aainstitutetext: Facultad de Física, Pontificia Universidad Catolica de Chile,
Av. Vicuña Mackenna 4860, Santiago, Chile bbinstitutetext: Departamento de
Física, Universidad de Concepción, Casilla, 160-C, Concepción,
Chileccinstitutetext: Instituto de Ciencias Exactas y Naturales, Universidad
Arturo Prat, Playa Brava 3256, 1111346, Iquique, Chileddinstitutetext:
Facultad de Ciencias, Universidad Arturo Prat, Avenida Arturo Prat Chacón
2120, 1110939, Iquique, Chileeeinstitutetext: Departamento de Ciencias
Físicas, Universidad Andres Bello, Sazié 2212, Santiago de Chile

# Holographic renormalization of Horndeski gravity

Nicolás Cáceres<EMAIL_ADDRESS>c,d Cristóbal Corral<EMAIL_ADDRESS>e
Felipe Diaz<EMAIL_ADDRESS>e Rodrigo Olea<EMAIL_ADDRESS>
###### Abstract

We study the renormalization of a particular sector of Horndeski theory. We
focus on the nonminimal coupling of a scalar field to the Gauss-Bonnet term
through an arbitrary function of the former plus a kinetic coupling to the
Einstein tensor. In the asymptotically AdS sector of the theory, we perform a
near-boundary expansion of the fields and we work out the asymptotic form of
the action and its variation. By assuming a power expansion of the scalar
coupling function and the Gauss-Bonnet term, we find specific conditions on
their coefficients such that the action and charges are finite. To accomplish
the latter, a finite set of intrinsic boundary terms has to be added. If the
nonminimal kinetic coupling is absent, the trace of the holographic stress-
energy tensor cannot be zero while the dual CFT remains unitary as the scalar
mass lies outside the Breitenlohner-Freedman bound. However, if one considers
the kinetic coupling to the Einstein tensor, we find that its contribution
allows one to recover the unitarity of the dual CFT, motivating the
introduction of that term from a holographic viewpoint.

##  1 Introduction

Scalar fields play a fundamental role in the description of systems in areas
as diverse as high-energy physics, gravity, cosmology, and condensed matter.
They are essential to account for spontaneous symmetry breaking by acquiring a
nonzero vacuum expectation value. This is the case of the behavior of
superconductors but also of the mass-generating mechanism in the electroweak
sector of the Standard Model of particle physics.

In the more recent framework of anti-de Sitter/Conformal Field Theory
(AdS/CFT) correspondence Maldacena:1997re ; Witten:1998qj ; Gubser:1998bc ;
Aharony:1999ti , coupling scalar fields to gravity introduces new sources and
operators in the dual boundary CFT. Indeed, holographic superconductors
Hartnoll:2008vx ; Albash:2008eh ; Nishioka:2009zj ; Montull:2009fe ;
Herzog:2010vz ; Cai:2015cya  and models featuring momentum dissipation
Andrade:2013gsa ; Gouteraux:2014hca ; Andrade:2016tbr  are examples of systems
with physical properties captured by scalar fields. However, in black hole
physics, different impersonations of the no-hair theorem pose strong
limitations to finding nontrivial scalar configurations in stationary
spacetimes Bekenstein:1971hc ; Bekenstein:1972ny ; Bekenstein:1972ky ;
Sudarsky:1995zg ; Bekenstein:1996pn . Nevertheless, it is possible to
circumvent these restrictions by introducing a cosmological constant and/or a
nonminimal coupling between the scalars fields and the geometry. This allows
to obtain black hole solutions in this class of scalar-tensor theories of
gravity Bocharova:1970skc ; Bekenstein:1974sf ; Martinez:1996gn ;
Martinez:1996uv ; Martinez:2002ru ; Martinez:2004nb ; Babichev:2013cya  (for a
review see Barack:2018yly ).

Finding sensible scalar-tensor theories whose dynamics and space of solutions
are of relevance for holography is per se an interesting quest. However, one
should start with those whose classical aspects have already been well
understood, in order to find a reasonable dual CFT at the boundary. In these
regards, a natural possibility is Horndeski gravity Horndeski:1974wa . This is
constructed as the most general scalar-tensor theory that: (i) is invariant
under diffeomorphisms, local Lorentz transformations and it is torsion
free,111For extensions of this theories in the presence of torsion see
Barrientos:2017utp , and Bahamonde:2022cmz ; Armaleo:2023rhj  for further
developments. (ii) is constructed out of the metric, a scalar field and
derivatives thereof, and (iii) possesses second-order field equations for the
dynamical fields, avoiding the presence of ghosts. Their separated sectors
have been studied in different contexts. In particular, the one that remains
invariant under a constant shift of the scalar field has received a lot of
attention in the last few years. This is usually referred to as the shift-
symmetric sector of Horndeski theory. A specific term that belongs to the
latter class is the nonminimal coupling between the Einstein tensor and the
kinetic term of the scalar field. Indeed, when general relativity is augmented
by this term, it has been shown that the space of solutions is endowed with
black holes Rinaldi:2012vy ; Babichev:2013cya ; Anabalon:2013oea ;
Cisterna:2014nua ; Bravo-Gaete:2014haa ; Babichev:2016rlq ; Stetsko:2018fzt ;
Baggioli:2021ejg , black strings Cisterna:2018jsx ; Cisterna:2018mww , boson
and neutron stars Cisterna:2015yla ; Brihaye:2016lin ; Cisterna:2016vdx ,
gravitational instantons Bardoux:2013swa ; Arratia:2020hoy , among many other
compact and extended objects (for a review see Babichev:2016rlq ).

In holography, models of momentum dissipation have been studied in shift-
symmetric sectors of Horndeski gravity, allowing one to compute the DC
conductivity analytically in the dual field theory Jiang:2017imk . Indeed, the
same sector admits asymptotically Lifshitz black holes, providing a bridge for
studying AdS/CMT correspondence Bravo-Gaete:2013dca . In the context of
quantum information theory, holographic entanglement entropy has been studied
in Ref. Caceres:2017lbr  and, based on the complexity = action conjecture
Brown:2015bva , the holographic complexity of AdS black holes with planar
transverse sections have been computed in Ref. Feng:2018sqm . The
thermodynamics of charged black holes in Horndeski gravity has been
investigated in Ref. Feng:2015wvb , showing that the usual viscosity bound can
be violated depending on the mass/charge ratio. Thus, different holographic
evidence has shown us that Horndeski gravity is a fruitful scenario for
studying holography (for additional references, for instance, see
Kuang:2016edj ; Jiang:2017imk ; Filios:2018xvy ; Li:2018kqp ; Liu:2018hzo ;
Feng:2018sqm ; Li:2018rgn ; Santos:2021orr ; Santos:2023mee ).

Another sector of Horndeski gravity that has drawn considerable attention in
recent literature is the nonminimal coupling of scalar fields to the Gauss-
Bonnet (GB) term. This interaction term appears naturally in the dimensional
reduction of string-inspired models of gravity, e.g. Lovelock theory Mueller-
Hoissen:1987mas ; Mueller-Hoissen:1987nvb ; Mueller-Hoissen:1989bdj ;
Charmousis:2008kc ; Charmousis:2012dw , and different black hole and wormhole
solutions have been found Mignemi:1992nt ; Kanti:1995vq ; Torii:1996yi ;
Yunes:2011we ; Sotiriou:2013qea ; Sotiriou:2014pfa ; Prabhu:2018aun ;
Antoniou:2019awm . A remarkable feature of this interaction is that a scalar
field coupled to the GB term can undergo a tachyonic instability, triggering a
spontaneous scalarization of black holes Doneva:2017bvd ; Silva:2017uqg ;
Ripley:2019aqj ; Silva:2020omi ; East:2021bqk ; Antoniou:2021zoy . Indeed, it
has been shown that such instability can end up with a scalarized black hole
possessing either rotation Cunha:2019dwb ; Berti:2020kgk ; Hod:2020jjy ;
Elley:2022ept  or charge Doneva:2018rou  (for a review see Doneva:2022ewd ).
From a holographic viewpoint, this phenomenon can be seen as a condensation
process in the dual field theory, producing a critical phase transition
characterized by a nonvanishing vacuum expectation value Guo:2020sdu .

In view of possible holographic applications of Horndeski gravity, finding a
suitable renormalization scheme in asymptotically AdS spacetimes for this
theory has become a crucial point. In Einstein-AdS gravity, there exists an
approach for rendering the Euclidean on-shell action and conserved charges
finite in asymptotically AdS spacetimes, known as holographic renormalization.
It is a scheme relevant in the context of gauge/gravity duality and it
consists on the addition of intrinsic boundary counterterms
Balasubramanian:1999re ; Chamblin:1999tk ; Emparan:1999pm ; deHaro:2000vlm ;
Skenderis:2002wp . A different possibility, referred to as topological
renormalization, consists in adding a single topological term to the bulk
action, which acts as a counterterm, cancelling divergences in the action and
its variation. This alternative has been widely explored in Refs. Aros:1999id
; Aros:1999kt ; Olea:2005gb ; Olea:2006vd ; Miskovic:2009bm  and it has been
proved to be equivalent to holographic renormalization for asymptotically AdS
spaces with conformally flat boundaries Anastasiou:2020zw . For a general
boundary geometry, topological renormalization needs to be supplemented by
additional counterterms, which can be obtained by a proper embedding of
Einstein gravity into conformal gravity as proposed in Ref. Anastasiou:2020mik
; Anastasiou:2021tlv ; Anastasiou:2022ljq ; Anastasiou:2023oro , in an
improved framework dubbed conformal renormalization (for the case including
scalar couplings, see Anastasiou:2022wjq ).

In Horndeski gravity, the variational principle with Dirichlet boundary
conditions for the fields was studied in full detail in Ref. Padilla:2012ze .
The holographic renormalization of a particular sector of the theory, i.e.,
the one considering the kinetic coupling to the Einstein tensor, was worked
out in Ref. Liu:2017kml . Additionally, in Ref. Feng:2015oea , when computing
the Euclidean on-shell action of the black hole in Ref. Anabalon:2013oea , it
was pointed out that background subtraction methods do not match the result of
holographic renormalization. Moreover, to first order in the saddle-point
approximation, none of these approaches reproduce the computation of the black
hole entropy via the Noether-Wald formalism Wald:1993nt ; Iyer:1994ys ;
Wald:1999wa . Thus, a careful analysis of renormalization schemes in Horndeski
gravity is a must in order to address this and other issues.

In this work, we analyze the renormalization of AAdS sector of Horndeski
gravity. Taking the idea of topological renormalization as motivation, we
consider an arbitrary coupling between the scalar field and the GB term. This
is a suitable starting point since, for a constant scalar coupling, one
recovers the standard results of holographic renormalization for pure gravity.
In the general case, for a power expansion of the scalar coupling, we require
specific conditions on their coefficients such that the action and charges are
finite. In particular, if one considers only a quadratic scalar term coupled
to the GB, the dual CFT cannot be unitary. Remarkably, we find that by
introducing the Einstein-kinetic coupling of Horndeski gravity, unitarity is
recovered.

This article is organized as follows. In Sec. 2, we shortly review the
holographic renormalization procedure of four-dimensional Einstein gravity
minimally coupled to a scalar field and fix notation. In Sec. 3, we move
forward and consider Einstein scalar-GB gravity. We show that an asymptotic
analysis restricts the possible couplings in order to have a consistent
holographic theory. In Sec. 4, we show that unitarity and conformal invariance
of the dual field theory can be restored if the kinetic coupling to the
Einstein tensor is introduced. Section 5 is devoted to summarizing our
results. Additionally, we include Appendix A for details about the boundary
conditions for the scalar field.

##  2 Holographic renormalization with minimally-coupled scalar fields

Anti-de Sitter is a maximally symmetric space with negative constant
curvature, whose conformal compactification results in a manifold with a
conformal boundary. The line element of AdSd+1 spacetime can be expressed in
Poincaré coordinates as

$\displaystyle
ds^{2}={\tilde{g}}_{\mu\nu}dx^{\mu}dx^{\nu}=\frac{\ell^{2}}{z^{2}}\left(dz^{2}+\eta_{ij}dx^{i}dx^{j}\right)~{},$
(1)

where $x^{i}=(t,x_{1},\dots,x_{d-1})$ are boundary coordinates, $\eta_{ij}$
represents the Minkowski metric, $z$ is the coordinate along the additional
bulk dimension, and $\ell$ stands for the AdS radius, which is related to the
cosmological constant by $\Lambda=-d(d-1)/2\ell^{2}$.

