Black holes in a new gravitational theory with trace anomalies

In a new gravitational theory with the trace anomaly recently proposed by Gabadadze, we study the existence of hairy black hole solutions on a static and spherically symmetric background. In this theory, the effective 4-dimensional action contains a kinetic term of the conformal scalar field related to a new scale $bar{M}$ much below the Planck mass. This property can overcome a strong coupling problem known to be present in general relativity supplemented by the trace anomaly as well as in 4-dimensional Einstein-Gauss-Bonnet gravity. We find a new hairy black hole solution arising from the Gauss-Bonnet trace anomaly, which satisfies regular boundary conditions of the conformal scalar and metric on the horizon. Unlike unstable exact black hole solutions with a divergent derivative of the scalar on the horizon derived for some related theories in the literature, we show that our hairy black hole solution can be consistent with all the linear stability conditions of odd- and even-parity perturbations.


I. INTRODUCTION
It is well known that the quantum field theory of gravitation gives rise to a trace anomaly which is absent at classical level. Quantum corrections to the graviton propagator arising from loops of massless particles (photons and fermions) were originally computed in Refs. [1,2] by using a dimensional regularization scheme. For the massless field system interacting with gravity, Capper and Duff [3] showed that the conformal invariance under Weyl scaling of the metric tensor g µν no longer holds at quantum level. In 4-dimensional spacetime, the regularized energy-momentum tensor develops nonvanishing trace anomalies which consist of curvature scalar quantities constructed from Riemann and Ricci tensors [4].
The general expression for the gravitational trace anomaly derived from one loop calculations in massless theories is given by T A = −αG + βW 2 + γ R, where G is a Gauss-Bonnet (GB) term, W 2 = C µνρσ C µνρσ is the squared of a Weyl tensor C µνρσ , and R is a Ricci scalar [5,6]. The constants α, β, γ are related to the numbers of real scalar fields, Dirac fermions, and vector fields present in conform field theory (CFT) [7][8][9][10][11][12]. Indeed, these coefficients exactly coincide with those obtained by using the AdS/CFT correspondence in strongly coupled large N CFT [13]. Decomposing the metric tensor into g µν = e 2σḡ µν , where e 2σ is a conformal factor and the metricḡ µν is restricted to have a fixed determinant, Riegert [14] derived an effective local action S A that generates the trace anomaly T A in the field equations of motion. This can be further promoted to consider a conformal geometrȳ g µν = e −2φ g µν , (1.1) without assuming any constraint onḡ µν [15,16]. The Weyl invariant combinationḡ µν = e −2φ g µν transforms as a metric under diffeomorphisms, with φ being a massless scalar field (dilaton). The metricḡ µν is invariant under Weyl transformations g µν → e 2σ g µν and φ → φ + σ. The diffeomorphism-invariant effective action S A of gravitational trace anomalies contains Galileon-type selfinteraction of the dilaton and derivative couplings to the Einstein tensor besides the couplings between φ and G, W 2 [14,15]. The same 4-dimensional action arises from a Wess-Zumino term for a gravitational theory nonlinearly realizing the conformal symmetry [17]. Moreover, the action same as the Riegert's one also appears in regularized 4-dimensional Einstein-Gauss-Bonnet (4DEGB) theory [18] after a regularized Kaluza-Klein reduction of the D-dimensional Einstein-GB gravity with a rescaling of the GB coupling constant α → α/(D − 4) [19][20][21][22]. In 4DEGB gravity, the scalar field φ corresponds to a radion mode characterizing the size of a maximally symmetric internal space. We note that the 4-dimensional action generating the GB trace anomaly belongs to a subclass of Horndeski theories with second-order field equations of motion [23][24][25][26].
General Relativity (GR) supplemented by the Riegert's action is plagued by a strong coupling problem at arbitrary low energy scales [27,28]. This is attributed to the fact that the conformal scalar field φ is not a propagating degree of freedom. In other words, the effective 4-dimensional action does not possess a kinetic term with the proper sign. In 4DEGB gravity, the same strong coupling problem was also recognized by studying linear perturbations around static and spherically symmetric black holes (BHs) [22,29] and neutron stars (NSs) [30]. In this case the kinetic term associated with the radion perturbation δφ vanishes everywhere. Moreover, there are ghost/Laplacian instabilities [22,29] for exact BH solutions derived in 4DEGB gravity [19,31].
