Probing the ATOMKI X17 vector boson using Dalitz decays $V\to Pe^+e^-$

Anh-Tuyet T. Nguyen; Chien-Thang Tran; Mikhail A. Ivanov

REVIEW 1 major objections 2 minor 61 references

Probing the ATOMKI X17 vector boson using Dalitz decays $V\to Pe^+e^-$

T0 review · 1 major / 2 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read With the X17 couplings favored by nuclear anomalies, the X17 contribution to the D*0 Dalitz decay is two orders of magnitude below the Standard Model, so the BESIII 3.5-sigma excess cannot be X17 alone.

desk verdict Independent CCQM Dalitz ratios for V→P e+e− are valuable, but the strong claim that X17 cannot explain the BESIII D*0 excess only holds under an unsupported universality assumption. read the letter →

arxiv 2506.23372 v2 pith:7ILQ2SUW submitted 2025-06-29 hep-ph

Chien-Thang Tran , Mikhail A. Ivanov , Anh-Tuyet T. Nguyen This is my paper

classification hep-ph PACS 13.25.Ft13.20.-v12.39.Ki

keywords X17bosonDalitzdecayvectormesoncovariantconfinedquarkmodeldominanceBESIIIanomalyATOMKItransitionformfactors

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

This paper asks whether the 17 MeV X17 vector boson invoked for the ATOMKI nuclear anomalies would leave a visible mark in the Dalitz decays $D^*_{(s)}\to D_{(s)}e^+e^-$, $B^*_{(s)}\to B_{(s)}e^+e^-$, and $J/\psi\to\eta_c e^+e^-$, and whether it can explain the BESIII $3.5\sigma$ excess in $D^{*0}\to D^0e^+e^-$. The hadronic form factors are computed in the covariant confined quark model over the full momentum range, without relying on vector meson dominance or heavy quark effective theory. In the Standard Model the predicted ratios $R_{ee}(V)\equiv\Gamma(V\to Pe^+e^-)/\Gamma(V\to P\gamma)$ agree with VMD-based calculations within a few percent, validating the model. For the X17-induced part the two approaches can differ by up to an order of magnitude, and under the ATOMKI-favored couplings the X17 contribution to $D^{*0}\to D^0e^+e^-$ is two orders below the Standard Model, so the BESIII excess cannot be explained by X17 alone.

What carries the argument

The load-bearing object is the ratio $R_{ee}(V)=\Gamma(V\to Pe^+e^-)/\Gamma(V\to P\gamma)$, assembled from the transition form factors $g_{VP\gamma}(q^2)$ and $g_{VPX}(q^2)$. Both are computed from triangle diagrams in the covariant confined quark model, whose Gaussian meson-quark vertex functions and infrared cutoff give the full $q^2$ dependence without vector meson dominance or heavy quark effective theory; the numerical results are interpolated with the double-pole form $g(0)/(1-aq^2+bq^4)$. Because the X17 is very narrow, its amplitude does not interfere with the photon amplitude, so $R_{ee}=R^\gamma_{ee}+R^X_{ee}$, and in the narrow-width approximation with $\mathrm{Br}(X\to e^+e^-)\approx1$ the electron coupling cancels, leaving predictions that depend only on the quark couplings and the hadronic form factors.

What would settle it

Measure the $e^+e^-$ invariant-mass distribution of $D^{*0}\to D^0e^+e^-$ at BESIII: if the $3.5\sigma$ excess persists but shows no narrow peak near 17 MeV, the X17-only explanation is falsified. A model-independent measurement of $R_{ee}(D^{*+}\to D^+e^+e^-)$ or $R_{ee}(D^{*+}_s\to D^+_s e^+e^-)$ that matches the VMD prediction rather than the CCQM's X17-enhanced value would also discriminate between the two form-factor treatments.

