emilio elizalde et al- casimir effect in de sitter and anti-de sitter braneworlds

8/3/2019 Emilio Elizalde et al- Casimir effect in de Sitter and Anti-de Sitter braneworlds

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YITP-02-59September 2002hep-th/0209242

Casimir effect in de Sitter and Anti-de Sitterbraneworlds

Emilio Elizalde1, Shinichi Nojiri2,Sergei D. Odintsov

3 and Sachiko Ogushi4

Department of Mathematics, Massachusetts Institute of Technology77 Massachusetts Avenue, Cambridge, MA 02139-4307

Department of Applied Physics, National Defence AcademyHashirimizu Yokosuka 239, JAPAN

Lab. for Fundamental StudyTomsk Pedagogical University, 634041 Tomsk, RUSSIA

Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto606-8502, JAPAN

ABSTRACT

We discuss the bulk Casimir effect (effective potential) for a conformal ormassive scalar when the bulk represents five-dimensional AdS or dS spacewith two or one four-dimensional dS brane, which may correspond to ouruniverse. Using zeta-regularization, the interesting conclusion is reached,that for both bulks in the one-brane limit the effective potential correspond-ing to the massive or to the conformal scalar is zero. The radion potential inthe presence of quantum corrections is found. It is demonstrated that both

the dS and the AdS braneworlds may be stabilized by using the Casimirforce only. A brief study indicates that bulk quantum effects are relevantfor brane cosmology, because they do deform the de Sitter brane. They mayalso provide a natural mechanism yielding a decrease of the four-dimensionalcosmological constant on the physical brane of the two-brane configuration.

PACS: 98.80.Hw, 04.50.+h, 11.10.Kk, 11.10.Wx

1On leave from: IEEC/CSIC, Edifici Nexus, Gran Capita 2-4, 08034 Barcelona, Spain;email: [email protected] [email protected]

2email: [email protected], [email protected]: [email protected] fellow, email: [email protected]


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Contents

1 Introduction 2

2 The Casimir effect for a de Sitter branein a five-dimensional Anti-de Sitter background 42.1 Effective potential for the brane . . . . . . . . . . . . . . . . 42.2 The one-brane limit (L ) . . . . . . . . . . . . . . . . . . 62.3 Small distance expansion . . . . . . . . . . . . . . . . . . . . 82.4 The dynamics of the brane . . . . . . . . . . . . . . . . . . . 102.5 Dynamics of two branes at small distance . . . . . . . . . . . . 132.6 Stabilization of the radion potential . . . . . . . . . . . . . . . 15

3 Casimir effect for the de Sitter branein a five-dimensional de Sitter background 19

3.1 Effective potential for the brane . . . . . . . . . . . . . . . . . 193.2 The dynamics of the brane . . . . . . . . . . . . . . . . . . . . 20

4 Effective potential for a massive scalar field in the AdS anddS bulks 234.1 Small mass limit (with L not large) . . . . . . . . . . . . . . . 254.2 Large mass limit (with L not small) . . . . . . . . . . . . . . . 264.3 Braneworld stabilization by the Casimir force . . . . . . . . . 27

5 Effective potential for a massive scalar without scalar-gravitationalcoupling 28

6 Discussion and conclusions 30

A Appendix 31

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1 Introduction

If our world is really multi-dimensional, as M-(string) theory predicts, thenone of most economical possibilities for its realization is the braneworld

paradigm. Indeed, in the case when string theory is taken in its exact vacuumstate, with the five-dimensional (asymptotically) Anti-de Sitter (AdS) sector,in a full ten-dimensional space, the corresponding effective five-dimensionaltheory represents some (gauged) supergravity. Adding the four-dimensionalsurface terms predicted by the AdS/CFT correspondence to such five-dimensionalAdS (super)gravity, one arrives at the dynamical four-dimensional boundary(brane) of this five-dimensional manifold. Depending on the structure of thesurface terms, the choice of (bulk and brane) matter, the assumptions aboutthe general structure of the brane and bulk manifold, fields content, etc., ourfour-dimensional universe can be realized in a particular way as such a brane.Brane universe can be consistent with observational data even when the ra-

dius of the extra dimension is quite significant. Moreover, the braneworldpoint of view of our universe may bring about a number of interesting mech-anisms to resolve such well-known problems as the cosmological constant andthe hierarchy problems.

As the braneworld corresponds to a five-dimensional (usually AdS) man-ifold with a four-dimensional dynamical boundary, it is clear that, when thefive-dimensional QFT is considered, the non-trivial vacuum energy (Casimireffect, see e.g. [1] for a recent review) should appear. Moreover, when thebrane QFT is considered, the non-trivial brane vacuum energy also appears.The bulk Casimir effect should conceivably play a quite remarkable role in

the construction of the consistent braneworlds. Indeed, it gives contributionto both the brane and the bulk cosmological constants. Hence, it is expectedthat it may help in the resolution of the cosmological constant problem.

For consistency, the five-dimensional braneworld should be stabilized (ra-dion stabilization) [2]. Then, the idea is that the bulk vacuum energy(Casimir contribution) may be used explicitly for realizing the radion sta-bilization in a number of models [3][14] (mainly with flat branes). An in-teresting connection between the bulk Casimir effect and supersymmetrybreaking in braneworld [15] or moving branes [16] also exists. On the otherhand, the brane Casimir effect may be used for a braneworld realization [17]of the anomaly-driven (also called Starobinsky) inflation [18].

The works mentioned above discuss mainly the Casimir effect in the sit-

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uation when the brane is flat space. But also the situation in which thebrane is more realistic, say a de Sitter (dS) universe, has been discussed inRefs. [4, 13]. It has been shown there that, in an AdS bulk, the Casimir en-ergy for the bulk conformal scalar field in a one-brane configuration is zero.

However, in situations where the bulk is different, a non zero contributionof the Casimir energy is not excluded and even a possibility may exist ofgravity trapping on the brane itself.

In the present work we study the bulk Casimir effect for a conformal ormassive scalar when the bulk is a five-dimensional AdS or a dS space andthe brane is a four-dimensional dS space. We show that zeta-regularizationtechniques at its full power [19] can be used in order to calculate the bulkeffective potential in such braneworlds, in a quite general setting. One inter-esting result we got is that, for both bulks (AdS and dS) under discussionwith one brane, the bulk effective potential is zero for a conformal as well asfor a massive scalar. Applications of our results to the stabilization of theradion and to the brane dynamics are presented as well.

The paper is organized as follows. The next section is devoted to thediscussion of a general effective potential (Casimir effect) for bulk conformalscalar on AdS when the brane is a de Sitter space. The small distancebehavior is investigated and the one-brane limit of the potential, which turnsout to be zero, is worked out. As an application, we discuss the role of theleading term of the effective potential to the brane dynamics. It is shown herethat the Casimir force only deforms the shape of the 4-dimensional sphereS4. The radion potential (in two limits), with account of the Casimir term, isfound and the stabilization of the braneworld is discussed. Using an explicit

short distance expansion for the effective potential, it is demonstrated thatthe brane may indeed be stabilized using the Casimir force only.In Sect. 3 similar questions are investigated for a conformal scalar when

the brane is S4, and bulk is a five-dimensional dS space. It is interesting thatthe effective potential turns out to be the same as in the case of the previoussection (AdS). Also, the one-brane limit of effective potential is again zero.From the study of brane dynamics it turns out that the role of the Casimirforce is again that of inducing some deformation of the S4 brane (especiallyclose to the poles).