In order to overcome the boundary singularity, one can employ a conformal
compactification of the spacetime through a Weyl rescaling denoted by
$g=z^{2}{\tilde{g}}$, where the conformal boundary is located at $z=0$. As a
result, the conformal metric $g$ smoothly extends to the boundary and defines
the boundary metric as

$\displaystyle g_{(0)}=\lim_{z\to 0}z^{2}{\tilde{g}}~{}.$ (2)

The same procedure can be extended for more general spaces Penrose:1985bww .
In particular, asymptotically AdS (AAdS) spacetime are defined in this way,
such that the bulk metric is written as

$\displaystyle
ds^{2}=\frac{\ell^{2}}{z^{2}}\left(dz^{2}+{\bar{g}}_{ij}(z,x^{i})dx^{i}dx^{j}\right)~{}.$
(3)

By construction, $\bar{g}_{ij}$ has a smooth limit as $z\to 0$, such that it
admits the Fefferman-Graham (FG) expansion Fefferman:1984asd

$\displaystyle\bar{g}_{ij}=g_{(0)ij}+zg_{(1)ij}+z^{2}g_{(2)ij}+\dots~{}.$ (4)

The coefficients $g_{(n)ij}$ with $n\neq 0$ can be obtained by solving
Einstein’s equations order by order. In doing so, the coefficients of the odd
powers of $z$ are set to zero. Here, for later convenience, we will choose the
radial coordinate as $\rho=z^{2}$. Then, the metric for AAdS spaces looks like

$\displaystyle
ds^{2}=\frac{\ell^{2}d\rho^{2}}{4\rho^{2}}+\frac{\ell^{2}}{\rho}\bar{g}_{ij}dx^{i}dx^{j}~{},$
(5)

where the asymptotic expansion of the boundary metric is given by

$\displaystyle\bar{g}_{ij}=g_{(0)ij}+\rho
g_{(2)ij}+\dots+\rho^{d/2}\left(g_{(d)ij}+h_{(d)ij}\log\rho\right)+\dots~{}.$
(6)

The coefficient $h_{(d)ij}$ only appears in even boundary dimensions and the
ellipsis denotes higher powers of $\rho$ which correspond to non-normalizable
modes.

The near-boundary form of the metric realizes the asymptotically AdS condition
on the Riemann tensor

$\displaystyle
R_{\mu\nu\lambda\sigma}=\frac{1}{\ell^{2}}\left({\tilde{g}}_{\mu\lambda}{\tilde{g}}_{\nu\sigma}-{\tilde{g}}_{\mu\sigma}{\tilde{g}}_{\nu\lambda}\right)+{\cal
O}\left(\rho\right)~{}.$ (7)

The FG expansion unveils the infrared divergences in the gravity action.222In
the AdS/CFT correspondence, the infrared behavior of the gravitational theory
is mapped to the physical properties of the gauge theory in the ultraviolet
Susskind:1998dq . In order to remove these divergences, one can introduce
counterterms, which are the result of solving the Einstein equations to
determine the expansion coefficients $g_{(n)ij}$, in terms of $g_{(0)}$.
Finally, one inverts the relations to obtain $g_{(n)ij}$ as intrinsic
covariant quantities from the point of view of the boundary metric, such that
divergences of the action are removed. This procedure is referred to as
_holographic renormalization_ and it was proposed in Refs. Henningson:1998gx ;
deHaro:2000vlm . It is a systematic method for extracting holographic
quantities such as correlation functions, Ward identities, and Weyl anomalies
(see Skenderis:2002wp  for a review).

In a similar way that the boundary of AdS defines a conformal class of
metrics, a bulk field does not induce a specific one on the boundary. Thus,
when imposing boundary conditions in AdS, they have to deal with a conformal
class rather than to any specific representative, key ingredient to set the
boundary value problem in AdS/CFT duality. For the metric tensor, it is
imperative to take a Dirichlet condition on $g_{(0)ij}$ instead of fixing the
boundary metric $h_{ij}=\bar{g}_{ij}/\rho$; the latter obviously exhibits a
divergent behavior. Therefore, the role of the counterterms is two-fold: (i)
they ensure the finiteness of the on-shell action, and (ii) they define a
well-posed variational principle for $g_{(0)ij}$.

With a renormalized action, one obtains the holographic stress tensor by
performing a variation with respect to the boundary metric. The Gubser-
Klebanov-Polyakov-Witten dictionary relates the low-energy limit of String
Theory (semiclassical regime of supergravity on an AAdS background) to the
generating functional of a gauge theory with conformal symmetry at its
boundary. The duality can be stated as

$\displaystyle{\rm exp}\left(iS_{\rm
grav}[\phi_{(0)}^{I}]\right)=\left\langle{\rm exp}\left(i\int_{\cal
M}d^{d}x~{}\phi_{(0)}^{I}{\cal O}_{I}\right)\right\rangle~{},$ (8)

where $S_{\rm grav}$ is a gravitational action with dynamical fields living on
an AAdSd+1 background, $\phi_{(0)}^{I}$ is the value of the fields at the
conformal boundary, and ${\cal O}_{I}$ is the collection of gauge operators
sourced by $\phi_{(0)}^{I}$. Then, a gravitational action on a
$(d+1)$-dimensional AAdS spacetime is related to the quantum effective action
of a $d$-dimensional CFT. Considering a scalar field in an Euclidean AdS
gravity, the quantum generating functional of the dual CFT living on a
background metric $g_{(0)}$ reads

$\displaystyle Z_{\rm CFT}[g_{(0)},\phi_{(0)}]=\left\langle{\rm
exp}\left[\int_{\cal
M}d^{d}x\sqrt{g_{(0)}}\left(\frac{1}{2}g_{(0)ij}T^{ij}+\phi_{(0)}{\cal
O}\right)\right]\right\rangle~{},$ (9)

where $T^{ij}$ is the CFT stress-energy tensor and ${\cal O}$ is a scalar
operator sourced by $\phi_{(0)}$. Then, one obtains a bulk/boundary relation,
that is,

$\displaystyle\langle T^{ij}\rangle=\frac{2}{\sqrt{-g_{(0)}}}\frac{\delta
S_{\rm ren}}{\delta g_{(0)ij}}=\lim_{\rho\to
0}\frac{2}{\sqrt{-\bar{g}}}\frac{\delta S_{\rm
ren}}{\delta\bar{g}_{ij}}=\lim_{\rho\to
0}\left(\frac{1}{\rho^{d/2-1}}T^{ij}[h]\right)~{},$ (10)

where $T^{ij}[h]$ is the stress tensor of the renormalized action. This tensor
is made of two parts: the first one is the canonical momentum which comes from
the addition of the Gibbons-Hawking-York (GHY) term to the Einstein-Hilbert
action Brown:1992br . The second contribution arises from the variation of the
counterterms introduced to renormalize the theory. The resulting holographic
stress-energy tensor is, indeed, finite and it corresponds to a boundary
operator that is dual to the bulk gravitational field. This mapping between
the asymptotic behavior of a bulk field and the source of a boundary quantum
operator in the dual CFT is a fundamental aspect of the holographic
dictionary.

For matter fields coupled to gravity, the procedure is analog. It involves the
asymptotic expansion of bulk fields near the conformal boundary. In a general
setting, the 1-point function of any field appears as the undetermined,
subleading term as described in Ref. Skenderis:2002wp . The coefficient is
determined by the boundary conditions imposed on the field, allowing for a
comprehensive understanding of the field’s behavior and its connection to the
boundary theory. Here, we focus on scalar operators that exhibit nontrivial
interactions with the boundary metric.

Let us examine first the four-dimensional Einstein-AdS gravity minimally
coupled to a massive scalar field. The action of such a theory is given by

$\displaystyle S_{\rm
min}=\frac{1}{2}\int_{\mathcal{M}}d^{4}x\sqrt{-g}\left(\frac{R+6\ell^{-2}}{\kappa}-\left(\partial\phi\right)^{2}-m^{2}\phi^{2}\right)-\frac{1}{\kappa}\int_{\partial{\cal
M}}d^{3}x\sqrt{-h}\,K~{},$ (11)

where ${\cal M}$ is an AAdS4 manifold, $\kappa$ is the gravitational constant
related to the Newton’s constant by $\kappa=8\pi G$, $R=g^{\mu\nu}R_{\mu\nu}$
is the Ricci scalar, $\ell$ is the AdS radius, $m$ is the mass parameter of
the scalar field $\phi$, and $K$ is the trace of the extrinsic curvature of
the boundary surface, denoted by ${\partial\cal M}$. An arbitrary variation of
the action gives

$\displaystyle\delta S_{\rm min}={}$ $\displaystyle\int_{\cal
M}d^{4}x\sqrt{-g}\left(\frac{1}{2\kappa}\left(E_{\mu\nu}-\kappa
T_{\mu\nu}\right)\delta
g^{\mu\nu}+\left(\Box\phi-m^{2}\phi\right)\delta\phi\right)$
$\displaystyle-\int_{\partial\cal M}d^{3}x\sqrt{-h}\left(\pi_{ij}\delta
h^{ij}+\pi_{\phi}\delta\phi\right)~{},$ (12)

where, in the bulk, we have defined

$\displaystyle E_{\mu\nu}={}$ $\displaystyle
R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}+\Lambda g_{\mu\nu}~{},$ $\displaystyle
T_{\mu\nu}={}$
$\displaystyle\nabla_{\mu}\phi\nabla_{\nu}\phi-\frac{1}{2}g_{\mu\nu}\left(\nabla^{\lambda}\phi\nabla_{\lambda}\phi+m^{2}\phi^{2}\right)~{},$
(13)

and the boundary variations define the canonical momenta associated with the
radial evolution of the metric and the scalar field by

$\displaystyle\pi_{ij}=\frac{1}{2\kappa}\left(K_{ij}-Kh_{ij}\right)\;\;\;\;\;\mbox{and}\;\;\;\;\;\pi_{\phi}=n^{\mu}\partial_{\mu}\phi~{},$
(14)

respectively. Thus, the field equations for the metric and the scalar field
can be read as

$\displaystyle E_{\mu\nu}-\kappa T_{\mu\nu}{}$ $\displaystyle=0~{},$
$\displaystyle\Box\phi-m^{2}\phi{}$ $\displaystyle=0~{}.$ (15)

In what follows, we shall consider the FG expansion with no logarithmic modes,
which describes a wide class of gravitational setups. In that case, the scalar
field can be expanded asymptotically as

$\displaystyle\phi(\rho,x^{i})=\rho^{(3-\Delta)/2}\bar{\phi}=\rho^{(3-\Delta)/2}\left(\phi_{(0)}+\rho\phi_{(2)}+\dots+\rho^{(2\Delta-3)/2}\phi_{(2\Delta-3)}+\dots\right)~{},$
(16)

where $\Delta$ is a constant to be determined. The holographic renormalization
method for the three-dimensional version of Eq.(11) with $\Delta=2$ was
studied in Ref. deHaro:2000vlm . Inserting the asymptotic expansion of the
fields into the equation of motion for the metric leads to the following
system of equations

$\displaystyle 0$ $\displaystyle={\cal
R}_{ij}(\bar{g})-\partial_{\rho}\bar{g}_{ij}-\bar{g}_{ij}\bar{g}^{mn}\partial_{\rho}\bar{g}_{mn}+{}\rho\left(2\bar{g}_{im}\partial_{\rho}\left(\bar{g}^{mn}\partial_{\rho}\bar{g}_{nj}\right)+\bar{g}^{mn}\partial_{\rho}\bar{g}_{mn}\partial_{\rho}\bar{g}_{ij}\right)$
$\displaystyle+{\kappa}\rho^{2-\Delta}\left(\tfrac{1}{2}m^{2}\ell^{2}\bar{\phi}^{2}\bar{g}_{ij}+\rho\partial_{i}\bar{\phi}\partial_{j}\bar{\phi}\right)~{},$
$\displaystyle 0$
$\displaystyle=\bar{g}^{mn}\partial_{\rho}\bar{g}_{ml}\bar{g}^{lp}\partial_{\rho}\bar{g}_{pn}+2\partial_{\rho}(\bar{g}^{mn}\partial_{\rho}\bar{g}_{mn})$
(17)
$\displaystyle+\kappa\rho^{1-\Delta}\left(\left(\tfrac{1}{2}m^{2}\ell^{2}+(3-\Delta)\right)\bar{\phi}^{2}+4\left((3-\Delta)\rho+\rho^{2}\right)\bar{\phi}\partial_{\rho}\bar{\phi}\right)$
$\displaystyle 0$
$\displaystyle=\nabla_{i}\left(\bar{g}^{jm}\partial_{\rho}\bar{g}_{jm}\right)-\nabla_{j}\left(\bar{g}^{jm}\partial_{\rho}\bar{g}_{mi}\right)+\kappa\rho^{2-\Delta}\left((3-\Delta)\bar{\phi}\partial_{i}\bar{\phi}+2\rho\partial_{\rho}\bar{\phi}\partial_{i}\bar{\phi}\right)~{},$

where ${\cal R}_{ij}(\bar{g})$ is the Ricci tensor of the metric $\bar{g}$.
The equation of motion for the scalar field, in turn, can be expressed as

