\chapter{The quark--gluon vertex}
\label{chap4}
\label{chap:qqg}

\section{Continuum symmetries and form factors}
\label{sec:vtx-continuum}

The continuum vertex function (fig.\ \ref{fig:vtx-illustrate}) can be
derived from the effective action $\Gamma$, cf.\ eq.~(\ref{eq:properfuncs})
\beq
\Lambda^a_\mu(x,y,z)^{ij}_{\alpha\beta} = \frac{\delta^3\Gamma}
{\delta\psi^i_\alpha(x)\delta A^a_\mu(y)\delta\psibar^i\beta(z)}
\eeq
and the momentum-space vertex is defined by
\beq
(2\pi)^4\delta^4(p+q-r)\Lambda^a_\mu(p,q) = 
\int d^4xd^4yd^4ze^{-i(px+qy-rz)}\Lambda^a_\mu(x,y,z)
\eeq

\begin{figure}[hbtp]
\begin{center}
\setlength{\unitlength}{1cm}
\setlength{\fboxsep}{0cm}
\begin{picture}(7,7)
\end{picture}
\end{center}
\caption{The quark--gluon vertex.}
\label{fig:vtx-illustrate}
\end{figure}

The only possible dependence this can have on the group coordinates
$a,i,j$ is proportional to $T^a_{ij}$.  We can therefore consider only
$\Lambda_\mu(p,q)$, defined by
\beq
(\Lambda^a_\mu)^{ij}_{\alpha\beta} = T^a_{ij}(\Lambda_\mu)_{\alpha\beta}
\eeq
%
Lorentz invariance and parity conservation require that this takes the
form 
\begin{eqnarray}
\lefteqn{\Lambda_\mu(p^2,q^2,pq) 
 = 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 $\Lambda^0_\mu =
ig_0\gamma_\mu$.  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.  

\section{Method for extracting the renormalised coupling}
\label{sec:vtx-method}

\subsection{Definitions and principles}
\label{sec:vtx-defs}

We define the lattice gluon field $A_{\mu}(x)$, which in the continuum
limit becomes $aA_{\mu}^{cont}(x)$, as
%
\beq
A_\mu(x) = \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

The momentum-space gluon field then becomes\footnote{This should really
be \(A_\mu(q)=\sum_{x}e^{-iq(x+\muhat/2)}A_\mu(x)\), but we are keeping
\(q_\mu=0\), so it makes no difference.}
\beqa
A_\mu(q) & = & \sum_x e^{-iqx} A_\mu(x) \nonumber \\
 & = & \frac{1}{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 Lie algebra-valued gluon fields can be computed by tracing with the
Gell-Mann matrices
\beq
\Tr\lambda^a A_\mu = \Tr\lambda^a\sum_b\frac{\lambda^b}{2}A_\mu^b =
\sum_b\delta^{ab}A_\mu^b = A_\mu^a
\eeq

We can define the configuration space quark--gluon vertex function as
%
\beq
V^a_\mu(x,y,z)^{ij}_{\alpha\beta} = 
\left < \psi^i_\alpha(x)\psibar^j_\beta(z)A^a_\mu(y)\right > 
  = \left < S^{ij}_{\alpha\beta}(x,z)A^a_\mu(y)\right >
\eeq
%
Fourier transforming this  and invoking translational invariance gives
us the full (unamputated) momentum space vertex function:
%
\beqa
\lefteqn{\sum_{x,y,z}
 e^{-i(px+qy-rz\!)}\!\left<\psi^i_\alpha\!(x)
A^a_\mu\!(y)\psibar^j_\beta\!(z)\right>} \nonumber \\
&=& \sum_ze^{-i(p+q-r)z}\sum_{x,y}e^{-i(p(x-z)+q(y-z))}
\left<\psi^i_\alpha\!(x-z)A^a_\mu\!(y-z)
\psibar^j_\beta\!(0)\right> \nonumber\\
&=& \sum_ze^{-i(p+q-r)z}\sum_{x,y}e^{-i(px+qy)}
\left<\psi^i_\alpha\!(x)A^a_\mu\!(y)
\psibar^j_\beta\!(0)\right> \nonumber\\
%\lefteqn{= V\delta(p\!+\!q\!-\!r)\sum_{x,y}
%\left<S^{ij}_{\alpha\beta}(x,\!0)A^a_\mu(y)\right>e^{-i(px+qy)}} \nonumber\\
&=& V\delta(p\!+\!q\!-\!r)\left<S^{ij}_{\alpha\beta}(p)A^a_\mu(q)\right>
\nonumber \\
&=& V\delta(p\!+\!q\!-\!r)\;G^a_\mu(p,q)^{ij}_{\alpha\beta}
\eeqa
%
and the proper vertex function
%
\beq
\Lambda_\mu^{a,\rm lat}(p,q) = \left<S(p+q)\right>^{-1}
\left<S(p)A^a_\mu(q)\right>\left<S(p)\right>^{-1}\left<D(q)\right>^{-1}
\eeq
%
where the momentum-space quark propagator $S(p)$ is
%
\beq
S(p) = \sum_x e^{-ipx}S(x,0)
\eeq

$D(q)$ is the gluon propagator,
%
\beq
D^{ab}_{\mu\nu}(q) = \delta^{ab}P_{\mu\nu}(q)D(q)
\eeq
and the projector $P_{\mu\nu}(q)$ is 
\beq
P_{\mu\nu}(q) = \delta_{\mu\nu}-(1-\xi)\frac{q_{\mu}q_{\nu}}{q^2}
\eeq

In the Landau gauge ($\xi$=0), $P_{\mu\nu}$ becomes the transverse
projector $T_{\mu\nu}$, and $D(q)$ can be determined by
\beq
D(q) = \frac{1}{3(N^2_C-1)}\sum_{\mu,a} D^{aa}_{\mu\mu}(q)
\eeq

At tree level, this becomes
%
\beq
\Lambda^{a(0)}_\mu(p,q) = -ig_0 T^a T_{\mu\nu}(q)\left ( \gamma_\nu
\cos\frac{(2p+q)_\nu}{2} - i\sin\frac{(2p+q)_\nu}{2}\right )
\label{eq:tree_lat}
\eeq

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

$F_3$ may be extracted by exploiting the symmetries of the problem. 
First, we isolate $F_3$ and $F_4$
%
\beq
\Tr \gamma_\mu\Lambda_\mu(p,q=0) = 4 F_3(p^2) + 4p_\mu^2 F_4(p^2)
\label{eq:formred}
\eeq
%
where no sum over $\mu$ is implied. We can then eliminate $F_4$ by
imposing an appropriate kinematics, e.g.\ $p_\mu=0, p_\nu\neq 0$ with
$\mu\neq\nu$.

