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arxiv: 2604.16130 · v1 · submitted 2026-04-17 · ✦ hep-ph

Recognition: unknown

Lepton masses and mixing in non-holomorphic modular A₄ with universal couplings

Mohammed Abbas
Authors on Pith no claims yet

Pith reviewed 2026-05-10 08:35 UTC · model grok-4.3

classification ✦ hep-ph
keywords modularneutrinochargedcouplingsleptonhandedmassnon-holomorphic
0
0 comments X

The pith

A modular A4 flavor model with universal couplings reproduces charged lepton masses via the modulus tau and predicts correlated neutrino observables for normal mass ordering and right-handed weight k_N = -1.

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

The authors build a model of lepton masses and mixing using a symmetry called modular A4 that is non-holomorphic, meaning it allows certain functions of the modulus field tau. They assume all couplings have the same strength but can differ in phase. For charged leptons, the observed mass hierarchy emerges automatically once tau sits near special fixed points in the complex plane and the right-handed leptons are given particular modular weights. For neutrinos, the same universal-coupling assumption plus a scan over phases and weights produces solutions only when the three neutrino masses follow normal ordering and the right-handed neutrinos carry weight -1. The resulting parameter space shows tight links between the three mixing angles, the effective mass for neutrinoless double-beta decay, and the total neutrino mass sum. These links are presented as testable predictions for upcoming oscillation, cosmology, and beta-decay experiments.

Core claim

The experimental charged lepton masses are reproduced with high precision for values of tau located near modular fixed points. Viable solutions arise only for normal neutrino mass ordering and a unique right handed neutrino modular weight, k_N=-1. The model yields strong correlations among mixing angles, the effective neutrinoless double beta decay parameter m_ee, and the total neutrino mass sum m_i.

Load-bearing premise

The assumption of universal couplings (equal magnitudes, different phases only) together with the specific modular weight assignments for right-handed charged leptons and neutrinos that are chosen to make the tau-driven hierarchy and the neutrino scan work.

Figures

Figures reproduced from arXiv: 2604.16130 by Mohammed Abbas.

Figure 1
Figure 1. Figure 1: FIG. 1: The allowed values of the phases [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Correlation between the ratio [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Allowed 3 [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Correlation between [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: The allowed values of the phases [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Correlation between ratio [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: Accepted points for NO with [PITH_FULL_IMAGE:figures/full_fig_p017_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: Correlation between [PITH_FULL_IMAGE:figures/full_fig_p018_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: Correlation between the phases [PITH_FULL_IMAGE:figures/full_fig_p019_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10: Correlation between the phases [PITH_FULL_IMAGE:figures/full_fig_p019_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11: Correlation between ratio [PITH_FULL_IMAGE:figures/full_fig_p020_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12 [PITH_FULL_IMAGE:figures/full_fig_p020_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13 [PITH_FULL_IMAGE:figures/full_fig_p021_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14: Correlation between [PITH_FULL_IMAGE:figures/full_fig_p021_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15: Correlation between [PITH_FULL_IMAGE:figures/full_fig_p022_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16: Correlation between [PITH_FULL_IMAGE:figures/full_fig_p022_16.png] view at source ↗
read the original abstract

We propose a non-holomorphic modular $A_4$ model under the assumption of universal couplings. In this framework, a charged lepton mass hierarchy is not created through parameter fine tuning or hierarchical Yukawa couplings, but instead is determined by the modulus $\tau$, with certain modular weight assignments of right handed charged leptons. The experimental charged lepton masses are reproduced with high precision for values of $\tau$ located near modular fixed points. In neutrino sector the couplings are imposed to be equal in magnitude with different relative phases. By fixing the modulus $\tau$ from charged lepton sector, we perform a comprehensive scan over the phase parameters and modular weight assignments of the right handed neutrinos. We find that viable solutions arise only for normal neutrino mass ordering and a unique right handed neutrino modular weight, $k_N=-1$. The model yields strong correlations among mixing angles, the effective neutrinoless double beta decay parameter $m_{ee}$, and the total neutrino mass $\sum m_i$. These results underscore the predictive quality of non-holomorphic modular symmetry with minimal parameter inputs and offer implications for neutrino experiments and cosmological observations that can be tested.

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

3 major / 2 minor

Summary. The paper proposes a non-holomorphic modular A4 model with universal couplings (equal magnitudes, phases only) for lepton Yukawa interactions. Charged-lepton masses are generated hierarchically via the modulus τ near fixed points with chosen modular weights for the right-handed fields, reproducing experimental values to high precision. Fixing τ from this sector, a scan over relative phases and right-handed neutrino modular weights yields viable neutrino solutions exclusively for normal ordering and k_N = -1, producing correlations among the PMNS angles, m_ee, and Σm_i.

