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Quantum M^2 ->2Lambda/3 discontinuity for massive gravity with a Lambda term

In a previous paper we showed that the absence of the van Dam-Veltman-Zakharov discontinuity as M^2 ->0 for massive spin-2 with a Lambda term is an artifact of the tree approximation, and that the discontinuity reappears at one loop, as a result of going from five degrees of freedom to two. In this paper we show that a similar classical continuity but quantum discontinuity arises in the"partially massless"limit M^2 ->2Lambda/3, as a result of going from five degrees of freedom to four.

In a previous paper [1], we showed that the absence [2][3][4] of the van Dam-Veltman-Zakharov discontinuity [5,6] for massive spin-2 with a Λ term is an artifact of the tree approximation, and that the discontinuity reappears at one loop. This result may be understood as follows. While a generic massive graviton propagates five degrees of freedom, gauge invariance ensures only propagation of the familiar two degrees of freedom of a massless graviton. Although the introduction of Λ = 0 allows for a smooth classical M 2 → 0 limit, the mismatch between two and five degrees of freedom cannot be eliminated altogether, and the discontinuity shows up at the quantum level.
Curiously, the presence of a cosmological constant allows for new gauge invariances of massive higher spin theories, yielding a rich structure of "partially massless" theories with reduced degrees of freedom [7]. In particular, for spin-2 a single gauge invariance shows up at the value M 2 = 2Λ/3, yielding a partially massless theory with four degrees of freedom [8][9][10]. In this paper we extend the result of [1] to the partially massless theory and show that a discontinuity first arises at the quantum level as M 2 → 2Λ/3.
We work in four dimensions with Euclidean signature (+ + ++). As in Refs. [4,1], we take the massive spin-2 theory to be given by linearized gravity with the addition of a Pauli-Fierz mass term. Thus our starting point is the action where S L is the Einstein-Hilbert action with cosmological constant , linearized about a background metric g µν satisfying the Einstein condition, R µν = Λg µν . Takingĝ µν = g µν + κh µν where κ 2 = 32πG, this linearized action for h µν is whereh µν = h µν − 1 2 g µν h σ σ . All indices are raised and lowered with respect to the metric g µν , and ∇ µ is taken to be covariant with ∇ µ g λσ = 0. Furthermore, the source term is given by As in Ref. [4], we apply the simplifying assumption that T µν is conserved with respect to the background metric, ∇ µ T µν = 0. S L and S T together correspond to the linearized massless theory coupled to a conserved source. Each term independently has a gauge symmetry described by a vector ξ µ (x): corresponding to diffeomorphism invariance of the Einstein theory. Introduction of the Pauli-Fierz spin-2 mass term, breaks the symmetry (4). However, at the critical value M 2 = 2Λ/3, there remains a residual symmetry parameterized by α(x). This gauge invariance was first noted in [8], and results in a partially massless de Sitter theory with four degrees of freedom and propagation along the light cone. It also requires that the coupling to matter be via a tracelees energy-momentum tensor. We wish to consider the generating functional Since the generic theory with mass term has broken gauge invariance and a quadratic action, it may be quantized in a straightforward manner. On the other hand, for the case M 2 = 2Λ/3, one would first gauge fix the symmetry (6) before proceeding. However, to make contact with previous results for the pure massless case, we find it useful to reintroduce the gauge symmetry (4) using a Stückelberg [11,2] formulation. This allows a uniform approach to quantization throughout the (Λ, M 2 ) plane, and provides connection to the operators appearing in Ref. [12] for the massless case, as well as the ones in Ref. [1] for the massive case.
For any value of M 2 > 0, we introduce an auxiliary vector field V µ to restore the gauge symmetry (4). We first multiply Z[g, T ] by an integration DV over all configurations of this decoupled field, and then perform the shift h µν → h µν − 2M −1 ∇ (µ V ν) . Since S L and S T are gauge invariant in themselves, the only effect of this shift is to make the replacement in (7). Thus S M becomes a "Stückelberg mass", and gauge invariance is restored, yielding the simultaneous shift symmetry For generic M 2 , this is the only symmetry of theory. However, for M 2 = 2Λ/3, the additional symmetry (6) remains even after the Stückelberg shift. Note that this symmetry is a combination of a Weyl scaling and diffeomorphism [with parameter ξ µ (x) = ∇ µ α(x)].
