Pith. sign in

REVIEW 74 references

Including order-q² relativistic corrections roughly doubles J/ψ→γη(') rates in pQCD and leaves the amplitudes almost independent of the light-cone DA and light-quark mass.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · grok-4.5

2026-07-30 14:25 UTC pith:DDMNPC5F

load-bearing objection Solid first O(q^{2}) pQCD calculation for these channels; J/ψ robustness and R1S preference for the smaller mixing angle hold up, while ψ(2S) rates and ηc-mixing interference are self-flagged as exploratory.

arxiv 2607.23737 v1 pith:DDMNPC5F submitted 2026-07-26 hep-ph

Radiative decays J/psi,\,psi(2S)rightarrowγη^((prime)) in perturbative QCD with relativistic corrections

Jun-Kang He , Chao-Jie Fan , Cong Wang This is my paper
classification hep-ph
keywords radiative charmonium decaysperturbative QCDrelativistic correctionsη–η' mixinglight-cone distribution amplitudesJ/ψψ(2S)ηc mixing
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper computes the OZI-forbidden radiative decays J/ψ and ψ(2S) → γη and γη' in perturbative QCD for the first time with consistent order-q² relativistic corrections in all three short-distance pieces: quark-antiquark, two-gluon, and QED. The dominant quark amplitude stays flat under changes of the light-cone distribution amplitude and the light-quark mass, both at leading order and after the q² correction, so the hard prediction is fixed almost entirely by decay constants and the charmonium wave function. For the J/ψ the relativistic piece roughly doubles the branching ratios and narrows the gap with data; the ratio of the two channels cleanly prefers the smaller of two standard η–η' mixing angles. For the ψ(2S) the same expansion is much larger, converges poorly, and already overshoots experiment, especially the tiny γη rate. The authors therefore add a coherent ηc-mixing amplitude and show that interference can restore agreement, though the fit is sensitive to the mixing parameters.

Core claim

A first complete order-q² pQCD calculation of J/ψ, ψ(2S) → γη(') shows that the dominant short-distance quark amplitude is insensitive (at the few-percent level) to both the light-cone DA and the light-quark mass through O(q²). The relativistic correction multiplies the J/ψ branching ratios by about two and makes the ratio R1S a sharp discriminator that favors the smaller η–η' mixing angle (~33.5°) over the larger lattice/FKS value (~39°). The same framework over-predicts the ψ(2S) rates, especially γη, already at leading order; coherent interference with an ηc-mixing contribution can bring those rates into line with data.

What carries the argument

The covariant Salpeter projector for the 1−− charmonium, Taylor-expanded through O(q²) and reduced to the two moments R(0) and ∇²R(0), convolved with the twist-2 quark and gluon light-cone DAs of the η(') and joined through the hard c c-bar → γ g* g* (and QED) kernels.

Load-bearing premise

That a low-order Taylor expansion in the heavy-quark relative momentum, kept only through q² and fixed by two moments from a Cornell potential, remains quantitatively trustworthy for the radially excited ψ(2S), even though that correction is as large as the leading term and the 2S node is not captured.

What would settle it

A precision lattice or higher-order calculation of the ψ(2S) → γη helicity amplitude that either confirms the large hard overshoot or shows that retaining the full relative-momentum dependence (instead of the q² truncation) removes the excess; alternatively, an independent extraction of the relative phase between the annihilation and ηc-mixing amplitudes that is stable under changes of the mixing angle.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • J/ψ → γη(') branching ratios and especially their ratio become reliable short-distance probes of η–η' mixing once the q² correction is included.
  • The smaller mixing angle (~33.5°) is preferred by hard exclusive charmonium decays over the larger anomaly-dominated value.
  • A pure hard mechanism cannot explain the anomalously small ψ(2S) → γη rate; an additional coherent contribution is required.
  • The same Salpeter-plus-DA framework can be reapplied to related channels (Dalitz decays, hc radiative modes) with controlled relativistic corrections.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the DA and mass insensitivity survives at still higher order, these radiative ratios become among the cleanest exclusive tests of the hard charm scale itself.
  • The poor 2S convergence suggests that any future global analysis of ψ(2S) exclusive rates will need the unexpanded relative-momentum dependence rather than moment truncations.
  • A lattice determination of the off-shell ηc–η(') mixing form factor at the physical photon energy would fix the relative phase and turn the interference fit into a genuine prediction.

Editorial analysis

A structured set of objections, weighed in public.

