Stability of Scalar Fields in Warped Extra Dimensions

Stability of Scalar Fields in Warped Extra Dimensions

S. Mert Aybat
Nikhef Theory Group, Science Park 105, 1098 XG Amsterdam, The Netherlands
Damien P. George
Nikhef Theory Group, Science Park 105, 1098 XG Amsterdam, The Netherlands
July 14, 2019
Abstract:

This work sets up a general theoretical framework to study stability of models with a warped extra dimension where scalar fields couple minimally to gravity. Our analysis encompasses Randall-Sundrum models with branes and bulk scalars, and general domain-wall models. We derive the Schrödinger equation governing the spin-0 spectrum of perturbations of such a system. This result is specialized to potentials generated using fake supergravity, and we show that models without branes are free of tachyonic modes. Turning to the existence of zero modes, we prove a criterion which relates the number of normalizable zero modes to the parities of the scalar fields. Constructions with definite parity and only odd scalars are shown to be free of zero modes and are hence perturbatively stable. We give two explicit examples of domain-wall models with a soft wall, one which admits a zero mode and one which does not. The latter is an example of a model that stabilizes a compact extra dimension using only bulk scalars and does not require dynamical branes.

Field Theories in Higher Dimensions, Supergravity Models
preprint: NIKHEF/2010-017

1 Introduction

Extra dimensions are a plausible extension to the Standard Model (SM), and a vast array of specific realizations of models with extra dimensions exist. String theory provides a lot of the motivation to include extra dimensions, and there are many string-inspired phenomenological models, for example Refs. [1, 2, 3, 4, 5], as well as purely field-theoretic constructions, for example Refs. [6, 7, 8].

Much work has been devoted to the study of the type I Randall-Sundrum (RS) model [4], where the electroweak hierarchy can be naturally generated by the warping of spacetime in the extra dimension. Such warping is a generic feature of models with an extra dimension: the warp factor of the background metric is sensitive to changes in the energy density as one moves along the extra dimension, and so non-trivial background sources will generically induce a warped metric. In the simplest 5D set-up, the line element of a warped space is

 ds2=e−2σ(y)ημνdxμdxν+dy2, (1)

where is the extra dimension and is the warp factor. Scalar fields with profile and fundamental branes with tension located at will provide a source of energy density and act to drive the warp factor:

 σ′′(y)∝∑iϕ′i(y)2+∑αλαδ(y−yα). (2)

Here, labels the scalars and labels the branes. Note that scalars can only provide ‘positive warping’, and only if they have a non-trivial profile in the extra dimension. Branes can provide both ‘positive warping’ and ‘negative warping’ since can be of either sign.

In the original RS set-up, the extra dimension was compactified on a circle with identified with . The two branes at the orbifold fixed points were required to have tensions of equal magnitude but opposite sign in order for the warp factor to join up consistently; the positive and negative warping needed to balance.111The model also required a bulk cosmological constant to keep constant between the two branes. This constant needed to be fine-tuned against the tension of the branes, a generic feature of warped extra dimensions with 4D Poincaré slices. The type II RS model [5] had the extra dimension infinite in size, and required only a single brane of positive tension at the origin. The warp factor is driven to at large distances from the origin, localizing 4D gravity to the brane. Since this model contains only a positive tension brane, one can replace it by a scalar field with a suitable -profile. Such domain-wall models also localize 4D gravity [9] and form the basis of a large area of research, see for example Refs. [7, 8].

Recently there has been interest in a new type of RS-like warped spacetime: a compact spacetime where the negative tension brane is replaced with a physical singularity. These soft-wall models were originally designed to yield linear Regge trajectories in the context of the AdS/CFT correspondence [10], but have since been the basis of actual models beyond the Standard Model (BSM) [11, 12, 13, 14, 15, 16, 17, 18], and also provide a holographic dual description of unparticle models [19, 20]. Our ultimate aim is take the soft-wall model and go one step further by removing the final, positive-tension brane at the origin, and replacing it with a suitable scalar field profile. The model will then describe a compact extra dimension without the use of fundamental dynamical branes, in other words there are no branes with tension and localized potential terms in the Lagrangian. This is interesting because it is a purely field theoretical construction and requires no appeal to string theory. 222The examples we will consider in the following sections, however, will have orbifold fixed points at the origin. These are properties of the geometry and one does not need to appeal to string theory to construct them.

