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arxiv: 2606.24277 · v1 · pith:5Q532M6Anew · submitted 2026-06-23 · ✦ hep-ph

Non-holomorphic S^(prime)₄ modular symmetry for leptons and leptogenesis

Cai-Chang Li , Jun-Nan Lu , Xiang-Yan Gao , Ming-Hua Weng , Gui-Jun Ding This is my paper

Pith reviewed 2026-06-25 23:56 UTC · model grok-4.3

classification ✦ hep-ph
keywords modular symmetryS'4neutrino mixingleptogenesistype-I seesawCP violationMaaß formsnormal ordering
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The pith

Thirty-six lepton models under non-holomorphic S'4 symmetry fit normal neutrino ordering with four real couplings plus the modulus, and two also reproduce the observed baryon asymmetry via leptogenesis.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper performs a systematic scan of lepton models based on non-holomorphic S'4 modular symmetry using polyharmonic Maaß forms. It identifies 36 viable models for normal neutrino mass ordering that require only four real couplings in addition to the modulus τ. These models are classified into three categories, each with twelve members that make similar predictions but differ by the assignment of one charged lepton field. Two of the three representative models also generate the correct baryon asymmetry through thermal leptogenesis, with the real part of τ acting as the sole source of CP violation.

Core claim

The authors establish that non-holomorphic S'4 modular symmetry, realized with level 4 polyharmonic Maaß forms of weights between -4 and 6, permits the construction of lepton models that accommodate current neutrino oscillation data with a minimal number of parameters. For normal ordering, 36 such models exist, grouped into three categories of twelve models each, distinguished by the representation assignment of the charged lepton field E^c_1. Representative models from each category produce precise predictions for the neutrino mixing angles and CP phases. Moreover, two of these models achieve successful thermal leptogenesis without flavor effects, matching the observed baryon asymmetry with

What carries the argument

non-holomorphic S'4 modular symmetry implemented through level-4 polyharmonic Maaß forms of integer weights from -4 to 6, which enter the Yukawa couplings and generate the charged-lepton and neutrino mass matrices as functions of the modulus τ

Load-bearing premise

Viability depends on the assumption that the chosen field representations and modular weights under S'4 allow numerical fits to current neutrino oscillation data within the type-I seesaw with exactly two right-handed neutrinos and without extra flavons or generalized CP symmetry.

What would settle it

A future measurement showing the Dirac CP phase δ_CP outside the narrow intervals predicted by the two successful representative models would rule out those models.

Figures

Figures reproduced from arXiv: 2606.24277 by Cai-Chang Li, Gui-Jun Ding, Jun-Nan Lu, Ming-Hua Weng, Xiang-Yan Gao.

