Confinement-deconfinement transition in an SU(2) Higgs theory

We study the confinement-deconfinement transition in $SU(2)$ gauge theory in the presence of bosons with vanishing bare mass 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 crossover for $N_{\tau }=2$, 4 and second order with the three-dimensional Ising universality class for $N_{\tau }=8$. Our results show that the second-order transition is accompanied by the realization of the $Z_2$ symmetry.