\vspace*{-35pt} {\normalsize \hfill {KEK-CP-116}} \\ \vspace*{20pt} % %% \vspace*{-6pt} % %% {\normalsize \hfill {Oct.\ 2000}} \\ Light hadron spectrum and quark masses
T.~Kaneko\address{High Energy Accelerator Research Organization (KEK), Tsukuba, Ibaraki 305-0801, Japan}
Recent developments in lattice QCD calculations of the light hadron spectrum and quark masses are reviewed.
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\title{
\vspace*{-35pt}
{\normalsize \hfill {\sf KEK-CP-116}} \\
\vspace*{20pt}
%   %% \vspace*{-6pt}
%   %% {\normalsize \hfill {\sf Oct.\ 2000}} \\
Light hadron spectrum and quark masses
}
\author{
T.~Kaneko\address{High Energy Accelerator Research Organization
                     (KEK), Tsukuba, Ibaraki 305-0801, Japan}
} 

\begin{document}

\begin{abstract}
  Recent developments in lattice QCD calculations
  of the light hadron spectrum and quark masses are reviewed.
\end{abstract}

\maketitle
\setcounter{footnote}{0}


%--- introduction -----------------------------------------------
\section{Introduction}
\label{sec:intro}


%--- lattice calculation of spectrum for test of QCD ---


One of the main goals of lattice QCD is to confirm the
validity of QCD as the theory of strong interactions.
Calculation of the hadron spectrum is fundamental in this respect. 
% Within the quenched approximation, large effort over the years 
In the quenched approximation, large effort over the years 
has culminated in the work by CP-PACS\cite{CP-PACS.Nf0} 
which revealed a systematic deviation 
of the quenched spectrum from experiment. 
%
%--- search for sea quark effects ---
%
Since then, the focus has shifted to full QCD including 
dynamical quarks, and a number of large-scale simulations have been 
executed in $N_f\!=\!2$ full QCD\cite{review.full}.  
A crucial issue here is how sea quark effects manifest in 
the spectrum.  Work to explore this question has continued this year. 


%--- Nf=3 ------------------------------------------


A realistic QCD simulation has to treat the strange quark dynamically. 
The algorithmic development to treat odd number of flavors has accelerated 
in the last few years\cite{review.algo}.  This has led to several attempts 
toward realistic calculations in $N_f\!=\!3$ QCD very recently.


%--- ChPT ------------------------------------------

One of the major uncertainties in lattice calculations
arises from the chiral extrapolation toward realistically 
light quark masses.
One possibility to control this extrapolation is to use 
chiral perturbation theory (ChPT)\cite{ChPT} as a guide.
Test of the validity of ChPT is an important step for this purpose, 
and full QCD data to examine this issue are becoming available. 


%--- improvement of simulation methods -------------

Another recent trend is the improvement of formulations of lattice QCD 
such as the domain-wall fermion\cite{review.GW} 
and fermions defined on anisotropic lattices\cite{aniso}. 
Quenched QCD presents a testing ground of these 
formulations, and some realistic simulations have been made.


%--- determination of fundamental param. -----------

Spectrum calculations allow a simultaneous determination of 
the strong coupling constant and quark masses 
which are 
the fundamental parameters of the Standard Model. 
Precise determination of these quantities is an important 
issue in lattice QCD\cite{review.alphas,review.mq}.
% The strong coupling constant and quark masses 
% are the fundamental parameters of the Standard Model.
% Precise determination of these quantities is an important 
% issue in lattice QCD\cite{review.alphas,review.mq}.
New studies in this area have also been reported at this conference.


%--- outline --------------------------------------

In this review, we present the status of 
lattice calculations of the light hadron spectrum and  
the fundamental parameters of QCD with focus on the points above.
%
We concentrate on results obtained with the Wilson-type quark actions, 
as those with the Kogut-Susskind (KS) action are covered  
in a separate talk\cite{KS.Toussaint}.
%
%The outline is as follows.
%
In Sec.~\ref{sec:Nf2},
we discuss sea quark effects in $N_f\!=\!2$ full QCD.
Recent developments toward $N_f\!=\!3$ simulations are reviewed 
in Sec.~\ref{sec:Nf3}, 
%
and verification of ChPT is discussed in Sec.~\ref{sec:ChPT}.
%
Section \ref{sec:Nf0} is devoted to recent developments
of spectroscopic studies in quenched QCD.
%         
The status of the strong coupling constant  
and quark masses are updated in Secs.~\ref{sec:alpha_s} 
and \ref{sec:mq}. 
%
A brief conclusion is given in Sec.~\ref{sec:conclusion}.


\begin{table*}[t]
\caption{
  Recent simulations in $N_f\!=\!2$ QCD.
  Values of $c_{\rm SW}$ for the clover quark action\cite{SWaction}
  are shown in brackets, where NP and TP stand for non-perturbative 
  and tadpole-improved values, respectively.
}
\begin{tabular}{llllllll}
\hline 
\hline 
group            & gauge        & quark                  
                 & $a_s$[fm]    & $L_s$[fm]             
                 & $m_{\rm PS,sea}/m_{\rm V,sea}$ & ref.         \\
\hline
SESAM            & plaquette    & Wilson                 
                 & 0.08         & 1.3          
                 & 0.68--0.83   & \cite{SESAM.Nf2}       \\
T$\chi$L         & plaquette    & Wilson                 
                 & 0.08         & 1.9           
                 & 0.57, 0.70   & \cite{TchiL.Nf2}       \\ 
SESAM-T$\chi$L   & plaquette    & Wilson             
                 & 0.09         & 1.5         
                 & 0.68--0.85   & \cite{SESAM.Nf2.beta55}\\ 
% Orth {\it et al.}& plaquette    & Wilson                 
%                  & 0.12         & 1.9         
%                  & 0.4          & \cite{GRAL.Nf2}        \\
\hline
UKQCD            & plaquette    & clover (1.76)     
                 & 0.12         & 1.0--1.9     
                 & 0.67--0.86   & \cite{UKQCD.Nf2.csw176}\\
UKQCD            & plaquette    & clover (NP) 
                 & 0.10         & 1.7
                 & 0.70--0.84   & \cite{UKQCD.Nf2.lat00,UKQCD.Nf2}\\
UKQCD            & plaquette    & clover (NP) 
                 & 0.10         & 1.6
                 & 0.58         & \cite{UKQCD.Nf2.lat00,UKQCD.Nf2}\\
QCDSF            & plaquette    & clover (NP)      
                 & 0.09         & 2.1,1.5      
                 & 0.69, 0.76   & \cite{QCDSF.Nf2.lat00} \\
JLQCD            & plaquette    & clover (NP)      
                 & 0.09         & 1.1--1.8      
                 & 0.60--0.80   & \cite{JLQCD.Nf2.lat00,JLQCD.Nf2}\\
\hline
CP-PACS          & RG-improved\cite{RGgauge}  
                 & clover (TP)  
                 & 0.11--0.22   & 2.5--2.6      
                 & 0.55--0.81   & \cite{CP-PACS.Nf2}     \\
\hline
\hline
Columbia         & plaquette    & KS
                 & 0.09         & 1.5           
                 & 0.57--0.70   & \cite{Coulumbia.Nf2}   \\
MILC             & plaquette    & KS
                 & 0.10--0.32   & 2.4--3.8      
                 & 0.3--0.8     & \cite{MILC.Nf2.stdKS}  \\
Columbia         & plaquette    & KS ($\xi\!\sim\!1.8$--5.0) 
                 & 0.23--0.34   & 3.7--5.4      
                 & 0.35--0.80   & \cite{Columbia.Nf2.aniso} \\
\hline
MILC             & Symanzik\cite{1loopSym}
                 & improved KS\cite{improvedKS}
                 & 0.13         & 2.6          
                 & 0.50         & \cite{MILC.Nf3}        \\
\hline
\hline
\end{tabular}
\label{tab:simulations.Nf2}
\vspace{-2mm}
\end{table*}


% {\bf non-perturbative improvement}
%
% non-perturbative $O(a)$ improvement and renormalization 
% are 
%
% improvement coefficients and renormalization factors 
% have been calculated using three methods, RI/MOM\cite{NP.RIMOM}
% Schr\"odinger functional(SF)\cite{NP.SF} and 
% from Ward identities\cite{NP.WI}.
%
% recent progress 
% 
% 1. $b_m$ and $b_A\!-\!b_P$ are calculated non-perturbatively 
%    in \cite{{NP.WI,NP-bm.SF,NP-cA.WI.LANL}.
%    Fully non-perturbative renormalization of quark masses
%    is now possible.
%         
% 2. Three groups reported new estimates of $c_A$.
%    Deviation between $c_A$ from SF and WI method at 
%    strong coupling is confirmed.
%
%     % Hernandez {\it et al.} estimates $c_A$ in SF scheme 
%     % at $\beta=5.85$\cite{NP-cA.SF}.
%   
%     % LANL and UKQCD collaboration determined $c_A$ 
%     % from Ward identities at 
%     % $\beta=5.93$-6.4\cite{NP-cA.WI.LANL,NP-cA.WI.UKQCD}.
%
%    A simple explanation of the difference is higher order 
%    correction ($O(a)$ in $c_A$) arising from the difference in 
%    the improvement conditions.
%    However, more detailed studies of the scaling property
%    is required to draw a definite conclusion.
%



\section{Light hadron spectrum in $N_f\!=\!2$ QCD}
\label{sec:Nf2}



\subsection{Recent simulations}

%--- status ---------------------------------------------

Recent simulations in two-flavor QCD 
are listed in Table~\ref{tab:simulations.Nf2}.
%
% status before lattice 2000
%
Most of the simulations with the Wilson-type quark action 
were made with the plaquette gauge action at a single lattice 
spacing $a\sim 0.1$~fm. 
An exception is the CP-PACS study\cite{CP-PACS.Nf2}
which explored the range $a \! \sim \! 0.1$--0.2~fm with the use of 
an improved gauge action.  
This work reported clear sea quark effects in the meson spectrum 
after the continuum extrapolation.
%
% CP-PACS\cite{CP-PACS.Nf2}
% JLQCD\cite{JLQCD.Nf2.lat00},
% SESAM\cite{SESAM.Nf2}, 
% UKQCD\cite{UKQCD.Nf2.csw176,UKQCD.Nf2.lat00}
% UKQCD-QCDSF Collaborations\cite{QCDSF.Nf2.lat00}
%
%
% KS
%
% In simulations with KS fermions,
% much efforts have been spent in investigation of 
% finite size effects and scaling violation.


