Minimal trinification

We study the trinified model, SU(3)C×SU(3)L×SU(3)R×Z3, 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→ν̄K+ and p→μ+K0 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.