# Higher order curvature corrections and holographic renormalization group
flow

Ahmad Ghodsi<EMAIL_ADDRESS>Department of Physics, Faculty of
Science,
Ferdowsi University of Mashhad, Mashhad, Iran  Malihe Siahvoshan
<EMAIL_ADDRESS>Department of Physics, Faculty of Science,
Ferdowsi University of Mashhad, Mashhad, Iran

We study the holographic renormalization group (RG) flow in the presence of
higher-order curvature corrections to the $(d+1)$-dimensional Einstein-Hilbert
(EH) action for an arbitrary interacting scalar matter field by using the
superpotential approach. We find the critical points of the RG flow near the
local minima and maxima of the potential and show the existence of the bounce
solutions. In contrast to the EH gravity, regarding the values of couplings of
the bulk theory, superpotential may have both upper and lower bounds.
Moreover, the behavior of the RG flow controls by singular curves. This study
may shed some light on how a c-function can exist in the presence of these
corrections.

###### Contents

  1. 1 Introduction
  2. 2 The general setup
  3. 3 Holographic RG flow: $\kappa_{1}=0$ theories
    1. 3.1 Critical points for $\kappa_{2}<0$
      1. 3.1.1 Local maxima of the potential
      2. 3.1.2 Local minima of potential
      3. 3.1.3 Bounces
    2. 3.2 Critical points for $\kappa_{2}>0$
      1. 3.2.1 Critical points for $W\neq W_{E}$
      2. 3.2.2 Critical points near $W=W_{E}$
  4. 4 Holographic RG flow: General case
    1. 4.1 Local maxima of potential
    2. 4.2 Local minima of potential
    3. 4.3 Bounces
      1. 4.3.1 $W^{\prime}=W^{\prime\prime}=0$
      2. 4.3.2 $W^{\prime}\neq 0$ and $W^{\prime\prime}=0$
  5. 5 Summary and conclusion

##  1 Introduction

The Wilsonian approach in renormalization of quantum field theories (QFTs) [1,
2], leads to the concept of the renormalization group (RG) by integrating out
the high energy degrees of freedom. The RG flow connects different scale-
invariant QFTs in the space of all the parameters (mass and couplings) of a
QFT with a UV cut-off.

The gauge/gravity duality [3] provides a useful tool for study of the
$d$-dimensional QFTs through the corresponding $(d+1)$-dimensional gravity
theories. It is specifically important when the quantum field theory is in the
strong coupling regime in which the perturbation theory does not work. Because
of this correspondence, the RG flow of the QFT on the boundary is related to
the various geometries of the bulk gravity and, a holographic dimension plays
the role of the RG scale in the dual QFT. Consequently, the RG flow between
two fixed points of the QFT is supposed to be given by a domain wall solution
on the gravity side which connects two $AdS$ geometries. Usually, the RG flow
obtains by perturbing the theory at a UV fixed point by a relevant or marginal
operator. Adding these operators leads to a line of fixed points or an IR
fixed point.

The main issue is the duality between the RG (Callan-Symanzik) equations and
equations of the motion of the bulk action. The reason is that the RG
equations are first-order differential equations and the corresponding
equations of motion in the Einstein-Hilbert (EH) gravity are from the second
order.

There are several approaches to solve this problem. The first attempts in [5,
4] showed the equivalency of the supergravity equation of motion and the RG
flow equation of the super Yang-Mills boundary theory. Afterward, by applying
the Hamilton-Jacobi formalism [6] found first-order equations of the classical
supergravity action with the same form of the RG equations of the dual field
theory, with even the correct contribution of the conformal anomalies. An
alternative method to lower the order of the differential equations is using
an auxiliary function (superpotential) [7]. It has been applied in [8, 9, 10].
Despite the naming in the holographic RG flow literature, this function does
not relate to the supersymmetric theories necessarily [11]. Moreover, due to
the monotonic behavior of the superpotential, at least in the EH gravity, this
function can be used as a candidate for $c$-function in the holographic study
of the $c$-theorem [5, 12].

In other approaches, [13] finds a relation between the holographic RG flow and
the Wilsonian RG flow in the corresponding field theory by using the role of
multi-trace operators. Paper [14] shows an equivalence between integrating out
the high energy degrees of freedom of the field theory and integrating out on
a range of the holographic coordinate in the gravity side.

Using the superpotential method, [16, 15] show a new possible exotic behavior
of the RG flow. It has been used for Einstein-Hilbert bulk gravity in [17]. In
the superpotential formalism, the exotic RG flow results from the new possible
zero points of the beta function (the bounce points). The sign of beta
function changes as it crosses this point, and unlike the usual UV/IR fixed
points of the theory, the value of the potential is not extremum here. In
other words, the beta function has a branch cut in the bounce points.

The result of [17] has been extended to the maximally symmetric solutions with
non-zero curvature in [18]. Following these works, [20, 19] have considered
the black hole solutions in Einstein-dilaton theory and studied the finite
temperature exotic RG flows. Also, the $c$-theorem along the exotic RG flow
has been studied in [21] for 3-dimensional Einstein-dilaton gravity by
considering the entanglement entropy as a holographic $c$-function. Among
other issues, [22] discusses the exotic behavior of the RG flow in the
anisotropic geometries in Einstein-Maxwell-scalar theories or equivalently in
the dual non-relativistic quantum field theories.

A notable suggestion about the holographic study of the RG flow is the
consideration of the higher curvature corrections to the gravity side.
Generally, the order of the equations of motion is more than two, so we may
encounter the holographic RG equations that are not the first order either.
The study of these equations and their solutions may help us to understand the
RG equations of the dual QFTs at the strong coupling regime.

The higher-order curvature terms appear in various theories. For example, in
supergravity or string theory, the action contains a series of higher
curvature terms. The first general non-trivial corrections are the quadratic
curvature terms

$\mathcal{L}_{corr}=\alpha_{1}R^{2}+\alpha_{2}R_{\mu\nu}^{2}+\alpha_{3}R_{\mu\nu\rho\sigma}^{2}\,.$
(1.1)

In general, the coefficients may depend on the other fields of the theory.
Moreover, in string theory, couplings are suppressed by string length scale,
$l_{s}$. If this length is smaller than the curvature length, $l_{s}^{2}R\ll
1$, then we are in the perturbative regime and we can truncate the series
expansion of the curvature corrections. Otherwise, all higher-order
corrections would also be relevant.

On the other hand, there are various constraints on the couplings of these
corrections. For instance, in five dimensional Gauss-Bonnet (GB) gravity
($\alpha_{1}=\alpha_{3}=-\alpha_{2}/4=\lambda_{GB}$), it has been shown that
to avoid the naked singularity, the GB coupling, $\lambda_{GB}$, has to be
limited to $\lambda_{GB}\leq\frac{1}{4}$. Moreover, the unitarity of the dual
boundary theory, demands that
$-\frac{7}{36}\leq\lambda_{GB}\leq\frac{9}{100}$, for example see [23, 24].

There are some evidence that the higher curvature terms affect the holographic
bounds. For instance, [25, 26] show that the viscosity bound is violated for
positive values of the GB coupling. They also observed that for
$\lambda_{GB}\geq\frac{9}{100}$ there are metastable quasiparticles on the
boundary CFT that can travel faster than the speed of light and violate
causality. The existence of the ghost modes and tachyonic excitations are
generic in these theories. To avoid causality violations, the higher-
derivative couplings should be suppressed by appropriate powers of the Planck
length $l_{P}$ or the string length scale $l_{s}$. At least for small values
of the couplings, the ghost or tachyonic modes are integrated out, and their
degrees of freedom are beyond the QFT cut-off.

In this paper, we consider constant coefficients for higher curvature terms
and do not impose any restrictive condition. Although, we can always apply the
above physical constraints, and our calculations would be trustful as far as
we restrict ourselves in the allowed regions of the couplings of the theory.

The holographic RG flow of the general quadratic curvature (GQC) gravity in a
simple toy model has been studied in [27]111For further properties of this
theory of gravity for example see [29, 28, 30]. It shows the existence of the
a-theorem for even-dimensional theories by finding the Wess-Zumino action,
which originates from the spontaneous breaking of the conformal symmetry, by
using a radial cut-off near the AdS boundary.

In this paper, we will develop our understanding of the holographic RG flow by
looking at the effects of the general quadratic curvature terms on the flow.
Similar to [17], we have considered a scalar matter field that minimally
coupled to gravity with an arbitrary potential. We define superpotential and
write the equations of motion in terms of it. As the gauge-gravity
correspondence suggests, the RG transformations of the boundary QFT are dual
to a domain wall solution that connects the UV/IR fixed points. The scale
factor of this geometry is related to the energy scale of the dual QFT, and
therefore to the beta function of a marginal/relevant operator on the
boundary. Finding the zeros of this beta function gives the critical points of
the corresponding dual QFT. We will explain these steps in more detail in
section 2.

Usually, the first choice to study the higher curvature corrections is the GB
gravity, in which the order of derivatives of the equations of motion is the
same as the EH action. Although generally, we have three independent couplings
($a_{1},a_{2},a_{3}$) in action (1.1), the equations of motion only depend on
two specific combinations of these couplings, say ($\kappa_{1},\kappa_{2}$).
The GB gravity stays on a specific simple class of the GQC gravity when
$\kappa_{1}=0$. The space of two-derivative theories is more significant, and
we will study this space in the first attempt. Then we extend our results for
the general case when the equations of motion are the fourth-order
derivatives.

This paper is organized as follows: In section 2 we provide a setup for GQC
gravity. In section3, we present the details of the calculations of the
holographic RG flow in two-derivative theories. We find the superpotential,
scalar field and, the scale factor near the possible critical points
perturbatively. We address the behavior of the holographic RG flow for GQC
gravity in section 4. The last section is the summary and conclusions.

##  2 The general setup

As was told in the introduction, we are going to study the effect of the
quadratic curvature terms in the bulk gravity on the holographic RG flow.
Nearly all steps of the calculations are based on the techniques of paper
[17]. Let’s consider a bulk action in a general $(d+1)$-dimensional space-
time, that in addition to the Hilbert-Einstein action, it includes the
following quadratic curvature terms with three independent couplings

$S_{bulk}=\int
d^{d+1}x\sqrt{-g}\Big{(}R+\mathit{a}_{1}R^{2}+\mathit{a}_{2}R_{\mu\nu}^{2}+\mathit{a}_{3}(R^{2}-4R_{\mu\nu}^{2}+R_{\mu\nu\rho\sigma}^{2})-\frac{1}{2}(\partial_{\mu}\phi)^{2}-V(\phi)\Big{)}\,.$
(2.1)

Moreover, we couple a real scalar matter field with an arbitrary potential to
gravity minimally. To study the holographic RG flow, we suppose that the
potential for all the possible values of the scalar field is negative
everywhere. It guarantees the existence of a dual conformal field theory at UV
or IR fixed points of the RG flow. In other words, at the gravity side, we
suppose that the scalar field has various stationary points, specifically at
the UV and IR fixed points. Consequently, at these points, the classical
solution of the equations of motion is the $AdS_{d+1}$ space-time.

By gauge-gravity correspondence, the RG transformations of the boundary
quantum field theories are dual to the diffeomorphisms of the bulk gravity.
Therefore, it would be reasonable to consider a solution for equations of
motion of the bulk fields that smoothly connects the AdS space-times at the UV
and IR fixed points. Therefore, we begin with the following metric, which in
particular is a holographic representation of the RG flow between the two
holographic dual boundary CFTs

$ds^{2}=e^{2A(r)}\big{(}-dt^{2}+\sum_{i=1}^{d-1}dx_{i}dx^{i}\big{)}+dr^{2}\,,\qquad\phi=\phi(r)\,.$
(2.2)

Here $r$ is the holographic coordinate, and we suppose that the boundary is a
$d$-dimensional Minkowski space. In this description, the warp factor is
related to the energy scale of the holographic dual quantum field theory

$\log\frac{E}{E_{0}}=A(r)\,.$ (2.3)

If we consider the relevant operator $\mathcal{O}$ on the boundary CFT, which
couples to the boundary value of the bulk scalar field, i.e.

$\mathcal{L}=\mathcal{L}_{CFT}+\phi(t,x^{i})\mathcal{O}(t,x^{i})\,,$ (2.4)

then it generates an RG flow, that goes away from the UV fixed point. The
coupling of this operator i.e. $g(E)=\phi(r,t,x^{i})$ is running according to
the RG flow equation and its corresponding beta function is given by

$\beta(g)=\frac{dg(E)}{d\log E}\,.$ (2.5)

Therefore we can compute the beta function by knowing the scale factor,
$A(r)$, so we need to find the solution of the bulk fields. For the field
configuration in (2.2), the equations of motion from the variation of the
action with respect to metric are given by

$\displaystyle\frac{1}{2}\phi^{\prime}(r)^{2}-V(\phi)$
$\displaystyle=-d\Big{(}-(d-1)A^{\prime}(r)^{2}+{\kappa_{2}}A^{\prime}(r)^{4}$
$\displaystyle~{}~{}~{}+{\kappa_{1}}\big{(}2dA^{\prime}(r)^{2}A^{\prime\prime}(r)-A^{\prime\prime}(r)^{2}+2A^{(3)}(r)A^{\prime}(r)\big{)}\Big{)}\,,$
(2.6a) $\displaystyle\frac{1}{2}\phi^{\prime}(r)^{2}+V(\phi)$
$\displaystyle=-d(d-1)A^{\prime}(r)^{2}-2(d-1)A^{\prime\prime}(r)$
$\displaystyle~{}~{}~{}+{\kappa_{2}}\big{(}dA^{\prime}(r)^{4}+4A^{\prime}(r)^{2}A^{\prime\prime}(r)\big{)}+{\kappa_{1}}\big{(}2A^{(4)}(r)$
$\displaystyle~{}~{}~{}+3dA^{\prime\prime}(r)^{2}+4dA^{(3)}(r)A^{\prime}(r)+2d^{2}A^{\prime}(r)^{2}A^{\prime\prime}(r)\big{)}\,,$
(2.6b)

where the above equations just depend on two combinations of the couplings of
the quadratic curvature terms i.e.

