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The center-symmetric Landau gauge meets the lattice

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Pith's one-line read In the center-symmetric Landau gauge, a single temporal link average becomes a local order parameter: in the confined phase it must equal $\mathrm{diag}(e^{-i2\pi/(3L_4)}, e^{i2\pi/(3L_4)}, 1)$, and any deviation signals deconfinement.

desk verdict First lattice test of the center-symmetric Landau gauge: a clean symmetry derivation and a parameter-free two-temperature check that supports the temporal link average as a local order parameter. read the letter →

arxiv 2412.07930 v1 pith:POV7YAC5 submitted 2024-12-10 hep-lat hep-phhep-th

Duifje Maria van Egmond , Orlando Oliveira , Urko Reinosa , Julien Serreau , Paulo J. Silva , Matthieu Tissier This is my paper · ORCID
classification hep-lathep-phhep-th PACS 11.15.Ha12.38.Aw12.38.Gc
keywords centersymmetryconfinement-deconfinementtransitioncenter-symmetricLandaugaugelatticetheoryPolyakovlooporderparameterGribovcopiesSU(3)Yang-Mills
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

This paper puts the continuum center-symmetric Landau gauge onto the lattice and shows that, in that gauge, the average of a single temporal link — a local quantity — is an order parameter for the confinement-deconfinement transition of SU(3) Yang-Mills theory. In the confined phase, an unbroken center symmetry forces the normalized temporal link average to equal the fixed matrix $\mathrm{diag}(e^{-i2\pi/(3L_4)}, \, e^{i2\pi/(3L_4)}, \, 1)$, so that any statistically significant deviation is a direct signal of deconfinement. This matters because the standard order parameter, the Polyakov loop, is nonlocal and awkward in continuum calculations, whereas the link average is local and renormalization-free. The subtlety, examined at length, is that this property holds only when both the gauge and the selection of Gribov copies keep the gauge-fixed ensemble invariant under the center transformation; Monte Carlo data at $\beta=6.0$ comply below $T_c$ ($64^3\times 8$) and show the predicted violation above $T_c$ ($64^3\times 6$).

What carries the argument

The load-bearing object is the gauge-fixing functional $\tilde F[U]=\sum_{n,\mu}\mathrm{Re}\,\mathrm{tr}\, g_c^\dagger(\hat\mu)U_\mu(n)$ with the fixed background $g_c(\hat\mu)=e^{-i(4\pi/3)(\hat\mu_4/L_4)\lambda_3/2}$, together with the center transformation $\tilde g(n)=e^{i\pi\lambda_4/2}e^{i\pi\lambda_1/2}e^{-i(n_4/L_4)\pi(\lambda_3+\lambda_8/\sqrt3)}$ that leaves it invariant. Because links transform linearly under $\tilde g$, the average obeys the exact identity (41) linking the ensembles $E_{gf}$ and $E^{\tilde g}_{gf}$; when those ensembles coincide approximately, that identity becomes a constraint fixing the matrix structure of $\langle U_\mu(n)\rangle$ down to a single real number $\eta$. The constraint is solved by decomposing the average along the weight vectors of $\mathrm{SU}(3)$ and using the Weyl-reflection form of $\tilde g$; the result is $\langle U_\mu(n)\rangle=\eta\,g_c(\hat\mu)$, whose temporal component yields Eq. (83). The paper proves in Appendix D that the basic steepest-ascent algorithm preserves the required ensemble invariance, and relies on numerical checks for the accelerated production algorithm.

What would settle it

Run the gauge fixing at $\beta=6.0$ on a $64^3\times 8$ lattice using the basic, provably $\tilde g$-symmetric steepest-ascent algorithm of Appendix D with high statistics: if the normalized temporal link average deviates from $\mathrm{diag}(e^{-i2\pi/(3L_4)}, e^{i2\pi/(3L_4)}, 1)$ beyond errors, the claimed constraint (60) does not force Eq. (83), and the central claim collapses. Conversely, the same run above $T_c$ ($L_4=6$) must show the deviation in one of the three center sectors; if the constraint still holds there, the link average is not tracking center-symmetry breaking. An independent check is a direct overlap measurement of the ensembles $E_{gf}$ and $E^{\tilde g}_{gf}$ below $T_c$: if they are not approximately equal, the derivation of the order-parameter property fails at its stated premise.

