Anomaly free Froggatt-Nielsen models of flavor

We introduce two anomaly free versions of Froggatt-Nielsen (FN) models, based on either $G_{\rm FN}=U(1)^3$ or $G_{\rm FN}=U(1)$ 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 $G_{\rm FN}$, are supplemented by chains of vector-like fermions charged under $G_{\rm FN}$. Unlike the traditional FN models the hierarchy of quark and lepton masses is obtained as an expansion in $M/\langle \phi\rangle$, where $M$ is the typical vector-like fermion mass, 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 ${\mathcal O}(10^7~{\rm GeV})$. If $G_{\rm FN}$ is gauged, the models can also be probed by searching for the flavorful $Z'$ gauge bosons. In principle, the $Z'$s can be very light, and can be searched for using precision flavor, astrophysics, and beam dump experiments.