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 $\beta >~\mathrm{0.5.}$ When $\beta <0.5\mathrm{}$ our argument is inconclusive; the semi-classical approximation breaks down in the spacetime region of interest.

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