Infrared behaviour of the ghost-gluon vertex in Landau gauge Yang-Mills theory

Axel Maas; Jochen Wambach; Reinhard Alkofer; Wolfgang Schleifenbaum

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Infrared behaviour of the ghost-gluon vertex in Landau gauge Yang-Mills theory

Wolfgang Schleifenbaum, Axel Maas, Jochen Wambach, Reinhard Alkofer

High Energy Physics - Phenomenology · Submitted 3 Nov 2004 · 34 references

Pith's verdict: A modest, useful one-iteration DSE evaluation of the ghost-gluon vertex that supports the near-bare picture, but its robustness claims lean on an unpublished diploma thesis.

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arXiv IDhep-ph/0411052 (version 2)

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CitationAxel Maas, Jochen Wambach, Reinhard Alkofer, Wolfgang Schleifenbaum. "Infrared behaviour of the ghost-gluon vertex in Landau gauge Yang-Mills theory." arXiv:hep-ph/0411052 (2004). https://pith.science/paper/4P47IYWD

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Worth your time if you work with the DSE truncation; skip it otherwise. The new content is concrete: they take the ghost-gluon vertex DSE, drop the four-point scattering kernel, feed in propagators from the standard truncated system, and do one iteration — in D=4 and D=3, with the bare vertex and with the STI-inspired ansatz of Eq. 8 as inputs. The output stays within roughly 20% of tree level for 1+A, with |B| below 0.2, and the STI input comes out closer to tree level than it went in. That contraction is the real evidence in the paper, and it deserves more emphasis than it gets. The appendix derivation is careful, and the SU(2) lattice comparison in Fig. 5 is handled honestly, including the color prefactor adjustment. The D=3 parallel is a nice consistency check. Where it is soft: the adequacy of one iteration is asserted, not shown. The line 'Fortunately, as we will see below, such a procedure is not necessary' carries a lot of weight, and what follows is two input choices, not a demonstration that the DSE map is contracting. The robustness claims — 'a systematic study of possible input vertex choices yields the same result' and the D=3 parallel — both point to ref. [31], a 2004 Darmstadt diploma thesis that a referee cannot check. Either fold the essentials into this paper or soften the wording. The circularity worry is real but smaller than it first looks: the propagator inputs did assume a bare vertex, but the STI test and the lattice comparison are independent anchors. The lattice data, however, are explicitly consistent with tree level within errors, so they cannot discriminate a 20% deviation; the paper should say that plainly instead of claiming agreement. The non-uniform zero-momentum limit is also asserted rather than demonstrated. None of this breaks the central claim, which is modest and correctly hedged. It is a useful brick in the ghost-dominance story. Send it out, and ask for the [31] material or a weakened claim, plus some estimate of the truncation error.

Referee report

Major objections (3)

[Section III, Eq. (7) and Fig. 1] The adequacy of a single iteration is asserted rather than demonstrated. The text states that 'a self-consistent solution ... is of significant technical complexity. Fortunately, as we will see below, such a procedure is not necessary,' but the two-input test of Fig. 4 shows only that two particular inputs map closer to tree level; it does not establish that the DSE map is contracting or that the one-iteration output coincides with the fixed point. The supporting 'systematic study of possible input vertex choices' and the D=3 parallel both rest on ref. [31], a diploma thesis that is not publicly verifiable. Either a second iteration (or an explicit contraction argument) should be shown, or the claim that the full solution 'will likely be very close to the tree-level form' should be softened to match what the two-input test actually supports.
[Section II and Section III] The propagator inputs of Eq. (5) and refs. [25, 26] were themselves obtained from a DSE system in which the ghost-gluon vertex was taken bare. A near-bare output therefore partly re-derives an assumption already present in the input, and the manuscript does not discuss this residual circularity. The STI-ansatz test of Fig. 4 and the lattice comparison of Fig. 5 are the genuinely independent anchors and should be foregrounded as such. A short explicit discussion of the circularity, and ideally a sensitivity check (for example, recomputing the propagators with the calculated vertex), would materially strengthen the validation claim.
[Section III, Figs. 2 and 3] The quantitative claim that deviations are 'clearly less than 20%' and that B is 'smaller than 0.2 for almost all momenta' carries no estimate of the truncation error from dropping the four-point function and taking the three-gluon vertex bare (footnote 4). Because the central quantitative statement is a bound, at least an order-of-magnitude estimate of the neglected contributions, for instance by varying the three-gluon vertex input or including an approximate four-point kernel, would make the bound meaningful. As it stands the 20% figure should be presented as indicative rather than as a controlled error bar.

