Metastable gravitons and infinite volume extra dimensions

29 June 2000

Ž .Physics Letters B 484 2000 112–118www.elsevier.nlrlocaternpe

Metastable gravitons and infinite volume extra dimensions

G. Dvali, G. Gabadadze, M. PorratiDepartment of Physics, New York UniÕersity, New York, NY 10003, USA

Received 29 February 2000; received in revised form 19 May 2000; accepted 23 May 2000Editor: M. Cvetic

Abstract

We address the issue of whether extra dimensions could have an infinite volume and yet reproduce the effects ofobservable four-dimensional gravity on a brane. There is no normalizable zero-mode graviton in this case, neverthelesscorrect Newton’s law can be obtained by exchanging bulk gravitons. This can be interpreted as an exchange of a singlemetastable 4D graviton. Such theories have remarkable phenomenological signatures since the evolution of the Universebecomes high-dimensional at very large scales. Furthermore, the bulk supersymmetry in the infinite volume limit might bepreserved while being completely broken on a brane. This gives rise to a possibility of controlling the value of the bulkcosmological constant. Unfortunately, these theories have difficulties in reproducing certain predictions of Einstein’s theoryrelated to relativistic sources. This is due to the van Dam–Veltman–Zakharov discontinuity in the propagator of a massivegraviton. This suggests that all theories in which contributions to effective 4D gravity come predominantly from the bulkgraviton exchange should encounter serious phenomenological difficulties. q 2000 Elsevier Science B.V. All rights reserved.

If Standard Model particles are localized on abrane, the volume of extra dimensions can be aslarge as a millimeter without conflicting to any

w xexperimental observations 1 . This is also true forwarped spaces in which the extra dimensions are

w x Žnon-compact but have a finite volume 2 for earlierw x.works on warped compactifications see 3,4 .

w x NIn the framework of Ref. 1 the volume V;Lof extra N space dimensions sets the normalizationof a four-dimensional graviton zero-mode. There-fore, the relation between the observable and the

Ž .E-mail address: [email protected] G. Gabadadze .

Žfundamental Planck scales M and M respec-P P f.tively reads as follows:

M 2 sM 2qN V . 1Ž .P P f

A similar relation holds for the Randall–SundrumŽ . w x y2RS scenario 2 , where the role of L is played bythe curvature of AdS . In this case the extra dimen-5

Žsion is not compact, nevertheless its length or vol-.ume is finite and is determined by the bulk cosmo-

'< <logical constant LA1r L . In the scenario of Ref.w x1 gravity becomes high-dimensional at distancesr<L with the corresponding change in Newton’slaw

1rr™1rr1qN . 2Ž .The same holds true for RS-type scenarios with

w xnon-compact extra dimensions 5 .

0370-2693r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved.Ž .PII: S0370-2693 00 00631-6

( )G. DÕali et al.rPhysics Letters B 484 2000 112–118 113

The purpose of the present letter is to studywhether the volume of extra space can be trulyinfinite while the four-dimensional Planck mass is

Ž .still finite. In this case the relation 1 should some-how be evaded. Since there are no normalizable zeromodes in such cases, the effects of 4D gravity mustbe reproduced by exchanging the continuum of bulkmodes. An example of this type was recently pro-

w xposed in Ref. 6 . The physical reason why such anexchange can indeed mimic the 1rr Newtonian lawcan be understood as follows. The four-dimensionalgraviton, although is not an eigenstate of the lin-earized theory, can still exist as a metastable reso-nance with a finite lifetime t . In such a case oneg

might hope that the exchange of this graviton ap-proximates Newton’s law at distances shorter thanthe graviton lifetime but changes the laws of gravityat larger scales. The question is whether the four-di-mensional effective theory obtained in this way isphenomenologically viable. In the present paper wewill argue that, despite the correct Newtonian limit,the infinite volume scenario has problems in repro-ducing other predictions of Einstein’s theory. Thisproblem is shared by any model in which the domi-nant contribution to 4D gravity comes from theexchange of continuum states. The reason behindsuch a discrepancy is the van Dam–Veltman–Zakharov discontinuity in the propagator of a mas-

w xsive spin-2 field in the massless limit 7 . Verybriefly, the physical effects of additional polariza-tions of a massive graviton survive in the masslesslimit and change dramatically predictions of the the-

w xory 7 . As a result, any theory which relies on theŽ .exchange of massive no matter how light gravitons,

will give rise to predictions which differ from thoseof General Relativity.

