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

TL;DR

This paper tests semi-local duality in a large N_C framework by analyzing pi-pi scattering amplitudes with dispersion relations, confirming duality holds at large N_C and providing insights into scalar meson structure.

Contribution

It introduces a method to study N_C dependence of hadron scattering amplitudes using dispersion relations, supporting semi-local duality at large N_C.

Findings

Semi-local duality is not violated at large N_C.

Contributions of certain resonances cancel in sum rules at large N_C.

The scalar meson structure insights are enhanced.

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.