[@USD] and little Higgs models [@ULT]. The $\mu $-problem and the dangerous dimension 5 operator can be solved naturally with an appropriate $U(1^{\prime })$ charge assignment. Furthermore, these $%
U(1)^{\prime }$ models can provide a new candidates for dark matter that are not excluded by direct dark matter searches and with interesting signatures at colliders [@UDM; @UDM2].

In the MSSM, the EWPT could be strongly first-order if the light stop is lighter than the top quark [@MSSMPT]. In the singlet extended MSSM [LLee]{}, such as the NMSSM [@NMSSM], the EWPT get stronger easily for a large range of parameters [@NMSSMPT]. In gauge extensions of the MSSM, such as UMSSM, the EWPT is also strongly first-order but with the price of introducing 3 new extra singlet scalars [@lang], or by adding new extra heavy singlet fermions [@ham].

The main reason that makes it less easier to have a strong EWPT in the UMSSM compared to NMSSM , is that the former contains a new gauge interaction which results in strong constraint on the mixing between the SM gauge boson $%
Z$ and the new one $Z^{\prime}$ [@MMz] $$2M_{ZZ^{\prime}}^{2}/(M_{Z^{\prime}Z^{\prime}}^{2}-M_{ZZ}^{2})<10^{-3},
\label{zmix}$$ and the bound on the heavy $Z^{\prime}$ mass [@Zplimit] $$M_{Z^{\prime}}>(500-800)~GeV,  \label{zpm}$$ which 