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Fermion Masses and Mixing in Pati-Salam Unification with $S_3$ Modular Symmetry

T0 review · 4 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The paper claims that S3 modular symmetry, imposed on a supersymmetric Pati–Salam GUT, lets three benchmark models fit all 16 observed fermion mass and mixing observables, preferring normal neutrino ordering.

desk verdict First S3-modular Pati-Salam models, but the fits are too underdetermined to support the predictivity story; worth reviewing if the code and numerics are supplied. read the letter →

arxiv 2501.00302 v2 pith:XXM7NAH2 submitted 2024-12-31 hep-ph

Mohamed Belfkir , Mohamed Amin Loualidi , Salah Nasri This is my paper · ORCID
classification hep-ph
keywords S3modularsymmetryPati-Salamunificationfermionmassesandmixingtype-Iseesawneutrinomassorderingneutrinolessdouble-betadecayformsoflevel2grandunifiedtheory
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

Inside the Pati–Salam grand unified framework, the paper constructs the first models where the flavor structure of quarks and leptons is governed by the S3 modular group. Three benchmark assignments of the matter fields to S3 doublets and singlets are proposed, with all Yukawa couplings given by modular forms of a single complex modulus τ. A χ² fit to sixteen measured quantities—charged fermion mass ratios, CKM and PMNS mixing parameters, and neutrino mass splittings—finds that all three models reproduce the data, prefer normal neutrino mass ordering, and fix the atmospheric angle in the lower octant. The models make distinct predictions for the effective Majorana mass, the sum of neutrino masses, and leptonic CP phases, which upcoming experiments can test.

What carries the argument

The engine is the level-2 finite modular group $\Gamma_2 \cong S_3$, whose weight-2 modular forms $Y_1(\tau)$ and $Y_2(\tau)$ form an $S_3$ doublet built from Dedekind eta functions. Tensor products generate weight-4 and weight-6 modular forms, which enter Yukawa couplings through the modular-invariant superpotential. The Pati–Salam matter content gives each generation only two multiplets, $F \sim (4,2,1)$ and $F^c \sim (\bar{4},1,2)$, so the same Yukawa matrices feed quarks and leptons; the VEV alignment of $\Sigma = (15,2,2)$ introduces the Georgi–Jarlskog factor $-3$ for leptons, splitting down-quark and charged-lepton masses. The $\Delta_R = (10,1,3)$ VEV generates right-handed neutrino masses and, through the type-I seesaw formula $m_{\nu} = -M_D^T M_R^{-1} M_D$, produces the light neutrino spectrum.

What would settle it

A decisive test is the atmospheric octant: all three models place the best fit in the lower octant, sin²θ23 ≈ 0.47. If a future global fit with improved atmospheric data finds sin²θ23 > 0.5, the central claim that these S3 assignments reproduce the observed mixing would fail. Independently, a 0νββ signal with mββ > 10 meV — above the best-fit values of 5.5, 1.6, and 8.9 meV — would also exclude all three benchmark points.

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Extended reading notes

Core claim

The central claim is that the $S_3$ modular symmetry, imposed on a supersymmetric Pati–Salam gauge theory, is capable of accounting for the masses and mixings of all three families of quarks and leptons from a small set of parameters. The paper demonstrates this with three renormalizable benchmark models that differ by the $S_3$ charges and modular weights of the matter multiplets $F$ and $F^c$. With the Higgs multiplets $\Phi$, $\Sigma$, and $\Delta_R$ taken as $S_3$ singlets and modular forms truncated at weight 6, the Yukawa matrices reduce to structures in which the quark and lepton sectors share the same couplings, split only by the Clebsch–Gordan factor $-3$ coming from the VEV of $\Sigma$ along the $SU(4)_C$ adjoint. The numerical fits yield $\chi^2_{\text{total}}$ values of 2.10, 0.76, and $1.8\times10^{-10}$ for the normal ordering, with the second and third models placing all sixteen observables inside their 1$\sigma$ ranges. The theory therefore predicts concrete values for $m_{\beta\beta}$, $\sum m_i$, $m_\beta$, and the CP phases, as well as Pati–Salam breaking scales that differ across the models.

Load-bearing premise

The fits rest on the hand-picked assignment of the matter fields to S3 doublets and singlets with specific modular weights shown in Tables I–III; if those assignments were fixed by some principle the claim would be stronger, but as it stands the success of the fit could be a product of the chosen ansatz rather than of modular symmetry itself.

