Instanton, monopole condensation and confinement

Abstract

The confinement mechanism in the nonperturbative QCD is studied in terms of topological excitation as QCDmonopoles and instantons. In the 't Hooft abelian gauge, QCD is reduced into an abelian gauge theory with monopoles, and the QCD vacuum can be regarded as the dual superconductor with monopole condensation, which leads to the dual Higgs mechanism. The monopole-current theory extracted from QCD is found to have essential features of confinement. We find also close relation between monopoles and instantons using the lattice QCD. In this framework, the lowest 0 ++ glueball (1.5 ∼ 1.7GeV) can be identified as the QCD-monopole or the dual Higgs particle. Dual Higgs Theory for NP-QCDQuantum chromodynamics (QCD) is established as the strong-interaction sector in the Standard Model, and the perturbative QCD provides the powerful and systematic method in analyzing high-energy experimental data. However, QCD is a 'black box' in the infrared region still now owing to the strong-coupling nature, although there appear rich phenomena as color confinement, dynamical chiral-symmetry breaking and topological excitation in the nonperturbative QCD (NP-QCD). In particular, confinement is the most outstanding feature in NP-QCD, and to understand the confinement mechanism is a central issue in hadron physics.In 1974, Nambu [1] presented an interesting idea that quark confinement and string picture for hadrons can be interpreted as the squeezing of the color-electric flux by the dual Meissner effect, which is similar to formation of the Abrikosov vortex in the type-II superconductor. This dual superconductor picture for the NP-QCD vacuum is based on the duality in the Maxwell equation, and needs condensation of color-magnetic monopoles, which is the dual version of electriccharge (Cooper-pair) condensation in the superconductivity.In 1981, 't Hooft [2] pointed out that colormagnetic monopoles appear in QCD as topological excitation in the abelian gauge [2,3], which diagonalizes a gauge-dependent variable X(s).Here, SU(N c ) gauge degrees of freedom is partially fixed except for the maximal torus subgroup U(1) Nc−1 and the Weyl group. In the abelian gauge, QCD is reduced into a U(1) Nc−1gauge theory, and monopoles with unit magnetic charge appear at hedgehog-like configurations according to the nontrivial homotopy group,In 90's, the Monte Carlo simulation based on the lattice QCD becomes a powerful tool for the analysis of the confinement mechanism using the maximally abelian (MA) gauge [7][8][9][10][11][12][13][14][15], which is a special abelian gauge minimizing the off-diagonal components of the gluon field. Recent lattice studies with MA gauge have indicated monopole condensation in the NP-QCD vacuum [7-9] and the relevant role of abelian degrees of freedom, abelian dominance [9-12], for NP-QCD. In the lattice QCD in MA gauge, monopole dominance for NP-QCD is also observed as the essential role of QCD-monopoles for the linear quark potential [10], chiral symmetry breaking [11,12] and instantons [6,13,14].In this paper, we study Q...

Abelian dominance for confinement and random phase property of off-diagonal gluons in the maximally abelian gauge

“…In the abelian gauge, there also remains the global Weyl symmetry as a "relic" of the nonabelian theory [46,47]. Here, the Weyl symmetry corresponds to the subgroup of SU(N c ) relating to the permutation of the basis in the fundamental representation.…”

“…Here, let us consider the SU(N c ) gauge-invariance condition on the operator defined in the abelian gauge [46]. To begin with, we investigate the gauge-transformation property of the gauge function Ω ∈ G/H which brings the abelian gauge (See.…”

“…Thus, we find a useful criterion on the SU(N c ) gauge invariance of the operator defined in the abelian gauge [46]: If the operator O Ω defined in the abelian gauge is H-invariant, O Ω is also invariant under the whole gauge transformation of G.…”

“…In the MA gauge, Φ(s) is diagonalized as Φ MA (s) = Φ diag (s)τ 3 with Φ diag (s) ∈ R, and there remain the local U(1) 3 symmetry and the global Weyl symmetry [46]. As a remarkable fact, Φ(s) does not obey the adjoint transformation in the MA gauge, and the sign of Φ diag (s) is not changed by the Weyl transformation by W in Eq.…”

“…Abelian dominance and monopole dominance for the nonperturbative phenomena are numerically observed in the MA gauge in the lattice QCD simulations [35,39,[44][45][46][47][48][49]. As for confinement, monopole dominance, which means the dominant role of the magnetic current k µ than that of the electric current j µ , seems trivial if abelian dominance holds, because j µ does not provide the electric confinement in 1+3 dimension.…”