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REVIEW 4 major objections 5 minor 45 references

Holographic quark masses and radiative decays of heavy vector mesons

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The authors extract charm and bottom constituent quark masses and several quarkonium decay widths from AdS/QCD plus the Segre formula, obtaining order-of-magnitude agreement with experiment but no precise match.

desk verdict The Segre-formula idea is genuinely new and the quark masses probably survive, but Eq. (17) has a normalization error by a factor πQ_q that shifts α_s and the absolute widths, so the paper needs a serious revision before the central claims can be trusted. read the letter →

arxiv 2505.14324 v2 pith:3ONUHYUY submitted 2025-05-20 hep-ph

Saulo Diles , Miguel Angel Martin Contreras , Alfredo Vega This is my paper · ORCID
classification hep-ph
keywords decayheavywidthquarkvectorconstituentderiveholographic
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

Bottom-up holographic QCD models describe mesons as waves in a curved five-dimensional space. The shape of the wave determines the meson mass and its electromagnetic decay constant. This paper takes that information and feeds it into an old quantum-mechanics formula, the Segre formula, which relates the wave function at the origin of a quark-antiquark pair to how quickly the binding energy changes with excitation level. By combining the two, the authors solve for the masses of the constituent charm and bottom quarks, getting about 1.97 and 5.21 GeV. They then use those masses to calculate how often heavy vector mesons decay into three gluons, three photons, or one photon plus two gluons, for both ground and radially excited states. Some quantities, like the strong coupling constant, are not really predicted: the authors set them by forcing the ground-state three-gluon width to match experiment. The decay widths they then compute for excited states agree with data in order of magnitude, but not precisely; the paper itself says the model is consistent rather than accurate. The approach is a phenomenological extension of known formulas rather than a derivation from first principles.
Extended reading notes

Core claim

The strongest claim is the abstract's statement: 'By applying the Segre formula from non-relativistic quantum mechanics, we derive new observables from holography: the constituent quark mass, the three-photon decay width, the effective fine structure constant of the strong interaction, and the mixed one-photon and two-gluon decay width,' with the quantitative result mc = 1.97 ± 0.28 GeV and mb = 5.21 ± 0.36 GeV. If correct, the holographic mass and decay-constant spectra would determine constituent quark masses and several heavy-quarkonium annihilation widths at the order-of-magnitude level.

Load-bearing premise

The load-bearing premise is that the holographic decay constant f_n can be identified, through the Van Royen-Weisskopf formula (Eq. 17), with the square of the non-relativistic quark-model wave function at the origin, and that this wave function obeys the Segre formula under the holographic Schrödinger equation. This bridge is stated in Section III and is assumed, not derived from AdS/CFT; if it fails, the quark masses and all decay widths derived from it lose their physical meaning.

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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

4 major / 5 minor

Summary. The manuscript proposes to bridge bottom-up AdS/QCD spectra of heavy vector mesons with the non-relativistic constituent quark model. Using the holographic Schrödinger potential generated by the Braga dilaton, the authors compute masses and decay constants for charmonium and bottomonium, parametrize the resulting radial Regge trajectories, and apply the Segre formula to extract the constituent quark masses, obtaining mc = 1.97 ± 0.28 GeV and mb = 5.21 ± 0.36 GeV. They then compute the three-gluon, three-photon, and one-photon-two-gluon annihilation widths of ground and excited states, fixing an effective strong coupling from the ground-state three-gluon width, and compare the results with PDG data in Table I. The paper concludes that AdS/QCD can generate these new observables at the order-of-magnitude level and claims a new paradigm for meson spectroscopy.

Significance. If correct, the paper would substantially extend the phenomenological reach of bottom-up AdS/QCD by connecting holographic decay constants to quark-model wave functions and by producing new decay-width predictions. The ratio-based quark-mass extraction in Eq. (20) is an appealing idea, and it is insensitive to the overall normalization of the Van Royen-Weisskopf relation. However, the quantitative claims in their present form are not reliable: the central bridge relation (17) is misnormalized, which shifts the fitted αs(c) and all absolute decay widths, and the numerical inputs needed to reproduce the results are not provided. The manuscript itself concedes in the conclusions that the model results are "not accurate," which tempers the "new paradigm" claim. With corrected normalization and fuller documentation of inputs, the conceptual framework could be salvageable.

