text:
\section{Introduction}
AdS/CFT correspondence has shed light on strongly coupled field theories such as quark gluon plasma, conformal fluid dynamics, condensed matter theories and so on. Together with these issues, holography itself is suggested as a wide field to study on, especially to look at its UV-IR structure(relation), by employing for example, sliding membrane, holographic renormalization group. One of the interesting branches of studies on holography is to develop holographic conformal $n$-point functions in momentum space. To get
conformal $n$-point functions (in momentum space), one needs to solve conformal Ward identities, among them the special conformal Ward identity is a second order differential equation with respect to their external momenta $\vec p_i$s. Somehow, it is rather difficult to get its solution, and so they usually take a specific limit to solve the equation.

The reasons why we look at the conformal correlation functions in momentum space are maybe the followings. One of them is that when one develop the correlation functions in momentum space, it is very clear how the momentum flows in the interaction vertices. One may construct or say analyze the correlation functions by a Feynman-like diagrammatic language. One may also develop a bootstrap or conformal block techniques in momentum space. The second reason could be holography. To understand conformal correlation functions, rather than directly solving the conformal Wanrd identites, maybe it would better emplying holography. It is hard to specify which conformal field theory corresponds to which gravity model in general though.

In this respect, there has been several attempts to study conformally coupled scalar theory in AdS space and its field theory dual\cite{Oh:2012bx,Jatkar:2013uga,Oh:2014nfa,Papadimitriou:2007sj,deHaro:2006ymc}
in the holographic framework. In \cite{Oh:2012bx,Jatkar:2013uga,Oh:2014nfa}, the authors obtain
the dual field theory correlation functions of certain composite operators in it by employing conformally coupled scalar theories in AdS$_{d+1}$. In \cite{Oh:2012bx,Jatkar:2013uga}, they concentrate on the 2-point correlation functions of a composite-single trace operator, $O_{\Delta}$ with its conformal dimension $\Delta=\frac{d+1}{2}$ in $d$-dimensional conformal field theory. In these researches, they do not consider any interactions higher than 2-point. The conformally coupled scalar theory contains its self-interaction term in it, which has a form of $\mathcal L_{int}=\frac{\lambda}{4}\phi^\frac{2(d+1)}{d-1}$, where the $\phi$ is the scalar field and the $\lambda$ is the self-interaction coupling. When $d=3$ it becomes $\phi^4$ interaction and if $d=5$, it is $\phi^3$ interaction. In the other cases, the exponent of the self-interaction term becomes fractional.

Holography of conformally coupled scalar model taking into account the interaction is considered and
it turns out that the results are matched with some of the results in conformal field theory computations \cite{Bzowski:2013sza,Bzowski:2019kwd,Maglio:2019grh,Oh:2020izq}.
In the field theory computations, the conformal correlation functions are obtained by directly solving conformal Ward identities\cite{Bzowski:2013sza,Bzowski:2019kwd,Maglio:2019grh,Oh:2020izq}.
In \cite{Bzowski:2013sza}, the authors get 2- and 3- point functions of operators with arbitrary scaling dimension sitting in arbitrary (Euclidean)spactial dimension. They compute correlation functions of mixtures of scalar, vector and (rank-2) tensoral operators. The form of the 2-point correlation functions are manifest. The 3-point function is a form of integration, which is so called triple-K integral meaning that its integrand contains a product of three Bessel-K functions. In \cite{Bzowski:2019kwd}, the authors suggest $n$-point conformal correlation function and it is a complex integral form. They mostly concentrate on 4-point function and introduce cross-ratios in momentum space. If the cross ratio is a specific form, then one may interpret the integration as 3-loop integral with certain propagators.

In \cite{Oh:2020izq}, the authors propose a fractional form of conformal $n$-point of scalar operators sharing the same scaling dimension $\Delta=\frac{d+1}{2}$ in $d$-dimensional Euclidean space. The exact form of the $n$-point function is $\langle O_1O_2...O_{n-1}O_n \rangle \sim \left(\sum_{i=1}^n|p_i|\right)^{-\alpha}$, where the $|p_i|=\sqrt{\vec p_i \cdot \vec p_i}$ is the absolute value of the $i$-th momentum coming into the vertex and $p_n=-\sum_{i=1}^{n-1} p_i$ due to momentum conservation. The $\alpha$ is a certain real number to be fixed by conformal Ward identities.

The conformally coupled scalar theory defined in $AdS_{d+1}$ space provides non-normalizable excitations which correspond to source terms of dual operators whose scaling dimension $\Delta=\frac{d+1}{2}$. The holographic 2-point, 3-point and 4-point functions computed by employing this model in AdS$_{d+1}$, AdS$_6$ and AdS$_4$ respectively present exactly the same forms of the conformal correlation functions suggested in \cite{Oh:2020izq}. In \cite{Oh:2014nfa}, the authors consider the self-interaction terms in $d=3$ case, and they compute 2-, 4- and 6- point functions of an operator $O_{\Delta=2}$ deforming the dual field theory. It turns out that the 2- and 4- point functions of the operator satisfy conformal Ward identities\cite{Oh:2020izq}

Now the frontier in this research field is getting multi-point functions beyond these.
In this note, we explore the conformally coupled scalar theory in AdS$_{6}$ and compute holographic multi-point functions. In Sect. \ref{Solutions of a conformal scalar field action}, we explore conformally coupled scalar theory in AdS$_{6}$ and the solution of its equation of motion.
We study this model in a new field frame where we use a field redefinition $\phi(x)=r^{\frac{d-1}{2}}f(x)$, where $\phi(x)$ is the original conformally coupled scalar field and the $f(x)$ is the field in the new field frame. The new field is effectively defined in flat Euclidean space whereas the $\phi(x)$ is defined in AdS space.

In sect.\ref{Solutions of a conformal scalar field action}, we solve the equation of motion by power expansion order by order in its coupling $\lambda$ and we get the explicit solutions upto order of $\lambda^3$. We also propose a diagrammatic way to get solution of order $n$ in $\lambda$ for the generic $n$.
It shows a certain pattern, by which one can construct $n$-th order solution in $\lambda$ graphically.

In sect.\ref{The boundary on-shell action}, we list the holographic 2-,3- and 4-point functions explicitly from the given solutions in sect.\ref{Solutions of a conformal scalar field action}.  In this section, we discuss the details for computing the boundary on-shell action as regularity and boundary conditions at the poincare horizon and AdS boundary respectively and the counter terms for the holographic calcualtions.

In sect.\ref{yes}, we develop the holographic Wilsonian renormalization group equations and find thier fixed points. The fixed points address the holographic correlation functions when their boundary condition is Dirichlet.

In sect.\ref{Holographic 4-point function and conformal Ward identities}, we examine the holographic 4-point function if it is really conformal. The form of the holographic $n$-point correlation functions are a function of absolute values of the external momenta $p_i$s or their linear combinations. Therefore, they are manifestly $SO(5)$ invariant, which is the rotation group in 5-dimensional Euclidean space. The rests to test are if they satisfy dilatation and special conformal Ward identities. The holographic 4-point function satisfies the dilatation Ward identity but it does not the special conformal Ward identity.

The special conformal Ward identity has a form of ${D^\kappa{\left(p_i,\frac{\partial}{\partial p_i}\right)}}{ \Psi(p_j)}=0$, where the $D^\kappa$ is a second order differential operator with respect to the momenta $p_i$s, $\kappa$ is a spatial index and $\Psi$ is conformal $n$-point correlation function. For the holographic 4-point function that we get from the conformally coupled scalar theory in AdS$_6$, an application of $D^\kappa$ on it gives ${D^\kappa\left({p_i,\frac{\partial}{\partial p_i}}\right)}{ \Phi(p_j)}=f^\kappa(p_j)$. The $f^\kappa(p_j)$ is a function of momenta $p_i$ and it vanishes when all of the external momenta $p_i$ are co-linear, which means that $\sum_j|p_j|=|\sum_jp_j|$. This co-linear limit is interesting if we compare the holographic correlation functions of the conformally coupled scalar theory in AdS space(especially AdS$_{6}$ and AdS$_4$) with the conformal correlation functions given in \cite{Oh:2020izq}. We remark that the co-linear limit of these two correlation functions coincide.
We finally examine $n$-point holographic correlation functions, and it turns out that in the colinear limit, the conformal correlation functions given in \cite{Oh:2020izq} coincide them. Therefore, we conclude that the conformally coupled scalar theory reproduces conformal $n$-point functions in the co-linear limit but in general they are not conformal.

\section{Solutions of a conformal scalar field action}
\label{Solutions of a conformal scalar field action}

We start with the conformally scalar field theory defined in (Euclidean)AdS spacetime, given by
\begin{equation}
\label{con-scalar-action}
S=\int_{r>\epsilon} dr d^dx \sqrt{g} \mathcal{L}(\phi ,\partial\phi)+S_B,
\end{equation}
where the spacetime is descrived by $d+1$-dimensional Euclidean AdS metric as follows:
\begin{equation}
\label{Euc-metric}
ds^2=g_{MN}dx^M dx^N=\frac{1}{r^2} \left( dr^2+\sum_{i=1}^{d}dx^i dx^i \right)
\end{equation}
The $S_B$ is a collection of boundary terms which is designed for a well defined variational problem of the theory as we will see.
The conformally coupled scalar field Lagrangian denstiy is given by
\begin{equation}
\label{Scalar-Lagrangian}
\mathcal L(\phi,\partial \phi)=\frac{1}{2}g^{MN}\partial_M\phi \partial_N\phi+\frac{1}{2}m^2 \phi^2 +\frac{\lambda}{4}\phi^\frac{2(d+1)}{d-1},
\end{equation}
where the Latin(capital) indices $M$, $N$... run from 1 to $d+1$, where the coordinate $x^{d+1}$ denotes the AdS radial variable $r$. The mass of the conformally coupled scalar is not arbitrary, which should be
\begin{equation}
\label{mass-condition}
m^2=-\frac{d^2-1}{4},
\end{equation}
where the mass term is originated from the background curvature scalar of the AdS space. Such a value of the scalar field mass implies two different properties of the theory. One is the alternative quantization scheme in holography. In AdS/CFT context, a non-normalizable excitations in dual gravity corresponds to a source term couples to a composite operator, $\mathcal O$ in the dual field theory defined on the AdS boundary and the $\mathcal O$ is proportional to the coefficient of a normalizable mode of excitations. One of the conditions that the operator $\mathcal O$ satisfies is that its correlation functions are unitary. It turns out that it does in its mass range of
\begin{equation}
\label{alt-condition}
-\frac{d^2}{4}\le m^2 \le -\frac{d^2}{4}+1,
\end{equation}
so  does the conformally coupled scalar theory.
Moreover, if the scalar field mass is in that range, the role of normalizable and non-normalizable excitations can be switched and the boundary field theory is a still unitary theory: alternative quantization.

Another property is that by employing a scale transformation of the scalar field as
\begin{equation}
\label{con-transform}
\phi(x^M)\equiv\Omega(r)f(x^M) {,\rm \ where\ } \Omega(r)=r^{\frac{d-1}{2}},
\end{equation}
the conformally coupled scalar field action is effectively defined in a flat background as follows:
\begin{eqnarray}
\label{trans-action}
S&=&\int_{r>\epsilon}drd^dx \left( \frac{1}{2}\delta^{MN}\partial_M f(x) \partial_N f(x)+\frac{\lambda}{4}f^\frac{2(d+1)}{d-1}(x) \right) \\ \nonumber
&+&\frac{d-1}{2}\int d^dx\left. \frac{f^2(x)}{2r} \right\vert_{\epsilon}^\infty+S_B
\end{eqnarray}
The action with a new field $f$ is a massless scalar field theory with a peculiar self interaction $\sim f^{\frac{2(d+1)}{d-1}}$. The power of the self interaction becomes an integral number when $d=3$ or $d=5$, corresponding to $f^4$ and $f^3$ self interaction respectively. We may study $f^3$ case extensively.