$\displaystyle 0={}$
$\displaystyle\left(\Delta(\Delta-3)-m^{2}\ell^{2}\right)\bar{\phi}+\rho\left(\bar{g}_{ij}\partial_{i}\partial_{j}\bar{\phi}-2(5-2\Delta)\partial_{\rho}\bar{\phi}+(3-\Delta)\partial_{\rho}(\log\bar{g})\bar{\phi}\right)$
$\displaystyle+\rho^{2}\left(2\partial_{\rho}(\log\bar{g})\partial_{\rho}\bar{\phi}+\partial_{\rho}^{2}\bar{\phi}\right)~{}.$
(18)

At zero-th order in the holographic coordinate, this equation gives rise to a
relation between the mass and $\Delta$ given by

$\displaystyle m^{2}\ell^{2}=\Delta(\Delta-3)~{}.$ (19)

As shown in Refs. Gubser:1998bc ; Witten:1998qj , this relation corresponds to
the one between the mass of a scalar field on an AdS background and the
conformal dimension of the dual operator at the boundary. The latter is
determined by the rescaling properties of a scalar operator in the CFT and it
can be obtained by analyzing the 1-point function of the holographic operator.
It is worth noticing that, in a unitary dual theory, the scaling dimension of
a scalar operator must be a positive integer, which defines constraints on the
allowed values of the mass of the corresponding bulk field. These constraints,
combined with the Breitenlohner-Freedman (BF) bound Breitenlohner:1982bm ;
Klebanov:1999tb , further restrict the mass to lie within a given interval,
that is,

$\displaystyle-\left(\frac{d}{2}\right)^{2}<m^{2}\ell^{2}<-\left(\frac{d}{2}\right)^{2}+1~{}.$
(20)

This bound serves to understand the interplay between bulk and boundary
physics in AdS/CFT correspondence, matching the stability of the bulk scalar
with the unitarity of the dual theory.

In order to determine the coefficients in the expansion (16), one needs to
solve the field equations (2) order by order in the holographic coordinate.
This process, however, cannot be carried out for a generic $\Delta$ and it
needs to be addressed case by case, as discussed in Ref. deHaro:2000vlm .
Looking at the scalar field expansion in Eq. (2), at first-order in the
holographic coordinate, there appears a second-order differential equation for
the scalar field which also involves first derivatives of the boundary metric.
In turn, in the equations of motion for the metric, the lowest-order term in
$\rho$ is proportional to $\rho^{2-\Delta}$. To ensure a consistent asymptotic
expansion, one should demand that $\Delta\leq 3$. For instance, when
$\Delta=2$, a self-interaction emerges between the boundary fields at the
leading order, producing a backreaction on the geometry of the dual CFT; this
represents a critical value of the conformal dimension. Alternatively, for
$\Delta=1$, the kinetic term backreacts on the boundary only at the next-to-
leading order. Similarly, when $\Delta=0$, the bulk scalar field becomes
massless and the backreaction mainly comes from the kinetic term of the
boundary scalar. These different choices of $\Delta$ lead to distinct
behaviors and interactions between the bulk and boundary fields, offering
valuable insights into the AdS/CFT correspondence.

Let us focus on the case $\Delta=2$ or, equivalently, $m^{2}\ell^{2}=-2$. This
value is, indeed, admissible by the BF bound. Then, we proceed to solve the
Einstein equations to determine the first coefficient in the metric expansion
in terms of the sources. The coefficient reads

$\displaystyle g_{(2)ij}=-{\cal
S}_{ij}(g_{(0)})-\frac{\kappa}{4}\phi^{2}_{(0)}g_{(0)ij}~{},$ (21)

where

$\displaystyle S_{ij}(g_{(0)}):={\cal
R}_{ij}(g_{(0)})-\frac{1}{4}g_{(0)ij}{\cal R}(g_{(0)})~{},$ (22)

is the Schouten tensor of the boundary metric $g_{(0)}$. An arbitrary
variation of the on-shell action and using the asymptotic expansion yields

$\displaystyle\delta S_{\rm min}={}$
$\displaystyle\frac{\ell^{2}}{2\kappa}\int_{\partial\cal
M}d^{3}x\sqrt{-g_{(0)}}\rho^{-1/2}\Bigg{[}\left(-\frac{2}{\rho}g_{(0)ij}-3g_{(2)ij}+{\cal
O}\left(\rho^{1/2}\right)\right)\delta g_{(0)}^{ij}$
$\displaystyle-\ell^{2}\left(\phi_{(0)}+{\cal
O}\left(\rho^{1/2}\right)\right)\delta\phi_{(0)}\Bigg{]}~{}.$ (23)

Therefore, in order to preserve a well-posed variational principle, one needs
to add only intrinsic boundary terms. Indeed, inverting the series, one finds
that the surface terms needed are

$\displaystyle S_{\rm ct}+S^{\prime}_{\phi}=\frac{1}{\kappa}\int_{\partial\cal
M}d^{3}x\sqrt{-h}\left(\frac{2}{\ell}+\frac{\ell}{2}{\cal
R}(h)\right)+\frac{1}{\kappa}\int_{\cal
M}d^{3}x\sqrt{-h}\left(\frac{\phi^{2}}{2\ell}\right),$ (24)

which renormalize the gravity sector Balasubramanian:1999re ; deHaro:2000vlm
and the counterterm for a massive scalar field on AdS which cures the
divergences coming from the scalar sector Klebanov:1999tb .333The covariant
counterterms for the massive scalar including logarithmic modes have been
found explicitly in Ref. deHaro:2000vlm , up to second order. Then, the action

$\displaystyle S_{\rm min}^{\rm ren}=S_{\rm min}+S_{\rm
ct}+S^{\prime}_{\phi}~{},$ (25)

is finite on shell. Nonetheless, it is possible to add an extra term

$\displaystyle
S^{\prime}_{\partial\phi}=\frac{\gamma}{2\kappa}\int_{\partial\cal
M}d^{3}x\sqrt{-h}\;\phi\,n^{\mu}\partial_{\mu}\phi~{},$ (26)

where $n^{\mu}$ is the outward-pointing unit normal to the boundary. Its
coupling $\gamma$, by the use of a Legendre transformation, redefines the mass
of the scalar field. This extra boundary term has been considered in Refs.
Lu:2013ura ; Lu:2014maa  to obtain the correct thermodynamics for a given
$\gamma$ which matches the ADM mass. At first glance, this term may be at odds
with a variational principle based on mixed boundary conditions (see Appendix
A). However, within a holographic framework, what is relevant is that the
variation of the action is finite and written down in terms of the variation
of the holographic sources 444A similar discussion, for the metric field,
leads to extrinsic counterterms in AdS gravity Anastasiou:2020zw . That is the
reason why, one can replace this extrinsic term by another which depends
explicitly on the sources and the boundary conditions Anabalon:2015xvl . Then,
the counterterm for the scalar field is given by

$\displaystyle S_{\phi}=\int_{\partial\cal
M}d^{3}x\sqrt{-h}\left(\frac{\phi^{2}}{2\ell}+\frac{W\left(\phi_{(0)}\right)}{\ell\phi_{(0)}^{3}}\phi^{3}\right)~{},$
(27)

where $W\left(\phi_{(0)}\right)$ is determined by the boundary conditions
imposed (see Appendix A). Then, the Euclidean renormalized action for the
minimally coupled scalar with $\Delta=2$ reads

$\displaystyle S_{\rm min}^{\rm ren}={}$ $\displaystyle S_{\rm min}+S_{\rm
ct}+S_{\phi}$ $\displaystyle={}$ $\displaystyle-\frac{1}{2}\int_{\cal
M}d^{4}x\sqrt{g}\left(\frac{R+6\ell^{-2}}{\kappa}-\left(\partial\phi\right)^{2}-m^{2}\phi^{2}\right)$
$\displaystyle+\frac{1}{\kappa}\int_{\partial\cal
M}d^{3}x\sqrt{h}\left[K-\frac{2}{\ell}-\frac{\ell}{2}{\cal
R}(h)-\kappa\left(\frac{\phi^{2}}{2\ell}+\frac{W\left(\phi_{(0)}\right)}{\ell\phi_{(0)}^{3}}\phi^{3}\right)\right]~{}.$
(28)

Additionally, the quasi-local stress-energy tensor is given by

$\displaystyle
T_{ij}[h]=-\frac{1}{\kappa}\left[K_{ij}-Kh_{ij}+\frac{2}{\ell}h_{ij}-\ell\left({\cal
R}_{ij}-\frac{1}{2}{\cal
R}h_{ij}\right)\right]-h_{ij}\left(\frac{\phi^{2}}{2\ell}+\frac{W\left(\phi_{(0)}\right)}{\ell\phi_{(0)}^{3}}\phi^{3}\right)~{},$
(29)

and it provides a regular holographic stress tensor through Eq. (10). In the
next section, we follow to same procedure to find the covariant counterterms
in theories of gravity with a non-minimally coupled scalar field.

##  3 Scalar-Gauss-Bonnet gravity

Let us consider scalar-GB gravity (sGB). This theory represents a particular
sector of Horndeski gravity and it involves the coupling of the GB invariant
with an arbitrary smooth function of the scalar field, building on top of the
action in Eq. (11) while omitting the GHY term. Specifically, the action for
sGB gravity is

$\displaystyle S_{\rm
sGB}=\frac{1}{2}\int_{\mathcal{M}}d^{4}x\sqrt{-g}\left(\frac{R+6\ell^{-2}}{\kappa}-\left(\partial\phi\right)^{2}-m^{2}\phi^{2}+2f(\phi){\cal
G}\right)~{},$ (30)

where the GB term is given by

$\displaystyle{\cal G}\equiv
R_{\mu\nu\lambda\sigma}R^{\mu\nu\lambda\sigma}-4R_{\mu\nu}R^{\mu\nu}+R^{2}~{}.$
(31)

The field equations of the theory from arbitrary variations with respect to
the metric and the scalar field, which yield

$\displaystyle R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R+\Lambda g_{\mu\nu}={}$
$\displaystyle\kappa\left(T_{\mu\nu}+C_{\mu\nu}\right)~{},$
$\displaystyle\Box\phi-m^{2}\phi={}$
$\displaystyle-f^{\prime}(\phi)\left(R_{\mu\nu\lambda\sigma}R^{\mu\nu\lambda\sigma}-4R_{\mu\nu}R^{\mu\nu}+R^{2}\right)~{},$
(32)

respectively, where $T_{\mu\nu}$ has been defined in Eq. (13) and

$\displaystyle C^{\mu}_{\nu}={}$
$\displaystyle-2\delta^{\mu\alpha\beta\gamma}_{\nu\lambda\sigma\tau}R_{\beta\gamma}^{\sigma\tau}\nabla^{\lambda}\nabla_{\alpha}f(\phi)\,.$
(33)

Let us consider a power-series expansion of the scalar coupling function

$\displaystyle f(\phi)=\sum_{n=0}^{\infty}f_{(n)}\phi^{n}~{}.$ (34)

Using the same asymptotic expansion as in the minimally coupled scalar [cf.
Eq. (16)], the scalar equation becomes

$\displaystyle
24f_{(1)}\rho^{(\Delta-3)/2}+\left(48f_{(2)}+\ell^{2}\left(\Delta(\Delta-3)-\ell^{2}m^{2}\right)\right)\bar{\phi}+\mathcal{O}\left(\rho^{(\Delta+1)/2}\right)=0~{}.$
(35)

The explicit form of the field equations in this frame turn rather lengthy and
we shall not present them here. However, it is worth noticing that the
leading-order analysis indicates that $\Delta>1$ is necessary in order to have
a nontrivial scalar source. Additionally, if $\Delta>3$, there is no
backreaction of the scalar field on the boundary metric, giving a trivial
scalar source. Consequently, similar to the minimally coupled case, we select
$\Delta=2$ for our analysis. This choice leads to more interesting dynamics,
including the interaction between the scalar field and the boundary metric.
Notice that, for this value of $\Delta$, the scalar field gives a finite
contribution to the on-shell action already at the cubic order. In this case,
the relation between $\Delta$ and the mass of the scalar field becomes

$\displaystyle
m^{2}\ell^{2}=\Delta(\Delta-3)+\frac{48}{\ell^{2}}f_{(2)}=-2+\frac{48}{\ell^{2}}f_{(2)}~{}.$
(36)

Then, in order to have a well-defined unitary quantum field theory at the
boundary, the BF bound now reads

$\displaystyle-\left(\frac{d}{2}\right)^{2}<\Delta\left(\Delta-3\right)+\frac{48}{\ell^{2}}f_{(2)}<-\left(\frac{d}{2}\right)^{2}+1~{}.$
(37)

For $\Delta=2$ and $D=d+1=4$, this bound is translated into a constraint on
the quadratic coupling of the scalar field to the GB, that is,

$\displaystyle-\frac{\ell^{2}}{192}<f_{(2)}<\frac{\ell^{2}}{64}~{}.$ (38)

The asymptotic analysis of Einstein equations requires that linear order in
the asymptotic expansion of the scalar vanishes, i.e., $f_{(1)}=0$.
Furthermore, the consistency of higher-order terms in the expansion requires
that either $f_{(3)}=0$ or that the boundary value of the scalar field
satisfies $\phi_{(0)}^{2}=0$. In our analysis, we will assume the former. This
choice simplifies the equations and allows us to focus on the relevant aspects
of the theory. It is worth noticing that ${\cal O}(\phi^{4})$ contributions in
the function $f(\phi)$ are finite when considering the on-shell action.
Therefore, they do not play a role in the discussion.

