Relativistic correction to $e^{+}e^{-}\to J/\psi+gg$ at $B$ factories and constraint on color-octet matrix elements

We calculate the relativistic correction to $J/\psi$ production in the color-singlet process $e^{+}e^{-}\to J/\psi+gg$ at B-factories. We employ the non-relativistic QCD (NRQCD) factorization approach, where the short-distance coefficients are calculated perturbatively and the long-distance matrix elements are extracted from the decays of $J/\psi$ into $e^{+}e^{-}$ and light hadrons. We find that the $O(v^2)$ relativistic correction can enhance the cross section by a factor of 20-30%, comparable to the enhancement due to the $O(\alpha_s)$ radiative correction obtained earlier. Combining the relativistic correction with the QCD radiative correction, we find that the color-singlet contribution to $e^{+}e^{-}\to J/\psi+gg$ can saturate the latest observed cross section $\sigma(e^{+}e^{-}\to J/\psi+X_{\mathrm{non-c\bar{c}}})=0.43 \pm0.09\pm0.09$ pb by Belle, thus leaving little room to the color-octet contributions. This gives a very stringent constraint on the color-octet contribution, and may imply that the values of color-octet matrix elements are much smaller than expected earlier by using the naive velocity scaling rules or extracted from fitting experimental data with the leading-order calculations.