Quantization Rules in Holographic QCD Models

Alfredo Vega; Luiz F. Ferreira

REVIEW 3 major objections 4 minor 80 references

Quantization Rules in Holographic QCD Models

T0 review · 3 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Generalized Bohr-Sommerfeld rules reproduce quasinormal-mode frequencies in holographic QCD models.

desk verdict Solid WKB/BS benchmark for holographic QNM widths, but Eq (21) has a missing factor 2 and needs a typo-fixing revision before the rules can be used as printed. read the letter →

arxiv 2505.09866 v1 pith:XM7KQWNH submitted 2025-05-15 hep-ph hep-th

Luiz F. Ferreira , Alfredo Vega This is my paper · ORCID

classification hep-phhep-th

keywords quasinormalmodesholographicQCDsoft-wallmodelWKBapproximationBohr-Sommerfeldquantizationglueballsdissociationfinitetemperature

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

This paper seeks to establish that quasinormal modes in holographic QCD—the complex frequencies that describe how thermal fluctuations decay in the dual gauge theory—can be computed from a generalized Bohr-Sommerfeld quantization rule, the same WKB phase-integral condition used for quasi-stationary states in quantum mechanics. The payoff would be a fast semiclassical route to the full complex spectrum: in the soft-wall model, the real part of scalar and vector quasinormal frequencies is reproduced to sub-percent accuracy after first-order corrections, and at low temperature an analytic formula gives that real part directly. The same framework also locates the temperature at which each mode moves above the potential barrier, which the authors read as a dissociation signal consistent with the spectral function's decreasing peak.

What carries the argument

The machinery is the generalized Bohr-Sommerfeld quantization condition for quasi-stationary states, applied to the Schrödinger-like equation obtained from the holographic field equation. The central object is the Langer-transformed potential $V_L(r_\ast) = V(r_\ast) + \frac{1}{4 r_\ast^2}$, whose extra term corrects the infinite barrier at the boundary and makes the WKB wave function vanish at $r_\ast = 0$. For modes near the barrier top, a parabolic approximation produces a gamma-function correction $\Xi(\lambda)$; for above-barrier modes, the condition is continued by $\lambda \to \lambda e^{-2\pi i}$, giving the contour form used in the tables. The imaginary part of the frequency comes from the Gamow formula, an exponential of the barrier integral, and first-order WKB corrections are included through an $I_1$ term. Together these pieces turn the quasinormal-mode boundary-value problem into phase integrals over turning points.

What would settle it

A direct check is to compute the ground-state quasinormal frequency at low temperature with a high-precision method that does not rely on WKB matching near the boundary and compare the imaginary part: the zeroth-order Gamow formula deviates from the shooting method by roughly ten percent in $\omega_I$ while the real part agrees to $10^{-3}$–$10^{-2}$ percent, so a more accurate independent calculation that confirms the shooting values would expose a systematic error in the Gamow exponential. A second check is the analytic low-temperature formula: at $T = 35$ MeV it gives $2.82466$ GeV for the scalar ground state while the shooting and numerical Bohr-Sommerfeld values are $2.82194$ GeV; if this gap does not close with higher-order terms, the claimed range of the analytic expansion fails.

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Extended reading notes

Core claim

The paper's central claim is that quasinormal modes in holographic QCD models are reproduced by generalized Bohr-Sommerfeld quantization rules for quasi-stationary states, with an infinite-barrier boundary handled by a Langer-type correction to the effective potential. Three temperature regimes are treated: modes trapped near the potential minimum, modes near the top of the finite potential barrier, and above-barrier modes obtained by analytic continuation. In each regime, the quantization condition yields frequencies that agree with the shooting method; with the first-order WKB correction, deviations in the real part remain below about one percent in most cases. In the soft-wall model at very low temperature, the real part of the scalar and vector quasinormal frequencies is given analytically as a temperature expansion. The shift of modes into the above-barrier region is interpreted as evidence of dissociation of the dual states, in line with the decreasing peak of the spectral function.

