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\title{A Study of the Quark--Gluon Vertex}

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


\begin{document}
\pagestyle{empty}

\begin{abstract} 

We present the first results from a study of the quark--gluon vertex
function in the Landau gauge.  The results are obtained for quenched QCD
with an $O(a)$-improved Wilson fermion action, at $\beta=6.0$.  We
discuss plans for further study, including extraction of a renormalised
QCD coupling from the momentum dependence of the vertex. 

\end{abstract}

\maketitle

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

\section{Introduction}

The study of quark and gluon correlation functions on the lattice is
emerging as a useful tool for increasing our understanding of QCD. 
Studies of the gluon propagator \cite{gprop1,gprop2} have given valuable
insight into the phenomena of confinement and dynamical mass generation. 
The three-gluon vertex has been shown \cite{ggg} to provide a method for
determining the running coupling from first principles, and preliminary
results for the quark propagator \cite{qprop} indicate that this will
yield useful information on chiral symmetry breaking.  Here, the methods
will be applied to the quark--gluon vertex. 

Studying the quark--gluon vertex on the lattice gives us yet another way
of determining the running coupling, similar to but independent of the
determination from the 3-gluon vertex, to complement other
determinations.  It may also provide input to Dyson--Schwinger equation
calculations, and give insight into the limits of perturbation theory
by indicating where the non-perturbative lattice results deviate from
perturbative calculations.

\section{The vertex function}

We can define the configuration space quark--gluon vertex function as
%
\beqa
V^a_\mu(x,y,z)^{ij}_{\alpha\beta} & = &
\left < \psi^i_\alpha(x)\psibar^j_\beta(z)A^a_\mu(y)\right > \nonumber \\
 & = & \left < S^{ij}_{\alpha\beta}(x,z)A^a_\mu(y)\right >
\eeqa
%
Fourier transforming this gives us the full (unamputated) momentum space
vertex function:
%
\beqa
\lefteqn{\int\!\!d^4\!xd^4\!yd^4\!z
 e^{-i(px+qy-rz\!)}\!\left<\psi^i_\alpha\!(x)
A^a_\mu\!(y)\psibar^j_\beta\!(z)\right>} \nonumber \\
&=&\!\!\!\int\!\!d^4\!ze^{-i(p+q-r)z}\!\!\int\!\!d^4\!xd^4\!ye^{-i(px+qy)}
\times \nonumber \\
& & \times \left<\psi^i_\alpha\!(x)A^a_\mu\!(y)
\psibar^j_\beta\!(0)\right> \nonumber\\
%\lefteqn{= \delta(p\!+\!q\!-\!r)\!\!\int\!\!d^4\!xd^4\!y
%\left<S^{ij}_{\alpha\beta}(x,\!0)A^a_\mu(y)\right>e^{-i(px+qy)}} \nonumber\\
&=&(2\pi)^4\delta(p\!+\!q\!-\!r)\left<S^{ij}_{\alpha\beta}(p)A^a_\mu(q)\right>
\nonumber \\
&=&(2\pi)^4\delta(p\!+\!q\!-\!r)\;G^a_\mu(p,q)^{ij}_{\alpha\beta}
\eeqa
%
and the amputated (OPI) vertex function
%
\beq
\Lambda_\mu^{a,\rm lat}(p,q) =\frac{ \left < S(p) A^a_\mu(q) \right >}
{\left<S(p+q)\right>\left<D(q)\right>\left<S(p)\right>}
\eeq
%
where $D(q) = \sum_{\mu,a} D^{aa}_{\mu\mu}(q)$ is the gluon propagator.

At tree level, this becomes
%
\beqa
\lefteqn{\Lambda^{a(0)}_\mu(p,q) = -ig_0 T^a (2\pi)^4 \times} \nonumber \\
& &\times \left ( \gamma_\mu
\cos\frac{(2p+q)_\mu}{2} - ir\sin\frac{(2p+q)_\mu}{2}\right )
\label{eq:tree_lat}
\eeqa

We will also define the colour-traced full vertex function
%
\beq
G_\mu(p,q) =
\frac{2}{N_C^2-1}\sum_{ij,a}T^a_{ij}G^a_\mu(p,q)^{ji}_{\alpha\beta}
\eeq

In the continuum, the off-shell amputated vertex
function will have the general form
\begin{eqnarray}
\lefteqn{\Lambda_\mu(p^2,q^2,pq) \equiv 
\frac{1}{N_C^2-1}\Tr_{col}T^a\Lambda^a_\mu(p,q)}\nonumber \\
 & = & 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 \label{eq:formfactors}\\
 & & + F_8 \sigma_{\mu\nu} p^\nu
+ F_9 \sigma_{\mu\nu} q^\nu \nonumber
\end{eqnarray}
%
where all the $F$'s depend only on the invariants $p^2, q^2$ and $pq$.

