Analyticity and the $N_c$ counting rule of $S$ matrix poles

Abstract

By studying $ππ$ scattering amplitudes in the large $N_c$ limit, we clarify the $N_c$ dependence of the $S$ matrix pole position. It is demonstrated that analyticity and the $N_c$ counting rule exclude the existence of $S$ matrix poles with ${\cal M}, Γ\sim O(1)$. Especially the properties of $σ$ and $f_0(980)$ with respect to the $1/N_c$ expansion are discussed. We point out that in general tetra-quark resonances do not exist.