Helicity Decomposition of Ghost-free Massive Gravity
aa r X i v : . [ h e p - t h ] A ug Helicity Decomposition of Ghost-freeMassive Gravity
Claudia de Rham, , Gregory Gabadadze and Andrew J. Tolley D´epartment de Physique Th´eorique and Center for Astroparticle Physics, Universit´e deGen`eve, 24 Quai E. Ansermet, CH-1211 Gen`eve Department of Physics, Case Western Reserve University, 10900 Euclid Ave, Cleveland,OH 44106, USA Center for Cosmology and Particle Physics, Department of Physics, New York University,NY, 10003, USA
Abstract.
We perform a helicity decomposition in the full Lagrangian of the class of MassiveGravity theories previously proven to be free of the sixth (ghost) degree of freedom via aHamiltonian analysis. We demonstrate, both with and without the use of nonlinear fieldredefinitions, that the scale at which the first interactions of the helicity-zero mode come inis Λ = ( M Pl m ) / , and that this is the same scale at which helicity-zero perturbation theorybreaks down. We show that the number of propagating helicity modes remains five in the fullnonlinear theory with sources. We clarify recent misconceptions in the literature advocatingthe existence of either a ghost or a breakdown of perturbation theory at the significantlylower energy scales, Λ = ( M Pl m ) / or Λ = ( M Pl m ) / , which arose because relevantterms in those calculations were overlooked. As an interesting byproduct of our analysis, weshow that it is possible to derive the St¨uckelberg formalism from the helicity decomposition,without ever invoking diffeomorphism invariance, just from a simple requirement that thekinetic terms of the helicity-two, -one and -zero modes are diagonalized. ontents Λ theory 164.2 Absence of ghosts nonlinearly or around arbitrary backgrounds 174.3 Coupling to matter 184.3.1 A first look at external sources 184.3.2 Consistency Relation for matter 204.3.3 Preserving approximate linearized diffeomorphism invariance 204.3.4 A specific example of consistent coupling to matter 214.4 Consistency of perturbation theory in the Λ theory 22 Λ A → ππ scattering Amplitude 245.2 ππ → ππ scattering Amplitude 255.3 Free theory at the scale Λ Λ and Λ : The true strong coupling scale 28 General Relativity (GR), with or without a cosmological constant, is the unique theory ofa single massless spin-2 field in four dimensions. Consistent with the requirements for amassless spin-2 representation of the Poincar´e group, it only excites the helicity-2 modesof the spin-2 field, while the helicity-1 and 0 modes are simply absent. This statement istrue to all orders in interactions, and is made manifest in the standard formulation of GRby the existence of a local symmetry: nonlinear diffeomorphism invariance. This symmetry– 1 –nsures that any potential additional helicity-1 and helicity-0 modes are pure gauge degreesof freedom.If the spin-2 field acquires a mass, then five polarizations (two helicity-2, two helicity-1and one helicity-0) are necessarily excited in a theory which preserves Lorentz invariance.At the same time, diffeomorphism invariance is broken by the mass term. This means theinteractions in a massive spin-2 theory, both with itself and with matter, are less constrainedthan in a massless one. As a consequence, it has frequently been argued that a sixth ghostlymode would arise in any attempt to define an interacting theory of a massive spin-2 field.The sixth mode shows up as higher derivative interactions for the required five polarizations.For a Fierz-Pauli theory, the mass term is defined such that this sixth mode is absent whenconsidering quadratic fluctuations around flat space-time, [1]. However the same does notnecessarily remain true around an arbitrary background, or at higher orders in perturbations,and until recently it was believed that massive gravity would inexorably excite a sixth (ghost)mode non-linearly. This was the state of affairs until a specific model of massive gravity wasproposed in [2] which generalizes the Fierz-Pauli mass to all orders, [3]. That model wasshown to be free of the ghost degree of freedom, in full generality in the ADM formalism [4]following earlier arguments in [2], and in the St¨uckelberg language both in the decouplinglimit [2, 3] and beyond [5].Massive gravity was recently reanalyzed using a standard helicity decomposition of themassive spin-2 field [6], and despite the above results it was suggested that a ghost, or failureof perturbation theory for the spin-2 field occurred at a surprisingly low scale in which thetheory was already known to be effectively free via the St¨uckelberg analysis and ghost freevia the ADM analysis. In this paper, we reconsider the helicity argument, and show thatwhen all the terms in the Lagrangian are properly accounted for at a given energy scale(including the ones left out in [6]), the massive gravity theory proposed in [2] is free fromboth ghost and low strong-coupling scale problems. We are able to reconfirm all previouslyknown results directly in the helicity variables without performing any field redefinitions andclarify how they are preserved even when considering the coupling to matter. We clarify theconnection between the St¨uckelberg and helicity decompositions and explain why the fieldredefinition between the two is well-defined and remains under perturbative control belowthe scale Λ = ( M Pl m ) / . We also show explicitly how the St¨uckelberg decomposition canbe derived from the helicity one by the need to diagonalize the kinetic terms of the differenthelicity modes around arbitrary backgrounds. These conclusions are in complete consistencywith the results previously obtained in unitary gauge which showed that the theory was ghostfree, [2, 4].Before focusing on this helicity argument, it is worth putting this paper in context andsummarizing the ups and downs in the development of massive gravity. A recent review ofmassive gravity can also be found in [7]. Readers familiar with those works may skip tosection 1.1. We recall here the different ways a generic model of massive gravity has beenargued to contain ghosts, and emphasize how the model presented in [2] evades all these ar-guments. For simplicity, the energy scale Λ n at which a given interaction occurs is expressedas Λ n = ( M Pl m n − ) /n . A. Absence of the Ghost in Unitary gauge :Since a massive spin-2 field has five polarizations, while a massless one only carries two, thelinear Fierz-Pauli theory has a discontinuity in the limit of vanishing mass as was first shown– 2 –y van Dam, Veltman, and Zakharov (vDVZ discontinuity), [8]. This discontinuity wasargued to rule out massive gravity on observational grounds. However, soon after Vainshteinshowed that theories of massive gravity could avoid the vDVZ discontinuity problem due tothe existence of nonlinear interactions, [9]. The question then was whether these nonlinearinteractions could consistently be incorporated into the Fierz-Pauli theory. It was thenrealized that massive gravity could potentially suffer from ghost-like pathologies due to apossible sixth degree of freedom emerging at the non-linear level, as first shown by Boulwareand Deser, who discovered that the lapse N always enters non-linearly for any mass termwhich is a function of the Fierz-Pauli combination f ( h µν − h ) or of the metric determinant,[10]. Whilst the results of Boulware and Deser are correct, they did not exhaust all possibleinteractions for massive gravity theories. Most importantly, they do not account for a subtletywhich allows the theory to maintain a Hamiltonian constraint even when the lapse entersnon-linearly. More precisely, in GR, the lapse N and the shift N i both enter linearly in theHamiltonian, and it is therefore transparent that this theory has a Hamiltonian constraint.In massive gravity, the lapse always enters non-linearly, so differentiating with respect to thelapse does not directly enforce a constraint, but rather an equation through which the lapseitself could in principle be determined. However, to be able to express all the shifts andthe lapse through these equations of motion, the system of corresponding equations shouldbe invertible, or equivalently, the determinant of the Hessian H ab = ∂ L m /∂N a ∂N b (where N a collectively denotes the shift and lapse) should not vanish. In the model of massivegravity proposed in [2], (called “Ghost-free Massive Gravity” in what follows) it has beenshown explicitly that this condition does not hold and the determinant of the Hessian is zero(equivalently it has been shown that the Hamiltonian is linear in the lapse after integrationover the shift), [2, 4]. As a consequence, not all the equations of motion are independent,but instead one of them gives rise to a constraint. This constraint is what projects out thepotential sixth degree of freedom, the so-called Boulware-Deser (BD) ghost. This suggeststhat one should look for a reformulation of the theory in which a non-linearly redefined lapseenforces a constraint from the outset [2, 4].The full unitary gauge ADM analysis performed [4] confirms the absence of ghosts inthe class of models proposed in [2]. However, in this approach the energy scale of the inter-actions of the different helicity modes is not transparent. This is the reason why alternativeapproaches that introduce auxiliary fields compensated by additional gauge symmetries havebeen utilized to analyze the interactions, principally to determine at which scale the helicity-zero mode becomes strongly coupled. The two main approaches are the St¨uckelberg approach,which introduces auxiliary fields compensated by full nonlinear diffeomorphisms, and the he-licity approach which introduces auxiliary fields compensated by only linear diffeomorphismsand an additional U (1) symmetry, described below. We should stress however that since theunitary gauge formulation is ghost free, any apparent ghost that appears via the introductionof auxiliary fields cannot actually be present. B. Absence of the Ghost in the St¨uckelberg language :In 2002, Arkani-Hamed, Georgi and Schwartz (AGS), suggested a complementarily approachto study massive gravity as an effective field theory and introduced four St¨uckelberg fields φ a to make the counting of degrees of freedom transparent, [11], (see also [12]). In this approach,the covariance of the theory is also explicit, but the power of this framework is in its ability– 3 –o identify the relevant degrees of freedom present in the theory, even when they arise atdifferent scales. This is possible by taking a specific limit of the theory in such a way thatthe different degrees of freedom decouple, hence called the “decoupling limit”.(a) In the decoupling limit : In this limit, AGS showed that the ghost could in principlebe pushed beyond the scale Λ , however shortly after, it was argued that the ghostwould inexorably reappear at the scale Λ , [13]. This was later shown not to be themost general answer, and there exists a class of theories (corresponding to Ghost-freeMassive Gravity) for which not only does the decoupling limit occur at the scale Λ ,but no ghost is present at that scale, [2, 3], the decoupling limit if therefore completelyhealthy.(b) Beyond the decoupling limit:
It the full theory the absence of the BD ghost wasshown up to and including the quartic order in nonlinearities in [2], and to all ordersin [4]. However, a analysis of the theory at energy scales beyond Λ indicated theappearance of ( ˙ φ ) which have been interpreted as the revival of the BD ghost atlarger scales, [14, 15], (notice that these are not the only terms appearing at the thatscale and that order).Whilst the presence of such a term is manifest, the conclusions of [14, 15] on theexistence of the BD ghost are not correct as these works do not account for the existenceof a constraint, which removes the ghost, [5]. The fact that ˙ φ comes in non-linearlyin the action does indeed imply that the equation of motion with respect to φ isdynamical as it involves some ¨ φ , however this does not yet imply that all the fourSt¨uckelberg fields are independent dynamical degrees of freedom. This would only bethe case if the determinant of the Hessian H ab = ∂ L m /∂ ˙ φ a ∂ ˙ φ b was nonzero. Howeverit has been shown explicitly that for the specific mass term considered in Ghost-freeMassive Gravity, this condition did not hold and therefore not all four St¨uckelbergfields are independent even beyond the decoupling limit, [5]. This theory thereforepropagates 3 degrees of freedom out of four φ a St¨uckelberg fields on top of the 2 usualtensor polarizations, leading to 5 degrees of freedom; this is the correct counting in theabsence of the BD ghost.The existence of this constraint manifests itself in various ways in classical solutionsdiscussed in [16–19].
