\vspace{-1.8cm} \hfill \null \hfill \hbox{\normalsize ADP-02-66/T506} \hskip1.5cm \phantom{?}\\ \vspace{+1.3cm} Quark propagator in a covariant gauge
F.D.R. Bonnet
D.B. Leinweber
A.G. Williams
J.M. Zanotti
and\
J.B. Zhang.
\vspace{+4mm}Dept. of Physics and Mathematical Physics and CSSM
Adelaide 5005.
Using mean--field improved gauge field configurations, we compare the results obtained for the quark propagator from Wilson fermions and Overlap fermions on a $\3$ lattice at a spacing of $a=0.125(2)$ fm. \vspace{1pc}
CairnPropProc.tex
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\nc{\ccmuofp}{{\cal{C}}_{\mu}(p)}
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\nc{\caofps}{{\cal{A}}^{2}(p)}
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\hfill \rm \null \hfill
 \hbox{\normalsize ADP-02-66/T506} \hskip1.5cm \phantom{?}\\
\vspace{+1.3cm}
Quark propagator in a covariant gauge}

\author{F.D.R. Bonnet,
        D.B. Leinweber,
        A.G. Williams,
        J.M. Zanotti, and\,
        J.B. Zhang.\\
\vspace{+4mm}Dept. of Physics and Mathematical Physics and CSSM, University of Adelaide, Adelaide 5005.}
\begin{document}

\begin{abstract}
Using mean--field improved gauge field configurations,
we compare the results obtained for the quark propagator from Wilson
fermions and Overlap fermions on a $\3$ lattice at a spacing of $a=0.125(2)$ fm.
\vspace{1pc}
\end{abstract}

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\section{INTRODUCTION}

In QCD, asymptotic freedom states that at short distances the effective coupling constant vanishes.
In that regime the interaction between quarks and gluons is largely reduced.
The dynamics of these particles can be studied through their propagators in
momentum space. The quark propagator is really a description of how the
quark propagates in the QCD vacuum and is one of the most fundamental building
blocks of QCD. By studying the momentum dependent quark mass function in the infrared region,
the scalar part of the propagator, we can gain some insights into the mechanism of
chiral symmetry breaking. Chiral symmetry is dynamically broken in the QCD vacuum.
This gives rise to mass generation in the infrared.

In the deep infrared region, artifacts associated with the finite size of the
lattice spacing become small. This is the most interesting region as nonperturbative physics
lies here. However, the ultraviolet behaviour of the propagator at large momenta will in general
strongly deviate from the correct continuum behaviour. This behaviour will be action dependent.

In this brief report we compare the quark propagators of Wilson and overlap fermions. Additional details
may be found in Ref.~\cite{bonnet}.

\subsection{The Quark Propagator}

In the continuum, it is possible to study dynamical chiral symmetry breaking
(DCSB) using the renormalized quark Dyson--Schwinger equation (quark--DSE) in Euclidean space.
In this equation, information about the renormalized dressed gluon propagator and dressed quark--gluon vertex
is embodied. The quark propagator has the general form
\bea
S(p)&=&\frac{1}{i\gamma\cdot{p}A(p^2;\zeta^{2})+B(p^2;\zeta^{2})}\nm\\
&=&\frac{Z(p^2;\zeta^{2})}{i\gamma\cdot{p}+M(p^2;\zeta^{2})},
\label{quarkprop}
\eea
where $M=B/A$ and $Z=1/A$. Here, the parameter $\zeta$ represents the renormalization point.
The functions $A(p^2;\zeta^{2})$ and $B(p^2;\zeta^{2})$ carry all the 
effects of vector and scalar quark dressing induced by the quark 
interactions with the gluon field.

Through this simple form we can extract the quark mass function $M(p)$ 
and renormalization function, $Z(p)$.

