$SU(2)$ Gauge - Higgs model that the ultraviolet cutoff is increased when one is moving along this line from the physical Higgs phase to the symmetric phase.

On the tree level the gauge boson mass in lattice units vanishes on the transition surface at small enough $\lambda$. This means that the tree level estimate predicts the appearance of an infinite ultraviolet cutoff at the transition point for small $\lambda$. At infinite $\lambda$ the tree level estimate gives nonzero values of lattice masses at the transition point. Our numerical investigation of $SU(2)\otimes U(1)$ model (at $\lambda = 0.0025,
0.009, 0.001$) and previous calculations in the $SU(2)$ Gauge Higgs model (both at finite $\lambda$ and at $\lambda = \infty$) showed that for the considered lattice sizes renormalized masses do not vanish and the transition is either of the first order or a crossover. (Actually, the situation, when the cutoff tends to infinity at the position of the transition point means that there is a second order phase transition.) The dependence on the lattice sizes for the $SU(2)$ Gauge Higgs model was investigated, for example, in [@10]. Namely, for $\beta = 8$, $\lambda \sim 0.00116$, where $M_H \sim M_W$, the correlation lengths were 