Before moving to the decoupling of massive gravity and bi-gravity, let us make a brief interlude concerning the correct identification of degrees of freedom. The Stückelberg trick used previously to identify the correct degrees of freedom works in all generality, but care must be used when taking a “decoupling limit” (i.e., scaling limit) as will be done in Section 8.2.
Imagine the following gauge field theory8.2*) still propagates no physical degree of freedom. 8.3*) propagates one degree of freedom. This is correct and thus means that (8.3*) is not a consistent decoupling limit of (8.2*) since the number of degrees of freedom is different already at the linear level. In the rest of this review, we will call a decoupling limit a specific type of scaling limit which preserves the same number of physical propagating degrees of freedom in the linear theory. As suggested by the name, a decoupling limit is a special kind of limit in which some of the degrees of freedom of the original theory might decouple from the rest, but the total number of degrees of freedom remains identical. For the theory (8.2*), this means that the scaling ought to be taken not with fixed but rather with fixed. This is indeed a consistent rescaling which leads to finite contributions in the limit ,
This procedure is true in all generality: a decoupling limit is a special scaling limit where all the fields in the original theory are scaled with the highest possible power of the scale in such a way that the decoupling limit is finite.
A decoupling limit of a theory never changes the number of physical degrees of freedom of a theory. At best it ‘decouples’ some of them in such a way that they are inaccessible from another sector.
Before looking at the massive gravity limit of bi-gravity and other decoupling limits of massive and bi-gravity, let us start by describing the different scaling limits that can be taken. We start with a bi-gravity theory where the two spin-2 fields have respective Planck scales and and the interactions between the two metrics arises at the scale . In order to stick to the relevant points we perform the analysis in four dimensions, but the following arguments extend trivially to arbitrary dimensions.
- Non-interacting Limit: The most natural question to ask is what happens in the limit where the interactions between the two fields are ‘switched off’, i.e., when sending the scale , (the limit is studied more carefully in Sections 8.3 and 8.4). In that case if the two Planck scales remain fixed as , we then recover two massless non-interacting spin-2 fields (carrying both 2 helicity-2 modes), in addition to a decoupled sector containing a helicity-0 mode and a helicity-1 mode. In bi-gravity matter fields couple only to one metric, and this remains the case in the limit , so that the two massless spin-2 fields live in two fully decoupled sectors even when matter in included.
- Massive Gravity: Alternatively, we may look at the limit where one of the spin-2 fields (say
) decouples. This can be studied by sending its respective Planck scale to infinity. The
resulting limit corresponds to a massive spin-2 field (carrying five dofs) and a decoupled massless
spin-2 field carrying 2 dofs. This is nothing other than the massive gravity limit of bi-gravity
(which includes a fully decoupled massless sector).
If one considers matter coupling to the metric which scales in such a way that a non-trivial solution for survives in the limit , we then obtain a massive gravity sector on an arbitrary non-dynamical reference metric . The dynamics of the massless spin-2 field fully decouples from that of the massive sector.
- Other Decoupling Limits Finally, one can look at combinations of the previous limits, and
the resulting theory depends on how fast compared to how fast . For
instance if one takes the limit and , while keeping both
and fixed, then we obtain what is called the -decoupling limit of bi-gravity
(derived in Section 8.4), where the dynamics of the two helicity-2 modes (which are both
massless in that limit), and that of the helicity-1 and -0 modes can be followed without keeping
track of the standard non-linearities of GR.
If on top of this -decoupling limit one further takes , then one of the massless spin-2 fields fully decoupled (no communication between that field and the helicity-1 and -0 modes). If, on the other hand, we take the additional limit on top of the -decoupling limit, then the helicity-0 and -1 modes fully decouple from both helicity-2 modes.
In all of these decoupling limits, the number of dofs remains the same as in the original theory, some fields are simply decoupled from the rest of the standard gravitational sector. These prevents any communication between these decoupled fields and the gravitational sector, and so from the gravitational sector view point it appears as if these decoupled fields did not exist.
It is worth stressing that all of these limits are perfectly sensible and lead to sensible theories, (from a theoretical view point). This is important since if one of these scaling limits lead to a pathological theory, it would have severe consequences for the parent bi-gravity theory itself.
Similar decoupling limit could be taken in multi-gravity and out of interacting spin-2 fields, we could obtain for instance decoupled massless spin-2 fields and decoupled dofs in the helicity-0 and -1 modes.
