# Strongly-Coupled Anisotropic Gauge Theories and Holography in 5D Einstien-
Gauss-Bonnet Gravity

S. N. Sajadi, H. R. Safari
School of Physics, Institute for Research in Fundamental Sciences (IPM),
P. O. Box 19395-5531, Tehran, Iran
Electronic address<EMAIL_ADDRESS>address<EMAIL_ADDRESS>
###### Abstract

In this paper we study uncharged, non-conformal and anisotropic systems with
strong interactions using the gauge-gravity duality by considering Einstein-
Quadratic-Axion-Dilaton action in five dimension. In fact we would like to
gain insight into the influence of higher derivative gravity on the QCD
system. At finite temperature, we obtain an anisotropic black brane solution
to a 5D Einstein-Gauss-Bonnet-Axion-Dilaton system. The system has been
investigated and the effect of the parameter of theory has been considered.
The blackening function supports the thermodynamical phase transition between
small/large and AdS/large black brane for suitable parameters. We also study
transport and diffusion properties, and observe in particular that the
butterfly velocity that characterizes both diffusion and growth of chaos
transverse to the anisotropic direction saturates a constant value in the IR
which can exceed the bound given by the conformal value. We also determine the
imaginary part of the heavy quark potential in a strongly coupled plasma dual
to Gauss-Bonnet gravity.

> IPM/P-2023/43

##  1 Introduction

Quantum Chromodynamics (QCD) is a non-abelian gauge theory that describes the
behavior of the strong nuclear force. One of the key features of QCD is the
idea of color confinement, which states that quarks and gluons cannot exist in
isolation, but are always bound together to form particles such as protons and
neutrons. Another important aspect of QCD is the concept of asymptotic
freedom, which describes how the strong force becomes weaker at higher
energies or shorter distances. The mechanism behind these two features is not
well understood, and remains an active area of research. A possible way to
study is gauge/gravity duality which is a duality between certain strongly
interacting quantum field theories in one dimension and a weakly interacting
gravitational theory in one higher dimension. In this framework, the extra
dimension is interpreted as a scale parameter that controls the energy scale
of the QCD theory[1, 2, 3, 4, 5]. Quark-gluon plasma (QGP)-is believed to have
existed in the early universe, in the cores of compact stars, and can also be
created in high-energy heavy ion collisions in anisotropic way- is a state of
matter in which, the quarks and gluons are no longer confined within the
hadrons, and exist as a deconfined. The anisotropic black hole solutions in
the gauge/gravity correspondence and applications to the QGP have been
investigated in [6]-[22]. Higher-order gravitational models have recently
received attention [23]-[25], because of string theory predicts that at low
energies Einstein’s equations are subject to first-order corrections due to
the interactions of the strings with the additional dimensions. The natural
correction of Lovelock gravity to the Einstein-Hilbert gravity appears in five
and higher dimensions and is given by a precise combination of quadratic
curvature terms yields the second-order field equations known as the Gauss-
Bonnet term [26], [27], [28]. The different aspect of this theory from
cosmology to black hole solutions have been studied in [29]-[34]. In this
paper, we extend the work of [6] to the Einstein Quadratic Gravity, which is
general relativity extended by quadratic curvature invariant in the action to
find the effect of higher derivative terms on QGP, with the difference that in
[6], they were fixed $V(\phi)$ and $Z(\phi)$, which were chosen in order to
obtain specific properties in the IR and UV. In the present paper, however, we
fix both $V(\phi)$ and $Z(\phi)$ from the equations of motion, because the
functionality of these potentials in terms of the couplings of QGP is unknown
to us.
The paper is organized as follows. In section 2 we construct the anisotropic
$5$-dimensional solution with an arbitrary dynamical exponent, an exponential
quadratic warp function, in the framework of EGB gravity. We obtained the
approximately solution for blackening function and other unknown quantities up
to the first order of the theoretical parameter. We have shown the behavior of
the quantities with plots and we discuss the thermodynamics of the constructed
background in section 2.1. In section 2.2, we obtain transport and diffusion
properties in anisotropic theories and observe in particular that the
butterfly velocity that characterizes both diffusion and growth of chaos
transverse to the anisotropic direction saturates a constant value in the IR
which can exceed the bound given by the conformal value. In section 2.3, the
imaginary part of the quark-antiquark potential in different directions. In
section 2.4, we obtained the Jet quenching related to the suppression of high-
energy jets of particles produced in high energy heavy-ion collisions due to
their interaction with QGP. We finish the paper with some concluding remarks
in section 3.

##  2 Basic Formalism

We consider a $5$-dimensional Einstein-Quadratic-Axion-Dilaton action as
follows

$S=\dfrac{1}{16\pi G_{5}}\int d^{5}x\sqrt{-g}L,$ (1)

where the Lagrangian is

$L=R+\gamma R_{abcd}R^{abcd}+\beta R_{ab}R^{ab}+\alpha
R^{2}-\dfrac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi+V(\phi)-\dfrac{1}{2}Z(\phi)\partial_{\mu}\chi\partial^{\mu}\chi,$
(2)

and $V(\phi)$ is the potential energy for the dilaton field $\phi$, $Z(\phi)$
is the coupling $\phi$ to the axion field $\chi$. $G_{5}$ is the Newton
constant in five dimensions and $\alpha$, $\beta$ and $\gamma$ are coupling
constants of theory. The Einstein and Quadratic terms describe the
gravitational interaction between matter in QCD, here the matter is quarks,
gluons and plasma. The dilaton field modifies the running of the QCD coupling
constant which reflected the non conformality of QCD in gravity. The axion,
which is dual to the gauge theory $\theta$-term (is a term in the QCD
Lagrangian that describes the possibility of CP violation in strong
interactions), is responsible for inducing the anisotropy [47],[48]. The
variation of the action (1) over metric $g_{\mu\nu}$, the scalar field $\phi$
and $\chi$ gives the field equations as follows

$\displaystyle
E_{\mu\nu}=G_{\mu\nu}+\alpha\left[2R(R_{\mu\nu}-\frac{1}{4}g_{\mu\nu}R)+2(g_{\mu\nu}\square-\nabla_{\mu}\nabla_{\nu})R\right]+\beta\left[(g_{\mu\nu}\square-\nabla_{\mu}\nabla_{\nu})R+\square
G_{\mu\nu}+2R^{\lambda\rho}(R_{\mu\lambda\nu\rho}\right.$
$\displaystyle\left.-\frac{1}{4}g_{\mu\nu}R_{\lambda\rho})\right]+\gamma\left[-\frac{1}{2}g_{\mu\nu}R_{\alpha\beta\gamma\eta}R^{\alpha\beta\gamma\eta}+2R_{\mu\lambda\rho\sigma}R_{\nu}{}^{\lambda\rho\sigma}+4R_{\mu\lambda\nu\rho}R^{\lambda\rho}-4R_{\mu\sigma}R^{\sigma}_{\nu}+4\square
R_{\mu\nu}-2\nabla_{\mu}\nabla_{\nu}R\right]$
$\displaystyle=\dfrac{1}{2}\nabla_{\alpha}\phi\nabla_{\beta}\phi+\dfrac{1}{2}g_{\alpha\beta}V(\phi)-\dfrac{1}{4}g_{\alpha\beta}\nabla_{\gamma}\phi\nabla^{\gamma}\phi+\dfrac{1}{2}Z(\phi)\nabla_{\alpha}\chi\nabla_{\beta}\chi-\dfrac{1}{4}g_{\alpha\beta}Z(\phi)\nabla_{\gamma}\chi\nabla^{\gamma}\chi,$
(3)
$\displaystyle\square\phi-\dfrac{1}{2}\nabla_{\alpha}\chi\nabla^{\alpha}\chi
Z^{\prime}(\phi)+V^{\prime}(\phi)=0,$ (4) $\displaystyle
Z(\phi)\square\chi+\nabla_{\alpha}\phi\nabla^{\alpha}\chi Z^{\prime}(\phi)=0,$
(5)

where $G_{\mu\nu}$ is the Einstein tensor. We use the metric ansatz
$g_{\mu\nu}$ in the following form:

