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A Solution to Coupled Dyson-Schwinger Equations for Gluons and Ghosts in Landau Gauge

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Lorenz von Smekal (1), Andreas Hauck (2), Reinhard Alkofer (2) ((1) Argonne National Laboratory, (2) Universitaet Tuebingen)

High Energy Physics - Phenomenology · Submitted 11 Jul 1997 · 54 references

Pith's verdict: The qualitative picture — ghost dominance, a vanishing gluon propagator, and an infrared fixed point — is the durable contribution; the specific numbers κ ≈ 0.92 and α_c ≈ 9.5 come out of the one-dimensional angle approximation and did not survive the full angular treatment.

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arXiv IDhep-ph/9707327 (version 3)
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Citation(2) Universitaet Tuebingen), Andreas Hauck (2), Lorenz von Smekal (1), Reinhard Alkofer (2) ((1) Argonne National Laboratory. "A Solution to Coupled Dyson-Schwinger Equations for Gluons and Ghosts in Landau Gauge." arXiv:hep-ph/9707327 (1997). https://pith.science/paper/SFWYAGIY
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The paper

The paper claims

Solving the coupled gluon and ghost equations of QCD in Landau gauge, the authors find that ghost loops, not gluon loops, control the far-infrared: the gluon propagator vanishes, the ghost propagator is enhanced, and the running coupling fl

The reading

This paper asks what gluons and ghosts actually do at long wavelengths in QCD, where perturbation theory gives no answer at all. The authors build a truncation of the coupled Dyson–Schwinger equations in Landau gauge in which the three-point vertices are not guessed but constructed from the Slavnov–Taylor identities that gauge invariance imposes on them, and then solve the resulting closed system for the gluon and ghost propagators in the pure gauge theory. The headline finding is a reversal of the earlier folklore: ghost loops, usually dismissed as numerically small, dominate the infrared, so that the gluon propagator vanishes as $D(k) = Z(k^2)/k^2$ with $Z(x) = (x)^{2-2/d}$-type falloff while the ghost propagator is enhanced relative to a free massless pole. Because the product $g^2 Z G^2$ is renormalization-group invariant in this scheme, the same infrared behaviour makes the running coupling freeze at a finite value rather than diverge, giving an infrared stable fixed point at $g_c^2$ with $4.5$ times the squared coupling of the strong force. A reader should care because this is a concrete, analytically controlled statement about the infrared limit of a confining gauge theory, with an irrational exponent that lattice simulations can check and with a direct consequence for whether gluons appear in the physical spectrum.

What carries the argument

The load-bearing device is a construction of the three-point vertices from Slavnov–Taylor identities rather than from bare or model Ansätze. The authors first derive a new BRS identity for the ghost–gluon vertex, neglecting irreducible ghost–ghost scattering, and solve it with a vertex dressed by the ghost renormalization function, $G_μ(p,q) = i q_μ G(k^2)/G(q^2) + $ longitudinal terms. That dressed vertex is what makes the three-gluon Slavnov–Taylor identity solvable without extra assumptions; the authors show explicitly in an appendix that a tree-level ghost–gluon vertex admits no solution with the correct symmetry, which is why the dressed form is needed. With these vertices the tower of equations closes at the level of propagators. To solve it, they use a modified angle approximation that replaces the angular dependence of the loop integrands by the external momentum, chosen to preserve both the leading infrared limit and the correct one-loop ultraviolet behaviour, and then match an asymptotic infrared series in powers of $x^{2.5}$, $x^{2.5}$ and $x^{1+2.5}$ to an iterative numerical solution, with a one-parameter family of solutions labelled by $t$ that is fixed by requiring a true (not merely local) infrared fixed point.

