${\mathbb{Z}}_3$ and $S_3$ gaugings of ${\mathbb{Z}}_N\times {\mathbb{Z}}_N^{\mathrm{\prime }}$ non-invertible symmetries: quark texture zeros and phenomenology

We investigate texture zeros in the quark Yukawa matrices arising from the ${\mathbb{Z}}_3$ and $S_3$ gauging of ${\mathbb{Z}}_N\times {\mathbb{Z}}_N^{\mathrm{\prime }}$, associated with the non-invertible $\mathrm{\Delta }(3N^2)$ and $\mathrm{\Delta }(6N^2)$ symmetries, respectively. We construct the equivalence classes and their fusion rules and use them as selection rules for the Yukawa interactions. We systematically classify the full-rank quark Yukawa textures generated by the two gaugings for $N=2,3,4,$ and $5$. The two gaugings yield the same texture spectrum for $N=2$ and $N=4$, whereas the $S_3$ gauging produces additional textures for $N=3$ and $N=5$. We further analyze the most restrictive texture pairs compatible with quark data and find that there are at most nine zeros in these texture pairs. For up- and down-quark mass matrices with a total of nine texture zeros, one sector is allowed to contain six zeros while the other contains three, or one contains five zeros while the other contains four. The former case has been studied in previous works. We perform numerical fits for the latter and find it consistent with experimental data. Some viable textures also admit real quark-mass determinants while retaining a physical CKM phase, suggesting a possible connection to the strong CP problem.