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$\eta$ and $\eta'$ meson production in $J/\psi$ radiative decays from lattice QCD

T0 review · 2 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper reports the first separate lattice QCD calculation of the transition form factors for $J/\psi \to \gamma\eta$ and $J/\psi \to \gamma\eta'$, finding values at $Q^2 = 0$ well below experimental extraction and a ratio…

desk verdict First separate lattice determination of J/psi radiative decays to eta and eta' with 2+1 flavors, technically careful, with one honest caveat about the charm-current-only approximation. read the letter →

arxiv 2506.09305 v1 pith:CBWQS74J submitted 2025-06-11 hep-lat hep-ph

Mischa Batelaan , Jozef J. Dudek , Robert G. Edwards This is my paper
classification hep-lathep-ph
keywords latticeQCDradiativedecaytransitionformfactoretamesoneta-primecharmoniumDalitzdisconnecteddiagrams
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 claims to compute, for the first time, the separate transition form factors for $J/\psi \to \gamma\eta$ and $J/\psi \to \gamma\eta'$ from lattice QCD at $m_\pi \sim 391$ MeV, using optimized meson operators to isolate the $\eta'$ as an excited state in the isoscalar pseudoscalar channel. The results cover 70 photon virtualities from deeply timelike to slightly spacelike, and yield the real-photon values $|F_{J/\psi\eta}(0)| = 0.00235(18)$ GeV$^{-1}$ and $|F_{J/\psi\eta'}(0)| = 0.00777(37)$ GeV$^{-1}$. If correct, these are the first separate first-principles determinations of these amplitudes available for comparison with experiment, where the $\eta'$ rate is about five times the $\eta$ rate.

What carries the argument

The calculation rests on three elements: optimized meson operators from a variational generalized eigenvalue problem on 2+1 flavor anisotropic clover lattices with 128 distillation vectors, which rapidly relax to the desired states and make the excited $\eta'$ accessible; a Wigner-Eckart averaging procedure that combines many lattice-rotation-equivalent three-point functions into one reduced correlation function, converting a noisy disconnected signal into a statistically precise one; and the assumption that the electromagnetic current couples only to the charm quark, $j_\mu = \tfrac{2}{3}e\,\bar c\gamma_\mu c$, so the amplitude reduces to a single Lorentz-invariant form factor $F_{\psi\eta^{(\prime)}}(Q^2)$. The form factor at 70 values of $Q^2$ is then fitted with dipole, exponential, and conformal $z$-polynomial parameterizations to extract the $Q^2 = 0$ point and the decay widths.

What would settle it

Compute the same matrix element with the full electromagnetic current (light, strange, and charm quark pieces included) on the same ensemble; if the resulting $|F(0)|$ values move toward the experimental ones, the charm-only truncation is the culprit. Alternatively, repeat at a lighter pion mass and a second lattice spacing; if the form factors do not move toward experiment, the discrepancy is not a light-quark mass artifact.

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Extended reading notes

Core claim

The central claim is that lattice QCD can access the $J/\psi \to \gamma\eta$ and $J/\psi \to \gamma\eta'$ transition amplitudes at realistic kinematics, and that on this ensemble the magnitudes of both form factors at $Q^2 = 0$ are significantly lower than those extracted from experimental radiative and Dalitz decay measurements. The $\eta$ and $\eta'$ are obtained separately by a variational analysis of a matrix of correlators built from optimized operators, so the $\eta'$ appears cleanly as the first excited state even in this disconnected process. The ratio $|F_{J/\psi\eta'}(0)|/|F_{J/\psi\eta}(0)| = 3.30(29)$ is found, somewhat larger than the experimental ratio of $2.44(4)$.

Load-bearing premise

The whole extraction assumes the photon in the decay couples only to the charm quark of the J/psi, ignoring photon coupling to the light and strange quarks inside the eta and eta'; if that approximation is inaccurate, the computed form factors do not describe the physical decay and the discrepancy with experiment could be partly an artifact.

Editorial extensions

If this is right

  • The paper provides the first separate first-principles numbers for the two radiative decays, giving a direct target for future lattice calculations at the physical quark mass.
  • The optimized-operator and averaging techniques are demonstrated to work for disconnected three-point functions, so they can be extended to $h_c \to \gamma\eta^{(\prime)}$ and $\psi' \to \gamma\eta^{(\prime)}$, where the charmonium state appears as an excited state.
  • The observed discrepancy with experiment motivates exploring the light-quark mass dependence, the lattice spacing dependence, and the possible role of lattice vacuum topology in these amplitudes.
  • The method is a precursor for computing $J/\psi$ radiative decays to two-hadron final states like $\pi\pi$ and $K\bar K$, which require finite-volume scattering formalism.

