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REVIEW 4 major objections 5 minor 38 references

The Coulomb flux tube revisited

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

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Pith's one-line read At finer lattice spacings, the Coulomb flux tube's transverse profile develops a power-law tail with exponent near 2, not the exponential decay seen on coarser lattices.

desk verdict New lattice data with a plausible but not yet demonstrated beta-dependence; the power-law tail claim needs a longer lever arm and a continuum extrapolation. read the letter →

arxiv 1908.08874 v1 pith:FYD3CSNP submitted 2019-08-23 hep-lat

Sebastian M. Dawid , Adam P. Szczepaniak This is my paper · ORCID
classification hep-lat
keywords CoulombgaugefluxtubechromoelectricfieldlatticeYang-Millspower-lawfalloffSU(2)theorypotentialsimulation
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 tries to settle a disagreement about the shape of the chromoelectric flux tube that forms between a static quark and antiquark in the Coulomb gauge of Yang-Mills theory. A previous lattice simulation reported that this 'bare' flux tube decays exponentially with distance from the quark–antiquark axis. The authors repeat that measurement at higher statistics and extend it to two additional lattice spacings, finding that as the lattice spacing shrinks (β = 2.3 to 2.7), the transverse profile is described increasingly well by a power law with fitted exponent b ≈ 2. This matches an analytic prediction for the large-distance falloff, and the authors conclude it is likely that a genuine Coulomb flux tube with power-law tail develops in the continuum limit. A correct resolution matters because it determines whether the bare Coulomb state is a localized string or has long-range Coulombic structure, and whether the earlier exponential result was a coarse-lattice artifact.

What carries the argument

The central object is the lattice observable $Q_T(R,y)$, defined as a normalized correlation between two temporal Wilson lines of length $T$ and a plaquette probing the longitudinal chromoelectric field at transverse distance $y$ from the axis; $T=1$ gives the bare-state quantity of the earlier study. The argument runs through a comparison of two fitted models: the power-law shape $Q_{\mathrm{PL}}(R,y) = 16aR^2/(R^2+4y^2)^b$, where $b$ is the exponent of interest, and the exponential shape $Q_{\mathrm{CG}}(R,y) = \exp(-A-By)$. A third exponential-with-flattening model (CCB) is used for the coarsest lattice. The diagnostic is the large-$y$ behavior: exponential models fall faster than any power law, so the fitted exponent $b\approx 2$ at $\beta=2.7$, in agreement with the analytic prediction, is taken as evidence that the continuum profile is a power law.

What would settle it

A measurement of Q_{T=1}(R,y) at a finer lattice spacing (e.g., β = 2.9 or larger) that extends the transverse region to y*a ≳ 0.5 fm and shows the profile declining faster than any power law, or a fit whose exponent b moves away from 2 rather than toward it, would falsify the power-law claim. Equivalently, a continuum extrapolation of the fitted exponent from β = 2.5 and 2.7 that does not converge to b = 2 would do so.

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Extended reading notes

Core claim

The central claim is that the normalized correlation $Q_{T=1}(R,y)$, which measures the longitudinal chromoelectric field component in the bare Coulomb state, changes its transverse falloff with the lattice coupling. At β = 2.3 the profile is well described by an exponential model for large quark separations; at β = 2.5 a power law with exponent $b \approx 2$ fits better than the exponential but with poor quality at large $R$; and at β = 2.7 the power law with $b \approx 2$ is clearly preferred for all $R$. The fitted exponent decreases from about 2.3–2.8 at small $R$ to about 2.1–2.2 at large $R$, converging to the value $b = 2$ predicted in the literature. The authors state that with increasing β it is likely that a genuine Coulomb flux tube with power-law falloff develops, and that the exponential result of the earlier single-β study was a finite-spacing effect.

Load-bearing premise

The load-bearing premise is that the transverse distances probed at β = 2.7 (y = 1 to 8 lattice units, i.e., 0.045 to 0.36 fm) are large enough to reveal the asymptotic power-law falloff, and that finite-volume and gauge-fixing systematics, which are not quantified, do not alter the qualitative comparison between exponential and power-law models.

Editorial extensions

If this is right

  • If the power-law tail is genuine, the bare Coulomb flux tube has no finite width: its field extends to arbitrarily large transverse distances, consistent with the long-range nature of the Coulomb interaction.
  • The earlier exponential result at β = 2.5 would be reinterpreted as a lattice-spacing artifact rather than a property of the continuum theory.
  • The convergence of the fitted exponent toward b = 2 with decreasing lattice spacing supports the analytic prediction that the bare Coulomb state is not the same as the minimal Wilson state, and that the difference becomes visible only as the continuum is approached.
  • The observed insensitivity of the profile shape to the Euclidean time T suggests that the bare state's flux-tube structure is already present at short times, and that larger volumes and finer lattices are needed to resolve its asymptotic form.

