A radiative seesaw in a non-holomorphic modular $S_3$ flavor symmetry

Hiroshi Okada; Yuta Orikasa

REVIEW 3 major objections 6 minor 5 cited by

A radiative seesaw in a non-holomorphic modular $S_3$ flavor symmetry

T0 review · 3 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read A radiative seesaw driven by a non-holomorphic modular S3 flavor symmetry can fit all measured neutrino observables and, for normal mass ordering with fermionic dark matter, predicts narrow windows for the Dirac CP phase, the neutrinoless…

desk verdict A serviceable modular-flavor application whose NH predictions are worth a look, but the IH sections are internally inconsistent with the paper's own DESI+CMB bound. read the letter →

arxiv 2501.15748 v1 pith:O3VIKSRY submitted 2025-01-27 hep-ph

Hiroshi Okada , Yuta Orikasa This is my paper · ORCID

classification hep-ph

keywords non-holomorphicmodularsymmetryS3flavorradiativeseesawMamodelneutrinomassdarkmatterleptonviolationforms

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

The paper constructs a model in which neutrino masses arise at one loop via the Ma radiative seesaw, with a non-holomorphic modular $S_3$ flavor symmetry fixing the lepton Yukawa structure. The authors perform a chi-square fit to the measured neutrino mixing angles and mass-squared differences from NuFit 6.0 and show that the model can accommodate all five observables. In the normal mass ordering with a fermionic dark-matter candidate, the fit becomes predictive: the Dirac CP phase $\delta_{\mathrm{CP}}$ is confined to about $140^\circ$–$240^\circ$, the neutrinoless double-$\beta$-decay mass $m_{ee}$ to $2$–$4.4$ meV, the sum of neutrino masses to $57$–$60$ meV, and the dark-matter mass to $10$–$3600$ GeV, with $\mathrm{BR}(\mu\to e\gamma)$ able to reach the current experimental upper bound. These are concrete, testable targets for upcoming neutrino, dark-matter, and flavor experiments.

What carries the argument

The load-bearing object is the non-holomorphic modular $S_3$ symmetry and its three modular forms $Y_2^{(2)}$, $Y_2^{(-2)}$, and $Y_2^{(0)}$ of weight 2, which respectively enter the charged-lepton, right-handed-neutrino, and Dirac-Yukawa sectors. These forms are taken from the explicit construction in ref. [22] and are not reproduced in the paper; they replace the flavon fields and fix the flavor structure through the modulus $\tau$. Neutrino masses are generated at one loop by the Ma radiative seesaw, with the lightest right-handed neutrino and the inert doublet $\eta$ running in the loop; the overall scale is set by $\kappa = M_2\gamma_\nu^2$. The $\chi^2$ scan varies the complex parameter $\widetilde{M}_1$, the Yukawa ratios $\tilde{\alpha}_\nu$ and $\tilde{\beta}_\nu$, and the coupling $\gamma_\nu$, then maps the allowed regions onto predictions for $\delta_{\mathrm{CP}}$, $m_{ee}$, $\sum m_\nu$, the dark-matter mass, and the flavor-violating branching ratios.

What would settle it

A cosmological measurement of the neutrino mass sum outside the $57$–$60$ meV window, under the normal mass ordering, would falsify the model's normal-hierarchy predictions.