To circumvent the above strong coupling problem, one may add a scalar kinetic term to the Riegert's action. However, adding terms depending on the scalar field explicitly violates the correct structure of gravitational trace anomalies. Instead, Gabadadze [28] recently proposed a new gravitational action of the form where g andḡ are the determinants of g µν andḡ µν = e −2φ g µν , respectively, andR is the Ricci scalar evaluated with g µν . The constant M is related to the reduced Planck mass M Pl as M = M Pl / √ 2, whereas the new mass scaleM is much smaller than M . The total effective action incorporating the trace anomaly is given by S = S RR + S A , where S A is the Riegert's action without a fixed determinant constraint onḡ µν . The Ricci scalarR can be expressed in terms of the sum of the term e 2φ R and derivatives of φ. Then, the action (1.2) is equivalent to where ϕ ≡M e −φ and the covariant derivative operator ∇ µ is associated with the metric g µν . Thus, the action S RR contains the scalar field kinetic term −6g µν ∇ µ ϕ∇ ν ϕ with the proper sign 1 . The last two terms in Eq. (1.3) respect the conformal invariance, while the first Einstein-Hilbert term explicitly breaks the conformal symmetry. The theory given by the action S = S RR + S A can be regarded as an effective field theory (EFT) valid below the scaleM . Introducing a canonical normalized field π =M φ, we see that nonlinear scalar derivative terms of π are strongly coupled above the scaleM , while they are weakly coupled belowM ( M ) [28]. Taking the limitM → 0 means that the theory is strongly coupled at any scales, as it happens in GR supplemented by the trace anomaly and in 4DEGB gravity.
In this paper, we will apply the new gravitational theory of Gabadadze to the investigation of static and spherically symmetric BHs with scalar hairs. It is anticipated that the presence of a canonical scalar kinetic term in S RR as well as the existence of a new scaleM should allow the possibility for overcoming the strong coupling and instability problems present for hairy BH solutions in 4DEGB gravity. Indeed, we will show that there is a new class of BH solutions where the scalar hair arises from the GB trace anomaly. If we impose a condition that the two metric componentsf andh associated withḡ µν are identical to each other, there is an exact BH solution analogous to those derived in 4DEGB gravity [19,31] and in gravitational theory with a conformal scalar field [16]. However, we will see that the solution consistent with all the field equations of motion and regular boundary conditions of φ,f ,h on the horizon satisfiesf (r) =h(r) at arbitrary distancesr, where the difference betweenf andh comes from the trace anomaly. Under the expansion of a small coupling constant α, we will derive analytic solutions tof ,h, and φ up to fourth order. We will also confirm that they are in very good agreement with numerically integrated solutions.
For our new BH solution, the radial field derivative φ (r) is a finite constant on the horizon (r =r h ). On the other hand, for the exact BH solution present in 4DEGB gravity [19,31], φ (r) diverges atr =r h . The latter property leads to the linear instability of BHs in the vicinity of the horizon [22,29]. This instability was shown for a timeindependent scalar field in full Horndeski theories [32] (including the shift-symmetric case [33]) by using the general results of BH perturbations formulated in Refs. [34][35][36]. We will consider odd-and even-parity perturbations about our new BH solution derived under the expansion of the small α and show that all the linear stability conditions can be consistently satisfied without the strong coupling problem. In particular, the propagation speeds of gravitational and scalar field perturbations are close to the speed of light with corrections induced by the GB trace anomaly. Thus, the new gravitational theory of Gabadadze gives rise to a linearly stable BH solution with the scalar hair induced by the trace anomaly.

II. GRAVITATIONAL ACTION WITH TRACE ANOMALIES
In this section, we first briefly review the Riegert's action [14] and then proceed to the explanation of the new gravitational action recently proposed by Gabadadze [28].

A. Riegert's action
As we already mentioned in Introduction, the trace anomaly obtained from closed loop calculations for massless fields in an external gravitational background has the following general expression [5,6] where the GB term G and the Weyl tensor squared W 2 are defined by with R µν and R µνρσ being the Ricci tensor and Riemann tensors, respectively. The coefficients α and β are independent of the scheme of renormalization, while γ is not [7][8][9][10][11][12]. In this regard, we do not consider the last term of Eq. (2.1) and set in the following discussion.
