Analysis of the $X(2370)$ as a glueball based on rigorous current-field duality

Zhi-Gang Wang

REVIEW 5 major objections 5 minor 1 cited by

Analysis of the $X(2370)$ as a glueball based on rigorous current-field duality

T0 review · 5 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Decay ratios peg X(2370) as a three-gluon glueball.

desk verdict A concrete three-gluon glueball decay construction undermined by an assumed factorization and a factor-of-ten error in the central ratio list. read the letter →

arxiv 2608.03362 v1 pith:RB72NSTK submitted 2026-08-04 hep-ph

Zhi-Gang Wang This is my paper · ORCID

classification hep-ph PACS 12.39.Mk14.20.Lq12.38.Lg

keywords glueballX(2370)pseudoscalarcurrent-fielddualityFierztransformationthree-bodystrongdecayspartialwidthratiosexotichadrons

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper argues that the X(2370), a particle with quantum numbers $J^{PC}=0^{-+}$ seen in radiative $J/\psi$ decays, is a glueball built from three valence gluons. To test this, it constructs a six-quark current through current-field duality, Fierz-transforms it into a sum of three-pseudoscalar-meson currents, and derives ratios among the partial decay widths of the glueball to three pseudoscalar mesons. The resulting ratios, $\Gamma(\pi\pi\eta):\Gamma(\pi\pi\eta'):\Gamma(KK\eta):\Gamma(KK\eta'):\Gamma(KK\pi):\Gamma(\eta\eta\eta):\Gamma(\eta\eta\eta') = 0.641:0.226:0.135:0.193:1.000:0.010:0.030$, are compatible with the measured product branching fractions. If the paper is right, the X(2370)'s decay pattern is set by its gluon substructure, supporting its identification as the $0^{-+}$ glueball.

What carries the argument

The load-bearing object is the pseudoscalar six-quark current $\tilde J(x)=f_{abc}\tilde J^a_{\mu\nu}\tilde J^{b\,\nu\rho}\tilde J^{c\,\mu}_{\rho}$, built from dual quark tensor currents $\tilde J^a_{\mu\nu}=\tfrac12\epsilon_{\mu\nu\alpha\beta}\bar q_j\tfrac{\lambda^a_{jk}}{2}\sigma^{\alpha\beta}q_k$ that carry the same quantum numbers as the dual gluon field-strength tensor. Coupling this current to the dual gluonic field via $L=\tilde g\tilde J\tilde G$ builds the glueball substructure into the Lagrangian. A Fierz transformation in Dirac-spinor and color spaces rewrites $\tilde J$ as a sum of products of standard pseudoscalar currents $J_\pi$, $J_K$, $J_5^q$, and $J_5^s$; the transition amplitudes to three mesons then factor into products of the single-meson couplings $\mu_\pi,\mu_K,\mu_\eta^q,\mu_\eta^s,\ldots$ times the common factor $\lambda_{\tilde G}\tilde g$. That factorization carries the argument: it turns the gluon substructure into a definite set of flavor-sensitive width ratios.

What would settle it

Measure the ratio $\Gamma(X\to KK\pi)/\Gamma(X\to\pi\pi\eta)$ in one experiment with controlled efficiencies: the paper predicts 1.56. A precise result away from 1.56 would refute the factorization prediction, as would a significant $X\to K^*(892)\bar K + c.c.$ signal, which the six-quark-current mechanism does not generate.

