Neutrino masses and proton decay modes in SU(3)×SU(3)×SU(3) trinification

A detailed study of the Higgs potential and the Higgs-fermion Yukawa couplings in the SU(3)×{}SU(3)×{}SU(3) trinification model proposed by de Ru{}jula, Georgi, and Glashow is carried out. Spontaneous symmetry breakdown to SU(3)×{}SU(2)×{}U(1) is analyzed, stability of the ground-state vacuum assured, and the Higgs- and gauge-boson mass matrices exhibited. The most economical method of generating all fermion masses and mixings is given. The large corrections to the masses of the neutral leptons are worked out at the one- and two-loop levels and shown to be stable. The new neutral leptons (neutrettos) turn out to be rather heavy ({~}100 TeV). The mass hierarchy of the neutrinos is ν_τ>ν_μ>νₑ. $nu sub tau--- can be a few keV, ${{ν}}_{{μ}}$ a few eV, and ${{ν}}ₑ~{10}^{{{-}}4}$ eV. The ${{ν}}ₑ$-${{ν}}_{{μ}}$ mixing angle is about ${10}^{{{-}}2}. The dominant proton-decay modes are p→{K}⁺ν{{}}_{{μ}}and p→{{π}}⁰μ⁺ with possibly comparable rates. Other modes are suppressed. For example, {Γ}(p{→}π⁺{ν}_μ)/ {Γ}(p{→}K⁺{ν}_μ){~}10^-2, {Γ}(p{→}e⁺{{π}}⁰)/Γ(p→{{μ}}⁺π⁰ ){~}(1/14), and p{→}K⁰{{μ}}⁺$ is absent.