Pith. sign in

REVIEW 2 major objections 4 minor 1 cited by

A simple fix for higher-order relativistic effects restores positive, data-matching three-gluon widths for 2S quarkonia after the usual q-hat-squared expansion turns negative.

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

2026-07-14 23:34 UTC pith:CEW2JMUY

load-bearing objection Clean analytic extension of their ground-state BS calculation to 2S three-gluon decays; the nodal cancellations kill pure q^{2} results, and a 1996-style rational fix restores data agreement after parameter fits. the 2 major comments →

arxiv 2603.10440 v2 pith:CEW2JMUY submitted 2026-03-11 hep-ph

Three-gluon decays of radially excited quarkonia psi(2S) and Upsilon(2S) with both relativistic and QCD radiative corrections

Chao-Jie Fan , Jun-Kang He This is my paper
classification hep-ph
keywords quarkoniumpsi(2S)Upsilon(2S)three-gluon decayBethe-Salpeterrelativistic correctionsradial nodeharmonic oscillator parameter
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.

Radially excited heavy quarkonia such as psi(2S) and Upsilon(2S) have a node in their radial wave function. That node causes destructive interference when the wave function is convolved with the multi-gluon hard kernel, so ordinary second-order relativistic corrections produce an unphysical negative three-gluon decay width. The authors work in the Bethe-Salpeter formalism with analytic harmonic-oscillator wave functions that keep the node, derive model-independent polarized-width relations from helicity and phase-space symmetries, and introduce a compact phenomenological replacement for the hard kernel that keeps the correct low-momentum limit while damping large-momentum contributions. Once both this improved relativistic factor and the known one-loop QCD corrections are included, the predicted branching fractions for V to three gluons and V to e-plus e-minus, together with their ratio R_V, match experiment. The same ratio yields a harmonic-oscillator parameter beta_V that sits at the lower end of previous phenomenological ranges, implying a more localized momentum-space wave function. The result shows that multi-gluon decays of excited quarkonia are far more sensitive to the detailed shape of the wave function than the corresponding leptonic decays, which already converge well at second order.

Core claim

For the 2S states the usual q-hat-squared relativistic expansion of the three-gluon width is unphysically negative because of nodal destructive interference; a compact phenomenological resummation that preserves the small-q-hat limit restores positivity and brings both the absolute widths and the ratio R_V into agreement with measured branching fractions.

What carries the argument

The phenomenological replacement (Eq. 2.26 and the polarized forms 3.12-3.15) that replaces the truncated factor (1 - kappa beta_V^2/M^2) by a rational expression whose Taylor expansion matches the second-order result at low momentum while remaining positive-definite at large momentum.

Load-bearing premise

The ad-hoc rational form used to stand in for all higher-order relativistic corrections is motivated by earlier phenomenology but is not derived from a controlled expansion or from first-principles factorization of the infrared pieces.

What would settle it

A lattice or NRQCD calculation of the three-gluon width that systematically includes the color-octet P-wave matrix elements and higher-order relativistic operators would either reproduce or contradict the numerical values obtained from the phenomenological form; disagreement at the 20-30 percent level would falsify the treatment.

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

If this is right

  • Three-gluon decays of 2S (and higher radial) quarkonia become reliable precision probes once the nodal interference is resummed, rather than remaining theoretically intractable.
  • The ratio R_V cleanly extracts the harmonic-oscillator parameter beta_V, supplying a tighter non-perturbative constraint for wave-function models of excited heavy quarkonia.
  • Polarized three-gluon widths, already classified into four symmetry-protected groups, can be used to predict transverse-spin correlations in baryon-pair final states such as V to p p-bar or Lambda Lambda-bar.
  • Leptonic widths remain trustworthy at second order, while gluonic widths require the improved treatment, clarifying which channels can safely use truncated expansions.