%
% new results
%
New results were reported by UKQCD\cite{UKQCD.Nf2} 
and JLQCD\cite{JLQCD.Nf2} at this conference.
The two simulations still use the plaquette gauge action and work 
at a single lattice spacing $a\!\sim\!0.1$~fm, 
but the quark action is fully $O(a)$ improved 
with $c_{\rm SW}$ determined non-perturvatively\cite{NP-csw.ALPHA}.

The UKQCD simulation shifts $\beta$ with sea quark mass, 
keeping the Sommer scale $r_0/a$\cite{r0} fixed. 
The JLQCD runs were performed at a fixed $\beta$.
%
% JLQCD studied finite size effects by simulating
% 3 different spatial volumes($L\!\sim\!1.1$, 1.4, 1.8~fm).
%
Another important difference 
is the range of quark mass covered in the two simulations: 
JLQCD explored light sea quark masses down to 
$m_{\rm PS,sea}/m_{\rm V,sea}\!\sim\!0.6$,
whilst the UKQCD's lightest point is around 
$m_{\rm PS,sea}/m_{\rm V,sea}\!\sim\!0.7$.\footnote[2]{
Although UKQCD made another simulation at a smaller sea quark mass
($m_{\rm PS,sea}/m_{\rm V,sea}\!\sim\!0.58$, $L_s\!\sim\!1.6$~fm),
finite size effects seem to be significant there\cite{JLQCD.Nf2}.}
We note that Orth {\it et al.} attempts to reduce quark masses 
to even lighter points, 
but actual simulations have not yet been started\cite{GRAL.Nf2}.

%
% For the KS fermions, MILC also reported new results of 
% the hadron spectrum\cite{MILC.Nf3}.
% Their simulations were performed at $a^{-1}\!\sim\!1.5$~GeV
% using improved glue and quarks.
% 
% Other simulations by Orth {\it et al.}\cite{GRAL.Nf2}
% and Columbia group\cite{Columbia.Nf2.aniso}
% have been just started and are at the stage of parameter tuning.
%



\subsection{Sea quark effects in meson spectrum}  
%\subsection{Meson masses at simulated points}  
         
Figure~\ref{fig:Nf2:mVvsmPS2} shows 
how the valence quark mass dependence of the vector meson mass 
varies as a function of sea quark mass 
in the JLQCD and UKQCD simulations.
%
In both UKQCD and JLQCD data, 
the set of points for the sea quark masses 
down to $m_{\rm PS,sea}/m_{\rm V,sea}\!\sim\!0.7$ almost overlap.  
On the other hand, the JLQCD data for a lighter sea quark 
$m_{\rm PS,sea}/m_{\rm V,sea}\!\sim\!0.6$ clearly show a larger slope, 
and are closer to the experimental points marked by asterisks.  

%i.e. larger slope in full QCD.
% This leads to a larger value of $K^*$-$K$ hyperfine splitting, 
% which is known to be small in quenched QCD.
%
%In the UKQCD data, on the other hand, 
%the sea quark effect is unclear,
%presumably because the sea quark mass is 
%limited in the heavier region $m_{\rm PS}/m_{\rm V}\!\geq\!0.7$,
%below which the sea quark effect become clear in the JLQCD data.
%
% other reasons
%  i) small statistical error in JLQCD data 
%       <- large volume, wave function for smearing 
% ii) smaller valence quark masses in JLQCD data (?)
%
%

The smallest sea quark mass in most of the previous simulations 
stopped at $m_{\rm PS,sea}/m_{\rm V, sea} \! \sim \! 0.7$, 
which explains why sea quark effects were not clearly seen 
in these simulations.  
%
% in CP-PACS data
%
The CP-PACS data did cover the range down to   
$m_{\rm PS,sea}/m_{\rm V, sea} \! \sim \! 0.6$.
The trend of sea quark effects as observed by JLQCD
is present at $a\!\sim\!0.1$~fm,
while it is less clear on coarser lattices 
due to larger error of $r_0$\cite{CP-PACS.mVvsmPS2}. 
% The trend of sea quark effect as observed in the JLQCD data,
% while present, 
% was less clear due to larger systematic error 
% arising from that of $r_0$. 

%
% It is also noteworthy that 
% this sea quark effect is weakened by finite size 
% effects\cite{JLQCD.Nf2}.


% \subsection{J parameter}


% The $J$ parameter\cite{J} can be used 
% as a quantitative measure of the sea quark effect.
%
% definition of $J$ parameter 
% $m_{\rm V} (d m_{\rm V}/d m_{\rm PS}^2)$ at 
% $m_{\rm V}/m_{\rm PS}\!=\!m_{K^*}/m_{K}$.
%
Figure~\ref{fig:Nf2:J} shows recent results of the $J$ parameter\cite{J}
calculated with differentiation in the valence quark mass 
as a function of the sea quark mass.
%
% From the first calculation of $J$\cite{J},
% it is well known that $J$ in quenched QCD 
% is smaller than its experimental value.
% New results from JLQCD, UKQCD and MILC are consistent 
% with this observation.
%
%In full QCD
The JLQCD data show 
%%%%%%
a trend that the value of $J$, which is consistent with that of quenched 
QCD for heavier sea quark masses, increases as the sea quark mass 
decreases.  The latter feature can be observed also in the UKQCD data.
%%%%%%
%a trend of increase of $J$ toward the chiral limit.
%A similar trend can be observed also in the UKQCD data.
% albeit with a larger error 
% due to the chiral extrapolation of the valence quark mass
% corresponding to kaons
% (see the right panel of Fig.~\ref{fig:Nf2:mVvsmPS2}).
As a result, 
the values extrapolated to the physical sea quark mass
are closer to experiment than in quenched QCD.

The $J$ parameter in full QCD 
can also be obtained using the experimental definition 
$J \equiv m_{K^*}(m_{K^*}\!-\!m_{\rho})/(m_{K}^2\!-\!m_{\pi}^2)$, 
or from the combined chiral extrapolation as a function of 
sea and valence quark masses.  The results are also plotted at 
$(r_0 m_{\rm PS,sea})^2\!=\!0$ in Fig.~\ref{fig:Nf2:J}.
% %
% %
% % 1. J from the experimental definition(SESAM,MILC)
% %    $J \equiv m_{K^*}(m_{K^*}\!-\!m_{\rho})/(m_{K}^2\!-\!m_{\pi}^2)$
% % 2. J from the combined chiral fit(CP-PACS,JLQCD,UKQCD)
% %
% %
They lead to the same conclusion 
that $J$ increases with dynamical quarks.
%
An exception is the SESAM point
presumably due to heavier sea quark masses
$m_{\rm PS,sea}/m_{\rm V,sea}\!\gtrsim\!0.7$
and scaling violation in their simulations.
%
Recent results with KS fermions also observed
similar sea quark effects\cite{MILC.Nf3}.


\begin{figure}[t]
\begin{center}
   \leavevmode
   \includegraphics*[width=66mm,clip]{./mVvsmPS2-r0.eps}
\end{center}
\vspace*{-15mm}
\caption{
   Vector meson mass as a function of pseudo-scalar meson mass squared,
   normalized in units of $r_0\!=\!0.49$~fm from JLQCD and UKQCD. 
%  Left and right figures show data from JLQCD and UKQCD,
%  respectively. 
%  Experimental values are plotted
%  using a phenomenological value .
}
\vspace*{-5mm}
\label{fig:Nf2:mVvsmPS2}
\end{figure}


\begin{figure}[t]
\vspace*{-2mm}
\begin{center}
   \leavevmode
   \includegraphics*[width=65mm,clip]{./J_mPS2-Nf2.eps}
\end{center}
\vspace*{-15mm}
\caption{
  Recent results of $J$ parameters
  in $N_f\!=\!2$ full QCD and quenched QCD. 
% Values extrapolated to the 
% physical quark mass are slightly shifted for better
% visibility. 
  Results presented this year are plotted 
  in filled symbols.
}
\vspace*{-6mm}
\label{fig:Nf2:J}
\end{figure}



% \subsection{Meson spectrum}

The results so far refer to a single lattice spacing 
of $a \! \sim \! 0.1$~fm. In Fig.~\ref{fig:Nf2:meson} 
we reproduce the previous results by CP-PACS\cite{CP-PACS.Nf2}
on the strange vector meson masses % ($m_{K^*}$ and $m_{\phi}$)
for a set of lattice spacings, 
and superimpose the new JLQCD points. % (circles).  
%
% Since JLQCD results is available 
% only at a single lattice spacing $a^{-1}\!\sim\!2$~GeV,
% the scaling violation is expected to be small 
% by use of the non-perturbatively $O(a)$-improved action.
%
The JLQCD points are consistent with the CP-PACS points
both for $N_f\!=\!2$ and $N_f\!=\!0$ (quenched), 
showing an increase of hyperfine splitting 
as dynamical quark effects are included.    

Strictly speaking comparison of the two data
should be performed in the continuum limit, albeit 
scaling violation in the JLQCD data may be expected to be small 
by the use of the non-perturbatively $O(a)$-improved action. 
This point should be checked in future studies.



\subsection{Baryons and finite size effects}  

Significant sea quark effects are also expected in the baryon spectrum,
as it sizably deviates from experiment in quenched QCD\cite{CP-PACS.Nf0}.
% Currently, 
However, no group has found clear evidence.
A possible reason is finite size effects,
which we expect to be more important for baryons than for mesons. 

Suppression of systematic errors to a few percent level 
is required for a convincing study of sea quark effects.
%
%
In order to estimate the lattice size needed 
to achieve this accuracy,
we fit hadron mass data of UKQCD\cite{UKQCD.Nf2.csw176}
and JLQCD\cite{JLQCD.Nf2} on several lattice volumes 
to an ansats $m(L)=m(L\!=\!\infty)+c/L^3$\cite{FSE.1/V}.
Using the fitted value of $m(L\!=\!\infty)$, 
we show in Fig.~\ref{fig:Nf2:FSE} the relative magnitude of 
finite size effects as a function of the spatial extent $L$
(divided by $r_0$)
at sea quark masses corresponding to $m_{\rm PS}/m_{\rm V}=0.7$ 
and 0.6.

