We consider the effects of a non-vanishing strange-quark mass in the determination of the full basis of dimension six matrix elements for $B_{s}$ mixing, in particular we get for the ratio of the $V-A$ Bag parameter in the $B_s$ and $B_d$ system: $overline{B}^s_{Q_1} / overline{B}^d_{Q_1} = 0.987^{+0.007}_{-0.009}$. Combining these results with the most recent lattice values for the ratio of decay constants $f_{B_s} / f_{B_d}$ we obtain the most precise determination of the ratio $xi = f_{B_s} sqrt{overline{B}^s_{Q_1}}/ f_{B_d} sqrt{overline{B}^d_{Q_1}} = 1.2014^{+0.0065}_{-0.0072}$ in agreement with recent lattice determinations. We find $Delta M_s=(18.5_{-1.5}^{+1.2})text{ps}^{-1}$ and $Delta M_d=(0.547_{-0.046}^{+0.035})text{ps}^{-1}$ to be consistent with experiments at below one sigma. Assuming the validity of the SM, our calculation can be used to directly determine the ratio of CKM elements $|V_{td} / V_{ts} | = 0.2045^{+0.0012}_{-0.0013}$, which is compatible with the results from the CKM fitting groups, but again more precise.
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