Weak Phase γ and Strong Phase δ from CP Averaged B→ ππ and πK Decays

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Assuming SU(3) symmetry for the strong phases in the four decay modes B π-π+, π⁰ π+, π- K+, π- K⁰ 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→ ππ and πK decays prefer a negative value of γδ. By combining with the other constraints from Vub, B⁰d,ₛ-B⁰d,ₛ mixings and indirect CP-violating parameter εₖ within the standard model, two favorable solutions for the phases γ and δ are found to lie in the region: 35∘γ 62∘ and 106∘ δ 180∘ or 86∘γ 151∘ and 0∘δ 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 to one solution with a much larger region for the phases γ and δ. Direct CP asymme try aε''⁽π⁻ ᵏ⁺⁾ in B π-K+ decay can be as large as the present experimental upper bound. Direct CP asymmetry aε''⁽π⁺π⁻⁾ in B π-π+ decay can reach up to about 40% at 1σ level.