Watch

Extended reading notes

Core claim

The central claim is that a single non-holomorphic modular $S_3$ symmetry, applied to the Ma radiative-seesaw particle content, is enough to reproduce the observed lepton mixing and mass splittings while also providing viable dark matter and respecting lepton-flavor-violation bounds. The charged-lepton mass matrix, the right-handed-neutrino mass matrix, and the Dirac Yukawa coupling are all built from three weight-2 modular forms $Y_2^{(2)}$, $Y_2^{(-2)}$, and $Y_2^{(0)}$ together with a small set of free parameters. A scan over these parameters with a $\chi^2$ statistic against the five oscillation observables yields four branches: normal or inverted ordering, each with a fermionic or bosonic dark-matter candidate. In the normal-ordering fermionic branch the fit localizes $\delta_{\mathrm{CP}}$ between about $140^\circ$ and $240^\circ$, $m_{ee}$ between $2$ and $4.4$ meV, the neutrino mass sum between $57$ and $60$ meV, and the dark-matter mass between $10$ and $3600$ GeV, while allowing $\mathrm{BR}(\mu\to e\gamma)$ to reach the current upper bound. In the inverted-ordering branches the mass sum is predicted near $100$ meV, which exceeds the DESI+CMB bound of $72$ meV that the paper itself cites, so those branches are in tension with that cosmological constraint.

Load-bearing premise

The predictions stand or fall on the explicit expressions and normalization conventions of the three non-holomorphic modular forms taken from ref. [22]; if those expressions differ, the fitted regions and all quoted ranges would shift.

Editorial extensions

If this is right

If the normal-ordering fermionic branch is realized, the Dirac CP phase lies in $140^\circ$–$240^\circ$, making it directly testable in long-baseline neutrino oscillation experiments.
The predicted $m_{ee}$ of $2$–$4.4$ meV is within the projected reach of next-generation neutrinoless double-beta-decay experiments such as KamLAND-Zen.
A neutrino mass sum of $57$–$60$ meV is a narrow, cosmologically testable band that upcoming DESI/CMB measurements can confirm or exclude.
The dark-matter mass window ($10$–$3600$ GeV for fermionic DM in the normal ordering, and about $534$ GeV for bosonic DM) gives a concrete target for direct and indirect dark-matter searches.
The $\mu\to e\gamma$ branching ratio can approach the current experimental upper bound in the fermionic normal-ordering branch, so MEG II could observe the signal.

Reading between the lines

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

Our reading of the quoted ranges: the inverted-ordering branches predict $\sum m_\nu \sim 100$ meV, which would conflict with the DESI+CMB bound of $72$ meV; if that bound stands, the model effectively selects the normal ordering.
Because the predictions depend on the adopted modular-form expressions, an independent derivation or numerical evaluation of the $S_3$ non-holomorphic modular forms would provide a cross-check and might shift the allowed windows.
The model forces the lightest neutrino mass to zero by using only two right-handed neutrinos; a three-right-handed-neutrino extension could test whether the predictions persist when all three active masses are nonzero.
The sharply predicted $\delta_{\mathrm{CP}}$ window could serve as a handle for leptogenesis scenarios, since a non-zero CP phase is a necessary ingredient; the paper does not pursue this connection.

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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

Quick take: the NH analysis is a usable, concrete application of non-holomorphic modular S3 to the Ma radiative seesaw; the IH analysis, as printed, is excluded by the very cosmological bound the paper itself quotes. That inconsistency is real and needs to be fixed before the IH sections can be taken seriously.

What the paper does well: it takes the recent non-holomorphic modular S3 framework from Qu–Ding and applies it to a well-known radiative seesaw, performing a genuine chi-square scan against NuFit 6.0. The NH fermionic-DM case gives a compact set of predictions: δCP roughly 140–240°, m_ee between 2 and 4.4 meV, sum of neutrino masses 57–60 meV, and a DM mass in the 10–3600 GeV range, with μ→eγ able to touch the current upper bound. Those are concrete, falsifiable statements, and the paper is honest about treating δCP as a prediction rather than an input. The LFV and DM constraints are imposed, not hand-waved.

The soft spot that matters: with two right-handed neutrinos, the lightest active neutrino mass is exactly zero, as the paper states. In the inverted hierarchy this gives m3 = 0 and therefore Σmν = m1 + m2 ≈ 100 meV, which is well above the DESI+CMB bound Σmν ≤ 72 meV that the authors quote and draw in Figs. 8 and 11. Yet the IH scan points sit right at that 100 meV region, and the text presents them as viable predictions. This is not a matter of modular-form conventions or fitting details; it follows directly from the model's particle content and the paper's own constraint. The IH sections either need to be withdrawn or explicitly flagged as excluded. The NH predictions remain unaffected, so the central NH claim survives.

