% %You can use \\ for explicit line-break The Determination of the Quark-Gluon Mixed Condensate $g\langle\bar{q}\sigma_{\mu\nu}G_{\mu\nu} q\rangle$ from Lattice QCD
Noriyoshi \textsc{Ishii}$^2$
Makoto \textsc{Oka}$^1$ and Hideo \textsc{Suganuma}$^1$
hongo-adelaide-qGq.tex
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%\pubinfo{Vol.~10X, No.~X, Mmmmm YYYY}%Editorial Office will fill in this.
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%KUNS-1325\\PTPTeX ver.0.8\\ August, 1997}
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\markboth{%     %running head for odd-page (authors' name)
T.~Doi, N.~Ishii, M.~Oka and H.~Suganuma
}{%             %running head for even-page (`short' title)
The Quark-Gluon Mixed Condensate 
$g\langle\bar{q}\sigma_{\mu\nu}G_{\mu\nu} q\rangle$
in Lattice QCD
}

\title{%        %You can use \\ for explicit line-break
The Determination of 
the Quark-Gluon Mixed Condensate 
$g\langle\bar{q}\sigma_{\mu\nu}G_{\mu\nu} q\rangle$
from Lattice QCD
}

%\subtitle{Subtitle}    %use this when you want a subtitle

\author{%       %Use \scshape  for the family name
Takumi \textsc{Doi}$^{1,}$\footnote{E-mail: doi@th.phys.titech.ac.jp}, 
Noriyoshi \textsc{Ishii}$^2$, 
Makoto \textsc{Oka}$^1$ and 
Hideo \textsc{Suganuma}$^1$
}

\inst{%         %Affiliation, neglected when [addenda] or [errata]
$^1$ Department of Physics, Tokyo Institute of Technology, Tokyo 152-8551, Japan\\
$^2$ Radiation Laboratory, The Institute of Physical and Chemical Research (RIKEN),
Wako, 351-0198, Japan
}

%\publishedin{%         %Write this ONLY in cases of addenda and errata
%Prog.~Theor.~Phys.\ \textbf{XX} (19YY), page.}

%\recdate{Mmmmm DD, YYYY}%            %Editorial Office will fill in this.

\abst{%         %this abstract is neglected when [addenda] or [errata]
%
%
We study the quark-gluon mixed condensate 
$g\langle\bar{q}\sigma_{\mu\nu}G_{\mu\nu} q\rangle$,
using the SU(3)$_c$ lattice QCD with the Kogut-Susskind fermion 
at the quenched level.
%
We generate 100 gauge configurations
on the $16^4$ lattice with $\beta = 6.0$, and perform the 
measurement of 
$g\langle\bar{q}\sigma_{\mu\nu}G_{\mu\nu} q\rangle$
at 16 points in each gauge configuration
for each current quark mass of $m_q=21, 36, 52$ MeV.
Using the 1600 data for each $m_q$, 
we find 
$m_0^2 \equiv 
g\langle\bar{q}\sigma_{\mu\nu}G_{\mu\nu} q\rangle / 
\langle\bar{q}q\rangle
\simeq 2.5$ GeV$^2$ 
at the lattice scale of
$a^{-1} \simeq 2 {\rm GeV}$ in the chiral limit.
The large value of 
$g\langle\bar{q}\sigma_{\mu\nu}G_{\mu\nu} q\rangle$
suggests its importance in
the operator product expansion in QCD.
%
We study also chiral restoration at finite temperature
in terms of 
$g\langle\bar{q}\sigma_{\mu\nu}G_{\mu\nu} q\rangle$,
which is another chiral order parameter.
We present the lattice QCD results of
$g\langle\bar{q}\sigma_{\mu\nu}G_{\mu\nu} q\rangle$
at finite temperature.
%
%
}

\begin{document}

\maketitle


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{The importance of the quark-gluon mixed condensate $\qGq$}
\label{sec:intro}


