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 dimensionless hair parameter $\eps$, which at fixed source is proportional to ${\lambda }_{-}T$, 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 is irrelevant (${\Delta }_{-}=4$), the second-order response develops a resonant logarithm: the condensate runs with a closed-form coefficient fixed by the cubic vertex. The metric and $\pp$ back-reaction is obtained in closed form; for arbitrary hair it diagonalizes into two Pöschl-Teller problems and locks the $\pp$ correction to the transverse metric correction with ratio $6/\sqrt{7}$. On AdS-Schwarzschild the first law and holographic renormalization agree and give an analytic equation of state, $T/T_{0}-1=-\tfrac{5\sqrt3}{8π}\eps^{2}$ at fixed entropy and $c_{s}^{2}=\tfrac12-\tfrac{5\sqrt3}{8π}\eps^{2}$, with $\eta /s=1/4\pi$ and $ζ/η=\tfrac14\eps^{2}$. We then add the ${\mathbb{Z}}_2$-odd third scalar $\chi$ of the truncation, dual to ${\Delta }_{\chi }=5$. On the hairy background its linear equation is a Heun equation with an apparent singularity, solved by a Clausen function. At second order $\chi$ necessarily sources $\phim$, yet neither cross-coupling makes either condensate run, a mutual non-renormalization that follows from a one-step resonance identity and the relation $2{\Delta }_{\chi }-{\Delta }_{-}=2d$.

arXiv comment30 pages; v2: minor change; v3: sections and references added, several corrections