\section{QUARK PROPAGATOR ON THE LATTICE}

On the lattice we expect the bare quark propagators, in momentum space,
to have a similar form as in the continuum~\cite{jon1,jon2,alkofer}. Hence, the dimensionless inverse lattice
bare quark propagator takes the general form of
\bea
S^{-1}(p)&\equiv&{i\left(\sum_{\mu}\cmuofp\gamma_{\mu}\right)+\bofp}\nm\\
&\equiv&\frac{i\left(\sum_{\mu}\ccmuofp\gamma_{\mu}\right)+\cbofp}{{\cal{C}}^{2}(p)+{\cal{B}}^{2}(p)},
\label{invquargen}
\eea
with ${\cal{C}}^{2}(p)=\sum_\mu(\ccmuofp)^{2}$. The discrete momentum values for a
$t$--antiperiodic lattice of size $N^{3}_{i}\times{N_{t}}$, with $n_i=1,..,N_i$ and $n_t=1,..,N_t$,
are given by $p_i=\frac{2\pi}{N_{i}a}\left(n_i-\frac{N_i}{2}\right)$ and
$p_t=\frac{2\pi}{N_{t}a}\left(n_t-\frac{1}{2}-\frac{N_t}{2}\right)$.
The quark propagator is
\be
S(p)\equiv{-i\left(\sum_{\mu}\ccmuofp\gamma_{\mu}\right)+\cbofp}.
\ee
Taking the trace we obtain
\bea
\ccmuofp&=&\frac{i}{4N_c}{\trace}[\gamma_\mu{S(p)}], \hspace{0.1cm}{\rm{and}}\hspace{0.1cm}\nm\\
\cbofp&=&\frac{1}{4N_c}{\trace}[S(p)],
\eea
which we use to construct the $\cmuofp$ and $\bofp$
\bea
\label{cmup}
\cmuofp&=&\frac{\ccmuofp}{{\cal{D}}(p)},\hspace{0.3cm}{\rm{and}}\hspace{0.3cm}\bofp=\frac{\cbofp}{{\cal{D}}(p)},
\eea
where ${\cal{D}}(p)={{\cal{C}}^{2}(p)+{\cal{B}}^{2}(p)}$. At tree--level the 
lattice quark propagator takes the same form as in Eq.~(\ref{quarkprop}) 
and we know that the free propagator
$(S^{(0)}(p))^{-1}{\equiv}(Z^{(0)}(p))^{-1}\left[ik\slh+M^{(0)}(p)\right]$, 
where $k_{\mu}\longrightarrow{p_{\mu}}$ as $p_{\mu}\longrightarrow{0}$. 
It is then possible
to extract the momentum directly from the lattice by calculating
\bea
q_\mu\equiv\comuofp&=&\frac{\ccomuofp}{({\cal{C}}^{(0)}(p))^{2}+({\cal{B}}^{(0)}(p))^{2}}.
\label{latmomt}
\eea
Results are displayed in Fig.~\ref{fig:pvsq}.

\begin{figure}[tbh]
\resizebox*{\columnwidth}{!}{\rotatebox{90}{\includegraphics{./pVSqsq.combM.ful.ps}}}
\caption{\label{fig:pvsq}The lattice momentum $q$ versus the discrete momentum $p$, both in GeV. Overlap ($\square$) and Wilson fermion ($\circ$) momenta are indicated.}
\end{figure}