$ds^{2}=A^{2}(z)\left(-g(z)dt^{2}+\dfrac{dz^{2}}{g(z)}+h^{2}(z)dx^{2}+dy_{1}^{2}+dy_{2}^{2}\right),\;\;\;\phi=\phi(z)$
(6)

where $A$, $g$ and $h$ are functions of the holographic coordinate $z$. The
warp function $A(z)$ is related to the energy scale of QCD by compactification
of extra dimension. $g(z)$ is the blackening function which is dual to thermal
states in the QCD. Also, $z=0$ and $z\to\infty$ correspond to boundary and
anisotropic IR region, respectively. Isotropy in the $y_{1}y_{2}$-directions
is respected, but not in the $x$-direction unless $h(z)=1$. Using the ansatz
of the metric, it is easy to obtain the equations of motion for the background
fields. The field equations for $\chi$ and $\phi$ are given by:

$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\dfrac{d^{2}\chi(x)}{dx^{2}}=0,$
(7) $\displaystyle
2h^{2}A^{3}\dfrac{dV}{d\phi}-A\dfrac{dZ}{d\phi}\left({\frac{d\chi}{dx}}\right)^{2}+6h^{2}g\phi^{\prime}A^{\prime}+2Agh\phi^{\prime}h^{\prime}+2Ah^{2}g\phi^{\prime\prime}+2Ah^{2}\phi^{\prime}g^{\prime}=0,$
(8)

where prime is differential with respect to $z$. From solving equation (7), we
have:

$\chi(x)=c_{1}x+c_{2}.$ (9)

For $\beta=-4\alpha$, $\gamma=\alpha$ (Einstien-Gauss-Bonnet Gravity) and
using (9) we get the other components of the field equation which was
presented in appendix (The components of the field equation).

We obtain the $Z(\phi)$ from $E_{y_{1}y_{2}}$ and $\phi^{\prime}$ from
$E_{zz}$. Inserting $Z(\phi)$ into the $E_{xx}$ we obtained $V(\phi)$.
Finally, inserting $Z(\phi)$, $V(\phi)$ and $\phi^{\prime}$ into the $E_{tt}$
a constrained differential equation obtains as follows

$\displaystyle
2hA^{2}g^{\prime\prime}+2A^{2}g^{\prime}h^{\prime}+6Ahg^{\prime}A^{\prime}+\dfrac{8\alpha}{A^{3}}\left[-hgAg^{\prime\prime}A^{\prime
2}-gA^{2}h^{\prime}g^{\prime}A^{\prime\prime}-A^{2}gA^{\prime}h^{\prime}g^{\prime\prime}+8ghg^{\prime}A^{\prime
3}\right.$ $\displaystyle\left.-2ghAA^{\prime}g^{\prime}A^{\prime}-hAA^{\prime
2}g^{\prime
2}-gA^{2}A^{\prime}g^{\prime}h^{\prime\prime}-A^{2}h^{\prime}A^{\prime}g^{\prime
2}-Agh^{\prime}g^{\prime}A^{\prime 2}\right]=0.$ (10)

To solve the above equation in IR region, we assume

$ds^{2}=(cz)^{k_{1}-2}\left[-g(z)dt^{2}+\dfrac{dz^{2}}{g(z)}+dy_{1}^{2}+dy_{2}^{2}+\dfrac{\mathfrak{c}dx^{2}}{(cz)^{k_{0}-2}}\right].$
(11)

For $k_{1}=0$ the solution exhibits a Lifshitz-like scaling, and for
$k_{1}\neq 0$ the metric (11) has the hyperscaling violation property. The
case $k_{0}=2$ and $\mathfrak{c}=1$ corresponds to isotropic solution. Using
(11), the differential equation (2) becomes

$\displaystyle
z^{2}(cz)^{k_{1}-2}\left[zg^{\prime\prime}-\dfrac{1}{2}(k_{0}-3k_{1}+4)g^{\prime}\right]+\alpha(k_{1}-2)(k_{0}-k_{1})\left[zgg^{\prime\prime}+g^{\prime}(zg^{\prime}-\dfrac{1}{2}g(k_{0}-k_{1}+4))\right]=0.$
(12)

To solve (12) with the generic constants $k_{0}$ and $k_{1}$, we work in a
perturbative manner as follows

$g(z)=g_{0}(z)+\alpha g_{1}(z)+\mathcal{O}(\alpha^{2}),$ (13)

therefore, one can obtain $g_{0}$ and $g_{1}$ as follows

$\displaystyle
g_{0}(z)=1-\left(\dfrac{z}{z_{h}}\right)^{3+\frac{k_{0}-3k_{1}}{2}}$ (14)

and

$\displaystyle
g_{1}(z)=1-\left(\dfrac{z}{z_{h}}\right)^{3+\frac{k_{0}-3k_{1}}{2}}+\dfrac{(k_{1}-2)(k_{0}-k_{1})(k_{0}-3k_{1}+6)}{k_{0}+6-5k_{1}}c^{2}(cz)^{-k}\left(\dfrac{z}{z_{h}}\right)^{3+\frac{k_{0}-3k_{1}}{2}}-$
$\displaystyle\dfrac{(k_{1}-2)(k_{0}-k_{1})(k_{0}+6-3k_{1})^{2}c^{-k_{1}+2}}{2(k_{0}+6-5k_{1})(k_{0}+6-4k_{1})}\dfrac{z^{3+\frac{k_{0}-3k_{1}}{2}}}{z_{h}^{3+\frac{k_{0}-k_{1}}{2}}}-\dfrac{(k_{1}-2)(k_{0}-k_{1})(k_{0}+6-3k_{1})c^{-k_{1}+2}}{2(k_{0}+6-4k_{1})}\dfrac{z^{6+k_{0}-4k_{1}}}{z_{h}^{k_{0}-3k_{1}+6}}$
(15)

here, we imposed the following conditions on the metric

$g(0)=1,\;\;\;\;g(z_{h})=0.$ (16)

(a) $\alpha=0.2,z_{h}=1,k_{1}=-2,c=1$ (b)
$\alpha=0.2,z_{h}=1,k_{1}=-2,k_{0}=3,c=1$

Figure 1: Plots of $g(z)$ in terms of $z$ for
$k_{0}={\color[rgb]{1,0,0}1},{\color[rgb]{0,0,1}-1},{\color[rgb]{0,0,0}-2},{\color[rgb]{0,1,0}-3},{\color[rgb]{1,1,0}-4},{\color[rgb]{.5,.5,.5}-5}$
(left) and for
$\alpha={\color[rgb]{1,0,0}0},{\color[rgb]{0,0,1}0.1},{\color[rgb]{0,0,0}0.2},{\color[rgb]{0,1,0}0.3},{\color[rgb]{1,1,0}0.4},{\color[rgb]{.5,.5,.5}0.5}$
(right).

The solutions we found do not interpolate between AdS in UV and anisotropic in
IR. These solutions have fixed $c$-function [10] in contrast to the full
solutions used in [6]. The behavior of the metric function is depicted in
Fig.(1). The main feature is that the metric function values decrease faster
for larger $\alpha$ and $k_{0}$. In the isotropic case ($k_{0}=2$) the metric
function values are larger than in the anisotropic ones ($k_{0}\neq 2$) (Fig.
1a). By inserting (13) in $E_{zz}$, one can obtain $\phi$ for $\alpha\ll 1$ as
follows:

$\displaystyle\phi(z)=$
$\displaystyle\dfrac{1}{2}\sqrt{6k_{1}^{2}-12k_{1}+4k_{0}-2k_{0}^{2}}\ln\left(\dfrac{z}{z_{h}}\right)+\dfrac{\alpha(k_{1}-2)^{2}(k_{0}+3k_{1}-2)(k_{1}-k_{0})c^{2-k_{1}}z_{h}^{-k_{1}}}{2\sqrt{2}k_{1}(k_{0}+6-5k_{1})\sqrt{3k_{1}^{2}-6k_{1}-k_{0}^{2}+2k_{0}}}$
$\displaystyle\left[3k_{1}-6-k_{0}+(k_{0}+6-5k_{1})\left(\dfrac{z}{z_{h}}\right)^{-k_{1}}+2k_{1}\left(\dfrac{z}{z_{h}}\right)^{3+\frac{k_{0}-5k_{1}}{2}}\right]+\mathcal{O}(\alpha^{2}),$
(17)

here, we imposed the condition $\phi(z_{h})=0$.

