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A radiative neutrino mass model with leptoquarks under non-holomorphic modular $A_4$ symmetry

T0 review · 3 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read A non-holomorphic modular A4 symmetry acting on a two-leptoquark radiative seesaw fits all measured quark and lepton flavor data and produces concrete predictions for the neutrino mass sum and neutrinoless double beta decay.

desk verdict A solid modular-flavor leptoquark model with a genuinely new quark-sector constraint, but the central neutrino-mass formula is misprinted and the numerics need transparency. read the letter →

arxiv 2504.21404 v2 pith:VP5X6SB4 submitted 2025-04-30 hep-ph

Takaaki Nomura , Hiroshi Okada , Xing-Yu Wang This is my paper · ORCID
classification hep-ph
keywords neutrinomassradiativeseesawleptoquarkmodularA4symmetrynon-holomorphicformsfermionmatricesneutrinolessdoublebetadecaysumofmasses
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 constructs a model in which a non-holomorphic modular $A_4$ symmetry dictates the Yukawa couplings of the Standard Model fermions, while two scalar leptoquarks generate neutrino masses at one loop. The authors scan the free parameters and report that the model can fit the measured quark masses and CKM mixings, the charged-lepton masses, and the neutrino oscillation data for both normal and inverted mass orderings. The quark-sector fit pins the modular parameter to $\mathrm{Im}[\tau]\sim 2.3$–$2.4$ with $|\mathrm{Re}[\tau]|\lesssim 0.4$, and the neutrino fit then gives definite allowed regions for the sum of neutrino masses and the neutrinoless double $\beta$ decay amplitude. If the construction is right, flavor in both sectors would trace back to one modulus $\tau$ and a handful of order-one couplings, with testable consequences for cosmology and double $\beta$ decay.

What carries the argument

The load-bearing machinery is the non-holomorphic modular $A_4$ framework and its Maaß-form triplets: fixed triplets of modular functions $Y_3^{(0)}$ and two independent $Y_3^{(6)}$ of the modulus $\tau$ that transform under $A_4$ and supply every Yukawa coupling in the Lagrangian. The two leptoquarks $\eta$ and $S$ mix through the term $H^\dagger\eta S$, giving mass eigenstates $\rho_{1/3}$ and $\chi_{1/3}$ with mixing angle $\alpha$; the one-loop neutrino mass formula $$M_\nu = \frac{f_1 g_1 N_c s_\$\alpha$ c_\$\alpha$}{2(4\pi)^2}\left(1-\frac{m_\$rho^{2}$}{m_\$chi^{2}$}\right)\sum_a \left[(\tilde F^T)_{ja} m_{d_a} \tilde G_{ai} + (\tilde G^T)_{ja} m_{d_a} \tilde F_{aj}\right] F_I(r_\rho, r_{d_a})$$ then converts the modular-symmetric Yukawa textures into the observed neutrino mass matrix. The machinery does two jobs at once: it reduces the Standard Model's arbitrary Yukawa matrices to a handful of parameters times fixed modular forms, and it explains the smallness of neutrino masses by loop suppression rather than by very heavy right-handed neutrinos.

What would settle it

A cosmological measurement that establishes $\sum m_\nu < 72\,\mathrm{meV}$ would exclude all fitted parameter sets, since the paper reports that every allowed point for both normal and inverted ordering violates the combined CMB and baryon-acoustic-oscillation bound. A second check: if future oscillation data pinned $\sin^2\theta_{12}$ below about 0.30 in the normal hierarchy, the NH allowed region would be excluded, because the paper finds a preference for larger solar mixing.

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Extended reading notes

Core claim

The paper's central claim is that a non-holomorphic modular $A_4$ symmetry can serve as the flavor principle behind all quark and lepton masses. In the setup, left-handed leptons, right-handed charged leptons, left-handed quarks and right-handed down quarks are $A_4$ triplets with modular weight zero, while the three right-handed up quarks are $A_4$ singlets of weight $-6$; the two scalar leptoquarks $\eta$ (an $SU(2)_L$ doublet) and $S$ (a singlet) carry no nontrivial $A_4$ charge. The neutrino mass matrix is produced radiatively by a one-loop diagram in which the mixed leptoquarks and down-type quarks run, with the loop integral controlled by the leptoquark mixing angle $\alpha$ and the masses $m_\rho$, $m_\chi$. A numerical scan over the remaining couplings finds parameter sets consistent with all measured fermion data. In the normal hierarchy the model slightly restricts the solar angle ($\sin^2\theta_{12}\gtrsim 0.30$–$0.32$) and disfavors $\delta_{CP}\in[-130^\circ,-70^\circ]$, while in the inverted hierarchy it predicts a narrow band $\sum m_\nu \sim [115,180]\,\mathrm{meV}$ and $\langle m_{ee}\rangle \sim [28,41]\,\mathrm{meV}$. In both cases the authors find that every allowed point violates the stricter combined cosmological bound on $\sum m_\nu$ of about $72\,\mathrm{meV}$, while some (normal) or all (inverted) points remain within reach of current double $\beta$ decay limits.