In Sect. 4 the effective potential for a massive scalar (also with scalar-gravitational coupling) is presented, for both a dS and an AdS bulk, when

the brane is S4. The small and large mass limits are found. The one-brane

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limit of the potential is again zero, even in the massive case, but the mainnon-zero correction to this limit is obtained explicitly. Brane stabilizationdue to the Casimir force for a massive scalar is discussed when the bulk isfive-dimensional dS.

In Sect. 5 the potential for a massive scalar without a scalar-gravitationalcoupling is briefly studied for dS and AdS braneworlds. It is shown that it isagain zero in the one-brane limit. Finally, a short summary and an outlookare presented in Sect. 6.

2 The Casimir effect for a de Sitter brane

in a five-dimensional Anti-de Sitter back-ground

2.1 Effective potential for the braneIn this section, we review the calculation of the effective potential for ade Sitter (dS) brane in a five-dimensional anti-de Sitter (AdS) background,following Refs. [3, 4, 13]. First, we start with the action for a conformallyinvariant massless scalar with scalar-gravitational coupling,

S= 12

d5x

gg + 5R(5)2

, (2.1)

where 5 = 3/16, R(5) being the five-dimensional scalar curvature. Thisaction is conformally invariant under the conformal transformations: 5

g = e(x) g = e

(x) , (2.2)

where 34

= .Let us recall the expression for the Euclidean metric of the five-dimensional

AdS bulk:

ds2 = gdxdx =

l2

sinh2 z

dz2 + d24

, (2.3)

d24 = d2 + sin2 d23 , (2.4)

5Note that there is a relation between and , namely

D24

= , and D depends

on the dimensions as D24(D1) , for the general D-dimensional bulk.

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where l is the AdS radius which is related to the cosmological constant of theAdS bulk, and d3 is the metric on the 3-sphere. Two dS branes, which arefour-dimensional spheres, are placed in the AdS background. If we put onebrane at z1, which is fixed, and the other brane at z2, the distance between

the branes is given by L = |z1 z2|. When z2 tends to , namely L = ,the two-brane configuration becomes a one brane configuration.

We can see that the action, Eq. (2.1), is conformally invariant under theconformal transformations for the metric Eq. (2.3) and the scalar field, whichare given by

g = sinh2 z l2g , = sinh

3/2 z l3/2 . (2.5)

The action is changed by the conformal transformation, Eqs. (2.5), as follows.

S=

1

2d5xg

g + 5R(5)2 . (2.6)

Then, the action is conformally invariant, which leads to the following La-grangian for a conformally invariant massless scalar with a scalar-gravitationalcoupling,

L =

2z + (4) + 5R

(4)

. (2.7)

where R(4) = 12. Since we are interested in the Casimir effect for the bulkscalar in the AdS background, we shall use this Lagrangian hereafter.

The one-loop effective potential can be written as [4, 13]

V = 12LVol(M4)

log det(L5/2) , (2.8)

where

L5 = 2z (4) 5R(4) = L1 + L4 . (2.9)

To calculate the effective potential in Eq. (2.8), we use -function regulariza-tion [19], as was done in Refs. [3, 4, 13]. Being precise, the very first step inthis procedure consists in the introduction of a mass parameter in order towork with dimensionless eigenvalues, thus we should write at every instance

L5/2, etc. However, as is often done for the sake of the simplicity of the

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notation, we will just keep in mind the presence of this factor, to recoverit explicitly only in the final formulas.

First, we assume that the eigenvalues of L1 and L4 are of the form2n,

2 0 (with n, = 1, 2, ) respectively. In terms of these eigen-

values, log det L5 can be rewritten as follows:

log det L5 = Tr log L5 = Tr log(L1 + L4) =n,

log(2n + 2) (2.10)

Since the -function for an arbitrary operator A is defined by

(s|A) m

(2m)s =

m

es log 2m , (2.11)

it turns out that Tr log L5 can be rewritten as

Tr log L5 = s(s|L5)|s=0 . (2.12)

Furthermore, the -function is related to the -function and heat kernelKt(A):

(s|A) = 1(s)

0

dt ts1Kt(A), Kt(A) =m

e2mt . (2.13)

L1 is a one-dimensional Laplace operator, and the boundary conditions resultin that the brane separation L can be taken as the width of a one-dimensionalpotential well. As a consequence, the eigenvalues of L1 are be given by

2n =

n

L

2, (2.14)

for finite L.

2.2 The one-brane limit (L )The above formula leads to the heat kernel Kt(L1)

Kt(L1) n

et(nL )

2

0dyet(

yL )

2

=L

2

t, (2.15)

where the large-L limit has been taken, namely, the continuous limit of n.The heat kernel for L5 is written in terms of Kt(L1) and Kt(L4), as

Kt(L5) = Kt(L1)Kt(L4) . (2.16)

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By using Eqs. (2.13), (2.15), and (2.16), we obtain (s|L5):

(s|L5) = 1(s)

0

dtts1Kt(L1)Kt(L4) ,

L2

s 12

(s)1

s 12

0

dtt(s1

2)1Kt(L4) + O

1L

=L

2

s 12

(s)

s 12|L4

+ O

1

L

. (2.17)

Combined with Eq. (2.12), we obtain the effective potential in the large Llimit:

V = 12LVol(M4)

(0|L5/2) + ln 2(0|L5/2)

=1

2LVol(M4)12 |L4/2+ O

1

L

. (2.18)

Note that the 2 factor has to be taken into account for obtaining the deriva-tive and, as discussed before, it is in fact everywhere present in each La-grangian and its eigenvalues (although it is usually not written down in orderto simplify the notation). For the spherical brane S4 whose radius is R, thefour-dimensional zeta function (s|L4) is given by

(s|L4) = g(s)R2s, (2.19)where

g(s) =1

6

l=0

(l + 1)(l + 2)(2l + 3)

l2 + 3l +9

4

s. (2.20)

This formula can be rewritten in terms of the values of the Hurwitz zeta-function H(s, a) at negative integer argument

H(m, a) = Bm+1(a)m + 1

(2.21)

where Bm(a) is a Bernoulli polynomial. Thus, (s|L4) is given by

(s|L4) = R2s

3

H

2s 3,3

2

1

4H

2s 1,3

2

, (2.22)

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which leads to

1

2|L4

=1

3R

H

4, 3

2

1

4H

2, 3

2

= 0 . (2.23)

As a result, the effective potential Eq. (2.18) becomes zero (as first has been

observed in [4] and has been confirmed in [13]) as L . This situationcorresponds to the case of a one-brane configuration.

