Electroweak phase transitions in the MSSM with an extra $U(1)$

We investigate the possibility of electroweak phase transition in the minimal supersymmetric standard model (MSSM) with an extra $U(1)$. This model has two Higgs doublets and a singlet, in addition to a singlet exotic quark superfield. We find that at the one-loop level this model may accommodate the electroweak phase transitions that are strongly first-order in a reasonably large region of the parameter space. In the parameter region where the phase transitions take place, we observe that the lightest scalar Higgs boson has a smaller mass when the strength of the phase transition becomes weaker. Also, the other three heavier neutral Higgs bosons get more large masses when the strength of the phase transition becomes weaker.