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Extended reading notes

Core claim

The paper's central claim is that the covariant confined quark model yields a complete set of Dalitz-decay form factors and ratios for seven vector-meson channels, and that these can be used to test the X17 hypothesis. With the ATOMKI-favored couplings $\varepsilon_u\approx\pm3.7\times10^{-3}$ and $\varepsilon_d\approx\mp7.4\times10^{-3}$, the X17 contribution to $D^{*0}\to D^0e^+e^-$ is $R^X_{ee}\simeq3\times10^{-5}$, two orders of magnitude below the Standard Model value $R^\gamma_{ee}\simeq6.45\times10^{-3}$, so the BESIII $3.5\sigma$ excess cannot be attributed to X17 alone. The same calculation identifies $D^{*+}\to D^+e^+e^-$ and $D^{*+}_s\to D^+_s e^+e^-$ as the most X17-sensitive channels, and shows that VMD-based X17 ratios can differ from the CCQM results by up to an order of magnitude. Only with the much larger best-fit coupling $\varepsilon_u=6.0\times10^{-2}$ from a Dalitz-data fit could the X17 saturate the BESIII excess, but that coupling conflicts with the ATOMKI constraints, exposing a tension between the two experimental inputs.

Load-bearing premise

The conclusions rest on X17 being a narrow vector boson with vector couplings to quarks and a nearly complete branching into $e^+e^-$, on the specific coupling values taken from the ATOMKI constraints, and on the quark-model assumptions (Gaussian vertex shapes and fitted quark masses and size parameters) used to compute the hadronic form factors.

Editorial extensions

If this is right

The Standard Model Dalitz ratios for all seven channels agree with VMD predictions at the few-percent level, so a precise measurement of any $R_{ee}(V)$ can test both frameworks wherever they differ.
Under the ATOMKI-favored couplings, $D^{*+}\to D^+e^+e^-$ and $D^{*+}_s\to D^+_s e^+e^-$ are the most X17-sensitive modes, and the predicted total ratio for $D^{*+}_s$ stays consistent with the CLEO measurement when the X17 contribution is included.
With the same couplings, the X17 contribution to $D^{*0}\to D^0e^+e^-$ is two orders of magnitude below the Standard Model, so the BESIII $3.5\sigma$ excess cannot be explained by X17 alone in that scenario.
Because the CCQM and VMD predictions for $R^X_{ee}$ differ by up to an order of magnitude in some channels, conclusions about X17 drawn from Dalitz decay data remain model-dependent until independent form-factor calculations converge.

Reading between the lines

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

Editorial inference: the decisive test is the $e^+e^-$ invariant-mass spectrum rather than the integrated ratio, since an X17 signal would appear as a narrow peak at $m_X\simeq17$ MeV while the current BESIII excess was extracted from the total rate.
Editorial inference: the same form-factor machinery could be extended to $\psi(2S)\to\eta_c e^+e^-$ or to muon Dalitz decays, giving additional channels where the X17 peak could be searched for above the QED background.
Editorial inference: the order-of-magnitude model spread in $R^X_{ee}$ suggests that a global fit treating hadronic form factors as nuisance parameters, rather than fixing them to either VMD or a single quark model, is needed before any Dalitz excess is interpreted as a new particle.
Editorial inference: if the D*0 excess is not X17, it would point to a deficiency in the Standard Model prediction for the $D^{*0}\to D^0\gamma$ transition form factor or to an experimental effect; comparing $R_{ee}(D^{*0})$ and the $q^2$ spectrum with the upcoming Super Charm-Tau factory would help settle it.

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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

Colleague,

The useful thing here is the CCQM computation of the V→P e+e− Dalitz form factors and ratios for D, B, and charmonium vector mesons, with and without a 17 MeV vector X17. The SM ratios agree with the VMD numbers to a few percent and with the CLEO D_s* measurement, so the hadronic baseline is credible. The X17 ratios, however, differ from the VMD-based Ref. [34] by up to an order of magnitude for D*+ and D*_s, which is a genuine discrepancy that a referee should ask the authors to explain more concretely.

