Effective-range-expansion study of near threshold heavy-flavor resonances

In this work we study the resonances near the thresholds of the open heavy-flavor hadrons using the effective-range-expansion method. The unitarity, analyticity and compositeness coefficient are also taken into account in our theoretical formalism. We consider the $Z_c(3900)$, $X(4020)$, ${\chi }_{c1}(4140)$, $\psi (4260)$ and $\psi (4660)$. The scattering lengths and effective ranges from the relevant elastic $S$-wave scattering amplitudes are determined. Tentative discussions on the inner structures of the aforementioned resonances are given.