Minimal trinification

Abstract

We study the trinified model, ${\mathrm{SU}(3)}_{C}\ifmmode\times\else\texttimes\fi{}{\mathrm{SU}(3)}_{L}\ifmmode\times\else\texttimes\fi{}{\mathrm{SU}(3)}_{R}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{3}$, with the minimal Higgs sector required for symmetry breaking. There are five Higgs doublets, and gauge-coupling unification results if all five are at the weak scale, without supersymmetry. The radiative seesaw mechanism yields sub-eV neutrino masses, without the need for intermediate scales, additional Higgs fields, or higher-dimensional operators. The proton lifetime is above the experimental limits, with the decay modes $p\ensuremath{\rightarrow}\overline{\ensuremath{\nu}}{K}^{+}$ and $p\ensuremath{\rightarrow}{\ensuremath{\mu}}^{+}{K}^{0}$ potentially observable. We also consider supersymmetric versions of the model, with one or two Higgs doublets at the weak scale. The radiative seesaw mechanism fails with weak-scale supersymmetry due to the nonrenormalization of the superpotential, but operates in the split-SUSY scenario.