Reformulations of the Yang-Mills theory toward quark confinement and mass gap

TL;DR

This paper introduces new reformulations of the SU(N) Yang-Mills theory using novel field variables, aiming to better understand quark confinement and the mass gap through gauge-invariant, non-Abelian monopoles and dual superconductivity.

Contribution

It develops a unified framework for reformulating Yang-Mills theory with new variables, extending previous approaches, and provides gauge-invariant definitions of magnetic monopoles without scalar fields.

Findings

Restricted field dominates quark confinement

Non-Abelian magnetic monopoles are gauge-invariantly defined

Applications include stability analysis, large N limit, and lattice simulations

Abstract

We propose the reformulations of the $SU(N)$ Yang-Mills theory toward quark confinement and mass gap. In fact, we have given a new framework for reformulating the $SU(N)$ Yang-Mills theory using new field variables. This includes the preceding works given by Cho, Faddeev and Niemi, as a special case called the maximal option in our reformulations. The advantage of our reformulations is that the original non-Abelian gauge field variables can be changed into the new field variables such that one of them called the restricted field gives the dominant contribution to quark confinement in the gauge-independent way. Our reformulations can be combined with the $SU(N)$ extension of the Diakonov-Petrov version of the non-Abelian Stokes theorem for the Wilson loop operator to give a gauge-invariant definition for the magnetic monopole in the $SU(N)$ Yang-Mills theory without the scalar field. In the so-called minimal option, especially, the restricted field is non-Abelian and involves the non-Abelian magnetic monopole with the stability group $U(N-1)$. This suggests the non-Abelian dual superconductivity picture for quark confinement. This should be compared with the maximal option: the restricted field is Abelian and involves only the Abelian magnetic monopoles with the stability group $U(1)^{N-1}$, just like the Abelian projection. We give some applications of this reformulation, e.g., the stability for the homogeneous chromomagnetic condensation of the Savvidy type, the large N treatment for deriving the dimensional transmutation and understanding the mass gap, and also the numerical simulations on a lattice which are given by Dr. Shibata in a subsequent talk.