N/D Description of Two Meson Amplitudes and Chiral Symmetry

The most general structure of an elastic partial wave amplitude when the unphysical cuts are 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($\chi$PT) 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 ${\mathcal{O}}(p^2)$ $\chi$PT 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 a mass around 1 GeV. We have also studied the size of the contributions of the unphysical cuts to the $\pi\pi$(I=0) and $K\pi$(I=1/2) elastic S-wave amplitudes from $\chi$PT and the exchange of resonances in crossed channels up to $\sqrt{s}\approx 800$ MeV. The loops are calculated as in $\chi$PT at next to leading order. We find a small correction from the unphysical cuts to our calculated partial waves.