text:
\section{Introduction}
The original Maldacena conjecture \cite{Maldacena:1997re} on the
AdS/CFT duality between ${\cal N}=4~SYM$ and string theory on
$AdS_5 \times S^5$ has been extended to a conjectured holographic
equivalence between four-dimensional (4D) strongly coupled QCD at
large $N_c$ and a 5D weakly interacting gauge theory coupled to
gravity on a 5D space $M_5$  not necessarily (even asymptotically)
$AdS_5$. New ingredients have to be introduced in order to comply
with the effective low energy description of a confining theory
such as QCD. One possibility is to modify the background geometry
of the dual gravity theory,  as in the Sakai-Sugimoto model
\cite{Sakai:2004cn}, or, as in the phenomenological Hard-Wall (HW)
models  of \cite{Erlich:2005qh} and \cite{Hirn:2005nr}, one can
cut-off the $AdS_5$ at a finite size, producing  confinement and
an infinite spectrum of Kaluza-Klein resonances. The two cited HW
models differ in the implementation of  spontaneous chiral
symmetry breaking ($\chi SB$), another fundamental ingredient of
low energy hadron dynamics. In \cite{Erlich:2005qh}, $\chi SB$ is
induced by a 5D scalar field, having the right properties to be
 the dual of the $\bar q q$ operator of QCD, whose
non-vanishing vacuum expectation value (VEV) is responsible for
$\chi SB$. In the approach of \cite{Hirn:2005nr}, $\chi SB$
follows from imposing appropriate infrared (IR) boundary
conditions.

The Soft-Wall (SW) model, proposed in \cite{Karch:2006pv}, is
instead a 5D holographic model in which the $AdS_5$ has no
cut-off, but confinement and an  infinite spectrum of Kaluza-Klein
resonances are due  a non-trivial dilaton background. The
phenomenological advantage of the SW model is the fact that,
contrary to the what happens for the KK states of the HW models,
the spectrum of resonances follows a linear Regge-trajectory,
which leads to a better agreement with the corresponding QCD
resonance spectrum in the intermediate energy regime.

In  any of  models above,  one boldly conjectures the
applicability of the following holographic recipe for the
calculation of correlation functions of the dual 4D theory.  For
every quantum operator ${\cal O}(x)$ in QCD, there exists a
corresponding 5D field $\phi(x,z)$, fixed  by the boundary
condition $\phi(x,0) \equiv \phi_0(x) $ on the ultraviolet (UV)
boundary of $M_5$. The generating functional of the correlation
functions of the 4D theory, is equal to  the 5D  action evaluated
\emph{on-shell}:
\begin{align}
\exp \left(W_4[\phi_0(x)]\right)\equiv\langle \mbox{exp}(i\int d^4
x \,\phi_0(x)\,{\cal O}(x))\rangle_{\rm QCD_4} =
\mbox{exp}\left(iS_{\rm M_5}[\phi_0(x)]\right) \label{holography}.
\end{align}

Varying $S_{\rm M_5}[\phi_0(x)]$   with respect to the sources
$\phi_0(x)$, and then setting them to zero, one gets the connected
$n$-point Green's functions of the 4D theory.

Having a 5D  holographic  description of low energy QCD makes
compelling the comparison with 4D low energy QCD models: VMD,
large $N_C$,  Chiral Quark Model, ... QCD at low energy is
described by chiral perturbation theory and thus the question
amounts to how good is the prediction for the Gasser-Leutwyler
coefficients, $L_i$. It is fair to say that all these models, 4D
and 5D, compare fairly well with the $L_i$'s phenomenological
values: a more specific question is if the presence of gluonic
condensate improves the agreement with phenomenology. Actually
this question has a very neat answer in the Chiral Quark Model
\cite{Manohar:1983md, Espriu:1989ff}: the agreement with
phenomenology improves \cite{Espriu:1989ff}. So we think that this
is a very well motivated question for holographic QCD.

In the holographic dictionary, 4D condensates,  \emph{i.e.}
non-vanishing VEV's of some  composite operators of the 4D theory,
are  related to the behavior near the UV boundary  of the dual 5D
fields \cite{Klebanov:1999tb}. The effect of a gluonic condensate
can be induced by a non-trivial dilaton field, or, equivalently,
by a suitable deformation of the original $AdS_5$ metric, as we
will discuss in detail.

We shall be concerned with the  correction to the two-point
functions  in presence of small deformations of the 5D metric
background. Here small means that we can treat them as
perturbations and we shall calculate the first order effect on the
two-point function of the theory. Although applicable also  to the
evaluation of  two-point functions of other 4D operators, we shall
consider only the case of conserved (flavor) currents, $ \langle
J_\nu (x) J_\mu (y) \rangle$. So, instead of  generic 5D actions,
we shall consider only the case of HW models with 5D vector gauge
fields, as they are the fields dual to these currents.

After a  a short review  of the holographic prescription in Sec.
2,  we compute   in Sec.3 the  first order correction to the
vector  two-point function and  show how to cast them in a very
compact form. With hindsight, we justify the result deriving it
directly from the holographic prescription. We  present also
another approach to evaluate directly  how the metric deformation
affects masses and  decay constants of the vector resonances,
which appear as intermediate states in the bound-state expansion
of the two-point function. We do it by developing  perturbation
theory for the resonance wave-functions. The resulting formulae
differ from those of usual  perturbation theory, since a
deformation of the metric affects also the scalar product of
wave-functions. The interest of this approach lies not only in the
fact that it provides a different route to the evaluation of the
perturbation of resonance masses and decay constants, which could
be alternatively extracted from the behavior of the perturbed
two-point around its singularities: in fact, since  it directly
provides the perturbed wave-functions, it allows, for instance,
the evaluation of the effects of the metric deformation on other
low energy coupling such as the two-pion-one-vector coupling. The
formal equivalence of the two methods is shown in the Appendix.

 In Sect.4, we consider first the case of an  HW model with flat 5D metric. It is
computationally simpler and we use it to illustrate our methods in the evaluation of the corrections
to resonance masses and decay constants: these results are original.
We then examine the
case of HW model where the extra-dimension is an $AdS_5$ slice
\cite{Erlich:2005qh}, \cite{Hirn:2005nr}. We
also consider some effects on  the axial vector resonance, for
which we adopt the description in \cite{Hirn:2005nr}. Moreover, perturbation
theory for the resonance wave-function allows us to evaluate also
the corrections to the two-pion-one-vector couplings. We want to
stress that our results represent a novelty, since no expressions
for the corrections to the vector decay constants and to the
two-pion-one-vector couplings have appeared in the literature even in
papers explicitly devoted to this effect in the $AdS_5$ HW model.
This was one of the motivations of our work and constitutes the core of our original analytic work.

 In
holographic models, the fact that all vector resonance wave
functions are Kaluza-Klein excitations of a 5D gauge field has the
consequence that they satisfy completeness relations from which
sum rules can be obtained. We address the issue of the saturation
of these sum rules by the first few resonances both in the
unperturbed and in the perturbed metric case. This relevant discussion is possible only after our novel calculation of the  corrections to the vector decay constants and to the
two-pion-one-vector couplings.

 In  Sec. 5   we present   numerical analyses and conclusions:  we evaluate  the effects  of the gluon condensate on low energy
parameters of QCD, in particular the Gasser-Leutwyler
coefficients, the pion decay constant and $\rho-$meson parameters.
Our discussion leads to a remarkable analogy between the  $AdS_5$ HW model and the
  Chiral Quark model.

\section{Two-point function of vector gauge fields in
Hard-Wall models}

We consider  a 5D  holographic Hard-Wall models  with the
extra-dimension $z$ restricted to a finite interval  $0\leq z\leq
z_0$ and  unperturbed metric written as a conformal factor times
the 5D flat metric as  follows:
\begin{align}
g_{MN}dx^Mdx^N = w(z)^2\left(\eta_{\mu \nu}dx^{\mu}dx^{\nu} -
dz^2\right) \label{metric} ,
\end{align}
where $ \eta_{\mu\nu} = {\rm Diag}\, (1,-1,-1,-1) $ and $ \mu, \nu
= (0,1,2,3) $, $ M, N = (0,1,2,3,z) $. The conformal factor $w(z)$
is left unspecified; in the case of $AdS_5$ it would be
$w_{AdS}(z)=1/z$. We consider $z=0$ as the UV-brane, where the
bulk fields have to coincide with the external sources of the 4D
theory and $z=z_0$ as the IR-brane where  suitable boundary
conditions are imposed. As we shall mainly consider vector gauge
fields, we impose  Neumann-type boundary conditions, which are
common to all the HW models mentioned in the introduction, where,
moreover, these b.c. force the absence of a massless resonance
mode.
 In the case of SW models, where  the fifth coordinate
 $z$ is no more restricted to a finite interval, the IR boundary
 condition  would be  replaced by a normalization condition.

 The quadratic part of the 5D action for the gauge fields is
given by:
\begin{equation}
\label{SAdS1} S_{M_5} = -\frac{1}{4g_5^2} \int d^4x \int_0^{z_0}
 dz\sqrt{g}\, g^{MN}g^{RS}F_{MR}F_{NS}
\end{equation}
with  linearized field strength $F_{MN}=\partial_M A_M-\partial_N
A_M$ and $g_5^2$ is a 5D coupling constant, which in $AdS_5$
models is usually fixed to $g_5^2=12\pi^2/N_c$, where $N_c$ is the
number of colors of the 4D dual gauge theory, by matching the
two-point function at large Euclidean momentum with the
perturbative result of QCD parton loop. Otherwise, $g_5^2$ could
be fixed by the requirement of obtaining the physical value of the
pion decay constant $f_{\pi}$.  Its actual value is irrelevant for
the general discussion of this and the next Sections, so we shall
momentarily suppress it.
 For the same reason, we  have also suppressed any flavor index of the
 5D  fields.
We shall restore the correct factors in Sec. 4.2, when we discuss
the 5D AdS slice, which is the playground of the HW models
\cite{Erlich:2005qh, Hirn:2005nr}. We shall need them in order to
make some numerical analysis of the effects of condensates and
compare with the existing literature.

It is convenient to work in $A_z = 0$ gauge. Then, the  5D gauge
fields $A_{\mu}(x,z)$   holographically correspond to  conserved
vector currents $J_{\mu}(x)$ of the dual 4D theory, and the
holographic formula will allow the calculation of the
 correlation function of two currents, in two steps.
First, one  solves the 5D equations of motion of the gauge field,
requiring  the solution to coincide on the UV boundary with the 4D
source $A_{\mu}(x)$ of the vector current. This is done by means
of  the bulk-to-boundary propagator.  Secondly,  the 5D action is
evaluated  on this solution and identified with the generating
functional of the 4D theory according to eq.(\ref{holography}).
Finally, by varying twice with respect to the boundary sources,
one obtains the holographic result for $<J_{\mu}(x)J_{\mu}(y)>$.

