pith. machine review for the scientific record. sign in

arxiv: 2605.11097 · v1 · submitted 2026-05-11 · ✦ hep-ph

Recognition: no theorem link

Two-loop neutrino mass model with modular S₄ symmetry

A. E. C\'arcamo Hern\'andez, Carlos A. Vaquera-Araujo, Daniel Salinas-Arizmendi, J. Echeverria-Puentes, Sergey Kovalenko, Vishnudath K. N.

Authors on Pith no claims yet

Pith reviewed 2026-05-13 02:33 UTC · model grok-4.3

classification ✦ hep-ph
keywords neutrino massmodular symmetryS4two-loopdark matterlepton flavor violationflavor symmetry
0
0 comments X

The pith

A modular S4 symmetry generates neutrino masses only at two loops while stabilizing dark matter candidates.

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

This paper constructs a neutrino mass model in which the modular S4 symmetry, combined with a Z3 symmetry, breaks spontaneously to leave a Z2 remnant. The Z2 symmetry prevents neutrino masses from appearing at tree level or one loop, forcing them to arise at two loops instead. The same Z2 stabilizes both scalar and fermionic dark matter candidates. The model fits the observed charged lepton masses and neutrino oscillation data for the normal ordering, while also predicting charged lepton flavor violation rates that could be seen in upcoming experiments.

Core claim

The model is based on the modular Γ4 ≃ S4 flavour symmetry supplemented by a discrete Z3 symmetry. Spontaneous breaking of the modular symmetry produces a remnant Z2 symmetry that guarantees the radiative origin of active neutrino masses at two loops and stabilizes dark matter candidates. The construction reproduces charged lepton masses and neutrino oscillation data for normal ordering, predicts observable rates for charged lepton flavour violation, and identifies parameter regions consistent with relic density, LFV constraints, and direct detection limits for both scalar and fermionic dark matter.

What carries the argument

The remnant Z2 symmetry that emerges after spontaneous breaking of the modular S4 symmetry, which both enforces two-loop neutrino masses and protects dark matter particles from decaying.

If this is right

  • Charged lepton flavor violation processes occur at observable rates.
  • Viable scalar dark matter candidates arise from singlet-doublet mixing.
  • A fermionic dark matter candidate is strongly linked to lepton flavor violation.
  • The model accommodates neutrino data specifically for normal ordering.

Where Pith is reading between the lines

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

  • Extensions to other modular symmetries could generate different loop orders for neutrino masses.
  • The correlation between fermionic dark matter and LFV rates offers a way to test the model in combined searches.
  • Similar remnant symmetries might apply to other flavor problems such as the hierarchy of quark masses.

Load-bearing premise

The modular S4 symmetry must break in a way that leaves a Z2 remnant capable of forbidding all lower-order contributions to neutrino masses.

What would settle it

Non-observation of the predicted charged lepton flavor violation signals at the sensitivity of experiments like MEG-II or Mu3e would rule out the viable parameter regions.

Figures

Figures reproduced from arXiv: 2605.11097 by A. E. C\'arcamo Hern\'andez, Carlos A. Vaquera-Araujo, Daniel Salinas-Arizmendi, J. Echeverria-Puentes, Sergey Kovalenko, Vishnudath K. N..

Figure 1
Figure 1. Figure 1: Scalar box diagram that induces the effective [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Two-loop radiative seesaw mechanism for the active neutrino sector. [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: (a) Allowed parameter space compatible with the CMS 3 [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Color map for the mixture of the lightest inert neutral scalar in the (∆ [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Left panel : Allowed values of the real and imaginary parts of the modular field [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Correlation of Br(µ → eγ) with the mass of the lightest heavy neutral lepton mN1 (left panel) and with the charged scalar mass mη± (right panel). The horizontal magenta line indicates the current MEG-II sensitivity [110]. 10−20 10−18 10−16 10−14 10−12 10−10 10−8 Br(µ → eγ) 10−22 10−20 10−18 10−16 10−14 10−12 10−10 10−8 Br( τ → eγ ) MEG-II BaBar superKEKB/Belle II 10−20 10−18 10−16 10−14 10−12 10−10 10−8 Br… view at source ↗
Figure 7
Figure 7. Figure 7: Correlations of Br(τ → eγ) and Br(τ → µγ) against Br(µ → eγ) [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Feynman diagrams for dark matter (DM = χ, N1, ΩR) annihilation channels into Standard Model species (SM = W±, Z, h, ℓ, ν, q) at tree-level. Diagram (a) corresponds to a contact interaction, whereas diagrams (b), (c), and (d) are mediated by a portal particle (PP), which may belong either to the SM or to the dark sector. DM N PP DM N [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Feynman diagram for elastic scattering between dark matter (DM = [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Dependence of DM relic density and DM-nucleon elastic scattering cross section on the lightest inert-scalar mass [PITH_FULL_IMAGE:figures/full_fig_p018_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Left panel: relic abundance Ωh 2 as a function of the fermionic dark matter mass mN1 , with the color code indicating Br(τ → µγ). Right panel: correlation between Ωh 2 and Br(τ → µγ), with the color code showing Br(µ → eγ). The observed relic abundance is achieved in the region where leptophilic annihilation of N1 is sufficiently efficient, while the electronic Yukawa entries remain suppressed enough to s… view at source ↗
Figure 12
Figure 12. Figure 12: Relic abundance Ωh 2 as a function of the fermionic dark matter mass mΩR in the metastable ΩR scenario. The observed Planck value is reached only in the narrow region where annihilation into Higgs bosons is efficient. abundance, whereas heavier ΩR masses typically lead to overabundance. Regarding direct detection, we can conclude the same behaviour as in the N1 case. Since the interaction of ΩR with quark… view at source ↗
read the original abstract

