5
6 September 2001 Physics Letters B 516 (2001) 156–160 www.elsevier.com/locate/npe Quantum M 2 2Λ/3 discontinuity for massive gravity with a Λ term M.J. Duff, James T. Liu, H. Sati Michigan Center for Theoretical Physics, Randall Laboratory, Department of Physics, University of Michigan, Ann Arbor, MI 48109-1120, USA Received 1 July 2001; accepted 16 July 2001 Editor: M. Cvetiˇ c Abstract In a previous paper we showed that the absence of the van Dam–Veltman–Zakharov discontinuity as M 2 0 for massive spin 2 with a Λ term is an artifact of the tree approximation, and that the discontinuity reappears at one loop, as a result of going from five degrees of freedom to two. In this Letter we show that a similar classical continuity but quantum discontinuity arises in the “partially massless” limit M 2 2Λ/3, as a result of going from five degrees of freedom to four. 2001 Published by Elsevier Science B.V. In a previous paper [1], we showed that the absence [2–4] of the van Dam–Veltman–Zakharov discontinu- ity [5,6] for massive spin 2 with a Λ term is an ar- tifact of the tree approximation, and that the discon- tinuity reappears at one loop. This result may be un- derstood as follows. While a generic massive gravi- ton propagates five degrees of freedom, gauge invari- ance ensures only propagation of the familiar two de- grees of freedom of a massless graviton. Although the introduction of Λ = 0 allows for a smooth classical M 2 0 limit, the mismatch between two and five de- grees of freedom cannot be eliminated altogether, and the discontinuity shows up at the quantum level. Curiously, the presence of a cosmological con- stant allows for new gauge invariances of massive higher spin theories, yielding a rich structure of “par- tially massless” theories with reduced degrees of free- dom [7]. In particular, for spin 2 a single gauge in- E-mail addresses: [email protected] (M.J. Duff), [email protected] (J.T. Liu), [email protected] (H. Sati). variance shows up at the value M 2 = 2Λ/3, yielding a partially massless theory with four degrees of freedom [8–10]. In this Letter we extend the result of [1] to the partially massless theory and show that a discontinuity first arises at the quantum level as M 2 2Λ/3. We work in four dimensions with Euclidean signa- ture (++++). As in Refs. [1,4], we take the massive spin 2 theory to be given by linearized gravity with the addition of a Pauli–Fierz mass term. Thus our starting point is the action (1) S [h µν ,T µν ]= S L [h µν ]+ S M [h µν ]+ S T [h · T ], where S L is the Einstein–Hilbert action with cosmo- logical constant S E =− 1 16πG d 4 x ˆ g ( R 2Λ ) , linearized about a background metric g µν satisfying the Einstein condition, R µν = Λg µν . Taking ˆ g µν = g µν + κh µν where κ 2 = 32πG, this linearized action 0370-2693/01/$ – see front matter 2001 Published by Elsevier Science B.V. PII:S0370-2693(01)00909-1

Quantum M2→2Λ/3 discontinuity for massive gravity with a Λ term

  • Upload
    mj-duff

  • View
    214

  • Download
    0

Embed Size (px)

Citation preview

Page 1: Quantum M2→2Λ/3 discontinuity for massive gravity with a Λ term

6 September 2001

Physics Letters B 516 (2001) 156–160www.elsevier.com/locate/npe

QuantumM2 → 2Λ/3 discontinuity for massive gravitywith aΛ term

M.J. Duff, James T. Liu, H. SatiMichigan Center for Theoretical Physics, Randall Laboratory, Department of Physics, University of Michigan,

Ann Arbor, MI 48109-1120, USA

Received 1 July 2001; accepted 16 July 2001Editor: M. Cvetic

Abstract

In a previous paper we showed that the absence of the van Dam–Veltman–Zakharov discontinuity asM2 → 0 for massivespin 2 with aΛ term is an artifact of the tree approximation, and that the discontinuity reappears at one loop, as a result of goingfrom five degrees of freedom to two. In this Letter we show that a similar classical continuity but quantum discontinuity arisesin the “partially massless” limitM2 → 2Λ/3, as a result of going from five degrees of freedom to four. 2001 Published byElsevier Science B.V.

