\chapter{Renormalisation of current operators}

\label{chap:op-renorm}
\label{chap3}

\section{Computational details}
\label{boring}

The results reported in this chapter have been obtained using the
Wilson gauge action (\ref{eq:wilson-gauge}) and SW fermion action
(\ref{eq:SW}), in the quenched approximation.  The configurations were
generated using the Hybrid Over-Relaxed algorithm, described in
\cite{qlhms}, on a $16^3\times 48$ lattice at $\beta=6.0$ and a
$24^3\times 48$ lattice at $\beta=6.2$.  At $\beta=6.2$, the
configurations were separated by 2400 sweeps, after an initial
thermalisation of 16800 sweeps.  The configurations at $\beta=6.0$ were
separated by 1200 sweeps, after a thermalisation of 10800 sweeps.

The propagators used are the same as in \cite{strange} for $\beta=6.2$
and \cite{hopping} for $\beta=6.0$, except for the tadpole improved and
heavy quark propagators in section \ref{sec:ZV}, which were generated
specifically for the purpose of this analysis.  They were all generated
using point sources and sinks.  The details of the algorithm used are
described in \cite{qlhms}.

Since we normally only compute propagators from the origin to any point
in space and time, the 3-point functions in (\ref{ratio-3pt}),
(\ref{WI-ZA-int}) and (\ref{WI-PS-int}) could not be evaluated directly
from the existing data.  Instead, `extended propagators' \(S_1(y,0) =
\sum_{x}S(y,x)\gamma_{5}S(x,0) \) and \( S_1^\prime(y,0;T) =
\sum_{\vec{x}}S(y,\vec{x},T)\gamma_{5}S(\vec{x},T,0) \) were generated
using
%
\beqa
M_{zy}S_1(y,0) & = & \gamma_5 S(z,0) \label{eq:S1-solve} \\
M_{zy}S_1^\prime(y,0) & = & \gamma_5 S(z,0)\delta_{t_zT} 
\label{eq:S1prime-solve} \\
\eeqa
The code to initialise the solver for (\ref{eq:S1-solve}) was written
specifically for this study.

The statistical analysis has been performed using minimum $\chi^2$ fits
to the timesliced data, using both correlated and uncorrelated
$\chi^2$'s.  The errors were obtained by a bootstrap procedure, using
100 bootstrap samples.

\section{The $\rho$ parameter}
\label{sec:rho}

Using the property of the Wilson propagator
\beq
S(0,x) = \gamma_5 S^{\dagger}(x,0)\gamma_5
\eeq
and the zero-momentum pseudoscalar ground state
\beq
|P\rangle = \sum_{\vec{x}} P(\vec{x},0)|0\rangle
\eeq
we find that $\rho$, defined in (\ref{eq:rhodef}), can be estimated by
\beq
\rho(t) = \frac{\partial_0\sum_{\vec{x}}S^{\dagger}(\vec{x},t;0)
\gamma_{0}S(\vec{x},t;0)}
{2\sum_{\vec{x}}S^{\dagger}(\vec{x},t;0)S(\vec{x},t;0)}
\label{eq:rhocompute}
\eeq
where $\partial_4$ is the symmetric derivative
\beq
\partial_{\mu}f(x) = \frac{f(x+\muhat)-f(x-\muhat)}{2}
\eeq

$\rho(t)$ was computed for 60 configurations at $\beta$=6.2, with
$\kappa$=0.14144 and 0.14262, and for 36 configurations at 
$\beta$=6.0, $\kappa$=0.1432.  The results are shown in figure
\ref{fig:rhoplots}.  From these plots we see that $\rho$ is constant
between $t=7$ and $t=23$ approximately.  Fitting $\rho(t)$ to a
constant between timeslices 7 and 23 also give the best values for
$\chi^2$ per degree of freedom (fitting between timeslices 7 and 22
gives a slightly better $\chi^2$ for the data at $\beta=6.0$, but this
is marginal), so these are the values quoted in table \ref{tab:rho} and
used subsequently.

