Particle Content of Quadratic and f( R μνσρ ) Theories in (A)dS
aa r X i v : . [ h e p - t h ] M a y Particle Content of Quadratic and f ( R µνσρ ) Theories in ( A ) dS Bayram Tekin ∗ Department of Physics,Middle East Technical University, 06800, Ankara, Turkey (Dated: October 14, 2018)We perform a complete decoupling of the degrees of freedom of quadratic gravity and the generic f ( R µνσρ ) theory about any one of their possible vacua, i.e. maximally symmetric solution, find themasses of the spin-2 and spin-0 modes in explicit forms. I. INTRODUCTION
The problem we shall address is simple to state: what is the perturbative particle spectrum of the generic gravitytheory defined by the action I = Z d n x √− g f ( g µν , R ρσµν ) , (1)about any one of its possible maximally symmetric solution? Here, f is assumed to be an analytic function of itsarguments, the inverse metric and the Riemann tensor. We shall consider spacetimes dimensions with n ≥ n = 2 case as a pure f ( R ) theory can also be included in the discussion with a redefinition of the cosmologicalconstant to appear below. The action is assumed to be diffeomorphism invariant (at least up to a boundary term).Depending on the powers of the Riemann tensor, the theory has generically many maximally symmetric solutionswhich can be found once the function f is given. For example, if the highest power is N , there are generically N vacua,modulo the assumption that the parameters of the theory satisfy certain constraints so that the effective cosmologicalconstant of these vacua are real. In any case, for the discussion to follow, all we need is that the theory has at leastone maximally symmetric solution, (anti)-de Sitter (A)dS spacetime, with an effective cosmological constant Λ whichis generically nonzero.The actual identification of the particle content with explicit expressions for the masses in (A)dS is easier said thandone as we shall work out in this work. The particle content of quadratic gravity in (A)dS will play a major role here.It turns out that, rather surprisingly, even though quadratic gravity has been studied for a long time-it is almost asold as General Relativity!- its full perturbative particle content in (A)dS with explicit expressions for the masses havenot been found. Stelle, in his groundbreaking works, [1, 2] gave the masses in four dimensional flat backgrounds. Inthe next section more discussion on the literature will be given. In any case, we will need the particle spectrum ofquadratic theory to answer the question posed above. The connection of the particle content of quadratic gravity andthe theory (1) will be clear in a moment. But first let us briefly argue why one would be interested in this theory.By now, it is no secret that General Relativity needs to be replaced by a, quantum-corrected, interim the-ory, below the Planck scale, with a higher derivative theory, in general, having an action of the form I = R d n x p − | g | f ( g µν , R µνσ,ρ , ∇ R µνσρ , ... ) with many powers of the Riemann tensor, its covariant derivatives and contrac-tions in a (most probably) diffeormophism invariant way. In addition, there might, of course, appear non-minimallycoupled fields, especially scalar fields, directly taking part in gravitation, ruining the equivalence principle. For thiscase we have not much to say here, but assuming that gravity is solely described by a classical pseudo-Riemannianspacetime which solves the field equations coming from a generic action, we can inquire the particle content of thetheory and the stability of a given solution. More specifically, one is usually interested in the linear stability of themaximally symmetric critical metrics ¯ g µν (flat, de Sitter or anti- de Sitter spaces) which are the potential vacua inthe absence of sources. As any physically viable theory should have a stable vacuum, this puts constraints on theform of possible low energy quantum gravity theories.Moreover, recently, in [3, 4], we have answered the following question in the affirmative : can one construct higherorder metric-based gravity theories that has only a single massless spin-2 excitation (no other local degrees of freedom)and a unique viable vacuum just like Einstein’s theory ? The theory obtained in these works is of the Born-Infeldtype as the uniqueness of the viable vacuum is a very strong condition. More generally, one can search for higherderivative metric-based theories with only a massless spin-2 graviton in their spectrum about (A)dS vacua. Then, ∗ Electronic address: [email protected] one cannot have derivatives of the Riemann tensor in the action, since generically these will yield extra degrees offreedom. Therefore, in this work we shall stick to the action of the form given in (1) whose field equations are stillfourth order and so generically have a massless spin-2, a massive spin-2 and a massive spin-0 excitation, just likequadratic gravity. The connection between the spectra and vacua of quadratic gravity and (1) are summarized inSection III.