\subsection{Small coupling expansion}
In this subsection, we will obtain the equation of motion of the conformally coupled scalar. The equation of motion is given by
\begin{equation}
\label{bulk-eom}
\delta^{MN}\partial_M\partial_Nf(x)-\frac{\lambda(d+1)}{2(d-1)}f^{\frac{d+3}{d-1}}(x)=0,
\end{equation}
where we get the equation of motion in the new field frame.
Since we are interested in the case of the self-interaction with an integral power, we choose $d=5$, namely $f^3$-interaction case in this note
\footnote{The $f^4(x)$ case is also extensively studied in \cite{Oh:2014nfa}.}
. In this case, the scalar field $\phi$ is defined in AdS$_6$ space time. The equation of motion becomes
\begin{equation}
\label{d5-case}
\left(\partial_r^2+\sum_{i=1}^{5}\partial_{i}^2\right)f(x)-\frac{3\lambda}{4}f^2(x)=0.
\end{equation}
Since the boundary directions described by the coordinates $\{x^i\}$ are flat and non-compact, we expect that the solution is a superpositon of plane waves along the boundary directions. Then, we try a solution as
\begin{equation}
\label{moment-field}
f(r,x^i)=\int\frac{d^5k}{{(2\pi)}^\frac{5}{2}} e^{-ik \cdot x}f_k(r),
\end{equation}
which is an effectively Fourier transform to the momentum space along the boundary directions.
The equation of motion in the momentum space is given by
\begin{equation}
\label{mfield-eom}
(\partial_r^2-{|q|}^2)f_q(r)-\frac{3\lambda}{4}\int\frac{d^5k}{{(2\pi)}^\frac{5}{2}}f_k(r)f_{q-k}(r)=0,
\end{equation}
and we solve this equation by using power expansion order by order in the small coupling $\lambda$ as
\begin{equation}
f_q(r)=\sum_{i=0}^\infty f^{(i)}(r),
\end{equation}
where $f^{(i)}(r)$ is $i$-th order solution in the small coupling, $\lambda$ expansion. In each order in $\lambda$, the equation of motion is given by
\begin{equation}
\label{eqobyo}
(\partial_r^2-{|q|}^2)f^{(n)}_q(r)-\frac{3\lambda}{4}\int\frac{d^5k}{{(2\pi)}^\frac{5}{2}}\sum_{i=0}^{n-1}f^{(i)}_k(r)f^{(n-i)}_{q-k}(r)=0,
\end{equation}
where we call the last terms in the equation source terms of the equation. The inhomogeneous part of solution comes from the source terms

The zeroth order equation of motion and its solution in the expansion are given by
\begin{equation}
\label{zero-odr}
(\partial_r^2-q^2)f_q^{(0)}(r)=0 \qquad   \longrightarrow \qquad f_q^{(0)}(r)=( F_qe^{-|q|r}+ g_qe^{|q|r}),
\end{equation}
where $F_q$ and $g_q$ are arbitrary boundary momentum dependent functions.
Since the solution being proportional to $e^{|q|r}$ is divergent as $r\rightarrow\infty$, which may cause infinite energy and momentum density and so it cause huge backreaction to the background space time. To avoid this(regularity condition), we set $g_q$ to zero.
To get the higher order solution, we take the following form of the trial solution:
\begin{equation}
\label{first-odr-form}
f_q(r)=f_q^{(0)}(r)+f_q^{(1)}(r)+O(\lambda^2)
\end{equation}
The equation of motion up to the first order in $\lambda$ is given by
\begin{eqnarray}
\label{first-eom}
(\partial_r^2-q^2)f_q^{(1)}(r)&=&\frac{3\lambda}{4}\int\frac{d^5k}{{(2\pi)}^{\frac{5}{2}}}f_k^{(0)}(r)f_{q-k}^{(0)}(r) \\ \nonumber
&=&\frac{3\lambda}{4}\int \frac{d^5k}{{(2\pi)}^{\frac{5}{2}}}F_k F_{q-k} e^{-(|k|+|q-k|)r},
\end{eqnarray}
and its solution is
\begin{equation}
\label{first-odr}
f_q^{(1)}(r)=\frac{3\lambda}{4{(2\pi)}^{\frac{5}{2}}}\int d^5k d^5p {\ } \delta^{(5)}(k+p-q)\Delta(k,p) F_k F_{p} e^{-(|k|+|p|)r}+F_1(q)e^{-|q|r},
\end{equation}
where
\begin{equation}
\label{zero-boundary-pro}
\Delta(p_1,p_2,...,p_n)\equiv\frac{1}{(\sum_{i=1}^n|p_i|)^2-(\sum_{i=1}^np_i)^2}.
\end{equation}
The $F_1(q)e^{-|q|r}$ is homogeneous solution of the first order equation of motion, namely it satisfies $(\partial_r^2-q^2)f_q^{(1)}(r)=0$. The coefficient $F_1(q)$ is an arbitrary function of the boundary momentum $q$, but there are several ways to determine it.
\begin{itemize}
\item One can simply choose it to vanish. The generic form of the solution in $n$-th order in $\lambda$ is given by
\begin{equation}
f^{(n)}_q(r)=f^{(n)}_{q, \rm inhomogeneous}(r)+F_n{(q)}e^{-|q|r},
\end{equation}
it means that all the $F_n(q)=0$ for $n>0$. We name such solutions $\bf Prime\  solutions(diagrams)$ for later use.
\label{vanish}
\item Another choice is that we choose it in such a way that it removes co-linear divergences in the solution.
\end{itemize}

The factor $\Delta(p_1,p_2,...,p_n)$ diverges when
\begin{equation}
\left|\sum_{i=1}^np_i\right|=\sum_{i=1}^n|p_i|,
\end{equation}
namely, it diverges when the directions of the boundary momenta are all the same. One can remove this divergence by a smart choice of the coefficient $F_1(q)$ as
\begin{equation}
F_1(q)=-\frac{3\lambda}{4{(2\pi)}^{\frac{5}{2}}}\int d^5k d^5p {\ } \delta^{(5)}(k+p-q)\Delta(k,p) F_k F_{p}
\end{equation}
Now, the first order solution becomes
\begin{equation}
\label{FIRST-ORDER-SOLLL}
=\lambda_5 \left(\prod_{i=1}^2\int d^5p_i F_{p_i}\right)\Delta_2(p_1,p_2;q)\left\{e^{-\sum_{i=1}^2|p_i|r}-e^{-\left|\sum_{i=1}^2p_i\right|r)}\right\}
\end{equation}
where
\begin{equation}
\label{zero-boundary-pro-general}
\Delta_n(p_1,p_2,...,p_n;q)\equiv\frac{\delta^{(5)}(\sum_{i=1}^n p_i-q)}
{(\sum_{i=1}^n|p_i|)^2-q^2}.
\end{equation}
This first order solution is expressed in a diagram in Fig\ref{fig2}-(b). One may define a new symbolic variable $E_R^{1,2}$ as
\begin{equation}
E_R^{1,2}=e^{1,2}-e^{12}\equiv \left\{e^{-\sum_{i=1}^2|p_i|r}-e^{-\left|\sum_{i=1}^2p_i\right|r)}\right\},
\end{equation}
where the subscript `$_R$' denotes ``regularized''.
For the future usage, we define
\begin{eqnarray}
E_R^{12...m,m+1m+2...n,...,q+1...s}&=&e^{12...m,m+1m+2...n,...,q+1...s}-e^{123...s} \\ \nonumber
&\equiv& \exp\left\{ -\left(  \left|\sum_{i=1}^m p_i\right|  +
\left|\sum_{j=m+1}^n p_j\right|+...+\left|\sum_{k=q+1}^s p_k\right|
\right)r \right\}- \exp\left\{ -  \left|\sum_{i=1}^s p_i\right|r \right\}
\end{eqnarray}
We express the regularized diagram in Fig\ref{fig22}.

\begin{figure}[b!]
\centering
\includegraphics[width=200.4mm]{rE1.pdf}
\caption{The zeroth and the first order solutions in $\lambda$}
\label{fig2}
\end{figure}

\begin{figure}[b!]
\centering
\includegraphics[width=120.2mm]{rE2.pdf}
\caption{The regularized diagram for the first order solution in $\lambda$}
\label{fig22}
\end{figure}

We keep performing such a process and get higher order solutions in the small coupling $\lambda$. The second order equation and solution in $\lambda$ are more complex. By using the following ansatz:
\begin{equation}
\label{second-odr-form}
f_q(r)=f_q^{(0)}(r)+f_q^{(1)}(r)+f_q^{(2)}(r)+O(\lambda^3),
\end{equation}
we get the second order equation as
\begin{eqnarray}
\label{second-eom}
\nonumber
(\partial_r^2-q^2)f_q^{(2)}(r)&=&\frac{3\lambda}{4}\int\frac{d^5k}{{(2\pi)}^{\frac{5}{2}}}\left[f_k^{(1)}(r)f_{q-k}^{(0)}(r)+f_k^{(0)}f_{q-k}^{(1)}(r)\right] =
2\cdot\frac{3\lambda}{4}\int\frac{d^5k}{{(2\pi)}^{\frac{5}{2}}}\left[f_k^{(1)}(r)f_{q-k}^{(0)}(r)\right]
\\ \nonumber
&=& 2\cdot{\left(\frac{3\lambda}{4}\right)}^2 \int\frac{d^5kd^5p}{{(2\pi)}^5}\frac{F_p^{}F_{k-p}^{}F_{q-k}^{}}{{(|p|+|k-p|)}^2-{|k|}^2}\{e^{-(|p|+|k-p|)\cdot r} - e^{-|k|\cdot r}\}e^{-|q-k|r}\\ \nonumber
\end{eqnarray}
One may realize that the first and the second terms are the same once one redefines the momenta in the second term as
$k\equiv q-k'$ and $p\equiv p'$ in the first line in (\ref{second-eom}).
The solution of the second order equation is given by
\begin{eqnarray}
\label{second-odr}
\nonumber
f_q^{(2)}(r)&=&2\cdot{\left(\frac{3\lambda}{4}\right)}^2 \int \frac{d^5k}{{(2\pi)}^5}\left(  \prod_{i=1}^2\int d^5 p_iF_{p_i}\right)\Delta_2(p_1,p_2;k)F_{q-k}
\left\{
 \frac{e^{-(|p_1|+|p_2|+|q-k|) r}-e^{-|q|r}}{{[(|p_1|+|p_2|+|q-k|)}^2-{|q|}^2]} \right.\\ \nonumber
&-& \left.\frac{e^{-(|k|+|q-k|)r}-e^{-|q|r}}{{[(|k|+|q-k|)}^2-{|q|}^2]} \right\}, \\ \nonumber
&=&2\cdot  \left(\prod_{i=1}^3\int d^5p_i F_{p_i
\right) (\lambda_5)^2\cdot\delta^{(5)}\left(\sum_{i=1}^3 p_i-q\right)\Delta(p_1,p_2)
\{\Delta(p_1,p_2,p_3)E_R^{1,2,3}  \\ \nonumber
&-&\Delta(p_1+p_2,p_3)E_R^{12,3}\},
\\ \nonumber
&=&2\cdot  \left(\prod_{i=1}^3\int d^5p_i F_{p_i
\right) (\lambda_5)^2\cdot\int d^5k_1\Delta_2(p_1,p_2;k_1)\{\Delta_3(p_1,p_2,p_3;q)E_R^{1,2,3}\\ \nonumber
&-&\Delta_2(p_1+p_2,p_3;q)E_R^{12,3}\}
\end{eqnarray}

As the last explicit example, we discuss the the third order equation of motion and the solution of it.  To get this, we let
\begin{equation}
\label{third-odr-form}
f_q(r)=f_q^{(0)}(r)+f_q^{(1)}(r)+f_q^{(2)}(r)+f_q^{(3)}(r)+O(\lambda^4).
\end{equation}
The third order equation of motion is
\begin{equation}
\label{third-eom}
(\partial_r^2-q^2)f_q^{(3)}(r)=\frac{3\lambda}{4}\int\frac{d^5k}{{(2\pi)}^{\frac{5}{2}}}\left[2\cdot f_k^{(2)}(r)f_{q-k}^{(0)}(r)+f_k^{(1)}(r)f_{q-k}^{(1)}(r)\right],
\end{equation}
and the third order solution is
\begin{eqnarray}
\label{third-odr}
\nonumber
f_q^{(3)}(r)
&=& (\lambda_5)^3\left(\prod_{i=1}^4\int d^5p_i F_{p_i}\right) \cdot\left(\prod_{j=1}^2\int d^5k_j\right)[ \Delta_2(p_1,p_2;k_1)\Delta_2(p_3,p_4;k_2)\{\Delta_4(p_1,p_2,p_3,p_4;q)E_R^{1,2,3,4} \\ \nonumber
&-&2\Delta_3(p_1,p_2,p_3+p_4)E_R^{1,2,34}+\Delta_2(p_1+p_2,p_3+p_4)E^{12,34}_R\} \\ \nonumber
&+&2^2\Delta_2(p_1,p_2;k_1)\Delta_3(p_1,p_2,p_3;k_2)\{\Delta_4(p_1,p_2,p_3,p_4;q)E^{1,2,3,4}_R-\Delta_2(p_1+p_2+p_3,p_4)E^{123,4}_R\} \\ \nonumber
&-&2^2\Delta_2(p_1,p_2;k_1)\Delta_2(p_1+p_2,p_3;k_2)\{\Delta_3(p_1+p_2,p_3,p_4;q)E^{12,3,4}_R-\Delta_2(p_1+p_2+p_3,p_4)E^{123,4}_R\}]
\end{eqnarray}

\begin{figure}[b!]
\centering
\includegraphics[width=128.4mm]{rrE1.pdf}
\caption{The prime diagrams for the zeroth(a) and the first9b) order solutions in $\lambda$}
\label{rrE1}
\end{figure}

\begin{figure}[b!]
\centering
\includegraphics[width=60.8mm]{rrE2.pdf}
\caption{The prime diagram for the second order solution in $\lambda$ }
\label{fig32}
\end{figure}

\begin{figure}[b!]
\centering
\includegraphics[width=130.07mm]{rE3.pdf}
\caption{The prime diagram for the third order solutions in $\lambda$}
\label{fig3}
\end{figure}

\begin{figure}[b!]
\centering
\includegraphics[width=170mm]{rE4.pdf}
\caption{The prime diagram for the fourth order solutions in $\lambda$ }
\label{fig4}
\end{figure}

\begin{figure}[b!]
\centering
\includegraphics[width=122.07mm]{rE5.pdf}
\caption{The prime diagram for the fifth order solutions in $\lambda$ }
\label{fig5}
\end{figure}

\paragraph{A pictorial way to express the solution in the small coupling expansion}
Even if perturbative solutions become more complex as they become higher order, the solutions that we obtain
show a certain pattern. Once we observe the solutions in detail,
we realize that we can express the solutions with a collection of certain diagrammatic elements.

To formulate each pictorial element, let us concentrate on the $\bf prime\ diagrams$ the we defined previously. Again, the prime diagrams are the solutions of the equation of motion(\ref{eqobyo}) with all the $F^{(n)}_i(p)=0$ except $n=0$ case, which are the coefficients of the homogeneous solutions in each order in power expansion in $\lambda$.

Let us look at the zeroth order solution(\ref{zero-odr}). The solution is $f_q^{(0)}(r)= F_q^{(0)}e^{-|q|r}$, and its near AdS boundary expansion is
\begin{equation}
f_q^{(0)}(r)= F_q^{(0)}-r|q|F_q^{(0)}+O(r^2){\rm \ \ \ as\ \ }r\rightarrow0,
\end{equation}
where the $F_q^{(0)}$ is the boundary value of the solution and it corresponds to a coefficient of the non-normalizable mode of the excitation(solutions). In  the context of AdS/CFT correspondence, this coefficient corresponds to a boundary source coupled to a composite operator in the dual conformal field theory.

