On Landau pole in the minimal 3-3-1 model

We show that in 3-3-1 models the existence of a Landau-like pole in the coupling constant related to the \(U(1)_X\) factor, \(g_X\), in a certain value of \(\sin^2\theta_W \), arises only assuming that the condition to match the gauge coupling constants of the standard model, \(g_{2L}\), with that of the 3-3-1 model, \(g_{3L}\), is valid for all energies. However, if we impose that this matching condition is valid only at a given energy, say \(\mu = M_Z\), the pole arises when \(\sin^2\theta_X(\mu_{LP})=1\), which is the only weak mixing angle in the models. The value of \(\mu_{LP} \) depends on the energy scales, \(\mu_m\) and \(\mu_{331}\), in which the matching and the 3-3-1 symmetry is fully realized, respectively. We also show that \(g_{2L} \) and \(g_{3L} \) have different running with energy. Therefore, differently from what is usually assumed in the literature, these couplings can not be considered equal for all energies. As a consequence, the fermion couplings with neutral vector bosons are different if we write them in terms of \(\sin\theta_X\) instead of \(\sin\theta_W \).