Four-Dimensional Planck Scale Is Not Universal in Fifth Dimension in Randall-Sundrum Scenario

It has recently been proposed that the hierarchy problem can be solved by considering the warped fifth dimension compactified on S1/Z2. Many studies in the context have assumed a particular choice for an integration constant σ0 that appears when one solves the five-dimensional Einstein equation. Since σ0 is not determined by the boundary condition of the five-dimensional theory, σ0 may be regarded as a gauge degree of freedom in a sense. To this time, all indications are that the four-dimensional Planck mass depends on σ0. In this paper, we carefully investigate the properties of the geometry in the Randall-Sundrum model, and consider in which location y the four-dimensional Planck mass is measured. As a result, we find a σ0-independent relation between the four-dimensional Planck mass MPl and the five-dimensional fundamental mass scale M, and remarkably enough, we can take M to the TeV region when we consider models with the Standard Model confined on a distant brane. We also confirm that the physical masses on the distant brane do not depend on σ0 by considering a bulk scalar field as an illustrative example. The resulting mass scale of the Kaluza-Klein modes is on the order of M.