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\begin{document}

\title{Confinement of ${\cal N}=1$ Super Yang-Mills from Supergravity}

\author{Xiao-Jun Wang}

\email{wangxj@itp.ac.cn} \affiliation{Institute of Theoretical
Physics, Beijing, 100080, P.R.China}

\author{Sen Hu}
\email{shu@ustc.edu.cn} \affiliation{Department of Mathematics,
University of Science and
Technology of China, \\
AnHui, HeFei, 230026, P.R. China}

\begin{abstract}
We calculate circular Wilson loop expectation value of pure ${\cal
N}=1$ super Yang-Mills from the Klebanov-Strassker-Tseytlin
solution of supergravity and the proposed gauge/gravity duality.
The calculation is performed numerically via searching world-sheet
minimal surface. It is shown that Wilson loop exhibits area law
for large radius which implies that ${\cal N}=1$ super Yang-Mills
is confined at large distance or low energy scale. Meanwhile,
Wilson loop exhibits logarithmic behavior for small radius and it
indicates asymptotical freedom of ${\cal N}=1$ super Yang-Mills at
short distance or high energy scale.
\end{abstract}

\pacs{11.15.-q,4.50.+h,4.65.+e}

\maketitle

\section{Introduction}

Physicists believe non-abelian gauge theory exhibits some dramatic
behaviors at long distances such as confinement and mass gap,
etc.. It is difficult to study these properties using the
traditional quantum field theory (QFT) method due to its
perturbative nature. However the remarkable success of AdS/CFT
correspondence provides a practical approach to study strong
coupling gauge theory\cite{Mald98,GKP98,Witten98}. The similar
correspondence is believed to exist for a broad class of theories
with less supersymmetry and non-conformal theories and we called
such correspondence gauge/gravity duality. We expect to get
strong-coupling properties of non-abelian gauge theory via a dual
super-gravity or a string theory which is weakly coupled. So far
several nonsingular supergravity (SUGRA) solutions have been
constructed. They preserve 1/2 or 1/4 of the maximal supersymmetry
and they are conjectured to be dual to $d=4$, ${\cal
N}=2$\cite{GKM01} or ${\cal N}=1$\cite{MN01,KT00,KS00} super
Yang-Mills (SYM) theories. In particular, the case of ${\cal N}=1$
super SYM is very interesting, because it possesses the common
features of non-abelian gauge theory at long distance and it is
hard to study by using usual QFT method. For this SYM, two dual
SUGRA solutions were found:
\begin{itemize}
\item The Maldacena-Nu$\td{\rm n}$ez (MN) solution\cite{MN01}:
It corresponds to a large ($N$) number of D5-branes wrapped on a
supersymmetric two-cycle inside a Calabi-Yau threefold. Meanwhile,
the little string theory onto D5-branes is reduced to pure
$d=4,\;\net$ $SU(N)$ SYM in the IR.
\item The Klebanov-Strassler-Tseytlin (KST)
solution~\cite{KT00,KS00}: It describes the geometry of the warped
deformed conifold when one places M D3-branes and N fractional
D3-branes at the apex of the conifold. It is dual to a certain
$\net$ supersymmetric $SU(N+M)\times SU(M)$ gauge theory. If $M$
is a multiple of $N$, then this theory flows to $SU(N)$ in the IR,
via a chain of duality cascade which reduces the rank of the gauge
group by $N$ units at each cascade jump. Thus at the end of the
duality cascade the gauge theory is effectively pure $\net$
$SU(N)$ SYM.
\end{itemize}
The purpose of this paper is to study confinement of $\net$ SYM
from KST solution. We will give a brief comments on MN solution at
the end of the paper.