Watch

Extended reading notes

Core claim

The central claim is that the decay of the X(2370) into three pseudoscalar mesons is controlled by the gluon substructure of the state. Starting from the pseudoscalar six-quark current $\tilde J(x)=f_{abc}\tilde J^a_{\mu\nu}(x)\tilde J^{b\,\nu\rho}(x)\tilde J^{c\,\mu}_{\rho}(x)$, with each $\tilde J^a_{\mu\nu}(x)=\tfrac12\epsilon_{\mu\nu\alpha\beta}\bar q_j(x)\tfrac{\lambda^a_{jk}}{2}\sigma^{\alpha\beta}q_k(x)$, and coupling it to the corresponding gluonic field $\tilde G(x)$ through $L=\tilde g\,\tilde J(x)\tilde G(x)$, the paper Fierz-rearranges the current into color-singlet currents for $\pi$, $K$, $\eta$, and $\eta'$. With each bilinear current coupling through the standard decay constants, every three-body amplitude becomes a product of three single-meson factors $\mu$ times one common residue-and-coupling factor $\lambda_{\tilde G}\tilde g$. Integrating the three-body phase space gives the width ratios in Eq.~(28), which the paper finds compatible with the measured product branching fractions and takes as support for assigning X(2370) to a $0^{-+}$ glueball with three valence gluons.

Load-bearing premise

The whole prediction assumes that the X(2370) decays to three pseudoscalars as three independent quark-antiquark pairs that each turn into a meson, with no correlated dynamics, form factors, or final-state rescattering; if those effects matter, the predicted ratios shift.

Editorial extensions

If this is right

If X(2370) is a three-gluon $0^{-+}$ glueball, its decays to three pseudoscalar mesons should obey the Eq.~(28) ratios, with $\Gamma(KK\pi)$ the largest of the listed channels and $\Gamma(\eta\eta\eta)$, $\Gamma(\eta\eta\eta')$ suppressed to about one percent of it.
The ratios are independent of the unknown absolute residue $\lambda_{\tilde G}$ and coupling $\tilde g$, since every partial width carries the same factor $\lambda_{\tilde G}^2\tilde g^2$; only the overall scale depends on those quantities.
The current-field-duality prediction restores sizeable $\pi\pi\eta'$ and $KK\eta'$ rates, which the paper argues the chiral-effective-Lagrangian treatment without substructure makes too small.
A measurement of the product branching fractions for the same channels can be compared with the ratios channel by channel, after accounting for the common $J/\psi\to\gamma X(2370)$ production factor.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A sharp observable test follows directly: in a single experiment with common efficiencies, the paper's numbers predict $\Gamma(KK\pi)/\Gamma(\pi\pi\eta)\approx1.56$; a precise result away from that would signal final-state interactions or a non-glueball interpretation.
The factorization assumption can also be probed with charge asymmetries: the paper's amplitudes give equal rates for $X\to K^+K^-\pi^0$ and $X\to K^0\bar K^0\pi^0$, so an observed asymmetry would reveal rescattering effects not in the factorized current.
Extending the same construction to other $J^{PC}$ glueball candidates (for example $2^{++}$ or $1^{-+}$) would produce analogous ratio predictions, offering a way to test gluon-substructure assignments beyond the X(2370).
Only ratios are fixed here; a value for $\lambda_{\tilde G}$ from QCD sum rules together with a fitted $\tilde g$ would turn the ratios into absolute widths that could be checked against the measured X(2370) width of roughly 170 MeV.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Dear colleague,

This one is worth a look for the construction, not for the conclusions. Wang builds a six-quark, three-gluon pseudoscalar current for the X(2370), Fierz-rearranges it into products of pseudoscalar meson currents, and derives a set of three-body decay ratios. That is a genuinely new specific proposal, and it is a real step beyond the chiral-Lagrangian treatments in that it tries to tie the decay pattern to the valence three-gluon structure. The Fierz decomposition in Eq. (11) is stated without derivation, but it is the kind of thing that could be checked.

The problem is that the central quantitative claim does not hold up. The transition amplitudes in Eqs. (18)-(21) are obtained by replacing a six-quark matrix element <0|Jπ Jπ J5|ππη> with a product of three single-current matrix elements. That is a vacuum-saturation ansatz. For a local operator creating three on-shell mesons it is not a theorem, and the paper gives no large-Nc argument, no OPE estimate, and no discussion of connected contractions or final-state rescattering. The '...' in Eq. (11) only makes it harder to know what was dropped.