II. FULL PARTICLE SPECTRUM OF QUADRATIC GRAVITY IN (A)DS
As stated above, to identify the particle spectrum ( i.e. calculate the masses) of generic f ( g µν , R ρσµν ) theory, thebest way is to first carry out the procedure for the quadratic gravity with the action I = Z d n x √− g (cid:20) κ ( R − ) + αR + βR µν + γ (cid:0) R µνσρ − R µν + R (cid:1)(cid:21) , (2)whose source-free field equations read [5]1 κ (cid:18) R µν − g µν R + Λ g µν (cid:19) + 2 αR (cid:18) R µν − g µν R (cid:19) + (2 α + β ) ( g µν (cid:3) − ∇ µ ∇ ν ) R +2 γ (cid:20) RR µν − R µσνρ R σρ + R µσρτ R σρτν − R µσ R σν − g µν (cid:0) R τλσρ − R σρ + R (cid:1)(cid:21) + β (cid:3) (cid:18) R µν − g µν R (cid:19) + 2 β (cid:18) R µσνρ − g µν R σρ (cid:19) R σρ = 0 . (3)We work with the mostly plus signature, therefore Λ > g µν denote amaximally symmetric solution whose curvatures are defined as¯ R µρνσ = 2Λ( n − n − (cid:0) ¯ g µν ¯ g ρσ − ¯ g µσ ¯ g ρν (cid:1) , ¯ R µν = 2Λ n − g µν , ¯ R = 2 n Λ n − . (4)Then for a vacuum, the field equations reduce to a quadratic equation that determines the effective cosmologicalconstant: Λ − Λ κ + k Λ = 0 , k ≡ ( nα + β ) ( n − n − + γ ( n −
3) ( n − n −
1) ( n − . (5)As noted in the Introduction, for Λ to be real, there is a constraint on the parameters of the theory, but this will notbe relevant to the ensuing discussion. We assume there is an (A)dS vacuum. Then considering generic perturbationsabout this vacuum defined as h µν ≡ g µν − ¯ g µν , one can show that the field equations at the linear order reduce to [5] c G Lµν + (2 α + β ) (cid:18) ¯ g µν ¯ (cid:3) − ¯ ∇ µ ¯ ∇ ν + 2Λ n − g µν (cid:19) R L + β (cid:18) ¯ (cid:3) G Lµν − n − g µν R L (cid:19) = 0 , (6)where the constant c in-front of the linearized Einstein tensor reads c ≡ κ + 4Λ nn − α + 4Λ n − β + 4Λ ( n −
3) ( n − n −
1) ( n − γ. (7)One cautionary remark is apt here: even though 1 /c may appear like the effective Newton’s constant in this theory, aswe shall see in a moment in the action formulation, this is not really correct. A further term will be added to c whichwill then yield the effective Newton’s constant of the theory. All the information about the particle content is in thelinearized fourth order equation (6), but it is clear that this is a complicated coupled equation of physical degrees offreedom as well as gauge degrees of freedom. We have to decouple the physical modes. The linearized version of thecosmological Einstein tensor is defined as G Lµν ≡ (cid:16) R µν − g µν R + Λ g µν (cid:17) L which reads G Lµν = R Lµν −
12 ¯ g µν R L − n − h µν . where the linearized Ricci tensor R Lµν and scalar curvature R L = ( g µν R µν ) L are given as R Lµν = 12 (cid:16) ¯ ∇ σ ¯ ∇ µ h νσ + ¯ ∇ σ ¯ ∇ ν h µσ − ¯ (cid:3) h µν − ¯ ∇ µ ¯ ∇ ν h (cid:17) , R L = − ¯ (cid:3) h + ¯ ∇ σ ¯ ∇ µ h σµ − n − h. Let us say a few words about what is already known in this theory: for flat spacetime, as we noted in four dimensions,8 degrees of freedom were identified in [1]. For the (A)dS backgrounds, in [6], the scattering amplitude at tree-levelbetween two sources in this theory (augmented with a Fierz-Pauli mass term) was computed from which in principleone can read the masses from the poles, but as the Lichnerowicz Laplacian is used in that work, it is not easy todirectly see all the masses, even though the computation is useful for unitarity and discontinuity analysis. In [7],for n = 3, a specific combination of the quadratic terms (8 α + 3 β = 0) was considered which has a massive spin-2excitation and the resulting theory is ”New Massive Gravity” (NMG). In [8], Einstein-Hilbert piece is amputated fromthe NMG and the resulting theory has a massless spin-2 excitation. The most general version of quadratic gravity in n = 3 was considered in [9]: the mass spectrum was found after a long computation. Specifically, h µν was decomposedinto its irreducible parts and the massive spin-0 and massless spin-2 modes were decoupled to calculate the masses. In[10] and [11], four and n dimensional versions of this theory for tuned parameters that eliminate the massive modeswere studied, hence the ”Critical Gravity” was obtained. Of course what makes the computation rather tricky is thefact that the background spacetime is not flat and the curvature contributes