Fierz-Pauli theory reloaded: from a theory of a symmetric tensor field to linearized massive gravity
aa r X i v : . [ g r- q c ] F e b Fierz-Pauli theory reloaded: from a theory of a symmet-ric tensor field to linearized massive gravity
Giulio Gambuti ,a and Nicola Maggiore , ,b Rudolf Peierls Centre for Theoretical Physics, Oxford University, U.K. Dipartimento di Fisica, Universit`a di Genova, Italy. Istituto Nazionale di Fisica Nucleare - Sezione di Genova, Italy.
Abstract :
Modifying gravity at large distances by means of a massive graviton may explain theobserved acceleration of the Universe without Dark Energy. The standard paradigm for MassiveGravity is the Fierz-Pauli theory, which, nonetheless, displays well known flaws in its massless limit.The most serious one is represented by the vDVZ discontinuity, which consists in a disagreementbetween the massless limit of the Fierz-Pauli theory and General Relativity. Our approach isbased on a field-theoretical treatment of Massive Gravity: General Relativity, in the weak fieldapproximation, is treated as a gauge theory of a symmetric rank-2 tensor field. This leads us topropose an alternative theory of linearized Massive Gravity, describing five degrees of freedom ofthe graviton, with a good massless limit, without vDVZ discontinuity, and depending on one massparameter only, in agreement with the Fierz-Pauli theory.
Keywords:
Linearized General Relativity; Massive Gravity; Fierz-Pauli Theory; Large DistancesModified Gravity.
E-mail: a [email protected], b [email protected]. Introduction
Motivations
It is an observational fact that the Universe is expanding at an accelerated rate [1, 2]. Withinthe theory of General Relativity (GR), the cosmological constant Λ explains this phenomenon.In a picture of the Universe seen as a perfect fluid, the cosmological constant gives rise to anaccelerated expansion by acting as a constant energy density (called Dark Energy) ρ ∼ Λ M P providing a negative pressure. A fine tuning of the constant Λ enables us to match the expansionpredicted by GR to the one which is observed. The estimate from observations is Λ M P ∼ − [3],where M P is the Planck mass, while the quantum field theoretical prediction on the cosmologicalconstant seen as the vacuum energy density gives Λ M P ∼ × . Unfortunately, these two estimatesdisagree by about 120 orders of magnitude. This huge tension is known as the cosmological constantproblem [4], which motivates alternative descriptions for Dark Energy, with the request that anycosmological model should reproduce an Universe which, at our epoch, is almost perfectly flat andfilled by matter and DE in the ratio of about 3/7, where the DE is effectively approximated by aconstant [5, 6, 7, 8, 9, 10]. Modifying gravity at large distances by means of a massive gravitonis a possible solution to the cosmological constant problem [11]. In fact, the Yukawa potential fora massive field at large distances goes like ∝ r e − αmr , where m is the mass of the field and α adimensional constant. At scales comparable to αm , the exponential factor suppresses the potentialand the strength of interactions as well. This is the reason why long-range forces are associated withmassless bosons and short-range forces with massive bosons. By means of a Yukawa-like potential,the gravitational effect of the vacuum energy density is exponentially suppressed at large scales,thus explaining the disagreement between the quantum field theory calculation and the observedcosmological value. Clearly, this exponential suppression of long-range gravitational interactionsis constrained by experimental evidence and, consequently, the mass of the graviton is subject torestrictive upper limits. Another way of understanding this point is to remember that a mass actsas a momentum space cutoff of interactions, effectively damping the effect of low frequency sources.The vacuum energy density can be considered constant, therefore its contribution to the expansionof the Universe is greatly reduced by the introduction of a mass for the graviton, again confirmingthat in a picture where the graviton is massive the cosmological constant problem might be fixedby a suitable graviton mass. Theories in which the graviton is massive are referred to as MassiveGravity (MG) theories. We nonetheless remark that the introduction of a mass for the gravitonas a potential solution of the cosmological constant problem was a hope more than a decade ago,but early experience with nonlinear MG (see, for example [12]) showed that this expectation is notrealized in the nonlinear theory where a cosmological constant bends spacetime in the same way asin GR. While a large vacuum energy can be canceled by a MG contribution, this still requires finetuning and does not amount to a degravitation mechanism. In some sense MG is a substantialmodification of GR, even if the mass of the graviton is extremely close to zero. In fact, when adding We thank the referee for this remark. mass, the two degrees of freedom (DOF) carried by the massless boson become 2 S + 1 massiveDOF. For the graviton, which is the spin S = 2 representation of the Poincar´e group, the twomassless DOF become five massive ones. Phenomenological Limits
