The Infrared Behavior of Gluon and Ghost Propagators in Landau Gauge QCD

Andreas Hauck; Lorenz von Smekal; Reinhard Alkofer

REVIEW

The Infrared Behavior of Gluon and Ghost Propagators in Landau Gauge QCD

Lorenz von Smekal, Andreas Hauck, Reinhard Alkofer

High Energy Physics - Phenomenology · Submitted 6 May 1997 · 26 references

Pith's verdict: The qualitative scaling picture is the real contribution and it holds up; the specific numbers κ ≈ 0.92 and α_c ≈ 9.5 come from the angle-approximated equations and should not be quoted as robust.

T0 · AI review

How this review was made: An AI referee (selfhosted/deepseek-v4.1-flash) read the paper on October 29, 2026 and answered a fixed list of questions about it. T0 is the starting level every paper gets. The levels describe how deeply Pith has checked the paper, never the authors.

T0 AI review An AI read the paper and answered a fixed list of questions.
T1 Audit Automatic checks that the references and files the paper relies on exist and match.
T2 Compute The paper's code or calculations re-run with the same results.
T3 Formal statement The claims written out in a formal math language a computer can read.
T4 Proof Those claims proved, and the proof checked by computer.

How Pith checks papers →

· 2 major objections · 6 minor · Ask about this paper · The compiled record · Read the full review →

Challenge this review Comment (0) Read on arXiv ↗

Cite ▾

arXiv IDhep-ph/9705242 (version 2)

Pith IDPKOEXDFZ, Pith's permanent record of this paper and its review

CitationAndreas Hauck, Lorenz von Smekal, Reinhard Alkofer. "The Infrared Behavior of Gluon and Ghost Propagators in Landau Gauge QCD." arXiv:hep-ph/9705242 (1997). https://pith.science/paper/PKOEXDFZ

Cite this reviewBibTeX for Pith's review of this paper

Claim paper

Why Pith says this

Highlight any text to comment on it.

Editor's note

read the letter

This is the paper that put ghosts into the Landau-gauge gluon DSE and got the opposite answer to everyone before them: the gluon propagator vanishes in the infrared, the ghost diverges faster than a massless pole, and the running coupling runs to a fixed point instead of blowing up. That qualitative flip is the contribution, and it is a real one — the Mandelstam-approximation literature had the gluon enhanced, and this paper shows that was an artifact of dropping ghosts. The argument that the three-gluon loop is IR-subdominant (∼x^κ against the ghost loop's ∼x^{-2κ}) is clean and, importantly, means the three-gluon vertex Ansatz of Eq. (5) does not drive κ. The lattice comparisons in Figs. 1–2 are a genuine external check, and the ghost fit beating the c/k² + d/k⁴ form of Suman–Schilling is a nice touch. The Z(x=1) ≈ 11.3 normalization is a matching choice, not a fit to the physics, so I would not hold that against them. The stress-test concern lands, though. The κ derivation in Eqs. (8)–(9) is done on the angle-approximated one-dimensional equations (6)–(7), and the approximation is roughest exactly where it matters, at q² ∼ k² → 0 in the ghost loop. Footnote 19 says as much: assessing sensitivity to the modified angle approximation is left to future work. Later full treatments of the same truncation give κ ≈ 0.6, which moves α_c a long way from 9.5. So the abstract's 'α_c ≃ 9.5' reads more definite than the derivation supports; the qualitative statement is safe, the number is not. Two smaller things: the Z₁ = Z₃/Ẑ₃ inconsistency is stated openly in the text and is a known cost of dropping four-gluon couplings at this order, so it is a caveat rather than a hidden flaw; and the classifier's 'limitations: none found' is simply wrong — the paper flags both the angle approximation and the renormalization-constant mismatch. Who should read it: anyone working on infrared QCD, confinement, or DSE truncations, and anyone who wants the origin of the scaling solution. It deserves referees.

Referee report

Major objections (2)

