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arxiv: 2604.06384 · v1 · submitted 2026-04-07 · ✦ hep-ph

An A4 model to accommodate maximal theta23 and maximal delta consistent with mu-tau reflection symmetry

Rupak Chakrabarty , Chandan Duarah This is my paper

Pith reviewed 2026-05-10 18:37 UTC · model grok-4.3

classification ✦ hep-ph
keywords A4 flavor symmetrymu-tau reflection symmetryneutrino mass matrixtype-I seesawDirac CP phaseatmospheric mixing anglediscrete symmetry model
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The pith

An A4 symmetry model with mu-tau reflection symmetry predicts maximal theta23 and maximal delta in its CP-symmetric limit.

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

The paper constructs a flavor model using the discrete group A4 x Z2 x Z4 together with the type-I seesaw to produce a light neutrino mass matrix that respects mu-tau reflection symmetry. In the generalized CP symmetry limit this texture forces the atmospheric mixing angle to exactly 45 degrees and the Dirac CP phase to 90 or 270 degrees. The construction still allows a non-zero reactor angle and is shown to be compatible with existing global oscillation data. When the strict CP limit is relaxed, two free parameters control small deviations of theta23 and delta from maximality while preserving the correct solar and reactor angles.

Core claim

In the generalized CP symmetry limit of the A4 x Z2 x Z4 model the light Majorana neutrino mass matrix obeys mu-tau reflection symmetry, which directly enforces theta23 = pi/4 and delta = pi/2 or 3pi/2 while permitting a non-zero theta13.

What carries the argument

The mu-tau reflection symmetric texture of the light Majorana neutrino mass matrix, realized through the A4 x Z2 x Z4 discrete symmetry in the type-I seesaw framework.

If this is right

  • Outside the strict CP limit the mixing angles and Dirac phase are controlled by two free parameters.
  • The model reproduces the observed values of theta12 and theta13 while allowing the small deviations of theta23 and delta preferred by global fits.
  • Numerical analysis confirms that viable parameter regions exist that match current three-neutrino oscillation data.

Where Pith is reading between the lines

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

  • Future precision data on theta23 and delta can test whether the model sits exactly at the CP-symmetric point or requires the two-parameter extension.
  • The same discrete symmetry structure could be used to constrain additional observables such as the effective Majorana mass in neutrinoless double-beta decay.
  • Embedding the A4 x Z2 x Z4 group into a larger unified theory might relate the neutrino predictions to charged-lepton or quark flavor observables.

Load-bearing premise

The light neutrino mass matrix is assumed to respect mu-tau reflection symmetry in the generalized CP symmetry limit of the A4 x Z2 x Z4 model.

What would settle it

A precise measurement showing theta23 far from 45 degrees together with delta far from 90 or 270 degrees would rule out the model's CP-symmetric limit.

Figures

Figures reproduced from arXiv: 2604.06384 by Chandan Duarah, Rupak Chakrabarty.

