Soft-Wall Stabilization

We propose a general class of five-dimensional soft-wall models with AdS metric near the ultraviolet brane and four-dimensional Poincar\'e invariance, where the infrared scale is determined dynamically. A large UV/IR hierarchy can be generated without any fine-tuning, thus solving the electroweak/Planck scale hierarchy problem. Generically, the spectrum of fluctuations is discrete with a level spacing (mass gap) provided by the inverse length of the wall, similar to RS1 models with Standard Model fields propagating in the bulk. Moreover two particularly interesting cases arise. They can describe: (a) a theory with a continuous spectrum above the mass gap which can model unparticles corresponding to operators of a CFT where the conformal symmetry is broken by a mass gap, and; (b) a theory with a discrete spectrum provided by linear Regge trajectories as in AdS/QCD models.


Introduction
Warped extra dimensions were introduced by Randall and Sundrum [1] (RS1) as a very elegant way of solving the hierarchy problem by means of the geometry of extra-dimensional theories. The original proposal consisted of a slice of AdS space bounded by two branes: the ultraviolet (UV) brane, located closely to the AdS boundary, and the infrared (IR) one. One of the most exciting aspects of RS1 theories is provided by the AdS/CFT correspondence [2], by which fields living in the UV brane are fundamental fields which can interact with a strongly coupled conformal sector (CFT), while fields living in the IR brane are the dual description of operators of the CFT. In this way a Higgs boson localized on the IR brane provides a dual description of the composite Higgs as a bound state of an extra strong interaction (technicolor). From the five dimensional (5D) point of view the Higgs boson mass is simply redshifted from its natural value, of the order of the Planck mass, to the TeV scale by the warp factor. The IR brane spontaneously breaks the conformal symmetry and its location needs to be stabilized by some dynamical mechanism.
A dynamical mechanism to stabilize the IR brane was proposed by Goldberger and Wise [3] (GW) by the introduction of a background scalar field propagating in the 5D bulk and acquiring a coordinate dependent vacuum expectation value (VEV). This triggered a 4D effective potential for the radion field with a minimum, stabilizing the brane separation at scales of order 1/TeV, determined by the values of the scalar field at both the UV and IR branes, with a modest ∼ 1% finetuning. Furthermore the back-reaction of the scalar field through the 5D Einstein equations generated a deviation from AdS of the metric far away from the UV brane, still preserving the main features of the AdS/CFT correspondence. In fact this is a general feature since AdS is the only 5D metric consistent with no (or constant) background scalar field, which means that any stabilizing bulk field is expected to back-react on the AdS metric.
Since the presence of bulk scalar fields, required to stabilize the brane distance, is expected to disturb the AdS metric in the far IR, one can introduce phenomenological models with background scalar fields and with the only constraint of describing the AdS geometry near the UV brane. One of them is of course RS1 with GW mechanism, but there are more general models 1 where the IR brane is replaced by a naked curvature singularity [5][6][7][8]: these models are called soft-wall models [9][10][11][12][13][14]. Soft-wall models possess AdS geometry near the UV brane and no IR brane, i.e. they have a non-compact extra dimension as those proposed by Randall and Sundrum with an infinite extra dimensional length [15] (RS2), but with additional background scalar fields. These fields back-react on the metric and generate a singularity at a finite value of the extra dimensions: the (finite) length of the extra dimension provides the location of the soft wall.
Soft wall models were first introduced to model the Regge behaviour of excited mesons in AdS/QCD models [9,10], as an alternative to RS1 for electroweak breaking models solving the hierarchy problem and providing experimental signatures at LHC [12,13] and also to provide a 5D setup to the theories of unparticles recently proposed by Georgi [16] in the presence of a mass gap [14,17]. In all these theories, as in GW models, there are background bulk scalar fields and thus a built-in mechanism to stabilize the extra dimension, i.e. the distance between the UV brane and the singularity. However in the proposed models this distance is naturally of the order of the AdS length and the hierarchy problem does not find a natural solution, i.e. for values of the scalar field at the UV brane of the order of the 5D Planck scale [13].
