A modular $A_4$ symmetric scotogenic model

Hiroshi Okada; Oleg Popov; Takaaki Nomura

REVIEW 2 major objections 6 minor 2 cited by

A modular $A_4$ symmetric scotogenic model

T0 review · 2 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read By replacing the Z2 of the minimal Scotogenic model with an A4 modular symmetry, this paper derives narrow, testable predictions for neutrino CP phases and neutrinoless double beta decay from a minimal parameter set.

desk verdict Genuinely minimal modular A4 scotogenic construction, but the inert doublet's modular weights make the mass-splitting quartic non-invariant, so the numerical predictions sit on a broken scalar sector. read the letter →

arxiv 1908.07457 v2 pith:BFNEMVFL submitted 2019-08-19 hep-ph

Takaaki Nomura , Hiroshi Okada , Oleg Popov This is my paper · ORCID

classification hep-ph

keywords A4modularsymmetryscotogenicmodelradiativeneutrinomassdarkmatterneutrinolessdoublebetadecayCPphasesinertHiggsdoubletflavor

topics Dark Matter

open problems Dark Matter

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

This paper proposes a compact extension of the Standard Model in which neutrino masses arise radiatively at one loop through the Scotogenic mechanism, with the usual stabilizing Z2 replaced by an A4 modular symmetry. The modular symmetry simultaneously acts as a flavor symmetry, as the scotogenic ingredient that forbids tree-level neutrino masses, and as the stabilizer of dark matter. With a handful of free parameters—three Yukawa couplings and one complex modulus—the model claims to reproduce all measured neutrino mixing angles and mass splittings under normal mass ordering, and to pin down the CP phases and neutrinoless double beta decay rate in narrow ranges. A sympathetic reader would care because these are experimentally testable predictions coming from symmetry alone rather than from a large parameter scan.

What carries the argument

The central object is the modular A4 symmetry acting through the weight-2 triplet modular form Y_3^(2) built from Dedekind eta functions; from it the paper constructs a weight-4 triplet Y_3^(4) and a weight-6 singlet Y_1^(6). These modular forms enter the Dirac Yukawa couplings and the right-handed neutrino Majorana mass matrix, and the parity-like behavior of odd versus even modular weight replaces the Z2 that stabilizes dark matter and forbids the (H†η) term. The argument is carried by the one-loop neutrino mass formula in which the inert-scalar mass splitting Δm² controls the loop factor, and by an inversion step that fixes that splitting from the atmospheric neutrino mass difference, leaving predictions that depend only on the modulus and a few Yukawa couplings.

What would settle it

A future measurement of the Dirac CP phase δ_CP that falls outside [100°, 120°] ∪ [230°, 250°], or a neutrinoless double beta decay bound below mee ≈ 0.002 eV with no signal, would contradict the model's allowed parameter region; evidence for inverted neutrino mass ordering would also rule it out.

Watch

Extended reading notes

Core claim

The central claim is that the minimal Scotogenic model can be realized with only modular A4 symmetry, with no extra Z2: the modular weight assignments of the right-handed neutrino triplet and the inert doublet forbid the dangerous (H†η) coupling, while a higher-weight modular singlet provides the quartic coupling that generates neutrino mass at one loop. After fixing the inert scalar mass m_R around 534 GeV and scanning the three Yukawa couplings, the model yields normal-hierarchy neutrino masses and mixings within the full 3σ ranges, with the sum of neutrino masses in [0.065, 0.070] eV, Dirac δ_CP in [100–120] and [230–250] degrees, Majorana phase α21 in [130–150] and [210–230] degrees, α31 in [165–190] degrees, lightest neutrino mass m1 in [0.0049, 0.0072] eV, and neutrinoless double beta decay effective mass mee in [0.002, 0.005] eV. The right-handed neutrino masses are predicted in the 40–750 TeV range, the inert scalar η_R at about 530 GeV is the dark matter candidate, and charged-lepton flavor violating rates are predicted far below current bounds.

Load-bearing premise

The load-bearing premise is that the inert-scalar mass splitting used in the numerical scan has the same sign as the splitting in the one-loop neutrino-mass formula; the paper writes these two with opposite signs, so if the scan followed the sign it displays in Sec. IV, the derived scalar mass and the predicted phase and 0νββ regions would shift.

