Consistent couplings between a massive spin-3/2 field and a partially massless spin-2 field

We revisit the problem of constructing consistent interactions between a massive spin-3/2 field and a partially massless graviton in four-dimensional (A)dS spacetime. We use the Stueckelberg formulation of the action principle for these fields and find two non-trivial cubic vertices with less than two derivatives, when going to the unitary gauge. One of the vertices is reminiscent of the minimal coupling of the massive spin-3/2 field to gravity, except that now the graviton is partially massless.


Introduction
The use of the Stueckelberg formulation for the problem of constructing consistent interactions between massive fields has proved very efficient, mainly through the works of Zinoviev and collaborators, see e.g.[1][2][3] and references therein.Some years ago in [4], Zinoviev constructed cubic couplings between a massive spin-3/2 field and a massive spin-2 graviton around (A)dS 4 background, with the assumption that the couplings should not bring more than one derivative.Then, the partially-massless limit for the graviton was considered, bringing the conclusion that no cubic vertex with one derivative survived in this limit.
We wish to revisit this question, this time taking the spin-2 field as partially massless from the very beginning of the analysis.Indeed, the operations of introducing interactions and taking a partially massless limit do not commute, in general.In fact, in this Letter we report about a coupling between a partially massless spin-2 field and a massive spin-3/2 field that seems to have gone unnoticed in previous investigations, as far as we could see.
In order to build consistent vertex involving massive fields (in the present case, the massive spin-3/2 field), we use the method proposed in [5] that combines the cohomological reformulation of the Noether method for gauge systems [6,7] with the Stueckelberg formulation for massive fields.The advantage of the method [5] is that it exploits the gauge structure of massless theories to describe interactions for massive fields.It proved useful in showing that dRGT gravity (see e.g.[8] for a review) can be recast in a frame where the Einstein-Hilbert structure disappears, leaving only a Born-Infeld-like theory, with vertices obtained by contraction of products of the manifestly gauge-invariant field strengths in the Stueckelberg formulation of the free massive theory.In particular, in [5] the full list of cubic vertices of massive dRGT gravity theory was recovered, showing the usefulness of the method.
In this Letter, we use the cohomological method of [5] for the search of consistent couplings between a massive spin-3/2 field and a partially massless (PM) spin-2 field, also called PM graviton.The spacetime backgrounds considered are the Anti-de Sitter (AdS) and the de Sitter (dS) geometries where the cosmological constant Λ is negative and positive, respectively.In such spacetimes, a PM graviton possesses four propagating degrees of freedom and is characterized by a well-suited mass directly related to the cosmological constant Λ , see [9][10][11] and references therein.In its PM phase, the graviton therefore propagates four degrees of freedom, exactly like the massive spin-3/2 field that carries four degrees of freedom.It is therefore natural to ask whether it is possible to elaborate a consistent gauge theory in which a PM graviton k µν and a massive spin-3/2 field ψ µ are involved.
We study the problem in both dS (Λ > 0) and AdS (Λ < 0) backgrounds at a stroke, through the use of a parameter σ that takes the value +1 in AdS and −1 in dS.

