It is one of the most challenging issues in elementary particle and nuclear physics to understand confinement of quarks and gluons in quantum chromodynamics (QCD). Among several scenarios of color confinement proposed since the discovery of QCD, Coulomb gauge QCD has recently been received much attention along lattice QCD simulations and a variational approach. In this thesis, we discuss the confinement mechanism in Coulomb gauge QCD using quenched lattice QCD simulations.
The confinement mechanism in Coulomb gauge was firstly studied by Gribov in his seminal work on gauge fixing ambiguities, and further elaborated by Zwanziger. Coulomb gauge is a physical gauge in the sense that unphysical degrees of freedom, such as longitudinal component of gluons, are integrated out and the color Gauss' law is formally solved. As a result, an instantaneous interaction shows up in the Hamiltonian in Coulomb gauge QCD, which plays a central role in the confinement mechanism in Coulomb gauge. In the Gribov-Zwanziger scenario, the path integral is dominated by the configurations near the Gribov horizon where the lowest eigenvalue of the Faddeev-Popov operator vanishes. Accordingly, the color-Coulomb instantaneous interaction is strongly enhanced and provides a confining force between color charges. On the other hand, the proximity of the Gribov horizon in the infrared directions suppresses the infrared components of the transverse gluon propagator and the would-be physical transverse gluons are confined in hadrons.
In order to explore the confinement mechanism in Coulomb gauge QCD, we calculate various quantities by lattice QCD simulations: the color-Coulomb potential, the eigenvalues of the Faddeev-Popov operator, the ghost propagator, the color-Coulomb propagator, and the equal-time transverse gluon propagator.
We show that the color-Coulomb potential between a quark and antiquark pair in the color-singlet channel rises linearly with distance and it is stronger than the static potential extracted from the Wilson loop. The string tension of the color-Coulomb potential is estimated to be about three times larger than that of the Wilson string tension. Therefore, the color-Coulomb instantaneous interaction is responsible for the confining force between color charges. The color-Coulomb potential shows linearly rising potential even in the deconfinement phase, which indicates that there exists remnant of confinement above the critical temperature of the confinement/deconfinement phase transition.
We find that the eigenvalues of the Faddeev-Popov operator is accumulated in the low-lying level, and the lowest eigenvalues decrease with increasing the lattice volume faster than that in the abelian theory. Effects of Gribov copies is observed in the low-lying modes of the Faddeev-Popov operator and the better gauge fixing gives larger eigenvalues. This implies that the configurations in the fundamental modular region which is free from the Gribov copies lie slightly away from the Gribov horizon.
The ghost propagator and the color-Coulomb propagator is singular than the simple pole in the infrared region, which supports the Gribov-Zwanziger confinement scenario. We further show that Gribov copy effects is visible in the infrared region on both quantities. For the ghost propagator, Gribov noise induces a systematic errors of a few percent. By contrast, the effect of Gribov copies causes substantial systematic errors on the color-Coulomb potential. The discrepancy between the results for first copies and for best copies reaches 200% at the smallest momentum we investigate. Our result of the color-Coulomb propagator is compatible with the linearly rising behavior of the color-Coulomb potential.
We calculate the equal-time transverse gluon propagator on large lattices, up to 11 [fm]. We find that the equal-time gluon propagator shows scaling violation; namely, the data for different lattice spacings do not fall on top of one curve. This problem is cured by discarding data at large momenta which suffer from discretization errors. We further show that the calculations on anisotropic lattices moderate scaling violation. In the infrared region, the transverse gluon propagator is strongly suppressed and shows a turnover at about 500 [MeV]. Fitting the power law ansatz to the data at small momenta predicts the vanishing gluon propagator at zero momentum, indicating the confinement of gluons.