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

Circularity Check

1 steps flagged

Main pQCD amplitudes and R1S are independently derived; partial circularity only in the exploratory ηc-mixing section, where relative phases are fitted to the γη rates then used to claim agreement.

specific steps
  1. fitted input called prediction [Sec. III D, Eqs. (44)–(48), Table IX]
    "Here Hηc=|Hηc|e^{iδnS}, with δ1S and δ2S treated as two free parameters, each common to the γη and γη′ channels of its charmonium, and fixed by a best fit to experiment. The fit is anchored to the two measured γη branching ratios... and returns δ1S≃192° and δ2S≃18°. Table IX collects the four resulting branching ratios. The interference brings the ψ(2S) into agreement with experiment in both channels"

    Two free phases are adjusted to the two measured γη branching ratios, so the post-fit match in those channels (e.g. ψ(2S)→γη: Bfit=0.0092 equals Bexp=0.0092) is forced by the fit rather than predicted. The coherent-sum “agreement with data” therefore partly renames a fitted input as a successful interference prediction. The γη′ entries are less tightly forced but still use the same fitted phases.

full rationale

The load-bearing pQCD results—kernel flatness through O(q²), the factor-of-two J/ψ enhancement, and R1S discrimination of the mixing angle—are computed from hard kernels, Salpeter moments, and external mixing/DA inputs, without being fitted to the radiative branching ratios they predict. Rψn(0) and ∇²R(0) come from a Cornell potential and are cross-checked on leptonic widths (different observables), which is standard and non-circular. The only clear circular step is in Sec. III D: two free phases δ1S, δ2S are fitted to the two measured γη branching ratios, after which the coherent sum is said to bring the ψ(2S) rates into agreement with data. That γη agreement is by construction of the fit; the γη' channels retain some predictive content but inherit the fitted phases. The paper itself labels this attempt tentative and mixing-sensitive. No self-definitional loop, uniqueness import, or renaming of a known result drives the central claims. Score 3 reflects one controlled fitted-input episode outside the main hard-mechanism results.

Axiom & Free-Parameter Ledger

6 free parameters · 7 axioms · 0 invented entities

The central hard-amplitude claims rest on standard pQCD factorization plus a controlled set of domain assumptions (instantaneous BS kernel, weak-binding mc=M/2, twist-2 only, Cornell moments). Absolute rates and the ηc-interference story additionally depend on free or fitted inputs: mixing angle/decay constants, DA Gegenbauer moments, αs scale, two relative phases, and the oscillator length used for the radial form factor.

free parameters (6)
  • ηc-mixing phases δ1S, δ2S = δ1S≃192°, δ2S≃18°
    Treated as free parameters common to each charmonium’s two channels and fixed by a best fit to the measured γη branching ratios (Sec. III D).
  • η–η' mixing angle ϕ and decay constants fq, fs = Lattice ϕ≃39.3°; η'TFF ϕ≃33.5°
    Phenomenological inputs taken from two external determinations (lattice ETMC and η'TFF); absolute BRs and especially R1S depend sharply on ϕ.
  • Quark and gluon DA Gegenbauer moments (aq2, aq4, ag2) = Models I–III of Table I
    Three representative models at μ0=1 GeV evolved by ERBL; varied to test sensitivity but still external nonperturbative inputs.
  • Strong coupling scale choice αs(μ=M/2) = αs(MJ/ψ/2)=0.294 (one-loop)
    One-loop running to μ=Mψn/2; absolute quark amplitude scales as αs^4, and the paper notes two-loop αs would raise J/ψ rates into better agreement.
  • Harmonic-oscillator length a for ηc radial form factor = a=1.62 GeV^{-1}
    Fixed from Cornell 1S rms radius to evaluate Fn(k) in the ηc-mixing amplitude.
  • Cornell-potential moments R(0) and ∇²R(0) = ∇²R/R = −0.53 GeV² (J/ψ), −1.59 GeV² (ψ(2S))
    Nonperturbative bound-state inputs taken from Eichten–Quigg frozen-αs Cornell potential; control both LO and O(q²) normalizations.
axioms (7)
  • domain assumption Instantaneous Bethe–Salpeter kernel and reduction to the equal-time Salpeter wave function with Dirac projector truncated at linear q̂
    Sec. II A; standard for heavy quarkonium but drops higher covariants and relative-energy dependence.
  • domain assumption Weak-binding replacement mc = M/2 inside all hard kernels
    Used throughout Sec. II for both LO and O(q²) kernels; simplifies propagators and is standard but not exact.
  • domain assumption Only leading-twist (twist-2) quark and gluon light-cone DAs retained; higher-twist omitted
    Eqs. (14), (29); controls the soft hadronization side of the factorization.
  • domain assumption Single-angle FKS quark-flavour mixing scheme relating flavour decay constants to fq, fs, ϕ
    Sec. III A, Eqs. (42)–(43); two numerical realizations are compared but the scheme structure is assumed.
  • ad hoc to paper Taylor expansion of the hard trace through O(q̂²) with isotropic angular average, neglecting O(q^4)
    Eqs. (8)–(13); load-bearing for the claimed relativistic correction, and the paper itself flags poor convergence for ψ(2S).
  • domain assumption ηc-mixing amplitude adds coherently to the hard annihilation amplitude with a single channel-independent phase per charmonium
    Sec. III D, Eq. (48); physically motivated by UA(1) anomaly but the relative phase is not predicted.
  • standard math Standard Passarino–Veltman reduction and one-loop QCD/QED Feynman rules in dimensional regularization / iε prescription
    Sec. II A numerical evaluation via FeynCalc and Package-X.

