Abstract
We present an exact treatment of the modulus stabilization condition with the general boundary conditions of the bulk scalar field in the Randall-Sundrum model. We find analytical expressions for the value of the modulus and the mass of the radion.
1. Introduction
The hierarchy problem is one of the most attractive open problems in the modern physics. Roughly speaking, the problem is the large discrepancy between the weak and the Planck scale. The problem was addressed by several theories, like supersymmetry, and higher dimensional theories; however, it has remained unsolved in the literature. Through the efforts in this direction, an impressive work is Randall-Sundrum (RS) model [1] which introduces a small extra dimension. The model has two branes which are called “Planck” and “TeV” branes; also it is assumed that a slice of spacetime exists between the branes. A five-dimensional solution to the Einstein field equations in RS model is given bywhere is the extra dimension coordinate, the coefficient is the compactification radius, and the parameter is related to the -D Planck mass, .
It would be interesting to check out the stability of the extra dimension, , in the RS model. Such an investigation was addressed by Goldberger and Wise (GW) [2]. The GW model contains a massive scalar field with usual kinetic term in the bulk and quartic interactions localized on the branes. The original work of [2] has several deficiencies, namely:(1)The bulk mass term breaks the conformal invariance of the theory.(2)In the limit of infinite quartic coupling considered in [2], it is not possible to unravel the complete structure of the critical points of the theory; hence they miss the source of instability as indicated by the existence of a closely spaced maximum.(3)The boundary conditions of the model are comprised of a pair of coupled cubic algebraic equations. In [2] these boundary conditions are not solved; instead they merely choose a specific configuration.(4)In addition they only consider the leading order term in their calculations. Hence their treatment of the subject matter is an approximate one.Therefore, many studies were done on this subject [3–12]. In [3–5], the authors have considered the stabilization of the modulus containing a scalar field, which interacts with the spacetime curvature , in the bulk. These theories have conformal invariance at certain value of the coupling constant of the curvature and the scalar field; hence they remedy objection 1, raised above.
An exact analysis of the mechanism has been discussed in [6] and the objections 2–4 have been addressed. However their treatment of the stabilized modulus is a numerical one. They also do not address the issue of the mass of radion at finite values of the quartic coupling constant. The issue of Goldberger-Wise mechanism with the general boundary conditions also has been discussed in [13]. Recently, the issue of stability of the Randall-Sundrum model has been discussed in the framework of AdS/CFT correspondence [14, 15].
The motivation for the present study is to discuss the relevance of going beyond the infinite quartic coupling limit, in physical terms, and to explore its phenomenological implications. The plan of this paper is as follows: in Section 2, we describe the model and calculate the effective potential, its extremization condition, and the value of the stabilized modulus. In Section 3, we study the modulus for the case where the quartic coupling is finite but very large. We obtain the mass of the radion in this limit as well. Finally in Section 4 we present our conclusions.
2. Review of the GW Mechanism
In this section, mainly based on [6], we present a brief review of the GW mechanism in the limit of finite quartic coupling. The action of the model is of the form:wherewhere is the five-dimensional cosmological constant, , are the visible and hidden brane tensions, and .
The -dependent vacuum expectation value is obtained from the equation of motion:where . Away from the boundaries () the solution iswhere .
If we insert this solution in (5) and integrate over we obtain the effective 4-dimensional potential for the modulus , namely, which is given byThe coefficients and are determined by imposing appropriate boundary conditions on the 3 branes. Putting (7) into (6) and matching delta functions, the conditions are obtained asFor arbitrary value of the boundary values of the scalar field at the two orbifold fixed points are and . Now and can be written, from (7), in terms of boundary values of the scalar field as follows:Putting above expressions for and into (9), we getUsing (10) and (12) into the extremization condition for the effective potential (), the modulus can be obtainedwhereFor , , and for , . So, and are minimum and maximum of the potential, respectively. Clearly, and are, respectively, corresponding to stability and instability of the modulus field or the radion field.
3. The Modulus and the Mass of Radion
In this section we consider some observable quantities such as the modulus and the mass of radion.
