# Black holes in a new gravitational theory with trace anomalies

Shinji Tsujikawa  Department of Physics, Waseda University, 3-4-1 Okubo,
Shinjuku, Tokyo 169-8555, Japan

###### Abstract

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.

††preprint: WUCG-23-05

##  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 $g_{\mu\nu}$ 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 $T_{A}=-\alpha{\cal
G}+\beta W^{2}+\gamma\square R$, where ${\cal G}$ is a Gauss-Bonnet (GB) term,
$W^{2}=C_{\mu\nu\rho\sigma}C^{\mu\nu\rho\sigma}$ is the squared of a Weyl
tensor $C_{\mu\nu\rho\sigma}$, and $R$ is a Ricci scalar [5, 6]. The constants
$\alpha$, $\beta$, $\gamma$ 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 the one
obtained by using the AdS/CFT correspondence in strongly coupled large $N$ CFT
[13]. Decomposing the metric tensor into
$g_{\mu\nu}=e^{2\sigma}\bar{g}_{\mu\nu}$, where $e^{2\sigma}$ is a conformal
factor and the metric $\bar{g}_{\mu\nu}$ is restricted to have a fixed
determinant, Riegert [14] derived an effective local action ${\cal S}_{A}$
that generates the trace anomaly $T_{A}$ in the field equations of motion.
This can be further promoted to consider a conformal geometry

$\bar{g}_{\mu\nu}=e^{-2\phi}g_{\mu\nu}\,,$ (1)

without assuming any constraint on $\bar{g}_{\mu\nu}$ [15, 16]. The Weyl
invariant combination $\bar{g}_{\mu\nu}=e^{-2\phi}g_{\mu\nu}$ transforms as a
metric under diffeomorphisms, with $\phi$ being a massless scalar field
(dilaton). The metric $\bar{g}_{\mu\nu}$ is invariant under Weyl
transformations $g_{\mu\nu}\to e^{2\sigma}g_{\mu\nu}$ and
$\phi\to\phi+\sigma$.

The diffeomorphism-invariant effective action ${\cal S}_{A}$ of gravitational
trace anomalies contains Galileon-type self-interaction of the dilaton and
derivative couplings to the Einstein tensor besides the couplings between
$\phi$ and ${\cal G},W^{2}$ [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 $\alpha\to\alpha/(D-4)$
[19, 20, 21, 22]. In 4DEGB gravity, the scalar field $\phi$ 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 $\phi$ 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 $\delta\phi$ 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

${\cal S}_{R\bar{R}}=M^{2}\int{\rm d}^{4}x\sqrt{-g}R-\bar{M}^{2}\int{\rm
d}^{4}x\sqrt{-\bar{g}}\bar{R}\,,$ (2)

where $g$ and $\bar{g}$ are the determinants of $g_{\mu\nu}$ and
$\bar{g}_{\mu\nu}=e^{-2\phi}g_{\mu\nu}$, respectively, and $\bar{R}$ is the
Ricci scalar evaluated with $\bar{g}_{\mu\nu}$. The constant $M$ is related to
the reduced Planck mass $M_{\rm Pl}$ as $M=M_{\rm Pl}/\sqrt{2}$, whereas the
new mass scale $\bar{M}$ is much smaller than $M$. The total effective action
incorporating the trace anomaly is given by ${\cal S}={\cal
S}_{R\bar{R}}+{\cal S}_{A}$, where ${\cal S}_{A}$ is the Riegert’s action
without a fixed determinant constraint on $\bar{g}_{\mu\nu}$. The Ricci scalar
$\bar{R}$ can be expressed in terms of the sum of the term $e^{2\phi}R$ and
derivatives of $\phi$. Then, the action (2) is equivalent to

${\cal S}_{R\bar{R}}=\int{\rm
d}^{4}x\sqrt{-g}\left(M^{2}R-\varphi^{2}R-6g^{\mu\nu}\nabla_{\mu}\varphi\nabla_{\nu}\varphi\right)\,,$
(3)

where $\varphi\equiv\bar{M}e^{-\phi}$ and the covariant derivative operator
$\nabla_{\mu}$ is associated with the metric $g_{\mu\nu}$. Thus, the action
${\cal S}_{R\bar{R}}$ contains the scalar field kinetic term
$-6g^{\mu\nu}\nabla_{\mu}\varphi\nabla_{\nu}\varphi$ with the proper sign. The
last two terms in Eq. (3) respect the conformal invariance, while the first
Einstein-Hilbert term explicitly breaks the conformal symmetry.

The theory given by the action ${\cal S}={\cal S}_{R\bar{R}}+{\cal S}_{A}$ can
be regarded as an effective field theory (EFT) valid below the scale
$\bar{M}$. Introducing a canonical normalized field $\pi=\bar{M}\phi$, we see
that nonlinear scalar derivative terms of $\pi$ are strongly coupled above the
scale $\bar{M}$, while they are weakly coupled below $\bar{M}~{}(\ll M)$ [28].
Taking the limit $\bar{M}\to 0$ 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 ${\cal
S}_{R\bar{R}}$ as well as the existence of a new scale $\bar{M}$ 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 components
$\bar{f}$ and $\bar{h}$ associated with $\bar{g}_{\mu\nu}$ 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 $\phi$, $\bar{f}$,
$\bar{h}$ on the horizon satisfies $\bar{f}(\bar{r})\neq\bar{h}(\bar{r})$ at
arbitrary distances $\bar{r}$, where the difference between $\bar{f}$ and
$\bar{h}$ comes from the trace anomaly. Under the expansion of a small
coupling constant $\alpha$, we will derive analytic solutions to $\bar{f}$,
$\bar{h}$, and $\phi$ 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 $\phi^{\prime}(\bar{r})$
is a finite constant on the horizon ($\bar{r}=\bar{r}_{h}$). On the other
hand, for the exact BH solution present in 4DEGB gravity [19, 31],
$\phi^{\prime}(\bar{r})$ diverges at $\bar{r}=\bar{r}_{h}$. The latter
property leads to the linear instability of BHs in the vicinity of the horizon
[22, 29]. This instability was shown for a time-independent 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 $\alpha$ 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].

###  II.1 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]

$T_{A}=-\alpha{\cal G}+\beta W^{2}+\gamma\square R\,,$ (4)

where the GB term ${\cal G}$ and the Weyl tensor squared $W^{2}$ are defined
by

$\displaystyle{\cal G}$ $\displaystyle\equiv$ $\displaystyle
R^{2}-4R_{\mu\nu}R^{\mu\nu}+R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\,,$ (5)
$\displaystyle W^{2}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{3}R^{2}-2R_{\mu\nu}R^{\mu\nu}+R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\,,$
(6)

with $R_{\mu\nu}$ and $R_{\mu\nu\rho\sigma}$ being the Ricci tensor and
Riemann tensors, respectively. The coefficients $\alpha$ and $\beta$ are
independent of the scheme of renormalization, while $\gamma$ is not [7, 8, 9,
10, 11, 12]. In this regard, we do not consider the last term of Eq. (4) and
set

$\gamma=0\,,$ (7)

in the following discussion.

To derive an action whose variation leads to the trace $T_{A}$, Riegert split
the metric tensor $g_{\mu\nu}$ into $g_{\mu\nu}=e^{2\phi}\bar{g}_{\mu\nu}$ and
imposed that the determinant of $\bar{g}_{\mu\nu}$ is fixed. It is also
possible to reformulate the construction of the action without putting a
constraint on $\bar{g}_{\mu\nu}$. In this case, the metric tensor
$\bar{g}_{\mu\nu}$ plays a dynamical role. Indeed, the Riegert’s action can be
reconstructed only by requiring the conformal invariance under simultaneous
Weyl transformations $g_{\mu\nu}\to e^{2\sigma}g_{\mu\nu}$ and
$\phi\to\phi+\sigma$ [15, 16]. Under these transformations, the infinitesimal
changes of $g_{\mu\nu}$ and $\phi$ are given, respectively, by
$\delta_{\sigma}g_{\mu\nu}=2\sigma g_{\mu\nu}$ and
$\delta_{\sigma}\phi=\sigma$. Under such infinitesimal Weyl transformations,
the action ${\cal S}_{A}[g,\phi]$, which depends on $g_{\mu\nu}$ and $\phi$,
varies by the amount

$\delta_{\sigma}{\cal S}_{A}=\int{\rm
d}^{4}x\left(2g_{\mu\nu}\frac{\delta{\cal S}_{A}[g,\phi]}{\delta
g_{\mu\nu}}+\frac{\delta{\cal
S}_{A}[g,\phi]}{\delta\phi}\right)\sigma=\int{\rm
d}^{4}x\sqrt{-g}\left(-T_{A}+\frac{1}{\sqrt{-g}}\frac{\delta{\cal
S}_{A}[g,\phi]}{\delta\phi}\right)\sigma\,,$ (8)

where $T_{A}$ is the trace given by

$T_{A}=-\frac{2}{\sqrt{-g}}g_{\mu\nu}\frac{\delta{\cal S}_{A}[g,\phi]}{\delta
g_{\mu\nu}}\,.$ (9)

The conformal invariance requires that $\delta_{\sigma}{\cal S}_{A}=0$ for
$\sigma\neq 0$ and hence

$\frac{\delta{\cal
S}_{A}[g,\phi]}{\delta\phi}=\sqrt{-g}\,T_{A}=\sqrt{-g}\left(-\alpha{\cal
G}+\beta W^{2}\right)\,.$ (10)

One can express ${\cal G}$, $W^{2}$ in $T_{A}$ and $\sqrt{-g}$ by using
corresponding quantities in the conformally transformed frame with the metric
$\bar{g}_{\mu\nu}=e^{-2\phi}g_{\mu\nu}$. In particular, the conformal
transformation of the GB trace anomaly gives rise to a derivative coupling
with the Einstein tensor $\bar{G}^{\mu\nu}$ and nonlinear derivative terms
like $8(\bar{\nabla}_{\mu}\phi\bar{\nabla}^{\mu}\phi)\bar{\square}\phi$ [37],
where we use an overbar for the quantities and derivatives in the frame with
the metric $\bar{g}_{\mu\nu}$. Then, Eq. (10) yields

$\displaystyle\frac{\delta{\cal S}_{A}[\bar{g},\phi]}{\delta\phi}$
$\displaystyle=$ $\displaystyle-\sqrt{-\bar{g}}\,\alpha[\bar{\cal
G}-8\bar{R}^{\mu\nu}\bar{\nabla}_{\mu}\phi\bar{\nabla}_{\nu}\phi+8\bar{G}^{\mu\nu}\bar{\nabla}_{\mu}\bar{\nabla}_{\nu}\phi+8(\bar{\nabla}_{\mu}\phi\bar{\nabla}^{\mu}\phi)\bar{\square}\phi-8\bar{\nabla}_{\mu}\bar{\nabla}_{\nu}\phi\bar{\nabla}^{\mu}\bar{\nabla}^{\nu}\phi+8(\bar{\square}\phi)^{2}$
(11)
$\displaystyle+16\bar{\nabla}_{\mu}\phi\bar{\nabla}_{\nu}\phi\bar{\nabla}^{\mu}\bar{\nabla}^{\nu}\phi]+\sqrt{-\bar{g}}\beta\bar{W}^{2}\,.$

The action ${\cal S}_{A}$ satisfying the relation (10) can be constructed by
considering a straight line path $\phi(\eta)=\eta\phi$ with $0\leq\eta\leq 1$,
which connects the values $\phi(0)=0$ and $\phi(1)=\phi$ [16]. Up to boundary
terms, the resulting action is given by

$\displaystyle{\cal S}_{A}[\bar{g},\phi]$ $\displaystyle=$
$\displaystyle\int{\rm d}^{4}x\int_{0}^{1}{\rm d}\eta\frac{\delta{\cal
S}_{A}[\bar{g},\phi]}{\delta\phi}\biggr{|}_{\phi\to\eta\phi}\phi$ (12)
$\displaystyle=$ $\displaystyle-\alpha\int{\rm
d}^{4}x\sqrt{-\bar{g}}\left(\phi\bar{\cal
G}-4\bar{G}^{\mu\nu}\bar{\nabla}_{\mu}\phi\bar{\nabla}_{\nu}\phi+8\bar{X}\bar{\square}\phi-8\bar{X}^{2}\right)+\beta\int{\rm
d}^{4}x\sqrt{-\bar{g}}\,\phi\bar{W}^{2}\,,$

where
$\bar{X}=-(1/2)\bar{g}^{\mu\nu}\bar{\nabla}_{\mu}\phi\bar{\nabla}_{\nu}\phi$.
This is equivalent to the action (8) of Ref. [14] originally derived by
Riegert, but now the metric tensor $\bar{g}_{\mu\nu}$ is not subject to the
fixed determinant constraint.