The non-perturbative nature of 
quantum chromodynamics (QCD) 
is characterized by its nontrivial vacuum structure
such as various condensates.
%In order to understand the non-perturbative structure
%of the QCD vacuum, it is important to study 
%various condensates such as $\qq$ and 
%$\langle G_{\mu\nu}G^{\mu\nu} \rangle$.
%
%
Among various condensates, we emphasize here the importance 
of the quark-gluon mixed condensate 
$\qGq \equiv 
{g\braket{\bar{q}\sigma_{\mu\nu}G_{\mu\nu}^A \frac{1}{2}\lambda^A q}}$.
%
%
First, the mixed condensate represents a direct correlation 
between quarks and gluons in the QCD vacuum. 
In this point, the mixed condensate differs from $\qq$ and 
$\langle G_{\mu\nu}G^{\mu\nu} \rangle$ even at the qualitative level.
%
%
Second, this mixed condensate is another chiral order parameter 
of the second lowest dimension
and it flips the chirality of the quark as 
%
\begin{eqnarray}
\qGq = g\braket{\bar{q}_R\ (\sigma_{\mu\nu}G_{\mu\nu})\ q_L}
+ g\braket{\bar{q}_L\ (\sigma_{\mu\nu}G_{\mu\nu})\ q_R}.
\end{eqnarray}
%
%
Third, the mixed condensate plays an important role in
various QCD sum rules, especially in the baryons\cite{Ioffe,Dosch},
the light-heavy mesons~\cite{Dosch2}
and the exotic mesons~\cite{Latorre}.
%
%
%
%
%
%
%
In the QCD sum rules,
the value $m_0^2 \equiv \qGq / \qq \simeq 0.8\pm 0.2\ {\rm GeV}^2$ has been 
proposed as a result of the 
phenomenological analyses~\cite{Bel,RRY2,Ovchi,Narison1}.
%
%
However, in spite of the importance of $\qGq$, 
there was only one preliminary
lattice QCD study~\cite{K&S}, 
which was performed with insufficient
statistics (only 5 data)
using a small ($8^4$) 
and coarse lattice ($\beta=5.7$).



Therefore, we present the calculation of $\qGq$ in 
lattice QCD with a larger $(16^4)$ and finer $(\beta=6.0)$
lattice and with high statistics (1600 data).
We perform the measurement of $\qGq$ as well as $\qq$
in the SU(3)$_c$ lattice at the quenched level,
using the Kogut-Susskind (KS) fermion to respect chiral symmetry.
%Since these condensates are chiral order parameters,
%we adopt the Kogut-Susskind (KS) fermion,
%which keeps the explicit chiral symmetry in the massless quark limit.
%
We generate 100 gauge configurations and pick up 16 space-time points 
for each configuration to calculate the condensates.
With this high statistics of 1600 data for each quark mass,
we perform a reliable estimate for 
the ratio $m_0^2\equiv \qGq/\qq$ at the lattice scale of 
$a^{-1} \simeq 2 {\rm GeV}$ 
in the chiral limit~\cite{DOIS:qGq}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Lattice Formalism}
\label{sec:formalism}


Since
both of the  condensates  $\qq$  and  $\qGq$
are chiral order parameters, 
they are sensitive to explicit chiral-symmetry breaking.
Therefore, we adopt the KS fermion, which
preserves the explicit chiral symmetry in massless quark limit, $m=0$.
On the other hand, the Wilson and  the clover fermions would not
appropriate for our study, because these fermions
explicitly break chiral symmetry even for $m=0$.

