Dimensional transmutation in the longitudinal sector of equivariantly gauge-fixed Yang-Mills theory

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.