Load-bearing premise

The load-bearing premise is that the non-holomorphic modular $A_4$ framework used here is valid and that its two weight-6 Maaß-form triplets for the up-quark sector are the complete set of such forms; the alternative natural assignment of the up-type singlets failed to fit data, so the successful scheme is partly reverse-engineered from observations.

Editorial extensions

If this is right

  • The quark sector fixes $\mathrm{Im}[\tau]\sim 2.3$–$2.4$, so any future measurement of lepton mixing or CP violation is a test of the same modulus, not an independent parameter.
  • If the fit is correct, neutrino masses are radiatively generated by leptoquark exchange; no right-handed neutrinos or high seesaw scale are required.
  • The inverted-hierarchy region predicts $\sum m_\nu\in[115,180]\,\mathrm{meV}$ and $\langle m_{ee}\rangle\in[28,41]\,\mathrm{meV}$, placing the model within reach of upcoming neutrinoless double beta decay experiments.
  • Every currently allowed parameter point violates the stricter $\sum m_\nu\lesssim 72\,\mathrm{meV}$ cosmological bound, so the model would be excluded if that bound is confirmed.
  • Both mass orderings remain viable, but the normal-hierarchy region has lower $\sum m_\nu$ and $\langle m_{ee}\rangle$ below current double beta limits, leaving some points for near-future tests.

Reading between the lines

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

  • A sharp test the authors do not state: since the quark sector fixes $\tau$, the predicted correlations among $\delta_{CP}$, $\alpha_{21}$, and $\sin^2\theta_{12}$ could be combined into a single frequentist test of the modular hypothesis, rather than separate scans.
  • The same modular-symmetric couplings that produce neutrino masses also generate charged-lepton flavor violation through leptoquark exchange; computing $\mu\to e\gamma$ rates for the allowed parameter points would give an independent, low-energy probe of the same flavor structure.
  • Because all fitted points hit the 72 meV cosmological bound, a natural extension would add a small additional contribution to neutrino masses from a different loop or tree-level operator, lowering $\sum m_\nu$ without changing the quark-sector fit.
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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

3 major / 4 minor

Summary. The paper constructs a radiative neutrino mass model with two scalar leptoquarks under a non-holomorphic modular A4 symmetry. SM leptons and quarks are assigned to A4 representations with various modular weights, and the resulting Yukawa structures fix the charged-fermion mass matrices. Neutrino masses are generated at one-loop order through leptoquark exchange, and the authors scan the model parameters to fit quark masses and CKM observables as well as neutrino oscillation data for both normal and inverted mass hierarchies. They then present predictions for the neutrino mass sum and the neutrinoless double beta decay amplitude.

Significance. If the numerical results are correct, the paper demonstrates a single modular-symmetry framework that jointly accommodates quark and lepton flavor data while generating radiative neutrino masses, with testable predictions for Σmν and m_ee. The construction is concrete, the group-theoretic setup is clearly stated, and the predictions are falsifiable: for example, the inverted-hierarchy points populate a narrow band Σmν ≈ 115–180 meV and m_ee ≈ 28–41 meV, which is in tension with the combined CMB+DESI bound but testable by next-generation experiments. However, the central one-loop formula contains an index error that makes the printed neutrino mass matrix ill-defined, and the quark-sector fit is not quantitatively documented. These issues must be resolved before the claimed simultaneous fit can be accepted.

