We develop a theoretical framework to investigate the two-body composite structure of a resonance as well as a bound state from its wave function. For this purpose, we introduce both one-body bare states and two-body scattering states, and define the
compositeness as a fraction of the contribution of the two-body wave function to the normalization of the total wave function. Writing down explicitly the wave function for a resonance state obtained with a general separable interaction, we formulate the compositeness in terms of the position of the resonance pole, the residue of the scattering amplitude at the pole and the derivative of the Green function of the free two-body scattering system. At the same time, our formulation provides the elementariness expressed with the resonance properties and the two-body effective interaction, and confirms the sum rule showing that the summation of the compositeness and elementariness gives unity. In this formulation the Weinbergs relation for the scattering length and effective range can be derived in the weak binding limit. The extension to the resonance states is performed with the Gamow vector, and a relativistic formulation is also established. As its applications, we study the compositeness of the $Lambda (1405)$ resonance and the light scalar and vector mesons described with refined amplitudes in coupled-channel models with interactions up to the next to leading order in chiral perturbation theory. We find that $Lambda (1405)$ and $f_{0}(980)$ are dominated by the $bar{K} N$ and $K bar{K}$ composite states, respectively, while the vector mesons $rho (770)$ and $K^{ast} (892)$ are elementary. We also briefly discuss the compositeness of $N (1535)$ and $Lambda (1670)$ obtained in a leading-order chiral unitary approach.
The structures of the hyperon resonance $Lambda (1405)$ and the scalar mesons $sigma$, $f_{0}(980)$, and $a_{0}(980)$ are investigated based on the coupled-channels chiral dynamics with finite volume effect. The finite volume effect is utilized to ex
tract the coupling constant, compositeness, and mean squared distance between two constituents of a Feshbach resonance state as well as a stable bound state. In this framework, the real-valued size of the resonance can be defined from the downward shift of the resonance pole according to the decreasing finite box size $L$ on a given closed channel. As a result, we observe that, when putting the $bar{K}N$ and $Kbar{K}$ channels into a finite box while other channels being unchanged, the poles of the higher $Lambda (1405)$ and $f_{0}(980)$ move to lower energies while other poles do not show downward mass shift, which implies large $bar{K}N$ and $Kbar{K}$ components inside higher $Lambda (1405)$ and $f_{0}(980)$, respectively. Extracting structures of $Lambda (1405)$ and $f_{0}(980)$ in our method, we find that the compositeness of $bar{K}N$ ($Kbar{K}$) inside $Lambda (1405)$ [$f_{0}(980)$] is 0.82-1.03 (0.73-0.97) and the mean distance between two constituents is evaluated as 1.7-1.9 fm (2.6-3.0 fm).
