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The interplay between Weak and Strong Phases and Direct CP Violation from the Charmless B-meson Decays

We present a general analysis on charmless B-meson decays $B\to ππ$ and $πK$. It is noticed that the final state interactions and inelastic rescattering effects must be significant in order to understand the consistency of the current data. By using general isospin decompositions, the isospin amplitudes and the corresponding strong phases could be extracted from a global $χ^2$ fit of the experimental data. We emphasize that in general there are two, rather than one, relative strong phases in the decomposition. In the assumption of two equalstrong phases as considered in the literature, the current data, especially the ones concerning $B\to π^0 K^{0(\pm)}$decays, will imply a large isospin amplitude $|a^c_{3/2}| > 40 $, which is larger by a factor of five than the one from the naive factorization estimation. When two different strong phases are considered, all the isospin amplitudes can become, within the $1σ$-level, comparable with the theoretical values. We also show that the difference between the two strong phases cannot be too large and will be restricted by the most recent upper bound of $B\to π^0π^0$ decay. In any case, the strong phases are found to be large and the branching ratio of $B\to π^0π^0$ is likely to be enhanced by an order of magnitude in comparison with the one obtained from the naive factorization approach. Direct CP violations in all decay modes are also calculated and found to be close to the sensitivity of the present experiments.