  Stimulated by the recent experimental results on the fully-charm tetraquark
states, we systematically calculate the mass spectra of the fully-charm
tetraquark states in $8_{[c\bar{c}]}\otimes8_{[c\bar{c}]}$ color configuration
via QCD sum rules. By constructing nine $8_{[c\bar{c}]}\otimes8_{[c\bar{c}]}$
type currents with quantum numbers $J^{PC}=0^{-+},0^{--},1^{-+},1^{+-},1^{--}$
and $2^{++}$, we perform analytic calculation up to dimension six in the
Operator Product Expansion (OPE). We find the fully-charm tetraquark states
with $J^{PC}=1^{+-},2^{++}$ lie around 6.48 $\sim$ 6.62 GeV while the
fully-charm tetraquark states with $J^{PC}=0^{-+},0^{--},1^{--},1^{-+}$ are
about 6.85 $\sim$ 7.02 GeV. Notably, the mass predictions for the
$c\bar{c}c\bar{c}$ tetraquarks, specifically those with $J^{PC}=2^{++}$, align
with the broad structure identified by LHCb. Moreover, the masses of
fully-charm tetraquarks with $J^{PC}=0^{-+}$ and $1^{-+}$ are anticipated to
match closely with the mass of X(6900), considering the margin of error. Such
findings hint at the presence of some $8_{[c\bar{c}]}\otimes8_{[c\bar{c}]}$
components within the di-$J/\psi$ structures observed by LHCb. The predictions
for tetraquark states with $J^{PC}=0^{--},1^{+-},1^{--}$ may be accessible in
the future BelleII, Super-B, PANDA, and LHCb experiments.