$\kappa_{1}=4da_{1}+(d+1)a_{2}\,,\qquad\kappa_{2}=(d-3)\Big{(}(d-1)(d-2)a_{3}+d((d+1)a_{1}+a_{2})\Big{)}\,.$
(2.7)

According to the above results, we divide our problem into two different
sections. In section 3, we suppose $\kappa_{1}=0$, then the equations of
motion are at most the second derivatives of the fields with respect to $r$.
Although the differential equations are in the same order as the Einstein-
Hilbert equations of motion, we experience new interesting situations. In
section 4, when $\kappa_{1}\neq 0$ the equations of motion have fourth-order
derivatives of the field. It makes things a little complicated. However, the
analysis of section 3 reveals more information and details for the latter
case.

##  3 Holographic RG flow: $\kappa_{1}=0$ theories

In this section, we consider a simplified version of the GQC and suppose that
$\kappa_{1}=0$. In particular, it contains the well known $(d+1)$-dimensional
Gauss-Bonnet gravity when $a_{1}=a_{2}=0$. Moreover, this theory in its
general form includes the critical and $f(R)$ theories of gravity up to the
quadratic curvature terms.

Following the Einstein-Hilbert case [17], we define a superpotential that
makes the equations of motion as a set of first-order differential
equations222Everywhere in this paper $\phi^{\prime}=\frac{d\phi(r)}{dr}$ and
$W^{\prime}=\frac{dW(\phi)}{d\phi}$.

$W(\phi(r))=-2(d-1)A^{\prime}(r)\,.$ (3.1)

If we rewrite the equations of motion (2.6) and (2.6b) in terms of the
superpotential and subtract/add these equations, then the set of the second
order differential equations will reduce to the following first order
equations

$\displaystyle\phi^{\prime}$
$\displaystyle=\Big{(}1-\frac{\kappa_{2}}{2(d-1)^{3}}W^{2}\Big{)}W^{\prime}\,,$
(3.2a) $\displaystyle V(\phi)$
$\displaystyle=-\frac{d}{4(d-1)}\Big{(}1-\frac{\kappa_{2}}{4(d-1)^{3}}W^{2}\Big{)}W^{2}+\frac{1}{2}\Big{(}1-\frac{\kappa_{2}}{2(d-1)^{3}}W^{2}\Big{)}^{2}{W^{\prime}}^{2}\,.$
(3.2b)

The holographic RG flow or the beta function for coupling $\phi$ of the
relevant operator $\mathcal{O}$ is obtained from equations (2.3) and (2.5) via

$\beta(\phi)=\frac{d\phi(r)}{dA(r)}=-2(d-1)\Big{(}1-\frac{\kappa_{2}}{2(d-1)^{3}}W^{2}\Big{)}\frac{W^{\prime}}{W}\,,$
(3.3)

where in the last equality we have substituted the equations (3.1) and (3.2a).
This equation shows that, by restricting to the region where $\kappa_{2}<0$,
the sign of beta function would only depend on the sign of
$\frac{W^{\prime}}{W}$. Furthermore, the monotonic behavior of the
superpotential along the RG flow would be guaranteed. This can be seen by

$\frac{dW(\phi(r))}{dr}=\Big{(}1-\frac{\kappa_{2}}{2(d-1)^{3}}W^{2}\Big{)}{W^{\prime}}^{2}\,.$
(3.4)

Based on the above relation, for a negative coupling $\kappa_{2}$, we can
define a monotonically decreasing holographic c-function constructed out of
the superpotential. For example in [27], for a specific constant $\alpha$, it
is proportional to $\frac{1}{W^{d+1}}(W^{2}+\alpha W^{4})$. However, for
positive values of $\kappa_{2}$, one needs to consider an upper bound either
for $W$ or $\kappa_{2}$, otherwise, this superpotential would not be a
monotonic function. Before proceeding further, we should emphasize two
important issues:

1\. There is a constraint on the superpotential from the equation of motion
(3.2b). If we write it as

$W^{\prime
2}=\frac{(d-1)^{2}}{2(\kappa_{2}W^{2}-2(d-1)^{3})^{2}}\Big{(}16(d-1)^{4}V+4d(d-1)^{3}W^{2}-d\kappa_{2}W^{4}\Big{)}\,,$
(3.5)

then the right hand side must be positive which it means that regarding the
sign of $\kappa_{2}$ there are upper or lower bounds on the value of the
superpotential. Since we have supposed $V(\phi)<0$ then

$\displaystyle W\geq B_{-}>0\,,\qquad\qquad\qquad\,\,\,\kappa_{2}<0\,,$ (3.6a)
$\displaystyle B_{+}\geq W\geq B_{-}>0\,,\qquad
0<\kappa_{2}<-\frac{d(d-1)^{2}}{4V(\phi)}\,,$ (3.6b)

where for simplicity, we have assumed a positive superpotential for all values
of the scalar field333In general, equations of motion are invariant under
$W\leftrightarrow-W$.. The upper and lower bounds are defined as

$B_{\pm}^{2}=\frac{2(d-1)^{3}}{\kappa_{2}}\Big{(}1\pm\sqrt{1+\frac{4\kappa_{2}}{d(d-1)^{2}}V(\phi)}\Big{)}\,.$
(3.7)

2\. The scalar curvature can be written in term of the superpotential

$\displaystyle\mathcal{R}$
$\displaystyle=\frac{d}{d-1}\Big{(}\phi^{\prime}W^{\prime}-\frac{d+1}{4(d-1)}W^{2}\Big{)}$
$\displaystyle=\frac{d}{d-1}\Big{(}\big{(}1-\frac{\kappa_{2}}{2(d-1)^{3}}W^{2}\big{)}{W^{\prime}}^{2}-\frac{d+1}{4(d-1)}W^{2}\Big{)}\,.$
(3.8)

The last equality shows that we should have a finite value for
$W^{\prime}(\phi(r))$ to have a regular geometry or nonsingular curvature.

###  3.1 Critical points for $\kappa_{2}<0$

The zero points of the beta function (3.3) for $\kappa_{2}<0$ are equivalent
to the points where $W^{\prime}$ vanishes. To find a relation between these
extremums of the superpotential and the scalar potential $V$, we can
differentiate (3.2b) with respect to the $\phi$

$V^{\prime}(\phi)=\big{(}1-\frac{\kappa_{2}W^{2}}{2(d-1)^{3}}\big{)}\Big{(}(1-\frac{\kappa_{2}W^{2}}{2(d-1)^{3}})W^{\prime\prime}-\frac{d}{2(d-1)}(1+\frac{2\kappa_{2}W^{\prime
2}}{d(d-1)^{2}})W\Big{)}W^{\prime}\,.$ (3.9)

In an extremum of the superpotential, if $W^{\prime\prime}$ is finite, then
$V^{\prime}=0$. When $W^{\prime\prime}$ diverges, the potential is not
necessarily an extremum. In the following subsections, we will analyze both of
these cases separately, i.e.

$\bullet$ $W^{\prime}=V^{\prime}=0$ and $W^{\prime\prime}$ is finite.

$\bullet$ $W^{\prime}=0$ and $W^{\prime\prime}$ diverges but $V^{\prime}$ is
finite.

We should emphasize that there is a lower bound on the superpotential
everywhere, as we showed in (3.6a) for negative values of $\kappa_{2}$.

####  3.1.1 Local maxima of the potential

Near a local maximum, the potential can be expanded as a sum of a cosmological
term, a mass term ($m^{2}>0$), and interacting terms

$V(\phi)=-\frac{d(d-1)}{L^{2}}-\frac{1}{2}m^{2}\phi^{2}+\mathcal{O}(\phi^{3})\,.$
(3.10)

For simplicity in the notation we have considered the extremum at $\phi=0$.
After substitution of the potential (3.10) in equation of motion (3.2b), two
independent solutions appear for superpotential

$W_{\pm}(\phi)=\frac{2(d-1)}{\tilde{L}}+\frac{\Delta_{\pm}}{2\tilde{L}}\phi^{2}+\mathcal{C}_{\pm}\phi^{z_{\pm}}+\cdots\,,$
(3.11)

where dots denote the sub-leading terms and $\mathcal{C}_{\pm}$ are the
constants of integration. The other parameters are defined as follow

$\displaystyle\tilde{L}$
$\displaystyle=\frac{L}{\sqrt{2}}\Big{(}1+\sqrt{1-\frac{4\kappa_{2}}{(d-1)L^{2}}}\Big{)}^{\frac{1}{2}}\,,$
(3.12a) $\displaystyle\Delta_{\pm}$
$\displaystyle=\frac{d\pm\sqrt{d^{2}-4\tilde{L}^{2}m^{2}}}{2\lambda}\,,\quad\lambda=1-\frac{2\kappa_{2}}{(d-1)\tilde{L}^{2}}\,.$
(3.12b)

The reality condition for $\Delta_{\pm}$ implies that the mass is bounded from
above, $0<m^{2}\leq\frac{d^{2}}{4\tilde{L}^{2}}$. Also $\kappa_{2}<0$
restricts $\lambda>1$ and therefore $L<\tilde{L}$. As we will show, the new
parameter $\tilde{L}$ is the radius of $AdS_{d+1}$ solution at the fixed
point. For negative values of $\kappa_{2}$, it is larger than the radius of
the $AdS_{d+1}$ solution of the Einstein-Hilbert action. The power of the
first leading terms are fixed by equations of motion

$z_{\pm}=\frac{d}{\lambda\Delta_{\pm}}\,,\qquad 0<z_{+}<2\,,\qquad 2<z_{-}\,,$
(3.13)

which consequently implies the boundary condition $\mathcal{C}_{+}=0$,
otherwise $\phi=0$ would not be a fixed point because $W^{\prime}$ diverges
and the geometry is not smooth here. In figure 1, the generic behavior of the
$W_{\pm}$ are depicted in green and red curves. These curves are limited to
the region above the bound curve (gray region) i.e. $W>B_{-}$.

Figure 1: The generic behavior of the superpotential near the local maximum of
the potential. The green curve is $W_{+}$ and the red one is for $W_{-}$. The
blue curves represent the bounce solutions. The gray region is the forbidden
area and is limited by the bound equation.

As we already mentioned, the RG flow is given by (3.4) and since
$\frac{dW}{dr}\geq 0$ in this case, therefore $W$ and $r$ have the same
increasing or decreasing behavior. The arrows in the figure 1, show the
increasing direction of the radial coordinate, or equivalently the RG flow
direction. The outgoing RG flow from $\phi=0$ indicates that this critical
point is a UV fixed point.

By solving the equation of motion (3.2a) around $\phi=0$ and by using the
superpotential solutions in (3.11), the scalar field solutions around the
critical point can be found

$\displaystyle\phi_{+}(r)=\phi_{+}e^{\frac{\lambda\Delta_{+}}{\tilde{L}}r}+\cdots\,,$
(3.14a)
$\displaystyle\phi_{-}(r)=\phi_{-}e^{\frac{\lambda\Delta_{-}}{\tilde{L}}r}+\frac{\mathcal{C}_{-}d\tilde{L}}{\Delta_{-}(d-2\lambda\Delta_{-})}\phi_{-}^{\Delta_{+}/\Delta_{-}}e^{\frac{\lambda\Delta_{+}}{\tilde{L}}r}+\cdots\,,$
(3.14b)

where $\phi_{\pm}$ are the constants of the integration. On the boundary field
theory side we can associate two sources $J_{\pm}$ to the operator
$\mathcal{O}$. In the standard quantization with using the above solution for
scalar field, these sources are

$J_{+}=0\,,\qquad J_{-}=\frac{\phi_{-}}{\tilde{L}^{\lambda\Delta_{-}}}\,.$
(3.15)

The vacuum expectation value of the operator $\mathcal{O}$ can be read as
follow

$\langle\mathcal{O}\rangle_{+}=\frac{2\lambda\Delta_{+}-d}{\tilde{L}^{\lambda\Delta_{+}}}\phi_{+}\,,\qquad\langle\mathcal{O}\rangle_{-}=\frac{d}{\tilde{L}^{\lambda\Delta_{+}}\Delta_{-}}\mathcal{C}\tilde{L}\phi_{-}^{\Delta_{+}/\Delta_{-}}\,.$
(3.16)

In the last step, we can read the geometry, near the fixed point, by using the
definition of the superpotential (3.1)

$\displaystyle A_{+}(r)=$
$\displaystyle-\frac{r-r_{*}}{\tilde{L}}-\frac{\phi_{+}^{2}}{8\lambda(d-1)}e^{\frac{2\lambda\Delta_{+}}{\tilde{L}}r}+\cdots,$
(3.17a) $\displaystyle A_{-}(r)=$
$\displaystyle-\frac{r-r_{*}}{\tilde{L}}-\frac{\phi_{-}^{2}}{8\lambda(d-1)}e^{\frac{2\lambda\Delta_{-}}{\tilde{L}}r}+\mathcal{O}(e^{\frac{dr}{\tilde{L}}})+\cdots\,.$
(3.17b)

Since $\lambda\Delta_{\pm}>0$, both geometries are asymptotically $AdS$ spaces
with radius $\tilde{L}$ when $r\rightarrow-\infty$. This is a UV fixed point
and we expect a dual CFT.

