NNLO QCD corrections to $B$-meson mixing

We report on the calculation of next-to-next-to-leading order (NNLO) QCD corrections to the width difference $\Delta \Gamma_s$ in the neutral $B$-meson mixing process $B^0_s - \bar{B}^0_s$. These contributions represent an important step in the task of reducing the existing large perturbative errors in the theory prediction for $\Delta \Gamma_s$ and approaching the current experimental uncertainties. We explain the theoretical framework employed in this computation and point out important subtleties in the treatment of evanescent operators and the renormalization. Part of our new results is already available in the literature, while the remaining pieces are expected to be published later this year.