GLUONS, QUARKS, AND THE TRANSITION FROM NONPERTURBATIVE TO PERTURBATIVE QCD
\underline{ANTHONY G.\ WILLIAMS}
FREDERIC D.R.\ BONNET
PATRICK O.\ BOWMAN
DEREK B.\ LEINWEBER
JON IVAR SKULLERUD
AND JAMES M.\ ZANOTTI
Lattice-based investigations of two fundamental QCD quantities are described, namely the gluon and quark propagators in Landau gauge. We have studied the Landau gauge gluon propagator using a variety of lattices with spacings from $a = 0.17$ to 0.41 fm to explore finite volume and discretization effects. We also introduce the general method of ``tree-level correction'' to minimize the effect of lattice artefacts at large momenta. We demonstrate that it is possible to obtain scaling behavior over a very wide range of momenta and lattice spacings and to explore the infinite volume and continuum limits of the Landau-gauge gluon propagator. These results confirm the earlier conclusion that the Landau gauge gluon propagator is infrared finite. We study the Landau gauge quark propagator in quenched QCD using two forms of the $\order (a)$-improved propagator with the Sheikholeslami-Wohlert quark action with the nonperturbative value for the clover coefficient $\csw$ and mean-field improvement coefficients in our improved quark propagators. We again implement an appropriate form of tree-level correction. We find good agreement between our improved quark propagators. The extracted value of infrared quark mass in the chiral limit is found to be $300\pm 30$ MeV. We conclude that the momentum regime where the transition from nonperturbative to perturbative QCD occurs is $Q^2\simeq 4$~GeV$^2$.
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\begin{document}

\title{GLUONS, QUARKS, AND THE TRANSITION FROM NONPERTURBATIVE
              TO PERTURBATIVE QCD
          }

\author{\underline{ANTHONY G.\ WILLIAMS}, FREDERIC D.R.\ BONNET,
          PATRICK O.\ BOWMAN,
          DEREK B.\ LEINWEBER, JON IVAR SKULLERUD, AND JAMES M.\ ZANOTTI}

\address{CSSM and Department of Physics and Mathematical
             Physics, University of Adelaide, Australia 5005\\
             E-mail: {\tt awilliam@physics.adelaide.edu.au}  }


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% You may repeat \author \address as often as necessary      %
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\maketitle\abstracts{
Lattice-based investigations of two
fundamental QCD quantities are described, namely
the gluon and quark propagators in Landau gauge.
We have studied the Landau gauge gluon
propagator using a variety of lattices with spacings from $a = 0.17$
to 0.41 fm to explore finite volume and discretization
effects.  We also introduce the general method of ``tree-level correction''
to minimize the effect of lattice artefacts at large momenta.
We demonstrate that it is possible to obtain scaling behavior over a
very wide range of momenta and lattice spacings and
to explore the infinite volume and continuum limits of the Landau-gauge
gluon propagator.  These results confirm
the earlier conclusion that the Landau gauge gluon propagator is
infrared finite.
We study the Landau gauge quark propagator in quenched QCD using two
forms of the $\order (a)$-improved propagator with the
Sheikholeslami-Wohlert quark action with the nonperturbative
value for the clover coefficient $\csw$ and mean-field 
improvement coefficients in our improved quark propagators.  
We again implement an appropriate form of tree-level correction.
We find good agreement between our improved quark propagators.
The extracted value of infrared quark
mass in the chiral limit is found to be $300\pm 30$ MeV.
We conclude that the momentum regime where
the transition from nonperturbative to perturbative QCD occurs
is $Q^2\simeq 4$~GeV$^2$.}


\section{Introduction}
\label{sec:intro}

Lattice gauge theory is currently the only known ``first principles''
approach to studying nonperturbative QCD.  
It is therefore important for lattice
QCD to provide constraints and guidance for the construction
of quark-based models\cite{DSE_review}
and to provide an indication of the momentum
regime at which we can expect perturbative QCD to become applicable.
The quark and gluon propagators are two of the most fundamental
quantities in QCD. There has been considerable interest in the
infrared behavior of the 
gluon propagator as a probe into the mechanism of confinement and by studying 
the scalar part of the quark propagator, the mass function, we can gain 
insight into the mechanisms of chiral symmetry breaking. 
Both are used as input for other quark-model
calculations.

\section{Gluon Propagator}

We use an ${\cal O}(a^2)$ tree-level, tadpole-improved action\cite{Weisz83}
and for the tadpole (mean-field)
improvement parameter we use the plaquette measure\cite{tadpole}.
A full description and discussion of the gluon propagator results
summarized here can be found
elsewhere.\cite{LandauGaugeDE,long_glu,big_vol_glu}

\begin{table}[tb]
\centering
\begin{tabular}{cccccc}
   & Dimensions	     & $\beta$ & $a$ (fm) & Volume $\text{(fm}^4\text{)}$ &
Configurations \\
\hline
1w & $16^3\times 32$ &   5.70  &   0.179  & $2.87^3 \times 5.73$  & 100 \\
1i & $16^3\times 32$ &   4.38  &   0.166  & $2.64^3 \times 5.28$  & 100 \\
2  & $10^3\times 20$ &   3.92  &   0.353  & $3.53^3 \times 7.06$  & 100 \\
3  & $8^3 \times 16$ &   3.75  &   0.413  & $3.30^3 \times 6.60$  & 100 \\
4  & $16^3\times 32$ &   3.92  &   0.353  & $5.65^3 \times 11.30$ & 100 \\
5  & $12^3\times 24$ &   4.10  &   0.270  & $3.24^3 \times 6.48$  & 100 \\
6  & $32^3\times 64$ &   6.00  &   0.099  & $3.18^3 \times 6.34$  &  75 \\
\end{tabular}
\caption{Details of the lattices used to calculate the gluon propagator.  
Lattices 1w and 1i have the same dimensions and approximately the same lattice
spacing, but were generated with the Wilson and improved actions respectively.
Lattice 6 was generated with the Wilson action.}
\label{table:latlist}
\end{table}

Gauge fixing on the lattice is achieved by maximizing a functional, the 
extremum of which implies the gauge fixing condition.  The usual Landau 
gauge fixing functional implies that $\sum_\mu \del_\mu A_\mu
= 0$ up to \oa{2}.  To ensure that gauge dependent quantities are also
\oa{2} improved, we implement the analogous \oa{2} improved gauge
fixing.\cite{LandauGaugeDE}
The dimensionless lattice gluon field $A_{\mu}(x)$ is calculated from
the link variables in the usual way, 
which agrees with the continuum to ${\cal O}(a^2)$.
We then calculate the scalar part of the propagator
\begin{equation}
D(x-y) = \sum_\mu \langle A_\mu(y) A_\mu(x) \rangle \, .
\end{equation}
To isolate the nonperturbative behavior of the gluon propagator, we
can divide the propagator by its lattice tree level form (i.e., that
of lattice perturbation theory).\cite{long_glu}
For the momentum space gluon propagator $D(q^2)$, we see
that in the continuum $q^2D(q^2)$ will approach a constant up to 
logarithmic corrections as $q^2\to\infty$ because of asymptotic
freedom.  The continuum tree-level propagator is $1/q^2$.
We also expect asymptotic freedom on the lattice despite
finite lattice spacing artefacts.  We {\em define} the lattice
$q_\mu$ such that the lattice $D^{\rm tree}(q)\equiv 1/q^2$,
and use this momentum throughout. This is referred to as tree-level
correction and we have seen that it significantly reduces discretization
arrors at large momenta.  For the two actions considered here,
this means that we work with the momentum variables defined as
%
\begin{equation}
q_{\mu}^W \equiv \frac{2}{a} \sin\frac{\qhat_{\mu} a}{2},
\hskip1cm
q_\mu^I \equiv \frac{2}{a}\sqrt{ \sin^2 
	\Bigl( \frac{\qhat_\mu a}{2} \Bigr)
	+ \frac{1}{3}\sin^4 \Bigl( \frac{\qhat_\mu a}{2} \Bigr) 
	} \, ,
\label{eq:latt_momenta}
\end{equation}
for the Wilson and improved actions respectively.
All figures (quark and gluon propagators) have a cylinder cut imposed upon 
them, i.e.\ all momenta must lie close to the lattice diagonal.
In Table~\ref{table:latlist} we show the various lattices that we have
studied for the gluon propagator.  In Fig.~\ref{fig:Comp1i_6} 
we plot $q^2 D(q^2)$ for a fine unimproved Wilson action and for
our finest improved action.  Despite having very different lattice
spacings the agreement is excellent for the entire intermediate
and high-momentum regime.  The small discrepancy in the deep infrared
due to finite volume effects is not apparent in this way of plotting
that data.  We plot $D(q^2)$ for five different lattice in
Fig.~\ref{fig:AllProps} and see pleasing agreement for the results.
Note that we are plotting bare quantities only and there is thus an
overall wavefunction renormalization for the gluon propagator
(i.e., $Z_3(\mu,a)$ for the renomalization point $\mu$).  The vertical
scale is thus unimportant and only the variation with momentum
is relevant.
This way of presenting the data shows that there is a small residual
finite volume dependence, where the infrared gluon propagator is
{\em decreasing} with increasing lattice volume $V$.  We have performed
a fit as a function of $1/V$ and have seen that the large volume
$\beta=3.92$ lattice gives results which are very close to the
infinite volume limit.
In Fig.~\ref{fig:pert_vs_latt} we plot $D(q^2)$ in the intermediate
and ultraviolet regime and compare with the three-loop perturbative
QCD form.  We see that the above $q\sim 2$~GeV the agreement is
excellent, but that below this momentum scale nonperturbative effects
are becoming apparent.