The factor $e^{-|q|r}$ is an evolution of the boundary value $F_q^{(0)}$ to the interior of AdS space along the its radial direction. Namely, it is the radial propagation of the solution. We express the boundary source $F_q^{(0)}$ by small cross and the radial propagator by dotted lines. The zeroth order solution is expressed at (a) in Figure.\ref{fig2}.

The first order solution in $\lambda$, $f^{(1)}_q$ given in (\ref{first-odr}),  is a little more complex than the zeroth order one. From the two different boundary source terms, $F^{(0)}_{p_1}$ and $F^{(0)}_{p_2}$, the fields propagate into the interior of AdS space and so its amplitude contains the factor of $e^{-|p_1|r}$ and $e^{-|p_2|r}$ in it. These are expressed by crosses and dotted lines as we do for the zeroth order solution. One more thing that we need to stress is that since the momenta $p_1$ and $p_2$ can be arbitrary, $\int d^5 p_1 d^5p_2$ factor should be inserted in the amplitude.

After the radial propagation, these two fields interact each other with  a coupling, $\lambda_5\equiv\frac{3\lambda}{4(2\pi)^{\frac{5}{2}}}$ and create another (scalar) field with momentum $q$. It is a 3-point interaction vertex.
The newly created field does not propagate in the radial direction but only along the boundary directions. The boundary propagator, $\Delta_2(p_1,p_2;q)$ is given in (\ref{zero-boundary-pro-general}) and the boundary propagator contains $\delta$-function for the momentum conservation. We express the boundary propagator by solid lines.
After all,  the first order solution can be demonstrated as given at (b) in the figure.(\ref{fig2}) in terms of the diagrammatic elements that we introduced.

The boundary propagator is not a usual two point amplitude since it depends on $p_1$ and and $p_2$ as well as $q=p_1+p_2$. Somehow it shows a non-local structure of the amplitude and the non-locality will be manifest as it goes higher orders.

\paragraph{General rules}  By looking at many of the higher order solutions together with the zeroth and the first order solutions, we find the following rules.
\begin{itemize}
\item The $n$-th order solution in the small coupling $\lambda$, $f^{(n)}(r)$ given in Fig.\ref{figcollection} contains $n+1$ boundary source terms $F_{p_i}$. All these source terms propagate in the radial direction therefore the solution contains the factor of $\prod_{i=1}^{n+1}\int d^5k_ie^{-|k_i|r}F_{k_i}$. These can be expressed in terms of collection of $n+1$ copies of the diagramatic element (but with different momentum for each dotted line) given at (a) in Fig. \ref{fig2}
\item After the radial propagation of the source terms, certain pairs of the radial propagators expressed in the dotted lines merge into a point where the propagators interact each other with a strength of the coupling constant of $\lambda_5$. This vertex produces another type of propagator, ``boundary propagator" $\Delta_2(p_1,p_2;q)$,
which is expressed in a solid line. The solid line will interact with a radial or boundary propagetor to produce another boundary propagator $\Delta_i(p_1,p_2,...,p_i;q)$.
\item There are 4 different types of interaction vertices given in Fig.\ref{fig1}. (a) in Fig.\ref{fig1} demonstrates that two radial propagators, $\prod_{i=1}^2\int d^5 p_iF_{p_i}e^{-|p_i|r}$ interact each other with coupling $\lambda_5$ to produce a boundary propagator  $\Delta_2(p_1,p_2;q)$. (b) in Fig.\ref{fig1} indicates that a radial propagator, $\int d^5 p_{n+1}F_{p_{n+1}}e^{-|p_{n+1}|r}$ and a boundary propagator, $\Delta_n(p_1,p_2,...,p_n;k)$ interact each other to produce another boundary propagator,  $\Delta_{n+1}(p_1,p_2,...,p_{n+1};q)$. The factor `2' inside the small circle at the interaction vertex means that we multiply a factor of ``2'' to the amplitude.
\item (c) and (d) in Fig.\ref{fig1} show that the two boundary propagators,  $\Delta_n(p_1,p_2,...,p_k;k_1)$ and  $\Delta_n(p_{k+1},p_{k+2},...,p_n;k_2)$ interact each other and produce another boundary propagator. At (c) in Fig.\ref{fig1}, the two boundary propagators coming into the vertex have different histories. In such a case, we mulpiply a factor of 2 to the amplitude. At (d) in Fig.\ref{fig1}, the two boundary propagators coming in share the same history. In this case, $n$ is an even number and $k=\frac{n}{2}$ and we muliply a factor of 1 to the amplitude.
\item The $n$-th order solution in $\lambda$ is a collection of all possible connections of the radial propagators via the 4-different types of the interaction vertices and finally produce a single boundary propagator as seen in Fig.\ref{figcollection}. For all the internal boundary propagator, $\Delta_n(p_1,p_2,...,p_k;k_1)$,  there should an integration, $\int dk_1$ inside the amplitude.
\item {\bf Caution}: We do not multi-count any amplitudes(or diagrams) which can be obtained by any permutations of the boundary momenta ``$p_i$"s. Namely, from an amplitude, if one gets another amplitude by an operation of such a permutation on it, then they are the same diagram and so we do not double count them.
\end{itemize}

\begin{figure}[b!]
\centering
\includegraphics[width=120mm]{rHow.pdf}
\caption{Each diagrammatic elements}
\label{fig1}
\end{figure}

\begin{figure}[b!]
\centering
\includegraphics[width=128.4mm]{figcol.pdf}
\caption{The general construction of the classical solutions}
\label{figcollection}
\end{figure}

\paragraph{Recovering the full solutions}
The prime solution that we obtained from the general rules addressed is given by
\begin{equation}
f_p(r)=\sum_{l=1}^\infty \lambda_5^{l-1}\left( \prod^l_{j=1}\int d^5p_{j}F_{p_j})\right)A_l(p_1,...,p_l;p),
\end{equation}
where
\begin{eqnarray}
\label{primedtt-solutions}
A_1(p_1;p)&=&\delta^{(5)}(p_1-p)e^1,{\ \ \ }A_2(p_1,p_2;p)=\Delta_2(p_1,p_2;p)e^{1,2}, \\ \nonumber
A_3(p_1,p_2,p_3;p)&=&2\cdot\mathcal P\left\{\int d^5 k_1 \Delta_2(p_1,p_2;k_1)\Delta_3(p_1,p_2,p_3;p)e^{1,2,3}\right\} \\ \nonumber
A_4(p_1,p_2,p_3,p_4;p)&=&\mathcal P\left\{\left(\prod_{j=1}^2\int d^5k_j\right)[ \Delta_2(p_1,p_2;k_1)\Delta_2(p_3,p_4;k_2)\Delta_4(p_1,p_2,p_3,p_4;q)\right. \\ \nonumber
&+&2\left.\cdot 2{\ }\Delta_2(p_1,p_2;k_1)\Delta_3(p_1,p_2,p_3;k_2)\Delta_4(p_1,p_2,p_3,p_4;q)]e^{1,2,3,4}\right\}, \\ \nonumber
{\rm \ and\  so\  on...}
\end{eqnarray}
and the symbol $\mathcal P$ denotes all possible permutations of the momenta $p_i$ in the expressions. {\bf How can we recover the full solution? A way is to add all possible terms which cancel the colinear divergences in the prime diagrams. }
\begin{itemize}
\item $A_1$ has no divergence in it(the $\delta$-function will be integrated). Then we leave it as it is.
\item $A_2$ diverges when the $p_1$ and $p_2$ are colinear. Then, we add a term as
\begin{equation}
\Delta_2(p_1,p_2;p)e^{1,2}\rightarrow \Delta_2(p_1,p_2;p)e^{1,2}-{\bf \Delta_2(p_1,p_2;p)e^{12}}=\Delta_2(p_1,p_2;p)E_R^{1,2},
\end{equation}
where the term that we add share the factor `$\Delta_2(p_1,p_2;p)$' with a different coefficient $e^{12}$. This term simply substract the divergence when $1,2=12$, which means that $|p_1|+|p_2|=|p_1+p_2|$.
\item $A_3$ has two different types of colinear divergences. When $1,2=12$, $\Delta_2(p_1,p_2;k_1)$ diverges and when $1,2,3=123$ meaning that $|p_1|+|p_2|+|p_3|=|p_1+p_2+p_3|$, $\Delta_3(p_1,p_2,p_3;p)$ is divergent. First, we remove the colinear divergence in
$\Delta_2(p_1,p_2;k_1)$. To do this, we perform
\begin{eqnarray}
 &{\ }&\Delta_2(p_1,p_2;k_1)\Delta_3(p_1,p_2,p_3;p)e^{1,2,3}  \\ \nonumber
&\rightarrow&
\Delta_2(p_1,p_2;k_1)\Delta_3(p_1,p_2,p_3;p)e^{1,2,3}-{\bf  \Delta_2(p_1,p_2;k_1)\Delta_2(p_1+p_2,p_3;p)e^{12,3}}
\end{eqnarray}
as we do in the $A_2$ case.
We compose the second term as follow. We keep the factor, $\Delta_2(p_1,p_2;k_1)$ in the prime diagram and change its coefficient. The coefficient is $\Delta_3(p_1,p_2,p_3;p)e^{1,2,3}$ and then we switch the variable $|p_1|+|p_2|$ by $|p_1+p_2|$ in it. Then, the coefficient becomes $\Delta_2(p_1+p_2,p_3;p)e^{12,3}$.

However, the expression still has the divergence in $\Delta_3(p_1,p_2,p_3;p)$, when $1,2,3=123$. To regularize the divergences from each term above, we add the other terms as below.
\begin{eqnarray}
&\rightarrow&
\Delta_2(p_1,p_2;k_1)\Delta_3(p_1,p_2,p_3;p)e^{1,2,3}-{\bf \Delta_2(p_1,p_2;k_1)\Delta_3(p_1,p_2,p_3;p)e^{123}}
 \\ \nonumber
&{\ }&-\Delta_2(p_1,p_2;k_1)\Delta_2(p_1+p_2,p_3;p)e^{12,3}
+{\bf \Delta_2(p_1,p_2;k_1)\Delta_2(p_1+p_2,p_3;p)e^{123}},
\end{eqnarray}
where the second term substract the colinear divergence of the first term and the fourth one does from the third one. The second term is designed by keeping the factor $\Delta_2(p_1,p_2;k_1)\Delta_3(p_1,p_2,p_3;p)$ and change its coefficient $e^{1,2,3}$ in a way that we switch the variable $|p_1|+|p_2|+|p_3|$ by $|p_1+p_2+p_3|$.  The third term is designed by keeping the factor $\Delta_2(p_1,p_2;k_1)\Delta_2(p_1+p_2,p_3;p)$ and change its coefficient $e^{12,3}$ in a way that we switch the variable $|p_1+p_2|+|p_3|$ by $|p_1+p_2+p_3|$ i it.

Finally the full solution is given by
\begin{equation}
A_3\rightarrow\Delta_2(p_1,p_2;k_1)\Delta_3(p_1,p_2,p_3;p)E_R^{1,2,3}
-\Delta_2(p_1,p_2;k_1)\Delta_2(p_1+p_2,p_3;p)E_R^{12,3}
\end{equation}
\item $A_4$ has two different terms in it. We perform the substraction process to each term and then we get the full solutions.
\end{itemize}

\section{The boundary on-shell action}
\label{The boundary on-shell action}

\subsection{Regularity and boundary conditions}
In our setting of the AdS geometry given in (\ref{Euc-metric}), the AdS boundary is at $r=0$ and the location of $r=\infty$ is for Poincare horizon, which is the deep interior of the AdS space.
At the zeroth order in  $\lambda$, we have the two independent solutions as
\begin{equation}
\phi_1(p_i,r)=F_pr^{\frac{d-1}{2}}e^{-|p|r} {\rm\ \ or\ \ }\phi_2(p_i,r)=g_pr^{\frac{d-1}{2}}e^{|p|r},
\end{equation}
where the coefficients $F_p$ and $g_p$ are arbitrary (boundary directional) momentum ``$p_i$'' dependent functions. The $\phi_1$ is regular everywhere but the $\phi_2$ does not. It shows divergence at $r=\infty$, the Poincare horizon unless the absolute value of the momentum $|p|$ vanishes. We stress that these solutions are entire solution for the conformally coupled scalar in zeroth order in $\lambda$, not being an approximate solutions with their near boundary expansions.

Let us argue this $r=\infty$ behavior of the solutions by computing the stress-energy tensor of conformally coupled scalar theory near the Poincare horizon. This stress energy tensor of the conformally coupled scalar theory is given by
\begin{equation}
T_{MN}=\partial_M\phi \partial_N \phi - g_{MN}\mathcal L,
\end{equation}
where the  $\mathcal L$ is the Lagrangian density of the conformally coupled scalar theory, given in (\ref{Scalar-Lagrangian}). We want to evaluate this in (boundary directional) momentum space, by Fourier transform given as (\ref{moment-field}) and which gives
\begin{eqnarray}
\nonumber
T^{MN}(p)&=&\int\frac{ d^dk}{(2\pi)^{\frac{d}{2}}}\left\{ g^{rM}g^{rN}\partial_r\phi_k \partial_r \phi_{p-k} +g^{\mu M}g^{\nu N} k_\mu(p-k)_\nu \phi_k \phi_{p-k}-\frac{g^{MN}}{2}\left( g^{\mu\nu}k_\mu(p-k)_\nu \phi_k \phi_{p-k}
\right.
\right. \\
&+&\left.\left.g^{rr}\partial_r\phi_k \partial_r \phi_{p-k} +m^2\phi_k \phi_{p-k}\right)\right\},
\end{eqnarray}
where $g^{MN}=r^2\delta^{MN}$, more precisely $g^{rr}=r^2$ and $g^{\mu M}=r^2 \delta^{\mu M}$.