pith-pipeline@v1.2.0-grok45-kimik3 · 36334 in / 4350 out tokens · 79364 ms · 2026-07-30T14:25:24.155194+00:00 · methodology

0 comments
read the original abstract

We present the first calculation of the radiative decays $J/\psi,\psi(2S)\rightarrow\gamma\eta^{(\prime)}$ in perturbative QCD that includes the order-$q^{2}$ relativistic corrections in all three short-distance contributions, namely the quark-antiquark, two-gluon, and QED contributions. The amplitudes are found to be remarkably insensitive to the light-cone distribution amplitude and to the light-quark mass, a robustness that persists through order $q^{2}$ and makes the predictions correspondingly reliable. The relativistic correction enhances the $J/\psi$ branching ratios by roughly a factor of two, narrowing their shortfall from experiment, whereas for the $\psi(2S)$ it is about twice as large as for the $J/\psi$ and the low-order expansion converges poorly. In two representative $\eta$--$\eta'$ mixing schemes, the ratio $\mathcal{R}_{1S}=\mathcal{B}(\gamma\eta')/\mathcal{B}(\gamma\eta)$ proves sharply sensitive to the mixing angle and favours the smaller of the two. The predicted $\psi(2S)$ rates lie well above the data in both channels, already at leading order, and most severely for the anomalously small $\gamma\eta$ channel. Such a discrepancy suggests that a mechanism beyond the hard perturbative process is at work. As a physically motivated attempt, we explore the $\eta_{c}$-mixing contribution, which adds coherently to the perturbative one and is comparable to it in the $\gamma\eta$ channel, and find that the interference can bring the $\psi(2S)$ rates into agreement with the data, although its extraction is limited by a strong sensitivity to the mixing parameters.

Figures

Figures reproduced from arXiv: 2607.23737 by Chao-Jie Fan, Cong Wang, Jun-Kang He.

Figure 1
Figure 1. Figure 1: FIG. 1. A representative Feynman diagram for the quark-content contribution to [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. A representative Feynman diagram for the gluon-content contribution to [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Feynman diagrams for the QED contribution [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

74 extracted references · 53 linked inside Pith

  1. [1]

    J. J. Aubertet al.(E598), Experimental Observation of a Heavy ParticleJ, Phys. Rev. Lett. 33, 1404 (1974)

    1974

  2. [2]

    Lattice” denotes the ETMC determination [58] and “η ′TFF

    Because theη (′) amplitudes are not precisely known, we use the three representative models of Table I, taken from theη (′) transition-form-factor analysis of Ref. [43]. At the reference scaleµ 0 = 1 GeV the three profiles differ markedly. Model I has small moments and is single-humped, close to the asymptotic amplitude 6u¯uof perturbative QCD [52, 53], a...