Using Fourier-transformed gauge fields, written as $\tilde
A_{\mu}(p,z) = \tilde{A}_{\mu}(p){\cal V}(p,z)$, where $\tilde
A_{\mu}(p,z)$ and $\tilde{A}_{\mu}(p)$ are the Fourier transforms
of $A_{\mu}(x,z)$ and the source $A_{\mu}(x)$ respectively, one
gets:
\begin{equation}
S_{{\rm AdS}_5}^{(2)} = -\int \frac{d^4p}{(2 \pi)^4}
\tilde{A}^{\mu}(p)\tilde{A}_{\mu}(p)\left( w(z)\partial_z
{\cal V}(p,z)\right)|_{z=\epsilon} \label{S2q}.
\end{equation}
 The
bulk-to-boundary propagator, ${\cal V}(p,z)$, satisfies the linearized
5D  equations of motion with 4D momentum, $p$:
\begin{equation}
\partial_z\left(w(z)\partial_z{\cal V}(p,z)\right)+p^2w(z){\cal V}(p,z)=0
\label{EOM}
\end{equation}
and the boundary conditions
\begin{equation}
{\cal V}(p,0)=1, \;\; \partial_z {\cal V}(p,z_0)=0\label{BC0}
\end{equation}
Varying (\ref{S2q}) with respect to the boundary sources gives the
scalar part of the two-point function:
\begin{equation}
\Sigma(p^2) = -\left(w(z)
\partial_z {\cal V}(p,z)\right)_{z = \epsilon \rightarrow 0} \label{sigma0}
\end{equation}
which is in general defined by
\begin{equation}
\int~d^4x \ e^{i p \cdot x}\langle J_{\mu}(x)J_{\nu}(0) \rangle =
\left( \eta_{\mu \nu} - \frac{{p_{\mu}p_{\nu}}}{p^2}
\right)\Sigma(p^2) \ .
\end{equation}
Let the functions $\varphi_n(z)$ form  the orthogonal basis of
normalizable eigenfunctions  satisfying
\begin{equation}
\partial_z\left(w(z)\partial_z\varphi_n(z)\right)+m_n^2w(z)\varphi_n(z)=0
\end{equation}
and the boundary conditions:
\begin{equation}
\varphi_n(0)=0, \;\; \partial_z \varphi_n(z_0)=0.\label{bceigen}
\end{equation}
They are normalized according to the scalar product
\begin{equation}
<\varphi_m,\varphi_n>\equiv
\int_0^{z_0}\varphi_m(z)\varphi_n(z)w(z)dz\label{scalarproduct}
\end{equation}
in order to get  from the 5D action a canonically normalized 4D
kinetic term for the Kaluza-Klein modes $A^{(n)}_{\mu}(x,z) =
A^{(n)}_{\mu}(x)\varphi_n(z)$. They also satisfy the completeness
relation
\begin{equation}
\sum_{n=1}^\infty w(z')\varphi_n(z)\varphi_n(z')=\delta(z-
z')\label{completeness}
\end{equation}
The two-point function $\Sigma(p^2)$ admits a bound-state
expansion given by
\begin{equation}
\Sigma(p^2)=\sum_{n=1}^\infty\frac{F^2_n}{p^2-m_n^2}\label{boundstates}
\end{equation}
with
\begin{equation}
F^2_n=\left(\left(w(z)
\partial_z \varphi_n(z)\right)_{z = \epsilon \rightarrow
0}\right)^2.
\end{equation}

Eq. (\ref{boundstates}) shows the pole structure of the two-point
function,  corresponding to an infinite sum over resonances which
 correspond  vector mesons with increasing masses $m_n$
and decay constants $F_n$.

It is also useful to introduce the Green function ${\cal G}(p, z,z')$,
which is the solution of the equation
\begin{equation}
\partial_z\left(w(z)\partial_z{\cal G}(p, z,z')\right)+p^2w(z){\cal G}(p,
z,z')=\delta(z-z')
\end{equation}
with the boundary conditions
\begin{equation}
{\cal G}(p, 0,z')=0, \;\; \partial_z {\cal G}(p, z_0,z')=0
\end{equation}
Then ${\cal G}(p, z,z')$ can be expanded as
\begin{equation}
{\cal G}(p,z,z')=\sum_{n=1}^\infty\frac{\varphi_n(z)\varphi_n(z')}{p^2-m_n^2}
\end{equation}
and the bulk-to-boundary propagator ${\cal V}(p,z)$ and the two-point
function $\Sigma(p^2)$ can be respectively written as:
\begin{equation}
{\cal V}(p,z)=\left(w(z')
\partial_{z'}{\cal G}(p,z,z')\right)_{z' = \epsilon \rightarrow 0}
\end{equation}
and
\begin{equation}
\Sigma(p^2)=\left(w(z)\partial_{z}w(z')
\partial_{z'}{\cal G}(p,z,z')\right)_{z =z' =\epsilon \rightarrow 0}
\end{equation}

\section{First order corrections  due to a metric perturbation}
\subsection{Correction to the two-point function}
Let us now suppose  the conformal factor in the metric
(\ref{metric}) to be  written as
\begin{equation}
w(z)=w_0(z)(1+h(z)),
\end{equation}
with $|h(z)|< 1$,  that can treated perturbatively, and $h(0)=0$,
so as  not to alter the behavior of the metric on the UV boundary
$z=0$. Then, writing the bulk-to-boundary as
 ${\cal V}(p,z)={\cal V}_0(p,z)+{\cal V}_1(p,z)$, one has that
the unperturbed ${\cal V}_0(p,z)$ obviously satisfies (\ref{EOM}) with
$w(z)$ replaced by $w_0(z)$ and the boundary conditions
(\ref{BC0}), while  the first order
 correction ${\cal V}_1(p,z)$  is a solution of
\begin{equation}
\partial_z\left(w_0(z)\partial_z{\cal V}_1(p,z)\right)+p^2w_0(z){\cal V}_1(p,z)=-w_0(z)D_1{\cal V}_0(p,z),
\end{equation}
where the differential operator $D_1$ is
\begin{equation}
D_1\equiv \partial_z h(z)\partial_z,\label{D1}
\end{equation}
 and satisfies homogeneous boundary conditions
\begin{equation}
{\cal V}_1(p,0)=0, \;\; \partial_z {\cal V}_1(p,z_0)=0.
\end{equation}
Let $f_1(p,z)$ and $f_2(p,z)$ be two independent solutions of the
differential equation for ${\cal V}_0(p,z)$, then, imposing the
boundary conditions (\ref{BC0}) it is easy to get
\begin{equation}
{\cal V}_0(p,z)=\frac{f'_2(p,z_0)f_1(p,z)-f'_1(p,z_0)f_2(p,z)}
{f'_2(p,z_0)f_1(p,\epsilon)-f'_1(p,z_0)f_2(p,\epsilon)},\label{V0}
\end{equation}
where prime denotes derivative with respect to $z$.  As we shall
see in the next Section, in the case of flat 5D slice ${\cal V}_0(p,z)$
is expressed in terms of trigonometric functions, while in  the
case of $AdS_5$ slice, $w_0(z)=1/z$ and one gets for ${\cal V}_0(p,z)$
the well known ratio of combinations of Bessel functions
\cite{Pomarol:1999ad, ArkaniHamed:2000ds}.

Standard methods, \emph{e.g.} the use of the Green function, lead
to the following expression for ${\cal V}_1(p,z)$
\begin{eqnarray}
{\cal V}_1(p,z)=\frac{f_2(p,\epsilon)f_1(p,z)-f_1(p,\epsilon)f_2(p,z)}
{f'_2(p,z_0)f_1(p,\epsilon)-f'_1(p,z_0)f_2(p,\epsilon)}\left({\cal I}_1
f'_2(z_0)-{\cal I}_2f'_1(z_0)\right) \nonumber\\
+\frac{1}{c}\displaystyle\int_\epsilon^{z}
\left(f_2(p,z)f_1(p,\zeta)-f_1(p,z)f_2(p,\zeta)\right)g(\zeta)w_0(\zeta)d\zeta
\label{V1}
\end{eqnarray}
where
\begin{equation}
g(z)\equiv -h'(z){\cal V}'_0(p,z)=-D_1{\cal V}_0(p,z),
\end{equation}
and
\begin{equation}
{\cal I}_i\equiv\frac{1}{c}\displaystyle\int_\epsilon^{z_0}f_i(p,\zeta)g(\zeta)w_0(\zeta)d\zeta,
\;\;i=1,2 \end{equation}
 and $c=w_0(z)W(z)$,
where $W(z)=f'_2(p,z)f_1(p,z)-f'_1(p,z)f_2(p,z)$ is the Wronskian
of the two functions $f_1(p,z)$ and $f_2(p,z)$.

Previous formulae have been written with an UV cut-off
$z=\epsilon>0$ to comply with UV divergences which my appear in
explicit examples, \emph{e.g.} $AdS_5$, due to the divergent
behavior of the conformal factor $w_0(z)$ as $z\rightarrow 0$.

From the expressions for ${\cal V}_0(p,z)$ and ${\cal V}_1(p,z)$ one obtains
the first order corrected   two point function
\begin{equation}
\Sigma(p^2)=\Sigma_0(p^2)+\Sigma_1(p^2)= -\left(w_0(z) \partial_z
{\cal V}_0(p,z)\right)_{z = \epsilon \rightarrow 0}-\left(w_0(z)
\partial_z {\cal V}_1(p,z)\right)_{z = \epsilon \rightarrow 0}
 \label{sigmacorr}
\end{equation}
where we have assumed that $w_0(z)h(z)\rightarrow 0$ for
$z\rightarrow 0$ to get rid of an additional term.

Explicit use of the Green function method to get the first order
correction to the two-point function due to metric perturbation in
the $AdS_5$ case was made in \cite{Erlich:2006hq}, where it is also
mentioned that the method   had been used by the authors of
\cite{Hirn:2005vk}. The expressions in \cite{Erlich:2006hq} are
readily obtained from (\ref{V1}) with $f_1(p,z)$ and $f_2(p,z)$
being, in that case, Bessel functions. We shall deal with the
$AdS_5$ HW model in Sec. 4.2.

Although (\ref{sigmacorr}) gives the solution to the problem of
finding the first order correction to the two-point function, due
to metric perturbations,  it rests on the choice of two solutions
$f_1(p,z)$ and $f_2(p,z)$ of eq.(\ref{EOM}). It would be more
satisfying if the first order correction could be written in term
of the unperturbed bulk-to-boundary  propagator (\ref{V0}) and of
the perturbation operator $D_1$ (\ref{D1}). Indeed, it is the
case. Using the expression (\ref{V0}) for ${\cal V}_0(p,z)$ and further
manipulating  the expression (\ref{sigmacorr}), the first order
correction to the two-point function can be written in the
following two equivalent forms obtained  one from the other by
integration by parts:
\begin{eqnarray}
\Sigma_1(p^2)=-<{\cal V}_0(p,z), D_1 {\cal V}_0(p,z)>_0=
\displaystyle\int_\epsilon^{z_0}w_0(z)h(z)\left(({\cal V}'_0(p,z))^2-p^2{\cal V}_0(p,z)^2\right)dz
\label{compact}
\end{eqnarray}
where $<f,g>_0$ denotes the scalar product (\ref{scalarproduct})
with respect to the unperturbed metric.

 Both expressions in (\ref{compact}) have interesting interpretations. The
 first shows  the  first order correction to the two-point
 function as the \lq\lq matrix element" of the perturbation operator (\ref{D1})
 on the unperturbed bulk-to-boundary propagator ${\cal V}_0(p,z)$, with
 the scalar product defined by the unperturbed metric.
  The second expression in (\ref{compact})
 shows that it is given by the 5D  action on-shell, but with
 the perturbed conformal factor of the metric $w_1(z)=w_0(z)h(z)$.

The last  result suggests the existence of  a simpler derivation.
In fact, as we shall see in a moment, it is a direct consequence
of the AdS/CFT recipe (\ref{holography}) of identifying the 4D
effective action with the 5D  action evaluated on-shell.

In momentum space, the $z$-dependent part of the 5D bulk action
on-shell is given by
\begin{equation}
S_{M_5}[{\cal V}]= -\displaystyle\int_{0}^{z_0}w(z) \left(\left(
{\cal V}'(p,z)\right)^2-p^2{\cal V}(p,z)^2\right)dz,
\end{equation}
where ${\cal V}(p,z)$ is the bulk-to-boundary propagator of 5D vectors
in the metric background with warp factor $w(z)$.  If we write
$w(z)=w_0(z)(1+h(z))$ and look for a perturbative solution of the
form ${\cal V}(p,z)={\cal V}_0(p,z)+{\cal V}_1(p,z)$ we have  at first order:
\begin{eqnarray}
&\hspace{-2cm}S_{M_5}[{\cal V}]=S_{M_5}^{(0)}[{\cal V}_0]+<\left[\Frac{\delta
S_{M_5}^{(0)}}{\delta
{\cal V}}\right]_{{\cal V}_0}, {\cal V}_1(p,z)>_0\label{Sexpa}\\
&-\displaystyle\int_{0}^{z_0}w_0(z)h(z) \left(\left(
{\cal V}'_0(p,z)\right)^2-p^2{\cal V}_0(p,z)^2\right)dz,\nonumber
\end{eqnarray}
On-shell,  the first term becomes a boundary term, i.e. the 0-th
order term in (\ref{sigmacorr}), while the second term,
proportional to the unperturbed equation of motion, vanishes and
we are left with just the last one, which is precisely what we got
in (\ref{compact}).