There is also a concrete arithmetic error. From the widths in Eqs. (24)-(27), Γ(KKη) = Γ(K+K−η)+Γ(K0K̄0η) = 3.61×10^-10, while Γ(KKπ) totals 2.68×10^-8. The ratio is 0.0135, not the 0.135 printed in Eq. (28). The factor of ten propagates into the second line as 0.44. This is the advertised ratio list, the quantitative core of the paper.

The comparison with BESIII is also selective. Normalizing Table 1 to the KKπ channel gives about 0.99 for ππη and 0.60 for ππη′, versus the paper's 0.641 and 0.226. 'Compatible' is doing a lot of work. And the measured quantities are product branching fractions that include the J/ψ→γX production, so a direct comparison with decay-width ratios assumes a common production rate.

Who is this for? Someone working on glueball spectroscopy who wants a concrete model with an explicit valence structure, and who is willing to treat the ratios as an ansatz to be tested rather than a derivation. As a proof of the glueball assignment, it fails.

My call: not acceptable as is. The construction is worth developing, but the factorization needs a real justification and the arithmetic needs fixing. If those were addressed, it could be a useful paper. I would send it to a serious referee before rejecting, mainly because the underlying question is timely and the construction is specific enough to be checked.

Best,

Referee Report

5 major / 5 minor

Summary. The paper assumes that the X(2370) with J^{PC}=0^{-+} is a three-gluon glueball and constructs a six-quark current (Eq. (6)) based on what it calls rigorous current-field duality. This current is coupled to a three-gluon field through the local Lagrangian in Eq. (8), then Fierz-rearranged into products of pseudoscalar currents (Eq. (11)). The three-meson decay amplitudes (Eqs. (18)-(21)) are then obtained by replacing the six-quark operator matrix element with a product of single-current couplings, and the partial widths are computed with the phase-space formula of Eq. (22). The central quantitative result is Eq. (28), a set of ratios among the X(2370) to three-pseudoscalar decay widths, which the authors compare with BESIII product branching fractions and use to support the glueball assignment.

Significance. If the calculation were sound, it would offer a concrete, structure-based connection between a three-gluon glueball and its three-pseudoscalar decay modes, and the comparison with BESIII would be a useful phenomenological input. The paper is also transparent about its input parameters (mixing angle, decay constants, quark condensate), which are taken from independent analyses rather than fitted to the X(2370) data. However, the central derivation rests on an unproven factorization of three-meson matrix elements, and the advertised ratio list in Eq. (28) contains a numerical inconsistency. The core claim of model independence is therefore not established, and the paper's main quantitative message is not reliable as it stands.

major comments (5)

[Section 2, Eq. (8)] The construction of the local interaction L = g~ J~(x) G~(x) is asserted rather than derived. Current-field duality is used in the literature to match quantum numbers of currents and fields, but it does not by itself imply a contact interaction between the six-quark current and the glueball field; the magnitude and even the local form of the coupling are assumptions. This step should be flagged as a model assumption, not as a rigorous consequence of duality.
[Section 2, Eq. (11)] The Fierz transformation is presented as 'straightforward but tedious' and the result contains an unspecified '...' set of terms. Without the full color and Dirac Fierz decomposition, one cannot check whether the omitted terms contribute to the same three-pseudoscalar channels and therefore whether the ratios in Eq. (28) are actually model-independent. The paper needs to provide the complete decomposition or an explicit argument that the omitted terms are negligible for the channels considered.
[Section 2, Eqs. (18)-(21)] The central amplitudes are obtained by factorizing a six-quark operator matrix element into a product of single-current matrix elements, for example <0|J_pi J_pi J_5|pi pi eta> replaced by <0|J_pi|pi><0|J_pi|pi><0|J_5|eta>. For a local operator creating three on-shell mesons this is not a theorem: crossed Wick contractions, connected contributions, momentum-dependent form factors, and final-state rescattering can contribute at comparable order. No large-N_c argument, OPE estimate, or comparison with a less factorized treatment is provided. This missing justification is load-bearing because the entire numerical comparison rests on this factorization.
[Section 3, Eq. (28)] The advertised ratio list is internally inconsistent. From Eqs. (24)-(27), Gamma(KK eta) = 3.6102 x 10^{-10} GeV^13 lambda_G^2 g^2 and Gamma(KK pi) = 2.6807 x 10^{-8} GeV^13 lambda_G^2 g^2, giving Gamma(KK eta)/Gamma(KK pi) = 0.0135, whereas Eq. (28) prints 0.135. The same factor-of-ten discrepancy appears in the second line (0.44 versus 0.0437). The central quantitative claim of the paper therefore cannot be used as evidence for the glueball assignment until this is corrected and the full set of ratios is rechecked.
[Section 3, Table 1 comparison] The comparison with BESIII data compares the theoretical width ratios to measured product branching fractions Br[J/psi -> gamma X(2370)] * Br[X(2370) -> PPP], which depend on the unknown production branching fraction and on reconstruction efficiencies. The paper does not explain how the common production factor cancels or how phase-space and efficiency differences are handled, so the phrase 'compatible with the experimental data' is not quantitatively justified even apart from the factorization issue.

