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

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