  3. [3]

    J. E. Augustinet al.(SLAC-SP-017), Discovery of a Narrow Resonance ine +e− Annihilation, Phys. Rev. Lett.33, 1406 (1974), [Adv. Exp. Phys.5,141(1976)]

    1974

  4. [4]

    Brambillaet al.(Quarkonium Working Group), Heavy quarkonium physics, CERN Yellow ReportCERN-2005-005, 10.5170/CERN-2005-005 (2005), arXiv:hep-ph/0412158 [hep-ph]

    N. Brambillaet al.(Quarkonium Working Group), Heavy quarkonium physics, CERN Yellow ReportCERN-2005-005, 10.5170/CERN-2005-005 (2005), arXiv:hep-ph/0412158 [hep-ph]

    Pith/arXiv arXiv 2005

  5. [5]

    Brambillaet al., Heavy quarkonium: progress, puzzles, and opportunities, Eur

    N. Brambillaet al., Heavy quarkonium: progress, puzzles, and opportunities, Eur. Phys. J. C71, 1534 (2011), arXiv:1010.5827 [hep-ph]

    Pith/arXiv arXiv 2011

  6. [6]

    M. B. Voloshin, Charmonium, Prog. Part. Nucl. Phys.61, 455 (2008), arXiv:0711.4556 [hep- ph]

    Pith/arXiv arXiv 2008

  7. [7]

    Eichten, S

    E. Eichten, S. Godfrey, H. Mahlke, and J. L. Rosner, Quarkonia and their transitions, Rev. Mod. Phys.80, 1161 (2008), arXiv:hep-ph/0701208

    Pith/arXiv arXiv 2008

  8. [8]

    V. A. Novikov, M. A. Shifman, A. I. Vainshtein, and V. I. Zakharov, A Theory of theJ/ψ→ η(η ′)γDecays, Nucl. Phys.B165, 55 (1980). 34

    1980

  9. [9]

    Kuang, Y.-P

    Y.-P. Kuang, Y.-P. Yi, and B. Fu, Multipole Expansion in Quantum Chromodynamics and the Radiative DecaysJ/ψ→γ+ηandJ/ψ→γ+π 0, Phys. Rev.D42, 2300 (1990)

    1990

  10. [10]

    Feldmann, P

    T. Feldmann, P. Kroll, and B. Stech, Mixing and decay constants of pseudoscalar mesons, Phys. Rev.D58, 114006 (1998), arXiv:hep-ph/9802409 [hep-ph]

    Pith/arXiv arXiv 1998

  11. [11]

    G` erard and A

    J.-M. G` erard and A. Martini, Ultimate survival in anomalousψ(2S) decays, Phys. Lett.B730, 264 (2014), arXiv:1312.3081 [hep-ph]

    Pith/arXiv arXiv 2014

  12. [12]

    Zhao, Understanding the radiative decays of vector charmonia to light pseudoscalar mesons, Phys

    Q. Zhao, Understanding the radiative decays of vector charmonia to light pseudoscalar mesons, Phys. Lett.B697, 52 (2011), arXiv:1012.1165 [hep-ph]

    Pith/arXiv arXiv 2011

  13. [13]

    Chao, Issue ofψ→γη,γη ′ Decays andη−η ′ Mixing, Phys

    K.-T. Chao, Issue ofψ→γη,γη ′ Decays andη−η ′ Mixing, Phys. Rev.D39, 1353 (1989)

    1989

  14. [14]

    Chao, Mixing ofη,η ′ withc¯c,b ¯bstates and their radiative decays, Nucl

    K.-T. Chao, Mixing ofη,η ′ withc¯c,b ¯bstates and their radiative decays, Nucl. Phys.B335, 101 (1990)

    1990

  15. [15]

    J. G. K¨ orner, J. H. K¨ uhn, M. Krammer, and H. Schneider, Zweig Forbidden Radiative Or- thoquarkonium Decays in Perturbative QCD,11th International Symposium on Lepton and Photon Interactions at High Energies Ithaca, New York, August 4-9, 1983, Nucl. Phys.B229, 115 (1983)

    1983

  16. [16]

    J. H. K¨ uhn, Light Cone Expansion and Scaling Laws for Radiative Orthoquarkonium Decays, Phys. Lett.B127, 257 (1983)

    1983

  17. [17]

    J. P. Ma, Reexamining radiative decays of 1 −− quarkonium intoη ′ andη, Phys. Rev.D65, 097506 (2002), arXiv:hep-ph/0202256 [hep-ph]

    Pith/arXiv arXiv 2002

  18. [18]

    Yang, Radiative decaysJ/ψ→η (′)γin perturbative QCD, (2004), arXiv:hep- ph/0404018 [hep-ph]

