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

Abstract

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 $B_d^0$--$\overline{B_d^0}$ 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.

16 pages, LaTeX, 4 figures

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