A new way of calculating the effective potential for a light radion

In Randall-Sundrum scenarios, with the Goldberger-Wise stabilization mechanism, a light radion effective action is defined by integrating out all heavy degrees of freedom in the path integral formalism. Performing the necessary constrained minimization of the full 5D action by the Lagrange multipliers method, we arrive to a new, much more economical, way of calculating the radion effective potential for different choices of the radion interpolating fields. The dependence of the effective potential on the choice of the radion interpolating field is investigated. The calculated exact effective potential is used as a reference for judging about the quality of various approximate methods used in the literature.


Introduction
Warped extra-dimensional models, the original Randall-Sundrum (RS) model [1], and its extensions (see for instance [2][3][4][5][6]), have been proposed as a solution to the hierarchy problem of the Standard Model (SM) -the quantum instability of the weak scale with respect to the Planck scale. In those scenarios the hierarchy between the Planck scale and the weak scale is generated by the anti-de Sitter warp factor present in the fourth spatial dimension. After integrating out the extra dimension, the four-dimensional spectrum contains towers of Kaluza-Klein modes of all SM particles propagating in the bulk and also a new degree of freedom, the radion, corresponding to the distance between the branes.
In the original, RS, version the radion remains massless, reflecting the fact that the distance between the branes is not dynamically stabilised. Thus, such model is not able to explain the hierarchy of the Planck and weak scales which may be obtained by arranging the correct separation between the branes. This issue is solved by the Goldberger-Wise mechanism to stabilise the size of the 4th spatial dimension, by adding a bulk scalar field with some field dependent potential in the bulk and on the branes [7]. The stabilisation mechanism generates a mass for the radion but it typically (or at least in large parameter ranges of those models) remains the lightest state in the physical, 4D spectrum [8].
Various potentials for the Goldberger-Wise scalar have been considered in the literature [7,9,10], categorized according to their back-reaction on the 5D metric, after solving the equations of motion for the scalar-gravity system. In models with weak back-reaction the original AdS metric is only weakly affected by the presence of the scalar field whereas in the case of strong back-reaction even a singularity in the 5D metric not far beyond the location of the IR brane can be generated (the so-called soft wall models).
Warped extra dimensional models are very interesting also in other contexts, beyond the original motivation of solving the hierarchy problem. According to the AdS/CFT correspondence [11], they provide a 5D holographic description of a conformal theory in four dimensions, with spontaneously broken conformal symmetry, and perturbed by close-tomariginal operators. The dynamics of the strongly coupled states in the 4D theory can be investigated perturbatively by means of the 5D theory. In that holographic interpretation, the radion is dual to the dilaton -a Goldstone boson of the spontaneously broken scale symmetry in 4D theory [12,13].
A very useful and intuitive concept in discussing the stabilisation mechanism and the 4D holographic interpretation of the RS-type models is the effective radion (dilaton) potential. Furthermore, it is particularly relevant for investigating the radion early universe cosmology, such as the character and the potential impact on the electroweak phase transition of the radion phase transition during which it acquires a vacuum expectation value (corresponding to the dilaton condensation in the holographic picture) [14].
The radion effective potentials are usually defined in the literature by solving the bulk equations of motion with only some of the boundary conditions fulfilled. Such modified background solutions have one degree of freedom which is considered to be related to the radion field [10,15,16]. Then the action calculated for a solution with a given value of that degree of freedom is taken as the minus effective potential for the radion field. In most of the papers investigating the RS scenarios, definitions of radion are not precisely related to the methods used to get the effective potential. Most often such definitions are based on intuitive relations to the distance between the branes and the approximations are not under a rigorous control or/and require a very high precision calculations.
In this technical paper we first provide a precise definition of the effective action and the effective potential for the lightest degrees of freedom by integrating out all heavy degrees of freedom in the path integral formalism. Then we preform the constrained minimization of the full 5D action using the Lagrange multipliers method. To apply this procedure one needs to chose 4D interpolating fields for the light states (the radion and the graviton). The obtained effective potential depends on the functional dependence of the 4D interpolating fields on the original 5D fields. However, for hierarchically heavier all other degrees of freedom one may expect that the dependence on the choice of the interpolating fields is weak. This, of course, depends on the parameters of the model, but can be checked a posteriori, by comparing the results for different interpolating fields.
The procedure presented in this paper leads to a new, much more efficient, way of calculating the radion effective potential for different choices of the radion interpolating field. This is done in section 2. In section 3 we discuss the advantages of some of the methods developed in section 2 and use them to calculate explicitly the exact effective potential for several choices of the radion interpolating field. In section 4 the calculated exact effective potentials are used as a reference for judging about the quality of various approximate methods used in the literature. The paper ends with several appendices.
In this paper we do not calculate the rescaling factors necessary to normalize the radion fields canonically [17]. However, even non-canonically normalized effective potential are useful for studying the stabilization mechanism and the character of the radion phase transition.
2 Effective action and potential for the radion

Background solution
We investigate models of gravity and a scalar field Φ with 5-dimensional (5D) space-time being an orbilofd equal to a warped product of 4-dimensional (4D) space-time and an interval: M 5 = M 4 × S 1 /Z 2 . The action involves a bulk potential V (Φ) and two brane potentials U i (Φ) and reads where κ 2 is the 5D Einstein's gravitational constant, related to the 5D Planck Mass M 5D as κ 2 = M −3 5D . We are interested in background solutions of the form (4) µν dx µ dx ν + dy 2 , (2.2) Φ = φ(y) .

(2.3)
We will consider two cases for the 4D metric: the flat Minkowski space-time and inflating, spatially flat de Sitter space-time. Both may be written using the following 4D metric 1 g (4) µν dx µ dx ν = −dt 2 + e 2Ht δ ij dx i dx j , (2.4) where Minkowski case corresponds to H = 0. In most of the cases considered in this paper we put H = 0, but for some analyses it will be useful to keep the possibility of H = 0 in the equations.
The variations of the action (2.1) for the ansatz (2.2)-(2.3) are explicitly written down in the appendix A. Their explicit form will be important below. The variations (A.1) and (A.2) contain delta-like contributions localized at both "end-of-the-world" branes at y = y 1 , y 2 . Thus, it is convenient to write the corresponding equations of motion separately for the bulk (i.e. for values of y different from the brane positions) and for the branes (obtained by integrating the full equations of motion over an infinitesimally short interval containing a given brane position). The bulk equations read 2 The brane equations of motion are usually written in the form of boundary conditions for derivatives of φ and A: lim where upper (lower) sign is for i = 1 (i = 2) and primes denote derivatives with respect to appropriate argument i.e. d/dy for A and φ and d/dφ in the case of potentials V and U i .
For general potentials V and U i the equations of motion (2.6)-(2.8) with the boundary conditions (2.9)-(2.10) have no solutions. This may be shown by the following reasoning. Let us start at the brane located at y 1 . There are two dynamical second order equations of motion, (2.6) and (2.7), for two functions: φ(y) and A(y). Thus, the initial conditions 1 We use the following convention for the space-time indices: M, N = 0, 1, 2, 3, 5; µ, ν = 0, 1, 2, 3; i, j = 1, 2, 3; x 0 = t; x 5 = y. 2 It is convenient to keep their form as they appear in the variations of the action (A.1)-(A.3). However, when the equation (2.8) is satisfied, it can be used to rewrite (2.7) in a simpler form as consist of fixing four values: φ(y 1 ), φ (y 1 ), A(y 1 ), and A (y 1 ). The third one corresponds just to the choice of units, so we use the convention A(y 1 ) = 0. The values of the three remaining functions at y 1 may be found by solving three equations: two boundary conditions (2.9) and (2.10) for i = 1 and the bulk equation (2.8). Three equations for three unknowns in general may be solved (there may be a discrete set of such solutions). Then we may integrate (for example numerically) the dynamical bulk equations of motion, (2.6) and (2.7), to find φ(y) and A(y) in the bulk. The problem appears when we try to fulfill the boundary conditions (2.9) and (2.10) at y = y 2 . 3 For general value of y non of these boundary conditions is fulfilled. For some discrete values of y one of them may be fulfilled. However, exact finetuning of the potential parameters (for example those in U 2 ) is necessary to have solutions to both boundary conditions (2.9) and (2.10) at the same value of y. This is the standard cosmological constant problem (the effective 4D cosmological constant must vanish for flat 4D sections of the 5D space-time, i.e. for H = 0, and must be equal 3H 2 in the case of inflating 4D sections).
Usually, from the 4D perspective such models are described in terms of KK towers of 4D fields. Such towers are obtained by expanding 5D perturbations around the background (2.2)-(2.3) in terms of 4D eigenmodes of the quadratic part of the Lagrangian. The lightest modes of the system are the lightest scalar KK mode (the radion) and a massless spin 2 field (the graviton). Reinserting the expansion in the action we could find the interactions among all the 4D fields. For some applications it is necessary to go farther and compute the effective action for the radion beyond the perturbative expansion in the number of fields around the vacuum. This is the case, for example, for the study of the cosmological evolution of these models, which includes possible phase transitions in the early universe [14][15][16][18][19][20][21][22], for which non-perturbative solutions interpolating between different vacua are crucial. For such studies it is more convenient to perform an expansion in the number of derivatives in the effective action. In this paper we will focus on the zero-derivative term of this expansion, the effective potential for the radion. Besides its importance for the applications mentioned so far, this object also helps to gain better understanding of the stabilization mechanism of these models [7,10,23].
In the following subsections we will present the definition and discuss the meaning of the effective potential of the radion and explore different ways to compute it. We will extend the approach usually considered in the literature [7,10,15,16,23].