Load-bearing premise

The central assumption is that the boundary treatment—the Langer-corrected potential plus the WKB matching that keeps only the normalizable $z^{5/2}$ solution near the boundary—stays valid for complex frequencies at finite temperature; if that matching is wrong, the quantization conditions would select the wrong modes even though the current shooting comparisons agree.

Editorial extensions

If this is right

The WKB formulas provide a fast semiclassical route to quasinormal frequencies in any holographic model whose effective potential has a well plus a finite barrier, with deviations below one percent for the real part in most tested cases.
At very low temperatures, the analytic expressions for the real part of scalar and vector frequencies allow direct reading of thermal mass shifts without solving the full differential problem.
The temperature at which a mode enters the above-barrier region can be read as a dissociation temperature; the paper lists values such as roughly 77 MeV for the scalar ground state and 105 MeV for the vector ground state.
In the high-temperature above-barrier regime, quasinormal frequencies approach the linear-in-$T$ scaling of conformal AdS$_5$, connecting the low-temperature non-conformal regime to the conformal limit.
Because the quantization rules work for scalar and vector fields in the soft-wall model and for a tangent-model charmonium potential, they should extend to other dilaton profiles and flavors.

Reading between the lines

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

A natural step the authors do not take is to use the quantization rules as a fitting engine, extracting the soft-wall model parameters such as the dilaton scale directly from measured or lattice thermal masses and widths without repeated full numerical integration.
The predicted low-temperature shift of the real part scales as $T^4$ then $T^8$, which is specific enough that an independent calculation of the thermal pole of the retarded correlator, or a lattice computation of thermal glueball masses, would test the soft-wall model's temperature dependence.
The dissociation temperatures read off from the above-barrier transition could be compared with the inflection point of the spectral-function peak; if the peak persists far beyond those temperatures, the dissociation criterion would need refinement.
The imaginary part agrees only to about ten percent at very low temperature while the real part agrees to parts in $10^{-4}$, suggesting the Gamow exponential factor is the most sensitive ingredient and is the most promising target for a next-order correction.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague, honestly, the paper is better than its central typo. The core result—that a generalized Bohr-Sommerfeld rule with a Langer correction reproduces shooting-method QNM frequencies in soft-wall holographic QCD—is demonstrated with real benchmarks. No fitted parameters, three regimes, and the first-order corrections push deviations below 1% in the real part and often below 0.1%. The analytic low-temperature expansions for the real part (E8, E14) are a nice addition. The dissociation-temperature picture is a bonus, not the main point.

The new content is the adaptation of the atomic-physics BS machinery to potentials with an infinite boundary barrier, which prior black-hole applications did not have. That adaptation is nontrivial because the WKB wave function near the boundary needs the Langer term 1/(4r_*^2), and the paper checks this by matching to the normalizable z^{5/2} falloff. The checks are plausible, and the numerical agreement supports them.

Now the soft spots. Eq (21), the main quantization rule, is missing a factor 2 in the tunneling exponent. As printed, a direct expansion gives an imaginary part proportional to e^{-G}, while the paper's own Gamow formula Eq (23) and all the tables require e^{-2G}. For the T=20 MeV ground state that is the difference between 10^-44 and 10^-88. So the printed rule cannot reproduce the tables. This looks like a typo—Eq (23) is the standard formula and the tables follow from it—but it has to be fixed. Relatedly, Appendix D writes the zero-temperature Langer potential as V_L = 2 + 1/(4z^2) + z^2; the quoted integral and spectrum correspond to V_L = 2 + 4/z^2 + z^2. Table VI also has a factor-10 error in several WKB imaginary parts (the deviations line up with 10^-4, not 10^-3). None of these undermine the method, but each would trip up a reader trying to apply it.

