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Probing Glueball content of X(2370) through heavy quarkonium radiative decays and electron-positron annihilation

This paper investigates the glueball content of the $X(2370)$ resonance by analyzing light-cone distribution amplitudes and a tetra-mixing scheme to predict branching ratios for heavy quarkonium radiative decays and electroproduction cross sections, offering testable theoretical predictions for current BESIII and Belle-II experiments.

Original authors: Ruilin Zhu, Qiang Zhao

Published 2026-08-25

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Ruilin Zhu, Qiang Zhao

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, where particles interact through the strong nuclear force, physicists have long searched for a ghostly kind of matter made entirely of force itself. In the standard model of particle physics, forces are carried by particles called gluons, which usually bind together to form protons and neutrons. However, theory suggests that gluons can also bind to each other, creating a particle made purely of glue, known as a glueball. Finding one would be a monumental achievement, offering a direct window into how the strong force holds the universe together and why quarks can never be found alone. For decades, these particles remained theoretical, hiding among a crowd of ordinary particles that look and act very similarly.

Recently, a particle named X(2370) has emerged as a prime suspect for being the lightest of these glueballs. Discovered in the debris of particle collisions, it has the right mass and the right quantum properties to be this elusive object. Yet, a simple explanation is not enough; because the strong force is so complex, a glueball might not exist in isolation. It could be a mixture, blending with ordinary particles made of quarks and antiquarks. The big question for physicists is not just whether the glueball exists, but how much of it is actually inside the X(2370). Is it a pure glueball, or is it a hybrid, and if so, how much of each ingredient is present?

A team of researchers has now taken a fresh look at this puzzle by analyzing how heavy particles called quarkonia decay. These heavy particles, such as the J/psi and the psi(2S), are essentially bound states of a charm quark and its antimatter partner. When they decay, they often emit a flash of light, a photon, and leave behind a lighter particle. The researchers focused on the specific process where these heavy particles shed a photon and transform into a pseudoscalar meson, a type of particle that includes the X(2370). By studying the rates at which these decays happen, the team could work backward to determine the internal composition of the resulting particles.

The core of their work involved a sophisticated mixing scheme, a mathematical framework that describes how different types of particles blend together. They treated the X(2370) as a product of four distinct components mixing together: a glueball, a particle made of charm quarks, and two types of particles made of lighter quarks. Using the latest experimental data on how often the J/psi particle decays into various known mesons, the team calculated the specific angles that define this mixture. They found that the data fits a model where the glueball component is significant, but not the only ingredient. The calculations revealed that the mixing between the glueball and the charm-quark particle is very small, but it is crucial for explaining the observed behavior.

One of the most striking findings concerns how the X(2370) behaves when produced by different parent particles. The researchers discovered that the probability of the J/psi decaying into an X(2370) and a photon is extremely sensitive to the exact details of the mixing. In fact, the calculations show that if the mixing angle were slightly different, the production rate would drop to nearly zero due to a cancellation effect, where the different components of the particle interfere with each other destructively. This sensitivity acts like a precise tuning fork; the fact that the X(2370) is observed at all, and with the specific rate measured by experiments, tells us exactly how the ingredients are mixed. The team also predicted that the psi(2S) particle, a heavier cousin of the J/psi, should produce the X(2370) at a different rate, with a similar sensitivity to the mixing angle but occurring at a different value.

Beyond the decay of heavy particles, the study also looked at how these particles are created when electrons and positrons smash into each other. The researchers calculated the likelihood of producing the X(2370) in these collisions, specifically when it is accompanied by a lighter particle called a rho meson. Their theoretical predictions for this process align well with existing measurements for other similar particles, giving confidence in their method. They provided specific predictions for the production rates of the X(2370) at various energy levels, which are now waiting to be tested by current experiments at facilities like BESIII and Belle-II.

The ultimate goal of this work is to move beyond speculation and pin down the true nature of the X(2370). The researchers conclude that the particle is indeed dominated by a glueball component, but it is a complex mixture rather than a pure state. Their analysis suggests that the glueball content is large enough to explain the particle's existence, yet small enough to allow for the mixing effects that make it observable. The findings offer a clear path forward: by measuring the decay rates of the psi(2S) and the production rates in electron-positron collisions, scientists can verify the mixing angles they have calculated. If the new data matches their predictions, it will provide definitive proof that the X(2370) is the long-sought lightest glueball, finally revealing the pure force of the strong interaction in a tangible form.

Technical Summary: Probing Glueball Content of X(2370) through Heavy Quarkonium Radiative Decays and Electron-Positron Annihilation

Problem Statement
The identification of the lightest pseudoscalar Glueball remains a central challenge in particle physics, crucial for understanding non-Abelian gauge fields and color confinement. Recent BESIII analyses suggest the $X(2370)$ resonance is the dominant constituent of this Glueball, based on its mass ( $2376.3 \pm 8.1$ MeV), spin-parity ( $J^{PC} = 0^{-+}$ ), and production rates. However, a discrepancy exists: the experimentally observed large decay rate for $J/\psi \to \gamma X(2370)$ (estimated around $10^{-3}$ ) exceeds the theoretical prediction for a pure pseudoscalar Glueball ( $\sim 2.3 \times 10^{-4}$ ) derived from Lattice QCD. This suggests that the $X(2370)$ may not be a pure Glueball state but rather a mixture involving quark-antiquark components. The paper aims to quantify the Glueball content within the $X(2370)$ by analyzing its production in heavy quarkonium radiative decays and electron-positron annihilation, treating the state within a mixing framework.