Such a step is not as straightforward as it may first seem, as it is crucial that one ensures the stability of the scalar field configuration. Part of this involves stabilizing the size of the compact extra dimension. For the RS model, one can utilize the Goldberger-Wise mechanism [21, 22] where a bulk scalar field stabilizes the set-up due to localized potentials on each brane. Here, the presence of branes and the ability to have a 4D potential for the 5D scalar localized to the brane is necessary for stabilization to work. Soft-wall models can also be stabilized in a similar way [17]; here, one needs a 4D potential localized to the brane at the origin. In order to build a soft-wall model without a brane, we shall need an alternative to the Goldberger-Wise mechanism. It is the aim of the present paper to show that, with the correct type of scalar background, one does not need any additional stabilization mechanism.

The techniques developed in this paper have wider application than just soft-wall models without a brane. The problem of stability of non-trivial scalar backgrounds is an import one, and in the case of a single scalar without gravity there exists a constructive way of finding the lowest energy, stable solution [23, 24]. For the case with gravity, things are not so simple, but some attempts have been made [25, 26, 27]. The problem is difficult because, as discussed above, non-trivial scalar profiles always induce warping of the metric, and, furthermore, modes of the scalars mix with spin-0 degrees of freedom in the metric. In the literature to date, all successful analyses have relied on the superpotential approach [28] also known as the fake supergravity approach [29]. Fake supergravity requires one to choose a scalar potential such that it is invariant under supergravity transformations, even though the whole theory itself is not locally supersymmetric. In practice this is very simple to achieve: one just needs to generate the full potential from a more primitive superpotential. Using this fake supergravity approach, one can easily solve the background equations of motion, including Einstein’s equations. It has been shown, using such a construction, that models with an arbitrary number of scalars are free of tachyonic spin-2 modes [28]. For a single scalar field, it has also been shown that this construction leads to models free of tachyonic modes in the spin-0 sector [30]. In this paper we extend this work to the case of an arbitrary number of background scalar fields. We find that the same result holds: there are no tachyonic modes in the spin-0 sector. Restricting ourselves to orbifold spaces that are symmetric under parity, we also show that set-ups where the scalar profiles are all odd do not contain spin-0 zero modes.

The paper is organized as follows. In Section 2 we discuss the gravity-free case as a warm-up and introduction to the superpotential approach. We show that a scalar field configuration has a mode spectrum which is always free of tachyons, and which always contains a zero mode of translation. In Section 3 we describe the model with gravity and the background solutions in the fake supergravity approach. The spectrum of spin-2 perturbations in this background are derived in Section 4. The usual result of a massless 4D graviton is reproduced. Section 5 contains the main results of this paper, and shows that, in the fake supergravity approach with an arbitrary number of background scalars, there are no tachyonic spin-0 perturbations. This analysis is valid for an extra dimension of general topology. In Section 6 we specialize to orbifold constructions with definite parity and discuss a criterion for determining the existence of spin-0 zero modes. We then analyze two specific models: one which supports such a zero mode, and one which does not. The latter is an example of a model with a stabilized compact extra dimension that does not require any dynamical branes. We conclude in Section 7.

2 Gravity-free case

As discussed in the introduction, non-trivial scalar profiles always induce a warped metric, and so realistic models must include gravity. Nevertheless, the gravity-free case is interesting to study before moving to the case with gravity, and serves also to introduce our notation. For one scalar field the general analysis has been completed in Refs. [23, 24] and for two scalar fields, using the superpotential approach, in Ref. [31]. In this section we show that, for scalar fields without gravity, the superpotential approach yields solutions which are globally stable, for given boundary conditions, and up to zero-mode translations of the configuration.

The action is

 S=∫d4xdy[−12∑i∂MΦi∂MΦi−V({Φi})], (3)

where the sum is over , and there are scalar fields. The potential is an arbitrary function of these scalars. The equations of motion are

 ∂M∂MΦi−Vi({Φi})=0. (4)

Here we place a subscript on to denote partial differentiation with respect to the field . This notation is used heavily throughout the paper; in general, for a function that depends on the scalar fields, we write

 Xij…k({Φi})≡∂∂Φi∂∂Φj…∂∂ΦkX({Φi}). (5)

We are interested in finding static background solutions for the scalars, configurations that depend only on the extra dimension: . In what follows, ‘configuration’ refers to the set of scalar fields taking on the static solutions given by .

We can restrict ourselves to potentials generated by a superpotential such that

 V({Φi})=∑i12[Wi({Φi})]2. (6)

The subscript denotes partial differentiation with respect to , as per Eq. (5). With this particular choice of potential, the Euler-Lagrange equations of motion, Eq. (4), are satisfied so long as the static configuration solve the first-order differential equation

 ϕ′i(y)=Wi({ϕi}). (7)

A prime denotes a derivative, and we are evaluating at . Note that to show this solves the equations of motion, and in many places throughout this paper, we use the relation

 dWdy=∑i∂W∂ΦidΦidy. (8)

It turns out that such background configurations are always globally stable, up to zero-mode translations. To prove this statement, we will look at three things: 1) the 4D energy density of the configuration; 2) perturbative modes; 3) the zero mode.

