331 Models and Grand Unification: From Minimal SU(5) to Minimal SU(6)

We consider the possibility of grand unification of the $\mathrm{ SU(3)_c \otimes SU(3)_L \otimes U(1)_X}$ model in an SU(6) gauge unification group. Two possibilities arise. Unlike other conventional grand unified theories, in SU(6) one can embed the $\mathrm{S}\mathrm{U}(3{)}_{\mathrm{c}}\otimes \mathrm{S}\mathrm{U}(3{)}_{\mathrm{L}}\otimes \mathrm{U}(1{)}_{\mathrm{X}}$ model as a subgroup such that different multiplets appear with different multiplicities. Such a scenario may emerge from the flux breaking of the unified group in an E(6) F-theory GUT. This provides new ways of achieving gauge coupling unification in $\mathrm{S}\mathrm{U}(3{)}_{\mathrm{c}}\otimes \mathrm{S}\mathrm{U}(3{)}_{\mathrm{L}}\otimes \mathrm{U}(1{)}_{\mathrm{X}}$ models while providing the radiative origin of neutrino masses. Alternatively, a sequential variant of the $\mathrm{S}\mathrm{U}(3{)}_{\mathrm{c}}\otimes \mathrm{S}\mathrm{U}(3{)}_{\mathrm{L}}\otimes \mathrm{U}(1{)}_{\mathrm{X}}$ model can fit within a minimal SU(6) grand unification, which in turn can be a natural E(6) subgroup. This minimal SU(6) embedding does not require any bulk exotics to account for the chiral families while allowing for a TeV scale $\mathrm{ SU(3)_c \otimes SU(3)_L \otimes U(1)_X}$ model with seesaw-type neutrino masses.