Horizontal symmetry and the fourth generation

A systematic investigation of all possible horizontal symmetries acting on four generations of quarks in a minimal left-right-symmetric model is carried out. There are only two consistent models with realistic constraints on the quark masses and mixing angles. It is shown that ${\mathrm{Z}}_4$ is the unique symmetry group leading to these models. The fourth-generation quark masses ${\mathrm{m}}_{\mathrm{b}\prime }$ and ${\mathrm{m}}_{\mathrm{t}\prime }$ are constrained to be (A) ${\mathrm{m}}_{\mathrm{b}}$/${\mathrm{m}}_{\mathrm{t}}$≃${\mathrm{m}}_{\mathrm{b}\prime }$/${\mathrm{m}}_{\mathrm{t}\prime }$, (B) ${\mathrm{m}}_{\mathrm{s}}$/${\mathrm{m}}_{\mathrm{c}}$≃(${\mathrm{m}}_{\mathrm{b}}$+${\mathrm{m}}_{\mathrm{b}\prime }$)/(${\mathrm{m}}_{\mathrm{t}}$+${\mathrm{m}}_{\mathrm{t}\prime }$). Thus, model (A) predicts ${\mathrm{m}}_{\mathrm{t}}$,${\mathrm{m}}_{\mathrm{b}\prime }$⩽43 GeV whereas model (B) has ${\mathrm{m}}_{\mathrm{b}\prime }$⩽58 GeV. The two models differ in the mixing of the fourth generation into the first three. A crucial test which can distinguish the two models is a direct measurement of the mixing matrix element |${\mathrm{V}}_{\mathrm{ub}}$|. Model (A) predicts it to be ∼${10}^{\mathrm{-}4}$, an order of magnitude smaller than the prediction of model (B).