major comments (3)
  1. [Eq. (16), Sec. II.D] As printed, Eq. (16) is not a valid matrix equation. The second term in the bracketed sum, (G̃^T)_{ja} m_{d_a} F̃_{aj}, has no free index i and is not symmetric under i ↔ j, so the right-hand side cannot define the (i,j) element of a Majorana mass matrix. Since Sec. III.B states that the numerical scan uses this formula, the NH and IH results in Figs. 3–8 are not reproducible from the manuscript as written. The authors must correct the index (presumably to (G̃^T)_{ia} m_{d_a} F̃_{aj}, i.e. G̃_{ai} F̃_{aj}) and re-run the analysis.
  2. [Eq. (15), Sec. II.D] The rotation in Eq. (15), ilde F_{ai} ≡ (V^T_{uL})_{aj} F_{ji}, uses the left-handed up-quark rotation matrix V_uL, but the interaction in Eq. (11) involves the left-handed quark doublet Q_L and down-type quarks propagate in the loop. If this is not a typo, the loop amplitude carries an incorrect CKM-type factor; if it is a typo and V_dL (or the appropriate component of the doublet rotation) is intended, the numerical results must be recomputed with the corrected rotation. The paper should clarify which quark mass-basis rotation enters the loop formula.
  3. [Sec. III.A, Eqs. (24)–(25)] The quark-sector fit is not quantitatively documented. The only result shown is Fig. 2, a region in the τ plane, with no Δχ² values, no best-fit point, no number of scanned points, and no per-observable comparison. The trace conditions (6)–(8) fix only Tr, Det, and the sum of principal minors of M†M, so the six quark masses are imposed by construction rather than independently predicted; the nontrivial content of the fit is the CKM matrix and the CP phase, and the paper should report how well those are reproduced. Without this information, the claim that the model fits all quark data is not quantitatively supported.
minor comments (4)
  1. [Eq. (1), Sec. II] In the second line of Eq. (1), the last term is written with f_A in the coupling proportional to d_R η L_L; from the definitions in Eq. (13) this should be g_A, not f_A.
  2. [Eq. (23), Sec. II.D] The phases α2 and α3 in the expression for ⟨m_ee⟩ are not defined; they presumably correspond to α21 and α31 introduced earlier, but this should be stated explicitly.
  3. [Sec. II.B, Eq. (5)] The modular forms Y_3^(6) are essential for the up-quark mass matrix, but their explicit components are not given; since the entire quark fit depends on them, the paper should either reproduce them from Ref. [2] or give the precise definition used in the numerical scan.
  4. [Fig. 2, Sec. III.A] Fig. 2 shows only a binary allowed region; a color scale indicating the Δχ² value inside the region would make the quality of the quark fit much more transparent.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular reduction: the model's fits and predictions are not equivalent to its inputs; apparent Eq. (16) issues are correctness concerns, not circularity.

full rationale

The derivation chain is not circular. The central neutrino-mass formula, Eq. (16), is imported from Ref. [13], which includes two of the present authors; however, it is a parameter-free one-loop expression whose stated assumptions (two scalar leptoquarks and a one-loop diagram) are reproduced in the paper, so under the review rules this citation is independent support and does not raise the circularity score. The overall scale κ is fitted to Δm²_atm via Eqs. (18)–(20), and κ then enters the displayed Σmν and ⟨m_ee⟩ formulas. This makes those observables dependent on the fitted splitting, but not equal to it by construction: the PMNS angles, phases, and, in the NH case, the lightest neutrino mass remain free or scan-dependent, so the plotted ranges are genuine model outputs rather than the fit itself. The assignment of uR to A4 singlets with modular weight −6 is explicitly presented as the minimal choice that reproduces the observed quark data (footnote 1), i.e., reverse-engineering, not a circular derivation. The quark-sector fit constraining Im τ to about 2.3–2.4 and the subsequent lepton-sector scan using that τ is a sequential fit; no output is fed back as an input to define the same quantity. The apparent index mismatch in Eq. (16), where the second term is labeled by j rather than (i,j), and the related V_uL/V_dL rotation question in Eq. (15), are correctness and reproducibility defects, not circular reductions: they do not make an output equal to an input by definition. Therefore no self-definitional, fitted-input-as-prediction, or self-citation-chain circularity is present.

Assumptions & free parameters 9 free parameters · 6 assumptions · 2 invented entities

The model introduces two new scalar leptoquarks and a large number of free couplings, a modulus, leptoquark masses, and a mixing angle. The symmetry restricts the forms of the mass matrices, but the values of the observables reported as 'predictions' follow from extensive parameter scanning rather than from parameter-free derivations.