Using the asymptotic expansion to solve the Einstein field equations order by
order one finds that

$\displaystyle g_{(2)ij}={}$ $\displaystyle-{\cal
S}\left(g_{(0)}\right)-\frac{\kappa}{4}\left(1-\frac{48}{\ell^{2}}f_{(2)}\right)\phi_{(0)}^{2}g_{(0)ij}~{},$
$\displaystyle\nabla^{i}g_{(3)ij}={}$
$\displaystyle\frac{2\kappa}{3\ell^{2}}\left\\{\left[32f_{(2)}\phi_{(1)}-8\kappa
f_{(2)}\left(1-\frac{48}{\ell^{2}}f_{(2)}\right)\phi_{(0)}^{2}\right]\partial_{j}\phi_{(0)}-\left(\ell^{2}-32f_{(2)}\right)\phi_{(0)}\partial_{j}\phi_{(1)}\right\\}~{},$
$\displaystyle{\rm Tr}~{}g_{(3)}={}$
$\displaystyle\frac{4\kappa}{3}\left(1-\frac{48}{\ell^{2}}f_{(2)}\right)\left(\frac{4\kappa}{3\ell^{2}}f_{(2)}\phi_{(0)}^{2}-\phi_{(1)}\right)\phi_{(0)}\,,$
(39)

which recovers the coefficient of the minimally coupled scalar theory if
$f_{(2)}=0$. An arbitrary on-shell variation of the action alongside the
asymptotic expansion of the fields yield

$\displaystyle\delta S_{\rm sGB}={}$
$\displaystyle\frac{1}{2\kappa}\int_{\partial{\cal
M}}d^{3}x\sqrt{-\bar{g}}\frac{1}{\rho^{3/2}}\Big{[}\left(\ell^{2}-8\kappa
f_{(0)}\right)\bar{g}^{ij}\delta\bar{g}_{ij}+\rho\Big{\\{}\left(16\kappa
f_{(0)}-2\ell^{2}\right)\bar{g}^{ij}\delta\left(\partial_{\rho}\bar{g}_{ij}\right)$
$\displaystyle-2\ell^{2}\kappa\bar{\phi}\delta\bar{\phi}-\left(8\kappa
f_{(0)}\left({\cal R}(\bar{g})^{ij}-\frac{1}{2}{\cal
R}(\bar{g})\bar{g}^{ij}+\bar{g}^{ij}\bar{g}^{mn}\partial_{\rho}\bar{g}_{mn}\right)+24f_{(2)}\kappa\bar{\phi}^{2}\bar{g}^{ij}\right.$
$\displaystyle+\ell^{2}\partial_{\rho}\bar{g}^{ij}\Big{)}\delta\bar{g}_{ij}\Big{\\}}+{\cal
O}\left(\rho^{2}\right)\Big{]}~{}.$ (40)

If the zeroth-order coefficient of the scalar function expansion is chosen as

$\displaystyle f_{(0)}=\frac{\ell^{2}}{8\kappa}~{},$ (41)

then, the Einstein-Hilbert sector becomes finite and a well-defined
variational principle is achieved in terms of the sources without the need of
a GHY term. This coupling coincides with the one obtained in Ref. Olea:2005gb
for pure Einstein-AdS gravity. In four dimensions, the GB term is purely
topological. This means that adding it to gravity action does not introduce
modifications to the bulk dynamics, even though it changes the value of the
on-shell action and conserved charges in a nontrivial way. Moreover, the
Einstein-Hilbert action coupled to the GB term with the coupling (41) on-shell
is a sector of conformal gravity as shown in Ref. Miskovic:2009bm . The
possibility to embed Einstein gravity in conformal gravity has shown to be
useful in renormalizing gravity coupled to conformally coupled scalar fields
Anastasiou:2022wjq .

The inclusion of the GB invariant together with the counterterm in Eq. (27) is
sufficient to have a renormalized on-shell action. To see this, let us first
define

$\displaystyle\overline{T}_{\mu\nu}:={}$
$\displaystyle\left(T_{\mu\nu}-\frac{1}{2}g_{\mu\nu}T_{\mu\nu}\right)+\left(C_{\mu\nu}-\frac{1}{2}g_{\mu\nu}C\right)~{}.$
(42)

This structure appears in the on-shell value of the Weyl tensor and the GB
density. Using the on-shell relation

$\displaystyle W_{\alpha\beta}^{\mu\nu}W_{\mu\nu}^{\alpha\beta}={}$
$\displaystyle
R_{\alpha\beta}^{\mu\nu}R_{\mu\nu}^{\alpha\beta}-\frac{24}{\ell^{2}}+\frac{4\kappa}{\ell^{2}}\overline{T}+\kappa^{2}\left(\delta^{\mu\nu}_{\alpha\beta}\delta^{\alpha\lambda}_{\mu\eta}\overline{T}^{\beta}_{\lambda}\overline{T}^{\eta}_{\nu}-4\overline{T}^{\mu}_{\nu}\overline{T}^{\nu}_{\mu}-\frac{2}{3}\overline{T}^{2}\right)~{},$
(43)

we can write the GB density in terms of the square of the Weyl tensor. Then,
the on-shell action can be written as

$\displaystyle S_{\rm sGB}={}$ $\displaystyle\int_{\cal
M}d^{4}x\sqrt{-g}\Bigg{[}-\frac{3}{\kappa\ell^{2}}+\frac{1}{2}\overline{T}+\left(f(\phi)-\frac{1}{2}\phi
f^{\prime}(\phi)\right)\Bigg{(}W^{\mu\nu}_{\alpha\beta}W^{\alpha\beta}_{\mu\nu}+\frac{24}{\ell^{2}}$
$\displaystyle-\frac{4\kappa}{\ell^{2}}\overline{T}+\kappa^{2}\left(\frac{5}{3}\overline{T}^{2}-\delta^{\mu\nu}_{\alpha\beta}\delta^{\alpha\lambda}_{\mu\eta}\overline{T}^{\beta}_{\lambda}\overline{T}^{\eta}_{\nu}\right)\Bigg{)}\Bigg{]}-\frac{1}{2}\int_{\partial\cal
M}d^{3}x\sqrt{-h}\,n^{\mu}\phi\partial_{\mu}\phi~{}.$ (44)

Notice that

$\displaystyle f(\phi)-\frac{1}{2}\phi
f^{\prime}(\phi)=f_{(0)}-\frac{3}{2}f_{(3)}\phi^{3}-2f_{(4)}\phi^{4}+\dots=\sum_{n=0}^{\infty}\left(1-\frac{n}{2}\right)f_{(n)}\phi^{n}~{},$
(45)

does not contain quadratic terms. Then, the scalar-GB coupling cannot be used
to remove quadratic self-interaction terms.

Since the Weyl square term is finite for AAdS spacetimes, we can choose the
value of Eq. (41) such that the first two bulk terms cancel. Then, we are left
only with quadratic terms in $\overline{T}$. These terms remain finite if the
action contains quadratic kinetic terms and/or quadratic self-interacting
potentials as in the case in sGB gravity. Hence, the only divergences that
cannot be eliminated by the GB density are those associated with the minimally
coupled scalar field, that can be renormalized using the counterterm in Eq.
(27). As a result, the renormalized action can be expressed as

$\displaystyle S_{\rm sGB}^{\rm ren}=S_{\rm sGB}+S_{\phi}~{},$ (46)

where $f(\phi)=\frac{\ell^{2}}{8\kappa}+f_{(2)}\phi^{2}$, even though higher-
order terms could be considered as they give finite contributions.

The holographic 1-point for the scalar field depends on the boundary
conditions and is controlled $W\left(\phi_{(0)}\right)$. For the holographic
stress-energy tensor we obtain

$\displaystyle\langle T_{ij}\rangle={}$
$\displaystyle-\ell^{2}\left[W\left(\phi_{(0)}\right)-\left(1-\frac{32}{\ell^{2}}f_{(2)}\right)\phi_{(0)}\phi_{(1)}+\frac{8\kappa}{\ell^{2}}f_{(2)}\left(1-\frac{{48}}{\ell^{2}}f_{(2)}\right)\phi_{(0)}^{3}\right]g_{(0)ij}$
$\displaystyle+\frac{3\ell^{2}}{2\kappa}g_{(3)ij}~{},$ (47)

whose trace yields

$\displaystyle\langle
T\rangle=-3\ell^{2}\left[W\left(\phi_{(0)}\right)-\frac{1}{3}\phi_{(0)}\phi_{(1)}-\frac{16\kappa}{3\ell^{2}}f_{(2)}\left(1-\frac{{48}}{\ell^{2}}f_{(2)}\right)\phi_{(0)}^{3}\right]~{}.$
(48)

Notice that if we consider mixed boundary conditions that respect conformal
invariance, i.e., $W\left(\phi_{(0)}\right)=C\phi_{(0)}^{3}$ , with $C$ some
constant (see Ref. Henneaux:2006hk ), one obtains

$\displaystyle\langle T\rangle=-16\kappa
f_{(2)}\left(1-\frac{{48}}{\ell^{2}}f_{(2)}\right)\phi_{(0)}^{3}~{}.$ (49)

Focusing on the nontrivial quadratic self-interaction, that is $f_{(2)}\neq
0$, we find that conformal invariance of the boundary CFT implies

$\displaystyle f_{(2)}=\frac{\ell^{2}}{{48}}~{},$ (50)

which is not admissible within the BF bound. Therefore, the theory that
consists solely of a scalar field with $\Delta=2$ and a quadratic self-
interacting potential coupled to the GB term does not satisfy the BF bound.
Consequently, it leads to a non-unitary dual CFT.

The above results indicate that the addition of the GB term in the bulk is
useful to renormalize the bulk theory, in a similar fashion as in the pure AdS
gravity case. This invariant, when expressed as a boundary term, can be
thought of as an extrinsic counterterm series. It is noteworthy that the
boundary contribution of the GB term in four dimensions cannot be seen as a
quasilocal stress tensor. However, it does contribute to the holographic
stress tensor and renders the variation of the action finite.

If we consider Dirichlet boundary conditions, denoted as
$W\left(\phi_{(0)}\right)=0$, the variation of the renormalized action
introduces an additional piece arising from the boundary value of the scalar
field. As a result, the vacuum expectation value of the boundary scalar
operator can be expressed as

$\displaystyle\langle{\cal O}\rangle=\frac{1}{\sqrt{-g_{(0)}}}\frac{\delta
S_{\rm H}^{\rm ren}}{\delta\phi_{(0)}}=\lim_{\epsilon\to
0}\left(\frac{1}{\epsilon^{\Delta-d}}\frac{1}{\sqrt{-h}}\frac{\delta S_{\rm
H}^{\rm
ren}}{\delta\phi}\right)=-\phi_{(1)}=-(2\Delta-d)\phi_{(2\Delta-d)}~{},$ (51)

which matches the holographic 1-point function of the scalar operator dual to
a scalar field on an AdS background with Dirichlet boundary conditions.

