From seesaw over-suppression to trimaximal mixing: why $A_4$ is the minimal resolution of the $Z_3$ neutrino failure

Abstract

We investigate whether the type-I seesaw mechanism can rescue the $Z_3$ Froggatt--Nielsen framework for neutrinos and find that it cannot. With right-handed Majorana masses carrying the $Z_3$ charge structure dictated by the Majorana bilinear -- where suppression powers follow $(q_i+q_j)\bmod 3$ -- the mass matrix contains an unsuppressed off-diagonal entry whose dominance in $M_R^{-1}$, combined with the hierarchical column texture of $M_D$, over-suppresses the two lightest neutrino masses to $\mathcal{O}(\varepsilon^3)$ while $m_3$ remains $\mathcal{O}(1)$. This pushes the solar-to-atmospheric mass ratio to a median $\Delta m^2_{21}/\Delta m^2_{31}\sim 4\times 10^{-11}$ -- eight orders of magnitude below the observed value of $0.030$. We prove this failure is universal across all six permutations of the charges $(2,1,0)$ and show analytically that the generic ratio scales as $\Delta m^2_{21}/\Delta m^2_{31}\sim\mathcal{O}(\varepsilon^6) \sim 10^{-11}$, with fewer than $0.01\%$ of parameter-space points exceeding $\varepsilon^2\approx 2\times 10^{-4}$. The PMNS angles remain Haar-random, carrying no information from the expansion parameter. We then show that $A_4$, the alternating group of order 12, is the minimal discrete symmetry resolving both failures. Its triplet representation provides two independent vacuum parameters controlling the solar and atmospheric mass scales separately, while constraining the PMNS matrix to the trimaximal TM$_1$ pattern. The TM$_1$ solar sum rule predicts $\sin^2\theta_{12}=0.318$ ($1.2\sigma$ from NuFit 6.0, $1.0\sigma$ from JUNO), and the atmospheric sum rule yields a parameter-free $(\sin^2\theta_{23},\,\cos\delta)$ correlation predicting $\delta\approx -71^\circ$, testable at DUNE and T2HK.