$Z$-boson decays into $S$-wave quarkonium plus a photon up to ${\cal O}(\alpha_{s} v^2)$ corrections

TL;DR

This paper calculates the decay widths of Z-boson into S-wave quarkonium plus a photon up to next-to-leading order corrections, highlighting the importance of various contributions for accurate predictions at future colliders.

Contribution

It provides the first comprehensive calculation including ${ m O}(rac{ ext{alpha}_s v^2)}$ corrections for Z-boson decays into S-wave quarkonium plus a photon.

Findings

Corrections are sizable and comparable, especially for charmonium.

All correction terms must be considered for accurate estimates.

Potential for observing these decays at high-luminosity Z-pole colliders.

Abstract

In this paper, we calculate the decay widths and branching fractions for the decays $Z \to H+ \gamma$ up to ${\cal O}(\alpha_{s} v^2)$ accuracy within the framework of nonrelativistic QCD, where $H$ stands for the $S$-wave quarkonium $\eta_c$, $J/\psi$, $\eta_b$ or $\Upsilon$, respectively. To compare with the leading-order terms, those corrections show good perturbative behavior as expected. It is found that contributions from the next-to-leading order QCD correction ${\cal O}(\alpha_{s}v^0)$, the relativistic correction ${\cal O}(\alpha^{0}_{s}v^2)$ and their joint correction ${\cal O}(\alpha_{s} v^2)$ are sizable and comparable to each other, especially for the charmonium case. Thus we need to take all of them into consideration for a sound estimation. For a high luminosity electron-positron collider running around the $Z$-pole, due to $Z$-boson resonance effect, sizable events could be produced from those rare decay channels.