2.3 Small distance expansion

Using the power of the zeta regularization formulas [19, 20], a much moreprecise (albeit involved) calculation can be carried out which respects atevery step the complete structure of the five-dimensional zeta function. Thatis, the full zeta function is preserved till the end, and the final expressionis given in terms of an expansion on the brane distance L over the branecompactification radius

R, valid for L/

R 1, which complements the one

for large brane distance obtained above. A detailed calculation follows.As to the specific zeta formulas employed, adhering to the classification

that has been given in [20], the case at hand is indeed to be found there (evenif at first sight it would not seem so). It corresponds to a two-dimensionalquadratic plus linear form with truncated spectrum. In fact, this is clearfrom the structure of the spectrum yielding the zeta function

(s|L5) = 2s

n,l=0

(2n + 2l )s, (2.24)

where is a dimensional regularization scale that renders the argument of

the zeta function dimensionless. In the case of the four-dimensional sphericalbrane of radius R considered above, this reduces to

(s|L5) = 2s

6

n,l=0

(l + 1)(l + 2)(2l + 3)

n

L

2+ R2

l2 + 3l +

9

4

s. (2.25)

This zeta function looks awkward, at first sight. But after some reshufflingit can be brought to exhibit the standard structure mentioned. Specifically,

(s|L5) =

R2s

62s

n,l=0

2l + 32l + 3

2

2

+2n2R2

L2

1s

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2n2

L2+

1

4

l +

3

2

2+

2n2R2L2

s

R2s

62s

[Z1(s) + Z2(s)] , (2.26)

where both Z1(s) and Z2(s) are obtained by taking derivatives (see [21] for adiscussion of this issue, nontrivial when asymptotic expansions are involved),with respect to x at x = 3/2, of a zeta function of the class just mentioned,e.g.

l=0

(l + x)2 + q

s, q

2n2R2L2

. (2.27)

In Refs. [20], explicit formulas for the analytical continuation of this class ofzeta functions are given. To be brief (and forgetting for the moment aboutthe n-sum, for simplicity), we just have to recall the celebrated asymptoticexpansion

n=0

(n + c)2 + q

1

2 c

qs +qs

(s)

n=1

(1)n(n + s)n!

qnH(2n, c)

+

(s 1/2)

2(s)q1/2s (2.28)

+ 2

s

(s)q1/4s/2

n=1ns1/2 cos(2nc)Ks1/2 (2nq) .

After some calculations, we get for Z1(s) and Z2(s)

Z1(s) = 12 s

2R2

L2

2s(2s 4) 1

(s 1)n=1

(1)n(n + s 2)n!

2R2L2

2ns(2s + 2n 4)H(2n, 3/2), (2.29)

Z2(s) =1

1 s 2R2

L2

1s

2R2L2

(2s

4) +1

4(2s

2)

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+1

(s)

n=1

(1)n(n + s 1)n!

2R2

L2

1ns(2.30)

2R2L2

(2s + 2n 4) + 14

(2s + 2n 2)

H(2n, 3/2).

Finally, for the derivative of the five-dimensional zeta function at s = 0,we obtain

(0|L5) = (4)

6

4R4L4

+(2)

12

2R2L2

+1

24

H(4, 3/2)

1

2H(2, 3/2)

ln

2R2L2

+(0)

6

H(4, 3/2)

1

2H(2, 3/2)

+

1

24H(4, 3/2)

+1

361

8H(4, 3/2) 1

3H(6, 3/2) L2

R2 + O L4

4R4

0.129652 R4

L4 0.025039 R

2

L2 0.002951 ln R

2

L2

0.017956 0.000315 L2

R2 + (2.31)

2.4 The dynamics of the brane

We now consider the dynamics of the dS brane, which is taken to be thefour-dimensional sphere S4, as in Ref. [4]. The bulk part is given by five-

dimensional Euclidean Anti-de Sitter space, Eq. (2.3), which can be rewrittenas

ds2AdS5 = dy2 + l2 sinh2

y

ld24 . (2.32)

One also assumes that the boundary (brane) lies at y = y0 and the bulkspace is obtained by gluing two regions, given by 0 y < y0 (see [17] formore details.)

We start with the action S which is the sum of the Einstein-Hilbert actionSEH, the Gibbons-Hawking surface term SGH [22], and the surface counter-term S1, e.g.

S = SEH + SGH + 2S1 (2.33)

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SEH =1

16G

d5x

g(5)

R(5) +

12

l2

(2.34)

SGH =1

8G

d4x

g(4)n (2.35)

S1 = 38Gl

d4xg(4) . (2.36)

Hereafter the quantities in the five-dimensional bulk spacetime are specifiedby the subindices (5) and those in the boundary four-dimensional spacetimeare by (4). The factor 2 in front of S1 in (2.33) is coming from the fact thatwe have two bulk regions, which are connected with each other by the brane.In (2.35), n is the unit vector normal to the boundary.

If we change the coordinate in the metric of S4, Eq. (2.4), to by

sin = 1cosh

, (2.37)

we obtain

d24 =1

cosh2

d2 + d23

. (2.38)

For later convenience, one can rewrite the metric of the five-dimensionalspace, Eqs. (2.32), (2.38), as follows:

ds2 = dy2 + e2A(y,)gdxdx , gdx

dx l2

d2 + d23

. (2.39)

From Eq. (2.39), the actions (2.34), (2.35), (2.36), have the following forms:

SEH = l4

V316G

dyd

82yA 20(yA)2 e4A+62A 6(A)2 + 6

e2Al2

+12

l2e4A

(2.40)

SGH =4l4V38G

de4AyA (2.41)

S1 = 3l3V3

8G

de4A . (2.42)

Here V3 =

d3 is the volume (or area) of the unit three-dimensional sphere.As it follows from the discussion in the previous subsections, there is a

gravitational Casimir contribution coming from bulk quantum fields. As one

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sees in the simple example of a bulk scalar, SCsmr (leading term) has typicallythe following form

SCsmr =cV3R5

dydeA. (2.43)

Here c is some coefficient, whose value and sign depend on the type of bulkfield (scalar, spinor, vector, graviton, ...) and on parameters of the bulktheory (mass, scalar-gravitational coupling constant, etc). In a previous sub-section we have found this coefficient for a conformal scalar. For the followingdiscussion it is more convenient to consider this coefficient to be some pa-rameter of the theory. Doing so, the results are quite common and may beapplied to an arbitrary quantum bulk theory. We also assume that there areno background bulk fields in the theory (except for the bulk gravitationalfield).

Adding the quantum bulk contribution to the action S in (2.33), one canregard

Stotal = S+ SCsmr (2.44)

as the total action. In (2.43), R is the radius of S4.In the bulk, one obtains the following equation of motion from SEH+SCsmr

by variation over A:

0 =242yA 48(yA)2 +

48

l2

e4A

+1

l2

122A 12(A)2 + 12

e2A +

16Gc

R5l4 eA . (2.45)

Let us discuss the solution in the situation when the scale factor depends onboth coordinates: y and . In Ref. [4], the solution of (2.45) given by anexpansion with respect to e

yl was found by assuming that yl is large:

eA =sinh ylcosh

32Gcl3

15R5 cosh4 e

4yl + O

e

5yl

(2.46)

for the perturbation from the solution where the brane is S4. On the braneat the boundary, one gets the following equation:

0 =

yA 1l

e4A. (2.47)

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Substituting the solutions (2.46) into (2.47), we find that

0

1

R

1 +

R2l2

+2Gl2c

3

R10

cosh5 1l

. (2.48)

Eq. (2.48) tells us that the Casimir force deforms the shape of S 4, since Rdepends on . The effect becomes larger for large . In the case of a S4 brane,the effect becomes large if the distance from the equator becomes large, since is related to the angle coordinate by (2.37). In particular, at the northand south poles ( = 0, ), cosh diverges and then R should vanish. Ofcourse, the perturbation would be invalid when cosh is large. Thus, wehave demonstrated that bulk quantum effects do support the creation of ade Sitter brane-world Universe.

