not far from the numerical result given in [@VZ2008]. In the $SU(2)$ Gauge Higgs model at $\lambda = \infty$ the critical $\gamma_c = 0.63$ for $\beta = 8$ [@14]. At this point the tree level estimate gives $\Lambda = 0.9$ Tev while the direct measurements give $\Lambda \in [0.8; 1.5]$ Tev for values of $\gamma \in [0.64; 0.95]$ [@14]. The investigations of the $SU(2)$ Gauge Higgs model showed that a consideration of finite $\lambda$ does not change much the estimate for the gauge boson mass. However, at finite $\lambda$ and values of $\gamma$ close to the phase-transition point the tree level formula does not work at all.

The tree level estimate for the critical $\gamma$ is $\gamma_c = (1-2\lambda)/4$. At small $\lambda$ this formula gives values that are close to the ones obtained by the numerical simulations [@12; @13; @14]. In particular, $\gamma_c \rightarrow 0.25$ ($\kappa_c \rightarrow 0.125$) at $\lambda << 1$. However, this formula clearly does not work for $\lambda > 1/2$. From [@Montvay; @12; @13; @14] we know that the critical coupling in the $SU(2)$ Gauge Higgs model is about $2 - 4$ times smaller for $\lambda =0$ than for $\lambda = \infty$.

Tree level estimate predicts that 