Bottomonium production associated with a photon at a high luminosity $e^{+}{e}^{-}$ collider with the one-loop QCD correction

We make a detailed discussion on the one-loop QCD correction to the bottomonium production associated with a photon, i.e., via the channel $e^{+}{e}^{-}\to {\gamma }^{*}/Z^0\to |H_{b\overline{b}}⟩+\gamma$, where $|H_{b\overline{b}}⟩$ stands for the color-singlet bottomonium state as ${\eta }_b$, $\mathrm{{\rm Y}}$, $h_b$ or ${\chi }_{bJ}$ ($J=0$, 1 or 2), respectively. At the super $Z$ factory with the collision energy $E_{cm}\sim m_Z$, by summing up the cross sections for all bottomonium states, we obtain a large one-loop QCD correction, i.e., $|R|\sim 30\%$. This ensures the necessity and importance of the one-loop QCD corrections for the present processes. Furthermore, for the ${\eta }_b$, $h_b$, and ${\chi }_{bJ}$ production, their cross sections are dominated by the $s$-channel diagrams and are enhanced by the $Z^0$ boson resonance effect when $E_{cm}\sim m_Z$; whereas, for the $\mathrm{{\rm Y}}$ production, such a resonance effect shall be smeared by a large $t(u)$-channel contribution that is dominant over the $s$-channel one. Theoretical uncertainties caused by slight change of $E_{cm}$, the $b$-quark mass, the renormalization scale, etc. have been presented. At the super $Z$ factory with a high luminosity up to $\mathcal{L}=10^{36}{\mathrm{cm}}^{-2}{\mathrm{s}}^{-1}$, the bottomonium plus one photon events are sizable, especially for ${\eta }_b$ and $\mathrm{{\rm Y}}$, which have large signal significance. Summing up all bottomonium states, we shall in total have $\sim 3.8\times 10^5$ bottomonium events in one operation year. So the super $Z$ factory shall provide a good platform for studying the bottomonium properties.

Figure 1

Tree-level Feynman diagrams for $e^{+}{e}^{-}\to {\gamma }^{*}/Z^0\to |H_{b\overline{b}}⟩+\gamma$, where $|H_{b\overline{b}}⟩$ denotes ${\eta }_b$, $\mathrm{{\rm Y}}$, $h_b$, and ${\chi }_{bJ}$ ($J=0$,1,2), respectively. The left two diagrams are $t(u)$ channel, and the leaving ones are $s$ channel.

Figure 2

One-loop QCD correction to the $s$-channel diagrams of $e^{+}{e}^{-}\to {\gamma }^{*}/Z^0\to |H_{b\overline{b}}⟩+\gamma$, where $|H_{b\overline{b}}⟩$ denotes ${\eta }_b$, $\mathrm{{\rm Y}}$, $h_b$, and ${\chi }_{bJ}$ $(J=0,1,2)$, respectively.

Figure 3

One-loop QCD correction to the $t(u)$-channel diagrams of $e^{+}{e}^{-}\to {\gamma }^{*}/Z^0\to |H_{b\overline{b}}⟩+\gamma$, where $|H_{b\overline{b}}⟩$ denotes ${\eta }_b$, $\mathrm{{\rm Y}}$, $h_b$, and ${\chi }_{bJ}$ $(J=0,1,2)$, respectively.

Figure 4

Total cross sections with the one-loop QCD correction to $e^{+}{e}^{-}\to Z^0/{\gamma }^{*}\to |H_{b\overline{b}}⟩+\gamma$ versus the collision energy $E_{cm}$. ${\mu }_R=2m_b$. Here, all $P$-wave bottomonium states’ cross sections have similar behaviors and have been summed up.

Figure 5

Differential cross sections for $e^{+}{e}^{-}\to {\gamma }^{*}/Z^0\to |H_{b\overline{b}}⟩+\gamma$ versus $\cos \theta$. Here, all $P$-wave bottomonium states’ cross sections have similar behaviors and have been summed up. We use the symbol tree to stand for the tree-level differential cross section and the symbol QCD to stand for the differential cross section with the one-loop QCD correction. For the ${\eta }_b$ production, its tree-level and one-loop curves almost coincide with each other. $\sqrt{s}=m_Z$, $m_b=4.9\mathrm{GeV}$, and ${\mu }_R=2m_b$.

Figure 6

Total cross sections for the bottomonium production via $e^{+}{e}^{-}\to {\gamma }^{*}/Z^0\to |H_{b\overline{b}}⟩+\gamma$ with the one-loop QCD correction versus the scale ${\mu }_R$, where all bottomonium states have been summed up. As a comparison, we present the cross sections under both the conventional scale setting and the improved scale setting, respectively. $\sqrt{s}=m_Z$ and ${\mu }_R=2m_b$.

Figure 7

Tree-level Feynman diagrams for $e^{+}{e}^{-}\to {\gamma }^{*}/Z^0\to q+\overline{q}+\gamma$, where $q$ stands for $u$, $d$, $s$, $c$, or $b$ quark, respectively.

Figure 8

A comparison of $s$- and $t(u)$-channel differential cross sections with the one-loop QCD correction for $\mathrm{\Upsilon }$ production via $e^{+}{e}^{-}\to {\gamma }^{*}/Z^0\to \mathrm{\Upsilon }+\gamma$. $E_{cm}=m_Z$, $m_b=4.9\mathrm{GeV}$, and ${\mu }_R=2m_b$.

Figure 9

A comparison of $s$- and $t(u)$-channel differential cross sections with the one-loop QCD correction for $J/\psi$ production via $e^{+}{e}^{-}\to {\gamma }^{*}/Z^0\to J/\psi +\gamma$. $E_{cm}=m_Z$, $m_c=1.5\mathrm{GeV}$, and ${\mu }_R=2m_c$.