\begin{figure}[ht]
\begin{center}
\epsfig{figure=Comp1i_6.ps,angle=90,width=8cm}
\end{center}
\caption{Comparison of the gluon propagator from the finest improved
lattice (lattice 1i, $\beta = 4.38$) and the finest Wilson lattice 
(lattice 6, $\beta=6.0$).  Data has been cylinder cut and the appropriate
tree-level corrections have been applied.  The data from lattice 6 is
half-cut whereas lattice 1i displays the full Brillouin zone.  We have
determined $Z_3(\text{improved}) / Z_3(\text{Wilson}) = 1.08$ by
matching the vertical scales of the data.}  
\label{fig:Comp1i_6}
\end{figure}

\begin{figure}[ht]
\begin{center}
\epsfig{figure=AllProps.ps,angle=90,width=8cm}
\end{center}
\caption{Comparison of the gluon propagator generated with an improved action
on five different lattices.  We find good agreement down to 
$q \simeq 500$ MeV.  At the lowest accessible momenta the data points drop
monotonically with increasing volume, but the lowest point (on the largest 
lattice) shows signs of having converged to its infinite volume value.
For comparison with perturbation theory, a plot of the continuum, tree-level 
gluon propagator (i.e., $1 / q^2$ appropriately scaled) has been included.}
\label{fig:AllProps}
\end{figure}


\begin{figure}[ht]
\begin{center}
\epsfig{figure=pert_vs_latt.ps,angle=90,width=8cm}
\end{center}
\caption{Comparison of the lattice gluon propagator with that obtained from
perturbation theory, in the ultraviolet to intermediate regime.  The continuum
expressions are tree-level (i.e., $1 / q^2$ appropriately scaled) and
the three-loop perturbative QCD expression.}
\label{fig:pert_vs_latt}
\end{figure}

\section{Quark Propagator}

The Landau gauge quark propagator results summarized here have been
presented and discussed in more detail
elsewhere.\cite{quark_prop}
All ${\cal O}(a)$ errors in the fermion action can be removed by adding
appropriate terms to the Lagrangian\cite{Luscher:1996sc,Dawson:1997gp}.
It is then usual to perform appropriate field
transformations to improve the quark operators
as well.\cite{Heatlie:1991kg}

%%%%%%%%%% Quark Prop Figs %%%%%%%%%%

\begin{figure}[ht]
\begin{center}
\epsfig{figure=Zr_np_scal_phys.ps,angle=0,width=7cm}
\end{center}
\caption{The tree-level corrected $Z(p)$ for $\csw=\text{NP}$ and
$\kappa=0.137$.
The figure shows $Z(p)$ vs.\ momentum in
physical units, and after rescaling (``renormalizing'') so that
$Z(2.1 \text{GeV})=1$. 
The infrared agreement after rescaling is very good and we take the tree-level
corrected $Z(p)$ from $S_R$ with the nonperturbative $\csw$ as the best
estimate for this quantity.
}
\label{fig:z_np_compare}
\end{figure}

\begin{figure}[ht]
\begin{center}
\setlength{\unitlength}{0.85cm}
\setlength{\fboxsep}{0cm}
\begin{picture}(14,7)
\put(0,0){\begin{picture}(7,7)\put(-0.9,-0.4){\ebox{6cm}{Mr_np_imp_hyb_phys.ps}}\end{picture}}
\put(7,0){\begin{picture}(7,7)\put(-0.9,-0.4){\ebox{6cm}{Mr_np_rot_hyb_phys.ps}}\end{picture}}
\end{picture}
\end{center}
\caption{
The tree-level corrected $M(p)$, for $\csw=\text{NP}$, using
$S_I$ (left) and $S_R$ (right).  We find good
agreement between the two data sets, both in the infrared and
ultraviolet.  The residual disagreement at intermediate momenta is a
pointer to lattice artifacts that we have not brought under full
control, even with nonperturbative improvement and tree-level
correction. 
}
\label{fig:m_np_phys}
\end{figure}

\begin{figure}[ht]
\begin{center}
\epsfig{figure=Mchiral_quad.ps,angle=0,width=6cm}
\end{center}
\caption{The tree-level corrected mass function from $S_I$ with
$\csw=\text{NP}$, with the bare mass extrapolated
to zero using a
quadratic fit.  The small dip at $p\sim1.6$ GeV is not statistically
significant and may be due to residual lattice artifacts.  The
non-zero values for $M(p)$ in the chiral limit are entirely due
to dynamical chiral symmetry breaking and provide a direct measure of
this effect.}
\label{fig:Mall-chiral}
\end{figure}


In the continuum, the quark propagator has the following general form,
%
\begin{equation}
S(p) = \frac{1}{i\pslash A^c(p) + B^c(p)} \equiv
\frac{Z^c(p)}{i\pslash+M^c(p)}.
\end{equation}
%
We expect the lattice quark propagator to have a similar form, but
with $\kslash$ replacing $\pslash$:
%
\begin{equation}
S(p) = \frac{Z(p)}{i\kslash + M(p)}
\end{equation}
%
where $k$ is a new `lattice momentum', 
$k_\mu = \frac{1}{a}\sin(\hat p_\mu a)$.
We do not have sufficient space here to describe the hybrid
tree-level correction that was used for the quark propagator
results presented here, but a detailed description has recently
been given.\cite{quark_prop}  We again use the cylinder cut
to further reduce hypercubic discretization artefacts.  
As for the gluon propagator the results for $Z(p)$ are for the
bare quantity only and contain an overall renormalization constant
$Z_2(\mu,a)$.  In Fig.~\ref{fig:z_np_compare}
the vertical scales have been adjusted so that the two sets of results
are renormalized and hence coincide at 2.1~GeV.
In this figure we see the charactreistic dip in the infrared for $Z(p)$,
which occurs also in model Dyson-Schwinger equation
studies\cite{DSE_review} of dynamical
chiral symmetry breaking.  This dip has essentially disappeared by
around 2~GeV.  The improved action correspondning to $S_R$ is the
one we prefer and it gives the more expected ultraviolet behavior
of the renormalized $Z(p)$, i.e., it tends towards a constant.  
In Fig.~\ref{fig:m_np_phys} we see the characteristic behavior
of the quark mass function familiar from quark model
studies\cite{DSE_review} and that the transition to the perturbative
regime is occuring at approximately 2 GeV.  Note that there is no
renormalization of the quark mass function.  At large momenta
the mass function should become the running quark mass of perturbative
QCD.  Finally in Fig.~\ref{fig:Mall-chiral} we present a simple quadratic
extrapolation to the chiral limit for the available $S_I$ data.
The slight dip is not statistically significant and is almost
certainly a residual lattice artefact.  The infrared mass (i.e., at
$p=0$) in the chiral limit is approximately $330\pm30$~MeV, which is
characteristic of the constituent quark mass scale.

\section{Summary and Conclusions}

The gluon propagator has been calculated on fine unimproved lattices
and on a variety of improved lattices with an
\oa{2} improved action in \oa{2} improved Landau gauge.  The infrared
behavior of this propagator strongly suggests the the Landau gauge
gluon propagator is infrared finite.  
We have ruled out the $1/q^{4}$ behavior popular in some Dyson-Schwinger
quark model studies\cite{DSE_review}
and indeed any infrared singularity appears to be very unlikely.
The possible effects of lattice Gribov copies remains a very interesting
question and we are currently carrying out similar studies across
a variety of lattices in Laplacian gauge, which is a Landau-like
smooth gauge fixing, but is free of Gribov copies.

