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

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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 Yang–Mills 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 gauge-invariant 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 group H̃ of G, while there is a unique option of H̃=U(1) for G=SU(2). The maximal stability group depends on the representation of the gauge group, to that the quark source belongs. For the fundamental quark for SU(3), the maximal stability group is U(2), which is different from the maximal torus group U(1)×U(1) suggested from the Abelian projection. Therefore, the chromomagnetic monopole inherent in the Wilson loop operator responsible for confinement of quarks in the fundamental representation for SU(3) is the non-Abelian magnetic monopole, which is distinct from the Abelian magnetic monopole for the SU(2) case. Therefore, we claim that the mechanism for quark confinement for SU(N) (N≥3) is the non-Abelian dual superconductivity caused by condensation of non-Abelian magnetic monopoles. We give some theoretical considerations and numerical results supporting this picture. Finally, we discuss some issues to be investigated in future studies.