Exploring $\mathrm{CP}$ violation through correlations in $\overrightarrow{B}\pi K,$ $B_d\to {\pi }^{+}{\pi }^{-},$ $B_s\to K^{+}{K}^{-}$ observable space

We investigate the allowed regions in the observable space of $\overrightarrow{B}\pi K,$ $B_d\to {\pi }^{+}{\pi }^{-},$ and $B_s\to K^{+}{K}^{-}$ decays, characterizing these modes in the standard model. After a discussion of a new kind of contour plot for the $\overrightarrow{B}\pi K$ system, we focus on the mixing-induced and direct $\mathrm{CP}$ asymmetries of the decays $B_d\to {\pi }^{+}{\pi }^{-}$ and $B_s\to K^{+}{K}^{-}.$ Using experimental information on the CP-averaged $B_d\to {\pi }^{\mp }{K}^{\pm }$ and $B_d\to {\pi }^{+}{\pi }^{-}$ branching ratios, the relevant hadronic penguin parameters can be constrained, implying certain allowed regions in observable space. In the case of $B_d\to {\pi }^{+}{\pi }^{-},$ an interesting situation arises now in view of the recent B-factory measurements of $\mathrm{CP}$ violation in this channel, allowing us to obtain new constraints on the CKM angle $\gamma$ as a function of the $B_d^0-\overline{B_d^0}$ mixing phase ${\phi }_d=2\mathrm{}\beta ,$ which is fixed through ${\mathcal{A}}_{\mathrm{CP}}^{\mathrm{mix}}{(B}_d\to J/\psi K_S)$ up to a twofold ambiguity. If we assume that ${\mathcal{A}}_{\mathrm{CP}}^{\mathrm{mix}}{(B}_d\to {\pi }^{+}{\pi }^{-})$ is positive, as indicated by recent Belle data, and that ${\phi }_d$ is in agreement with the “indirect” fits of the unitarity triangle, also the corresponding values for $\gamma$ around $60°$ can be accommodated. On the other hand, for the second solution of ${\phi }_d,$ we obtain a gap around $\gamma \sim 60°.$ The allowed region in the space of