\subsection{Mass and Renormalization Functions}

The dimensionless inverse lattice bare quark propagator takes the form
\bea
S^{-1}(p)&\equiv&{ia{q\slh}A(p)+B(p)}\nm\\
&=&{[Z^{\rm{L}}(p)]^{-1}}\left({ia{q\slh}+M^{\rm{L}}(p)}\right),
\eea
where $M^{\rm{L}}(p)=B(p)/A(p)$ is the lattice quark mass function and $Z^{\rm{L}}(p)=1/A(p)$ the
lattice renormalization function. The functions $\aofp$ and $\bofp$ may be written as:
\bea
\label{ap}
\aofp&=&\frac{\caofp}{{\cal{D}}(p)},\hspace{0.3cm}{\rm{and}}\hspace{0.3cm}\bofp=\frac{\cbofp}{{\cal{D}}(p)},
\eea
where ${\cal{D}}(p)={\caofps{q^2}+\cbofps}$. Hence an equivalent definition for the quark propagator
is $S(p)\equiv{-ia{q\slh}\caofp+\cbofp}$. Extracting the functions $\caofp$ and $\cbofp$ is done via:
\bea
\label{curlap}
\caofp&=&\frac{i}{4N_caq^2}\trace[q\slh{S(p)}],\hspace{0.5cm}{\rm{and}}\hspace{0.5cm}\nm\\
\cbofp&=&\frac{1}{4N_c}\trace[S(p)].
\eea
\begin{figure}[tbh]
\resizebox*{\columnwidth}{!}{\rotatebox{90}{\includegraphics{./uncorrected.qsq.hlf.ps}}\rotatebox{90}{\includegraphics{./uncorrected.qsq.sca.hlf.ps}}}
\caption{\label{fig:uncorwil}The uncorrected mass function $\mofp$ for Wilson fermions.}
\end{figure}

In Fig.~\ref{fig:uncorwil} we show the uncorrected mass function for Wilson
fermions, where we can see a large divergence in the ultraviolet region.

\subsection{Tree--Level Correction}

The tree-level multiplicative correction is given by
\bea
\acofp=\frac{\aofp}{\aoofp}1\hspace{0.1cm}{\rm{,}}\hspace{0.1cm}\bcofp=\frac{\bofp}{\boofp}m_q,
\label{acbc}
\eea
where $\aoofp$ and $\boofp$ are obtained from $S^{(0)}(p)$. Then tree-level corrected mass
and renormalization functions are constructed using Eq.(\ref{acbc})
\bea
\mcofp=\left(\frac{\bcofp}{\acofp}\right)\hspace{0.1cm}{\rm{,}}\hspace{0.1cm}\zcofp=\frac{1}{\acofp}.
\eea

\section{FERMIONS ON THE LATTICE}

\subsection{Wilson Fermions}

The Wilson fermion~\cite{wilson} action on the lattice is defined as,
\bea
S_{W}[U,\overline{\psi},\psi]=\sum_{xy}\overline{\psi}(x)D_{W}(x,y)\psi(y),\nm
\label{wilfermact}
\eea
with $D_{W}(x,y)$ the usual Wilson fermion operator.
The lattice bare mass is related to the hopping parameter via 
$\kappa={1}/{(2m_q a +8r)}$ and
$m_q\equiv({1}/{2a})({1}/{\kappa}-{1}/{\kappa_{c}})$.

\subsection{Overlap Fermions}

The overlap fermion
\cite{Neu95} 
realizes exact chiral symmetry on the lattice
\bea
%D(m_0)&=&1+\frac{am_0}{2}+\left(1-\frac{am_0}{2}\right){\gamma_5}\xi(H)\nm\\
D(\mu)&=&\frac{1}{2}\left[1+\mu+(1-\mu)\frac{D_{W}}{\sqrt{D^{\dagger}_{W}D_{W}}}\right]\nm
\eea
with $0{\leq}\mu\leq{1}$ describing fermions with a positive mass from 0 to $\infty$.
The hopping parameter is given by $\kappa={1}/{(-2m+8r)}$ and 
$m\equiv({1}/{2\kappa_{c}}-{1}/{2\kappa})$.
To describe a single massless Dirac fermion for $D(0)$ we must have $0\,{\leq}m\leq{2}$
at tree--level. The overlap propagator is given by
\bea
\tilde{D}^{-1}(\mu)=(1-\mu)^{-1}[{D}^{-1}(\mu)-1],
\eea
and is related to the continuum propagator by ${D}^{-1}_{c}(m_q)=\calz_{\psi}\tilde{D}^{-1}(\mu)$, and similarly
for the quark mass $m_q=\calz^{-1}_{m}\mu$.