Definition of the radion effective action
It is known that in order to extract an effective action for low energy degrees of freedom of a model, it is not necessary to know the exact light one-particle mass eigenstates [24].
In the case of one such light state |l , one can calculate the corresponding effective action using some simple field related to the light degree of freedom in question. The necessary condition is that the state obtained by acting with such a field on the vacuum must not be orthogonal to the light one-particle mass eigenstate: l|Σ|0 = 0. Formally, it is enough that this matrix element is just non-vanishing but it practical applications it is better if it is not very small, as we will discuss later. In the case of more light states the necessary condition becomes somewhat more complicated. Namely, for n light particles |l i one needs n fields Σ j , which must fulfill the condition that the (n × n)-dimensional matrix l i |Σ j |0 is invertible: (2.11) The simple fields Σ j , which to some extend should approximate operators related to the light mass eigenstates, are usually called in the literature the interpolating fields [24]. Once the interpolating fields are chosen, we can integrate out the remaining degrees of freedom to obtain the low energy effective action. Different interpolating fields generate different effective actions, but all of them must be equivalent in the sense of being related by some field redefinitions. The well known equivalence theorem for field redefinitions then ensures that all these actions will produce the same S-matrix elements [25][26][27].
Explicit calculations show that in the case of choosing non-diagonalized mass interpolating fields for the light states, the effective action after integrating out the remaining degrees of freedom contains a tower of higher-derivative bilinears terms. However, these bilinears are suppressed by powers of the mass of the lightest state which was integrated out. For a 4D scalar field ψ, the first possible higher-derivative bilinear (beyond the kinetic term) in the effective action, ψ∂ 4 ψ, will be suppressed by M −2 , where M is the smallest mass integrated out. Of course, we can choose, using appropriate field redefinitions, an operator basis which eliminates these higher-derivative bilinears but at the same time changes the remaining terms (in particular, the potential). We would then recover the effective action we would obtain using mass diagonalized interpolating fields. The effects of the field redefinition that eliminates those bilinears will affect the potential by terms of order O(m 2 /M 2 ) or higher, where m is the scale of the light states. If the mass gap between the light states and the integrated out heavy states is large, such field redefinition would give contributions that in many cases may be neglected.
The lightest modes of the model under consideration are the graviton and the radion. We assume that the first massive KK graviton is much heavier than the radion (this is really the case for many 5D models and for all models we will use as examples illustrating our general considerations). Therefore, following the logic explained above, we will define some interpolating 4D graviton and radion fields and integrate out the remaining degrees of freedom. Different choices of these interpolating fields should give similar results due to big mass gap above the radion mass. In realistic models, one expects to have additionally all the Standard Model particles with masses lighter or comparable to the radion mass, but here we will assume that the mixing with them is negligible, so we are only interested in the radion-gravity sector. However, we stress that more general scenarios, where for instance the Higgs-radion mixing is important, could also be treated within the same formalism presented here.
Letĥ µν [g, Φ] andχ[g, Φ] be our interpolating fields for the graviton and the radion: 4D fields that can be defined as functionals of the 5D elementary fields of our system (2.1). We use the following notation: the hat inĥ orχ denotes thatĥ andχ are functionals depending on some arguments. The same letters without a hat, h and χ, will be some specific configurations for these fields. We will restrict to such functionals of 5D fields which may be written as: where H µν and Y are functions of y, the fields g M N and Φ, and perhaps y-derivatives of them, that transform covariantly under 4D diffeomorphisms: (x µ , y) → (x µ (x), y). Therefore,ĥ µν [g, Φ] andχ[g, Φ] will also transform covariantly under 4D diffeomorphisms. Furthermore,ĥ must have all the properties that define a 4D Lorentzian metric for all configurations of the 5D fields g M N , Φ. Of course, our interpolating fields (2.12) and (2.13) should also fulfill the condition (2.11).
We calculate the effective action integrating out all the fields except the 4D fieldsĥ µν andχ. This can be expressed as (2.14) Here, the action S inside the path integral is the action (2.1), and the δ-distributions fix the fieldsχ[g M N , Φ] andĥ µν [g M N , Φ] to have values equal χ and h µν , respectively. Of course, (2.14) has to be understood as a symbolic equation. However, it has a clear and rigorous interpretation at the tree level (using the saddle point approximation to compute the path integral): The effective action is therefore a functional of 4D fields h µν and χ. The minimization in this equation is performed over all possible 5D configurations whose image under the functionalsĥ µν andχ is h µν and χ, respectively. This constrained minimization can be performed by the Lagrange multiplier method. The equations for the constrained system become δS δΦ(x , y) where λ χ (x) and λ h µν (x) are the Lagrange multipliers: new 4D fields whose values have to be determined by solving the system. These fields will therefore introduce some breaking of the EoMs (2.6)-(2.10) that will allow the interpolating fieldsĥ µν andχ to take desired configurations (all 5D EoMs are satisfied everywhere for the background solution which leads to specific fixed values ofĥ µν andχ). The effective Lagrangian then becomes where L 5D is the 5D Lagrangian of (2.1), andĝ µν sol [h µν , χ],Φ sol [h µν , χ] the solution to the system (2.16)-(2.19) given the 4D fields h µν and χ. Let us stress that we have added hats toĝ µν sol andΦ sol to indicate they are functionals (in this case of 4D fields h and χ), as opposite to g µν and Φ which are specific 5D field configurations. It is possible to give a clear meaning to the Lagrange multipliers: they are the variations of the effective action under the corresponding interpolating fields. This can be checked explicitly. Let α i (x) = h µν (x), χ(x) be the interpolating fields, and ω n (x, y) = g µν (x, y), Φ(x, y) the 5D fields. Then, where λ i (x) = λ h µν (x), λ χ (x) and we have again used the hat notation. In the first line we have used (2.20) (integrated over x), in the second line, (2.17), and in the third one, the fact thatα i [ω sol [α]] = α, and therefore,