The Appendix B matching is the part I'd push on: it is done via low-temperature expansion and asymptotic analysis, and the claim that the non-normalizable term is absent for complex omega at finite T is stated more than proven. Given the numerical agreement, I'm willing to believe it, but a referee should ask for a more careful argument or a numerical check at moderate T.

Who is this for? Anyone computing QNM widths in soft-wall or similar holographic models who wants a fast semi-analytic alternative to shooting. It deserves a serious referee; the typos are easy to fix and the core is solid. I'd send it to review, with a request for code/data, not desk-reject.

Referee Report

3 major / 4 minor

Summary. The paper derives generalized Bohr–Sommerfeld (WKB) quantization rules for quasi-stationary states in holographic QCD, treating the effective Schrödinger potential with an infinite boundary barrier through a Langer-type correction. The rules are applied to scalar and vector fluctuations in the soft-wall AdS black hole model and to a charmonium-like tangent model, and the resulting quasinormal frequencies are compared with a shooting method in three regimes: below the barrier, near the top, and above the barrier. The authors report sub-percent agreement in the real parts and a few-percent agreement in the imaginary parts in most regimes, and they provide analytic low-temperature expansions for the real parts as well as a discussion of dissociation signaled by the above-barrier transition.

Significance. If the claims survive revision, the paper provides a fast semiclassical route to quasinormal-mode spectra in holographic QCD, with analytic control at low temperature and a physical criterion for dissociation. The numerical comparisons are genuine benchmarks: no quantity is fitted to the quasinormal frequencies, and the first-order corrected rules agree with the shooting method to better than about 1% in the real part in nearly all tables (e.g., Tables III, IV, and VII), with imaginary parts agreeing to a few percent in the under-barrier and above-barrier regimes. The extension to vector fields and to a charmonium-like potential demonstrates that the method is not limited to the scalar soft-wall case. However, the printed central quantization rule and several appendix formulas contain internal inconsistencies that currently prevent an independent reader from reproducing the claimed results.

major comments (3)

[Eqs. (21) and (23)] The generalized Bohr–Sommerfeld rule is printed as an equality between the well integral and π(n+1/2) − (i/4) exp(i ∫_{r1}^{r2} p_L dr'), with no factor 2 in the barrier phase. Expanding this equation for small ε_I gives an imaginary part proportional to e^{−γ}, where γ = ∫_{r1}^{r2} √(V_L−ε_R) dr' is the single-pass barrier action. The Gamow formula (23), however, contains e^{2i∫...}, i.e. e^{−2γ}, and the tabulated values require the two-pass exponent: at T = 20 MeV, γ ≈ 100, so e^{−γ} would give |ω_I| ∼ 10^{−44} rather than the printed 1.48 × 10^{−88}. Since Eq. (23) is stated to follow from Eq. (21), the central derivation is internally inconsistent. The manuscript provides no code or data to indicate whether Tables I–IV were generated from Eq. (21), from Eq. (23), or from an unstated corrected rule; this inconsistency must be resolved before the quantization rule can be used as stated.
[Appendix D, scalar zero-temperature potential] Equation (D3) defines VL(z) = 2c^2 + 1/(4z^2) + c^4 z^2, but adding the Langer term 1/(4r_*^2) to the zero-temperature potential (D1), V(z) = 2c^2 + 15/(4z^2) + c^4 z^2, gives VL(z) = 2c^2 + 4/z^2 + c^4 z^2. The quoted integral −3π/2 + m^2π/4 and the claimed spectrum m^2 = 4c^2(n+2) correspond to the corrected potential, not to the displayed one. With the displayed VL and c = 1, the WKB action is (π/4)(m^2 − 3), which yields m^2 = 4n + 5 rather than the soft-wall spectrum. This error occurs in the place where the paper demonstrates the necessity of the Langer correction and must be corrected.
[Appendix E, turning-point expressions] The turning point z1 in Eq. (E6) contains √((ε_R − 6c^2)(ε_R − 2c^2)) in the numerator; consistency with the defining equation ε_R − VL = 0 and with the z0 expression in Eq. (E5) requires √((ε_R − 6c^2)(ε_R + 2c^2)). As printed, the two turning points are roots of different quadratic equations, so the subsequent integrals and the analytic formula (E8) cannot be reproduced from the displayed expressions. The corresponding vector-field turning points in Eqs. (E11) and (E12) should be checked for the same type of error.