Extended reading notes

Core claim

The central claim is that in Landau-gauge QCD without quarks the infrared behaviour of the propagators is governed entirely by ghost contributions, and is characterized by a single irrational exponent $2.5$ that the authors compute analytically. Writing $Z(x)G(x) = (x)^{2.5}$ for small $x = k^2$, they derive $G(x) = (x)^{-2.5}$ and $Z(x) = (x)^{2.5}$, with the exponent fixed by matching the ghost loop against the three-gluon loop in the gluon equation, giving $2.5 = (61 - 2.5)/19$, so that $D(k) = Z(k^2)/k^2$ vanishes as $(k^2)^{2.5-1}$ and the ghost propagator $D_G(k) = -G(k^2)/k^2$ is enhanced as $(1/k^2)^{1+2.5}$. The exponent is independent of the number of colours. From the same behaviour the renormalization-group-invariant product $g^2 Z G^2$ tends to a finite constant, so the running coupling of the non-perturbative subtraction scheme approaches an infrared stable fixed point at $2.5$ and $4.5$ times the strong coupling, and the authors note that this is a counterexample to the assertion that ghosts have negligible influence on strong-interaction physics. They further show, by Fourier-transforming the gluon propagator in time, that it takes negative values and therefore violates reflection positivity, which means it has no Lehmann representation and cannot describe a physical asymptotic gluon state.

Load-bearing premise

The reduction of the four-dimensional loop integrals to one-dimensional equations by the modified angle approximation, which replaces the angular dependence of the integrands with the external momentum and whose error the authors justify qualitatively but never bound.

Editorial extensions

If this is right

  • The gluon propagator vanishes at long range rather than diverging like $σ/k^4$, so the string-tension interpretation of the earlier gluon-only truncation does not survive once ghosts are included.
  • The running coupling has a genuine infrared fixed point at $α_c = g_c^2/(4π) ≈ 9.5$, scaling as $1/N_c$, which is the behaviour expected from large-$N_c$ counting.
  • The gluon propagator violates reflection positivity and therefore has no Lehmann representation, which the authors read as a signal of gluon confinement.
  • Because the combination $g^2 G^2(k^2) Z(k^2)/k^2$ is free of the unphysical infrared divergences that appear in the individual vertex functions, it is the natural effective quark interaction in Landau gauge, and the authors verify that its Fourier transform has no zeros.
  • The ghost propagator's non-integer infrared exponent $1+2.5$ with $2.5 ≈ 0.92$ is a sharp prediction that can be compared directly with lattice data.

Reading between the lines

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

  • The exponent $2.5 ≈ 0.92$ sits just below $1$, which is the boundary the authors themselves derive from the requirement that the vertex functions remain consistent; a lattice measurement of $2.5$ noticeably above $1$ would put the truncation in a qualitatively different regime rather than merely shifting a number.
  • The fixed-point value $α_c ≈ 9.5$ is quoted for the pure gauge theory; adding quark loops changes the one-loop coefficient $β_0$ and hence the relation between $α_c$ and $2.5$, so the number should be read as a benchmark for the quenched case rather than a QCD prediction.
  • The agreement with lattice ghost data, obtained with algorithms that suppress Gribov copies, suggests that the copy ambiguity is a small perturbation on these particular correlators; that is a checkable statement, since a lattice study varying the copy-fixing algorithm should leave the fitted exponent stable.
  • If the gluon propagator really vanishes in the infrared, the appropriate effective long-distance degree of freedom in Landau gauge is the ghost-dressed combination $g^2 G^2 D$, which suggests that model builders using a phenomenological gluon propagator with a $1/k^4$ infrared enhancement are using the wrong object.
Original abstract

A truncation scheme for the Dyson-Schwinger equations of QCD in Landau gauge is presented which implements the Slavnov-Taylor identities for the 3-point vertex functions. Neglecting contributions from 4-point correlations such as the 4-gluon vertex function and irreducible scattering kernels, a closed system of equations for the propagators is obtained. 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 running coupling of the non-perturbative subtraction scheme approaches an infrared stable fixed point at a critical value of the coupling alpha_c of approx. 9.5. The gluon propagator is shown to have no Lehmann representation. The results for the propagators obtained here compare favorably with recent lattice calculations.

The compiled record

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What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.80 · human participants 0.01 · controlled comparison 0.04 · randomized assignment 0.01 · states quantitative result 0.98 · compares with prior results 0.94 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v3. Results and the main claim are underlined; .