Editorial extensions

If this is right

If the model is correct, the sum of neutrino masses is narrowly confined to 0.065–0.070 eV, below the current cosmological bound but potentially testable by future cosmological surveys.
The predicted Dirac CP phase is restricted to two narrow windows, so a precise measurement by long-baseline experiments can distinguish this modular A4 model from other one-loop modular A4 constructions.
The neutrinoless double beta decay effective mass is predicted to be 0.002–0.005 eV, within the projected sensitivity of next-generation experiments, meaning a null result near the lower end would constrain the model.
Charged-lepton flavor violating rates are predicted to be orders of magnitude below present limits, making the model effectively invisible to current cLFV searches.
The dark matter candidate is the inert scalar with mass around 530 GeV, whose relic density can be accommodated through gauge interactions with coannihilation, so the model ties dark matter to the neutrino mass mechanism.

Reading between the lines

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

Editorial extension: The sign-convention ambiguity in the inert-scalar mass splitting is not merely typographical; because m_I is derived from Δm², re-running the scan with the opposite sign would shift the derived scalar mass and change the plotted phase and 0νββ regions, so the paper's predictions should be read as contingent on that convention choice.
Editorial extension: Because the model assumes normal neutrino mass ordering throughout, an inverted-ordering discovery would immediately exclude its parameter region; the modular A4 construction itself might still be adapted, but the quoted phase ranges would not survive.
Editorial extension: The narrow allowed region for the modulus τ (Re[τ]≈0.43–0.45, Im[τ]≈0.65–0.67) suggests that the model effectively selects a small patch of the modular field space; varying the modular weight assignments or adding higher-weight forms could reveal whether this patch is robust or an artifact of the scan.
Editorial extension: A future measurement of δ_CP near 90° or 270° would fall outside the predicted windows and would disfavor this specific symmetry assignment, offering a relatively near-term experimental discriminator.

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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

Short version: this is a genuinely new and cleaner way to build a scotogenic model with modular A4, but the modular weight assignments in the scalar sector are inconsistent: the same field that makes the neutrino Yukawa invariant makes the (H†η)^2 quartic non-invariant, and vice versa. Since that quartic is the one that splits the inert scalar masses and generates the radiative neutrino mass, the model as written doesn't have a working mass mechanism. The reader's flagged sign inconsistency in Δm^2 is real but harmless; the bigger issue is the one the reader didn't catch.

What's good: the group-theoretic setup is standard and the numerical work is honest. The authors show that a single modular A4 with higher-weight forms can replace Z2, simplify the scalar sector relative to refs. [6] and [11], and give concrete targets for δ_CP, α_21, α_31, m_1, and m_ee. The loop formulas and LFV constraints are correctly quoted, and the cLFV rates are far below present bounds. Those parts are useful.

The soft spots: besides the sign typo (Eq. III.7 vs. III.9), the scan fixes m_R ≈ 534 GeV by hand, doesn't specify how the phases of α_ν, β_ν, γ_ν are scanned, and takes the DM relic density from the standard inert doublet model without calculation. Those are moderate. The load-bearing problem is the modular weight tension I opened with. Under the usual convention where a conjugate field carries opposite modular weight, Table I gives η* weight -3, so η has +3. The Yukawa then has weight 0+4-1-3=0 and is invariant, but Y_1^(6)(H†η)^2 has weight 6+6=12. If you flip the assignment to make the quartic invariant, the Yukawa acquires weight 6. No single assignment works. Since the quartic is exactly what splits m_R and m_I, the one-loop neutrino mass vanishes. The authors need to either change the field convention, add another scalar, or explicitly state a non-standard modular-weight rule for conjugates.

Who this is for: modular flavor builders and anyone tracking scotogenic model space. It deserves peer review—this kind of construction error is exactly what a referee should catch, and the idea is repairable. But as it stands, the numerical predictions are sitting on top of a broken scalar sector. I'd wait for a revised version before citing it.

Referee Report

2 major / 6 minor

Summary. This paper proposes a non-supersymmetric extension of the Standard Model in which neutrino masses arise radiatively through the one-loop scotogenic mechanism, with an A4 modular symmetry replacing the usual Z2 stabilising symmetry. The right-handed neutrinos are an A4 triplet with modular weight -1, the inert doublet (via its conjugate) carries weight -3, and the couplings are built from modular forms of weights 2, 4 and 6. The authors derive the one-loop neutrino mass matrix, include charged-lepton flavour violation bounds, and perform a numerical scan over the modulus and Yukawa parameters. They report that all three mixing angles and the mass-squared splittings can be reproduced at 3 sigma under normal hierarchy, and they state as predictions narrow ranges for the Dirac phase, the two Majorana phases, the lightest neutrino mass, and the neutrinoless double beta decay effective mass. They also identify the real component of the inert scalar as a dark matter candidate with mass around 530 GeV.

