Hunting for states in the recent LHCb di-$J/\psi$ invariant mass spectrum

Partial wave analysis is performed, with effective potentials as dynamical inputs, to scrutinize the recent LHCb data on the di-$J/\psi$ invariant mass spectrum. Coupled-channel effects are incorporated in the production amplitude via final state interactions. The LHCb data can be well described. A dynamical generated pole structure, which can be identified as the $X(6900)$ state, is found. Our analysis also provides hints for the existence of three other possible states: a bound state $X(6200)$, a broad resonant state $X(6680)$ and a narrow resonant state $X(7200)$. The $J^{PC}$ quantum numbers of the $X(6680)$ and $X(6900)$ states should be $2^{++}$, while the $X(6200)$ and $X(7200)$ states prefer $0^{++}$. To determine the above states more precisely, more experimental data for the channels, such as $J/\psi\psi(2S)$, $J/\psi\psi(3770)$, di-$\psi(2S)$, are required.