Reading between the lines

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

  • One could test the power-law claim further by simulating at β ≈ 2.9 on a larger volume, where transverse distances beyond 0.5 fm become accessible; the fitted exponent should move toward 2 if the continuum tail is genuine.
  • The same large-y analysis could be applied to other components of the energy density, such as the action density or the magnetic field, to check whether the power-law tail is a universal feature of the Coulomb flux tube or specific to the longitudinal electric field.
  • A dedicated study of the near-axis region (small y), where the simple models fail, could separate a non-perturbative core from the asymptotic tail and reduce the fitting ambiguity that currently forces the exclusion of y = 0, 1, and sometimes 2.
  • If the continuum Coulomb flux tube is indeed power-law, the 'string' picture of confining tubes is modified: the Coulomb part of the potential would receive contributions from transverse separations that grow with R, which might have observable consequences for the inter-quark potential at large distances.
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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 paper reports lattice measurements of the transverse profile Q_T=1(R,y) of the longitudinal chromoelectric field in Coulomb-gauge SU(2) Yang-Mills theory at beta=2.3, 2.5, and 2.7 on 32^4 lattices, extending the earlier work of Ref. [30] to larger statistics and additional couplings. The authors fit the profile with a power law Q_PL(R,y)=16aR^2/(R^2+4y^2)^b, an exponential Q_CG=exp(-A-By), and a modified exponential Q_CCB, and find that at beta=2.5 and 2.7 the large-y data favor a power law with fitted exponent b close to 2, in agreement with the analytic prediction of Ref. [8], while at beta=2.3 the profile is better described by an exponential. They also study the Euclidean-time dependence of Q_T and observe that the functional form does not visibly change with T. They conclude that, with increasing beta, a power-law Coulomb flux tube likely develops and that the profile may evolve from a Wilson-like to a Coulomb-like shape in the continuum limit, while cautioning that the small-y behavior is not reproduced by the power-law model. No continuum extrapolation is attempted, and finite-volume and gauge-fixing systematics are assumed not to affect the conclusion.

Significance. If correct, the result would overturn the exponential picture of the bare Coulomb flux tube reported in Ref. [30] and would provide lattice support for long-range Coulomb-gauge interactions and for the analytic prediction b about 2 of Ref. [8]. The paper's strengths are its transparent presentation: the complete Q_T=1 data set is shown in the appendix, fit parameters and chi2/dof values are tabulated for all R, jackknife errors are used, and the authors explicitly flag the fit-interval sensitivity and the lack of systematic-error estimates. These strengths make the measurement reproducible and the analysis auditable. The significance is, however, moderated by the fact that the central continuum statement rests on a short transverse lever arm (0.36 fm at beta=2.7) and on fits that require exclusion of small-y points; the result is therefore best read as a strong motivation for a dedicated continuum study rather than as an established continuum-limit result.