To derive an action whose variation leads to the trace T A , Riegert split the metric tensor g µν into g µν = e 2φḡ µν and imposed that the determinant ofḡ µν is fixed. It is also possible to reformulate the construction of the action without putting a constraint onḡ µν . In this case, the metric tensorḡ µν plays a dynamical role. Indeed, the Riegert's action can be reconstructed only by requiring the conformal invariance under simultaneous Weyl transformations g µν → e 2σ g µν and φ → φ + σ [15,16]. Under these transformations, the infinitesimal changes of g µν and φ are given, respectively, by δ σ g µν = 2σg µν and δ σ φ = σ. Under such infinitesimal Weyl transformations, the action S A [g, φ], which depends on g µν and φ, varies by the amount where T A is the trace given by The conformal invariance requires that δ σ S A = 0 for σ = 0 and hence One can express G, W 2 in T A and √ −g by using corresponding quantities in the conformally transformed frame with the metricḡ µν = e −2φ g µν . In particular, the conformal transformation of the GB trace anomaly gives rise to a derivative coupling with the Einstein tensorḠ µν and nonlinear derivative terms like 8(∇ µ φ∇ µ φ)¯ φ [37], where we use an overbar for the quantities and derivatives in the frame with the metricḡ µν . Then, Eq. (2.7) yields The action S A satisfying the relation (2.7) can be constructed by considering a straight line path φ(η) = ηφ with 0 ≤ η ≤ 1, which connects the values φ(0) = 0 and φ(1) = φ [16]. The resulting action is given by We can choose other paths connecting two points φ(0) = 0 and φ(1) = φ, but the resulting action is equivalent to Eq. (2.9) [38]. The action (2.9) coincides with Eq. (8) of Ref. [14] originally derived by Riegert, but now the metric tensorḡ µν is not subject to the fixed determinant constraint.
The Einstein-Hilbert action in GR is expressed as where the second equality holds up to boundary terms. As we already mentioned, GR supplemented by the trace anomaly action (2.9) is an inconsistent EFT. We observe that the action (2.10) expressed in terms of the metricḡ µν contains an apparent kinetic term of the scalar field, but it has a negative kinetic energy. This kinetic term can be eliminated by the field redefinitionḡ µν → e −2φ g µν , but nonlinear scalar field derivatives survive in the action of . Hence such a theory is plagued by the strong coupling problem. It is worth mentioning that the action same as (2.9) also appears as a result of the regularized Kaluza-Klein reduction of D (> 4)-dimensional Einstein-Gauss-Bonnet (EGB) theory on a (D−4)-dimensional maximally symmetric space with a vanishing spatial curvature. In this scenario, the D-dimensional metric can be written in the form ds 2 D = ds 2 4 + e −2φ dσ 2 D−4 , where ds 2 4 and dσ 2 D−4 are the line elements of 4-dimensional spacetime and internal space, respectively. Here, the scalar field φ corresponds to the size of internal space, which only depends on the 4-dimensional coordinate. The D-dimensional action of EGB theory is given by where the subscript "D" represents D-dimensional quantities, andα is the GB coupling constant. Performing the volume integral of (2.11) under the above metric ansatz, we can express S EGB in terms of the 4-dimensional curvature quantities R, G µν , G, and the scalar field φ and its derivatives. In this process, the integration constant is absorbed into M D to define the 4-dimensional reduced Planck mass M Pl = √ 2M . We add a counter term −M 2 d 4 x √ −gαG to the action (2.11) and rescale the coupling constant asα → α/(D − 4) [18]. Taking the D → 4 limit in the end, we obtain the reduced 4-dimensional action The terms proportional to α in Eq. (2.12) are exactly the same as those associated with the GB trace anomaly in the action (2.9). The action (2.12) has an Einstein-Hilbert term, but there is no kinetic term of the scalar field φ. Thus, 4DEGB gravity also suffers from the strong coupling problem, as recognized in Refs. [22,27,29].