Editorial extensions

If this is right

  • All three models favor normal neutrino mass ordering: the inverted ordering gives larger χ² values and leaves some observables outside their 3σ ranges.
  • If the claim is correct, modular symmetry can replace flavon alignment in grand unified flavor model building, removing the need for a separate flavor-breaking scalar sector.
  • Models I and III predict mββ ≈ 5.5 meV and 8.9 meV, within reach of upcoming tonne-scale neutrinoless double-beta decay experiments, while model II predicts 1.6 meV, too small to detect.
  • The predicted sum of neutrino masses, roughly 59–70 meV, sits below the current cosmological bound and inside the sensitivity range planned for next-generation cosmological surveys.
  • The leptonic Dirac CP phase is predicted around 1.13π–1.20π, giving future long-baseline and reactor oscillation experiments a concrete way to distinguish the three models.

Reading between the lines

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

  • The paper does not scan the full space of possible S3 charge and modular-weight assignments, so the three benchmark models are existence proofs rather than an exhaustive classification; a systematic scan might uncover other viable assignments, some with different phenomenological signatures.
  • Model I yields a low Pati–Salam breaking scale, c₁v_R ≈ 7.9×10³ GeV, which suggests that if that model is realized, right-handed neutrinos and the associated new gauge bosons could be light enough for collider searches; the other models place the scale far above current reach.
  • A sharper test would combine the predicted correlation between the atmospheric octant and δ_CP with precise neutrino mass measurements: each model occupies a distinct region in the (sin²θ23, mββ) plane, so a future measurement of both would separate them without ambiguity.
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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

4 major / 5 minor

Summary. The paper proposes three supersymmetric Pati-Salam models with a level-2 modular S3 symmetry, assigning the matter multiplets F_i and F^c_i to S3 doublets or singlets and using modular forms of weights up to 6 to build Yukawa couplings. The models are designed to fit 16 observables—charged fermion mass ratios, quark mixing angles and CP phase, and neutrino oscillation parameters from NuFIT 6.0—at the GUT scale. The authors perform a numerical chi-square minimization and present best-fit parameter values, chi-square statistics, and predictions for the neutrino mass sum, effective masses m_beta and m_beta_beta, and CP phases. They find that all three models prefer normal neutrino mass ordering and predict a lower-octant atmospheric angle.

Significance. If the numerical results were independently reproducible, the paper would provide a useful first example of S3 modular symmetry combined with Pati-Salam unification, complementing the existing A4 modular PS model. The explicit Yukawa matrices and the correlation plots give concrete benchmarks for future model building. The main value is the model construction and the demonstration that the chosen S3 assignments can accommodate the measured fermion masses and mixings. However, the significance is weakened by the large number of free parameters relative to observables, the absence of code or data files, and apparent transcription errors in the one-TeV check, which currently prevent the reader from assessing whether the claimed fits are meaningful.

major comments (4)
  1. [Section IV, Table V] The parameter count is 26 for model III and 24 for model II, while the fit uses 16 observables. The reported chi2_total of 1.8e-10 for model III means the model reproduces the central values almost exactly; with ten more free parameters than data points, this near-perfect chi2 is a typical consequence of overfitting rather than evidence that the S3 assignments are predictive. The paper should quote a goodness-of-fit statistic that accounts for the number of parameters (for example, chi2 per degree of freedom or an information criterion), and should demonstrate that the fit is stable and that alternative, less flexible assignments cannot achieve comparable agreement.
  2. [Eq. (V.39)] The numerical input values for the 1 TeV check are internally inconsistent: md/ms is listed with the same central value as me/mu, ms/mb is given as 0.05046 (which is essentially the md/ms value from Table IV), and mu/mc appears twice with two different values (0.00204 and 0.01368). Because the authors use these values to claim agreement at M_SUSY=1 TeV, these errors prevent the reader from trusting the 1 TeV fits and suggest a problem in the table-composition or fitting pipeline.
  3. [Section V] The fitting procedure is not reproducible from the manuscript. The paper names FlavorPy and lmfit but provides no code, seed values, parameter bounds, or numerical precision information; the only record of the fit is the bare best-fit values in Table V. Without the ability to rerun the minimization, the claims of global minima and the reported chi2 values cannot be verified. This is a load-bearing omission for a paper whose central claim is a quantitative fit.
  4. [Section V, Eqs. (V.34)-(V.35)] The quantities labeled as predictions (m_beta, m_beta_beta, the sum of neutrino masses, and the leptonic Dirac phase) are computed directly from the fitted neutrino masses and PMNS elements, so they are not independent tests of the models unless the paper shows that the fitted parameter space is strongly constrained by the non-oscillation data. The paper should state more carefully which of these are genuinely predicted from the model structure and which are derived from inputs that were already adjusted to the oscillation data.
minor comments (5)
  1. [Table V] The header contains the typo "chi2 totlal" and should read "chi2 total".
  2. [Table V] In the Model I (NO) column, the parameter b3/a1 is written as "0635.17e4.246pi i" with an extra leading zero; this should be cleaned up.
  3. [Abstract] The phrase "with the atmospheric mixing angle lie in the lower octant" should be "with the atmospheric mixing angle lying in the lower octant".
  4. [Section VI] The listing of right-handed neutrino masses for model I gives M2 and M3 both approximately 1.4e6 GeV; a brief comment on the near-degeneracy and its consequences would be helpful.
  5. [Appendix A] It would be useful to define the modular form components Y1 and Y2 explicitly before using them in the tables, since the notation Y(4)_1 and Y(4)_2 appears before the appendix details are introduced.

