  In this paper we test the semi-local duality based on the method of Ref.[1]
for calculating final-state interactions at varying number of colors ($N_C$).
We compute the amplitudes by dispersion relations that respect analyticity and
coupled channel unitarity, as well as accurately describing experiment. The
$N_C$ dependence of the $\pi\pi\to\pi\pi$ scattering amplitudes is obtained by
comparing these amplitudes to the one of chiral perturbation theory. The
semi-local duality is investigated by varying $N_C$. Our results show that the
semi-local duality is not violated when $N_C$ is large. At large $N_C$, the
contributions of the $f_2(1270)$, the $f_0(980)$ and the $f_0(1370)$ cancel
that of the $\rho(770)$ in the finite energy sum rules, while the $f_0(500)$
has almost no effect. This gives further credit to the method developed in
Ref.[1] for investigating the $N_C$ dependence of hadron-hadron scattering with
final-state interactions. This study is also helpful to understand the
structure of the scalar mesons.

High Energy Physics - Phenomenology;Nuclear Theory