Bottomonium production associated with a photon at a high luminosity $e^{+}{e}^{-}$ collider with next-to-leading order QCD corrections

Abstract

We make a detailed discussion on the one-loop QCD correction to the bottomonium production associated with a photon, i.e. via the channel $e^{+}{e}^{-}\to {\gamma }^{\ast }/Z^0\to |H_{b\bar{b}}⟩+\gamma$, where $|H_{b\bar{b}}⟩$ stands for the color-singlet bottomonium state as ${\eta }_b$, ${\rm Y}$, $h_b$ 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. Further more, 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 $E_{cm}\sim m_Z$. While, for the ${\rm Y}$ production, such resonance effect shall be smeared by a large $t(u)$-channel contribution that dominant over the $s$-channel one. Theoretical uncertainties caused by slight change of $E_{cm}$, the $b$-quark mass, the renormalization scale and 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 ${\rm Y}$, which have large signal significance. Summing up all bottomonium states, we shall totally 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.

Comments: 15 pages, 9 figures. Discussions improved, to be published in Phys.Rev.D