Understanding the 1P- and 2S-wave nucleon resonances within the extended Lee-Friedrichs Model

We present a unified desciption of the low-lying $1P$- and $2S$-wave nucleon resonance within the framework of an extended Lee-Friedrichs scheme. By incorporating the coupled-channel dynamics between bare quark-model states and the $\pi N$, $\pi \mathrm{\Delta }$ and $\eta N$ meson-baryon continua, we examine the mass shifts and structural properties of these excited states. We demonstrate that when the model parameters are calibrated to match the $1P$-wave spectrum and their widths, the pole associated with the bare $2S$ state is naturally shifted downward to the mass region of physical Roper resonance–$N(1440)$, thereby offering a dynamical explanation for the long-standing level-inversion problem. An approximate analysis of compositeness and elementariness reveals that the Roper resonance contains a significant meson-baryon continuum states, consistent with the picture of a bare core heavily dressed by meson-baryon cloud. Simultaneously, the pole positions and properties of five $1P$-wave resonances–$N(1535)$, $N(1650)$, $N(1520)$, $N(1700)$ and $N(1675)$ are successfully reproduced. Our results highlight the essential role of coupled-channel effects in shaping the nucleon spectrum and provide a consistent microscopic insight into the interplay between internal quark degrees of freedom and external hadronic fields.