Significance. If the model is consistent, its practical significance is moderate but real: it gives a very economical radiative neutrino mass framework with a modular flavour symmetry and makes falsifiable statements about CP violation and neutrinoless double beta decay. The paper should be credited for providing explicit analytic expressions for the one-loop neutrino mass matrix, for incorporating the current cLFV and cosmological bounds, and for stating concrete numerical predictions for phases and mee that could be tested in forthcoming experiments. At the same time, the numerical procedure is a fit to the measured mixing parameters rather than an ab initio prediction; the genuinely predictive content is concentrated in the CP phases and in the mee vs m1 correlation. The main conceptual novelty, namely replacing the ad hoc Z2 by an oddness in modular weight, is interesting but, as explained below, is undermined in the present text by an inconsistency in the modular weight assignments of the scalar quartic interaction.

major comments (2)

[Sec. II, Table I and Eq. (II.1); Sec. II quartic term] The modular weight assignments of the inert doublet are inconsistent when the stated scalar quartic is included. Table I assigns modular weight -k = -3 to η*, so the combination \tilde η = iσ2 η* that appears in the neutrino Yukawa (II.1) has weight -3 and Eq. (II.1) is invariant. Under the standard convention used here, the physical field η then has weight +3. The quartic written in Sec. II as Y_1^{(6)}(H†η)^2 then has field-weight 2×(0+3)=6 plus the modular-form weight 6, giving total weight 12 rather than zero. If instead one assigns weight -3 to η to make this quartic invariant, then \tilde η has weight +3 and the neutrino Yukawa in Eq. (II.1) has total weight 4-1+3=6, again not zero. No single assignment of modular weight to the inert doublet makes both Eq. (II.1) and the quoted quartic term invariant. Since this quartic is precisely the term that splits the real and imaginary neutral scalar masses and thereby generates the one-loop neutrino mass, the model as written is not fully modular invariant. Please specify the correct invariant operator (for instance, whether the quartic should involve \tilde η or the conjugate modular form) and demonstrate that the loop calculation and the numerical predictions are unchanged.
[Sec. IV, numerical scan] The scan description states only that the absolute values of αν, βν, γν are in [0.1, 1] and that M1 is of order 100 TeV, but it does not state how the phases of these Yukawa couplings are treated. The predicted Dirac phase δCP and the Majorana phases α21, α31 in Figs. 2 and 3 depend on the relative phases entering yη. If the couplings are taken real, this assumption should be stated explicitly and justified; if they are scanned as complex, then the scan ranges and the resulting phase distributions should be shown so that the quoted phase intervals are reproducible. This is a load-bearing point for the main predictions of the paper.

minor comments (6)

[Eq. (III.7), Eq. (III.9), Sec. IV] The sign convention for Δm^2 is inconsistent: Eq. (III.7) defines Δm^2 = m_R^2 - m_I^2, while Eq. (III.9) and the text after it use m_I = sqrt(m_R^2 + Δm^2). The normalized quantities used for the mixing and CP-phase predictions are independent of Δm^2 because the overall factor cancels in kα/k3, but the notation should be made consistent for the reader.
[Sec. V, item 1] The conclusion text reads '[0.065,0.0070] eV' for the allowed sum of neutrino masses; this should be '[0.065,0.070] eV' to agree with Fig. 1.
[Abstract] There are typos in the abstract: 'extention' should be 'extension' and 'scanario' should be 'scenario'.
[References] In the reference list, the entry before reference [23] contains a raw LaTeX citation key 'citeLiu:2019khw' instead of a formatted bibliography entry.
[Sec. IV] The numerical analysis would be more reproducible if the paper stated the scan ranges for the modulus τ, the number of scanned points, and the precise treatment of the 3σ constraints; in particular, the text says 'we assume mR ≈ mI ≈ mη±' but then computes mI from Δm^2, which should be clarified.
[Sec. II, DM stability] The paper states that odd modular weight replaces the Z2 of the scotogenic model, but since the modulus τ acquires a VEV and breaks the modular symmetry, the exact residual symmetry that stabilises the dark matter candidate should be identified explicitly.

Circularity Check

1 steps flagged · score 4.0 of 10

Oscillation-data 'predictions' are scan constraints, but CP phases and 0νββ mass are genuine unconstrained outputs.

fitted input called prediction [Abstract; Sec. III after Eq. (III.10); Sec. IV Eq. (IV.1) and Fig. 1 caption]
"the model makes predictions for neutrino oscillation data, Majorana and Dirac phases, dark matter characteristics, and neutrinoless double beta decay. ... We show numerical analysis to satisfy all of the constraints that we discussed above ... we provide the experimentally allowed ranges for neutrino mixings and mass difference squares at 3σ range [45] as follows: Δm2atm = [2.431−2.622]×10−3 eV2, Δm2sol = [6.79−8.01]×10−5 eV2, sin2θ13 = [0.02044−0.02437], sin2θ23 = [0.428−0.624], sin2θ12 = [0.275−0.350]. ..."

