arxiv: 2605.01757 · v1 · submitted 2026-05-03 · ✦ hep-ph · hep-lat

Recognition: unknown

Production Rate of Glueball-like X(2370) in J/psi Radiative Decay

Ying Chen , Long-Cheng Gui , Geng Li , Wei Sun

Authors on Pith no claims yet

Pith reviewed 2026-05-09 17:16 UTC · model grok-4.3

classification ✦ hep-ph hep-lat

keywords X(2370)pseudoscalar glueballJ/ψ radiative decaymixing anglecharmoniumbranching ratiolattice QCD

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The pith

Mixing with charmonium can make the glueball candidate X(2370) appear at much higher rates in J/ψ radiative decays.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper investigates X(2370) as a possible mixture of the lowest pseudoscalar glueball and the charmonium state η_c, linked by a small mixing angle. In this setup the branching fraction for J/ψ decaying radiatively to X(2370) grows sharply because the charmonium component supplies a large transition strength while the lower mass of X(2370) opens more phase space. The branching fraction for the pure glueball is only about 2.3 times 10 to the minus 4 according to lattice calculations, yet the admixture can push the observed rate well above that value. Current data favor a mixing angle of roughly one degree, and the picture leaves the rate for η_c production almost unchanged. A reader cares because the result offers a concrete way to connect an observed resonance to the still-unconfirmed glueball states predicted by QCD.

Core claim

X(2370) and η_c are admixtures of the pseudoscalar glueball G_{0^-} and the charmonium state c c-bar (0^-) with a small mixing angle sinθ. Although Br(J/ψ → γ η_c) remains insensitive to this small angle, Br(J/ψ → γ X(2370)) is strongly enhanced by the larger kinematic factor for the lighter X(2370) and the large transition form factor of the charmonium component, so that the rate can exceed the pure-glueball prediction of 2.3(8)×10^{-4} depending on the value of sinθ.

What carries the argument

The small mixing angle sinθ between the pseudoscalar glueball and the charmonium state that transfers the large transition form factor of the charmonium component into the production amplitude for X(2370).

If this is right

Br(J/ψ → γ X(2370)) can exceed the pure-glueball lattice value once sinθ is nonzero but still small.
BESIII data currently prefer sinθ of order 1 degree.
Further measurements of X(2370) decay modes can tighten the allowed range for sinθ under the mixing assumption.
Br(J/ψ → γ η_c) shows little sensitivity to the same small mixing angle.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

If the mixing picture holds, similar small admixtures may appear in other glueball candidates and alter how lattice results are compared with experiment.
Precision studies of the relative rates to X(2370) and η_c could map out the mixing angle without needing new lattice runs.
The mechanism suggests that observed 'glueball-like' states may routinely contain quarkonium fractions large enough to dominate certain production channels.

Load-bearing premise

That X(2370) really is a glueball-charmonium admixture with a mixing angle of order one degree and that the charmonium transition form factor stays large.

What would settle it

A measurement that finds Br(J/ψ → γ X(2370)) near or below 2.3×10^{-4} while other decay properties of X(2370) remain similar to those of η_c would remove the room for the mixing enhancement.

Figures

Figures reproduced from arXiv: 2605.01757 by Geng Li, Long-Cheng Gui, Wei Sun, Ying Chen.

**Figure 1.** Figure 1: ), where gg(· · ·) denotes intermediate multi-gluon states (at least two gluons). If X(2370) (|X⟩) and ηc (|ηc⟩) are the corresponding two mass eigenstates after the mixing, one has |X⟩ |ηc⟩ = cos θ − sin θ sin θ cos θ |G⟩ |cc¯⟩ , (3) where θ is the mixing angle. Subsequently, the on-shell form factors M(P ) (0) for J/ψ → γX(ηc) processes are expressed as M(X) (0) = M(G) (0) cos θ − M(cc¯) (0) … view at source ↗

