QCD improved estimates to rare B-Decays at the heavy b-quark limit

The renormalization effects from the b-quark scale down to the non-perturbative QCD regime are studied for rare $B$-decays at the heavy b-quark limit. Phenomenological consequences of these effects are investigated. We find that the anomalous scaling behavior plays a positive role in making non-perturbative model calculations consistent with recent CLEO measurements of $B\to K^{\ast }\gamma \mathrm{.}$