We compute the 4D energy density of the system by integrating the 5D stress energy over . We shall assume that takes the form given by Eq. (6) and that the scalar fields are functions of only, given by . In particular we do not require the scalar fields to satisfy Eq. (7). We obtain

 E =∫T00dy =∫[12∑i(ϕ′i)2+12∑i[Wi({ϕi})]2]dy =∫dW+12∑i∫[ϕ′i−Wi({ϕi})]2dy. (9)

The first term in the last line here is just a surface term; it is difference of evaluated at the boundaries of the extra dimension, and depends only on the values of at these boundaries. Given a particular , which completely determines , and choices for the boundary values of , the total energy density of the configuration is minimized precisely when Eq. (7) is satisfied. Hence, for given boundary conditions, the superpotential approach yields configurations which globally minimize the energy.

Since the configuration minimizes the energy, the solution must be perturbatively stable, and we can demonstrate this explicitly. This will be useful as a precursor to the case with gravity. Expanding in linear perturbations about the background

 Φi(xμ,y)=ϕi(y)+φi(xμ,y), (10)

the equations of motion, Eq. (4), reduce to

 (11)

Here, repeated indices are to be summed over, and indicates that is a function of and is to be evaluated using the background solutions ; that is, . Performing a Fourier transform, , we obtain a set of , coupled, time-independent Schrödinger equations. We can write these Schrödinger equations as [31]

 (∂yδik+Wik|bg)(−∂yδkj+Wkj∣∣bg)φj=Eφi, (12)

and so, using the results of Appendix A, we see the system admits solutions only with . This is true so long as the perturbations vanish at the boundaries of the extra dimension which is a valid assumption.

Now let us consider the existence of the zero-mode perturbations with . For such a mode to exist, Eq. (12) implies that , which admits the solution , with a real, non-zero constant. This is the familiar result that the zero mode of translation is the first derivative of the background configuration. Note that when there are multiple background fields, the zero mode is the mode where all perturbations are excited simultaneously with profiles proportional to .

As a quick example, consider , which yields the familiar potential . Solutions are and the well-known kink: . To get the anti-kink solution, one needs to start with instead. This demonstrates the fact that the superpotential encodes for both and the static solution, and that different superpotentials can yield the same potential. Also noteworthy is the fact that, for this choice of , is a solution of the equation of motion but cannot be obtained from any superpotential. This is because the solution is unstable, and we have shown that the superpotential approach always yields globally stable solutions.

3 Warped Background Configuration

We now come to the main topic of the paper and consider a general 5D theory with gravity coupled minimally to scalar fields, including the possibility of fundamental brane terms. The corresponding action is given by

 S=∫d4xdy[√−g(M3R+L% matter)−√−g4λ]. (13)

We are using a signature and is the 5D Planck mass. The scalar-matter Lagrangian and brane terms are given respectively by

 Lmatter =−12∑igMN∂MΦi∂NΦi−V({Φi}), (14) λ =λ({Φi})=∑αλα({Φi})δ(y−yα). (15)

The sum over is from 1 to , and is the potential that in general depends on all scalars. The subscript indexes the branes, are their locations, and the scalar potentials localized to the branes, which includes the brane’s tension. Our aim is to study the general stability conditions for non-trivial background configurations of this model. In this section we discuss background solutions, and then in the following two sections analyze spin-2 and spin-0 perturbations.

For 4D Poincaré slices, the background metric ansatz is

 ds2=e−2σ(y)ημνdxμdxν+dy2, (16)

where index the 4D subspace. Einstein’s equations are

 GMN=12M3TMN. (17)

Assuming the scalar fields only depend on the extra dimension , and denoting such a background configuration by , the Euler-Lagrange and Einstein’s equations yield equations for functions:

 3σ′′=12M3(∑iϕ′2i+λ({ϕj})), (18) 12σ′2−3σ′′=−12M3[2V({ϕj})+λ({ϕj})], ϕ′′i−4σ′ϕ′i−Vi({ϕj})−λi({ϕj})=0.

The first two equations are in fact the sum and difference of the and components of Einstein’s equations. The notation means that the potential is to be evaluated with the background fields . Also, we place a subscript on and to denote partial differentiation with respect to the field , as per Eq. (5). It is easy to show that one of these equations can be obtained using two others, therefore the equations are not all independent.

Solutions to the above equations can be obtained using the so-called fake supergravity approach. In this approach a superpotential is introduced so that

 V({Φi})=∑i12[Wi({Φi})]2−13M3[W({Φi})]2, (19)

where is obtained using Eq. (5). It was shown in Ref. [28] that the system of equations, Eqs. (18), can be written in terms of the superpotential and solutions are given by

 σ′(y) =16M3W({ϕi}), (20) ϕ′i(y) =Wi({ϕi}).