The natural criterion for confinement of pure Yang-Mills theory is
that the expectation value of large Wilson loop satisfies area
law\cite{WTM}. The evaluation on Wilson loop of ${\cal N}=4$ SYM
from supergravity in $AdS$ space has been studied in various
aspects. It was proposed in\cite{Mald98b,RY98} that Wilson loops
of the CFT can be described in AdS by
 \beq{1}
   <W({\cal C})>\;\sim\;\lim_{\Phi\to\infty}e^{-(S_{\Phi}-l\Phi)},
 \eeq
where $S$ is the proper area of a string world-sheet whose
boundary lies on the loop ${\cal C}$ of the boundary of AdS, $l$
is the total length of the Wilson loop and $\Phi$ is the mass of
the $W$ boson. The presence of the term $l\Phi$ is to subtract
divergence from infinity mass of the $W$ boson. It corresponds to
UV divergence in QFT. The evaluation for the rectangular Wilson
loop was performed in refs.\cite{Mald98,RY98} and for simpler
circular loop it was performed in refs.\cite{BCFM99,DGO99}. There
are no any surprising result due to conformal invariance. The more
non-trivial results are from calculation of operator product
expansion for the Wilson loop when probed from a distance much
larger than the size of the loop\cite{BCFM99}, and from
calculation on Wilson loop correlator\cite{GOphase}. The extension
of Eq.~(\ref{1}) to KST background is in principle direct.
However, since KST background is much more complicated than AdS
space, we can not expect to obtain any analytic results. Although
there is a simple scaling argument that large Wilson loops of
$\net$ SYM resulted by KST solution satisfies the area
law\cite{HKO01}, a direct verification is still lacked so far. In
this paper we use numerical tools to search minimal surface of
string world-sheet in KST background whose boundary lies on the
Wilson loop of the boundary of KST space. In order to simplify the
calculation, we consider circular Wilson loop of $\net$ SYM in
Euclidean space. The proper area of string world-sheet can be
expressed as a function of radius of circular loop. We will
evaluate both cases of small and large radius. It is expected that
behavior of Wilson loop exhibits the following properties,
\begin{itemize}
\item The confinement of $\net$ SYM at large distance.
\item The asymptotical freedom of $\net$ SYM at short distance.
\item Phase transition from short distance to large distance.
\end{itemize}
This study can also be treated as an examination for duality
between KST SUGRA solution and $\net$ SYM.

The paper is organized as follows. In Sec.\ref{sec2}, we extend
calculation on Wilson loop in AdS space to KST background and
derive Euler-Lagrangian equation which governs minimal surface of
string world-sheet. In Sec.\ref{sec3}, we present some numerical
results, such as curve on expectation of Wilson loop vs. radius of
the loop. The physical results are obtained via those numerical
results. A brief summary will be devoted in Sec.\ref{sec4} and
several details on numerical evaluation are presented in the
Appendix.

