Towards Extractions of the CKM Angle $\gamma$ from $B_{u,d}\to\pi K$ and Untagged $B_{s}\to K\overline{K}$ Decays

It has recently been pointed out that combined $B_d\to\pi^\mp K^\pm$ and $B^\pm\to\pi^\pm K$ branching ratios provide interesting constraints on the angle $\gamma$ of the unitarity triangle. We show that it is also possible to obtain bounds on $\gamma$ from the time evolution of untagged $B_s\to K^+K^-$ and $B_s\to K^0\bar{K^0}$ decays, provided the $B_s$ system exhibits a sizeable width difference $\Delta\Gamma_s$. These bounds can be shown to be even stronger than those arising from the combined $B_{u,d}\to\pi K$ branching ratios. In order to go beyond these bounds and to extract $\gamma$ separately from the $B_{u,d}\to\pi K$ and $B_s\to K\bar{K}$ modes, the magnitudes of the corresponding colour-allowed $\bar b\to\bar uu\bar s$ current--current amplitudes have to be known, leading to hadronic uncertainties. We point out that such an input can be avoided by using rather mild SU(3) flavour symmetry arguments to relate the observables of the $B_{u,d}\to\pi K$ and untagged $B_s\to K\bar{K}$ transitions. The impact of new-physics contributions to $B^0_s$--$\bar{B^0_s}$ mixing for the $B_s\to K\bar{K}$ decays is also discussed.