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Radiative decays of the $h_c$ meson from lattice QCD

T0 review · 2 major / 6 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The first lattice-QCD calculation of the h_c radiative decays finds both real-photon decay rates below the measured values, with the η′/η amplitude ratio matching the J/ψ pattern.

desk verdict First lattice QCD results for h_c radiative decays, internally consistent and credible; the main caveat is excited-state systematics from the three-point fit forms. read the letter →

arxiv 2607.25707 v1 pith:6WICJYUM submitted 2026-07-28 hep-lat

Mischa Batelaan , Jozef J. Dudek , Robert G. Edwards This is my paper · ORCID
classification hep-lat PACS 12.38.Gc13.40.Hq
keywords latticeQCDcharmoniumh_cmesonradiativedecayelectricdipoleformfactorlongitudinaleta-primeDalitz
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 reports the first lattice QCD determination of the form factors controlling the radiative decays $h_c \to \gamma\eta$ and $h_c \to \gamma\eta'$, separating the electric dipole ($E_1(Q^2)$) and longitudinal ($C_1(Q^2)$) multipoles for this $J^{PC}=1^{+-}$ charmonium state. Best estimates of $|E_1(0)| = 0.0195(19)$ GeV and $0.0556(30)$ GeV translate through the decay-rate formula into radiative widths that lie significantly below the BESIII measurements — the same suppression previously found for $J/\psi \to \gamma\eta^{(\prime)}$ on the same lattices — while the $\eta'/\eta$ amplitude ratio of $2.9(3)$ matches the $J/\psi$ value of $3.3(3)$. The paper reads this as evidence that the discrepancy lives in the $\eta$–$\eta'$ sector of this ensemble, most plausibly in how the gauge-field topology sampling affects their flavor-singlet content, and it supplies concrete predictions for the as-yet-unmeasured Dalitz decays $h_c \to \ell^+\ell^-\eta^{(\prime)}$.

What carries the argument

The load-bearing object is the covariant decomposition of the vector-current matrix element into two invariant form factors, the electric dipole $E_1(Q^2)$ and the longitudinal $C_1(Q^2)$, with kinematic weights fixed by the helicities and momenta; by construction the longitudinal piece vanishes for a real photon, so the decay rate is controlled by $E_1(0)$ alone. On the lattice, each computed matrix element is a known linear combination of these two unknowns, so measuring many momentum, irrep, and helicity combinations at the same $Q^2$ and solving the overconstrained system $\Gamma = K \cdot F$ separates the multipoles. The inputs to that system come from three-point correlation functions fitted simultaneously across four source–sink separations $\Delta t/a_t = \{16, 20, 24, 28\}$, using fit forms of a constant plus at most one source and one sink exponential, combined by AIC model averaging; optimized charmonium operators from a two-point generalized eigenvalue problem are what isolate the $h_c$ in irreps where it is an excited state above the $J/\psi$. Dipole, exponential, and conformal-$z$ polynomial parameterizations then carry the discretely sampled form factors to $Q^2 = 0$.

What would settle it

Compute $|E_1(0)|$ for both channels on a second lattice ensemble with a different spacing, with physical pion mass, or with well-characterized gauge-field topology sampling: if the values rise toward the BESIII decay widths, the suppression is an artefact of these lattices, while if they stay low the suppression is a physical feature of the $\eta/\eta'$ system. A cheaper probe is to redo the three-point fits with multi-particle operators added and check whether the extracted matrix elements near $Q^2 = 0$ move outside their quoted errors, which would put the blame on the time-window fits rather than on the $\eta$–$\eta'$ physics.

Watch

Extended reading notes

Core claim

An axial meson such as the $h_c$ ($J^{PC} = 1^{+-}$) decaying to a pseudoscalar through the vector current is described by two invariant form factors, an electric dipole $E_1(Q^2)$ and a longitudinal $C_1(Q^2)$, whose kinematic weights depend on the helicities selected. Using three-point correlation functions on three-flavor lattices with $m_\pi \sim 391$ MeV, optimized operators that isolate the $h_c$ even in irreps where it sits above the $J/\psi$, and an overconstrained linear system $\Gamma = K \cdot F$ solved at each virtuality, the paper determines both form factors for $h_c \to \gamma\eta$ at 53 retained $Q^2$ points and for $h_c \to \gamma\eta'$ at 63 points. The central results are the real-photon couplings $|E_1^{h_c \to \gamma\eta}(0)| = 0.0195(19)$ GeV and $|E_1^{h_c \to \gamma\eta'}(0)| = 0.0556(30)$ GeV, which through Eqn. (2) give radiative decay rates significantly below the BESIII experimental values. The paper argues the suppression is common to both $J/\psi$ and $h_c$ decays to $\eta^{(\prime)}$, because the $\eta'/\eta$ ratio $2.9(3)$ is compatible with the same-lattice $J/\psi$ ratio $3.3(3)$ and a dimensionless electric-versus-magnetic dipole comparison agrees with experiment for both mesons; the likely common cause is a property of the $\eta$ and $\eta'$ on this lattice, most plausibly their SU(3) flavor-singlet content and its sensitivity to gauge-field topology. First results for $\psi' \to \gamma\eta^{(\prime)}$, extracted with the same operators, are compatible with zero.

