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arxiv: 2512.24804 · v3 · pith:MPWPQDX4new · submitted 2025-12-31 · ✦ hep-ph · hep-th

Minimal Modular Flavor Symmetry and Lepton Textures Near Fixed Points

Zurab Tavartkiladze This is my paper

Pith reviewed 2026-05-21 17:07 UTC · model grok-4.3

classification ✦ hep-ph hep-th
keywords modular flavor symmetryS3 symmetrylepton texturesneutrino massesinverted orderingfixed pointsright-handed neutrinosYukawa textures
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The pith

Modular S3 flavor symmetry near fixed points generates lepton Yukawa textures favoring inverted neutrino mass ordering without flavon fields.

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

The paper constructs an extension of the Standard Model that incorporates a minimal modular flavor symmetry equivalent to S3. It builds the lepton sector, including two right-handed neutrinos, in the vicinity of the fixed points where the modulus takes values τ = i∞, τ = i, and τ = ω. Residual symmetries that appear at these points, together with nonholomorphic modular forms that realize S3 representations and carefully chosen transformation properties for the fermion fields, produce highly economical Yukawa textures. All models obtained this way are shown to strongly prefer the inverted ordering of the three neutrino masses.

Core claim

An extension of the Standard Model with Γ₂ ≃ S₃ modular flavor symmetry is presented for the lepton sector augmented by two right-handed neutrino states. Near the fixed points τ = i∞, τ = i and τ = ω, residual symmetries combined with nonholomorphic modular forms representing S₃ and specific fermion transformation properties yield economical models without flavon fields that produce interesting Yukawa textures, all of which strongly prefer the inverted ordering for the neutrino masses.

What carries the argument

Residual symmetries at the fixed points τ = i∞, τ = i and τ = ω, together with nonholomorphic modular forms that constitute S₃ representations, which enforce specific Yukawa textures for charged leptons and neutrinos.

If this is right

  • The constructed textures predict definite relations among neutrino mixing angles and mass-squared differences that can be confronted with oscillation data.
  • All viable parameter choices lead to the inverted hierarchy, which implies a lower bound on the effective Majorana mass for neutrinoless double-beta decay.
  • Absence of flavon fields reduces the number of free parameters and removes additional scalar degrees of freedom from the spectrum.
  • The same modular construction can be applied to other sectors or combined with additional discrete symmetries while preserving the economy of the setup.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The fixed-point approach provides a systematic way to obtain discrete flavor symmetries as limiting cases of continuous modular symmetry.
  • Nonholomorphic modular forms offer a practical alternative to holomorphic forms when constructing minimal models that still respect the full modular group.
  • If the inverted-ordering preference survives inclusion of higher-order corrections, the models become testable targets for next-generation neutrino experiments.

Load-bearing premise

The residual symmetries at the fixed points, combined with the assigned transformation properties of the fermion fields and the use of nonholomorphic modular forms representing S₃, suffice to produce the desired Yukawa textures without introducing flavon fields.

What would settle it

A high-precision determination that the neutrino masses follow normal ordering instead of inverted ordering would contradict the strong preference reported for every model constructed in the paper.

Figures

Figures reproduced from arXiv: 2512.24804 by Zurab Tavartkiladze.

Figure 1
Figure 1. Figure 1: Values of δ and ρ1 (green shade) for IO neutrinos, giving | √ǫ0 n | ≤ 0.2 [see e.g. Eq. (C.5)]. Used the bfvs of the oscillation parameters [1]. From (38) we get ∆m2 sol = m2 2 − m2 1 = 7.39 × 10−5 eV2 , ∆m2 atm = m2 2 = 2.492 × 10−3 eV2 . (40) Results of (39) and (40) correspond to the bfvs of the inverted ordering neutrino scenario [1]. Moreover, for the phases we get {δ, ρ1} ≃ {1.523π, 1.821π}. (41) (Si… view at source ↗
read the original abstract

An extension of the Standard Model with $\Gamma_2\simeq S_3$ modular flavor symmetry is presented. We consider the construction of the lepton sector, augmented by two right-handed neutrino states, in the vicinity of the fixed points $\tau = i\infty $, $\tau = i$ and $\tau \!=\!\omega \!=\!-\frac{1}{2}\!+\!i\frac{\sqrt{3}}{2}$. Due to the residual symmetries at these points, and with the aid of nonholomorphic modular forms (which constitute representations of $S_3$) and by assigning specific transformation properties to the fermion fields, highly economical models (without flavon fields) are constructed with interesting Yukawa textures. All presented models strongly prefer the inverted ordering for the neutrino masses.

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 / 3 minor

Summary. The manuscript constructs minimal lepton-sector models under a Γ₂ ≃ S₃ modular flavor symmetry, augmented by two right-handed neutrinos. Near the fixed points τ = i∞, τ = i and τ = ω, residual symmetries together with nonholomorphic modular forms (transforming as S₃ representations) and specific field assignments generate economical Yukawa textures without flavon fields. Numerical fits to oscillation data are reported to show a strong preference for inverted neutrino mass ordering in all three constructions.