As we have already mentioned,  one can use (\ref{compact}), for
instance, to evaluate sub-leading corrections to the asymptotic
expansion of $\Sigma(Q^2)$ for large Euclidean momentum
$Q^2=-p^2$. More interestingly, one can use it in order to extract
corrections to the parameters, masses and decay constants,
appearing in the bound-state decomposition of the two-point
function, by studying the structure of pole singularities of the
two-point function.  In fact, the correction alters both the
position of the poles and the corresponding residues. There are,
however, some little subtleties in this procedure which we shall
illustrate in Sec.4.

\subsection{Corrections to the resonance wave functions}

The two-point function can be written as the bound-state
decomposition (\ref{boundstates}), where masses and decay
constants are  related to the mass eigenvalues and to the values,
weighted by the metric conformal factor, of the derivatives of the
wave functions $\varphi_n(z)$ at the origin. In presence of a
deformation of the metric,  wave functions are modified too. First
order corrections can be evaluated using perturbation theory for
eigenfunction. The derivation is straightforward, but leads to
final formulae  slightly different from those familiar from
quantum mechanics. The reason is that the deformation of the
metric does not only affect the wave-function equations, but also
the scalar product entering in the normalization condition
(\ref{scalarproduct}). This leads to a new diagonal term in the
first order correction of $\varphi_n(z)$, which is normally absent
in perturbation theory.

If  $\lambda^{(0)}_n\equiv -(m_n^{(0)})^2$ and
$\varphi_n^{(0)}(z)$ denote the eigenvalues and the
eigenfunctions of the unperturbed case $w(z)=w_0(z)$, then  the
first order corrections
$\lambda_n=\lambda^{(0)}_n+\lambda^{(1)}_n$ and $\varphi_{
n}(z)=\varphi_{ n}^{(0)}(z)+\varphi_{ n}^{(1)}(z)$ are given by
the following expressions
\begin{eqnarray}
\lambda^{(1)}_n&=&<\varphi_{ n}^{(0)}, D_{1}\; \varphi_{
n}^{(0)}>_0
\label{correlambda}\\
\varphi_{ n}^{(1)}(z)&=&-\frac{1}{2}< \varphi_{ n}^{(0)},
\varphi_{ n}^{(0)}h>_0 \varphi_{ n}^{(0)}(z)\nonumber\\
&+&\displaystyle\sum_{m\neq n}\frac{< \varphi_{ m}^{(0)}, D_{1}\;
\varphi_{ n}^{(0)}>_0}{\lambda_m^{(0)}-\lambda_n^{(0)}}
\varphi_{m}^{(0)}(z),\label{correeigen}
\end{eqnarray}
where $D_1$ is defined in (\ref{D1}). The first term in
(\ref{correeigen}) is the one due to the fact that the deformation
of the metric affects also the scalar product. In explicit
examples, it is numerically  relevant.

From (\ref{correlambda}) and (\ref{correeigen}) one gets the
 the corrections to masses and coupling constants:
\begin{eqnarray}
m_n&=&m_n^{(0)}\left(1-\Frac{1}{2}\Frac{<\varphi_{ n}^{(0)},
D_{1}\;
\varphi_{ n}^{(0)}>_0}{(m_n^{(0)})^2}\right)\label{pertmass}\\
F_n &=&F_n^{(0)}\left(1+\Frac{(\varphi_{ n}^{(1)})'(z)}{(\varphi_{
n}^{(0)})'(z)}\right)_{z=\epsilon\rightarrow 0}\label{pertdecay}
\end{eqnarray}

It may seem not evident  that the corrections to masses
 and decay constants obtained here generate,
through the bound-state decomposition (\ref{boundstates}), the
same correction to the two-point (\ref{compact}) deduced in the
previous Section, but this is the case, as explicitly shown in the
Appendix.

\section{Perturbing  the metric of 5D Hard-Wall models}

\subsection{5D flat Hard Wall model}

The case of gauge vector field in a flat extra-dimension slice $0<
z< z_0$, \emph{i.e.} with warp factor $w(z)=1$,  has a nice
interpretation as the continuum limit of deconstruction models
\cite{Son:2003et}. Its simplicity allows for analytic calculations
of the first order perturbative corrections to masses and coupling
constants due to a metric perturbation. We report here the
corresponding formulae. Moreover, eq.(\ref{compact}) can be used
to extract the sub-leading terms in the expansion of the two-point
function for large Euclidean momenta $p^2=-Q^2$ which is also an
important effect of the metric deformation establishing an
holographic link between metric deformations of the 5D theory and
non-vanishing condensates of the 4D dual theory. In fact, in the
4D gauge gauge theory, sub-leading terms of the two-point function
in the deep Euclidean kinematical region are produced by
non-perturbative effects of condensates \cite{Shifman:1978bx}.

Turning to the case of flat 5D extra-dimension on a finite
interval, the normalized wave-functions are
\begin{equation}
\varphi_n^{(0)}(z)=\sqrt{\Frac{2}{z_0}}\sin\left(\kappa_{0,n}\frac{z}{z_0}
\right),\;\;{\rm with} \;\;\kappa_{0,n}=\frac{2n+1}{2}\pi,
\end{equation}
with eigenvalues
\begin{equation}
\lambda_n^{(0)}=-(m_n^{(0)})^2=-\left(\frac{\kappa_{0,n}}{z_0}\right)^2.
\end{equation}
The bulk-to-boundary propagator is the solution of (\ref{EOM}),
with  $w_0(z)=1$ and satisfying the boundary conditions
(\ref{BC0}):
\begin{equation}
{\cal V}_0(z,p)=\left(\frac{\sin(pz_0)}{\cos(pz_0)}\sin(pz)+\cos(p
z)\right)
\end{equation}
Using (\ref{sigma0}), the two-point function can be written
\begin{equation}
\Sigma_0(p^2)=-{\cal V}'_0(0,p)=-p\tan(p z_0).
\end{equation}
The bulk-to-boundary  can also be expanded in series of
eigenfunctions as follows
\begin{equation}
{\cal V}_0(z,p)=-\frac{2}{z_0^2}\sum_{n=0}^\infty \,
\Frac{\kappa_{0,n}}{p^2-\left(\Frac{\kappa_{0,n}}{z_0}
\right)^2}\,\sin\left(\kappa_{0,n}\Frac{z}{z_0}\right).
\end{equation}
Then, the two point function admits the bound-states decomposition
\begin{equation}
\Sigma_0 (p^2)=\frac{2}{z_0^3}\sum_{n=0}^\infty
\Frac{\kappa_{0,n}^2}{p^2-\left(\Frac{\kappa_{0,n}}{z_0}\right)^2}.
\end{equation}
and one  can read the values of the unperturbed masses of the
vector resonances from the poles and their coupling constants from
the corresponding residues at the pole:
\begin{equation}
m_n^{(0)}=\Frac{\kappa_{0,n}}{z_0},
\;\;F_n^{(0)}=\sqrt{\Frac{2}{z_0^3}}\,\kappa_{0,n}.
\end{equation}
 We now use  the results of Secs. 3,  to evaluate the
the correction to the two-point function and the effects on
resonance masses and decay constants, due to a  deformation of the
flat metric of the form
\begin{equation}
 h(z)=\eta\,(z/z_0)^4.\label{pertz4}
\end{equation}
 As we
shall see in the  in the next section, this choice of the
deformation is suggested by the fact that,  in the AdS/CFT
correspondence, it would reproduce the  effects of a  4D operator
of conformal dimension four such as the gluon condensate. We
obviously assume in our perturbative approach  the dimensionless
constant $\eta$ to be  small, and consider only corrections of
first order in $\eta$. Notice that $h(0)=0$.

Inserting ${\cal V}_0(z,p)$ above  into the expression (\ref{compact}),
the resulting integrals can be easily evaluated and one gets for
the first order correction of the two-point function
\begin{equation}
\Sigma_1(p^2)=-\Frac{\eta}{4z_0^4p^3}\left(\Frac{2pz_0(2p^2z_0^2-3)}{\cos^2(pz_0)}+
 6\tan(pz_0)\right)\label{Sigma1flat}
\end{equation}
It is convenient to use the identity $a+\eta \;b\sim a/(1-\eta\;
b/a)$, valid to first order in $\eta$, and rewrite  1st order
corrected two-point function in the form
\begin{equation}
\Sigma_0(p^2)+\Sigma_1(p^2)\sim
-\Frac{p\sin(pz_0)}{\cos(pz_0)-\eta\Frac{1}{4z_0^4p^4}
\left(\Frac{2pz_0(2p^2z_0^2-3)}{\sin(pz_0)}+
 6\cos(pz_0)\right)}\label{Sigma0and1flat}
\end{equation}
Perturbed masses and decay constants can be extracted from
(\ref{Sigma0and1flat}) by looking for the zeroes of the
denominator, \emph{i.e.} the positions of the poles, and then
extracting  the corresponding residues.

Explicitly, once that the two-point function has been put in the
form of a ratio of two functions $\Sigma(p^2)={\cal N}(p)/{\cal D}(p)$ as
above, residues are evaluated by expanding, around the poles,
\emph{i.e.} around the zeros $m_n$ of the denominator ${\cal D}(q)$,
and thus
\begin{equation}
F_n^2=2 m_n \Frac  {{\cal N}(m_n)}{{\cal D}'(m_n)}.
\end{equation}
 In a perturbative approach,
${\cal D}(p)={\cal D}_0(p)+\eta \,{\cal D}_1(p)$ and,  to first order in $\eta$,
the following expressions for masses and residues  are obtained
\begin{equation}
m_n=m_n^{(0)}+\eta\,
m_n^{(1)}=m_n^{(0)}\left(1-\eta\,\Frac{{\cal D}_1(m_n^{(0)})}{{\cal D}'_0(m_n^{(0)})\,m_n^{(0)}}\right)
\end{equation}
and
\begin{equation}
F_n=F_n^{(0)}\left(1+\Frac{\eta}{2}\left(\Frac{m_n^{(1)}}{m_n^{(0)}}+
\Frac{{\cal N}'_0(m_n^{(0)})}{{\cal N}_0(m_n^{(0)})}m_n^{(1)}-
\left(\Frac{{\cal D}_0'(m_n^{(0)}){\cal D}_1'(m_n^{(0)})-{\cal D}_0''(m_n^{(0)}){\cal D}_1(m_n^{(0)})}
{{\cal D}_0'(m_n^{(0)})^2} \right)\right)\right)
\end{equation}

In the case of 5D flat  slice, previous formulae boil down to:
\begin{equation}
m_n=m_n^{(0)}\left(1-\frac{\eta}{2}\frac{2\kappa_{0,n}^2-3}{\kappa_{0,n}^4}\right)
\end{equation}
\vskip .05cm
\begin{equation}
F_n=F_n^{(0)}\left(1-\frac{\eta}{2}\frac{1}{\kappa_{0,n}^2}\right)
\end{equation}
Corrections to masses and decay constants can also be obtained
using the perturbation theory for the bound-state eigenfunctions
in Sec.3. It is not difficult to check that the  corrections to
the masses coincide with those obtained from perturbation formula
(\ref{correlambda}) for the eigenvalues. We have also checked that
the corrections to the decay constants coincides numerically with
the ones obtained using perturbation formulae (\ref{correeigen})
and taking first derivatives of the perturbed eigenfunctions
evaluated at $z=0$ as in (\ref{pertdecay}).

Finally, using (\ref{Sigma1flat}) with Euclidean momentum
$p^2=-Q^2$ one has in the large-$Q^2$ that the two-point function
receives a subleading contribution from the metric deformation
given by
\begin{equation}
\Sigma(-Q^2)\approx Q\left(1+\Frac{3\eta}{2}\Frac{1}{z_0^4
Q^4}\right).
\end{equation}
\subsection{AdS$_5$  Hard-Wall model}

As explained in the Introduction, the case of gauge vector field
in a 5D AdS slice $0< z< z_0$, i.e. with warp factor $w(z)=1/z$,
is the one originally proposed as  holographic dual of QCD. There,
the AdS/CFT dictionary, relating 5D fields  and 4D operators  is
supposed to hold and it  gives a good understanding of what 4D
effect one is actually evaluating. Referring to the original
papers for the details, we limit ourself to repeat the same sort
of calculations we did in the previous section for the 5D flat
space, and concentrate again on the effects of a perturbation of
the form (\ref{pertz4}), whose link to 4D condensates can be given
a much solid ground in the AdS case \cite{Hirn:2005vk}.