to the masses of the particles. Here weremedy the gap on this and give a relatively concise derivation of the spectrum in n dimensions for generic values ofthe parameters in (A)dS.Directly extending the quadratic gravity action up to O ( h ) is a very cumbersome task and it is rather a longexercise to put the final result in an explicitly gauge invariant form. Therefore, the best way to proceed is to use”inverse” calculus of variations and get the action that yields the linearized field equations (6): let that action be I ( h ) = R d n x √− g L . Then the second order Lagrangian is obtained by first multiplying the linearized field equationsby − h µν and integrating the result over spacetime to arrive at (after dropping the boundary terms) L = − (cid:16) c + 2Λ β ( n − n − (cid:17) h µν G Lµν + β G Lµν G µνL + (cid:16) α + β (4 − n )4 (cid:17) R L . (8)Up to a boundary term, this O ( h ) action is invariant under background diffeomorphisms of the form δ ξ h µν =¯ ∇ µ ξ ν + ¯ ∇ ν ξ µ since both the linearized Einsten tensor and the linearized curvature scalar are gauge-invariant. Onecan fix the gauge at this stage, but we shall proceed without a choice of gauge. The minus factor in front of theEinsteinian piece is important as it is chosen to give the correct kinetic energy for the massless spin-2 graviton. Orequivalently, if we couple the theory to matter, that is the correct sign, from which we can also identify the effectiveNewton’s constant as 1 κ eff ≡ κ + 4Λ( nα + β ) n − n −
3) ( n − n −
1) ( n − γ, (9)which has the earlier noted shift from the constant c . In what follows we shall make frequent use of integration byparts an the ”Hermitian” property of the operator that defined as G Lµν ≡ ( O h ) µν . So, not to clutter the notation, wework with Lagrangian but drop the boundary terms. To be able to identify the physical modes, let us introduce twoauxiliary fields f µν and ϕ to recast the Lagrangian as L = − κ eff (cid:16) h µν + f µν (cid:17) G Lµν ( h ) − βκ (cid:16) f µν f µν − f (cid:17) + ϕR L − b ϕ , (10)where f ≡ ¯ g µν f µν and the constant b is found as b ≡ n − α ( n −
1) + βn . (11)So, integrating out the auxiliary fields in (10) gives us back our original action (8). To get rid of the ϕR L term let usdefine a new field ˜ f µν as f µν = ˜ f µν − κ eff n − ϕ ¯ g µν , f = ˜ f − nκ eff n − ϕ, (12)which then reduces (10) to L = − κ eff (cid:16) h µν + ˜ f µν (cid:17) G Lµν ( h ) − βκ (cid:16) ˜ f µν ˜ f µν − ˜ f (cid:17) − n − n − βκ eff ϕ ˜ f + (cid:16) n ( n − β ( n − − b (cid:17) ϕ . (13)As ϕ appears without derivatives, we can integrate it out to arrive at L = − κ eff (cid:16) h µν + ˜ f µν (cid:17) G Lµν ( h ) − βκ (cid:16) ˜ f µν ˜ f µν − ˜ f (cid:17) − βκ ξ ˜ f , (14)where the constant ξ is given as ξ ≡ α ( n −
1) + βn αn + β ) . (15)A further field definition is needed to decouple h µν and ˜ f µν . By inspection one observes that the following definitiondoes the job h µν ≡ ˜ h µν − ˜ f µν . (16)With this our second order Lagrangian reduces to the decoupled form L = − κ eff h µν G Lµν ( h ) + 12 κ eff f µν G Lµν ( f ) − βκ (cid:16) f µν f µν − f (cid:17) − βκ ξf , (17)where we removed all the tildes for notational simplicity. As the first term is just the linearized Einstein theory withan effective Newton’s constant, as long as κ eff >
0, it describes a massless unitary spin-2 excitation, which is theEinsteinian mode, that is the massless graviton. Immediately, it is also clear that the second term has the wrong sign,so there will be a massive ghost. It is also clear that when ξ = 0, the f µν part is just the Fierz-Pauli massive gravity(with the wrong kinetic sign of course). For ξ = 0 as in our case, there is an additional massive mode which we haveto decouple. To be able to read the masses, let us vary the action with respect to f µν to get G Lµν ( f ) − βκ eff (cid:16) f µν − ¯ g µν f (cid:17) − βκ eff ξ ¯ g µν f = 0 , (18)whose trace yields R L ( f ) + 1( n − βκ eff (cid:16) − n + nξ (cid:17) f = 0 . (19) G Lµν ( f ) satisfies the background Bianchi identity, hence double-divergence of (18) yields( ξ −
