Anomaly free Froggatt-Nielsen models of flavor

We introduce two anomaly free versions of Froggatt-Nielsen (FN) models, based on either $G_{\mathrm{F}\mathrm{N}}=U(1{)}^3$ or $G_{\mathrm{F}\mathrm{N}}=U(1)$ horizontal symmetries, that generate the SM quark and lepton flavor structures. The structure of these “inverted” FN models is motivated by the clockwork mechanism: the chiral fields, singlets under $G_{\mathrm{F}\mathrm{N}}$, are supplemented by chains of vector-like fermions charged under $G_{\mathrm{F}\mathrm{N}}$. Unlike the traditional FN models the hierarchy of quark and lepton masses is obtained as an expansion in $M/\langle \phi\rangle$,where$M$isthetypicalvector-likefermionmass,and$\langle \phi\rangle$ the flavon vacuum expectation value. The models can be searched for through deviations in flavor observables such as $K-\bar{K}$ mixing, $\mu\to e$ conversion, etc., where the present bounds restrict the masses of vector-like fermions to be above $\mathcal{O}({10}^7\mathrm{G}\mathrm{e}\mathrm{V})$. If $G_{\rm FN}$isgauged,themodelscanalsobeprobedbysearchingfortheflavorful$Z’$gaugebosons\mathrm{.}Inprinciple,the$Z’$s can be very light, and can be searched for using precision flavor, astrophysics, and beam dump experiments.