The four-gluon and ghost-gluon vertices in the Landau gauge from lattice simulations

Manuel Cola\c{c}o; Nuno Brito; Orlando Oliveira; Paulo J. Silva

REVIEW 3 major objections 4 minor 12 references

The four-gluon and ghost-gluon vertices in the Landau gauge from lattice simulations

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper claims that in collinear kinematics the four-gluon vertex has a form factor $F^{(2)}$ that grows toward the infrared while $F^{(0)}$ stays constant, and that the soft-gluon ghost-gluon form factor agrees with earlier lattice and…

desk verdict A modest but honest lattice proceedings update on two Yang-Mills vertices; the main worry is an imprecise kinematic condition that could let disconnected diagrams in, and it should be tightened before the results are used as a reference. read the letter →

arxiv 2505.23476 v1 pith:QIHCLYIB submitted 2025-05-29 hep-lat

Nuno Brito , Manuel Colaço , Orlando Oliveira , Paulo J. Silva This is my paper

classification hep-lat PACS 12.38.Gc11.15.Ha

keywords four-gluonvertexghost-gluonLandaugaugelatticeQCDcollinearkinematicssoftgluonlimitinfraredbehaviorYang-Millstheory

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 reports lattice measurements of two fundamental vertices of pure Yang-Mills theory in the Landau gauge: the four-gluon vertex at collinear kinematics (all external momenta proportional) and the ghost-gluon vertex in the soft-gluon limit (vanishing gluon momentum). For the four-gluon vertex, the authors update earlier results with larger statistical samples on $32^4$ and $48^4$ lattices, and find that one of the three form factors, $F^{(2)}$, grows as the momentum invariant $s = \sum_i p_i^2/4$ goes to zero while $F^{(0)}$ stays essentially constant. For the ghost-gluon vertex, they show that the form factor $H_1$ extracted with lattice and continuum tensor structures agrees over a large momentum range, and that the results are consistent with previous lattice determinations. These vertices control the infrared coupling of gluons and ghosts, so a reliable measurement of their momentum dependence constrains how continuum treatments of nonperturbative QCD should be built. Because all data come from a single lattice spacing, the paper stops short of claiming a continuum-limit result.

What carries the argument

The load-bearing mechanism is the kinematic simplification of collinear momenta: when $p_i \propto p_j$, the full four-gluon Green function receives contributions only from the one-particle-irreducible four-gluon vertex, so no subtraction of three-gluon or disconnected diagrams is required, and the tensor basis collapses to the three operators $\tilde\Gamma^{(0)}$, $\tilde\Gamma^{(1)}$, $\tilde\Gamma^{(2)}$ of Eq. (1), whose amputated form factors $F^{(i)}$ are measured. For the ghost-gluon vertex, the mechanism is Landau-gauge orthogonality of the gluon propagator, which removes the $H_2$ form factor and leaves a single scalar $H_1$, extracted by Lorentz-color contraction and evaluated with both the lattice vertex $\Gamma^{\mathrm{Lat}}_\mu$ and the continuum vertex $\Gamma^{\mathrm{Cont}}_\mu$ to monitor discretization effects.

What would settle it

Measuring the four-gluon form factors on a finer lattice spacing (for example $\beta = 6.4$) and with external momenta that are only nearly collinear would settle it: if the infrared rise of $F^{(2)}$ disappears or depends sharply on the small deviation from collinearity, the reported behavior is an artifact of the kinematic assumption or of discretization rather than a property of the vertex.

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Extended reading notes

Core claim

The paper's central claim is that the infrared behavior of the four-gluon vertex can be measured reliably in the collinear kinematics where all external momenta are proportional, and that in this regime the amputated form factors show a clear hierarchy $F^{(0)}, F^{(2)} \gg F^{(1)}$, with $F^{(0)}$ essentially constant and $F^{(2)}$ increasing as $s = \sum_i p_i^2/4$ approaches zero. For the one-particle-irreducible ghost-gluon vertex in the soft-gluon limit (gluon momentum taken to zero), the claim is that $H_1$ computed with the lattice and continuum versions of the tensor structure agree up to roughly 3 GeV, indicating that finite-size effects are under control, and that the result is consistent with previous lattice determinations. The authors present these as ongoing calculations with large statistical ensembles but a single lattice spacing, so they do not claim a continuum-limit determination.

Load-bearing premise

The central measurement assumes that when the four external momenta are all parallel, the full four-gluon correlation function receives contributions only from the genuine four-gluon vertex, so no subtraction of three-gluon or disconnected pieces is needed.

