NNLO fermionic corrections to the charm quark mass dependent matrix elements in B̄ → X_sγ

We compute the virtual (α_s²) fermionic contributions to the charm quark mass dependent matrix elements of the X _sγ decay. In the case of a massless quark loop insertion into the gluon propagator, our result obtained as an expansion in z ≤ m _c²/m_b² and an exact expression in terms of multi-fold MB integrals, confirms the findings of Bieri, Greub and Steinhauser [22]. In the case of a b-quark loop insertion, we observe large differences between the actual massive case and the massless approximation. These turn out, as expected, to be less pronounced for the charm quark.