⚛️ phenomenology

Paired charmonium and bottomonium production in rare exclusive decays of Z boson

This paper investigates the rare exclusive decays of the Z boson into paired charmonium and bottomonium states using a relativistic quark model to construct decay amplitudes and calculate decay widths with and without relativistic corrections.

Original authors: F. A. Martynenko, A. P. Martynenko, A. A. Skakun

Published 2026-09-18

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CC BY 4.0

Original authors: F. A. Martynenko, A. P. Martynenko, A. A. Skakun

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Deep within the subatomic world, particles known as quarks rarely travel alone. They are bound together by the strong force, a powerful glue that prevents them from existing in isolation, forming composite particles called hadrons. Among these, the most famous are the mesons, which consist of a quark and its antimatter partner, an antiquark. When these partners are heavy, such as the charm or bottom quarks, they form stable, exotic atoms known as quarkonia. Scientists study how these particles are created and destroyed to test the fundamental rules of the universe, specifically the Standard Model, which acts as the master blueprint for all known matter and forces. While much attention has recently focused on the Higgs boson, another heavy particle called the Z boson offers a unique laboratory. The Z boson is a carrier of the weak nuclear force, and when it decays, it can occasionally split its energy into pairs of these heavy quark-antiquark bound states. Understanding exactly how often this happens, and through what specific pathways, allows physicists to probe the limits of their theories and search for new physics hiding in the details.

A team of researchers at Samara University in Russia has taken a closer look at these rare events, specifically focusing on the production of paired charmonium and bottomonium mesons during the decay of a Z boson. In their study, they investigated the specific mechanisms by which a single Z boson can transform into two heavy mesons at once. This is not a simple process; it involves complex interactions where the initial energy of the Z boson is converted into quarks and antiquarks, which then bind together. The researchers examined three distinct ways this could happen. The first involves the exchange of a gluon, the particle that carries the strong force. The second involves the exchange of a photon, the particle of light and electromagnetism. The third, more complex route involves a temporary connection to a Higgs boson. By calculating the probabilities for each of these pathways, the team aimed to determine which mechanism dominates the process and how the speed of the quarks affects the outcome.

A crucial aspect of their work was moving beyond simple, static calculations to include the effects of relativity. In the world of heavy quarks, the particles move at significant fractions of the speed of light, meaning that standard, non-relativistic physics is not accurate enough. The researchers used a sophisticated relativistic quark model to account for the fact that these quarks are not stationary points but are in constant, rapid motion relative to one another. They found that ignoring these relativistic effects leads to significant errors in predicting how often these decays occur. When they included the corrections for the quarks' high speeds and the way their internal wave functions change as they move, the predicted rates shifted noticeably. For instance, in the case of producing a pair of bottomonium mesons, the inclusion of these relativistic details changed the calculated probability by a factor of nearly three compared to the simpler, non-relativistic estimate.

The study revealed that among the various ways a Z boson can decay into a pair of mesons, the mechanism involving the exchange of a photon is the most significant contributor to the total rate for mixed vector-pseudoscalar pairs (such as J/Ψ + ηc or Υ + ηb). However, for pairs of identical vector mesons (like Υ + Υ), the gluon-exchange mechanism actually provides the dominant contribution, as the photon-mediated process is suppressed in this specific configuration. The researchers also looked at a more exotic loop mechanism involving W bosons, which are other carriers of the weak force, but found that this pathway is heavily suppressed and contributes very little to the overall rate. Their calculations provide specific numbers for how likely these rare decays are, offering precise targets for future experiments. For example, they calculated that the probability of a Z boson decaying into a pair of J/psi mesons (a type of charmonium) is roughly one in ten billion, while the probability for a pair of upsilon mesons (a type of bottomonium) is slightly higher, though still incredibly rare.

These findings are important because they refine the theoretical predictions that experimentalists at facilities like the Large Hadron Collider and future electron-positron colliders will use to search for these events. The researchers emphasize that to get reliable predictions, one cannot simply treat the quarks as slow-moving particles; the relativistic nature of their motion is essential. Their work also highlights that the structure of the decay amplitudes changes fundamentally when relativistic effects are included, with some terms that vanish in simple models becoming significant in the full picture. While the current results are theoretical calculations based on established models, they serve as a necessary guide for the next generation of experiments. By providing a more accurate map of these rare decays, the study helps physicists distinguish between standard behavior and potential signs of new, undiscovered particles or forces that might appear in the gaps between theory and observation.

Technical Summary: Paired Charmonium and Bottomonium Production in Rare Exclusive Decays of Z Boson

Problem Statement
Following the discovery of the Higgs boson, research into the Standard Model has increasingly focused on processes involving the production of quark and lepton bound states. These reactions serve as critical tests for quarkonium production models and offer a pathway to search for undiscovered states. While experimental upper bounds exist for the exclusive pair production of heavy quarkonia (charmonium and bottomonium) in $Z$ -boson and Higgs boson decays, theoretical calculations for these processes require continuous improvement. Specifically, there is a need to refine decay width calculations by moving beyond non-relativistic approximations and rigorously accounting for relativistic effects and various competing decay mechanisms.

