Exploring ππ scattering in the 1/Ncpicture

In the large ${\mathit{N}}_{\mathit{c}}$ approximation to QCD, the leading ππ scattering amplitude is expressed as the sum of an infinite number of tree diagrams. We investigate the possibility that an adequate approximation at energies up to somewhat more than 1 GeV can be made by keeping diagrams which involve the exchange of resonances in this energy range in addition to the simplest chiral contact terms. In this approach crossing symmetry is automatic but individual terms tend to drastically violate partial wave unitarity. We first note that the introduction of the ρ meson in a chirally invariant manner substantially delays the onset of drastic unitarity violation which would be present for the current algebra term alone. This suggests a possibility of local (in energy) cancellation which we then explore in a phenomenological way. We include exchanges of leading resonances up to the 1.3 GeV region. However, unitarity requires more structure which we model by a four derivative contact term or by a low-lying scalar resonance which is presumably subleading in the 1/${\mathit{N}}_{\mathit{c}}$ expansion, but may nevertheless be important. The latter two flavor model gives a reasonable description of the phase shift ${\mathrm{\delta }}_0^0$ up until around 860 MeV, before the effects associated with the KK¯ threshold come into play.