$Z^{{}^{\mathrm{\prime }}0}{Z}^{{}^{\mathrm{\prime }}0}$, $K^{+}{K}^{-}$ and $K^0{\bar{K}}^0$ boson production with definite helicity amplitudes through $e^{+}{e}^{-}$ collisions in the 3-3-1 models without exotic electric charges

A detailed analysis of $Z^{{}^{\mathrm{\prime }}0}{Z}^{{}^{\mathrm{\prime }}0}$, $K^{+}{K}^{-}$ and $K^0{\bar{K}}^0$ pair production in $e^{+}{e}^{-}$ collisions is presented by using helicity amplitudes. The trilinear bosons couplings in the $SU(3){C}\otimes SU(3){L}\otimes U(1)_{X}$ models without exotic electric charges are also calculated. We carry out the mentioned analysis for two models, one of them is a one family model which is an $E_6$ subgroup \cite{b1} and the other one is a three family model with right handed neutrinos\cite{b2,b3}. These models do not contain exotic electric charges. For them, we give explicit formulae and the corresponding numerical estimates of the cross-sections and angular distributions occurred in the processes $e^{+}{e}^{-}\to Z^{{}^{\mathrm{\prime }}0}{Z}^{{}^{\mathrm{\prime }}0}$, $e^{+}{e}^{-}\to K^{+}{K}^{-}$ and $e^{+}{e}^{-}\to K^0{\bar{K}}^0$ present in our models. We suppose these processes are invariant under $C$, $P$ and $T$ transformation.