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

TL;DR

This study applies NRQCD factorization to the associated production of charmonia+bottomonia in $e^+e^-$ annihilation at projected $Z$-factory energies (CEPC/FCC-ee), including both color-singlet and color-octet channels and relativistic corrections up to $O(v^2)$. It finds that color-octet fragmentation channels overwhelmingly dominate the cross sections across most processes, offering a strong avenue to constrain CO LDMEs and test HQSS relations, with relativistic corrections reducing LO predictions by about a factor of two. The authors provide detailed predictions for cross sections and event yields at CEPC and FCC-ee, discuss uncertainties from scale, masses, and LDMEs, and emphasize the potential to discriminate COM contributions in both near-threshold and high-energy regimes. They also highlight the need for higher-order QCD corrections (e.g., NNLO) to further refine the theoretical predictions.

Abstract

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.

Figures (9)

• Figure 1: Feynman diagrams for $e^-(p_1)+e^+(p_2)\rightarrow \gamma^*/Z^0\rightarrow H_1(p_3)+H_2(p_4)$ at the tree level. $H_1, H_2$ is charmonia and bottomonia, respectively. The permutation diagrams can be obtained by reversing the quark line. (a) and (b) diagrams belong to CO (QCD) channels, (c) and (d) diagrams belong to CS (EW) channels.
• Figure 2: (Color online) Cross section $~\sigma~$ versus c.m. energy $E_{cm}$$(E_{cm}=\sqrt{s})$. The solid line represents LO and dashed line represents NLO($v^2$) result. The red line represent CS channel, green line represent total CO channels, black line represent the sum of CS and CO. And there are only CO channels for the last $\eta_Q+\chi_{Q'J}, \eta_Q+h_Q', \eta_c+\eta_b$ processes.
• Figure 3: $\sigma_{CO}/\sigma_{Total}$ as a function of c.m. energy $E_{cm}$.
• Figure 4: (Color online) The K factors ($\sigma_{\text{NLO}(v^2)}/\sigma_{\text{LO}}$) as functions of c.m. energy $E_{cm}$.
• Figure 5: (Color online) The differential cross sections $d\sigma/dcos\theta$ at $\sqrt{s}$=$m_Z$. The solid line represents LO and dashed line represents NLO($v^2$) result.