However, what we are interested in here are not the solutions of the system of equations. Instead, assuming solutions exist, we would like to see if the system is stable or not.

When studying the gravity-free case in the previous section, we were able to obtain the result that the superpotential approach yields globally stable configurations, Eq (9). Unfortunately, for the case with gravity included the corresponding analysis does not tell us much. The 4D energy density is (recall that we do not assume the relations in Eq (20))

 E =∫T00dy =∫e−2σ[12∑i(ϕ′i)2+12∑i[Wi({ϕi})]2−13M3[W({ϕi})]2]dy =∫[e−2σW({ϕi})]′dy+12∑i∫e−2σ[ϕ′i−Wi({ϕi})]2dy +2∫e−2σW({ϕi})[σ′−16M3W({ϕi})]dy. (21)

The first term here is just a surface term, which generally vanishes for a Randall-Sundrum-like configuration. The second term is a non-negative integral which is minimized when the scalar fields obey the first order fake supergravity equation (as in the gravity-free case). The final term is the difficult term. Although it vanishes when satisfies its first order equation, it is not obvious that this minimizes . So we cannot conclude that the fake supergravity approach yields a globally stable background configuration.

To proceed we shall study local, perturbative stability of a configuration with gravity coupled to scalars. The spin-2 and spin-0 fluctuations of the metric and the scalar fields are treated separately in the following two sections. For the initial stages, our analysis will be for an arbitrary scalar potential . Later on we will need to specialize to the fake supergravity approach.

4 Spin-2 Perturbations

The general ansatz which takes into account both spin-0 and spin-2 perturbations is

 ds2=e−2σ(y)[(1−2F(xμ,y)ημν+hμν(xμ,y)]dxμdxν+[1+G(xμ,y)]2dy2, (22) Φi(xμ,y)=ϕi(y)+φi(xμ,y).

Here, we have chosen to work in the axial gauge, , with transverse traceless part . The (spatial) components of Einstein’s equations enforce which we take from now on. The Einstein tensor has the following non-zero components

 Gμν =3e−2σημν(2σ′2−σ′′−F′′+6σ′F′−12σ′2F+6σ′′F) (23) +e−2σ(−12e2σ□hμν−12h′′μν+2σ′h′μν+3(2σ′2−σ′′)hμν), Gμ5 =3∂μ(F′−2σ′F), G55 =6σ′2−3e−2σ□F+12σ′F′.

The stress-energy tensor is

 Tμν =e−2σημν[−12(1−6F)(ϕ′i)2−(V+λ)|bg+(2V+4λ)|bgF−(Vi+λi)|bgφi−ϕ′iφ′i] (24) +e−2σhμν[−12(ϕ′i)2−(V+λ)|bg−(Vi+λi)|bgφi], Tμ5 =∂μ(ϕ′iφi), T55 =12(ϕ′i)2−V|bg−4V|bgF−Vi|bgφi+ϕ′iφ′i.

Here, repeated indices are to be summed over. Taking the 4-trace of the part of Einstein’s equations shows that the spin-2 and spin-0 perturbations completely decouple from one another. See Ref. [22] for a discussion of the gauge degrees of freedom and the decoupling of spin-2 and spin-0 sectors. The spin-2 perturbations are the easiest to analyze, and we look at them first, returning to the spin-0 sector in the following section. The Euler-Lagrange equations for the scalar fields do not contain ; these equations are given in the next section.

Using the background equations, Eq. (18), the equation for is

 −e2σ□hμν−h′′μν+4σ′h′μν=0. (25)

There is always a zero mode, , which is normalizable. It is the well-known 4D massless graviton [5]. In conformal coordinates defined by with rescaled we have

 −~h′′μν+(94σ′2−32σ′′)=□~hμν. (26)

This is a Schrödinger-like equation. It can be rewritten in a self-adjoint form as

 (∂z+Sh)(−∂z+Sh)~hμν=□~hμν, (27)

with

 Sh=−32σ′. (28)

Appealing to Appendix A, we see that there are no tachyonic modes in the spin-2 sector.

5 Spin-0 Perturbations

In the previous section we have seen that the spin-2 sector decouples from the spin-0 sector, independent of the number of scalar fields involved. We now consider spin-0 perturbations around the background solutions and . The metric ansatz is a restricted version of Eq. (22) with  [22]

 ds2=e−2σ(y)[1−2F(xμ,y)]ημνdxμdxν+[1+2F(xμ,y)]2dy2, (29) Φi(xμ,y)=ϕi(y)+φi(xμ,y).

