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

Abstract

The first column of the lepton mixing matrix carries two independent magnitude conditions: the trimaximal norm $|U_{e1}{|}^2=2/3$ and the $\mu$--$\tau$ balance $|U_{\mu 1}|=|U_{\tau 1}|$. We parameterize their violation by ${\epsilon }_1\equiv (3/2)|U_{e1}{|}^2-1$ and ${\epsilon }_2\equiv 3(|U_{\tau 1}{|}^2-|U_{\mu 1}{|}^2)$ and show that current data treat the two differently in central value: including the first JUNO measurement, ${\epsilon }_1=+0.014\pm 0.010$ is consistent with zero, while ${\epsilon }_2=+0.29_{-0.13}^{+0.07}$ is large but disfavors zero at only $1.5\sigma$ on the global-fit profile likelihood, the driving phase being still poorly measured. This asymmetry selects the $\pi /12$ model. Its neutrino sector is TM${}_1$ with the electron-row condition $|U_{e2}|/|U_{e3}|=2+\sqrt{3}$ that fixes ${\sin }^2{\theta }_{13}=(2-\sqrt{3})/12=0.02233$ and ${\sin }^2{\theta }_{12}=0.31811$. A charged-lepton 2--3 rotation $R_{23}({\theta }_e,\alpha )$ leaves the electron row exactly invariant and generates ${\epsilon }_2=\sin 2{\theta }_e\cos \alpha$. In an $S_4\times C_4\times C_3\times C_2$ realization, the rotation arises from a single additional flavon whose alignment forces $\alpha =0$; a natural angle ${\theta }_e\simeq 2^{\circ }$ then reproduces the departure of ${\theta }_{23}$ from maximal while pinning the CP phase to $\delta =272^{\circ }\pm 2^{\circ }$, near-maximal CP violation with ${\epsilon }_2=-0.066$. The current ${\theta }_{23}$--$\delta$ measurements thereby become a sharp test: the predicted $\delta$ sits $1.7\sigma$ from the present central value on the same profile likelihood, the predicted ${\epsilon }_2$ has the opposite sign to the measured one, and the Hyper-Kamiokande design precision of $20^{\circ }$ on $\delta$ resolves the question at $3\sigma$. The mass predictions are inherited unchanged, $\Sigma m_{\nu }=65.6$ meV and $m_{\beta \beta }=5.4$ meV, at the current DESI bound in $\Lambda$CDM.

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