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REVIEW 3 major objections 4 minor 25 references

A holographic AdS/QCD model with a WKB and Langer-corrected potential reproduces the nonlinear Regge trajectory m_n^2 = β(n+c_0)^{2/3}+c_1 for quarkonia, fitting experimental masses to within about 0.7%.

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2026-08-03 14:23 UTC pith:MOCUU63B

load-bearing objection A useful bottom-up AdS/QCD model that derives the quarkonia n^(2/3) Regge trajectory with explicit c0 and c1 via Langer-corrected WKB and fits masses extremely well, but the WKB truncation error for the low-n fitted states is uncontrolled and the QSSE mapping is asserted rather than derived. the 3 major comments →

arxiv 2512.20371 v2 pith:MOCUU63B submitted 2025-12-23 hep-ph

Non linear Regge trajectories of quarkonia from holography

Nelson R. F. Braga , Yan F. Ferreira This is my paper
classification hep-ph
keywords AdS/QCDquarkoniaRegge trajectoriesWKB approximationLanger correctioncharmoniumbottomoniumheavy vector mesons
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 claims that a holographic model of heavy quark–antiquark states (vector mesons) can reproduce the nonlinear Regge trajectories expected from a Coulomb-plus-linear potential: squared masses grow as the 2/3 power of the radial excitation number, not linearly. The authors build a five-dimensional AdS model with a dilaton background chosen so that the Schrödinger-like potential contains a quark-mass shift, a linear term, and an adjustable 1/√z term. Using the WKB quantization condition with the Langer correction, they derive a closed-form spectrum m_n^2 = [(3π/2)(n+1+A/2)]^{2/3} κ^2 + M_q^2. Fitting to measured charmonium and bottomonium masses gives errors around 0.6–0.7%, and the corresponding decay constants are reasonable. If correct, the result connects bottom-up holography to potential-model descriptions of heavy quarkonia.

Core claim

The central claim is that the holographic potential V(z)=3/(4z^2)+Aκ^2/√(κz)+M_q^2+κ^3 z, treated with first-order WKB and the Langer correction, yields the closed-form spectrum m_n^2 = [(3π/2)(n+1+A/2)]^{2/3} κ^2 + M_q^2 (Eq. 59). This is a nonlinear Regge trajectory with exponent 2/3, an adjustable intercept c_0=1+A/2, and a quark-mass shift M_q^2. The derivation expands the WKB phase integral in powers of 1/m and keeps terms through order m^0; the Langer correction replaces the singular 3/(4z^2) term by 1/z^2, making the quantization condition valid. Fitting to measured states gives NRMSE of 0.71% for charmonium and 0.59% for bottomonium in the two-parameter case (A fixed at −1/2), with κ

What carries the argument

The central object is the Schrödinger-like potential V(z)=3/(4z^2)+Aκ^2/√(κz)+M_q^2+κ^3 z in the holographic radial coordinate z. The WKB quantization condition with the Langer transformation z=e^{x/κ}, ψ=e^{x/2}ψ̃ turns the singular 3/(4z^2) term into 1/z^2, making the phase integral well-defined. The key step is an asymptotic expansion of the phase integral in powers of 1/m, keeping terms through O(m^0); this yields the closed-form mass formula and shows that the 1/√z term controls the intercept c_0=1+A/2.

Load-bearing premise

The derivation expands the WKB phase integral in powers of 1/m and discards all terms beyond m^0 without quantifying their size for the low-n states (n=0–5) used in the fits; if those neglected terms are not small, the closed formula (59) may contain an uncontrolled approximation error absorbed into the fitted parameters.

What would settle it

Numerically solve the one-dimensional Schrödinger equation −ψ''+Vψ=m^2ψ for the potential (52) with the fitted parameters (e.g., κ=1.41 GeV, M_c=2.60 GeV, A_c=−1.71 for charmonium) and compare the exact eigenvalues with the WKB formula (59). If the differences are larger than the reported 0.7% NRMSE, the WKB truncation is the reason.

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

If this is right

  • Quarkonia squared masses are predicted to follow m_n^2 = [(3π/2)(n+1+A/2)]^{2/3} κ^2 + M_q^2 for all radial excitations, with no linear-in-n component—a sharp, testable departure from the linear Regge behavior of light mesons.
  • With A fixed at −1/2, the holographic spectrum coincides exactly with the Regge trajectory derived from the quadratic spinless Salpeter equation with a Coulomb-plus-linear potential, giving a direct dictionary between κ, M_q and the constituent quark mass and string tension.
  • The model yields decay constants as a byproduct; in the three-parameter fit, bottomonium decay constants have a normalized root-mean-square error around 9% and charmonium around 22%, without spoiling the mass fit.
  • The construction demonstrates a practical inverse problem solution: for any desired spectrum of the form β(n+c_0)^{2/3}+c_1, there exists a dilaton background and a corresponding Schrödinger potential that generates it.

