Virtual $O(\a_s)$ corrections to the inclusive decay $b \to s \gamma$

We present in detail the calculation of the $O(\a_s)$ virtual corrections to the matrix element for $b \to s \g$. Besides the one-loop virtual corrections of the electromagnetic and color dipole operators $O_7$ and $O_8$, we include the important two-loop contribution of the four-Fermi operator $O_2$. By applying the Mellin-Barnes representation to certain internal propagators, the result of the two-loop diagrams is obtained analytically as an expansion in $m_c/m_b$. These results are then combined with existing $O(\a_s)$ Bremsstrahlung corrections in order to obtain the inclusive rate for $B \to X_s \g$. The new contributions drastically reduce the large renormalization scale dependence of the leading logarithmic result. Thus a very precise Standard Model prediction for this inclusive process will become possible once also the corrections to the Wilson coefficients are available.