We work to linear order in the perturbations and . The equations for these perturbations consist of two of Einstein’s equations and the Euler-Lagrange equations:

 6M3(F′−2σ′F)=ϕ′iφi, (30) 6M3(−e2σ□F−2σ′F′+F′′)=2ϕ′iφ′i+2λ|% bgF+λi|bgφi, (31) e2σ□φi+φ′′i−4σ′φ′i−6F′ϕ′i−(4Vi+2λi)|bgF−(Vij+λij)∣∣bgφj=0. (32)

Note that one of the redundant Einstein’s equations has been omitted and it can be easily obtained by using Eq. (30) and Eq. (31).

We now go to conformal coordinates to eliminate the factor from the ’s, and rescale the fields by a factor of to help eliminate first derivatives. That is, we make the following change of variables and field redefinitions

 dy=e−σ(y)dz,σ(y)=σ(z(y)),ϕi(y)=ϕi(z(y)), (33) F(y)=e32σ(z(y))~F(z),φi(y)=e32σ(z(y))~φi(z).

In terms of the conformal coordinate and the new fields and , Eqs. (30) through (32) become, respectively,

 6M3(−12σ′~F+~F′)=ϕ′i~φi, (34) 6M3[−□~F+~F′′+2σ′~F′+(34σ′2+32σ′′)~F]=2ϕ′i~φ′i+2e−2σλ|bg~F +(3σ′ϕ′i+e−2σλi|bg)~φi, (35) □~φi+~φ′′i+(−94σ′2+32σ′′)~φi−3ϕ′i(3σ′~F+2~F′) −e−2σ(4Vi+2λi)|bg~F−e−2σ(Vij+λij)∣∣bg% ~φj=0. (36)

We can make judicious use of Eq. (34), and its derivative, to eliminate all first derivatives of and in the other two equations. We also make use of the background equation for . This allows us to obtain

 6M3(~F′−12σ′~F)−ϕ′i~φi=0, (37) (38) □~φi+~φ′′i+(−94σ′2+32σ′′)~φi+(−4ϕ′′i+2e−2σλi|bg)~F −1M3ϕ′iϕ′j~φj−e−2σ(Vij+λij)∣∣bg~φj=0. (39)

This is a set of equations for functions of .

5.1 Simplifying the perturbation equations

The main aim of this section is to manipulate and simplify the above equations in order to obtain a set of coupled Schrödinger-like equations, which can be solved to determine the spin-0 spectrum. Before simplifying the equations, we need to understand precisely their redundancy. Let , and be equal to the left-hand-sides of Eqs. (37) through (5) respectively. Then the Euler-Lagrange and Einstein’s equations amount to

 α=β=γi=0. (40)

One can show that the following relation holds in the bulk:

 □α+α′′−(94σ′2+32σ′′)α−β′+12σ′β+ϕ′iγi=0. (41)

Thus, there is a certain amount of redundancy in the equations. The redundancy is quantified precisely by Eq. (41). In particular, if we solve then we automatically have and solving for of the ’s is enough to solve for the system (that is, we have eliminated one of the scalars ). Counting the number of free integration constants (ICs), we have 2 for and together, and for of the ’s. This is a total of free ICs.

We can make other choices for eliminating an equation. For example, if we choose to solve all and , then we will have an equation left over for :

 □α+α′′−(94σ′2+32σ′′)α=0. (42)

Solving only solves such that it is a solution to the above differential equation. Counting the total number of free ICs: 2 each from all of the ’s and 2 from . However, 2 of these ICs have to be used in order to pick the solution of Eq. (42) and so we end up with a total of free ICs for the system, the same as in the alternative choice above.

We can also choose to solve and with free ICs. Then Eq. (41) becomes and we need to have 1 constraining IC from demanding that we pick out the solution from this differential equation. Thus there are again only free ICs for the system.

These arguments hold for solutions of the system in the bulk. Without branes, the system is completely specified by integration constants. If there are branes in the set-up, the situation is the same, as the values of fields on the brane are continuous and hence the same as the values in the bulk just to the sides of the brane(s). Derivatives of the fields are discontinuous on the brane, and hence undefined. Instead, one has freedom to choose the field derivatives on one side of the brane or the other, and then uses the ‘jump’ conditions to determine the derivative on the opposite side. The jump conditions are found by integrating Eqs. (38) and (5) over each brane; see [22] for details.

We choose to solve the system , as it leaves us with the most ‘symmetric’ system of equations. Solutions to this particular set of equations will include all solutions of the true system (by that we mean Eq. (40)), as well as some additional solutions which do not satisfy all the boundary conditions of Einstein’s equations. For us, this is all we shall require, as we are going to show that the extended set of solutions does not contain certain modes (tachyonic and/or zero modes), and so the full system cannot therefore contain such modes.

