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Fluctuations Destroying Long-Range Order in SU(2) Yang-Mills Theory

We study lattice SU(2) Yang-Mills theory with dimension $d\ge 4$. The model can be expressed as a $(d-1)$-dimensional O(4) non-linear $σ$-model in a $d$-dimensional heat bath. As is well known, the non-linear $σ$-model alone shows a phase transition. If the quark confinement is a consequence of absence of a phase transition for the Yang-Mills theory, then the fluctuations of the heat bath must destroy the long-range order of the non-linear $σ$-model. In order to clarify whether this is true, we replace the fluctuations of the heat bath with Gaussian random variables, and obtain a Langevin equation which yields the effective action of the non-linear $σ$-model through analyzing the Fokker-Planck equation. It turns out that the fluctuations indeed destroy the long-range order of the non-linear $σ$-model within a mean field approximation estimating a critical point, whereas for the corresponding U(1) gauge theory, the phase transition to the massless phase remains against the fluctuations.