Smaller issues: the explicit modular forms Y^(2)_2, Y^(-2)_2, and Y^(0)_2 are only referenced from Qu–Ding, not displayed, which makes the scan harder to reproduce. The chi-square definition, the scan size, and how the sigma-level color bands are assigned are also missing. For a model paper this is not fatal, but it is the kind of detail a referee should ask for. The DM treatment is approximate in places (e.g., assuming degenerate inert scalars for FDM, and simply rescaling a known result for BDM), but adequate for the paper's purpose.

Novelty is modest: this is a direct extension of the same authors' earlier holomorphic S3 paper to the non-holomorphic version, with new numerical predictions. That is enough to make it citable within the modular-flavor community, and it deserves a serious referee rather than a desk rejection. The referee should insist on fixing the IH presentation and on providing enough detail to reproduce the NH scan.

Referee Report

3 major / 6 minor

Summary. The paper constructs a radiative seesaw model based on the Ma model, augmented with a non-holomorphic modular S3 flavor symmetry. Active neutrino masses are generated at one loop through an inert scalar doublet and two right-handed neutrinos, which forces one active neutrino mass to be exactly zero. The authors carry out a chi-square scan over the model parameters to fit the three neutrino mixing angles and two mass-squared differences from NuFit 6.0, and they derive predictions for the Dirac and Majorana phases, the neutrinoless double beta decay effective mass, the sum of neutrino masses, the dark matter mass, and lepton flavor violation rates, in four cases: normal or inverted hierarchy with fermionic or bosonic dark matter. For the normal hierarchy, the paper reports rather sharp predictions, including a Dirac CP phase between about 140 and 240 degrees, m_ee between about 2 and 4.4 meV, a neutrino mass sum between 57 and 60 meV, and a fermionic dark matter mass between 10 and 3600 GeV, with BR(mu -> e gamma) reaching the current experimental upper bound. For the inverted hierarchy, the paper reports m_ee at about 50 meV and a neutrino mass sum at about 100 meV, without noting that this sum exceeds the DESI+CMB bound of 72 meV that the same section quotes and plots.

Significance. If the normal-hierarchy predictions survive scrutiny, the model is a complete and testable framework: it unifies a radiative seesaw with a modular flavor symmetry, treats the Dirac CP phase and mass sum as predictions rather than inputs, and makes falsifiable statements for upcoming experiments on neutrinoless double beta decay, dark matter, and charged lepton flavor violation. The paper also applies current LFV and DM constraints rather than ignoring them. However, the inverted-hierarchy scenario is internally inconsistent: because the model has only two right-handed neutrinos, the lightest neutrino mass is zero, and in the inverted hierarchy the predicted mass sum is about 100 meV, which is above the DESI+CMB bound of 72 meV that the authors themselves adopt. This discrepancy is not acknowledged, so the IH sections as written overstate the viable parameter space. The NH results are not affected by this issue, but their numerical values depend on explicit modular form expressions that are not displayed in the paper, which is a separate load-bearing gap.

major comments (3)