minor comments (5)

[Abstract and Introduction] The phrase 'consists of three valence gluons' should be 'consisting of three valence gluons' or 'composed of three valence gluons'.
[Section 2, after Eq. (10)] The sentence 'the field G~(x) couples potentially to the glueball G~(p)' would be clearer as 'the field G~(x) can couple to the glueball state'.
[Section 2, Eq. (11)] The notation for the quark currents J_q^5 and J_s^5 is introduced above Eq. (11), but the reader has to infer which currents are the light-quark and strange-quark ones; a single defining sentence after Eq. (12) would improve readability.
[Section 3, Eq. (28)] The normalization of the second ratio line is unexplained; the reader must divide by 3.25 to recover the first line. Please state the normalization convention explicitly.
[References] There are several forthcoming or 2026-dated references (e.g., Refs. [6], [7], [43], [45]) with no journal or arXiv identifiers; please provide the full bibliographic information or clearly mark them as preprints.

Circularity Check

1 steps flagged · score 4.0 of 10

Partial circularity: the predicted decay ratios reduce by construction to the Fierz decomposition of the assumed six-quark current; the comparison with BESIII data is a consistency check of the current ansatz, not an independent model-free confirmation.

self definitional [Section 2, Eqs. (11)-(21), after Eq. (17)]
"It is obvious that the current ~J(x) couples potentially to three pseudoscalar mesons according to current-meson duality. Then it is easy to obtain the transition amplitudes T routinely, T_{~G→π0π0η} = √2 μπ μπ μη^q λ_{~G} ~g, ..."

The amplitudes in Eqs. (18)-(21) are not computed from the three-meson matrix element of the six-quark operator; they are obtained by replacing each pseudoscalar current J_P appearing in the Fierz decomposition of ~J (Eq. 11) with its one-meson matrix element μ_P from Eq. (17). Thus each T(~G→PPP) is, by construction, exactly the corresponding Fierz coefficient times a product of the μ_P's and the common residue-coupling factor λ_{~G}~g. The ratios in Eq. (28) therefore are the ratios of the Fierz terms of the particular six-quark current ~J chosen at the start; no other dynamics enter.

full rationale

The decay ratios are not fitted to the X(2370) branching-fraction data: the μ values, f_q, f_s, the mixing angle φ, and quark masses come from PDG and Feldmann-Kroll-Stech analyses, and the overall λ_{~G}~g factor cancels in the ratios. No parameter is tuned to BESIII data, and the Fierz coefficients are algebraically determined by the chosen current, so in that narrow sense the ratios are genuine outputs. The circular element is that the six-quark current ~J is selected to represent the assumed three-gluon glueball, and the three-meson amplitude is then defined by replacing the currents in the Fierz decomposition (11) with their one-meson matrix elements (17). Hence the predicted ratios are by construction the Fierz content of the input current; there is no independent calculation of the connected three-meson matrix element, and the 'model-independent' claim rests on an unproved vacuum-saturation/duality identification. The paper's self-citations to Refs. [38]-[41] and [44] are routine methodological references and are not load-bearing for the central result. Separately, and not counted as circularity, Eq. (28) is internally inconsistent: from Eqs. (24)-(27), Γ(KKη)/Γ(KKπ) = 3.610e-10 / 2.681e-8 = 0.0135, not the printed 0.135 (and 0.44 in the scaled line); this arithmetic error further weakens the quantitative comparison with BESIII. Overall, the central claim has independent numerical content but is partially circular in the sense that the output ratios reduce by construction to the assumed current's Fierz decomposition, giving a partial circularity score of 4.

