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$\eta$ and $\eta'$ production in $J/\psi$ radiative decays from quantum chromodynamics

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

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Pith's one-line read Lattice QCD yields first-principles amplitudes for J/psi decaying to eta and eta'.

desk verdict A solid first 2+1 flavor lattice QCD calculation of J/psi -> gamma eta(eta'), but the charm-only current approximation needs quantitative support before the absolute amplitudes are taken as benchmarks. read the letter →

arxiv 2506.09306 v1 pith:FXXC5G6Z 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 latticeQCDradiativedecayJ/psieta-eta'mixingtransitionformfactorvariationalanalysisdisconnecteddiagramcharmonium
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 the first calculation within lattice QCD, with two light and one strange dynamical quark flavor, of the radiative decays $J/\psi \to \gamma \eta$ and $J/\psi \to \gamma \eta'$. It reports the transition form factors over a wide range of photon virtuality and determines the real-photon amplitudes at a pion mass of about 391 MeV, finding that $\eta'$ production is enhanced over $\eta$ by a factor $3.30(29)$. A sympathetic reader would care because these decay amplitudes are a direct, non-perturbative prediction of QCD for processes that are measured experimentally and are relevant to light-meson and glueball spectroscopy. The paper also shows that the variational and averaging technology makes this noisy 'disconnected' sector tractable, which is the prerequisite for studying radiative production of light-meson resonances in $J/\psi$ decays.

What carries the argument

The load-bearing machinery is the variational construction of optimized meson operators: in a basis of fermion-bilinear interpolating operators, a matrix of two-point correlation functions is solved variationally at each momentum and each irreducible representation of the (boosted) lattice symmetry group, producing operators with dominant overlap with the $\eta'$ excited state and cleaner $\eta$ and $J/\psi$ signals. On top of this, a two-stage correlator-averaging procedure reduces 1748 non-zero three-point correlators to 70 well-determined values of $F(Q^2)$: first, Wigner–Eckart averaging of correlators related by exact lattice symmetries; second, an over-constrained linear least-squares fit across different irreps assuming approximate Lorentz symmetry via subduction. The electromagnetic current is evaluated in its charm-quark part only, renormalized using the $\eta_c$ form factor at $Q^2=0$ from the same lattice action, and the real-photon amplitudes are extracted by parameterizing the $Q^2$ dependence with convergent conformal-mapping forms. The single identity carrying the physical interpretation is the ratio $|F_{\psi\eta'}(0)|/|F_{\psi\eta}(0)|$, which exposes the flavor-singlet content of the $\eta'$.

What would settle it

Recompute the amplitudes on the same lattice including the full electromagnetic current (light and charm quarks) and compare |F(0)| for both mesons; if the values move toward the experimental ones, the charm-only approximation was responsible, whereas if they do not, a measurement of the topological-charge distribution or a physical-pion-mass ensemble would discriminate the quark-mass and axial-anomaly explanations.

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Extended reading notes

Core claim

This paper presents the first lattice QCD calculation, with two degenerate light quark flavors and a heavier strange flavor, of the radiative decays $J/\psi \to \gamma \eta$ and $J/\psi \to \gamma \eta'$. It computes the transition form factor $F(Q^2)$ defined by the matrix element $\langle \eta^{(\prime)}(p') | j^\mu_{\rm em}(0) | J/\psi(p,\lambda)\rangle = \epsilon^{\mu\nu\rho\sigma} p'_\nu p_\rho \epsilon_\sigma(p,\lambda) F_{\psi\eta^{(\prime)}}(Q^2)$, obtaining it at 70 discrete values of the photon virtuality $Q^2$ from the timelike region (Dalitz decay) to the real-photon point. At $m_\pi \simeq 391$ MeV the real-photon amplitudes are $|F_{\psi\eta}(0)| = 0.00235(18)\,\mathrm{GeV}^{-1}$ and $|F_{\psi\eta'}(0)| = 0.00777(37)\,\mathrm{GeV}^{-1}$, with ratio $3.30(29)$, where the $\eta'$ is isolated as the first-excited pseudoscalar isoscalar state using variationally optimized operators. The paper reports that both amplitudes lie below the experimentally measured values and discusses as candidate origins a strong light-quark-mass dependence or an under-realized axial $U(1)$ anomaly on this ensemble.

Load-bearing premise

The calculation assumes the photon couples only to the charm quark in the J/psi, so only the charm piece of the electromagnetic current is evaluated; if the neglected light-quark current contributes at the few-percent level, the absolute amplitudes and possibly the eta'/eta ratio would shift.