major comments (4)
  1. [Sec. III.A, Table IV] The beta=2.7 power-law claim relies on a very short lever arm: with a=0.045 fm the fit interval y in [1,8] covers only 0.045-0.36 fm, which is of the same order as the fitted exponential scales at beta=2.3 (Table III gives lambda about 1.3-2.1 lattice units, i.e. about 0.22-0.35 fm). The data therefore do not expose the asymptotic transverse tail, and a slowly decaying exponential or a transient cannot be excluded; a continuum-limit claim needs y_perp values well beyond the intrinsic scale.
  2. [Sec. III.A, Tables II and IV, Appendix A] The fitted exponent b is not robust against the choice of fit interval. Table IV states that b depends significantly on y_min and approaches 2 only as the low-y points are excluded, and at beta=2.5 the PL fits for R=5-8 have chi2/dof=10, 13, 19, 15 (Table II) despite the exclusion of y=0,1, and sometimes 2, admitted in Appendix A. This means the nominal b about 2 is controlled by a few large-y points; the paper should provide a y_min scan for fixed R and show that the power law is stable before claiming agreement with Ref. [8].
  3. [Sec. II.A] The assumption that finite-volume and gauge-fixing effects do not affect the main conclusion is load-bearing because the paper's conclusion is a statement about the continuum limit. The study uses one volume (32^4), one gauge-fixing criterion, and only three beta values, with no continuum extrapolation; the observed beta-dependence could include discretization or volume effects. At minimum the authors should estimate these systematics, for example with a second volume at beta=2.7 and a stricter gauge-fixing consistency check, and ideally add a fourth beta value or perform a continuum extrapolation of the fitted exponent b.
  4. [Sec. III.A, Fig. 4 and Table IV] At beta=2.7 the fitted exponent is strongly R-dependent for small separations (b=2.75(5) at R=1, 2.46(3) at R=2, decreasing to 2.09(3) at R=8), whereas the cited prediction of Ref. [8] is b about 2 independent of R. The text's statement that the profile falls as 1/y^4 in agreement with Ref. [8] is therefore only true at the largest R; the R-dependence of b needs to be addressed before the agreement is claimed.
minor comments (5)
  1. [Sec. IV] The sentence beginning with "with increasing beta the genuine, it is likely..." is ungrammatical, and "concussion" should be "conclusion"; please rewrite this passage.
  2. [Figure captions 9-11] "sperations" should be "separations" in the captions of Figs. 9, 10, and 11.
  3. [Sec. III.A and Fig. 3] The vertical-axis label in Fig. 3 is beta Q(R,y) while the text and fits use Q_T=1(R,y) and Q(R,y); please make the notation consistent and state which quantity is plotted.
  4. [Fig. 11 caption] The caption notes that the point (R=5, y=8, T=4) is negative but the text does not discuss this; because Q_T is a normalized expectation value, a negative value should be commented on, whether it is a statistical fluctuation, a subtraction artifact, or a finite-volume effect.
  5. [Sec. III.A, Eq. (8)] The sign of the -2 nu/lambda term in the CCB model is only explained in a footnote; please make the model definition self-contained and clarify the relation to the corresponding expression in Ref. [27].

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the lattice fits are compared against an independent analytic prediction; interval choices are a data-analysis caveat, not a constructed equivalence.

full rationale

The paper's derivation chain is an empirical lattice measurement: define the correlation Q_T(R,y) in Eq. (3), compute it at beta = 2.3, 2.5, 2.7, fit competing models (CG, CCB, PL), and compare the fitted large-y power b with the analytic prediction b ~ 2 of Ref. [8]. No fitted parameter is defined in terms of the target conclusion. The PL form in Eq. (6) is an ansatz with free exponent b; the value b ~ 2 is an output of the fit, not imposed. The use of Ref. [8] is a genuine external benchmark: it is an analytic Dyson-Schwinger prediction made independently of these lattice data, so citing it is self-citation but not load-bearing circularity under rule 4. The admitted sensitivity of b to the fit interval (Tabs. II and IV) and the removal of small-y points (App. A) weaken the asymptotic claim and are correctness risks, but they are explicitly disclosed and do not make the comparison equivalent to its input. The absence of a continuum extrapolation and the Sec. II.A assumption that finite-volume and gauge-fixing errors do not affect the conclusion are also caveats, not circular reductions. Accordingly, the central result is self-contained against an independent theoretical prediction, and no circular step is exhibited.

Assumptions & free parameters 5 free parameters · 7 assumptions · 0 invented entities

The analysis rests on standard lattice QCD assumptions plus one explicit no-systematic-error assumption. The central numerical evidence comes from fits with several free parameters, and the claimed agreement with b=2 is a fitted exponent rather than a parameter-free prediction.

free parameters (5)
  • PL amplitude a and exponent b (Eq. 6) = Tables II, IV: e.g. b=2.114(4) at beta=2.5,R=2; b=2.75(5) at beta=2.7,R=1
    Fitted to Q_T=1(y) data; the b approximately 2 result is the main evidence for power-law falloff, not a parameter-free prediction.
  • CG exponential parameters A, B (Eq. 7) = Table I: e.g. A=6.58(2), B=0.554(8) at beta=2.5,R=7
    Fitted to test the exponential model from Ref. [30].
  • CCB parameters nu, lambda (Eq. 8) = Table III: e.g. nu=2.587(4), lambda=1.973(3) at beta=2.3,R=7
    Fitted to describe beta=2.3 exponential-like profiles.
  • Coulomb string tension sigma_C at beta=2.7 = 0.0409(3) in lattice units
    Extracted from fit V_C(R)=sigma_C R + beta/R + gamma; used in the theoretical prediction for Q(y=0).
  • Fit interval lower cutoff y_min = 1, 2, or 3 depending on R and model
    Points y=0,1 and sometimes y=2 were excluded to obtain acceptable chi2; the choice affects fitted exponents.
assumptions (7)
  • domain assumption Wilson action and heat-bath algorithm sample the correct SU(2) Yang-Mills ensemble.
    Standard lattice QCD setup, cited to [33,35]; no independent verification in this paper.
  • domain assumption Coulomb gauge fixing with Delta F < 10^-7 is sufficient and Gribov copies do not affect the result.
    Stated in Sec. II.A; no Gribov-copy analysis is performed.
  • domain assumption Q_T defined in Eq. (3) measures the x-component of the longitudinal chromoelectric field in the bare QQbar state.
    Inherited from Ref. [30] and Ref. [6]; this interpretation is central to the observable.
  • domain assumption Spatial links are close to identity in Coulomb gauge, so the Wilson loop can be approximated by two temporal Wilson lines.
    Used in Sec. III.B to connect Q_T to the minimal-energy flux tube.
  • ad hoc to paper Finite lattice volume and gauge fixing quality do not affect the main conclusion.
    Explicitly assumed in Sec. II.A without systematic study.
  • domain assumption Lattice spacings from Ref. [34] are correct for beta=2.3, 2.5, 2.7.
    Used to convert lattice units to fm.
  • domain assumption The asymptotic transverse falloff b=2 predicted in Ref. [8] is valid.
    External analytic prediction used as benchmark; not rederived here.

