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

Kaluza-Klein modes of $U(1)$ gauge vector field on brane with codimension-$d$