Stueckelberg-like Kaluza–Klein Modes in 6D Brane World

We investigate the Kaluza–Klein modes of a bulk U(1) gauge field coupled to a dilaton in six-dimensional brane-world scenarios. We demonstrate that four-dimensional gauge invariance is preserved via a Stueckelberg-like mechanism driven by scalar KK modes arising from the extra-dimensional components. By introducing a bulk gauge-fixing constraint to determine the scalar dynamics and mapping the equations of motion into Schrödinger-like forms, we numerically analyze the mass spectra for two distinct brane solutions. Crucially, we clarify that the physical scalar degree of freedom arises from a gauge-invariant coupling between two underlying scalar modes. Our numerical results reveal that this mixing effect lifts the spectral degeneracy, establishing a distinct mass hierarchy where the massive vector modes are consistently heavier than the physical scalar modes.