our numerical results. Finally, we summarize our results. The different field-dependant masses used in the estimation of the effective potential are given in Appendix A.

The UMSSM model
===============

The $U(1)^{\prime }$-MSSM (or UMSSM) is based on the gauge group $%
G=SU(3)_{c}\times SU(2)_{L}\times U(1)_{Y}\times U(1)^{\prime }$ with the couplings $g_{3}$, $g_{2}$, $g_{1}$ and $g^{\prime }$, respectively [UMSSM]{} and the superpotential is given by $$W=\lambda S\epsilon _{ij}H_{1}^{i}H_{2}^{j}+Y_{U}\epsilon
_{ij}Q^{i}U^{c}H_{2}^{j}+Y_{D}\epsilon
_{ij}Q^{i}D^{c}H_{1}^{j}+Y_{L}\epsilon _{ij}L^{i}E^{c}H_{1}^{j},$$where $\epsilon _{ij}$ is the antisymmetry $2\times 2$ tensor, $Y_{U}$, $%
Y_{D}$ and $Y_{L}$ are Yukawa couplings, and $\lambda $ is a coupling constant in which $\lambda \left\langle S\right\rangle $ generates the $\mu $-term in the MSSM. The particle content of this model is given by the left-handed chiral superfields $L\sim (1,2,-1/2,Q_{L})$, $E^{c}\sim
(1,1,1,Q_{E})$, $Q_{i}\sim (3,2,1/6,Q_{Q})$, $U^{c}\sim (\bar{3}%
,1,-2/3,Q_{U})$, $D^{c}\sim (\bar{3},1,1/3,Q_{D})$, $H_{1}\sim
(1,2,-1/2,Q_{1})$, $H_{2}\sim (1,2,1/2,Q_{2})$ and $S\sim (1,1,0,Q_{S})$, where the $U(1)^{\prime }$ charges, $Q$’s, are model dependent. For instance, in a class of $E_{6}$ gauge models, the group can be broken in two steps to its $SO(10)$ and $SU(5)$ subgroups:

$$E_{6}\rightarrow SO(10)\times U(1)_{\psi}\rightarrow SU(5)\times U(1)_{\psi
}\times U(1)_{\chi},$$

and in this case, the $U(1)^{\prime}$ generator is given in terms of the $%
U(1)_{\psi}$ and $U(1)_{\chi}$ generators and mixing angle $\theta_{E_{6}}$ [@UMSSM]

$$Q^{\prime }=Q_{\psi 