Photons and gravitons as Goldstone bosons and the cosmological constant

PHYSICAL REVIEW D 66, 045015 ~2002!

Photons and gravitons as Goldstone bosons and the cosmological constant

Per Kraus* and E. T. Tomboulis†

Department of Physics and Astronomy, UCLA, Los Angeles, California 90095-1547~Received 24 April 2002; published 19 August 2002!

We reexamine a scenario in which photons and gravitons arise as Goldstone bosons associated with thespontaneous breaking of Lorentz invariance. We study the emergence of Lorentz invariant low energy physicsin an effective field theory framework, with non-Lorentz invariant effects arising from radiative corrections andhigher order interactions. Spontaneous breaking of the Lorentz group also leads to additional exotic but weaklycoupled Goldstone bosons, whose dispersion relations we compute. The usual cosmological constant problemis absent in this context: being a Goldstone boson, the graviton can never develop a potential, and the existenceof a flat spacetime solution to the field equations is guaranteed.

DOI: 10.1103/PhysRevD.66.045015 PACS number~s!: 11.15.Ex, 11.10.Ef

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I. INTRODUCTION

Massless particles can arise by a variety of mechanisgauge symmetry, chiral symmetry, supersymmetry, andspontaneous breaking of global symmetry. The masslessof the photon and graviton is typically associated with texistence of gauge symmetry; here we explore an alternaoption in which it is associated with the spontaneous breing of Lorentz invariance, with the ‘‘gauge’’ fields arising aGoldstone bosons. This basic idea has a long history, daback to the 1963 work of Bjorken@1#. For related work, see@2,3,4,5#.

At the level of low energy effective field theory, the usuargument in favor of exact gauge invariance is based on Lentz invariance. Some form of gauge invariance is requin order to obtain an interacting, unitary, Lorentz invariatheory of massless particles with spin 1 or 2. In a manifesLorentz invariant formulation a violation of gauge invarianwill typically imply a noncancellation of timelike and longitudinal modes, yielding a nonunitarySmatrix in the physicalsector. In a noncovariant gauge fixed formulation unitaritymanifest, but the action is required to be formally gauinvariant in order to recover Lorentz invariance of theSma-trix.

Especially in the case of gravity, there are various movations for exploring alternatives. First, it is a basic fact thwe inhabit a universe that is not Lorentz invariant at larscales due to cosmological expansion. Second, in the cagravity, gauge invariance is insufficient to guarantee theistence of massless gravitons propagating in Minkowspacetime, since a potentialL4A2g is allowed. Since a nonzero cosmological constant leads to a non-Lorentz invarvacuum, the original motivation for gauge invariance is nactually realized.

As an alternative we turn to effective actions that yiemassless photons and gravitons as Goldstone bosonspontaneously broken Lorentz invariance. Of course, thefective theory must be consistent with the observed accuof Lorentz invariance at distances small compared to the

*Email address: [email protected]†Email address: [email protected]

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vature scale of the universe. This is achieved by effecttheories consisting of three sorts of terms: gauge invarkinetic terms, non-gauge invariant potential terms, and smcorrections to these. The simplest example isL5(Fmn)2

2V(AmAm). As we will discuss, it is not too hard to generasuch effective actions from some more conventional undlying dynamics. All terms in the action are taken to be Loentz invariant; however, the potential is assumed to giveto a constant non-Lorentz invariant vacuum expectatvalue. The broken Lorentz generators imply the existencemassless Goldstone bosons. In the absence of the potethe vacuum expectation value would have no physical effbeing pure gauge; in particular, exact Lorentz invarianwould be maintained. With the potential included gaugevariance is broken and with it exact Lorentz invariancHowever, by definition the Goldstone bosons do not appin the potential; the expansion of the potential is in termsmassive fields. Since Lorentz invariance is broken onlythe potential, the breaking will be suppressed by the invemass of the heavy fields. So at low energies we will haveapproximately Lorentz invariant theory of Goldstone bosoThe expansion of the potential to quadratic order in flucttions simply acts a gauge fixing term, so the low enetheory is approximately that of photons or gravitons innoncovariant gauge.

Actually, there are additional interesting complicationsthe quantum theory, where loops induce small non-Loreinvariant kinetic terms. This leads to the appearance of ational non-Lorentz invariant but ‘‘weakly coupled’’ Goldstone bosons whose effects we discuss. Directly relatethis is that it is crucial to study whether the form of oeffective action is stable under radiative corrections, as thwill generically induce all possible operators consistent wthe symmetries. We study this question carefully, and fithat with reasonable starting assumptions the resultingenergy physics appears approximately Lorentz invariant.

Motivated by the cosmological constant problem, a mtitude of authors have experimented with modifying gravin various ways~see @6# for a review of some attempts!.From the point of view of low energy effective field theorthe problem is that general covariance allows one—and oone—potential term,L4A2g, and so unless this term vanishes there exists no solution to the equations of motion wconstant fields. But in a scenario in which the graviton is

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PER KRAUS AND E. T. TOMBOULIS PHYSICAL REVIEW D66, 045015 ~2002!