[Abstract (s4) and Eq. (10)] The abstract and Eq. (10) quote α_c ≈ 9.5 as a definite result, but the derivation of κ in Eqs. (8)–(9) is carried out on the angle-approximated one-dimensional equations (6)–(7), not on the full four-dimensional DSEs (1)–(2). Footnote 19 acknowledges that “it will nevertheless be important to assess the sensitivity of the results to the modified angle approximation in future.” Because α_c = (16π/9)(1/κ − 1/2)^{−1} is steep in κ, a shift from κ ≈ 0.92 to κ ≈ 0.6 would give α_c ≈ 4.7, roughly half the quoted value. The abstract and Eq. (10) should state that the numerical value is obtained within the angle approximation, and the qualitative fixed-point statement should be separated from the quantitative one.
[Eqs. (6)–(7)] The angle approximation is applied to the ghost loop that dominates the infrared. For q² < k² the replacement G((k−q)²) → G(k²) is used; with G ~ x^{−κ} and κ ≈ 0.92 the argument (k−q)² varies by a large factor across the integration range, so the approximation is least controlled precisely where the integrand is largest. The argument that the three-gluon loop is infrared-subdominant (~x^κ versus ~x^{−2κ}) addresses the vertex Ansatz, not the angle approximation itself. A concrete test would be to repeat the infrared analysis on the unapproximated ghost loop, or to report how κ moves when the angle approximation is relaxed; without this, the claim that κ ≈ 0.92 is a property of the truncation rather than of the approximation is not established.

Minor objections (6)

[Eqs. (8)–(9)] The text states that matching to Eq. (6) yields “a quadratic equation with a unique solution κ = (61 − √1897)/19” but does not display the quadratic. Since the root is the paper's central quantitative result, the quadratic should be written explicitly so readers can verify it and see why the second root is excluded (presumably because it exceeds 2).
[Fig. 2 and surrounding text] The text concludes that lattice calculations “confirm the existence of an infrared enhanced ghost propagator of the form D_G ~ 1/(k²)^{1+κ} with 0 < κ < 1.” The data in Fig. 2 span a limited momentum range, and the comparison fit c/k² + d/k⁴ is itself dominated by the 1/k⁴ term at low k, so the figure does not sharply discriminate κ ≈ 0.92 from κ = 1. The word “confirm” overstates what Fig. 2 shows; “is consistent with” would be more accurate.
[Text following Eq. (7)] The truncation's violation of the Slavnov–Taylor identity Z1 = Z3/Ẑ3 is acknowledged in the paragraph after Eq. (7), but it is a consistency issue for the vertex Ansatz and deserves a more prominent statement, since the three-gluon vertex Eq. (5) is built from that identity.
[Reflection-positivity claim] The statement that the Euclidean gluon correlation function violates reflection positivity is supported by citation to [20], which is listed as an e-print. A one- or two-line summary of the argument, or a pointer to the specific equation in [20], would let readers assess the claim without leaving the letter.
[Fig. 1] The gluon normalization Z(x=1) ≈ 11.3 is the only quantity adjusted to match the lattice; the text should say so explicitly, so that the comparison is not mistaken for a parameter-free prediction.
[Eq. (4) and footnote 15] The ghost-gluon vertex is asserted to be fully determined at this truncation level, with no undetermined transverse terms. Since this is a strong claim relative to the fermion case discussed just below, a sentence explaining why Bose symmetry combined with the Slavnov–Taylor identity removes the ambiguity would strengthen the presentation.

How Pith checks papers → · Full audit of this paper →

The paper

The paper claims

By keeping the ghost loop that earlier treatments discarded, the authors argue that Landau-gauge QCD has a vanishing gluon propagator, a ghost propagator that diverges faster than a massless pole, and a running coupling that stops running.

The reading

In Landau-gauge QCD the gluon and ghost propagators are the two elementary correlation functions that any non-perturbative treatment has to get right at low momentum. Earlier Dyson-Schwinger studies dropped the ghost loop and concluded that the gluon propagator blows up in the infrared. This paper keeps the ghost in the coupled system, builds the three-gluon and ghost-gluon vertices out of the Slavnov-Taylor identities that gauge invariance imposes, and solves the resulting truncated equations. The payoff is an analytic statement about the $k^2 to 0$ limit: the gluon propagator goes to zero, the ghost propagator diverges faster than a massless particle's, and the running coupling flattens onto a fixed point $alpha_c simeq 9.5$ instead of running away. If that is right, the infrared enhancement of Landau-gauge QCD lives in the ghost sector, and a gluon propagator that vanishes is a sign that the gluon is not a propagating asymptotic particle at all.

What carries the argument

The load-bearing device is the infrared power Ansatz $Z(x)G(x) to c, x^{kappa}$ applied to a one-dimensional reduction of the two coupled integral equations. The reduction replaces $G((k-q)^2)$ and $Z((k-q)^2)$ by their values at $k^2$ for loop momenta below $k^2$ and by their values at $q^2$ above it, which preserves the $q^2 to 0$ limit and therefore the infrared exponent. The vertices that feed these equations are not free: the ghost-gluon vertex is taken as $G_mu(p,q) = i q_mu G(k^2)/G(q^2) + i p_mu (G(k^2)/G(p^2) - 1)$, and the three-gluon vertex is built from the same functions through the Slavnov-Taylor construction with coefficients $A_pm$. That closed set of functions is what makes the exponent computable rather than fitted: the ghost loop's $x^{-2kappa}$ term, balanced against the $x^{2kappa}$ form the ghost equation dictates, is the equation that fixes $kappa$.

