D-dimensional Randall-Sundrum models from Brans-Dicke theory and Kaluza-Klein modes

We investigate the spectroscopy of scalar and vector Kaluza-Klein modes that arise in a deformed Randall-Sundrum model that is constructed from Brans-Dicke theory. The non-minimal coupling in the Brans Dicke theory translates into a deformation of the Randall-Sundrum geometry that depends on the Brans-Dicke parameter $\omega$. We find that the $\omega$ parameter has a non-trivial effect in the spectroscopy of scalar and vector Kaluza-Klein modes. Our results suggest the interpretation of $\omega$ as a fine-tuning parameter.

. This model considers a configuration of two 4D branes in a 5D space-time with negative cosmological constant. The hierarchy problem between the Planck and electro-weak scale is solved by the warp factor present in the 5d metric. An important problem in the Randall-Sundrum scenario is the fixing of the extra dimension size L. The first attempt to fix L was to consider a five dimensional scalar field with brane potentials [5]. Including the backreaction of this field on the metric led to a five dimensional scalar-tensor model [6] that differs from the original Randall-Sundrum solution. Recently, a five dimensional Brans-Dicke model with branes was proposed in [7]. Working in the Jordan-Fierz frame, the 5D Brans-Dicke action can lead to metric solutions very similar to the original Randall-Sundrum metric. The model of [7] includes backreaction and the solution is stable because the size of the extra dimension is fixed by the scalar field.
In this paper we construct D-dimensional Randall-Sundrum models from Brans-Dicke theory. We consider a BPS-like mechanism that translates the second-order differential equations coming from the Brans-Dicke action into first-order ones. This way we find a special class of scalar potentials that simplifies the background solutions. A particular choice of the scalar potential leads to a Randall-Sundrum solution for the metric which can be stabilized following a procedure similar to [7]. We analyze the possible implications of the D dimensional Brans-Dicke parameter by performing a Kaluza-Klein decomposition of a massless scalar fluctuation living in the bulk. We find an interesting dependence of the D −1 dimensional scalar masses on the D dimensional Brans-Dicke parameter. We also discuss the effect of the Brans-Dicke parameter on the Kaluza-Klein modes arising on a recent Higgless model for electroweak symmetry breaking [8]. Our results suggest the possibility of considering the Brans-Dicke parameter as a fine-tuning for the W and Z resonances.
We begin in Sec. II with a review of the Randall-Sundrum metric. In Sec. III we show how this metric arises from the Brans-Dicke theory via a BPS-like mechanism. In Sec. IV we analyze the Kaluza-Klein modes coming from the decomposition of a massless scalar fluctuation while in Sec. V we discuss the gauge field Kaluza-Klein modes of a Higgless electroweak model. We end with conclusions in Sec. VI.

II. THE RANDALL-SUNDRUM METRIC IN D-DIMENSIONS
The Anti-de-Sitter space-time is a maximally symmetric solution of the Einstein equations with negative cosmological constant Λ. This space -time can be interpreted as a hyperboloid of radius ℓ related to the cosmological constant by −Λ ℓ 2 = (D − 1)(D − 2). The Poincaré chart cuts the hyperboloid in two regions (see [9] for details). The metric of each region can be written as where k = 1/ℓ, dx 2 = D−2 i=1 dx 2 i and z > 0 (or z < 0). The Randall-Sundrum metric can be constructed by considering two slices of the z > 0 region. For this purpose, it is convenient to define a new coordinate Ω by z = 1 k e k|Ω| . The two AdS slices are given by 0 < Ω ≤ L and −L ≤ Ω < 0 and can be joined at Ω = 0. The relation between z and Ω is plotted in Fig.1.
Identifying Ω with −Ω we get the orbifold space The metric (2) naturally satisfies this condition.
The Randall-Sundrum metric was obtained from Einstein equations coming from a Ddimensional gravitational action with negative cosmological constant in the presence of two (D-1)-branes located at Ω = 0 and Ω = L with opposite tensions. We will see in the next section how this metric also arises from a D-dimensional Brans-Dicke theory.

