Dimensional reduction gauge and effective dimensional reduction in the four-dimensional Yang-Mills theory

TL;DR

This paper introduces a new gauge fixing in 4D Yang-Mills theory that effectively reduces the theory to lower dimensions, revealing dominant roles of certain gauge fields in quark confinement and providing insights into the mass spectrum.

Contribution

The paper proposes a novel 'dimensional reduction (DR) gauge' in 4D Yang-Mills theory and demonstrates its effectiveness in simplifying the theory to 2D systems, highlighting the roles of specific gauge fields.

Findings

Suppression of transverse gauge fields in the DR gauge.

Dominance of $A_t$ and $A_z$ in quark confinement.

Estimated spatial mass of transverse gauge fields ~$1.7$ GeV.

Abstract

Motivated by one-dimensional color-electric flux-tube formation in four-dimensional (4D) QCD, we investigate a possibility of effective dimensional reduction in the 4D Yang-Mills (YM) theory. We propose a new gauge fixing of "dimensional reduction (DR) gauge" defined so as to minimize $R_{\mathrm{DR}}~\equiv~\int d^{4}s ~ \mathrm{Tr} \left[ A_{x}^{2}(s) + A_{y}^{2}(s) \right]$, which has a residual gauge symmetry for the gauge function $\Omega (t,z)$ like 2D QCD on the $t$-$z$ plane. We investigate effective dimensional reduction in the DR gauge using SU(3) quenched lattice QCD at $\beta = 6.0$. The amplitude of $A_{x}(s)$ and $A_{y}(s)$ are found to be strongly suppressed in the DR gauge. We consider "$tz$-projection" of $A_{x,y}(s) \to 0$ for the gauge configuration generated in the DR gauge, in a similar sense to Abelian projection in the maximally Abelian gauge. By the $tz$-projection in the DR gauge, the interquark potential is not changed, and $A_{t}(s)$ and $A_{z}(s)$ play a dominant role in quark confinement. In the DR gauge, we calculate a spatial correlation $\langle \mathrm{Tr} A_{\perp}(s) A_{\perp}(s+ra_{\perp}) \rangle ~ (\perp = x,y)$ and estimate the spatial mass of $A_{\perp}(s) ~ (\perp = x,y)$ as $M \simeq 1.7 ~ \mathrm{GeV}$. It is conjectured that this large mass makes $A_{\perp}(s)$ inactive and realizes the dominance of $A_{t}(s)$ and $A_{z}(s)$ in infrared region in the DR gauge. We also calculate the spatial correlation of two temporal link-variables and find that the correlation decreases as $\exp (-mr)$ with $m \simeq 0.6 ~ \mathrm{GeV}$. Using a crude approximation, the 4D YM theory is reduced into an ensemble of 2D YM systems with the coupling of $g_{\rm 2D} = g m$.