C. (Non-)existence of the Ghost in the Helicity language :Finally, despite the proof for the absence of any ghost in the model [2] both in unitary gaugein the ADM language and the St¨uckelberg approach (first in the decoupling limit and thenlater beyond it), it has recently been argued, that the ghost actually manifests itself in thehelicity decomposition, at the scale Λ already at cubic order in perturbations, and thenat the even lower scale Λ at quartic order in perturbations, [6]. Whilst this result is inclear contradiction with the results present in the literature, we here analyze this argumentin more detail and show once again the absence of ghost in the helicity language when allterms in the Lagrangian are properly accounted for (including terms previously left out in [6]).– 4 – .1 Relationship between the Helicity and St¨uckelberg Decompositions Since this paper is concerned with the helicity decomposition, let us clarify the relationshipbetween this and the now familiar St¨uckelberg decomposition. Both decompositions rely onsplitting the massive spin-2 field up into a would-be helicity-2 ˆ h µν , helicity-1 A µ and helicity-0 modes π . In other words, both decompositions express the spin-2 metric perturbation inthe form h µν = M Pl ( g µν − η µν ) = ˆ h µν + 12 m ∂ ( µ A ν ) + 13 m ∂ µ ∂ ν π + D µν , (1.1)where the final piece D µν is chosen to diagonalize the kinetic term of the different helicities toavoid kinetic mixing. In both decompositions the helicity-1 and helicity-0 terms are identifiedin the same way π helicity = π Stueckelberg (1.2) A helicity µ = A Stueckelberg µ . (1.3)The difference lies entirely in the final term D µν , or equivalently in the identification of thehelicity-2 mode. In the helicity decomposition this is chosen to diagonalize only the freetheory (Fierz-Pauli) kinetic term defined around Minkowski spacetime D helicity µν = 16 πη µν . (1.4)By contrast, in the St¨uckelberg decomposition, one chooses D µν to diagonalize the kineticmixing term between the different helicities for all terms that arise at energy scales below Λ around an arbitrary background . Direct calculation (performed below) shows that thisgives D Stueckelberg µν = 16 πη µν + M Pl (cid:18) Ψ µα Ψ να − η µν − m ∂ ( µ A ν ) − m ∂ µ ∂ ν π (cid:19) (1.5)= 16 η µν π + 136Λ ∂ µ ∂ α π∂ ν ∂ α π + 112Λ ∂ ( µ A α ∂ ν ) ∂ α π + 14Λ ∂ µ A α ∂ ν A α , where Ψ µν = η µν + M Pl m ∂ µ A ν + M Pl m ∂ µ ∂ ν π . The St¨uckelberg decomposition does notcompletely diagonalize the kinetic terms, it leaves a mixing that arises at the scale Λ . Onecould go further and diagonalize at these scales, however it is known that for the generic twoparameter allowed massive gravity models the diagonalization needed is nonlocal at thesescales (although it is local for special cases).In the St¨uckelberg decomposition, the diagonalization term D µν is defined nonlinearlyin the helicity fields. More precisely ( D Stueckelberg µν − D helicity µν ) is quadratic in the fields. Thisdoes not mean that performing this field redefinition implies perturbation theory for a spin-2field is breaking down. A direct application of perturbation theory in the helicityvariables without the use of this field redefinition shows that for energy scalesbelow Λ perturbation theory always converges . Since there is no fundamental need to In most presentations of the St¨uckelberg approach, the diagonalization term D µν = πη µν is not includeduntil the very end of the calculation. For ease of comparison with the helicity calculation we include it fromthe outset. None of the conclusions of standard St¨uckelberg presentations are affected by this. – 5 –iagonalize the kinetic terms (it simply simplifies calculations) we will find that the helicityand St¨uckelberg decompositions agree identically on the energy scale at which interactionscome in. In particularly, both decompositions will agree that the first interactions that ariseare at the scale Λ and that the helicity-0 mode becomes strongly coupled precisely at thisscale. This is all much easier to see in the St¨uckelberg language because of the choice to diag-onalize all kinetic mixings below the scale Λ around an arbitrary background. Nevertheless,with some effort (see below) all the results are confirmed by the helicity decomposition.In summary, it is the helicity-2 mode (and not the helicity-0) which is defined differentlyin each language h Stueckelberg µν = h helicity µν + 136Λ ∂ µ ∂ α π∂ ν ∂ α π − ∂ ( µ A α ∂ ν ) ∂ α π + 14Λ ∂ µ A α ∂ ν A α . (1.6)Crucially this field redefinition is trivially invertible to all orders and includes only a finitenumber of terms. Furthermore since the second term is nonlinear, this redefinition does notchange the asymptotic states in scattering amplitudes, and thus both h Stueckelberg µν and h helicity µν deserve the right to be called the helicity-2 field. The fact that the redefinition includes termsat the scales Λ and Λ may cause one to suspect that it disguises interactions at these scalesor that perturbation theory breaks down at these scales. We shall find by direct calculationin the helicity language that neither of these two fears is upheld and that perturbation theoryof the entire spin-2 field only breaks down at the scale Λ . Thus the apparent hierarchy ofnew interactions at the scales Λ , Λ and Λ is a fake one.The rest of this paper is organized as follows: In section 2 we present a scalar fieldtoy-model that summarizes the essence of the argument. We then move onto discussing theallowed two parameter family of ghost-free models of interacting spin-2 fields, i.e. massivegravity, in section 3.1 before proving the absence of ghost at the scale Λ in section 4, thenat the scale Λ in section 5 and finally all the way up to the scale Λ and beyond in section6. We emphasize how the St¨uckelberg decomposition emerges from this framework withoutever evoking diffeomorphism invariance by the simple requirement of diagonalizing the kineticterm of the different helicity modes. Throughout we emphasize why these results are leftunchanged by the coupling to matter. Before jumping into the subtleties of massive gravity, let us consider a representative scalarfield model which will capture the essential points. The essential problem we are faced withis that when massive gravity is analyzed in the helicity decomposition, it appears to generatehigher derivative interactions at scales which are significantly lower than when the same anal-ysis is performed in the St¨uckelberg decomposition. In [6] this fact was used to argue eitherfor the presence of a ghost at these low scales or for the breakdown of spin-2 perturbationtheory. Since the relationship between the helicity and St¨uckelberg decompositions is just afield redefinition (described above), the two methods could only disagree if the field redefi-nition were ill-defined . The key to seeing that the field redefinition is well-defined is that it It is worth emphasizing however that since both descriptions are equivalent in Unitary gauge, they cannot disagree on the existence of a ghost, even if the field redefinition were ill-defined, they could only disagree onthe scale of strong coupling. – 6 –s not a redefinition of a single field in terms of another single field, but is rather a mixingof two fields which arise when the kinetic term of one field is diagonalized. As such we willbe able to see both in the helicity and St¨uckelberg decompositions that there are the samenumber of degrees of freedom even in the presence of a source. This guarantees the absenceof ghosts. Furthermore we will see that perturbation theory remains under control since theperturbative expansion always converges below the scale Λ . This property is again tied tothe fact that the field redefinition between the helicity and St¨uckelberg decompositions is onethat mixes two fields in a manner characteristic of a kinetic term diagonalization.The field redefinition that relates the helicity and St¨uckelberg decompositions allows usto write a theory which is already perturbatively unitary in a more manifest way, and thiscan be performed to all orders in the fields expansion in a way that preserves the consistencyand validity of perturbation theory. In other words whilst the field redefinition from thehelicity to the St¨uckelberg decomposition is helpful, it is not fundamentally necessary. Wecan, and will below, understand all the physics directly in terms of the helicity variables andwithout the use of field redefinitions see that perturbation theory does not break down untilthe scale Λ in complete agreement with the St¨uckelberg decomposition.We summarize this procedure in a simple two scalar field model. In this example, aswe have explained, it is essential to maintain a coupling between two different fields, (as isnaturally the case in massive gravity). We consider a field φ (which mimics the helicity-0mode in massive gravity) and another field ψ (that mimics the helicity-2). As a simple butrepresentative example, one can consider the (assumed to be exact) tree-level Lagrangian L = 12 φ (cid:3) φ + 12 ψ (cid:3) ψ + (cid:3) ψ ( ∂ α ∂ β φ ) Λ + ( ∂ α ∂ β φ ) (cid:3) ( ∂ µ ∂ ν φ ) . (2.1)By integrating w.r.t. the field ψ , one gets ψ = − ( ∂ α ∂ β φ ) / Λ + ψ , where (cid:3) ψ = 0. Substi-tuting this back into the Lagrangian, the latter reduces (up to a total derivative) to a freetheory for φ . Hence, two initial data is needed to determine φ , and another two to determine ψ , leaving no room for any extra ghostly degree of freedom.On the other hand, looking at (2.1) in its entirety, it does not appear to be obviouslysafe: A priori the conjugate momentum to ψ does not seem well defined, however this can beresolved using a standard Ostrogradsky argument as is performed in the appendix A. Thefact that this theory involves higher derivative interactions seems to imply, at first glance, thebreakdown of perturbative unitarity at the scale Λ. Usually this is indicative of the presenceof ghosts at that scale since in most theories the breakdown of perturbative unitarity alsosignals the breakdown of genuine unitarity.However as we shall see, here unitarity is perfectly intact. To see this let us calculatesome scattering amplitudes. Whilst the 3-point function h ψφφ i vanishes on-shell from mo-mentum conservation, the 4-point function h φ i on the other hand receives a contributionfrom the vertex ∂ φ / Λ which taken alone would indeed break perturbative unitarity atthe scale Λ. However, in addition, there exist three other diagrams that contribute to thisamplitude, which propagate a virtual ψ , through a decay channel of the form φφ → ψ → φφ .Such a diagram involves two vertexes, each of the form ∂ ψφ / Λ , which by themselves alsobreak unitarity. However, one can easily see that the sum of all four diagrams vanishes.– 7 – φ φ φ φ (cid:0) ∂ Λ (cid:1) + (cid:2) ψφ φ φ φ ∂ Λ ∂ Λ + (cid:3) ψφ φ φ φ + (cid:4) ψφ φ φ φ = 0 Figure 1 . Tree-level diagrams contributing to the 4-point function h φ i at the scale Λ. The contribution from the first diagram in Fig.1 is given by M (1) h φ i = − i Λ (cid:0) ( p · p ) + 2 ↔ ↔ (cid:1) δ (4) ( X p j ) , (2.2)where p a is the momentum of each external φ particle, a = 1 , . . . ,