(a) $z_{h}=1,\alpha=0.2,k_{1}=-2,c=1$ (b) $z_{h}=1,k_{0}=1,k_{1}=-2,c=1$

Figure 2: Plot of $\phi$ in terms of $z$ for
$k_{0}={\color[rgb]{1,0,0}1},{\color[rgb]{0,0,1}-1},{\color[rgb]{0,0,0}-2},{\color[rgb]{0,1,0}-3}$
(left) and for
$\alpha={\color[rgb]{1,0,0}0},{\color[rgb]{0,0,1}0.1},{\color[rgb]{0,0,0}0.2},{\color[rgb]{0,1,0}0.3}$
(right).

In figure (2), the scalar field in terms of $z$ for different values of
parameters have been shown. As can be seen the imaginary part of scalar field
inside and outside the black brane has a zero value and is stable. By
increasing $k_{0}$ and $\alpha$ in $0<z<z_{h}$, the scalar field increase. By
inserting (13) in $E_{y_{1}y_{1}}$, one can obtain $Z(\phi)$ for $\alpha\ll 1$
as follows:

$\displaystyle{Z(\phi)=}$
$\displaystyle\dfrac{1}{2c_{1}}(k_{0}-2)(3k_{1}-k_{0}-6)c^{2-k_{0}}z_{h}^{-k_{0}}e^{\frac{-2\phi
k_{0}}{\sqrt{3k_{1}^{2}-6k_{1}-k_{0}^{2}+2k_{0}}}}+\alpha\hat{Z}+\mathcal{O}(\alpha^{2}),$
(18)

$\hat{Z}$ provided in appendix (The GB Correction to Potentials).

Figure 3: Plot of $Z(\phi)$ in terms of $\phi$ for
$k_{0}={\color[rgb]{1,0,0}1},{\color[rgb]{0,0,1}-1},{\color[rgb]{0,0,0}-2},{\color[rgb]{0,1,0}-3},k_{1}=-2,\alpha=0.2,c=1$
(left) and for
$k_{0}=1,\alpha={\color[rgb]{1,0,0}0},{\color[rgb]{0,0,1}0.1},{\color[rgb]{0,0,0}0.2},{\color[rgb]{0,1,0}0.3},{\color[rgb]{1,1,0}0.4}$(right).

In figure (3), the behavior of $Z(\phi)$ in terms of $\phi$ are shown. As
$\alpha$ increases and $k_{0}$ decreases, $Z$ decrease. Finally from $E_{xx}$,
one can get $V(\phi)$. In the case of $\alpha\ll 1$, one can get

$\displaystyle
V(\phi)=-\dfrac{3}{4}(k_{1}-2)(k_{0}+6-3k_{1})c^{2-k_{1}}z_{h}^{-k_{1}}e^{\frac{-2k_{1}\phi}{\sqrt{6k_{1}^{2}+4k_{0}-2k_{0}^{2}-12k_{1}}}}+\alpha\hat{V}+\mathcal{O}(\alpha^{2}),$
(19)

$\hat{V}$ provided in appendix (The GB Correction to Potentials). As can be
seen the leading term of potential ($\alpha=0$) is exponential which shows
that we are in IR regime and the solution is not asymptotically AdS. In Fig.
(4), the behavior of scalar potential in terms of $z$ for finite $\alpha$ has
been shown. As can be seen from the plots for $\alpha\neq 0$, it is clear
there are multiple minima, and the solution can flow in the UV, yielding AdS
asymptotics. In the both panels, between $0<z<z_{h}$ by increasing $k_{0}$ and
$\alpha$, the potential increase. For the case of $k_{1}=0$ and
$k_{0}=2+\epsilon$, the potential becomes

$V\sim 12c^{2}+12c^{2}(1-2c^{2})\alpha+\dfrac{3}{2}\epsilon
c^{2}+\mathcal{O}(\epsilon\alpha).$ (20)

(a) $z_{h}=1,\alpha=0.2,k_{1}=-2,c=1$ (b) $z_{h}=1,k_{0}=1,k_{1}=-2,c=1$

Figure 4: Plot of $V$ in terms of $z$ for
$k_{0}={\color[rgb]{1,0,0}1},{\color[rgb]{0,0,1}-1},{\color[rgb]{0,0,0}-2},{\color[rgb]{0,1,0}-3}$,${\color[rgb]{1,1,0}-4}$,${\color[rgb]{.5,.5,.5}-5}$
(left) and for
$\alpha={\color[rgb]{1,0,0}0},{\color[rgb]{0,0,1}0.1},{\color[rgb]{0,0,0}0.2},{\color[rgb]{0,1,0}0.3},{\color[rgb]{1,1,0}0.4},{\color[rgb]{.5,.5,.5}0.5}$
(right).

###  2.1 Thermodynamics of the background

In this subsection, we explore the thermodynamics of the black hole solution
(13). In order to investigate the thermodynamic properties of the black hole
we need to obtain some relevant thermodynamic quantities. The temperature of
the black hole is obtained as follows:

$T=\left|\dfrac{g^{\prime}}{4\pi}\right|=\dfrac{|k_{0}+6-3k_{1}|}{8\pi
z_{h}}+\alpha\left[\dfrac{1}{8\pi
z_{h}}+\dfrac{c^{-k_{1}+2}(k_{0}+6-3k_{1})^{3}(k_{1}-2)(k_{0}-k_{1})}{16\pi
z_{h}^{k_{1}+1}(k_{0}+6-5k_{1})(k_{0}+6-4k_{1})}\right]+\mathcal{O}(\alpha^{2}).$
(21)

The temperature monotonically decreases with the increase of the horizon if
$k_{1}<-2$. For $k_{1}\geq-2$, the temperature has a minimum. In Fig. (9)a,
the variation of temperature with respect to the horizon radius $z_{h}$ for
different values of $k_{0}$ is shown. As can be seen there exists a minimum
temperature $T_{min}$ below which no black hole solution exist (thermal gas).
However, for $T>T_{min}$, there are two black hole solutions, a large and a
small one (deconfined quark gluon plasma phase). The small black hole phase
for which $T$ increases with $z_{h}$ whereas the large black hole phase for
which $T$ decreases with $z_{h}$. For $\alpha=0$, one can get the result of
[6].

Figure 5: Plot of $T$ in terms of $z_{h}$ for
$k_{1}=-2,c=1,k_{0}={\color[rgb]{1,0,0}1},{\color[rgb]{0,0,1}-1},-2,{\color[rgb]{0,1,0}-3}$
(left). Plot of $C_{V}$ and $T$ in terms of $z_{h}$ for $c=1,2,3,4$ (middle).
Plot of $F$ in terms of $T$ for different values of the anisotropy parameter
$c={\color[rgb]{1,0,0}1},{\color[rgb]{0,0,1}2},3$ (right).

In Fig. (9)b, to study the stability of the solutions, we have shown the
behavior of heat capacity $C_{V}$ and temperature. As can be seen the large
black hole has positive heat capacity and therefore is stable and small black
hole is unstable and thus not physical.
Following the Wald formula, one can easily read the black hole entropy density
$S$, which is defined as

$\displaystyle S=-\dfrac{1}{8}\int_{\Sigma}d^{n-2}x\sqrt{\eta}\dfrac{\delta
L}{\delta R_{\mu\alpha\beta\nu}}\epsilon_{\mu\alpha}\epsilon_{\beta\nu},$ (22)

where

$\displaystyle\dfrac{\delta L}{\delta R_{\mu\alpha\beta\nu}}=$
$\displaystyle\left(\dfrac{1}{2}+\alpha
R\right)\left(g^{\mu\beta}g^{\alpha\nu}-g^{\mu\nu}g^{\alpha\beta}\right)+\dfrac{1}{2}\beta\left(R^{\mu\beta}g^{\alpha\nu}-R^{\alpha\beta}g^{\mu\nu}-R^{\mu\nu}g^{\alpha\beta}+R^{\alpha\nu}g^{\mu\beta}\right)$
$\displaystyle+2\gamma R^{\mu\alpha\beta\nu},$ (23)

and $\epsilon_{\mu\nu}=-2\sqrt{-\zeta}\delta^{t}_{[\mu}\delta^{z}_{\nu]}$.
Here

$\sqrt{-\zeta}=A^{2},\;\;\;\;\sqrt{-\eta}=A^{3}h$ (24)

at the horizon the entropy becomes

$\displaystyle s=\left.\dfrac{1}{4}hA^{3}\right|_{z=z_{h}}-\left.\dfrac{\alpha
g[3hA^{\prime
2}+2Ah^{\prime}A^{\prime}]}{4A}\right|_{z=z_{h}}=\left.\dfrac{1}{4}hA^{3}\right|_{z=z_{h}}=\dfrac{\sqrt{\mathfrak{c}}}{4}(cz_{h})^{\frac{3k_{1}-k_{0}-4}{2}}.$
(25)

The scaled entropy density as a function of scaled temperature is shown in
Fig. (6)a. The red lines correspond to the large stable solution and blue
lines are for the small unstable solution [41].