The main soft spot is the D*0 conclusion. The paper states, under Table VI, that ε_c = ε_u and ε_s = ε_b = ε_d. That universality is not constrained by the ATOMKI data, which fix only combinations of ε_u and ε_d; charm is open. Under that assumption the X17 contribution to D*0→D0e+e− is two orders below the SM and the BESIII excess is not explained. But the paper's own Table IX, using the non-universal best-fit of Ref. [34], reproduces the BESIII central value with a large ε_u. So the Summary's claim that the excess 'cannot be accounted for by the X17 alone' is too broad. What is established is that the ATOMKI-favored protophobic universal couplings cannot do it. That is still a worthwhile negative statement, but it must be scoped precisely.

Two smaller issues. The 10% model uncertainty is quoted globally, not propagated to the ratios, so the error on the X17 sensitivities, which matter for experimental planning, is not quantified. And no code or data are shipped, which makes the order-of-magnitude gap with Ref. [34] hard to investigate. The neglect of resonance diagrams is argued qualitatively; for these small q² windows it looks harmless, but a numerical estimate would be cleaner.

Presentational point: Table IX is a demonstration of tension, not an independent prediction, because the couplings are fitted to the same BESIII data. The paper mostly frames it that way, but a reader could over-read it as a constraint.

Who is this for? Experimentalists at BESIII and Super Charm-Tau who want an independent hadronic calculation to plan Dalitz searches, and theorists working on X17 interpretations. It is a serious, if niche, paper. It deserves a genuine referee: the formalism is standard, the parameter tables are explicit, and the central CCQM-vs-VMD discrepancy is worth resolving in the literature. My recommendation is to send it out, with a request that the authors restrict the X17-exclusion claim to the universality assumption and give some error estimate on the ratios.

Referee Report

1 major / 2 minor

Summary. The paper studies the Dalitz decays V -> P e+ e- for V = D*, D*_s, B*, B*_s, J/psi and P the corresponding pseudoscalar meson, both in the Standard Model and in the presence of a hypothetical X17 vector boson associated with the ATOMKI anomalies. The hadronic form factors are computed in the covariant confined quark model (CCQM), without invoking VMD or HQET, and are presented as double-pole parametrizations with explicit coefficients. The authors give SM decay widths and ratios R_ee(V) = Gamma(V -> P e+ e-)/Gamma(V -> P gamma), then add the X17 contribution for several coupling scenarios: the ATOMKI protophobic combination, the Denton-Gehrlein sets with and without isospin effects, and the Lee et al. best-fit non-universal couplings. They compare with VMD-based predictions and with CLEO and BESIII data, and conclude that under the ATOMKI-favored couplings the X17 contribution to D*0 -> D0 e+ e- is too small to explain the BESIII 3.5 sigma excess.

Significance. If the calculations are correct, the SM predictions provide a useful independent cross-check of VMD-based form factors, and the explicit parameter tables make the calculation reproducible in principle. The agreement with VMD for the SM ratios R_gamma_ee, at the few-percent level, is a genuine strength of the paper. The X17 analysis is also timely given the recent BESIII measurement and the upcoming Super Charm-Tau factory program. However, the central negative claim about D*0 -> D0 e+ e- depends on an unproven flavor-universality assumption, and the quoted X17 ratios lack propagated uncertainties, so the new-physics conclusions are more fragile than the SM part of the paper.

major comments (1)

[Section III] The neglect of resonance diagrams in Section III is justified only by the qualitative statement that the V-P mass splittings are small compared with the vector-resonance masses. This argument does not by itself bound the size of the resonance-pole amplitudes at q^2 = 0 or q^2 = m_X^2, which are the kinematical points relevant for the ratios R_ee. Since one of the paper's goals is to provide an independent alternative to the VMD calculation, the authors should either quantify the neglected contributions by computing a representative resonance diagram, or soften the claim that the X17 ratios differ from the VMD predictions by up to an order of magnitude.

minor comments (2)

[Section V] The Summary states that among the two most sensitive modes, D*0 -> D0 e+ e- 'has not been observed', but it was measured by BESIII in Ref. [27], and the paper compares with that measurement in Tables VI-IX. This sentence should be corrected.
[Section IV] The sentence in Section IV that the X17 contribution to D*0 is 'smaller than the SM contribution by two orders of magnitude' in Table VI is referring to the ratio R_X/R_gamma, not to the width itself; the wording could be made more precise to avoid confusion.