Assumptions & free parameters 4 free parameters · 4 assumptions · 1 invented entities

The calculation depends heavily on phenomenological inputs fitted to meson data (mixing angle, decay constants, quark masses) and on strong modeling assumptions (current-field duality, vacuum saturation, Fierz truncation). The only truly new object is the specific form of the six-quark current and the resulting effective interaction.

free parameters (4)

eta-eta' mixing angle phi = 39.3 degrees
Taken from the Feldmann-Kroll-Stech scheme [42]; used in the pseudoscalar density matrix elements in Eq. (17).
Quark condensate <q qbar> = (240 MeV)^3
Standard value from [44,45], used with the Gell-Mann-Oakes-Renner relation to set the light quark mass m_q.
Strange-to-light quark mass ratio m_s/m_q = 27.30
From PDG [43]; sets the relative size of mu_s versus mu_q in the eta and eta' couplings.
Decay constants f_q and f_s = f_q = 1.07 f_pi, f_s = 1.34 f_pi
From the FKS phenomenological analysis [42]; determine the absolute scale of the eta and eta' couplings.

assumptions (4)

ad hoc to paper The six-quark current \tilde J(x) and the dual three-gluon field \tilde G(x) have the same quantum numbers and can be coupled by the local effective Lagrangian L = g~ \tilde J \tilde G (Eq. 8).
This current-field duality is asserted as rigorous but is a modeling assumption; no proof is given that the overlap with the three-gluon glueball dominates.
ad hoc to paper The Fierz transformation in Eq. (11) is complete for the pseudoscalar-meson channels, with all other terms negligible or absent.
The derivation is not shown and the '...' terms are not specified; the calculation assumes only the displayed pseudoscalar currents contribute.
domain assumption Vacuum saturation: the matrix element of the product of three currents between vacuum and three mesons factorizes into the product of single-current matrix elements.
Used to obtain amplitudes in Eqs. (18)-(21); standard in sum-rule analyses but a strong approximation for three-body decays.
domain assumption The U_A(1) anomaly subtraction in Eq. (16) correctly yields the pseudoscalar density matrix elements for eta and eta'.
Relies on the standard PCAC/anomaly framework and the FKS mixing scheme.

invented entities (1)

Local glueball-quark contact interaction L = g~ \tilde J(x) \tilde G(x)
purpose: To mediate the X(2370) to three pseudoscalar meson decays from the assumed three-gluon substructure.
No direct evidence is offered that the decay amplitude is dominated by this local current-field duality term; it is introduced ad hoc in Eq. (8).

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Pith. "Pith review of Analysis of the $X(2370)$ as a glueball based on rigorous current-field duality." pith.science (2026). https://pith.science/paper/RB72NSTK

@misc{pith2026260803362,
  author       = {Pith},
  title        = {Pith review of: Analysis of the $X(2370)$ as a glueball based on rigorous current-field duality},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RB72NSTK}},
  note         = {Machine review of arXiv:2608.03362}
}

abstract

In this work, we take the $X(2370)$ with $J^{PC}=0^{-+}$ as a glueball consists of three valence gluons, and construct a six-quark current based on rigorous current-field duality to obtain the glueball-quark Lagrangian. Then we perform Fierz transformation to bosonize the quark current into a series of three pseudoscalar mesons. At last, we obtain ratios among the partial decay widths of the glueball to three pseudoscalar mesons in a model-independent way, which are compatible with the experimental data from the BESIII Collaboration and support assigning the $X(2370)$ as a glueball.

