Pith. sign in

REVIEW 2 major objections 5 minor 4 cited by

No-group Scotogenic Model

T0 review · 2 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read This paper claims that replacing the ad hoc $\mathbb{Z}_2$ of the scotogenic model with a non-invertible $\mathbb{Z}_{11}$ symmetry yields the minimal consistent model, with a one-zero neutrino mass texture and dark matter stabilized by…

desk verdict A clean one-zero texture from a non-invertible Z_11 rule, but the accidental Z2 protecting dark matter is not shown to survive radiative corrections, so DM stability is an unproven premise. read the letter →

arxiv 2507.10299 v1 pith:5JUY2547 submitted 2025-07-14 hep-ph hep-exhep-th

Takaaki Nomura , Oleg Popov This is my paper · ORCID
classification hep-phhep-exhep-th PACS 14.60.Pq95.35.+d
keywords scotogenicmodelnon-invertiblesymmetryno-groupneutrinomassone-zerotexturedarkmatterstabilitychargedleptonflavorviolationmixing
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 tries to establish that the scotogenic mechanism—tiny one-loop neutrino masses plus an inert dark-matter doublet—can be realized with a non-invertible (no-group) $\mathbb{Z}_M$ symmetry in place of the usual ad hoc $\mathbb{Z}_2$. The minimal working case is $M=11$, with the inert doublet $\eta$ assigned to class $[g^5]$; this assignment produces a Yukawa texture that fits the measured lepton mixing angles and forbids the lepton-number-violating $\bar{\ell}L\eta$ operators that would let the dark matter decay. The resulting neutrino mass matrix has a one-zero texture, and the position of the zero determines which charged-lepton flavor-violating decay is absent at one loop. If the construction is right, it is the minimal no-group scotogenic model meeting the stated criteria and a new way to tie dark matter stability to a flavor symmetry.

What carries the argument

The machinery is the non-invertible (no-group) selection rule based on conjugacy classes $[g^k]$ under $\mathbb{Z}_M$, together with the one-loop scotogenic mass formula. A term is allowed only if the product of its class factors contains $[g^0]$; applying this to $\bar{L}N\eta$, $\bar{L}\ell H$, and $\bar{\ell}L\eta$ gives the Yukawa texture that the model needs. The one-loop expression $(m_\nu)_{ij}\simeq \sum_\alpha y^\eta_{i\alpha} M_\alpha F_\alpha (y^\eta)^T_{\alpha j}/(4\pi)^2$, with $F_\alpha$ the standard loop function of inert scalar and Majorana masses, converts that texture into a one-zero Majorana mass matrix, and the diagonal charged-lepton sector makes the PMNS matrix equal to the neutrino diagonalizing matrix.

What would settle it

A specific test would be a complete calculation of the radiatively corrected effective potential: if loop corrections generate an $\eta^\dagger H$ mixing term or a lepton-number-violating coupling at any order, the inertness of $\eta$ and the stability of dark matter collapse. A model-level test is fitting the one-zero texture $(m_\nu)_{13}=0$ to future neutrino oscillation data; if that texture is excluded at more than $3\sigma$ for the class assignment used here, the construction fails.

Watch

Extended reading notes

Core claim

The central claim is that a non-invertible $\mathbb{Z}_{11}$ symmetry, whose selection rules are fixed by conjugacy classes $[g^k]$ with product $[g^k][g^{k'}] = [g^{k+k'}] + [g^{M-k+k'}]$, can fully replace the hand-assigned $\mathbb{Z}_2$ parity of the original scotogenic model. With leptons in classes $\{[g^0],[g^1],[g^2]\}$, singlets $N$ in $\{[g^3],[g^4],[g^5]\}$, and $\eta$ in $[g^5]$, the Yukawa matrix $y_\eta$ has exactly the five non-zero entries needed to fit neutrino oscillation data, the charged-lepton mass matrix remains diagonal, and an accidental $\mathbb{Z}_2$ coincides with the scotogenic parity so that $\eta$ stays inert and dark matter is stable. The paper argues $M=11$ is minimal for these criteria, that the neutrino mass matrix is forced to a one-zero texture, and that permuting the class assignments moves the zero while changing which flavor-violating decay is forbidden.

