Hayward Boson Stars Overview

• Hayward boson stars are horizonless, regular configurations of a complex scalar field in Einstein gravity modified by a nonlinear electrodynamics sector that reproduces the Hayward metric in the scalar-free limit.
• They are constructed in models ranging from asymptotically flat to AdS settings, employing Einstein–Klein–Gordon–NLED frameworks that reveal phenomena like frozen endpoints and solitonic self-interactions.
• While exact Hayward profiles emerge in these NLED-supported systems, semiclassical Hamiltonian analyses suggest that a globally exact Hayward metric is model-dependent and may not exist under positivity constraints.

Hayward boson stars are static, spherically symmetric, horizonless, regular self-gravitating configurations built from a complex scalar field in Einstein gravity, with a nonlinear electrodynamics sector chosen so that the electrovacuum geometry is the regular Hayward spacetime. The recent literature treats this label in several closely related settings: asymptotically flat Einstein–Klein–Gordon–NLED solutions, frozen $\omega \to 0$ limits, and asymptotically AdS solitonic generalizations. A distinct Hamiltonian semiclassical-gravity analysis reaches a different conclusion: in that framework, exact Hayward-type boson stars do not arise, even though regular boson stars with de Sitter-like cores do exist (Chicaiza-Medina et al., 16 Aug 2025, Yue et al., 2023, Liu et al., 11 Dec 2025, Javed et al., 4 Oct 2025).

1. Terminology and model classes

In the NLED-based literature, a Hayward boson star is a self-gravitating, horizonless, regular configuration of a complex scalar field whose gravitational field is modified by a magnetic-monopole nonlinear electrodynamics sector engineered to reproduce the Hayward regular geometry in the scalar-free limit. The scalar is uncharged and interacts with the electromagnetic sector only via gravity. Two extensions are prominent: frozen Hayward-boson stars, defined by the $\omega \to 0$ endpoint of solution families, and solitonic Hayward–boson stars in asymptotically AdS spacetime, where the scalar has a soliton self-interaction (Chicaiza-Medina et al., 16 Aug 2025, Yue et al., 2023, Liu et al., 11 Dec 2025).

A separate usage appears in Hamiltonian semiclassical gravity. There, one quantizes a real scalar field, replaces matter phase-space functions by expectation values in a fixed state, and solves an effective constrained system. That framework admits regular boson stars and regular black holes, but it excludes Hayward and Bardeen metrics as exact solutions under the positivity assumptions imposed on the matter expectation values (Javed et al., 4 Oct 2025).

Setting	Matter sector	Characteristic statement
Asymptotically flat HyBS	Complex massive scalar plus Hayward-supporting NLED	Boson stars exist only if $\sqrt{\beta}\,Q > 1.49661$
Frozen HyBS	Free complex massive scalar plus Hayward NED	For sufficiently large $q$, families extend to $\omega \to 0$
AdS SHBS / FSHBS	Complex scalar with soliton potential plus Hayward NED and $\Lambda<0$	Frozen solutions occur only for $q \ge q_c$
Hamiltonian semiclassical boson stars	Real quantum scalar in effective constraints	Exact Hayward-type boson stars do not exist

2. Einstein–Klein–Gordon–NLED construction

The asymptotically flat model is defined by

$$\mathcal{S} =\int d^4 x\,\sqrt{-g}\left[ \frac{R}{16\pi} -\mathcal{L}(F) -\nabla_\alpha \Psi^*\nabla^\alpha \Psi -U(|\Psi|) \right],$$

with $U(|\Psi|)=\mu^2|\Psi|^2$ and

$$\mathcal{L}(F)=\frac{3}{\beta\pi}\,(\beta F)^{3/2}\,\left[1+(\beta F)^{3/4}\right]^2 .$$

The metric and matter ansätze are

$\omega \to 0$0

$\omega \to 0$1

Since the scalar is uncharged, the electrodynamics equations decouple from the Klein–Gordon equation (Chicaiza-Medina et al., 16 Aug 2025).