We observe that a lattice size of $L \sim 2$~fm suffices 
for mesons to keep finite size effects under a few percent level for 
sea quark masses down to $m_{\rm PS}/m_{\rm V}\sim 0.6$. 
On the other hand, 
calculations of the baryon masses need 
$L \sim 3$~fm at the same sea quark masses 
and the required size increases rapidly
as the sea quark mass decreases.
Further 
% investigations on such large volumes 
studies on such large volumes 
will be necessary to identify sea quark effects on the baryon spectrum.


\begin{figure}[t]
\begin{center}
   \leavevmode
   \includegraphics*[width=65mm,clip]{./meson-K.eps}
\end{center}
\vspace*{-16mm}
\caption{
  Strange vector meson masses with $m_K$ as input
  in $N_f\!=\!2$(filled symbols)
  and quenched QCD(open symbols).
%  New JLQCD results are plotted in circles.
}
\vspace*{-6mm}
\label{fig:Nf2:meson}
\end{figure}

         
% {\bf Edinburgh plots}
%
% At the last conference, it is pointed out that 
% baryon masses from CP-PACS and MILC collaboration
% do not agree each other even at the finite sea quark
% mass in the continuum limit.


\subsection{Glueball masses}

An unambiguous experimental identification of the glueball 
is still missing, 
and lattice QCD can provide helpful theoretical estimates.
Calculations in quenched QCD have been performed for a long time 
and recent results for $J^{PC}\!=\!0^{++}$ and $2^{++}$ 
states are in good agreement with each other\cite{review.glue.lat99}.
The focus has now shifted to calculations in full QCD
and mixings with quarkonia.
%
% Recently SESAM\/T$\chi$L calculated glueball masses 
% in their $N_f\!=\!2$ full simulations.
%
% They observe strong finite size effects in glueball 
% masses\cite{SESAM.Nf2.glue}, which is plausible 
%from the expectation that the scalar glueball state 
% can mix with the single torelon states in full QCD\cite{review.glue.lat94}.
% 

UKQCD\cite{UKQCD.Nf2.glue} and SESAM-T$\chi$L\cite{SESAM.Nf2.glue}
reported new results this year.
Both groups find a decrease of $m_{0^{++}}$ by about 20\%
with dynamical quarks.
The observed sea quark mass dependence, however, is quite different:
SESAM-T$\chi$L finds a significant slope, 
while UKQCD data are almost flat.
%
It is, thus, too early to draw a definite conclusion
on the sea quark effect.  Scaling violation should be 
also studied carefully, as it is known to be large in 
quenched QCD. 
A similar indication was already 
seen by SESAM-T$\chi$L\cite{SESAM.Nf2.glue} in full QCD.

% % Effects of the mixing with the scalar meson states
% % are also important issues.
%
% An preliminary results on the calculation of the scalar 
% meson in $N_f\!=\!2$ full QCD was also presented 
% in Ref.\cite{SCALAR.Nf2}, which is an important subject 
% to resolve the mixing with glueballs.


\begin{figure}[t]
\begin{center}
   \leavevmode
   \includegraphics*[width=65mm,clip]{./FSE.eps}
\end{center}
\vspace*{-15mm}
\caption{
  Mass shift at finite lattice size $L$ as a function of 
  $L/r_0$. 
%  Left and right figure show data for 
%  pseudo-scalar meson and octet baryon.
  The dashed and dotted lines shows fit curve for data 
  at sea quark mass corresponding to $m_{\rm PS}/m_{\rm V}\!=\!0.7$
  and 0.6, respectively.
}
\vspace*{-6mm}
\label{fig:Nf2:FSE}
\end{figure}



%\section{$N_f\!=\!3$ full QCD}
\section{QCD with dynamical up, down  and strange quarks} 
\label{sec:Nf3}

Realistic simulations of QCD require up, down and strange 
quarks to be treated dynamically.  While such studies already exist 
for the KS quark action using an approximate R algorithm\cite{KS.Toussaint,MILC.Nf3}, 
exact algorithms for odd number of Wilson-type quarks
were developed only recently\cite{review.algo}. 
% In particular algorithms for the $O(a)$-improved Wilson fermion 
% action have been implemented this year.   
% 
% simulations in Nf=2+1 QCD with clover action 
% is possible already in Takaishi-Forcrand's PHMC.
%

This year JLQCD initiated a study of $N_f=3$ full QCD 
using an exact algorithm 
for the $O(a)$-improved Wilson fermion\cite{JLQCD.Ishikawa}. 
As a first step, they studied the phase structure 
on the $(\beta, \kappa)$ plane 
and found an unexpected first-order phase transition
with the standard plaquette gauge 
and the $O(a)$-improved Wilson quark action,
as shown in Fig.~\ref{fig:Nf3:PD}\cite{JLQCD.Okawa}.
%
The transition points do not move 
between $8^3\times16$ and $12^3\times32$ lattices, 
and they disappear at $\beta\!\geq\!5.0$.
Hence this is a bulk transition at zero temperature, 
and is a lattice artifact. 
% $\beta\!\sim\!5.0$,
% which corresponding to $a^{-1}\!\sim\!2.6$~G,
%

Their finding poses a serious practical problem, 
because the lattice spacing estimated from $r_0$ 
equals $a^{-1} \! \sim \! 2.6$~GeV
at the end-point of the first-order transition at $\beta\!\sim\!5.0$.  
Large-scale simulations at such fine lattice spacings are
not practical with the computer power currently available.
%

They found, however, that this unphysical phase transition 
disappears if improved gauge actions are employed. 
For realistic simulations, 
it is a natural choice to use improved gauge actions 
at moderately large values of lattice spacing such as 
$a \! \sim \! 0.1$--0.2~fm.


%
% Another attempt using the standard Wilson quark action 
% have been initiated by Gebert {\it et al.} and current status 
% was presented at this conference\cite{Gebert.Nf3}.
%

\begin{figure}[t]
\begin{center}
   \leavevmode
   \includegraphics*[width=65mm,clip]{./PD.eps}
\end{center}
\vspace*{-15mm}
\caption{
  Phase diagram in $N_f\!=\!3$ full QCD with 
  the plaquette gauge and $O(a)$-improved Wilson quark.
}
\vspace*{-6mm}
\label{fig:Nf3:PD}
\end{figure}



\section{Test of chiral perturbation theory}
\label{sec:ChPT}

%Chiral perturbation theory(ChPT) may be used 
%as a guide in the chiral extrapolation of lattice data,
%as it is an effective theory of low energy QCD.
%However, such a strategy is not justified 
%until the applicability of ChPT in the quark mass region 
%where lattice simulations are performed is confirmed.
%
% In quenched QCD, ChPT predict that the pseudo-scaler meson mass
% squared has the logarithmic singularity % $\ln\left[ m_q \right]$
% (``quenched chiral log'')\cite{qChPT}.
% The existence of quenched chiral log have been tested by many 
% groups and clear evidences were observed in recent simulations 
% with small valence quark masses\cite{CP-PACS.Nf0,MILC.Nf3,qChPT.
% delta.Bardeen,mud.Dong}.
% Recent estimates of the overall size of the quenched chiral log 
% $\delta$ are scattered in a moderately narrow range 
% $0.06-0.14$\cite{MILC.Nf3,qChPT.delta}.
% There is no firm evidence for the coupling constant 
% $\alpha_{\phi}$ from numerical simulations.
 

Chiral perturbation theory (ChPT)\cite{ChPT} predicts 
the presence of logarithm singularities 
in the quark mass dependence of physical quantities,
{\it e.g.}, for the pseudo-scalar meson mass $m_{\rm PS}$, 
\bea
   \frac{m_{\rm PS}^2}{2B_0m_q}
      &=&
      1 + \frac{1}{N_f} y \ln \left[ y \right] 
      + O(y)
%        + y \left\{ \left( 2\alpha_8 - \alpha_5 \right) 
%
%   \right. \nonumber \\
%      & & 
%      \left.
%        +N_f\left( 2\alpha_6 - \alpha_4 \right) \right\},
\label{eq:ChPT:mPS2}
\eea
where $y \equiv 2B_0m_q/(4\pi f)^2$. %and 
%$\alpha_i$ are the low-energy constants in the chiral Lagrangian.
%
A test of this relation is shown in Fig.~\ref{fig:ChPT:JLQCD},
where lattice data for two-flavor QCD from JLQCD\cite{JLQCD.Nf2}
are plotted together with the fit curves of Eq.~(\ref{eq:ChPT:mPS2}) 
assuming $f$ to be a free parameter or fixed to $f=93$~MeV.
%
The lattice results are not consistent with the curvature
characterized by the chiral logarithm.
% Other tests with the partially quenched ChPT (PQChPT)\cite{PQChPT}
% also suggest the absence of the chiral logarithm.

\begin{figure}[t]
\vspace*{-2mm}
\begin{center}
   \leavevmode
   \includegraphics*[width=66mm,clip]{./B.eps}
\end{center}
\vspace*{-15mm}
\caption{
Test of chiral perturbation theory.
% extrapolation of $(r_0 m_{PS})^2 / (2 r_0 m_q)$.
% The solid (dashed) line is a fit curve with $f$ unfixed (fixed to
% the experimental value of 93~MeV).
}
\vspace*{-6mm}
\label{fig:ChPT:JLQCD}
\end{figure}


Recently UKQCD\cite{ChPT.Nf2.UKQCD}
attempted to extract the low energy constants
of chiral lagrangian from a fit of lattice data 
to the partially quenched ChPT formula\cite{PQChPT}
using ratios proposed by ALPHA\cite{ChPT.Nf0.ALPHA}.
The curvature expected from the chiral logarithm is also not clear
in their data.

