  From the paper [JHEP 01 (2019) 021], it is known that the effective action of
a massless $U(1)$ gauge vector field on a codimension-2 brane is gauge
invariant due to the coupling between the vector Kaluza-Klein (KK) modes with
two types of scalar KK modes. It is interesting to generalize this result to a
brane world model with an arbitrary number of extra dimensions. In this work,
we first investigate the case with three extra dimensions. After KK
decomposition, there are three types of scalar KK modes. In addition to the
mutual coupling between these scalar modes, there are also coupling between the
scalar and the vector KK modes. The coupling constants are not all independent.
The relationships between the coupling constants enable us to obtain a gauge
invariant effective action, from which we can see that the masses of the vector
KK modes depend on all the three extra dimensions. The masses of the scalar
modes, however, depend only on two of the three extra dimensions. Then we
generalize the results into branes with codimension $d$ ($d=1, 2...$), and find
that $d$ will directly affect the masses of the KK modes. But there is always a
gauge invariant effective action for the massive vector KK modes.