We shall consider the five-dimensional Einsteingravity coupled to an arbitrary energy-momentumtensor T which is independent of four space-timeA B

coordinates x . We will assume that T resultsm A B

from a certain combination of branes, bulk cosmo-logical constant L and classical field configurationswhich preserve four-dimensional Poincare invari-´ance. Einstein’s equations G sT qg L giveA B A B A B

rise to a metric of the following form

ds2 sA z ds2 ydz 2 , 3Ž . Ž .Ž .4

where z is an extra coordinate, and we assume thatthe four-dimensional metric ds2 is flat. The volume4

of the extra space in this construction is determinedby an integral

q`5r2A z dz . 4Ž . Ž .H

y`

Ž . ŽFor instance, in the RS framework A z s 1q< <.y2H z , where H is proportional to the square root

'< <of the cosmological constant HA L . This warpŽ .factor gives rise to a finite expression in 4 . An

infinite volume theory is obtained if, for instance, theŽ .value of A z tends to a nonzero constant as z goes

to "`. If the volume is finite, there is a normaliz-able zero-mode graviton in the spectrum of fluctua-

w xtions about this background 2 . Indeed, let usparametrize the four-dimensional graviton fluctua-tions as follows:

2 m n 2ds sA z h qh x , z dx dx ydz . 5Ž . Ž . Ž .Ž .mn mn

The corresponding linearized Schrodinger equation¨Ž . y3r4Ž . Ž . Ž0. Ž .for the excitation h x, z sA z C z h xmn mn m

takes the form:

3 AXX AX 22 2yE q y C z sm C z . 6Ž . Ž . Ž .z 2½ 5ž /4 A 4 A

Where h mnE E hŽ0.sym2 hŽ0. and primes denote dif-m n

ferentiation with respect to z. This equation has azero-mode solution for a generic form of a warp

Ž .factor A z :

C z sA3r4 z . 7Ž . Ž . Ž .zm

This implies that the z dependent part of the fluctua-Ž . Ž .tions h x, z is just a constant, i.e., h x, z sconst.P

Ž . Ž .exp ipx . If the integral in 4 diverges, the volumeis infinite and the zero-mode is not normalizable.Thus, the spectrum in this case would consist ofcontinuum states only. One should notice, however,that even in this case the correct Newtonian limitmay be recovered in some approximation by ex-changing the continuum of non-localized states! Thephysical reason for such a behavior is that the 4Dlocalized graviton, although not an eigenstate, canstill exist as a metastable resonance with an expo-nentially long lifetime, t . Therefore, the exchangeg

of this resonance could give rise to a correct Newto-nian potential at intermediate distances. Note that

( )G. DÕali et al.rPhysics Letters B 484 2000 112–118114

this is just a reformulation of the fact that the towerof the bulk states conspire in such a way that theNewtonian potential is seen as an approximation. Tomake these discussions a bit precise let us turn to thepropagator of a massive metastable graviton. This isgiven as follows:

G mna b xŽ .4

dp41 ma nb mb nas g g qg gŽ .ŽH 242pŽ .

1 Ž .mn a b 2 2 yi p xy g g qOO p D p ,m ,G e , 8Ž . Ž .Ž .. 03

Ž 2 2 .where D p , m , G stands for the scalar part of a0Ž 2 2 . Ž 2massive resonance propagator, D p ,m ,G s p0

2 .y1ym q im G . G denotes the width of the reso-0 0

nance. The momentum dependent part in the tensorstructure gives zero contribution when the propaga-tor is convoluted with a conserved energy-momen-tum tensor, thus, this part will be omitted. In order tomake a contact with continuum modes, let us use thefollowing spectral representation for D:

1 r sŽ .2 2D p , m , G s ds , 9Ž .Ž . H0 22p syp q ie

Ž .where s denotes the Mandelstam variable and r sis a spectral density. If the assumption of the reso-

