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.

New solutions on the scalar -isoscalar ππ phase shifts are analysed together with previous KK results using a separable potential model of three coupled channels (ππ , KK and an effective 2π2π system). Model parameters are fitted to two sets of solutions obtained in a recent analysis of the CERN-Cracow-Munich measurements of the π − p ↑ → π + π − n reaction on a polarized target. A relatively narrow (90 -180 MeV) scalar resonance f 0 (1400 − 1460) is found, in contrast to a much broader (Γ ≈ 500 MeV) state emerging from the analysis of previous unpolarized target data.

A separable potential formalism is used to describe the ππ and KK interactions in the I G (J P C ) = 0 + (0 ++ ) states in the energy range from the ππ threshold up to 1.4 GeV. Introduction of relativistic propagators into a system of Lippmann-Schwinger equations leads to a very good description of the data (χ 2 = 0.93 per one degree of freedom). Three poles are found in this energy region: f 0 (500) (M = 506 ± 10 MeV, Γ = 494 ± 5 MeV), f 0 (975) (M = 973±2 MeV, Γ = 29±2 MeV) and f 0 (1400) (M = 1430±5 MeV, Γ = 145 ± 25 MeV). The f 0 (975) state can be interpreted as a KK bound state. The f 0 (500) state may be associated with the often postulated very broad scalar resonance under the KK threshold (sometimes called σ or ǫ meson). The scattering lengths in the ππ and KK channels have also been obtained. The relativistic approach provides qualitatively new results * Unité de Recherche des Universités Paris 11 et Paris 6 associée au CNRS

Properties of scalar-isoscalar mesons are analysed in an unitary model using separable interactions in three decay channels: ππ , KK and an effective 2π2π. We obtain different solutions by fitting various data on the ππ and KK phase shifts and inelasticities including the CERN-Cracow-Munich measurements of the π − p ↑ → π + π − n reaction on a polarized target. Analytical structure of the meson-meson multichannel amplitudes is studied with a special emphasis on the role played by the S-matrix zeroes. S-matrix poles, located in the complex energy plane not too far from the physical region, are interpreted as scalar resonances. We see a wide f 0 (500) , a narrow f 0 (980) and a relatively narrow f 0 (1400) . In one of our solutions a resonance at about 1700 MeV is also found. Total, elastic and inelastic channel cross sections, branching ratios and coupling constants are evaluated and compared with available data. We construct an approximation to our model and show that the Breit-Wigner approach has a limited phenomenological applicability.

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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.

Roy's equations, which incorporate crossing symmetry of the ππ scattering amplitudes, are used to resolve the present ambiguity between two solutions for the scalar-isoscalar phase shifts below 1 GeV. It is shown that the "down-flat" solution satisfies well Roy's equations and consequently crossing symmetry while the other solution called "up-flat" does not and thus should be eliminated. * Unité de Recherche des Universités Paris 6 et Paris 7, associée au CNRS