Mean field analysis of theSO(3) lattice gauge theory at finite temperature

We study the finite temperature properties of the $\mathrm{SO}(3)$ lattice gauge theory using mean field theory. The main result is the calculation of the effective action at finite temperature. The form of the effective action is used to explain the behavior of the adjoint Wilson line in numerical simulations. Numerical simulations of the $\mathrm{SO}(3)$ lattice gauge theory show that the adjoint Wilson line has a very small value at low temperatures; at high temperatures, metastable states are observed in which the adjoint Wilson line takes positive or negative values. The effective action is able to explain the origin of these metastable states. A comparison of the effective actions of the $\mathrm{SU}(2)$ and the $\mathrm{SO}(3)$ lattice gauge theories explains their different behavior at high temperatures. Mean field theory also predicts a finite temperature phase transition in the $\mathrm{SO}(3)$ lattice gauge theory.