Pith. sign in

REVIEW 3 major objections 3 minor 29 references

On the Yang-Mills propagator at strong coupling

T0 review · 3 major / 3 minor · reviewed 2026-08-11 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read At leading order in the strong-coupling expansion, the Yang-Mills gluon propagator becomes a contact term proportional to $\delta^{(4)}(x-y)$, signalling that gluons do not propagate and suggesting a mass-dimension-two gluon condensate.

desk verdict The paper's contact-term gluon propagator rests on a wrong O(N) group integral (Eq. 49, missing a 2/N factor), so the headline result is unsupported, though the qualitative picture might survive a corrected calculation. read the letter →

arxiv 2412.12124 v1 pith:JSUCBJLA submitted 2024-12-03 hep-th

Yves Gabellini , Thierry Grandou , Ralf Hofmann This is my paper · ORCID
classification hep-th PACS 12.38.Cy
keywords Yang-Millseffectivelocalitygluonpropagatorstrongcouplingfunctionalmethodsrandommatrixtheorycondensatemassgap
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Effective locality is the property that, once all virtual gauge-field exchanges are summed, non-abelian Green's functions reduce to local contact interactions. This paper applies that property to pure Yang-Mills theory in Minkowski spacetime and asks what the gluon propagator looks like when the coupling is strong. Working at leading order in $1/g$, the authors derive a propagator proportional to $\delta^{(4)}(x-y)\delta_{ab}g_{\mu\nu}$: a contact term with no spacetime propagation. They interpret this as evidence for a mass-dimension-two gluon condensate whose scale is set by a meshing parameter $\Delta$, and they show the result is consistent with the Yang-Mills scale and with existing estimates of a gluon mass. If the derivation holds, effective locality gives a calculable, non-perturbative route to the mass gap and confinement.

What carries the argument

The carrying object is the effective-locality gluon generating functional $Z_{YM}[j]$, Eq. (13), obtained through Halpern's field-strength representation. The functional integral over the antisymmetric tensor fields $\chi^a_{\mu\nu}$ is re-expressed in terms of the $32\times 32$ real symmetric matrix $M=\frac{i}{8}\chi^a\otimes T^a$, whose spectrum comes in opposite pairs; the measure image theorem converts $d[\chi]$ into the Vandermonde-weighted eigenvalue measure times the Haar measure on the orthogonal group. The decisive identity is the orthogonal-group average in Eq. (49), which yields a factor $\delta(\xi_k)$ for each paired eigenvalue; after analytic continuation and use of the semicircle law to bound $C_1,C_2$, the final contact amplitude is finite and proportional to $\sqrt{\Delta}$.

What would settle it

Evaluate the orthogonal-group integral in Eq. (49) numerically at $N=32$ or exactly for small $N$ with the paper's measure: if the delta factors $\delta(\xi_k)$ do not appear, the leading-order delta-function propagator is not established. A lattice measurement of the strong-coupling gluon propagator near $x=y$ would also test whether the contact amplitude scales as $\sqrt{\Delta}$ as claimed.

Watch

Extended reading notes

Core claim

The paper's central result is that at leading order in the strong-coupling expansion the non-perturbative gluon propagator is $$\langle T A^a_\mu(x) A^b_\nu(y)\rangle = \frac{\pi N}{2g}\frac{C_1}{C_2}\sqrt{\frac{\$\Delta$}{4N_c}}\,\delta_{ab}g_{\mu\nu}\$delta^{{(4)}}$(x-y) + O($g^{{-2}}$),$$ with $C_1/C_2$ a finite, computable ratio of Vandermonde integrals and $\Delta$ the spacetime meshing parameter. Because the only spacetime dependence is a delta function, the authors conclude that gluonic degrees of freedom do not propagate in this regime, and because the colour and Lorentz structure is $\delta_{ab}g_{\mu\nu}$, they identify the amplitude as a gluon condensate of mass dimension two. The associated one-point function $\langle A^a_\mu(x)\rangle$ vanishes at the same order. The derivation uses a strong-coupling simplification of the effective-locality generating functional, followed by a random-matrix reduction and an analytic continuation that leaves a finite constant ratio.

Load-bearing premise

The whole result hinges on the integration formula in Eq. (49), which claims that averaging the inverse of the diagonalized matrix over all rotations leaves a delta function at each eigenvalue, because without that delta factor the propagator would not become a contact term.

