A nonperturbative determination of $C_A$ and the scaling of $f_\pi$ and $m^{\overline{MS}}$
UKQCD Collaboration
We calculate the $O(a)$ improvement coefficient for the axial-vector current using the nonperturbative method proposed by the LANL group. Results for the coefficient in the range $\beta=5.93$ to 6.2 are presented. We find $c_A$ is close to the 1-loop tadpole-improved perturbative value. In addition, scaling of the pseudoscalar decay constant and renormalised quark mass is improved compared to that obtained using the (larger) $c_A$ values obtained by the ALPHA collaboration.
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\hyphenation{author another created financial paper re-commend-ed}

\title{A nonperturbative determination of $C_A$ and the scaling of $f_\pi$ and
$m^{\overline{MS}}$} \author{UKQCD Collaboration, presented by S.~Collins\address{Dept. of Physics and Astronomy,
    Glasgow University, Glasgow, G12 8QQ, Scotland}\thanks{In collaboration with C.~Davies~(Glasgow University), G.~P.~Lepage~(Cornell University),
and J.~Shigemitsu~(Ohio-State University)}}

\begin{document}

\begin{abstract}
We calculate the $O(a)$ improvement coefficient for the axial-vector
current using the nonperturbative method proposed by the LANL group.
Results for the coefficient in the range $\beta=5.93$ to 6.2 are
presented. We find $c_A$ is close to the 1-loop tadpole-improved
perturbative value. In addition, scaling of the pseudoscalar decay
constant and renormalised quark mass is improved compared to that
obtained using the (larger) $c_A$ values obtained by the ALPHA
collaboration.
\end{abstract}

\maketitle

\section{Introduction}
Symanzik improvement offers reduced discretisation errors for masses
and matrix elements at finite lattice spacing but requires the
calculation of the coefficients which accompany the improvement terms.
In the light quark regime these coefficients can be calculated
nonperturbatively by imposing the chiral Ward identities on the
lattice.

$c_A$ is the $O(a)$ improvement coefficient for the axial-vector
current.
\begin{equation}
A_4 \rightarrow A_4^I = A_4 + ac_A\partial_4 + O(a^2)
\label{imp}
\end{equation}
It has been calculated nonperturbatively by the ALPHA
collaboration~\cite{alpha} and LANL group~\cite{lanl}. The results are
summarized in figure~\ref{ca}. There is a significant discrepancy in
the values of $c_A$ at smaller $\beta$. This might be accounted for by
the $O(a)$ ambiguity that exists in $c_A$ when it is determined
nonperturbatively.  Nevertheless, larger values of $c_A$ lead to
stronger lattice spacing dependence of $f_\pi$ and the quark mass, and
increase the difficulty of performing a continuum extrapolation. In
addition, the ALPHA results have a very different $\beta$ dependence
compared to the 1-loop perturbative values, which suggests they
contain a large nonperturbative contribution. Our aim was to
investigate how well $c_A$ is determined using the LANL method, which
only requires conventional analysis, rather than Schr\"odinger
functional techniques of the ALPHA collaboration.

