Virtual O(alpha(s)) corrections to the inclusive decay b->s gamma

We present in detail the calculation of the O(alpha(s)) virtual corrections to the matrix element for b-->s 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 m(c)/m(b). These results are then combined with existing O(alpha(s)) bremsstrahlung corrections in order to obtain the inclusive rate for B-->X(s) 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.