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Branching Fractions and Direct CP Violation in Charmless Three-body Decays of B Mesons

Charmless three-body decays of B mesons is studied using a simple model based on the framework of the factorization approach. Hadronic three-body decays receive both resonant and nonresonant contributions. Dominant nonresonant contributions to tree-dominated three-body decays arise from the $b\to u$ tree transition which can be evaluated using heavy meson chiral perturbation theory valid in the soft meson limit. For penguin-dominated decays, nonresonant signals come mainly from the penguin amplitude governed by the matrix elements of scalar densities $<M_1M_2|\bar q_1 q_2|0>$. We use the measurements of $\bar B^0\to K_SK_SK_S$ to constrain the nonresonant component of $<K\bar K|\bar ss|0>$. The intermediate vector meson contributions to three-body decays are identified through the vector current, while the scalar meson resonances are mainly associated with the scalar density. While the calculated direct CP violation in $B^-\to K^+K^-K^-$ and $B^-\to π^+π^-π^-$ decays agrees well with experiment in both magnitude and sign, the predicted CP asymmetries in $B^-\to π^- K^+K^-$ and $B^-\to K^-π^+π^-$ are wrong in signs when confronted with experiment. It has been conjectured recently that a possible resolution to this CP puzzle may rely on final-state rescattering of $π^+π^-$ and $K^+K^-$. Assuming a large strong phase associated with the matrix element $<Kπ|\bar sq|0>$ arising from some sort of power corrections, we fit it to the data of $K^-π^+π^-$ and find a correct sign for $π^- K^+K^-$.