Fermionic next-to-next-to leading logarithmic corrections to b→sγ -: art. no. 114019

In this paper we take the first step towards a complete next-to-next-to-leading logarithmic (NNLL) calculation of the inclusive decay rate for B-->X(s)gamma. We consider the virtual corrections of the order of alpha(s)(2)n(f) to the matrix elements of the operators O-1, O-2 and O-8 and evaluate the real and virtual contributions to O-7. These corrections are expected to be numerically important. We observe a strong cancellation between the contributions from the current-current operators and O-7 and obtain, after applying naive non-Abelianization, a reduction of the branching ratio of 3.9% (for mu=3.0 GeV) and an increase of 3.4% (for mu=9.6 GeV).