$c_A$ is determined nonperturbatively using the PCAC relation. In
euclidean space, for zero momentum
\begin{equation}
<\partial_4A_4(x)J> = 2m_{PCAC}<P(x)J>\label{pcac}
\end{equation}
where $m_{PCAC}$ is the PCAC quark mass, $P=\bar{\psi}\gamma_5\psi$
and $J$ is any operator with the pseudoscalar quantum numbers.
Defining $r_J(t)=\frac{<\partial_4 A_4(t) J>}{<P(t)J>}$,
equation~\ref{pcac} becomes on the lattice
\begin{eqnarray}
 r_J(t) = m(t) &=& 2m_{PCAC}+O(a).
\end{eqnarray}
The size of the $O(a)$ terms depends on the number of states
contributing to $r_J(t)$. At early times~($t_{ex}$), the significant
contribution from excited states means the $O(a)$ term is larger than
at later times when the ground state dominates~($t_{gs}$). The
improvement of equation~\ref{imp} removes the $O(a)$ terms for all
states and hence the two masses
\begin{eqnarray}
r_J(t_{gs}) + ac_As_J(t_{gs}) & = & m_{imp}(t_{gs}) = \nonumber \\ & & 2m_{PCAC} + O(a^2).\\
r_J(t_{ex}) + ac_As_J(t_{ex}) & = & m_{imp}(t_{ex}) = \nonumber \\ & & 2m_{PCAC} + O'(a^2).
\end{eqnarray}
where $s_J(t) = \frac{<\partial_4^2 P(t) J>}{<P(t)J>}$, are equal
up to higher order terms. We can use these equations to solve for $c_A$:
\begin{eqnarray}
 -\frac{1}{a} \frac{r(t_{ex}) - r(t_{gs})}{s(t_{ex})-s(t_{gs})} & \equiv & c_A \label{ratca}
\end{eqnarray}
The LANL method involves performing a fit to
$r_J(t)+ac_As_J(t)=constant=2m$ over the fitting range $t_{ex}$ to
$t_{gs}$ and is numerically equivalent to equation~\ref{ratca}; $c_A$
and $2m$ are extracted from the fit. The advantage of performing a fit
is that one can test the ansatz with the $\chi^2$.

The $O(a)$ ambiguity in $c_A$ is due to $O(a^2)$ terms in the
axial-vector current, the pseudoscalar current and the light quark
action. We define $c_A$ in the chiral limit, but at finite quark mass
there is an additional source of $O(a)$ error if standard, symmetric,
$O(a^2)$ lattice temporal derivatives are used. This error can be
removed by using improved $O(a^4)$ lattice derivatives. While, both
choices of lattice derivative lead to consistent values for $c_A$ in
the chiral limit, we found it advantageous to use improved derivatives
as this weakened the dependence of $c_A$ on the quark mass and made
the chiral extrapolation easier. We also found a wider window in
$t_{ex}$ from which $c_A$ can be extracted.

\begin{figure}
\begin{center}
\includegraphics[width=5.0truecm,angle=-90]{all4.ps}
\vspace{-1cm}
\caption{Summary of $c_A$ determinations. LANL-2 and LANL-3 refer
to $c_A$ determined using 2-point and 3-point lattice derivatives
respectively, as described in reference~\protect\cite{lanl}.}
\label{ca}
\end{center}
\vskip -0.75truecm
\end{figure}

\section{Results for $c_A$}
We calculated $c_A$ on the UKQCD Collaboration quenched data set, at
$\beta=5.93$, $6.0$ and $6.2$; simulation details can be found
in~\cite{ukqcd}. The best analysis was possible at
$\beta=5.93$.

\begin{table}
\begin{center}
\begin{tabular}{cc|c}\hline
$t_{ex}$ & $\kappa=0.1327$ & $\kappa=\kappa_c$ \\\hline
\multicolumn{3}{c}{$FL$}\\\hline
4  & -0.037(2)& -0.050(3) \\
5  & -0.035(4) & -0.052(8) \\
6  & -0.029(9) & -0.022(19) \\
7  & -0.025(15) & - \\
8 &  -0.066(264) & -\\\hline
\multicolumn{3}{c}{$LL$}\\\hline
6  & -0.024(6) & -0.032(14) \\
7  & -0.022(11) & - \\
8  & -0.046(30) & - \\\hline
\end{tabular}
\caption{$c_A$ values extracted for $\beta=5.93$ at finite
quark mass and extrapolated to the chiral limit. For all results shown
$Q$ for the fit is $>0.01$. At $\kappa_c$ and higher $t_{ex}$ the
statistical errors dominate and the results are not
shown. $t_{gs}=14$.}
\label{tab}
\end{center}
\vskip -0.75truecm
\end{table}

Table~\ref{tab} presents the results for $c_A$ at $\beta=5.93$ for
``fuzzed'' source~\cite{ukqcd} and $LL$ correlators, where improved
lattice derivatives have been used. We see that there is agreement
between the results from different smearings and the results are
stable with $t_{ex}$. However, there is only a small window of
timeslices for which $Q$ for the fit is reasonable and before noise
dominates. No significant change in $c_A$ was found when $t_{gs}$
was varied.

