  We investigate conformally coupled quantum matter fields on spherically
symmetric, continuously self-similar backgrounds. By exploiting the symmetry
associated with the self-similarity the general structure of the renormalized
quantum stress-energy tensor can be derived. As an immediate application we
consider a combination of classical, and quantum perturbations about exactly
critical collapse. Generalizing the standard argument which explains the
scaling law for black hole mass, $M \propto |\eta-\eta^*|^\beta$, we
demonstrate the existence of a quantum mass gap when the classical critical
exponent satisfies $\beta \geq 0.5$. When $\beta < 0.5$ our argument is
inconclusive; the semi-classical approximation breaks down in the spacetime
region of interest.
