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Phenomenology of Inverse Seesaw Using $S_3$ Modular Symmetry

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Pith's one-line read By imposing the smallest modular group S3 on an inverse seesaw model, this paper derives a six-parameter neutrino sector with definite predictions: a massless lightest neutrino, inverted mass ordering, a mass sum near 100 meV, and a…

desk verdict A minimal and clean modular S3 inverse seesaw model whose IO-only claim rests on a finite scan and whose m_ee numbers disagree between abstract and text; worth peer review with revisions. read the letter →

arxiv 2504.12954 v2 pith:DJLRGU2Q submitted 2025-04-17 hep-ph

Mitesh Kumar Behera , Pawin Ittisamai , Chakrit Pongkitivanichkul , Patipan Uttayarat This is my paper · ORCID
classification hep-ph
keywords inverseseesawmodularS3symmetryneutrinomassinvertedorderingneutrinolessdoublebetadecayleptonflavorviolationsupersymmetryforms
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 minimal supersymmetric inverse seesaw model in which the smallest modular group, $S_3$, plays the role of flavor symmetry, with Yukawa couplings promoted to modular forms. The resulting light neutrino mass matrix depends on only six real parameters, and the paper claims this minimality turns into definite predictions: the lightest neutrino is massless, neutrino masses follow inverted ordering, the sum of neutrino masses is about 100 meV, and the beta-decay endpoint effective mass is about 50 meV. The effective mass for neutrinoless double beta decay is found in a narrow window (38-48 meV in the numerical scan; 38-58 meV as quoted in the abstract), within reach of next-generation experiments. The same structure also predicts radiative lepton flavor violating decays that respect current bounds. If correct, the model would turn neutrino oscillation data alone into a fixed absolute mass scale, with concrete targets for cosmology and double-beta decay searches.

What carries the argument

The load-bearing object is the approximate inverse seesaw formula $m_\nu \simeq M_D^T M_{NS}^{-1} M_S M_{NS}^{-1} M_D$, which in this model reduces to a rank-two matrix proportional to $\kappa$, with entries built from the modular forms $Y_\pm = Y_1 \pm iY_2$ and two independent Yukawa ratios $\tilde{\alpha}_D$ and $\tilde{\gamma}_D$. Modular $S_3$ symmetry, the smallest finite modular group, elevates the Yukawa couplings to modular forms, so no flavon fields or vacuum alignment is needed; the complex modulus $\tau$ and its phase are physical parameters. The rank-two structure forces a massless lightest neutrino, while the six-parameter dependence and a scan over $\tau$ in the fundamental domain with $\kappa$ in $0.1$ meV-$10$ eV select inverted ordering and fix the absolute mass scale. The same Yukawa couplings enter the one-loop formula for $\ell \to \ell' \gamma$, tying the flavor-violating predictions to the same parameters that fit neutrino oscillations.

What would settle it

A future determination that neutrino masses follow normal ordering, or a measurement of the neutrino mass sum $\sum_i m_i$ incompatible with the predicted 100 meV scale, would contradict the model. So would a neutrinoless double $\beta$ decay search finding $m_{ee}$ outside the $38$-$48$ meV window, or an extended scan with wider parameter ranges that locates a viable normal-ordering solution.

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Extended reading notes

Core claim

The paper's central claim is that the inverse seesaw mechanism combined with modular $S_3$ flavor symmetry yields a neutrino sector with a rank-two light-neutrino mass matrix: one neutrino is exactly massless, and fitting the six oscillation parameters at the $3\sigma$ level leaves only the inverted ordering. Because the lightest mass vanishes, the absolute scale is fixed by the atmospheric splitting, giving $\sum_i m_i \simeq 100$ meV and the $\beta$-decay endpoint effective mass $m_{\text{eff}}^{\nu_e} \simeq 50$ meV. The neutrinoless double $\beta$ decay mass $m_{ee}$ is not fixed by oscillation data alone but is predicted in the narrow range $38$-$48$ meV in the detailed scan ($38$-$58$ meV in the abstract), just above current limits and within the projected reach of next-generation experiments. The model additionally predicts one-loop radiative decays $\ell \to \ell' \gamma$; for a benchmark with TeV-scale masses, $\mu \to e \gamma$ sits just below the current combined limit while the tau decay modes are well below their limits.

