Studies on Polyakov and Nambu-Goto Random Surface Path Integrals on QCD (SU(infinity)): Interquark Potential and Phenomenological Scattering Amplitudes

Luiz C L Botelho

REVIEW 4 major objections 5 minor 12 references

Studies on Polyakov and Nambu-Goto Random Surface Path Integrals on QCD (SU(infinity)): Interquark Potential and Phenomenological Scattering Amplitudes

T0 review · 4 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Adding an extrinsic-curvature term to the Nambu-Goto string makes the interquark force diverge as separation grows.

desk verdict Genuinely new R ln R confining-potential claim, but the derivation rests on an unvalidated Epstein limit and a cutoff substitution; not established, though worth a referee's look. read the letter →

arxiv 1909.03807 v1 pith:GFROHQU2 submitted 2019-08-20 hep-th

Luiz C L Botelho This is my paper · ORCID

classification hep-th MSC 81T3081T1381V05 PACS 11.25.-w12.38.-t

keywords interquarkpotentialNambu-GotostringextrinsiccurvaturequarkconfinementPolyakovlarge-NQCDpathintegralWilsonloop

topics Yang-Mills Mass Gap

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 argues that a Nambu-Goto string (a bosonic string whose action is the surface area of the worldsheet) augmented by an extrinsic-curvature term (the square of the surface mean curvature) produces, in a one-loop computation on a rectangle, an interquark potential whose large-distance part contains a term proportional to $R \ln R$. Because the derivative of $R \ln R$ is $\ln R$, the force pulling a quark and an antiquark together grows without bound as their separation $R$ increases, which the author reads as genuine quark confinement rather than the constant-force confinement of the plain Nambu-Goto string. The same logarithmic-growth structure is claimed to reappear in Polyakov's non-critical string at large spacetime dimension and in a curved-background version, suggesting a common mechanism. If correct, the result would mean that a purely stringy description of large-$N$ QCD confines static color charges more strongly than the usual linear potential, and that point-particle QCD descriptions are only phenomenological at large distances.

What carries the argument

The workhorse is the extrinsic string action $S = \frac{1}{2\pi\alpha'}\int \sqrt{h}\,d^2\xi + \gamma^2\int \sqrt{h}(\Delta_h X^{\mu})^2\,d^2\xi$, evaluated at one loop on an $R\times T$ rectangle with Dirichlet boundary conditions. Expanding around the static classical string reduces the path integral to a Gaussian, and the fluctuation determinant factorizes as $\det\bigl(\gamma^2(-\partial^2)(\gamma^{-2}+(-\partial^2))\bigr)^{-(D-2)/2} = \det(-\partial^2)^{-(D-2)/2}\det(\gamma^{-2}+(-\partial^2))^{-(D-2)/2}$. The second, massive factor is evaluated using an Epstein-type lattice sum whose large-$R$ limit is treated distributionally in Appendix A; the parameter integral develops a $\ln E_{\rm QCD}$ divergence that is absorbed into a renormalized coupling $\gamma_{\rm ren}$, and the surviving term carries $R\ln R$. That logarithmic term is the mechanism that turns a constant string tension into an unbounded force.

What would settle it

Compute the exact functional determinant of $\gamma^2(-\partial^2)(\gamma^{-2}+(-\partial^2))$ with Dirichlet boundary conditions on an $R\times T$ rectangle, take $T\to\infty$ and then $R\to\infty$, and read off the coefficient of $R\ln R$ in $-\ln Z/T$; if it is not $\frac{D-2}{4\pi\gamma_{\rm ren}^4}$, the paper's central claim fails. A large-$N$ lattice measurement of the static quark force at large separation would also distinguish a $\ln R$ growth from the constant-force linear potential of the plain Nambu-Goto string.

