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REVIEW 3 major objections 5 minor 23 references

Center-symmetric Landau gauge, the deconfinement transition and the gluon propagator as seen in lattice QCD

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper reports lattice evidence that, in the center-symmetric Landau gauge, the normalized link average and the difference between color-3 and color-8 gluon propagators serve as order parameters for the deconfinement transition.

desk verdict First lattice data for the center-symmetric Landau gauge gluon propagator: promising, honest, but missing Gribov-copy checks and propagator error bars. read the letter →

arxiv 2505.16940 v1 pith:XDZOWGAA submitted 2025-05-22 hep-lat hep-phhep-th

Duifje Maria van Egmond , Orlando Oliveira , Urko Reinosa , Julien Serreau , Paulo J. Silva , Matthieu Tissier This is my paper · ORCID
classification hep-lathep-phhep-th MSC 81T2581V05 PACS 11.15.Ha12.38.Gc
keywords center-symmetricLandaugaugedeconfinementtransitiongluonpropagatorPolyakovlooplatticeQCDcentersymmetrylinkaverageSU(3)theory
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

The paper reports the first lattice computation of the gluon propagator in the center-symmetric Landau gauge and argues that two gauge-fixed observables act as order parameters for the deconfinement transition in pure SU(3) gauge theory. Below the critical temperature, the average temporal link, normalized by the cube root of its determinant, sits at the center-symmetric value and the color-3 and color-8 gluon propagators coincide. Above $T_c$, the link average moves away from that value, configurations cluster into three $\mathbb{Z}_3$-related sectors, and $D^{33}$ separates from $D^{88}$. The numerical agreement with the analytical predictions in the confined phase is the main evidence, and the same observables are proposed as practical probes of center-symmetry breaking on the lattice.

What carries the argument

The object that carries the argument is the gauge-fixing functional $F=\sum_{x,\mu}\operatorname{Re}\operatorname{Tr}[U_0^\dagger(\mu)U_\mu(x)]$, where $U_0^\dagger(\mu)$ encodes the center-symmetric background $e^{i a g \bar{A}_{0,\mu}}$ with $\bar{A}_{0,\mu}=(T/g)(4\pi/3)t_7\delta_{\mu 0}$. Maximizing $F$ over gauge orbits selects a representative field configuration in each of the three $\mathbb{Z}_3$ sectors, and $F$ is invariant under the particular center transformation that rotates the Polyakov loop by $2\pi/3$. The two proposed order parameters are the normalized link average and the color-resolved gluon propagator difference $D^{33}_L-D^{88}_L$; both are predicted analytically to take the center-symmetric value in the confined phase and to deviate in the deconfined phase, and the lattice data are compared with those predictions.

What would settle it

Perform the same gauge fixing on identical ensembles with many random initial gauge seeds: if the normalized link average or the $D^{33}_L-D^{88}_L$ difference changes between distinct local maxima of $F$ within the same $\mathbb{Z}_3$ sector by more than the quoted statistical errors, the claimed order-parameter behavior is an artifact of the chosen extremum. A cleaner test is to bracket $T_c$ with several $N_t$ values: if the jump in these observables does not occur at the temperature where the Polyakov loop susceptibility peaks, the identification with the deconfinement transition fails.

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Extended reading notes

Core claim

The central claim is that a lattice version of the center-symmetric Landau gauge, defined by maximizing $\sum_{x,\mu}\operatorname{Re}\operatorname{Tr}[U_0^\dagger(\mu)U_\mu(x)]$ with the background phase $U_0^\dagger(\mu)=\exp(i a g \bar{A}_{0,\mu}\delta_{\mu 4})$, can be fixed in a way that makes the deconfinement transition visible directly in gluonic two-point data. The authors compute the link average $\langle U_4\rangle/(\det\langle U_4\rangle)^{1/3}$ and the longitudinal gluon propagator $D_L(p^2)$ for color indices 3 and 8 on $64^3\times N_t$ lattices at $\beta=6.0$. For $T=243$ MeV (below $T_c$) the link average matches the center-symmetric prediction and $D^{33}=D^{88}$ within errors; for $T=324$ MeV (above $T_c$) the link average deviates and $D^{33}$ and $D^{88}$ split. This is presented as evidence that both quantities can serve as order parameters for the deconfinement phase transition.