minor comments (4)

[Table VI, n = 1 block] For T = 55, 56, and 57 MeV, the WKB imaginary parts are printed as 7.96662×10^−3, 1.25368×10^−2, and 1.85427×10^−2, respectively, while the shooting-method values are 8.47488×10^−4, 1.32819×10^−3, and 1.95701×10^−3. The reported deviations of 6.00%, 5.61%, and 5.25% are consistent only if the WKB entries are 7.96662×10^−4, 1.25368×10^−3, and 1.85427×10^−3; the exponents in the table appear to be off by one order of magnitude.
[Sec. IV.A, Tables X and XI] The text refers to Table X as the 'second mode' and then as the 'ground state' for n = 1, and Table XI as the 'third mode' and then as the 'ground state' for n = 2; these labels should be made consistent.
[Eq. (19)] The exponential factors in Eq. (19) contain ℏ inconsistently: some terms have i/ℏ, while others have i without ℏ. Since the paper sets ℏ = 1 throughout, either remove all ℏ factors or include them uniformly.
[Fig. 9] The caption assigns the same color (purple) to the n = 6 and n = 7 modes, which makes the figure harder to read; a distinct color would improve clarity.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the WKB quantization rules are imported from the atomic-physics literature and benchmarked against an independent shooting method, with no fitted parameter renamed as a prediction.

full rationale

The paper's central claim is a benchmark comparison between two independent methods. The effective soft-wall potential V(r*) is obtained from the bulk action and metric through a Liouville transformation, the generalized Bohr–Sommerfeld and Gamow formulas are standard WKB results taken from Refs. [33–38], and the shooting method is a separate numerical solution of the equation of motion with quasinormal-mode boundary conditions. No parameter in the WKB calculation is adjusted to the shooting results: the only scale, c, is set to 1 as the soft-wall model parameter, and the tangent-model parameters in Eq. (43) are prior model inputs, not fits to the computed frequencies. The analytic low-temperature expressions in Appendix E are derived by inserting the derived potential into the quantization condition and expanding in temperature, and they are compared with, not fitted to, the numerical values. Appendix B verifies that the WKB wave function satisfies the same boundary conditions used by the shooting method, which is consistency between methods rather than circularity. The internal inconsistency between Eq. (21) and Eq. (23) noted elsewhere is a correctness or typographical issue, not a circular reduction, and therefore does not affect the circularity score.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central calculation uses one hand-set model scale, c=1, from the soft-wall model, plus charmonium parameters imported from earlier work for the tangent model; none are fitted to quasinormal frequencies. The WKB rules themselves are imported from atomic physics and are the main external input. No new entities are introduced.

free parameters (2)

soft-wall dilaton scale c = 1 (set by hand, GeV)
Appears in the dilaton Phi=c^2 z^2 and in the zero-temperature spectrum m_n^2=4c^2(n+2); the paper sets c=1 for convenience and does not fit it to quasinormal data, so the central comparisons use this fixed scale.
charmonium model parameters (k_c, sqrt(Gamma_c), M_c) = 1.2 GeV, 0.55 GeV, 2.2 GeV
Taken from the authors' earlier charmonium model and used for the tangent-model application; they are inputs from prior literature, not fitted in this paper.