In what follows we focus on massive gravity limit of bi-gravity when .
In the following two sections we review the decoupling arguments given previously in the literature, (see for instance [154*]). We start with the theory of bi-gravity presented in Section 5.4 with the action (5.43*)6.3*) and where . We also allow for the coupling to matter with different species living on each metrics.
We now consider matter fields such that is a solution to the equations of motion (so for instance there is no overall cosmological constant living on the metric ). In that case we can write that metric as6.3*). That massive gravity Lagrangian remains fully non-linear in this limit and is expressed in terms of the full metric and the reference metric . While the metric is ‘frozen’ in this limit, we emphasize however that the massless spin-2 field is itself not frozen – its dynamics is captured through the kinetic term , but that spin-2 field decouple from its own matter sector , (although this can be accommodated for by scaling the matter fields accordingly in the limit so as to maintain some interactions).
At the level of the equations of motion, in the limit we obtain the massive gravity modified Einstein equation for , the free massless linearized Einstein equation for which fully decouples and the equation of motion for all the matter fields on flat spacetime, (see also Ref. ).
. Here again the Lagrangian for massive gravity is given in (6.3*) with now . The massive gravity action remains fully non-linear in the limit and is expressed solely in terms of the full metric and the reference metric , while the excitations for the massless graviton remain dynamical but fully decouple from the massive sector.
As is already clear from the previous discussion, to recover massive gravity on a non-trivial reference metric as a limit of bi-gravity, one needs to scale the Matter Lagrangian that couples to what will become the reference metric (say the metric for definiteness) in such a way that the Riemann curvature of remains finite in that decoupling limit. For a macroscopic description of the matter living on this is in principle always possible. For instance one can consider a point source of mass living on the metric . Then, taking the limit while keeping the ratio fixed, leads to a theory of massive gravity on a Schwarzschild reference metric and a decoupled massless graviton. However, some care needs to be taken to see how this works when the dynamics of the matter sourcing is included.
As soon as the dynamics of the matter field is considered, one has to send the scale of that field to infinity so that it maintains some nonzero effect on in the limit , i.e.,8.2.1 where the matter field scale as and the decoupling limit of bi-gravity on an arbitrary reference metric derived here.
As an example, suppose that the Lagrangian for the matter (for example a scalar field) sourcing the metric is
As a result, massive gravity with an arbitrary reference metric can be seen as a consistent limit of bi-gravity in which the additional degrees of freedom in the metric and matter that sources the background decouple. Thus all solutions of massive gravity may be seen as decoupling limits of solutions of bi-gravity. This will be discussed in more depth in Section 8.4. For an arbitrary reference metric which can be locally written as a small departures about Minkowski the decoupling limit is derived in Eq. (8.81*).
Having derived massive gravity as a consistent decoupling limit of bi-gravity, we could of course do the same for any multi-metric theory. For instance, out of -interacting fields, we could take a limit so as to decouple one of the metrics, we then obtain the theory of -interacting fields, all of which being massive and one decoupled massless spin-2 field.
We now turn to a different type of decoupling limit, whose aim is to disentangle the dofs present in massive gravity itself and analyze the ‘irrelevant interactions’ (in the usual EFT sense) that arise at the lowest possible scale. One could naively think that such interactions arise at the scale given by the graviton mass, but this is not so. In a generic theory of massive gravity with Fierz–Pauli at the linear level, the first irrelevant interactions typically arise at the scale . For the setups we have in mind, . But we shall see that interactions arising at such a low-energy scale are always pathological (reminiscent to the BD ghost [111*, 173*]), and in ghost-free massive gravity the first (irrelevant) interactions actually arise at the scale .
We start by deriving the decoupling limit in the absence of vectors (helicity-1 modes) and then include them in the following section 8.3.4. Since we are interested in the decoupling limit about flat spacetime, we look at the case where Minkowski is a vacuum solution to the equations of motion. This is the case in the absence of a cosmological constant and a tadpole and we thus focus on the case where in (6.3*).
In GR, the interactions of the helicity-2 mode arise at the very high energy scale, namely the Planck scale. In massive gravity a new scale enters and we expect some interactions to arise at a lower energy scale given by a geometric combination of the Planck scale and the graviton mass. The potential term (6.3*) includes generic interactions between the canonically normalized helicity-0 (), helicity-1 (), and helicity-2 modes () introduced in (2.48*)
Clearly ,the lowest interaction scale is which arises for an operator of the form . If present such an interaction leads to an Ostrogradsky instability which is another manifestation of the BD ghost as identified in [173*].