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Extended reading notes

Core claim

The central claim, stated on the paper's own terms, is that in the center-symmetric Landau gauge the link average $\langle U_\mu(n)\rangle$ is a local order parameter for center symmetry. The gauge is defined by maximizing $\tilde F[U]=\sum_{n,\mu}\mathrm{Re}\,\mathrm{tr}\, g_c^\dagger(\hat\mu)U_\mu(n)$ with a fixed center-symmetric background $g_c(\hat\mu)=e^{-i(4\pi/3)(\hat\mu_4/L_4)\lambda_3/2}$, a functional left invariant by the specific center transformation $\tilde g$ of Eq. (49). In the symmetric phase, where the gauge-fixed ensemble obeys $E_{gf}^{\tilde g}\simeq E_{gf}$, the transformation law (41) becomes the constraint $\langle U_\mu(n)\rangle = \tilde g(n)\langle U_\mu(n)\rangle \tilde g^\dagger(n+\hat\mu)$, and combining it with global color and charge-conjugation invariance yields the unique solution $\langle U_\mu(n)\rangle = \eta\, g_c(\hat\mu)$ with $\eta$ real. For temporal links this is Eq. (83): the average normalized by its determinant equals $\mathrm{diag}(e^{-i2\pi/(3L_4)}, e^{i2\pi/(3L_4)}, 1)$. The numerical data support the prediction: below $T_c$ the normalized average matches the required matrix, while above $T_c$ it deviates in three distinct ways that correspond to the three center sectors of the broken phase.

Load-bearing premise

The order-parameter property rests on the gauge-fixed ensemble being approximately invariant under the center transformation $\tilde g$ in the confined phase; this is proven only for the basic steepest-ascent algorithm, while the production algorithm inherits it by assumption, supported by numerical checks.

Editorial extensions

If this is right

  • The normalized temporal link average of Eq. (83) supplies a local, renormalization-free proxy for the Polyakov loop: any deviation of the matrix elements from $e^{\pm i2\pi/(3L_4)}$ and $1$ signals deconfinement, observable on a single site or averaged over the lattice.
  • The same mechanism extends to two-link correlators and, in the continuum limit, to the gluon and ghost propagators in this gauge, giving momentum-space quantities that behave as order parameters — quantities continuum methods can compute.
  • Each center sector of the deconfined phase is tagged by which diagonal element of the link average sits on $e^{i2\pi k/3}$, so the gauge-fixed link average labels the broken-symmetry ensembles just as the Polyakov loop phase does.
  • The data reveal an unexplained fact: in the deconfined phase the normalized link average $M$ is unitary, since one diagonal element is of modulus one and charge conjugation forces the other two to have equal modulus; the paper derives $\det M=1$ and $|M_{11}|=|M_{22}|=|M_{33}|=1$ from this, but not the origin of the modulus-one element.

Reading between the lines

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

  • If the link average is a true order parameter, a fine scan of the normalized average across $T_c$ (several $L_4$ values at fixed $\beta$) should show it locking onto the Eq. (83) form exactly in the confined phase and leaving it sharply at $T_c$; the paper's two-point comparison ($L_4=6$ vs $8$) does not yet establish the sharpness of the transition this observable sees.
  • The empirical unitarity of $M$ in the deconfined phase, if generic, suggests the vanishing of color-off-diagonal components of the link average is stronger than statistical — it may follow from the combined color-and-$\hat C$ symmetry structure, a line the paper leaves open.
  • A natural extension the authors mention but do not pursue on the lattice: with dynamical quarks, center symmetry is explicitly broken and the transition becomes a crossover; a local order-parameter-like quantity with a well-defined crossover temperature could be useful in finite-density simulations where the Polyakov loop is sign-problematic.
  • The twisted-link formulation of Sec. VI A implies a particularly clean test: in the confined phase $\langle\hat U_\mu(n)\rangle = \eta\,\mathbb{1}$, so components such as $\mathrm{Im}\langle\hat U_4^{3}\rangle$ and $\mathrm{Re}\langle\hat U_4^{8}\rangle$ vanish; counting how often individual configurations deviate from this pattern gives a per-configuration order-parameter diagnostic that the pape
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper proposes a lattice implementation of the center-symmetric Landau gauge for SU(3) pure gauge theory at finite temperature. It derives symmetry constraints for the temporal link average; in particular, Eq. (83) states that in the confined phase the ratio M = <U4>/(det<U4>)^{1/3} must equal the fixed diagonal matrix diag(e^{-i2π/(3L4)}, e^{i2π/(3L4)}, 1). The paper tests this prediction in Monte Carlo simulations at beta=6.0 on 64^3 x 8 (below Tc) and 64^3 x 6 (above Tc) lattices, finding agreement with the predicted pattern below Tc and violations above Tc. The authors also discuss the role of Gribov copies, show that a basic steepest-ascent algorithm preserves the center transformation, and provide numerical evidence that the production algorithm approximately does so.

