Compositeness of baryonic resonances: Application to the Δ(1232), N(1535), and N(1650) 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 Delta(1232), N(1535), and N(1650) 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 Delta(1232) resonance is evaluated in the pi N single-channel scattering, and we find that the pi N component inside Delta(1232) in the present framework is non-negligible, which supports the previous work. On the other hand, the compositeness for the N(1535) and N(1650) resonances is evaluated in a coupled-channel approach, resulting that the pi N, eta N, K Lambda, and K Sigma components are negligible for these resonances.