Model dependent analyses on the $N_c$ dependence of the $\sigma$ pole trajectory

We explore the nature of the$\sigma$ or $f_0(600)$ meson using large $N_c$ technique, assuming that the $\sigma$ pole dominates low energy physics in the IJ=00 channel. We trace the $N_c$ dependence of the $\sigma$ pole position using [1,1] Pad\'e approximation of SU(3) chiral amplitude, [1,2] Pad\'e approximation of SU(2) chiral amplitudes, and the PKU parametrization form. We find that in all 3 cases the $\sigma$ pole moves to the real axis of the complex $s$-plane. Finally we construct a pedagogical example using the SU(2) Pad\'e amplitude to show that when the analyticity property of the scattering amplitude is violated, it is possible that the $\sigma$ pole resides on the complex $s$ plane when $N_c$ is large.