Even if that very interaction is absent there is actually an infinite set of dangerous interactions of the form which arise at the scale , with
Any interaction with or automatically leads to a larger scale, so all the interactions arising at a scale between (inclusive) and are of the form and carry an Ostrogradsky instability. For DGP we have already seen that there is no interactions at a scale below . In what follows we show that same remains true for the ghost-free theory of massive gravity proposed in (6.3*). To see this let us identify the interactions with and arbitrary power for .
We now express the potential term introduced in (6.3*) using the metric in term of the helicity-0 mode, where we recall that the quantity is defined in (6.7*), as , where is the ‘Stückelbergized’ reference metric given in (2.78*). Since we are interested in interactions without the helicity-2 and -1 modes (), it is sufficient to follow the behaviour of the helicity-0 mode and so we havetotal derivatives. So even though the ghost-free theory of massive gravity does in principle involve some interactions with higher derivatives of the form it does so in a very precise way so that all of these terms combine so as to give a total derivative and being harmless.22
As a result the potential term constructed proposed in Part II (and derived from the deconstruction framework) is free of any interactions of the form . This means that the BD ghost as identified in the Stückelberg language in [173*] is absent in this theory. However, at this level, the BD ghost could still reappear through different operators at the scale or higher.
Since there are no operators all the way up to the scale (excluded), we can take the decoupling limit by sending , and maintaining the scale fixed.
The operators that arise at the scale are the ones of the form (8.18*) with either and arbitrary or with and arbitrary . The second case scenario leads to vector interactions of the form and will be studied in the next Section 8.3.4. For now we focus on the first kind of interactions of the form , (see also refs. [137*] and )
Since we are dealing with the decoupling limit with the metric is flat and all indices are raised and lowered with respect to the Minkowski metric. These tensors can be written more explicitly as follows
From the expression of these tensors in terms of the fully antisymmetric Levi-Cevita tensors, it is clear that the tensors are transverse and that the equations of motion of with respect to both and never involve more than two derivatives. This decoupling limit is thus free of the Ostrogradsky instability which is the way the BD ghost would manifest itself in this language. This decoupling limit is actually free of any ghost-lie instability and the whole theory is free of the BD even beyond the decoupling limit as we shall see in depth in Section 7.
Not only does the potential term proposed in (6.3*) remove any potential interactions of the form which could have arisen at an energy between and , but it also ensures that the interactions that arise at the scale are healthy.
As already mentioned, in the decoupling limit the metric reduces to Minkowski and the standard Einstein–Hilbert term simply reduces to its linearized version. As a result, neglecting the vectors for now the full -decoupling limit of ghost-free massive gravity is given by
As was already the case at the linearized level for the Fierz–Pauli theory (see Eqs. (2.47*) and (2.48*)) the kinetic term for the helicity-0 mode appears mixed with the helicity-2 mode. It is thus convenient to diagonalize these two modes by performing the following shift, [412*] 6.9*) – (6.13*), or more explicitly in (6.14*) – (6.18*), leading to the explicit form for the Galileon Lagrangians 2.48*).
In general, the last coupling between the helicity-2 and helicity-0 mode cannot be removed by a local field redefinition. The non-local field redefinition2.64*), fully diagonalizes the helicity-0 and -2 mode at the price of introducing non-local interactions for .
Note however that these non-local interactions do not hide any new degrees of freedom. Furthermore, about some specific backgrounds, the field redefinition is local. Indeed focusing on static and spherically symmetric configurations if we consider and given by[61*] for more details on spherically symmetric configurations with the -coupling.
As can be seen from the relation (8.19*), the scale associated with interactions mixing two helicity-1 fields with an arbitrary number of fields , ( and arbitrary ) is also . So at that scale, there are actually an infinite number of interactions when including the mixing with between the helicity-1 and -0 modes (however as mentioned previously, since the vector field always appears quadratically it is always consistent to set them to zero as was performed previously).
The full decoupling limit including these interactions has been derived in Ref. [419*], (see also Ref. ) using the vielbein formulation of massive gravity as in (6.1*) and we review the formalism and the results in what follows.
