A new method to study the number of colors in the final-state interactions of hadrons

We match the $\pi \pi \to \pi \pi$ scattering amplitudes of Chiral Perturbation Theory with those from dispersion relations that respect analyticity and coupled channel unitarity, as well as accurately describing experiment. Their dependence on the number of colors ($N_C$) is obtained. By varying $N_C$ the trajectories of the poles and residues (the couplings to $\pi \pi$) of light mesons, the $\sigma$, $f_0(980)$, $\rho (770)$ and $f_2(1270)$ are investigated. Our results show that the method proposed is a reliable way to study the $N_C$ dependence in hadron-hadron scattering with final-state interactions.