assumptions (5)

domain assumption WKB approximation is valid for the effective holographic potential, with slowly varying potential between turning points.
The quantization rules rest on standard WKB matching across turning points; the paper cites Refs. [33-38] and applies them in Section III.
domain assumption The Langer transformation, adding 1/(4 r_*^2) to the potential, correctly handles the infinite boundary barrier.
The paper modifies the potential following Lange(r) and requires that the WKB wave function vanish at the origin; Section III.A and Appendix B.
domain assumption Quasinormal mode boundary conditions are ingoing at the horizon and Dirichlet at the boundary.
Standard holographic prescription used in Section II.B and enforced in the WKB matching of Appendix B.
domain assumption The low-temperature expansion of the tortoise coordinate and potential, truncated at O(T^4) or O(T^8), is accurate enough for the analytic formulas.
Equations (12)-(13) and Appendix E use these expansions; Table XII shows visible degradation at 35 MeV for the ground state, so this expansion is a load-bearing restriction.
domain assumption The tangent-model dilaton and charmonium parameters from Refs. [51-53] correctly describe heavy vector mesons.
Used in Section IV.A to compute quasinormal modes of the tangent model; imported from the authors' earlier work.

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Cite this review

Pith. "Pith review of Quantization Rules in Holographic QCD Models." pith.science (2026). https://pith.science/paper/XM7KQWNH

@misc{pith2026250509866,
  author       = {Pith},
  title        = {Pith review of: Quantization Rules in Holographic QCD Models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XM7KQWNH}},
  note         = {Machine review of arXiv:2505.09866}
}

read the original abstract

In this paper, we investigate quasinormal modes in holographic QCD models from the perspective of the WKB approximation. We derive the generalized Bohr Sommerfeld quantization rules for quasi-stationary states in holographic QCD models. As a simple application of these formulas, we compute the quasinormal modes of scalar and vector fields in the soft wall model, where an analytic expression for the real part of the frequency can be found in the low temperature regime. Additionally, this study provides useful insights into the dissociation process in holographic QCD models.

Figures

Figures reproduced from arXiv: 2505.09866 by the authors.

**Figure 2.** The effective potential (10) for the scalar glueball as a function of the tortoise coordinate r∗ at various temperatures [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** The effective potential (20) is shown as a function of the tortoise coordinate r∗ at T = 20 MeV . The red dashed line denotes the real part of the frequency squared, with ωR = 2.82029 GeV. According to [38, 39], the next step involves matching the WKB wave functions across the different regions. However, special care must be taken when performing this matching due to the presence of an infinite barrier near the bou… view at source ↗

Figures from the paper (9 more)

**Figure 4.** Figure 4: The effective potential (20) as function of the tortoise coordinate r∗ at T = 73 MeV . The red dashed line indicates the value of the real part of the frequency squared, with ωR = 2.73944 GeV. The WKB wave solutions considered in this regime are given by:  …

**Figure 5.** Figure 5: The effective potential (20) as a function of the tortoise coordinate r∗ at T = 64 MeV . The red dashed line represents the real part of the frequency squared, with ωR = 4.0329 GeV. widths, often exceeding the spacing between neighboring states. Consequently, these sta…

**Figure 6.** Figure 6: The spectral function of the scalar glueballs at various temperatures. [PITH_FULL_IMAGE:figures/full_fig_p016_6.png]

**Figure 7.** Figure 7: The quasinormal frequencies for the ground state at [PITH_FULL_IMAGE:figures/full_fig_p020_7.png]

**Figure 8.** Figure 8: The quasinormal frequencies for the second mode at [PITH_FULL_IMAGE:figures/full_fig_p020_8.png]

**Figure 9.** Figure 9: The quasinormal frequencies for modes ranging from [PITH_FULL_IMAGE:figures/full_fig_p021_9.png]

**Figure 10.** Figure 10: The frequencies for the first three modes of the vector field in the soft wall model, computed via [PITH_FULL_IMAGE:figures/full_fig_p022_10.png]

**Figure 11.** Figure 11: The holographic potential (10) as function of the tortoise coordinate r∗ at temperature T¯ = 0.02 for the scalar glueball. The holographic potential is calculated using the same procedure described for the vector case above, following the formula (40). However, the di…

**Figure 12.** Figure 12: The holographic potential for the scalar field at zero temperature as a function of the holographic [PITH_FULL_IMAGE:figures/full_fig_p029_12.png]

Discussion (0). Continue with ORCID to comment.