The action for the KS fermion is described by 
spinless Grassmann fields $\bar{\chi},\chi$
and the gauge link-variable $U_\mu \equiv \exp[ -iagA_\mu ]$.
When the gauge field is set to be zero,
%In the absence of the gauge field, 
the SU(4)$_f$ quark-spinor field $q$ with spinor $i$ and flavor $f$
is expressed 
by $\chi$ as 
%
%
\begin{eqnarray}
%
\label{eq:q-ks_trans}
%
%
q_i^f (x) 
&=&  \frac{1}{8}
\sum_{\rho}\
( \Gamma_\rho )_{if}\ 
\chi (x+\rho ), \ \ 
%
\Gamma_\rho \equiv \gamma_1^{\rho_1} \gamma_2^{\rho_2} \gamma_3^{\rho_3} \gamma_4^{\rho_4}, \ \ \rho \equiv (\rho_1,\rho_2,\rho_3,\rho_4)
%
%
\end{eqnarray}
%
%
where $\rho$ with $\rho_\mu \in \{0,1\}$ runs over the 16 sites 
in the $2^4$ hypercube.
% and $i, f$ denote the spinor and the flavor indices, respectively.
When the gluon field is turned on, 
we insert
additional link-variables in Eq.~(\ref{eq:q-ks_trans})
in order to respect the gauge covariance.
%
%
%
Hence, the flavor-averaged condensates are expressed as 
%
%
\begin{eqnarray}
%
\label{eq:condensates-qq-def} 
%&&
%\lefteqn{
&&a^3 \qq 
= - \frac{1}{4}\sum_f {\rm Tr}\left[ \braket{q^f(x) \bar{q}^f(x)} \right] 
%%%\right]_{\stackrel
%%%	{\mbox{$\scriptstyle \mbox{\tiny color}$}}
%%%	{\mbox{$\scriptstyle \mbox{\tiny spinor}$}}}
%
%
= - \frac{1}{2^8}\sum_\rho
       {\rm Tr}\left[ \Gamma_\rho \Gamma_\rho^\dag\ 
	\braket{\chi(x+\rho ) \bar{\chi}(x+\rho )} \right], \\
%
\label{eq:condensates-qGq-def} 
%&&
%\lefteqn{
&&a^5 \qGq
= - \frac{1}{4}\sum_f \sum_{\mu,\nu}{\rm Tr}
	\left[ \braket{q^f(x) \bar{q}^f(x)} \sigma_{\mu\nu} G_{\mu\nu}\right]  \nonumber \\
%
%&& 
%\qquad 
&&= - \frac{1}{2^8} \sum_{\mu,\nu} \sum_\rho
{\rm Tr}\left[\
  {\cal U}_{\pm\mu,\pm\nu}(x+\rho)\ 
\Gamma_{\rho'} \Gamma_{\rho}^\dag\
\braket{\chi (x+\rho') 
\bar{\chi}(x+\rho)} \
\sigma_{\mu\nu}\ 
  G_{\mu\nu}^{\rm lat}(x+\rho)\ 
\right], \nonumber \\ 
%
&& \\
%
&&  \qquad\qquad \rho' \equiv \rho \pm \mu \pm \nu, \nonumber
%
%
\end{eqnarray}
%
%
where the sign $\pm$ is taken such that the sink point 
$(x+\rho') = (x+\rho\pm\mu\pm\nu )$
belongs to the same hypercube of the source point $(x+\rho)$.
%
Here, in order to respect the gauge covariance,
we have used, in Eq.~(\ref{eq:condensates-qGq-def}),
%
\begin{eqnarray}
{\cal U}_{\pm\mu,\pm\nu}(x) \equiv 
        \frac{1}{2}\left[\ U_{\pm\mu} (x) U_{\pm\nu} (x\pm\mu ) 
                + U_{\pm\nu} (x) U_{\pm\mu} (x\pm\nu )\ \right],
\label{eq:qGq-def-U}
\end{eqnarray}
%
where the definition of $U_{-\mu} (x) \equiv U^\dag_{\mu}(x-\mu)$ is used.



%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
On the gluon field strength $G_{\mu\nu}$,
we adopt the clover-type definition on the lattice,
%
%
\begin{eqnarray}
G_{\mu\nu}^{\rm lat}(s) = \frac{i}{16} \sum_A \lambda^A \ {\rm Tr}
\left[ 
 \lambda^A\{U_{\mu\nu}(s) +U_{\nu\,-\!\mu}(s)+U_{-\!\mu\,-\!\nu}(s)+U_{-\!\nu\,\mu}(s) \}
-\lambda^A \{\mu \leftrightarrow \nu \} 
\right], \nonumber \\
\label{eq:clover}
\end{eqnarray}
%
%
%
which contains no ${\cal O}(a)$ discretization error.
This is also an advanced point which is absent in Ref.~\citen{K&S}.





