Weak Phase $γ$ and Strong Phase $δ$ from CP Averaged $B\to ππ$ and $πK$ Decays

Assuming SU(3) symmetry for the strong phases in the four decay modes $B\rarrow \pi^-\pi^+, \pi^0 \pi^+, \pi^- K^+, \pi^- \bar{K}^0$ 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}, $B^{0}_{d,s}-\bar{B}^{0}_{d,s}$ mixings and indirect CP-violating parameter $\epsilon_K$ within the standard model, two favorable solutions for the phases $\gamma$ and $\delta$ are found to lie in the region: $35^{\circ}\alt\gamma\alt 62^{\circ}$ and $106^{\circ}\alt \delta \alt 180^{\circ}$ or $86^{\circ}\alt\gamma\alt 151^{\circ}$ and $0^{\circ}\alt\delta\alt 75^{\circ}$ within 1$\sigma$ 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 $a_{\epsilon''}^{(\pi^- K^+)}$ in $B\rarrow \pi^-K^+$ decay can be as large as the present experimental upper bound. Direct CP asymmetry $a_{\epsilon''}^{(\pi^+\pi^-)}$ in $B\rarrow \pi^-\pi^+$ decay can reach up to about 40% at 1$\sigma$ level.