Where Pith is reading between the lines

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

  • The same nodal-resummation device should be tested on other multi-gluon or multi-photon exclusive decays of radial excitations (e.g., eta_c(2S) to ggg or chi_cJ to ggg) where similar cancellations are expected.
  • If the extracted lower beta_V values are confirmed, many light-front and potential-model calculations that employ larger beta may be systematically over-estimating the high-momentum tails of 2S wave functions.
  • The sharp contrast between leptonic and gluonic convergence supplies a diagnostic: any process whose amplitude samples the full radial wave function rather than only the origin will require analogous higher-order treatment for 2S states.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper computes three-gluon and leptonic decays of the radially excited vector quarkonia ψ(2S) and Υ(2S) in the Bethe-Salpeter formalism under the covariant instantaneous ansatz. Analytic 2S harmonic-oscillator Salpeter wave functions that incorporate the radial node are constructed, model-independent polarized-width relations are derived from helicity-flip and phase-space symmetries, and both q̂^{2}-order relativistic corrections and O(α_s) QCD radiative corrections are included. Because the pure q̂^{2} truncation produces an unphysical negative Γ(ψ(2S)→ggg), a phenomenological rational resummation of the hard kernel (Eq. (2.26) and the polarized analogues (3.12)–(3.15)) is introduced that preserves the low-momentum limit. With N_V fixed to the experimental leptonic branching ratios and β_V chosen near lattice nodal positions (or extracted from R_V), the resulting B(V o ggg), B(V o e^{+}e^{-}) and R_V agree with experiment (Table 1); the extracted β_V values lie at the lower end of typical phenomenological ranges.

Significance. If the phenomenological treatment of higher-order relativistic pieces is reliable, the work supplies the first fully analytic Bethe-Salpeter calculation of V o ggg for 2S states that simultaneously incorporates the radial node, polarized-width symmetries, and both relativistic and QCD corrections. The sharp contrast between the rapid convergence of the leptonic width and the slow convergence of the multi-gluon width is a useful diagnostic of nodal interference. The extraction of β_V from the ratio R_V (which cancels N_V) offers an independent, process-specific constraint on the momentum-space width of the 2S wave function that can be compared with lattice and potential-model determinations. These results would therefore constitute a concrete benchmark for non-perturbative descriptions of excited heavy quarkonia.

major comments (2)
  1. The central numerical claim (Table 1 agreement for ψ(2S)) rests on the uncontrolled replacement of the truncated factor (1-κ β_V^{2}/M^{2}) by the rational form in Eq. (2.26) and the polarized expressions (3.12)–(3.15). The paper itself notes that a strict O(q̂^{4}) expansion reintroduces infrared divergences that must be absorbed into poorly known color-octet matrix elements; the rational ansatz simply sidesteps that problem without a truncation-error estimate or a matching to NRQCD factorization. Because the 2S node already generates large cancellations inside the multi-gluon convolution, any ad-hoc high-momentum suppression can restore positivity and produce numbers that match experiment after N_V and β_V are fixed. A quantitative assessment of the residual uncertainty (e.g., by varying the functional form or by comparing with a partial O(q̂^{4}) calculation that absorbs the IR pieces
  2. N_V is fixed by fitting the experimental leptonic branching ratio (Sec. 3.2), while β_V is either chosen for consistency with the lattice nodal position or extracted from the experimental R_V. The subsequent prediction of B(V o ggg) therefore has limited independent predictive power; the cleaner observable is R_V itself. The manuscript should clarify more sharply which quantities are genuine predictions versus which are used for parameter determination, and should quote the residual theoretical uncertainty on R_V that arises from the phenomenological higher-order factor.
minor comments (4)
  1. The numerical values of the process-dependent coefficients C_V = 3.7 (ψ(2S)) and 4.9 (Υ(2S)) are taken from Ref. [34] without a brief reminder of their origin or of the scale at which they were evaluated; a short clarifying sentence would help the reader.
  2. Figure 1 shows the dependence of R_V on β_V but does not display the corresponding curves obtained with the pure q̂^{2} truncation; adding those curves would make the dramatic effect of the phenomenological resummation visually immediate.
  3. The notation for the complementary-error-function expressions (3.12)–(3.15) is dense; defining the auxiliary parameters a_i more prominently (or moving them to an appendix) would improve readability.
  4. A few typographical inconsistencies appear (e.g., occasional missing spaces around β_V, and the arXiv identifier in the header). A careful proof-reading pass is recommended.

Circularity Check

1 steps flagged

N_V is fitted to experimental leptonic BRs so that the reported B(V o e^{+}e^{-}) is forced by construction; R_V cancels N_V and is a cleaner (but still β_V-dependent) observable whose agreement is not fully independent of the phenomenological higher-order ansatz.

specific steps
  1. fitted input called prediction [Sec. 3.2, paragraph before Table 1 and Table 1 itself]
    "The constants N_ψ(2S)=0.232 MeV^{-1/2} for ψ(2S) and N_Υ(2S)=0.167 MeV^{-1/2} for Υ(2S) are determined by fitting to the experimental branching ratios of the corresponding leptonic decays V→e^{+}e^{-}. With these inputs, we present our predictions for the branching ratios B(V→ggg), B(V→e^{+}e^{-}), and their ratio R_V in Table 1."