Figures

Figures reproduced from arXiv: 2608.03362 by the authors.

**Figure 1.** The mechanism for the decay chain J/ψ → γX(2370) → γP P P, where the P denotes the pseudoscalar mesons. and pseudoscalar field Ge(x) = fabcGea µν (x)Geνρ b (x)Gecµ ρ (x), (7) have the same quantum numbers. We construct the effective Lagrangian L(x) = egJe(x)Ge(x), (8) to describe the glueball-quark interactions, which symbolizes the under-structures of the glueball definitely. Such a method can be applied directly t… view at source ↗

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

The decay properties of two- and three- gluon glueballs
hep-ph 2026-08 conditional novelty 5.0 of 10

The authors derive relative branching ratios for two- and three-gluon glueball decays with Fierz rearrangement and a QCD sum rule check, favoring f0(1710) and eta(2370) as glueball candidates and identifying K*(892) a...

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Works this paper leans on

45 extracted references · 39 canonical work pages · cited by 1 Pith paper

[1]

Ablikim et al, Phys

M. Ablikim et al, Phys. Rev. Lett. 106 (2011) 072002

work page 2011
[2]

Ablikim et al, Eur

M. Ablikim et al, Eur. Phys. J. C80 (2020) 746

work page 2020
[3]

Ablikim et al, Phys

M. Ablikim et al, Phys. Rev. D103 (2021) 012009

work page 2021
[4]

Ablikim et al, Phys

M. Ablikim et al, Phys. Rev. Lett. 129 (2022) 022002

work page 2022
[5]

Ablikim et al, Phys

M. Ablikim et al, Phys. Rev. Lett. 132 (2024) 181901. 7

work page 2024
[6]

Ablikim et al, arXiv: 2605.26495 [hep-ex]

M. Ablikim et al, arXiv: 2605.26495 [hep-ex]

work page arXiv
[7]

Lightest $0^{-+}$ Glueball as Dominant Constituent of $X(2370)$

M. Ablikim et al, arXiv:2607.20366 [hep-ex]

work page Pith review arXiv
[8]

L. C. Gui, J. M. Dong, Y. Chen and Y. B. Yang, Phys. Rev. D100 (2019) 054511

work page 2019

Show all 45 references

[9]

Cao et al, Phys

J. Cao et al, Phys. Rev. D110 (2024) 054046

2024
[10]

X. T. Li, G.Y. Wang and Q. A. Zhang, Eur. Phys. J. C85 (2025) 365

2025
[11]

Y. Chen, L. C. Gui, G. Li and W. Sun, arXiv: 2605.01757 [hep-ph]

arXiv
[12]

W. H. Tan and H. X. Chen, Phys. Rev. D113 (2026) 114030

2026
[13]

J. S. Yu, Z. F. Sun, X. Liu and Q. Zhao, Phys. Rev. D83 (2011) 114007

2011
[14]

L. M. Wang, Q. S. Zhou, C. Q. Pang and X. Liu, Phys. Rev. D102 (2020) 114034

2020
[15]

C. Deng, J. Ping, Y. Yang and F. Wang, Phys. Rev. D86 (2012) 014008

2012
[16]

R. R. Dong, N. Su, H. X. Chen, E. L. Cui and Z. Y. Zhou, Eur. Ph ys. J. C80 (2020) 749

2020
[17]

Q. N. Wang, D. K. Lian and W. Chen, Phys. Rev. D112 (2025) 034010

2025
[18]

G. S. Bali et al, Phys. Lett. B309 (1993) 378

1993
[19]

C. J. Morningstar and M. J. Peardon, Phys. Rev. D60 (1999) 034509

1999
[20]

Chen et al, Phys

Y. Chen et al, Phys. Rev. D73 (2006) 014516

2006
[21]

Gregory et al, JHEP 10 (2012) 170

E. Gregory et al, JHEP 10 (2012) 170

2012
[22]

Mathieu, N

V. Mathieu, N. Kochelev and V. Vento, Int. J. Mod. Phys. E18 (2009) 1

2009
[23]

