\documentstyle[preprint,prd,aps]{revtex}
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\begin{document}


%\draft
\preprint{Edinburgh Preprint: 94/545}

\title{Current Renormalisation Constants with an $O(a)$-improved Fermion
Action}
\author{UKQCD Collaboration}

\author{D.S.~Henty, R.D.Kenway, B.J.~Pendleton, J.I.~Skullerud}
\address{Department of Physics and Astronomy, The University of
Edinburgh, Edinburgh EH9~3JZ, Scotland}

%\date{\today}
\maketitle

\begin{abstract} 

Using chiral Ward identities, we determine the renormalisation constants
of bilinear quark operators for the Sheikholeslami-Wohlert action
lattice at $\beta=6.2$.  The results are obtained with a high degree of
accuracy.  For the vector current renormalisation constant we obtain
$Z_V=0.817\pm 2\pm 8$, where the first error is statistical and the
second is due to mass dependence of $Z_V$.  This is close to the
perturbative value of 0.83.  For the axial current renormalisation
constant we obtain $Z_A =
1.045\mbox{\scriptsize${}^{\makebox[1.1em][c]{$+$}\makebox[1em][r]{10}}
_{\makebox[1.1em][c]{$-$}\makebox[1em][r]{14}}$}$, significantly higher
than the value obtained in perturbation theory.  This is shown to reduce
the difference between lattice estimates and the experimental values for
the pseudoscalar meson decay constants, but a significant discrepancy
remains.  The ratio of pseudoscalar to scalar renormalisation constants,
$Z_P/Z_S$, is less well determined, but seems to be slightly lower than
the perturbative value.

\end{abstract}
\pacs{}

\narrowtext
\section{Introduction}

In a recent paper \cite{strange}, we reported values for the decay
constants $f_\pi$ and $f_K$ from lattice calculations that were
considerably lower than the experimental values.  This has been a
persistent feature of lattice calculations of pseudoscalar decay
constants \cite{APE1,Weingarten}.  However, it has not been possible to
attribute this discrepancy to any particular systematic error in the
calculations (eg quenching), because of uncertainties in the
renormalisation of the current operators involved. 

A fully non-perturbative determination of renormalisation constants for
finite operators can be achieved using chiral Ward identities
\cite{Boch,MPSV}, thereby bypassing the need for perturbative
calculation of these quantities.  A preliminary calculation at
$\beta=6.0$, presented in \cite{MPSV}, indicated that the axial vector
current renormalisation, in particular, differs significantly from its
perturbative value.  In this paper, we present results for the vector,
axial-vector, pseudoscalar and scalar renormalisation constants at
$\beta=6.2$, and update the values for decay constants given in
\cite{strange}. 

\section{Lattice Ward Identities and Renormalisation Constants}
\label{theory}

We define the (non-conserved) lattice vector current, the axial current
and the pseudoscalar and scalar densities as follows:
%
\begin{eqnarray}
V^{La}_{\mu}(x) & = & \overline{\psi}(x)\gamma_{\mu}\frac{1}{2}\lambda^a\psi(x) \\
A^{La}_{\mu}(x) & = & \overline{\psi}(x)\gamma_{\mu}\gamma_5\frac{1}{2}\lambda^a\psi(x) \\
P^{a}(x) & = & \overline{\psi}(x)\gamma_5\frac{1}{2}\lambda^a\psi(x) \\
S^{a}(x) & = & \overline{\psi}(x)\frac{1}{2}\lambda^a\psi(x)
\end{eqnarray}

In the continuum limit, these operators are related to operators obeying
the correct current algebra by multiplicative renormalisation constants
$Z_V^L, Z_A^L$ etc, so that $V_{\mu} = Z_V^L V^L_{\mu}$
etc.~\cite{MPSV}. 