%%%We show the diagrams corresponding to 
%%%Eqs.~(\ref{eq:condensates-qq-def}),(\ref{eq:condensates-qGq-def}),(\ref{eq:clover}) 
%%%in figure~\ref{fig:diags}.
%%%
%%%
%%%\begin{figure}[tbh]
%%%\begin{center}
%%%\includegraphics[scale=0.8]{qq_qGq.diagrams.eps}
%%%\caption{\label{fig:diags}
%%%The diagrammatic representation of $\qq$ (left diagram) and
%%%$\qGq$ (middle and right diagrams).
%%%The solid line with the arrow denotes the propagator of $\chi$, and
%%%the wavy line the gauge link.
%%%}
%%%\end{center}
%%%\end{figure}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{The lattice QCD results}
\label{sec:results}



We calculate the condensates $\qq$ and $\qGq$ using the SU(3)$_c$ lattice QCD at the quenched level.
We perform the Monte Carlo simulation 
with the standard Wilson action at $\beta=6.0$ 
on the $16^4$ lattice.
%We use the standard Wilson action for $\beta=6.0$ 
%on the $16^4$ lattice.
%
%
The lattice unit $a\simeq 0.10\, {\rm fm}$ is obtained so as to reproduce 
the string tension $\sigma = 0.89\, {\rm GeV/fm}$~\cite{rabbit:3Q}.
%
%
We use the quark mass $m = 21, 36, 52$ MeV,
i.e., 
$ma = 0.0105,\ 0.0184,\ 0.0263$.
For the fields $\chi$, $\bar{\chi}$, the anti-periodic condition 
is imposed.
%The dependence on the boundary condition will be discussed later.
%
%
%
%
%We take the source point $(x+\rho)$ in 
%Eqs.~(\ref{eq:condensates-qq-def}),(\ref{eq:condensates-qGq-def}) as follows.
We measure the condensates 
on 16 different physical space-time points $x$ in each
configuration as $x=(x_1,x_2,x_3,x_4)$ with $x_\mu \in \{0, 8\}$ 
in the lattice unit.
%Note that $\rho$ is taken such that $\rho$ runs over the all sites 
%in the hypercube for each given $x$.
%
%
For each $m$, we calculate the flavor-averaged condensates,
and average them over the 16 space-time points and 
100 gauge configurations.
%Statistical errors are calculated using the jackknife error estimate.



Figure~\ref{fig:plot} shows the values of the
bare condensates $a^3\qq$ and $a^5\qGq$ against the quark mass $ma$.
We emphasize that the jackknife errors are almost negligible, 
due to the high statistics of $1600$ data for each quark mass.
%
From Fig.~\ref{fig:plot}, both $\qq$ and $\qGq$ show 
a clear linear behavior against the quark mass $m$.
Therefore, we fit the data with a linear function and determine 
the condensates in the chiral limit.
The results are summarized in Table~\ref{tab:mass-beta-6.0}.


\begin{figure}
%\begin{minipage}[t]{155mm}
%\begin{center}
\includegraphics[scale=0.6]{qq2_qGq2.plot.eps}
%\vspace*{-8mm}
\caption{\label{fig:plot}
The bare condensates $a^3\qq$ and $a^5\qGq$ plotted against the quark mass $ma$.
%at $\beta=6.0$.
The dashed lines denote the best linear extrapolations, 
and the cross symbols correspond to the values in the chiral limit. 
The jackknife errors are hidden in the dots.
}
%\vspace*{-3mm}
%\end{center}
%\end{minipage}
%\hspace{\fill}
\end{figure}