In this paper [18] we will propose a set of one-parameter (ν) 5D models with a background scalar field propagating in the bulk of the extra dimension and with the following properties: • The metric is AdS near the UV brane.
• The hierarchy between the AdS length and the soft-wall length (the electroweak length) is naturally stabilized for values of the scalar field at the UV brane of the order of the 5D Planck scale. The naturalness of this result comes from a double exponential suppression.
• For values of ν < 2 the naked singularity does not contribute to the vacuum energy and thus it does not need to be resolved to satisfy Einstein equations. Moreover the potential is bounded from above in the solution and the singularity is a physical one according to Ref. [7].
• For 1 < ν < 2 the spectrum of fluctuations behave as in RS1 models with Standard Model (SM) fields propagating in the bulk: for scalar field fluctuations localized near the singularity the hierarchy problem is automatically solved.
• For ν = 1 one can model unparticles with a mass gap provided by the inverse length of the extra dimension. Fluctuations have a continuous spectrum above the mass gap.
• For 0 < ν < 1 unparticles without a mass gap emerge: the spectrum is continuous above zero mass.
The paper is organized as follows: In Section 2 we review the general construction of backgrounds and the associated question of self vs. fine-tuning of the cosmological constant. The general conditions for models with physical singularities are summarized. In Section 3 we consider the background solutions for a particularly simple class of models satisfying all physical requirements on the singularity and describing the electroweak/Planck hierarchy without fine-tuned parameters. Fluctuations on the background for the graviton, radion and the scalar field are studied in Section 4 where a complete numerical analysis and some analytical approximations are provided. A general class of models having all the good required properties, included the hierarchy determination, is presented in Section 5. One of these models has the mass of excitations as m 2 n ∼ n and it is thus a good candidate to model the Regge behaviour in AdS/QCD models. Finally our conclusions and outlook are drawn in Section 6.

The 5D Scalar Gravity System
In this section we would like to review the construction of backgrounds with 4D Poincare invariance proposed in Ref. [19,20], and the associated question of self vs. fine-tuning of the cosmological constant (CC) [5][6][7][8]21,22]. Let us consider 5D gravity with a scalar field, and look for the most general solutions to this system that preserve 4D Poincare invariance, i.e. a background of the form with a "mostly plus" flat Mikowskian metric η µν = diag(−, +, +, +, +) and an arbitrary warp factor A(y). We will introduce a single brane sitting at y = 0 and impose the orbifold Z 2 symmetry y → −y under which A and φ are even. We will thus consider the bulk plus brane action We have introduced arbitrary bulk and brane potentials V and λ, and M denotes the 5D Planck mass which we will set to unity for the remainder of the paper. Note the noncanonical form of the kinetic term for the scalar that will simplify future formulas. The bulk equations of motion (EOM) that follow from the action in Eq. (2.2) are (2.5) This system has three integration constants 2 . One of them is A(0) that remains totally free. The other two can be fixed from the boundary conditions (BC) that follow from the boundary pieces of the EOM, where φ 0 = φ(0). Using Eqs. (2.6) and (2.7) in Eq. (2.5) determines φ 0 , which could be used to replace (2.7). The authors of Ref. [20] introduced the following "trick" to obtain solutions to this system by defining the so-called "superpotential" via the differential equation and writing while the boundary conditions are satisfied if Again, the system of Eqs. (2.9)-(2.11) has three integration constants and in principle every solution to Eqs. (2.3)-(2.5) can be constructed in this way [20]. One integration constant is the trivial additive constant A(0) that does not enter in Eq. (2.12). We are left with the integration constant in Eq. (2.9) and the value φ 0 to fix the two constraints Eq. (2.12). The equation for W is a complicated nonlinear differential equation, and in practice it is often easier to start with a particular superpotential satisfying the boundary conditions and deduce the potential needed to reproduce it. A peculiarity of the scalar-gravity system with one brane is the appearance of naked curvature singularities at finite proper distance. In particular, it can easily be checked that if the superpotential W grows faster than φ 2 at large φ, the profile φ(y) diverges at finite value of y ≡ y s . Moreover the 5D curvature scalar along the fifth dimension can be written as so that the curvature in general diverges at y = y s . The interpretation is that spacetime ends at y s .