The Einstein-Hilbert action in GR is expressed as

${\cal S}_{\rm GR}=M^{2}\int{\rm d}^{4}x\sqrt{-g}R=M^{2}\int{\rm
d}^{4}x\sqrt{-\bar{g}}\,e^{2\phi}\left(\bar{R}+6\bar{g}^{\mu\nu}\bar{\nabla}_{\mu}\phi\bar{\nabla}_{\nu}\phi\right)\,,$
(13)

where the second equality holds up to boundary terms. As we already mentioned,
GR supplemented by the trace anomaly action (12) is an inconsistent EFT. We
observe that the action (13) expressed in terms of the metric
$\bar{g}_{\mu\nu}$ 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 redefinition $\bar{g}_{\mu\nu}\to e^{-2\phi}g_{\mu\nu}$, but nonlinear
scalar field derivatives survive in the action of ${\cal S}_{A}[g,\phi]$.
Hence such a theory is plagued by the strong coupling problem.

It is worth mentioning that the action same as (12) 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 ${\rm d}s_{D}^{2}={\rm
d}s_{4}^{2}+e^{-2\phi}{\rm d}\sigma_{D-4}^{2}$, where ${\rm d}s_{4}^{2}$ and
${\rm d}\sigma_{D-4}^{2}$ are the line elements of 4-dimensional spacetime and
internal space, respectively. Here, the scalar field $\phi$ 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

${\cal S}_{\rm EGB}=M_{D}^{2}\int{\rm
d}^{D}x\sqrt{-g_{D}}\left(R_{D}+\hat{\alpha}{\cal G}_{D}\right)\,,$ (14)

where the subscript $``D$” represents $D$-dimensional quantities, and
$\hat{\alpha}$ is the GB coupling constant. Performing the volume integral of
(14) under the above metric ansatz, we can express ${\cal S}_{\rm EGB}$ in
terms of the 4-dimensional curvature quantities $R$, $G^{\mu\nu}$, ${\cal G}$,
and the scalar field $\phi$ and its derivatives. In this process, the
integration constant is absorbed into $M_{D}$ to define the 4-dimensional
reduced Planck mass $M_{\rm Pl}=\sqrt{2}M$. We add a counter term
$-M^{2}\int{\rm d}^{4}x\sqrt{-g}\,\hat{\alpha}{\cal G}$ to the action (14) and
rescale the coupling constant as $\hat{\alpha}\to\alpha/(D-4)$ [18]. Taking
the $D\to 4$ limit in the end, we obtain the reduced 4-dimensional action

${\cal S}_{\rm 4DEGB}=M^{2}\int{\rm d}^{4}x\sqrt{-g}\left[R-\alpha(\phi{\cal
G}-4G^{\mu\nu}\nabla_{\mu}\phi\nabla_{\nu}\phi+8X\square\phi-8X^{2})\right]\,.$
(15)

The terms proportional to $\alpha$ in Eq. (15) are exactly the same as those
associated with the GB trace anomaly in the action (12). The action (15) has
an Einstein-Hilbert term, but there is no kinetic term of the scalar field
$\phi$. Thus, 4DEGB gravity also suffers from the strong coupling problem, as
recognized in Refs. [27, 22, 29].

###  II.2 Gabadadze’s action

To circumvent the strong coupling problem present in GR supplemented by the
trace anomaly action (12), Gabadadze [28] proposed the action

${\cal S}={\cal S}_{R\bar{R}}+{\cal S}_{A}\,,$ (16)

where ${\cal S}_{R\bar{R}}$ and ${\cal S}_{A}$ are given, respectively, by
Eqs. (2) and (12), and the new mass scale $\bar{M}$ is assumed to be much
smaller than $M$. In terms of the metric
$\bar{g}_{\mu\nu}=e^{-2\phi}g_{\mu\nu}$, the action (16) can be expressed as

${\cal S}=\int{\rm
d}^{4}x\sqrt{-\bar{g}}\left[M^{2}e^{2\phi}\left(\bar{R}-12\bar{X}\right)-\bar{M}^{2}\bar{R}-\alpha\left(\phi\bar{\cal
G}-4\bar{G}^{\mu\nu}\bar{\nabla}_{\mu}\phi\bar{\nabla}_{\nu}\phi+8\bar{X}\bar{\square}\phi-8\bar{X}^{2}\right)+\beta\phi\bar{W}^{2}\right]\,.$
(17)

Note that GR with the trace anomaly action corresponds to the limit
$\bar{M}\to 0$. In terms of the metric $g_{\mu\nu}$, the action ${\cal
S}_{R\bar{R}}$ in Eq. (3) contains a kinetic term
$-6g^{\mu\nu}\nabla_{\mu}\varphi\nabla_{\nu}\varphi$ with a correct sign
(i.e., no ghost) for the canonically normalized scalar field
$\varphi=\bar{M}e^{-\phi}$. The appearance of this correct sign is the result
of introducing the action $-\bar{M}^{2}\int{\rm
d}^{4}x\sqrt{-\bar{g}}\,\bar{R}$. The second and third terms of (3) correspond
to the action of a conformally invariant scalar field, whose invariance is
broken by the Einstein-Hilbert term.

Under the transformations $g_{\mu\nu}\to
g_{\mu\nu}(1+\mu\varphi/M)^{2}/(1-\mu^{2})$ and $\varphi\to\varphi+\mu
M(1-\varphi^{2}/M^{2})/(1+\mu\varphi/M)$, where $\mu$ is an arbitrary
constant, the action (3) is invariant [28]. There is also an invariant
combination of the metric tensor
$\hat{g}_{\mu\nu}=g_{\mu\nu}(1-\varphi^{2}/M^{2})$. If the matter fields in
standard model of particle physics are coupled to gravity through the metric
$\hat{g}_{\mu\nu}$, the corresponding actions in the matter sector are also
invariant under such transformations. Provided that $\bar{M}\ll M$, one has
$\hat{g}_{\mu\nu}\simeq g_{\mu\nu}$ and hence the matter fields are
approximately coupled to the metric tensor $g_{\mu\nu}$.

The flat space expansion of (17) can be performed by substituting
$g_{\mu\nu}=e^{2\phi}\bar{g}_{\mu\nu}=\eta_{\mu\nu}(1-\varphi^{2}/M^{2})^{-1}$,
where $\eta_{\mu\nu}$ is the Minkowski metric. Introducing a canonically
normalized field $\pi\equiv\bar{M}\phi$, the action (17) contains the
derivative terms

${\cal S}\supset\int{\rm
d}^{4}x\left[-6e^{-2\pi/\bar{M}}(\partial\pi)^{2}+\alpha\left\\{\frac{4(\partial\pi)^{2}\square\pi}{\bar{M}^{3}}+\frac{2(\partial\pi)^{4}}{\bar{M}^{4}}\right\\}\right]\,,$
(18)

where we used the notation
$(\partial\pi)^{2}=\partial_{\mu}\pi\partial^{\mu}\pi$ and the partial
derivative $\partial_{\mu}=\partial/\partial x^{\mu}$. Nonlinear derivative
terms are suppressed below the scale $\bar{M}$ in comparison to the canonical
kinetic term, while the theory is strongly coupled above the scale $\bar{M}$.
Hence the theory given by the action (17) is an EFT with trace anomaly
corrections valid below the scale $\bar{M}~{}(\ll M)$. Unlike the Riegert’s
theory, the metric $\bar{g}_{\mu\nu}=e^{-2\phi}g_{\mu\nu}$ is not subject to a
fixed determinant constraint. Then, the Gabadadze’s theory has two tensor
propagating degrees of freedom besides one scalar mode $\phi$.

Varying the action (17) with respect to $\bar{g}_{\mu\nu}$, it follows that

$\left(M^{2}e^{2\phi}-\bar{M}^{2}\right)\bar{G}_{\mu\nu}=-M^{2}e^{2\phi}\left[2\bar{\nabla}_{\mu}\phi\bar{\nabla}_{\nu}\phi-2\bar{\nabla}_{\mu}\bar{\nabla}_{\nu}\phi+2\bar{g}_{\mu\nu}\left(\bar{\square}\phi-\bar{X}\right)\right]-\alpha\bar{{\cal
H}}_{\mu\nu}-2\beta\phi\left(2\bar{\nabla}^{\rho}\bar{\nabla}^{\sigma}+\bar{R}^{\rho\sigma}\right)\bar{C}_{\mu\rho\nu\sigma}\,,$
(19)

where

$\displaystyle\bar{{\cal H}}_{\mu\nu}$ $\displaystyle\equiv$
$\displaystyle-4\bar{X}\bar{G}_{\mu\nu}+4\left(\bar{\nabla}_{\alpha}\phi\bar{\nabla}_{\mu}\phi-\bar{\nabla}_{\alpha}\bar{\nabla}_{\mu}\phi\right)\left(\bar{\nabla}^{\alpha}\phi\bar{\nabla}_{\nu}\phi-\bar{\nabla}^{\alpha}\bar{\nabla}_{\nu}\phi\right)+4\left(\bar{\nabla}_{\mu}\phi\bar{\nabla}_{\nu}\phi-\bar{\nabla}_{\nu}\bar{\nabla}_{\mu}\phi\right)\bar{\square}\phi$
(20)
$\displaystyle+\bar{g}_{\mu\nu}\left[2(\bar{\square}\phi)^{2}-4\bar{X}^{2}+2\bar{\nabla}_{\beta}\bar{\nabla}_{\alpha}\phi\left(2\bar{\nabla}^{\alpha}\phi\bar{\nabla}^{\beta}\phi-\bar{\nabla}^{\beta}\bar{\nabla}^{\alpha}\phi\right)\right]+4\bar{{\cal
P}}_{\mu\alpha\nu\beta}\left(\bar{\nabla}^{\alpha}\phi\bar{\nabla}^{\beta}\phi-\bar{\nabla}^{\beta}\bar{\nabla}^{\alpha}\phi\right)\,,$

with

$\bar{{\cal
P}}_{\mu\alpha\nu\beta}\equiv\bar{R}_{\mu\alpha\nu\beta}+\bar{g}_{\mu\beta}\bar{R}_{\alpha\nu}-\bar{g}_{\mu\nu}\bar{R}_{\alpha\beta}+\bar{g}_{\alpha\nu}\bar{R}_{\mu\beta}-\bar{g}_{\alpha\beta}\bar{R}_{\mu\nu}+\frac{1}{2}\left(\bar{g}_{\mu\nu}\bar{g}_{\alpha\beta}-\bar{g}_{\mu\beta}\bar{g}_{\alpha\nu}\right)\bar{R}\,.$
(21)