Editorial extensions

If this is right

  • The measured ratio $|F_{\psi\eta'}(0)|/|F_{\psi\eta}(0)| = 3.30(29)$ confirms that the flavor-singlet, gluon-rich component of the $\eta'$ drives its enhanced production in charmonium radiative decay.
  • The $Q^2$ dependence of the $\eta$ and $\eta'$ form factors is consistent with a common shape, supporting the hypothesis that the production amplitude factorizes from the light-meson final-state dynamics.
  • The computed amplitudes lie below the experimental central values, with the paper attributing the gap to either strong light-quark-mass dependence or an under-realized axial $U(1)$ anomaly on this lattice ensemble.
  • The demonstrated signal quality for the disconnected diagram means the same methods can be extended to other charmonium radiative transitions, to $h_c$ and $\psi(2S)$, and to processes like $J/\psi \to \gamma\,(\pi\pi,\,K\bar K)$ where light-meson resonances appear.

Reading between the lines

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

  • A direct test of the charm-only current approximation is to compute the light-quark part of the electromagnetic current on this same ensemble; if the omitted piece is not negligible, the amplitudes and possibly the $\eta'/\eta$ ratio would move toward the experimental values.
  • The paper's observation that the $\eta'$ mass, 945(9) MeV, is unexpectedly low at this pion mass suggests a measurable accompanying prediction: the topological-charge distribution on this ensemble should show a suppressed topological susceptibility, which would simultaneously explain the low mass and the low radiative amplitudes.
  • If the common $Q^2$ shape for $\eta$ and $\eta'$ persists for resonant final states, the factorization hypothesis could be used to analyze $J/\psi \to \gamma\,(\pi\pi,\,K\bar K)$ data in terms of a single universal charm-quark production amplitude multiplied by a light-meson transition form factor.
  • A finer lattice spacing at the same quark masses would discriminate whether part of the deficit relative to experiment is a discretization artifact rather than a physical quark-mass or topology effect.
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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. This letter reports the first 2+1-flavor lattice QCD calculation of the transition form factors for J/ψ → γ η and J/ψ → γ η′, performed on a single anisotropic ensemble with a_s ≈ 0.12 fm, a_t^{−1} ≈ 5.7 GeV, and m_π ≈ 391 MeV. The η and η′ are isolated with variationally optimized operators (the η′ as the first excited state in the isoscalar pseudoscalar channel), and a two-stage averaging procedure — Wigner-Eckart subduction of symmetry-related correlators, followed by an over-constrained fit imposing a single F(Q^2) per kinematical point — yields 70 values of Q^2 for each channel. Three-point functions are computed at source-sink separations Δt/a_t = 12, 16, 20, 24, with time-window fits combined via AIC weighting. The real-photon amplitudes are extracted from several Q^2 parameterizations and from linear interpolation near Q^2 = 0, giving |F_{J/ψη}(0)| = 0.00235(18) GeV^{−1}, |F_{J/ψη′}(0)| = 0.00777(37) GeV^{−1}, and a ratio 3.30(29). Only the charm-quark part of the electromagnetic current is evaluated, with justification deferred to a companion write-up; the resulting values lie below the BESIII/PDG rates, and the computed ratio is larger than the experimental ratio.

Significance. If the omitted light- and strange-quark pieces of the electromagnetic current are indeed negligible at the few-percent level, this is a landmark calculation: it demonstrates that the η′ can be accessed as an excited state with variational operators, that the disconnected-process signal can be controlled by symmetry-based averaging, and that a wide range of Q^2 can be covered with a single ensemble. The technical care is evident and is a genuine strength: multiple source-sink separations, AIC-based averaging over time windows, consistency between irreps and between neighboring Q^2 values, agreement among several functional forms for the Q^2 dependence, and an explicit statement of the pion-mass and single-spacing limitations. The principal caveat is that the computation evaluates the charm-current contribution to the amplitude rather than the full electromagnetic amplitude, and the size of the omitted disconnected light-current contributions is not estimated in the letter.

