Jaume Garriga and Ariel Megevand- Coincident brane nucleation and the neutralization of Lambda

8/3/2019 Jaume Garriga and Ariel Megevand- Coincident brane nucleation and the neutralization of Lambda

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Coincident brane nucleation and the neutralization of

Jaume Garriga Departament de Fsica Fonamental, Universitat de Barcelona, Diagonal 647, 08028 Barcelona, Spain,

Institut de Fsica dAltes Energies, Campus UAB, 08193 Bellaterra (Barcelona), Spain,

and Institute of Cosmology, Department of Physics and Astronomy, Tufts University, Medford, Massachusetts 02155, USA

Ariel Megevand Departament de Fsica Fonamental, Universitat de Barcelona, Diagonal 647, 08028 Barcelona, Spain

and Institut de Fsica dAltes Energies, Campus UAB, 08193 Bellaterra (Barcelona), Spain

Received 6 November 2003; published 12 April 2004

Nucleation of branes by a four-form field has recently been considered in string motivated scenarios for the

neutralization of the cosmological constant. An interesting question in this context is whether the nucleation of

stacks of coincident branes is possible, and if so, at what rate does it proceed. Feng et al. have suggested that,

at high ambient de Sitter temperature, the rate may be strongly enhanced, due to large degeneracy factors

associated with the number of light species living on the worldsheet. This might facilitate the quick relaxation

from a large effective cosmological constant down to the observed value. Here, we analyze this possibility in

some detail. In four dimensions, and after the moduli are stabilized, branes interact via repulsive long range

forces. Because of that, the Colemande Luccia CdL instanton for coincident brane nucleation may not exist,

unless there is some short range interaction that keeps the branes together. If the CdL instanton exists, we find

that the degeneracy factor depends only mildly on the ambient de Sitter temperature, and does not switch off

even in the case of tunneling from flat space. This would result in catastrophic decay of the present vacuum.

If, on the contrary, the CdL instanton does not exist, coincident brane nucleation may still proceed through a

static instanton, representing pair creation of critical bubblesa process somewhat analogous to thermal

activation in flat space. In that case, the branes may stick together due to thermal symmetry restoration, and the

pair creation rate depends exponentially on the ambient de Sitter temperature, switching off sharply as the

temperature approaches zero. Such a static instanton may be well suited for the saltatory relaxation scenario

proposed by Feng et al.

DOI: 10.1103/PhysRevD.69.083510 PACS numbers: 98.80.Cq

I. INTRODUCTION

It has long been recognized that the effective cosmologi-cal constant e f f may have contributions from a four-formfield F, and that in such case

e f f F2/2 1

may vary in space and time due to brane nucleation events.This has led to various proposals for solving the cosmologi-cal constant problem, starting with the pioneering work ofBrown and Teitelboim 1. These authors considered a cos-mological scenario where e f f is initially very large andpositive, due to a large F2 term. The additive constant in1 is assumed to be negative, but not fine-tuned in any way,so its absolute value is expected to be of the order of somecutoff scale to the fourth power. During the cosmologicalevolution, e f f is neutralized through successive nucle-ation of closed 2-branes charged with respect to the formfield, which decrease the value of F, until eventually e f f isrelaxed down to the small observed value, ob s .

One problem with the original scenario is that neutraliza-tion must proceed in very small steps, so that any initiallylarge e f f can be brought to ob s without overshooting intonegative values. For that, the charge of the branes should betiny, ensuring that e f fob s at each step. Also, the nucle-ation rate must be very small, or else the present vacuum

would quickly decay. These two constraints make the relax-ation process extremely slow on a cosmological time scale.

Meanwhile, ordinary matter in the universe is exponentiallydiluted by the quaside Sitter expansion, resulting in a dis-appointing empty universe.

Recently, Feng, March-Russell, Sethi, and WilczekFMSW 2 have suggested that nucleation of coincidentbranes may offer a solution to the empty universe prob-lem. Their proposal can be summarized as follows. In thecontext of M theory, a stack of k coincident D-branes sup-ports a number of low energy degrees of freedom, corre-sponding to a U(k) super Yang-Mills SYM theory livingon the worldsheet. Consequently, the nucleation rate of co-incident branes should be accompanied by large degeneracyfactors, and could in principle be enhanced with respect tothe nucleation of single branes. The charge of a stack of

branes can be very large even if the individual charges aresmall, facilitating quick jumps from e f f to ob s . In thisway, neutralization might proceed very rapidly, perhaps in

just a few multiple steps of the right size. Finally, thestability of the present vacuum could be due to gravitationalsuppression of the nucleation rate 1,3.

FMSW argued, rather heuristically, that the nucleationrate of coincident branes should be enhanced by a factor ofthe form

DeS, 2

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where S is the entropy of the worldsheet SYM fields. Thisentropy was estimated through simple thermodynamic argu-ments, as

Sg*

R 2T2, 3

where g*

is the effective number of worldsheet field degreesof freedom, and R is the size of the brane at the time of

nucleation. However it remained unclear in 2 which tem-perature T should be used for the worldsheet degrees of free-dom. Brane nucleation takes place in an ambient de SitterdS space characterized by a Gibbons-Hawking temperature

To e f f1/2 . The region inside the closed brane has a smaller

value of the effective cosmological constant, and is thereforecharacterized by a smaller temperature Ti . Feng et al. con-sidered two alternative possibilities for the temperature ofthe worldsheet degrees of freedom: T1To and T2(ToTi)

1/2. The proposed enhancement of the nucleationrate and the resulting cosmological scenarios are quite differ-ent in both cases, and therefore it seems important to try andclarify the issue of which temperature is the relevant one.

The purpose of this paper is to present a more formalderivation of the nucleation rate corresponding to multiplebrane nucleation. As we shall see, the temperature relevantfor the worldsheet degrees of freedom is in fact determinedby the internal geometry of the worldsheet 4. For theColemande Luccia CdL instanton, this worldsheet is a21 dimensional de Sitter space of radius R, and the corre-sponding temperature is TR1. When substituted into thenaive expression 3, this leads to Sg

*, independent of R

and hence of the ambient dS temperatures. As we shall see,the actual result has a certain dependence on R due to theanomalous infrared behavior of light fields in the lower di-mensional de Sitter space, but this results only in a rather

mild dependence on the ambient dS temperatures.We shall also see that de Sitter space allows for a static

instanton that may be quite relevant to the nucleation of co-incident branes. This is analogous to the instanton for ther-mal activation in flat space. It has a higher Euclidean actionthan the CdL solution, and hence ignoring the degeneracyfactor it seems to represent a subdominant channel of decay.However, we shall argue that, depending on the short dis-tance behavior of the interactions amongst the branes, theCdL instanton for coincident brane nucleation may simplynot exist, and in this situation the static instanton may be therelevant one. In several respects, the static instanton appearsto be better suited to the neutralization scenario proposed byFeng et al. than the CdL one.

The paper is organized as follows. In Sec. II we reviewdifferent proposals for neutralization ofe f f via brane nucle-ation. Section III contains a discussion of coincident branesin 4 spacetime dimensions. These are obtained from dimen-sional reduction of type IIA supergravity in ten dimensions.In the 4 dimensional picture, the gravitational and four-formforces are both repulsive. However, the two are exactly bal-anced by the attractive force mediated by the scalar dilaton.In Secs. IV and V we discuss the stabilization of the dilaton,which is required in a more realistic scenario. After the dila-ton acquires a mass, the remaining long range forces are

repulsive, rendering the stack of coincident branes unstable,or metastable at best. This has important implications, sincethe instanton for nucleation of coincident branes will onlyexist provided that some mechanism causes an attractive in-terbrane force at short distances.

Section IV contains a description of the CdL instanton fornucleation of coincident branes, highlighting a few limitingcases of interest. In Sec. VII we discuss the corresponding

degeneracy factor in the nucleation rate, and we show that itsdependence on the ambient de Sitter temperatures is rathermild. We also include a heuristic interpretation of this resultbased on the observation that the relevant temperature for theworldsheet degrees of freedom is determined by the inverseof the radius of the instanton. Implications for the scenario of2 are briefly discussed.

Section VIII is devoted to a study of the static instan-ton, where the worldsheet has the topology S 2S1, andwhere the intrinsic temperature is comparable to To . In thiscase, the dependence of the nucleation rate on the ambientdS temperature is exponential. Coincident brane nucleationcan be unsuppressed at large To but strongly suppressed at

present. Our conclusions are summarized in Sec. IX. Sometechnical discussions are left to the Appendices.To conclude this Introduction, a disclaimer may be useful.

For most of the paper, we shall work directly in four dimen-sions, and our discussion will be certainly less than rigorousfrom the string theory point of view. In particular, we shallmodel the degrees of freedom of a stack of k coincident2-branes by a weakly coupled U(k) gauge theory on theworldsheet. This may or may not correspond to a true dimen-sional reduction from M theory, but it should at least repre-sent some of the broad features of the degeneracy factors.

