  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.
