Compositeness of baryonic resonances: Application to the $\mathrm{\Delta }\left(1232\right),N\left(1535\right)$, and $N\left(1650\right)$ resonances

We present a formulation of the compositeness for baryonic resonances to discuss the meson-baryon molecular structure inside the resonances. For this purpose, we derive a relation between the residue of the scattering amplitude at the resonance pole position and the two-body wave function of the resonance in a sophisticated way, and we define the compositeness as the norm of the two-body wave functions. As applications, we investigate the compositeness of the $\mathrm{\Delta }\left(1232\right),N\left(1535\right)$, and $N\left(1650\right)$ resonances from precise $\pi N$ scattering amplitudes in a unitarized chiral framework with the interaction up to the next-to-leading order in chiral perturbation theory. The $\pi N$ compositeness for the $\mathrm{\Delta }\left(1232\right)$ resonance is evaluated in the $\pi N$ single-channel scattering, and we find that the $\pi N$ component inside $\mathrm{\Delta }\left(1232\right)$ in the present framework is non-negligible, which supports the previous work. On the other hand, the compositeness for the $N\left(1535\right)$ and $N\left(1650\right)$ resonances is evaluated in a coupled-channel approach, resulting that the $\pi N,\eta N,K\mathrm{\Lambda }$, and $K\mathrm{\Sigma }$ components are negligible for these resonances.

Figure 1

Feynman diagrams for the interaction kernel: (a) Weinberg-Tomozawa term, (b) next-to-leading order term, (c) $s$- and $u$-channel $N\left(940\right)$ exchange terms, and (d) $s$- and $u$-channel $\mathrm{\Delta }\left(1232\right)$ exchange terms. The solid, dashed, and double lines represent baryons, mesons, and $\mathrm{\Delta }\left(1232\right)$, respectively. The dots and square represent the $\mathcal{O}\left(p^1\right)$ and $\mathcal{O}\left(p^2\right)$ vertices from chiral perturbation theory, respectively.

Figure 2

Scattering amplitude $P_{33}$ with parameter sets Naive and Constrained fitted to the WI 08 solution [34]. Two theoretical curves are very similar. The number of the plotted data points is $1/2$ of the total in the fits for a better visualization.

Figure 3

Loop function $G_{\pi N,L=1}$ around the energy from the nucleon mass $M_N$ to the $\pi N$ threshold. The imaginary part of the loop function takes the same value in both of the parameter sets Naive and Constrained.

Figure 4

Scattering amplitude $S_{11}$ with parameters fitted to the WI 08 solution [34]. The number of the plotted data points is $1/5$ of the total in the fits for a better visualization. The vertical dotted lines indicate the $\eta N,K\mathrm{\Lambda }$, and $K\mathrm{\Sigma }$ thresholds, respectively.