Within the nonrelativistic quantum chromodynamics framework, we make a comprehensive study of the exclusive production of excited charmonium and bottomonium in $e^{+}{e}^{-}\to {\gamma }^{*}/Z^0\to |(Q\overline{Q})[n]⟩+\gamma$ ($Q=c$ or $b$ quarks) at a future $Z$ factory, where the [$n$] represents the color-singlet $n{{}^{1}S}_0$, $n{{}^{3}S}_1$, $n{{}^{1}P}_0$, and $n{{}^{3}P}_J$ ($n=1$, 2, 3, 4; $J=0$, 1, 2) Fock states. The “improved trace technology” is adopted to derive the analytic expressions at the amplitude level, which is useful for calculating the complicated $nP$-wave channels. Total cross sections, differential distributions, and uncertainties are discussed in system. According to our study, production rates of heavy quarkonia of high excited Fock states are considerable at a future $Z$ factory. The cross sections of charmonium for $2S$-, $3S$-, $4S$-, $1P$-, $2P$-, $3P$-, and $4P$-wave states are about 53.5%, 30.4%, 23.7%, 13.7%, 6.8%, 9.2%, and 9.2% of that of the $1S$ state, respectively. And cross sections of bottomonium for $2S$-, $3S$-, $4S$-, $1P$-, $2P$-, $3P$-, and $4P$-wave states are about 39.3%, 12.3%, 14.3%, 7.1%, 3.1%, 2.7%, and 3.1% of that of the $1S$ state, respectively. The main uncertainties come from the radial wave functions at the origin and their derivatives at the origin under different potential models. Then, this super $Z$ factory should be a good platform to study the properties of the high excited charmonium and bottomonium states.

• Received 8 December 2021
• Accepted 12 January 2022

DOI:https://doi.org/10.1103/PhysRevD.105.016026

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Published by the American Physical Society