rewrite the lattice action in an appropriate way. Namely, we define the scalar field $\tilde{\Phi} =
\sqrt{\frac{\gamma}{2}} \Phi$. We have:

$$\begin{aligned}
 S & = & \beta \!\! \sum_{\rm plaquettes}\!\!
 ((1-\mbox{${\small \frac{1}{2}}$} \, {\rm Tr}\, U_p )
 + \frac{1}{{\rm tg}^2 \theta_W} (1-\cos \theta_p))+\nonumber\\
 && + \sum_{xy} |\tilde{\Phi}_x - U_{xy} e^{i\theta_{xy}}\tilde{\Phi}_y|^2 + \sum_x (\mu^2 |\tilde{\Phi}_x|^2 +
 \tilde{\lambda} |\tilde{\Phi}_x|^4) + \omega , \label{S2}\end{aligned}$$

where $\mu^2 = - 2(4+(2\lambda-1)/\gamma)$, $\tilde{\lambda} =
4\frac{\lambda}{\gamma^2}$, and $\omega = \lambda V$. Here $V = L^4$ is the lattice volume, and $L$ is the lattice size.

For negative $\mu^2$ we fix Unitary gauge $\tilde{\Phi}_2=0$, ${\rm Im}\, \tilde{\Phi}_1
= 0$, and introduce the vacuum value of $\tilde{\Phi}$: $v =
\frac{|\mu|}{\sqrt{2\tilde{\lambda}}}$. We also introduce the scalar field $\sigma$ instead of $\tilde{\Phi}$: $\tilde{\Phi}_1 = v + \sigma$. We denote $V_{xy} =
(U^{11}_{xy}e^{i\theta_{xy}} - 1)$, and obtain: $$\begin{aligned}
 S & = & \beta \!\! \sum_{\rm plaquettes}\!\!
 ((1-\mbox{${\small \frac{1}{2}}$} \, {\rm Tr}\, U_p )
 + \frac{1}{{\rm tg}^2 \theta_W} (1-\cos \theta_p))+\nonumber\\
 && + \sum_{xy} ((\sigma_x - \sigma_y)^2 + |V_{xy}|^2 v^2)  + \sum_x 2|\mu|^2 \sigma_x^2 \nonumber\\
 && + \sum_{xy} ((\sigma^2_y+2v \sigma_y)|V_{xy}|^2 - 2(\sigma_x - \sigma_y){\rm Re} V_{xy} (\sigma_y +v) ) + \nonumber\\
 && + \sum_x  \tilde{\lambda} \sigma_x^2 (\sigma_x^2 + 4 v \sigma_x) +  \tilde{\omega} , \label{S2}\end{aligned}$$ 