####  3.1.2 Local minima of potential

We can perform the same analysis as the previous section if the potential is
locally near its minima ($m^{2}>0$)

$V(\phi)=-\frac{d(d-1)}{L^{2}}+\frac{1}{2}m^{2}\phi^{2}+\mathcal{O}(\phi^{3}),$
(3.18)

We expect the same solutions for superpotential similar to the (3.11), with an
exception that $\Delta_{\pm}$ now are equal to

$\Delta_{\pm}=\frac{d\pm\sqrt{d^{2}+4\tilde{L}^{2}m^{2}}}{2\lambda}.$ (3.19)

Since we have considered $m^{2}>0$, the $\phi^{z}$ term is not the first
leading term because

$z=z_{\pm}=\frac{d}{\lambda\Delta_{\pm}}\,,\qquad 0<z_{+}<1\,,\qquad
z_{-}<0\,,$ (3.20)

therefore we should choose $\mathcal{C}_{\pm}=0$. Finally, the solutions for
scalar field and scale factor are given by

$\displaystyle\phi_{\pm}(r)$
$\displaystyle=\phi_{\pm}e^{\frac{\lambda\Delta_{\pm}}{\tilde{L}}r}+\cdots\,,$
(3.21a) $\displaystyle A_{\pm}(r)$
$\displaystyle=-\frac{r-r_{*}}{\tilde{L}}-\frac{\phi_{\pm}^{2}}{8\lambda(d-1)}e^{\frac{2\lambda\Delta_{\pm}}{\tilde{L}}r}+\cdots\,.$
(3.21b)

In figure 2, we have sketched the generic behavior of the superpotential
(again green and red curves for $W_{\pm}$). The arrows on the curves show the
increasing direction of the holographic coordinate $r$. So $\phi=0$ is an IR
fixed point for $W_{-}$ solution, and at the same time, it is a UV fixed point
for the $W_{+}$ branch. Acctually, $\Delta_{+}$ is positive and $\Delta_{-}$
is negative always, so the UV fixed point sits at $r\rightarrow-\infty$ but
the IR fixed point is located at $r\rightarrow+\infty$. In other words, the
endpoint of an RG flow is the starting point of another RG flow (see the same
behavior and its related discussions for Einstein-Hilbert action in [17]). On
the boundary field theory side the associate sources and vacuum expectation
values of the operator $\mathcal{O}$ are as follow

$J_{\pm}=0\,,\qquad\langle\mathcal{O}\rangle_{\pm}=\frac{2\lambda\Delta_{\pm}-d}{\tilde{L}^{\lambda\Delta_{\pm}}}\phi_{\pm}\,.$
(3.22)

Figure 2: The generic behavior of the superpotential near the local minimum of
the potential. The green curve is $W_{+}$ and the red one is for $W_{-}$. The
blue curves represent the bounce solutions.

####  3.1.3 Bounces

As we already discussed, there is another possibility for a critical point.
The initial conditions for derivatives of the superpotential and potential at
this point are given by

$W^{\prime}(\phi_{B})=0,\qquad V^{\prime}(\phi_{B})\neq
0,\qquad\text{Divergent }W^{\prime\prime}(\phi_{B}).$ (3.23)

It means that near the so-called bounce points [17], the superpotential is
still at the extremum, however for a general potential with $V^{\prime}\neq
0$, equation (3.9) leads to a divergent $W^{\prime\prime}$. By differentiating
the equation of motion (3.2b) with respect to $\phi$ and imposing the above
conditions of the bounce point we get

$V^{\prime}(\phi_{B})\simeq(1-\frac{\kappa_{2}}{2(d-1)^{3}}W^{2})^{2}W^{\prime}W^{\prime\prime}\,.$
(3.24)

Consider the following superpotential for some arbitrary constants $W_{B}$ and
$\mathcal{C}_{z}$

$W=W_{B}+\mathcal{C}_{z}(\phi-\phi_{B})^{z}+\dots\,.$ (3.25)

By definition (equation (3.5)), the lower bound is a set of points where
$W^{\prime}=0$ for a general potential function. Therefore, all the critical
points belong to this set, and the extremum points of the potential are just
specific points on this curve. Therefore, $W_{B}$ is the value of the
superpotential at the bounce point, which is an arbitrary point on the lower
bound curve (3.6a)

$W_{B}=B_{-}(\phi_{B})=\sqrt{\frac{2(d-1)^{3}}{\kappa_{2}}}\Big{(}1-\sqrt{1+\frac{4\kappa_{2}}{d(d-1)^{2}}V(\phi_{B})}\Big{)}^{\frac{1}{2}}\,.$
(3.26)

Substituting the ansatz (3.25) into the (3.24) and keeping the leading terms,
we find that $z=\frac{3}{2}$ and the constant $\mathcal{C}_{z}$ is equal to

$\begin{split}\mathcal{C}_{z}^{2}=\frac{8}{9}\Big{(}1-\frac{\kappa_{2}}{2(d-1)^{3}}B_{-}(\phi_{B})^{2}\Big{)}^{-2}V^{\prime}(\phi_{B})=\frac{8}{9}\Big{(}1+\frac{4\kappa_{2}}{d(d-1)^{2}}V(\phi_{B})\Big{)}^{-1}V^{\prime}(\phi_{B})\,,\end{split}$
(3.27)

where in the second equality we have used the equation (3.26). Finally, we
have two distinct solutions for the superpotential near the bounce point

$\begin{split}W_{\uparrow,\downarrow}(\phi)&=B_{-}(\phi_{B})\pm\frac{2}{3}\Big{(}1-\frac{\kappa_{2}}{2(d-1)^{3}}B_{-}(\phi_{B})^{2}\Big{)}^{-1}\sqrt{2V^{\prime}(\phi_{B})}(\phi-\phi_{B})^{\frac{3}{2}}+\dots\,.\end{split}$
(3.28)

The corresponding (blue) curves for $\uparrow$ and $\downarrow$ solutions are
depicted in figures 1 and 2. It should be noted that for positive(negative)
$V^{\prime}$, $\phi$ is greater (smaller) than $\phi_{B}$ i.e. the bounces are
reaching the bounds from the right(left) hand side. These superpotentials,
then lead to the following solution for scalar field near the bounce point.
From equation (3.2a) and the conditions of the bounce point we know that the
$r$-derivative of the $\phi$ must be zero (since $W^{\prime}=0$ at this
point), so we find

$\begin{split}\phi_{\uparrow,\downarrow}(r)&=\phi_{B}+\frac{9}{16}\Big{(}1-\frac{\kappa_{2}}{2(d-1)^{3}}B_{-}(\phi_{B})^{2}\Big{)}^{2}\mathcal{C}_{z}^{2}(r-r_{B})^{2}+\mathcal{O}(r-r_{B})^{3}\\\
&=\phi_{B}+\frac{V^{\prime}(\phi_{B})}{2}(r-r_{B})^{2}+\mathcal{O}(r-r_{B})^{3}\,,\end{split}$
(3.29)

where $r_{B}$ is the corresponding radius of the bounce point and up (down)
solution corresponds to $r>r_{B}\,(r<r_{B})$. The second line above is
simplified by using the value of $\mathcal{C}_{z}$ and shows that at least at
the first non-trivial order of the equations, the value of $\kappa_{2}$ does
not play a role in the scalar field solution.

The next step would be the determination of the geometry at this point. Using
the definition of the superpotential in (3.1) while we have the functionality
of the superpotential and the scalar field near the bounce point we get

$\begin{split}A(r)=A_{B}-\frac{B_{-}(\phi_{B})}{2(d-1)}(r-r_{B})-\frac{V^{\prime}(\phi_{B})^{2}}{24(d-1)}\Big{(}1-\frac{\kappa_{2}B_{-}(\phi_{B})^{2}}{2(d-1)^{3}}\Big{)}^{-1}(r-r_{B})^{4}+\dots\,,\end{split}$
(3.30)

which is the same solution for the both branches of $W_{\uparrow,\downarrow}$.
As $r\rightarrow r_{B}$, the above geometry tends to a regular geometry. The
behavior of the beta function (3.3) near the bounce point is given by

$\begin{split}\beta_{\uparrow,\downarrow}(\phi)=\pm\frac{2(d-1)}{B_{-}(\phi_{B})}\sqrt{V^{\prime}(\phi_{B})}\sqrt{\phi-\phi_{B}}+\mathcal{O}(\phi-\phi_{B})^{2}\,.\end{split}$
(3.31)

Although the beta function is zero, it is not a fixed point, and actually, the
RG flow does not reverse its direction at the bounce point. It should be
emphasized that the geometry in (3.30), unlike the local maxima and minima
points, is a $(d+1)$-dimensional asymptotically flat space.

###  3.2 Critical points for $\kappa_{2}>0$

In this section, we consider $\kappa_{2}>0$ and repeat all the previous steps
in $\kappa_{2}>0$. The beta function in equation (3.3) for $\kappa_{2}>0$ now
has two types of fixed points. In addition to the extremum points of the
superpotential, there are specific values for the supoerpotential at

$W_{E}=\pm\sqrt{\frac{2(d-1)^{3}}{\kappa_{2}}}\,,$ (3.32)

where the beta function vanishes. We should be careful since the beta function
would be zero at $W=W_{E}$ only if $W^{\prime}$ gets a finite value. Therefore
we divide our analysis into two subsections when $W$ is equal to $W_{E}$ or
not.

Before we proceed, it is worth reminding that according to the relation (3.6b)
the superpotential is limited between two regions i.e. $B_{+}\geq W\geq
B_{-}$, provided that the coupling $\kappa_{2}$ has an upper bound

$0<\kappa_{2}\leq-\frac{d(d-1)^{2}}{4V_{Max}}\,,$ (3.33)

where $V_{Max}$ is supposed to be a global maximum of the potential, i.e.
$V(\phi)\leq V_{Max}<0$.

####  3.2.1 Critical points for $W\neq W_{E}$

Away from $W=W_{E}$, we have various critical points:

$\bullet$ Local maxima of the potential: Near $\phi=\phi_{max}$ (see figure
3(a)), the solutions of the superpotential similarly are written as (3.11)
with a slightly change in the parameters of the solutions. Unlike the negative
coupling solutions, there are two acceptable value for $\tilde{L}$ for each
branches. By choosing $\phi_{max}=0$ we have

$\displaystyle W_{\pm}^{u}(\phi)$
$\displaystyle=\frac{2(d-1)}{\tilde{L}_{u}}+\frac{\Delta_{\pm}^{u}}{2\tilde{L}_{u}}\phi^{2}+\mathcal{C}_{\pm}^{u}\phi^{u_{\pm}}+\cdots\,,$
(3.34a) $\displaystyle W_{\pm}^{d}(\phi)$
$\displaystyle=\frac{2(d-1)}{\tilde{L}_{d}}+\frac{\Delta_{\pm}^{d}}{2\tilde{L}_{d}}\phi^{2}+\mathcal{C}_{\pm}^{d}\phi^{v_{\pm}}+\cdots\,,$
(3.34b)

where $\pm$ define the branches and $u$ and $d$ indices denote the solutions
according to their starting points on the upper or lower bounds. The
parameters of the solutions are defined as follow

$\displaystyle\tilde{L}_{(u,d)}=\frac{L}{\sqrt{2}}\Big{(}1\mp\sqrt{1-\frac{4\kappa_{2}}{(d-1)L^{2}}}\Big{)}^{\frac{1}{2}}\,,\qquad\Delta_{\pm}^{(u,d)}=\frac{d\pm\sqrt{d^{2}-4\tilde{L}_{(u,d)}^{2}m^{2}}}{2\lambda_{(u,d)}}\,,$
(3.35a)
$\displaystyle\lambda_{(u,d)}=1-\frac{2\kappa_{2}}{(d-1)\tilde{L}_{(u,d)}^{2}}\,,\qquad
u_{\pm}=\frac{d}{\Delta_{\pm}^{u}\lambda_{u}}\,,\qquad
v_{\pm}=\frac{d}{\Delta_{\pm}^{d}\lambda_{d}}\,.$ (3.35b)

The reality conditions of these parameters restrict the mass of the scalar
field and the value of $\kappa_{2}$ coupling to

$0<m\leq\frac{d}{2\tilde{L}_{(u,d)}}\,,\qquad
0<\kappa_{2}\leq\frac{d-1}{4}L^{2}\,.$ (3.36)

Therefore the upper bound on $\kappa_{2}$ is determined via
$\min\big{(}\frac{d-1}{4}L^{2},-\frac{d(d-1)^{2}}{4V_{Max}}\big{)}$. The
parameters of the solutions obey the following inequalities

$\displaystyle 0<\tilde{L}_{u}\leq\frac{L}{\sqrt{2}}\leq\tilde{L}_{d}<L\,,$
(3.37a)
$\displaystyle-1\leq\lambda_{u}<0\leq\lambda_{d}<1\longrightarrow\Delta_{\pm}^{u}<0<\Delta_{\pm}^{d}\,,$
(3.37b) $\displaystyle
1<u_{+}<2<u_{-}\longrightarrow\mathcal{C}_{+}^{u}=\mathcal{C}_{+}^{d}=0\,.$
(3.37c)

By using the above results, we have sketched the superpotential in figure
3(a), in terms of the scalar field near $\phi=\phi_{max}$. In this figure, the
gray regions are the forbidden areas of the superpotential due to the upper
and lower bounds. The blue dashed line represents the critical value
$W=W_{E}$. For those solutions above the $W_{E}$ line,
$\tilde{L}=\tilde{L}_{u}$ and for those under the line
$\tilde{L}=\tilde{L}_{d}$. In figure 3(a), all green curves are sketched for
$W_{+}$ solution and the red ones for $W_{-}$. The analysis of RG flow shows
that in $\phi=\phi_{max}$, there are two UV fixed points related to the up and
down solutions of the superpotential, and the RG flow goes out of the local
maxima points.

(a)

(b)

Figure 3: (a): The generic behavior of the superpotential near a local maximum
$\phi_{max}$ or minimum $\phi_{min}$ of the potential and its RG flow are
depicted as green and red curves. The local maxima are the UV fixed points,
but the local minima are the UV fixed points for $W_{-}$ branch and IR fixed
points for $W_{+}$. The bounce solutions also exist everywhere away from the
extrema (purple curves). The blue dashed line is $W=W_{E}$, which is the
location of curvature singularity. (b) In a specific point where the two
bounds meet each other, the singularity removes, and an IR fixed point appears
at the cross points.

$\bullet$ Local minima of the potential: Similar to the local maxima, there
are two sets of solutions together with two values for $\tilde{L}$. We have
the same solutions as (3.34a) and (3.34b) for $W>W_{E}$ and $W<W_{E}$ together
with the same parameters as (3.35a) and (3.35b) but with a change
$m^{2}\rightarrow-m^{2}$. The leading terms of the superpotential near
$\phi=\phi_{min}=0$ is given by

$W_{\pm}^{(u,d)}(\phi)=\frac{2(d-1)}{\tilde{L}_{(u,d)}}+\frac{\Delta_{\pm}^{(u,d)}}{2\tilde{L}_{(u,d)}}\phi^{2}+\cdots\,.$
(3.38)

A generic behavior of the superpotential is depicted in the figure 3(a) near
the point $\phi=\phi_{min}$. It should be noted that in this case,

$\Delta_{+}^{u}<0<\Delta_{-}^{u}\,,\qquad\Delta_{-}^{d}<0<\Delta_{+}^{d}\,.$
(3.39)

The critical point above the line of $W=W_{E}$ is an IR fixed point for
$W_{-}^{u}$ branch (red curve) and a UV fixed point for $W_{+}^{u}$ branch
(green curve). The same UV/IR behavior holds for $W_{\pm}^{d}$ below the line
of $W=W_{E}$.