    N_V multiplies both Γ(ggg) and Γ(e^{+}e^{-}) identically. Fitting N_V so that the theoretical B(V o e^{+}e^{-}) reproduces the experimental value makes the subsequent 'prediction' of that same branching ratio tautological by construction; only the ratio R_V (which cancels N_V) and the absolute B(ggg) retain independent content.

full rationale

The derivation of the hard kernel, polarized widths, and model-independent helicity/phase-space relations is self-contained and non-circular. The only clear circularity is the standard but explicit fitting of the overall normalization N_V to the experimental leptonic branching ratios, after which B(V o e^{+}e^{-}) is presented as a 'prediction' that necessarily matches by construction (Table 1). R_V cancels N_V exactly and therefore supplies an independent handle on β_V; the paper both chooses β_V from lattice/phenomenology and later extracts it from R_exp, which is legitimate parameter determination rather than a closed loop. The rational resummation (2.26)/(3.12)–(3.15) is an external phenomenological ansatz (motivated by Ref. [15]), not a self-citation or definitional identity, so it raises correctness risk but not circularity score. No uniqueness theorems or load-bearing self-citations force the central claim. Overall mild fitted-input circularity on one observable; central physics content remains independent.

Axiom & Free-Parameter Ledger

3 free parameters · 5 axioms · 1 invented entities

The central numerical claims rest on the covariant instantaneous ansatz, a phenomenological harmonic-oscillator scalar wave function that encodes the 2S node by hand, a power-counting truncation of Dirac structures, factorization of relativistic and O(α_s) QCD corrections, literature values for the radiative coefficients C_V, and—most critically—an ad-hoc rational form that replaces the divergent or negative higher-order expansion. Two free parameters (N_V, β_V) are fixed or extracted from data. No new particles or forces are postulated.

free parameters (3)
  • N_V (normalization of Salpeter wave function) = N_ψ(2S)=0.232 MeV^{-1/2}, N_Υ(2S)=0.167 MeV^{-1/2}
    Fixed by fitting the experimental leptonic branching ratio B(V→e⁺e⁻) for each state; enters every absolute width.
  • β_V (harmonic-oscillator parameter) = β_ψ(2S)≈420 MeV (range 360–430 MeV), β_Υ(2S)≈650 MeV (range 540–670 MeV)
    Chosen for consistency with lattice nodal position (ψ) or literature range (Υ), then re-extracted from experimental R_V; controls the momentum-space width and the size of relativistic corrections.
  • α_s(M_V/2) = α_s(ψ)=0.31, α_s(Υ)=0.21
    Running coupling evaluated at half the meson mass via two-loop RGE; enters both gluonic and leptonic widths.
axioms (5)
  • domain assumption Covariant instantaneous ansatz: interaction kernel independent of the relative energy q_K
    Standard in the authors’ prior B-S work; reduces the 4-D equation to a 3-D Salpeter equation (Sec. 2.1).
  • ad hoc to paper Scalar wave function f(q̂) is a 2S harmonic-oscillator form with an explicit node factor (1−2q̂²/3β_V²)
    Phenomenological choice stated in Eq. (2.8); not solved from a dynamical kernel.
  • domain assumption Dirac structures of the Salpeter wave function truncated at O(q̂²) according to power counting
    Follows Refs. [26,30] and the authors’ ground-state paper; Eq. (2.7).
  • domain assumption Relativistic and O(α_s) QCD corrections factorize
    Standard working assumption at the present accuracy (Sec. 3.2).
  • ad hoc to paper The rational form 1/(1+c q̂²/M²) (and its polarized analogues) correctly captures the dominant higher-order relativistic pieces while preserving the low-momentum limit
    Introduced in Eq. (2.26) following Chao et al.; not derived from a systematic expansion.
invented entities (1)
  • Phenomenological higher-order relativistic factor for the multi-gluon hard kernel no independent evidence
    purpose: Restore positivity of Γ(ψ(2S)→ggg) and improve high-momentum convergence without introducing infrared-sensitive color-octet matrix elements
    The form is new to this calculation (adapted from an older leptonic/gluonic paper); no independent experimental handle outside the same decay rates it is fitted to describe.

