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Residual gauge symmetry and color confinement in the Yang-Mills theory

Naoki Fukushima, Kei-Ichi Kondo

Abstract

We examine the restoration of the residual gauge symmetry in the Yang-Mills theory to be regarded as a confinement criterion. For this purpose we restrict the four-dimensional $SU(2)$ Yang-Mills instantons to those with spatial spherical symmetry $SO(3)$, which automatically causes the dimensional reduction of the four-dimensional $SU(2)$ Yang-Mills theory to the two-dimensional $U(1)$ gauge-scalar theory as implemented explicitly by the Witten transformation. In this setting, we show that the restoration of the residual gauge symmetry occurs due to condensations of instantons and anti-instantons, although the residual gauge symmetry was spontaneously broken in the perturbative vacuum. This result demonstrates that the true confinement phase is a disordered phase in which all internal electric symmetries of the gauge field remain unbroken.

Residual gauge symmetry and color confinement in the Yang-Mills theory

Abstract

We examine the restoration of the residual gauge symmetry in the Yang-Mills theory to be regarded as a confinement criterion. For this purpose we restrict the four-dimensional Yang-Mills instantons to those with spatial spherical symmetry , which automatically causes the dimensional reduction of the four-dimensional Yang-Mills theory to the two-dimensional gauge-scalar theory as implemented explicitly by the Witten transformation. In this setting, we show that the restoration of the residual gauge symmetry occurs due to condensations of instantons and anti-instantons, although the residual gauge symmetry was spontaneously broken in the perturbative vacuum. This result demonstrates that the true confinement phase is a disordered phase in which all internal electric symmetries of the gauge field remain unbroken.

Paper Structure

This paper contains 15 sections, 59 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Gauge fixing and the residual gauge symmetry
  3. $SU (2)$ Yang-Mills theory and gauge fixing
  4. Symmetry in the quantum gauge theory
  5. Residual gauge symmetry in the preceding work
  6. Symmetric instanton and dimensional reduction
  7. Witten transformation (Ansatz)
  8. Dimensional reduction through the Witten transformation
  9. Dimensional reduction of the residual gauge symmetry
  10. gauge fixing and the residual gauge symmetry from new perspective
  11. Restoration of the residual gauge symmetry in the dimensional reduction
  12. Spontaneously breaking of the global $U (1)$ symmetry in the perturbative vacuum
  13. Restoration of the $U (1)$ gauge symmetry and confinement
  14. Conclusion and discussion
  15. Implications of the $D=2$ symmetry restoration for the original $D = 4$ $SU (2)$ Yang-Mills theory

Figures (3)

  • Figure 1: The numerical solution $\theta$ of the non-linear differential equation \ref{['eq:gfc-perturbative']} converging to $\pi$ for the initial conditions at $r=0$: $\theta (0) = 0$ and $\dot{\theta} (0) = 0 \ (\text{red}), \ 0.05 \ (\text{black}), \ 0.1 \ (\text{blue})$.
  • Figure 2: The $(t , r)$ cross section of the cylinder in $\mathbb{R}^4$ and the location $(a_r , a_t)$ of an instanton.
  • Figure 3: The decomposition of the integration region with a given (fixed) $x$ into two parts in \ref{['eq:Vv']}. ( i) The center $a_s$ of a 1-instanton is inside the region $v$ centered at $x$; $|a_s - x| < \lambda$: The instanton centered at $a_s$ includes the point $x$ in the inside. ( ii) The center $a_s$ of a 1-instanton is outside the region $v$ centered at $x$; $|a_s - x| > \lambda$: The instanton centered at $a_s$ does not include the point $x$ in the inside.