Now, to eveluate the stress energy tensor, consider the general solution of $\phi$, which is given by a linear combination of the $\phi_1$ and $\phi_2$. Near the Poincare horizon, the $\phi_2$ and its derivatives are more dominant than those of $\phi_1$. Therefore,
the solution $\phi$ becomes $\phi\rightarrow \phi_1$ as $r\rightarrow\infty$. First, let us compute $T^{rr}$.
\begin{eqnarray}
\nonumber
T^{rr}(p)&=&\int\frac{d^d k}{(2\pi)^{d/2}}\phi_2(k)\phi_2({p-k})\left\{ \frac{r^4}{2}\left(|k||p-k|-k\cdot(p-k)\right) +\frac{(d-1)}{4}r^3(|k|+|p-k|)\right. \\
&+&\left.\frac{d(d-1)}{4}r^2\right\}
\end{eqnarray}
We also compute
$T^{\tau\tau}$, which is given by
\begin{eqnarray}
T^{\tau\tau}(p)&=&\int\frac{d^d k}{(2\pi)^{d/2}}\phi_2(k)\phi_2({p-k})
\left\{ \frac{r^4}{2}\left( (E_p-E_k)E_k-|k||p-k|-k\cdot(p-k)\right) \right. \\ \nonumber
&+&\frac{(d-1)}{4}r^3(|k|+|p-k|)+\left.\frac{d-1}{4}r^2\right\},
\end{eqnarray}
where $E_k$ is an energy of an excitation in the boundary field theory, which can be defined in the context of radial quantization, and the $\tau$ is the radial variable.

In the stress energy tensor, there is a factor of $\phi_2(p)\phi_2(p-k)$ in it, which diverges near the Poincare horizon as $\phi_2(p)\phi_2(p-k)\sim \exp\left\{(|k|+|p-k|)r\right\}$ as $r\rightarrow\infty$. Therefore, this causes an infinite stress energy tensor, which implies a large back reaction to the background geometry near the Poincare horizon whereas the AdS boundary does not change. The geometry can change into a black brane(but we do not examine the system for such a case in detail). If we assume as such, we may have different kinds of states(may be thermal states) in the dual field theory corresponding to the black brane solution and the holographic correlation function must not be the same with those from pure AdS space.

In this paper, what we want to do is to compute holographic correlation functions from a model of conformally coupled scalar theory in the background of pure AdS space. Therefore, we keep the Poincare horizon as that of AdS space.

\paragraph{Divergences near AdS boundary and the counter terms}
To evaluate holographic correlation functions from gravity models, one need to regularize the boundary on-shell action near AdS boundary at $r=\epsilon$ where $\epsilon<<1$, showing divergences as the boundary approaches $\epsilon\rightarrow0$ in general.

For this, the holographic renormalization process is needed. Practically, we add counter terms in the action. In our case, it is
\begin{equation}
S_{ct}=\frac{d-1}{2}\int d^d x \sqrt{\gamma}\phi^2(x),
\end{equation}
where $\gamma$ is determinant of the $\gamma_{\mu\nu}$, which is defined as
\begin{equation}
\gamma_{\mu\nu}=\frac{\partial x^M}{\partial x^\mu}\frac{\partial x^N}{\partial x^\nu}g_{MN}
\end{equation}
at $r=\epsilon$. This counter term respects AdS boundary diffeomorphism invariance and precisely cancels the divergence pieces from the boundary on-shell action that we will compute.
One can see if this counter term works correctly in the new field frame. By employing the field transformation(\ref{con-transform}), one sees that another boundary term
appers, which is the first term in the second line in (\ref{trans-action}). The counter term precisely cancel this term. Then, in this field frame, the solutions of the equation of motion and the boundary on-shell action from them become regular manifestly.

\paragraph{Boundary conditions on the AdS boundary}
Since the mass of conformally coupled scalar is in a window as $-\frac{d^2}{4}\le m^2 \le -\frac{d^2}{4}+1$, various quantization schemes are possible. Accordingly, in the dual gravity theory, various boundary conditions are suggested for these, Dirichlet, Neumann and Mixed boundary conditions.
In our computation, we restrict ourselves imposing Dirichlet boundary condition on the AdS boundary. This is for a holographic computations of correlation functions of an operator, of which conformal dimansion $\Delta=\frac{d+1}{2}$ coupled with a source $J$.

\subsection{Computation of the boundary on-shell action}
The form of the on-shell action is given by
\begin{eqnarray}
S_{os}&\equiv&S^{(1)}_{os}+S^{(2)}_{os}=\frac{1}{2}\int \left[\prod_{i=1}^2 d^5 k_i\right]f_{k_1}(r)\partial_r f_{k_2}(r)\delta^{(5)}\left.\left(\sum_{i=1}^2 k_i\right)\right|^{r=\infty}_{r=0} \\ \nonumber
&-&\frac{\lambda_5}{6}\int^\infty_0 dr  \left[\prod_{i=1}^3 f_{k_i}(r) d^5 k_i\right]\left(\sum_{i=1}^3 k_i\right),
\end{eqnarray}
where the $f_k(r)$ is the solution that we obtaine in the previous section. We note that the solution $f_k(r)$ has no colinear divergences in it and so the on-shell action is finite too except the infra-red point where all the momenta $k_i=0$. The process computing the boundary on-shell action to get $n$-point function of the composit operator which corresponds this solution is somewhat long and tedious. We just address a few of them below.
\begin{itemize}
\item $\langle O_\Delta(k_1) O_{\Delta}(k_2) \rangle=\frac{|k_1|}{2}\delta^{(5)}(k_1+k_2)$.
\item $\langle O_\Delta(k_1) O_{\Delta}(k_2) O_{\Delta}(k_3)\rangle=
\frac{\lambda_5}{3\cdot(\sum_{i=1}^3|k_i|)}\delta^{(5)}(k_1+k_2+k_3)$.
\item
\begin{eqnarray}
\nonumber
\langle O_\Delta(k_1) O_{\Delta}(k_2) O_{\Delta}(k_3) O_{\Delta}(k_4)\rangle=-\frac{\lambda^2_5}{6\cdot(\sum_{i=1}^4|k_i|)}\left(\frac{1}{(|k_1|+|k_2|+|k_1+k_2|)(|k_3|+|k_4|+|k_3+k_4|)}\right. \\ \nonumber
+\left.({k_1 \leftrightarrow k_3})+({k_1 \leftrightarrow k_4})\right)\delta^{(5)}(k_1+k_2+k_3+k_4)
\end{eqnarray}
\item And so on.
\end{itemize}

\section{Holographic correlation functions as fixed points of holographic Winsonain RG
}
\label{yes}
\subsection{Holographic renormalization and Hamilton Jacobi equation}
In this section, we would like to compute holographic Wilsonian renormailization group flows of a certain 5-dimensional conformal field theory when its bulk dual is given by 6-dimensional conformally coupled scalar theory in AdS$_{6}$. When the conformally coupled scalar theory(\ref{Scalar-Lagrangian}) is defiend in the AdS$_6$, its enjoys 3-point self interaction. The action is given by
\begin{eqnarray}
 \nonumber
\label{phi3-bulk-action}
S&=&\int_{r>\epsilon} dr \left[  \frac{1}{2} \int d^5k d^5 k^\prime \delta^{(5)}(k+k^\prime)     \left( \partial_r f_k \partial_r f_{k^\prime} + k^2 f_k f_{k^\prime} \right)   +\frac{\lambda}{4(2\pi)^{5/2}}\int \prod_{i=1}^3 d^5 k_i f_{k_i}\delta^{(5)}\left(\sum_{j=1}^3 k_j\right)      \right]  \\
&+&S^\prime_B(\epsilon),
\end{eqnarray}
where $k_i$ are 5-dimensional boundary directional momenta, $S_B^\prime$ is the boundary deformation, i.e. the boundary generating functional on the $r=\epsilon$ hypersurface. We define this action in the momentum space of $k_i$ by using the fourier transform that we define in the last section.

The canonical momentum and the equation of motion
are given by
\begin{equation}
\Pi_k\equiv\partial_r f_{-k}=\frac{\delta S^\prime_B}{\delta f_k}, {\ \ \ }\partial_r \Pi_k=k^2f_k + \frac{3\lambda}{4}\int \frac{d^5k^\prime}{(2\pi)^{5/2}}f_{k^\prime} f_{k-k^\prime},
\end{equation}
where the first equation ensures that a variation with respect to the field $f_k$ is well posed.
The fact that the total action $S$ does not depend on the radial cut-off $\epsilon$ leads the Hamilton-Jacobi equation(HJ-equation)\cite{
Oh:2012bx,Jatkar:2013uga,Heemskerk:2010hk,Faulkner:2010jy,Oh:2013tsa,Oh:2015xva,Moon:2017btx} which is given by
\begin{eqnarray}
\partial_\epsilon S^\prime_B(\epsilon)&=&-\frac{1}{2}\int d^5 k \left( \frac{\delta S^\prime_B}{\delta f_k(\epsilon)} \right)\left( \frac{\delta S^\prime_B}{\delta f_{-k}(\epsilon)}\right) +   \frac{1}{2} \int d^5k d^5 k^\prime \delta^{(5)}(k+k^\prime)      k^2 f_k f_{k^\prime}  \\ \nonumber
 &+&\frac{\lambda}{4}\int \prod_{i=1}^3 d^5 k_i f_{k_i}\delta^{(5)}\left(\sum_{j=1}^3 k_j\right)
\end{eqnarray}
To get the solution of the equation, we suggest the following trial soltuion:
\begin{eqnarray}
S^\prime_B(\epsilon)&=&\Lambda(\epsilon)+\int J_k(\epsilon) f_{-k}(\epsilon)d^5 k +   \left[\int\prod_{i=1}^2 d^5 k_i f_{k_i}(\epsilon)\right]D^{(2)}_{k_1k_2}(\epsilon)\delta^{(5)}\left(\sum_{j=1}^2 k_j\right)\\ \nonumber
&+&\sum_{n=1}^{\infty}\lambda^n\left[\int\prod_{i=1}^{n+2} d^5 k_i f_{k_i}(\epsilon)\right]D^{(n+2)}_{k_1,...,k_{n+2}}(\epsilon)\delta^{(5)}\left(\sum_{j=1}^{n+2} k_j\right),
\end{eqnarray}
We plug such ansatz into the HJ-equation and obtain the following series of the equations by the power expansion in $\lambda$.
The ansatz is comprised of the series of the fields $f_k$ with arbitrary coefficients, $D^{(n)}$. The HJ-equation is an identical equation for the fields.
The equations are given by
\begin{eqnarray}
\label{floweq1}
\partial_\epsilon \Lambda(\epsilon)&=&-\frac{1}{2} \int d^5 k J_k(\epsilon)J_{-k}(\epsilon), \\
\label{floweq2}
\partial_\epsilon J_k(\epsilon)&=&-2J_k(\epsilon)D^{(2)}_{k,-k}(\epsilon), \\ \nonumber
\\ \nonumber
\label{floweq3}
\partial_\epsilon D^{(2)}_{(p,-p)}(\epsilon)&=&-\frac{1}{2}(4D^{(2)}_{(p,-p)}(\epsilon)D^{(2)}_{(-p,p)}(\epsilon)-p^2)-3\lambda\int d^5k J_{-k}(\epsilon)D^{(3)}_{(p,-p+k,-k)}(\epsilon) \\
\label{floweq4}
\partial_\epsilon D^{(3)}_{(k_1,k_2,k_3)}(\epsilon)&=&\frac{1}{4(2\pi)^{5/2}}-2\left( \sum_{j=1}^3 D^{(2)}_{k_j,-k_j}\right)(\epsilon) D^{(3)}_{k_1,k_2,k_3}(\epsilon)
\\ \nonumber
&-&4\lambda\int d^5k J_{-k}(\epsilon)D^{(4)}_{(k_1,k_2,-k_1-k_2+k,-k)}(\epsilon)  \\
\label{floweqn}
\partial_\epsilon D^{(n)}_{(k_1,...,k_{n})}(\epsilon)&=&
-2\left(\sum_{j=1}^nD^{(2)}_{(k_j,-k_j)}\right)(\epsilon)D^{(n)}_{(k_1,...,,k_{n-1},-\sum_{j=1}^{n-1}k_j)}(\epsilon)\\ \nonumber
-\frac{1}{2}\sum_{n^\prime=1}^{n-3}(n^\prime+2)(n-n^\prime)&{\mathcal P}&\left \{
D^{(n^\prime+2)}_{(k_1,...,k_{n^\prime+1},-\sum_{j=1}^{n^\prime+1}k_j)}(\epsilon)
D^{(n-n^\prime)}_{(k_{n^\prime+2},...,k_{n-1},-\sum_{j=1}^{n-1}k_j,\sum_{j=1}^{n^\prime+1}k_j)}(\epsilon)\right\}, \\ \nonumber
&-&\lambda(n+1)\int d^5k J_{-k}(\epsilon)D^{(n+1)}_{(k_1,...,k_{n-1},k-\sum_{j=1}^{n-1}k_j,-k)}
\\
\nonumber
{\ \ \rm \ for\ \ }n\geq 4,
\end{eqnarray}

\paragraph{$J_k=0$ solution:}
The simplest solutions are able to obtained when $J_k=0$ identically, which means that there is no single trace deformation at all. In such case, the boundary cosmological constant, $\Lambda=0$.
The other parts of the solutions are given by
\begin{eqnarray}
\label{d-trace-sol}
D^{(2)}_{p,-p}(\epsilon)&=&\frac{1}{2}\frac{\partial_\epsilon f_p(\epsilon)}{f_p(\epsilon)}, \\
\label{t-trace-sol}
D^{(3)}_{(k_1,k_2,k_3	)}(\epsilon)&=&\frac{1}{4(2\pi)^{5/2}} \frac{\int^\epsilon \left( f_{k_1}(\epsilon^\prime)f_{k_2}(\epsilon^\prime)f_{k_3}(\epsilon^\prime)\right) d\epsilon^\prime+C^{(3)}_{k_1,k_2,k_3}}{f_{k_1}(\epsilon)f_{k_2}(\epsilon)f_{k_3}(\epsilon)}, \\
\label{n-trace-sol}
D^{(n)}_{(k_1,...,k_{n-1},-\sum_{j=1}^{n-1}k_j)}(\epsilon)&=&\frac{C^{(n)}}{\prod_{i=1}^n f_{k_i}(\epsilon)}-\frac{1}{2}\int ^\epsilon d\epsilon^\prime \left(\frac{\prod_{j=1}^n f_{k_j}(\epsilon^\prime)}{\prod_{l=1}^n f_{k_l}(\epsilon)}\right) \\ \nonumber
\times\sum_{n^\prime=1}^{n-3}(n^\prime+2)(n-n^\prime)&{\mathcal P}&\left\{
D^{(n^\prime+2)}_{(k_1,...,k_{n^\prime+1},-\sum_{j=1}^{n^\prime+1}k_j)}(\epsilon)
D^{(n-n^\prime)}_{(k_{n^\prime+2},...,k_{n-1},-\sum_{j=1}^{n-1}k_j,\sum_{j=1}^{n^\prime+1}k_j)}(\epsilon)\right\},
\end{eqnarray}
where
\begin{equation}
f_p(\epsilon)=C_p \cosh(|p|\epsilon+\theta_p),
f_p(\epsilon)=C_p \sinh (|p|\epsilon+\theta_p), {\rm \ \ or \ \ }f_p(\epsilon)=C_p e^{\pm|p|\epsilon}
\end{equation}
where $C_p$ and $\theta_p$ are arbitrary, momentum $p$ dependent constants.

