Virtual O(a.) corrections to the inclusive decay b + S? )
We present in detail the calculation of the O(as) virtual corrections to the matrix element for b + s-y. Besides the one-loop virtual corrections of the electromagnetic and color dipole operators 07 and 08, we include the important two-loop contribution of the four-Fermi operator 02. By applying the Mellin-Barnes representation to certain internal propagators, the result of the two-loop diagrams is obtained analytically as an expansion in rnc/m6. These results are then combined with existing O(a, ) Bremsstrahlung corrections in order to obtain the inclusive rate for B + X~y. 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.