Fate of $θ_{12}$ under $μ-τ$ Reflection Symmetry in Light of the First JUNO Results

The recent JUNO measurements of $θ_{12}$ and $Δm^2_{21}$ open a new avenue for probing flavor symmetric structures in the lepton sector. Motivated by this, we study a model in which $μ-τ$ reflection symmetry naturally emerges from an underlying $A_4$ flavor symmetry within a type-II seesaw framework. Beyond its standard predictions of $θ_{23}=45^{\circ}$ and $δ_{\rm CP}=\pm π/2$, the framework yields testable predictions for $θ_{12}$ that can be probed by JUNO. Two viable scenarios arise, one predicting $\sin^2θ_{12} \gsim 0.335$, which is strongly disfavored by the latest JUNO results. Correlations between $θ_{12}$ and model parameters further enhance the model’s predictivity. Future measurements at DUNE and T2HK will provide complementary tests of this scenario.

💡 Research Summary

The paper investigates how the first results from the Jiangmen Underground Neutrino Observatory (JUNO) on the solar mixing angle θ₁₂ and the solar mass‑squared difference Δm²₂₁ constrain a class of lepton‑flavor models that combine μ‑τ reflection symmetry with an underlying A₄ discrete symmetry, realized within a type‑II seesaw framework. In the standard μ‑τ reflection scenario the atmospheric mixing angle is fixed to θ₂₃ = 45° and the Dirac CP phase to δ_CP = ±π/2, while θ₁₂ and θ₁₃ remain free. By embedding the model in A₄, the authors introduce two SU(2)_L triplet scalars Δ_i (singlets of A₄) and Δ′ (an A₄ triplet). The vacuum expectation values (VEVs) of these scalars are aligned as ⟨Δ_i⟩ = u_i and ⟨Δ′⟩ = u′√3(±1, ω, ω²) with ω³ = 1. This alignment yields a neutrino mass matrix that is exactly μ‑τ reflection symmetric:

M_ν = ⎛ A C C*
   C B D
   C* D B* ⎞,

where A and D are real, B and C are complex, and C and D are related, leaving only four independent real parameters (A, D, |B|, arg B). The sign choice of the Δ′ VEV leads to two phenomenologically distinct cases:

• Case‑I (D = +β u′√3)
• Case‑II (D = ‑β u′√3).

Both cases automatically predict θ₂₃ = 45° and δ_CP = ±π/2, independent of the remaining parameters. However, the A₄ structure imposes non‑trivial correlations among the solar parameters (θ₁₂, Δm²₂₁) and the model parameters. The authors perform a comprehensive numerical scan over A, |D|, |B| in the range 10⁻⁴–10⁻¹ and the phase of B from 0 to 2π, imposing the 3σ global‑fit constraints from the AHEP analysis (Δm²₂₁, Δm²₃₁, sin²θ₁₃, sin²θ₁₂). They then compare the resulting predictions with the JUNO measurement sin²θ₁₂ = 0.3092 ± 0.0087 (1σ) and Δm²₂₁ = (7.50 ± 0.12) × 10⁻⁵ eV².

The scan reveals that Case‑I accommodates the full experimentally allowed region. The ratios r₁ = |D/A|, r₂ = |D/B|, r₃ = |A/B| are constrained to roughly 1.3–2.4, 0.57–0.70, and 0.28–0.42 respectively, and the predicted values of sin²θ₁₂ span the entire 3σ interval from both the global fit and JUNO. Consequently, Case‑I remains a viable realization of μ‑τ reflection symmetry within the A₄ framework.

In contrast, Case‑II predicts a lower bound sin²θ₁₂ ≳ 0.335, arising from the specific sign of D. This bound lies well above the JUNO 3σ upper limit (≈0.327) and therefore excludes the majority of the parameter space. While Case‑II can still be compatible with the broader AHEP global fit at the 2σ level, the precision of JUNO’s first data set essentially rules it out. The authors illustrate these findings with scatter plots showing the correlations between sin²θ₁₂ and the ratios r₁, r₂, r₃, as well as the (sin²θ₁₂, Δm²₂₁) plane, overlaying the 1σ–3σ contours from both the global fit and JUNO.

The paper concludes that the A₄‑based μ‑τ reflection model is highly predictive: it not only fixes the atmospheric mixing and CP phase but also ties the solar mixing angle to a small set of underlying parameters. The latest JUNO results dramatically shrink the viable parameter space, leaving only the “positive‑D” (Case‑I) scenario. Future long‑baseline experiments such as DUNE and T2HK, with their sensitivity to θ₂₃ octant and δ_CP, will provide complementary tests of the remaining model predictions. Moreover, the identified correlations among the model parameters (r₁, r₂, r₃) and observable quantities offer a clear roadmap for confronting the theory with forthcoming precision data, potentially allowing a decisive validation or falsification of the A₄‑induced μ‑τ reflection symmetry.