We have used two different definitions of the ${\cal O}(a)$ improved
quark propagator, corresponding to the quark propagators
denoted $S_I$ and $S_R$.
We make use of asymptotic freedom to factor out the
tree level behaviour, replacing it with the `continuum' tree level
behaviour $Z(p)=1, M(p)=m$.  This
tree-level correction dramatically improves the data.  
We find that $M(0)$ approaches a value of $300\pm30$ MeV in the
chiral limit, which is very much in keeping with the concept
of a ``constituent quark mass'' and agrees with the infrared values
of the quark mass commonly used in model studies.\cite{DSE_review}
We also find a significant dip in the value for $Z(p)$ at low
momenta.  This is again entirely consistent with what
is found in model studies of dynamical chiral symmetry breaking
\cite{DSE_review}.  An examination of
Figs.~\ref{fig:Comp1i_6}, \ref{fig:pert_vs_latt}, \ref{fig:z_np_compare},
\ref{fig:m_np_phys}, and \ref{fig:Mall-chiral} provide a clear indication
that perturbative QCD behavior is not becoming dominant in the gluon
and quark propagators until we reach momenta of order
$Q^2\simeq 4$~GeV$^2$.



%%%%\section*{Acknowledgments}
%%%%
%%%This work was supported by the Australian Research Council and by
%%%grants of supercomputer time on the CM-5 made available through the
%%%South Australian Centre for Parallel Computing (SACPC).



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




\section*{References}
\begin{thebibliography}{99}

\bibitem{DSE_review} C.D.\ Roberts and A.G.\ Williams,
        Prog.\ Part.\ Nucl.\ Phys.\ {\bf 33}, 477 (1994).

\bibitem{Weisz83} P.\ Weisz, Nucl.\ Phys.\ B {\bf 212}, 1 (1983).

\bibitem{tadpole} G.P.~Lepage \& P.B.~Mackenzie,
        Phys.\ Rev.\ D {\bf 48}, 2250 (1993).

\bibitem{LandauGaugeDE} F.D.R.~Bonnet, P.O.~Bowman, D.B.~Leinweber, 
	A.G.~Williams \& D.G.~Richards, 
	Aust.\ J.\ Phys.\ {\bf 52}, 939 (1999);
        Nucl.\ Phys.\ B (Proc. Suppl.) {\bf 83--84}, 905--907 (2000).

\bibitem{long_glu} 
        D.B.~Leinweber, J.I.\ Skullerud, A.G.\ Williams, and
        C.\ Parrinello, Phys.\ Rev. D {\bf 58}, 031501 (1998);
        {\em ibid.} {\bf 60}, 094507 (1999);
        Erratum -- {\em ibid.} {\bf 61}, 079901 (2000);
        D.B.\ Leinweber, C.\ Parrinello, J.I.\ Skullerud, and A.G.\ Williams
        Nucl.\ Phys.\ B (Proc.\ Suppl.) {\bf 73}, 629-631 (1999);
        {\em ibid.}, 626-628 (1999).

\bibitem{big_vol_glu} 
        F.D.R.~Bonnet, P.O.~Bowman, D.B.~Leinweber, 
	and A.G.~Williams, Phys.\ Rev.\ D {\bf 62}, 051501 (2000);
        {\em ibid.} {\bf 64}, 034501 (2001).

\bibitem{Luscher:1996sc} M.\ L\"{u}scher {\it et al.}, 
	Nucl.\ Phys.\ B {\bf 478}, 365 (1996).

\bibitem{Dawson:1997gp} C.\ Dawson {\it et al.}, Nucl.\ Phys.\ B
	(Proc. Suppl.) {\bf 63}, 877 (1998).

\bibitem{Heatlie:1991kg} G.\ Heatlie {\it et al.}, 
        Nucl.\ Phys.\ B {\bf 352}, 266 (1991).

\bibitem{quark_prop} J.I.\ Skullerud and A.G.\ Williams,
        Nucl.\ Phys.\ B (Proc. Suppl.) {\bf 83--84}, 209--211 (2000);
        Phys.\ Rev.\ D {\bf 63}, 054508 (2001);
        J.I.\ Skullerud, D.B.\ Leinweber, and A.G.\ Williams,
        to appear in Phys.\ Rev.\ D, hep-lat/0102013.