Having three integration constants but only two constraints, it seems that one can obtain flat 4D solutions with fairly generic brane and bulk potentials without fine-tuning. This miraculous self-tuning property of the scalar-gravity system was first pointed out in [5,6] and it was further scrutinized in several papers [7,8,21,22]. In particular, the authors of Refs. [21,22] pointed out that the on-shell Lagrangian, integrated over the fifth dimension can be written as 14) (we have set A(0) = 0). They then make particular choices for V and λ and show that the result is non-vanishing. The interpretation of this apparent contradiction to the existence of a flat background is simple: having dynamically generated a new boundary at the singularity, we must ensure that the boundary pieces of the equations of motion vanish at y = y s . If this is not the case, the resulting "solution" does in fact not extremize the action, resulting in a nonzero 4D CC. This can actually be seen in rather general terms. Making use of the equations for the superpotential, we write leading to (2.16) The first two terms cancel if Eq. (2.12) is satisfied, while the last one depends on the particular form of the superpotential. In order for the last term to vanish, W needs to grow more slowly than e 2φ at large φ: this can be seen by using the field φ itself as a coordinate. The position of the singularity moves to φ(y s ) = ∞ and the equation for A(φ) becomes A ′ = W/W ′ . It follows that the last term in Eq. (2.16) goes to a constant for W ∼ e 2φ , while it goes to infinity (zero) when W grows faster (slower) than e 2φ . We thus arrive at a simple criterion for the existence of singular solutions: A singularity with φ(y s ) → ∞ is allowed if and only if W (φ) grows asymptotically more slowly than e 2φ . (2.17) Notice that a potential (V ) growing more slowly than e 4φ is only necessary but not sufficient for the validity of (2.17), the trivial counterexample being V ≡ 0 which has the general solution W = c e 2φ . It is instructive to compare our criterion with the one found in Ref. [7] where AdS-CFT duality was used to classify physical singularities. According to Ref. [7] admissible singularities are those whose potential is bounded above in the solution. Inspection of Eq. (2.9) shows that singularities fulfilling (2.17) have a potential that goes to −∞, while those that fail (2.17) go to +∞. Although we here employ a much more basic condition (a consistent solution to the Einstein equations), which in particular can be applied to theories without any field theory dual, it is good to know that our allowed solutions have potentially consistent interpretations as 4D gauge theories at finite temperature. However achieving a superpotential that grows more slowly than e 2φ requires a hidden fine-tuning of the cosmological constant. To see this it suffices to consider a potential that behaves asymptotically as we can express the solutions for w as the roots of where c is an integration constant. For ν > 2 this implies that w asymptotes to a constant at large φ and for b > 0. However for 0 < ν < 2, w generically behaves as Only if we adjust c → ∞ we can achieve that w behaves as in Eq. (2.21). In this case, b has to be negative in order to have a real solution for W . The generic solution to Eq. (2.9) thus grows as W ∼ e νφ for ν ≥ 2 and W ∼ e 2φ for ν ≤ 2. However, it is possible to arrange for W ∼ e νφ in the latter case by picking a particular value for the integration constant in Eq. (2.9) 3 . There are thus two possible scenarios.
• The superpotential W grows as e 2φ or faster and the equation of motion are not satisfied at the singularity. The only consistent way out is to resolve the singularity, for instance by introducing a second brane located at y s . In that case fine-tuning of the CC is restored as we introduce two more conditions analogous to Eqs. (2.6) and (2.7) or Eq. (2.12) respectively, but do not increase the number of free parameters [21].