%%%%%%%%%%%%%%%%%%%

%\vspace*{-3mm}


\begin{table}[thb]
%\begin{center}
\caption{
The numerical results of $a^3\qq$ and $a^5\qGq$ for various $ma$.
The last column denotes their values in the chiral limit obtained 
by the linear chiral extrapolation.
\label{tab:mass-beta-6.0}
}
\begin{tabular}{ccccc}
\hline\hline
	   &  $ma=0.0263$     &  $ma=0.0184$     &  $ma=0.0105$      &  chiral limit\\
\hline
$a^3\qq$   &  $-0.04240(16)$  &  $-0.03247(15)$ &  $-0.02212(16)$  & 
$-0.00872(17)$ \\
$a^5\qGq$  &  $-0.01882(15)$  &  $-0.01498(14)$ &  $-0.01088(14)$  & 
$-0.00565(14)$ \\
\hline
\end{tabular}
%\end{center}
\end{table}

%\vspace*{-5mm}



We check the reliability of our lattice QCD results
by considering the finite volume artifact.
%
%
In order to estimate this artifact,
%we consider the finite volume artifact.
%As indicated by the Banks-Casher formula,
%the total volume $V$ should be large enough before the quark mass goes to zero.
we carry out the same calculation imposing the periodic boundary condition 
on the Grassmann fields $\chi$ and $\bar{\chi}$,
instead of the anti-periodic boundary condition,
keeping the other parameters same.
%we calculate the condensates imposing the periodic boundary 
%condition on $\chi, \bar{\chi}$,
%instead of the anti-periodic condition as before.
%If the physical volume is not large enough,
%the results would change.
% by using the different boundary condition.
%
We obtain that the results with different 
boundary conditions almost coincide within 
about 1\% difference, and thus we
conclude that the physical volume
$V \sim (1.6\ {\rm fm})^4$ in this simulation is large enough to 
avoid the finite volume artifact~\cite{DOIS:qGq}.
%
%





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\subsection{Determination of $m_0^2=\qGq/\qq$}
%\label{sec:M0}

The values  of the  condensates in the  continuum limit  
are to be obtained after the renormalization,
which, however, suffers from the
uncertainty of the non-perturbative effect.
%
%
As a more reliable quantity,
we provide   the  ratio  $m_0^2  \equiv  \qGq   /  \qq$,
which is free  from the  uncertainty from  the wave
function  renormalization  of the  quark.
%
%
%In  addition, the  dependence of  $m_0^2$  on the  lattice spacing  is
%reduced  to $a^2$,  while  $\qq$, $\qGq$  are  proportional to  $a^3$,
%$a^5$, respectively.
%
%
%We also note  that $m_0^2$ itself represents the importance 
%of $\qGq$ in OPE.
%because $\qGq$ appears as the next-to-leading 
%term to $\qq$ in  OPE in usual QCD sum rules and 
%can be parameterized by $m_0^2$.
%
%without referring to the absolute  value.







Now, we present the estimate of $m_0^2$ using the bare results
in SU(3)$_c$ lattice QCD as
%
%
%
%
%We find
%\vspace*{-2mm}
\begin{eqnarray}
%
m_0^2 \equiv \qGq / \qq  \simeq  2.5\ {\rm GeV}^2 \qquad (\beta = 6.0 \ \ {\rm or} \ \ a^{-1} \simeq 2{\rm GeV}).
%\vspace*{-2mm}
\end{eqnarray}
%
The large value of $m_0^2$ suggests the importance of
the   mixed  condensate   in  OPE.
%
%
Although we  do not include  renormalization effect, 
this  result
itself is determined very precisely~\cite{DOIS:qGq}.




%\newpage





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Discussions and Outlook}
\label{sec:summary}