We propose a two-loop radiative neutrino mass model based on the modular $\Gamma_4 \simeq S_4$ flavour symmetry supplemented by a discrete $Z_3$ symmetry. After spontaneous modular symmetry breaking, a remnant $Z_2$ symmetry guarantees both the radiative origin of active neutrino masses and stabilizes dark matter candidates. The model successfully reproduces charged lepton masses and neutrino oscillation data for normal ordering. It also predicts observable rates for charged lepton flavour violation (LFV). Due to the singlet-doublet mixing the model provides a viable scalar dark matter candidate. A fermionic dark matter candidate, strongly linked to LFV, is also present. We identify parameter regions consistent with relic density, LFV constraints, and direct detection limits, providing testable benchmark configurations.

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

3 major / 2 minor

Summary. The paper proposes a two-loop radiative neutrino mass model based on modular S4 (Gamma4) flavor symmetry supplemented by a Z3 discrete symmetry. Spontaneous breaking of the modular symmetry is claimed to leave a remnant Z2 that both enforces the two-loop origin of neutrino masses and stabilizes dark matter candidates. The model is stated to reproduce charged lepton masses and neutrino oscillation data for normal ordering, while predicting observable charged lepton flavor violation (LFV) rates. Viable scalar (singlet-doublet mixed) and fermionic DM candidates are identified in regions consistent with relic density, LFV bounds, and direct detection limits.

Significance. If the remnant Z2 protection is rigorously verified, the construction provides a unified modular-symmetry explanation for radiative neutrino masses, flavor mixing, and DM stability, with testable LFV predictions. The modular S4 approach is current and the two-loop structure is a non-trivial feature. However, the extensive fitting of Yukawa couplings, VEVs, and other parameters to neutrino data, relic density, and LFV constraints introduces significant circularity, making the LFV and DM 'predictions' largely post-fit rather than independent outputs. This reduces the model's falsifiability and overall impact unless the symmetry protection is shown to be parameter-independent.

major comments (3)
  1. [Model construction / symmetry breaking] Model construction and symmetry breaking section: The central claim that the remnant Z2 (arising after the modulus tau acquires its VEV) forbids all one-loop neutrino mass operators must be demonstrated explicitly. List the S4 representations, Z3 charges, and modular weights of the lepton doublets, right-handed neutrinos, and scalars; then enumerate all possible dimension-5 and one-loop effective operators allowed before and after breaking to confirm none survive. Without this explicit check, the two-loop suppression and DM stabilization rest on an unverified assumption.
  2. [Neutrino mass generation] Neutrino mass generation section (likely around the two-loop diagrams): The abstract asserts reproduction of oscillation data for normal ordering, but the fitting procedure for the free Yukawa couplings and VEVs must be shown to leave the two-loop diagrams as the dominant contribution without additional fine-tuning or cancellation. Specify which parameters are fixed by the modular weights versus those adjusted post-hoc.
  3. [DM and LFV phenomenology] DM and LFV phenomenology section: The claimed link between the fermionic DM candidate and LFV processes requires quantitative demonstration that the parameter regions satisfying the relic density (and direct detection) automatically produce LFV rates within current bounds without further adjustment. If the regions are selected after fitting, the predictivity for future LFV searches is weakened.
minor comments (2)
  1. [Model construction] Clarify the notation for modular weights and the specific form of the modular forms used in the Yukawa couplings; ensure consistency between text and any tables of field assignments.
  2. [Phenomenology] In the numerical results, provide a table or plot showing the range of predicted LFV branching ratios for the benchmark points that satisfy all constraints, rather than only stating 'observable rates'.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major point below with clarifications and indicate revisions where they strengthen the manuscript without altering its core results.

read point-by-point responses

  1. Referee: Model construction and symmetry breaking section: The central claim that the remnant Z2 (arising after the modulus tau acquires its VEV) forbids all one-loop neutrino mass operators must be demonstrated explicitly. List the S4 representations, Z3 charges, and modular weights of the lepton doublets, right-handed neutrinos, and scalars; then enumerate all possible dimension-5 and one-loop effective operators allowed before and after breaking to confirm none survive. Without this explicit check, the two-loop suppression and DM stabilization rest on an unverified assumption.

    Authors: We agree that an explicit operator enumeration will make the remnant Z2 protection fully transparent. In the revised manuscript we will add a dedicated table listing the S4 representations, Z3 charges and modular weights of all lepton doublets, right-handed neutrinos and scalar fields. We will then enumerate every dimension-5 and one-loop effective operator allowed by the unbroken symmetries and show that each is forbidden once the modulus VEV breaks S4 to the remnant Z2, while the two-loop diagrams remain permitted. This addition confirms the protection is symmetry-based and parameter-independent. revision: yes

  2. Referee: Neutrino mass generation section (likely around the two-loop diagrams): The abstract asserts reproduction of oscillation data for normal ordering, but the fitting procedure for the free Yukawa couplings and VEVs must be shown to leave the two-loop diagrams as the dominant contribution without additional fine-tuning or cancellation. Specify which parameters are fixed by the modular weights versus those adjusted post-hoc.

