Covariant Approach to the No-ghost Theorem in Massive Gravity

We discuss the no-ghost theorem in the massive gravity in a covariant manner. Using the BRST formalism and St\"{u}ckelberg fields, we first clarify how the Boulware-Deser ghost decouples in the massive gravity theory with Fierz-Pauli mass term. Here we find that the crucial point in the proof is that there is no higher (time) derivative for the St\"{u}ckelberg `scalar' field. We then analyze the nonlinear massive gravity proposed by de Rham, Gabadadze and Tolley, and show that there is no ghost for general admissible backgrounds. In this process, we find a very nontrivial decoupling limit for general backgrounds. We end the paper by demonstrating the general results explicitly in a nontrivial example where there apparently appear higher time derivatives for St\"{u}ckelberg scalar field, but show that this does not introduce the ghost into the theory.


Introduction
Recently, there has been renewed interest in the search for the modification of gravity at large distances by adding mass terms for the graviton. The motivation for this comes from both theoretical and observational considerations.
On the theoretical side, it is interesting to explore the possibility of formulating a massive spin-2 field theory. In general relativity, which describes the massless spin-2 field, the four constraints of the theory together with the invariance under the four general coordinate transformations remove eight of the modes from the ten degrees of freedom in the metric, and the number of propagating modes reduces to the physical two modes of the massless graviton. When the mass term is added, the four constraints remove four propagating modes, but the general covariance is broken. Thus there remain six degrees of freedom in general. Five of these are the modes of the massive spin-2 graviton, but it turns out that the sixth scalar mode is a ghost with a negative metric. A unique mass term that does not contain this ghost is known as a Fierz-Pauli mass term [1].
However, it has been pointed out that this theory suffers from the problem that the helicity-0 state couples to the trace of the matter stress-energy tensor with the same strength as the helicity-2 state [2,3]. This means that this massive gravity does not continuously reduce to general relativity in the massless limit. This is called the van Dam-Veltman-Zakharov discontinuity. It was then argued by Vainshtein that the discontinuity could be avoided by the nonlinear interaction [4]. Unfortunately, the very nonlinearity that cures the discontinuity problem re-introduces the ghost into the theory, known PTEP 2014, 043B04 T. Kugo and N. Ohta the Stückelberg fields and BRST formalism in this theory. Here we count the degrees of freedom, and clarify that the necessary and sufficient condition for the theory to be ghost-free is that there is no higher (time) derivative of the Stückelberg fields. The discussion is completed in Sect. 2.2, where we compute the propagators for all the fields in the theory and show that all the ghost degrees of freedom cancel against the Faddeev-Popov ghosts, and there remain only physical (D−2)(D+1) 2 (five for four dimensions) degrees of freedom for spin-2.
In Sect. 3, we come to the main theme of this paper: to prove the no-ghost theorem in nonlinear massive gravity on arbitrary backgrounds. In Sect. 3.1, we first discuss how to diagonalize the general background metric in order to properly take its square root, which is necessary to rewrite the mass term suitable for examining the spectrum. We then use this result in Sect. 3.2 to compute the generating function of the mass terms. In Sect. 3.3, we find that there is an important hidden U (1) gauge invariance that ensures the decoupling of the ghost. In this process, we find that the way how to introduce the Stückelberg fields in a general background is significantly modified from its counterpart for a flat background, and the associated decoupling limit is also quite nontrivial. We then show that there are no higher derivative terms for the Stückelberg fields. Combined with the above result in the Fierz-Pauli mass term, this implies that there remains no BD ghost in this massive gravity. In Sect. 4, we go on to discuss an explicit and nontrivial example for a background metric with shift. We show that, naively, higher time derivatives on the Stückelberg scalar field seem to appear, but our definition of the Stückelberg fields avoids the trouble, so that there is no BD ghost in the theory.