II. NEUTRALIZATION VERSUS RANDOMIZATION

In four dimensions, a four-form can be written as FFg where g is the determinant of the metric and is the Levi-Civita symbol. This field has no propagatingdegrees of freedom, since in the absence of sources the equa-tion of motion d*F0 implies that F is a constant. Thissimply gives a contribution to the effective cosmologicalconstant that gets added to the true cosmological constant, orvacuum energy density ,

e f fF2

2. 4

Four-forms may couple to charged 2-dimensional ex-

tended sources, or 2-branes, through a term of the form

qA, 5

where the integral is over the worldsheet of the extendedobject and A is the 3-form potential (FdA). In this case,F changes by

Fq 6

across a brane of charge q. Consequently, F can decaythrough nucleation of closed spherical branes. The process is

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analogous to pair creation in the presence of an electric field,and very similar to false vacuum decay in field theory 5.The closed brane is the boundary of a newly formed truevacuum bubble, where the field strength differs from theoriginal value by the amount 6. After nucleation, the radiusof the bubble grows with constant proper acceleration, andthe volume occupied by the new phase keeps increasing.Further nucleation events take place in the region with a low

F, lowering e f f even further. In the absence of gravity, thisneutralization would proceed as long as Fq /2, wipingout any large initial value of F2 and leaving us with a large

negative cosmological constant e f ff inalO(q2) . Of

course, this is not what we want.Gravitational effects can improve the picture dramatically

1. In particular, tunneling from flat or antide Sitter spacewith a negative cosmological constant is forbidden pro-vided that the squared tension of the branes is sufficientlylarge compared with the jump e f fqF in energy densityacross the brane,

2 4/3qFMp2 . 7

Here is the brane tension and Mp21/(8G) is the square

of the reduced Planck mass. This gravitational shutdownof brane nucleation could be useful, since an initially largee f f may eventually get neutralized to a value that ismuch smaller than in absolute value 1. Suppose that thetrue cosmological constant is negative 0. As long ase f f0, branes keep nucleating. But once a vacuum withe f f0 is reached, the process stops provided that Eq. 7 issatisfied. After that, the vacuum becomes absolutely stable.Brown and Teitelboim conjectured that we may live in onesuch vacuum, where the effective cosmological constant isexpected to be of the order of the energy density gap be-

tween neighboring vacua

e f ffinal FFq1/2. 8

In this vacuum, F1/2, and the huge negative bare cos-mological constant is almost completely cancelled by the F2

contribution in the final state.For sufficiently small charge q, Eq. 8 leads to a suppres-

sion of e f ff inal relative to the true vacuum energy ,

which might be helpful in solving the old fine tuningproblem of the cosmological constant. Particle physics mod-els suggest (TeV)4. Hence, we need

q1035 eV2, 9

so that the final value e f ff inal is consistent with the observed

value e f fob s1011(eV) 4. The constraint 9 seems rather

demanding, since in the context of supergravity we wouldexpect q to be closer to the Planck scale. This is the so-calledgap problem. FMSW argued that the smallness of thecharge could be due to the wrapping of branes on degener-ating cycles in the extra dimensions. A successful implemen-tation of this idea has not yet been presented, but some plau-sibility arguments have been given in Ref. 2. Alternatively,in a different context, it has been suggested that branes with

a very tiny charge q may arise due to symmetries of the

theory. An explicit example was given in Ref. 6, where thebranes are not fundamental objects but domain walls of a

broken discrete symmetry. This same symmetry suppresses

the coupling of the domain walls to the four-form F without

any fine-tuning of parameters see also Ref. 7 for a fullerdiscussion.

A more severe problem of the Brown-Teitelboim neutral-

ization scenario is the empty universe problem, which we

discussed in the Introduction. By combining the condition

9 with the stability condition 7, it can be shown that thetime required to reach the value 8 is huge compared withthe age of the universe 1. By the time the effective cosmo-logical constant would be wiped out, all other forms of mat-

ter would have also been diluted exponentially, in clear con-

tradiction with observations. Furthermore, the endpoint of

neutralization would be a state with vanishing or negative

effective cosmological constant, whereas the observed value

e f fob s is positive.

One way around the empty universe problem is to con-

sider a slightly different scenario, where the effective cosmo-logical constant is randomized rather than neutralizedduring inflation. Assume, for simplicity, that the energy scale

of inflation is much larger than e f f . In an inflationary

phase, brane nucleation processes may increase as well as

decrease the value of F2 4. Thus, e f f will randomly fluc-tuate up and down the ladder as a result of brane nucleation.

Inflation is generically eternal to the future and there is an

unlimited amount of time available for the randomization

process to take place before thermalization 4. Assume thatthe tunneling barriers are sufficiently high, so that no nucle-

ation events happen in the last 60 e-foldings of inflation, or

during the hot phase after thermalization up to the present

time. In this scenario there is no empty universe problem: thelocal value ofe f f is decided many e-foldings before the end

of inflation, and a wide range of values of e f f will be foundin distant regions of the universe, separated from each other

by distances much larger than the present Hubble radius.

Some of these regions will just happen to have a very tiny

e f f . In combination with anthropic selection effects, thisapproach may be used to explain the smallness of the ob-

served effective cosmological constant 4,6,8,9. This mayalso explain the so-called cosmic time coincidence, or why

do we happen to live at the time when an effective cosmo-

logical constant starts dominating 4,9,10.Bousso and Polchinski 11 have proposed a somewhat

related randomization scenario, which, moreover, does notrely on branes with the exceedingly small charge q satisfying9. In the context of string theory one may expect not justone but many different four-form fluxes Fi coupled to braneswith different charges q i . Each one of these fluxes is quan-tized in units of the charge, so that Fin iq i . In this case, thecondition for a generic negative cosmological constant to be

compensated for by the fluxes is e f f i1J (q i

2n i

2)/2

ob s . The larger the number of different fluxes, thedenser is the discretuum of possible values of e f f , and theeasier it is to find a set of values of n i such that the above

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inequality is satisfied. A sufficiently dense discretuum is typi-cally obtained provided that the number of fluxes is suffi-ciently large J100, even if the individual charges q i arePlanckian a smaller number of fluxes J6 may be enoughin a scenario with large extra dimensions, where the chargesq i are suppressed with respect to the Planck scale by a largeinternal volume effect. In the Bousso-Polchinski model,e f f is typically very large, and drives an exponential expan-

sion at a very high energy scale. Suppose we start with asingle exponentially expanding domain characterized by aset of integers n i. Whenever a brane of type j nucleates inthis region, the integer nj will change by one unit inside ofthe brane. The newly formed region will itself expand expo-nentially, creating a huge new domain. The nucleation offurther branes within this region will cause an endless ran-dom walk of the values ofn i, which will sample the wholediscretuum of values of e f f . Eventually, a bubble willnucleate where e f f is comparable to the observed value.The nucleation of this last bubble is still a high energy pro-cess, which kicks some inflaton field off its minimum, andstarts a short period of inflation within this last bubble. This

period of ordinary inflation is necessary in order to pro-duce the entropy we observe, thereby avoiding the emptyuniverse problem. Of course, some anthropic input is stillnecessary in this approach in order to explain why, out of thediscretuum of possibilities, we live in a vacuum with smalle f f .

Nucleation of coincident branes would drastically modifythe neutralization scenario of Brown and Teitelboim, as wellas the randomization scenarios sketched above. As proposedin 2, in the case of neutralization, this modification maylead to a solution of the empty universe problem. In therandomization scenarios there is no such problem, and it isunclear whether an enhancement of the multiple brane nucle-

ation rate is desirable at all. This enhancement would triggerlarge jumps in the effective cosmological constant, makingthe calculation of its spatial distribution more complicatedthan that for single brane nucleation. Thus, it is of interest tounderstand the conditions under which multiple brane nucle-ation is allowed, and what are the degeneracy factors thatmight enhance their nucleation rate relative to the nucleationof single branes. Before addressing this issue, it will be con-venient to present a short discussion of coincident branes.

III. COINCIDENT BRANES IN 4D

In the context of string theory, one may consider stacks ofk coincident D-p-branes. Each brane in the stack has chargeq with respect to the form field A. Thus, in four noncompactdimensions, a pair of parallel 2-branes repel each other witha constant force per unit area given by

fqq2/2, 10

due to the four-form field interaction. In the ten dimensionaltheory, the repulsive force due to A is balanced with othercontributions from the closed string sector, such as the gravi-ton and dilaton. As a result, there are no net forces amongstthe different branes on the stack.

It is in principle possible to maintain this delicate balanceby suitable compactification from 10 to 4 dimensions. In 4D,the branes look like domain walls, and their interactionthrough the ordinary graviton leads to a mutual repulsiveforce given by 12

f32/4Mp

2 . 11

Hence, both forces given by 10 and 11 tend to push thebranes apart from each other. On the other hand, some of thehigher dimensional gravitational degrees of freedom are rep-resented by scalars in 4D, and these, together with the dila-ton, lead to attractive forces.

