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Compositeness of baryonic resonances: Applications to the Delta(1232), N(1535), and N(1650) resonances

We present a formulation of the compositeness for baryonic resonances in order 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 $Δ(1232)$, $N (1535)$, and $N (1650)$ resonances from precise $πN$ scattering amplitudes in a unitarized chiral framework with the interaction up to the next-to-leading order in chiral perturbation theory. The $πN$ compositeness for the $Δ(1232)$ resonance is evaluated in the $πN$ single-channel scattering, and we find that the $πN$ component inside $Δ(1232)$ in the present framework is nonnegligible, which supports the previous work. On the other hand, the compositeness for the $N (1535)$ and $N (1650)$ resonances is evaluated in a coupled-channels approach, resulting that the $πN$, $ηN$, $K Λ$ and $K Σ$ components are negligible for these resonances.