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\title{A lattice study of the
quark propagator and vertex function}

\author{Jon Ivar Skullerud
%(with David Henty, Claudio Parrinello)
\address{Department of Physics and Astronomy, The University of Edinburgh,
Edinburgh EH9 3JZ, Scotland} (UKQCD Collaboration)}

\begin{document}
\pagestyle{empty}

\begin{abstract} 

We report on the status of a study of the quark propagator and
quark--gluon vertex in momentum space.  Quark propagators have been
generated at $\beta=6.0$ using the $O(a)$-improved
Sheikholeslami--Wohlert (SW) action and fixed to the Landau gauge.  The
first results for the quark pole mass and field renormalisation constant
are reported, and plans for future work are presented.  \end{abstract}

\maketitle

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

\section{Introduction}

The study of quark and gluon correlation functions on the lattice
provides a powerful tool for increasing our understanding of QCD. 
Firstly, it enables us to study non-perturbatively the fundamental
degrees of freedom in QCD, ie the quark and gluon fields.  Secondly, it
allows for a more direct study of phenomena like confinement,
hadronisation and dynamical chiral symmetry breaking.  Thirdly, it
provides an opportunity to extract basic QCD parameters like the running
coupling and quark masses from first principles.  Here, we will report
on work that is in progress to study the quark propagator and
quark--gluon vertex.

%The quark propagator.  
%This is interesting in itself: 
% A basic component of the theory 
% Gives insight into chiral symmetry breaking and dynamic masses

\section{The quark propagator}

The quark propagator may be studied both in time and in momentum space. 
The first approach is suited to investigating certain aspects of
dynamical mass generation, while the second approach allows us to
extract field renormalisation and the quark pole mass.

\subsection{Quark propagator in time}
%Exponential fall-off of quark propagator in time
%\cite{soni}.

The zero-momentum quark propagator in a covariant (or related) gauge is
expected to show the following behaviour \cite{soni}:

\begin{eqnarray}
S(t) & = & \left < \sum_{\vec{x}}\overline{\psi}(\vec{x},t) 
\psi(0,0)\right> \nonumber \\
     & = & A(t) + \gamma_4 B(t)
\end{eqnarray}
where $A(-t) = -A(t)$ and $B(-t) = B(t)$, with the asymptotic behaviour
\begin{eqnarray}
A(t)  & \sim & A e^{-m_s t} \\
B(t) & \sim & B e^{-m_v t}
\end{eqnarray}

The `scalar' and `vector' quark masses $m_s$ and $m_v$ should be equal,
as should the amplitudes $A$ and $B$.

\subsection{Propagator in momentum space}

At tree level, the continuum momentum space propagator is

\beq
S(p) = \frac{1}{i\pslash + m}
\eeq

On the lattice, we would therefore expect the inverse propagator for
some appropriate range of momenta ($1/L \ll p \ll 1/a$) to behave
according to
%
\beq
\Gamma(p) = Z_\psi(pa) \left ( i \pslash + m(pa) \right )
\eeq
%
where $m(pa)$ is the pole mass, which can be compared with
calculations in perturbation theory and from Schwinger--Dyson equations.


\section{Vertex function}

We can define a quark--gluon vertex on the lattice with quark momenta
$p-q$ and $p$, and gluon momentum $q$ as
%
\beq
\Gamma_\mu^{lat}(p,q) =\frac{ \left < S(p) A_\mu(q) \right >}
{\left<S(p-q)\right>\left<G(q)\right>\left<S(p)\right>}
\eeq
%
where $G(q)$ is the gluon propagator.

In the continuum, the off-shell amputated vertex
function will have the general form
\begin{eqnarray}
\lefteqn{\Gamma_\mu(p^2,q^2,pq)} \nonumber \\
\mbox{ } & = & F_1 p_\mu + F_2 q_\mu + F_3 \gamma_\mu \nonumber \\
 & & + F_4 \pslash p_\mu + F_5 \pslash q_\mu + F_6 \qslash p_\mu 
+ F_7 \qslash q_\mu \\
 & & + F_8 \sigma_{\mu\nu} p^\nu
+ F_9 \sigma_{\mu\nu} q^\nu \nonumber
\end{eqnarray}

At tree level, this reduces to $\Gamma^0_\mu = g_0\gamma_\mu$.
From this we can see that the form factor containing the running
coupling is $F_3$, while all the other form factors are expected to be
finite.  We can use the symmetries of the problem, including current
conservation, to extract $F_3(q^2;p=0)$.
We then define the renormalised coupling as
\beq
g_R(q^2) = Z_\psi Z_A^{1/2} F_3(q^2)
\label{eq:gr}
\eeq

From this, we can compute $g_R(q^2)$ at various $\beta$ values, and
relate this to the running coupling calculated in other schemes, eg,
$g_R^{\overline{MS}}(q^2)$ or the lattice 3-gluon
coupling \cite{claudio}. 

\section{Computation and results}

We have analysed propagators generated from 36 configurations at
$\beta=6.0$ with the SW action, with a lattice size of $16^3\times 48$,
at $\kappa = 0.1432, 0.1440, 0.1450$ (corresponding to $m_{PS}a =
0.386$, 0.311 and 0.257 respectively).  The propagators have been fixed
to landau gauge, with accuracy $\frac{1}{VN_c}\sum_x \left |\partial_\mu
A_\mu(x)\right|^2 < 10^{-6}$, and Fourier transformed. 