Finally, we define the renormalised coupling as
\beq
g_R(p^2) = i \Zquark(p)\Zgluon^{1/2}(p) F_3(p^2)
\label{eq:gr}
\eeq
where $\Zquark$ and $\Zgluon$ are the renormalisation constants for the quark
and gluon fields.  $\Zquark$ may be determined from (\ref{eq:Zquark-def}),
while $\Zgluon$ is defined by
%
\beq
D_{\mu\nu}(q^2)|_{q^2=p^2} = T_{\mu\nu}(p)\Zgluon(p)\frac{1}{p^2} 
\label{eq:Zgluon-def}
\eeq

From this, we can compute $g_R(q^2)$, which should be 
independent of the bare coupling $g_0^2=6/\beta$, 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}. 
It is worth noting that, since this scheme is defined in the continuum
as well as on the lattice, the matching with $g_R^{\overline{MS}}(q^2)$
can be performed entirely within continuum perturbation theory,
provided there is a region where perturbation theory is valid and
lattice artefacts are insignificant.

\subsection{Computational details}
\label{sec:vtx-boring}

All the results in this chapter have been obtained with the Wilson
gauge action at $\beta=6.0$ on a $16^3\times 48$ lattice.  The gauge
fields were generated with a Hybrid Over-Relaxed algorithm, with
configurations separated by 800 sweeps.
The quark propagators have been generated using a
tadpole improved SW fermion action, with $c_{SW}=1.479$, for one value
of $\kappa/u_0=0.1370$, which corresponds to a pseudoscalar mass of
830 MeV.

The gauge fields have been fixed to Landau gauge, using a Fourier
accelerated algorithm \cite{CTHD-fouracc} to deal with low-momentum
modes.  The Landau gauge condition has been achieved to an accuracy of
\beq
\frac{1}{VN_C}\sum_{x,\mu}\left|\partial_\mu A_\mu\right|^2 < 10^{-12}
\eeq

It is necessary to be careful when Fourier transforming the quark
propagators.  Having chosen
antiperiodic boundary conditions in the time direction,
we have to choose half-integer values for the momenta in the time
direction
\beq
p_{t}a = 2\pi(n_t+\half)/L_t \hspace{1cm} n_t=0,1,\ldots,L_t-1
\eeq
%
while the spatial momenta (and the gauge momenta in all directions) have
integer values,
\beq
p_{i}a = 2\pi n_i/L_i \hspace{1cm} n_i = 0,1,\ldots,L_i-1
\eeq
The time Fourier transformation for the quarks
cannot be performed directly using a standard FFT routine; instead,
$S(t)$ is extended onto a lattice of length $2L_t$, using
\(S(t+L_t)=-S(t)\), and picking only the odd momenta.  Alternatively, and
equivalently, we could perform a standard FFT on \(
\tilde{S}(t)=e^{-i\pi t/L_t}S(t) \).

The analysis is performed as in chapter \ref{chap3}, using least
$\chi^2$ fits and with the errors obtained from bootstrap samples. 
However, the large datasets involved give problems.  To circumvent
these, the data have been divided into 4 batches of 83 configurations
each, with 100 bootstrap samples generated for the average \( \bra
K_\mu\ket = i\Tr\gamma_{\mu}\Lambda_{\mu}(p,q) \) and the covariance
matrix \( \bra K_{\mu}(\vec{p},p_t;q) K_{\mu}(\vec{p},p_t^\prime;q)\ket
\) (so that the correlation between $K_\mu$ for different $p_t$ can be
taken into account) for each batch, with different seeds used for the
random number generator for each batch.  A combined estimate for
$K_\mu$ and $K_\mu K_\mu^\prime$ is then obtained from \[ K_\mu =
\frac{1}{4}\sum_{i=1}^{4}K_{\mu}^i \] for each bootstrap sample of
$K_\mu$, and similarly for the covariance matrix.  This is a good
estimate provided the average and covariance for each batch provide 
reasonable estimates. 

\section{Determination of $\Zquark$ and $\Zgluon$}
\label{sec:zgluon-zquark}

The quark field renormalisation constant $\Zquark$ has been determined
using equation (\ref{eq:Zquark-def}).  The results are plotted as a
function of $|pa|$ in fig.\ \ref{fig:zquark-all}.  We see that $\Zquark$
has large ambiguities.  A natural assumption is that these ambiguities
arise from violations of rotational symmetry, and this is confirmed by
fig.\ \ref{fig:zquark-select}, where it is plotted for momenta only in
certain directions.

\begin{figure}[tbp]
\centering
\ewxy{plots/zquark_all.ps}{220pt}
\vspace{-8mm}
\caption{$\Zquark$ as a function of $|pa|$, for 80 configurations}
\label{fig:zquark-all}
\end{figure}

\begin{figure}[tbp]
\begin{center}
\setlength{\unitlength}{0.95cm}
\setlength{\fboxsep}{0cm}
\begin{picture}(14,14)
\put(0,7){\begin{picture}(7,7)\put(-0.9,-0.4){\shit{8.0cm}{plots/zquark_onaxis_pt.ps}}\end{picture}}
\put(7,7){\begin{picture}(7,7)\put(-0.9,-0.4){\shit{8.0cm}{plots/zquark_onaxis_spatial.ps}}\end{picture}}
\put(0,0){\begin{picture}(7,7)\put(-0.9,-0.4){\shit{8.0cm}{plots/zquark_offaxis.ps}}\end{picture}}
\put(7,0){\begin{picture}(7,7)\put(-0.9,-0.4){\shit{8.0cm}{plots/zquark_psmall.ps}}\end{picture}}
\end{picture}
\end{center}
\vspace{-0.8cm}
\caption{$\Zquark$ as a function of $|pa|$, for 80 configurations,
selected momenta.  (a) Momenta only in the time direction; (b) momenta 
on-axis in spatial directions, with $n_t=\pm\half$; (c) off-axis momenta
only; (d) $|n_i|\le 2; |n_t|\le\frac{5}{2}$}
\label{fig:zquark-select}
\end{figure}