Formulae become somewhat more complicated due to the appearance of
Bessel functions both in the expression of the unperturbed
bulk-to-boundary propagator and in the eigenfunctions. First order
correction is obtained in terms of  integrals containing  Bessel
functions, which, in principle, could be written in terms of
higher transcendental functions, but for practical purposes, can
be evaluated numerically, once the value of dimensional parameter
$z_0$, giving the size of the 5D slice and the 4D energy scale at
which 4D conformal invariance is broken, is fixed.

In the AdS case the  eigenfunction of the $n$-th bound-state is
given by
\begin{equation}
\varphi_n^{(0)}(z)=\Frac{\sqrt{2}}{|J_1\left(\gamma_{0,n}\right)|}\Frac{z}{z_0}
J_1(\gamma_{0,n}\Frac{z}{z_0}) ,\label{eigenAdS}
\end{equation}
with $\gamma_{0,n}$ being the $n$-th zero of the Bessel function
of order zero, \emph{i.e.} $J_0(\gamma_{0,n})=0$.

The  eigenvalues are
\begin{equation}
\lambda_n^{(0)}=-(m_n^{(0)})^2=-\left(\Frac{\gamma_{0,n}}{z_0}\right)^2.
\end{equation}
The bulk-to-boundary propagator is the solution of (\ref{EOM}),
with $w_0(z)=1/z$ and satisfying the boundary conditions
(\ref{BC0}) is:
\begin{equation}
{\cal V}_0(p,z)=\Frac{z}{\epsilon}\Frac{Y_0(p z_0)J_1(p z)-J_0(p
z_0)Y_1(p z)} {Y_0(p z_0)J_1(p\epsilon)-J_0(p z_0)Y_1(p
\epsilon)},\label{V0AdS}
\end{equation}
There is a divergence as the UV cut-off $\epsilon$ is sent to
zero. A singular behavior is shown by the  two-point function:
\begin{equation}
\Sigma_0(p^2)\approx p^2 \log
\left(\Frac{p\epsilon}{2}\right)-\Frac{\pi p^2}{2} \Frac{Y_0( p
z_0)}{J_0(p z_0)}.\label{Sigma0AdS}
\end{equation}
The logarithmic behavior in the UV is actually welcome since it is
identified with the contribution of the parton loop at high
momenta. The finite part in (\ref{Sigma0AdS}) contains the
information on the resonance poles, which,  are located at the
zeroes of $J_0(p z_0)$. From the residues  one gets the decay
constants:
\begin{equation}
F_n^{(0) 2}=\Frac{\pi\gamma_{0,n}^3}{z_0^4}
\Frac{Y_0(\gamma_{0,n})}{J_1(\gamma_{0,n})}.\label{res0AdS}
\end{equation}
We can  explicitly show the bound-state expansion of the two-point
function by means of the Kneser-Sommerfeld formula
\cite{Grigoryan:2007vg} (valid for $ z \leq z_0 $)
\begin{equation}
\label{KS} \frac{Y_0(p z_0)J_0(p z) - J_0(p z_0)Y_0(p
z)}{J_0(pz_0)} \nonumber  = -\frac4\pi  \, \sum_{n = 1}^{\infty}
\frac{J_0(\gamma_{0,n}z/{z_0})} {[J_1(\gamma_{0,n})]^2(p^2z^2_0 -
\gamma^2_{0,n})} \ ,
\end{equation}
The limit  $ z \rightarrow 0 $  gives  a logarithmically divergent
series, which however correctly describes the behavior around the
poles:
\begin{equation}
\label{vecor} \Sigma_0(p^2) = \frac{2 p^2}{z^2_0}\sum_{n =
1}^{\infty} \Frac{[J_1(\gamma_{0,n})]^{-2}}{p^2
-\left(\Frac{\gamma_{0,n}}{z_0}\right)^2} ,
\end{equation}

From this expression one  reads the values of the (unperturbed)
masses of the vector  resonances from the poles and their coupling
constants from the corresponding residues at the pole. The
agreement with the value (\ref{res0AdS}), is provided by the
identity $\gamma_{0,n}J_1(\gamma_{0,n})Y_0(\gamma_{0,n})=2/\pi$.

 We now use  the results of Sec.3,  to calculate the effects
 of the   perturbation  of
a metric of the form (\ref{pertz4}) on the two-point function, and
on resonance masses and decay constants. This choice of the
deformation is suggested by the fact that,  in the AdS/CFT
correspondence, it would reproduce the effects of a  4D operator
of conformal dimension four such as the gluon condensate. We again
assume in our perturbative approach the dimensionless constant
$\eta$ to be a small, and consider only first order corrections in
$\eta$.

Inserting ${\cal V}_0(z,p)$ of eq.(\ref{V0AdS}) into the expression
(\ref{compact}) one obtains integrals containing Bessel functions
and, taking the limit $\epsilon\rightarrow 0$, the first order
correction of the two-point function  can be written
\begin{equation}
\Sigma_1(p^2)=\eta\Frac{\pi^2}{p^2z_0^4}\left[\Frac{Y_0(p
z_0)^2}{J_0(p z_0)^2}{\cal I}_{J_1J_0}(p z_0)-\Frac{Y_0(p z_0)}{J_0(p
z_0)}({\cal I}_{J_1Y_0}(p z_0)+{\cal I}_{J_0Y_1}(p z_0))+{\cal I}_{Y_0Y_1}(p
z_0)\right],\label{Sigma1AdS}
\end{equation}
where we defined the following integral of Bessel functions
$Z_n(x)$:
\begin{equation}
{\cal I}_{Z_mZ_n}(x)\equiv\int_0^xZ_m(y)\,Z_n(y)\,y^4\,dy.
\end{equation}
Proceeding as in the flat case, we can extract perturbed masses
and decay constants from poles and  residues of the perturbed
two-point function. The final results are
\begin{equation}
m_n=m_n^{(0)}+\eta\, m_n^{(1)}=m_n^{(0)}\left(1-\eta\Frac{4
{\cal I}_{J_1J_0}(\gamma_{0,n})}{\gamma_{0,n}^6 J_1(\gamma_{0,n})^2
}\right)\label{masscorrAdS}
\end{equation}
\vskip .05cm
\begin{eqnarray}
&F_n^2 =F_n^{(0)^2}\left[1+\eta\left(-\Frac{\pi}{\gamma_{0,n}^4}(
{\cal I}_{J_0Y_1}(\gamma_{0,n})+{\cal I}_{Y_0J_1}(\gamma_{0,n}))
-\Frac{1}{2}\Frac{m_n^{(1)}}{m_n^{(0)}}\right.\right.\label{decaycorrAdS}\\
&\left.\left.
+\Frac{\pi}{\gamma_{0,n}^4}\Frac{Y_1(\gamma_{0,n})}{J_1(\gamma_{0,n})}
{\cal I}_{J_0J_1}(\gamma_{0,n})
+\Frac{1}{4}\Frac{m_n^{(1)}}{m_n^{(0)}}\gamma_{0,n}\Frac{J_2(\gamma_{0,n})}{J_1(\gamma_{0,n})}
-\Frac{1}{2}\Frac{m_n^{(1)}}{m_n^{(0)}}\gamma_{0,n}\Frac{Y_1(\gamma_{0,n})}{Y_0(\gamma_{0,n})}
\right) \right]\nonumber
\end{eqnarray}
As an independent check, we can apply the  perturbation theory of
Sec.3.2 to the AdS bound-states (\ref{eigenAdS}). Using
eq.(\ref{pertmass})  one recovers the same expression
(\ref{masscorrAdS}) for  mass corrections; through the
perturbation series (\ref{correeigen})
 and from eq.(\ref{pertdecay}), one finds
numerical agreement with the values of the corrections to the
decay constants one gets from (\ref{decaycorrAdS}).

In Table.1, these corrections are shown for the first few vector
and axial vector resonances. In order to get these numerical
values,  we had to restore in our formulae the 5D coupling
constant $g_5^2$ and fix it to $12\pi^2/N_c$ by matching the
logarithmic  term of the two-point function to the value of the
QCD parton loop. We have also posed $z_0=3.1 \times 10^{-3}\,{\rm
MeV}^{-1}$ in order to have that the mass of the first vector
resonance coincide with the value of the $\rho$ meson mass of
about $776\, \rm{MeV}$.

The values for the axial-vectors have been obtained working in the
HW model of  \cite{Hirn:2005nr}, where they are 5D gauge fields
satisfying the same 5D field equations  as the vectors, but
different IR boundary conditions at $z_0$, {\emph{i.e.}
$\varphi_n^A(z_0)=0$, in order to break chiral symmetry. This
leads to different unperturbed masses, decay constant,
wave-function and bulk-to-boundary propagator; however our
perturbative treatment of the correction due to the metric
deformation (\ref{pertz4}) can be done analogously to what we have
explicitly  illustrated in the case of vectors.
\begin{table}[tpb]%
\label{deltaMandF}\centering %
\begin{tabular}{|c|c|c|c|c|}
\hline
& & &  &  \\
Resonance  & $m_n^{(0)}$ (MeV) & $(\delta m_n/m_n^{(0)})/\eta$ &
$10^{-3}\;F_n^{(0)}$ (MeV$^2$)
 & $(\delta F_n/F_n^{(0)})/\eta$\\
& & &  &  \\
\hline\hline
$\rho^1$  & 776 & -0.15 & 109 & -0.23\\
\hline
$\rho^2$  & 1781 & -0.041 & 380 & -0.044\\
\hline
$\rho^3$  & 2792 & -0.017 & 747 & -0.018\\
\hline \hline
$a_1^1$  & 1236 & 0.045 & 223 & 0.049\\
\hline
$a_1^2$  & 2263 & 0.013 & 548 & 0.011\\
\hline
$a_1^3$  & 3282 & 0.006 & 955 & 0.009\\
\hline
\end{tabular}
{\caption{\small Corrections to vector and axial vector masses and
decay constants due to the  deformation $h(z)=\eta\, (z/z_0)^4$ of
the metric of a 5D AdS slice.} }
\end{table}

The mass corrections of the first vector and axial vector
resonance, \emph{i.e.} the $\rho$ and the $A_1$, agree with those
reported in Ref.\cite{Hirn:2005vk} once our parameter $\eta$ is
written as in terms of their $o_4$, both related to the gluon
condensate as follows:
\begin{equation}
\eta=\Frac{9\pi^2}{4} \ o_4 = \Frac{3\pi}{16 N_c}z_0^4\,\langle
\alpha_s  G_{\mu\nu}G^{\mu\nu}\rangle. \label{frometatoo4}
\end{equation}}
 We also notice that $\eta$ is almost equal to the
parameter $g$ of Ref.\cite{Bijnens:1992uz}, which expresses the
gluon condensate effects in the Extended Nambu-Jona Lasinio model
of low energy hadron phenomenology: one has $\eta=(9/8)(M_Q z_0)^4
g$, where $M_Q$ is the constituent quark mass of that model.

The relation between  $\eta$  and  the gluon condensate can be
established  by the fact that in presence of the metric
perturbation a non-vanishing sub-leading inverse power correction
appears in the two-point function at large Euclidean momentum.
Explicitly in this limit and using $p^2=-Q^2$  in (\ref{Sigma1AdS})   the two-point
function receives a sub-leading contribution from the metric
deformation  given by
\begin{equation}
\Sigma(-Q^2)\approx -\Frac{N_c}{24\pi^2}Q^2\left[\log
\left(\Frac{Q\epsilon}{2}\right)^2-\Frac{16\,\eta}{3 z_0^4
Q^4}\right].\label{2ptHW}
\end{equation}
The coefficient of the sub-leading term can then be  matched
with the one produced in QCD by a non-vanishing  gluon condensate:
\begin{equation}
  \Sigma \left( - Q^2 \right) = - \frac{N_c}{24 \pi^2} Q^2
  \left[\log\left(\frac{Q^2}{\mu^2}\right)
 - \frac{(\pi \alpha_s  \left\langle G_{\mu \nu}
  G^{\mu \nu} \right\rangle/N_c)}{Q^4}+ \mathcal{O} \left( \frac{1}{Q^6} \right) \right],
  \label{QCD2pointsub}
\end{equation}
From (\ref{2ptHW}) and (\ref{QCD2pointsub}) it follows that $\eta$
has the same sign of the gluon condensate, which experimentally
results to be positive (see, for instance, the up-to-date
discussion in \cite{Narison:2009vy} and references therein). The
use of the value, $\langle \alpha_s  G_{\mu\nu}G^{\mu\nu}\rangle
\sim 6.8\cdot 10^{-2}\ {\rm GeV}^4 $, consistent with the present
determination (see \cite{Espriu:1989ff}) for the gluon condensate,
leads to   $\eta\sim 1.26$ in eq.(\ref{frometatoo4}); however the
proper value to use for holographic models should only come from a
fit of the full holographic predictions to the data but,
anticipating the phenomenological discussion in the next
Section, we can say that it is close to a reasonable value.