Goldstone boson this problem does not arise, since a Gstone boson can never acquire a potential. So even if sscalar field undergoes a phase transition, contributing a tV(f0)A2g to the effective action, one knows that thvacuum expectation value for massive fields can be shisuch that the Goldstone boson gravitons remain massTherefore, there willalwaysexist an exact vacuum solutiowith constant fields, on which propagate massless graviwith approximately Lorentz invariant physics. This theguarantees the existence of the sort of solution one wwithout fine-tuning. However, one should note that themay also exist other solutions with spacetime varying fielA complete solution to the problem should address why a~or nearly flat! spacetime solution is preferred; we discuthis in Sec. III.

The first part of this paper is devoted to the detailed stuof the photon as a Goldstone boson, but this is essentiawarmup for the more interesting case of gravity, which pvides our main motivation. There does not seem to beobvious advantage in producing the photon as a Goldsboson, and we have made no attempt at a realistic modeincluding the other standard model fields. We should astress that we will work in an effective field theory framwork in which the graviton is to be thought of as a composof more fundamental degrees of freedom. It may be womentioning why our scenario is not in conflict with the therem of Weinberg and Witten@7#, which rules out ‘‘compositegravitons’’ in a broad class of models. Specifically, the therem states that a Lorentz invariant theory with a Loreinvariant vacuum and a Lorentz covariant energy-momentensor cannot have a massless spin 2 particle in its spectThere is no conflict here since our vacuum will not be Loentz invariant.

This paper is organized as follows. In Sec. II we studydetail the example of the photon as a Goldstone boson.photon example illustrates most of the important generaltures and is much simpler computationally than the gravcase. We also discuss the relation to previous work onsubject. In Sec. III we discuss the graviton and the cosmlogical constant problem in this context. Section IV has sofinal comments, and the Appendix contains technical resfor the gravity case.

II. PHOTON AS GOLDSTONE BOSON

We begin by writing down a certain effective action forvector field Am coupled to matter. The motivation for thiform will be discussed subsequently. The action is tothought of as an effective action defined at a UV cutoff scL. We thus consider the Lagrangian

L5NH 21

4FmnFmn2V~AmAm!

1higher derivativesJ 1Lmatter~f,Am!

1O~N0!. ~2.1!

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In the above,N is some large number, and we have writtout the leadingN terms in the action. The action is Loreninvariant—all indices being raised and lowered with tMinkwoski metric—but not gauge invariant. In particulathe potentialV(AmAm) is not gauge invariant; a crucial poinis that this is theonly non-gauge invariant term at leadinorder in N. In particular, the higher derivatives terms aterms like (FmnFmn)2, ]aFmn]aFmn , etc. Similarly, the ge-neric matter fieldsf are gauge invariantly coupled toAm inLmatter(f,Am). Apart from these stipulations the action is gneric in the sense that all dimensionless couplings~apartfrom N! are of order unity; that is, all dimensionful quantitieare of the order of the cutoffL to the appropriate power.

Since in our effective action we have explicitly excludevarious terms which are consistent with the symmetriesthe theory, two questions immediately come to the fore: hmight an action of this form be generated from some undlying dynamics, and is the assumed structure of the acstable under radiative corrections? We now address thesepoints in turn.

A. Effective action from fermions

To show why an action of the form~2.1! is fairly natural,we show how it can be generated by integrating out solarge number of fermion species. This will be a generalition of the original mechanism proposed by Bjorken@1#,which considered four fermion interactions. So considerNspecies of Dirac fermionsc i . We imagine these fermionbeing coupled to gauge fields which acquire masses at sL. Integrating out the massive gauge bosons will yieldinfinite set of fermion interactions, and we will focus on thfollowing subset:

Lc5c̄ i~ i ]/ 2m!c i1N(n51

`

l2n

~ c̄ igmc i !

2n

N2n . ~2.2!

Here, summation over flavor indicesi and spacetime indicem is implied. We wrote the action to have a U(N) flavorsymmetry. The couplingsl2n are of order unity times theappropriate power ofL,

l2n;L426n. ~2.3!

Factors ofN have been inserted in order to give a well dfined largeN limit. In particular, the normalized bilinearOm5(1/N)c̄ ig

mc i then have correlators scaling asN0, andthe action written in terms ofO has an overall factor ofN.

We will employ the standard trick of rendering the actioquadratic in fermions by introducing an auxiliary fieldAm.We therefore consider

Lc,A5c̄ i~ i 2]/ 2m!c i2NV~AmAm!. ~2.4!

The potentialV is a power series inAmAm with coefficientschosen such that by solving the algebraic equations oftion for Am and substituting back in we recover Eq.~2.2!.The most familiar case corresponds to a pure four ferminteraction, with only l2 nonvanishing, in which caseV(AmAm)5AmAm/4l2 . The quantum version of this theor

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PHOTONS AND GRAVITONS AS GOLDSTONE BOSONS . . . PHYSICAL REVIEW D 66, 045015 ~2002!

is defined by a path integral~with a cutoff! over the fieldsc iandAm. The idea is to imagine first doing the path integoverc i to yield an effective action forAm. Sincec i is mini-mally coupled toAm, provided we choose a gauge invariacutoff the terms in the effective action generated in this wwill be gauge invariant. Furthermore, since there areN spe-cies of fermions the effective action will have an overfactor ofN. Therefore, the form of the effective action is thof the first set of terms in Eq.~2.1!.