Expansion of the effective action
Using the diffeomorphism invariance, we can write the effective Lagrangian using the expansion where the ellipsis represents terms of higher order in derivatives and/or in the curvature. This expansion defines the effective potential V eff (χ), the kinetic term C eff (χ), and the kinetic mixing with gravity K eff (χ) for the radion.
The above effective Lagrangian may be rewritten in a more canonical way after some field redefinitions. Namely, the kinetic mixing with gravity can be eliminated through a Weyl transformation of the metric. We can always go to the Einstein frame by doing the transformationh where χ * is some reference value of the radion. The effective Lagrangian is theñ , (2.27) and the 4D Planck mass is given by M 2 P = K eff (χ * ). In some cases it is simpler to take as the reference value for the radion, χ * , its vacuum value, but this is not strictly necessary. It is desirable to work in the Einstein frame because it diagonalizes the kinetic terms of the graviton and the radion, so it describes in a more natural way their dynamics. For instance, when higher curvature terms are negligible, the 4D homogeneous solution for the system, i.e., the vacuum solution, is in general not given by the minimum of V eff , but by the minimum ofṼ eff . 4 Additionally, going to the Einstein frame is a way to define univocally the potential up to field redefinitions of χ (assuming we neglect higher curvature terms) for a given interpolating radion field. These field redefinition ambiguities for χ can also be fixed if we use the canonically normalized fieldχ that makes the kinetic term canonical, C eff = 1.
The effective potential V eff (χ) can be calculated computing the effective Lagrangian for the 4D flat metric h µν ∝ η µν and homogeneous configurations of χ(x) = χ. Then, the terms involving curvature tensors and derivatives vanish and we have V eff (χ) = −L eff (η µν , χ). In this case, the equations to find the 5D fieldsĝ sol andΦ sol are Minkowski invariant, so we set H = 0 in the ansatz (2.2)-(2.4). The Lagrange multipliers obtained with these configurations can be related to V eff (χ) through (2.21) because higher derivative terms in χ and h µν vanish. Using 4D Lorentz covariance to rewrite the Lagrange multiplier λ h µν as These formulae provide a method to compute the effective potential that will be developed further below. In particular, the development of (2.28) for a specific choice of interpolating fields we will describe below gives the way of computing the effective potential for soft wall models that has been used in the literature [10,15,16]. However, (2.29) provides a new method for the same computation which is much more economical in numeric calculations. Additionally, this philosophy brings new possible definitions for the effective potential (considering other interpolating fields), which could be more convenient in some situations.
Notice that if χ is kept homogeneous, χ(x) = χ, but the metric h µν is chosen of the form h µν ∝ g (4) µν of (2.4) with H = 0, instead of (2.28) we have, where again λ h µν is written as λ h µν = λ h h µν . This form of the Lagrange multiplier is required because h µν describes an Einstein manifold, and therefore, every 2-covariant tensor in this manifold must be proportional to the metric. It is possible to obtain an expression for the kinetic mixing with gravity K eff . The derivation is shown in the appendix B and here we just give the result: where A sol is the solution for given homogeneous χ and h µν = η µν .
To properly determine the dynamics of the radion, we also need to compute the kinetic term C eff (χ) and the higher orders in (2.23). If we are interested in regimes and configurations of fields without high derivatives (with respect to x) of the fields and without highly curved spacetimes, the higher orders can be neglected, but knowing the kinetic term is necessary. This work is however mainly devoted to the effective potential and its calculation. A thorough analysis of the kinetic term will be presented in a follow-up article. In any case, the effective potential already provides valuable physical information about the system, e.g. the critical temperature at which the system phase changes [15,16].

Interpolating fields
To compute the effective potential we must specify the interpolating fields to be used. In this subsection we will list some possibilities and discuss them.

The UV metric and the IR warp factor as interpolating fields
The way to calculate the radion effective potential most often used in the literature [10,15,16] may be justified by our procedure with the following interpolating fields for the metric and the radionĥ where the ansatz (2.2) was used in the last line. We have added the subscript g to the interpolating radion field χ g to distinguish it from other choices. The above choice may be justified by its simplicity. Both interpolating radion and graviton fields depend only on the fields evaluated at the branes. The fact that we have taken as interpolating graviton field the metric on the UV brane, and not on the IR brane, has to do with the resulting K eff function. Choice (2.32) leads to a very weak dependence of K eff on χ as we will see below.
For the interpolating fields (2.32) and (2.33), the Lagrange multipliers introduce modifications of EoMs (2.6)-(2.10) due to the r.h.s. of (2.16) and (2.17). The non-vanishing variations of the interpolating fields are: Therefore, using (2.17) and (A.2), one can see that the UV and IR Israel junction conditions (2.10) are replaced with The Lagrange multipliers break the Israel junction conditions, and so they allow to find solutions to the EoMs (2.16) and (2.17) for given values of the interpolation graviton and radion fields (2.18). Following (2.28), the effective potential is proportional to λ h . It follows from (2.36) and (2.37) that this effective potential may be written as a sum of two terms: (2.40) The first of the above terms is exclusively evaluated in the UV brane while the second one is evaluated in the IR brane (we use convention in which A(y 1 ) = 0). In the literature these terms are usually called, respectively, the ultraviolet and the infrared contribution to the effective potential [10]. We use similar notation, V UV eff and V IR eff , but we want to stress that these terms do not represent contributions to the radion potential physically localized at the branes. Decomposition of V eff into such two terms is just a consequence of using formula (2.28) and the interpolating fields (2.32) and (2.33). As we will show later, other decompositions appear for other methods of finding V eff and/or other choices of the interpolating fields 5 . Let us first use the method based on (2.29).
As we already mentioned, relation (2.29) provides a new way of calculating the effective potential using λ χ . Equation (2.29) determines V eff up to some overall shift. This shift may be easily obtained from the condition that the effective potential vanishes for the radion corresponding exactly to the background position of the brane -for which all EoMs and BCs are fulfilled. For the choice of the interpolating fields given in (2.32)-(2.33), λ χ can be calculated from the breaking of the IR Israel junction condition (2.37): We stress that this relation between the derivative of the full effective potential and the IR part had not been appreciated before, and it constitutes one of the main results of this article. In addition, it provides a much more efficient (and much faster) method of numerical computation of the effective potential.
With the choice of the interpolating graviton field (2.32), for the systems we are interested in and which will be discussed in the next section, the kinetic mixing between radion and gravity, described by K eff (χ), depends very weakly on χ. In such systems the warp factor in the ultraviolet region may be approximated by A(y) ≈ A(y 1 ) + k(y − y 1 ) with some constant k, and then from (2.31) we obtain (2.42) 5 Even using the method corresponding to (2.32)-(2.33) and defining the effective potential as the integral of the 5D potential over the 5-th dimension, as done e.g in [10], one gets contributions to V eff from both branes and from the bulk. The result may be written as a difference of two boundary terms if integration by parts is applied. However, integration by parts is just a tool to calculate integrals in which "contribution" from a given boundary is known only up to an arbitrary constant.
Therefore, the Weyl transformation necessary to eliminate the radion-gravity mixing (2.26) to go to the Einstein frame does not change substantially the potential (only terms suppressed by χ 2 as compared to the leading terms of the potential).

Other interpolating fields
One can also use interpolating fieldsĥ µν [g, Φ] andχ[g, Φ] different than (2.32)-(2.33). Let us first change our choice for the graviton interpolating field while keeping (2.33) for the radion. We choose the metric evaluated at the infrared brane: Then, the application of the methods explained above gives results which, when compared to (2.38)-(2.40), may be written asV where bars denote quantities obtained with (2.43), as opposite to the unbarred ones, obtained with (2.32). Notice that now the derivative of the potential is related to V UV eff while before it was related to V IR eff . Nevertheless, equality (2.45) is equivalent to (2.41). Also, althoughV eff differs from V eff , the effective potential in the Einstein frame is exactly the same for both choices. This is somewhat peculiar for this choice as a consequence of the fact that with both interpolating graviton fields we are breaking the same equations of motion. In general, for other radion interpolating fields we expect slightly different effective potentials in the Einstein frame when we change the graviton interpolating field. In any case the difference between them has to be always suppressed by the mass of the lightest heavy state that has been integrated out.
Let us take again (2.32) as the graviton interpolating field but explore other choices for the radion interpolating field. Some possibilities are: the physical distance between braneŝ 47) and the value of the field φ on the IR brane 6 Non-vanishing variations of these interpolating fields, for our ansatz (2.2)-(2.3), are For both choices, λ h gives the only contribution to the UV Israel junction condition breaking: giving the effective potential with only the UV contribution: We use the superscript (y) or (φ) to indicate that the effective potential was obtained usinĝ χ y orχ φ , respectively, as the interpolating radion field. In the case ofχ y , the Lagrange multiplier λ χ modifies the constraint equation (2.8) to Any value of y may be used in the above equation because the dependence of the l.h.s. on y vanishes if the remaining bulk EoMs are satisfied. Equation (2.28) gives different expression for the effective potential which in the present case reads and the r.h.s. may be evaluated at any value of y. Usingχ φ as the interpolating field, we find the following modification of the boundary condition (2.9) at the IR brane Thus, the derivative of the effective potential may be written as (2.56) The equations presented in this subsection provide different ways to compute the effective potential for different interpolating fields. Of course, the number of possible interpolating fields is infinite. Here we have just reviewed some simple choices. In the next section we will illustrate our general results using some specific models as examples. In general, the numerical computations will be necessary. We will show how some methods of calculating the effective potential offer advantages when compared to other approaches.

