Why Quarks and Leptons Demand Different Symmetries: A Systematic $Z_3$ Froggatt-Nielsen Analysis

Abstract

We present a systematic analysis of a minimal supersymmetric $Z_3$ discrete flavor symmetry as a solution to the fermion mass hierarchy problem. With generation-dependent $Z_3$ charges on the right-handed chiral superfields and a single flavon chiral superfield, holomorphy of the superpotential restricts the Yukawa operators so that a single expansion parameter $\epsilon \simeq 0.015$ structurally accounts for the hierarchical pattern of quark and charged lepton mass ratios with $\mathcal{O}(1)$ Yukawa couplings. A Monte Carlo scan over $10^5$ random $\mathcal{O}(1)$ coefficient sets confirms that adjacent-generation mass ratios generically fall within the experimental ranges. The CKM mixing angles are reproducible with specific coefficient choices ($\chi^2/\text{dof} \simeq 1.6$) but are not structurally predicted. Extended to neutrinos within a type-I seesaw, the framework fails decisively on two fronts. First, the mass spectrum is far too hierarchical: $\Delta m_{21}^2/\Delta m_{31}^2 \lesssim 10^{-4}$, two orders of magnitude below the observed $0.030$. Second, the PMNS mixing angles are generically $\mathcal{O}(1)$ random -- consistent with Haar-distributed unitaries -- providing no mechanism to predict the observed pattern. When $M_R$ carries the $Z_3$ charge structure dictated by the Majorana charge algebra, an unsuppressed off-diagonal entry combines with the hierarchical column texture of the Dirac mass: the seesaw congruence transformation over-suppresses both light masses $m_1, m_2$ to $\mathcal{O}(\epsilon^3)$, deepening the ratio $\Delta m_{21}^2/\Delta m_{31}^2$ to $\mathcal{O}(\epsilon^6) \sim 10^{-11}$. These results motivate a sectorial view of flavor where different fermion sectors arise from distinct symmetry mechanisms.