mixing between the gauge boson $Z$ and the new one $%
Z^{\prime }$ (\[zmix\]), and the lower bound on the $Z^{\prime }$ mass (\[zpm\]). The condition (\[zpm\]), could be achieved by considering relatively large $\upsilon _{x}$ or large $\left( g^{\prime }Q^{\prime
}\right) $. The condition (\[zmix\]) could be fulfilled if the mixing term $M_{ZZ^{\prime }}^{2}$ is vanishing, i.e.,$$Q_{1}=Q_{2}\tan ^{2}\beta ,  \label{zmixing}$$which leads to a fine tuning in the values of $Q_{1,2}$ and $tan\beta $. The second possibility is making $M_{Z^{\prime }Z^{\prime }}^{2}>>M_{ZZ}^{2},\
M_{ZZ^{\prime }}^{2}\,$, which roughly means$$g^{\prime }\left\vert Q_{S}\right\vert {\upsilon _{x}}\gtrsim (500-800)~GeV.
\label{supmixing}$$

In our search for the parameter’s space that fulfills the strong first-order phase transition criterion, $\upsilon \left( T_{c}\right) /T_{c}>1$, we will focus on the two following regions :

**(1)** Moderate values for the parameters $Q_{1,2}$ and $tan\beta $, where (\[zmix\]) is nearly satisfied. In this case, the singlet vev $%
\upsilon _{x}$ can be of order $\upsilon $ or relatively smaller.

**(2)** The two terms $M_{ZZ}^{2}$ and $M_{ZZ^{\prime }}^{2}$ in ([Mzz]{}), are suppressed with respect to the mass term $M_{Z^{\prime
}Z^{\prime }}^{2}$. In this case, the values of $U^{\prime }(1)$ charge and the vev of the singlet, $Q_{s}$ and $\upsilon _{x}$, must be large enough. We should 