(a) $\alpha=0.2,k_{1}=-2,c=1$ (b) $\alpha=0.2,k_{1}=-2,c=1$ (c)
$\alpha=0.2,k_{1}=-2,c=1$

Figure 6: Plot of $s/T^{3}$ (left), $c_{s}^{2}/T^{3}$(middle) and
$C_{V}/T^{3}$ (right) in terms of $T/T_{min}$ for $k_{0}=1,-1,-3$.

The free energy density $F$ can be calculated from the entropy density $s$ by
integrating as follows

$\displaystyle F=$ $\displaystyle\int-
sdT=\dfrac{(cz_{h})^{-\frac{1}{2}(k_{0}+4-3k_{1})}}{16\pi
z_{h}}+\dfrac{\alpha\sqrt{\mathfrak{c}}(cz_{h})^{\frac{1}{2}(3k_{1}-k_{0}-6)}}{16\pi
z_{h}c^{2}}+\dfrac{\alpha\sqrt{\mathfrak{c}}(k_{0}-3k_{1}+6)^{2}(k_{1}+1)(cz_{h})^{\frac{1}{2}(k_{1}-k_{0}-6)}}{64\pi(k_{0}-k_{1}+6)(k_{0}-4k_{1}+6)(k_{0}+6-4k_{1})}$
$\displaystyle[11k_{0}^{3}+(68-65k_{1})k_{0}^{2}+(131k_{1}^{2}-188k_{1}+12)k_{0}+60k_{1}+92k_{1}^{2}-81k_{1}^{3}]+\mathcal{O}(\alpha^{2}).$
(26)

The $y_{1}y_{2}$-component of the pressures is related to the free energy as
follows: $P_{y_{1}y_{2}}=-F$. The other components of pressure can be obtained
by an asymptotic expansion in the near boundary regime and reading the
expectation value of the energy-momentum tensor components, by this expansion
[39],[40]. The energy is given as follows:

$\displaystyle E=$ $\displaystyle
F+Ts=\dfrac{(k_{0}-3k_{1}+4)(cz_{h})^{\frac{3k_{1}-k_{0}-4}{2}}\sqrt{\mathfrak{c}}}{32\pi
z_{h}}+{\dfrac{(k_{0}-3k_{1}+4)\sqrt{\mathfrak{c}}\alpha}{128\pi
z_{h}^{3}c^{2}(k_{0}-k_{1}+6)(k_{0}-4k_{1}+6)(k_{0}-5k_{1}+6)}}$
$\displaystyle[-4(k_{0}-k_{1}+6)(k_{0}-4k_{1}+6)(k_{0}+6-5k_{1})(cz_{h})^{\frac{3k_{1}-k_{0}}{2}}+c^{2}(cz_{h})^{\frac{k_{1}-k_{0}}{2}}(k_{0}+6-3k_{1})^{2}[11k_{0}^{3}+$
$\displaystyle(68-65k_{1})k_{0}^{2}+(131k_{1}^{2}-188k_{1}+12)k_{0}+60k_{1}+92k_{1}^{2}-81k_{1}^{3}]]+\mathcal{O}(\alpha^{2}).$
(27)

The unstable-stable nature of the small-large black hole phases can also be
seen from the free energy behaviour shown in Fig. 6. We see that the free
energy of the small black hole phase is always larger than the large black
hole and thermal AdS phases, indicating the unstable nature of this small
black hole phase. Importantly, upon varying the Hawking temperature, a phase
transition from the large black hole phase to thermal AdS phase takes place at
a critical temperature $T_{c}$. This is the famous black hole-thermal AdS
Page-Hawking phase transition.
The dependence of $T_{c}$ on $\alpha$ for different values of $k_{0}$ has been
shown in Fig. 7. There exists a naked singularity of the zero temperature
solution.

Figure 7: Plot of $T_{c}$ in terms of $\alpha$ for $c=1$,
$({\color[rgb]{1,0,0}k_{0}={2},k_{1}=0})$,
$({\color[rgb]{0,0,1}k_{0}={0},k_{1}=0})$ and
$({\color[rgb]{0,0,0}k_{0}={-4},k_{1}=-2})$.

The sound velocity $c_{s}^{2}$ which can directly measure the conformality of
the system, can be obtained from the temperature and entropy:

$\displaystyle c_{s}^{2}=$ $\displaystyle\dfrac{d\log T}{d\log
s}=\dfrac{2}{k_{0}+4-3k_{1}}-{\dfrac{\alpha
k_{1}(k_{0}+6-3k_{1})c^{-k_{1}+2}z_{h}^{-k_{1}}}{2(k_{0}+6-5k_{1})(k_{0}+6-4k_{1})(k_{0}+4-3k_{1})}}\times$
$\displaystyle[11k_{0}^{3}+(68-35k_{1})k_{0}^{2}+(131k_{1}^{2}-188k_{1}+12)k_{0}-81k_{1}^{3}+92k_{1}^{2}+60k_{1}]+\mathcal{O}(\alpha^{2}).$
(28)

For isotropic case ($k_{0}+4-3k_{1}=6$), $c_{s}^{2}=1/3$, the system is
conformal, for anisotropic ($k_{0}+4-3k_{1}\neq 6$), $c_{s}^{2}\neq 1/3$ the
system is non-conformal. The numerical result of the square of the sound
velocity is shown in 6b. At $T_{min}$, the sound velocity square is around $0$
which is in agreement with lattice data $0.05$. At high temperature, the sound
velocity square goes to $0.33$ for $k_{0}=-4$, which means that the system is
asymptotically conformal. The heat capacity is given as

$\displaystyle C_{V}=$ $\displaystyle
T\dfrac{ds}{dT}=\dfrac{(k_{0}+4-3k_{1})\sqrt{\mathfrak{c}}c^{2}(cz_{h})^{\frac{3k_{1}-k_{0}-4}{2}}}{8}+{\dfrac{\alpha\sqrt{\mathfrak{c}}k_{1}(k_{0}+4-3k_{1})(k_{0}-3k_{1}+6)(cz_{h})^{\frac{-k_{0}-4+k_{1}}{2}}}{32(k_{0}-4k_{1}+6)(k_{0}-5k_{1}+6)}}$
$\displaystyle\times[11k_{0}^{3}+(68-35k_{1})k_{0}^{2}+(131k_{1}^{2}-188k_{1}+12)k_{0}-81k_{1}^{3}+92k_{1}^{2}+60k_{1}]+\mathcal{O}(\alpha^{2})=\dfrac{s}{c_{s}^{2}}.$
(29)

The numerical result of the specific heat is shown in Fig. (6)c. It can be
clearly seen that the specific heat $C_{V}$ diverges at $T_{min}$. At
$T\to\infty$, the scaled specific heat $C_{V}/T^{3}$ approaches to the zero.
For a null vector $\xi^{\mu}$ the null energy condition is given as

$\displaystyle T_{\mu\nu}\xi^{\mu}\xi^{\nu}\geq 0,$ (30)

where $T_{\mu\nu}$ is the energy-momentum tensor which is given by the right-
hand side of equation (3). The magnitude of the null vector is zero. Thus

$g_{\mu\nu}\xi^{\mu}\xi^{\nu}=0,$ (31)

and using the geodesic equation leads to

$\xi^{\mu}=\dfrac{1}{A}\left[\dfrac{1}{g},\sqrt{1-\dfrac{g}{h}-2g},\dfrac{1}{h},1,1\right],$
(32)

using (30), considering the leading terms of expansion, from the positivity of
heat capacity, and boundness of the sound/butterfly velocities, one can obtain
the conditions on $k_{0}$ and $k_{1}$ as follows:

$\displaystyle(2-k_{0})(k_{0}-3k_{1}+6)\geq 0,$ (33) $\displaystyle
3k_{1}^{2}-6k_{1}+2k_{0}-k_{0}^{2}\geq 0,$ (34) $\displaystyle
k_{0}-3k_{1}+4\geq 0,$ (35) $\displaystyle k_{0}+6-5k_{1}\geq 0.$ (36)

Combining these inequalities, we observe that for $-6\leq k_{0}\leq 2$ the
value of $k_{1}$ is bounded from above $k_{1}^{(-)}$, and from below
$k_{1}^{(+)}$ as

$k_{1}^{(\pm)}=1\pm\dfrac{\sqrt{9-6k_{0}+3k_{0}^{2}}}{3}.$ (37)

The ranges $k_{0}>2$ and $k_{0}<-6$ are forbidden. The allowed range of
parameters has been shown in figure 8.

