Primary Scalar Hair in Black Hole Models

• Primary scalar hair is an independent scalar charge appearing in the asymptotic structure of black-hole solutions, distinct from mass or angular momentum effects.
• It evades classical no-hair theorems by exploiting modified gravity frameworks such as shift-symmetric beyond Horndeski and conformal scalar theories.
• Observational probes like black hole shadows, quasinormal modes, and thermodynamic anomalies provide practical tests for detecting primary scalar hair.

Searching arXiv for recent and foundational papers on primary scalar hair. Primary scalar hair is a class of black-hole or compact-object scalar-field configurations in which the asymptotic scalar charge is an independent parameter of the solution, not fixed by the mass, angular momentum, electromagnetic charges, or imposed boundary data. In the literature surveyed here, the distinction between primary and secondary hair is operationally sharp: secondary hair denotes a nontrivial scalar profile whose amplitude is determined by other charges or couplings, whereas primary hair denotes an additional integration constant that survives in the asymptotics and labels inequivalent solutions. Contemporary work places primary scalar hair in several settings—Einstein–Weyl–Maxwell–conformal scalar theory (Zou et al., 2020), shift-symmetric beyond Horndeski and DHOST theories (Bakopoulos et al., 2023, Baake et al., 2023, Charmousis et al., 28 Mar 2025), asymptotically AdS and supergravity constructions (Anabalon et al., 2012, Mohanty et al., 7 Apr 2026), and phenomenological applications ranging from strong lensing and shadows to quasinormal spectra and EMRIs (Kuntz et al., 2020, Erices et al., 2024, Fathi, 26 Feb 2025, Konoplya et al., 15 Jun 2026).

1. Concept and taxonomy

Primary scalar hair is defined by the presence of an independent scalar charge in the asymptotic solution. In the four-dimensional Einstein–Weyl–Maxwell–conformal scalar construction, this distinction is explicit: the scalar field behaves as

$\phi(r)=\frac{\sqrt{3}\,Q_s}{r}+O(1/r^2),$

and the coefficient $Q_s$ is not fixed by the ADM mass $m$ or the electric charge $Q$, but is dialed by an independent near-horizon datum and identified at infinity (Zou et al., 2020). By contrast, the charged BBMB solution recovered in the $m_2^2\to\infty$ limit has scalar amplitude $\sqrt{3(m^2-Q^2)}$, so the scalar is fully determined by $m$ and $Q$, and is therefore secondary rather than primary (Zou et al., 2020).

The same taxonomy is adopted in effective descriptions of compact-object spacetimes. In the EMRI EFT framework, primary scalar hair is defined as an extra integration constant in the spherically symmetric background solution that controls the $1/r$ falloff of the scalar field and induces post-Newtonian deviations in the metric not constrained by GR’s mass-only dependence (Kuntz et al., 2020). Secondary hair instead denotes scalar profiles whose effective charge is fixed by the mass or by asymptotic conditions, as in the linear scalar Gauss–Bonnet example where $P=2\bar{\alpha}/M$ (Kuntz et al., 2020).

The distinction is not merely semantic. It determines whether the scalar sector adds an independent thermodynamic variable, whether the first law should contain an additional work term, and whether deviations from Kerr or Schwarzschild can be parameterized by an extra observable beyond the usual conserved charges. It also clarifies several borderline cases. Time-dependent or environment-induced scalar configurations around Schwarzschild black holes produced by oscillating scalar backgrounds are not primary hair in the standard sense, because their amplitude is fixed by external boundary conditions rather than by an intrinsic conserved charge (Clough et al., 2019). Related “Jacobson-type” $Q_s$0 tails induced by ambient scalar dark matter are best regarded as boundary-induced or environmental primary hair, independent of $Q_s$1 and $Q_s$2 but not an intrinsic integration constant of an isolated asymptotically flat stationary black hole (Hui et al., 2019).