A possible reason behind these disagreements 
is that sea quark masses in  current simulations are too large.  
In fact, 
the coefficient of the chiral logarithm is modified
when the effect of flavor singlet meson is included into 
ChPT\cite{pqChPT.with.eta}; 
it can take much smaller values than $1/N_f$, 
if the pesudo-scalar meson is not so light
compared to the flavor singlet meson\cite{JLQCD.Nf2}.
A recent work by Nelson {\it et al.}\cite{ChPT.Nf3.Nelson}
using KS fermions suggests a similar conclusion.
The curvature is visible in their data at small sea quark masses, 
which are possible with KS fermions.


% Once the chiral logarithm is reproduced by lattice QCD,
Another interesting subject of ChPT is 
determination of the low energy constants $\alpha_i$ 
from lattice data\cite{mu0,ChPT.const}.
%
In particular, 
$\alpha_8$ is an important quantity 
since it is related to the possibility of $m_u\!=\!0$\cite{mu0}.
UKQCD  attempted to extract $\alpha_i$'s in two-flavor QCD, 
and obtained $\alpha_8\!=\!0.79(^{+5}_{-7})(21)$, 
which disfavors $m_u\!=\!0$.
%
Nelson {\it et al.} obtained a consistent conclusion 
in three flavor QCD.
%
%
These results, however, should be confirmed with lighter sea quark masses,
for which ChPT at the one-loop level becomes more reliable.




%--- development in simulation technique, etc -----

\section{Developments in quenched QCD}
%\section{Quenched QCD as a testing ground of new formulations}
\label{sec:Nf0}


\subsection{Test of improved formulations}


Quenched QCD is now used to develop and test 
improved formulations of lattice QCD.
Domain-wall fermion formalism is an important case,
as it is expected to 
realize exact chiral symmetry at finite lattice spacings.
Anisotropic lattices may provide a way 
to achieve simulations with large lattice cut-offs 
on large spatial volumes.


\begin{figure}[t]
\begin{center}
   \leavevmode
   \includegraphics*[width=66mm,clip]{./Nf0.eps}
\end{center}
\vspace*{-15mm}
\caption{
  Continuum extrapolation of $m_{K^*}/m_{\rho}$
  obtained with domain-wall fermion(squares) 
  and on anisotropic lattice(triangles).
%  Circles are CP-PACS's results with the Wilson action.
%  The experimental value is represented by a star symbol.
}
\vspace*{-6mm}
\label{fig:Nf0:mKs}
\end{figure}


Figure~\ref{fig:Nf0:mKs} shows 
results of $m_{K^*}/m_{\rho}$ mass ratio calculated 
with the domain-wall fermions\cite{DW.RBC},
and on anisotropic lattices\cite{aniso.Matsufuru}.
Clearly these formulations reproduce, and confirm, the small 
hyperfine splitting in the quenched meson spectrum 
obtained by the precison calculation by CP-PACS
using the conventional Wilson quark action\cite{CP-PACS.Nf0}.
%Therefore, these are now applicable also for computations of 
%other observables.


Recently, the Wilson quark action with a chirally twisted 
mass term\cite{tmQCD.Aoki}
% $i \bar{\psi} \gamma_5 \tau^3 \psi$
% where the $\tau^3$ is a Pauli matrix acting in flavor space,
has drawn attention as an alternative formulation to suppress 
exceptional configurations\cite{tmQCD}.
Some evidences of such suppression % of the exceptional configuration 
have been presented 
at this conference\cite{tmQCD.Frezzotti.lat01,tmQCD.McNeile.lat01}.
% by monitoring the decrease of error of the pseudo-scalar 
% meson correlator with statistics.
%
It will be interesting to apply this action
to the non-perterburtive determination of improvement and 
renormalization constants at strong coupling,
where exceptional configurations cause
a serious problem\cite{NP-csw.ALPHA.Nf0,NP-Z.RG}. 


\subsection{Negative parity baryon}

An interesting subject in recent quenched simulations 
is to explore the rich structure of the excited baryons.
There already exist several studies 
for the negative parity baryon $N^*$ ($J^{P}\!=\!1/2^-$)
at finite lattice spacings\cite{Nf0.N-}.
The first systematic study of scaling violation 
and finite size effects 
has been performed this year by the QCDSF-UKQCD-LHPC collaboration
using the $O(a)$-improved Wilson action\cite{QCDSF.Nf0.N-}.
Their result $m_{N^*}/m_{N}\!=\!1.50(3)(8)(15)$ 
in the continuum limit is consistent with the experimental value 1.63.

It is also interesting to explore other excitations.
A preliminary result
for the excited decuplet baryons, $J^{P}\!=\!3/2^+$ and $3/2^-$,
was presented by Lee {\it et al.} at this conference\cite{Lee.Nf0.N-}.
% whcih shows a reasonable agreement with experiment 
% at simulated quark masses\cite{Lee.Nf0.N-}.

%   {\bf MEM}
%
%   The maximum entropy method : as an alternative method 
%   for the calculations of the spectrum.
%




\section{Strong coupling constant}
\label{sec:alpha_s}

%Precise determination of the strong coupling $\alpha_s$ 
%is important for phenomenological applications 
%and search for new physics.

Recent results of $\alpha_{\MSbar}^{(N_f=5)}(M_z)$ 
from lattice QCD are summarized in Table~\ref{tab:alpha_s}.
%
% There a variety of method to compute $alpha_s$ in lattice 
% QCD, (i) $alpha_P$ from the plaquette\cite{alphaP},
% (ii) $alpha_{q\bar{q}}$ from the force $dV/dR$\cite{alphaqq}, 
% (iii) $\alpha_{\rm SF}$ from the Schr\"odinger functional\cite{alphaSF},
% and (iv) from $\alpha_{\rm lat}$ in MOM scheme from the green 
% functions\cite{alphaGF}. 
%
% 
Recent values using the Wilson-type quark action for sea 
quarks\cite{alphas.SESAM,alphas.UKQCD} are significantly smaller
compared to a standard lattice result previously obtained by Davies 
{\it et al.}\cite{alphas.NRQCD} using KS fermions.
%
% SESAM with Wilson action\cite{alphas.SESAM}
% UKQCD with $O(a)$-improved Wilson fermion\cite{alphas.UKQCD}
% QCDSF-UKQCD\cite{alphas.QCDSF}.
% Boucaud {\it et al.} also obtained 0.113(3)(4)
%
In particular this year's result of the QCDSF-UKQCD collaboration
is smaller despite a consistent use of two-loop perturbation theory 
and a continuum extrapolation\cite{alphas.QCDSF}.
%
% albeit a large error bar from 
% non-perturbative effects in running of $\alpha_s$ =(3) 
% and calibration of the lattice scale =(4)
% 

The origin of 
the difference between results with the KS and Wilson fermions 
should be clarified.  
Since the discrepancy is already present 
in the value of $\alpha_P$ at the cutoff scale in two-flavor QCD,
systematic uncertainties such as scaling violation and higher order 
corrections in perturbation theory 
should be studied carefully in future.
%
%

Calculations of $\alpha_s$ through various definitions
are also helpful to resolve this problem.
%
% $\alpha_{q\bar{q}}$ from the force $dV/dR$\cite{alphaqq}, 
% or $\alpha_{\rm SF}$ from the Schr\"odinger functional\cite{alphaSF},
%
Boucaud {\it et al.} presented a preliminary result 
from lattice calculations of the gluon green functions\cite{alphas.Boucaud}.
%
%
A non-perturbative determination of $\alpha_s$ 
in the Schr\"odinger functional scheme\cite{alphaSF} 
was performed by ALPHA\cite{alphas.ALPHA} in two-flavor QCD,
and determination of the lattice energy scale 
is in progress\cite{alphas.ALPHA}.




\section{Quark masses}
\label{sec:mq}

% Calculation of the light quark masses 
% have been pursued for a long time in quenched QCD\cite{review.mq}.
By last year results for the light quark mass $m_{ud}$ 
in quenched QCD converged within an accuracy of about 
15\%\cite{review.mq}.  For $m_s$ there turned out an additional 
uncertainty of about 20\% 
due to the choice of the input meson mass
({\it e.g.} $m_K$ or $m_{\phi}$)\cite{CP-PACS.Nf0}.
A noticable trend in this year's calculation 
is the use of the domain-wall and overlap fermions.
% having good chiral symmetry.
Calculations of the charm quark mass $m_c$ 
have also been pursued in quenched QCD. 

In $N_f\!=\!2$ full QCD,
CP-PACS reported results of a systematic study already last year 
\cite{mq.CP-PACS}.
They found that sea quark effects reduce $m_{ud}$ and $m_s$ 
by about 25\%, 
and the discrepancy in $m_s$ with $m_K$ or $m_{\phi}$ as input
is remarkably reduced.
This year there has been less progress in full QCD.


\begin{table}[t]
\setlength{\tabcolsep}{0.2pc}
\caption{
  Recent results of $\alpha_{\MSbar}^{(N_f=5)}(M_z)$.
%  The standard and the non-perturbatively $O(a)$-improved
%  Wilson quark actions are denoted as ``W'' and ``C(NP)'', 
%  respectively.
  All results except Ref.\cite{alphas.Boucaud} are calculated 
  through $\alpha_P$\cite{alphaP}. 
}
\begin{tabular}{lll}
\hline 
\hline 
group   &   quark   & $\alpha_{\MSbar}^{(N_f\!=\!5)}$  \\
\hline 
SESAM\cite{alphas.SESAM}       & Wilson     & 0.1118(17)       \\
QCDSF-UKQCD\cite{alphas.QCDSF} & clover(NP) & 0.1076(20)(18)   \\
Boucaud {\it et al.}\cite{alphas.Boucaud} & Wilson & 0.113(3)(4)\\
\hline 
Davies {\it et al.}\cite{alphas.NRQCD} & KS & 0.1174(24)  \\
\hline
\hline
\end{tabular}
\label{tab:alpha_s}
\vspace*{-5mm}
\end{table}

%\subsection{$m_s$ in quenched QCD}
\subsection{Strange quark mass in quenched QCD}

A new estimate of $m_s$ was obtained by CP-PACS 
using the domain-wall fermion\cite{ms.CP-PACS}.
Their calculations were performed at $a^{-1}\!\sim\!2$ and 3~GeV
using one-loop perturbative renormalization constants.
%
% $m_s^{\MSbar}(2~{\rm GeV})\!=\!98.7(2.1)(5.6)$ 
% in the continuum limit.
%
% A caclualtion using the overlap fermion
% was performed by Giusti {\it et al.} 
% at $a^{-1}\!\sim\!2$~GeV\cite{ms.Giusti}.
Giusti {\it et al.} calculated $(m_{s}+m_{ud})$ 
at $a^{-1}\!\sim\!2$~GeV using the overlap fermion\cite{ms.Giusti}.
Their renomalization constants are evaluated 
non-perturbatively in the RI/MOM scheme.
% Their results of $m_s$ is determined from $(m_{s}+m_{ud})$ 
% and the mass ratio $m_s/m_{ud}\!=\!24.4(1.5)$ from 
% one-loop ChPT.
%
% Their results of the renormalization group invariant mass
% $(m_{s}+m_{ud})\!=\!120(7)(21)$ corresponds to 
% $m_s^{\MSbar}(2~{\rm GeV})\!=\!102(6)(18)$, if the mass ratio
% $m_s/m_{ud}\!=\!24.4(1.5)$ from ChPT is applied.
%


Recent results with $m_K$ as input are plotted 
in Fig.~\ref{fig:mq:ms}.
A reasonable agreement among the new and previous results 
is observed.
Therefore, the average 
$m_{s}^{\MSbar}(\mbox{2~GeV})=105 \pm 15~{\rm MeV}$
does not change significantly since last year\cite{review.mq}.

