Flavor in Ninths and a Discrete Gauge Origin of the QCD Axion

Abstract

Quark and lepton hierarchies are organized by rational powers of a single parameter (the quark masses and mixings in ninths, the charged-lepton mass ratios in sixths). We argue that this structure points to a discrete ${\mathbb{Z}}_{18}$ origin of Froggatt--Nielsen symmetry, fixed as the least common denominator $\mathrm{lcm}(9,6)=18$, with its ${\mathbb{Z}}_9$ subgroup governing the quark lattice and the full ${\mathbb{Z}}_{18}\cong {\mathbb{Z}}_9\times {\mathbb{Z}}_2$ required to accommodate the half-integer lepton charges. The flavon is the unique ${\mathbb{Z}}_{18}$-charged scalar with a large VEV, so its phase is the Peccei--Quinn axion; the \emph{same} symmetry then solves the strong $CP$ problem and forbids every Planck-suppressed PQ-violating operator below dimension eighteen. Our central result is that, because the order of ${\mathbb{Z}}_{18}$ is fixed by the flavor data rather than chosen to protect the axion, the resulting high axion quality (the residual shift of the axion vacuum from $\bar{\theta }=0$ being only $|\Delta \bar{\theta }|\lesssim 3\times 10^{-27}$) is a prediction of the flavor structure rather than an imposed ingredient. The axion is a generation-dependent Type-II DFSZ axion with domain-wall number $N_{\mathrm{DW}}=6$. The decisive phenomenological consequence is that the same non-universal charges that reproduce the fermion masses enhance the axion--photon coupling to an observable level, $C_{a\gamma }\simeq 0.6$--$1.0$ (well above the higgsino-suppressed MSSM value), placing the dark-matter axion at $m_a\simeq 7$--$12\mu$eV within reach of upcoming haloscope searches.