Reading between the lines

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

  • If the charm-current dominance approximation is the main source of the discrepancy, then including the light- and strange-quark pieces of the electromagnetic current on the same ensemble should move $|F(0)|$ toward experimental values; this is a concrete check the authors did not perform.
  • The unexpectedly light $\eta'$ mass on this ensemble, noted in the paper, suggests that limited sampling of QCD vacuum topology could suppress the gluonic production amplitude; a dedicated topology-controlled ensemble would test whether this is the cause.
  • The paper's finding that a dipole form fails for $J/\psi \to \gamma\eta$ across the accessible $Q^2$ range warns that older extractions of $|F(0)|$ that assumed dipole behavior for single-channel pseudoscalar data should be revisited with the exponential or $z$-polynomial forms used here.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The manuscript reports a lattice QCD calculation of the J/ψ→γ η and J/ψ→γ η′ transition form factors on a single 2+1-flavor anisotropic ensemble with m_π ≈ 391 MeV. Using distillation, optimized operators obtained from a variational analysis (which allow the η′ to be accessed as an excited state), and a Wigner-Eckart averaging procedure over many equivalent three-point functions, the authors extract F_{J/ψη}(Q^2) and F_{J/ψη′}(Q^2) at 70 values of photon virtuality ranging from deeply timelike to slightly spacelike. The electromagnetic current is approximated by the charm-quark vector current. Parameterizations of the Q^2 dependence give |F_{J/ψη}(0)| = 0.00235(18) GeV^{-1} and |F_{J/ψη′}(0)| = 0.00777(37) GeV^{-1}, which are factors of roughly 1.6–2 below values extracted from BESIII/PDG data; the ratio F_{η′}/F_η = 3.30(29) is compared with the experimental 2.44(4). Possible sources of the discrepancy (light-quark mass, lattice spacing, topology) are discussed.

Significance. If the results are correct, this is the first separate first-principles determination of these two amplitudes and a methodological milestone: it demonstrates that the disconnected, excited-state-dominated η′ channel can be computed with good statistical control using optimized operators and symmetry averaging. The paper is careful with fit systematics (AIC weighting, multiple source-sink separations, anisotropy variation, improved-current checks), and the Q^2 coverage is unprecedented for this process. The main caveat is that the computed amplitude is not the full physical electromagnetic amplitude because only the charm-quark part of j_μ is included; this conditionality must be resolved or quantified before the claimed discrepancy with experiment can be interpreted.

major comments (2)
  1. [Sec. II, Eqs. (1)-(2), and Fig. 1] The calculation uses j_μ = (2/3)e \bar c γ_μ c as the full electromagnetic current, but the physical current also contains u, d, s terms. The isoscalar light-quark and strange-quark pieces can contribute to η and η′ without violating isospin, so the measured 150-fold suppression of J/ψ→γ π^0 does not bound them. The paper gives no numerical estimate of the neglected diagrams (the contraction with the current inserted on the light/strange quark line in Fig. 1), and the charm-loop sum rule of Ref. [12]—which has the same approximation—underestimates both experimental widths, with the η′ particularly discrepant. Since all quoted F(Q^2) values and the experimental comparison inherit this approximation, the central result is conditional. I ask the authors to provide a quantitative estimate or an upper bound on the light-quark current contribution, or to recompute the matrix element with the full current, before drawing conclusions about the discrepancy with experiment.
  2. [Sec. V.B and Appendix A] The O(a)-improved spatial and temporal vector currents give noticeably different J/ψ decay constants (Fig. 15), yet the main form-factor analysis uses only the improved spatial current. The temporal-current form factors in Fig. 22 are stated to agree with the spatial ones 'within large statistical uncertainties', but those uncertainties are too large to set a meaningful bound on a residual current-discretization systematic. Please quantify the spatial/temporal difference for the transition form factors, or include an estimate of this effect in the systematic error on F(0).
minor comments (4)
  1. [Sec. V.A] There is a typo in 'obatined' (should be 'obtained').
  2. [Sec. V, Eq. (5)] The notation for the subduced current helicity operator is dense; a brief explanation of the S and D matrices immediately before Eq. (5) would improve readability.
  3. [Sec. VI.B, Eq. (11)] The ratio |F_{J/ψη′}(0)|/|F_{J/ψη}(0)| is quoted with an uncertainty, but the covariance between the two form-factor determinations is not given; a correlated estimate would be useful if available.
  4. [Abstract and Sec. VII] The abstract states that the form-factor magnitudes are 'significantly lower' than experimental values; because this comparison is conditional on the charm-only current approximation (and on m_π ≈ 391 MeV), consider wording that makes these conditions explicit in the abstract.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the transition form factors are computed from lattice three-point functions, and the comparison with experiment is an external benchmark, not a re-insertion of inputs.