1) ¯ (cid:3) f + ¯ ∇ µ ¯ ∇ ν f µν = 0 . (20)Making use of this in the trace equation and using the definition of R L , one arrives at a massive scalar wave equationsatisfied by the trace of the f field: ξ ¯ (cid:3) + 2Λ n − − − n + nξ ( n − βκ eff ! f = 0 , (21)from which we can read the mass of the scalar mode as m s = − ξ n − − − n + nξ ( n − βκ eff ! , (22)which of course decouples from the spectrum for the Fierz-Pauli tuning ξ = 0. It is then easy to see that the trace-freepart of (18) yields the usual Fierz-Pauli massive graviton with the mass-square m g = − βκ eff . (23)Let us summarize the particle content of n -dimensional quadratic gravity in (A)dS : there is a unitary masslessspin-2 mode, that is the usual graviton, there is a massive spin-zero mode whose mass-square is given as (22) whichshould satisfy the Breitenlohner-Freedman bound in AdS , namely m s ≥ n − n − Λ to be non-tachyonic, and there isa massive spin-2 ghost with the mass-square given as (23). All together in n dimensions the quadratic gravity has n ( n − + ( n +1)( n − + 1 = n ( n −
2) degrees of freedom. As concrete examples let us consider the three and the fourdimensional cases. n = 3 : The masses of the spin-2 and spin-0 modes respectively read m g = − κβ − (cid:18) αβ (cid:19) Λ , m s = 1(8 α + 3 β ) κ − α + β )(8 α + 3 β ) Λ , (24)which are the same as the ones found with the canonical method in [9]. Note that altogether, these are the 3 degreesof freedom in 3 dimensions since there is no massless graviton in the generic theory. On the other hand, the choice8 α + 3 β = 0 leads to the decoupling of the scalar mode, yielding the NMG theory [7] with a massive graviton. n = 4 : The masses of the spin-2 and spin-0 modes respectively read m g = − κβ − (cid:18) αβ (cid:19) Λ , m s = 12(3 α + β ) κ . (25)Together with the massless spin-2 graviton, these modes exhaust the 8 degrees of freedom whose flat space versionsin four dimensions were given by Stelle [1]. It is interesting to note that, four dimensions is rather unique in the sensethat it is the only dimension for which the mass of the scalar field is not shifted due to the cosmological constant.Also, it is clear that for 3 α + β = 0, the scalar mode decouples, which corresponds to the Weyl-square correctedEinstein’s theory. It is a little cumbersome-looking, but it pays to write the masses in generic n -dimensions in termsof the parameters of the Lagrangian: the massive spin-2 mode has the mass-square m g = − βκ −
4Λ ( n − β + αn ) + γ ( n − n − β ( n − n − , (26)while the massive spin-0 has m s = n − κ (4 α ( n −
1) + βn ) + 4Λ( n − (cid:16) ( n − β + αn ) + γ ( n − n − (cid:17) ( n − n − α ( n −
1) + βn ) , (27)from which one can study various specific theories. For example, as n → ∞ , both masses remain intact. For thepure Einstein-Gauss-Bonnet theory, they become infinite and decouple, leaving only the Einsteinian massless modeas expected, since the Einstein-Gauss-Bonnet theory is a second order theory.As we have seen in the above construction, the massive spin-2 mode is a ghost, therefore as long as it is inthe spectrum, the theory is problematic at the linear level. Namely, the vacuum is not stable against the copiousproduction of these states which lower the energy. For the tuned case of letting both masses go to infinity, onearrives at the ”critical gravity” [10, 11] in AdS. But at exactly in this point of the parameter space, there ariseasymptotically non-AdS logarithmic modes [12, 13]. These solutions are of the wave type and they are valid both asexact and as perturbative solutions and as perturbative modes, they are ghosts [14]. There is no consistent truncationof them that yields a nontrivial theory. There is an important digression that we would like to make here: in sometheories, a perturbative solution cannot be obtained from the linearization of an exact solution, a phenomenon called”linearization instability”. If linearization instability exists, one might have a hope of obtaining a consistent theory asthe log-modes can be truncated as is the case in three dimensional chiral gravity [15–17]. This says that the dangerousperturbative log-modes of chiral gravity do not come from the linearization of exact solutions. This is not the casein critical gravity as these modes are also exact solutions. Therefore, for n ≥ β = 0 to avoid themassive spin-2 ghost. Let us now study the generic gravity. III. FULL PARTICLE SPECTRUM OF f ( R µνσρ ) GRAVITY IN (A)DS