The explicit form of the solution of the double trace operator $D^{(2)}_{p,-p}$ is given by
\begin{eqnarray}
D^{(2)}_{p,-p}(\epsilon)=\frac{|p|}{2}\tanh(|p|\epsilon+\theta_p), {\rm \ \ where\ \ } |D^{(2)}_{p,-p}(\epsilon)|\leq\frac{|p|}{2}, \\ \nonumber
D^{(2)}_{p,-p}(\epsilon)=\frac{|p|}{2}\coth(|p|\epsilon+\theta_p), {\rm \ \ where\ \ } |D^{(2)}_{p,-p}(\epsilon)|\geq\frac{|p|}{2}, \\ \nonumber
{\rm \ \ or \ \ }D^{(2)}_{p,-p}(\epsilon)=\pm\frac{|p|}{2}
\end{eqnarray}
It shows a trivial $IR$ fixed point when $\epsilon \rightarrow \infty$ and its $IR$ value
is $D^{(2)}_{p,-p}(\infty)=\frac{|p|}{2}$. We note that the double trace operator is the square of the norm of the single trace operator, $O$. Therefore, if the double trace deformation is negative, it may spoil the unitarity of the deformed theory.

The multi-trace operators present diverse behaviors. Let us check the triple trace operator. Its explicit solution is given by
\begin{eqnarray}
D^{(3)}_{k_1,k_2,k_3}(\epsilon)&=&\frac{1}{4(2\pi)^{5/2}\prod_{i=1}^3\cosh(|k_i|\epsilon+\theta_i)}\left( \frac{C_p^{(3)}}{\prod_{i=1}^3C_{k_i}}
+\frac{1}{4}\frac{\sinh\left(\sum_{j=1}^3(|k_j|\epsilon+\theta_{k_j})\right)}{\sum_{l=1}^3|k_l|} \right. \nonumber \\ \nonumber
 &+&\left.\frac{1}{4}\sum_{j=1}^3\frac{\sinh\left(\sum_{l=1}^3(|k_l|\epsilon+\theta_{k_l})-2(|k_j|\epsilon+\theta_{k_j})\right)}{\sum_{m=1}^3|k_m|-2|k_j|} \right)
\\ \nonumber
D^{(3)}_{k_1,k_2,k_3}(\epsilon)&=&\frac{1}{4(2\pi)^{5/2}\prod_{i=1}^3\sinh(|k_i|\epsilon+\theta_i)}\left( \frac{C_p^{(3)}}{\prod_{i=1}^3C_{k_i}}
+\frac{1}{4}\frac{\cosh\left(\sum_{j=1}^3(|k_j|\epsilon+\theta_{k_j})\right)}{\sum_{l=1}^3|k_l|} \right. \nonumber \\ \nonumber
 &-&\left.\frac{1}{4}\sum_{j=1}^3\frac{\cosh\left(\sum_{l=1}^3(|k_l|\epsilon+\theta_{k_l})-2(|k_j|\epsilon+\theta_{k_j})\right)}{\sum_{m=1}^3|k_m|-2|k_j|} \right),
\end{eqnarray}
The both of the solutions show their fixed point as
\begin{equation}
D^{(3)}_{k_1,k_2,k_3}(\infty)=\frac{1}{4(2\pi)^{5/2}(\sum_{i=1}^3|k_i|)}
\end{equation}
The multi trace operator solutions can be given by solving the equations that we addressed previously and its analytic forms are given in appendix.

\paragraph{Fixed points of multi trace operators in $J_k=0$ case:}
The fixed points of the multi trace operators can be obtained by requesting the equations(\ref{floweq3}),(\ref{floweq4}) and (\ref{floweqn}) at the fixed points. When the equation(\ref{floweq3}) is examined to find a fixed points under a condition that its left hand side vanishes i.e. $\partial_{\epsilon^\star} D^{(2)}_{p,-p}(\epsilon^\star)=0$, then the fixed points are given by
\begin{equation}
D^{(2)}_{p,-p}(\epsilon^\star)=\pm \frac{|p|}{2}
\end{equation}
By looking at the given solution of the double trace operator(\ref{d-trace-sol}),
$D^{(2)}_{p,-p}(\epsilon^\star)= \frac{|p|}{2}$ is a fixed point at $\epsilon^\star=\infty$(so called IR fixed point) and
$D^{(2)}_{p,-p}(\epsilon^\star)= -\frac{|p|}{2}$ is a fixed point at $\epsilon^\star=-\infty$(a fixed point in relatively UV region).

The equation(\ref{t-trace-sol}) shows fixed points of the triple trace operator as
\begin{equation}
D^{(3)}_{k_1,k_2,k_3}(\epsilon^\star_\pm)=\pm\frac{1}{4(2\pi)^{5/2}(\sum_{i=1}^3|k_i|)},
\end{equation}
where $\epsilon_\pm^\star=\pm\infty$. In general the fixed points of the $n$-trace operators are given by
the following condition
\begin{eqnarray}
\nonumber
D^{(n)}_{(k_1,...,k_{n})}(\epsilon^\star_\pm)&=&\mp\frac{1}{2(\sum_{i=1}^{n}|k_i|)}
\sum^{n-3}_{n^\prime=1}(n^\prime+2)(n-n^\prime){\mathcal P}\left\{D^{(n^\prime+2)}_{(k_1,...,k_{n^\prime+1},-\sum_{j=1}^{n^\prime+1}k_j)}(\epsilon^\star_\pm)\right.\\
&\times& \left.D^{(n-n^\prime)}_{(k_{n^\prime+2},...,k_{n-1},-\sum_{j=1}^{n-1}k_j,\sum_{j=1}^{n+1}k_j)}(\epsilon^\star_\pm)\right\}
\end{eqnarray}
By utilizing such relation, for example,  one can derive the fixed points of the quadraple-trace operator as
\begin{eqnarray}
\nonumber
D^{(4)}_{k_1,k_2,k_3,k_4}(\epsilon^\star_\pm)&=&\mp\frac{3}{2^5(2\pi)^5(\sum_{i=1}^4|k_i|)}\left(\frac{1}{(|k_1|+|k_2|+|k_1+k_2|)(|k_3|+|k_4|+|k_3+k_4|)}\right. \\
&+&\left.({k_1 \leftrightarrow k_3})+({k_1 \leftrightarrow k_4})\right)
\end{eqnarray}

\paragraph{$J_k\neq0$ solutions}
In the case that we turn on the single trace operator, $J_k\neq0$, it is not easy to get their solutions since the equations become integral equations as well as the radial evolution of the $n$-ple trace operator depends on the $n+1$-ple trace operator through the terms $\sim \int d^5 kJ_k D^{(n+1)}_k$ in the equations.
One way to solve such equations is by an assumption that the single trace term $J_k$ is parameterically small. We introduce a dimensionless small parameter $\alpha$, and assume that the single trace term is suppressed by the parameter.

The solutions of the single trace and the boundary cosmological constant are given by
\begin{equation}
J_k=\frac{\alpha B_k}{f_k(\epsilon)}, {\rm \ \ and \ \ }
\Lambda(\epsilon)=-\alpha^2\frac{1}{2}\int d^5k\frac{B_k B_{-k}}{f_k(\epsilon)f_{-k}(\epsilon)},
\end{equation}
where $B_k$ is arbitary boundary momentum $k_i$ dependent function. To get the $n$-ple trace operators, we try
the following form of ansatz,
\begin{equation}
\label{idontknow}
\mathbb D^{(n)}_{k_1,...,k_{n-1},-\sum^{n-1}_{j=1}k_j}(\epsilon)\equiv D^{(n)}_{k_1,...,k_{n-1},-\sum^{n-1}_{j=1}k_j}(\epsilon)+\alpha C^{(n)}_{k_1,...,k_{n-1},-\sum^{n-1}_{j=1}k_j}(\epsilon)+O(\alpha^2)
\end{equation}
where $\mathbb D^{(n)}(\epsilon)$ is the full solution which is a form of power expansion order by order in small parameter $\alpha$. $D^{(n)}(\epsilon)$ is the $n$-ple trace operator that we obtained in the previous section and $C^{(n)}(\epsilon)$ is the first order solution in the $\alpha$ respectively.

It is very unlikely to get analytic solutions of $C^{(n)}(\epsilon)$. However, one certain statement is that the fixed points of the $n$-ple trace operator does not change by the first order solutions in $\alpha$ at $r=\epsilon^\star$, as long as the zeroth order solution
of the $n+1$-ple trace operator's fixed points are finite at the given radial velue.

\section{Holographic 4-point function and conformal Ward identities}
\label{Holographic 4-point function and conformal Ward identities}

Conformal 2- and 3-point correlation functions are determined due to the global conformal invariance.
In the momentum space as well as in the position space, such correlators  are investigated.
In \cite{Bzowski:2013sza}, the authors compute the 2- and 3-point functions in momentum space by imploying Fourier transform
from them in position space. The 2-point function the they obtain is given by
\begin{equation}
\langle O(p_1)O(p_2) \rangle=(2\pi)^d \delta^{(d)}(p_1+p_2) \langle\langle O(p_1)O(p_2) \rangle\rangle,
\end{equation}
where
\begin{equation}
\langle\langle O(p_1)O(p_2) \rangle\rangle=\frac{C_2 \pi^{d/2}2^{d-2\Delta}\Gamma\left( \frac{d-2\delta}{2} \right)}{\Gamma(\Delta)}|p|^{2\Delta-d}.,
\end{equation}
The $C_2$ is the coefficient of the 2-point function in position space, $\Delta$ is the conformal dimension of the operator $O(x)$, $\Gamma$ is the gamma function and the
$d$ is the dimensionality of the spacetime.
In the previous researches devoting to conformally coupled scalar in AdS spacetime and its holographic duals\cite{Jatkar:2013uga,Oh:2014nfa},
the authors compute the conformal 2-point correlation functions for the cases that $2\Delta-d=1$.
The 2-point functions have
the forms of
\begin{equation}
\langle\langle O(p_1)O(p_2) \rangle\rangle = C|p|,
\end{equation}
upto a coefficient $C$.

The form of the 3-point function is an integral form being given by
\begin{equation}
\langle\langle O(p_1)O(p_2)O(p_3) \rangle\rangle = C_3{p^{\Delta_1-\frac{d}{2}}_1}{p^{\Delta_2-\frac{d}{2}}_2}{p^{\Delta_3-\frac{d}{2}}_3}\int^\infty_0
dx{\ } x^{\frac{d}{2}-1}\left\{ \prod^3_{j=1} K_{\Delta_j-\frac{d}{2}}(p_j x) \right\},
\end{equation}
where the coefficient $C_3$ is
\begin{equation}
C_3=\frac{C_{123}\pi^d 2^{4+\frac{3d}{4}-\Delta_t}}{\Gamma\left(\frac{\Delta_t-d}{2}\right)
\Gamma\left(\frac{\Delta_1+\Delta_2-\Delta_3}{2}\right)
\Gamma\left(\frac{\Delta_2+\Delta_3-\Delta_1}{2}\right)
\Gamma\left(\frac{\Delta_3+\Delta_1-\Delta_2}{2}\right)},
\end{equation}
and the $K_\nu$ represents the modified Bessel-K function.
The form of the 3-point function is not analytic form yet(except some special cases), which is so called``triple-K integral"\cite{Bzowski:2013sza}.

In \cite{Jatkar:2013uga,Oh:2014nfa} and this note, the authors suggest the forms of the 3-point functions from a holographic computations. They study conformally coupled scalar theory in AdS background again. It turns out the model enjoies a reasonable self-interaction for the cases that $d=5$ and $d=3$(which are the boundary spacetime dimensions of the AdS space)
\footnote{In other cases, the self-interaction is not an integral power of the fields.}
. In such cases, the conformal dimensions corresponding of the exitations in tis AdS space are
$\Delta=3$ and $\Delta=2$ respectively.