Load-bearing premise

The claim that the model permits only inverted neutrino ordering rests on a numerical scan over finite, hand-chosen parameter ranges; no analytic proof rules out normal ordering outside those ranges.

Editorial extensions

If this is right

  • The absolute neutrino mass scale is fixed by oscillation data alone: the model predicts $\sum_i m_i \simeq 100$ meV and a beta-decay endpoint mass of about 50 meV, almost independently of the free parameters.
  • The effective neutrinoless double beta decay mass is confined to a narrow window, 38-48 meV in the scan (38-58 meV in the abstract), placing the model inside the projected reach of next-generation experiments.
  • The radiative decays $\mu \to e \gamma$, $\tau \to e \gamma$, and $\tau \to \mu \gamma$ are predicted below current bounds over a range of $\tan\beta$ for a TeV-scale benchmark, with $\mu \to e \gamma$ just below the combined current limit.
  • If future data establish normal neutrino mass ordering, the model is ruled out, since the parameter scan admits only inverted ordering.
  • The charged lepton sector is fixed by three parameters reproducing the electron, muon, and tau masses, so the model has no extra freedom to adjust the predicted correlations among the CP phase, $\text{Im}(\tau)$, and $m_{ee}$.

Reading between the lines

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

  • Because the lightest neutrino is exactly massless, the model sits at a structural limit: any measurement of the absolute neutrino scale that excludes a massless state would count against the entire class of rank-two inverse seesaw models, not just this parameter choice.
  • The 'inverted ordering only' result is only as strong as the numerical search; an analytic proof or an expanded scan outside the quoted parameter ranges would determine whether it is a property of the model or an artifact of the scan.
  • The scaling of the flavor-violating widths with $\mu_S^{-2}$ means that lowering the lepton-number-violating scale while keeping neutrino masses fixed would push $\mu \to e \gamma$ upward; upcoming searches could therefore probe the same parameter that sets the neutrino mass scale.
  • The correlation between $\text{Im}(\tau)$ and $m_{ee}$ suggests that the viable region is effectively low-dimensional, so a joint measurement of the cosmological mass sum and neutrinoless double beta decay could test the correlation directly.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 6 minor

Summary. This paper constructs a minimal supersymmetric inverse seesaw model with an S3 modular flavor symmetry. The light neutrino mass matrix is derived from a small superpotential and depends on six real parameters: the modulus τ, an overall scale κ, a real Yukawa ratio α̃_D, and a complex Yukawa ratio γ̃_D. The matrix has rank two, giving a massless lightest neutrino. A numerical scan over hand-chosen parameter ranges finds only inverted mass ordering, leading to predictions Σm_i ≈ 100 meV, m_eff_νe ≈ 50 meV, and m_ee in the range 38–48 meV (abstract: 38–58 meV). Radiative lepton flavor violating decays are computed and shown to be compatible with current bounds for a benchmark with TeV-scale masses.

Significance. If the inverted-ordering exclusivity holds, the model is a compact and falsifiable framework: the massless lightest neutrino and the narrow m_ee window are testable by cosmological surveys and 0νββ experiments, and the LFV rates are within an order of magnitude of MEG sensitivity. The derivation of the inverse seesaw mass matrix is standard and internally coherent, and the charged lepton sector is handled consistently through traces. The structural predictions (rank-2 matrix, inverted ordering) are model-derived rather than fitted, which is a strength. However, the numerical values of Σm_i and m_eff_νe are not independent predictions; they follow from the ordering and the measured atmospheric splitting, so the model's novel predictive content is the ordering itself and the m_ee window.