Minor objections (5)

[Eq. (9)] The extraction divides by the Gram determinant Delta = q^2 k^2 - (q.k)^2, which vanishes when q is parallel to k; in that kinematics the q_mu and k_mu structures of Eq. (6) degenerate and A and B cannot be separated. The statement that the sub-20% bound holds 'for all momenta allowed by momentum conservation' should therefore be qualified to exclude the degenerate configuration.
[Fig. 5 and abstract] The lattice data are for SU(2) and only for vanishing gluon momentum, with the smallest available momentum 366 MeV, whereas the DSE result is shown down to 0.01 GeV. The abstract's 'in agreement with recent lattice data' should be qualified to the momentum range and kinematics actually covered by the lattice, and the color-stripping procedure (changing the loop prefactor to f^{abc}Nc/2) should be stated explicitly in the caption or text.
[Section III, end] The statement that A and B have a 'seemingly non-uniform limit' when all three momenta vanish is explained by the ordering of limits, but no explicit limiting values along different paths are given. A one-line demonstration would make the regularity claim for the full vertex transparent.
[Section III, Figs. 2 and 3] The phrase 'smaller than 0.2 for almost all momenta' should specify which momenta exceed this value, so that the reader can judge whether the exceptions are kinematically relevant.
[Reference [31]] Several robustness claims (the systematic input-variation study and the D=3 parallel) are delegated to a diploma thesis. If these are important to the argument, consider making the relevant material available in a citable, publicly accessible form.

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The paper

The paper claims

The ghost-gluon vertex stays close to its bare form at every momentum, including the infrared.

The reading

In Landau-gauge Yang–Mills theory the ghost-gluon vertex is special: perturbation theory says it never needs its own ultraviolet renormalization, and it has been conjectured that in the extreme infrared it reduces to its bare (tree-level) form. That conjecture matters because almost every Dyson–Schwinger calculation of the gluon and ghost propagators, and much of the hadron physics built on them, simply assumes the bare vertex. This paper tests the assumption by inserting numerically determined non-perturbative gluon and ghost propagators into the one-loop Dyson–Schwinger equation for the vertex itself and iterating once, in four and in three Euclidean dimensions. The answer is that the transverse part $1+A$ stays within roughly 20 percent of unity for every momentum configuration allowed by momentum conservation, and the longitudinal function $B$ stays below about 0.2 and finite everywhere, with no sign of an infrared singularity. The result agrees with recent lattice data for the same quantity and, if it holds, closes a gap in the chain of approximations behind the ghost-dominance picture of confinement.

What carries the argument

The machinery is a single iteration of the ghost-gluon vertex Dyson–Schwinger equation, truncated by dropping the ghost-gluon scattering kernel and the four-point function, with the three-gluon vertex taken bare. The two loop integrals (one with two ghost-gluon vertices, one with a three-gluon vertex) are evaluated using non-perturbative gluon and ghost dressing functions: in $D=4$ a pointwise fit to the coupled propagator equations with infrared power laws $Z(p^2) 招sim (p^2)^{2招kappa+2-D/2}$ and $G(p^2) 招sim (p^2)^{-招kappa}$, $招kappa 招approx 0.595$, and in $D=3$ the numerical propagator solutions with $招kappa 招approx 0.398$. The transverse part is projected out with a Gram determinant, $k^2/(ig_D 招Delta), q^招nu (招delta_{招mu招nu}-k_招mu k_招nu/k^2) 招Gamma^{abc}_招mu = f^{abc}(1+A)$, and the same loop is rerun with the Slavnov–Taylor ansatz $q_招mu(G(k^2)/G(q^2)+G(k^2)G(p^2)-1)$ as input to check stability of the conclusion against the choice of input vertex.