As previously said, long range forces are usually carried by massless bosons because the Yukawaexponential factor limits the range of interactions mediated by massive particles. Nevertheless,choosing a sufficiently small value of the graviton mass, the interactions at scales of the observableUniverse remain essentially identical to those of ordinary gravity, while only interactions at greaterscales are affected. Indeed, the observed accelerated rate of the Universe gives an upper limit to themass of the graviton, which in [13] is shown to be of order 10 − eVc , roughly an order of magnitudesmaller than the Hubble parameter at our epoch. With such a limit, for interactions at distancesmuch smaller than the radius of the observable Universe (which is estimated to be about 46 . × ly[14]), the Yukawa exponential drops to one, and the classical Newton potential ∝ r is recovered, asdesired. Still, this cosmological upper limit becomes irrelevant when the mass is acquired throughthe condensation of some additional scalar field (see [13]). Another constraint on the mass of thegraviton was recently obtained by the LIGO-Virgo collaboration through the analysis of binaryblack holes merger signals, measuring the phase shift between components of different frequenciesof the gravitational waves detected by the interferometers [15]. In this way, the upper limit for themass of the graviton is estimated to be 4 . × − eVc , which is several orders of magnitude higherthan the limit derived in [13] by cosmological considerations. State of the Art
In 1939 Fierz and Pauli (FP) proposed a relativistic theory for a massive particle with arbitraryspin f , described by a symmetric rank- f tensor field [16]. They showed that “in the particular caseof spin two, rest-mass zero, the equations agree in the force-free case with Einstein’s equations forgravitational waves in GR in first approximation; the corresponding group of transformations arisesfrom the infinitesimal coordinate transformations”. This is what is known as the FP theory, whichis considered as the standard paradigm for Linearized MG (LMG). Much later, it has been realizedby van Dam, Veltman and Zakharov [17, 18] that the massless limit of the FP theory exhibits adiscontinuity with GR, known as vDVZ discontinuity. In particular, in [19] it is shown that theinteraction of light with a massive body, like a star for instance, is 25% smaller in the masslesslimit of the FP theory than in GR. Consequently, the angle of deviation of light is also differentby 25% in the two theories, a difference which can be observed by means of the time delay in thegravitational lensing phenomenon. The problems of the FP theory at small scales can be solvedby the Vainshtein mechanism [20], which nonetheless requires a nonlinear modification of the FPtheory. In principle, this measurable disagreement between GR and the FP theory of LMG wouldallow us to distinguish between a strictly vanishing mass and a very small one. Such a tension iscrucial, since GR has been extensively tested and as a consequence we might rule out the FP theory s a theory of LMG. It was also pointed out in [21] that in order for Linearized Gravity (LG) todescribe a pure spin two system, it is necessary to use a particular FP mass term, referred to asthe FP tuning. Otherwise, there will be admixtures of lower spin, in general with negative-energy,called Boulware-Deser ghost. Moreover, Boulware and Deser argued that the ghost will reappearif nonlinear extension of LG are considered. Some possibilities to circumvent these problems havebeen proposed in [22, 23, 24]. A detailed review on MG can be found in [25]. Strategy
The approach adopted in this paper is based on a field-theoretical treatment of LG. We consider LGas a gauge theory of a symmetric rank-2 tensor field, the gauge symmetry being the infinitesimaldiffeomorphism invariance [26]. Before adding a mass term to the action, which breaks the gaugesymmetry, the massless theory should be well defined, in the sense that a well defined partitionfunction must exist. In gauge field theory this is achieved by gauge fixing the action. This procedure(gauge fixing the massless action first, adding a breaking mass term after) leads to a gauge theoryof LMG describing five DOF for the massive graviton, with a good massless limit and without thevDVZ discontinuity. In the FP theory, instead, the mass term is added directly to the invariantaction and effectively plays the double and unnatural role of both mass term and gauge fixing.Basically, this is the reason why the FP theory displays an ill-defined massless limit, which showsitself in a divergent propagator and, physically, in the vDVZ discontinuity with GR. We stress thatany theory of LMG should display a good massless limit since the phenomenological limits on themass of the graviton are such that the mass term must be seen as a small perturbation of masslessLG and, equivalently, as a small breaking of diffeomorphism invariance.
Summary
This paper is organized as follows: in Section 2 the gauge fixed massive theory is presented. Wealso review the FP theory and the problems related to its massless limit which motivate this work,and we remark that, curiously, the original 1939 paper by Fierz and Pauli does not exactly describewhat is generally known as the FP theory. In Section 3 we show, in a gauge independent way, thatthe propagating DOF of the ten components of the rank-2 symmetric tensor describing the massivegraviton are indeed five, and depend on one mass parameter only, in agreement with the FP theory.In Section 4 we compute the propagator of the theory, we show that its massless limit is regularand that the same observable which is used to unveil the vDVZ discontinuity presents an uniquepole in the same mass parameter which characterizes the five DOF of the theory. In Section 5 weshow the absence of the vDVZ discontinuity, for any gauge choice. Our results are summarized anddiscussed in the concluding Section 6.List of acronyms: DOF: Degree(s) of Freedom; EOM: Equation(s) of Motion; FP: Fierz-Pauli;GR: General Relativity; LG: Linearized Gravity; LMG: Linearized Massive Gravity; MG: MassiveGravity; vDVZ: van Dam, Veltman and Zacharov. The massive action