Watch

Extended reading notes

Core claim

The central claim is eq. (41): for the extrinsic Nambu-Goto string, the one-loop vacuum energy of a static quark-antiquark pair gains the terms $$V_{\rm extrinsic}(R) = -\frac{D-2}{6}\frac{\pi}{R} + \frac{D-2}{2}\frac{1}{\gamma_{\rm ren}^2} - \frac{D-2}{4\pi}\frac{1}{\gamma_{\rm ren}^4} R\ln(4\pi $e^{{-\hat{\gamma}}$}) + \frac{D-2}{4\pi}\frac{1}{\gamma_{\rm ren}^4} R\ln R + \frac{D-2}{16\$pi^{{5/2}}$}\frac{1}{\gamma_{\rm ren}^2} R,$$ to be added to the pure Nambu-Goto linear term. The $R\ln R$ term makes the force between static color charges grow as $\ln R$ at large $R$, so the potential confines with a force that diverges as $R\to\infty$, which the author calls "real quark confinement" in contrast to the weak, constant-force confinement of the pure Nambu-Goto string. The same structural term appears in the Polyakov string in the large-$D$ limit, eq. (64), and in the Nambu-Goto string on a de Sitter-like background, eq. (72), indicating a common origin in the determinant of a fourth-order fluctuation operator. The paper also argues that adding a fourth-order two-dimensional-gravity term to Polyakov's Liouville action does not alter the tachyonic poles of closed-string amplitudes, and that a Nambu-Goto area functional solves the Migdal-Makeenko loop equation for the Wilson loop in $SU(\infty)$ QCD.

Load-bearing premise

The whole growing-force conclusion rests on one step: a divergent integral over an auxiliary parameter $x$ is stopped at a lower bound $E_{\rm QCD}$, and the logarithm this produces is absorbed into a redefined coupling constant. The paper gives no proof that this cutoff procedure, or the similar distributional limit used for the Epstein sum in Appendix A, is valid; if that step fails, the $R\ln R$ term and the growing force disappear.

Editorial extensions

If this is right

If eq. (41) is correct, the interquark force at large separation is $\frac{D-2}{4\pi\gamma_{\rm ren}^4}\ln R$, so confinement is stronger than the constant-force Nambu-Goto confinement and cannot be reproduced by a point-particle QCD potential.
The same $R\ln R$ structure in eqs. (64) and (72) means the growing-force signature is not an accident of the extrinsic Nambu-Goto action but reappears in Polyakov's non-critical string at large $D$ and in curved-background versions.
The extrinsic-curvature term leaves the closed-string scattering amplitudes structurally unchanged: the Virasoro-Shapiro/Veneziano poles survive, so the tachyon is not removed; the paper concludes that fermionic or supersymmetric degrees of freedom are needed for a physically sensible string.
The Nambu-Goto area functional is shown to solve the Migdal-Makeenko loop equation for $SU(\infty)$, tying the string tension to the large-$N$ QCD coupling and making string wave functionals candidate nonperturbative vacuum states of large-$N$ QCD.

Reading between the lines

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

If the cutoff procedure survives a rigorous treatment, the force law $F(R)=\sigma + c\ln R$ is a sharp, testable signature: large-$N$ lattice simulations could distinguish it from the constant-force linear potential by fitting the coefficient of $\ln R$ at large separation.
The same determinant technique should apply to other higher-order string actions: any action whose fluctuation determinant contains a massive factor with a mass set by a length scale will generically produce logarithmic corrections to the interquark potential, making $R\ln R$ a plausible universal feature of extrinsic-curvature string theories.
The relation between $\gamma_{\rm ren}$ and the QCD coupling suggests a renormalization-group picture in which the running of the extrinsic coupling with the cutoff $E_{\rm QCD}$ encodes dimensional transmutation, with the coefficient of $R\ln R$ determined by the beta function of the large-$N$ theory.
A direct numerical check is available: compute the exact vacuum energy of the fourth-order determinant on large rectangles and verify whether the large-$R$ coefficient of $R\ln R$ matches eq. (41); a mismatch would localize the error in the distributional Epstein limit.

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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

Here's my take: the paper contains a concrete new prediction—an R ln R term in the large-distance interquark potential from an extrinsic-curvature string—that would give a logarithmically growing confining force. If right, it distinguishes the model from Nambu-Goto. But the derivation does not survive close reading. The central steps in Appendix A and eq. (40) are asserted, not proved, and the stress-test note is accurate.

