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\begin{document}

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\preprint{\vbox{\noindent Submitted to 
                          {\it ???}
\hfill ADP-98-??/T???\\}} 

% \draft command makes pacs numbers print
%\draft

\title{The Nonperturbative Gluon Propagator in the Infrared Regime}

\author{Derek Leinweber\footnote{
E-mail:~dleinweb@physics.adelaide.edu.au ~$\bullet$~ Tel:
+61~8~8303--3548 ~$\bullet$~ Fax: +61~8~8303--3551 \hfill\break
\null\quad\quad
WWW:~http://www.physics.adelaide.edu.au/theory/staff/leinweber/}, 
Jon Ivar Skullerud\footnote{
E-mail:~jskuller@physics.adelaide.edu.au ~$\bullet$~ Tel:
+61~8~8303--3542 ~$\bullet$~ Fax: +61~8~8303--3551 \hfill\break
\null\quad\quad
WWW:~http://www.physics.adelaide.edu.au/$^\sim$jskuller/} 
and Tony Williams\footnote{
E-mail:~awilliam@physics.adelaide.edu.au ~$\bullet$~ Tel:
+61~8~8303--3546 ~$\bullet$~ Fax: +61~8~8303--3551 \hfill\break
\null\quad\quad
WWW:~http://www.physics.adelaide.edu.au/theory/staff/williams.html}}
\address{Special Research Centre for the Subatomic Structure of Matter
and The Department of Physics and Mathematical Physics, University of
Adelaide, Adelaide SA 5005, Australia}
\author{Claudio Parrinello}
\address{University of Liverpool}
\author{(UKQCD Collaboration)}
%\address{ (UKQCD Collaboration)}

\maketitle

\begin{abstract} 
The gluon propagator is calculated in quenched QCD on two different
lattice sizes ($16^3 \times 48$ and $32^3 \times 64$) at $\beta =
6.0$.  The physical volume dependence of the propagator in Landau
gauge is studied.  The smaller lattice is instrumental in revealing
finite volume and anisotropic lattice artifacts.  Methods for
minimizing these artifacts are developed and applied to the larger
lattice data.  New structure seen in the infrared region survives
these conservative cuts to the lattice data.  This structure serves to
exclude a number of models that have appeared in the literature.  Fits
to simple analytical forms capturing the momentum dependence of the
nonperturbative gluon propagator are also reported.
\end{abstract}

\vspace{1cm}
\pacs{PACS numbers: 
 12.38.Gc  % Lattice QCD calculations          
 }  
\parskip=2mm


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

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

The infrared behavior of the nonperturbative gluon propagator has been
the subject of extensive research and debate.  Knowledge of this
behavior is central to understanding the mechanisms of confinement in
quantum chromodynamics (QCD).  Moreover, it is of great importance in
various model-based approaches.  For example, some studies using
Dyson--Schwinger equations \cite{mandelstam,bp,bbz} have indicated
that an infrared enhanced gluon propagator may be required for
confinement; however, others \cite{cornwall,gribov,vsmekal} have
pointed to the possibility of a dynamically generated gluon mass, or
other forms leading to an infrared finite (or even vanishing)
propagator.  

   Computing the gluon propagator directly on the lattice should
provide an opportunity to resolve these contradictory claims.
However, previous lattice studies \cite{bernard,marenzoni} have been
unable to access the ``deep'' infrared region where the most
interesting nonperturbative behavior is expected.  Significant finite
volume effects introduced through zero momentum components of the
gluon propagator four-momenta prevent the study of the infrared
behavior of the propagator on a small lattice.

   The main aim of this study is to obtain a definite behavior for
the gluon propagator for momenta smaller than 1 GeV, where the
interesting physics resides.  In the following, we will report such
results for momenta as small as 0.4 GeV.  This momenta is small enough
to reveal new structure in the gluon propagator which points the
direction to the resolution of the infrared behavior of the
gluon propagator.