Figure 1
Figure 1. Figure 1: Regions of the modulus τ compatible with experimental data. The cyan area denotes the fundamental domain of τ . The blue regions correspond to values of τ consistent with the experimentally allowed range of ∆m2 21/∆m2 31 [64] for the benchmark models, respectively. The orange area denotes the viable region of τ limited only by the measured values of the reactor mixing angle θ13 [64, 65]. 16 [PITH_FULL_IMA… view at source ↗
Figure 2
Figure 2. Figure 2: Allowed regions for the lepton input parameters for case C (−4,2,−1) (0,0,3) − S (−2,−4). Different color shadings correspond to the 1σ, 2σ, and 3σ confidence levels. high-precision measurements of θ23 and δCP could help distinguish between the three lepton fla￾vor models. Furthermore, the allowed ranges of sin2 θ13 and sin2 θ12 span almost the entire 3σ intervals for the models C (−4,2,−1) (0,0,3) − S (−2… view at source ↗
Figure 3
Figure 3. Figure 3: Allowed regions for the lepton observables for case C (−4,2,−1) (0,0,3) − S (−2,−4). Different color shadings correspond to the 1σ, 2σ, and 3σ confidence levels. is provided by the KamLAND-Zen experiment with mββ < (28 − 122) meV at 90% C.L. [78]. Future tonne-scale 0νββ-decay experiments such as LEGEND-1000 [79] and nEXO [80] are ex￾pected to reach sensitivities of mββ ∼ (9 − 21) meV and mββ ∼ (4.7 − 20.3… view at source ↗
Figure 4
Figure 4. Figure 4: Matrix of the correlations among the model parameters and lepton observables for the lepton model C (−4,2,−1) (0,0,3) − S (−2,−4) . In our setup, the light neutrino masses are generated through the minimal type-I seesaw mechanism involving two RH neutrinos. Remarkably, both the Dirac neutrino mass matrix MD and the heavy Majorana mass matrix MN are uniquely fixed by the modulus τ , up to the overall normal… view at source ↗
Figure 5
Figure 5. Figure 5: Allowed regions for the lepton input parameters for case C (−4,−3,−1) (0,1,3) − S (−2,4). Different color shadings correspond to the 1σ, 2σ, and 3σ confidence levels. depends on τ up to the overall scale Λ, as shown in Eq. (3.8). Hence the mass ratio M2/M1 of the two RH neutrinos is entirely determined by the complex modulus τ . When evaluated at the corresponding best-fit values of τ , this ratio is consi… view at source ↗
Figure 6
Figure 6. Figure 6: Allowed regions for the lepton observables for case C (−4,−3,−1) (0,1,3) − S (−2,4). Different color shadings correspond to the 1σ, 2σ, and 3σ confidence levels. metry Y∆ and the abundance YN1 of the lightest RH neutrino [92] dYN1 dz = K zf1(z)K1(z) K2(z)  Y eq N1 − YN1  , dY∆ dz = K zK1(z) K2(z)  ε1f1(z)  Y eq N1 − YN1  − f2(z)Y eq N1 Y∆ Y eq ℓ  , (5.4) where the dimensionless variable z = M1/T is i… view at source ↗
Figure 7
Figure 7. Figure 7: Matrix of the correlations among the input parameters and lepton observables for the lepton model C (−4,−3,−1) (0,1,3) − S (−2,4) . where m∗ SM ≃ 1.08 × 10−3 eV in the SM. The effective neutrino mass me 1, which depends on the total decay rate of N1, is given by me 1 ≡ (λλ† )11v 2 M1 , (5.7) where the parameter v = 174 GeV represents the VEV of the Higgs field. It is straightforward to verify that the wash… view at source ↗
Figure 8
Figure 8. Figure 8: Allowed regions for the lepton input parameters for case C (−4,0,1) (0,0,1) − S (−2,4). Different color shadings correspond to the 1σ, 2σ, and 3σ confidence levels. 5.1 Numerical analysis The 36 phenomenologically viable models listed in tables 3 and 4 naturally fall into three distinct categories, each comprising twelve models that have the same set of neutrino mass matrices and yield nearly identical pre… view at source ↗
Figure 9
Figure 9. Figure 9: Allowed regions for the lepton observables for case C (−4,0,1) (0,0,1) −S (−2,4). Different color shadings correspond to the 1σ, 2σ, and 3σ confidence levels. Consequently, the Dirac Yukawa coupling matrix entering Eq. (5.3) is given by λ = M′ D/v. For the three benchmark models, all input parameters are fixed at their best-fit values given in table 3, ensuring consistency with current neutrino oscillation… view at source ↗
Figure 10
Figure 10. Figure 10: Matrix of the correlations among the input parameters and lepton observables for the lepton model C (−4,0,1) (0,0,1) − S (−2,4) . 1012 1013 1014 1015 10-13 10-12 10-11 10-10 10-9 10-8 10-7 [PITH_FULL_IMAGE:figures/full_fig_p026_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Correlation between the baryon asymmetry YB and the RH neutrino mass M1 for the representative models C (−4,2,−1) (0,0,3) − S (−2,−4) and C (−4,0,1) (0,0,1) − S (−2,4). The black horizontal line represents the observed baryon asymmetry YB = 8.703 × 10−11 . 26 [PITH_FULL_IMAGE:figures/full_fig_p026_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: The evolution of YB with M1/T for the representative models C (−4,2,−1) (0,0,3) − S (−2,−4) and C (−4,0,1) (0,0,1) −S (−2,4) denoted by blue solid and red dashed lines respectively, the left and right panels are for vanishing and thermal initial RH neutrino abundances respectively. The black horizontal line represents the observed baryon asymmetry YB = 8.703 × 10−11 . in agreement with observational data.… view at source ↗
read the original abstract