\begin{figure}[t]
\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{8.5cm}{plots/rho_6.2.ps}}\end{picture}}
\put(7,0){\begin{picture}(7,7)\put(-0.9,-0.4){\shit{8.5cm}{plots/rho_6.0.ps}}\end{picture}}
\end{picture}
\end{center}
\vspace{-0.7cm}
\caption{$\rho$ as a function of $t$ (in lattice units), for $\beta=6.2$
(left) and $\beta=6.0$ (right).}
\label{fig:rhoplots}
\end{figure}

\begin{table}
\centering
\begin{tabular}{|c|c|c|c|}\hline
$\beta$ & $\kappa$ & $m_{PS}^2$ (GeV$^2$) & $\rho a$ \\ \hline
6.2 & 0.14262 & 0.208(7) & 0.00903\err{12}{18} \\
 & 0.14144 & 0.663(7) & 0.03141\err{11}{20} \\
\hline
6.0 & 0.1432 & 0.615(10) & 0.03849\err{28}{37} \\
\hline
\end{tabular}
\caption{Values of the $\rho$ parameter, in lattice units, as
a function of the quark mass.}
\label{tab:rho}\end{table}


\section{Vector current}
\label{sec:ZV}

\subsection{Standard SW action}
\label{sec:ZV-standard}

$Z_V$ was determined from eq.~(\ref{ratio-3pt}), using 10
configurations, at three values for the quark mass for $\beta=6.2$
(corresponding to $\kappa=0.14144$, $\kappa=0.14226$ and
$\kappa=0.14262$), and at $\kappa=0.1432$ for $\beta=6.0$.  The results
are presented as a function of $t$ in figs.\ref{fig:zv_tslice} and
\ref{fig:zv_tslice_6.0}.  We see that the values for $Z_V$ are roughly
independent of $t$, as one would expect from
(\ref{eq:ZV-ratio-derive}).  The best values, obtained by fitting to
timeslices 5--19, are given in table~\ref{tab:zv}.  The errors from the
variation between the timeslices are obtained from fits to 100
bootstrap samples of timeslices within this fitting range.

The results are plotted as a function of the square of the mass of the
pseudoscalar meson (proportional to the quark mass) in
fig.\ref{fig:zv_chiral}.  We see that the results show a clear (linear)
dependence on the quark mass.  Since this renormalisation scheme is not
explicitly mass independent, one would expect that the improved,
renormalised  current takes the form \cite{alpha}
\(V_R=Z_V(1+b_Vam)V\).  The parameter $b_V$ is currently not known,
even to tree level. The mass dependence is also
consistent with the expectation that the leading corrections to these
calculations should be of $O(\alpha_s m_0 a)$.  Perturbation theory at
one-loop level \cite{Gabrielli,Borrelli} with a mean-field rescaled
coupling constant $\beta^\prime=\beta/u_0^4$ gives $Z_V^L=0.842$
which is quite close to, but still incompatible with, these
non-perturbative values.  The result for $\beta$=6.0 is also in good
agreement with the values quoted in \cite{Vladikas}, which are obtained
using a slightly different method.

$Z_V^L$ has also been determined from eq.~(\ref{WI-ZA-int}), with
$\nu=\rho=0$ (using that \(\bra V_\nu(y)V_\rho(0)\ket=0 \) to simplify
the expression),  at $\kappa$=0.14144 ($\beta$=6.2) and $\kappa$=0.1432
($\beta$=6.0).  This gives $Z_V^L = 0.817\err{8}{10}$ at $\beta$=6.2,
and $Z_V^L=0.801\err{10}{8}$ at $\beta$=6.0.  Both these values are
within $2\sigma$ of the results obtained from eq.~(\ref{ratio-3pt}). 

\begin{figure}[t]
\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/zv_6.2.ps}}\end{picture}}
\put(6.8,0){\begin{picture}(7,7)\put(-0.9,-0.4){\shit{8.2cm}{plots/zv_6.2_ward.ps}}\end{picture}}
\end{picture}
\end{center}
\vspace{-0.7cm}
\caption{$Z_V^L$ as a function of $t$, using the ratio
(\ref{ratio-3pt}) (left), and for
$\kappa=0.14144$, using the Ward identity (\ref{WI-ZA-int}) (right).}
\label{fig:zv_tslice}
\end{figure}

