We study the possibility of the scalar charmonium and glueball mixing in $e^+ e^-$ annihilation at $sqrt{s}=10.6$ GeV. The effects can be used to explain the unexpected large cross section ($12pm 4$ fb) and the anomalous angular distribution ($alpha= -1.1^{+0.8}_{-0.6}$) of the exclusive $e^+e^-to J/psichi_{c0}$ process observed by Belle experiments at KEKB. We calculate the helicity amplitudes for the process $e^+ e^- to J/psi H(0^{++})$ in NRQCD, where $H(0^{++})$ is the mixed state. We present a detailed analysis on the total cross section and various angular asymmetries which could be useful to reveal the existence of the scalar glueball state.
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