Instantons and Gribov copies in the maximally Abelian gauge

The purpose of this paper is to review the recent progress in understanding quark confinement. The emphasis of this review is placed on how to obtain a manifestly gauge-independent picture for quark confinement supporting the dual superconductivity in the Yang-Mills theory, which should be compared with the Abelian projection proposed by 't Hooft. The basic tools are novel reformulations of the YangMills theory based on change of variables extending the decomposition of the SU(N) Yang-Mills field due to Cho, Duan-Ge and Faddeev-Niemi, together with the combined use of extended versions of the Diakonov-Petrov version of the non-Abelian Stokes theorem for the SU(N) Wilson loop operator.Moreover, we give the lattice gauge theoretical versions of the reformulation of the Yang-Mills theory which enables us to perform the numerical simulations on the lattice. In fact, we present some numerical evidences for supporting the dual superconductivity for quark confinement. The numerical simulations include the derivation of the linear potential for static interquark potential, i.e., non-vanishing string tension, in which the "Abelian" dominance and magnetic monopole dominance are established, confirmation of the dual Meissner effect by measuring the chromoelectric flux tube between quark-antiquark pair, the induced magnetic-monopole current, and the type of dual superconductivity, etc. In addition, we give a direct connection between the topological configuration of the Yang-Mills field such as instantons/merons and the magnetic monopole.We show especially that magnetic monopoles in the Yang-Mills theory can be constructed in a manifestly gauge-invariant way starting from the gauge-invariant Wilson loop operator and thereby the contribution from the magnetic monopoles can be extracted from the Wilson loop in a gauge-invariant way through the non-Abelian Stokes theorem for the Wilson loop operator, which is a prerequisite for exhibiting magnetic monopole dominance for quark confinement. The Wilson loop average is calculated according to the new reformulation written in terms of new field variables obtained from the original Yang-Mills field based on change of variables. The Maximally Abelian gauge in the original Yang-Mills theory is also reproduced by taking a specific gauge fixing in the reformulated Yang-Mills theory. This observation justifies the preceding results obtained in the maximal Abelian gauge at least for gaugeinvariant quantities for SU(2) gauge group, which eliminates the criticism of gauge artifact raised for the Abelian projection. The claim has been confirmed based on the numerical simulations.However, for SU(N) (N ≥ 3), such a gauge-invariant reformulation is not unique, although the extension along the line proposed by Cho, Faddeev and Niemi is possible. In fact, we have found that there are a number of possible options of the reformulations, which are discriminated by the maximal stability 1 groupH of G, while there is a unique option ofH = U(1) for G = SU(2). The maximal stability group depends...

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“…Then the Cartan metric is diagonal: 83) and the structure constant f ABC differ from f AB C by the factor 3:…”

Section: Explicit Construction For Su(3)mentioning

confidence: 99%

Quark confinement: Dual superconductor picture based on a non-Abelian Stokes theorem and reformulations of Yang–Mills theory

et al. 2015

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“…There are also studies from a perspective of the KugoOjima (KO) criterion and the Gribov problem. The KO criterion has been generalized to the MA gauge in a very recent paper [16] and there are also the Gribov copies in the MA gauge, located on the other side of the horizon [22][23][24]. The gluon propagator in the MA gauge has been also studied on the basis of the original Gribov's work and the horizon term in the MA gauge, corresponding to that in the Landau gauge has been studied [2,3,22].…”

Section: Introductionmentioning

confidence: 99%

Gribov-Zwanziger action in SU(2) maximally Abelian gauge withU(1)3Landau gauge

Gongyo

Iida

2014

Phys. Rev. D

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We construct the local Gribov-Zwanziger action for SU(2) Euclidean Yang-Mills theories in the maximally Abelian (MA) gauge with U(1)3 Landau gauge fixing based on the Zwanziger's work in the Landau gauge. By the restriction of the functional integral region to the Gribov region in the MA gauge, we give the nonlocal action. We localize the action with new fields and obtain the action with the shift of the new scalar fields, which has the terms, corresponding to the localized action of the horizon function in the MA gauge. The diagonal gluon propagator in the MA gauge at tree level behaves like the propagator from Gribov-Zwanziger action in the Landau gauge and shows the violation of Kallen-Lehmann representation.

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“…The present work deals with the study of the zero modes of the Faddeev-Popov operator in the maximal Abelian gauge [5] in various Euclidean space-time dimensions and for the gauge group SU (2). We shall provide explicit examples of classes of normalizable zero modes and associated gauge configurations by taking into account two boundary conditions for the gauge fields, namely: i) the finite Euclidean Yang-Mills action,…”

Section: Introductionmentioning

confidence: 99%

Study of the zero modes of the Faddeev–Popov operator in the maximal Abelian gauge

et al. 2014

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A study of the zero modes of the Faddeev-Popov operator in the maximal Abelian gauge is presented in the case of the gauge group SU (2) and for different Euclidean space-time dimensions. Explicit examples of classes of normalizable zero modes and corresponding gauge field configurations are constructed by taking into account two boundary conditions, namely: i) the finite Euclidean Yang-Mills action, ii) the finite Hilbert norm.

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Instantons and Gribov copies in the maximally Abelian gauge

Cited by 38 publications

References 60 publications

Quark confinement: Dual superconductor picture based on a non-Abelian Stokes theorem and reformulations of Yang–Mills theory

Quark confinement: Dual superconductor picture based on a non-Abelian Stokes theorem and reformulations of Yang–Mills theory

Gribov-Zwanziger action in SU(2) maximally Abelian gauge withU(1)3Landau gauge

Study of the zero modes of the Faddeev–Popov operator in the maximal Abelian gauge

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