  Recently, the CMS and ATLAS collaborations have reported their $J/\psi J/\psi
$ and $J/\psi \psi^\prime$ invariant mass distributions, respectively, for
searching for fully charmed tetraquarks. Both of them reported the existence of
a peak structure around $7.2~\gev$. In this article, we exhibit the role of the
$\eta_c(2S)\eta_c(2S)$ channel, which is close to this peak position, by
studying the $J/\psi J/\psi $ and $J/\psi \psi^\prime$ invariant mass
distributions for the potential quantum numbers $J^{PC}=0^{++}$ or $2^{++}$ by
considering both the coherence and incoherence with the background
contribution. All these frameworks can describe the experimental data very
well, however with different pole structures. For instance, in the case of
$2^{++}$ description of these structures, there always exists a pole slightly
below the $J/\psi J/\psi$ threshold. For the $0^{++}$ case, a similar pole can
also be found below the $J/\psi J/\psi$ threshold however with a much lower
mass. More importantly, the number of the poles is found to be case-dependent.
For the $0^{++}$ ($2^{++}$) case, the peak structure around $7.2~\gev$ can
(cannot) be produced due to the presence (absence) of the
$\eta_c(2S)\eta_c(2S)$ channel. Although the pole positions are case-dependent,
the relation between the peak structure in the $J/\psi J/\psi$ invariant mass
distribution and the dip structure in the $J/\psi \psi^\prime$ invariant mass
distribution around $7.2~\gev$ is unambiguous. We suggest experimentalists to
detailed scan both the $J/\psi J/\psi$ and the $J/\psi \psi^\prime$ invariant
mass distributions, especially around $7.2~\gev$ to probe the nature of the
third fully charmed state.
