  In this article, we take the $Y(4260/4220)$ as the vector tetraquark state
with $J^{PC}=1^{--}$, and construct the
$C\gamma_5\otimes\stackrel{\leftrightarrow}{\partial}_\mu\otimes \gamma_5C$
type diquark-antidiquark current to study its mass and pole residue with the
QCD sum rules in details by taking into account the vacuum condensates up to
dimension 10 in a consistent way. The predicted mass
$M_{Y}=4.24\pm0.10\,\rm{GeV}$ is in excellent agreement with experimental data
and supports assigning the $Y(4260/4220)$ to be the
$C\gamma_5\otimes\stackrel{\leftrightarrow}{\partial}_\mu\otimes \gamma_5C$
type vector tetraquark state, and disfavors assigning the $Z_c(4100)$ to be the
$C\gamma_5\otimes\stackrel{\leftrightarrow}{\partial}_\mu\otimes \gamma_5C$
type vector tetraquark state. It is the first time that the QCD sum rules have
reproduced the mass of the $Y(4260/4220)$ as a vector tetraquark state.