For comparison with the standard value in the QCD sum rule,
we change the renormalization point from  $\mu\simeq\pi/a$ to $\mu\simeq 1$
GeV corresponding to the QCD sum rule.
Following Ref.~\citen{K&S},  we  first  take  the lattice results  of  the
condensates  as  the starting  point  of  the  flow, then  rescale  the
condensates perturbatively.
We adopt the anomalous dimensions at the one-loop level~\cite{Narison2},
and choose the parameters $\Lambda_{\rm QCD} = 200-300 \mbox{MeV}$ and
$N_f=0$ corresponding to quenched lattice QCD.
We obtain
$ m_0^2  \big|_{\mu=1{\rm GeV}} \equiv  \qGq/\qq  \big|_{\mu=1{\rm GeV}} \sim  3.5-3.7 \  
{\rm GeV}^2$.
%at $\mu=1$GeV.
%
%
%from $\qq  \big|_{\mu} \sim -  (0.0477 - 0.0506 )  \ {\rm GeV}^3 
%=  - (0.36-0.37  {\rm GeV})^3$ and 
%
%$\qGq \big|_{\mu}  \sim - (0.176- 0.177)  \ {\rm GeV}^5$.
%
%
%
Comparing with the standard value of  $m_0^2 = 0.8 \pm 0.2$ GeV$^2$ in the
QCD sum rule,  our calculation results in a  rather large value. (Note
that  the instanton  model has  made  a slightly  larger estimate  as
$m_0^2 \simeq 1.4$ GeV$^2$ at $\mu\simeq 0.6$ GeV~\cite{Polyakov}.)
%
For a more definite determination of $m_0^2$, the non-perturbative
renormalization scheme may be desired.
%which is also  expected to  improve the value  of $\qq$  simultaneously.
%although this scheme would require a significant calculation cost.
% \cite{Martinelli}.


\begin{figure}[tbh]
%\begin{minipage}[t]{155mm}
\begin{center}
\includegraphics[scale=0.29]{qGq.finite_T.ratio.eps}
%\vspace*{-8mm}
\caption{\label{fig:qGq_finite_T}
The quark-gluon mixed condensate 
$\qGq_T/\qGq_{T=0}$
%
%$g\langle \bar q \sigma_{\mu\nu}G q (T)\rangle/g\langle \bar q \sigma_{\mu\nu}G q (T=0)\rangle$ 
%
%normalized by the zero-temperature value 
plotted against the temperature $T$.
The jackknife errors are hidden in the triangles.
The vertical dashed line denotes the critical temperature $T_c \simeq 280{\rm MeV}$ at the quenched level.
}
\end{center}
%\vspace*{-8mm}
%\end{minipage}
%\hspace{\fill}
\end{figure}



We  again  emphasize  that  the  mixed condensate  $\qGq$  plays  the
important roles  in various contexts in quark  hadron physics.  
Hence, it is preferable to perform  further studies. 
In particular, the  thermal  effects  are interesting in relation 
to chiral restoration, because the  mixed  condensate is  another 
chiral  order parameter.  
Considering also the on-going experiments for the finite-temperature QCD in the RHIC project, 
we investigate the thermal effects on the mixed condensate $\qGq$.
We perform the calculation using the $16^3\times N_t$ lattices 
with $N_t=16, 12, 10, 8, 6, 4$ at $\beta=6.0$.
Figure~\ref{fig:qGq_finite_T} shows the lattice QCD results for the mixed 
condensate at finite temperature.
We find a drastic change of the mixed condensate around the critical temperature $T_c$, 
which reflects chiral-symmetry restoration~\cite{DOIS:T}.



In summary, 
we  have studied  the quark-gluon  mixed condensate  $\qGq$  using 
SU(3)$_c$ lattice QCD with  the KS fermion at the quenched
level.
For each  quark mass of  $m_q=21, 36, 52$  MeV, we have  generated 100
gauge configurations  on the $16^4$ lattice with  $\beta =6.0$.
%
%
Using the 1600 data for each $m_q$,  
we have found $m_0^2 \equiv \qGq / \qq
\simeq 2.5$ GeV$^2$ in the chiral limit at the lattice scale corresponding to 
$\beta=6.0$ or $a^{-1} \simeq$ 2 GeV.
%
We have also investigated $\qGq$ at finite temperature in lattice QCD.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%





\section*{Acknowledgements}

We would like to thank Dr. H. Matsufuru for his useful comments on 
the programming technique. 
This work is supported in part by the Grant for Scientific Research 
(No.11640261, No.12640274 and No.13011533) 
from the Ministry of Education, Culture, Science and Technology, Japan.
T.D. acknowledges the support of 
the JSPS Research Fellowships for Young Scientists.
The Monte Carlo simulations have been performed on the 
NEC SX-5 supercomputer at Osaka University.


