Integration of Kaluza-Klein modes in Yang-Mills theories

TL;DR

This paper derives a low-energy effective Lagrangian for a five-dimensional Yang-Mills theory compactified on an orbifold, integrating out Kaluza-Klein modes and demonstrating the renormalizability of their one-loop contributions.

Contribution

It introduces a gauge-fixing procedure that allows integrating out KK modes, resulting in a gauge-invariant effective Lagrangian involving dimension-six operators.

Findings

Effective Lagrangian invariant under standard gauge transformations.

Explicit demonstration of one-loop renormalizability of KK contributions.

Dependence of the effective theory on gauge-fixing of KK excitations.

Abstract

A five dimensional pure Yang--Mills theory, with the fifth coordinate compactified on the orbifold $S^1/Z_2$ of radius R, leads to a four dimensional theory which is governed by two types of local gauge transformations, namely, the well known standard gauge transformations (SGT) dictated by the $SU_4(N)$ group under which the zero Fourier modes transform as gauge fields, and a set of nonstandard gauge transformations (NSGT) determining the gauge nature of the Kaluza--Klein (KK) excitations. By using a SGT-covariant gauge-fixing procedure for removing the degeneration associated with the NSGT, we integrate out the KK excitations and obtain a low-energy effective Lagrangian expansion involving all of the independent canonical-dimension-six operators that are invariant under the SGT of the $SU_4(N)$ group and that are constituted by light gauge fields (zero modes), exclusively. It is shown that this effective Lagrangian is invariant under the SGT, but it depends on the gauge-fixing of the gauge KK excitations. Our result shows explicitly that the one-loop contributions of the KK excitations to light (standard) Green's functions are renormalizable.