We present an analysis of the preliminary experimental data of direct $J/\psi$ production in $e^{+}{e}^{-}$ processes at low energies. We find that the color-octet contributions are crucially important to the cross section in this energy region, and their inclusion produces a good description of the data. By fitting to the data, we extract the individual values of two color-octet matrix elements: $〈{\mathcal{O}}_8^{\psi }{(}^1{S}_0)〉\approx 1.1\times {10}^{-2}$ ${\mathrm{GeV}}^3$; $〈{\mathcal{O}}_8^{\psi }{(}^3{P}_0)〉{/m}_c^2\approx 7.4\times {10}^{-3}$ ${\mathrm{GeV}}^3$. We discuss the allowed range of the two matrix elements constrained by the theoretical uncertainties. We find that $〈{\mathcal{O}}_8^{\psi }{(}^1{S}_0)〉$ is poorly determined because it is sensitive to the variation of the choice of $m_c,$ ${\alpha }_s$ and $〈{\mathcal{O}}_1^{\psi }{(}^3{S}_1)〉$. However, $〈{\mathcal{O}}_8^{\psi }{(}^3{P}_0)〉{/m}_c^2$ is quite stable [about $\left(6\mathrm{}-9\right)\times {10}^{-3}$ ${\mathrm{GeV}}^3$] when the parameters vary in reasonable ranges. The uncertainties due to large experimental errors are also discussed.

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