Load-bearing premise

The load-bearing premise is that the accidental symmetry that keeps the second scalar doublet inert and the dark matter stable survives quantum corrections, even though the paper itself notes that non-invertible symmetries are broken by loop effects.

Editorial extensions

If this is right

  • No ad hoc $\mathbb{Z}_2$ is needed: dark matter stability emerges from an accidental parity of the no-group class assignment.
  • The neutrino mass matrix is forced to one-zero form, and the paper shows that such forms fit current Nufit 6.0 oscillation data within $3\sigma$ for both hierarchies.
  • Exactly one charged-lepton flavor-violating decay ($\mu\to e\gamma$, $\tau\to e\gamma$, or $\tau\to\mu\gamma$) is forbidden at one loop, with the forbidden mode correlated to the position of the zero in $m_\nu$.
  • In the inverted hierarchy the model predicts $|m_{ee}|$ in the range between the current KamLAND-Zen limits, making near-future neutrinoless double beta decay experiments a direct probe.
  • The cosmological bound on the sum of neutrino masses $\sum m_\nu$ already selects among the allowed parameter points, with the normal hierarchy faring better than the inverted one in the presented scan.

Reading between the lines

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

  • If radiative corrections do break the accidental $\mathbb{Z}_2$, a small $\eta^\dagger H$ mixing term would be induced; a two-loop calculation would show whether the minimal $M=11$ model is radiatively stable or needs an additional protecting symmetry.
  • Beyond the paper's own scan, the one-zero position versus forbidden decay correlation is a general feature of the no-group product rule, so the same class-assignment logic could be applied to other radiative seesaw models or to non-minimal field content.
  • A future observation of, say, $\tau\to\mu\gamma$ at non-zero rate while $\tau\to e\gamma$ remains below current bounds would single out one of the class assignments listed in the paper's Table II, giving an experimental fingerprint of the symmetry.
  • A future global fit that rejects all one-zero textures would exclude this $M=11$ construction, since the texture is forced by the symmetry rather than chosen by hand.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper constructs a scotogenic model in which the usual ad hoc Z2 symmetry is replaced by a non-invertible Z_M symmetry. The authors search over M and class assignments, identify M = 11 with η assigned to class [g5] as minimal, and derive a Yukawa texture that yields a one-zero neutrino mass matrix. They then scan the model parameters to fit the neutrino oscillation observables from NuFIT 6.0, present correlations among CP phases, sum of neutrino masses, and |mee|, and compute charged-lepton flavor-violating branching ratios. The paper claims that the setup guarantees dark-matter stability and the scotogenic structure via the no-group symmetry.

Significance. If the central construction is radiatively stable, the paper would be a useful addition to the recent literature on phenomenological applications of non-invertible symmetries: it provides a minimal scotogenic realization with a one-zero texture and a correlated prediction for which cLFV mode is forbidden. The class-product algebra in Appendix A and the one-loop neutrino mass formula are standard and are applied cleanly, and the numerical scan is transparent and easy to reproduce. The main significance is however conditional on the fate of the accidental Z2 under radiative corrections, which the paper does not analyze; this is the key point that the revision must address.