The main ambiguity is between on-axis and off-axis points, as we can see
by comparing fig.\ \ref{fig:zquark-select} (a)--(c).  The values for
spatial on-axis points are lower than those which are on-axis in the
time direction, because it is not possible to set $p_t=0$.  Comparing
(c) and (d), we see there is also an ambiguity at high momenta, which
accounts for the `spurs', and is almost removed entirely by selecting
only low-lying momentum values, as in (d).

These violations are primarily of $\Op(pa)$, since the SW action is
only $\Op(a)$-improved for on-shell quantities.  Following the
prescription of \cite{Heatlie}, one could `rotate' the propagators at
either end with \( (1+\frac{1}{4}(\Dslash+m_0)) \) (cf.\ sections
\ref{sec:WI-clover} and \ref{sec:ZV-tadpole}), and there are
indications that this would help.  An earlier study, using the same
(rotated) propagators as were used at $\beta=6.0$ in section
\ref{sec:ZV}, had no ambiguity between on-axis and off-axis points, but
still large high-momentum ambiguities, as figure \ref{fig:zquark-rot}
shows.  However, attempts to add these rotations by hand by replacing
$\Dslash$ with $\pslash$ have not been successful.

\begin{figure}[tbp]
\ewxy{plots/zquark_rot.ps}{220pt}
\vspace{-0.8cm}
\caption{$\Zquark$ as a function of $|pa|$, for 8 configurations,
using rotated propagators, $c_{SW}=1, \kappa=0.1432$}
\label{fig:zquark-rot}
\end{figure}

$\Zquark$ has been fitted to a phenomenological curve (to be used as
input when computing the renormalised coupling),
$\Zquark=c_0+c_1\ln(pa)+c_2(pa)^2$, using only the on-axis points in
the time direction.  The best estimates for the parameters are given in
table \ref{tab:zquark}.

\begin{table}[htbp] 
\centering 
\begin{tabular}{|c|c|}\hline 
$c_0$ & 0.960\er{8}{9} \\ 
$c_1$ & $-0.065$\err{9}{14} \\ 
$c_2$ & 0.054\er{7}{6} \\ \hline 
\end{tabular} 
\caption{Parameters for the quark field renormalisation constant
$\Zquark$, from 83 configurations, 100 bootstrap samples.}
\label{tab:zquark} 
\end{table}

The gluon field renormalisation constant $\Zgluon$ has been determined
from (\ref{eq:Zgluon-def}), and is shown as a function of $qa$ in
figure \ref{fig:zgluon-all}.  This has been fitted to a
phenomenological curve $\Zgluon(qa)=A_0+A_1\ln(qa)+A_2(qa)^2$ for
$0.45<qa<1.57$, and to a curve $\Zgluon(qa)=a_1(qa)+a_2(qa)^2+a_3(qa)^3$
for $qa<0.45$.  The best estimates for the parameters are given in table
\ref{tab:zgluon}.   A high accuracy is obtained already for 20
configurations, and only minimal violations of rotational
symmetry are observed.

\begin{table}[htbp]
\centering
\begin{tabular}{|c|c|c|c|}\hline
\multicolumn{2}{|c|}{$pa<0.45$} & \multicolumn{2}{|c|}{$pa>0.45$}\\ \hline
$a_1$ & 6.0\err{1.4}{0.9} & $A_0$ & 2.12\er{2}{3} \\
$a_2$ & 38\er{6}{9} & $A_1$ & $-2.25$\er{6}{8} \\
$a_3$ & $-70$\err{13}{9} & $A_2$ & 0.347\err{25}{20} \\
\hline
% z (small p) = ax + bx^2 + c x^3
% a = 5.96443    (+ 1.40476    - 0.885563  )
% b = 37.5444    (+ 5.52195    - 8.94382   )
% c = -70.3126   (+ 13.0975    - 8.55259   )
%chisq (small p) = 8.03 / 3
%2.11894    (+ 0.0199214  - 0.0257474 )
%-2.25494   (+ 0.0595621  - 0.0818724 )
%0.346565   (+ 0.0252528  - 0.0196616 )
%chi^2 (high p) = 339.489 / 105
\end{tabular}
\caption{Parameters for the gluon renormalisation constant $\Zgluon$,
from 20 configurations, 100 bootstrap samples.}
\label{tab:zgluon}
\end{table}

\begin{figure}[tbp]
\ewxy{plots/zgluon.ps}{250pt}
\vspace{-8mm}
\caption{$\Zgluon$ as a function of $|qa|$, for 20 configurations}
\label{fig:zgluon-all}
\end{figure}

It should be emphasised that these fits are only performed to facilitate
the computation of the running coupling $g_R(p)$, and no physical
significance should be attached to the phenomenological parameters
quoted. 