We take $c_A=-0.032(14)$ as our final value for this $\beta$. In
choosing a value with a conservative error we aim to take into account
the residual $O(a)$ ambiguity in $c_A$ and some of the other
systematic errors.  In principle all results in the table are valid
estimates of $c_A$. However, choosing a result with a much smaller
error would require much more care in keeping the improvement
conditions applied fixed when $c_A$ is recalculated at other $\beta$s;
thus ensuring, for example, any residual $O(a)$ error in $c_A$
vanishes when the continuum limit is taken. In particular, with the
LANL method one would need to keep the relative contribution of ground
state to excited states at $t_{ex}$~(in physical units) constant and
this requires tuning of the smearing function, which was
not possible in this study.

Repeating the analysis at $\beta=6.0$ we found evidence of significant
finite volume dependence. $c_A$ decreased from $-0.039(6)$ at
$\kappa=0.13344$ on a $16^3$ lattice to $-0.003(10)$ on a $32^3$
lattice. The $16^3$ lattice is $16\%$ smaller in extent than that used
at $\beta=5.93$. Due to problems with chiral extrapolations, our final
results as a function of $t_{ex}$ either had too small a statistical
error to reflect the change in volume, or had too large an error to be
useful. Hence, we disregard the results at this $\beta$ in the later
analysis. At $\beta=6.2$, the lattice size is only $6\%$ smaller, and
we obtain $c_A=-0.031(5)$. We assume the latter error is sufficient to
compensate for the small change in volume from $\beta=6.2$ to $5.93$.


\section{Scaling of $f_\pi$ and $m^{\overline{MS}}$}
\begin{figure}
\begin{center}
\epsfxsize=6.0truecm\epsffile{fscalb.ps}
\vspace{-1cm}
\caption{Scaling of the renormalised $PS$ decay constant in units of $r_0$.}
\label{fscal}
\end{center}
\vskip -0.75truecm
\end{figure}

We calculate the renormalised pseudoscalar decay constant, given by
\begin{eqnarray}
f^{ren}  =  Z_A(1+ab_Am_q)(f^{(0)}+ac_Af^{(1)})& &\\
f^{(0)}=<0|A_4|P>,  f^{(1)}=<0|\partial_4P|P> & &
\end{eqnarray}
using our results for $c_A$ and nonperturbative determinations of
$Z_A$ and the perturbative results for $b_A$; $f^{ren}$ is extracted
at a reference mass $(r_0M_{PS})^2=3.0$. This is compared in
figure~\ref{fscal} with the results using the ALPHA collaboration
values for $c_A$. We see that scaling is much better when our smaller
values of $c_A$ are used. The simultaneous fit to all 4 points only
has a marginal $\chi^2=2.4$. More points and higher orders in $a$
would be needed to include the ALPHA $c_A$ values for $f^{ren}$.


\begin{figure}
\begin{center}
\epsfxsize=6.0truecm\epsffile{mscalb.ps}
\vspace{-1cm}
\caption{Scaling of the renormalised quark mass in units of $r_0$.}
\label{mscal}
\end{center}
\vskip -0.75truecm
\end{figure}


The renormalised quark mass, at the same reference mass, was
calculated using the method of reference~\cite{squark}. The
renormalisation-group invariant mass is computed from the PCAC mass using
the nonperturbatively determined renormalisation factor $Z_M$:
$\hat{M}=Z_Mm_{PCAC}^{imp}$ The $\overline{MS}$ mass at $2$~GeV is
then obtained using 4-loop perturbation theory. Figure~\ref{mscal}
shows that the scaling is better for smaller values of $c_A$; the
$\chi^2$ for the combined fit is $3.3$.