\end{thebibliography}

\end{document}

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%%% 1.19 14.11.93
%%%   mistaken placement of calculation of EPS@Height resulted in errors
%%% 1.20 defaults to no compression
%--------------------------------------------------------
%%%
%%% code rearranged by Sebastian Rahtz (spqr@uk.ac.york.minster)
%%% from work by
%%% Trevor Darrell; Tom Rokicki; Daniel Smith; H. Payne, Robert Russell
%%%
%%% This is a version of Darrell's `psfig' macros which are designed
%%% to be used with dvips or dvitops, and the `epsf' macros (these are
%%% automaticaly included by the style file). I did this because the
%%% \special commands used by `epsf' appear to be more robust and
%%% portable than the raw \specials used by `psfig', but I preferred
%%% the psfig interface. This system preserves the psfig interface,
%%% but mainly uses epsf macros to carry out the job. The user
%%% interface is identical to \psfig, but you just use \epsfig
%%% instead. Rotation uses the original psfig macros: I could not work
%%% out how to do rotation with the epsf macros.
%%%
%%% The same applies to `rheight' and `rwidth'
%%%
%%% If you *really* want old psfig behaviour regardless, use \psfig. It is
%%% still there. Some of the features have gone, however. \figurepath
%%% seemed redundant, since it could be controlled by the TEXINPUTS
%%% variable anyway.
%%%
%%% As an added feature, if FILE cannot be found, a file called FILE.bb
%%% is searched for. This should simply contain a %%BoundingBox line. A perl
%%% script "epsbb" is provided to generate these automatically.  It is
%%% assumed that if this exists, then the actual postscript is in UNIX
%%% compress format in FILE.Z or in GNU gzip format in FILE.gz or FILE.z.
%%%
%%% The macros then read the BB from FILE.bb, but insert "`epsfig FILE`" in
%%% the  output for dvips to interpret (don't worry, it knows about this).
%%% "Epsfig" is another perl script which  searches the TEXINPUTS path for
%%% FILE or FILE.{gz,z,Z}.
%%%
%%% As an added bonus if you say \pscompress the -c switch will be passed to
%%% epsfig, causing automatic FILE.bb generation and compression the first
%%% time dvips is run.  Files in the system directories will NOT be
%%% compressed.
%%% If you are *not* on a Unix system and use this, deliberately or by
%%% mischance, you will have problems!
%%%
%%% This file can be included in a dumped format, or used as a LaTeX
%%% style file, or used in plain TeX. It does not load itself twice (I hope)
%%%
%--------------------------------------------------------------------
%--------------------------------------------------------------------
%%% previous notes. this copy of psfig.tex inherited from versions 1.8
%--------------------------------------------------------------------
%%% All psfig/tex software, documentation, and related files
%%% in this distribution of psfig/tex are
%%% Copyright 1987, 1988, 1991 Trevor J. Darrell
%
%%% Permission is granted for use and non-profit distribution of psfig/tex
%%% providing that this notice be clearly maintained. The right to
%%% distribute any portion of psfig/tex for profit or as part of any commercial
%%% product is specifically reserved for the author(s) of that portion.
%%%
%%% Thanks to Greg Hager (GDH) and Ned Batchelder for their contributions
%%% to this project.
%%%--------------------------------------------------------------------------
%%% GDH 7/26/87 -- changed so that it first looks in the local directory,
%%% then in a specified global directory for the ps file.
%%% SPQR 12/91 removed that last change. it follows TEXINPUTS anyway
%%% RPR 6/25/91 -- changed so that it defaults to user-supplied name if
%%% boundingbox info is specified, assuming graphic will be created by
%%% print time.
%%% SPQR 12/91 removed that last change. use LaTeX error message instead
%%% SPQR 12/91 -- `no file found' produces LaTeX error message.
%%%                If you recover from that the file name is set in draft
%%%                mode in place of the file.
%%%
%%% Modified by J. Daniel Smith on 9 October 1990 to accept the
%%% %%BoundingBox: comment with or without a space after the colon.  Stole
%%% file reading code from Tom Rokicki's EPSF.TEX file (see below).
%%% --- SPQR 12.91 removed most of this to use epsf directly
%%%
%%% More modifications by J. Daniel Smith on 29 March 1991 to allow the
%%% the included PostScript figure to be rotated.  The amount of
%%% rotation is specified by the "angle=" parameter of the \epsfig command.
%%%
%%% Modified by Robert Russell on June 25, 1991 to allow users to specify
%%% .ps filenames which don't yet exist, provided they explicitly provide
%%% boundingbox information via the \epsfig command. Note: This will only work
%%% if the "file=" parameter follows all four "bb???=" parameters in the
%%% command. This is due to the order in which psfig interprets these params.
%%% ---- SPQR 12.91 removed all this
%%%
%%% 3 Jul 1991  JDS     check if file already read in once
%%%
%-----------------------------USAGE---------------------------------------
% From: gvr@cs.brown.edu (George V. Reilly)
%
% \psdraft      draws an outline box, but doesn't include the figure
%               in the DVI file.  Useful for previewing.
%
% \psfull       includes the figure in the DVI file (default).
%
% \psscalefirst width= or height= specifies the size of the figure
%               before rotation.
% \psrotatefirst (default) width= or height= specifies the size of the
%                figure after rotation.  Asymetric figures will
%                appear to shrink.
%
% usage: \epsfig{file=, figure=, height=, width=,
%                       bbllx=, bblly=, bburx=, bbury=,
%                       rheight=, rwidth=, clip=, angle=, silent=}%
%
%       "file" is the filename.  If no path name is specified and the
%               file is not found in the current directory,
%               it will be looked for in directory \psfigurepath.
%       "figure" is a synonym for "file".
%       By default, the width and height of the figure are taken from
%               the BoundingBox of the figure.
%       If "width" is specified, the figure is scaled so that it has
%               the specified width.  Its height changes proportionately.
%       If "height" is specified, the figure is scaled so that it has
%               the specified height.  Its width changes proportionately.
%       If both "width" and "height" are specified, the figure is scaled
%               anamorphically.
%       "bbllx", "bblly", "bburx", and "bbury" control the PostScript
%               BoundingBox.
%       "rheight" and "rwidth" are the reserved height and width
%               of the figure, i.e., how big TeX actually thinks
%               the figure is.  They default to "width" and "height".
%       The "clip" option ensures that no portion of the figure will
%               appear outside its BoundingBox.  "clip=" is a switch and
%               takes no value, but the `=' must be present.
%       The "angle" option specifies the angle of rotation (degrees, ccw).
%       The "silent" option makes \psfig work silently.
%
%--------------------------------------------------------------------------
\newcount\EPS@Height \newcount\EPS@Width \newcount\EPS@xscale
\newcount\EPS@yscale
\def\psfigdriver#1{%
  \bgroup\edef\next{\def\noexpand\tempa{#1}}%
    \uppercase\expandafter{\next}%
    \def\LN{DVITOLN03}%
    \def\DVItoPS{DVITOPS}%
    \def\DVIPS{DVIPS}%
    \def\emTeX{EMTEX}%
    \def\OzTeX{OZTEX}%
    \def\Textures{TEXTURES}%
    \global\chardef\fig@driver=0
    \ifx\tempa\LN
        \global\chardef\fig@driver=0\fi
    \ifx\tempa\DVItoPS
        \global\chardef\fig@driver=1\fi
    \ifx\tempa\DVIPS
        \global\chardef\fig@driver=2\fi
    \ifx\tempa\emTeX
        \global\chardef\fig@driver=3\fi
    \ifx\tempa\OzTeX
        \global\chardef\fig@driver=4\fi
    \ifx\tempa\Textures
        \global\chardef\fig@driver=5\fi
  \egroup
\def\psfig@start{}%
\def\psfig@end{}%
\def\epsfig@gofer{}%
\ifcase\fig@driver
% case 0 ln03
\typeout{WARNING! ****
 no specials for LN03 psfig}%
%
\or % case 1: dvitops
\def\psfig@start{}%
\def\psfig@end{\special{dvitops: import \@p@sfilefinal \space
\@p@swidth sp \space \@p@sheight sp \space fill}%
\if@clip \typeout{Clipping not supported}\fi
\if@angle \typeout{Rotating not supported}\fi
}%
\let\epsfig@gofer\psfig@end
%
\or %case2 dvips
\def\psfig@start{\special{ps::[begin]  \@p@swidth \space \@p@sheight \space%
        \@p@sbbllx \space \@p@sbblly \space%
        \@p@sbburx \space \@p@sbbury \space%
        startTexFig \space }%
        \if@clip
                \if@verbose
                        \typeout{(clipped to BB) }%
                \fi
                \special{ps:: doclip \space }%
        \fi
        \if@angle              % moved after \if@clip ... \fi -pks-
                \special {ps:: \@p@sangle \space rotate \space}
        \fi
        \special{ps: plotfile \@p@sfilefinal \space }%
        \special{ps::[end] endTexFig \space }%
}%
\def\psfig@end{}%
\def\epsfig@gofer{\if@clip
                        \if@verbose
                           \typeout{(clipped to BB)}%
                        \fi
                        \epsfclipon
                  \fi
                  \epsfsetgraph{\@p@sfilefinal}%
}%
%
\or % case 3, emTeX
\typeout{WARNING. You must have a .bb info file with the Bounding Box
  of the pcx file}%
\def\psfig@start{}%
% when emtex imports a pcx file, it is *below* where we are now.
% so raise the box
\def\psfig@end{\typeout{pcx import of \@p@sfilefinal}%
\if@clip \typeout{Clipping not supported}\fi
\if@angle \typeout{Rotating not supported}\fi
\raisebox{\@p@srheight sp}{\special{em: graph \@p@sfilefinal}}}%
\def\epsfig@gofer{}%
%
\or % case 4, OzTeX
\def\psfig@start{}%
\def\psfig@end{%
\EPS@Width\@p@swidth
\EPS@Height\@p@sheight
\divide\EPS@Width by 65781  % convert sp to bp
\divide\EPS@Height by 65781
\special{epsf=\@p@sfilefinal
\space
width=\the\EPS@Width
\space
height=\the\EPS@Height
}%
\if@clip \typeout{Clipping not supported}\fi
\if@angle \typeout{Rotating not supported}\fi
}%
\let\epsfig@gofer\psfig@end
\or % case 5, Textures
\def\psfig@end{
% what is scaling of picture
         \EPS@Width=\@bbw  
         \divide\EPS@Width by 1000
         \EPS@xscale=\@p@swidth \divide \EPS@xscale by \EPS@Width
         \EPS@Height=\@bbh  
         \divide\EPS@Height by 1000
         \EPS@yscale=\@p@sheight \divide \EPS@yscale by\EPS@Height
% take largest calculated scale
  \ifnum\EPS@xscale>\EPS@yscale\EPS@xscale=\EPS@yscale\fi
\if@clip
   \if@verbose
      \typeout{(clipped to BB)}%
   \fi
   \epsfclipon
\fi
\special{illustration \@p@sfilefinal\space scaled \the\EPS@xscale}%
}%
\def\psfig@start{}%
\let\epsfig\psfig
\else
\typeout{WARNING. *** unknown  driver - no psfig}%
\fi
}%
% copies of bits of LaTeX we need....
\newdimen\ps@dimcent
%
% \fbox is defined in latex.tex; so if \fbox is undefined, assume that
% we are not in LaTeX.
% Perhaps this could be done better???
\ifx\undefined\fbox
% \fbox code from modified slightly from LaTeX
\newdimen\fboxrule
\newdimen\fboxsep
\newdimen\ps@tempdima
\newbox\ps@tempboxa
\fboxsep = 0pt
\fboxrule = .4pt
\long\def\fbox#1{\leavevmode\setbox\ps@tempboxa\hbox{#1}\ps@tempdima\fboxrule
    \advance\ps@tempdima \fboxsep \advance\ps@tempdima \dp\ps@tempboxa
   \hbox{\lower \ps@tempdima\hbox
  {\vbox{\hrule height \fboxrule
          \hbox{\vrule width \fboxrule \hskip\fboxsep
          \vbox{\vskip\fboxsep \box\ps@tempboxa\vskip\fboxsep}\hskip
                 \fboxsep\vrule width \fboxrule}%
                 \hrule height \fboxrule}}}}%
\fi
\ifx\@ifundefined\undefined
\long\def\@ifundefined#1#2#3{\expandafter\ifx\csname
  #1\endcsname\relax#2\else#3\fi}%
\fi
%
\@ifundefined{typeout}%
{\gdef\typeout#1{\immediate\write\sixt@@n{#1}}}%
{\relax}%
%
%%% check to see if macros already loaded in (maybe some other file says
%%% "\input epsfig") ...
\@ifundefined{epsfig}{}{\typeout{EPSFIG --- already loaded}\endinput}%
%
%%%
%%% we need Rokicki's EPSF macros anyway, unless they are already loaded
%
\@ifundefined{epsfbox}{\input epsf}{}%
%
%% SPQR 12.91 handling of errors using standard LaTeX error
%% mechanism. In case we are plain TeX we first define the
%% error routines...
\ifx\undefined\@latexerr
        \newlinechar`\^^J
        \def\@spaces{\space\space\space\space}%
        \def\@latexerr#1#2{%
        \edef\@tempc{#2}\expandafter\errhelp\expandafter{\@tempc}%
        \typeout{Error. \space see a manual for explanation.^^J
         \space\@spaces\@spaces\@spaces Type \space H <return> \space for
         immediate help.}\errmessage{#1}}%
\fi
%------------------------
%% a couple of LaTeX error messages
\def\@whattodo{You tried to include a PostScript figure which
cannot be found^^JIf you press return to carry on anyway,^^J
The failed name will be printed in place of the figure.^^J
or type X to quit}%
\def\@whattodobb{You tried to include a PostScript figure which
has no^^Jbounding box, and you supplied none.^^J
If you press return to carry on anyway,^^J
The failed name will be printed in place of the figure.^^J
or type X to quit}%
%------------------------
%
%------------------------------------------------------------------------
%------------------------------------------------------------------------
%%% @psdo control structure -- similar to Latex @for.
%%% I redefined these with different names so that psfig can
%%% be used with TeX as well as LaTeX, and so that it will not
%%% be vunerable to future changes in LaTeX's internal
%%% control structure,
%
\def\@nnil{\@nil}%
\def\@empty{}%
\def\@psdonoop#1\@@#2#3{}%
\def\@psdo#1:=#2\do#3{\edef\@psdotmp{#2}\ifx\@psdotmp\@empty \else
    \expandafter\@psdoloop#2,\@nil,\@nil\@@#1{#3}\fi}%
\def\@psdoloop#1,#2,#3\@@#4#5{\def#4{#1}\ifx #4\@nnil \else
       #5\def#4{#2}\ifx #4\@nnil \else#5\@ipsdoloop #3\@@#4{#5}\fi\fi}%
\def\@ipsdoloop#1,#2\@@#3#4{\def#3{#1}\ifx #3\@nnil
       \let\@nextwhile=\@psdonoop \else
      #4\relax\let\@nextwhile=\@ipsdoloop\fi\@nextwhile#2\@@#3{#4}}%
\def\@tpsdo#1:=#2\do#3{\xdef\@psdotmp{#2}\ifx\@psdotmp\@empty \else
    \@tpsdoloop#2\@nil\@nil\@@#1{#3}\fi}%
\def\@tpsdoloop#1#2\@@#3#4{\def#3{#1}\ifx #3\@nnil
       \let\@nextwhile=\@psdonoop \else
      #4\relax\let\@nextwhile=\@tpsdoloop\fi\@nextwhile#2\@@#3{#4}}%
%%%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% file reading stuff from epsf.tex
%%%   EPSF.TEX macro file:
%%%   Written by Tomas Rokicki of Radical Eye Software, 29 Mar 1989.
%%%   Revised by Don Knuth, 3 Jan 1990.
%%%   Revised by Tomas Rokicki to accept bounding boxes with no
%%%      space after the colon, 18 Jul 1990.
%%%   Portions modified/removed for use in PSFIG package by
%%%      J. Daniel Smith, 9 October 1990.
%%%   Just the bit which knows about (atend) as a BoundingBox
%
%%%    hacked back a bit by SPQR 12/91
%
\long\def\epsfaux#1#2:#3\\{\ifx#1\epsfpercent
   \def\testit{#2}\ifx\testit\epsfbblit
        \@atendfalse
        \epsf@atend #3 . \\%
        \if@atend
           \if@verbose
                \typeout{epsfig: found `(atend)'; continuing search}%
           \fi
        \else
                \epsfgrab #3 . . . \\%
                \epsffileokfalse\global\no@bbfalse
                \global\epsfbbfoundtrue
        \fi
   \fi\fi}%
%
%%% Determine if the stuff following the %%BoundingBox is `(atend)'
%%% J. Daniel Smith.  Copied from \epsf@grab above.
%
\def\epsf@atendlit{(atend)}
%
\def\epsf@atend #1 #2 #3\\{%
   \def\epsf@tmp{#1}\ifx\epsf@tmp\empty
      \epsf@atend #2 #3 .\\\else
   \ifx\epsf@tmp\epsf@atendlit\@atendtrue\fi\fi}%
 
