 Sec. 3 and study the dynamics of their perturbations. For the case of a weight zero scalar field ϕ, ∆ ϕ = 0, the symmetry takes the form δDϕ = x µ ∂ µ ϕ, (2.23) and the invariant Lagrangian reduces to the shift-symmetric action (see also [36][37][38][39]) which is furthermore invariant under the full conformal group SO(D, 2). Again, we refer to [11] for more details on this theory. We found one more scalar generator of degree N = 1, denoted W, whose commutator with P µ gives the degree zero vector generator V µ (f ). The corresponding transformation takes the form: This is in fact another infinite dimensional family of scalar symmetries since g(φ) is an arbitrary differentiable function. In agreement with what was found in [11], these generators restrict the Lagrangian of (2.12) to the shift-symmetric Cuscuton action Before moving on to higher m generators, notice that when restricted to g(φ) = d φ the generator W (g) becomes the dilation operatorD and one finds a scaling Cuscuton action where the scalar field has scaling dimension ∆ = d. In D = 1 spacetime dimension the Cuscuton term √ X is a total derivative so we can ignore it. The commutation relation with V µ is in this case: These are all the scalar symmetries of degree N = 1. For m = 1 generators, it is impossible to match the Lorentz indices on both sides of the commutator with P µ to yield V µ or P µ. For m = 2 generators, in principle one can write down transformations of the field whose commutation with translations gives back P µ. However we have checked that -except for the Lorentz transformation δ M µν φ -none of these transformations are symmetries of the action (2.1). On the other hand, we have found an m = 2 symmetry generator T µν whose commutation relation with P µ gives back with T µν = T [µν]. Without loss of generality we can set c = 1 and the associated symmetry transformation reads where f (φ) is again an arbitrary differentiable function. The P (X, φ) action invariant under this symmetry is again the Cuscuton (2.12). The Cuscuton algebra is then: Interestingly the symmetry generators V µ and T µν are both of the form That is they are given by an arbitrary function of the field φ multiplying the Poincaré generators. We shall see in Sec. 4.1 that this is in fact true for any background spacetimes. The Cuscuton action (2.12) is invariant under symmetries of the form "f (φ)×background isometries". For generators with m ≥ 3 the commutation with P µ in (2.3) returns an S 0 m+1 generator. But by the argument of Sec. 2.2 there are no S 0 n for n ≥ 2 generators and since the δ S 1 m φ transformation can only depend on one x µ term we can see that such generator S 1 m must take the form of an S 1 1 or S 1 0 generator multiplied by the appropriate factor of η's. Since we have already classified such generators we conclude that we have all possible N = 1 symmetry generators.

Degree two
By the closure of the algebra, the commutator of any N = 2 symmetry with translations must give either a dilationD or a Lorentz transformation. The only other possibility for the commutator would be to return W (g) or T µν but the action (2.26) has no more free parameters and so it cannot be constrained further. 4 In this case we found two m = 1 vector symmetry generators. The first one corresponds to the generator of special conformal transformations, for non-zero weight ∆ = 0 it reads The corresponding invariant action is 5 This situation is different from the shift-symmetric case where there exists no non-trivial conformally invariant action for fields with non-zero weight [11]. In D = 4, we recover the usual λφ 4 conformal theory. In the D = 2 action, one could make a field redefinition φ = e χ to get a canonical kinetic term in (2.35) but then the scaling symmetry would be less obvious so we preferred to leave the action as is. For fields with weight ∆ ϕ = 0, the special conformal transformations read 36) and the invariant action is again (2.24). The second symmetry of degree N = 2 that we found has also a vector generator: where R D is some constant of dimension of length. The corresponding invariant action is The minus sign in the square root (and the corresponding plus sign in the transformation) represents the action for a flat D-brane living in an AdS D+1 spacetime [40] with radius R D. In this case φ non-linearly realizes the Conformal algebra SO(2, D) (the isometry group of AdS D+1 ) where the generator A + µ of (2.37) is the generalization of rotation in the extra dimension. The action (2.38) is also invariant under non-linearly realized dilation (2.39) In (2.38), the plus sign can also be interpreted as a brane embedding but this time in an ambient de Sitter space with two time directions [41]. Because of this, perturbations around homogeneous backgrounds will generically have super-luminal speed of propagation. More generally this theory will fail to satisfy positivity bounds coming from unitarity of the UV completion [42]. We will review later in Sec. 4 that both these theories have known extensions to (A)dS backgrounds corresponding to different brane geometries and embeddings. When considering the higher-derivatives terms invariant under the symmetry (2.37), which were first computed in [43], it is interesting to note that the actions (2.35) and (2.38) are in fact related by a non-linear, field-dependent change of coordinates and field redefinition [44]. They correspond in fact to the non-linear realization of a different basis for the conformal algebra and thus encode the same physics [45]. However for the truncation to leading-order in derivatives that we consider here, they are two distinct theories 6. Following arguments similar to those of sections 2.2 and 2.3 one can show that there are no S 2 m symmetry generator for m ≥ 2, thus this completes the list of possibilities for N = 2 generators.

