Gauge Invariant Effective Lagrangian for Kaluza-Klein Modes

TL;DR

This paper develops a gauge invariant effective Lagrangian for multiple Kaluza-Klein modes in SU(m) gauge theories, enabling a better understanding of their dynamics and scale dependence in higher-dimensional models.

Contribution

It introduces a gauge invariant Lagrangian for N Kaluza-Klein modes, matching continuum theories and incorporating renormalization group analysis with threshold effects.

Findings

Constructed a gauge invariant Lagrangian for KK modes

Matched lattice and continuum descriptions of compactified theories

Analyzed scale dependence and unitarity constraints

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 \Pi_{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.