    Authors: The modular weights and S4 assignments fix the allowed Yukawa structures and modular forms, leaving only the modulus tau and a small set of scalar VEVs as free parameters. By construction the unbroken symmetries forbid all lower-order neutrino mass operators, so the two-loop diagrams are the sole contribution; no cancellations are imposed by hand. The numerical fit to normal-ordering data is performed within these symmetry-allowed ranges without additional tuning. We will insert a clarifying table that distinguishes symmetry-fixed quantities from the fitted parameters. revision: partial

  3. Referee: DM and LFV phenomenology section: The claimed link between the fermionic DM candidate and LFV processes requires quantitative demonstration that the parameter regions satisfying the relic density (and direct detection) automatically produce LFV rates within current bounds without further adjustment. If the regions are selected after fitting, the predictivity for future LFV searches is weakened.

    Authors: The scan first fits the symmetry-allowed parameters to neutrino and charged-lepton data. Only afterward are relic-density and direct-detection constraints applied to identify viable regions for the fermionic DM candidate. These same regions automatically satisfy current LFV bounds because the DM annihilation channels share the same Yukawa couplings that mediate LFV. No extra tuning is performed to enforce LFV compliance. The resulting benchmark points therefore constitute genuine predictions for future LFV searches. We will add explicit statements and reference the existing figures to emphasize this sequential, adjustment-free consistency. revision: no

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained model construction

full rationale

The paper defines a Lagrangian with modular S4 and Z3 symmetries, assigns representations and weights to fields, invokes spontaneous breaking to a remnant Z2, and states that this Z2 forbids tree-level and one-loop neutrino mass terms while allowing two-loop ones. It then numerically fits a finite set of free parameters (Yukawa couplings, VEVs, modulus tau) to reproduce charged-lepton masses and neutrino oscillation data, after which it computes LFV branching ratios and DM relic density for the same parameter points. None of these steps reduces by construction to its own inputs: the symmetry-allowed operators are fixed by group theory before any fit, the two-loop suppression is a direct consequence of the charge assignments rather than a redefinition of the data, and the LFV/DM outputs are distinct observables computed from the fitted parameters. No self-citation is used to justify uniqueness or to smuggle an ansatz; the central claims rest on explicit symmetry analysis and numerical reproduction of external data. The structure is therefore a standard, non-circular model-building exercise.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 2 invented entities

The central claim rests on the assumption that modular S4 forms and the chosen Z3 charges produce the required remnant Z2 after breaking, plus a large number of Yukawa couplings and vacuum expectation values that are adjusted to data.

free parameters (1)
  • Yukawa couplings and VEVs
    Multiple couplings and vacuum expectation values are adjusted to reproduce charged lepton masses, neutrino oscillation parameters, and dark matter relic density.
axioms (2)
  • domain assumption Modular S4 symmetry with specific modular forms and weights
    Invoked to organize the flavor structure of the Yukawa matrices.
  • ad hoc to paper Spontaneous breaking leaves a remnant Z2 symmetry
    Assumed to enforce the two-loop suppression of neutrino masses and DM stability.
invented entities (2)
  • scalar DM candidate from singlet-doublet mixing no independent evidence
    purpose: Provide a viable scalar dark matter particle
    Introduced via the field content and mixing; no independent evidence outside the model.
  • fermionic DM candidate linked to LFV no independent evidence
    purpose: Provide a second DM candidate whose interactions are tied to flavor violation
    Introduced as part of the particle content; no independent evidence outside the model.

pith-pipeline@v0.9.0 · 5460 in / 1529 out tokens · 56613 ms · 2026-05-13T02:33:46.866997+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

126 extracted references · 126 canonical work pages

  1. [1]

    Pure cases To characterize the phenomenology of the DM candidate in the pure limits, we will discuss in general terms the main annihilation channels, direct detection channels and their behaviour in both regimes. We begin by noting that the exact pure limits are not phenomenologically viable, since the active neutrino mass mechanism is directly controlled...

    work page

  2. [2]

    This mixed region interpolates between the phenomenology of the pure limits, opening viable regions compatible with both relic abundance and direct detection constraints

    Mixed case and numerical results The richest scalar DM phenomenology is found in the intermediate mixing region between doublet and singlet compo- nents, where both the scalar portal and gauge interactions can be simultaneously active. This mixed region interpolates between the phenomenology of the pure limits, opening viable regions compatible with both ...

    work page 2018

  3. [3]

    The relevant question is whether the two states can be dynamically distinguished by the thermal bath at freeze-out

    The role of the radiative splitting The distinctive feature of this model is that at tree-level, the light inert CP-even and CP-odd states are exactly degenerate but a mass splitting between them is radiatively generated. The relevant question is whether the two states can be dynamically distinguished by the thermal bath at freeze-out. If the splitting re...

    work page 2018

  4. [4]

    From the trees to the forest: a review of radiative neutrino mass models,

    Y. Cai, J. Herrero-Garc´ ıa, M. A. Schmidt, A. Vicente, and R. R. Volkas, “From the trees to the forest: a review of radiative neutrino mass models,”Front. in Phys.5(2017) 63,arXiv:1706.08524 [hep-ph]

    work page arXiv 2017

  5. [5]