major comments (2)
  1. [Section 2.1 (after Table I) and Introduction] The defining property of the model—inertness of η and dark-matter stability—is attributed to an accidental Z2 that coincides with the scotogenic Z2, yet the Introduction states that non-invertible Z_M symmetries are broken by radiative corrections (refs. [32,36-38]). No loop-level analysis is given for the accidental Z2. If radiative corrections to the non-invertible selection rule generate any operator that is odd under that Z2, e.g. \bar\ell L η or η†H, then the Z2 is explicitly broken, η can mix with the Higgs or decay to leptons, and the scotogenic one-loop neutrino mass mechanism together with the dark-matter candidate is lost. The authors must either prove that the accidental Z2 is exact to all orders, compute the leading loop-induced coefficients and show they are harmless, or modify the model. The criterion in Section 2.1 that stability is 'guaranteed, at least renormalized level' is not substantiated by the present analysis.
  2. [Section 3, Eqs. (14)-(19)] The numerical treatment double-counts the atmospheric mass splitting. Equation (15) determines M1 from the experimental Δm²_atm, and then Δm²_atm is included among the four observables in the χ² of Eq. (19). With M1 fixed in this way, the theoretical Δm²_atm equals the input value by construction, so its contribution to χ² is trivial. The confidence-level statements in Figs. 1-4 should be based on the genuinely fitted observables, and the procedure for scanning M1 and Δm²_atm should be described precisely; otherwise the quoted Δχ² ranges are not well defined.
minor comments (5)
  1. [Throughout] There are several typographical errors: 'discss' should be 'discuss' in the Introduction, 'mehcanism' should be 'mechanism' and 'lows' should be 'rows' in Section 2.1, 'ad.hoc.' should be 'ad hoc' in Section 2, and 'storongest' should be 'strongest' in the Fig. 3 caption.
  2. [Figs. 1-2] The color legend says red and blue points correspond to Δχ² values within 5σ-3σ and within 3σ, but the numerical thresholds for these confidence levels are not stated; please add the relevant Δχ² values or a color scale.
  3. [Section 2.4, Table II] The notation 'µ(τ) → e(µ)γ' in Table II is ambiguous; it should be written as separate modes, e.g. µ → eγ, τ → eγ, and τ → µγ, with the correspondence to the zero-texture position made explicit.
  4. [Section 3, Eq. (13)] The Majorana phase convention should be stated more precisely: the PMNS matrix should be written as U = U_ν diag(1, e^{iα21/2}, e^{iα31/2}) so that the phases appearing in |mee| are unambiguous.
  5. [Abstract and Section 4] The abstract and summary state that dark-matter stability and the scotogenic structure are 'achieved via the new non-group symmetry', whereas the body attributes stability to an accidental Z2. This attribution should be corrected throughout the paper.

Circularity Check

0 steps flagged · score 0.0 of 10

No load-bearing circularity: the one-zero texture follows from the chosen Z11 class assignment, and the numerical scan does not rename fitted inputs as predictions.

full rationale

The central derivation is self-contained. The Yukawa texture in Eq. (8) for eta in class [g5] is obtained by applying the class-multiplication rule of Eqs. (A2)-(A3) to the class assignments in Table I; the one-zero neutrino texture (m_nu)_13=(m_nu)_31=0 then follows from Eq. (11) and the loop formula Eq. (12), with no neutrino-oscillation data used to fix the zero. The charged-lepton mass matrix is diagonal by the same assignment, and the cLFV selection in Table II is a direct consequence of the same y_eta texture. The numerical analysis fits sin^2 theta12, sin^2 theta13, Delta m^2_atm and Delta m^2_sol with a scan over the free couplings, masses and phases, and M1 is normalized to Delta m^2_atm through Eq. (15); the plotted correlations for Dirac and Majorana phases, sum of masses and |m_ee| are marginals over the surviving scan points, not fitted values relabeled as predictions. The paper's own caveat that non-invertible symmetries can be broken by radiative corrections (Sec. 1, citing refs. [32,36-38]) and the absence of a loop-level proof that the accidental Z2 survives are genuine robustness gaps for the dark-matter-stability claim, but they are omissions rather than reductions of a claimed output to an input. The self-citations in the introductory survey are contextual and are not the load-bearing justification for the M=11 minimality or for the texture derivation.