Load-bearing premise

The gauge-fixing algorithm is assumed to land on a representative maximum of the gauge-fixing functional, yet no check is made that different starting points give the same link average and propagators; if the result depends on which maximum is chosen, the order-parameter interpretation above $T_c$ is not robust.

Editorial extensions

If this is right

  • The normalized link average and the $D^{33}_L-D^{88}_L$ difference can be computed in ordinary lattice simulations and function as order parameters without relying on the Polyakov loop.
  • Below $T_c$, matching $D^{33}=D^{88}$ and the link-average value against analytic predictions confirms the center-symmetric background picture; above $T_c$, their deviation is a signal of $\mathbb{Z}_3$ breaking.
  • The gluon propagator can be decomposed into longitudinal and transverse parts with the same procedure as standard Landau gauge, making these observables cheap to add to existing analyses.
  • The three-sector clustering of $(A_4^3,A_4^8)$ above $T_c$ gives a per-configuration view of which $\mathbb{Z}_3$ sector is selected.

Reading between the lines

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

  • Beyond the paper: if the same two observables stay well-defined with dynamical quarks, they could serve as diagnostics of the deconfinement crossover, but the explicit breaking of center symmetry by quarks would likely smear the sharp jump seen here.
  • Beyond the paper: scanning $\beta$ at fixed $N_t$ to bracket $T_c$ more finely would turn the link-average jump into a quantitative determination of $T_c$; the two temperatures reported here indicate but do not establish this.
  • Beyond the paper: the lack of a Gribov-copy check means the clearest next test is seed dependence; if different local maxima give different order-parameter values, a selection rule (such as the global maximum) must be specified.
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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

3 major / 5 minor

Summary. The manuscript reports a lattice implementation of the center-symmetric Landau gauge, defined by maximizing the gauge-fixing functional (5) built from a center-symmetric background field. Using Wilson gauge action ensembles at beta=6.0 on a 64^3 x 8 lattice at T=243 MeV (below the deconfinement temperature) and on a 64^3 x 6 lattice at T=324 MeV (above), the authors compare the link average with analytical predictions from Refs. [13,14] and compute the bare longitudinal gluon propagator for color components 3 and 8. They find that below T_c the link average matches the center-symmetric prediction and that D33 and D88 coincide, while above T_c the link average deviates and D33 and D88 separate. On this basis they conclude in Section 6 that the link average and the difference of the color components of the gluon propagator can serve as order parameters for the deconfinement transition.

Significance. If established, the proposed observables would provide local, gauge-fixed order parameters for the SU(3) deconfinement transition and would strengthen the connection between the center-symmetric Landau gauge formalism and lattice data. The paper has concrete strengths: the analytical predictions from Refs. [13,14] are parameter-free, the lattice data are an independent numerical test of those predictions, the sector-rotation consistency check in Figures 1 and 2 is a useful internal cross-check, and the numerical values in Eqs. (12)-(14) agree with the analytic prediction to about 0.1% below T_c. However, the evidence is preliminary in ways that matter for the central claim: there are only two temperatures and one spatial volume, no continuum extrapolation is attempted, the propagator data in Figure 4 are shown without error bars, and no Gribov-copy analysis is provided for the non-convex gauge-fixing functional. These limitations are partly acknowledged by the authors' characterization of the work as the 'first steps' of a program, but they are not fully reflected in the strength of the wording in Section 6, where the order-parameter interpretation is stated without qualification.

