No-Go Theorems for Unitary and Interacting Partially Massless Spin-Two Fields
aa r X i v : . [ h e p - t h ] S e p AEI-2014-027
No-Go Theorems for Unitary and InteractingPartially Massless Spin-Two Fields
Euihun JOUNG, a Wenliang LI a and Massimo TARONNA b,c a AstroParticule et Cosmologie
10 rue Alice Domon et L´eonie Duquet, 75205 Paris Cedex 13, France b Max-Planck-Institut f¨ur Gravitationsphysik (Albert-Einstein-Institut)Am M¨uhlenberg 1, 14476 Golm, Germany c Kavli Institute for Theoretical Physics China, CASBeijing 100190, China
E-mail: [email protected] , [email protected] , [email protected] Abstract:
We examine the generic theory of a partially massless (PM) spin-two fieldinteracting with gravity in four dimensions from a bottom-up perspective. By analyzingthe most general form of the Lagrangian, we first show that if such a theory exists, its deSitter background must admit either so (1 ,
5) or so (2 ,
4) global symmetry depending on therelative sign of the kinetic terms: the former for a positive sign the latter for a negativesign. Further analysis reveals that the coupling constant of the PM cubic self-interactionmust be fixed with a purely imaginary number in the case of a positive sign. We concludethat there cannot exist a unitary theory of a PM spin-two field coupled to Einstein gravitywith a perturbatively local Lagrangian. In the case of a negative sign we recover conformalgravity. As a special case of our analysis, it is shown that the PM limit of massive gravityalso lacks the PM gauge symmetry. Unit´e Mixte de Recherche 7164 du CNRS ontents
B.1 Cubic interactions of PM field 14B.2 Global symmetries 15– i –
Introduction
In de Sitter space (dS) space, unitary spin-two modes have a mass gap, as opposed tothose in the flat space or anti-de Sitter space. The lightest massive spin-two modes do notcorrespond to the massless graviton but to a special massive field called partially massless spin-two [1, 2]. This lower bound is also known as
Higuchi bound [3]. The partially-masslessspin-two (PM) field has one less degree of freedom (DoF) than a generic massive spin-twofield due to the decoupling of the scalar mode: for example in four dimensions, it has fourDoFs instead of the five of the usual massive field.The PM field is gaining renewed interest in the context of the massive gravity theoryof [4–6] and the bimetric gravity theory of [7, 8]. With a suitable choice of parameters,these theories can be linearized around dS space and describe the propagation of massivespin-two modes. One of the natural questions is the following: when the mass is tuned tothat of PM , can the resulting theory consistently describe the dynamics of an interactingPM field? In other words, does the scalar DoF decouple from the theory in the PM limit?In the free theory of the PM field ϕ µν , the decoupling of the scalar DoF is due to theemergence of a gauge symmetry of the form, δ ϕ µν = (cid:18) ¯ ∇ µ ¯ ∇ ν + Λ3 ¯ g µν (cid:19) α , (1.1)where ¯ g µν and ¯ ∇ µ are the metric and covariant derivative of dS space with cosmologicalconstant Λ . If the PM limit of massive or bimetric gravity is consistent, then they shouldalso admit a PM gauge symmetry which extends the free one (1.1) to the interacting level.While the emergence of such a gauge symmetry has not yet been reported, there have beenmany discussions on the possible (in-)consistencies of this limit: see [10–13] for positiveand [14–18] for negative results. One of the aims of the present work is to provide a definiteanswer to this question.Another playground for the PM field is conformal gravity (CG), which has six propa-gating DoFs [19–21]. Two DoFs correspond to the usual graviton while the additional fourDoFs organize themselves into a PM representation around dS space (see e.g. [22, 23]). Inorder to see this point, it is convenient to recast the action into S CG = Z d x √− g (cid:20) − Λ6 ( R − L PM ( ϕ, ∇ ϕ, g, R ) (cid:21) , (1.2)by introducing an auxiliary field ϕ µν which can then be interpreted as a PM field. Here, L PM is a Lagrangian whose quadratic part coincides with that of the free PM field, whilethe higher power parts involve interactions of ϕ µν — see [23] for more details.CG is non-unitary because of the wrong relative sign between the Einstein-Hilbertterm and L PM . Nevertheless, as far as the number of DoFs is concerned, CG provides aconsistent theory of PM plus gravity. One can see this from the presence of the PM gaugesymmetry in CG, which is nothing but a disguised version of Weyl symmetry. Since the The possibility of having partially