Load-bearing premise

The results assume that the fitted time dependence of the three-point correlation functions — a constant plus at most one source and one sink exponential, combined by AIC model averaging — removes all contamination from states other than the desired one, including multi-particle states and the nearby J/ψ and ψ′, an assumption that the consistency checks in Appendices A and B never probe with multi-particle operators.

Editorial extensions

If this is right

  • The real-photon decay rates $\Gamma(h_c \to \gamma\eta)$ and $\Gamma(h_c \to \gamma\eta')$ computed on these lattices lie significantly below the BESIII measurements, so on this ensemble the lattice description of the $\eta$–$\eta'$ system, rather than the charmonium transition, is the prime suspect for the gap.
  • The $\eta'/\eta$ amplitude ratio of $2.9(3)$ for $h_c$ is compatible with the $3.3(3)$ found for $J/\psi$ on the same lattices, consistent with both amplitudes being governed by the SU(3) flavor-singlet content of the final-state mesons produced through a gluonic intermediate state.
  • A dimensionless ratio of the $h_c$ electric-dipole strength to the $J/\psi$ magnetic-dipole strength agrees with experiment for both $\eta$ and $\eta'$, supporting the paper's picture that the charmonium annihilation factorizes from the $\eta/\eta'$ production.
  • The timelike $Q^2$ dependence of both $E_1$ and $C_1$ determines the Dalitz decays $h_c \to \ell^+\ell^-\eta^{(\prime)}$, with $E_1$ dominant for the $\eta$ final state but both multipoles significant across the range, giving angular and $q^2$ distributions that experiment can test.
  • First lattice results for $\psi' \to \gamma\eta^{(\prime)}$ are statistically compatible with zero across the sampled $Q^2$ range, so the data cannot yet constrain the measured $\psi'$ radiative widths.

Reading between the lines

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

  • If the suppression is really a topology-sampling artefact, the extracted $E_1(0)$ values should shift when the gauge ensemble is binned or reweighted by topological charge; that correlation is directly testable on the existing configurations.
  • The same mechanism would predict a similar suppression for any light-meson final state produced through a gluonic intermediate state on these lattices, which could be checked by computing other flavour-singlet-dominated radiative channels.
  • The simultaneous-$\Delta t$ fitting with AIC model averaging is a transferable recipe for three-point function analyses beyond radiative transitions, wherever multiple source–sink separations are computed but currently fitted independently.
  • A lattice with physical pion mass and well-sampled topology that recovers the experimental rates would confirm the paper's conclusion, while one that still falls short would instead indicate a genuinely physical suppression of the gluonic production of $\eta/\eta'$ — with consequences for how charmonium radiative decays are used as a light-meson factory.
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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 / 6 minor

Summary. The manuscript presents a lattice QCD calculation of the radiative transition form factors for h_c -> gamma eta and h_c -> gamma eta'. It uses anisotropic three-flavor ensembles with m_pi ~ 391 MeV, optimized operators to access the h_c in boosted irreps where it can be an excited state, three-point correlation functions at several source-sink separations with simultaneous time-window fits and AIC model averaging, and a linear-system solution to separate the electric dipole (E1) and longitudinal (C1) form factors at each Q^2. The authors report best estimates |E1^{eta}(0)| = 0.0195(19) GeV and |E1^{eta'}(0)| = 0.0556(30) GeV, which imply radiative widths below the BESIII values, with a suppression pattern similar to their earlier J/psi -> gamma eta(eta') calculation, and an eta'/eta ratio of 2.9(3) compatible with the J/psi ratio. They also present first lattice results for psi' -> gamma eta(eta'), finding signals statistically compatible with zero.