The scan over τ, αν, βν, γν, M1 is accepted only when the resulting sin2θ12, sin2θ23, sin2θ13 and Δm2sol fall inside the quoted 3σ windows, while Δm2atm is imposed as an input (Eq. IV.1). The agreement of the plotted mixings with data is therefore a selection constraint on the parameters, not a predicted consequence; the abstract's 'predictions for neutrino oscillation data' is a fitted input relabeled as an output. The CP phases, m1, and mee in Figs. 2-3 are not among the imposed constraints and retain independent predictive content, so the circularity is confined to the oscillation-data part.

full rationale

The paper's numerical method is a parameter scan: modular field τ, Yukawa couplings (αν, βν, γν), M1, and scalar masses are varied, and points are kept only when the resulting mixings and solar mass splitting lie in the 3σ ranges of Eq. (IV.1), with Δm_atm taken as input. Reproducing those measured quantities is thus by construction, and the abstract's phrasing 'predictions for neutrino oscillation data' overstates the derivation. The distinctive outputs — δ_CP, α21, α31, m1, and mee in Figs. 2-3 — are not imposed by any of these constraints and are genuine model predictions. The Δm^2 sign mismatch around Eq. (III.7) and Eq. (III.9)/(Sec. IV) changes the inferred m_I value, but the normalized matrix used for the phase scan is independent of Δm^2, so it is a convention typo rather than a circular reduction. The self-citations [6,11] supply a construction template and cLFV comparisons, not the numerical claim; the odd-modular-weight statement is parameter-free and independently checkable. The DM mass m_R≈534 GeV is chosen by hand and only checked against relic-density literature [46], so 'dark matter characteristics' is also an input/assumption rather than a derived prediction. The modular-weight consistency of Eq. (II.1) with Y_1^(6)(H†η)^2 is a model-construction concern, not a circularity. Overall, circularity is limited to the fitted-input-overstatement part; score 4.

Assumptions & free parameters 6 free parameters · 6 assumptions · 1 invented entities

The model introduces no new particles beyond the standard scotogenic right-handed neutrinos and inert doublet. Its predictive content comes from the modular weight assignments and from the choice of scanned parameter ranges. Several of the central numerical outputs depend on parameters that are fitted or set by hand, which is common in model-building but should be acknowledged when calling them predictions.

free parameters (6)

τ (complex modulus) = Re[τ] ≈ 0.43-0.45, Im[τ] ≈ 0.65-0.67
A free parameter of the modular group; the allowed region after all constraints is narrow.
αν, βν, γν = absolute values in [0.1, 1]
Yukawa coefficients in the Dirac mass matrix; scanned to fit oscillation data; phases not clearly specified.
M1 = order 100 TeV (eigenvalues DN1-DN3 reach 40-750 TeV)
Overall scale of right-handed neutrino masses; set by hand, not predicted.
m_R (inert scalar mass) = 534 ± 8.5 GeV
Chosen by hand to match inert doublet dark matter relic density; not derived from the neutrino sector.
Δm^2 (inert scalar mass splitting) = determined by Eq. (III.9) from Δm_atm^2
Inverted from the one-loop neutrino mass formula; sign convention is ambiguous in the text.
Higgs portal coupling = small (not quantified)
Assumed small to evade dark matter direct detection; not computed.

assumptions (6)

standard math Modular forms of weight 2, 4, and 6 with the stated A4 transformation properties exist and are given by Eq. (II.2) and (II.3).
Standard modular form theory used as the starting point for all modular flavor models.
domain assumption A Lagrangian term is allowed only if its total modular weight vanishes, with modular forms carrying positive weight and fields carrying the weights in Table I.
This invariance rule is the basis for forbidding (H†η) and is not proved in the paper.
domain assumption Neutrino masses follow the normal hierarchy with Δm_atm^2 as an input.
Section III assumes normal hierarchy; inverted ordering is not studied.
domain assumption The right-handed neutrino masses (40-750 TeV) are much larger than the inert scalar mass (~530 GeV), so the first-order approximation in Δm^2 is valid.
Used in Eq. (III.7) to derive the simplified loop factor.
ad hoc to paper The inert scalar η_R is a thermal dark matter candidate whose relic density can be set by gauge interactions and coannihilation at m_R ~ 530 GeV, with a small Higgs portal coupling.
The authors do not compute relic density or direct detection in this paper; they rely on the canonical inert doublet model result from ref. [46].
domain assumption The cosmological bound on the sum of neutrino masses, Tr[Dν] ≲ 0.12 eV, applies.
Taken from Planck 2018 and used as a constraint in the scan.

invented entities (1)

Modular-weight parity (oddness of modular weight replacing Z2)
purpose: Forbids the (H†η) term and stabilizes the dark matter candidate without a separate Z2 symmetry.
It is a bookkeeping assignment in Table I; no independent observable outside the model is provided.