    Y.-D. Yang, Radiative decaysJ/ψ→η (′)γin perturbative QCD, (2004), arXiv:hep- ph/0404018 [hep-ph]

    arXiv 2004

  19. [19]

    G. Li, T. Li, X.-Q. Li, W.-G. Ma, and S.-M. Zhao, Revisiting the OZI-forbidden radiative decays of orthoquarkonia, Nucl. Phys.B727, 301 (2005), arXiv:hep-ph/0505158 [hep-ph]

    Pith/arXiv arXiv 2005

  20. [20]

    B. A. Li, Υ(1s)→γ(η ′, η) decays, Phys. Rev.D77, 097502 (2008), arXiv:0712.4246 [hep-ph]

    Pith/arXiv arXiv 2008

  21. [21]

    Gao, Y.-J

    Y.-J. Gao, Y.-J. Zhang, and K.-T. Chao, Radiative decays of charmonium into light mesons, Chin. Phys. Lett.23, 2376 (2006), arXiv:hep-ph/0607278 [hep-ph]

    Pith/arXiv arXiv 2006

  22. [22]

    He and Y.-D

    J.-K. He and Y.-D. Yang, Revisiting the radiative decaysJ/ψ→γη (′) in perturbative QCD, Nucl. Phys. B943, 114627 (2019), arXiv:1903.11430

    Pith/arXiv arXiv 2019

  23. [23]

    Z.-G. He, Y. Fan, and K.-T. Chao, Relativistic corrections toJ/ψexclusive and inclusive dou- ble charm production atBfactories, Phys. Rev. D75, 074011 (2007), arXiv:hep-ph/0702239

    Pith/arXiv arXiv 2007

  24. [24]

    G. T. Bodwin, D. Kang, T. Kim, J. Lee, and C. Yu, Relativistic corrections toe+e− →J/ψ+η c in a potential model, AIP Conf. Proc.892, 315 (2007), arXiv:hep-ph/0611002

    Pith/arXiv arXiv 2007

  25. [25]

    Jiang, C.-J

    H.-M. Jiang, C.-J. Fan, J.-K. He, and C. Kong, Heavy quarkonium decayV→gggwith both relativistic and QCD radiative corrections, Phys. Rev. D112, 114014 (2025), arXiv:2509.16604 [hep-ph]

    arXiv 2025

  26. [26]

    Kivel, Relativistic corrections toJ/ψ→p¯pdecay, Phys

    N. Kivel, Relativistic corrections toJ/ψ→p¯pdecay, Phys. Rev. D107, 054026 (2023), arXiv:2211.13603. 35

    Pith/arXiv arXiv 2023

  27. [27]

    Kivel, Relativistic corrections toψ(nS)→ρπexclusive decays and their role in the under- standing of theρπ-puzzle, Phys

    N. Kivel, Relativistic corrections toψ(nS)→ρπexclusive decays and their role in the under- standing of theρπ-puzzle, Phys. Rev. D107, 094015 (2023), arXiv:2301.03884

    Pith/arXiv arXiv 2023

  28. [28]

    Fan and J.-K

    C.-J. Fan and J.-K. He, Three-gluon decays of radially excited quarkoniaψ(2S) and Υ(2S) with both relativistic and QCD radiative corrections, Phys. Rev. D114, 014004 (2026), arXiv:2603.10440 [hep-ph]

    Pith/arXiv arXiv 2026

  29. [29]

    E. N. Elekina and A. P. Martynenko, Relativistic effects in the double S- and P-wave char- monium production ine +e− annihilation, Phys. Rev. D81, 054006 (2010), arXiv:0910.0394

    Pith/arXiv arXiv 2010

  30. [30]

    He and C.-J

    J.-K. He and C.-J. Fan, Revisiting the P-wave charmonium radiative decaysh c →γη (′) with relativistic corrections, Phys. Rev. D103, 114006 (2021), arXiv:2003.05634 [hep-ph]

    Pith/arXiv arXiv 2021

  31. [31]

    Ablikimet al.(BESIII), Measurement of branching fractions forψ(3686)→γη ′, γηand γπ 0, Phys

    M. Ablikimet al.(BESIII), Measurement of branching fractions forψ(3686)→γη ′, γηand γπ 0, Phys. Rev. D96, 052003 (2017), arXiv:1708.03103 [hep-ex]

    Pith/arXiv arXiv 2017

  32. [32]

    Navaset al.(Particle Data Group), Review of Particle Physics, Phys

    S. Navaset al.(Particle Data Group), Review of Particle Physics, Phys. Rev. D110, 030001 (2024)

    2024

  33. [33]

