Weak Phase $\gamma$ and Strong Phase $\delta$ from CP Averaged $B\to \pi \pi$ and $\pi K$ Decays

Assuming SU(3) symmetry for the strong phases in the four decay modes $B\text{rarrow}{\pi }^{-}{\pi }^{+},{\pi }^0{\pi }^{+},{\pi }^{-}{K}^{+},{\pi }^{-}{\bar{K}}^0$ and ignoring the relative small electroweak penguin effects in those decays, the weak phase $\gamma$ and the strong phase $\delta$ can be determined in a model independent way by the CP-averaged branching ratios of the four decay modes. It appears that the current experimental data for $B\to \pi \pi$ and $\pi K$ decays prefer a negative value of $\cos \gamma \cos \delta$. By combining with the other constraints from $V_{ub}$, $B_{d,s}^0-{\bar{B}}_{d,s}^0$ mixings and indirect CP-violating parameter ${ϵ}_K$ within the standard model, two favorable solutions for the phases $\gamma$ and $\delta$ are found to lie in the region: ${35}^{\circ }\text{alt}\gamma \text{alt}{62}^{\circ }$ and $106^{\circ}\alt \delta \alt 180^{\circ}$or$86^{\circ}\alt\gamma\alt 151^{\circ}$and$0^{\circ}\alt\delta\alt 75^{\circ} $within1$\sigma$ standard deviation. It is noted that if allowing the standard deviation of the data to be more than 1$\sigma$, the two solutions could approach to one solution with a much larger region for the phases $\gamma$ and $\delta$. Direct CP asymme try $a_{{ϵ}^{\mathrm{\prime }\mathrm{\prime }}}^{({\pi }^{-}{K}^{+})}$ in $B\text{rarrow}{\pi }^{-}{K}^{+}$ decay can be as large as the present experimental upper bound. Direct CP asymmetry $a_{{ϵ}^{\mathrm{\prime }\mathrm{\prime }}}^{({\pi }^{+}{\pi }^{-})}$ in $B\text{rarrow}{\pi }^{-}{\pi }^{+}$ decay can reach up to about 40% at 1$\sigma$ level.