  In this article, we construct both the
$[sc]_T[\bar{s}\bar{c}]_A+[sc]_A[\bar{s}\bar{c}]_T$ type and
$[sc]_T[\bar{s}\bar{c}]_V-[sc]_V[\bar{s}\bar{c}]_T$ type axialvector currents
with $J^{PC}=1^{++}$ to study the mass of the $X(4140)$ with the QCD sum rules.
The predicted masses support assigning the $X(4140)$ to be the
$[sc]_T[\bar{s}\bar{c}]_V-[sc]_V[\bar{s}\bar{c}]_T$ type axialvector tetraquark
state. Then we study the hadronic coupling constant $g_{XJ/\psi\phi}$ with the
QCD sum rules based on solid quark-hadron duality, and obtain the decay width
$\Gamma(X(4140)\to J/\psi \phi)=86.9\pm22.6\,{\rm{MeV}}$, which is in excellent
agreement with the experimental data $83\pm 21^{+21}_{-14} {\mbox{ MeV}}$ from
the LHCb collaboration.
