Abstract
We suggest searching for excited charmonium states in ${e}^{+}{e}^{\ensuremath{-}}$ annihilation via double charmonium production at $\sqrt{s}=10.6\text{ }\text{ }\mathrm{GeV}$ with $B$ factories, based on a more complete leading order calculation including both QCD and QED contributions for various processes. In particular, for the $C=+$ states, the ${\ensuremath{\chi}}_{c0}(nP)\text{ }(n=2,3)$ and ${\ensuremath{\eta}}_{c}(mS)\text{ }(m=3,4)$ may have appreciable potentials to be observed, while for the $C=\ensuremath{-}$ states, the ${\ensuremath{\eta}}_{c}{h}_{c}$ production and especially the ${\ensuremath{\chi}}_{c1}{h}_{c}$ production might provide opportunities for observing the ${h}_{c}$ with higher statistics in the future. A brief discussion for the X(3940) observed in the double charmonium production is included.
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