\section{Wilson Loop of $\net$ SYM from KST solution}
\label{sec2}

The Wilson loop operator in $\net$ SYM is
 \beq{2}
  W({\cal C})=\frac{1}{N}Tr P e^{i\oint_{\cal C} A},
 \eeq
where ${\cal C}$ denotes a closed loop in spacetime, and the trace
is taken over the fundamental representation of the gauge group.
The expectation value of the Wilson loop can be calculated by
super-gravity in terms of Eq.~(\ref{1}) directly, with action $S$
given by
 \beq{3} S=\frac{1}{2\pi\alpha'}\int d^2\sigma\sqrt{-g}, \eeq
where $\sigma^1$ and $\sigma^2$ parameterize string world-sheet
swept by the fundamental string,
 \beq{4} g_{\alpha\beta}=G_{MN}\pa_\alpha X^M\pa_\beta X^N, \eeq
is the induced metric on the world-sheet. In this paper, we focus
our attention on nonsingular KST background such that $G_{MN}$ is
given by
 \beq{5}
 ds_{10}^2&=&h^{-1/2}(\tau)dx_ndx_n+h^{1/2}(\tau)ds_6^2,
    \nonumber\\
 h(\tau)&=&\alpha\int_\tau^\infty dx
     \frac{x\coth{x}-1}{\sinh^2{x}}(\sinh{(2x)}-2x)^{1/3}
    \nonumber \\
    &=&\alpha I(\tau),
 \eeq
where $\tau$ is the radial parameter of the transverse space,
$\alpha=2^{2/3}(g_sN\alpha')^2\ep^{-8/3}$. $ds_6^2$ is the metric
of the deformed conifold\cite{DC},
 \beq{6}
  ds_6^2=\frac{1}{2}\ep^{4/3}K(\tau)
  \left[\frac{1}{3K^3(\tau)}(d\tau^2+(g^5)^2)
     +\cosh^2{(\frac{\tau}{2})}\sum_{i=3}^4(g^i)^2
   +\sinh^2{(\frac{\tau}{2})}\sum_{i=1}^2(g^i)^2
 \right] \eeq
where $\ep$ parameterizes length of the deformed conifold and
possesses mass dimension $-3/2$, $K(\tau)$ is defined by
 \beq{7}
 K(\tau)=\frac{(\sinh{(2\tau)}-2\tau)^{1/3}}{2^{1/3}\sinh{\tau}},
 \eeq
and the 1-form basis $g^i$ parameterize transverse
five-dimensional space. To simplify calculations we do the
following:
\begin{itemize}
\item Set angle parameters of the transverse space as constants
$g^i\equiv 0$. The Wilson loop in ${\cal N}=4$ SYM with
non-constant angle parameters has been considered in
ref.\cite{TZ02}. When we extend this consideration to $\net$ SYM
calculations become very difficult.
\item To define a new dimensionless radial parameter
$r=e^{-\tau/3}$, $0\leq r\leq 1$. The boundary of KST space is
$r=0$.
\item We consider that $\net$ SYM is defined in Euclidean space,
and the circular Wilson loop is located at $x^1-x^2$ plane. Since
we are aiming at finding a minimal surface, the string world-sheet
can be considered as a rotational surface. Then we define
 \beq{8} x^1&=&\sqrt{\frac{3}{2}\alpha}\ep^{2/3}f(r)\cos{\theta},
   \nonumber \\
      x^2&=&\sqrt{\frac{3}{2}\alpha}\ep^{2/3}f(r)\sin{\theta},
 \eeq
with a dimensionless function $f(r)$. Consequently the string
world-sheet is parameterized by $r$ and $\theta$,
 \beq{9}
   ds^2=\frac{3}{2}\sqrt{\alpha}\ep^{4/3}I^{-1/2}(r)\left\{
    [f'^2(r)+\frac{1}{r^2}I(r)K^{-2}(r)]dr^2+f^2(r)d\theta^2\right\},
 \eeq
where prime denotes derivative on $r$.
\end{itemize}

Using the above notions, the action defined in Eq.~(\ref{3}) is
rewritten as
 \beq{10}
  S=\frac{3}{2^{2/3}}g_sN\int dr\cdot I^{-1/2}
   f\sqrt{f'^2+G(r)},
 \eeq
with
 \beq{11} G(r)=\frac{1}{r^2}I(r)K^{-2}(r). \eeq
From action~(\ref{10}) we obtain the Euler-Lagrangian equation
which governs minimal surface of string world-sheet,
 \beq{12}
   f''-\frac{1}{2}(\ln{IG})'f'-\frac{I'}{2IG}f'^3-
   \frac{1}{f}(f'^2+G)=0.
 \eeq
The minimal surface condition requires the initial conditions of
Eq.~(\ref{12}) be
 \beq{13} f(0)=R,\hspace{0.5in}f'(0)=0. \eeq
In order to show that this initial condition is taken care of
properly, let us find asymptotical solution of Eq.~(\ref{12}) for
$r\to 0$. Noticing the expressions,
 \beq{14}\left.\begin{array}{ccc}
   \displaystyle I(r)&\stackrel{r\to 0}{\lraw}&
  -\frac{3}{16\cdot 2^{1/3}} r^4(1+12\ln{r})+O(r^{10}) \\
   & & \\
   K(r)&\stackrel{r\to 0}{\lraw}& 2^{1/3}r+O(r^7)
   \end{array}\right\}
 \quad \Rightarrow \quad G(r)\stackrel{r\to 0}{\lraw}
   -\frac{3}{32}(1+12\ln{r}), \eeq
at limit $r\to 0$, the Euler-Lagrangian equation~(\ref{12})
approaches to
 \beq{15} ff''-(\frac{2}{r}+\frac{1}{r\ln{r}})ff'
   +\frac{9}{8}\ln{r}+\frac{3}{32}=0. \eeq
Considering the initial condition~(\ref{13}), we obtain
asymptotical solution of the above equation as follows
 \beq{16} f(r\to 0)&=&R+\frac{9}{16R}r^2\ln{r}
  -\frac{15}{64R}r^2+O(r^3), \nonumber \\
  f'(r\to 0)&=&\frac{9}{8R}r\ln{r}+\frac{3}{32R}r+O(r^2).
 \eeq