At tree-level \cite{Edw99} it is found that
$\calz_{\psi}^{(0)}=\calz_{m}^{-1}=2m$, and the free propagator becomes
\bea
\left[{D}^{(0)}_{c}(m_q)\right]^{-1}=\left[\calz_{\psi}^{(0)}\tilde{D}^{(0)}(\mu)\right]^{-1},\nm
\eea
and when the interactions are turned on we have
\bea
\left[{D}_{c}(m_q)\right]^{-1}=\left[\calz_{\psi}\tilde{D}(\mu)\right]^{-1},\nm
\eea
where $\calz_{\psi}=\calz_{m}^{-1}$. 

\section{NUMERICAL RESULTS}

Using fifty improved gauge field configurations, on a $12^3\times{24}$ lattice with $a=0.125(1)$ fm, the
overlap quark propagator is calculated for ten different masses,
$m_{q}=\calz_{\psi}\mu=\{126,147,168,210,252,315,420,524,629,734\}$ MeV. Results are illustrated
in Figs.~\ref{fig:ovrm} and~\ref{fig:ovrz}.
\begin{figure}[tb]
\resizebox*{\columnwidth}{!}{\rotatebox{90}{\includegraphics{./Mcor.combM.qsq.ovr.ful.ps}}}
\caption{\label{fig:ovrm}The mass function $\mofp$ for overlap fermions (full data) for the ten different masses.}
\end{figure}
\begin{figure}[tbh]
\resizebox*{\columnwidth}{!}{\rotatebox{90}{\includegraphics{./Zcor.combM.qsq.Norm.ovr.ful.ps}}}
\caption{\label{fig:ovrz}The renormalization function for overlap fermions (full data) for the ten different masses. The function is renormalized at $\zeta=8.2$ GeV.}
\end{figure}
For Wilson fermions, using a hundred configurations, five
masses are considered, namely $m_{q}=\{221,181.5,138.3,99.91,62.04\}$ MeV. Results are displayed in
Figs.~\ref{fig:extwilm} and~\ref{fig:extwilz}.
\begin{figure}[tbh]
\resizebox*{\columnwidth}{!}{\rotatebox{90}{\includegraphics{./Mcor.combM.psq.wil.hlf.ps}}}
\caption{\label{fig:extwilm}The tree--level corrected mass function $\mcofp$ for Wilson fermions (half cut data).}
\end{figure}
\begin{figure}[tbh]
\resizebox*{\columnwidth}{!}{\rotatebox{90}{\includegraphics{./Zcor.combM.psq.wil.hlf.ps}}}
\caption{\label{fig:extwilz}The tree--level corrected renormalization function for Wilson fermions (half cut data).}
\end{figure}

A linear extrapolation is used to compare the two actions. In Fig.~\ref{fig:extwilovrm}
\begin{figure}[tbh]
\resizebox*{\columnwidth}{!}{\rotatebox{90}{\includegraphics{./Mext.psq.Proc.2pi.ps}}}
\caption{\label{fig:extwilovrm}Comparison of the linearly extrapolated mass function $M$ for both Wilson fermions ($\times$) and overlap fermions ($\square$), (cylinder cut data).}
\end{figure}
we show the linearly extrapolated quark mass function for both actions plotted versus the discrete
momentum $p$. When plotting $M$ and $Z$ versus the lattice momentum, $q$, the points are pushed away
from the origin in the case of overlap fermions as opposed to Wilson fermions where the points are pulled
towards the origin. The overlap action (plotted as $\square$) produces in the deep infrared $M(0)=297(11)$ MeV.
Comparing with the Wilson fermion action ($\times$) we see a clear superiority of the overlap action over the
Wilson action. Wilson fermions show a significant dip between 0.8 GeV and 2.0 GeV.