Editorial extensions

If this is right

  • Gluon propagation is absent at leading order in the strong-coupling regime, and the first correction enters only at $O(g^{-2})$.
  • The propagator's colour-Lorentz structure $\delta_{ab}g_{\mu\nu}$ identifies the contact term with a condensate of mass dimension two, the same object as the Landau-gauge gluon and ghost mass.
  • The amplitude vanishes with the meshing parameter $\Delta$, recovering the perturbative short-distance behaviour expected from asymptotic freedom.
  • The vanishing one-point function $\langle A^a_\mu(x)\rangle=0$ confirms that the strong-coupling vacuum carries no preferred Lorentz or colour direction.
  • The method gives a systematic strong-coupling expansion, so subleading orders could reveal residual propagation effects, possibly of a glueball-like kind.

Reading between the lines

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

  • If the delta-function form is taken at face value, it implies that the strong-coupling vacuum cannot resolve points closer than the meshing scale $\Delta$; this is a concrete reinterpretation that could be probed by seeing whether the same $\Delta$ controls higher $n$-point functions.
  • The random-matrix technology used here is portable: the same spectral and orthogonal-group decomposition could be applied to the effective-locality quark sector, where one would expect a comparable condensate scale to emerge from the same $\Delta$.
  • A finite-$N$ numerical evaluation of Eq. (49) would settle whether the delta factor is a genuine property or an artefact of the analytic continuation; this test is within reach because $N=32$ is small enough for high-precision quadrature.
  • The paper's numerical match, $1/\sqrt{\Delta}\approx (230\,\mathrm{MeV})^2$ at $g_R\simeq 4$, invites comparison in other gauge choices; a gauge-independent condensate would require showing that the contact coefficient is independent of the $\zeta$-gauge fixing used in Eq. (7).
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 3 minor

Summary. This paper claims to compute the non-perturbative gluon propagator in strongly coupled Yang-Mills theory starting from the effective-locality generating functional constructed in Ref. [8]. After writing the field-strength tensor as a real symmetric 32x32 matrix M(x), the authors perform an orthogonal-group integration and obtain a leading-order contact term proportional to sqrt(Delta) delta^(4)(x-y) delta_ab g_munu, which is interpreted as a gluon condensate of mass dimension two controlled by a 'meshing parameter' Delta. The paper also presents a random-matrix estimate of the ratio of the two relevant integrals and a numerical comparison in which the meshing parameter is matched to a lattice value of the Yang-Mills scale.

Significance. If the derivation were valid, a pure contact-term gluon propagator at strong coupling would be a striking result with potential implications for the mass gap and confinement. The paper is clearly organized and it is useful that the authors identify the technical bottleneck, namely the Vandermonde integral and the need for random-matrix bounds. However, the central mathematical step, Eq. (49), is incorrect, and the numerical comparison is a fit rather than a test. The manuscript does not provide machine-checked proofs, reproducible code, parameter-free derivations, or independent falsifiable predictions; its main output rests on an unsupported group-integral identity.

major comments (3)
  1. [Eq. (49), Section III.B.2] Equation (49) is the load-bearing step of the derivation, but the claimed O(N) integration is not correct. For the diagonal D^{-1} used in Eq. (48), the left-hand side is sum_k xi_k^{-1} O_{ki}O_{kj}. The normalized Haar average of O_{ki}O_{kj} is delta_{ij}/N for each k, so the full integral equals (vol(O_N)/N) delta_{ij} sum_{k=1}^N 1/(xi_k+i epsilon), not -i pi delta_{ij} vol(O_N) sum_{k=1}^{N/2} delta(xi_k). With the paired spectrum xi_{N-i+1}=-xi_i this standard result becomes -(2 pi i vol(O_N)/N) delta_{ij} sum_{p=1}^{N/2} delta(xi_p), which differs from Eq. (49) by a factor 2/N and by the way the eigenvalue sum is handled. This is not a cosmetic prefactor: the delta(xi_1) factor in Eq. (54), which produces the nonzero integral C1 and the sqrt(Delta) dependence in Eq. (53), is an artifact of the incorrect identity. As written, Eq. (53) is unsupported.
  2. [Section III.D, Eqs. (71) and (66)] The numerical agreement with the Yang-Mills scale is obtained by tuning. In Eq. (71) the authors choose g_R=10 and extract 1/sqrt(Delta) = (230 MeV)^2, and in the following paragraph they state that the same value is obtained at g_R ~ 4 when Eq. (66) is used. Since g_R is a free input and the lattice value of Lambda_YM is the target of the comparison, the proximity of the numbers is a consequence of this tuning, not a prediction. The relation Delta Lambda^4_YM >= 1 invoked after Eq. (52) is also assumed rather than derived. The paper should present these numbers as an illustration, not as independent confirmation of the result.
  3. [Section III.B.5, Eq. (65)] The random-matrix estimate that fixes the numerical coefficient is uncontrolled. The step leading to Eq. (65) replaces (Theta_k^2 - Theta_l^2)^2 by (Theta_k - Theta_l)^2 in C1 and C2, and then uses Wigner's semicircle law to obtain |C1/C2| ~ (1/pi) sqrt(2/N). This replacement is not justified by the gaussian suppression, and the rigorous bound in Eq. (64) is far too wide to serve as a check. Since Eq. (66), and therefore the g_R ~ 4 extraction in Section III.D, depends on this coefficient, the numerical prefactor in the final result is not established.
minor comments (3)
  1. [Conclusion and Section III.B.4] There are several typographical errors: 'wether' should be 'whether', 'conurations' should be 'configurations', and 'untractable' should be 'intractable'.
  2. [Section III.B.3] After the rescaling below Eq. (49), the same symbol xi_i is retained for the dimensionless integration variables; this makes Eqs. (50)-(55) difficult to follow, especially the argument of the exponential and the delta(xi_1) factor.
  3. [Section III.B.7, Eq. (70)] The replacement of delta^(4)(x-y) by delta^(4)(X-Y) with X = Delta \hat{X} is introduced without a precise definition of the rescaling, so the relationship between Eq. (53) and Eq. (70) is ambiguous and should be clarified.