major comments (2)
  1. [V, Eqs. (24)–(27)] The statement 'from the scan, we find that our model can only accommodate IO neutrino masses' (Sec. V) is supported only by a finite numerical scan over the hand-chosen ranges in Eqs. (24)–(27): τ in the fundamental domain, κ in 0.1 meV–10 eV, α̃_D and |γ̃_D| in 10^-3–10^3, and φ_γD in [0,2π]. No analytic proof or completeness argument is given, and the modular forms vary rapidly near the fundamental-domain boundary, so the scan boundaries are not protected by symmetry. A normal-ordering solution outside this box, or in a thin region missed by the sampling, would change Σm_i from about 100 meV to about 59 meV and shift the m_ee window, invalidating the paper's central phenomenological package. Please provide an analytic no-go for normal ordering or demonstrate exhaustive coverage (e.g., a denser scan with a documented global-search strategy), and qualify the exclusivity claim accordingly.
  2. [Abstract vs Section V/Conclusion] The abstract states m_ee ∈ [38,58] meV, while Section V (paragraph after Eq. (29)) and the Conclusion both state 38 meV ≤ m_ee ≤ 48 meV. This is a direct inconsistency in a headline numerical prediction. The authors should correct the abstract or the body and ensure all reported ranges are consistent.
minor comments (6)
  1. [VI (Conclusion)] The text refers to the 'Simon Observatory'; the correct name is the 'Simons Observatory'.
  2. [V, near Eq. (30)] The sentence 'we get tanβ2∼ 1/µS' appears to contain a typo; based on Eq. (13) with κ fixed, the intended relation is likely β_D^2 ∼ 1/µS. Please clarify.
  3. [Eq. (13)] The notation 'sin2β' should be written as 'sin 2β' to avoid ambiguity with sin^2 β.
  4. [V, Eqs. (28)–(29)] It would be clearer to state explicitly that Σm_i ≈ 100 meV and m_eff_νe ≈ 50 meV are consequences of the inverted ordering and the measured Δm_atm^2, rather than independent model predictions; the model's predictive content is the ordering and the massless lightest neutrino.
  5. [IV] The assumptions that sneutrinos do not mix and that H̃_u does not mix with other charginos are stated, but their quantitative impact on the computed LFV branching ratios is not estimated; a brief discussion would help the reader judge the robustness of the benchmark results.
  6. [V, Fig. 2] The x-axis label of Fig. 2 appears to be missing the symbol for κ before '[eV]'; please check the figure rendering.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the model's predictions follow from its rank-2 neutrino mass structure and a scan over both mass orderings; the finite scan coverage is a robustness concern, not a circular reduction.

full rationale

The derivation chain is self-contained against external neutrino data and is not circular. The rank-2 light neutrino mass matrix in Eq. (12) is obtained from the inverse-seesaw formula with the model's field content and modular Yukawa structure; the massless lightest neutrino follows from the rank, not from an input assumption about the sum of masses or m_ee. The scan in Section V fits the six model parameters to the six oscillation observables in Table II without imposing a mass ordering or any 0νββ constraint, and the claim that only inverted ordering survives is a scan output rather than a fitted input. Equations (28)-(29) are then testable sum rules relating Σm_i and m_effνe to the measured Δm_atm^2 once m_lightest=0 and IO are established, which is a legitimate cross-prediction against cosmology and beta-decay experiments, not a renamed fitted parameter. The m_ee range 38-48 meV and the LFV branching ratios are determined by the surviving model parameter points and external benchmark masses, not fitted to the observables they predict. Self-citations such as Ref. [55] are contextual literature references and are not load-bearing for any of the paper's quantitative claims. The abstract's 38-58 meV versus Section V's 38-48 meV is an internal numerical inconsistency, and the hand-chosen scan ranges are a completeness/robustness concern, but neither is an instance of a claim reducing by construction to its own input.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The core neutrino mass predictions rest on the S3 modular assignments in Table I, the inverse seesaw hierarchy, and a scan with hand-chosen ranges; the LFV rates additionally rely on a benchmark supersymmetric spectrum and the assumed absence of sneutrino and chargino mixing.

