Perturbative static quark potential in Maximal Abelian gauge

Matthias Berwein; Yukinari Sumino

REVIEW 3 major objections 4 minor 23 references

Perturbative static quark potential in Maximal Abelian gauge

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The static quark potential's Abelian projection is computed to two loops in the Maximal Abelian gauge, with explicit coefficients and a running gauge parameter, showing that the diagonal-gluon part of the potential depends on the…

desk verdict A genuine two-loop technical result with new Abelian-projection coefficients, but the exact MA gauge condition cannot be imposed and the paper half-admits it; worth refereeing if reframed as an MA-type gauge calculation. read the letter →

arxiv 1908.08753 v2 pith:DBQOTIC2 submitted 2019-08-23 hep-ph hep-lathep-th

Matthias Berwein , Yukinari Sumino This is my paper · ORCID

classification hep-phhep-lathep-th

keywords staticquarkpotentialMaximalAbeliangaugeprojectiontwo-loopperturbationtheoryparameterrenormalizationSU(N)latticecomparisondominance

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

The Maximal Abelian (MA) gauge treats diagonal gluons specially, and its Abelian projection keeps only those diagonal gluons in the static quark-antiquark potential. This paper establishes the two-loop perturbative result for both the full potential and the Abelian projection in this gauge, with explicit coefficients through order $\alpha_s^2$. The full potential comes out gauge-invariant and agrees with the known result, while the Abelian projection depends on the off-diagonal gauge-fixing parameter $\xi$ already at one loop; at two loops $\xi$ itself must be renormalized, and the paper gives its running. A comparison with lattice data at $r \lesssim 0.5$ fm shows reasonable agreement for the full potential but a short-distance discrepancy for the Abelian projection, which the paper attributes to the fact that the exact MA gauge condition (bare $\xi=0$) cannot be maintained while keeping the projection finite. This fixes what the diagonal-gluon sector of the potential does perturbatively at short distances, which is the sector that lattice studies associate with confinement.

What carries the argument

The central machinery is the Faddeev-Popov gauge-fixed Lagrangian (7), which enforces the off-diagonal MA condition with parameter $\xi$ and the diagonal covariant condition with parameter $\eta$. The Abelian projection is implemented by including only diagonal gluons in the Wilson line, and the logarithm of the Wilson loop is evaluated diagram-by-diagram through the exponentiation theorem, derived here via the replica trick. Color coefficients are computed with Fierz identities that split SU(N) generators into diagonal and off-diagonal sectors, and all two-loop integrals are reduced with the Laporta algorithm and integration-by-parts identities to five known master integrals. The renormalization of $\xi$, whose running is Eq. (20), is the new element that removes the ultraviolet divergences of the Abelian projection at two loops.

What would settle it

Perform the same two-loop calculation of the Abelian-projected potential in a BRST-invariant generalized Maximal Abelian gauge that includes the extra four-ghost interaction; if the dependence on the gauge-fixing parameters differs from Eqs. (15)-(19), then the Faddeev-Popov Lagrangian (7) is not the correct two-loop representative of the MA gauge.

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Extended reading notes

Core claim

The paper's central claim is that the Abelian projection of the static quark potential in the MA gauge is, to two loops, $$V_{\rm AP}(|q|)=-\frac{4\pi $C_F^{{\rm AP}}$\alpha_s(q)}{$q^{2}$}\left[1+\frac{\alpha_s}{4\pi}($a_1^{{(AP)}}$+$b_1^{{(AP)}}$\xi+$c_1^{{(AP)}}$\$xi^{2}$)+\left(\frac{\alpha_s}{4\pi}\right)^2($a_2^{{(AP)}}$+$b_2^{{(AP)}}$\xi+$c_2^{{(AP)}}$\$xi^{2}$+$d_2^{{(AP)}}$\$xi^{3}$)+O(\$alpha_s^{3}$)\right],$$ with the coefficients given in Eqs. (15)-(19). The full potential has the same form with only $a_1,a_2$ nonzero. The gauge parameter runs as in Eq. (20), and because $\zeta_0^{(-1)}=3\neq 0$, the bare $\xi$ cannot be set to zero with the Abelian projection kept finite, so the two-loop result necessarily lives at $\xi\neq 0$ rather than at the exact MA gauge condition. The paper concludes that this is the likely source of the short-distance discrepancy with lattice data for the projected potential, since the full potential at the same distances already agrees reasonably with the lattice.