 It should be noticed that, in the HW model
of \cite{Hirn:2005nr}, where chiral symmetry breaking is imposed
through different boundary conditions for vector and  axial vector
gauge fields, the deformation of the metric produces the same
sub-leading term at large Euclidean momentum both for the vector
and the axial-vector two-point functions. The effects of different
IR b.c. is indeed exponentially suppressed at large Euclidean
momentum. This is in agreement with the non-chiral nature of the
gluon condensate in QCD.

Let us  come to a consistency check of our results. In any HW
model, completeness property of the resonance wave-functions give
rise to sum rules involving vector and axial vector meson masses
and decay constant and other parameters of the low energy chiral
lagrangian. Explicitly, one has the following sum rules
\cite{Hirn:2005nr},\cite{Hirn:2007bb}

\begin{eqnarray}
\sum_n  g_{n}^2&=& 8\, L_1\label{Sum1}
\\
\sum_n f_{n} g_{n}&=&2L_9\label{Sum2}
\\
\sum_n  g_{n}^2 m_{n}^2&=&\Frac{f_{\pi}^2}{3}\label{Sum3},
\\
\sum_n f_{n} g_{n}m_{n}^2&=&f_{\pi}^2\label{Sum4},
\end{eqnarray}
where $L_1$ and $L_9$ are two of the Gasser-Leutwyler coefficients
of the $O(p^4)$ terms of the chiral Lagrangian \footnote{The HW
models of \cite{Hirn:2005nr} automatically enforce, in the
$SU_L(3)\times SU_R(3)$ case, the relations $L_3=6\,L_1=3\,L_2$
and $L_9+L_{10}=0$}, $f_{\pi}=92.3$ MeV is the pion decay
constant, $f_{n}=F_n/m_n^2$ and $g_{n}$ are the pion decay
constants and two-pion-one-vector couplings as they are usually
written in the Chiral Lagrangian formulation of spin-1 vector
field sector,
\begin{eqnarray}
{\cal L}_V \, &=& \, - \, \Frac{f_V}{2 \sqrt{2}} \langle \, V_{\mu
\nu} \, f_{+}^{\mu \nu} \, \rangle \, - \, \Frac{i g_V}{2
\sqrt{2}} \, \langle V_{\mu \nu} \, [ \, u^{\mu} \, , \, u^{\nu}
\, ] \rangle + \ldots \label{spinone}
\end{eqnarray}
where $V_{\mu\nu}$ is the $V_\mu$ field strength, $u_{\mu} \, = \,
i \, u^{\dagger} \, D_{\mu} \, U \, u^{\dagger}$, with $U=u^2$
being the chiral field and $f_{+}^{\mu \nu} \, = \, u \, F_L^{\mu
\nu} \, u^{\dagger} \, +\, u^{\dagger} \, F_R^{\mu \nu} \, u $,
with $F_{L,R}^{\mu \nu}$ external sources. For the lowest
resonance,  the $\rho$, one would have $f_V\equiv f_\rho\equiv
f_1$ and $g_V\equiv g_\rho\equiv g_1$. Our notations in
Eq.(\ref{spinone}) agree with those of Ref.\cite{Hirn:2005nr}.

Notice that in writing the sum rules (\ref{Sum1}-\ref{Sum4}) we
are assuming $\chi$SB and  the pion wave function of
 in the 5D holographic model of QCD of Ref.\cite{Hirn:2005nr}.  A
different approach is followed in Ref.\cite{Erlich:2005qh}.

The physical 4D meaning of the sum rules above is that they ensure
soft high energy behavior of some quantities relevant for physical
processes  such as the pion elastic scattering amplitude
(\ref{Sum1}), the vector form factor (\ref{Sum2},\ref{Sum4}). The
sum rule (\ref{Sum3}) is the analog of the KSFR for the infinite
tower of resonances one obtains in these HW model. Note that in
models with vector meson dominance, corresponding to just the
first resonance, the natural value KSFR is a  factor \lq2\rq   \
instead of  \lq 3\rq \  which is what one gets instead in these HW
models.

In the expression of the $L_i$ there enters the pion wave function
$\alpha(z)$ which, for a metric of the form (\ref{metric}) is
solution of the equation
\begin{equation}
\partial_z\left(w(z)\partial_z\alpha(z)
\right)=0,
\end{equation}
with boundary conditions
\begin{equation}
\alpha(0)=1,\;\; \alpha(z_0)=0.
\end{equation}
In presence of a deformation of the original AdS metric, the
function $\alpha(z)$ will also get corrections. At first order in
$\eta$ one has
\begin{equation}
\alpha(z)=1-\Frac{z^2}{z_0^2}-\Frac{\eta}{3}\left(\Frac{z^2}{z_0^2}-
\Frac{z^6}{z_0^6}\right)\label{pionwave}
\end{equation}

The  value of  $g_n$ is obtained as a 5D  overlapping integral
containing $\alpha(z)$ and the vector wave-function
$\varphi_n(z)$, \emph{i.e}:
\begin{equation}
g_n=\frac{\sqrt{2}}{2g_5}\int_0^{z_0}w(z)(1-\alpha(z)^2)\varphi_n(z)dz.
\end{equation}

As we may apply the perturbation formulae (\ref{correeigen}) to
get the correction to the vector wave functions $\phi_n(z)$, we
are able to evaluate numerically the corrections to the couplings
$g_n$'s. Their values for the firs few resonances are shown  in
the table and the plot of Fig. 1.
\begin{figure}[h]
\begin{minipage}[b]{6cm}
{
 \hspace{1cm}\vspace{.5cm}
\begin{tabular}{|c|c|c|}
\hline
 &  &  \\
Resonance  & $g_{n}^{(0)}$  & $(\delta
g_{n}/g_{n}^{(0)})/\eta$ \\
 &  &  \\
\hline\hline
$\rho^1$  & 0.0538 & 0.34 \\
\hline
$\rho^2$  & -0.0019 & -2.31 \\
\hline
$\rho^3$  & 0.0003 & -3.72\\
\hline
\end{tabular}
}
\end{minipage}
\hskip3cm
\begin{minipage}[b]{8cm}
\includegraphics[width=7cm,height=4cm]{deltagVsugV}
 \end{minipage}
{\caption{\small Corrections to the one-vector-two-pions couplings
for the first few vector resonances  due to the deformation
$h(z)=\eta\, (z/z_0)^4$ of the metric of a 5D AdS slice.} }
 \end{figure}

We now report the expressions of the Gasser-Leutwyler coefficients
of the chiral lagrangian obtained in the AdS HW model
\cite{Hirn:2005nr}, at first order in the metric perturbation of
eq.(\ref{pertz4}). Using $1/g_5^2=N_c/12\pi^2$, one has:

\begin{eqnarray}
f_\pi&=&\sqrt{\Frac{1}{g_5^2}\int_0^{z_0}w(z)(\alpha'(z))^2 dz}=
\Frac{\sqrt{N_c}}{\sqrt{6}\pi
z_0}\left(1+\Frac{1}{6}\eta\right)\label{fpipert}
\\
L_1&=&\Frac{1}{32 g_5^2}\int_0^{z_0}w(z)(1-\alpha(z)^2
)^2dz=\Frac{11
N_c}{9216\pi^2}\left(1+\Frac{2}{3}\eta\right)\label{L1pert}
\\
L_9&=&\Frac{1}{4 g_5^2}\int_0^{z_0}w(z)(1-\alpha(z)^2)dz=\Frac{
N_c}{64\pi^2}\left(1+\Frac{25}{54}\eta\right)\label{L9pert}
\\
L_2&=&2L_1,\;L_3=-6L_1,\;L_{10}=-L_9. \label{L10pert}
\end{eqnarray}

Notice that the smallness of the numerical coefficients in front
of $\eta$ in (\ref{fpipert}-\ref{L9pert}) makes these corrections
perturbative even for $\eta\sim O(1)$. When the perturbation
parameter $\eta$ is written in terms of $o_4$, following
(\ref{frometatoo4}), our  expressions (\ref{fpipert}-\ref{L9pert})
coincide with the ones in Ref.\cite{Hirn:2005vk}.

An important result of holographic models is that the sum rules
(\ref{Sum1}-\ref{Sum4}) still hold with the perturbed values of
the $L_i$'s, of $f_\pi$ and of the single resonances masses $m_n$,
decay constants $f_n$ and coupling to two pions $g_n$. As such, we
can also  address an issue similar to the one raised in
Ref.\cite{Chivukula:2004kg} in the context of deconstruction
models with unperturbed metrics. The authors  wondered how fast is
the convergence of the sum over the resonances  in the first and
second Weinberg sum rule (WSR): these sums, compared to eqs.
(\ref{Sum1}-\ref{Sum4})   involve also axial vectors. Their
findings was that actually a large number of terms  was needed;
for instance in the case of deconstruction with \lq\lq $\cosh z
$\rq\rq background, the one leading to AdS in the continuum limit,
and twenty lattice sites, practically all resonances had to be
considered in order to get a small deviation in  the WSR.

Authors of Ref. \cite{Becciolini:2009fu} address a  different
question in  QCD  deconstruction: how many sites are needed to
have a  a good phenomenological picture. They find that already a
model with three or four sites reproduces the relevant features of
holographic models of QCD.

These two issues led us to investigate how good is the convergence
of the sum rules in a perturbed metrics  in Eqs.
(\ref{Sum1}-\ref{Sum4}), i.e. how many resonances are needed to
achieve a good descriptions of Eqs. (\ref{fpipert}-\ref{L9pert}).
This  is even more reasonable, if some pattern  of low resonance
dominance shows up in the theory. Indeed, for all sum rules the
convergence remains quite good even after the corrected values of
$m_n$, $f_n$ and $g_n$ are inserted, and what is more important,
the numerical values converge rapidly with just few terms in the
sum to those obtained from the perturbations of $f_{\pi}$, $L_1$
and $L_9$, what gives us an important check. Actually, a dominance
of the few lowest resonances appear, as it is illustrated in Fig.
2, where saturation of each sum rule by the lowest resonances is
shown in both the unperturbed and in the first order corrected
case.

\begin{figure}[!t]\centering \epsfysize=14cm\epsfxsize=14cm%
\epsfbox{AllSums.eps} {\caption{\small The four plots show the
result of adding the first $K$ terms in the two sum rules
(\ref{Sum1}-\ref{Sum4}) in the HW model of Ref.
\cite{Hirn:2005nr}, both in the case of pure AdS metric (dots) and
in presence of first order corrections in the  metric deformation
(\ref{pertz4}) (squares). To put both in the same plot, the
differences in the unperturbed and in the perturbed case have been
magnified by suitable factors, as indicated. One easily recognizes
that the sum rules are dominated by the contribution of the first
few resonances also in presence of the metric perturbation. In the
plots, the perturbative corrections
 in eqs. (\ref{fpipert}),(\ref{L1pert}) and (\ref{L9pert}) and
 for $m_n$, $f_n$ and $g_n$  (re-scaled by $\eta$) have been denoted
 by $\delta f_{\pi}$, $\delta L_1$ and $\delta L_9$ etc.. }}\label{psiv}
\end{figure}
\newpage
\section{Conclusions}

Having evaluated in eqs.(\ref{fpipert}-\ref{L9pert}), the gluonic
contributions to the  $L_i$'s and to  $f_\pi$ it is  interesting to check if they lead to
a better agreement with the known  experimental values.
Since
 we have been able to evaluate the effects of the
gluonic condensate on masses, decay constants and
vector-to-two-pion couplings of  single vector resonances we can
add  also these parameter to the discussion.