major comments (4)
  1. [Section III, Eq. (17)] The Van Royen-Weisskopf relation is misquoted. Combining the standard definitions in Eqs. (1) and (16) with the usual nonrelativistic width Γ(V→e+e-) = 16π α_em^2 Q_q^2 |Ψ(0)|^2/M_n^2 yields f_n^2 = 12 |Ψ_n(0)|^2/M_n, not 12π Q_q |Ψ_n(0)|^2/M_n. The extra factor πQ_q, about 2.09 for charm and 1.05 for bottom, enters every absolute use of the bridge: Eq. (26), Eq. (27), Eq. (30), Eq. (34), Eq. (35), and Table I. Since α_s(c) is fitted to the ground-state three-gluon width through Eq. (27), the reported α_s(c) = 0.293 is too high by the cube root of πQ_c, about 1.28, so the corrected value is near 0.23. The absolute decay widths in Table I shift accordingly, and the quoted Γ(J/ψ→3γ) = 0.96 eV moves outside the measured 1.07 ± 0.20 eV. The quark masses obtained from Eq. (20) are ratios of wave functions and are unaffected by this factor, but the central numerical claims of the paper must be recomputed.
  2. [Section III, Eq. (22) and Table I] The numerical results cannot be reproduced from the manuscript alone. The Regge parameters a, b, ν in Eq. (22), the fitted dilaton parameters {κ, α, M, Γ} for charmonium and bottomonium, and the values of f_n and M_n entering Table I are not shown. In particular, fitting the three-parameter Regge form to only the first two states of each family is underdetermined unless additional states are used from the model output. The paper should provide the full input spectra, the fitted parameter values, and the error-propagation procedure that leads to the quoted uncertainties in Eq. (24) and Table I.
  3. [Section IV.A, Eq. (27)] The procedure fixes α_s from the experimental ground-state three-gluon width, so the 1S three-gluon width is an input rather than a prediction; this is acknowledged by the asterisk in Table I. The abstract and introduction nevertheless present the effective strong coupling as one of the observables "derived" from holography. This overstates the predictive content. The paper should state explicitly that only the excited-state ggg widths and the γγγ and γgg channels are genuine predictions once α_s is fixed by the ground state.
  4. [Section IV.C, Eq. (34)] The three-photon width formula is internally inconsistent with the paper's own bridge relation. Substituting Eq. (26) into Eq. (33) gives a coefficient proportional to Q_q^5/π, whereas Eq. (34) omits the 1/π and uses Q_q^5; Eq. (33) itself contains Q_q^6. These discrepancies are not discussed and change the quoted three-photon widths by a factor of order unity. The derivation of Eq. (34) from Eq. (33) should be shown explicitly and the numerical results in Table I corrected accordingly.
minor comments (5)
  1. [Section II, Eq. (1)] The power of Q_q in Eq. (1) is typeset ambiguously; it should read Q_q^2 to match the standard relation Γ(V→e+e-) = 4π α_em^2 Q_q^2 f^2/(3M).
  2. [Section V, Eq. (21)] The relativistic correction factor (1 + E_n/m_q)[1 + E_n/(4m_q)]^{1/2} is quoted from Ref. [24] without derivation, although it is used to argue that the sum-rule and WKB results agree at large n; a brief derivation or a more detailed explanation of its regime of validity would improve the presentation.
  3. [Section IV, Eqs. (31) and (35)] The numerical values λ_c = 3.7, λ_b = 4.9, σ_c = 6.7, and σ_b = 7.4 are introduced without derivation or explicit citation; the paper should state where these values come from and how they depend on the number of light flavors.
  4. [Throughout] The notation for decay channels and meson states is used inconsistently, for example ΓVn→e+e− in Eq. (1) versus Γ(q qbar → e+ e−) in the text, and the same symbol Γ is reused for different channels; a consistent notation would make the paper easier to follow.
  5. [Section VI] The abstract's phrase "new paradigm of meson spectroscopy in AdS/QCD" is too strong in light of the paper's own concluding statement that the results are "not accurate." The claims should be softened or accompanied by a quantitative criterion for what counts as acceptable order-of-magnitude agreement.
Assumptions & free parameters 8 free parameters · 6 assumptions · 0 invented entities

The central claim rests on a chain of inputs: a dilaton model fitted to experimental spectra in prior work, a Regge parametrization with unavailable fitted constants, an assumed wave-function mapping between holographic and non-relativistic descriptions, and strong-coupling values fixed by one measured width. These are the uncharged items the reader must accept.

