Controlling Rescattering Effects in Constraints on the CKM Angle $γ$ arising from $B\toπK$ Decays

It has recently been pointed out that the observables of the decay $B^+\to K^+\bar{K^0}$ and its charge conjugate allow us to take into account rescattering effects in constraints on the CKM angle $γ$ arising from $B^\pm\toπ^\pm K$ and $B_d\toπ^\pm K^\pm$ modes, and that they play an important role to obtain insights into final-state interactions. In this paper, the formalism needed to accomplish this task is discussed in detail. Furthermore, using a transparent model to describe the rescattering processes, as well as electroweak penguins, we calculate the quantities parametrizing the $B^+\toπ^+ K^0$ and $B_d^0\toπ^- K^+$ decay amplitudes for specific examples, and illustrate the constraints on $γ$ arising from the corresponding observables. Although this model is very crude, it shows nicely the power of $B^\pm\to K^\pm K$ both to include the rescattering effects in the bounds on $γ$ and to obtain insights into final-state interactions. Moreover this model exhibits the interesting feature that the combined branching ratio BR$(B^\pm\to K^\pm K)$ may be considerably enhanced through rescattering processes, as was recently pointed out within a general framework.

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