We make a comparative and comprehensive study on the charmonium exclusive productions at the $e^+e^-$ collider with the collision energy either round the $Z^0$-boson mass for a super $Z$ factory or equals to 10.6 GeV for the $B$ factories as Belle and BABAR. We study the total cross sections for the charmonium production via the exclusive processes $e^+e^- \to \gamma^*/Z^0 \to H_{ 1}+H_{ 2}$ and $e^+e^- \to \gamma^*/Z^0 \to H_{ 1} +\gamma$, where $H_{ 1}$ and $H_{ 2}$ represent the dominant color-singlet $S$-wave and $P$-wave charmonium states respectively. Total cross sections versus the $e^+e^-$ collision energy $\sqrt{ s}$, together with their uncertainties, are presented, which clearly show the relative importance of these channels. At the $B$ factory, the production channels via the virtual $\gamma^*$ propagator are dominant over the channels via the $Z^0$ propagator by about four orders. While, at the super $Z$ factory, due to the $Z^0$-boson resonance effect, the $Z^0$ boson channels shall provide sizable or even dominant contributions in comparison to the channels via the $\gamma^*$ propagator. Sizable exclusive charmonium events can be produced at the super $Z$ factory with high luminocity up to $10^{ 36}{ \rm cm}^{ -2}{ \rm s}^{ -1}$, especially for the channel of $e^+e^- \to Z^0 \to H_{ 1} +\gamma$, e.g. by taking $m_c=1.50\pm0.20$ GeV, we shall have $(5.0^{ +0.8}_{ -0.6})\times10^4$ $J/\psi$, $(7.5^{ +1.1}_{ -0.9})\times10^3$ $\eta_c$, $(6.2^{ +3.3}_{ -1.9})\times10^3$ $h_{ c}$, $(3.1^{ +1.7}_{ -0.9})\times10^2$ $\chi_{ c0}$, $(2.2^{ +1.0}_{ -0.4})\times10^3$ $\chi_{ c1}$, and $(7.7^{ +4.1}_{ -2.4})\times10^2$ $\chi_{ c2}$ events by one operation year. Thus, in addition to the $B$ factories as BABAR and Belle, such a super $Z$ factory shall provide another useful platform for studying the heavy quarkonium properties and for testing QCD theories.
Categories: hep-ph