Abstract
Within the nonrelativistic quantum chromodynamics framework, we make a comprehensive study of the exclusive production of excited charmonium and bottomonium in ${e}^{+}{e}^{\ensuremath{-}}\ensuremath{\rightarrow}{\ensuremath{\gamma}}^{*}/{Z}^{0}\ensuremath{\rightarrow}|(Q\overline{Q})[n]⟩+\ensuremath{\gamma}$ ($Q=c$ 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.
Highlights
In comparison to the hadronic colliders like Large Hadron Collider (LHC), an electron-positron collider has some advantages, as it provides a cleaner hadronic background and the collision energy and polarization of incoming electron and positron beams can be well controlled
The heavy quarkonium provides an ideal platform to investigate the properties of bound states, which is a multiscale problem for probing quantum chromodynamics (QCD) theory at all energy regions
Taking J=ψ as an example, the cross section of the inclusive production in eþe− → J=ψ þ X is measured by the Bell experiment [4], the two-photon scattering in eþe− → eþe−J=ψ þ X is studied by the DELPHI experiment at LEP II [5], the photoproduction in ep → J=ψ þ X is explored by Zeus and H1 experiments at HERA [6,7], the hadroproduction in pp → J=ψ þ X is studied by a CDF experiment at Tevatron [8], and the hadroproduction in pp → J=ψ þ X is widely explored by ATLAS, CMS, ALICE, and LHCb experiments at the LHC [9,10,11,12]
Summary
In comparison to the hadronic colliders like Large Hadron Collider (LHC), an electron-positron collider has some advantages, as it provides a cleaner hadronic background and the collision energy and polarization of incoming electron and positron beams can be well controlled. We shall concentrate our attention on the production of both ground and high Fock states of both charmonium and bottomonium in eþe− → γÃ=Z0 → jðQQ Þ1⁄2ni þ γðQ 1⁄4 c; bÞ at a future super Z factory, where [n] is short for the color-singlet 1⁄2n1S0, 1⁄2n3S1, 1⁄2n1P0, and 1⁄2n3PJ Fock states (n 1⁄4 1, 2, 3, 4; J 1⁄4 0, 1, 2). The analysis on differential distributions and the uncertainties shall be discussed This would be a helpful support for the experimental exploration on production of those high excited heavy charmonium and bottomnium at future super Z factory or GigaZ mode at CEPC.
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