$\bullet$ Bounces: For positive values of $\kappa_{2}$, the bounces are
sitting on both the upper and lower bounds. The solutions are

$W^{(u,d)}_{\uparrow,\downarrow}(\phi)=B_{\pm}(\phi_{B})\pm\frac{2}{3}\Big{(}1-\frac{\kappa_{2}}{2(d-1)^{3}}B_{\pm}(\phi_{B})^{2}\Big{)}^{-1}\sqrt{2V^{\prime}(\phi_{B})}(\phi-\phi_{B})^{\frac{3}{2}}+\dots\,,$
(3.40)

where the upper and lower bounds, $B_{\pm}$, are given in equation (3.7). Two
examples of the bounce solution are sketched, as the purple curves, in figure
3(a).

####  3.2.2 Critical points near $W=W_{E}$

As we discussed earlier, the beta function in (3.3) or equivalently

$\beta(\phi)=-2(d-1)\Big{(}1-\frac{W^{2}}{W_{E}^{2}}\Big{)}\frac{W^{\prime}}{W}\,,$
(3.41)

tends to zero at a specific critical value of the superpotential,
$W_{E}=\sqrt{\frac{2(d-1)^{3}}{\kappa_{2}}}$, if $W^{\prime}$ has a finite
value. On the other hand the equation of motion (3.2b) can be written as
follow

$V(\phi)=-\frac{d}{4(d-1)}\Big{(}1-\frac{1}{2}\frac{W^{2}}{W_{E}^{2}}\Big{)}W^{2}+\frac{1}{2}\Big{(}1-\frac{W^{2}}{W_{E}^{2}}\Big{)}^{2}{W^{\prime}}^{2}\,.$
(3.42)

Therefore for a finite $W^{\prime}$ at $W=W_{E}$ we get

$V(\phi_{c})=V_{c}=-\frac{d(d-1)^{2}}{4\kappa_{2}}\,.$ (3.43)

If we insert $V_{c}$ into the equation of the bounds in (3.7), then
$W_{E}=B_{+}=B_{-}$. It means that this critical point exists only if the two
bounds meet each other at $W=W_{E}$. However, if $V(\phi)\neq V_{c}$, then
$W^{\prime}$ tends to infinity, and therefore, we would have a curvature
singularity. This singularity is shown as a blue dashed line in figures 3(a)
and 3(b). The RG flow never crosses the line of $W=W_{E}$, except at the
specific points where the two bounds meet each other, see figure 3(b). As we
see in (3.4), above and below the line of $W=W_{E}$, the RG flow changes its
sign

$W>W_{E}:\,\,\frac{dW}{dr}<0\,,\qquad W<W_{E}:\,\,\frac{dW}{dr}>0\,,$ (3.44)

It means that the RG flow stops on the cross points of the bounds at
$\phi=\phi_{c}$. If there is no crossing point, then the RG flow tends to the
$W=W_{E}$ line asymptotically from above and below as $r$ goes to $\pm\infty$.
To find the solutions of superpotential near the crossing points, we consider
the following ansatz for potential and superpotential

$\displaystyle
V(\phi)=V_{c}+V_{1}(\phi-\phi_{c})+V_{2}(\phi-\phi_{c})^{2}+\cdots\,,$ (3.45a)
$\displaystyle
W(\phi)=W_{E}+\sum_{i}\mathcal{C}_{i}(\phi-\phi_{c})^{z_{i}}\,.$ (3.45b)

For a finite value of $W^{\prime}(\phi_{c})$, the solution exists if $V_{1}=0$
and $V_{2}>0$. It means that $\phi_{c}$ is a local minimum of the potential.
The superpotential is

$\displaystyle W$
$\displaystyle=W_{E}+\mathcal{C}_{1}(\phi-\phi_{c})+\mathcal{C}_{2}(\phi-\phi_{c})^{2}+\cdots\,,$
(3.46a) $\displaystyle\mathcal{C}_{1}$ $\displaystyle=\pm
W_{E}\Big{(}\frac{d}{8(d-1)}\big{(}-1+\sqrt{1+32\frac{V_{2}}{W_{E}^{2}}\frac{(d-1)^{2}}{d^{2}}}\big{)}\Big{)}^{\frac{1}{2}}\,,$
(3.46b) $\displaystyle\mathcal{C}_{2}$
$\displaystyle=\frac{2(d-1)V_{2}}{dW_{E}-3\sqrt{32(d-1)^{2}V_{2}+d^{2}W_{E}^{2}}}\,,$
(3.46c)

and the scalar field and scale factor are given by

$\displaystyle\phi_{\pm}(r)=\phi_{c}+\phi_{\pm}e^{-2\frac{\mathcal{C}_{1}^{2}}{W_{E}}r}\pm\frac{|\mathcal{C}_{1}|dW_{E}}{12(d-1)\mathcal{C}_{1}^{2}+dW_{E}^{2}}(\phi_{\pm}e^{-2\frac{\mathcal{C}_{1}^{2}}{W_{E}}r})^{2}+\cdots\,,$
(3.47a) $\displaystyle
A_{\pm}(r)=-\frac{W_{E}}{2(d-1)}(r-r_{*})\pm\frac{W_{E}}{2|\mathcal{C}_{1}|}\phi_{\pm}e^{-2\frac{\mathcal{C}_{1}^{2}}{W_{E}}r}+\cdots\,.$
(3.47b)

The last equation obviously shows that $\phi_{c}$ is an IR fixed point as
$r\rightarrow+\infty$ and the radius of $AdS$ space is given by
$\tilde{L}_{E}=\frac{2(d-1)}{W_{E}}$. Moreover, by using the definition of the
superpotential in (3.1), $A_{+}(A_{-})$ belongs to the region above (below)
the line of $W=W_{E}$ as we discussed in (3.44). Depending on how the RG flow
approaches $\phi=\phi_{c}$ from the left or right, there might be four
solutions, two correspond to the up and two for the down points of $W=W_{E}$.
These solutions are depicted by brown curves in figure 3(b).

##  4 Holographic RG flow: General case

In this section, we consider the general quadratic curvature action in which
both $\kappa_{1}$ and $\kappa_{2}$ are non-zero. Although in these theories
usually, one may find the spin two ghost modes (for example see [31, 32, 33])
and the holographic dual quantum field theories suffer the lack of unitarity,
nevertheless we can consider the coefficient to be small (suppressed by
including a Planck or string length scale). We hope that the study of the
holographic RG flow would give interesting information on the RG equations of
the dual QFTs at strong coupling regime.

To find the critical points of this theory, we use the same definition of the
superpotential in (3.1), although it does not reduce the differential
equations to the first-order ones. By writing the equations of motion in terms
of the superpotential, we encounter with the higher-order derivatives of
$\phi(r)$ which we can eliminate by using the equation of motion of the scalar
field

$\phi^{\prime\prime}(r)=V^{\prime}(\phi(r))-dA^{\prime}(r)\phi^{\prime}(r)\,.$
(4.1)

This equation is coming either from the variation of the Lagrangian with
respect to the $\phi(r)$ or equivalently by a simple combination of the
equations of motion (2.6) and (2.6b) and their derivatives. Doing this, we get
the following set of equations

$\displaystyle V$
$\displaystyle-\frac{1}{16(d-1)^{4}}\Big{(}d\kappa_{2}W^{4}-4d(d-1)^{3}W^{2}+4(d-1)^{2}\big{(}2(d-1)^{2}-d\kappa_{1}{W^{\prime}}^{2}\big{)}{\phi^{\prime}}^{2}$
$\displaystyle\qquad\qquad\qquad\,\,\,+8d(d-1)^{2}\kappa_{1}W\big{(}V^{\prime}W^{\prime}+{\phi^{\prime}}^{2}W^{\prime\prime}\big{)}\Big{)}=0\,,$
(4.2a) $\displaystyle V$
$\displaystyle+\frac{1}{16(d-1)^{4}}\Big{(}-d\kappa_{2}W^{4}+4(d-1)W^{2}\big{(}d(d-1)^{2}+2\kappa_{2}W^{\prime}\phi^{\prime}\big{)}-8d(d-1)^{2}\kappa_{1}W(V^{\prime}W^{\prime}$
$\displaystyle\qquad\qquad\qquad\,\,\,-{\phi^{\prime}}^{2}W^{\prime\prime})+4(d-1)^{2}\phi^{\prime}\big{(}-d\kappa_{1}{W^{\prime}}^{2}\phi^{\prime}-4(d-1)W^{\prime}(d-1-\kappa_{1}V^{\prime\prime})$
$\displaystyle\qquad\qquad\qquad\,\,\,+2(d-1)\big{(}(d-1)\phi^{\prime}+6\kappa_{1}V^{\prime}W^{\prime\prime}+2\kappa_{1}{\phi^{\prime}}^{2}W^{(3)}\big{)}\big{)}\Big{)}=0\,.$
(4.2b)

To extract the value of $\phi^{\prime}$ it is enough to subtract the above
equations

$\displaystyle\phi^{\prime}$
$\displaystyle=\frac{\zeta_{1}-\zeta_{2}}{4\kappa_{1}(d-1)W^{(3)}}=\frac{-2\zeta_{3}}{(d-1)(\zeta_{1}+\zeta_{2})}\,,$
(4.3)

where for simplicity in the notation and analysis it would be better to work
with the following variables

$\displaystyle\zeta_{1}$
$\displaystyle=\sqrt{\zeta_{2}^{2}-8\kappa_{1}\zeta_{3}W^{(3)}}\,,$ (4.4a)
$\displaystyle\zeta_{2}$
$\displaystyle=-d\kappa_{1}{W^{\prime}}^{2}+2((d-1)^{2}+d\kappa_{1}WW^{\prime\prime})\,,$
(4.4b) $\displaystyle\zeta_{3}$
$\displaystyle=W^{\prime}\big{(}\kappa_{2}W^{2}-2(d-1)^{2}(d-1-\kappa_{1}V^{\prime\prime})\big{)}+6(d-1)^{2}\kappa_{1}V^{\prime}W^{\prime\prime}\,.$
(4.4c)

Notice that the expression (4.3) reduces to the equation (3.2a) as
$\kappa_{1}\rightarrow 0$. In the next step, we add two equations (4.2) and
(4.2). This gives a relation between superpotential and potential, which we
need to solve for finding the solutions

$\displaystyle\Big{[}4\kappa_{1}\zeta_{2}\zeta_{3}W^{(3)}+2\kappa_{1}^{2}\Big{(}16(d-1)^{4}V+dW\big{(}4(d-1)^{3}W-\kappa_{2}W^{3}$
$\displaystyle-8(d-1)^{2}\kappa_{1}V^{\prime}W^{\prime}\big{)}\Big{)}{W^{(3)}}^{2}-\zeta_{2}^{3}+\zeta_{1}\zeta_{2}^{2}\Big{]}/\big{(}\kappa_{1}W^{(3)}\big{)}^{2}=0\,.$
(4.5)

This is a polynomial equation for $W^{(3)}$ and to have a real solution for
$W$ we should have the following condition

$\zeta_{2}\big{(}16(d-1)^{4}V+dW(4(d-1)^{3}W-\kappa_{2}W^{3}-8(d-1)^{2}\kappa_{1}V^{\prime}W^{\prime})\big{)}\geq
0\,.$ (4.6)

This condition is precisely the reality condition for $\phi^{\prime}(r)$ when
we find it from equation (4.2). As we see, by keeping $\kappa_{1}=0$, we come
back to the bound conditions that we found in section 3, i.e. (3.6a), (3.6b)
and (3.7). Therefore the above relation describes two possible bounds on the
value of the superpotential $W$ and its derivatives. We should note that there
is another condition for the reality of $\zeta_{1}$ in equation (4.4a) which
we have used to find the condition (4.6), this puts a constraint on the values
of $W^{(3)}$. Consequently, this constraint together with (4.6) determine the
allowed values of $W^{\prime}$, $W^{\prime\prime}$ and $W^{(3)}$.

It is important to check another condition that connect with the curvature
singularity. Similar to the $\kappa_{1}=0$ case, the curvature is given by

$\mathcal{R}=\frac{d}{d-1}\Big{(}\phi^{\prime}W^{\prime}-\frac{d+1}{4(d-1)}W^{2}\Big{)}\,,$
(4.7)

however, the singularity occurs when $\phi^{\prime}W^{\prime}$ diverges. If we
suppose that $W^{\prime}$ has a non-zero finite value then from equation of
motion (4.2) we realize that the divergence of $\phi^{\prime}$ happens at
$\zeta_{2}=0$ or

$-d\kappa_{1}{W^{\prime}}^{2}+2((d-1)^{2}+d\kappa_{1}WW^{\prime\prime})=0\,.$
(4.8)

Therefore, the $\zeta_{2}=0$ bound in (4.6) actually is a singularity bound.
This singularity can be avoided either when $W^{\prime}\rightarrow 0$ faster
than the divergence behavior of $\phi^{\prime}$ or when $\phi^{\prime}$ itself
becomes finite. The latter happens if the (4.8) bound intersect with the
second bound in relation (4.6), i.e.