Circularity Check

1 steps flagged · score 6.0 of 10

The CCQM Dalitz ratios are independent predictions, but Table IX's X17 'explanation' of the BESIII D*0 excess reuses couplings fitted to that same excess, making that particular step circular.

fitted input called prediction [Section IV, paragraph introducing Table IX, and Table IX itself]
"Finally, in Table IX, we present our predictions for the decays D∗0 → D0e+e−, D∗+s → D+se+e−, and J/ψ → ηce+e− using the best-ﬁt couplings εu = 6.0 × 10−2, εc = 6.4 × 10−3, and εs = −2.0 × 10−3 recently obtained in Ref. [34]. Note that these values were determined by ﬁtting the VMD predictions for the Dalitz decays of D∗s, D∗0, ψ(2S), and φ to available experimental data."

The couplings used in Table IX were obtained in Ref. [34] by fitting to experimental data that include the D*0 → D0 e+e− decay, i.e. the very BESIII measurement that Table IX then 'explains'. The table shows Rtot_ee(D*0) = 11.68e-3 versus the BESIII value 11.08 ± 0.90e-3 and the text states that 'the X17 contributes largely to the decay D∗0 → D0e+e− and can explain the BESIII excess'. Because the input coupling εc was adjusted using this same Dalitz channel, the agreement is a reconstruction of the fitted data point, not an independent prediction. The paper is transparent that the values were fitted, but the language 'we present our predictions' and 'can explain the BESIII excess' presents the fitted result as a derived consequence.

full rationale

The central Standard Model calculation is not circular: the CCQM form factors and the ratios Rγ_ee are computed from a fixed Lagrangian with parameters fitted to unrelated observables (Tables II-III), and the ratios are compared with VMD predictions and with CLEO/BESIII data without tuning those parameters to the target channels. The ATOMKI and NA48/2 constraints behind Tables VI-VIII are external inputs independent of the Dalitz data, so the small X17 contribution to D*0 → D0 e+e− under the universal-coupling assumption is a genuine prediction. The single circular element is Table IX, where the 'best-fit couplings' of Ref. [34] were fitted to data including the D*0 Dalitz decay and then used to 'explain' the same BESIII excess; that agreement is enforced by construction. The Summary statement that the BESIII excess 'cannot be accounted for by the X17 alone' should therefore be read as conditional on the assumption εc = εu, which ATOMKI and NA48/2 do not determine; with the fitted εc of Ref. [34] the excess is reproduced. Overall, the paper is mostly self-contained against external benchmarks, with one partial circularity at the Table IX consistency check, giving score 6.

Assumptions & free parameters 7 free parameters · 6 assumptions · 0 invented entities

The paper introduces no new particle, symmetry, or mediator; X17 is adopted from prior ATOMKI literature, and the CCQM parameters are inherited from earlier fits. The double-pole interpolation coefficients in Table IV are numerical fits to the computed form factors and are not treated as independent physics parameters. The main model dependence is the CCQM vertex ansatz, the fitted parameter set, and the external X17 coupling scenarios.

free parameters (7)

Constituent quark masses mu/d, ms, mc, mb = 0.241, 0.428, 1.672, 5.046 GeV
Best-fit CCQM parameters from Table III, fitted to meson data in prior CCQM studies.
Meson size parameters Lambda_H = 1.529 to 3.777 GeV for D, Ds, B, Bs, eta_c, D*, Ds*, B*, Bs*, J/psi
Best-fit values in Table II; each meson has its own Gaussian vertex width.
Confinement infrared cutoff lambda = 0.181 GeV
Fitted universal cutoff in Eq. (25) that enforces quark confinement in the CCQM.
X17 quark couplings epsilon_u, epsilon_d (ATOMKI protophobic central) = ±3.7e-3, ∓7.4e-3
Taken from Feng et al. constraints (Eqs. 31-32) and used in Table VI; assumes universal couplings epsilon_c=epsilon_u and epsilon_s=epsilon_b=epsilon_d.
X17 quark couplings epsilon_u, epsilon_d (Denton-Gehrlein, no isospin) = ±5.0e-4, ∓2.9e-3
Used in Tables IV and VII; taken from Ref. [61] with isospin corrections neglected.
X17 quark couplings epsilon_u, epsilon_d (Denton-Gehrlein, with isospin) = ±9.0e-4, ∓2.5e-3
Used in Table VIII; taken from Ref. [61] with isospin mixing and breaking effects included.
X17 quark couplings epsilon_u, epsilon_c, epsilon_s (Lee et al. best fit) = 6.0e-2, 6.4e-3, -2.0e-3
Fitted by Ref. [34] to BESIII, CLEO, and other Dalitz data; used in Table IX as a consistency check rather than an independent prediction.

