Complete next-to-next-to-leading order QCD corrections to the decay matrix in 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 afs 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 mc/mb. We provide numerical results for ∆Γ and ∆Γ/∆M , both for the B$_d$ and B$_s$ system, including a detailed uncertainty analysis. Using the experimental value for the mass difference ∆M$_s$ we predict ∆Γ$_s$ = (0.078 ± 0.015) ps$^{−1}$. For the CP asymmetries we find as f$_s$ = (2.27 ± 0.13) × 10$^{−5}$ and ad fs = −(5.19 ± 0.30) × 10$^{−4}$. Furthermore, we show that the ratios (∆Γ$_s$/∆M$_s$)/(∆Γ$_d$/∆M$_d$) and ∆Γ$_d$/∆Γ$_s$ can be predicted with high precision. The former quantity permits the prediction ∆Γd = (0.00215 ± 0.00013) ps−1 from the measurements of ∆M$_{d,s}$ and ∆Γ$_s$. We further discuss the impact of ∆Γ$_d$/∆Γ$_s$ 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.