We now consider the case when the bulk quantum effects are the leadingones. From Eq. (2.48), one obtains

R8 4Glc3

cosh5 . (2.49)

Here we only consider the leading term with respect to c, which correspondsto the large R approximation. Thus, we have demonstrated that bulk quan-tum effects do not violate (in some cases they even support) the creation ofa de Sitter brane living in a five-dimensional AdS background.

2.5 Dynamics of two branes at small distance

In this subsection, we consider the dynamics of two dS branes when thedistance between them is small. Before including the Casimir effect, weconsider the following actions.

S = SEH +a=

(SGH + 2S1) (2.50)

SEH =1

16G

d5x

g(5)

R(5) +

12

l2

(2.51)

SGH = 1

8G

d4x

g(4)n (2.52)

S1 = 3

8Gl

d

4

xg(4) . (2.53)

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Here the index a = distinguishes the two branes and we assume that theradius R+ (R) corresponds to the larger (smaller) brane. The bulk spaceis AdS again and, on the branes, we obtain the following equations:

1R

1 + R2

l2= 1

l. (2.54)

The left-hand side in (2.54) is a monotonically decreasing function with re-spect to R. Since the left-hand side becomes + when R 0 and 1

lwhen

R +, there is a solution whenl > l+ > l . (2.55)

We now include the Casimir effect. First, we consider the backreaction tothe bulk geometry. As we assume the distance between the branes is small,the radius of the branes are almost constant. The distance L in (2.31) is given

by |z+ z|, the energy density by the Casimir effect would be proportionalto e

5A

L5 . Then the effective action would be

SCsmr =cV3L5

dydeA. (2.56)

Therefore, as in the previous section, the bulk geometry would be deformedas

eA =sinh ylcosh

32Gcl3

15L5cosh4 e

4yl + O

e

5yl

. (2.57)

In this case, the equation of the brane corresponding to (2.48), has the fol-

lowing form

0 1

R

1 +R2

l2 2Gl

2c

3L10cosh5 1

l

. (2.58)

Eq. (2.58) tells us that the Casimir force deforms the shape of S4 and theeffect becomes larger for large , again, as in the previous section. Weshould note, however, the signs of the contribution from the Casimir effectare different for the larger and smaller branes. Then if the radius of thelarger brane becomes large (small), that in the smaller one it becomes small(large). It is interesting that if larger brane is physical universe, this may

serve as dynamical mechanism of decreasing of the cosmological constant.

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2.6 Stabilization of the radion potential

In this subsection, we consider the stabilization of the radion potential fol-lowing Ref. [2]. As first setup, we prepare the suitable metric and action forthe discussion of the stabilization of the radion potential.

ds2 = e2krc||dxdx r2cd2 (2.59)Here is the coordinate on five-dimensions and x are the coordinates onthe four-dimensional surfaces of constant , and with (x, )and (x, ) identified. The coordinate z in the metric (2.3) corresponds toekrc/k in Eq. (2.59), and the distance between two branes L corresponds to(ekrc ekrc)/k.

We assume that a potential can arise classically from the presence of abulk scalar with interaction terms that are localized at the two 3-branes. Theaction of the model with scalar field is given by

Sb = 12

dx4

dG

GABAB m22

, (2.60)

where GAB with A, B = , as in Eq. (2.59). The interaction terms on thehidden and visible branes (at = 0 and = respectively) are also givenby

Sh =

d4xghh(2 v2h)2 , (2.61)

Sv =

d4xgvv(2 v2v)2 , (2.62)

where gh and gv are the determinants of the induced metric on the hidden

and visible branes respectively.The general solution for which only depends on the coordinate is

taken from the equation of motion of the action with respect to , to havethe following form:

() = e2[Ae + Be ] , (2.63)

where = krc|| and =

4 + m2/k2. Substituting this solution (2.63)into the action and integrating over yields an effective four-dimensionalpotential for rc which has the form [2]

V(rc) = k(+ 2)A2(e2krc

1) + k(

2)B2(1

e2krc)

+ve4krc(()2 v2v)2 + h((0)2 v2h)2 (2.64)

15


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The unknown coefficients A and B are determined by imposing appropriateboundary conditions on the 3-branes. Recalling Ref. [2], the coefficients Aand B are given by

A = vve(2+)krc

vhe2krc

, (2.65)B = vh(1 + e

2krc) vve(2+)krc , (2.66)

for large krc limit. Here we take (0) = vh and () = vv.For the large krc limit, which corresponds to the large L limit, since

L = (ekrc ekrc)/k, we assume that the effective potential includes theterm induced by the Casimir effect as

Ldiscussed in subsection 2.2, where

is some constant. Thus, we shall add this term to the potential (2.64)hereafter.6 Then the effective potential in the large krc limit becomes

V(rc) = kv2h + 4ke

4krc vv vhe

krc2

1 +

4

kvhe(4+)krc(2vv vhekrc) + k(ekrc ekrc) . (2.67)

We suppose that m/k 1 so that = 2 + with m2/4k2 a smallquantity.

With the purpose of obtaining the minimum of the potential, we differ-entiate it with respect to rc :

d

drcV(rc) = 16k2e4krc

vv vhekrc

2 1 +

4

+8vhk2e4krc

vv vhekrc 1 + 4+k2vh(4 + )e

(4+)krc(2vv vhekrc)+k22v2he

(4+)krcekrc k2(ekrc + ekrc)

(ekrc ekrc)2 .(2.68)

The minimum of the potential is reached for

rc = r0 + rc ,

r0 =

4

k

m2ln

vhvv

. (2.69)

6Note that a Casimir term may be induced by other bulk fields.

16


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Substituting Eq. (2.69) into ddrc

V(rc) = 0 in Eq. (2.68), we get rc as

rc = (ekr0 + ekr0)e(4+)kr0

16kvhvv(ekr0 ekr0)2 +1

4k

(e5kr0 + e3kr0)16kvhvv(ekr0 ekr0)2 , (2.70)

where terms of order 2 are neglected. The role of Casimir effect is in onlyto shift slightly the minimum.

In the small krc limit, which corresponds to the small L limit as well, thecoefficients A and B in the radion potential (2.64) are changed as follows

A =1

2krc{vv(1 + krc( 2)) vh} (2.71)

B =1

2krc {vv(1 + krc(

2)) + vh(1 + 2krc)

}. (2.72)

In this limit, we suppose that m/k 1, so that m/k. The effectivepotential might include the term induced by the Casimir effect as

L5discussed

in subsection 2.3, where is some constant. Then, the radion potential inthe small krc limit is

V(rc) = 2mrck

m

k+ 2

A2 + 2mrck

m

k 2

B2 +

L5

1rc

(vv vh)2 + (2rc)5

, (2.73)

being here L 2rc. To obtain the minimum of the potential, we differenti-ate Eq. (2.73) with respect to rc:

d

drcV(rc) = 1

r2c(vv vh)2 5

(2)5r6c. (2.74)

Then, if 0, the minimum of the potential is reached at

rc = 12(vv vh)1/2

5

2

1/4. (2.75)

Therefore, the role of the Casimir effect in brane stabilization is seen to be

essential. Let us give some numbers. If vv, vh (1019GeV)3

2 and

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(1019GeV)1

, we have that rc (1019GeV)1 and krc could be of O(1).Thus, it is not so unnatural for the hierarchy problem.