Now suppose we had included the other fermion termsEq. ~2.2! that would certainly arise upon integrating out masive gauge fields. This will introduce other bilinears in ttheory, e.g.,c̄ ig

mnc i , c̄ igm]nc i , etc. The above procedur

should then be generalized by introducing a new auxilifield for each bilinear. Integrating out the fermions thyields an effective action for a set of interacting auxiliafields. The analysis rapidly becomes complicated; howeone expects on general grounds that the auxiliary fieldsacquire mass terms of order of the cutoff and so canneglected at lower energies. By contrast, in our scenariotain components ofAm will remain massless since they wicorrespond to Goldstone bosons.

To reiterate somewhat, in our approach where we consAm as the only auxiliary field it is consistent to omit termlike (c̄ ig

mc i)2c̄ j]/ c j which might seem to lead to non

gauge invariant terms likef (AmAm)FabFab in the effectiveaction forAm. If such fermionic terms are to be included onshould introduce a new scalar auxiliary field for the bilinec̄ i]/ c j , and we are not doing this for the reasons staabove.

We have thus demonstrated one possible way of geneing an effective action with the structure~2.1!, though thereare presumably other ways as well. For the most partconsider Eq.~2.1! in its own right, without reference to itsorigin.

B. Coupling to matter

By coupling a matter fieldf to the fermions via con-served currents,Jm(f)c̄ igmc i , we will generate the mattecouplings given in Eq.~2.1!. An easy way to accomplish thiis to modify Eq. ~2.2! by taking the mass for some of thfermions to be much less than the cutoffL. In this case wewould keep the light fermions in the low energy effectiaction rather than integrating them out. From Eq.~2.4! wesee that these fermions will be minimally coupled toAm.Whether we use this or some other mechanism to genethe matter couplings, it will be important that the mattaction is gauge invariant, at least up to the level of tderivative terms.

C. Stability under radiative corrections

Treating Eq.~2.1! as an effective field theory at scaleL,to extract out low energy physics we still need to integrout the fluctuations ofAm. Since the potential term violategauge invariance, once we start computing loop diagrampossible Lorentz invariant, but not necessarily gauge invant, terms will generically be generated. Some of these tewould lead, after spontaneous symmetry breaking, to la

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violations of Lorentz invariance at low energies and so neto be suppressed. The need to suppress such terms ismotivation for introducing the large numberN. Given theform ~2.1!, the loop expansion is an expansion in 1/N, so thedangerous terms will only arise at orderN0.

So at orderN0 we need to consider all possible Loreninvariant terms generated by computing loop diagramsAm. At energies low compared to the cutoff we can restrattention to terms with at most two derivatives. We furthassume symmetry under charge conjugationC, acting as signreversal onAm. This forbids single derivative terms. Termwith no derivatives just give a small correction to the potetial in Eq. ~2.1!, which we are taking to be arbitrary, so wneed consider only two derivative terms. Up to integratiby parts, there are seven independent terms:

~1! f 1~A2!]mAn]mAn,

~2! f 2~A2!]mAn]nAm,

~3! f 3~A2!AmAa]mAn]aAn,

~4! f 4~A2!AnAa]mAn]aAm,

~5! f 5~A2!AnAa]mAn]mAa ,

~6! f 6~A2!AmAnAa]m]nAa ,

~7! f 7~A2!AmAnAaAb]mAn]aAb . ~2.5!

HereA2[AmAm . As always, we assume that all dimensioful couplings in f i are of order unity times the appropriapower of L. As we will see, after spontaneous symmebreaking some of these terms will lead to low energy viotions of Lorentz invariance at order 1/N.

D. Spontaneous symmetry breaking

Our potential will generically have the form

V~A2!5L4(n51

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VnS A2

L2D n

, ~2.6!

with the coefficientsVn of order unity.Vn can be determinedin terms of thel2n appearing in Eq.~2.2!. We will assumethat the potential leads to spontaneous symmetry breaki

Âm&5cLnm , ~2.7!

where for definitenessnm is a spacelike unit vector andc isof order unity. This expectation value spontaneously breLorentz invariance at the cutoff scaleL. Nevertheless, wewill see that low energy physics is approximately Loreninvariant.

We now expand the action around the vacuum~2.7! bywriting

Am5cLnm1am . ~2.8!

To quadratic order inam the potential becomes

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V5V~2c2L2!11

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where

a51

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Shifting the vacuum has a trivial effect on theAm kineticterms since, being gauge invariant, these depend onlyderivatives ofAm. Hence in the kinetic terms we can jureplaceAm→am. We can similarly make this replacementthe matter Lagrangian after performing a compensagauge rotation of the matter fields:

Lmatter~f,Âm&1am!5Lmatter~f8,am!, ~2.11!

wheref8 is a gauge transformation off. We will henceforthdrop the prime onf. Therefore, the action takes the form

L5NH 21

4FmnFmn2

1

2a~n•a!2

1higher derivatives1O~a3!J1Lmatter~f,am!1O~N0!. ~2.12!

The (n•a)2 term plays the role of an axial gauge fixinterm. It gives a mass of orderL to a spacelike component oam . So to the above order our action takes the form ofaxial gauge fixed photon coupled gauge invariantly to matNeglecting the higher derivative terms, the photon propator is given by

2i

Np2 S hmn21

n•p~nmpn1nnpm!

2pmpn

~n•p!2 ~ap22n2! D . ~2.13!

As usual, only the first term contributes when the propagais sandwiched between conserved current, and the resuLorentz invariant. Corrections to this Lorentz invariant resare suppressed byp/L and/or 1/N, as will be discussed below.