Assumptions & free parameters 8 free parameters · 6 assumptions · 0 invented entities

The model contains no new particles; the invented content is the non-invertible class assignment itself, which acts as a bookkeeping symmetry. The many continuous Yukawa couplings and mass ratios listed above carry most of the predictive freedom.

free parameters (8)
  • yη13, yη22, yη23, yη31, yη32 = 10^-8 to 1 (scan range)
    Five real entries of the neutrino Dirac Yukawa matrix; scanned to fit oscillation data.
  • α, β = [-π, π] (scan range)
    Two physical phases in yη; scanned.
  • M2/M1, M3/M1 = 1 to 10 (scan range)
    Mass ratios of the three singlet fermions; free parameters.
  • mηR/M1 = 1 to 10 (scan range)
    Mass scale of the inert neutral scalar; free parameter.
  • Delta m^2 (dimensionless ηR-ηI splitting) = 10^-6 to 0.1 (scan range)
    Sets the loop function Fα; free parameter.
  • M1 = fixed via Eq (15) to Δm²atm
    Overall neutrino mass scale is normalized to the atmospheric mass splitting, so absolute masses are fitted, not predicted.
  • yℓ11, yℓ22, yℓ33 = fixed to me, mμ, mτ
    Diagonal charged lepton Yukawa couplings; determined by experiment.
  • Scalar potential couplings (μη², λ0, λHη, λ'Hη) = not scanned directly
    They fix the η mass spectrum; only combinations such as Δm² enter the observables.
assumptions (6)
  • standard math Selection rule Eq (A3) for class products fully determines allowed couplings.
    Assumed to be the correct theory of non-invertible selection rules as established in refs [22-36].
  • domain assumption The scotogenic one-loop formula Eq (12) is valid with diagonal MN and inert η.
    Standard result from the original scotogenic model (ref [1]); used without re-derivation.
  • domain assumption Experimental inputs from NuFit 6.0, MEG, BaBar, KamLAND-Zen, Planck, and DESI are accurate.
    External data used for the χ² and constraints.
  • ad hoc to paper An accidental Z2 symmetry remains exact after quantum corrections and protects dark matter.
    The paper asserts this after Table I but does not analyze loop-generated operators; its own cited refs [32,36-38] state that no-group symmetries are radiatively broken.
  • domain assumption One-zero textures of the neutrino mass matrix can fit current data.
    Inherited from ref [40]; used to argue the texture is viable.
  • domain assumption The minimality search over M is exhaustive for the stated criteria.
    The paper checks representative assignments for M=4..11; no formal proof covers all permutations and class orderings.

how reviews work

0 comments
Cite this review

Pith. "Pith review of No-group Scotogenic Model." pith.science (2026). https://pith.science/paper/5JUY2547

@misc{pith2026250710299,
  author       = {Pith},
  title        = {Pith review of: No-group Scotogenic Model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5JUY2547}},
  note         = {Machine review of arXiv:2507.10299}
}
abstract

In the present work, the scotogenic model is constructed applying non invertible $Z_M$ symmetries. The stability of dark matter and the scotogenic structure of the neutrino mass matrix is achieved via the new non-group symmetry. The non-group Scotogenic model is given with minimalistic content giving a one-zero structure of the neutrino mass matrix, and numerical analysis of the lepton mixing angles and physics are presented. Other relevant constraints are also studied.

Figures

Figures reproduced from arXiv: 2507.10299 by the authors.

Figure 1
Figure 1. FIG. 1: Values of [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Values of [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Values of [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Values of [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 4 Pith papers

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

  1. Radiative lepton model in a non-invertible fusion rule

    hep-ph 2025-10 conditional novelty 6.0 of 10

    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.

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

    hep-ph 2025-08 unverdicted novelty 6.0 of 10

    A Z3 Tambara-Yamagami fusion rule is used to build an inverse seesaw model with radiatively generated Majorana masses and a dark matter candidate, fitted to neutrino oscillation data.