At tree level, this reduces to $\Lambda^0_\mu =
\frac{i}{2}g_0\gamma_\mu$.  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.  $F_9$ will also be of interest,
since it is related to the chromomagnetic moment of the quarks,

$F_3$ may be extracted by exploiting the symmetries of the problem. 
First, we isolate $F_3$ and $F_7$
%
\beq
\Tr \gamma_\mu\Lambda_\mu(p\to 0,q) = 4 F_3(q^2) + 4q_\mu^2 F_7(q^2)
\label{eq:formred}
\eeq
%
We can then eliminate $F_7$ by imposing an appropriate kinematics, e.g.\
$q=q_\nu$ with $\mu\neq\nu$.

Finally, we define the renormalised coupling as
\beq
g_R(q^2) = -2i Z_\psi Z_A^{1/2} F_3(q^2)
\label{eq:gr}
\eeq
where $Z_\psi$ and $Z_A$ are the renormalisation constants for the quark
and gluon fields, defined by
%
\beqa
D_{\mu\nu}(p^2)|_{p^2=q^2} & = & T_{\mu\nu}(q)Z_A\frac{1}{q^2} \\
\Tr \pslash S^{-1}(p)|_{p^2=q^2} & = & i Z_\psi q^2
\eeqa
%
where $T_{\mu\nu}$ is the projection onto transverse fields.

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{ggg}. 
In this connection, it is worth noting that the matching with
$g_R^{\overline{MS}}(q^2)$ can be performed entirely within continuum
perturbation theory \cite{nprenorm}. 

\section{Computation and results}

In the initial study which we will report on here, we have so far looked
only at the full (unamputated) vertex function, in configuration space
as well as in momentum space.

36 configurations have been analysed at $\beta=6.0$ with a lattice size
of $16^3\times 48$.  The propagators have been generated using the
Sheikholeslami--Wohlert action, at $\kappa = 0.1432$ (corresponding to
$m_{PS}a = 0.386$).  The gauge fields and propagators have been fixed to
Landau gauge, with accuracy $\frac{1}{VN_c}\sum_{x,\mu} \left
|\partial_\mu A_\mu(x)\right|^2 < 10^{-6}$. 

For the vertex function in configuration space, the quantity
$V_\mu(x_A,x_\psi) = \Tr_{col} \lambda^a V_\mu^a(0,x_A,x_\psi)$ has been
calculated.  A clear signal has been found for $\Tr V_\mu$ and
$\Tr\gamma_\mu V_\mu$.  The latter is shown as a function of $x_\psi$
and $x_A$ in fig.~\ref{fig:V2xy}.

\begin{figure}[htb]
\ewxy{A2x_psiy.ps}{190pt}
\vspace{-12mm}
\caption{Vertex function as a function of $x_\psi = (0,y_\psi,0,0)$ and
different values of $x_A = (x_A,0,0,0)$, for 36 configurations,
at $\kappa=0.1432$.}
\label{fig:V2xy}
\end{figure}

We have also calculated the `timesliced' vertex functions
%
\beqa
\lefteqn{G_\mu(\vec{p},\vec{q},t,t_A)
=\Tr_{col}\lambda^a\int d^3xd^3y
e^{-i(\vec{p}\vec{x}+\vec{q}\vec{y})}\times}\nonumber\\
&\times&\left<S(\vec{x},t;0)A^a_\mu(\vec{y},t_A)\right>
\eeqa
%
and 
%
\beq
G_\mu(\vec{p},q,t) = \int dt_Ae^{iq_t t_A}
G_\mu(\vec{p},\vec{q},t,t_A)
\eeq
%
These both fall off exponentially with $t$ (after antisymmetry in $t$
has been imposed), as one should expect.  Fig.~\ref{fig:G3_tAvar} shows
$\gamma_\mu G_\mu(\vec{p},\vec{q},t,t_A)$ as a function of $t$ for
different values of $t_A$, while fig.~\ref{fig:G3_varmom} shows
$\gamma_\mu G_\mu(\vec{p},q,t)$ as a function of $t$ for different
values of the momenta. 