pith-pipeline@v1.1.0-grok45 · 21403 in / 3498 out tokens · 31103 ms · 2026-07-14T23:34:42.132441+00:00 · methodology

0 comments
read the original abstract

For the radially excited heavy quarkonia $V=\psi(2S)$ and $\Upsilon(2S)$, the nodal structure of the wave function renders the three-gluon decay $V\to ggg$ acutely sensitive to relativistic corrections, a longstanding challenge for reliable theoretical predictions. Within the Bethe-Salpeter formalism under the covariant instantaneous ansatz, we construct analytic harmonic-oscillator wave functions incorporating the $2S$ node and derive model-independent relations among the polarized decay widths from helicity-flip and phase-space symmetries. Motivated by the strikingly slow $\hat{q}^{2}$-order convergence driven by destructive interference at the node, we introduce a concise phenomenological treatment of the higher-order contributions that preserves the correct low-momentum limit. Including both relativistic and QCD radiative corrections, our predictions for $\Gamma(V\to ggg)$, $\Gamma(V\to e^{+}e^{-})$ and $R_{V}$ agree well with experiment, and the extracted $\beta_{V}$ lies at the lower end of typical phenomenological ranges, reflecting a more localized momentum-space wave function.

Figures

Figures reproduced from arXiv: 2603.10440 by Chao-Jie Fan, Jun-Kang He.

Figure 1
Figure 1. Figure 1: Dependence of the ratio RV on the harmonic oscillator parameter βV . The green curve shows our theoretical prediction. The yellow band represents the experimental value with its 1σ uncertainty. The intersection determines the extracted βV range. The functional dependence of the ratio RV on the harmonic oscillator parameter βV is dis￾played in [PITH_FULL_IMAGE:figures/full_fig_p020_1.png] view at source ↗

discussion (0)

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

Forward citations

Cited by 1 Pith paper

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

  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

    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

Works this paper leans on

38 extracted references · 31 linked inside Pith · cited by 1 Pith paper

  1. [1]

    G. T. Bodwin, E. Braaten, and G. P. Lepage,Rigorous QCD analysis of inclusive annihilation and production of heavy quarkonium,Phys. Rev.D51(1995) 1125–1171, [hep-ph/9407339]. [Erratum: Phys. Rev.D55,5853(1997)]. [2]Quarkonium Working GroupCollaboration, N. Brambilla et al.,Heavy Quarkonium Physics, [hep-ph/0412158]. [3]H1Collaboration, C. Adloff et al.,Di...

    Pith/arXiv arXiv 1995

  2. [2]

    H. S. Shao, H. Han, Y. Q. Ma, C. Meng, Y. J. Zhang, and K. T. Chao,Yields and polarizations of promptJ/ψandψ(2S)production in hadronic collisions,JHEP05 (2015) 103, [arXiv:1411.3300]

    Pith/arXiv arXiv 2015

  3. [3]

    Fu, X.-j

    H.-f. Fu, X.-j. Chen, and G.-L. Wang,Annihilation Rates of Heavy1 −− S-wave Quarkonia in Salpeter Method,Phys. Lett. B692(2010) 312–316, [arXiv:1006.3898]

    Pith/arXiv arXiv 2010

  4. [4]

    Z.-K. Geng, T. Wang, Y. Jiang, G. Li, X.-Z. Tan, and G.-L. Wang,Relativistic effects in the semileptonicB c decays to charmonium with the Bethe-Salpeter method,Phys. Rev. D 99(2019), no. 1 013006, [arXiv:1809.02968]

    Pith/arXiv arXiv 2019

  5. [5]

    Nochi, T

    K. Nochi, T. Kawanai, and S. Sasaki,Bethe-Salpeter wave functions ofη c(2S)andψ(2S) states from full lattice QCD,Phys. Rev. D94(2016), no. 11 114514, [arXiv:1608.02340]

    Pith/arXiv arXiv 2016

  6. [6]

    Bertone, J.-P

    V. Bertone, J.-P. Lansberg, and K. Lynch,Impact of relativistic corrections to high-pT prompt-psi(2S) production at hadron colliders, [arXiv:2510.06456]

    Pith/arXiv arXiv

  7. [7]

    Copeland, L

    M. Copeland, L. Dai, Y. Fu, and J. Roy,ψ(2S)production in jets using NRQCD, [arXiv:2508.00814]. 23

    Pith/arXiv arXiv

  8. [8]

