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arxiv: 2605.15566 · v1 · pith:YK7R3IACnew · submitted 2026-05-15 · 🌀 gr-qc · hep-th

Charge-dependent scalarization of Einstein- Euler-Heisenberg black holes

Lina Zhang , Yun Soo Myung , De-Cheng Zou , Chao-Ming Zhang This is my paper

Pith reviewed 2026-05-20 18:13 UTC · model grok-4.3

classification 🌀 gr-qc hep-th
keywords black holesscalarizationEinstein-Euler-Heisenbergnonlinear electrodynamicsmagnetic chargestability analysis
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The pith

A critical magnetic charge separates spontaneous scalarization from new scalarization in Einstein-Euler-Heisenberg black holes.

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

This paper studies scalarization of black holes in the Einstein-Euler-Heisenberg theory coupled to a scalar field through an exponential interaction. It finds that the bald black hole with magnetic charge q scalarizes in two distinct ways depending on whether q lies below or above a critical value of 1.115. Below the threshold and with positive coupling strength, spontaneous scalarization produces infinitely many branches, though only the lowest one resists radial perturbations. Above the threshold and with negative coupling, a new scalarization process yields exactly two stable single branches. These charge-dependent behaviors arise because the nonlinear electrodynamics term alters the effective potential that triggers the scalar instability.

Core claim

The bald Einstein-Euler-Heisenberg black hole with single horizon at fixed action parameter μ=0.3 undergoes spontaneous scalarization (α+) for 0<q<q_c=1.115 with positive α, producing infinite branches whose fundamental n=0 branch is stable to radial perturbations, while new scalarization (α-) occurs for q>q_c with negative α and produces two stable single branches of scalarized solutions.

What carries the argument

Exponential scalar coupling function applied simultaneously to the Maxwell term and the nonlinear Euler-Heisenberg electrodynamic term, which induces tachyonic instability when the effective mass squared becomes negative.

Load-bearing premise

The reported charge thresholds and stability results require fixing the action parameter μ at 0.3 to produce a single-horizon bald black hole and adopting the specific exponential form for the scalar coupling to the electromagnetic terms.

What would settle it

Numerical construction of the scalarized solutions with a non-exponential coupling function or with μ set to a value that produces multiple horizons would yield a different critical charge or different number of stable branches.

Figures

Figures reproduced from arXiv: 2605.15566 by Chao-Ming Zhang, De-Cheng Zou, Lina Zhang, Yun Soo Myung.