massless fields in the context of massive gravity has been firstdiscussed in 3D massive gravity [9]. – 1 –elative negative sign makes the theory non-unitary, one may think of naively flipping thissign. The redefinition ϕ µν → ± i ϕ µν might do the job, but it would introduce an imaginarypart in L PM , due to the presence of odd-power interactions involving PM fields. Hence,there is no obvious simple way to obtain a unitary theory of PM plus gravity out of CG.In the search of a consistent theory of interacting PM field, it is instructive to try tobuild it starting from the free theory order by order in powers of the PM field. Abouttwo-derivative cubic interactions, the task has been carried out in [24] while the cubicinteractions with general number of derivatives have been analyzed in a wider context in[27, 28] for the fields of arbitrary spins and arbitrary masses in generic dimensions. Let usbriefly summarize the results: • Two-derivative couplings:
The cubic self-interaction of PM field is unique andexists only in four dimensions. It is associated with Abelian conserved charges. More-over, it coincides with the coupling that can be extracted from the CG action (1.2)[23]. The PM–PM–graviton interaction, which corresponds to the gravitational min-imal coupling, is consistent provided that the graviton also transform under PMtransformation. It is associated with non-Abelian conserved charges. • Higher-derivative couplings:
Any of the cubic self-interactions of PM fieldwith more than two derivatives is not associated with a conserved charge.In the present paper, we prove that there cannot exist a unitary interacting theoryfor PM plus gravity. Our proof is based on the following two consequences of the gaugeinvariance condition: • Global symmetry:
Conserved charges must form a Lie algebra. • Admissibility:
The linearized theory must carry a unitary representation of theglobal symmetry algebra.We first show, by demanding PM field to be gravitationally interacting, that the firstcondition is automatically satisfied, and that the corresponding global symmetry turnsout to be so (1 ,
5) containing the dS isometry algebra so (1 , λ ,does not enter in the structure constants of so (1 ,
5) . Moving to the second requirement, admissibility condition , we demonstrate that the linearized field can carry a representationof so (1 ,
5) only when the PM self-interaction coupling constant λ satisfies λ + 8 π G N = 0 , (1.3)where G N is the Newton’s gravitational constant. This shows that the gauge invarianceof PM plus gravity action requires the PM self interaction to have an imaginary couplingconstant λ = ± i √ π G N , which manifestly violates the unitarity. In the case of PMtheory without gravity, which can be achieved by taking G N → λ = 0so the theory cannot have a cubic interaction. In particular, this implies that the PM See also [25, 26] for the other related results of the same author. – 2 –imit of massive/bimetric gravity cannot lead to a gauge invariant Lagrangian theory, andconsequently it should suffer from a kind of Boulware-Deser ghost problem [29]: the scalarDoF does not decouple from the theory. Moreover, one can notice that this choice ofimaginary λ coincides with the redefinition ϕ µν → ± i ϕ µν of CG: in such a case, the globalsymmetry so (1 ,
5) is replaced by the conformal algebra so (2 , In search for a theory of PM plus gravity, one may begin with the most general form ofthe action S . It has two parts: S = S EH + S PM , (2.1)where the gravity sector S EH is given by Einstein-Hilbert term: S EH [ g ] = 12 κ Z d x √− g ( R − , (2.2)with κ = 8 π G N , while the PM part S PM is not fixed for the moment except that it isgiven through a quasi-local Lagrangian L PM of manifestly diffeomorphism-invariant form: S PM [ ϕ, g ] = Z d x √− g L PM ( ϕ, ∇ ϕ, g, R, . . . ) , (2.3)where, . . . means that there may be higher derivatives of ϕ µν or curvature R µνρσ . Let usemphasize that this ansatz also covers the bimetric gravity of [7, 8]: see Appendix A formore details.Besides the diffeomorphism symmetries, we also require the action to be invariantunder PM gauge symmetries: δ α S = 0 , (2.4)where δ α is the nonlinearly deformed PM transformation which we aim to determine to-gether with L PM . For further analysis of this gauge invariance condition, it is convenientto expand the action and the PM gauge transformations in powers of the PM field ϕ µν as S EH = S (0) , S PM = S (2) + S (3) + · · · , δ α = δ (0) α + δ (1) α + · · · , (2.5)where δ ( n ) α = Z d x √− g (cid:20) ( δ α ϕ µν ) ( n ) δδϕ µν + ( δ α g