    He, X.-B

    Z.-G. He, X.-B. Jin, and B. A. Kniehl,Relativistic corrections to prompt double charmonium hadroproduction near threshold,Phys. Rev. D109(2024), no. 9 094013, [arXiv:2402.07773]

    Pith/arXiv arXiv 2024

  9. [9]

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

    N. Kivel,Relativistic corrections toψ(nS)→ρπexclusive decays and their role in the understanding of theρπ-puzzle,Phys. Rev. D107(2023), no. 9 094015, [arXiv:2301.03884]

    Pith/arXiv arXiv 2023

  10. [10]

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

    N. Kivel,Relativistic corrections toJ/ψ→p¯pdecay,Phys. Rev. D107(2023), no. 5 054026, [arXiv:2211.13603]

    Pith/arXiv arXiv 2023

  11. [11]

    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(2025), no. 11 114014, [arXiv:2509.16604]

    arXiv 2025

  12. [12]

    Chao, H.-W

    K.-T. Chao, H.-W. Huang, and Y.-Q. Liu,Gluonic and leptonic decays of heavy quarkonia and the determination of alpha-s (m(c)) and alpha-s (m(b)),Phys. Rev.D53 (1996) 221–230, [hep-ph/9503201]

    Pith/arXiv arXiv 1996

  13. [13]

    E. E. Salpeter and H. A. Bethe,A Relativistic equation for bound state problems,Phys. Rev.84(1951) 1232–1242

    1951

  14. [14]

    E. E. Salpeter,Mass corrections to the fine structure of hydrogen - like atoms,Phys. Rev. 87(1952) 328–342

    1952

  15. [15]

    Mengesha and S

    E. Mengesha and S. Bhatnagar,Cross section for double charmonium production in electron-positron annihilation at energy √s= 10.6 GeV,Int. J. Mod. Phys.E20(2011) 2521, [arXiv:1105.4944]

    Pith/arXiv arXiv 2011

  16. [16]

    Negash and S

    H. Negash and S. Bhatnagar,Mass spectrum and leptonic decay constants of ground and radially excited states ofη c andη b in a Bethe-Salpeter equation framework,Int. J. Mod. Phys.E24(2015), no. 04 1550030

    2015

  17. [17]

    Negash and S

    H. Negash and S. Bhatnagar,Spectroscopy of ground and excited states of pseudoscalar and vector charmonium and bottomonium,Int. J. Mod. Phys.E25(2016) 1650059, [arXiv:1508.06131]. 24

    Pith/arXiv arXiv 2016

  18. [18]

    Bhatnagar and L

    S. Bhatnagar and L. Alemu,Approach to calculation of mass spectra and two-photon decays ofc cmesons in the framework of Bethe-Salpeter equation,Phys. Rev.D97(2018), no. 3 034021, [arXiv:1610.03234]

    Pith/arXiv arXiv 2018

  19. [19]

    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(2019) 054034, [arXiv:1901.01888]

    Pith/arXiv arXiv 2019

  20. [20]

    He and C.-J

    J.-K. He and C.-J. Fan,Revisiting theP-wave charmonium radiative decaysh c →γη (′) with relativistic corrections,Phys. Rev. D103(2021), no. 11 114006, [arXiv:2003.05634]

    Pith/arXiv arXiv 2021

  21. [21]

    A. N. Mitra and S. Bhatnagar,Hadron - quark vertex function. Interconnection between 3-D and 4D wave function,Int. J. Mod. Phys. A07(1992) 121–134

    1992

  22. [22]

    Bhatnagar, S.-Y

    S. Bhatnagar, S.-Y. Li, and J. Mahecha,power counting of various Dirac covariants in hadronic Bethe-Salpeter wave functions for decay constant calculations of pseudoscalar mesons,Int. J. Mod. Phys.E20(2011) 1437, [arXiv:0912.3081]

    Pith/arXiv arXiv 2011

  23. [23]

    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(2014) 014034, [arXiv:1307.4044]

    Pith/arXiv arXiv 2014

  24. [24]

    C. H. Llewellyn-Smith,A relativistic formulation for the quark model for mesons,Annals Phys.53(1969) 521–558

    1969

  25. [25]

    Wang,Decay constants of heavy vector mesons in relativistic Bethe-Salpeter method,Phys

    G.-L. Wang,Decay constants of heavy vector mesons in relativistic Bethe-Salpeter method,Phys. Lett.B633(2006) 492, [math-ph/0512009]

    Pith/arXiv arXiv 2006

  26. [26]