For the short rc case, we may not include the scalar field in (2.60)but instead we may include the next-to-leading order of the effective poten-

tial (2.31), induced by the Casimir effect, although the next-to-leading termshould be neglected for the flat brane corresponding to R +:

VC(rc) =1

(2rc)5 +

2

(2rc)3 . (2.76)

From (2.31), we see that 1 > 0 and 2 < 0. As a consequence, in the abovepotential, there is a minimum at

rc =1

2

51

32 0.4675l . (2.77)

The result in (2.31) is not for flat brane but for de Sitter brane and onlyincluding the contribution from massless scalar. We also put a length pa-rameter l in (2.77). Then the numerical value in (2.77) would be changed buthopefully the main structure would not be changed. We conclude, therefore,that with the only consideration of the Casimir effect, the brane might getstabilized, which is a nice result.7

As we will see later in (4.10), when one considers the massive scalar withsmall mass, there appears the correction to the effective potential. Moti-vated with such result, one considers the following correction to the effectivepotential, which corresponds to the leading term in (4.10) when L is small:

VC(rc) =

3m2

2rc . (2.78)

Here m expresses the mass of the scalar field. The result in (4.10) suggeststhat 3 is negative. By assuming that the correction term (2.78) is dominantcompared with the third (logarithmic) term in (2.31), the minimum in (2.77)is shifted as

rc =1

2

51

32

1 +

513m2

1822+ O

m4

. (2.79)

Then the contribution from small mass has a tendency to make the distancebetween the two branes smaller.

7Note however that thermal effects [5] may significantly change the above discussion.

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3 Casimir effect for the de Sitter brane

in a five-dimensional de Sitter background

3.1 Effective potential for the braneNext, we use the Euclideanised form of the five-dimensional de Sitter (dS)metric for a four-dimensional dS brane as follows:

ds2 = l2

d2 + sin2 d24

, (3.1)

=l2

cosh2 z

dz2 + d24

,

where l is the dS radius, which is related to the cosmological constant of thedS bulk.

We place two dS branes which are four-dimensional spheres, as in the

AdS bulk case in a dS background as the one depicted in Fig. 1. Since theparameter in Eq. (3.1) takes values between 0 and , the parameter z takesvalues between and . As in the AdS bulk case, the distance betweenthe branes can be defined as L = |z1 z2|. When z2 is placed at , namelyL = , the two-brane configuration becomes a one-brane configuration, asseen in Fig. 1.

LS5

S4

L

S4S4

Figure 1: The two dS branes are placed in the dS5 background. The two-brane configuration becomes a one-brane configuration as L .

The Casimir effect for the bulk scalar in dS background can be calculated

by using the same method as in AdS bulk.

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Namely, the Lagrangian for a conformally invariant massless scalar withscalar-gravitational coupling, is obtained by conformal transformation of theaction, Eq. (2.1), for the metric and the scalar field given by

g = cosh2

z l

2

g , = cosh

3/2

z l3/2

. (3.2)Then the Lagrangian is of the same form of Eq. (2.7).

The one-loop effective potential is calculated by means of -function reg-ularization techniques. Then, the calculated result for the effective potentialin the large L limit is of the same form of Eq. (2.18). Since the effectivepotential in Eq. (2.18) becomes zero at L , the effective potential of theone-brane configuration becomes zero. Note that this means that the effec-tive potential for B5, which is the right part in Fig. 1, is zero. Concerningthe small distance expansion, for a potential corresponding to a conformallyinvariant scalar we have an expression as Eq. (2.31). No essential difference

is encountered in this case.

3.2 The dynamics of the brane

The dynamics of dS brane in a five-dimensional Euclidean de Sitter bulk canbe considered in a similar way as for the AdS bulk. The brane is de Sitter,and is taken to be a four-dimensional sphere S4, as in the previous section.The five-dimensional Euclidean de Sitter space Eq. (3.1) can be rewritten as

ds2dS5 = dy2 + sin2

y

ld24 . (3.3)

Here, we adopt Eq. (2.38) for the metric of S4. We assume that the branelies at y = y0 and that the bulk is obtained by gluing two regions given by0 y < y0.

The total action S is the sum of the Einstein-Hilbert action SEH, theGibbons-Hawking surface term SGH, and the surface counter term S1: likein the AdS bulk case:

S = SEH + SGH + 2S1 . (3.4)

The Einstein-Hilbert action SEH is

SEH =

1

16G

d5

xg(5) R(5) 12

l2

(3.5)

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The Gibbons-Hawking surface term SGH and the surface counter term S1 areof the same forms as in Eqs. (2.35), (2.36).

For later convenience, we rewrite the metric of the five-dimensional dSspace, Eqs. (3.3), (2.38), as follows:

ds2 = dy2 + e2A(y,)gdxdx , gdx

dx l2

d2 + d23

(3.6)

By using Eq. (3.6), the action Eq. (3.5) becomes

SEH =l4V3

16G

dyd

82yA 20(yA)2

e4A

+62A 6(A)2 + 6

e2Al2

12l2

e4A

. (3.7)

which is similar to the AdS bulk case, Eq. (2.40), except for the last term.i.e. the cosmological constant. The Gibbons-Hawking surface term, SGH, and

the surface counter term, S1, Eqs. (2.35), (2.36), have also the same form ofEqs. (2.41), (2.42). We also consider the gravitational Casimir contributiondue to bulk quantum fields. So we add the action of the Casimir effect, SCsmr,(2.43) to the total action S (3.4).

In the bulk, we obtain the following equation of motion from SEH + SCsmrby variation over A:

0 =242yA 48(yA)2

48

l2

e4A

+1

l2

122A 12(A)2 + 12

e2A +

16Gc

R5l4 eA . (3.8)

For the AdS bulk case, the solution of (3.8) can be found as an expansionwith respect to e

yl , assuming that y

lis large. But for the dS bulk case, we

cannot adopt the same method, since the function sin yl cannot be regarded

as an expansion with respect to eyl . Thus, we assume the solution to have

the following form

eA =sin y

l

cosh + A . (3.9)

Substituting Eq. (3.9) into Eq. (3.8), we obtain

0 =

1

l2

6sin yl

cosh +

cosh

sin yl 2

cosh sin yl

A

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4l

cos ylcosh

y(A) 2sinyl

cosh 2y(A)

cosh l2 sin y

l

2(A) 4Gc

3

R5l4

cosh

sin yl

3. (3.10)

We now investigate the behavior of Eq. (3.10) at the north and south poles( = 0, ), that is, as cosh diverges. In this case, Eq. (3.10) is approximatedas

0 e

2l2 sin ylA e

2l2 sin yl2(A)

4Gc

3R5l4

e

2sin yl

3, (3.11)

and then

A 2(A) Gc

3

R5l2

e2

sin2 yl

. (3.12)

Here, we have used the approximation cosh e2 . From Eq. (3.12), weassume

A = e2

sin2 yl, (3.13)

where is the constant which is obtained by substituting Eq. (3.13) intoEq. (3.12), thus

= Gc9R

5l2. (3.14)

The region of the equator = /2, namely, cosh 1 + 12

2, Eq. (3.10),is approximated as

0

1

l2

6sin

y

l+

2

sin yl

A

+4

lcos

y

ly(A) + 2 sin

y

l2y(A)

1 1

22

. (3.15)

On the brane at the boundary, we get the same equation Eq. (2.47):

0 =

yA 1

l

e4A

. (3.16)

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Finally, by substituting the solutions (3.9) into (3.16), we find

0 =1

l cosh

cos

y

l sin y

l

+ y(A) . (3.17)

In the region at the north and south poles, cosh e||/2, if we assumey = 4 l + y, from Eq. (3.17), y is obtained by

y =

2Gc

9R5l e3|| . (3.18)

Thus, the deformation of the brane becomes large at the north and southpole.