Editorial extensions

If this is right

If $F^{(2)}$ indeed grows toward the infrared, the four-gluon vertex has a nontrivial momentum dependence that any functional or perturbative description of Yang-Mills dynamics in the deep infrared must reproduce.
The hierarchy $F^{(0)}, F^{(2)} \gg F^{(1)}$ implies that the $\tilde\Gamma^{(0)}$ and $\tilde\Gamma^{(2)}$ tensor structures dominate collinear kinematics, simplifying the modelling of this vertex in continuum functional approaches.
The agreement of the lattice $H_1$ with previous lattice and continuum determinations supports the standard soft-gluon truncations used in studies of ghost and gluon propagators.
For momenta up to about 3 GeV, the ghost-gluon vertex shows no significant finite-size effects on these ensembles, validating the volume strategy for future analyses.
Larger ensembles and additional lattice spacings are required to firm up the size of the infrared rise of $F^{(2)}$ and to control the $k \gtrsim 3$ GeV region for the ghost-gluon vertex.

Reading between the lines

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

A direct extension the authors leave implicit: if $F^{(2)}$ keeps rising toward zero momentum, the four-gluon vertex may become as important as the three-gluon vertex in the infrared dynamics of Yang-Mills theory, a possibility that continuum functional studies could test by feeding in the measured lattice form factors.
The collinear kinematic trick could be stress-tested on the same ensembles by measuring a slightly non-collinear momentum configuration and checking that the extracted form factors do not drift, which would probe the assumption that only the one-particle-irreducible four-gluon diagram contributes.
A run at a finer lattice spacing with the same physical volume would allow a first continuum extrapolation of $F^{(2)}$'s infrared growth; if the rise persists, it is a genuine nonperturbative signal rather than a discretization artefact.

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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

This is a conference proceedings, not a full paper: it updates the same group's earlier lattice calculations of the four-gluon vertex (collinear kinematics) and adds a ghost-gluon vertex measurement in the soft-gluon limit. What's genuinely new is the higher statistics, the deeper infrared extension, and a clean consistency check between continuum and lattice tensor structures for H1. The qualitative findings—F^(0) roughly constant, F^(2) growing toward the IR, and H1 agreeing with earlier lattice and continuum determinations—are plausible and in line with independent calculations. The authors are also upfront about the single lattice spacing and the absence of a continuum limit. That honesty deserves credit.

The soft spots are real but mostly fixable. First, there are no numerical tables, only figures; for a proceedings that's tolerable, but it means the reader can't easily reuse the data. Second, the finite-size discussion is thin: citing the gluon propagator literature to argue that volume effects are small for 32^4 and 48^4 is reasonable, but it's not a substitute for checking the vertex directly at another volume. Third, and more concerning, is the statement in Sec. 1 that for collinear momenta with p_i ∝ p_j and p_i ≠ p_j only the 1PI four-gluon diagram contributes. The stress-test note is correct: that condition does not exclude p_i = -p_j, where disconnected two-gluon contributions survive. If any of the momentum tuples used for Fig. 1 contained opposite pairs, the reported IR growth of F^(2) could be contaminated. The paper defers the details to ref. [1], so the proceedings is not self-contained on this point. I suspect this is a writing lapse rather than a real error—the authors likely did avoid opposite pairs—but the whole kinematic simplification hinges on the exact tuple set, and it needs to be stated precisely.

Who gets value from this? People working on Yang-Mills vertices and continuum functional methods, as a quick cross-check. It is not a replacement for the fuller papers the authors have already published or are writing. It deserves a serious referee, mostly to force the kinematic condition to be stated correctly and to ask for the momentum configurations. I'd send it back with that request rather than reject it.

Recommendation: conditional acceptance after a minor revision that spells out the momentum tuples and the no-opposite-pair condition, and ideally adds one table with the raw numbers.

Referee Report

3 major / 4 minor

Summary. The manuscript reports lattice determinations of two Landau-gauge Yang-Mills vertices: the four-gluon vertex in collinear kinematics and the ghost-gluon vertex in the soft-gluon limit. The four-gluon form factors F^(0), F^(1), F^(2) are extracted from amputated Green functions on 32^4 and 48^4 ensembles at beta=6.0, with the claims that F^(0) is essentially constant and F^(2) grows in the infrared. The ghost-gluon form factor H1 is computed with both lattice and continuum tensor structures and the two are reported to agree up to about 3 GeV, with agreement with previous lattice results. The paper is a short proceedings contribution that explicitly states that no continuum limit is attempted.