    Minimal realizations of Dirac neutrino mass from generic one-loop and two-loop topologies atd= 5,

    S. Jana, P. K. Vishnu, and S. Saad, “Minimal realizations of Dirac neutrino mass from generic one-loop and two-loop topologies atd= 5,”JCAP04(2020) 018,arXiv:1910.09537 [hep-ph]

    work page arXiv 2020

  6. [6]

    How many 1-loop neutrino mass models are there?,

    C. Arbel´ aez, R. Cepedello, J. C. Helo, M. Hirsch, and S. Kovalenko, “How many 1-loop neutrino mass models are there?,”JHEP08(2022) 023,arXiv:2205.13063 [hep-ph]

    work page arXiv 2022

  7. [7]

    Quark Mixings and Mass Hierarchy From Radiative Corrections,

    B. S. Balakrishna, A. L. Kagan, and R. N. Mohapatra, “Quark Mixings and Mass Hierarchy From Radiative Corrections,”Phys. Lett. B205(1988) 345–352

    work page 1988

  8. [8]

    Radiative Quark and Lepton Masses Through Soft Supersymmetry Breaking,

    E. Ma, “Radiative Quark and Lepton Masses Through Soft Supersymmetry Breaking,”Phys. Rev. D39(1989) 1922

    work page 1989

  9. [9]

    One Loop Induced Fermion Masses and Exotic Interactions in a Standard Model Context,

    E. Ma, D. Ng, J. T. Pantaleone, and G.-G. Wong, “One Loop Induced Fermion Masses and Exotic Interactions in a Standard Model Context,”Phys. Rev. D40(1989) 1586

    work page 1989

  10. [10]

    Hierarchical Radiative Quark and Lepton Mass Matrices,

    E. Ma, “Hierarchical Radiative Quark and Lepton Mass Matrices,”Phys. Rev. Lett.64(1990) 2866–2869

    work page 1990

  11. [11]

    Pathways to naturally small neutrino masses,

    E. Ma, “Pathways to naturally small neutrino masses,”Phys. Rev. Lett.81(1998) 1171–1174,arXiv:hep-ph/9805219

    work page arXiv 1998

  12. [12]

    Radiative seesaw mechanism at weak scale,

    Z.-j. Tao, “Radiative seesaw mechanism at weak scale,”Phys. Rev. D54(1996) 5693–5697,arXiv:hep-ph/9603309

    work page arXiv 1996

  13. [13]

    Verifiable radiative seesaw mechanism of neutrino mass and dark matter,

    E. Ma, “Verifiable radiative seesaw mechanism of neutrino mass and dark matter,”Phys. Rev. D73(2006) 077301, arXiv:hep-ph/0601225

    work page arXiv 2006

  14. [14]

    Radiative Neutrino Mass, Dark Matter and Leptogenesis,

    P.-H. Gu and U. Sarkar, “Radiative Neutrino Mass, Dark Matter and Leptogenesis,”Phys. Rev. D77(2008) 105031, arXiv:0712.2933 [hep-ph]

    work page arXiv 2008

  15. [15]

    Fermion Triplet Dark Matter and Radiative Neutrino Mass,

    E. Ma and D. Suematsu, “Fermion Triplet Dark Matter and Radiative Neutrino Mass,”Mod. Phys. Lett. A24(2009) 583–589,arXiv:0809.0942 [hep-ph]. 29

    work page arXiv 2009

  16. [16]

    WIMP dark matter as radiative neutrino mass messenger,

    M. Hirsch, R. A. Lineros, S. Morisi, J. Palacio, N. Rojas, and J. W. F. Valle, “WIMP dark matter as radiative neutrino mass messenger,”JHEP10(2013) 149,arXiv:1307.8134 [hep-ph]

    work page arXiv 2013

  17. [17]

    A new radiative neutrino mass generation mechanism with higher dimensional scalar representations and custodial symmetry,

    A. Aranda and E. Peinado, “A new radiative neutrino mass generation mechanism with higher dimensional scalar representations and custodial symmetry,”Phys. Lett. B754(2016) 11–13,arXiv:1508.01200 [hep-ph]

    work page arXiv 2016

  18. [18]

    Radiative Neutrino Masses in the Singlet-Doublet Fermion Dark Matter Model with Scalar Singlets,

    D. Restrepo, A. Rivera, M. S´ anchez-Pel´ aez, O. Zapata, and W. Tangarife, “Radiative Neutrino Masses in the Singlet-Doublet Fermion Dark Matter Model with Scalar Singlets,”Phys. Rev. D92no. 1, (2015) 013005, arXiv:1504.07892 [hep-ph]

    work page arXiv 2015

  19. [19]

    The Inert Zee Model,

    R. Longas, D. Portillo, D. Restrepo, and O. Zapata, “The Inert Zee Model,”JHEP03(2016) 162,arXiv:1511.01873 [hep-ph]

    work page arXiv 2016

  20. [20]

    Verifiable Associated Processes from Radiative Lepton Masses with Dark Matter,

    S. Fraser, E. Ma, and M. Zakeri, “Verifiable Associated Processes from Radiative Lepton Masses with Dark Matter,” Phys. Rev. D93no. 11, (2016) 115019,arXiv:1511.07458 [hep-ph]

    work page arXiv 2016

  21. [21]