Computation of the radion effective potential
In subsection 2.4 we have obtained several formulae to obtain the effective potential for the radion using several interpolating radion fields. In this section we will apply these results to the numerical calculation of the effective radion potential in a few models.
We will consider two different types of bulk potentials for the scalar bulk field Φ. First, a quadratic potential (used for example in [10,15,28]): This is a typical potential used in the Goldberger-Wise mechanism [7]. In order to obtain a large hierarchy between the Planck and TeV scales in a natural way, one usually considers small values of | | (≤ 0.1). In this paper we will additionally focus on positive , which typically results in asymptotically AdS spaces with strong back-reaction only close to the IR brane. We also consider the following exponential bulk potential (used in [9,16] with the assumption 3γ 2 < 4κ 2 for which the potential is monotonously decreasing. 8 Both potentials have in common that they are bounded from above by −6k 2 /κ 2 but are not bounded from below. This in principle is not problematic for (3 + 1)-D Poincaré invariant solutions of five-dimensional gravity coupled to scalars [29]. Besides, in asymptotically AdS spaces, unitarity bounds allow for negative masses squared as long as the Breitenlohner-Freedman bound is satisfied (for the potential (3.1) this leads to the bound < 1) [30].
For the brane potentials we consider quadratic ones: where Λ 1,2 are the tensions of the UV and IR branes, respectively. In most of the cases we will take the so called stiff wall approximation, where ξ 1,2 → ∞. In such a case, the boundary conditions for φ (2.9) and A (2.10) simplify to The apparently capricious form of the exponential potential follows from the simple superpotential where the plus (minus) sign is taken for the subscript i = 1 (i = 2). Also, if φ(y) satisfies the boundary condition (2.9), A typical solution of the EoMs (2.6)-(2.10) for this kind of bulk potentials is shown in figure 1. One can see that in the UV region (small values of y), the metric is very close to the AdS case (for which A = const), but close to the IR brane, large deviations from the AdS metric appear. These are the so called soft wall models, which have been vastly studied in detail due to their exceptional suitability for constructing phenomenologically interesting models [5,6,[31][32][33][34].

Warped factor as interpolating radion field
From now on we use the interpolating graviton field given by (2.32). For the interpolating radion field, we focus first onχ g defind in (2.33). The effective potential for this case is given by (2.38). In order to evaluate this formula one has to find a class of solutions to the EoMs (2.6)-(2.9) which leaves one free degree of freedom (e.g. the distance between the branes). Then, (2.38) may be used to compute the effective potential as a function of this degree of freedom. In addition, it is necessary to find the value of the radion field χ g as a function of the same degree of freedom. Combining these results one can write the obtained effective potential as a function of the radion. However, in general it is not possible to perform this procedure analytically so the use of numerical computations is necessary.
In figure 2 we show an example of the effective potential for both scalar potentials considered, (3.1) and (3.3), showing at the same time the IR and UV parts defined in (2.39)-(2.40). As one can see, both contributions are comparable in the region of the minimum, and therefore both are necessary to compute accurately the effective potential. Although the IR part has qualitatively the same shape as the full potential, the UV contribution is necessary to move the minimum of the potential to the point where V IR eff = 0, as it is required by one of the equations of motion: the IR Israel condition (2.10) 9 . This shows the importance of V UV eff in this region. Under the assumptions for the parameters of the potential we consider, there is no case or limit where the UV part can be neglected, or just contributes a constant to the full potential. 10 It has been suggested that tuning the UV brane parameters to send it to infinity (which, from the 4D point of view corresponds to send the effective 4D M P to infinity) would make V UV eff to vanish. In the light of the previous argument, it cannot be the case. In the appendix C we present a calculation where we send carefully the UV brane to infinity to find the surviving contribution to the potential in such limit, and we check that V UV eff does not vanish.
It is obvious form equations (2.39) and (2.40) that, due to strong warping, V IR eff is exponentially suppressed with respect to V UV eff . This suppression must by of around 50 or more orders of magnitude in phenomenologically realistic models. On the other hand, the two terms contributing to V UV eff are O(1). Therefore, if V UV eff and V IR eff are similar in the region of interest, V UV eff must be the result of a large cancellation between the two terms in V UV eff . Indeed, this makes the UV part of the potential much more difficult to compute numerically than the IR part: one has to solve the differential equations of motion keeping a very large number of significant digits (more than 50 as we have argued). This can be done, but increases the computational time. The solution to this technical difficulty is to use (2.41) to compute the full potential only from the IR breaking of the boundary conditions. Formula (2.41) allows us to find the derivative of the effective potential so an integration is necessary. As a result we obtain the potential up to an arbitrary integration constant. This constant has to be chosen in such a way that value of the potential vanishes at the minimum so that the effective 4D cosmological constant is zero. This choice corresponds to fine-tune the UV brane tension Λ 1 . In all cases studied along the paper, Λ 1 will be the fine-tuned parameter to obtain a vanishing 4D cosmological constant. In figure 2 we can see the perfect agreement between the two calculation methods explained above. We emphasize that the calculation with (2.38) has required to solve differential equations with more than 70 digits of significance, fact that has increased significantly the computational time for this method. However, for the calculation using (2.41), only a few digits of significance are needed. This is a central result of this paper: this new method associated with (2.41) offers a clear improvement in the numeric calculation of the effective potential. 9 We assume here that the configuration we consider is stable, i.e. the distance between the branes is stabilized, so the value of the radion corresponding to this configuration is at the minimum of the effective potential. 10 Only when = 0, and the potential V eff (χ) ∝ χ 4 + O(χ 6 ) and the minimum of the potential is in χ = 0, the UV part of the potential will be a constant plus negligible corrections. This case however is not relevant to construct phenomenologically viable models.

Different choices for the interpolating radion field
At the beginning of subsection 2.2, it was argued that almost every choice of the 4D interpolating fields works equally well to describe the light degrees of freedom of the system (i.e. the graviton and the radion), as long as the condition (2.11) is fulfilled. One should not use interpolating fields which are (almost) "orthogonal" to the corresponding light states. All interpolating fields produce equivalent effective actions related by field redefinitions. However, the truncation of the higher derivative and curvature terms remove contributions that could be reintroduced in the effective potential by field redefinitions. These terms are suppressed by powers of the smallest mass of heavy states which have been integrated out, so, for systems where the KK modes are much heavier than the radion, our effective potential for different interpolating fields should lead to very similar results (at least in some region close to the minimum).
As we have argued in subsection 2.3, to define unambiguously the potential, we should go to the Einstein frame. However, for our choice of the graviton interpolating field (2.32) the corresponding transformations result in corrections (2.42) which are subleading and typically very small so we will neglect them. The only remaining ambiguity comes from possible redefinitions of the radion field χ →χ. To fix this ambiguity, and make all kinetic terms similar, we will express all different effective potentials as function of the same 4D field, independently of the interpolating radion field used. This is equivalent to apply a field redefinition after integrating out the KK modes. One can show that after this field redefition, all kinetic terms are equal up to terms that are suppressed by the mass of the lighest KK mode we have integrated out.
We will analyze the effective potential for the interpolating radion fieldsχ y andχ φ (see subsection 2.4.2). Computing V (y) eff and V (φ) eff we encounter similar technical features and problems as for interpolating fieldχ g discussed in the previous subsection. Although it seems more straightforward to use (2.52), it has the same numerical difficulties as (2.38) for the same reasons: we need to solve the equations of motion with the accuracy of more than 50 significant digits. It is much more economical to use (2.54) and (2.56), formulae that depend only on quantities evaluated at the IR brane, and for which we need to solve the equations of motion with the accuracy of only a few significant digits.
To check the level of agreement between different potentials, we plot in figures 3 and 4 the effective potentials V (y) eff and V (φ) eff together with V eff computed with the radion interpolating fieldχ g . We express all potentials as functions of χ g = e A(y 1 )−A(y 2 ) .
In figure 3 one can see how well the potentials V eff and V (y) eff close to their minima agree. This is a clear indication that the assumption of the radion being very light as compared to other massive KK states is absolutely justified. However, V (φ) eff does not agree with the other two. This is a consequence of a poor choice of the interpolating radion field. We have checked numerically that, in general, the contribution of the zero scalar mode toχ φ is suppressed with respect the scalar KK mode contributions, soχ φ is an "almost orthogonal" direction to the zero mode. Also,χ φ cannot be used to deform the system making χ g arbitrary large or small because the appropriate system of EoMs has no solutions for such values of χ g (this is illustrated in figures 3 and 4 where curves describing V (φ) ef f do not extend over the whole range of the radion field χ g ).
In figure 4 we show the same potentials for two different ranges of χ g . On the left panel one can see that both V eff and V (y) eff agree very well reproducing the small barrier close to χ g = 0, but V (φ) eff fails and ends up at a value χ g > 0. However, differences between V eff and V (y) eff appear for bigger ranges of the values of the radion. We can see them on the right panel of figure 4. We should conclude that these disagreements indicate that this low energy description of the problem based only on the radion and the graviton and the first terms of the expansion of the effective action breaks down. One should consider finer descriptions if such large values of the radion are to be considered.

