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Towards Extractions of the CKM Angle $γ$ from $B_{u,d}\toπK$ and Untagged $B_s\to K\bar{K}$ Decays

The decays $B_d\toπ^\mp K^\pm$ and $B^\pm\toπ^\pm K$ provide interesting constraints on the angle $γ$ of the unitarity triangle of the CKM matrix. It is shown that bounds on $γ$ can also be obtained 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 $ΔΓ_s$. A detailed discussion of rescattering processes and electroweak penguin effects, which limit the theoretical accuracy of these constraints, and of methods to control them through experimental data is given. Moreover, strategies are pointed out to go beyond these bounds by relating the $B_{u,d}\toπK$ and untagged $B_s\to K\bar{K}$ decays through the SU(3) flavour symmetry of strong interactions. If a tagged, time-dependent measurement of the $B_s\to K^+K^-$ and $B_s\to K^0\bar{K^0}$ modes should become possible, $γ$ could be determined from the corresponding observables in a way that makes use of only the SU(2) isospin symmetry and takes into account rescattering effects ``automatically''. The impact of new-physics contributions to $B^0_s - \bar{B^0_s}$ mixing is also analysed, and interesting features arising in such a scenario of physics beyond the Standard Model are pointed out.