Associated Production of Charmonia-Bottomonia with Color-Octet Channels at the Z Factory, CEPC and FCC-ee

Within the nonrelativistic QCD (NRQCD) framework, we investigate the associated production of charmonia+bottomonia at future super $Z$ factory and at the CEPC/FCC-ee. The color-octet(CO) channels in the $γ^*/Z^0$-propagated process are considered besides the color-singlet(CS) channels. We find that the contributions of the CO states to the total cross section are dominant for almost all processes from the production threshold to the $Z^0$ mass. Thus, the comparison between the theoretical results and future data may give constraints to the CO matrix elements. In addition, we calculate the relativistic corrections to both the CS and CO channels, which decrease the cross sections significantly, with the $K$ factor $\simeq0.5$. The predicted events of ($J/ψ+Υ$, $Υ+η_c$) production, with $J/ψ, Υ$ reconstructed by lepton pair and $η_c$ reconstructed by hadronic decays, are (1,1) and (13,10) at the CEPC(2-yr) and FCC-ee(4-yr), respectively.

💡 Research Summary

The paper presents a comprehensive study of the associated production of charmonium and bottomonium states in electron–positron annihilation at future high‑luminosity facilities, namely a super‑Z factory, the Circular Electron‑Positron Collider (CEPC) and the Future Circular Collider in electron‑positron mode (FCC‑ee). Using the non‑relativistic QCD (NRQCD) factorization framework, the authors calculate the cross sections for a wide set of final‑state combinations such as $J/ψ+Υ$, $J/ψ+η_b$, $χ_{cJ}+Υ$, $η_c+Υ$, $h_c+Υ$, etc. Both color‑singlet (CS) and color‑octet (CO) channels are included, with the latter being mediated by QCD interactions at order $α^2α_s^2$, while CS contributions arise only through electroweak (EW) processes at order $α^4$.

A key finding is that, from the production threshold up to the $Z^0$ pole, CO channels dominate the total cross section for almost every process, contributing 70–95 % of the rate. This dominance stems from the severe suppression of the CS mechanism in $e^+e^-$ collisions, where the photon or $Z$ boson must fragment into a heavy‑quark pair in a color‑singlet configuration. The authors also incorporate relativistic corrections of order $v^2$ (the square of the relative velocity of the heavy quarks inside the quarkonium). By expanding the amplitudes in the internal momentum $q$ and performing the appropriate angular integrations, they find that these corrections reduce the LO cross sections by roughly a factor of two (a $K$‑factor ≈ 0.5). This effect is larger than the typical uncertainties associated with the long‑distance matrix elements (LDMEs).

The numerical analysis adopts state‑of‑the‑art input parameters: $\alpha_s=0.26$, $m_c=1.5$ GeV, $m_b=4.7$ GeV, $M_Z=91.1876$ GeV, and the standard set of LDMEs extracted from previous hadron‑collider fits. Heavy‑quark spin symmetry relations are used to relate matrix elements of different spin and orbital configurations. With these inputs, the authors obtain LO cross sections of order $0.5$ fb for $J/ψ+Υ$ and $Υ+η_c$ at the $Z$‑pole energy. Integrating over the projected luminosities—$L≈10^{34}$ cm⁻² s⁻¹ for CEPC (2 years) and $L≈10^{36}$ cm⁻² s⁻¹ for FCC‑ee (4 years)—they predict observable event yields: roughly one $J/ψ+Υ$ and one $Υ+η_c$ event at CEPC, and about 13 and 10 events respectively at FCC‑ee. The reconstruction strategy assumes leptonic decays of $J/ψ$ and $Υ$ (into $e^+e^-$ or $\mu^+\mu^-$) and hadronic decays of $η_c$, leading to realistic detection efficiencies.

Uncertainties are evaluated by varying the renormalization scale, the LDMEs, and the relativistic correction parameters $v_c^2=0.23$, $v_b^2=0.10$. The total theoretical error is estimated at the 30 % level, dominated by the poorly known CO LDMEs. Nonetheless, the predicted event numbers are large enough to allow a meaningful comparison with future data, which would in turn constrain the CO matrix elements and test the universality hypothesis of NRQCD.

In conclusion, the study demonstrates that color‑octet mechanisms are the primary driver of double‑quarkonium production at future $e^+e^-$ colliders operating at the $Z$ resonance, and that relativistic effects significantly suppress the rates. The results provide a clear physics case for measuring these processes at CEPC and FCC‑ee, offering a novel laboratory for probing the interplay of perturbative QCD, heavy‑quark dynamics, and electroweak interactions. The authors suggest further work including next‑to‑leading‑order QCD corrections, the impact of initial‑state radiation, and extensions to other quarkonium combinations such as $χ_{cJ}+χ_{bJ}$, which would enrich the phenomenological program of the next generation of electron–positron colliders.