}\cos \theta _{E_{6}}+Q_{\chi }\sin \theta _{E_{6}}.$$

Although the $SU(3)_{C}\times SU(2)_{L}\times U(1)_{Y}\times U(1)^{\prime }$ model is motivated in the $E_{6}$ framework, we will not single out a particular charge pattern and we only require that the model be anomaly free. However, with the above particle content, one can easily show that the invariance of the Yukawa terms in the superpotential under $U(1)^{\prime }$ and the absence of the $SU(3)_{C}-SU(3)_{C}-U(1)^{\prime }$ anomaly implies that $Q_{s}=0$, and the $\mu $ problem arises again. Thus, anomaly cancelation requires exotic representations beyond those of the MSSM. The simplest extension for the anomaly to vanish is to assume three generations of heavy (few TeV) vectorlike pairs of chiral fields $K_{i}$ and $K_{i}^{c}$ which transform as $(3,1)$ and $(\overline{3},1)$ under $SU(3)_{C}\times
SU(2)_{L}$ and opposite hypercharges [@AFREE].

The effective potential
-----------------------

In case where both squarks or/and sneutrinos do not develop vevs, the scalar potential is a combination of the so-called $D$, $F$ and soft terms, which are given by $$\begin{aligned}
V_{D}& =\frac{g_{2}^{2}+g_{1}^{2}}{8}\left(
H_{2}^{+}H_{2}-H_{1}^{+}H_{1}\right) ^{2}+\frac{g_{2}^{2}}{2}\left\vert
H_{1}^{+}H_{2}\right\vert ^{2}+\frac{g^{\prime 2}}{2}\left\vert
Q_{1}H_{1}^{+}H_{1}+Q_{2}H_{2}^{+}H_{2}+Q_{S}\left\vert S\right\vert
^{2}\right\vert ^{2},  \notag \\
V_{F}& =\left\vert \lambda \right\vert ^{2}\{\left\vert \epsilon
_{ij}H_{1}^{i}H_{2}^{j}\right\vert ^{2}+\left\vert S\right\vert ^{2}\left[
H_{1}^{+}H_{1}+H_{2}^{+}H_{2}\right] \},  \notag \\
V_{soft}&
=m_{1}^{2}H_{1}^{+}H_{1}+m_{2}^{2}H_{2}^{+}H_{2}+m_{S}^{2}\left\vert
S\right\vert ^{2}+\{A_{\lambda }S\epsilon _{ij}H_{1}^{i}H_{2}^{j}+h.c\}.
\label{tree}\end{aligned}$$Here $m_{1}^{2}$, $m_{2}^{2}$, $m_{S}^{2}$ 