\begin{figure}[htb]
%\vspace{-5mm}
\ewxy{G3_tAvar.ps}{190pt}
\vspace{-12mm}
\caption{Zero-momentum vertex function as a function of time, for
different values of the gluon time $t_A$.}
\label{fig:G3_tAvar}
\end{figure}

\begin{figure}[htb]
%\vspace{-5mm}
\ewxy{G3_pvar.ps}{190pt}
\vspace{-12mm}
\caption{$\Tr\gamma^3 G_3(\vec{p},q,t)$ as a function of $t$, for
different values of $\vec{p}$ and $q$.}
\label{fig:G3_varmom}
\end{figure}

$\Tr\gamma_\mu G_\mu(p,q)$ is shown as a function of $p=(0,0,0,p_t)$ for
$q=0$ (with $\mu=3$) in fig.~\ref{fig:mom_vtx}.  We see that there is a
clean signal which falls off with $p$ much the same way as the quark
propagator \cite{qprop}.  For non-zero spatial $p$ and non-zero $q_t$ it
falls off, but remains clean, while for non-zero spatial $q$ it
deteriorates so rapidly that no conclusions can be drawn about its
behaviour. 

\begin{figure}[hbt] 
%\vspace{-14mm} 
\ewxy{G3_mom_zero.ps}{190pt}
\vspace{-12mm} 
\caption{$\Tr\gamma^3 G_3(p,q)$ for $q=0$, $p=(\vec{p},p_t)$, as a
function of $p_t a$.} 
\label{fig:mom_vtx} 
\end{figure}

It turns out that $\gamma_\mu G_\mu$ falls off smoothly with $p$ and $q$
for $p=p_\nu$ ($q=q_\nu$) with $\mu\neq\nu$.  This is not the case for
$\mu=\nu$; however, the signal here is too poor to make it possible to
draw any conclusions about the actual behaviour of the vertex function
in this case.  It may be related to the $\cos(2p+q)_\mu$ term in
equation~(\ref{eq:tree_lat}), or to the $F_4$ and $F_7$ form factors in
equations (\ref{eq:formfactors}) and (\ref{eq:formred}),
but this cannot be stated with any certainty at this stage. 

\section{Discussion and conclusions}

We have found a clear signal for the quark--gluon vertex in both
configuration space and momentum space.  However, the momentum-space
vertex is still very noisy.  This may be either because our
statistics are too low, or because our gauge fixing is not accurate
enough.  Early results using gauge fixing with an accuracy of
$10^{-12}$, as well as experience from study of the 3-gluon vertex
\cite{ggg}, lend support to the latter hypothesis, but future work will
involve both better gauge fixing accuracy and higher statistics.

\section{Future work}

When the uncertainties connected with the gauge fixing accuracy have
been brought under control, we will go on to analyse the amputated
vertex.  The plan thereafter is to repeat this on a larger ensemble of
configurations at $\beta=6.2$, and possibly also at $\beta=6.4$.  This
should enable us to come closer to a realistic determination of
$g_R(q^2)$ from the quark--gluon vertex. 

\section*{Acknowledgements} 

This work has been supported by Norwegian Research Council grant
100229/432.  I wish to thank Claudio Parrinello, David Henty and David
Richards for fruitful discussions. 

\begin{thebibliography}{9}
\bibitem{gprop1} C.~Bernard, C.~Parrinello, A.~Soni, Phys.\ Rev.\ {\bf
D~49}, 1585 (1994)
\bibitem{gprop2} P.~Marenzoni {\em et al.}, Phys.~Lett.~{\bf B 318}, 511
(1993) 
\bibitem{ggg} C.~Parrinello, Phys.~Rev.~{\bf D~50}, 4247 (1994);
D.~S.~Henty and C.~Parrinello, to be published.
\bibitem{qprop} J.~I.~Skullerud,  Nucl.~Phys.~{\bf B 42} (Proc.\ suppl.),
364 (1995) 
\bibitem{nprenorm} G.~Martinelli, C.~Pittori,
C.~T.~Sachrajda, M.~Testa and A.~Vladikas,
Nucl.\ Phys.\ {\bf B~445} 81 (1995)

\end{thebibliography}

\end{document}