    Bhatnagar, J

    S. Bhatnagar, J. Mahecha, and Y. Mengesha, Relevance of various Dirac covariants in hadronic Bethe-Salpeter wave functions in electromagnetic decays of ground state vector mesons, Phys. Rev. D90, 014034 (2014), arXiv:1307.4044 [hep-ph]

    Pith/arXiv arXiv 2014

  34. [34]

    Gebrehana, S

    E. Gebrehana, S. Bhatnagar, and H. Negash, Analytic approach to calculations of mass spectra and decay constants of heavy-light quarkonia in the framework of Bethe-Salpeter equation, Phys. Rev. D100, 054034 (2019), arXiv:1901.01888 [hep-ph]

    Pith/arXiv arXiv 2019

  35. [35]

    V. L. Chernyak and A. R. Zhitnitsky, Asymptotic Behavior of Exclusive Processes in QCD, Phys. Rept.112, 173 (1984)

    1984

  36. [36]

    Kroll and K

    P. Kroll and K. Passek-Kumeriˇ cki, The two gluon components of theηandη ′ mesons to leading twist accuracy, Phys. Rev.D67, 054017 (2003), arXiv:hep-ph/0210045 [hep-ph]

    Pith/arXiv arXiv 2003

  37. [37]

    Mertig, M

    R. Mertig, M. B¨ ohm, and A. Denner, FEYN CALC: Computer algebraic calculation of Feyn- man amplitudes, Comput. Phys. Commun.64, 345 (1991)

    1991

  38. [38]

    Shtabovenko, R

    V. Shtabovenko, R. Mertig, and F. Orellana, New Developments in FeynCalc 9.0, Comput. Phys. Commun.207, 432 (2016), arXiv:1601.01167 [hep-ph]

    Pith/arXiv arXiv 2016

  39. [39]

    Shtabovenko, R

    V. Shtabovenko, R. Mertig, and F. Orellana, FeynCalc 9.3: New features and improvements, Comput. Phys. Commun.256, 107478 (2020), arXiv:2001.04407 [hep-ph]

    Pith/arXiv arXiv 2020

  40. [40]

    H. H. Patel, Package-X: A Mathematica package for the analytic calculation of one-loop inte- grals, Comput. Phys. Commun.197, 276 (2015), arXiv:1503.01469 [hep-ph]

    Pith/arXiv arXiv 2015

  41. [41]

    H. H. Patel, Package-X 2.0: A Mathematica package for the analytic calculation of one-loop integrals, Comput. Phys. Commun.218, 66 (2017), arXiv:1612.00009 [hep-ph]

    Pith/arXiv arXiv 2017

  42. [42]

    He and C.-J

    J.-K. He and C.-J. Fan, QCD analysis of electromagnetic Dalitz decaysJ/ψ→η (′)ℓ+ℓ−, Phys. Rev. D105, 094034 (2022), arXiv:2005.13568 [hep-ph]

    Pith/arXiv arXiv 2022

  43. [43]

    Ball and G

    P. Ball and G. W. Jones,B→η (′) Form Factors in QCD, JHEP08, 025, arXiv:0706.3628 [hep-ph]. 36

    Pith/arXiv arXiv

  44. [44]

    S. S. Agaev, V. M. Braun, N. Offen, F. A. Porkert, and A. Sch¨ afer, Transition form factors γ∗γ→ηandγ ∗γ→η ′ in QCD, Phys. Rev.D90, 074019 (2014), arXiv:1409.4311 [hep-ph]

    Pith/arXiv arXiv 2014

  45. [45]

    Krammer, A Polarization Prediction From Two Gluon Exchange for 1 −− (Q ¯Q)→γ2 ++ (q¯q), Phys

    M. Krammer, A Polarization Prediction From Two Gluon Exchange for 1 −− (Q ¯Q)→γ2 ++ (q¯q), Phys. Lett.B74, 361 (1978)

    1978

  46. [46]

    Billoire, R

    A. Billoire, R. Lacaze, A. Morel, and H. Navelet, The Use of QCD in OZI Violating Radiative Decays of Vector Mesons, Phys. Lett.B80, 381 (1979)

    1979

  47. [47]

    E. J. Eichten and C. Quigg, Mesons with Beauty and Charm: New Horizons in Spectroscopy, Phys. Rev. D99, 054025 (2019), arXiv:1902.09735 [hep-ph]