$16(d-1)^{4}V+dW\big{(}4(d-1)^{3}W-\kappa_{2}W^{3}-8(d-1)^{2}\kappa_{1}V^{\prime}W^{\prime}\big{)}=0\,.$
(4.9)

We can solve (4.8) exactly. The curve of curvature singularities is given by

$W_{S}=c(\phi-\phi_{0})^{2}-\frac{(d-1)^{2}}{2cd\kappa_{1}}\,.$ (4.10)

If we consider a solution of the superpotential around an arbitrary point
$\phi$ then, we expect that this solution respects the above bounds. For
example, let us suppose that RG flow starts or ends at the second bound (4.9).
If we insert (4.10) into the (4.9) at this point we can choose boundary
conditions that fix the constants of integration, $\phi_{0}$ and $c$ in (4.10)
and therefore fix the singularity curve. For a generic potential one choice
would be $\phi_{0}=0$ and $c=-\frac{(d-1)\tilde{L}}{4d\kappa_{1}}$. Therefore,
in this example the singularity curve is

$\displaystyle W_{S}$
$\displaystyle=\frac{2(d-1)}{\tilde{L}}-\frac{(d-1)\tilde{L}}{4d\kappa_{1}}\phi^{2}\,,$
(4.11a) $\displaystyle\tilde{L}$
$\displaystyle=\sqrt{\frac{-d(d-1)}{2V(0)}}\Big{(}1\pm\sqrt{1+\frac{4\kappa_{2}V(0)}{d(d-1)^{2}}}\Big{)}^{\frac{1}{2}}\,.$
(4.11b)

The critical points of the RG flow can be found by analyzing the zero points
of the beta function. We have the same definition of the beta function as in
section 3

$\beta=\frac{d\phi(r)}{dA(r)}=\frac{\zeta_{2}-\zeta_{1}}{2\kappa_{1}WW^{(3)}}=\frac{\zeta_{3}}{4W(\zeta_{1}+\zeta_{2})}\,.$
(4.12)

So if we suppose a finite value for superpotential and its derivatives, the
condition to have a critical point is coming from $\zeta_{3}=0$

$W^{\prime}\big{(}\kappa_{2}W^{2}+2(d-1)^{2}(\kappa_{1}V^{\prime\prime}-d+1)\big{)}+6(d-1)^{2}\kappa_{1}V^{\prime}W^{\prime\prime}\big{|}_{\phi=\phi_{c}}=0\,.$
(4.13)

If we insert this condition into the equation of motion (4) and suppose that
$\zeta_{2}\geq 0$, then find the equation (4.9) at the critical point
$\phi=\phi_{c}$. It means that this family of the critical points of (4.13)
lives on the bound of (4.9) even when $\zeta_{2}=0$. We had observed this
behavior already in section 3 when the line of $W=W_{E}$, which was the
location of curvature singularities, met the upper or lower bounds. With the
help of (4.9) and (4.13) we can predict the conditions of existence of a
critical point:

1) Local maxima and minima of the potential: Here, we assume that
$W^{\prime}=V^{\prime}=0$, then $W^{2}=B_{\pm}^{2}$ gives the location of the
bounds where $B_{\pm}$ are those in equation (3.7). This condition is similar
to the previous section and corresponds to the critical points near the local
maxima and minima of the potential. We should emphasize that the RG flow
starts or ends on the bound (4.9) and always avoids hitting the singularity
bound (4.8).

2) Bounces: If we consider $V^{\prime}\neq 0$ it may have various situations:

$\bullet$ $W^{\prime}=W^{\prime\prime}=0$ then the bound is located at
$W_{B}^{2}=B_{\pm}^{2}$.

$\bullet$ $W^{\prime},W^{\prime\prime}\neq 0$ therefore the bound is modified
according to (4.9).

$\bullet$ $W^{\prime}\neq 0$ but $W^{\prime\prime}=0$. There is a new value
for superpotential at the bounce point

$W_{B}=\pm(d-1)\sqrt{\frac{-2(\kappa_{1}V^{\prime\prime}-d+1)}{\kappa_{2}}}\,.$
(4.14)

All the above cases are not necessarily every possibility of the critical
points because we supposed that every derivative of the superpotential is
finite. For example, the last equality of (4.12) suggests that for a finite
non-zero value of $\zeta_{3}$ when $W^{(3)}$ is diverging, the beta function
tends to zero.

Finally, the direction of the RG flow is controlled by the sign of
$W^{\prime}(r)$

$\frac{dW(\phi(r))}{dr}=W^{\prime}\phi^{\prime}=\frac{d-1}{d}\mathcal{R}+\frac{d+1}{4(d-1)}W^{2}$
(4.15)

It is not clear that, in general, this function remains monotonic from one
fixed point to another one along with the RG flow. Although we have supposed a
negative value of the potential, it is not apparent that the curvature of the
domain wall geometry remains negative or not as the holographic coordinate
changes. The reason backs to the existence of the higher curvature correction
terms in the Lagrangian. A conflict between these terms and potential may
change the sign of the curvature. Nevertheless, we can put a bound on
superpotential at each point if we want to have a monotonic function

$\displaystyle|W|<\frac{2(d-1)}{L_{c}}\rightarrow\frac{dW}{dr}<0\,,\qquad|W|>\frac{2(d-1)}{L_{c}}\rightarrow\frac{dW}{dr}>0\,,$
(4.16)

where $L_{c}$ is a variable critical length at each point of the holographic
coordinate $r$, and it defines by the curvature of space-time

$L_{c}=2\sqrt{\frac{d(d+1)}{-\mathcal{R}}}\,.$ (4.17)

So as far as the superpotential does not cross the critical value
$\frac{2(d-1)}{L_{c}}$ there is a c-function which may constructed from
superpotential. In the following sections, we will discuss the direction of
the RG flow near the critical points.

###  4.1 Local maxima of potential

As a first type of the critical points, we examine the points near the local
maxima of the potential. We consider again the following potential

$V(\phi)=-\frac{d(d-1)}{L^{2}}-\frac{1}{2}m^{2}\phi^{2}+O(\phi^{3})\,,\qquad
m^{2}>0\,.$ (4.18)

By solving the equation of motion (4) we obtain

$\displaystyle\qquad\qquad\qquad
W_{\pm}(\phi)=\frac{2(d-1)}{\tilde{L}}+\frac{\Delta_{\pm}}{2\tilde{L}}\phi^{2}+\mathcal{C}_{\pm}\phi^{z_{\pm}}+\cdots\,,$
(4.19a)
$\displaystyle\qquad\qquad\qquad\tilde{L}=\tilde{L}_{\pm}=\frac{L}{\sqrt{2}}\Big{(}1\pm\sqrt{1-\frac{4\kappa_{2}}{(d-1){L}^{2}}}\Big{)}^{\frac{1}{2}}\,,$
(4.19b)
$\displaystyle\Delta_{\pm}=\frac{2(d-1)\tilde{L}^{4}m^{2}}{d\big{(}(d-1)\tilde{L}^{2}-2\kappa_{2}\big{)}\pm\big{(}2\kappa_{2}-(4\kappa_{1}m^{2}+d-1)\tilde{L}^{2}\big{)}\sqrt{d^{2}-4m^{2}\tilde{L}^{2}}}\,.$
(4.19c)

These solutions reduce to (3.12b) when $\kappa_{1}=0$, and are real for
$\kappa_{2}<\frac{(d-1)L^{2}}{4}$. The scalar field mass has an upper bound
$m<\frac{d}{2\tilde{L}}$. Moreover for $\kappa_{2}<0$ just the
$\tilde{L}=\tilde{L}_{+}$ is a valid choice 444In the following computations
we keep $\tilde{L}=\tilde{L}_{\pm}$ and analyze the results for both positive
and negative values of $\kappa_{2}$ at the same time.. By solving the equation
of motion to the leading term for $W_{+}$ solution we find three different
values for $z_{+}$ as follows

$\displaystyle u_{0}=\frac{2d}{d+\Delta_{0}}\,,\qquad
u_{\pm}=\frac{d\pm\sqrt{d^{2}+\delta}}{d+\Delta_{0}}\,,$ (4.20)

where we have defined the following parameters that control the behavior of
the solutions

$\displaystyle\delta=4\frac{(d-1)\tilde{L}^{2}-2\kappa_{2}}{\kappa_{1}}\,,\qquad\Delta_{0}=\sqrt{d^{2}-4\tilde{L}^{2}m^{2}}\,.$
(4.21)

A simple algebraic analysis shows that we have the following regions for
$u_{0}$ and $u_{\pm}$. Note that only those regions give the leading terms
that $z_{+}$ restricts to $2<z_{+}$

$\displaystyle u_{-}<0\,,\qquad\qquad
1<u_{0}<2\,,\qquad\quad\,u_{0}<u_{+}\,,\qquad\qquad\quad\,\delta>0\,,$ (4.22a)
$\displaystyle 0<u_{-}<1\,,\qquad\frac{1}{2}<u_{+}\\!<2\,,\qquad
1<u_{0}<2\,,\qquad-d^{2}<\delta<0\,.$ (4.22b)

Also the leading term for $W_{-}$ branch has the following values for $z_{-}$

$\displaystyle v_{0}=\frac{2d}{d-\Delta_{0}}\,,\qquad
v_{\pm}=\frac{d\pm\sqrt{d^{2}+\delta}}{d-\Delta_{0}}\,,$ (4.23)

where these parameters are restricted

$\displaystyle
v_{-}<0\,,\qquad\qquad\,\,\,2<v_{0}<v_{+}\,,\qquad\quad\quad\quad\,\delta>0\,,$
(4.24a) $\displaystyle
0<v_{-}<v_{+}<v_{0}\,,\qquad\,\,2<v_{0}\,,\qquad-d^{2}<\delta<0\,.$ (4.24b)

According to the value of $\delta$, we have different solutions for
superpotential. We have listed the various regions for $\delta$ in the first
row of table 1. In this table we have defined

$\displaystyle\delta_{\pm}=4\Delta_{0}(d\pm\Delta_{0})\,.$ (4.25)

The correct solutions of $W_{\pm}$ in each region are presented in the second
and third rows of table 1.

$\delta$ interval | $\delta\\!<\\!-\delta_{-}\,,\Delta_{0}\\!<\\!\frac{d}{2}$ | $\delta\\!<\\!-\delta_{-}\,,\frac{d}{2}\\!<\\!\Delta_{0}$ | $-\delta_{-}\\!<\\!\delta\\!<0$ | $0<\\!\delta\\!<\delta_{+}$ | $\delta_{+}\\!<\delta$
---|---|---|---|---|---
$W_{+}$ | $W_{+}^{2}$ | $W_{+}^{2}$ | $W_{+}^{2}$ | $W_{+}^{2}$ | $W_{+}^{1}$
$W_{-}$ | $W_{-}^{1}$ | $W_{-}^{2}$ | $W_{-}^{3}$ | $W_{-}^{1}$ | $W_{-}^{1}$
Table 1: Superpotential and corresponding regions near the local maxima of the
potential.

Every solution in this table is one of the following expressions

$\displaystyle
W^{1}_{+}=\frac{2(d-1)}{\tilde{L}}+\frac{\Delta_{+}}{2\tilde{L}}\phi^{2}+\mathcal{C}_{+}\phi^{u_{+}}+\cdots\,,$
(4.26a) $\displaystyle
W^{2}_{+}=\frac{2(d-1)}{\tilde{L}}+\frac{\Delta_{+}}{2\tilde{L}}\phi^{2}+\cdots\,,$
(4.26b) $\displaystyle
W^{1}_{-}=\frac{2(d-1)}{\tilde{L}}+\frac{\Delta_{-}}{2\tilde{L}}\phi^{2}+\mathcal{C}_{-}\phi^{v_{0}}+\cdots\,,$
(4.26c) $\displaystyle
W^{2}_{-}=\frac{2(d-1)}{\tilde{L}}+\frac{\Delta_{-}}{2\tilde{L}}\phi^{2}+\mathcal{C}_{-}\phi^{v_{-}}+\cdots\,,$
(4.26d) $\displaystyle
W^{3}_{-}=\frac{2(d-1)}{\tilde{L}}+\frac{\Delta_{-}}{2\tilde{L}}\phi^{2}+\mathcal{C}_{-}\phi^{v_{+}}+\cdots\,.$
(4.26e)

Here by using the relations (4.21) and (4.25) we can rewrite $\Delta_{\pm}$ in
equation (4.19c) as

$\Delta_{\pm}=\frac{-L^{2}\delta(d\pm\Delta_{0})}{2(L^{2}-2\tilde{L}^{2})(\delta\mp\delta_{\pm})}\,.\\\
$ (4.27)

We should also remember that for $\kappa_{2}>0$ both $\tilde{L}=\tilde{L}_{+}$
and $\tilde{L}=\tilde{L}_{-}$ are valid; therefore, we have two distinct sets
of upper and lower bounds on the superpotential. The superpotential solution
must respect the bounds in (4.6). Since we have assumed $V^{\prime}=0$, the
shape of the second bound (4.9) is given by $W=B_{\pm}$ in (3.7). To analyze
and draw the shape of the superpotential easier, it would be better to find
the shape of the bounds approximately. If we insert a series solution similar
to the superpotential, into the equation (4.6) we will get the following
results

$\displaystyle
W_{b}=\frac{2(d-1)}{\tilde{L}}+\frac{\Delta_{b}}{2\tilde{L}}\phi^{2}+\cdots\,,\qquad\Delta_{b}=\Delta_{(1,2)}\,,$
(4.28a)
$\displaystyle\Delta_{1}=-\frac{(d-1)\tilde{L}^{2}}{2d\kappa_{1}}\,,\qquad\Delta_{2}=\frac{(d-1)\tilde{L}^{4}m^{2}}{d(-2\kappa_{2}+\tilde{L}^{2}(d-1+2\kappa_{1}m^{2}))}\,.$
(4.28b)

To simplify our analysis we write

$\displaystyle\Delta_{1}=\frac{L^{2}\delta}{8d(L^{2}-2\tilde{L}^{2})}\,,\quad\Delta_{2}=\frac{L^{2}\delta\delta^{*}}{8d(L^{2}-2\tilde{L}^{2})(\delta-\delta^{*})}\,,\qquad\delta^{*}=2(\Delta_{0}^{2}-d^{2})\,.$
(4.29)

Since $0<\Delta_{0}<d$, we have $\delta^{*}<-\delta_{-}<0<\delta_{+}$.
Moreover, $0<\tilde{L}_{-}\leq\frac{L}{\sqrt{2}}\leq\tilde{L}_{+}<L$.
Therefore in various regions of $\delta$ (the first column of table 2), the
signs of $\Delta_{b}$’s determine the region of forbidden area, bounded by the
(4.28a) curves.

