Constraining Penguin Contributions and the CKM Angle $\gamma$ through $B_d\to\pi^+\pi^-$

The decays $B_d\to\pi^+\pi^-$ and $B_s\to K^+K^-$ provide an interesting strategy to extract the CKM angle $\gamma$ at ``second-generation'' $B$-physics experiments of the LHC era. A variant for ``first-generation'' experiments can be obtained, if $B_s\to K^+K^-$ is replaced by $B_d\to\pi^\mp K^\pm$. We show that the most recent experimental results for the CP-averaged $B_d\to\pi^+\pi^-$ and $B_d\to\pi^\mp K^\pm$ branching ratios imply a rather restricted range for the corresponding penguin parameters, and upper bounds on the direct CP asymmetries $A_{CP}^{dir}(B_d\to\pi^+\pi^-)$ and $A_{CP}^{dir}(B_d\to\pi^\mp K^\pm)$. Moreover, we point out that interesting constraints on $\gamma$ can be obtained from the CP-averaged $B_d\to\pi^+\pi^-$ and $B_d\to\pi^\mp K^\pm$ branching ratios, if in addition mixing-induced CP violation in the former decay is measured, and the $B^0_d$--$\bar{B^0_d}$ mixing phase is fixed through $B_d\to J/\psi K_S$. An extraction of $\gamma$ becomes possible, if furthermore direct CP violation in $B_d\to\pi^+\pi^-$ or $B_d\to\pi^\mp K^\pm$ is observed.