Towards $\bar{B}\rightarrow X_s\gamma$ at the NNLO in QCD without interpolation in $m_c$

Strengthening constraints on new physics from the $\bar{B}\rightarrow X_s\gamma$ branching ratio requires improving accuracy in the measurements and the Standard Model predictions. To match the expected Belle-II accuracy, Next-to-Next-to-Leading Order (NNLO) QCD corrections must be calculated without the so-far employed interpolation in the charm-quark mass $m_c$. In the process of evaluating such corrections at the physical value of $m_c$, we have finalized the part coming from diagrams with closed fermion loops on the gluon lines that contribute to the interference of the current-current and photonic dipole operators. We confirm several published results for corrections of this type, and supplement them with a previously uncalculated piece. Taking into account the recently improved estimates of non-perturbative contributions, we find $B_{s \gamma} = (3.40 \pm 0.17) * 10^{-4}$ and $R_\gamma = B_{(s+d) \gamma}/B_{c l\bar{\nu}} = (3.35 \pm 0.16) * 10^{-3}$ for $E_\gamma > 1.6 GeV$ in the decaying meson rest frame.