Complete next-to-next-to-leading order QCD corrections to the decay matrix in $\mathit{B}$-meson mixing at leading power

We compute next-to-next-to-leading order corrections to the decay width difference of mass eigenstates and the charge-parity (CP) asymmetry $a_{\mathrm{f}\mathrm{s}}$ in flavour-specific decays of neutral $B$ mesons. We include both current-current and penguin operators at three-loop order. All input integrals in the transition amplitude are reduced to a small set of master integrals which depend on the ratio of the charm and bottom quark masses. The latter are computed using semi-analytic methods which provide deep expansions around properly selected values of $m_c\mathrm{/}{m}_b$. We provide numerical results for $\mathrm{\Delta }\mathrm{\Gamma }$ and $\mathrm{\Delta }\mathrm{\Gamma }\mathrm{/}\mathrm{\Delta }M$, both for the $B_d$ and $B_s$ system, including a detailed uncertainty analysis. Using the experimental value for the mass difference $\mathrm{\Delta }{M}_s$ we predict $\mathrm{\Delta }{\mathrm{\Gamma }}_s=(0.078\pm 0.015)p{s}^{-1}$. For the CP asymmetries we find $a_{\mathrm{f}\mathrm{s}}^s=(2.27\pm 0.13)\times {10}^{-5}$ and $a_{\mathrm{f}\mathrm{s}}^d=-(5.19\pm 0.30)\times {10}^{-4}$. Furthermore, we show that the ratios $(\mathrm{\Delta }{\mathrm{\Gamma }}_s\mathrm{/}\mathrm{\Delta }{M}_s)\mathrm{/}(\mathrm{\Delta }{\mathrm{\Gamma }}_d\mathrm{/}\mathrm{\Delta }{M}_d)$ and $\mathrm{\Delta }{\mathrm{\Gamma }}_d\mathrm{/}\mathrm{\Delta }{\mathrm{\Gamma }}_s$ can be predicted with high precision. The former quantity permits the prediction $\mathrm{\Delta }{\mathrm{\Gamma }}_d=(0.00215\pm 0.00013)p{s}^{-1}$ from the measurements of $\mathrm{\Delta }{M}_{d,s}$ and $\mathrm{\Delta }{\mathrm{\Gamma }}_s$. We further discuss the impact of $\mathrm{\Delta }{\mathrm{\Gamma }}_d\mathrm{/}\mathrm{\Delta }{\mathrm{\Gamma }}_s$ on the CKM unitarity triangle and present ready-to-use formulae which permit improved predictions once updated results for the operator matrix elements are available.