The π/12 model: trimaximal first-column lepton mixing with charged-lepton μ--τ breaking
Abstract
The first column of the lepton mixing matrix carries two independent magnitude conditions: the trimaximal norm |Ue1|2=2/3 and the μ--τ balance |Uμ1|=|Uτ1|. Parameterizing their violation by ε1≡3/2|Ue1|2−1 and ε2≡3(|Uτ1|2−|Uμ1|2), we show that current data treat the two differently in central value: with the first JUNO measurement included, ε1=+0.014±0.010 is consistent with zero, while ε2=+0.29+0.07−0.13 is large but disfavors zero at only 1.5σ on the profile likelihood of the global fit, since the phase that drives it remains poorly measured. This asymmetry selects a specific framework, the π/12 model. Its neutrino sector is TM1, supplemented by the electron-row condition |Ue2|/|Ue3|=2+√3 that fixes sin2θ13=(2−√3)/12=0.02233 and sin2θ12=0.31811. A charged-lepton 2--3 rotation R23(θe,α) leaves the electron row exactly invariant and generates ε2=sin2θecosα. In a concrete S4× C4× C3× C2 realization, a single additional flavon alignment forces α=0; a natural angle θe≃2∘ then reproduces the observed departure of θ23 from maximal while pinning δ=272∘±2∘, near-maximal CP violation with ε2=−0.066. The result is a sharp test: the predicted δ sits 1.7σ from the present central value on the same profile likelihood, the predicted ε2 has the opposite sign to the measured one, and the Hyper-Kamiokande design precision of 20∘ on δ resolves the question at the 3σ level. The mass predictions are inherited unchanged, Σmν=65.6 meV and mββ=5.4 meV, placing the model at the current DESI bound in ΛCDM.
Related
- Charged Lepton Flavor Violation: A Theory Overview
- The conformal origin of the lepton flavor mixing matrix
- Searching for Charged Lepton Flavor Violation
- Effective flavor interaction for the charged-lepton mass hierarchy and the Koide relation
- Charged-Lepton Koide Geometry from a Green-Dressed Compact Family Cycle
BibTeX
@misc{barger2026π12,
title = {{The $π/12$ model: trimaximal first-column lepton mixing with charged-lepton $μ$--$τ$ breaking}},
author = {Vernon Barger},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.08394},
eprint = {2608.08394},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.08394},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The π/12 model: trimaximal first-column lepton mixing with charged-lepton μ--τ breaking
AU - Vernon Barger
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2608.08394
UR - https://arxiv.org/abs/2608.08394
AB - The first column of the lepton mixing matrix carries two independent magnitude conditions: the trimaximal norm |U_e1|^2=2/3 and the μ--τ balance |U_μ1|=|U_τ1|. Parameterizing their violation by ε_1≡3/2|U_e1|^2−1 and ε_2≡3(|U_τ1|^2−|U_μ1|^2), we show that current data treat the two differently in central value: with the first JUNO measurement included, ε_1=+0.014±0.010 is consistent with zero, while ε_2=+0.29^+0.07_−0.13 is large but disfavors zero at only 1.5σ on the profile likelihood of the global fit, since the phase that drives it remains poorly measured. This asymmetry selects a specific framework, the π/12 model. Its neutrino sector is TM_1, supplemented by the electron-row condition |U_e2|/|U_e3|=2+√3 that fixes sin^2θ_13=(2−√3)/12=0.02233 and sin^2θ_12=0.31811. A charged-lepton 2--3 rotation R_23(θ_e,α) leaves the electron row exactly invariant and generates ε_2=sin2θ_ecosα. In a concrete S_4× C_4× C_3× C_2 realization, a single additional flavon alignment forces α=0; a natural angle θ_e≃2^∘ then reproduces the observed departure of θ_23 from maximal while pinning δ=272^∘±2^∘, near-maximal CP violation with ε_2=−0.066. The result is a sharp test: the predicted δ sits 1.7σ from the present central value on the same profile likelihood, the predicted ε_2 has the opposite sign to the measured one, and the Hyper-Kamiokande design precision of 20^∘ on δ resolves the question at the 3σ level. The mass predictions are inherited unchanged, Σm_ν=65.6 meV and m_ββ=5.4 meV, placing the model at the current DESI bound in ΛCDM.
ER -