If we take instead $f_{(3)}\neq 0$ and $\phi_{(0)}^{2}=0$ , we found that
asymptotic analysis of the field equations gives

$\displaystyle g_{(2)}={}$ $\displaystyle-{\cal
S}\left(g_{(0)}\right)_{ij}~{},$ $\displaystyle\nabla^{i}_{(0)}g_{(3)ij}={}$
$\displaystyle\frac{2\kappa}{3\ell^{2}}\left[32f_{(2)}\phi_{(1)}\partial_{j}\phi_{(0)}-\left(\ell^{2}-32f_{(2)}\right)\phi_{(0)}\partial_{j}\phi_{(1)}\right]~{},$
$\displaystyle{\rm Tr}~{}g_{(3)}={}$
$\displaystyle-\frac{4\kappa}{3\ell^{2}}\left(\ell^{2}-48f_{(2)}\right)\phi_{(0)}\phi_{(1)}~{}.$
(52)

Using the same counterterm as in the previous case, given in Eq. (27), we find
that

$\displaystyle\langle
T_{ij}\rangle=\left(\ell^{2}-32f_{(2)}\right)\phi_{(0)}\phi_{(1)}g_{(0)ij}+\frac{3\kappa}{2\ell^{2}}g_{(3)ij}~{},$
(53)

which, independent of the boundary conditions, is traceless. Moreover,
considering Dirichlet boundary conditions one gets

$\displaystyle\langle{\cal
O}\rangle=-\phi_{(1)}=-(2\Delta-d)\phi_{(2\Delta-d)}~{},$ (54)

just as in the previous scenario.

Another interesting scenario to explore involves setting $\Delta=3$. In this
case, the scalar field behaves as

$\displaystyle\phi(\rho,x^{i})=\phi_{(0)}(x^{i})+\rho\phi_{(2)}(x^{i})+\rho^{3/2}\phi_{(3)}(x^{i})+\dots~{},$
(55)

near the boundary. The relation between the mass and $\Delta$ becomes

$\displaystyle m^{2}\ell^{2}=\frac{48}{\ell^{2}}f_{(2)}~{}.$ (56)

Solving the field equations order by order, we can derive several conditions
on the scalar couplings $f_{(n)}$. First, the zeroth-order equations yield
$f_{(1)}=0=f_{(3)}$, or alternatively, $f_{(1)}=0$ together with
$\phi_{(0)}^{2}=0$. Assuming the former condition, the Einstein equations
impose $f_{(2)}=0$. Then, we obtain

$\displaystyle g_{(2)ij}={}$ $\displaystyle-{\cal
S}(g_{(0)})_{ij}-\frac{\kappa}{4}g_{(0)ij}\partial^{m}\phi_{(0)}\partial_{m}\phi_{(0)}+\kappa\partial_{i}\phi_{(0)}\partial_{j}\phi_{(0)}~{},$
$\displaystyle\phi_{(2)}={}$
$\displaystyle\frac{1}{2}\Box_{(0)}\phi_{(0)}~{},$ $\displaystyle{\rm
Tr}~{}g_{(3)}={}$ $\displaystyle 0~{},$
$\displaystyle\nabla^{i}_{(0)}g_{(3)ij}={}$
$\displaystyle\frac{2\kappa}{\ell^{2}}\phi_{(1)}\partial_{j}\phi_{(0)}~{}.$
(57)

Arbitrary variations of the on-shell action make evident that selecting the
zeroth-order coupling in Eq. (41) eliminates the leading-order divergences.
However, to address the remaining divergences associated with the scalar
field, we need to introduce an appropriate counterterm. We have determined
that including the intrinsic counterterm

$\displaystyle S_{\rm sGB}^{\rm ct}=-\frac{\ell}{2}\int_{\partial\cal
M}d^{3}x\sqrt{-h}h^{ij}\partial_{i}\phi\partial_{j}\phi~{},$ (58)

together with the choice in Eq. (41), renders the renormalized action

$\displaystyle S_{\rm sGB}^{\rm ren}=S_{\rm sGB}+S_{\rm sGB}^{\rm ct}~{},$
(59)

finite on shell. This approach allows us to handle and regularize the
divergences encountered in the theory. The resulting holographic stress tensor
is given by

$\displaystyle\langle T^{ij}\rangle=\frac{3\ell^{2}}{2\kappa}g_{(3)}^{ij}~{}.$
(60)

Remarkably, this holographic stress-energy tensor is traceless as it can be
seen from Eq. (57). Additionally, we find a non-zero vacuum expectation value
for the boundary scalar operator when we consider Dirichlet boundary
conditions

$\displaystyle\langle{\cal
O}\rangle=-3\phi_{(3)}=-(2\Delta-d)\phi_{(2\Delta-d)}~{}.$ (61)

This expectation value is akin to the behavior observed for the scalar field
in AdS. These results provide insights into the behavior of bulk fields and
their dual operators in the context of the AdS/CFT correspondence.

Finally, considering $\phi_{(0)}^{2}=0$ while keeping $f_{(3)}$ unconstrained,
the holographic data now reads

$\displaystyle g_{(2)ij}={}$ $\displaystyle-{\cal
S}(g_{(0)})_{ij}-\frac{\kappa}{4}g_{(0)ij}\partial_{m}\phi_{(0)}\partial^{m}\phi_{(0)}~{},$
$\displaystyle\phi_{(2)}={}$
$\displaystyle\frac{1}{2}\left(1-\frac{72}{\ell^{2}}f_{(3)}\phi_{(0)}\right)^{-1}\Box_{(0)}\phi_{(0)}~{},$
$\displaystyle\rm{Tr}~{}g_{(3)}={}$
$\displaystyle\frac{64\kappa}{\ell^{2}}f_{(2)}\phi_{(0)}\phi_{(3)}~{},$
$\displaystyle\nabla^{i}_{(0)}g_{(3)ij}={}$
$\displaystyle\frac{2\kappa}{3\ell^{2}}\left[\left(3+\frac{32}{\ell^{2}}f_{(2)}\right)\phi_{(3)}\partial_{j}\phi_{(0)}+\frac{32}{\ell^{2}}f_{(2)}\phi_{(0)}\partial_{j}\phi_{(3)}\right]~{}.$
(62)

In this case, the holographic stress tensor is

$\displaystyle\langle
T_{ij}\rangle=\frac{3\ell^{2}}{2\kappa}g_{(3)ij}-\frac{32}{\ell^{2}}f_{(2)}\phi_{(0)}\phi_{(3)}~{},$
(63)

which is traceless as a consequence of Eq.(3).

##  4 Kinetic coupling to the Einstein tensor

A particular sector of the Horndeski theory, which has been widely studied,
considers the nonminimal coupling of the scalar field to the Einstein tensor.
This term belongs to the Horndeski class of gravity theories. In the case
under consideration here, a scalar coupling of the GB term is also included

$\displaystyle S_{\rm H}=\int_{\cal
M}d^{4}x\sqrt{-g}\left(\frac{R-2\Lambda}{2\kappa}-\frac{1}{2}\left(\partial\phi\right)^{2}-\frac{1}{2}m^{2}\phi^{2}+f(\phi){\cal
G}+\frac{\eta}{2}G_{\mu\nu}\nabla^{\mu}\phi\nabla^{\nu}\phi\right)~{},$ (64)

The field equations can be obtained by demanding arbitrary variations for the
metric and scalar field, giving

$\displaystyle G_{\mu\nu}+\Lambda g_{\mu\nu}={}$
$\displaystyle\kappa\left(T_{\mu\nu}+C_{\mu\nu}+H_{\mu\nu}\right)~{},$ (65)
$\displaystyle\nabla_{\mu}\left[\left(g^{\mu\nu}-\eta
G_{\mu\nu}\right)\nabla_{\nu}\phi\right]={}$ $\displaystyle
m^{2}\phi-f^{\prime}(\phi){\cal G}~{},$ (66)

respectively, where $T_{\mu\nu}$ and $C_{\mu\nu}$ have been defined in Eqs.
(13) and (33), and

$\displaystyle H_{\mu\nu}={}$
$\displaystyle\frac{\eta}{4}\Big{[}\delta^{\sigma\alpha\beta}_{\mu\lambda\rho}\nabla_{\sigma}\phi\nabla_{\nu}\phi
R_{\alpha\beta}^{\lambda\rho}+\delta^{\rho\lambda\sigma}_{\alpha\beta\mu}\nabla_{\rho}\phi\nabla^{\alpha}\phi
R^{\beta}_{~{}\nu\lambda\sigma}$
$\displaystyle+2g_{\mu\nu}G_{\alpha\beta}\nabla^{\alpha}\phi\nabla^{\beta}\phi+2\delta^{\alpha\beta\lambda}_{\rho\sigma\nu}g_{\lambda\mu}\nabla^{\sigma}\phi\nabla_{\beta}\phi\left(\nabla_{\alpha}\phi\nabla^{\rho}\phi\right)\Big{]}~{},$
(67)

is the contribution to the field equations coming from the scalar-kinetic
coupling to the Einstein tensor.

For a massless scalar field, the absence of the GB coupling implies that this
theory is endowed with a shift symmetry in field space, namely, the field
equations remain invariant under $\phi\to\phi+\tilde{\phi}_{0}$, where
$\tilde{\phi}_{0}$ is a constant. This the typical symmetry exhibited by
Galileons Nicolis:2008in ; Deffayet:2009wt ; Deffayet:2009mn .555If the scalar
coupling to the GB is linear, the theory still has shift symmetry. However,
this possibility was excluded by the dynamics as shown in the previous
section. In that case, however, there exists a no-hair theorem which prevents
from finding black hole solutions with a nontrivial scalar field Hui:2012qt .
Nevertheless, a suitable on-shell condition on the metric can be imposed such
that the no-hair theorem is circumvented, allowing for asymptotically locally
flat black holes Rinaldi:2012vy . Later, the same idea was extended to the
case with the cosmological constant Anabalon:2013oea  and with Maxwell fields
Cisterna:2014nua . In this section, we consider the coupling between the
scalar field and the GB term such that the shift symmetry is broken, to see
how the BF bound is modified with respect to the one found in the previous
section.

###  4.1 $\Delta=2$

Performing the asymptotic analysis in the presence of the kinetic coupling to
the Einstein tensor, we find that the mass relation is modified according to

$\displaystyle
m^{2}\ell^{2}=\Delta(\Delta-3)\left(1-\frac{3}{\ell^{2}}\eta\right)+\frac{48}{\ell^{2}}f_{(2)}\,.$
(68)

In this case, the BF bound turns out to be

$\displaystyle-\frac{\ell^{2}}{192}\left[9-4\Delta(3-\Delta)\left(1-\frac{3}{\ell^{2}}\eta\right)\right]<f_{(2)}<-\frac{\ell^{2}}{192}\left[5-4\Delta(\Delta-3)\left(1-\frac{3}{\ell^{2}}\eta\right)\right]\,,$
(69)

and, focusing on the case $\Delta=2$, it becomes

$\displaystyle-\frac{\ell^{2}}{192}\left(1+\frac{24}{\ell^{2}}\eta\right)<f_{(2)}<\frac{\ell^{2}}{64}\left(1-\frac{8}{\ell^{2}}\eta\right)~{}.$
(70)

The boundary scalar equations impose that $f_{(1)}=0$ , together with either
$f_{(3)}=0$ or $\phi_{(0)}^{2}=0$. We choose $f_{(3)}=0$ to ensure that the
boundary value of the scalar field remains unconstrained, displaying a
nontrivial interaction with the boundary metric. Then, solving the equations
of motion order by order, the coefficients are found to be