• The superpotential grows as e νφ with ν < 2, or slower. The equations of motion are satisfied at the singularity, and there is no need to resolve it. The price one pays is the adjustment of the integration constant in Eq. (2.9). In this case we lose one of our parameters needed to satisfy the boundary condition Eq. (2.12), resulting in a fine-tuning of the brane tension.
It is important to realize that either fine-tuning precisely corresponds to the finetuning of the CC. In the second possibility above this is particularly obvious: the superpotential is completely specified by the bulk potential and the boundary condition at φ → ∞. Eq. (2.12) is then simply the minimization of the 4D potential under the condition that V 4 (φ) vanishes at the minimum φ = φ 0 . In fact the brane potential λ(φ) should be determined by physics localized at the UV brane interacting with the (dilaton) field φ. For example if the Standard Model Higgs field H is localized at the UV brane it will generate a brane potential as λ(φ, H) which will in turn provide the effective brane potential λ(φ) ≡ λ(φ, H ) after electroweak symmetry breaking. So after the electroweak phase transition 4 there will be a φ-dependent vacuum energy which will require re-tuning the cosmological constant to zero and possibly a shift in the minimum of Eq. (2.23).
What matters to us here is that there exist consistent solutions to the equations of motion in the full closed interval [0, y s ] that, although requiring a fine-tuning of the CC, do not demand the introduction of a second brane or any other means of resolving the singularity.
where k is some arbitrary dimensionful constant of the order of the 5D Planck scale, and ν < 2. The solution can be immediately written down At the point y = y s we encounter a naked curvature singularity as explained in Section 2. For y ≪ y s , i.e. near the boundary at y = 0, the geometry is AdS 5 . The bulk potential which corresponds to the superpotential (3.1) is given by • For ν ≤ 2 the potential is bounded from above. More precisely it satisfies the condition necessary [7] for the corresponding bulk geometry to support finite temperature in the form of black hole horizons 5 . Moreover for ν < 2, as we have seen in the previous section, the equations of motion are satisfied at the singularity and there is no need to resolve it.
• For ν > 2 the equations of motion are not satisfied at the singularity and the latter would need to be resolved to fine-tune to zero the four-dimensional cosmological constant. Finally the potential is not bounded from above and finite temperature is not supported in the dual theory.
The location of the singularity depends exponentially on the brane value of φ, As we will see in the next section the relevant mass scale for the 4D spectrum is not the inverse volume but rather the "warped down" quantity All we need in order to create the electroweak hierarchy is thus φ 0 < 0 but otherwise of order unity. This can be achieved with a fairly generic brane potential, for instance by chosing a suitable λ(φ) such that the second of Eq. (2.12) is satisfied for our superpotential 6 . For negative φ 0 , the ratio of scales k/ρ exhibits a double exponential behaviour and we can create a huge hierarchy with very little fine-tuning. In Fig. 1 we plot ρ/k as a function of |φ 0 | for different values of ν and also as a function of ν for a fixed value ky s = 30 which generates a hierarchy of about fourteen orders of magnitude.
A comment about the choice of our superpotential is in order here. Its particular form, Eq. (3.1), guarantees full analytic control over our solution. A more detailed analysis of other possibilities will be postponed to Section 5. It will be useful in the following to define the metric also in conformally flat coordinates defined by the line element where A(z) ≡ A[y(z)], the relationship between z and y coordinates being given by exp[A(y)]dy = dz. One easily finds, for ν > 0, that where z 0 corresponds to the location of the UV brane that we assume to be at z 0 = 1/k and Γ(a, x) is the incomplete gamma function. Since we are taking e kys ≫ 1 and hence k/ρ ≫ 1 we can approximate Γ(1 − 1/ν 2 , ky s ) ≃ ρ/k and (3.9) simplifies to ρz ≃ Γ(1 − 1/ν 2 , ky s − ky) . (3.10) For ν > 1 the singularity at y s translates into a singularity at z s given by For 0 < ν ≤ 1 the singularity at y s translates into a singularity at z s → ∞.