\section{Conclusions}
We implemented the LANL method for determining $c_A$ in the range
$\beta=5.93$ to $6.2$. Figure~\ref{ca} shows our results are
consistent with those of the LANL group. In particular, $c_A$ does not
change rapidly with $\beta$ and is close to the 1-loop
tadpole-improved value~(using $\alpha_P(1/a)$). At $\beta=6.0$,
evidence of a finite volume dependence in $c_A$ indicated care must be
taken in keeping the physical volume in the simulation fixed when
changing $\beta$ with this method.

The scaling of both $f_\pi$ and $m^{\overline{MS}}$ are improved using
our, smaller, values of $c_A$ compared to the values obtained by the
ALPHA collaboration.

S.~Collins acknowledges the support of a Royal Society of Edinburgh
fellowship. 
%In addition, this work has been supported by the European
%Community's Human potential programme under HPRN-CT-2000-00145
%Hadrons/LatticeQCD???

\begin{thebibliography}{9}
\bibitem{alpha} ALPHA collaboration, M. L\"uscher et. al., Nucl. Phys. B491 323 (1997).
\bibitem{lanl} T. Bhattacharya et. al., Phys. Lett. B461 79-88 (1999),
T. Bhattacharya et. al., Phys,~Rev.~D63 074505 (2001). R.~Gupta, these
proceedings.
\bibitem{ukqcd} UKQCD collaboration, K.~Bowler et. al, Phys. Rev. D62 054506 (2000), C.~Allton et. al., hep-lat/0107021.
\bibitem{squark} J.~Garden et. al., Nucl.~Phys.~B571, 237-256(2000).
\end{thebibliography}
\end{document}
The use of Symanzik-improved lattice actions and matrix elements is
widespread. However, with each improvement term added the
corresponding coefficient must be determined to enable discretisation
effects to be reduced to the desired level. Considering the light
hadron spectrum and matrix elements, the relevant $O(a)$ improvement
coefficients are, for the most part, only known to $1$-loop in
perturbation theory. A nonperturbative determination of these
coefficients is desirable and may be simpler than performing higher
loop perturbative calculations.

The ALPHA collaboration, using Schr\"odinger functional techniques,
have calculated several $O(a)$ improvement coefficients
nonperturbatively. In most cases agreement is found with $1$-loop
(tadpole-improved) perturbation theory, or the discrepancy is
consistent with estimates of the size of omitted higher orders in
$\alpha$~(albeit in some cases assuming a slow convergence of the
perturbative series). However, for the $O(a)$ improvement of the
axial-vector current 
\begin{equation}
A_{\mu}\rightarrow A_{\mu}^I = A_{\mu} + c_A\partial_{\mu} P + O(a^2)\label{improv}
\end{equation}
where $A_\mu =
\bar{\psi}\gamma_\mu\gamma_5\psi$ and $P=\bar{\psi}\gamma_5\psi$, $c_A$ is found to be many times larger than the 1-loop perturbative
value even at reasonably high $\beta$, as shown in
figure~\ref{falpha}, taken from reference~\cite{alpha}.

\begin{figure}
\begin{center}
\epsfig{file=cafig.ps,width=70mm,
clip= , bbllx=151pt,bblly=556pt,bburx=437pt,bbury=702pt} 
\vskip-1cm
\caption{$C_A$ extracted by the ALPHA collaboration. The circles indicate the
results of reference~\protect\cite{alpha}, the dotted line the results of 1-loop perturbation theory and the crosses the results from 1-loop
tadpole-improved perturbation theory.\label{falpha}}
\end{center}
\vskip-0.5cm
\end{figure}

$c_A$ appears in the expression for the pion decay constant, $f_\pi$,
\begin{equation}
f_\pi = (1+b_A)Z_A(f_\pi^{(0)} + c_A af_\pi^{(1)} +  O(a^2)
\end{equation}
and the bare quark mass~(from the PCAC relation)
\begin{equation}
2m = \frac{\partial_\mu<A_4 P> + c_Aa\partial^2_\mu<PP>}{2<PP>} +  O(a^2)
\end{equation}

Choosing the perturbative or nonperturbative value for $c_A$ leads to
significantly different values for these quantities at finite $\beta$.
Bhattacharya et. al. proposed an alternative method for determining
$c_A$, and other $O(a)$ coefficients in reference~\cite{lanl}. Here, we
apply their method, but with higher statistics than reported in
reference~\cite{lanl}. In addition, we extract $c_A$ at finite, as well as
zero, momentum.

