Eur. Phys. J. C (2024) 84: 1000
https://doi.org/10.1140/epjc/s10052-024-13292-6

Regular Article

Production of double P-wave heavy quarkonia at a super Z factory

¹ Chongqing College of Mobile Telecommunications, 401520, Chongqing, China
² Chongqing Key Laboratory of Public Big Data Security Technology, 401520, Chongqing, China
³ School of Cybersecurity, Chengdu University of Information Technology, 610225, Chengdu, Sichuan, China

^a xiaosueer@163.com

Received: 22 April 2024
Accepted: 23 August 2024
Published online: 5 October 2024

Abstract

The production of double P-wave heavy quarkonia is studied systematically within the framework of non-relativistic quantum chromodynamics at the exclusive process of the electron–positron collision annihilation at the CM energy $\sqrt{s}=91.1876$ GeV of the $Z^0$ pole, i.e., double P-wave charmonia, double P-wave $B_c$ mesons, and double P-wave bottomonia in $e^{+}{e}^{-}\to {\gamma }^{\ast }/Z^0\to |(Q\overline{Q^{\prime }})[n]⟩+|(Q^{\prime }\overline{Q})[n^{\prime }]⟩$ ($Q/Q^{\prime }=c$- or b-quarks) at a future super Z factory, where [n] or $[n^{\prime }]$ stands for the color-singlet ${[}^1{P}_1]$ and ${[}^3{P}_J]$ ($J=0,1,2$) Fock states. As an improved trace technology is useful for calculating the complicated double P-wave channels, the analytical result can be obtained at the amplitude level. According to our study, the generation rates for the double P-wave heavy quarkonia at the future super Z factory are considerable. The values obtained for the total cross sections of the double P-wave charmonium for $\sigma (|(c\overline{c})[1]⟩+|(c\overline{c})[1]⟩)=4.{191}_{-0.498}^{+0.489}\times {10}^{-3}fb$, the total cross sections of the double P-wave $B_c$ mesons for $\sigma (|(c\overline{b})[1]⟩+|(b\overline{c})[1]⟩)=0.{2045}_{-0.0080}^{+0.0084}fb$, and the total cross sections of the double P-wave bottomonium for $\sigma (|(b\overline{b})[1]⟩+|(b\overline{b})[1]⟩)=5.{244}_{-0.136}^{+0.134}\times {10}^{-3}fb$. The main uncertainties come from the mass of heavy quarkonia and the corresponding first-derivative radial wave functions at the origin under the Buchmüller–Tye (BT) potential. The number of events in the production of double P-wave charmonium, $B_c$-meson, and bottomonium states via $e^{-}{e}^{+}\to {\gamma }^{\ast }/Z^0\to |(Q{\overline{Q}}^{\prime })[n]⟩+|(Q^{\prime }\overline{Q})[n^{\prime }]⟩$ at the integrated luminosity $\mathcal{L}\approx {10}^{34}c{m}^{-2}{s}^{-1}$ at the super Z factory is almost unobservable.