Ž .nance dominance is made, then r s is approxi-mated by a sharply peaked function around the reso-nance mass ssm2 . In what follows we assume that0

the resonance lifetime is very big and we neglect theŽeffects of a nonzero resonance width these will

.modify gravity laws at very large distances only .Exchanging such a particle between two static sourcesone obtains the potential

y s r'eV r ; r s ds . 10Ž . Ž . Ž .H

r

This expressions reproduces a standard 1rr interac-Ž .y1tion at distances r< m in the single, narrow-0

Ž . Ž 2 .resonance approximation, r s Ad sym . On the0Ž .other hand, we can expand the spectral density r s

into the complete set of bulk modes

`2 2< <r s s c 0 d sym dm . 11Ž . Ž . Ž . Ž .H m

0

Ž .Here, c 0 denote the wave functions of the bulkm

modes at the point zs0. Using this expression forthe spectral density one finds the following potential

`ym re

2< <V r ; c 0 dm . 12Ž . Ž . Ž .H mr0

This is nothing but the potential mediated by thecontinuum of the bulk modes. Thus, the effect of themetastable graviton, when it exists, can be read offthe expression which includes all the bulk modes.These two descriptions are complementary to eachother. In the case when the resonance exists, thecontinuum modes can conspire in such a way thatŽ .12 yields the 1rr law in a certain approximation.The inverse statement is also likely to be true. In theAppendix we will show explicitly the presence of aresonance state in a model with infinite volume extradimension and 1rr potential produced by the bulk

w xmodes 6 . As we mentioned above, such modelgives Newtonian gravity only at intermediate dis-tances. At large distances, the five-dimensional laws

Žof gravity should be restored due to the metastable.nature of the resonance . This phenomenon could

have dramatic cosmological and astrophysical conse-quences. Indeed, at large cosmic scales the time-de-

Ž .pendence of the scale factor R t in Freedman–Ro-bertson–Walker metric would dramatically changedue to the change in the laws of gravity1.

Another interesting comment concerns bulk su-persymmetry. Since the volume of the extra dimen-sion is infinite, it might be possible to realize thefollowing scenario. The bulk is exactly supersym-metric and SUSY is completely broken on a braneŽthis could be a non-BPS brane which is stable for

w x.some topological reasons 9 . The transmission ofSUSY breaking from the brane worldvolume to thebulk is suppressed by the volume of the bulk and isvanishing. In such a case one could imagine a setupwhere the bulk cosmological constant is zero due tothe bulk SUSY2.

1 A different possibility to modify the long distance gravity duew xto an additional massive graviton was proposed earlier in Ref. 8 .

In the view of phenomenological problems discussed below, thisgraviton should be very weakly coupled.

2 Note that local SUSY does not necessarily imply vanishing ofthe vacuum energy. However, this can be accomplished by impos-ing on a model additional global symmetries.


Having these attractive features of the theorieswith truly infinite extra dimensions discussed, wemove to some phenomenological difficulties of thesemodels. In fact, we will argue below that thesetheories cannot reproduce other predictions of Ein-stein’s general relativity. The reason is that all thespin-2 modes that dominantly contribute to the four-dimensional gravity in this case are massive modes.

w xIt has been known for a long time 7 that propagatorof massive spin-2 states has no continuous masslesslimit. As a result the effects of the massless spin-2graviton are different from the massive one, nomatter how small the mass is. Let us show how thisaffects the phenomenology of infinite volume theo-ries. The four-dimensional gravity on a brane isreproduced by an exchange of the continuum of bulkgravitons. At a tree level this gives

`X X X X4 4 mna bG dm dx d x T x G xyx T x ,Ž . Ž . Ž .H H5 mn m a b

0

13Ž .Ž . X Ž X .where T x and T x are the energy-momentummn mn

tensors for two gravitating sources. For m/0 thegraviton propagator is given by

mna b Ž X .G xy xm

1 14 ma nb mb na mn a b Ž .dp g g q g g y g g q OO pŽ .2 3sH 4 2 2p y m y ieŽ .2p

