Abstract
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.
Highlights
The 1=numbers of colors (NC) expansion [1,2] provides an effective diagnostic to differentiate the ordinary from the nonordinary quark-antiquark structure of the mysterious scalars; see e.g., [3,4,5,6,7]
They found that the f0ð500Þ should contain a subdominant qq component and this ensures that the semilocal duality is fulfilled up to NC 1⁄4 15–30
In this paper we have studied the semilocal duality for large NC
Summary
The 1=NC expansion [1,2] provides an effective diagnostic to differentiate the ordinary from the nonordinary quark-antiquark structure of the mysterious scalars; see e.g., [3,4,5,6,7]. I.e., at NC 1⁄4 3, there should be local duality [8,9,10,11,12,13] This means that Regge exchange in the crossed channel is dual to the contribution of resonances in the direct channel. In the high-energy region the overlap of the resonances is much stronger, leading to a smooth amplitude Such a smooth amplitude is similar to the one generated by Regge poles in the t-channel. The FESR are tested by tuning NC up to 30 or 100 They found that the f0ð500Þ (often called the σ) should contain a subdominant qq component and this ensures that the semilocal duality is fulfilled up to NC 1⁄4 15–30.
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