The resulting radial ODE system is

$\omega \to 0$2

$\omega \to 0$3

$\omega \to 0$4

It is also convenient to write

$\omega \to 0$5

with

$\omega \to 0$6

In the electrovacuum limit $\omega \to 0$7, regularity at the origin and asymptotic flatness imply $\omega \to 0$8, and the mass function integrates to

$\omega \to 0$9

which is the standard Hayward mass function, with

$\sqrt{\beta}\,Q > 1.49661$0

This embeds the Hayward regular core directly into the matter sector rather than imposing it kinematically (Chicaiza-Medina et al., 16 Aug 2025).

The AdS generalization keeps the same structural idea but replaces the free scalar by a solitonic potential,

$\sqrt{\beta}\,Q > 1.49661$1

and uses

$\sqrt{\beta}\,Q > 1.49661$2

In that setting the Hayward sector is again generated by a magnetic monopole and the scalar-free limit reduces to the pure Hayward solution (Liu et al., 11 Dec 2025).

3. Boundary conditions, existence criterion, and asymptotically flat families

Regularity at the center requires

$\sqrt{\beta}\,Q > 1.49661$3

while asymptotic flatness imposes

$\sqrt{\beta}\,Q > 1.49661$4

The ADM mass is

$\sqrt{\beta}\,Q > 1.49661$5

and the global $\sqrt{\beta}\,Q > 1.49661$6 symmetry yields a Noether charge interpreted as particle number (Chicaiza-Medina et al., 16 Aug 2025).

The central existence result is

$\sqrt{\beta}\,Q > 1.49661$7

This is the horizonless condition for the electrovacuum Hayward background. When it fails, the electrovacuum spacetime contains at least one horizon, and a simple integral identity derived from

$\sqrt{\beta}\,Q > 1.49661$8

excludes nontrivial stationary scalar configurations outside a nonextremal horizon. In this setup, Hayward boson stars therefore exist only when the underlying Hayward electrovacuum is itself horizonless (Chicaiza-Medina et al., 16 Aug 2025).

Ground-state families were constructed for fixed $\sqrt{\beta}\,Q > 1.49661$9 and representative charges $q$0. The qualitative behavior parallels mini-boson stars at small $q$1 but is systematically reshaped by the Hayward sector: $q$2 rises to a maximum and then decreases, $q$3 spirals at small $q$4 and progressively unwinds as $q$5 increases, and the $q$6 curves turn back toward the electrovacuum Hayward values rather than extending to arbitrarily large radii. The paper emphasizes that the ADM mass of a Hayward boson star is always larger than the electrovacuum Hayward mass $q$7, since $q$8 (Chicaiza-Medina et al., 16 Aug 2025).

$q$9	$\omega \to 0$0	$\omega \to 0$1
$\omega \to 0$2	$\omega \to 0$3	$\omega \to 0$4
$\omega \to 0$5	$\omega \to 0$6	$\omega \to 0$7
$\omega \to 0$8	$\omega \to 0$9	$\Lambda<0$0
$\Lambda<0$1	$\Lambda<0$2	$\Lambda<0$3
$\Lambda<0$4	$\Lambda<0$5	$\Lambda<0$6

These values are quoted in the dimensionless units defined by the rescalings

$\Lambda<0$7

They exhibit the main trend emphasized in the paper: increasing the magnetic charge lowers the maximum mass and drives the family toward the underlying Hayward scale (Chicaiza-Medina et al., 16 Aug 2025).

