Bs mixing observables and Vtd/Vts from sum rules

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.