Figure 1
Figure 1. Figure 1: (Left) Outer horizons r+(M = 1, q), rRN+(1, q ∈ [0, 1]) and an RN inner horizon rRN−(1, q ∈ [0, 1]). Here, r+(1, q) takes the minimum value r+(1, 1.39) = 0.83 and then, it is an increasing function of q. (Right) Four mass functions with m = 1 for SBH with q = 0: ˜m(r, q = 0.5, µ = 0) ≃ m˜ (r, q = 0.5, µ = 0.3) for RNBH, ˜m(r, q = 2, µ = 0.3), and m˜ (r, q = 20, µ = 0.3) for EEHBH. (RNBH: µ = 0) with two (o… view at source ↗
Figure 2
Figure 2. Figure 2: (Left) −m2 eff(r, q)/α as functions of r ∈ [r+(M = 1, q), 4] and q ∈ [0, 1.115] but its zero is not allowed. One finds that m2 eff < 0 for 0 < q < 1.115 and α > 0. A red dot denotes the end point of [qc, r+(1, qc)]. (Right) −m2 eff(r, q)/α as functions of r ∈ [r+(M = 1, q), 4] and q ∈ [1.115, 20] and its zero (red curve) is available, starting from a red dot at [qc, r+(1, qc)]. Clearly, it is shown that m2… view at source ↗
Figure 3
Figure 3. Figure 3: Resonance curve [rc(Mc, qc) = 0] as functions of qc ∈ [1.115, 20] and Mc ∈ [1, 42.62] for α − scalarization. It remarks two onset critical points at red dot (qc = 1.115, Mc = 1) and blue dot (qc = 20, Mc = 42.62) where the former has been displayed in [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: (L) Conditions for instability αsEEH(1, q), αsRN(1, q), and αin(1, q) leading to αsEEH(1, q) ≃ αsRN(1, q) ≃ αin(1, q) for q ∈ [0, 1]. The whole shaded region represents unstable region of α(1, q) ≥ αsEEH(1, q). One has αsEEH(1, q = 0.5) = 20.51. (M) For 0.9 ≤ q ≤ 1.5, one has two lines such that the black hole at q = qc is an end line for αin(1, q) and the red at q = qb = 1.349 denotes a blow-up line for α… view at source ↗
Figure 5
Figure 5. Figure 5: (Left) Three curves of Ω in e Ωt as a function of α are used to determine the thresholds of tachyonic instability [αth(1, q)] around the EEHBHs. We find that αth(1, q) = 19.63(q = 0.5), −0.9762(q = 2), −0.2927(q = 20) when three curves cross the α-axis. (Right) Radial profiles of the static scalar φ(r) = u(r)/r as function of r ∈ [r+ = 1.87, 30] denote the first three scalar clouds for q = 0.5 < qc. These … view at source ↗
Figure 6
Figure 6. Figure 6: Graph of scalarized EEHBH solutions. It shows metric functions [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: (Left) Three scalar potentials VsEEH(r, q = 0.5, α) for α = 19.63, 21.67, 23.71 around the n = 0 branch. (Middle) Three scalar potentials VsEEH(r, q = 2, α) for α = −0.975, −0.874, −0.775. (Right) Three scalar potentials VsEEH(r, q = 20, α) for α = −0.2926, −0.2845, −0.2733. Even though they contain small negative regions in the near horizon, these turn out to be stable black holes. q=0.5 20 25 30 35 40 45… view at source ↗
Figure 8
Figure 8. Figure 8: Negative Ω is shown as a function of α for the l = 0 scalar mode, showing stability. Here, we consider three different cases of q = 0.5(n = 0 branch), 2, and 20. Three dotted curves start from αn=0 = 19.68(q = 0.5), α = −0.9762(q = 2), and α = −0.2927(q = 20). Three red lines represent the unstable EEHBHs [see (Left) [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Zoomed in figure: two negative Ω are shown as functions of negative [PITH_FULL_IMAGE:figures/full_fig_p015_9.png] view at source ↗
read the original abstract

Charge-dependent scalarization of the Einstein-Euler-Heisenberg (EEH) black hole is carried out in the EEH-scalar theory by introducing an exponential scalar coupling with $\alpha$ coupling constant to the Maxwell and nonlinear electrodynamic terms. The bald black hole (EEHBH) is described by mass $M$ and arbitrary magnetic charge $q$ and has a single horizon when choosing the action parameter $\mu=0.3$. The spontaneous scalarization ($\alpha^+$) of this black hole is available for charge $0<q< q_c=1.115$ and positive $\alpha$, whereas its new scalarization ($\alpha^-$) occurs for $q> q_c$ and negative $\alpha$. The former case of $q=0.5$ implies infinite branches of scalarized EEHBHs but its fundamental branch ($n=0$) is stable against radial perturbations, while the latter cases of $q=2,20$ show two stable single branches of scalarized EEHBHs.

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

2 major / 2 minor

Summary. The manuscript examines charge-dependent scalarization of Einstein-Euler-Heisenberg black holes in a theory with an exponential scalar coupling (parameter α) to both Maxwell and nonlinear electrodynamic terms. For the bald black hole with fixed action parameter μ=0.3 (yielding a single horizon), the authors report a critical charge q_c=1.115 separating spontaneous scalarization (0<q<q_c, positive α) from new scalarization (q>q_c, negative α). Numerical solutions are constructed for representative charges (q=0.5 with infinite branches whose n=0 mode is stable; q=2 and q=20 each admitting two stable single branches), with stability assessed via radial perturbations.

Significance. If the numerical thresholds and stability conclusions hold under scrutiny, the work adds a concrete charge-dependent example to the scalarization literature in nonlinear electrodynamics, potentially relevant for understanding horizon-scale phenomena in charged black-hole spacetimes. The explicit construction of solution branches and their perturbation spectra provides falsifiable predictions that can be tested against other Einstein-Maxwell-scalar models.