As we will see in the next section the case 0 ≤ ν < 1 provides continuous spectra without any mass gap 7 , i.e. typically it leads from a 4D perspective to unparticles [16]. The case ν = 1 corresponds to continuous spectra with a mass gap provided by ρ in Eq. (3.7) leading in 4D to unparticles with mass gaps [14,17]. Finally the case 1 < ν < 2 corresponds to discrete spectra with typical level spacing controlled by ρ, as in AdS models with two branes [1].

Fluctuations and the 4D Spectrum
In this section we study the fluctuations of the metric and scalar around the classical background solutions. A general ansatz to describe all gravitational excitations of the model is, with the appropiate gauge choice [23] φ(x, y) = φ(y) + ϕ(x, y), (4.1) where φ(y) is the background solution given in Eq. (3.2). The Einstein equations that arise from this ansatz have the spin-two fluctuations decoupled from the spin-zero fluctuations, so we can proceed to study them independently.

The Graviton
Let us first consider the graviton as the transverse traceless fluctuations of the metric ds 2 = e −2A(y) (η µν + h µν (x, y))dx 2 + dy 2 , where h µ µ = ∂ µ h µν = 0. In order to respect the orbifold symmetry and to keep the possibility of a constant profile zero mode, we will consider h(y) = h(−y) which leads to the boundary condition at the brane h ′ (0) = 0. The part of the action quadratic in the graviton fluctuations becomes In addition, one has to impose that the solutions are normalizable, i.e. It is now convenient to change to conformally flat coordinates, as defined in (3.9). In this frame, rescaling the field byh(z) = e −3A(z)/2 h(z), Eq. (4.6) can be written as a Schroedinger-like equation, where a dot denotes derivation with respect to z, and the potential is given by The boundary equations are written in the z-frame as   In the case of study, it is only possible to obtain an analytic expression for the potential in the y-frame, where it reads (4.13) It is however possible to invert numerically the coordinate change (3.9), and so to plot (4.13). Its behaviour for different values of ν is shown in Fig. 2. One can distinguish three possible situations 8 : • ν < 1 [ Fig. 2(a)] In this case z extends to infinity where V h → 0. The mass spectrum is continuous from m = 0, leading to unparticles without a mass 8 Similar potentials were considered in Ref. [24].
gap. However, conformal symmetry is broken due to the occurrence of the scale y s .
The potential diverges at z s changing sign at ν 2 = 5/2, but this does not have observable consequences in the mass spectrum as we will see.
Equations (4.6) and (4.9) do not have analytic solutions. However, for ν > 1 one can find approximations for the wavefunction in the regions near the brane and near the singularity. Let us first consider the region near the brane (ky ≃ 0). Assuming ky s ≫ 1 the potential (4.13) is approximated as where the coordinate change is given by (3.10), which is approximated for ν > 1 as kz ≃ e ky . One can see that (4.14) corresponds to an AdS metric. With this approximated potential, Eq. (4.9) is solved bỹ The two coefficients can be determined by the normalization and the boundary condition (4.11) at z 0 , which yields since we expect the first mass modes to be of order m ≃ (z s − z 0 ) −1 , and in our approximation k(z s − z 0 ) ≫ 1.
Let us now move on to consider the region next to the singularity (y ≃ y s ). In this case the potential is approximated by where we have used that, for ν > 1, the coordinate change (3.9) is approximated by With this approximation, Eq. (4.9) yields the solutioñ and ∆z ≡ z s − z. The two integration constants can be obtained by imposing the boundary condition at the singularity and normalizability and by matching this solution to the solution for the intermediate region between the brane and the singularity. Near the singularity (4.20) behaves likẽ where numerical factors are being absorbed in the constants c i . We have included the next to leading order in the expansion of J α as we need it for computing the boundary condition, which reads Again numerical factors have been absorbed in c ′ i . Note that the boundary condition is only satisfied when c Y = 0, and that this condition also ensures that the solution (4.22) is normalizable when ν 2 < 2.