    Alkofer, P

    R. Alkofer, P. Watson, and H. Weigel,Mesons in a Poincare covariant Bethe-Salpeter approach,Phys. Rev. D65(2002) 094026, [hep-ph/0202053]

    Pith/arXiv arXiv 2002

  27. [27]

    Bhatnagar and S.-Y

    S. Bhatnagar and S.-Y. Li,Generalized structure of hadron-quark vertex function in Bethe- Salpeter framework: Applications to leptonic decays of V-mesons,J. Phys.32 (2006) 949–961, [hep-ph/0512352]. 25

    Pith/arXiv arXiv 2006

  28. [28]

    G. T. Bodwin and A. Petrelli,Order-v 4 corrections toS-wave quarkonium decay,Phys. Rev. D66(2002) 094011, [hep-ph/0205210]. [Erratum: Phys.Rev.D 87, 039902 (2013)]. [32]BESIIICollaboration, M. Ablikim et al.,Polarization and Entanglement in Baryon-Antibaryon Pair Production in Electron-Positron Annihilation,Nature Phys.15 (2019) 631–634, [arXiv:1808.08917...

    Pith/arXiv arXiv 2002

  29. [29]

    Kwong, P

    W. Kwong, P. B. Mackenzie, R. Rosenfeld, and J. L. Rosner,Quarkonium Annihilation Rates,Phys. Rev.D37(1988) 3210. [35]Particle Data GroupCollaboration, S. Navas et al.,Review of particle physics,Phys. Rev. D110(2024), no. 3 030001

    1988

  30. [30]

    Weng, L.-Y

    X.-Z. Weng, L.-Y. Xiao, W.-Z. Deng, X.-L. Chen, and S.-L. Zhu,Three body open flavor decays of higher charmonium and bottomonium,Phys. Rev. D99(2019), no. 9 094001, [arXiv:1811.09002]. [37]CLEOCollaboration, D. Besson et al.,Measurement of the direct photon momentum spectrum in upsilon(1S), upsilon(2S), and upsilon(3S) decays,Phys. Rev. D74(2006) 012003, ...

    Pith/arXiv arXiv 2019

  31. [31]

    Peng and B.-Q

    T. Peng and B.-Q. Ma,Heavy quarkonium 2S states in light-front quark model,Eur. Phys. J. A48(2012) 66, [arXiv:1204.0863]

    Pith/arXiv arXiv 2012

  32. [32]

    Choi, C.-R

    H.-M. Choi, C.-R. Ji, Z. Li, and H.-Y. Ryu,Variational analysis of mass spectra and decay constants for ground state pseudoscalar and vector mesons in the light-front quark model,Phys. Rev. C92(2015), no. 5 055203, [arXiv:1502.03078]

    Pith/arXiv arXiv 2015

  33. [33]

    A. J. Arifi, H.-M. Choi, C.-R. ji, and Y. Oh,Mixing effects on 1S and 2S state heavy mesons in the light-front quark model,Phys. Rev. D106(2022), no. 1 014009, [arXiv:2205.04075]. 26

    Pith/arXiv arXiv 2022

  34. [34]

    Acharyya, S

    R. Acharyya, S. Puhan, H. Dahiya, and N. Kumar,Spectroscopy of excited quarkonium states in the light-front quark model*,Chin. Phys. C49(2025), no. 2 023104, [arXiv:2408.07715]

    Pith/arXiv arXiv 2025

  35. [35]

    Godfrey and K

    S. Godfrey and K. Moats,Bottomonium Mesons and Strategies for their Observation, Phys. Rev. D92(2015), no. 5 054034, [arXiv:1507.00024]

    Pith/arXiv arXiv 2015

  36. [36]

    Asghar and N

    I. Asghar and N. Akbar,Spectrum and decay properties of bottomonium mesons,Eur. Phys. J. A60(2024), no. 3 58, [arXiv:2309.15438]

    Pith/arXiv arXiv 2024

  37. [37]

    Gao, J.-X

    X.-L. Gao, J.-X. Cui, Y.-H. Zhou, and Z.-Y. Zhou,Updated analysis of charmonium states in a relativized quark potential model, [arXiv:2504.14575]

    Pith/arXiv arXiv

  38. [38]

    Ahmad, I

    Z. Ahmad, I. Asghar, B. Masud, and M. A. Sultan,Charmonium spectrum and its decay properties,Eur. Phys. J. A61(2025), no. 9 200, [arXiv:2508.17841]. 27

    Pith/arXiv arXiv 2025