%\appendix
%\section{First Appendix} %Empty argument \section{} yields `Appendix'. 
%
%\section{Second Appendix}


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%    \JL : general journals                 \andvol : Vol (Year) Page
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%    \AJ   : Astrophys. J.           \NC         : Nuovo Cim.
%    \ANN  : Ann. of Phys.           \NPA, \NPB  : Nucl. Phys. [A,B]
%    \CMP  : Commun. Math. Phys.     \PLA, \PLB  : Phys. Lett. [A,B]
%    \IJMP : Int. J. Mod. Phys.      \PRA - \PRE : Phys. Rev. [A-E]     
%    \JHEP : J. High Energy Phys.    \PRL        : Phys. Rev. Lett.
%    \JMP  : J. Math. Phys.          \PRP        : Phys. Rep.
%    \JP   : J. of Phys.             \PTP        : Prog. Theor. Phys.     
%    \JPSJ : J. Phys. Soc. Jpn.      \PTPS       : Prog. Theor. Phys. Suppl.
% Usage:
%  \PRD{45,1990,345}          ==> Phys.~Rev.\ \textbf{D45} (1990), 345
%  \JL{Nature,418,2002,123}   ==> Nature \textbf{418} (2002), 123
%  \andvol{B123,1995,1020}    ==> \textbf{B123} (1995), 1020
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  

\bibitem{Ioffe}     {B. L. Ioffe,
                            Nucl. Phys.  {\bf B188} (1981), 317
			[Errata; \textbf{B191} (1981), 591].}

\bibitem{Dosch}     {H.G. Dosch, M. Jamin and S. Narison,
                        Phys. Lett. {\bf B220} (1989), 251. }

\bibitem{Dosch2}     {H.G. Dosch and S. Narison,
                        Phys. Lett. {\bf B417} (1998), 173
                        and references therein.}

\bibitem{Latorre}    {J.I. Latorre, P. Pascual and S. Narison,
                        Z. Phys. {\bf C34} (1987), 347.}

\bibitem{Bel}       {V.M. Belyaev and B.L. Ioffe,
                        Sov. Phys. JETP {\bf 56} (1982), 493.}

\bibitem{RRY2}     {L.J. Reinders, H.R. Rubinstein, and S. Yazaki,
                        Phys. Lett. B {\bf 120} (1983), 209.}

\bibitem{Ovchi}     {A.A. Ovchinnikov et al.,
                        Sov. J. Nucl. Phys. {\bf 48} (1988), 721.
                        (Yad. Fiz. {\bf 48} (1988), 1135).}

\bibitem{Narison1}     {S. Narison,
                        Phys. Lett. {\bf B210} (1988), 238.}



\bibitem{K&S}       {M. Kremer and G. Schierholz,
                        Phys. Lett. {\bf B194} (1987), 283.}

\bibitem{DOIS:qGq}  {T. Doi, N. Ishii, M. Oka and H. Suganuma,
			Phys. Rev. {\bf D} (2003) in press,
			hep-lat/0211039.}

\bibitem{rabbit:3Q}    {T.T. Takahashi, H. Suganuma, Y. Nemoto 
                        and H. Matsufuru,
                        Phys. Rev. {\bf D65} (2002), 114509.}

%\bibitem{Martinelli} {G. Martinelli, C. Pittori, C.T. Sachrajda,
%			M. Testa, and A. Vladikas,
%			Nucl. Phys. {\bf B445} 81 (1995),.}

\bibitem{Narison2}  { S. Narison and R. Tarrach,
                        Phys. Lett. {\bf B 125} (1983), 217.}

\bibitem{Polyakov}    {M.V. Polyakov and C. Weiss,
                        Phys. Lett. {\bf B387} (1996), 841.}

\bibitem{DOIS:T}  {T. Doi, N. Ishii, M. Oka and H. Suganuma,
                        Nucl. Phys. {\bf A} (2003) in press, hep-lat/0212006;
                        Proc. Int. Conf. on
                        ``Quark Confinement and the Hadron Spectrum V'',
                        Italy, 2002, hep-lat/0212025.}


