Gauge Invariant Effective Lagrangian for Kaluza-Klein Modes

TL;DR

This work addresses how to describe Kaluza-Klein (KK) modes of a bulk gauge theory in a manifestly gauge-invariant 3+1D framework. It develops a transverse-lattice or moose construction: an aliphatic theory with $SU(m)^{N+1}$ linked by $N$ bifundamental fields reproduces the KK spectrum and interactions of a truncated 4+1 continuum theory, with precise mappings $N+1 = M_s R$, $g_L = \sqrt{M_s/M}$, and $v = \sqrt{M_s M}$; the zero mode remains massless and the KK spacing matches $1/R$. The authors show that one-loop running in this setup exhibits power-law running via threshold contributions from KK modes, and that the aliphatic model closely approximates the continuum spectrum (deviations $\lesssim 10\%$) for large $N$. Fermions and scalars in the bulk are accommodated, yielding chiral zero modes and KK towers with calculable couplings to gauge fields. The framework preserves gauge invariance and offers a renormalizable, gauge-invariant route to KK phenomenology, with potential extensions to gravity and higher dimensions, and connections to deconstruction/more general moose constructions.

Abstract

We construct a manifestly gauge invariant Lagrangian in 3+1 dimensions for N Kaluza-Klein modes of an SU(m) gauge theory in the bulk. For example, if the bulk is 4+1, the effective theory is Π_{i=1}^{N+1} SU(m)_i with N chiral (\bar{m},m) fields connecting the groups sequentially. This can be viewed as a Wilson action for a transverse lattice in x^5, and is shown explicitly to match the continuum 4+1 compactifed Lagrangian truncated in momentum space. Scale dependence of the gauge couplings is described by the standard renormalization group technique with threshold matching, leading to effective power law running. We also discuss the unitarity constraints, and chiral fermions.

Figures (3)

• Figure 1: The geometric interpretation for the aliphatic model as a transverse lattice in the $x^5$ dimension with continuum theory in $4+1$. The number of branes in the foliation is $N+1 = (R/a) + 1$.
• Figure 2: The geometric interpretation for the plaquette model with two extra dimensions. The Eguchi-Kawai model corresponds to a single plaquette. At each circle, there is a 3-brane with one $SU(3)$ symmetry.
• Figure 3: $F_{lin}$ (solid line) and $F_{ali}$ (dashed line) as functions of $n_{max}$. $N=20$ is chosen for the plot.