  In this paper, the color-octet long distance matrix elements are studied
though the inclusive $J/\psi$ production in $e^{+}e^{-}$ annihilation within
the non-relativistic QCD frame. The calculations are up-to next-to-leading
order with the radiative corrections and relativistic corrections at the
B-factory energy region and the near-threshold region of $4.6\sim5.6~{\rm
GeV}$. A constraint of the long distance matrix elements
($\langle^1S_0^{8}\rangle$, $\langle^3P_0^{8}\rangle$) is obtained. Through our
estimation, the P-wave color-octet matrix element ( $\langle0|^3P^8_0|0\rangle$
) should be the order of $0.005m_c^2~{\rm GeV}^3$ or less. The constraint
region is not compatible with the values of the long distance matrix elements
fitted at hadron colliders.