Circularity Check

1 steps flagged · score 6.0 of 10

The Λ_YM 'prediction' is a fit: g_R is chosen and Δ is extracted from the lattice condensate value, so Eq. (71)'s agreement is by construction.

  1. fitted input called prediction [Section III.D (remarks after Eq. (69), numerical estimate around Eq. (71))]
    "The only mass scale one can think of in a quantum Yang–Mills theory is that of the asymptotic freedom ΛYM parameter say, which is a renormalisation group invariant. There shouldn’t be any inconsistency in assuming the relation Λ ≃ ΛYM ... To begin with, if in (69) the strong coupling constant gR is taken at a value of 10, then it is quite amusing to discover that the inverse squared root of the meshing parameter ∆ comes out to be 1√∆ = (230 MeV)2 ... close enough to the YM scale of ΛYM = 259 MeV found by a full lattice QCD calculation at zero flavor number nf = 0."

    The numerical 'prediction' of the Yang–Mills scale is not an independent output. The lattice value g_R^2⟨A^2⟩_R = (2.76 GeV)^2 is an input, and the strong coupling g_R is chosen freely (10 when using Eq. (69), about 4 when using the main result approximated by Eq. (66)) to convert the calculated contact term into a value of 1/√Δ. Since Λ_YM had already been assumed to be the relevant mass scale and is the same number used in the comparison, Eq. (71) merely re-expresses the input condensate and the chosen coupling in different units. The claimed agreement with Λ_YM is therefore forced by the parameter choice, not derived from the theory.

full rationale

The formal functional derivation of the contact-term gluon propagator (Eqs. (13)–(53)) is largely self-contained: the generating functional is re-derived in outline in Section II, and the heavy use of the authors' earlier effective-locality papers is not by itself circularity under the review rules. The genuinely circular element is the numerical scale comparison in Section III.D. There the lattice value of g_R^2⟨A^2⟩_R is taken as input, the Yang–Mills scale Λ_YM is assumed to be the only relevant mass scale, and g_R is chosen freely so that the extracted meshing parameter gives 1/√Δ ≈ 230 MeV, close to the lattice Λ_YM = 259 MeV. Because Δ is fixed by the input condensate plus the chosen g_R, the agreement in Eq. (71) is a consistency check with a free parameter, not a prediction. This is a partial circularity: the central claim of a non-propagating, contact-type strong-coupling propagator does not itself reduce to the fit, but its advertised numerical confirmation does. The separate concern about Eq. (49)—that standard O(N) Haar integration would not produce a sum of delta functions δ(ξ_k) in that form—is a mathematical validity issue rather than a circularity, and is not added to the score, though it would further undermine the support for Eq. (53). Overall: one fitted input presented as a numerical prediction, giving a score of 6.