4. Frozen Hayward-boson stars and AdS solitonic generalizations

In the Einstein–Hayward–scalar model with a free, complex, massive scalar, frozen Hayward-boson stars are defined by the $\Lambda<0$8 limit of boson-star families. In that limit the scalar field piles up at a radius $\Lambda<0$9, the metric function $q \ge q_c$0 attains a very small minimum at $q \ge q_c$1 without crossing zero, and $q \ge q_c$2 becomes nearly zero for $q \ge q_c$3. The solutions remain horizonless, but an observer at infinity sees an arbitrarily large redshift from the interior. The scalar thus disrupts the formation of an event horizon and replaces it by a critical surface (Yue et al., 2023).

The same work studies ground and excited states, classified by the number $q \ge q_c$4 of radial nodes of the scalar profile. The critical magnetic charge for the onset of frozen behavior increases with excitation number,

$q \ge q_c$5

For $q \ge q_c$6 and $q \ge q_c$7, the ground, first, and second excited states all have

$q \ge q_c$8

and they share the same critical radius $q \ge q_c$9. The paper reports this equality of $$\mathcal{S} =\int d^4 x\,\sqrt{-g}\left[ \frac{R}{16\pi} -\mathcal{L}(F) -\nabla_\alpha \Psi^*\nabla^\alpha \Psi -U(|\Psi|) \right],$$0 across the near-frozen families under the same parameters (Yue et al., 2023).

The AdS extension introduces a solitonic self-interaction and a cosmological constant. Its defining potential is

$$\mathcal{S} =\int d^4 x\,\sqrt{-g}\left[ \frac{R}{16\pi} -\mathcal{L}(F) -\nabla_\alpha \Psi^*\nabla^\alpha \Psi -U(|\Psi|) \right],$$1

and the corresponding frozen objects are termed frozen solitonic Hayward-boson stars. They exist only above a critical magnetic charge $$\mathcal{S} =\int d^4 x\,\sqrt{-g}\left[ \frac{R}{16\pi} -\mathcal{L}(F) -\nabla_\alpha \Psi^*\nabla^\alpha \Psi -U(|\Psi|) \right],$$2, and the threshold depends on both $$\mathcal{S} =\int d^4 x\,\sqrt{-g}\left[ \frac{R}{16\pi} -\mathcal{L}(F) -\nabla_\alpha \Psi^*\nabla^\alpha \Psi -U(|\Psi|) \right],$$3 and $$\mathcal{S} =\int d^4 x\,\sqrt{-g}\left[ \frac{R}{16\pi} -\mathcal{L}(F) -\nabla_\alpha \Psi^*\nabla^\alpha \Psi -U(|\Psi|) \right],$$4. For $$\mathcal{S} =\int d^4 x\,\sqrt{-g}\left[ \frac{R}{16\pi} -\mathcal{L}(F) -\nabla_\alpha \Psi^*\nabla^\alpha \Psi -U(|\Psi|) \right],$$5, the quoted values are

$$\mathcal{S} =\int d^4 x\,\sqrt{-g}\left[ \frac{R}{16\pi} -\mathcal{L}(F) -\nabla_\alpha \Psi^*\nabla^\alpha \Psi -U(|\Psi|) \right],$$6

For fixed $$\mathcal{S} =\int d^4 x\,\sqrt{-g}\left[ \frac{R}{16\pi} -\mathcal{L}(F) -\nabla_\alpha \Psi^*\nabla^\alpha \Psi -U(|\Psi|) \right],$$7, $$\mathcal{S} =\int d^4 x\,\sqrt{-g}\left[ \frac{R}{16\pi} -\mathcal{L}(F) -\nabla_\alpha \Psi^*\nabla^\alpha \Psi -U(|\Psi|) \right],$$8 is non-monotonic in $$\mathcal{S} =\int d^4 x\,\sqrt{-g}\left[ \frac{R}{16\pi} -\mathcal{L}(F) -\nabla_\alpha \Psi^*\nabla^\alpha \Psi -U(|\Psi|) \right],$$9, with a maximum near $U(|\Psi|)=\mu^2|\Psi|^2$0 (Liu et al., 11 Dec 2025).