assumptions (6)

domain assumption Gaussian vertex function Phi_M(-p^2)=exp(p^2/Lambda_M^2) with compositeness condition Z_M=0 defines the meson-quark interaction vertices.
Invoked in Eqs. (2)-(4); the predicted form factors depend on this ansatz and on the fitted Lambda_M values.
domain assumption Quark confinement is implemented by a universal infrared cutoff lambda in the Fock-Schwinger integrals, Eq. (25).
This cutoff is a fitted model ingredient; it is not derived from QCD and affects all numerical form factors.
domain assumption X17 exists and is a narrow vector boson with vector couplings to quarks and electrons, Eqs. (10)-(11).
Adopted from the ATOMKI-motivated literature; the paper does not derive or test the spin, parity, or Lorentz structure of X17.
ad hoc to paper The X17 boson decays dominantly into e+e-, so Gamma_X approximately equals Gamma(X to e+e-), Eq. (19), which makes epsilon_e cancel.
This simplifying assumption maximizes the X17 contribution; if Br(X to e+e-) is smaller, all quoted R_X values are upper bounds.
standard math The narrow-width approximation, Eq. (18), replaces the X17 Breit-Wigner propagator by a delta function.
Valid for a narrow X17 but neglects off-shell and interference effects; stated explicitly in Section III.
domain assumption Vector-resonance intermediate states (V' to gamma*) are negligible because the mass splittings Delta m(VP) are much smaller than the vector resonance masses.
Asserted in Section III with a qualitative argument; the stated 10% error estimate is not derived in detail.

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Pith. "Pith review of Probing the ATOMKI X17 vector boson using Dalitz decays $V\to Pe^+e^-$." pith.science (2026). https://pith.science/paper/7ILQ2SUW

@misc{pith2026250623372,
  author       = {Pith},
  title        = {Pith review of: Probing the ATOMKI X17 vector boson using Dalitz decays $V\to Pe^+e^-$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7ILQ2SUW}},
  note         = {Machine review of arXiv:2506.23372}
}

abstract

Recent anomalies observed in $e^+e^-$ nuclear transitions of $^8 \textrm{Be}$, $^4 \textrm{He}$, and $^{12} \textrm{C}$ by the ATOMKI collaboration may hint at the existence of a vector boson with a mass around 17 MeV, referred to as X17. If it exists, this boson would also affect similar processes in particle physics, including the Dalitz decays of vector mesons. Recently, the BESIII collaboration measured the Dalitz decay $D^{*0}\to D^0e^+e^-$ for the first time and reported a $3.5\sigma$ excess over the theoretical prediction based on the vector meson dominance (VMD) model. This excess may be another signal of the X17. In this study, we investigate the possible effects of the X17 on the Dalitz decays $D^*_{(s)}\to D_{(s)}e^+e^-$, $B^*_{(s)}\to B_{(s)}e^+e^-$, and $J/\psi\to \eta_ce^+e^-$. The required hadronic form factors are calculated within the framework of our covariant confined quark model, without relying on heavy quark effective theory or the VMD model. We present predictions for the Dalitz decay widths and the ratios $R_{ee}(V)\equiv \Gamma(V\to Pe^+e^-)/\Gamma(V\to P\gamma)$ within the Standard Model and in several new physics scenarios involving modifications due to the X17. Our results are compared with other theoretical calculations.

