Insights into the inner structures of the fully charmed tetraquark state $X(6900)$

The recently discovered fully charmed tetraquark candidate $X(6900)$ is analyzed within the frameworks of effective-range expansion, compositeness relation and width saturation, and a coupled multichannel dynamical study. By taking into account constraints from heavy-quark spin symmetry, the coupled-channel amplitude including the $J/\psi J/\psi,~ \chi_{c0}\chi_{c0}$ and $\chi_{c1}\chi_{c1}$ is constructed to fit the experimental di-$J/\psi$ event distributions around the energy region near $6.9$ GeV. Another dynamical two-coupled-channel amplitude with the $J/\psi J/\psi$ and $\psi(3770) J/\psi$ is also considered to describe the same datasets. The three different theoretical approaches lead to similar conclusions that the two-meson components do not play dominant roles in the $X(6900)$. Our determinations of the resonance poles in the complex energy plane from the refined coupled-channel study are found to be consistent with the experimental analyses. The coupled-channel amplitudes also have another pole corresponding to a narrow resonance $X(6825)$ that we predict sitting below the $\chi_{c0}\chi_{c0}$ threshold and of molecular origin. We give predictions to the line shapes of the $\chi_{c0}\chi_{c0}$ and $\chi_{c1}\chi_{c1}$ channels, which could provide a useful guide for future experimental measurements.