Load-bearing premise

The result stands or falls on whether the gauge-fixed Lagrangian used here, with a renormalized off-diagonal gauge parameter, really represents the Maximal Abelian gauge at two loops; the paper itself notes that the exact MA condition would require the bare $\xi$ to vanish, which its result shows is incompatible with a finite Abelian projection.

Editorial extensions

If this is right

The Abelian-projected static quark potential is now known through order $\alpha_s^2$ in the MA gauge, making direct perturbative-versus-lattice comparisons possible for the diagonal-gluon sector at short distances.
The full potential is independent of both $\xi$ and $\eta$ at two loops, as required by Wilson-loop gauge invariance, while the Abelian projection depends on $\xi$ but is independent of $\eta$.
At short distances the Abelian projection is Coulombic with logarithmic corrections, and its main difference from the full potential is an overall factor from the ratio of quadratic Casimirs; no monopole-induced linear term appears in perturbation theory.
Because $\zeta_0^{(-1)}\neq 0$, any two-loop calculation of the Abelian projection with finite counterterms necessarily uses $\xi\neq 0$, so the exact MA gauge condition cannot be realized perturbatively at this order.
The comparison with quenched SU(3) lattice data gives reasonable agreement for the full potential up to $r\lesssim 0.2$-$0.25$ fm, while the Abelian projection disagrees at short distances, consistent with the gauge-condition issue identified in the paper.

Reading between the lines

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

A natural test is to fit the lattice Abelian-projected potential to the two-loop form (12) with $\xi$ treated as a free running parameter; a good fit would confirm that the discrepancy is the $\xi\neq 0$ effect, while a poor fit would point to nonperturbative or monopole contributions missing from the calculation.
Repeating the computation in a BRST-invariant generalized MA gauge with the extra four-ghost coupling would localize how much of the $\xi$-dependence is an artifact of the Faddeev-Popov choice; agreement with (15)-(19) would validate the shortcut, while disagreement would identify where the gauge-fixing scheme matters.
If the $\xi$-dependence persists in higher orders, then Abelian dominance at short distances is not a purely gauge-invariant statement: the diagonal-gluon potential would depend on how strictly the MA gauge condition is enforced, and the nonperturbative monopole contribution could be the only piece that restores scheme independence.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

You should look at this paper if you care about gauge dependence of perturbative potentials, but go in knowing that the title overclaims. The genuinely new content is the two-loop Abelian-projection coefficients in Eqs. (15)–(19), with the explicit ξ dependence of the projected potential. That is a real first: nobody had computed this in a maximal-Abelian-type gauge before. The full potential is reproduced from the known result, which is a useful check, and the separate consistency with the off-diagonal self-energy and with Gracey's anomalous dimension gives me reasonable confidence the coefficients are right, even though the derivation itself is deferred to "elsewhere." The authors also deserve credit for being upfront about the renormalization problem at the end: ζ_0^{(-1)} = 3 means the bare ξ cannot be set to zero while keeping the result finite, so the exact MA gauge condition is not met. That is not a hidden flaw; it is printed in the last paragraph. But it is a load-bearing caveat. The object computed is a one-parameter family of ξ-deformed gauges, and the identification with the MA gauge that motivates the monopole picture is exactly what fails. The lattice comparison in Fig. 1 uses renormalized ξ(μ)=0, but since ξ runs, that is not the same as the exact gauge. So the central claim should be restated: these are the two-loop coefficients for a generalized MA-type gauge, not for the strict MA gauge. As a technical contribution that is still valuable, but the paper as written invites a reader to infer more than is established. The derivation being absent is the other soft spot; the benchmark checks partially compensate, but I would want the detailed calculation or a public code before fully trusting the color structure. The citation pattern looks fine; the self-citation to Sumino's scale-setting is relevant, and Gracey's paper is properly credited. If I were the editor I would send this to a competent referee with the instruction to focus on the gauge-condition interpretation and to ask for the computation details. It is a borderline but legitimate technical result that deserves a fair hearing, provided the authors accept a revision that either supplies the bare-ξ=0 limit or renames the gauge.