 
%%% End of file reading stuff from epsf.tex
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% trigonometry stuff from "trig.tex"
\chardef\trig@letter = 11
\chardef\other = 12
 
\newif\ifdebug %%% turn me on to see TeX hard at work ...
\newif\ifc@mpute %%% don't need to compute some values
\newif\if@atend
\c@mputetrue % but assume that we do
 
\let\then = \relax
\def\r@dian{pt }%
\let\r@dians = \r@dian
\let\dimensionless@nit = \r@dian
\let\dimensionless@nits = \dimensionless@nit
\def\internal@nit{sp }%
\let\internal@nits = \internal@nit
\newif\ifstillc@nverging
\def \Mess@ge #1{\ifdebug \then \message {#1} \fi}%
 
{ %%% Things that need abnormal catcodes %%%
        \catcode `\@ = \trig@letter
        \gdef \nodimen {\expandafter \n@dimen \the \dimen}%
        \gdef \term #1 #2 #3%
               {\edef \t@ {\the #1}%%% freeze parameter 1 (count, by value)
                \edef \t@@ {\expandafter \n@dimen \the #2\r@dian}%
                                   %%% freeze parameter 2 (dimen, by value)
                \t@rm {\t@} {\t@@} {#3}%
               }%
        \gdef \t@rm #1 #2 #3%
               {{%
                \count 0 = 0
                \dimen 0 = 1 \dimensionless@nit
                \dimen 2 = #2\relax
                \Mess@ge {Calculating term #1 of \nodimen 2}%
                \loop
                \ifnum  \count 0 < #1
                \then   \advance \count 0 by 1
                        \Mess@ge {Iteration \the \count 0 \space}%
                        \Multiply \dimen 0 by {\dimen 2}%
                        \Mess@ge {After multiplication, term = \nodimen 0}%
                        \Divide \dimen 0 by {\count 0}%
                        \Mess@ge {After division, term = \nodimen 0}%
                \repeat
                \Mess@ge {Final value for term #1 of
                                \nodimen 2 \space is \nodimen 0}%
                \xdef \Term {#3 = \nodimen 0 \r@dians}%
                \aftergroup \Term
               }}%
        \catcode `\p = \other
        \catcode `\t = \other
        \gdef \n@dimen #1pt{#1} %%% throw away the ``pt''
}%
 
\def \Divide #1by #2{\divide #1 by #2} %%% just a synonym
 
\def \Multiply #1by #2%%% allows division of a dimen by a dimen
       {{%%% should really freeze parameter 2 (dimen, passed by value)
        \count 0 = #1\relax
        \count 2 = #2\relax
        \count 4 = 65536
        \Mess@ge {Before scaling, count 0 = \the \count 0 \space and
                        count 2 = \the \count 2}%
        \ifnum  \count 0 > 32767 %%% do our best to avoid overflow
        \then   \divide \count 0 by 4
                \divide \count 4 by 4
        \else   \ifnum  \count 0 < -32767
                \then   \divide \count 0 by 4
                        \divide \count 4 by 4
                \else
                \fi
        \fi
        \ifnum  \count 2 > 32767 %%% while retaining reasonable accuracy
        \then   \divide \count 2 by 4
                \divide \count 4 by 4
        \else   \ifnum  \count 2 < -32767
                \then   \divide \count 2 by 4
                        \divide \count 4 by 4
                \else
                \fi
        \fi
        \multiply \count 0 by \count 2
        \divide \count 0 by \count 4
        \xdef \product {#1 = \the \count 0 \internal@nits}%
        \aftergroup \product
       }}%
 