\section{The $F_3$ form factor and the running coupling}
\label{sec:vtx-results}

\subsection{The unamputated vertex}
\label{sec:vtx-full}

$\Tr\gamma_\mu G_\mu(p,q)$ is shown as a function of $p=(0,0,0,p_t)$ for
$q=0$ (with $\mu=1$) in fig.~\ref{fig:mom_vtx}.  We see that there is a
clean signal which falls off smoothly with $p$.

\begin{figure}[tbp] 
%\vspace{-14mm} 
\ewxy{plots/G1_3700_83cfg_q0000_p000t_g1.ps}{300pt}
\vspace{-0.8cm} 
\caption{$\Tr\gamma^1 G_1(p,q)$ for $q=0$, $p=(0,p_t)$, as a
function of $p_t a$, for 83 configurations.} 
\label{fig:mom_vtx} 
\end{figure}

Fig.\ \ref{fig:G1_q12} shows $i\Tr\gamma_1 G_1$ for $n_{q_y}=1$ and 2. 
For $n=1$ there is still a clear signal, although the amplitude has
fallen off from $q=0$ to $qa=\pi/8$ as it does from $p_t\sim 0$ to $p_t
a\sim\pi/8$.  However, for $n_q=2$, as fig.\ \ref{fig:G1_q12} (b) shows,
noise dominates, and insofar as there is a signal, it is negative.  This
indicates that a renormalisation scheme where $F_3$ is calculated as a
function of $q^2$, keeping $p$=0, is not feasible.

\begin{figure}[tbp] 
\begin{center}
\setlength{\unitlength}{0.95cm}
\setlength{\fboxsep}{0cm}
\begin{picture}(14,7)
\put(0,0){\begin{picture}(7,7)\put(-0.9,-0.4){\shit{8.0cm}{plots/G1_3700_83cfg_q0100_p000t_g1.ps}}\end{picture}}
\put(7,0){\begin{picture}(7,7)\put(-0.9,-0.4){\shit{8.0cm}{plots/G1_3700_83cfg_q0200_p000t_g1.ps}}\end{picture}}
\end{picture}
\end{center}
\vspace{-0.7cm}
\caption{$\Tr\gamma_1 G_1(p,q)$ for $q\ne 0$, $p=(0,p_t)$, as a
function of $p_t a$, for 83 configurations.} 
\label{fig:G1_q12}
\end{figure}

Fig.\ \ref{fig:G1-various} shows $i\Tr\gamma_1 G_1(p,q)$ for various
non-zero $p$ and $q$.  The most interesting of these are (a) and (b),
which show what happens to $\Tr\gamma_{\mu}G_{\mu}(p,q)$ when $p_\mu\ne 0$
or $q_\mu\ne 0$.  Fig.\ \ref{fig:G1-various} (a) shows that the signal
is strongly suppressed when $q_\mu\ne 0$, and (b) indicates a highly
non-trivial behaviour for $p_\mu\ne 0$.  This indicates how the form
factors for the proper vertex $F_4$--$F_7$ in (\ref{eq:formfactors})
come into play for these kinematics, something which is also manifest in
the full vertex.

Fig.\ \ref{fig:G1-various} (c) and (d) show that there is hardly any
deterioration in the signal as $p$ is increased in a perpendicular
direction.  As expected, the amplitude falls off with $p$.  As (e) and
(f) show, there is also a clear signal for $q_t\ne 0$, although we here
see a more erratic behaviour for $p_t\sim 0$, as one would expect from
fig.\ \ref{fig:G1_q12}. 
% The signal is clearest for $p\sim -q/2$.

\begin{figure}[tbp] 
\begin{center}
\setlength{\unitlength}{0.95cm}
\setlength{\fboxsep}{0cm}
\begin{picture}(14,20)
\put(0,14){\begin{picture}(7,7)\put(-0.9,-0.4){\shit{8.0cm}{plots/G1_3700_83cfg_q1000_p000t_g1.ps}}\end{picture}}
\put(7,14){\begin{picture}(7,7)\put(-0.9,-0.4){\shit{8.0cm}{plots/G1_3700_83cfg_q0000_p100t_g1.ps}}\end{picture}}
\put(0,7){\begin{picture}(7,7)\put(-0.9,-0.4){\shit{8.0cm}{plots/G1_3700_83cfg_q0000_p002t_g1.ps}}\end{picture}}
\put(7,7){\begin{picture}(7,7)\put(-0.9,-0.4){\shit{8.0cm}{plots/G1_3700_83cfg_q0000_p004t_g1.ps}}\end{picture}}
\put(0,0){\begin{picture}(7,7)\put(-0.9,-0.4){\shit{8.0cm}{plots/G1_3700_83cfg_q0003_p000t_g1.ps}}\end{picture}}
\put(7,0){\begin{picture}(7,7)\put(-0.9,-0.4){\shit{8.0cm}{plots/G1_3700_83cfg_q0006_p000t_g1.ps}}\end{picture}}
\end{picture}
\end{center}
\vspace{-0.7cm}
\caption{$\Tr\gamma_1G_1(p,q)$ for various combinations of $p$ and $q$,
as a function of $p_t$.}
\label{fig:G1-various}
\end{figure}

\subsection{The proper vertex}
\label{sec:vtx-proper}

The proper vertex has been computed as a function of $p_t$ for different
values of $\vec{p},q$ in batches of 83 configurations, as described in
section \ref{sec:vtx-boring}.  Fig.\ \ref{fig:vtx-batches} shows the
vertex for $q=0,\vec{p}=0$ for each of the 4 batches, and for the
combined estimate.  It is evident that the data are distorted by large
correlations, which are only evened out by the large number of
configurations in the total ensemble.  It also confirms that the sample
size in each batch is large enough (at least for these momenta) to give
a sufficiently reliable estimate of the vertex to justify the method
used.  But the spread between the data for different batches indicates
that using much smaller samples could invalidate the assumptions the
method is based on.

\begin{figure}[tbp] 
\begin{center}
\setlength{\unitlength}{0.95cm}
\setlength{\fboxsep}{0cm}
\begin{picture}(14,20)
\put(0,14){\begin{picture}(7,7)\put(-0.9,-0.4){\shit{8.0cm}{plots/vertex_plot_1_3700_83cfg_q0000_p000t_gamma1_00.ps}}\end{picture}}
\put(7,14){\begin{picture}(7,7)\put(-0.9,-0.4){\shit{8.0cm}{plots/vertex_plot_1_3700_83cfg_q0000_p000t_gamma1_01.ps}}\end{picture}}
\put(0,7){\begin{picture}(7,7)\put(-0.9,-0.4){\shit{8.0cm}{plots/vertex_plot_1_3700_83cfg_q0000_p000t_gamma1_02.ps}}\end{picture}}
\put(7,7){\begin{picture}(7,7)\put(-0.9,-0.4){\shit{8.0cm}{plots/vertex_plot_1_3700_83cfg_q0000_p000t_gamma1_03.ps}}\end{picture}}
\put(3.5,0){\begin{picture}(7,7)\put(-0.9,-0.4){\shit{8.0cm}{plots/vertex_plot_1_3700_332cfg_q0000_p000t_gamma1.ps}}\end{picture}}
\end{picture}
\end{center}
\vspace{-0.8cm}
\caption{$K_1(p,q)$ for $q=0$, $p=(0,p_t)$, as a
function of $p_t a$, for 83 configurations (a)--(d), and for all 332
configurations (e).} 
\label{fig:vtx-batches}
\end{figure}