Methodology
The authors investigate the processes of paired charmonium ( $c\bar{c}$ ) and bottomonium ( $b\bar{b}$ ) production in $Z$ -boson decays using a relativistic quark model. The study employs a quasipotential approach to construct relativistic decay amplitudes, explicitly incorporating the relative momenta of the heavy quarks.

The methodology involves:

Formalism Construction: The authors derive decay amplitudes for the production of $S$ -wave vector ( $V$ ) and pseudoscalar ( $P$ ) states. The wave functions of the bound states are treated as products of rest-frame wave functions and spin projection operators accurate to all orders in $|p|/m$ , where $p$ is the relative momentum and $m$ is the heavy quark mass.
Mechanism Analysis: Three primary production mechanisms are examined:
- Quark-Gluon Mechanism: Involves the perturbative production of heavy quark-antiquark pairs with gluon emission, followed by non-perturbative formation.
- Quark-Photon Mechanism: Involves a photon emitted by a quark converting into a vector meson. The authors note that while the coupling constant is smaller ( $\alpha$ vs. $\alpha_s$ ), the contribution is enhanced by the mass ratio $M_Z^2/M_{meson}^2$ .
- ZZH Mechanism: A tree-level process involving $Z$ -boson and Higgs boson exchange, which is generally suppressed by mass factors in the denominator.
- W-Boson Loop Mechanism: A loop-induced process involving $W$ -bosons and virtual photons, considered for the production of mixed $1S$ and $2S$ vector states.
Relativistic Corrections: The calculation extracts second-order relativistic corrections ( $p^2/m^2$ ) in the relative momenta. These corrections are parameterized by integrals involving the radial wave functions ( $\omega_1, \tilde{\omega}_1$ ) and are averaged over angles. The authors utilize the Form package for trace calculations of Dirac factors.
Decay Width Calculation: The differential and total decay widths are calculated by summing over vector meson polarizations and averaging over $Z$ -boson polarizations, incorporating the derived relativistic parameters.

Key Contributions and Results
The paper provides analytical formulas for decay widths and numerical estimates for various exclusive decay channels of the $Z$ boson.

Dominant Mechanisms: The analysis reveals that the quark-photon mechanism generally provides the leading-order contribution to the total decay width for mixed vector-pseudoscalar pairs and charmonium channels, driven by the enhancement factor $M_Z^2/M_{meson}^2$ . However, for the bottomonium vector-vector pair ( $\Upsilon + \Upsilon$ ), the quark-gluon mechanism remains dominant. The calculated branching ratio for the quark-gluon mechanism ( $Z \to b\bar{b}g \to \Upsilon + \Upsilon$ ) is $0.35 \times 10^{-11}$ , whereas the quark-photon mechanism ( $Z \to b\bar{b}\gamma \to \Upsilon + \Upsilon$ ) yields a significantly smaller value of $0.27 \times 10^{-12}$ .
Impact of Relativistic Corrections: The inclusion of relativistic corrections significantly alters the numerical results compared to non-relativistic approximations. For instance, in the $Z \to J/\Psi + \eta_c$ channel, the branching ratio decreases from $0.31 \times 10^{-12}$ (non-relativistic) to $0.11 \times 10^{-12}$ (relativistic). Similar reductions are observed for bottomonium channels.
Specific Decay Channels: The authors calculate branching ratios for:
- Mixed vector-pseudoscalar pairs ( $J/\Psi + \eta_c$ , $\Upsilon + \eta_b$ ).
- Vector-vector pairs ( $J/\Psi + J/\Psi$ , $\Upsilon + \Upsilon$ ).
- Mixed $1S$ and $2S$ vector pairs via the $W$ -loop mechanism.
Suppressed Channels: The ZZH mechanism and the $W$ -loop mechanism for charmonium pairs are found to be strongly suppressed. The $W$ -loop contribution is particularly small due to the power-law suppression by the electromagnetic coupling $\alpha^5$ , despite potential mass enhancements.
Numerical Data: Table II presents the calculated branching ratios ( $Br_{nr}$ and $Br_{rel}$ ) for all considered mechanisms, demonstrating that relativistic effects can change the magnitude of the decay widths by factors of 2 to 3.

Significance and Claims
The authors claim that their work extends previous studies on Higgs boson decays to the $Z$ -boson sector, providing a more rigorous theoretical framework for rare exclusive decays. The significance of the paper lies in three main features:

Mechanism Interplay: It highlights that the dominant production mechanism is not immediately obvious and depends on the interplay of interaction constants and mass factors, varying by specific decay channel.
Necessity of Relativistic Effects: The paper asserts that a reliable description of heavy quark bound state formation is virtually impossible without accounting for significant relativistic effects. The structure of the decay amplitudes themselves changes when relativistic corrections are included (e.g., certain terms vanish in the non-relativistic limit but contribute significantly when corrections are applied).
Theoretical Precision: The study emphasizes that while the results show order-of-magnitude agreement with previous non-relativistic calculations, the inclusion of relativistic corrections is essential for obtaining reliable predictions. The authors note that the main theoretical uncertainty remains the one-loop corrections to the interaction amplitudes, which could be as large as 30%.

The paper concludes that the large number of $Z$ -bosons produced at facilities like LEP, FCC-ee, Tevatron, HL-LHC, and FCC-hh makes them a promising source for studying these rare decays, offering new opportunities to probe non-perturbative QCD interactions and the dynamics of bound state formation.

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