4 Effective potential for a massive scalar field

in the AdS and dS bulks

Until now we have dealt with a massless scalar. In this section we willconsider a massive scalar field in AdS and dS backgrounds. Let us start withthe action for a massive scalar with scalar-gravitational coupling,

S= 12

d5x

gg m22 + 5R(5)2

, (4.1)

For the AdS background with the metric Eq. (2.3), under the conformaltransformations (2.5), the action changes as

S= 12

d5x

gg m2l2 sinh2 z2 + 5R(5)2

, (4.2)

which yields the Lagrangian for the massive scalar field with scalar-gravitationalcoupling in an AdS background as

L =

2z + (4) m2l2 sinh2 z + 5R(4)

. (4.3)

In the above Lagrangian, there appears a singularity at z = 0. The pointz = 0 corresponds to , where the warp factor blows up to infinity. Thenby putting a brane as the boundary of the bulk, say putting a brane at

z = z0 < 0 (or z0 > 0) and considering the region z < z0 (or z > z0 as bulk

23


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space, the singularity does not appear. And as we can see in Appendix A, ifwe include the singular point z = 0, a half of the solutions are excluded butthere remain other half of solutions. From this Lagrangian, we can calculatethe one-loop effective potential like in the case of a massless scalar field. The

form of the effective potential from the massive scalar field is given by

V =1

2LVol(M4)log det(L5/

2) ,

L5 2z + m2l2 sinh2 z (4) 5R(4) = L1 + L4 , (4.4)

where the mass term is included in L1. The eigenvalue ofL1 is different fromthat in Eq. (2.14), for finite L, since L1 in Eq. (4.4) is the one-dimensionalSchrodinger operator with the potential term m2l2 sinh2 z. But this poten-tial term, which is positive valued and has no bound state, becomes zeroin the limit z2

, that is, when the distance between branes L becomes

. In this case, the eigenvalue of L1 reduces to the same form of Eq. (2.14)and thus the effective potential becomes zero at the limit of a one-braneconfiguration.

For the case of a dS background, Eq. (3.1), the conformal transformations,Eqs. (3.2) change the action (4.1) as follows:

S= 12

d5x

gg m2 cosh2 z2 + 5R(5)2

. (4.5)

Then, the Lagrangian for a massive scalar field in the dS background is givenby

L = 2z + (4) m2 cosh2 z + 5R(4) . (4.6)Similarly, the effective potential for the massive scalar field in the dS bulkcan be calculated as in Eqs. (2.8), (4.4), by using the operators:

L5 2z + m2 cosh2 z (4) 5R(4) = L1 + L4 , (4.7)

where the mass term is included in L1. The potential term ofL1, m2 cosh2 z,

has always a positive value and no bound state like in the AdS case. Itbecomes zero in the limit z2 as well. Therefore, the effective potentialfor the massive scalar field in a dS background also becomes zero in the limit

of a one-brane configuration.

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4.1 Small mass limit (with L not large)

Continuing with the massive scalar field, and for a de Sitter brane in an AdSbulk, in the case of the two brane configuration we just need to supplementthe calculation carried out in Appendix A, which can be done exactly, withthe boundary conditions imposed on the two branes. We thus obtain amodification of a perfectly solvable model which appears in several textbooks(namely, an hyperbolic variant of the celebrated Poschl-Teller potential),albeit with reverse sign and supplemented with the infinite well created bythe branes (as in the massless case). Since we shall deal with the low andhigh mass approximations, the WKB method turns out to be well suited tocarry out the analysis.

Setting the branes at z = L/2 (for the sake of symmetry) we get thefollowing results. In the small mass limit, we obtain a modification of theeigenvalues of the L1 Lagrangian, in the form

2n 2n2

2L2+ m2l2

tanh(L/2)

L/2. (4.8)

Carrying this into the zeta function, after a further approximation one getsthat the elementary zeta functions in the formulas are modified in the way,e.g.

(2s) (2s) s(2s + 2) + s(1 + s)2

(2s + 4)2 + O(m6),

m2l22L2

2

tanh(L/2)

L/2

. (4.9)

Thus, in the case here considered, when m is small and L is not very large, forthe derivative of the zeta function at z = 0 we obtain the following additionalterms (l22 = 1):

(0|L5) a + a22

48

2

144

a2

2+ [2(4, 3/2) (2, 3/2)]

4

4370[2(4, 3/2) (2, 3/2)] 2 + O(m6), (4.10)

a

2

R2

L2,

m2l2

2

tanh(L/2l)

L/2l.

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These terms have just to be added to the derivative of the zeta function atz = 0, Eq. (2.31), corresponding to the de Sitter brane in AdS bulk, in orderto obtain the corresponding effective potential. In a full-fledged analysis ofthe different contributions to the effective potential, one has to take into

account the relative importance of the different dimensionless ratios involvedhere. The working hypothesis has been that m2 was small. In fact, we seefrom the final result that m2 most naturally goes with l2, which also servesas a unit for L and, indirectly, for R. The ordering in Eq. (4.10) assumesthat a 1, < 1, but a lot more information can be extracted from thissmall-mass expansion.

The calculation in the same case of a massive scalar field but for a deSitter brane in a dS bulk (two and one brane configurations) proceeds in aquite similar fashion. Only, an additional coordinate change is required atthe beguining, to deal with the problem of the singularity of the potentialof the Schrodinger equation at z = 0 in the initial coordinates, as carefullyexplained in the Appendix.

4.2 Large mass limit (with L not small)

In this case the calculation turns out to be more involved. The eigenvaluesget modified as follows:

2n 2n2l2

L2+

2 arctan(sinh L/2l)

sinh(L/2l)m2l2 +

nml2

L sinh(L/2l)+ (4.11)

However, we will be interested in the dominant contribution only. Thus, inthe approximation which is opposite to the previous one, namely when m2

is large and L is not very small, we get a simple modification of the relevantzeta function, of the form

(s|L5) = L2l

(s 1/2)(s)

s 12

L4 + 2m2 arctan(sinh(L/2l))sinh(L/2l)

+ (4.12)

And this leads to the following result, for the derivative of the zeta functionat z = 0, which is valid for sufficiently large scalar mass and L:

(0|L5) = 4m2l3

3Rarctan(sinh L/2l)

sinh(L/2l) + (4.13)

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Again, this is the additional contribution to the derivative of the full zetafunction at z = 0, the same Eq. (2.23) but corresponding to the de Sitter case.However, as this derivative was equal to zero in the massless case, the aboveexpression yields now the whole value of the derivative and, correspondingly,

of the effective potential. Note in fact that this reduces to zero, exponentiallyfast, in the one-brane limit (L ), in perfect accordance with Eq. (2.23).Also in this case we are allowed to play with the relative values of the differentdimensionless fractions appearing in our expression.