\section{The gluon propagator on the lattice}
\label{sec:gluon-lat}

The lattice gluon field $A_{\mu}(x)$ may be obtained from the gauge
links $U_\mu$ via
%
\beq
A_\mu(x+\muhat/2) = \frac{1}{2ig_0}\left(U_\mu(x)-U^{\dagger}_\mu(x)\right)
 - \frac{1}{6ig_0}\Tr\left(U_\mu(x)-U^{\dagger}_\mu(x)\right) \, ,
\eeq
%
accurate to $\cal O(a^2)$. In momentum space,
%
\beqa
A_\mu(q) & = & \sum_x e^{-iq \cdot (x+\muhat/2)} A_\mu(x) \nonumber \\
 & = & \frac{e^{-iq_\mu/2}}{2ig_0}\left(U_\mu(q)-U^{\dagger}_\mu(-q)\right)
 - \frac{1}{6ig_0}\Tr\left(U_\mu(q)-U^{\dagger}_\mu(-q)\right) \, .
\eeqa
%
The gluon propagator $D_{\mu\nu}^{ab}(q)$ is defined by
%
\beq
D^{ab}_{\mu\nu}(q) \equiv \bra A_\mu^a(q)A_\nu^b(-q)\ket \, .
\eeq
In Landau gauge
\beq
D_{\mu\nu}^{ab}(q) =
(\delta_{\mu\nu}-\frac{q_{\mu}q_{\nu}}{q^2})D^{ab}(q^2) \, .
\eeq
%
On a finite lattice, the available momentum values are given by
\beq
q_\mu a = \sin\frac{2\pi n_\mu}{L_\mu}, \qquad
n_\mu=0,\ldots,L_\mu-1
\eeq
where $L_\mu$ is the length of the box in the $\mu$ direction.

The Landau gauge is obtained by maximizing $F = \sum_{\mu,x}\Re\Tr
U_\mu(x)$. 

\section{Simulation parameters}

We have analyzed 75 configurations at $\beta=6.0$, on a volume of
$32^3\times 64$.  For comparison, we have also studied an ensemble of
125 configurations on a smaller volume of $16^3\times 48$.  In both
cases, we have fixed to Landau gauge with an accuracy of
$\frac{1}{VN_C}\sum_{\mu,x}|\partial_{\mu}A_{\mu}|^2 <10^{-12}$.  To
double-check the gauge fixing we also looked at $A^{00}_4(\vec{x},t)$,
which should be constant in time.

Present results for A(t)...

Further details about the generation of the configurations can be
found in \cite{ukqcd-data}.

\section{FINITE SIZE EFFECTS AND ANISOTROPIES}
\label{sec:artefacts}

\subsection{Small Lattice Analysis}

On the small lattice, we see significant anisotropy in the data which
has its origin in both finite volume and finite lattice spacing
artifacts.  Finite size effects are expected to be largest where the
momentum along one or more of the Cartesian directions is zero.  Since
the length of the lattice in the time direction is 3 times that of the
spatial directions, we notice a clear difference between points which
correspond to momenta directed in Cartesian spatial directions from
those in the time direction.  Fig.~\ref{fig:small_comp} shows the
gluon propagator for momenta directed throughout the lattice.  In
preparing this data a $Z_3$ averaging was done where for example the
momentum along (t,x,y,z) = (1,2,1,1) is averaged with (1,1,2,1) and
(1,1,1,2).


\begin{figure}[t]
\begin{center}
\epsfysize=11.6truecm
\leavevmode
\setbox\rotbox=\vbox{\epsfbox{SmallLattComp.ps}}\rotl\rotbox
\end{center}
\caption{The gluon propagator from our small lattice multiplied by
$q^2 a^2$ plotted as a function of momenta $q\, a$.  Values for each
momentum direction are plotted.  Only a $Z_3$ averaging has been
performed.  Filled squares denote momenta directed along spatial
Cartesian directions, while filled triangles denote momenta directed
along the time axis.  Other momenta are indicated by open circles.
Finite volume errors are expected to be largest for the time axis
momenta, as the three shorter spatial directions have momentum
components equal to zero.}
\label{fig:small_comp}
\end{figure}


At small momenta the discrepancies between the filled squares
(denoting momenta directed along spatial Cartesian directions), and
the filled triangles (denoting momenta directed along the time axis),
clearly display the effects of finite volume artifacts.  

There are also anisotropies at larger momenta, which are a result of
finite lattice spacing errors breaking the O(4) symmetry down to the
hyper-cubic symmetry.  This lattice action anisotropy is displayed in
Fig.~\ref{fig:small_comp} by the filled squares and triangles denoting
momenta directed along Cartesian directions lying below the majority
of points from other non Cartesian momentum directions\footnote{In
Fig.~\protect\ref{fig:small_comp} it appears that this anisotropy
disappears as one goes to larger momenta.  However, this is not the
case.  Only momentum components up to $q_\mu a = 4 \times 2\pi/16$
have been selected in preparing Fig.~\protect\ref{fig:small_comp}.  As
such, the larger momenta are not accessed through Cartesian directions
alone, and such points are absent in this figure.}.