Circularity Check

1 steps flagged · score 4.0 of 10

Partial circularity: the 16 fitted observables (notably δ_CP) are relabeled as 'predictions' in Table V, while the genuinely new mββ, Σm_i, and Majorana-phase outputs are not fitted inputs.

  1. fitted input called prediction [Section V, Eq. (V.36) and Table V; also Abstract]
    "The χ2 statistic is defined as χ2 = X i ( Pi(¯x) − µi σi )2 ... The Pi(¯x) is the prediction of the physical parameters by the model ... We represent our results in Table V ... Predictions for observables, including fermion mass ratios, flavor mixing parameters, the effective Majorana neutrino mass mββ ..., are also presented."

    The 16 observables in Table IV are exactly the µi entering Eq. V.36, and minimizing χ2 forces Pi(¯x) close to µi. Table V then lists these same quantities—me/mµ, mµ/mτ, sin2θl12, sin2θl13, sin2θl23, δlCP, etc.—as 'predictions.' In particular δlCP is an explicit fit input with 1σ range 212+26−41, so the abstract's 'Predicted leptonic CP-violating phases' is, for the Dirac phase, the fitted value relabeled. This is fitted-input-called-prediction by construction. The mββ, Σmi, and Majorana-phase outputs are not among the µi and depend on unconstrained m1 and αi, so they are genuine model predictions; hence the circularity is partial and confined to the presentation of the fitted observables.

full rationale

The core model-building chain is not circular: the S3 modular forms are standard mathematical objects (eta-function quotients), the tensor-product rules are explicitly given in Appendix A, and the Yukawa matrices in Eqs. IV.28, IV.30, and IV.32 are derived from the stated representation and modular-weight assignments in Tables I-III. No load-bearing step is justified by a self-citation, and no uniqueness theorem is imported from the authors' prior work; in fact the paper contains no self-citations. The benchmark models are presented as explicit ansätze with hand-picked S3 assignments and weights, and the numerical analysis honestly fits the 16 observables via χ2 minimization. The derived quantities mβ, mββ, Σmi, and the Majorana phases are not among the fitted inputs: absolute neutrino mass scale m1 and Majorana phases are not constrained by the oscillation data used, so reporting them as predictions is legitimate model output. The main circularity is terminological: Table V and the accompanying text call the 16 fitted observables 'predictions,' and the Dirac CP phase in the abstract's 'Predicted leptonic CP-violating phases' is a fitted input. This is a real but partial instance of fitted-input-called-prediction. Concerns about 26 parameters for 16 observables and χ2 ≈ 10−10 for model III are robustness/overfitting issues, not circularity, and do not by themselves raise the circularity score.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new particles or forces; it relies on standard PS multiplets (Phi, Sigma, Delta_R) and the type-I seesaw. The central claim rests on roughly 21 to 26 fitted free parameters per model, the assumed VEV alignment of the PS Higgses, and the hand-picked S3 charge assignments. These are the main entries in the ledger, with the model selection and VEV alignment being the least motivated from first principles.