\begin{figure}[t]
\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/zv_6.0.ps}}\end{picture}}
\put(6.8,0){\begin{picture}(7,7)\put(-0.9,-0.4){\shit{8.2cm}{plots/zv_6.0_ward.ps}}\end{picture}}
\end{picture}
\end{center}
\vspace{-0.7cm}
\caption{$Z_V^L$ as a function of $t$, for $\beta=6.0$,
$\kappa=0.1432$, using the ratio (\ref{ratio-3pt}) (left), and
using the Ward identity (\ref{WI-ZA-int})  (right).}
\label{fig:zv_tslice_6.0}
\end{figure}

\begin{table}
\centering
\begin{tabular}{|c|c|c|c|}\hline
$\beta$ & $\kappa$ & $m_{PS}^2$ & $Z_V^L$ \\ \hline
6.2 & 0.14262 & 0.208 & 0.82139\err{41}{12}\err{25}{25} \\
& 0.14226 & 0.341 & 0.82453\err{24}{22}\err{24}{23} \\
& 0.14144 & 0.663 & 0.83136\err{23}{16}\err{22}{23} \\
\hline
6.0 & 0.1432 & 0.615 & 0.81683\err{45}{33} \\
\hline
\end{tabular}
\caption{Values of the renormalisation constant $Z_V^L$ as
a function of the quark mass.
The first set of errors are the statistical errors, while
the second set are the errors due to the variation between the
timeslices.}
\label{tab:zv}\end{table}

\begin{figure}
\setlength{\epsfxsize}{0.5\textwidth}
\centerline{\epsfbox{plots/zv_chiral.ps}}
\vspace{-0.7cm}
\caption{$Z_V^L$ as a function of the mass of the pseudoscalar meson. 
The heavier point at $\beta$=6.0, taken from \cite{MPSV}, is included for
comparison.}
\label{fig:zv_chiral}
\end{figure}

\subsection{Tadpole improved action}
\label{sec:ZV-tadpole}

In an attempt to investigate the effect and viability of tadpole
improvement, $Z_V$ was computed for mesons composed of two heavy quarks
and of one heavy and one light quark, for a `standard' and a tadpole
improved SW action at $\beta=6.0$.  The tadpole improvement was
implemented by replacing $U_\mu(x)$ with $U_\mu(x)/u_0$, with
$u_0=0.87779$, before computing the propagators.

It was expected that tadpole improvement would bring about better
agreement between non-perturbative perturbative results.  In
particular, since to one-loop order $Z_V$ is supposed to be
mass-independent, tadpole improvement should reduce
the mass dependence observed here and in \cite{UKQCD-IW}.

Another effect of tadpole improvement should be that the current quark
mass used in the definition of improved operators is more well-defined. 
If we write 
\beq
\hat{\Gamma} = (1+(1-z)ar\tilde{m}_0)
\left(1+z\frac{ar}{2}\Dslashbak\right)
\Gamma\left(1-z\frac{ar}{2}\Dslash\right)
\label{eq:rotate}
\eeq
the equations of motion can be used to eliminate either $\tilde{m}_0
(z=1)$ or $\Dslash (z=0)$.  The latter would be considerably cheaper
computationally, but depends on a reliable estimate of the `bare'
(subtracted) quark mass $\tilde{m}_0$.  Using a tadpole improved
action will bring $\kappa_c$ closer to its tree-level value of $1/8$,
and tadpole perturbation theory could then provide an estimate of
$\tilde{m}_0$, or the dependence of quantities like $\tilde{Z_V}$ on
$\tilde{m}_0$. 

In this study, \(\tilde{Z_V}=\tilde{C}^{2pt}/\tilde{C}^{3pt}\), where
$\tilde{C}^{2pt,3pt}$ are the unrotated (unimproved) correlators, is
computed, and $Z_V^{z=0}$ is obtained from
\beq
Z_V^{z=0} = (1+am_p)\tilde{Z_V}
\eeq
where \(m_p=\ln(1+m_0-m_c)=\ln(1+\frac{1}{2\kappa}
-\frac{1}{2\kappa_c})\) is the (heavy) quark pole mass.