$\displaystyle g_{(2)ij}={}$ $\displaystyle-{\cal
S}(g_{(0)})_{ij}-\frac{\kappa}{4}\left(1-\frac{48}{\ell^{2}}f_{(2)}-\frac{5}{\ell^{2}}\eta\right)\phi_{(0)}^{2}g_{(0)ij}~{},$
$\displaystyle{\rm Tr}~{}g_{(3)}={}$
$\displaystyle\frac{4\kappa}{3}\left[\frac{4\kappa}{3\ell^{2}}f_{(2)}\left(1-\frac{5}{\ell^{2}}\eta-\frac{48}{\ell^{2}}f_{(2)}\right)\phi_{(0)}^{3}-\left(1-\frac{6}{\ell^{2}}\eta-\frac{48}{\ell^{2}}f_{(2)}\right)\phi_{(0)}\phi_{(1)}\right]~{},$
$\displaystyle\nabla^{i}_{(0)}g_{(3)ij}={}$
$\displaystyle\frac{2\kappa}{3\ell^{2}}\Bigg{\\{}\left[\left(32f_{(2)}+2\eta\right)\phi_{(1)}-8\kappa
f_{(2)}\left(1-\frac{48}{\ell^{2}}f_{(2)}-\frac{5}{\ell^{2}}\eta\right)\phi_{(0)}^{2}\right]\partial_{j}\phi_{(0)}$
$\displaystyle-\left(\ell^{2}-32f_{(2)}-5\eta\right)\phi_{(0)}\partial_{j}\phi_{(1)}\Bigg{\\}}~{}.$
(71)

Let us consider the variational problem. Using the asymptotic expansion and
taking an arbitrary variation of the fields, we find

$\displaystyle\delta S_{\rm H}={}$ $\displaystyle\int_{\partial\cal
M}d^{3}x\sqrt{-\bar{g}}\rho^{-\frac{3}{2}}\Big{\\{}\left(\ell^{2}-8\kappa
f_{(0)}\right)\bar{g}^{ij}\left(\delta\bar{g}_{ij}+2\rho\delta\partial_{\rho}\bar{g}_{ij}\right)$
$\displaystyle-2\rho\Big{[}\left(\ell^{2}-3\eta\right)\bar{\phi}\delta\bar{\phi}-\left(4\kappa
f_{(0)}\left(\frac{1}{2}{\cal
R}(\bar{g})\bar{g}^{ij}+\bar{g}^{ij}\bar{g}^{mn}\partial_{\rho}\bar{g}_{mn}-{\cal
R}(\bar{g})_{mn}\bar{g}^{im}\bar{g}^{jn}\right)\right.$
$\displaystyle\left.\left.\left.-2\kappa\left(\eta+12f_{(2)}\right)\bar{\phi}^{2}\bar{g}^{ij}+\frac{\ell^{2}}{2}\bar{g}^{im}\bar{g}^{jn}\partial_{\rho}\bar{g}_{mn}\right)\delta\bar{g}_{ij}\right]+{\cal
O}(\rho^{2})\right\\}~{}.$ (72)

Notice that, in this case, Eq. (41) also removes the divergences coming from
the gravitational sector. Additionally, we must take into account the
counterterm in Eq. (27) associated with the scalar field. However, this
counterterm should involve the kinetic coupling to the Einstein tensor. Then,
the renormalized action turns out to be

$\displaystyle S_{\rm H}^{\rm ren}=S_{\rm
H}+\frac{1}{\ell}\left(1-\frac{3}{\ell^{2}}\eta\right)S_{\phi}~{}.$ (73)

Similar to the previous section, we can write the GB in terms of the square of
the Weyl tensor and check that the on-shell action in Eq. (73) is, indeed,
finite. For the holographic stress-energy tensor, we find that it is given by

$\displaystyle\langle T_{ij}\rangle={}$
$\displaystyle-\ell^{2}g_{(0)ij}\Bigg{[}\left(1-\frac{3}{\ell^{2}}\eta\right)W\left(\phi_{(0)}\right)-\left(1-\frac{32}{\ell^{2}}f_{(2)}-\frac{5}{\ell^{2}}\eta\right)\phi_{(0)}\phi_{(1)}$
$\displaystyle+8\kappa
f_{(2)}\left(1-\frac{5}{\ell^{2}}\eta-\frac{48}{\ell^{2}}f_{(2)}\right)\phi_{(0)}^{3}\Bigg{]}+\frac{3\ell^{2}}{2\kappa}g_{(3)ij}~{}.$
(74)

Imposing suitable boundary conditions, i.e., $W(\phi)=C\phi_{(0)}^{3}$ , the
trace of the latter becomes

$\displaystyle\langle T\rangle={}$ $\displaystyle-16\kappa
f_{(2)}\left(1-\frac{48}{\ell^{2}}f_{(2)}-\frac{5}{\ell^{2}}\eta\right)\phi_{(0)}^{3}~{}.$
(75)

Therefore, conformal invariance is preserved in the boundary field theory if
the trace vanishes. This implies that $f_{(2)}=0$ , or

$\displaystyle
f_{(2)}=\frac{\ell^{2}}{48}\left(1-\frac{5}{\ell^{2}}\eta\right)~{}.$ (76)

The latter condition is admissible by the BF bound if

$\displaystyle-\frac{5\ell^{2}}{4}<\eta<-\frac{\ell^{2}}{4}~{}.$ (77)

Then, $\eta$ has to be negative and it has to lie within the BF bound defined
in the previous equation. Notice that taking $\eta=0$ is not admissible. This
shows how the kinetic coupling to the Einstein tensor can be used to fix the
unitarity problem in sGB with $\Delta=2$. Moreover, if we consider Dirichlet
boundary conditions, i.e. $W\left(\phi_{(0)}\right)=0$, we find an additional
contribution to the vacuum expectation value of the boundary scalar coming
from the kinetic coupling, namely,

$\displaystyle\langle{\cal
O}\rangle=-\left(1-\frac{3}{\ell^{2}}\eta\right)\phi_{(1)}=-\left(1-\frac{3}{\ell^{2}}\eta\right)(2\Delta-d)\phi_{(2\Delta-d)}~{}.$
(78)

This modifies the result of the minimally coupled scalar field by a factor of
$(1-3\eta\ell^{-2})$. Moreover, diffeomorphism invariance of $g_{(0)}$ at the
boundary implies a holographic Ward identity. While the coefficient in the
expansion at a holographic order, $g_{(3)}$, can not be determined, we can
determine its trace and divergence throughout the field equations. Then, the
Ward identity reads

$\displaystyle\nabla^{i}_{(0)}\langle
T_{ij}\rangle=\left(1-\frac{3}{\ell^{2}}\eta\right)\phi_{(1)}\partial_{j}\phi_{(0)}=-\langle{\cal
O}\rangle\partial_{j}\phi_{(0)}~{},$ (79)

which matches the result of Ref. deHaro:2000vlm  for the minimally coupled
scalar field. This computation of the holographic energy tensor-momentum
tensor is not associated to a quasi-local stress tensor on the gravity side.
However, the coupling of matter renders its divergence different from zero, as
it is related to the flow of momentum out of the boundary Brown:1992br .

Let us examine Horndeski theory by considering a different possibility. For
instance, if we fix $\Delta=2$ with $\phi_{(0)}^{2}=0$ while keeping $f_{(3)}$
arbitrary, we obtain

$\displaystyle g_{(2)ij}={}$ $\displaystyle-{\cal S}(g_{(0)})_{ij}~{},$
$\displaystyle{\rm Tr}~{}g_{(3)}={}$
$\displaystyle-\frac{4\kappa}{3}\left(1-\frac{48}{\ell^{2}}f_{(2)}-\frac{6}{\ell^{2}}\eta\right)\phi_{(0)}\phi_{(1)}~{},$
$\displaystyle\nabla_{(0)}^{i}g_{(3)ij}={}$
$\displaystyle\frac{2\kappa}{3\ell^{2}}\left[\left(32f_{(2)}+2\eta\right)\phi_{(1)}\partial_{j}\phi_{(0)}-\left(\ell^{2}-32f_{(2)}-5\eta\right)\phi_{(0)}\partial_{j}\phi_{(1)}\right]~{}.$
(80)

The variation of the on-shell action gives the same kind of divergences as in
the previous case. Therefore, the renormalized action must be the one in Eq.
(73). The holographic stress tensor in this case becomes

$\displaystyle\langle
T_{ij}\rangle=\ell^{2}\left(1-\frac{32}{\ell^{2}}f_{(2)}-\frac{5}{\ell^{2}}\eta\right)\phi_{(0)}\phi_{(1)}g_{(0)ij}+\frac{3\ell^{2}}{2\kappa}g_{(3)ij}~{},$
(81)

which is always traceless as $\phi_{(1)}$ is proportional to positive powers
of $\phi_{(0)}$ when considering mixed boundary conditions. Therefore,
considering $\phi_{(0)}$ to be infinitesimal, one has a well-defined boundary
CFT with a continuous $\eta\to 0$ limit. Finally, using Dirichlet boundary
conditions we find that the vacuum expectation value and the Ward identity
associated to the boundary scalar operator are the same as in the case with
$f_{(3)}=0$ and $\phi_{(0)}$ unfixed.

###  4.2 $\Delta=3$

Consider now the case $\Delta=3$. For this choice, the mass simply becomes
$m^{2}\ell^{4}=48f_{(2)}$ and the scalar equation imposes either
$f_{(1)}=0=f_{(3)}$ or $f_{(1)}=0=\phi_{(0)}^{2}$. Moreover, the zeroth order
of the Einstein equations restricts further the theory with $f_{(2)}=0$ or
$\phi_{(0)}^{2}=0$ . Focusing first on the case when $\phi_{(0)}$ is arbitrary
and solving for the coefficients of the metric, we obtain

$\displaystyle g_{(2)ij}={}$ $\displaystyle-{\cal
S}(g_{(0)})_{ij}-\frac{\kappa}{4}\left(1-\frac{3}{\ell^{2}}\eta\right)g_{(0)ij}\partial_{m}\phi_{(0)}\partial^{m}\phi_{(0)}+\kappa\left(1-\frac{2}{\ell^{2}}\eta\right)\partial_{i}\phi_{(0)}\partial_{j}\phi_{(0)}~{},$
$\displaystyle\phi_{(2)}={}$
$\displaystyle\frac{1}{2}\Box_{(0)}\phi_{(0)}~{},$ $\displaystyle{\rm
Tr}~{}g_{(3)}={}$ $\displaystyle 0\,,$
$\displaystyle\nabla^{i}_{(0)}g_{(3)ij}={}$ $\displaystyle
2\kappa\left(1-\frac{3}{\ell^{2}}\eta\right)\phi_{(3)}\partial_{j}\phi_{(0)}~{}.$
(82)

An arbitrary on-shell variation of the action with $f_{(0)}=\ell^{2}/8\kappa$
yields

$\displaystyle\delta S_{\rm H}={}$
$\displaystyle-\frac{1}{2}\int_{\partial\cal
M}d^{3}x\sqrt{-\bar{g}}\rho^{-1/2}\Big{[}\kappa\eta\left(\bar{g}^{mi}\bar{g}^{nj}\partial_{i}\bar{\phi}\partial_{j}\bar{\phi}\delta\bar{g}_{mn}-\bar{g}^{ij}\partial_{m}\bar{\phi}\partial^{m}\bar{\phi}\delta\bar{g}_{ij}\right)$
$\displaystyle+\ell^{2}\left({\cal R}(\bar{g})^{ij}-\frac{1}{2}{\cal
R}(\bar{g})\bar{g}^{ij}+\bar{g}^{im}\bar{g}^{jn}\partial_{\rho}\bar{g}_{mn}\right)\delta\bar{g}_{ij}+4\kappa\left(\ell^{2}-3\eta\right)\partial_{\rho}\bar{\phi}\delta\bar{\phi}\Big{]}~{}.$
(83)

In order to eliminate the divergencies in this case, we notice that the same
counterterm as in the sGB theory renders the theory finite but with a
different coupling, that is,

$\displaystyle S_{\rm H}^{\rm
ct}=-\frac{\ell}{2}\left(1-\frac{3}{\ell^{2}}\eta\right)\int_{\partial\cal
M}d^{3}x\sqrt{-h}h^{ij}\partial_{i}\phi\partial_{j}\phi~{}.$ (84)

Then, the action

$\displaystyle S_{\rm H}^{\rm ren}=S_{\rm H}+S_{\rm H}^{\rm ct}~{},$ (85)

is on-shell finite. Although the counterterm in Eq. (84) is necessary, most of
the aforementioned solutions have a scalar profile that depends on the radial
coordinate only, making the counterterm identically zero. Thus, the particular
value of Eq. (41) is enough to remove the divergences of the theory.
Nevertheless, it has been found that the theory admits regular solutions if
one considers time-dependent scalar field Babichev:2013cya , such that the
scalar does not inherit the spacetime symmetries, but the stress-tensor does.
Moreover, there are interesting solutions of scalar-tensor gravity theories
containing scalar fields that depend on both the radial and boundary
coordinates such as accelerating black holes Lu:2014ida ; Cisterna:2021xxq ;
Cisterna:2023qhh ; Barrientos:2023tqb , whose holographic properties remain to
be fully understood, and instantons deHaro:2006wy ; deHaro:2000vlm , that can
be used to explore vacuum decay of the boundary conformal theory
Papadimitriou:2007sj . In this case, the holographic stress-energy tensor
becomes

$\displaystyle\langle T_{ij}\rangle=\frac{3\ell^{2}}{2\kappa}g_{(3)}^{ij}~{},$
(86)

which is traceless. Moreover, the vacuum expectation value of the boundary
source becomes

$\displaystyle\langle{\cal
O}\rangle=-3\left(1-\frac{3}{\ell^{2}}\eta\right)\phi_{(3)}=-\left(1-\frac{3}{\ell^{2}}\eta\right)(2\Delta-d)\phi_{(2\Delta-d)}~{}.$
(87)

This shows how the couplings of the theory modify the value of the 1-point
functions of the dual operators. Moreover, the holographic Ward identity
becomes

$\displaystyle\nabla^{i}_{(0)}\langle
T_{ij}\rangle=3\left(1-\frac{3}{\ell^{2}}\eta\right)\phi_{(3)}\partial_{j}\phi_{(0)}=-\langle{\cal
O}\rangle\partial_{j}\phi_{(0)}~{},$ (88)

just as in the previous cases.

