Running coupling in Yang-Mills theory - a flow equation study -

Holger Gies (CERN)

REVIEW

Running coupling in Yang-Mills theory - a flow equation study -

Holger Gies (CERN)

High Energy Physics - Theory · Submitted 28 Feb 2002 · 44 references

Pith's verdict: A careful ERG calculation with a genuinely new cutoff technique, but the IR fixed point rests on a resummation whose error at the relevant coupling is not controlled.

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arXiv IDhep-th/0202207 (version 2)

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CitationHolger Gies (CERN). "Running coupling in Yang-Mills theory - a flow equation study -." arXiv:hep-th/0202207 (2002). https://pith.science/paper/KIA3T75J

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Gies takes the exact RG flow for Yang-Mills, truncates the gauge-invariant part to an arbitrary function W(θ) of F², and puts the full Γ^(2) into the cutoff argument — the 'spectrally adjusted' cutoff, including the ∂_t Γ^(2) terms that Reuter-Wetterich dropped. That is the real contribution, and it is a good one: the one-loop coefficient comes out universal as it must, and the two-loop coefficient moves from 113% of the perturbative value (without the ∂_t Γ^(2) terms) to 99% with them, for SU(2) and the exponential cutoff. The IR fixed point values (α* ≈ 11.3 for SU(2), 7.7 ± 2 for SU(3)) are new and worth having on record. The paper is also honest about its approximations: the truncation to W(θ), the neglected gauge-noninvariant remainder, the neglected ghost running, and the resummation limits are all stated in the text, not buried in an appendix. That kind of candor is not universal in this literature and deserves credit. The soft spot is the one the stress-test flags, and it lands. η_a, built from the leading-growth coefficients, is positive for all G and never vanishes. The zero of η = η_a + η_b at G* ≈ 0.45 exists only because η_b — which the paper itself calls subleading, η_b ~ G^{1/2} versus η_a ~ G^{3/2} — is negative and large enough to cancel η_a there. The asymptotic ordering that justifies the label 'subleading' only holds for G ≫ 1; at G ≈ 0.45 the two pieces are numerically comparable, so the dropped sub-subleading terms are not parametrically suppressed at the zero. The paper acknowledges it could not systematize those terms and gives no error bar for SU(2). So the fixed point is a plausible indication from a truncation whose error at the relevant coupling is uncontrolled, not a robust prediction. The author's own hedge in Sect. 6 ('If the fixed point exists and our truncation even covers the true mechanism...') is consistent with that reading. Two smaller things: only the exponential cutoff could be resummed, and it was selected partly because it gives good two-loop agreement, so cutoff independence is not demonstrated; and the SU(3) result carries a ±2 from an unresolved color-factor ambiguity (App. E), which the paper is upfront about. None of this is fatal. The technique is sound, the checks are honest, and the paper is clear about what it does and does not control. It is a useful benchmark for anyone working on the IR behavior of Yang-Mills in the ERG, and a natural reference for the spectrally adjusted cutoff. Worth a serious referee.

Referee report

Major objections (4)