In addition, the minimal surface equation also implies that $f(r)$
must vanish at a point $r=r_0$ with $0<r<r_0< 1$, but $f'(r)$ is
divergent at this point. Numerical evaluation will not be valid
near this point, and we have to find an asymptotical analytic
solution for $r\to r_0$,
 \beq{17}
   f(r\to r_0)=c(r_0-r)^{1/2}+O((r_0-r)^{3/2}), \eeq
with constant $c^2=8I(r_0)G(r_0)/I'(r_0)$.

Another aspect which needs to be dealt with manually is
renormalization of divergence. It is caused by integration in
Eq.~(\ref{10}) when $r\to 0$. This divergence should be subtracted
from numerical result on proper area of string world-sheet.
Explicitly, using Eqs.~(\ref{14}) (\ref{16}) and taking a cut-off
$\lambda\to 0$, we obtain the divergence as follows
 \beq{18} \int dr\cdot I^{-1/2}f\sqrt{f'^2+G(r)}=
 \frac{R}{2^{1/3}\lambda}+\frac{9}{2^{1/3}\cdot 32R}\lambda+O(\lambda^2)
 +{\rm other\;\;terms\;\;independent\;\;of}\;\;\lambda.
 \eeq
Here we include sub-sub leading term which is proportional to
$\lambda/R$ such that it will strictly vanish at limit $\lambda\to
0$. In numerical evaluation, however, the cut-off $\lambda$ is no
longer very small. Then this term will play a role for small $R$.

According to the above discussions the Euler-Lagrangian
equation~(\ref{12}) can be solved numerically. The consistency of
numerical calculations will be checked via matching initial
conditions (\ref{13}) and (\ref{17}). Consequently, the proper
area~(\ref{10}) of string world-sheet can be obtained after
subtracting $\lambda$-dependent terms in Eq.~(\ref{18}). Since
difference equation associated to equation~(\ref{12}) is
ill-defined for $r\to r_0$, some tricks on numerical evaluation
are needed. We put all details on numerical evaluation in appendix
of this paper.

\section{Numerical results and discussions}
\label{sec3}

We first show some numerical results:
\begin{itemize}
\item Fig.~1 includes some curves of $f(r)$ vs. $r$, which
represents some solutions of Euler-Lagrangian equation~(\ref{12})
with different initial values $R$.
\begin{figure}[hptb]
\parbox{3.5in}{
\label{f1}
   \centering
   \includegraphics[width=3in]{fig1.eps}
\begin{minipage}{3in}
   \caption{Some solutions of Euler-Lagrangian equation~(\ref{12})
   with different initial values. The dash line denotes $R=0.32$,
   the dot line denotes $R=2.09$ and solid line denotes $R=5.06$.}
\end{minipage}}
\parbox{3.5in}{\label{f2}
   \centering
   \includegraphics[width=3in]{fig2.eps}
\begin{minipage}{3in}
   \caption{Curve of $r_0$ vs. $R$, where $r_0$ is
zero of $f(r)$ and $R$ is initial value.}
\end{minipage}}
\end{figure}
\item Fig.~2 denotes curve of $r_0$ vs. $R$, where $r_0$ is
an zero of $f(r)$ and govern integral region in Eq.~(\ref{10}),
i.e., $0\leq r\leq r_0$.
\item Fig.~3 and fig.~4 show that the resulted numerical data
and fitting curves on proper area $A$ of string world-sheet vs.
radius $R$ of Wilson loop at large distance and short distance
respectively. The unit of area $A$ is taken over $3g_sN/2^{2/3}$.
\begin{figure}[hptb]
\parbox{3.5in}{
\label{f3}
   \centering
   \includegraphics[width=3in]{fig3.eps}
\begin{minipage}{3in}
   \caption{Curves of proper area $A$ (in unit $3g_sN/2^{2/3}$) of
   string world-sheet vs. radius $R$ of Wilson loop at larger
   distance. The diamonds denote numerical results, and solid line is
   the fitting function, $A=-1.5126+0.2941R+1.04R^2-2.162\ln{R}$
   with $\chi^2=0.001277$.}
\end{minipage}}
\parbox{3.5in}{
\label{f4}
   \centering
   \includegraphics[width=3in]{fig4.eps}
\begin{minipage}{3in}
   \caption{Curves of proper area $A$ (in unit $3g_sN/2^{2/3}$) of
   string world-sheet vs. radius $R$ of Wilson loop at shorter
   distance. The diamonds denote numerical results, and solid line is
   the fitting function, $A=-1.511+0.829R+0.0753\ln{R}$
   with $\chi^2=3\times 10^{-5}$.}
\end{minipage}}
\end{figure}
\end{itemize}

