Normalization of the Perturbative QCD Corrections for $B\to X_s \, \gamma$ Decay

Abstract

We study the normalization of perturbative QCD corrections to the inclusive $B \rightarrow X_s \gamma$ decay. We propose to set the renormalization scale using the Brodsky-Lepage-Mackenzie (BLM) method. In the proposed method the scale is determined by absorbing the vacuum polarization correction from light fermions to renormalization scale but not including the anomalous dimensions. The BLM scale depends in general on the renormalization scheme and the factorization scale. We find that the BLM scale is insensitive to the factorization scale. In the heavy-quark potential scheme, we find that the BLM scale is $\mu_{BLM} \approx (0.315 - 0.334) m_b$ when the factorization scale varies from $m_b/2$ to $2 m_b$.