The Compactification of QCD$_4$ to QCD$_2$ in a Flux Tube

TL;DR

This paper demonstrates how QCD in four dimensions can be approximately reduced to a two-dimensional form within a flux tube, revealing relationships between coupling constants and mass generation mechanisms.

Contribution

It provides a detailed derivation of the compactification process from QCD$_4$ to QCD$_2$ and explores how confinement influences particle masses and interactions in the reduced model.

Findings

QCD$_4$ can be approximated as QCD$_2$ in a flux tube environment.

The relation between 4D and 2D coupling constants is established.

Mass contributions from confinement are derived for quarks and gluons.

Abstract

We show from the action integral that in the special environment of a flux tube, QCD$_4$ in (3+1) dimensional space-time can be approximately compactified into QCD$_2$ in (1+1) dimensional space-time. In such a process, we find out how the coupling constant $g_{2D}$ in QCD$_2$ is related to the coupling constant $g_{4D}$ in QCD$_4$. We show how the quark and the gluon in QCD$_2$ acquire contributions to their masses arising from their confinement within the tube, and how all these quantities depend on the excitation of the partons in the transverse degrees of freedom. The compactification facilitates the investigation of some dynamical problems in QCD$_4$ in the simpler dynamics of QCD$_2$ where the variation of the gluon fields leads to a bound state.