E. Goldstone bosons

After spontaneous symmetry breaking we have ‘‘massparticles’’1 by virtue of Goldstone’s theorem. In particular,spacelike vector breaks the Lorentz group according to

SO~3,1!→SO~2,1!. ~2.14!

1We employ quotation marks since we can no longer use the sdard definition of particles as being irreducible representationthe ~now spontaneously broken! Poincare´ group. But it should beclear what we mean when we use the particle terminology.

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There are thus three broken generators corresponding torotations and a boost. The corresponding three Goldstbosons are the three components ofam orthogonal tonm. Wechoose a basis of two transverse components and a timcomponent:

transverse: em~1,2! , k•e~1,2!5n•e~1,2!50,

timelike: em~0!5km1~n•k!nm ,

obeys n•e~0!50. ~2.15!

At low energies only the Goldstone bosons are relevantleading order inN the gauge invariant form of the kinetiterms implies that only the transverse Goldstone bospropagate, giving us the conventional Lorentz invariant eltrodynamics. However, at orderN0 the timelike componentwill also propagate, and this will lead to interesting effect

We now concentrate on the low energy physics ofGoldstone bosons coupled to matter. The Goldstone boscan be thought of as coordinates on the coset SO~3, 1!/SO~2,1!. Since the Goldstone bosons label flat directions ofpotential, the cubic and higher order terms from the expsion of the potential all involve the massive componen•a, and so are irrelevant at low energies. Now considerorderN0 terms~2.5!. Note that after spontaneous symmetbreaking the terms~4!–~7! expanded to quadratic order ifluctuations will all involve at least one factor ofn•a. There-fore, only terms~1!–~3! are relevant for low energy physicsFurthermore, one linear combination of~1! and~2! is propor-tional to (Fmn)2 and so just provides a small correction to torder N value of this term. Hence we can omit one linecombination, say~2!, and focus only on terms~1! and~3!. Itis also convenient at this point to rescaleam→am/AN to putthe gauge kinetic term in standard form. Therefore to orN0, and discarding terms suppressed at low energies byp/L,the effective action is

L521

4FmnFmn2

1

2a~n•a!21

1

2

c1

N]man]man

11

2

c2

Nnanb]aam]bam1Lmatter~f,am /AN!.

~2.16!

Here we defined the order unity numerical coefficientsc1,2 as

f 1~Â&2!51

2c1 , c2L2f 3~Â&2!5

1

2c2 . ~2.17!

We also kept the term (n•a)2 as a convenient way of implementing the axial gauge condition.

Equation~2.16! clearly shows the need for a 1/N suppres-sion of the third and fourth terms in order to have appromate Lorentz invariance at low energies. It is easiest to copute the propagator from the first two terms and to thinkthe third and fourth terms as interactions. Then it is clear twhen we compute interaction between conserved matter

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rents we will get the standard QED results at leading ordwith non-Lorentz invariant corrections occurring at ord1/N.

F. Low energy spectrum

We now determine the dispersion relations for the thGoldstone bosons by solving the linearized equations oftion. The latter are

S 11c1

N D ]m]man2]n]mam1c2

Nnanb]a]ban

21

anmamnn50. ~2.18!

First consider the transverse modes. Using the ansatz@seeEq. ~2.15!#

an5en~1,2!e2 ik•x ~2.19!

we find the dispersion relation

transverse: S 11c1

N D k21c2

N~n•k!250. ~2.20!

This corresponds to an anisotropic speed of light. The spof light parallel tonm differs from that orthogonal tonm byan amount of order 1/AN.

Now consider the timelike mode. We use the ansatz

an5~kn1gnn!e2 ik•x. ~2.21!

We find

g5n•k1O~ak2!, ~2.22!

and the dispersion relation

timelike: k22N

c1~n•k!250. ~2.23!

In the dispersion relation we have dropped terms down1/N or ak2. The dispersion relation~2.23! is non-Lorentzinvariant at leading order. The timelike modes propagatethe ordinary speed of light in directions orthogonal tonm, butat a speed of orderAN in the direction parallel tonm. We areassuming thatc1.0.

The physics of the transverse modes is thus standard usmall corrections, while that of the timelike mode is quexotic. The reason for this is that the timelike mode doespropagate with respect to the leadingN gauge invariant ki-netic terms of Eq.~2.1!. It only acquires a kinetic term aorder N0, and these terms are non-Lorentz invariant afspontaneous symmetry breaking.

While exotic, the timelike mode leads to acceptably smeffects for sufficiently largeN. Its contribution to the inter-action between conserved currents is suppressed byN,since as we have discussed we can use the standard

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gauge propagator at leading order and regard correctioncoming from interaction vertices with coefficients of ord1/N.

The timelike mode is also suppressed by phase spacesiderations. For fixed available energyk0 the dispersion re-lation forces

uk•nu,Ac1 /Nk0. ~2.24!

Consider putting the system in a box of sizeL. Then forN.(k0L)2 only the zero momentum mode parallel tonm sur-vives, yielding a phase space suppression proportiona1/L.

