# Towards gauge coupling unification in left-right symmetric $\mathrm{SU(3)_c \times SU(3)_L \times SU(3)_R \times U(1)_{X}}$ theories

## TL;DR

This paper explores gauge coupling unification in a specific left-right symmetric model with an extended gauge group, showing unification is possible at TeV scales with certain particle content, and discusses potential experimental implications.

## Contribution

It provides the first detailed analysis of gauge coupling evolution in the SU(3)^3 x U(1) model with free generator normalization, demonstrating unification feasibility at TeV scales.

## Key findings

- Unification achievable at a few TeV scale with leptonic octets.
- Renormalization group equations analyzed for generic normalization.
- Potential for observable lepton flavor violation at the LHC.

## Abstract

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 $\mathrm{SU(3)_c \times SU(3)_L \times SU(3)_R \times U(1)_{X}}$ symmetry breaking scale $M_X$ 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.