The linearly extrapolated renormalization function is shown in Fig.~\ref{fig:extwilovrz}.
\begin{figure}[tbh]
\resizebox*{\columnwidth}{!}{\rotatebox{90}{\includegraphics{./plot.Zext.psq.Proc.2pi.ps}}}
\caption{\label{fig:extwilovrz}Comparison of the linearly extrapolated renormalization function for Wilson fermions ($\times$) and overlap fermions ($\square$), (cylinder cut data).}
\end{figure}
In the deep infrared for the overlap action we have $Z(0)=0.48(2)$ with the renormalization point located
at $\zeta=3.9$ GeV.

\section{SUMMARY}

Tree-level correction represents a crucial step and a powerful tool for correcting
the Wilson action results. The division method produces a smooth quark mass function throughout
the momentum spectrum. We are able to gain some insights into the analytic structure of the overlap quark
propagator: $\boofp=\calz_{\psi}^{(0)}\mu$ and $\aoofp=1$. The overlap produces very good results
for $M(p)$ and $Z(p)$. No tree-level correction is required for overlap fermions.

\begin{thebibliography}{9}
\bib{bonnet}F.\ D.\ R.\ Bonnet, P.\ O.\ Bowman, D.\ B.\ Leinweber, A.\ G.\ Williams \& J.\ B.\ Zhang, Submitted to Phys. Rev. D, hep-lat/0202003.
\bib{jon1}J-I.\ Skullerud \& A.\ G.\ Williams, Phys. Rev. D63 (2001) 054508.
\bib{jon2}J-I.\ Skullerud, D.\ B.\ Leinweber \& A.\ G.\ Williams, Phys. Rev. D64 (2001) 074508.
\bib{alkofer}R.\ Alkofer \& L- von Smekal, Phys. Rep. 353 (2001) 281.
\bib{wilson}K.\ G.\ Wilson, in New Phenomena in Subnuclear Physics, Part A, A. Z
ichichi (ed.), Plenum Press, NY, (1975) 69.
\bib{Neu95} H. Neuberger {\it et al.}, Nucl. Phys. {\bf B443}, 305 (1995).
\bib{Edw99} R. Edwards, {\it et al.}, Phys. Rev. D {\bf 59}, 094510 (1999).
\end{thebibliography}

\end{document}

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\begin{table*}[htb]
\caption{The next-to-leading order (NLO) results
{\em without} the pion field.}
\label{table:1}
\newcommand{\m}{\hphantom{$-$}}
\newcommand{\cc}[1]{\multicolumn{1}{c}{#1}}
\renewcommand{\tabcolsep}{2pc} % enlarge column spacing
\renewcommand{\arraystretch}{1.2} % enlarge line spacing
\begin{tabular}{@{}lllll}
\hline
$\Lambda$ (MeV)           & \cc{$140$} & \cc{$150$} & \cc{$175$} & \cc{$200$} \\
\hline
$r_d$ (fm)                & \m1.973 & \m1.972 & \m1.974 & \m1.978 \\
$Q_d$ ($\mbox{fm}^2$)     & \m0.259 & \m0.268 & \m0.287 & \m0.302 \\
$P_D$ (\%)                & \m2.32  & \m2.83  & \m4.34  & \m6.14  \\
$\mu_d$                   & \m0.867 & \m0.864 & \m0.855 & \m0.845 \\
$\mathcal{M}_{\mathrm{M1}}$ (fm)   & \m3.995 & \m3.989 & \m3.973 & \m3.955 \\
$\mathcal{M}_{\mathrm{GT}}$ (fm)   & \m4.887 & \m4.881 & \m4.864 & \m4.846 \\
$\delta_{\mathrm{1B}}^{\mathrm{VP}}$ (\%)   
                          & $-0.45$ & $-0.45$ & $-0.45$ & $-0.45$ \\
$\delta_{\mathrm{1B}}^{\mathrm{C2:C}}$ (\%) 
                          & \m0.03  & \m0.03  & \m0.03  & \m0.03  \\
$\delta_{\mathrm{1B}}^{\mathrm{C2:N}}$ (\%) 
                          & $-0.19$ & $-0.19$ & $-0.18$ & $-0.15$ \\
\hline
\end{tabular}\\[2pt]
The experimental values are given in ref. \cite{Eato75}.
\end{table*}

