Improved analysis of rare $Z$-boson decays into a heavy vector quarkonium plus lepton pair

TL;DR

This work refines SM predictions for rare $Z\to V\ell^+\ell^-$ decays, where $V$ is a heavy vector quarkonium such as $J/\Psi$, $\Psi(2S)$, or $\Upsilon(nS)$. It carries out a complete tree-level calculation of all SM diagrams, finding that charmonium channels are almost fully described by the electromagnetic fragmentation contribution, while bottomonium channels receive ${\sim}4$–$9\%$ enhancements from non-fragmentation diagrams. The paper provides precise branching ratios, e.g., ${\cal B}(Z\to J/\Psi\ell^+\ell^-)=(7.78\pm0.14)\times10^{-7}$ and ${\cal B}(Z\to \Upsilon(1S)\ell^+\ell^-)=(2.18\pm0.04)\times10^{-8}$, and analyzes differential distributions showing no SM forward-backward asymmetry ${A_{\rm FB}}$, with CP-violating effects only arising from hypothetical anomalous neutral gauge couplings. It also discusses potential CP-violating observables that could be probed in future high-statistics $Z$ factories like FCC-ee or CEPC, where large samples would enable stringent tests of the SM and constraints on new physics.

Abstract

We improve the theoretical predictions for rare $Z$-boson decays, $Z\to V\ell^+\ell^-$ ($\ell=e$ or $μ$), where $V$ denotes a heavy vector quarkonium including $J/Ψ$, $Ψ(2S)$, and $Υ(nS)$ with $n=1,2,3$. These processes are thought to be dominated by the electromagnetic fragmentation transition, i.e., $Z\to γ^*\ell^+\ell^-$ followed by $γ^*\rightarrow V$. The present study includes all of the relevant tree-level Feynman diagrams, which contribute to these decays in the standard model. Our analysis shows that, for the charmonium final states, the fragmentation transition almost saturates the whole contribution and the other diagrams can be neglected; while for the bottomonium final states, the inclusion of other diagrams can increase their branching fractions by $4\%\sim 9\%$. Further investigation of the differential distributions, especially the angular distributions, indicates that forward-backward asymmetries for final leptons in these processes would be zero in the standard model. Therefore, in future experimental facilities with large number of $Z$-boson events accumulated, studies of these rare $Z$ decays may help both to test the standard model and to probe its interesting extensions.

Figures (6)

• Figure 1: The dominant Feynman diagrams for $Z\to V \ell^+ \ell^-$ decays. The virtual photon $\gamma^*$ could also be emitted from the $\ell^+$ line.
• Figure 2: The Feynman diagram contributing to $Z\to V\ell^+\ell^-$ via $Z\to V\gamma^*$ transitions. The virtual photon $\gamma^*$ could also be emitted from the left quark line.
• Figure 3: The normalized invariant mass distribution of $Z\to\Upsilon(1S) e^+ e^-$ decay, where $m_{ee}=\sqrt{s}$ is the dilepton invariant mass. The red-dashed line denotes the contribution from Fig. 1 only while the solid line gives the total contribution.
• Figure 4: The normalized angular distributions of $Z\to J/\Psi \ell^+ \ell^-$ (left plot) and $Z\to \Upsilon(1S)\ell^+\ell^-$ (right plot) decays. The red-dashed line denotes the contribution from Fig. 1 only.
• Figure 5: The Feynman diagram contributing to CP-violating amplitude in $Z\to V\ell^+\ell^-$ decays by the effective $Z\gamma G^*$ vertex.