  In equivariantly gauge-fixed SU(N) Yang--Mills theories, the gauge symmetry
is only partially fixed, leaving a subgroup $H\subset SU(N)$ unfixed. Such
theories avoid Neuberger's nogo theorem if the subgroup $H$ contains at least
the Cartan subgroup $U(1)^{N-1}$, and they are thus non-perturbatively well
defined if regulated on a finite lattice. We calculate the one-loop beta
function for the coupling $\tilde{g}^2=\xi g^2$, where $g$ is the gauge
coupling and $\xi$ is the gauge parameter, for a class of subgroups including
the cases that $H=U(1)^{N-1}$ or $H=SU(M)\times SU(N-M)\times U(1)$. The
coupling $\tilde{g}$ represents the strength of the interaction of the gauge
degrees of freedom associated with the coset $SU(N)/H$. We find that
$\tilde{g}$, like $g$, is asymptotically free. We solve the
renormalization-group equations for the running of the couplings $g$ and
$\tilde{g}$, and find that dimensional transmutation takes place also for the
coupling $\tilde{g}$, generating a scale $\tilde{\Lambda}$ which can be larger
than or equal to the scale $\Lambda$ associated with the gauge coupling $g$,
but not smaller. We speculate on the possible implications of these results.