[Sec. IV.B, Figs. 8 and 11] The inverted-hierarchy scan points are excluded by the very DESI+CMB bound the paper quotes. The model has only two right-handed neutrinos, and the text states that the lightest neutrino mass eigenvalue is zero (Sec. II, after Eq. (II.11)); in the inverted hierarchy this gives m3 = 0, so the sum of neutrino masses is m1 + m2, which is about 2*sqrt(Delta m^2_atm) ≈ 100 meV. The paper quotes the DESI+CMB bound sum m_nu <= 72 meV in Sec. II and draws it as a vertical line in Figs. 8 and 11, yet the IH points in those figures are localized near 100 meV, to the right of the line, and the text reports these values without noting the exclusion. This is a load-bearing internal inconsistency in the claimed IH predictions. The IH sections should either be withdrawn or explicitly labeled as excluded by the quoted bound, and the abstract and conclusions should be adjusted accordingly.
[Sec. II, Eqs. (II.3), (II.5), (II.7)] The numerical results depend entirely on the explicit expressions of the non-holomorphic modular forms Y_2^(2), Y_2^(-2), and Y_2^(0) for S3, which are taken from ref. [22] but are not displayed or tested in this paper. Different normalizations or phase conventions of these modular forms could shift the fitted regions and all the predictions in Sec. IV. The authors should provide the explicit expressions (in the text or an appendix) and state the convention they use, so that the scan is reproducible and the basis matches that of ref. [22].
[Sec. IV] The chi-square analysis is not fully specified. The text does not define the chi-square function, the covariance matrix used for the NuFit 6.0 data, the number of degrees of freedom, or the criterion that defines the colored sigma intervals in Figs. 1-12. Without this information, the quality of the fit cannot be assessed and the colored regions are not reproducible. Please provide the explicit chi-square definition and the mapping from chi-square values to the sigma labels.

minor comments (6)

[Sec. II] In the paragraph following Eq. (II.1), the phrase 'is given by is given by' is duplicated; please remove the repetition.
[Eq. (II.6)] The last term of the Dirac Yukawa Lagrangian is written with beta_l, but Eq. (II.7) and the text indicate that this coefficient should be beta_nu (or beta_tilde_nu) to match the definition in the Yukawa matrix.
[Sec. II after Eq. (II.11)] The notation P Dnu for the sum of neutrino masses is nonstandard and easy to misread; please use sum m_nu and define it as sum_i D_nu_i at first occurrence.
[Sec. III.A] The phrase 'v1/2 rel ≈ 0.3' appears to be a typo; the standard notation would be v_rel ≈ 0.3 for the relative velocity in the p-wave cross-section expansion.
[Eq. (II.8)] The summation in the LFV branching ratio formula is written with an 'X' placeholder; please write out the sum over alpha = 1-3 explicitly.
[Abstract] The sentence 'We achieve chi-square analysis and demonstrate some predictions' is grammatically awkward; please rephrase to something like 'We perform a chi-square analysis and demonstrate predictions'.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the claimed predictions are scan outputs, not fitted inputs; only the kappa normalization of Delta m^2_atm and non-load-bearing self-citations keep the score slightly above zero.

full rationale

The central derivation is self-contained with respect to the claimed predictions. The model parameters are scanned and the chi-square uses the three mixing angles and two mass-squared differences from NuFIT 6.0 as inputs. Equations (II.15)-(II.16) fix kappa from Delta m^2_atm, so the atmospheric splitting is reproduced by construction; this is a fit, not a claimed prediction, and the paper does not list Delta m^2_atm among its demonstrated predictions. The genuine predictions (delta_CP, alpha_21, m_ee, sum of neutrino masses, DM mass, LFV rates) are outputs of the scan and are not used as inputs. The non-holomorphic modular forms are imported from Qu and Ding [22] as an external framework; this is a model assumption with possible convention sensitivity, but it is not a circular reduction and is not a same-author citation. Several Okada/Orikasa citations appear in the literature review ([14], [23], [26], [27]) and in standard references, but none carries a load-bearing argument that replaces an independent derivation. For bosonic DM, the relic-density calculation is explicitly imported from [38], with the paper stating 'we just fit the DM mass to be in the range of 534 +/- 8.5 GeV'; this is an acknowledged simplification, not a circular prediction. A separate, non-circular correctness concern is that with two right-handed neutrinos the lightest neutrino mass is exactly zero, so for inverted hierarchy the sum of neutrino masses is approximately 100 meV; the paper's own IH panels in Figs. 8 and 11 sit to the right of the DESI+CMB bound sum <= 72 meV that the same figures display. That is an internal consistency issue, not a circularity of the derivation chain.

