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We investigate decoupling of heavy Kaluza-Klein modes in an Abelian Higgsmodel with space-time topologies $\mathbb{R}^{3,1} \times S^{1}$ and$\mathbb{R}^{3,1} \times S^{1}/\mathbb{Z}_{2}$. After integrating out heavy KKmodes we find the effective action for the zero mode fields. We find that inthe $\mathbb{R}^{3,1} \times S^{1}$ topology the heavy modes do not decouple inthe effective action, due to the zero mode of the 5-th component of the 5-dgauge field $A_{5}$. Because $A_{5}$ is a scalar under 4-d Lorentztransformations, there is no gauge symmetry protecting it from getting mass and$A_{5}^{4}$ interaction terms after loop corrections. In addition, aftersymmetry breaking, we find new divergences in the $A_{5}$ mass that did notappear in the symmetric phase. The new divergences are traced back to thegauge-goldstone mixing that occurs after symmetry breaking. The relevance ofthese new divergences to Symanzik's theorem is discussed. In order to get amore sensible theory we investigate the $S^{1}/\mathbb{Z}_{2}$compactification. With this kind of compact topology, the $A_{5}$ zero modedisappears. With no $A_{5}$, there are no new divergences and the heavy modesdecouple. We also discuss the dependence of the couplings and masses on thecompactification scale. We derive a set of RG-like equations for the running ofthe effective couplings with respect to the compactification scale. It is foundthat magnitudes of both couplings decrease as the scale $M$ increases. Theeffective masses are also shown to decrease with increasing compactificationscale. All of this opens up the possibility of placing constraints on the sizeof extra dimensions.

A Kaluza-Klein Model with Spontaneous Symmetry Breaking: Light-Particle Effective Action and its Compactification Scale Dependence