to its own inputs. The abstract claims that a Z3 Tambara-Yamagami fusion rule forbids tree-level Majorana masses for N_R and N_L and that these masses are generated at one loop after dynamical breaking. This is a standard symmetry-forbids / breaking-generates mechanism, not a self-definitional identity: forbidding a term by a selection rule and later generating it through breaking are distinct conditions, not the same statement. The only numerical claim is described as 'the best fit value of the neutrino sector for normal and inverted hierarchies referring recent experimental results' — i.e., an explicit fit to external data, not a parameter-free prediction from the symmetry. Thus there is no exhibited 'fitted input called prediction'. No citations, self or otherwise, are legible, so no self-citation chain can be identified as load-bearing. Because the hard rule requires quoting a specific reduction and none is available, I find no substantiated circularity. I note that the unreadable full text is a serious limitation for any verification of anomaly cancellation, loop finiteness, or vacuum structure, but that is a correctness/evidence concern, not a circularity finding.

Assumptions & free parameters 2 free parameters · 3 assumptions · 2 invented entities

The model rests on an ad hoc non-invertible symmetry and new neutral fields, with no independent evidence provided in the abstract. The neutrino parameters are fitted to data, and the loop masses are unspecified. These are the costs the model pays for its claimed naturalness.

free parameters (2)

Neutrino sector fit parameters = not reported in abstract
The abstract reports a best fit of the neutrino sector to recent oscillation data, implying parameters such as Yukawa couplings or mass scales are fitted.
Loop-suppressed Majorana mass scale = not reported in abstract
The one-loop induced Majorana masses depend on the new boson and fermion masses and couplings, which are likely free parameters of the model.

assumptions (3)

ad hoc to paper Z3 Tambara-Yamagami fusion rule is an exact symmetry of the theory
Introduced to forbid tree-level Majorana masses; no independent evidence for this symmetry in nature is given in the abstract.
ad hoc to paper The symmetry is dynamically broken at one-loop level
The abstract states this as the mechanism without providing the potential or dynamics that triggers the breaking.
domain assumption The added neutral boson and fermion preserve Standard Model gauge invariance and do not create anomalies
Any BSM extension must satisfy this; the abstract does not discuss anomaly cancellation.

invented entities (2)

Neutral boson (scalar or vector)
purpose: mediates the one-loop Majorana mass and can be a dark matter candidate
Introduced ad hoc; no experimental signature or theoretical constraint is mentioned in the abstract.
Neutral fermion beyond N_R and N_L
purpose: participates in the loop and can be a dark matter candidate
Introduced to realize the loop; no evidence outside the model is given.

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Cite this review

Pith. "Pith review of A natural realization of inverse seesaw model in a non-invertible selection rule." pith.science (2026). https://pith.science/paper/HIDXKFCP

@misc{pith2026250816174,
  author       = {Pith},
  title        = {Pith review of: A natural realization of inverse seesaw model in a non-invertible selection rule},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HIDXKFCP}},
  note         = {Machine review of arXiv:2508.16174}
}

abstract

We propose natural hierarchies among neutral fermions in a framework of inverse seesaw, imposing a $Z_3$ Tambara-Yamagami fusion rule which is applied to our phenomenology. Under the symmetry, the Majorana mass terms for $N_R$ and $N_L$ are forbidden at tree level. However they are generated at one-loop level where the symmetry is dynamically broken. In order to realize such loop corrections, we introduce neutral boson and fermion either of which can be an appropriate dark matter candidate. Finally, we show the best fit value of the neutrino sector for normal and inverted hierarchies referring recent experimental results.

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Forward citations

Cited by 2 Pith papers

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

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A radiative lepton mass model with a Z2-gauged Z5 non-invertible fusion rule can fit neutrino oscillation data and predicts charged-lepton EDMs that indirectly bound 0νββ to ≲28–33.5 meV.
Non-Invertible Selection Rules on Heterotic Non-Abelian Orbifolds
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Non-Abelian orbifolds in heterotic string theory produce non-invertible coupling selection rules, since twisted sectors are labeled by conjugacy classes whose products contain multiple classes and yield characteristic...