In this exploratory work, one component $\Lambda_1$ of the proper vertex
has been chosen for detailed study.  Rotational invariance would imply
that the only thing we lose by this is statistics, since the other
components are expected to behave in the same fashion.  This is true
within errors, as figure \ref{fig:vtx23} shows.  We have already seen in
section \ref{sec:vtx-full} that additional form factors complicate the
picture of $\Tr\gamma_{\mu}G_{\mu}$, and therefore also $K_\mu(p,q)$,
when $p_\mu\ne 0$.  This means that we should not expect $K_4$ to behave
similarly to the other three components, since $p_4=p_t$ is
necessarily non-zero.  Fig.\ \ref{fig:vtx4} confirms this --- indeed,
apart from $p_t$ close to zero, no signal can be observed.

\begin{figure}[tbp] 
\begin{center}
\setlength{\unitlength}{0.95cm}
\setlength{\fboxsep}{0cm}
\begin{picture}(14,7)
\put(0,0){\begin{picture}(7,7)\put(-0.9,-0.4){\shit{8.0cm}{plots/vertex_plot_2_3700_83cfg_q0000_p000t_gamma2_00.ps}}\end{picture}}
\put(7,0){\begin{picture}(7,7)\put(-0.9,-0.4){\shit{8.0cm}{plots/vertex_plot_3_3700_83cfg_q0000_p000t_gamma3_00.ps}}\end{picture}}
\end{picture}
\end{center}
\vspace{-0.8cm}
\caption{$K_2(p,q)$ and $K_3(p,q)$ 
for $q=0$, $p=(0,p_t)$, as a
function of $p_t a$, for 83 configurations.} 
\label{fig:vtx23}
\end{figure}

\begin{figure}[tbp] 
\centering
\ewxy{plots/vertex_plot_4_3700_83cfg_q0000_p000t_gamma4_00.ps}{200pt}
\vspace{-0.8cm}
\caption{$K_4(p,q)$ for $q=0$, $p=(0,p_t)$, as a
function of $p_t a$, for 83 configurations.} 
\label{fig:vtx4}
\end{figure}

One would expect that $K_\mu$ is symmetric in $p\to -p$ and $q\to -q$,
at least when $p$ and $q$ are perpendicular, and $p_\mu=q_\mu=0$. 
Fig.\ \ref{fig:vtx_signsymmetry} shows that some violations of
reflection symmetry may be present, but they are all within a few
standard deviations.  It should be emphasised that these are the only
values of the momenta for which any significant violation of reflection
symmetry can be detected.  When computing the running coupling and
$\Lambda_{QCD}$, all equivalent momenta have been averaged.

\begin{figure}[tbp] 
\begin{center}
\setlength{\unitlength}{0.95cm}
\setlength{\fboxsep}{0cm}
\begin{picture}(14,20)
\put(0,14){\begin{picture}(7,7)\put(-0.9,-0.4){\shit{8.0cm}{plots/vertex_plot_1_3700_332cfg_q0000_p00-1t_gamma1.ps}}\end{picture}}
\put(7,14){\begin{picture}(7,7)\put(-0.9,-0.4){\shit{8.0cm}{plots/vertex_plot_1_3700_332cfg_q0000_p001t_gamma1.ps}}\end{picture}}
\put(0,7){\begin{picture}(7,7)\put(-0.9,-0.4){\shit{8.0cm}{plots/vertex_plot_1_3700_332cfg_q0100_p00-1t_gamma1.ps}}\end{picture}}
\put(7,7){\begin{picture}(7,7)\put(-0.9,-0.4){\shit{8.0cm}{plots/vertex_plot_1_3700_332cfg_q0100_p001t_gamma1.ps}}\end{picture}}
\put(0,0){\begin{picture}(7,7)\put(-0.9,-0.4){\shit{8.0cm}{plots/vertex_plot_1_3700_332cfg_q0-100_p00-1t_gamma1.ps}}\end{picture}}
\put(7,0){\begin{picture}(7,7)\put(-0.9,-0.4){\shit{8.0cm}{plots/vertex_plot_1_3700_332cfg_q0-100_p001t_gamma1.ps}}\end{picture}}
\end{picture}
\end{center}
\vspace{-0.8cm}
\caption{$K_1(p,q)$ for $q_y=\pm 1$, $p_z=\pm 1$, as a
function of $p_t a$, for 332 configurations. (a) $q=0, n_{p_z}=-1$ (b) $q=0,
n_{p_z}=1$ (c) $n_{q_y}=1,n_{p_z}=-1$ (d) $n_{q_y}=1, n_{p_z}=1$ (e) 
$n_{q_y}=-1, n_{p_z}=-1$ (f) $n_{q_y}=-1, n_{p_z}=1$ }
\label{fig:vtx_signsymmetry}
\end{figure}

In fig.\ \ref{fig:vtx_q0} $K_1$ is shown as a function of $|pa|$ for
$q=0$.  This is equal to the form factor $F_3$, and as the figures
show, it is a well-defined function of $p$ (within the statistical
errors) at least for $pa<1-1.2$.  Fig.\ \ref{fig:vtx_qy1} shows $K_1$
for $n_{q_y}=1$.  This shows a similar dependence on $p$ as $F_3(p)$,
although the errors are larger, as one would expect.