We take the trace of Eq. (19) by exerting $\bar{g}^{\mu\nu}$ and exploit the
relation $12\bar{X}-6\bar{\square}\phi=e^{2\phi}R-\bar{R}$. On using the
property $\bar{g}^{\mu\nu}\bar{\cal
P}_{\mu\alpha\nu\beta}=-\bar{G}_{\alpha\beta}$, it follows that
$\bar{g}^{\mu\nu}\bar{{\cal H}}_{\mu\nu}=(e^{4\phi}{\cal G}-\bar{\cal G})/2$.
We also note that the divergence of the Weyl tensor vanishes, such that
$\bar{g}^{\mu\nu}\bar{C}_{\mu\rho\nu\sigma}=0$. Then, the trace of Eq. (19) is
expressed as

$2\bar{M}^{2}\bar{R}-\alpha\bar{\cal G}=e^{4\phi}\left(2M^{2}R-\alpha{\cal
G}\right)\,.$ (22)

Varying the action ${\cal S}_{R\bar{R}}=\int{\rm
d}^{4}x\sqrt{-\bar{g}}\left[M^{2}e^{2\phi}(\bar{R}-12\bar{X}\right)-\bar{M}^{2}\bar{R}]$
with respect to $\phi$, we obtain

$\frac{\delta{\cal
S}_{R\bar{R}}}{\delta\phi}=2\sqrt{-\bar{g}}\,M^{2}e^{2\phi}\left(\bar{R}+12\bar{X}-6\bar{\square}\phi\right)=2\sqrt{-g}M^{2}R\,.$
(23)

From Eq. (10), the variation of the trace anomaly action is given by
$\delta{\cal S}_{A}/\delta\phi=\sqrt{-g}\left(-\alpha{\cal G}+\beta
W^{2}\right)$. Then, varying the action ${\cal S}={\cal S}_{R\bar{R}}+{\cal
S}_{A}$ with respect to $\phi$ leads to

$2M^{2}R-\alpha{\cal G}+\beta W^{2}=0\,,$ (24)

which is the scalar field equation in the frame with the metric $g_{\mu\nu}$.
From Eqs. (22) and (24), we obtain

$2\bar{M}^{2}\bar{R}-\alpha\bar{\cal G}+\beta\bar{W}^{2}=0\,.$ (25)

The spacetime geometry associated with the metric $\bar{g}_{\mu\nu}$ is
modified by the GB and Weyl terms through the trace anomaly Eq. (25). While
Eq. (24) contains the mass $M$ of order the Planck scale $M_{\rm Pl}$, it is
replaced by the new mass scale $\bar{M}$ in Eq. (25).

The Weyl curvature term in Eq. (17) gives rise to the field equations of
motion higher than second order. If the Weyl trace anomaly is absent, i.e.,

$\beta=0\,,$ (26)

then the action (17) belongs to a subclass of Horndeski theories [23] given by

$\displaystyle{\cal S}$ $\displaystyle=$ $\displaystyle\int{\rm
d}^{4}x\sqrt{-\bar{g}}\,\biggl{[}\bar{G}_{2}-\bar{G}_{3}\bar{\square}\phi+\bar{G}_{4}R+\bar{G}_{4,\bar{X}}\left\\{(\bar{\square}\phi)^{2}-(\bar{\nabla}_{\mu}\bar{\nabla}_{\nu}\phi)(\bar{\nabla}^{\mu}\bar{\nabla}^{\nu}\phi)\right\\}$
(27)
$\displaystyle+\bar{G}_{5}\bar{G}^{\mu\nu}\bar{\nabla}_{\mu}\bar{\nabla}_{\nu}\phi-\frac{1}{6}\bar{G}_{5,\bar{X}}\left\\{(\bar{\square}\phi)^{3}-3(\bar{\square}\phi)\,(\bar{\nabla}_{\mu}\bar{\nabla}_{\nu}\phi)(\bar{\nabla}^{\mu}\bar{\nabla}^{\nu}\phi)+2(\bar{\nabla}^{\mu}\bar{\nabla}_{\alpha}\phi)(\bar{\nabla}^{\alpha}\bar{\nabla}_{\beta}\phi)(\bar{\nabla}^{\beta}\bar{\nabla}_{\mu}\phi)\right\\}\biggr{]}\,,$

where $\bar{G}_{j,\bar{X}}\equiv\partial\bar{G}_{j}/\partial\bar{X}$ (with
$j=4,5$), and

$\displaystyle\bar{G}_{2}=-12M^{2}\bar{X}e^{2\phi}+8\alpha\bar{X}^{2}\,,\qquad\bar{G}_{3}=8\alpha\bar{X}\,,$
$\displaystyle\bar{G}_{4}=M^{2}e^{2\phi}-\bar{M}^{2}+4\alpha\bar{X}\,,\qquad\bar{G}_{5}=4\alpha\ln|\bar{X}|\,.$
(28)

In particular, the linearly coupled GB Lagrangian $-\alpha\phi\bar{\cal G}$
can be accommodated by the quintic Horndeski function
$\bar{G}_{5}=4\alpha\ln|\bar{X}|$ [25, 38]. In this case, the field equations
of motion are kept up to second order in both $\bar{g}_{\mu\nu}$ and $\phi$.

##  III Hairy black hole

We study the existence of hairy BH solutions for the action (17) with
$\beta=0$, i.e.,

${\cal S}=\int{\rm
d}^{4}x\sqrt{-\bar{g}}\left[\left(M^{2}e^{2\phi}-\bar{M}^{2}\right)\bar{R}-12M^{2}e^{2\phi}\bar{X}-\alpha\left(\phi\bar{\cal
G}-4\bar{G}^{\mu\nu}\bar{\nabla}_{\mu}\phi\bar{\nabla}_{\nu}\phi+8\bar{X}\bar{\square}\phi-8\bar{X}^{2}\right)\right]\,,$
(29)

which is equivalent to the Horndeski action (27) with the coupling functions
(28). 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
tensor $\bar{g}_{\mu\nu}$, the static and spherically symmetric spacetime is
given by the line element

${\rm d}\bar{s}^{2}=-\bar{f}(\bar{r}){\rm d}t^{2}+\bar{h}^{-1}(\bar{r}){\rm
d}\bar{r}^{2}+\bar{r}^{2}\left({\rm d}\theta^{2}+\sin^{2}\theta\,{\rm
d}\varphi^{2}\right)\,,$ (30)

where the metric components $\bar{f}$ and $\bar{h}$ depend on the radial
coordinate $\bar{r}$. The scalar field is assumed to be a function of
$\bar{r}$ alone, i.e., $\phi=\phi(\bar{r})$.

We also write the line element associated with the metric
$g_{\mu\nu}=e^{2\phi}\bar{g}_{\mu\nu}$ as

${\rm d}s^{2}=-f(r){\rm d}t^{2}+h^{-1}(r){\rm d}r^{2}+r^{2}\left({\rm
d}\theta^{2}+\sin^{2}\theta\,{\rm d}\varphi^{2}\right)\,,$ (31)

which is related to (30) according to ${\rm d}\bar{s}^{2}=e^{-2\phi}{\rm
d}s^{2}$. There are the following relations

$f=\bar{f}e^{2\phi}\,,\qquad
h=\bar{h}\left[1+\bar{r}\phi^{\prime}(\bar{r})\right]^{2}\,,\qquad
r=\bar{r}e^{\phi}\,,$ (32)

where a prime represents the derivative with respect to $\bar{r}$. In the
following, we will obtain the solutions to $\bar{f}$, $\bar{h}$, and $\phi$ as
functions of $\bar{r}$. Using the correspondence (32), we will also derive the
functions $f(r)$ and $h(r)$ in the line element (31).

The differential equations for the metric components $\bar{f}(\bar{r})$ and
$\bar{h}(\bar{r})$ are given by

$\displaystyle\frac{\bar{f}^{\prime}}{\bar{f}}$ $\displaystyle=$
$\displaystyle\frac{\bar{M}^{2}(\bar{h}-1)-M^{2}e^{2\phi}[\bar{h}(3\bar{r}\phi^{\prime}+1)(\bar{r}\phi^{\prime}+1)-1]-\alpha\bar{h}\phi^{\prime
2}[\bar{h}\\{\bar{r}\phi^{\prime}(3\bar{r}\phi^{\prime}+8)+6\\}-2]}{\bar{h}[M^{2}\bar{r}(\bar{r}\phi^{\prime}+1)e^{2\phi}+2\alpha\phi^{\prime}(\bar{h}\\{3+\bar{r}\phi^{\prime}(\bar{r}\phi^{\prime}+3)\\}-1)-\bar{M}^{2}\bar{r}]}\,,$
(33)
$\displaystyle\frac{\bar{h}^{\prime}}{\bar{h}}-\frac{\bar{f}^{\prime}}{\bar{f}}$
$\displaystyle=$ $\displaystyle\frac{2(\phi^{\prime
2}-\phi^{\prime\prime})[M^{2}\bar{r}^{2}e^{2\phi}+2\alpha(\bar{r}^{2}\bar{h}\phi^{\prime
2}+2\bar{r}\bar{h}\phi^{\prime}+\bar{h}-1)]}{\bar{r}M^{2}e^{2\phi}(\bar{r}\phi^{\prime}+1)-\bar{r}\bar{M}^{2}+2\alpha\phi^{\prime}(\bar{r}^{2}\bar{h}\phi^{\prime
2}+3\bar{r}\bar{h}\phi^{\prime}+3\bar{h}-1)}\,.$ (34)

From Eq. (25), we obtain

$\displaystyle\left[2\bar{r}^{2}\bar{f}\bar{h}\bar{f}^{\prime\prime}-\bar{r}\bar{f}^{\prime}\left(\bar{r}\bar{h}\bar{f}^{\prime}-\bar{r}\bar{f}\bar{h}^{\prime}-4\bar{f}\bar{h}\right)+4\bar{f}^{2}(\bar{h}-1)+4\bar{r}\bar{f}^{2}\bar{h}^{\prime}\right]\bar{M}^{2}$
$\displaystyle+2\alpha\left[2\bar{f}\bar{h}(\bar{h}-1)\bar{f}^{\prime\prime}-\bar{f}^{\prime}(\bar{h}^{2}\bar{f}^{\prime}-3\bar{f}\bar{h}\bar{h}^{\prime}-\bar{f}^{\prime}\bar{h}+\bar{f}\bar{h}^{\prime})\right]=0\,.$
(35)

In the absence of the GB trace anomaly, i.e., $\alpha=0$, there is the
following exact solution to Eqs. (33)-(35):

$\bar{f}(\bar{r})=c_{0}\left(1-\frac{m_{0}}{\bar{r}}\right)^{2}\,,\qquad\bar{h}(\bar{r})=\left(1-\frac{m_{0}}{\bar{r}}\right)^{2}\,,\qquad
e^{\phi(\bar{r})}=\pm\frac{\bar{M}}{M}\frac{m_{0}}{\bar{r}-m_{0}}\,,$ (36)

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 [39] and by Bekenstein [40] (BBMB). However, it is
unstable against monopole perturbations [41, 42, 43] as well as perturbations
for general multipoles $l$ [35]. This instability is also related to the
divergence of $\phi^{\prime}(\bar{r})=1/(m_{0}-\bar{r})$ on the horizon
located at $\bar{r}=m_{0}$ [33, 32]. In Appendix A, we will show that the
solution (36) is unstable against linear perturbations for the radius
$\bar{r}>2m_{0}$.