major comments (2)
  1. [Calculation (charm-current paragraph, footnote 1); Results; Outlook] The central quantitative results are obtained with only the charm-quark part of the electromagnetic current. The omitted light- and strange-quark pieces, <η^(′)|(2/3) ū γ_μ u − (1/3) d̄ γ_μ d − (1/3) s̄ γ_μ s |J/ψ>, are disconnected diagrams whose size is not estimated anywhere; the sentence 'To a very good approximation, ... we will compute only the charm part of the electromagnetic current' defers the justification entirely to the 'associated longer write-up' (footnote 1). For the η′, which has a substantial strange-quark component, and for the η, whose octet component carries the net charge factor, these contributions need not be negligible at the level of the quoted 4–8% errors, and they enter with different charge factors for the two mesons, so both |F(0)| values and the ratio 3.30(29) (quoted against the PDG value 2.44(4)) could shift. Because the abstract claims a 'first principles calculation of the radiative decays', and because Fig. 3 and the Outlook compare the computed values with the BESIII/PDG rates and interpret the disagreement as a 'mystery', the manuscript currently presents the charm-current contribution as the full physical amplitude. The authors should either include in the letter a quantitative estimate or upper bound of the light-current contribution (for example, by evaluating the light-quark disconnected diagram on a subset of configurations, or by a model-based estimate anchored to the known flavor content of η and η′), or reframe the abstract, the results, and the Outlook explicitly and consistently as the charm-current contribution to the transition form factors, with the comparison to experiment qualified accordingly. This is load-bearing because it affects the absolute amplitudes, the ratio, and the central comparison with experiment.
  2. [Calculation (renormalization paragraph); Results (final F(0) values)] The overall normalization of every extracted form factor is set by the multiplicative renormalization of the O(a)-improved charm vector current, which is fixed by matching to the η_c form factor at Q^2 = 0 from Ref. [28], plus a tree-level improvement term that is not specified in the letter. The precision of this renormalization is not stated, and its uncertainty is not propagated into |F_{J/ψη}(0)| and |F_{J/ψη′}(0)|; the quoted errors appear to include statistics and anisotropy variation only, yet the F(0) values are compared with experiment at the 5–8% level. The renormalization uncertainty should be quantified, or at least argued to be negligible, in the error budget, since it rescales the absolute values that enter the central claims.
minor comments (4)
  1. [Calculation; Figure 2] The sign convention for Q^2 used in Eq. (1) and in the figures is not stated; since the text refers to the timelike region (the Dalitz decay) while the figure axes run over negative Q^2, the convention should be given explicitly in the text.
  2. [Outlook] The 'somewhat unexpectedly small' η′ mass, 945(9) MeV, is used to motivate a possible underestimate of the axial U(1) contribution, but the letter does not show or cite where this mass is determined; a one-line statement of the extraction, or a reference to the spectrum analysis, is needed for this claim to be evaluable.
  3. [Calculation] The two-stage averaging procedure is described in detail, but the 'novel' aspect would be easier to judge if the statistical gain were quantified (for example, the variance reduction or the effective number of independent correlators) relative to un-averaged correlators.
  4. [Results] Minor typography and completeness issues: χ2/Ndof should be typeset as χ²/N_dof; the width formula following Eq. (1) has no reference, and the factor 4/27 (which combines the charm charge and phase space) is not derived; the acronym PDG is introduced only in the caption of Fig. 3.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the form factors are extracted from lattice three-point functions and compared with experiment only after extraction.

full rationale

The derivation chain is self-contained. The transition form factor is defined by Eq. (1) and extracted from three-point correlation functions via a two-stage averaging procedure that uses only lattice data. 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 are obtained by fitting parameterizations to the computed Q^2 dependence and by linear interpolation near Q^2 = 0; BESIII and PDG values are shown only for comparison (Fig. 3), not as inputs. The only externally supplied number entering the computation is the multiplicative current renormalization, 'set by using the eta_c form-factor at Q^2 = 0 determined in a previous work with the same lattice action [28]'; this calibrates the charm vector current and does not encode the J/psi -> gamma eta^(prime) amplitude, so it is not a fitted input to the claimed prediction. The approximation of keeping only the charm part of the electromagnetic current (footnote 1, with justification deferred to the longer write-up) is a physical assumption that narrows the quantity actually computed; it is a limitation of the prediction, not a circular definition. Similarly, the Outlook's remark that the eta' mass 945(9) MeV may indicate an underestimated axial U(1) contribution identifies a possible systematic, not a circularity. No step in the paper reduces Eq. (1) to an input by construction, and no load-bearing uniqueness theorem or ansatz is imported from the authors' prior work.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The central numbers rest on a standard lattice QCD setup, several model assumptions (charm-only current, approximate Lorentz symmetry, parameterization), and calibration constants from prior same-group work. No new particles or forces are introduced. The main uncontrolled systematic is the single lattice spacing, unphysical pion mass, and the unquantified light-quark current contribution.

