CP violating phase γ and the partial widths asymmetry in B − → π + π − K − and B − → K + K − K − decays

Motivated by recent Belle measurement of the branching ratios B → ππK and B → KKK and B → χc0K, we investigate the CP violating asymmetry in the partial widths for the decays B → ππK and B → KKK, which results from the interference of the nonresonant decay amplitude with the resonant amplitude B → χc0K → ππK and B → χc0K → KKK. By taking γ ≃ 58 we predict that the partial widths asymmetry is reaching 10% for the B → ππK decay and 16% for the B → KKKdecay. The extraction of the CP violation phases α, β and γ has stimulated many studies during last decade ((see e.g. [1]). Among them the phase γ is most difficult to constrain. Numerous proposals were aimed to determine its size (see e.g. [2]). The partial width asymmetry in charmless three-body decays offers a chance to get more information on sin γ [3] [10]. The basic idea was that the nonzero asymmetry results from the interference of the nonresonant three-body decay amplitude and the resonant B → χc0π → πππ decay amplitude [3, 4, 8, 10]. All these approaches were based on a theoretical prediction [3] for the decay rate of B → χc0π which is difficult to obtain in the factorization model. Instead of relying on this prediction for this decay rate, we can now use the recent Belle collaboration measurement of the decay rate [11, 12] BR(B → χc0K) = (6.0 −1.8)× 10. (1) which is surprisingly large, comparable to the B → J/ψK decay rate. In addition, the Belle collaboration has anounced the measurement of the decay rates BR(B → KKK) = (37.0± 3.9± 4.4)× 10, (2) and BR(B → ππK) = (58.5± 7.1± 8.8)× 10. (3) Although these measurements do not distinguish yet the contributions to the branching ratios of the nonresonant decays, they constrain their magnitudes. On the other hand, the charmless two-body decays of B mesons are stimulating many research projects. The factorization approximation is usually used and recently great improvement has been made in the understanding of B → ππ and B → πK decay modes (e.g. [13]). The decay mechanism of the nonleptonic charmless three-body B decays is much less understoood. The intermediate resonance states might be simpler to understand due to better knowledge of the two-body B decays. Due to the lack of understanding of the three-body nonresonant decay modes of B mesons, we will adopt the existing approaches [4, 8]. The first is the use of the factorization approach. Then we reduce the problem to the calculation of the matrix elements of currents (or density operators). In the