**Figure 2.** Figure 2: FIG. 2. Schematic diagrams for view at source ↗

**Figure 3.** Figure 3: FIG. 3. Schematic diagram for the contribution of light view at source ↗

**Figure 4.** Figure 4: In other words, a small mixing angle sin θ can lead to a large production rate. The preliminary results of BESIII [5] indicate that, X(2370) and ηc appear simultaneously in the five three-pseudoscalar systems in the J/ψ radiative decay, namely, K0 SK0 S η, K0 SK0 Sπ 0 , π0π 0η as well as π +π −η ′ and K0 SK0 S η ′ . Especially, there is one clear structure (likely X(2370)) appearing in the energy region f… view at source ↗

read the original abstract

$X(2370)$ falls in the mass region of the lowest pseudoscalar glueball predicted by lattice QCD studies and its decay properties are similar to those of $\eta_c$. A previous lattice QCD study finds that the pseudoscalar glueball ($G_{0^-}$) and the lowest pseudoscalar charmonium ($c\bar{c}(0^-)$) can mix with a small mixing angle $\sin\theta$. It is therefore possible that $X(2370)$ and $\eta_c$ are admixtures of $G_{0^-}$ and $c\bar{c}(0^-)$. In this picture, although $\mathrm{Br}(J/\psi\to\gamma\eta_c)$ is insensitive to the small $\sin\theta$, $\mathrm{Br}(J/\psi\to \gamma X(2370))$ can be enlarged drastically by the mixing due to the much larger kinematic factor for $J/\psi\to \gamma X(2370)$ and the much larger transition form factor for $J/\psi\to \gamma (c\bar{c}(0^{-}))$. Depending on the value of $\sin\theta$, $\mathrm{Br}(J/\psi\to \gamma X(2370))$ can be much larger than that of the pure pseudoscalar glueball, namely, $2.3(8)\times 10^{-4}$ that is predicted by a quenched lattice QCD calculation. Present results by BESIII favor a small mixing angle of $\mathcal{O}(1^\circ)$, which can be further constrained by more measurements of $X(2370)$ decays if the mixing picture applies here.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Mixing with a small angle can enhance the J/ψ to gamma X(2370) branching ratio above the lattice glueball baseline, provided the charmonium form factor does not suppress too much at higher recoil.

read the letter

The punchline is that this paper calculates how a small mixing angle can make Br(J/ψ → γ X(2370)) substantially larger than the pure pseudoscalar glueball value from quenched lattice QCD, which is 2.3(8)×10^{-4}. The boost comes from the larger phase space for the heavier X(2370) and the bigger transition form factor inherited from the charmonium component. The work applies a mixing framework that has been used before for masses and decays to this specific production channel. It incorporates the kinematic enhancement and contrasts the form factors for the glueball and ccbar parts. The result is a prediction that depends on sinθ, and they note that current BESIII measurements favor a small angle of order one degree. This turns an abstract lattice state into something that can be compared directly to radiative decay data. It does a decent job of connecting the lattice baseline to an experimental observable without fitting to the same data, so the circularity is low. The claim is new in its quantitative application to this branching ratio. The soft spot is the form factor assumption. The stress-test note is on target: the radiative transition amplitude is an overlap that should decrease with larger photon recoil momentum, here about 640 MeV versus 115 MeV for η_c. The paper treats the ccbar form factor as remaining large after rescaling by kinematics, but without an explicit model or calculation showing that suppression is negligible, the enhancement factor could be smaller than stated. Error propagation on the mixing angle is also not detailed. This paper is for people in glueball phenomenology and charmonium spectroscopy who want to interpret X(2370) as a mixed state. A reader looking for testable predictions from lattice glueball ideas will get value from the branching ratio estimates. It deserves a serious referee. The logic is straightforward and the inputs are from independent sources, so reviewers can assess whether the form factor treatment holds up.

Referee Report

1 major / 2 minor

Summary. The manuscript proposes that the X(2370) resonance is an admixture of the pseudoscalar glueball G_{0^-} and the charmonium state c c-bar(0^-) with a small mixing angle sinθ of O(1°). Under this mixing, Br(J/ψ → γ X(2370)) is enhanced relative to the pure-glueball baseline of 2.3(8)×10^{-4} from quenched lattice QCD because of the larger kinematic factor at the higher mass and the larger transition form factor inherited from the c c-bar component; BESIII data are said to favor the small mixing angle.