\begin{figure}[tbp] 
\begin{center}
\setlength{\unitlength}{1cm}
\setlength{\fboxsep}{0cm}
\begin{picture}(13.7,7)
\put(0,0){\begin{picture}(7,7)\put(-0.9,-0.4){\shit{8.2cm}{plots/vertex_q0_py.ps}}\end{picture}}
\put(7,0){\begin{picture}(7,7)\put(-0.9,-0.4){\shit{8.2cm}{plots/vertex_q0_pz.ps}}\end{picture}}
\end{picture}
\end{center}
\vspace{-0.8cm}
\caption{$F_3(p)$ as a function of $|pa|$ for $p=(0,p_y,0,p_t)$ (left) and
$p=(0,0,p_z,p_t)$ (right).}
\label{fig:vtx_q0}
\end{figure}

\begin{figure}[tbp] 
\begin{center}
\setlength{\unitlength}{0.95cm}
\setlength{\fboxsep}{0cm}
\begin{picture}(14,7)
\put(0,0){\begin{picture}(7,7)\put(-0.9,-0.4){\shit{8.0cm}{plots/vertex_allp_qy1.ps}}\end{picture}}
\put(7,0){\begin{picture}(7,7)\put(-0.9,-0.4){\shit{8.0cm}{plots/vertex_allp_qy-1.ps}}\end{picture}}
\end{picture}
\end{center}
\vspace{-0.8cm}
\caption{$K_1(p,q)$ as a function of $|pa|$ for $n_{q_y}=1$ (left) and
$n_{q_y}=-1$ (right). }
\label{fig:vtx_qy1} 
\end{figure}

Fig.\ \ref{fig:vtx_qy2} shows $K_1(q_y a=\pi/4)$ as a function of $|pa|$. 
Although any signal here is swamped by the noise, it is still possible
to detect a $p$ dependence similar to the dependence for $q=0$ and
$q=\pm 1$.

\begin{figure}[tbp] 
%\vspace{-14mm} 
\ewxy{plots/vertex_allp_qy2.ps}{250pt}
\vspace{-0.8cm} 
\caption{$K_1(p,q)$ as a function of $|pa|$ for $n_{q_y}=2$}
\label{fig:vtx_qy2} 
\end{figure}

$K_1$ is also shown as a function of $p_t a$ for several values of
$q_t\ne 0$ in fig.\ \ref{fig:vtx_qt}.  Since $p$ and $q$ are parallel in
this case, it is difficult to give a physical interpretation of this,
but it can be noted that
\begin{itemize}
\item The signal is most distinct for $p_t=-q_t/2$.  This indicates that
the leading dependence of the vertex on $p$ and $q$ is as a function of
$p+q/2$, as indicated by the tree level expression (\ref{eq:tree_lat}).
\item There is a clear signal for all values of $q_t$.  This means that
it may be feasible to choose an alternative renormalisation scheme,
using the symmetric point $q_\nu = -2p_\nu$.  This has the added
advantage that finite size effects should play no role, and it is also a
fairly commonly used scheme in perturbation theory.
\end{itemize}

\begin{figure}[tbp] 
\begin{center}
\setlength{\unitlength}{0.95cm}
\setlength{\fboxsep}{0cm}
\begin{picture}(14,14)
\put(0,7){\begin{picture}(7,7)\put(-0.9,-0.4){\shit{8.0cm}{plots/vertex_plot_1_3700_83cfg_q0003_p000t_gamma1_00.ps}}\end{picture}}
\put(7,7){\begin{picture}(7,7)\put(-0.9,-0.4){\shit{8.0cm}{plots/vertex_plot_1_3700_83cfg_q0005_p000t_gamma1_00.ps}}\end{picture}}
\put(0,0){\begin{picture}(7,7)\put(-0.9,-0.4){\shit{8.0cm}{plots/vertex_plot_1_3700_83cfg_q0008_p000t_gamma1_00.ps}}\end{picture}}
\put(7,0){\begin{picture}(7,7)\put(-0.9,-0.4){\shit{8.0cm}{plots/vertex_plot_1_3700_83cfg_q00011_p000t_gamma1_00.ps}}\end{picture}}
\end{picture}
\end{center}
\vspace{-0.7cm}
\caption{$K_1(p,q)$ for $q=q_t$, $p=(0,p_t)$, as a function of $p_t a$.}
\label{fig:vtx_qt}
\end{figure}

\subsection{The running coupling and the $\Lambda$ parameter}
\label{sec:gR}

We see from fig.\ \ref{fig:vtx_q0} that
$F_3$ is a well-defined, decreasing function of $p$, although the
uncertainties are still quite large.  Fig.\ \ref{fig:F3-restricted}
shows $F_3$ for a restricted set of momenta ($p_\nu a\le 1.2;
|pa|\le 1.5$), where equivalent momenta are averaged.  This removes
almost all the remaining ambiguities.

\begin{figure}[tbp] 
%\vspace{-14mm} 
\ewxy{plots/vertex_q0_psmall.ps}{250pt}
\vspace{-0.8cm} 
\caption{$F_3(p)$ as a function of $p$, with equivalent momenta averaged.}
\label{fig:F3-restricted} 
\end{figure}

Using the values for $\Zquark$ and $\Zgluon$ in
\ref{sec:zgluon-zquark}, we obtain $g_R(p)$, which is shown in fig.\
\ref{fig:gren}.  These results are close to those obtained from the
3-gluon vertex \cite{ggg}.  (The dip at low $p$ is entirely due to the
renormalisation factor $\Zgluon$.) We find that $\alpha_S (2{\rm GeV})=
0.27\pm 0.06\pm 0.03$, where the first error is statistical, and the
second is the systematic error from the ambiguity in $\Zquark$.  The
scale has been set to $a^{-1}=2$GeV, from the string tension.

\begin{figure}[tbp] 
\begin{center}
\setlength{\unitlength}{0.95cm}
\setlength{\fboxsep}{0cm}
\begin{picture}(14,7)
\put(0,0){\begin{picture}(7,7)\put(-0.9,-0.4){\shit{8.0cm}{plots/gren_psmall.ps}}\end{picture}}
\put(7,0){\begin{picture}(7,7)\put(-0.9,-0.4){\shit{8.0cm}{plots/gren_allp.ps}}\end{picture}}
\end{picture}
\end{center}
\vspace{-0.8cm}
\caption{$g_R(p)$ as a function of $p$ (GeV), for $p_\nu < 2.4$GeV
(left), and for a wider range of momenta (right).}
\label{fig:gren} 
\end{figure}

