Scale Invariance vs. Conformal Invariance: Holographic Two-Point Functions in Horndeski Gravity

We consider Einstein-Horndeski gravity with a negative bare constant as a holographic model to investigate whether a scale invariant quantum field theory can exist without the full conformal invariance. Einstein-Horndeski gravity can admit two different AdS vacua. One is conformal, and the holographic two-point functions of the boundary energy-momentum tensor are the same as the ones obtained in Einstein gravity. The other AdS vacuum, which arises at some critical point of the coupling constants, preserves the scale invariance but not the special conformal invariance due to the logarithmic radial dependence of the Horndeski scalar. In addition to the transverse and traceless graviton modes, the theory admits an additional trace/scalar mode in the scale invariant vacuum. We obtain the two-point functions of the corresponding boundary operators. We find that the trace/scalar mode gives rise to an non-vanishing two-point function, which distinguishes the scale invariant theory from the conformal theory. The two-point function vanishes in $d=2$, where the full conformal symmetry is restored. Our results indicate the strongly coupled scale invariant unitary quantum field theory may exist in $d\ge 3$ without the full conformal symmetry. The operator that is dual to the bulk trace/scalar mode however violates the dominant energy condition.

The Poincaré invariance is the underlying symmetry of any relativistic quantum field theories (QFT). Interestingly the Poincaré and scale transformations form a subgroup, which leads to an important question whether there exists a scale invariant quantum field theory (SQFT) that is not a (fully) conformal field theory (CFT). After decades of research, a definite answer to this question for the general situation remains elusive. The subject has been reviewed in [1] not so recently. In d = 2 dimensions, unitary scale invariant theories that have the discrete spectrum and the finite two-point function of energy-momentum tensor are necessarily conformal [2] 1 . Examples of SQFTs without a full conformal symmetry in d = 2 violating these assumptions can be constructed, see, e.g. [4][5][6]. In other dimensions, for example, in d = 3 and d ≥ 5, some so called "free Maxwell theories" were constructed and demonstrated that they were SQFTs but not CFTs [7]. However, the situation is much subtler in d = 4. The perturbative approach can be used to demonstrate the enhancement of conformal symmetry from scale invariance near the fixed point [8], and a number of such perturbative examples were studied extensively [9][10][11]. It was argued that even beyond perturbative region, SQFT should also be CFT [8,12,13]. However, no well-defined proof is available yet for the non-perturbative statement and a complete answer is far from clear in d = 4.
The AdS/CFT correspondence [14] provides a powerful tool to study certain stronglycoupled CFTs. It is also natural to adopt the holographic technique for the SQFTs [15]. In fact the holographic approach may be exactly the right tool to address whether SQFTs without the full conformal invariance can exist, since this may be an intrinsic non-perturbative problem. Indeed, although anti-de Sitter (AdS) spacetimes with full conformal group arise naturally and commonly as vacua in bulk gravity theories, geometries that preserve both the Poincaré and scale invariance, but not full conformal invariance, are hard to come by. 2 The difficulty is a reflection of the fact that an SQFT is likely a CFT. However, concrete such an example does exist and it is provided by Einstein-Horndeski gravity coupled to a negative cosmological constant. Horndeski terms are higher-derivative invariant polynomials that are built from the Riemann curvature tensor and the 1-form of an axion [16,17], analogous to the Gauss-Bonnet combination [18]. It turns out that in addition to the usual AdS vacuum with the vanishing Horndeski scalar, the theory also admits the planar AdS at some critical point of the coupling constants, where the Horndeski scalar is non-vanishing and the special conformal invariance of the AdS is broken by the scalar. Black holes of Horndeski gravity at the critical point were also constructed, e.g. [19,20].
This result strongly suggests that Einstein-Horndeski gravity on the critical AdS vacuum may provide a consistent holographic dual for some strongly coupled SQFT that is not conformal. An important test to distinguish a CFT and SQFT is to examine the trace of the stress tensor, which vanishes for CFT, but not for SQFT. The holographic dictionary [29,30] provides a powerful technique to calculate the two-point functions of the energy-momentum tensor of strongly-coupled field theory using classical gravity. In this paper, we employ this technique to calculate the holographic two-point functions in Einstein-Horndeski gravity in both the conformal AdS and scale-invariance AdS vacua. The holographic two-point functions in the conformal invariant vacuum satisfies as one would expect. However, the above quantities do not vanish for the scale invariant vacuum since there is in addition a trace/scalar mode.
The paper is organized as follows. In section 2, we review the Horndeski gravity and its vacuum solutions. One vacuum has full conformal symmetry, while the other at the critical point exhibits only the scale invariance. In section 3, we readily obtain the two-point functions of energy-momentum tensor in conformal vacuum, the result makes no difference compared to pure Einstein gravity. In section 4, we consider the scale invariant vacuum, and we obtain the linear perturbation solutions. We find in addition to the graviton modes, extra trace mode is also available. Furthermore, we obtain the holographic counterterms at the critical point. With the counterterms in hand, we employ the holographic dictionary to derive the two-point functions of the boundary energy-momentum tensor associated with the graviton modes and moreover, the one-point function formulae associated with the trace mode. In section 5, we analyze the ambiguity in determining the source of the trace mode.
To resolve the ambiguity, we come up with an algebraic proposal to obtain the two-point functions. We also apply the extended metric basis method to verify the two-point functions we derive. In section 6, we discuss two-point functions in d = 2, and it turns out the trace of energy-momentum tensor indeed gives no contribution to the two-point functions. The paper is summarized in section 7. In Appendix A, we exhibits the algebraic proposal for the diagonal part of the graviton modes in Einstein gravity, and demonstrate explicitly that the algebraic proposal yields the right answer.

