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 at the 1$\sigma$ level of the data: $35^{\circ}<\gamma<55^{\circ}$ and $115^{\circ}<\delta<136^{\circ}$ or $105^{\circ}<\gamma<151^{\circ}$ and $51^{\circ}<\delta<65^{\circ} $. It is noted that if allowing the standard deviation of the data to be more than 1$\sigma$, the two solutions could approach to one solution with a much larger region for the phases $\gamma$ and $\delta$. Direct CP asymmetries in the decay modes $B\rarrow \pi^-\pi^+$ and $\pi^-K^+$ are correspondingly predicted and could be as large as 50$% $ and 40% respectively.