The main conclusions of this paper are included in fig.~3 and
fig.~4. In order to precisely describe behavior of
$A\sim\ln{<W({\cal C})}>$ with variation of radius $R$ of Wilson
loop, we fit numerical data with several different functions and
pick up functions with smallest $\chi^2$ to approximate Wilson
loop.

The resulted functions for long distance (larger $R$) are listed
in Eq.~(\ref{19a})-(\ref{19c}), which fit the numerical data in
fig.~3. \\
 \alphaeqn
 \stepcounter{saveeqn}
\setcounter{equation}{0}
\renewcommand{\theequation}{\mbox{\arabic{saveeqn}-\alph{equation}}}
\parbox{1in}{\mbox{}}\hfill\parbox{5.5in}{
\parbox{3in}{\begin{eqnarray*}
  A&=&-1.5126+0.2941R+1.04R^2-2.162\ln{R}  \\
 A&=&-0.7835-1.0274R+1.1367R^2   \\
 A&=&-0.0464-1.717R+1.342R^2-0.014R^3
 \end{eqnarray*}}
 \parbox{2.5in}{
 \beq{19}\label{19a}&&\chi^2=0.0013 \\ \label{19b}&& \chi^2=0.0046 \\
 \label{19c}&& \chi^2=0.0017
 \eeq}}\stepcounter{saveeqn}
 \setcounter{equation}{0} \\

We can see that Eq.~(\ref{19a}) is the best approximation on
Wilson loop at large distance. It is obvious that Wilson loop
exhibits area law for large $R$. In Eq.~(\ref{19a}) we also meet
linear term and logarithmic term of $R$ which are unexpected.
Those extra terms may be induced by the following reasons:
\begin{enumerate}
\item Due to error bar of numerical evaluation, the subtraction on
divergence in Eq.~(\ref{18}) is not precise.
\item Other error bar of numerical evaluation.
\item Due to lack of powerful computers, in our numerical
evaluation $R$ is not very large. Then the resulted Wilson loop
does not close to pure confinement phase. These extra may implies
existence of some ``mixing-phases''.
\end{enumerate}
However it is unambiguous that Wilson loop are dominant by
$R^2$-term for very large $R$. Consequently $\net$ SYM is confined
at long distance.

Now let us consider the case of short distance (smaller $R$). The
numerical data with a fitting curve are shown in fig.~4, and
fitting functions are listed as follows,
\\ \parbox{1in}{\mbox{}}\hfill\parbox{5.5in}{
\parbox{3in}{\begin{eqnarray*}
 A&=&-1.511+0.829R+0.0753\ln{R}   \\
 A&=&-1.682+1.128R-0.131R^2
 \end{eqnarray*}}
 \parbox{2.5in}{
 \beq{20}\label{20a}&&\chi^2=6.4\times 10^{-4} \\
 \label{20b}&& \chi^2=0.001
 \eeq}} \\
 \reseteqn
\renewcommand{\theequation}{\arabic{equation}}
It is unambiguous that Eq.~(\ref{20a}) is the best fit. Therefore
Wilson loop of $\net$ SYM exhibits logarithmic behavior at short
distance. It indicates that $\net$ SYM is asymptotical free or it
approaches to Coulomb phase at short distance. In addition, we see
that the constants in Eq.~(\ref{19a}) and in Eq.~(\ref{20a}) are
almost the same. Then those constants are harmless. It should be
associated to renormalization or normalization of Wilson loop.
This also indicates fitting in Eq.~(\ref{19a}) and in
Eq.~(\ref{20a}) are consistent.