4, and the external legsare computed on-shell such that p a = 0. The three last diagrams rely on the exchange of avirtual ψ with h ψ i p = − ip − , giving the following contribution M ( n ) h φ i = − (cid:0) ( p · p n ) h ψ i p + p n ( p + p n ) (cid:1) δ (4) ( X j p j ) , n = 2 , , . (2.3)Computing the external legs on-shell such that ( p + p n ) = 2 p · p n , the sum of all diagramsis manifestly zero , M h φ i = X n =1 M ( n ) h φ i = 0 . (2.4)The same remains true for any n -point function h φ n i , (we could also consider here n -pointfunctions with external ψ but such diagrams cancel automatically since (cid:3) ψ vanishes on-shell.) Now, the fact that this theory is well-defined and does not propagate any processesthat violate unitarity should not come as a surprise as the cubic term in (2.1) simply indi-cates that we were not dealing with the properly diagonalized variables, at least around anarbitrary background. Instead, if we were to perform the well-defined, all orders invertiblefield redefinition, ψ = ¯ ψ − ( ∂ α ∂ β ¯ φ ) , φ = ¯ φ , (2.5)one would simply uncover a free theory, L = 12 ¯ φ (cid:3) ¯ φ + 12 ¯ ψ (cid:3) ¯ ψ , (2.6)which is manifestly healthy. One might worry that field redefinitions as (2.5) are prohibitedas they change the order of the field. Such an argument would of course contradict thefact that the S-matrix is independent of the choice of variable. Fortunately, the result withor without field redefinition is of course just the same, as we have explicitly shown here,one simply needs to work harder to derive the correct physical outcomes. Furthermore aswe shall show below perturbation theory remains under control even at scales above Λ and It is easy to check that the same results hold even if the external legs are taken off-shell in which case(2.2) needs to be modified accordingly. – 8 –ven when working with the original variables ψ, φ . To complete the analogy with massivegravity, the φ, ψ variables represent the helicity-0 and helicity-2 modes defined in the helicitydecomposition, whereas ¯ φ, ¯ ψ represent their cousins in the St¨uckelberg decomposition.To make the unitarity of the theory manifest in this example, we have not redefinedthe field φ itself, but have instead diagonalized the field ψ by a linear shift which dependson φ . This distinction is crucial. If we were dealing with a single field theory, such a situ-ation would not have been possible and any interactions at the scale Λ would always havesignaled the breakdown of perturbation theory regardless of the ability to redefine them away.To give an example of what we are not doing and where concerns about field redefi-nitions containing derivatives are well founded, consider a single field model which takes theform L = 12 χ (cid:3) χ + 1Λ χ ( (cid:3) χ ) + 12Λ χ (cid:3) χ (cid:3) ( χ (cid:3) χ ) (2.7)This model certainly contains more solutions than allowed for a single field theory with awell-defined Cauchy problem. To see this, expand the theory around a background solution χ = µ , where µ ≪ Λ is a constant. To quadratic order the action is L (2) = 12 δχ (cid:3) δχ + 1Λ µδχ (cid:3) δχ + 12Λ µ δχ (cid:3) δχ , (2.8)hence the equation of motion for the perturbations is (cid:3) (cid:16) µ Λ (cid:3) (cid:17) δχ = 0 , (2.9)which exhibits a healthy massless mode and a ghostly mode with mass m = Λ /µ . Thisis just a special case of the general result made apparent through a standard Ostrogradskyanalysis.However, we may also choose to perform the field redefinition ¯ χ = χ + χ ( (cid:3) χ ) / Λ andsee that the theory appears to become free L = 12 ¯ χ (cid:3) ¯ χ . (2.10)However this is clearly fake. The original theory has more solutions than the redefined one.The original theory has a ghostly pole around the background solution χ = ¯ χ = µ . In termsof ¯ χ the ghost seems to have disappeared, but in reality the field redefinition has simplydisguised it. The field redefinition becomes ill-defined at the scale Λ, i.e. when χ ∼ Λ and ∂ ∼ Λ since at that point perturbation theory breaks down since χ ( (cid:3) χ ) / Λ ∼ χ . This meansthat it is not possible express χ in terms of ¯ χ via a convergent perturbation expansion. Thisproblem becomes manifest if the coupling to an external source is through χ . Thus whilstbelow the scale Λ we may work with the variable ¯ χ , the ghost reappears at the scale Λ.For massive gravity the situation is completely different. The number of degrees offreedom even in the presence of matter remains 5 in both variables. Thus the field redefini-tion between the helicity and St¨uckelberg variables could never have disguised any degrees offreedom. This is backed up by the fact that the field redefinition remains under perturbativecontrol below the scale Λ . We shall demonstrate this in the next section, but doing so let– 9 –s first end this section by commenting on order by order field redefinitions.Let us consider a slightly different example. Suppose a theory in which the scalar sectorto third order in an expansion in χ takes the form L = 12 χ (cid:3) χ + 1Λ χ ( (cid:3) χ ) + O ( χ ) . (2.11)Can we say that there exists a ghost at the scale Λ? Here the answer is clearly no, pre-cisely because to the same order the cubic term is removable by the same field redefinition¯ χ = χ + χ ( (cid:3) χ ) / Λ . This means that to order ( E/ Λ) there are no interactions. Whilstthis term does give unitarity violation at higher order ( E/ Λ) we cannot say that there istruly unitarity violation at this scale until we have computed the Lagrangian to sufficientlyhigh order in the field expansion so that all terms that can contribute at the same orderare accounted form. In other words, by neglecting interactions that can contribute at thesame order in an expansion in E/ Λ we might wrongly conclude the existence of a ghost or abreakdown in perturbation theory where there is not one.
Ostrogradsky’s famous argument, adopted to the case considered here, is that higher deriva-tive interactions inevitably lead to ghost is predicated on the observation that theories withhigher derivatives require additional initial data to specify the Cauchy problem. This extrainitial data needed is always found to be precisely the initial data defining the ghost degreeof freedom. It is easy to see that the model (2.1) has a well-defined Cauchy problem without the need to specify additional initial data. This is why there is no ghost in this model despitethe appearance of higher derivatives in the Lagrangian.Let us derive the equations of motion from (2.1). There are two equations: E φ = (cid:3) φ + 2Λ ∂ µ ∂ ν (cid:20) ∂ µ ∂ ν φ (cid:18) (cid:3) ψ + 1Λ (cid:3) (( ∂ α ∂ β φ ) ) (cid:19)(cid:21) = 0 (2.12) E ψ = (cid:3) ψ + 1Λ (cid:3) (( ∂ α ∂ β φ ) ) = 0 . These equations appear to have higher derivatives, suggesting that they contain ghosts. Butit is easy to see that this is not the case. The first equation in (2.12), with the account ofthe second one, reduces to a trivial second order equation. More formally speaking, taking asimple combination these two equations give E φ − ∂ µ ∂ ν [ ∂ µ ∂ ν φ E ψ ] = (cid:3) φ = 0 . (2.13)The key point is that this equation is a standard second order differential equation for φ , andso requires just the usual two pieces of initial data. The remaining equation to solve, E ψ = 0does contain higher derivatives in φ , but these are already fixed by the solution of the firstequation . Thus there are only two remaining pieces of initial data needed, giving a total of Naturally, we are considering a situation with no discontinuity, so the equations of motion are satisfied atleast in an infinitesimal region prior where the Cauchy surface is taken. – 10 –our: The same as in a theory of two free scalar fields. So even without the use of any fieldredefinition, and even at the nonlinear level, since the theory requires the same amount ofinitial data as usual, there can be no ghosts.The above observation also explains why there is no difficulty in performing the fieldredefinition (2.5), even though this redefinition requires double derivatives of φ . This isbecause all double derivatives are completely fixed by the usual Cauchy data along with thewell-defined equation of motion for φ . The only subtlety in the previous arguments has to do with the coupling to sources, be theyexternal or additional fields. In the case of massive gravity to be treated in what follows thisis concerned with the issue of how to couple with matter . The essential point here is that ageneric coupling to sources could reintroduce the ghosts/perturbation theory problems at thescale Λ . For instance this can be seen easily by naively coupling to an external source by theaddition of a term in the Lagrangian of the form L source = J φ φ + J ψ ψ . (2.14)This coupling introduces problems at the scale Λ as can be seen by recomputing the equationof motion for φ in the presence of the source E φ − ∂ µ ∂ ν [ ∂ µ ∂ ν φ E ψ ] = (cid:3) φ + J φ − ∂ µ ∂ ν [ ∂ µ ∂ ν φ J ψ ] = 0 . (2.15)This equation now explicitly includes higher derivatives at the scale Λ and the previouslyvanquished Ostrogradsky ghost reappears.The resolution of this problem is to be more careful in the choice of couplings to sources.At the classical level, we can choose to couple how we see fit, at the quantum level we mustensure that the choice made classically is preserved under quantum corrections. It is easy tosee that this can be achieved in our toy scalar example. For instance for an external source,instead of the previous coupling we choose to couple as L source = ¯ J φ φ + ¯ J ψ (cid:18) ψ + 1Λ ( ∂ α ∂ β φ ) (cid:19) , (2.16)we easily see that the equation of motion for φ now becomes simply E φ − ∂ µ ∂ ν [ ∂ µ ∂ ν φ E ψ ] = (cid:3) φ + ¯ J φ = 0 , (2.17)and we recover the result that the Ostrogradsky ghost is vanquished. All of this is mademanifest by the field redefinition, which shows that the theory is equivalent to a free theorywith external sources L = 12 ¯ φ (cid:3) ¯ φ + 12 ¯ ψ (cid:3) ¯ ψ + ¯ J φ ¯ φ + ¯ J ψ ¯ ψ. (2.18)But the point is, since this is equivalent to a free theory, this form of coupling to φ and ψ , eventhough apparently fine tuned in the original variables, is nevertheless preserved to all orders We point out however that the apparent presence of a ghost in [14] and [6] is not attributed to the presenceof matter. – 11 –nder quantum corrections! For instance, even if the external sources ¯ J φ and ¯ J ψ are madeexplicit functions of matter fields, there are no interactions at the scale Λ to reintroduce thepathological couplings at any order in a loop expansion. Although this toy model is special,it is clear that it is always possible to:1. Choose couplings to sources which preserves the consistency of perturbation theory andabsence of ghosts at the scale Λ;2. This choice can be made so that it is preserved under quantum corrections.In the case of massive gravity the implications of this are that it is possible to chooseconsistent couplings to matter which preserve the absence of ghosts/strong cou-pling up to the scale Λ . This choice of couplings can be made so that it is preservedunder quantum corrections . We should hardly be surprised by this result. It is wellunderstood that in GR the couplings to matter must respect diffeomorphism invariance oth-erwise ghosts will appear at a scale well below the Planck scale. Thus it is no surprise thatwe have to impose similar restrictions in massive gravity even though diffeomorphism invari-ance is broken. Of course in GR this situation is safer since in the absence of an anomaly,diffeomorphism invariance guarantees that the matter couples in the correct way to all ordersin quantum corrections. In massive gravity, at present we can guarantee this at least withinthe regime of validity of the effective field theory. It is clear that all the current formalisms todescribe massive gravity do not yet present all its special properties in a manifest way. It islikely that there exists another formulation of the allowed ghost-free massive gravity modelsthat will make the consistency of coupling to matter explicit. Finally we conclude this scalar field toy-model by showing why perturbation theory is well-defined here even when the field redefinition is large. The redefinition (2.5) may still givea lingering doubt that since the second term becomes as large as the first at energy scales ∂ ∼ Λ and φ ∼ ψ ∼ Λ, then perturbation theory is breaking down. This was the argumentsuggested in [6] that the perturbation theory for the spin-2 field was breaking down, appliedin the present toy-model. However, this is not the condition for the validity of perturbationtheory. Rather the condition is that the perturbative expansion converges (at least is anasymptotic series, although in the present case this is not a concern). The second term maybe as large or larger than the first as long as the higher order terms become sufficiently smallthat the whole perturbative expansion converges. Fortunately this is trivial to see here, notonly does the expansion converge, it terminates. Recognizing that the expansion parameteris Λ − , i.e. ( E/ Λ) we may express the general solutions of the equations of motion (2.12)in the form φ = ∞ X n =0 n φ ( n ) (2.19) ψ = ∞ X n =0 n ψ ( n ) . (2.20)– 12 –t is easy to see that φ ( n ) = 0 for n ≥ ψ ( n ) = 0 for n ≥ φ (0) = φ h − (cid:3) − ¯ J φ (2.21) ψ (0) = ψ h − (cid:3) − ¯ J ψ (2.22) ψ (1) = − ( ∂ α ∂ β φ (0) ) , (2.23)where φ h and ψ h are homogeneous solutions (cid:3) φ h = (cid:3) ψ h = 0.Thus whilst it may be true that above the scale Λ the second order perturbation maybe larger than the first, i.e. ψ (1) ≫ ψ (0) , perturbation theory is still well-defined because thethird and all higher perturbations are zero. Thus the perturbative expansion always convergesand is moreover exact at first order. In the full massive gravity theory, this argument can beused to see that perturbation theory only breaks down, i.e. that the perturbative expansionfails to converge, only once we hit the scale Λ where the helicity-0 mode becomes stronglycoupled. In fact in massive gravity, the situation is even clearer. At the same scales at whichthe second order perturbation becomes larger than the first it is still true that h µν /M Pl ≪ In this paper we will consider only the two parameter family of the ghost-free models pro-posed in [2] and proven ghost free in [4]. This is what we mean by ‘Ghost-free MassiveGravity’. We consider the metric g µν , and in what follows, we use the notation where squarebrackets [ . . . ] represent the trace of a tensor contracted using the Minkowski metric, e.g. [ K ] = η µν K µν and [ K ] = η αβ η µν K αµ K βν , while angle brackets h . . . i represent the trace withrespect to the physical metric g µν , so that h H i = g µν H µν and h H i = g αβ g µν H αµ H βν .The action for the allowed interacting theories of massive spin-2 fields written in adiffeomorphism invariant way are then of the form [2], L = 2 M √− gR + L m , (3.1)where R is the scalar curvature and the mass term is given by L m = 2 m M √− g (cid:16) L (2)der ( K ) + α L (3)der ( K ) + α L (4)der ( K ) (cid:17) (3.2)with the interaction terms are given by L (2)der = [ K ] − [ K ] , L (3)der = [ K ] − K ][ K ] + 2[ K ] , (3.3) L (4)der = [ K ] − K ][ K ] + 8[ K ][ K ] + 3[ K ] − K ] , and K is defined via 2 K µν − K µα K αν = H µν or equivalently K µν = δ µν − p ∂ µ φ a ∂ ν φ b η ab , with H µν = g µν − η ab ∂ µ φ a ∂ ν φ b . (3.4)– 13 –he φ a ’s are four St¨uckelberg fields introduced to provide a diffeomorphism invariant formal-ism. Throughout the rest of the paper we shall work in unitary gauge for which φ a = x a andso H µν = g µν − η µν = h µν /M Pl .The allowed mass terms derived in [2] can be shown to arise as the expansion of thedeterminant √− g det[ g µν + λ K µν ] to fourth order in λ [16, 20, 21]. In particular, it is easy toshow that up to a total derivative, a linear combination of the above three interaction termsin (3.3), is equivalent to a single term [ K ] in combination with a cosmological constant. Thisis referred to as the minimal model in [21].With these terms it was shown that no ghosts are present in the models, neither in thedecoupling limit [2, 3], nor in the full theory by performing a Hamiltonian analysis via theADM formulation [2, 4], nor beyond the decoupling limit in the St¨uckelberg formulation [5]. The essence of the helicity decomposition is the observation that at physical momenta k ≫ m the spin-2 representation may naturally be decomposed into two helicity-2 modes, twohelicity-1 modes, and a single helicity-0 mode. The decomposition that serves to diagonalizethe quadratic order kinetic term is, written in terms of the metric g µν = η µν + h µν /M Pl where h µν = ˜ h µν + 12 m ∂ ( µ A ν ) + 13 (cid:18) m Π µν + 12 π η µν (cid:19) , (3.5)with Π µν = ∂ µ ∂ ν π . (Here ˜ h µν is what was earlier referred to as h helicity µν ). An important pointto note, is that whilst the last term is subdominant to the one that precedes it, it is crucial toinclude it in order to diagonalize the kinetic term. Without it there will be a kinetic mixingbetween π and ˜ h µν . The fact that subdominant terms must be included to remove the kineticmixing will play an important role in the following discussion.This decomposition can for example be useful in considering scattering amplitudes, sinceit correctly identifies the asymptotic helicity-2 mode ˜ h µν , two helicity-1 modes A µ , helicity-0mode π . The decomposition also respects linearized diffeomorphisms and a U(1) symmetrywhich guarantees the correct counting of degrees of freedom of the free theory.As in the St¨uckelberg decomposition, the 1 /m in the normalization of the helicity-0mode implies its interactions come in at a different scale than those of the helicity-2. As suchwe can take a decoupling limit in which M Pl → ∞ and m → n = ( M Pl m n − ) /n fixed to isolate the dominant interactions.To start with, it will be useful to perform an expansion in h µν , such that we have, L m = − m h µν − h ) + m M Pl (cid:0) k h µν + k hh µν + k h (cid:1) + O ( h ) , (3.6)with k = − ( k + k ) , k = − (1 + 3 α /
4) and k = (1 + α ) / , (3.7)and all indices are raised with η µν . Similarly, the Einstein-Hilbert (EH) term, is of the form2 M √− gR = 12 h µν ˆ E αβµν h αβ + 12 M Pl h µν h ∂ µ h α [ β ∂ ν h βα ] − ∂ α h β [ β ∂ α h µ ] ν + 2 ∂ α h µ ( ν ∂ β h αβ ) (3.8) − ∂ ( α h αµ ∂ β h βν ) − η µν (cid:16) ∂ α h γ [ β ∂ α h βγ ] + 2 ∂ α h γ ( γ ∂ β ) h αβ + 4( ∂ α h αβ ) (cid:17) i + O ( h ) , – 14 –here ˆ E is the linearized Einstein tensor,ˆ E αβµν h αβ = (cid:3) h µν − ∂ α ∂ ( µ h αν ) + ∂ µ ∂ ν h − η µν ( (cid:3) h − ∂ α ∂ β h αβ ) , (3.9)and we define ( a, b ) = ab + ba while [ a, b ] = ab − ba . In terms of the helicity decomposition(3.5), the full quadratic action is then L (2) = 12 ˜ h µν ˆ E αβµν ˜ h αβ − F µν + 112 π (cid:3) π (3.10) − mA µ (cid:16) ∂ ν ˜ h µν − ∂ µ ˜ h (cid:17) + m π∂ µ A µ − m (cid:16) ˜ h µν − ˜ h (cid:17) + 12 m π ˜ h + 16 m π . Λ In this section we shall derive the decoupling limit obtained keeping the scale Λ fixed andshow that the resulting theory is a ghost-free free theory. The most dangerous interactionsthat could arise in a massive theory of gravity occur at the scale Λ . At cubic order, theinteractions that arise at the scale Λ are that of the form ( ∂ π ) arising both from the EHterm and the mass term, as well as ˜ h∂ ( ∂ π )( ∂ π ) arising solely from the EH term. Thissecond kind of interactions, play an essential role in the consistency of the theory. The mostrelevant interactions at cubic order are, L (3)Λ = − ˜ h µν (cid:18) V µν − V η µν (cid:19) (4.1)+ 127Λ (cid:18) πV + k [Π ] + k [Π][Π ] + k [Π] (cid:19) , with V µν = ∂ µ ∂ α ∂ β π∂ ν ∂ α ∂ β π − ∂ α ∂ µ ∂ ν π∂ α (cid:3) π . (4.2)and V = η µν V µν . As a consistency check, one can see that V µν satisfies the transverse relation ∂ µ V µν = ∂ ν V , which is a simple consequence of linearized diffeomorphism invariance. (Inthe decoupling limit, full diffeomorphism invariance always reduces to the linearized diffeo-morphism invariance, but this is true only in the decoupling limit [5, 17]). One can also checkthat V µν is nothing but the Lichnerowicz operator applied on Π µν = ∂ µ ∂ α π∂ ν ∂ α π , V µν − V η µν = 12 ˆ E αβµν Π αβ , (4.3)which will have important consequences for the consistency of the theory, as we will seebelow. After integration by parts, we simply have the relation πV = 12 ([Π] − [Π][Π ]) . (4.4)Unsurprisingly, the mass term in (3.2) is precisely such that the last line (4.1) is a totalderivative,14 πV + k [Π ] + k [Π][Π ] + k [Π] = 3 + 2 α (cid:0) [Π] − ] + 2[Π ] (cid:1) . (4.5)– 15 –owever, it is not possible, nor is it desirable to cancel the ˜ hV interactions, because theyplay a crucial role in ensuring the consistency of the theory. This leads to the cubic vertexat the scale Λ L (3)Λ = − (cid:2) Π (cid:3) µν ˆ E αβµν ˜ h αβ . (4.6)The full decoupling theory at the scale Λ also includes quartic interactions of the form L (4)Λ = 13 Λ Π µν ( V µν − V η µν ) = 13 Λ Π µν ˆ E µναβ Π αβ . (4.7)Putting this together, the exact decoupling theory obtained in the limit M Pl → ∞ keepingΛ fixed is L decΛ = 12 (cid:18) ˜ h µν − Π µν (cid:19) ˆ E αβµν ˜ h αβ − Π αβ ! − F µν + 112 π (cid:3) π . (4.8)We shall analyze the physics of this action below. Λ theory A cursory glance at this action (4.8) would seem to imply the presence of strong coupling, orunitarity violation at the scale Λ . However it is easy to see that this is not the case, sincethere is something remarkably special about this theory that is not apparent at first. To seewhat this is, let us calculate scattering amplitudes. The cubic vertex L (3)Λ in (4.6) can beused in diagrams but only if ˜ h is computed off-shell (no external leg in ˜ h ). There is thereforeno non-trivial 3-point function at the scale Λ . For the 4-point function, on the other hand,the situation is different and let us calculate the 2 → L (4)Λ . Taking this term alone one would find that the scattering amplitude violatesperturbative unitarity at the scale Λ . Thus, if the decoupling theory only included the term L (4)Λ , it would certainly be true that the theory would violate tree level unitarity at the scaleΛ . However, there are 3 other Feynman diagrams that contribute at tree level at the sameorder corresponding to s-channel, t-channel and u-channel exchange of a virtual helicity-2mode. Individually each of these diagrams also violates perturbative unitarity at the scale Λ .However the sum of all the diagrams which arise at the same order 1 / Λ is easily shown to bezero, in just the same way as was performed in the preamble scalar field toy-model (see Fig.1).Since only the total amplitude can be measured, and the total scattering amplitude iszero, we find that there is no unitarity violation at the scale Λ . This in turn implies thereare no new degrees of freedom, or ghosts at this scale. This argument can easily be extendedto all n -point functions and to all loops, and one may confirm by direct although laboriouscalculation that the S-matrix is trivial h f | ˆ S − ˆ I | i i = 0 . (4.9) If we were to compute the 3-point scattering amplitude on top of a background, for which we could take˜ h off-shell, it is true that this vertex could give rise to a non-trivial contribution to the 3-point function, butit would be canceled by the contribution from the matter. – 16 –here is fortunately a much simpler way to see the triviality of the S-matrix. By performingthe well-defined, invertible field redefinition˜ h µν = ¯ h µν + 136Λ Π µν , (4.10)we can easily see that the decoupling theory reduces to L decΛ = 12 ¯ h µν ˆ E αβµν ¯ h αβ − F µν + 112 π (cid:3) π , (4.11)where the free kinetic term is now written in terms of ¯ h µν . Thus since the theory is free inthis decoupling limit we can see easily why the S-matrix turned out to be trivial to all orders.One might worry that such redefinitions are not appropriate since they mix orders of thespin-2 field and hence imply that at least one of the degrees of freedom is strongly coupled.However, this is clearly incorrect, since we did not need to perform this field redefinition tosee that the S-matrix was trivial. It was possible to see it in the original variables. Since theS-matrix is trivial, all the degrees of freedom are not only weakly coupled at these scales,they are precisely free in the limit. This fact is true regardless of the variables used. TheS-matrix is of course only an on-shell quantity, however we shall see below that perturbationtheory is under control even off-shell.Although not necessary, the field redefinition (4.10) is desirable to perform since itdiagonalizes the kinetic mixing of the helicity-2 and -0 modes at cubic order. Remarkably itis precisely the same contribution we derive using the St¨uckelberg decomposition. However,we did not have to invoke the St¨uckelberg decomposition nor diffeomorphism invariance tosee this. It follows simply from diagonalization of the kinetic term. Furthermore since theredefinition is quadratic in the fields, it does not change the asymptotic states, and so both˜ h µν and ¯ h µν deserve to be called the helicity-2 mode. Note also that the additional pieceseen in (4.10) is a modification of the subleading part of the decomposition since it is downby m /k relative to the leading order helicity-0 contribution ∂ π/m when π ∼ ∂ ∼ Λ . The previous arguments guarantee the consistency of the theory defined perturbativelyaround h µν = 0. However, sometimes it is possible for ghosts to arise when expanded aroundan arbitrary background. Again, if we neglect the ˜ hV interaction this is true of the theory(4.8). However, when all interactions are included it is easy to show that this is not the case.To see this, let us derive the full equations of motion, in the original helicity variables. In anarbitrary gauge the equations of motion that follow from (4.8) are E π = (cid:3) π − ∂ ω ∂ ν (cid:20) ∂ µ ∂ ω π ˆ E αβµν (cid:18) ˜ h αβ − Π αβ (cid:19)(cid:21) = 0 , (4.12) E A µ = ∂ µ F µν = 0 , (4.13) E h µν = ˆ E αβµν (cid:18) ˜ h αβ − Π αβ (cid:19) = 0 . (4.14)As usual these equations are seemingly higher derivative, and so a cursory analysis wouldseem to suggest the presence of a ghost. However, again this can be shown not to be the– 17 –ase. As before taking the combination E π + 13Λ ∂ ω ∂ ν (cid:2) ∂ µ ∂ ω π E h µν (cid:3) = (cid:3) π = 0 . (4.15)This gives a well-defined equation of motion for π whose solution, once known, can be sub-stituted back into the equation (4.14) for ˜ h µν . This analysis is completely analogous to whatis performed in the St¨uckelberg language beyond the decoupling limit, see Ref. [5].The number of degrees of freedom is one half the total amount of initial data neededto solve these equations. As usual the vector part gives two helicity-1 modes. Crucially theequation for the helicity-0 is second order and so gives rise to only one independent degreeof freedom (two pieces of initial data). Finally, although the right hand side of the equationfor the helicity-2 mode (4.14) contains higher derivatives, the helicity-0 mode is already de-termined by its own well-defined equation. Thus the only remaining initial data to specifyis that for ˜ h µν . As usual, the gauge condition removes 4 degrees of freedom, but if say wechoose de Donder gauge, since it does not fix all the freedom, we can use the remainingnon-uniqueness to reduce down to two independent polarizations which describe the realhelicity-2 component.All of the above is trivial once we work with the field redefined coordinates for whichthe equations of motion take the form (cid:3) π = 0 ,∂ µ F µν = 0 , (4.16)ˆ E αβµν ¯ h αβ = 0 . It is however important to make clear that the field redefinition does not disguise any degreesof freedom, which is why we have labored the argument. The counting can be performedcorrectly without performing any field redefinitions.Note that again had we discarded the ˜ h∂ ( ∂ π ) interaction, as was done in [6], theequation of motion for the helicity-0 mode would have been higher order, and it would havebeen necessary to supply more initial data to solve it. This would indeed indicate the presenceof a ghost at the scale Λ . Only once all the interaction terms are taken into account can wecorrectly diagnose the absence of a ghost. Let us now come to the coupling to matter. We already know from our simple two scalartoy model that this deserves some care. For instance, let us suppose as a first attempt thatwe just couple to an external source S µν which we shall not assume to be conserved. This isimportant since diffeomorphism invariance is broken by the mass term, there is no reason toassume that the external source for a massive spin-2 field is conserved. Adding the term∆ L matter = 12 h µν S µν , (4.17)– 18 –nd taking the Λ decoupling limit modifies the equations of motions to E π = (cid:3) π − ∂ ω ∂ ν (cid:20) ∂ µ ∂ ω π ˆ E αβµν (cid:18) ˜ h αβ − Π αβ (cid:19)(cid:21) + 1 m ∂ µ ∂ ν S µν + 12 S = 0 (4.18) E A ν = ∂ µ F µν + 12 m ∂ µ S µν = ∂ µ F µν = 0 , (4.19) E h µν = ˆ E αβµν (cid:18) ˜ h αβ − Π αβ (cid:19) + 12 S µν = 0 . (4.20)The absence of a source in the helicity-1 equation need a qualification: In the decoupling limit m → S µν = S (0) µν + mS (1) µν + m S (2) µν + ... , where S (0) µν is conserved, ∂ µ S (0) µν = 0, while in general S (1 , µν is not. However, we will show below that ∂ ν S (1) µν ∼ m ,which is a sufficient condition to guarantee that the source in the helicity-1 equation E A ν vanishes in the decoupling limit, and hence trivially satisfies the Bianchi identity to thatorder. In what follows, for all nonconserved currents on the r.h.s. of the helicity-1 equationthis decomposition will be implied. Thus, the helicity-1 and helicity-2 equations have well-defined Cauchy problems.The problem lies entirely with the helicity-0 equation which can be rearranged to takethe form E π + 13Λ ∂ ω ∂ ν (cid:2) ∂ µ ∂ ω π E h µν (cid:3) = (cid:3) π + 1 m ∂ µ ∂ ν S µν + 12 S + 16Λ ∂ ω ∂ ν [ ∂ µ ∂ ω π S µν ] = 0 . (4.21)If no conditions are placed on the external source, this equation includes higher derivatives of π at the scale Λ so the ghost or perturbation theory problems would seem to be reintroduced.However, in any realistic case the matter source itself will depend on the spin-2 field. Is itpossible to couple the entire spin-2 field h µν to matter in such away that perturbation theory iswell-defined at the scale Λ (and even up to the scale Λ ) and there are no ghosts? The clue tothe resolution is that the source in (4.19) appears to diverge as m → i.e. in the decouplinglimit that has been taken. Thus in order to have a consistent decoupling limit it is necessaryfor the divergence of the source to behave as ∂ µ S µν ∼ m →