full rationale

The derivation is self-contained. The transition form factors F_{J/psi eta(eta')}(Q^2) are obtained by computing three-point correlation functions of a vector current with optimized meson operators on a 2+1 flavor ensemble, and the resulting 70 lattice values are fit with dipole, exponential, and z-polynomial forms to interpolate F(0). No experimental decay amplitude or branching fraction enters the lattice computation as an input. The renormalization constants Z_s^V and Z_t^V are taken from Ref. [40], where they are fixed by the independent eta_c charge-normalization condition, not by J/psi -> gamma eta(eta'); ensemble and action parameters come from previous spectrum studies that are also independent of the decay amplitudes. The charm-only truncation j_mu = \bar c gamma_mu c is a physical approximation argued from the narrow J/psi width and the suppression of J/psi -> gamma pi^0; it is an assumption about the current, not a re-insertion of the target result. Self-citations to Hadron Spectrum methodology (distillation, operator construction, Wigner-Eckart averaging, AIC fits) are procedural and not load-bearing for the central physics claim. The comparison with PDG/BESIII extractions is an external benchmark, and the paper explicitly flags the observed discrepancy and possible systematic causes (quark mass, lattice spacing, topology). The paper's own stated limitations are acknowledged systematic uncertainties, not circular reductions. No step reduces by definition or by fitted input to the claimed prediction.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The calculation rests on standard lattice QCD assumptions (QCD, discretization, continuum/infinite-volume limits), on the physical approximation that the photon couples only to the charm quark, and on analysis assumptions (optimized operator dominance, effective Lorentz symmetry). The only newly fitted numbers are the anisotropy and the parameterization coefficients; the renormalization constants and scale are external inputs from prior work.

free parameters (4)
  • Anisotropy xi = 3.51 (range 3.45-3.59)
    Fitted to the relativistic dispersion relation of charmonium and eta/eta' states (Eqn. 4, Figs. 3-4). Used to convert lattice units to GeV and in the O(a)-improvement term for the vector current.
  • Charm quark mass = Tuned to reproduce eta_c mass
    The non-dynamical charm quark mass is tuned so that the eta_c mass matches experiment (Sec. III), affecting the J/psi mass and the charm propagators.
  • Vector current renormalization constants Z_V^s, Z_V^t = 1.145(3), 1.253(3)
    Taken from Ref. [40] on the same lattice; multiply the local current to renormalize it. External input not rederived in this paper.
  • Q^2 parameterization coefficients (dipole F0, Lambda; exponential F0, beta, alpha; z-polynomial a_n; linear slope s) = See Appendix D tables (e.g., dipole eta: F0=0.002720(57) GeV^-1, Lambda=2.626(14) GeV; linear eta: F0=0.00235(13)…
    Fitted to the computed lattice form-factor data to extract F(0) and describe the Q^2 dependence. The spread across functional forms is included in the systematic uncertainty on F(0).
assumptions (6)
  • domain assumption QCD is the correct underlying theory and the lattice discretization (anisotropic clover, 2+1 flavors) with distillation approximates it, with the continuum limit and infinite volume limits assumed to be recoverable.
    Standard working assumptions of lattice QCD; the paper uses a single finite volume (m_pi L ~ 5) and one lattice spacing, so continuum/infinite-volume extrapolations are not performed.
  • domain assumption The electromagnetic current couples dominantly to the charm quark in J/psi -> gamma eta(eta'), i.e., j_em = (2/3)e ar c gamma_mu c (Sec. II).
    Used to define the matrix element and decay rate (Eqns. 1-2). Argued from the narrow J/psi width, the suppression of J/psi -> gamma pi^0, and the smoothness of the experimental Dalitz form factor; not proven to all orders.
  • domain assumption The optimized operators obtained from the variational analysis of two-point functions interpolate the desired states with negligible excited-state contamination in the three-point functions (footnote 9, Sec. V).
    Required for the fit form ilde{C}(t, Delta t) = J + A_src e^{-delta E_src t} + A_snk e^{-delta E_snk (Delta t - t)} to converge to the matrix element. Checked indirectly through Delta t = 12-24 consistency.
  • domain assumption Effective Lorentz symmetry holds for the lattice matrix elements after subduction, i.e., different irrep choices at the same Q^2 yield the same form factor (Sec. V.A, Appendix C).
    Required for the second-stage averaging via Eqn. 7. The paper shows compatibility of different irrep embeddings (Fig. 6) and calls discretization effects modest.
  • domain assumption The renormalization constants Z_V^s and Z_V^t from Ref. [40] are correct and applicable to this ensemble.
    Used to renormalize the local vector current; the paper does not re-derive them.
  • domain assumption Scale setting via the Omega baryon mass yields a^{-1}_t = 5.666 GeV (Sec. III).
    Converts lattice units to physical units; standard but an external input.

