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

From the paper of Fu et al. (2019) [44], 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 are contributed by all the three extra dimensions. The masses of the scalar modes, however, are contributed only by two of the three extra dimensions. Then we generalize our results to branes with codimension d (d = 1, 2...). (C) 2020 The Author(s). Published by Elsevier B.V.