Pole analysis on unitarized $SU(3)\times SU(3)$ one loop $\chi$PT amplitudes

We analyze $\pi \pi -K\bar{K}$ and $\pi \eta -K\bar{K}$ couple channel [1,1] matrix Pad'e amplitudes of $SU(3)\times SU(3)$ chiral perturbation theory. By fitting phase shift and inelasticity data, we determine pole positions in different channels ($f_0(980)$, $a_0(980)$,$f_0(600)$, $K_0^{\ast }(800)$, $K^{\ast }(892)$, $\rho (770)$) and trace their $N_c$ trajectories. We stress that a couple channel Breit–Wigner resonance should exhibit two poles on different Riemann sheets and meet each other on the real axis when $N_c=\mathrm{\infty }$. Poles are hence classified using this criteria and we conclude that $K^{\ast }(892)$ and $\rho (770)$ are unambiguous Breit–Wigner resonances. For scalars the situation is much less clear. We find that $f_0(980)$ is a molecular state rather than a Breit–Wigner resonance, while $a_0(980)$, though behaves oddly when varying $N_c$, does maintain a twin pole structure.