Einstein-Horndeski gravity and AdS vacua
In this paper, we consider Einstein gravity with a bare cosmological constant Λ 0 , extended with the lowest-order Horndeski term. The bulk action is given by In this action, the Newton's constant is set to unity. The remaining nontrivial parameter is the bare cosmological constant Λ 0 and the ratio γ/α. This is because a constant scaling of the Horndeski scalar χ yields a homogeneous scaling of both the coupling constants (α, γ).
Note also that χ is axionic-like and the Lagrangian is invariant under the constant shift of χ.
The covariant equations of motion of (2.1) can be found in, e.g. [20,21]. They are given by the Einstein equation E µν = 0 and the scalar equation E = 0, where [21] It is easy to see that the theory admits the AdS vacuum of radius ℓ, namely This vacuum involves only the Einstein gravity sector, and the Horndeski term can be treated perturbatively for small γ. The linearized perturbation of the scalar χ has the kinetic term The ghost-free condition requires that At the saturation point of the above inequality, which is referred to as the "critical point" in [26], the theory admits a new AdS vacuum, whose radius is not governed by the bare cosmological constant, but by the ratio γ/α instead: Note that the χ solution is parameterized to be the same as in [26]. In this vacuum solution, the Horndeski term is an integral part and cannot be viewed as a small perturbation. In particular, when ℓ is large, corresponding to small curvature in gravity, we must have large γ; on the other hand, the small γ implies small ℓ and the corresponding large spacetime curvature. A further important feature is that the full AdS conformal symmetry of the metric is broken by the Horndeski axion down to the subgroup of Poincaré symmetry together with the scale invariance, namely 3 The special conformal transformation invariance of the conformal group is broken. It should be pointed out right away that under the above scale transformation, the axion χ undergoes a constant shift. We take the view that since χ appears in the theory only through a derivative, vacua with χ and χ + c for any constant c should be identified as the same. The holographic dual is thus expected to be a relativistic quantum field theory with the scale rather than the full conformal invariance.
Holographic conformal anomaly for Einstein-Horndeski gravity (2.1) was recently ob-tained [26]. Specifically, the a-charges are generic AdS vacuum : critical AdS vacuum : Note that we have stripped off the overall (inessential) purely numerical constants in presenting the a-charges. It turns out that the a-theorem cannot be established for the generic AdS vacuum, but it can be for the critical vacuum [26].