\def\r@duce{\ifdim\dimen0 > 90\r@dian \then   % sin(x) = sin(180-x)
                \multiply\dimen0 by -1
                \advance\dimen0 by 180\r@dian
                \r@duce
            \else \ifdim\dimen0 < -90\r@dian \then  % sin(x) = sin(360+x)
                \advance\dimen0 by 360\r@dian
                \r@duce
                \fi
            \fi}%
 
\def\Sine#1%
       {{%
        \dimen 0 = #1 \r@dian
        \r@duce
        \ifdim\dimen0 = -90\r@dian \then
           \dimen4 = -1\r@dian
           \c@mputefalse
        \fi
        \ifdim\dimen0 = 90\r@dian \then
           \dimen4 = 1\r@dian
           \c@mputefalse
        \fi
        \ifdim\dimen0 = 0\r@dian \then
           \dimen4 = 0\r@dian
           \c@mputefalse
        \fi
%
        \ifc@mpute \then
                % convert degrees to radians
                \divide\dimen0 by 180
                \dimen0=3.141592654\dimen0
%
                \dimen 2 = 3.1415926535897963\r@dian %%% a well-known constant
                \divide\dimen 2 by 2 %%% we only deal with -pi/2 : pi/2
                \Mess@ge {Sin: calculating Sin of \nodimen 0}%
                \count 0 = 1 %%% see power-series expansion for sine
                \dimen 2 = 1 \r@dian %%% ditto
                \dimen 4 = 0 \r@dian %%% ditto
                \loop
                        \ifnum  \dimen 2 = 0 %%% then we've done
                        \then   \stillc@nvergingfalse
                        \else   \stillc@nvergingtrue
                        \fi
                        \ifstillc@nverging %%% then calculate next term
                        \then   \term {\count 0} {\dimen 0} {\dimen 2}%
                                \advance \count 0 by 2
                                \count 2 = \count 0
                                \divide \count 2 by 2
                                \ifodd  \count 2 %%% signs alternate
                                \then   \advance \dimen 4 by \dimen 2
                                \else   \advance \dimen 4 by -\dimen 2
                                \fi
                \repeat
        \fi
                        \xdef \sine {\nodimen 4}%
       }}%
 
%%% Now the Cosine can be calculated easily by calling \Sine:
%%%  cos(x) = sin(90-x)
\def\Cosine#1{\ifx\sine\UnDefined\edef\Savesine{\relax}\else
                             \edef\Savesine{\sine}\fi
        {\dimen0=#1\r@dian\multiply\dimen0 by -1
         \advance\dimen0 by 90\r@dian
         \Sine{\nodimen 0}%
         \xdef\cosine{\sine}%
         \xdef\sine{\Savesine}}}
%%% end of trig stuff
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\def\psdraft{\def\@psdraft{0}}%
\def\psfull{\def\@psdraft{1}}%
\psfull
\newif\if@compress
\def\pscompress{\@compresstrue}
\def\psnocompress{\@compressfalse}
\@compressfalse
\newif\if@scalefirst
\def\psscalefirst{\@scalefirsttrue}%
\def\psrotatefirst{\@scalefirstfalse}%
\psrotatefirst
%
\newif\if@draftbox
\def\psnodraftbox{\@draftboxfalse}%
\@draftboxtrue
%
\newif\if@noisy
\@noisyfalse
\newif\ifno@bb
%%% These are for the option list.
%%% A specification of the form a = b maps to calling \@p@@sa{b}%
\newif\if@bbllx
\newif\if@bblly
\newif\if@bburx
\newif\if@bbury
\newif\if@height
\newif\if@width
\newif\if@rheight
\newif\if@rwidth
\newif\if@angle
\newif\if@clip
\newif\if@verbose
\newif\if@prologfile
\def\@p@@sprolog#1{\@prologfiletrue\def\@prologfileval{#1}}%
\def\@p@@sclip#1{\@cliptrue}%
%
\newif\ifepsfig@dos  % only single suffix possible
\def\epsfigdos{\epsfig@dostrue}%
\epsfig@dosfalse
%%% if this is true, the original Darrell macros and specials are used
\newif\ifuse@psfig
%
\def\ParseName#1{\expandafter\@Parse#1}%
\def\@Parse#1.#2:{\gdef\BaseName{#1}\gdef\FileType{#2}}%

\def\@p@@sfile#1{%
  \ifepsfig@dos
     \ParseName{#1:}%
  \else
     \gdef\BaseName{#1}\gdef\FileType{}%
  \fi
  \def\@p@sfile{NO FILE: #1}%
  \def\@p@sfilefinal{NO FILE: #1}%
  \openin1=#1
  \ifeof1\closein1\openin1=\BaseName.bb
    \ifeof1\closein1
      \if@bbllx                 % No postscript file but bb given explicitly.
        \if@bblly\if@bburx\if@bbury
          \def\@p@sfile{#1}%
          \def\@p@sfilefinal{#1}%
        \fi\fi\fi
      \else                     % No bounding box found.
        \@latexerr{ERROR. PostScript file #1 not found}\@whattodo
        \@p@@sbbllx{100bp}%
        \@p@@sbblly{100bp}%
        \@p@@sbburx{200bp}%
        \@p@@sbbury{200bp}%
        \psdraft
      \fi
    \else                       % Postscript file is compressed.
      \closein1%
      \edef\@p@sfile{\BaseName.bb}%
      \typeout{using BB from \@p@sfile}%
      \ifnum\fig@driver=3
        \edef\@p@sfilefinal{\BaseName.pcx}%
      \else
        \ifepsfig@dos
          \edef\@p@sfilefinal{"`gunzip -c `texfind \BaseName.{z,Z,gz}"}%
        \else
          \edef\@p@sfilefinal{"`epsfig \if@compress-c \fi#1"}%          
        \fi
      \fi
    \fi
  \else\closein1                % Postscript file is not compressed.
    \edef\@p@sfile{#1}%
    \if@compress  
      \edef\@p@sfilefinal{"`epsfig -c #1"}%
    \else
      \edef\@p@sfilefinal{#1}%
    \fi
  \fi%
}