Figure 8: The allowed region for $k_{1}$ and $k_{0}$ are shown in red color.

###  2.2 Transport and Diffusion

Here, we look at the macroscopic and the microscopic motion of QGP using
holography. Transport is related to the movement of particles in response to
an external force. Transport in the QGP is characterized by its viscosity,
which is a measure of the resistance of the plasma to flow. The shear
viscosity to entropy density ratio ($\eta/s$) is a quantity of particular
interest, as it is a fundamental property that governs the evolution of the
plasma. The holographic value of the shear viscosity $4\pi\eta/s=1$. This
universal value is violated in anisotropic systems for the shear component
parallel to the anisotropic direction, $\eta_{||}$, while the component
transverse to the anisotropic direction $\eta_{\bot\bot}$ remains universal. A
calculation shows that $\eta_{||}$ can be obtained in the IR limit

$\displaystyle\dfrac{\eta_{\parallel}}{s}=\dfrac{1}{4\pi}\dfrac{g_{\bot\bot}}{g_{\|}}=\dfrac{(cz_{h})^{k_{0}-2}}{\mathfrak{c}}.$
(38)

The shear viscosity is parametrized according to the properties of the IR
geometry, i.e. the scaling exponents $k_{0}$ and $k_{1}$. For the case of
$k_{1}=0$ and $k_{0}=2-\epsilon$, equation (38) becomes

$\dfrac{4\pi\eta_{\parallel}}{s}=1-\ln\left(\dfrac{c}{\pi
T}\right)\epsilon-\alpha\epsilon+\mathcal{O}(\alpha^{2}\epsilon^{2}),\;\;\;\epsilon\ll
1,\;\;\;\alpha\ll 1,$ (39)

where $\epsilon$ is deviation from the isotropy. We observe that the shear
viscosity in the anisotropic direction is generally below the universal value
$\eta/s=1/4\pi$, which is attained only in the UV regime. Also, the GB term
decreases the shear viscosity. Another interesting phenomenon is momentum
diffusion, which is related to the random motion of particles due to
fluctuations in the local environment. In holographic theories, diffusion is
characterized by the time scale $\tau=1/T$ and the butterfly velocity $v_{B}$,
both entering in the diffusion constant as $D\sim v_{B}^{2}/T$. These
quantities can be computed holographically through the near horizon dynamics.
In anisotropic theories, there are two notions of butterfly velocities,
$v_{B\parallel}$ and $v_{B\perp}$, corresponding to the parallel and
transverse directions, respectively. On the background one finds

$\displaystyle\mu_{\perp}^{2}$
$\displaystyle=\left.\dfrac{g^{\prime}(3k_{1}-k_{0}-4)}{4z}\right|_{z=z_{h}}=\dfrac{(k_{0}+4-3k_{1})(k_{0}+6-3k_{1})}{8z_{h}^{2}}-\dfrac{(k_{0}+6-3k_{1})(k_{0}+4-3k_{1})\alpha}{32z_{h}^{2}(k_{0}+6-5k_{1})(k_{0}+6-4k_{1})}$
$\displaystyle[(k_{0}+6-3k_{1})c^{2-k_{1}}(k_{0}^{3}+k_{0}^{2}(68-65k_{1})+k_{0}(12+131k_{1}^{2}-188k_{1})+92k_{1}^{2}+60k_{1}-81k_{1}^{3})-$
$\displaystyle 4(k_{0}+6-4k_{1})(k_{0}+6-5k_{1})]+\mathcal{O}(\alpha^{2}),$
(40) $\displaystyle\mu_{\parallel}^{2}$
$\displaystyle=\dfrac{\mathfrak{c}\mu_{\perp}^{2}}{(cz_{h})^{k0-2}}.$ (41)

The corresponding butterfly velocities $v_{Bi}^{2}=(2\pi T)^{2}/\mu_{i}^{2}$
in the IR regime becomes

$\displaystyle v_{B\perp}^{2}$
$\displaystyle=\dfrac{k_{0}+6-3k_{1}}{2k_{0}+8-6k_{1}}+\dfrac{\alpha(k_{0}+6-3k_{1})}{8(k_{0}+6-4k_{1})(k_{0}+6-5k_{1})(k_{0}+4-3k_{1})}[(k_{0}+6-3k_{1})z_{h}^{-k_{1}}c^{-k_{1}+2}$
$\displaystyle(11k_{0}^{3}+k_{0}^{2}(76-69k_{1})+k_{0}(60-244k_{1}+147k_{1}^{2})-93k_{1}^{3}+12k_{1}+140k_{1}^{2})+4(k_{0}+6-5k_{1})$
$\displaystyle(k_{0}+6-4k_{1})]+\mathcal{O}(\alpha^{2}),$ (42) $\displaystyle
v_{B\parallel}^{2}$
$\displaystyle=\dfrac{(cz_{h})^{k_{0}-2}}{\mathfrak{c}}v_{B\perp}^{2}.$ (43)

For $k_{0}=2,k_{1}=0$, the velocities becomes $2/3+\alpha$. $v_{B\perp}^{2}\to
3/4$ as $k_{0}=k_{1}=0$. For other values $k_{0}=2-\epsilon$ and $k_{1}=0$, we
have

$\dfrac{3}{2}v_{B\perp}^{2}=1+\alpha+\epsilon\left[0.04-(1+\alpha)\ln\left(\dfrac{c}{\pi
T}\right)-\alpha\right]+\mathcal{O}(\alpha^{2}\epsilon^{2}).\\\ $ (44)

For allowed values of parameters, $v_{B\parallel}^{2}$ and $v_{B\perp}^{2}$
are positive and $v_{B\parallel}^{2}$ smaller than 2/3 but the value of
$v_{B\perp}^{2}$ in the IR regime can exceed the conformal value 2/3. For
conformal but anisotropic case in GB theory provided in appendix The result
about conformal-anisotropic solution.

###  2.3 The Imaginary Part of the Potential

Here, we study the imaginary part of the static potential associated to the
thermal width in finite temperature strongly coupled anisotropic plasma
[35],[36],[38]. The imaginary part of the quark-antiquark potential arises due
to the presence of virtual quark-antiquark pairs, which can be created and
annihilated in the interaction between two heavy quarks, which can be
extracted from the correlation functions of heavy quark-antiquark pairs. The
real part of the static potential is related to the energy of the system,
while the imaginary part is related to the decay rate of the system. The
imaginary part of potential is given by

$ImV_{Q\bar{Q}}=-\dfrac{1}{2\sqrt{2}\lambda}\sqrt{M_{0}}\left[\dfrac{V^{\prime}_{0}}{V^{\prime\prime}_{0}}-\dfrac{V_{0}}{V^{\prime}_{0}}\right]$
(45)

where

$\displaystyle M=$ $\displaystyle-
g_{tt}g_{yy}=A^{4}(z)g(z)(\sin^{2}{\theta}+h^{2}(z)\cos^{2}{\theta}),$
$\displaystyle V=$ $\displaystyle-g_{tt}g_{zz}=A^{4}(z),$ (46)

and $V_{0}=V(z_{0})$, and $z_{0}$ is the minimum of string. For generic values
of $k_{0}$, $k_{1}$ and $\theta$, the imaginary part of potential has a huge,
complicated expression. Therefore, we consider the special cases: for
$\theta=\pi/2$ and $k_{0}=8$ and $k_{1}=2+\epsilon$ (conformal-anisotropic)
becomes

$\displaystyle ImV_{Q\bar{Q}}\sim$
$\displaystyle-\dfrac{\sqrt{2}(-1+3Z_{0}^{4})}{48\pi\lambda
TZ_{0}}+\dfrac{\sqrt{2}\epsilon}{576\pi\lambda
Z_{0}T}[12(3Z_{0}^{4}-1)\ln\left(\dfrac{\pi T}{c}\right)+6(15Z_{0}^{4}-2)\ln
Z_{0}+$
$\displaystyle(52Z_{0}^{4}-18Z_{0}^{8}-5)]-\dfrac{\sqrt{2}(-1+3Z_{0}^{2})\alpha}{48\pi\lambda
TZ_{0}}+\mathcal{O}(\alpha\epsilon).$ (47)

