Complete next-to-next-to-leading order QCD corrections to the decay matrix in $\mathbf{B}$-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 $a_{\mathrm{fs}}$ 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 $ΔΓ_s=(0.078 \pm 0.015) ~\mbox{ps}^{-1}$. For the CP asymmetries we find $a_{\mathrm{fs}}^s=(2.27\pm 0.13)\times 10^{-5}$ and $a_{\mathrm{fs}}^d=-(5.19\pm 0.30)\times 10^{-4}$. 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 $ΔΓ_d=(0.00215\pm 0.00013)~\mbox{ps}^{-1}$ 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.

31 pages

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