Towards low-dimensionalization of four-dimensional QCD

Inspired by the one-dimensional color-electric flux-tube in a hadron, we propose a possible way of low-dimensionalization of 4D QCD. As a strategy, we use gauge degrees of freedom and propose a new gauge fixing of ``dimensional reduction (DR) gauge". The DR gauge is defined so as to minimize $R_{\rm DR} \equiv \int d^4s~{\rm Tr}~[A^2_x(s)+A^2_y(s)]$, which preserves the 2D SU($N_{c}$) gauge symmetry. We investigate low-dimensionalization properties of 4D DR-gauged QCD in SU(3) lattice QCD at $\beta$ = 6.0. In the DR gauge, the amplitudes of two gluon components $A_{x}(s)$ and $A_{y}(s)$ are found to be strongly suppressed, and these components have a large effective mass of $M_{\perp} \simeq 1.7$ GeV. In the DR gauge, the static interquark potential is well reproduced only with the two components $A_{t}(s)$ and $A_{z}(s)$, while $A_{x}(s)$ and $A_{y}(s)$ seem to be inactive. We investigate the spatial correlation of two $t$-directed gluons and find that the correlation decreases as $e^{-mr}$ with $m \simeq$ 0.64 GeV, corresponding to the correlation length $\xi \equiv 1/m \simeq$ 0.31 fm. We reduce 4D QCD in the DR gauge to 2D QCD with the coupling $g_{2D} = g/\xi$, which approximately reproduces the string tension.