Two-point functions in the conformal vacuum
The linear perturbation of the generic AdS vacuum (2.3) involves the graviton and axion modes. Taking the AdS spacetime to be the planar type, the perturbations are where ∂ i = η ij ∂ j = η ij ∂/∂x j . The perturbation satisfies where the Laplacian¯ is defined with respect to the AdS vacuum. The scalar field satisfies Thus the linear equation of the graviton is identical to that of AdS vacuum in Einstein gravity, and the solution can be expressed in terms of the Hankel's function This leads to the holographic two-point function for the boundary energy-momentum tensor [32] T where I ijkl (x) is the boundary spacetime tensor defined by The coefficient C T can be expressed as [33] .
In other words, C T is the dimensionful quantity with the overall purely numerical factors stripped off. It is easy to see the identity This identity was seen and conjectured to hold for all the higher-order massless gravities [33].

Linear modes and boundary terms in the scale invariant vacuum
We now consider the linear perturbation of the AdS vacuum at the critical point (2.6). The ansatz is again the linearized FG type, namely The linearized equations are complicated and they become simpler when we examining the transverse traceless modes and the trace mode separately.

Graviton mode
The massless spin-2 graviton mode satisfies further the transverse and traceless conditions.
The scalar equation is then automatically satisfied at the linear level. The tensor E µν becomes where the metric and covariant derivatives are defined on the AdS background, and The linearized equation E ri = 0 implies that Thus c r = 0 and ψ = c i x i . It follows that the equation E ij = 0 does not involve ψ and we where is defined with respect to the metric η ij . The absence of ghost excitations requires that the coefficient of the time-derivative termḧ ij be non-negative, namely The solution is given by

The trace mode
In addition to the transverse and traceless graviton mode obtained in the previous subsection, we find the theory admits an additional scalar mode that consists of the metric trace and also the Horndeski axion excitation. Taking the Lorenz gauge p µ = (−E, 0, 0, . . . , 0), the ansatz is given by The perturbative functions (h 0 , h, ψ) depend on the bulk radius r and boundary time t only. Thus any constant r slice of the spacetime is an FLRW cosmological metric. For this reason we may also call this the cosmological mode. The kinetic term of the linearized bulk Lagrangian is where κ eff is given in (4.7). This Lagrangian is analogous to the kinetic term of the linearized FLRW model withh corresponding to the scale factor and ψ corresponding to the matter scalar field. The absence of the ghost excitation requires that κ eff ≥ 0.
The full linearized equations can be solved by where ψ satisfies The general solution is given by Performing a Lorentz transformation in the boundary, we obtain the Lorentz covariant expression for the trace/scalar mode, namely where the Lorentz covariant linear perturbation is and (h 0 , h) are given by (4.11) with (4.13) where Et is replaced by p i x i . It is easy to verify that this mode is neither transverse nor traceless; it is the Lorentz covariantization of the boundary cosmological mode.

Boundary action
In order to derive the boundary properties from the bulk perturbations, it is necessary to construct the boundary action. The boundary action contains two parts. The first part is the Gibbons-Hawking surface term, which is given by where n µ is the unit vector normal to the surface and K is the trace of the second fundamental form K µν = h ρ µ ∇ ρ n ν and h µν = g µν − n µ n ν . The first term in the square bracket is the contribution from the Einstein-Hilbert term [34]. The γ-dependent terms are associated with the Horndeski term in the bulk action. Note that in this subsection only, we use the standard convention h µν for the boundary metric. It was referred to as β µν in subsection 4.1. There should be no confusion between the h µν here and the metric perturbation h ij in the rest of the paper.
The second part is the boundary counterterm that is necessary for asymptotic AdS spacetimes. We find, up to the quadratic curvature term and scalar term, that it is given by where R is the Ricci scalar of the boundary metric, and the derivatives are associated with the boundary metric. The coefficients of the counterterms for the generic AdS vacuum was know, given by For the critical AdS vacuum, we find that they become (4.19)

