A theory for multiple partially massless spin-2 fields

Nicolas Boulanger, Cédric Deffayet, 3 Sebastian Garcia-Saenz, and Lucas Traina Service de Physique de l’Univers, Champs et Gravitation, Université de Mons, UMONS Research Institute for Complex Systems, Place du Parc 20, 7000 Mons, Belgium Sorbonne Université, UPMC Paris 6 and CNRS, UMR 7095, Institut d’Astrophysique de Paris, GReCO, 98bis Boulevard Arago, 75014 Paris, France IHES, Le Bois-Marie, 35 Route de Chartres, 91440 Bures-sur-Yvette, France


I. INTRODUCTION
Partially massless (PM) particles arise as special representations of the (anti-)de Sitter ((A)dS) group, with physical properties intermediate between those of a massless and generic massive particles: they possess a nonzero mass proportional to the (A)dS curvature scale, and also a gauge symmetry that reduces the number of propagating degrees of freedom [1,2]. The simplest PM field has spin s = 2, and the possible relevance of a PM graviton in the contexts of gravitational physics and cosmology has recently attracted much attention [3][4][5][6]. At a more formal level, PM fields are of interest in the endeavor to understand the structure of higher-spin gauge theories for which the irreducible representations of the (A)dS group play a prominent role [7][8][9][10][11].
Although interactions for PM fields of spin s > 2 remain largely unexplored, the spin-2 case has proved amenable to some detailed analyses (see e.g. [12][13][14]). An important outcome of these studies is the no-go theorem stating the absence of two-derivative interaction vertices for a single PM graviton [15][16][17], suggesting that any non-trivial PM spin-2 theory must contain additional fields. In fact, a consistent theory-in the sense that interactions preserve the counting of degrees of freedomthat contains a PM spin-2 particle does exist: conformal gravity, which can be regarded as a non-linear model for a massless and a PM spin-2 field in (A)dS [18,19]. Conformal gravity can moreover be deformed so as to include interactions among several conformal gravitons [20,21], thus leading a non-linear theory for arbitrarily many massless-PM pairs. (The cubic sector of this generalized model has been rediscovered recently in [22].) To the best of our knowledge, these are the only instances of Lagrangian theories coupling PM fields in a non-trivial way. They necessitate massless spin-2 fields in the spectrum.
A natural starting point when attempting a bottomup construction of interactions involving PM particles is to consider a multiplet of PM spin-2 fields h a µν , with a = 1, . . . , N . Indeed, asking the same question in relation to massless spin-1 fields leads one to Yang-Mills theory [23], which overcomes the obstructions encountered by the single-field Maxwell theory, namely the impossibility of generalizing the gauge algebra and gauge symmetry in order to allow for non-trivial interactions. For a collection of PM spin-2 fields the problem was first addressed in [24], where it was established that the gauge algebra of the PM symmetry does not admit any nonabelian extension to first non-trivial order in the fields. This however does not rule out the existence of non-linear deformations of the gauge symmetry or the possibility of constructing interactions. In addition, it does not rule out more general non-abelian deformations of the gauge algebra at higher order in the fields. In the present work we revisit these two questions and give the following answers: (i) given two or more PM spin-2 fields, there exist two-derivative cubic interaction vertices that require a field-dependent deformation of the PM gauge symmetry and are consistent at the fully non-linear level; and (ii) there is no possible non-abelian deformation of the gauge algebra-it remains abelian to all orders in perturbations.