While the timelike mode gives small corrections to tinteraction between conserved currents it could have mdramatic consequences given its unusual dispersion relaWe should emphasize that the result~2.23! does not necessarily imply faster than light2 signal propagation, since Eq~2.23! is only valid for long wavelengths. Also, even if Eq~2.23! could be extrapolated to short wavelengths so tsignals could propagate faster than light, there would beconflict with causality since Lorentz invariance has bespontaneously broken by a preferred frame. It would beteresting to study the physics of the timelike mode in modetail.

G. Summary and relation to previous work

Let us summarize what has been accomplished. We hshown that spontaneous breaking of Lorentz symmetrylead to an approximately Lorentz invariant low energy theoof massless photons coupled to matter. This is possible incontext of a theory in which gauge invariance is violatedleading order only by a potential term. Since the Goldstoboson photons do not appear in the potential, the Loreviolating condensate leads to only small corrections tolow energy physics of the photons. On the other hand,existence of three broken Lorentz generators implies theistence of a third Goldstone boson whose physics is not eapproximately Lorentz invariant. However, its effects asuppressed by 1/N and phase space considerations. Algether, Lorentz invariance appears as an approximate smetry of the low energy world.

This is a good place to compare and contrast with preous work on this subject, in particular the original workBjorken @1#. The main difference is that we have takenmodern effective field theory point of view, emphasized ththe violation of Lorentz invariance is real, and pointed othe existence of an extra Goldstone boson. Earlier wstarted from a four Fermi interaction, i.e., just keeping tterml2 in Eq. ~2.2!. The trouble with this is that it is incompatible with spontaneous symmetry breaking, since it cosponds to a potentialV;AmAm with no higher order terms. Areflection of this is that the condensate was never actucomputed in earlier work, but was eitherassumedto arisesomehow, or emerged after formal manipulations with div

2Here we refer to the speed of light as the speed of the transvmodes after spontaneous symmetry breaking.

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gent integrals. The claim was then made that the phyafter spontaneous symmetry breaking was the usualexactlyLorentz invariant quantum electrodynamics. This concluswas again dependent on manipulating divergent quantiAs far as we can tell, the origin of this claim is that if ontakes the pure four Fermi interaction and assumes~wrongly!that this leads to spontaneous symmetry breaking, thenpandingV;AmAm around the new vacuum would yield aaxial gauge fixing term and nothing more. This of courgives the usual QED in the axial gauge. But from our poof view it is clear what would actually happen in this theoThe four Fermi theory leads either to an instability or tostable vacuum with a massive vector fieldAm . In neithercase does one find QED. On the other hand, the probdisappears once one includes the higher order fermion teas we have done; this is also the natural starting point frthe view of effective field theory.

Since earlier work took the point of view that the sponneous breaking of Lorentz invariance was somehow fitious, the existence of the extra Goldstone boson wasnoted.

Before turning to gravity we should also note that somethe above criticisms were commented on recently by Bjor@8#. In particular it was noted that the four Fermi theoryitself is inadequate, and that some real violations of Loreinvariance should be expected once quantum effectstaken into account. These points were also examinedBanks and Zaks in@9# in the context of non-Abelian gaugfields; they also concluded that there is a real violationLorentz invariance. We hope to have resolved these isshere.

III. GRAVITON AS GOLDSTONE BOSON

We now turn to our main interest: producing a gravitona Goldstone boson. Fortunately, the analysis closely parathe photon case, and so we can draw on our experiencethat example to navigate in the more complicated gravtional setting. The main difference is in the different patteof spontaneous Lorentz breaking as well as in the connecto the cosmological constant problem. The cosmologconstant provides a motivation for modifying the low enereffective theory of general relativity, and indeed we will sthat the problem is avoided in the sense that the Goldsboson graviton remains massless even in the presencvacuum energy.

To adapt the previous approach to the case of gravityconsider an effective action in direct analogy with Eq.~2.1!,

L5N$L2A2gR~g!2L4V~h!

1higher derivatives%1Lmatter~f,g!

1O~N0!. ~3.1!

Here h is defined via expansion of the metric around flspacetime:

gmn5hmn1hmn . ~3.2!

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Flat spacetime thus plays a preferred role in this conteindeed we think of the underlying dynamics as that ofnongravitational theory in Minkowski spacetime. The Apendix discusses one possibility for such an underlytheory. The Einstein-Hilbert term in Eq.~3.1! is standardwhile the potential is some generic Lorentz invariant funtion of hmn with indices contracted withhmn . Odd powers ofh are allowed; for instance, the termhm

m can appear in theexpansion ofV(h). As in the photon case the higher derivtive terms are generally covariant, as is the matter actGeneral covariance is violated only by the potential.

Note that the observed Newton’s constant will be

GN;1

NL2 . ~3.3!

Therefore, the cutoffL is smaller by a factor of 1/AN com-pared to the usual Planck scale. Indeed, this lowered valuthe cutoff is responsible for suppressing loop corrections.Nis a large number but presumably need not be more than4

or so. Thus the cutoff can still be well above observaenergy scales.

We consider potentials leading to a vacuum expectavalue for hmn . By performing a Lorentz transformation wcan bring the expectation value to the form

^hmn&5S h̄00

h̄11

h̄22

h̄33

D . ~3.4!

For h̄mm all nonvanishing and distinct, the Lorentz group wbe completely broken:

SO~3,1!→nothing. ~3.5!