0. The fact that at leadingorder the stress energy must be conserved is a reflection of the fact that in the m → h µν → ˜ h µν + ∂ µ χ ν + ∂ ν χ µ such that the helicity-1 and helicity-0 modes are invariant (since their variation is of order m ). This symmetry that arises in this limit then forces that its source is conserved (at leadingorder) just as in GR and so ∂ µ S (0) µν = 0. Of course, this symmetry is not exact, and so thesource fails to be conserved at order m such that ∂ µ S µν /m is finite in the limit m → even in the presence of asource. The implication is that the helicity-1 and helicity-0 modes are sourced by termswhich are subleading in the decoupling limit from the point of view of the matter fields.These subleading terms account for the non-conservation of the source due its backreaction, i.e. the backreaction of gravity, on the equations of motion for the matter field. However thisin turn implies that ∂ µ S µν /m must itself depend on the spin-2 field. The question we mustaddress then is: Is it possible in a realistic matter system to couple to matter in such a waythat the failure of conservation of the source because of the gravitational backreaction ontothe matter implies that the net source in (4.21) does not dependent on higher derivatives ofthe helicity-0 mode? – 19 – .3.2 Consistency Relation for matter In practice this is all extremely easy to achieve. For instance, suppose the matter couples tothe spin-2 field in a way that preserves diffeomorphism invariance. This is a choice that canalways be made classically. S µν then takes the same expression it would do in GR, i.e. S µν isjust proportional to the usual stress energy of the matter field S µν = √− g T µν /M Pl . However,as is well known, this stress energy is not ordinarily conserved but is rather covariantlyconserved as a direct consequence of the backreaction of gravity on the matter: √− g D µ T µν = ∂ µ ( √− g T µν ) + √− g Γ νµω T µω = 0 , (4.22)so that we can write the covariant conservation law for S µν as ∂ µ ( S µν ) = − Γ νµω S µω . (4.23)Here the Christoffel symbol should be evaluated on the full metric, however in the context ofthe Λ decoupling limit it is sufficient to evaluate it with only the leading part of the helicityzero mode contribution to the metric included δg µν = M Pl m ∂ µ ∂ ν π since this is the termthat dominates in the Λ decoupling limit. This givesΓ νµω = 16 M Pl m ∂ ν ∂ ω ∂ µ π + . . . . (4.24)Thus to leading order in the decoupling limit we have1 m ∂ µ ∂ ν S µν = − m ∂ ω (cid:0) Γ ωµν S µν (cid:1) (4.25)= − ∂ ω ( ∂ ω ∂ µ ∂ ν πS µν ) + . . . (4.26)= − ∂ ω ∂ ν ( ∂ ω ∂ µ πS µν ) + O (Λ − ) + . . . (4.27)where in the last step we made use of the fact that the failure of conservation of S µν is a termonly relevant at a higher scale. Putting this together, at the scale Λ the correct equationof motion for the helicity-0 mode, in the decoupling limit m → fixed is theperfectly well-defined equation E π + 13Λ ∂ ω ∂ ν (cid:2) ∂ µ ∂ ω π E h µν (cid:3) = (cid:3) π + 12 S = 0 . (4.28)Thus even in the presence of matter the equations of motion remain second order, and thetotal number of propagating modes in the gravity sector remains as 5. Furthermore as withprevious arguments it is trivial to see that perturbation theory remains well behaved at thescale Λ since it terminates at first order, as in the scalar toy model. We made this argument assuming that matter couples in a way that preserves diffeomorphisminvariance. Since diffeomorphism invariance is broken in massive gravity we can in principleimagine other couplings to matter. However it is easy to see that regardless of the couplingto matter it is never possible to reintroduce problems at the scale Λ . The reason is that aswe have already explained, in the limit m → S µν must be conserved ∂ µ S µν → M Pl h µν T µν back into the Lagrangian then causes the matter fields to besourced by its own stress energy in such away that the stress energy fails to be conserved atorder 1 /M Pl . However, the way it fails to be conserved to this order is just the same as inGR and so the above argument applies. In other words, any couplings to matter which donot respect diffeomorphism invariance would have to come in at order h µν or higher, and sowould not contribute in the Λ decoupling limit. In fact this argument extends up to thescale Λ since this is the scale at which h µν ∼ in couplingto matter, the approximate symmetry ˜ h µν → ˜ h µν + ∂ µ χ ν + ∂ ν χ µ is sufficient to protect thecouplings to matter even at the quantum level in such a way that below the scale Λ theequation of motion for the helicity modes are all well-defined, propagating no more that 5degrees of freedom, and perturbation theory remains well behaved in the presence of matter.Notice that at the scale Λ the resulting theory of massive gravity is similar to that of DGP,suggesting that the treatment of both theories could be thought in similar terms. One simple example of how one can consistently couple to matter working in the helicityvariables is provided by a point particle of mass m p described by the trajectory X µ ( τ ) withaction S matter = − m p Z d τ (cid:18) − e g µν d X µ d τ d X ν dτ + 12 e (cid:19) (4.29)where τ denotes the proper time along the trajectory, and e is the einbein which accountsfor the worldline reparameterization invariance. To have a consistent Λ decoupling limit wescale m p → ∞ such that m p /M Pl is finite. In this case S µν is given by S µν = m p M Pl Z d τ δ ( x − X ) 1 e d X µ d τ d X ν d τ . (4.30)By the previous arguments, one can show that using the equations of motion for the particle,ensures that the coupling to π in the Λ decoupling limit is carried entirely by S . However,we can also see this more directly by performing a field redefinition of the matter variables.Working with the full action and making the helicity decomposition we have g µν d X µ d τ d X ν dτ = η µν + ˆ h µν M Pl + 12 mM Pl ∂ ( µ A ν ) + 13 m M Pl ∂ µ ∂ ν π + 16 M Pl πη µν ! d X µ dτ d X ν dτ = (cid:18) η µν + ¯ h µν M Pl + 136Λ Π µν + 12 mM Pl ∂ ( µ A ν ) + 13 m M Pl ∂ µ ∂ ν π + 16 M Pl πη µν (cid:19) d X µ dτ d X ν d τ . Let us focus on one set of terms in this expression. After integration by parts we see that Z d τ e (cid:18) mM Pl ∂ ( µ A ν ) + 13 m M Pl ∂ µ ∂ ν π (cid:19) d X µ d τ d X ν d τ = Z d τ d e d τ (cid:18) mM Pl A ν + 13 m M Pl ∂ ν π (cid:19) d X ν d τ . (4.31)Similarly Z d τ e Π µν d X µ dτ d X ν d τ = Z d τe d ∂ µ π d τ d ∂ µ π d τ . – 21 –ow let us perform the following well-defined, invertible field definition Y µ = X µ + m A µ + m ∂ µ π . (4.32)It is easy to see that up to terms which vanish in the Λ decoupling limit the matter actionbecomes S matter = − m p Z d τ (cid:18) − e (cid:18) η µν + ¯ h µν M Pl + 16 M Pl πη µν (cid:19) d Y µ d τ d Y ν d τ + 12 e (cid:19) + O (1 / Λ ) . (4.33)Since as we have already discussed the kinetic terms of ¯ h µν and π decouple at this order, itis transparent from the above action that that π only couples to S and that the equationsof motion are second order. The field redefinition of the matter variables remains underperturbative control since Y µ − X µ vanishes as m in the Λ decoupling limit. As usual,we did not need to perform the field redefinition to see that the equations of motion of thespin-two field and the particle were both well-defined, however it certainly simplifies theanalysis.As discussed earlier, the above result would be identical had we modified the action forthe particle by adding terms which violate full diffeomorphism invariance. For example, ifwe add terms of the form∆ S matter = − m p Z d τ e F µναβ (cid:18) X ω , e dX ω dτ (cid:19) h µν M Pl h αβ M Pl + . . . (4.34)where F µναβ is a tensor constructed locally out of the particles position and velocity. Thischoice is a natural one since we still require the equations of motion for the particle toremain second order. This term clearly violates full diffeomorphism invariance, neverthelessit is harmless at this order because it vanishes in the Λ (and Λ ) decoupling limit with thenatural assumption that F , being dimensionless, is kept fixed in this limit. Similar reasoningapplies to terms with more powers of the spin-2 field, or more derivatives. Thus the factthat in massive gravity the coupling to matter is less constrained is not in itself necessarilya problem.To reiterate, the linearized diffeomorphism invariance which always arises accidently inthe decoupling limit, forces the leading order coupling to matter M Pl h µν T µν to be the same asin GR, and is sufficient to guarantee the absence of ghosts/breakdown of perturbation theorybelow the scale Λ at least. At the scale Λ the situation becomes more subtle because at thisscale the helicity-0 mode becomes strongly coupled. However, even at this scale, whilst loopsfrom the helicity-0 mode will in principle generate matter couplings which do not respect fulldiffeomorphism invariance, these will inevitably be suppressed by additional powers of M pl and thus occur at a scale higher that Λ . Since the helicity decomposition is a bad one inthis regime, it is beyond the scope of this work to understand the full quantum consistencyof couplings to matter in massive gravity. Λ theory Finally let us come to the issue of the scale at which perturbation theory breaks down. Forscattering processes we have seen that the theory is free at the scale Λ and so perturbationtheory is valid (as we will see below up to the scale Λ ). However, let us consider whathappens for spherically symmetric configurations in the presence of a source of mass M ,– 22 –ocalized at r = 0. Following standard reasoning in the weak field region the solutions for˜ h and π take the form ∼ MM Pl r . In the original helicity variables, the Λ terms do have animportant role to play in the classical solutions since at some characteristic distance, theterm ( ∂∂π ) / Λ ∼ M M Λ r becomes comparable to ˜ h ∼ MM Pl r . Thus in these variables theapparent interactions seem to become important at a scale r ∼ ( M Λ − ) / . However thisdoes not signal the breakdown of perturbation theory for the spin-2 field. As in the two scalartoy-model, the perturbative expansion actually terminates at first order in perturbations inthe Λ decoupling limit. Thus not only is perturbation theory well behaved, it isexact at first order! This is all made clear by actually solving the equations of motion in the presence ofmatter, which by the above arguments are seen to be of the form E π + 13Λ ∂ ω ∂ ν (cid:2) ∂ µ ∂ ω π E h µν (cid:3) = (cid:3) π + 12 M Pl T = 0 (4.35) E A ν = ∂ µ F µν = 0 , (4.36) E h µν = ˆ E αβµν (cid:18) ˜ h αβ − Π αβ (cid:19) + 12 M Pl T µν = 0 . (4.37)The equation (4.35) may be solved exactly without any perturbative corrections π = π − M Pl (cid:3) T , (4.38)where π as a solution of the homogeneous equation (cid:3) π = 0. The vector equation may besolved similarly, e.g. by choosing Lorentz gauge ∂ µ A µ = 0 A ν = A ν , (4.39)for which (cid:3) A ν = 0 and finally by choosing de Donder gauge ∂ µ (˜ h µν − η µν ˜ h ) = 0 andsubstituting in the solution for