For $\theta=0$, we have

$\displaystyle ImV_{Q\bar{Q}}$
$\displaystyle\sim-\dfrac{(1+\alpha)\sqrt{2}(6Z_{0}^{8}-9Z_{0}^{4}+1)}{12\pi\lambda
Z_{0}T(-1+3Z_{0}^{4})(-1+7Z_{0}^{4})}-\dfrac{21\epsilon}{\sqrt{2}\pi
Z_{0}\lambda
T(1-7Z_{0}^{4})^{2}(1-3Z_{0}^{2})^{2}}[-(\frac{83}{42}Z_{0}^{12}+$
$\displaystyle\frac{19}{126}Z_{0}^{4}-Z_{0}^{16}-Z_{0}^{8}-\frac{1}{126})\ln(\frac{\pi
T}{c})-\frac{17Z_{0}^{4}-1}{756}+\frac{41Z_{0}^{16}}{168}-\frac{17Z_{0}^{12}}{42}+\frac{11Z_{0}^{8}}{72}]+\mathcal{O}(\epsilon^{2}\alpha^{2}).$
(48)

For $k_{1}=0$, $k_{0}=2+\epsilon$ and $\theta=0,\pi/2$ we have

$\displaystyle ImV_{Q\bar{Q}}$ $\displaystyle\sim-\dfrac{\sqrt{2}\pi
T(-5+3Z_{0}^{4})}{80\lambda c^{2}Z_{0}^{3}}-\dfrac{\sqrt{2}\alpha\pi
T(-15Z_{0}^{12}+30c^{2}Z_{0}^{12}+25Z_{0}^{8}-100c^{2}Z_{0}^{8}+132c^{2}Z_{0}^{4}-50c^{2})}{400\lambda
c^{2}Z_{0}^{11}}$ $\displaystyle+\dfrac{\sqrt{2}\pi\epsilon
T(25-8Z_{0}^{4}-15Z_{0}^{8}+100Z_{0}^{4}\ln(Z_{0}))}{3200\lambda
c^{2}Z_{0}^{7}}+\mathcal{O}(\epsilon\alpha),$ (49)

where $Z_{0}=z_{h}/z_{0}$.

###  2.4 Jet Quenching

Jet quenching is related to the suppression of high-energy jets of particles
produced in high-energy heavy-ion collisions due to their interaction with
QGP. The jet quenching parameter for a parton (quark or gluon) moving along
the $p$ direction while the broadening happens along the $k$ direction is
given by [37]:

$\displaystyle
q_{p}(k)=\dfrac{\sqrt{2}}{\pi\alpha}\left(\int_{0}^{z_{h}}dz\dfrac{1}{g_{kk}}\sqrt{\dfrac{g_{zz}}{g_{--}}}\right)^{-1},\;\;\;g_{--}=\dfrac{1}{2}(g_{tt}+g_{pp}).$
(50)

First assume the quark moves along the anisotropic direction and the momentum
broadening occurs along the transverse one. Then

$g_{pp}=g_{xx},\;\;\;\;g_{kk}=g_{y_{1}y_{1}}$ (51)

for this case

$\dfrac{q_{\parallel}(\perp)}{q_{0}}\sim
1+\dfrac{\epsilon\sqrt{c_{1}}}{c^{3}}\left[-0.25+\ln\left(\dfrac{0.32c}{T}\right)\right]+0.5\dfrac{\sqrt{c_{1}\alpha}}{c^{3}}+\mathcal{O}(\epsilon\alpha).$
(52)

The second case is

$g_{pp}=g_{y_{1}y_{1}},\;\;\;\;g_{kk}=g_{xx},$ (53)

for this case $q_{\perp}(\parallel)$ we have

$\dfrac{q_{\perp}(\parallel)}{q_{0}}\sim
1+\dfrac{\epsilon{c_{1}}}{c^{6}}\left[-0.14+\ln\left(\dfrac{0.32c}{T}\right)\right]+0.75\dfrac{{c_{1}\alpha}}{c^{6}}+\mathcal{O}(\epsilon\alpha).$
(54)

Finally, we look at $q_{\perp}(\perp)$ where the quark motion and the momentum
broadening happen along the transverse directions as follows

$g_{pp}=g_{y_{1}y_{1}},\;\;\;\;\;g_{kk}=g_{y_{2}y_{2}}.$ (55)

Therefore

$\dfrac{q_{\perp}(\perp)}{q_{0}}\sim
1+\epsilon\left[-0.88+0.95\ln\left(\dfrac{0.32c}{T}\right)\right]+0.7\alpha+\mathcal{O}(\epsilon\alpha).$
(56)

To summarize we find that the jet quenching is in generally enhanced in
presence of anisotropy compared to the isotropic case and that its value
depends on the direction of the moving quark and the direction which the
momentum broadening occurs. But the effect of GB term is more respect to the
anisotropy directions. More particularly

$q_{\perp}(\parallel)>q_{\parallel}(\perp)>q_{\perp}(\perp)>q_{0}$ (57)

Figure 9: Plot of ${\color[rgb]{1,0,0}q_{\perp}(\parallel)/q_{0}}$,
${\color[rgb]{0,0,0}q_{\perp}(\perp)/q_{0}}$ and
${\color[rgb]{0,0,1}q_{\parallel}(\perp)/q_{0}}$ in terms of $c/T$ for
$\alpha=0.1,\epsilon=0.01$.

##  3 Conclusion

In this work, we extended the strongly coupled anisotropic systems at finite
temperature to quadratic gravity to gain insight into the influence of gravity
on QCD. To do so, we considered an anisotropic black hole metric as a solution
to a system of $5D$ Quadratic-Axion-Dilaton Gravity. The anisotropic
background is specified by an arbitrary exponent ($k_{0}$ and $k_{1}$), a non-
zero dilaton field, a non-zero axion field. The field equations for the
considered theory are coupled and bulky differential equations for six unknown
functions. Therefore, obtaining the solution to such field equations is too
hard, this is why we considered the special case of field equation, i.e.
$\gamma=\alpha,\beta=-4\alpha$ (Einstien-Gauss-Bonnet gravity). The
differential equation for the metric function in EGB gravity is a nonlinear
second-order equation that has been solved in a perturbative manner. In
addition to the small/large phase transition, the blackening function
supported the phase transition from the large black hole phase to the thermal
AdS phase at a critical temperature. Holographically, this phase transition
corresponds to the confinement-deconfinement phase transition in QCD. In each
case, we investigated the anisotropy influence and the effect of parameters of
theory on the thermodynamic properties of our background, in particular, on
the small/large black holes phase transition diagram. Finally, we study
transport and diffusion properties in anisotropic theories and observe in
particular that the butterfly velocity that characterizes both diffusion and
growth of chaos transverse to the anisotropic direction saturates a constant
value in the IR limit which can exceed the bound given by the conformal value.
For future work, one can look at the effect of GB term on the entropic
$c$-function following [10] the pressure along the different directions, the
velocity law [42], bound states in baryons [43] and on the rotation of the
mesons [44].

Acknowledgements
We would like to thank Dimitrios Giataganas for his fruitful comments which
help us to improve the manuscript. We also thank the anonymous referee for
her/his valuable comments.