free parameters (9)
  • modulus τ = Im τ ≈ 2.3-2.4, |Re τ| < 0.4 (quark fit)
    Program parameter of the modular symmetry; scanned and fixed by the quark-sector fit (Eq. 5, Fig. 2), then used in the lepton sector.
  • up-Yukawa ratios ᾱ, β̄, γ̄ = ranges [10^-5, 10^2], random phases
    Ratios α2/α1, β2/β1, γ2/γ1 in Eq. (5); scanned to fit CKM and up-quark masses.
  • up-Yukawa scales α1^u, β1^u, γ1^u = fitted to up-quark masses
    Determined by the trace relations (6)-(8).
  • down-Yukawa couplings yd1, yd3, yd3' = fitted to down-quark masses
    Determined by the trace relations (6)-(8).
  • charged-lepton couplings yℓ1, yℓ3, yℓ3' = fitted to charged-lepton masses
    Determined by the trace relations (10).
  • neutrino-related couplings f1, fS, fA, g1, gS, gA = magnitudes in [10^-5, 1], random phases
    Enter the neutrino mass matrix via F and G in Eqs. (12)-(13); κ=f1g1 is later scaled to the atmospheric splitting.
  • leptoquark mixing angle α = s_α in [10^-3, 1]
    Mixing between η and S in Eq. (3); a free parameter entering the loop formula.
  • leptoquark masses mρ, mχ = [10^3, 10^5] GeV
    Loop masses in Eq. (16); scanned freely.
  • overall scale κ = f1g1 = fitted to Δm²_atm
    Dimensionless overall factor in Mν = κ M̃ν; fixed by Eq. (18) or (19).
assumptions (6)
  • domain assumption The non-holomorphic modular forms Y3^(0) and Y3^(6) are Maaß forms transforming as A4 triplets with the given weights, as established in Ref [2].
    The entire mass matrix ansatz relies on these modular form components; Sec. II, Eq. (1).
  • domain assumption The weight-6 modular form space is two-dimensional, giving two independent forms Y3^(6)_1 and Y3^(6)_2 used in Eq. (5).
    The up-quark mass matrix needs two independent modular form triplets; if the space has a different dimension, the ansatz changes.
  • domain assumption All scalar potential parameters in Eq. (2) are real.
    Avoids extra CP phases in the scalar sector and simplifies the mixing.
  • domain assumption The one-loop neutrino mass formula (16) from Ref [13] is applicable here with the color factor N_c and the loop function FI.
    The paper cites rather than re-derives; any error propagates into all numerical results.
  • standard math The observed quark masses, CKM elements, lepton masses, and neutrino oscillation parameters from PDG and NuFit 6.0 are correct.
    The fit uses these as external input; standard experimental data are treated as benchmarks.
  • ad hoc to paper uR can be assigned to A4 singlets with modular weight -6 despite the other fermions having weight 0; this is the minimal assignment that fits data.
    Footnote 1 states the natural universal assignment was tried and failed; the chosen assignment is data-motivated, not derived from the symmetry.
invented entities (2)
  • Scalar leptoquark η (3, 2, 1/6)
    purpose: Couples quarks and leptons and propagates in the one-loop neutrino mass diagram; also affects flavor observables and collider signals.
    Its mass and couplings are scanned freely over TeV-scale ranges; no benchmark, production cross-section, or flavor observable prediction is given that would test it independently.
  • Scalar leptoquark S (3bar, 1, 1/3)
    purpose: Together with η generates the radiative neutrino mass and mediates quark-lepton transitions.
    Same as η: free parameters, no external falsifiable handle; the model only claims these particles can be made heavy enough to evade bounds.

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Cite this review

Pith. "Pith review of A radiative neutrino mass model with leptoquarks under non-holomorphic modular $A_4$ symmetry." pith.science (2026). https://pith.science/paper/VP5X6SB4

@misc{pith2026250421404,
  author       = {Pith},
  title        = {Pith review of: A radiative neutrino mass model with leptoquarks under non-holomorphic modular $A_4$ symmetry},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VP5X6SB4}},
  note         = {Machine review of arXiv:2504.21404}
}
abstract

We investigate a radiative seesaw model with two leptoquarks under non-holomorphic modular $A_4$ symmetry. The leptons and quarks belong to non-trivial representations of the modular $A_4$ and the structures of their mass matrices are restricted. Neutrino masses are generated at one-loop level via leptoquark inside loop diagram where structures of relevant Yukawa interactions are determined by the modular $A_4$ symmetry. We scan the free parameters in the model and try to fit all the observed data for both lepton and quark sectors. For allowed parameters, we show some predictions regarding neutrino observables such as sum of neutrino mass and neutrinoless double beta decay.

Figures

Figures reproduced from arXiv: 2504.21404 by the authors.

Figure 1
Figure 1. FIG. 1: The one-loop diagram generating neutrino masses. [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: The region of modulus [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: The neutrino mixing angles obtained from the allowed parameter points in NH case where [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: FIG. 4: The values of Dirac CP phase and Majorana phases in NH case. [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: Neutrino mass related observables in NH case. Left: the obtained points on [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: The neutrino mixing angles obtained from the allowed parameter points in IH case where [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: The values of Dirac CP phase and Majorana phase in IH case. [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8: Neutrino mass related observables in IH case. Left: the obtained points on [PITH_FULL_IMAGE:figures/full_fig_p014_8.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

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    I. Esteban, M. C. Gonzalez-Garcia, M. Maltoni, I. Martinez-Soler, J. a. P. Pinheiro, and T. Schwetz (2024), 2410.05380. 17

    2024 arXiv

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.