Dimensional reduction gauge and effective dimensional reduction in 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 𝑅DR≡∫𝑑4⁢𝑠⁢Tr⁡[𝐴2𝑥⁡(𝑠)+𝐴2𝑦⁡(𝑠)], 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 𝐴𝑥⁡(𝑠) and 𝐴𝑦⁡(𝑠) are found to be strongly suppressed in the DR gauge. We consider “𝑡⁢𝑧-projection” of 𝐴𝑥, 𝑦⁡(𝑠)→0 for the gauge configuration generated in the DR gauge, in a similar sense to Abelian projection in the maximally Abelian gauge. By the 𝑡⁢𝑧-projection in the DR gauge, the interquark potential is not changed, and 𝐴𝑡⁡(𝑠) and 𝐴𝑧⁡(𝑠) play a dominant role in quark confinement. In the DR gauge, we calculate a spatial correlation ⟨Tr⁡𝐴⊥⁡(𝑠)⁢𝐴⊥⁡(𝑠+𝑟⁢𝑎⊥)⟩⁢(⊥=𝑥, 𝑦) and estimate the spatial mass of 𝐴⊥⁡(𝑠)⁢(⊥=𝑥, 𝑦) as 𝑀≃1.7 GeV. It is conjectured that this large mass makes 𝐴⊥⁡(𝑠) inactive and realizes the dominance of 𝐴𝑡⁡(𝑠) and 𝐴𝑧⁡(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⁡(−𝑚⁢𝑟) with 𝑚≃0.6 GeV. Using a crude approximation, the 4D YM theory is reduced into an ensemble of 2D YM systems with the coupling of 𝑔2⁢D=𝑔⁢𝑚.