In a previous paper [1], we showed that the absence[2–4] of the van Dam–Veltman–Zakharov discontinu-ity [5,6] for massive spin 2 with aΛ term is an ar-tifact of the tree approximation, and that the discon-tinuity reappears at one loop. This result may be un-derstood as follows. While a generic massive gravi-ton propagates five degrees of freedom, gauge invari-ance ensures only propagation of the familiar two de-grees of freedom of a massless graviton. Although theintroduction ofΛ �= 0 allows for a smooth classicalM2 → 0 limit, the mismatch between two and five de-grees of freedom cannot be eliminated altogether, andthe discontinuity shows up at the quantum level.

Curiously, the presence of a cosmological con-stant allows for new gauge invariances of massivehigher spin theories, yielding a rich structure of “par-tially massless” theories with reduced degrees of free-dom [7]. In particular, for spin 2 a single gauge in-

E-mail addresses: [email protected] (M.J. Duff),[email protected] (J.T. Liu), [email protected] (H. Sati).

variance shows up at the valueM2 = 2Λ/3, yielding apartially massless theory with four degrees of freedom[8–10]. In this Letter we extend the result of [1] to thepartially massless theory and show that a discontinuityfirst arises at the quantum level asM2 → 2Λ/3.

We work in four dimensions with Euclidean signa-ture(++++). As in Refs. [1,4], we take the massivespin 2 theory to be given by linearized gravity with theaddition of a Pauli–Fierz mass term. Thus our startingpoint is the action

(1)S[hµν,Tµν] = SL[hµν] + SM [hµν] + ST [h · T ],whereSL is the Einstein–Hilbert action with cosmo-logical constant

SE = − 1

16πG

∫d4x

√g

(R − 2Λ

),

linearized about a background metricgµν satisfyingthe Einstein condition,Rµν = Λgµν . Taking gµν =gµν + κhµν whereκ2 = 32πG, this linearized action

0370-2693/01/$ – see front matter 2001 Published by Elsevier Science B.V.PII: S0370-2693(01)00909-1

Page 2: Quantum M2→2Λ/3 discontinuity for massive gravity with a Λ term

M.J. Duff et al. / Physics Letters B 516 (2001) 156–160 157

for hµν is

SL =∫

d4x√g

[1

2hµν(−gµρgνσ✷ − 2Rµρνσ )h

ρσ

(2)− ∇ρhρµ∇σ hσµ

],

wherehµν = hµν − 12gµνh

σσ . All indices are raised

and lowered with respect to the metricgµν , and∇µ istaken to be covariant with∇µgλσ = 0. Furthermore,the source term is given by

(3)ST =∫

d4x√g hµνT

µν.

As in Ref. [4], we apply the simplifying assumptionthatTµν is conserved with respect to the backgroundmetric,∇µT

µν = 0.SL and ST together correspond to the linearized

massless theory coupled to a conserved source. Eachterm independently has a gauge symmetry describedby a vectorξµ(x):

(4)hµν → hµν + 2∇(µξν),

corresponding to diffeomorphism invariance of theEinstein theory. Introduction of the Pauli–Fierz spin-2mass term,

(5)SM = M2

2

∫d4x

√g

[hµνhµν − (

hµµ

)2],

breaks the symmetry (4). However, at the critical valueM2 = 2Λ/3, there remains a residual symmetry

(6)hµν → hµν + 2∇(µ∇ν)α + 2

3Λgµνα

parameterized byα(x). This gauge invariance was firstnoted in [8], and results in a partially massless de Sittertheory with four degrees of freedom and propagationalong the light cone. It also requires that the couplingto matter be via a traceless energy–momentum tensor.

We wish to consider the generating functional

(7)Z[g,T ] =∫

Dhe−(SL[h]+SM [h]+ST [h·T ]).

Since the generic theory with mass term has brokengauge invariance and a quadratic action, it may bequantized in a straightforward manner. On the otherhand, for the caseM2 = 2Λ/3, one would first gaugefix the symmetry (6) before proceeding. However, to

make contact with previous results for the pure mass-less case, we find it useful to reintroduce the gaugesymmetry (4) using a Stückelberg [2,11] formula-tion. This allows a uniform approach to quantizationthroughout the(Λ,M2) plane, and provides connec-tion to the operators appearing in Ref. [12] for themassless case, as well as the ones in Ref. [1] for themassive case.

