Production of Prompt Charmonia in $e^+e^-$ Annihilation at $\sqrt{s}\approx 10.6$ GeV

The production of prompt $J/\psi$, $\psi(2S)$, $\chi_{c1}$ and $\chi_{c2}$ is studied using a $32.4 fb^{-1}$ data sample collected with the Belle detector at the $\Upsilon(4S)$ and 60 MeV below the resonance. The yield of prompt $J/\psi$ mesons in the $\Upsilon(4S)$ sample is compatible with that of continuum production; we set an upper limit ${\cal B}(\Upsilon(4S) \to J/\psi X) < 1.9 \times 10^{-4}$ at the 95% confidence level, and find $\sigma(e^{+}e^{-} \to J/\psi X)=1.47\pm 0.10 \pm 0.13$ pb. The cross-sections for prompt $\psi(2S)$ and direct $J/\psi$ are measured. The $J/\psi$ momentum spectrum, production angle distribution and polarization are studied.