Penguin Topologies, Rescattering Effects and Penguin Hunting with $B_{u,d}\to K\bar{K}$ 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 $B^{+}\to \{{\pi }^0{K}^{+}\}\to {\pi }^{+}{K}^0$. 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 $B_{u,d}\to K\bar{K}$ 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.