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Cite this review

Pith. "Pith review of The Coulomb flux tube revisited." pith.science (2026). https://pith.science/paper/FYD3CSNP

@misc{pith2026190808874,
  author       = {Pith},
  title        = {Pith review of: The Coulomb flux tube revisited},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FYD3CSNP}},
  note         = {Machine review of arXiv:1908.08874}
}
abstract

We perform $SU(2)$ Yang-Mills lattice simulation of the electric field distribution in the Coulomb gauge for different values of $\beta$ to further investigate the nature of the Coulomb flux tube.

Figures

Figures reproduced from arXiv: 1908.08874 by the authors.

Figure 1
Figure 1. Schematic picture of the arrangement of links which, [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Dependence of QT =1 on transverse distance y at fixed quark-antiquark separation R (in lattice units). Red, solid line is an exponential fit (CG), and green, dashed line is a power-law (PL) fit, both described in Sec. III A. Data points were obtained from 30 000 gauge-fixed lattice configurations at β = 2.5, for lattice volume V = 324 . 1 See Eq. (33) in Sec. III of Ref. [24]. profile changes from power-law to expon… view at source ↗
Figure 3
Figure 3. Dependence of βQT =1 on transverse distance y⊥ = ay at fixed quark-antiquark separation r = aR [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (7 more)
Figure 5
Figure 5. Figure 5: The log-log plot of QT =1 at the middle point y = 0 as a function of the quark-antiquark separation R. Green, dashed line is the Q = a/Rb fit, with parameters b = 3.4(1) and a = 0.20(2). The R = 1 point was excluded from the fit, because it does not satisfy µR 1 condit…
Figure 7
Figure 7. Figure 7: Dependence of QT (R, y) on transverse distance y at fixed quark-antiquark separation R for different T’s. Lines joining the points are included to guide the eye. Data points were obtained from around 11 000 gauge-fixed lattice config￾urations at β = 2.5, for lattice vo…
Figure 6
Figure 6. Figure 6: Dependence of the string tension on T. It was obtained following the procedure of Ref. [6], for β = 2.5 and around 1700 lattice configurations. The horizontal line is the asymptotic value from Ref. [24] IV. CONCLUSIONS In this paper we have extended the calculation of …
Figure 8
Figure 8. Figure 8: Results for QT =1(R, y) for different quark separations R, obtained for β = 2.3 and 11 000 lattice configurations. The red line is the best CCB fit to the data. It can be seen, how the results start agreeing with this model as quark and anti-quark are separated further…
Figure 9
Figure 9. Figure 9: Results for Q1(R, y) for different quark sperations R, obtained for β = 2.5 and 30 000 lattice configurations. The red, solid line is the best exponential (CG) fit to the data, performed following the procedure of Ref. [30] for R 6 5. The green, dashed line is the best…
Figure 10
Figure 10. Figure 10: Results for Q1(R, y) for different quark sperations R, obtained for β = 2.7 and around 11 000 lattice configurations. The green, dashed line is the best PL fit. We also present the CCB fit (red, solid line) for R = 7 and R = 8, performed in the interval y ∈ [2, 8] and…
Figure 11
Figure 11. Figure 11: Results for QT (R, y) for different quark sperations R and different T’s, obtained for β = 2.5 and around 11 000 lattice configurations. Lines joining the data points are included to guide the eye. The data point for R = 5, y = 8 and T = 4 was negative, so is not visi…

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