Perturbative Double-Scalar Hair on Planar Black Holes in AdS4-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 λ−T, and solve the coupled Einstein-scalar system to order eps2. 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 (Δ−=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/√7. On AdS-Schwarzschild the first law and holographic renormalization agree and give an analytic equation of state, T/T0−1=−5√3/8πeps2 at fixed entropy and cs2=1/2−5√3/8πeps2, with η/s=1/4π and ζ/η=1/4eps2. We then add the ℤ2-odd third scalar χ of the truncation, dual to Δχ=5. On the hairy background its linear equation is a Heun equation with an apparent singularity, solved by a Clausen function. At second order χ 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 2Δχ−Δ−=2d.
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BibTeX
@misc{yun2026perturbative,
title = {{Perturbative Double-Scalar Hair on Planar Black Holes in AdS$_4$-Einstein-Scalar Gravity}},
author = {Sangheon Yun},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.01332},
eprint = {2609.01332},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.01332},
}
RIS
TY - UNPB
TI - Perturbative Double-Scalar Hair on Planar Black Holes in AdS_4-Einstein-Scalar Gravity
AU - Sangheon Yun
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2609.01332
UR - https://arxiv.org/abs/2609.01332
AB - 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 λ_−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 (Δ_−=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/√7. On AdS-Schwarzschild the first law and holographic renormalization agree and give an analytic equation of state, T/T_0−1=−5√3/8πeps^2 at fixed entropy and c_s^2=1/2−5√3/8πeps^2, with η/s=1/4π and ζ/η=1/4eps^2. We then add the ℤ_2-odd third scalar χ of the truncation, dual to Δ_χ=5. On the hairy background its linear equation is a Heun equation with an apparent singularity, solved by a Clausen function. At second order χ 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 2Δ_χ−Δ_−=2d.
ER -