$\delta$ interval | $\tilde{L}=\tilde{L}_{-}$ | $\tilde{L}=\tilde{L}_{+}$ | Fig.
---|---|---|---
$(-d^{2},2\delta^{*})$ | $0>\Delta_{2}>\Delta_{-}>\Delta_{+}>\Delta_{1}$ | $\Delta_{1}>\Delta_{+}>\Delta_{-}>\Delta_{2}>0$ | 4(a)
$(2\delta^{*},\delta^{*})$ | $0>\Delta_{1}>\Delta_{+}>\Delta_{-}>\Delta_{2}$ | $\Delta_{2}>\Delta_{-}>\Delta_{+}>\Delta_{1}>0$ | 4(b)
$(\delta^{*},-\delta_{-})$ | $\Delta_{2}>0>\Delta_{1}>\Delta_{+}>\Delta_{-}$ | $\Delta_{-}>\Delta_{+}>\Delta_{1}>0>\Delta_{2}$ | 4(c)
$(-\delta_{-},0)$ | $\Delta_{-}>\Delta_{2}>0>\Delta_{1}>\Delta_{+}$ | $\Delta_{+}>\Delta_{1}>0>\Delta_{2}>\Delta_{-}$ | 4(d)
$(0,\delta_{+})$ | $\Delta_{+}>\Delta_{1}>0>\Delta_{2}>\Delta_{-}$ | $\Delta_{-}>\Delta_{2}>0>\Delta_{1}>\Delta_{+}$ | 4(e)
$(\delta_{+},\infty)$ | $\Delta_{1}>0>\Delta_{2}>\Delta_{-}>\Delta_{+}$ | $\Delta_{+}>\Delta_{-}>\Delta_{2}>0>\Delta_{1}$ | 4(f)
Table 2: The shape of upper and lower bounds depends on the signs of
$\Delta_{b}$. The orientation of $W_{\pm}$ curves also can be read from this
table.

The shape of $W_{\pm}$ curves also depends on the relation between
$\Delta_{\pm}$ with $\Delta_{b}$. We have summarized these information in the
second and third columns of table 2 for $\tilde{L}=\tilde{L}_{\pm}$. Note that
for the upper bound $\tilde{L}=\tilde{L}_{-}$ and for lower bound
$\tilde{L}=\tilde{L}_{+}$. For every region, we can draw a generic behavior of
the superpotential. We have addressed every related configuration in the last
column of the table 2 to figures 4(a) to 4(f). To find which $W_{\pm}$
corresponds to which figure, we should use the table 1 according to the region
of $\delta$. To compute the scalar field, we insert each function of $W_{\pm}$
into the equation (4.3) and then solve the differential equation

$\displaystyle\phi^{1}_{+}(r)$
$\displaystyle=\phi_{+}e^{\frac{d}{\tilde{L}u_{0}}r}+\frac{\alpha(u_{0},u_{+},u_{-})\mathcal{C}_{+}\tilde{L}u_{0}}{du_{+}}(\phi_{+}e^{\frac{d}{\tilde{L}u_{0}}r})^{u_{+}+1}+\cdots\,,$
(4.30a) $\displaystyle\phi^{2}_{+}(r)$
$\displaystyle=\phi_{+}e^{\frac{d}{\tilde{L}u_{0}}r}+\cdots\,,$ (4.30b)
$\displaystyle\phi^{1}_{-}(r)$
$\displaystyle=\phi_{-}e^{\frac{d}{\tilde{L}v_{0}}r}+\frac{\alpha\,\mathcal{C}_{-}\tilde{L}v_{0}}{d(v_{0}-2)}(\phi_{-}e^{\frac{d}{\tilde{L}v_{0}}r})^{v_{0}-1}+\cdots\,,$
(4.30c) $\displaystyle\phi^{2}_{-}(r)$
$\displaystyle=\phi_{-}e^{\frac{d}{\tilde{L}v_{0}}r}+\frac{\alpha(v_{0},v_{-},v_{+})\mathcal{C}_{-}\tilde{L}v_{0}}{dv_{-}}(\phi_{-}e^{\frac{d}{\tilde{L}v_{0}}r})^{v_{-}+1}+\cdots\,,$
(4.30d) $\displaystyle\phi^{3}_{-}(r)$
$\displaystyle=\phi_{-}e^{\frac{d}{\tilde{L}v_{0}}r}+\frac{\alpha(v_{0},v_{+},v_{-})\mathcal{C}_{-}\tilde{L}v_{0}}{dv_{+}}(\phi_{-}e^{\frac{d}{\tilde{L}v_{0}}r})^{v_{+}+1}+\cdots\,,$
(4.30e)

(a)

(b)

(c)

(d)

(e)

(f)

Figure 4: The generic behavior of superpotential near the local maxima of the
potential. The green (red) curves belong to $W^{+}$ ($W^{-}$) branch. The
shaded regions are the forbidden areas restricted by the blue bound
(singularity curve) and the black (second) bound.

where in above relations we have used the following definitions

$\displaystyle\alpha(a,b,c)$
$\displaystyle=\frac{1}{2d(d-1)\kappa_{1}(4-4a+bc)^{2}}\Big{(}d^{2}\kappa_{1}(4-4a+bc)(2-3b+b^{2})$
$\displaystyle+2\kappa_{2}a\big{(}8+2(4+c)b+(c-6)b^{2}+2b^{3}+2a(b^{2}-3b-4)\big{)}\Big{)}\,,$
(4.31a) $\displaystyle\alpha$
$\displaystyle=\frac{d^{2}\kappa_{1}v_{0}v_{+}v_{-}(4-2v_{0}+v_{+}v_{-})}{(4v_{0}-v_{+}v_{-}-4)(2\kappa_{2}v_{0}^{2}-d^{2}\kappa_{1}v_{+}v_{-})}\,.$
(4.31b)

In the last step, we can find the scale factors associated with the above
solutions

$\displaystyle A^{1}_{+}(r)$
$\displaystyle=-\frac{r-r_{*}}{\tilde{L}}+\frac{u_{0}\Delta_{+}}{8d(1-d)}\big{(}\phi_{+}e^{\frac{d}{u_{0}\tilde{L}}r}\big{)}^{2}+\frac{\mathcal{C}_{+}\tilde{L}u_{0}}{2d(1-d)u_{+}}\big{(}\phi_{+}e^{\frac{d}{u_{0}\tilde{L}}r}\big{)}^{u_{+}}+\cdots\,,$
(4.32a) $\displaystyle A^{2}_{+}(r)$
$\displaystyle=-\frac{r-r_{*}}{\tilde{L}}+\frac{u_{0}\Delta_{+}}{8d(1-d)}\big{(}\phi_{+}e^{\frac{d}{u_{0}\tilde{L}}r}\big{)}^{2}+\cdots\,,$
(4.32b) $\displaystyle A^{1}_{-}(r)$
$\displaystyle=-\frac{r-r_{*}}{\tilde{L}}+\frac{v_{0}\Delta_{-}}{8d(1-d)}\big{(}\phi_{-}e^{\frac{d}{v_{0}\tilde{L}}r}\big{)}^{2}+\frac{\mathcal{C}_{-}\tilde{L}(1+\frac{v_{0}\alpha\Delta_{-}}{d(v_{0}-2)})}{2d(1-d)}\big{(}\phi_{-}e^{\frac{d}{v_{0}\tilde{L}}r}\big{)}^{v_{0}}+\cdots\,,$
(4.32c) $\displaystyle A^{2}_{-}(r)$
$\displaystyle=-\frac{r-r_{*}}{\tilde{L}}+\frac{v_{0}\Delta_{-}}{8d(1-d)}\big{(}\phi_{-}e^{\frac{d}{v_{0}\tilde{L}}r}\big{)}^{2}+\frac{\mathcal{C}_{-}\tilde{L}v_{0}}{2d(1-d)v_{-}}\big{(}\phi_{-}e^{\frac{d}{v_{0}\tilde{L}}r}\big{)}^{v_{-}}+\cdots\,,$
(4.32d) $\displaystyle A^{3}_{-}(r)$
$\displaystyle=-\frac{r-r_{*}}{\tilde{L}}+\frac{v_{0}\Delta_{-}}{8d(1-d)}\big{(}\phi_{-}e^{\frac{d}{v_{0}\tilde{L}}r}\big{)}^{2}+\frac{\mathcal{C}_{-}\tilde{L}v_{0}}{2d(1-d)v_{+}}\big{(}\phi_{-}e^{\frac{d}{v_{0}\tilde{L}}r}\big{)}^{v_{+}}+\cdots\,.$
(4.32e)

As a result, the local maxima is a UV fixed point at $r\rightarrow-\infty$.
This is clear also from the computation of the beta function near the extremum
point

$\displaystyle\beta^{1,2}_{+}(r)=-\frac{d}{u_{0}}\phi_{+}e^{\frac{d}{u_{0}\tilde{L}}r}+\cdots\,,\qquad\beta^{1,2,3}_{-}(r)=-\frac{d}{v_{0}}\phi_{-}e^{\frac{d}{v_{0}\tilde{L}}r}+\cdots\,,$
(4.33)

where all the beta functions are vanishing at the fixed point. This is exactly
what we expect from equation (4.13) where for all solutions of the
superpotential at the critical point, $W^{\prime}$ vanishes and
$W^{\prime\prime}$ is finite.

###  4.2 Local minima of potential

For the local maxima of the potential, we consider again the following
potential

$V(\phi)=-\frac{d(d-1)}{L^{2}}+\frac{1}{2}m^{2}\phi^{2}+O(\phi^{3})\,,\qquad
m^{2}>0\,.$ (4.34)

The solutions of the equation of motion (4) are similar to the local maxima
solutions with a change in the sign of $m^{2}$, specifically

$\displaystyle\Delta_{\pm}=\frac{-2(d-1)\tilde{L}^{4}m^{2}}{d\big{(}(d-1)\tilde{L}^{2}-2\kappa_{2}\big{)}\pm\big{(}2\kappa_{2}+(4\kappa_{1}m^{2}-d+1)\tilde{L}^{2}\big{)}\sqrt{d^{2}+4m^{2}\tilde{L}^{2}}}\,.$
(4.35)

The computation of the leading term for $W_{+}$ branch leads to three
different values for $z_{+}$ with different ranges

$\displaystyle
u_{0}=\frac{2d}{d+\Delta_{0}}\,,\qquad\qquad\,\,\,\,u_{\pm}=\frac{d\pm\sqrt{d^{2}+\delta}}{d+\Delta_{0}}\,,$
(4.36a) $\displaystyle u_{-}<0<u_{0}<1\,,\qquad\quad
u_{0}<u_{+}\,,\qquad\qquad\qquad\,\,\,\delta>0\,,$ (4.36b) $\displaystyle
0<u_{-}<u_{+}<u_{0}<1\,,\qquad\qquad\qquad\qquad-d^{2}<\delta<0\,,$ (4.36c)

and we have defined

$\displaystyle\delta=4\frac{(d-1)\tilde{L}^{2}-2\kappa_{2}}{\kappa_{1}}\,,\qquad\Delta_{0}=\sqrt{d^{2}+4\tilde{L}^{2}m^{2}}\,.$
(4.37)

The leading terms of $W_{-}$ branch have also three values for $z_{-}$ with
the following constraints

$\displaystyle v_{0}=\frac{2d}{d-\Delta_{0}}\,,\qquad
v_{\pm}=\frac{d\pm\sqrt{d^{2}+\delta}}{d-\Delta_{0}}\,,$ (4.38a)
$\displaystyle v_{+}<v_{0}<0<v_{-}\,,\qquad\qquad\quad\,\,\delta>0\,,$ (4.38b)
$\displaystyle v_{0}<v_{-}<v_{+}<0\,,\,\qquad\,-d^{2}<\delta<0\,.$ (4.38c)

There exist three regions for $\delta$ with a specific solution of $W_{\pm}$
in each region. These are shown in the first row of table 3 where
$\delta_{\pm}$ have the same definition as (4.25).

Interval | $-d^{2}\\!<\\!\delta\\!<\\!-\delta_{-}$ | $-\delta_{-}\\!<\\!\delta\\!<\delta_{+}$ | $\delta_{+}\\!<\delta$
---|---|---|---
$W_{+}$ | $W_{+}^{2}$ | $W_{+}^{2}$ | $W_{+}^{1}$
$W_{-}$ | $W_{-}^{2}$ | $W_{-}^{1}$ | $W_{-}^{1}$
Table 3: Superpotential and corresponding regions near the local minima of the
potential.

The $W_{\pm}$ solutions are presented in the second and third rows of table 3.
Every solution in this table has one of the following values

$\displaystyle
W^{1}_{+}=\frac{2(d-1)}{\tilde{L}}+\frac{\Delta_{+}}{2\tilde{L}}\phi^{2}+C_{+}\phi^{u_{+}}+\cdots\,,$
(4.39a) $\displaystyle
W^{2}_{+}=\frac{2(d-1)}{\tilde{L}}+\frac{\Delta_{+}}{2\tilde{L}}\phi^{2}+\cdots\,,$
(4.39b) $\displaystyle
W^{1}_{-}=\frac{2(d-1)}{\tilde{L}}+\frac{\Delta_{-}}{2\tilde{L}}\phi^{2}+C_{-}\phi^{v_{-}}+\cdots\,,$
(4.39c) $\displaystyle
W^{2}_{-}=\frac{2(d-1)}{\tilde{L}}+\frac{\Delta_{-}}{2\tilde{L}}\phi^{2}+\cdots\,.$
(4.39d)

To draw the RG flow curves near the local minima of the potential, similar to
the local maxima, we find the shape of the bounds from equation (4.6). Here
again, at the critical point, the upper or lower bounds are given by $B_{\pm}$
in equation (3.7). A little away from this point, we should solve the
constraint (4.6). If we insert a series solution into the equation (4.6), we
obtain the following results

$\displaystyle
W_{b}=\frac{2(d-1)}{\tilde{L}}+\frac{\Delta_{b}}{2\tilde{L}}\phi^{2}+\cdots\,,\qquad\Delta_{b}=\Delta_{(1,2)}$
(4.40a)
$\displaystyle\Delta_{1}=-\frac{(d-1)\tilde{L}^{2}}{2d\kappa_{1}}\,,\qquad\Delta_{2}=\frac{-(d-1)\tilde{L}^{4}m^{2}}{d(-2\kappa_{2}+\tilde{L}^{2}(d-1-2\kappa_{1}m^{2}))}\,.$
(4.40b)

To simplify our analysis let’s write

$\displaystyle\Delta_{1}=\frac{L^{2}\delta}{8d(L^{2}-2\tilde{L}^{2})}\,,\qquad\qquad\quad\Delta_{2}=\frac{-L^{2}\delta\delta^{*}}{8d(L^{2}-2\tilde{L}^{2})(\delta+\delta^{*})}\,,$
(4.41a)
$\displaystyle\Delta_{\pm}=\frac{-L^{2}\delta(d\pm\Delta_{0})}{2(L^{2}-2\tilde{L}^{2})(\delta\mp\delta_{\pm})}\,,\qquad\delta_{\pm}=4\Delta_{0}(d\pm\Delta_{0})\,,\qquad\delta^{*}=2(d^{2}-\Delta_{0}^{2})\,,$
(4.41b)

where
$0\\!<\\!-\delta^{*}\\!<\\!-\delta_{-}\\!<\\!-2\delta^{*}\\!<\\!\delta_{+}$.
For various values of $\delta$ (the first column of table 4), the signs of
$\Delta_{b}$’s determine the region of forbidden areas bounded by the (4.40a)
curves. The orientation of $W_{\pm}$ curves also depend on the relation
between $\Delta_{\pm}$ with $\Delta_{b}$. We have summarized these information
in the second and third columns of table 4 for $\tilde{L}=\tilde{L}_{\pm}$.