    Type II Radiative Seesaw Model of Neutrino Mass with Dark Matter,

    S. Fraser, C. Kownacki, E. Ma, and O. Popov, “Type II Radiative Seesaw Model of Neutrino Mass with Dark Matter,” Phys. Rev. D93no. 1, (2016) 013021,arXiv:1511.06375 [hep-ph]

    work page arXiv 2016

  22. [22]

    Radiative linear seesaw model, dark matter, andU(1) B−L,

    W. Wang and Z.-L. Han, “Radiative linear seesaw model, dark matter, andU(1) B−L,”Phys. Rev. D92(2015) 095001, arXiv:1508.00706 [hep-ph]

    work page arXiv 2015

  23. [23]

    Radiative Seesaw-type Mechanism of Fermion Masses and Non-trivial Quark Mixing,

    C. Arbel´ aez, A. E. C´ arcamo Hern´ andez, S. Kovalenko, and I. Schmidt, “Radiative Seesaw-type Mechanism of Fermion Masses and Non-trivial Quark Mixing,”Eur. Phys. J. C77no. 6, (2017) 422,arXiv:1602.03607 [hep-ph]

    work page arXiv 2017

  24. [24]

    Radiative Type III Seesaw Model and its collider phenomenology,

    F. von der Pahlen, G. Palacio, D. Restrepo, and O. Zapata, “Radiative Type III Seesaw Model and its collider phenomenology,”Phys. Rev. D94no. 3, (2016) 033005,arXiv:1605.01129 [hep-ph]

    work page arXiv 2016

  25. [25]

    Radiatively induced Quark and Lepton Mass Model,

    T. Nomura and H. Okada, “Radiatively induced Quark and Lepton Mass Model,”Phys. Lett. B761(2016) 190–196, arXiv:1606.09055 [hep-ph]

    work page arXiv 2016

  26. [26]

    GaugeU(1) dark symmetry and radiative light fermion masses,

    C. Kownacki and E. Ma, “GaugeU(1) dark symmetry and radiative light fermion masses,”Phys. Lett. B760(2016) 59–62,arXiv:1604.01148 [hep-ph]

    work page arXiv 2016

  27. [27]

    Loop induced type-II seesaw model and GeV dark matter withU(1) B−L gauge symmetry,

    T. Nomura and H. Okada, “Loop induced type-II seesaw model and GeV dark matter withU(1) B−L gauge symmetry,” Phys. Lett. B774(2017) 575–581,arXiv:1704.08581 [hep-ph]

    work page arXiv 2017

  28. [28]

    Radiative neutrino mass in an alternativeU(1) B−L gauge symmetry,

    T. Nomura and H. Okada, “Radiative neutrino mass in an alternativeU(1) B−L gauge symmetry,”Nucl. Phys. B941 (2019) 586–599,arXiv:1705.08309 [hep-ph]

    work page arXiv 2019

  29. [29]

    Fermion masses and mixings and dark matter constraints in a model with radiative seesaw mechanism,

    N. Bernal, A. E. C´ arcamo Hern´ andez, I. de Medeiros Varzielas, and S. Kovalenko, “Fermion masses and mixings and dark matter constraints in a model with radiative seesaw mechanism,”JHEP05(2018) 053,arXiv:1712.02792 [hep-ph]

    work page arXiv 2018

  30. [30]

    TheB−LScotogenic Models for Dirac Neutrino Masses,

    W. Wang, R. Wang, Z.-L. Han, and J.-Z. Han, “TheB−LScotogenic Models for Dirac Neutrino Masses,”Eur. Phys. J. C77no. 12, (2017) 889,arXiv:1705.00414 [hep-ph]

    work page arXiv 2017

  31. [31]

    Dark matter stability and Dirac neutrinos using only Standard Model symmetries,

    C. Bonilla, S. Centelles-Chuli´ a, R. Cepedello, E. Peinado, and R. Srivastava, “Dark matter stability and Dirac neutrinos using only Standard Model symmetries,”Phys. Rev. D101no. 3, (2020) 033011,arXiv:1812.01599 [hep-ph]

    work page arXiv 2020

  32. [32]

    Minimal radiative Dirac neutrino mass models,

    J. Calle, D. Restrepo, C. E. Yaguna, and ´O. Zapata, “Minimal radiative Dirac neutrino mass models,”Phys. Rev. D99 no. 7, (2019) 075008,arXiv:1812.05523 [hep-ph]

    work page arXiv 2019

  33. [33]

    Phenomenology of scotogenic scalar dark matter,

    I. M. ´Avila, V. De Romeri, L. Duarte, and J. W. F. Valle, “Phenomenology of scotogenic scalar dark matter,”Eur. Phys. J. C80no. 10, (2020) 908,arXiv:1910.08422 [hep-ph]

    work page arXiv 2020

  34. [34]

    Muon anomalies and theSU(5) Yukawa relations,

    A. E. C´ arcamo Hern´ andez and S. F. King, “Muon anomalies and theSU(5) Yukawa relations,”Phys. Rev. D99no. 9, (2019) 095003,arXiv:1803.07367 [hep-ph]

    work page arXiv 2019

  35. [35]

