*Phys.Rev. D *97 (2018) no.12 arXiv link

**a-theorem for Horndeski gravity at the critical point**

**Yue-Zhou Li**^{1}, H. Lu^{1}

*1. Department of Physics, Tianjin University, Tianjin 300350, China*

**Abstract:**

We study holographic conformal anomalies and the corresponding a-theorem for Einstein gravity extended with Horndeski terms that involve up to and including linear curvature tensors. We focus on our discussion in D=5 bulk dimensions. For the generic Horndeski coupling, the a-charge is the same as that in Einstein gravity, but the inclusion of the Horndeski term violates the a-theorem. However, there exists a critical point of the Horndeski coupling, for which the theory admits nearly anti–de Sitter (AdS) spacetimes with nonvanishing Horndeski scalar. The full AdS isometry is broken down by the logarithmic scalar hair to the Poincaré group plus the scale invariance. We find that in this case the a-charge depends on the AdS radius ℓ and the integration constant χs of the Horndeski scalar. In addition, we find that two new central charges emerge, that are absent in gravities with minimally coupled matter. We call them b-charges. These b-charges also depend on ℓ and χs. We construct an a-function for fixed ℓ but with the running Horndeski scalar χ replacing the constant χs, and establish the holographic a-theorem using the null energy condition in the bulk. Furthermore, we find that there exist analogous monotonous b-functions as well. We also obtain the a-charge and the a-theorem in general odd bulk dimensions.