For the case that $d=5$, the 3-point function from the model(they perform minimal substraction and add no other terms on the conformal boundary) is
\begin{equation}
D^{(3)}_{k_1,k_2,k_3}(\epsilon^\star_\pm)=\pm\frac{1}{4(2\pi)^{5/2}(\sum_{i=1}^3|k_i|)},
\end{equation}
as addressed in this note, where absolutely the sum of all the momenta needs to be conserved. One can show that when $\Delta_1=\Delta_2=\Delta_3=3$ and $d=5$, the triple-K integral becomes the same form with this upto the overall coefficient.

In the case that $\Delta=2$ and $d=3$\cite{Oh:2014nfa}, they find that the holographic computation of the 3-point function vanishes(again their prescription is minimal subtraction and adding no other manipulations on the conformal boundary). The triple-K integral suggest that
\begin{equation}
\langle\langle O(p_1)O(p_2)O(p_3) \rangle\rangle \sim \int^\infty_0 \frac{dx}{x}\exp\left\{{-\left(\sum_{i=1}^3|p_i|\right)x}\right\},
\end{equation}
which has no momentum dependence and so it is a number.

There days, there are some of attampts to compute 4-point correlation function in momentum space\cite{Bzowski:2019kwd}.

\paragraph{Conformal Ward identities}
Dilatation Ward identity is $\mathcal D\left( p_j,\frac{\partial}{\partial p_j} \right)\Phi(p_1,...,p_n)=0$, where $\Phi$ is the $n$-point correlation function in momentum space, the momenta are constrained to accept momentum conservation $\sum_{j=1}^n \vec p_j=0$. The differential operator $\mathcal D$ is given by
\begin{equation}
\mathcal D=p^a \frac{\partial}{\partial p_a}+\Delta^\prime,
\end{equation}
where the $\Delta^\prime=-\sum_{i=1}^n\Delta_i + (n-1)d$.

Special conformal Wand identity is $\mathcal K^k\left(p_j,\frac{\partial}{\partial p_j}\right) \Phi(p_1,...,p_{n})=0$,
where $\Phi$ is the $n$-point correlation function in momentum space, the momenta are constrained to accept momentum conservation $\sum_{j=1}^n \vec p_j=0$. In fact, the operator $\mathcal K^k$ is an differential operator with respect to $n-1$ independent momenta. More preceisely, $\mathcal K^k\equiv\sum_{j=1}^{n-1} \mathcal K_j^k$ and
\begin{equation}
\mathcal K^k_{j}=2(\Delta_j-d)\frac{\partial}{\partial p^k_j} + p_j^k\frac{\partial^2}{\partial p^a_j \partial p_j^a} - 2p_j^a\frac{\partial}{\partial p^k_j \partial p_j^a},
\end{equation}
where the $\Delta_j$ is the conformal dimension of the j-th operator.

If one applies this operator to a product of a certain two arbitrary functions of momenta, then it must be by chain rule as
\begin{eqnarray}
\mathcal K^k[A(p_l)B(p_m)]=\left\{ \mathcal K^k A(p_l) \right\} B(p_m) + A(p_l)\left\{ \mathcal K^k B(p_m) \right\} \\ \nonumber
+\sum_{j=1}^{n-1}\left( 2 p^k_j \frac{\partial A}{\partial p^a_j} \frac{\partial B}{\partial p^a_j}
-2 p^a_j \frac{\partial A}{\partial p^k_j} \frac{\partial B}{\partial p^a_j}
-2 p^a_j \frac{\partial A}{\partial p^a_j} \frac{\partial B}{\partial p^k_j}
\right)
\end{eqnarray}

\paragraph{4-point functions from conformally coupled scalar}
The holographic 4-point function is obtained from the conformally coupled scalar theory in AdS space\cite{Jatkar:2013uga,Oh:2014nfa}. For $d=3$ and $\Delta=2$ scalar operators, the 4-point function obtained from the model is
\begin{equation}
\langle O(p_1)O(p_2)O(p_3)O(p_4) \rangle=u^{-1}(4)\delta^{(4)}\left(\sum_{j=1}^n p_j\right),
\end{equation}
where
\begin{equation}
u(n)=\sum_{j=1}^{n-1}|p_j|+\left|\sum_{j=1}^{n-1}p_j\right|.
\end{equation}
Namely, the $u(4)=|p_1|+|p_2|+|p_3|+|p_1+p_2+p_3|$.

In fact, the object $u^{-1}(n)$ can be a candidate for $n$-point conformal scalar correlator for specific conformal dimensions. When one applies the special conformal Wand identity on the $u^{-1}(n)$, one gets in the end,
\begin{equation}
\label{u(n)-SCWI}
\mathcal K^k u^{-1}(n)=\frac{d-2\Delta+1}{u^2(n)}\left(\sum_{j=1}^{n-1}\frac{p_j^k}{|p_j|}\right)
+\frac{2(d-\Delta)(n-1)-d-3}{u^2(n)}
\left(\frac{\sum_{j=1}^{n-1} p_j^k}{\sum_{l=1}^{n-1}|p_l|}\right)
\end{equation}
For the $u^{-1}(n)$ to become a conformal correlator, the right hand side of the (\ref{u(n)-SCWI}) should vanish. Then,  we have two algebraic equations. After a little computations, the equations become
\begin{equation}
d=2\Delta-1,{\ \ \ \rm and \ \ } \Delta=\frac{2}{n-2}+1.
\end{equation}
Together with this, one needs to consider the dilatation Ward identity too. This gives a more algebraic constraint, which is
\begin{equation}
\Delta=\frac{(n-1)d-1}{n}.
\end{equation}
The possible and reasonable(both the spacetime dimension, $d$ and the $n$ are integral) solutions are listed below:
\begin{itemize}
\item $d=2$, $\Delta=\frac{3}{2}$ and $n=6$, so the $u^{-1}(6)$ is a conformal 6-point function for the scalar operators with conformal dimension $\Delta=\frac{3}{2}$ in $d=2$.
\item $d=3$, $\Delta=2$ and $n=4$, so the $u^{-1}(4)$ is a conformal 4-point function for the scalar operators with conformal dimension $\Delta=2$ in $d=3$. This is what we get in holographic 4-point function from the conformally coupled scalar model.
\item $d=5$, $\Delta=3$ and $n=3$, so the $u^{-1}(3)$ is a conformal 3-point function for the scalar operators with conformal dimension $\Delta=3$ in $d=5$. This is what we get in holographic 3-point function from the conformally coupled scalar model in this note.
\end{itemize}

The 4-point function that we get in this note for the $5$-dimensional conformal field theory( obtained from the conformally coupled scalar model in AdS$_6$) is given by
\begin{eqnarray}
\label{4-finction-point}
D^{(4)}_{k_1,k_2,k_3,k_4}(\epsilon^\star_\pm)&=&\mp\frac{3}{2^5(2\pi)^5u(4)u(3)v(4,1)}+({k_1 \leftrightarrow k_3})+({k_2 \leftrightarrow k_3}),
\end{eqnarray}
where
\begin{equation}
v(E,n)=\left| \sum_{i=1}^{E-1-n}P_i \right|+\sum_{j=E-n}^{E-1}|p_j|+\left| \sum_{k=1}^{E-1}p_k \right|.
\end{equation}
The first term in (\ref{4-finction-point}) is s-channel, the second is t- and the last term is u- channel in order.

When one applies the special conformal Ward identity on the first term of the holographic 4-point function, which gives
\begin{equation}
\label{SCWI-result}
\mathcal K^k D^{(4)}_{k_1,k_2,k_3,k_4} = 2\frac{\{u(3)-u(4)\}\{u(4)+v(4,1)\}}{u^2(3)u^2(4)v^2(4,1)}\left\{ \frac{(p_1+p_2)^k}{|p_1+p_2|} +  \frac{(p_1+p_2+p_3)^k}{|p_1+p_2+p_3|}\right\}
\end{equation}
\paragraph{Discussion}
The result above that the (s-channel)4-point function is not a conformal correlation function since it does not satisfy the special confromal Ward identity in general. However, in a certain limit, it does. The right hand side of (\ref{SCWI-result}) is proportional to the factor, $u(3)-u(4)=|p_1+p_2|-|p_3|-|p_1+p_2+p_3|$. Considering momentum conservation, this becomes $|p_3+p_4|-|p_3|-|p_4|$, which vanishes when $\vec p_3$ and $\vec p_4$ are colinear. There is another factor $\left\{ \frac{(p_1+p_2)^k}{|p_1+p_2|} +  \frac{(p_1+p_2+p_3)^k}{|p_1+p_2+p_3|}\right\}=\hat n_{12}-\hat n_4$, where $\hat n_{12}$ is a unit vector alnog $\vec p_1+\vec p_2$ and $\hat n_4$ is a nunit vector along $\vec p_4$. If a condition
$\hat n_{12}=\hat n_4$, then the right hand side of (\ref{SCWI-result}) vanishes. This means that
$\vec p_1+\vec p_2$ and $\vec p_4$ are colinear.

If one sum up all posible channels, then the above argument is not hold. The only possible limit which makes
the 4-point function conformal, is that $\vec p_i$ for $i=1,2,3$ and $4$ are colinear. In this limit, the 4-point function effectively becomes
\begin{equation}
\label{u3}
\sim\frac{1}{u^3(4)},
\end{equation}
which is expected in \cite{Oh:2020izq}. In \cite{Oh:2020izq}, it is proved that $n$-point correlation function among the scalar operators $O_{\Delta}$, where $\Delta=\frac{d+1}{2}$ is given by
\begin{equation}
\Phi(p_1,p_2,...,p_n)=\frac{1}{\left(
\sum_{i=1}^{n-1}|p_i|+\left| \sum_{j=1}^{n-1} p_j \right|
\right)^{(n-1)d-\frac{n}{2}(d+1)}},
\end{equation}
and the (\ref{u3}) is recovered when $n=4$ and $d=5$.
Again, $d$ is spatial dimensionality that the theory defined on and the space is Eucliean.

The colinear limit of the 4-point function may be related to the colinear divergences of the solution of the equation of motion of the conformally coupled scalar theory in AdS space. To remove the colinear divergences of the soluton, we add the homogeneous solutions of the equation of motion to regulariz it. In the prevous computation, we choose such a scheme. In such scheme, it is very clear that the holographic correlation function from the conformally coupled scalar theory is free from the colinear divergneces. However, we point out the forms of the boundary correlation functions are scheme-independent.

We also point out that one may recognize that in the colinear limit, the $n$-point holographic correlation functions that we get, or the multi-trace deformations in the fixed points $\epsilon=\epsilon^\star_+$ of the scalar operators $O$ effectively become the conformal correlation functions given in \cite{Oh:2020izq}.
More precisely,
\begin{equation}
\langle O({p_1})O({p_2})..O({p_n})\rangle=\mathcal D^{(n)}_{p_1,p_2,...,p_n}(\epsilon^\star_+)\sim\frac{C_n}{\left(   \sum_{i=1}^{n-1}|p_i|+|\sum_{i=1}^{n-1}p_i| \right)^{2n-5}},
\end{equation}
where the colinear limit denotes that all the external momenta $p_i$s for $i=1,...,n$ are alined in the same direction and so it is satisfied that $\sum_{i=1}^{n}|p_i|=|\sum_{i=1}^{n}p_i|$. Therefore, in some sense, one may say that the conformally coupled scalar theory in $AdS_6$ produces conformal correlation functions of a scalar operator $O$ with $\Delta=3$ in 5-dimensinal Euclidean space as its dual conformal field theory.

We also expect that the conformally coupled scalar theory in $AdS_4$ will produce $n$-point conformal correlation functions as a well-posed dual gravity theory of a certain $3$-dimensional CFT deformed with a scalar operator $O$ with its conformal dimension is $\Delta=2$ in a colinear limit. It could be an excercise for my students.

\section*{Acknowledgement}
J.H.O would like to thank his $\mathcal W.J.$ and $\mathcal Y.J.$ This work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIP) (No.2016R1C1B1010107). This work is also partially supported by Research Institute for Natural Sciences, Hanyang University.

