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

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 $Ω(t,z)$ like 2D QCD on the $t$-$z$ plane. We investigate effective dimensional reduction in the DR gauge using SU(3) quenched lattice QCD at $β= 6.0$. 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 $g_{\rm 2D} = g m$.