Assumptions & free parameters 4 free parameters · 7 assumptions · 2 invented entities

The central claims rest on the authors' own effective locality framework, several unproven mathematical steps, and two tunable parameters (g_R and Δ) that are fitted to reproduce a known scale. The most serious issue is the unsupported orthogonal group integration identity in Eq. (49).

free parameters (4)
  • g_R (strong renormalized coupling) = 10 (or 4 in an alternative estimate)
    Chosen in Section III.B.7 so that the extracted 1/√Δ = (230 MeV)^2 matches Λ_YM ≈ 259 MeV. The paper treats this as a strong coupling value, but the specific choice is adjusted to reproduce the desired scale.
  • Δ (meshing parameter) = 1/√Δ ≈ (230 MeV)^2 from the lattice comparison
    Undetermined by the formalism; its value is fixed by matching the lattice gluon condensate. The paper only derives the inequality ΔΛ^4_YM ≥ 1, leaving the actual value as an input extracted from data.
  • Λ (mass scale) = cancels in the final result
    Introduced in Eq. (50) for dimensional reasons and set to Λ_YM by assumption, but it drops out of the ratio I1/I2, so it does not affect the final expression.
  • ∆Λ^4 (dimensionless combination) = constrained only by ∆Λ^4 ≥ 1
    The paper treats it as a pure number related to the spacetime meshing; its precise value is not predicted.
assumptions (7)
  • domain assumption The generating functional Z_YM[j] in Eq. (13), derived in the authors' earlier paper Ref. [8], is correct.
    The whole calculation starts from Eq. (13). The derivation is only summarized and relies on the authors' own prior work, with no independent verification cited.
  • domain assumption The measure image theorem transforms the functional measure d[χ] into a finite-dimensional matrix measure with a Vandermonde determinant (Eq. 23).
    This step is used to replace the functional integral over χ by an integral over eigenvalues of the matrix M. Its rigorous applicability here is assumed.
  • domain assumption The strong-coupling expansion in powers of g^{-1} is valid, and at leading order the exponent in Eq. (44) can be set to unity.
    The paper keeps only the leading term in 1/g and discards subleading contributions. No proof of convergence or control of higher orders is given.
  • ad hoc to paper The orthogonal group integral identity in Eq. (49) yields -iπ δ_{ij} vol(O_N) Σ δ(ξ_k).
    This identity is stated without derivation and appears inconsistent with standard O(N) integration of O_{ki} O_{kj}, which gives vol(O_N) δ_{ij}/N. It is load-bearing because it creates the delta function that makes C1 nonzero.
  • domain assumption The analytic continuation ξ_k → √i Θ_k via Cauchy's theorem is valid for the divergent integrals in (54) and (55).
    The paper relies on analytic continuation of Meijer functions from Ref. [14] to justify the contour rotation. The treatment of distributions and divergent integrals is not fully rigorous.
  • domain assumption Wigner's semicircle law, valid at N → ∞, applies to this N = 32 matrix ensemble and controls the integrals.
    The paper uses Wigner's law to bound and estimate C1 and C2, appealing to universality, but N=32 is finite and the corrections are not computed.
  • ad hoc to paper The meshing parameter Δ is a physical scale, and δ^{(4)}(x-y) should be reinterpreted as δ^{(4)}(X-Y) on rescaled spacetime points.
    This interpretive step is introduced to resolve the apparent contradiction between √Δ and the delta function; it is not independently motivated.
invented entities (2)
  • Meshing parameter Δ
    purpose: Regulates the functional measure and sets the non-perturbative effective locality distance scale; determines the magnitude of the gluon condensate.
    The value of Δ is not predicted; it is extracted by matching a lattice value after choosing g_R. No independent observable is given.
  • Rescaled spacetime points X, Y (fuzzed points)
    purpose: Reinterprets the product √Δ δ^{(4)}(x-y) as √Δ δ^{(4)}(X-Y) on a new spacetime manifold, resolving the apparent contradiction between the meshing scale and point-localized contact term.
    This is a qualitative interpretive addition without quantitative experimental or lattice consequences beyond the original fit.

how reviews work

0 comments
Cite this review

Pith. "Pith review of On the Yang-Mills propagator at strong coupling." pith.science (2026). https://pith.science/paper/JSUCBJLA

@misc{pith2026241212124,
  author       = {Pith},
  title        = {Pith review of: On the Yang-Mills propagator at strong coupling},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JSUCBJLA}},
  note         = {Machine review of arXiv:2412.12124}
}
read the original abstract

About twelve years ago the use of standard functional manipulations was demonstrated to imply an unexpected property satisfied by the fermionic Green's functions of QCD. This non-perturbative phenomenon is dubbed Effective Locality. In a much simpler way than in QCD, the most remarkable and intriguing aspects of Effective Locality have been presented in a recent letter in the Yang-Mills theory on Minkowski spacetime. While quickly recalled in the current paper, these results are used to calculate the problematic gluonic propagator in the Yang-Mills non-perturbative regime.