[Sect. 4, Eqs. (54)–(61) and the paragraph after Eq. (57)] The existence of the fixed point rests on the resummation truncation. η_a is positive for all G and never vanishes; the zero of η = η_a + η_b at G* ≈ 0.45 is produced by η_b, which the paper classifies as subleading in large-order growth (Eq. 60: η_b ~ G^{1/2} versus η_a ~ G^{3/2}). At G* the two pieces are numerically comparable (η_a(G*) ≈ 1), so the asymptotic hierarchy that justifies dropping sub-subleading terms is not operative where the zero sits. The paper acknowledges at the end of Sect. 4 that it could not systematize the sub-subleading terms and gives no error bar for SU(2). Concrete request: include the next order in the large-m expansion, or vary the resummation prescription, and report the shift in G*; if the zero moves outside the quoted range, the central claim should be restated as a bound rather than a value.
[Sect. 4, Eq. (56) and Appendix F] η_b, the piece that creates the zero, is computed from an integral with a pole from the Nielsen-Olesen unstable mode, handled by rotating the t contour by π/4 and taking the real part. Since η_b is load-bearing for the fixed point, the paper should demonstrate that the real part is insensitive to the rotation angle and to the analytic-continuation prescription, not merely state that the imaginary part is irrelevant. A short table of Re η_b versus rotation angle would settle this.
[Sect. 4, footnote 7 and the exponential-cutoff selection] The exponential cutoff is selected because it gives 99% two-loop agreement, but the paper itself notes that the two-loop coefficient is cutoff-scheme dependent and that comparing to a mass-independent-scheme result is not strictly legitimate. The selection is made at weak coupling and then used to justify the strong-coupling extrapolation to G ≈ 0.45, and the paper states it could not find another cutoff shape for which the resummation could be done. The fixed point's existence has therefore not been shown to be cutoff-independent. Concrete request: exhibit at least one other cutoff shape for which the asymptotic series can be resummed, even with worse two-loop agreement, and show that the zero persists.
[Sect. 2, Eq. (10) and the neglect of Γ^gauge_k] The modified Ward identity forces Γ^gauge_k ≠ 0, and the paper assumes without quantification that it does not strongly modify the leading growth of the coefficients. This is acknowledged as 'a nontrivial assumption.' Since the whole flow rests on it, a quantitative estimate — even a crude one from a simple ansatz for Γ^gauge_k — would materially strengthen the result. As it stands, the reader cannot tell whether the fixed point survives a partial resolution of the Ward identity.

Minor objections (8)

[Abstract versus Eq. (63)] '7.7 ± 2' reads as a computed central value with a symmetric error, but 7.7 is the midpoint of the interval [5.7, 9.7] obtained from two extremal color directions (Appendix E); the abstract should say so explicitly.
[Sect. 6, the α* ~ 1/N observation] With the SU(3) spread [5.7, 9.7], the relation 11.3 × 2/3 ≈ 7.5 holds only for the midpoint, not across the interval; the claim should be softened accordingly.
[Fig. 1] α_s(M_Z) = 0.117 is the five-flavor QCD value, used here for pure SU(2) Yang-Mills; the caption should state explicitly that this is illustrative and not a physical initial condition for pure glue.
[Eq. (44) and surrounding text] '113% agreement' is an odd way to express a 13% deviation; rephrase as '13% deviation' or similar.
[Sect. 4, Eq. (48)] The reader needs a clearer statement of which terms are retained in a_m^{l.g.} and why the 1/Γ(2m) term is kept despite being subleading; the current text explains this only in passing.
[Appendix D, Eq. (D.4)] The representation of h̃(s) as a sum of δ functions with an s-derivative acting on the remaining integrand is unusual and deserves one explanatory sentence.
[Sect. 3, Eq. (31)] The arguments of f_1, f_2, f_3 are introduced in Eq. (28) and then reused with different arguments; a short glossary would help the reader track them.
[Sect. 1 and Sect. 3] Typos: 'exakt RG equation' (Sect. 1), 'see. Eq. (D.4)' (Sect. 3), 'App. (A)' versus 'App. A' (Sect. 3), and inconsistent spacing in reference [25].

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The paper

The paper claims

The paper argues that the gauge coupling of four-dimensional Yang-Mills theory stops running at an infrared fixed point, with $α_* �simeq 11.3$ for SU(2) and $α_* �simeq 7.7 �pm 2$ for SU(3).

The reading

The paper asks whether the gauge coupling of pure Yang-Mills theory can be followed from the ultraviolet all the way to zero momentum without a Landau pole, and whether it settles at a finite infrared fixed point. The author uses the exact renormalization-group flow equation for the effective average action, truncates the gauge-invariant part to an arbitrary function $W_k(θ)$ of $θ=�frac14 F^a_{μν}F^{aμν}$, and extracts the coupling from the flow of the linear term. The distinctive ingredient is a cutoff whose argument is the full second functional derivative of the effective action, so that the cutoff tracks the changing spectrum as modes are integrated out. The result is a $\beta$ function that reproduces perturbation theory at weak coupling and, after resumming an asymptotic series, gives an infrared-stable zero of the anomalous dimension at $α_* �simeq 11.3$ for SU(2) and $α_* �simeq 7.7 �pm 2$ for SU(3). If correct, this supplies a continuum nonperturbative definition of a freezing coupling and a benchmark for other infrared studies of Yang-Mills theory.

