Permutation symmetry and the origin of fermion mass hierarchy

A realization of the ‘‘flavor-democracy’’ approach to quark and lepton masses is provided in the context of the standard model with a horizontal ${\mathit{S}}_3$ permutation symmetry. In this model, t and b quarks pick up mass at the tree level, c, s-quark and τ-lepton masses arise at the one-loop level, u, d, and μ masses at the two-loop level, and the electron mass at the three-loop level, thus reproducing the observed hierarchial structure without fine tuning of the Yukawa couplings. The pattern of quark mixing angles also emerges naturally, with ${\mathit{V}}_{\mathit{u}\mathit{s}}$,${\mathit{V}}_{\mathit{c}\mathit{b}}$≊O(ε), ${\mathit{V}}_{\mathit{u}\mathit{b}}$≊O(${\mathrm{\epsilon }}^2$), where ε is a loop expansion parameter.