In addition to the Stückelberg fields associated with local covariance, in the vielbein formulation one also needs to introduce 6 additional Stückelberg fields associated to local Lorentz invariance, . These are non-dynamical since they never appear with derivatives, and can thus be treated as auxiliary fields which can be integrated. It is however useful to keep them in the decoupling limit action, so as to retain a closes-form expression. In terms of the Lorentz Stückelberg fields, the full decoupling limit of massive gravity in four dimensions at the scale is then (before diagonalization) [419*]6.28*).
The auxiliary Lorentz Stückelberg fields carries all the non-linear mixing between the helicity-0 and -1 modes,[139*], (see also Refs. [364*, 456*]).
This decoupling limit includes non-linear combinations of the second-derivative tensor and the first derivative Maxwell tensor . Nevertheless, the structure of the interactions is gauge invariant for , and there are no higher derivatives on in the equation of motion for , so the equations of motions for both the helicity-1 and -2 modes are manifestly second order and propagating the correct degrees of freedom. The situation is more subtle for the helicity-0 mode. Taking the equation of motion for that field would lead to higher derivatives on itself as well as on the helicity-1 field. Since this theory has been proven to be ghost-free by different means (see Section 7), it must be that the higher derivatives in that equation are nothing else but the derivative of the equation of motion for the helicity-1 mode similarly as what happens in Section 7.2.
When working beyond the decoupling limit, the even the equation of motion with respect to the helicity-1 mode is no longer manifestly well-behaved, but as we shall see below, the Stückelberg fields are no longer the correct representation of the physical degrees of freedom. As we shall see below, the proper number of degrees of freedom is nonetheless maintained when working beyond the decoupling limit.
In Section 8.3, we have introduced four Stückelberg fields which transform as scalar fields under coordinate transformation, so that the action of massive gravity is invariant under coordinate transformations. Furthermore, the action is also invariant under global Lorentz transformations in the field space,[173*], (see also [111*]). However, beyond the DL, the helicity-0 mode of the graviton does not behave as a scalar field and neither does the in the split of the Stückelberg fields. So beyond the DL there is no reason to anticipate that captures a whole degree of freedom, and it indeed, it does not. Beyond the DL, the equation of motion for will typically involve higher derivatives, but the correct requirement for the absence of ghost is different, as explained in Section 7.2. One should instead go back to the original four scalar Stückelberg fields and check that out of these four fields only three of them be dynamical. This has been shown to be the case in Section 7.2. These three degrees of freedom, together with the two standard graviton polarizations then gives the correct five degrees of freedom and circumvent the BD ghost.
Recently, much progress has been made in deriving the decoupling limit about arbitrary backgrounds, see Ref. .
Before deriving the decoupling limit of massive gravity on (Anti) de Sitter, we first need to analyze the linearized theory so as to infer the proper canonical normalization of the propagating dofs and the proper scaling in the decoupling limit, similarly as what was performed for massive gravity with flat reference metric. For simplicity we focus on dimensions here, and when relevant give the result in arbitrary dimensions. Linearized massive gravity on (A)dS was first derived in [307*, 308]. Since we are concerned with the decoupling limit of ghost-free massive gravity, we follow in this section the procedure presented in [154*]. We also focus on the dS case first before commenting on the extension to AdS.
At the linearized level about dS, ghost-free massive gravity reduces to the Fierz–Pauli action with , where is the dS metric with constant Hubble parameter ,2.80*), although now considered about the dS metric,
The most important difference from linearized massive gravity on Minkowski is that the properly canonically normalized helicity-0 mode is now instead[358, 430], unlike what was found on Minkowski, see Section 2.2.3, which confirms the Newtonian approximation presented in .
While this observation is correct on AdS, in the dS one cannot take the massless limit without simultaneously sending at least the same rate. As a result, it would be incorrect to deduce that the helicity-0 mode decouples in the massless limit of massive gravity on dS.
