The Study of Color Octet Matrix Elements Through \(J/\psi\) and \(\Upsilon\) Production in \(e^{+}e^{-}\) Annihilation }

The next-to-leading radiative corrections and relativistic corrections to the inclusive heavy quarkonium production in \(e^{+}e^{-}\rightarrow J/\psi+LH\) and \(e^{+}e^{-}\rightarrow \Upsilon+LH\) are calculated within non-relativistic QCD frame. And the cross section with different the long distance matric elements has been given. We find that the inclusive heavy quarkonium production process in \(e^{+}e^{-}\) annihilation can give a strong restrictions on the color octet matrix elements. And the long distance matric elements of P-wave channel for \(J/\psi(\psi^\prime)\) would be very small. More measurement of \(J/\psi(\psi^\prime)\) and \(\Upsilon\) production associated with light hadron at BESIII, Belle, and BelleII can give more information about the color octet matrix elements.