A. Dimensional reduction

Let us first consider the case of D-2-branes in 10 dimen-sional type IIA supergravity. The relevant part of the action isgiven by see e.g. 13

S 10M10

8

2 d10xG e2R4 2 1

24!F 2

T2

d3G e q2

d3A. 12

Here G and G are the determinants of the 10 dimen-sional metric GAB and of the metric induced on the world-sheet , respectively, whereas R is the Ricci scalar corre-sponding to GAB . The carets on the four-form, the gaugepotential, and the corresponding charge are introduced in or-der to distinguish them from the four dimensional ones thatwill be used below, and which differ from those by constantnormalization factors. Compactifying on a Calabi-Yau mani-fold K through the ansatz

ds102e26gdx

dx e2dK62 ,

where the Greek indices run from 0 to 3, we readily obtainthe following four-dimensional action:

S 4Mp

2

2 d4xg R 2

72

6

72

1

24!e2

F 2

T2

d3e q2

A. 13

Here Mp2M10

8 V6, where V6 is the coordinate volume of the

manifold K, R is the Ricci scalar for the metric g, and wehave introduced two linear combinations of the internal vol-ume modulus and the dilaton

29, .

The field decouples from the branes, and shall be ignoredin what follows. In Ref. 2, a different expression was givenfor the dimensionally reduced action, because no moduluswas introduced for the size of the internal space. However, aswe shall see, such modulus is necessary for the cancellationof forces among the branes.


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A more direct route to the 4-dimensional theory 13,which will be slightly more convenient for our discussion, isto start directly from 11-dimensional supergravity and com-pactify on a 7 dimensional internal space. The action ineleven dimensions is given by

S 11M11

9

2 d11xGR 1

24!F2

T2

d3G q2A 14

where T2 q22M113 . Introducing the ansatz

ds 112e7

gdx

dxe 2

d72 , 15

with a Ricci flat internal manifold, we find

S 4Mp

2

2 d4xg R 2

72

1

24!e2

F2 ,

16

where Mp2M11

9V7 and

21

2.

Not surprisingly, this has the same form as 13, since afterall the 10-dimensional Lagrangian can be obtained from the11-dimensional one by compactifying on a circle.

Equation 16 is a particular case of the slightly moregeneral action in four dimensions:

S 41

2d4xg

Mp

2R21

4!

e2F2

d3eq

A. 17

The parameter characterizes the scalar charge of the brane.As we shall see below, linearizing in around 0, it canbe easily shown that the scalar force is given by

fee2/2, 18

where we have introduced the scalar charge

e/Mp . 19

Thus, from 10, 11, and 18, the branes will be in indif-ferent equilibrium provided that the following relation holds:

Q 2 e2q 2

2

32

4Mp20. 20

From 19, this condition can be rewritten as

q 2 32

1/2

Mp. 21

Note that the case of type IIA supergravity discussed above,

corresponds to 7/2, q(2/Mp)q2, and T2 notethat A(2/Mp)A]. Therefore 21 is satisfied providedthat T2 q2, the usual Bogomolnyi-Prasad-SommerfieldBPS condition.

B. Multiple brane solutions

Multiple brane solutions to 17 can easily be constructedprovided that 21 is satisfied. In the bulk, the equation ofmotion for A leads to

FFe2g ,

where F is a constant and is the Levi-Civita symbol. At thebranes, this constant jumps by the amount

Fq .

It can be checked that the remaining equations of motion forthe scalar and the gravitational field follow from the action

S 4i i d4xg

Mp2

2R2Vi

d3e, 22

where

ViFi

2

2e2, 23

and the sum is over the regions with different values of F.With the metric ansatz

ds 2w 2z ab dxadxbdz 2,

where Latin indices run from 0 to 2, the solution is given by

ec i2

23/21/2

Fi

Mpz , 24

and

wz e/2. 25

Here c i are integration constants. These, and the sign optionin 24, must be chosen so that the junction conditions for the

gravitational field and for the scalar field are satisfied at thebranes. For the gravity part, the condition is 14

Kab 4Geab , 26

where Kab is the difference of extrinsic curvature on thetwo sides of the brane and ab is the worldsheet metric. Inthe present case, this reduces to

ww

2Mp

2e. 27


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For the scalar field, the junction condition that follows fromthe equation of motion at the brane reduces to

Mp2

e, 28

which is consistent with 27 and 25. For instance, a solu-tion with two branes separated by a distance d is given byflat space (w1,0) in the region between the branes, andby

e12

Mp2

z d/2 z d/2 29

in the exterior region see Fig. 1. The solution correspondsto two branes of charge q interpolating between regions withFq and Fq , separated by a region with F0.

Note that the solution 29 contains a singularity at a dis-

tance zMp2/2 where the warp factor w and the volume

of the internal space vanish. In what follows, however, weshall not be interested in flat, infinite branes, such as the onesdiscussed above. Rather, we shall be interested in compactinstanton solutions of finite size, in a theory with a stabilizeddilaton. In this case, the singularities due to linear potentialsof the form 29 should not arise, but the solutions discussedabove should remain a good description in the vicinity of thebranes. Another consequence of stabilizing the dilaton is thatthe perfect balance of forces amongst the branes will bespoiled at distances larger than the inverse mass of themoduli, as we now discuss.

IV. STABILIZING THE MODULI

As it stands, the dimensionally reduced supergravity La-grangian 13 or its generalization 17 is not useful fordiscussing the neutralization of the cosmological constant.The Lagrangian does not include the bare cosmological term, which is precisely the subject of our interest, and the termproportional to F2 does not behave as an effective cosmo-logical constant, but rather as an exponential potential for amodulus which is not flat enough to mimick the vacuum

energy1. A more realistic model is obtained by introducing astabilization mechanism that fixes the expectation value of, and gives it some mass m. Once is stabilized the F2

term does behave as a contribution to e f f . Stabilization isalso desirable because the dilaton and the radion modulicorresponding to the size of extra dimensions mediate sca-lar interactions of gravitational strength, which are severelyconstrained by observations. The study of mechanisms for

stabilization is currently an active topic of research see e.g.Refs. 16,17 and references therein. Although the details ofstabilization will not be too important in our subsequent dis-cussion, it may be nevertheless illustrative to have in mind aspecific toy mechanism for which we do not claim any rig-orous justification in the context of string theory.

The general problem is that the potential 23 has no mini-mum and leads to a runaway dilaton. In order to create aminimum, let us consider two contributions that may beadded to 23. First of all, instead of using a Ricci flat inter-nal manifold in 15 we may compactify on an Einsteinmanifold, with

Rab

6Kg ab

. 30

Upon dimensional reduction, the curvature K contributes anexponential term to the effective potential for . However,this will still not be sufficient for stabilizing the internal vol-ume in an interesting way. In fact, as shown by Maldacenaand Nunez 18, there are no static compactifications of theclassical supergravity Lagrangian with a positive effectivefour-dimensional cosmological constant e f f0, and so farwe have added nothing to the classical Lagrangian. Thus, inorder to implement the four dimensional situation of our in-terest, a third term related to quantum corrections has to beconsidered. Following Candelas and Weinberg 19, we mayconsider the Casimir energy of bulk fields we are of course

assuming that supersymmetry is broken, so that the Casimircontributions of bosons and fermions do not cancel eachother exactly.

In the example considered by Candelas and Weinberg19, besides the Casimir energy term, a higher dimensionalcosmological term 4n was used, and the internal manifoldwas taken to be a space of constant positive curvature ( K1) . In 11D supergravity a cosmological constant is notallowed, so here we use the F2 flux instead. Also, we will

1Exponential potentials such as the one appearing in 22 have

been thoroughly studied in the literature, and it is known that they

can drive cosmological solutions with a power-law scale factor15. Such attractor solutions are approached for a wide range of

initial conditions, and the resulting expansion can be accelerating or

decelerating, depending on whether 21/2 or 21/2. For 2

3/2 the cosmological scale factor approaches a(t)t1/22, where

t is cosmological time. This solution corresponds to an effective

equation of state p(42/3)1, where the ratio of kinetic and

potential energies of the scalar field remains constant. In our case,

from 21, we need 23/2 and therefore the kinetic term becomes

completely dominant in the long run, which leads to p.

Hence, by itself, the F2 term does not behave like an effective

cosmological constant.

FIG. 1. Configuration of two parallel branes.


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need to compactify on a negatively curved internal manifold(K1) , or else the dilaton would be stabilized at negativee f f .

