Production double heavy quarkonium via $e^+e^-\to \gamma^*/Z^0 \to|(Q\bar{Q'})[n]\rangle +|(Q'\bar{Q})[n']\rangle$ at Z mass pole

TL;DR

This paper calculates the production rates of double excited heavy quarkonium states at a future Z factory using nonrelativistic QCD, revealing significant cross sections especially for double excited Bc-mesons.

Contribution

It provides the first comprehensive calculation of double excited heavy quarkonium production at the Z pole, including analytic amplitude expressions using improved trace technology.

Findings

Double excited charmonium cross section: ~0.0117 fb

Double excited bottomonium cross section: ~0.1132 fb

Double excited Bc-mesons cross section: ~3.692 fb

Abstract

The exclusive production of double excited quarkonium is comprehensive studied, i.e., the production of double excitedcharmonium, double excited bottomonium, and double excited $Bc$-mesons via $e^+e^-\to \gamma^*/Z^0 \to|(Q\bar{Q'})[n]\rangle +|(Q'\bar{Q})[n']\rangle$ ($Q/Q'=c$- or $b$-quarks) at a future $Z$ factory under the nonrelativistic quantum chromodynamics framework, where the $[n]$~/$[n']$ represents the color-singlet heavy quarkonium states $[^1S_0]\rangle, ~[^3S_1]\rangle, ~[^1P_1]\rangle$, and $[^3P_J]\rangle$ ($J=0,1,2$). The "improved trace technology" is adopted for calculating the complicated $P$-wave channels for derive the analytic expressions at the amplitude level. According to our study, the production rates of double heavy quarkonium are considerable at the future $Z$ factory. We obtain the cross sections for the production of double excited charmonium for $\sigma{(|(c\bar{c})[n]\rangle+|(c\bar{c})[n']\rangle)_{total}}=1.167^{+0.113}_{-0.164}\times 10^{-2}~fb$, the cross sections of double excited bottomonium for $\sigma{(|(b\bar{b})[n]\rangle+|(b\bar{b})[n']\rangle)_{total}}=0.1132^{+0.0139}_{-0.0130}~fb$, and the cross sections of double excited $Bc$-mesons for $\sigma{(|(c\bar{b})[n]\rangle+|(b\bar{c})[n']\rangle)_{total}}=3.692^{+0.124}_{-0.097}~fb$. The main uncertainties come from the mass of the heavy quarkonium and the radial wave functions at the origin and their derivatives at the origin under different potential models. The numerical results show that such the super $Z$ factory should be a good platform to study the properties of the double excited charmonium, the double excited bottomonium , especially the double excited $Bc$-mesons.