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Pith. "Pith review of A modular $A_4$ symmetric scotogenic model." pith.science (2026). https://pith.science/paper/BFNEMVFL

@misc{pith2026190807457,
  author       = {Pith},
  title        = {Pith review of: A modular $A_4$ symmetric scotogenic model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BFNEMVFL}},
  note         = {Machine review of arXiv:1908.07457}
}

abstract

We propose a minimal extention of the Standard Model where neutrino masses are generated radiatively at one-loop level via Scotogenic scanario. The model is augmented with $A_4$ modular symmetry as a scotogenic and flavor symmetry. With minimal number of parameters, the model makes predictions for neutrino oscillation data, Majorana and Dirac phases, dark matter characteristics, and neutrinoless double beta decay.

Figures

Figures reproduced from arXiv: 1908.07457 by the authors.

**Figure 2.** shows phases of δ ℓ CP (red color) and α21(blue color) in terms of α31. This figure implies that Dirac CP is allowed by the range [100-120, 230-250] [deg], α21 is [130-150, 210-230] [deg], and α31 is [165-190] [deg] [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** FIG. 3: The lightest neutrino mass versus the effective mass fo [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: FIG. 4: The correlations among masses of [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]

**Figure 5.** Figure 5: FIG. 5: The BRs for cLFV processes [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]

Discussion (0). Continue with ORCID to comment.

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

A Predictive Non-Holomorphic Modular $A_4$ Linear Seesaw Framework Testable at DUNE
hep-ph 2026-02 conditional novelty 6.0 of 10

A non-holomorphic modular A4 linear seesaw model with six singlet fermions and one flavon reproduces observed neutrino mixing and predicts absolute mass and 0νββ ranges that DUNE and other experiments can test.
Neutrino mass model with a modular $S_4$ symmetry
hep-ph 2019-08 conditional novelty 4.0 of 10

A modular S4 radiative seesaw model fits normal-hierarchy neutrino data and predicts a total neutrino mass of roughly 58-62 meV and a double-beta decay mass of 1-4 meV.

Reference graph

Works this paper leans on

53 extracted references · 10 canonical work pages · cited by 2 Pith papers

[6]

Dirac CP is allowed by the range [100-120, 230-250] [deg], α21 is [130-150, 210-230] [deg], and α31 is [165-190] [deg]

work page
[11]

Kobayashi, N

T. Kobayashi, N. Omoto, Y. Shimizu, K. Takagi, M. Tanimot o and T. H. Tatsuishi, JHEP 1811, 196 (2018) doi:10.1007/JHEP11(2018)196 [arXiv:1808.03 012 [hep-ph]]

work page doi:10.1007/jhep11(2018)196 2018
[1]

The typical region of modulus τ is found in rather narrow modular ﬁeld space as 0.43 ≲ Re[τ ] ≲ 0.45 and 0.65 ≲ Im[τ ] ≲ 0.67

work page
[2]

We also show correlation among the mass eigenvalues in Fig

The Majorana mass eigenvalues are in the range of DN1 = [40 − 230] TeV, D N2 = [80 − 520] TeV, D N3 = [100 − 750] TeV. We also show correlation among the mass eigenvalues in Fig. 4. Note th at the mass scale is larger than the previous model in ref. [6] which is due to simpler loop structure requiring heavier masses of Ni

work page
[3]

6 where normal hierarchy is assumed here and ∆ m2 atm is the atmospheric neutrino mass dif- ference square

It is then found that k2 3 is given by k2 3 = ∆m2 atm ˜m2 ν3 − ˜m2 ν1 , (III.8) 2 Advantage of this approximation is that ˜kα does not depend on ∆ m. 6 where normal hierarchy is assumed here and ∆ m2 atm is the atmospheric neutrino mass dif- ference square. Thus, comparing Eq.(III.7) and Eq.(III.8), we ca n rewrite ∆ m2 by other parameters as follows: ∆m2...

work page
[4]

5, therefore following upper bounds are realized: BR(µ →eγ) ≲ 6.0×10−15, BR(τ →eγ) ≲ 1.2×10−15, BR(τ →µγ) ≲ 1.7×10−14