This latter problem may be easily avoided by simply selecting points
lying along a particular direction in momentum space.  In order to
minimize lattice artifacts for large momenta introduced by
%
\begin{eqnarray}
p_\mu a &\to& 2 \sin({p_\mu a \over 2}) \, , \nonumber \\
        &=& p_\mu^2 a^2 - {1\over 12} p_\mu^4 a^4 + \cdots \, ,
\end{eqnarray}
%
we select momenta lying within a cylinder directed along the diagonal
$(t,x,y,z) = (1,1,1,1)$ of the lattice.  This allows one to access the
largest of momenta with the smallest of components.  We found the
selection of a cylinder with a radius of two spatial momenta units ($2
\times 2\pi/L_s$, where $L_s$ is the number of sites along a spatial
Cartesian direction) provides a reasonable number of points falling
along a single curve for large momenta.  The data surviving this cut
is displayed in Fig.~\ref{fig:small_comp_cyl2}.

\begin{figure}[p]
\begin{center}
\epsfysize=11.6truecm
\leavevmode
\setbox\rotbox=\vbox{\epsfbox{SmallLattCompCyl2.ps}}\rotl\rotbox
\end{center}
\caption{The gluon propagator from our smaller lattice multiplied by
$q^2 a^2$.  The points displayed in this plot lie within a cylinder of
radius $2 \times 2\pi/16$ directed along the diagonal $(t,x,y,z) =
(1,1,1,1)$ of the lattice.}
\label{fig:small_comp_cyl2}
\end{figure}

\begin{figure}[p]
\begin{center}
\epsfysize=11.6truecm
\leavevmode
\setbox\rotbox=\vbox{\epsfbox{SmallLattCompAng20Cyl2.ps}}\rotl\rotbox
\end{center}
\caption{The gluon propagator from our smaller lattice multiplied by
$q^2 a^2$.  The points displayed in this plot lie within a cylinder of
radius $2 \times 2\pi/16$ directed along the diagonal of the lattice
and within a cone of $20^\circ$ measured from the diagonal at the
origin.}
\label{fig:small_comp_ang20_cyl2}
\end{figure}

However, this cut does not address the large finite volume errors
surviving in Fig.~\ref{fig:small_comp_cyl2}.  To remove these
problematic momenta, we consider a further cut designed to remove
momenta which have one or more vanishing components.  As such we keep
momentum directions that lie within a certain solid angle of the
diagonal.  We found that a cone of angle $20^\circ$ measured from the
diagonal at the origin was sufficient to provide a set of points lying
along a smooth curve.  Fig.~\ref{fig:small_comp_ang20_cyl2} displays
this data.  After these conservative cuts, there is relatively little
structure left in the nonperturbative regime.  Armed with knowledge of
how to obtain reliable lattice data, we now turn our attention to the
gluon propagator data obtained from our larger lattice.


\subsection{Large Lattice Analysis}

Fig.~\ref{fig:large_comp} displays the gluon propagator data for all
momenta directions.  Again, only a $Z_3$ averaging has been done.
Examination of the infrared region indicates that finite volume
artifacts are very small on the larger lattice.  In particular, the
agreement between the spatial Cartesian (filled squares) and time-like
Cartesian points (filled triangles) at $qa = 0.20$ indicates that
finite size effects are relatively small on this lattice.  Such an
observation is consistent with topological studies of the QCD vacuum
which provide some insight into the typical scale of QCD vacuum
fluctuations.  


\begin{figure}[t]
\begin{center}
\epsfysize=11.6truecm
\leavevmode
\setbox\rotbox=\vbox{\epsfbox{LargeLattComp.ps}}\rotl\rotbox
\end{center}
\caption{The gluon propagator from our larger lattice multiplied by
$q^2 a^2$ plotted as a function of momenta $q\, a$.  Values for each
momentum direction are plotted.  Only a $Z_3$ averaging has been
performed.  Plotting symbols are as in
Fig.~\protect\ref{fig:small_comp}.  Finite volume errors are greatly
reduced here, as displayed by the overlap of spatial and time oriented
Cartesian directions.  However, significant anisotropy is displayed
for larger momenta.}
\label{fig:large_comp}
\end{figure}


At large momenta, significant lattice action anisotropy is seen
similar to that displayed in Fig.~\ref{fig:small_comp}.  Of course,
this is as one expects as the action and lattice spacing are common to
both lattices.  To side step this problem, we adopt the same cut where
all momenta must lie within a cylinder of radius two-spatial momentum
units directed along the lattice diagonal.
Fig.~\ref{fig:large_comp_cyl2} displays the momenta surviving this
cut.