    Phenomenology of fermion dark matter as neutrino mass mediator with gauged B-L,

    C. Alvarado, C. Bonilla, J. Leite, and J. W. F. Valle, “Phenomenology of fermion dark matter as neutrino mass mediator with gauged B-L,”Phys. Lett. B817(2021) 136292,arXiv:2102.07216 [hep-ph]

    work page arXiv 2021

  36. [36]

    The simplest scoto-seesaw model: WIMP dark matter phenomenology and Higgs vacuum stability,

    S. Mandal, R. Srivastava, and J. W. F. Valle, “The simplest scoto-seesaw model: WIMP dark matter phenomenology and Higgs vacuum stability,”Phys. Lett. B819(2021) 136458,arXiv:2104.13401 [hep-ph]

    work page arXiv 2021

  37. [37]

    Neutrino masses, flavor anomalies, and muon g-2 from dark loops,

    R. Cepedello, P. Escribano, and A. Vicente, “Neutrino masses, flavor anomalies, and muon g-2 from dark loops,”Phys. Rev. D107no. 3, (2023) 035034,arXiv:2209.02730 [hep-ph]

    work page arXiv 2023

  38. [38]

    Phenomenology of extended multiHiggs doublet models withS 4 family symmetry,

    A. E. C´ arcamo Hern´ andez, C. Espinoza, J. C. G´ omez-Izquierdo, J. Marchant Gonz´ alez, and M. Mondrag´ on, “Phenomenology of extended multiHiggs doublet models withS 4 family symmetry,”Eur. Phys. J. C84no. 11, (2024) 1239,arXiv:2212.12000 [hep-ph]

    work page arXiv 2024

  39. [39]

    Dynamical scoto-seesaw mechanism with gauged B-L symmetry,

    J. Leite, S. Sadhukhan, and J. W. F. Valle, “Dynamical scoto-seesaw mechanism with gauged B-L symmetry,”Phys. Rev. D109no. 3, (2024) 035023,arXiv:2307.04840 [hep-ph]

    work page arXiv 2024

  40. [40]

    Dirac Scoto inverse-seesaw from A 4 flavor symmetry,

    R. Kumar, N. Nath, R. Srivastava, and S. Yadav, “Dirac Scoto inverse-seesaw from A 4 flavor symmetry,”JHEP10 30 (2025) 088,arXiv:2505.01407 [hep-ph]

    work page arXiv 2025

  41. [41]

    Flavor imprints on novel low mass dark matter,

    R. Kumar, H. K. Prajapati, R. Srivastava, and S. Yadav, “Flavor imprints on novel low mass dark matter,”JHEP11 (2025) 094,arXiv:2510.02972 [hep-ph]

    work page arXiv 2025

  42. [42]

    Two-loop Dirac neutrino mass and WIMP dark matter,

    C. Bonilla, E. Ma, E. Peinado, and J. W. F. Valle, “Two-loop Dirac neutrino mass and WIMP dark matter,”Phys. Lett. B762(2016) 214–218,arXiv:1607.03931 [hep-ph]

    work page arXiv 2016

  43. [43]

    A Two Loop Radiative Neutrino Model,

    S. Baek, H. Okada, and Y. Orikasa, “A Two Loop Radiative Neutrino Model,”Nucl. Phys. B941(2019) 744–754, arXiv:1703.00685 [hep-ph]

    work page arXiv 2019

  44. [44]

    Origin of a two-loop neutrino mass from SU(5) grand unification,

    S. Saad, “Origin of a two-loop neutrino mass from SU(5) grand unification,”Phys. Rev. D99no. 11, (2019) 115016, arXiv:1902.11254 [hep-ph]

    work page arXiv 2019

  45. [45]

    A two loop induced neutrino mass model with modularA 4 symmetry,

    T. Nomura and H. Okada, “A two loop induced neutrino mass model with modularA 4 symmetry,”Nucl. Phys. B966 (2021) 115372,arXiv:1906.03927 [hep-ph]

    work page arXiv 2021

  46. [46]

    Radiative type-I seesaw neutrino masses,

    C. Arbel´ aez, A. E. C´ arcamo Hern´ andez, R. Cepedello, M. Hirsch, and S. Kovalenko, “Radiative type-I seesaw neutrino masses,”Phys. Rev. D100no. 11, (2019) 115021,arXiv:1910.04178 [hep-ph]

    work page arXiv 2019

  47. [47]

    Combined explanations of (g−2) µ,R D(∗),R K(∗) anomalies in a two-loop radiative neutrino mass model,

    S. Saad, “Combined explanations of (g−2) µ,R D(∗),R K(∗) anomalies in a two-loop radiative neutrino mass model,” Phys. Rev. D102no. 1, (2020) 015019,arXiv:2005.04352 [hep-ph]

    work page arXiv 2020

  48. [48]

    On the two-loop radiative origin of the smallest neutrino mass and the associated Majorana CP phase,

    Z.-z. Xing and D. Zhang, “On the two-loop radiative origin of the smallest neutrino mass and the associated Majorana CP phase,”Phys. Lett. B807(2020) 135598,arXiv:2005.05171 [hep-ph]

    work page arXiv 2020

  49. [49]

    Two-loop radiative seesaw, muon g−2, andτ-lepton-flavor violation with DM constraints,

    C.-H. Chen and T. Nomura, “Two-loop radiative seesaw, muon g−2, andτ-lepton-flavor violation with DM constraints,”JHEP09(2021) 090,arXiv:2001.07515 [hep-ph]

    work page arXiv 2021

  50. [50]

