Size measurement of dynamically generated hadronic resonances with finite volume effect

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 extract 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 $K\bar{K}$ 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 $K\bar{K}$ 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 $\bar{K}N$ ($K\bar{K}$) 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).