If one makes the definitions

 ~φi(z) =M3/2ψi(z), (43) ~F(z) =χ(z)/√12, (44)

then the equations , Eqs. (38), and (5), become

 □χ+χ′′−(V00+B00)χ−(V0i+B0i)ψi=0, (45) □ψi+ψ′′i−(Vij+Bij)ψj−(V0i+B0i)χ=0,

where

 V00 =94σ′2+52σ′′, (46) V0i =2√3M3ϕ′′i, Vij =(94σ′2−32σ′′)δij+1M3ϕ′iϕ′j+e−2σVij∣∣bg,

and the brane terms are

 B00 =13M3e−2σλ|bg, (47) B0i =1√3M3e−2σλi|bg, Bij =e−2σλij∣∣bg.

Note the symmetry of the cross-coupling in Eq. (45). We can use Eq. (45) to solve for the physical spin-0 spectrum, which includes the eigenvalues and corresponding extra-dimensional profiles.

A physical mode is defined as having the same dependence in and , but possibly different dependence. Separation of variables then proceeds by defining

 χ(xμ,z)=f0(z)η(xμ), (48) ψi(xμ,z)=fi(z)η(xμ),

where , are the extra-dimensional profiles and the 4D mode. To find the spectrum of mass states of , we perform a Fourier transform on the variable : , where is the eigenvalue, proportional to the mass squared of the mode. Eq. (45) now becomes a coupled eigenvalue equation of the form

 H(f0fi)=(−∂2z+V00+B00V0j+B0jV0i+B0i−∂2zδij+Vij+Bij)(f0fj)=E(f0fi). (49)

In the matrix multiplication there is a sum over repeated indices from 1 to , and we shall use such a notation in subsequent equations.

This equation is a system of coupled Schrödinger equations, with a symmetric coupling potential. The symmetry implies that the eigenvalues of the system will be real, so long as certain boundary conditions are satisfied. This equation is one of the main results of the paper. It allows one to determine the spectrum of physical spin-0 modes with scalars coupled to gravity. It is valid for an arbitrary potential , with or without branes at the edges of the extra dimension, and for generic topology of the extra dimension.

5.2 The case without branes

We are now going to specialize to the case where the brane terms are absent; . A particular example of such models is one where a domain wall replaces the fundamental brane [9]. In this case we know exactly how the perturbations behave at the boundaries, and can proceed to determine the spectrum. We shall show, using the fake supergravity approach, that there are no tachyonic modes for these models. We present conditions for the non-existence of tachyonic modes for models with fundamental branes in the following section.

In order to prove stability for a system with scalar fields and the gravitational perturbation , we need to show that the eigenvalues are strictly positive for the given boundary conditions. Our analysis so far has been for a general potential and general background configuration of scalar fields. If we specialize to configurations generated by the fake supergravity approach, we can prove that the eigenvalues of this system are non-negative, and in some cases strictly positive.

The key for proving such positivity is the observation that one can write the perturbation potential as

 V=(V00V0jV0iVij)=S2+S′, (50)

where

 (51)

Then,

 H=(∂z+S†)(−∂z+S). (52)

We have shown in Appendix A that the eigenvalues satisfy whenever

 Ψ†(−∂z+S)Ψ|boundary=0, (53)

where .

As is obvious from the above equation, boundary conditions play a crucial role on the positivity of the eigenvalues and therefore on the stability of the system. Here we discuss the boundary conditions with general for the following specific topologies:

Full-interval space: On a full interval there are no restrictions, such as parity or Dirichlet or Neumann boundary conditions, on the perturbation wave function . We only require that the perturbations are normalizable over the full space. The boundaries are at and , which can be at infinity or at finite values as in the case of a soft-wall model, where space ends at a physical singularity. As seen in Eq. (33) solutions in conformal coordinates are related to the ones in coordinates through the warp factor which diverges at large . We see that in order to have normalizable solutions, the rescaled perturbations in conformal coordinates must vanish at the boundaries. Therefore, is satisfied.

Half-interval orbifold space: Taking a full-interval space and identifying with yields an orbifold, with effective boundaries at and . Note that for such a topology there will be a non-dynamical brane at the origin, but this will not contribute to the brane terms in the effective potential for the perturbations. Furthermore, we do not need to appeal to string theory for information on the physics of this non-dynamical brane; it is just an orbifold fixed plane. The boundary conditions for at are now equivalent to a choice of parity for the warp factor and the scalars. The warp factor must be even in order to localize gravity. Scalar fields that are odd vanish at , even scalars have vanishing at . It is also easy to verify that has odd parity, and therefore it also vanishes at . As such, the condition is always satisfied. For the boundary, the scalar perturbations vanish as for the full-interval case discussed above due to the normalizable condition on the perturbations.