The versatility of negatively curved compactifications hasbeen stressed in Ref. 20. In particular, they have the inter-esting property of rigidity, which means they do not lead toother moduli besides the size of the internal space. Compacthyperbolic manifolds CHM can be obtained from the maxi-

mally symmetric negatively curved space H7 through iden-tifications by a discrete isometry group . The volume ofH7 / is given by

V7rc7

e, 31

where rc is the curvature radius of the manifold, related to

the curvature parameter in 30 by K1/rc2 . The factor e

depends on the topology, and it is bounded below but notabove. If L is the largest distance around the manifold, thenfor Lrc /2 we have e

e 6L/rc. The Kaluza-Klein KKspectrum in this manifold is believed to have a mass gap,

bounded below by m KKe

rc1

. From the 11 dimensionalpoint of view the Casimir energy density scales like

C(11)Cm KK

11Ce11

rc11 . 32

The factor C can be estimated by naive dimensional analysisas C, where 103 is the number of physical polariza-tions of bulk fields, and is some small one-loop factor.This factor will depend on the precise topology of the com-pact hyperbolic manifold, but it could plausibly be in therange 102104. Hence, the parameter C could be oforder one. Explicit calculations for different choices of the

manifold H7 / have not been performed, and are well be-yond the scope of the present paper see e.g. Ref. 21 andreferences therein. In what follows we shall leave C un-specified, assuming that a compactification exists where C1. Multiplying the higher dimensional energy density C

(11)

given in 32 by the size of the internal manifold, V7e7, and

by a factor e14

that arises from the four-dimensional vol-ume element in going to the Einstein frame, the effectivepotential that appears in Eq. 22 gets replaced by

Vi21M119 V7Ke

9CV7rc

11e18

Fi2

2e21

,

33

where we have also added the curvature contribution. Here

(2/21), where 7/2.We can always adopt the convention that rc1/M11 ,

since a change of rcerc can be reabsorbed by a shift in

and a constant re-scaling of the four-dimensionalmetric ge

7g in this frame, the curvature of themanifold is of the order of the higher dimensional Planck

scale for 1). Since Fin iq , where n i is an integer andq(2/Mp)q222M11

3 /Mp , we have

ViMp2M11

2 21e9Ce1842n i2M11Mp 4

e21 .34

As illustrated by Eq.31

a large value of V7 in units of rc1/M11 can be obtained by using a manifold with suffi-

ciently complicated topology. Since Mp2M11

9V7M11

2e,

the factor (M11/Mp)4e2 in the last term of 34 can be

rather small. The scale M11 could be as low as the TeV, inwhich case that factor can be as low as 1064. Moreover, asemphasized in Ref. 20, this can be achieved for CHM evenif the linear size L of the internal manifold is not very muchlarger than rc . In Fig. 2 we plot the effective potential forC20, e103 and values of n i485, 487,489 and 491.

When a single brane nucleates, it changes n i by one unit,and hence changes the value of the effective potential at theminimum changing therefore the effective cosmologicalconstant. If the discretuum of values of Fi were sufficientlydense, then there would always be one of the minima of theeffective potential where the vacuum energy is sufficientlysmall to match observations. In the case we have consideredhere, the discretuum is not dense enough. The cancellationbetween the last term in 33 and the other two requires n i(Mp /M11)

2, and so the gap between levels near Vi0 can

be estimated as ViM114 , which is far too large. This situ-

ation can be remedied by considering 5M branes wrappedaround 3-cycles in the internal space. As emphasized byBousso and Polchinski 11, these are coupled with differentcharges to additional fluxes, and a large number of fluxeswill result in a much denser spectrum. Alternatively, FMSW

have suggested that the branes may wrap a degeneratingcycle 2, in which case the individual charges might them-selves be exponentially smaller than the fundamental scale,resulting also in a sufficiently dense discretuum.

Before closing this section, we should note that the loca-tion where the modulus sits is basically determined by thecompetition between the two first terms in 33, correspond-ing to curvature and Casimir energy. The physical curvatureradius of the internal space is therefore stabilized at rphysrce

M111

C1/(D2) , where D11 is the spacetime di-mension. Thus, unless the constant C in Eq. 32 is exceed-

FIG. 2. The effective potential 34, for different values of the

integer n i which characterizes the quantized flux of the four-form F.


083510-7


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ingly large, the compactification scale is comparable to theinverse of the higher dimensional cut-off scale M11 . In sucha case the semiclassical analysis that we have entertainedabove is not justified, since higher order corrections will be

just as important as the one loop effect that we have in-cluded. This appears to be a generic problem when we try tostabilize by making the curvature and the Casimir termscomparable, as in the Candelas and Weinberg example 19.

There, the problem was not quite as poignant, since the con-stant C could be made very large by adding a sufficient num-ber of fields also, as it is clear from the above estimate ofrphys , the problem is somewhat milder if the number of extradimensions is smaller. Here we shall not dwell on this prob-lem, since the main purpose of the above discussion is just toillustrate the role of the four-form in obtaining a discretuumof states. For more fundamental approaches to moduli stabi-lization in the present context, the reader is referred to Refs.16 and references therein. Our ensuing discussion will belargely independent of the details of the stabilization mecha-nism.

V. INTERBRANE FORCES

Inevitably, the stabilization of spoils the perfect balanceof forces. At distances larger than the inverse of the mass ofthe modulus m1, the remaining gravitational and four-forminteractions are both repulsive, and lead to a linear potentialper unit surface of the form

V d q2

2

32

4Mp2d dm1 35

where d is the distance between the branes.To investigate the behavior of the interaction potential at

distances shorter than m1, let us first consider the situationwhere gravity and the 3-form gauge potential A are ignored,and the branes interact only through a scalar field of massm. The action is given by

SMp

2

2 d4x2m 22

d3e.

36

The solution with a single brane on the plane z0 has thecusp profile

0 emz , 37

where 0 is a solution of

0 e0

2m Mp2

e

2m Mp. 38

The energy per unit area of this configuration is given by

1e0

Mp2

2 dz2m 22e0m Mp202 .

39

For small charge and tension, we have

1e 2

4m e22m.

Due to the scalar field dressing, the effective tension of thebrane, denoted by 1, is smaller than the parameter thatappears in the action. This effect becomes more dramatic ifwe place a large number k of branes on top of each other.Since both and e scale like k, the effective tension of thestack is given by

kkk e2

4m k e 22m, 40

which grows with kbut less than linearly. In the limit of verylarge k, using 38 with replaced by k we have

km2

e 2ln2 ke

2

2m k e 22m 41

and so the tension almost saturates, growing only logarith-mically with k. This last expression should not be taken too

literally, however, since in 36 we are neglecting gravita-tional effects. As shown above, nonlinear effects of gravity

become important at a distance given by Mp2/2k , which

in the limit given by 41 is smaller than m1. Hence thecusp solution 37, which has typical width m1, will re-ceive sizable gravitational corrections in the limit of large k.Nevertheless, it still seems likely that the effective tension ofthe stack of branes will grow with k much slower than lin-early. Physically, the reason is that the scalar charge of thestack increases with k. According to 28, this means that thecusp in the field on the branes grows stronger, which inturn means that the value of the field 0 on the branes has tobe further displaced into negative values. Hence, the brane

contribution to the effective tension ke0 shows only avery modest growth with k, and the tension for large k is infact dominated by the potential and gradient energy of thescalar near the brane.

Let us now look at the interaction potential between twobranes separated from each other. For simplicity, we shallrestrict our attention to the case of small scalar charge, e 2

2m. Then, the last term in 36 is well approximated byits linearized expression:

SMp

2

2 d4x2m 22

d3eMp.

42

Placing the two branes at zd/2, the solution for the sca-lar field has the Golden Gate profile shown in Fig. 3:

e

m Mpemd/2 cosh mz, z d/2

e

m Mpcosh md/2emz, z d/2 . 43

The energy per unit area of this configuration, as a functionof the interbrane distance, is given by


083510-8


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2 d21e 2

2memd. 44

From this expression, and adding the long range contribu-tions from gravity and the four-form, the interaction potentialper unit surface is given by

V d q2

2

32

4Mp2d e

2

2memd.

At short distances, this takes the form

V de2

2mQ 2d

me 2

4d2 45

where the parameter

Q 2 e2q 2

2

32

4Mp2 , 46

was introduced in 20. As shown in Sec. III, dimensionalreduction from the supergravity Lagrangian with BPScharges gives Q 20. In this case, the linear term in 45disappears. The quadratic term is negative, which means thatthe stack of coincident branes is unstable and tends to dis-solve. Thus the stabilization of the modulus seems to makethe superposition of branes an unstable configuration.

As we shall see in more detail in Sec. VII, stacks ofbranes in marginal or unstable equilibrium will not be appro-priate for constructing instantons, since in particular thesewould have too many negative modes. The instanton is onlymeaningful if the branes attract each other. The above analy-sis shows that at the classical level, the branes with BPSvalues of the charges would not attract each other, and con-sequently the nucleation of coincident branes is not allowedat least in the semiclassical description.

There may be several escape routes to this conclusion. Forinstance, after supersymmetry breaking, the charges of thebranes get renormalized, and it is possible that the correctedcharges satisfy Q 20. In this case the branes in the stackwould attract each other with a linear potential. Other mecha-nisms by which nearby branes attract each other are conceiv-able, but here we shall not try to pursue their study. It shouldbe emphasized, however, that this remains an important open

question that needs to be addressed in order to justify thesemiclassical description of coincident brane nucleation. InSec. VII, where we discuss the degeneracy factor, we shallsimply postulate that an attractive interaction exists at shortdistances.

VI. COLEMANDE LUCCIA CDL INSTANTONS

The brane nucleation rate per unit volume is given by anexpression of the form 5

DeB, 47

where BSE(I)SE(B). Here SE(I) is the Euclidean actionof the instanton corresponding to the decay of the four-formfield, and SE(B) is the action for the background solutionbefore nucleation. The prefactor D will be discussed in thenext section. The formal expression 47 can be used both inflat space and in curved space. Also, it can be used at zero orat finite temperature. The difference is in the type of instan-ton and background solutions to be used in each case.