Extended reading notes

Core claim

The paper's claim is that the coupled gluon-ghost Dyson-Schwinger system, truncated so that the three-gluon and ghost-gluon vertices are fixed by their Slavnov-Taylor identities while four-gluon couplings and the ghost-gluon and ghost-ghost scattering kernels are dropped, has a self-consistent infrared solution with a definite power-law exponent. Assuming $Z(x)G(x) to c, x^{kappa}$ for $x = k^2 to 0$, the ghost equation forces $G(x) propto x^{-kappa}$ and $Z(x) propto x^{2kappa}$, and the gluon equation is then dominated not by the three-gluon loop, which contributes only order $x^{kappa}$, but by the ghost loop, which contributes order $x^{-2kappa}$. Matching the two determinations of $Z$ leaves a quadratic equation with a single root below 2, $kappa = (61 - sqrt{1897})/19 simeq 0.92$. Because $2kappa - 1$ is positive, the gluon propagator $Z(x)/x$ vanishes at small spacelike momentum; because $1 + kappa$ exceeds 1, the ghost propagator $G(x)/x^2$ diverges more strongly than a massless pole; and because $Z G^2$ is scale invariant in this limit, the running coupling tends to $alpha_c = frac{16pi}{9}(frac{1}{kappa} - frac{1}{2})^{-1} simeq 9.5$. The numerical solutions of the same truncated equations are then set against lattice data for both propagators, and the gluon correlation function is noted to violate reflection positivity, which the authors read as confined gluons.

Load-bearing premise

The exponent $kappa$ is extracted from a one-dimensional angle approximation of the integral equations together with vertex Ansätze that drop all transverse three-gluon structures, and the paper itself flags in footnote 19 that the sensitivity to that modified angle approximation still has to be assessed and notes that the transverse terms are discarded because the Slavnov-Taylor identity cannot constrain them.

Editorial extensions

If this is right

The gluon propagator vanishes at small spacelike momentum as a positive power of $k^2$ (roughly $(k^2)^{0.84}$), rather than being infrared enhanced as earlier ghost-free treatments had it.
The ghost propagator behaves as $(k^2)^{-1-kappa}$ with $kappa simeq 0.92$, i.e. it diverges more strongly than a massless particle pole.
The running coupling of the subtraction scheme approaches a fixed point $alpha_c simeq 9.5$ in the infrared, in qualitative disagreement with the infrared singular coupling of the Mandelstam approximation.
Normalizing at $alpha_S(M_Z) = 0.118$ gives $alpha_S(M_tau) = 0.38$, close to the measured value, so the scheme is not ruled out by intermediate-scale data.
The gluon correlation function violates reflection positivity, so it admits no Lehmann representation, which the authors interpret as a signal of gluon confinement.
The size of the fixed-point coupling is taken as an indication that the quark Dyson-Schwinger equation will generate dynamical chiral symmetry breaking once quarks are included.

Reading between the lines

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

The number $kappa = (61 - sqrt{1897})/19$ is the paper's sharpest prediction and the easiest to attack, because it is fixed by the truncation rather than by gauge invariance; a lattice extraction of the small-momentum slope of the ghost propagator would test it without needing to reproduce the whole scheme.
The agreement with lattice configurations restricted to the fundamental modular region is offered as evidence that Gribov copies barely affect Landau-gauge propagators, which would decouple the gauge-fixing ambiguity question from the propagator question for practical purposes.
If the gluon propagator really vanishes at long wavelength, then phenomenological models that mediate quark binding through a massive or infrared-enhanced gluon exchange are using the wrong low-energy variable, and the ghost-gluon system becomes the natural carrier of the interaction.
The authors' own remark that keeping $Z_1$ ultraviolet finite is inconsistent with the gauge-invariance relation $Z_1 = Z_3/tilde{Z}_3$ suggests a diagnostic: the transverse, Slavnov-Taylor-unconstrained pieces of the three-gluon vertex are exactly what could restore that relation, so their coefficient is a measurable bookkeeping device for the size of the truncation error.