Typical scale of cLFVs tends to be very small in our analyses as sh own in Fig. 5, therefore following upper bounds are realized: BR(µ →eγ) ≲ 6.0×10−15, BR(τ →eγ) ≲ 1.2×10−15, BR(τ →µγ) ≲ 1.7×10−14. These values are smaller than the models in ref. [6, 11] which is due to h eavier mass scale of Ni. Note that DM candidate in our scenario is inert scalar ηR ...

work page
[5]

Three mixings cover all the experimental results by 3 σ interval, but the sum of neu- trino masses are the narrow range [0.065,0.0070] eV that is below the upper bound of cosmological data of 0.12 eV

work page
[7]

We found the following regions; 0 .0049 ≲ m1 ≲ 0.0072 eV and 0 .002 ≲ ⟨mee⟩ ≲ 0.005 eV which can be seen from Fig. 3. These predictions will be tested in the near future. In fact, compa ring with previous one-loop models with modular A4 in ref. [6, 11], we ﬁnd prediction for CP-phases are diﬀerent although that for neutrino mass and mixing are similar. Th...

work page

Show all 53 references

[8]

de Adelhart Toorop, F

R. de Adelhart Toorop, F. Feruglio and C. Hagedorn, Nucl. Phys. B 858, 437 (2012) [arXiv:1112.1340 [hep-ph]]

2012 arXiv
[9]

Feruglio, doi:10.1142/9789813238053 0012 arXiv:1706.08749 [hep-ph]

F. Feruglio, doi:10.1142/9789813238053 0012 arXiv:1706.08749 [hep-ph]

arXiv
[10]

J. C. Criado and F. Feruglio, arXiv:1807.01125 [hep-ph]

arXiv
[12]

Okada and M

H. Okada and M. Tanimoto, Phys. Lett. B 791, 54 (2019) doi:10.1016/j.physletb.2019.02.028 [arXiv:1812.09677 [hep-ph]]

2019 arXiv
[13]

Nomura and H

T. Nomura and H. Okada, arXiv:1904.03937 [hep-ph]

1904 arXiv
[14]

Okada and M

H. Okada and M. Tanimoto, arXiv:1905.13421 [hep-ph]

1905 arXiv
[15]

F. J. de Anda, S. F. King and E. Perdomo, arXiv:1812.05620 [hep-ph]

arXiv
[16]

P. P. Novichkov, S. T. Petcov and M. Tanimoto, arXiv:1812 .11289 [hep-ph]
[17]

Nomura and H

T. Nomura and H. Okada, arXiv:1906.03927 [hep-ph]

1906 arXiv
[18]

Okada and Y

H. Okada and Y. Orikasa, arXiv:1907.13520 [hep-ph]

1907 arXiv
[19]

Kobayashi, K

T. Kobayashi, K. Tanaka and T. H. Tatsuishi, Phys. Rev. D 98 (2018) no.1, 016004 [arXiv:1803.10391 [hep-ph]]

2018 arXiv
[20]

Kobayashi, Y

T. Kobayashi, Y. Shimizu, K. Takagi, M. Tanimoto, T. H. T atsuishi and H. Uchida, Phys. Lett. B 794, 114 (2019) doi:10.1016/j.physletb.2019.05.034 [arXiv: 1812.11072 [hep-ph]]

2019 arXiv
[21]

Kobayashi, Y

T. Kobayashi, Y. Shimizu, K. Takagi, M. Tanimoto and T. H . Tatsuishi, arXiv:1906.10341 [hep-ph]

1906 arXiv
[22]

Okada and Y

H. Okada and Y. Orikasa, arXiv:1907.04716 [hep-ph]

1907 arXiv
[23]

J. T. Penedo and S. T. Petcov, Nucl. Phys. B 939, 292 (2019) doi:10.1016/j.nuclphysb.2018.12.016 [arXiv:1806.11040 [hep-ph]]

2019 arXiv
[24]

P. P. Novichkov, J. T. Penedo, S. T. Petcov and A. V. Titov , JHEP 1904, 005 (2019) doi:10.1007/JHEP04(2019)005 [arXiv:1811.04933 [hep-ph ]]

2019 arXiv
[25]

Kobayashi, Y

T. Kobayashi, Y. Shimizu, K. Takagi, M. Tanimoto and T. H . Tatsuishi, arXiv:1907.09141 [hep-ph]

1907 arXiv
[26]

P. P. Novichkov, J. T. Penedo, S. T. Petcov and A. V. Titov , arXiv:1812.02158 [hep-ph]. 12

arXiv
[27]