B. Gabadadze's action
To circumvent the strong coupling problem present in GR supplemented by the trace anomaly action (2.9), Gabadadze [28] proposed the action where S RR and S A are given, respectively, by Eqs. (1.2) and (2.9), and the new mass scaleM is assumed to be much smaller than M . In terms of the metricḡ µν = e −2φ g µν , the action (2.13) can be expressed as (2.14) Note that GR with the trace anomaly action corresponds to the limitM → 0. In terms of the metric g µν , the action S RR in Eq. (1.3) contains a kinetic term −6g µν ∇ µ ϕ∇ ν ϕ with a correct sign (i.e., no ghost) for the canonically normalized scalar field ϕ =M e −φ . The appearance of this correct sign is the result of introducing the action −M 2 d 4 x √ −ḡR. The second and third terms of (1.3) correspond to the action of a conformally invariant scalar field, whose invariance is broken by the Einstein-Hilbert term.
Under the transformations g µν → g where µ is an arbitrary constant, the action (1.3) is invariant [28]. There is also an invariant combination of the metric tensor g µν = g µν (1 − ϕ 2 /M 2 ). If the matter fields in standard model of particle physics are coupled to gravity through the metricĝ µν , the corresponding actions in the matter sector are also invariant under such transformations. Provided thatM M , one hasĝ µν g µν and hence the matter fields are approximately coupled to the metric tensor g µν . The flat space expansion of (2.14) can be performed by substituting g µν = e 2φḡ µν = η µν (1 − ϕ 2 /M 2 ) −1 , where η µν is the Minkowski metric. Introducing a canonically normalized field π ≡M φ, the action (2.14) contains the derivative terms where we used the notation (∂π) 2 = ∂ µ π∂ µ π and the partial derivative ∂ µ = ∂/∂x µ . Nonlinear derivative terms are suppressed below the scaleM in comparison to the canonical kinetic term, while the theory is strongly coupled above the scaleM . Hence the theory given by the action (2.14) is an EFT with trace anomaly corrections valid below the scaleM ( M ). Unlike the Riegert's theory, the metricḡ µν = e −2φ g µν is not subject to a fixed determinant constraint. Then, the Gabadadze's theory has two tensor propagating degrees of freedom besides one scalar mode φ.
Varying the action (2.14) with respect toḡ µν , it follows that We take the trace of Eq. (2.16) by exertingḡ µν and exploit the relation 12X − 6¯ φ = e 2φ R −R. On using the propertyḡ µνP µανβ = −Ḡ αβ , it follows thatḡ µνH µν = (e 4φ G −Ḡ)/2. We also note that the divergence of the Weyl tensor vanishes, such thatḡ µνC µρνσ = 0. Then, the trace of Eq. (2.16) is expressed as From Eq. (2.7), the variation of the trace anomaly action is given by δS A /δφ = √ −g −αG + βW 2 . Then, varying the action S = S RR + S A with respect to φ leads to  .14) gives rise to the field equations of motion higher than second order. If the Weyl trace anomaly is absent, i.e., then the action (2.14) belongs to a subclass of Horndeski theories [23] given by whereḠ j,X ≡ ∂Ḡ j /∂X (with j = 4, 5), and In particular, the linearly coupled GB Lagrangian −αφḠ can be accommodated by the quintic Horndeski function G 5 = 4α ln |X| [25,39]. In this case, the field equations of motion are kept up to second order in bothḡ µν and φ.

III. HAIRY BLACK HOLE
We study the existence of hairy BH solutions for the action (2.14) with β = 0, i.e., which is equivalent to the Horndeski action (2.24) with the coupling functions (2.25). For completeness we need to take into account the Weyl trace anomaly term, but in this paper we would like to clarify whether or not the GB trace anomaly can induce linearly stable hairy BH solutions. In terms of the metric tensorḡ µν , the static and spherically symmetric spacetime is given by the line element where the metric componentsf andh depend on the radial coordinater. The scalar field is assumed to be a function ofr alone, i.e., φ = φ(r).