The weak field expansion of GR around the flat Minkowskian background η µν = diag ( − , , ,
1) isgiven by the LG action S LG [ h ] = Z d x h h∂ h − h µν ∂ µ ∂ ν h − h µν ∂ h µν + h µν ∂ ν ∂ ρ h µρ i , (2.1)where h µν ( x ) is a symmetric rank-2 tensor field representing the graviton, and h ( x ) ≡ η µν h µν ( x ) isits trace. The action S LG [ h ] (2.1) is the most general functional invariant under the infinitesimaldiffeomorphism transformation δh µν ( x ) = ∂ µ ξ ν ( x ) + ∂ ν ξ µ ( x ) , (2.2)where ξ µ ( x ) is a local vector parameter. The transformation (2.2) represents the gauge symmetryof the action S LG (2.1). The most general mass term which can be added to the invariant action S LG (2.1), respectingLorentz invariance and power counting, is S m [ h ; m , m ] = 12 Z d x ( m h µν h µν + m h ) , (2.3)where m and m are massive parameters. The presence of a mass term breaks the diffeomorphisminvariance (2.2), as usual in any gauge field theory. It can be shown (see for instance [25]) that theaction S = S LG + S m . (2.4)describes the propagation of five DOF only if m + m = 0 , (2.5)otherwise a sixth ghost mode with negative energy appears [21], and the theory does not describea massive graviton. The choice (2.5) is generally referred to as FP tuning, and the FP theory isdefined by the action S F P [ h ; m ] ≡ S LG [ h ] + S m [ h ; m , − m ] . (2.6)Following [11], we now show that the theory described by the action S F P (2.6) does indeed displayfive DOF. The Equations of Motion (EOM) obtained from (2.6) read δS F P δh µν = ∂ h µν − ∂ α ∂ µ h αν − ∂ α ∂ ν h αµ + η µν ∂ α ∂ β h αβ + ∂ µ ∂ ν h − η µν ∂ h − m ( h µν − η µν h ) = 0 , (2.7)which, saturated with ∂ µ , yield the constraint ∂ µ h µν − ∂ ν h = 0 . (2.8) lugging (2.8) into (2.7) we get ∂ h µν − ∂ µ ∂ ν h − m ( h µν − η µν h ) = 0 . (2.9)Saturating (2.9) with η µν we find h = 0 , (2.10)which, together with (2.8), implies ∂ µ h µν = 0 . (2.11)Therefore, the EOM (2.7) imply the following set of equations( ∂ − m ) h µν ( x ) = 0 (2.12) ∂ µ h µν ( x ) = 0 (2.13) h ( x ) = 0 . (2.14)Eq. (2.12) is the Klein-Gordon equation for the field h µν ( x ), while (2.13) and (2.14) represent fiveconstraints (transversality and tracelessness) which reduce the ten independent components of h µν to five. These five components carry the five massive DOF of the graviton. The propagator of the FP theory (2.6) is G F Pµν,αβ ( p ) = 2 p + m (cid:20)
12 ( P µα P νβ + P να P µβ ) − P µν P αβ (cid:21) , (2.15)where P µν is the transverse massive projector defined as P µν = η µν + p µ p ν m . (2.16)A crucial remark is that the propagator (2.15) exists only thanks to the presence of the mass term(2.3) on the FP point (2.5). As a consequence of this fact, it is apparent from (2.16) that the FPtheory has a divergent massless limit. Moreover, the massless limit of the FP theory is flawed bythe vDVZ discontinuity [17, 18], which basically consists in the fact that the correlator involvingtwo energy-momentum tensors, computed in the FP theory in the limit m →
0, does not matchthe GR prediction. We sketch here the proof (for details see [11, 19]). The gravitational interactionbetween two non relativistic energy-momentum tensors T (1) µν and T (2) µν (which are conserved, i.e.p ν ˜ T (1) µν = p ν ˜ T (2) µν = 0) is described by the introduction in the action of an interaction term of thetype S int = λ Z d x h µν T µν , (2.17)where T µν denotes a generic energy-momentum tensor, coupled to h µν ( x ) through a constant λ ,which we call λ LG and λ F P for LG and FP theory, respectively. Therefore, the interaction strength etween T (1) µν and T (2) µν can be computed by contraction with the propagator of the graviton. In LG,the propagator G µν,αβLG is obtained from the LG action (2.1) after a gauge fixing, as done in [11]. TheFP propagator G µν,αβF P , on the other hand, is given by (2.15). In the non relativistic limit, only the00-components of the energy-momentum tensors are non negligible. Two cases are considered: inthe first, T (1) µν and T (2) µν are associated with massive objects and therefore have non vanishing trace.In the second case, T (1) µν still has a non vanishing trace whereas T (2) µν is traceless, representing, forinstance, electromagnetic radiation ( e.g. light). Concerning the first case, the interaction strengthin LG is λ LG ˜ T (1) µν G µν,αβLG ˜ T (2) αβ = λ LG ˜ T (1)00 ˜ T (2)00 p , (2.18)while, in the second, i.e. when T (2) µν is traceless, we have λ LG ˜ T (1) µν G µν,αβLG ˜ T (2) αβ = λ LG ˜ T (1)00 ˜ T (2)00 p . (2.19)On the other hand, the interaction strengths corresponding to (2.18) and (2.19) obtained using theFP propagator (2.15), in the massless limit m → λ F P ˜ T (1) µν G µν,αβF P ˜ T (2) αβ = 43 λ F P ˜ T (1)00 ˜ T (2)00 p (2.20)and λ F P ˜ T (1) µν G µν,αβF P ˜ T (2) αβ = λ F P ˜ T (1)00 ˜ T (2)00 p . (2.21)The vDVZ discontinuity arises from the observation that it is not possible to choose an unique FPcoupling λ F P which matches both (2.18) with (2.20) and (2.19) with (2.21): the value λ F P = λ LG matches the first pair but not the second. The interaction strength between a massive body and anelectromagnetic wave in the massless limit of the FP theory turns out to be of the LG prediction,hence the discontinuity. In [27] Fierz studied the general relativistic theory of force-free particles with any (integer or half-integer) spin f , described by symmetric “world” tensors A ik...l ( x ) . The starting point in [27]. Thestarting point in [27] is the request that the fields A ik...l ( x ) satisfy the massive wave equation (whichFierz called of the “Schr¨odinger-Gordon type”) ∂ A ik...l = κ A ik...l , (2.22)where κ is a constant with the dimension of the inverse of a length (to which corresponds the mass m = ¯ hκc ). In addition, the fields A ik...l ( x ) are asked to satisfy the additional constraints (called In this subsection we respectfully maintain the notation used in the original paper by Fierz and Pauliin 1939. In particular, the indices ( i, j ) refer to spacetime, contrarily to the rest of the paper, where greekletters are used. secondary conditions”) A ii...l = 0 (2.23) ∂ i A ik...l = 0 . (2.24)As explained in [27], the secondary conditions (2.23) and (2.24) are introduced in order to ensurethat only particles with the spin f and not also those of smaller spins could be assigned to thetensor field, and that if the field A ik...l ( x ) describes a particle with spin f satisfying the massive“Schr¨odinger-Gordon” wave equation (2.22), then the number of linearly independent plane wavesis 2 f + 1, which differ by the orientation of the spin. Shortly after, in [16] Fierz and Pauli appliedthe general formalism described in [27] to the particular case of spin f = 2, simply taken as anexample. Hence, they looked for the theory of a symmetric tensor field A ij ( x ) satisfying the massivewave equation ∂ A ij = κ A ij , (2.25)together with the secondary conditions A ii = 0 (2.26) ∂ i A ij = 0 . (2.27) Imposing the condition of tracelessness (2.26) by hand , they derived the above equations (2.25) and(2.27) from the EOM of the following Lagrangian L orig = κ A ij A ij + ∂ l A ij ∂ l A ij + a ∂ i A ik ∂ j A j k + a κ C + a ∂ i C∂ i C + ∂ i A ij ∂ j C , (2.28)where an additional scalar field C ( x ) is introduced, and a i , i = 1 , , a = − a = a = − . (2.30)In (2.28) the introduction of the scalar field C ( x ) is an artifice which enables one to derive thetransversality condition (2.27). In a more modern language, we would call this scalar field aNakanishi-Lautrup Lagrange multiplier [28, 29] introduced to implement the gauge condition (2.27).It is therefore interesting (and surprising !) to remark that, in the original article [16], Fierz andPauli wrote an action for the graviton which included both a mass term and a gauge fixing term,implemented by means of the Lagrange multiplier C ( x ). It appears that the problem of treatingthe mass term as a gauge fixing was absent in the original formulation. Moreover, the masslesssector of the lagrangian L orig (2.28) does not coincide with the LG action S LG (2.1), since the trace A ii ( x ) (a.k.a. h ( x )) is set to zero a priori . As a consequence, the only mass parameter appearing in(2.28) is the one coupled to A ij A ij (a.k.a. h µν h µν ), which therefore corresponds to m in (2.3). Itseems that, despite the fact that the original FP approach to LMG was realized in term of a gauge xed action, later developments left the gauge fixing term behind, leaving room for the divergentmassless limit and for the vDVZ discontinuity.In this paper we adopt a conservative policy, recovering the original FP approach to LMG, with afew important differences. One of the reasons of the divergent massless limit of the FP theory is that the mass term S m (2.3)serves as gauge fixing as well, in the sense that its presence is necessary to define the propagator(2.15), as shown in Section 2.2. The standard way to proceed in gauge field theory, instead, isfirst to gauge fix the invariant action, in order to have a well defined partition function Z [ J ]. Thislatter generates all the correlation functions of the theory, starting from the 2-points green function,a.k.a. the propagator. Only after having obtained that, the fields of the theory can be given massesthrough various procedures. In LMG this is achieved by adding to the gauge fixed action the massterm S m (2.3) [30]. So, let us first proceed by gauge fixing the invariant action S LG [ h ] (2.1). Thegauge field is represented by a rank-2 symmetric tensor h µν ( x ). Hence, the most general covariantgauge fixing is ∂ µ h µν + κ∂ ν h = 0 , (2.31)which, by means of the usual Faddeev-Popov (ΦΠ) exponentiation [31], yields the gauge fixingaction term S gf [ h ; k, κ ] = − k Z d x h ∂ µ h µν + κ∂ ν h i . (2.32)Notice that the gauge fixing term (2.32) depends on two gauge parameters k and κ , which playdifferent roles. In fact, k determines how the gauge fixing condition (2.31) is enforced. It can beseen as a kind of primary gauge fixing parameter, which corresponds to the standard gauge fixingparameter of Yang-Mills theory: k = 0 corresponds to the Landau gauge, for instance. On the otherhand, the parameter κ fine-tunes the class of gauge fixing identified by k . It plays a secondary role.As an example, the harmonic, or Lorenz, gauge is obtained with the choice κ = − /
2. Hence, itmakes sense to talk about harmonic-Landau gauge, for instance, meaning by that the choice k = 0and κ = − /
2. Once the action S LG [ h ] (2.1) has been gauge fixed by the gauge fixing term (2.32),we can add the mass term (2.3), so that our starting point for a theory of LMG is given by theaction S LMG = S LG + S gf + S m . (2.33)The action (2.33) is the starting, rather than the arrival, point, because the road ahead of us is stilllong. We have indeed to face the problems which affect the FP theory: in particular the masslesslimit and the absence of the vDVZ discontinuity. But, first, we have to deal with the main featureof LMG: the five DOF which must characterize a spin-2 massive particle. Our comparison is the P theory, which reaches this goal with one mass parameter only, because of the FP tuning (2.5).The action (2.33), instead, depends on two masses, for now. This will be done in the next Section.