A truncation scheme for the Dyson-Schwinger equations of QCD in Landau gauge is presented which implements the Slavnov-Taylor identities for the 3-point vertex functions. Neglecting contributions from 4-point correlations such as the 4-gluon vertex function and irreducible scattering kernels, a closed system of equations for the propagators is obtained. 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 running coupling of the non-perturbative subtraction scheme approaches an infrared stable fixed point at a critical value of the coupling alpha_c of approx. 9.5. The gluon propagator is shown to have no Lehmann representation. The results for the propagators obtained here compare favorably with recent lattice calculations.

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 A truncation scheme for the Dyson-Schwinger equations of QCD in Landau gauge is presented which implements the Slavnov-Taylor identities for the 3-point vertex functions. contribution 0.57 0.92 0.08 asserted 1.00
s1 Neglecting contributions from 4-point correlations such as the 4-gluon vertex function and irreducible scattering kernels, a closed system of equations for the propagators is obtained. method 0.95 0.90 0.05 asserted 1.00
s2 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. result 0.93 0.96 0.60 asserted 1.00
s3 The running coupling of the non-perturbative subtraction scheme approaches an infrared stable fixed point at a critical value of the coupling alpha_c of approx. 9.5. result 1.00 0.90 0.18 asserted 1.00
s4 The gluon propagator is shown to have no Lehmann representation. result 0.95 0.91 0.12 asserted 1.00
s5 The results for the propagators obtained here compare favorably with recent lattice calculations. interpretation 0.54 0.68 0.08 asserted 0.79

What kind of paper the classifier reports this is: theoretical result 0.86 · evidence from proof or derivation 0.81.

What was examined, and what was not

sources read
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not examined
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  • figures and tables
  • references and citing sentences
  • relations to other papers
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  • P abstract
not run
  • S sections
  • 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. Diagrammatic representation of the gluon, ghost a
Figure 2. Diagrammatic representation of the gluon and ghos
Figure 3. Diagrammatic representation of the gluon Dyson–S
Figure 4. The gluon renormalisation function gZ(k 2 ) = 8πσ/k2 +gC(k 2 ) for eq. (37). The dashed lines show the corresponding function gC(k 2 ) for eqs. (37) and (34). ization group invariant product gZ(k 2 ) which is equivalent to the running coupling g¯(t, g) of the scheme, gZ(k 2 ) = ¯g(tk, g) , tk = 1 2 ln k 2 /µ2 . (40) The scaling behavior of the solution near the ultraviolet fixed point is determined by the coefficients β0 = 25/2 and γ 0 A = 25/4 for eq. (34) vs. β0 = 14 and γ 0 A = 7 for (37) which are reasonably close to the perturbative values for Nf = 0, i.e., β0 = 11 and γ 0 A =…
Figure 5. The dismissed contribution (49) (dashed line) com
Figure 6. The numerical solutions of F(x) and R(x) for t = 0 and b = 1 together with their asymptotic expansions to order N = 1 as well as N = 2 at small x. below). Furthermore, it can be verified numerically, that a solution for a value of b 6= 1 for fixed t is identical to a solution for b = 1 and t ′ = t bν/κ, if x is substituted by x ′ = x/b1/κ. This is the numerical manifestation of the scale invariance mentioned above (for λ = 1/b1/κ). Note that under scale transformations (107) the constant A trivially transforms according to its dimension, A → A′ = A/λ, without any adjustments from the
Figure 7. The numerical solutions of F(x) and R(x) with b = 1 for different values of the parameter t = {−4, −2, −1, 0, 1, 2, 4, 8, 16} (solid lines represent t = 0 solutions). 8 Numerical Results and Perturbative Limit The numerical results were obtained with the order N = 1 in the asymptotic expan￾sion for which no dependence on x0 was observed for x0 < 0.1. Higher values for the matching point are possible at higher orders. This shows that the expansion to next to leading order in the infrared (N = 1) is sufficient to find numerically stable results for a satisfactory regime of matching…
Figure 8. The renormalization functions Z(x) and G(x) obtained for t = 0 and their leading logarithmic forms (with γ → 1 − 2δ for the gluon, and γ → δ for the ghost). logarithms. This will, however, at the same time affect the perturbative form of the other renormalization function in the opposite direction, e.g., for λ approximately 4 ∼ 5 the perturbative logarithm of the gluon is hardly distinguishable from Z(x) for x > 10 on the scales as used in figure 8, while a similarly good fit of the one–loop form to G(x) is obtained for 0.2 ∼ 0.3. To fit the perturbative logarithms at high momenta…
Figure 9. The renormalization functions Z(x) and G(x) for t = {−4.2, 0, 16}. Their leading logarithmic form at large x is not given (see fig. 8), it is indistinguishable from the t = −4.2 results for x > 10 (and with λ in 1.5 — 2). no dependence on this cutoff. There are no ultraviolet boundary conditions imposed on the solutions. Fixing the infrared boundary condition by choosing a value for the parameter t (and the scale by setting b = 1) we have no further influence on the high momentum behavior of the solutions which selfconsistently results from the iterative process alone. Deviations…
Figure 10. The functions F(x) and R(x) for t = 0 compared to the respective results obtained from setting Z1 = 1. question is the one for the momentum scale above which the results should follow their leading logarithms in phenomenological applications. Fixing the momentum scale in our calculations from the rough estimate σ = λΛ 2 QCD given above is not satisfactory. Not only that the parameter λ is poorly determined, also the value of ΛQCD, appropriate for the present subtraction scheme (and Nf = 0), is not a phenomenologically well known quantity. We discuss in more detail in the next sectio
Figure 11. The running coupling αS(µ) for t = 0 (left), and the corresponding β– function (right) in comparison with its leading perturbative form (one–loop) and the β–function as a result of the Mandelstam approximation from ref. [9]. improve the ultraviolet behavior, noting that R(x) as obtained for Z1 = 1 does not approach a momentum independent constant in the ultraviolet on any reasonable scale. 9 Discussion of the Running Coupling The running coupling of the present scheme is given in eq. (71) by αS(µ) = g 2 (µ) 4π = 1 4πβ0 F(s) , (112) where s = µ 2/σ involves the scale σ related
Figure 12. The numerical result for the gluon propagator fro
Figure 13. The numerical result for the ghost propagator fro
Figure 14. The one–dimensional Fourier transform of the glu