The boundary conditions provide the quantization of the mass eigenstates for ν > 1. In order to compute the mass spectrum for the graviton one should match the solutions at the ends of the space with a solution for the intermediate region.
Unfortunately, for the parameter range we are interested in we do not have good analytic control for this region. However we can extract a generic property of the spectrum by looking at the potential Eq. (4.13) and using the form of the coordinate transformation Eq (3.10) to deduce that, assuming e kys ≫ 1, the potential has the form 9 V where v h is some dimensionless function of the dimensionless variable ρz. In other words we have eliminated the two scales k, y s in favour of the single scale ρ given in Eq. (3.6). The spectrum is therefore of the form m n (ν, k, y s ) = µ n (ν) ρ(ν, k, y s ) , (4.25) where the pure numbers µ n only depend on the parameter ν but not on the parameters k or y s . Moreover one can find an expression for the spacing of the mass eigenstates by approximating the potential as an infinite well, which is valid for m 2 ≫ V h . The result of this approximation is Note that the mass spectrum is linear (m n ∼ n), and that as one approaches ν = 1 lim ν→1 ∆m = 0, (4.27) recovering the expected continuous spectrum at this value (for ν < 1 the spectrum is continuous too, since (4.26) is only valid for ν > 1). The numerical result for the mass eigenvalues is shown in Fig. 3 where these behaviours can be observed. Some profiles for the graviton computed numerically using the equation of motion (4.6) and the boundary conditions (4.7) are shown in Fig. 4.

The Radion-Scalar system
Now we consider the spin-zero fluctuations of the system. This is φ(x, y) = φ(y) + ϕ(x, y), (4.28) With an appropiate gauge choice, the equations of motion for the y-dependent part of the KK modes form a coupled system with only one degree of freedom. The derivation of the equations is given with detail in [23], and the result is (4.32) The boundary equations on the brane depend on the brane tension λ(φ). The precise form of the dependence can be found in [23]. At the singularity, similarly to the graviton case, one gets the boundary equation where the field has been rescaled byφ(z) ≡ e −3A/2 ϕ(z). It is convenient, as for the graviton, to use conformally flat coordinates. Rescaling the field byF (z) = e −3A(z)/2 F (z)/φ(z), (4.32) can be written as the Schroedinger equation (4.36) The relation between the rescaled fieldF and the scalar field ϕ is In the y-frame, Eq. (4.36) is given by (4.38) This potential has similar form to the graviton potential, and the three situations presented above also apply for the radion (with the same mass gap for ν = 1). A difference is that this potential does not change the sign of divergence for ν > 1 but, as said before, this does not have any observable consequences.
Let us now proceed to find the approximation for the wavefunction near the UV brane. Taking ky ≃ 0 and ky s ≫ 1 and using (4.15), (4.38) is given by and hence the solution to (4.35) is The coefficients c i are to be determined using the boundary conditions at the brane [23]. As an example, using the condition 10 ϕ(y = 0) = 0 (which we will use for the numerical computation) yields Next to the singularity, using (4.19) the potential is approximated by that gives the solutioñ The behaviour of this solution near the singularity is . (4.45) Using (4.37) we can compute the behaviour of the field and apply the normalizability condition (4.34), Again, the condition c Y = 0 is sufficient to ensure both the fulfillment of the boundary conditions and the normalizability. The scaling of the mass eigenvalues Eq (4.25) and the approximation (4.26) for the spacing of the mass modes also holds for the radion. The numerically obtained values for the masses are shown in Fig. 5. In comparison to the graviton mass modes of Fig. 3, note that the first mode for the radion is lighter than the first massive mode of the graviton. This can be understood recalling that the radion does not have a zero mode 11 . Some profiles of the scalar fluctuations of the fieldφ are shown in Fig. 6.