Main results
The main results we report in this Letter consist in the construction of two vertices expressed in the Stueckelberg formulation for both the PM spin-2 and the massive spin-3/2 fields.In the unitary gauge where the Stueckelberg fields are set to zero, the first vertex is proportional to where the spinor field ψ µ satisfies the Majorana reality condition and denotes the field for the massive spin-3/2 particle (the spinor indices are left implicit), the Lorentz-covariant derivative for the background geometry 2 is denoted by the symbol ∇ µ , and k µν represents the PM spin-2 field.The components of the (A)dS background metric will be denoted g µν .As usual, the four Dirac matrices are denoted by γ a , a = 0, 1, 2, 3 , and γ µ := e µ a γ a featuring the components of the background (A)dS vierbein.
The other coupling is more interesting.In the unitary gauge, it reads where the real parameter m is the mass of the spin-3/2 field in AdS, in the sense that the limit m → 0 is the limit where the spin-3/2 field enjoys a gauge symmetry that removes the helicity ±1/2 degrees of freedom, leaving only the helicity ±3/2 degrees of freedom on-shell.The tensor T µν is traceless and divergenceless on-shell: where a weak equality is an equality that holds on the solutions of the field equations for the free theory.The above vertex ℓ (2) = k µν T µν induces a deformation of the gauge transformations on the physical fields (k µν , ψ µ ) in the unitary gauge, given by where we recall that the free, quadratic action From the knowledge of the quadratic and cubic actions S 0 [k, ψ] and 2) in the unitary gauge, one readily finds the consistency of the deformation reported above: 2 We use conventions whereby the Lorentz-covariant derivative satisfies , where σ = ±1 .In other terms, the cosmological constant is Λ = −3 σ λ 2 in four dimensions, where σ = −1 corresponds to dS 4 and σ = 1 to AdS 4 .On a Dirac spinor ψ , we have As far as the deformation ℓ (2) is concerned, notice from ( 4) and ( 5) that the transformation of the massive spin-3/2 field can be written as δ 1 ψ µ = − ψ ν δ 0 k µν , from which it is tempting to view the contravariant spinor ψ µ as a gauge-invariant quantity, defining the covariant field as , where κ is the deformation parameter that we take with units of length, that defines the perturbative expansion ) .In this sense, like in Riemannian geometry, it would appear that a metric g µν := g µν − κ k µν + O(κ 2 ) could be defined in terms of the (A)dS background metric and the PM spin-2 field k µν .
In the following section, we present the two deformations reported above in their Stueckelberg form, and explain how one can reach the unitary gauge at first order in perturbation, thereby reproducing the main results presented above.

Consistent couplings in the Stueckelberg formulation
In this section we first spell out the free model, then exhibit the first order interactions we found and finally explain how one can reach the unitary gauge at first order in deformation.

The free model
We want to investigate the couplings between a massive spin-3/2 field and a PM spin-2 field.Our starting point will be the Stueckelberg formulation for these models.The Stueckelberg action for a massive spin-3/2 field and a PM spin-2 field in (A)dS 4 reads where For the spin-2 sector, we use the conventions of [12].The vector field B µ and the Majorana spinor χ are, respectively, the Stueckelberg companions of k µν and ψ µ .The action (7) is invariant under the following Abelian gauge transformations: It is useful to introduce the gauge-invariant quantities in terms of which the free action ( 7) can be written in a manifestly gauge invariant way: where The equations of motion obtained by extremizing the above action with respect to the fields k µν and ψ µ are, respectively, These equations are useful in checking the consistency of the vertex ℓ (2) .

Interactions to first order
We now report our main findings obtained following the method proposed in [5] for constructing interactions of massive fields in the Stueckelberg formulation.For the sake of conciseness, in this Letter we refrain from reviewing this method and instead spell out the results that can be checked without referring to the formalism developed in [5].
• In the Stueckelberg formulation, the deformation L 1 of the Lagrangian that corresponds to the first vertex ℓ (1) presented in the Introduction, reads up to trivial field redefinitions.The coefficient in front of it is, at this stage, arbitrary.
• In the Stueckelberg formulation, the deformation L (2) 1 of the Lagrangian that corresponds to the second vertex ℓ (2) presented in the Introduction reads Contrarily to the free Stueckelberg theory where the flat limit is smooth, in the interacting case one cannot take the limit λ → 0 , as the vertex is non-analytical in the constant λ .
• The above vertex induces a deformation of the gauge transformations given by The corresponding gauge algebra is The redefinition of the gauge parameters that trivializes the gauge algebra is In order to express our results in the unitary gauge, we first need to explain how to reach the unitary gauge, in perturbation.