Significance. If the numerical scan results hold, the construction demonstrates that non-holomorphic modular symmetry plus a single universal-coupling assumption can yield a predictive lepton flavor model with very few free parameters. The reported correlations would constitute falsifiable predictions for oscillation experiments, 0νββ searches, and cosmological neutrino-mass bounds, strengthening the case for modular symmetry approaches in flavor model building.

major comments (3)
  1. [§3] §3 (charged-lepton sector): the assertion of 'high-precision' reproduction of the three charged-lepton masses for τ near fixed points is not accompanied by explicit numerical values of τ, the resulting mass eigenvalues, or the fractional deviations from experiment; without these, the precision claim cannot be verified.
  2. [§4] §4 (neutrino scan): the statement that viable solutions exist 'only' for normal ordering and k_N = -1 requires the scan range, number of sampled points, and precise viability criteria (e.g., 3σ intervals on Δm²_{21}, Δm²_{31}, and mixing angles) to be reported; absent this information the exclusivity of the reported solutions cannot be assessed.
  3. [§5] §5 (correlations): the reported correlations among mixing angles, m_ee, and Σm_i are derived from the same numerical scan that selects viable points after fixing τ from the charged-lepton fit; the paper should clarify whether these correlations survive when the neutrino data are varied within their experimental uncertainties or whether they are artifacts of the fitting procedure.
minor comments (2)
  1. [Abstract] The abstract and introduction should explicitly list the modular weights assigned to the right-handed charged leptons, as these are central to the τ-driven hierarchy.
  2. [§2] Notation for the universal coupling magnitude and the relative phases should be defined once in a dedicated subsection rather than introduced piecemeal.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments, which help improve the clarity and verifiability of our results. We address each major comment point by point below and will revise the manuscript to incorporate additional details where appropriate.

read point-by-point responses

  1. Referee: [§3] §3 (charged-lepton sector): the assertion of 'high-precision' reproduction of the three charged-lepton masses for τ near fixed points is not accompanied by explicit numerical values of τ, the resulting mass eigenvalues, or the fractional deviations from experiment; without these, the precision claim cannot be verified.

    Authors: We agree that explicit numerical values are necessary to substantiate the high-precision claim. In the revised manuscript, we will add a table in Section 3 listing the specific τ values near the fixed points used, the resulting charged-lepton mass eigenvalues, and the fractional deviations from experimental values. This addition will allow direct verification without changing the underlying model or results. revision: yes

  2. Referee: [§4] §4 (neutrino scan): the statement that viable solutions exist 'only' for normal ordering and k_N = -1 requires the scan range, number of sampled points, and precise viability criteria (e.g., 3σ intervals on Δm²_{21}, Δm²_{31}, and mixing angles) to be reported; absent this information the exclusivity of the reported solutions cannot be assessed.

    Authors: We concur that reporting the scan details is essential for assessing the robustness and exclusivity of the solutions. In the revised Section 4, we will specify the scanned ranges for the relative phases and right-handed neutrino modular weights, the total number of sampled points, and the precise viability criteria based on the 3σ experimental intervals for the neutrino oscillation parameters. This will clarify the basis for finding viable solutions exclusively in normal ordering with k_N = -1. revision: yes

  3. Referee: [§5] §5 (correlations): the reported correlations among mixing angles, m_ee, and Σm_i are derived from the same numerical scan that selects viable points after fixing τ from the charged-lepton fit; the paper should clarify whether these correlations survive when the neutrino data are varied within their experimental uncertainties or whether they are artifacts of the fitting procedure.

    Authors: The correlations arise from the model's structure with universal couplings and τ fixed by the charged-lepton sector; they are exhibited by points that satisfy the neutrino data within the current 3σ uncertainties and are not fitting artifacts. In the revised Section 5, we will add explicit clarification that the scan selects points consistent with experimental ranges and that the correlations are robust within those bounds, while noting that future shifts in data uncertainties could affect the viable region but preserve the model's predictive correlations for testing. revision: yes

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The construction rests on the assumption of universal couplings, specific modular weight choices for right-handed fields, and the existence of a modulus tau whose value near fixed points generates the observed hierarchies; no independent evidence for these choices is supplied beyond the fit itself.

free parameters (2)
  • modulus tau
    Chosen near modular fixed points to reproduce charged-lepton masses; its precise value is fitted rather than derived from first principles.
  • relative phases of universal couplings
    Scanned to obtain viable neutrino solutions after tau is fixed from the charged-lepton sector.
axioms (2)
  • domain assumption Non-holomorphic modular A4 symmetry governs the lepton superpotential
    Invoked to organize the flavor structure and allow tau-dependent Yukawa couplings.
  • ad hoc to paper All couplings have equal magnitude (universal couplings)
    Imposed to avoid hierarchical Yukawas and to reduce the number of free parameters.

pith-pipeline@v0.9.0 · 5496 in / 1567 out tokens · 27804 ms · 2026-05-10T08:35:38.026208+00:00 · methodology

discussion (0)

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Reference graph

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