Gauge symmetry breaking via Euclidean time component of gauge fields

We study gauge theories with/without an extra dimension at finite temperature, in which there are two kinds of order parameters of gauge symmetry breaking. One is the zero mode of the gauge field for the Euclidean time direction, and the other is that for the direction of the extra dimension. We evaluate the effective potential for the zero modes in the one-loop approximation and investigate the vacuum configuration in detail. Our analyses show that gauge symmetry can be broken only through the zero mode for the direction of the extra dimension, and no nontrivial vacuum configuration of the zero mode for the Euclidean time direction is found.