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

Abstract

We compute next-to-next-to-leading order corrections to the decay width difference of mass eigenstates and the charge-parity (CP) asymmetry $a_{\rm fs}$ 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 $m_c/m_b$. We provide numerical results for $\Delta\Gamma$ and $\Delta\Gamma/\Delta M$, both for the $B_d$ and $B_s$ system, including a detailed uncertainty analysis. Using the experimental value for the mass difference $\Delta M_s$ we predict $\Delta\Gamma_s=(0.078 \pm 0.015) ~\mbox{ps}^{-1}$. For the CP asymmetries we find $ a_{\rm fs}^s = (2.27 \pm 0.13 ) \times 10^{-5}$ and $a_{\rm fs}^d = -(5.19 \pm 0.30)~\times~10^{-4}$. Furthermore, we show that the ratios $(\Delta\Gamma_s/\Delta M_s) / (\Delta\Gamma_d/\Delta M_d)$ and $\Delta\Gamma_d / \Delta\Gamma_s$ can be predicted with high precision. The former quantity permits the prediction $\Delta\Gamma_d=(0.00215\pm 0.00013)~\mbox{ps}^{-1}$ from the measurements of $\Delta M_{d,s}$ and $\Delta\Gamma_s$. We further discuss the impact of $\Delta\Gamma_d/\Delta\Gamma_s$ 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.