=eyi pŽ xyxX. , 14Ž .whereas for ms0 we have

mna b Ž X .G xy x0

1 14 ma nb mb na mn a b Ž .dp g g q g g y g g q OO pŽ .2 2sH 4 2p y ieŽ .2p

=eyi pŽ xyxX. . 15Ž .As we see, the tensor structures in the two cases aredifferent. In the massless limit, the propagator ex-hibits the celebrated van Dam–Veltman–Zakharovdiscontinuity. This is due to the difference in thenumber of degrees of freedom for massive and mass-less spin-2 fields. In our case this difference is verytransparent, KK gravitons at each mass level ‘‘eatup’’ three extra degrees of freedom of g and g5m 55

components of the higher dimensional metricŽ .‘‘graviphotons’’ and ‘‘graviscalars’’ respectively .

Since we choose a model in which there is nonormalizable-zero mode, the whole answer is givenby the bulk continuum. Let us show that the 4Dgravity which is obtained in this way cannot repro-duce observable effects of General Relativity. Let usfirst consider the Newtonian limit. In this case, wetake two static point-like sources

T x sm d d d x ,Ž . Ž .mn 1 m0 n 0

T X xX sm d d d xX yr . 16Ž . Ž . Ž .mn 2 m0 n 0

For this setup the bulk graviton exchange gives

eym r22 < <m m G dm c 0 . 17Ž . Ž .H1 2 5 m3 r

Since the leading behavior of the integral for theparticular case at hand is 1rr,

eym r a2< <dm c 0 ; q . . . , 18Ž . Ž .H mr r

Žthe correct Newtonian limit may be reproduced a is.some normalization constant . On the other hand,

since the exchange of one normalizable masslessgraviton would give

1 m m1 2G , 19Ž .N 2 r

we have to set

3 GNa G s . 20Ž .5 4

This identification provides the correct Newtonianpotential for static sources. So far so good. Unfortu-nately, the problem arises when one tries to accountfor moving sources. To see this let us take one of thesources to be a moving point-like particle of massm and proper time t . The energy-momentum tensor2

for this particle is written as:

T X xX sm dt x x d xX yx t . 21Ž . Ž . Ž .Ž .Hmn 2 m n˙ ˙

The result of the bulk graviton exchange then gives

1 mG m m dt x x y x xŽ .˙ ˙H5 1 2 0 0 m3 ˙

=eym r Žt .

2< <dm c 0 , 22Ž . Ž .H mr tŽ .


Ž . Ž .where rsx t . With the identification 20 , in theleading order this yields

13 1 mG m m dt x x y x x . 23Ž .Ž .˙ ˙HN 1 2 0 0 m4 3 ˙ r tŽ .

On the other hand the exchange of a normalizablegraviton zero-mode produces the following result

11 mG m m dt x x y x x . 24Ž .Ž .˙ ˙HN 1 2 0 0 m2 ˙ r tŽ .

This shows the discrepancy between the predictionsof the two theories. In particular, the same procedureapplied to the problem of bending of light by the Sungives the discrepancy by the factor 3r4. Indeed, forthe bending of light in the gravitational field of theSun, the tree-level bulk graviton exchange gives:

G M T k ,q ,e ,e XŽ .5 Sun 00 m n

=d k yqŽ .0 0 2< <dm c 0Ž .H m2 2kyq ym y i´Ž .

3 G M T d k yqŽ .N Sun 00 0 0,y q . . . , 25Ž .24 < <kyq

Ž X .where T k,q,e ,e is the component of the en-00 m n

ergy-momentum tensor for photons in the momen-Ž . Ž X .tum representation, and k e and q e are them n

Ž .momenta polarizations of initial and final photons.This is just 3r4 of the result of the 4D masslessgraviton exchange.