4.3 Braneworld stabilization by the Casimir force

In [12], the brane stabilization via study of radion potential in the LorentziandeSitter bulk space was discussed in direct analogy with AdS case. Thebranes are spacelike and the distance between two branes is time-like and wedenote the distance by T. As in (2.76-2.79), we now consider the contribution

from the Casimir effect to the stabilization. For simplicity, we do not includethe massive scalar field as in (2.60) but we take the next-to-leading order ofthe effective potential (2.31), induced by the Casimir effect, and we assume:

VC(T) =dS1T5

+dS2T3

. (4.14)

If dS1 > 0 and dS2 < 0 as in (2.31), there is a minimum at

T =

5dS13dS2

. (4.15)

Then even for the branes in the deSitter bulk, only by the Casimir effect, thebrane might get stabilized.

As in (4.10), when we consider the Casimir effect from the massive scalarwith small mass, we may consider the following correction to the effectivepotential:

VC(T) =3m

2

T. (4.16)

Here m expresses the mass of the scalar field. Then the minimum in (4.15)is shifted as

rc =

5dS1

3dS2

1 +

5dS1 dS3 m

2

18dS2 2 + O m4

. (4.17)

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Then again the contribution from small mass has a tendency to make the dis-tance between the two branes smaller. Thus, the possibility of dS braneworldstabilization occurs in the same way as with AdS bulk.

5 Effective potential for a massive scalar with-

out scalar-gravitational coupling

In this section we will consider a more simple case, which does not include ascalar-gravitational coupling term, 5R

(5)2. The action is simply

S= 12

d5x

gg m22

. (5.1)

This action is not conformally invariant under the conformal transformations(2.2), which change it as

S = 12

d5x

ge3 g

e

3

2

e3

2

m2e22

=1

2

d5x

g

g 9

4g

2 + 3g m2e22

.(5.2)

where we take = 2 and = 32 for simplicity. The third term in Eq. (5.2)can be rewritten as

g =1

2D

21

22(5) (5.3)

and using partial integration, we obtain

S = 12

d5x

gg

9

4g +

3

2(5)

2 m2e22

.

(5.4)

If we now introduce the AdS background, which has the metric Eq. (2.3),under the conformal transformations (2.5), namely e2 = l2 sinh2 z, theaction changes as

S = 12

d5x

gg

9

4+

15

4sinh2 z

2 m2l2 sinh2 z2

.

(5.5)

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This action leads the Lagrangian for the massive scalar field without scalar-gravitational coupling in an AdS background as

L =

2z + (4)

9

4+

15

4sinh2 z

m2l2 sinh2 z

. (5.6)

Note that the third term in Eq. (5.6),

9

4+

15

4sinh2 z

, (5.7)

corresponds to

5

R(4) R(5)e2

, (5.8)

coming from Eqs. (2.1), (2.7), where e2 = l2 sinh2 z, because if we put5 = 3/16, R(4) = 12, R(5) = 20l2 , which are the scalar curvatures of S4and AdS5, respectively, into Eq. (5.8), then Eq. (5.8) coincides with Eq. (5.7)exactly.

The one-loop effective potential can be written as

V =1

2LVol(M4)log det(L5/

2) ,

L5 = 2z (4) +

9

4+

15

4sinh2 z

+ m2l2 sinh2 z = L1 + L4 ,

L1 = 2z +15

4sinh2 z + m2l2 sinh2 z, L4 =

9

4 (4) . (5.9)

Then, the eigenvalue of L1 agrees with Eq. (2.14) in the limit when thedistance between the two brane becomes infinity, L , because the po-tential terms of (5.9), 154 sinh

2 z + m2l2 sinh2 z, become zero in this limit.Therefore, the effective potential for the massive scalar field without scalar-gravitational coupling in an AdS background becomes zero in the limit of theone-brane configuration.

Similarly, the Lagrangian for the massive scalar field without scalar-gravitational coupling in a dS background can be seen to be

L =

2z + (4)

9

4 3

4cosh2 z

m2l2 cosh2 z

. (5.10)

In the limit L , the eigenvalue of L1 and the heat kernel Kt(L1) havethe same form of Eqs. (2.14), (2.15) as in the AdS case. Thus, the effectivepotential becomes zero too, in the limit when the distance between the two

branes becomes infinite.

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6 Discussion and conclusions

To summarize, in this paper we have shown how one can bring the calculationof the effective potential for a massive or conformal bulk scalar, in an AdS or

dS braneworld with a dS brane, down to well-known cases corresponding tozeta-function expansions [19]. In this way, a complete and detailed analysisof the different situations can be given, and corrections to the limiting casesare obtainable at any order. As our four-dimensional universe is (or willbe) in a dS phase, our results have, potentially, very interesting applicationsto primordial cosmology. What is also important, our method and resultshere open the door to corresponding calculations for other quantum fieldsas spinors, vectors, graviton, gravitino, etc. As we see it, this will onlyneed some more involved calculations, but no new conceptual problems areexpected, at least at the level of the one-loop efective potential. In the caseof several spin fields, the bulk Casimir effect may also be found in this way, at

least in principle, for supersymmetric theories, including supergravity too. Itis quite possible then, that a five-dimensional AdS gauged supergravity can beconstructed, with AdS being the vacuum state but still having a dynamicallyrealized de Sitter brane, which represents our observable universe.

Another issue where bulk quantum effects may play a dominant role in-volves moving, curved branes. The corresponding bulk effective potentialmight sometimes be a measure of supersymmetry breaking, and thus be ofprimordial cosmological importance in the study of the very early brane uni-verse.

Finally, the bulk effective potential in realistic SUSY theories gives a non-

trivial contribution to the effective cosmological constant, in five as well asin four dimensions. Hence, it is conceivable to use it in a relaxation of thecosmological constant problem.

Acknowledgements

EE is indebted with the Mathematics Department, MIT, and specially withDan Freedman for warm hospitality. Very interesting discussions with BobJaffe and collaborators at CTF, MIT, on the Casimir effect are acknowl-edged. SDO thanks A. Starobinsky and S. Zerbini for helpful discussions

on related questions and the IEEC, where this work was initiated, for warm

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hospitality. The research by EE is supported in part by DGI/SGPI (Spain),project BFM2000-0810, and by CIRIT (Generalitat de Catalunya), contract1999SGR-00257. The research by SN is supported in part by the Ministryof Education, Science, Sports and Culture of Japan under the grant number

13135208. The research by SO is supported in part by the Japanese Societyfor the Promotion of Science under the Postdoctoral Research Programme.

