Non-Invertible Froggatt-Nielsen Symmetry

Abstract: We investigate non-invertible completions of anomalous flavor symmetries and their application to Dirac neutrino masses. We formulate symmetry defects dressed by topological quantum field theories for flavor symmetries including finite non-Abelian flavor transformations with rational anomaly characters and modular flavor symmetries with the rational part of modular anomaly cocycles, for gauge sectors with the required exact higher-form symmetry. As a concrete example, we construct a Froggatt-Nielsen (FN) model in which an order-four subgroup $$\mathbb Z_4^{\rm FN}\subset U(1)<em>{\rm FN}$$ is completed into a non-invertible defect through its mixed anomaly with an intermediate hidden $U(1)_H$ gauge sector in the infrared monopole-decoupling limit. The non-invertible FN symmetry can be explicitly broken at the energy scale where monopoles become dynamical. This breaking from the infrared viewpoint can be realized in an ultraviolet hidden $SU(2)_H$ theory as instanton effects, where a constrained instanton with topological charge one generates a fermion mixing term that violates FN charge by one unit. Its coefficient contains the nonperturbative factor $ζ_H\sim\exp[-8π<sup>2/g_H<sup>2(v_H)]$. When independent FN-breaking terms are subleading, integrating out the heavy fermions gives $(Y</em>ν)_{ij}\proptoζ_Hε<sup>{\ell_i+n_j}$ in a rotated basis for the right-handed neutrinos, to leading order in the instanton-induced fermion mixing. The nonperturbative factor provides the exponential suppression of the neutrino Yukawa couplings, while the FN charges determine the powers of $ε$. For the chosen FN charges and generic order-one coefficients, $m_2/m_3\simε$, favoring normal ordering. An independent $\mathbb Z_8<sup>D$ gauge symmetry forbids Majorana masses. The same FN charge assignment reproduces the leading quark and charged-lepton Yukawa hierarchies.