Approximate methods for the effective potential
In the previous sections we have developed a new machinery to find the effective potential for the radion numerically in an efficient way. In this section we use these new methods to compute the effective potential and use it as a reference to analyze and judge the quality of several approximations that have been used in the literature. In particular, for the   quadratic bulk potential (3.1) we will analyze two approximations: the small back-reaction approximation, widely used in the literature since the Goldberger-Wise mechanism was proposed [7], and the approximation used in [10]. For the exponential bulk potential (3.3) we will analyze the approximation developed in [16].

Small back-reaction
The small back-reaction limit has been extensively used in the literature to deal with Randall-Sundrum type models, where the spacetime geometry is nearly that of a slice of an AdS space and the back-reaction is assumed to be small 11 . This is the approximation used in the first studies of phase transitions which can take place in such models [14,18]. However, it is interesting to check how far we can trust this approximation.
Small back-reaction requires the deviation from the AdS geometry to be small. Pure AdS space is obtained when we neglect the scalar field contribution to the energy momentum tensor and the bulk and brane potentials have the form with take the plus (minus) sign for i = 1 (i = 2). The resulting warp factor is equal to that of the AdS space:, A = k(y − y 1 ) ≡ A (0) . To treat perturbatively the introduction of the bulk scalar field, and the change in the brane potentials, we introduce an expansion parameter r. Pure AdS solution corresponds to r = 0, but at the end of the day, r it will be sent to 1. We do the substitution in the action (2.1), and express the actual potentials as where Notice that Λ i = rδΛ i + U There is a number of ways to calculate the potential in this approximation (see for example [7]). Here we will follow our approach of section 2. Then, the small back-reaction approximation can be formally obtained from our equations expanding in r around r = 0.
In the following subsections we derive the effective potential using bothχ g andχ y as radion interpolating fields and show that the small back-reaction limit of the effective potentials in both cases is the same. As usual, the metric field will be (2.32).

Warped factor as radion interpolating field
To obtain the effective potential forχ g as the interpolating field we need to solve the system (2.6)-(2.9) (with H = 0) with the boundary conditions for A given by (2.36) and (2.37) 11 In some papers back-reaction is even completely ignored (instead of (2.10)). The background solutions may be expanded as The zeroth orders are where and the integration constants y 1 , C − and C + are found imposing the boundary conditions. The warp factor (4.8) gives the geometry of an AdS space. The next order for A can be obtained from (2.53), To this order, the interpolating radion field and the derivative of the potential (2.41) are given by: After imposing the boundary conditions, working in the stiff wall approximation (3.5)-(3.7), and setting r = 1, the potential reads Of course, the same potential is found if we use (2.38) instead of (2.41). The terms of the order O(χ 8 g ) are absolutely negligible for the relevant case of large hierarchy for which χ g 1. Moreover, they disappear if we take the limit M P → ∞ as discussed in appendix C (with Φ (1,0) = v 1 ).
The above effective potential is bounded from below if In such a case, it has a minimum at with the value As Goldberger and Wise showed [7], this minimum sets naturally a large hierarchy between the Planck scale and the TeV scale if 0.1.
From (4.14) one can see that for the quadratic bulk potential, the radion effective potential has the form

Physical distance between branes as radion interpolating field
If we useχ y as the interpolating radion field, we need to solve the system (2.6)-(2.10) (with H = 0) with the bulk constraint (2.8) replaced with (2.53), and the UV boundary condition (2.10) replaced with (2.51). Using the expansion (4.6)-(4.7) and the boundary conditions to order O(r 0 ), we obtain again (4.8) and (4.9). However, the next order of A changes due to the modified UV boundary condition (2.53). Taking into account the new constraint equation (2.53) and the IR boundary condition for A (2.10) (that now is indeed satisfied), we obtain (4.19) Now we can apply any of the formulae, (2.52) or (2.54), to obtain the potential. Taking into account that we recover the same expressions (4.13) and (4.14). Although the intermediate solutions are different, we obtain the same effective potential as when using χ g :

Applicability of the small back-reaction approximation
To obtain the limits of applicability of this approximation, the authors of [18] proposed to compare the leading (in the parameter r) contribution to the energy-momentum tensor of the scalar field T Φ M N (which is O(r 1 )) to the corresponding contribution of the bulk 5D cosmological constant T Λ 5 M N (which is O(r 0 )): For the cases we are considering, 0, the system deviates from the AdS geometry in the IR, so we will compare both tensors in the IR brane. In those cases, the strongest constraint comes from the comparison of the 55-component of the tensor. The corresponding condition T Φ 55 T Λ 5 55 may be written as Of course, the result depends on the value of χ g . Notice in particular that the condition will be always violated if χ g → 0. Moreover, one can trust this approximation only if the condition is satisfied in big enough range around the minimum of the potential. Using the expression (4.13) one can find that the condition for the applicability of the approximation may be written as (no expansion in ∆ − ) Thus, the detuning of the IR brane tension, as compared to the pure AdS case, is a crucial factor to determine if the small back-reaction approximation is applicable.

Comparison with exact potential
In this subsection, we will compare the exact effective potential and its small back-reaction approximation. The quality of the approximation (in this subsection by approximation we always mean the small back-reaction approximation) will be judged by the comparision with numerical calculations of the exact effective potential obtained with the methods explained in the previous sections. In the rest of this section we will useχ g as the radion interpolating field for all calculations. Figure 5 we plot χ g,min (left panel) and (V eff (0) − V eff (χ g,min ))/χ 4 g,min (right panel) calculated exactly numerically and obtained using the approximation. Since there are many independent parameters, we have choose a benchmark model depending only on one parameter as an representative example. We work in the stiff wall approximation and fix k = 1, κ = 0.5, = 0.01. The values of Λ 2 and v 1 are chosen as the following functions of v 2 (the form of the above relations among parametrs will be justified in the next subsection) while v 2 is scanned in the range 0 ÷ 2κ −1 . Thus, when v 2 = 0, δΛ 2 = 0 and we have the exact AdS solution. The bigger |δΛ 2 | and v 2 are, the larger is the back-reaction.
On the left panel of figure 5, one can see how the estimation of χ g,min fails as soon as the system slightly departs from pure AdS. This makes the prediction for the position of the minimum, obtained using the approximation, trustable only when κ 2 |δΛ 2 |/6k 1. This agrees with our general condition (4.25). However, the right panel of figure 5 shows that the approximation works much better in the calculation of (V eff (0) − V eff (χ g,min ))/χ 4 g,min : it gives a good estimation of the correct result for κ 2 |Λ 2 |/6k < 1.5 and v 2 < κ −1 . From the form of the effective potential (4.18) it follows that Then, figure 5 suggests that the approximation gives good estimation of the value of χ∂ χ F in much larger range of the parameter space than it does in the case of F (χ g ) itself (which affects directly the value of χ g,min ).
Notice that the exact value of χ g,min is relatively easy to compute numerically: one has only to solve the equation of motion (2.6)-(2.7) with the correct boundary conditions (2.9)-(2.10) and evaluate χ g . This motivates a way to improve the small back-reaction approximation in a hybrid pseudo-analytic way. The prefactor F (χ g ) in the effective potential (4.18) may be shifted by a constant in such a way that χ g,min computed in the approximation is moved to match the numerically computed value. This can be done making in (4.14) the substitution As an example, in figure 6, we compare the small back-reaction approximations and its modification proposed above with the exact effective potential for a particular choice of the parameters. We have chosen one of the bechmark models of (4.26)-(4.27) with κ 2 |Λ 2 |/6k ∼ 1.06 to be in the region where the back-reaction approximation does not give a correct estimation of the full potential, but the hybrid method does. On the right panel of figure 6 one can check that the small back-reaction approximation in this region of parameters indeed estimates correctly χ g ∂ χg F (χ g ) in (4.18), but leads to a shift in F (χ g ).