## The result about conformal-anisotropic solution

From the equation (21), to have a result of Schwartzschild-AdS5 we should have

$\displaystyle k_{0}-3k_{1}-2=0.$ (58)

To obtain the results of conformal and anisotropic, we write

$k_{0}=8,\;\;\;\;k_{1}=2+\epsilon\;\;\;\;\;\epsilon\ll 1$ (59)

for which $\epsilon$ is the deviation from the isotropicity. Under the
conditions (59), we presented some results below:

$T=\dfrac{1+\alpha}{\pi z_{h}}+\epsilon\left[-\dfrac{3}{8\pi
z_{h}}-\alpha\left(-\dfrac{3}{8\pi z_{h}}+\dfrac{8}{\pi
z_{h}^{3}}\right)\right]+\mathcal{O}(\alpha^{2},\epsilon^{2}).$ (60)

$\displaystyle s\sim$
$\displaystyle\dfrac{\sqrt{\mathfrak{c}}\pi^{3}T^{3}}{4c^{3}}-\dfrac{3\sqrt{\mathfrak{c}}\alpha\pi^{3}T^{3}}{4c^{3}}+\dfrac{\epsilon\sqrt{\mathfrak{c}}\pi^{3}T^{3}}{8c^{3}}\left[\dfrac{9}{4}+\ln\left(\dfrac{c}{\pi
T}\right)\right]+$
$\displaystyle\dfrac{\epsilon\alpha\sqrt{\mathfrak{c}}\pi^{3}T^{3}}{8c^{3}}\left[3\ln\left(\dfrac{\pi
T}{c}\right)-\dfrac{23}{4}\right]+\mathcal{O}(\alpha^{2},\epsilon^{2}).$ (61)

$\displaystyle E=$
$\displaystyle\dfrac{3\pi^{3}\sqrt{\mathfrak{c}}T^{4}}{16c^{3}}-\dfrac{9\alpha\pi^{3}\sqrt{\mathfrak{c}}T^{4}}{16c^{3}}-\dfrac{3\epsilon\pi^{3}\sqrt{\mathfrak{c}}T^{4}}{32c^{3}}\left[\ln\left(\dfrac{\pi
T}{c}\right)-\dfrac{13}{2}\right]-$
$\displaystyle\dfrac{9\alpha\epsilon\pi^{3}\sqrt{\mathfrak{c}}T^{4}}{32c^{3}}\left[\ln\left(\dfrac{c}{\pi
T}\right)+\dfrac{11}{6}\right]+\mathcal{O}(\alpha^{2},\epsilon^{2}).$ (62)

$\displaystyle\dfrac{\eta_{||}}{s}\sim\dfrac{c^{6}}{\mathfrak{c}\pi^{6}T^{6}}+\dfrac{6\alpha
c^{6}}{\mathfrak{c}\pi^{6}T^{6}}-\dfrac{9\epsilon
c^{6}}{4\mathfrak{c}\pi^{6}T^{6}}+\mathcal{O}(\alpha^{2},\epsilon^{2})$ (63)

$\displaystyle\mu_{\perp}^{2}$ $\displaystyle\sim
6\pi^{2}T^{2}-6\alpha\pi^{2}T^{2}(1+32\pi^{2}T^{2})-\dfrac{3\epsilon\pi^{2}T^{2}}{4}+\mathcal{O}(\alpha\epsilon)$
$\displaystyle\mu_{\parallel}^{2}$
$\displaystyle\sim\dfrac{6\mathfrak{c}\pi^{8}T^{8}}{c^{6}}+\dfrac{51\epsilon\mathfrak{c}\pi^{8}T^{8}}{4c^{6}}-\dfrac{6\mathfrak{c}\pi^{8}T^{8}}{c^{6}}(7+232\pi^{2}T^{2})+\mathcal{O}(\epsilon\alpha).$
(64)

$\displaystyle v_{B\perp}^{2}$
$\displaystyle\sim\dfrac{2}{3}+\dfrac{\epsilon}{12}+\alpha\Big{[}\dfrac{3}{2}+\dfrac{464\pi^{2}T^{2}}{3}\Big{]}+\mathcal{O}(\alpha\epsilon),$
$\displaystyle v_{B\parallel}^{2}$
$\displaystyle\sim\dfrac{2c^{6}}{3\mathfrak{c}\pi^{6}T^{6}}+\alpha\Big{[}\dfrac{14c^{6}}{3\mathfrak{c}\pi^{6}T^{6}}+\dfrac{464c^{6}}{3\mathfrak{c}\pi^{4}T^{4}}\Big{]}-\dfrac{17\epsilon
c^{6}}{12\mathfrak{c}\pi^{6}T^{6}}+\mathcal{O}(\alpha\epsilon),$ (65)

where we have used

$z_{h}\sim\dfrac{1+\alpha}{\pi T}-\dfrac{3\epsilon}{8\pi
T}-\dfrac{3\epsilon\alpha}{8\pi T}+\mathcal{O}(\alpha^{2},\epsilon^{2}).$ (66)

We see that in the limit $\epsilon\to 0$ the results approach Schwartzschild-
AdS5, as expected.

## The components of the field equation

Here, the components of the field equation are presented:

$\displaystyle
E_{zz}=-6A^{2}hA^{\prime\prime}-2A^{3}h^{\prime\prime}+12AhA^{\prime
2}-hA^{3}\phi^{\prime 2}+\dfrac{\alpha}{A^{3}}\left[{24hgAA^{\prime
2}A^{\prime\prime}}+{16gA^{2}h^{\prime}A^{\prime}A^{\prime\prime}}-{48ghA^{\prime
4}}\right.$ $\displaystyle\left.\;\;\;\;\;\;-{32gAh^{\prime}A^{\prime
3}}+{8gA^{2}A^{\prime 2}h^{\prime\prime}}\right]=0,$ (67) $\displaystyle
E_{tt}=-48AhgA^{\prime\prime}-6hA^{2}g^{\prime\prime}-12A^{2}gh^{\prime\prime}-48AhA^{\prime}g^{\prime}-48gAA^{\prime}h^{\prime}-24ghA^{\prime
2}+10hA^{4}V(\phi)$
$\displaystyle\;\;\;\;-12A^{2}g^{\prime}h^{\prime}-3hgA^{2}\phi^{\prime
2}-\dfrac{3c_{1}^{2}A^{2}Z(\phi)}{h}+\dfrac{\alpha}{A^{4}}\left[192Ahg^{2}A^{\prime\prime}A^{\prime
2}+96g^{2}A^{2}A^{\prime}A^{\prime\prime}h^{\prime}+48hgA^{2}g^{\prime}A^{\prime}A^{\prime\prime}\right.$
$\displaystyle\left.\;\;\;\;+16A^{3}gA^{\prime\prime}h^{\prime}g^{\prime}+24hgA^{2}g^{\prime\prime}A^{\prime
2}+16gA^{3}A^{\prime}h^{\prime}g^{\prime\prime}+16gA^{3}A^{\prime}g^{\prime}h^{\prime\prime}+48g^{2}A^{2}h^{\prime\prime}A^{\prime
2}+96hAgA^{\prime 3}g^{\prime}\right.$
$\displaystyle\left.\;\;\;\;+{112gA^{2}g^{\prime}h^{\prime}A^{\prime
2}}+24hA^{2}g^{\prime 2}A^{\prime 2}-144hg^{2}A^{\prime 4}+16A^{3}g^{\prime
2}A^{\prime}h^{\prime}\right]=0,$ (68) $\displaystyle
E_{xx}=-12A^{2}gA^{\prime\prime}-2A^{3}g^{\prime\prime}-12A^{2}A^{\prime}g^{\prime}+2A^{5}V(\phi)-gA^{3}\phi^{\prime
2}+\dfrac{c_{1}A^{3}Z(\phi)}{h^{2}}+\dfrac{\alpha}{A^{3}}\left[16gA^{2}A^{\prime}g^{\prime}A^{\prime\prime}\right.$
$\displaystyle\left.\;\;\;\;\;+48Ag^{2}A^{\prime
2}A^{\prime\prime}+16gA^{2}g^{\prime}A^{\prime 3}+8A^{2}A^{\prime 2}g^{\prime
2}+8gA^{2}A^{\prime 2}g^{\prime\prime}-48g^{2}A^{\prime 4}\right]=0,$ (69)
$\displaystyle
E_{y_{1}y_{1}}=-6hgA^{2}A^{\prime\prime}-6hA^{2}A^{\prime}g^{\prime}-6gA^{2}h^{\prime}A^{\prime}-hA^{3}g^{\prime\prime}-\dfrac{c_{1}^{2}A^{3}Z(\phi)}{2h}-\dfrac{1}{2}hgA^{3}\phi^{\prime
2}+hA^{5}V(\phi)-$
$\displaystyle\;\;\;\;\;\;\;2A^{3}g^{\prime}h^{\prime}-2A^{3}gh^{\prime\prime}+\dfrac{\alpha}{A^{3}}\left[24hAg^{2}A^{\prime
2}A^{\prime\prime}+8hA^{2}gA^{\prime}g^{\prime}A^{\prime\prime}+16A^{2}g^{2}h^{\prime}A^{\prime}A^{\prime\prime}+4gA^{3}h^{\prime}g^{\prime}A^{\prime\prime}-\right.$
$\displaystyle\left.\;\;\;\;\;\;24hg^{2}A^{\prime
4}+4gA^{3}A^{\prime}h^{\prime}g^{\prime\prime}+4ghA^{2}A^{\prime
2}g^{\prime\prime}+4hA^{2}g^{\prime 2}A^{\prime
2}+16gA^{2}h^{\prime}g^{\prime}A^{\prime
2}+4gA^{3}A^{\prime}g^{\prime}h^{\prime\prime}+\right.$
$\displaystyle\left.\;\;\;\;\;\;8g^{2}A^{2}A^{\prime
2}h^{\prime\prime}+4A^{3}A^{\prime}h^{\prime}g^{\prime
2}+8ghAg^{\prime}A^{\prime 3}-8Ag^{2}h^{\prime}A^{\prime 3}\right]=0.$ (70)

