there is the second order phase transition. This means that according to the tree level estimate the value of the cutoff at the transition point is infinite. Our numerical simulations, however, show that the cutoff remains finite and the transition is, most likely, a crossover at the considered values of $\theta_W$, $\lambda$ and $\beta$.

Nambu monopoles
===============

In this section we remind the reader what is called Nambu monopole [@Nambu]. First let us define the continuum Electroweak fields as they appear in the Weinberg-Salam model. The continuum scalar doublet is denoted as $\Phi$. The $Z$-boson field $Z^{\mu}$ and electromagnetic field $A_{\rm EM}^{\mu}$ are defined as $$\begin{aligned}
 Z^{\mu} = - \frac{1}{\sqrt{\Phi^+ \Phi}} \Phi^+ A^{\mu} \Phi -  B^{\mu},
\nonumber\\
 A_{\rm EM}^{\mu} =  2 B^{\mu}  + 2 \,{\rm sin}^2\, \theta_W
 Z^{\mu},\label{FSM}\end{aligned}$$ where $A^{\mu}$ and $B^{\mu}$ are the corresponding $SU(2)$ and $U(1)$ gauge fields of the Standard Model.

After fixing the unitary gauge $\Phi_2=const.$, $\Phi_1 = 0$ we have $$\begin{aligned}
 Z^{\mu} =  \frac{g_z}{2}[\frac{\tilde{A_3}^{\mu}}{g_2}{\rm cos}\theta_W -  \frac{\tilde{B}^{\mu}}{g_1}{\rm sin}\theta_W] = \frac{1}{2}\tilde{Z}^{\mu},
\nonumber\\
 A_{\rm EM}^{\mu} =  e[\frac{\tilde{A_3}^{\mu}}{g_2}{\rm sin}\theta_W +  \frac{\tilde{B}^{\mu}}{g_1}{\rm cos}\theta_W] = \tilde{A}^{\mu},\end{aligned}$$ where $\frac{\tilde{A}_3}{g_2} = \frac{1}{g_2}{\rm Tr}\, A \sigma^3$, $\frac{\tilde{B}}{g_1} = 2 B/g_1$, $\frac{\tilde{Z}}{g_z}$, $\frac{\tilde{A}}{e}$ - conventional Standard Model fields, and $g_z = \sqrt{g_1^2+g_2^2}$.

Nambu monopoles 