Excited heavy quarkonium production via $Z^0$ decays at a high luminosity collider

We present a systematic study of the production of the heavy quarkonium, i.e., $|(c\overline{c})[n]⟩$, $|(b\overline{c})[n]⟩$ (or $|(c\overline{b})[n]⟩$), and $|(b\overline{b})[n]⟩$ quarkonium [$|(Q\overline{Q^{\prime }})[n]⟩$ quarkonium for short], through $Z^0$ boson semiexclusive decays with new parameters [1] for the heavy quarkonium under the framework of the nonrelativistic QCD, where $[n]$ stands for $n{{}^{1}S}_0$, $n{{}^{3}S}_1$, $n{{}^{1}P}_0$, $n{{}^{3}P}_J$ [$n=1,\dots ,6$; $J=(0,1,2)$]. “Improved trace technology” is adopted to derive the simplified analytic expressions at the amplitude level, which shall be useful for dealing with these decay channels. If all higher $|(Q\overline{Q^{\prime }})[n]⟩$ quarkonium states decay to the ground state $|(Q\overline{Q^{\prime }})[1{{}^{1}S}_0]⟩$ with 100% efficiency via electromagnetic or hadronic interactions, we obtain $\mathrm{\Gamma }(Z^0\to |(c\overline{c})[1{{}^{1}S}_0]⟩)=1476\mathrm{KeV}$, $\mathrm{\Gamma }(Z^0\to |(b\overline{c})[1{{}^{1}S}_0]⟩)=1485\mathrm{KeV}$, $\mathrm{\Gamma }(Z^0\to |(b\overline{b})[1{{}^{1}S}_0]⟩)=127.5\mathrm{KeV}$. At the LHC and ILC with the luminosity $\mathcal{L}\propto {10}^{34}{\mathrm{cm}}^{-2}{\mathrm{s}}^{-1}$, sizable heavy quarkonium events can be produced through $Z^0$ boson decays; i.e., about $5.9\times {10}^5$ $(c\overline{c})$, $6.0\times {10}^5$ $(b\overline{c})$ [or $(c\overline{b})$], $5.1\times {10}^4$ $(b\overline{b})$ events per year can be obtained.

Figure 1

Feynman diagrams for the process $Z^0(k)\to |(Q\overline{Q^{\prime }})[n]⟩(q_3)+\overline{Q}(q_2)+Q^{\prime }(q_1)$, where $|(Q\overline{Q^{\prime }})[n]⟩$ stands for $|(Q\overline{Q^{\prime }})[n{{}^{1}S}_0]⟩$, $|(Q\overline{Q^{\prime }})[n{{}^{3}S}_1]⟩$, $|(Q\overline{Q^{\prime }})[n{{}^{1}P}_1]⟩$, $|(Q\overline{Q^{\prime }})[n{{}^{3}P}_J]⟩$ quarkonium (with $n=1,\dots ,6$; $J=[0,1,2]$), respectively. $Q$ or $Q^{\prime }$ is the $c(\overline{c})$ or $b(\overline{b})$ quark, respectively.

Figure 2

Differential decay widths $d\mathrm{\Gamma }/d{s}_1$ (Left) and $d\mathrm{\Gamma }/d{s}_2$ (Right) for $Z^0\to |(c\overline{c})[n]⟩+\overline{c}c$, where the diamond line, the dotted line, the solid line, the dash-dotted line, and the crossed line are for $|(c\overline{c})[n]⟩$, where $n=1,\cdots ,5$, respectively. $|(c\overline{c})[n]⟩$ stands for the summed decay width of $[n1{{}^{1}S}_0]$, $[n{{}^{3}S}_1]$, $[n{{}^{1}P}_1]$, and $[n{{}^{3}P}_J]$ ($J=0,1,2$) at the same $n$th level.

Figure 3

Differential decay widths $d\mathrm{\Gamma }/d{s}_1$ (Left) and $d\mathrm{\Gamma }/d{s}_2$ (Right) for $Z^0\to |(b\overline{c})[n]⟩+\overline{b}c$, where the diamond line, the dotted line, the solid line, and the dash-dotted line are for $|(b\overline{c})[n]⟩$, where $n=1,\cdots ,4$, respectively. $|(b\overline{c})[n]⟩$ stands for the summed decay width of $[n{{}^{1}S}_0]$, $[n{{}^{3}S}_1]$, $[n{{}^{1}P}_1]$, and $[n{{}^{3}P}_J]$ ($J=0,1,2$) at the same $n$th level.

Figure 4

Differential decay widths $d\mathrm{\Gamma }/d{s}_1$ (Left) and $d\mathrm{\Gamma }/d{s}_2$ (Right) for $Z^0\to |(b\overline{b})[n]⟩+\overline{b}b$, where the diamond line, the dotted line, the solid line, the dash-dotted line, and the crossed line are for $|(b\overline{b})[n]⟩$, where $n=1,\dots ,5$, respectively. $|(b\overline{b})[n]⟩$ stands for the summed decay width of $[n{{}^{1}S}_0]$, $[n{{}^{3}S}_1]$, $[n{{}^{1}P}_1]$, and $[n{{}^{3}P}_J]$ ($J=0,1,2$) at the same $n$th level.

Figure 5

Differential decay widths $d\mathrm{\Gamma }/d\cos {\theta }_{12}$ (Left) and $d\mathrm{\Gamma }/d\cos {\theta }_{13}$ (Right) for $Z^0\to |(c\overline{c})[n]⟩+\overline{c}c$, where the diamond line, the dotted line, the solid line, the dash-dotted line, and the crossed line are for $|(c\overline{c})[n]⟩$, where $n=1,\dots ,5$, respectively. $|(c\overline{c})[n]⟩$ stands for the summed decay width of $[n{{}^{1}S}_0]$, $[n{{}^{3}S}_1]$, $[n{{}^{1}P}_1]$, and $[n{{}^{3}P}_J]$ ($J=0,1,2$) at the same $n$th level.

Figure 6

Differential decay widths $d\mathrm{\Gamma }/d\cos {\theta }_{12}$ (Left) and $d\mathrm{\Gamma }/d\cos {\theta }_{13}$ (Right) for $Z^0\to |(b\overline{c})[n]⟩+\overline{b}c$, where the diamond line, the dotted line, the solid line, and the dash-dotted line are for $|(b\overline{c})[n]⟩$, where $n=1,\dots ,4$, respectively. $|(b\overline{c})[n]⟩$ stands for the summed decay width of $[n{{}^{1}S}_0]$, $[n{{}^{3}S}_1]$, $[n{{}^{1}P}_1]$, and $[n{{}^{3}P}_J]$ ($J=0,1,2$) at the same $n$th level.

Figure 7

Differential decay widths $d\mathrm{\Gamma }/d\cos {\theta }_{12}$ (Up) and $d\mathrm{\Gamma }/d\cos {\theta }_{13}$ (Down) for $Z^0\to |(b\overline{b})[n]⟩+\overline{b}b$, where the diamond line, the dotted line, the solid line, the dash-dotted line, and the crossed line are for $|(b\overline{b})[n]⟩$, where $n=1,\dots ,5$, respectively. $|(b\overline{b})[n]⟩$ stands for the summed decay width of $[n{{}^{1}S}_0]$, $[n{{}^{3}S}_1]$, $[n{{}^{1}P}_1]$, and $[n{{}^{3}P}_J]$ ($J=0,1,2$) at the same $n$th level.