Significance. If the reported qualitative behaviors survive a complete analysis, the results are a useful cross-check for continuum functional methods and for the understanding of confining correlation functions: an IR-enhanced F^(2) is a nontrivial signature of the four-gluon vertex, and a soft-ghost-gluon vertex consistent with earlier determinations constrains truncation schemes. The paper's explicit strengths are the large statistics (~9000 configurations for the four-gluon vertex for each volume; thousands for the ghost-gluon vertex) and the direct comparison with independent continuum calculations [9,10] and previous lattice data [8]. The manuscript is honest about the lack of a continuum limit, but the kinematic justification of the four-gluon extraction and the finite-size argument need to be tightened before the central claims can be considered established.

major comments (3)

[Sec. 1, four-gluon kinematics] The condition 'p_i ∝ p_j and p_i ≠ p_j' does not exclude p_i = -p_j, and for a tuple such as (p, 2p, -p, -2p) the disconnected diagrams do not vanish; after amputation they can project onto the F^(i) basis. The text defers the required no-opposite-pair condition to Ref. [1] and does not specify the momentum tuples used for Fig. 1. If opposite pairs are present, the reported infrared growth of F^(2) would mix 1PI four-gluon physics with two-point contributions. Please state the exact momentum sets used and either prove or cite a proof that all disconnected contributions vanish for those sets.
[Sec. 3, ghost-gluon finite-size conclusion] The agreement between H1 obtained with the lattice and continuum tensor structures is a check of discretization/improvement effects, not of finite-volume effects; it does not by itself support the sentence that 'finite size effects are under control'. The comparison of 32^4 and 48^4 data, or a direct reference to a dedicated volume study, would be the appropriate check. The paper should either provide this comparison or qualify the statement.
[Figs. 1-2 and Secs. 1,3] The central qualitative claims — F^(0) constant, F^(2) infrared growth, and agreement of H1 with previous results — are supported only by figures and verbal statements. No numerical table, fit, or explicit statistical error is given. A short table of representative values with errors, or a fit parametrization, is needed to make the claims checkable, particularly for the infrared behavior of F^(2), which is the paper's main new result.

minor comments (4)

[Sec. 3] The text contains a typo: 'Oure result' should read 'Our result' or 'Our results'.
[Sec. 1, Eq. (1)] The notation 'eΓ' is not defined; if it denotes the amputated vertex or a particular tensor basis, please state this explicitly.
[Sec. 2] The formula for H1 is written with all Lorentz and color indices omitted; please spell out the contraction or give the explicit expression with indices.
[Abstract/Sec. 3] The abstract describes the computations as being 'addressed', while Sec. 3 says the calculations are 'on-going'; the wording should be harmonized.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper is a direct lattice measurement with no fitted parameters or back-substitution.

full rationale

The paper reports lattice extractions of the collinear four-gluon form factors and the soft-gluon ghost-gluon vertex. The four-gluon F^(i) are obtained by amputating the measured full Green function and contracting with the three tensor structures of Eq. (1); the ghost-gluon H1 is read directly from the Lorentz-color contraction in Sec. 2. Neither quantity is fitted to data nor derived from the same quantity by construction. The kinematic statement that only the one-particle-irreducible four-gluon diagram contributes for proportional momenta is a diagrammatic fact whose derivation is delegated to [1]; even though [1] shares authors, this is a methodological citation, not a claim that the present data reproduce the cited paper's output. The qualitative conclusions (F^(0) roughly constant, F^(2) infrared growth) follow from the plots of raw amputated form factors, and the ghost-gluon result is compared against independent continuum and lattice determinations [8-10]. The possible subtlety concerning opposite momenta (p_i = -p_j) would be a correctness/systematics issue about whether the kinematic condition excludes disconnected diagrams, not a circularity, since it does not make the output equal the input. No equation in the paper reduces to a prior result by construction, and no fitted parameter is renamed as a prediction.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper adds no free parameters or invented entities. The lattice spacing and ensembles are inputs from [4]. The central assumptions are the kinematic isolation of the 1PI four-gluon vertex, the standard Landau-gauge contraction for H1, and the transfer of finite-size conclusions from the propagator to the vertices, all flagged above.

