Perturbative Double-Scalar Hair on Planar Black Holes in AdS$_{4}$ Einstein-Scalar Gravity

Abstract

We construct a two-scalar hairy planar black hole in four-dimensional Einstein gravity coupled to two scalars whose potential descends from the ABJM consistent truncation of eleven-dimensional supergravity. Starting from the known single-scalar hairy black brane, we switch on the second scalar $\phim$ perturbatively in its amplitude $\eps$ and solve the coupled Einstein-scalar system to order $\eps^{2}$. The linearized $\phim$ profile is an exact hypergeometric function, independent of the $\pp$ hair, whose horizon-regular branch always carries a source, so there is no spontaneous $\phim$ hair. Because the dual operator $\mathcal{O}_{-}$ is irrelevant ($\Delta=4$), the second-order response develops a logarithmically running condensate with a closed-form, scheme-independent slope $\tfrac{\sqrt{21}}{70}C_{s}^{2}$. The renormalized free energy is scheme dependent, but the geometric temperature shift $\delta T/T_{0}\approx-0.345\,\eps^{2}$ at fixed entropy is not. The full metric and $\pp$ back-reaction is obtained in closed form, diagonalizing into two P\"oschl-Teller problems with $\delta\pp=\tfrac{6}{\sqrt7}\,\delta H$. Three transport coefficients of the dual fluid remain exactly conformal to this order, while the sound speed, like the free energy, is scheme dependent.