Gauge invariance and quantization of Yang-Mills theories in extra dimensions

TL;DR

This paper reexamines the gauge structure and quantization of 5D Yang-Mills theories compactified on an orbifold, analyzing different scenarios for gauge parameters and deriving the effective quantum Lagrangian with implications for radiative corrections.

Contribution

It provides a detailed BRST-based analysis of gauge invariance and constraints in 4D effective theories from 5D Yang-Mills models, clarifying the role of Kaluza-Klein modes and gauge fixing.

Findings

4DET is gauge invariant when using curvatures for compactification.

Two types of gauge transformations govern the 4DET, from zero and excited KK modes.

The effective quantum Lagrangian describes interactions between light and heavy physics.

Abstract

The gauge structure of the four dimensional effective theory (4DET) arising from a pure Yang-Mills theory in five dimensions compactified on the orbifold $S^1/Z_2$ is reexamined on the basis of the BRST symmetry. The two scenarios that can arise are analyzed: if the gauge parameters propagate in the bulk, the excited Kaluza-Klein (KK) modes are gauge fields, but they are matter vector fields if these parameters are confined in the 3-brane. In the former case, it is shown that the 4DET is gauge invariant only if the compactification is carried out by using curvatures instead of gauge fields as fundamental objects. It is shown that the 4DET is governed by two types of gauge transformations, one determined by the KK zero modes of the gauge parameters and the other by the excited KK modes. The Dirac's method and the proper solution of the master equation are employed to show that the theory is subject to first class constraints. A gauge-fixing procedure to quantize the KK modes, that is covariant under the first type of gauge transformations, is introduced. The ghost sector induced by these gauge-fixing functions is derived on the basis of the BRST formalism. The effective quantum Lagrangian that links the interactions between light physics and heavy physics is presented. Concerning the radiative corrections of the excited KK modes on the light Green functions, the predictive character of this Lagrangian is stressed. In the case of the gauge parameters confined to the 3-brane, the known result in the literature is reproduced with some minor variants, although it is emphasized that the excited KK modes are not gauge fields, but matter fields that transform under the adjoint representation of $SU_4(N)$. The Dirac's method is employed to show that this theory is subject to both first and second class constraints.