The renormalisation constants for $V^{L}_{\mu}$ can easily be determined
by evaluating

\begin{equation}
\label{ratio-3pt}
Z_{V}^{L} = \frac{\sum_{\vec{x}} 
\langle P^{\dag}(\vec{0},0) P(\vec{x},T) \rangle}
{\sum_{\vec{x},\vec{y}} \langle P^{\dag}(\vec{0},0) 
V^{L}_4(\vec{y},t) P(\vec{x},T) \rangle} 
\end{equation}

Inserting a complete set of states and noting that the matrix element of
$V^L_4(0)$ between a degenerate pseudoscalar meson state $P_n$ is $\langle
P_n \,|\, V^{L}_4 \,|\, P_n \rangle = 2E_n/Z^L_V$ , we see that this should
give a precise estimate provided the effect of the off-diagonal matrix
elements $\langle P_m \,|\, V^{L}_4 \,|\, P_n \rangle$ can be neglected. 


For the axial case, there is no conserved current, or any other ``easy''
way of determining the renormalisation constants, but they can be
obtained using chiral Ward Identities.  Using the arguments of
\cite{Boch,MPSV}, we obtain the following identities for the
Sheikholeslami--Wohlert (SW) \cite{SW} action:

\begin{eqnarray}
\label{WI-ZA-int}
\lefteqn{2\rho\sum_{x,\vec{y}}\langle P^a(x)A^{Lb}_{\nu}(y)V^{Lc}_{\rho}(0)\rangle}
\nonumber \\
& = & -i (\frac{Z_V^L}{{Z_A^L}^2}-\rho ra) 
f^{abd}\sum_{\vec{y}}\langle V^{Ld}_{\nu}(y) V^{Lc}_{\rho}(0)\rangle \nonumber \\
 & & -i (\frac{1}{Z_V^L}-\rho ra)
f^{acd}\sum_{\vec{y}}\langle A^{Lb}_{\nu}(y)A^{Ld}_{\rho}(0)\rangle 
\end{eqnarray}
%
and
%
\begin{eqnarray}
\label{WI-PS-int}
\lefteqn{2\rho\sum_{x,\vec{y}}\langle P^a(x)S^b(y)P^c(0)\rangle}
\nonumber \\
& = & (\frac{Z_P}{Z_A^L Z_S}-\rho ra) 
d^{abd}\sum_{\vec{y}}\langle P^d(y) P^c(0)\rangle \nonumber \\
 & & (\frac{Z_S}{Z_A^L Z_P}-\rho ra)
d^{acd}\sum_{\vec{y}}\langle S^b(y)S^d(0)\rangle 
\end{eqnarray}
%
where
\begin{equation}
\rho = \frac{\langle 0 \,|\, \partial_4 A^{La}_4\,|\, P^a\rangle}
{2 \langle 0\,|\, P^a\,|\, P_a\rangle}
\end{equation}
%
and $d^{abc}$ is defined by
%
\begin{equation}
\left\{\lambda^a,\lambda^b\right\} = 2 d^{abc}\lambda^c +
\frac{4}{N_f}\delta^{ab} 
\end{equation}

\section{Computation and results}
\label{results}

We have performed simulations at $\beta=6.2$ on a $24^3\times 48$
lattice.  We have generated propagators using the SW action, with two
quark masses, corresponding to $\kappa=0.14144$ and 0.14262, where
$\kappa=1/2(m_0+4r)$ and $r=1$.  60 configurations have been analysed at
$\kappa=0.14144$, and 26 configurations at $\kappa=0.14262$ (for
details, see \cite{qlhms}).  The statistical errors are calculated with
a bootstrap procedure, using 100 bootstrap samples. 

%%%% The vector renormalisation contant $Z_V^L$

$Z_V$ was determined from eq.~(\ref{ratio-3pt}), using 10
configurations, at three values for the quark mass (corresponding to
$\kappa=0.14144$, $\kappa=0.14226$ and $\kappa=0.14262$).  The results
are presented as a function of $t$ in fig.\ref{fig-zv6.2}.  We see that
the values for $Z_V$ are roughly independent of $t$.  Our 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 the fit 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, consistent with the expectation that the
leading corrections to our calculations should be of $O(\alpha_s m_0 a)$. 
Perturbation theory at one-loop level \cite{Gabrielli,Borrelli} with a
``boosted'' coupling constant \cite{LM} gives $Z_V^L\approx 0.83$, which
is quite close to our non-perturbative values. 

