Bremsstrahlung corrections to the decayb→sγ

We calculate the $O\left({\alpha }_s\right)$ gluon bremsstrahlung corrections to the inclusive decay $b\to s\gamma$, involving the full operator basis ${\hat{O}}_1-{\hat{O}}_8$. Confirming and extending earlier calculations of Ali and Greub, we give formulas for the total decay width as well as the perturbative photon spectrum, regarding the former as a necessary part of the forthcoming complete NLO analysis. We explore in detail the renormalization scale dependence of our results and find it considerably increased.