Extended reading notes

Core claim

The paper reports that the full non-perturbative ghost-gluon vertex in Landau gauge is numerically very close to its tree-level form across the whole momentum range, in both $D=4$ and $D=3$. Writing the vertex as $ig_D(q_招mu(f^{abc}+A^{abc})+k_招mu B^{abc})$, the authors compute the scalar functions $A$ and $B$ from a truncated Dyson–Schwinger equation in which the propagators are dressed with previously obtained non-perturbative solutions and the internal vertices are left bare. The transverse combination $1+A$ deviates from 1 by less than about 20 percent in all kinematic regions studied, and $B$ remains smaller than 0.2 and finite. Repeating the calculation with a Slavnov–Taylor-motivated vertex ansatz as input, the output vertex comes out closer to tree level than the input was, which the authors read as evidence that the self-consistent solution is near the bare vertex. Comparison with lattice data for $1+A(0;p^2,p^2)$ in SU(2) shows agreement within the lattice systematic errors, with a predicted slight decrease at small momenta. In particular, the authors see no sign of a singular infrared behaviour of the vertex.

Load-bearing premise

A single iteration of the vertex equation, with the ghost-gluon scattering kernel and four-point function dropped and the three-gluon vertex left bare, is assumed to land close enough to the true self-consistent solution that the reported 20 percent deviation reflects the full vertex rather than the truncation.

Editorial extensions

If this is right

The bare-vertex truncation used in the propagator Dyson–Schwinger equations is numerically justified, so the propagator solutions and the confinement picture built on them do not need vertex corrections at leading order.
Zwanziger's hypothesis of a bare ghost-gluon vertex in the infrared is satisfied, which supports the Gribov-horizon and Kugo–Ojima lines of argument for confinement.
Because the vertex stays finite and close to bare, the non-perturbative running coupling $招alpha(p^2)=招alpha(招mu^2)G^2(p^2,招mu^2)Z(p^2,招mu^2)$ needs no vertex function at any scale, extending the perturbative non-renormalization statement into the infrared.
The result is expected to survive the inclusion of matter fields, since the input propagators are only weakly sensitive to quark contributions and no new terms enter the truncated equation, in contrast to the quark-gluon vertex where large deviations are found.

Reading between the lines

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

If the vertex really is bare to within 20 percent everywhere, then the product $G^2(p^2,招mu^2)Z(p^2,招mu^2)$ should be renormalization-point independent as a fully non-perturbative statement, which is a sharper check than any single propagator comparison.
The same one-iteration strategy applied to the quark-gluon vertex would give a controlled contrast: the same truncation that lands near tree level for ghosts should reproduce the reported large deviations for quarks, or else the truncation itself is suspect.
The predicted slight decrease of $1+A$ at small momenta is a concrete target for lattice work at finer spacing and larger volume, where the smallest accessible momentum falls below the 366 MeV quoted here.
If a future self-consistent solution including the ghost-gluon scattering kernel found leading-order contributions from it, the error would propagate into every hadron-physics calculation that uses these propagators as input.

Original abstract

A semi-perturbative calculation of the ghost-gluon vertex in Landau gauge Yang-Mills theory in four and three Euclidean space-time dimensions is presented. Non-perturbative gluon and ghost propagators are employed, which have previously been calculated from a truncated set of Dyson--Schwinger equations and which are in qualitative and quantitative agreement with corresponding lattice results. Our results for the ghost-gluon vertex show only relatively small deviations from the tree-level one in agreement with recent lattice data. In particular, we do not see any sign for a singular behaviour of the ghost-gluon vertex in the infrared.

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What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.33 · human participants 0.01 · controlled comparison 0.41 · randomized assignment 0.01 · states quantitative result 0.09 · compares with prior results 0.98 · 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 semi-perturbative calculation of the ghost-gluon vertex in Landau gauge Yang-Mills theory in four and three Euclidean space-time dimensions is presented. Non-perturbative gluon and ghost propagators are employed, which have previously been calculated from a truncated set of Dyson--Schwinger equations and which are in qualitative and quantitative agreement with corresponding lattice results. Our results for the ghost-gluon vertex show only relatively small deviations from the tree-level one in agreement with recent lattice data. In particular, we do not see any sign for a singular behaviour of the ghost-gluon vertex 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 semi-perturbative calculation of the ghost-gluon vertex in Landau gauge Yang-Mills theory in four and three Euclidean space-time dimensions is presented.	contribution 0.99	0.95	0.04	asserted 0.95
s1	Non-perturbative gluon and ghost propagators are employed, which have previously been calculated from a truncated set of Dyson--Schwinger equations and which are in qualitative and quantitative agreement with corresponding lattice results.	method 0.94	0.14	0.04	asserted 0.96
s2	Our results for the ghost-gluon vertex show only relatively small deviations from the tree-level one in agreement with recent lattice data.	result 1.00	0.95	0.41	asserted 1.00
s3	In particular, we do not see any sign for a singular behaviour of the ghost-gluon vertex in the infrared.	result 0.93	0.94	0.75	asserted 0.97

What kind of paper the classifier reports this is: theoretical result 0.85 · evidence from simulation or computation 0.92.

