1. 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 the anti-de Sitter (AdS
) and the de Sitter (dS
) backgrounds, with the assumption that the couplings should not bring more than one derivative. Then, the partially massless limit for the graviton was considered, resulting in the conclusion that no cubic vertex with one derivative survived in this limit.
The partially massless graviton does not exist around a Minkowski background. In order to propagate, such a particle requires a background that is either anti-de Sitter (AdS) or de Sitter (dS), depending on whether the cosmological constant
of the maximally-symmetric space is negative of positive, respectively. In such maximally symmetric spacetimes with nonvanishing cosmological constant, a partially massless graviton possesses four propagating degrees of freedom and is characterized by a well-suited mass directly related to the cosmological constant
, as we recall shortly and as discussed at length in [
5,
6,
7], for example. Obviously, the dS
background is particularly relevant for early-Universe cosmology. The terminology “Partially Massless” (sometimes shortened by the acronym “PM” in the rest of this paper) is justified because the four helicity degrees of freedom of such a particle are intermediate between the five degrees of freedom of a massive spin-2 field and the two helicity
degrees of freedom of the massless graviton. The partially massless graviton possesses four propagating degrees of freedom corresponding to helicities
and
. The corresponding PM field
satisfies the Klein-Gordon-like equation
where □ is the Laplace–Beltrami operator in (A)dS
. We note that the mass term is proportional to the cosmological constant
of the maximally symmetric background.
A physical motivation behind the search for interactions between a PM graviton and a real, massive spin-3/2 field is that there is an equal number of physical degrees of freedom for a partially massless graviton and a real, massive spin-3/2 field, suggesting a possible supersymmetry mixing these two fields. An interacting theory for the PM graviton and a real massive spin-3/2 field would therefore be reminiscent of supergravity, for a partially massless graviton instead of the massless graviton and for a massive spin-3/2 field instead of the massless gravitino of supergravity. On a more phenomenological side, the results of black-hole mergers [
8] have put a lower bound on the mass of the graviton that does not prevent the graviton from being partially massless. The point is that the mass of the partially massless graviton is fixed by the cosmological constant, as recalled in Equation (
1), and that the cosmological constant of our Universe is observed to be very small indeed [
9]. Numerically, we have
, while it has been observed that the graviton squared mass
is bounded from above by the numerical value of the order
. This leaves enough room for a partially massless graviton, with fourteen orders of magnitude.
We wish to revisit the question of possible couplings between the PM graviton and a real, massive spin-3/2 field, 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 paper, we report 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 [
10] that combines the cohomological reformulation of the Noether method for gauge systems [
11,
12] with the Stueckelberg formulation for massive fields. The Stuckelberg formulation [
13] of theories for massive fields proves to be very useful, in the sense that it brings gauge invariance that controls the degrees of freedom and hence the possible interactions to be added to the free theory. The Stueckelberg formalism has been widely used since its invention; see [
14] for a pedagogical review of the Stueckelberg formalism. The advantage of the method [
10] is that it exploits the gauge structure of massless theories to describe interactions for massive fields. It proved useful in showing that the de Rham, Gabadadze, and Tolley (dRGT) gravity (see, e.g., [
15] 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 [
10], the full list of cubic vertices of massive dRGT gravity theory was recovered, showing the usefulness of the method.
In this paper, we use the cohomological method of [
10] for the search of consistent couplings between a massive spin-
field and a partially massless (PM) spin-2 field, also called PM graviton. As we wrote above, the spacetime backgrounds considered in this paper are the anti-de Sitter (AdS) and the de Sitter (dS) geometries where the cosmological constant
is negative and positive, respectively. 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
and a massive spin-
field
are involved.
We study the problem in both dS () and AdS () backgrounds at a stroke, through the use of parameter that takes the value of in AdS and in dS.
2. Main Results
The main results we report in this paper 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
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 (we use conventions whereby the Lorentz-covariant derivative satisfies
, where
. In other terms, the cosmological constant is
in four dimensions, where
corresponds to dS
and
to AdS
. On a Dirac spinor
, we have
) is denoted by symbol
, and the symmetric rank-two tensor field
represents the PM spin-2 field. Throughout this paper, spacetime indices between square brackets are antisymmetrised with strength one. For example, one has
and
. The components of the (A)dS background metric are denoted as
. As usual, the four Dirac matrices are denoted by
,
, and
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
is the limit where the spin-3/2 field enjoys a gauge symmetry that removes the helicity
degrees of freedom, leaving only the helicity
degrees of freedom on shell. Tensor
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
induces a deformation of the gauge transformations on the physical fields
in the unitary gauge, given by
where we recall that the free, quadratic action
is invariant under [
5,
6,
7]
From the knowledge of the quadratic and cubic actions
and
in the unitary gauge, we readily find the consistency of the deformation reported above:
As far as deformation
is concerned, we note from (
5) and (
6) that the transformation of the massive spin-3/2 field can be written as
, from which it is tempting to view the contravariant spinor
as a gauge-invariant quantity, defining the covariant field as
, where
is the deformation parameter that we take with units of length that defines perturbative expansion
,
such that
. In this sense, like in Riemannian geometry, it would appear that metric
could be defined in terms of the (A)dS background metric and the PM spin-2 field
.
In the following section, we present the two deformations reported above in their Stueckelberg form, and explain how the unitary gauge at first order in perturbation can be reached, thereby reproducing the main results presented above.
4. Conclusions and Outlook
The first vertex,
, that we presented above in (
14) does not deform Stueckelberg gauge transformations (
9). It is exactly invariant under the latter transformations. The second vertex,
, given in (
15) is more interesting in the sense that it truly deforms the gauge transformations given in (
9). The first term on the right-hand side of (
16) is reminiscent of the local supersymmetry transformations in the AdS background, offering the minimal deformation of the mass-like term on the right-hand side of
; see (
9). Correspondingly, the first term on the right-hand side of (
15) is the minimal deformation of the mass-like term for the spinor
in the free action (
11). However, on the contrary to the situation in supergravity theories, there is no deformation proportional to the linearised “spin-connection”
in (
15) or in (
16).
It will be interesting to investigate the consistent interactions among the fields of the enlarged spectra given in [
17]. There, it was shown that the doublet
consisting of a PM spin-2 and a massive spin-3/2 field studied in the present paper must be completed with a massless spin-
doublet in order to carry the action of supersymmetry. We hope to report soon on the interactions among these four fields. In paper [
18], on the other hand, it was found that the partially massless doublet
can be completed with a massless doublet
in order to carry the action of supersymmetry. We intend to investigate the consistent couplings among those fields in future work.