% alternative syntax: figure=
\let\@p@@sfigure\@p@@sfile
%
\def\@p@@sbbllx#1{%
                                            \@bbllxtrue
                \ps@dimcent=#1
                \edef\@p@sbbllx{\number\ps@dimcent}%
                \divide\ps@dimcent by65536
                \global\edef\epsfllx{\number\ps@dimcent}%
}%
\def\@p@@sbblly#1{%
                \@bbllytrue
                \ps@dimcent=#1
                \edef\@p@sbblly{\number\ps@dimcent}%
                \divide\ps@dimcent by65536
                \global\edef\epsflly{\number\ps@dimcent}%
}%
\def\@p@@sbburx#1{%
                \@bburxtrue
                \ps@dimcent=#1
                \edef\@p@sbburx{\number\ps@dimcent}%
                \divide\ps@dimcent by65536
                \global\edef\epsfurx{\number\ps@dimcent}%
}%
\def\@p@@sbbury#1{%
                \@bburytrue
                \ps@dimcent=#1
                \edef\@p@sbbury{\number\ps@dimcent}%
                \divide\ps@dimcent by65536
                \global\edef\epsfury{\number\ps@dimcent}%
}%
\def\@p@@sheight#1{%
                \@heighttrue
                \global\epsfysize=#1
                \ps@dimcent=#1
                \edef\@p@sheight{\number\ps@dimcent}%
}%
\def\@p@@swidth#1{%
                \@widthtrue
                \global\epsfxsize=#1
                \ps@dimcent=#1
                \edef\@p@swidth{\number\ps@dimcent}% 
}%
\def\@p@@srheight#1{%
                \@rheighttrue\use@psfigtrue
                \ps@dimcent=#1
                \edef\@p@srheight{\number\ps@dimcent}%
}%
\def\@p@@srwidth#1{%
                \@rwidthtrue\use@psfigtrue
                \ps@dimcent=#1
                \edef\@p@srwidth{\number\ps@dimcent}%
}%
\def\@p@@sangle#1{%
                \use@psfigtrue
                \@angletrue
                \edef\@p@sangle{#1}%
}%
\def\@p@@ssilent#1{%
                \@verbosefalse
}%
\def\@p@@snoisy#1{%
                \@verbosetrue
}%
\def\@cs@name#1{\csname #1\endcsname}%
\def\@setparms#1=#2,{\@cs@name{@p@@s#1}{#2}}%
%
%%% initialize the defaults (size the size of the figure)
%
\def\ps@init@parms{%
                \@bbllxfalse \@bbllyfalse
                \@bburxfalse \@bburyfalse
                \@heightfalse \@widthfalse
                \@rheightfalse \@rwidthfalse
                \def\@p@sbbllx{}\def\@p@sbblly{}%
                \def\@p@sbburx{}\def\@p@sbbury{}%
                \def\@p@sheight{}\def\@p@swidth{}%
                \def\@p@srheight{}\def\@p@srwidth{}%
                \def\@p@sangle{0}%
                \def\@p@sfile{}%
                \use@psfigfalse
                \@prologfilefalse
                \def\@sc{}%
                \if@noisy
                        \@verbosetrue
                \else
                        \@verbosefalse
                \fi
                \@clipfalse
}%
%
%%% Go through the options setting things up.
%
\def\parse@ps@parms#1{%
                \@psdo\@psfiga:=#1\do
                   {\expandafter\@setparms\@psfiga,}%
\if@prologfile
\special{header=\@prologfileval}%
\fi
}%
%
%%% Compute bb height and width
%
\def\bb@missing{%
        \if@verbose
            \typeout{psfig: searching \@p@sfile \space  for bounding box}%
        \fi
        \epsfgetbb{\@p@sfile}%
        \ifepsfbbfound
            \ps@dimcent=\epsfllx bp\edef\@p@sbbllx{\number\ps@dimcent}%
            \ps@dimcent=\epsflly bp\edef\@p@sbblly{\number\ps@dimcent}%
            \ps@dimcent=\epsfurx bp\edef\@p@sbburx{\number\ps@dimcent}%
            \ps@dimcent=\epsfury bp\edef\@p@sbbury{\number\ps@dimcent}%
        \else
            \epsfbbfoundfalse
        \fi
}
%
% rotate point (#1,#2) about (0,0).
% The sine and cosine of the angle are already stored in \sine and
% \cosine.  The result is placed in (\p@intvaluex, \p@intvaluey).
\newdimen\p@intvaluex
\newdimen\p@intvaluey
\def\rotate@#1#2{{\dimen0=#1 sp\dimen1=#2 sp
%               calculate x' = x \cos\theta - y \sin\theta
                  \global\p@intvaluex=\cosine\dimen0
                  \dimen3=\sine\dimen1
                  \global\advance\p@intvaluex by -\dimen3
%               calculate y' = x \sin\theta + y \cos\theta
                  \global\p@intvaluey=\sine\dimen0
                  \dimen3=\cosine\dimen1
                  \global\advance\p@intvaluey by \dimen3
                  }}%
\def\compute@bb{%
                \epsfbbfoundfalse
                \if@bbllx\epsfbbfoundtrue\fi
                \if@bblly\epsfbbfoundtrue\fi
                \if@bburx\epsfbbfoundtrue\fi
                \if@bbury\epsfbbfoundtrue\fi
                \ifepsfbbfound\else\bb@missing\fi
                \ifepsfbbfound\else
                \@latexerr{ERROR. cannot locate BoundingBox}\@whattodobb
                        \@p@@sbbllx{100bp}%
                        \@p@@sbblly{100bp}%
                        \@p@@sbburx{200bp}%
                        \@p@@sbbury{200bp}%
                        \no@bbtrue
                        \psdraft
                \fi
                %
%
% store height/width of original (unrotated) bounding box
                \count203=\@p@sbburx
                \count204=\@p@sbbury
                \advance\count203 by -\@p@sbbllx
                \advance\count204 by -\@p@sbblly
                \edef\ps@bbw{\number\count203}%
                \edef\ps@bbh{\number\count204}%
                 \edef\@bbw{\number\count203}%
                \edef\@bbh{\number\count204}%
               \if@angle
                        \Sine{\@p@sangle}\Cosine{\@p@sangle}%
 
{\ps@dimcent=\maxdimen\xdef\r@p@sbbllx{\number\ps@dimcent}%
 
\xdef\r@p@sbblly{\number\ps@dimcent}%
 
\xdef\r@p@sbburx{-\number\ps@dimcent}%
 
\xdef\r@p@sbbury{-\number\ps@dimcent}}%
%
% Need to rotate all four points and take the X-Y extremes of the new
% points as the new bounding box.
                        \def\minmaxtest{%
                           \ifnum\number\p@intvaluex<\r@p@sbbllx
                              \xdef\r@p@sbbllx{\number\p@intvaluex}\fi
                           \ifnum\number\p@intvaluex>\r@p@sbburx
                              \xdef\r@p@sbburx{\number\p@intvaluex}\fi
                           \ifnum\number\p@intvaluey<\r@p@sbblly
                              \xdef\r@p@sbblly{\number\p@intvaluey}\fi
                           \ifnum\number\p@intvaluey>\r@p@sbbury
                              \xdef\r@p@sbbury{\number\p@intvaluey}\fi
                           }%
%                       lower left
                        \rotate@{\@p@sbbllx}{\@p@sbblly}%
                        \minmaxtest
%                       upper left
                        \rotate@{\@p@sbbllx}{\@p@sbbury}%
                        \minmaxtest
%                       lower right
                        \rotate@{\@p@sbburx}{\@p@sbblly}%
                        \minmaxtest
%                       upper right
                        \rotate@{\@p@sbburx}{\@p@sbbury}%
                        \minmaxtest
 
\edef\@p@sbbllx{\r@p@sbbllx}\edef\@p@sbblly{\r@p@sbblly}%
 
\edef\@p@sbburx{\r@p@sbburx}\edef\@p@sbbury{\r@p@sbbury}%
                \fi
                \count203=\@p@sbburx
                \count204=\@p@sbbury
                \advance\count203 by -\@p@sbbllx
                \advance\count204 by -\@p@sbblly
                \edef\@bbw{\number\count203}%
                \edef\@bbh{\number\count204}%
}%
%
% \in@hundreds performs #1 * (#2 / #3) correct to the hundreds,
%       then leaves the result in @result
%
\def\in@hundreds#1#2#3{\count240=#2 \count241=#3
                     \count100=\count240        % 100 is first digit #2/#3
                     \divide\count100 by \count241
                     \count101=\count100
                     \multiply\count101 by \count241
                     \advance\count240 by -\count101
                     \multiply\count240 by 10
                     \count101=\count240        %101 is second digit of #2/#3
                     \divide\count101 by \count241
                     \count102=\count101
                     \multiply\count102 by \count241
                     \advance\count240 by -\count102
                     \multiply\count240 by 10
                     \count102=\count240        % 102 is the third digit
                     \divide\count102 by \count241
                     \count200=#1\count205=0
                     \count201=\count200
                        \multiply\count201 by \count100
                        \advance\count205 by \count201
                     \count201=\count200
                        \divide\count201 by 10
                        \multiply\count201 by \count101
                        \advance\count205 by \count201
                        %
                     \count201=\count200
                        \divide\count201 by 100
                        \multiply\count201 by \count102
                        \advance\count205 by \count201
                        %
                     \edef\@result{\number\count205}%
}%
\def\compute@wfromh{%
                % computing : width = height * (bbw / bbh)
                \in@hundreds{\@p@sheight}{\@bbw}{\@bbh}%
                \edef\@p@swidth{\@result}%
}%
\def\compute@hfromw{%
                % computing : height = width * (bbh / bbw)
                \in@hundreds{\@p@swidth}{\@bbh}{\@bbw}%
                \edef\@p@sheight{\@result}%
}%
\def\compute@handw{%
                \if@height
                        \if@width
                        \else
                                \compute@wfromh
                        \fi
                \else
                        \if@width
                                \compute@hfromw
                        \else
                                \edef\@p@sheight{\@bbh}%
                                \edef\@p@swidth{\@bbw}%
                        \fi
                \fi
}%
\def\compute@resv{%
                \if@rheight \else \edef\@p@srheight{\@p@sheight} \fi
                \if@rwidth \else \edef\@p@srwidth{\@p@swidth} \fi
}%
%
% Compute any missing values
\def\compute@sizes{%
        \if@scalefirst\if@angle
% at this point the bounding box has been adjsuted correctly for
% rotation.  PSFIG does all of its scaling using \@bbh and \@bbw.  If
% a width= or height= was specified along with \psscalefirst, then the
% width=/height= value needs to be adjusted to match the new (rotated)
% bounding box size (specifed in \@bbw and \@bbh).
%    \ps@bbw       width=
%    -------  =  ----------
%    \@bbw       new width=
% so `new width=' = (width= * \@bbw) / \ps@bbw; where \ps@bbw is the
% width of the original (unrotated) bounding box.
        \if@width
           \in@hundreds{\@p@swidth}{\@bbw}{\ps@bbw}%
           \edef\@p@swidth{\@result}%
        \fi
        \if@height
           \in@hundreds{\@p@sheight}{\@bbh}{\ps@bbh}%
           \edef\@p@sheight{\@result}%
        \fi
        \fi\fi
        \compute@handw
        \compute@resv
}
%