Model	Critical threshold	Reported endpoint behavior
Frozen HyBS in asymptotically flat space	$U(|\Psi|)=\mu^2|\Psi|^2$1 for $U(|\Psi|)=\mu^2|\Psi|^2$2	Branches extend to $U(|\Psi|)=\mu^2|\Psi|^2$3
FSHBS in AdS with $U(|\Psi|)=\mu^2|\Psi|^2$4	$U(|\Psi|)=\mu^2|\Psi|^2$5 for $U(|\Psi|)=\mu^2|\Psi|^2$6	Frozen endpoint only for $U(|\Psi|)=\mu^2|\Psi|^2$7

The frozen AdS limit is quantitatively sharp. At $U(|\Psi|)=\mu^2|\Psi|^2$8 one finds $U(|\Psi|)=\mu^2|\Psi|^2$9 for $$\mathcal{L}(F)=\frac{3}{\beta\pi}\,(\beta F)^{3/2}\,\left[1+(\beta F)^{3/4}\right]^2 .$$0 and $$\mathcal{L}(F)=\frac{3}{\beta\pi}\,(\beta F)^{3/2}\,\left[1+(\beta F)^{3/4}\right]^2 .$$1, while no true horizon forms because both $$\mathcal{L}(F)=\frac{3}{\beta\pi}\,(\beta F)^{3/2}\,\left[1+(\beta F)^{3/4}\right]^2 .$$2 and $$\mathcal{L}(F)=\frac{3}{\beta\pi}\,(\beta F)^{3/2}\,\left[1+(\beta F)^{3/4}\right]^2 .$$3 remain strictly positive. Continuously decreasing $$\mathcal{L}(F)=\frac{3}{\beta\pi}\,(\beta F)^{3/2}\,\left[1+(\beta F)^{3/4}\right]^2 .$$4 can disrupt the frozen state: at fixed $$\mathcal{L}(F)=\frac{3}{\beta\pi}\,(\beta F)^{3/2}\,\left[1+(\beta F)^{3/4}\right]^2 .$$5 and $$\mathcal{L}(F)=\frac{3}{\beta\pi}\,(\beta F)^{3/2}\,\left[1+(\beta F)^{3/4}\right]^2 .$$6, sufficiently negative $$\mathcal{L}(F)=\frac{3}{\beta\pi}\,(\beta F)^{3/2}\,\left[1+(\beta F)^{3/4}\right]^2 .$$7 removes the $$\mathcal{L}(F)=\frac{3}{\beta\pi}\,(\beta F)^{3/2}\,\left[1+(\beta F)^{3/4}\right]^2 .$$8 endpoint and the $$\mathcal{L}(F)=\frac{3}{\beta\pi}\,(\beta F)^{3/2}\,\left[1+(\beta F)^{3/4}\right]^2 .$$9–$\omega \to 0$00 curve regains a spiral. The role of the self-interaction is frequency-dependent: at high frequency, increasing $\omega \to 0$01 can drive the solution toward the pure Hayward geometry, whereas at low frequency increasing $\omega \to 0$02 reduces both $\omega \to 0$03 and $\omega \to 0$04 and strengthens the near-frozen behavior (Liu et al., 11 Dec 2025).

5. Geodesic structure, light rings, and optical appearance

The optical properties of Hayward boson stars are controlled by

$\omega \to 0$05

Light rings satisfy

$\omega \to 0$06

with stability determined by the sign of the second derivative of the effective potential. In the AdS solitonic system, light rings appear in pairs: an inner stable ring and an outer unstable ring. For $\omega \to 0$07, the branch structure can even generate a finite-frequency window with four light rings, i.e. two pairs; that window expands with increasing $\omega \to 0$08 and shrinks as $\omega \to 0$09 decreases or $\omega \to 0$10 increases. For $\omega \to 0$11, the typical frozen branch contains one pair that merges and disappears at a critical frequency (Liu et al., 11 Dec 2025).