In this section, we shall concentrate in the maximally

symmetric instanton. In flat space, this represents the decayof a metastable vacuum at zero temperature. The Euclideansolution can be described as follows 5. At infinity, the fieldstrength takes the value Fo , which plays the role of a falsevacuum. Near the origin, we have FFiFoq. This playsthe role of a true vacuum phase. Both phases are separatedby the Euclidean worldsheet of the brane, which is a three-sphere of radius

R3

. 48

Here is the tension of the brane and

1

2 Fo

2Fi

2 q Fo q/2 49

is the jump in the energy density across the brane. The dif-ference in Euclidean actions between the instanton and thebackground solution is given in this case by 5

B (flat)272

2

4

3. 50

If instead of considering a single brane, we are looking at thenucleation of k coincident branes, an analogous solution ofthe Euclidean equations of motion should be considered. The

only difference is that the effective charge is a factor of khigher, and the effective tension is also higher by approxi-mately the same factor. Using 40, we should replace

kkq Fo kq/2 ,

kk ke2/4m , 51

in Eq. 50 for the bounce action. As shown in Sec. V, theapproximate form ofk is valid when the second term in ther.h.s. of 51 is small compared with the first. In the modelconsidered in Sec. IV, the modulus is stabilized with a mass

FIG. 3. Profile of a massive dilaton in the presence of two

branes.


083510-9


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of order mM11 . This may be regarded as a conservativeupper bound for the general case. Since e/Mp , the con-dition (ke 2/4m) requires

k1

2 M11

3

Mp

M11

2

. 52

For larger k, gravitational corrections to the brane profilebecome important on a length scale comparable to the in-verse mass of the dilaton, as we discussed in the precedingsection. In this case the instanton solution will depend on thedetailed dynamics of the dilaton near the brane. The investi-gation of these dilatonic instantons is per se an interestingproblem, which we leave for further research. Here, for sim-

plicity, we shall assume a scenario where either M113 ,

due perhaps to the wrapping of branes on a degeneratingcycle, or where the extra dimensions are relatively large, sothat MpM11 , and we shall restrict attention to instantonswhere the number of branes is bounded by 52. This gives

Bk(flat)

272

2

k4

q 3 Fo kq/23

. 53

Nucleation in flat space is impossible when the number ofbranes is too large, since otherwise we would be jumping tovacua with a higher energy density. Therefore, we must re-strict ourselves to k2Fo /q . In fact, the minimum value ofFi is achieved for the largest k satisfying k(Fo /q)(1/2). Note that the action increases faster than linearly aswe increase the number of branes k, but we still have

B k(flat)kB ( f lat )

throughout this range.When gravity is taken into account, the maximally sym-

metric instanton was given by Coleman and De Luccia 3. Itis constructed by gluing two different de Sitter solutions thatis, two four-spheres at the worldsheet of the brane, which isstill a three-sphere. The instanton is sketched in Fig. 4. The

four-spheres have the radii Ho1 and Hi

1 , where

Ho2

e f f

3MP2

, Hi2

e f f

3MP2

, 54

are the Hubble rates of the de Sitter phases before and afterthe nucleation event, respectively. Here, and in what follows,we are assuming that the effective cosmological constant isstill positive after nucleation, since these are the final stateswe are interested in. The bubble radius at the time of nucle-ation which coincides with the radius of the three-sphere is

bounded by 0RHo1 . Analytic expressions for R are

given in Refs. 1,3. The general expression is cumbersomeand not particularly illuminating, so we shall concentrate ona few limiting cases of interest.

As discussed in Sec. III, the gravitational field of a braneis repulsive, and is characterized by an acceleration of

order /MP2 . This gravitational field will be negligible pro-

vided that the corresponding Rindler radius or inverse of theacceleration is much larger than the radius R of the Euclid-

ean worldsheet, which in turn is smaller than Ho1 :

MP2Ho . 55

In this regime, we can distinguish two cases. For Ho /1, the radius of the Euclidean worldsheet is much smallerthan the de Sitter radius, and the flat space expression 50

holds. In the opposite limit, Ho /1, we have (Ho1

R)Ho1 and

B(wall )22Ho3 . 56

The vacuum energy difference is unimportant in this case,and the action coincides with that for domain wall nucleation22.

Finally, the gravitational field of the brane is importantwhen

MP2Ho . 57

In this case, the radius of the worldsheet is given by R

(1/2G), and

B (wall)

GHo2

. 58

In this limit, the action of the instanton is much smaller thanthe action of the background, and this is the reason why 58is independent of the tension. The same arguments apply ofcourse to the instantons with coincident branes, and the cor-responding expressions for the action and radii in the differ-ent regimes are summarized in Table I.

FIG. 4. Colemande Luccia instanton, which is obtained by

gluing together two different four-spheres of radii Ho1 and Hi

1 at

the worldsheet of the brane, itself a three-sphere.

TABLE I. Values of the radius R of the Colemande Luccia

instanton and the corresponding bounce action B in different limits.


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VII. THE PREFACTOR FOR CDL INSTANTONS

The prefactor D in Eq. 47 is given by 5

DZ

ZB. 59

Here Z and ZB are the Gaussian integrals of small fluctua-

tions around the instanton and the background solutions re-spectively. Expanding all brane and bulk fields which wegenerically denote by ) around the instanton config-uration I as Ijj , we have SESEIS (2 )I,j , where S

(2 ) includes the terms quadraticin j . At the one loop order we have

Z jDj eS(2 ).In the functional integral, there are some directions that cor-respond to spacetime translations of the instanton. Theprimes in the numerator of 59 and in the preceding equa-tion indicate that the translational zero modes are excludedfrom the integration, and replaced by the correspondingspacetime volume. The latter is subsequently factored out inorder to obtain a nucleation rate per unit time and volume.

The degrees of freedom that live on the brane will make acontribution to the numerator but not to the denominator.Consider, for instance, a free bosonic field of mass m living on the worldsheet. Its contribution to the prefactor is

D ZeW De1/2(2m2 )d3.

Here the integral in the exponent is over the Euclideanworldsheet of the brane. If we have k2 of such fields, their

effect on the nucleation rate is to replace

B kBkk2W , 60

in the naive expression for the nucleation rate, eBk, areplacement which can become very important as we in-crease the number of fields. This is, in essence, the observa-tion made by Feng et al. that the large number of worldsheetfields might strongly affect the nucleation rate.

A. Scalars

The Euclidean worldsheet in the Colemande Luccia in-stanton is a 3-sphere. Determinantal prefactors due to scalar

fields were considered in some detail in Ref. 23. They aregiven by

WR 2 y2/2ln siny

1

2

0

y

x lnsinx dx ,

61

where y 21m 2

R 2 and R is the usual Riemann zeta func-tion. For instance, the contribution of a conformally coupled

scalar field can be obtained by taking m 2(3/4)R2, which

gives

WcR 2 1

8ln 2

7

162R3 0.0638.

Hence, the effective degeneracy factor contributed by a con-formal scalar field is given by

D ceWc0.941. 62

The first thing to note is that this factor is not an enhance-ment, but a suppression. Hence, the prefactor cannot simplybe thought of as the exponential of an entropy. More gener-ally, from Eq. 61, the prefactor is independent of the ex-pansion rate of the ambient de Sitter space. This has impli-cations for the scenario proposed by FMSW, as we shalldiscuss in the next section.

In general, the degeneracy factor will depend on R and onthe mass of the particle. For light minimally coupled scalars,Eq. 61 gives

D s

eR(2)

1/2m R m R

1 . 63

There can be a strong enhancement in the nucleation rate ifthere are very light scalar fields. In the limit m 0 thefactor goes to infinity. This is because a massless scalar has anormalizable zero mode on the sphere, corresponding to thesymmetry const. In this case, the zero mode mustbe treated as a collective coordinate. The nucleation rate isproportional to the range of the field , because thebubbles can nucleate with any average value of the scalarfield with equal probability 23

Ds m 20 lim

m20

m D s m R3 1/2

eR(2)R 1/2. 64

Finally, for large mass the expression 61 leads to

Dsexpm 3

R 3/6 m R1 . 65

The exponent of this expression can be interpreted as arenormalization of the tension of the stack of branes, due tothe heavy scalars living on it 24. Indeed, the effective po-tential for a scalar field in 21 dimensions in the flat space

limit is proportional to m 3 , and the factor of R3 is just due

to the volume of the worldsheet. The factor D s is plotted inFig. 5 for different values of m R.

Note that there is an enhancement both at small and atlarge mass, but the two have very different origin. The largevalue of 63 for light fields can be interpreted as a phasespace enhancement. As we shall discuss in Sec. VII D, quan-tum fluctuations of fields living on the worldsheet of thebrane are characterized by a temperature T1/2R . Thecorresponding fluctuations in the potential term are of order

m 2 2T3, which corresponds to a root mean squared ex-

pectation value for of the order


083510-11


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1

mR 3/2. 66

Equation 63 is recovered if we insert the range 66 in Eq.64. The lighter the field, the larger is the phase space factor, and the larger is the nucleation rate. For m R1 thisargument cannot be used, since the field does not behave aseffectively massless. In that limit the field decouples, as itshould, and its effect is felt as a renormalization of the pa-rameters in the Lagrangian. For a scalar field, the leadingeffect is to renormalize the tension of the brane, making itlower see Appendix A. This causes an exponential en-hancement of the nucleation rate.