% Recent results of $m_s$ with $m_K$ as input are plotted 
% in Fig.~\ref{fig:mq:ms}.
% We obtain the average 
% $m_{s}^{\MSbar}(\mbox{2~GeV})=105 \pm 15 \pm 30~{\rm MeV}$,
% where the first error is spread of results with 
% $m_{K}$-input and second one is due to the choice of 
% the input to fix $m_s$\cite{CP-PACS.Nf0}.
% This does not change significantly since last year\cite{review.mq},
% because of the reasonable agreement among the new and previous 
% results of $m_s$.




%\subsection{ $m_{ud}$ in quenched QCD}
\subsection{Light quark mass in quenched QCD}

% The average of up and down quark mass $m_{ud}$ have been 
% also studied in lattice QCD for a long time.
% In the calculation of $m_{ud}$, 
% the systematic errors 
% from the chiral extrapolation and finite size effects 
% can be more important than those for $m_{s}$.
% Therefore, we consider a ratio $m_s/m_{ud}$, 
% rather than $m_{ud}$ itself as in Ref.\cite{review.mq}.
% It is expected that systematic uncertainties are 
% partially canceled in this ratio.
We consider the ratio $m_s/m_{ud}$, 
rather than $m_{ud}$ itself, since 
systematic uncertainties partially cancel in this ratio.
%
A new result $m_s/m_{ud}=26.2(2.3)$ by CP-PACS\cite{ms.CP-PACS} 
as well as previous results are in good agreement with 
the prediction of one-loop ChPT :  24.4(1.5)\cite{mq.ChPT}.
We therefore apply the ratio from ChPT to the average of $m_s$ 
to obtain 
$m_{ud}^{\MSbar}(\mbox{2~GeV})=4.3 \pm 0.7~{\rm MeV}$.
% $m_{ud}^{\MSbar}(\mbox{2~GeV})=4.5 \pm 0.7 \pm 1.4~{\rm MeV}$.
% \bea
% m_{ud}^{\MSbar}(\mbox{2~GeV})=4.5 \pm 0.7~{\rm MeV}.
% \label{eq:mq:mud}
% \eea

Dong {\it et al.} calculated $m_{ud}$ 
with the overlap fermion\cite{mud.Dong}.
Their estimate $m_{ud}^{\MSbar}(\mbox{2~GeV})\!=\!5.3(0.6)(0.7)$~MeV
agrees with the above average.




\begin{figure}[t]
\vspace*{-1mm}
\begin{center}
   \leavevmode
   \includegraphics*[width=72mm,clip]{./ms-K.eps}
\end{center}
\vspace*{-15mm}
\caption{
  Continuum extrapolation of strange quark mass
  with $m_K$ as input in $\MSbar$ scheme at scale $\mu\!=\!2$~GeV.
  Filled and open symbols show new and previous results,
  respectively.
  The choice of the quark action is written in brackets,
  where ``W''=Wilson, ``C''=clover, ``KS''=Kogut-Susskind and 
  ``DWF''=domain-wall fermions.
}
\vspace*{-6mm}
\label{fig:mq:ms}
\end{figure}


% \subsection{ $m_{c}$ in quenched QCD}
\subsection{Charm quark mass in quenched QCD}

There have been few studies for calculating 
the charm quark mass $m_c$\cite{mc.previous}.
The reason is that $O((am_q)^n)$ scaling violation
is expected to be large in relativistic formulations,
while the charm quark is too light 
to be treated in the non-relativistic approximation.

New estimates of $m_c$ have been reported by two groups this year, both 
using the $O(a)$-improved action to reduce scaling violation.
%
Becirevic {\it et al.} performed a simulation 
at a fixed lattice spacing $a^{-1}\!\sim\!3$~GeV\cite{mc.Becirevic}.
The results significantly differ for $m_c$ obtained from vector(VWI) 
and axial vector Ward identities(AWI).
They take an average %of the two determination(VWI and AWI)
to quote $m_c^{\MSbar}(m_c)\!=\!1.26(3)(12)$. 
The difference between the two methods(AWI and VWI) and the average 
is treated as a systematic error. 
%
% due to the scaling violation,
% which is found to be the most dominant error.
%
%

ALPHA performed simulations
in a wide range of lattice spacing $a^{-1}\!\sim\!2$--4~GeV 
and performed the continuum extrapolation\cite{mc.ALPHA}.
%
They observed a nice convergence of $m_c$ 
determined from AWI and VWI toward the continuum limit,
and obtained $m_c^{\MSbar}(m_c)\!=\!1.314(40)(20)(7)$ 
in the continuum limit.

These results are plotted in Fig.~\ref{fig:mq:mc}  
together with previous results\cite{mc.previous} 
and an average by PDG\cite{PDG}.
%
The two new results are consistent with previous ones
as well as the PDG's average.

A new estimate using the heavy quark action in the FNAL interpretation 
\cite{Fermilab} 
was presented by Juge {\it et al.} at this conference\cite{mc.Juge}.
%
While the estimation of the systematic error is still in progress, 
their preliminary result is also around 1.3~GeV 
and consistent with the above new results.
   

\begin{figure}[t]
\vspace*{-4.5mm}
\begin{center}
   \leavevmode
   \includegraphics*[width=59mm,clip]{./mc.eps}
\end{center}
\vspace*{-16mm}
\caption{
  Recent results of charm quark mass.
  An average quoted by PDG\cite{PDG}
  is also plotted.
}
\vspace*{-6mm}
\label{fig:mq:mc}
\end{figure}



\subsection{Results in $N_f\!=\!2$ full QCD}

SESAM-T$\chi$L 
attempted a continuum extrapolation 
of $m_{ud}$ and $m_s$ from previous\cite{mq.SESAM.prev}
and new simulations at $\beta\!=\!5.5$\cite{SESAM.Nf2.beta55}. 
Their range of the lattice spacing is, however, narrow
and far from the continuum limit.
This leads to a large uncertainty in their results,
$m_{ud}^{\MSbar}(2~{\rm GeV})\!=\!4.5(1.7)$~MeV and 
$m_{s}^{\MSbar}(2~{\rm GeV})\!=\!92(83)$~MeV.
%
In near future, 
we expect results from the UKQCD and JLQCD simulations
with the $O(a)$-improved Wilson fermion.

         

\section{Conclusions}
\label{sec:conclusion}

% This year lattice QCD calculation of the light hadron spectrum
% made progress in a number of aspects both in quenched 
% and full QCD.

In quenched QCD, improved formulations,  
such as the domain-wall and overlap fermions 
and anisotropic lattice actions,
have been tested by calculating 
the light hadron spectrum and quark masses,
and good consistency with previous results has been observed.
These formulations are ready for applications
to precise determinations of other observables.
% It will be also worthwhile to apply these formulations
% to full QCD.

In $N_f\!=\!2$ full QCD,
% calculations have been pushed substantially toward
% precise determination of the light hadron spectrum.
sea quark effects on the meson spectrum,
which were previously observed by CP-PACS,
have been confirmed using 
the $O(a)$-improved Wilson and an improved KS fermions.
%
However we still need further studies %investigations 
for sea quark effects on the baryon and glueball spectrum.
%
Exploring small sea quark masses with Wilson-type fermions
is also an important issue 
to resolve the poor consistency between lattice data and ChPT.

Pursuit of realistic simulations in three flavor QCD 
is one of the most important issue in spectroscopic studies.
This year significant progress to this end has been made 
by several groups using both of the KS and Wilson fermions.
Further systematic studies will be expected in near future.
   
And finally, there has been few progress in non-perturbative 
calculation of the improvement coefficient $c_A$ and renormalization 
factors in full QCD.
These calculations are urgently required for precise 
determinations of the quark masses and decay constants.



\vspace{-2mm}
\section*{Acknowledgments}


I thank   
   Y.~Aoki,  G.S.~Bali, C.~Bernard, S.~Collins, 
   G.~Di~Carlo, S.J.~Dong, N.~Eicker, 
   R.~Frezzotti, C.~Gebert, L.~Giusti,
   A.~Hart, S.~Hashimoto,  S.~Hauswirth, J.~Hein, 
   K.~Ide, A.C.~Irving, K-I.~Ishikawa, 
   K.~Jansen, 
   F.X.~Lee, L.~Levkova, X.Q.~Luo, T.~Manke, 
   H.~Matsufuru, C.~McNeile, H.~Moutarde, A.~Nakamura, 
   M.~Okawa, K.~Orginos, O.~Philipsen, 
   C.~Rebbi, J.~Rolf, G.~Rossi, S.~Sint, 
   H.~Wittig, T.~Yamazaki, and A.~Ukawa
for communications and useful discussions.
%
I also thank 
  S.~Hashimoto, Y.~Iwasaki, M.~Okawa and A.~Ukawa
for valuable suggestions on the manuscript.
%
This work is supported by the JSPS Research Fellowship.