Figures

Figures reproduced from arXiv: 2412.12124 by the authors.

Figure 1
Figure 1. FIG. 1: The integration contour displaying the analyticity domain of the integrals of (54) and (55) [PITH_FULL_IMAGE:figures/full_fig_p019_1.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

29 extracted references · 29 canonical work pages

  1. [8]

    Effective Locality in the pure gluon sector

    Fried, H.M.; Gabellini, Y.; Grandou, T. Effective Locality in the pure gluon sector. Mod. Phys. Lett. A 2023, 38, 2350163

    work page 2023

  2. [1]

    Gauge Invariant Summation of All QCD Virtual Gluon Exchanges

    Fried, H.M.; Gabellini, Y.; Grandou, T.; Sheu, Y.–M. Gauge Invariant Summation of All QCD Virtual Gluon Exchanges. Eur. Phys. J. C 2010, 65, 395

    work page 2010

  3. [2]

    On the non–perturbative realization of QCD gauge– invariance

    Fried, H.M.; Grandou, T.; Hofmann, R. On the non–perturbative realization of QCD gauge– invariance. Mod. Phys. Lett. A 2017, 32, 1730030

    work page 2017

  4. [3]

    Monopole Condensation, And Confinement In N = 2 Supersymmetric Yang–Mills Theory

    Seiberg, N.; Witten, E. Monopole Condensation, And Confinement In N = 2 Supersymmetric Yang–Mills Theory. Nucl. Phys. B 1994, 426, 19; Erratum: Nucl. Phys. B 1994, 430, 485

    work page 1994

  5. [4]

    Instantons in field strength formulated Yang–Mills theories

    Reinhardt, H.; Langfeld, K.; von Smekal, L. Instantons in field strength formulated Yang–Mills theories. Phys. Lett. B 1993, 300, 111

    work page 1993

  6. [5]

    Casimir operator dependences of non–perturbative fermionic QCD amplitudes

    Fried, H.M.; Grandou, T.; Hofmann, R. Casimir operator dependences of non–perturbative fermionic QCD amplitudes. Int. J. Mod. Phys. A 2016, 31, 1650120

    work page 2016

  7. [6]

    Cubic Casimir operator of SUc(3) and confinement in the nonrelativistic quark model

    Dmitrasinovic, V. Cubic Casimir operator of SUc(3) and confinement in the nonrelativistic quark model. Phys. Lett. B 2001, 499, 135. 27

    work page 2001

  8. [7]

    QCD Effective Locality: A Theoretical and Phenomenological Review

    Fried, H.M.; Gabellini, Y.; Grandou, T.; Tsang, P.H. QCD Effective Locality: A Theoretical and Phenomenological Review. Universe 2021, 7, 481

    work page 2021

Show all 29 references
  1. [9]

    Basics of Functional Methods and Eikonal Models ; Editions Fronti` eres: Paris, France, 1990

    Fried, H.M. Basics of Functional Methods and Eikonal Models ; Editions Fronti` eres: Paris, France, 1990

    1990

  2. [10]

    Vacuum condensate of mass dimension 2 as the origin of mass gap and quark confinement

    Kondo, K.-I. Vacuum condensate of mass dimension 2 as the origin of mass gap and quark confinement. Phys. Lett. B 2001, 514, 335

    2001

  3. [11]

    Field–strength formulation of quantum chromodynamics

    Halpern, M.B. Field–strength formulation of quantum chromodynamics. Phys. Rev. D 1977, 16, 1798

    1977

  4. [12]

    On a Method of Calculating Quantum Distribution Functions

    Stratonovich, R.L. On a Method of Calculating Quantum Distribution Functions. Soviet Physics Doklady 1958, 2, 416. Hubbard, J. Calculation of Partition Functions. Phys. Rev. Lett. 1959, 3, 77

    1958

  5. [13]

    Quantization of non–Abelian gauge theories

    Gribov, V.N. Quantization of non–Abelian gauge theories. Nucl. Phys. B 1978, 139, 1

    1978

  6. [14]

    Non–Perturbative QCD Amplitudes at Quenched and Eikonal Approximations

    Fried, H.M.; Grandou, T.; Sheu, Y.-M. Non–Perturbative QCD Amplitudes at Quenched and Eikonal Approximations. Ann. Phys. 2014, 344, 78, Appendix A

