Towards $\overline{B}\to {X}_s\gamma$ at the NNLO in QCD without interpolation in m$_{c}$

Strengthening constraints on new physics from the $\overline{B}\to {X}_s\gamma$ 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 mc, 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$_{sγ}$ = (3.40±0.17)×10$^{-4}$ and R$_{γ}$≡B$_{(s+d)γ}$/B$_{cℓ\overline{v}}$ = (3.35 ± 0.16) × 10$^{-3}$ for E$_{γ}$> 1.6 GeV in the decaying meson rest frame.