\section{Appendices}
\subsection{Appendix A: Application of special conformal Ward identity to the holographic 4-point function}

\begin{eqnarray}
\mathcal K^k\left(\frac{1}{u(4)u(3)v(4,1)} \right)=\frac{1}{u(3)v(4,1)}\mathcal K^k\left(
\frac{1}{u(4)} \right)+\frac{1}{u(3)u(4)}\mathcal K^k\left(
\frac{1}{v(4,1)} \right) \\ \nonumber
+ {\ \ \rm A-type\ terms\ \ } + {\ \ \rm B-type\ terms\ \ } + {\ \ \rm C-type\ terms\ \ },
\end{eqnarray}
where
\begin{eqnarray}
&{\ }&{\ \ \rm A-type\ terms\ \ }= 2\sum_{l=1}^3p_l^k \frac{1}{u(3)} \frac{\partial}{\partial p_l^a }\left( \frac{1}{u(4)} \right)\frac{\partial}{\partial p_l^a }\left( \frac{1}{v(4,1)} \right) \\ \nonumber
&-&2\sum_{l=1}^3p_l^a \frac{1}{u(3)} \frac{\partial}{\partial p_l^k }\left( \frac{1}{u(4)} \right)\frac{\partial}{\partial p_l^a }\left( \frac{1}{v(4,1)} \right)
-2\sum_{l=1}^3p_l^a \frac{1}{u(3)} \frac{\partial}{\partial p_l^a }\left( \frac{1}{u(4)} \right)\frac{\partial}{\partial p_l^k }\left( \frac{1}{v(4,1)} \right),
\end{eqnarray}
\begin{eqnarray}
&{\ }&{\ \ \rm B-type\ terms\ \ }= 2\sum_{l=1}^3p_l^k \frac{1}{u(4)} \frac{\partial}{\partial p_l^a }\left( \frac{1}{u(3)} \right)\frac{\partial}{\partial p_l^a }\left( \frac{1}{v(4,1)} \right) \\ \nonumber
&-&2\sum_{l=1}^3p_l^a \frac{1}{u(4)} \frac{\partial}{\partial p_l^k }\left( \frac{1}{u(3)} \right)\frac{\partial}{\partial p_l^a }\left( \frac{1}{v(4,1)} \right)
-2\sum_{l=1}^3p_l^a \frac{1}{u(4)} \frac{\partial}{\partial p_l^a }\left( \frac{1}{u(3)} \right)\frac{\partial}{\partial p_l^k }\left( \frac{1}{v(4,1)} \right),
\end{eqnarray}
and
\begin{eqnarray}
&{\ }&{\ \ \rm C-type\ terms\ \ }= 2\sum_{l=1}^3p_l^k \frac{1}{v(4,1)} \frac{\partial}{\partial p_l^a }\left( \frac{1}{u(3)} \right)\frac{\partial}{\partial p_l^a }\left( \frac{1}{u(4)} \right) \\ \nonumber
&-&2\sum_{l=1}^3p_l^a \frac{1}{v(4,1)} \frac{\partial}{\partial p_l^k }\left( \frac{1}{u(3)} \right)\frac{\partial}{\partial p_l^a }\left( \frac{1}{u(4)} \right)
-2\sum_{l=1}^3p_l^a \frac{1}{v(4,1)} \frac{\partial}{\partial p_l^a }\left( \frac{1}{u(3)} \right)\frac{\partial}{\partial p_l^k }\left( \frac{1}{u(4)} \right).
\end{eqnarray}

Each term is given by
\begin{equation}
\mathcal K^k\left(
\frac{1}{u(4)} \right)=\frac{4}{u^2(4)} \frac{(p_1+p_2+p_3)^k}{|p_1+p_2+p_3|},
\end{equation}
which is given in (\ref{u(n)-SCWI}).

\begin{equation}
\mathcal K^k\left(
\frac{1}{v(4,1)} \right)=\frac{4}{v^2(4,1)}\left\{ \frac{(p_1+p_2)^k}{|p_1+p_2|} +  \frac{(p_1+p_2+p_3)^k}{|p_1+p_2+p_3|}\right\}
\end{equation}

\begin{equation}
{\ \ \rm A-type\ terms\ \ }=\frac{2}{u(3)u^2(4)v^2(4,1)}\left\{  \frac{(p_1+p_2)^k}{|p_1+p_2|}(v(4,1)-u(4)) -  \frac{(p_1+p_2+p_3)^k}{|p_1+p_2+p_3|}(v(4,1)+u(4))   \right\}
\end{equation}

\begin{equation}
{\ \ \rm B-type\ terms\ \ }=-\frac{2}{u^2(3)v^2(4,1)}\left(  \frac{(p_1+p_2)^k}{|p_1+p_2|} +  \frac{(p_1+p_2+p_3)^k}{|p_1+p_2+p_3|}   \right)
\end{equation}

\begin{equation}
{\ \ \rm C-type\ terms\ \ }=-\frac{2}{u^2(3)u(4)v(4,1)}\left(  \frac{(p_1+p_2)^k}{|p_1+p_2|} +  \frac{(p_1+p_2+p_3)^k}{|p_1+p_2+p_3|}   \right)
\end{equation}

redpajama_set_name: RedPajamaArXiv


He decided to start his own kitchen cabinetry business because he wanted to bring his practical yet creative approach to kitchen design to benefit his clients. He relished the challenge of giving his customers exactly what they wanted within their budger.
Bayram Gulsen takes deep satisfaction in making his clients completely satisfied and happy with their purchases. He continues to improve the design and selection processes to make it easier for clients to make decisions, and makes sure everyone on his team is onboard with these core principles.


text:
Q: Android circular progress with transparent background I have a fullscreen dialog that contain only a ProgressBar. The background of the dialog is transparent with the good color, but behind the progressbar, even if I set it's background to transparent there will an overlay...
How can I get this progressBar to have the same transparent background as the dialog ?
(I blanked private part informations)

Yes I used a custom view, but i tested with both default ProgressBar and the custom one, I get the same behavior.
I also tested with a DialogFragment, same result
<?xml version="1.0" encoding="utf-8"?>
<LinearLayout xmlns:android="http://schemas.android.com/apk/res/android"
    xmlns:app="http://schemas.android.com/apk/res-auto"
    android:layout_width="match_parent"
    android:layout_height="match_parent"
    android:background="@color/black_transparent"
    android:gravity="center"
    android:orientation="vertical">

    <com.wang.avi.AVLoadingIndicatorView
        android:id="@+id/progressBar"
        android:layout_width="@dimen/pop_button_height"
        android:layout_height="@dimen/pop_button_height"
        android:background="@color/black_transparent"
        android:visibility="visible"
        app:indicatorColor="@color/colorPrimaryDark"
        app:indicatorName="BallSpinFadeLoaderIndicator" />

</LinearLayout>

Creation of the dialog in an Activity :
    mProgressDialog = new Dialog(this);
    View view = LayoutInflater.from(this).inflate(R.layout.dialog_loading_overlay, null);
    mProgressDialog.setCancelable(false);
    mProgressDialog.requestWindowFeature(Window.FEATURE_NO_TITLE);
    mProgressDialog.getWindow().setBackgroundDrawable(new ColorDrawable(android.graphics.Color.TRANSPARENT));
    mProgressDialog.setContentView(view);
    mProgressDialog.show();

A: Try this, it may works for you
mProgressDialog.getWindow().setBackgroundDrawable(new ColorDrawable(android.graphics.Color.TRANSPARENT));

Edit
Remove android:background="@color/black_transparent" in AVLoadingIndicatorView and try
Optional
If you want to put the background for that (If any warning came) use this android:background="@android:color/transparent"

redpajama_set_name: RedPajamaStackExchange


Do you dream of meeting real mermaids and touch their little silver tails? Does the astonishing variety of the sea creatures attract you? Well, then the new Just Cavalli spring/summer 2014 collection showcased during Milan Fashion Week is right for you. Frisky, bold and wild, the new Just Cavalli line of Roberto Cavalli is definitely not for the faint-hearted, as it requires tons of self-confidence for wearing these brightly printed pieces!
The frantic combination of fishy colors and abstract prints is just perfect for a crazy all-night-long party. The mixture of rainbow-y colors, hot cuts, bizarre shapes and cute outlines does help the models be like mermaids, and brings back the sweet memory of the romantic cruise you always wanted to have.
The designer has created a bright, flashy and colorful collage of artistic, abstract, leopard prints that surely remind you of an awesome beach or luxurious yacht party. The very first look is a unique jumpsuit that has all the imaginable colors and comes matched with a cute blue, white and green swimming suit worn underneath. With this apparel you can easily grab a glass of whiskey and walk on the hot sand watching the black sea. Another stylish look from the sophisticated fashioner is the orange sweatshirt with tiny white tiles on it, and blue, orange skinny pants that perfectly show your exclusively unique character and your love of life.
There are also some loose shirts in the Just Cavalli spring/summer 2014 collection that can be worn not only on vacation but also during your harsh work time. The simple chiffon shirt with creamy tiles and light green prints is elegant and sassy. It will give you an extravagant look at business meetings, formal parties or while touring around in any fashionable city.
The tight black pants with a shirt colorful chiffon top are just glamorous and enticing. It is an ideal outfit for hot Spanish and Brazilian dances.
Overly, the Just Cavalli spring/summer 2014 collection is blooming and positive with a lot of flowers and adorable abstract prints.


Throw your baggage in the back, roll down the windows and blast the stereo. Getting around Steuben, United States of America, with your own rental car is hassle free with CarRentals.com.
After your plane hits the tarmac at Hancock County - Bar Harbor Airport, picking up a car is simple. Hertz Trenton sits just a quick stroll from the terminal.
If a day trip sounds like an appealing idea, consider exploring Hancock. It's 17 miles from Steuben. Let your GPS guide you to Tidal Falls and Pirate's Cove Adventure Golf, the most prominent attractions here. You can also spend a few hours at the Garland Farm. 20 miles away, Seal Harbor is another destination that is well worth the price of gas. Upon arriving, you won't regret paying a visit to Egg Rock Lighthouse and Ocean Path Trailhead. If that isn't enough, why not also stop by the Kebo Valley Golf Club?
Open the windows and feel that country air on your skin as you cruise through the scenic coastal plains of Maine.
If you're eager to hit the wide open road, a road trip might be in order. A rewarding 160-mile drive away, North Conway is a captivating city to explore. Cranmore Mountain Resort is a prominent sight that should definitely be on your itinerary.
So what are you waiting for? As soon as you've got the keys in your hand to of your shiny new rental, there'll be no limit to the freedom you'll have!


Thanks for coming through as usual–haven't heard the show yet but this site has been holding me down for years!
can u please download the sheryl underwood show??
I second the Sheryl Underwood Show request. Please Cap't.


Energy Recovery Inc Introduces ERI(TM) AquaBold(TM) and ERI(TM) AquaSpire(TM) High-Pressure Pumps for Desalination | Energy Recovery Inc.
SAN LEANDRO, Calif., Feb 24, 2011 (BUSINESS WIRE) -- Energy Recovery Inc (NASDAQ: ERII), a leader in the design and development of energy recovery devices for desalination, today launched its new Aqua line of high-pressure desalination pumps - the ERI(TM) AquaBold(TM) multi-stage reverse osmosis (RO) high-pressure pump and the ERI(TM) AquaSpire(TM) single-stage pumps. The pumps, which ERI obtained with its acquisition of Pump Engineering, L.L.C. and re-engineered, offer best-in-class efficiencies (nearly 90 percent), and complement ERI's product suite of isobaric and centrifugal energy recovery technologies.
Designed for maximum performance and power savings, the ERI AquaBold and ERI AquaSpire pumps are also easy to operate with no mechanical wear of the components, and are manufactured for a 20+ year service life with little to no scheduled maintenance required. Additionally, when combined, ERI's pumps and energy recovery devices offer the highest efficiency packaged solution with minimum power consumption for desalination plants around the world.
"ERI has always been dedicated to rigorous R&D efforts in order to find new and better ways to improve the desalination process and enhance our product offering. Our new ERI Aqua line of pumps stems from our ongoing product development and is a great complement to our existing energy recovery devices," said Timothy Dyer, chief technical officer at ERI. "ERI continues to lead the global effort to make seawater desalination affordable, and we are proud to offer our customers easy-to-use products that will provide them with the most savings and best performance possible."
The ERI AquaBold high-pressure seawater pump has been optimized for quality, reliability and durability, and offers enhanced corrosion resistance. A unique new internal component protection system (ICPS) acts as an "anti-crash" feature similar to those found in cars that protect internal components in the event of damage to the pump, offering an elevated level of reliability. Additionally, the new and improved bearing and debris durability increases the lifecycle of the pump and reduces the need for maintenance. The ERI AquaBold and ERI AquaSpire pumps include innovative, process lubricated thrust bearing technology, helping to eliminate efficiency losses.
The ERI AquaSpire high-pressure single-stage seawater or brackish water pump contains ERI's patented Volute Insert Technology(TM), which allows the custom hydraulics of the pump to be adjusted should the system conditions change in the future, such as plant expansions. Whereas other pumps require completely new casings made of the original pump material, this new feature enables customers to simply change out the lightweight internal volutes to bring the pump back to its Best Efficiency Point (BEP).
ERI acquired the pump product line in December of 2009 from Pump Engineering and has since optimized and enhanced the custom-designed pumps to reduce energy consumption and increase efficiency in specific process conditions. ERI solutions offer simplicity, durability, reliability and small footprint - all hallmarks of ERI products.
Energy Recovery, Inc. (NASDAQ:ERII) designs and develops energy recovery devices that help make desalination affordable by significantly reducing energy consumption. Energy Recovery technologies include the PX Pressure Exchanger(TM) (PX(TM)) device for desalination and the Turbocharger hydraulic turbine energy recovery device and pump for desalination, gas and liquid processing applications. In total, Energy Recovery reduces the carbon footprint of desalination, saving more than 970 MW of energy and offsetting CO2 emissions by more than 5.2 million tons per year. The company is headquartered in the San Francisco Bay Area with offices in Detroit and in key desalination centers worldwide, including Madrid, Shanghai, and the United Arab Emirates. For more information about Energy Recovery, Inc. please visit www.energyrecovery.com.


text:
Michigan officials ousted in campus-death cover-up
YPSILANTI, Mich. -- Three Eastern Michigan University administrators have been forced out, months after top school officials were accused of covering up the rape and slaying of a student by publicly ruling out foul play. President John Fallon was fired, and Vice President of Student Affairs Jim Vick and Public Safety Director Cindy Hall also lost their jobs at the 23,500-student university, the chairman of the school's governing board said Monday. Laura Dickinson, 22, was discovered Dec. 15 in her dorm room. University officials told her parents and the media that she died of asphyxiation but that there was no sign of foul play, despite evidence to the contrary. It was not until another student was arrested in February and charged with murder that her family and students learned she had been raped and killed.

redpajama_set_name: RedPajamaCommonCrawl


lularoe lucy size chart Inspirational Randy T Size Chart LuLaRoe Styles and Sizing Info. Inspirational 58 Best LuLaRoe size chart images. New Lindsey sizing LulaRoe Pinterest. New LuLaRoe Joy Sizing Chart with Price Lula. Awesome Pin on Itty Bitty LuLaRoe. . . .