π we obtain˜ h µν = ˜ h µν − M Pl (cid:3) (cid:18) T µν − η µν T (cid:19) + 118Λ (cid:3) V µν | π = π − M Pl 1 (cid:3) T . (4.40)This completes the exact solutions for an arbitrary source. These solutions are also the onesthat arise at first order in perturbation theory since the contribution to ˜ h µν is linear in 1 / Λ .This we see that indeed first order perturbation theory is exact, and so perturbation theoryis certainly under control.Another way of understanding this result is when the Λ terms become important, thedimensionless spin-2 field h µν /M Pl ∼ ( mr s ) / ≪ r s is the Schwarzschild radius ofthe source. Since for realistic values this is negligible, we see that perturbation theory ofthe spin-2 field is easily under control at this scale. Pursuing this argument to its end, werecover the known result that the actual scale of the breakdown of perturbation theory, i.e. the scale at which the perturbative expansion fails to converge, is the scale Λ , the correctstrong coupling scale of the helicity-0 mode, since it is only at this scale that h µν ∼ Relevant interactions at the scale Λ Now that we have checked the consistency of the theory at the scale Λ , let us go beyondthat scale and look for the relevant interactions all the way up to the scale Λ .There are potentially dangerous interactions that arise at this order from mixing thevectors and scalars. For instance in [6] it was pointed out we could get already at cubic orderterms of the form ( ∂A )( ∂ π ) . Besides these, there is an entire zoo of interactions enteringat the scale Λ . As in the previous section, it is essential to keep track of every single oneof them before determining the consistency of the theory, since all the Feynman diagramsat the same scale contribute to the amplitude of scattering processes. Many of them takenalone would break perturbative unitarity, but as we shall see below, for the specific massterm considered in (3.2), the sum of all these diagrams does not violate unitarity and leadsto a trivial S-matrix at that scale.The interactions entering at the scale Λ are as follows:1. At the cubic level, ∂A ( ∂ π ) from the mass term and ∂ ˜ h∂A∂ π , ∂A ( ∂ π ) and ∂ ( ∂A ) from the EH term.2. At the quartic level, ( ∂ π ) from the mass term and ∂ ˜ h ( ∂ π ) , ( ∂ π ) or ( ∂A ) ∂ ( ∂ π ) from the EH term.3. At the quintic level, ( ∂A ) ∂ ( ∂ π ) from the EH term.4. And finally, at the sextic level, ( ∂ π ) ∂ ( ∂ π ) from the EH term.Furthermore, there are also two interactions in the intermediate region between Λ and Λ ,the first one occurs at quartic order and is of the form ( ∂A ) ∂ ( ∂ π ) coming at the scaleΛ / and the second one occurs at quintic order and is of the form ∂ ( ∂ π ) with the slightlyhigher scale Λ / .With the correct choice of mass (3.2), many of these terms combine to give rise to atotal derivative. The rest of them can be treated in a similar manner to the terms that ariseat the scale Λ . One can easily see that up to total derivatives, one has L (3)Λ , ∂ ( ∂A ) = 0 . (5.1)As for the other interactions, we can start by focusing on terms of the form ( ∂A )( ∂ π ) atthe cubic order. A → ππ scattering Amplitude Although on-shell the 3-point function h Aππ i vanishes in the decoupling limit by energy/momentumconservation, it does not necessarily do so off-shell, e.g. when looking at perturbations arounda given background. Since its off-shell behaviour is important in higher order diagrams letus consider the off-shell form of its amplitude. This has a contribution from the followinginteraction L (3)Λ , ( ∂A )( ∂ π ) = 136Λ A µ h s ∂ µ [Π] + s ∂ µ [Π ] + s ∂ α (cid:3) π∂ µ ∂ α π i , (5.2)with s = − (1 + 4 k + 12 k ), s = (1 − k − k ) and s = − (6 k + 4 k ). With the coefficient k ’s given in (3.7) this simplifies to L (3)Λ , ( ∂A )( ∂ π ) = 136Λ A µ h ∂ µ [Π ] − ∂ ν Π µν i . (5.3)– 24 –he existence of this interaction, taken alone, would imply the breakdown of perturbativeunitarity at the scale Λ . Furthermore, since this interaction involves three derivatives on π ,the 3-point scattering process A → ππ would appear to contain a ghost. If this was the endproduct, this scattering process would not have a well-posed Cauchy problem and the theorywould have a ghost. Fortunately, there is another class of diagrams that contributes to thisamplitude which arises from the tree level process A → ˜ h → ππ .The first decay A → ˜ h is governed by the interaction − mA µ (cid:16) ∂ ν ˜ h µν − ∂ µ ˜ h (cid:17) in (3.10).A priori this term is negligible as it suppressed by a factor m/k . However the second decay˜ h → ππ is governed by the interaction − ˜ h µν ˆ E αβµν Π αβ found in (4.1) and is enhanced bya factor Λ /k ∼ k/m . While the first decay, is very unlikely to occur at an energy scaleΛ or below, the second decay on the other hand is extremely fast, such that this channel isactually as efficient as the direct channel A → ππ . Combining these two diagrams together,it is easy to see that the resulting amplitude of the 3-point function h Aππ i vanishes at treelevel at the scale Λ or below. (cid:1) A µ π π ∂ + (cid:2) ˜ h µν A µ π π m∂ − ∂ = 0 Figure 2 . Tree-level diagrams contributing to the 3-point function h Aππ i at the scale Λ . Of course the fact that we must work hard to see this cancelation is just a reflectionof the fact that to look at physics at the scale Λ , it is better to work with the correctlydiagonalized helicity-2 mode defined by the scale Λ . Had we done this from the outsetneither of these processes would have arisen in the first place. ππ → ππ scattering Amplitude Another type of interaction that arises on-shell and deserves attention is the quartic one ofthe form ( ∂ π ) , which gives a contribution L (4)Λ , ( ∂ π ) = − × Λ (cid:0) [Π ] − [Π ] (cid:1) . (5.4)Here again, this interaction has non-vanishing higher derivative terms which would seemto suggest a problem. However, at the same scale the channel ππ → ˜ h → ˜ h → ππ givesprecisely an opposite contribution which cancels (5.4). This channel can be understood asfollows: First both π ’s decay into a helicity-2 mode via the interaction − ˜ h µν ˆ E αβµν Π αβ found in (4.1) at the scale Λ . Then the interaction − m (cid:16) ˜ h µν − ˜ h (cid:17) in (3.10) is suppressedby m (strictly speaking this should be part of the helicity-2 propagator, but since this hasbeen neglected for the previous interactions, this has to be included as an “additional” inter-action at this level). Finally, the helicity-2 decays back into 2 helicity-0 via the inverse process˜ h → ππ which occurs at the scale Λ . Putting all this together, this gives rise to three new di-agrams with an effective scale (Λ /m ) / = Λ which should therefore be considered at the– 25 –ame time as (5.4). We can easily see that the sum of all these four diagrams precisely cancels.As we can see, this is a very cumbersome approach to compute these scattering processes,and the most direct way to deal with this is simply to diagonalize the helicity-2 mode at thecubic level through the redefinition (4.10). We emphasize however once again, that whilstit makes more sense to work in terms of the diagonalized field, no physics is hidden in thisredefinition, and we have gone to great length to show explicitly how to understand thephysics if we were to chose to work in terms of the undiagonalized field. We now turn backto a more conventional approach and work in terms of the ¯ h µν defined in (4.10). Λ Except for the three interactions ∂ ˜ h∂A∂ π , ( ∂A ) ∂ ( ∂ π ) and ( ∂A ) ∂ ( ∂ π ) all the otherinteractions arising at or below the scale Λ are nothing else but part of the diagonalizedhelicity-2 ¯ h µν interactions. More precisely, L (3)Λ , ( ∂A )( ∂ π ) ⊂ − mA µ (cid:0) ∂ ν ¯ h µν − ∂ µ ¯ h (cid:1) (5.5) L (4)Λ , ( ∂ π ) ⊂ − m (cid:0) ¯ h µν − ¯ h (cid:1) (5.6) L (4)Λ , ( ∂ ˜ h )( ∂ π ) ⊂ M Pl m ¯ h∂ ¯ h ( ∂ π ) (5.7) L (5)Λ , ( ∂A )( ∂ π ) ⊂ M Pl m ∂A ¯ h∂ ¯ h (5.8) L (6)Λ , ( ∂ π ) ∂ ( ∂ π ) ⊂ M Pl ¯ h ∂ ¯ h (5.9) L (5)Λ / , ∂ ( ∂ π ) ⊂ M Pl m ¯ h∂ ¯ h ( ∂ π ) , (5.10)so after appropriate diagonalization of the helicity-2, the previous terms do not appear in thetheory. In other words, if we were to keep working in terms of ˜ h µν , any diagram involving theprevious vertices precisely cancels with another diagram such that the resulting scatteringamplitude in question vanishes. Once we work with ¯ h µν , the only three remaining terms thatarise between the scale Λ and Λ included are given by, L (3)Λ , ∂ ¯ h∂A∂ π = − ¯ h µν ˆ E αβµν P αβ (5.11) L (4)Λ , ( ∂A ) ∂ ( ∂ π ) = 18Λ P µν ˆ E αβµν P αβ (5.12) L (4)Λ / , ( ∂A ) ∂ ( ∂ π ) = 112 × Λ P µν ˆ E αβµν Π αβ , (5.13)with P µν = ∂ ( µ A α Π ν ) α . All these interactions therefore combine to form L = − F µν + 112 π (cid:3) π + 12 (cid:18) ¯ h µν − P µν (cid:19) ˆ E αβµν (cid:18) ¯ h αβ − P αβ (cid:19) + · · · , (5.14)where the dots denote terms that vanish in the decoupling limit if ¯ h is kept finite. It iseasy to see that the second line does lead to higher derivative interactions terms, which by– 26 –hemselves break unitarity, but the combined sum of the diagrams from interactions on thesecond line is of course fine as has already been discussed at length previously. The remainingcoupling between ¯ h µν and P µν at quartic order, is nothing else but the indication that ¯ h hasstill not been fully diagonalized, and the properly diagonalized helicity-2 mode (up to thescale Λ ) is instead ˘ h µν = ˜ h µν − Π µν − P µν . (5.15)Working in terms of the helicity-2 mode ˘ h , which is the one whose kinetic terms is diagonalizedaround an arbitrary background at least up to the scale Λ , the decomposition of field readsas follows: h µν = ˘ h µν + 16 πη µν + M Pl (Ψ µα Ψ αν − η µν ) − ∂ µ A α ∂ ν A α , (5.16)with Ψ µα = η µα + 12 M Pl m ∂ µ A α + 16 M Pl m ∂ µ ∂ α π . (5.17)and the Lagrangian can again be seen to be that of a free theory L = − F µν + 112 π (cid:3) π + 12 ˘ h µν ˆ E αβµν ˘ h αβ + O (Λ − nn ) , (5.18)where corrections come in at a larger energy scale n < We may now simply apply all the arguments used at the scale Λ to see that perturbationtheory is under control at the scale Λ . As before the point is that the π equation of motioncan be seen to be (cid:3) π = 0 (5.19)in the absence of a source. As before, taking into account the conditions that must be satisfiedby any allowed coupling to matter we infer that in the presence of a source the equationsalso remain second order (cid:3) π = − M Pl T. (5.20)This means it is always possible to determine π to all orders in perturbation theory. Giventhis solution we may then solve the well-defined second order equations for the helicity-2 andhelicity-1 modes and as before perturbation theory is seen to be exact to first order in 1 / Λ and 1 / Λ something which is made manifest by the field redefinition (5.15). Λ Λ and Λ Having shown that no pathological interactions arise at energy scales k lower or equal to Λ ,let us consider the region Λ < k < Λ . Since the last term in (5.16) is negligible at these– 27 –cales, we may simply omit it and check the consistency of the theory without it. As a simpleconsequence of local diffeomorphism invariance, the EH term can be computed without thecontribution from Ψ, M √− gR | g µν = M Pl (˘ h µν + πη µν )+Ψ µα Ψ αν = (6.1) M √− gR | g µν = η µν + M Pl (˘ h µν + πη µν ) + terms which vanish below Λ , so between the scale Λ and Λ there cannot be any additional contribution from the EHterm. Any contribution should therefore come from the mass term. Fortunately only twoclasses of interactions are relevant at these scales, namely terms of form ( ∂ π ) n arising at thescale Λ (3 n − / ( n − (with n > ∂A )( ∂ π ) n arising at the scaleΛ (3 n − / ( n − (with n > ∂ π ) n and we will not reproducethis calculation here. The second type of interactions, ( ∂A )( ∂ π ) n on the other hand deservesspecial care. Upon close inspection, one can check that all the terms of that form resum toform the following all-orders Lagrangian L ( ∂A )( ∂ π ) n = M Pl m ( ∂ µ A ν + ∂ ν A µ ) X µν , (6.2)with X µν = 1 M Pl m (Π η µν − Π µν ) + 1 + 3 α M m (cid:18) Π µν − Π Π µν + 12 (cid:0) [Π] − [Π ] (cid:1) η µν (cid:19) (6.3)+ α + 4 α M m Π µν − Π Π µν + 12 Π µν (cid:0) [Π] − [Π ] (cid:1) −
