Weak phase $\gamma$ and strong phase $\delta$ from $\mathrm{CP}$ averaged $\overrightarrow{B}\pi \pi$ and $\pi K$ decays

Assuming SU(3) symmetry for the strong phases in the four decay modes $\overrightarrow{B}{\pi }^{-}{\pi }^{+},$ ${\pi }^0{\pi }^{+},$ ${\pi }^{-}{K}^{+},{\pi }^{-}{K}^0$ and ignoring the relative small electroweak penguin effects in those decays, the weak phase γ and the strong phase δ can be determined in a model independent way by the $\mathrm{CP}$ averaged branching ratios of the four decay modes. It appears that the current experimental data for $\overrightarrow{B}\pi \pi$ and $\pi K$ decays prefer a negative value of $\cos \gamma \cos \delta .$ By combining with the other constraints from $V_{\mathrm{ub}},$ $B_{d,s}^0-B_{d,s}^0$ mixing, and the indirect $\mathrm{CP}$-violating parameter ${\epsilon }_K$ within the standard model, two favorable solutions for the phases $\gamma$ and δ are found to lie in the region $35°\lesssim \gamma \lesssim 62°$ and $106°\lesssim \delta \lesssim 180°$ or $86°\lesssim \gamma \lesssim 151°$ and $0°\lesssim \delta \lesssim 75°$ within 1σ standard deviation. It is noted that if allowing the standard deviation of the data to be more than 1σ, the two solutions could approach one solution with a much larger region for the phases γ and δ. Direct $\mathrm{CP}$ asymmetry $a_{{\epsilon }^{″}}^{\left({\pi }^{-}{K}^{+}\right)}$ in $\overrightarrow{B}{\pi }^{-}{K}^{+}$ decay can be as large as the present experimental upper bound. Direct $\mathrm{CP}$ asymmetry $a_{{\epsilon }^{″}}^{\left({\pi }^{+}{\pi }^{-}\right)}$ in $\overrightarrow{B}{\pi }^{-}{\pi }^{+}$ decay can reach up to about $40\%$ at the 1σ level.