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

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 $R_{\mathrm{DR}}\equiv \int d^{4}s\mathrm{Tr}\left[A_x^2\right(s)+A_y^2(s\left)\right]$, which has a residual gauge symmetry for the gauge function $\Omega (t,z)$ like 2D QCD on the $t\text{−}z$ plane. We investigate effective dimensional reduction in the DR gauge using SU(3) quenched lattice QCD at $\beta =6.0$. The amplitude of $A_x\left(s\right)$ and $A_y\left(s\right)$ are found to be strongly suppressed in the DR gauge. We consider "$tz$-projection" of $A_{x,y}\left(s\right)\to 0$ 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 $A_t\left(s\right)$ and $A_z\left(s\right)$ play a dominant role in quark confinement. In the DR gauge, we calculate a spatial correlation $⟨\mathrm{Tr}{A}_{\perp }\left(s\right)A_{\perp }(s+r{a}_{\perp })⟩(\perp =x,y)$ and estimate the spatial mass of $A_{\perp }\left(s\right)(\perp =x,y)$ as $M\simeq 1.7\mathrm{GeV}$. It is conjectured that this large mass makes $A_{\perp }\left(s\right)$ inactive and realizes the dominance of $A_t\left(s\right)$ and $A_z\left(s\right)$ 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 $g_{2D}=gm$.