What carries the argument

The load-bearing device is the spectrally adjusted cutoff: instead of using only the Laplacian or momentum squared as the cutoff argument, the author inserts $�Gamma_k^{(2)}$ evaluated at the background field, so the regulator adjusts to the spectral flow of the theory. This is meant to condense information that ordinary cutoffs spread over the flow of infinitely many higher operators into the flow of the gauge coupling itself. The price is a highly nonlinear flow equation; the author rewrites it in a propertime representation, expands in $θ$ and the coupling, and then reconstructs an integral representation from the leading and subleading growth of the resulting asymptotic series. The exponential cutoff is chosen because it makes that resummation tractable and gives the closest two-loop benchmark.

Extended reading notes

Core claim

Within the truncation, the flow equation for $W_k(θ)$ is derived and the running coupling is encoded in the anomalous dimension $η=-�partial_t�ln Z_F$, with $β(g^2)=η g^2$ in four dimensions. The paper's central claim is that $η(G)$, as a function of $G=g^2/[2(4π)^2]$, has an infrared-stable zero, so the coupling stops running and approaches $α_*$. The author checks that the one-loop coefficient is universal and that the exponential cutoff gives 99% agreement with the two-loop coefficient for SU(2) and 95% for SU(3), compared with 113% when the $�partial_t�Gamma_k^{(2)}$ terms are dropped. For SU(3) the quoted interval $α_* �simeq [5.7,9.7]$ reflects unresolved color factors $τ_m$, not a numerical error.

Load-bearing premise

The load-bearing premise is that the Borel-type resummation built from only the leading and subleading growth of the asymptotic series locates the true zero of $η(G)$, and that the exponential cutoff does not move that zero at strong coupling.

Editorial extensions

If this is right

If the result holds, pure SU(2) and SU(3) Yang-Mills theory have an infrared-stable fixed point in this continuum truncation, so the coupling freezes instead of hitting a Landau pole.
The beta function agrees with perturbation theory up to $α_s �sim 1$, which would explain why perturbative QCD remains accurate to its usual validity limit.
The fixed-point values satisfy $α_* �sim 1/N$ between the central SU(3) value and the SU(2) value, matching a scaling reported in Dyson-Schwinger studies of the ghost-gluon vertex.
Including dynamical quarks is expected to shift $α_*$ downward while preserving the fixed point, so full QCD would also freeze in the infrared.

Reading between the lines

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

A natural next check is whether the same $η$ zero survives a derivative expansion or a truncation that keeps other gauge invariants such as $(F^a_{μν}�tilde F^{aμν})^2$; the paper's truncation omits them by construction.
If the spectrally adjusted cutoff is the main reason the flow remains integrable to $k�to0$, the method should also remove Landau-pole behavior in QED-like theories or in scalar models with known infrared fixed points.
The SU(3) interval could be narrowed by computing the full symmetric color tensor $τ_i$ rather than sampling the two extremal color directions; that would turn the quoted $�pm 2$ into a genuine prediction or exclusion.
Connecting the fixed-point coupling to a lattice static-quark definition would let the result be checked against existing nonperturbative data, although the paper explicitly leaves that dictionary open.

Original abstract

The effective average action of Yang-Mills theory is analyzed in the framework of exact renormalization group flow equations. Employing the background-field method and using a cutoff that is adjusted to the spectral flow, the running of the gauge coupling is obtained on all scales. In four dimensions and for the gauge groups SU(2) and SU(3), the coupling approaches a fixed point 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, 2,905 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.09 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.11 · compares with prior results 0.05 · 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.