For any value ofM2 > 0, we introduce an auxiliaryvector fieldVµ to restore the gauge symmetry (4). Wefirst multiply Z[g,T ] by an integration

∫DV over

all configurations of this decoupled field, and thenperform the shifthµν → hµν − 2M−1∇(µVν). SinceSL andST are gauge invariant in themselves, the onlyeffect of this shift is to make the replacement

(8)SM [hµν] → SM

[hµν − 2M−1∇(µVν)

]in (7). ThusSM becomes a “Stückelberg mass”, andgauge invariance is restored, yielding the simultaneousshift symmetry

hµν → hµν + 2∇(µξν),

(9)Vµ → Vµ + Mξµ.

For genericM2, this is the only symmetry of theory.However, forM2 = 2Λ/3, the additional symmetry(6) remains even after the Stückelberg shift. Note thatthis symmetry is a combination of a Weyl scaling anddiffeomorphism [with parameterξµ(x) = ∇µα(x)].Since the latter has been restored by the addition ofVµ, we are now able to disentangle the two. Theresulting gauge symmetry for the partially masslesstheory may be written as

hµν → hµν + 2∇(µξν)(x) + 2

3Λgµνα(x),

(10)Vµ → Vµ + M[ξµ(x) − ∇µα(x)

],

with parametersξµ(x) for diffeomorphisms andα(x)for Weyl rescalings.

For the partially massless theory, there are five de-grees of freedom to gauge fix. As in [1], we makeuse of diffeomorphisms to identifyV with the longi-tudinal part ofh, i.e., MVµ = ∇ρhρµ. Additionally,the conformal rescaling may be used to makehµν

traceless. This choice is made in order to simplify therelevant operators appearing in the action, and is ac-complished by adding to the action the gauge-fixing

Page 3: Quantum M2→2Λ/3 discontinuity for massive gravity with a Λ term

158 M.J. Duff et al. / Physics Letters B 516 (2001) 156–160

terms

Sgf =∫

d4x√g

(∇ρhρµ − MVµ

)(∇σ hµσ − MVµ

)(11)+ 2

∫d4x

√g h2.

In conjunction with this gauge fixing, it is necessaryto include a Faddeev–Popov determinant connectedwith the variation of the gauge condition under (10).It is straightforward to show that the appropriatedeterminant is

(12)Det

([�(1

2,12) − 4Λ/3

]0

2∇µ 8Λ/3

)corresponding to the set of gauge parameters(ξµ,α).So up to an overall (infinite) constant piece, Det[8Λ/3],the relevant Faddeev–Popov term is Det[�(1

2,12) −

4Λ/3], where the second-order vector spin operatoris defined by�

(12,

12

)ξµ ≡ −✷ξµ + Rµνξ

ν [12], andwe have exploited the Einstein condition for the back-ground metric. Note that, after gauge fixing, there re-mains a coupling proportional tohσ

σ∇ · V whichcan be eliminated by making the change of variablesVµ → Vµ + (

M

−4Λ+2M2

)∇µhσσ .

To highlight the tensor structure of the gauge-fixedaction, we decompose the metric fluctuationhµν intoits traceless and scalar parts:φµν ≡ hµν − 1

4gµνhσσ ,

andφ ≡ hσσ . The source may similarly be split into its

irreducible componentsjµν andj , so thatTµν = jµν +14gµνj . The gauge-fixed partially massless action thenbecomes

S =∫

d4x√g

[1

2φµν

(�(1,1)− 4

)φµν

+ V µ

(�

(12,

12

) − 4

)Vµ

(13)− (∇ · V )2 + φµνjµν + 1

4φj

].

The second-order spin operators are the scalar Lapla-cian �(0,0) ≡ −✷ and the Lichnerowicz operatorfor symmetric rank-2 tensors�(1,1)φµν = −✷φµν +Rµτφ

τν + Rντφ

τµ − 2Rµρντφ

ρτ [12].The Stückelberg field,Vµ, in (13) appears as a

massive spin-1 field in the Einstein background withan effective massm2 = −4Λ/3. We now restorevector gauge invariance by repeating the Stückelbergformalism. Thus we introduce a scalar fieldχ and

make the change of variablesVµ → Vµ − M−1∇µχ .By construction, the resulting action is now invariantunder the gauge transformation

(14)Vµ → Vµ + ∇µζ, χ → χ +Mζ.

