<jats:title>Abstract</jats:title> <jats:p>We discuss the vacuum structure of the gauge–Higgs unification theory, which is one of the attractive candidates for physics beyond the standard model. This scenario has a remarkable feature: it has infinitely degenerate vacua due to the characteristic periodic potential of the Higgs field, to be identified with the extra space component of the higher-dimensional gauge field. We address the question of whether forming the superposition of such degenerate vacua, like the θ-vacuum in quantum chromodynamics (QCD), is necessary or not, in order to realize the true vacuum state of the theory. We derive a 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 two-dimensional U(1) model, the derived configuration to describe the transition is shown to have finite Euclidean action, and accordingly the “θ-vacuum” and the resultant “θ-term” are formulated. In a realistic five-dimensional 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, superposition of the degenerate vacua is not necessary.</jats:p>