major comments (3)
  1. [Section 2 and Section 5] The gauge-fixing functional (5) is non-convex, and the Fourier-accelerated steepest descent method described in the last paragraph of Section 2 selects one local maximum without any Gribov-copy control. Since both the link average (Section 4) and the D33/D88 propagator comparison (Figure 4) are evaluated on gauge-fixed configurations, the close agreement below T_c in Eqs. (12)-(14) may partly reflect the bias of the functional toward the center-symmetric background rather than an independent physical signal, and the D33/D88 splitting above T_c could be influenced by sector-dependent stationary points. The authors should provide a Gribov-copy analysis—for instance, comparing observables obtained from multiple random gauge starts or from several distinct local maxima—or otherwise demonstrate that the reported results are insensitive to the choice of extremum.
  2. [Section 5, Figure 4] The propagator plots in Figure 4 show data points without statistical error bars, yet the central claim that D33=D88 below T_c and that D33 separates from D88 above T_c is made from a visual comparison. Without errors, the reader cannot assess the statistical significance of the splitting, and with only two temperatures and one lattice volume the evidence is too thin to support an order-parameter claim. The authors should report jackknife or bootstrap errors on the propagator points and, ideally, add results at additional volumes or temperatures to show that the effect survives beyond a single ensemble.
  3. [Section 4] The link-average evidence is based on exactly two ensembles: 64^3 x 8 at T=243 MeV and 64^3 x 6 at T=324 MeV. The Monte Carlo history in Figure 3, which is used to argue that the link average tracks the confinement-deconfinement transition, comes from a different simulation taken from Ref. [2], not from the present setup. Finite-volume effects can mix center sectors, as Figure 3 itself shows, and this could bias the averaged link values in an uncontrolled way. The authors should either perform a volume-dependence study or explicitly discuss the expected size of finite-volume corrections before concluding that the link average is an order parameter.
minor comments (5)
  1. [Equation (6)] The definition of u0^dagger(mu) in Eq. (6) is written in a compressed notation; please spell out the exponent explicitly in terms of the lattice spacing a, the temperature T = 1/(a N_t), and the generators, so that the lattice implementation is unambiguous.
  2. [Figure 4] The axis annotations 'aa=3' and 'aa=8' should read 'a=3' and 'a=8' or be replaced by an explicit statement that these are color indices; the vertical axis label D_L(p^2) should also state whether the plotted quantity is the bare or renormalized propagator and in which units.
  3. [Equation (11)] Since det<U4> is 1 for an SU(3) matrix, the denominator in Eq. (11) is trivial; please clarify the intended normalization and define all symbols used in the displayed expression.
  4. [References] References [19]-[23] contain garbled or incomplete DOI strings; please update them with the correct identifiers or remove the broken URL fragments.
  5. [Section 6] The conclusion that the link average and the D33/D88 difference are order parameters is stated more strongly than the preliminary two-temperature, single-volume, no-error-bar data warrant; consider wording such as 'candidate order parameters' or explicitly list the required checks in the concluding paragraph.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the lattice data provide an independent numerical test of parameter-free analytical predictions from prior work; self-citations are minor and not load-bearing.

full rationale

The paper's claimed chain is: (i) define a center-symmetric Landau gauge through the background field (Eqs. 3-6); (ii) measure the link average and the gluon propagator in that gauge (Secs. 4-5); (iii) compare with analytical results from earlier work by the same group (refs. [13,14]) and conclude that these quantities are order parameters for deconfinement (Sec. 6). I find no circular reduction that would make the comparison vacuous. The analytical predictions, such as ⟨g A4^3⟩=4π/3 and D33=D88 in the symmetric phase, are parameter-free results of the previous continuum studies; they are not obtained by fitting the present lattice data. The gauge-fixing functional (5) is indeed constructed with the center-symmetric background phase (Eq. 6), so the link average below Tc is partly anchored toward the value quoted as the 'theoretical prediction' (Eq. 11). But the maximization is not a definitional identification: the shared gauge degrees of freedom and the spatial-link part of the functional compete with the temporal alignment, the agreement below Tc is within errors and not exact, and the deviation observed above Tc and the D33/D88 splitting are not imposed by the gauge condition. The absence of a Gribov-copy analysis (end of Sec. 2) is a genuine limitation for the order-parameter interpretation, because uncontrolled local maxima could bias the gauge-fixed observables; however, that is a robustness/correctness concern, not a circularity of the derivation. The self-citations [12-14,16] carry the analytical expectations, but they are external published derivations and the lattice simulation provides an independent check. Consequently the paper earns a low circularity score.