Before embarking in the numerical analysis, we  should stress
however  that the model we are considering has intrinsic
approximations that cannot be underestimated when trying to
compare it with the real dynamics of low energy QCD. As in any
Large-$N_c$ model of QCD we are neglecting order ${\cal O}(1/N_c)$ effects, such as $\pi\pi$-loops. We
also know that the model does not reproduce the Regge behavior. The same
mechanism of chiral symmetry breaking is strongly model dependent.
We have followed the one
proposed in Ref.\cite{Hirn:2005nr}, where  the pion is introduced
as a 5D \emph{non-local} object linked to a Wilson line between
the two boundary of the $AdS_5$-slice.
This leads to the
expression (\ref{pionwave}) for the pion wave function.
 In Refs.
\cite{Erlich:2005qh}, although the description of vector
resonances is the same, the spontaneous chiral symmetry breaking
is triggered by a 5D scalar field, dual to the $\bar{q}q$ operator
and not by IR boundary conditions.
We shall try to take care of these caveats in the numerical
discussion that follows.
As  in Ref.\cite{Hirn:2005nr} we work in the chiral limit.

We want to  distinguish between the predictions of the model for
those observables which involve the spectrum of resonance as a
whole as in the case for the Gasser-Leutwyler coefficients and the
pion decay constant and the predictions for parameters related to
a single resonance. Moreover, motivated by the previous
discussion, we shall estimate the systematic error of the model
for the prediction of the pion decay constant and of single
resonance parameters to be of about $10-20\%$. The numerical
analysis which we have performed gives indication in favor of
this approach.

We refer to Table 2 and Figure 3 to illustrate the results of the
analysis:   five Gasser-Leutwyler
coefficients, the pion decay constant and the mass, the decay
constant and the coupling to two pions of the lowest vector
resonance, identified with the $\rho$ meson, are the observables
that we are using in different fits.

\begin{table}[!ht]%
\label{GLcoeff}\centering %
\begin{tabular}{|c|c|c|c|c|}
\hline
 & & & &  \\
   &Vanishing gluon  & Fit of $L_i$ &
   Fit of $L_i$,  $f_\pi (20\%)$&
      \\
  & condensate&  and $f_\pi (20\%)$ & and ($m_{\rho}$, $f_{\rho}$, $g_{\rho}$)$(10\%)$
           & Experiment \\
  &$\eta=0, \, z_0(m_{\rho}^{phys})$ & $(\chi^2/ {\rm d.o.f.})=(3.2/4)$ &  $(\chi^2/{\rm d.o.f.})=(8.0/ 7)$&  \\
    & & & &  \\
      \hline\hline
    $ 10^3 L_1 $&$\  0.36 $& $\ 0.54$ & $\ 0.58$& $\ \ 0.4 \pm 0.3$\\
     \hline
    $ 10^3 L_2 $&$\ 0.72 $&$\ 1.08$& $\ 1.16$ & $\ 1.35 \pm 0.3$\\
     \hline
    $ 10^3 L_3   $&$ - 2.2  $& $ -3.2$ & $ -3.5$& $ - 3.5 \pm 1.1$\\
     \hline
    $ 10^3 L_9   $&$\ \ 4.7 $ & $\ \ 6.3 $& $\ \ 6.7 $& $\ \ \ 6.9 \pm 0.7$\\
     \hline
    $ 10^3 L_{10}  $&$ -4.7$ & $ -6.3 $& $ -6.7 $&$\ - 5.5 \pm 0.7$\\
     \hline
         $ f_{\pi} \, \text{(MeV)}  $&$\ 72.8$ & $\ 92.4 $& $\ 95.6 $&$\ 92.3 \pm 0.3$\\
     \hline
     $m_{\rho} \, \text{(MeV)}  $ & $\ 776^{\,\sharp}$ &  $\ 783^{\,\dag} $& $\ 769$&$\ 775.8 \pm 0.5$\\
     \hline
         $ f_{\rho}   $&$\ 0.18$ & $\ 0.19^{\,\dag} $& $\ 0.19 $&$\ 0.20 \pm 0.02^*$\\
     \hline
             $ g_{\rho}   $&$\ 0.054$ & $\ 0.067^{\,\dag} $& $\ 0.070 $&$\ 0.087 \pm 0.009^*$\\
     \hline

\end{tabular}
{\caption{\small Theoretical predictions and experimental values
for the five Gasser-Leutwyler coefficients $L_1,\, L_2, L_3, L_9,
L_{10}$ , the pion decay constant $f_\pi$ and the mass $m_\rho$,
the decay constant $f_\rho$ and the coupling $g_\rho$ of $\rho$ to
two pions. The first column refers to the values obtained in the
absence of gluon condensate, $\eta=0$, with $N_c=3$,  and with IR
scale of the model $z_0$ fixed by using the physical value
$m_\rho=776$ MeV (the $^\sharp$ denotes that there $m_\rho$ is an
input parameter). The second and third column show the result of
fitting the experimental values of the $L_i$, with their
experimental values, together with  $f_\pi$ (third column) and
together also with $m_\rho$, $f_\rho$ and $g_\rho$ (The values in
the second column have a $^{\,\dag}$ to remember that they have
been evaluated after the fit of the other observable had been
done). As explained in the text, to $f_\pi$ and to the physical
quantities of the $\rho$ meson have been assigned theoretical
errors of $20\%$ and $10\%$ respectively. The $^*$ on the values
of $f_\rho$ and $g_\rho$ in the last column indicate that errors
have been estimated in the Chiral Lagrangian theoretical
framework.}}
\end{table}

As benchmarks, in the  first column of Table 2 are shown
 the values that one obtains in the absence of
gluon condensate,  $\eta=0$,  with $N_c=3$, \emph{i.e.} correct
matching in the UV of the  vector two-point function with the QCD
parton loop, and with IR scale of the model $z_0$ fixed by using
the physical value $m_\rho=776$ MeV as input. The values of the
$L_i$'s appear to be underestimated.

As an attempt to improve the  predictions of the model, in absence
of gluon condensate, one could abandon the UV matching and let
$N_c$ vary. This was done in Ref. \cite{Hirn:2005nr}, where $N_c$
was fixed by using the value of $f_\pi$ as input, with $z_0$ still
fixed by the physical value of $m_\rho$. There resulted an
improved agreement with experimental values, at the cost of having
$N_c=4.3$, signalling that important physical effects were still
not captured by the holographic model without the condensate.

Let us now discuss the inclusion of the gluon condensate. As we
explained before, we shall focus on the prediction of the model
for observables which involve the complete spectrum of resonance
as is the case for the Gasser-Leutwyler coefficients and the pion
decay constant. In the second column of Table 2, we report the
values obtained by the fit of the $L_i$ and  $f_\pi$, using the
expressions (\ref{fpipert}-\ref{L10pert}), with an estimated
theoretical error on $f_\pi$ of $20\%$.  The values of $\eta$ and
$z_0$ which minimize the $\chi^2$ are $\eta=0.73$ and
$z_0=2.7\times 10^{-3}$ MeV$^{-1}$ (not far from the value
$3.1\times 10^{-3}$ MeV$^{-1}$ which would be obtained by fitting
$m_\rho$ without condensate), with $\chi^2$/{\rm d.o.f.}=3.2/4.
 The values obtained for $\eta$ and $z_0$ have been used to
compute the values of $m_\rho$, $f_\rho$ and $g_\rho$. The
$1\sigma$ region for $(z_0,\eta)$ is light shaded in the first
contour plot of Fig.3.

The third column of Table 2 shows the values obtained by adding
also the values of the physical parameters of the lowest vector
resonance, the $\rho$, with an estimated theoretical error of
$10\%$. The values of $\eta$ and $z_0$ which minimize the $\chi^2$
are $\eta=0.89$ and $z_0=2.7\times 10^{-3}$ MeV$^{-1}$, with
$\chi^2$/{\rm d.o.f.}=8.0/ 7. The $1\sigma$ region for
$(z_0,\eta)$ is light shaded in the second contour plot of Fig.3.
The determination of the best-$\chi^2$ values is sharpened at the
cost of having an higher $\chi^2$ per d.o.f.

\begin{figure}[!ht]\centering \epsfysize=8cm\epsfxsize=16cm%
\epsfbox{TuttiChiq.eps} {\caption{\small Central values and
 $1\sigma$ regions for the values of the IR scale $z_0$
 and the condensate parameter $\eta$ are represented by a black dot and a light shaded
 region respectively  for the two fits  in Table 2. The left plot, with central
 values $\eta=0.73$ and $z_0=2.7\times 10^{-3}$ MeV$^{-1}$,
  corresponds  to the second column in Table 2: fit of the $L_i$'s and
  $f_\pi$. The right plot, with central values $\eta=0.89$ and $z_0=2.7\times 10^{-3}$ MeV$^{-1}$,
  corresponds to third column in Table 2: fit of the $L_i$'s, $f_\pi$,
  $m_\rho$, $f_\rho$ and $g_\rho$.
  The dark shaded band in both plots is the $1\sigma$ region for $\eta$ obtained
  by a fit of $L_i$'s values only, with the central value
  $\eta=0.73$ shown by the horizontal line.}}
\end{figure}

Using Eq.(\ref{frometatoo4}) one can derive from $\eta=0.89$ and
$z_0=2.7\times 10^{-3}$ MeV$^{-1}$ a prediction for the condensate
$\langle \alpha_s
G_{\mu\nu}G^{\mu\nu}\rangle=(8.7^{\,+9.4}_{\,-4.2})\times
10^{-2}\,{\rm GeV}^4$ which is compatible with the average
experimental value $\langle \alpha_s
G_{\mu\nu}G^{\mu\nu}\rangle_{\rm exp.}=(6.8\pm 1.13)\times
10^{-2}\,{\rm GeV}^4$ reported in Ref.\cite{Narison:2009vy}.

We believe that this simple analysis already shows that gluon
condensate corrections evaluated through metric deformation in HW
models leads to results qualitatively in agreement to what we
expect from QCD. We notice also that the trends of the numerical
 values of the Gasser-Leutwyler coefficients, that we have found,
in the presence of gluon condensate, and also those of vector and
axial vector masses and decay constants agree with those evaluated
in the ENJL model in \cite{Bijnens:1992uz}.

Now we want to observe another hint towards the connection between
the Chiral Quark Model \cite{Manohar:1983md,Espriu:1989ff} and
the AdS Hard-Wall model. As remarked in Ref.
\cite{deRafael:1995zv},  it just happens in the Chiral Quark Model
that, with neglect of the gluonic couplings,  for some of the
${\cal O}(p^4)$ terms in the QCD effective action ($L_1,\ L_2,\
L_3, \  L_9,\ L_{10}$) one gets finite results in the UV and IR
domains, {\it i.e.} cut-off independent values; of course this
does not hold once the  gluonic corrections are turned on. The
same is happening for the same $L_i$'s in the AdS HW model: no
dependence on $z_0$. Here we want to stress that this gives
further  support to the AdS picture vs the flat metric where the
same $L_i$'s show an explicit dependence on the IR cut-off, $z_0$.

In this paper we have  considered  the holographic description of
the effects of a gluon condensate in terms of a suitable
deformation of the  metric of a 5D Hard Wall model.  We have not
considered the effects of the $\langle \bar q q\rangle$ condensate
since our focus was on low energy QCD physics, once $\chi$SB has
been already triggered by some mechanism. Since the estimated
value for the gluon condensate is phenomenologically acceptable,
our treatment seems appropriate.  Moreover, the quark condensate
should not  be considered  in the comparison  with the Chiral
Quark model.

Comparing with previous literature, we have computed for the first
time  the corrections to the resonance decay constants, $f_n$ and
vector-to two-pion couplings, $g_n$ and we have shown the validity
of the sum rules even in presence of gluon condensate: this has
been the core of our calculation. We have also discussed the case
of flat metric. We have found substantial evidence  toward an
analogy of the AdS HW holographic model and the Chiral Quark
model. We have also performed a novel numerical analysis, which
leads to a satisfactory value for the gluon condensate, showing
that the model is able to capture the relevant physics properties.