G. S. Bali et al, Phys. Rev. D62 (2000) 054503

2000
[24]

Athenodorou and M

A. Athenodorou and M. Teper, JHEP 11 (2020) 172

2020
[25]

Vaccarino and D

A. Vaccarino and D. Weingarten, Phys. Rev. D60 (1999) 114501

1999
[26]

G. Hao, C. F. Qiao and A. L. Zhang, Phys. Lett. B642 (2006) 53

2006
[27]

Huang, H

T. Huang, H. Y. Jin and A. L. Zhang, Phys. Rev. D59 (1999) 034026

1999
[28]

Tang and C

L. Tang and C. F. Qiao, Nucl. Phys. B904 (2016) 282

2016
[29]

H. X. Chen, W. Chen and S. L. Zhu, Phys. Rev. D104 (2021) 094050

2021
[30]

S. Q. Zhang, B. D. Wan, L. Tang and C. F. Qiao, Phys. Rev. D106 (2022) 074010

2022
[31]

C. F. Qiao and L. Tang, Phys. Rev. Lett. 113 (2014) 221601

2014
[32]

Harnett, R

D. Harnett, R. T. Kleiv, K. Moats and T. G. Steele, Nucl. Phys. A850 (2011) 110

2011
[33]

A. L. Zhang and T. G. Steele, Nucl. Phys. A728 (2003) 165

2003
[34]

X. Sun, L. Y. Dai, S. Q. Kuang, W. Qin and A. P. Szczepaniak, Phy s. Rev. D105 (2022) 034010

2022
[35]

W. I. Eshraim, S. Janowski, F. Giacosa and D. H. Rischke, Phys. Rev. D87 (2013) 054036

2013
[36]

W. I. Eshraim, Phys. Rev. D100 (2019) 096007. 8

2019
[37]

W. I. Eshraim, Eur. Phys. J. C83 (2023) 262

2023
[38]

Z. G. Wang, Phys. Rev. D111 (2025) 114009

2025
[39]

Z. G. Wang, Eur. Phys. J. C74 (2014) 2874

2014
[40]

Z. G. Wang, Commun. Theor. Phys. 63 (2015) 466

2015
[41]

Z. G. Wang, Int. J. Mod. Phys. A35 (2020) 2050003

2020
[42]

Feldmann, Int

T. Feldmann, Int. J. Mod. Phys. A15 (2000) 159

2000
[43]

Takahashi et al, Int

F. Takahashi et al, Int. J. Mod. Phys. A41 (2026) 2630011

2026
[44]

Z. G. Wang, Front. Phys. 21 (2026) 016300

2026
[45]

Colangelo and A

P. Colangelo and A. Khodjamirian, hep-ph/0010175. 9

arXiv

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

Ablikim et al, Phys

M. Ablikim et al, Phys. Rev. Lett. 106 (2011) 072002

work page 2011

[2] [2]

Ablikim et al, Eur

M. Ablikim et al, Eur. Phys. J. C80 (2020) 746

work page 2020

[3] [3]

Ablikim et al, Phys

M. Ablikim et al, Phys. Rev. D103 (2021) 012009

work page 2021

[4] [4]

Ablikim et al, Phys

M. Ablikim et al, Phys. Rev. Lett. 129 (2022) 022002

work page 2022

[5] [5]

Ablikim et al, Phys

M. Ablikim et al, Phys. Rev. Lett. 132 (2024) 181901. 7

work page 2024

[6] [6]

Ablikim et al, arXiv: 2605.26495 [hep-ex]

M. Ablikim et al, arXiv: 2605.26495 [hep-ex]

work page arXiv

[7] [7]

Lightest $0^{-+}$ Glueball as Dominant Constituent of $X(2370)$

M. Ablikim et al, arXiv:2607.20366 [hep-ex]

work page Pith review arXiv

[8] [8]

L. C. Gui, J. M. Dong, Y. Chen and Y. B. Yang, Phys. Rev. D100 (2019) 054511

work page 2019