Where Pith is reading between the lines

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

  • The same WKB-plus-Langer expansion could be applied to potentials with a z^α term instead of κ^3 z, yielding a family of trajectories m_n^2 ∼ (n+c)^{2α/(2+α)} that interpolate between linear and other powers; the paper only hints at this generalization.
  • The truncation at O(m^0) is untested for the low-n states actually fitted; solving the Schrödinger equation numerically with the fitted potential would quantify the missing WKB corrections and could shift the reported 0.6–0.7% errors.
  • The parameter dictionary M_q=2m_q and κ=(8m_qσ)^{1/3} suggests a way to translate potential-model string tensions and quark masses into the holographic inputs needed to compute thermal dissociation, connecting the spectrum fit to quark–gluon-plasma phenomenology.
  • The fitted values of A (−1.71 for charmonium, −2.14 for bottomonium) differ from the QSSE value −1/2, implying that the effective potential has a larger 1/√z term than the Cornell-inspired form; this could be tested by comparing to lattice-computed potentials.

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

3 major / 4 minor

Summary. The paper proposes a bottom-up AdS/QCD model for heavy vector mesons (charmonium and bottomonium). Starting from the soft-wall dilaton background, the authors engineer a Schrödinger potential V(z)=3/(4z^2)+Aκ^2/√(κz)+M_q^2+κ^3z (Eq. 52) and analyze it with a first-order WKB quantization supplemented by the Langer correction. Expanding the WKB phase integral in powers of 1/m and truncating at O(m^0), they derive Eq. (59): m_n^2 = [(3π/2)(n+1+A/2)]^{2/3}κ^2 + M_q^2, which is identified with the QSSE/Cornell trajectory m_n^2 = β(n+c_0)^{2/3}+c_1. The parameters κ, M_q, and A are fitted to experimental masses, yielding NRMSE values around 0.6–0.7% for the mass spectra; decay constants are computed as a secondary result with larger errors. The paper also compares the model with the QSSE result and interprets M_q and κ in terms of constituent quark masses and string tension.

Significance. If the derivation and the numerical fits were fully controlled, the paper would be a useful addition to holographic QCD: it gives an analytic, bottom-up realization of n^{2/3} Regge trajectories, recovers the soft-wall spectrum in the α=2 limit as a nontrivial check, and makes a transparent contact with the QSSE/Cornell approach. The authors are candid that decay constants are a byproduct. However, the central formula rests on an unquantified semiclassical truncation, and the reported parameters do not appear consistent with Eq. (59). The significance is therefore conditional on resolving these issues.

major comments (3)
  1. [Sec. IV.A, Eqs. (30)–(41) and IV.C, Eqs. (53)–(58)] The WKB phase integral is evaluated by expanding √(1−U/m^2) and discarding all terms beyond O(m^0), with the leading-order selection made 'by inspection' (Eqs. 35, 38, 56, 57). No estimate is given for the neglected O(m^{-1}) terms. For the fitted low-lying states the expansion parameter is not small: e.g., with the three-parameter charmonium fit (κ_c=1.41 GeV, M_c=2.60 GeV, A_c=−1.71), the non-constant part of the ground-state squared mass from Eq. (59) is only about 1.5 GeV^2, and U/m^2 equals unity at the turning points by definition. Omitted terms can enter at the same formal order as the retained −Aπ/2 and −π/2 contributions in Eq. (58). The authors should estimate the truncation error, e.g. by evaluating the full WKB integral numerically or by comparing Eq. (59) with exact eigenvalues of Eq. (7) for the same potential, before using Eq. (60) to identify c0 and c1.
  2. [Tables I–III and Eq. (59)] Direct substitution of the fitted parameters into Eq. (59) does not reproduce the masses listed in Tables II and III. For bottomonium with A_b=−2.14, κ_b=2.10 GeV, M_b=9.27 GeV, the argument n+1+A/2 at n=0 is −0.07, so Eq. (59) has no real positive solution; yet Table III reports a 1S mass of 9541 MeV. For charmonium with A_c=−1.71, κ_c=1.41 GeV, M_c=2.60 GeV, Eq. (59) gives a ground-state mass of about 2.88 GeV, not the quoted 3125 MeV. This suggests that the numerical masses were obtained from an exact solution of the Schrödinger equation for the potential (52), not from Eq. (59), or that one of the two is misreported. The manuscript must state explicitly which equation generated the values in Tables II and III and reconcile the WKB formula with the fitted parameters.
  3. [Sec. IV.C and V, Eq. (60)] The parameter A is introduced specifically to make c0 adjustable, but the three-parameter fit returns A_b=−2.14, which gives c0=1+A/2=−0.07. This is incompatible with the Regge form (51) for n=0: the 2/3 power of a negative number is not a standard positive intercept, and Eq. (58) would have a negative left-hand side for n=0. If the exact numerical spectrum is being used for the fit, then Eq. (60) is only an asymptotic large-m identification and should not be presented as an exact consequence. The claimed match to the QSSE value c0=3/4 (Eq. 65) is therefore not established for the actual fitted parameter set.
minor comments (4)
  1. [Sec. VI.A] Please specify the numerical procedure used to generate the masses and decay constants: is Eq. (7) solved exactly for the potential (52), or is Eq. (59) used for masses and Eq. (72) evaluated with numerical wavefunctions? This is essential for reproducing the results.
  2. [Sec. VI.B, Tables IV–V] For charmonium the decay-constant NRMSE is 22.4% (three-parameter model) and 37.9% (two-parameter model). Calling this 'reasonable agreement' is an overstatement; the abstract should be tempered, or the fit improved, given that decay constants are presented as a byproduct.
  3. [Sec. IV.A] The 'by inspection' leading-order selections (Eqs. 35, 38, 56, 57) should be expanded with at least the first correction to each contribution, so the reader can verify that the omitted terms are indeed beyond O(m^0).
  4. [General] Minor typographical issues: 'reparoduce' in Sec. VI.B; 'the slope of the slope of the Regge trajectory' in Sec. II.B; Eq. (5) has an awkward prime notation. These do not affect the physics.

