Eigenvalue conditions and asymptotic freedom ofSO(N)gauge theories

Abstract

We report on the study of a class of $\mathrm{SO}\mathrm{}\left(N\right)\mathrm{}$ grand unified gauge theories that truly have only one coupling constant in the theory. The Yukawa and the different quartic self-coupling constants in the Higgs potential are totally fixed by the eigenvalue conditions required for asymptotic freedom. We discuss briefly the phenomenological implications of a particular SO(12) solution which, at the first stage of the hierarchy of spontaneous symmetry breakdown, exhibits a U(2) × SO(8) residual symmetry, which contains SU(2) × SU(4) × U(1) × U(1).