Summarizing, we have shown that in theorieswith truly infinite extra dimensions the correct four-dimensional Newtonian gravity can be obtained atintermediate distances due to a metastable resonancegraviton. This description is complementary to theexact summation of continuum modes. Due to thefinite lifetime of the resonance the laws of gravityare modified at large distances. This would give riseto interesting cosmological consequences. Moreover,these models could allow to preserve bulk supersym-metry while it is completely broken on a brane.Unfortunately, these models encounter a number ofphenomenological difficulties. The effects of addi-tional polarization degrees of freedom of massivegravitons survive even in the massless limit and lead

to substantial discrepancies with the predictions ofGeneral Relativity. It might be possible to cure thesediscrepancies by introducing new very unconven-tional interactions. The addition of dilaton-typescalars coupled to T m seems to make things worse.m

Acknowledgements

We wish to thank Ian Kogan and Valery Rubakovfor comments. The work of G.D. is supported in partby David and Lucile Packard Foundation Fellowshipfor Science and Engineering and by Alfred P. SloanResearch Fellowship. That of G.G. is supported byNSF grant PHY-94-23002. M.P. is supported in partby NSF grant PHY-9722083.

Note added

After this paper was prepared for submission thew xwork 10 appeared. The authors of this work have

also realized that metastable gravitons can be respon-sible for the 1rr law in theories with infinite extradimensions. However, the generic phenomenologicaldifficulties of this class of theories which is a crucial

w xpart of our work have not been addressed in 10 .After this work appeared on the net, we were

informed by V. Rubakov that in the revised versionw xof Ref. 6 the role of a metastable graviton was also

elucidated and its decay width was calculated.The results obtained in this work assume that 5D

gravity couples universally to T and that the theorymn

has no unconÕentional or unphysical states such asghosts or tachyons. If these states were present, aswe mentioned in the text, the results could be modi-fied, but then it is hard to make sense of the theory.Within a few weeks after this work appeared on thehep-th bulletin board some interesting developments

w x Ž w x.followed. First, in Ref. 11 see also 12 it waspointed out that if one does calculations in the RSgauge one should take into account the brane bend-

w xing term 13 which can cancel the extra polariza-tions discussed in the present work. Shortly after, we

w xargued in Ref. 14 that the brane bending term ofw x w xRefs. 11 and 12 is nothing but the manifestation

of a scalar ghost field which could exist in thew xmodel of Ref. 6 due to the violation of the null


w x3energy condition pointed out in 15 . To make themodel consistent and stable the ghost field with thenegative kinetic term should be projected out from

w xthe physical spectrum of the theory 14 . If this isdone one returns back to the phenomenological prob-lems with extra polarizations discussed in the present

Ž w x.work see also 14 . A model without ghosts wherethe extra polarizations are manifest was proposed

w xrecently in Ref. 17 .We would like to thank C. Csaki, J. Erlich, and´

T.J. Hollowood for useful communications on theseissues.

Appendix A.

Below we show the presence of a resonance in asystem with an infinite extra dimension. The particu-lar example which we consider is the one studied in

w xRef. 6 . The five-dimensional interval defining thebackground and four-dimensional graviton fluctua-

Žtions is set as follows here we choose to use anon-conformaly flat metric in order to be consistent

w x.with the conventions of 6 :2 m n 2ds s A y h qh x , y dx dx ydy .Ž . Ž .mn mn

A.1Ž .

There is a 3-brane with positive tension T which islocated at ys0. In addition, there are two 3-braneswith equal negative tensions yTr2 located at adistance y to the left and right of the positive-ten-c

< <sion brane. For y -y the space is AdS and thec 5Ž . Ž .warp-factor is normalized as A y sexp y2 Hy .

< <Furthermore, for y ) y , the space becomesc

Minkowskian and the corresponding warp-factor is a2 Ž .constant, c 'exp y2 Hy . Thus, at large dis-c< <tances, i.e. y 4y , the system reduces to a singlec

tensionless brane embedded in five-dimensionalMinkowski space-time. For simplicity of presenta-tion in what follows we will deal with the positive

Žsemi-axis only, i.e., yG0 the negative part of the

3 w xThe scalar ghost of the model of 6 has been identifiedŽ .subsequently in the harmonic gauge with a radion field which

has the negative kinetic term in the corresponding brane construc-w xtion 16 .