Absence of ghosts in the Fierz-Pauli mass term
In this section, we first discuss the no-ghost theorem in massive gravity with the Fierz-Pauli mass term in arbitrary dimensions D. Let us consider the action where κ 2 is the D-dimensional gravitational constant, m and a are constants. Here h μν is the fluctuation of the metric around the background spacetime g μν =ḡ μν + κh μν , (2.2) and h ≡ḡ μν h μν . We use the conventions in Ref. [21] and set κ = 1 henceforth. In the rest of this section, we consider the flat backgroundḡ μν = η μν for simplicity. At first sight, one expects that this theory contains (D−2)(D+1) 2 (five for four dimensions) degrees of freedom, corresponding to the massive spin-2 field. However, it has been shown that this massive gravity contains an additional (sixth in four dimensions) degree of freedom unless a = 1, known as the BD ghost [5]. We first recapitulate how to understand this situation. PTEP 2014, 043B04 T. Kugo and N. Ohta replacement reduces at the linearized level simply to [22] where m is a mass scale. The metric is invariant under the transformation In order to quantize the theory, we gauge fix the theory and introduce the Faddeev-Popov ghosts and anti-ghosts corresponding to the invariance (2.4). They are the vector ghost c μ and anti-ghostc μ for ξ μ , and the scalar ghost c and anti-ghostc for . Now the physical degrees of freedom in the theory are counted as D(D+1) 2 (10 for four dimensions) from h μν , D (4 for four dimensions) from A μ and 1 from π , minus D × 2 (4 × 2 for four dimensions) from the vector ghost and anti-ghost, minus 2 from the scalar ghost and anti-ghost. This leaves us with (D−2)(D+1) 2 (5 for four dimensions) degrees of freedom, the right number for massive spin-2 fields. However, this cannot be true in general. It is known that the theory (2.1) describes D(D−1) 2 (6 for four dimensions) degrees of freedom unless a = 1 and one of them is a ghost. What is wrong with this counting then?
We can see the origin of the problem if we substitute (2.3) into the action: The quadratic part of the mass terms of the Lagrangian in (2.1) takes the form The last term here indicates that the field π has two degrees of freedom unless a = 1, so the above counting is not correct. If and only if a = 1, the above counting is correct and we are left with (5 for four dimensions) degrees of freedom. Note that the terms involving only π vanish in this case, and this corresponds to the requirement of no ghost in the decoupling limit. The mixing with the metric fluctuation gives the dynamics to π . This can be checked by taking the determinant of the kinetic matrix (containing only second derivatives) to see if it does not vanish identically.
Alternatively, we can see that the shift h μν → h μν + 2 D−2 η μν π in the Einstein term cancels the mixing and produces a normal kinetic term for π . Here we have defined At this stage, we have (2.8) Here and henceforth in this section, all the h μν and h denote the new gravity fields after the above shifting: We now discuss the gauge fixing of the theory and examine explicitly what spectrum we have. In order to resolve the field mixing terms, we adopt the so-called R ξ gauges. The gauge fixing and 4/21 Faddeev-Popov terms are concisely written as where α, β, x, y, z are gauge parameters, and the (fermionic) BRST transformations are defined as Using (2.10), we find the gauge fixing and Faddeev-Popov terms as follows: Here B μ and B are the shifted B μ and B fields to complete the squares: (2.14) Now we determine the gauge parameters x, y, and z so as to cancel the various field transition terms as follows. Note that this gauge fixing term (2.12) is arranged to cancel the term −m A μ h μ in (2.8).
In order to cancel the A μ ∂ μ h term in (2.8) by the corresponding terms from the gauge fixing term (2.12), we should have To cancel the A μ ∂ μ π term in (2.8) by a term from (2.12), we set Finally, the hπ mixing term in (2.8) can be canceled by that from (2.12) by choosing The resulting total quadratic Lagrangian is  Using these, we find that the quadratic term in the gravity field h μν is written in the form where is given by the inverse of the kinetic operator: This condition requires so that we find We see that most of the terms have gauge-dependent masses, which should cancel with the Faddeev-Popov ghost.