Significance. This is, to my knowledge, the first lattice QCD determination of the h_c radiative multipole form factors, and it demonstrates a viable method for extracting matrix elements of an excited charmonium state in flight and for separating two form factors at fixed Q^2. The internal validation is in many respects careful: overlapping momentum points agree, the E1/C1 systems are overconstrained at most Q^2, ground-state and excited-state h_c extractions are compared in Appendix A, simultaneous and independent Delta-t fits are compared in Appendix B, and the Q^2 dependence is tested against several parameterizations. If the systematic concerns raised below are resolved, the results will provide a valuable nonperturbative benchmark for charmonium radiative transitions and for the ongoing discussion of the discrepancy between lattice determinations and BESIII for J/psi -> gamma eta(eta'). The paper is clearly a strong technical contribution from an experienced collaboration.

major comments (2)
  1. [Section III, Eq. (6), and Appendices A and B] The extraction of the matrix elements J is the pivot of the entire analysis, but the excited-state systematics are not fully controlled. Equation (6) allows only one source and one sink exponential in addition to a constant; for the h_c helicity +/-1 channels the h_c is an excited state above the J/psi, and the eta' is itself an excited state in the pseudoscalar channel, so residual contamination from the J/psi, psi', or multi-particle states such as eta-pi cannot be represented if the contamination has more than one comparable exponential. Appendix A checks ground-state versus excited-state h_c extractions for h_c -> gamma eta, and Appendix B checks simultaneous versus independent Delta-t fits, but both use the same fit family, so a common bias would not be visible, and no analogous check is shown for h_c -> gamma eta'. Because J feeds directly into E1(Q^2), C1(Q^2), and E1(0), this is a load-bearing systematic. The authors should either add fits with additional exponentials or operators, or at least demonstrate stability of E1(0) under a wider class of fit forms and time windows, with the h_c -> gamma eta' channel explicitly included.
  2. [Section V, Figure 13, and Appendix D] The quoted 'best estimate' uncertainties do not appear to cover the spread among the parameterizations. For h_c -> gamma eta, the z-poly order-1 fit gives |E1(0)| = 0.0128(11) GeV while the best estimate is 0.0195(19) GeV and the dipole and z-poly order-2 fits give about 0.0197 GeV; the spread is roughly 0.007 GeV, several times the quoted error of 0.0019 GeV. For h_c -> gamma eta', the z-poly order-1 value is 0.0445(21) versus the best estimate 0.0556(30), again a spread of about 0.011 GeV. The text says that variation over parameterization choice is considered in the best estimate, but neither the combination algorithm nor the error budget is specified. Please state explicitly how the central values and errors are constructed (e.g., AIC-weighted average, an envelope over the fits, or a stated choice of representative parameterization), and either enlarge the quoted errors to cover the parameterization dependence or provide a quantitative justification for excluding the low z-poly order-1 values.
minor comments (6)
  1. [Figure 5] The axis label 'square_i K_E1 K_C1' in the upper panel appears garbled; it should indicate the kinematic factors K_E1 and K_C1, and the lower panel's 'K_i' should be 'K_i'.
  2. [Figure 13 and Appendix D] Figure 13 labels entries as 'z poly-4' although Appendix D states that z-polynomial fits were considered up to cubic order; please clarify whether a quartic fit was performed and, if so, tabulate its parameters.
  3. [Section III] The statement that the 30 largest AIC values are retained and used in a model average would benefit from a specification of the total number of fits considered and the exact weighting formula.
  4. [Section IV and Appendix D] The two exponential forms are defined in Appendix D, but the figure captions in Section IV refer to 'exp fit order 1' and 'exp fit order 2' without explicit definitions; please label them consistently in the main text.
  5. [Section II, Figure 1] The eigenvalue plateaus for the h_c are visibly noisier and shorter than those for the J/psi; a quantitative statement of the selected t-range used in the GEVP and the resulting overlap factors would help the reader judge the quality of the optimized-operator projection.
  6. [Section V] The conclusion that the discrepancy is 'most likely' due to properties of the eta and eta' on this particular lattice is a plausible conjecture but is not tested here; consider softening the wording to reflect the alternatives listed in the introduction (quark-mass dependence, discretization, topological sampling).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: E1(0) is obtained from lattice three-point correlators via an independent linear extraction, with parameterized Q2 extrapolation and an external experimental comparison; self-citations are to prior methodological work that does not assume the target result.