%--- the bibliography ---
\vspace{-2mm}
\input{ref}

\end{document}



 


ref.tex
\begin{thebibliography}{9}
%\vspace{-1mm}


%--- introduction ---------------------------------------------

\bibitem{CP-PACS.Nf0}
S.~Aoki {\it et al.}
(CP-PACS Collaboration), 
Phys. Rev. Lett. 84 (2000) 238.

% \bibitem{UKQCD.Nf0}
% C.R.~Allton {\it et al.} (UKQCD Collaboration), 
% Phys. Rev. D62 (2000) 054506.

% \bibitem{QCDSF.Nf0}
% M.~G\"ockeler {\it et al.},
% Phys.Rev. D57 (1998) 5562-5580 

% \bibitem{APE.Nf0}
% D.~Becirevic {\it et al.} (APE Collabortion),
% Phys. Lett. B444 (1998) 401.

% \bibitem{ALPHA.Nf0}
% J.~Garden {\it et al.} (ALPHA and UKQCD Collaboration),
% Nucl. Phys. B571 (2000) 237.

% \bibitem{JLQCD.Nf0}
% S.~Aoki {\it et al.} (JLQCD Collaboration),
% Phys. Rev. Lett. 82 (1999) 4392.

\bibitem{review.full}
For a recent review, see
S.~Aoki,
Nucl. Phys. B (Proc. Suppl.) 94 (2001) 3.

\bibitem{review.algo}
M.~Peardon, review in these proceedings.

\bibitem{ChPT}
J.~Gasser and H.~Leutwyler, 
Ann. Phys. 158 (1984) 142;
Nucl. Phys. B250 (1985) 465.

\bibitem{review.GW}
%P.M.~Vranas, 
%Nucl.Phys. B (Proc. Suppl.) 94 (2001) 177;
Y.~Kikukawa, 
review in these proceedings and references therein;
P. Hernandez, {\it ibid.}

\bibitem{aniso}
T.R.~Klassen, 
Nucl. Phys. B (Proc. Suppl.) 73 (1999) 918.
% P.~Chen, 
% Phys. Rev. D64 (2001) 034509;
% A.~Ali Khan {\it et al.} (CP-PACS Collaboration),
% Nucl. Phys. B (Proc. Suppl.) 94 (2001) 325,
% these proceedings;
% T.~Umeda {\it et al.},
% Int. J. Mod. Phys. A 16 (2001) 2215.

\bibitem{review.alphas}
% For reviews, see
% P.~Weisz,
% Nucl. Phys. B (Proc. Suppl.) 47 (1996) 71;
For a recent review, see
J.~Shigemitsu,
Nucl. Phys. B (Proc. Suppl.) 53 (1997) 16.

\bibitem{review.mq}
For a recent review, see 
V.~Lubicz, 
Nucl. Phys. B (Proc. Suppl.) 94 (2001) 116.

\bibitem{KS.Toussaint}
D.~Toussaint, review in these proceedings.


%--- section 2 ----------------------------------------------------

\bibitem{SWaction}
B.~Sheikholeslami and R.~Wohlert,
Nucl. Phys. B259 (1985) 572.



\bibitem{SESAM.Nf2}
N.~Eicker {\it et al.} (SESAM Collaboration), 
Phys. Rev. D59 (1999) 014509.

\bibitem{TchiL.Nf2}
T.~Lippert {\it et al.} (SESAM-T$\chi$L Collaboration), 
Nucl. Phys. B (Proc. Suppl.) 60A (1998) 311.

\bibitem{SESAM.Nf2.beta55}
N.~Eicker (SESAM-T$\chi$L Collaboration), 
these proceedings.



\bibitem{UKQCD.Nf2.csw176}
C.R.~Allton {\it et al.} (UKQCD Collaboration),
Phys. Rev. D60 (1999) 034507.

\bibitem{UKQCD.Nf2.lat00}
A.C.~Irving (UKQCD Collaboration),
Nucl. Phys. B (Proc. Suppl.) 94 (2001) 242.

\bibitem{UKQCD.Nf2}
C.R.~Allton {\it et al.} (UKQCD Collaboration), 
hep-lat/0107021;
these proceedings.

\bibitem{QCDSF.Nf2.lat00}
D.~Pleiter (QCDSF-UKQCD Collaboration), 
Nucl. Phys. B (Proc. Suppl.) 94 (2001) 265;
H.~St\"uben (QCDSF-UKQCD Collaboration), 
{\it ibid.} 94 (2001) 273.

\bibitem{JLQCD.Nf2.lat00}
S.~Aoki {\it et al.} (JLQCD Collaboration), 
Nucl. Phys. B (Proc. Suppl.) 94 (2001) 233.

\bibitem{JLQCD.Nf2}
S.~Hashimoto (JLQCD Collaboration), these proceedings.

\bibitem{RGgauge}
Y.~Iwasaki, 
Nucl. Phys. B258 (1985) 141;
Univ. of Tsukuba report UTHEP-118(1983), unpublished.

\bibitem{CP-PACS.Nf2}
A.~Ali Khan {\it et al.} (CP-PACS Collaboration),
Nucl. Phys. B (Proc. Suppl.) 94 (2001) 229;
hep-lat/0105015.

\bibitem{Coulumbia.Nf2}
C.~Sui {\it et al.},
Nucl. Phys. B (Proc. Suppl.) 73 (1999) 228;
see also R.D.~Mawhinney,
{\it ibid.} 83-84 (2000) 57.


\bibitem{MILC.Nf2.stdKS}
C.~Bernard {\it et al.} 
(MILC Collaboration),
Nucl. Phys. B (Proc. Suppl.) 73 (1999) 198;
{\it ibid.} 60A (1998) 297 
and references therein.

\bibitem{Columbia.Nf2.aniso}
L.~Levkova, 
these proceedings.

\bibitem{1loopSym}
P.~Weisz, 
Nucl. Phys. B212 (1983) 1;
M.~L\"uscher and P.~Weisz, 
Comm. Math. Phys. 97 (1985) 59.

\bibitem{improvedKS}
S.~Naik, 
Nucl. Phys. B316 (1989) 238;
K.~Orginos, D.~Toussaint and R.L.~Sugar,
Phys. Rev. D60 (1999) 054503;
Nucl. Phys. (Proc. Suppl.) 83-84 (2000) 878.

\bibitem{MILC.Nf3}
C.~Bernard {\it et al.} 
(MILC Collaboration), 
Phys. Rev. D64 (2001) 054506.



\bibitem{NP-csw.ALPHA}
K.~Jansen and R.~Sommer (ALPHA Collaboration), 
Nucl. Phys. B530 (1998) 185.


\bibitem{r0}
R.~Sommer, 
Nucl. Phys. B411 (1994) 839.

\bibitem{GRAL.Nf2}
B.~Orth, these proceedings.






\bibitem{CP-PACS.mVvsmPS2}
CP-PACS Collaboration, unpublished.

\bibitem{J}
P.~Lacock and C.~Michael, 
Phys. Rev. D52 (1995) 5213.

\bibitem{FSE.1/V}
M. Fukugita {\it et al.},
Phys. Lett. B294 (1992) 380;
Phys. Rev. D47 (1993) 4739.


\bibitem{review.glue.lat99}
For a recent review, see
D.~Toussaint, 
Nucl. Phys. B (Proc. Suppl.) 83-84 (2000) 151.

\bibitem{UKQCD.Nf2.glue}
A.~Hart and M.~Teper (UKQCD Collaboration),
hep-lat/0108022;
A.~Hart (UKQCD-QCDSF Collaboration),
these proceedings.

\bibitem{SESAM.Nf2.glue}
G.S.~Bali {\it et al.} (SESAM-T$\chi$L Collaboration),
Phys. Rev. D62 (2000) 054503;
G.S.~Bali, hep-ph/0110254.

% \bibitem{review.glue.lat94}
% C.~Michael, Nucl. Phys. B (Proc. Suppl.) 
% 42 (1995) 147.

% \bibitem{SCALAR.Nf2}
% A.~Nakamura, these proceedings.



%--- section 3 --------------------------------------------------

\bibitem{JLQCD.Ishikawa}
K-I.~Ishikawa (JLQCD Collaboration), these proceedings.

\bibitem{JLQCD.Okawa}
M.~Okawa (JLQCD Collaboration), these proceedings.

% \bibitem{Gebert.Nf3}
% C.~Gebert, these proceedings.


%--- section 4 --------------------------------------------------

% \bibitem{qChPT}
% S.~Sharpe, 
% Phys. Rev. D 46 (1992) 3146;
% C.~Bernard and M.~Goltermann D 46 (1992) 853.

\bibitem{ChPT.Nf2.UKQCD}
A.C.~Irving {\it et al.} (UKQCD Collaboration), 
Phys. Lett. B518 (2001) 243. 


\bibitem{PQChPT}
S.R.~Sharpe, 
Phys. Rev. D56 (1997) 7052.

\bibitem{ChPT.Nf0.ALPHA}
J.~Heitger, R.~Sommer and H.~Wittig (ALPHA Collaboration),
Nucl. Phys. B588 (2000) 377.

\bibitem{pqChPT.with.eta}
M.F.L.~Goltermann and K.C.~Leung, 
Phys. Rev. D57 (1998) 5703.

\bibitem{ChPT.Nf3.Nelson}
D.~Nelson, these proceedings.

\bibitem{mu0}
A.G.~Cohen, D.B.Kaplan and A.E.Nelson,
JHEP 11 (1999) 027.

\bibitem{ChPT.const}
S.~Sharpe and N.~Shoresh,
Nucl. Phys. B (Proc. Suppl.) 83-84 (2000) 968.
% Phys. Rev. D62 (2000) 094503.


% \bibitem{qChPT.delta.Bardeen}
% W.Bardeen {\it et al.},
% Phys. Rev. D 62 (2000) 114505



%--- section 5 -------------------------------------------------


% \bibitem{NP.RIMOM}
% G.~Martinelli {\it et al.},
% Nucl. Phys. B445 (1995) 81.

% \bibitem{NP.SF}
% M.~L\"uscher {\it et al.},
% Nucl. Phys. B478 (1996) 365;
% Nucl. Phys. B491 (1997) 344.

