TTP97-41 Penguin Topologies, Rescattering Effects and Penguin Hunting with $B_{u,d}\to K\overline{K}$ and $B^\pm\to\pi^\pm K$

In the recent literature, constraints on the CKM angle $\gamma$ arising from the branching ratios for $B^\pm\to\pi^\pm K$ and $B_d\to\pi^\mp K^\pm$ decays received a lot of attention. An important theoretical limitation of the accuracy of these bounds is due to rescattering effects, such as $B^+\to\{\pi^0K^+\}\to\pi^+K^0$. We point out that these processes are related to penguin topologies with internal up quark exchanges and derive $SU(2)$ isospin relations among the $B^+\to\pi^+K^0$ and $B_d^0\to\pi^-K^+$ decay amplitudes by defining ``tree and ``penguin amplitudes in a proper way, allowing the derivation of generalized bounds on the CKM angle $\gamma$. We propose strategies to obtain insights into the dynamics of penguin processes with the help of the decays $B_{u,d}\to K\overline{K}$ and $B^\pm\to\pi^\pm K$, derive a relation among the direct CP-violating asymmetries arising in these modes, and emphasize that rescattering effects can be included in the generalized bounds on $\gamma$ completely this way. Moreover, we have a brief look at the impact of new physics.