Assumptions & free parameters 6 free parameters · 4 assumptions · 0 invented entities

The model is a standard Ma-model extension; all input parameters are scanned or fitted, and the key structural input is the non-holomorphic modular S3 assignment, which is taken from the cited literature without derivation in this paper.

free parameters (6)

beta_l, gamma_l, delta_l (charged lepton Yukawa coefficients) = fixed numerically to reproduce electron, muon, tau masses
These three real parameters enter the charged-lepton mass matrix (Eq. II.3). They are fitted to three lepton masses, so they carry no predictive content.
tau (complex modulus) = scanned over the fundamental region; allowed ranges shown in Figs. 1, 4, 7, 10
The modulus tau determines the modular form values and hence the flavor structure. It is scanned, not predicted.
M_tilde_1 (complex right-handed neutrino mass ratio) = scanned with |M_tilde_1| in [1e-2, 1e2] and Arg in [-pi, pi]
This parameter enters the right-handed neutrino mass matrix (Eq. II.5) and controls the texture of the seesaw.
alpha_tilde_nu, beta_tilde_nu (Dirac Yukawa ratios) = scanned in [1e-3, 1e3]
These ratios enter the Dirac Yukawa matrix (Eq. II.7) and influence the neutrino mass matrix and LFV rates.
gamma_nu (overall Dirac Yukawa coupling) = scanned in [1e-10, 1]; constrained by LFV and relic density
Sets the overall scale of the neutrino-DM interaction. It controls LFV branching ratios and the fermionic DM relic density.
m_tilde_R, m_tilde_I (inert scalar mass ratios) = m_tilde_R in [1e-2, 1e2] GeV; m_tilde_I = m_tilde_eta+
These determine the loop function in the neutrino mass formula (Eq. II.11) and the mass splittings relevant for DM coannihilations.

assumptions (4)

domain assumption The non-holomorphic modular symmetry framework of Qu and Ding (ref [22]) is valid, and the modular forms Y^(2)_2, Y^(-2)_2, Y^(0)_2 for S3 take the values used in the mass matrices.
All numerical results depend on these modular forms, but the paper does not display them; it cites ref [22] and proceeds.
domain assumption Two right-handed neutrinos are sufficient, forcing the lightest active neutrino mass to be exactly zero.
The model contains only two right-handed neutrinos (Table I), leading to one massless neutrino eigenstate. This is an assumption about the new physics content.
domain assumption The scalar potential of the model is identical to the Ma model (Eq. II.1), including the same mass spectrum.
The paper states the potential is exactly the same as in the Ma model; any modification would change the loop functions in Eq. (II.11).
domain assumption For the fermionic DM relic density, the approximation m_eta+ = m_etaR = m_etaI and p-wave domination in Eq. (III.1) is accurate enough.
This approximation avoids coannihilation and oblique parameter constraints, and it affects the inferred DM mass range.

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Pith. "Pith review of A radiative seesaw in a non-holomorphic modular $S_3$ flavor symmetry." pith.science (2026). https://pith.science/paper/O3VIKSRY

@misc{pith2026250115748,
  author       = {Pith},
  title        = {Pith review of: A radiative seesaw in a non-holomorphic modular $S_3$ flavor symmetry},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/O3VIKSRY}},
  note         = {Machine review of arXiv:2501.15748}
}

abstract

We study a non-holomorphic modular $S_3$ flavor symmetry in which we analyze neutrino sector, dark matter, and lepton flavor violations. The active neutrino mass is generated via one-loop level. We achieve chi-square analysis and demonstrate some predictions in cases of normal hierarchy with fermionic or bosonic dark matter and inverted hierarchy of fermionic or bosonic dark matter.

Figures

Figures reproduced from arXiv: 2501.15748 by the authors.