% \bibitem{NP.WI}
% T.~Bhattacharya {\it et al.},
% Phys.Rev. D63 (2001) 074505.


% \bibitem{NP-cA.SF}
% P.~Hernandez {\it et al.}, 
% JHEP 0107 (2001) 018.
%
% \bibitem{NP-cA.WI.LANL}
% R.~Gupter, these proceedings.
%
% \bibitem{NP-cA.WI.UKQCD}
% S.~Collins, these proceedings.
%       


% \bibitem{NP-Z.GW}
% 



\bibitem{DW.RBC}
Y.~Aoki (RBC Collaboration), these proceedings. 

\bibitem{aniso.Matsufuru}
H.~Matsufuru, T.~Onogi and T.~Umeda, 
hep-lat/0107001;
these proceedings.


\bibitem{tmQCD.Aoki}
S.~Aoki, 
Phys. Rev. D30 (1984) 2653.

\bibitem{tmQCD}
W.~Bardeen {\it et al.}, 
Phys. Rev. D59 (1999) 014507;
R.~Frezzotti {\it et al.},
Nucl. Phys. B (Proc. Suppl.) 83-84 (2000) 941;
JHEP 0108 (2001) 058.

\bibitem{tmQCD.Frezzotti.lat01}
R.~Frezzotti {\it et al.},
these proceedings.

\bibitem{tmQCD.McNeile.lat01}
C.~McNeile (UKQCD Collaboration),
these proceedings.

\bibitem{NP-csw.ALPHA.Nf0}
M.~L\"uscher {\it et al.}, Nucl. Phys. B491 (1997) 323.

\bibitem{NP-Z.RG}
K.~Ide (CP-PACS Collaboration), these proceedings.


\bibitem{Nf0.N-}
F.X.~Lee and D.B.~Leinweber,
Nucl. Phys. B (Proc. Suppl.) 73 (1999) 258;
F.X.~Lee,
{\it ibid.} 94 (2001) 251;
S.~Sasaki, T.~Blum and S.~Ohta,
hep-lat/0102010.


\bibitem{QCDSF.Nf0.N-}
M.~G\"ockeler {\it et al.} 
(QCDSF-UKQCD-LHPC Collaboration), 
hep-lat/0106022.

\bibitem{Lee.Nf0.N-}
F.X.~Lee, these proceedings.




%--- section 6 -------------------------------------------------------

\bibitem{alphaP}
G.P.~Lepage and P.B.~Mackenzie, 
Phys. Rev. D48 (1993) 2250.

\bibitem{alphas.SESAM}
A.~Spitz {\it et al.} (SESAM Collaboration), 
Phys. Rev. D60 (1999) 074502.

\bibitem{alphas.QCDSF}
S.~Booth {\it et al.} (QCDSF-UKQCD Collaboration), 
Phys. Lett. B519 (2001) 229;
G.~Schierholz (QCDSF-UKQCD Collaboration),
these proceedings.

\bibitem{alphas.Boucaud}
Ph.~Boucaud~{\it et al.},~hep-ph/0107278;
H.~Moutarde, these proceedings.

\bibitem{alphas.NRQCD}
C.T.H.~Davies {\it et al.},
Phys. Rev. D56 (1997) 2755.



\bibitem{alphas.UKQCD}
L.~Marcantonio~{\it et al.}~(UKQCD~Collaboration), 
Nucl. Phys. B (Proc. Suppl.) 94 (2001) 363.


% \bibitem{alphaGF}
% B.~Alles {\it et al.},
% Nucl. Phys. B502 (1997) 325.

\bibitem{alphaSF}
M.~L\"uscher {\it et al.}, 
Nucl. Phys. B413 (1994) 481.

\bibitem{alphas.ALPHA}
A.~Bode {\it et al.} (ALPHA Collaboration),
Phys. Lett. B515 (2001) 49;
U.~Wolff (ALPHA Collaboration), these proceedings.

% \bibitem{alphaqq}
% C.~Michael, 
% Phys. Lett. B283 (1992) 103.


%--- section 7 -------------------------------------------------------


% \bibitem{review.NP}
% For a recent review see,
% S.~Sint,
% Nucl.Phys. B (Proc. Suppl.) 94 (2001) 79.

\bibitem{mq.CP-PACS}
A.~Ali Khan {\it et al.} (CP-PACS Collaboration),
Phys. Rev. Lett. 85 (2000) 4674.

\bibitem{ms.CP-PACS}
A.~Ali Khan {\it et al.} (CP-PACS Collaboration),
hep-lat/0105020.

\bibitem{ms.Giusti}
L.~Giusti, C.~Hoelbling and C.~Rebbi, hep-lat/0108007;
these proceedings.

\bibitem{mq.ChPT}
H.~Leutwyler, 
Phys. Lett. B378 (1996) 313.

% \bibitem{ms.ALPHA}
% ALPHA and UKQCD Collaboration,
% Nucl. Phys. B571 (2000) 237.

\bibitem{mud.Dong}
S.J.~Dong, these proceedings.

%\bibitem{NP-bm.SF}
%M.~Guagnelli {\it et al.} (ALPHA Collaboration),
%Nucl. Phys. B595 (2001) 44.

\bibitem{mc.previous}
A.~Bochkarev and P.~de~Forcrand, 
Nucl. Phys. B477 (1996) 489;
V.Gim\'enez {\it et al.},
Nucl. Phys. B540 (1999) 472;
A.S.~Kronfeld, 
Nucl. Phys. B (Proc. Suppl.) 63 (1998) 311.
% for updates of these results, 
% see also Ref.\cite{mc.Becirevic,mc.Juge}.

\bibitem{mc.Becirevic}
D.~Be\'cirevi\'c, V.~Lubicz and G.~Martinelli,
hep-ph/0107124. 

\bibitem{mc.ALPHA}
J.~Rolf and S.~Sint (ALPHA Collaboration), these proceedings.

\bibitem{PDG}
D.E.~Groom {\it et al.} (Particle Data Group), 
Eur. Phys. J. C15 (2000) 1. 

\bibitem{Fermilab}
A.X.~El-Khadra,~A.S.~Kronfeld and \ P.B.~Mackenzie,
Phys.~Rev. D 55~(1997)~3933.

\bibitem{mc.Juge}
K.J.~Juge, these proceedings.

\bibitem{mq.SESAM.prev}
N.~Eicker {\it et al.} (SESAM Collaboration), 
Phys. Lett. B407 (1997) 290.

\end{thebibliography}
 
hep-lat0111005.extracted_bib
\begin{thebibliography}{9}
%\vspace{-1mm}


%--- introduction ---------------------------------------------

\bibitem{CP-PACS.Nf0}
S.~Aoki {\it et al.}
(CP-PACS Collaboration), 
Phys. Rev. Lett. 84 (2000) 238.

% \bibitem{UKQCD.Nf0}
% C.R.~Allton {\it et al.} (UKQCD Collaboration), 
% Phys. Rev. D62 (2000) 054506.

% \bibitem{QCDSF.Nf0}
% M.~G\"ockeler {\it et al.},
% Phys.Rev. D57 (1998) 5562-5580 

% \bibitem{APE.Nf0}
% D.~Becirevic {\it et al.} (APE Collabortion),
% Phys. Lett. B444 (1998) 401.

% \bibitem{ALPHA.Nf0}
% J.~Garden {\it et al.} (ALPHA and UKQCD Collaboration),
% Nucl. Phys. B571 (2000) 237.

% \bibitem{JLQCD.Nf0}
% S.~Aoki {\it et al.} (JLQCD Collaboration),
% Phys. Rev. Lett. 82 (1999) 4392.

\bibitem{review.full}
For a recent review, see
S.~Aoki,
Nucl. Phys. B (Proc. Suppl.) 94 (2001) 3.

\bibitem{review.algo}
M.~Peardon, review in these proceedings.

\bibitem{ChPT}
J.~Gasser and H.~Leutwyler, 
Ann. Phys. 158 (1984) 142;
Nucl. Phys. B250 (1985) 465.

\bibitem{review.GW}
%P.M.~Vranas, 
%Nucl.Phys. B (Proc. Suppl.) 94 (2001) 177;
Y.~Kikukawa, 
review in these proceedings and references therein;
P. Hernandez, {\it ibid.}

\bibitem{aniso}
T.R.~Klassen, 
Nucl. Phys. B (Proc. Suppl.) 73 (1999) 918.
% P.~Chen, 
% Phys. Rev. D64 (2001) 034509;
% A.~Ali Khan {\it et al.} (CP-PACS Collaboration),
% Nucl. Phys. B (Proc. Suppl.) 94 (2001) 325,
% these proceedings;
% T.~Umeda {\it et al.},
% Int. J. Mod. Phys. A 16 (2001) 2215.

\bibitem{review.alphas}
% For reviews, see
% P.~Weisz,
% Nucl. Phys. B (Proc. Suppl.) 47 (1996) 71;
For a recent review, see
J.~Shigemitsu,
Nucl. Phys. B (Proc. Suppl.) 53 (1997) 16.

\bibitem{review.mq}
For a recent review, see 
V.~Lubicz, 
Nucl. Phys. B (Proc. Suppl.) 94 (2001) 116.

\bibitem{KS.Toussaint}
D.~Toussaint, review in these proceedings.


%--- section 2 ----------------------------------------------------

\bibitem{SWaction}
B.~Sheikholeslami and R.~Wohlert,
Nucl. Phys. B259 (1985) 572.



\bibitem{SESAM.Nf2}
N.~Eicker {\it et al.} (SESAM Collaboration), 
Phys. Rev. D59 (1999) 014509.

\bibitem{TchiL.Nf2}
T.~Lippert {\it et al.} (SESAM-T$\chi$L Collaboration), 
Nucl. Phys. B (Proc. Suppl.) 60A (1998) 311.

\bibitem{SESAM.Nf2.beta55}
N.~Eicker (SESAM-T$\chi$L Collaboration), 
these proceedings.



\bibitem{UKQCD.Nf2.csw176}
C.R.~Allton {\it et al.} (UKQCD Collaboration),
Phys. Rev. D60 (1999) 034507.

\bibitem{UKQCD.Nf2.lat00}
A.C.~Irving (UKQCD Collaboration),
Nucl. Phys. B (Proc. Suppl.) 94 (2001) 242.