    2014

  7. [15]

    Random Matrices; Academic Press: Cambridge, MA, USA, 1967

    Mehta, M.L. Random Matrices; Academic Press: Cambridge, MA, USA, 1967

    1967

  8. [16]

    Johnson and M.L

    G.W. Johnson and M.L. Lapidus, The Feynman Integral and Feynman ’s Operational Calculus, Oxford University Press (2000)

    2000

  9. [17]

    Linear Algebra ; Pearson Education, Inc: Upper Saddle River, NJ, USA, 2003

    Friedberg, S.; Insel, A.; Spence, L. Linear Algebra ; Pearson Education, Inc: Upper Saddle River, NJ, USA, 2003

    2003

  10. [18]

    Volumes of Orthogonal Groups and Unitary Groups

    Zhang, L. Volumes of Orthogonal Groups and Unitary Groups. arXiv 2017, arXiv:math- ph/1509.00537v5

    2017 arXiv

  11. [19]

    Report on the Status of the Yang–Mills Millennium Prize Problem 2004, https://ncatlab.org/nlab/files/Douglas-StatusOfYMMillenniumProb.pdf

    Douglas, M. Report on the Status of the Yang–Mills Millennium Prize Problem 2004, https://ncatlab.org/nlab/files/Douglas-StatusOfYMMillenniumProb.pdf

    2004

  12. [20]

    Quantum Field Theory in a Nutshell ; Princeton University Press: Princeton, NJ, USA, 2010; chap.I.3

    Zee, A. Quantum Field Theory in a Nutshell ; Princeton University Press: Princeton, NJ, USA, 2010; chap.I.3

    2010

  13. [21]

    Effective Locality and Chiral Symmetry Breaking in QCD

    Grandou, T.; Tsang, P.H. Effective Locality and Chiral Symmetry Breaking in QCD. Mod. Phys. Lett. A 2019, 34, 1950335

    2019

  14. [22]

    Wigner law for matrices with dependent entries–a per- turbative approach

    Krajewski, T.; Tanasa, A.; Vu, D.L. Wigner law for matrices with dependent entries–a per- turbative approach. J. Phys. A: Math. Theor 2017, 50, 16LT02. 28

    2017

  15. [23]

    See Ref.[15], Appendix A8

  16. [24]

    Comparison of QCD curves with elastic pp scattering data

    Fried, H.M.; Tsang, P.H.; Gabellini, Y.; Grandou, T.; Sheu, Y.-M. Comparison of QCD curves with elastic pp scattering data. Int. J. Mod. Phys. A 2019, 34, 1950236

    2019

  17. [25]

    Des ph´ enom` enes critiques aux champs de jauge; InterEditions/Editions du CNRS: Paris, France, 1988; p

    Le Bellac, M. Des ph´ enom` enes critiques aux champs de jauge; InterEditions/Editions du CNRS: Paris, France, 1988; p. 91

    1988

  18. [26]

    Lattice calculation of 1 /p2 corrections to αs and of Λ QCD in the MOM scheme

    Boucaud, Ph.; Burgio, G.; Di Renzo, F.; Leroy, J.P.; Micheli, J,; Parrinello, C.; P` ene, O.; Pittori, C.; Rodriguez-Quintero, J.; Roiesnel, C.; Sharkey, K. Lattice calculation of 1 /p2 corrections to αs and of Λ QCD in the MOM scheme. JHEP 0004: 006 2000

    2000

  19. [27]

    An exact, finite, gauge– invariant, non–perturbative approach to QCD renormalization

    Fried, H.M.; Tsang, P.H.; Gabellini, Y.; Grandou, T.; Sheu, Y.-M. An exact, finite, gauge– invariant, non–perturbative approach to QCD renormalization. Ann. Phys. 2015, 359 1

    2015

  20. [28]

    Determination of the Lambda parameter from full lattice QCD

    G¨ ockeler, M.; Horsley, R.; Iving, A.C.; Pleiter, D.; Rakow, P.E.L.; Schierholz, G.; St¨ uben, H. Determination of the Lambda parameter from full lattice QCD. Phys. Rev. D 2006, 73, 014513

    2006

  21. [29]

    The Thermodynamics of Quantum Yang Mills Theory ; World Scientific: Singa- pore, 2012

    Hofmann, R. The Thermodynamics of Quantum Yang Mills Theory ; World Scientific: Singa- pore, 2012. 29

    2012

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.