Another interesting issue is that a phase transition occurs when
$R$ varies from small ones to large ones. This phase transition
occurs between the phase of asymptotical freedom and the phase of
confinement of non-abelian gauge theory. The transition point lies
in region $0.5<R<2$ in unit $3g_sN/2^{2/3}$. Consequently from
Eq.~(\ref{8}) we can define $\Lambda_{\rm SYM}=\ep^{2/3}/\alpha'$
as a scale associating to confinement. Using the results listed in
ref.\cite{KS00,HKO01}, we have:
\begin{itemize}
\item The masses of glueball and Kaluza-Klein (KK) states scale as
 \beq{21} m_{\rm glueball}\;\sim\; m_{\rm KK}\;\sim \;
  \frac{\ep^{2/3}}{g_sN\alpha'}\;\sim\;\Lambda_{\rm SYM}/g_sN. \eeq
We usually expect $m_{\rm glueball}\;\sim\; m_{\rm
KK}>\Lambda_{\rm SYM}$ such that it requires smaller
$g_s\sim\gym^2$ at large $N$. In other words, the glueball is
formed at near perturbative region. Using NSVZ $\beta$
function\cite{NSVZ} for $\net$ SYM, we have
 \beq{22} \frac{1}{\gym^2 N}=\frac{\frac{3}{16\pi^2}
  (\ln{\mu^2/\Lambda_{\rm SYM}^2}+c_0)}
  {1-2\frac{\ln{(\ln{\mu^2/\Lambda_{\rm SYM}^2}+c_0)}}
  {\ln{\mu^2/\Lambda_{\rm SYM}^2}+c_0}}, \eeq
where $c_0$ is a small constant regularizing divergence at
$\mu=\Lambda_{\rm SYM}$ and $\mu$ is the scale that glueball or KK
states are produced (roughly we can set $\mu\simeq m_{\rm
glueball}$). Since the singularity in KST solution is removed
through the blowing-up of the $S^3$ of $T^{1,1}$ we may include
relevant coefficients to Eq.~(\ref{21}),
 \beq{23}m_{\rm KK}\simeq 3I^{-1/2}(s_0)\Lambda_{\rm SYM}/g_sN, \eeq
where $I(s_0)$ was defined Eq.~(\ref{5}) and $s_0$, which denotes
the lowest energy detected by certain Wilson loop, is an zero of
$f(s)$. Considering radius/energy-scale relation for KST
background\cite{WH02}, in our nations $s\sim \mu/\mu_0$, where the
typical scale $\mu_0\simeq \Lambda_{\rm SYM}$ consequently it
corresponds to $R_0\simeq 1$ and $\bar{s}_0\simeq 0.8$. Then
defining $x=m_{\rm KK}/\Lambda_{\rm SYM}\simeq \mu/\Lambda_{\rm
SYM}$, we achieve the following equation
 \beq{24} x\simeq 3I^{-1/2}(\bar{s}_0/x)\frac{\frac{3}{16\pi^2}
 \ln{x^2}}{1-2\frac{\ln{\ln{x^2}}}{\ln{x^2}}}. \eeq
The solution of the above equation is $x\simeq 2.6$. We then
obtain
 \beq{25} m_{\rm glueball}\;\simeq\; m_{\rm KK}\simeq 2.6
  \Lambda_{\rm SYM}. \eeq
\item The mass of the baryon scales as
 \beq{26} M_b\;\sim\; N\frac{\ep^{2/3}}{\alpha'}\;\sim\;
 N\Lambda_{\rm SYM}. \eeq
\item The gluino condensate of $\net$ SYM scales as
 \beq{27} <\lambda\lambda>\;\sim\; N\frac{\ep^2}{\alpha'^3}
        \;\sim\; N\Lambda_{\rm SYM}^3. \eeq
\end{itemize}

In QCD we knew that there is a so-called ``confinement scale'',
$\Lambda_{\rm QCD}\simeq 0.3GeV$. Extending the above results to
QCD we have
 \beq{28}
  m_{\rm glueball}&\simeq& 2.6\Lambda_{\rm QCD}\simeq 0.8GeV,
  \nonumber \\
  M_{\rm baryon}&\simeq& N_c \Lambda_{\rm QCD}\simeq 0.9GeV, \\
  <\bar{\psi}\psi>&\simeq& \Lambda_{\rm QCD}^3\simeq (0.3GeV)^3,
   \nonumber \eeq
where we use gluino condensation to mimic quark condensation in
QCD, but we ignore $N_c$ in Eq.~(\ref{27}) since those chiral
quarks are fundamental representation of the gauge group. It is
surprising that the above results agree with phenomenology results
of QCD well. It apparently means that pure $\net$ SYM mimics some
low energy behaviors of QCD even without chiral multiplets.