We also write the line element associated with the metric g µν = e 2φḡ µν as which is related to (3.2) according to ds 2 = e −2φ ds 2 . There are the following relations where a prime represents the derivative with respect tor. In the following, we will obtain the solutions tof ,h, and φ as functions ofr. Using the correspondence (3.4), we will also derive the functions f (r) and h(r) in the line element (3.3). The differential equations for the metric componentsf (r) andh(r) are given bȳ h[M 2r (rφ + 1)e 2φ + 2αφ (h{3 +rφ (rφ
From Eq. (2.22), we obtain In the absence of the GB trace anomaly, i.e., α = 0, there is the following exact solution to Eqs. (3.5)-(3.7): where c 0 and m 0 are constants. For a conformal scalar field with the Einstein-Hilbert action, the same type of solution was obtained by Bocharova, Bronnikov, and Melnikov [40] and by Bekenstein [41] (BBMB). However, it is unstable against monopole perturbations [42][43][44] as well as perturbations for general multipoles l [35]. This instability is also related to the divergence of φ (r) = 1/(m 0 −r) on the horizon located atr = m 0 [32,33]. In Appendix A, we will show that the solution (3.8) is unstable against linear perturbations for the radiusr > 2m 0 . For α = 0, there is also the GR branch given bȳ wherer h is the horizon radius. This solution is stable against linear perturbations. Thus, instead of (3.8), the Schwarzschild branch without a scalar hair (3.9) should be selected as a linearly stable BH. The no-hair property of BHs for α = 0 changes in the presence of the GB trace anomaly. Let us derive the solutions tof ,h, and φ under the expansion of a small coupling constant α. In the limit that α → 0, the solutions need to recover the Schwarzschild metric without the scalar hair, i.e., Eq. (3.9). Outside the horizon characterized by the radiusr h , we search for solutions in the forms wheref i ,h i , and φ i are functions ofr. Substituting Eq. (3.10) into Eqs. (3.5)-(3.7), we obtain the differential equations forf i ,h i , and φ i at each order in α i . We can also derive a second-order differential equation for φ(r) by differentiating Eq. (3.5) with respect tor and eliminate the derivativesf ,h , andf by using Eqs. (3.5), (3.6), and (3.7). At first order in the expansion of φ(r), we have The integrated solution to φ 1 (r) contains two integration constants. They are determined by imposing the regular boundary conditions φ 1 (r h ) = constant on the horizon and φ 1 (∞) → 0 at spatial infinity. The finiteness of φ (r) atr =r h is not only required for the validity of the expansion (3.10) but also for avoiding the instability of linear perturbations [32,33]. Then, the resulting integrated solution to Eq. (3.11) yields Substituting this solution into Eq. (3.6) and using Eq. (3.5), we obtain the differential equation forh 1 (r) as The integrated solution respecting the finiteness ofh 1 (r) on the horizon is given bȳ which approaches 0 at spatial infinity. From Eq. (3.5), the differential equation forf 1 (r) is The integrated solution satisfying the boundary conditionf 1 (∞) = 0 is which decreases asf 1 (r) ∝r −1 at large distances. Similarly, we derive the solutions tof i (r),h i (r), and φ i (r) at each order in α i by imposing the regular boundary conditions explained above. As we will see in Sec. IV, the expansion up to the order of i = 4 is required for the purpose of studying the propagation of linear perturbations correctly. Due to the complexity of functionsf i (r),h i (r), φ i (r) for i ≥ 3, we only write the solutions up to the i = 2 order as Outside the horizon, the leading-order trace anomaly corrections tof (r) andh(r) are largest aroundr =r h , which are of order α/(M 2r2 h ). For the validity of the expansion (3.10), we then require that Provided thatM M , the second-order corrections to Eqs. (3.17) and (3.18) are at most of orderα 2 . From Eq. (3.19), the first-and second-order corrections to φ(r) are at most of ordersα andα 2 , respectively. In the limit thatM → 0, the condition (3.20) is violated and hence the expanded solutions (3.17)- (3.19) are invalid in GR supplemented by the trace anomaly action S A . For the mass scaleM larger than 1/r h , the inequality (3.20) is satisfied for α 1. This means thatM can be chosen down to the order 1/r h . ForM 1/r h , the EFT is valid for the length scale larger than r h .