\section{$C_A$ from the PCAC relation}
The PCAC relation in euclidean space
\begin{equation}
<\partial_{\mu} A_{\mu}(x) P> = 2m <P(x) P>\label{pcac}
\end{equation}
should hold for all $x$ on the lattice up to discretisation
terms. For simplicity we assign
\begin{eqnarray}
r(t) & = & \frac{<\partial_{\mu} A_{\mu}(t) P>}{<P(t)P>} \\
s(t) & = & \frac{<\partial_{\mu}^2 P(t) P>}{<P(t)P>}\label{es}
\end{eqnarray}
Thus, 
\begin{eqnarray}
2m & = & r(t) + \mbox{O(a)}\\
2m & = & r(t) + c_A s(t) + O(a^2). \label{im}
\end{eqnarray}
Equation~\ref{im} only holds if the $O(a)$-improvement term~(the
clover term) is included in the light quark action and the
coefficient, $c_{SW}$, is determined nonperturbatively.  The method of
Bhattacharya et. al. is to determine $c_A$ by minimizing the
dependence of $r(t)+c_As(t)$ on $t$ and, hence, reduce the
discretisation errors in this quantity.  Using this method, we
investigate the effect of improving $\partial_{\mu}$ on the
determination of $c_A$; discretisation errors in the lattice
derivatives may be the dominant source of lattice spacing errors in
the bare quark mass, and hence, the value obtained for $c_A$ may be
artificially high. 

Normally, the symmetric lattice derivatives
\begin{eqnarray}
\partial_{\mu} \rightarrow \Delta_{\mu}^{(+-)} & = & \frac{1}{2}(\delta_{\vec{x},\vec{x}+\hat{\mu}}-\delta_{\vec{x},\vec{x}-\hat{\mu}})\label{norm}\\
\partial^2_{\mu} \rightarrow \Delta^{(2)}_{\mu} & = & \delta_{\vec{x},\vec{x}+\hat{\mu}} - 2\delta_{\vec{x},\vec{x}} + \delta_{\vec{x},\vec{x}-\hat{\mu}}\nonumber
\end{eqnarray}
are used, which contain $O(a^2)$ errors. We will consider the effect of using
\begin{eqnarray}
\tilde{\Delta}_{\mu}^{(+-)} & = & \Delta_{\mu}^{(+-)} -\frac{1}{6}\Delta^+\Delta^{(+-)}\Delta^{-}\label{impd}\\
\tilde{\Delta}^{(2)}_{\mu} & =& \Delta_{\mu}^{(2)} -\frac{1}{12}\left[\Delta^+\Delta^-\right]^2\nonumber
\end{eqnarray}
which are correct up to $O(a^4)$. One can continue to correct to
$O(a^6)$, we denote these derivatives $\tilde{\Delta'}_{\mu}^{(+-)}$
and $\tilde{\Delta'}^{(2)}_{\mu}$.  As an additional constraint on
$c_A$, we investigate whether Lorentz invariance is restored to
sufficient accuracy i.e. that consistent values of $m$ are obtained at
zero and finite momentum.