Being dimensionless, we expecth̄mm of order unity. There-fore there will be six Goldstone bosons corresponding tosix broken Lorentz generators. The Lorentz generatorJmn

acts on ^hmn& by exciting themÞn components. So theGoldstone bosons are the six off-diagonal components ofsymmetric matrixhmn . Fluctuations of the diagonal components will generically correspond to massive fields. We wultimately associate two of the Goldstone bosons withtwo physical polarizations of the graviton, while the remaing four will appear in analogy with the timelike Goldstonboson in the photon case.

We now consider fluctuations around the vacuum by wing

hmn5^hmn&1h̃mn . ~3.6!

The expansion of the potential will correspond to mass tefor the diagonal components ofh̃mn . The precise form of thismass matrix is not important, so we will write

5-6

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nal/e-up-x-of.

itt of

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naltan-ex-oxi-le


V~h!5const1L4 (a50

3

~ f an~a!m n~a!

n h̃mn!2

1O~ h̃3!. ~3.7!

Here

n~a!m 5da

m , ~3.8!

and f a are numbers of order unity.We can simplify the action by performing a general co

dinate transformation to put the background metric backstandard form. In particular, introduce new coordinatesx8m

such that

]x8m

]xa

]x8n

]xb hmn5hab1^hab&. ~3.9!

The metric appearing in the action will then behmn1h̃mn8where

]x8m

]xa

]x8n

]xb h̃mn8 ~x8!5h̃ab~x!. ~3.10!

We similarly act with a coordinate transformation on tmatter fields; e.g., for a scalar

f8~x8!5f~x!. ~3.11!

After changing the integration variable tox8 the potentialterm is modified while the general covariant terms arecourse invariant. Given the form~3.10!, the modification ofthe potential can be absorbed in a redefinition of the cstantsf a . So after the coordinate transformation our actitakes the form

L5N$L2A2gR~g!2L4V~ h̃8!

1higher derivatives%1Lmatter~f8,g!

1O~N0!, ~3.12!

with

gmn5hmn1h̃mn8 ~3.13!

and

V~ h̃8!5const1L4 (a50

3

~ f an~a!m n~a!

n h̃mn8 !2

1O~ h̃83!. ~3.14!

We henceforth relabel fields:h̃8→h, f8→f.We think of the quadratic terms in the potential as gau

fixing terms, corresponding to the gauge

gmm5hmm , no sum. ~3.15!

This defines an acceptable noncovariant gauge. The gravpropagator in this gauge is extremely complicated and

04501

-n

f

-

e

on-

wieldy, and so we will not display it here. Fortunately, foleading order calculations we only need to know that it hthe structure of the standard covariant propagator

2 i

p2 ~hmahnb1hmbhna2hmnhab! ~3.16!

plus terms with at least one factor ofp ~vector! in the nu-merator. Sandwiched between conserved energy-momentensors, the latterp terms vanish, and so we recover thstandard Lorentz invariant result. Of course this is noprise, since Eq.~3.15! represents a valid gauge choice.

The low energy physics is therefore quite similar to whwe found in the photon example. We will have two gravitostates which propagate at the speed of light, up to a smanisotropic correction. Further, there are four additioGoldstone bosons that acquire kinetic terms at order 1N.These will have highly non-Lorentz invariant dispersion rlations, but their couplings to conserved currents are spressed by 1/N. Working out these dispersion relations eplicitly would be quite involved given the large numberterms in the action at order 1/N and the proliferating indicesHowever from our discussion of the photon exampleshould be clear that the essential physics is independenthese details.

The cosmological constant

Notice that we have obtained an approximately Loreinvariant theory of gravity without making any specific asumptions about the form of the potentialV(h). Therefore,we see that if the potential is suddenly modified, say bmatter phase transition, then the vacuum expectation valuh can simply shift to the new minimum, leaving us agawith an approximately Lorentz invariant theory. In particulathe termA2gVmatter(f0) can be added to our previous potential and the analysis proceeds as before. We have thfore evaded the usual cosmological constant problem.usual problem arises because of general covariance: onsingle potential term is allowed,L4A2g, and a nonzerovalue of this term is incompatible with a Lorentz invariasolution. If one is willing to violate general covariance bwriting a more general potential then this conclusion nenot follow. Indeed, if the graviton is a Goldstone boson ois guaranteed to find a solution with constant fields anmassless graviton. What is perhaps surprising is thatphysics around such a non-Lorentz invariant solution isproximately Lorentz invariant, as we have seen.

On the other hand, the above discussion does not imdiately imply that the approximately Lorentz invariant soltions are theonly solutions. Indeed, at least in the weak fieapproximation we will find additional approximately de Ster and anti–de Sitter solutions. This follows from the fathat at low energies and for weak fields our theory is thatstandard gravity in a noncovariant gauge, plus additioweakly coupled Goldstone bosons. If one now takes a sdard solution with a given energy-momentum tensor andpresses it in our gauge, this must continue to be an apprmate solution of our theory. The existence of multip

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solutions is hardly surprising given that we are solving~atleading order! second order differential equations.3 Note thatthe approximately Lorentz invariant solution with constafields is guaranteed to be an exact solution as it just cosponds to extremizing the potential, while the other solutiowith spacetime varying fields need not be exact. In a mconventional field theory context one would expect a sotion with time dependent fields to eventually settle down tstatic solution by radiating energy. One might expectsame here, with the time dependent fields of the de Stype solutions radiating away, leading to the solution wconstant fields. A realistic proposal in this framework muinvolve showing how to make the transition from an expaning radiation or matter dominated universe to the appromately Lorentz invariant solution discussed above. We hto return to this in future work.