full rationale

The derivation chain is self-contained and non-circular. The paper computes three-point correlation functions C(t, Delta t) with optimized h_c and eta^(prime) operators (Eq. 4), fits their residual time dependence using the simultaneous AIC model-average described in Section III, assembles the resulting matrix elements into the linear system Gamma = K*F (Eq. 5), and solves for E1(Q2) and C1(Q2) at each discrete virtuality. The real-photon values E1(0) are then obtained by evaluating parameterizations fitted to the finite-Q2 lattice data at Q2 = 0, or by local linear interpolation between points straddling Q2 = 0. E1(0) is therefore an extrapolated boundary value of fitted curves, not a fitted input renamed as a prediction, and the decay rate comparison via Eq. 2 is an external benchmark against BESIII data. The citations to Refs. [3,4] are to the same group's earlier lattice framework, including the operator basis, anisotropic lattices, improved vector current, and Wigner-Eckart averaging; these are published methodological results whose stated assumptions do not include the h_c -> gamma eta^(prime) form factors, so they provide independent support rather than a circular premise. No uniqueness theorem is imported from the authors' prior work, and no target quantity enters the definition of the inputs. The remaining concern about excited-state contamination from the limited fit family in Eq. 6 and the possible influence of J/psi, psi-prime, or multi-particle states is a systematic uncertainty, not a circularity: a bias in J would propagate to the final numbers, but the paper does not define J in terms of E1(0) or define E1(0) in terms of the experimental rates. Thus no circular step can be exhibited.

Assumptions & free parameters 8 free parameters · 6 assumptions · 0 invented entities

The numerical results inherit the lattice framework (ensembles, action, improved current, operator basis) from Refs [3,4] and rest on the assumed two-form-factor Lorentz decomposition, the assumption of small discretization effects, and the reliability of the optimized-operator plus fitting procedure. The Q^2 dependence is encoded in several parameterizations whose parameters are fitted to the lattice data; the full set is in Appendix D. No new particles or forces are introduced.

free parameters (8)
  • E1(hc->gamma eta): dipole F0 = F0 = 0.01969(52) GeV
    Fitted to lattice E1 data; used for one estimate of E1(0).
  • E1(hc->gamma eta): dipole Lambda = Lambda = 2.783(25) GeV
    Pole mass in dipole fit; fitted to the lattice form factor data.
  • E1(hc->gamma eta): z-poly order 2 coefficients a0, a1, a2 = a0 = 0.03673(68) GeV, a1 = -0.334(15) GeV, a2 = 1.58(29) GeV
    Conformal-mapped polynomial fit; yields E1(0) = 0.0197(16) GeV.
  • E1(hc->gamma eta'): dipole F0 = F0 = 0.0595(10) GeV
    Fitted to lattice E1 data for eta-prime.
  • E1(hc->gamma eta'): dipole Lambda = Lambda = 2.705(13) GeV
    Pole mass in dipole fit for eta-prime channel.
  • E1(hc->gamma eta'): z-poly order 2 coefficients a0, a1, a2 = a0 = 0.1316(24) GeV, a1 = -1.412(70) GeV, a2 = 6.35(69) GeV
    Yields E1(0) = 0.0554(24) GeV.
  • C1(hc->gamma eta): z-poly order 2 coefficients a0, a1, a2 = a0 = 0.0667(19) GeV, a1 = -0.747(38) GeV, a2 = 3.31(73) GeV
    Fitted longitudinal form factor used for Dalitz predictions.
  • C1(hc->gamma eta'): z-poly order 2 coefficients a0, a1, a2 = a0 = 0.2897(80) GeV, a1 = -3.52(22) GeV, a2 = 17.0(28) GeV
    Fitted longitudinal form factor used for Dalitz predictions.
assumptions (6)
  • domain assumption The Lorentz covariant decomposition of the h_c -> gamma eta(eta-prime) matrix element into two form factors, E1 and C1 (Eqn. 1), is complete.
    Taken from Refs [8,9]; if additional form factors were needed, the extraction of E1 and C1 from lattice matrix elements would be incomplete. Section III.
  • domain assumption Discretization effects are at most modest, so that the continuum Lorentz decomposition can be applied to lattice matrix elements in different irreps after subduction.
    Explicitly assumed in Section III; without it, combining different irreps at the same Q^2 to solve for E1 and C1 is unjustified.
  • domain assumption The optimized operator technology reliably projects out the h_c, including in irreps where the h_c is an excited state above the J/psi and psi-prime.
    Central to accessing the h_c; tested in Appendix A by comparing ground-state and excited-state extractions, but the test only covers h_c -> gamma eta, not eta-prime.
  • domain assumption The fit forms in Eqn. 6 (constant plus up to one source and one sink exponential) and the AIC model-averaging procedure remove excited-state contamination in the three-point functions.
    If higher states or multi-particle states contribute in the time windows, the extracted matrix elements J would be biased. Appendix B compares independent Delta-t fits but does not include multi-particle states.
  • domain assumption The lattice ensembles and improved vector current described in Ref [4] correctly reproduce QCD in this channel.
    The calculation inherits the action, current renormalization, and gauge ensembles from prior work; no independent verification is provided here.
  • domain assumption The Akaike Information Criterion weighting of fit forms and time-windows gives a conservatively estimated uncertainty.
    This is a methodological assumption about the model-averaging procedure; the paper adopts it for all extracted matrix elements and derived quantities.

