The team worked within the Einstein–Euler–Heisenberg-scalar (EEHS) theory, which extends Einstein–Maxwell gravity by adding a nonlinear electrodynamics term rooted in the pioneering work of Hans Euler and Werner Heisenberg. In 1936, Euler and Heisenberg computed how quantum fluctuations of the vacuum polarize in strong electromagnetic fields, producing effective corrections to Maxwell’s classical equations. These corrections are controlled by a parameter, denoted here by the Greek letter mu, that encodes the strength of the nonlinear term proportional to the square of the electromagnetic invariant. When this quantum-inspired correction is included, the resulting Einstein–Euler–Heisenberg (EEH) black hole, first obtained by Yajima and Tamaki, is described by its mass, its magnetic charge, and the parameter mu. Crucially, the authors chose mu equal to 0.3, a value for which the black hole possesses a single event horizon and, remarkably, faces no upper limit on its charge, unlike the familiar Reissner–Nordström solution of classical electrodynamics.

Into this setting the researchers introduced a scalar field coupled exponentially to both the Maxwell term and the nonlinear electrodynamics term, with a coupling constant alpha. When the scalar field vanishes, the solution reduces exactly to the bald EEH black hole. The central question was whether, and under what conditions, that bald configuration becomes unstable to the growth of scalar hair. The answer hinges on an effective mass term that the scalar field experiences in the black hole’s background. If this effective mass squared becomes negative in a region outside the horizon, the scalar behaves like a tachyon: tiny fluctuations grow exponentially rather than decay, triggering spontaneous scalarization. This is the same tachyonic instability mechanism that underlies the celebrated scalarization of neutron stars in Gauss–Bonnet gravity and of charged black holes in Einstein–Maxwell-scalar theory.

The key discovery of the new work is a sharp divide in behavior, governed by a critical onset charge. Using a resonance condition derived from the merging of turning points of the scalar’s binding potential, the authors found that this critical charge equals the square of the outer horizon radius divided by the square root of twice mu. For a black hole of unit mass and mu equal to 0.3, the critical charge evaluates to approximately 1.115. Below this threshold, scalarization occurs only for positive values of the coupling constant alpha. Above it, a qualitatively new kind of scalarization emerges, requiring negative alpha. The authors label these two regimes alpha-plus and alpha-minus scalarization, and the transition between them is dictated entirely by the black hole’s charge, a feature that had not been seen in previous studies of this class of theories.

The physical origin of this divide lies in the structure of the effective mass. In the near-horizon region, the effective mass squared is proportional to minus alpha times a combination of the charge and the nonlinear correction, a quantity that changes sign depending on how large the charge is relative to the horizon radius. For charges below the critical value, this combination is positive throughout the exterior region, so a positive coupling alpha drives it negative and destabilizes the bald black hole. For charges above the critical value, the combination itself can vanish and turn negative, so the roles reverse and only a negative coupling produces the tachyonic instability. The authors verified this picture with a sufficient instability criterion based on an integral of the perturbation potential, showing that the instability threshold for the EEH black hole closely matches that of the Reissner–Nordström black hole for charges below unity, but departs dramatically at larger charges, even blowing up and flipping sign at a charge of about 1.349.

To quantify the onset precisely, the team computed the exact threshold coupling at which the instability begins, by solving the linearized scalar equation for exponentially growing modes with the correct boundary conditions: a power-law behavior at the horizon and a decaying solution at infinity. For a charge of 0.5, the threshold is a positive alpha of about 19.63. For a charge of 2, it is a negative alpha of about minus 0.976, and for a charge of 20, about minus 0.293. In each case the black hole is unstable, and therefore prone to growing hair, whenever the coupling exceeds the threshold. Below the critical charge, the authors also identified an infinite tower of static scalar clouds, labeled by a node number n, each corresponding to a distinct bifurcation point. For the charge of 0.5, the first three clouds appear at couplings of roughly 19.63, 101.9, and 250.0, seeding infinite branches of scalarized black holes.

With the onset analysis complete, the researchers constructed fully nonlinear scalarized black hole solutions by numerically integrating the coupled field equations, using a shooting method that matches a near-horizon expansion to an asymptotically flat far region. Each scalarized solution carries a secondary scalar charge, read off from the falloff of the scalar field at infinity, in addition to the ADM mass and the magnetic charge. The branches of solutions emerge continuously from the bald black holes at the threshold couplings, with the scalar value at the horizon growing smoothly away from zero. For the sub-critical charge of 0.5, the first three branches were tracked, while for the super-critical charges of 2 and 20, only a single scalarized branch was found for each within the explored parameter ranges, a consequence of the absence of higher scalar clouds in the negative-coupling regime.

Stability is what separates a viable black hole solution from a mathematical curiosity, so the authors subjected their scalarized solutions to radial perturbations. After a careful decoupling procedure that eliminates the metric perturbations, the scalar perturbation obeys a Schrödinger-like equation with an effective potential determined by the background geometry and the scalar hair itself. Solving the resulting eigenvalue problem with quasinormal mode boundary conditions, the team computed purely imaginary frequencies, where a negative imaginary part signals exponential damping and a positive one signals catastrophic growth. The verdict was encouraging across the board: the fundamental branch for the charge of 0.5, and the single branches found for the charges of 2 and 20, all exhibit damped modes and are stable against the s-wave, l equals zero scalar perturbations considered. The excited branches with one or more nodes are expected to be unstable, mirroring earlier findings in related models.

The broader significance of this result is twofold. First, it demonstrates that quantum electrodynamics corrections, encoded in the Euler–Heisenberg nonlinearity, can qualitatively reshape the landscape of hairy black hole solutions, splitting it into two distinct regimes separated by a critical charge. This contrasts sharply with the Einstein–Maxwell-scalar case with the same exponential coupling, where over-charged black holes with charge exceeding mass appear in spontaneous scalarization; in the EEH theory, the single-horizon structure permits unlimited charge and produces the positive-to-negative coupling flip instead. Second, the work adds to a growing catalog of mechanisms by which black holes can evade the no-hair theorem, each with potentially observable consequences in gravitational wave signatures and shadow measurements. As precision tests of strong-field gravity advance, theories that blend classical relativity with genuine quantum-field-theoretic corrections, and black holes that grow hair only when their charge crosses a precise threshold, may prove to be more than mathematical elegance: they could become part of the toolkit for probing the quantum structure of the vacuum itself.

Subject of Research: Spontaneous scalarization of charged black holes in Einstein–Euler–Heisenberg nonlinear electrodynamics theory

Article Title: Charge-dependent scalarization of Einstein–Euler–Heisenberg black holes

Article References: Charge-dependent scalarization of Einstein–Euler–Heisenberg black holes. (n.d.). https://doi.org/10.1140/epjc/s10052-026-16436-y

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-16436-y

Keywords: black holes, spontaneous scalarization, Einstein–Euler–Heisenberg theory, nonlinear electrodynamics, no-hair theorem, tachyonic instability, scalar field, critical charge, quasinormal modes, general relativity, quantum electrodynamics corrections, theoretical physics