\section*{Acknowledgements}

We thank Fabio Ambrosino,  Domenico Cuozzo and Guglielmo De Nardo
for discussions and help. This work has been supported in part by
the European Commission (EC) RTN FLAVIAnet under Contract No.
MRTN-CT-2006-035482 and by the Italian Ministry of Education and
Research (MIUR), project 2005-023102.

redpajama_set_name: RedPajamaArXiv


text:
Monitoring Accountability Panel insists fulfilment of UNHRC resolution
The Sri Lanka Monitoring Accountability Panel, expressing concern at recent rejections by Sri Lanka's president on the need for an international component to the accountability process, called on the UN Rights Chief to insist that Sri Lanka implements the UNHRC resolution.
The Sri Lanka Monitoring Accountability Panel, consists of a panel of international legal experts including Richard J Rogers, Justice Akit Prakash Shah, Marie Guiraud, Peter Haynes QC and Heather Ryan.
Mr Zeid Ra'ad Al Hussein
UN High Commissioner for Human Rights
Sri Lanka Monitoring Accountability Panel
Contact: Richard J Rogers
richardrogers@globaldiligence.com
The Sri Lanka Monitoring Accountability Panel ('MAP') has been established to provide independent monitoring, advice, and recommendations on the transitional justice mechanisms in Sri Lanka. We focus on accountability from a victims' perspective. Please see http://war-victims-map.org/ for more information.
We understand that you will visit Sri Lanka in the coming days to discuss the progress made by the Sri Lankan Government to address the mass atrocities and other human rights abuses committed during, and since, the armed conflict. The types of judicial and non-judicial measures necessary to deal effectively with these violations were outlined in the OHCHR Report, dated 28 September 2015. The subsequent Human Rights Council Resolution (A/HRC/30/L.29) – which was co-sponsored by the Sri Lankan Government – reiterated the need to establish a special judicial mechanism to investigate and prosecute "violations and abuses of human rights and violations of international humanitarian law." The Resolution affirmed "the importance of participation in a Sri Lankan judicial mechanism, including the special counsel's office, of Commonwealth and other foreign judges, defence lawyers and authorized prosecutors and investigators."
We note with concern the recent statement by President Maithripala Sirisena (BBC) interview, 21 January 2016) that he will "never agree to international involvement" and that "[w]e have more than enough specialists, experts and knowledgeable people in our country to solve our internal issues". This statement goes against both the word and spirit of the Resolution. It is also patently incorrect – as clearly outlined in successive reports by independent UN experts, the Sri Lankan justice system does not have the requisite independence, impartiality or expertise to administer fair and effective war crimes prosecutions. The Sri Lanka Government must not be allowed to backtrack on its commitments or use technical legal excuses to block full international judicial and prosecutorial participation.
If the accountability mechanisms are to "uphold the rule of law and to build confidence in the people of all communities of Sri Lanka" there must be a meaningful consultation process. The victims' voice must be heard. And their views on the need for full participation of foreign judges and prosecutors within the judicial process must be accommodated.
We urge you to insist that the Sri Lankan Government conducts broad consultations with all stakeholders and respects
Strengthening Investigative Journalism in Sri Lanka
Transitional justice is not a matter of offering redress of only one side

redpajama_set_name: RedPajamaCommonCrawl


This year has brought stability in key performance indicators of Latvijas Finieris – resource availability, production and sales volumes as well as finance. In 2011 it is planned to reach a turnover of 117 million lats by selling 206 000 cubic metres of products thus exceeding the results of previous year by 20% and 5%, respectively.
Increase in sales volumes and turnover has been possible due to raising the share of higher value added products. This year the market share of Latvijas Finieris in the European birch plywood market has remained stable within 14% range.
Continuing investment policy of Latvijas Finieris has secured the availability and effective utilisation of the most important raw material – birch roundwood – on a truly Baltic scale, which will make it possible to increase plywood production volumes in the future.
Considering development of the banking sector in the European Union, a decrease of available funds for business is possible in these countries. This could cause some fluctuations in the purchasing power of Latvijas Finieris export markets during the next year, but conservative estimates show that the demand will increase moderately in the longer term.
During the year Latvijas Finieris has hired 100 new employees. In total there are more than 2 000 people employed by the company in Latvia.


text:
Is it legal for a landlord to keep first months rent without the tenant signing any lease or legal document?
UPDATED: Aug 3, 2011Fact Checked
Home » Ask a Lawyer » Is it legal for a landlord to keep first months rent without the tenant signing any lease or legal document?
I paid security for an apartment after being approved, and that was all that was required of me. About a month after I was planning to move in but still hadn't moved from my previous state due to issues with my job, I paid first months rent because my landlord said that I had to to because he was losing out on money. Later on I had to withdraw my app, and he refused to give me back the 1st months rent because he was "broke" and I had to wait until he had the money. I never signed a lease or agreed to month-month rent verbally. It has now been 6 months. Is it legal for him to keep 1st month?
Asked on August 3, 2011 Pennsylvania
The issue will be whether or not, on the basis of your behavior, a court could find that you did in fact agree to a month-to-month rental. Contracts can be formed on the basis of behavior by a party, which can be considered to show acceptance of them; it is not alway necessary for an explicit oral or written agreement. It may well be that act of paying the first month's rent will be taken as agreement to rent the premises--after all, no one pays rent unless they are renting space--and therefore, even without your explicit oral or written agreement, a month-to-month lease was formed. In that instance, the landlord could certainly keep the first month's rent. The fact that you did not move in is irrelevant; it is agreement to lease or rent a premises, not physically moving in, that creates a lease and the landlord-tenant relationship.
Do I have any legal recourse regarding an injury that I believe was caused by forcing extreme labor on my work assignments?
If I was convicted of a non-dangerous felony 2 years ago and completed probation but my fiance owns guns and we live togetherI can i get in trouble?
How to get my in-law's name off of our deed?
Is it legal for a manager or area manager to seek retribution against employees who notified HR about not getting paid?


On July 7, "2019 Asia Model Festival of Mongolia with Mycinomobox" was held in the UB Place Royal Room in Ulaanbaatar, Mongolia."FACE of ASIA" is an international exchange event that is one of the Asian Model Festival's "Asia Model Awards," "Asia American Festival," and "Face of Asia" hosted by Asian Model Festival and "Face of Asia," which are co-hosted by Asian Model Festival.
In particular, the "mygenomebox" (Chairman Shin Sang-chul) that participated as the main sponsor of the event introduced the application of E1DIA (EDGC), which launched a number of health and beauty applications on the global gene-sharing economic platform, and attracted a lot of favorable reviews and interests from the local community. The judges of the event were Yang Sang-sik, chairman of the Asia Model Festival Organizing Committee, Shin Sang-chul, chairman of Maine Numun Box, chairman of Mongolia's Lotteria, President Lee Joon-ho of Comedy Holdings, lawyer Park Bum-il, CEO of Asia Model Agency Sertoya, lawyer of M&Z Law Group, Miss Mong-Golia Association, Webzol.
Two final men and three women were selected for the event through free dress screening, swimsuits, traditional dress screening, and dress screening. The final selection will be given the opportunity to act as an exclusive model for the sponsor, a drama shoot and an advertisement model, and will participate as a Mongolian representative in the "2019 Face of Asia" to be held in Korea next year. The winner of this year's Face of Mongolia for women's division were candidate number 3, Suvdansondor; candidate number 7, B.Tegshjargal; and candidate number 19 B.Odgerel. For men's division are candidate number 3, E.A.Megalanbat and candidate number 4, B.Mandakhbayar were selected for the final model representatives. Among them, the women candidate number 3, Kh.Suvdansondor won the Grand Prix with the Miami Box Award as the final winner.


text:
Deputato laburista dal 1945 e ministro dei Carburanti dal 1947, nel 1950 fu nominato ministro degli affari economici. Nell'ottobre dello stesso anno succedette a Stafford Cripps come cancelliere dello Scacchiere, ma nel 1951 il tracollo dei laburisti lo allontanò dalla carica.

Nel 1955 subentrò a Herbert Stanley Morrison ed assunse la direzione del Labour Party, mantenuta poi fino al 1963.

Altri progetti

Collegamenti esterni

Deputati britannici
Segretari di Stato britannici
Politici del Partito Laburista (Regno Unito)

redpajama_set_name: RedPajamaWikipedia


text:
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As the complexity of HVAC systems began to increase in the 1960s, a small group of engineers founded the Associated Air Balance Council (AABC) as a non-profit association in 1965. Early on, they recognized the need for independence and stringent membership requirements to ensure objectivity, integrity, and technical expertise, and they made these the guiding principles of the association. AABC also developed the first comprehensive industry standards for test and balance.
ASHRAE is the global leader, the foremost source of technical and educational information, and the primary provider of opportunity for professional growth in the arts and sciences of heating, ventilating, air conditioning and refrigerating.
Synergy Test and Balance utilizes the expertise of Facility Commissioning Group (FCG) based in Lexington, KY for nationally recognized AABC Commissioning Group (ACG) certified Commissioning Authorities and Technicians.
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13th August, 2018: Tata Chemicals Group (the "Company") today declared its Consolidated Financial Results for the first quarter ended 30 June, 2018. The Company reported income from operations on consolidated basis at Rs. 2,769 Cr, up 10% and Rs. 996 Cr, up 15% on a standalone basis against the same quarter last year.
The Company's results by reporting segment showed Income from operations for Basic Chemistry Products at Rs 63 Cr, up by 3%, Consumer products segment at Rs 108 Cr, up by 33% and Specialty Products Rs 116 Cr, up by 25%.
Mr. R. Mukundan, Managing Director, Tata Chemicals Ltd., said, "We are pleased to share a good overall performance across all businesses. India's Basic Chemistry products business registered a robust performance on the back of improved operational efficiencies, a better sales mix and better realisations. On the global front, performance of the North American operations reflects planned shut down and Kenyan operations were impaired by heavy rains.
The company is poised for growth with a transformation agenda built on Innovation, Sustainability and Digitisation. While we continue our drive for optimization and efficiency in the Chemicals business, our future growth drivers are going to be the Consumer and Specialty businesses."


I've analyzed the pros and cons of CISA review courses based on readers' experience and comments from various IT audit forums. With this information, you can find the CISA review course that works best for you.
Let Me Find the Best CISA Review Course for You!
The CISA Review Manual serves as the foundation for many candidates' CISA review, but it is dry and boring. It also lacks explanation and examples.
For a more efficient way to understand the concepts, I recommend using one of these supplements.
* Conditions apply. Please check the respective review course providers for details.
Renowned info security expert Allen Keele operates this popular review course.
This is the only full CISA review course, as it includes video instruction, hundreds of practice questions, and exam simulations for an affordable price.
SuperReview affords great customer service with direct phone and email communication available.
You can use this course to help you fulfill the CPE requirements.
Other courses come with such a guarantee, so SuperReview's lack of it is a letdown.
SuperReview doesn't supply as many practice questions as Surgent CISA Review.
I believe you could supplement or even replace the CISA review manual with this excellent, fully integrated course.
Simplilearn is one of the largest online training platforms, as they cover over 40 certifications. So far, more than 4,800 students have used their CISA review course.
SimpliLearn has created a very clear and organized video instruction series.
With this guarantee, you can try out the course for up to 7 days. As long as you don't access more than 50% of course content, you can receive a full refund. (Please read the terms and conditions for details).
The course presents quizzes after each study module but doesn't have a database of practice questions (questions only in mock exams).
SimpliLearn is the second most expensive, but it needs a little more value for the price.
The Simplilearn course is a good choice if you're looking for a further explanation of the exam concepts via video.
Surgent provides exam prep for various accounting and tax certifications. It prides itself on its proprietary adaptive learning technology that feeds students questions according to their weaknesses.
The adaptive learning system is one of the most efficient ways for students to learn and can really help retakers improve their scores.
Surgent offers a pass-or-get-your-money-back guarantee.
This course is not an integrated review system. It does not come with videos, audios, flashcards, or personalized customer support. Additionally, the reference book only covers basic concepts.
I find Surgent to be a great supplement to the CISA Review Manual, especially for those who understand the concepts and only need more practice questions.
I believe most candidates would agree that CISA SuperReview has the best video lectures, practice questions, and CPE credits. On top of that, it includes excellent customer support with access to Allen himself. Everything you need comes in one nice package. What do you think? Drop a note here, and we can discuss.
hello, i got gleim resources for cia, so the question is, can i use gleim for cisa and if the answer yes, for which domain?
Hi! Yes, you can use it to supplement some CISA study areas, but there's not a huge overlap overall. Therefore, our team highly encourages candidates to purchase exam-specific review. You can learn all about cisa courseshere.
Hi Stephanie, thanks for the nice sharing. But, i think maybe i got it wrong? the prices showing up there are actually before discount. Not after discount.
This shouldn't be the case. Please email stephanie[at]ipasstheciaexam.com so we can help!


text:
Q: Is it proper to use "the" before the name of a government organization?
Possible Duplicate:
Definite article with proper nouns, titles followed by a common noun
Using the definite article with acronyms and initialisms

When I listen to major news programs, often I notice that they seem to intentionally omit "the" before the name of the government organizations. For example:

We contacted E.P.A. for comment but they refused our requests.

or

Others consider the actions of treasury to be detrimental to the economy.

