We present in detail the calculation of the $O({\ensuremath{\alpha}}_{s})$ virtual corrections to the matrix element for $b\ensuremath{\rightarrow}s\ensuremath{\gamma}$. In addition to 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 $\frac{{m}_{c}}{{m}_{b}}$. These results are then combined with existing $O({\ensuremath{\alpha}}_{s})$ bremsstrahlung corrections in order to obtain the inclusive rate for $B\ensuremath{\rightarrow}{X}_{s}\ensuremath{\gamma}$. 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 the corrections to the Wilson coefficients are also available.
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