\section{ Estimating $C_A$ using excited states}
The configurations and light quark propagators were provide by the
UKQCD collaboration.  The simulation parameters are given in
table~\ref{sim}. For more details see reference~\cite{ukqcd}. Note that the
$C_{SW}$ values correspond to those determined by the ALPHA
collaboration.

\begin{table}
\begin{center}
\begin{tabular}{ccccc}\hline
$\beta$ & Volume & $n_{confs}$ & $C_{SW}$ & $\kappa_l$  \\\hline
6.0 & $16^3\times 48$ & 496 & 1.77 & 0.13344 \\
6.2 & $24^3\times 48$ & 214 & 1.61 & 0.13460 \\\hline
$\beta$ & $\kappa_c$ & $\kappa_s(K)$ &  $a^{-1}(r_0)$\\\hline
6.0 & $0.135252(^{+16}_{-9})$ & $0.13401(^{+2}_{-2})$ & 2.12 & \\
6.2 & $0.135815(^{+17}_{-14})$ & $0.13495(^{+2}_{-2})$  & 2.91 & \\\hline
\end{tabular}
\caption{Simulation details.\label{sim}}
\end{center}
\vskip-0.5cm
\end{table}

\begin{figure}
\begin{center}
\epsfig{file=figr.ps,width=60mm}
\vskip-1cm
\caption{The ratio of correlators $r(t).$\label{figr}}
\end{center}
\vskip-0.5cm
\end{figure}


Figure~\ref{figr} shows the ratio of correlators, $r(t)$, for the data
set at $\beta=6.0$ for correlators local at the source and sink~(LL)
and those fuzzed at the source and local at the sink~(FL). The
$O(a^2)$ lattice derivatives, equation~\ref{norm}, were used. Towards
the center of the lattice $r(t)$ tends to a constant, $2m\sim
0.1096$. However, in the LL case, close to the origin there are
significant deviations from this value due to discretisation effects.
The much wider plateau in $r(t)$ for the FL correlators~(where the
fuzzing has been chosen to more or less eliminate excited state
contributions) show that the discretisation effects in $r(t)$ LL are
associated with excited states.  We can determine $c_A$ by adding the
term $c_A s(t)$ to reduce the discretisation effects at small $t$. To
do this we perform a {\it correlated} fit to $r(t)$ using $r(t) = 2m -
c_A s(t)$, to extract $2m$ and $c_A$. The values of $c_A$ obtained are
detailed in the next section.



\begin{figure}
\begin{center}
\epsfig{file=comp_mass.ps,width=60mm}
\vskip-1cm
\caption{The $O(a)$-improved bare quark mass as determined using from
LL correlators.\label{cmass}}
\end{center}
\vskip-0.5cm
\end{figure}


\begin{figure}
\begin{center}
\epsfig{file=delta_mass_2_LL_beta60np.ps,width=60mm}
\vskip-1cm
\caption{The difference in the bare quark mass determined at zero and
finite momentum, where standard lattice derivatives are
used.\label{diffa}}
\end{center} 
\vskip-0.5cm
\end{figure}

\begin{figure}
\begin{center}
\epsfig{file=delta_mass_4_LL_beta60np.ps,width=60mm}
\vskip-1cm
\caption{The difference in the bare quark mass determined at zero and
finite momentum, where improved lattice derivatives are
used.\label{diffb}}
\end{center} 
\vskip-0.5cm
\end{figure}


To see how well the discretisation effects are reduced we plug these
values back into $r(t)+c_As(t)$ for LL correlators and compare with
the unimproved case. At $\beta=6.0$ we find that the plateau in $2m$
can be extended from $t=11$ to $t=6$ when $c_A=-0.10$ and $O(a^2)$
lattice derivatives are used. However, as shown in figure~\ref{cmass},
if one improves $\partial_\mu$, the discretisation effects can be
removed by the same amount but lower values of $c_A$ are
required. Clearly the value of $c_A$ and the form of the derivatives
significantly affects the value of $m$ extracted.  Furthermore, we
find that consistent values of $m$ at zero and finite momentum can
only be obtained if $O(a^4)$ or $O(a^6)$ derivatives are
used. Figures~\ref{diffa} and~\ref{diffb} show the difference in the
bare quark mass at zero and finite momentum for standard and improved
derivatives.