free parameters (4)
  • Complex modulus tau = Re tau from 0.0005 to 0.41, Im tau from 0.87 to 1.84 (per model, Table V)
    The VEV of tau sets all modular-form values; it is a free input varied in the fit.
  • Coupling ratios a_i/a1, b_i/a1, c_i/c1 = See Table V, e.g., a2/a1 ~ 2548.6 in model I NO, b1/a1 ~ 1340 e^{1.839 pi i} in model I NO
    The free couplings of each renormalizable Yukawa operator, normalized to a1; 15 to 20 ratios per model are fitted to the 16 observables.
  • VEV ratio r2 = 0.90 e^{5.490 pi i} (model I NO), 0.941 e^{-0.425 pi i} (model II NO), 0.0538 e^{0.891 pi i} (model III NO)
    Ratio of PS Higgs VEVs entering up-type versus down-type mass matrices; fitted to data.
  • Overall mass scales a1 v_u, a1 r1 v_d, (a1 v_u)^2/(c1 v_R) = e.g., (a1 v_u)^2/(c1 v_R) = 1.161 meV (model I NO), 130.34 meV (model II NO), 0.7466 meV (model III NO)
    Overall scales set the absolute charged-fermion and neutrino masses; fitted to data.
assumptions (5)
  • domain assumption Modular invariance of the superpotential with Yukawa couplings transforming as level-2 modular forms under Gamma_2 congruent to S3 (Section III, Eqs. III.18-III.21).
    The central framework postulate; not derived from string theory or a UV completion.
  • domain assumption Pati-Salam matter embedding and the minimal Higgs content Phi, Sigma, Delta_R with VEV alignment <Phi> proportional to 1 and <Sigma> proportional to diag(1,1,1,-3) (Section II, Eq. II.9).
    The alignment yields the -3 Clebsch-Gordan factor for leptons (Georgi-Jarlskog relations); the scalar potential is not analyzed to show this is the minimum.
  • domain assumption Type-I seesaw with light neutrino mass m_nu = -M_D^T M_R^{-1} M_D and M_R = Y^{10R} <Delta_R> (Section II, Eqs. II.10-II.13).
    The standard seesaw mechanism; assumes no other neutrino mass sources contribute.
  • ad hoc to paper The Higgs multiplets Phi, Sigma, Delta_R are S3 singlets, k_Phi not equal to k_Sigma, and only modular forms of weight <= 6 are included (Section IV).
    These restrictions are imposed to reduce model complexity; they are not derived from symmetry or dynamics.
  • domain assumption Input observables at the GUT scale (tan beta = 10, M_SUSY = 500 GeV) from Ref. [38] and NuFIT v6.0 (Table IV).
    The fit trusts these external determinations of GUT-scale mass ratios and mixing angles.

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Cite this review

Pith. "Pith review of Fermion Masses and Mixing in Pati-Salam Unification with $S_3$ Modular Symmetry." pith.science (2026). https://pith.science/paper/XXM7NAH2

@misc{pith2026250100302,
  author       = {Pith},
  title        = {Pith review of: Fermion Masses and Mixing in Pati-Salam Unification with $S_3$ Modular Symmetry},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XXM7NAH2}},
  note         = {Machine review of arXiv:2501.00302}
}
abstract

Modular invariance has recently paved new promising directions in flavor model building. Motivated by this development, we present in this work the first implementation of the $S_3$ modular symmetry within the Pati-Salam unification framework, addressing the flavor structure of quarks and leptons. Assigning left- and right-handed matter fields as $S_3$ doublets or singlets, we propose three benchmark models that achieve compelling fits to sixteen observables including charged fermion mass ratios and flavor mixing parameters. Light neutrino masses arise via the type-I seesaw mechanism, and neutrino oscillation parameters are explored in light of the latest NuFIT-6.0 results. All models favor a normal neutrino mass ordering, with the atmospheric mixing angle lie in the lower octant. For models I and III, the effective Majorana mass $m_{\beta\beta}$ is within the reach of upcoming neutrinoless double-beta decay experiments, while it is too small to be detected in model II. Predicted leptonic CP-violating phases, the sum of active neutrino masses, and Majorana phases span wide but distinctive ranges, enabling future experiments to test and differentiate the proposed models.

Figures

Figures reproduced from arXiv: 2501.00302 by the authors.

Figure 1
Figure 1. FIG. 1. Allowed region of [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Same as figure 1 but for model II. [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Same as figure 1 but for model III. [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗

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Forward citations

Cited by 2 Pith papers

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