assumptions (4)

domain assumption The Wilson action ensembles at beta = 6.0 have inverse lattice spacing a^{-1} = 1.943 GeV and represent pure SU(3) Yang-Mills in the Landau gauge.
The ensembles are taken from reference [4]; the paper performs no new ensemble generation or scale setting.
domain assumption For collinear external momenta, only the one-particle-irreducible four-gluon diagram contributes to the full four-point Green function, so the amputated correlator directly gives the 1PI form factors.
Sec. 1 invokes this simplification and cites [1] for the proof, making the four-gluon extraction depend on it.
domain assumption In the Landau gauge, the H2 form factor of the ghost-gluon vertex decouples due to gluon propagator transversality, and H1 can be isolated with the given Lorentz-color contraction using lattice or continuum tree-level vertices.
Sec. 2 states this property and gives the contraction formula.
domain assumption Finite-size effects for the four-gluon and ghost-gluon vertices are small for these ensembles, inferred from prior gluon propagator studies.
Sec. 3 cites [11] and uses agreement between lattice and continuum tensor versions to argue volume effects are controlled, without a dedicated finite-size study of the vertices.

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Cite this review

Pith. "Pith review of The four-gluon and ghost-gluon vertices in the Landau gauge from lattice simulations." pith.science (2026). https://pith.science/paper/QIHCLYIB

@misc{pith2026250523476,
  author       = {Pith},
  title        = {Pith review of: The four-gluon and ghost-gluon vertices in the Landau gauge from lattice simulations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QIHCLYIB}},
  note         = {Machine review of arXiv:2505.23476}
}

abstract

The computation of the four-gluon and ghost-gluon vertices in the Landau gauge using high statistical lattice ensembles for $32^4$ and $48^4$ volumes is addressed. For the four-gluon vertex, our previous results for the collinear kinematics are updated allowing to get a better coverage of the IR region. Furthermore, the one-particle irreducible ghost-gluon Green function in the soft gluon limit is computed covering, with precision, a large momentum region.

Figures

Figures reproduced from arXiv: 2505.23476 by the authors.

**Figure 2.** The ghost-gluon vertex data. Previous lattice studies of the gluon propagator suggest that the finite size effects are small for the type of ensembles considered herein [11]. In what concerns the four-gluon vertex, although being a difficult lattice calculation, the signalto-noise is reasonably good below 𝑝 < 1.5 GeV. Moreover, the numerical simulations considered here agree, at least at a qualitative level, with r… view at source ↗

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Reference graph

Works this paper leans on

12 extracted references · 10 canonical work pages

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Four-gluon vertex from lattice QCD,

M. Colaço, O. Oliveira and P. J. Silva, “Four-gluon vertex from lattice QCD,” Phys. Rev. D 109, 074502 (2024)

work page 2024

[2] [8]

Muller-Preussker, A

M. Muller-Preussker, A. Sternbeck, A. Schiller, and I. L. Bogolubsky, Landau gauge gluon and ghost propagators from lattice QCD, Braz. J. Phys. 37, 193 (2007)

work page 2007

[3] [2]

The four-gluon vertex from lattice QCD

O. Oliveira, M. Colaço and P. J. Silva, “The four-gluon vertex from lattice QCD,” PoS LATTICE2024, 391 (2025) doi:10.22323/1.466.0391 [arXiv:2501.17650 [hep-lat]]

work page Pith review arXiv 2025

[4] [3]

Gribov copies, lattice QCD and the gluon propagator,

P. J. Silva , O. Oliveira, “Gribov copies, lattice QCD and the gluon propagator,” Nucl. Phys. B690, 177-198 (2004)

work page 2004

[5] [4]

LatticeGluonandGhostPropagators,andtheStrong CouplinginPureSU(3)Yang-MillsTheory: FiniteLatticeSpacingandVolumeEffects,

A.G.Duarte,O.OliveiraandP.J.Silva,“LatticeGluonandGhostPropagators,andtheStrong CouplinginPureSU(3)Yang-MillsTheory: FiniteLatticeSpacingandVolumeEffects,”Phys. Rev. D94, 014502 (2016)

work page 2016

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High statistical computation of the Landau gauge ghost-gluon vertex

N.Brito,O.OliveiraandP.J.Silva,“HighstatisticalcomputationoftheLandaugaugeghost- gluonvertex,”PoSLATTICE2024,469(2025)doi:10.22323/1.466.0469[arXiv:2411.17280 [hep-lat]]

work page Pith review arXiv 2025

[7] [6]

Cucchieri , A

A. Cucchieri , A. Maas, and T. Mendes, Three-point vertices in Landau-gauge Yang-Mills theory, Phys. Rev. D 77, 094510 (2008)

work page 2008

[8] [7]

Maas, Constraining the gauge-fixed Lagrangian in minimal Landau gauge, SciPost Phys

A. Maas, Constraining the gauge-fixed Lagrangian in minimal Landau gauge, SciPost Phys. 8, 071 (2020)

work page 2020