Fermionic next-to-next-to leading logarithmic corrections tob→sγ

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 $\overrightarrow{B}{X}_s\gamma .$ 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 $\mu =3.0\mathrm{}\mathrm{GeV})$ and an increase of 3.4% (for $\mu =9.6\mathrm{}\mathrm{GeV}).\mathrm{}$