A realistic theory of MG needs five propagating massive DOF. The easiest way to see this is to noticethat the massive graviton is a spin S = 2 particle, which displays 2 S + 1 independent components.Hence, given that the graviton is described by a symmetric rank-2 tensor h µν ( x ), only five out of itsten components correspond to physical DOF. Therefore, a necessary condition for a gauge theory ofa massive rank-2 symmetric tensor to be promoted to a theory of LMG is to recover the five linearequations represented by the constraints of tracelessness (2.26) (or (2.14)) and of transversality(2.27) (or (2.13)), in order to lower the number of independent components of h µν ( x ) from ten tofive. The realization of this necessary condition, in the framework of a well defined gauge fieldtheory of a symmetric rank-2 tensor, is the main aim of this paper. To reach our goal, we shall nowconsider the EOM of the action (2.33) and we shall manage to restrict the mass parameters m and m to the cases in which we can find enough constraints to ensure the propagation of five massiveDOF. Moreover, in order to deal with a physically consistent theory, we must also require that thepropagating DOF do not depend on the gauge parameters k and κ , allowing instead, of course, adependence on the mass parameters m and m .The action S LMG (2.33) in momentum space reads: S LMG = Z d p ˜ h µν Ω µν,αβ ˜ h αβ , (3.1)where the kinetic operator Ω isΩ µν,αβ = 12 (cid:20) m − (cid:18) κ k (cid:19) p (cid:21) η µν η αβ + 12 (cid:16) − κk (cid:17) ( η µν e αβ + η αβ e µν ) p +12 ( p + m ) I µν,αβ − (cid:18) k (cid:19) ( e µα η νβ + e να η µβ + e µβ η να + e νβ η µα ) p , (3.2) I is the rank-4 tensor identity I µν,ρσ = 12 ( η µρ η νσ + η µσ η νρ ) (3.3)and e µν is the transverse projector e µν = p µ p ν p . (3.4)From the action S LMG (3.1) we get the momentum space EOM δSδ ˜ h µν = − (cid:18) κ k (cid:19) η µν p ˜ h + (cid:16) − κk (cid:17) p µ p ν ˜ h + (cid:16) − κk (cid:17) η µν p α p β ˜ h αβ + p ˜ h µν − (cid:18) k (cid:19) (cid:0) p µ p α ˜ h να + p ν p α ˜ h µα (cid:1) + m ˜ h µν + m η µν ˜ h = 0 . (3.5) n order to study the propagating DOF, we saturate the EOM (3.5) with η µν and e µν (3.4), to get η µν : h ( m + 4 m ) − (cid:16) κk (1 + 4 κ ) (cid:17) p i ˜ h + (cid:18) − k (1 + 4 κ ) (cid:19) p e µν ˜ h µν = 0 (3.6) e µν : h m − κk (1 + κ ) p i ˜ h + (cid:20) m − k (1 + κ ) p (cid:21) e µν ˜ h µν = 0 . (3.7)From these two equations we deduce that, if m = 0 (we will consider the case m = 0 later), theonly solution is ˜ h = 0 e µν ˜ h µν = 0 , (3.8)(3.9)since (3.6) and (3.7) form a homogeneous system of two linear equations which has a non trivialsolution only if the determinant of the coefficients matrix vanishes. For m = 0 this determinantcannot vanish, independently of the choice of κ , k and m . Substituting (3.8) and (3.9) into theEOM (3.5) we get ( p + m )˜ h µν − (cid:18) k (cid:19) (cid:16) p µ p α ˜ h να + p ν p α ˜ h µα (cid:17) = 0 , (3.10)which, saturated with p ν and using again (3.9), yields (cid:18) m − k p (cid:19) p ν ˜ h µν = 0 . (3.11)The above equation is satisfied if p = 2 km (3.12)or p ν ˜ h µν = 0 , (3.13)but the requirement that the physical masses should not depend on the gauge parameters, im-plies that the only allowable solution of (3.11) is (3.13). This, together with (3.8), gives the fiveconstraints ˜ h = 0 p µ ˜ h µν = 0 , (3.14)(3.15)which ensure the propagation of five DOF. Conditions (3.14) and (3.15) inserted into the EOM(3.5) give the massive propagation of h µν ( p + m )˜ h µν = 0 , (3.16)which, interestingly, does not depend on the mass parameter m . If, on the other hand, m = 0,the two equations (3.6) and (3.7) have non trivial solutions only if κ = − k = − , (3.17) hich, plugged back into (3.6) and (3.7) with m = 0, yield h = 0 . (3.18)Substituting m = 0 and h = 0 in the EOM (3.5) we notice that all the dependence on the massparameters vanishes. The resulting EOM would therefore describe the propagation of a masslessfield or, alternatively, the propagation of a field with a mass dependent on the gauge parameters k and κ . Both cases do not represent acceptable descriptions of a LMG theory, therefore we concludethat it must be m = 0 . (3.19)To summarize, we showed that the gauge fixed action S LMG (2.33), with the mass term S m (2.3),displays five DOF, provided that m = 0, for any value of m and for arbitrary gauge parameters k and κ . The momentum space propagator G µν,αβ ( p ) is defined by the conditionΩ µν αβ G αβ,ρσ = I µν,ρσ , (4.1)where Ω µν,αβ ( p ) is the kinetic operator introduced in (3.2) and I µν,ρσ is the rank-4 tensor identity(3.3). In order to find G µν,αβ ( p ) it is convenient to introduce the rank-2 projectors e µν (3.4) and d µν ≡ η µν − e µν (4.2)which are idempotent and orthogonal e µλ e λµ = e µν , d µλ d λν = d µν , e µλ d λν = 0 . (4.3)With these rank-2 projectors we can construct a basis formed by five rank-4 tensors which wecollectively denote