Figures

Figures reproduced from arXiv: 2506.23372 by the authors.

**Figure 2.** FIG. 2. Feynman diagrams for Dalitz decays [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]

**Figure 5.** Figure 5: FIG. 5. Normalized form factors squared as functions of the d [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]

**Figure 6.** Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]

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Works this paper leans on

61 extracted references · 24 canonical work pages

[34]

F. F. Lee, L. T. T. Uyen, and G. L. Lin, arXiv:2501.13530

work page Pith review arXiv
[1]

A. J. Krasznahorkay, M. Csatl´ os, L. Csige et al. , Phys. Rev. Lett. 116, 042501 (2016), arXiv:1504.01527

arXiv 2016
[2]

A. J. Krasznahorkay, M. Csatl´ os, L. Csige et al. , Acta Phys. Polon. B 50, 675 (2019)

work page 2019
[3]

M. E. Rose, Phys. Rev. 76, 678 (1949) [Erratum: Phys. Rev. 78, 184(E) (1950)]

work page 1949
[4]

Schl¨ uter, G

P. Schl¨ uter, G. Soﬀ, and W. Greiner, Phys. Rept. 75, 327 (1981)

work page 1981
[5]

A. J. Krasznahorkay, M. Csatl´ os, L. Csige et al. , Phys. Rev. C 104, 044003 (2021), arXiv:2104.10075

arXiv 2021
[6]

A. J. Krasznahorkay, A. Krasznahorkay, M. Begala et al. , Phys. Rev. C 106, L061601 (2022), arXiv:2209.10795

arXiv 2022
[7]

Banerjee et al

D. Banerjee et al. (NA64 Collaboration), Phys. Rev. Lett. 120, 231802 (2018), arXiv:1803.07748

arXiv 2018

Show all 61 references

[8]

Banerjee et al

D. Banerjee et al. (NA64 Collaboration), Phys. Rev. D 101, 071101 (2020), arXiv:1912.11389

2020 arXiv
[9]

Depero et al

E. Depero et al. (NA64 Collaboration), Eur. Phys. J. C 80, 1159 (2020), arXiv:2009.02756

2020 arXiv
[10]

Afanaciev et al

K. Afanaciev et al. (MEG II Collaboration), arXiv:2411.07994

arXiv
[11]

T. Anh, T. D. Trong, A. J. Krasznahorkay et al. , Universe 10, 168 (2024), arXiv:2401.11676

2024 arXiv
[12]

Bossi, R

F. Bossi, R. De Sangro, C. Di Giulio et al. , arXiv:2505.24797
[13]

Bertelli et al

S. Bertelli et al. (PADME Collaboration), JHEP 06, 040 (2025), arXiv:2503.05650

2025 arXiv
[14]

J. L. Feng, B. Fornal, I. Galon et al. , Phys. Rev. Lett. 117, 071803 (2016), arXiv:1604.07411

2016 arXiv
[15]

J. L. Feng, B. Fornal, I. Galon et al. , Phys. Rev. D 95, 035017 (2017), arXiv:1608.03591

2017 arXiv
[16]

Zhang and G

X. Zhang and G. A. Miller, Phys. Lett. B 813, 136061 (2021), arXiv:2008.11288

2021 arXiv
[17]

Kozaczuk, D

J. Kozaczuk, D. E. Morrissey, and S. R. Stroberg, Phys. R ev. D 95, 115024 (2017), 19 arXiv:1612.01525

2017 arXiv
[18]

Barducci and C

D. Barducci and C. Toni, JHEP 02, 154 (2023) [Erratum: JHEP 07, 168(E) (2023)] arXiv:2212.06453

2023 arXiv
[19]

Ellwanger and S

U. Ellwanger and S. Moretti, JHEP 11, 039 (2016), arXiv:1609.01669

2016 arXiv
[20]

D. V. Kirpichnikov, V. E. Lyubovitskij, and A. S. Zhevla kov, Phys. Rev. D 102, 095024 (2020), arXiv:2002.07496

2020 arXiv
[21]

