  In this article, we take the $X(4140)$ as the diquark-antidiquark type
$cs\bar{c}\bar{s}$ tetraquark state with $J^{PC}=1^{++}$, and study the mass
and pole residue with the QCD sum rules in details by constructing two types
interpolating currents. The numerical results
$M_{X_{L,+}}=3.95\pm0.09\,\rm{GeV}$ and $M_{X_{H,+}}=5.00\pm0.10\,\rm{GeV}$
disfavor assigning the $X(4140)$ to be the $J^{PC}=1^{++}$ diquark-antidiquark
type $cs\bar{c}\bar{s}$ tetraquark state. Moreover, we obtain the masses of the
$J^{PC}=1^{+-}$ diquark-antidiquark type $cs\bar{c}\bar{s}$ tetraquark states
as a byproduct. The present predictions can be confronted to the experimental
data in the future.