We perform a comprehensive and systematic investigation of lepton models based on the non-holomorphic $S^{\prime}_{4}$ modular symmetry, by using level 4 polyharmonic Maa{\ss} forms spanning integer weights from $-4$ to $6$. The light neutrino masses are generated by the type-I seesaw mechanism with two right-handed neutrinos, no flavon fields other than the modulus $\tau$ is introduced, and the generalized CP symmetry is not imposed. An exhaustive numerical analysis yields 36 viable models with only four real couplings besides the modulus $\tau$ when neutrino masses are normal ordering. They are classified into three categories, each containing twelve models which yield quite similar predictions for lepton observables and are distinguished by the assignment of $E^c_1$. Furthermore, we perform a detailed numerical analysis for one representative model from each category. These representative models are found to yield very sharp predictions for neutrino masses and mixing parameters, and they are distinguished by the predictions for the atmospheric mixing angle $\theta_{23}$, the Dirac CP phase $\delta_{CP}$ and the Majorana CP phase $\alpha_{21}$. Furthermore, we find that only two of these three representative models accommodate successful thermal leptogenesis in the unflavored regime, reproducing the observed baryon asymmetry with the identical parameter values that satisfy neutrino oscillation data. In these models, the real part of the modulus $\tau$ is the unique source of CP violation in both lepton mixing and leptogenesis.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper performs a systematic study of lepton flavor models based on non-holomorphic S'_4 modular symmetry, employing level-4 polyharmonic Maass forms with integer weights from -4 to 6. Light neutrino masses arise via the type-I seesaw with exactly two right-handed neutrinos; no additional flavon fields or generalized CP symmetry are introduced. An exhaustive numerical scan over four real couplings plus the modulus τ identifies 36 viable models for normal neutrino mass ordering. These are grouped into three categories of twelve models each, distinguished by the assignment of E^c_1. Three representative models are examined in detail, producing sharp predictions for the neutrino mass-squared differences, mixing angles, and CP phases. Two of the three representatives also reproduce the observed baryon asymmetry via thermal leptogenesis in the unflavored regime, using the identical parameter values that fit oscillation data, with Re(τ) identified as the sole source of CP violation.

Significance. If the numerical results prove robust, the work supplies a concrete, flavon-free realization of modular symmetry for leptons that simultaneously addresses neutrino data and leptogenesis. The use of Maass forms across a range of weights, the classification into categories with similar predictions, and the explicit demonstration that the same fitted parameters can yield successful unflavored leptogenesis constitute a useful addition to the modular-flavor literature. The claim that Re(τ) is the unique CP source is a falsifiable prediction that can be tested by future precision measurements of δ_CP and the Majorana phases.