II. DEFORMATION ANALYSIS
Our starting point is the action for a collection of N free PM spin-2 fields h a µν , where F a µνρ := ∇ µ h a νρ − ∇ ν h a µρ , F a µ := g νρ F a µνρ , and k ab is an internal metric that may be chosen to be a diagonal matrix with entries +1 and −1. 1 In a dS background, a unitary theory corresponds to k ab = δ ab , although we will see that this choice does not admit fully consistent cubic interactions. The tensor F a µνρ is an abelian field strength in the sense that it is invariant under the PM gauge symmetry, The goal of the deformation analysis is to extend the action (1) with non-trivial interactions in a consistent manner, i.e. while maintaining the number of gauge symmetries. In general, this may require modifying the gauge transformation law in (2) with field-dependent terms, schematically δ ǫ = δ ǫ + · · · , in a way that the non-linear action, S = S 0 +S 1 +S 2 +· · · , respects the gauge invariance, that is δ ǫ S = 0. In our case, since S 0 is quadratic in the fields, S 1 will encode the cubic interaction vertices, S 2 the quartic ones, and so on. In practice we will introduce a bookkeeping parameter α to perform the perturbative expansion, so that S n and δ (n) ǫ are each proportional to α n .
Instead of solving directly for the deformations S n and δ (n) ǫ , what is known as the Noether procedure as spelled out in [25], we make use its cohomological reformulation [26]. This method is specially well suited to deal with ambiguities related to trivial deformations arising from redefinitions of the fields and gauge parameters. We refer the reader to the Section 2 of [27] and to [28] for pedagogical introductions. The case of deformations of massive theories was analyzed in the same framework in [29].

III. FIRST-ORDER DEFORMATIONS
At the first order in the deformation procedure we seek to extend the gauge algebra, gauge symmetry and classical action to leading non-trivial order in the fields.

A. Gauge algebra
The consistency requirement that gauge transformations must form an algebra can be written as where "trivial" denotes gauge transformations that vanish on-shell and that leave the action identically invariant, and for some functional χ that depends on the gauge label the coordinates, and they are raised and lowered with the background space metric gµν . We initially work in an arbitrary number D of spacetime dimensions and use the mostly-plus metric signature. Covariant derivatives are compatible with the metric tensor g of the (A)dS background. The curvature scalar of the background is −σ/L 2 , with σ = +1 for AdS and σ = −1 for dS.
parameters ǫ 1 and ǫ 2 as well as possibly on the fields. An important advantage of the cohomological approach is that χ, and hence the structure of the algebra, can be strongly constrained from algebraic considerations with no a priori knowledge of the possible deformations of the gauge symmetry itself. At zeroth order in our deformation parameter α we obviously have [δ ǫ1 , δ ǫ2 ] h a µν = 0 + O(α), stating that the algebra of the free theory is abelian. At first order we find the unique candidate extension to be given by χ as in Eq. (2) and where m a bc = m a [bc] and n a bc = n a [bc] are otherwise arbitrary at this stage. They correspond to the structure constants of the gauge algebra. In fact, it can be shown that there are no terms of order α 2 in the above expression for χ . That χ is field-independent is not an assumption and can be proved indeed. The proof is rather technical and will be presented elsewhere, but it is similar to the corresponding proof given in Section 7 of [27] for the case of massless spin-2 fields and combines it with embedding-space techniques as used e.g. in [16,30,31].
Further constraints on m a bc and n a bc arise by demanding that the algebra is realized on the fields by some infinitesimal gauge symmetry. We find that this requirement is very strong and leads to the result m a bc = 0 , n a bc = 0 .
This implies that the PM spin-2 gauge algebra does not allow for any non-abelian extension. This no-go result was first established in [24], although the present derivation is stronger in that no assumption is needed on the number of derivatives entering in the algebra or on the (in)dependence of χ on the fields-our result remains true to all orders in perturbations.

B. Gauge symmetry
An abelian gauge algebra does not imply the absence of non-trivial extensions of the gauge symmetry. For instance, the Chapline-Manton and Freedman-Townsend theories of differential form fields belong to this class [32][33][34]. In the present setting, abelian deformations of the PM gauge symmetry are simple to classify, since they are constructed solely out of the field strength tensor F a µνρ and its derivatives. At first order we restrict our attention to contractions that are linear in F a µνρ , finding the following six candidate structures: where the constants u a (i)bc and v a (i)bc are arbitrary at this stage of the calculation. We remark that this ansatz is the most general one containing two derivatives. Although this is a restrictive assumption, it is enough for our purposes as it will allow us to classify all cubic interaction vertices that have no more than two derivatives. Notice that, a priori, gauge transformation terms bringing more than two derivatives could be required in order to produce a vertex with two derivatives or less. However, we checked that this is not the case for the couplings of PM spin-2 fields.