  3. A radiative lepton model in a non-invertible fusion rule

    hep-ph 2025-07 conditional novelty 6.0 of 10

    Electron and muon masses are generated at one loop through loop-level breaking of the Ising fusion rule, while neutrino masses arise from an unbroken Z2-like sector that stabilizes loop particles.

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

    hep-ph 2025-07 reject novelty 4.0 of 10

    A Ma-model extension with a non-invertible symmetry generates neutrino masses at three loops and yields S0 or eta_R dark matter candidates, with scanned parameter regions shown.

Reference graph

Works this paper leans on

53 extracted references · 51 canonical work pages · cited by 4 Pith papers

  1. [1]

    The origin of neutrino masses, which requires Beyond the Standard Model (BSM) physics, has been a cherished goal for many

    INTRODUCTION N eutrino masses and mixing angles have been a puzzle to physicists for the recent 30 years, since the neutrino oscillations observation in the 90’s. The origin of neutrino masses, which requires Beyond the Standard Model (BSM) physics, has been a cherished goal for many. There have been many proposals on how neutrino masses are produced. Som...

    work page

  2. [2]

    In the scenario the dis- crete Z2 symmetry in the original scotogenic model is converted into a non-invertible ZM symmetry

    MODEL SETUP In this section, we discuss setup of a scotogenic model with a non-invertible symmetry. In the scenario the dis- crete Z2 symmetry in the original scotogenic model is converted into a non-invertible ZM symmetry. As in the original scotogenic model, we introduce second Higgs doublet η and SM siglet fermionsNα. The relevant La- grangian for the ...

    work page

  3. [3]

    We write the inert scalar doubletη by η = 1√ 2 (ηR + iηI ) η− , (3) where η− has electric charge −1 and the other compo- nents are electrically neutral. After electroweak symmetry arXiv:2507.10299v1 [hep-ph] 14 Jul 2025 2 breaking, squared masses of inert scalar bosons are m2 ηR[I] = µ2 η + v2 2 (λHη + λ′ Hη − [+]2λ0), (4a) m2 η± = µ2 η + v2λHη 2 , (4b) w...

    arXiv 2025

  4. [4]

    For illustration, we discuss the structure ofyη in Eq

    NUMERICAL ANALYSIS In this section, we carry out numerical analysis for neutrino mass matrix and cLFV. For illustration, we discuss the structure ofyη in Eq. (11) inducing one-zero texture of mν with vanishing (mν)13(31) element. We randomly scan our Yukawa couplings with two phases {yη 13, yη 22, yη 23, yη 31, yη 32, α, β} and mass related pa- rameters {...

    work page

  5. [5]

    The three generations of stan- dard model leptons and singlet Majorana fermions are distinguished by the assignment of conjugacy classes un- der the ZM symmetry

    SUMMARY AND DISCUSSION In this work, we have discussed scotogenic models with a no-group ZM symmetry. The three generations of stan- dard model leptons and singlet Majorana fermions are distinguished by the assignment of conjugacy classes un- der the ZM symmetry. We have then searched for the minimal case that realize scotogenic neutrino mass gener- ation...

    work page

  6. [6]

    Verifiable radiative seesaw mechanism of neutrino mass and dark matter.Phys

    Ernest Ma. Verifiable radiative seesaw mechanism of neutrino mass and dark matter.Phys. Rev., D73:077301, 2006

    work page 2006

  7. [7]

    K. S. Babu, Ernest Ma, and J. W. F. Valle. Underlying A(4) symmetry for the neutrino mass matrix and the quark mixing matrix.Phys. Lett. B, 552:207–213, 2003

    work page 2003

  8. [8]

    Rajasekaran

    Ernest Ma and G. Rajasekaran. Softly broken A(4) sym- metry for nearly degenerate neutrino masses.Phys. Rev. D, 64:113012, 2001

    work page 2001

Show all 53 references
  1. [9]