We have also used eq.~(\ref{WI-ZA-int}), with $\nu=\rho=0$, to check the
consistency of our results at $\kappa=0.14144$, using the value for
$Z_V^L$ quoted in table~\ref{tab:zv} and the value for $Z_A^L$ given in
table~\ref{tab:za} as input.  This gives $Z_V^L = 
0.817\mbox{\scriptsize${}^{\makebox[1.1em][c]{$+$}\makebox[1em][r]{8}}
 _{\makebox[1.1em][c]{$-$}\makebox[1em][r]{10}}$}$,
which is within $2\sigma$ of the result obtained from
eq.~(\ref{ratio-3pt}). 

%%%% Axial current renormalisation $Z_A^L$

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 for $\kappa=0.14144$ are plotted against $t$ in
fig.\ref{fig-za6.2}.  We see that, apart from the effect of the contact
terms on the first few timeslices, they show virtually no dependence on
$t$.  Our best estimates are given in table~\ref{tab:za}. 

Within the statistical errors, these results show no dependence of $Z_A$
on the quark mass.  The comparison with results from perturbation theory
is more interesting: one-loop calculations with a ``boosted'' coupling
constant give $Z_A\approx 0.97$, which is considerably lower than our
non-perturbative results.  The discrepancy is higher at lower $\beta$,
as expected; in \cite{MPSV} $Z_A$ at $\beta=6.0$ was found to be 1.09. 

In table~\ref{tab:decayconsts} we show how the values for the decay
constants reported in \cite{strange} change when we use the results
given above for the renormalisation constants.  For $Z_V$, we have
extrapolated the values in table~\ref{tab:zv} to the limit of zero quark
mass, giving $Z_V=0.817\pm 2$, with an additional uncertainty due to the
quark mass dependence of $Z_V$ of $\pm 0.008$, which corresponds the
difference between the value at our largest quark mass and the value for
zero quark mass.  For $Z_A$, we have taken a best estimate, combining
our results at the two $\kappa$-values, of 
$Z_A=1.045\mbox{\scriptsize${}^{\makebox[1.1em][c]{$+$}\makebox[1em][r]{10}}
 _{\makebox[1.1em][c]{$-$}\makebox[1em][r]{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 $f_K$.  $f_\pi$ 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. 


%%%% Pseudoscalar and scalar density renormalisation --- $Z_P/Z_S$

The ratio of pseudoscalar to scalar renormalisation constant is
determined using (\ref{WI-PS-int}).  The results 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 here. 
Perturbative calculations with a ``boosted'' coupling constant give
$Z_P/Z_S = 0.68$.  As can be seen, our 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).  Comparison with the
result reported in \cite{MPSV} at $\beta=6.0$ shows that, as in the case
with $Z_A$, the discrepancy decreases with increasing $\beta$. 

\section{Conclusions}
\label{conclusion}

In this paper we have reported the determination of lattice
renormalisation constants using chiral Ward identities with the
Sheikholeslami-Wohlert action.  Our results are obtained with good
accuracy, yielding values for $Z_V$ close to the values from
perturbation theory (but increasing with increasing quark mass), while
the value for $Z_A$ is considerably higher than perturbative results. 
For $Z_P/Z_S$ the uncertainties are larger, but the results we can
have confidence in lie slightly lower than perturbative values.  The
result for $Z_A$ brings our estimates for $f_\pi$ and $f_K$ considerably
closer to experimental values --- within $3\sigma$ for $f_\pi$. 


\acknowledgements

This work was supported by SERC grants GR/J21347 and GR/J98202 and
carried out on a Meiko i860 Computing Surface supported by SERC grant
GR/G32779, Meiko Limited, and the University of Edinburgh.  JIS
acknowledges the support of the Norwegian Research Council.  DSH
akcnowledges the support of PPARC through a Personal Fellowship.  We are
grateful to David Richards for help with the code, and to Chris
Sachrajda for valuable discussions. 