Faddeev-Popov ghost propagator.
The kinetic term of the Faddeev-Popov ghosts is: We find the propagator from the inverse of this: (2.35)

Vector propagator.
The kinetic term of the vector field A μ is given by 36) whose inverse gives the vector propagator: (2.37) Note that the massless singularities contained in d μν and e μν have actually been canceled in this vector propagator. This should be so since those singularities are of course not physical but an artifact 7 T. Kugo and N. Ohta of our computational device using projection operators. The same cancellations of massless singularities have also occurred in the above tensor propagators, which the reader can confirm by using the above expressions for the tensor propagator. Summarizing, we have the following propagators after suitable normalization: h μν -sector: (2.38) A μ -π -sector: We see that almost all modes cancel out with the Faddeev-Popov ghosts, and we are left with (five for four dimensions) modes of the symmetric transverse-traceless tensor h μν with mass m.

Absence of ghosts in nonlinear massive gravity
We now consider four-dimensional theory for nonlinear massive gravity as formulated by dRGT [6,7]. For simplicity, here we discuss only four-dimensional theory, but the generalization to arbitrary dimensions is straightforward. The action is given by [6,7] where L mass is given by Here c 3 , c 4 are parameters and where f μν is a fiducial metric that can be chosen to be a flat metric η μν . Actually, we would like to keep the general coordinate invariance by introducing the Stückelberg field Y M . Following Ref. [ Here Y M is a coordinate in the "target space" and we can set it to where μ and M represent the "worldsheet" and "target space" indices, respectively. The original dRGT formulation corresponds to taking the target space metric G M N to be flat Minkowski's η M N , as we follow henceforth. We then have where we freely raise and lower the index We should note that, although we use the formulation in which the general coordinate invariance is recovered, our following discussions proceed with this choice of fiducial metric; we restrict ourselves to the frame where the Stückelberg fields φ μ have no vacuum expectation value.
We are interested in the question of whether there are higher time-derivative terms in the Stückelberg fields. To study this, we introduce the metric fluctuation around the general background g μν : However, since we are interested only in the question of whether there remains a BD ghost that exists in the Stückelberg modes, we can simply set the graviton fluctuation to zero: and study the spectrum. What we have to show now is that there are no higher derivative kinetic terms for the Stückelberg fields. If this is confirmed, the preceding discussion shows that we have only five degrees of freedom and there is no BD ghost.

Diagonalizing the background
The expansion of the square root ḡ −1 f around the general backgroundḡ is very complicated in general, if not impossible. For example, one cannot simply make an expansion like unless the matrices A and B commute with each other. We can make a general expansion around a unit matrix as with a binomial coefficient n C 1/2 , but then the term (A − 1 + B) n is not so simple:  (3.14) in matrix form. The c-number partḡ −1 η can generally be made diagonal by a matrix V . This is true when all the eigenvectors of the 4 × 4 matrixḡ −1 η are independent and not degenerate. Degeneracy of the eigenvectors may occur at measure-zero points in the functional space of the background metric g μν . Moreover, as we shall see later in an explicit example, we suspect that such degeneracy occurs at the metricḡ μν , which corresponds to a rather singular and unphysical background. Therefore, we confine ourselves to the cases where the matrixḡ −1 η can be made diagonal.
