Mapping and Probing Froggatt-Nielsen Solutions to the Quark Flavor Puzzle

The Froggatt-Nielsen (FN) mechanism is an elegant solution to the flavor problem. In its minimal application to the quark sector, the different quark types and generations have different charges under a $U(1)_X$ flavor symmetry. The SM Yukawa couplings are generated below the flavor breaking scale with hierarchies dictated by the quark charge assignments. Only a handful of charge assignments are generally considered in the literature. We analyze the complete space of possible charge assignments with $|X_{q_i}| \leq 4$ and perform both a set of Bayesian-inspired numerical scans and an analytical spurion analysis to identify those charge assignments that reliably generate SM-like quark mass and mixing hierarchies. The resulting set of top-20 flavor charge assignments significantly enlarges the viable space of FN models but is still compact enough to enable focused phenomenological study. We demonstrate that these distinct charge assignments result in the generation of flavor-violating four-quark operators characterized by significantly varied strengths, potentially differing substantially from the possibilities previously explored in the literature. Future precision measurement of quark flavor violating observables may therefore enable us to distinguish among otherwise equally plausible FN charges, thus shedding light on the UV structure of the flavor sector.