\begin{figure}[p]
\begin{center}
\epsfysize=11.6truecm
\leavevmode
\setbox\rotbox=\vbox{\epsfbox{LargeLattCompCyl2.ps}}\rotl\rotbox
\end{center}
\caption{The gluon propagator from our smaller lattice multiplied by
$q^2 a^2$.  The points displayed in this plot lie within a cylinder of
radius $2 \times 2\pi/32$ directed along the diagonal of the
lattice.}
\label{fig:large_comp_cyl2}
\end{figure}

\begin{figure}[p]
\begin{center}
\epsfysize=11.6truecm
\leavevmode
\setbox\rotbox=\vbox{\epsfbox{LargeLattCompAng20Cyl2.ps}}\rotl\rotbox
\end{center}
\caption{The gluon propagator from our larger lattice multiplied by
$q^2 a^2$.  The points displayed in this plot lie within a cylinder of
radius $2 \times 2\pi/32$ directed along the diagonal of the lattice
and within a cone of $20^\circ$ measured from the diagonal at the
origin.}
\label{fig:large_comp_ang20_cyl2}
\end{figure}


To remove any possibility of remaining finite size effects, we also
consider the additional angular cut, requiring that all points must
lie within a cone of opening angle $20^\circ$ from the diagonal at the
origin.  The data surviving this cautiously conservative cut is
illustrated in Fig.~\ref{fig:large_comp_ang20_cyl2}.  It is interesting
to see that the turn over in $q^2a^2 \, D(q^2)$ in the nonperturbative
regime survives this extreme cut.


\subsection{Numerical results}
\label{sec:numbers}

Here we consider a few simple analytical forms which capture the
essence of the nonperturbative gluon propagator.  The lattice data for
the fit is taken from the large lattice with the cylindrical cut.
Equivalent squared four-momenta surviving the cut are averaged.

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

Even with the conservative cuts described in
section~\ref{sec:artefacts}, there is a definite turnover in the
behavior of $q^2 a^2 \, D(q)$ around $q^2 = 1\ {\rm GeV}^2$, which
points to an infrared finite gluon propagator.  This data suggests the
exclusion of models like those proposed in \cite{bbz}, that lead to an
infrared enhanced propagator.  Other models that have been proposed in
the past, including those of \cite{marenzoni} and \cite{gribov}, are
also ruled out.  Our best fit indicate ????

Future work will focus on a detailed analysis of various analytic
modeling forms for the gluon propagator \cite{next}.  There, a
stability analysis of the fit parameters is central to identifying the
model best able to capture the essence of the gluon propagator in both
the nonperturbative and the well known perturbative regimes.  A study
of the effects of Gribov copies \cite{cucchieri}, and a complete
analysis of the tensor structure of the gluon propagator are also
under consideration.

Finally, it would be very interesting to study the gluon propagator
using the improved actions which are currently revolutionizing the
lattice QCD field.  With lattice spacings of 0.4 fm possible, a modest
$16^4$ lattice would be 6.4 fm on a side allowing the investigation
of very soft momenta.  Moreover, the restoration of rotational
symmetry demonstrated in improved glue studies should aid in largely
removing the anisotropy problems encountered here.


\section*{Acknowledgments} 

Financial support from the Australian Research Council is gratefully
acknowledged. 

\begin{thebibliography}{99}

\bibitem{mandelstam} S.~Mandelstam, \rf{\pr}{D 20}{3223}{1979}

\bibitem{bp} N.~Brown and M.R.~Pennington, \rf{\pr}{D 39}{2723}{1989}

\bibitem{bbz} M.~Baker, J.S.~Ball, F.~Zachariassen, \rf{\np}{B 186}{531}{1981}

\bibitem{cornwall} J.~Cornwall, \rf{\pr}{D 26}{1453}{1982}

\bibitem{gribov} V.N.~Gribov, \rf{\np}{B 139}{19}{1978}; D.~Zwanziger,
\rf{\np}{B 378}{525}{1992}

\bibitem{vsmekal} L.~von Smekal, A.~Hauck, R.~Alkofer,
\rf{\prl}{79}{3591}{1997}

\bibitem{bernard} C.~Bernard, C.~Parrinello, A.~Soni,
\rf{\pr}{D49}{1585}{1994} 

\bibitem{marenzoni} P.~Marenzoni, G.~Martinelli, N.~Stella, 
\rf{\np}{B 455}{339}{1995}; P.~Marenzoni, G.~Martinelli, N.~Stella,
M.~Testa, \rf{\pl}{B 318}{511}{1993}

\bibitem{ukqcd-data} UKQCD Collaboration, uuu

\bibitem{next} S.~Bilson-Thompson, D.~Leinweber, C.~Parrinello,
J.I.~Skullerud, A.G.~Williams, in preparation


\end{thebibliography}

\end{document}


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