A complementary AdS study analyzes these structures through ray tracing of thin accretion disks. In the non-frozen regime, the absence of an event horizon allows null geodesics to execute multiple windings and re-emerge, producing multiple photon rings within the central shadow region. In the frozen regime, rays that would otherwise generate deeper internal images cross the critical horizon and do not return, so the image resembles that of a Schwarzschild black hole, with no additional photon rings appearing within the shadow region (Zhao et al., 13 Jul 2025).

The negative cosmological constant modifies both orbital and imaging diagnostics. In the quoted examples with $\omega \to 0$12 and $\omega \to 0$13, the ISCO radius is $\omega \to 0$14 at $\omega \to 0$15 and $\omega \to 0$16 at $\omega \to 0$17. The same study reports that $\omega \to 0$18 and $\omega \to 0$19 increase as $\omega \to 0$20 decreases, that the inner stable light ring moves inward while the outer unstable light ring moves outward as $\omega \to 0$21 becomes more negative, and that the smaller ISCO can obscure internal photon rings at some viewing angles (Zhao et al., 13 Jul 2025).

These results give Hayward boson stars a dual phenomenology. Non-frozen configurations are horizonless ultracompact objects with internal ring structure in their images, whereas frozen configurations are quasi-black-hole solutions whose exterior observables can become very close to those of Hayward or Schwarzschild black holes. The distinction is therefore not the presence of a regular core alone, but whether the family terminates in a critical-horizon state (Yue et al., 2023, Zhao et al., 13 Jul 2025).

6. Exact Hayward profiles, Hayward-like cores, and the semiclassical objection

The Hamiltonian semiclassical analysis of spherically symmetric gravity coupled to a real quantum scalar field formulates the problem in ADM variables, replaces matter phase-space functions by expectation values in a fixed state, and solves an effective static ODE system in generalized Painlevé–Gullstrand gauge. In that framework, one finds regular boson stars and regular black holes with asymptotically flat, de Sitter, and anti-de Sitter behavior. Near the center, regular solutions satisfy

$\omega \to 0$22

so the core is de Sitter-like, with the same $\omega \to 0$23 scaling as the Hayward solution (Javed et al., 4 Oct 2025).

The same paper nevertheless proves that Hayward and Bardeen metrics are not solutions of its effective equations. In Eddington–Finkelstein form with $\omega \to 0$24, the two independent radial equations reduce to

$\omega \to 0$25

$\omega \to 0$26

Adding them yields

$\omega \to 0$27

which contradicts the imposed positivity conditions

$\omega \to 0$28

The paper therefore concludes that a “Hayward Boson Star,” understood as a boson star exactly described by the Hayward-type metric, does not exist in that semiclassical scalar-field framework (Javed et al., 4 Oct 2025).

This identifies an important distinction in the literature. In Einstein–Klein–Gordon–NLED models, “Hayward boson star” denotes a boson star living in a theory whose scalar-free sector is exactly the Hayward geometry. In the Hamiltonian semiclassical model, regular boson stars can have Hayward-like central scaling, but the full self-consistent metric cannot coincide globally with the closed-form Hayward profile. The shared $\omega \to 0$29 core behavior does not imply global equivalence of the solutions (Javed et al., 4 Oct 2025).

The current state of the subject therefore contains both a positive and a negative result. Hayward boson stars, frozen Hayward-boson stars, and AdS solitonic Hayward–boson stars exist in the NLED-supported Einstein–scalar systems studied in 2023–2025. By contrast, exact Hayward boson stars do not emerge in the Hamiltonian semiclassical theory with a real scalar and positive localized source expectations. The phrase “Hayward boson star” is thus model-specific rather than universally interchangeable across regular-core constructions (Chicaiza-Medina et al., 16 Aug 2025, Yue et al., 2023, Liu et al., 11 Dec 2025, Javed et al., 4 Oct 2025).