For $\alpha=0$, there is also the GR branch given by

$\bar{f}(\bar{r})=\bar{h}(\bar{r})=1-\frac{\bar{r}_{h}}{\bar{r}}\,,\qquad\phi(\bar{r})=\phi_{0}={\rm
constant}\,,$ (37)

where $\bar{r}_{h}$ is the horizon radius. This solution is stable against
linear perturbations. Thus, instead of (36), the Schwarzschild branch without
a scalar hair (37) should be selected as a linearly stable BH.

The no-hair property of BHs for $\alpha=0$ changes in the presence of the GB
trace anomaly. Let us derive the solutions to $\bar{f}$, $\bar{h}$, and $\phi$
under the expansion of a small coupling constant $\alpha$. In the limit that
$\alpha\to 0$, the solutions need to recover the Schwarzschild metric without
the scalar hair, i.e., Eq. (37). Outside the horizon characterized by the
radius $\bar{r}_{h}$, we search for solutions in the forms

$f(\bar{r})=\left(1-\frac{\bar{r}_{h}}{\bar{r}}\right)\left[1+\sum_{i=1}\bar{f}_{i}(\bar{r})\alpha^{i}\right]\,,\qquad
h(\bar{r})=\left(1-\frac{\bar{r}_{h}}{\bar{r}}\right)\left[1+\sum_{i=1}\bar{h}_{i}(\bar{r})\alpha^{i}\right]\,,\qquad\phi(\bar{r})=\phi_{0}+\sum_{i=1}\phi_{i}(\bar{r})\alpha^{i}\,,$
(38)

where $\bar{f}_{i}$, $\bar{h}_{i}$, and $\phi_{i}$ are functions of $\bar{r}$.
Substituting Eq. (38) into Eqs. (33)-(35), we obtain the differential
equations for $\bar{f}_{i}$, $\bar{h}_{i}$, and $\phi_{i}$ at each order in
$\alpha^{i}$. We can also derive a second-order differential equation for
$\phi(\bar{r})$ by differentiating Eq. (33) with respect to $\bar{r}$ and
eliminate the derivatives $\bar{f}^{\prime}$, $\bar{h}^{\prime}$, and
$\bar{f}^{\prime\prime}$ by using Eqs. (33), (34), and (35). At first order in
the expansion of $\phi(\bar{r})$, we have

$\phi_{1}^{\prime\prime}(\bar{r})+\frac{2\bar{r}-\bar{r}_{h}}{\bar{r}(\bar{r}-\bar{r}_{h})}\phi_{1}^{\prime}(\bar{r})-\frac{\bar{r}_{h}^{2}(M^{2}-\bar{M}^{2}e^{-2\phi_{0}})}{M^{2}\bar{M}^{2}\bar{r}^{5}(\bar{r}-\bar{r}_{h})}=0\,.$
(39)

The integrated solution to $\phi_{1}(\bar{r})$ contains two integration
constants. They are determined by imposing the regular boundary conditions
$\phi_{1}^{\prime}(\bar{r}_{h})={\rm constant}$ on the horizon and
$\phi_{1}(\infty)\to 0$ at spatial infinity. The finiteness of
$\phi^{\prime}(\bar{r})$ at $\bar{r}=\bar{r}_{h}$ is not only required for the
validity of the expansion (38) but also for avoiding the instability of linear
perturbations [33, 32]. Then, the resulting integrated solution to Eq. (39)
yields

$\phi_{1}(\bar{r})=-\frac{(6\bar{r}^{2}+3\bar{r}_{h}\bar{r}+2\bar{r}_{h}^{2})(M^{2}-\bar{M}^{2}e^{-2\phi_{0}})}{18M^{2}\bar{M}^{2}\bar{r}_{h}\bar{r}^{3}}\,.$
(40)

Substituting this solution into Eq. (34) and using Eq. (33), we obtain the
differential equation for $\bar{h}_{1}(\bar{r})$ as

$\bar{h}_{1}^{\prime}(\bar{r})+\frac{\bar{h}_{1}(\bar{r})}{\bar{r}-\bar{r}_{h}}-\frac{\bar{r}^{2}+\bar{r}_{h}\bar{r}-5\bar{r}_{h}^{2}}{3\bar{M^{2}}\bar{r}^{4}(\bar{r}-\bar{r}_{h})}=0\,.$
(41)

The integrated solution respecting the finiteness of $\bar{h}_{1}(\bar{r})$ on
the horizon is given by

$\bar{h}_{1}(\bar{r})=-\frac{(\bar{r}+2\bar{r}_{h})(\bar{r}+5\bar{r}_{h})}{18\bar{M}^{2}\bar{r}_{h}\bar{r}^{3}}\,,$
(42)

which approaches 0 at spatial infinity.

From Eq. (33), the differential equation for $\bar{f}_{1}(\bar{r})$ is

$\bar{f}_{1}^{\prime}(\bar{r})+\frac{23\bar{r}^{2}+22\bar{r}_{h}\bar{r}+18\bar{r}_{h}^{2}}{18\bar{M}^{2}\bar{r}_{h}\bar{r}^{4}}=0\,.$
(43)

The integrated solution satisfying the boundary condition
$\bar{f}_{1}(\infty)=0$ is

$\bar{f}_{1}(\bar{r})=\frac{23\bar{r}^{2}+11\bar{r}_{h}\bar{r}+6\bar{r}_{h}^{2}}{18\bar{M}^{2}\bar{r}_{h}\bar{r}^{3}}\,,$
(44)

which decreases as $\bar{f}_{1}(\bar{r})\propto\bar{r}^{-1}$ at large
distances.

Similarly, we derive the functions $\bar{f}_{i}(\bar{r})$,
$\bar{h}_{i}(\bar{r})$, and $\phi_{i}(\bar{r})$ at each order in $\alpha^{i}$
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 functions for $i\geq 3$, we only write the solutions up to the
$i=2$ order as

$\displaystyle\bar{f}(\bar{r})$ $\displaystyle=$
$\displaystyle\left(1-\frac{\bar{r}_{h}}{\bar{r}}\right)\biggl{[}1+\frac{23\bar{r}^{2}+11\bar{r}_{h}\bar{r}+6\bar{r}_{h}^{2}}{18\bar{M}^{2}\bar{r}_{h}\bar{r}^{3}}\alpha-\frac{1}{32400M^{2}\bar{M}^{4}\bar{r}_{h}^{3}\bar{r}^{6}}\\{\bar{M}^{2}(38731\bar{r}^{5}+26371\bar{r}_{h}\bar{r}^{4}+22721\bar{r}_{h}^{2}\bar{r}^{3}$
(45)
$\displaystyle-769\bar{r}_{h}^{3}\bar{r}^{2}-5572\bar{r}_{h}^{4}\bar{r}-9300\bar{r}_{h}^{5})e^{-2\phi_{0}}-M^{2}(6979\bar{r}^{5}+26539\bar{r}_{h}\bar{r}^{4}+28489\bar{r}_{h}^{2}\bar{r}^{3}+33379\bar{r}_{h}^{3}\bar{r}^{2}$
$\displaystyle+13492\bar{r}_{h}^{4}\bar{r}+4500\bar{r}_{h}^{5})\\}\alpha^{2}+{\cal
O}(\alpha^{3})\biggr{]}\,,$ $\displaystyle\bar{h}(\bar{r})$ $\displaystyle=$
$\displaystyle\left(1-\frac{\bar{r}_{h}}{\bar{r}}\right)\biggl{[}1-\frac{(\bar{r}+2\bar{r}_{h})(\bar{r}+5\bar{r}_{h})}{18\bar{M}^{2}\bar{r}_{h}\bar{r}^{3}}\alpha-\frac{1}{32400M^{2}\bar{M}^{4}\bar{r}_{h}^{3}\bar{r}^{6}}\\{\bar{M}^{2}(21211\bar{r}^{5}+13231\bar{r}_{h}\bar{r}^{4}+11041\bar{r}_{h}^{2}\bar{r}^{3}$
(46)
$\displaystyle-33019\bar{r}_{h}^{3}\bar{r}^{2}-36964\bar{r}_{h}^{4}\bar{r}-40300\bar{r}_{h}^{5})e^{-2\phi_{0}}+M^{2}(\bar{r}-\bar{r}_{h})$
$\displaystyle\times(1901\bar{r}^{4}-4178\bar{r}_{h}\bar{r}^{3}-15747\bar{r}_{h}^{2}\bar{r}^{2}-4976\bar{r}_{h}^{3}\bar{r}-1300\bar{r}_{h}^{4})\\}\alpha^{2}+{\cal
O}(\alpha^{3})\biggr{]}\,,$ $\displaystyle\phi(\bar{r})$ $\displaystyle=$
$\displaystyle\phi_{0}-\frac{(6\bar{r}^{2}+3\bar{r}_{h}\bar{r}+2\bar{r}_{h}^{2})(M^{2}-\bar{M}^{2}e^{-2\phi_{0}})}{18M^{2}\bar{M}^{2}\bar{r}_{h}\bar{r}^{3}}\alpha-\frac{1}{32400M^{4}\bar{M}^{4}\bar{r}_{h}^{3}\bar{r}^{6}}[3M^{4}(740\bar{r}^{5}-130\bar{r}_{h}\bar{r}^{4}-20\bar{r}_{h}^{2}\bar{r}^{3}$
(47)
$\displaystyle+1285\bar{r}_{h}^{3}\bar{r}^{2}+688\bar{r}_{h}^{4}\bar{r}+300\bar{r}_{h}^{5})-200M^{2}\bar{M}^{2}e^{-2\phi_{0}}(33\bar{r}^{5}+18\bar{r}_{h}\bar{r}^{4}+16\bar{r}_{h}^{2}\bar{r}^{3}+33\bar{r}_{h}^{3}\bar{r}^{2}+15\bar{r}_{h}^{4}\bar{r}+6\bar{r}_{h}^{5})$
$\displaystyle+\bar{M}^{4}e^{-4\phi_{0}}(4380\bar{r}^{5}+3990\bar{r}_{h}\bar{r}^{4}+3260\bar{r}_{h}^{2}\bar{r}^{3}+2745\bar{r}_{h}^{3}\bar{r}^{2}+936\bar{r}_{h}^{4}\bar{r}+300\bar{r}_{h}^{5})]\alpha^{2}+{\cal
O}(\alpha^{3})\,.$

Outside the horizon, the leading-order trace anomaly corrections to
$\bar{f}(\bar{r})$ and $\bar{h}(\bar{r})$ are largest around
$\bar{r}=\bar{r}_{h}$, which are of order
$\alpha/(\bar{M}^{2}\bar{r}_{h}^{2})$. For the validity of the expansion (38),
we then require that

$\hat{\alpha}\equiv\frac{\alpha}{\bar{M}^{2}\bar{r}_{h}^{2}}\ll 1\,.$ (48)

Provided that $\bar{M}\ll M$, the second-order corrections to Eqs. (45) and
(46) are at most of order $\hat{\alpha}^{2}$. From Eq. (47), the first- and
second-order corrections to $\phi(\bar{r})$ are at most of orders
$\hat{\alpha}$ and $\hat{\alpha}^{2}$, respectively. In the limit that
$\bar{M}\to 0$, the condition (48) is violated and hence the expanded
solutions (45)-(47) are invalid in GR supplemented by the trace anomaly action
${\cal S}_{A}$. For the mass scale $\bar{M}$ larger than $1/\bar{r}_{h}$, the
inequality (48) is satisfied for $\alpha\ll 1$. This means that $\bar{M}$ can
be chosen down to the order $1/\bar{r}_{h}$. For $\bar{M}\simeq
1/\bar{r}_{h}$, the EFT is valid for the length scale larger than
$\bar{r}_{h}$.