    Discrete Flavor Symmetries and Models of Neutrino Mixing.Rev

    Guido Altarelli and Ferruccio Feruglio. Discrete Flavor Symmetries and Models of Neutrino Mixing.Rev. Mod. Phys., 82:2701–2729, 2010

    2010

  2. [10]

    Hernandez and A

    D. Hernandez and A. Yu. Smirnov. Lepton mixing and discrete symmetries. Phys. Rev. D, 86:053014, 2012

    2012

  3. [11]

    King and Christoph Luhn

    Stephen F. King and Christoph Luhn. Neutrino Mass and Mixing with Discrete Symmetry.Rept. Prog. Phys., 76:056201, 2013

    2013

  4. [12]

    Neutrino Mass and Mixing: from Theory to Experiment

    StephenF.King, AlexanderMerle, StefanoMorisi, Yusuke Shimizu, and Morimitsu Tanimoto. Neutrino Mass and Mixing: from Theory to Experiment. New J. Phys., 16:045018, 2014

    2014

  5. [13]

    Tatsuo Kobayashi, Hiroshi Ohki, Hiroshi Okada, Yusuke Shimizu, and Morimitsu Tanimoto.An Introduction to Non-Abelian Discrete Symmetries for Particle Physicists. 1 2022

    2022

  6. [14]

    Ferruccio Feruglio.Are neutrino masses modular forms?, pages 227–266. 2019

    2019

  7. [15]

    Modu- lar flavor symmetric models

    Tatsuo Kobayashi and Morimitsu Tanimoto. Modu- lar flavor symmetric models. Int. J. Mod. Phys. A, 39(09n10):2441012, 2024

    2024

  8. [16]

    Gui-Jun Ding and Stephen F. King. Neutrino mass and mixing with modular symmetry.Rept. Prog. Phys., 87(8):084201, 2024

    2024

  9. [17]

    Tat- suishi

    Tatsuo Kobayashi, Kentaro Tanaka, and Takuya H. Tat- suishi. Neutrino mixing from finite modular groups.Phys. Rev. D, 98(1):016004, 2018

    2018

  10. [18]

    A modular A4 symmetric model of dark matter and neutrino.Phys

    Takaaki Nomura and Hiroshi Okada. A modular A4 symmetric model of dark matter and neutrino.Phys. Lett. B, 797:134799, 2019

    2019

  11. [19]

    A modular A4 symmetric scotogenic model.Phys

    Takaaki Nomura, Hiroshi Okada, and Oleg Popov. A modular A4 symmetric scotogenic model.Phys. Lett. B, 803:135294, 2020

    2020

  12. [20]

    Dirac Radiative Neu- trino Mass with Modular Symmetry and Leptogenesis

    Arnab Dasgupta, Takaaki Nomura, Hiroshi Okada, Oleg Popov, and Morimitsu Tanimoto. Dirac Radiative Neu- trino Mass with Modular Symmetry and Leptogenesis. 11 2021

    2021

  13. [21]

    Modular flavor models with positive modular weights: a new lepton model building.JHEP, 01:121, 2024

    Tatsuo Kobayashi, Takaaki Nomura, Hiroshi Okada, and Hajime Otsuka. Modular flavor models with positive modular weights: a new lepton model building.JHEP, 01:121, 2024

    2024

  14. [22]

    Non-holomorphic modular flavor symmetry.JHEP, 08:136, 2024

    Bu-Yao Qu and Gui-Jun Ding. Non-holomorphic modular flavor symmetry.JHEP, 08:136, 2024

    2024

  15. [23]

    Lepton seesaw model in a modularA4 symmetry

    Takaaki Nomura and Hiroshi Okada. Lepton seesaw model in a modularA4 symmetry. 9 2024

    2024

  16. [24]

    Non- holomorphic modular A5 symmetry for lepton masses and mixing

    Cai-Chang Li, Jun-Nan Lu, and Gui-Jun Ding. Non- holomorphic modular A5 symmetry for lepton masses and mixing. JHEP, 12:189, 2024

