  In this paper, we discuss the vacuum structure of the gauge-Higgs unification
theory, which is one of the attractive candidates of physics beyond the
standard model. This scenario has a remarkable feature, namely it has
infinitely degenerate vacua due to the characteristic periodic potential of the
Higgs filed, to be identified with the extra space component of the higher
dimensional gauge field. We address a question, whether to form the
superposition of such degenerate vacua, like the $\theta$-vacuum in QCD, is
necessary or not, in order to realize the true vacuum state of the theory. We
derive gauge field configuration which describes the transition between
neighboring vacua, like the instanton (or anti-instanton) solution in QCD, and
the corresponding Euclidean action in two models. In a simplified 2-dimensional
U(1) model, the derived configuration to describe the transition is shown to
have finite Euclidean action, and accordingly the "$\theta$-vacuum" and the
resultant "$\theta$-term" are formulated. In a realistic 5D U(1) model,
however, the gauge field configuration to describe the transition is shown to
have infinite Euclidean action and therefore the tunneling probability between
the degenerate vacua vanishes. Thus the superposition of the degenerate vacua
is not necessary.