3.1. The Modulus
Now we study the stable and unstable values of the modulus for the original mechanism at finite coupling. The analytic expression for the modulus was obtained and analyzed in [6], but we want to investigate the modulus by taking the parameter as a variable. In the large limit, the values of the stable and unstable modulus when , areIn order to consider the value of the stabilized modulus for the large but finite value of the quartic coupling constant, expansion of boundary scalar field should be consideredThe values of the modulus at finite value of the quartic coupling are [6]whereThe values of the stable and unstable modulus are obtained by solving (17) which arewhereUsing above equation, we can analyze the modulus by taking the parameter as a variable. Figure 1 shows the variation of the stable modulus versus the parameter . In this figure the value of and . The solid curve corresponds to case of infinite coupling. The dashed curve corresponds to the case where and the dotted curve corresponds to the case where . As seen above, is another value of the modulus which, because of , corresponds to the maximum of the potential. Due to the maximality, leads to instability, that is, for any small perturbation the system will roll down to the minimum. Figure 2 shows the variation of the unstable modulus versus the parameter . The value of and . The solid curve corresponds to case of infinite coupling. The dashed curve corresponds to the case where and the dotted curve corresponds to the case where . From Figures 1 and 2, we observe that by increasing the value of the parameters and , we obtain more deviation from usual Randall-Sundrum case. As we know the value of may solve hierarchy problem. From this point of view, Figure 1 shows that the parameter in the finite coupling case can be smaller in comparison with the infinite coupling case.
It is appropriate to study the difference of the modulus at finite and infinite values of the quartic coupling constant; the result iswhere in the series expansion we have kept the linear terms of the parameters and .
3.2. Mass of the Radion
The phenomenology of radion has been addressed in [8, 11, 16]. The canonically normalized radion field is , where is another scale of order of Planck mass andThe radion mass is defined the second derivative of with respect to the canonically normalized radion field evaluated at its minimum. HenceIn order to calculate the mass of radion, we calculate the second derivative of the effective potential and we getIn the limit of , , , and , we findIn [16] the scale factor is ; moreover in their work . Now by using these values for the scale factor and the modulus , from (19) and (20), we obtainwhich is identical to the result of [16]. This in turn validates our results for the second derivative of the effective potential against a similar calculations presented in [6].
Similarly we can calculate the mass of radion in the case where the quartic coupling constants are finite; the result isSince and , the radion mass at finite coupling is when . This is in agreement with previous works in the context of radion phenomenology. As seen above, the radion mass has dependency on the parameter as . If one takes into account backreaction of the stabilizing field on the background geometry, the radion mass turns out to have dependence [17]. This discrepancy of dependence between these two approaches may come from the fact that during all calculations we have assumed that () and we have neglected higher order terms of . It would be of interest to take and study similarities and differences between the effective potential and the exact gravity-scalar approaches.
It is appropriate to study the logarithmic ratio of these masses which we denote by defined by
Figure 3 shows the variation of the versus the parameter . In this figure the value of and . The dashed curve corresponds to the case where and the dotted curve corresponds to the case where . As seen in Figure 3, the mass of the radion at finite value of could be much larger than the original value reported in [16] (the infinite coupling case). From phenomenological point of view, this is a reasonable result because it shows that the mass of the radion is governed by the strength of radion coupling. Moreover, since the radion mass increases as the finite quartic coupling becomes smaller, it could be considered in the context of high-mass radion or Higgs-radion mixed scenarios [18–24].
4. Conclusions
We have made a critical assessment of the GW mechanism for the stabilization of modulus. We have managed to extend the work initiated in [6] to the phenomenology of radion. The limit studied by Goldberger and Wise, because the boundary conditions become very simple in this case, corresponds to the limit of infinite quartic coupling of the scalar potential term on the boundary branes. We have succeeded in refining this aspect of the mechanism. We have found correction for the value of the modulus and the mass of the radion. Instead of brane potential with quartic coupling, it is also possible to consider brane potential of the quadratic form. We plan to report on these issues in the future. As final note, it is possible to stabilize the modulus by a massless scalar nonminimally coupled to gravity. In such work, it would be of interest to study the limit of finite coupling.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.