**Figure 2.** FIG. 2: Allowed regions for Majorana phase [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

**Figure 3.** FIG. 3: Allowed region of [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗

Figures from the paper (9 more)

**Figure 4.** Figure 4: FIG. 4: Allowed regions for input parameters [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]

**Figure 5.** Figure 5: FIG. 5: Allowed regions for Majorana phase [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]

**Figure 6.** Figure 6: FIG. 6: Allowed region of [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]

**Figure 7.** Figure 7: FIG. 7: Allowed regions for input parameters [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]

**Figure 8.** Figure 8: FIG. 8: Allowed regions for Majorana phase [PITH_FULL_IMAGE:figures/full_fig_p014_8.png]

**Figure 9.** Figure 9: FIG. 9: Allowed region of [PITH_FULL_IMAGE:figures/full_fig_p015_9.png]

**Figure 10.** Figure 10: FIG. 10: Allowed regions for input parameters [PITH_FULL_IMAGE:figures/full_fig_p015_10.png]

**Figure 11.** Figure 11: FIG. 11: Allowed regions for Majorana phase [PITH_FULL_IMAGE:figures/full_fig_p016_11.png]

**Figure 12.** Figure 12: FIG. 12: Allowed region of [PITH_FULL_IMAGE:figures/full_fig_p017_12.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

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A Type-III seesaw model based on non-holomorphic modular symmetry fits NuFIT 6.0 neutrino data and produces baryogenesis via leptogenesis with Y_B-L about 1e-9.

Reference graph

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

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work page Pith review arXiv 2024

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Fermionic DM In Fig. 1, we show the allowed ranges for our input parameters; τ (left-side), ˜α and ˜β(center), and ˜M1(right-side) where the blue, green, yellow, and red points respectively represent the intervals of < 1σ, 1σ − 2σ, 2σ − 3σ, and 3 σ − 5σ. The left-side figure tells us −0.2 ≲ Re[τ ] ≲ 0.14 and 1 .17 ≲ Im[τ ] ≲ 1.26. The center-side figure s...

work page

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4, we show the allowed ranges for our input parameters; τ (left-side), ˜α and ˜β(center), and ˜M1(right-side) where the color legends are the same as Fig

Bosonic DM In Fig. 4, we show the allowed ranges for our input parameters; τ (left-side), ˜α and ˜β(center), and ˜M1(right-side) where the color legends are the same as Fig. 1. The left-side figure tells us −0.1 ≲ Re[τ ] ≲ 0.1 and 1.165 ≲ Im[τ ] ≲ 1.22. The center-side figure suggests 2 ≲ |˜α| ≲ 3.5 and 0 .5 ≲ ˜β ≲ 0.65 which are almost the same region as...

work page

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7, we show the allowed ranges for our input parameters; τ (left-side), ˜α and ˜β(center), and ˜M1(right-side) where the color legends are the same as Fig

Fermionic DM In Fig. 7, we show the allowed ranges for our input parameters; τ (left-side), ˜α and ˜β(center), and ˜M1(right-side) where the color legends are the same as Fig. 1. The left- side figure tells us |Re[τ ]| ≲ 0.5 and 1 .30 ≲ Im[τ ] ≲ 1.50. The center-side figure suggests 11 FIG. 5: Allowed regions for Majorana phase α21 in terms of δCP(up-left...

work page

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10, we show the allowed ranges for our input parameters; τ (left-side), ˜α and ˜β(center), and ˜M1(right-side) where the color legends are the same as Fig

Bosonic DM In Fig. 10, we show the allowed ranges for our input parameters; τ (left-side), ˜α and ˜β(center), and ˜M1(right-side) where the color legends are the same as Fig. 1. The left- side figure tells us |Re[τ ]| ≲ 0.5 and 1 .33 ≲ Im[τ ] ≲ 1.50. The center-side figure suggests 13 FIG. 8: Allowed regions for Majorana phase α21 in terms of δCP(up-left)...

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T. Kobayashi, H. Ohki, H. Okada, Y. Shimizu, and M. Tanimoto, An Introduction to Non- Abelian Discrete Symmetries for Particle Physicists(2022), ISBN 978-3-662-64678-6, 978-3- 662-64679-3. 16 FIG. 12: Allowed region of γν in terms of DM mass where the color legends are the same as Fig. 1

work page 2022