µν ) ( n − δδg µν (cid:21) , (2.6) By quasi-local Lagrangian, we mean that there exists an expansion parameter such that every truncationof the Lagragian to a finite power of this parameter contains finitely many derivatives. The number ofderivatives of the full Lagrangian may be unbounded. – 3 –hile the superscript ( n ) means that the corresponding term involves the n th power of ϕ µν . Then, the PM gauge invariance condition (2.4) provides an infinite set of equations: δ (1) α S (0) = 0 , (2.7) δ (0) α S (2) + δ (2) α S (0) = 0 , (2.8) δ (0) α S (3) + δ (1) α S (2) + δ (3) α S (0) = 0 , (2.9) · · · . The first condition (2.7) simply tells us that ( δ α g µν ) (0) = 0 — the metric does not transformunder PM at the lowest order — whereas the other conditions constrain possible forms of L PM and δ α . The advantage of the expansion (2.5) is that we can attack the gaugeinvariance conditions one by one from the lowest level. In the following, we shall analyzethe second and third conditions (2.8 , 2.9). The quadratic part of the gauge invariance condition (2.8) reads Z d x √− g (cid:18) ( δ α ϕ µν ) (0) (cid:20) δS (2) δϕ µν (cid:21) + ( δ α g µν ) (1) G µν Λ (cid:19) = 0 , (2.10)where G µν Λ ≡ R µν − g µν R/ g µν is the cosmological Einstein tensor. The lowest-orderPM gauge transformation is given by the covariantization of the free PM transformation(1.1) around the dS background:( δ α ϕ µν ) (0) = (cid:18) ∇ µ ∇ ν + Λ3 g µν (cid:19) α , (2.11)up to trivial transformations proportional to G µν Λ . Eq. (2.8) can be solved by properlycovariantizing the free action of the PM field around dS background. The solution for S (2) reads S (2) = σ Z d x √− g (cid:20) − ∇ α ϕ µν ∇ α ϕ µν + ∇ α ϕ µν ∇ ν ϕ µα − ∇ µ ϕ ∇ ν ϕ µν + 12 ∇ µ ϕ ∇ µ ϕ + Λ (cid:18) ϕ µν ϕ µν − ϕ (cid:19) − m PM ϕ µν ϕ µν − ϕ ) + L (2)m . r . ( ϕ, ∇ ϕ ) (cid:21) , (2.12)where the mass of the PM field is given by m PM = Λ , and L (2)m . r . is proportional to G µν Λ : L (2)m . r . ( ϕ, ∇ ϕ ) = G µν Λ ( a ϕ µρ ϕ ρν + b g µν ϕ ρσ ϕ ρσ + c ϕ µν ϕ + d g µν ϕ ) , (2.13)hence arbitrary for the moment. Different choices of L (2)m . r . are all physically equivalent asthey are related by a field redefinition: g µν → g µν + (cid:16) ˜ a ϕ µρ ϕ ρν + ˜ b g µν ϕ ρσ ϕ ρσ + ˜ c ϕ µν ϕ + ˜ d g µν ϕ (cid:17) . (2.14)In eq. (2.12), we have also introduced a sign factor σ in order to keep track of the role ofthe relative sign between the graviton and PM kinetic terms:– 4 – σ = +1 : the kinetic terms of gravity and PM field have a relatively positive sign,hence the theory may eventually be unitary; • σ = − δ α g µν ) (1) for each choice of L (2)m . r . . Without loss of generality, we choose L (2)m . r . with( a, b, c, d ) = (2 , − , − , ) , to end up with a relatively simple form for ( δ α g µν ) (1) α :( δ α g µν ) (1) = 2 σ κ (cid:0) ∇ ( µ ϕ ν ) ρ − ∇ ρ ϕ µν (cid:1) ∂ ρ α , (2.15)where we use the weight-one normalization convention for (anti-)symmetrization: T ( µν ) =( T µν + T νµ ) / T [ µν ] = ( T µν − T νµ ) / covariantizing the free PM action — the metric tensor transforms under PM gauge transformations as(2.15), up to field redefinitions. We turn to the cubic part of the gauge invariance condition (2.9): Z d x √− g (cid:18) ( δ α ϕ µν ) (0) (cid:20) δS (3) δϕ µν (cid:21) + ( δ α ϕ µν ) (1) (cid:20) δS (2) δϕ µν (cid:21) + ( δ α g µν ) (2) G µν Λ (cid:19) = 0 . (2.16)In this case, we aim to identify S (3) together with ( δ α ϕ µν ) (1) and ( δ α g µν ) (2) . Similar tothe quadratic part, one can solve the condition (2.16) by properly covariantizing the PMcubic self-interaction derived for the dS background [24]. Checking its gauge invariance ongeneral backgrounds, we get S (3) = λ Z d x √− g h Λ (cid:0) ϕ µρ ϕ µν ϕ νρ − ϕ µµ ϕ νρ ϕ νρ + ϕ µµ ϕ ν ν ϕ ρρ (cid:1) − ϕ µν ∇ µ ϕ ρσ ∇ ν ϕ ρσ + 2 ϕ µν ∇ µ ϕ ρρ ∇ ν ϕ σσ − ϕ µν ∇ ν ϕ σσ ∇ ρ ϕ µρ − ϕ µν ∇ ν ϕ µρ ∇ ρ ϕ σσ + 2 ϕ µν ∇ ρ ϕ σσ ∇ ρ ϕ µν − ϕ µµ ∇ ρ ϕ σσ ∇ ρ ϕ ν ν − ϕ µν ∇ ρ ϕ µν ∇ σ ϕ ρσ + 2 ϕ µµ ∇ ρ ϕ ν ν ∇ σ ϕ ρσ + 6 ϕ µν ∇ ν ϕ ρσ ∇ σ ϕ µρ + 2 ϕ µν ∇ ρ ϕ νσ ∇ σ ϕ µρ − ϕ µν ∇ σ ϕ νρ ∇ σ ϕ µρ − ϕ µµ ∇ ρ ϕ νσ ∇ σ ϕ νρ + ϕ µµ ∇ σ ϕ νρ ∇ σ ϕ νρ i . (2.17)See subsection B.1 for the ambient-space formulation of the cubic interactions along thelines of [27]. By plugging the solution (2.17) into the condition (2.16), the gauge transfor-mations ( δ α ϕ µν ) (1) and ( δ α g µν ) (2) can be determined straightforwardly. In particular, the– 5 –xpression for ( δ α ϕ µν ) (1) will be important for the forthcoming analysis and it is given by ( δ α ϕ µν ) (1) = 2 σ λ (cid:0) ∇ ( µ ϕ ν ) ρ − ∇ ρ ϕ µν (cid:1) ∂ ρ α . (2.20)Let us remind the reader that the expression (2.17) for S (3) is the covariantization of theunique two-derivative self-interaction which exists only in four dimensions. On the otherhand, higher-derivative PM self-interactions are shown [27, 28] to not affect the form of( δ α ϕ µν ) (1) . Therefore, the expression (2.20) provides the only possible form for the ϕ -linearpart of nonlinear PM gauge transformation, up to redefinitions of ϕ µν which are physicallyirrelevant.Notice that the cubic-order gauge-invariance condition (2.9) does not constrain thecoupling constant λ at all. The coupling constants can be determined by the quartic orhigher-order consistency conditions. Hence, in principle, we may have to proceed to higherorders to see the eventual (in-)consistency of the PM-plus-gravity theory. However, thereexist other consequences of gauge invariance that cubic couplings must satisfy. They canbe examined without analyzing quartic interactions. In the following, we shall explain thispoint and solve the correponding conditions. Until now, we have analyzed the gauge invariance of the PM plus gravity action up to thecubic order in the PM field. In general, when an action S , involving a set of bosonic fields χ i , admits gauge symmetries, then the gauge symmetries must form an (open) algebra: δ ε δ η − δ ε δ η = δ [ η,ε ] + (trivial) , (3.1)where δ ε stands for δ ε = δ ε χ i δδχ i in deWitt notation. The gauge-algebra bracket [ η, ε ] mightin principle also depend on fields: [ η, ε ] = f ( η, ε, χ i ) , while the term “(trivial)” denotesany trivial symmetry generated by an arbitrary antisymmetric matrix C ij = − C ji as(trivial) = C ij ( η, ε ) δSδχ i δδχ j . (3.2)In the following, we will seek the consequences of the above condition for the PM plusgravity theory. The expression for ( δ α g µν ) (2) can be equally determined, though we shall not use it in later analysis.It takes the following relatively simple form,( δ α g µν ) (2) = 8 κ λ ( ϕ ρσ ∇ ( µ ϕ ν ) σ − ϕ ( µσ ∇ ν ) ϕ ρσ + ϕ ( µσ ∇ ρ ϕ ν ) σ − ϕ ρσ ∇ σ ϕ µν ) ∂ ρ α , (2.18)after the redefinition, g µν → g µν + κλ (12 ϕ µρ ϕ νρ ϕ σσ − ϕ µρ ϕ νσ ϕ ρσ + 4 ϕ µν ϕ ρσ ϕ ρσ + g µν ϕ ρα ϕ ρσ ϕ σα − ϕ µν ϕ ρρ ϕ σσ − g µν ϕ ρρ ϕ σα ϕ σα + g µν ϕ ρρ ϕ σσ ϕ αα ) . (2.19) – 6 – .1 Algebra of gauge symmetries In the previous sections, we have identified the PM gauge transformations up to linearorder in the PM fields: see (2.11), (2.15) and (2.20). This makes it possible to identifythe ϕ µν -independent part of the brackets by explicitly evaluating the commutator of twosuccessive gauge transformations. First, the diffeomorphisms give rise to the usual Liederivative: [ ξ , ξ ] = ( ξ ν ∇ ν ξ µ − ξ ν ∇ ν ξ µ ) ∂ µ . (3.3)Next, the commutators between diffeomorphism and PM transformations give( δ α δ ξ − δ ξ δ α ) g µν = O ( ϕ ) , ( δ α δ ξ − δ ξ δ α ) ϕ µν = (cid:18) ∇ µ ∇ ν + Λ3 g µν (cid:19) ( ξ σ ∂ σ α ) + O ( ϕ ) . (3.4)From the above, one can extract the corresponding bracket as[ ξ , α ] = ξ µ ∂ µ α + O ( ϕ ) . (3.5)Finally, there is the commutator of two PM transformations: its action on g µν is given by( δ α δ α − δ α δ α ) g µν = − σ κ h ∇ µ ( ∂ ρ α ∇ ν ∂ ρ α − ∂ ρ α ∇ ν ∂ ρ α )+ ∇ ν ( ∂ ρ α ∇ µ ∂ ρ α − ∂ ρ α ∇ µ ∂ ρ α ) i + O ( ϕ ) . (3.6)For the action on ϕ µν , let us notice that the transformation (2.20) involves the covarianti-zation of the tensor, C µν,ρ = ¯ ∇ ( µ ϕ ν ) ρ − ¯ ∇ ρ ϕ µν , (3.7)which is invariant under the free PM symmetry (1.1). Indeed, one can show that( δ α δ α − δ α δ α ) ϕ µν = O ( ϕ ) . (3.8)The absence of ϕ µν -independent part in the above commutator implies that the bracketbetween two PM transformations does not give a PM transformation, while eq. (3.6) showsthat it results in a diffeomorphism:[ α , α ] = − σ κ ( ∂ ρ α ∇ µ ∂ ρ α − ∂ ρ α ∇ µ ∂ ρ α ) ∂ µ + O ( ϕ ) . (3.9)So far, we have determined the ϕ µν -independent part of the gauge-algebra brackets. Dueto the (possible) field-dependent pieces, the full gauge-algebra brackets do not define aLie algebra. However, their restriction to the Killing fields, namely the global-symmetrybrackets, must define a Lie algebra. We will discuss this point in more detail in the nextsection. – 7 – .2 Lie algebra of global symmetries Once we get the field-independent part of the gauge-algebra brackets, it is already sufficientto fully determine the global-symmetry structure constants. Similarly to the gauge sym-metries, the global-symmetry transformations must be closed; what is more is that theymust also form a Lie algebra. Hence, this point — whether the brackets indeed satisfy