Original abstract

A solvable systematic truncation scheme for the Dyson-Schwinger equations of Euclidean QCD in Landau gauge is presented. It implements the Slavnov-Taylor identities for the three-gluon and ghost-gluon vertices, whereas irreducible four-gluon couplings as well as the gluon-ghost and ghost-ghost scattering kernels are neglected. The infrared behavior of gluon and ghost propagators is obtained analytically: The gluon propagator vanishes for small spacelike momenta whereas the ghost propagator diverges stronger than a massless particle pole. The numerical solutions are compared with recent lattice data for these propagators. The running coupling of the renormalization scheme approaches a fixed point, $\alpha_c \simeq 9.5$, in the infrared.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,334 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.93 · human participants 0.01 · controlled comparison 0.06 · randomized assignment 0.01 · states quantitative result 0.98 · compares with prior results 0.93 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

A solvable systematic truncation scheme for the Dyson-Schwinger equations of Euclidean QCD in Landau gauge is presented. It implements the Slavnov-Taylor identities for the three-gluon and ghost-gluon vertices, whereas irreducible four-gluon couplings as well as the gluon-ghost and ghost-ghost scattering kernels are neglected. The infrared behavior of gluon and ghost propagators is obtained analytically: The gluon propagator vanishes for small spacelike momenta whereas the ghost propagator diverges stronger than a massless particle pole. The numerical solutions are compared with recent lattice data for these propagators. The running coupling of the renormalization scheme approaches a fixed point, $\alpha_c \simeq 9.5$, in the infrared.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	A solvable systematic truncation scheme for the Dyson-Schwinger equations of Euclidean QCD in Landau gauge is presented.	contribution 0.71	0.92	0.06	asserted 1.00
s1	It implements the Slavnov-Taylor identities for the three-gluon and ghost-gluon vertices, whereas irreducible four-gluon couplings as well as the gluon-ghost and ghost-ghost scattering kernels are neglected.	method 0.78	0.82	0.04	asserted 1.00
s2	The infrared behavior of gluon and ghost propagators is obtained analytically: The gluon propagator vanishes for small spacelike momenta whereas the ghost propagator diverges stronger than a massless particle pole.	result 1.00	0.94	0.82	asserted 1.00
s3	The numerical solutions are compared with recent lattice data for these propagators.	method 0.55	0.86	0.03	asserted 0.97
s4	The running coupling of the renormalization scheme approaches a fixed point, $\alpha_c \simeq 9.5$, in the infrared.	result 1.00	0.89	0.18	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.80 · evidence from proof or derivation 0.74.

What was examined, and what was not

sources read

arXiv listing abstract v2

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** FIG. 1. The numerical result for the gluon propagator from

**Figure 2.** FIG. 2. The numerical result for the ghost propagator from

Figures reproduced from arXiv:hep-ph/9705242 by the authors.

References

Works cited (26)

[1]

W. Marciano and H. Pagels, Phys. Rep. 36, 137 (1978)

View on Pith 1978
[2]

S. Mandelstam, Phys. Rev. D 20, 3223 (1979)

View on Pith 1979
[3]

D. Atkinson et al., J. Math. Phys. 22, 2704 (1981); D. Atkinson, P. W. Johnson and K. Stam, ibid. 23, 1917 (1982)

View on Pith 1981
[4]

N. Brown and M. R. Pennington, Phys. Rev. D 39, 2723 (1989)

View on Pith 1989
[5]

A. Hauck, L. v. Smekal and R. Alkofer, e-print, hep- ph/9604430

View on Pith
[6]

M. Baker, J. S. Ball and F. Zachariasen, Nucl. Phys. B186, 531/560 (1981); W. J. Schoenmaker, ibid. B194, 535 (1982); J. R. Cudell and D. A. Ross, ibid. B358, 247 (1991)

View on Pith 1981
[7]

K. B¨ uttner and M. R. Pennington, Phys. Rev. D 52, 5220 (1995)

View on Pith 1995
[8]

R. Alkofer, M. R. Pennington, L. v. Smekal and P. Wat- son, work in progress

View on Pith
[9]

See, e.g., C. D. Roberts and A. G. Williams, Prog. Part. Nucl. Phys. 33, 477 (1994), and references therein

View on Pith 1994
[10]

See also “Dynamical Symmetry Breaking in Quantum Field Theories”, V. A. Miranski, World Scientiﬁc, 1993

View on Pith 1993
[11]

An infrared enhanced gluon propagator was found in Landau gauge in Mandelstam approximation [2–5] as well as in some studies of its simpliﬁed axial gauge DSE [6,7]

View on Pith
[12]

L. G. Vachnadze, N. A. Kiknadze and A. A. Khelashvili, Theor. Math. Phys. 102, 34 (1995)

View on Pith 1995
[13]

We use positive deﬁnite metric, gµν = δµν . Color indices are suppressed and the number of colors is ﬁxed, Nc = 3

View on Pith
[14]

J. C. Taylor, Nucl. Phys. B33, 436 (1971)

View on Pith 1971
[15]