C. Cubic action
Although the result in Eq. (6) is consistent from the point of view of the gauge algebra, there is of course no guarantee that there exists a local action that realizes this symmetry in full. By demanding consistency with the existence of non-trivial cubic interactions with no more than two derivatives we obtain that the constants u a

(i)bc
and v a (i)bc are all forced to vanish with the exception of v a (1)bc ≡ f a b,c , that is and f ab,c must be symmetric under the exchange of the first two indices. Moreover, this non-trivial possibility is only available when the spacetime dimension is D = 4. The cubic vertex is given explicitly by where Given the symmetries of the constants f ab,c , we have that the number of independent non-trivial deformations of the free PM spin-2 theory is given by 1 2 N 2 (N + 1) at this order in the analysis.
The consistency of the cubic action (8) is rather easy to check (the strength of our result lies in having proved its uniqueness): given that J µν a is manifestly invariant under the undeformed PM symmetry (2), it suffices to observe that it defines a conserved current in the sense that where "≈" means equality modulo the equations of motion of the free theory. The current J µν a actually satisfies stronger conditions: it is identically traceless in D = 4, i.e. g µν J µν a = 0 , and it is (covariantly) conserved in the usual sense, i.e. ∇ ν J µν a ≈ 0 . These properties stem from the fact that J µν a is related to the Noether current associated with some rigid symmetries of the free PM theory. Explicitly, defining it is straightforward to verify that J µ ab is a true Noether current in the sense that ∂ µ J µ ab ≈ 0 , and again only in D = 4 dimensions. The functionǭ a in Eq. (11) is by definition a Killing parameter of the free theory, i.e., a solution of ∇ µ ∇ νǭa − σ L 2 g µνǭa = 0. 2 The corresponding rigid symmetry of the quadratic theory is obtained by considering (7) where the gauge parameters ǫ a are replaced byǭ a .

IV. HIGHER-ORDER CONSISTENCY
Having found the most general first-order deformation of the PM spin-2 gauge symmetry and classical action (assuming up to two-derivative interactions), we now turn to the question of its consistency at higher orders in perturbations.

A. Consistency of the deformed gauge symmetry
The statement that a gauge symmetry must be consistent with an algebra leads to further constraints at higher orders in the deformation analysis. For instance, in Yang-Mills theory, the consistency of the extended gauge transformation law implies the Jacobi identity on the structure constants [36,37]. A similar quadratic constraint also applies to the coefficients of the first-order extension of the PM gauge symmetry that we derived in the previous section, Eq. (7). We find that the constants f ab,c must satisfy Two simple conclusions readily follow. The first is that in the case of one field (N = 1) one immediately gets that f 11,1 = 0, implying the failure of the field-dependent PM gauge symmetry to extend beyond lowest order. This is the well known no-go result on the absence of deformations for a single PM spin-2 field. The second remark is that solutions to (12) do not exist when k ab = δ ab . Indeed, if this were the case, taking c = a and d = b (with no summation) in Eq. (12) leads to the conclusion that f ea,b = 0. It follows that the consistency 2 Explicit expressions for the Killing parameters associated to the PM spin-2 theory have been found in [35] for D = 4 dimensions (although the procedure may be readily generalized to arbitrary D).
of the deformed gauge symmetry can only be achieved provided that at least one or more of the fields enter in the action with a "wrong-sign" kinetic term. Thus we conclude that any non-trivial theory (subject to our assumptions) of multiple PM spin-2 particles must be non-unitary.

B. Consistency of the deformed action
The consistency of the deformed classical action is simply the requirement of gauge invariance at higher orders in perturbations. In our setting, this is the statement that the extended action S 0 + S 1 be invariant under the deformed symmetry δ where the ellipses contain terms that vanish on the free equations of motion, and which may be removed by extending the gauge symmetry with appropriate O(α 2 ) terms. On the other hand, the expression shown represents an obstruction to the invariance of the action that may in principle be removed by the inclusion of quartic interactions to the action, i.e. by a new deformation term S 2 . Although it would be interesting to pursue this route, it goes beyond our present scope as it would require a cumbersome classification of candidate quartic operators as well as consistency checks at O(α 3 ).
We will instead impose the vanishing of the obstruction in Eq.
In conclusion, the unique first-order deformations to the PM gauge symmetry (Eq. (7)) and PM action (Eq. (8)) remain consistent at the complete non-linear level provided (a) that no further deformations are introduced beyond O(α), and (b) that the constants f ab,c satisfy the constraints in Eqs. (12) and (14). Understanding the space of solutions of the quadratic constraints (12) and (14) (which depends of course on the signature of the internal metric k ab and on the number N of particle species) is a highly non-trivial task that will be addressed elsewhere. However we can show that there is a unique solution (modulo trivial rescalings) for N = 2 and k ab = diag(+1, −1), and have found particular solutions for all N ≥ 3 and different choices of k ab that give rise to three-particle interactions. They are given explicitly in the Appendix.