Significance. If the central claim holds, this is a genuine advance: it provides a local, gauge-dependent order parameter for center symmetry that can be computed with standard lattice techniques and that may be useful in continuum frameworks where the Polyakov loop is difficult to handle. The derivation of Eq. (83) is elegant, and the predicted phase pattern exp(±i 2π/(3L4)) is parameter-free and falsifiable. The paper is transparent about the Gribov-copy caveats and includes an explicit equivariance proof for the basic algorithm (App. D). The numerical test is a direct and encouraging confrontation of the theory with data, albeit with limited statistics and no continuum extrapolation.

major comments (2)
  1. [Sec. IV E and App. D] The equivariance proof in App. D covers only the basic steepest-ascent algorithm, whereas the production runs use Fourier accelerated steepest descent as stated in Sec. V D. The grouping argument in Sec. IV E is heuristic and does not exclude the possibility that a non-equivariant copy-selection map artificially produces the symmetric-phase pattern. Since the derivation of Eq. (83) depends on the approximate invariance of the gauge-fixed ensemble under tilde-g, this gap is load-bearing for the numerical verification. I recommend either proving that the production algorithm preserves tilde-g equivariance (or a closely related variant), or providing a dedicated numerical test with more configurations, a second beta value, and a comparison of full configuration-level distributions, not only the averages.
  2. [Sec. V D] The numerical evidence is based on 100 configurations per ensemble at a single lattice spacing (beta = 6.0), with no continuum extrapolation. The agreement with Eq. (83) below Tc is encouraging, but the statistical and systematic errors are large enough that the claim that 'any deviation from the RHS signals the dynamical breaking of center-symmetry' is not yet a robust, quantitative statement. A scan over beta or an explicit test of the L4 dependence at fixed T would substantially strengthen the paper.
minor comments (4)
  1. [Sec. IV D] Footnote 13 contains the typo 'Strictily', and Sec. III D contains 'reasonning'; both should be corrected.
  2. [Appendix C] Some matrix entries in Eqs. (C2), (C5), and (C6) contain notations like '0.0522(21) i' and '0.02(21)2' that appear to be formatting errors; these should be cleaned up.
  3. [Sec. V C] The statement that the LHS of Eq. (83) 'does not require renormalization' would benefit from a one-sentence justification, since the unrenormalized product may have operator-mixing subtleties that are worth addressing.
  4. [Fig. 4] The caption does not define which ensemble is which color; consider adding a legend or explicit color labels.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Eq. (83) is derived from the gauge functional and the center-symmetry constraint; the unproven equivariance of the accelerated algorithm is an acknowledged evidence gap, not a fit disguised as prediction.