What it does well: it lays out the standard 1-loop determinant machinery for Nambu-Goto and Polyakov strings, and the structure of the three potentials (eqs. 41, 64, 72) is coherent. The claim that adding a fourth-order 2D gravity term to the Polyakov action does not eliminate the tachyon is plausible, and the paper's no-go conclusion aligns with earlier results. I also take the author's word that eqs. (41), (64), and (72) are not in the prior literature. So there is something here.

Where it falls apart: the new term comes from eq. (40), where a formally divergent integral is regulated by introducing an ad hoc lower cutoff E_QCD and absorbing ln(E_QCD) into gamma^2. No justification is given for the replacement, and the displayed algebra is internally inconsistent—the R^2 coefficient changes between the preceding line and the final expression, and no R^2 term survives in eq. (41). Appendix A's Epstein-function limit is explicitly called a 'formal result' and rests on a 'yet undiscovered asymptotic distributional theory'; the heat-kernel trace on C^2([0,1]) with Dirichlet boundary conditions uses eigenvalues n^2 rather than (n*pi)^2, which would change the numerical coefficient in eq. (39). Also, the Luscher coefficient in eq. (18) is a factor of two off the standard value, and the key identification of gamma^2 with the QCD(SU(infinity)) coupling is supported only by self-citations. These aren't cosmetic issues—they are exactly where the new physics enters.

Is the paper coherent? Yes, mostly. It is a review-like exposition with a bold conjecture attached. The author flags some steps as formal, which is honest, but honesty doesn't fix the math.

Bottom line: the R ln R prediction is not established. I would not cite it as a result. But it is a concrete, checkable claim, and a good referee could quickly confirm whether the heat-kernel/Epstein step is salvageable. If the paper lands on an editor's desk, I would send it out for review rather than desk-reject; the referee will likely reject it, but there is enough substance to justify the time.

Referee Report

4 major / 5 minor

Summary. The manuscript studies path-integral formulations of the Nambu-Goto and Polyakov bosonic strings and applies them to the interquark potential in large-N QCD. It computes one-loop determinants on rectangular worldsheets and derives vacuum energies that include a Lüscher-type term and, in the extrinsic-string model, an R ln R term in the potential, which the author claims produces a growing confining force at large separation. The paper also proposes a fourth-order two-dimensional gravity modification of the Polyakov string, evaluates scattering amplitudes in that model, and constructs area-functionals as solutions of the Migdal-Makeenko loop equation for SU(∞) QCD.

Significance. If the central calculation were correct, the claimed R ln R term in Eq. (41) would be a striking result: it would imply a stronger-than-linear confinement of static quarks in a string description of QCD. The paper also contains useful formal exercises, including exact determinant evaluations and an explicit Gaussian path-integral treatment of a fourth-order Polyakov model. However, the central claim is not established. The derivation depends on an unproved distributional limit of an Epstein function, on an ad hoc infrared cutoff in the Feynman-parameter integration, and on an internal algebraic inconsistency in Eq. (40). In addition, the baseline Lüscher coefficient in Eq. (18) differs by a factor of 2 from the standard literature value, which indicates an overall normalization problem in the determinant computations on which the main result relies. The paper's identification of the extrinsic string coupling with the QCD(SU∞) coupling is asserted rather than derived and is supported mainly by self-citations.

major comments (4)