For the data without tadpole improvement, $\kappa$ values of 0.133,
0.125 and 0.111 were used for the heavy quarks.   For the tadpole
improved data, $\kappa=0.11411, 0.1095$ and 0.096 were used for the
heavy quarks, and  $\kappa=0.1210$ and 0.1218 for the light quarks. 
The lightest quark was only used to create degenerate light mesons, in
order to obtain an estimate for $\kappa_c$ --- all the
heavy--light mesons had $\kappa_l=0.1210$.  The pseudoscalar
mass was extracted using uncorrelated fits to timeslices 10 to 19 (5 to
19 for the tadpole improved $z=1$ data).  Using the data from the two
light quarks, the critical hopping parameter was found to be
$\kappa_c=0.12235$. 

The results are presented in tables \ref{tab:zv_heavy_notad} and
\ref{tab:zv_tadpole}.
All the rotated results are obtained with a sample of 10
configurations. The heavy--heavy and light--light unrotated results are
also obtained with 10 configurations, while the heavy--light results
are from 15 configurations.  All the data are shown as a function of the
heavy quark pole mass $m_p$ in fig.\ref{fig:zv_chiral_tadpole}. 


\begin{figure}[tp]
\begin{center}
\setlength{\unitlength}{1cm}
\setlength{\fboxsep}{0cm}
\begin{picture}(13.7,19)
\put(0,13){\begin{picture}(7,7)\put(-0.9,-0.4){\shit{8cm}{plots/ZV_plot_ratio_4320_2500_10cfg_5_19_00.ps}}\end{picture}}
\put(7,13){\begin{picture}(7,7)\put(-0.9,-0.4){\shit{8cm}{plots/ZV_plot_ratio_2500_2500_10cfg_5_19_00.ps}}\end{picture}}
\put(0,6.5){\begin{picture}(7,7)\put(-0.9,-0.4){\shit{8cm}{plots/ZV_plot_ratio_2100_950_10cfg_5_19_00.ps}}\end{picture}}
\put(7,6.5){\begin{picture}(7,7)\put(-0.9,-0.4){\shit{8cm}{plots/ZV_plot_ratio_950_950_10cfg_5_19_00.ps}}\end{picture}}
\put(0,0){\begin{picture}(7,7)\put(-0.9,-0.4){\shit{8cm}{plots/ZV_plot_norot_2100_0950_15cfg_5_19_00.ps}}\end{picture}}
\put(7,0){\begin{picture}(7,7)\put(-0.9,-0.4){\shit{8cm}{plots/ZV_plot_norot_0950_0950_10cfg_5_19_00.ps}}\end{picture}}
\end{picture}
\end{center}
\vspace{-0.8cm}
\caption{The vector current renormalisation constant as a function of
$t$, for $\beta=6.0$, heavy--light mesons (left) and heavy--heavy
mesons (right).  The top figures are data without tadpole improvement,
the middle figures for tadpole improved data with $z=1$, and the bottom
figures show $\tilde{Z_V}$ for tadpole improved data with $z=0$.}
\label{fig:zv_tadpole}
\end{figure}

\begin{figure}[t]
\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{8.5cm}{plots/zv_tadpole_rot_light.ps}}\end{picture}}
\put(7,0){\begin{picture}(7,7)\put(-0.9,-0.4){\shit{8.5cm}{plots/ZV_plot_norot_2100_2100_10cfg_5_19_00.ps}}\end{picture}}
\end{picture}
\end{center}
\vspace{-1cm}
\caption{$Z_V^L$ as a function of $t$ for rotated (left) and unrotated
(right) light mesons.}
\label{fig:zv_light_tadpole}
\end{figure}

\begin{table}[htb]
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline
 & \multicolumn{2}{|c|}{Heavy--heavy} & \multicolumn{2}{|c|}{Heavy--light} \\
\cline{2-5}
$\kappa_h$ & $m_{PS}a$ & $Z_V$ & $m_{PS}a$ & $Z_V$ \\ \hline
0.133 & 1.020\er{5}{4} & 0.9022\er{4}{5} & 0.733\err{10}{6} & 0.8921\er{6}{7} \\
0.125 & 1.409\er{4}{4} & 0.9602\er{3}{5} & 0.947\err{10}{7} & 0.9458\er{6}{8} \\
0.111 & 2.013\er{3}{3} & 1.0391\er{4}{3} & 1.274\err{11}{8} & 1.0178\err{11}{15} \\
\hline
\end{tabular}
\end{center}
\caption{$Z_V$ for heavy--heavy and heavy--light mesons, using the
standard SW action.}
\label{tab:zv_heavy_notad}
\end{table}

