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

TL;DR

This paper studies how heavy Kaluza-Klein modes affect the effective action in a compactified Abelian Higgs model, revealing non-decoupling effects and the impact of different topologies on the theory's divergences and couplings.

Contribution

It demonstrates the non-decoupling of KK modes in certain topologies and analyzes the scale dependence of couplings and masses, providing insights into extra dimension constraints.

Findings

Heavy modes do not decouple in $ eals^{3,1} imes S^{1}$ topology due to $A_5$ zero mode.

Disappearance of $A_5$ zero mode in $S^{1}/ eals_{2}$ compactification removes divergences.

Couplings and masses decrease with increasing compactification scale.

Abstract

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.