2. Mechanisms that evade no-hair expectations

The constructions realizing primary scalar hair exploit loopholes in classical no-hair arguments. In shift-symmetric beyond Horndeski and DHOST theories, the scalar is often taken to be linearly time dependent,

$Q_s$3

while the metric remains static. Because only derivatives of $Q_s$4 enter the action, the stress tensor is stationary even though the scalar itself is not. This mechanism underlies the exact asymptotically flat black holes with primary scalar hair found in the shift-symmetric subclass of beyond Horndeski theories (Bakopoulos et al., 2023), the exact “endorsement” framework for adding primary hair to Schwarzschild, Reissner–Nordström, Lovelock, and quasitopological black holes (Baake et al., 2023), and the DHOST families analyzed through disformal maps and axial perturbations (Charmousis et al., 28 Mar 2025).

A structurally different mechanism appears in Einstein–Weyl–Maxwell–conformal scalar theory, where a Weyl-squared term modifies the near-horizon data and promotes the BBMB scalar from secondary to primary hair. The action

$Q_s$5

with positive $Q_s$6 admits asymptotically flat charged solutions whose scalar charge $Q_s$7 is independent of $Q_s$8 and $Q_s$9; in the $m$0 limit one recovers the charged BBMB branch with only secondary hair (Zou et al., 2020).

AdS and supergravity examples realize primary hair without relying on asymptotic flatness. In the $m$1 truncation of gauged $m$2 supergravity, exact static AdS$m$3 black holes exist in three branches—hairless, secondary hair, and primary hair—distinguished by symmetry-breaking patterns in the scalar sector. In the primary-hair branch, the scalar amplitude is controlled by a continuous parameter that does not affect the conserved charges; in the uncharged case the total mass vanishes for any value of the hair parameter (Anabalon et al., 2012). In asymptotically AdS planar geometries, analytic Ricci-flat black hole and soliton families also exist in which the scalar hair parameter $m$4 is independent of the mass integration constant $m$5, providing another explicit realization of primary hair (Mohanty et al., 7 Apr 2026).

A further class uses nontrivial horizon topology. In Einstein–Gauss–Bonnet theory at the Chern–Simons point, asymptotically locally AdS$m$6 black holes with Nil, Solv, and $m$7 Thurston horizons are supported by a scalar profile $m$8, where $m$9 is an independent scalar integration constant. Here $Q$0 is the genuine primary hair, while the gravitational parameter $Q$1 changes the local geometry of the conformal boundary and therefore is not interpreted as hair (Guajardo et al., 2024).

3. Representative solution families

Several exact or numerical families have become reference points for the subject.

In Einstein–Weyl–Maxwell–conformal scalar theory, static spherical solutions are constructed with

$Q$2

The on-shell condition $Q$3 reduces the scalar equation to $Q$4, and asymptotically flat solutions with $Q$5 carry independent $Q$6. The near-horizon expansion has a double zero of $Q$7, so the solutions are extremal, $Q$8, and the scalar diverges at the horizon, as in BBMB, although the asymptotic scalar charge is now independent (Zou et al., 2020).

In shift-symmetric beyond Horndeski, one exact asymptotically flat family is generated by

$Q$9

with scalar

$m_2^2\to\infty$0

This family depends on $m_2^2\to\infty$1 and $m_2^2\to\infty$2 independently and reduces to Schwarzschild when $m_2^2\to\infty$3. For a special relation between $m_2^2\to\infty$4 and $m_2^2\to\infty$5, the central singularity disappears and one obtains regular black holes or solitons (Bakopoulos et al., 2023).

The exact “endorsement” framework generalizes this logic. In quadratic beyond Horndeski theories with

$m_2^2\to\infty$6

and a specific relation

$m_2^2\to\infty$7

the scalar equation reduces to a first-order algebraic constraint for $m_2^2\to\infty$8, while the metric takes a Schwarzschild–(A)dS form plus a $m_2^2\to\infty$9-dependent inhomogeneous piece. This permits adding primary hair to Einstein, Einstein–Maxwell, Lovelock, and quasitopological black holes in arbitrary dimension (Baake et al., 2023).