The effective average action of Yang-Mills theory is analyzed in the framework of exact renormalization group flow equations. Employing the background-field method and using a cutoff that is adjusted to the spectral flow, the running of the gauge coupling is obtained on all scales. In four dimensions and for the gauge groups SU(2) and SU(3), the coupling approaches a fixed point 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	The effective average action of Yang-Mills theory is analyzed in the framework of exact renormalization group flow equations.	contribution 0.61	0.82	0.05	asserted 0.99
s1	Employing the background-field method and using a cutoff that is adjusted to the spectral flow, the running of the gauge coupling is obtained on all scales.	method 1.00	0.90	0.12	asserted 1.00
s2	In four dimensions and for the gauge groups SU(2) and SU(3), the coupling approaches a fixed point in the infrared.	result 1.00	0.88	0.73	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.82 · evidence from proof or derivation 0.63.

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sources read

arXiv listing abstract v2

not examined

full text
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references and citing sentences
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P abstract

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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.** Running coupling αs versus momentum scale k in GeV for gauge group SU(2), using the initial value αs(MZ) ≃ 0.117. The solid line represents the result of our calculation in comparison with one-loop perturbation theory (dashed line). from k to k − ∆k. It appears natural that a truncated flow can be controlled better if each momentum shell covers an equal part of the spectrum of quantum fluctuations. The spectrum itself is not fixed, but k dependent; lower modes get dressed by integrating out higher modes. In order to adapt the cutoff function to this spectral flow, we insert the…

**Figure 2.** Anomalous dimension η = β(g 2 )/g2 for SU(2) Yang-Mills theory in d = 4 versus G = g 2/[2(4π) 2 ]. The long-dashed line represents the contribution ηa, the short-dashed line ηb, as defined in Eqs. (54) and (56); the solid line is the sum of both. Without going into details, we note that there exist fits for S(−iq) similarly to Eq. (58) involving a square-root behavior for large q (large G). It turns out that the G3/2 coefficient vanishes exactly, so that8 η N=2 b ∼ +G 1/2 . (60) Obviously, ηb is subleading for large G, which agrees with the fact that it arises from subleading parts…

**Figure 3.** Anomalous dimension η = β(g 2 )/g2 for SU(3) Yang-Mills theory in d = 4 versus G = g 2/[2(4π) 2 ]. The black lines correspond to η N=3 8 , the grey lines to η N=3 3 , as defined in Eq. (62). The meaning of the dashed lines is as in

Figures reproduced from arXiv:hep-th/0202207 by the authors.

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Functional Renormalization Group meets Computational Fluid Dynamics: RG flows in a multi-dimensional field space
cond-mat.stat-mech 2024-12 conditional novelty 6.0 of 10

A two-dimensional Kurganov-Tadmor finite-volume scheme accurately solves FRG flow equations for effective potentials in multi-dimensional field space, benchmarked against exact zero-dimensional path integrals and appl...
Non-perturbative propagators in quantum gravity
hep-th 2021-05 conditional novelty 6.0 of 10

Mode-resolved functional renormalisation group flows show that the graviton's spin-two and spin-zero propagators have qualitatively different momentum dependence, and the resulting form factors have alternating deriva...
Viability of quantum-gravity induced ultraviolet completions for matter
gr-qc 2017-05 conditional novelty 6.0 of 10

Within a truncated functional renormalization group flow, the existence of interacting matter imposes a weak-gravity bound and a Yukawa-relevance bound that carve out viable regions of the coupling space of asymptotic...
The Renormalization Group flow of unimodular f(R) gravity
gr-qc 2015-01 conditional novelty 6.0 of 10

Unimodular f(R) gravity admits an asymptotically safe fixed point with two relevant directions, one fewer than standard f(R) gravity, and tolerates the Standard Model matter content.
Higher order quark-mesonic scattering processes and the phase structure of QCD
hep-ph 2014-03 conditional novelty 6.0 of 10

