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Uniqueness of infrared asymptotics in Landau gauge Yang-Mills theory

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Christian S. Fischer, Jan M. Pawlowski

High Energy Physics - Theory · Submitted 1 Sep 2006 · 55 references

Pith's verdict: A real structural result — the DSE/FRG squeeze to κ_{2,1}=0 and κ_{0,2}=-2κ_{2,0} is clean — but the 'uniqueness' and the exclusion of Boucaud et al. rest on an acknowledged no-cancellation premise.

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arXiv IDhep-th/0609009 (version 1)
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CitationChristian S. Fischer, Jan M. Pawlowski. "Uniqueness of infrared asymptotics in Landau gauge Yang-Mills theory." arXiv:hep-th/0609009 (2006). https://pith.science/paper/A35QI6XJ
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The paper

The paper claims

Landau-gauge Yang-Mills theory has one infrared power-law solution, not two, once its two equation hierarchies are required to agree.

The reading

In Landau-gauge Yang-Mills theory the ghost propagator, the gluon propagator, and every proper vertex are expected to behave as pure power laws in the deep infrared, and for a decade two competing assignments have been on the table: the scaling solution, in which the ghost dressing diverges and the gluon propagator vanishes, and a finite alternative in which both stay bounded. This paper argues that only the first survives. The authors do not solve either the Dyson-Schwinger hierarchy or the functional renormalisation-group hierarchy; instead they exploit a structural difference between them — Dyson-Schwinger equations contain bare vertices, flow equations contain only dressed ones — and demand that one set of power laws satisfy both. Consistency then forces the ghost-gluon vertex to carry zero anomalous dimension and ties the gluon exponent to minus twice the ghost exponent, a relation that propagates to every vertex with $n$ ghost legs and $m$ gluon legs as $Z^{(2n,m)}_0(p^2) = (p^2)^{(n-m)K}$ with $K>0$.

What carries the argument

The engine is the contrast between the two functional equations. The Dyson-Schwinger equation for a vertex is a sum of diagrams each carrying one classical vertex from the bare action; the flow equation is a sum of one-loop diagrams with regulator insertions and only fully dressed vertices. Because the flow depends on no bare quantity, a vertex that appears in a flow must have an anomalous dimension compatible with the dressed vertices alone, whereas the same vertex appearing in a Dyson-Schwinger equation is constrained by the bare vertex it multiplies. The load-bearing input is Taylor's non-renormalisation theorem for the ghost-gluon vertex, which supplies $K_{2,1} < 0$ and lets the inequality chain close. The authors also use a single-mode regulator that has support only at momenta near the cutoff scale, so that the momentum dependence of the effective action can be read off directly from the flow at $k^2=p^2$; they note in a footnote that such a regulator cannot be used to solve the theory by successively integrating out modes, only to study fixed-point structure.

Extended reading notes

Core claim

The paper's claim is that the infrared asymptotics of Landau-gauge Yang-Mills theory are uniquely determined, not merely one consistent option among several. Writing the anomalous exponent of a proper vertex with $n$ ghost, $n$ anti-ghost and $m$ gluon legs as $K_{2n,m}$, the authors derive from the ghost-gluon vertex a chain of inequalities that closes: the flow equation bounds $K_{2,1} < 0$ from the non-renormalisation of that vertex, the bound $K_{2,1} < 2K_{2,0}+K_{0,2}$ is shown to be saturated, and the Dyson-Schwinger bound $K_{2,1} < K_{0,2}+2K_{2,0}$ then collapses to $K_{2,1} < 2K_{2,1}$, giving $K_{2,1}=0$ and $K_{0,2}=-2K_{2,0}$. The same style of argument on the three- and four-gluon vertices gives $K_{0,3} = 3K_{0,2}/2$ and $K_{0,4} = 2K_{0,2}$, and an induction then rules out any vertex that would scale more singularly than $(n-m)K$. The concrete casualty is the finite-propagator proposal, in which $p^2D(p^2) = (p^2)^1$ and the ghost dressing stays constant: inserting either of the three vertices that would be needed to rescue it into the gluon flow forces $K_{0,2} <-1$, contradicting $K_{0,2}=-1$.

Load-bearing premise

The whole uniqueness argument assumes that no cancellations occur between different diagrams, or inside a single diagram, in the infrared, so that the most singular diagram sets the leading behaviour; the authors acknowledge this at the end of section II and again in section V and argue that such cancellations would need symmetry protection and are highly unlikely, but they do not prove their absence.