theJacobi identity and define a Lie algebra — provides us with a simple necessary conditionfor the consistency of the theory.In order to see this point more clearly, let us briefly move back to the general discussionspresented at the beginning of section 3. We shall now analyze the closure of the symmetryalgebra perturbatively. One considers the expansions: S = S [2] + S [3] + · · · , δ ε = δ [0] ε + δ [1] ε + · · · , [ η, ε ] = [ η, ε ] [0] + [ η, ε ] [1] + · · · , C ij = C [0] ij + C [1] ij + · · · , (3.10)where the superscript [ n ] stands for the total power of fields χ i involved. Then, the lowest-order part of the closure condition (3.1) reads simply δ [0] ε δ [1] η − δ [0] η δ [1] ε = δ [0] [ η,ε ] [0] . (3.11)At the next-to-lowest order, it gives δ [1] ε δ [1] η − δ [1] η δ [1] ε + δ [0] ε δ [2] η − δ [0] η δ [2] ε = δ [1] [ η,ε ] [0] + δ [0] [ η,ε ] [1] + C [0] ij ( η, ε ) δS [2] δχ i δδχ j . (3.12)Restricting gauge parameters to Killing fields, the above two conditions (3.11) and (3.12)provide simple but important consistency requirements for the theory. The Killing fields ¯ ε are defined by the solutions of the Killing equations: δ [0] ¯ ε = 0 . (3.13)The first condition (3.11) becomes δ [0] [¯ η, ¯ ε ] [0] = 0 , (3.14)meaning that the global symmetry is closed under the bracket [[¯ η, ¯ ε ]] := [¯ η, ¯ ε ] [0] . The secondcondition (3.12) reduces to δ [1] ¯ ε δ [1] ¯ η − δ [1] ¯ η δ [1] ¯ ε = δ [1] [[¯ η, ¯ ε ]] + δ [0] [¯ η, ¯ ε ] [1] + C [0] ij (¯ η, ¯ ε ) δS [2] δχ i δδχ j , (3.15)meaning that δ [1] ¯ ε provides a representation of the Lie algebra of the global symmetries onthe space of fields.Having the above general lessons in mind, let us come back to the PM-plus-gravitytheory and consider the dS metric g µν = ¯ g µν and ϕ µν = 0 as the background. The globalsymmetries of this background are the subset of gauge symmetries which leave it invariant.The gauge parameters of the global transformations are defined as the solutions of thefollowing Killing equations: (cid:2) δ ¯ ξ g µν (cid:3) bg = 2 ¯ ∇ ( µ ¯ ξ ν ) = 0 , (cid:2) δ ¯ α ϕ µν (cid:3) bg = (cid:18) ¯ ∇ µ ¯ ∇ ν + Λ3 ¯ g µν (cid:19) ¯ α = 0 , (3.16)– 8 –here [ · ] bg means the evaluation g µν = ¯ g µν and ϕ µν = 0 . From the lowest part of the gauge-algebra brackets (3.3), (3.5) and (3.9), we get the following brackets of global symmetries:[[ ¯ ε , ¯ ε ]] = 2 (cid:18) ¯ ξ ν [2 ∂ ν ¯ ξ µ − σ κ Λ3 ¯ α [2 ∂ µ ¯ α (cid:19) ∂ µ + 2 ¯ ξ µ [2 ∂ µ ¯ α , (3.17)where we have conveniently packed the parameters as ¯ ε = ¯ ξ µ ∂ µ + ¯ α . The ¯ ξ µ -transformationsform the isometry algebra of dS space, while the ¯ α -transformation extends the isometry toa larger global symmetry. In order to identify such global symmetry, we need to solve theKilling equations (3.16). For that, it is convenient to reformulate them in the ambient-spaceformalism through the standard embedding: ξ µ ( x ) = ℓ Ξ M ( X ) ∂ µ X M X , α ( x ) = ℓ A ( X ) √ X , (3.18)where ℓ = 3 / Λ and the X M ’s are the coordinates of the ambient space containing dS spaceas a hyperboloid: dS = (cid:8) X ∈ R , (cid:12)(cid:12) X = ℓ (cid:9) , dX = dR + R ℓ ¯ g µν dx µ dx ν . (3.19)In terms of the ambient space fields, the Killing equations simply read ∂ ( M ¯Ξ N ) = 0 , ∂ M ∂ N ¯ A = 0 , (3.20)and the solutions are given by¯Ξ M ∂ M = W AB M AB , ¯ A = V A K A . (3.21)Here W AB = − W BA and V A are arbitrary parameters while M AB and K A are the globalsymmetry generators: M AB = 2 X [ A ∂ B ] , K A = X A . (3.22)To recapitulate, the global symmetries are generated by the Killing fields:¯ ξ µ = W AB (cid:18) ℓ X [ A ∂ µ X B ] X (cid:19) , ¯ α = V A (cid:18) ℓ X A √ X (cid:19) . (3.23)Using these explicit form of the generators and eq. (3.17), one can calculate their bracketsand get (cid:2)(cid:2) M AB , M CD (cid:3)(cid:3) = η AD M BC + η BC M AD − η AC M BD − η BD M AC , (cid:2)(cid:2) M AB , K C (cid:3)(cid:3) = η BC K A − η AC K B , (cid:2)(cid:2) K A , K B (cid:3)(cid:3) = − Λ3 σ κ M AB . (3.24)The structure constants do not involve the PM self-interaction coupling constant λ as itwas manifest already from eq. (3.17). This means that the PM cubic self-interaction is– 9 –belian. One can easily check that the above brackets (3.24) define a simple Lie algebrafor any value of the relative sign σ between the kinetic terms: • for σ = +1 , they define so (1 ,
5) , • for σ = − so (2 ,
4) .These algebras contain the isometry algebra so (1 ,