In DHOST models studied through axial perturbations, the hairy black holes form a one-parameter deformation of Schwarzschild,

$\sqrt{3(m^2-Q^2)}$0

where $\sqrt{3(m^2-Q^2)}$1 is the hair parameter tied to the time slope $\sqrt{3(m^2-Q^2)}$2. For $\sqrt{3(m^2-Q^2)}$3, the scalar shift symmetry gives a finite Noether charge independent of $\sqrt{3(m^2-Q^2)}$4, so the hair is genuinely primary (Charmousis et al., 28 Mar 2025).

Rotating counterparts have also been constructed. In beyond Horndeski gravity with shift symmetry and scalar ansatz $\sqrt{3(m^2-Q^2)}$5, a rotating Kerr-like metric is obtained by a revised Newman–Janis procedure, with a scalar-hair parameter $\sqrt{3(m^2-Q^2)}$6 independent of $\sqrt{3(m^2-Q^2)}$7 and $\sqrt{3(m^2-Q^2)}$8. The resulting spacetime reduces to Kerr for $\sqrt{3(m^2-Q^2)}$9, while negative $m$0 enlarges the shadow and positive $m$1 shrinks it (Nozari et al., 18 Feb 2026).

4. Asymptotics, regularity, and horizon behavior

A defining feature of primary hair is that the scalar charge appears in the asymptotics as a new independent coefficient. In asymptotically flat constructions this is usually a $m$2 term. The EWMCS solutions exhibit

$m$3

so $m$4 affects both the scalar and metric tails (Zou et al., 2020). In EMRI EFT language, such an asymptotic parameter modifies the PN coefficients $m$5 and hence alters the background geodesics and waveform phasing (Kuntz et al., 2020).

Regularity properties vary across models. The EWMCS black holes are extremal with a double-zero horizon and divergent scalar at the horizon, which makes the Wald entropy ill-defined because the conformally coupled scalar contribution diverges there (Zou et al., 2020). By contrast, the beyond Horndeski exact solutions constructed via $m$6 are designed so that the scalar field is regular on future or past horizons depending on the sign choice in $m$7; the time dependence effectively selects a horizon orientation (Bakopoulos et al., 2023). The exact endorsement framework likewise emphasizes that imposing $m$8 and $m$9 yields everywhere regular scalar profiles and finite invariants outside the horizon (Baake et al., 2023).

Regular black holes and solitons form an important subtopic. In beyond Horndeski, special relations between mass and scalar-hair parameters can remove the central singularity and produce globally regular solitonic geometries (Bakopoulos et al., 2023). In the thermodynamic analysis of primary and secondary scalar hair, a singular primary-hair black hole reduces, under a specific fine-tuning of $Q$0 in terms of $Q$1, to a regular Bardeen-type solution with secondary rather than primary hair (Myung et al., 5 May 2025). In planar AdS constructions, both the scalar and curvature invariants remain regular everywhere in the black hole and soliton sectors (Mohanty et al., 7 Apr 2026).

Negative results are also informative. In the EWMCS system no asymptotically flat scalar-hairy black holes were found for $Q$2; numerically the metric function diverges at large radius over a wide parameter survey, which strongly suggests that the primary-hair branch exists only for positive Weyl coupling (Zou et al., 2020).

5. Thermodynamics and conserved quantities

Primary scalar hair forces a re-examination of black-hole thermodynamics because the scalar parameter is an additional state variable. In general one expects a first law of the form

$Q$3

but the practical realization depends on the theory and on whether the scalar is regular at the horizon.

The EWMCS case is thermodynamically subtle. The Hawking temperature vanishes because the horizon is extremal, $Q$4, while the divergent scalar field makes the Wald entropy ill-defined. The work identifies these issues but does not complete a thermodynamic first-law analysis (Zou et al., 2020).

A more systematic thermodynamic treatment has been developed in shift-symmetric beyond Horndeski gravity. Using the Euclidean method, the free energy of primary-hairy black holes is obtained as

$Q$5

which leads to the first law

$Q$6

This provides a direct way to extract the scalar chemical potential $Q$7 in theories where the scalar contains a linear time dependence and conventional Hamiltonian methods are cumbersome (Bakopoulos et al., 2024).

