Fermionic NNLL corrections to b -> s γ

Abstract

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 corrections are expected to be numerically important. We observe a strong cancelation 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).

38 pages, result extended to allow for an explicit photon energy cut-off, appendix and references added, final result and conlclusions unchanged