Unlike the BBMB solution, the field derivative $\phi^{\prime}(\bar{r})$ is
finite on the horizon. At large distances, the scalar field solution up to the
order of $\alpha$ is given by

$\phi(\bar{r})=\phi_{0}-\frac{q_{s}}{\bar{r}}\qquad{\rm
for}\qquad\bar{r}\gg\bar{r}_{s}\,,$ (49)

where $q_{s}$ is regarded as a scalar charge given by

$q_{s}=\frac{M^{2}-\bar{M}^{2}e^{-2\phi_{0}}}{3M^{2}\bar{M}^{2}\bar{r}_{h}}\alpha\,.$
(50)

The GB trace anomaly gives rise to a hairy BH solution possessing the scalar
charge $q_{s}$.

Figure 1:  We plot $\bar{h}$,
$\bar{\Delta}=\bar{r}_{h}(\bar{h}^{\prime}/\bar{h}-\bar{f}^{\prime}/\bar{f})$,
and $\phi^{\prime}$ versus $\bar{r}/\bar{r}_{h}$ for $\hat{\alpha}=0.02$ and
$\bar{M}=10^{-38}M$ (black lines). The numerical integration is performed
outward by using Eqs. (45)-(47) as boundary conditions at
$\bar{r}/\bar{r}_{h}=1.0001$ with $\phi_{0}=0$. The colored dashes lines are
those derived by using the solutions (45)-(47) expanded up to fourth order in
$\alpha^{i}$, which agree well with the numerical results.

From Eqs. (45) and (46) we find that $\bar{f}(\bar{r})$ and $\bar{h}(\bar{r})$
are not identical to each other for $\alpha\neq 0$. This is different from an
exact BH solution obtained by assuming $\bar{f}(\bar{r})=h(\bar{r})$ in a
similar conformally invariant theory [16]. In Appendix B, we will derive such
a solution by imposing the condition $\bar{f}(\bar{r})=h(\bar{r})$ in Eq.
(35). This condition demands that the right hand side of Eq. (34) 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. (33). In other words, the solutions satisfying all the
equations of motion and regular boundary conditions on the horizon are of the
forms (45)-(47). They are also different from the hairy BH solution present
for a canonical scalar field $\phi$ linearly coupled to the GB term [44, 45]
(see also Refs. [46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 33, 32] for related
works). In the latter theory the first order metric components $\bar{f}_{1}$
and $\bar{h}_{1}$ vanish [45, 33], but this is not the case for our theory due
to the existence of additional nonlinear derivative terms in Eq. (29).

In Fig. 1, we show numerically integrated solutions to $\bar{h}$,
$\bar{\Delta}\equiv\bar{r}_{h}(\bar{h}^{\prime}/\bar{h}-\bar{f}^{\prime}/\bar{f})$,
and $\phi^{\prime}$ for $\hat{\alpha}=0.02$ and $\bar{M}=10^{-38}M$. This
choice of $\bar{M}$ corresponds to the length scale of order $1/\bar{M}\simeq
10$ km. Provided that $\bar{M}\ll M$, the leading-order trace anomaly
corrections to Eqs. (45)-(47) are determined by the coupling
$\hat{\alpha}=\alpha/(\bar{M}^{2}\bar{r}_{h}^{2})$. For $\bar{M}\simeq
1/\bar{r}_{h}$, the corrections to $\bar{f}(\bar{r})$, $\bar{h}(\bar{r})$, and
$\phi(\bar{r})$ are at most of order $\alpha$. In Fig. 1, we observe that the
numerical results are in good agreement with the analytic solutions (45)-(47)
expanded up to fourth order in $\alpha^{i}$. The field derivative
$\phi^{\prime}$, which is finite on the horizon, decreases as
$\phi^{\prime}(\bar{r})\propto\bar{r}^{-2}$ in the regime
$\bar{r}\gg\bar{r}_{h}$. Outward from the horizon, $\bar{h}$ continues to
increase toward the asymptotic value 1. The quantity $\bar{\Delta}$, which
characterizes the difference between $\bar{h}$ and $\bar{f}$, is largest
around $\bar{r}=\bar{r}_{h}$. From Eqs. (45) and (46), we obtain the
asymptotic behavior $\bar{\Delta}\simeq 4\alpha/(3\bar{M}^{2}\bar{r}^{2})$ in
the regime $\bar{r}\gg\bar{r}_{h}$, whose dependence can be also confirmed in
Fig. 1.

The metric components $f$ and $h$ in the frame with the metric tensor
$g_{\mu\nu}$ can be obtained as functions of $r$ by using the correspondence
(32). Taking $\phi_{0}=0$, the resulting forms of $f(r)$ and $h(r)$ correspond
to those derived by the replacements $\bar{r}\to r$, $\bar{r}_{h}\to r_{h}$,
$\bar{M}\to M$, and $M\to\bar{M}$ in Eqs. (45) and (46), i.e.,

$\displaystyle f(r)$ $\displaystyle=$
$\displaystyle\left(1-\frac{r_{h}}{r}\right)\biggl{[}1+\frac{23r^{2}+11r_{h}r+6r_{h}^{2}}{18M^{2}r_{h}r^{3}}\alpha-\frac{1}{32400\bar{M}^{2}M^{4}r_{h}^{3}r^{6}}\\{M^{2}(38731r^{5}+26371r_{h}r^{4}+22721r_{h}^{2}r^{3}$
(51)
$\displaystyle-769r_{h}^{3}r^{2}-5572r_{h}^{4}r-9300r_{h}^{5})-\bar{M}^{2}(6979r^{5}+26539r_{h}r^{4}+28489r_{h}^{2}r^{3}+33379r_{h}^{3}r^{2}$
$\displaystyle+13492r_{h}^{4}r+4500r_{h}^{5})\\}\alpha^{2}+{\cal
O}(\alpha^{3})\biggr{]}\,,$ $\displaystyle h(r)$ $\displaystyle=$
$\displaystyle\left(1-\frac{r_{h}}{r}\right)\biggl{[}1-\frac{(r+2r_{h})(r+5r_{h})}{18M^{2}r_{h}r^{3}}\alpha-\frac{1}{32400\bar{M}^{2}M^{4}r_{h}^{3}r^{6}}\\{M^{2}(21211r^{5}+13231r_{h}r^{4}+11041r_{h}^{2}r^{3}$
(52)
$\displaystyle-33019r_{h}^{3}r^{2}-36964r_{h}^{4}r-40300r_{h}^{5})+\bar{M}^{2}(r-r_{h})$
$\displaystyle\times(1901r^{4}-4178r_{h}r^{3}-15747r_{h}^{2}r^{2}-4976r_{h}^{3}r-1300r_{h}^{4})\\}\alpha^{2}+{\cal
O}(\alpha^{3})\biggr{]}\,.$

The leading-order trace anomaly corrections to $f(r)$ and $h(r)$ are subject
to strong suppression by the appearance of the mass $M=M_{\rm Pl}/\sqrt{2}$ in
their denominators. As we already mentioned, the matter fields feel the
gravitational force through the metric tensor $\hat{g}_{\mu\nu}\simeq
g_{\mu\nu}$. Let us consider a test particle with mass $m_{g}$ on the metric
background $g_{\mu\nu}$. Defining the gravitational potential $\Psi(r)$ as
$f(r)=1+2\Psi(r)$ in the regime away from the horizon, the force exerting on
the particle is given by

$F(r)=-m_{g}\frac{{\rm d}\Psi(r)}{{\rm d}r}=-\frac{m_{g}}{2}\frac{{\rm
d}f(r)}{{\rm
d}r}=-\frac{m_{g}r_{h}}{2r^{2}}\left[1-\left(\frac{23}{18}-\frac{4r_{h}}{3r}-\frac{5r_{h}^{2}}{6r^{2}}-\frac{4r_{h}^{3}}{3r^{3}}\right)\frac{\alpha}{M^{2}r_{h}^{2}}+{\cal
O}(\alpha^{2})\right]\,,$ (53)

where we used the expanded solution (51), and $r$ is related to $\bar{r}$
according to

$r=\bar{r}\left[1-\left(1-\frac{\bar{M}^{2}}{M^{2}}\right)\left(\frac{\bar{r}_{h}}{3\bar{r}}+\frac{\bar{r}_{h}^{2}}{6\bar{r}^{2}}+\frac{\bar{r}_{h}^{3}}{9\bar{r}^{3}}\right)\hat{\alpha}+{\cal
O}(\hat{\alpha}^{2})\right]\,.$ (54)

In terms of the distance $\bar{r}$ and horizon radius $\bar{r}_{h}$, the force
(53) can be expressed as

$\displaystyle F(r)$ $\displaystyle=$
$\displaystyle-\frac{m_{g}e^{\phi}}{2}\frac{\bar{f}^{\prime}+2\bar{f}\phi^{\prime}}{1+\bar{r}\phi^{\prime}}$
(55) $\displaystyle=$
$\displaystyle-\frac{m_{g}\bar{r}_{h}}{2\bar{r}^{2}}\left[1-\left\\{\frac{11}{18}-\frac{2\bar{r}_{h}}{3\bar{r}}-\frac{\bar{r}_{h}^{2}}{3\bar{r}^{2}}-\frac{2\bar{r}_{h}^{3}}{9\bar{r}^{3}}+\frac{\bar{M}^{2}}{M^{2}}\left(\frac{2}{3}-\frac{2\bar{r}_{h}}{3\bar{r}}-\frac{\bar{r}_{h}^{2}}{2\bar{r}^{2}}-\frac{10\bar{r}_{h}^{3}}{9\bar{r}^{3}}\right)\right\\}\hat{\alpha}+{\cal
O}(\hat{\alpha}^{2})\right]\,.$

The $\alpha$-dependent terms in Eq. (53) 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
$(\bar{M}/M)^{2}\hat{\alpha}$, so it is even more suppressed than
$\hat{\alpha}$ due to the small ratio $(\bar{M}/M)^{2}\ll 1$. In Eq. (55), on
the other hand, the leading-order trace anomaly correction relative to
$-m_{g}\bar{r}_{h}/(2\bar{r}^{2})$ is of order $\hat{\alpha}$. This difference
arises from the fact that the relative difference between $r$ and $\bar{r}$ is
at most of order $\hat{\alpha}$, see Eq. (54). When we express $r$ in terms of
$\bar{r}$, the order $\hat{\alpha}$ correction, which is much larger than
$(\bar{M}/M)^{2}\hat{\alpha}$, appears in Eq. (55). Since the test particle
feels gravity associated with the metric $g_{\mu\nu}$ and distance $r$, we
need to interpret that the fifth force is suppressed by the factor
$(\bar{M}/M)^{2}\hat{\alpha}$ 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
$(\bar{M}/M)^{2}$ is significantly smaller than 1 for $\bar{M}$ of order
$1/r_{h}$. This is not necessarily the case for microscopic BHs with smaller
horizon sizes, in which case the ratio $(\bar{M}/M)^{2}$ can be larger.