Significance. If the mixing picture and the form-factor scaling are valid, the work supplies a concrete, falsifiable link between the observed production rate of X(2370) and lattice glueball predictions, while offering a method to constrain the mixing angle through additional decay measurements. The use of an independent quenched-lattice baseline (rather than a fit to the same data) avoids circularity in the enhancement estimate and is a positive feature.

major comments (1)

[Abstract and the paragraph introducing the mixing-enhanced branching ratio] The central claim that the branching ratio is 'enlarged drastically' rests on the statement that the transition form factor for J/ψ → γ (c c-bar(0^-)) 'remains large' at the X(2370) mass and is only rescaled by kinematics. No explicit evaluation or reference is supplied showing that the momentum-space overlap integral does not fall appreciably when the photon recoil increases from ~115 MeV (η_c) to ~640 MeV (X(2370)). This assumption is load-bearing for whether |sinθ|^2 times the kinematic ratio can exceed the pure-glueball value.

minor comments (2)

[Abstract] The abstract states that 'present results by BESIII favor a small mixing angle of O(1°)' without quoting the specific observable, the fit procedure, or the resulting uncertainty on sinθ.
[Abstract] Notation for the mixing angle alternates between θ and sinθ in the abstract; a single consistent symbol and a brief definition would improve clarity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive major comment. The positive assessment of the work's significance and the avoidance of circularity in using the quenched lattice baseline are appreciated. We address the major comment below.

read point-by-point responses

Referee: [Abstract and the paragraph introducing the mixing-enhanced branching ratio] The central claim that the branching ratio is 'enlarged drastically' rests on the statement that the transition form factor for J/ψ → γ (c c-bar(0^-)) 'remains large' at the X(2370) mass and is only rescaled by kinematics. No explicit evaluation or reference is supplied showing that the momentum-space overlap integral does not fall appreciably when the photon recoil increases from ~115 MeV (η_c) to ~640 MeV (X(2370)). This assumption is load-bearing for whether |sinθ|^2 times the kinematic ratio can exceed the pure-glueball value.

Authors: We agree that the assumption concerning the transition form factor for the charmonium component is central to the enhancement claim and that the manuscript would benefit from a more explicit justification. The reasoning in the paper is that the c c-bar wave-function overlap integral, governed by the compact size of the charmonium system, varies only mildly over the photon-energy range from ~115 MeV to ~640 MeV, so that the dominant effect remains the kinematic prefactor. However, we acknowledge that no dedicated evaluation or supporting reference was supplied. In the revised manuscript we will expand the relevant paragraph (both in the abstract and in the main text) to include a short discussion of this point, citing potential-model and lattice studies of radiative charmonium transitions that indicate only modest suppression of the form factor in this kinematic window. This addition will make the argument self-contained while leaving the overall conclusions unchanged. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation uses independent lattice inputs and phenomenological mixing without self-referential reduction.

full rationale

The paper takes the pure-glueball branching ratio 2.3(8)×10^{-4} from an external quenched lattice QCD calculation and the small mixing angle sinθ from a prior lattice study as fixed inputs. The claimed enhancement of Br(J/ψ→γX(2370)) is then obtained by adding the ccbar admixture contribution scaled by the kinematic factor (p_γ ≈ 640 MeV) and the assumed transition form factor; neither quantity is fitted to the target branching ratio nor defined in terms of the output. BESIII data are invoked only to bound sinθ after the fact, not to force the prediction. No equation reduces to its own input by construction, no parameter is renamed as a prediction, and no uniqueness theorem or ansatz is smuggled via self-citation. The chain remains self-contained against the cited external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The claim rests on three external inputs: the quenched lattice branching ratio for the pure glueball, the assumption that the charmonium transition form factor is much larger than the glueball one, and the small-mixing-angle ansatz itself. No new free parameters are introduced beyond the mixing angle whose value is taken from BESIII data.

free parameters (1)

sinθ
Mixing angle between glueball and charmonium states; its small value is inferred from existing BESIII results rather than fitted inside this calculation.

axioms (2)

domain assumption Quenched lattice QCD gives Br(J/ψ → γ G_{0^-}) = 2.3(8)×10^{-4}
Used as the pure-glueball baseline; invoked in the abstract without re-derivation.
domain assumption The transition form factor for J/ψ → γ (c c-bar(0^-)) is much larger than for the pure glueball
Central to the enhancement argument; stated as a premise.

pith-pipeline@v0.9.0 · 5599 in / 1569 out tokens · 36117 ms · 2026-05-09T17:16:15.078189+00:00 · methodology

discussion (0)

Reference graph

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