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

TL;DR

This paper systematically analyzes a minimal supersymmetric $Z_3$ flavor symmetry to explain fermion mass hierarchies, finding success for quarks and charged leptons but significant issues with neutrino masses and mixing angles.

Contribution

It introduces a $Z_3$ symmetry framework that structurally accounts for quark and charged lepton hierarchies, but reveals limitations in explaining neutrino properties.

Findings

Reproduces quark and charged lepton mass ratios within experimental ranges.

Successfully fits CKM mixing angles with specific coefficient choices.

Fails to predict realistic neutrino mass spectrum and mixing angles.

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.