  We study the pure-gauge sector of an $SU(N)$ gauge theory, equivariantly
gauge fixed to $SU(N-1)\times U(1)$, which is an asymptotically free non-linear
sigma model in four dimensions. We show that dimensional transmutation takes
place in the large-$N$ limit, and elaborate on the relevance of this result for
a speculative scenario in which the strong longitudinal dynamics gives rise to
a novel Higgs-Coulomb phase.