Holographic one-point functions
Having obtained the full action, namely we can define the one-point function of the holographic energy-momentum tensor [35] T The two-point function can be obtained from dividing the one-point function from the leading expression of the mode h ij . This approach is not yet covariant and a full covariant approach based on writing the modes in the metric basis was given in [33,36,37]. In this paper, we shall not go through this detail but simply present the results. The two-point function of the boundary energy-momentum tensor associated with the bulk graviton is given immediately by the same expression (3.6), with C T = N 2 C T and (4.24) By applying (4.21) and (4.22),we find that the one-point function of the energy-momentum tensor takes the form The vanishing of T tt implies that the trace/scalar mode violates the dominant energy condition, but it can satisfy the other energy conditions. The result can be straightforwardly covariantized and then we have The one-point function of χ can be defined similarly For now, we have It is of importance to keep in mind that from (4.11) and (4.24) we have (4.29) It follows that we have and they are linear functions of the source h (0) (or ψ (0) .) It follows from (4.26) that the trace of the energy-momentum tensor is shows that there is no local expression for a virial current, which was typically introduced in a scale invariant theory to form a conserved current associated with the scaling symmetry.
This seems to suggest that even the scaling symmetry may be broken. It is thus necessary to study the two-point functions associated with the trace scalar mode and we find that the scaling symmetry does preserve. In the next subsection, we carrying out the computation of the two-point functions, including T T , T χ = χT and χχ .

Two-point functions in scale invariant vacuum
The two-point functions of energy-momentum tensor associated with graviton modes can be obtained readily and it is given in (3.6) with the coefficient C T given in (4.23). In this section, we focus on the two-point functions associated with the trace mode. However, as the on-shell solution (4.11) suggests, all seemingly different sources, h ij and ψ (0) actually belong to the same singlet, so do the responses h which is a necessary information to derive the corresponding two-point functions. The analogous stituation can be found even in graviton modes, which is discussed in Appendix A. Following the procedure presented in Appendix A for an simper example, we shall rewrite the one-point functions such that the distinctions between different contributions become clear.

An algebraic proposal
It follows from (4.15) that the covariant source h (0)kl is given by we can rewrite the one-point functions of the previous subsection as In other words, we split the one-point functions as the linear combinations of the metric and scalar sources as if they have different origins, with the coefficients a i , b i to be determined.
Requiring T χ = χT implies a 2 = b 1 . The two-point functions can then be deduced, yielding To obtain a i and b i , we note they must be such that (5.3) reproduces the results (4.25), (4.26) and (4.28). It follows from the identity where T and χ should be evaluated using (4.25) and (4.28). Furthermore, (5.4) forms a that must have one zero eigenvalue since the system has only one independent mode. Hence, we have the constraint Thus we have three equations for (a 1 , a 2 , b 2 ). Solving (5.6) and (5.7), we can obtain a i and b i . They are given by The zero eigenvector is given by Thus the operator N ij is null with N ij · · · = 0, and decouples from the physical spectrum, implying that trace/scalar operators have only one nontrivial combination. The result is consistent with (4.30), reflecting that this is the right approach. In Appendix A, we use the same procedure to obtain the correct two-point functions of the diagonal part of the energy momentum tensor associated with the graviton mode. In the next subsection we verify the result by means of the similar analysis developped in [33,36,37] and encoding the trace/scalar mode in the extended metric basis.

Extended metric basis
In this subsection, we aim to validate the algebraic proposal in the previous subsection by following and generalizing the explicit analysis in [33,36,37]. Considering we have only one single mode, h where we have introduced the extended vielbein e a with a = 0, . . . , d, and the factor "2" is for latter convenience. We can also define the extended one-point function With this notation, the two-point functions are given by To proceed, after using (5.5), we note (5.13) Therefore, we have We end up with Then we can immediately obtain the two-point functions by reading off the components of (5.15), namely Provided with (4.30) and the value of C in (5.13), we reproduce the results that were given by (5.4) and (5.8). Obviously, (5.6) and (5.7) are satisfied.

Explicit results of the two-point functions
We now present the explicit two-point functions for the trace/scalar mode and these are T ij T kl , T ij χ = χT ij and χχ . It should be understood that the combination (5.9) is a null operator and non-physical.