While the fifth force exerting on the test particle can be suppressed for
astrophysical BHs, the scalar field (47) receives the trace anomaly correction
at most of order $\hat{\alpha}$ (even if we express $\phi$ with respect to
$r$). During the inspiral phase of a BH binary system, the scalar radiation
arising from the perturbation of $\phi$ 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 (45)-(47) obtained under the expansion
of a small coupling constant $\alpha$ in the frame with the metric
$\bar{g}_{\mu\nu}$. The perturbations on the static and spherically symmetric
background can be decomposed into odd- and even-parity modes [56, 57]. 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 $\rho$ and pressure $P$ and obtained all the linear
stability conditions applicable to NSs as well (see also Ref. [58]). We will
exploit those results in the following discussion.

In Horndeski theories given by the action (29), the odd-parity sector has a
gravitational wave mode $\chi$ arising from the metric perturbation. For this
perturbation $\chi$, there are neither ghost nor Laplacian instabilities under
the following conditions

$\displaystyle\bar{{\cal G}}$ $\displaystyle\equiv$ $\displaystyle
2\bar{G}_{4}+2\bar{h}\phi^{\prime 2}\bar{G}_{4,\bar{X}}-\bar{h}\phi^{\prime
2}\left(\bar{G}_{5,\phi}+{\frac{\bar{f}^{\prime}\bar{h}\phi^{\prime}\bar{G}_{5,\bar{X}}}{2\bar{f}}}\right)=2(M^{2}e^{2\phi}-\bar{M}^{2})+4\alpha\frac{\bar{h}}{\bar{f}}\phi^{\prime}(\bar{f}\phi^{\prime}+\bar{f}^{\prime})>0\,,$
(56) $\displaystyle\bar{{\cal F}}$ $\displaystyle\equiv$ $\displaystyle
2\bar{G}_{4}+\bar{h}\phi^{\prime 2}\bar{G}_{5,\phi}-\bar{h}\phi^{\prime
2}\left(\frac{1}{2}\bar{h}^{\prime}\phi^{\prime}+\bar{h}\phi^{\prime\prime}\right)\bar{G}_{5,\bar{X}}=2(M^{2}e^{2\phi}-\bar{M}^{2})+4\alpha(2\bar{h}\phi^{\prime\prime}+\bar{h}^{\prime}\phi^{\prime}-\bar{h}\phi^{\prime
2})>0\,,$ (57) $\displaystyle\bar{{\cal H}}$ $\displaystyle\equiv$
$\displaystyle 2\bar{G}_{4}+2\bar{h}\phi^{\prime
2}\bar{G}_{4,\bar{X}}-\bar{h}\phi^{\prime
2}\bar{G}_{5,\phi}-\frac{\bar{h}^{2}\phi^{\prime
3}\bar{G}_{5,\bar{X}}}{\bar{r}}=2(M^{2}e^{2\phi}-\bar{M}^{2})+4\alpha\frac{\bar{h}}{\bar{r}}\phi^{\prime}(\bar{r}\phi^{\prime}+2)>0\,.$
(58)

Substituting the solutions (45)-(47) into Eq. (56), the no ghost parameter in
the odd-parity sector yields

$\bar{{\cal
G}}=2(M^{2}e^{2\phi}-\bar{M}^{2})\left[1-\frac{6\bar{r}^{2}+3\bar{r}_{h}\bar{r}+2\bar{r}_{h}^{2}}{9\bar{M}^{2}\bar{r}_{h}\bar{r}^{3}}\alpha+{\cal
O}(\alpha^{2})\right]\,.$ (59)

Provided that $\hat{\alpha}\ll 1$, the no-ghost condition $\bar{{\cal G}}>0$
is satisfied if

$M^{2}e^{2\phi}-\bar{M}^{2}>0\,,$ (60)

which automatically holds for $\bar{M}\ll M$. Up to linear order in
$\alpha^{i}$, the expressions of $\bar{{\cal F}}$ and $\bar{{\cal H}}$
coincide with $\bar{{\cal G}}$. However, the difference appears at the order
of $\alpha^{2}$. The squared propagation speeds of odd-parity gravitational
perturbation $\chi$ along the radial and angular directions are given,
respectively, by

$\displaystyle\bar{c}_{r,{\rm odd}}^{2}=\frac{\bar{{\cal G}}}{\bar{{\cal
F}}}=1+\frac{4(\bar{r}-\bar{r}_{h})(2\bar{r}^{2}+3\bar{r}_{h}\bar{r}+4\bar{r}_{h}^{2})e^{-2\phi_{0}}}{3M^{2}\bar{M}^{2}\bar{r}_{h}\bar{r}^{6}}\alpha^{2}+{\cal
O}(\alpha^{3})\,,$ (61) $\displaystyle\bar{c}_{\Omega,{\rm
odd}}^{2}=\frac{\bar{\cal G}}{\bar{\cal
H}}=1-\frac{2(2\bar{r}-3\bar{r}_{h})(\bar{r}^{2}+\bar{r}_{h}\bar{r}+\bar{r}_{h}^{2})e^{-2\phi_{0}}}{3M^{2}\bar{M}^{2}\bar{r}_{h}\bar{r}^{6}}\alpha^{2}+{\cal
O}(\alpha^{3})\,.$ (62)

Since the trace anomaly corrections in Eqs. (61) and (62) are at most of order
$(\bar{M}^{2}/M^{2})\hat{\alpha}^{2}$, both $\bar{c}_{r,{\rm odd}}^{2}$ and
$\bar{c}_{\Omega,{\rm odd}}^{2}$ are very close to 1 in the vicinity of the
horizon. Thus, the Laplacian stability conditions $\bar{c}_{r,{\rm
odd}}^{2}>0$ and $\bar{c}_{\Omega,{\rm odd}}^{2}>0$ are automatically
satisfied for $\hat{\alpha}\ll 1$. We also note that the $\alpha^{2}$-order
corrections to Eqs. (61) and (62) decrease in proportion to $\bar{r}^{-3}$ at
large distances, so $\bar{c}_{r,{\rm odd}}^{2}$ and $\bar{c}_{\Omega,{\rm
odd}}^{2}$ rapidly approach 1 far away from the horizon.

In the even-parity sector, the ghost does not appear under the condition

$\bar{{\cal K}}\equiv 2\bar{{\cal P}}_{1}-\bar{{\cal F}}>0\,,$ (63)

where

$\displaystyle\bar{{\cal P}}_{1}$ $\displaystyle\equiv$
$\displaystyle\frac{\bar{h}\bar{\mu}}{2\bar{f}\bar{r}^{2}\bar{{\cal
H}}^{2}}\left(\frac{\bar{f}\bar{r}^{4}\bar{{\cal
H}}^{4}}{\bar{\mu}^{2}\bar{h}}\right)^{\prime}\,,$ (64)
$\displaystyle\bar{\mu}$ $\displaystyle\equiv$ $\displaystyle
4\bar{r}\bar{G}_{4}+2\bar{r}^{2}\phi^{\prime}\bar{G}_{4,\phi}+2\bar{h}\bar{r}\phi^{\prime
2}(4\bar{G}_{4,\bar{X}}-3\bar{G}_{5,\phi})+\bar{h}\phi^{\prime
3}[(1-5\bar{h})\bar{G}_{5,\bar{X}}+\bar{r}^{2}(\bar{G}_{3,\bar{X}}-2\bar{G}_{4,\phi\bar{X}})$
(65)
$\displaystyle+\bar{h}\phi^{\prime}(\bar{h}\phi^{\prime}\bar{G}_{5,\bar{X}\bar{X}}-4\bar{r}G_{4,\bar{X}\bar{X}}+2\bar{r}\bar{G}_{5,\phi\bar{X}})]\,.$

Using the expanded solutions (45)-(47), it follows that

$\bar{{\cal
K}}=\frac{2(M^{2}e^{2\phi_{0}}-\bar{M}^{2})(\bar{r}^{2}+\bar{r}_{h}\bar{r}+\bar{r}_{h}^{2})^{2}e^{-2\phi_{0}}}{3M^{2}\bar{M}^{2}\bar{r}_{h}^{2}\bar{r}^{6}}\alpha^{2}+{\cal
O}(\alpha^{3})\,.$ (66)

Under the condition (60), the leading-order term of $\bar{{\cal K}}$ 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
term $\bar{{\cal K}}$, in which case the strong coupling problem is absent.

For even-parity modes, there are two perturbations $\psi$ and $\delta\phi$
arising from the gravitational and scalar field sectors, respectively [34, 35,
36]. The radial propagation speed squared of $\psi$ is identical to
$\bar{c}_{r,{\rm odd}}^{2}=\bar{{\cal G}}/\bar{{\cal F}}$ in the odd-parity
sector. The other radial propagation speed squared
$\bar{c}_{r,\delta\phi}^{2}$ can be obtained by setting $\rho=0=P$ in Eq.
(5.30) of Ref. [36]. On using Eqs. (45)-(47), we obtain

$\bar{c}_{r,\delta\phi}^{2}=1-\frac{4(2\bar{r}^{2}+3\bar{r}_{h}\bar{r}+4\bar{r}_{h}^{2})(M^{4}+M^{2}\bar{M}^{2}e^{-2\phi_{0}}+\bar{M}^{4}e^{-4\phi_{0}})(\bar{r}-\bar{r}_{h})}{9M^{4}\bar{M}^{4}\bar{r}_{h}\bar{r}^{6}}\alpha^{2}+{\cal
O}(\alpha^{3})\,.$ (67)

To derive this expression, we need to resort to the background solutions of
$\bar{f}(\bar{r})$, $\bar{h}(\bar{r})$, and $\phi(\bar{r})$ expanded up to
fourth order in $\alpha^{i}$. In other words, the third order expansion leads
to a result different from Eq. (67), but using the solutions higher than
fourth order gives the same expression as Eq. (67). The trace anomaly
correction in $\bar{c}_{r,\delta\phi}^{2}$ is at most of order
$\hat{\alpha}^{2}$ and it decreases in proportion to $\bar{r}^{-3}$ at large
distances. Provided that $\hat{\alpha}\ll 1$, the radial Laplacian stability
of $\delta\phi$ is always ensured.

