We study the contribution of the strong interactions between the two pions in S and P waves to the weak B ! K decay amplitudes. The interference between these two waves is analyzed in the effectivemass range of the 770 0 and f 0 980 resonances. We use a unitary and KK coupled-channel model to describe the S-wave interactions and a Breit-Wigner function for the P-wave amplitude. The weak B-decay amplitudes, obtained from QCD factorization, are supplemented with charming penguin contributions in both waves. The four complex parameters of these long-distance terms are determined by fitting the model to the BABAR and Belle data on B ;0 ! ÿ K ;0 branching fractions, CP asymmetries, effective-mass and helicity-angle distributions. This set of data, and, in particular, the large direct CP asymmetry for B ! 770 0 K decays, is well reproduced. The interplay of charming penguin amplitudes and the interference of S and P waves describes rather successfully the experimental S and A values of the CP-violating asymmetry for both B 0 ! f 0 980K 0 S and B 0 ! 770 0 K 0 S decays.

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We study CP violation and the contribution of the strong kaon-pion interactions in the three-body B ! K þ À decays. We extend our recent work on the effect of the two-pion S-and P-wave interactions to that of the corresponding kaon-pion ones. The weak amplitudes have a first term derived in QCD factorization and a second one as a phenomenological contribution added to the QCD penguin amplitudes. The effective QCD coefficients include the leading order contributions plus next-to-leading order vertex and penguins corrections. The matrix elements of the transition to the vacuum of the kaon-pion pairs, appearing naturally in the factorization formulation, are described by the strange K scalar (S-wave) and vector (P-wave) form factors. These are determined from Muskhelishvili-Omnès coupled channel equations using experimental kaon-pion T-matrix elements, together with chiral symmetry and asymptotic QCD constraints. From the scalar form factor study, the modulus of the K Ã 0 ð1430Þ decay constant is found to be ð32 AE 5Þ MeV. The additional phenomenological amplitudes are fitted to reproduce the K effective mass and helicity angle distributions, the B ! K Ã ð892Þ branching ratios and the CP asymmetries of the recent data from Belle and BABAR collaborations. We use also the new measurement by the BABAR group of the phase difference between the B 0 and " B 0 decay amplitudes to K Ã ð892Þ. Our predicted B AE ! K Ã 0 ð1430Þ AE , K Ã 0 ð1430Þ ! K AE Ç branching fraction, equal to ð11:6 AE 0:6Þ Â 10 À6 , is smaller than the result of the analyzes of both collaborations. For the neutral B 0 decays, the predicted value is ð11:1 AE 0:5Þ Â 10 À6 . In order to reduce the large systematic uncertainties in the experimental determination of the B ! K Ã 0 ð1430Þ branching fractions, a new parametrization is proposed. It is based on the K scalar form factor, well constrained by theory and experiments other than those of B decays.

We study CP violation and the contribution of the strong pion-pion interactions in the three-body B ± → π ± π ∓ π ± decays within a quasi twobody QCD factorization approach. The short distance interaction amplitude is calculated in the next-to-leading order in the strong coupling constant with vertex and penguin corrections. The meson-meson final state interactions are described by pion non-strange scalar and vector form factors for the S and P waves and by a relativistic Breit-Wigner formula for the D wave. The pion scalar form factor is calculated from a unitary relativistic coupled-channel model including ππ, KK and effective (2π)(2π) interactions. The pion vector form factor results from a Belle Collaboration analysis of τ − → π − π 0 ν τ data. The recent B ± → π ± π ∓ π ± BABAR Collaboration data are fitted with our model using only three parameters for the S wave, one for the P wave and one for the D wave. We find not only a sizable contribution of the S wave just above the ππ threshold but also under the ρ(770) peak a significant interference, mainly between the S and P waves. For the B to f 2 (1270) transition form factor, we predict F Bf2 (m 2 π ) = 0.098 ± 0.007. Our model yields a unified unitary description of the contribution of the three scalar resonances f 0 (600), f 0 (980) and f 0 (1400) in terms of the pion non-strange scalar form factor.

A coupled channel model of the a 0 (980) and a 0 (1450) resonances has been constructed using the separable πη and KK interactions. We have shown that two S-matrix poles corresponding to the a 0 (980) meson have significantly different widths in the complex energy plane. The KK to πη branching ratio, predicted in our model near the a 0 (1450) mass, is in agreement with the result of the Crystal Barrel Collaboration. The KK interaction in the S-wave isovector state is not sufficiently attractive to create a bound a 0 (980) meson.