major comments (2)
  1. [Numerical analysis section (abstract and results section)] The central claim of exactly 36 viable models (abstract; numerical-analysis section) rests on an exhaustive scan whose sampling method, scanned ranges for the four real couplings and for Re(τ)/Im(τ) inside the fundamental domain, grid density or Monte-Carlo statistics, and precise χ² viability threshold are not specified. Without these details the enumerated count, the classification into three equal categories of twelve, and the subsequent selection of the three representative models for leptogenesis cannot be independently verified and may shift under alternative sampling choices.
  2. [Leptogenesis section] The assertion that only two of the three representative models accommodate successful unflavored leptogenesis (abstract; leptogenesis section) is obtained after fitting the same four couplings plus τ to oscillation data. Because the scan details are missing, it is unclear whether the parameter space has been exhaustively explored or whether local minima or degenerate solutions have been missed; this directly affects the robustness of the statement that Re(τ) is the unique CP source for both mixing and leptogenesis.
minor comments (2)
  1. [Abstract] The abstract states that the models use “only four real couplings besides the modulus τ”; it would improve clarity to give their explicit Lagrangian notation or coupling labels at first mention.
  2. [Model-construction section] Notation for the Maass forms (weights, transformation properties under S'_4) should be collected in a single table or subsection for easy reference when comparing the three categories.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed review and the positive evaluation of our work's significance. We address the two major comments point by point below. We agree that additional details on the numerical analysis are needed and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Numerical analysis section (abstract and results section)] The central claim of exactly 36 viable models (abstract; numerical-analysis section) rests on an exhaustive scan whose sampling method, scanned ranges for the four real couplings and for Re(τ)/Im(τ) inside the fundamental domain, grid density or Monte-Carlo statistics, and precise χ² viability threshold are not specified. Without these details the enumerated count, the classification into three equal categories of twelve, and the subsequent selection of the three representative models for leptogenesis cannot be independently verified and may shift under alternative sampling choices.

    Authors: We agree that the specific details of the numerical scan are not provided in the current manuscript, which is necessary for full reproducibility and verification of the 36 models. We will revise the paper to include these details in the numerical analysis section, specifying the sampling method, the ranges scanned for the couplings and τ, the grid density or Monte Carlo statistics used, and the exact χ² threshold for viability. revision: yes

  2. Referee: [Leptogenesis section] The assertion that only two of the three representative models accommodate successful unflavored leptogenesis (abstract; leptogenesis section) is obtained after fitting the same four couplings plus τ to oscillation data. Because the scan details are missing, it is unclear whether the parameter space has been exhaustively explored or whether local minima or degenerate solutions have been missed; this directly affects the robustness of the statement that Re(τ) is the unique CP source for both mixing and leptogenesis.

    Authors: We acknowledge that the robustness of the leptogenesis results and the claim regarding Re(τ) as the unique CP source depend on the completeness of the scan. By incorporating the detailed description of the numerical procedure in the revision (as noted in the response to the first comment), we will clarify that the scan was exhaustive within the specified method, supporting the identification of the two successful models. revision: yes

Circularity Check

0 steps flagged

No significant circularity: numerical scan fits parameters to neutrino data then independently checks leptogenesis consistency

full rationale

The paper constructs models via modular form assignments under non-holomorphic S'_4, then performs numerical fitting of four real couplings plus τ to oscillation data under type-I seesaw. It reports 36 viable models for normal ordering and checks that two representative models also reproduce the observed baryon asymmetry using exactly those fitted values. This is a standard consistency test against an external observable (baryon asymmetry), not a reduction by construction. No self-definitional equations, fitted inputs renamed as predictions, or load-bearing self-citations appear in the abstract or described chain. The count of 36 models and viability thresholds are reproducibility issues, not circularity. The derivation remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

2 free parameters · 3 axioms · 0 invented entities

The central claims rest on standard domain assumptions of modular symmetry and seesaw mechanism plus four fitted real couplings and the modulus τ; no new entities are postulated.

free parameters (2)
  • four real couplings
    Explicitly stated as the only free parameters besides τ that are fitted to neutrino data.
  • modulus τ
    The modulus is scanned and fitted to reproduce oscillation data and enable leptogenesis.
axioms (3)
  • domain assumption Type-I seesaw mechanism with two right-handed neutrinos generates the light neutrino masses
    Stated directly in the abstract as the mechanism employed.
  • domain assumption No flavon fields other than the modulus τ are introduced
    Explicitly stated in the abstract.
  • domain assumption Generalized CP symmetry is not imposed
    Explicitly stated in the abstract.

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discussion (0)

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Reference graph

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