\bibitem{UKQCD.Nf2}
C.R.~Allton {\it et al.} (UKQCD Collaboration), 
hep-lat/0107021;
these proceedings.

\bibitem{QCDSF.Nf2.lat00}
D.~Pleiter (QCDSF-UKQCD Collaboration), 
Nucl. Phys. B (Proc. Suppl.) 94 (2001) 265;
H.~St\"uben (QCDSF-UKQCD Collaboration), 
{\it ibid.} 94 (2001) 273.

\bibitem{JLQCD.Nf2.lat00}
S.~Aoki {\it et al.} (JLQCD Collaboration), 
Nucl. Phys. B (Proc. Suppl.) 94 (2001) 233.

\bibitem{JLQCD.Nf2}
S.~Hashimoto (JLQCD Collaboration), these proceedings.

\bibitem{RGgauge}
Y.~Iwasaki, 
Nucl. Phys. B258 (1985) 141;
Univ. of Tsukuba report UTHEP-118(1983), unpublished.

\bibitem{CP-PACS.Nf2}
A.~Ali Khan {\it et al.} (CP-PACS Collaboration),
Nucl. Phys. B (Proc. Suppl.) 94 (2001) 229;
hep-lat/0105015.

\bibitem{Coulumbia.Nf2}
C.~Sui {\it et al.},
Nucl. Phys. B (Proc. Suppl.) 73 (1999) 228;
see also R.D.~Mawhinney,
{\it ibid.} 83-84 (2000) 57.


\bibitem{MILC.Nf2.stdKS}
C.~Bernard {\it et al.} 
(MILC Collaboration),
Nucl. Phys. B (Proc. Suppl.) 73 (1999) 198;
{\it ibid.} 60A (1998) 297 
and references therein.

\bibitem{Columbia.Nf2.aniso}
L.~Levkova, 
these proceedings.

\bibitem{1loopSym}
P.~Weisz, 
Nucl. Phys. B212 (1983) 1;
M.~L\"uscher and P.~Weisz, 
Comm. Math. Phys. 97 (1985) 59.

\bibitem{improvedKS}
S.~Naik, 
Nucl. Phys. B316 (1989) 238;
K.~Orginos, D.~Toussaint and R.L.~Sugar,
Phys. Rev. D60 (1999) 054503;
Nucl. Phys. (Proc. Suppl.) 83-84 (2000) 878.

\bibitem{MILC.Nf3}
C.~Bernard {\it et al.} 
(MILC Collaboration), 
Phys. Rev. D64 (2001) 054506.



\bibitem{NP-csw.ALPHA}
K.~Jansen and R.~Sommer (ALPHA Collaboration), 
Nucl. Phys. B530 (1998) 185.


\bibitem{r0}
R.~Sommer, 
Nucl. Phys. B411 (1994) 839.

\bibitem{GRAL.Nf2}
B.~Orth, these proceedings.






\bibitem{CP-PACS.mVvsmPS2}
CP-PACS Collaboration, unpublished.

\bibitem{J}
P.~Lacock and C.~Michael, 
Phys. Rev. D52 (1995) 5213.

\bibitem{FSE.1/V}
M. Fukugita {\it et al.},
Phys. Lett. B294 (1992) 380;
Phys. Rev. D47 (1993) 4739.


\bibitem{review.glue.lat99}
For a recent review, see
D.~Toussaint, 
Nucl. Phys. B (Proc. Suppl.) 83-84 (2000) 151.

\bibitem{UKQCD.Nf2.glue}
A.~Hart and M.~Teper (UKQCD Collaboration),
hep-lat/0108022;
A.~Hart (UKQCD-QCDSF Collaboration),
these proceedings.

\bibitem{SESAM.Nf2.glue}
G.S.~Bali {\it et al.} (SESAM-T$\chi$L Collaboration),
Phys. Rev. D62 (2000) 054503;
G.S.~Bali, hep-ph/0110254.

% \bibitem{review.glue.lat94}
% C.~Michael, Nucl. Phys. B (Proc. Suppl.) 
% 42 (1995) 147.

% \bibitem{SCALAR.Nf2}
% A.~Nakamura, these proceedings.



%--- section 3 --------------------------------------------------

\bibitem{JLQCD.Ishikawa}
K-I.~Ishikawa (JLQCD Collaboration), these proceedings.

\bibitem{JLQCD.Okawa}
M.~Okawa (JLQCD Collaboration), these proceedings.

% \bibitem{Gebert.Nf3}
% C.~Gebert, these proceedings.


%--- section 4 --------------------------------------------------

% \bibitem{qChPT}
% S.~Sharpe, 
% Phys. Rev. D 46 (1992) 3146;
% C.~Bernard and M.~Goltermann D 46 (1992) 853.

\bibitem{ChPT.Nf2.UKQCD}
A.C.~Irving {\it et al.} (UKQCD Collaboration), 
Phys. Lett. B518 (2001) 243. 


\bibitem{PQChPT}
S.R.~Sharpe, 
Phys. Rev. D56 (1997) 7052.

\bibitem{ChPT.Nf0.ALPHA}
J.~Heitger, R.~Sommer and H.~Wittig (ALPHA Collaboration),
Nucl. Phys. B588 (2000) 377.

\bibitem{pqChPT.with.eta}
M.F.L.~Goltermann and K.C.~Leung, 
Phys. Rev. D57 (1998) 5703.

\bibitem{ChPT.Nf3.Nelson}
D.~Nelson, these proceedings.

\bibitem{mu0}
A.G.~Cohen, D.B.Kaplan and A.E.Nelson,
JHEP 11 (1999) 027.

\bibitem{ChPT.const}
S.~Sharpe and N.~Shoresh,
Nucl. Phys. B (Proc. Suppl.) 83-84 (2000) 968.
% Phys. Rev. D62 (2000) 094503.


% \bibitem{qChPT.delta.Bardeen}
% W.Bardeen {\it et al.},
% Phys. Rev. D 62 (2000) 114505



%--- section 5 -------------------------------------------------


% \bibitem{NP.RIMOM}
% G.~Martinelli {\it et al.},
% Nucl. Phys. B445 (1995) 81.

% \bibitem{NP.SF}
% M.~L\"uscher {\it et al.},
% Nucl. Phys. B478 (1996) 365;
% Nucl. Phys. B491 (1997) 344.

% \bibitem{NP.WI}
% T.~Bhattacharya {\it et al.},
% Phys.Rev. D63 (2001) 074505.


% \bibitem{NP-cA.SF}
% P.~Hernandez {\it et al.}, 
% JHEP 0107 (2001) 018.
%
% \bibitem{NP-cA.WI.LANL}
% R.~Gupter, these proceedings.
%
% \bibitem{NP-cA.WI.UKQCD}
% S.~Collins, these proceedings.
%       


% \bibitem{NP-Z.GW}
% 



\bibitem{DW.RBC}
Y.~Aoki (RBC Collaboration), these proceedings. 

\bibitem{aniso.Matsufuru}
H.~Matsufuru, T.~Onogi and T.~Umeda, 
hep-lat/0107001;
these proceedings.


\bibitem{tmQCD.Aoki}
S.~Aoki, 
Phys. Rev. D30 (1984) 2653.

\bibitem{tmQCD}
W.~Bardeen {\it et al.}, 
Phys. Rev. D59 (1999) 014507;
R.~Frezzotti {\it et al.},
Nucl. Phys. B (Proc. Suppl.) 83-84 (2000) 941;
JHEP 0108 (2001) 058.

\bibitem{tmQCD.Frezzotti.lat01}
R.~Frezzotti {\it et al.},
these proceedings.

\bibitem{tmQCD.McNeile.lat01}
C.~McNeile (UKQCD Collaboration),
these proceedings.

\bibitem{NP-csw.ALPHA.Nf0}
M.~L\"uscher {\it et al.}, Nucl. Phys. B491 (1997) 323.

\bibitem{NP-Z.RG}
K.~Ide (CP-PACS Collaboration), these proceedings.


\bibitem{Nf0.N-}
F.X.~Lee and D.B.~Leinweber,
Nucl. Phys. B (Proc. Suppl.) 73 (1999) 258;
F.X.~Lee,
{\it ibid.} 94 (2001) 251;
S.~Sasaki, T.~Blum and S.~Ohta,
hep-lat/0102010.


\bibitem{QCDSF.Nf0.N-}
M.~G\"ockeler {\it et al.} 
(QCDSF-UKQCD-LHPC Collaboration), 
hep-lat/0106022.

\bibitem{Lee.Nf0.N-}
F.X.~Lee, these proceedings.




%--- section 6 -------------------------------------------------------

\bibitem{alphaP}
G.P.~Lepage and P.B.~Mackenzie, 
Phys. Rev. D48 (1993) 2250.

\bibitem{alphas.SESAM}
A.~Spitz {\it et al.} (SESAM Collaboration), 
Phys. Rev. D60 (1999) 074502.

\bibitem{alphas.QCDSF}
S.~Booth {\it et al.} (QCDSF-UKQCD Collaboration), 
Phys. Lett. B519 (2001) 229;
G.~Schierholz (QCDSF-UKQCD Collaboration),
these proceedings.

\bibitem{alphas.Boucaud}
Ph.~Boucaud~{\it et al.},~hep-ph/0107278;
H.~Moutarde, these proceedings.

\bibitem{alphas.NRQCD}
C.T.H.~Davies {\it et al.},
Phys. Rev. D56 (1997) 2755.



\bibitem{alphas.UKQCD}
L.~Marcantonio~{\it et al.}~(UKQCD~Collaboration), 
Nucl. Phys. B (Proc. Suppl.) 94 (2001) 363.


% \bibitem{alphaGF}
% B.~Alles {\it et al.},
% Nucl. Phys. B502 (1997) 325.

\bibitem{alphaSF}
M.~L\"uscher {\it et al.}, 
Nucl. Phys. B413 (1994) 481.

\bibitem{alphas.ALPHA}
A.~Bode {\it et al.} (ALPHA Collaboration),
Phys. Lett. B515 (2001) 49;
U.~Wolff (ALPHA Collaboration), these proceedings.

% \bibitem{alphaqq}
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%--- section 7 -------------------------------------------------------


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\end{thebibliography}