N/D Description of Two Meson Amplitudes and Chiral Symmetry

The most general structure of an elastic partial wave amplitude when the left hand cut is neglected is deduced in terms of the N/D method. This result is then matched to lowest order, ${\mathcal{O}}(p^2)$, Chiral Perturbation Theory and to the exchange (consistent with chiral symmetry) of resonances in the s-channel. The extension of the method to coupled channels is also given. Making use of the former formalism, the $\pi\pi$ and $K\pi$(I=1/2) P-wave scattering amplitudes are described without free parameters when taking into account relations coming from the 1/$N_c$ expansion and unitarity. Next, the scalar sector is studied and good agreement with experiment up to $\sqrt{s}=1.4$ GeV is found. It is observed that the $a_0(980)$, $\sigma$ and $\kappa(900)$ resonances are meson-meson states originating from the unitarization of the ${\mathcal{O}}(p^2)$ Chiral Perturbation Theory amplitudes. On the other hand, the $f_0(980)$ is a combination of a strong S-wave meson-meson unitarity effect and of a preexisting singlet resonance with around 1 GeV.