{
  "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 $\\eps^{2}$. 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 $6/\\sqrt7$. 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 $η/s=1/4π$ and $ζ/η=\\tfrac14\\eps^{2}$. We then add the $\\mathbb{Z}_{2}$-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 $2Δ_χ-Δ_{-}=2d$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.01332",
  "author_total": 1,
  "authors": [
    {
      "name": "Sangheon Yun"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0vk1454we4nbx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.01332",
  "fulltext_url": null,
  "id": "0vk1454we4nbx",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.01332",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "gr-qc"
    ],
    [
      "arxiv",
      "hep-th"
    ],
    [
      "openalex",
      "3106|Black Holes and Theoretical Physics"
    ]
  ],
  "openalex": "W7207880380",
  "pdf_url": "https://arxiv.org/pdf/2609.01332",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0vk1454we4nbx.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:54Z",
      "native_id": "2609.01332",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.01332"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:gr-qc",
      "subfield": "gravitation"
    },
    {
      "field": "physics",
      "origin": "arxiv:hep-th",
      "subfield": "high-energy"
    }
  ],
  "title": "Perturbative Double-Scalar Hair on Planar Black Holes in AdS$_{4}$-Einstein-Scalar Gravity",
  "title_text": "Perturbative Double-Scalar Hair on Planar Black Holes in AdS_4-Einstein-Scalar Gravity",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:54Z",
  "url": "https://research.draconitic.com/paper/0vk1454we4nbx",
  "venue": "arXiv",
  "versions": []
}