4) generated by M AB as a subalgebra.Hence, we conclude that any unitary theory of gravitationally interacting PM fields, if sucha theory exists, must have the global symmetry so (1 , . We are now at the point to examine the condition (3.15), namely the admissibility condition ,which implies that the linearized theory must carry a representation of the global symmetry.The admissibility condition plays an important role in higher-spin field theories [30] as wellas in supergravities. In the case of PM plus gravity, it will also turn out to be a decisivecondition.In order to examine the admissibility condition for the system under consideration,one first needs to linearize the transformations with respect to the metric perturbation h µν = g µν − ¯ g µν as δ ¯ ε h µν = δ ¯ ε g µν = δ [1] ¯ ε h + O ( h, ϕ ) ,δ ¯ ε ϕ µν = δ [1] ¯ ε ϕ + O ( h, ϕ ) , (3.25)where the superscript [1] means that the corresponding terms are linear in h µν or ϕ µν .First, from the diffeomorphism symmetry, we get δ [1] ¯ ξ h µν = 2 ¯ ∇ ( µ ¯ ξ ρ h ν ) ρ + ¯ ξ ρ ¯ ∇ ρ h µν , δ [1] ¯ ξ ϕ µν = 2 ¯ ∇ ( µ ¯ ξ ρ ϕ ν ) ρ + ¯ ξ ρ ¯ ∇ ρ ϕ µν , (3.26)which tell how the so (1 ,
4) charges act on the fields. Then, from eq. (2.11), (2.15) and(2.20) , we get the PM transformations as δ [1] ¯ α h µν = − σ κ ∂ ρ ¯ α (cid:0) ∇ ( µ ϕ ν ) ρ − ¯ ∇ ρ ϕ µν (cid:1) , (3.27) δ [1] ¯ α ϕ µν = 2 λ σ ∂ ρ ¯ α (cid:0) ¯ ∇ ( µ ϕ ν ) ρ − ¯ ∇ ρ ϕ µν (cid:1) − ∂ ρ ¯ α (cid:0) ∇ ( µ h ν ) ρ − ¯ ∇ ρ h µν (cid:1) + Λ3 ¯ α h µν . (3.28)This indicates how the PM charges K A act on the fields. For more explicit expressions, wecan replace ¯ ξ µ and ¯ α with the solutions (3.23) of the Killing equations. Such expressionscan be found in subsection B.2 where we carry out the derivation in the ambient-spaceformalism. In fact, the brackets (3.24) also encode the properties of the other cubic interactions. First, the gravita-tional self-interaction is associated with the bracket [[
M, M ]] = M : the fact that it does not vanish impliesthat the interaction is non-Abelian. Second, the gravitational minimal-coupling of PM field is associatedwith the brackets [[ M, K ]] = K and [[ K, K ]] = M : it is also a non-Abelian interaction. Finally, the absenceof [[ K, K ]] = K structure means that the PM self-interaction is Abelian. – 10 –ith (3.27) and (3.28), we are ready to compute the LHS of eq. (3.15), which is thecommutator between two PM transformations. After straightforward calculations andimposing the global symmetry condition on gauge parameters, we obtain the commutatorof two PM transformations as (cid:0) δ [1] ¯ α δ [1] ¯ α − δ [1] ¯ α δ [1] ¯ α (cid:1) h µν = 2 ¯ ∇ ( µ A ρ h ν ) ρ + A ρ ¯ ∇ ρ h µν + 2 ¯ ∇ ( µ B ν ) , (cid:0) δ [1] ¯ α δ [1] ¯ α − δ [1] ¯ α δ [1] ¯ α (cid:1) ϕ µν = 2 ¯ ∇ ( µ A ρ ϕ ν ) ρ + A ρ ¯ ∇ ρ ϕ µν + ( λ + σκ ) C µν , (3.29)where A µ , B µ and C µν are given by A µ = 2 σ κ Λ3 ¯ α [1 ∂ µ ¯ α , (3.30) B µ = − σ κ (cid:20) ∂ ρ ¯ α [1 ∂ σ ¯ α ( ¯ ∇ ρ h σµ − σ λ ¯ ∇ ρ ϕ σµ ) + 2 Λ3 ¯ α [1 ∂ ρ ¯ α h ρµ (cid:21) , (3.31) C µν = 4 ∂ ρ α [1 ∂ σ α ¯ ∇ ( µ | ¯ ∇ σ ϕ | ν ) ρ + 4 Λ α [1 ∂ ρ α ( ¯ ∇ ( µ ϕ ν ) ρ − ¯ ∇ ρ ϕ µν ) . (3.32)Let us analyze each term in (3.29) to see whether they are compatible with the RHS ofeq. (3.15): • First, the terms involving A µ in (3.29) take the form of a Lie derivative. Moreoverone can show that the form (3.30) of A µ coincides with the bracket (3.17): A µ ∂ µ = [[ ¯ α , ¯ α ]] . (3.33)Hence, these terms correspond to the δ [1] [[ ¯ α ¯ α ]] contribution in the RHS of eq. (3.15). • Second, the terms involving B µ in (3.29) take the form of linearized diffeomorphism,hence corresponding to the δ [0] [ ¯ α ¯ α ] [1] contribution in the RHS of eq. (3.15) with[ ¯ α , ¯ α ] [1] = B µ ∂ µ . (3.34)The above relation can be explicitly checked by extracting δ [2] α h µν from eq. (2.18). • Finally, there remains the C µν term in (3.29), which does not correspond to any of thecontributions in the RHS of eq. (3.15). Therefore, in order the admissibility conditionto be satisfied, we must require that the coefficient of the C µν term vanishes: λ + σ κ = 0 . (3.35)This determines the coupling constant λ for the PM self-interaction in terms of thegravitational constant κ = 8 π G N as λ = ±√− σ κ . Now one has two options for atheory of PM-plus-gravity theory depending on the relative sign σ between the kineticterms: – for σ = −