X µν,αβ ≡ ( A, B, C, D, E ) µν,αβ (4.4)with the symmetry properties X µν,αβ = X νµ,αβ = X µν,βα = X αβ,µν . (4.5) n terms of the projectors e µν and d µν the operators X µν,αβ read A µν,αβ = d µν d αβ B µν,αβ = e µν e αβ (4.7) C µν,αβ = 12 (cid:18) d µα d νβ + d µβ d να − d µν d αβ (cid:19) (4.8) D µν,αβ = 12 ( d µα e νβ + d µβ e να + e µα d νβ + e µβ d να ) (4.9) E µν,αβ = η µν η αβ , (4.10)and have the following properties: • decomposition of the identity I µν,αβ (3.3) : A µν,αβ + B µν,αβ + C µν,αβ + D µν,αβ = I µν,αβ ; (4.11) • idempotency : X ρσµν X ρσ,αβ = X µν,αβ ; (4.12) • orthogonality of A , B , C and D : X µν,αβ X ′ αβ ρσ = 0 if ( X, X ′ ) = E and X = X ′ ; (4.13) • contractions with E : A µν,αβ E αβ ρσ = d µν η ρσ B µν,αβ E αβ ρσ = e µν η ρσ C µν,αβ E αβ ρσ = D µν,αβ E αβ ρσ = 0 . (4.16)The kinetic operator Ω µν,αβ ( p ) (3.2) can be written in terms of the rank-4 projectors (4.6)-(4.10) :Ω µν,αβ = tA µν,αβ + uB µν,αβ + vC µν,αβ + zD µν,αβ + wE µν,αβ , (4.17)where, after a lenghty but straightforward calculation, the coefficients are given by = (cid:18) κ k − (cid:19) p + 12 m (4.18) u = − k (2 κ + 1) p + 12 m (4.19) v = 12 ( p + m ) (4.20) z = − k p + 12 m (4.21) w = − κk (1 + κ ) p + 2 m . (4.22)Similarly, we can expand the propagator G µν,αβ ( p ) : G µν,αβ = ˆ tA µν,αβ + ˆ uB µν,αβ + ˆ vC µν,αβ + ˆ zD µν,αβ + ˆ wE µν,αβ , (4.23)where ˆ t , ˆ u , ˆ v , ˆ z and ˆ w are functions of the momentum p and depend on the gauge parameters k and κ appearing in S gf (2.32) and on the masses m and m of the mass term S m (2.3). Solvingthe defining equation (4.1), we find :ˆ t = 2(1 + κ )(1 + 4 κ ) p − k ( m + 4 m )DN( m , m , k, κ, p ) (4.24)ˆ u = 2 (cid:2) κ (1 + 4 κ ) + 2 k (cid:3) p − k ( m + 4 m )DN( m , m , k, κ, p ) (4.25)ˆ v = 2 p + m (4.26)ˆ z = − kp − km (4.27)ˆ w = 8 km − κ (1 + κ ) p DN( m , m , k, κ, p ) , (4.28)where we used the shorthand notation for the denominatorDN( m , m , k, κ, p ) ≡− κ ) p + h (1 + 2 κ + 4 κ + 2 k ) m + (3 + 2 k ) m i p − km m − km . (4.29)The propagator (4.23) displays poles that depend on the gauge parameters k and κ and might evenbe tachyonic. This does not come as a surprise, given that the theory describes the dynamics of arank-2 symmetric tensor field, and the pole structure of its propagator is more complicated than the sual scalar, spinor or vector cases. Neither it should be seen as a problem, since in the previousSection we proved that only five of the ten components of h µν represent independent DOF, whichsatisfy the massive wave equation of the Klein-Gordon type (3.16) which depends on the mass m only, in agreement with the FP theory. Hence, looking at the whole propagator is neither helpful norcorrect in order to identify the physical pole in this case, since we may allow for non physical poleslocated in non physical sectors of the propagator. Rather, what should be done in order to select thephysical pole, is to look to the observables related to the propagator (4.23). An important exampleis the one already considered in the analysis of the vDVZ discontinuity: the scattering amplitude oflight and a massive object, mediated by the gravitational interaction, which is responsible for theobserved time delay in gravitational lensing. Formally, this observable can be traced back to themore general interaction amplitude of two conserved energy-momentum tensors T (1) µν and T (2) µν , ofwhich the one corresponding to light (which in the following we choose to be T (2) µν ) is traceless :˜ T (1) µν G µν,αβ ˜ T (2) αβ . (4.30)Substituting the propagator (4.23) into (4.30), we get˜ T (1) µν (cid:16) ˆ tA µν,αβ + ˆ uB µν,αβ + ˆ vC µν,αβ + ˆ zD µν,αβ + ˆ wE µν,αβ (cid:17) ˜ T (2) αβ . (4.31)The above expression contains contractions of the tensor projectors (4.6)-(4.10) with ˜ T (1) µν and ˜ T (2) µν .These are greatly simplified thanks to the fact that the energy-momentum tensors are conserved: p ν ˜ T (1) µν = p ν ˜ T (2) µν = 0. In particular:˜ T (1) µν A µν,αβ ˜ T (2) αβ = 13 ˜ T (1) µν η µν η αβ ˜ T (2) αβ (4.32)˜ T (1) µν B µν,αβ ˜ T (2) αβ = 0 (4.33)˜ T (1) µν C µν,αβ ˜ T (2) αβ = 12 ˜ T (1) µν ( η µα η νβ + η µβ η να − η µν η αβ ) ˜ T (2) αβ (4.34)˜ T (1) µν D µν,αβ ˜ T (2) αβ = 0 (4.35)˜ T (1) µν E µν,αβ ˜ T (2) αβ = 14 ˜ T (1) µν η µν η αβ ˜ T (2) αβ . (4.36)Eqs. (4.32)-(4.36) can be further simplified by using the fact that ˜ T (2) µν is traceless ( η µν ˜ T (2) µν = 0),which means that the only non-vanishing contraction left is˜ T (1) µν C µν,αβ ˜ T (2) αβ = 12 ˜ T (1) µν ( η µα η νβ + η µβ η να ) ˜ T (2) αβ . (4.37)Therefore the scattering amplitude (4.30) reduces to˜ T (1) µν (cid:16) ˆ v I µν,αβ (cid:17) ˜ T (2) αβ , (4.38) ccording to which only the pole contained in the coefficient ˆ v (4.26) plays a physical role. Reas-suringly, that pole is p = − m , (4.39)which confirms that the theory describes a graviton with mass m , in agreement with the massivewave equation (3.16), which was obtained by studying the EOM deriving from the action S LMG (2.33), and also with the FP theory, (2.12). Notice that the same argument may be repeatedto show that also the scattering amplitude between two radiation-like objects, both described byconserved and traceless energy-momentum tensors, isolates (4.39) as the unique physical pole of thepropagator (4.23).