One can then choose a gauge-condition to simplify theshifted action. It is useful to associate the longitudinalcomponent ofV with χ according toM∇ · V =(−2Λ+M2)χ . This is done by adding a gauge-fixingterm

(15)S′gf =

∫d4x

√g

(∇ · V − −2Λ +M2

)2

,

along with a corresponding scalar Faddeev–Popovdeterminant

(16)Det[�(0,0)− 2Λ + M2].

The final completely gauged-fixed action for thepartially massless graviton now takes the form

S =∫

d4x√g

[1

2φµν

(�(1,1)− 4

)φµν

+ V µ

(�

(12,

12

) − 4

)Vµ

− 2χ

(�(0,0)− 4

(17)+ φµνjµν + 1

4φj

].

Along with the addition to the two Faddeev–Popovdeterminants (12) and (16), this provides a completedescription ofZ, including couplings to the back-ground metric. This is to be compared with the genericmassive case where the corresponding action is givenby [1]

S =∫

d4x√g

×[

1

2φµν

(�(1,1)− 2Λ+ M2)φµν

− 1

8

−2Λ+ 3M2

−2Λ+ M2 φ(�(0,0)− 2Λ+ M2)φ

+ V µ(�

(12,

12

) − 2Λ+ M2)Vµ

+ −2Λ + M2

M2χ

(�(0,0)− 2Λ+ M2)χ

(18)+ φµνjµν + 1

4φj

].

Page 4: Quantum M2→2Λ/3 discontinuity for massive gravity with a Λ term

M.J. Duff et al. / Physics Letters B 516 (2001) 156–160 159

Note that in the partially massless case the tracemodeφ has disappeared except for its coupling to thetrace of the energy–momentum tensor. Withφ nowacting as a Lagrange multiplier, this indicates that thetheory couples to conformal matter. To compare themassive and partially massless theories at the classicallevel, therefore, let us assume that the massive theoryalso couples to matter withT µ

µ = 0 as well as∇µT

µν = 0. Then the tree-level amplitude for thecurrentTµν can be read from the action (18) directlyand is given by

A[T ] = 1

2T µν

(�(1,1)− 2Λ + M2)−1

Tµν,

since there are sources for neitherVµ nor χ . Thusat tree level, there is no discontinuity in taking theM2 → 2Λ/3 limit. We note here that there wouldbe sources for the Stückelberg fields if one were torelax the assumption of a conserved stress tensor or atraceless stress tensor. In this case, one needs only toaccount for the shifts inhµν andVµ to see howTµν

contributes to sources forVµ andχ .For the partially massless case, (17), we integrate

over all species to find the first quantum correction

Z[g,T ] ∝ e−A[T ] Det

[�

(12,

12

) − 4

]

× Det

[�(0,0)− 4

]

× Det

[�(1,1)− 4

]−1/2

× Det

[�

(12,

12

) − 4

]−1/2

(19)× Det

[�(0,0)− 4

]−1/2

,

where the operator�(1,1) − 43Λ arises in the trace-

lessφµν sector so its determinant refers to tracelessmodes only. This allows us to compute the one-loopcontribution

&(1)[g] = − lnZ[g,0]= −1

2lnDet

[�

(12,

12

) − 4

]

+ 1

2lnDet

[�(1,1)− 4

]

(20)− 1

2lnDet

[�(0,0)− 4

]to the effective action for the Einstein backgroundgµν . This is now to be compared with the one loopcontribution in the generic massive case [1]

&(1)[g] = − lnZ[g,0]= −1

2ln Det

[�

(12,

12

) − 2Λ +M2](21)+ 1

2lnDet

[�(1,1)− 2Λ + M2].

The difference in these two expressions reflects thefact that 5 degrees of freedom are being propagatedaround the loop in the massive case and only 4 in thepartially massless case. Denoting the dimension of thespin (A,B) representation byD(A,B) = (2A + 1)×(2B + 1), we countD(1,1) − D(1/2,1/2) = 5 forthe massive case, whileD(1,1) − D(1/2,1/2) −D(0,0) = 4 for the partially massless one.

