Modulus Stabilization with Bulk Fields

We propose a mechanism for stabilizing the size of the extra dimension in the Randall-Sundrum scenario. The potential for the modulus field that sets the size of the fifth dimension is generated by a bulk scalar with quartic interactions localized on the two 3-branes. The minimum of this potential yields a compactification scale that solves the hierarchy problem without fine tuning of parameters. ∗[email protected] †[email protected] 1 The Standard Model for strong, weak, and electromagnetic interactions based on the gauge group SU(3)× SU(2)× U(1) has been extremely succesful in accounting for experimental observations. However, it has several unattractive features that suggest new physics beyond that incorporated in this model. One of these is the gauge hierarchy problem, which refers to the vast disparity between the weak scale and the Planck scale. In the context of the minimal Standard Model, this hierarchy of scales is unnatural since it requires a fine tuning order by order in perturbation theory. A number of extensions have been proposed to solve the hierarchy problem, notably technicolor [1] (or dynamical symmetry breaking) and low energy supersymmetry [2]. Recently, it has been suggested that large compactified extra dimensions may provide an alternative solution to the hierarchy problem [3]. In these models, the observed Planck mass MP l is related to M, the fundamental mass scale of the theory, by M 2 P l = M Vn, where Vn is the volume of the additional compactified dimensions. If Vn is large enough, M can be of the order of the weak scale. Unfortunately, unless there are several large extra dimensions, a new hierarchy is introduced between the compactification scale, μc = V −1/n n , and M. Randall and Sundrum [4] have proposed a higher dimensional scenario to solve the hierachy problem that does not require very large extra dimensions. This model consists of a spacetime with a single S/Z2 orbifold extra dimension. Three-branes with opposite tensions reside at the orbifold fixed points and together with a finely tuned cosmological constant serve as sources for five-dimensional gravity. The resulting spacetime metric contains a redshift factor which depends exponentially on the radius rc of the compactified dimension: ds = eημνdx dx − r cdφ, (1) where k is a parameter which is assumed to be of order M , x are Lorentz coordinates on the four-dimensional surfaces of constant φ, and −π ≤ φ ≤ π with (x, φ) and (x,−φ) identified. The two 3-branes are located at φ = 0 and φ = π. A similar scenario to the one described in ref. [4] is that of Horava and Witten [5], which arises within the context of M-theory. Supergravity solutions similar to Eq. (1) are presented in ref. [6]. In ref. [7], it is shown how this model may be obtained from string theory compactifications. The non-factorizable geometry of Eq. (1) has several important consequences. For instance, the four-dimensional Planck mass is given in terms of the fundamental scale M by