A no-hair theorem for spherically symmetric black holes in R2 gravity

In a recent paper Canate (Class Quantum Gray 35:025018, 2018) proved a no hair theorem to static and spherically symmetric or stationary axisymmetric black holes in general f (R) gravity. The theorem applies for isolated asymptotically flat or asymptotically de Sitter black holes and also in the case when vacuum is replaced by a minimally coupled source having a traceless energy momentum tensor. This theorem excludes the case of pure quadratic gravity, f (R) = R-2. In this paper we use the scalar tensor representation of general f (R) theory to show that there are no hairy black hole in pure R-2 gravity. The result is limited to spherically symmetric black holes but does not assume asymptotic flatness or de-Sitter asymptotics as in most of the no-hair theorems encountered in the literature. We include an example of a static and spherically symmetric black hole in R-2 gravity with a conformally coupled scalar field having a Higgs-type quartic potential.