Unlike the BBMB solution, the field derivative φ (r) is finite on the horizon. At large distances, the scalar field solution up to the order of α is given by where q s is regarded as a scalar charge given by The GB trace anomaly gives rise to a hairy BH solution possessing the scalar charge q s . From Eqs. (3.17) and (3.18) we find thatf (r) andh(r) are not identical to each other for α = 0. This is different from an exact BH solution obtained by assumingf (r) = h(r) in a similar conformally invariant theory [16]. In Appendix B, we will derive such a solution by imposing the conditionf (r) = h(r) in Eq. (3.7). This condition demands that the right hand side of Eq. (3.6) vanishes. However, the scalar field solution derived in this way is not consistent with the regular boundary condition on the horizon as well as the other background equations like Eq. (3.5). In other words, the solutions satisfying all the equations of motion and regular boundary conditions on the horizon are of the forms (3.17)- (3.19). They are also different from the hairy BH solution present for a canonical scalar field φ linearly coupled to the GB term [45,46] (see also Refs. [32,33,[47][48][49][50][51][52][53][54][55][56][57][58] for related works). In the latter theory the first-order metric componentsf 1 andh 1 vanish [33,46], but this is not the case for our theory due to the existence of additional nonlinear derivative terms in Eq. (3.1).
In Fig. 1 h ). ForM 1/r h , the corrections tof (r),h(r), and φ(r) are at most of order α. In Fig. 1, we observe that the numerical results are in good agreement with the analytic solutions (3.17)- (3.19) expanded up to fourth order in α i . The field derivative φ , which is finite on the horizon, decreases as φ (r) ∝r −2 in the regimer r h . Outward from the horizon,h continues to increase toward the asymptotic value 1. The quantity∆, which characterizes the difference betweenh andf , is largest aroundr =r h . From Eqs. (3.17) and (3.18), we obtain the asymptotic behavior∆ 4α/(3M 2r2 ) in the regimer r h , whose dependence can be also confirmed in Fig. 1. The metric components f and h in the frame with the metric tensor g µν can be obtained as functions of r by using the correspondence (3.4). Taking φ 0 = 0, the resulting forms of f (r) and h(r) correspond to those derived by the replacementsr → r,r h → r h ,M → M , and M →M in Eqs. (3.17) and (3.18), i.e., The leading-order trace anomaly corrections to f (r) and h(r) are subject to strong suppression by the appearance of the mass M = M Pl / √ 2 in their denominators. As we already mentioned, the matter fields feel the gravitational force through the metric tensorĝ µν g µν . Let us consider a test particle with mass m g on the metric background g µν . Defining the gravitational potential Ψ(r) as f (r) = 1 + 2Ψ(r) in the regime away from the horizon, the force exerting on the particle is given by where we used the expanded solution (3.23), and r is related tor according to In terms of the distancer and horizon radiusr h , the force (3.25) can be expressed as The α-dependent terms in Eq. (3.25) correspond to fifth-force corrections to the gravitational force F g = −m g r h /(2r 2 ). The leading-order fifth force relative to F g is at most of order (M /M ) 2α , so it is even more suppressed thanα due to the small ratio (M /M ) 2 1. In Eq. (3.27), on the other hand, the leading-order trace anomaly correction relative to −m grh /(2r 2 ) is of orderα. This difference arises from the fact that the relative difference between r andr is at most of orderα, see Eq. (3.26). When we express r in terms ofr, the orderα correction, which is much larger than (M /M ) 2α , appears in Eq. (3.27). Since the test particle feels gravity associated with the metric g µν and distance r, we need to interpret that the fifth force is suppressed by the factor (M /M ) 2α relative to the gravitational force F g = −m g r h /(2r 2 ). If we consider astrophysical BHs whose horizon radii r h are larger than the order 10 km, then the ratio (M /M ) 2 is significantly smaller than 1 forM of order 1/r h . This is not necessarily the case for microscopic BHs with smaller horizon sizes, in which case the ratio (M /M ) 2 can be larger.
While the fifth force exerting on the test particle can be suppressed for astrophysical BHs, the scalar field (3.19) receives the trace anomaly correction at most of orderα (even if we express φ with respect to r). During the inspiral phase of a BH binary system, the scalar radiation arising from the perturbation of φ may leave some signatures of the scalar charge in observed gravitational waveforms. This will deserve for a further detailed study.