\section{Results for $C_A$}
Results are shown in figures~\ref{caresa} and~\ref{caresb}.
The main points are:
\begin{itemize}
\item A reliable estimate for $c_A$ is indicated by
a stable value with $t_{min}$. At $\beta=6.0$, the statistical errors
grow rapidly for $t_{min}>8$, although the results are reasonably
stable for $t_{min}=6-8$. The situation is similar at $\beta=6.2$.
\item Clearly $c_A$ decreases when the temporal derivatives are
improved. The decrease is less going from derivatives correct to
$O(a^4)$ to those correct to $O(a^6)$, as expected.
\item $c_A$ obtained using improved derivatives is below the
value obtained by the ALPHA collaboration. For the $O(a^6)$-correct
derivatives $c_A$ is close to the 1-loop tadpole-improved
value~(although the perturbation theory has not been performed with
improved derivatives). In principle, our results can differ with those
of reference~\cite{alpha} by $O(\Lambda_{QCD}a)\sim .300/2.1 =.14$ at $\beta=6.0$ and
$.10$ at $\beta=6.2$.
\item The results can be compared 
to those obtained by Bhattacharya et. al.~\cite{lanl}. They obtain
$c_A=-0.02(2)(2)$ at $\beta=6.0$ and $-0.033(4)(3)$ at $\beta=6.2$ for
$\kappa_c$ with a significantly smaller number of configurations. Our
results are at finite $\kappa$ around $\kappa_{strange}$. A
preliminary study indicates that the light quark mass dependence of
$c_A$ is small.
\end{itemize}

\begin{figure}[t]
\begin{center}
\epsfig{file=comp_ca_ti_12_2_4_6_LL_beta60np_cbt.ps,width=60mm}
\vskip-1cm
\caption{The results for $c_A$ at $\beta=6.0$ obtain using correlated fits with
$Q>.10$~(the dashed points have $0.10>Q>.01$). The data has been
averaged over positive and negative~(i.e. $t>25$) timeslices. The
errors are bootstrapped over 100 bootstrap samples. The result of the
ALPHA collaboration is shown as a horizontal line, where the dashed
lines indicate the numerical error. The tadpole-improved $1$-loop
perturbative result is also indicated, the error is taken to be
$1\alpha_p^2(\pi/a)$.\label{caresa}}
\end{center}
\vskip-0.5cm
\end{figure}

\begin{figure}[t]
\begin{center}
\epsfig{file=comp_ca_ti_14_2_4_6_LL_beta62np_cbt.ps,width=60mm}
\vskip-1cm
\caption{The results for $c_A$ at $\beta=6.2$. The figure is labelled in the same way as figure~\protect\ref{caresa}\label{caresb}}
\end{center}
\vskip-0.5cm
\end{figure}

\section{Conclusions}
We have applied the method of reference~\cite{lanl} to extract the $O(a)$
improvement coefficient, $c_A$. We find that the accuracy and
reliability of this method is limited by the small range of timeslices
over which a stable value of $c_A$ is found. Nevertheless, we have
shown that the value of $c_A$ obtained depends significantly on the
discretisation chosen for the lattice derivatives appearing in the
PCAC relation. $c_A$ reduces as the lattice derivatives are improved,
and lies between the value obtained by the ALPHA collaboration and the
1-loop tadpole-improved value. Furthermore, consistent values for the
bare quark mass at zero and finite momentum can only be obtained for
derivatives correct to $O(a^4)$ and above.

These results suggest a similar study should be performed within the
Schr\"odinger functional approach. In the future, we plan to
investigate the quark mass dependence of $c_A$ and check the scaling
properties of the renormalised light quark mass and $f_\pi$ obtained
using our values of $c_A$. 
   
\end{document}

hep-lat0109036.extracted_bib
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\bibitem{ukqcd} UKQCD collaboration, K.~Bowler et. al, Phys. Rev. D62 054506 (2000), C.~Allton et. al., hep-lat/0107021.
\bibitem{squark} J.~Garden et. al., Nucl.~Phys.~B571, 237-256(2000).
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