To be more precise, the linearized action (8.62*) is free from ghost and tachyons only if which corresponds to GR, or if , which corresponds to the well-know Higuchi bound [307*, 190*]. In spacetime dimensions, the Higuchi bound is . In other words, on dS there is a forbidden range for the graviton mass, a theory with or with always excites at least one ghost degree of freedom. Notice that this ghost, (which we shall refer to as the Higuchi ghost from now on) is distinct from the BD ghost which corresponded to an additional sixth degree of freedom. Here the theory propagates five dof (in four dimensions) and is thus free from the BD ghost (at least at this level), but at least one of the five dofs is a ghost. When , the ghost is the helicity-0 mode, while for , the ghost is he helicity-1 mode (at quadratic order the helicity-1 mode comes in as ). Furthermore, when , both the helicity-2 and -0 are also tachyonic, although this is arguably not necessarily a severe problem, especially not if the graviton mass is of the order of the Hubble parameter today, as it would take an amount of time comparable to the age of the Universe to see the effect of this tachyonic behavior. Finally, the case (or in spacetime dimensions), represents the partially massless case where the helicity-0 mode disappears. As we shall see in Section 9.3, this is nothing other than a linear artefact and non-linearly the helicity-0 mode always reappears, so the PM case is infinitely strongly coupled and always pathological.
- : Helicity-1 modes are ghost, helicity-2 and -0 are tachyonic, sick theory
- : General Relativity: two healthy (helicity-2) degrees of freedom, healthy theory,
- : One “Higuchi ghost” (helicity-0 mode) and four healthy degrees of freedom (helicity-2 and -1 modes), sick theory,
- : Partially Massless Gravity: Four healthy degrees (helicity-2 and -1 modes), and one infinitely strongly coupled dof (helicity-0 mode), sick theory,
- : Massive Gravity on dS: Five healthy degrees of freedom, healthy theory.
- As one can see from Figure 4*, in the case where (corresponding to massive gravity
on AdS), one can take the massless limit while keeping the AdS length scale fixed in
that limit. In that limit, the helicity-0 mode decouples from external matter sources and there
is no vDVZ discontinuity. Notice however that the helicity-0 mode is nevertheless still strongly
coupled at a low energy scale.
When considering the decoupling limit , of massive gravity on AdS, we have the choice on how we treat the scale in that limit. Keeping the AdS length scale fixed in that limit could lead to an interesting phenomenology in its own right, but is yet to be explored in depth.
- In the dS case, the Higuchi forbidden region prevents us from taking the massless limit while
keeping the scale fixed. As a result, the massless limit is only consistent if
simultaneously as and we thus recover the vDVZ discontinuity at the linear level in
When considering the decoupling limit , of massive gravity on dS, we also have to send . If in that limit, we then recover the same decoupling limit as for massive gravity on Minkowski, and all the results of Section 8.3 apply. The case of interest is thus when the ratio remains fixed in the decoupling limit.
When taking the decoupling limit of massive gravity on dS, there are two additional contributions to take into account:
- First, as mentioned in Section 8.3.5, care needs to be applied to properly identify the helicity-0 mode on a curved background. In the case of (A)dS, the formalism was provided in Ref. [154*] by embedding a -dimensional de Sitter spacetime into a flat -dimensional spacetime where the standard Stückelberg trick could be applied. As a result the ‘covariant’ fluctuation defined in (2.80*) and used in (8.59*) needs to be generalized to (see Ref. [154*] for details)
- Second, as already encountered at the linearized level, what were total derivatives in Minkowski (for instance the combination ), now lead to new contributions on de Sitter. After integration by parts, . This was the origin of the new kinetic structure for massive gravity on de Sitter and will have further effects in the decoupling limit when considering similar contributions from , where are defined in (6.12*, 6.13*) or more explicitly in (6.17*, 6.18*).
Taking these two effects into account, we obtain the full decoupling limit for massive gravity on de Sitter,8.52*), and are the Galileon Lagrangians as encountered previously. Notice that while the ratio remains fixed,this decoupling limit is taken with , so all the fields in (8.66*) live on a Minkowski metric. The constant coefficients depend on the free parameters of the ghost-free theory of massive gravity, for the theory (6.3*) with and , we have 8.39*) as in flat space and obtain the following semi-diagonalized decoupling limit, 8.52*), and the new coefficients cancel identically for , and , as pointed out in [154*], and the same result holds for bi-gravity as pointed out in [301*]. Interestingly, for these specific parameters, the helicity-0 loses its kinetic term, and any self-mixing as well as any mixing with the helicity-2 mode. Nevertheless, the mixing between the helicity-1 and -0 mode as presented in (8.52*) are still alive. There are no choices of parameters which would allow to remove the mixing with the helicity-1 mode and as a result, the helicity-0 mode generically reappears through that mixing. The loss of its kinetic term implies that the field is infinitely strongly coupled on a configuration with zero vev for the helicity-1 mode and is thus an ill-defined theory. This was confirmed in various independent studies, see Refs. [185*, 147*].
