Monopoles of the Dirac type and color confinement in QCD: First results of $SU(3)$ numerical simulations without gauge fixing

If non-Abelian gauge fields in $SU(3)$ QCD have a line-singularity leading to noncommutativity with respect to successive partial-derivative operations, the non-Abelian Bianchi identity is violated. The violation as an operator is shown to be equivalent to violation of Abelian-like Bianchi identities. Then there appear eight Abelian-like conserved magnetic monopole currents of the Dirac type in $SU(3)$ QCD. Exact Abelian (but kinematical) symmetries appear in non-Abelian $SU(3)$ QCD. Here we try to show, using lattice Monte Carlo simulations of $SU(3)$ QCD, the Abelian dual Meissner effect due to the above Abelian-like monopoles are responsible for color confinement in $SU(3)$ QCD. If this picture is correct, the string tension of non-Abelian Wilson loops is reproduced fully by that of the Abelian Wilson loops. This is called perfect Abelian dominance. Furthermore, since the linear potential in Abelian Wilson loops is caused by the solenoidal monopole currents, the Abelian string tension is fully reproduced by that of Abelian monopole potentials. It is called perfect monopole dominance. In this report, the perfect Abelian dominance is shown to exist with the help of the multilevel method but without introducing additional smoothing techniques like partial gauge fixings, although lattice sizes studied are not large enough to study the infinite volume limit. Perfect monopole dominance on ${24}^3\times 4$ at $\beta =5.6$ is also shown without any additional gauge fixing but with a million thermalized configurations. The dual Meissner effect around a pair of static quark and antiquark is studied also on the same lattice. Abelian electric fields are squeezed due to solenoidal monopole currents and the penetration length for an Abelian electric field of a single color is the same as that of non-Abelian electric field. The coherence length is also measured directly through the correlation of the monopole density and the Polyakov loop pair. The Ginzburg-Landau parameter indicates that the vacuum type is the weak type I (dual) superconductor. Although the scaling and the infinite-volume limits are not studied yet, the results obtained above without any additional assumptions as well as more clear previous $SU(2)$ results seem to suggest strongly the above Abelian dual Meissner picture of color confinement mechanism.

Figure 1

Convergence history of the non-Abelian PLCF at $\beta =5.6$ on ${16}^3\times 16$ lattice.

Figure 2

Convergence history of the Abelian PLCF at $\beta =5.6$ on ${16}^3\times 16$ lattice.

Figure 3

The static quark potentials from non-Abelian and Abelian PLCF at $\beta =5.6$ on ${12}^3\times 12$ lattice.

Figure 4

The static quark potentials from non-Abelian and Abelian PLCF at $\beta =5.7$ on ${12}^3\times 12$ lattice.

Figure 5

The static quark potentials from non-Abelian and Abelian PLCF at $\beta =5.8$ on ${12}^3\times 12$ lattice.

Figure 6

The static quark potentials from non-Abelian and Abelian PLCF at $\beta =5.6$ on ${16}^3\times 16$ lattice.

Figure 7

The $SU(3)$ static quark potentials from PLCF at $\beta =5.60$ on ${24}^3\times 4$ lattice (up). For comparison, the $SU(2)$ static quark potentials from PLCF at $\beta =2.43$ on ${24}^3\times 8$ lattice (down) cited from Ref. [15].

Figure 8

The cylindrical coordinate.

Figure 9

The Abelian color electric field around static quarks for $d=5$ at $\beta =5.6$ on ${24}^3\times 4$ lattices.

Figure 10

The profile of monopole current $k_{\varphi },k_z,k_r$ with $d=3$ at $\beta =5.6$ on ${24}^3\times 4$ lattices.

Figure 11

The dual Ampère’s law with $d=3$ at $\beta =5.6$ on ${24}^3\times 4$ lattices.

Figure 12

The squared monopole density with $d=5$ at $\beta =5.6$ on ${24}^3\times 4$ lattices.