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

The central results rest on the discretization choice for the gauge condition, the invariance properties of the gauge-fixing functional, the analytic background-field predictions from the authors' prior work, and the scale setting that places the two ensembles on opposite sides of Tc. No numbers are fitted to the lattice data.

assumptions (5)
  • domain assumption The gauge-fixing functional (5) with R0(μ)=exp(i a A0bar δ_{μ4}) correctly implements the continuum center-symmetric Landau gauge condition (4).
    Stated in Section 2 without proof of equivalence; all lattice results depend on this discretization.
  • domain assumption The center transformation (9) leaves the gauge-fixing functional invariant, so the three center sectors are gauge-equivalent.
    Section 3 states this invariance; it is used to classify configurations into sectors and to rotate data back to sector 0.
  • domain assumption The Fourier-accelerated steepest descent method finds the intended maximum of the gauge-fixing functional, and Gribov copies do not affect the observables.
    No Gribov-copy study is presented; this is the paper's weakest premise (Section 2, last paragraph).
  • domain assumption The analytical predictions of refs. [13,14] for the link average and for D33=D88 in the symmetric phase are correct.
    The paper relies on these prior results for its comparison and for the expected behavior of the propagator.
  • domain assumption The ensembles at β=6.0 with Nt=8 and Nt=6 are respectively below and above the deconfinement temperature.
    The temperature identification uses standard scale setting; the paper does not independently determine Tc on these lattices.

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Cite this review

Pith. "Pith review of Center-symmetric Landau gauge, the deconfinement transition and the gluon propagator as seen in lattice QCD." pith.science (2026). https://pith.science/paper/XDZOWGAA

@misc{pith2026250516940,
  author       = {Pith},
  title        = {Pith review of: Center-symmetric Landau gauge, the deconfinement transition and the gluon propagator as seen in lattice QCD},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XDZOWGAA}},
  note         = {Machine review of arXiv:2505.16940}
}
read the original abstract

We address the lattice computation of the gluon propagator in the center-symmetric Landau gauge. After discussing a proper lattice implementation of the center-symmetric Landau gauge, we compare the lattice data with analytical results, and we identify various signatures of center symmetry breaking.

Figures

Figures reproduced from arXiv: 2505.16940 by the authors.

Figure 1
Figure 1. Plotting ( 3 4 (0), 8 4 (0)) — below : 643 × 8, T=243 MeV. -2 -1,5 -1 -0,5 0 0,5 1 1,5 2 A 3 (x=0) -2 -1 0 1 2 A 8(x=0) sector 0 sector 1 sector 2 (a) Original data. -2 -1,5 -1 -0,5 0 0,5 1 1,5 2 A 3 (x=0) -2 -1,5 -1 -0,5 0 0,5 1 1,5 2 A 8(x=0) sector 0 sector 1 sector 2 After rotation (b) Rotated data [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Plotting ( 3 4 (0), 8 4 (0)) — above : 643 × 6, T=324 MeV. In the continuum, there is a prediction for the gluon field such that h3 4 ()i = 4 3 , see [13, 14], which becomes h3 4 ()i = 4 3 . This can also be studied through the link average: h4 ()i (det h4()i)1/3 = − [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Monte Carlo history of a lattice simulation close to . whereas non-diagonal elements are zero within errors. The theoretical prediction for the first element is 0.9659258263 − 0.2588190451, so the numerical values are pretty close to it. Charge conjugation imposes that the second diagonal element is the complex conjugate of the first element, while we find the third element to be close to 1. So the numerical simulat… view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Bare gluon propagator for color indices 3 (black) and 8 (red), below (left) and above (right), for the various center sectors. [3] O. Oliveira and P. J. Silva, Eur. Phys. J. C 79 (2019) no.9, 793. [4] V. Paiva, P. J. Silva and O. Oliveira, JHEP 05 (2024), 164. [5] A. C…

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Reference graph

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