\begin{sidewaystable}
\caption{The next-to-leading order (NLO) results
{\em without} the pion field.}
\label{table:2}
\newcommand{\m}{\hphantom{$-$}}
\newcommand{\cc}[1]{\multicolumn{1}{c}{#1}}
\renewcommand{\arraystretch}{1.2} % enlarge line spacing
\begin{tabular*}{\textheight}{@{\extracolsep{\fill}}lllllllllllll}
\hline
& $\Lambda$ (MeV) & \cc{$140$} & \cc{$150$} & \cc{$175$} & \cc{$200$} & \cc{$225$} & \cc{$250$} &
\cc{Exp.} & \cc{$v_{18}$~\cite{v18}} &  \\
\hline
%b
 & $r_d$ (fm)                        & \m1.973 & \m1.972 & \m1.974 & \m1.978 & \m1.983 & \m1.987 & 1.966(7) & \m1.967 & \\[2pt]
 & $Q_d$ ($\mbox{fm}^2$)             & \m0.259 & \m0.268 & \m0.287 & \m0.302 & \m0.312 & \m0.319 & 0.286    & \m0.270 & \\[2pt]
 & $P_D$ (\%)                        & \m2.32  & \m2.83  & \m4.34  & \m6.14  & \m8.09  & \m9.90  & $-$      & \m5.76  & \\[2pt]
 & $\mu_d$                           & \m0.867 & \m0.864 & \m0.855 & \m0.845 & \m0.834 & \m0.823 & 0.8574   & \m0.847 & \\[5pt]
 & $\mathcal{M}_{\mathrm{M1}}$ (fm)             & \m3.995 & \m3.989 & \m3.973 & \m3.955 & \m3.936 & \m3.918 & $-$      & \m3.979 & \\[5pt]
 & $\mathcal{M}_{\mathrm{GT}}$ (fm)             & \m4.887 & \m4.881 & \m4.864 & \m4.846 & \m4.827 & \m4.810 & $-$      & \m4.859 & \\[2pt]
 & $\delta_{\mathrm{1B}}^{\mathrm{VP}}$ (\%)   & $-0.45$ & $-0.45$ & $-0.45$ & $-0.45$ & $-0.45$ & $-0.44$ & $-$      & $-0.45$ & \\[2pt]
 & $\delta_{\mathrm{1B}}^{\mathrm{C2:C}}$ (\%) & \m0.03  & \m0.03  & \m0.03  & \m0.03  & \m0.03  & \m0.03  & $-$      & \m0.03  & \\[2pt]
 & $\delta_{\mathrm{1B}}^{\mathrm{C2:N}}$ (\%) & $-0.19$ & $-0.19$ & $-0.18$ & $-0.15$ & $-0.12$ & $-0.10$ & $-$      & $-0.21$ & \\
\hline
\end{tabular*}\\[2pt]
The experimental values are given in ref. \cite{Eato75}.
\end{sidewaystable}

Horizontal lines should be placed above and below table headings, above
the subheadings and at the end of the table above any notes. Vertical
lines should be avoided.

If a table is too long to fit onto one page, the table number and
headings should be repeated above the continuation of the table. For
this you have to reset the table counter with
\verb|\addtocounter{table}{-1}|. Alternatively, the table can be turned
by $90^\circ$ (`landscape mode') and spread over two consecutive pages
(first an even-numbered, then an odd-numbered one) created by means of
\verb|\begin{table}[h]| without a caption. To do this, you prepare the
table as a separate \LaTeX{} document and attach the tables to the
empty pages with a few spots of suitable glue.