A Appendix

We consider the following Schrodinger equation d

2

dz2+

m2l2

sinh2 z

= . (A.1)

This equation is the z-dependent part of the Klein-Gordon equation in AdS5

and = sinh3

2 z corresponds to the original scalar field in the action. Thelimit z = corresponds to the infinity in AdS5 at which the warp factorvanishes, and z = 0 corresponds to the infinity where the warp factor growsup to infinity. In (A.1) there appears a singularity at z = 0. As the pointz = 0 corresponding to , by putting a brane as the boundary of the bulk,say putting a brane at z = z0 < 0 (or z0 > 0), and considering the regionz < z0 (or z > z0) as bulk space, the singularity does not appear.

With the redefinitions

= sinh1

2 z , x = cosh z , (A.2)

Eq. (A.1) can be rewritten as

0 =

x2 1 d2

dx2+ 2x

d

dx 1

4+

m2l2 + 14x2 1

, (A.3)

whose solutions are given by the associated Legendre functions P (x), whichare defined in terms of the Gauss hypergeometric function:

P (z) =1

(1 )

x + 1

x 1

2

F, + 1, 1 ; 1 x

2

. (A.4)

The parameters and are here given by

2

= l2

m2

+1

4 , (+ 1) = 1

4 or = 1

4

2 . (A.5)

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When x is large, P (x) behaves as

P (x) 1

+ 12

(2x)

(

+ 1)

++ 1

2

(

) (2x)+1

. (A.6)

Since x12 , then in order that is regular there, we have the constraintthat

4 0 or 0 , (A.7)which is identical with what we have in the massless case. When we includethe point z = 0, which corresponds to x = 1, when

x 1 z 0,

Eq. (A.4) becomes singular for positive as (x1)2 z. As z 12 z1

2 = z

1

2(1

1+4l2m2), the positive branch of should be excluded and we

must have =

l2m2 + 14

.If we do not include the brane, the spectrum for the massive case is not

changed. In order to investigate the effect of the mass, we put a brane atx = x0 1 (or z = z0). On the brane, we impose the Neumann boundarycondition for :

d

dz= 0 ,

d

dx= 0

. (A.8)

For simplicity, we consider the model where the bulk space includes the point

x = 1 (z = 0); hence =

l2m2 + 14

. We write and in (A.5) as

= 12

, = 12

+ i . (A.9)

Then we have = 2. By using (A.6), we find, for large x,

(x) (i)(i + k)

(2x)i +(i)

(i + k) (2x)i . (A.10)

Then the boundary condition (A.8) yields

(i)

(i + k)(2x0)

i (i)(i + k) (2x0)

i . (A.11)

If we assume and k to be small, the Gamma function can be approximated

by (i) 1i and (i+k) 1i+k . Then, Eq. (A.11) can be rewritten

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as

ln

1 + i k

1 i k

= i ln(2x0) + 2in (n = 0, 1, 2, ) . (A.12)

For large x0, the solution for n = 0 is given by

ln(2x0), (A.13)

for non-vanishing k (m = 0), which gives the following lower bound for :

= 2

ln(2x0)

2

2

z20. (A.14)

We now consider the equation for the dS case: d

2

dz2+

m2l2

cosh2 z

= . (A.15)

This equation is the z-dependent part of the Klein-Gordon equation in S5or Euclidean de Sitter space, and = cosh

3

2 z corresponds to the originalscalar field in the action. The limit ofz = corresponds to the south andnorth poles in S5. With the following redefinitions,

= cosh1

2 z , x = cosh z , (A.16)

Eq. (A.15) can be rewritten as

0 =

x2 + 1 d2

dx2+ 2x

d

dx 1

4+

m2l2 + 14

x2 + 1

. (A.17)

If we replace x by x = iy, the above equation (A.17) turns into

0 =

y2 1 d2

dy2+ 2x

d

dx 1

4 m

2l2 + 14

y2 1

. (A.18)

Finally, if we choose, as in (A.5),

2 =

l2m2 +1

4

, (+ 1) = 1

4or =

1 42

, (A.19)

the solution of Eq. (A.18) or (A.17) is given by the associated Legendrefunctions P (ix), again. Note that in (A.19) is imaginary, in general.Anyhow, in order that be regular there, we must impose again the same

constraint (A.7).

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References

[1] K. Milton, The Casimir effect: Physical Manifestations of Zero-PointEnergy, World Sci., Singapore, 2001.

[2] W.D. Goldberger and M.B. Wise, Phys.Rev.Lett.83(1999) 4922-4925,hep-ph/9907447.

[3] J. Garriga, O. Pujolas and T. Tanaka, Nucl.Phys. B605(2001)192-214,hep-th/0004109; hep-th/0111277.

[4] S. Nojiri, S.D. Odintsov and S. Zerbini, Class. Quant. Grav.17 (2000)4855-4866, hep-th/0006115.

[5] I. Brevik, K.A. Milton, S. Nojiri and S.D. Odintsov, Nucl.Phys.B599(2001) 305-318, hep-th/0010205.

[6] B. Grinstein, D.R. Nolte and W. Skiba,Phys.Rev.D63 (2001)105016,hep-th/0012202.

[7] R. Hofmann, P. Kanti, M. Pospelov, Phys.Rev.D63 (2001)124020, hep-ph/0012213.

[8] S. Nojiri, S.D. Odintsov and S. Ogushi, Phys.Lett.B506 (2001)200-206,hep-th/0102082.

[9] I.Z. Rothstein, Phys.Rev.D64 (2001)084024, hep-th/0106022.

[10] A. Flachi, I.G. Moss and D.J. Toms, Phys.Rev.D64(2001)105029, hep-th/0106076.

[11] H. Kudoh and T. Tanaka, Phys.Rev. D65 (2002) 104034, hep-th/0112013.

[12] S. Nojiri, S.D. Odintsov and A. Sugamoto, Mod.Phys.Lett.A17 (2002)1269-1276, hep-th/0204065.

[13] W. Naylor and M. Sasaki, hep-th/0205277.

[14] A.A. Saharian and M.R. Setare, hep-th/0207138.

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36/36

[15] T. Gherghetta and A. Pomarol, Nucl.Phys.B602 (2001) 3-22, hep-ph/0012378.

[16] A.L. Maroto, hep-th/0207207.

[17] S.W. Hawking, T. Hertog and H.S. Reall, Phys.Rev. D62 (2000)043501, hep-th/0003052; S. Nojiri and S.D. Odintsov, Phys.Lett. B484(2000)119, hep-th/0004097.

[18] A. Starobinsky, Phys.Lett. B91 (1980) 99.

[19] E. Elizalde, S.D. Odintsov, A. Romeo, A.A. Bytsenko, and S. Zerbini,Zeta Regularization Techniques with Applications (World Scientific, Sin-gapore, 1994); E. Elizalde, Ten Physical Applications of Spectral ZetaFunctions (Springer-Verlag, Berlin, 1995); A.A. Bytsenko, G. Cognola,L. Vanzo and S. Zerbini, Phys. Reports 266 (1996) 1.

[20] E. Elizalde, Commun. Math. Phys. 198 (1998) 83; J. Phys. A34 (2001)3025.

[21] E. Elizalde, Math. Computation 47 (1986) 175; J. Phys. A18 (1985)1637; J. Math. Phys. 34 (1993) 3222.

[22] G. Gibbons and S. Hawking, Phys.Rev. D15 (1977) 2752.

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emilio elizalde et al- casimir effect in de sitter and anti-de sitter braneworlds

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