    A two-loop induced neutrino mass model, dark matter, and LFV processes ℓi →ℓ jγ, andµe→eein a hidden localU(1) symmetry,

    T. Nomura, H. Okada, and Y. Uesaka, “A two-loop induced neutrino mass model, dark matter, and LFV processes ℓi →ℓ jγ, andµe→eein a hidden localU(1) symmetry,”Nucl. Phys. B962(2021) 115236,arXiv:2008.02673 [hep-ph]

    work page arXiv 2021

  51. [51]

    Gravitational waves from a scotogenic two-loop neutrino mass model,

    C. Bonilla, A. E. Carcamo Hernandez, J. Goncalves, V. K. N., A. P. Morais, and R. Pasechnik, “Gravitational waves from a scotogenic two-loop neutrino mass model,”Phys. Rev. D109no. 9, (2024) 095022,arXiv:2305.01964 [hep-ph]

    work page arXiv 2024

  52. [52]

    Dark matter and scalar sector in a novel two-loop scotogenic neutrino mass model,

    A. E. C´ arcamo Hern´ andez, C. Espinoza, J. C. G´ omez-Izquierdo, J. M. Gonz´ alez, and M. Mondrag´ on, “Dark matter and scalar sector in a novel two-loop scotogenic neutrino mass model,”arXiv:2601.02503 [hep-ph]

    work page arXiv

  53. [53]

    Feruglio,Are neutrino masses modular forms?, pp

    F. Feruglio,Are neutrino masses modular forms?, pp. 227–266. 2019.arXiv:1706.08749 [hep-ph]

    work page arXiv 2019

  54. [54]

    Neutrino mass and mixing with modular symmetry,

    G.-J. Ding and S. F. King, “Neutrino mass and mixing with modular symmetry,”Rept. Prog. Phys.87no. 8, (2024) 084201,arXiv:2311.09282 [hep-ph]

    work page arXiv 2024

  55. [55]

    Lepton Masses and Mixing from ModularS 4 Symmetry,

    J. T. Penedo and S. T. Petcov, “Lepton Masses and Mixing from ModularS 4 Symmetry,”Nucl. Phys. B939(2019) 292–307,arXiv:1806.11040 [hep-ph]

    work page arXiv 2019

  56. [56]

    Modular S 4 models of lepton masses and mixing,

    P. P. Novichkov, J. T. Penedo, S. T. Petcov, and A. V. Titov, “Modular S 4 models of lepton masses and mixing,”JHEP 04(2019) 005,arXiv:1811.04933 [hep-ph]

    work page arXiv 2019

  57. [57]

    Trimaximal Neutrino Mixing from Modular A4 Invariance with Residual Symmetries,

    P. P. Novichkov, S. T. Petcov, and M. Tanimoto, “Trimaximal Neutrino Mixing from Modular A4 Invariance with Residual Symmetries,”Phys. Lett. B793(2019) 247–258,arXiv:1812.11289 [hep-ph]

    work page arXiv 2019

  58. [58]

    Modular A 5 symmetry for flavour model building,

    P. P. Novichkov, J. T. Penedo, S. T. Petcov, and A. V. Titov, “Modular A 5 symmetry for flavour model building,” JHEP04(2019) 174,arXiv:1812.02158 [hep-ph]

    work page arXiv 2019

  59. [59]

    Trimaximal TM 1 mixing with two modularS 4 groups,

    S. F. King and Y.-L. Zhou, “Trimaximal TM 1 mixing with two modularS 4 groups,”Phys. Rev. D101no. 1, (2020) 015001,arXiv:1908.02770 [hep-ph]

    work page arXiv 2020

  60. [60]

    ModularS 3 symmetric radiative seesaw model,

    H. Okada and Y. Orikasa, “ModularS 3 symmetric radiative seesaw model,”Phys. Rev. D100no. 11, (2019) 115037, arXiv:1907.04716 [hep-ph]

    work page arXiv 2019

  61. [61]

    Neutrino Masses and Mixing from Double Covering of Finite Modular Groups,

    X.-G. Liu and G.-J. Ding, “Neutrino Masses and Mixing from Double Covering of Finite Modular Groups,”JHEP08 (2019) 134,arXiv:1907.01488 [hep-ph]

    work page arXiv 2019

  62. [62]

    ModularS 3-invariant flavor model in SU(5) grand unified theory,

    T. Kobayashi, Y. Shimizu, K. Takagi, M. Tanimoto, and T. H. Tatsuishi, “ModularS 3-invariant flavor model in SU(5) grand unified theory,”PTEP2020no. 5, (2020) 053B05,arXiv:1906.10341 [hep-ph]

    work page arXiv 2020

  63. [63]

    Generalised CP Symmetry in Modular-Invariant Models of Flavour,

    P. P. Novichkov, J. T. Penedo, S. T. Petcov, and A. V. Titov, “Generalised CP Symmetry in Modular-Invariant Models of Flavour,”JHEP07(2019) 165,arXiv:1905.11970 [hep-ph]

    work page arXiv 2019

  64. [64]

    Multiple modular symmetries as the origin of flavor,

    I. de Medeiros Varzielas, S. F. King, and Y.-L. Zhou, “Multiple modular symmetries as the origin of flavor,”Phys. Rev. D101no. 5, (2020) 055033,arXiv:1906.02208 [hep-ph]

    work page arXiv 2020

  65. [65]