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Ungmennafélagið Fjölnir, commonly known as Fjölnir, is a multi-sport club from Iceland. The club is located in Grafarvogur, Reykjavík. The club was founded in 1988 under the original name ; however, because another team already had the abbreviation UMFG, the name was changed to , commonly referred to as . A total of nine sports are practised at the club: football, basketball, handball, taekwondo, karate, tennis, swimming, athletics and gymnastics. Chess is also played at the club. Each one of these sports has their own department with their own board but all are under the main board and the club office.

Football

Men's football

Current squad

Out on loan

Trophies and achievements
1. deild karla (1):
2013

Basketball

Men's basketball

Trophies and achievements
Icelandic Second Division:
2001–02

Women's basketball

Trophies and achievements
Division I (2):
2007, 2010

Ice hockey
On 28 September 2018, Skautafélagið Björninn merged into Fjölnir who overtook all its departments, assets and debts.

References

External links

Official website
 KKÍ: Fjölnir – kki.is

Football clubs in Iceland
Football clubs in Reykjavík
Multi-sport clubs in Iceland
Sports clubs established in 1988
Sport in Reykjavík
1988 establishments in Iceland
Ice hockey teams in Iceland
Ice hockey clubs established in 1990
Icelandic Hockey League

redpajama_set_name: RedPajamaWikipedia


UPDATE 12/18: We updated our King Tide photos blog post with some non-King Tide comparative photos ... like we should have done to begin with.
Thursday morning at 10:46 a.m., was when the North Coast was predicted to have its highest tide of 2012, i.e. THE KING TIDE. So the Journal went out and took pictures. Yup. Lots o' water (9.5 ft at the Eureka Slough Bridge).
If you happened to be out and about gawking at high tideage too, and you'd like your pictures to do more than to claim bytes on your hard drive, Humboldt Baykeeper is soliciting shots for their 2012 King Tide Photo Initiative. For more details email KingTidePhotos@gmail.com or call 268-8897.
You know how, when you ride the Madaket, you have to descend down that plank. Well, that plank was just 'bout level today.
UPDATE: Facebook commentor and tide lover Matthew Knight would like us all to share in his private hell. Everyone sing along!


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Feeding your friends and foes – why the gut microbiome has become a nutrition focus
The following blog is a summary of key learning points from a talk given at Digestive Health and Wellbeing: The Patient Journey in 2021. This blog highlights the importance of nutrition in the shaping of the gut microbiome, including a focus on prebiotics and probiotics.
Meet the speaker
Dr Gemma Walton is a lecturer in metagenomics at the University of Reading. In the arena of gut health, her primary area of interest is the gut's microbial community. In particular, Dr Walton studies how it supports health and how we can change this community through dietary intervention. Examples include the inclusion of prebiotics and probiotics as part of a balanced diet.
Dr Walton has studied the gut microbial community through experimental models of the large intestine, as well as through human intervention studies. These approaches allow greater exploration of how dietary changes could influence the community of bacteria in the gut, as well as how this could ultimately impact health and wellbeing.
Why the gut microbiome has become a nutrition focus
Dr Walton's presentation takes a closer look at why the gut microbiome has become so central to our understanding of nutrition. The talk considers what gut microbiota do and how changes to our diet can alter our microbial communities. Dr Walton focuses on existing regimens and technologies, such as prebiotic and probiotic interventions, as well as postbiotics, as approaches to adjuvant therapies.
The gut microbial community, which consists of trillions of bacteria, is increasingly recognised to play a significant role in host health. The ever-shifting community develops alongside us from birth, influenced by our lifestyles and importantly by what we eat. With the number of microbial genes outnumbering human genes, it is not surprising that this community can have a significant impact on our health. We can alter this community through changes in our diet, and as a result, impact our wellbeing.
Bacteria that inhabit the gut microbiome can use our non-digested foods for various purposes. From these substances, bacteria create end products, such as short chain fatty acids, that influence our health. As a result, our microbiota is important for immune support and pathogen protection. Our gut bacterial communities have an impact all over the body, influencing areas such as skin health, obesity and satiety, coughs and colds, to name just a few. There is strong evidence to suggest that gut microbial communities could have direct effects on host health.
Development of the gut microbiome
The human gut microbiome develops from birth. We have initial colonisers that are dependent on birth mode and from there, the microbiome goes on to respond to the diet. Clear differences are seen in the gut microbiome composition of breast-fed infants versus formula fed, for example. There are other factors affecting bacteria in the gut, such as physiochemical changes in response to digestion of food and microbial effects. This includes competition of microbes, as well as immunity and inhibition through microbial end products.
The end products of non-digestible carbohydrates fermentation can have numerous effects in the body. The process provides around 10% of human energy requirements and produces end products that can be metabolised further by gut bacteria. In contrast, end products of protein fermentation are generally associated with negative effects. Protein is important in the diet as it contributes to cell growth and repair, but too much can increase the fermentation of protein and undesirable end products.
While a high fibre diet is understood to be beneficial for health, not all carbohydrates are used by the body in the same way. This is why a tailored approach to dietary interventions may be appropriate. Use of prebiotics to supplement the diet may be beneficial because it is a selectively fermented non-digestible compound, proven to be beneficial to the host (Gibson et al., 2017). Studies into the role of prebiotics have revealed advantages such as reduced allergy development, demonstrating that microbiome composition is particularly important in the early stages of life (Cukrowska et al., 2020).
In contrast to prebiotics, probiotics are live microorganisms, that when administered in adequate quantities, elicit a beneficial effect on the host. Probiotics are strain-specific, known by their health effects. There is a common misconception that all fermented foods are probiotic, which is not correct. Benefits of probiotic interventions include systemic effects as a result of short chain fatty acid production, support of barrier function through the digestive system, positive modulation of the microbial community and reduced passage of inflammatory compounds into systemic circulation.
We know that the gut microbiome community is important for supporting key areas of health and wellbeing. We also know that the microbial community can be affected by diet. Since changing the gut microbiome is associated with positive health outcomes, prebiotics and probiotics can be used as tools to modify microbial communities and elicit positive advantages in health and wellbeing. Looking further ahead, there are promising areas for future research for the systemic effects of the gut microbiota on the body.


text:
Securing America: Keeping U.S. Embassies Safe [VIDEO]
Securing America
Anti-American protests have spread across the Middle East. What can be done to secure U.S. embassies and consulates in this volatile region?
FOX News Radio's Courtney Kealy reports:
Securing America.
Gil Torrez, a former FBI agent that has has spent time in different U.S. embassies saying there are many ways that security is taken into consideration.
(Torrez) "We have many many embassies around the world so each one has a different threat level assessments are made, so you know the State Department is heavily involved in this."
After the attacks in the 1980's in Beirut, Lebanon against U.S. Marines and the U.S. embassy. The embassy is now outside the city ringed with fences and multi-layers of protection. So too is the U.S. embassy in Sanaa, Yemen. The U.S. embassy in Baghdad, Iraq is the largest and most expensive at a cost of over $500 million dollars.
Courtney Kealy, FOX News Radio.
WATCH to learn more on the U.S. embassy attacks:
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I am using professional grade oil paints o mixed media and 100% linen or cotton quality canvases or mixed so you will have your original oil painting for a lifetime to enjoy.
This artwork is sold and shipped to you by Françoise Dugourd-Caput, and will be packaged in a reinforced envelope. Artworks are dispatched within 8 working days from France.


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Joining family of eight-month-old on transplant list to help raise awareness of having conversations about organ donation | News
Joining family of eight-month-old on transplant list to help raise awareness of having conversations about organ donation
Posted Wednesday, 19 June 2019 by Jon Harris
The parents of an eight-month-old currently on the transplant list awaiting a donor heart are urging families to have conversations and share their wishes about organ donation to help offer the precious 'Gift of Life'.
Like many others, Riona and Damien Grant, never expected that they would find themselves in such a position. Their son Ollie was born with the rare and complex condition hypoplastic left heart syndrome – a defect that affects blood flow through the heart – diagnosed during the pregnancy.
It's meant that Ollie has only spent 16 days of his life in his family home in County Down, Northern Ireland.
During the first week of his life he was airlifted to Birmingham Children's Hospital where he was cared for by the specialist cardiac team for a number of months when he underwent several surgical procedures. Ollie has recently returned to the Royal Belfast Hospital for Sick Children after his condition stabilised.
Despite this positive news, Ollie needs a heart transplant and, like others across the country, parents Riona and Damien, are hoping for a life-saving call.
They're now encouraging all families to have conversations about becoming organ donors and, importantly, sharing their wishes with loved ones.
Dad, Damien, said registering their children was something they had never thought about or knew was possible.
He explained:
"Organ donation is so important, yet so many people don't talk about it with their loved ones. But from the point of view of a child needing a transplant, maybe people would consider it.
"We have been on that tightrope so many times. Ollie is all smiles, but underneath he has a fatal heart condition. That's always in the back of your mind and it means you can never settle."
Mum, Riona, added:
"We're hoping by sharing our story we can raise awareness by getting families talking about organ donation and get more people, both children and adults, signed to the register – not just for Ollie but for the many others who need a life-saving transplant."
Around 180 children across the UK are waiting for a life-saving organ transplant.
Children are waiting longer than before for transplants, with some sadly dying, as the number of young donors currently remains static. It's particularly difficult to find a donor for children and babies in need of a heart as a donor of a similar size is needed.
Those children who need an urgent heart transplant will wait on average two and a half times as long as adults on the urgent waiting list.
Mr Tim Jones, Consultant Paediatric Surgeon and Clinical Lead for Cardiac Surgery at Birmingham Children's Hospital, has been part of the team caring for Ollie. He echoed the family's message.
"For children like Ollie, transplantation is a final option, which means it's almost impossible to describe how important finding a suitable donor is.
"It's difficult to imagine being in the heart-breaking position of losing a child, along with considering what happens afterwards. In a time of such immense sadness there might be the option of offering the precious gift of life to others.
"Like, Damien and Riona, we'd urge all families to have what could be difficult conversations and make their decisions known now. For the vast majority, they will never find themselves in such a position but, in such circumstances, talking earlier and being clear on wishes could make such a difference.
"We know that parents who've agreed to donate their child's organs can gain great comfort from knowing they've helped to save other lives."
To find out more about organ donation, please visit their website www.organdonation.nhs.uk.
From spring 2020, all adults in England will be considered as having agreed to donate their own organs when they die unless they record a decision not to donate or are in one of the excluded groups.
Findings from NHS Blood and Transplant show that while eight in ten people in England say they would definitely want to donate or would consider donating, only just over a third of adults have told their partner or family that they want to donate their organs after they die. Regardless of the organ donation decision made, the most important thing is to make sure loved ones are aware of the decision.
To find out more about this change in law, please visit www.organdonation.nhs.uk/uk-laws/organ-donation-law-in-england.
Leading the way nationally and globally
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Decorate your dealership vehicle lot with informative and promotional flags, banners, balloons and more. We sell a wide variety of products which can be used to clearly display certain brands, price discounts or just simply to add some decoration to your lot. Get noticed with our range of car dealership decorations.
Fluorescent Antenna Pennants. A Variety Of Colors For You To Choose From.
Checkered Antenna Pennants. Pkg. 12 Per Package.


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9/23/13 - Harrisburg City, PA - Agate Street
Around 2320 hours Box 1-3 sent companies to the 2500 block of Agate Street for a reported house on fire. County advised responding units they were getting multiple calls. Captain 1 (Enterline) reported a column of smoke showing from the State Street Bridge and requested the balance of the first alarm to the scene. Wagon 3 and Tower 2 arrived on scene with a row of 2 story, brick dwellings and companies immediately went in service with heavy water lines. With fire already throughout 2 dwellings and exposing the rest of the row, Captain 1 requested the second alarm which brought county units to the scene and recalled city firefighters to man reserve apparatus. A step gun, hand lines, and master stream from Tower 1 poured water onto the structures creating heavy smoke conditions from multiple structures. The vacant buildings were previously inspected by the fire department and marked accordingly indicating no interior operations and utilities were secured. The main body of fire was knocked down in about 20 minutes with units operating an extended period of time making sure all pockets of fire were extinguished. Seven homes, two of which were occupied, were damaged in the fire. The fire is being investigated as suspicious in nature. No injuries were reported.


This Total Impact 4U Flight Case has come to us via a part exchange and comes with a 3 month warranty.
The item is in good to average cosmetic condition as there are some marks on the edges and some light, tiny marks/chips on the black finish. There is also one small scratch on the item.
This Total Impact 4U Flight Case does not include original packaging but has been professionally packaged to ensure safe delivery to your door.


At Desimone Jewelers we use the most innovative technology to create our one of a kind masterpieces.Our Laser Welding machine, The LaserStar, is the most advanced machine of it's kind. Lou DeSimone is a certified master welder and uses the LaserStar on a daily basis.It allows for perfect welds and when creating custom jewelry this is a must! LASER is an acronym for "Light Amplification by the Stimulated Emission of Radiation" which produces a sharp, focused light beam that melts a very small area of metal. The benefit of this technology is that very little heat is generated at the weld point, allowing users to easily weld .5 to 1.0mm (.020" - .040") away from the most complicated and intricate component parts without damaging heat sensitive materials. Operators hold parts in their hands while viewing the application through a stereo-microscope in the welding chamber. Internal cross-hair allows the operator to easily align and weld the parts at the correct location.
Crafford-LaserStar Technologies' development of the "free-moving" concept enable users to eliminate costly fixturing devices, benefit from pin-point accuracy, increase the range of assembly and repair applications, and minimize the potential hazards of heat damage. The resulting weld is considerably stronger than a traditional solder joint.
The Revo Milling System, operated by Will DeSimone is another amazing piece of technology used by the DeSimone Team. This high tech machine, run by AutoCad creates wax molds in small portion of the time it would take to create by hand. Another advantage is the Wax Mold is Millled perfectly the first time and allows for the mold to be casted much quicker. There are many benefits to using this machine but what's most important is, this machine helps to make our designs perfect!
Now you can have those items that you didn't think could be fixed, fixed with laser perfection!