%------------------------------------------------------------------
%%% by the way, possible parameters to the PSfile= command in dvips are:
%%%                    llx
%%%                    lly
%%%                    urx
%%%                    ury
%%%                    rwi
%       hoffset The horizontal offset (default 0)
%       voffset The vertical offset (default 0)
%       hsize   The horizontal clipping size (default 612)
%       vsize   The vertical clipping size (default 792)
%       hscale  The horizontal scaling factor (default 100)
%       vscale  The vertical scaling factor (default 100)
%       angle   The rotation (default 0)
%------------------------------------------------------------------
\long\def\graphic@verb#1{\def\next{#1}%
  {\expandafter\graphic@strip\meaning\next}}
\def\graphic@strip#1>{}
\def\graphic@zapspace#1{%
  #1\ifx\graphic@zapspace#1\graphic@zapspace%
  \else\expandafter\graphic@zapspace%
  \fi}
\def\psfig#1{%
\edef\@tempa{\graphic@zapspace#1{}}%
\ifvmode\leavevmode\fi\vbox {%
        %
        \ps@init@parms
        \parse@ps@parms{\@tempa}%
        %
        \ifnum\@psdraft=1
                \typeout{[\@p@sfilefinal]}%
                \if@verbose
                        \typeout{epsfig: using PSFIG macros}%
                \fi
                \psfig@method
        \else
                \epsfig@draft
        \fi
}
}%
\def\graphic@zapspace#1{%
  #1\ifx\graphic@zapspace#1\graphic@zapspace%
  \else\expandafter\graphic@zapspace%
  \fi}
\def\epsfig#1{%
\edef\@tempa{\graphic@zapspace#1{}}%
\ifvmode\leavevmode\fi\vbox {%
        %
        \ps@init@parms
        \parse@ps@parms{\@tempa}%
        %
        \ifnum\@psdraft=1
          \if@angle\use@psfigtrue\fi
          {\ifnum\fig@driver=1\global\use@psfigtrue\fi}%
          {\ifnum\fig@driver=3\global\use@psfigtrue\fi}%
          {\ifnum\fig@driver=4\global\use@psfigtrue\fi}%
          {\ifnum\fig@driver=5\global\use@psfigtrue\fi}%
                \ifuse@psfig
                        \if@verbose
                                \typeout{epsfig: using PSFIG macros}%
                        \fi
                        \psfig@method
                \else
                        \if@verbose
                                \typeout{epsfig: using EPSF macros}%
                        \fi
                        \epsf@method
                \fi
        \else
                \epsfig@draft
        \fi
}%
}%
 
 
\def\epsf@method{%
        \epsfbbfoundfalse
        \if@bbllx\epsfbbfoundtrue\fi
        \if@bblly\epsfbbfoundtrue\fi
        \if@bburx\epsfbbfoundtrue\fi
        \if@bbury\epsfbbfoundtrue\fi
        \ifepsfbbfound\else\epsfgetbb{\@p@sfile}\fi
        \ifepsfbbfound
           \typeout{<\@p@sfilefinal>}%
           \epsfig@gofer
        \else
          \@latexerr{ERROR - Cannot locate BoundingBox}\@whattodobb
          \@p@@sbbllx{100bp}%
          \@p@@sbblly{100bp}%
          \@p@@sbburx{200bp}%
          \@p@@sbbury{200bp}%
                \count203=\@p@sbburx
                \count204=\@p@sbbury
                \advance\count203 by -\@p@sbbllx
                \advance\count204 by -\@p@sbblly
                \edef\@bbw{\number\count203}%
                \edef\@bbh{\number\count204}%
          \compute@sizes
          \epsfig@@draft
       \fi
}%
\def\psfig@method{%
        \compute@bb
        \ifepsfbbfound
          \compute@sizes
          \psfig@start
          % Create the vbox to reserve the space for the figure%
          \vbox to \@p@srheight sp{\hbox to \@p@srwidth 
            sp{\hss}\vss\psfig@end}%
        \else
           \epsfig@draft
        \fi
}%
%
% draft figure, just reserve the space and print the
% path name.
\def\epsfig@draft{\compute@bb\compute@sizes\epsfig@@draft}%
%
\def\epsfig@@draft{%
\typeout{<(draft only) \@p@sfilefinal>}%
\if@draftbox
        % Verbose draft: print file name in box
        \hbox{{\fboxsep0pt\fbox{\vbox to \@p@srheight sp{%
        \vss\hbox to \@p@srwidth sp{ \hss 
           \expandafter\Literally\@p@sfilefinal\@nil
                          \hss }\vss
        }}}}%
\else
        % Non-verbose draft
        \vbox to \@p@srheight sp{%
        \vss\hbox to \@p@srwidth sp{\hss}\vss}%
\fi
}%
\def\Literally#1\@nil{{\tt\graphic@verb{#1}}}
% default to dvips
\psfigdriver{dvips}%
% and to DOS file conventions
%%% \epsfigdos
\epsfigRestoreAt
 


hep-ph0107029.extracted_bib
\begin{thebibliography}{99}

\bibitem{DSE_review} C.D.\ Roberts and A.G.\ Williams,
        Prog.\ Part.\ Nucl.\ Phys.\ {\bf 33}, 477 (1994).

\bibitem{Weisz83} P.\ Weisz, Nucl.\ Phys.\ B {\bf 212}, 1 (1983).

\bibitem{tadpole} G.P.~Lepage \& P.B.~Mackenzie,
        Phys.\ Rev.\ D {\bf 48}, 2250 (1993).

\bibitem{LandauGaugeDE} F.D.R.~Bonnet, P.O.~Bowman, D.B.~Leinweber, 
	A.G.~Williams \& D.G.~Richards, 
	Aust.\ J.\ Phys.\ {\bf 52}, 939 (1999);
        Nucl.\ Phys.\ B (Proc. Suppl.) {\bf 83--84}, 905--907 (2000).

\bibitem{long_glu} 
        D.B.~Leinweber, J.I.\ Skullerud, A.G.\ Williams, and
        C.\ Parrinello, Phys.\ Rev. D {\bf 58}, 031501 (1998);
        {\em ibid.} {\bf 60}, 094507 (1999);
        Erratum -- {\em ibid.} {\bf 61}, 079901 (2000);
        D.B.\ Leinweber, C.\ Parrinello, J.I.\ Skullerud, and A.G.\ Williams
        Nucl.\ Phys.\ B (Proc.\ Suppl.) {\bf 73}, 629-631 (1999);
        {\em ibid.}, 626-628 (1999).

\bibitem{big_vol_glu} 
        F.D.R.~Bonnet, P.O.~Bowman, D.B.~Leinweber, 
	and A.G.~Williams, Phys.\ Rev.\ D {\bf 62}, 051501 (2000);
        {\em ibid.} {\bf 64}, 034501 (2001).

\bibitem{Luscher:1996sc} M.\ L\"{u}scher {\it et al.}, 
	Nucl.\ Phys.\ B {\bf 478}, 365 (1996).

\bibitem{Dawson:1997gp} C.\ Dawson {\it et al.}, Nucl.\ Phys.\ B
	(Proc. Suppl.) {\bf 63}, 877 (1998).

\bibitem{Heatlie:1991kg} G.\ Heatlie {\it et al.}, 
        Nucl.\ Phys.\ B {\bf 352}, 266 (1991).

\bibitem{quark_prop} J.I.\ Skullerud and A.G.\ Williams,
        Nucl.\ Phys.\ B (Proc. Suppl.) {\bf 83--84}, 209--211 (2000);
        Phys.\ Rev.\ D {\bf 63}, 054508 (2001);
        J.I.\ Skullerud, D.B.\ Leinweber, and A.G.\ Williams,
        to appear in Phys.\ Rev.\ D, hep-lat/0102013.

\end{thebibliography}