full rationale

The central derivation is self-contained. The gauge functional F is defined in Eq. (47) with the background gc, the center transformation tilde g is given in Eq. (49), and its invariance is verified in the paper (Eq. (53), with the explicit algebra in App. B). The constraint Eq. (60) follows from the stated dynamical assumption that the gauge-fixed ensemble is approximately tilde-g-invariant in the symmetric phase; solving that constraint gives Eq. (81) and, after normalizing by the determinant, the predicted matrix Eq. (83). The prefactor eta is not predicted, but it cancels in the order parameter M, so no fitted parameter is renamed as a prediction. The low-temperature numerical matrices, e.g. Eqs. (98)-(100), are compared with, not fitted to, the predicted phases exp(+-i 2pi/(3 L4)). The paper explicitly acknowledges the one real gap: the production Fourier-accelerated steepest-descent algorithm is not proven to preserve the tilde-g symmetry, unlike the basic steepest-ascent algorithm in App. D, and it then tests the symmetry directly in Figs. 5-6 and Sec. VI C. That is an honest evidence limitation, not a circular reduction. Citations to the authors' continuum papers [7,16,17] motivate the gauge choice and provide Weyl-group conventions, but the load-bearing invariance identities and the uniqueness of the solution are derived within the present paper, so the prior work is not the basis of the claim.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central prediction is parameter-free, but the order-parameter property rests on assumptions about the Monte Carlo sampling and Gribov-copy selection. One undetermined prefactor eta is measured and then cancelled in the normalized order parameter. No new physical entities are introduced; the center-symmetric Landau gauge is a method, not a new particle or force.

free parameters (1)
  • eta (link-average prefactor) = approx 0.855 to 0.881 depending on temperature and ensemble
    The center-symmetry constraint fixes the temporal link average to eta times a fixed matrix, with eta an undetermined complex coefficient that is real by charge conjugation. It is measured from the data and then cancelled in the normalized order parameter M, so it is not used to make the central prediction.
assumptions (4)
  • domain assumption The Wilson action and integration measure are invariant under center transformations, and the Monte Carlo ensemble approximates this invariance when center symmetry is unbroken.
    Used in Secs. III C and IV D to convert the identity (41) into the constraint (60) for the link average.
  • domain assumption With a center-symmetric gauge functional and a suitable Gribov-copy selection, the gauge-fixed ensemble is approximately invariant under tilde g in the symmetric phase.
    Motivated by App. D for the basic steepest-ascent algorithm and tested numerically in Sec. V D, Figs. 5 and 6. If false, the link average would not satisfy the center-symmetry constraint and would not work as an order parameter.
  • domain assumption The lattice spacing calibration 1/a = 1.943 GeV at beta = 6.0 and the pure-gauge transition temperature Tc approx 270 MeV place the L4=8 and L4=6 lattices below and above the transition.
    Used in Sec. V D to interpret the two simulation points as confined and deconfined; the input is taken from previous lattice calibration [35].
  • standard math Standard root and weight identities for SU(3), including the Weyl reflection composition R_{alpha13} R_{alpha12} = R_{2pi/3} and the action of Weyl transformations on diagonal generators.
    Used in Sec. V C to prove the invariance of the gauge functional and to solve the center-symmetry constraint for the diagonal link average.

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Pith. "Pith review of The center-symmetric Landau gauge meets the lattice." pith.science (2026). https://pith.science/paper/POV7YAC5

@misc{pith2026241207930,
  author       = {Pith},
  title        = {Pith review of: The center-symmetric Landau gauge meets the lattice},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/POV7YAC5}},
  note         = {Machine review of arXiv:2412.07930}
}
read the original abstract

A lattice implementation of the recently introduced center-symmetric Landau gauge is discussed and its predictions confronted with numerical Monte Carlo simulations. It is shown that the link average and the link correlators computed in that gauge are order parameters of the confinement-deconfinement transition at nonzero temperature. Strictly speaking, this requires a specific treatment of the Gribov copies that we discuss in detail. The numerical simulations comply with the theoretical predictions for the link average computed below and above the deconfinement temperature. Our results show that, within appropriately chosen gauges, one can construct local order parameters for center symmetry, as proxies for the non-local Polyakov loop.

Figures

Figures reproduced from arXiv: 2412.07930 by the authors.