The squared propagation speeds of $\psi$ and $\delta\phi$ in the angular
direction are $\bar{c}_{\Omega\pm,{\rm
even}}^{2}=-B_{1}\pm\sqrt{B_{1}^{2}-B_{2}}$, 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 have

$\bar{c}_{\Omega\pm,{\rm
even}}^{2}=1\pm\frac{2\sqrt{3}\,\bar{r}_{h}e^{-\phi_{0}}}{M\bar{M}\bar{r}^{3}}\alpha+{\cal
O}(\alpha^{2})\,.$ (68)

Hence the leading-order trace anomaly correction in $\bar{c}_{\Omega\pm,{\rm
even}}^{2}$ is suppressed by the order $(\bar{M}/M)\hat{\alpha}$ and it
decreases in proportion to $\bar{r}^{-3}$ for increasing $\bar{r}$.

We have thus shown that, under the condition $M^{2}e^{2\phi}-\bar{M}^{2}>0$,
our hairy BH solution arising from the trace anomaly satisfies all the linear
stability conditions for $\hat{\alpha}\ll 1$ and $\bar{M}\ll M$. 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 $-\bar{M}^{2}\int{\rm
d}^{4}x\sqrt{-\bar{g}}\bar{R}$ 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 tensor
$\bar{g}_{\mu\nu}=e^{-2\phi}g_{\mu\nu}$ invariant under Weyl transformations
$g_{\mu\nu}\to e^{2\sigma}g_{\mu\nu}$ and $\phi\to\phi+\sigma$, the total
action ${\cal S}={\cal S}_{R\bar{R}}+{\cal S}_{A}$ is expressed in the form
(17). This is the EFT valid below the mass scale $\bar{M}$, in which regime
nonlinear derivatives of the conformal scalar field are weakly coupled. Thanks
to the presence of the action $-\bar{M}^{2}\int{\rm
d}^{4}x\sqrt{-\bar{g}}\bar{R}$, 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. (19), we derived the covariant gravitational field equations by varying
the action (17) with respect to $\bar{g}_{\mu\nu}$. The trace of this equation
can be expressed in the simple form (22). Varying the action with respect to
$\phi$ leads to the scalar field equation $2M^{2}R-\alpha{\cal G}+\beta
W^{2}=0$ and hence the combination with Eq. (22) gives
$2\bar{M}^{2}\bar{R}-\alpha\bar{\cal G}+\beta\bar{W}^{2}=0$. Thus, the GB and
Weyl terms explicitly affect the spacetime geometry in both frames with the
metrics $g_{\mu\nu}$ and $\bar{g}_{\mu\nu}$. So long as the Weyl trace anomaly
is absent, i.e., $\beta=0$, the resulting theory is equivalent to a subclass
of Horndeski theories given by the action (27) with the coupling functions
(28). 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 ($\alpha=0$) is restricted to a no-hair
Schwarzschild solution given by Eq. (37). In this case there exists the other
hairy BH solution (36) originally found by BBMB for a conformally invariant
scalar, but this solution is known to be unstable (see Appendix A). For
$\alpha\neq 0$, we derived the solutions to $\bar{f}(\bar{r})$,
$\bar{h}(\bar{r})$, and $\phi(\bar{r})$ expanded with respect to a small
coupling constant $\alpha$ in the frame with the metric $\bar{g}_{\mu\nu}$.
The BH solution expanded up to second order in $\alpha^{i}$, which is
consistent with regular boundary conditions on the horizon and at spatial
infinity, is given by Eqs. (45)-(47), with the BH scalar charge (50).

The two metric components $\bar{f}(\bar{r})$ and $\bar{h}(\bar{r})$ are not
proportional to each other for our BH solution, so it is different from the
exact solution derived by imposing the condition
$\bar{f}(\bar{r})=c_{0}\bar{h}(\bar{r})$ (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 (45)-(47), which is valid for $\hat{\alpha}\ll 1$, exhibits good
agreement with the numerical results. We also derived the metric components
$f(r)$ and $h(r)$ in the frame with the metric $g_{\mu\nu}$ as (51)-(52) 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 (45)-(47)
against linear perturbations on the background metric (30). In the odd-parity
sector, the stability conditions are given by Eqs. (56)-(58). Provided that
$M^{2}e^{2\phi}-\bar{M}^{2}>0$, they can be satisfied for the small coupling
constant in the range $\hat{\alpha}\ll 1$. 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 conditions
$\hat{\alpha}\ll 1$ and $\bar{M}\ll M$. 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 (50),
it may be possible to probe its signature from gravitational waves emitted
from the binary system containing BHs (see e.g., Refs. [59, 60, 61, 62]).
Moreover, the Weyl trace anomaly may also give rise to an additional scalar
hair for BHs and possibly for NSs. These issues are left for future works.

## Acknowledgements

I thank Gregory Gabadadze for giving me insightful and positive comments. The
author is supported by the Grant-in-Aid for Scientific Research Fund of the
JSPS Nos. 19K03854 and 22K03642.

##  Appendix A Instability of the BBMB solution

For $\alpha=0$, BBMB originally found an exact BH solution with a nonvanishing
field derivative in the frame given with the metric tensor $g_{\mu\nu}$ [39,
40]. In the frame with $\bar{g}_{\mu\nu}$, the BBMB solution corresponds to
Eq. (36), which we will derive from now on. We will also show that it is
unstable against linear perturbations for some ranges of $\bar{r}$ outside the
horizon.

The BBMB solution has the metric components where $\bar{f}(\bar{r})$ is
proportional to $\bar{h}(\bar{r})$, i.e.,
$\bar{f}(\bar{r})=c_{0}\bar{h}(\bar{r})$, where $c_{0}$ is a constant. In this
case, Eq. (35) is integrated to give

$\bar{f}(\bar{r})=c_{0}+\frac{C_{1}}{\bar{r}}+\frac{C_{2}}{\bar{r}^{2}}\,,$
(69)

where $C_{1}$ and $C_{2}$ are integration constants. Since we are considering
the case $\bar{h}^{\prime}/\bar{h}=\bar{f}^{\prime}/\bar{f}$, the scalar field
satisfies the relation $\phi^{\prime 2}-\phi^{\prime\prime}=0$ from Eq. (34).
Then, we obtain the following integrated solution

$\phi^{\prime}(\bar{r})=-\ln\left(C_{3}\bar{r}+C_{4}\right)\,,$ (70)

where $C_{3}$ and $C_{4}$ are constants. The other equations of motion are
satisfied for

$C_{2}=\frac{C_{1}^{2}}{4c_{0}}\,,\qquad
C_{4}=\frac{C_{1}C_{3}}{2c_{0}}\,,\qquad
C_{1}=\pm\frac{2Mc_{0}}{C_{3}\bar{M}}\,.$ (71)

Setting $C_{3}=M/(\bar{M}m_{0})$, the resulting BH solution is given by Eq.
(36).

For the exact BH solution (36), the quantities (56)-(58) and (63) reduce,
respectively, to

$\bar{{\cal G}}=\bar{{\cal F}}=\bar{{\cal
H}}=\frac{2\bar{M}^{2}\bar{r}(2m_{0}-\bar{r})}{(\bar{r}-m_{0})^{2}}\,,\qquad\bar{{\cal
K}}=\frac{6\bar{M}^{2}m_{0}^{2}\bar{r}(2m_{0}-\bar{r})}{(\bar{r}^{2}-3m_{0}\bar{r}+3m_{0}^{2})^{2}}\,,$
(72)

which are all negative for $\bar{r}>2m_{0}$. Hence the exact BH solution is
unstable at the distance $\bar{r}>2m_{0}$.

On using the transformation properties (32), the metrics and scalar field
reduce to the forms

$f(r)=\frac{c_{0}\bar{M}^{2}}{M^{2}}\left(1-\frac{m}{r}\right)^{2}\,,\qquad
h(r)=\left(1-\frac{m}{r}\right)^{2}\,,\qquad\varphi=\bar{M}e^{-\phi}=\pm\frac{Mm}{r-m}\,,$
(73)

where

$r=m\frac{\bar{r}}{\bar{r}-m_{0}}\,,\qquad m=\frac{\bar{M}}{M}m_{0}\,.$ (74)

One can choose $c_{0}=M^{2}/\bar{M}^{2}$ by using the time reparametrization
freedom of $f$, in which case $f(r)=h(r)=(1-m/r)^{2}$. On using the first
relation of Eq. (74), it follows that the BH instability region
$\bar{r}>2m_{0}$ translates to $r<2m$. Indeed, this result agrees with the
stability analysis performed in the frame with the metric $g_{\mu\nu}$ [35].

##  Appendix B Exact solution with $\bar{f}$ proportional to $\bar{h}$

For $\alpha\neq 0$, we derive an exact BH solution by imposing the condition
$\bar{f}(\bar{r})=c_{0}\bar{h}(\bar{r})$, where $c_{0}$ is a constant. Then,
from Eq. (35), we obtain the simple differential equation

$\bar{h}^{\prime\prime}=-\frac{2[\bar{M}^{2}(2\bar{r}\bar{h}^{\prime}+\bar{h}-1)+\alpha\bar{h}^{\prime
2}]}{\bar{M}^{2}\bar{r}^{2}+2\alpha(\bar{h}-1)}\,.$ (75)

This is integrated to give

$\bar{h}(\bar{r})=1-\frac{\bar{M}^{2}\bar{r}^{2}}{2\alpha}\left[1-\sqrt{1-\frac{4\alpha(2m\bar{r}-q)}{\bar{M^{2}}\bar{r}^{4}}}\right]\,,$
(76)

where $m$ and $q$ are integration constants. Note that we have chosen the
asymptotically flat branch where $\bar{h}(\bar{r})$ approaches 1 in the limit
$\bar{r}\to\infty$, such that
$\bar{h}(\bar{r})=1-2m/\bar{r}+q/\bar{r}^{2}+{\cal O}(1/\bar{r}^{4})$. This
solution is analogous to the one derived in Ref. [16] in gravitational
theories with a conformal scalar field. Since we are now imposing the
condition $\bar{f}(\bar{r})=c_{0}\bar{h}(\bar{r})$, the right hand side of Eq.
(34) vanishes. This gives a constraint on the scalar field equation of motion.
For example, there is a possibility for $\phi$ satisfying the relation
$\phi^{\prime 2}-\phi^{\prime\prime}=0$. However, we recall that the hairy BH
solution (45)-(46) derived by using regular boundary conditions for both the
metric and scalar field on the horizon does not satisfy the relation
$\bar{f}(\bar{r})=c_{0}\bar{h}(\bar{r})$. This means that imposing the
condition $\bar{f}(\bar{r})=c_{0}\bar{h}(\bar{r})$ is not consistent with all
the field equations of motion and regular boundary conditions at
$\bar{r}=\bar{r}_{h}$, so Eq. (76) is not actually a desirable solution to the
system.