Degree three and higher
We have seen that the generators of order N = 2 have commutation relations with P µ of the form whereD is the generator of dilations. Then let us assume an order N = 3 symmetry generator existed. Its commutator with P µ would yield an order N = 2 generator, that is either K µ or A µ. Since the warped DBI algebra involving A µ is the same as the conformal algebra involving K µ, namely SO(2, D) without loss of generality we can limit ourselves to just consider K µ. The commutation relation can be written in complete generality as: where we followed the same notation as [11]: α = {α 1,..., α m } and S 1 m+1 is an order 1 or 0 generator (Dilation, Lorentz transformations or Translations). Then since the structure is that of the conformal algebra as in [11], the theorem proved there applies here too. That is, in D ≥ 3 the tensors t αµσ must vanish and thus S 3 m is not a degree three symmetry, contradicting out starting assumption. As noted in [11] and as well-known from CFT, the case D = 2 is special: the above theorem does not apply and there exist degree 3 symmetry generators. However the only invariant Lagrangians are those of (2.35) and (2.38), which have no free parameters and therefore cannot be constrained further. We conclude that there cannot be any new symmetric Lagrangians in any number of spacetime dimensions D ≥ 2. Our classification of symmetric P (X, φ) theories on flat space is thus complete. The theories listed in Table 1 exhaust all possibilities for enhanced continuous symmetries of Poincaré-invariant scalar fields theories to leading order in derivatives. In the rest of this paper we discuss some aspects of Scaling theories (2.22) and comment on the extension of our classification to cosmological backgrounds.

Scaling theories on Minkowski spacetime
In this section, we discuss the scaling-invariant theories in more detail. We first look at the dynamics of perturbations around time-dependent backgrounds. In particular, we derive constraints on the arbitrary function h(X) in (2.22) that guarantees a healthy dynamics for the fluctuations. Then we look at the implications of scale invariance for perturbations that non-linearly realize broken timetranslational invariance. The additional dilation invariance constrains the coefficients of this EFT to satisfy infinitely many recursion relations, which we derive. These results can be interpreted as constraints on the flat-space, decoupling limit of the EFT of Inflation [31].

Dynamics of perturbations
Consider a homogeneous background solution φ =φ(t). To expand the action (2.22) around this background, we find it easier to make a field redefinition:. (3.1) Note that we assume ∆ = 0, otherwise the invariant theory is automatically shift-symmetric. In this case ϕ has scaling dimension ∆ ϕ = −1 and the action takes the simpler form We rescale ϕ to make it dimensionless with an arbitrary energy scale µ: We now expand around a homogeneous solution ϕ(x, t) =φ(t) + π(x, t) whereφ(t) is a solution of the equation of motion: Using this, we can show that the action for π(x, t) starts at the