Referee Report

3 major / 4 minor

Summary. The paper computes the static quark potential and its Abelian projection in SU(N) gauge theory up to two loops using Faddeev-Popov gauge fixing with a Maximal-Abelian-type gauge condition parametrized by an off-diagonal gauge parameter ξ and a diagonal one η. The full potential is found to be ξ- and η-independent and to reproduce the known result, while the Abelian projection acquires a ξ dependence, with coefficients given in Eqs. (15)–(19) and the running of ξ given in Eqs. (20)–(21). The results are compared with lattice data at short distances. The paper explicitly acknowledges that the exact Maximal Abelian gauge condition cannot be imposed at two loops because the renormalization of ξ prevents setting the bare ξ to zero.

Significance. If the central claim were fully established, this would be the first two-loop perturbative computation of the Abelian-projected static potential in a Maximal-Abelian-type gauge, providing a quantitative reference for lattice studies of Abelian dominance and for the role of the gauge parameter. The paper includes useful cross-checks: the full potential agrees with known results, and the Abelian-projection coefficients are consistent with an independent off-diagonal gluon self-energy calculation and with Gracey's anomalous dimension. However, the stated interpretation of the result as the MA-gauge Abelian projection is not supported, because the computation is performed in a one-parameter family of ξ-deformed gauges rather than in the exact MA gauge. The comparison with lattice data is therefore not a test of the MA-gauge Abelian projection as such.

major comments (3)

[Section 4, final paragraph and Eq. (21)] The central claim that Eqs. (15)–(19) give the Abelian projection of the static potential in the Maximal Abelian gauge is not established. The exact MA gauge condition corresponds to a delta-function enforcement of Eq. (3), i.e., the limit of vanishing bare ξ. The paper itself states that ζ_0^{(-1)}=3 (Eq. (21)) makes it impossible to set the bare ξ to zero while keeping the Abelian-projected potential finite, and that the MA gauge condition is only satisfied if the bare ξ exactly vanishes. Hence the computed object is the Abelian projection in a one-parameter family of MA-type gauges with renormalized ξ, not in the MA gauge. The title, abstract, and Section 1 should be revised to state this qualification, or a renormalization scheme that allows bare ξ=0 must be provided.
[Figure 1 and the comparison with lattice data] The comparison of the perturbative result with the lattice data of Ref. [20] is presented as a comparison of the Abelian-projected potential in the MA gauge. But the lattice result is obtained in the exact MA gauge defined by minimization of Eq. (1), whereas the perturbative result uses Faddeev-Popov gauge fixing with a renormalized ξ=0, which, by the paper's own admission, does not satisfy the exact MA gauge condition. Setting the renormalized ξ=0 does not make the gauge the exact MA gauge. The agreement or disagreement shown in Fig. 1 therefore concerns a different gauge and should not be used to support the central claim.
[Section 4, cross-checks] The two cross-checks reported in Section 4 do not validate the identification of the Abelian projection with the MA gauge. The check that the full potential is ξ-independent tests only the Wilson-loop expectation value of the full color trace, which is a gauge-invariant quantity; it has no sensitivity to the Abelian projection. The check against the off-diagonal gluon self-energy tests a two-point function, not the Wilson-loop color structure (the definition of the Abelian projection in Eq. (8) with only diagonal gluons). These checks confirm internal consistency of the calculation but do not resolve the question of whether the computed Abelian projection belongs to the exact MA gauge.

minor comments (4)

[Abstract and title] If the central claim is restated as a result for an MA-type gauge, the title and abstract should be adjusted accordingly, because 'Maximal Abelian gauge' currently suggests the exact gauge condition is satisfied.
[Section 4, renormalized ξ in Fig. 1] The text says 'We set the renormalized ξ=0 in the Abelian projection of the potential' but does not specify the value of η used in the comparison. Since the result is independent of η, this is not a technical problem, but a brief statement would improve clarity.
[General presentation] There are several typographical errors, including 'A ramaki' in the affiliation line and 'T he' in the abstract. The caption of Fig. 1 calls the red lines 'Abelian-projected potentials in the MA gauge,' which is misleading in light of the qualifications in the final paragraph.
[Section 4, Eq. (20)] The running of ξ is given to one-loop order, but the potential is computed to two loops. The text should clarify the order counting: whether the O(α_s^2) terms in Eq. (20) are irrelevant for the two-loop potential or whether they are assumed to be dropped, so that the reader understands the accuracy of the input.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the two-loop Abelian-projected coefficients are new outputs benchmarked against external results, with a self-admitted MA-gauge validity caveat that is a correctness issue, not a circularity.

