Advances in High Energy Physics

Volume 2015, Article ID 590980, 10 pages

http://dx.doi.org/10.1155/2015/590980

## Vacuum Expectation Value Profiles of the Bulk Scalar Field in the Generalized Randall-Sundrum Model

^{1}Department of Physics, Faculty of Basic Sciences, University of Mazandaran, P. O. Box 47416-95447, Babolsar, Iran^{2}Young Researchers and Elite Club, Islamic Azad University, Bojnord Branch, Bojnord, Iran

Received 3 January 2015; Revised 15 March 2015; Accepted 6 April 2015

Academic Editor: Sergio Palomares-Ruiz

Copyright © 2015 A. Tofighi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

In the generalized Randall-Sundrum warped brane-world model the cosmological constant induced on the visible brane can be positive or negative. In this paper we investigate profiles of vacuum expectation value of the bulk scalar field under general Dirichlet and Neumann boundary conditions in the generalized warped brane-world model. We show that the VEV profiles generally depend on the value of the brane cosmological constant. We find that the VEV profiles of the bulk scalar field for a visible brane with negative cosmological constant and positive tension are quite distinct from those of Randall-Sundrum model. In addition we show that the VEV profiles for a visible brane with large positive cosmological constant are also different from those of the Randall-Sundrum model. We also verify that Goldberger and Wise mechanism can work under nonzero Dirichlet boundary conditions in the generalized Randall-Sundrum model.

#### 1. Introduction

Extra dimension is an important subject in the realm of theoretical physics that provides many creative ways to solve some problems in physics such as hierarchy problem. Unlike the model of Arkani-Hamed et al. [1], Randall and Sundrum (RS) proposed an alternative scenario [2] to solve the hierarchy problem that does not require large extra dimensions. They assumed an exponential function of the compactification radius called a warp factor in a -dimensional anti-de Sitter space-time compactified on a orbifold. Two 3-branes are located at the orbifold fixed point (visible brane) and (hidden brane). Due to the nonfactorizable geometry of metric all fundamental scalar masses are subject to an exponential suppression on the visible brane.

In the original RS model the cosmological constant induced on the visible brane is zero and the brane tension is negative. This model has been generalized such that the cosmological constant on the brane as well as brane tension can be positive or negative [3]. In the RS model the size of extra dimension, , is not determined by the dynamic of the model. For this scenario to be relevant, it is necessary to find a mechanism for generating a potential to stabilize the value of . This mechanism was proposed by Goldberger and Wise (GW) [4] so that the dynamic of a five-dimensional bulk scalar field in such model could stabilize the size of extra dimension. In the GW mechanism the potential for the radion that sets the size of the fifth dimension is generated by a bulk scalar with quartic interactions localized on the two -branes. The minimum of this potential yields a compactification scale that solves the hierarchy problem without fine-tuning of parameters. The backreaction of the bulk scalar field was neglected in the original GW mechanism but it was studied in [5] later. Some studies about this mechanism can be found in [6–8].

Recently, GW mechanism of radius stabilization in the brane-world model with nonzero brane cosmological constant has been considered [9]. It was shown that, for a generalized RS model, the modulus stabilization condition explicitly depends on the brane cosmological constant. Furthermore Haba et al. [10] have analyzed profiles of vacuum expectation value (VEV) of the bulk scalar field under the general boundary conditions (BCs) on the RS warped compactification. They have investigated GW mechanism in several setups with the general BCs of the bulk scalar field. Also they showed that triplet Higgs in the bulk left-right symmetric model with custodial symmetry can be identified by the Goldberger-Wise scalar.

The motivation for the present study is to study the VEV profile of the scalar field, including a brane cosmological constant and with different combinations of Dirichlet and Neumann boundary conditions at the two branes. We also want to know if this scalar can stabilize the size of the warped extra dimension. We note that the present accelerating phase of the universe is due to the presence of small positive cosmological constant with a tiny value of in Planck unit. Hence it is desirable to consider the effect of nonzero cosmological constant on the brane.

This paper is organized as follows. In Section 2, we briefly summarize the brane-world model with nonzero brane cosmological constant. In Section 3 we study the behavior of the bulk scalar field under four BCs on the general RS warped model in the case without brane localized potential and in the next section we analyze profiles of VEV of the bulk scalar field in the case with brane localized scalar potential. Also we investigate the GW mechanism under the BCs in the generalized brane-world scenario. Finally in Section 5 we conclude with the summary of our results.