G. J. Ding, S. F. King and X. G. Liu, arXiv:1903.12588 [he p-ph]

1903 arXiv
[28]

A. Baur, H. P. Nilles, A. Trautner and P. K. S. Vaudrevang e, arXiv:1901.03251 [hep-th]

1901 arXiv
[29]

de Medeiros Varzielas, S

I. de Medeiros Varzielas, S. F. King and Y. L. Zhou, arXiv :1906.02208 [hep-ph]. citeLiu:2019khw

1906 arXiv
[30]

X. G. Liu and G. J. Ding, arXiv:1907.01488 [hep-ph]

1907 arXiv
[31]

Altarelli and F

G. Altarelli and F. Feruglio, Rev. Mod. Phys. 82 (2010) 2701 [arXiv:1002.0211 [hep-ph]]

2010 arXiv
[32]

Ishimori, T

H. Ishimori, T. Kobayashi, H. Ohki, Y. Shimizu, H. Okada and M. Tanimoto, Prog. Theor. Phys. Suppl. 183 (2010) 1 [arXiv:1003.3552 [hep-th]]

2010 arXiv
[33]

Ishimori, T

H. Ishimori, T. Kobayashi, H. Ohki, H. Okada, Y. Shimizu and M. Tanimoto, Lect. Notes Phys. 858 (2012) 1, Springer

2012
[34]

Hernandez and A

D. Hernandez and A. Y. Smirnov, Phys. Rev. D 86 (2012) 053014 [arXiv:1204.0445 [hep-ph]]

2012 arXiv
[35]

S. F. King and C. Luhn, Rept. Prog. Phys. 76 (2013) 056201 [arXiv:1301.1340 [hep-ph]]

2013 arXiv
[36]

S. F. King, A. Merle, S. Morisi, Y. Shimizu and M. Tanimot o, arXiv:1402.4271 [hep-ph]

arXiv
[37]

S. F. King, Prog. Part. Nucl. Phys. 94 (2017) 217 doi:10.1016/j.ppnp.2017.01.003 [arXiv:1701.04413 [hep-ph]]

2017 arXiv
[38]

S. T. Petcov, Eur. Phys. J. C 78 (2018) no.9, 709 [arXiv:1711.10806 [hep-ph]]

2018 arXiv
[39]

A. Baur, H. P. Nilles, A. Trautner and P. K. S. Vaudrevang e, arXiv:1908.00805 [hep-th]

1908 arXiv
[40]

Ma, Phys

E. Ma, Phys. Rev. D 73, 077301 (2006) doi:10.1103/PhysRevD.73.077301 [hep-ph/ 0601225]

2006 doi
[41]

Hirsch, S

M. Hirsch, S. Morisi, E. Peinado and J. W. F. Valle, Phys. Rev. D 82, 116003 (2010) doi:10.1103/PhysRevD.82.116003 [arXiv:1007.0871 [hep- ph]]

2010 arXiv
[42]

J. M. Lamprea and E. Peinado, Phys. Rev. D 94, no. 5, 055007 (2016) doi:10.1103/PhysRevD.94.055007 [arXiv:1603.02190 [hep -ph]]

2016 arXiv
[43]

L. M. G. De La Vega, R. Ferro-Hernandez and E. Peinado, Ph ys. Rev. D 99, no. 5, 055044 (2019) doi:10.1103/PhysRevD.99.055044 [arXiv:1811.106 19 [hep-ph]]

2019 doi
[44]

P. P. Novichkov, J. T. Penedo, S. T. Petcov and A. V. Titov , arXiv:1905.11970 [hep-ph]

1905 arXiv
[45]

S. Baek, T. Nomura and H. Okada, Phys. Lett. B 759, 91 (2016) doi:10.1016/j.physletb.2016.05.055 [arXiv:1604.03738 [hep-ph]]

2016 arXiv
[46]

A. M. Baldini et al. [MEG Collaboration], Eur. Phys. J. C 76, no. 8, 434 (2016) [arXiv:1605.05081 [hep-ex]]

2016 arXiv
[47]

Renga [MEG Collaboration], Hyperﬁne Interact

F. Renga [MEG Collaboration], Hyperﬁne Interact. 239, no. 1, 58 (2018) [arXiv:1811.05921 [hep-ex]]. 13

2018 arXiv
[48]

Aubert et al

B. Aubert et al. [BaBar Collaboration], Phys. Rev. Lett. 104 (2010) 021802 [arXiv:0908.2381 [hep-ex]]

2010 arXiv
[49]

