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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 $B\rarrow π^-π^+, π^0 π^+, π^- K^+, π^- \bar{K}^0$ 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 CP-averaged branching ratios of the four decay modes. It appears that the current experimental data for $B\to ππ$ and $πK$ decays prefer a negative value of $\cosγ\cosδ$. By combining with the other constraints from $V_{ub}$, $B^{0}_{d,s}-\bar{B}^{0}_{d,s}$ mixings and indirect CP-violating parameter $ε_K$ within the standard model, two favorable solutions for the phases $γ$ and $δ$ are found to lie in the region: $35^{\circ}\altγ\alt 62^{\circ}$ and $106^{\circ}\alt δ\alt 180^{\circ}$ or $86^{\circ}\altγ\alt 151^{\circ}$ and $0^{\circ}\altδ\alt 75^{\circ} $ within 1$σ$ standard deviation. It is noted that if allowing the standard deviation of the data to be more than 1$σ$, the two solutions could approach to one solution with a much larger region for the phases $γ$ and $δ$. Direct CP asymme try $a_{ε''}^{(π^- K^+)}$ in $B\rarrow π^-K^+$ decay can be as large as the present experimental upper bound. Direct CP asymmetry $a_{ε''}^{(π^+π^-)}$ in $B\rarrow π^-π^+$ decay can reach up to about 40% at 1$σ$ level.