Double heavy quarkonia production with color-octet channels at Z factory and at the CEPC/FCC-ee

Abstract

Within the NRQCD framework, we calculate the exclusive production of double heavy quarkonium(double charmonium and double bottomonium) at future super $Z$ factory and at the CEPC/FCC-ee. The color-octet(CO) channels in the $\gamma^*/Z^*$-propagated process are considered along with the color-singlet(CS) channels. We found that the contributions of CO states to the total cross section are significant or dominant for many processes within energy region at $Z$ factory and at the CEPC/FCC-ee. The experimental measurements will help us to verify the CO mechanism. Among these CO channels, the gluon fragmentation into $^3S_1^{8}$ states is most important. Thus, the comparison between the theoretical results and future data will give a strong constraint to the matrix elements $\langle\mathcal{O}\left(^3S_1^{[8]}\right)\rangle$. Additionally, we consider the relativistic corrections to both the CS and CO channels which decrease the cross sections significantly. Specially, the $K$ factors are about $0.5$ for most charmonium channels. We get estimates of the events for double heavy quarkonium production. The final events of $J/\psi+\eta_c$, $J/\psi+J/\psi$, $\Upsilon+\eta_b$, $\Upsilon+\Upsilon$ production would be (22, 570, 71, 61) and (206, 5343, 665, 576) at the CEPC (2-year) and at the FCC-ee (4-year) for the $Z$ factory mode, respectively.