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Why Quarks and Leptons Demand Different Symmetries: A Systematic Z3 Froggatt-Nielsen Analysis

Navid Ardakanian

Abstract

We present a systematic analysis of a minimal Z_3 discrete flavor symmetry as a solution to the fermion mass hierarchy problem. Using a Froggatt-Nielsen mechanism with generation-dependent Z_3 charges assigned to the right-handed fermions, we show that a single expansion parameter epsilon ~ 0.015 structurally accounts for the hierarchical pattern of quark and charged lepton mass ratios with O(1) Yukawa couplings. A Monte Carlo scan over 10^5 random O(1) coefficient sets confirms that adjacent-generation mass ratios generically fall within the experimentally measured ranges. By contrast, the CKM mixing angles, while reproducible with specific O(1) coefficient choices (chi^2/dof ~ 1.6), are not structurally predicted by the symmetry. When the same framework is extended to neutrinos within a type-I seesaw, it fails decisively on two fronts. First, the mass spectrum is far too hierarchical: the model predicts Delta m^2_{21}/Delta m^2_{31} < 10^{-4}, at least two orders of magnitude below the observed ratio of 0.030. Second, the PMNS mixing angles are generically O(1) random, consistent with Haar-distributed unitaries. When M_R carries the Z_3 charge structure dictated by the correct Majorana charge algebra, the mass spectrum failure deepens catastrophically through a pseudo-Dirac mechanism. These results motivate a sectorial view of flavor where different fermion sectors arise from distinct symmetry mechanisms.

Why Quarks and Leptons Demand Different Symmetries: A Systematic Z3 Froggatt-Nielsen Analysis

Abstract

We present a systematic analysis of a minimal Z_3 discrete flavor symmetry as a solution to the fermion mass hierarchy problem. Using a Froggatt-Nielsen mechanism with generation-dependent Z_3 charges assigned to the right-handed fermions, we show that a single expansion parameter epsilon ~ 0.015 structurally accounts for the hierarchical pattern of quark and charged lepton mass ratios with O(1) Yukawa couplings. A Monte Carlo scan over 10^5 random O(1) coefficient sets confirms that adjacent-generation mass ratios generically fall within the experimentally measured ranges. By contrast, the CKM mixing angles, while reproducible with specific O(1) coefficient choices (chi^2/dof ~ 1.6), are not structurally predicted by the symmetry. When the same framework is extended to neutrinos within a type-I seesaw, it fails decisively on two fronts. First, the mass spectrum is far too hierarchical: the model predicts Delta m^2_{21}/Delta m^2_{31} < 10^{-4}, at least two orders of magnitude below the observed ratio of 0.030. Second, the PMNS mixing angles are generically O(1) random, consistent with Haar-distributed unitaries. When M_R carries the Z_3 charge structure dictated by the correct Majorana charge algebra, the mass spectrum failure deepens catastrophically through a pseudo-Dirac mechanism. These results motivate a sectorial view of flavor where different fermion sectors arise from distinct symmetry mechanisms.
Paper Structure (29 sections, 28 equations, 2 figures, 2 tables)

This paper contains 29 sections, 28 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Theoretical Framework
  3. Model construction
  4. Effective Yukawa Lagrangian
  5. Structural predictions from the column texture
  6. Quark Sector
  7. Experimental input
  8. Determination of the expansion parameter
  9. Yukawa coefficient ratios and naturalness
  10. Monte Carlo scan: mass hierarchy success
  11. Explicit best-fit Yukawa matrices
  12. CKM mixing and the column texture
  13. Predictive correlations
  14. Lepton Sector and the Failure for Neutrinos
  15. Charged leptons
  16. ...and 14 more sections

Figures (2)

  • Figure 1: Comparison between experimental fermion mass ratios and $Z_3$ model predictions. Red squares show the "bare" $Z_3$ scaling ($\varepsilon^2$ and $\varepsilon$); orange diamonds show the predictions after fitting $\mathcal{O}(1)$ coefficients. The $Z_3$ model correctly accounts for the hierarchy through powers of $\varepsilon$, with the remaining spread absorbed by natural $\mathcal{O}(1)$ factors.
  • Figure 2: Distribution of the neutrino mass ratio $\Delta m_{21}^2/\Delta m_{31}^2$ from Monte Carlo scans over $\mathcal{O}(1)$ coefficient sets. With a diagonal $M_R$ (blue), the median is $\sim 10^{-4}$, two orders of magnitude below the observed value of $0.030$ (dashed line). With the $Z_3$-charged $M_R$ (red), the pseudo-Dirac mechanism suppresses the ratio to $\sim 10^{-11}$. In both cases, zero realisations out of $10^5$ reach the experimental value.