link variables), and the scalar doublet $ \Phi_{\alpha}, \;(\alpha = 1,2)$ defined on sites.

The action is taken in the form $$\begin{aligned}
 S & = & \beta \!\! \sum_{\rm plaquettes}\!\!
 ((1-\mbox{${\small \frac{1}{2}}$} \, {\rm Tr}\, U_p )
 + \frac{1}{{\rm tg}^2 \theta_W} (1-\cos \theta_p))+\nonumber\\
 && - \gamma \sum_{xy} Re(\Phi^+U_{xy} e^{i\theta_{xy}}\Phi) + \sum_x (|\Phi_x|^2 +
 \lambda(|\Phi_x|^2-1)^2), \label{S}\end{aligned}$$ where the plaquette variables are defined as $U_p = U_{xy} U_{yz} U_{wz}^* U_{xw}^*$, and $\theta_p = \theta_{xy} + \theta_{yz} - \theta_{wz} - \theta_{xw}$ for the plaquette composed of the vertices $x,y,z,w$. Here $\lambda$ is the scalar self coupling, and $\gamma = 2\kappa$, where $\kappa$ corresponds to the constant used in the investigations of the $SU(2)$ gauge Higgs model. $\theta_W$ is the Weinberg angle.

Bare fine structure constant $\alpha$ is expressed through $\beta$ and $\theta_W$ as $\alpha = \frac{{\rm tg}^2 \theta_W}{\pi \beta(1+{\rm tg}^2
\theta_W)}$. In order to demonstrate this we consider naive continuum limit of (\[S\]). We set $$\begin{aligned}
  \quad U_{x,\mu} = e^{iA_{\mu}(x)a}, \quad e^{i\theta_{x,\mu}} = e^{iB_{\mu}(x)a}\end{aligned}$$ Here $a$ is the lattice spacing. The field $B_{\mu}=\frac{\tilde{B_{\mu}}}{2}$, where $\tilde{B_{\mu}}$ - is the conventional $U(1)$ field while $A_{\mu}$ is the conventional $SU(2)$ field. In continuum limit (\[S\]) must become $$\begin{aligned}
 S_g & = & \int d^4x
 \{\frac{1}{2g_2^2}  {\rm 