\subsection{Useful table packages}

Modern \LaTeX{} comes with several packages for tables that
provide additional functionality. Below we mention a few. See
the documentation of the individual packages for more details. The
packages can be found in \LaTeX's \texttt{tools} directory.

\begin{description}
  
\item[\texttt{array}] Various extensions to \LaTeX's \texttt{array}
  and \texttt{tabular} environments.
  
\item[\texttt{longtable}] Automatically break tables over several
  pages. Put the table in the \texttt{longtable} environment instead
  of the \texttt{table} environment.
  
\item [\texttt{dcolumn}] Define your own type of column. Among others,
  this is one way to obtain alignment on the decimal point.

\item[\texttt{tabularx}] Smart column width calculation within a
  specified table width.
  
\item[\texttt{rotating}] Print a page with a wide table or figure in
  landscape orientation using the \texttt{sidewaystable} or
  \texttt{sidewaysfigure} environments, and many other rotating
  tricks. Use the package with the \texttt{figuresright} option to
  make all tables and figures rotate in clockwise. Use the starred
  form of the \texttt{sideways} environments to obtain full-width
  tables or figures in a two-column article.

\end{description}

\subsection{Line drawings}

Line drawings may consist of laser-printed graphics or professionally
drawn figures attached to the manuscript page. All figures should be
clearly displayed by leaving at least one line of spacing above and
below them. When placing a figure at the top of a page, the top of the
figure should align with the bottom of the first text line of the other
column.

Do not use too light or too dark shading in your figures; too dark a
shading may become too dense while a very light shading made of tiny
points may fade away during reproduction.

All notations and lettering should be no less than 2\,mm high. The use
of heavy black, bold lettering should be avoided as this will look
unpleasantly dark when printed.

\subsection{PostScript figures}

Instead of providing separate drawings or prints of the figures you
may also use PostScript files which are included into your \LaTeX{}
file and printed together with the text. Use one of the packages from
\LaTeX's \texttt{graphics} directory: \texttt{graphics},
\texttt{graphicx} or \texttt{epsfig}, with the \verb|\usepackage|
command, and then use the appropriate commands
(\verb|\includegraphics| or \verb|\epsfig|) to include your PostScript
file.

The simplest command is: \newline
\verb|\includegraphics{file}|, which inserts the
PostScript file \texttt{file} at its own size. The starred version of
this command: \newline
\verb|\includegraphics*{file}|, does the same, but clips
the figure to its bounding box.

With the \texttt{graphicx} package one may specify a series of options
as a key--value list, e.g.:
\begin{tabular}{@{}l}
\verb|\includegraphics[width=15pc]{file}|\\
\verb|\includegraphics[height=5pc]{file}|\\
\verb|\includegraphics[scale=0.6]{file}|\\
\verb|\includegraphics[angle=90,width=20pc]{file}|
\end{tabular}

See the file \texttt{grfguide}, section ``Including Graphics Files'',
of the \texttt{graphics} distribution for all options and a detailed
description.

The \texttt{epsfig} package mimicks the commands familiar from the
package with the same name in \LaTeX2.09. A PostScript file
\texttt{file} is included with the command
\verb|\psfig{file=file}|.

Grey-scale and colour photographs cannot be included in this way,
since reproduction from the printed CRC article would give
insufficient typographical quality. See the following subsections.

\begin{figure}[htb]
\vspace{9pt}
\framebox[55mm]{\rule[-21mm]{0mm}{43mm}}
\caption{Good sharp prints should be used and not (distorted) photocopies.}
\label{fig:largenenough}
\end{figure}
%
\begin{figure}[htb]
\framebox[55mm]{\rule[-21mm]{0mm}{43mm}}
\caption{Remember to keep details clear and large enough.}
\label{fig:toosmall}
\end{figure}

\subsection{Black and white photographs}

Photographs must always be sharp originals ({\em not screened
versions\/}) and rich in contrast. They will undergo the same reduction
as the text and should be pasted on your page in the same way as line
drawings.

