Confinement-Deconfinement transition in $SU(2)+$Higgs Theory

We study the confinement-deconfinement transition in $SU(2)$ gauge theory in the presence of massless bosons using lattice Monte Carlo simulations. The nature of this transition depends on the temporal extent ($N_{\tau }$) of the Euclidean lattice. We find that the transition is a cross-over for $N_{\tau }=2,4$ and second order with $3D$ Ising universality class for $N_{\tau }=8$. Our results show that the second order transition is accompanied by realization of the $Z_2$ symmetry.