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, bounds on the CKM angle $\gamma$ arising from combined branching ratios for $B_d\to\pi^\mp K^\pm$ and $B^\pm\to\pi^\pm K$ decays received a lot of attention. If the ratio $R$ of these branching ratios is found experimentally to be smaller than 1 -- present CLEO results give $R=0.65\pm0.40$ -- these bounds are free of hadronic uncertainties at the level of 1%. It has recently been argued that rescattering effects, such as $B^+\to\{\pi^0K^+\}\to\pi^+K^0$, could spoil this approach. We point out that this is actually not the case, since these processes are related to penguin topologies with internal up quark exchanges. Analogously, rescattering processes of the kind $B^+\to\{\bar{D^0}D_s^+\}\to\pi^+K^0$ are related to penguin topologies with internal charm quark exchanges. Moreover we propose strategies to obtain insights into the dynamics of penguin processes with the help of the decays $B_{u,d}\to K\bar{K}$ and $B^\pm\to\pi^\pm K$, derive a relation among direct CP-violating asymmetries arising in these modes, and have a brief look at new physics.