Figures reproduced from arXiv:hep-ph/9707327 by the authors.

References
How the literature uses it

57 arXiv papers cite this in 130 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 3 papers
States a conflicting result 3 papers

Papers citing it · 82 sentences without a confident label

Uses its method or tool 5 papers
  • This is done by replacing all dressing functions in the loop diagrams by their leading perturbative expressions [ref] : [equation] where $t_x=ln(x/)$ and $Λ_QCD^2=s e^-1/ω$; $δ=-9/44$ and $γ=-13/22$ are the propagator anomalous dimensions at one-loop level, $ω=11 N_c α(s)/12/π=β_0 g^2(s)$ with $α(s)=g^2(s)/(4π)$ denoting the strong coupling at some sufficiently large reference scale $s$ in the UV, $N_c$ is the number of colors and $β_0=11N_c/3/(4π)^2$ is the one-loop coefficient of the $β$-function.

    Spurious divergences in Dyson-Schwinger equations · Spurious divergences in the gluon propagator DSE · 0.93

  • Having computed the gauge field propagator [ref] and the ghost propagator on the lattice, the running coupling $α^b$ can be extracted for every value of $b=1,2,…,N^2-1$ in the miniMOM scheme [ref] as [equation] where $μ$ is the renormalization scale.

    The spectrum of an SU(3) gauge theory with a fundamental Higgs field · Techniques for gauge-variant quantities · 1.00

  • The models include terms for the renormalization group improvement necessary for the one-loop truncated gluon propagator DSE [ref] .

    Spectral dimension as a tool for analyzing non-perturbative propagators · Details on propagator solutions: One-loop truncated gluon propagator DSE with different models and subtraction methods ( · 0.86

Uses its data or values 2 papers
Compares with it 1 paper
Cites it as background 28 papers

Discussion

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