Spontaneous breaking of a global symmetry in the Minimum Supersymmetric $SU(3{)}_C\times SU(3{)}_L\times U(1{)}_N$ Model

We present a preliminar study of the scalar potential of the usual scalars in the Minimal Supersymmetric $SU(3{)}_C\times SU(3{)}_L\times U(1{)}_N$ Model. We will consider the case in which all the usual neutral scalars fields of this model obtain vacuum expectation values. When we allow in our superpotential, only interactions that respect the invariance of the quantum number, $\mathcal{F}\equiv B+L$, where $B$ is the baryon number while $L$ is the total lepton number. We obtain, as in the minimal $SU(3{)}_C\times SU(3{)}_L\times U(1{)}_N$ model without supersymmetry, a Majoron. However, by allowing the violation of this quantum number, we can remove the Majoron in a mechanism similar to that recently presented in the Supersymmetric version of the scheme of Gelmini-Roncadelli. The mass spectrum obtained is in agreement with current experimental data.