Let α(n) (n = 1, 2, 3, 4) be the roots of the characteristic equation det[x1 −ḡ −1 η] = 0, and V n be eigenvectors of the matrixḡ −1 η belonging to the eigenvalue α(n): Note that we use Roman letters to denote the eigenvector labels in distinction to the original vector indices denoted by Greek letters. Since the matrixḡ −1 η satisfies it is made diagonal as Noting thatḡ is real symmetric, we can show that the matrix V satisfies Indeed, using Eq. (3.16) and also its transpose, we can show If all the eigenvalues are different from one another, this implies that V T ηV is diagonal, so that we can realize V T ηV = 1 by the normalization condition for the eigenvectors. Even if some eigenvalues are degenerate, we can realize it as the orthonormalization condition in each common eigenvalue sector. Performing a similar transformation to (3.14) with the matrix V , and using the relation (3.18), we find It is important to notice here that both the "vector" indices μ of ∂ μ and of the Stückelberg field φ μ are commonly transformed by the matrix V : so that We should emphasize here that the derivatives ∂ μ are only acting on the Stückelberg field and never differentiate the "rotation matrix" elements V μ m , even if the V μ m are written after ∂ μ . 10 Now the c-number part A (0) of this matrix is diagonal and its square root is simply given by It is more convenient to make the matrix (3.22) symmetric, so we further perform a similarity transformation by B (0) , and call the resultant symmetric matrix A: The matrices A (1) and A (2) are the linear and quadratic terms, respectively, in the Stückelberg field φ and their matrix elements are given more explicitly by Here the double-barred quantitiesφ and∂ are defined as and the same for∂ m with the understanding that the derivative acts only on φ but neither on √ α(m) nor on V μ m . Remember that the "vector" index of the barred quantities∂ andφ defined in (3.21) now stands for the rotated one by V , and that of the double-barred quantities∂ andφ for the "rotated" one by V B (0) .
Before beginning a detailed computation, let us look at the "decoupling limit" at this stage. Our inspection of the expressions (3.25) finds it natural to define a decoupling limit by the following replacement, similar to the decoupling limit in the flat background case: It should be noted that the coefficients (V B (0) V −1 ) ν μ here must be real in order for this replacement to make sense. This is because φ μ and ∂ ν π are real fields. Fortunately, from (3.17) and (3.23), we have V A (0) V −1 =ḡ −1 η and hence so that V B (0) V −1 is a real matrix as long as ḡ −1 η is real. But the latter is the very condition that the present dRGT theory has a Hermitian mass term, so that it holds as long as the present theory makes sense. We also note that this decoupling limit is quite nontrivial because it mixes time and spatial derivatives by the coefficients (V B (0) V −1 ) ν μ in general. This happens when the background metricḡ has 11/21 Downloaded from https://academic.oup.com/ptep/article-abstract/2014/4/043B04/1620633 by guest on 29 July 2018 a time-space component (shift). We will see this in more detail in an explicit example later. On the flat backgroundḡ = η, this of course reduces to the usual one φ μ → ∂ μ π , and they are not mixed.
In this decoupling limit on the general background, we havē That is, the derivative operator∂ m here is understood to act only on the field π but not on the coefficients (V B (0) ) ν n , and then∂ m and∂ n are commutative on π . If we define a symmetric matrix by mn =∂ m∂n π, (3.31) then, from Eq. (3.25), we have in this limit where (∂π -term) denotes the first-order derivative terms of the π field. Namely, if we keep only the second-order derivative terms of π , neglecting the first-order derivative terms, then the matrix A takes a very simple form: That is, as far as the second-order derivative terms ∂∂π are concerned, in this decoupling limit and so there appear no quadratic terms of the Stückelberg field . This is a very similar situation to the flat background case, where actually it gave dRGT the motivation for taking the square root form for the mass term. This form (3.34) of √ A guarantees that the dRGT mass terms generated by det[1 + λ √ A] clearly have total derivative forms in the decoupling limit as far as the higher derivative terms ∂∂π are concerned.
