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 $.