B. Uk fields

Besides scalar fields, gauge bosons and fermions may liveon a brane. At low energies, the field content on a stack ofcoincident branes will be model dependent. The idea is that itwill correspond to a gauge theory whose symmetry is en-hanced when branes are coincident, giving rise to a largenumber of light species on the worldsheet. The details of thetheory, however, will depend on whether we start from 2Dbranes that descend directly from the ten dimensions, offrom higher dimensional p-branes wraped on (p2)-cycles.They will also depend on the details of compactification.Rather than building a particular scenario from first prin-ciples, here we shall try to gain some intuition by consider-ing a toy model directly in four dimensions. The degrees offreedom on the stack of k coincident branes will be bluntlymodeled by a weakly coupled SUSY U(k) gauge theory onthe 21 dimensional worldsheet. This contains a U(k)gauge field, (k21) scalar degrees of freedom in the adjointrepresentation of SU(k), and a scalar singlet; plus the corre-sponding fermionic degrees of freedom. The action for theworldsheet fields is given by

S SY M d3 12 Tr FFTrDD

V , 67

where the ellipsis indicate the terms containing fermions.

Here F is the field strength of the gauge field AA a

a , where a are the generators of U(k), normalized

by Tr(ab)ab /2, and aa , where a are real sca-lar fields (a ,b1, . . . ,k2). By analogy with the well knowncase of D-branes in 10 dimensions, we shall assume that ifthe coincident branes are flat as in the case when there is noexternal four-form field, then the theory is supersymmetricand all degrees of freedom are massless. The scalar field isa Hermitian matrix and can always be diagonalized by asuitable U(k) gauge transformation, diag(1 , . . . ,k).The eigenvalues i , i1, . . . , k are then interpreted as thepositions of the different parallel branes along an axis per-pendicular to them if the codimension of the branes were

higher, there would be additional scalar matrices represent-ing the positions of the branes along the additional orthogo-nal directions, but here we are interested in the case of codi-mension one. In the supersymmetric case, the potential forthe scalar field vanishes, V()0. However, for nonflatbranes, the displacement of the stack of branes no longerbehaves as an exactly massless field 23,25, but one whichcouples to a combination of the worldsheet and extrinsiccurvatures, as well as to the background four-form field.Also, after moduli stabilization, the forces amongst branesare nonvanishing, and this also contributes to the potentialfor the relative displacements of the different branes. Thus,as we shall see, there will be a nonvanishing potentialV()0 in the physical situation of our interest.

When the positions are not coincident, the U(k) symme-try breaks to a smaller group because some of the gauge

bosons acquire masses mA2(g2/2)(ij)

2. There are al-ways at least k massless vectors corresponding to Maxwelltheory on the individual branes on the stack and the remain-ing k2k have double degeneracy. For example, in the caseof a single brane, the scalar field will represent the Goldstonemode of the broken translational symmetry, associated totransverse displacement of the brane. For the case of twobranes, there are two such scalars. One of them, (12)/2, will corresponds to simultaneous motion of bothbranes, and is a singlet under SU(2). The other one, (

1

2)/2, will correspond to relative motion of the

branes and it transforms under SU(2). When the two branesmove apart, acquires an expectation value and two of thegauge bosons get a mass, breaking the symmetry U(2)U(1 )U(1).

The case of interest to us is not a flat brane, but the world-volume of the 2-brane in the CdL instanton, which forms a3-sphere of radius R. In this situation, we do not expect thetheory to be supersymmetric in particular, corrections to theeffective action will appear at one loop, which will be relatedin fact to the determinantal prefactor in the nucleation rate .The case of a single brane is very similar to the case of a

FIG. 5. Contribution of a scalar field to the prefactor D in the

nucleation rate 47, as a function of its mass m measured in units

of the inverse radius of the instanton. At low mass, the enhance-

ment is due to large phase space: bubbles can nucleate with values

of the field in the range 66, which becomes larger for smaller

masses. The enhancement at large mass can be understood as a

finite renormalization of the brane tension.


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vacuum bubble, and in that case we know that the transversedisplacements correspond to a scalar field of negative masssquared 23,25

m

23R2. 68

The origin and precise value of this mass term can be under-stood geometrically, since it leads to four normalizable zero

modes which are the spherical harmonics with l1. Thesecorrespond to the four space-time translational modes of theinstanton, which have to be treated as collective coordinates.This scalar field has also a single negative mode, which isthe constant l0 mode. A negative mode is precisely what isneeded for an instanton to contribute to the imaginary part ofthe vacuum energy, and hence to contribute to false vacuumdecay 5. Integrating out the transverse displacement of thebrane gives a determinantal prefactor of the form 23

D2R2

4eR (2) , 69

where VT is the spacetime volume. The prefactor in thenucleation rate 47 per unit time and volume is obtainedafter dividing by . The above argument neglects gravity,

and it is a good approximation when 0Mp2Ho . The case

of strong gravity 0Mp2Ho is far more complicated, since

one has to integrate over fluctuations of the gravitationalfield in the bulk, and is left for further research.

If there are 2 coincident branes, then there are two inde-pendent transverse displacements corresponding to the ei-genvalues 1 and 2. The center of mass displacement behaves just like in the case of a single brane. The or-thogonal combination represents the brane separation,and the two-brane instanton will only be relevant if this sec-

ond combination acquires a positive mass through somemechanism, so that there is a single negative mode, not two,and four normalizable zero modes in total. In other words,the branes must attract each other. As shown in Sec. V, if thebranes have BPS charges, they in fact tend to repel eachother once the dilaton is stabilized, and the configurationwith coincident branes is unstable. In this case, we do notexpect that there will be any instanton representing thenucleation of multiple branes.

One possible way out of this conclusion is to assume thatthe charges are different from their BPS values, due to su-persymmetry breaking effects, so that the sum Q 2 defined inEq. 46 is positive. In that case, the two branes attract eachother with a linear potential. The canonical field is re-lated to the interbrane distance d by 1/2d. Hence, theinterbrane potential 45 takes the form

VQ21/2

me 2

4

2 . 70

This potential is attractive at small distances, and has a maxi-mum at mQ

21/2/me2. Classically, the branes willattract at short distances. However, there is a danger that theywill be separated by quantum fluctuations. As we discussedin the preceding subsection and we will argue more at

length in Sec. VII D quantum fluctuations of fields living onthe worldsheet of the brane are characterized by a tempera-ture T1/2R . The fluctuations in the potential are of orderVT3, and these correspond to a root mean squared expec-tation value for of the order

1/2/R 3Q 2. 71

The stability of the two-brane instanton requires that m . Otherwise, unsuppressed quantum fluctuations takethe field over the barrier and the distance between the branesstarts growing without bound. This requires

Q 4me 2

R3. 72

If this condition is satisfied, then is trapped near theorigin, and the branes stay together. It can be shown that forQ 4R5 the field behaves as approximately massless inthe range given by 71, so from 64 its contribution to theprefactor can be estimated as

DR1/2

1/2

R 5/2Q2. 73

As in the case of the massive field discussed in the precedingsection, this expression is only justified when D1. IfQ

2

is too large, then the field will not behave as massless in therange 71, and we expect that for Q 4R5 the sole effectof the field will be to renormalize the coefficients of opera-tors such as the brane tension in the classical Lagrangian. Aninconvenient feature of the linear potential 70 is that isnonanalytic at the origin, and hence an explicit calculation inthe limit of large slope is not straightforward. Moreover, we

cannot write down an expression for it in terms of the matrixoperator , but just in terms of its eigenvalues i .Another possibility, which is more tractable from the for-

mal point of view, is to assume that there is an attractiveinterbrane potential which is quadratic at short distances.That is, as in 70 with Q 20 but with a positive coefficientin front of the second term. In terms of the eigenvalues i ,which represent the displacements of the branes, we assumethe following expression for the potential

Vj m2

2

1

2km

2 i j

ij2 , 74

where k1/2 i1

k ik1/2Tr() and m

23R2 as

given in Eq. 68, while m2 0 is a new parameter which

characterizes the attractive interaction at short distances. Interms of the field , we can write the potential as

Vm

2 k1Tr2m

2 Tr2k1Tr2

1

2m

2 12

1

2m

2 b2

k2

b2 , 75

where in the last equality we have expanded aa in thebasis of generators a , and we have used 1(2k)

1/21 and


083510-13


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Tr b0 for b2, . . . ,k2. In the symmetric phase and as-

suming mR1) each one of the adjoint fields b will con-tribute a determinantal prefactor of the form

DeR(2)

1/2mR, mR1 76

where we have used 63. This shows a somewhat milderdependence in R than in the case of a linear interaction be-tween branes, given in Eq. 73, but still of power law form.

In the limit of large mass Dexp(m3 R3 /6), which as

discussed before amounts to a finite renormalization of thebrane tension.

Aside from scalars, we should also consider the contribu-tions from gauge bosons and fermions. For the case of a3-sphere, these have been studied in Ref. 24. For vectors ofmass mA , the result is

WA1

2log g

2R

42log sinhmAR

mAR

0

mAR

y 2d

dy

logsinhy dyR 2 2R 0 , 77

where g is the gauge coupling, which is dimensionful inthree dimensions. When the branes are coincident, the theoryis in the symmetric phase and the gauge bosons are massless.A massless gauge boson gives a contribution of the form

DAeWAgR 1/2eR (2) mA0 , 78

which again behaves as a power of R. A Dirac fermion ofmass m yields the contribution 24

W1

4log coshmR 0

mR

u2 tanhu du

2R 2,1/2 1

2R 0,1/2. 79

For the massless case, the R dependent terms vanish and wehave

D 21/4e (3/2)R (2) m 0 , 80

which is a constant independent of the radius R.