Other soft walls with a hierarchy
The particular form of W , Eq. (3.1), guarantees full analytic control over our solution but may seem a little ad hoc. It is natural to ask what are the essential ingredients of our stabilization mechanism and whether it is possible to generalize it to other potentials or superpotentials. The location of the singularity, and hence the size of the extra dimension, is given by Here and in the following we will assume that W is a monotonically increasing function of φ, i.e. W ′ (φ) > 0. The integral is finite whenever W diverges faster than W ∼ φ 2 . However, the inverse volume y It is easy to warp the geometry near the brane without affecting y s by adding a positive constant of O(k) to the superpotential, leading to Notice that A(y) is a monotonically increasing function of y, such that kz s > e kys . (5.4) One sees that the KK scale is warped down with respect to the compactification scale, a phenomenon well known in RS models with two branes [1]. In order to obtain, e.g., the TeV from the Planck scale we need This is not hard to achieve in a natural manner. In our model, Eq. (3.1), it works so well because the exponential behavior that was introduced for large values of φ is also valid at O(1) negative values and dominates the integral, leading to Eq. (3.5).
Moreover there are many cases where z s is infinite, even though y s is finite. There can still be mass gaps or even a discrete spectrum, but z s is clearly inadequate to characterize the energy levels. One such example is the case W = ke φ that leads to a mass gap. Let us be slightly more general and consider the class of superpotentials given by with φ 1 < φ 0 . This superpotential is monotonically increasing for β ≥ 0 and has infinite z s for β ≤ 1 2 , so we will assume 0 ≤ β ≤ 1 2 . The volume y s is approximately so, again, ky s is (mildly) exponentially enhanced when |φ 0 | = O(1 − 10), φ 0 < 0. In order to estimate the spectrum, we need the asymptotic behavior of the warp factor in conformally flat coordinates. For large z, it is given by Combining this with Eq. (5.7) we find a strong suppression of ρ w /k resulting just from O(1) numbers. The quantity ρ w sets the scale for the KK spectrum in this case. In fact metrics of the form Eq. (5.8) have been studied in Ref. [13]. A WKB approximation shows that the spectrum can be approximated by We see that ρ w indeed sets the scale of the 4D masses, which are hence parametrically suppressed with respect to k. The complete superpotential that accomplishes a hierarchy and leads to the spectrum Eq (5.13) is In particular the case β = 1/4 generates the linear Regge trajectory spectrum m 2 n ≃ ρ 2 w n appropriate for AdS/QCD models as in Ref. [9]. In this case one would obtain the linear confinement behaviour of e.g. ρ-mesons by considering an additional piece in our action d 5 x √ −g e − 1 2 φ L mesons . The fact that asymptotically A(z) ∼ φ(z) ∼ z 2 guarantees that the resonances of the vector mesons follow the same linear law as the ones for the scalars and tensors.
Let us conclude this section by noting that there are certainly other ways to obtain the mild hierarchy ky s , including moderate fine-tunings of parameters. What is completely generic, though, is the fact that adding warping as in Eq. (5.3) leaves ky s manifestly unchanged but suppresses the masses by an additional warp factor e kys .

Conclusions and Outlook
In this paper we have studied the stabilization of soft walls, i.e. 4+1 dimensional geometries with 4D Poincaré invariance, that are only bounded by a single three-brane but that nevertheless exhibit a finite volume for the extra dimensional coordinate. The second brane is typically replaced by a naked curvature singularity at a finite proper distance. In particular we have studied how these soft walls arise in models with a single scalar field, and classified the type of models that can be realized as full solutions to the Einstein equations without destabilizing contributions at the singularities. We have proven that all admissible solutions result in a fine-tuning of the cosmological constant.
Our main objective has been to show how to stabilize the position of the singularity at parametrically large values compared to the 5D Planck length. We have employed the superpotential method of [19,20] and proposed a family of models that accomplishes this goal. Our stabilizing superpotential allows three types of 4D spectra: continuous, continuous with a mass gap, and discrete. The 4D mass scale ρ, controlling the mass gap in the continuous case and the spacing in the discrete one, depends in a double exponential manner on the value of the scalar field at the brane, Eq (3.7), and can thus be naturally suppressed with respect to the 5D mass scale k without fine-tuning.