Asymptotic matching
If the parameter in the quadratic potential (3.1) is small one can use the method of asymptotic matching to obtain the solution to the equation of motion beyond the small back-reaction approximation. This method, used in [10], consists in finding analytical solutions to the equations of motion asymptotically in the deep UV and IR regions. Then, a consistency condition at some intermediate region is used to match the integration constants of both asymptotic solutions. As a result one obtains a full approximate solution. In the  . The blue line gives the exact potential V eff (χ g ) − V eff (0) calculated numerically. The dashed red line (indistinguishable from the blue one) is the hybrid approximation: (4.14) with the substitution (4.29) after computing numerically χ g,min . On the left we plot the effective potential using linear scales for the axes. On the right, we plot F (χ g ) = (V eff (χ g ) − V eff (0))/χ 4 g using a logarithmic scale for the horizontal axis.
stiff wall limit, such approximate background solution is given by .

(4.31)
These analytic expressions allow to obtain the IR part (2.40) of the effective potential (2.38) and the result reads This expression mixes to different definition of the radion: χ g related to the chosen interpolating field andχ y = e −k(y 2 −y 1 ) . The latter should be expressed in terms of χ g by inverting the relation (4.33) Such IR potential vanishes, and therefore, minimizes the full effective potential, according to (2.41), atχ The above approximate solution, obtained using the method proposed in [10], is not precise enough to calculate the V UV eff with accuracy sufficient to get a good approximation of the full effective potential (2.38) (as we argued before, many significant digits are required in the calculation of V UV eff ). Below we propose two improvements of this asymptotic matching method. First, instead the usual definition of the effective potential (2.38), one may apply our proposal to calculate the derivative of the effective potential solely from the IR part. Thus, using (2.41), we may integrate (4.32) to obtain the full effective potential. Unfortunately, this integration cannot be done analytically. However, for small , we see that V IR eff in (4.32) has a form similar to that of V eff in (4.18): where, similarly as in the case of F (χ g ), derivatives of F IR are small: . Using (2.41), we see that the functions F (χ g ) (for V eff ) and F IR (χ g ) (for V IR eff ) are related by This differential equation can be solved perturvatively in giving From (4.32) we obtain so the full effective potential may be approximated as where againχ y should be replaced with χ g with the help of (4.33). The total effective potential V eff differes from V IR eff only by terms of order O( ). However, such terms are crucial to correctly locate the minimum of the potential (see figure 2). The inclusion of these terms in the approximate expression of the potential is a new result of this article. The depth of the potential is (4.42) Notice that from (4.40) one can see the main difference between the effective potential for small back-reaction and large back-reaction models. In the first ones, v i ≈ 0, so χ∂ χ F (χ) is not only suppressed by small but also by small v i . The function F (χ) stays small along a big range of χ including the region χ = O(1). This is a consequence of some fine-tuning of the parameters of the theory: the small back-reaction limit requires fine-tuning of the IR brane tension expressed in (4.25). For large back-reaction cases, F (χ) is not suppressed (1)). Its slow logarithmic change with χ suppressed by generates the large hierarchy (i.e. χ g,min 1), in the line of the so called Pomarol-Rattazzi-Contino mechanism [10,35].
It follows from (4.32) that, in order to have a bounded from below effective potential, the parameters have to satisfy the condition Otherwise, V IR eff , and therefore, dV eff /dχ g , would be negative when χ g → ∞. This bound justifies our choice (4.26) of the benchmark model in subsection 4.1.4. In (4.26) we used the r.h.s. of (4.43) changing v 2 → (3/4)v 2 and v 1 → 0 in order not to saturate the bound but to stay close to it and make |Λ 2 | large. The choice of (4.27) comes from imposing k(y 2 − y 1 ) = 30 in (4.35). Now we use the same benchmark model defined by (4.26) and (4.27) to judge the asymptotic matching approximation, with and without our improvement. In figure 7 we plot the value of χ g,min (left panel) and (V eff (0)−V eff (χ g,min ))/χ 4 g,min (right panel) calculated exactly with the methods explained in the previous sections and within this approximation. On the left panel one can see that the equation (4.32) for computing χ g,min gives inaccurate results, but correct in the order of magnitude for the whole range we have explored (which includes very large deviation from AdS geometry when |Λ 2 |κ 2 /6k > 2). On the right panel of figure 7 we however see that the value obtained with the approximation for (V eff (0) − V eff (χ g,min ))/χ 4 g,min matches very well with the exact result. As in the previous subsection, this suggests that also this approximation reproduces very well the quantity χ g ∂ χg F (χ g ), but not the absolute value of F (χ g ), which is good enough just to reproduce the order of magnitude of χ g,min . Therefore, we can improve these results with our new hybrid analytic-numeric method, computing numerically the value ofχ y,min before applying the approximate formulae for the effective potential. Then we apply a shift in F (χ g ) to match the value ofχ y,min . This is equivalent to replacing the IR brane tension in (4.41) according to In figure 8 we plot the exact effective potential and several approximations for a particular set of parameters. We show that the application of (4.41) gives results of the same order of magnitude, but inaccurate. However, the application of the hybrid method reproduces very well the exact result. On the right panel of figure 8 we also check that the non-improved approximation indeed estimates correctly χ g ∂ χg F (χ g ) (4.40), but produces a shifted F (χ g ).

