  In this paper, we study the $X(3860)$ production associated with $J/\psi$ via
$e^{+}e^{-}$ annihilation at the next-to-leading-order (NLO) QCD accuracy,
within the nonrelativistic QCD (NRQCD) framework. Under the hypothesis of
$J^{PC}_{X(3860)}=0^{++}$, the predicted total cross sections agree well with
the Belle's measurements at $\sqrt{s}=\Upsilon(4S,5S)$, while the theoretical
results given by the $2^{++}$ hypothesis significantly undershoot the data.
This is consistent with the Belle's conclusion that the $0^{++}$ hypothesis is
favored over the $2^{++}$ hypothesis for $X(3860)$. Despite the agreement of
the total cross sections, the NRQCD predictions seem to be incompatible with
the Belle-measured $J/\psi$ angular distributions. We also perform the NLO
calculations of $\sigma_{e^{+}e^{-} \to J/\psi+X(3940)}$ on the assumption of
$J^{PC}_{X(3940)}=0^{-+}$, discovering that the NRQCD predictions that coincide
with the light-cone results are in good agreement with the experiment.