For every region, we can draw a generic behavior of superpotential. We have
addressed the related figures in the last column of table 4. To find which
$W_{\pm}$ corresponds to each figure we should use table 3 according to the
region of $\delta$.

$\delta$ interval | $\tilde{L}=\tilde{L}_{-}$ | $\tilde{L}=\tilde{L}_{+}$ | Fig.
---|---|---|---
$(-d^{2},0)$ | $\Delta_{2}>\Delta_{-}>0>\Delta_{+}>\Delta_{1}$ | $\Delta_{1}>\Delta_{+}>0>\Delta_{-}>\Delta_{2}$ | 5(a)
$(0,-\delta^{*})$ | $\Delta_{1}>\Delta_{+}>0>\Delta_{-}>\Delta_{2}$ | $\Delta_{2}>\Delta_{-}>0>\Delta_{+}>\Delta_{1}$ | 5(b)
$(-\delta^{*},-\delta_{-})$ | $\Delta_{2}>\Delta_{1}>\Delta_{+}>0>\Delta_{-}$ | $\Delta_{-}>0>\Delta_{+}>\Delta_{1}>\Delta_{2}$ | 5(c)
$(-\delta_{-},-2\delta^{*})$ | $\Delta_{-}>\Delta_{2}>\Delta_{1}>\Delta_{+}>0$ | $0>\Delta_{+}>\Delta_{1}>\Delta_{2}>\Delta_{-}$ | 5(d)
$(-2\delta^{*},\delta_{+})$ | $\Delta_{+}>\Delta_{1}>\Delta_{2}>\Delta_{-}>0$ | $0>\Delta_{-}>\Delta_{2}>\Delta_{1}>\Delta_{+}$ | 5(e)
$(\delta_{+},\infty)$ | $\Delta_{1}>\Delta_{2}>\Delta_{-}>0>\Delta_{+}$ | $\Delta_{+}>0>\Delta_{-}>\Delta_{2}>\Delta_{1}$ | 5(f)
Table 4: The shape of upper and lower bounds depends on the signs of
$\Delta_{b}$. The orientation of $W_{\pm}$ curves also can be read from this
table.

By insertion of the values of $W_{\pm}$ from (4.39a) to (4.39d) into the
equation (4.3) and solving the differential equation we obtain the scalar
field solution

$\displaystyle\phi^{1}_{+}(r)$
$\displaystyle=\phi_{+}e^{\frac{d}{\tilde{L}u_{0}}r}+\frac{\alpha(u_{0},u_{+},u_{-})\mathcal{C}_{+}\tilde{L}u_{0}}{du_{+}}(\phi_{+}e^{\frac{d}{\tilde{L}u_{0}}r})^{u_{+}+1}+\cdots\,,$
(4.42a) $\displaystyle\phi^{2}_{+}(r)$
$\displaystyle=\phi_{+}e^{\frac{d}{\tilde{L}u_{0}}r}+\cdots\,,$ (4.42b)
$\displaystyle\phi^{1}_{-}(r)$
$\displaystyle=\phi_{-}e^{\frac{d}{\tilde{L}v_{0}}r}+\frac{\alpha(v_{0},v_{-},v_{+})\mathcal{C}_{-}\tilde{L}v_{0}}{dv_{-}}(\phi_{-}e^{\frac{d}{\tilde{L}v_{0}}r})^{v_{-}+1}+\cdots\,,$
(4.42c) $\displaystyle\phi^{2}_{-}(r)$
$\displaystyle=\phi_{-}e^{\frac{d}{\tilde{L}v_{0}}r}+\cdots\,,$ (4.42d)

where the coefficients can be read from equation (4.31). In addition, any
scale factor associated to each superpotential is given by

$\displaystyle A^{1}_{+}(r)$
$\displaystyle=-\frac{r-r_{*}}{\tilde{L}}+\frac{u_{0}\Delta_{+}}{8d(1-d)}\big{(}\phi_{+}e^{\frac{d}{u_{0}\tilde{L}}r}\big{)}^{2}+\frac{\mathcal{C}_{+}\tilde{L}u_{0}}{2d(1-d)u_{+}}\big{(}\phi_{+}e^{\frac{d}{u_{0}\tilde{L}}r}\big{)}^{u_{+}}+\cdots\,,$
(4.43a) $\displaystyle A^{2}_{+}(r)$
$\displaystyle=-\frac{r-r_{*}}{\tilde{L}}+\frac{u_{0}\Delta_{+}}{8d(1-d)}\big{(}\phi_{+}e^{\frac{d}{u_{0}\tilde{L}}r}\big{)}^{2}+\cdots\,,$
(4.43b) $\displaystyle A^{1}_{-}(r)$
$\displaystyle=-\frac{r-r_{*}}{\tilde{L}}+\frac{v_{0}\Delta_{-}}{8d(1-d)}\big{(}\phi_{-}e^{\frac{d}{v_{0}\tilde{L}}r}\big{)}^{2}+\frac{\mathcal{C}_{-}\tilde{L}v_{0}}{2d(1-d)v_{-}}\big{(}\phi_{-}e^{\frac{d}{v_{0}\tilde{L}}r}\big{)}^{v_{-}}+\cdots\,,$
(4.43c) $\displaystyle A^{2}_{-}(r)$
$\displaystyle=-\frac{r-r_{*}}{\tilde{L}}+\frac{v_{0}\Delta_{-}}{8d(1-d)}\big{(}\phi_{-}e^{\frac{d}{v_{0}\tilde{L}}r}\big{)}^{2}+\cdots\,.$
(4.43d)

Since $u_{0}>0$ and $v_{0}<0$, at the local minima, the $W_{+}$ branch tends
to a UV fixed point as $r\rightarrow-\infty$, in contrary, this point is an IR
fixed point for $W_{-}$ branch as $r\rightarrow+\infty$.

The behavior of the beta function near the UV/IR fixed point is as follow

$\displaystyle\beta^{1,2}_{+}(r)=-\frac{d}{u_{0}}\phi_{+}e^{\frac{d}{u_{0}\tilde{L}}r}+\cdots\,,\qquad\beta^{1,2}_{-}(r)=-\frac{d}{v_{0}}\phi_{-}e^{\frac{d}{v_{0}\tilde{L}}r}+\cdots\,,$
(4.44)

where the both functions vanish at the fixed point. This is an expected result
from equation (4.13), where at the critical point, $W^{\prime}$ is zero while
$W^{\prime\prime}$ is finite.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 5: The generic behavior of superpotential near the local minima of the
potential. The green (red) curves belong to $W^{+}$ ($W^{-}$) branch.

###  4.3 Bounces

To have a general analysis of other possible solutions, let’s consider the
following ansatz near a critical point $\phi=\phi_{B}$ as a starting point

$\displaystyle V(\phi)$ $\displaystyle=\sum_{n=0}V_{n}(\phi-\phi_{B})^{n}\,,$
(4.45a) $\displaystyle W(\phi)$
$\displaystyle=W_{B}+\sum_{i=0}^{5}\mathcal{C}_{i}(\phi-\phi_{B})^{\frac{i+1}{2}}+\mathcal{C}_{z}(\phi-\phi_{B})^{z}+\cdots\,,$
(4.45b)

where we have assumed that $V^{\prime}\neq 0$ at $\phi_{B}$ which means that
the potential is no longer located near a local minimum or maximum of the
potential. Moreover, although in this ansatz $W^{\prime}$ diverges at
$\phi_{B}$ but one can check that at the same time $\phi^{\prime}\rightarrow
0$, so the curvature remains finite. If we put the above ansatz into the
equation of motion (4) then we can read the unknown coefficients. For example
the first three coefficients are

$\displaystyle\mathcal{C}_{1}$
$\displaystyle=\mathcal{C}_{0}\Big{(}\frac{\mathcal{C}_{0}}{4W_{B}}+\frac{dW_{B}}{3\sqrt{2V_{1}}(d-1)}\Big{)}+\frac{16(d-1)^{4}V_{0}+4d(-1+d)^{3}W_{B}^{2}-d\kappa_{2}W_{B}^{4}}{8d(d-1)^{2}\kappa_{1}V_{1}W_{B}}\,,$
(4.46a) $\displaystyle\mathcal{C}_{2}$
$\displaystyle=\mathcal{C}_{0}\Big{(}\frac{12(d-1)^{2}(2d-2-\kappa_{1}V_{2})+(d^{2}\kappa_{1}-12\kappa_{2})W_{B}^{2}}{72(d-1)^{2}\kappa_{1}V_{1}}+\frac{\mathcal{C}_{0}d}{4\sqrt{2V_{1}}(d-1)}\Big{)}\,,$
(4.46b) $\displaystyle\mathcal{C}_{3}$
$\displaystyle=-\frac{\mathcal{C}_{0}^{2}(d^{2}\kappa_{1}+20\kappa_{2})W_{B}-20\mathcal{C}_{1}\bigl{(}2(d-1)^{2}(d-1-2\kappa_{1}V_{2})-\kappa_{2}W_{B}^{2}\bigr{)}}{240(d-1)^{2}\kappa_{1}V_{1}}$
(4.46c)
$\displaystyle+\frac{\mathcal{C}_{0}d\Big{(}108\mathcal{C}_{1}(d-1)^{2}\kappa_{1}V_{1}-6(d-1)^{2}(5(d-1)-4\kappa_{1}V_{2})W_{B}-(d^{2}\kappa_{1}-15\kappa_{2})W_{B}^{3}\Big{)}}{540\sqrt{2V_{1}^{3}}(d-1)^{3}\kappa_{1}}\,,$

where we have supposed that $\mathcal{C}_{0}\neq 0$ in finding the above
results. Here for $\phi>\phi_{B}$ we assumed that $V_{1}>0$, otherwise for
$\phi<\phi_{B}$, we should consider $V_{1}<0$. It is related to how the bounce
is reaching the bound from the left or the right.

In this general solution, although all derivatives of $W$ diverge at the
critical point, the bounce solution respects the condition (4.6). Moreover, we
can write all the coefficients in terms of $\mathcal{C}_{1}$ and
$\mathcal{C}_{3}$ or $\phi$ and $\phi^{2}$ coefficients, in this way, the
location of the bound, $W_{B}$, is specified by two boundary conditions.

Now we can compute the scalar field and scale factor and we get

$\displaystyle\phi(r)$
$\displaystyle=\phi_{B}+\frac{1}{2}V_{1}(r-r_{B})^{2}+\mathcal{O}(r-r_{B})^{3}\,,$
(4.47a) $\displaystyle A(r)$
$\displaystyle=A_{B}-\frac{W_{B}}{2(d-1)}(r-r_{B})-\frac{\mathcal{C}_{0}\sqrt{2V_{1}}}{8(d-1)}(r-r_{B})^{2}+\cdots\,.$
(4.47b)

This shows that their values are independent of two branches of the
superpotential. The behavior of the beta function near the bounce point is
given by

$\beta(\phi)=-sign(\mathcal{C}_{0})\frac{2(d-1)\sqrt{2V_{1}}}{W_{B}}(\phi-\phi_{B})^{\frac{1}{2}}+\mathcal{O}(\phi-\phi_{B})\,.$
(4.48)

We can trace the direction of RG flow by computing

$\frac{dW(r)}{dr}=\sqrt{\frac{V_{1}}{2}}\mathcal{C}_{0}+\cdots\,.$ (4.49)

For a fixed value of $\mathcal{C}_{0}$, as the holographic coordinate changes
from values below the $r=r_{B}$ to the values above it, the direction of the
RG flow does not change. It means that $r_{B}$ is a bounce point.

As we told, in the above solutions, we have considered $\mathcal{C}_{0}\neq
0$. To find other possible bounces with other boundary conditions, we should
solve the equations of motion from the beginning. These solutions are those
that we indicated for the bounce points above the equation (4.15). In the
following sections, we find these solutions.

####  4.3.1 $W^{\prime}=W^{\prime\prime}=0$

As we mentioned earlier, the boundary location is determined when we impose
boundary conditions on $W^{\prime}$ and $W^{\prime\prime}$. To maintain on the
bound of the previous section, from equation (4.9) we conclude that
$W^{\prime}=0$ at this point. On the other hand, regarding our discussion
below the equation (4.13), we should also insert $W^{\prime\prime}=0$ at this
point. Imposing both conditions on (4.45b) or equivalently on its
coefficients, we can read the superpotential of the bounce solution

$W(\phi)=B_{\pm}\pm\frac{\sqrt{2}(d-1)}{15\kappa_{1}\sqrt{V_{1}}}(\phi-\phi_{B})^{\frac{5}{2}}+\frac{dB_{\pm}}{90\kappa_{1}V_{1}}(\phi-\phi_{B})^{3}+\cdots\,,$
(4.50)

where $B_{\pm}$ is given in equation (3.7). According to this solution, the
scalar field and scale factor near the critical point behave as

$\displaystyle\phi(r)$
$\displaystyle=\phi_{B}+\frac{1}{2}V_{1}(r-r_{B})^{2}+\mathcal{O}(r-r_{B})^{3}\,,$
(4.51a) $\displaystyle A(r)$
$\displaystyle=A_{B}-\frac{B_{\pm}}{2(d-1)}(r-r_{B})-\frac{V_{1}^{2}}{720\kappa_{1}}(r-r_{B})^{6}+\cdots\,.$
(4.51b)

Therefore the direction of the RG flow is

$\frac{dW}{dr}=\frac{V_{1}^{2}(d-1)}{12}(r-r_{B})^{4}+\cdots\,.$ (4.52)

Again below and above the critical point $r=r_{B}$, the sign of
$W^{\prime}(r)$ does not change as we expect from a bounce point.

