We improve our description of scattering data by imposing additional requirements on our previous fits, in the form of once-subtracted Roy-like equations, while extending our analysis up to 1100 MeV. We provide simple and ready to use parametrizations of the amplitude. In addition, we present a detailed description and derivation of these once-subtracted dispersion relations that, in the 450 to 1100 MeV region, provide an additional constraint which is much stronger than our previous requirements of forward dispersion relations and standard Roy equations. The ensuing constrained amplitudes describe the existing data with rather small uncertainties in the whole region from threshold up to 1100 MeV, while satisfying very stringent dispersive constraints. For the S0 wave, this requires an improved matching of the low and high energy parametrizations. Also for this wave we have considered the latest low energy K '4 decay results, including their isospin violation correction, and we have removed some controversial data points. These changes on the data translate into better determinations of threshold and subthreshold parameters which remove almost all disagreement with previous chiral perturbation theory and Roy equation calculations below 800 MeV. Finally, our results favor the dip structure of the S0 inelasticity around the controversial 1000 MeV region.

We use our latest dispersive analysis of ππ scattering data and the very recent K(ℓ4) experimental results to obtain the mass, width, and couplings of the two lightest scalar-isoscalar resonances. These parameters are defined from their associated poles in the complex plane. The analytic continuation to the complex plane is made in a model-independent way by means of once- and twice-subtracted dispersion relations for the partial waves, without any other theoretical assumption. We find the f(0)(600) pole at (457(-13))+14))-i(279(-7)(+11)) MeV and that of the f(0)(980) at (996 ± 7)-i(25(-6)(+10)) MeV, whereas their respective couplings to two pions are 3.59(-0.13)(+0.11) and 2.3 ± 0.2 GeV.

We complete and improve the fits to experimental scattering amplitudes, both at low and high energies, that we performed in the previous papers of this series. We then verify that the corresponding amplitudes satisfy analyticity requirements, in the form of partial wave analyticity at low energies, forward dispersion relations (FDR) at all energies, and Roy equations below KK threshold; the first by construction, the last two, inside experimental errors. Then we repeat the fits including as constraints FDR and Roy equations. The ensuing central values of the various scattering amplitudes verify very accurately FDR and, especially, Roy equations, and change very little from what we found by just fitting data, with the exception of the D2 wave phase shift, for which one parameter moves by 1:5 . These improved parametrizations therefore provide a reliable representation of pion-pion amplitudes with which one can test various physical relations. We also present a list of low energy parameters and other observables.

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.

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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.