Figure 1
Figure 1. Figure 1: Correlation between sin2 θ13 and the parameter θ. The horizontal shaded band denotes the current 3σ allowed experimental range of sin2 θ13, while the red dashed line indicates the best-fit value. 12 [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Correlation between sin2 θ12 and the model parameter θ. The horizontal shaded band denotes the current 3σ allowed experimental range of sin2 θ12, while the red dashed line indicates the best-fit value. Figs. 1 and 2 illustrate the correlations of sin2 θ13 and sin2 θ12 with the model parameter θ, respectively. These analyses follow from the analytical relations given in Eqs. (41) and (42), where the phase p… view at source ↗
Figure 3
Figure 3. Figure 3: Correlation between sin2 θ13 and the model parameter ψ. The horizontal shaded band denotes the current 3σ allowed experimental range of sin2 θ13, while the red dashed line indicates the best-fit value. 13 [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Correlation between sin2 θ12 and the model parameter ψ. The horizontal shaded band denotes the current 3σ allowed experimental range of sin2 θ12, while the red dashed line indicates the best-fit value. Figs. 3 and 4 display the correlations of sin2 θ13 and sin2 θ12 with the phase parameter ψ, respectively. These correlation plots are obtained using the analytical expressions presented in Eqs. (41) and (42)… view at source ↗
Figure 5
Figure 5. Figure 5: In this case, the range 34.1 ◦ ≤ θ ≤ 55.9 ◦ , as ψ of the model parameter θ yields sin2 θ23 in the second octant, whereas 124.2 ◦ ≤ θ ≤ 145.9 ◦ , as ψ corresponds to the first octant. The color gradient along the vertical direction indicates that the deviation of sin2 θ23 from its maximal value (sin2 θ23 = 0.5) increases with decreasing ψ within the interval 338.3 ◦ ≤ ψ ≤ 360◦ [PITH_FULL_IMAGE:figures/ful… view at source ↗
Figure 6
Figure 6. Figure 6: Correlation between sin2 θ23 and θ for 158.2 ◦ ≤ ψ ≤ 201.7 ◦ . The horizontal red dashed line represents the maximal value of sin2 θ23. The color gradient along the vertical direction indicates that the deviation of sin2 θ23 from its maximal value (sin2 θ23 = 0.5) increases with increasing ψ. 15 [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Correlation between sin2 θ23 and θ for 338.3 ◦ ≤ ψ ≤ 360◦ . The horizontal red dashed line represents the maximal value of sin2 θ23. The color gradient along the vertical direction indicates that the deviation of sin2 θ23 from its maximal value (sin2 θ23 = 0.5) increases with increasing ψ [PITH_FULL_IMAGE:figures/full_fig_p017_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Correlation between δ and θ for 0.0 ◦ ≤ ψ ≤ 21.8 ◦ . The black and red solid lines denote the maximal values of δ, namely 270◦ and 90◦ , respectively, while the blue and yellow lines indicate the best-fit values in the IO without and with SK data, respectively. The colour gradient along the vertical axis shows increasing ψ. 16 [PITH_FULL_IMAGE:figures/full_fig_p017_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Correlation between δ and θ for 158.2 ◦ ≤ ψ ≤ 201.7 ◦ . The black and red solid lines denote the maximal values of δ, namely 270◦ and 90◦ , respectively, while the blue and yellow lines indicate the best-fit values in the IO without and with SK data, respectively. The colour gradient along the vertical axis shows increasing ψ [PITH_FULL_IMAGE:figures/full_fig_p018_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Correlation between δ and θ for 338.3 ◦ ≤ ψ ≤ 360◦ . The black and red solid lines denote the maximal values of δ, namely 270◦ and 90◦ , respectively, while the blue and yellow lines indicate the best-fit values in the IO without and with SK data, respectively. The colour gradient along the vertical axis shows increasing ψ. In a similar way, we study the prediction of δ as a function of θ using Eq. (44). … view at source ↗
Figure 8
Figure 8. Figure 8: From this figure, we observe that the model prediction of [PITH_FULL_IMAGE:figures/full_fig_p018_8.png] view at source ↗
Figure 11
Figure 11. Figure 11: Correlation between the mass eigenvalue |m2| and the neutrino mass matrix element |(Mν)11| for the normal mass ordering (NO). δ < 270◦ is given by 124.2 ◦ ≤ θ ≤ 145.9 ◦ and 158.2 ◦ ≤ ψ ≤ 180◦ . Within these suitable parameter ranges, we also look for specific values of the model parameters θ and ψ that yield mixing angles and the Dirac CP phase in best agreement with the global data. As an example, we cho… view at source ↗
Figure 12
Figure 12. Figure 12: Correlation between the mass eigenvalue |m2| and λ1 for the normal mass ordering (NO). scenario including SK data. The corresponding prediction for the atmospheric mixing angle is sin2 θ23 ≈ 0.494, which lies in the first octant and is reasonably close to its best-fit value 0.470. Furthermore, the predicted Dirac CP phase is δ ≈ 226◦ , which is fairly close to the global best-fit value 212◦ . For the rang… view at source ↗
Figure 13
Figure 13. Figure 13: Correlation between the mass eigenvalue |m2| and λ2 for the normal mass ordering (NO) [PITH_FULL_IMAGE:figures/full_fig_p022_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Correlation of the mass eigenvalue |m2| with the neutrino mass matrix element |(Mν)11| in the inverted ordering (IO) case [PITH_FULL_IMAGE:figures/full_fig_p023_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Correlation between the mass eigenvalue |m2| and λ1 for the inverted mass ordering (IO) [PITH_FULL_IMAGE:figures/full_fig_p024_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Correlation between the mass eigenvalue |m2| and λ2 for the inverted mass ordering (IO). 23 [PITH_FULL_IMAGE:figures/full_fig_p024_16.png] view at source ↗
read the original abstract

In this work, we construct an A4-based flavor symmetry model within the framework of the type-I seesaw mechanism to realize a light neutrino mass matrix consistent with mu-tau reflection symmetry. The entire framework is based on the Standard Model gauge symmetry extended by the discrete group A4 x Z2 x Z4. In general, the elements of the light Majorana neutrino mass matrix are complex. The mu-tau reflection symmetric texture of the mass matrix can be realized in a generalized CP symmetry limit. In this symmetry limit, the model predicts a maximal atmospheric mixing angle theta23 = pi/4 and a maximal Dirac CP phase delta = pi/2 or 3pi/2. These features are consistent with current experimental observations, including a near-maximal value of theta23, a non-zero reactor angle, and a preference for delta close to 270 degrees, as indicated by the T2K and NOvA experiments. Non-maximal values of theta23 and delta can be accommodated when one does not restrict to the CP symmetry limit. The model predictions for the mixing angles and the Dirac CP phase delta are then controlled by two parameters. We perform a numerical analysis to identify the allowed values of the model parameters consistent with current global three-neutrino oscillation data. The model successfully reproduces the desired deviations of theta23 and delta from their maximal values, consistent with global fit data, while simultaneously accommodating the observed values of theta12 and theta13.