Therefore, similarly to the flat case, we expect that the original Stückelberg "vector" field φ μ appears only in the following "gauge-invariant" tensor combination in the quadratic terms in the mass term: This combination of∂ andφ is suitable because of the form (3.27) of the decoupling limitφ μ → ∂ μ π . We shall now show that this is indeed the case if we neglect some lower-order derivative terms. 12

Computing the general mass terms
Let us compute the generating function of the general mass terms: Since this is invariant under the similarity transformation, we can use the expression A in (3.24) for the matrix ḡ −1 f : The square root of the matrix A can be calculated order by order in the Stückelberg field φ thanks to the fact that the matrix B (0) is diagonal. The matrix equation for X can be solved explicitly [23]. The solution X to this equation, denoted formally as (B (0) * ) −1 C, is given explicitly by This formula enables us to find the square root of A: Substituting the expression (3.25), we find  (2) , We now simplify each term. First consider tr β (2) Averaging with the term obtained by exchanging the dummy indices m ↔ n, we get The contribution of the second term in the bracket here is combined with the tr[β (1) β (1) ] term to yield which partially cancels the first term in (3.46). We are thus left with The first term takes a "gauge-invariant" form while the second term does not. The latter term is, however, almost "canceled" by the remaining term in (3.44): If we could do a partial integration with respect to the differential operators∂ m and∂ n here, this term would really cancel the second term in (3.48). But there are various x-dependent factors √ α(m) and V μ m in front of the differential operators, the cancellation is not complete, and the terms with lower derivative terms of the form φ∂φ or φφ remain. 14 The final quadratic terms are thus given by (3.50)

Gauge invariance and the no-ghost theorem
As anticipated from the consideration of the decoupling limit, the resultant generic mass term is almost "gauge invariant" under δφ m =∂ m , or, more precisely, Actually, it is not exactly invariant since the coefficients V μ m and √ α(m) are x-dependent and the derivatives do not commute with them. So we find that it is convenient to introduce the Stückelberg "scalar" field π by (3.52) Then the U (1) gauge invariance under becomes exact since the change cancels between the A μ and ∂π terms, leaving φ μ intact. It is important to make this U (1) gauge invariance exact; this is because it is lifted to the BRST invariance to define the physical subspace in covariant gauges so that it must be an exact gauge symmetry of the total action. The above-mentioned approximate "gauge invariance" under (3.51), on the other hand, guarantees that the higher derivative terms in the kinetic term of the π field cancel. This is essentially due to the fact that the Stückelberg field expression (3.52) for φ μ is defined in accordance with the decoupling limit (3.27).
Let us now explicitly show that the higher derivative terms of the π field indeed cancel in the kinetic term (3.50).
First, consider the first term in (3.50) written in terms of F mn =∂ mφn −∂ nφm . Note that the Stückelberg expression (3.52) for φ gives where∂ m∂n π is defined in (3.30) and the coefficient C ρ mn of the ∂π term is given by Recalling that∂ m∂n π defined in (3.30) is symmetric under m ↔ n, we see that the second-order derivative terms∂∂π cancel in Next, consider the second term in (3.50). In order to do the partial integration carefully, let us make explicit the factors contained in the definitions of barred quantities: We define the coefficient C μν , which will frequently appear below: Noting V −1 = V T η, we can rewrite the Stückelberg field expression (3.52) in the form (3.59) We find∂ and, similarly,∂ Consequently, the second term in (3.48) can be put, after performing partial integrations twice, into the form where c stands for all the prefactors in front of this term in the action (including det √ḡ ). Now the first term on the right-hand side of (3.62) contains only φs with no derivatives, so that it contains at most first-order derivatives of the π fields. The second term seems to contain ∂φ, which gives the second-order derivative of π since ∂ μ φ β = ∂ μ A β + ∂ μ (η βρ C ρν ∂ ν π). (3.63) Nevertheless, we now show that these second-order derivative terms of π vanish. Since the first-order derivative of the "vector" field A μ is in any case contained in the action, we can forget about it here. Keeping only the π field in φ, we find that the second term of (3.62) becomes 2φ α ∂ ν (cC μα C νβ ) · ∂ μ φ β | π 2 terms = 2η αδ C δρ ∂ ρ π · ∂ ν (cC μα C νβ ) · η βγ ∂ μ C γ τ · ∂ τ π + η βγ C γ τ · ∂ μ ∂ τ π . (3.64) The first term is harmless with only the first derivatives on the π s, but the last term is the dangerous one containing the second derivative ∂∂π, which we write in the form the last term of (3.64) = 2d μνρ ∂ ν π · ∂ μ ∂ ρ π ≡ L , (3.65) by introducing a coefficient By performing a partial integration for ∂ μ , we can rewrite (3.65) as L = −2d μνρ ∂ μ ∂ ν π · ∂ ρ π − 2(∂ μ d μνρ )∂ ν π · ∂ ρ π. (3.67) Averaging these two expressions (3.65) and (3.67), we have Noticing that the coefficient of the first term d μρν − d μνρ ≡ 2d μ [ρν] is antisymmetric under ρ ↔ ν, we can make a partial integration to put it into the first-order derivative terms: We have thus shown that all the π field terms can be put solely into first-order derivative terms. So the quadratic part in the fields of the mass term takes the usual form L(ϕ, ∂ϕ), containing only up to first-order derivatives for all the fields ϕ = {h μν , A μ , π}. 1 The U (1) gauge invariance is exact and all the fields appear only up to the first-order derivative in the quadratic kinetic term. On any background metric, the particle modes are determined by the quadratic terms. Combined with our previous counting of physical degrees of freedom, this implies that there appears no BD ghost mode in this theory on the general background metric, and completes our proof of the no-ghost theorem.

