Fermionic NNLL corrections to $b \to s \gamma$

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 $B \to X_s\gamma$. We consider the virtual corrections of order $\alphas^2 n_f$ 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 results provide an estimate of the ${\cal O}(\as^2)$ correction terms and they are expected to be numerically important. We observe a strong cancelation between the contributions from the current-current operators and $O_7$ and obtain a reduction of the branching ratio of $\approx4.5%$ (for $\mu=3.0$ GeV) and an increase of $\approx3.1%$ (for $\mu=9.6$ GeV)