On the High $T$ Phase Transition in the Gauged $SU(2)$ Higgs Model

We study the effective field theory of a weakly coupled 3+1d gauged $\phi^4$ type model at high temperature. Our model has $4N$ real scalars ($N$ complex Higgs doublets) and a gauge group $SU(2)$ which is spontaneously broken by a nonzero scalar field $vev$ at zero temperature. We find, for sufficiently large $N$, that the transition from the high temperature symmetry restoring phase to the low temperature phase can be either first order or second order depending on the ratio of the gauge coupling to the scalar self coupling.