Finally, for an arbitrary $f_{(2)}$ and setting the source of the scalar
operator to be infinitesimal, i.e. $\phi_{(0)}^{2}=0$, we find that the
coefficients can be solved as

$\displaystyle g_{(2)ij}={}$ $\displaystyle-{\cal
S}_{(0)ij}+\kappa\left(1-\frac{2}{\ell^{2}}\eta\right)\partial_{i}\phi_{(0)}\partial_{j}\phi_{(0)}$
$\displaystyle+\frac{\kappa}{\ell^{2}}\left(24f_{(2)}\phi_{(0)}\phi_{(2)}-\frac{1}{4}\left(\ell^{2}-3\eta\right)\partial^{m}\phi_{(0)}\partial_{m}\phi_{(0)}\right)g_{(0)ij}$
$\displaystyle\phi_{(2)}={}$
$\displaystyle\frac{1}{2}\left(\frac{\ell^{2}-3\eta}{\ell^{2}-3\eta-72f_{(2)}\phi_{(0)}}\right)\Box_{(0)}\phi_{(0)}$
$\displaystyle{\rm Tr}~{}g_{(3)}={}$
$\displaystyle\frac{64\kappa}{\ell^{2}}f_{(2)}\phi_{(0)}\phi_{(3)}~{},$
$\displaystyle\nabla^{i}_{(0)}g_{(3)ij}={}$
$\displaystyle\frac{2\kappa}{3\ell^{2}}\left[\left(3+\frac{32}{\ell^{2}}f_{(2)}-\frac{9}{\ell^{2}}\eta\right)\phi_{(3)}\partial_{j}\phi_{(0)}+\frac{32}{\ell^{2}}f_{(2)}\phi_{(0)}\partial_{j}\phi_{(3)}\right]~{},$
(89)

Choosing $f_{(0)}$ as in Eq. (41) and considering the same counterterm as in
Eq. (84), the on-shell action becomes finite and the variational principle is
well-posed. Then, the holographic stress-energy tensor becomes

$\displaystyle\langle
T_{ij}\rangle=-\frac{32}{\ell^{2}}f_{(2)}\phi_{(0)}\phi_{(3)}g_{(0)ij}+\frac{3\ell^{2}}{2\kappa}g_{(3)ij}~{},$
(90)

which is traceless by virtue of Eq. (4.2). If we consider Dirichlet boundary
conditions, the scalar operator acquires a nontrivial vacuum expectation value
given by

$\displaystyle\langle{\cal
O}\rangle=-3\left(1-\frac{3}{\ell^{2}}\eta\right)\phi_{(3)}=-\left(2\Delta-d\right)\left(1-\frac{3}{\ell^{2}}\eta\right)\phi_{(2\Delta-d)}~{},$
(91)

and a Ward identity given by

$\displaystyle\nabla^{i}_{(0)}\langle
T_{ij}\rangle=3\left(1-\frac{3}{\ell^{2}}\eta\right)\phi_{(3)}\partial_{j}\phi_{(0)}=-\langle{\cal
O}\rangle\partial_{j}\phi_{(0)}~{},$ (92)

just as in the case with an arbitrary $\phi_{(0)}$.

###  4.3 Minimal Horndeski theory

Finally, we discuss on the particular case $f_{(n)}=0=m^{2}$ , $\forall
n\in{\mathbb{N}}_{>0}$ Charmousis:2011ea ; Charmousis:2011bf , which has
received a lot of attention in a holographic context Filios:2018xvy ;
Kuang:2016edj ; Li:2018kqp ; Liu:2018hzo ; Feng:2018sqm ; Li:2018rgn ;
Jiang:2017imk  as it contains analytic solutions Arratia:2020hoy ;
Anabalon:2013oea ; Cisterna:2014nua ; Cisterna:2015yla ; Cisterna:2016vdx ;
Brihaye:2016lin ; Babichev:2013cya ; Babichev:2016rlq ; Cisterna:2017jmv ;
Stetsko:2018fzt ; Cisterna:2018mww . The action in this case is

$\displaystyle S_{\rm Hmin}=\int_{\cal
M}d^{4}x\sqrt{-g}\left(\frac{R-2\Lambda}{2\kappa}-\frac{1}{2}\left(\alpha
g_{\mu\nu}-\eta G_{\mu\nu}\right)\nabla^{\mu}\phi\nabla^{\nu}\phi+f_{(0)}{\cal
G}\right)~{},$ (93)

where we have included an arbitrary constant for the canonical kinetic term;
this is achieved by rescaling the scalar field and the metric.666Consider
$\phi\to\alpha^{-1}\phi$ and $g_{\mu\nu}\to\alpha g_{\mu\nu}$, such that
$R\to\alpha^{-1}R$. The Einstein tensor is scale-invariant, i.e.,
$G_{\mu\nu}\to G_{\mu\nu}$ and, rescaling,
$\Lambda\to\alpha\Lambda~{},~{}\kappa\to\alpha\kappa$ , the minimal Horndeski
theory with canonical kinetic term, i.e., $\alpha=1$ , becomes that of Eq.
(93). Additionally, we have included the GB term as it will serve as a
counterterm in the scalar-tensor theories we are interested in. Recall that
this sector of Horndeski theory is endowed with a shift symmetry as the scalar
field appears only through derivatives in the action.

The field equations correspond to a subset of Eq. (65); explicitly, they are

$\displaystyle G_{\mu\nu}+\Lambda g_{\mu\nu}={}$
$\displaystyle\kappa\left(T_{\mu\nu}^{\rm min}+H_{\mu\nu}\right)~{},$ (94)
$\displaystyle\nabla^{\mu}\left[\left(\alpha g_{\mu\nu}-\eta
G_{\mu\nu}\right)\nabla^{\nu}\phi\right]={}$ $\displaystyle 0~{},$ (95)

where $T_{\mu\nu}^{\rm min}=\tfrac{\alpha}{2}T_{\mu\nu}$ with the latter
defined in Eq. (13) by setting $m=0$. Using the FG expansion, we find that the
scalar field equation becomes

$\displaystyle\Delta\left(\Delta-3\right)\left(\ell^{2}\alpha-3\eta\right)\bar{\phi}+{\cal
O}\left(\rho\right)=0~{},$ (96)

which is satisfied either if $\Delta=3$ or $\eta=\frac{1}{3}\ell^{2}\alpha$ to
all orders without fixing the conformal weight. If the latter point in the
parameter space is assumed, then the action is renormalized simply by the GB
term if one fixes $f_{(0)}$ as in Eq. (41). Nevertheless, this is a critical
point of the theory. As shown in Refs. Jiang:2017imk ; Feng:2018sqm ;
Li:2018kqp , the theory admits a solution that is nearly AdS with a nontrivial
scalar field whose integration constant appears in the same footing as the
cosmological constant. Thus, it is convenient to introduce an effective
cosmological constant, say $\Lambda_{\rm eff}$, that leads to $\Lambda$ when
$\phi$ vanishes.

In an Einstein-AdS background, the scalar sector of the minimal Horndeski
theory becomes simply a minimally coupled massless scalar, that is,

$\displaystyle{\cal
L}_{\phi}=-\frac{1}{2}\left(\alpha-\frac{3}{\ell^{2}}\eta\right)\left(\partial\phi\right)^{2}~{}.$
(97)

Thus, the absence of ghosts implies the inequality

$\displaystyle\alpha-\frac{3}{\ell^{2}}\eta\geq 0\,.$ (98)

Notice that, if $\alpha\geq 0$, this condition is always fulfilled as long as
the parameter $\eta$ lies within the bound of Eq. (77). This implies both
unitarity and the absence of ghosts. If $\alpha<0$, on the other hand, there
is still a region in the parameter space where the condition (98) is
satisfied. The case $\alpha=0$ is also allowed by the Eq. (77). On the other
hand, the bound is saturated at the critical point in which the scalar
contribution vanishes. These kinds of critical points were also studied in the
pure GB gravity Fan:2016zfs  and the black hole solution of Ref.
Anabalon:2013oea  simply becomes the Schwarzschild-AdS black hole with a
vanishing scalar field. In this case, the holographic stress-energy tensor is
traceless and it equals that of the pure gravity case. Nonetheless, it is
possible to obtain an exact global AdS background with a nontrivial scalar.
The latter breaks the AdS isometries and it contains a logarithmic mode in the
FG expansion. This indicates that the dual theory is scale invariant but not
conformally invariant (see Ref. Nakayama:2013is  for details).

As logarithmic modes are beyond the scope of this paper, we will move forward
and consider $\Delta=3$ with $\alpha\neq 3\eta\ell^{-2}$. Solving Einstein’s
equations order by order, we obtain $f_{(1)}=0=f_{(3)}$ as before, and the
coefficients of the metric expansion are now solved as

$\displaystyle g_{(2)ij}={}$ $\displaystyle-{\cal
S}_{ij}\left(g_{(0)}\right)-\frac{\kappa}{4\ell^{2}}\left(\ell^{2}\alpha-3\eta\right)g_{(0)ij}\partial_{m}\phi_{(0)}\partial^{m}\phi_{(0)}$
$\displaystyle+\frac{\kappa}{\ell^{2}}\left(\ell^{2}\alpha-2\eta\right)\partial_{i}\phi_{(0)}\partial_{j}\phi_{(0)}~{},$
$\displaystyle{\rm Tr}~{}g_{(3)}={}$ $\displaystyle 0~{},$
$\displaystyle\nabla^{i}_{(0)}g_{(3)ij}={}$
$\displaystyle\frac{2\kappa}{\ell^{2}}\left(\alpha-\frac{3}{\ell^{2}}\eta\right)\phi_{(3)}\partial_{j}\phi_{(0)}~{}.$
(99)

Considering an on-shell variation of the action with the GB coupling found in
Eq. (41), we arrive at

$\displaystyle\delta S_{\rm Hmin}={}$
$\displaystyle\frac{1}{2}\int_{\partial\cal
M}d^{3}x\sqrt{-g_{(0)}}\left[\rho^{-\frac{1}{2}}\left(\ell^{2}\alpha-3\eta\right)\left(\frac{1}{2}g_{(0)}^{ij}\phi_{(0)}^{2}-\partial^{i}\phi_{(0)}\partial^{j}\phi_{(0)}\right)\right.$
$\displaystyle\left.-\frac{3\ell^{2}}{2\kappa}g^{ij}_{(3)}\right]\delta
g_{(0)ij}~{}.$ (100)

This is finite if we choose the value in Eq. (41) but the critical value of
$\eta$ is still needed. Nonetheless, the latter can be rendered arbitrary if
we add a suitable counterterm, that is,

$\displaystyle S_{\rm
ct}=-\frac{1}{2\ell}\left(\ell^{2}\alpha-3\eta\right)\int_{\partial\cal
M}d^{3}x\sqrt{-h}h^{ij}\partial_{i}\phi\partial_{j}\phi~{}.$ (101)

With this counterterm, the action is finite. Then, the holographic stress-
energy tensor is given by

$\displaystyle\langle T_{ij}\rangle=\frac{3\ell^{2}}{2\kappa}g_{(3)ij}~{},$
(102)

and the vacuum expectation value for the boundary scalar operator, and the
holographic Ward identity become

$\displaystyle\langle{\cal
O}\rangle=-3\left(\alpha-\frac{3}{\ell^{2}}\eta\right)\phi_{(3)}=-\left(\alpha-\frac{3}{\ell^{2}}\eta\right)(2\Delta-d)\phi_{(2\Delta-d)}~{},$
(103)

and

$\displaystyle\nabla^{i}\langle
T_{ij}\rangle=3\left(\alpha-\frac{3}{\ell^{2}}\eta\right)\phi_{(3)}\partial_{j}\phi_{(0)}=-\langle{\cal
O}\rangle\partial_{j}\phi_{(0)}~{},$ (104)

respectively. Notice that, at the critical point $\alpha=3\eta\ell^{-2}$, the
vacuum expectation value of the scalar field vanishes and the holographic
stress-energy tensor becomes covariantly conserved. In this case, the known
scalar-field solutions either vanish or do not backreact and the action
becomes simply the one of Einstein’s gravity. This shows that taking the limit
to this critical value is consistent also at the level of 1-point functions.