Including a meson-field-dependent Yukawa coupling and high-order meson self-interactions in the quark-meson model shifts the chiral phase boundary and places the critical endpoint at (T, mu) = (50, 291) MeV.
The local potential approximation in the background field formalism
hep-th 2013-12 conditional novelty 6.0 of 10

Imposing the modified shift Ward identity alongside the functional renormalization group flow makes field-dependent cutoff operators in the local potential approximation reproduce exactly the standard fixed points and...
Off-diagonal heat-kernel expansion and its application to fields with differential constraints
math-ph 2011-12 conditional novelty 6.0 of 10

The heat-kernel coefficients for divergence-free vector fields acquire infrared poles in even dimensions that cancel only on Einstein backgrounds, so the transverse-field decomposition used in gravitational functional...
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.
Asymptotic freedom of Yang-Mills theory with gravity
hep-th 2011-01 conditional novelty 6.0 of 10

The gravitational contribution to the Yang-Mills beta function is regulator-dependent, vanishes for symmetry-preserving regulators, and is negative semi-definite for all regulators, so asymptotic freedom survives gravity.
Ising exponents from the functional renormalisation group
hep-th 2010-09 conditional novelty 6.0 of 10

A fourth-order background-field functional renormalisation group computation yields 3d Ising exponents nu = 0.6247, eta = 0.0313, omega = 0.865, with systematic errors estimated by averaging over independent projectio...
The QCD Phase Boundary from Quark-Gluon Dynamics
hep-ph 2008-10 conditional novelty 6.0 of 10

Functional RG flow from the Z mass scale to the chiral regime gives a curvature of the one-flavor QCD phase boundary between 0.4 and 1.3, with the spread set by the choice of running coupling.
A Manifestly Gauge Invariant and Universal Calculus for SU(N) Yang-Mills
hep-th 2006-02 conditional novelty 6.0 of 10

An all-orders diagrammatic calculus for manifestly gauge invariant Exact Renormalization Group flows cancels explicit seed-action and cutoff-covariantization dependence from the SU(N) Yang-Mills beta function.
Universality of spontaneous chiral symmetry breaking in gauge theories
hep-th 2002-09 conditional novelty 6.0 of 10

A truncated renormalization-group analysis identifies an infrared-attractive fixed point in the scalar sector that makes one-flavor QCD with a fundamental chiral scalar indistinguishable from ordinary QCD over a wide ...
An asymptotically safe solution to the U(1) triviality problem
hep-th 2017-02 conditional novelty 5.0 of 10

Quantum gravity fluctuations within asymptotic safety shift the U(1) gauge sector's Gaussian fixed point to an interacting one and can render the gauge coupling asymptotically free, provided the gravitational coupling...
Continuum limit in matrix models for quantum gravity from the Functional Renormalization Group
gr-qc 2013-09 conditional novelty 5.0 of 10

A functional renormalization group with matrix size as the cutoff reproduces known perturbative beta functions for 2d quantum gravity matrix models and supplies a recipe for tensor models.
The phase diagram of quantum gravity from diffeomorphism-invariant RG-flows
hep-th 2012-03 conditional novelty 5.0 of 10

Within a diffeomorphism-invariant functional RG truncation, quantum gravity has an attractive UV fixed point at (g_N, lambda) = (1.692, 0.144) and an IR fixed point at (0, 1/2).
On the Nature of the Phase Transition in SU(N), Sp(2) and E(7) Yang-Mills theory
hep-ph 2010-07 conditional novelty 5.0 of 10

Using a functional-RG Polyakov-loop potential built from zero-temperature propagators, the authors confirm first-order deconfinement for SU(N≥3) and Sp(2) and predict a weaker first-order transition for E(7).
Renormalization Group Flow in Scalar-Tensor Theories. I
hep-th 2009-11 conditional novelty 5.0 of 10

In a two-derivative truncation of scalar-tensor gravity, the only continuous fixed point in any dimension d > 2 is one where all scalar self-couplings vanish while Newton's constant and the cosmological constant stay nonzero.
Asymptotically free scalar curvature-ghost coupling in Quantum Einstein Gravity
hep-th 2009-07 conditional novelty 5.0 of 10

