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In this paper, we obtain a gauge invariant effective action for a bulkmassless $U(1)$ gauge vector field on a brane with codimension two by using ageneral Kaluza-Klein (KK) decomposition for the field. It suggests that thereexist two types of scalar KK modes to keep the gauge invariance of the actionfor the massive vector KK modes. Both the vector and scalar KK modes can bemassive. 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 extradimensions. 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, wederive two Schr\"{o}dinger-like equations for the vector KK modes, for whichthe effective potentials are just the functions of the warp factor.

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