It remains to check that there is no conspiracyamong the eigenvalues of these operators that wouldmake these two expressions coincide. To show this, itsuffices to calculate the coefficients in the heat-kernelexpansion for the graviton propagator associated withSL + SM , and compare it with the massive case givenin Ref. [1]. The coefficient functionsb(Λ)

k in theexpansion

(22)Tr e−�(Λ)t =∞∑k=0

t(k−4)/2∫

d4x√g b

(Λ)k ,

were calculated in Ref. [12] for general “spin oper-ators” �(Λ)(A,B) ≡ �(A,B) − 2Λ in an EinsteinbackgroundRµν = Λgµν . So, adapted for the genericoperators�(Λ,M) ≡ �(A,B)− 2Λ+M2 appearingin Eq. (21), but still in the same Einstein backgroundRµν = Λgµν , the results are:

180(4π)2b(Λ,M)4 (1,1)

= 189RµνρσRµνρσ − 756Λ2 + 810M4,

180(4π)2b(Λ,M)4

( 12,

12

)= −11RµνρσR

µνρσ + 984Λ2 − 1200ΛM2

+ 360M4,

180(4π)2b(Λ,M)4 (0,0)

(23)= RµνρσR

µνρσ + 636Λ2 − 480ΛM2 + 90M4.

Page 5: Quantum M2→2Λ/3 discontinuity for massive gravity with a Λ term

160 M.J. Duff et al. / Physics Letters B 516 (2001) 156–160

For the partially massless four degrees of freedomtheory,M2 = 2Λ/3, we obtain

180(4π)2b(Λ)4 (total)

= 180(4π)2[b(Λ)4 (1,1) − b

(Λ)4

( 12,

12

) − b(Λ)4 (0,0)

](24)= 199RµνρσR

µνρσ − 1096Λ2,

which differs from the result for theM2 → 2Λ/3 limitof the massive case,

180(4π)2b(Λ,M)4 (total)

= 180(4π)2[b(Λ,M)4 (1,1)− b

(Λ,M)4

( 12,

12

)](25)→ 200RµνρσR

µνρσ − 740Λ2.

Even for a de Sitter background with constant curva-ture

Rµνρσ = 1

3Λ(gµνgρσ − gµρgνσ ),

(26)RµνρσRµνρσ = 8

3Λ2,

there is no cancellation.Thus we conclude that the absence of a disconti-

nuity between theM2 → 2Λ/3 andM2 = 2Λ/3 re-sults for massive spin 2 is only a tree-level phenom-enon, and that the discontinuity itself persists at oneloop. That the full quantum theory is discontinuousis not surprising considering the different degrees offreedom for the two cases. Just as theM2 → 0 limitis discontinuous at the quantum level as a result ofgoing from five degrees of freedom to two, so theM2 → 2Λ/3 limit is discontinuous as a result of goingfrom five degrees of freedom to four.

Acknowledgements

We wish to thank S. Deser for enlightening discus-sions and F. Dilkes for initial collaboration. This re-search was supported in part by DOE Grant DE-FG02-95ER40899.

References

[1] F.A. Dilkes, M.J. Duff, J.T. Liu, H. Sati, QuantumM → 0discontinuity for massive gravity with aΛ term, hep-th/0102093.

[2] A. Higuchi, Forbidden mass range for spin 2 field theory in deSitter space–time, Nucl. Phys. B 282 (1987) 397.

[3] I.I. Kogan, S. Mouslopoulos, A. Papazoglou, Them → 0 limitfor massive graviton indS4 andAdS4: How to circumvent thevan Dam–Veltman–Zakharov discontinuity, Phys. Lett. B 503(2001) 173, hep-th/0011138.

[4] M. Porrati, No van Dam–Veltman–Zakharov discontinuity inAdS space, Phys. Lett. B 498 (2001) 92, hep-th/0011152.

[5] H. van Dam, M. Veltman, Massive and massless Yang–Millsand gravitational fields, Nucl. Phys. B 22 (1970) 397.

[6] V.I. Zakharov, JETP Lett. 12 (1970) 312.[7] S. Deser, A. Waldron, Gauge invariances and phases of

massive higher spins in (A)dS, hep-th/0102166.[8] S. Deser, R.I. Nepomechie, Gauge invariance versus massless-

ness in de Sitter space, Ann. Phys. 154 (1984) 396.[9] S. Deser, A. Waldron, Partial masslessness of higher spins in

(A)dS, hep-th/0103198.[10] S. Deser, A. Waldron, Stability of massive cosmological

gravitons, hep-th/0103255.[11] E.C.G. Stückelberg, Helv. Phys. Acta 30 (1957) 209.[12] S.M. Christensen, M.J. Duff, Quantizing gravity with a cos-

mological constant, Nucl. Phys. B 170 (1980) 480.