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 $|\Delta\bar\theta|\lesssim 3\times10^{-27}$) 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 $N_{\rm DW}=6$. 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, $C_{a\gamma}\simeq0.6$--$1.0$ (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.