  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.