[Section 3, Eqs. (38)-(40) and Appendix A] The central R ln R term in Eq. (41) rests on the 'formal' large-R replacement Eq. (39) for the Epstein sum, which Appendix A introduces as a 'yet undiscovered asymptotic distributional theory' and does not prove. More concretely, the heat-kernel trace in Eq. (A-1) is evaluated for the operator -d^2/dz^2 on C^2([0,1]) with Dirichlet conditions, whose eigenvalues are (nπ)^2, not n^2; using the stated eigenvalues changes the powers of π in Eqs. (38)-(39). Additionally, the large-a asymptotic is inserted before the x-integration in Eq. (40), although the asymptotic is not uniform near x=0, where the divergence is regulated by an ad hoc cutoff E_QCD. No proof is given that this cutoff procedure is well-defined or that it preserves the coefficient of the R ln R term.
[Eq. (40)] The displayed algebra in Eq. (40) is internally inconsistent: the penultimate line gives a coefficient proportional to R^2/(16π^{7/2}γ^4), while the final equality contains 1/(16π^{5/2}γ^2), and the resulting R^2 term disappears without explanation from the potential in Eq. (41). This makes it impossible to verify the claimed cancellation of the R^2 contribution and the final renormalized potential.
[Section 2, Eq. (18)] The Lüscher coefficient in Eq. (18) is -π(D-2)/(6R), which is a factor of 2 larger than the standard result -π(D-2)/(12R) for the static quark-antiquark potential in a Nambu-Goto string. Since Eq. (18) is the baseline for the later extrinsic-string computation, this discrepancy indicates a normalization error in the determinant evaluation that also affects the coefficients in Eq. (41).
[Note Added item 2 and Section 6] The identification of the extrinsic string coupling γ^2 with the QCD(SU∞) coupling is asserted in Note Added item 2 by referencing two papers, one by the author, while Section 6, Eq. (84), proposes a different relation g^2_∞(a^2)/a^2 = 1/(2πα′). Without a derivation connecting these identifications, the QCD interpretation of the R ln R term in Eq. (41) is not supported even if the formal string calculation were correct.

minor comments (5)

[Abstract and throughout] The abstract contains the typo 'news path integral studies'; it should read 'new path integral studies'. There are also repeated misspellings such as 'Mandelstan' for Mandelstam and 'Virassoro' for Virasoro.
[Section 5, Eqs. (60)-(64)] The text refers to a '1/D expansion' but uses D → -∞ in the large-R potential (Eq. (64)); the direction of the limit should be stated consistently.
[Eq. (65)] The symbol D is used both as the spacetime dimension and as a two-dimensional integration domain in Eq. (65), which is confusing.
[Eqs. (64) and (72)] The expressions contain the typo '16πs/2' where the context and Eq. (41) suggest '16π^{5/2}'.
[Eq. (34)] The determinant identity in Eq. (34) is not written clearly: the factors '-(D-2)/2 det' appear as multiplicative coefficients inside what should be a product of powers of determinants.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular reduction found: Eq. (41) follows from a standard determinant evaluation; self-citations are interpretive rather than load-bearing, while the main risk is an unproved (but non-circular) Epstein asymptotic.

full rationale

The central result, V_extrinsic(R) in Eq. (41), is obtained by evaluating the one-loop determinant in Eq. (33) with the massive-determinant formula of Eq. (35) and taking the T to infinity limit. The R ln R term is a direct algebraic consequence of the (R/(2 pi gamma^2))^2 ln(4 pi e^-gamma_hat / R) term already present in Eq. (35); it is not put in by hand or fitted to an assumed potential. Eq. (40)'s cutoff integral is presented as a renormalization of gamma^2, and although the Appendix A large-a Epstein asymptotics is explicitly flagged as a 'formal' result resting on a 'yet undiscovered asymptotic distributional theory,' this is a missing mathematical justification, not a circular equivalence: the potential is not an input to the calculation that is later re-identified as an output. The self-citations in Note Added item 2 and Appendix C are used for the QCD/string coupling identification and for the 1/D method, but the cited equivalence is also supported by the externally authored Karanikos-Ktorides paper, and neither identification is used to fit the R-dependence of Eq. (41). No parameter is fitted to the target potential, no uniqueness theorem is imported from the authors' prior work to force a choice, and no known result is merely renamed. The paper's central derivation is therefore self-contained; the score of 2 reflects only the unusually heavy, but non-load-bearing, reliance on the author's own previous publications for physical interpretation.