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full text
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P abstract

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T body sentences
C citations
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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 truncated DSE for the ghost-gluon vertex. Dotted

**Figure 2.** FIG. 2: The normalized transverse part 1 +

**Figure 3.** FIG. 3: Same as Fig. 2 for

**Figure 4.** FIG. 4: Same as Fig. 2 but here with eq. (8) as input for the ghos

Figures reproduced from arXiv:hep-ph/0411052 by the authors.

References

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How the literature uses it

49 arXiv papers cite this in 80 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

In all cases considered, the ghost-gluon vertex is found to be essentially tree-level-like, while the three-gluon vertex is suppressed at intermediate momenta.

Three-point vertices in Landau-gauge Yang-Mills theory · 0.87

States a conflicting result 1 paper

We find that there is indeed a massless pole in the ghost-gluon vertex, however in our approximation with the assumption of complete infrared dominance of the ghost this pole is only present for the scaling solution.

On mass generation in Landau-gauge Yang-Mills theory · 0.80

Papers citing it · 57 sentences without a confident label

Uses its method or tool 2 papers

This procedure is described in detail in [ref] . * Selfconsistency effects of the four-gluon vertex will be neglected. * The dressed ghost-gluon vertex will be replaced by the bare vertex [ref] . * We employ an ansatz for the three-gluon vertex, which is constructed such that it has the infrared behaviour as given in [ref] and guarantees the correct UV-behaviour of the four-gluon DSE as known from perturbation theory.

The four-gluon vertex and the running coupling in Landau gauge Yang-Mills theory · The DSE for the four-gluon vertex · 0.97
One can actually analytically calculate $C$ by means of the DSE for the ghost-gluon vertex [ref] , see Fig. [ref],

Nonperturbative aspects of Yang-Mills theory · Ghost-gluon vertex · 0.89

Compares with it 3 papers

We point out that in the soft-gluon limit this $B_1(r,p,q)$ reproduces the lattice data of [ref] (see Fig. 14 of [ref] ), and agrees qualitatively with various continuum studies [ref] .

Gluon mass scale through the Schwinger mechanism · Computing W(r2) · 0.98
It suffices to mention that the results for $B_1(r,p,q)$ deviate only moderately from its tree-level value, in agreement with several previous continuum studies [ref] , and reproduce the available lattice data from Ref. [ref] .

Evidence of the Schwinger mechanism from lattice QCD · Ghost-gluon kernel contribution · 0.88
As in four dimensions, this result coincides with predictions obtained using functional methods [ref] .

Three-point vertices in Landau-gauge Yang-Mills theory · Vertices in three dimensions · 0.84

Cites it as support 7 papers

Since neither the DSE studies [ref] nor the lattice calculations [ref] show any infrared divergences, the dressing function of the ghost-gluon vertex must be some finite function.

Nonperturbative aspects of Yang-Mills theory · Ghost-gluon vertex · 0.96
As an approximation this is well justified by the overall comparatively small deviations of the fully momentum dependent vertex from its tree-level form [ref] which induce only minor changes in the propagators likewise [ref] .

Spurious divergences in Dyson-Schwinger equations · The gluon propagator DSE · 1.00
This property has already been conjectured by Taylor in the early seventies [ref] and has recently been verified numerically in continuum as well as lattice calculations [ref] .

The running coupling from the four-gluon vertex in Landau gauge Yang-Mills theory · The DSE for the four-gluon vertex · 0.98

Cites it as background 9 papers

The groundwork was laid by various calculations of vertices, e.g., [ref] .

On non-primitively divergent vertices of Yang-Mills theory · Introduction · 0.99
Some attention have been recently paid to the ghost-gluon vertex DSE [ref] .] orders:

On the leading OPE corrections to the ghost-gluon vertex and the Taylor theorem · The Taylor kinematics and the asymmetric gluon-ghost vertex · 0.92
Non-perturbative modifications of the vertex were investigated with various approaches: Monte-Carlo simulations [ref] , semi-perturbative DSE calculations [ref] , OPE analysis [ref] and based on this indirect studies via effects in the ghost DSE [ref] .

Going beyond the propagators of Landau gauge Yang-Mills theory · Ghost-gluon vertex · 0.97

Discussion

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