free parameters (5)
  • τ complex modulus = τ ≈ 1.2 i, Im τ in 1.14 to 1.24
    Varied within the fundamental domain; controls all modular form values and hence neutrino masses and mixing.
  • κ overall neutrino mass scale = around tens of meV via oscillation fit
    Determines the absolute scale of the light neutrino mass matrix; scan range is 0.1 meV to 10 eV.
  • α̃_D real Yukawa ratio = α_D/β_D ≈ 1
    A real parameter in the light neutrino mass matrix, fixed by the oscillation fit.
  • γ̃_D complex Yukawa ratio = |γ_D/β_D| ≈ 0.1, phase in 2.4 to 3.9
    A complex ratio that provides one of the CP violating sources; magnitude and phase are fit to oscillation data.
  • LFV benchmark parameters = M = m_H+ = m_ÑI = m_H̃u+ = 1 TeV, μ_S = 10 eV, tan β = 1, 2, 50
    Not needed for the neutrino mass fit, but required to compute ℓ → ℓ'γ rates; results scale with μ_S^2.
assumptions (5)
  • standard math S3 modular forms transform under the specified representations and the modular group acts as assumed.
    Invoked in Section II and Appendix A; relies on known S3 tensor products and modular form expansions.
  • domain assumption The inverse seesaw hierarchy ||M_R||, ||M_S|| << ||M_D|| << ||M_NS|| holds.
    Used to derive Eq. (12); if violated, the light neutrino mass formula and rank-2 structure receive corrections.
  • ad hoc to paper Only the superpotential terms in Eq. (1) are present and no additional allowed operators contribute.
    The minimality claim depends on omitting higher-weight or additional S3 singlet terms; no UV completion is provided.
  • ad hoc to paper Sneutrinos do not mix and there is no chargino mixing in the LFV calculation.
    Stated in Section IV; simplifies the LFV computation but is an assumption about the supersymmetric spectrum.
  • domain assumption The NuFIT 6.0 3σ ranges are the correct global fit for neutrino oscillation parameters.
    Used in Section V to define the viable parameter space of the model.

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Cite this review

Pith. "Pith review of Phenomenology of Inverse Seesaw Using $S_3$ Modular Symmetry." pith.science (2026). https://pith.science/paper/DJLRGU2Q

@misc{pith2026250412954,
  author       = {Pith},
  title        = {Pith review of: Phenomenology of Inverse Seesaw Using $S_3$ Modular Symmetry},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DJLRGU2Q}},
  note         = {Machine review of arXiv:2504.12954}
}
abstract

Describing neutrino masses using the inverse seesaw mechanism with discrete flavor symmetry imposed through modular forms provides a testable framework at TeV scales with fewer parameters. However, $S_3$, the smallest modular group, remains relatively underexplored. In this work, we construct the minimal supersymmetric inverse seesaw model based on the modular $S_3$ flavor symmetry. In our model, the light neutrino mass matrix depends on 6 real parameters: the complex modulus, an overall scale for light neutrino mass, a real ratio and a complex ratio of Yukawa coupling. Thanks to its minimality, our model offers various definite predictions: the lightest neutrino is massless, the neutrino masses are inverted ordering, the sum of the three light neutrino masses ($\sum_i m_i$) is 100 meV, the effective mass for the end point of the beta decay spectrum is 50 meV, the effective mass for neutrinoless double beta decay ($m_{ee}$) is in the range $38-58$ meV. In particular, the predicted values for $\sum_i m_i$ and $m_{ee}$ from our model are within reach of the next generation experiments. Our model also predicts radiative lepton flavor violating decays $\ell\to\ell'\gamma$ which are compatible with experimental constraints.

Figures

Figures reproduced from arXiv: 2504.12954 by the authors.

Figure 1
Figure 1. FIG. 1: The space of modulus [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: The relations between the overall neutrino mass scale [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: The relations between the free parameters Im( [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4: The Br( [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]

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Forward citations

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