  We investigate decoupling of heavy Kaluza-Klein modes in an Abelian Higgs
model 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 KK
modes we find the effective action for the zero mode fields. We find that in
the $\mathbb{R}^{3,1} \times S^{1}$ topology the heavy modes do not decouple in
the effective action, due to the zero mode of the 5-th component of the 5-d
gauge field $A_{5}$. Because $A_{5}$ is a scalar under 4-d Lorentz
transformations, there is no gauge symmetry protecting it from getting mass and
$A_{5}^{4}$ interaction terms after loop corrections. In addition, after
symmetry breaking, we find new divergences in the $A_{5}$ mass that did not
appear in the symmetric phase. The new divergences are traced back to the
gauge-goldstone mixing that occurs after symmetry breaking. The relevance of
these new divergences to Symanzik's theorem is discussed. In order to get a
more sensible theory we investigate the $S^{1}/\mathbb{Z}_{2}$
compactification. With this kind of compact topology, the $A_{5}$ zero mode
disappears. With no $A_{5}$, there are no new divergences and the heavy modes
decouple. We also discuss the dependence of the couplings and masses on the
compactification scale. We derive a set of RG-like equations for the running of
the effective couplings with respect to the compactification scale. It is found
that magnitudes of both couplings decrease as the scale $M$ increases. The
effective masses are also shown to decrease with increasing compactification
scale. All of this opens up the possibility of placing constraints on the size
of extra dimensions.
