Production of excited heavy quarkonia in $e^+e^- \to \gamma^*/Z^0 \to |(Q\bar{Q})[n]\rangle +\gamma$ at super $Z$ factory

TL;DR

This paper studies the production of excited heavy quarkonia in electron-positron collisions at future Z factories using nonrelativistic QCD, providing detailed cross sections and distributions for various excited states.

Contribution

It introduces an improved trace technology for analytic amplitude calculations and offers comprehensive predictions for excited quarkonium production at super Z factories.

Findings

Production rates of high excited states are significant at future Z factories.

Cross sections for various excited states are quantified relative to ground states.

Main uncertainties stem from wave functions at the origin under different potential models.

Abstract

Within the nonrelativistic quantum chromodynamics framework, we make a comprehensive study on the exclusive production of excited charmonium and bottomonium in $e^+e^-\to \gamma^*/Z^0 \to|(Q\bar{Q})[n]\rangle +\gamma$ ($Q=c$ or $b$ quarks) at future $Z$ factory, where the $[n]$ represents the color-singlet $n^1S_0,~n^3S_1,~n^1P_0$ and $n^3P_J$ ($n=1,2,3,4; J=0,1,2$) 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 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, such super $Z$ factory should be a good platform to study the properties of the high excited charmonium and bottomonium states.