The Smarr problem is more intricate. For primary-hair and Bardeen-type secondary-hair black holes in beyond Horndeski gravity, the traditional thermodynamic approach does not produce a consistent Smarr relation. A resolution is obtained by promoting the dimensionful couplings to thermodynamic variables, following a method introduced in 2024. In this extended framework one finds, for both primary- and secondary-hair solutions,

$Q$8

with the scalar-hair contribution incorporated consistently into the thermodynamic scaling (Myung et al., 5 May 2025). This suggests that in higher-derivative scalar-tensor theories the appropriate thermodynamic state space is broader than the naive $Q$9 one.

AdS examples exhibit further structure. In gauged $1/r$0 supergravity, the uncharged theory admits a triple point in thermodynamic phase space where the hairless, secondary-hair, and primary-hair branches coexist. The free energies can be written explicitly as functions of temperature, and the transition between hairless and secondary-hair branches is third order, while the primary-hair branch is thermodynamically competitive only at the coexistence point (Anabalon et al., 2012). In planar AdS black holes and solitons with primary hair, the hairy soliton is the ground state, and there is a first-order phase transition between hairy black hole and hairy soliton controlled by the ratio of Euclidean periods. Increasing the hair parameter enlarges the temperature window in which the soliton phase dominates (Mohanty et al., 7 Apr 2026).

An unusual case is provided by the AdS$1/r$1 Einstein–Gauss–Bonnet solutions with Thurston horizons, for which Regge–Teitelboim analysis yields vanishing mass and entropy. Despite this, the scalar amplitude $1/r$2 remains a genuine primary hair because it labels inequivalent solutions without altering the asymptotic boundary geometry in the way $1/r$3 does (Guajardo et al., 2024).

6. Phenomenology and observational probes

Primary scalar hair modifies observables in a model-dependent but often sign-definite way. Strong-field imaging provides one major window. In rotating beyond Horndeski black holes, the scalar-hair parameter $1/r$4 alters the shadow: negative $1/r$5 enlarges the shadow and reduces its oblateness, while positive $1/r$6 shrinks it and enhances distortion. Imposing the Event Horizon Telescope bounds $1/r$7 and $1/r$8 for M87* leaves viable $1/r$9 regions, though positive $P=2\bar{\alpha}/M$0 is more tightly restricted (Nozari et al., 18 Feb 2026).

For static beyond Horndeski primary-hair black holes, shadow and thermodynamic analyses show a similar sign dependence governed by $P=2\bar{\alpha}/M$1. When $P=2\bar{\alpha}/M$2, increasing the scalar hair enlarges the shadow; when $P=2\bar{\alpha}/M$3, it reduces the shadow. Joint constraints from thermodynamic stability and the M87* and Sgr A* shadows isolate only restricted regions in parameter space (Erices et al., 2024). Strong-lensing observables sharpen these constraints further. Using Bozza’s strong-deflection formalism and EHT data, recent work found that the observed ring may be more naturally associated with the secondary image of a thin accretion disk than with the mathematical shadow edge, yielding a conservative M87* interval approximately

$P=2\bar{\alpha}/M$4

However, the same parameter region appears to suffer from local instabilities, indicating tension between lensing compatibility and dynamical viability in that specific model (Fathi, 26 Feb 2025).

Ringdown and Hawking radiation provide a complementary probe. Axial perturbations of DHOST black holes with primary scalar hair can be mapped to a Regge–Wheeler problem in an effective metric whose causal structure differs from the background metric. This implies distinct luminous and gravitational horizons, and quasinormal frequencies depend on the hair parameter through the effective potential (Charmousis et al., 28 Mar 2025). A more recent analysis of asymptotically flat beyond-Horndeski black holes with primary scalar hair found that fundamental QNMs shift moderately, while higher overtones are highly sensitive to near-horizon deformations, showing overtone rearrangements and, in some regimes, echo-like signals due to double-barrier effective potentials. The same structures generate resonant tunneling in graybody factors and oscillatory Hawking emission spectra (Konoplya et al., 15 Jun 2026).

