The $\pi/12$ model: trimaximal first-column lepton mixing with charged-lepton $\mu$--$\tau$ breaking

Technical Summary: The $\pi/12$ Model

Problem Statement
Recent measurements, particularly the first JUNO reactor data, have shifted the status of constant lepton mixing patterns from theoretical preferences to experimentally discriminated hypotheses. While the Tribimaximal (TBM) mixing pattern and its TM2 variant are now disfavored by global fits, the Trimaximal-1 (TM1) pattern remains compatible with current data at the $1.4\sigma$ level. However, a critical asymmetry has emerged within the TM1 framework: the first column of the lepton mixing matrix $U$ contains two distinct conditions that data now treat differently.

1. The Norm Condition: $|U_{e1}|^2 = 2/3$ (trimaximal norm).
2. The Balance Condition: $|U_{\mu1}| = |U_{\tau1}|$ ($\mu$–$\tau$ balance).

Current global fits (NuFIT 6.1) show that the norm condition is consistent with zero deviation ($\varepsilon_1 \approx +0.014$), while the balance condition exhibits a significant central value deviation ($\varepsilon_2 \approx +0.29$), though the statistical significance of this deviation is currently limited to $1.5\sigma$ due to the poor measurement of the CP-violating phase $\delta$. The paper seeks a framework that preserves the successful electron-row predictions of TM1 while naturally accommodating the observed departure from $\mu$–$\tau$ balance.

Methodology and Framework Construction
The author proposes the $\pi/12$ model, a specific realization of TM1 mixing supplemented by a charged-lepton sector rotation. The methodology proceeds as follows:

• Neutrino Sector (Undressed Base): The model assumes an exact TM1 neutrino mixing matrix generated at the "Krishnan point" within an $S_4 \times C_4 \times C_3 \times C_2$ flavor symmetry framework. This base point fixes the reactor angle $\theta_{13}$ and solar angle $\theta_{12}$ to specific radical values derived from the internal angle $\theta = -\pi/12$. Crucially, this base point enforces $\mu$–$\tau$ reflection symmetry, resulting in a maximal atmospheric angle ($\theta_{23} = \pi/4$) and a maximal CP phase ($\delta = \pm \pi/2$), implying $\varepsilon_2 = 0$.
• Charged-Lepton Sector (Dressing): To break the balance condition while preserving the norm condition, the model introduces a single charged-lepton rotation in the 2–3 sector, $R_{23}(\theta_e, \alpha)$. Because this rotation acts trivially on the first row of the mixing matrix, all electron-row predictions (including $|U_{e1}|^2$, $|U_{e2}|/|U_{e3}|$, and the effective Majorana mass $m_{\beta\beta}$) remain exactly invariant.
• Symmetry Origin: The rotation angle $\theta_e$ and phase $\alpha$ are constrained by the flavor symmetry. The model posits that a single additional flavon field $\xi$, aligned in a "conjugate-trimaximal" direction, generates a purely real rotation ($\alpha = 0$). This alignment is enforced by the $C_3$ charge assignments and a renormalizable potential, making the real-rotation limit a structural prediction rather than an arbitrary assumption.

Key Contributions and Predictions
The $\pi/12$ model yields four parameter-free predictions for the electron row and a single free parameter ($\theta_e$) that governs the atmospheric sector:

1. Electron Row Predictions:

   • $\sin^2 \theta_{13} = (2 - \sqrt{3})/12 \approx 0.02233$.
   • $\sin^2 \theta_{12} = \cos^2(\pi/12) / (2 + \cos^2(\pi/12)) \approx 0.31811$.
   • Ratio $|U_{e2}|/|U_{e3}| = 2 + \sqrt{3}$.
   • Effective Majorana mass $m_{\beta\beta} = 5.4$ meV.
   • Status: All four are currently consistent with experimental data within $1.5\sigma$.