\label{sec:lambdaQCD}

The QCD scale parameter $\Lambda$ has been computed according to the
2-loop formula (\ref{eq:Lambda_QCD}).  The results are shown in fig.\
\ref{fig:Lambda_QCD}.  It is difficult to draw any firm conclusions from
this, since the data become very noisy at the point where 2-loop
perturbation theory might become valid.  However, it seems that
$\Lambda$ reaches a plateau at about $p$=1.5 GeV, giving an estimate for
$\Lambda_{QCD}$ of about 300--600MeV in this renormalisation scheme.

\begin{figure}[tbp] 
\begin{center}
\setlength{\unitlength}{0.95cm}
\setlength{\fboxsep}{0cm}
\begin{picture}(14,7)
\put(0,0){\begin{picture}(7,7)\put(-0.9,-0.4){\shit{8.0cm}{plots/Lambda_QCD_psmall.ps}}\end{picture}}
\put(7,0){\begin{picture}(7,7)\put(-0.9,-0.4){\shit{8.0cm}{plots/Lambda_QCD_allp.ps}}\end{picture}}
\end{picture}
\end{center}
\vspace{-0.8cm}
\caption{$\Lambda_{QCD}(p)$ (GeV) as a function of $p$ (GeV), for $p_\nu
< 2.4$GeV (left), and for a wider range of momenta (right).}
\label{fig:Lambda_QCD} 
\end{figure}

\section{Other form factors}
\label{sec:otherforms}

A useful check on the symmetries of the vertex is to see whether form
factors other than $F_3$ exhibit the behaviour expected of them
according to (\ref{eq:formfactors}).  As a preliminary exercise, I have
calculated $\Tr\Lambda_1(p,q=0)$ for varying $p_1$ and $p_4$.  At tree
level, this is equal to \(g_0 \sin p_1 \), and in general it should be
proportional to $p_1$ or $\sin{p_1}$.  As figure \ref{fig:F1} shows,
this is indeed the case, at least qualitatively.  It could also be
argued that there is a constant term of $\sim 0.02$, and a slight linear
dependence on $p_t$, but the uncertainties are so large that this cannot
be ascertained with any confidence.

I have also computed $\Tr\gamma_5\Lambda_1$ and
$\Tr\gamma_5\gamma_1\Lambda_1$, which should be zero because of parity
conservation.  This is confirmed.

\begin{figure}[tbp] 
\begin{center}
\setlength{\unitlength}{0.95cm}
\setlength{\fboxsep}{0cm}
\begin{picture}(14,20)
\put(0,14){\begin{picture}(7,7)\put(-0.9,-0.4){\shit{8.0cm}{plots/vertex_plot_1_3700_80cfg_q0000_p-100t_gamma0.ps}}\end{picture}}
\put(7,14){\begin{picture}(7,7)\put(-0.9,-0.4){\shit{8.0cm}{plots/vertex_plot_1_3700_80cfg_q0000_p100t_gamma0.ps}}\end{picture}}
\put(0,7){\begin{picture}(7,7)\put(-0.9,-0.4){\shit{8.0cm}{plots/vertex_plot_1_3700_80cfg_q0000_p-200t_gamma0.ps}}\end{picture}}
\put(7,7){\begin{picture}(7,7)\put(-0.9,-0.4){\shit{8.0cm}{plots/vertex_plot_1_3700_80cfg_q0000_p200t_gamma0.ps}}\end{picture}}
\put(0,0){\begin{picture}(7,7)\put(-0.9,-0.4){\shit{8.0cm}{plots/vertex_plot_1_3700_80cfg_q0000_p-300t_gamma0.ps}}\end{picture}}
\put(7,0){\begin{picture}(7,7)\put(-0.9,-0.4){\shit{8.0cm}{plots/vertex_plot_1_3700_80cfg_q0000_p300t_gamma0.ps}}\end{picture}}
\end{picture}
\end{center}
\vspace{-0.7cm}
\caption{$\Tr\Lambda_1(p,q)$ for $n_{p_x}=\pm 1,2,3$, as a
function of $p_t a$, for 80 configurations.}
\label{fig:F1}
\end{figure}

\section{Outlook}
\label{sec:outlook}

\subsection{Sources of error}
\label{sec:vtx-errors}

As mentioned in section \ref{sec:zgluon-zquark}, off-shell Green
functions will contain residual $\Op(a)$ errors even when they are
computed using the ($\Op(a)$-improved) SW action.  We have already seen
that this gives uncertainties of 5\% in the estimate for $\Zquark$,
from violations of rotational symmetry.  Although no such violations
are observed for the quark--gluon vertex, at least not for the
kinematics used in this study, it is not unreasonable to assume that it
also contains $\Op(a)$ errors.  Indeed this may be one reason why the
signal deteriorates as rapidly with $q$ as it does, to the extent that
for $q_{\nu}a\geq 0.75$, with $p_\nu=0$, all that can be
observed is noise.

It is slightly puzzling that $\Op(qa)$ errors are so large when
$\Op(pa)$ errors are relatively small.  One explanation of this may be
that the $\Op(pa)$ effect is cancelled when amputating with $S(p)$ on
both sides of the vertex, while $S(p+q)$ only enters once.  This is
supported by the plots (fig.\ \ref{fig:vtx_qt}) for $q_t\neq 0$, where
the errors are smallest for $p+q/2\approx 0$.

Considering the tree level expressions (\ref{eq:free-wilson}) and
(\ref{eq:tree_lat}) for the quark propagator and the vertex function, it
may be noted that the cosine term in the vertex will not contribute in
the kinematics chosen for this study.  The sine term in the quark
propagator may on the other hand give $\Op(a)$ contributions that are
not taken care of by the amputation and the determination of $\Zquark$.