Reference graph

Works this paper leans on

37 extracted references · 4 canonical work pages · cited by 2 Pith papers

[1]

author R. N. Mohapatra and author J. W. F. Valle , journal Phys. Rev. D volume 34 , pages 1642 ( year 1986 )

work page 1986
[2]

Wyler and author L

author D. Wyler and author L. Wolfenstein , journal Nucl. Phys. B volume 218 , pages 205 ( year 1983 )

work page 1983
[3]

Fernandez-Martinez , author J

author E. Fernandez-Martinez , author J. Hernandez-Garcia , and author J. Lopez-Pavon , journal JHEP volume 08 , pages 033 ( year 2016 ), 1605.08774

arXiv 2016
[4]

author N. R. Agostinho , author G. C. Branco , author P. M. F. Pereira , author M. N. Rebelo , and author J. I. Silva-Marcos , journal Eur. Phys. J. C volume 78 , pages 895 ( year 2018 ), 1711.06229

arXiv 2018
[5]

Das , author T

author A. Das , author T. Nomura , author H. Okada , and author S. Roy , journal Phys. Rev. D volume 96 , pages 075001 ( year 2017 ), 1704.02078

arXiv 2017
[6]

Nomura and author H

author T. Nomura and author H. Okada , journal Phys. Rev. D volume 99 , pages 055027 ( year 2019 a ), 1807.04555

arXiv 2019
[7]

author U. K. Dey , author T. Nomura , and author H. Okada , journal Phys. Rev. D volume 100 , pages 075013 ( year 2019 ), 1902.06205

work page Pith review arXiv 2019
[8]

Modular $A_4$ symmetric inverse seesaw model with $SU(2)_L$ multiplet fields

author T. Nomura and author H. Okada ( year 2020 ), 2007.15459

work page Pith review arXiv 2020

Show all 37 references

[9]

Nomura and author H

author T. Nomura and author H. Okada , journal Phys. Lett. B volume 792 , pages 424 ( year 2019 b ), 1809.06039

2019 arXiv
[10]

Nomura and author H

author T. Nomura and author H. Okada ( year 2024 ), 2403.14193

2024
[11]

Nomura , author H

author T. Nomura , author H. Okada , and author P. Sanyal , journal Eur. Phys. J. C volume 82 , pages 697 ( year 2022 ), 2103.09494

2022 arXiv
[12]

Okada and author T

author H. Okada and author T. Toma , journal Phys. Rev. D volume 86 , pages 033011 ( year 2012 ), 1207.0864

2012 arXiv
[13]

Kobayashi , author H

author T. Kobayashi , author H. Okada , and author H. Otsuka ( year 2025 a ), 2505.14878

2025
[14]

Okada and author Y

author H. Okada and author Y. Shigekami ( year 2025 ), 2507.16198

2025 arXiv
[15]

Choi , author H

author Y. Choi , author H. T. Lam , and author S.-H. Shao , journal Phys. Rev. Lett. volume 129 , pages 161601 ( year 2022 ), 2205.05086

2022 arXiv
[16]

Cordova , author S

author C. Cordova , author S. Hong , author S. Koren , and author K. Ohmori , journal Phys. Rev. X volume 14 , pages 031033 ( year 2024 a ), 2211.07639

2024 arXiv
[17]

Cordova and author K

author C. Cordova and author K. Ohmori , journal Phys. Rev. X volume 13 , pages 011034 ( year 2023 ), 2205.06243

2023 arXiv
[18]

Cordova , author S

author C. Cordova , author S. Hong , and author S. Koren ( year 2024 b ), 2402.12453

2024 arXiv
[19]

Kobayashi , author H

author T. Kobayashi , author H. Otsuka , and author M. Tanimoto , journal JHEP volume 12 , pages 117 ( year 2024 ), 2409.05270

2024 arXiv
[20]

Kobayashi and author H

author T. Kobayashi and author H. Otsuka , journal JHEP volume 11 , pages 120 ( year 2024 ), 2408.13984

2024 arXiv
[21]

Kobayashi , author Y

author T. Kobayashi , author Y. Nishioka , author H. Otsuka , and author M. Tanimoto , journal JHEP volume 05 , pages 177 ( year 2025 b ), 2503.09966

2025 arXiv
[22]

Suzuki and author L.-X

author M. Suzuki and author L.-X. Xu ( year 2025 ), 2503.19964

2025
[23]

Liang and author T

author Q. Liang and author T. T. Yanagida ( year 2025 ), 2505.05142

2025 arXiv
[24]

Kobayashi , author H

author T. Kobayashi , author H. Otsuka , author M. Tanimoto , and author H. Uchida ( year 2025 c ), 2505.07262

