The Shape of Gravity

In a nontrivial background geometry with extra dimensions, gravitational effects will depend on the shape of the Kaluza-Klein excitations of the graviton. We investigate a consistent scenario of this type with two positive tension three-branes separated in a five-dimensional Anti-de Sitter geometry. The graviton is localized on the “Planck” brane, while a gapless continuum of additional gravity eigenmodes probe the infinitely large fifth dimension. Despite the background five-dimensional geometry, an observer confined to either brane sees gravity as essentially four-dimensional up to a position-dependent strong coupling scale, no matter where the brane is located. We apply this scenario to generate the TeV scale as a hierarchically suppressed mass scale. Arbitrarily light gravitational modes appear in this scenario, but with suppressed couplings. Real emission of these modes is observable at future colliders; the effects are similar to those produced by six large toroidal dimensions. Extra dimensions provide an alternative route to addressing the hierarchy problem. This is because the Planck scale, describing the strength of the graviton coupling at low energies, is a derived scale. In a simple factorizable geometry, the Planck scale of a four-dimensional world is related to that of a higher dimensional world simply by a volume factor. The large Planck scale indicates weak graviton coupling which is in turn a consequence of the large volume over which the graviton can propagate [1]. In this scenario, a large hierarchy only arises in the presence of a large volume for the compactified dimensions, which is very difficult to justify. A more compelling alternative has been suggested in Ref. [2]. The idea of this paper was that the weak graviton coupling arises because of an interesting shape of the graviton wave function in the extra dimensions. The graviton is localized away from the 3+1-dimensional world on which the Standard Model resides. The large value of the Planck scale arises because of the small amplitude for the graviton to coincide with our “brane”. In Ref. [3], it was shown that the geometry of a single brane with cosmological energy densities tuned to guarantee Poincare invariance takes the form: ds = eημνdx dx + dy , (1) where μ, ν parameterize the four-dimensional coordinates of our world, and y is the coordinate of a fifth dimension. The remarkable aspect of the above geometry is that it gives rise to a localized graviton field. Mechanisms for confining matter and gauge fields to a smaller dimensional subspace were already known. The new feature here is that the background geometry gives rise to a single gravitational bound state. This mode plays the role of the graviton of a four-dimensional world, and is responsible for reproducing four-dimensional gravity. In Ref. [3], the Kaluza-Klein (KK) spectrum reflecting the large extra dimension was derived and it was argued that the additional continuum gapless spectrum of states gives rise to very suppressed corrections to conventional four-dimensional gravity, suppressed by (energy/MP l) . However, from the perspective of generating the mass hierarchy between the Planck and weak scales, the important aspect of this geometry is the correspondence between location in the fifth dimension and the overall mass scale. This can be understood by the fact that the warp factor is a conformal factor so far as a four-dimensional world located at a fixed y location is concerned. Mass factors are rescaled by this factor, so that a natural scale for mass parameters might be MP l = 10 19 GeV on a brane at the origin, but is MP lexp−k|y| for physics confined to a location y. This exponential could be the source of the hierarchy between the electroweak scale of order TeV and the Planck scale which is approximately 10 times bigger. Notice that the generation of this hierarchy only requires an exponential of order 30. In Ref. [2], this observation was exploited by introducing an orbifold geometry, and located a positive energy brane at one point and a negative energy brane at the second orbifold point. If the standard model is located on this second, negative energy brane, the amplitude of the graviton is exponentially suppressed and a hierarchy is generated. The potential disadvantage of this setup is the necessity for the negative energy object and the orbifold geometry. Although not ruled out, it is desirable to have an alternative setup