\end{thebibliography}
\end{document}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%











hep-lat0303014.extracted_bib
\begin{thebibliography}{99}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Some macros are available for the bibliography:
%  o for general use
%    \JL : general journals                 \andvol : Vol (Year) Page
%  o for individual journal 
%    \AJ   : Astrophys. J.           \NC         : Nuovo Cim.
%    \ANN  : Ann. of Phys.           \NPA, \NPB  : Nucl. Phys. [A,B]
%    \CMP  : Commun. Math. Phys.     \PLA, \PLB  : Phys. Lett. [A,B]
%    \IJMP : Int. J. Mod. Phys.      \PRA - \PRE : Phys. Rev. [A-E]     
%    \JHEP : J. High Energy Phys.    \PRL        : Phys. Rev. Lett.
%    \JMP  : J. Math. Phys.          \PRP        : Phys. Rep.
%    \JP   : J. of Phys.             \PTP        : Prog. Theor. Phys.     
%    \JPSJ : J. Phys. Soc. Jpn.      \PTPS       : Prog. Theor. Phys. Suppl.
% Usage:
%  \PRD{45,1990,345}          ==> Phys.~Rev.\ \textbf{D45} (1990), 345
%  \JL{Nature,418,2002,123}   ==> Nature \textbf{418} (2002), 123
%  \andvol{B123,1995,1020}    ==> \textbf{B123} (1995), 1020
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  

\bibitem{Ioffe}     {B. L. Ioffe,
                            Nucl. Phys.  {\bf B188} (1981), 317
			[Errata; \textbf{B191} (1981), 591].}

\bibitem{Dosch}     {H.G. Dosch, M. Jamin and S. Narison,
                        Phys. Lett. {\bf B220} (1989), 251. }

\bibitem{Dosch2}     {H.G. Dosch and S. Narison,
                        Phys. Lett. {\bf B417} (1998), 173
                        and references therein.}

\bibitem{Latorre}    {J.I. Latorre, P. Pascual and S. Narison,
                        Z. Phys. {\bf C34} (1987), 347.}

\bibitem{Bel}       {V.M. Belyaev and B.L. Ioffe,
                        Sov. Phys. JETP {\bf 56} (1982), 493.}

\bibitem{RRY2}     {L.J. Reinders, H.R. Rubinstein, and S. Yazaki,
                        Phys. Lett. B {\bf 120} (1983), 209.}

\bibitem{Ovchi}     {A.A. Ovchinnikov et al.,
                        Sov. J. Nucl. Phys. {\bf 48} (1988), 721.
                        (Yad. Fiz. {\bf 48} (1988), 1135).}

\bibitem{Narison1}     {S. Narison,
                        Phys. Lett. {\bf B210} (1988), 238.}



\bibitem{K&S}       {M. Kremer and G. Schierholz,
                        Phys. Lett. {\bf B194} (1987), 283.}

\bibitem{DOIS:qGq}  {T. Doi, N. Ishii, M. Oka and H. Suganuma,
			Phys. Rev. {\bf D} (2003) in press,
			hep-lat/0211039.}

\bibitem{rabbit:3Q}    {T.T. Takahashi, H. Suganuma, Y. Nemoto 
                        and H. Matsufuru,
                        Phys. Rev. {\bf D65} (2002), 114509.}

%\bibitem{Martinelli} {G. Martinelli, C. Pittori, C.T. Sachrajda,
%			M. Testa, and A. Vladikas,
%			Nucl. Phys. {\bf B445} 81 (1995),.}

\bibitem{Narison2}  { S. Narison and R. Tarrach,
                        Phys. Lett. {\bf B 125} (1983), 217.}

\bibitem{Polyakov}    {M.V. Polyakov and C. Weiss,
                        Phys. Lett. {\bf B387} (1996), 841.}

\bibitem{DOIS:T}  {T. Doi, N. Ishii, M. Oka and H. Suganuma,
                        Nucl. Phys. {\bf A} (2003) in press, hep-lat/0212006;
                        Proc. Int. Conf. on
                        ``Quark Confinement and the Hadron Spectrum V'',
                        Italy, 2002, hep-lat/0212025.}


\end{thebibliography}