In [20] we derive a Slavnov–Taylor identity for the ghos t– gluon vertex from the usual BRS invariance. This to- gether with the symmetry of the ghost–gluon vertex fully determines its form at the present level of truncation. There are no undetermined transverse terms in this case

View on Pith
[16]

The ghost–gluon vertex (4) is consistent with a ghost– gluon scattering kernel of tree–level structure in the STI of the 3–gluon vertex, which can then be constructed from procedures developed and used previously (U. Bar– Gadda, Nucl. Phys. B163, 312 (1980; S. K. Kim and M. Baker, ibid. B164, 152 (1980); J. S. Ball and T.- W. Chiu, Phys. Rev. D 22, 2550 (1980))

View on Pith 1980
[17]

N. Brown and N. Dorey, Mod. Phys. Lett. A6, 317 (1991)

View on Pith 1991
[18]

D. C. Curtis and M. R. Pennington, Phys. Rev. D 42, 4165 (1990)

View on Pith 1990
[19]

A similar assumption underlies the Mandelstam approx- imation. For q2 > k 2 it can be further justiﬁed from a detailed study of the solutions in the ultraviolet [20]. It will nevertheless be important to assess the sensitivity of the results to the modiﬁed angle approximation in future

View on Pith
[20]

L. v. Smekal, A. Hauck and R. Alkofer, e–print, hep– ph/9707327

View on Pith
[21]

Particle Data Group, Phys. Rev. D 54, 77 (1996)

View on Pith 1996
[22]

C. Bernard, C. Parrinello and A. Soni, Phys. Rev. D 49, 1585 (1994); D. S. Henty et al., ibid. 54 (1996), 6923

View on Pith 1994
[23]

P. Marenzoni, G. Martinelli, and N. Stella, Nucl. Phys. B455, 339 (1995)

View on Pith 1995
[24]

H. Suman and K. Schilling, Phys. Lett. B373, 314 (1996)

View on Pith 1996
[25]

This is supported by the qualitative similarity of our s o- lutions to the infrared behavior obtained from studies of the inﬂuence of a complete gauge ﬁxing by D. Zwanziger, Nucl. Phys. B378, 525 (1992); ibid. B412, 657 (1994)

View on Pith 1992
[26]

We are indebted to F. Coester for pointing this out. 0 200 400 600 800 1000 1200 1400 1600 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 x = k 2 a 2 , with a-1 = 2GeV finite size Z(x)/x Lattice-Fit Marenzoni et. al. 24 3 x 48 16 3 x 48 FIG. 1. The numerical result for the gluon propagator from Dyson–Schwinger equations (solid line) compared to lattic e data from ...

View on Pith

Forward citations

Cited by 20 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Correlation functions of Landau gauge Yang-Mills theory
hep-ph 2020-03 conditional novelty 7.0 of 10

A parameter-free simultaneous solution of the coupled equations of motion for Landau gauge Yang-Mills correlation functions reproduces lattice propagators and vertices and removes the quadratic-divergence ambiguity in...
Quark Confinement from Color Confinement
hep-th 2007-08 conditional novelty 7.0 of 10

A simple inequality on the infrared exponents of gluon and ghost propagators is shown to imply quark confinement, and the resulting Polyakov-loop potential predicts first-order deconfinement for SU(3) and second-order...
Scalar diquarks in the QCD vacuum
hep-ph 2026-06 unverdicted novelty 6.0 of 10

A first-principles FRG approach to two-flavor QCD derives low-energy constants for the pion, sigma-meson and scalar diquark without parameters beyond QCD itself, including new diquark properties for color-superconduct...
Reconstructing the graviton
hep-th 2021-02 conditional novelty 6.0 of 10

The fluctuation graviton's spectral function in asymptotically safe gravity is reconstructed as positive, while the background graviton's is proved to have negative parts and vanishing total weight.
Reconstructing the gluon
hep-ph 2018-04 conditional novelty 6.0 of 10

A new analytic relation fixes the low-frequency shape of the gluon spectral function from the infrared behavior of the Euclidean propagator, and a basis built on this relation reconstructs the spectrum from numerical ...