Discussions
It is instructive to see explicitly the general result in the previous section for a concrete nontrivial background example. Let us consider the following background metricḡ μν , discussed by dRGT [10]: This is the metric with the lapse N = 1 and the shift vector N i = 2l i . Since the space metric γ i j is taken to be δ i j , we can freely rotate the spatial axis such that the shift vector points in the x 1 direction: For this background metricḡ μν , we have where the blank entries are all zeros. The characteristic equation for the first nontrivial 2 × 2 matrix in the (x 0 , x 1 ) subspace is The metric is flat for l = 0. For the reason to become clear shortly, we consider only the case |l| < 1.
The eigenvalues are then complex: The eigenvectors for these two eigenvalues in the (x 0 , x 1 ) subspace are conveniently chosen as Note that α and a are unimodular: αα * = 1 = aa * , and satisfy √ α The other two eigenvalues and eigenvectors in the (x 2 , x 3 ) directions are trivial. Hence the matrix V that diagonalizes the matrixḡ −1 η in (4.3) and the diagonalized matrix are given by Note that this matrix V is properly normalized so as to satisfy Eq. (3.18): Now the barred derivatives∂ m = V μ m ∂ μ defined in Eq. (3.21) are explicitly read as and, of course,∂ 2 = ∂ 2 ,∂ 3 = ∂ 3 . The barred fieldsφ m = V μ m φ μ are similar: Our result for the general mass term det[1 + λB]| quad was given in Eq. (3.50). If we keep only the nontrivial terms∂ mφn with (m, n) = (1, 0) and (0, 1), it gives Substituting Eqs. (4.10) and (4.11), and using Eq. (4.7), we find that this reduces to whereφ ≡ ∂ 0 φ, φ ≡ ∂ 1 φ. Note that the second term has lost the x μ -dependent coefficients and the overall factor √ −ḡ = 1 in front is also x μ -independent, so the second term can be partial-integrated away. Note also that the first term contains the square ofφ 0 , which would yield the square of the second-order time derivativeπ if we had introduced the Stückelberg scalar field π in the same manner as the flat background case: (4.14) As was argued in Ref. [10], this term is actually harmless becauseφ 0 comes into the action only with the particular combination (φ 1 − lφ 0 ) withφ 1 and does not give rise to another degree of freedom PTEP 2014, 043B04 T. Kugo and N. Ohta than φ 1 . In our discussions, we can see the absence of the ghost in a better way. It is important to remember that the proper way of introducing π in the general background is not (4.14) but (4.15) as given in Eq. (3.52). The coefficient which is indeed real, as it should be. Therefore our definition of the π field yields If we substitute this into the first term in (4.13) and concentrate on the second-order derivative terms of π (forgetting about the terms with the coefficients differentiated), we havė Thus we explicitly see that all the second-order derivative terms of π disappear, as was shown generally in the previous section. This is due to the "gauge invariance" of the F mn term under δφ m =∂ m π . This also clearly shows the importance and nontriviality of our definition of the Stückelberg π field or decoupling limit in the general curved spacetime. When l becomes 1, our expression for the quadratic term of the mass term diverges (see Eq. (4.13)). What happens there?
