Further study of $c\bar{c}c\bar{c}$ system within a chiral quark model

Inspired by the recent Altas and CMS experiments on the invariant mass spectrum of $J/\psi J/\psi$, we systematically study the $c\bar{c}c\bar{c}$ system of $J^{P}=0^{+}$. In the framework of chiral quark model, we have carried out bound-state calculation and resonance-state calculation respectively by using Real-scaling method. The results of bound-state calculation show that there are no bound states in the $c\bar{c}c\bar{c}$ with $0^{+}$ system. The resonance-state calculation shows that there are four possible stable resonances: $R(6920)$, $R(7000)$, $R(7080)$ and $R(7160)$. $R(6920)$ and $R(7160)$ are experimental candidates for $X(6900)$ and $X(7200)$, whose main decay channel is $J/\psi J/\psi$. It is important to note that the another major decay channel of $R(7160)$ is $\chi_{c0} \chi_{c0} $, and the $\chi_{c0} \chi_{c0} $ is also the main decay channel of $R(7000)$, $R(7080)$. Therefore, we propose to search experimentally for these two predicted resonances in the $\chi_{c0} \chi_{c0}$ invariant mass spectrum.