These just sound wrong to me. I think it should be "the E.P.A" or "the treasury". However I most often hear this on very credible news programs (The PBS News Hour and Frontline come to mind) so I am sure they know what they're doing...
Is it proper to use "the" before the name of a government organization, or is it optional?

A: There are several reasons to drop the article. One is that some institutions, governmental or not, are never referred to using the definite article. Another is that the article can get dropped as a consequence of familiarity with and/or personification of the institution. Of course, NewsHour and the like may simply impose a stylistic preference to remove nonessential words.

We often apply the definite article to the names of governmental organizations when the type of organization is part of the name (ministry, office, committee, et al). This is natural, since it sounds like we are specifying one organization of a type— National Park Service, which happens to be the name of the national park service; likewise the National Health Service or the Department for International Development. Where the organization is "branded," however, this is not the case, and we do not use the article: Parks Canada, Medicare, USAID (even though we would write out the United States Agency for International Development).
As with the names of countries and geographic features, there are no absolute rules in naming institutions. It is simply Gosbank but always the Bundesbank; someone attended North Carolina State University but attended the University of North Carolina (and yet attended UNC). With proper nouns, whichever usage becomes popular is that which becomes accepted, and sometimes (e.g. [the] Ohio State University) it is a muddle.

Dropping the article is quite common in some other professional communication. People who work with a particular organization may personify it, especially in internal circles, and as English does not use articles for personal names, the article may get dropped.
This is more obvious where an unofficial nickname or abbreviation is used. A large company, say ABC Inc., might organize its employees into divisions. A press release or business school case study about them would write out the full name: the ABC Marketing Division and the ABC Product Development Division.
Suppose you worked for ABC, and you interact regularly with those divisions as well as with some outside actors: the Las Vegas Chamber of Commerce, the Greater Boston Chamber of Commerce, and the U.S. Bureau of Indian Affairs. Your internal emails will get simpler over time, indicating each group with the minimum of identifying information: there's good news out of Marketing, you should talk to PDD, send the proposal to Vegas but not Boston, we're waiting on approval from BIA. Obvious all those groups are made of up many different people doing many different things, but in terms of their interaction with you, they might as well be monolithic. It doesn't matter if it's Sandy from PDD or Chris from PDD, you just need someone from PDD.

A: I think it should be based on the full name. For example, as you say "We contacted the EPA..." but "we contacted IBM .....".
From this discussion,
 Note the example of "followed FDA standards" versus "contacted the FDA".
I reversed the edit as my opinion is not based on that discussion.

redpajama_set_name: RedPajamaStackExchange


Having a true fund accounting™ solution can make a world of difference to your nonprofit organization in ways you may not have even considered. Like small businesses, small nonprofits generate profit and pay expenditures; however, they also have social and legal responsibilities that extend beyond the balance sheet.
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Trevor has been sculpting full time since 2003. He is of New Zealand Māori and European descent and is strongly influenced by his dual heritage and the impact of both cultures in contemporary New Zealand. Participating in country wide symposia, and previously organising the Rotorua event, he won the 2016 Rotorua Sculpture Symposium with his piece Synergy. Trevor has exhibited throughout the North Island, has permanent sculptures around central Rotorua and completed a solo exhibition called Two Views at the Elim Gallery in 2012. In December 2013, he opened Trevor Nathan Sculpture Gardens and Gallery which is now his main focus.
Thirty years of farming in North Hokianga has given Trisha the practical skills needed for sculpture. Now that her three sons are grown and the farm sold she has the time to devote to her art. Trisha enjoys working in many mediums from hard and soft stones to wood, Darjit and ferro cement. After two years of study with the Learning Connexion she has successfully taken part in sculpture symposia in Whangarei, Matamata and Auckland. She enjoys being versatile and doing a variety of work. Trisha believes art has to improve your day either through its boldness of its form or the inspiration of its subject.
Oriah Rapley is a full time practising artist based in Taranaki. Oriah has pieces in national and International collections and her work has been in selected exhibitions including Homework exhibition at Puke Ariki, NZ Sculpture Onshore and Sculpture on the Peninsula. She has attended many symposia around New Zealand including Te Kupenga International Symposium.
Oriah's work is in constant refinement towards the purity of form, and acknowledges freedom of choice and embodiment. Using symbols such as the feather, waka and bird to provide subtle layers interwoven around movement, growth, exploration, expansion, freedom, expression, empowerment and transformation.
Anna Korver is an established New Zealand artist based in Taranaki. She finished her BFA in sculpture from the University of Canterbury in 2003 and has been working and exhibiting nationally and internationally ever since. Anna was a finalist in the Wallace Awards twice and has attended over 50 national and international symposia. Previously the co-owner of The Korver Molloy Gallery and Sculpture Park for 4 years, Anna has been a trustee on the board of Eco Artists New Zealand for 10 years and an executive member of Te Kupenga Stone Sculpture Society and selector for the 2018 International Symposium.
Jocelyn Pratt is an established New Zealand sculptor presenting modernist, interpretive works representative of natural forms. Jocelyn favours working with local resources, exploring and enhancing the natural qualities of stone with polish and texture techniques. Jocelyn's works are tactile and suggestive of a restful gentle form, a juxtaposition of the natural solidity and strength of the stone and concept. Many of Jocelyn's works are evocative of her surrounding environment, nature, culture, and socio-political effects on the environment and people. Having participated in over 80 symposia and numerous exhibitions, Jocelyn's sculptures are in both public and private collections nationally and internationally.
With no previous formal art training, Claire began learning stone carving with andesite boulders at Te Kupenga Stone Sculpture Society in New Plymouth nearly ten years ago. Her approach is to explore the individual piece of raw natural material, to respect and feature its natural characteristics and proportions, rather than seeing it purely as a medium for her own ideas. Claire has evolved two distinct styles – either heavily geometric or curvaceous. By attending over a dozen symposia around the country and exhibiting in sculpture parks and exhibitions, such as NZ Sculpture Onshore and Shapeshifter at The Dowse, Claire is gaining a widening experience and reputation.
Christchurch-born Bodhi has been a professional artist for as long as he can remember. He attended design school in Wellington at the start of his career and elements of design do come into his work, however Bodhi's style echoes more a classical training in the Fine Arts. Bodhi has exhibited both overseas and throughout New Zealand, and has attended up to six sculpture events per year for the past two decades. Having lived in Raumati South for over twenty years, he has a significant relationship with nature, and his work has an organic quality often involving sinuous lines and curvaceous forms.
Born and bred in Turangi, all Natanahira wanted to do when he was young was join the armoured division in Waiouru. The idea of being a tank driver was a passion for him, but instead took up the second, safer option of art. Natanahira attended the New Zealand Māori Arts & Crafts Institute in Rotorua where his career as an artist blossomed and he has now been sculpting for over thirty years. He has taken part in numerous sculpture symposia in the Bay of Plenty and Waikato, including the 2014 Rotorua Sculpture Symposium, and also travelled to Nepal in 2016 to represent Aotearoa.
Steve Molloy is a multi-award winning New Zealand artist and photographer, and is the owner and director of the Molloy Gallery in New Plymouth. He works predominantly in object and installation sculpture, photography and painting, exhibiting in galleries as well as attending symposia nationally and internationally. Steve's photography has been shown in magazines, newspapers, commercial businesses and galleries throughout the world and his sculptures are in public places in New Zealand and overseas. His work is uniquely defined by the process of incorporating fragmentation, movement, miniaturisation, space and illusion as a tool to view normal scenes and objects in an abstract way.
Having completed a two year Craft Design Māori Certificate at Waiariki Polytechnic (now Toi Ohomai) in Rotorua, Jamie gained design and practical skills which have held him in good stead over the years. He then went traveling for a few years before completing a Fine Arts Degree in Whanganui, majoring in sculpture, and at the same time setting up his business, Jamie Pickernell Art Furniture. Moving back to Rotorua twenty years ago, he set up his home and workshop, and makes his living through outdoor sculpture exhibitions and commissions for sculpture, furniture and architectural hardware.
Simone is a Swiss sculptor who has been living and creating in New Zealand since 2009. She first came in contact with hard stone at the 'Scuola di Scultura di Peccia' (Peccia Sculpture School) in Switzerland in 2002. Simone chooses to create in an abstract and minimalist style allowing the viewer to connect with her forms in their own way. Her art is a reflection of her passion for nature and she aims to create sculptures which will stir the viewer's soul, and entice them to touch. Simone has exhibited her work in New Zealand and Switzerland and has participated in several symposia throughout New Zealand.
Rory was raised in the Highlands of Scotland, in an area full of Pictish-Celtic stonework and archaeology. His schoolbook doodles attracted the attentions of his art teacher and he was introduced to the book "Celtic Art – Methods of Construction" by George Bain. Coincidentally Mr Bain had taught at his school fifty years previously and was a revivalist and promoter of the Celtic Arts. Rory studied metal craft and stone masonry in Germany prior to settling in New Zealand and now lives in Hokitika on the West Coast. His fascination with Celtic art and its iconography continues to influence his work and has exhibited extensively with several public art objects on display around New Zealand.
ex was born and raised in Mt Maunganui; the second of three sons. Following a childhood love of carving and clay work, he later began bone carving. In 2000 he returned to sculpture, working predominantly in stone, exploring bold flowing forms with an emphasis on shadow and balance. In 2010 he graduated with a Diploma in Art from the Bay Of Plenty Polytechnic and while studying he attended the Whakatane Sculpture Symposium. This led to work exploring human form and nature, studying the people of New Zealand who arrived, stayed and left. The latest of his work is a journey into new mediums and practices and enjoying the challenges and achievements it brings.
Andrew studied at Ilam School of Fine Arts in Canterbury and has been sculpting in wood and stone for more than 30 years. Balancing full time work and his passion for expression has allowed him to look openly at all forms. His love of nature and all things universal offers no limits on what will be unveiled for his next masterpiece. Andrew has thoroughly enjoyed being part of symposia across New Zealand including Golden Bay, Nelson and Tauranga. Exhibited widely, Andrew has been well received by the art fraternity including being awarded Sculptor of the Year by Trends Magazine.
Te Arawa carver, Peter Akurangi, works in a variety of wood, stone and bone to produce intricate and detailed carvings. Peter started his carving passion over 20 years ago while attending a carving course in Wellington. Peter's mastery and passion for his art continues to grow. Many of his designs evolve from a traditional foundation, incorporating historical designs pertaining to the area. Native wood or Oamaru stone are his preferred materials.
Susan Dinkelacker is a wood carver from a family of woodworkers who developed her carving style and skills between helicopter flights. Her "day job" for the last few decades has been as a career helicopter pilot, racking up over 10,000 flying hours all over the world.
Specialising in 3D sculptures, usually in wood, Susan strives to reveal the intriguing inner character ofwhatever she carves – be they birds, creatures or the human form – they are designed to delight.
Susan's sculptures are displayed at Helena Bay Gallery and at the Quarry Arts Centre in Whangarei, as well as in private collections in New Zealand, the Netherlands, Malaysia, UK, Germany, Korea and the USA. Susan exhibited in the Quarry Arts Outdoor Sculpture Exhibition 2018, and recently took part in the Whangarei Sculpture Symposium 2018 and the Kawerau Woodbugs Sculpture Symposium 2018.


I believe Members of Parliament should be as open and accessible as possible. That's why you can now book one-on-one appointments online.
Just use the form below to tentatively schedule a meeting and we'll be in touch to lock it in.


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