Two-body wave functions and compositeness from scattering amplitudes: II. Application to the physical $N ^{\ast}$ and $\Delta ^{\ast}$ resonances

TL;DR

This paper investigates the meson-baryon molecular components of $N^*$ and $\Delta^*$ resonances using scattering amplitudes, revealing the dominant molecular nature of the Roper resonance $N(1440)$ and characterizing its structure.

Contribution

It applies a coupled-channels meson exchange model to analyze the compositeness of resonances, providing new insights into their meson-baryon molecular components and structure.

Findings

The Roper resonance $N(1440)$ is mainly a $\pi N$ and $\sigma N$ molecular state.

Most other $N^*$ and $\Delta^*$ resonances are not dominated by meson-baryon molecules.

The spatial separation in the $N(1440)$ resonance exceeds 1 fm in relevant channels.

Abstract

The meson-baryon molecular components for the $N^{\ast}$ and $\Delta ^{\ast}$ resonances are investigated in terms of the compositeness, which is defined as the norm of the two-body wave function from the meson-baryon scattering amplitudes. The scattering amplitudes are constructed in a $\pi N$-$\eta N$-$\sigma N$-$\rho N$-$\pi \Delta$ coupled-channels problem in a meson exchange model together with several bare $N^{\ast}$ and $\Delta ^{\ast}$ states, and parameters are fitted so as to reproduce the on-shell $\pi N$ partial wave amplitudes up to the center-of-mass energy 1.9 GeV with the orbital angular momentum $L \le 2$. As a result, the Roper resonance $N (1440)$ is found to be dominated by the $\pi N$ and $\sigma N$ molecular components while the bare-state contribution is small. The squared wave functions in coordinate space imply that both in the $\pi N$ and $\sigma N$ channels the separation between the meson and baryon is about more than 1 fm for the $N (1440)$ resonance. On the other hand, dominant meson-baryon molecular components are not observed in any other $N^{\ast}$ and $\Delta ^{\ast}$ resonances in the present model, although they have some fractions of the meson-baryon clouds.