Aghanim et al

N. Aghanim et al. [Planck Collaboration], arXiv:1807.06209 [astro-ph.CO]

arXiv
[50]

Gando et al

A. Gando et al. [KamLAND-Zen Collaboration], Phys. Rev. Lett. 117, no. 8, 082503 (2016) Addendum: [Phys. Rev. Lett. 117, no. 10, 109903 (2016)] doi:10.1103/PhysRevLett.117.109903, 10.1103/PhysRevL ett.117.082503 [arXiv:1605.02889 [hep-ex]]

2016 arXiv
[51]

Hambye, F.-S

T. Hambye, F.-S. Ling, L. Lopez Honorez and J. Rocher, JH EP 0907, 090 (2009) Erratum: [JHEP 1005, 066 (2010)] doi:10.1007/JHEP05(2010)066, 10.1088/1126 -6708/2009/07/090 [arXiv:0903.4010 [hep-ph]]

2009 arXiv
[52]

Esteban, M

I. Esteban, M. C. Gonzalez-Garcia, A. Hernandez-Cabez udo, M. Maltoni and T. Schwetz, JHEP 1901, 106 (2019) doi:10.1007/JHEP01(2019)106 [arXiv:1811.05 487 [hep-ph]]

2019 doi
[53]

Arhrib, Y

A. Arhrib, Y. L. S. Tsai, Q. Yuan and T. C. Yuan, JCAP 1406, 030 (2014) doi:10.1088/1475- 7516/2014/06/030 [arXiv:1310.0358 [hep-ph]]. 14

2014 arXiv

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [6]

Dirac CP is allowed by the range [100-120, 230-250] [deg], α21 is [130-150, 210-230] [deg], and α31 is [165-190] [deg]

work page

[2] [11]

Kobayashi, N

T. Kobayashi, N. Omoto, Y. Shimizu, K. Takagi, M. Tanimot o and T. H. Tatsuishi, JHEP 1811, 196 (2018) doi:10.1007/JHEP11(2018)196 [arXiv:1808.03 012 [hep-ph]]

work page doi:10.1007/jhep11(2018)196 2018

[3] [1]

The typical region of modulus τ is found in rather narrow modular ﬁeld space as 0.43 ≲ Re[τ ] ≲ 0.45 and 0.65 ≲ Im[τ ] ≲ 0.67

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[4] [2]

We also show correlation among the mass eigenvalues in Fig

The Majorana mass eigenvalues are in the range of DN1 = [40 − 230] TeV, D N2 = [80 − 520] TeV, D N3 = [100 − 750] TeV. We also show correlation among the mass eigenvalues in Fig. 4. Note th at the mass scale is larger than the previous model in ref. [6] which is due to simpler loop structure requiring heavier masses of Ni

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[5] [3]

6 where normal hierarchy is assumed here and ∆ m2 atm is the atmospheric neutrino mass dif- ference square

It is then found that k2 3 is given by k2 3 = ∆m2 atm ˜m2 ν3 − ˜m2 ν1 , (III.8) 2 Advantage of this approximation is that ˜kα does not depend on ∆ m. 6 where normal hierarchy is assumed here and ∆ m2 atm is the atmospheric neutrino mass dif- ference square. Thus, comparing Eq.(III.7) and Eq.(III.8), we ca n rewrite ∆ m2 by other parameters as follows: ∆m2...

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[6] [4]

5, therefore following upper bounds are realized: BR(µ →eγ) ≲ 6.0×10−15, BR(τ →eγ) ≲ 1.2×10−15, BR(τ →µγ) ≲ 1.7×10−14

Typical scale of cLFVs tends to be very small in our analyses as sh own in Fig. 5, therefore following upper bounds are realized: BR(µ →eγ) ≲ 6.0×10−15, BR(τ →eγ) ≲ 1.2×10−15, BR(τ →µγ) ≲ 1.7×10−14. These values are smaller than the models in ref. [6, 11] which is due to h eavier mass scale of Ni. Note that DM candidate in our scenario is inert scalar ηR ...

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[7] [5]

Three mixings cover all the experimental results by 3 σ interval, but the sum of neu- trino masses are the narrow range [0.065,0.0070] eV that is below the upper bound of cosmological data of 0.12 eV

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[8] [7]

We found the following regions; 0 .0049 ≲ m1 ≲ 0.0072 eV and 0 .002 ≲ ⟨mee⟩ ≲ 0.005 eV which can be seen from Fig. 3. These predictions will be tested in the near future. In fact, compa ring with previous one-loop models with modular A4 in ref. [6, 11], we ﬁnd prediction for CP-phases are diﬀerent although that for neutrino mass and mixing are similar. Th...

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