####  4.3.2 $W^{\prime}\neq 0$ and $W^{\prime\prime}=0$

In this case, the shape of the bound near the critical point $\phi_{B}$ is
given by $W_{B}$ in (4.14). If we start from the following ansatz

$W_{B}(\phi)=W_{B}+\mathcal{C}_{1}(\phi-\phi_{B})+\mathcal{C}_{z}(\phi-\phi_{B})^{z}+\cdots\,,$
(4.53)

a perturbative analysis around the $\phi_{B}$ shows that $z=\frac{5}{2}$. This
can be confirmed by equation of motion for scalar field (4.1). Then the
coefficients are given by

$\displaystyle\mathcal{C}_{1}$
$\displaystyle=\frac{16(d-1)^{4}V_{0}+4(d-1)^{3}dW_{B}^{2}-d\kappa_{2}W_{B}^{4}}{8(d-1)^{2}d\kappa_{1}V_{1}W_{B}}\,,$
(4.54a) $\displaystyle\mathcal{C}_{z}$
$\displaystyle=\pm\frac{2(d-1)^{2}-\mathcal{C}_{1}^{2}d\kappa_{1}}{15(d-1)\kappa_{1}\sqrt{V_{1}}}\,.$
(4.54b)

Now if we compute the leading term of the scalar field we find that

$\displaystyle\phi(r)$
$\displaystyle=\phi_{B}+V_{1}(r-r_{B})^{2}+\mathcal{O}(r-r_{B})^{3}\,.$
(4.55a)

However, for $V_{1}\neq 0$ equation (4.1) in addition to value of $z$ predicts
the coefficient of the first term that should be $\frac{1}{2}V_{1}$. This
behavior is happening in every bounce point that we have found up to now
except this last one solution. In other words such a solution with
$W^{\prime}\neq 0$ and $W^{\prime\prime}=0$ does not exist.

##  5 Summary and conclusion

In this work, in the context of gauge-gravity duality, we study the
holographic RG flow of a CFT that is perturbed by a marginal/relevant
operator. The bulk action contains the Einstein-dilaton gravity with an
arbitrary scalar field potential together with the general quadratic curvature
corrections (2.1). Since the equations of motion are generally differential
equations with more than two derivatives, see (2.6) and (2.6b), we may
encounter the holographic RG equations that are not the first order either.
The study of these equations and their solutions may help us to understand the
RG equations of the dual QFTs at their strong coupling regimes.

We do not impose any condition on the couplings, but we emphasize that our
calculations are valid as far as we include all possible constraints or
stability conditions on higher derivative theories. At least for small values
of the couplings, we expect that one may avoid the ghost or tachyonic modes,
which are integrated out and are beyond the QFT cut-off.

The equations of motion are classified by two combinations of the couplings,
$\kappa_{1}$ and $\kappa_{2}$, in equation (2.7). As long as $\kappa_{1}$ is
zero, we have a second-order differential equation of motion. The GB gravity
is a particular case in this class.

The sign of $\kappa_{2}$ plays an important role in the analysis of the
critical points. For the negative values of $\kappa_{2}$ in section 3.1, we
observe similar behavior for holographic RG flow like the known EH case in
[17]. The superpotential solutions near the local maxima (figure 1) or minima
(figure 2) of the potential are parameterizing by some new coefficients as a
function of $\kappa_{2}$. For example, the radii of $AdS$ solutions at the
fixed points get modified according to (3.12). Moreover, the superpotential
and its bound, $B_{-}$, in (3.7), are shifted according to the value of this
coupling. In section 3.1.3, similar to EH case, we observe the bounce solution
(3.28) in this theory. For negative values of $\kappa_{2}$, one can show that
the superpotential has the monotonic behavior (3.4), and so it can be used to
construct a c-function in this theory.

For positive values of $\kappa_{2}$, the situation is different. In addition,
to the former lower bound on the superpotential, there is an upper bound, see
figure 3(a). According to the potential functionality, the values of the
superpotential are more restricted, and they may confine in a specific
region(s).

For both positive and negative values of $\kappa_{2}$, the curvature
singularity removes if we restrict $W^{\prime}$ to the finite values. However,
for positive values of $\kappa_{2}$, a new condition should be added. The
$W=W_{E}$ line (3.32) is a boundary for the RG flows that divides the space of
superpotential into two distinct upper and lower sub-spaces. We can find the
superpotential near the local minima and maxima of the potential and the
bounce solution on both the upper and lower bounds of the $W=W_{E}$ line. On
this line, we have a curvature singularity. To avoid crossing this line and to
have a smooth geometry, the RG flow should change its direction.

For each positive value of $\kappa_{2}$, there is a lower bound for
superpotential, which above that it has a monotonic behavior (3.4). If this
situation holds, we expect the RG flow to reach asymptotically to the
$W=W_{E}$ line at infinity. Otherwise, as we have shown in section (3.2.2),
there are IR fixed points at the intersection of the upper and lower bounds,
figure 3(b), and therefore the flow ends on these points.

In regions below the $W=W_{E}$, the RG flow from a UV fixed point to an IR
fixed point occurs such that $\tilde{L}_{UV}>\tilde{L}_{IR}$ where
$\tilde{L}$’s are the radii of $AdS$ spaces at the fixed points. In the upper
regions $\tilde{L}_{UV}<\tilde{L}_{IR}$. The same behavior has been reported
already in [27] via an ansatz for holographic RG flow. In this paper, the
authors show that the a-anomaly, which is proportional to the number of
degrees of freedom, decreases from UV to IR fixed points. In other words,
$a^{*}_{UV}>a^{*}_{IR}$, in both situations above. If we consider a monotonic
superpotential, then we expect to have the same argument here.

In section 4, we consider the non-zero values of the $\kappa_{1}$. Although
the equations of motion are more involved, we can find the perturbed solutions
of the superpotential near the critical points of the theory. As we expect,
since the equations of motion are fourth-order differential equations, there
is more diversity in the solutions at fixed points that depend on the various
parameters of the theory. The generic behavior of these solutions for fixed
points near the local maxima of the potential, are shown in figures 4(a) to
4(f) and for local minima, in figures 5(a) to 5(f). It is a general property
that the local maxima of the potential are the UV fixed points, and the local
minima could be either UV or IR fixed points.

According to the sign of $\kappa_{2}$, we may have two upper and lower bounds
on the superpotential. However, the equations of the bounds are changed. We
have a couple of constraints on superpotential and its derivative, (4.6). The
fixed singularity line $W=W_{E}$ in the previous case replaces by parabolic
curves (4.10). These singular curves restrict the orientation of the
superpotential. In other words, besides the value of the superpotential, the
direction of the RG flow is controlled by the singular curves everywhere.

In the general theory, it is not clear that the superpotential remains a
monotonic function. Nevertheless, the equation (4.15) shows that it controls
by the curvature of the domain wall solution. In other words, we can define a
critical length at each point of the RG flow, and violation of a monotonous
superpotential is related to the length scale in equation (4.17).

## Acknowledgment

We would like to thanks E. Kiritsis for reading the manuscript and for his
valuable comments. This work is supported by Ferdowsi University of Mashhad
under the grant 2/54017 (1399/12/16).

## References

  * [1] K. G. Wilson, “Renormalization group and critical phenomena. 1. Renormalization group and the Kadanoff scaling picture,” Phys. Rev. B 4 (1971) 3174.
  * [2] K. G. Wilson, “Renormalization group and critical phenomena. 2. Phase space cell analysis of critical behavior,”  Phys. Rev. B 4 (1971) 3184.
  * [3] J. M. Maldacena, “The Large N limit of superconformal field theories and supergravity,” Adv. Theor. Math. Phys. 2 (1998), 231-252 [arXiv:hep-th/9711200].
  * [4] E. T. Akhmedov, “A Remark on the AdS / CFT correspondence and the renormalization group flow,” Phys. Lett. B 442 (1998), 152-158 [arXiv:hep-th/9806217].
  * [5] D. Z. Freedman, S. S. Gubser, K. Pilch and N. P. Warner, “Renormalization group flows from holography supersymmetry and a c theorem,” Adv. Theor. Math. Phys. 3 (1999), 363-417 [arXiv:hep-th/9904017].
  * [6] J. de Boer, E. P. Verlinde and H. L. Verlinde, “On the holographic renormalization group,” JHEP 08 (2000), 003 [arXiv:hep-th/9912012].
  * [7] A. Ceresole and G. Dall’Agata, “Flow Equations for Non-BPS Extremal Black Holes,” JHEP 03 (2007), 110 [arXiv:hep-th/0702088].
  * [8] E. Kiritsis and V. Niarchos, “The holographic quantum effective potential at finite temperature and density,” JHEP 08 (2012), 164 [arXiv:1205.6205][hep-th].
  * [9] J. Bourdier and E. Kiritsis, “Holographic RG flows and nearly-marginal operators,” Class. Quant. Grav. 31 (2014), 035011 [arXiv:1310.0858][hep-th].
  * [10] E. Kiritsis, W. Li and F. Nitti, “Holographic RG flow and the Quantum Effective Action,” Fortsch. Phys. 62 (2014), 389-454 [arXiv:1401.0888][hep-th].
  * [11] K. Skenderis and P. K. Townsend, “Gravitational stability and renormalization group flow,” Phys. Lett. B 468 (1999), 46-51 [arXiv:hep-th/9909070].
  * [12] L. Girardello, M. Petrini, M. Porrati and A. Zaffaroni, “Novel local CFT and exact results on perturbations of N=4 superYang Mills from AdS dynamics,” JHEP 9812 (1998) 022 [arXiv:hep-th/9810126].
  * [13] I. Heemskerk and J. Polchinski, “Holographic and Wilsonian Renormalization Groups,” JHEP 06 (2011), 031 [arXiv:1010.1264][hep-th].
  * [14] T. Faulkner, H. Liu and M. Rangamani, “Integrating out geometry: Holographic Wilsonian RG and the membrane paradigm,” JHEP 08 (2011), 051 [arXiv:1010.4036][hep-th].
  * [15] U. Gursoy, E. Kiritsis, L. Mazzanti and F. Nitti, “Holography and Thermodynamics of 5D Dilaton-gravity,” JHEP 05 (2009), 033 [arXiv:0812.0792][hep-th].
  * [16] U. Gursoy and E. Kiritsis, “Exploring improved holographic theories for QCD: Part I,” JHEP 02 (2008), 032 [arXiv:0707.1324][hep-th].
  * [17] E. Kiritsis, F. Nitti and L. Silva Pimenta, “Exotic RG Flows from Holography,” Fortsch. Phys. 65 (2017) no.2, 1600120 [arXiv:1611.05493][hep-th].
  * [18] J. K. Ghosh, E. Kiritsis, F. Nitti and L. T. Witkowski, “Holographic RG flows on curved manifolds and quantum phase transitions,” JHEP 05 (2018), 034 [arXiv:1711.08462][hep-th].
  * [19] Y. Bea and D. Mateos, “Heating up Exotic RG Flows with Holography,” JHEP 08 (2018), 034 [arXiv:1805.01806][hep-th].
  * [20] U. Gürsoy, E. Kiritsis, F. Nitti and L. Silva Pimenta, “Exotic holographic RG flows at finite temperature,” JHEP 10 (2018), 173 [arXiv:1805.01769][hep-th].
  * [21] C. Park and J. Hun Lee, “Exotic RG flow of entanglement entropy,” Phys. Rev. D 101 (2020) no.8, 086008 [arXiv:1910.05741]hep-th].
  * [22] S. Cremonini, L. Li, K. Ritchie and Y. Tang, “Constraining nonrelativistic RG flows with holography,” Phys. Rev. D 103 (2021) no.4, 046006 [arXiv:2006.10780][hep-th].
  * [23] A. Buchel, J. Escobedo, R. C. Myers, M. F. Paulos, A. Sinha and M. Smolkin, “Holographic GB gravity in arbitrary dimensions,” JHEP 1003, 111 (2010) [arXiv:0911.4257][hep-th].
  * [24] S. Qolibikloo and A. Ghodsi, “More on phase transition and Rényi entropy,” Eur. Phys. J. C 79 (2019) no.5, 406 [arXiv:1811.04980][hep-th].
  * [25] M. Brigante, H. Liu, R. C. Myers, S. Shenker and S. Yaida, “Viscosity Bound Violation in Higher Derivative Gravity,” Phys. Rev. D 77 (2008), 126006 [arXiv:0712.0805][hep-th].
  * [26] M. Brigante, H. Liu, R. C. Myers, S. Shenker and S. Yaida, “The Viscosity Bound and Causality Violation,” Phys. Rev. Lett. 100 (2008), 191601 [arXiv:0802.3318][hep-th].
  * [27] A. Ghodsi and M. Siahvoshan, “A Holographic Study of the $a$-theorem and RG Flow in General Quadratic Curvature Gravity,” Eur. Phys. J. C 79 (2019) no.10, 820 [arXiv:1907.03497][hep-th].
  * [28] A. Ghodsi and M. Moghadassi, “Holographic entanglement entropy from minimal surfaces with/without extrinsic curvature,” JHEP 02, 037 (2016) [arXiv:1508.02527][hep-th].
  * [29] A. Ghodsi, S. Qolibikloo and S. Karimi, “Holographic complexity in general quadratic curvature theory of gravity,” Eur. Phys. J. C 80, no.10, 920 (2020) [arXiv:2005.08989][hep-th].
  * [30] G. Anastasiou, I. J. Araya, J. Moreno, R. Olea and D. Rivera-Betancour, “Renormalized holographic entanglement entropy for Quadratic Curvature Gravity,” [arXiv:2102.11242][hep-th].
  * [31] D. G. Boulware and S. Deser, “String Generated Gravity Models,” Phys. Rev. Lett 55, 2656 (1985).
  * [32] I. Gullu and B. Tekin, “Massive Higher Derivative Gravity in D-dimensional Anti-de Sitter Spacetimes,” Phys. Rev. D 80, 064033 (2009) [arXiv:0906.0102][hep-th].
  * [33] A. Ghodsi and F. Najafi, “Ricci cubic gravity in d dimensions, gravitons and SAdS/Lifshitz black holes,” Eur. Phys. J. C 77, no.8, 559 (2017) [arXiv:1702.06798][hep-th].