Confinement and the Global $SU(3)$ Color Symmetry

TL;DR

This paper discusses how the global $SU(3)$ color symmetry leads to confinement in QCD, deriving the potential for color singlet systems and establishing minimal sizes and allowed multi-quark configurations.

Contribution

It provides a theoretical derivation of confinement and minimal hadron sizes based on the global $SU(3)$ symmetry and vacuum field effects, identifying allowed minimal color singlet systems.

Findings

Color charge self-energy is infinite, implying confinement.

Derived Cornell-type potential with Casimir scaling for singlet systems.

Minimal hadron size and energy due to uniform color field.

Abstract

The global $SU(3)$ color symmetry and its physical consequences are discussed. The N\"{o}ther current is actually governed by the conserved matter current of color charges if the color field generated by this charge is properly polarized. The color field strength of a charge can have a uniform part due to the nontrivial QCD vacuum field and the nonzero gluon condensate, which implies that the self-energy of a system with a net color charge is infinite and thereby cannot exist as a free state. This is precisely what the color confinement means. Accordingly, the Cornell type potential with the feature of the Casimir scaling is derived for a color singlet system composed of a static color charge and an anti-charge. The uniform color field also implies that a hadron has a minimal size and a minimal energy. Furthermore, the global $SU(3)$ color symmetry requires that the minimal irreducible color singlet systems can only be $q\bar{q}$, $qqq$, $gg$, $ggg$, $q\bar{q}g$, $qqqg$ and $\bar{q}\bar{q}\bar{q}g$, etc., as such a multi-quark systems can only exist as a molecular configurations if there are no other binding mechanisms.