full rationale

No circular step is present. The calculation derives the two-loop potential from the gauge-fixed Lagrangian (7) using standard Wilson-loop exponentiation, Laporta reduction to five master integrals taken from [16], and Fierz-based color algebra; none of these inputs contains the target coefficients (15)-(19). The full potential is cross-checked against the known result, and the Abelian projection is a new output. The comparison in Fig. 1 uses external inputs (alpha_s=0.165, r0*Lambda_MS=0.574 from [21], lattice data from [20]) plus an arbitrary additive constant; no shape parameter is fitted, so the plotted curves are not forced by the data. The running of xi in Eq. (20) is fixed by renormalization and agrees with the independent calculation [19]; citations [6] and [21] are by one of the authors but are only used as benchmarks/inputs, not to justify the central derivation. The paper's own final paragraph concedes that zeta0^(-1)=3 (Eq. (21)) prevents taking the bare-xi=0 limit, so the exact MA gauge condition is not satisfied at two loops. This is a real validity limitation of the central claim's gauge identification, located at the end of Section 4, but it is not circularity: the computed object is well defined as an MA-type gauge with renormalized xi, and its coefficients are not equivalent to any input by construction. Accordingly, no circularity is charged; the limitation should be weighed as a correctness/interpretation risk elsewhere.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new entities. Its free parameters are external inputs and a hand-chosen ξ; the key axioms are the validity of the FP gauge fixing and the reuse of known master integrals. The main caveat is that the exact MA gauge condition cannot be imposed, so the calculation is in a softer gauge.

free parameters (3)

renormalized ξ(μ) for the Abelian projection = set to 0 in the plot; running coefficient ζ_0^(-1)=3 prevents setting bare ξ=0
The abelian projected potential depends on the gauge parameter ξ. The paper chooses the renormalized value ξ=0 for the lattice comparison, but this is a hand-chosen condition that cannot be maintained at all scales with the exact MA gauge condition.
α_s(μ) and Λ_MS/μ = α_s(μ)=0.165, Λ_MS/μ=0.07
These values are taken from the literature to set the scale in the comparison with lattice data. They are not fitted to the abelian projection data in this paper, but the comparison depends on them.
arbitrary additive constant per potential in Fig. 1 = one constant per curve
Each potential is shifted by a constant to align with the lattice data. This is a free parameter in the comparison, so only the shape of the potential is being tested.

assumptions (4)

domain assumption Faddeev-Popov gauge fixing with two parameters (ξ,η) is a valid starting point for the two-loop MA-gauge calculation, despite known renormalizability concerns about BRST terms.
Invoked in Section 2 when the authors choose FP over BRST and assert problematic diagrams do not enter at two loops.
standard math Master integrals from the known static potential calculation [16] can be reused for the integration-by-parts reduction.
Used in Section 4 to reduce the many integrals to five master integrals.
standard math The replica-trick exponentiation theorem for Wilson lines holds with a split into diagonal and off-diagonal color indices.
Used in Section 3 to compute logarithmic color factors; the authors note the theorem does not depend on the diagonal/off-diagonal distinction.
domain assumption The perturbative expansion in α_s is reliable at r ≲ 0.5 fm for the potentials compared with lattice data.
The comparison in Fig. 1 assumes the two-loop result is relevant in this range.

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Cite this review

Pith. "Pith review of Perturbative static quark potential in Maximal Abelian gauge." pith.science (2026). https://pith.science/paper/DBQOTIC2

@misc{pith2026190808753,
  author       = {Pith},
  title        = {Pith review of: Perturbative static quark potential in Maximal Abelian gauge},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DBQOTIC2}},
  note         = {Machine review of arXiv:1908.08753}
}

abstract

We calculate the static quark potential for an SU(N) gauge theory in the Maximal Abelian gauge as well as its Abelian projection up to two loops in perturbation theory. We discuss its renormalization properties. The result is compared with a recent lattice result at $r \lesssim 0.5$ fm.

