Maximal Abelian Flavor Symmetries

Abstract: A framework, MAFS, is introduced that provides an approximate description of the hierarchies of quark and lepton masses and mixing angles in terms of a set of small parameters, $ε_a$, one for each fermion multiplet. MAFS is an alternative to the Froggatt-Nielsen mechanism and has a unique application in any theory, as there are no fermion charges to choose. It becomes more powerful as the number of multiplets is reduced. In $SU(5)$ unified theories, 15 observed mass ratios and mixing angles are described, at the factor of two level, by five small $ε_a$ parameters. Even though quarks and leptons are unified, the observed hierarchical pattern of quark masses and mixings {\it requires} large neutrino mixing angles and small neutrino mass hierarchies. In an $SO(10)$ unified theory, MAFS successfully describes the 15 observed flavor hierarchies with just three small $ε_a$, taking values of $0.01, 0.02$ and $0.002$. The observed cosmological baryon asymmetry results approximately from leptogenesis using MAFS in $SU(5)$, without the need for any additional small parameter; while in $SO(10)$, a further small parameter of about 0.2 appears necessary.