As long as the condition l 2 < 1 is satisfied, the characteristic equation (4.4) has two roots α and α * , and the matrixḡ −1 η is diagonalizable. But when l becomes as large as 1, the complex eigenvalues α and α * become degenerate and take the value −1, and the corresponding eigenvectors V 1 and V 2 also degenerate, i.e., N V 1 ∝ N V 2 . This implies that the eigenvectors do not span a complete set, so that the matrixḡ −1 η is non-diagonalizable. At l = 1,ḡ −1 η can be brought at most into a Jordan standard form: The form of the quadratic kinetic term for the Stückelberg fields, which was derived in the previous section assuming diagonalizability, diverges in the limit l → 1. Fortunately, the mass term 1 can be calculated exactly in this example if we retain only the ∂ μ φ ν terms with μ, ν = 0, 1. 2 This is fine since we are mainly interested in the time If we look at Eq. (4.21) for l = 1, we see that theφ μ = 0 point becomes the branch point of the square root, so that the expansion itself of the mass term in powers of the Stückelberg fields φ does not make sense. The origin of the square root here is, of course, the square root factor g −1 f of the dRGT mass term. So even if we do not introduce the Stückelberg fields φ (i.e., setting f μν = η μν ), this singularity at g =ḡ with l = 1 is the singularity of the Lagrangian itself and the metric fluctuation h μν around the background g =ḡ does not make sense. This does not allow for any particle interpretation.
Also, beyond |l| = 1, the background value inside the square root in Eq. (4.21) is negative, and again this implies that the the square root factor g −1 f in the dRGT mass term comes to have a complex value at the background g =ḡ so that the dRGT Lagrangian itself becomes non-Hermitian and no longer gives a well defined theory. This is the reason why we have to restrict the shift vector to |l| < 1; in this region our discussions work perfectly well and there is no ghost in this massive gravity. This must be the general situation; as long as the dRGT mass term defines a Hermitian Lagrangian, then the matrixḡ −1 f is diagonalizable and the general no-ghost proof in the previous section will apply.
In summary, we have discussed the no-ghost theorem in massive gravity. We start with a discussion of the simple gravity theory with the Fierz-Pauli mass term and analyze the spectrum in a covariant manner. Naively we have six degrees of freedom since the general coordinate invariance is broken in the presence of the mass term. However, we have shown that one of the modes, the BD ghost, decouples for the special choice of the mass term. By introducing the Stückelberg fields, which recover the general coordinate invariance, and using the BRST formalism, we have then clarified how the various modes in the theory cancel each other, leaving the correct five degrees of freedom. The crucial point in this formulation is that there remains no higher (time) derivative on the Stückelberg fields.
We then proceed to a discussion of the nonlinear dRGT massive gravity on arbitrary backgrounds. Because of the complicated nature of the square root form of the mass term, it is rather cumbersome to identify fluctuations around arbitrary backgrounds, but we were able to do it by diagonalizing the background. We have then shown that there remains no higher (time) derivatives on the Stückelberg fields, and hence the theory is ghost free. In this process, we have identified the correct way to introduce the Stückelberg fields on general backgrounds, and found that the associated decoupling limit is also quite nontrivial, naively mixing time and space derivatives. Nonetheless, we have shown that this does not cause trouble with the ghost. Rather, this is necessary in order for the ghost to decouple. This is further confirmed by an explicit example.
Recently, it has been shown that this class of massive gravity can be derived from the fivedimensional Einstein gravity by deconstruction [24]. It would be interesting to extend that approach to supergravity and study the structure of the theory.
properties of the Pauli matrices, one can easily find the square root as