2025 arXiv
[25]

Kobayashi , author H

author T. Kobayashi , author H. Mita , author H. Otsuka , and author R. Sakuma ( year 2025 d ), 2506.10241

2025 arXiv
[26]

Nomura and author H

author T. Nomura and author H. Okada ( year 2025 ), 2506.16706

2025 arXiv
[27]

Dong , author T

author J. Dong , author T. Jeric , author T. Kobayashi , author R. Nishida , and author H. Otsuka ( year 2025 ), 2507.02375

2025 arXiv
[28]

Nomura and author O

author T. Nomura and author O. Popov ( year 2025 ), 2507.10299

2025 arXiv
[29]

Chen , author C.-Q

author J. Chen , author C.-Q. Geng , author H. Okada , and author J.-J. Wu ( year 2025 ), 2507.11951

2025 arXiv
[30]

Kobayashi , author H

author T. Kobayashi , author H. Otsuka , and author T. T. Yanagida ( year 2025 e ), 2508.12287

2025 arXiv
[31]

Suzuki , author L.-X

author M. Suzuki , author L.-X. Xu , and author H. Y. Zhang ( year 2025 ), 2508.14970

2025 arXiv
[32]

Chang , author Y.-H

author C.-M. Chang , author Y.-H. Lin , author S.-H. Shao , author Y. Wang , and author X. Yin , journal JHEP volume 01 , pages 026 ( year 2019 ), 1802.04445

2019 arXiv
[33]

Ma , journal Phys

author E. Ma , journal Phys. Rev. D volume 73 , pages 077301 ( year 2006 ), hep-ph/0601225

2006 arXiv
[34]

Kanemura , author S

author S. Kanemura , author S. Matsumoto , author T. Nabeshima , and author N. Okada , journal Phys. Rev. D volume 82 , pages 055026 ( year 2010 ), 1005.5651

2010 arXiv
[35]

Aghanim et al

author N. Aghanim et al. ( collaboration Planck ), journal Astron. Astrophys. volume 641 , pages A6 ( year 2020 ), note [Erratum: Astron.Astrophys. 652, C4 (2021)] , 1807.06209

2020 arXiv
[36]

Abe et al

author S. Abe et al. ( collaboration KamLAND-Zen ) ( year 2024 ), 2406.11438

2024
[37]

Esteban , author M

author I. Esteban , author M. C. Gonzalez-Garcia , author M. Maltoni , author I. Martinez-Soler , author J. a. P. Pinheiro , and author T. Schwetz , journal JHEP volume 12 , pages 216 ( year 2024 ), 2410.05380

2024 arXiv

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [1]

author R. N. Mohapatra and author J. W. F. Valle , journal Phys. Rev. D volume 34 , pages 1642 ( year 1986 )

work page 1986

[2] [2]

Wyler and author L

author D. Wyler and author L. Wolfenstein , journal Nucl. Phys. B volume 218 , pages 205 ( year 1983 )

work page 1983

[3] [3]

Fernandez-Martinez , author J

author E. Fernandez-Martinez , author J. Hernandez-Garcia , and author J. Lopez-Pavon , journal JHEP volume 08 , pages 033 ( year 2016 ), 1605.08774

arXiv 2016

[4] [4]

author N. R. Agostinho , author G. C. Branco , author P. M. F. Pereira , author M. N. Rebelo , and author J. I. Silva-Marcos , journal Eur. Phys. J. C volume 78 , pages 895 ( year 2018 ), 1711.06229

arXiv 2018

[5] [5]

Das , author T

author A. Das , author T. Nomura , author H. Okada , and author S. Roy , journal Phys. Rev. D volume 96 , pages 075001 ( year 2017 ), 1704.02078

arXiv 2017

[6] [6]

Nomura and author H

author T. Nomura and author H. Okada , journal Phys. Rev. D volume 99 , pages 055027 ( year 2019 a ), 1807.04555

arXiv 2019

[7] [7]

author U. K. Dey , author T. Nomura , and author H. Okada , journal Phys. Rev. D volume 100 , pages 075013 ( year 2019 ), 1902.06205

work page Pith review arXiv 2019

[8] [8]

Modular $A_4$ symmetric inverse seesaw model with $SU(2)_L$ multiplet fields

author T. Nomura and author H. Okada ( year 2020 ), 2007.15459

work page Pith review arXiv 2020