Having explicitly shown that the boundary terms vanish, which is expected for a Hermitian Hamiltonian corresponding to a physical problem, we can conclude that without dynamical branes models with scalars coupled to gravity do not have any tachyonic modes. Note that our results so far apply to both of these scenarios while in the following sections for explicit examples with two or more scalar fields we will concentrate on the half-interval case.

5.3 The case with fundamental branes

For completeness we would also like to give sufficient conditions for the non-existence of tachyonic modes for models with scalars coupled to gravity in the presence of fundamental branes with . The profiles for the scalar fields for models with fundamental branes satisfy the effective Schrödinger equation given by Eq. (49) where the brane terms are given in Eq. (47). Repeating the analysis of Appendix A with brane terms we find that the effective Schrödinger equation can be written as

 ∫dz|SΨ|2+∫dz∂z(Ψ†SΨ)+∫dz(Ψ†BΨ)=E∫dz|Ψ|2, (54)

where with given by Eq. (51) and the brane terms are

 (55)

where the sum over is over all fundamental branes in the model. For the sufficient condition is then modified to be

 Ψ†(−∂z+S)Ψ|boundary+Ψ†BΨ≥0. (56)

Analyzing this requirement for different models with different brane potential terms is outside the scope of this work.

6 Analyzing Zero-Modes

As we have seen in the previous section, the eigenvalues of the spin-0 sector are guaranteed to be non-negative when constructing configurations using the fake supergravity approach. We must still analyze the existence of zero mode solutions to show complete stability. In particular this will ensure that the size of the extra dimension is stabilized.

In this section, using the formalism we have developed, we will first show that for the case the eigensystem reduces to a single Schrödinger-like equation, and we determine the mass gap of the spin-0 spectrum. For configurations with scalar fields, analyzing the general properties of the mass spectrum is a very difficult problem. As such, we restrict our attention to orbifold spaces where the fields have definite parity and discuss a criterion for the possible existence, or lack, of zero modes. Guided by this criterion, we analyze the zero modes in two example models with specific superpotentials. We shall explicitly construct a model with scalar fields that does not have a zero mode and therefore is stable.

6.1 The N=1 case

In this section we will analyze the system with one scalar coupled to gravity and show that it is stable with positive mass eigenvalues. The analysis is valid for general topologies. Stability for models with one scalar field and no branes have been analyzed previously in Refs. [28, 30]. For models with a soft-wall and a fundamental brane stability has been proven in Ref. [17] and for RS models with two fundamental branes in Refs. [21, 22].

We start by analyzing the Einstein constraint equation, Eq. (37), and the second-order equation for from Eq. (49). They are, respectively,

 −f′0+S00f0+S0ifi=0, (57) −f′′0+V00f0+V0ifi=Ef0 . (58)

As before, repeated , and later , indices are to be summed. Combining them in an obvious way yields

 −f′′0+S0iV0iS0jS0jf′0+(V00−S00S0iV0iS0jS0j)f0+(V0i−S0iS0kV0kS0jS0j)fi=Ef0. (59)

We can eliminate the term, and obtain a Schrödinger-like equation, by defining

 f0=√S0iS0ig. (60)

This gives

 −g′′+(A2−A′+S0iS0i)g+1√S0iS0iBifi=Eg, (61)

where

 A =S0iSijS0jS0kS0k, (62) Bi =V0i−S0iS0kV0kS0jS0j. (63)

Equation (61) can be used to prove the non-existence of a zero mode for a theory with one scalar field. For we have:

 A =S11, (64) Bi =0. (65)

Following the arguments in Appendix A, we see that for wave functions , the eigenvalues of this system are non-negative since the boundary terms vanish as we have discussed previously. Equation (61) simplifies to

 (∂z−A)(−∂z−A)g+S0iS0ig=Eg. (66)

Now we will use the same trick as in the analysis of Appendix A and multiply Eq. (66) from the left by and integrate over the extra dimension . Up to a surface term that vanishes we have

 ∫dz∣∣∣(−ddz−A)g∣∣∣2+∫dz|S01g|2=E∫dz|g|2. (67)

Consider the existence of a zero mode, with . In order to satisfy Eq. (67) for a generic superpotential , and for the given boundary conditions by which the surface terms vanish, both of the terms on the left-hand side must vanish simultaneously. For a generic this means that the field , and hence , has to vanish. This shows that for nontrivial there does not exist any zero mode.

