Narrow Results

In the previous paper,¹ we derived the Abelian projected effective gauge theory as a low energy effective theory of the SU(N) Yang–Mills theory by adopting the maximal Abelian gauge. At that time, we have demonstrated the multiplicative renormalizability of the propagators for the diagonal gluon and the dual Abelian antisymmetric tensor field. In this paper, we show the multiplicative renormalizability of the Green's functions also for the off-diagonal gluon. Moreover, we complement the previous results by calculating the anomalous dimension and the renormalization group functions which are undetermined in the previous paper.

We construct the most general gauge fixing and the associated Faddeev–Popov ghost term for the SU(2) Yang–Mills theory, which leaves the global U(1) gauge symmetry intact (i.e. the most general Maximal Abelian gauge). We show that the most general form involves eleven independent gauge parameters. Then we require various symmetries which help to reduce the number of independent parameters for obtaining the simpler form. In the simplest case, the off-diagonal part of the gauge fixing term obtained in this way is identical to the modified maximal Abelian gauge term with two gauge parameters which was proposed in the previous paper from the viewpoint of renormalizability. In this case, moreover, we calculate the beta function, anomalous dimensions of all fields and renormalization group functions of all gauge parameters in perturbation theory to one-loop order. We also discuss the implication of these results to obtain information on low-energy physics of QCD.

We propose a novel type of color magnetic condensation originating from magnetic monopoles so that it provides the mass of off-diagonal gluons in the Yang-Mills theory. This dynamical mass generation enables us to explain the infrared Abelian dominance and monopole dominance by way of a non-Abelian Stokes theorem, which supports the dual superconductivity picture of quark confinement. Moreover, we show that the instability of Savvidy vacuum disappears by sufficiently large color magnetic condensation.

We demonstrate the monopole condensation in SU(3) QCD. We first discuss the gauge independent and Weyl symmetric Abelian (Cho-Duan-Ge) decomposition of the SU(3) QCD, and present a new gauge invariant integral expression of the one-loop effective action which has no infrared divergence. Integrating it gauge invariantly imposing the color reflection invariance ("the C-projection") we show that the effective potential generates the stable monopole condensation which generates the mass gap.

We investigate Abelian and monopole contributions to spatial string tension in the deconfined phase of finite temperature SU(2) gauge theory without imposing any gauge fixing conditions. Lattice calculations of non-Abelian and Abelian spatial string tensions from the Wilson action at gauge coupling $\beta =2.74$ and lattice volume $24^3\times N_t$$(N_t=\{24,8,6,4,2\})$ show that these string tensions agree with each other within error bars at any adopted value of $N_t$, which implies Abelian dominance. From measurements of non-Abelian, Abelian and monopole forces that arise from the corresponding spatial string tension, furthermore, we find the tendency that the monopole contribution to the spatial string tension can be almost as large as the non-Abelian and Abelian ones. The temperature dependence of the calculated non-Abelian and Abelian spatial string tensions allows us to conclude that the concept of dimensional reduction holds both for non-Abelian and Abelian sectors at temperatures higher than twice the critical temperature.

Under the assumption of the Abelian dominance in QCD, we show that chiral condensate is locally present around a QCD monopole. The appearance of the chiral condensate around a GUT monopole was shown in the previous analysis of the Rubakov effect. We apply a similar analysis to the QCD monopole. It follows that the condensation of the monopole carrying the chiral condensate leads to the chiral symmetry breaking as well as quark confinement. To realize the result explicitly, we present a phenomenological linear sigma model coupled with the monopoles, in which the monopole condensation causes the chiral symmetry breaking as well as confinement. The monopoles are assumed to be described by a model of dual superconductor. Because the monopoles couple with mesons, we point out the presence of an observable color singlet monopole coupled with the mesons.

We show that the monopole condensation is responsible for the confinement. To demonstrate this we present a new gauge invariant integral expression of the one-loop QCD effective action which has no infra-red divergence, and show that the color reflection invariance ("the C-projection") assures the gauge invariance and the stability of the monopole condensation.

We demonstrate the monopole condensation in SU(3) QCD. We first discuss the gauge independent and Weyl symmetric Abelian (Cho-Duan-Ge) decomposition of the SU(3) QCD, and present a new gauge invariant integral expression of the one-loop effective action which has no infrared divergence. Integrating it gauge invariantly imposing the color reflection invariance (“the C-projection”) we show that the effective potential generates the stable monopole condensation which generates the mass gap.