We now proceed to derive the -decoupling limit of bi-gravity, and we will see how to recover the decoupling limit about any reference metric (including Minkowski and de Sitter) as special cases. As already seen in Section 8.3.4, the full DL is better formulated in the vielbein language, even though in that case Stückelberg fields ought to be introduced for the broken diff and the broken Lorentz. Yet, this is a small price to pay, to keep the action in a much simpler form. We thus proceed in the rest of this section by deriving the -decoupling of bi-gravity and start in its vielbein formulation. We follow the derivation and formulation presented in [224*]. As previously, we focus on -spacetime dimensions, although the whole formalism is trivially generalizable to arbitrary dimensions.6.28*).
We now introduce Stückelberg fields for diffs and for the local Lorentz. In the case of massive gravity, there was no ambiguity in how to perform this ‘Stückelbergization’ but in the case of bi-gravity, one can either ‘Stückelbergize the metric or the metric . In other words the broken diffs and local Lorentz symmetries can be restored by performing either one of the two replacements in (8.69*),8.71*) but keep in mind that this freedom has deep consequences for the theory, and is at the origin of the duality presented in Section 10.7.
Since we are interested in the decoupling limit, we now perform the following splits, (see Ref.  for more details),
- Mixing of the helicity-0 mode with the helicity-1 mode , as derived in (8.52*),
- Mixing of the helicity-0 mode with the helicity-2 mode , as derived in (8.40*),
- Mixing of the helicity-0 mode with the new helicity-2 mode ,
noticing that before field redefinitions, the helicity-0 mode do not self-interact (their self-interactions are constructed so as to be total derivatives).
As already explained in Section 8.3.6, the first contribution ❶ arising from the mixing between the helicity-0 and -1 modes is the same (in the decoupling limit) as what was obtained in Minkowski (and is independent of the coefficients or ). This implies that the can be directly read of from the three last lines of (8.52*). These contributions are the most complicated parts of the decoupling limit but remained unaffected by the dynamics of , i.e., unaffected by the bi-gravity nature of the theory. This statement simply follows from scaling considerations. In the decoupling limit there cannot be any mixing between the helicity-1 and neither of the two helicity-2 modes. As a result, the helicity-1 modes only mix with themselves and the helicity-0 mode. Hence, in the scaling limit (8.74*, 8.75*) the helicity-1 decouples from the massless spin-2 field.
Furthermore, the first line of (8.52*) which corresponds to the dynamics of and the helicity-0 mode is also unaffected by the bi-gravity nature of the theory. Hence, the second contribution ❷ is the also the same as previously derived. As a result, the only new ingredient in bi-gravity is the mixing ❸ between the helicity-0 mode and the second helicity-2 mode , given by a fixing of the form .
Unsurprisingly, these new contributions have the same form as ❷, with three distinctions: First the way the coefficients enter in the expressions get modified ever so slightly ( and ). Second, in the mass term the space-time index for ought to dressed with the Stückelberg field,
Taking these three considerations into account, one obtains the decoupling limit for bi-gravity,
Notice as well the presence of a tadpole for if . When this tadpole vanishes (as well as the one for ), one can further take the limit keeping all the other ’s fixed as well as , and recover straight away the decoupling limit of massive gravity on Minkowski found in (8.52*), with a free and fully decoupled massless spin-2 field.
In the presence of a cosmological constant for both metrics (and thus a tadpole in this framework), we can also take the limit and recover straight away the decoupling limit of massive gravity on (A)dS, as obtained in (8.66*).
This illustrates the strength of this generic decoupling limit for bi-gravity (8.78*). In principle we could even go further and derive the decoupling limit of massive gravity on an arbitrary reference metric as performed in [224*]. To obtain a general reference metric we first need to add an external source for that generates a background for . The reference metric is thus expressed in the local inertial frame as8.2.3.
We can then perform the scaling limit , while keeping the ’s and the scale fixed as well as the field and the fixed tensor . The decoupling limit is then simply given by8.81*) (just as observed for the decoupling limit on AdS). These new interactions are ghost-free and look like Galileons for conformally flat , with constant, but not in general. In particular, the interactions found in (8.81*) would not be the covariant Galileons found in [166, 161, 157*] (nor the ones found in [237*]) for a generic metric.