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

Abstract

The gauge structure of the four-dimensional effective theory arising from a pure $S{U}_{5}(N)$ Yang-Mills theory in five dimensions compactified on the orbifold ${S}^{1}/{Z}_{2}$ is reexamined on the basis of Becchi-Rouet-Stora-Tyutin symmetry. In this context, 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 four-dimensional theory is gauge invariant only if the compactification is carried out by using curvatures instead of gauge fields as fundamental objects. Then, it is shown that the four-dimensional theory is governed by two types of gauge transformations, one determined by the KK zero modes of the gauge parameters, ${\ensuremath{\alpha}}^{(0)a}$, and another by the excited KK modes, ${\ensuremath{\alpha}}^{(n)a}$. The Dirac method and the proper solution of the master equation in the context of the field-antifield formalism are employed to show that the theory is subject to first-class constraints. A gauge-fixing procedure to quantize the KK modes ${A}_{\ensuremath{\mu}}^{(n)a}$ that is covariant under the first type of gauge transformations, which embody the standard gauge transformations of $S{U}_{4}(N)$, is introduced through gauge-fixing functions transforming in the adjoint representation of this group. The ghost sector induced by these gauge-fixing functions is derived on the basis of the Becchi-Rouet-Stora-Tyutin formalism. The effective quantum Lagrangian that links the interactions between light physics (zero modes) and heavy physics (excited KK gauge modes) is presented. Concerning the radiative corrections of the excited KK modes on the light Green's functions, the predictive character of this Lagrangian at the one-loop level 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 exited KK modes are not gauge fields but matter fields that transform under the adjoint representation of $S{U}_{4}(N)$. The Dirac method is employed to show that this theory is subject to both first- and second-class constraints, which arise from the zero and excited KK modes, respectively.