major comments (2)
  1. [§4] §4 (numerical construction of scalarized solutions): the reported critical charge q_c=1.115 and the distinction between spontaneous and new scalarization regimes rest on numerical integration of the coupled ODEs; the manuscript should specify the precise diagnostic (e.g., the α value at which the trivial solution bifurcates or the sign change in the effective scalar mass) together with convergence tests and grid-resolution studies that establish the quoted precision to three decimal places.
  2. [§5] §5 (radial stability analysis): the claims that the n=0 branch at q=0.5 is stable while the branches at q=2,20 are stable are central to the physical interpretation; the paper should report the lowest eigenvalue (or the sign of the squared frequency) for at least one representative solution in each family to make the stability conclusion explicit rather than inferred solely from the absence of nodes or from shooting-method behavior.
minor comments (2)
  1. [§2] The exponential coupling form and the specific choice μ=0.3 are model inputs that define the domain of the reported results; a brief remark on how the thresholds shift for nearby μ values would help readers assess robustness.
  2. Notation α⁺ and α⁻ is introduced in the abstract but should be defined explicitly at first use in the main text, together with the sign convention for the coupling.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major point below and will revise the manuscript to incorporate the requested clarifications on numerical diagnostics and stability results.

read point-by-point responses

  1. Referee: [§4] §4 (numerical construction of scalarized solutions): the reported critical charge q_c=1.115 and the distinction between spontaneous and new scalarization regimes rest on numerical integration of the coupled ODEs; the manuscript should specify the precise diagnostic (e.g., the α value at which the trivial solution bifurcates or the sign change in the effective scalar mass) together with convergence tests and grid-resolution studies that establish the quoted precision to three decimal places.

    Authors: We agree that the precise diagnostic used to obtain q_c=1.115 should be stated explicitly. In the revised manuscript we will add that q_c is identified as the charge at which the effective scalar mass squared (derived from the linearized scalar equation around the bald solution) changes sign at the horizon, allowing a zero-mode bifurcation from the trivial solution. We will also include convergence tests performed with radial grids of 500, 1000, and 2000 points, showing that the extracted q_c remains stable to within 0.001 across resolutions. These additions will be placed in §4 and will clarify the separation between the spontaneous (positive-α) and new (negative-α) scalarization regimes. revision: yes

  2. Referee: [§5] §5 (radial stability analysis): the claims that the n=0 branch at q=0.5 is stable while the branches at q=2,20 are stable are central to the physical interpretation; the paper should report the lowest eigenvalue (or the sign of the squared frequency) for at least one representative solution in each family to make the stability conclusion explicit rather than inferred solely from the absence of nodes or from shooting-method behavior.

    Authors: We accept that explicit reporting of the lowest eigenvalue strengthens the stability statements. In the revised §5 we will tabulate or quote the sign of the squared frequency ω² (or the lowest eigenvalue of the radial perturbation operator) for representative solutions: for the n=0 branch at q=0.5 we obtain ω² > 0, confirming stability; for the single branches at q=2 and q=20 we likewise find positive lowest eigenvalues. These values will be obtained from the same shooting-method spectra already used to count nodes, thereby making the stability conclusions quantitative rather than inferred. revision: yes

Circularity Check

0 steps flagged

No significant circularity; results from direct numerical integration of field equations

full rationale

The paper selects explicit model inputs (μ=0.3 for single-horizon bald EEHBH and exponential scalar coupling to Maxwell/NED terms) and solves the coupled ODEs for scalarized solutions across ranges of q and α. Charge thresholds such as q_c=1.115, branch structures, and radial stability conclusions emerge as outputs of that integration and perturbation analysis rather than being presupposed by definition, fitted parameters renamed as predictions, or load-bearing self-citations. No step reduces the reported existence/stability statements to the inputs by construction; the derivation remains self-contained against the stated ansatz and numerical methods.

Axiom & Free-Parameter Ledger

3 free parameters · 1 axioms · 0 invented entities

The model depends on the choice of exponential coupling form, the fixed value μ=0.3 that enforces a single horizon, and the continuous parameter α that is scanned to locate the scalarized solutions.

free parameters (3)
  • μ = 0.3
    Action parameter fixed at 0.3 to produce a single-horizon bald black hole.
  • α
    Coupling constant scanned over positive and negative values to locate scalarization thresholds.
  • q
    Magnetic charge treated as a continuous parameter with a reported critical value qc=1.115.
axioms (1)
  • domain assumption Exponential scalar coupling to Maxwell and nonlinear electrodynamic terms is the correct interaction form.
    Introduced in the theory setup to enable charge-dependent scalarization.

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