Decoupling of Heavy Kaluza-Klein Modes In Models With Five-Dimensional Scalar Fields

We investigate the decoupling of heavy Kaluza-Klein modes in $\phi^{4}$ theory and scalar QED with space-time topology $\mathbb{R}^{3,1} \times S^{1}$. We calculate the effective action due to integrating out heavy KK modes. We construct generalized RGE's for the couplings with respect to the compactification scale $M$. With the solutions to the RGE's we find the $M$-scale dependence of the effective theory due to higher dimensional quantum effects. We find that the heavy modes decouple in $\phi^{4}$ theory, but do not decouple in scalar QED. This is due to the zero mode of the 5-th component $A_{5}$ of the 5-d gauge field. 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 light of these unpleasant features, we explore $S^{1}/\mathbb{Z}_{2}$ compactifications, which eliminate $A_{5}$, allowing for the heavy modes to decouple at low energies. We also explore the possibility of decoupling by including higher dimensional operators. It is found that this is possible, but a high degree of fine tuning is required.