Figure 1
Figure 1. FIG. 1. Left plot: The distances [PITH_FULL_IMAGE:figures/full_fig_p014_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Location in the complex plane of the first (square), second (triangles) and third (circles) diagonal elements of [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Spread of the real and imaginary parts of [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: In the symmetric phase, the system explores [PITH_FULL_IMAGE:figures/full_fig_p018_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. The real and imaginary parts of the configurations [PITH_FULL_IMAGE:figures/full_fig_p021_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. The distance between the sets [PITH_FULL_IMAGE:figures/full_fig_p022_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Transformation of a Weyl chamber under a center transformation. The colored chamber represents the various locations [PITH_FULL_IMAGE:figures/full_fig_p023_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. The three diagonal components (from left to right but one) of the link elevated to the power [PITH_FULL_IMAGE:figures/full_fig_p027_8.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Center-symmetric Landau gauge, the deconfinement transition and the gluon propagator as seen in lattice QCD

    hep-lat 2025-05 conditional novelty 6.0 of 10

    Lattice simulations in the center-symmetric Landau gauge show that the link average and the D33-D88 gluon propagator difference signal center-symmetry breaking across the deconfinement transition.

Reference graph

Works this paper leans on

45 extracted references · 38 canonical work pages · cited by 1 Pith paper

  1. [1]

    Center-symmetric Landau gauges Center-symmetric Landau gauges are obtained when choosing particular backgrounds ¯Ac,µ, known as center- symmetric, which are invariant under particular center transformations ˜g, that is ¯A˜g c,µ = ¯Ac,µ , (A4) In this case, Aµ and A˜g µ are both configurations in the same background gauge, of associated background ¯Ac,µ. O...

    work page

  2. [2]

    Connection to the Landau gauge Coming back to the family of background Landau gauges, let us now show that, in the case of a constant background, of which (A6) is a particular case, the gauge condition (A1) over gauge fields Aµ that are periodic along the temporal direction, is equivalent to the stan- dard Landau gauge condition but over gauge fields obey...

    work page

  3. [3]

    Notes on the deconfining phase transi- tion,

    R. D. Pisarski, “Notes on the deconfining phase transi- tion,” hep-ph/0203271

    arXiv

  4. [4]

    An introduction to the confinement prob- lem,

    J. Greensite, “An introduction to the confinement prob- lem,” Lect. Notes Phys. 821, 1 (2011)

    work page 2011

  5. [5]

    R. V. Gavai, Nucl. Phys. B 215, 458-469 (1983)

    work page 1983

  6. [6]

    R. V. Gavai and F. Karsch, Phys. Lett. B 125, 406-410 (1983)

    work page 1983

  7. [7]

    Celik, J

    T. Celik, J. Engels and H. Satz, Phys. Lett. B 125, 411- 414 (1983)

    work page 1983

  8. [8]

    Svetitsky, Phys

    B. Svetitsky, Phys. Rept. 132, 1-53 (1986). 28 In fact it could happen that one reaches more generally an ex- tremum. But looking for maxima is just a numerical convenience and, if one ends up with some extrema, there is no problem in including them in the definition of the gauge-fixed configurations

    work page 1986

Show all 45 references
  1. [9]

    D. M. van Egmond and U. Reinosa, Phys. Rev. D 108, no.5, 054029 (2023)

    2023

  2. [10]

    A. M. Polyakov, Phys. Lett. 72B, 477 (1978)

    1978

  3. [11]

    T. K. Herbst, J. Luecker and J. M. Pawlowski, [arXiv:1510.03830 [hep-ph]]

    arXiv

  4. [12]

    Braun, H

    J. Braun, H. Gies and J. M. Pawlowski, Phys. Lett. B 684, 262-267 (2010)

    2010

  5. [13]

    Braun, A

    J. Braun, A. Eichhorn, H. Gies and J. M. Pawlowski, Eur. Phys. J. C 70, 689-702 (2010)

    2010

  6. [14]

    L. F. Abbott, Nucl. Phys. B 185, 189-203 (1981)

    1981

  7. [15]

    L. F. Abbott, Acta Phys. Polon. B 13, 33 (1982)

    1982

  8. [16]

    Fister and J

    L. Fister and J. M. Pawlowski, Phys. Rev. D 88, 045010 (2013)

    2013

  9. [17]

    Perturbative aspects of the deconfinement transition – Physics beyond the Faddeev-Popov model,

    U. Reinosa, “Perturbative aspects of the deconfinement transition – Physics beyond the Faddeev-Popov model,” Lect. Notes Phys. 1006 (2022)

    2022

  10. [18]

    D. M. van Egmond, U. Reinosa, J. Serreau and 28 M. Tissier, SciPost Phys. 12, no.3, 087 (2022)

    2022

  11. [19]