The first thing to note is that taking the Lagrangian density as a function of the Riemann tensor with two up andtwo down indices is better as one can do away with the inverse metric: L = f (cid:0) R µνρσ (cid:1) . (28)Now the usual route to the particle spectrum of this theory is again to find the O ( h ) expansion of this action aboutany one of its potential vacua ¯ g µν . But as we have shown in sufficient detail in [18–21], for this purpose and for findingthe vacua of the theory, it is actually best to construct a quadratic action that has the same vacua and the spectrumis this theory. Namely, we need to construct the following action f quad-equal (cid:0) R µνρσ (cid:1) = 1 κ ( R − ) + αR + βR µν R νµ + γ (cid:0) R µνρσ R ρσµν − R µν R νµ + R (cid:1) , (29)whose vacua and degrees of freedom match the theory we want to explore (28). Namely, we have to relate theparameters in this theory to the values of the f function and its derivatives. As explained in detail in the abovequoted works, this can be done by the following Taylor series expansion f quad-equal (cid:0) R µνρσ (cid:1) ≡ X i =0 i ! " ∂ i f∂ ( R µνρσ ) i ¯ R µνρσ (cid:0) R µνρσ − ¯ R µνρσ (cid:1) i . (30)Therefore, given f (cid:0) R µνρσ (cid:1) defining the theory, and denoting the background Riemann tensor as¯ R µνρσ = 2Λ( n − n − (cid:16) δ µρ δ νσ − δ µσ δ νρ (cid:17) , (31)one has to compute the following two derivatives and contractions (cid:20) ∂f∂R µνρσ (cid:21) ¯ R µνρσ R µνρσ ≡ ζR, (32)12 " ∂ f∂R µνρσ ∂R αβλγ ¯ R µνρσ R µνρσ R αβλγ ≡ αR + βR λσ R σλ + γ (cid:0) R µνρσ R ρσµν − R µν R νµ + R (cid:1) , (33)which determine the constants ζ , α , β , γ . The constant ζ appears in the bare Newton’s constant of the equivalentquadratic theory as 1 κ = ζ − (cid:18) n − nα + β ) + 4Λ ( n − n − γ (cid:19) , (34)while the bare cosmological constant of the equivalent theory reads as (see [18–21] for further details)Λ κ = − f (cid:0) ¯ R µνρσ (cid:1) + Λ nn − ζ − n ( n − ( nα + β ) − n ( n − n −
1) ( n − γ. (35)So, these parameters are sufficient to determine the quadratic theory that has the same particle spectrum as the f ( R µνσρ ) theory and since we found the spectrum of the former, we can simply read the spectrum of the latter. Givena generic f ( R µνσρ ), let us summarize the recipe: compute the first and second derivative with respected to R µνσρ anduse (33) to obtain ζ , α , β and γ . Then use (34) to determine κ . Compute f (cid:0) ¯ R µνρσ (cid:1) and use (35) to determine Λ .Now, one can use (5) to to determine the possible effective cosmological constants. Once this is done, the massesgiven in the previous section for quadratic gravity yield the masses of the massive spin-2 and massive spin-0 modesin the generic f ( R µνσρ ) gravity. What is rather remarkable is that one actually has to do 3 basic computations: thevalue of the Lagrangian density, the first and second derivatives of the Lagrangian density with respect to the up-updown-down Riemann tensor evaluated at the (A)dS background. IV. CONCLUSIONS
Using auxiliary fields we have decoupled the free particle spectrum of general quadratic gravity in n dimensionalconstant curvature backgrounds and calculated the masses of the massive spin-2 and and massive spin-0 modes whosespecial forms only have appeared before, even though quadratic gravity has been of interest for along time. Thenfinding a quadratic action that has the same free particle spectrum and vacuum equation as the generic f ( R µνσρ ) gravity,we found the particle spectrum of the latter theory. Therefore once an explicit form of the action is given, no matterhow complicated the action is, as long as it depends on the powers of the Riemann tensor (and its contractions, theRicci tensor and the scalar curvature) our formulas give the masses of the gravitons about the (A)dS backgrounds.Perturbative stability of vacuum of the generic theory is similar to the quadratic case that we have discussed in thetext: the massive spin-2 mode is a ghost hence it should not appear in the spectrum. We have given an example of anon-trivial theory in the Born-Infeld form in [3, 4] which does not also have the spin-0 mode, see also another recentexample [22]. Acknowledgments
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