\begin{table}
\centering
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
& & \multicolumn{3}{|c|}{$z=0$}&\multicolumn{2}{|c|}{$z=1$} \\ \cline{3-7}
$\kappa_1$ & $\kappa_2$ & $m_{PS}a$ & $\tilde{Z_V}$ & $Z_V$ & $m_{PS}a$ & $Z_V$ \\ \hline
0.12100 &0.12100 & 0.334(8) & 0.8957\er{3}{4} & 0.8576\er{3}{4} & 0.332(7) & 0.9437\er{4}{5} \\
0.12180 &0.12180 & & & & 0.21(1) & 0.9367\er{8}{7}\\
\hline
0.12100 &0.11411 & 0.665(8) & 1.133\er{5}{6} & 0.900\er{5}{6} & 0.671(8) & 0.964\er{2}{2} \\
& 0.10950 & 0.834(8) & 1.360\er{2}{3} & 0.977\er{2}{3} & 0.841(9) & 1.016\er{1}{1} \\
 & 0.09600 & 1.24(1) & 2.054\er{6}{6} & 1.172\er{6}{6} & 1.25(1) & 1.079\er{2}{3} \\
\hline
0.11411 &0.11411 & 0.932(7) & 1.179\er{1}{2} & 0.9365\err{11}{14} & 0.933(6) & 1.0010\er{5}{6} \\
0.10950 &0.10950 & 1.237(5) & 1.385\er{1}{1} & 0.9953\err{12}{10}& 1.238(4) & 1.0334\er{4}{6} \\
0.09600 & 0.09600 & 1.982(3) & 2.107\er{1}{1} & 1.2023\err{10}{11} & 1.983(3) & 1.1016\er{4}{7} \\
\hline
\end{tabular}
\caption{$Z_V$ for light--light, heavy--light, and heavy--heavy
mesons, using rotated  and unrotated
tadpole improved propagators.  The errors are purely statistical.}
\label{tab:zv_tadpole}
\end{table}


\begin{figure}
\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{8.5cm}{plots/chiral_tadpole_compare.ps}}\end{picture}}
\put(7,0){\begin{picture}(7,7)\put(-0.9,-0.4){\shit{8.5cm}{plots/chiral_rotate_compare.ps}}\end{picture}}
\end{picture}
\end{center}
\vspace{-0.6cm}
\caption{$Z_V^L$ as a function of the heavy quark pole mass, comparing
tadpole improved with unimproved data (left), and $z=0$ with $z=1$
(right).}
\label{fig:zv_chiral_tadpole}
\end{figure}

As fig.\ref{fig:zv_chiral_tadpole} shows, tadpole improvement does
reduce the mass dependence of $Z_V$, although there is still a
significant mass dependence left.  This is not surprising, since the SW
action is only tree level improved, and $\Op(\alpha_s ma)$ errors may
still be significant.  Also, the mass dependence of the
$\Op(a)$-improved vector current is not known at tree level, as mentioned
previously. 

Using the two lightest quark masses, $Z_V(m=0)$ is found to be 0.932
for the tadpole improved action with $z$=1.  This agrees poorly with
the perturbative value \cite{ns} of 0.99, but the ratio
$Z_V^{z=0}/Z_V^{z=1}$ is in good agreement with perturbation theory.

The comparison of the data at $z=0$ and $z=1$ is inconclusive, but
indicates that it is possible to use either prescription.  A more
recent and detailed study of these mass effects can be found in
\cite{par}. 

\section{Axial current}
\label{sec:ZA}

The axial vector renormalisation constant $Z_A^L$ is determined using
eq.~(\ref{WI-ZA-int}), with $\nu=\rho=i$ and summing over $i=1,2,3$,
using the values for $Z_V^L$ quoted in table~\ref{tab:zv} as input. 
The results are obtained at $\beta$=6.2, using 60 configurations for
$\kappa$=0.14144 and 26 configurations for $\kappa$=0.14262, and at
$\beta$=6.0 using 36 configurations for $\kappa$=0.1432. The results
are plotted against $t$ in fig.\ref{fig:za6.2} for $\beta$=6.2, and in
fig.\ref{fig:za6.0} for $\beta$=6.0.  We see that, apart from the
effect of the contact terms on the first few timeslices, they show
virtually no dependence on $t$, especially for $\kappa$=0.14144.  Since
the Ward identities are time independent, this is to be expected.  For
the two other $\kappa$ values, the data becomes unreliable at high $t$.
 The best estimates are obtained from fitting to timeslices 7--22 for
$\kappa$=0.14144 and to timeslices 6--16 for $\kappa$=0.14262 and
0.1432. The results are given in table~\ref{tab:za}. 