    D. M. van Egmond and U. Reinosa, Phys. Rev. D 109, no.3, 036002 (2024)

    2024

  12. [20]

    D. M. van Egmond and U. Reinosa, Phys. Rev. D 106, no.7, 074005 (2022)

    2022

  13. [21]

    Curci and R

    G. Curci and R. Ferrari, Nuovo Cim. A 32, 151-168 (1976)

    1976

  14. [22]

    Tissier and N

    M. Tissier and N. Wschebor, Phys. Rev. D 82, 101701 (2010)

    2010

  15. [23]

    Tissier and N

    M. Tissier and N. Wschebor, Phys. Rev. D 84, 045018 (2011)

    2011

  16. [24]

    Pel´ aez, U

    M. Pel´ aez, U. Reinosa, J. Serreau, M. Tissier and N. Wschebor, Rept. Prog. Phys. 84 (2021) no.12, 124202

    2021

  17. [25]

    Cucchieri and T

    A. Cucchieri and T. Mendes, PoS F ACESQCD, 007 (2010)

    2010

  18. [26]

    C. S. Fischer, A. Maas and J. A. Muller, Eur. Phys. J. C 68, 165-181 (2010)

    2010

  19. [27]

    A. Maas, J. M. Pawlowski, L. von Smekal and D. Spiel- mann, Phys. Rev. D 85, 034037 (2012)

    2012

  20. [28]

    Mendes and A

    T. Mendes and A. Cucchieri, PoS LA TTICE2014, 183 (2015)

    2015

  21. [29]

    P. J. Silva, O. Oliveira, P. Bicudo and N. Cardoso, Phys. Rev. D 89, no.7, 074503 (2014)

    2014

  22. [30]

    Aouane, V

    R. Aouane, V. G. Bornyakov, E. M. Ilgenfritz, V. K. Mitrjushkin, M. Muller-Preussker and A. Stern- beck, Phys. Rev. D 85, 034501 (2012)

    2012

  23. [31]

    Fister and J

    L. Fister and J. M. Pawlowski, [arXiv:1112.5440 [hep- ph]]

    arXiv

  24. [32]

    Quandt and H

    M. Quandt and H. Reinhardt, Phys. Rev. D 92, no.2, 025051 (2015)

    2015

  25. [33]

    P. J. Silva and O. Oliveira, Phys. Rev. D 93, no.11, 114509 (2016)

    2016

  26. [34]

    Gattringer, Phys

    C. Gattringer, Phys. Lett. B 690, 179-182 (2010)

    2010

  27. [35]

    R. G. Edwards et al. , Nucl. Phys. B Proc. Suppl. 140, 832 (2005)

    2005

  28. [36]

    Pippig, SIAM J

    M. Pippig, SIAM J. Sci. Comput. 35, C213 (2013)

    2013

  29. [37]

    Oliveira and P

    O. Oliveira and P. J. Silva, Phys. Rev. D 86, 114513 (2012) doi:10.1103/PhysRevD.86.114513 [arXiv:1207.3029 [hep-lat]]

    2012 arXiv

  30. [38]

    M. A. Shifman, A. I. Vainshtein and V. I. Zakharov, Nucl. Phys. B 147, 385-447 (1979)

    1979

  31. [39]

    Martinelli and C

    G. Martinelli and C. T. Sachrajda, Nucl. Phys. B 478, 660-686 (1996)

    1996

  32. [40]

    Horsley, G

    R. Horsley, G. Hotzel, E. M. Ilgenfritz, R. Millo, H. Perlt, P. E. L. Rakow, Y. Nakamura, G. Schierholz and A. Schiller, Phys. Rev. D 86, 054502 (2012)

    2012

  33. [41]

    https://doi.org/10.54499/UIDB/04564/2020

    2020 doi

  34. [42]

    https://doi.org/10.54499/UIDP/04564/2020

    2020 doi

  35. [43]

    https://doi.org/10.54499/CERN/FIS-PAR/0023/2021

    2021 doi

  36. [44]

    https://doi.org/10.54499/CEECIND/00488/2017/- CP1460/CT0030

    2017 doi

  37. [45]

    https://doi.org/10.54499/2021.09759.CPCA

    2021

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