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Massive $U(1)$ gauge field and their accompanying scalars in brane world

In brane-world scenarios, the effective action of a massless bulk \(U(1)\) gauge field preserves gauge invariance via couplings between massive vector Kaluza-Klein (KK) modes and scalar KK modes. In this work, we extend this framework by introducing a term \((\nabla^M X_M)^2\) into the massless bulk \(U(1)\) gauge action. This modification explicitly breaks the full gauge redundancy while preserving a residual gauge symmetry both in the bulk and on the brane. In this setup, the scalar KK modes can acquire masses from the background geometry. Notably, we find that on the 5D brane, these scalar KK modes are lighter than the vector KK modes. In contrast, on the 6D brane, two types of scalar modes emerge; the mixed interactions between them give rise to oscillations among these scalar modes.