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

1 major / 1 minor

Summary. The manuscript constructs an A4 × Z₂ × Z₄ flavor symmetry model in the type-I seesaw framework that realizes a light neutrino mass matrix with exact μ-τ reflection symmetry (M_eμ = M_eτ*, M_μμ = M_ττ*, M_μτ real) in a generalized CP limit. In this limit the model predicts maximal θ₂₃ = π/4 and maximal δ = ±π/2. Deviations from maximality are controlled by two additional parameters; a numerical scan is performed to identify values consistent with global three-neutrino oscillation data while keeping θ₁₂ and θ₁₃ inside experimental ranges.

Significance. If the explicit group assignments, flavon VEVs, and generalized CP transformations are correctly implemented, the work supplies a concrete discrete-symmetry realization of μ-τ reflection symmetry whose maximal predictions are parameter-free in the symmetry limit. The two-parameter breaking then provides a controlled way to accommodate the observed near-maximal θ₂₃ and δ ≈ 3π/2 preference reported by T2K and NOvA. This is a standard and useful approach in flavor model building.

major comments (1)
  1. [Abstract and numerical analysis] Abstract and numerical analysis: the manuscript asserts that the numerical scan 'successfully reproduces the desired deviations of θ₂₃ and δ … consistent with global fit data,' yet reports neither the minimum χ² value, the best-fit ranges of the two deviation parameters, nor any comparison with the standard three-neutrino parametrization. Because the central claim includes successful accommodation of the data, these quantitative results are load-bearing and must be supplied.
minor comments (1)
  1. [Abstract] The abstract and introduction should explicitly state that the maximal predictions hold only in the generalized CP limit and that the two parameters are introduced to break this limit.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment of our work and the recommendation for minor revision. We address the single major comment below by agreeing to supply the requested quantitative details.

read point-by-point responses

  1. Referee: [Abstract and numerical analysis] Abstract and numerical analysis: the manuscript asserts that the numerical scan 'successfully reproduces the desired deviations of θ₂₃ and δ … consistent with global fit data,' yet reports neither the minimum χ² value, the best-fit ranges of the two deviation parameters, nor any comparison with the standard three-neutrino parametrization. Because the central claim includes successful accommodation of the data, these quantitative results are load-bearing and must be supplied.

    Authors: We agree that the central claim of successful data accommodation requires explicit quantitative support. In the revised manuscript we will add to the numerical analysis section (and reference from the abstract) the minimum χ² value obtained in the scan, the best-fit ranges (or allowed intervals at 1σ/3σ) for the two deviation parameters, and a direct comparison of the resulting mixing angles and δ with the standard three-neutrino global-fit values. These additions will make the numerical results fully transparent and load-bearing. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper constructs specific A4 x Z2 x Z4 field assignments, flavon VEVs, and generalized CP transformations so the type-I seesaw yields the mu-tau reflection texture exactly in the CP limit. Maximal theta23 = pi/4 and delta = +/- pi/2 then follow by ordinary matrix diagonalization of that texture, a standard mathematical result independent of the model. Deviations are introduced by two explicit breaking parameters whose values are scanned numerically against oscillation data; the paper describes these as accommodations controlled by parameters rather than parameter-free predictions. No load-bearing self-citation, self-definition, or renaming of known results occurs. The derivation chain is self-contained and does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on the assumption of discrete symmetries A4 x Z2 x Z4 together with mu-tau reflection symmetry and the type-I seesaw mechanism, plus two free parameters adjusted to match oscillation data.

free parameters (1)
  • two parameters controlling deviations
    Control the deviations of theta23 and delta from their maximal values and are fitted numerically to global three-neutrino oscillation data.
axioms (2)
  • domain assumption mu-tau reflection symmetry on the light neutrino mass matrix
    Imposed to produce the texture that yields maximal mixing in the CP limit.
  • domain assumption A4 x Z2 x Z4 flavor symmetry
    Discrete symmetry group extended to the Standard Model to constrain the Yukawa couplings and mass matrices.

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