C. Nucleation rate

Collecting all one-loop contributions, the prefactor in Eq.47 due to the weakly coupled U(k) gauge theory in theunbroken phase is given by

DD

D

k21DAk2D

k2. 81

Using 69, 76, 78, and 80 we are led to a nucleationrate per unit volume of the form

k1/2mk

2

4R 3 e 7R (2) g 22m

2R

k2/2

eBk mR1 ,

82

where B k is the corresponding bounce action for the nucle-ation of k coincident branes. Note that the R dependence ofthe prefactor is simply as a power law. Here we have used

the form 76 for the scalar contribution D , correspondingto an attractive interaction amongst branes which is quadraticat short distances, with a curvature of the potential charac-terized by some mass parameter m .

2 The prefactor in Eq.82 has the exponential dependence on k2 which counts thenumber of worldsheet field degrees of freedom, while theEuclidean action Bk behaves approximately linearly with k.Hence, as suggested in Ref. 2, the prefactor can be quiteimportant in determining the nucleation rate.

In the scenario proposed in Ref. 2 it was also desirablethat the enhancement in the nucleation rate would switch offat the present time, in order to prevent the vacuum fromdecaying further. Unfortunately, the expression 82 does not

seem to have this property. The prefactor depends only onthe radius R, which is itself a function of various parameters,such as the brane tension, the charge and the ambient expan-sion rate, as summarized in Table I. According to 82, anenhancement of the nucleation rate of coincident branes willoccur for

RMin 3kk

,Ho1 ,

4Mp2

k g/m2, 83

where we have used the results of Table I in the first step.

Consider first the situation where kMp2Ho . Note that

even in the regime when RH01 , the dependence of the

degeneracy factor on the corresponding dS temperature isonly power law, and not exponential as suggested in Ref. 2.Also, it is clear the stability of our vacuum is not guaranteedby the smallness of the present expansion rate. The enhance-ment will persist provided that k /k(g/m)

2, even if theambient de Sitter temperature vanishes. More worrisome isthe fact that for sufficiently large k we enter the regime

where kMp2Ho . In that case, we have

RMp2/k ,

which can get smaller and smaller as we increase the numberof coincident branes, eventually leading to a catastrophic de-cay rate, regardless of the value of Ho .

The expression 82 for the nucleation rate is valid formR1. In the opposite limit, mR1, the scalars de-couple, contributing a finite renormalization of the param-eters in the action such as the brane tension and inducedNewtons constant. For completeness, this is discussed in

2If we assume instead a linear interaction at short distances, we

should use 73 and the behavior changes to DR (9/2)2k2, but in

any case the dependence is still a power of the radius R.


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Appendix A, where it is shown that the renormalization ofparameters can have a very significant impact on the nucle-ation rate.

D. Discussion: Temperature of a vacuum bubble

In Ref. 2, the prefactor in the nucleation rate for thenucleation of coincident branes was estimated as an entropy

enhancement

DeS,

where, from dimensional analysis, the entropy was estimatedas ST2R2 per field degree of freedom. The factor R 2 is dueto the area of the bubble, and T is some effective tempera-ture. Although the interpretation of the prefactor as the ex-ponential of an entropy should not be taken too literally, letus try and phrase the results of the preceding subsection inthis intuitive language.

A particle detector following a geodesic in a de Sitterspace responds as if it was at rest in a thermal bath in flatspace, at the temperature ToHo/2. It should be kept in

mind, however, that the dS invariant quantum state is in facta pure state, and hence rather different from a true thermalstate. For instance, any two detectors in geodesic relativemotion observe the same temperature, with a perfectly iso-tropic distribution. This is a consequence of de Sitter invari-ance, and is in contrast with the situation in a thermal bath inflat space, where moving observers detect a temperatureblue-shift in the direction of their motion relative to the bath.

The fields living on a nucleated brane will experiencesome thermal effects too. The bubble is embedded in a dSspace characterized by a temperature To . The interior of thebubble is also a dS space characterized by a different expan-sion rate, with corresponding temperature Ti . The existence

of two different de Sitter spaces in contact with the brane ledthe authors of Ref. 2 to consider two different possibilitiesfor the effective temperature of the fields on the brane: TTo and T(ToTi)

1/2. However, there is in fact no ambigu-ity in the temperature of such fields 4, which is determinedas follows.

The worldsheet of the brane is an S 3 of radius R, theEuclidean de Sitter space in 21 dimensions. If interactionswith bulk fields are neglected, brane fields are only sensitiveto the geometry of the worldsheet, and do not know aboutthe properties of the ambient space. In this approximation,the relevant temperature is clearly the intrinsic temperatureof the lower dimensional de Sitter space,

TR1/2R. 84

This conclusion remains unchanged when we include inter-actions with bulk fields. The simplest way to see this is toconsider the limiting case where gravity can be ignored andthe nucleation takes place in a flat space. There, the ambienttemperature vanishes To0, but the fields on the brane willfeel the temperature TR , because the nucleated brane ex-pands with constant acceleration a1/R . An acceleratingobserver in the Minkowski vacuum will detect a Rindlertemperature TRa/2, which happens to coincide with the

intrinsic worldsheet temperature. Hence, the de Sittervacuum in the 21 dimensional worldsheet is in equilibriumwith the Minkowski vacuum in the bulk. This conclusion isquite general, and applies also to bubbles nucleating in deSitter. The CdL instanton has an O(4) symmetry under Eu-clidean rotations. This becomes an O(3,1) symmetry afteranalytic continuation into Lorentzian time. The quantumstate after bubble nucleation is expected to inherit this sym-

metry 5,26,27, and the only way to achieve it is if the fieldson the brane are in their intrinsic de Sitter vacuum, which ischaracterized by temperature TR .

Note that TR is a relatively high temperature. The radius

of the instanton is always smaller than Ho1 see Table I, and

therefore TR is strictly larger than To and Ti . Nevertheless,

the product k2TR2R2k2, and hence the entropy enhance-

ment, is independent of the ambient de Sitter temperature.As shown in the previous subsections, the independence ofthe prefactor D on the ambient expansion rate is only ap-proximate, due to the anomalous behavior of light fields inde Sitter space. This introduces a dependence of the effectiveaction We f flogD on the radius R of the instanton which

in turn may depend on Ho in certain regimes. This depen-dence, however, is quite different from the one proposed inRef. 2, where it was suggested that the nucleation rate ofcoincident branes would be enhanced at high Ho , and wouldswitch off at low Ho due to the drop in ambient temperature.What we find instead is that, if the CdL instanton for nucle-ation of coincident branes really exists, then the correspond-ing degeneracy factor does not necessarily switch off.

We shall return to a discussion of this point in the con-cluding section. Before that, let us turn our attention to adifferent instanton, which may be relevant to the FMSW sce-nario.

VIII. PAIR CREATION OF CRITICAL BUBBLES

Euclidean de Sitter space is compact in all spacetime di-rections, and as we just discussed it behaves in some re-spects as a system at finite temperature. One may then askwhether there are instantons similar to the thermal ones inflat space. These correspond to static bubbles, in unstableequilibrium between expansion and collapse.

Static instantons with O(3 ) symmetry have previouslybeen considered in a variety of contexts, notably for the de-scription of false vacuum decay in the presence of a blackhole see e.g. Refs. 28,29 and references therein. The par-ticular instanton we shall consider corresponds to pair cre-

ation of critical bubbles in de Sitter, and to our knowledge itdoes not seem to have received much attention in the past.This is perhaps not surprising, since its action is higher thanthat of the action for the maximally symmetric CdL instan-ton. However, if the CdL instanton does not exist for coinci-dent branes, the static one may turn out to be relevant oncethe degeneracy factors are taken into account.

A. The instanton solution

The energy of a critical bubble is different from zero, andconsequently, the metric outside of the bubble is no longer


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pure de Sitter, but Schwarzschildde Sitter SdS. The in-stanton is a solution of the Euclidean equations of motion,with two metrics glued together at the locus of the wall,which is a surface of constant rin the static chart of SdS seeFig. 6. The metric outside is given by

ds2fo rdt2fo

1 rdr2r2d2, 85

where d2d2sin2d2, and

fo r 1 2G Mr Ho2r2 . 86The metric inside is given by

ds2C2fi rdt2fi

1 rdr2r2d2, 87

where

fi r1Hi2

r2 , 88

corresponding to a de Sitter solution. The constant C isdetermined by the condition that on the brane i.e., at rR) the two metrics must agree, which leads to Cfo(R)/fi(R)

1/2.The parameters Mand R depend on k , Ho and Hi . Their

values are determined by the junction conditions at the brane14,

Kab 4Gkab , 89

where Kab is the difference in the extrinsic curvature Kab(1/2)f1/2rg ab on the two sides and ab is the worldsheetmetric. Equation 89 gives rise to the junction conditions,

g4Gk , g0, 90

where we have introduced the new function g(r)f1/2(r)/r. Using Eqs. 86 and 88, we have

gog o1

r3

3G M

r4, g ig i

1

r3. 91

Hence, using 90, g o(R)g i(R)3M/4kR4, and

then g i(R) and g o(R) are easily obtained from Eqs. 91:

g iR 4kR

3M, g oR g iR 1 3G MR . 92

From 86 and 88 we have

g o2R

1

R2

2G M

R 3Ho

2 , g i2R

1

R 2Hi

2 . 93

Inserting 92 in 93 we finally obtain a quadratic equationfor g i(R)x. The solution is

x

4k

3k

16Mp2

4k

3k

16Mp2

2

Hi2

2

1/2

, 94

where we have used Ho2Hi

28G/3/3Mp

2 . Then theparameters M and R are given in terms of x by

R2x 2Hi2 , M4kR/3x. 95

This concludes the construction of the instanton solution forgiven values of the physical parameters k , Ho and Hi .