1, we get λ = ±√ κ which coincides with the coupling constant of thePM self-interaction in CG; One can examine also the other commutators, but they do actually satisfy the admissibility condition(3.15) without constraining any coupling constant. – 11 – for σ = +1, the coupling constant λ becomes purely imaginary. This choicesimply corresponds to the ϕ µν → ± i ϕ µν redefinition of CG.In this section, we have shown how the admissibility condition allows us to determine theAbelian interaction — the cubic self-interaction of PM field. In particular, this demon-strates that the relatively positive kinetic terms in PM plus gravity theory cannot becompatible with a real Lagrangian. In this paper, we have investigated the most general form of Lagrangian for PM field andgravity with a positive cosmological constant. By examining its gauge symmetries, we haveshown that • There cannot exist a real-valued Lagrangian whose kinetic terms have the relativelypositive sign for the PM field and graviton. In other words, there cannot exist aunitary theory for PM plus gravity. • The PM cubic self-interaction is entirely fixed by gauge invariance. For the relativelynegative sign of the kinetic terms, the linearization of this theory admits the globalsymmetry of so (2 , • The case of PM theory without gravity is covered by taking the limit κ → so (1 , λ to vanish; therefore, no two-derivative cubic interaction isconsistent in the pure PM theory. This rules out the PM limit of massive gravityfrom the possible consistent theories of the PM field due to the presence of its two-derivative cubic interaction inherited from the Einstein- Hilbert term.Our results imply in particular that the PM limit of the bimetric gravity cannot have theputative gauge symmetries of PM field. Besides, for the relative negative sign of kineticterms, CG has been recovered as the result of the analysis up to the cubic order in ϕ µν .However, we argue that the conclusion can be valid beyond the cubic order, without anyassumption on the number of derivatives for the following reasons: • The inclusion of higher-derivative interactions cannot change the conclusion sincethey do not affect the form of PM transformations [27], on which our analysis isbased on. • The validity beyond the cubic order can be argued from the two points: first,
CGis the unique Weyl invariant as well as the unique invariant theory of so (2 , second, the interaction structure of CG is the unique one up to fourderivatives [32]. – 12 –e expect that a similar construction might be possible in higher dimensions where thecorresponding CG equations still have a factorized form on any Einstein background [33]involving some massive modes. Acknowledgments
EJ thanks J. Mourad for useful discussions. MT thanks Stefan Theisen for useful discus-sions. MT is also grateful to KITPC for the hospitality and support during the final stagesof this work.
A PM bimetric gravity as a theory of gravity plus matter
In this section, we show how the bimetric gravity of [7, 8] can be recast into the form (2.1).For simplicity, let us consider the model with a particular choice of parameters, which wasconsidered as the unique candidate for the PM plus gravity theory in that context. Itsaction takes the following form after a rescaling of the metric f µν → (cid:0) m g m f (cid:1) f µν , S [ g, f ] = 12 κ Z d x (cid:20) √− g ( R ( g ) − p − f ( R ( f ) − − Λ3 V ( g, f ) (cid:21) , (A.1)where the potential term V ( g, f ) is given by V ( g, f ) = √− g (cid:2) (Tr S ) − Tr S (cid:3) , ( S ) µν = g µρ f ρν . (A.2)The constants κ and Λ are related to the parameters used in [12] as follows,2 κ = 1 m g , Λ = 3 m m f β . (A.3)Then one can redefine its two tensors into the physical metric G µν and the massive spin-twofield ϕ µν as g µν = 12 ( G µν + 2 ϕ µν ) , f µν = 12 ( G µν − ϕ µν ) , (A.4)in order to recover an action S [ G, ϕ ] of the form (2.1). This action can be derived by ex-panding S [ g, f ] around the background g µν = G µν / f µν and by replacing the fluctuation by the PM field ϕ µν : the ϕ n -order part of the action is given by S ( n ) [ G, ϕ ] = 1 n ! (cid:20)Z ϕ µν (cid:18) δδg µν − δδf µν (cid:19)(cid:21) n S [ g, f ] (cid:12)(cid:12)(cid:12) g µν = G µν = f µν . (A.5)For example, the zeroth order action S (0) gives the Einstein-Hilbert action: S (0) = S [ g, f ] (cid:12)(cid:12)(cid:12) g µν = G µν = f µν = 12 κ Z d x √− G (cid:2) R ( G ) − (cid:3) . (A.6)At the first order, we get S (1) = 0 due to the symmetry, S [ g, f ] = S [ f, g ] . (A.7)– 13 –he quadratic action S (2) linearized around dS space was already computed in [12], whichconfirms that it coincides with the PM free Lagrangian. Therefore, one can see that theaction of the bimetric gravity can be recast into the stardard form of the gravity plus matter ,which our analysis concerned. Moreover, one can also see, from the symmetry (A.7), thatall odd powers of PM field do not appear in the action S [ G, ϕ ] .