    Pith/arXiv arXiv 2019

  48. [48]

    Eichten, K

    E. Eichten, K. Gottfried, T. Kinoshita, K. D. Lane, and T. M. Yan, Charmonium: The Model, Phys. Rev. D17, 3090 (1978), [Erratum: Phys. Rev. D 21, 313 (1980)]

    1978

  49. [49]

    E. J. Eichten and C. Quigg, Quarkonium wave functions at the origin: an update, (2019), arXiv:1904.11542 [hep-ph]

    Pith/arXiv arXiv 2019

  50. [50]

    G. T. Bodwin, D. Kang, and J. Lee, Potential-model calculation of an orderv 2 NRQCD matrix element, Phys. Rev. D74, 014014 (2006), arXiv:hep-ph/0603186 [hep-ph]

    Pith/arXiv arXiv 2006

  51. [51]

    G. T. Bodwin, H. S. Chung, D. Kang, J. Lee, and C. Yu, Improved determination of color- singlet nonrelativistic QCD matrix elements for S-wave charmonium, Phys. Rev. D77, 094017 (2008), arXiv:0710.0994 [hep-ph]

    Pith/arXiv arXiv 2008

  52. [52]

    P. B. Mackenzie and G. P. Lepage, QCD Corrections to the Gluonic Width of the Υ Meson, Phys. Rev. Lett.47, 1244 (1981)

    1981

  53. [53]

    G. P. Lepage and S. J. Brodsky, Exclusive Processes in Perturbative Quantum Chromody- namics, Phys. Rev. D22, 2157 (1980)

    1980

  54. [54]

    A. V. Efremov and A. V. Radyushkin, Factorization and Asymptotical Behavior of Pion Form-Factor in QCD, Phys. Lett. B94, 245 (1980)

    1980

  55. [55]

    V. L. Chernyak and A. R. Zhitnitsky, Exclusive Decays of Heavy Mesons, Nucl. Phys.B201, 492 (1982), [Erratum: Nucl. Phys.B214,547(1983)]

    1982

  56. [56]

    G. S. Bali, V. M. Braun, S. B¨ urger, M. G¨ ockeler, M. Gruber, F. Hutzler, P. Korcyl, A. Sch¨ afer, A. Sternbeck, and P. Wein (RQCD), Light-cone distribution amplitudes of pseu- doscalar mesons from lattice QCD, JHEP08, 065, [Addendum: JHEP 11, 037 (2020)], arXiv:1903.08038 [hep-lat]

    Pith/arXiv arXiv 2020

  57. [57]

    Baker, D

    E. Baker, D. Bollweg, P. Boyle, I. Clo¨ et, X. Gao, S. Mukherjee, P. Petreczky, R. Zhang, and Y. Zhao, Lattice QCD calculation of the pion distribution amplitude with domain wall fermions at physical pion mass, JHEP07, 211, arXiv:2405.20120 [hep-lat]

    Pith/arXiv arXiv

  58. [58]

    Zhong, Z.-H

    T. Zhong, Z.-H. Zhu, and H.-B. Fu, Constraints ofξ-moments computed using QCD sum rules on pion distribution amplitude models, Chin. Phys. C47, 013111 (2023), arXiv:2209.02493 [hep-ph]

    Pith/arXiv arXiv 2023

  59. [59]

    Ottnadet al.(Extended Twisted Mass),η,η ′ mesons from lattice QCD in fully physical conditions, Eur

    K. Ottnadet al.(Extended Twisted Mass),η,η ′ mesons from lattice QCD in fully physical conditions, Eur. Phys. J. A61, 169 (2025), arXiv:2503.09895 [hep-lat]. 37

    Pith/arXiv arXiv 2025

  60. [60]

    Escribano, P

    R. Escribano, P. Masjuan, and P. Sanchez-Puertas,ηandη ′ transition form factors from rational approximants, Phys. Rev.D89, 034014 (2014), arXiv:1307.2061 [hep-ph]

    Pith/arXiv arXiv 2014

  61. [61]

    G. T. Bodwin, E. Braaten, and G. P. Lepage, Rigorous QCD analysis of inclusive annihilation and production of heavy quarkonium, Phys. Rev. D51, 1125 (1995), [Erratum: Phys. Rev. D 55, 5853 (1997)], arXiv:hep-ph/9407339

    Pith/arXiv arXiv 1995

  62. [62]