free parameters (5)
  • light quark mass (pion mass) = m_pi ~ 391 MeV
    Chosen heavier than physical to reduce computational cost; all quoted form factors are at this unphysical mass, so the values are not direct predictions at the physical point.
  • strange quark mass = tuned to approximately physical
    Set by tuning to the physical strange quark mass; affects the eta/eta' mixing and form factors.
  • anisotropy xi = 3.51(+0.08/-0.06)
    Determined from fits to meson dispersion relations; enters the conversion of lattice momenta to physical Q^2 and the systematic error estimate.
  • vector current renormalization = absorbed into calibration to eta_c form factor from Ref. [28]
    The multiplicative renormalization of the charm vector current is fixed using a previous lattice calculation of the eta_c form factor at Q^2=0 on the same action; an error there propagates into all extracted absolute values.
  • Q^2 parameterization coefficients (z-polynomial, dipole, Gaussian) = not tabulated in the letter
    Fitted to the 70 form-factor points to interpolate F(0); the final best estimate also uses linear interpolation and is stable across forms except the dipole for eta.
assumptions (5)
  • domain assumption QCD with 2+1 flavors, discretized on an anisotropic lattice with a_s ~0.12 fm and 288 configurations, provides a valid approximation to continuum QCD at the scales of charmonium and light mesons.
    The calculation uses a single lattice spacing and one volume; continuum and infinite-volume extrapolations are not performed, so the quoted numbers inherit any discretization and finite-volume errors.
  • domain assumption The eta' can be isolated as the first excited state in the pseudoscalar isoscalar channel using variationally optimized operators built from fermion bilinears.
    Relies on the variational method of Refs. [11,12,29-31]; the letter shows consistency with dispersion relations but the operator basis details are in the longer write-up.
  • ad hoc to paper Approximate Lorentz symmetry on the boosted lattice allows correlators in different lattice irreps to be combined under a single form factor F(Q^2) via subduction and an over-constrained linear least-squares fit.
    Introduced specifically for this averaging procedure; Figure 1 shows internal consistency, but it is an assumption about lattice artifacts rather than a proven symmetry of the discretized theory.
  • domain assumption The charm-quark part of the electromagnetic current dominates the radiative decay matrix element; light-quark current contributions are neglected.
    Invoked in the paragraph after Eq. (1) with justification deferred to a longer write-up; it is a physical approximation, not derived in this letter.
  • domain assumption The form factor's Q^2 dependence is smooth and can be described by conformal-mapping polynomials with the nearest singularity at Q^2 = -m_J/psi^2.
    Used for the z-poly parameterizations; the letter reports chi^2/Ndof near unity, so the assumption is consistent with the data, but it is not unique.

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Cite this review

Pith. "Pith review of $\eta$ and $\eta'$ production in $J/\psi$ radiative decays from quantum chromodynamics." pith.science (2026). https://pith.science/paper/FXXC5G6Z

@misc{pith2026250609306,
  author       = {Pith},
  title        = {Pith review of: $\eta$ and $\eta'$ production in $J/\psi$ radiative decays from quantum chromodynamics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FXXC5G6Z}},
  note         = {Machine review of arXiv:2506.09306}
}
abstract

We present a first principles calculation within quantum chromodynamics (QCD) of the radiative decays of the $J/\psi$ into the light pseudoscalar mesons $\eta$ and $\eta'$. Within a lattice computation we obtain the transition form-factors as a function of photon virtuality from the timelike region, accessible experimentally via the 'Dalitz' decay $J/\psi \to e^+ e^-\, \eta^{(\prime)} $, through to the real photon point corresponding to $J/\psi \to \gamma\, \eta^{(\prime)} $. This is the first calculation in lattice QCD with two (heavier than physical) degenerate flavors of light quark and a heavier strange quark, in which the $\eta'$ appears as the first-excited state with pseudoscalar isoscalar quantum numbers. We access it reliably by using variationally optimized operators, use of which also improves the purity of the $J/\psi$ and $\eta$ signals, reducing systematic uncertainties. High quality results at a large number of kinematic points are obtained in a typically noisy disconnected process by using a novel correlator averaging procedure. Our results show the expected enhanced production of the $\eta'$ over the $\eta$ in this process, and suggest that the demonstrated lattice technology is suitable for future calculations considering processes in which light meson resonances are produced.

Figures

Figures reproduced from arXiv: 2506.09306 by the authors.

Figure 1
Figure 1. FIG. 1. Examples of timeslice fitting and correlator averaging [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Form–factors for [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Form–factors for [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

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Forward citations

Cited by 1 Pith paper

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

  1. Radiative decays $J/\psi,\,\psi(2S)\rightarrow\gamma\eta^{(\prime)}$ in perturbative QCD with relativistic corrections

    hep-ph 2026-07 conditional novelty 6.0 of 10

    Order-q² relativistic corrections in pQCD roughly double J/ψ→γη(') rates and favor a smaller mixing angle, while ψ(2S) rates overshoot data and may require coherent ηc mixing.

Reference graph

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Pith tools

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