In Section 2.2 we reviewed the vDVZ discontinuity, which concerns the mismatch between thegravitational interaction of two energy-momentum tensors in the pure massless theory described bythe action S LG (2.1) and in the massless limit of FP theory S F P (2.6). In [32] it has been shownthat in a particular gauge (namely the k = − , κ = − harmonic one) the theory described bythe action S LMG (2.33) is free of the vDVZ discontinuity. In this Section we generalize the resultof [32] to all possible gauge choices, i.e. for every value of the gauge parameters k and κ . Withthe propagator G µν,αβ (4.23) derived from the gauge fixed action S LMG (2.33), hence for generic k and κ gauge parameters, we can compute the interaction amplitudes between two non-relativisticconserved energy-momentum tensors T (1) µν and T (2) µν and compare them to the corresponding LGamplitudes, (2.18) and (2.19). As explained in Section 2.2, the interaction of the graviton fieldwith an external energy-momentum tensor is obtained by means of the interaction term S int (2.17).To distinguish the gauge fixed theory S LMG (2.33) from the LG and FP theories, we denote itscoupling constant with λ LMG . Following the steps described in [19], if neither T (1) µν nor T (2) µν aretraceless, which corresponds to the scattering between two massive bodies, the resulting amplitudein the massless limit ( m , m ) → λ LMG ˜ T (1) µν G µν,αβ ˜ T (2) αβ = λ LMG ˜ T (1)00 ˜ T (2)00 p , (5.1)whereas, if T (2) µν is traceless, corresponding to the scattering between light and a massive body, likein the gravitational lensing, we have λ LMG ˜ T (1) µν G µν,αβ ˜ T (2) αβ = λ LMG ˜ T (1)00 ˜ T (2)00 p . (5.2)It is straightforward to see that if we set λ LMG = λ LG (5.3)the LG amplitudes (2.18) and (2.19) exactly match (5.1) and (5.2) respectively, which proves theabsence of the vDVZ discontinuity in a gauge independent way . We stress that, although the massive ropagator G µν,αβ (4.23) depends on the gauge parameters k and κ through the coefficients (4.24)-(4.28), the massless limit of the interaction amplitudes (5.1) and (5.2) is gauge independent andcoincides with the LG prediction. This is related to the fact that the massless limit of the action S LMG (2.33) is the LG gauge fixed action S LG + S gf , (5.4)where S LG and S gf are given by (2.1) and (2.32), respectively. Hence, in the massless limit, thepropagators of the two theories coincide. This is the basic reason of the absence of the vDVZdiscontinuity. In other words, the vDVZ discontinuity is a direct consequence of the structure ofthe FP action, where the mass term S m (2.3) is added directly to the invariant action S LG (2.1),therefore acting effectively as a gauge fixing term. Without the mass term, the invariant action doesnot have a propagator. Therefore, the mass parameters, which also play the role of gauge fixingparameters, cannot be physical. In the massless limit, the FP theory is not dynamical ( i.e. doesnot have a propagator), and therefore it is not surprising that a discontinuity arises. Throughoutthis paper, we repeatedly claimed that the vDVZ discontinuity may be related to the nature ofthe FP mass term as a gauge fixing term. The massive gravity propagator in (2.15) is not welldefined in the zero mass limit due to the lack of gauge fixing in the massless theory. However, thevDZV discontinuity appears in the analysis of the propagator contracted with energy-momentumtensors like in (4.30). This quantity is gauge invariant in linearized GR (for conserved sources).Hence one might expect that the ill defined terms in the zero mass limit of massive FP propagatordrop out of equations like (4.30) which contribute to the vDVZ discontinuity (this quantity indeedremains finite in the zero mass limit). The apparent contradiction between this observation andthe explanation provided in this paper is explained by observing that when we say that the massterm S m (2.3) at the FP point (2.5) has the role of a gauge fixing in the FP theory, we are onlyreferring to the fact that it enables us to invert the quadratic part of the action. In fact, not allquadratic terms which make it possible to find a propagator are also legitimate gauge fixing terms.The gauge fixing procedure is much more than this. Indeed, it consists in limiting the functionalintegral in order to choose one representative for each gauge orbit (modulo Gribov copies). This isrealized through the highly non trivial ΦΠ trick. In the case of an abelian theory such as linearizedgravity, where the ghost sector decouples, the ΦΠ gauge fixing is realized by adding to the actionthe square of a functional