16 ([Π] − ] + 2[Π ]) η µν ! . Since X µν is identically transverse ∂ µ X µν = 0, the final interaction term (6.2) that couldsurvive below Λ is actually a total derivative and gives no contribution (in any case, we seethat this would never give anything beyond n = 3). The theory at hand is therefore free allthe way up to (but not including) the scale Λ . In other words we have finally shown that thecorrect scale of interactions of the helicity modes is the scale Λ . Thus the hierarchy of scalesthat seems to arise between Λ and Λ is entirely fake, no physics actually occurs at thesescales, no interactions occur at these scales, no ghosts appear at these scales, perturbationtheory of the spin-2 field remains under control at these scales, and last but not least, thereare no quantum corrections at these scales ! Λ : The true strong coupling scale Finally we come to the scale Λ . At this scale the helicity-0 mode has genuine interactions,and it is easy to show that the helicity-0 becomes strongly coupled at this scale. Tree levelunitarity is violated at this scale. This is not a new statement, but it follows equally in thehelicity decomposition. When this occurs the spin-2 perturbation theory breaks down aswell, and so this is also the scale at which the helicity decomposition is no longer a usefulone. However, it certainly does not mean that there are ghosts, it simply means that thehelicity decomposition is not a good way to analyze the system at these scales. As has al-ready been shown in great detail [3], the St¨uckelberg decomposition demonstrates that forthe special two parameter family of mass terms, the absence of ghosts at the scale Λ and theADM decomposition proves it to all orders (at least classically) [4]. For these reasons there– 28 –s no need to pursue the helicity decomposition to this scale, since it becomes redundant toprevious analysis.Despite the fact that the results are already known, let us sketch a few of the details tomake clear the comparison between the helicity and St¨uckelberg decompositions at the scaleΛ . It is worth pointing out that at the scale Λ , the cubic interaction ∂ ˘ h ( ∂A ) from theEH is relevant and ought to be diagonalized, L (3)Λ , ∂ ˘ h ( ∂A ) = − ˘ h αβ ˆ E µναβ ∂ µ A γ ∂ ν A γ . (6.4)To diagonalize this interaction it is therefore necessary to make the field redefinitionˆ h µν = ˘ h µν − ∂ µ A γ ∂ ν A γ , (6.5)and as result we recover the fact that after this diagonalization, the helicity-2 mode at thatlevel is nothing else but what we would have inferred from the St¨uckelberg decomposition h µν = h Stueckelberg µν + 16 πη µν + M Pl (Ψ µα Ψ αν − η µν ) . (6.6) We have derived here the St¨uckelberg decomposition without ever invokingdiffeomorphism invariance!
To be clear we are not dealing with the St¨uckelberg decom-position in its standard presentation, but rather what the St¨uckelberg decomposition lookslike when translated back into unitary gauge. The relationship between the above definedunitary gauge metric and the metric in St¨uckelberg gauge, in which the 4 St¨uckelberg fieldsare taken to be dynamical, is given by g µν = g Sαβ ∂ µ Φ α ∂ ν Φ β (6.7)With g Sαβ = η αβ + h Sαβ and Φ A = x A + M Pl m (cid:0) A α + m ∂ α π (cid:1) we obtain h µν = M Pl ( g µν − η µν ) = M Pl (Ψ µα Ψ µα − η µν ) + h Sαβ ∂ µ Φ α ∂ ν Φ β (6.8)As long as we are working at energy scales below Λ this is equivalent to h µν = M Pl ( g µν − η µν ) = M Pl (Ψ µα Ψ µα − η µν ) + h Sµν (6.9)Finally writing h Sµν = h Stueckelberg µν + πη µν gives the desired expression h µν = h Stueckelberg µν + 16 πη µν + M Pl (Ψ µα Ψ µα − η µν )= h Stueckelberg µν + 12 m ∂ ( µ A ν ) + 13 m ∂ µ ∂ ν π + D Stueckelberg µν . (6.10)In [6], it was suggested that their apparent discrepancy between the St¨uckelberg lan-guage and the helicity decomposition is related to the fact that the helicity-0 mode in bothlanguages is related via a nonlinear expression π helicity η µν = π Stueckelberg η µν + 1Λ (cid:16) ∂ µ ∂ α π Stueckelberg ∂ ν ∂ α π Stueckelberg − · · · (cid:17) , (6.11)– 29 –here in their logic, π helicity identifies the helicity-0 mode in the helicity decomposition while π Stueckelberg is the scalar that appears in the St¨uckelberg language. By this point it is howeverobvious that this is by no mean the correct identification to be made. In reality, the mode π Stueckelberg is nothing else but the very same helicity-0 mode π helicity that appears in thehelicity decomposition. Instead, the St¨uckelberg and helicity decomposition are reconciledvia the correct identification of the helicity-2 part of the massive spin-2 field. As we haveseen, up to the scale Λ , the correctly diagonalized helicity-2 mode is not h helicity µν but instead h Stueckelberg µν given in (6.6). Of course there is no problem whatsoever in expressing h helicity µν interms of h Stueckelberg µν , their relation is perfectly well-defined and invertible. In this article we have considered the helicity decomposition of the two parameter allowedghost-free interacting models of massive spin-2 fields, i.e. massive gravity [2]. We find that,despite the apparent hierarchy of scales due to the appearance of terms in the Lagrangianat the intermediate scales Λ and Λ , the first interactions of the helicity modes arise at thescale Λ . We demonstrate this result without the use of field redefinitions. We show thatthe equations of motion of the helicity modes have a well-defined Cauchy problem in thepresence of matter, and that the symmetry that arises in the m → or Λ . Thisresult confirms the absence of ghosts in these decoupling limits. We show that perturbationof the spin-2 field remains well-defined below the scale Λ confirming that Λ is the truestrong coupling scale of the helicity-0 mode. Our results are completely consistent with theghost-free proof of these models using the ADM formalism [2, 4] and are consistent with howthe ghosts can be seen to be absent in the St¨uckelberg formalism [5]. Acknowledgments
We would like to thank Clare Burrage, Lavinia Heisenberg and Kurt Hinterbichler for usefuldiscussions and insightful comments. CdR is funded by the SNF and the work of GG wassupported by NSF grant PHY-0758032. AJT would like to thank the Universit´e de Gen`evefor hospitality whilst this work was being completed.
A Phase Space degrees of freedom
In this appendix we show how the Hamiltonian of the scalar field model (2.1) is perfectlywell defined and propagates only four degrees of freedom in phase space. To simplify we onlyfocus on the time dependencies, such that the Lagrangian is of the form L = 12 (cid:18) ˙ ψ + 2Λ ¨ φ ... φ (cid:19) + 12 ˙ φ . (A.1)To understand better the degrees of freedom, it is helpful to rewrite this as a constrainedsystem L = 12 (cid:18) ˙ ψ + 2Λ µ ˙ µ (cid:19) + 12 ˙ φ + ρ (cid:16) µ − ¨ φ (cid:17) . (A.2)– 30 –here the auxiliary field ρ enforces µ = ¨ φ making transparent the equivalence with theprevious system. Now on integration by parts this is equivalent to L = 12 (cid:18) ˙ ψ + 2Λ µ ˙ µ (cid:19) + 12 ˙ φ + ρµ + ˙ ρ ˙ φ (A.3)A priori this extends the phase space to a eight dimensional one (2 times too large),signaling the presence of a ghost, however the theory at hand is (as we know) extremelyspecial in that it propagates a constraint. One can start by defining the conjugate momentaassociated to φ , ψ , ρ and µ , P ψ = ˙ ψ + 2Λ µ ˙ µ (A.4) P φ = ˙ φ + ˙ ρ , (A.5) P ρ = ˙ φ (A.6) P µ = 2Λ µ (cid:18) ˙ ψ + 2Λ µ ˙ µ (cid:19) (A.7)Already it is clear that there is a constraint C = P µ − µP ψ = 0. Taking this into accountthe Hamiltonian is H = (cid:20) P ψ + P φ P ρ − P ρ − ρµ (cid:21) + λ (cid:18) P µ − µP ψ (cid:19) (A.8)However there is also a secondary constraint C = i [ C , H ] = ρ = 0 and a tertiary constraint C = i [ C , H ] = P ρ − P φ = 0. Putting this together we get H = (cid:20) P ψ + 12 P φ (cid:21) + λ (cid:18) P µ − µP ψ (cid:19) + λ ( ρ ) + λ ( P ρ − P φ ) (A.9)with λ , λ , λ Lagrange multipliers. Although C and C are second class constraints since[ C , C ] = i , C is clearly first class and thus removes two degrees of freedom. It generatesthe symmetry ψ → ψ + 2 µα ( t ), µ → µ − Λ α ( t ) for infinitessimal α ( t ). Thus there are atotal of 8 − − × ψ = ¯ ψ − ¨ φ / Λ , however since this is a non-canonical transformation it wasimportant to check that this redefinition could be consistently performed.Interestingly the special coupling to sources considered in Section 2.2 is precisely thecoupling L source = ¯ J φ φ + ¯ J ψ (cid:18) ψ + 1Λ ( ∂ α ∂ β φ ) (cid:19) = ¯ J φ φ + ¯ J ψ (cid:18) ψ + 1Λ µ (cid:19) (A.10)which preserves the first class symmetry. This is yet another way to understand how thecorrect number of physical degrees of freedom can be maintained at the quantum level evenin the presence of sources since it will be protected by this gauge symmetry. References [1] M. Fierz and W. Pauli, “On relativistic wave equations for particles of arbitrary spin in anelectromagnetic field,” Proc. Roy. Soc. Lond. A , 211 (1939). – 31 –
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