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Cite this review

Pith. "Pith review of $\eta$ and $\eta'$ meson production in $J/\psi$ radiative decays from lattice QCD." pith.science (2026). https://pith.science/paper/CBWQS74J

@misc{pith2026250609305,
  author       = {Pith},
  title        = {Pith review of: $\eta$ and $\eta'$ meson production in $J/\psi$ radiative decays from lattice QCD},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CBWQS74J}},
  note         = {Machine review of arXiv:2506.09305}
}
abstract

We report on the computation of amplitudes describing the radiative decay processes $J/\psi \to \gamma \, \eta$ and $J/\psi \to \gamma \, \eta'$ from first principles via lattice QCD. Using lattices with two degenerate flavors of light quark and a heavier strange quark where $m_\pi \sim 391$ MeV, we compute three-point correlation functions using optimized meson operators with a range of momenta. The use of optimized operators allows access to the $\eta'$, even though it appears as an excited state, lying above the ground-state $\eta$ in the isoscalar pseudoscalar channel. Statistically precise signals are obtained in this disconnected process by averaging over large numbers of kinematically equivalent correlators. We determine transition form-factors as a function of photon virtuality across the timelike region, which describe also the 'Dalitz' processes $J/\psi \to e^+ e^- \, \eta^{(\prime)}$. Significantly lower magnitudes of transition form-factor for both the $\eta$ and $\eta'$ are found than those extracted from experimental data, and possible explanations for this observation are presented.

Figures

Figures reproduced from arXiv: 2506.09305 by the authors.

Figure 1
Figure 1. FIG. 1: Wick diagram structure required to compute [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. we show results from the case n = [111]. The upper two panels show the eigenvalues corresponding to the ground-state J/ψ in the cc A¯ 1 and E2 irreps, while the lower two panels show two eigenvalues in the isoscalar A2 irrep, corresponding to the η (ground state) and η ′ (first excited state). In practice we make use of energies constructed from a weighted average over several time– windows via the Akaike Informatio… view at source ↗
Figure 3
Figure 3. FIG. 3: Energies of [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (18 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Energies of [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: An example [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: Example of correlator averaging procedure for [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: Contributions to a three-point function for [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8: Form–factor for [PITH_FULL_IMAGE:figures/full_fig_p010_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9: Form–factor for [PITH_FULL_IMAGE:figures/full_fig_p010_9.png]
Figure 10
Figure 10. Figure 10: includes the results of dipole fits to both form–factors. For the J/ψ → γ η case, where our data goes deeper into the timelike region, spanning a larger range in Q2 , this form does not describe the data well, with a χ 2/Ndof = 2.11, and a systematic trend visibly dif…
Figure 11
Figure 11. Figure 11: FIG. 11: Form–factors for [PITH_FULL_IMAGE:figures/full_fig_p012_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12: Form–factor for [PITH_FULL_IMAGE:figures/full_fig_p013_12.png]
Figure 13
Figure 13. Figure 13: FIG. 13: Form–factor for [PITH_FULL_IMAGE:figures/full_fig_p013_13.png]
Figure 14
Figure 14. Figure 14: FIG. 14: Dispersion relation of [PITH_FULL_IMAGE:figures/full_fig_p016_14.png]
Figure 15
Figure 15. Figure 15: FIG. 15: Decay constant of the [PITH_FULL_IMAGE:figures/full_fig_p017_15.png]
Figure 16
Figure 16. Figure 16: FIG. 16 [PITH_FULL_IMAGE:figures/full_fig_p018_16.png]
Figure 17
Figure 17. Figure 17: shows the rescaled correlators, C˜(t, ∆t = 16at) for J/ψ → γ η with operators in the same irreps and momenta as the previous example. The left panel shows the correlator for the (real-valued) reference row–combination (orange squares) and the result of the average ove…
Figure 19
Figure 19. Figure 19: FIG. 19: Form–factor at [PITH_FULL_IMAGE:figures/full_fig_p019_19.png]
Figure 20
Figure 20. Figure 20: FIG. 20: As for Fig [PITH_FULL_IMAGE:figures/full_fig_p019_20.png]
Figure 21
Figure 21. Figure 21: FIG. 21: Form–factors for [PITH_FULL_IMAGE:figures/full_fig_p020_21.png]
Figure 22
Figure 22. Figure 22: we show the form–factor results using the im￾proved spatial and temporal vector currents, which are observed to agree within the large statistical uncertainties on the temporal results. The increased uncertainty is mainly due to the vastly reduced number of irrep and …

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.