T T
In momentum space, we have for d is odd, and for d is even.
In the configuration space, two-point function is given by whereΘ ij is given byΘ We can obtain the explicit structure of the two-point function of energy-momentum tensor associated with the trace/scalar mode by evaluatingΘ ijΘkl 1 x 2(d−2) , we then obtain where

χχ
In momentum space, the two-point function is given by for d is odd, and for d is even. In configuration space, we have immediately

T χ = χT
In momentum space, explicitly, we have for d is odd, and for d is even. In configuration space, two-point function is It is worth noting thatΘ To conclude this section, we stress that all the two-point functions respect the covariance of the scaling symmetry, indicating the theory is scale invariant.
6 Additional comments in d = 2 As was discussed in the introduction, an scale invariant theory in d = 2 should be enhanced to be fully conformal, satisfying the theorem proved in [2]. For the D = 3 Einstein-Horndeski theory, we expect that the holographic two-point functions in d = 2 for the trace/scalar modes vanish identically. Indeed, these two-point functions presented in (5.3) all have an overall factor (d − 2). The traceless energy-momentum tensor on the other hand has nonvanishing two-point functions and they were presented in section 3.6. The overall coefficient, when specialized in d = 2, is given by which is precisely the holographic central charge. Although this result fits the expectation, it is derived from holographic dictionary between the bulk graviton and boundary stress tensor in general dimensions. However, there is no local graviton in D = 3 gravity.
In order to derive the two-point function, we follow the procedure presented in [38].
The remaining equations of motion give rise to The resulting one-point functions are given by These two-point functions are all vanishing up to the contact terms, i.e.
To compute the traceless part of two-point functions, for example, T zz T zz , we set ψ = 0 and turn off the irrelevant sources h zz . The remaining equations of motion reduce to From the first equation above, we have immediately (6.9) (Here we made use of the formula ∂z 1 z = 2πδ 2 (z).) In addition, we find that the one-point function is Therefore, we have where the numerical-stripped coefficient is This result is different from (6.1) which was obtained from specializing the general results to d = 2. Instead we haveC The discrepancy at the higher orders of γ requires further investigation.

Conclusion
In this paper, we obtained the holographic two-point functions of Einstein-Horndeski gravity with negative cosmological constant. Einstein-Horndeski gravity admits the AdS vacuum with full AdS conformal symmetry, and it is denoted as the conformal vacuum in this paper. In addition, the theory admits a scale invariant AdS vacuum whose full conformal symmetry is broken by the Horndeski scalar which exhibits the log r behavior. Therefore, the theory should have some SQFT dual and can naturally serve as the holographic model to investigate the difference between SQFT and CFT.
We obtained the holographic two-point functions of the energy-momentum tensor in the conformal vacuum, and they are the same as those in pure Einstein-AdS gravity. Our   [39], where a vector field and the curvature tensor are non-minimally coupled. It is of great interest to investigate the same issue in these theories where there are also scale-invariant but not conformal AdS vacua.

A An example in diagonal graviton modes
To illustrate the algebraic proposal in subsection 5.1, we consider the diagonal graviton modes as an example since they are not all linearly independent, but satisfying the traceless condition. For simplicity, we focus on pure Einstein gravity in D = 4, d = 3, and consider following the following diagonal transverse perturbation ds 2 = ℓ 2 r 2 dr 2 + r 2 η ij dx i dx j + r 2 h 1 (r, t)dx 2 1 + h 2 (r, t)dx 2 2 . (A.1) The solution is given by In other words, h 1 and h 2 are linearly dependent. The one-point functions can be found in [33] and are given by From the solution (A.2), it is clear that there is only one mode rather than two, which is similar to our case in trace/scalar mode of Einstein-Horndeski gravity at the critical point. We rewrite the one-point functions as where a 2 = b 1 is due to T 1 T 2 = T 2 T 1 . We can now immediately read out the two-point functions where N 1 is i 8π , which is exactly the same as N 1 in [33] Therefore, the two-point functions are given by On the other hand, the two-point function of Einstein gravity in momentum space is explicitly obtained, and it is given by We focus on p = (E, 0, 0), and hence Θ 11 = Θ 22 = E 2 , Θ 12 = Θ 21 = 0 , It is now clear that our results (A.10) match the exact results obtained in [33].