Figures

Figures reproduced from arXiv: 1908.08753 by the authors.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

23 extracted references · 14 canonical work pages

[20]

Three-quark potential and Abelian dominance of confinement in SU(3) QCD

N. Sakumichi, H. Suganuma, Three-quark potential and Abelian dom- inance of conﬁnement in SU(3) QCD, Phys. Rev. D92 (2015) 034511 . arXiv:1501.07596, doi:10.1103/PhysRevD.92.034511

work page Pith review arXiv 2015
[1]

’t Hooft, Topology of the Gauge Condition and New Conﬁnement Phases in Nonabelian Gauge Theories, Nucl

G. ’t Hooft, Topology of the Gauge Condition and New Conﬁnement Phases in Nonabelian Gauge Theories, Nucl. Phys. B190 (1981) 455–

work page 1981
[2]

A. S. Kronfeld, M. L. Laursen, G. Schierholz, U. J. Wiese, Monop ole Condensation and Color Conﬁnement, Phys. Lett. B198 (1987) 51 6–520. doi:10.1016/0370-2693(87)90910-5

work page doi:10.1016/0370-2693(87)90910-5 1987
[3]

A. S. Kronfeld, G. Schierholz, U. J. Wiese, Topology and Dynam- ics of the Conﬁnement Mechanism, Nucl. Phys. B293 (1987) 461–47 8. doi:10.1016/0550-3213(87)90080-0

work page doi:10.1016/0550-3213(87)90080-0 1987
[4]

J. D. Stack, S. D. Neiman, R. J. Wensley, String tension from mon opoles in SU(2) lattice gauge theory, Phys. Rev. D50 (1994) 3399–3405. arXiv:hep-lat/9404014, doi:10.1103/PhysRevD.50.3399

work page Pith review arXiv 1994
[5]

Perfect Abelian dominance of quark confinement in SU(3) QCD

N. Sakumichi, H. Suganuma, Perfect Abelian dominance of quark conﬁnement in SU(3) QCD, Phys. Rev. D90 (11) (2014) 111501. arXiv:1406.2215, doi:10.1103/PhysRevD.90.111501

work page Pith review arXiv 2014
[6]

Anzai, Y

C. Anzai, Y. Kiyo, Y. Sumino, Static QCD potential at three- loop order, Phys. Rev. Lett. 104 (2010) 112003. arXiv:0911.4335, doi:10.1103/PhysRevLett.104.112003

arXiv 2010
[7]

A. V. Smirnov, V. A. Smirnov, M. Steinhauser, Three-loop static potential, Phys. Rev. Lett. 104 (2010) 112002. arXiv:0911.4742, doi:10.1103/PhysRevLett.104.112002

arXiv 2010

Show all 23 references

[8]

H. Min, T. Lee, P. Y. Pac, Renormalization of Yang-Mills Theory in the Abelian Gauge, Phys. Rev. D32 (1985) 440. doi:10.1103/PhysRevD.32.440

1985 doi
[9]

Kondo, T

K.-I. Kondo, T. Shinohara, Abelian dominance in low-energy gluody - namics due to dynamical mass generation, Phys. Lett. B491 (2000 ) 263–

2000
[10]

Pineda, J

A. Pineda, J. Soto, Eﬀective ﬁeld theory for ultrasoft moment a in NRQCD and NRQED, Nucl. Phys. Proc. Suppl. 64 (1998) 428–432. arXiv:hep-ph/9707481, doi:10.1016/S0920-5632(97)01102-X . 11

1998 arXiv
[11]

Brambilla, A

N. Brambilla, A. Pineda, J. Soto, A. Vairo, Potential NRQCD: An Eﬀective theory for heavy quarkonium, Nucl. Phys. B566 (2000) 2 75. arXiv:hep-ph/9907240, doi:10.1016/S0550-3213(99)00693-8

2000 arXiv
[12]

V. S. Dotsenko, S. N. Vergeles, Renormalizability of Phase Fact ors in the Nonabelian Gauge Theory, Nucl. Phys. B169 (1980) 527–546. doi:10.1016/0550-3213(80)90103-0

1980 doi
[13]

J. G. M. Gatheral, Exponentiation of Eikonal Cross-sections in Nonabelian Gauge Theories, Phys. Lett. B133 (1983) 90–94. doi:10.1016/0370-2693(83)90112-0

1983 doi
[14]

Gardi, E

E. Gardi, E. Laenen, G. Stavenga, C. D. White, Webs in mul- tiparton scattering using the replica trick, JHEP 11 (2010) 155. arXiv:1008.0098, doi:10.1007/JHEP11(2010)155

2010 arXiv
[15]

Laporta, High precision calculation of multiloop Feynman integr als by diﬀerence equations, Int

S. Laporta, High precision calculation of multiloop Feynman integr als by diﬀerence equations, Int. J. Mod. Phys. A15 (2000) 5087–515 9. arXiv:hep-ph/0102033, doi:10.1142/S0217751X00002157