We can push the analysis a little further and provide a lower bound on the mass gap to the first spin-0 state. Dropping one of the terms on the left-hand side of Eq. (67) gives the following inequality:

 E∫dz|g|2 ≥∫dz|S01g|2 (68) ≥min[S201]∫dz|g|2, (69)

where is the minimum of over . Thus we obtain a bound on the mass of the first spin-0 state:

 E≥min[S201]. (70)

6.2 The N≥2 case

Set-ups with more than one scalar field can in general posses a zero mode. Nevertheless, we shall provide a simple criterion that can be used to find models which do not have a zero mode. From now on we must specialize to half-interval orbifold spaces so that the scalar fields have definite parity. This enables us to formulate the following criterion:

Zero-mode criterion: For a system of definite parity with scalar fields that couple to gravity, the number of independent normalizable zero modes with is at most equal to the number of even fields.

This criterion is essentially a statement about integration constants. In the fake supergravity approach, the background configuration is given by the solutions to the first order equations, Eq (20). This is a system of linear differential equations for functions, which require Dirichlet boundary conditions for a unique solution. The value of can always be chosen to be since any other constant shift can be obtained by a redefinition of the coordinates . The restriction to orbifold models further eliminates those integration constants associated with odd-parity scalars, since their field value must also vanish at . Unique solutions to the fake supergravity equations are then parametrized by the integration constants of the even-parity fields. The final point is that zero modes move us continuously through this space of solutions, so there cannot be more zero modes than the number of even fields.

For our above argument to hold, we must show that the zero modes do indeed take us from one solution to the next. Looking at Eq. (88) it is easy to see that a solution with is satisfied when , that is

 −f′0+S00f0+S0ifi=0, (71) −f′j+S0jf0+Sijfi=0,

where the matrix elements are given explicitly in Eq. (51). Now suppose there are normalizable zero mode solutions which satisfy the above equations. Working in coordinates, we now add these zero mode solutions as perturbations to the background configuration to define the following new fields

 ¯σ(¯y) =σ(y)+ϵe3/2σ√12f(0)0(y), (72) ¯ϕi(¯y) =ϕi(y)+ϵe3/2σM3/2f(0)i(y),

where the new coordinate is expressed in terms of the zero mode perturbations as

 d¯y=dy(1+2ϵe3/2σ√12f(0)0). (73)

in the above equations is a small parameter that parametrizes the small perturbations.

Solving for the derivatives of the zero mode solutions, and , in terms of the zero mode solutions themselves, by using Eq. (71), it is easy to show that the new fields defined in Eq. (72) satisfy

 dd¯y¯σ(¯y) =16M3W({¯ϕi}), (74) dd¯y¯ϕi(¯y) =Wi({¯ϕi}),

up to first order in the perturbation parameter . We see that these new fields are themselves background solutions with the same superpotential . This demonstrates that the zero mode solutions for the system of scalar fields coupled to gravity translates from one background solution to another, taking us from one set of integration constants in the frame to another set of integration constants in the frame. Note that this result is valid up to first order in , so the zero mode solutions have to be strictly much smaller than the background solutions, restricting our reasoning to normalizable zero modes. Since any given zero mode is associated with continuously changing a Dirichlet boundary condition, the number of physical zero modes cannot be more than the number of even-parity scalars. This completes the proof of our criterion.

6.3 Explicit Examples

In this section we look at two specific domain wall models with scalar fields coupled to gravity. We have already shown that the spin-0 spectrum has strictly non-negative mass eigenmodes, and proven a criterion relating the existence of zero modes to the parities of the background scalar profiles. The two models to be presented will form explicit realizations of this criterion: one has an even scalar profile and a zero mode, the other all odd profiles and no normalizable zero mode. Aside from this difference, both models are qualitatively the same. The field will play the role of a dilaton and has a background solution which diverges at finite , generating a physical singularity and cutting off the extra dimension, effectively compactifying it. This is known as a soft wall [10, 11, 13, 17]. The second field takes the form of a kink, creating a domain-wall whose purpose is to replace the positive tension brane in usual soft-wall set-ups. The models we consider are on a half-interval orbifold space with definite parity, and the domain-wall sits at the origin, acting as an effective boundary of the extra dimension. A domain-wall soft-wall model is an appropriate name for this type of set-up. In this section we are concerned primarily in the stability of such models, and do not discuss any other phenomenology. Whether or not these models solve the hierarchy problem is an interesting question which we intend to address in future work.

Example 1

The first model we consider has the following superpotential

 W(Φ1,Φ2)=(aΦ2−bΦ32)eνΦ1, (75)

where and are the dilaton and the kink fields respectively. , and are parameters in the model. We write the fields in terms of their background solutions and perturbations as

 Φ1(x,y) =ϕ1(y)+φ1(x,y), (76) Φ2(x,y) =ϕ2(y)+φ2(x,y).

The background fields satisfy

 ddyϕ1 =ν(aϕ2−bϕ32)eνϕ1, (77)