#### 2. The Generalized Randall-Sundrum Model

One of the many different possibilities that can be explored to solve the hierarchy problem is Randall-Sundrum (RS) model [2]. The RS model proposes that space-time is described by a 5D anti-de Sitter (AdS) metric. Some good reviews on this subject can be found in, for example, [11–13]. In the Randall-Sundrum model the visible brane tension is negative and the cosmological constant on the visible brane is assumed to be zero. It has been shown in [3] that one can indeed generalize the model with nonzero cosmological constant on the brane and still can have Planck to TeV scale warping from the resulting warp factor. In this section we want to study the warped brane-world model with nonzero brane cosmological constant briefly. Other studies for this model have been reported in [14–16].

The action of a bulk scalar field, , on the warp brane-world model that was suggested by Randall and Sundrum iswhere , . The general form of the warped metric for a five-dimensional space-time is given bywhere stands for four-dimensional curved brane. The brane tension is defined bywhere is the Planck mass in five dimensions. We assume that the scalar field is a function of the extra dimension only that can be defined as [10]and, for simplicity, we take the bulk potential asA scalar mass on the visible brane gets warped through the warp factor where the warp factor index, , is set to to achieve the desired warping from Planck to TeV scale. The magnitude of the induced cosmological constant on the brane in the generalized RS model is nonvanishing and is given by in Planck unit. For the negative value of the cosmological constant, , on the visible brane which leads to AdS-Schwarzschild space-times the solution of the warp factor can be written as [9]where and . It was shown that real solution for the warp factor exists if and only if . This leads to an upper bound for the magnitude of the cosmological constant as . When , the visible brane tension is zero. For AdS brane, there are degenerate solutions of whose values will depend on and . The brane tension is negative for some of these solutions and is positive for others. Next, for AdS brane we want to investigate the solution of the equation of motion which is obtained from the action given by (1). As it has been discussed in [3], to resolve the gauge hierarchy problem without introducing any intermediate scale, the brane cosmological constant should be tuned to a very small value; therefore in this case the warp factor in (6) can be given by [9]We use background field method to study behaviors of the bulk scalar field of the generalized Randall-Sundrum model. In this method one separates the field into classical and quantum fluctuation parts. The configuration of the classical field obeys an equation of motion [10].

With the above warp factor equation of motion for the classical field can be written down:where stands for and . We solve the above equation and obtain the solution for the scalar field aswhereand and . In the above solution and are arbitrary constant which are evaluated by using the appropriate boundary conditions at the locations of the brane.

For the positive induced brane cosmological constant which corresponds to dS-Schwarzschild space-time the warp factor turns out to bewhere and . In this case there is no bound on the value of , and the positive cosmological constant can be of arbitrary magnitude. For dS brane, the brane tension is negative for the entire range of values of the positive cosmological constant. Also for a small the warp factor can be written down:By using the above warp factor the solution of the equation of motion for the dS brane isIt is obvious that by taking and we can obtain the above solution from AdS one. Notice that from now on we will use to represent induced brane cosmological constant. With this generalized RS warped model we now investigate the profile of the bulk scalar field under boundary conditions in the next section.

#### 3. VEV Profiles in the Absence of Brane Localized Potential

In this section, we study the VEV profiles of the bulk scalar field in a case without brane localized scalar potential in the generalized Randall-Sundrum model. By utilizing (2), the action can be rewritten asThe above action was defined on a line segment as . If we write the bulk scalar as (4) then the variation of the action is given bywhere . The VEV profile is obtained by the action principal ; that is,Notice that the above equation of motion has been solved in the previous section for dS and AdS brane in (9) and (13), respectively. The boundary conditions at and read as either Dirichletor Neumannwhere plus and minus signs are for and , respectively. As mentioned in [10] we can have four choices of combination of Dirichlet and Neumann boundary conditions shown by , , , and at and . Here by using the solution of the equation of motion that was given by (9) and (13) we want to verify the profile of the bulk scalar field under the four boundary conditions in the generalized Randall-Sundrum model with nonzero brane cosmological constant.

##### 3.1. Case

We discuss a case in which both boundary conditions on the and branes are the Dirichlet type boundary conditions. The most general form of the Dirichlet BC is andwhere is taken as and . For AdS brane these boundary conditions can be rewritten by (9) asThe above equations lead toTherefore, under boundary conditions, in the generalized RS model with negative brane cosmological constant the VEV profile isnotice that for (i.e., the RS case) which corresponds to and the above equations lead to the results that were proposed by Haba et al. [10] for BCs.

In Figure 1 the VEV profile for is shown for the AdS brane. In this figure we have assumed the brane tension of visible brane to be positive (see (A.4) for details). we find that the pattern of localization is different from that of the RS case. It is seen that the position of the IR brane shifts to the larger value of so the drastic change of the VEV profile occurs later than RS case. For this figure we have taken , , , GeV.