Superpotential techniques
For the exponential potential (3.3), for which we know analytically a superpotential, [16] offers an alternative way to compute the effective potential for the radion ifĥ µν andχ g , defined in (2.32) and (2.33), are used as intepolating fields for graviton and radion, respectively. The superpotential W is related to the bulk scalar potential V by the following differential equation Then, functions φ(y) and A(y) with derivatives given by On the left we plot the effective potential using linear scales for the axes. On the right, we plot F (χ g ) = (V eff (χ g ) − V eff (0))/χ 4 g using a logarithmic scale for the horizontal axis.
In order to get the radion effective potential one has to know bulk solutions with some of boundary conditions violated. Thus, in the case of superpotential method one has to know the whole, depending on one integration constant, family of superpotentials solving differential equation (4.45). In general it is impossible to find analytically all such solutions. Instead, it was proposed in [16] to expand the unknown superpotentials in some (small) parameter s and to solve the equation (4.45) perturbatively using this expansion around the one known analytic solution: Then, the corresponding solutions of (4.46) may also be expanded in s: φ(y) = φ 0 + sφ 1 (y) + s 2 φ 2 (y) + . . . , (4.50) Thus, one obtaines a system of iterative differential equations for W 1 , W 2 , φ 1 , φ 2 , etc. The explicit formulae can be found in [16]. 12 Working in the stiff wall limit (3.5)-(3.7) (this limit is not necessary for this method, but simplifies the following discussion), after breaking the boundary conditions (4.48), one finds the following expression for the effective potential Notice that the value of s does not independ on Λ 2 . Therefore, Λ 2 enters the calculation only through its explicit term of the effective potential formula (4.52). We have used here (2.38) for the calculation because we have semi-analytic expressions. Thus, we avoid the integration in χ g which is necessary when using (2.41).
Let us now discuss this method in more detail using the specific bulk potential (3.3) proposed in [16]. This form of the potential follows from the analytic superpotential, which can be chosen to be W 0 : (4.55) 12 In every equation for Wn, a new integration constant appears. They can be fixed arbitrarily because a change of them is translated into a reparametrization of the parameter s. We will use the convention Wn(v1) = 0 for n > 1.
For this superpotential and the IR boundary condition (4.54), the solutions φ 0 and A 0 to (4.46) are Knowing the exact form of W (φ), i.e. knowing all orders in s (if the convergence radius of the expansion is big enough), one would get exact solutions for φ(y) and A(y), (4.50) and (4.51). However, in practice this expansion is truncated at some (rather low) order. Therefore, only the solution when s = 0 gives an exact solution. This solution corresponds to a specific value of the interpolating radion field, which we denote by χ g,0 = exp(A(y 1 ) − A(y 2 )), and in general it does not correspond to the minimum of the effective potential. At this point (χ g,0 ), the method reproduces the exact value of the potential. It is characterized by Assuming that s is a regular function of χ g : we see that the truncation in (4.49) to order O(s n ) eliminates terms O(χ g − χ g,0 ) n+1 in (4.52). Thus, the truncated version of (4.52) will reproduce the effective potential in the neighborhood of χ g,0 only up to order O(χ g − χ g,0 ) n .
For physical applications, one should be able to reproduce the effective potential to a good accuracy at least in the neighborhood of its minimum. But, as we argued above, the superpotential method gives good accuracy for χ g close to χ g,0 and not necessarily close to χ g,min . Notice that χ g,0 does not depend on Λ 2 , while χ g,min does. Therefore, the condition for χ g,min to coincide with χ g,0 can be achieved by tuning the IR brane tension to be To compare results of exact calculation with those obtained using the lowest orders of expansion in s in the superpotential method, we take now a particular case where the condition (4.61) is satisfied. In the left panel of figure 9 we plot the exact effective potential and the one obtained with the approximation up to order 1, 2 and 3 in the expansion in s. We can see that indeed the approximations reproduce the exact effective potential in a region around χ g,0 = χ g,min . We also see that including higher orders does not improve substantially the approximation: the second order destabilizes the potential and the third order slightly improves the first one close to χ g,0 = χ g,min , but then it diverges faster. On the right panel of figure 9 we plot F (χ g ) ≡ (V eff (χ g ) − V eff (0))/χ 4 g . One can see that higher orders in s worsen the approximation of F (χ g ) for χ g > χ g,0 and do not improve it for χ g < χ g,0 (except a small region close to χ g,0 ).
A remarkable consequence of the choice of parameters (4.61) is that V eff (χ g,min ) − V eff (0) = 0. Systems where V eff (χ g,min ) = V eff (0) seems to be not interesting for constructing viable models because they cannot trigger a phase transition to a confined phase (χ g,min = 0). To make V eff (χ g,min ) < V eff (0), one has to decrease IR brane tension (see (4.52)), Λ 2 < Λ * 2 . If we leave the remaining parameters unchanged, this just corresponds to a vertical shift in F (χ g ) (see the right panel of figure 9). For the potential itself, it will push χ g,min to larger values χ g,min > χ g,0 (while χ g,0 is unchanged). As may be seen in the right panel of figure 9, larger values of χ g (> χ g,0 ) lead soon to a poor estimation of F (χ g ), which implies a poor estimation of the effective potential in a neighborhood of the minimum (but it will still describe correctly a neighborhood of χ g,0 ). In figure 10 we plot the exact and approximate effective potential for a Λ 2 < Λ * 2 . As expected, the superpotential method approximation gives a poor estimation of the potential in the region of the minimum. To check how much the approximation disagrees with the numerically computed exact result as we decrease Λ 2 , in figure 11 we plot χ g,min and V eff (0)−V eff (χ g,min ) as functions of Λ 2 (< Λ * 2 ) with the remaining parameters fixed as in the previous figures of this subsection. We keep only the first order in s because, as discussed before, higher orders in s do not improve the approximation away from χ g,0 . Figure 11 shows that the results of superpotential approximate method may differ from the exact ones by, sometimes many,  10 -52 Figure 11. Superpotential approximation. The choice of parameters is κ = 0.5, k = 1, γ = 0.1, v 1 = −15 and v 2 = −3.3. For these ones, κ 2 Λ * 2 /6k = −1.719. On the left, we plot the value of χ g,min as function of Λ 2 < Λ * 2 . On the right, we plot the difference between the minimum of the potential and the origin, V eff (0) − V eff (χ g,min ) as function of Λ 2 < Λ * 2 . In both cases, the blue line is the exact calculation and the black line the approximate calculation keeping only orders O(s). orders of magnitude.