Circularity Check

0 steps flagged

No significant circularity: the target trajectory is explicit model input and the mass agreement is a fit, while the WKB derivation is self-contained.

full rationale

Derivation chain checked. The target form (1)/(51), m_n^2 = β(n+c0)^{2/3} + c1, is taken explicitly from the external QSSE paper [2]; the holographic model is built to reproduce it, not to predict its functional form. Each potential term in (52) is a deliberate building block: 3/4z^2 is fixed by the AdS5 conformal term plus the Langer correction; M_q^2 is added in Sec. III to generate c1 by a direct Schrodinger inverse-problem construction; κ^3 z is chosen in Sec. IVB to produce n^{2/3}; and Aκ^2/√(κz) is introduced specifically to promote c0 to a free parameter. The WKB evaluation (53)-(58) is a real calculation: the leading (2/3)m^3/κ^3 term, the -Aπ/2 term, and the -π/2 term are obtained by expanding turning-point contributions, and the α=2 limit reproduces the exact soft-wall spectrum (11), a genuine internal consistency check. Eq. (59) is therefore the exact WKB expression for the engineered potential, with identifications (60) following from comparison with (51); this is model construction, not a hidden identification of input and output. The experimental comparison is presented as a fit: parameters κ, M_q, A are minimized against PDG masses (and in the three-parameter case also decay constants), so the ~0.6-0.7% mass agreement has no independent predictive content by itself—but the paper does not claim otherwise. The decay constants in the two-parameter case are not used in the fit and are genuine outputs (albeit with large errors); in the three-parameter case they partly enter the fit and so their agreement is partly a fit target. Self-citations ([7]-[14]) provide background models, the decay-constant formula, and a technical normalization point; none is load-bearing for Eq. (59), and no uniqueness theorem is invoked. The main caveat, the uncontrolled O(m^0) truncation of the WKB expansion, is an approximation-validity/correctness risk, not a circularity: it does not make Eq. (59) equivalent to Eq. (51) by construction beyond the explicit model-building input. Verdict: no significant circularity.

Axiom & Free-Parameter Ledger

6 free parameters · 5 axioms · 2 invented entities

The model's central result rests on (i) the AdS/QCD action, (ii) the WKB+Langer treatment whose 1/m expansion is truncated, and (iii) an engineered potential containing an ad hoc 1/√z term. Three parameters per quarkonium (κ, M_q, A) are fitted to data; the two-parameter case fixes A=-1/2 from matching to the QSSE model. A new dilaton background is introduced, and derived quark masses/string tensions are within a factor ~1.5–2 of standard values.