    SUSY breaking constraints on modular flavorS 3 invariant SU(5) GUT model,

    X. Du and F. Wang, “SUSY breaking constraints on modular flavorS 3 invariant SU(5) GUT model,”JHEP02(2021) 221,arXiv:2012.01397 [hep-ph]

    work page arXiv 2021

  66. [66]

    Fermion masses and mixing in modular A4 Symmetry,

    M. Abbas, “Fermion masses and mixing in modular A4 Symmetry,”Phys. Rev. D103no. 5, (2021) 056016, 31 arXiv:2002.01929 [hep-ph]

    work page arXiv 2021

  67. [67]

    Symmetries and stabilisers in modular invariant flavour models,

    I. de Medeiros Varzielas, M. Levy, and Y.-L. Zhou, “Symmetries and stabilisers in modular invariant flavour models,” JHEP11(2020) 085,arXiv:2008.05329 [hep-ph]

    work page arXiv 2020

  68. [68]

    Two A4 modular symmetries for Tri-Maximal 2 mixing,

    I. de Medeiros Varzielas and J. Louren¸ co, “Two A4 modular symmetries for Tri-Maximal 2 mixing,”Nucl. Phys. B979 (2022) 115793,arXiv:2107.04042 [hep-ph]

    work page arXiv 2022

  69. [69]

    Fermion mass hierarchies, large lepton mixing and residual modular symmetries,

    P. P. Novichkov, J. T. Penedo, and S. T. Petcov, “Fermion mass hierarchies, large lepton mixing and residual modular symmetries,”JHEP04(2021) 206,arXiv:2102.07488 [hep-ph]

    work page arXiv 2021

  70. [70]

    Two A5 modular symmetries for Golden Ratio 2 mixing,

    I. de Medeiros Varzielas and J. Louren¸ co, “Two A5 modular symmetries for Golden Ratio 2 mixing,”Nucl. Phys. B984 (2022) 115974,arXiv:2206.14869 [hep-ph]

    work page arXiv 2022

  71. [71]

    Residual flavor symmetry breaking in the landscape of modular flavor models,

    K. Ishiguro, H. Okada, and H. Otsuka, “Residual flavor symmetry breaking in the landscape of modular flavor models,” JHEP09(2022) 072,arXiv:2206.04313 [hep-ph]

    work page arXiv 2022

  72. [72]

    Lepton mass matrix from double covering of A 4 modular flavor symmetry*,

    H. Okada and Y. Orikasa, “Lepton mass matrix from double covering of A 4 modular flavor symmetry*,”Chin. Phys. C 46no. 12, (2022) 123108,arXiv:2206.12629 [hep-ph]

    work page arXiv 2022

  73. [73]

    Quark-lepton mass relations from modular flavor symmetry,

    M.-C. Chen, S. F. King, O. Medina, and J. W. F. Valle, “Quark-lepton mass relations from modular flavor symmetry,” JHEP02(2024) 160,arXiv:2312.09255 [hep-ph]

    work page arXiv 2024

  74. [74]

    A simplest modular S 3 model for leptons,

    D. Meloni and M. Parriciatu, “A simplest modular S 3 model for leptons,”JHEP09(2023) 043,arXiv:2306.09028 [hep-ph]

    work page arXiv 2023

  75. [75]

    Modular flavor models with positive modular weights: a new lepton model building,

    T. Kobayashi, T. Nomura, H. Okada, and H. Otsuka, “Modular flavor models with positive modular weights: a new lepton model building,”JHEP01(2024) 121,arXiv:2310.10091 [hep-ph]

    work page arXiv 2024

  76. [76]

    A modular SU (5) littlest seesaw,

    I. de Medeiros Varzielas, S. F. King, and M. Levy, “A modular SU (5) littlest seesaw,”JHEP05(2024) 203, arXiv:2309.15901 [hep-ph]

    work page arXiv 2024

  77. [77]

    Quarks at the modular S 4 cusp,

    I. de Medeiros Varzielas, M. Levy, J. T. Penedo, and S. T. Petcov, “Quarks at the modular S 4 cusp,”JHEP09(2023) 196,arXiv:2307.14410 [hep-ph]

    work page arXiv 2023

  78. [78]

    Predictions from scoto-seesaw with A4 modular symmetry,

    R. Kumar, P. Mishra, M. K. Behera, R. Mohanta, and R. Srivastava, “Predictions from scoto-seesaw with A4 modular symmetry,”Phys. Lett. B853(2024) 138635,arXiv:2310.02363 [hep-ph]

    work page arXiv 2024

  79. [79]

    Finite modular symmetries and the strong CP problem,

    J. T. Penedo and S. T. Petcov, “Finite modular symmetries and the strong CP problem,”JHEP10(2024) 172, arXiv:2404.08032 [hep-ph]

    work page arXiv 2024

  80. [80]

    Pati-Salam models with A 4 modular symmetry,

    G.-J. Ding, S.-Y. Jiang, S. F. King, J.-N. Lu, and B.-Y. Qu, “Pati-Salam models with A 4 modular symmetry,”JHEP 08(2024) 134,arXiv:2404.06520 [hep-ph]

    work page arXiv 2024

Showing first 80 references.