## References

  * Capper  _et al._ [1974] D. M. Capper, M. J. Duff, and L. Halpern,  _Phys. Rev. D_ 10, 461 (1974).
  * Capper and Duff [1974a] D. M. Capper and M. J. Duff,  _Nucl. Phys. B_ 82, 147 (1974a).
  * Capper and Duff [1974b] D. M. Capper and M. J. Duff,  _Nuovo Cim. A_ 23, 173 (1974b).
  * Deser  _et al._ [1976] S. Deser, M. J. Duff, and C. J. Isham,  _Nucl. Phys. B_ 111, 45 (1976).
  * Duff [1977] M. J. Duff,  _Nucl. Phys. B_ 125, 334 (1977).
  * Birrell and Davies [1984] N. D. Birrell and P. C. W. Davies,  _Quantum Fields in Curved Space_, Cambridge Monographs on Mathematical Physics (Cambridge Univ. Press, Cambridge, UK, 1984).
  * Dowker and Critchley [1977] J. S. Dowker and R. Critchley,  _Phys. Rev. D_ 16, 3390 (1977).
  * Christensen and Fulling [1977] S. M. Christensen and S. A. Fulling,  _Phys. Rev. D_ 15, 2088 (1977).
  * Brown [1977] L. S. Brown,  _Phys. Rev. D_ 15, 1469 (1977).
  * Brown and Cassidy [1977] L. S. Brown and J. P. Cassidy,  _Phys. Rev. D_ 15, 2810 (1977).
  * Davies  _et al._ [1977] P. C. W. Davies, S. A. Fulling, S. M. Christensen, and T. S. Bunch,  _Annals Phys._ 109, 108 (1977).
  * Hawking [1977] S. W. Hawking,  _Commun. Math. Phys._ 55, 133 (1977).
  * Henningson and Skenderis [1998] M. Henningson and K. Skenderis,  _JHEP_ 07, 023 (1998), arXiv:hep-th/9806087 .
  * Riegert [1984] R. J. Riegert,  _Phys. Lett. B_ 134, 56 (1984).
  * Komargodski and Schwimmer [2011] Z. Komargodski and A. Schwimmer,  _JHEP_ 12, 099 (2011), arXiv:1107.3987 [hep-th] .
  * Fernandes [2021] P. G. S. Fernandes,  _Phys. Rev. D_ 103, 104065 (2021), arXiv:2105.04687 [gr-qc] .
  * Gabadadze and Tukhashvili [2020] G. Gabadadze and G. Tukhashvili,  _Phys. Rev. D_ 102, 024054 (2020), arXiv:2005.01729 [hep-th] .
  * Glavan and Lin [2020] D. Glavan and C. Lin,  _Phys. Rev. Lett._ 124, 081301 (2020), arXiv:1905.03601 [gr-qc] .
  * Lu and Pang [2020] H. Lu and Y. Pang,  _Phys. Lett. B_ 809, 135717 (2020), arXiv:2003.11552 [gr-qc] .
  * Kobayashi [2020] T. Kobayashi,  _JCAP_ 07, 013 (2020), arXiv:2003.12771 [gr-qc] .
  * Fernandes  _et al._ [2022] P. G. S. Fernandes, P. Carrilho, T. Clifton, and D. J. Mulryne,  _Class. Quant. Grav._ 39, 063001 (2022), arXiv:2202.13908 [gr-qc] .
  * Tsujikawa [2022] S. Tsujikawa,  _Phys. Lett. B_ 833, 137329 (2022), arXiv:2205.09932 [gr-qc] .
  * Horndeski [1974] G. W. Horndeski,  _Int. J. Theor. Phys._ 10, 363 (1974).
  * Deffayet  _et al._ [2011] C. Deffayet, X. Gao, D. A. Steer, and G. Zahariade,  _Phys. Rev. D_ 84, 064039 (2011), arXiv:1103.3260 [hep-th] .
  * Kobayashi  _et al._ [2011] T. Kobayashi, M. Yamaguchi, and J. Yokoyama,  _Prog. Theor. Phys._ 126, 511 (2011), arXiv:1105.5723 [hep-th] .
  * Charmousis  _et al._ [2012] C. Charmousis, E. J. Copeland, A. Padilla, and P. M. Saffin,  _Phys. Rev. Lett._ 108, 051101 (2012), arXiv:1106.2000 [hep-th] .
  * Bonifacio  _et al._ [2020] J. Bonifacio, K. Hinterbichler, and L. A. Johnson,  _Phys. Rev. D_ 102, 024029 (2020), arXiv:2004.10716 [hep-th] .
  * Gabadadze [2023] G. Gabadadze, arXiv:2301.13265 [hep-th] .
  * Kase and Tsujikawa [2023] R. Kase and S. Tsujikawa, arXiv:2301.10362 [gr-qc] .
  * Minamitsuji and Tsujikawa [2022] M. Minamitsuji and S. Tsujikawa,  _Phys. Rev. D_ 106, 064008 (2022), arXiv:2207.04461 [gr-qc] .
  * Fernandes  _et al._ [2021] P. G. S. Fernandes, P. Carrilho, T. Clifton, and D. J. Mulryne,  _Phys. Rev. D_ 104, 044029 (2021), arXiv:2107.00046 [gr-qc] .
  * Minamitsuji  _et al._ [2022a] M. Minamitsuji, K. Takahashi, and S. Tsujikawa,  _Phys. Rev. D_ 106, 044003 (2022a), arXiv:2204.13837 [gr-qc] .
  * Minamitsuji  _et al._ [2022b] M. Minamitsuji, K. Takahashi, and S. Tsujikawa,  _Phys. Rev. D_ 105, 104001 (2022b), arXiv:2201.09687 [gr-qc] .
  * Kobayashi  _et al._ [2012] T. Kobayashi, H. Motohashi, and T. Suyama,  _Phys. Rev. D_ 85, 084025 (2012), [Erratum: Phys. Rev. D 96, 109903 (2017)], arXiv:1202.4893 [gr-qc] .
  * Kobayashi  _et al._ [2014] T. Kobayashi, H. Motohashi, and T. Suyama,  _Phys. Rev. D_ 89, 084042 (2014), arXiv:1402.6740 [gr-qc] .
  * Kase and Tsujikawa [2022] R. Kase and S. Tsujikawa,  _Phys. Rev. D_ 105, 024059 (2022), arXiv:2110.12728 [gr-qc] .
  * Dabrowski  _et al._ [2009] M. P. Dabrowski, J. Garecki, and D. B. Blaschke,  _Annalen Phys._ 18, 13 (2009), arXiv:0806.2683 [gr-qc] .
  * Langlois  _et al._ [2022] D. Langlois, K. Noui, and H. Roussille,  _JCAP_ 09, 019 (2022), arXiv:2204.04107 [gr-qc] .
  * Bocharova  _et al._ [1970] N. M. Bocharova, K. A. Bronnikov, and V. N. Melnikov,  _Mosk. Univ. Fiz. Astron._ 6, 706 (1970).
  * Bekenstein [1974] J. D. Bekenstein,  _Annals Phys._ 82, 535 (1974).
  * Bronnikov and Kireev [1978] K. A. Bronnikov and Y. N. Kireev,  _Phys. Lett. A_ 67, 95 (1978).
  * McFadden and Turok [2005] P. L. McFadden and N. G. Turok,  _Phys. Rev. D_ 71, 086004 (2005), arXiv:hep-th/0412109 .
  * Konoplya and Molina [2005] R. A. Konoplya and C. Molina,  _Phys. Rev. D_ 71, 124009 (2005), arXiv:gr-qc/0504139 .
  * Sotiriou and Zhou [2014a] T. P. Sotiriou and S.-Y. Zhou,  _Phys. Rev. Lett._ 112, 251102 (2014a), arXiv:1312.3622 [gr-qc] .
  * Sotiriou and Zhou [2014b] T. P. Sotiriou and S.-Y. Zhou,  _Phys. Rev. D_ 90, 124063 (2014b), arXiv:1408.1698 [gr-qc] .
  * Kanti  _et al._ [1996] P. Kanti, N. E. Mavromatos, J. Rizos, K. Tamvakis, and E. Winstanley,  _Phys. Rev. D_ 54, 5049 (1996), arXiv:hep-th/9511071 .
  * Kanti  _et al._ [1998] P. Kanti, N. E. Mavromatos, J. Rizos, K. Tamvakis, and E. Winstanley,  _Phys. Rev. D_ 57, 6255 (1998), arXiv:hep-th/9703192 .
  * Torii  _et al._ [1997] T. Torii, H. Yajima, and K.-i. Maeda,  _Phys. Rev. D_ 55, 739 (1997), arXiv:gr-qc/9606034 .
  * Chen  _et al._ [2007] C.-M. Chen, D. V. Gal’tsov, and D. G. Orlov,  _Phys. Rev. D_ 75, 084030 (2007), arXiv:hep-th/0701004 .
  * Guo  _et al._ [2008] Z.-K. Guo, N. Ohta, and T. Torii,  _Prog. Theor. Phys._ 120, 581 (2008), arXiv:0806.2481 [gr-qc] .
  * Pani and Cardoso [2009] P. Pani and V. Cardoso,  _Phys. Rev. D_ 79, 084031 (2009), arXiv:0902.1569 [gr-qc] .
  * Kleihaus  _et al._ [2011] B. Kleihaus, J. Kunz, and E. Radu,  _Phys. Rev. Lett._ 106, 151104 (2011), arXiv:1101.2868 [gr-qc] .
  * Ayzenberg and Yunes [2014] D. Ayzenberg and N. Yunes,  _Phys. Rev. D_ 90, 044066 (2014), [Erratum: Phys.Rev.D 91, 069905 (2015)], arXiv:1405.2133 [gr-qc] .
  * Maselli  _et al._ [2015] A. Maselli, P. Pani, L. Gualtieri, and V. Ferrari,  _Phys. Rev. D_ 92, 083014 (2015), arXiv:1507.00680 [gr-qc] .
  * Kleihaus  _et al._ [2016] B. Kleihaus, J. Kunz, S. Mojica, and E. Radu,  _Phys. Rev. D_ 93, 044047 (2016), arXiv:1511.05513 [gr-qc] .
  * Regge and Wheeler [1957] T. Regge and J. A. Wheeler,  _Phys. Rev._ 108, 1063 (1957).
  * Zerilli [1970] F. J. Zerilli,  _Phys. Rev. Lett._ 24, 737 (1970).
  * Kase  _et al._ [2020] R. Kase, R. Kimura, S. Sato, and S. Tsujikawa,  _Phys. Rev. D_ 102, 084037 (2020), arXiv:2007.09864 [gr-qc] .
  * Will [1994] C. M. Will,  _Phys. Rev. D_ 50, 6058 (1994), arXiv:gr-qc/9406022 .
  * Yunes and Pretorius [2009] N. Yunes and F. Pretorius,  _Phys. Rev. D_ 80, 122003 (2009), arXiv:0909.3328 [gr-qc] .
  * Higashino and Tsujikawa [2023] Y. Higashino and S. Tsujikawa,  _Phys. Rev. D_ 107, 044003 (2023), arXiv:2209.13749 [gr-qc] .
  * Quartin  _et al._ [2023] M. Quartin, S. Tsujikawa, L. Amendola, and R. Sturani, arXiv:2304.02535 [astro-ph.CO] .