The analysis of tagged $B_s\to J/\psi \varphi$ decays determines the $CP$ phase ${\varphi }_s$ in $B_s-{\overline{B}}_s$ mixing with a twofold ambiguity. The solutions differ in the sign of $\cos {\varphi }_s$ which equals the sign of the width difference $\Delta {\Gamma }_s$ among the two $B_s$ mass eigenstates. We point out that this ambiguity can be removed with the help of $B_s\to D_{s}K$ decays. We compare untagged and tagged strategies and find the tagged analysis more promising. The removal of the sign ambiguity in $\Delta {\Gamma }_s$ can be done with relatively low statistics and could therefore be a target for the early stage of $B_s\to D_{s}K$ studies.

• Received 30 December 2007

DOI:https://doi.org/10.1103/PhysRevD.77.054010