\section{Summary and Comments}
\label{sec4}

Circular Wilson loop of $\net$ SYM is studied from the KST
supergravity solution. The Wilson loop exhibits area law at long
distance and logarithmic law at short distance. Therefore, $\net$
SYM lies in confinement phase at low energy scale, and it is
asymptotical free or it approaches to Coulomb phase at high
energy scale. Phase transition occurs when energy scale varies.
We also discuss glueball mass, baryon mass and gluino
condensation. It is shown that pure $\net$ SYM mimics some low
energy behaviors of QCD even without chiral multiplets.

We would like to make a comment on MN background. In principle,
the above calculations of this paper can be done to MN background
directly. The Euler-Lagrangian equation in MN solution, however,
does not possesses initial condition like Eq.~(\ref{13}). Instead,
it will be $f\sim$ fixed, $f'\to\infty$ at boundary. Since another
initial condition at an zero $r=r_0$ of $f(r)$ is $f\to 0$ fixed,
$f'\to -\infty$, the numerical evaluation on solution of
Euler-Lagrangian is very difficult according to a criterion in
Appendix. The study on this issue, however, is valuable because it
can help us to understand which phase of $\net$ SYM in UV is dual
to MN background.

\appendix*
\section{Numerical Evaluations}

The main difficulty on numerical evaluations are from near-zero
point behavior of $f(r)$, i.e., $f'(r\to r_0)\to -\infty$. It
makes Euler-Lagrangian equation~(\ref{12}) be stiff for $r\to
r_0$. If we use Eq.~(\ref{13}), or more precisely, Eq.~(\ref{16})
with very small $r$ as initial conditions of numerical evaluation,
due to inevitable error on initial condition in numerical
evaluation this error will be enlarged rapidly for $r\to r_0$.
Consequently we can not obtain a convergent solution.
Fortunately, since we can analytically obtain solution of
Eq.~(\ref{12}) for $r\to r_0$, we can use Eq.~(\ref{17}) as
initial condition of numerical evaluation. In this case, the
error bar induced by numerical approximation on initial condition
is controllable, and it will decrease when $r$ is evaluated from
$r_0$ to zero. So that we obtain the convergent result.

In our evaluation, we adopt the forth order Runge-Kutta method. We
take initial condition at $r_0-r=10^{-6}$ and interval $2\times
10^{-9}$. According to Eq.~(\ref{17}), the error bar on initial
condition is about $10^{-6}$. When we let initial value vary
$0.1\%$, the resulted $f(0)=R$ and area only vary $0.0001\%$. It
indicates that the numerical evaluation is indeed convergent, and
subleading order on initial condition~(\ref{17}) can be ignored
consistently.

Another detail is to fix the value of cut-off $\lambda$ defined in
Eq.~(\ref{18}). Theoretically, function $f'(r)$ is single-valued
in region $0\leq r\leq r_0$, and increases from $-\infty$ to zero
when $r$ varies from $r_0$ to zero. When string world-sheet lies
across branes, i.e., in region $r<0$, $f'(r)$ is again
single-valued and increases from $-\infty$ to zero when $r$
varies from $-r_0$ to zero. It means that $f'(r=0)$ is maximal
value of $f'(r)$ if we extend space to $-r_0\leq r\leq r_0$.
Numerically, however, we can not expect to obtain the maximal
value of $f'(r)$ at $r=0$ exactly and maximal value of $f'(r)$ is
not precise zero. Then we can take maximal-value point $f'(r)$ as
a natural cut-off. In our evaluation, the value of the cut-off
$\lambda$ varies from $0.0015$ to $0.0033$ when $R=f(0)$ varies
from $0.1$ to $5.5$. It implies that we should indeed consider the
second term in Eq.~(\ref{18}) for smaller $R$ in order to achieve
high precision fit.

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