The equation fo(r)0 has three real solutions for

27Ho2M2G 21. One of them, say r , is negative and the

other two are positive. The two positive roots correspond tothe black hole and cosmological horizons. We call them re-spectively rs and r . Therefore we can write

fo rHo

2

r rr rrs rr. 96

Of course, in our instanton the horizon at rs is not present,since the exterior metric is matched to an interior metric atsome rRrs see Fig. 6. For rR the metric is just a ballof de Sitter in the static chart, and it is regular down to thecenter of symmetry at r0. In general, the size of the cos-mological horizon is given by

Hor2

3cos

3 , 97

where we have introduced the angle

FIG. 6. Static instanton in de Sitter space. The left figure shows the geometry induced on the plane r,t, while keeping angular coordinates

fixed, whereas the right figure shows the geometry induced on the plane r,, keeping and tfixed. The vertical direction corresponds to the

coordinate r, common to both pictures. The cosmological horizon is at rr , the brane is at rR , and r0 is the center of the static

bubble of the new phase.


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arctan 127Ho

2M2G 2

1. 98

In the limit M0 the angle /2, and Hor1.According to Eq. 92, at the brane we have fo(R)

x2(R3G M)2, so the equation fo(R)0 has a doublezero instead of two different roots. This means that the radius

of the instanton will coincide with the radius of one of thehorizons only in the special case where both horizons havethe same size, rsrR3G M. In the limit rsr theexterior metric becomes the Nariai solution 30,31, which

has r(3Ho)1. Note that in the limit 3G M

(3Ho)1, 0 and Hor1/3, as expected.

Like in the case of instantons describing the production ofblack holes 30 or monopoles 22 in de Sitter metric, theinstanton presented here describes the creation of pairs ofbubbles. As we shall see, the Euclidean solution is periodicin the time direction, so that time runs on a circle S1 see Fig.6. The geometry at the time of nucleation is obtained byslicing the compact instanton through a smooth spacelike

surface that cuts the S1 factor at two places, say, t0 andt. The resulting geometry contains two different bubblesseparated by a distance comparable to the inverse expansionrate.

B. Temperature and action

In order to calculate the temperature of the worldsheet inthe static instanton we must first find the time periodicity .This is determined by the regularity of the Euclidean metricat the cosmological horizon. For rr , we have

fo rA2 1 r

r , 99

where

A 2Ho2 rr rrs3Ho

2r

21. 100

In terms of the new coordinates

2r

A1 r

r,

A 2

2rt, 101

the metric 85 for rr reads

ds 2

2d

2d

2r

2d2

,102

so it is clear that is an angle, 02, and t varies inthe range 0t4r/A

2. Therefore the value of is

4r

3Ho2

r

21

2r

2

r3G M. 103

The temperature of the worldsheet instanton is given by the

proper time periodicity R0fo

1/2(R)dtfo1/2(R)

C fi1/2(R). Hence, the inverse temperature is given by

R2xr2

R3G M

r3G M. 104

We shall also be interested in the Euclidean action, whichturns out to have a rather simple expression in terms of r .This is derived in Appendix B, where it is shown that thedifference in Euclidean actions between the instanton and the

background solutions is given by

B

GHo2

1r

2 Ho2 . 105

C. Some limiting cases

Let us start with the case of low tension branes, k/Mp2

Ho ,HoHi . In this case the parameter x is large com-pared with Ho , Rx

1 is small, and G M Ho1. In thislimit the angle in Eq. 98 is close to /2 and Hor1. We have

B2

Ho

16k3

32, R

2

Ho. 106

This is just the flat space expression for the energy of acritical bubble, multiplied by Euclidean time periodicity ofthe low curvature de Sitter space in which this bubble isembedded.

Next, we may consider the case of intermediate tension

HoHik/Mp2Ho ,Hi . In this case, xHi /2, R

(3x)1, Hor1G M Ho , with G M Ho1, and wehave

B 16

2

33k

Ho3

, R2

3Ho. 107

In this case, the difference in pressure between inside andoutside of the brane is insignificant compared with the branetension term, which is balanced against collapse by the cos-mological expansion.

Finally, in the limit of very large k we find that 3 G Mbecomes larger than R, namely, 3 G M4R /3. This meansthat fo(R) vanishes for some value kmax , given belowin Eq. 118, so it is not sensible to consider the limit of verylarge k but just the limit kmax . As we have mentionedsee the discussion below Eq. 96, the exterior metric inthis limit corresponds to the Nariai solution, with rsr(3Ho)

1. Replacing this value in 105 we find readily

B2

3GHo2

. 108

It is interesting to compare this value of B with the corre-sponding one for the nucleation of black holes in the same deSitter universe. This is described by the Nariai instanton30, which has the bounce action


083510-17


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BN

3GHo2

. 109

Note that the difference BBN/3GHo2A bh/4G , where

A bh4rs2 is the area of the black hole horizon in the Nariai

solution. Hence, the probability of nucleating black holesdivided by the probability of nucleating brane configurations

characterized by the same mass parameter is just the expo-nential of the black hole entropy, as expected from generalconsiderations in this argument, we are of course neglectingthe entropy stored in the field degrees of freedom living onthe branes, which will only show up when the determinantalprefactor in the nucleation rate is evaluated.

Let us consider the value of R in the limit kma x .This is a singular limit in Eq. 104 due to the simultaneousvanishing of numerator and denominator. Thus we will needto change to more appropriate coordinates. The fact that rsr does not mean, though, that both horizons coincide,since the coordinates r,t become inadequate in this case.Near this limit the metric outside takes the form 85, with

fo rA2 1 r

r 1 r

r

2

, 110

and rr , plus higher orders in A. The constant A is thesame parameter defined in 100, but in this limit tends to

zero, A 23Ho(rrs). Now we define new coordinates and by

cos12

A 2 1 r

r , A

2

2t, 111

so that the metric becomes

ds 2sin2d2r

2 d2r

2 d2. 112

In these coordinates the cosmological horizon is at 0 andthe black hole horizon is at . Now in the limit A0 we

just replace r(3Ho)1.

We must determine the position R of the brane, which isgiven as before by the matching conditions 89, where nowthe metric outside is 112. So, on the brane, we have

ds k2sin2 Rd

2r

2d2 113

fiR dt2R 2d2 114

and the extrinsic curvature on the outside of the brane is

(1/2)g ab , with g00sin2 and g r

2 , i.e., K00(1/r)g 00 cot,K0. The curvature inside is as

before K00g 00rfi1/2 and Kg fi

1/2/r, with fi(r)(1

Hi2r2), so the Israel conditions give

1

rcotRfi

1/2R4Gk , 115

fi1/2R /R4Gk . 116

These equations are easily solved and give

sinR 3Ho2Hi

2

6Ho2Hi

21/2

, 117

kmax2Mp23Ho

2Hi

2, 118

so Hi must be less than 3Ho . Now regularity at the cos-mological horizon 0 implies that 0/r2, so Rsin(R)2r . Hence,

R2

3Ho 3Ho

2Hi

2

6Ho2Hi

21/2

. 119

Thus, also in this case, the effective temperature of the fielddegrees of freedom living on the worldsheet will be of orderH0 or higher.

D. The prefactor for static instantons

In flat space, and at finite temperature Tk/k , the in-stanton which is relevant for vacuum decay is static and

spherically symmetric in the spatial directions. The fluctua-tions are periodic in Euclidean time, with periodicity 1/T. The worldsheet of the brane has the topology S 1

S2, where the S 1 is the direction of imaginary time, andthe S 2 is the boundary of the critical bubble, a closedbrane in unstable equilibrium between expansion and col-lapse this is in contrast with the zero temperature instanton,where the worldsheet is a 3-sphere. The radius of the criticalbubble is given by

R2k

k

,

and the difference of the instanton action and the action forthe background is given by

BE(0 ),

where E(0 )(4/3)kR2 is the classical energy of the criti-

cal bubble. The one loop quantum correction can be writtenas see e.g. Ref. 32

WFETS . 120

Here F denotes the free energy and E is the correction tothe energy of the critical bubble due to the field . This

includes the zero point energy of in the presence of thebubble, as well as the thermal contribution

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