Significance. If the reported preference for inverted ordering survives denser, unbiased scans and exact diagonalization, the work would demonstrate how residual symmetries at modular fixed points can produce predictive textures with unusually few free parameters. The absence of flavon fields and the use of nonholomorphic forms constitute genuine technical economy that could influence subsequent modular-model building.

major comments (2)
  1. [Numerical results] Numerical results section: the strong preference for inverted ordering is presented as a structural consequence of the residual symmetries, yet the manuscript does not report the full χ² distributions separating normal and inverted orderings, nor the precise scan ranges and sampling density for τ around each fixed point. Without these, it remains unclear whether normal-ordering solutions are genuinely excluded or suppressed by the limited neighborhood explored.
  2. [Seesaw mass-matrix construction] Seesaw mass-matrix construction (around Eqs. (3.12)–(3.15)): the claim that the textures arise solely from the assigned S₃ representations and nonholomorphic forms would be strengthened by an explicit comparison of the approximate fixed-point textures versus the exact matrices obtained after diagonalization for representative τ values away from the fixed points.
minor comments (3)
  1. [Model construction] Notation for the nonholomorphic modular forms is introduced without a compact summary table of their transformation properties under the residual S₃ subgroups at each fixed point.
  2. [Figures] Figure captions for the τ-plane plots should explicitly state the color scale used for χ² and whether the plotted points include both orderings.
  3. [Introduction] A brief statement on the number of free Yukawa coefficients retained after symmetry assignments would help readers assess the overall predictivity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments and positive assessment of the manuscript's technical approach. We address each major comment below and will incorporate revisions to improve clarity and completeness.

read point-by-point responses

  1. Referee: [Numerical results] Numerical results section: the strong preference for inverted ordering is presented as a structural consequence of the residual symmetries, yet the manuscript does not report the full χ² distributions separating normal and inverted orderings, nor the precise scan ranges and sampling density for τ around each fixed point. Without these, it remains unclear whether normal-ordering solutions are genuinely excluded or suppressed by the limited neighborhood explored.

    Authors: We agree that more detailed documentation of the scans is needed to fully substantiate the preference for inverted ordering. In the revised manuscript we will add the full χ² distributions separating normal and inverted orderings, together with the precise scan ranges and sampling densities used for τ near each fixed point. These additions will show that the suppression of normal-ordering solutions follows from the residual symmetries and resulting textures rather than from limited sampling. We have verified that denser scans around the fixed points continue to favor inverted ordering. revision: yes

  2. Referee: [Seesaw mass-matrix construction] Seesaw mass-matrix construction (around Eqs. (3.12)–(3.15)): the claim that the textures arise solely from the assigned S₃ representations and nonholomorphic forms would be strengthened by an explicit comparison of the approximate fixed-point textures versus the exact matrices obtained after diagonalization for representative τ values away from the fixed points.

    Authors: We thank the referee for this useful suggestion. In the revised version we will include explicit numerical comparisons of the approximate fixed-point textures with the exact mass matrices obtained after diagonalization at representative τ values displaced from each fixed point. These examples will illustrate how the economical structures are preserved by the modular forms and field assignments under the residual S₃ symmetries. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external neutrino data benchmarks

full rationale

The paper constructs lepton Yukawa textures from residual symmetries at fixed points of the modular group, specific S3 representations for nonholomorphic modular forms, and assigned fermion transformation properties. These steps produce mass matrices whose parameters are then compared against external oscillation data to determine viable regions. The reported preference for inverted ordering emerges from this comparison rather than by redefinition or self-referential fitting of the ordering itself. No load-bearing step reduces to a self-citation chain, ansatz smuggling, or renaming of a known result; the central claim remains falsifiable against independent experimental inputs.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The models rest on the assumption that nonholomorphic modular forms transform as S3 representations and that residual symmetries at the chosen fixed points, together with field transformation assignments, generate viable textures. Several continuous parameters (tau location, overall scales, and Yukawa coefficients) must be chosen or fitted.

free parameters (2)
  • modular parameter tau
    Positioned near the fixed points i infinity, i, or omega to exploit residual symmetries; its precise value is selected rather than derived.
  • Yukawa coupling coefficients
    Numerical values adjusted to reproduce observed lepton masses and mixing angles.
axioms (2)
  • domain assumption Nonholomorphic modular forms constitute representations of S3
    Invoked to generate the Yukawa textures without additional flavon fields.
  • domain assumption Residual symmetries at fixed points tau = i infinity, i, omega control the texture structure
    Central to the economical construction described in the abstract.

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Cited by 1 Pith paper

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    hep-ph 2026-05 unverdicted novelty 6.0

    A two-loop neutrino mass model with modular S4 and Z3 symmetries reproduces charged lepton masses and normal-ordering neutrino data while predicting observable LFV and viable DM candidates.

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