Relativistic Friedrichs-Lee model and quark-pair creation model

In this paper, we present how the Friedrichs-Lee model could be extended to the relativistic scenario and be combined with the relativistic quark pair creation model in a consistent way. This scheme could be applied to study the "unquenched" effect of the meson spectra. As an example, if the lowest $J^{PC}=0^{++}$ $(u\bar u+d\bar d)/\sqrt{2}$ bound state in the potential model is coupled to the $\pi\pi$ continuum, two resonance poles could be found from the scattering amplitude for the continuum states. One of them could correspond to the $f_0(500)/\sigma$ and the other probably $f_0(1370)$. This scheme might shed more light on why extra states could appear in the hadron spectrum other than the prediction of the quark potential model.