V. DISCUSSION
In the Introduction we recalled the existence of the fully nonlinear theories of multi-conformal gravitons ob-tained in [20,21], corresponding to models that couple a set of massless spin-2 fields to the same number of PM spin-2 fields. Given that the family of theories we have uncovered here are built on the cubic vertex of conformal gravity, this shows a posteriori that there does exist a fully nonlinear truncation of the conformal multigraviton models of [20,21] to an interacting PM theory. This truncation should be done in conjunction with the imposition of the constraint that the internal structure constants a a bc of the multi-conformal graviton theories should satisfy our quadratic constraints (12) and (14). This consistent truncation thereby avoids the no-go results obtained in [16,38], thanks to our allowing for several conformal gravitons from the beginning.
More interestingly, our findings show that there exist other branches of consistent theories of interacting PM fields, that cannot be attained by a truncation of the multi-conformal graviton models. This happens when the structure constants f ab,c entering our model are not totally symmetric. Indeed, the aforementioned theories of conformal multi-gravity necessitate completely symmetric internal structure constants a abc = a (abc) , while the present construction allows for coupling constants f ab,c which are not. One such solution with a mixedsymmetric f ab,c is given in the Appendix.
Our results are relevant in that they provide a proof of principle that it is possible to construct non-trivial theories of finitely many PM fields that are fully consistent from the point of view of the gauge structure. It is indeed an outstanding field theoretical problem to determine what theories containing PM fields may in principle be written purely on the basis of consistency, and our construction is a step forward in this program. Regarding the applicability of our model of multiple PM spin-2 particles, it would be interesting to see whether it could be embedded into a higher-spin theory that should in turn provide an extension of the higher-spin model proposed in [7] and discussed later in [8,9]. This theory is of interest as it has been conjectured to be dual to the O(N ) model at a multicritical isotropic Lifshitz point (see [39] for a review). The non-unitarity of the dual theory is not necessarily a pathology in this context, as non-unitary conformal theories are of potential interest in condensed matter physics [40]. It could actually be more likely, in connection to what we explained previously, that our model could be embedded into an extended version of conformal higher-spin gravity (see e.g. [41,42]), the existence of which is an interesting question on its own.

ACKNOWLEDGMENTS
We would like to thank Andrea Campoleoni, Kurt Hinterbichler and Per Sundell for useful comments and discussions. The work of NB is partially supported by an F.R.S.-FNRS PDR grant "Fundamental issues in extended gravity" № T.0022. 19 An explicit solution of the quadratic constraints, Eqs. (12) and (14) is given by for the choice of internal metric k ab = diag(+1, . . . , +1, −1) , and where (n g ) abc ∈ {0, 1, 2, 3} denotes the number of times that the index "N " (corresponding to the "ghostly" field in our convention) appears in f ab,c ; for instance f N N,N = (N − 1) 3/2 .
For N = 2 fields these two solutions reduce to f ab,c = 1 , ∀ a, b, c ∈ {1, 2} , with metric k ab = diag(+1, −1). In this case we can moreover show that this solution is unique modulo rescalings of the fields and gauge parameters. We remark that for N ≥ 3 the constants (A1) lead to cubic vertices that couple three distinct fields, so that it is not a trivial extension of the N = 2 solution.

Mixed symmetric solutions
For N = 2 the unique solution to the constraints was totally symmetric under the exchange of the three indices. However, for N ≥ 3 , there also exist solutions for mixed-symmetric constants f ab,c . For example, when N = 3 and the metric is k ab = diag(+1, +1, −1) , one such solution is given by