Assumptions & free parameters 7 free parameters · 6 assumptions · 1 invented entities

The central claim rests on several unproven or weakly justified assumptions: the validity of the 'distributional' Epstein-function limit, the legitimacy of the cutoff in eq. (40), the identification of the extrinsic coupling with the QCD coupling via self-cited references, and the replacement of a nonlocal fourth-order gravity term by a local one. These are cataloged in the ledger; the paper does not provide independent evidence for any of them.

free parameters (7)

alpha' (string tension)
The Regge slope is a model input, not fitted to data; it sets the QCD string tension if one accepts the QCD-string identification, but no value is given.
gamma^2 (bare extrinsic coupling)
Introduced in eq. (27) as the coefficient of the extrinsic curvature term; its value is not determined and it is later renormalized.
gamma^2_ren (renormalized extrinsic coupling)
Defined after absorbing the cutoff ln(E_QCD) in eq. (40); the paper does not specify how it would be fixed.
E_QCD (infrared cutoff on Feynman parameter x)
Ad hoc cutoff introduced in eq. (40) to render a divergent integral finite; the result depends logarithmically on it, and the dependence is absorbed into the coupling.
mu_R (cosmological constant / mass in Polyakov 1/D expansion)
Appears in eq. (62) and (64); a free parameter of the 2D gravity model.
M^2 (effective mass in curved-space version)
Defined in eq. (71) in terms of the cosmological constant Lambda; not fixed by data.
b (QCD coupling per area in Section 6) = b = g^2_infty(a)/a^2
Introduced in eq. (80) to make the area functional satisfy the loop equation; its value is tied to the large-N coupling, not derived.

assumptions (6)

domain assumption The 'weighted' Feynman measure dh_mu[X] preserves local diffeomorphism invariance of the worldsheet (eq. 5).
The measure is defined as a product over points with weight h^{1/4}; it is taken as the path-integral measure without proof of consistency.
ad hoc to paper Dimensional regularization sets the tadpole delta function delta^{(2)}(0) = 0 (Section 2).
Used to drop divergent tadpole terms in eqs. (22) and (25); this is a formal device not justified in the paper.
ad hoc to paper The 'distributional' large-a limit of the Epstein function in Appendix A (eqs. 38-39) is valid.
The interchange of sum, integral, and asymptotic limit is asserted without proof and is used to evaluate the divergent x-integral in eq. (40).
ad hoc to paper The equivalence between the extrinsic string coupling and the QCD(SU∞) coupling holds via refs. [1] and the author's own previous papers.
Stated in Note Added item 2; this identification is needed to interpret the derived potential as the QCD interquark potential.
domain assumption The fourth-order 2D gravity term in the Polyakov path integral (eq. 43) can be replaced by the local term (∂²φ)² in the weak-field expansion (eqs. 47-48).
The replacement substitutes the Green's function by a delta function; this is a weak-coupling approximation whose validity is not discussed.
domain assumption The string surfaces in the evaluations have trivial topology (no handles), so the Gauss-Bonnet constraint can be ignored.
Stated in Section 2, where the topological constraint is disregarded for trivial topological surfaces.

invented entities (1)

Fourth-order 2D quantum gravity term in the Polyakov action
purpose: Proposed to improve the ultraviolet behavior of the covariant Polyakov string while remaining exactly solvable
Introduced in eq. (43) as a new term; the paper itself concludes it does not alter the tachionic spectrum, so it produces no falsifiable prediction.

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Cite this review

Pith. "Pith review of Studies on Polyakov and Nambu-Goto Random Surface Path Integrals on QCD (SU(infinity)): Interquark Potential and Phenomenological Scattering Amplitudes." pith.science (2026). https://pith.science/paper/GFROHQU2

@misc{pith2026190903807,
  author       = {Pith},
  title        = {Pith review of: Studies on Polyakov and Nambu-Goto Random Surface Path Integrals on QCD (SU(infinity)): Interquark Potential and Phenomenological Scattering Amplitudes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GFROHQU2}},
  note         = {Machine review of arXiv:1909.03807}
}

read the original abstract

We present news path integral studies on the Polyakov Non-Critical and Nambu-Goto critical string theories and its applications to QCD SU((infinity)) interquark potential .

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

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Botelho, Methods of Bosonic and Fermionic Path Integra ls, Nova Science Pub- lisher, Inc., 2009

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