##  5 Discussion

In this work, we have analyzed holographic properties of different gravity
theories of the Horndeski class. The aspects of the renormalization of both
the action and its variation, for AAdS spaces, are worked out in great detail.
To this end, we introduced the GB term nonminimally coupled to an arbitrary
function of the scalar field. One of the main results is that the asymptotic
analysis of the field equations restricts considerably the form of such a
function. In particular, we found that the linear-scalar coupling to the GB
term is not allowed in AAdS spaces. Moreover, if the scalar-GB term is the
only nonminimal coupling in the bulk, we found that the boundary field theory
cannot be unitary and conformal at the same time, since the scalar mass lies
outside the BF bound. This argument limits the possible holographic
applications of this bulk gravity theory.

If the scalar-kinetic coupling to the Einstein tensor is introduced, we found
that the issue mentioned above can be solved, rendering the boundary field
theory holographically consistent. We have analyzed different possibilities
and, in all cases, we obtained the counterterms that render the theory finite,
the vacuum expectation value of the scalar operator at the boundary, and the
holographic stress-energy tensor. Furthermore, we have studied the holographic
Ward identity associated with coordinate transformations at the boundary and
identify how the nonminimal coupling of Horndeski theory produces an anomalous
term.

The minimal Horndeski gravity theory considered in Eq. (93) encompasses
numerous black hole solutions. However, only a limited subset among them
possesses exact solutions for the scalar field, as usually only its radial
derivative is known, rather than the full analytic expression. Nonetheless,
there are a few cases in which this issue can be circumvented. In these
situations, the solutions feature either negligible backreaction or modify the
geometry as an effective cosmological constant. For instance, in Ref.
Anabalon:2013oea , an analytic solution corresponding to a topological
Schwarzschild black hole with a flat transverse section and such effective
cosmological constant was found. There exists an analytic solution for the
scalar field, even though there are new logarithmic divergences near the
boundary. For those cases, the additional divergence introduced in the action
(93) can be thought as coming from an effective cosmological, which appears as
modified by the scalar field, such that can be properly renormalized. This
approach requires a fine-tuned coupling for the GB term as in Eq. (41) but in
terms of an effective cosmological constant. As a result, introducing the GB
term with a different coupling is sufficient to renormalize the additional
divergences in the action, even in the presence of logarithmic modes of the
scalar field.

Interesting questions remain open. In particular, extending this analysis to
include the logarithmic modes is certainly very important, since some of the
known analytic solutions in the literature are endowed with this asymptotic
behavior. This shall be discussed elsewhere. Additionally, it is well known
that in even boundary dimensions, the holographic trace anomaly is related to
the logarithmic modes of the metric Henningson:1998gx . The role of
nonminimally coupled scalar fields in the holographic trace anomaly is indeed
worth studying, alongside the generalization of the counterterms found here to
higher dimensions. On the other hand, the results found here are useful for
studying holographic measurements such as entanglement entropy and
superconductivity, which may be the subject of future work.

###### Acknowledgements.

We thank Giorgos Anastasiou, Adolfo Cisterna, José Figueroa, Julio Oliva, and
Leonardo Sanhueza for helpful discussions. This work is partially funded by
Anillo Grant ANID /ACT210100 _Holography and its applications to High Energy
Physics, Quantum Gravity and Condensed Matter Systems_. The work of NC is
supported by Beca Doctorado Nacional (ANID) 2023 Scholarship No. 21231876. The
work of CC is partially supported by Agencia Nacional de Investigacioón y
Desarrollo (ANID) through FONDECYT grants No 11200025, 1230112, and 1210500.
FD is supported by Beca Doctorado nacional (ANID) 2021 Scholarship No.
21211335, and FONDECYT Regular grant No. 1210500. The work of RO has been
funded by the FONDECYT Regular Grants 1230492 and 1231779.

##  Appendix A Scalar field boundary conditions

In order to stress the importance of boundary conditions in the dual field
theory, let us analyze the case of a massive scalar field on global AdS as an
example. The corresponding action can be expressed as

$\displaystyle S=\frac{1}{2}\int_{{\cal
M}}d^{d+1}x\sqrt{g}\left[\left(\partial\phi\right)^{2}+m^{2}\phi^{2}\right]~{},$
(105)

where ${\cal M}$ is an Euclidean AdSd+1 background with spacetime coordinates
$x^{\mu}=(z,x^{i})$. Its line element in the Poincaré patch is given by

$\displaystyle
ds^{2}=\frac{\ell^{2}}{z^{2}}\left(dz^{2}+\delta_{ij}dx^{i}dx^{j}\right)~{}.$
(106)

An arbitrary variation of the on-shell action with respect to the dynamic
field yields

$\displaystyle\delta S={}$ $\displaystyle-\int_{\cal
M}d^{d+1}x\sqrt{g}\left(\nabla^{2}-m^{2}\right)\phi\delta\phi-\int_{\partial\cal
M}d^{d}x\sqrt{h}\left(n^{\mu}\nabla_{\mu}\phi\right)\delta\phi$
$\displaystyle={}$ $\displaystyle-\int_{\partial\cal
M}d^{d}x\left(\frac{z}{\ell}\right)^{1-d}\partial_{z}\phi\,\delta\phi~{}.$
(107)

Depending on the boundary conditions, an extra boundary term must be added
such that the on-shell action possesses a minimum. For Dirichlet boundary
conditions, the variation of the scalar field vanishes at the boundary, so
there is no need to add such a term. For Neumann boundary conditions, however,
the normal derivative of the scalar field is fixed. Thus, one needs to add

$\displaystyle S_{\rm N}=\int
d^{d}x\left(\frac{z}{\ell}\right)^{1-d}\phi\partial_{z}\phi~{},$ (108)

to the bulk action, such that the variational principle is well-posed. Then,
the source of the boundary scalar operator is given by the normal derivative
of the field. This corresponds to the radial canonical momentum associated
with the scalar field. Additionally, it is possible to impose mixed boundary
conditions that involve a relationship between the scalar field and its normal
derivative. The latter specifies the behavior of the boundary scalar operator
according to

$\displaystyle\psi:=\phi+\lambda n^{\mu}\partial_{\mu}\phi~{},$ (109)

where $\lambda$ is a non-zero real number. Then, one needs to consider a
boundary term of the form

$\displaystyle S_{\rm M}=-\frac{1}{2}\int
d^{d}x~{}\left(\frac{z}{\ell}\right)^{1-d}\psi\partial_{z}\phi~{},$ (110)

and the source is now related to $\psi$ .

On the Euclidean AdSd+1 background, the field equation for the scalar field
can be written as

$\displaystyle
z^{d+1}\partial_{z}\left(z^{1-d}\partial_{z}\phi\right)+z^{2}\delta^{ij}\partial_{i}\partial_{j}\phi=m^{2}\ell^{2}\phi~{}.$
(111)

Near the boundary, one can check that the solution is,777If $\Delta=d/2$, one
needs to consider solutions with logarithmic behavior.

$\displaystyle\phi(z,x^{i})\sim
z^{d-\Delta}\phi_{(0)}(x^{i})+z^{\Delta}\phi_{(1)}(x^{i})~{},$ (112)

where

$\displaystyle\Delta=\frac{d}{2}\pm\sqrt{\frac{d^{2}}{4}+m^{2}\ell^{2}}~{}.$
(113)

In Ref. Witten:1998qj , it was shown that $\Delta$ corresponds to the
conformal weight of a scalar operator of a $d$-dimensional CFT. This
connection implies that for the scaling weight to be a real quantity, the
condition $m^{2}\ell^{2}>-d^{2}/4$ must be satisfied. Remarkably, this
condition is compatible with the BF bound Breitenlohner:1982bm , suggesting
that a scalar field in AdS can possess a negative mass while being stable. The
functions $\phi_{(0)}$ and $\phi_{(1)}$ are the two linearly independent
solutions of the second-order field equations. The leading-order term can be
either singular if $\Delta>d$, trivial if $\Delta<d$, or constant if
$\Delta=d$. Then, depending on the mass of the scalar field, the bulk geometry
is modified while preserving the asymptotic structure.

As it can be seen from Eq. (113), the conformal weight $\Delta$ is a positive
real number such that $\Delta>d-\Delta$. Therefore, $\phi_{(0)}$ is associated
with non-normalizable modes at the boundary. To identify the source of the
boundary scalar operator, one needs to consider

$\displaystyle\varphi(x^{i})=\lim_{z\to 0}z^{\Delta-d}\phi(z,x^{i})~{},$ (114)

which is always finite. Boundary conditions fix a function of $\phi_{(0)}$ and
$\phi_{(1)}$ at the boundary, that corresponds to the source of the dual
scalar operator. The relation between the modes reduces the degrees of freedom
in the dual theory by one half. For instance, Dirichlet boundary conditions
fix the source to be $\phi_{(0)}$ and the remaining degree of freedom
corresponds to the normalizable mode. As shown in Ref. Papadimitriou:2004ap ,
the non-normalizable mode $\phi_{(1)}$ does not transform properly under Weyl
rescalings. Therefore, from a holographic viewpoint, one needs to consider the
renormalized radial momentum $\hat{\pi}_{\phi}$, i.e., the first regular
coefficient in the asymptotic expansion of the canonical momentum
$\pi_{\phi}$. The latter usually differ from $\phi_{(1)}$ by a local
functional of $\phi_{(0)}$. Then, for Neumann boundary conditions, one fixes
$\hat{\pi}_{\phi}$ rather than $\phi_{(1)}$. This shows that the leading and
sub-leading coefficients in the asymptotic expansion are canonically
conjugated variables (see Ref. Papadimitriou:2007sj  details).

In general, one introduces a boundary term that depends on the scalar field
and derivatives thereof, whose explicit form depends on the boundary
conditions. As shown before, the field equations fix the relation between
derivatives of both the field and the boundary term via boundary conditions.
The extra boundary contribution to the gravity action implies a modification
of the boundary theory. In the case of mixed boundary conditions, this
corresponds to modifying the holographic CFT by multi-trace operators if the
deformation function is built not only by the fields but also by the operators
Witten:2001ua . Then, the vacuum expectation value of the dual operator with
conformal dimension $d-\Delta$ corresponds to $\phi_{(0)}$ and its current
related to $\phi_{(1)}$. Since the on-shell action is identified with the
generating functional of connected correlators of the dual CFT, say $\Gamma$,
the addition of the new term modifies the dual theory as

$\displaystyle\Gamma[\phi_{(0)}]\to\Gamma[J]-\int
d^{d}x\sqrt{g_{(0)}}J\phi_{(0)}~{},$ (115)

where $J$ is the current which depends on $\phi_{(0)}$ and $\hat{\pi}_{\phi}$
and is fixed by boundary condition in the string theory side. Following Ref.
Anabalon:2015xvl , we have encoded the deformations of the boundary theory in
$W\left(\phi_{(0)}\right)$, which must be fixed such that the variational
principle is consistent with the corresponding boundary conditions. Then, the
boundary CFT is deformed and the boundary conditions impose Witten:2001ua

$\displaystyle J=\frac{dW\left(\phi_{(0)}\right)}{d\phi_{(0)}}~{}.$ (116)

Using Neumann or mixed boundary conditions corresponds to modifying the
conformal vacuum at the boundary. In the case of mixed boundary conditions,
the multi-trace deformations could break the conformal invariance as they
modify the $n$-point functions (see Ref. Minces:1999eg ). Then, the
deformations could be marginal, relevant, or irrelevant depending on the mass
of the bulk scalar field. Moreover, multi-trace deformations have been
associated to multi-particle states in the dual gravity theory Aharony:2001pa
.

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