## The GB Correction to Potentials

Here, we present the GB correction to equations (18) and (19) as follows:

$\displaystyle\hat{Z}=$
$\displaystyle\dfrac{-(k_{0}-2)}{4c_{1}k_{1}z_{h}^{6}(k_{0}-4k_{1}+6)(k_{0}+6-5k_{1})(-3k_{1}^{2}+6k_{1}+k_{0}^{2}-2k_{0})}\Big{[}8k_{1}(k_{1}-2)(k_{0}+6-4k_{1})$
$\displaystyle
z_{h}^{3+\frac{3k_{1}}{2}-\frac{k_{0}}{2}}c^{4-k_{0}-k_{1}}e^{\frac{(k_{0}+5k_{1}-6)\sqrt{6k_{1}^{2}-12k_{1}-2k_{0}^{2}+4k_{0}}(2\phi+\ln(z_{h})\sqrt{6k_{1}^{2}-12k_{1}-2k_{0}^{2}+4k_{0}})}{24k_{1}-12k_{1}^{2}+4k_{0}^{2}-8k_{0}}}\Big{(}-\frac{3k_{1}^{5}}{4}+(-\frac{3}{2}+$
$\displaystyle\frac{45k_{0}}{4})k_{1}^{4}+(-\frac{17k_{0}^{2}}{4}-41k_{0}+15)k_{1}^{3}+(37k_{0}-\frac{13}{4}k_{0}^{3}-18+21k_{0}^{2})k_{1}^{2}+k_{0}^{2}(4k_{0}-24+k_{0}^{2})k_{1}$
$\displaystyle+6k_{0}^{2}-\frac{k_{0}^{4}}{2}-2k_{0}^{3}\Big{)}+(k_{0}+6-5k_{1})(k_{0}+6-4k_{1})(k_{1}-2)^{2}z_{h}^{6}c^{4-k_{0}-k_{1}}$
$\displaystyle
e^{\frac{(k_{0}+k_{1})\sqrt{6k_{1}^{2}-12k_{1}-2k_{0}^{2}+4k_{0}}(2\phi+\ln(z_{h})\sqrt{6k_{1}^{2}-12k_{1}-2k_{0}^{2}+4k_{0}})}{12k_{1}-6k_{1}^{2}+2k_{0}^{2}-4k_{0}}}(-6k_{1}^{4}+(48+15k_{0})k_{1}^{3}-(7k_{0}^{2}+40k_{0}+72)k_{1}^{2}$
$\displaystyle+(12k_{0}^{2}-3k_{0}^{3}+36k_{0})k_{1}+k_{0}^{2}(k_{0}+6)(k_{0}-2))+(7k_{1}^{2}-(9k_{0}+10)k_{1}+2k_{0}^{2}+14k_{0}+12)(k_{0}+$
$\displaystyle
6-5k_{1})^{2}(k_{1}-2)k_{1}(6k_{1}+k_{0}^{2}-2k_{0}-3k_{1}^{2})c^{4-k_{0}-k_{1}}z_{h}^{3k_{1}-k_{0}}$
$\displaystyle
e^{\frac{(-3+2k_{1})\sqrt{6k_{1}^{2}-12k_{1}-2k_{0}^{2}+4k_{0}}(2\phi+\ln(z_{h})\sqrt{6k_{1}^{2}-12k_{1}-2k_{0}^{2}+4k_{0}})}{6k_{1}-3k_{1}^{2}+k_{0}^{2}-2k_{0}}}-(-4k_{1}+k_{0}+6)(k_{0}+6-3k_{1})$
$\displaystyle
e^{\frac{k_{0}\sqrt{6k_{1}^{2}-12k_{1}-2k_{0}^{2}+4k_{0}}(2\phi+\ln(z_{h})\sqrt{6k_{1}^{2}-12k_{1}-2k_{0}^{2}+4k_{0}})}{12k_{1}-6k_{1}^{2}+2k_{0}^{2}-4k_{0}}}(k_{0}(k_{1}-2)^{2}(k_{0}+6-3k_{1})(k_{0}+3k_{1}-2)$
$\displaystyle(k_{0}-k_{1})z_{h}^{6-k_{1}}c^{4-k_{0}-k_{1}}-2k_{1}c^{-k_{0}+2}z_{h}^{6}(k_{0}+6-5k_{1})(6k_{1}-3k_{1}^{2}+k_{0}^{2}-2k_{0}))\Big{]},$
(71)

$\displaystyle\hat{V}=$
$\displaystyle-\dfrac{c^{2}(k_{1}-2)e^{-\frac{(k_{0}-k_{1}+6)\sqrt{6k_{1}^{2}-12k_{1}-2k_{0}^{2}+4k_{0}}\phi}{k_{0}^{2}+6k_{1}-3k_{1}^{2}-2k_{0}}}}{8(k_{0}^{2}+6k_{1}-3k_{1}^{2}-2k_{0})(k_{0}-5k_{1}+6)(k_{0}-4k_{1}+6)z_{h}^{6}}\Big{[}4c^{-2k_{1}+2}z_{h}^{-2k_{1}+6}(k_{0}-4k_{1}+6)$
$\displaystyle
e^{\dfrac{(k_{0}+5k_{1}+6)\sqrt{6k_{1}^{2}-12k_{1}-2k_{0}^{2}+4k_{0}}\phi}{2k_{0}^{2}+12k_{1}-6k_{1}^{2}-4k_{0}}}\Big{(}9k_{1}^{6}-(45+\frac{9}{2}k_{0})k_{1}^{5}+(\frac{21}{2}k_{0}^{2}-12k_{0}+90)k_{1}^{4}+$
$\displaystyle(-\frac{9}{2}k_{0}^{3}-33k_{0}^{2}-108+102k_{0})k_{1}^{3}+(35k_{0}^{3}-\frac{7}{2}k_{0}^{4}+72-104k_{0}^{2}-120k_{0})k_{1}^{2}+(80k_{0}^{2}+k_{0}^{5}+$
$\displaystyle
4k_{0}^{4}-52k_{0}^{3})k_{1}-k_{0}^{2}(k_{0}+6)(k_{0}-2)^{2}\Big{)}-3c^{-2k_{1}+2}z_{h}^{-2k_{1}+6}(k_{1}-2)^{2}(k_{0}+6-4k_{1})(k_{0}-k_{1}-$
$\displaystyle
2)(k_{0}-5k_{1}+6)(k_{0}^{2}-2k_{0}+3k_{1}^{2}-6k_{1})e^{\frac{(k_{1}+k_{0}+6)\sqrt{6k_{1}^{2}-12k_{1}-2k_{0}^{2}+4k_{0}}\phi}{6k_{1}-3k_{1}^{2}+k_{0}^{2}-2k_{0}}}-3(k_{0}+6-4k_{1})(k_{0}+$
$\displaystyle
6-3k_{1})e^{\frac{(k_{0}+6)\sqrt{6k_{1}^{2}-12k_{1}-2k_{0}^{2}+4k_{0}}\phi}{k_{0}^{2}+6k_{1}-3k_{1}^{2}-2k_{0}}}+(k_{0}+6-5k_{1})(-3k_{1}^{2}+6k_{1}+k_{0}^{2}-2k_{0})c^{-2k_{1}+2}z_{h}^{-2k_{1}+6}$
$\displaystyle\Big{(}3k_{1}^{4}+(42-27k_{0})k_{1}^{3}+(-216+46k_{0}+34k_{0}^{2})k_{1}^{2}+(-78k_{0}^{2}+44k_{0}+312-11k_{0}^{3})k_{1}+k_{0}^{4}-$
$\displaystyle 144+11k_{0}^{3}+40k_{0}^{2}-72k_{0}\Big{)}\Big{]}.$ (72)

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