Absence of isolated critical points with nonstandard critical exponents in the four-dimensional regularization of Lovelock gravity

Abstract

Hyperbolic vacuum black holes in Lovelock gravity theories of odd order $N$ are known to have the so-called isolated critical points with nonstandard critical exponents (as $\alpha = 0$, $\beta = 1$, $\gamma = N-1$, and $\delta = N$), different from those of mean-field critical exponents (with $\alpha = 0$, $\beta = 1/2$, $\gamma = 1$, and $\delta = 3$). Motivated by this important observation, here, we explore the consequences of taking the $D \to 4$ limit of Lovelock gravity and the possibility of finding nonstandard critical exponents associated with isolated critical points in four-dimensions by use of the four-dimensional regularization, recently proposed by Glavan and Lin \cite{Glavan2020}. It is shown that the regularized $4D$ Einstein-Lovelock gravity theories of odd order $N > 3$ do not possess any physical isolated critical point. In fact, the critical (inflection) points of equation of state always occurs for the branch of black holes with negative entropy. The situation is quite different for the case of the regularized $4D$ Einstein-Lovelock gravity with cubic curvature corrections ($N=3$). In this case ($N=3$), although the entropy is non-negative and the equation of state of hyperbolic vacuum black holes has a nonstandard Taylor expansion about its inflection point, but there is no criticality associated with this special point. At this point, the physical properties of the black hole system change drastically, e.g., both the mass and entropy of the black hole vanishes, meaning that there do not exist degrees of freedom in order for a phase transition to occur. These results are in strong contrast to those findings in Lovelock gravity.Comment: 8 pages, 4 figures, 1 tabl