Since the gluon is at zero momentum in the chosen kinematics, one would
expect finite volume effects to play an important role.  However,
studies of the 3-gluon vertex \cite{ggg} indicate that finite volume
effects are not significant, even when zero-momentum gluons are
involved.  Still, it would be desirable to perform the simulation on a
larger lattice in order to have a better resolution of the momentum in
directions other than the time direction.

The gauge is fixed to an accuracy near machine precision.  This should
be an indication that errors due to inaccurate gauge fixing are small
compared to other errors.  I have not conducted any investigation of
this kind of errors, but early studies of the quark propagator showed
that gauge dependent quantities did not change significantly when the
accuracy was changed from $10^{-6}$ to $10^{-11}$.  

Another potential source of uncertainties is Gribov copies.  One would
expect these to play some role, but studies of, among other things,
gauge dependent determinations of current renormalisation constants
\cite{Paciello} indicate that the errors due to Gribov copies are not
large, and would not affect the qualitative results in this study.

The statistical errors are quite large, even with 332 configurations. 
This is not unexpected for gauge dependent, non-zero momentum 3-point
functions.  There seems to be little prospect of improving on the
statistical errors at $\beta=6.0$.  Data taken at $\beta=6.2$ would be
expected to be less noisy, both because of self-averaging and because
the Green functions are more well-defined at weak couplings.

A problem is that the perturbative window --- where the effects of
finite lattice spacing are small, and perturbation theory is still valid
--- is small.  This means that comparisons with perturbation theory, or
with lattice results obtained using different methods, is difficult. 
Simulations at higher $\beta$ would reduce this problem as well.

\subsection{Comparisons and applications}
\label{sec:compare}

Matching these results for $g_R$ to the $\bar{MS}$ scheme, and thereby
to any other renormalisation scheme, is a fairly straightforward
calculation in continuum perturbation theory.  However, this has not
yet been done, and no such calculations, using this specific scheme,
are to my knowledge presented in literature.  Without matching to other
schemes, any comparison with other calculations or with experimental
results are fairly meaningless --- although  it is encouraging that the
results for $g_R$ are quite close to those of \cite{ggg}, where a
similar renormalisation scheme was used.

This does not affect the main aim of this study, which has been to
demonstrate the feasibility of the method, and despite the difficulties
related to being restricted to a fermion action which is only on-shell
improved, and the poor quality of the data for $q \neq 0$, this has been
achieved.  

Another potential application of this kind of study is to use the
vertex as input in Dyson--Schwinger equations.  The DSE for the quark
self-energy \( \Sigma(p)=\Sigma^\prime(p)-\Sigma^\prime(p=\mu) \) is 
% 
\beq 
\Sigma^\prime(p) = i\frac{4}{3}Z_\Lambda g^2\int\frac{d^4q}{(2\pi)^4}  
\gamma_\mu S(q)D^{\mu\nu}(p-q)\Lambda_\nu(p,q) 
\eeq 
% 
where $Z_\Lambda \Lambda_\mu(p,q)$ is the main unknown quantity.
Usually, some ansatz is made about the form of it, after invoking the
constraints of the Slavnov--Taylor identities.  The remaining
undetermined functions are determined using the DSE for the
quark--gluon vertex --- but that requires knowledge of the quark--quark
scattering kernel, and only the asymptotic form of this is known.  If
the form of $\Lambda_\mu$ could be determined from the lattice, that
would provide more solid input, or test the validity of the usual
assumptions.

The methods employed in this study can easily be extended to the study
of couplings between quarks and composite gluonic objects.  Of
particular phenomenological interest is the pomeron, which is assumed
to be a 2-gluon $J^{PC}=J^{++}$ object (with $J$ undetermined).  Work
is underway to investigate different models for this \cite{pomeron}. 
The methods are also similar to those involved in the non-perturbative
renormalisation scheme discussed in section~\ref{sec:quarkGF-renorm},
which can be used to supplement the methods used in
chapter~\ref{chap3}. It is not surprising, however that the
uncertainties in determinations of renormalisation constants using this
method \cite{quarkGF-renorm} are larger than those from the Ward
identity and ratio methods discussed there.

\subsection{Conclusions and suggestions for further work}
\label{sec:further-work}

The results presented in this chapter demonstrate that it is possible to
extract the renormalised QCD running coupling non-perturbatively by
studying the quark--gluon vertex on the lattice.  Values for
$\alpha_s$(2GeV) and $\Lambda_{QCD}$ are found that are in reasonable
agreement with results from other calculations and measurements.  This
is however only a qualitative result, since the perturbative matching to
other schemes has yet to be done.

There is also a good prospect for computing the other form factors, thus
determining non-perturbatively the complete (proper) quark--gluon
vertex.  A good signal for $F_1$ has already been found, and a more
indepth study would attempt to quantify this and the remaining form
factors.  This would be useful especially for use in Dyson--Schwinger
equations. 

It is unsatisfactory that the gluon momentum has been fixed to zero when
calculating physical quantities in this study.  It would therefore be
desirable to investigate the feasibility of using a renormalisation
scheme where $p=-r=-q/2=\mu$.  There are indications that this might
give useful results, especially from the data at $q_t\ne 0$, but other
data, for $q$ in other directions are not so encouraging.

The errors are still quite large, and apart from statistices, the most
significant contribution is $\Op(a)$ errors from the fermion action. It
is clear that in order to obtain reliable results, it is important to
bring these under control.  Repeating these calculations with off-shell
improved quark propagators would be a step in that direction. 
Reduction of the effects of finite lattice spacing would also be
achieved by performing the calculations at higher $\beta$.  This would
have the added advantage of widening the perturbative window and
(presumably) reducing statistical errors.

Although finite size effects are not expected to be large, it would be
useful to quantify them by repeating the study on a lattice of different
size.

Of more fundamental interest would be a study of the gauge dependence of
the vertex, including the effect of Gribov copies.  Since any general
covariant gauge, apart from the Landau gauge with Gribov copies, requires
evaluation of multiple gauge transformations per configuration, this is
considerably more expensive computationally, and is not a realistic
prospect for the near future.