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Pith. "Pith review of Radiative decays of the $h_c$ meson from lattice QCD." pith.science (2026). https://pith.science/paper/6WICJYUM

@misc{pith2026260725707,
  author       = {Pith},
  title        = {Pith review of: Radiative decays of the $h_c$ meson from lattice QCD},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6WICJYUM}},
  note         = {Machine review of arXiv:2607.25707}
}
abstract

We explore, for the first time in lattice QCD, the radiative decays to $\gamma \eta$ and $\gamma \eta'$ of the $J^{PC}=1^{+-}$ charmonium state, $h_c$. This work expands upon a previous calculation of $J/\psi$ decays to the same final states, incorporating novel technology to access the $h_c$ where it is an excited state and to separate the independent electric dipole and longitudinal form--factors. Results for the radiative decay rates, computed on three-flavor lattices with $m_\pi \sim $ 391 MeV show a suppression relative to experiment that is similar to that observed in the prior calculation of $J/\psi \to \gamma \eta^{(\prime)}$. The timelike $Q^2$ dependence of both form--factors together influence the Dalitz decays, $h_c \to \ell^+ \ell^- \eta^{(\prime)}$, which are as yet unmeasured experimentally.

Figures

Figures reproduced from arXiv: 2607.25707 by the authors.

Figure 1
Figure 1. FIG. 1: Eigenvalues of generalized eigenvalue [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. For each meson we observe no systematic depar￾ture from a continuum–like relativistic dispersion relation, and we extract broadly consistent estimates of the lattice anisotropy, ξ, from each. In order to access the region around Q2 = 0 in the processes hc → γη(′) , owing to the heavier hc, we need to consider η (′) with higher momentum than in the previous J/ψ calculation [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 4
Figure 4. FIG. 4: An example of the result of simultaneous ∆ [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figures from the paper (13 more)
Figure 5
Figure 5. Figure 5: FIG. 5: The upper panel shows the real (open symbols) [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: Electric dipole (top) and longitudinal (bottom) form–factors for [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]
Figure 8
Figure 8. Figure 8: FIG. 8: Electric dipole (top) and longitudinal (bottom) [PITH_FULL_IMAGE:figures/full_fig_p008_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9: Electric dipole (top) and longitudinal (bottom) form–factors for [PITH_FULL_IMAGE:figures/full_fig_p009_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10: Electric dipole (top) and longitudinal (bottom) [PITH_FULL_IMAGE:figures/full_fig_p010_10.png]
Figure 12
Figure 12. Figure 12: we show the extracted form–factor in the case of using just a single source–sink separation, ∆t/at = 20, where we see that across the entire Q2 range, the signal is broadly compatible with zero7 . Form–factor values for real photons corresponding to the measured BESII…
Figure 13
Figure 13. Figure 13: FIG. 13: Electric dipole form–factor at [PITH_FULL_IMAGE:figures/full_fig_p011_13.png]
Figure 14
Figure 14. Figure 14: FIG. 14: Form-factors, [PITH_FULL_IMAGE:figures/full_fig_p012_14.png]
Figure 15
Figure 15. Figure 15: FIG. 15: Electric dipole (top) and longitudinal (bottom) form-factors for [PITH_FULL_IMAGE:figures/full_fig_p015_15.png]
Figure 16
Figure 16. Figure 16: FIG. 16: (left) Description of the same data as plotted in Figure [PITH_FULL_IMAGE:figures/full_fig_p016_16.png]
Figure 17
Figure 17. Figure 17: FIG. 17: Correlators previously shown in Figure [PITH_FULL_IMAGE:figures/full_fig_p016_17.png]
Figure 18
Figure 18. Figure 18: FIG. 18: Electric dipole (top) and longitudinal (bottom) form-factors for [PITH_FULL_IMAGE:figures/full_fig_p017_18.png]
Figure 19
Figure 19. Figure 19: FIG. 19: As Figure [PITH_FULL_IMAGE:figures/full_fig_p017_19.png]

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