Puli¸ ce, Chin

B. Puli¸ ce, Chin. J. Phys. 71, 506-517 (2021), arXiv:1911.10482

2021 arXiv
[22]

Seto and T

O. Seto and T. Shimomura, JHEP 04, 025 (2021), arXiv:2006.05497

2021 arXiv
[23]

Nomura and P

T. Nomura and P. Sanyal, JHEP 05, 232 (2021), arXiv:2010.04266

2021 arXiv
[24]

Di Luzio, P

L. Di Luzio, P. Paradisi, and N. Selimovic, arXiv:2504. 14014
[25]

Barducci, D

D. Barducci, D. Germani, M. Nardecchia et al. , JHEP 04, 035 (2025), arXiv:2501.05507

2025 arXiv
[26]

D. S. M. Alves, D. Barducci, G. Cavoto et al. , Eur. Phys. J. C 83, 230 (2023)

2023
[27]

Ablikim et al

M. Ablikim et al. (BESIII Collaboration), Phys. Rev. D 104, 112012 (2021), arXiv:2111.06598

2021
[28]

J. J. Sakurai, Annals Phys. 11, 1 (1960)

1960
[29]

L. G. Landsberg, Phys. Rept. 128, 301 (1985)

1985
[30]

J. Fu, H. B. Li, X. Qin, and M. Z. Yang, Mod. Phys. Lett. A 27, 1250223 (2012), arXiv:1111.4055

2012 arXiv
[31]

Ablikim et al

M. Ablikim et al. (BESIII Collaboration), Phys. Rev. D 99, 012013 (2019), arXiv:1809.00635

2019 arXiv
[32]

G. L. Castro and N. Quintero, Phys. Rev. D 103, 093002 (2021), arXiv:2101.01865

2021 arXiv
[33]

K. Ban, Y. Jho, Y. Kwon et al. , JHEP 04, 091 (2021), arXiv:2012.04190

2021 arXiv
[35]

Salam, Nuovo Cim

A. Salam, Nuovo Cim. 25, 224-227 (1962)

1962
[36]

Weinberg, Phys

S. Weinberg, Phys. Rev. 130, 776-783 (1963)

1963
[37]

Faessler, T

A. Faessler, T. Gutsche, M. A. Ivanov et al. , Eur. Phys. J. direct 4, 18 (2002), arXiv:hep- ph/0205287

2002
[38]

M. A. Ivanov, J. G. K¨ orner, and P. Santorelli, Phys. Rev . D 73, 054024 (2006), arXiv:hep- ph/0602050

2006
[39]

M. A. Ivanov, J. G. K¨ orner, P. Santorelli et al. , Particles 3, 193 (2020), arXiv:2009.00306

2020 arXiv
[40]

M. A. Ivanov, J. G. K¨ orner, J. N. Pandya et al. , Front. Phys. (Beijing) 14, 64401 (2019) arXiv:1904.07740

2019 arXiv
[41]

C. T. Tran, M. A. Ivanov, P. Santorelli et al. , Chin. Phys. C 49, 013111 (2025), 20 arXiv:2408.13776

2025 arXiv
[42]

C. T. Tran, M. A. Ivanov, N. D. Nguyen et al. , J. Phys. Conf. Ser. 3040, 012003 (2025) arXiv:2506.21902

2025 arXiv
[43]

Gutsche, M

T. Gutsche, M. A. Ivanov, J. G. K¨ orneret al., Phys. Rev. D 87, 074031 (2013), arXiv:1301.3737

2013 arXiv
[44]

Gutsche, M

T. Gutsche, M. A. Ivanov, J. G. K¨ orner et al. , Phys. Rev. D 98, 053003 (2018), arXiv:1807.11300

2018 arXiv
[45]

Groote, M

S. Groote, M. A. Ivanov, J. G. K¨ orner et al. , Phys. Rev. D 103, 093001 (2021), arXiv:2102.12818

2021 arXiv
[46]

Dubnicka, A

S. Dubnicka, A. Z. Dubnickova, M. A. Ivanov et al. , Phys. Rev. D 84, 014006 (2011), arXiv:1104.3974

2011 arXiv
[47]