2. Atmospheric Sector and CP Phase:

   • The model predicts a linear relationship between the rotation angle $\theta_e$ and the atmospheric angle $\theta_{23}$, and a quadratic protection of the CP phase $\delta$ near $270^\circ$.
   • Matching the measured $\sin^2 \theta_{23} = 0.470$ yields a rotation angle $\theta_e \approx -1.89^\circ$, leading to a predicted $\delta \approx 271.6^\circ$ and $\varepsilon_2 = -0.066$.
   • A global fit minimizing $\Delta\chi^2$ yields a slightly different best-fit angle $\theta_e \approx -2.23^\circ$, resulting in a predicted $\delta \approx 271.6^\circ \pm 2^\circ$ (near-maximal CP violation).
   • Tension: This prediction sits $1.7\sigma$ away from the current NuFIT central value of $\delta \approx 212^\circ$ on the profile likelihood. The model predicts a negative $\varepsilon_2$, whereas the current data central value is positive.

3. Mass Spectrum:

   • The neutrino mass spectrum is inherited unchanged from the base model: $m_1 \approx 5.2$ meV, $m_2 \approx 10.1$ meV, $m_3 \approx 50.4$ meV.
   • Sum of Masses: $\Sigma m_\nu = 65.6$ meV.
   • Sign Pattern: The Majorana phases are fixed to $\alpha_{21}=0, \alpha_{31}=\pi$, leading to a specific sign pattern $(+, +, -)$.

Results and Experimental Tests
The paper identifies five distinct experimental tests for the framework:

1. JUNO Endgame: A final precision of $\pm 0.003$ on $\sin^2 \theta_{12}$ will test the electron-row predictions. If the current central value of 0.309 holds, this projection places the first row at a $-3.1\sigma$ deviation, providing a sharp test of the norm condition.
2. CP Phase Measurement: The predicted $\delta \approx 272^\circ$ is distinct from the current central value. The Hyper-Kamiokande experiment, with a projected precision of $\sigma(\delta) \approx 20^\circ$, is expected to resolve this tension at the $3\sigma$ level within a decade.
3. Octant Correlation: In the real-rotation limit, the sign of $\theta_e$ correlates the octant of $\theta_{23}$ with the phase. A negative $\theta_e$ (as fitted) requires $\theta_{23}$ in the first octant ($\sin^2 \theta_{23} < 0.5$) and $\delta$ slightly above $270^\circ$.
4. Cosmology: The predicted mass sum $\Sigma m_\nu = 65.6$ meV sits at the current DESI bound ($\Sigma m_\nu < 64.2$ meV at 95% C.L. in $\Lambda$CDM). The model is viable only if dynamical dark energy is present or if the cosmological bound relaxes.
5. Neutrinoless Double Beta Decay: The prediction $m_{\beta\beta} = 5.4$ meV is below current experimental reach but within the target range of next-generation experiments.

Significance and Claims
The paper argues that the $\pi/12$ model is a "sharply testable" framework. Its primary significance lies in its falsifiability:

• It converts the current imprecise measurements of $\theta_{23}$ and $\delta$ into a definitive test. The model predicts a specific, near-maximal CP phase that is currently in mild tension with data but will be decisively confirmed or excluded by upcoming experiments (Hyper-Kamiokande, DUNE).
• It explains the observed asymmetry between the norm and balance conditions of the first column as a natural consequence of a charged-lepton rotation that preserves the electron row but breaks $\mu$–$\tau$ balance.
• The model distinguishes itself from competing explanations (such as renormalization group running) by predicting a hierarchical mass spectrum rather than quasi-degenerate masses.
• The author notes a parallel between the lepton sector and the quark sector, where a Fritzsch–Xing phase is also consistent with a maximal value, suggesting a potential structural origin for near-maximal phases in both sectors, though this remains a parallel observation rather than a derived unification in this work.

The paper concludes that while the current fit quality is modest (goodness of fit $\approx 19\%$), the framework's economy (one parameter replacing the entire $\theta_{23}$–$\delta$ sector) and its complete falsifiability within the next decade constitute the primary argument for taking it seriously.