free parameters (6)
  • κ_c (charmonium dilaton scale) = 1.47 GeV (2-param) / 1.41 GeV (3-param)
    Fitted to experimental charmonium masses (and decay constants in the 3-param case).
  • M_c (charmonium mass shift) = 2.10 GeV (2-param) / 2.60 GeV (3-param)
    Fitted to experimental charmonium masses; identified with 2m_c.
  • κ_b (bottomonium dilaton scale) = 2.21 GeV (2-param) / 2.10 GeV (3-param)
    Fitted to experimental bottomonium masses (and decay constants in the 3-param case).
  • M_b (bottomonium mass shift) = 8.87 GeV (2-param) / 9.27 GeV (3-param)
    Fitted to experimental bottomonium masses; identified with 2m_b.
  • A_c (charmonium c0-adjusting coefficient) = -1/2 (fixed) / -1.71 (fitted)
    In the two-parameter fit A=-1/2 is set to match QSSE c0=3/4; in the three-parameter fit it is fitted to masses and decay constants.
  • A_b (bottomonium c0-adjusting coefficient) = -1/2 (fixed) / -2.14 (fitted)
    Same as A_c for bottomonium.
axioms (5)
  • domain assumption AdS/QCD vector meson action and AdS5 background (Eq. 2)
    The model assumes the standard bottom-up holographic description of vector mesons as 5D gauge fields with a dilaton background.
  • domain assumption WKB approximation with Langer correction and 1/m expansion truncated at O(m^0) is accurate for the states fitted
    Used throughout Sec. IV to derive the spectrum (Eq. 59). No error estimate is given for the neglected terms at low n; the soft-wall α=2 case is used only as a consistency check.
  • ad hoc to paper The potential (52) is the correct effective potential, including the ad hoc 1/√z term
    The Aκ^2/√(κz) term is introduced specifically to adjust c0 (Sec. IVC). Its form is not derived from QCD or from the QSSE model.
  • ad hoc to paper The Airy solution with c2=0 in Eq. (47) is chosen to avoid a null mass state
    This choice fixes the large-z behavior of the dilaton (Eq. 49) and is a modeling decision rather than a uniqueness result.
  • domain assumption Decay constant formula (72) and g5 normalization from pQCD matching (73)
    Standard in AdS/QCD; used to compute decay constants in Sec. VI B.
invented entities (2)
  • New dilaton background φ_II(z) independent evidence
    purpose: Generates the n^(2/3) Regge trajectory (Eq. 59) for charmonium and bottomonium.
    The dilaton is not directly observable, but the model's mass and decay constant predictions are falsifiable and compared to PDG data.
  • 1/√z term in the potential (coefficient A) independent evidence
    purpose: Adjusts the intercept c0 in the Regge trajectory.
    The term is an invented model ingredient whose only handle is its effect on the mass spectrum; the fitted A values indirectly constrain it.

pith-pipeline@v1.3.0-alltime-deepseek · 12421 in / 18412 out tokens · 158350 ms · 2026-08-03T14:23:23.712876+00:00 · methodology

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read the original abstract

We propose a holographic model for quarkonia using the WKB approximation with the Langer correction to properly reproduce nonlinear Regge trajectories of the form $m_n^2 = \beta (n + c_0)^{2/3} + c_1$. This form is expected from previous studies involving the solution of Cornell potential for heavy quark-antiquark interactions using a model based on the quadratic form of the spinless Salpeter-type equation (QSSE). The model fits experimental masses with very good accuracy. The corresponding decay constants also show a reasonable agreement with the results obtained from experimental data.

Figures

Figures reproduced from arXiv: 2512.20371 by Nelson R. F. Braga, Yan F. Ferreira.

Figure 1
Figure 1. Figure 1: Charmonium and bottomonium masses. Experimental data and results from the model with 3 parameters, that are very close to the case with two parameters. Using Eqs. (63) and (64) and the values of the parameters κ and Mq with A = −1/2, we find the charm and bottom quark masses and the string tensions for each meson in this case. From charmonium parameters, we have mc = Mc 2 = 1.05 GeV and σc = κ 3 c 4Mc = 0.… view at source ↗
Figure 2
Figure 2. Figure 2: Comparison of charmonium experimental decay constants to the decay constants predicted by the model with 2 parameters, and the ones given by the best fit of the model with 3 parameters. C. Dilatons and potentials In [PITH_FULL_IMAGE:figures/full_fig_p022_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Dilatons for the model with 2 and 3 parameters with the parameter values presented in Table I. 0. 0.5 1. 1.5 2. 2.5 3. 6 8 10 12 14 0. 0.5 1. 1.5 2. 2.5 3. 80 90 100 110 120 [PITH_FULL_IMAGE:figures/full_fig_p023_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Schrödinger-like potentials for the model with 2 and 3 parameters with the parameter values presented in Table I. 23 [PITH_FULL_IMAGE:figures/full_fig_p023_4.png] view at source ↗

discussion (0)

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Reference graph

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