Goerke, T

F. Goerke, T. Gutsche, M. A. Ivanov et al., Phys. Rev. D 94, 094017 (2016), arXiv:1608.04656

2016 arXiv
[48]

Gutsche, M

T. Gutsche, M. A. Ivanov, J. G. K¨ orner et al. , Phys. Rev. D 96, 114004 (2017), arXiv:1710.02357

2017 arXiv
[49]

N. R. Soni, A. N. Gadaria, J. J. Patel et al. , Phys. Rev. D 102, 016013 (2020), arXiv:2001.10195

2020 arXiv
[50]

Navas et al

S. Navas et al. (Particle Data Group), Phys. Rev. D 110, 030001 (2024)

2024
[51]

Branz, A

T. Branz, A. Faessler, T. Gutsche et al. , Phys. Rev. D 81, 034010 (2010), arXiv:0912.3710

2010 arXiv
[52]

L. M. Gu, H. B. Li, X. X. Ma et al. , Phys. Rev. D 100, 016018 (2019), arXiv:1904.06085

2019 arXiv
[53]

Y. Tan, Z. Zhang, and X. Zhou, Int. J. Mod. Phys. A 37, 2250075 (2022), arXiv:2111.04932

2022 arXiv
[54]

Cronin-Hennessy et al

D. Cronin-Hennessy et al. (CLEO Collaboration), Phys. Rev. D 86, 072005 (2012), arXiv:1104.3265

2012 arXiv
[55]

C. T. Tran, M. A. Ivanov, P. Santorelli et al. , Chin. Phys. C 48, 023103 (2024), arXiv:2311.15248

2024 arXiv
[56]

Arnaldi et al

R. Arnaldi et al. (NA60 Collaboration), Phys. Lett. B 677, 260 (2009), arXiv:0902.2547

2009 arXiv
[57]

M. N. Achasov et al. , Phys. Usp. 67, 55 (2024)

2024
[58]

Fornal, Int

B. Fornal, Int. J. Mod. Phys. A 32, 1730020 (2017), arXiv:1707.09749

2017 arXiv
[59]

J. L. Feng, T. M. P. Tait, and C. B. Verhaaren, Phys. Rev. D 102, 036016 (2020), arXiv:2006.01151

2020 arXiv
[60]

J. R. Batley et al. (NA48/2 Collaboration), Phys. Lett. B 746, 178-185 (2015), arXiv:1504.00607

2015 arXiv
[61]

P. B. Denton and J. Gehrlein, Phys. Rev. D 108, 015009 (2023), arXiv:2304.09877 21

2023 arXiv

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [34]

F. F. Lee, L. T. T. Uyen, and G. L. Lin, arXiv:2501.13530

work page Pith review arXiv

[2] [1]

A. J. Krasznahorkay, M. Csatl´ os, L. Csige et al. , Phys. Rev. Lett. 116, 042501 (2016), arXiv:1504.01527

arXiv 2016

[3] [2]

A. J. Krasznahorkay, M. Csatl´ os, L. Csige et al. , Acta Phys. Polon. B 50, 675 (2019)

work page 2019

[4] [3]

M. E. Rose, Phys. Rev. 76, 678 (1949) [Erratum: Phys. Rev. 78, 184(E) (1950)]

work page 1949

[5] [4]

Schl¨ uter, G

P. Schl¨ uter, G. Soﬀ, and W. Greiner, Phys. Rept. 75, 327 (1981)

work page 1981

[6] [5]

A. J. Krasznahorkay, M. Csatl´ os, L. Csige et al. , Phys. Rev. C 104, 044003 (2021), arXiv:2104.10075

arXiv 2021

[7] [6]

A. J. Krasznahorkay, A. Krasznahorkay, M. Begala et al. , Phys. Rev. C 106, L061601 (2022), arXiv:2209.10795

arXiv 2022

[8] [7]

Banerjee et al

D. Banerjee et al. (NA64 Collaboration), Phys. Rev. Lett. 120, 231802 (2018), arXiv:1803.07748

arXiv 2018