Kaluza-Klein model with spontaneous symmetry breaking: Light-particle effective actionand its compactification scale dependence

We investigate decoupling of heavy Kaluza-Klein (KK) modes in an Abelian Higgs model with space-time topologies R3,1xS1 and R3,1xS1/Z2. After integrating out only the heavy KK modes we find the one-loop, light-particle (irreducible) effective action (LPEA) for the zero-mode fields. We find that in the R3,1xS1 topology the heavy modes do not decouple in this low-energy effective action, due to the zero mode of the 5th component of the 5D gauge field A5. Because A5 is a scalar under 4D Lorentz transformations, there is no gauge symmetry protecting it from getting mass and A54 interaction terms after loop corrections. In addition, after symmetry breaking, we find that the effective action has new divergences in the A5 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. We find that when considering low-energy physical processes, however, the divergences of the zero-mode loop diagrams will cancel the divergences in the effective action, rendering the radiatively corrected couplings finite. Although, this clears up the extra divergences in the A5 sector, the gauge coupling still has a different compactification scale dependence in the A5 then it does in the Aμ sector, leading to an explicit violation of decoupling. If instead of the LPEA one considers the Wilsonian effective action by integrating out zero modes of momenta |p|>M (M is the mass of the lowest KK excitation) in addition to the heavy modes, then decoupling is manifest. However, as is well known the price is the difficulty in maintaining 4D Lorentz and gauge invariance. In order to get a more sensible effective theory in the LPEA formalism, we investigate the S1/Z2 compactification. With this kind of compact topology, the A5 zero mode disappears. With no A5, there are no new divergences and the heavy modes decouple. We also discuss the dependence of the couplings and masses on the compactification scale, and derive a set of renormalization group-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.