RANDALL-SUNDRUM GEOMETRY
In this section we will use a BPS-like mechanism to solve the field equations of motion coming from a D-dimensional Brans-Dicke theory with two (D-1)-brane potentials. In this theory there is a scalar field non-minimally coupled to gravity. The total action is given by The action (3) leads to the following background equations We consider the following ansatz for the metric and scalar field : whereΦ(Ω) and σ(Ω) are even functions in Ω. The background equations above then translates into a system of second order differential equations Finding a solution of these differential equations is in general complicated for an arbitrary potentialṼ (Φ). We could also invert the problem and solve the equations for the scalar field solution and potential once we know the metric . In this work we use a BPS-like mechanism that simplifies the background equations and leads to a special class of potentials. The Randall-Sundrum solution for the metric arises from a particular potential belonging to this class.
If we substitute the ansatz (8) in the lagrangian density of eq. (3) we find In order to have periodicity in the coordinate Ω and justify the presence of the δ(Ω − L) function the lagrangian density has to be integrated from −L + ǫ to L + ǫ and make ǫ → 0 at the end. The lagrangian density can be rewritten in the following form where we have introduced an arbitrary odd function The last two terms in (13) are total derivatives so they vanish using the periodicity of Ω.
The first two terms are square terms which are zero wheñ Assuming that the equations above are satisfied by the scalar field and the metric we find that the following class of bulk potentials with the brane conditions lead to a vanishing action. Using (14) the brane conditions read and similar for the derivatives inΦ. It is straightforward to show that the system of equations (15)-(20) give background solutions that also satisfy the background equations (9)-(11).
This way we find a BPS-like mechanism that gives background solutions for second order differential equations by solving first order equations that appear inside the square terms in the lagrangian density. Because the square terms appear with opposite signs there is no a Bogomolnyi bound. This mechanism is similar to that found in ref. [6]. Note that for the case D = 5 our equations (15)-(20) reduce to those obtained in [7]. with The value of C was chosen for convenience. The brane potentials in this case arẽ Note that although we have obtained the Randall-Sundrum metric (2) Randall-Sundrum scenario can be obtained in the limit ω → ∞ in which the scalar field becomes trivial, as discussed in [7].

The Einstein frame
If we perform the following background transformations : with we go from the Jordan-Fierz frame (in which the Brans-Dicke theory is originally formulated) to the Einstein frame. These background transformations are known in the literature as conformal transformations [10]. Note that this transformation imposes a reality condition for the Brans-Dicke parameter: w > − D−1 D−2 . The total action (3) becomes In the Einstein frame, the background solutions of (22) become In terms of the coordinate z = 1 k e −σ(Ω) the metric reads This metric can be interpreted as a deformed Randall-Sundrum metric where the deformation is given by f ω (z). Note that the Planck brane is localized at z = 1/k while the TeV brane is localized at z = (1/k)e kL . In the limit ω → ∞ the deformation factor f ω (z) goes to 1 and we recover the original Randall-Sundrum metric.

IV. SPECTROSCOPY OF SCALAR KALUZA-KLEIN MODES
Now we consider the compactification of a massless scalar field fluctuation in the Einstein frame. This frame is well motivated for many reasons being the most important the positive sign of the energy density [10]. A scalar field fluctuation can be described by the following This action can be decomposed as where h(Ω) is defined by g µν = η µν h(Ω). The Kaluza-Klein decomposition of ϕ(x, Ω) is If the modes χ n (Ω) satisfy the relations then we get the D-1 dimensional action for φ n (x) : As in usual Kaluza-Klein compactifications, the bulk field φ(x, Ω) manifests to a D − 1 dimensional observer as an infinite "tower" of scalars φ n (x) with masses m n .
The tower of masses m n can be obtained by solving the equation (34) which can be rewritten as where It is convenient to solve this equation in terms of the coordinate z = 1 where u = (D−2)ω+D−1