Next we have studied in detail the spectra resulting from fluctuations around our family of solutions. As with the case of the background, we have paid close attention to the boundary conditions at the singularity, projecting out solutions that would give contributions at the dynamically generated boundary. We have given analytical forms of the wavefunctions near the brane and the singularity, as well as numerical values for the lowest lying excitations and their profiles.
We have also given a constructive recipe of how to obtain superpotentials that accomplish the stabilization of the hierarchy, a desired spectrum, and an "end-ofthe-world singularity" that is consistent with the equations of motion. In a first step, one chooses the asymptotic (i.e. large φ) behavior of W . This will determine the asymptotic form of the spectrum. To ensure consistency of the equations of motion, the divergence should be milder than W ∼ e 2φ . The different asymptotic forms of W and corresponding spectra are summarized 12 in Tab. 1. In a second step, one completes W for smaller values of φ in such a way as to accomplish a mild hierarchy of the proper distance y s with respect to the fundamental 5D scale k, given by the simple relation Eq. (5.5). Notice that many of the interesting spectra require some kind of exponential behavior at large φ, such that this region does not contribute at all to ky s . At the same time one minimizes the effective 4D potential, Eq. (2.23), at the brane at y = 0 to find the vacuum value of φ 0 ≡ φ(0).
continuous continuous discrete spectrum w/ mass gap m n ∼ n 2β m n ∼ n consistent yes no solution Table 1: Spectra resulting from different asymptotic forms of the superpotential. In the first row we give the asymptotic behavior of W (φ), with the strength of the divergence increasing from left to right (> means "diverges faster than", etc). Second and third row show the finiteness of y s and z s , with the behavior changing at W ∼ φ 2 and W ∼ e φ φ 1 2 respectively. The third row shows the spectrum, while in the last one we indicate the consistency of the solution.
In a third and final step, one adds a constant of O(k) to the superpotential. This adds strong warping near the UV brane, but has no effect whatsoever on the determination of ky s and φ 0 . We have shown that in this way one can warp down the parameter setting the overall scale for the spectrum by a factor e kys , leading to the desired hierarchy.
There are a number of phenomenological applications which are outside the scope of the present paper but which are worth of future investigations. For the range of the parameter 1 < ν < 2 these applications are common with two brane models, as RS1, but with some peculiarities. In particular graviton (and radion) KK modes are at the TeV scale and they can be produced and decay at LHC by their interaction with matter ∼ h µν T µν , so they are expected to be produced through gluon annihilation [26]. Since there is no IR brane, for soft-wall models to solve the gauge hierarchy problem the Higgs boson (either a scalar doublet or the fifth component of a gauge field in a gauge-Higgs unified model) has to propagate in the bulk and it has to be localized near the singularity for its mass to feel the warping. On the other hand fermions with sizable Yukawa couplings (third generation fermions) should be localized near the singularity as well while first and second generation fermions can propagate at (or near the) UV brane. As we have seen that the first graviton KK mode is localized near the singularity, once produced it is expected to decay into either Higgs or tt pairs. For ν = 1 the mass spectrum of fields propagating in the bulk is a continuum above an O(T eV ) mass gap. This continuum (endowed with a given conformal dimension) can interact with SM fields propagating in the UV brane as operators of a CFT, where the conformal invariance is explicitly broken at a scale given by the mass gap, and can model and describe the unparticle phenomenology. In particular the Higgs embedded into such 5D background can describe the unHiggs theory of Ref. [27] in the presence of a mass gap. In all those cases the strength of electroweak constraints should be an issue. Finally, the case where the spectrum is described by linear Regge trajectories (ν = 1, β = 1/4) can give rise to a phenomenological description of AdS/QCD, similar to that of Ref. [9], where the QCD scale can be naturally stabilized by the scalar field.