\begin{figure}[t]
\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{8.5cm}{plots/za_6.2_4144.ps}}\end{picture}}
\put(7,0){\begin{picture}(7,7)\put(-0.9,-0.4){\shit{8.5cm}{plots/za_6.2_4262.ps}}\end{picture}}
\end{picture}
\end{center}
\vspace{-0.7cm}
\caption{$Z_A^L$ as a function of $t$ for $\kappa=0.14144$ (left) and
$\kappa=0.14262$ (right).}
\label{fig:za6.2}
\end{figure}

\begin{figure}
\setlength{\epsfxsize}{0.5\textwidth}
\centerline{\epsfbox{plots/za_6.0_4320.ps}}
\vspace{-0.7cm}
\caption{$Z_A^L$ as a function of $t$ for $\kappa=0.1432, \beta=6.0$}
\label{fig:za6.0}
\end{figure}

\begin{table}
\centering
\begin{tabular}{|c|c|c|c|c|}\hline
$\beta$ & $\kappa$ & $m_{PS}^2$ & $Z_A^L$ & $Z_P/Z_S$ \\ \hline
6.2 & 0.14262 & 0.208 & 1.040\err{10}{9} & 0.693\err{28}{40} \\
& 0.14144 & 0.663 & 1.047\er{8}{8} & 0.649\er{9}{8} \\ \hline
6.0 & 0.1432 & 0.615 & 1.097\err{19}{17} & 0.607\err{13}{28} \\
\hline
\end{tabular}
\caption{Values of the renormalisation constants $Z_A^L$ and $Z_P/Z_S$
as functions of the quark mass.}
\label{tab:za}\end{table}

Within the statistical errors, these results show no dependence of
$Z_A$ on the quark mass.  This is confirmed by simulations at
$\beta$=6.0 \cite{rome-WI}.  The comparison with results from
perturbation theory is more interesting: one-loop calculations with a
mean-field rescaled coupling constant give $Z_A\approx 0.97$, which is
considerably lower than these non-perturbative results.  The
discrepancy is higher at lower $\beta$, as expected, and the value for
$\beta$=6.0 is consistent with the value of 1.09 quoted in \cite{MPSV}
(and within $2\sigma$ of the updated value \cite{Paciello} of 1.06(2)).

\section{Pseudoscalar and scalar densities}
\label{sec:ZP}

The ratio of pseudoscalar to scalar renormalisation constant is
determined from (\ref{WI-PS-int}), using the same ensembles as for
$Z_A$.  $Z_P/Z_S$ is plotted as a function of $t$ in
figs.\ref{fig:zp6.2} and \ref{fig:zp6.0}.  For $\beta$=6.2,
$\kappa$=0.14144 a fit to timeslices 7--24 was used, while for
$\kappa$=0.14262 the fit was to timeslices 16--24, and at $\beta$=6.0,
$\kappa$=0.1432, a fit range of 8--20 was used. The best estimates
are given in table~\ref{tab:za}.  The uncertainty in these results is
too large to determine whether there is any dependence on the quark
mass.  Perturbative calculations with a mean-field rescaled
coupling constant give $Z_P/Z_S$=0.70.  As can be seen, the result
for the heavier quark mass (which is the more accurate) is slightly
lower than this, while the lighter quark mass gives a value compatible
with perturbative results (although the errors here are still quite
large).  At $\beta$=6.0, the perturbative estimate is $Z_P/Z_S$=0.68,
confirming that, as in the case with $Z_A$, the discrepancy decreases
with increasing $\beta$.