.whole y axis is restored by reflection symmetry .Choosing the traceless covariant gauge for graviton

Ž m m .fluctuations h s0, E h s0 the Einstein equa-m mn

tions take the form:

CXX y4H 2Cqm2e2 H yCs0, 0-y-y ,c

m2XX

C q Cs0, y)y . A.2Ž .c2c

Where we have introduced the y dependent part ofŽ . Ž . Ž .the fluctuations as follows h x, y 'C y exp ipx .

Furthermore, the mass-shell condition for gravitonfluctuations is defined as p2 sm2. The equationspresented above should be accompanied by the Israelmatching conditions at the points where the branesare located. For the particular case at hand these

w xconditions take the form 6

CX q2 HCs0, ys0 ;X <C s2 HC , ysy . A.3Ž .jump c

Ž .The solutions to Eqs. A.2 are combinations ofBessel functions for 0-y-y , and exponentials forc

y)y :c

m mH y H yC y sA J e qB N e ,Ž .m m 2 m 2ž / ž /H H

0-y-y , A.4Ž .c

mC y sC exp i yyyŽ . Ž .m m cž /c

mq D exp yi yyy ,Ž .m cž /c

y)y . A.5Ž .c

The constant coefficients A , B , C and D arem m m mŽ .determined by using the matching conditions A.3

Ž .along with the normalization equation . The pres-ence of a resonance state requires that the coefficient

Žof the incoming wave in the solution D in thism.case vanishes at a point in the complex m plane.

This determines a resonance. Calculating D andm

putting D s0 one finds:m

K r I reH y q I r K reH yŽ . Ž .Ž . Ž .1 2 1 2

s I r K reH y yK r I reH y , A.6Ž . Ž . Ž .Ž . Ž .1 1 1 1

where we have introduced a new variable r'

yimrH. This relation can be solved for small values


of r. The result is r ,y2exp y3Hy . There-Ž .c

fore, the resonance width is proportional to

G A H exp y3Hy . A.7Ž . Ž .c

In the limit y ™`, the resonance width goes to zeroc

and one recovers a zero-mode graviton localized onw xa positive tension brane 2 .

References

w x1 N. Arkani-Hamed, S. Dimopoulos, D. Dvali, Phys. Lett. BŽ . Ž .429 1998 263; hep-phr9803315; Phys. Rev. D 59 1999

0860; hep-phr9807344.w x Ž .2 L. Randall, R. Sundrum, Phys. Rev. Lett. 83 1999 3370;

Ž .hep-phr9905221; Phys. Rev. Lett. 83 1999 4690; hep-thr9906064.

w x Ž .3 M. Visser, Phys. Lett. B 159 1985 22; hep-thr9910093.

w x4 M. Gogberashvili, hep-phr9812296; hep-phr9812365.w x5 N. Arkani-Hamed, S. Dimopoulos, D. Dvali, N. Kaloper,

hep-thr9907209.w x6 R. Gregory, V.A. Rubakov, S.M. Sibiryakov, hep-

thr0002072.w x Ž .7 H. van Dam, M. Veltman, Nucl. Phys. B 22 1970 397; V.I.

Ž .Zakharov, JETP Lett. 12 1970 312.w x8 I.I. Kogan, S. Mouslopoulos, A. Papazoglou, G.G. Ross, J.

Santiago, hep-thr9912552.w x Ž .9 G. Dvali, M. Shifman, Nucl. Phys. B 504 1997 127.

w x10 C. Csaki, J. Erlich, T.J. Hollowood, hep-thr0002161.´w x11 C. Csaki, J. Erlich, T.J. Hollowood, hep-thr0003020.´w x12 R. Gregory, V.A. Rubakov, S.M. Sibiryakov, hep-

thr0003045.w x13 J. Garriga, T. Tanaka, hep-thr9911055; S.B. Giddings, E.

Katz, L. Randall, hep-thr0002091.w x14 G. Dvali, G. Gabadadze, M. Porrati, hep-thr0003054.w x15 E. Witten, hep-phr0002297.w x16 L. Pilo, R. Rattazzi, A. Zaffaroni, hep-thr0004028.w x17 G. Dvali, G. Gabadadze, M. Porrati, hep-thr0005016.

Documents

Metastable gravitons and infinite volume extra dimensions