F µ [ h αβ ], where F must carry the same number of Lorentz indices as thegauge parameter of the theory, defined in our case in (2.2). Since the mass term cannot be writtenas the square of any such functional, it follows that S m is not a true gauge fixing term and thereforeno contradiction arises when comparing the linearized GR and FP propagators. In this paper we presented a gauge field theory of a massive symmetric tensor field describing amassive spin-2 particle. We summarize here the main features which allow to interpret this theory s an alternative to the FP theory of LMG.1. finite massless limit The theory described by the action S LMG (2.33) yields the propagator G µν,αβ (4.23) which isregular in the limits ( m , m ) → five massive DOF In Section 3 we showed that only five of the ten components of the massive symmetric tensor h µν ( x ) actually represent DOF, as required for a massive graviton . The remarkable fact isthat the mass associated with the propagation of the five DOF is m , while the value of m is irrelevant, under this respect. Although obtained in a quite different way, this result is inagreement with the FP theory, which is characterized by one mass parameter only ;3. gauge independence In gauge field theory, any claim concerning physical properties should not depend on a par-ticular gauge choice. In our case, this requirement is met in the determination of the physicalDOF of the theory: none of the steps leading to the five constraints represented by (3.14)and (3.15) rely on particular values of the gauge parameters k and κ ;4. continuity with LG Any candidate theory for LMG must display a finite massless limit, which implies a nondivergent propagator, and, in this limit, it should also provide physical predictions in agree-ment with GR. Physically, LMG should represent only a small correction of GR, thereforethe mass term (2.3) should be seen as a perturbation. In particular, the effects of a mass ofthe graviton should become relevant only at very large distances, while at smaller scales GRpredictions must be restored. As discussed in Section 2.2, a serious flaw of the FP theory ofLMG is the vDVZ discontinuity [17, 18]. This well known issue is fixed by the Vainshteinmechanism [11, 20, 25], which recovers the non linearities of GR in order to shield the effectof the extra scalar DOF introduced by the St¨uckelberg formalism [33] in the massless limitof the FP theory. Our gauge fixed theory of LMG restores the continuity with GR in a morenatural way, without the introduction of additional fields, cutoffs or non linearities of thetheory. In [32] it has been shown that in the particular ( k = κ = − ) harmonic gauge, thetheory described by the action S LMG (2.33) does not display the vDVZ discontinuity. InSection 5 this result has been generalized to any gauge choice. The fact that the gauge fixedmassive theory described by the action S LMG (2.33) is not affected by the vDVZ discontinuitywith GR encourages to believe that the way of approaching LMG presented in this paper iscorrect. Moreover, although the absence of the vDVZ discontinuity is a specific test and itdoes not prove in general that every measurable quantity is continuous with GR, the way thetheory itself was constructed might imply this result. In fact, the evaluation of any observ-able in quantum field theory needs a gauge fixing in order to define a propagator, which isguaranteed by the ΦΠ procedure, adopted in this paper. The key feature of our approach isthat the massive action S LMG (2.33) in the zero-mass limit becomes exactly the ΦΠ gauge xed LG action, i.e. the one we would have used to calculate the propagator and every otherquantity in massless LG as well. Therefore, while it does not come as a surprise that thegravitational coupling between non relativistic matter and light turns out to be continuouswith GR, we may also infer that every other observable is, indeed, continuous with LG, inthe massless limit. Moreover, the striking difference between the FP theory of LMG, wherethe vDVZ discontinuity is present, and the massive ΦΠ gauge fixed theory described in thispaper, where it is absent, gives evidence of the fact that, as anticipated, the FP theory is nota sub-case of our theory despite the fact that our approach contains the FP tuning (2.5) ;5. role of the mass parameters We stress as a remarkable fact that, although by means of a completely different approach, werecover here the main result of the FP theory, i.e. that only the mass parameter m associatedto h µν h µν in the action S m (2.3) appears in the propagation of the physical DOF through themassive wave equation (3.16), which is the starting request (2.25) of the original FP theory[16]. Therefore, the mass of the graviton should be identified by m . Acknowledgments
We gratefully acknowledge Gianluca Gemme for his phenomenological hints and enlightening ob-servations.
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