Show 15 more citing papers

Non-perturbative finite-temperature Yang-Mills theory
hep-ph 2017-08 conditional novelty 6.0 of 10

Functional renormalisation group calculations produce a comprehensive set of finite-temperature Landau-gauge Yang-Mills correlation functions, agreeing with lattice data at the 10 percent level for magnetic quantities.
Non-perturbative quark, gluon and meson correlators of unquenched QCD
hep-ph 2017-06 conditional novelty 6.0 of 10

A self-consistent FRG solution of two-flavour QCD correlation functions reproduces lattice gluon and quark propagators and yields a momentum-dependent quark-gluon vertex, with the semi-perturbative running constrained...
Landau gauge Yang-Mills correlation functions
hep-ph 2016-05 accept novelty 6.0 of 10

Coupled flow equations for the gluon and ghost propagators and the ghost-gluon, three-gluon and four-gluon vertices of Landau gauge Yang-Mills theory reproduce lattice correlation functions with the classical action a...
On the influence of three-point functions on the propagators of Landau gauge Yang-Mills theory
hep-th 2012-11 conditional novelty 6.0 of 10

Including the ghost-gluon vertex dynamically and a new Bose-symmetric three-gluon vertex model brings Dyson-Schwinger propagators of Landau gauge Yang-Mills theory closer to lattice results in the mid-momentum regime.
Gauge bosons at zero and finite temperature
hep-ph 2011-06 conditional novelty 6.0 of 10

Gauge bosons in Yang-Mills theory remain confined at all temperatures despite Stefan-Boltzmann bulk thermodynamics, and the same framework yields a constructive resolution of the Linde problem.
The quark-gluon vertex in Landau gauge QCD: Its role in dynamical chiral symmetry breaking and quark confinement
hep-ph 2008-04 conditional novelty 6.0 of 10

The quark-gluon vertex's infrared singularity, generated self-consistently by dynamical chiral symmetry breaking, produces a linear confining potential; restoring chiral symmetry removes the singularity and leaves a C...
Infrared singularities in Landau gauge Yang-Mills theory
hep-th 2008-01 conditional novelty 6.0 of 10

Landau gauge Yang-Mills theory admits two infrared fixed points: a ghost-dominated scaling solution with new kinematic vertex singularities, and a decoupling solution with a massive gluon and no vertex enhancement.
The Landau gauge gluon and ghost propagators in 4D SU(3) gluodynamics in large lattice volumes
hep-lat 2007-10 conditional novelty 6.0 of 10

On lattices up to (13.2 fm)^4, the Landau-gauge gluon propagator flattens at low momentum instead of falling to zero, and the ghost dressing function shows no clean infrared power law.
Constraining bulk-to-boundary correlators under Poincar\'e symmetry
hep-th 2026-01 conditional novelty 5.0 of 10

Poincaré symmetry plus null-infinity fall-off conditions force scalar bulk-to-boundary correlators to 1/(u+n·x)^Δ and fermionic ones to a sum of 1/(u+n·x)^Δ and /n/(u+n·x)^(Δ+1) branches.
Quark Propagator at one-loop in the Refined Gribov-Zwanziger framework
hep-th 2025-05 conditional novelty 5.0 of 10

At one loop in the Refined Gribov-Zwanziger framework, the quark mass function agrees roughly with lattice QCD after fitting the gluon propagator, but the dressing function disagrees in the infrared.
Lattice gluodynamics computation of Landau-gauge Green's functions in the deep infrared
hep-lat 2009-01 conditional novelty 5.0 of 10

Lattice SU(3) gluon and ghost propagators on volumes up to (16 fm)^4 flatten in the infrared, matching decoupling-type Dyson-Schwinger and functional renormalization group solutions.
On the Faddeev-Popov operator eigenspectrum in topological background fields
hep-th 2006-03 conditional novelty 4.0 of 10

In SU(2) gauge theory, an instanton background adds three zero-modes and a center vortex of flux 2n+1 adds 4n zero-modes to the Faddeev-Popov operator.
Light-Front Holographic QCD and Emerging Confinement
hep-ph 2014-07 conditional novelty 3.0 of 10

A harmonic light-front confinement potential, fixed by a conformal-algebra construction and mapped to a soft-wall AdS dilaton, yields a massless pion and linear Regge trajectories with one fitted scale.
AdS/CFT and Light-Front QCD
hep-ph 2008-02 conditional novelty 3.0 of 10

A holographic model identifies the fifth AdS coordinate with the light-front impact separation of quarks, producing analytic hadron wavefunctions that fit spectra and form factors with one or two fitted scales.
Infrared Properties of QCD from Dyson-Schwinger equations
hep-ph 2006-05 accept novelty 3.0 of 10

Landau-gauge Dyson-Schwinger studies indicate an infrared-finite running coupling, a vanishing gluon propagator, and dynamically generated quark mass, though lattice and continuum results still differ at low momentum.

How the literature uses it

91 arXiv papers cite this in 148 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers whose main result relates to this one's

Found among the papers closest to this one by title and main claim. Pith's classifier compared each pair's main results and a second model family confirmed the relation; checked against two further families, shown conflicts agreed 118 times in 120 and shown same results 218 times in 239. The quoted sentence is the other paper's main claim.