IV. LINEAR STABILITY OF HAIRY BLACK HOLE
Finally, we study the linear stability of hairy BHs derived in Sec. III. For this purpose, we exploit the solutions (3.17)- (3.19) obtained under the expansion of a small coupling constant α in the frame with the metricḡ µν . The perturbations on the static and spherically symmetric background can be decomposed into odd-and even-parity modes [59,60]. In Refs. [34,35], the stability conditions of BHs against odd-and even-parity linear perturbations were derived except for the angular stability of even-parity modes. In Ref. [36], the authors incorporated a perfect fluid with the background density ρ and pressure P and obtained all the linear stability conditions applicable to NSs as well (see also Ref. [61]). We will exploit those results in the following discussion.
In Horndeski theories given by the action (3.1), the odd-parity sector has a gravitational wave mode χ arising from the metric perturbation. For this perturbation χ, there are neither ghost nor Laplacian instabilities under the following conditions Provided thatα 1, the no-ghost conditionḠ > 0 is satisfied if which automatically holds forM M . Up to linear order in α i , the expressions ofF andH coincide withḠ. However, the difference appears at the order of α 2 . The squared propagation speeds of odd-parity gravitational perturbation χ along the radial and angular directions are given, respectively, bȳ Since the trace anomaly corrections in Eqs. (4.6) and (4.7) are at most of order (M 2 /M 2 )α 2 , bothc 2 r,odd andc 2 Ω,odd are very close to 1 in the vicinity of the horizon. Thus, the Laplacian stability conditionsc 2 r,odd > 0 andc 2 Ω,odd > 0 are automatically satisfied forα 1. We also note that the α 2 -order corrections to Eqs. (4.6) and (4.7) decrease in proportion tor −3 at large distances, soc 2 r,odd andc 2 Ω,odd rapidly approach 1 far away from the horizon. In the even-parity sector, the ghost does not appear under the condition Under the condition (4.5), the leading-order term ofK is positive and hence the ghost is absent in the even-parity sector as well. We also note that the trace anomaly correction generates the nonvanishing kinetic termK, in which case the strong coupling problem is absent. For even-parity modes, there are two perturbations ψ and δφ arising from the gravitational and scalar field sectors, respectively [34][35][36]. The radial propagation speed squared of ψ is identical toc 2 r,odd =Ḡ/F in the odd-parity sector. The other radial propagation speed squaredc 2 r,δφ can be obtained by setting ρ = 0 = P in Eq. ( To derive this expression, we need to resort to the background solutions off (r),h(r), and φ(r) expanded up to fourth order in α i . In other words, the third order expansion leads to a result different from Eq. (4.12), but using the solutions higher than fourth order gives the same expression as Eq. (4.12). The trace anomaly correction inc 2 r,δφ is at most of orderα 2 and it decreases in proportion tor −3 at large distances. Provided thatα 1, the radial Laplacian stability of δφ is always ensured.
The squared propagation speeds of ψ and δφ in the angular direction arec 2 Ω±,even = −B 1 ± B 2 1 − B 2 , where the explicit expressions of B 1 and B 2 are given, respectively, by Eqs. (5.37) and (5.38) of Ref. [36]. For our hairy BH solution, we havec  Hence the leading-order trace anomaly correction inc 2 Ω±,even is suppressed by the order (M /M )α and it decreases in proportion tor −3 for increasingr.
We have thus shown that, under the condition M 2 e 2φ −M 2 > 0, our hairy BH solution arising from the trace anomaly satisfies all the linear stability conditions forα 1 andM M . All the propagation speeds discussed above are close to 1 with small corrections induced by the GB trace anomaly.

V. CONCLUSIONS
In a new gravitational theory with the trace anomaly recently proposed by Gabadadze, we studied the existence of hairy BH solutions on the static and spherically symmetric background. Introducing the action −M 2 d 4 x √ −ḡR besides the Einstein-Hilbert term allows a possibility for avoiding the strong coupling problem present in GR supplemented by the trace anomaly. In terms of the metric tensorḡ µν = e −2φ g µν invariant under Weyl transformations g µν → e 2σ g µν and φ → φ + σ, the total action S = S RR + S A is expressed in the form (2.14). This is the EFT valid below the mass scaleM , in which regime nonlinear derivatives of the conformal scalar field are weakly coupled. Thanks to the presence of the action −M 2 d 4 x √ −ḡR, the scalar field acquires a kinetic term without the ghost. We note that 4DEGB gravity is plagued by the strong coupling problem because of the absence of such a healthy kinetic term of the radion field.