    Fan and J.-K

    C.-J. Fan and J.-K. He, Radiative decays ofh c to the light mesonsη (′): A perturbative QCD calculation, Phys. Rev. D100, 034005 (2019), arXiv:1906.07353 [hep-ph]

    Pith/arXiv arXiv 2019

  63. [63]

    A. Ali, J. Chay, C. Greub, and P. Ko, Contribution ofb→sggthrough the QCD anomaly in exclusive decaysB ± →(η ′, η)(K±, K∗±) andB 0 →(η ′, η)(K0, K∗0), Phys. Lett. B424, 161 (1998), arXiv:hep-ph/9712372 [hep-ph]

    Pith/arXiv arXiv 1998

  64. [64]

    Be´ cirevi´ c, G

    D. Be´ cirevi´ c, G. Duplanˇ ci´ c, B. Klajn, B. Meli´ c, and F. Sanfilippo, Lattice QCD and QCD sum rule determination of the decay constants ofη c,J/ψandh c states, Nucl. Phys. B883, 306 (2014), arXiv:1312.2858 [hep-ph]

    Pith/arXiv arXiv 2014

  65. [65]

    Beneke and M

    M. Beneke and M. Neubert, Flavor singletBdecay amplitudes in QCD factorization, Nucl. Phys. B651, 225 (2003), arXiv:hep-ph/0210085 [hep-ph]

    Pith/arXiv arXiv 2003

  66. [66]

    Brambilla, Y

    N. Brambilla, Y. Jia, and A. Vairo, Model-independent study of magnetic dipole transitions in quarkonium, Phys. Rev. D73, 054005 (2006), arXiv:hep-ph/0512369

    Pith/arXiv arXiv 2006

  67. [67]

    Akbar, B

    N. Akbar, B. Shafaq, S. Zahra, and A. Mir, Mass spectrum, root-mean-square radii, form factors, and charge radii of mesons, J. Korean Phys. Soc.86, 1037 (2025), arXiv:2303.07394 [hep-ph]

    Pith/arXiv arXiv 2025

  68. [68]

    R. E. Mitchellet al.(CLEO),J/ψandψ(2S) Radiative Transitions toη c, Phys. Rev. Lett. 102, 011801 (2009), [Erratum: Phys. Rev. Lett. 106, 159903 (2011)], arXiv:0805.0252 [hep-ex]

    Pith/arXiv arXiv 2009

  69. [69]

    Ablikimet al.(BESIII), Measurements of the mass and width of theη c usingψ ′ →γη c, Phys

    M. Ablikimet al.(BESIII), Measurements of the mass and width of theη c usingψ ′ →γη c, Phys. Rev. Lett.108, 222002 (2012), arXiv:1111.0398 [hep-ex]

    Pith/arXiv arXiv 2012

  70. [70]

    S. F. Radford and W. W. Repko, Note on recent measurements of theψ(1S)→γη c(1S) and ψ(2S)→γη c(1S) branching ratios, Phys. Rev. D78, 057501 (2008), arXiv:0805.3833 [hep-ph]

    Pith/arXiv arXiv 2008

  71. [71]

    W.-J. Deng, H. Liu, L.-C. Gui, and X.-H. Zhong, Charmonium spectrum and their electro- magnetic transitions with higher multipole contributions, Phys. Rev. D95, 034026 (2017), arXiv:1608.00287 [hep-ph]

    Pith/arXiv arXiv 2017

  72. [72]

    Delaney, C

    J. Delaney, C. E. Thomas, and S. M. Ryan, Radiative Transitions in Charmonium from Lattice QCD, JHEP05, 230, arXiv:2301.08213 [hep-lat]

    Pith/arXiv arXiv

  73. [73]

    Batelaan, J

    M. Batelaan, J. J. Dudek, R. G. Edwards,et al.(Hadron Spectrum),ηandη ′ meson production inJ/ψradiative decays from lattice QCD, Phys. Rev. D112, 074505 (2025), arXiv:2506.09305 [hep-lat]

    Pith/arXiv arXiv 2025

  74. [74]

    Batelaan, J

    M. Batelaan, J. J. Dudek, R. G. Edwards,et al.(Hadron Spectrum),ηandη ′ production in J/ψradiative decays from quantum chromodynamics, Phys. Rev. Lett.135, 161904 (2025), arXiv:2506.09306 [hep-lat]

    Pith/arXiv arXiv 2025