$U(1)$ gauge vector field on a codimension-2 brane

TL;DR

This paper derives a gauge-invariant effective action for a massless U(1) gauge field in a six-dimensional brane-world, revealing the structure of KK modes and their masses influenced by extra dimensions.

Contribution

It introduces a general KK decomposition for gauge fields on a codimension-2 brane, showing the existence of two types of scalar KK modes maintaining gauge invariance.

Findings

Existence of two scalar KK modes related to vector modes.

Masses of KK modes depend on extra-dimensional parameters.

Derived Schrödinger-like equations for vector KK modes.

Abstract

In this paper, we obtain a gauge invariant effective action for a bulk massless $U(1)$ gauge vector field on a brane with codimension two by using a general Kaluza-Klein (KK) decomposition for the field. It suggests that there exist two types of scalar KK modes to keep the gauge invariance of the action for the massive vector KK modes. Both the vector and scalar KK modes can be massive. The masses of the vector KK modes $m^{(n)}$ contain two parts, $m_{1}^{(n)}$ and $m_{2}^{(n)}$, due to the existence of the two extra dimensions. The masses of the two types of scalar KK modes $m_{\phi}^{(n)}$ and $m_{\varphi}^{(n)}$ are related to the vector ones, i.e., $m_{\phi}^{(n)}=m_{1}^{(n)}$ and $m_{\varphi}^{(n)}=m_{2}^{(n)}$. Moreover, we derive two Schr\"{o}dinger-like equations for the vector KK modes, for which the effective potentials are just the functions of the warp factor.