In Eq. (2.16), we derived the covariant gravitational field equations by varying the action (2.14) with respect toḡ µν . The trace of this equation can be expressed in the simple form (2.19). Varying the action with respect to φ leads to the scalar field equation 2M 2 R − αG + βW 2 = 0 and hence the combination with Eq. (2.19) gives 2M 2R − αḠ + βW 2 = 0. Thus, the GB and Weyl terms explicitly affect the spacetime geometry in both frames with the metrics g µν andḡ µν . So long as the Weyl trace anomaly is absent, i.e., β = 0, the resulting theory is equivalent to a subclass of Horndeski theories given by the action (2.24) with the coupling functions (2.25). In this paper, we studied whether the GB trace anomaly induces a hairy BH solution without instabilities.
In Sec. III, we first showed that the linearly stable BH solution in theories without the GB trace anomaly (α = 0) is restricted to a no-hair Schwarzschild solution given by Eq. (3.9). In this case there exists the other hairy BH solution (3.8) originally found by BBMB for a conformally invariant scalar, but this solution is known to be unstable (see Appendix A). For α = 0, we derived the solutions tof (r),h(r), and φ(r) expanded with respect to a small coupling constant α in the frame with the metricḡ µν . The BH solution expanded up to second order in α i , which is consistent with regular boundary conditions on the horizon and at spatial infinity, is given by Eqs. (3.17)- (3.19), with the BH scalar charge (3.22).
The two metric componentsf (r) andh(r) are not proportional to each other for our BH solution, so it is different from the exact solution derived by imposing the conditionf (r) = c 0h (r) (see Appendix B). Indeed, the latter is not consistent with all the field equations of motion and regular boundary conditions on the horizon. As we observe in Fig. 1, our analytic hairy BH solution (3.17)- (3.19), which is valid forα 1, exhibits good agreement with the numerical results. We also derived the metric components f (r) and h(r) in the frame with the metric g µν as (3.23)-(3.24) and computed the fifth force exerted on a test particle away from the horizon.
In Sec. IV, we studied the stability of the hairy BH solution (3.17)-(3.19) against linear perturbations on the background metric (3.2). In the odd-parity sector, the stability conditions are given by Eqs. (4.1)-(4.3). Provided that M 2 e 2φ −M 2 > 0, they can be satisfied for the small coupling constant in the rangeα 1. Under this inequality, the no-ghost condition for even-parity perturbations is also consistently satisfied without the strong coupling problem. The radial and angular propagation speeds of three dynamical perturbations in the odd-and even-parity sectors are close to 1 with small corrections induced by the trace anomaly. Thus, there are no Laplacian instabilities for our hairy BH solution under the conditionsα 1 andM M . These properties are different from those for exact BH solutions known in 4DEGB gravity, which are plagued by the strong coupling as well as the linear instability problems [22,29].
Since the BH solution obtained in this paper possesses the scalar charge (3.22), it may be possible to probe its signature from gravitational waves emitted from the binary system containing BHs (see e.g., Refs. [62][63][64][65]). While we have neglected effects of the Weyl trace anomaly on the BH solutions and its stabilities, it may also give rise to an additional scalar hair for BHs and possibly for NSs. Since the Weyl term gives rise to the field equations of motion containing derivatives higher than second order, the BH stability conditions derived for Horndeski theories cannot be literally applied to Weyl gravity theories. In the EFT scheme, there should be some way of properly dealing with such higher-order derivative terms. It will be certainly of interest to study whether the linearly stable BH solutions exist or not in the presence of full trace anomaly terms. The detailed studies of such issues are left for future works.
where C 1 and C 2 are integration constants. Since we are considering the caseh /h =f /f , the scalar field satisfies the relation φ 2 − φ = 0 from Eq. (3.6). Then, we obtain the following integrated solution φ (r) = − ln (C 3r + C 4 ) , where C 3 and C 4 are constants. The other equations of motion are satisfied for