    2024

  17. [25]

    A radiative neutrino mass model with leptoquarks under non-holomorphic modular A4 symmetry

    Takaaki Nomura, Hiroshi Okada, and Xing-Yu Wang. A radiative neutrino mass model with leptoquarks under non-holomorphic modular A4 symmetry. 4 2025

    2025

  18. [26]

    Non- holomorphic modular A4 symmetric scotogenic model

    Takaaki Nomura, Hiroshi Okada, and Oleg Popov. Non- holomorphic modular A4 symmetric scotogenic model. Phys. Lett. B, 860:139171, 2025

    2025

  19. [27]

    Pedro R. S. Gomes. An introduction to higher-form symmetries. SciPost Phys. Lect. Notes, 74:1, 2023

    2023

  20. [28]

    ICTP lectures on (non-)invertible generalized symmetries

    Sakura Schafer-Nameki. ICTP lectures on (non-)invertible generalized symmetries. Phys. Rept., 1063:1–55, 2024

    2024

  21. [29]

    Bottini, Ludovic Fraser- Taliente, Liam Gladden, Dewi S

    Lakshya Bhardwaj, Lea E. Bottini, Ludovic Fraser- Taliente, Liam Gladden, Dewi S. W. Gould, Arthur Platschorre, and Hannah Tillim. Lectures on general- ized symmetries. Phys. Rept., 1051:1–87, 2024

    2024

  22. [30]

    What’s Done Cannot Be Undone: TASI Lectures on Non-Invertible Symmetries

    Shu-Heng Shao. What’s Done Cannot Be Undone: TASI Lectures on Non-Invertible Symmetries. 8 2023

    2023

  23. [31]

    Yukawa textures from non-invertible symmetries

    Tatsuo Kobayashi, Hajime Otsuka, and Morimitsu Tani- moto. Yukawa textures from non-invertible symmetries. JHEP, 12:117, 2024

    2024

  24. [32]

    Non-invertible Peccei- Quinn symmetry, natural 2HDM alignment, and the visi- ble axion

    Antonio Delgado and Seth Koren. Non-invertible Peccei- Quinn symmetry, natural 2HDM alignment, and the visi- ble axion. JHEP, 02:178, 2025

    2025

  25. [33]

    Quantum aspects of non-invertible flavor symmetries in intersecting/magnetized D-brane models.JHEP, 04:183, 2025

    Shuta Funakoshi, Tatsuo Kobayashi, and Hajime Otsuka. Quantum aspects of non-invertible flavor symmetries in intersecting/magnetized D-brane models.JHEP, 04:183, 2025

    2025

  26. [34]

    More about quark Yukawa textures from selection rules without group actions.JHEP, 05:177, 2025

    Tatsuo Kobayashi, Yume Nishioka, Hajime Otsuka, and Morimitsu Tanimoto. More about quark Yukawa textures from selection rules without group actions.JHEP, 05:177, 2025

    2025

  27. [35]

    Yanagida

    Qiuyue Liang and Tsutomu T. Yanagida. Non-invertible symmetry as an axion-less solution to the strong CP problem. Phys. Lett. B, 868:139706, 2025

    2025

  28. [36]

    Lepton mass textures from non- invertible multiplication rules

    Tatsuo Kobayashi, Hajime Otsuka, Morimitsu Tanimoto, and Haruki Uchida. Lepton mass textures from non- invertible multiplication rules. 5 2025

    2025

  29. [37]

    Radiative neutrino mass models from non-invertible se- lection rules

    Tatsuo Kobayashi, Hiroshi Okada, and Hajime Otsuka. Radiative neutrino mass models from non-invertible se- lection rules. 5 2025

    2025

  30. [38]

    Matter symmetries in supersymmetric standard models from non-invertible selection rules