2000 arXiv
[16]

Schroder, The Static potential in QCD, Ph.D

Y. Schroder, The Static potential in QCD, Ph.D. thesis, Hambur g U. (1999). URL http://www-library.desy.de/cgi-bin/showprep.pl?desy-thesis99-021

1999
[17]

Fischler, Quark - anti-Quark Potential in QCD, Nucl

W. Fischler, Quark - anti-Quark Potential in QCD, Nucl. Phys. B1 29 (1977) 157–174. doi:10.1016/0550-3213(77)90026-8

1977 doi
[18]

Billoire, How Heavy Must Quarks Be in Order to Build Coulombic q anti-q Bound States, Phys

A. Billoire, How Heavy Must Quarks Be in Order to Build Coulombic q anti-q Bound States, Phys. Lett. 92B (1980) 343–347 . doi:10.1016/0370-2693(80)90279-8

1980 doi
[19]

J. A. Gracey, Three loop MS-bar renormalization of QCD in the ma x- imal Abelian gauge, JHEP 04 (2005) 012. arXiv:hep-th/0504051, doi:10.1088/1126-6708/2005/04/012

2005 arXiv
[21]

Sumino, Static QCD potential at r < 1/Λ QCD: Perturbative expan- sion and operator-product expansion, Phys

Y. Sumino, Static QCD potential at r < 1/Λ QCD: Perturbative expan- sion and operator-product expansion, Phys. Rev. D76 (2007) 11 4009. arXiv:hep-ph/0505034, doi:10.1103/PhysRevD.76.114009. 12

2007 arXiv
[274]

arXiv:hep-th/0004158, doi:10.1016/S0370-2693(00)01028-5

arXiv
[478]

doi:10.1016/0550-3213(81)90442-9

doi

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [20]

Three-quark potential and Abelian dominance of confinement in SU(3) QCD

N. Sakumichi, H. Suganuma, Three-quark potential and Abelian dom- inance of conﬁnement in SU(3) QCD, Phys. Rev. D92 (2015) 034511 . arXiv:1501.07596, doi:10.1103/PhysRevD.92.034511

work page Pith review arXiv 2015

[2] [1]

’t Hooft, Topology of the Gauge Condition and New Conﬁnement Phases in Nonabelian Gauge Theories, Nucl

G. ’t Hooft, Topology of the Gauge Condition and New Conﬁnement Phases in Nonabelian Gauge Theories, Nucl. Phys. B190 (1981) 455–

work page 1981

[3] [2]

A. S. Kronfeld, M. L. Laursen, G. Schierholz, U. J. Wiese, Monop ole Condensation and Color Conﬁnement, Phys. Lett. B198 (1987) 51 6–520. doi:10.1016/0370-2693(87)90910-5

work page doi:10.1016/0370-2693(87)90910-5 1987

[4] [3]

A. S. Kronfeld, G. Schierholz, U. J. Wiese, Topology and Dynam- ics of the Conﬁnement Mechanism, Nucl. Phys. B293 (1987) 461–47 8. doi:10.1016/0550-3213(87)90080-0

work page doi:10.1016/0550-3213(87)90080-0 1987

[5] [4]

J. D. Stack, S. D. Neiman, R. J. Wensley, String tension from mon opoles in SU(2) lattice gauge theory, Phys. Rev. D50 (1994) 3399–3405. arXiv:hep-lat/9404014, doi:10.1103/PhysRevD.50.3399

work page Pith review arXiv 1994

[6] [5]

Perfect Abelian dominance of quark confinement in SU(3) QCD

N. Sakumichi, H. Suganuma, Perfect Abelian dominance of quark conﬁnement in SU(3) QCD, Phys. Rev. D90 (11) (2014) 111501. arXiv:1406.2215, doi:10.1103/PhysRevD.90.111501

work page Pith review arXiv 2014

[7] [6]

Anzai, Y

C. Anzai, Y. Kiyo, Y. Sumino, Static QCD potential at three- loop order, Phys. Rev. Lett. 104 (2010) 112003. arXiv:0911.4335, doi:10.1103/PhysRevLett.104.112003

arXiv 2010

[8] [7]

A. V. Smirnov, V. A. Smirnov, M. Steinhauser, Three-loop static potential, Phys. Rev. Lett. 104 (2010) 112002. arXiv:0911.4742, doi:10.1103/PhysRevLett.104.112002

arXiv 2010