free parameters (8)
  • Braga dilaton parameters for charmonium {kappa, alpha, M, Gamma} = not stated in paper
    These fix the holographic potential and hence M_n and f_n for charmonium. They were fitted to experimental charmonium masses and e+e- widths in Ref. [20], which the paper uses without restating them.
  • Braga dilaton parameters for bottomonium {kappa, alpha, M, Gamma} = not stated in paper
    These fix the holographic potential and hence M_n and f_n for bottomonium. They were fitted to experimental bottomonium masses and e+e- widths in Ref. [20], which the paper uses without restating them.
  • Regge parametrization a_c, b_c, nu_c = not stated in paper
    Used in Eq. (22) to compute dM_n/dn in the Segre extraction of the charm constituent mass; values are fitted to the holographic mass spectrum but not tabulated.
  • Regge parametrization a_b, b_b, nu_b = not stated in paper
    Used in Eq. (22) to compute dM_n/dn in the Segre extraction of the bottom constituent mass; values are fitted to the holographic mass spectrum but not tabulated.
  • alpha_s(c) (first order) = 0.280 ± 0.007
    Determined by forcing the J/psi to ggg width to the experimental value via Eq. (31); not independently predicted.
  • alpha_s(b) (first order) = 0.191 ± 0.009
    Determined by forcing the Upsilon(1S) to ggg width to the experimental value via Eq. (31); not independently predicted.
  • lambda_q first-order correction = lambda_c = 3.7, lambda_b = 4.9
    Enters Eq. (31) for the three-gluon width and is treated as a family-dependent input rather than derived in this paper.
  • sigma_q first-order correction = sigma_c = 6.7, sigma_b = 7.4
    Enters Eq. (35) for the gamma-gg width and is treated as a family-dependent input rather than derived in this paper.
assumptions (6)
  • domain assumption Bottom-up AdS/QCD with the soft-wall dilaton correctly encodes heavy vector meson masses and decay constants.
    Section II assumes the holographic dictionary and the dilaton model from Ref. [20]; no independent check is provided in this paper.
  • ad hoc to paper The holographic decay constant f_n maps to the non-relativistic wave function at the origin through the Van Royen-Weisskopf formula, Eq. (17).
    This mapping is the bridge that lets the Segre formula be applied to holographic spectra; it is assumed rather than derived from AdS/CFT.
  • domain assumption Heavy quarkonium is a non-relativistic bound state satisfying M_n = 2 m_q + E_n and the Segre formula, Eq. (15).
    Used in Section III; the non-relativistic approximation is imported from constituent quark models.
  • ad hoc to paper The radial Regge trajectory has the form M_n^2 = a(n+b)^nu with constant parameters.
    Eq. (22). The functional form and fitted parameters are chosen to represent the computed spectrum; the exponent nu is not derived from the model.
  • domain assumption The first-order corrections to the wave function at the origin are independent of n.
    Eqs. (29)-(30) and (35) use n-independent correction factors, which is necessary for the quark mass to be invariant under higher-order corrections.
  • domain assumption The values of alpha_s(q), lambda_q, and sigma_q are consistent with the renormalization scheme used in Refs. [32, 35].
    The paper adopts these values and factors without deriving them from the holographic model.

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Pith. "Pith review of Holographic quark masses and radiative decays of heavy vector mesons." pith.science (2026). https://pith.science/paper/3ONUHYUY

@misc{pith2026250514324,
  author       = {Pith},
  title        = {Pith review of: Holographic quark masses and radiative decays of heavy vector mesons},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3ONUHYUY}},
  note         = {Machine review of arXiv:2505.14324}
}
read the original abstract

Holographic models of QCD provide the spectrum of heavy vector meson masses and electromagnetic decay constants through bulk computations of the current-current correlation function. Conversely, the phenomenology of heavy vector mesons is articulated by the constituent heavy quark model utilizing a non-relativistic approximation. By applying the Segre formula from non-relativistic quantum mechanics, we derive new observables from holography: the constituent quark mass, the three-photon decay width, the effective fine structure constant of the strong interaction, and the mixed one-photon and two-gluon decay width. We also derive the three-gluon decay width, the three-photon decay width, and the mixed one-photon and two-gluon decay width for the radially excited states of heavy quarkonia and compare them with available experimental data. The present results reveal a new paradigm of meson spectroscopy in AdS/QCD.

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