Theoretical update of Bs-Bs-bar mixing

Abstract

We update the theory predictions for the mass difference $\dm_s$, the width difference $\dg_s$ and the CP asymmetry in flavour-specific decays, $a_{\rm fs}^{s}$, for the \bbs system. In particular we present a new expression for the element $\Gamma_{12}^s$ of the decay matrix, which enters the predictions of $\dg_s$ and $a_{\rm fs}^{s}$. To this end we introduce a new operator basis, which reduces the troublesome sizes of the $1/m_b$ and $\alpha_s$ corrections and diminishes the hadronic uncertainty in $\dg_s/\dm_s$ considerably. Logarithms of the charm quark mass are summed to all orders. We find $\dg_s/\dm_s= (49.7 \pm 9.4) \cdot 10^{-4}$ and $\dg_s =(f_{B_s}/240 {\rm MeV})^2 [(0.105 \pm 0.016) B + (0.024 \pm 0.004) \tilde{B}_S' - 0.027 \pm 0.015] {ps}^{-1}$ in terms of the bag parameters $B$, $\tilde{B}_S'$ in the NDR scheme and the decay constant $f_{B_s}$. The improved result for $\Gamma_{12}^s$ also permits the extraction of the CP-violating \bbms phase from $a_{\rm fs}^{s}$ with better accuracy. We show how the measurements of $\Delta M_s$, $\Delta\Gamma_s$, $a_{\rm fs}^{s}$, $A_{\rm CP}^{\rm mix}(B_s\to J/\psi \phi)$ and other observables can be efficiently combined to constrain new physics. Applying our new formulae to data from the D{\O}experiment, we find a 2$\sigma$ deviation of the \bbms phase from its Standard Model value. We also briefly update the theory predictions for the \bbd system and find $\dg_d/\dm_d = \lt(52.6 \epm{11.5}{12.8} \rt) \cdot 10^{-4}$ and $a_{\rm fs}^d = \lt(-4.8\epm{1.0}{1.2} \rt) \cdot 10^{-4}$ in the Standard Model.