\subsection{Colour photographs}

Sharp originals ({\em not transparencies or slides\/}) should be
submitted close to the size expected in publication. Charges for the
processing and printing of colour will be passed on to the author(s) of
the paper. As costs involved are per page, care should be taken in the
selection of size and shape so that two or more illustrations may be
fitted together on one page. Please contact the Author Support
Department at Elsevier (E-mail: \texttt{authorsupport@elsevier.nl})
for a price quotation and layout instructions before producing your
paper in its final form.

\section{EQUATIONS}

Equations should be flush-left with the text margin; \LaTeX{} ensures
that the equation is preceded and followed by one line of white space.
\LaTeX{} provides the document class option {\tt fleqn} to get the
flush-left effect.

\begin{equation}
H_{\alpha\beta}(\omega) = E_\alpha^{(0)}(\omega) \delta_{\alpha\beta} +
                          \langle \alpha | W_\pi | \beta \rangle 
\end{equation}

You need not put in equation numbers, since this is taken care of
automatically. The equation numbers are always consecutive and are
printed in parentheses flush with the right-hand margin of the text and
level with the last line of the equation. For multi-line equations, use
the {\tt eqnarray} environment.

For complex mathematics, use the \AmS math package. This package
sets the math indentation to a positive value. To keep the equations
flush left, either load the \texttt{espcrc} package \emph{after} the
\AmS math package or set the command \verb|\mathindent=0pt| in the
preamble of your article.

\begin{thebibliography}{9}
\bibitem{Scho70} S. Scholes, Discuss. Faraday Soc. No. 50 (1970) 222.
\bibitem{Mazu84} O.V. Mazurin and E.A. Porai-Koshits (eds.),
                 Phase Separation in Glass, North-Holland, Amsterdam, 1984.
\bibitem{Dimi75} Y. Dimitriev and E. Kashchieva, 
                 J. Mater. Sci. 10 (1975) 1419.
\bibitem{Eato75} D.L. Eaton, Porous Glass Support Material,
                 US Patent No. 3 904 422 (1975).
\end{thebibliography}

References should be collected at the end of your paper. Do not begin
them on a new page unless this is absolutely necessary. They should be
prepared according to the sequential numeric system making sure that
all material mentioned is generally available to the reader. Use
\verb+\cite+ to refer to the entries in the bibliography so that your
accumulated list corresponds to the citations made in the text body. 

Above we have listed some references according to the
sequential numeric system \cite{Scho70,Mazu84,Dimi75,Eato75}.
\end{document}

hep-lat0202011.extracted_bib
\begin{thebibliography}{9}
\bib{bonnet}F.\ D.\ R.\ Bonnet, P.\ O.\ Bowman, D.\ B.\ Leinweber, A.\ G.\ Williams \& J.\ B.\ Zhang, Submitted to Phys. Rev. D, hep-lat/0202003.
\bib{jon1}J-I.\ Skullerud \& A.\ G.\ Williams, Phys. Rev. D63 (2001) 054508.
\bib{jon2}J-I.\ Skullerud, D.\ B.\ Leinweber \& A.\ G.\ Williams, Phys. Rev. D64 (2001) 074508.
\bib{alkofer}R.\ Alkofer \& L- von Smekal, Phys. Rep. 353 (2001) 281.
\bib{wilson}K.\ G.\ Wilson, in New Phenomena in Subnuclear Physics, Part A, A. Z
ichichi (ed.), Plenum Press, NY, (1975) 69.
\bib{Neu95} H. Neuberger {\it et al.}, Nucl. Phys. {\bf B443}, 305 (1995).
\bib{Edw99} R. Edwards, {\it et al.}, Phys. Rev. D {\bf 59}, 094510 (1999).
\end{thebibliography}