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 dge 3 without the full conformal symmetry. The operator that is dual to the bulk trace/scalar mode however violates the dominant energy condition.
Highlights
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 the Horndeski axion excitation
Einstein-Horndeski gravity admits the anti-de Sitter (AdS) vacuum with full AdS conformal symmetry, and it is denoted as the conformal vacuum in this paper
The theory admits a scale invariant AdS vacuum whose full conformal symmetry is broken by the Horndeski scalar which exhibits the log r behavior
Summary
We consider Einstein gravity with a bare cosmological constant 0, extended with the lowest-order Horndeski term. It is easy to see that the theory admits the AdS vacuum of radius , namely dsD2 = dsA2 dSd+1 , D = d + 1 This vacuum involves only the Einstein gravity sector, and the Horndeski term can be treated perturbatively for small γ. At the saturation point of the above inequality, which is referred to as the “critical point” in [27], the theory admits a new AdS vacuum, whose radius is not governed by the bare cosmological constant, but by the ratio γ /α instead: dsD2 =. 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. It turns out that the a-theorem cannot be established for the generic AdS vacuum, but it can be for the critical vacuum [27]
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