States the same result 1 paper

For the pure gauge theory without quarks this system of equations for the propagators of gluons and ghosts is solved in an approximation which allows for an analytic discussion of its solutions in the infrared: The gluon propagator is shown to vanish for small spacelike momenta whereas the ghost propagator is found to be infrared enhanced.

The Infrared Behavior of Gluon, Ghost, and Quark Propagators in Landau Gauge QCD · 0.92

States a conflicting result 2 papers

In particular, the infrared exponent of the ghost propagator seems to be smaller than the one predicted analytically and the gluon propagator seems to display a (finite) nonzero value at zero momentum.

What's up with IR gluon and ghost propagators in Landau gauge? A puzzling answer from huge lattices · 0.85
We get a finite propagator at zero momentum, the ghost propagator going to infinity as $1/p^{2+2\kappa}$ with $\kappa=0$.

Infrared Gluon and Ghost Propagators · 0.81

Papers citing it · 97 sentences without a confident label

Builds on it 1 paper

*Conclusions: We have studied generic features of the IR limit of $SU(N)$ Yang-Mills theory established in [ref] in more detail.

The infrared behavior of Landau gauge Yang-Mills theory in d=2, 3 and 4 dimensions · Introduction · 0.95

Uses its method or tool 2 papers

We will use [ref] what we call the “Taylor coupling constant" which, thanks to Taylor's theorem [ref] , is only dependent on the gluon and ghost propagators.

Vacuum expectation value of A^2 from LQCD · The strong coupling constant · 0.97
The renormalization of the vertex is performed within the MiniMOM scheme [ref] .

On Dyson-Schwinger studies of Yang-Mills theory and the four-gluon vertex · Truncation · 0.98

Uses its data or values 1 paper

The exponents of the corresponding power laws, [equation] were shown to be related by [ref] [equation] The value obtained for $$ was $0.595353$ [ref] .

Nonperturbative properties of Yang-Mills theories · Landau gauge Yang-Mills theory · 0.88

Compares with it 1 paper

The definitions above should be contrasted with the so-called MiniMOM [ref] or Taylor coupling [ref] : [equation] It agrees with $α_ghg(p^2)$ for a bare ghost-gluon vertex.

Nonperturbative properties of Yang-Mills theories · Resummed perturbative behavior · 0.88

Cites it as support 9 papers

Solutions of the Dyson-Schwinger equations for the three-gluon and four-gluon couplings [ref] and phenomenological studies [ref] of QCD couplings based on physical observables such as $τ$ decay [ref] and the Bjorken sum rule [ref] , show that the QCD $β$ function vanishes and $α_s(Q^2)$ become constant at small virtuality; i.e., effective charges develop an “infrared fixed point." Recent lattice simulations [ref] and nonperturbative analyses [ref] have also indicated an infrared fixed point for QCD.

AdS/CFT and Light-Front QCD · Introduction · 1.00
A direct consequence of this behavior are positivity violations in the gluon propagator and therefore the confinement of transverse gluons [ref] .

The quark-gluon vertex in Landau gauge QCD: Its role in dynamical chiral symmetry breaking and quark confinement · Yang-Mills sector · 1.00
QCD is not conformal but there is in fact much empirical evidence from lattice gauge theory [ref] , Dyson Schwinger theory [ref] , and empirical effective charges [ref] , that the QCD $β$-function vanishes in the infrared [ref] .

The AdS/QCD Correspondence and Exclusive Processes · AdS/QCD · 0.99

Cites it as background 31 papers

In general DSEs and RGEs allow two different types of solutions in Landau gauge [ref] , called the scaling solution, in which the dressing functions are related via scaling relations [ref] , and the decoupling solution [ref] .

On the infrared scaling solution of SU(N) Yang-Mills theories in the maximally Abelian gauge · Introduction · 0.94
The scaling solution, on the other hand, features an IR divergent ghost dressing and an IR vanishing gluon propagator [ref] .

Going beyond the propagators of Landau gauge Yang-Mills theory · Two-point functions · 0.89
The recent progress is due to the discovery of consistent asymptotic solutions of the whole tower of Dyson-Schwinger (DS) equations and more recently of functional renormalization group (FRG) equations in the deep infrared (IR) limit [ref] .