Reaching the unitary gauge at first order in deformation
The starting point is a free theory S 0 [ϕ i , χ I ] with a spectrum of fields (ϕ i , χ I ) such that the latter are Stueckelberg companions of the former.In other words, the action S 0 = d n x L 0 is invariant under the gauge transformations where we use De Witt's condensed notation.The gauge invariance under the Stueckelberg gauge parameters ε I imply the Noether identities where the operator R + 0 i I denotes the adjoint of R 0 i I .We assume we also have a consistent, first order deformation of the action and gauge transformations, i.e., a functional S 1 [ϕ i , χ I ] = d n x L 1 and gauge transformation laws such that Upon expanding the latter equation using ( 21)-(25) gives the following Noether identity associated with the gauge parameters ε I : Inserting this expression for δL 1 δχ I in the equation ( 26) yields where Equation ( 28) expresses the ǫ α -gauge invariance of the action to first order in perturbation -here g denotes the coupling constant used in perturbation.This equation is valid for an arbitrary field configuration, in particular it is valid when we set the Stueckelberg fields χ I to zero.Using the following obvious equality gives In its turn, the latter equation expresses the gauge invariance, to first order in perturbation, of the reduced action Š[ϕ i ] = S 0 [ϕ i , χ I = 0] + g S 1 [ϕ i , χ I = 0] under the gauge transformations In the particular case studied in this Letter where we have the physical fields ϕ i = {k µν , ψ µ } and the Stueckelberg fields χ I = {B µ , χ} with the gauge transformations at zeroth and first order given in ( 8) and ( 15)-( 17), respectively, we find the equation ( 6): the reduced action is invariant under (4)- (5) as announced in the Introduction, where we renamed the scalar parameter π into ξ , absorbing in it the constant factor − σ λ .From the above-derived formula (33) for the gauge transformations δ ǫ ϕ i that leave invariant the reduced action, one can make an observation on the corresponding transformations of the Stueckelberg field strengths As is well-known, these quantities are invariant under the Stueckelberg transformations Under the complete transformation laws (21), ( 22), ( 24) and ( 25), the Stueckelberg field strengths Φ i transform as If, on the right-hand-side of the above formula, one sets χ I = 0 and ε I = − 1 m I R 0 I α ǫ α for the residual ε I (ǫ) parameters that preserve the unitary gauge χ I = 0 at zeroth order in perturbation, it turns out that one exactly recovers the expression for the δ ǫ ϕ i transformations (33).
In other words, one could turn the argument around and get a heuristic way of producing the right-hand side of formula (33): by demanding that the operations of setting the fields χ I to zero and performing gauge transformations commute on the Stueckelberg field strengths, i.e., imposing ( δΦ i )| χ=0,ε=ε(ǫ) = δ ǫ ( Φ i | χ=0 ) .

Conclusions and outlook
The first vertex L (1) 1 that we presented above in (13) does not deform the Stueckelberg gauge transformations (8).It is exactly invariant under the latter transformations.The second vertex L (2) 1 given in ( 14) is more interesting in the sense that it truly deforms the transformations (8).The first term of on the right-hand side of (15) is reminiscent of the local supersymmetry transformations in AdS background, giving the minimal deformation of the mass-like term on the right-hand side of δ 0 ψ µ = ∇ µ θ + ω 2 γ µ θ , see (8).Correspondingly, the first term on the right-hand side of ( 14) is the minimal deformation of the mass-like term for the spinor Ψ µ in the free action (10).However, contrarily to the situation in supergravity theories, there is no deformation proportional to the linearised "spin-connection" ∇ [µ k ν]ρ in (14) or in (15).
It will be interesting to investigate the consistent interactions among the fields of the enlarged spectra given in [13].There it was shown that the doublet (k µν , ψ µ ) consisting of a PM spin-2 and a massive spin-3/2 field studied in the present Letter must be completed with a masseless spin-(3/2, 1) doublet in order to carry the action of supersymmetry.We hope to report soon on the interactions among these four fields.In the paper [14], on the other hand, it was found that the partially massless doublet (5/2, 2) can be completed with a massless doublet (2, 3/2) in order to carry the action of supersymmetry.We intend to investigate the consistent couplings among those fields in a future work.