Since the latter has been restored by the addition of V µ , we are now able to disentangle the two. The resulting gauge symmetry for the partially massless theory may be written as with parameters ξ µ (x) for diffeomorphisms and α(x) for Weyl rescalings. For the partially massless theory, there are five degrees of freedom to gauge fix. As in [1], we make use of diffeomorphisms to identify V with the longitudinal part ofh, i.e. MV µ = ∇ ρh ρµ . Additionally, the conformal rescaling may be used to make h µν traceless.
This choice is made in order to simplify the relevant operators appearing in the action, and is accomplished by adding to the action the gauge-fixing terms In conjunction with this gauge fixing, it is necessary to include a Faddeev-Popov determinant connected with the variation of the gauge condition under (10). It is straightforward to show that the appropriate determinant is corresponding to the set of gauge parameters (ξ µ , α To highlight the tensor structure of the gauge-fixed action, we decompose the metric fluctuation h µν into its traceless and scalar parts: φ µν ≡ h µν − 1 4 g µν h σ σ , and φ ≡ h σ σ . The source may similarly be split into its irreducible components j µν and j, so that T µν = j µν + 1 4 g µν j. The gauge-fixed partially massless action then becomes The second-order spin operators are the scalar Laplacian ∆(0, 0) ≡ −✷ and the Lichnerowicz operator for symmetric rank-2 tensors ∆(1, 1)φ µν = −✷φ µν + R µτ φ τ ν + R ντ φ τ µ − 2R µρντ φ ρτ [12].
The Stückelberg field, V µ , in (13) appears as a massive spin-1 field in the Einstein background with an effective mass m 2 = −4Λ/3. We now restore vector gauge invariance by repeating the Stückelberg formalism. Thus we introduce a scalar field χ and make the change of variables V µ → V µ −M −1 ∇ µ χ. By construction, the resulting action is now invariant under the gauge transformation One can then choose a gauge-condition to simplify the shifted action. It is useful to associate the longitudinal component of V with χ according to M∇ · V = (−2Λ + M 2 )χ. This is done by adding a gauge-fixing term along with a corresponding scalar Faddeev-Popov determinant The final completely gauged-fixed action for the partially massless graviton now takes the formS Along with the addition to the two Faddeev-Popov determinants (12) and (16), this provides a complete description of Z, including couplings to the background metric. This is to be compared with the generic massive case where the corresponding action is given by [1] Note that in the partially massless case the trace mode φ has disappeared except for its coupling to the trace of the energy-momentum tensor. With φ now acting as a Lagrange multiplier, this indicates that the theory couples to conformal matter. To compare the massive and partially massless theories at the classical level, therefore, let us assume that the massive theory also couples to matter with T µ µ = 0 as well as ∇ µ T µν = 0. Then the tree-level amplitude for the current T µν can be read from the action (18) directly and is given by since there are sources for neither V µ nor χ. Thus at tree level, there is no discontinuity in taking the M 2 → 2Λ/3 limit. We note here that there would be sources for the Stückelberg fields if one were to relax the assumption of a conserved stress tensor or a traceless stress tensor. In this case, one needs only to account for the shifts in h µν and V µ to see how T µν contributes to sources for V µ and χ.
For the partially massless case, (17), we integrate over all species to find the first quantum correction Even for a de Sitter background with constant curvature R µνρσ = 1 3 Λ(g µν g ρσ − g µρ g νσ ) , there is no cancellation. Thus we conclude that the absence of a discontinuity between the M 2 → 2Λ/3 and M 2 = 2Λ/3 results for massive spin-2 is only a tree-level phenomenon, and that the discontinuity itself persists at one loop. That the full quantum theory is discontinuous is not surprising considering the different degrees of freedom for the two cases. Just as the M 2 → 0 limit is discontinuous at the quantum level as a result of going from five degrees of freedom to two, so the M 2 → 2Λ/3 limit is discontinuous as a result of going from five degrees of freedom to four.

ACKNOWLEDGMENTS
We wish to thank S. Deser for enlightening discussions and F. Dilkes for initial collaboration. This research was supported in part by DOE Grant DE-FG02-95ER40899.