B Cubic-interaction analysis in ambient-space formulation
In subsection 2.2 and subsection 3.3, we provided the form of the PM cubic self-interaction(2.17), and computed the commutator of the linearized transformations (3.29). In thisappendix, we present the ambient-space version of such calculations, which makes the dSsymmetry and the link to the previous works [27, 28] more transparent. For that, we firstrecast everything into the objects in the ambient space (3.19): the metric perturbation h µν and the PM field ϕ µν correspond to the ambient-space fields H ( X, U ) and Φ(
X, U ) as h µν ( x ) = ∂ µ X M ∂ ν X N X ∂ H ( X, U ) ∂U M ∂U N , ϕ µν ( x ) = ∂ µ X M ∂ ν X N √ X ∂ Φ( X, U ) ∂U M ∂U N , (B.1)while the corresponding gauge parameters are already given in eq. (3.18). Here, we havealso introduced the auxiliary variables U M to handle the tensor indices conveniently. B.1 Cubic interactions of PM field
Following [34] (see [35] for more details), the part of the cubic interactions not involvingtraces and divergences can be obtained as a function of six independent scalar contractions: Z i = ∂ U i +1 · ∂ U i − , Y i = ∂ U i · ∂ X i +1 [ i ≃ i + 3] . (B.2)About the two-derivative cubic interactions considered in this paper, it reads Z dS (cid:20) κ (cid:2) ( Y Z + Y Z + Y Z ) + 3 Λ Z Z Z (cid:3) H H H + σ (cid:2) ( Y Z + Y Z + Y Z ) + Λ Z Z Z (cid:3) H Φ Φ + λ Y Z + Y Z + Y Z ) Φ Φ Φ (cid:21) Xi = XUi =0 , (B.3)where H i and Φ i stand for H ( X i , U i ) and Φ( X i , U i ) . This form of the action is invariantunder the gauge transformation, δ H = U · ∂ X Ξ , δ Φ = ( U · ∂ X ) A , (B.4)modulo the terms involving divergences and traces. Several comments are in order: • First, the H H H term corresponds to the usual gravitational self-interactions.– 14 – Second, the H Φ Φ term corresponds the the gravitational minimal coupling of thePM field. In fact, as far as the TT part is concerned [27], there exists one moretwo-derivative coupling of the form:( Y Z + Y Z + Y Z )( Y Z − Y Z − Y Z ) . (B.5)However, there do not exist divergence and trace pieces that can uplift the abovecoupling to a fully gauge invariant one. • Finally, the Φ Φ Φ term corresponds to the PM self interaction, which exists only infour dimensions. The work [27] does not contain this coupling since it only concernsthe interactions existing in generic dimensions. A particular dimensional dependencein the latter work was hidden in the variable ˆ δ , which can be eventually replaced as Z dS ˆ δ n I ∆ = (∆ + d − d − · · · (∆ + d − n + 1) ℓ n Z dS I ∆ , (B.6)where the integrand I ∆ satisfies ( X · ∂ X − ∆) I ∆ = 0 . Taking into account the aboveidentity, one can easily find the two-derivative Φ Φ Φ coupling. Let us also rectifyone claim in the literature — the footnote 7 of [23] and the footnote 11 of [27]: thefour-derivative PM self-interaction does not reduce to the two-derivative one due tothe Gauss-Bonnet identity, but it actually coincides with the Gauss-Bonnet term,hence vanishing identically in four dimensions. B.2 Global symmetries
After identifying the cubic interactions in the ambient-space form, one can systematicallyextract the corresponding deformations of gauge transformations following [36]. Uponrestricting on global symmetries adopting a ℓ = 1 convention, we obtain the following setof deformations from the expression (B.3) and the results obtained in [36]: δ [1] H = − Π (cid:20) U · ∂ U (cid:18) U · ∂ U ∂ U · ∂ X − U · ∂ U ∂ U · ∂ X (cid:19) Ξ H + σ κ ( U · ∂ U ) (cid:18) ∂ X · ∂ X − (cid:19) A Φ (cid:21) , (B.7) δ [1] Φ = − Π (cid:20) U · ∂ U (cid:18) U · ∂ U ∂ U · ∂ X − U · ∂ U ∂ U · ∂ X (cid:19) Ξ Φ + 14 ( U · ∂ U ) (cid:18) ∂ X · ∂ X − (cid:19) A H + λ U · ∂ U ) ∂ X · ∂ X A Φ (cid:21) , (B.8)where Π is the operator adjusting tangent or radial contributions so that the resultingdeformations of the gauge transformations remain compatible with the tangentiality andhomogeneity constraints. Making use of the solution (3.21), the M AB -transformations ofthe isometry algebra so (1 ,
4) with parameters W AB become δ [1] W H = ( X · W · ∂ X + U · W · ∂ U ) H ,δ [1] W Φ = ( X · W · ∂ X + U · W · ∂ U ) Φ . (B.9)– 15 –n the other hand, the K A -transformations, associated to the additional generators of theglobal symmetries with parameters V A , take the form, δ [1] V H = − σ κ (cid:0) X (cid:1) (cid:18) U · ∂ X V · ∂ U − V · ∂ X + 2 X · UX V · ∂ U (cid:19) (cid:0) X (cid:1) − Φ , (B.10) δ [1] V Φ = −
12 ( U · ∂ X V · ∂ U − V · ∂ X ) H + λ (cid:0) X (cid:1) ( U · ∂ X V · ∂ U − V · ∂ X ) Φ . (B.11)Using these expressions of the gauge transformations, one can easily compute the rele-vant commutators. In particular, we are interested in the commutator between two K A -transformations δ [1] V [2 δ [1] V . After a straightforward computation, one gets δ [1] V [2 δ [1] V H = − σ κ (cid:0) V [1 · X V · ∂ X + V [1 · U V · ∂ U (cid:1) H + U · ∂ X B , (B.12) δ [1] V [2 δ [1] V Φ = (2 σ κ + 3 λ ) (cid:0) V [1 · X V · ∂ X + V [1 · U V · ∂ U (cid:1) Φ + ( λ + σ κ ) C , (B.13)with B = σ κ (cid:0) X V [1 · ∂ X V · ∂ U + V [1 · X V · ∂ U (cid:1) H − σ κ λ (cid:0) X (cid:1) V [1 · ∂ X V · ∂ U Φ , C = h U · ∂ X V [1 · ∂ X V · ∂ U X − U · ∂ X V [1 · X V · ∂ U + X · U V [1 · ∂ X V · ∂ U + V [1 · X V · ∂ X − V [1 · U V · ∂ U i Φ . (B.14)Comparing the above expression with the bracket,[[ V · K , V · K ]] = − σ κ V · M · V , (B.15)and imposing closure one can conclude that the C term has to vanish: λ + σ κ = 0 , (B.16)while the B term corresponds to the contribution of δ [0] [¯ η, ¯ ε ] [2] . References [1] S. Deser and R. I. Nepomechie,
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