and taking only the thermal correction of $Z^{\prime }$. Then at lower temperatures, it is not sure that the system moves directly from $%
(0,0,0)$ to ($\upsilon _{1},\upsilon _{2},\upsilon _{x}e^{i\theta _{0}}$), or via an intermediate step: $(0,0,0)\overset{T_{c}^{\prime }}{\rightarrow }%
(0,0,x)\overset{T_{c}}{\rightarrow }(\upsilon _{1},\upsilon _{2},\upsilon
_{x}e^{i\theta _{0}})$. This will depend, in general, on the theory parameters, especially the value of the singlet vev. If it is comparable to the EW vev $\upsilon $, the phase transition could occur once, however if it is much larger than $\upsilon $, the phase transition will occur in two steps.

Since the singlet dynamics does not affect the $SU(2)$ sphaleron processes, we will not be interested in distinguishing between the one- and two-steps symmetry breaking. We will treat our field dynamics using the effective potential where the singlet is replaced by a temperature-dependant vev, i.e., $$\mathcal{V}\left( h_{1},h_{2},T\right) =V_{eff}\left(
h_{1},h_{2},x(T),T\right) .$$At higher temperatures, the effective potential admits only one minimum where $\upsilon _{1,2}\left( T\right) =0$. As the Universe cools down, the effective potential acquires a new minimum $\upsilon _{1,2}\left( T\right)
\neq 0$; but it is not the absolute one. At such temperature, the critical temperature $T_{c}$, the two minima get degenerate$$V_{eff}\left( \upsilon _{1}^{c},\upsilon _{2}^{c},\upsilon
_{x}^{c},T_{c}\right) =V_{eff}\left( 0,0,x^{c},T_{c}\right) 