ω+1
. This equation has a zero mode solution corresponding to m n = 0 of the form For m n > 0 the solution is a combination of BesselJ and BesselY functions of argument m n z. In terms of Ω the solution reads where σ = −k|Ω| and ν = (D−1)w+D 2(w+1) = (u + 1)/2 and N n is a normalization constant.
Besides the condition w > − D−1 D−2 , it is interesting to note that in order to find finite ν we need ω = −1. The limit ω → ∞ leads to the result found in [11] for the massless case. Our modes solutions are even functions in Ω. To guarantee the continuity at the orbifold points Ω = 0 and Ω = L we impose Neumann boundary conditions. The boundary condition at where we have defined x nν = (m n /k)e kL . The boundary condition at Ω = π gives the important equation The Kaluza-Klein modes x nν are obtained by solving this equation. We choose kL = 12 as considered in the original Randall-Sundrum model. We present in Fig. 2 our results for the first modes as functions of the Brans-Dicke parameter ω in the particular case D = 5 .
We see from that figure that the modes grow rapidly when ω → −1 and approach constant functions for large ω (for instance x 1ν → 3.83 for large ω). This way the distance between these modes is preserved at large ω. The Fig. 3 shows how the first mode x 1ν increases with the dimension.
The normalization constant N n appearing in the modes solutions can be calculated by performing the integral of eq. (33). This integral is not simple in general because involves products of BesselJ and BesselY functions. However, for the lower modes the dominant con-tribution to the integral comes from the square of BesselJ. For these cases the normalization constant can be approximated by where we have supposed in this approximation that kL is large as expected for the resolution of the hierarchy problem [7]. In our case the metric contains an extra degree of freedom which is the Brans-Dicke parameter. As we saw in the last section, this parameter acts as a fine-tuning for the Kaluza-Klein masses arising from scalar fluctuations. We will see in this section how the W ± and Z resonances of the Higgless model will depend on the Brans-Dicke parameter as well.
We begin with the SU(2) L × SU(2) R × U(1) B−L action where with M = {z, µ}. We denote as g 5 the coupling constant of SU(2) L , SU(2) R andg 5 the U(1) coupling constant. In order to cancel the interaction terms between the z and µ components we must add gauge fixing terms of the form The bulk fields can be decomposed in the following way The boundary conditions on the Planck brane z = 1 k arẽ These conditions lead to the symmetry breaking SU(2) R ×U(1) B−L → U(1) Y . The boundary condition at the TeV brane z = 1 k e kL are and the following equation of motion whereū = ω+4/3 ω+1 . The solution to equation (59) is where with x nν = (m n /k) e kL ,ᾱ = {ν,ν − 1} and we assumed that g 2 5 > 0. Similarly, substituting (50) into (53) and (56) , we find the W ± mass equation The mass equations (61)  This behavior is shown in Figure 4 for kL = 12 andg 2 5 /g 2 5 = 0.426. Note that when ω → −1 the first mode vanishes while the higher modes diverge. Another interesting result is the evolution of the quotient m 2 W /m 2 Z with the Brans-Dicke parameter where m W and m Z are the masses of the W and Z resonances. This quotient is lower than 1 for the first and third modes and increases when decreasing ω while for the second mode it is greater than 1 and decreases when decreasing ω. This behavior is shown in figure 5 for kL = 12 andg 2 5 /g 2 5 = 0.426. These values were chosen to obtain a realistic value for the quotient m 2 W /m 2 Z in the limit ω → ∞. Indeed, in this limit we obtain the result m 2 W /m 2 Z ∼ 0.764 that can be compared with the asymptotic expression (67) where θ W is the Weinberg angle. The relation (67) is characteristic of Higgless models that preserve the SU(2) custodial symmetry.

VI. CONCLUSIONS
In this paper we have constructed D-dimensional Randall-Sundrum models from Brans-Dicke theory by using a BPS-like mechanism for solving the background equations. We have also studied the Kaluza-Klein decomposition of massless scalar and gauge fields and showed how the Kaluza-Klein modes depend on the Brans-Dicke parameter ω. In particular, we saw how the Brans-Dicke parameter act as a fine-tuning parameter for the W and Z resonances of a Higgless electroweak model.
We have considered in our analysis of scalar and vector Kaluza-Klein modes a wide range of values for the Brans-Dicke parameter ω. We also assumed that kL is large as is expected for solving the Planck-weak hierarchy problem. However, it is important to remark that stability of this model requires the addition of scalar field potentials on the Planck and TeV branes. As mentioned in [7], after introducing stabilizing potentials a large value of ω is needed in order to avoid a new hierarchy for the scalar field.
In the Brans-Dicke theory the presence of a background scalar field was crucial. A possible future investigation would be studying the effect of other background fields like the Kalb-Ramond field which is motivated by String Theory (see for instance [12,13]).
Another interesting feature to be explored is the effect of the Brans-Dicke parameter on scalar and gauge field interactions and in the presence of fermionic fields. .