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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 andcoupled channel unitarity, as well as accurately describing experiment. The$N_C$ dependence of the $\pi\pi\to\pi\pi$ scattering amplitudes is obtained bycomparing these amplitudes to the one of chiral perturbation theory. Thesemi-local duality is investigated by varying $N_C$. Our results show that thesemi-local duality is not violated when $N_C$ is large. At large $N_C$, thecontributions of the $f_2(1270)$, the $f_0(980)$ and the $f_0(1370)$ cancelthat 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 inRef.[1] for investigating the $N_C$ dependence of hadron-hadron scattering withfinal-state interactions. This study is also helpful to understand thestructure of the scalar mesons.

Test of semi-local duality in a large $N_C$ framework