In this paper we test the semilocal duality based on the method of [L. Y. Dai and U.-G. Meißner, Phys. Lett. B 783, 294 (2018)] for calculating final-state interactions at varying numbers 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 ones of chiral perturbation theory. The semilocal duality is investigated by varying $N_C$. Our results show that the semilocal 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 [L. Y. Dai and U.-G. Meißner, Phys. Lett. B 783, 294 (2018)] 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.

• Received 20 August 2018

DOI:https://doi.org/10.1103/PhysRevD.98.074033

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP³.

Published by the American Physical Society