\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{8.5cm}{plots/zp_6.2_4144.ps}}\end{picture}}
\put(7,0){\begin{picture}(7,7)\put(-0.9,-0.4){\shit{8.5cm}{plots/zp_6.2_4262_allt.ps}}\end{picture}}
\end{picture}
\end{center}
\vspace{-0.7cm}
\caption{$Z_P/Z_S$ as a function of $t$ for $\kappa=0.14144$ (left) and
$\kappa=0.14262$ (right).}
\label{fig:zp6.2}
\end{figure}

\begin{figure}
\setlength{\epsfxsize}{0.6\textwidth}
\centerline{\epsfbox{plots/zp_6.0_4320.ps}}
\vspace{-0.7cm}
\caption{$Z_P/Z_S$ as a function of $t$ for $\kappa=0.1432, \beta=6.0$}
\label{fig:zp6.0}
\end{figure}

\section{Effect on decay constants}
\label{sec:decayconst}

Table~\ref{tab:decayconsts} shows how the values for the decay
constants reported in \cite{strange}, for $\beta$=6.2, change when we
use the results given above for the renormalisation constants.  For
$Z_V$, the values in table~\ref{tab:zv} have been extrapolated to
the limit of zero quark mass, giving $Z_V=0.817(2)$.  When
determining $f_{K^*}$ and $f_\phi$, an additional uncertainty, due to
the quark mass dependence, of $\pm 0.008$, is added, corresponding to
the difference between the values at the largest quark mass and in the
chiral limit.  For $Z_A$, a best estimate is obtained by combining the
results at the two $\kappa$-values, of $Z_A=1.045\err{10}{14}$, with
the errors corresponding to the spread between the highest and lowest
estimate.  We see that all the decay constants move closer to the
experimental values, but that a significant discrepancy still remains,
especially for $f_\phi$ and $f_K$.  This may be partly related to
uncertainties in the determination of $\kappa_s$. $f_\pi$ turns out to
be about $3\sigma$ away from its experimental value.  The APE
collaboration has found $f_{\pi}/(m_{\rho}Z_A) = 0.186(20)$ at
$\beta=6.2$ \cite{APE2}, which gives a value for $f_\pi/m_\rho$
compatible with experiment.  Recently obtained results with a tadpole
improved action \cite{sinead} give $f_\pi/(Z_Am_\rho)$=0.162\er{5}{4}.
Using the perturbative value for $Z_A$, which for the tadpole improved
action is 0.93 at $\beta=6.2$, this still gives a value for $f_\pi/m_\rho$ of
0.151\er{5}{4} which is $4\sigma$ lower than the experimental value.
It is possible that a non-perturbative determination of $Z_A$ will
improve on this, although it is again unlikely that it will bridge the
gap between the lattice estimate and experiment.

%%{{{  tab:decayconsts (from the "strange" paper)
\begin{table}
\begin{center}
\begin{tabular}{|c||c|c|c|}
\hline
 & old estimates & updated estimates & experiment \\
\hline
$f_\pi$           & 102\er{6}{7} MeV  & 110\er{7}{8} MeV   & 132 MeV \\
$f_K$             & 123\er{5}{6} MeV  & 133\er{7}{7} MeV   & 160 MeV \\
\hline
$1/f_\rho$        & 0.316\err{7}{13}  & 0.311\err{7}{13}   & 0.28 \\
$1/f_{K^*}$       & 0.298\er{5}{9}    & 0.293\er{5}{9}\er{1}{1}     &  \\
$1/f_\phi$        & 0.280\er{3}{6}    & 0.276\er{3}{6}\er{1}{1}     & 0.23 \\
\hline
$f_\pi/m_\rho$    & 0.138\er{6}{9}    & 0.149\err{6}{10}   & 0.172 \\
$f_K/m_\rho$      & 0.160\er{7}{8}    & 0.172\er{8}{9}     & 0.208 \\
$f_K/m_{K^*}$     & 0.144\er{4}{6}    & 0.155\er{5}{7}     & 0.179 \\
\hline
\end{tabular}
 %
\caption{Values of decay constants in physical units, using perturbative
and non-perturbative values for the renormalisation constants.  The
second set of errors in the vector meson decay constants are systematic
uncertainties due to the quark mass dependence of $Z_V$.}
 %
\label{tab:decayconsts}
\end{center}
\end{table}
%%}}}

All the work reported in this chapter was performed two years or more
ago.  Since then, there have been a number of new developments, both
theoretically and in algorithms and computer power.  This means that
the results reported here are to a large extent already out of date,
and must be updated with new results for the new actions that are now
being used.
