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arxiv: 2606.05766 · v1 · pith:G7B27L2Unew · submitted 2026-06-04 · 🌀 gr-qc · hep-th

Scalarized extremal black holes in the Einstein-Maxwell-scalar theory with two U(1) fields

Xiao Yan Chew , Yun Soo Myung This is my paper

Pith reviewed 2026-06-28 00:47 UTC · model grok-4.3

classification 🌀 gr-qc hep-th
keywords scalarized extremal black holessecondary scalar hairEinstein-Maxwell-scalar theorytwo U(1) fieldsN=4 supergravityentropy function approach
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The pith

Two scalarized extremal black holes with constant secondary scalar hair are found in the Einstein-Maxwell-scalar theory with two U(1) fields.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines scalarized extremal black holes in a theory with two U(1) fields and specific scalar couplings inspired by N=4 supergravity. It identifies two such black holes that carry constant secondary scalar hair. These solutions are recovered using both the standard scalarization method and the entropy function approach. A sympathetic reader would care because this suggests that primary scalar hair may be difficult to obtain in extremal black hole solutions within this framework.

Core claim

Two scalarized extremal black holes are found with constant secondary scalar hair. These are exactly obtained from the standard scalarization and entropy function approach. This may imply that it is not easy to find extremal black holes with primary scalar hair.

What carries the argument

The Einstein-Maxwell-scalar theory with two different scalar couplings to two U(1) fields, together with the scalarization and entropy function methods.

If this is right

  • The black holes carry constant secondary scalar hair.
  • The solutions match results from the standard scalarization technique.
  • The entropy function approach independently produces the same black holes.
  • Primary scalar hair is difficult to realize for extremal solutions in this setup.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Analogous secondary-hair solutions may appear in other supergravity truncations with similar couplings.
  • Primary hair could emerge if the extremality condition is relaxed.
  • The result limits the range of scalar hair types possible at the extremal limit.

Load-bearing premise

The specific scalar couplings to the two U(1) fields correctly capture the bosonic sector of N=4 supergravity and that the standard scalarization and entropy function methods fully apply without missing primary hair solutions.

What would settle it

Discovery of an extremal black hole with primary scalar hair in this theory, or failure of the standard methods to recover the reported solutions.

read the original abstract

We study scalarized extremal black holes in the Einstein-Maxwell-scalar theory with two different scalar couplings to two U(1) fields. This theory is inspired by the bosonic sector of $N=4$ supergravity. Two scalarzied extremal black holes are found with constant secondary scalar hair. We confirm that these are exactly obtained from the standard scalarization and entropy function approach. This may imply that it is not easy to find extremal black holes with primary scalar hair.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The manuscript studies scalarized extremal black holes in an Einstein-Maxwell-scalar theory with two distinct scalar couplings to two U(1) fields, motivated by the bosonic sector of N=4 supergravity. It reports finding two such black holes that possess constant secondary scalar hair and states that these solutions are exactly recovered via the standard scalarization procedure and the entropy function formalism. The authors suggest that this outcome indicates primary scalar hair is difficult to obtain for extremal black holes in the model.

Significance. If the standard scalarization and entropy-function methods are shown to be exhaustive within the adopted near-horizon ansatz, the result supplies a concrete illustration that secondary hair can be recovered while primary hair is absent, thereby furnishing a data point relevant to no-hair conjectures in supergravity-inspired scalar-tensor theories. The absence of free parameters in the reported solutions is a positive feature.

major comments (1)
  1. [Sections describing the entropy function approach and the scalarization procedure] The central claim that primary scalar hair is 'not easy to find' rests on the unexamined premise that the standard scalarization and entropy-function procedures are exhaustive. The manuscript must demonstrate that the near-horizon ansatz (metric, gauge fields, and scalar profile) has been varied sufficiently to exclude non-constant scalar profiles or independent horizon values that could support primary hair; without this check the implication does not follow from the two recovered solutions.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comment. We address the major point below.

read point-by-point responses

  1. Referee: The central claim that primary scalar hair is 'not easy to find' rests on the unexamined premise that the standard scalarization and entropy-function procedures are exhaustive. The manuscript must demonstrate that the near-horizon ansatz (metric, gauge fields, and scalar profile) has been varied sufficiently to exclude non-constant scalar profiles or independent horizon values that could support primary hair; without this check the implication does not follow from the two recovered solutions.

    Authors: We agree that the tentative suggestion in the abstract and conclusion—that primary scalar hair may be difficult to obtain—relies on the standard scalarization and entropy-function methods applied within the conventional near-horizon ansatz for extremal black holes. This ansatz typically assumes an AdS₂ × S² geometry with constant scalar and gauge-field profiles to satisfy the equations of motion and the attractor mechanism. We did not perform an exhaustive variation of the ansatz to rule out all possible non-constant scalar profiles or independent horizon values. To address the concern, we will revise the relevant sections to explicitly state the assumptions underlying the ansatz, provide a brief justification for restricting to constant scalars (consistent with extremality), and moderate the language of the implication to clarify that our findings hold within the standard framework rather than claiming exhaustiveness. This constitutes a partial revision. revision: partial

Circularity Check

0 steps flagged

No circularity: solutions verified against independent standard methods

full rationale

The paper reports finding two specific scalarized extremal black hole solutions with constant secondary hair in the given model, then states that these match exactly the output of the pre-existing standard scalarization procedure and entropy-function formalism. No equation is shown reducing a claimed prediction to a fitted input by construction, no load-bearing uniqueness theorem is imported from the authors' own prior work, and no ansatz is smuggled via self-citation. The central claim is a verification plus an inference about the difficulty of primary hair; that inference rests on an assumption of method completeness rather than on any definitional loop or self-referential fit. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies insufficient information to identify free parameters, axioms, or invented entities.

pith-pipeline@v0.9.1-grok · 5608 in / 942 out tokens · 31037 ms · 2026-06-28T00:47:34.170642+00:00 · methodology

discussion (0)

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Reference graph

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