Gauge invariance, quantization and integration of heavy modes in a gauge Kaluza-Klein theory

TL;DR

This paper develops a gauge-invariant approach to quantize and integrate out heavy Kaluza-Klein modes in a gauge theory with extra dimensions, deriving an effective Lagrangian with dimension-six operators and analyzing their physical effects.

Contribution

It introduces a novel covariant expansion method for KK theories, preserving gauge invariance, and explicitly derives the effective Lagrangian including nonrenormalizable operators.

Findings

KK excited modes contribute dominantly at low energies.

The effective Lagrangian includes gauge-invariant dimension-six operators.

One-loop contributions of KK modes are shown to be renormalizable.

Abstract

We start from a pure Yang-Mills theory defined on a spacetime with one universal extra dimension that we orbifold-compactify. We obtain a Kaluza-Klein (KK) theory by expanding in KK towers covariant objects rather than fields, as such an approach yields a four-dimensional description possessing an interesting gauge structure in which two sorts of gauge transformations leave the theory invariant. One type are the standard gauge transformations (SGT), under which the KK zero modes behave as gauge fields. The other transformations receive the name of nonstandard gauge transformations (NSGT), and under them some of the the KK excited modes are gauge fields. We quantize the KK excited modes within the BRST approach, which includes the elimination of the gauge symmetries associated to the KK excitations through a gauge-fixing procedure that preserves gauge invariance with respect to the SGT. We present the most general Faddeev-Popov ghost sector. We integrate out the KK excited modes and derive an effective Lagrangian containing the explicit expressions of the coefficients multiplying all the independent nonrenormalizable operators of canonical dimension six that are allowed by the SU(N) gauge group and by Lorentz invariance. We first perform the calculation in the Feynman-`t Hooft (FtH) gauge and then in the general $R_\xi$ gauge. We find for the latter case a gauge-dependent result. The derivation of the effective Lagrangian explicitly proves that the contributions of KK excited modes to one-loop light Green's functions are renormalizable. Finally, we compare, at the four-dimensional level, the effects of the extra dimension with the contributions of a presumed fundamental theory describing nature at energies higher than those corresponding to the extra-dimensional physics. We find that the effects of the KK excited modes are the dominant ones.