Froggatt-Nielsen mechanism with non-invertible symmetry

We formulate the Froggatt-Nielsen mechanism in the framework of non-invertible symmetries obtained by ${\mathbb{Z}}_2$ gauging of ${\mathbb{Z}}_N$ or $U(1)$. While the non-invertible selection rules can constrain the texture-zero structure of fermion mass matrices, they do not by themselves determine the magnitudes of the allowed couplings. Introducing a flavon field allows the fusion rules to fix the leading powers of the small parameter $\epsilon =⟨S⟩\mathrm{/}\mathrm{\Lambda }$, thereby determining both the texture zeros and the orders of magnitude of the nonzero entries up to $\mathcal{O}(1)$ coefficients. We have constructed explicit quark and lepton models and study their predictions for the fermion mass and mixing hierarchies through analytical estimates and numerical analysis. In particular, a benchmark model with nine texture zeros in the quark sector can well accommodate the quark masses and mixing parameters with non-Hermitian Yukawa matrices. Its lepton sector admits a minimal type-I seesaw realization with normal neutrino mass ordering and one massless neutrino. These results provide a common symmetry framework for fermion mass hierarchies and texture zeros.