    Tatsuo Kobayashi, Hironobu Mita, Hajime Otsuka, and Riku Sakuma. Matter symmetries in supersymmetric standard models from non-invertible selection rules. 6 2025

    2025

  31. [39]

    Radiative lepton seesaw model in a non-invertible fusion rule and gauged B − L symmetry

    Takaaki Nomura and Hiroshi Okada. Radiative lepton seesaw model in a non-invertible fusion rule and gauged B − L symmetry. 6 2025

    2025

  32. [40]

    On discrete gauging and non- invertible selection rules

    Jun Dong, Tim Jeric, Tatsuo Kobayashi, Ryusei Nishida, and Hajime Otsuka. On discrete gauging and non- invertible selection rules. 7 2025

    2025

  33. [41]

    Phenomenological im- plications of a class of non-invertible selection rules

    Motoo Suzuki and Ling-Xiao Xu. Phenomenological im- plications of a class of non-invertible selection rules. 3 2025

    2025

  34. [42]

    Heckman, Jacob McNamara, Miguel Mon- tero, Adar Sharon, Cumrun Vafa, and Irene Valenzuela

    Jonathan J. Heckman, Jacob McNamara, Miguel Mon- tero, Adar Sharon, Cumrun Vafa, and Irene Valenzuela. Fate of stringy noninvertible symmetries.Phys. Rev. D, 110(10):106001, 2024

    2024

  35. [43]

    Justin Kaidi, Yuji Tachikawa, and Hao Y. Zhang. On a class of selection rules without group actions in field theory and string theory.SciPost Phys., 17(6):169, 2024

    2024

  36. [44]

    It is also possible to take only two generations ofN where the neutrino mass matrix in the case is mentioned in ref. [32]

  37. [45]

    E. I. Lashin and N. Chamoun. The One-zero Textures of Majorana Neutrino Mass Matrix and Current Experimen- tal Tests.Phys. Rev. D, 85:113011, 2012. 7

    2012

  38. [46]

    Abe et al

    S. Abe et al. Search for Majorana Neutrinos with the Complete KamLAND-Zen Dataset. 6 2024

    2024

  39. [47]

    An explanation of one-loop induced h→ µτ decay

    Seungwon Baek, Takaaki Nomura, and Hiroshi Okada. An explanation of one-loop induced h→ µτ decay. Phys. Lett. B, 759:91–98, 2016

    2016

  40. [48]

    A. M. Baldini et al. Search for the lepton flavour violat- ing decay µ+ → e+γ with the full dataset of the MEG experiment. Eur. Phys. J. C, 76(8):434, 2016

    2016

  41. [49]

    Searches for Lepton Flavor Viola- tion in the Decaysτ ± → e±γ and τ ± → µ±γ

    Bernard Aubert et al. Searches for Lepton Flavor Viola- tion in the Decaysτ ± → e±γ and τ ± → µ±γ. Phys. Rev. Lett., 104:021802, 2010

    2010

  42. [50]

    The quest for µ → eγ: present and future

    Francesco Renga. The quest for µ → eγ: present and future. Hyperfine Interact., 239(1):58, 2018

    2018

  43. [51]

    Ivan Esteban, M. C. Gonzalez-Garcia, Michele Maltoni, Ivan Martinez-Soler, João Paulo Pinheiro, and Thomas Schwetz. NuFit-6.0: updated global analysis of three- flavor neutrino oscillations.JHEP, 12:216, 2024

    2024

  44. [52]

    Aghanim et al

    N. Aghanim et al. Planck 2018 results. VI. Cosmological parameters. Astron. Astrophys., 641:A6, 2020. [Erratum: Astron.Astrophys. 652, C4 (2021)]

    2021

  45. [53]

    DESI2024VI:cosmologicalconstraints from the measurements of baryon acoustic oscillations

    A.G.Adameetal. DESI2024VI:cosmologicalconstraints from the measurements of baryon acoustic oscillations. JCAP, 02:021, 2025

    2025

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.