Towards gauge coupling unification in left-right symmetric $\mathrm{S}\mathrm{U}(3{)}_{\mathrm{c}}\times \mathrm{S}\mathrm{U}(3{)}_{\mathrm{L}}\times \mathrm{S}\mathrm{U}(3{)}_{\mathrm{R}}\times \mathrm{U}(1{)}_{\mathrm{X}}$ theories

We consider the possibility of gauge coupling unification within the simplest realizations of the $\mathrm{SU(3)_c \times SU(3)_L \times SU(3)R \times U(1){X}}$ gauge theory. We present a first exploration of the renormalization group equations governing the “bottom-up” evolution of the gauge couplings in a generic model with free normalization for the generators. Interestingly, we find that for a $\mathrm{SU(3)_c \times SU(3)_L \times SU(3)R \times U(1){X}}$symmetrybreakingscale$M_X$ as low as a few TeV one can achieve unification in the presence of leptonic octets. We briefly comment on possible grand unified theory frameworks which can embed the $\mathrm{SU(3)_c \times SU(3)_L \times SU(3)R \times U(1){X}}$ model as well as possible implications, such as lepton flavour violating physics at the LHC.