N/Ddescription 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 (χPT) 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 ππ and $K\pi \hspace{0.5em}\mathrm{}(I=1/2\mathrm{})\mathrm{}$ $P$-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 $\sqrt{s}=1.4\mathrm{}\mathrm{GeV}$ is found. It is observed that the $a_0\mathrm{}\left(980\right),$ σ, and κ(900) resonances are meson-meson states originating from the unitarization of the $\mathcal{O}{(p}^2)$ χPT amplitudes. On the other hand, the $f_0\mathrm{}\left(980\right)\mathrm{}$ is a combination of a strong $S$-wave meson-meson unitarity effect and a preexisting singlet resonance with a mass around 1 GeV. We also study the size of the contributions of the unphysical cuts to the ππ $\mathrm{}(I=0\mathrm{})\mathrm{}$ and $K\pi \hspace{0.5em}\mathrm{}(I=1/2\mathrm{})\mathrm{}$ elastic $S$-wave amplitudes from χPT and the exchange of resonances in crossed channels up to $\sqrt{s}\approx 800\mathrm{MeV}.$ The loops are calculated as in χPT at next to leading order. We find a small correction from the unphysical cuts to our calculated partial waves.