\subsection{Propagator in time}

A complete analysis has been done of the propagators on 9
configurations.  The values for the scalar and vector masses are given
in table~\ref{tab:tmass}. 

\begin{figure}[t]
\vspace{-5mm}
\ewxy{tprop.ps}{250pt}
\vspace{-12mm}
\caption{Scalar and vector propagators in time, for 36 configurations,
at $\kappa=0.1432$.}
\end{figure}

\begin{table}[b]
\caption{Values for the scalar and vector mass of the quark at
$\beta=6.0$, for different values of $\kappa$.}
\begin{tabular}{lccc}
\hline
$\kappa$ & $(m_{PS}a)^{^2}$ & $m_s a$ & $m_v a$ \\ \hline
0.1432 & 0.1490 & 0.265\er{7}{9} & 0.262\err{10}{14} \\ 
0.1440 & 0.0967 & 0.238\er{8}{7} & 0.232\err{9}{10} \\
0.1445 & 0.0660 & 0.215\er{7}{7} & 0.209\err{11}{9} \\ \hline
$\kappa_c$ & 0 & $ 0.170\err{14}{10}$ & 0.162\err{16}{11} \\ \hline
\end{tabular}
\label{tab:tmass}
\end{table}

This gives a value for $m_q$ in the chiral limit of $340 \pm 40$ MeV, in
good agreement with the $350 \pm 40$ MeV reported in \cite{soni}. 
For propagators with finite spatial momenta inserted, the signal
deteriorates rapidly with increasing momentum, so no meaningful
information can be extracted with the current statistics. %(9 configs)

\subsection{Propagator in momentum space}

Preliminary analysis has been done on 15 configurations at all 3
$\kappa$ values.  We have computed and fitted \[ Z_\psi(p)= -i\Tr
\pslash\Gamma(p)/p^2\] and \[ m(p) = \Tr\Gamma(p)/Z_\psi(p)\] $Z_\psi$
is shown as a function of $pa$ in fig~\ref{fig:renorm}.  $Z_\psi(p)$ has
been fitted to a logarithmic function $Z_\psi(p) = Z + k\ln pa$, while
$m(p)$ has been fitted to a polynomial function $m(p) = M + b_1 p + b_2
p^2$, where $b_2$ turns out to be consistent with zero.  The best fit
values (setting $b_2=0$) are given in table~\ref{tab:bestfits}. 

% and $b_1$ %shows no dependence on $\kappa$. $M$ decreases with increasing
%$\kappa$, as expected. 

\begin{figure}[t]
\vspace{-5mm}
\ewxy{renorm_log.ps}{250pt}
\vspace{-12mm}
\caption{$Z_\psi(p)$ as a function of $p$ for 15 configurations at
$\kappa=0.1432$.  The line represents a fit to a logarithmic function.}
\label{fig:renorm}
\end{figure}

\begin{table}[t]
\vspace{-3mm}
\caption{Best fit values for the inverse quark propagator.}
\label{tab:bestfits}
\begin{tabular}{lcccc}
\hline
$\kappa$ & $Z$ & $k$ & $M$ & $b_1/a$ \\ \hline
0.1432 & 1.97 & -0.67 & 0.077 &
-0.243 \\
0.1440 & 1.92 & -0.59 & 0.058 &
-0.235 \\
0.1445 & 1.87 & -0.51 & 0.050 &
-0.234 \\ \hline
\end{tabular}
\vspace{-5mm}
\end{table}

%\begin{figure}
%\ewxy{mass.ps}{200pt}
%\caption{$m(p)$ as a function of $p$ at $\kappa=$ ???}
%\label{fig:mass}
%\end{figure}
%\ewxy{mass_chiral.ps}{200pt}

\subsection{Future work}

The analysis of the quark propagator at $\beta=6.0$ discussed here is
currently being extended to include study of the vertex function.  When
this is completed, the propagator and vertex function will be computed
also at $\beta=6.2$ and 6.4 on hypercubic lattices, in order
both to get a proper understanding of discretisation and finite size
errors and possible power divergences in the quark mass, and to obtain a
reliable determination of the running coupling.

\section{Discussion and conclusions}

It is encouraging that we get such a clean and unambigouous signal for
the quark propagator with a small number of configurations.  This gives
us confidence that we will be able to extract a reasonable signal also
for the vertex funtion.  In particular, we see from
fig.~\ref{fig:renorm} that the quark field renormalisation constant,
which enters into equation~(\ref{eq:gr}) for the running coupling, can be
precisely determined.

However, more analytical and computational work is required in order to
fully understand the significance of the results reported here.  In
particular, the pole mass seems to contain a number of ambiguities,
although on the positive side it must be noted that it does decrease
with increasing $\kappa$, as required.  Firstly, the relation between
the mass extracted from the fall-off of the propagator in time and the
pole mass in momentum space is unclear.  Secondly, the linear dependence
of the pole mass on $pa$ is not well understood.  Hopefully, more light
can be cast on both these issues by studies at different $\beta$ values. 

\section*{Acknowledgements} 

This work has been supported by a Norwegian
Research Council grant.  I wish to thank Claudio Parrinello and David
Henty for fruitful discussions. 

\begin{thebibliography}{9}

\bibitem{soni} C.Bernard, D.Murphy,~A.Soni, K.Yee. Nucl.\ Phys.~{\bf B}
(Proc.~Suppl.) 17 (1990), 593--598.
\bibitem{claudio} C.Parrinello, these proceedings.

\end{thebibliography}

\end{document}
