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

Navid Ardakanian

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 $Δm^2_{21}/Δm^2_{31}\sim 4\times 10^{-11}$ -- 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 $Δm^2_{21}/Δm^2_{31}\sim\mathcal{O}(\varepsilon^6) \sim 10^{-11}$, 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 $\sin^2θ_{12}=0.318$ ($1.2σ$ from NuFit 6.0, $1.0σ$ from JUNO), and the atmospheric sum rule yields a parameter-free $(\sin^2θ_{23},\,\cosδ)$ correlation predicting $δ\approx -71^\circ$, testable at DUNE and T2HK.

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 Froggatt--Nielsen framework for neutrinos and find that it cannot. With right-handed Majorana masses carrying the charge structure dictated by the Majorana bilinear -- where suppression powers follow -- the mass matrix contains an unsuppressed off-diagonal entry whose dominance in , combined with the hierarchical column texture of , over-suppresses the two lightest neutrino masses to while remains . This pushes the solar-to-atmospheric mass ratio to a median -- eight orders of magnitude below the observed value of . We prove this failure is universal across all six permutations of the charges and show analytically that the generic ratio scales as , with fewer than of parameter-space points exceeding . The PMNS angles remain Haar-random, carrying no information from the expansion parameter. We then show that , 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 pattern. The TM solar sum rule predicts ( from NuFit 6.0, from JUNO), and the atmospheric sum rule yields a parameter-free correlation predicting , testable at DUNE and T2HK.
Paper Structure (45 sections, 2 theorems, 40 equations, 1 figure, 7 tables)

This paper contains 45 sections, 2 theorems, 40 equations, 1 figure, 7 tables.

Table of Contents

  1. Introduction
  2. The $Z_3$ seesaw framework
  3. The $Z_3$ Froggatt--Nielsen column texture
  4. Type-I seesaw
  5. The $Z_3$ charge structure of $M_R$
  6. Key observation.
  7. Inverse seesaw reconstruction
  8. Exact reconstruction formula
  9. Scan setup and results
  10. The seesaw over-suppression mechanism
  11. Forward scan setup
  12. Main result: catastrophic suppression of the solar mass splitting
  13. Origin of the over-suppression
  14. Analytical bound on the mass ratio
  15. Numerical verification of the eigenvalue splitting
  16. ...and 30 more sections

Key Result

Theorem 4.1

For a $Z_3$ FN seesaw with charges $(2,1,0)$, column-texture $M_D$, $Z_3$-charged $M_R$, $\varepsilon \simeq 0.015$, and $\mathcal{O}(1)$ coefficients with magnitudes in $[0.3, 3.0]$ and random phases, the solar-to-atmospheric mass ratio satisfies with the probability of exceeding $\mathcal{O}(\varepsilon^2) \approx 2.3 \times 10^{-4}$ being less than $10^{-4}$.

Figures (1)

  • Figure 1: The TM$_1$ atmospheric sum rule in the $(\sin^2\theta_{23},\,\cos\delta)$ plane. The blue curve is a parameter-free prediction for $\sin^2\theta_{13}=0.0222$, with $\sin^2\theta_{12} = 0.3182$ taken from the TM$_1$ solar sum rule eq. \ref{['eq:TM1_solar']} (not from the NuFit best fit). At maximal atmospheric mixing, the model predicts maximal CP violation ($\delta=\pm 90^\circ$). The red band shows the current $1\sigma$ range for $\theta_{23}$ from NuFit 6.0.

Theorems & Definitions (4)

  • Theorem 4.1: Seesaw over-suppression
  • proof
  • Lemma 5.1: Unavoidable unsuppressed off-diagonal entry
  • proof