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Results For comparison, we compute the quasinormal frequencies using the shooting method [42] (see Appendix A for details). The discrepancy between the two approaches is quantified by defining the deviation in the real part of the frequency as follows: ∆R = ωR−ωsm R ωsm R × 100% (24) 10 and similarly for the imaginary part: ∆I = ωI−ωsm I ωsm I × 100% (25)...

work page
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Higher order corrections Although the WKB approximation shows good agreement with the shooting method, the results can still be improved by incorporating corrections to the approximation. Let us point out that the Langer transformation introduces corrections to the wave function near the boundary, which 11 n = 2 Shooting Method 0th Order – BS Rule Deviati...

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(D1) We plot in Fig

Scalar Field At zero temperature, the holographic potential (10) for the scalar field takes the following form: V (z) = 2c2 + 15 4z2 +c4z2. (D1) We plot in Fig. 12 the holographic potential as a function of the holographic coordinate. The 0 5 10 15 0 50 100 150 200 z(GeV) V(z)(GeV 2) Figure 12: The holographic potential for the scalar field at zero temper...

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Vector Field As in the case of scalar field we can use the BS quantization rule: Z z1 z0 p m2n−VL(z)dz =π n + 1 2 . (D8) Then, by considering the Langer potential in the case of the vector field, VL(z) = 1 z2 +c4z2, (D9) with the turning points given by z0 = q m2n− p −4c4 +m4n √ 2c2 , z1 = q m2n + p −4c4 +m4n √ 2c2 , (D10) and by solving Eq. (D8), one fin...

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Pith tools

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

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Results For comparison, we compute the quasinormal frequencies using the shooting method [42] (see Appendix A for details). The discrepancy between the two approaches is quantified by defining the deviation in the real part of the frequency as follows: ∆R = ωR−ωsm R ωsm R × 100% (24) 10 and similarly for the imaginary part: ∆I = ωI−ωsm I ωsm I × 100% (25)...

work page

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Higher order corrections Although the WKB approximation shows good agreement with the shooting method, the results can still be improved by incorporating corrections to the approximation. Let us point out that the Langer transformation introduces corrections to the wave function near the boundary, which 11 n = 2 Shooting Method 0th Order – BS Rule Deviati...

work page

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(D1) We plot in Fig

Scalar Field At zero temperature, the holographic potential (10) for the scalar field takes the following form: V (z) = 2c2 + 15 4z2 +c4z2. (D1) We plot in Fig. 12 the holographic potential as a function of the holographic coordinate. The 0 5 10 15 0 50 100 150 200 z(GeV) V(z)(GeV 2) Figure 12: The holographic potential for the scalar field at zero temper...

work page

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Vector Field As in the case of scalar field we can use the BS quantization rule: Z z1 z0 p m2n−VL(z)dz =π n + 1 2 . (D8) Then, by considering the Langer potential in the case of the vector field, VL(z) = 1 z2 +c4z2, (D9) with the turning points given by z0 = q m2n− p −4c4 +m4n √ 2c2 , z1 = q m2n + p −4c4 +m4n √ 2c2 , (D10) and by solving Eq. (D8), one fin...

work page

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H.-P. Nollert, Class. Quant. Grav.16, R159 (1999). 34 n=0 Shooting Method Numerical BS rule Analytic BS rule T(MeV) ωR (GeV) ωR (GeV) ωR (GeV) 20 1.99988 1.99988 1.99988 25 1.99970 1.99970 1.99970 30 1.99937 1.99937 1.99937 35 1.99882 1.99882 1.99883 Table XIII: The real part of the frequencies for the ground state of the vector meson particle withc = 1, ...

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