In EMRIs, primary scalar hair enters a model-independent EFT through the background PN coefficients and operator functions in the perturbation action. In the odd sector, the dissipated power is

$P=2\bar{\alpha}/M$5

where the coefficients are explicit functions of the EFT parameters induced by hair (Kuntz et al., 2020). The even sector, not completed there, is expected to carry the leading scalar-radiation channel, in particular a dipole contribution at $P=2\bar{\alpha}/M$6PN for primary-hair scenarios. This suggests that EMRI phasing in LISA could be a particularly sensitive probe of primary scalar charges (Kuntz et al., 2020).

Holographic transport provides an instructive counterexample to naive expectations. In beyond-Horndeski planar black holes with primary hair and axionic momentum dissipation, the DC conductivity does not depend explicitly on the hair parameter $P=2\bar{\alpha}/M$7, even though both the scalar and the metric depend on it. The conductivity

$P=2\bar{\alpha}/M$8

depends on the horizon radius $P=2\bar{\alpha}/M$9, which is indirectly shifted by hair, but contains no explicit scalar-hair term. This nondependence appears robust across a broader shift-symmetric class (Hernandez-Vera, 2024).

7. Extensions beyond black holes and current open problems

Primary scalar hair is not limited to black holes. Static, spherically symmetric neutron stars with primary scalar hair have recently been constructed in a subfamily of DHOST theories by solving modified TOV equations. Positive scalar charges make stars more compact than in GR and, beyond a critical threshold, generate singular behavior. Since the mass–radius relation is altered for both polytropic and realistic equations of state, neutron-star observations may place stringent constraints on the same beyond-GR parameters that govern black-hole scalar hair (Boumaza et al., 2 Jul 2026).

Open problems cluster around stability, rotation, and thermodynamics. Many explicit backgrounds exist without a full perturbative analysis. The EWMCS primary-hair branch lacks a dedicated linear-stability study (Zou et al., 2020). The exact beyond-Horndeski endorsement framework and the planar AdS constructions likewise leave perturbative stability for future work (Baake et al., 2023, Mohanty et al., 7 Apr 2026). Even when an axial sector is under control, as in DHOST black holes, the polar sector and scalar-led modes remain essential for a complete assessment (Charmousis et al., 28 Mar 2025).

Rotation is still comparatively underdeveloped. Although rotating beyond Horndeski black holes with primary hair and shadow phenomenology are now available (Nozari et al., 18 Feb 2026), most exact constructions remain static. Extending the endorsement mechanism or the DHOST families to fully rotating, asymptotically flat solutions with controlled perturbation theory remains a central goal.

Thermodynamics is also unresolved in several sectors. Divergent horizon scalars obstruct the usual Wald analysis in BBMB-like cases (Zou et al., 2020). The need to promote couplings to thermodynamic variables in order to recover consistent Smarr relations suggests that scalar-hairy higher-derivative gravities may require a more general notion of thermodynamic extensivity than is standard in Einstein gravity (Myung et al., 5 May 2025).

Finally, the observational status of primary scalar hair remains mixed. Some beyond-Horndeski parameter regions are compatible with current EHT shadow data (Erices et al., 2024, Nozari et al., 18 Feb 2026), but strong-lensing analyses indicate that astrophysically allowed regions may coincide with local instabilities in simple models (Fathi, 26 Feb 2025). This suggests that future progress will likely come from combining multiple channels—shadows, strong lensing, EMRI phasing, ringdown spectroscopy, and neutron-star structure—to distinguish viable primary-hair theories from those that are only background-level solutions.

In current usage, primary scalar hair therefore denotes more than a scalar profile: it denotes an independent scalar degree of freedom visible in the asymptotic data and, in the viable cases, in the observable phenomenology of compact objects. The recent literature has moved the concept from isolated exact examples to a broader framework spanning higher-derivative gravity, AdS thermodynamics, compact-star structure, and precision strong-field observables (Zou et al., 2020, Bakopoulos et al., 2023, Baake et al., 2023, Charmousis et al., 28 Mar 2025).