Conclusions
There is a continuous theoretical and experimental interest in the warped extra-dimensional models as the framework for an extension of the Standard Model. An important ingredient of such models is the Goldberger-Wise mechanism for stabilizing the distance between the UV and IR branes. The models can address the hierarchy problem, can serve as perturbative tools to study strongly interacting scale invariant structures and may have important collider and cosmological implications. A very useful concept for investigating the low energy four-dimensional effective theory describing the gravitational and the scalar sector in the limit of heavy (decoupled) Kaluza-Klein modes is the radion effective potential.
In this technical paper, we have reviewed the concept of the radion effective potential and its subtleties. First, we have given a precise definition of the radion effective potential as a construct obtained by integrating out heavy degrees of freedom in the path integral for the 5D theory and discussed its dependence on the interpolating field for the radion. Next, using the Lagrange multiplier method we have found a new way of calculating the exact effective potential for a given interpolating field. The functional derivative of the effective potential with respect to the radion field is exactly given in terms of the values of the metric and the scalar field only on the IR brane. One avoids numerically very difficult procedure of solving equations of motion with extremely high precision, to control a delicate cancellation between the contributions from the IR and UV branes in the standard approach. Often, this problem is bypassed by using some approximate methods whose quality is not easy to quantify. Having numerically easy method to calculate the exact effective potential, we have investigated its dependence on the definition of the radion interpolating field for the chosen parameters of the 5D theory. A weak dependence is an a posteriori useful check of both, the choice of the interpolating fields and the underlying assumption about the mass gap between the zero and KK modes, underlying the effective field theory approach.
In the second part of the paper we have used our exact results for the radion effective potential calculated for various choices of the 5D parameters, to judge the quality of the approximate methods used in the literature. Several new observations have been made concerning the dependence of the quality of those approximate methods on the parameter ranges of the 5D theory and as the function of the background values of the radion field in V eff (χ).
In this paper we have not included the canonical normalization of the radion field, which requires an independent calculation of the next term in the effective action. This is of course necessary to get the final shape of the effective potential but even non-canonically normalized it is physically interesting as it gives the correct pattern of extrema. Thus, for instance, one can discuss the character of the radion phase transition once the temperature effects are included. Those topics will be discussed in the forthcoming publication [17].
The tools developed in this paper can be extended in a number of ways. An effective action for more fields besides the radion and graviton can be defined. The additional fields could be the Higgs boson, to study models with Higgs-radion mixing, or even higher scalar KK modes. The last extension would open the possibility to analyze the goodness of the approximation that considers the radion as the only relevant field for the phase transition mentioned above. The improvement of considering extra degrees of freedom can be compared with the one-field effective theory and quantified. Also, these methods can be applied to a broader class of models that have not been considered here. For example models that replace the IR brane with a naked singularity which have been also studied because of their interesting phenomenological features [5,9,36]. let us consider an infinitesimal diffeomorphism generated by any vector field ε µ , δ ε x µ = ε µ . With δ X we denote the infitesimal variation of the object X under the diffeomorphism generated by ε. Because of the scalar and tensor character of the effective Lagrangian and the metric, respectively, we have where the connection ∇ (h) is the Levi-Civita connection of h µν . Then, the infinitesimal transformation of the effective Lagrangian satisfies Also, this infinitesimal transformation can be expanded using (2.20) and performing a variation of the 5D Lagrangian under the infinitesimal diffeomorphism generated byε M , withε µ = ε µ andε 5 = 0, Here, g sol , Φ sol and A sol are the 5D solutions given χ and h µν , and R µν [g (4) ] is the Ricci tensor of the metric g (4) µν . In the first line we express the variation of the Lagrangian as the equation of motion plus a total derivative term and we use δεΦ sol = 0. It is easy to see that for the ansatz (2.2) and the vectorε M , the tensor inside the total derivative has nonvanishing components when some index is 5 only if M = 5. Then, the contribution of these components vanishes under the integral in y. All the indices can therefore be substituted by 4D indices. In the second line we use (2.17), we express the infinitesimal transformation of χ and h using (2.12) and (2.13), and we work out the total derivative term in 4D. In the third line we use that χ is constant so δ ε χ = ε µ ∂ µ χ = 0, the transformation of the metric (B.2), and R µν [g (4) ] = 3H 2 g (4) µν . Taking the r.h.s. of (B.4) and the third line of (B.5) we finally obtain where in the second line we have used (2.23). Combining (2.30) and (B.6), in the H → 0 limit, we can extract the function K eff (χ) as given in (2.31).
C Effective potential in the 4D M P → ∞ limit Under the AdS/CFT duality [11], the models we investigate describe a strongly coupled conformal field theory that is also coupled at the Planck scale with dynamical gravity and a weakly couple sector (also called elementary sector [37]). The UV brane is responsible of this interaction and explicitly breaks the conformal symmetry through irrelevant terms [12,13,38]. The radion is holographically dual to a dilaton, and its existence points out that the conformal symmetry is spontaneously broken around the IR scale of the model. The IR brane is actually the origin of such breaking, and it is hologrophically related to an operator of the CFT of arbitrary high dimension developing a vacuum expectation value [12,13]. If the distance between the branes is large, and so the hierarchy between the Planck scale and IR scale, the model will behave in a approximate conformal regime below the Planck scale (explicit breaking) and above the IR scale (spontaneous breaking). The naturality of the apparition of a light dilaton without fine-tuning in explicitly broken CFTs is discussed in [10,35] and it can be achieved by the so-called Contino-Pomarol-Rattazzi mechanism. The effective potential of the radion receives contribution, not only from the strongly coupled sector, but also from the elementary sector. If the hierarchy between the scales is large, the elementary sector and gravity will decouple from the CFT. As we will see, in the limit in which the 4D Planck scale is sent to infinity, the contribution of the elementary sector will be reduced exclusively to a deformation of the CFT through a unique operator, the dual operator to the bulk scalar field.
Sending the effective 4D Planck mass to infinity is equivalent to eliminating the UV brane by sending it to y → −∞. The resulting dimensional-reduced 4D theory will not contain dynamical gravity, so it can live in a Minkowski space without requiring the finetuning necessary to adjust the 4D cosmological constant. To calculate the radion effective potential, let us look for solutions of the form (2.2)-(2.3) with g (4) µν = η µν . The holographic coordinate y will run from the IR brane at y 2 to −∞. The EoMs are (2.6)-(2.8) with H = 0. Let us assume the bulk potential has an extremum, that, without lost of generality is located at Φ = 0 where the potential take a negative value, where ∆ ± = 2 ± 2 √ 1 − . 13 All the coefficients in this expansion are determined by the EoMs if we fix Φ (1,0) and Φ (0,1) . The coefficients Φ (n,m) and A (n,m) depend on Φ (1,0) and Φ (0,1) only if n > 0 and m > 0, respectively. Also, if Φ (1,0) = 0 or Φ (0,1) = 0, all coefficients Φ (n,m) and A (n,m) with n > 0 or m > 0, respectively, will vanish. This expansion only make sense if > 0, so ∆ ± > 0. In that case, Φ → 0 and A(y) ∼ ky when y → −∞. The space-time will be asymptotically AdS (AAdS) with the conformal boundary at y → −∞. If < 0, then ∆ − < 0 and ∆ + > 0, and only solutions with Φ (1,0) = 0 will behave like that.
Following the AdS/CFT dictionary, these models are holographically dual to a CFT deformed by Φ (1,0) O, where O is a relevant (irrelevant) operator if > 0 ( < 0) with dimension ∆ + [39,40]. The coefficient Φ (0,1) is related with the vacuum expectation value of O [40]. 14 We will only consider models with > 0. They have a well controlled y → −∞ limit. Their dual CFTs are then deformed by a relevant operator and have a well-behaved UV limit.
The counting of the degrees of freedom for the background solution is as follows. After imposing the bulk equations (2.6)-(2.8), the solution depends on three integration constants: Φ (1,0) , Φ (0,1) and A 0 in (C.2) and (C.3). Also, the position of the IR brane, y 2 , will be an extra parameter to fix. The constant A 0 can be set to zero using the shift symmetry of the EoMs, that can be seen as a choice of units. In the IR brane we have two boundary conditions, (2.9) and (2.10). These leave only one degree of freedom, and relate Φ (1,0) , Φ (0,1) and y 2 . If we fix y 2 , we can read Φ (1,0) and Φ (0,1) in the UV. Notice that a shift in y 2 → y 2 + a would affect the other coefficients by Φ (1,0) → e −∆ − ka Φ (1,0) , Φ (0,1) → e −∆ + ka Φ (0,1) .
(C.6) Therefore, unless there is some symmetry or fine-tuning, we would not expect a vanishing Φ (1,0) for any finite y 2 : the dual CFT is necessary deformed by the dual operator O. The model does not have UV brane, so no UV boundary conditions (2.9) and (2.10) have to be imposed. Instead, for a scalar field in an AAdS space, we must impose one additional 13 For 4D non-homogeneous backgrounds, every term has to be substituted by the next tower of terms Φ (n,m) → Φ Additionally, we are assuming that every term in the sum has a different exponent. Otherwise, polynomials of y may also appear in front of the exponential terms (for instance, if n∆− + m∆+ = n ∆− + m ∆+ for two different pairs of (n, m) = (n , m )). 14 In general, it is possible to define two different quantum theories for a scalar living in an AAdS space, the standard and alternate quantizations [41]. The one we use here is the standard one, and it exists for all values of ≤ 1. In the alternate one, the dual theory and the dictionary change. The role of Φ (1,0) and Φ (0,1) is exchanged, so Φ (0,1) is related to the source and Φ (1,0) to the expectation value. The dimension of the dual operator is now ∆−. The alternate quantization however only exists for values 3/4 < < 1. Values of > 1 are not allowed for a field in an AAdS space due to the Breitenlohner-Freedman bound [30,41].
boundary condition associated to the asymptotic behavior of the field in the UV [42]. We take the value of Φ (1,0) as boundary condition. This is a very natural choice because it defines the source of the deformation of the dual CFT. Because we have changed a model with UV brane and two UV boundary conditions to a model without UV brane and one UV boundary condition, no fine-tuning is necessary to obtain solutions with flat 4D sections as we anticipated.
To compute the radion effective potential for this model, let us specify the interpolating fields we use:ĥ The effective potential is given by the minus on shell action of homogeneous and flat configurations. However, it is well known that on shell actions in AAdS space are divergent due to the infinite volume of the AdS space. These divergences are actually related holographically to the UV divergences of the dual quantum field theory [43]. It is necessary a renormalization procedure to remove them and to obtain a finite renormalized action S ren . There are several approaches for this under the name of holographic renormalization [44][45][46]. Here, we will follow the most standard one [47]. It consists in: (1) computing the regularized on shell action S reg integrating until some finite y 1 , (2) adding a suitable counterterm action S c.t localized at y 1 , and then (3)  Here, g and Φ are the 5D solutions to the EoMs (breaking the IR boundary condition (2.10)) andg(y 1 ) is the metric in the hypersurface y = y 1 inherited from g. No new UV boundary conditions that affect the on shell fields g and Φ have to be consider: g and Φ are kept unaltered during this procedure and they are the solution when the value Φ (1,0) is fixed in the y 1 → −∞ limit. The effective potential is Inserting the asymptotic expansions (C.2) and (C.3) in the second line of (C.10), we can extract the required form for S c.t. to cancel the divergences: For non-homogeneous configurations or more general metric backgrounds, additional terms have to be introduced [48]. In the y 1 → −∞ limit we obtain The effective potential is therefore written as a contribution from the fields evaluated at the IR brane plus a non-vanishing contribution given by the asymptotic behavior of the fields in the UV.