Methodology
The authors employ Quantum Chromodynamics (QCD) factorization theory to model the production of pseudoscalar mesons. The core methodological steps include:

Light-Cone Distribution Amplitudes (LCDA): The paper defines the LCDA for the pseudoscalar Glueball, acknowledging that at the quantum loop level, two-gluon and quark-antiquark pair contributions naturally mix. A two-dimensional LCDA vector $\Phi(u)$ is constructed to describe the mixing between the gluonic component ( $\phi_g$ ) and the quark-antiquark component ( $\phi_q$ ).
Tetra-Mixing Scheme: The study revises the $\eta - \eta' - G - \eta_c$ tetra-mixing scheme. This framework treats the physical states as a superposition of flavor-singlet/octet quark-antiquark states ( $|q\bar{q}\rangle, |s\bar{s}\rangle$ ), a gluonium state ( $|gg\rangle$ ), and the charmonium state ( $|c\bar{c}\rangle$ ). The mixing matrix is parameterized by angles $\phi, \theta_{g,c}, \phi_{gc}$ and coefficients $a_{g,c}$ .
Factorization and Calculation: The amplitude for radiative decays (e.g., $J/\psi \to P + \gamma$ ) is factorized into a hard kernel and the LCDA. The authors calculate branching ratios using experimental inputs for $J/\psi$ decays to $\eta, \eta', \eta_c$ , and the Lattice QCD prediction for a pure Glueball.
Parameter Extraction: By comparing theoretical calculations with experimental branching ratios, the authors extract the unknown mixing parameters ( $\theta_{g,c}, a_{g,c}$ ) and the Glueball-charmonium mixing angle ( $\phi_{gc}$ ).
Predictions: The derived parameters are used to predict branching ratios for $\psi(2S)$ and $\Upsilon(nS)$ radiative decays, as well as electroproduction cross-sections ( $e^+e^- \to P + V$ ) for pseudoscalar mesons, including the $X(2370)$ .

Key Contributions and Results

Mixing Parameter Extraction: Using the $\eta - \eta' - G - \eta_c$ scheme and experimental data, the authors identify two sets of parameter values (I and II) that yield consistent results. The analysis confirms that kinematic enhancement factors play a significant role in the branching ratios, particularly for $\eta$ and $\eta'$ production.
Sensitivity of $X(2370)$ Production: The branching ratios for $J/\psi \to X(2370) + \gamma$ $J / ψ \to X (2370) + γ$ and $\psi(2S) \to X(2370) + \gamma$ $ψ (2 S) \to X (2370) + γ$ are found to be highly sensitive to the Glueball-charmonium mixing angle ( $\phi_{gc}$ $ϕ_{g c}$ ).
- Destructive Interference: Occurs at $\phi_{gc} \approx 0.01$ for $J/\psi$ decays and $\phi_{gc} \approx 0.11$ for $\psi(2S)$ decays.
- Constructive Interference: Observed in $J/\psi \to X(2370) + \gamma$ when $\phi_{gc} > 0.02$ .
Validation via $\psi(2S)$ : The theoretical predictions for $\psi(2S)$ radiative decays into pseudoscalar mesons show good agreement with existing experimental data, validating the mixing scheme.
New Predictions:
- $\Upsilon$ Decays: The paper provides theoretical predictions for $\Upsilon(1S, 2S) \to \eta_c, G, \eta', \eta + \gamma$ , noting that these channels have not yet been unambiguously measured.
- Electroproduction: Theoretical calculations for $e^+e^- \to \eta^{(')} + \rho^0$ generally agree with experimental data around 3 GeV, though deviations exist at lower and higher energies (attributed to potential hadronic resonance contributions).
- $X(2370)$ Electroproduction: Specific cross-section predictions are provided for $e^+e^- \to X(2370) + \rho^0$ at center-of-mass energies $\sqrt{S} = 4, 5,$ and $10.58$ GeV, with values ranging from approximately $4$ to $12$ pb.

Significance and Claims
The paper claims that the large production rate of $X(2370)$ in $J/\psi$ radiative decays can be explained by a mixing scheme where the Glueball component is significant but mixed with charmonium ( $\eta_c$ ) and light quark states. The authors assert that their theoretical framework successfully reproduces known experimental data for $\eta, \eta', \eta_c$ and $\psi(2S)$ decays, thereby supporting the hypothesis that $X(2370)$ possesses a large pseudoscalar Glueball component.

The significance of the work lies in providing testable predictions for current and future experiments. The authors state that the predicted branching ratios for $\Upsilon$ decays and the electroproduction cross-sections for $X(2370)$ can be readily tested by the BESIII and Belle-II collaborations. They conclude that further experimental measurements in these channels will be essential to definitively reveal the intrinsic nature of the $X(2370)$ and constrain the Glueball mixing parameters.

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