IV. CONCLUDING REMARKS

Building on the work of Bjorken, we have shown thmassless photons and gravitons can be produced as Gstone bosons associated with the spontaneous breakinLorentz invariance, and with low energy physics appearLorentz invariant to high accuracy. The most dramatic effof the Lorentz breaking is the existence of additional weacoupled Goldstone bosons obeying highly non-Lorentzvariant dispersion relations. These fields would be difficuldetect as they couply weakly to conserved currents. A rageneral framework for studying Lorentz violating extensioof the standard model has been developed~see, e.g.,@12#!,and it might be useful to study some of our results in tlanguage.

We find it interesting that the observed low energy phics of gravity can be produced in the context of an effectfield theory that differs markedly from general relativity, anwhich does not suffer from the usual cosmological constproblem. While it remains to be seen whether a theory oftype could be incorporated into a truly fundamental framwork or be developed into a realistic cosmology, it seemsbe an idea worth pursuing.

ACKNOWLEDGMENTS

This work was supported by NSF grants PHY-00995and PHY-9819686.

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APPENDIX

In this appendix we indicate one way the gravitationeffective action~3.1! may emerge from some underlyinconventional dynamics. There may be others. Here weploy the same mechanism as in the photon case, Eq.~2.1!.Thus we considerN fermions coupled to gauge fields thacquire masses. We then imagine integrating out the fiefrom some initialL0 down to a scaleL obtaining the effec-tive action

c̄ i~ i ]”2M !c i14p2N(k

Ck~L0 ,L!Ok~L! ~A1!

involving an infinite set of fermion interactions. Restrictinconsideration first to the subset consisting only of powers

Omn51

Nc̄ i

i

2~gm]W n2gm]Q n!c i , ~A2!

we introduce the symmetric auxiliary fieldhmn to render theeffective action quadratic in the fermions, so that we mwrite it in the form @cf. Eq. ~2.4!#

Lc,h5~hmn1hmn!c̄ i

i

2~gm]W n2gm]Q n!c i

2M c̄ ic i2NL4

4p2 V~h!. ~A3!

All indices are raised and lowered by the flat metrichmn .Integrating out the fermions, the effective action from t

resulting determinant can be expressed, as usual, as theover all fermion one loop diagrams with externalh legs.Note, in particular, that the diagram with one externalh leg isin general nonvanishing. This reflects the fact that Eq.~A2!has a nonvanishing expectation~proportional tohmn! even inordinary perturbation theory on a Lorentz invariant vacuui.e., interactions built from Eq.~A2! shift the classical back-ground. Correspondingly, all terms, including a linear terare included in the general potentialV(h) in Eq. ~3.1! ~cf.discussion in the text!.

Explicit evaluation of the fermion-loop graphs with onand two externalh legs gives, after a lengthy computatiothe contribution to the effective action toO(h2):

t

L~2!5NI4F2hmm1

1

2hmnhmn1

1

2~hm

m!2G1N

6I 2@~]lhmn!22~]nhm

m!212]mhmn]nhll22~]mhmn!2#

2N

20I 0F ~!hmn!22

1

3~!hm

m!222~]m]lhln!212

3]m]nhmn!hl

l12

3~]m]nhmn!2G1¯ . ~A4!

3It is also reminiscent of brane world scenarios for addressing the cosmological constant problem@10,11#. But there the Lorentz invariansolution on the brane is tied up with a naked singularity away from the brane, and so need not exist.

5-8

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Equation~A4! displays explicitly the leading terms, i.e., thlocal, cutoff dependent~‘‘divergent’’ ! part of the result of theloop integrations. The ellipses denote the subleading piefrom the finite, nonlocal parts, which can be expandedpowers of!/M2, and contribute to higher derivative inteactions relevant only for short distance behavior nearcutoff. Dimensional regularization, under the usual corspondence lnL↔(1/e1const), gives

I n51

~4p!2 Mn lnS L2

M2D , n50,2,4. ~A5!

In Pauli-Villars regularization, which appears more physiin the present context, one has

I n51

~4p!2 (k51

3

ckMkn lnS Mk

2

M2D , n50,2,4. ~A6!

Three regulator massesMk , of order of the cutoffL, arerequired here with coefficientsck satisfying (k51

3 ckMkn

1Mn50, for n50, 2, 4.Equation~A4! is now seen to be the flat space expans

to second order of the gravitational action

L5A2gNF I 411

6I 2R

21

20I 0S RmnRmn2

1

3R2D G ~A7!

with the metric expressed in terms of the vierbein4

gmn5eamean, ea

m5dam1ha

m , ~A8!

and

hmn5damhan. ~A9!

Taking hmn to be symmetric, as done in the above calcution, amounts to the~standard! local Lorentz gauge fixing toa symmetric vierbein.@It is known that the antisymmetricpart in fact decouples in Eq.~A7!.# Thus our effective gravi-tational action~3.1! is reproduced to this order.

To see how this comes about, note that the result~A4! isprecisely what one obtains after integrating out the fermfields in the Lagrangian forN fermions in curved space:

L5e~122w!Feamc̄ i

i

2~ga¹Wm2¹Qmga!c i

2M c̄ ic i G , ~A10!

to second order in the expansion about flat space~A8!. In Eq.~A10!, e5deteam , and

4Curved and flat indices are denoted by Greek and Latin lettrespectively.