Editorial extensions

If this is right

  • The Kugo-Ojima and Gribov-Zwanziger horizon conditions hold, so the ghost dressing diverges and the gluon propagator vanishes in the infrared, giving a well-defined global colour charge and positivity violation for transverse gluons.
  • The running couplings built from the ghost-gluon, three-gluon and four-gluon vertices all approach infrared fixed points, since the vertex exponents combine with $K_{0,2}=-2K$ to cancel momentum dependence.
  • Truncations of the gluon Dyson-Schwinger equation that keep only the gluonic one-loop diagram, or that treat all diagrams as equally leading, miss the dominant infrared contribution and are excluded.
  • Lattice extrapolations that report a finite gluon propagator together with a finite ghost dressing cannot survive the infinite-volume continuum limit, though a finite gluon propagator with a diverging ghost remains compatible.
  • The same bookkeeping extends to Yang-Mills-Higgs theory in the symmetric phase, where the Higgs propagator is predicted to have a constant dressing and the four-Higgs coupling to have zero anomalous dimension.

Reading between the lines

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

  • The argument is a consistency argument, not a construction: it says which power laws can coexist, not that a solution with those power laws exists. A constructive check — truncating both hierarchies at a finite order and verifying that the scaling solution is reached numerically — would be the natural next step and is not attempted here.
  • If the uniqueness claim holds, the practical bottleneck for lattice comparison shifts entirely to the ghost sector: the gluon exponent $K_{0,2}=-2K$ is only as good as the ghost measurement it is tied to, so a lattice determination of $K$ at the few-percent level would test the whole chain rather than one propagator.
  • The structural observation — that a Dyson-Schwinger equation's bare vertex forces at least one dressed vertex in the theory to have zero anomalous dimension — looks portable to other gauge-fixed theories with a non-renormalised vertex, and could be used as a cheap filter on candidate scaling solutions before any numerical work.
  • The paper's exclusion of the finite-propagator scenario rests on the absence of cancellations between diagrams. A symmetry that protects such a cancellation would be visible as an exact linear relation among diagrams in the flow of the ghost-gluon vertex; searching for one directly is a sharper test than accumulating lattice evidence.
Original abstract

We uniquely determine the infrared asymptotics of Green functions in Landau gauge Yang-Mills theory. They have to satisfy both, Dyson-Schwinger equations and functional renormalisation group equations. Then, consistency fixes the relation between the infrared power laws of these Green functions. We discuss consequences for the interpretation of recent results from lattice QCD.

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, 3,456 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.09 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.07 · compares with prior results 0.44 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

We uniquely determine the infrared asymptotics of Green functions in Landau gauge Yang-Mills theory. They have to satisfy both, Dyson-Schwinger equations and functional renormalisation group equations. Then, consistency fixes the relation between the infrared power laws of these Green functions. We discuss consequences for the interpretation of recent results from lattice QCD.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We uniquely determine the infrared asymptotics of Green functions in Landau gauge Yang-Mills theory. result 0.76 0.93 0.93 asserted 1.00
s1 They have to satisfy both, Dyson-Schwinger equations and functional renormalisation group equations. method 0.77 0.18 0.06 asserted 0.98
s2 Then, consistency fixes the relation between the infrared power laws of these Green functions. result 0.92 0.76 0.20 asserted 1.00
s3 We discuss consequences for the interpretation of recent results from lattice QCD. interpretation 0.98 0.76 0.05 unclear 0.78

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.97.

What was examined, and what was not

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  • figures and tables
  • references and citing sentences
  • relations to other papers
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  • 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: Functional DSE for the effective action. Filled circle
Figure 2. FIG. 2: Functional flow for the effective action. Filled circle
Figure 3. FIG. 3: Dyson-Schwinger equations for the gluon and ghost pr
Figure 4. FIG. 4: Dyson-Schwinger equation for the ghost-gluon verte
Figure 5. FIG. 5: Functional renormalisation group equations for the
Figure 6. FIG. 6: Functional renormalisation group equations for the
Figure 7. FIG. 7: Functional renormalisation group equations for the
Figure 8. FIG. 8: Functional renormalisation group equations for the

Figures reproduced from arXiv:hep-th/0609009 by the authors.

References
How the literature uses it

59 arXiv papers cite this in 134 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

Papers citing it · 99 sentences without a confident label

Builds on it 1 paper
  • In fact, the complete construction of the infrared behavior of Green's functions in SU($N$) gauge theory [ref] can be carried over to arbitrary (semi-)simple gauge groups, as the color structure separates trivially, as long as only the tree-level color structure is present.

    A first look at Landau-gauge propagators in G2 Yang-Mills theory · The ghost propagator · 0.93

Uses its method or tool 2 papers
Compares with it 3 papers
Cites it as support 8 papers
  • This bound is furthermore required to be satisfied due to the following argument: an IR fixed point for the background-field running coupling is observed in the background-field gauge in a derivative expansion [ref] as well as for the running coupling fixed at the ghost-gluon vertex in the Landau gauge in a vertex expansion [ref] .

    Gluon condensation and scaling exponents for the propagators in Yang-Mills theory · relation between the $β$ function and the critical exponents · 0.99

  • This solution agrees with the Slavnov-Taylor identities and is the unique scaling solution in the infrared [ref] .

    Infrared behavior of QCD from the Dyson-Schwinger formalism · Infrared Yang-Mills theory from DSEs · 0.96

  • 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

Cites it as background 17 papers

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