The scalar curvature-ghost coupling in Quantum Einstein Gravity is asymptotically free and RG relevant at the non-Gaussian fixed point, which stays stable when this ghost operator is added to the truncation.
Investigating the Ultraviolet Properties of Gravity with a Wilsonian Renormalization Group Equation
hep-th 2008-05 conditional novelty 5.0 of 10

In truncations of gravity up to eighth-order polynomials in the Ricci scalar, the renormalization group flow has a nontrivial fixed point whose ultraviolet critical surface is three-dimensional, with the surface equat...
Exploratory study of three-point Green's functions in Landau-gauge Yang-Mills theory
hep-lat 2006-05 conditional novelty 5.0 of 10

Lattice data in 3d SU(2) Landau-gauge Yang-Mills show the ghost-gluon vertex staying near its tree-level value while the three-gluon vertex shrinks at small momenta, broadly consistent with functional-method assumptions.
Geometrical effective action and Wilsonian flows
hep-th 2003-10 conditional novelty 5.0 of 10

A gauge invariant exact renormalization group flow equation is derived for Yang-Mills theory using the Vilkovisky-DeWitt geometrical effective action, with gauge symmetry encoded in modified Nielsen identities.
Ghost anomalous dimension in asymptotically safe quantum gravity
hep-th 2010-01 accept novelty 4.0 of 10

The ghost anomalous dimension in asymptotically safe quantum gravity is negative (about -0.78 in the Landau-deWitt gauge), and its backreaction preserves the non-Gaussian UV fixed point.
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

36 arXiv papers cite this in 44 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

The running coupling as extracted from either of these vertex functions possesses an infrared fixed point.

Vertex functions and infrared fixed point in Landau gauge SU(N) Yang-Mills theory · 0.88

Papers citing it · 28 sentences without a confident label

Uses its method or tool 6 papers

Inserting the regulator eq:Rk in the flow eq:FRG and setting $ϕ=ϕ̅$ we arrive at [ref] [equation] where $Γ_k[ϕ]=Γ_k[ϕ,ϕ]$, see eq:Gammainv.

Asymptotic freedom of Yang-Mills theory with gravity · Asymptotic freedom and gravitational interactions · 0.88
We choose a spectrally and RG-adjusted regulator [ref] ,

Ghost anomalous dimension in asymptotically safe quantum gravity · Ghost anomalous dimension · 1.00
Moreover, the regulator can be adjusted to the flow of the spectrum of the full propagator [ref] , by choosing $y = Γ_^(2)/Z_N κ̅^2 k^2$ evaluated on the background field.

Asymptotically free scalar curvature-ghost coupling in Quantum Einstein Gravity · Computational method · 0.97

Cites it as support 1 paper

This behaviour is suggested by the scaling solutions for the gluon and ghost propagators in the Landau gauge in the Dyson-Schwinger equations (DSE) approach [ref] , in the functional renormalization group (FRG) approach [ref] , stochastic quantization [ref] , and by Gribov-Zwanziger approach [ref] .

pQCD running couplings finite and monotonic in the infrared: when do they reflect the holomorphic properties of spacelike observables? · Introduction · 0.97

Cites it as background 9 papers

In the past 20 years, tremendous progress has been made within the functional RG framework in the development of a “top-down” approach to QCD, see, e.g., Refs. [ref] .

Fierz-complete NJL model study III: Emergence from quark-gluon dynamics · Introduction · 0.96
While vacuum Yang-Mills correlation functions have been studied intensively in the past two decades [ref] , results for the finite-temperature correlation functions are scarce, see [ref] for propagator studies.

Non-perturbative finite-temperature Yang-Mills theory · Introduction · 0.94
The latter class is often employed in studies of gauge theories, see, e.g., Refs. [ref] .

Renormalization Group Studies of Dense Relativistic Systems · Massive fermions · 0.90

Discussion

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