Lattice gluodynamics computation of Landau-gauge Green's functions in the deep infrared · Introduction · 0.96

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

W. Marciano and H. Pagels, Phys. Rep. 36, 137 (1978)

View on Pith 1978

[2] [2]

S. Mandelstam, Phys. Rev. D 20, 3223 (1979)

View on Pith 1979

[3] [3]

D. Atkinson et al., J. Math. Phys. 22, 2704 (1981); D. Atkinson, P. W. Johnson and K. Stam, ibid. 23, 1917 (1982)

View on Pith 1981

[4] [4]

N. Brown and M. R. Pennington, Phys. Rev. D 39, 2723 (1989)

View on Pith 1989

[5] [5]

A. Hauck, L. v. Smekal and R. Alkofer, e-print, hep- ph/9604430

View on Pith

[6] [6]

M. Baker, J. S. Ball and F. Zachariasen, Nucl. Phys. B186, 531/560 (1981); W. J. Schoenmaker, ibid. B194, 535 (1982); J. R. Cudell and D. A. Ross, ibid. B358, 247 (1991)

View on Pith 1981

[7] [7]

K. B¨ uttner and M. R. Pennington, Phys. Rev. D 52, 5220 (1995)

View on Pith 1995

[8] [8]

R. Alkofer, M. R. Pennington, L. v. Smekal and P. Wat- son, work in progress

View on Pith

[9] [9]

See, e.g., C. D. Roberts and A. G. Williams, Prog. Part. Nucl. Phys. 33, 477 (1994), and references therein

View on Pith 1994

[10] [10]

See also “Dynamical Symmetry Breaking in Quantum Field Theories”, V. A. Miranski, World Scientiﬁc, 1993

View on Pith 1993

[11] [11]

An infrared enhanced gluon propagator was found in Landau gauge in Mandelstam approximation [2–5] as well as in some studies of its simpliﬁed axial gauge DSE [6,7]

View on Pith

[12] [12]

L. G. Vachnadze, N. A. Kiknadze and A. A. Khelashvili, Theor. Math. Phys. 102, 34 (1995)

View on Pith 1995

[13] [13]

We use positive deﬁnite metric, gµν = δµν . Color indices are suppressed and the number of colors is ﬁxed, Nc = 3

View on Pith

[14] [14]

J. C. Taylor, Nucl. Phys. B33, 436 (1971)

View on Pith 1971

[15] [15]

In [20] we derive a Slavnov–Taylor identity for the ghos t– gluon vertex from the usual BRS invariance. This to- gether with the symmetry of the ghost–gluon vertex fully determines its form at the present level of truncation. There are no undetermined transverse terms in this case

View on Pith

[16] [16]

The ghost–gluon vertex (4) is consistent with a ghost– gluon scattering kernel of tree–level structure in the STI of the 3–gluon vertex, which can then be constructed from procedures developed and used previously (U. Bar– Gadda, Nucl. Phys. B163, 312 (1980; S. K. Kim and M. Baker, ibid. B164, 152 (1980); J. S. Ball and T.- W. Chiu, Phys. Rev. D 22, 2550 (1980))

View on Pith 1980

[17] [17]

N. Brown and N. Dorey, Mod. Phys. Lett. A6, 317 (1991)

View on Pith 1991

[18] [18]

D. C. Curtis and M. R. Pennington, Phys. Rev. D 42, 4165 (1990)

View on Pith 1990

[19] [19]

A similar assumption underlies the Mandelstam approx- imation. For q2 > k 2 it can be further justiﬁed from a detailed study of the solutions in the ultraviolet [20]. It will nevertheless be important to assess the sensitivity of the results to the modiﬁed angle approximation in future

View on Pith

[20] [20]

L. v. Smekal, A. Hauck and R. Alkofer, e–print, hep– ph/9707327

View on Pith

[21] [21]

Particle Data Group, Phys. Rev. D 54, 77 (1996)

View on Pith 1996

[22] [22]

C. Bernard, C. Parrinello and A. Soni, Phys. Rev. D 49, 1585 (1994); D. S. Henty et al., ibid. 54 (1996), 6923

View on Pith 1994

[23] [23]

P. Marenzoni, G. Martinelli, and N. Stella, Nucl. Phys. B455, 339 (1995)

View on Pith 1995

[24] [24]

H. Suman and K. Schilling, Phys. Lett. B373, 314 (1996)

View on Pith 1996

[25] [25]

This is supported by the qualitative similarity of our s o- lutions to the infrared behavior obtained from studies of the inﬂuence of a complete gauge ﬁxing by D. Zwanziger, Nucl. Phys. B378, 525 (1992); ibid. B412, 657 (1994)

View on Pith 1992

[26] [26]

We are indebted to F. Coester for pointing this out. 0 200 400 600 800 1000 1200 1400 1600 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 x = k 2 a 2 , with a-1 = 2GeV finite size Z(x)/x Lattice-Fit Marenzoni et. al. 24 3 x 48 16 3 x 48 FIG. 1. The numerical result for the gluon propagator from Dyson–Schwinger equations (solid line) compared to lattic e data from ...

View on Pith