Bounds for $Z^\prime$ Mass in 3-3-1 Models from $e^+$ $e^-$ Collisions at ILC and CLIC Energies

We obtain bounds for the mass of the extra neutral gauge boson, ${Z^\prime}$, predicted by two versions of ${SU (3)_C \times SU (3)_L \times U (1)_X}$ model, namely one corresponding to its minimal version and another where the neutrinos are allowed to have right-handed projection. We explore $\sqrt s$ from 0.5 TeV to 5 TeV region, that will be accessible in next linear colliders (ILC and CLIC). We used the process $e^+ + e^- \longrightarrow f^+ + f^-$, with $f=\mu$, $c$ and $b$ to obtain the energy dependent lower bounds for $M_{Z^\prime}$, within 95% C.L., by performing a $\chi ^{2}$ fit of the difference between the final lepton angular distribution calculated from the SM and that predicted by the 3-3-1 models, for a zero mixing angle. In addition we show that, as the angular distributions depends on $Z^\prime$ couplings, the obtained bounds allow one for disentangle the studied models. Finally, using the model with right-handed neutrinos, we calculate the total cross section for $e^+ + e^- \longrightarrow e^+ + e^- + e^+ + e^-$. We concluded that $Z^\prime$ contribution is very small when compared with the bilepton contribution of the minimal version and with the SM cross section.