04501

esn

e-

l

n

-

n

¹�

m[ ]�

m7i

2vmabS

ab ~A11!

with Lorentz generatorsSab5( i /4)@ga,gb#. The spinorcmay, in general, be taken to transform under general coonate transformations as a density of weightw. The result ofthe explicit computation is in fact found to be independentw ~see, for example,@13#!. Indeed, note that, in termof the functional integration, the factore(122w) can beabsorbed by the change of variables:c→e(122w)/2c8,c̄→e(122w)/2c̄8. The spin connectionvmab is given in termsof the vierbein by

vmab51

2em

a ~Tamn2Tmna2Tnam!,

Tmna 5~em

r ens2en

rems !]ser

a . ~A12!

Equation ~A12! implies that the connection terms in Eq~A10!, in the expansion~A8!, do not give ahc̄c vertex, butonly ‘‘seagull’’ h2c̄c and higherh powers vertices. This ismost easily seen by rewriting the connection terms in E~A10!, after a little rearrangement, in the form

emmc̄

1

4vmabe

mabcgcg5c.

Furthermore, it follows from this form that all fermion onloop diagrams with two externalh legs do not receive anycontribution from the spin connection interaction. The resto O(h2) thus agrees with that obtained from Eq.~A3!.

The vertices from the spin connection terms will, however, contribute to the diagrams with three or more exterlegs, reproducing Eq.~A7! to all orders, as dictated by thgeneral coordinate invariance of Eq.~A10!. It may thereforeappear that, in addition to Eq.~A2!, one would need an in-finite set of different operators5 from Eq.~A1!, with preciselyspecified coefficients, to be included in Eq.~A3! in order togenerate Eq.~A7!. This, however, is not the case. In thcontext of Eq.~A1!, the connection arises naturally when E~A3! is extended to include the set of powers of the opera

Omkl5

i

4Nc̄ i$gm ,@gk,gl#%c i , ~A13!

in addition to those of Eq.~A2!. Introducing the correspond

s,5Note that Eq.~A12! contains botheam and its inverse.

5-9

ngg

ft

-

atice-

s.ons

ust

der

f

n-

ns.d toe


ing independent auxiliary fieldvklm , Eq. ~A3! is extended to

Lc,h5~hmn1hmn!cW i

i

2

3S gm]W n2gm]Q n11

8vnkl$gm ,@gk,gl#% Dc i

2M c̄ ic i2NL4

4p2 V~h,v!. ~A14!

The fermionic part of Eq.~A14! is equivalent to Eq.~A10! inthe first order~Palatini! formulation.

Now, in the first order formalism, the nonpropagaticonnection in Eq.~A10! serves as a constraint field enforcinvanishing of torsionem

mOmab generated by the fermions.Equation~A14! differs from Eq.~A10! by the presence o

a potential inv. Hence, variation of the connection will noimply vanishing torsion, andv will not be expressible en

04501

tirely in terms of the vierbein as in Eq.~A12!.6 For illustra-tion purposes, we may adopt a model where the quadrterms in the potential forv are suppressed. Then, upon intgrating out the fermions, Eq.~A7! and the effective gravita-tional action~3.1! are reproduced to within small deviationThis is easily seen to be stable under radiative correctifrom graviton loops.

It is perhaps worth pointing out again that, as we saw, jthe fermionic self-interactions of products of Eq.~A2! in flatspacetime already suffice to fully reproduce to second orthe Einstein-Hilbert~plus R2 terms! parts in our effectivegravitational action~3.1!, i.e., reproduce the full content olinearized general relativity.

6This is analogous to considering Eq.~A10! in the first orderformalism not just by itself, but with the addition of the EinsteiHilbert action. The latter provides a potential for the connectionv,thus leading to the usual result of torsion generated by fermioThis is, of course, the usual situation. In the above we were leconsider Eq.~A10! by itself in the context of generating thEinstein-Hilbert action from Eq.~A1!.

n-

@1# J. D. Bjorken, Ann. Phys.~N.Y.! 24, 174 ~1963!.@2# G. S. Guralnik, Phys. Rev.136, B1404~1964!.@3# P. R. Phillips, Phys. Rev.146, 966 ~1966!.@4# H. C. Ohanian, Phys. Rev.184, 1305~1969!.@5# D. Atkatz, Phys. Rev. D17, 1972~1978!.@6# S. Weinberg, Rev. Mod. Phys.61, 1 ~1989!.@7# S. Weinberg and E. Witten, Phys. Lett.96B, 59 ~1980!.@8# J. Bjorken, hep-th/0111196.

@9# T. Banks and A. Zaks, Nucl. Phys.B184, 303 ~1981!.@10# S. Kachru, M. Schulz, and E. Silverstein, Phys. Rev. D62,

045021~2000!.@11# N. Arkani-Hamed, S. Dimopoulos, N. Kaloper, and R. Su

drum, Phys. Lett. B480, 193 ~2000!.@12# D. Colladay and V. A. Kostelecky, Phys. Rev. D58, 116002

~1998!.@13# D. M. Capper and M. J. Duff, Nucl. Phys.B82, 147 ~1974!.

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Documents

Photons and gravitons as Goldstone bosons and the cosmological constant