# Scale invariance vs. conformal invariance: holographic two-point functions in Horndeski gravity

## Abstract

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.

## 1 Introduction

Conformal groups are generated by three types of transformations: (1) Poincaré transformations, (2) a scale (dilatation) transformation and (3) special conformal transformations. 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 \(d\ge 5\), 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, 15] provides a powerful tool to study certain strongly-coupled CFTs. It is also natural to adopt the holographic technique for the SQFTs [16]. 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 [17, 18], analogous to the Gauss-Bonnet combination [19]. 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. [20, 21].

*a*-theorem can only be established for the scale invariant AdS vacuum, but not for the conformal AdS vacuum [27]. 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 [31, 32] 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

The paper is organized as follows. In Sect. 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 Sect. 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 Sect. 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 Sect. 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 Sect. 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 Sect. 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.

## 2 Einstein-Horndeski gravity and AdS vacua

^{3}

*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.

*a*-charges are

*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 [27].

## 3 Two-point functions in the conformal vacuum

## 4 Linear modes and boundary terms in the scale invariant vacuum

*r*; the constant is absorbed by the background \(\chi _0\).

### 4.1 Graviton mode

### 4.2 The trace mode

*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

*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.

### 4.3 Boundary action

*K*is the trace of the second fundamental form \(K_{\mu \nu } = h_{\mu }^\rho \nabla _\rho n_\nu \) and \(h_{\mu \nu } = g_{\mu \nu } - n_\mu n_\nu \). The first term in the square bracket is the contribution from the Einstein-Hilbert term [36]. The \(\gamma \)-dependent terms are associated with the Horndeski term in the bulk action. Note that in this subsection only, we use the standard convention \(h_{\mu \nu }\) for the boundary metric. It was referred to as \(\beta _{\mu \nu }\) in Sect. 4.1. There should be no confusion between the \(h_{\mu \nu }\) here and the metric perturbation \(h_{ij}\) in the rest of the paper.

### 4.4 Holographic one-point functions

*d*increases, more and more higher-order counterterms are necessary to cancel the divergence as \(r\rightarrow \infty \). However, only the coefficient \(c_0\) contributes to the overall coefficient of the two-point functions.

## 5 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 \(C_{\mathrm{T}}\) 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, \(h^{(0)}_{ij}\) and \(\psi ^{(0)}\) actually belong to the same singlet, so do the responses \(h^{(d)}_{ij}\) and \(\psi ^{(d)}\). In fact they are the Lorentz covariantization of the cosmological mode. It is thus not apparent at the first sight which source contributes the specific percentage of the response in the one-point functions (4.25), (4.26) and (4.28), 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.

### 5.1 An algebraic proposal

### 5.2 Extended metric basis

*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.

### 5.3 Explicit results of the two-point functions

We now present the explicit two-point functions for the trace/scalar mode and these are \(\langle {\hat{T}}_{ij}{\hat{T}}_{kl}\rangle \), \(\langle {\hat{T}}_{ij} \mathcal{O}_\chi \rangle = \langle \mathcal{O}_\chi {\hat{T}}_{ij}\rangle \) and \(\langle \mathcal{O}_\chi \mathcal{O}_\chi \rangle \). It should be understood that the combination (5.9) is a null operator and non-physical.

**5.3.1** \(\langle {\hat{T}}{\hat{T}}\rangle \)

*d*is odd, and

*d*is even.

**5.3.2**\(\langle \mathcal{O}_\chi \mathcal{O}_\chi \rangle \)

*d*is odd, and

*d*is even. In configuration space, we have immediately

**5.3.3**\(\langle {\hat{T}}\mathcal{O}_\chi \rangle =\langle \mathcal{O}_\chi {\hat{T}}\rangle \)

*d*is odd, and

*d*is even. In configuration space, two-point function is

## 6 Additional comments in \(d=2\)

## 7 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 focus was on the scale invariant vacuum. We found that the perturbations around the scale invariant vacuum have nontrivial trace/scalar mode in addition to the graviton modes. The solution is the Lorentz covariantization of the boundary cosmological mode and it can contribute to the two-point functions. We obtained the holographic counterterms associated with the scale invariant vacuum, and then we derived the two-point functions of energy-momentum tensor associated with both the graviton modes and the trace/scalar mode. The non-vanishing of the two-point function of the trace/scalar mode is a distinguishing feature of SQFTs from CFTs.

The situation becomes more subtle in \(D=3\), \(d=2\) case. As expected the two-point function of the trace/scalar mode vanishes in \(d=2\), indicating that the scale invariant theory is fully conformal. The central charge derived from the holographic two-point function of the energy-momentum tensor differs from holographic anomalous *a*-charge beyond the linear order of the Horndeski coupling constant \(\gamma \). This discrepancy clearly warrants further investigation.

Our investigation of the scale invariant AdS vacuum in Einstein-Horndeski gravity, which is ghost free, indicates that strongly coupled scale invariant quantum field theory might exist without the full conformal invariance. Furthermore, the operator that is dual to the trace/scalar bulk mode however violates the dominant energy condition. Its bulk origin as the cosmological mode suggests that the boundary scalar operator may serves as an inflaton in cosmology. However, multiple subtleties remain that raise further questions. In our holographic construction, the boundary field theory must include a scalar operator that is the holographic dual of the Horndeski axion. It is this conformal primary operator that serves the purpose of violating the special conformal symmetry, via the trace equation (4.31). To make the bulk theory quantum complete by introducing additional necessary fields will not alter this fact unless the scale-invariant AdS vacuum no longer exists at all at the full quantum level. On the other hand, in our construction, there is no apparent local virial current operator that is typically arising in an SQFT, indicating that the scale symmetry may be violated as well. However, we find that the holographic two-point functions all respect the covariance of the scaling symmetry. It is thus intriguing to speculate whether there should be a generalized theory of Einstein-Horndeski gravity in which the local virial current operator is visible in the spectrum. Indeed it should be pointed that the constant shift symmetry of the Horndeski theory can be gauged to give rise to Einstein-vector theories [41], 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.

## Footnotes

- 1.
- 2.
Although Einstein gravity with minimally coupled vector fields can admit AdS that is not fully conformal, the theories typically violate the null-energy condition [16].

- 3.
We do do not consider in this paper the scale transformation that leaves the equations of motion invariant, but not the action. Such scale invariance was referred to as the “trombone” symmetry in [33].

## Notes

### Acknowledgements

We are grateful to Hong-Da Lü and Zhao-Long Wang for useful discussions. H.-Y.Z. is grateful to the Center of Joint Quantum Studies for hospitality. This work is supported in part by NSFC Grants nos. 11875200 and 11475024.

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