\begin{thebibliography}{99}

\bibitem{strange} UKQCD collaboration (C.R.Allton et al.),
{\em Phys.Rev.\ }{\bf D49}, 474 (1994)
\bibitem{APE1} APE collaboration (C.R.Allton et al.)
{\em Nucl.Phys.\ }{\bf B413}, 461 (1994)
\bibitem{Weingarten}  F.Butler, H.Chen, J.Sexton, A.Vaccarino, 
D.Weingarten, {\em Nucl.Phys.\ }{\bf B421}, 217 (1994)
\bibitem{Boch} M.Bochicchio et al., {\em Nucl.Phys.\ }{\bf B262}, 331 (1985)
\bibitem{MPSV} G.Martinelli, S.Petrarca, C.T.Sachrajda and A.Vladikas,
{\em Phys.Lett.\ }{\bf B311}, 241 (1993)
\bibitem{SW} B.Sheikholeslami and R.Wohlert,
{\em Nucl.Phys.\ }{\bf B259}, 572 (1985)
\bibitem{qlhms} UKQCD collaboration (C.R.Allton et al.),
{\em Nucl.Phys.\ }{\bf B407}, 331 (1993)
\bibitem{Gabrielli} E.Gabrielli, G.Heatlie, G.Martinelli, C.Pittori and
C.T.Sachrajda, {\em Nucl.Phys.\ }{\bf B362}, 475 (1991)
\bibitem{Borrelli} A.Borrelli, C.Pittori, R.Frezzotti and E.Gabrielli,
{\em Nucl.Phys.\ }{\bf B409}, 382 (1993)
\bibitem{LM} G.P.Lepage and P.B.MacKenzie, {\em Nucl.Phys.\ }{\bf B (Proc.\
Suppl.) 20}, 173 (1991)
\bibitem{APE2} APE collaboration (C.R.Allton et al.)
{\em Phys.Lett.\ }{\bf B326}, 295 (1994)

\end{thebibliography}

\begin{figure}
\caption{$Z_V^L$ as a function of $t$.}
\label{fig-zv6.2}
\end{figure}

\begin{figure}
\caption{$Z_V^L$ as a function of the mass of the pseudoscalar meson. 
The scale is taken from the string tension \protect\cite{qlhms}.}
\label{fig-zv_chiral}
\end{figure}

\begin{figure}
\caption{$Z_A^L$ as a function of $t$ for $\kappa=0.14144$}
\label{fig-za6.2}
\end{figure}

\begin{table}
\begin{tabular}{ccc}\hline
$\kappa$ & $m_{PS}^2$ & $Z_V^L$ \\ \hline
0.14262 & 0.208 & 
0.82139{\scriptsize${}^{\makebox[1.1em][c]{$+$}\makebox[1em][r]{41}}
 _{\makebox[1.1em][c]{$-$}\makebox[1em][r]{12}}
{}^{\makebox[1.1em][c]{$+$}\makebox[1em][r]{25}}
 _{\makebox[1.1em][c]{$-$}\makebox[1em][r]{25}}$} \\
0.14226 & 0.341 & 
0.82453{\scriptsize${}^{\makebox[1.1em][c]{$+$}\makebox[1em][r]{24}}
 _{\makebox[1.1em][c]{$-$}\makebox[1em][r]{22}}
{}^{\makebox[1.1em][c]{$+$}\makebox[1em][r]{24}}
 _{\makebox[1.1em][c]{$-$}\makebox[1em][r]{23}}$} \\
0.14144 & 0.663 & 
0.83136{\scriptsize${}^{\makebox[1.1em][c]{$+$}\makebox[1em][r]{23}}
 _{\makebox[1.1em][c]{$-$}\makebox[1em][r]{16}}
{}^{\makebox[1.1em][c]{$+$}\makebox[1em][r]{22}}
 _{\makebox[1.1em][c]{$-$}\makebox[1em][r]{23}}$} \\
\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{table}
\begin{tabular}{cccc}\hline
$\kappa$ & $m_{PS}^2$ & $Z_A^L$ & $Z_P/Z_S$ \\ \hline
0.14262 & 0.208 & 
1.040{\scriptsize${}^{\makebox[1.1em][c]{$+$}\makebox[1em][r]{10}}
 _{\makebox[1.1em][c]{$-$}\makebox[1em][r]{9}}$} & 
0.693{\scriptsize${}^{\makebox[1.1em][c]{$+$}\makebox[1em][r]{28}}
 _{\makebox[1.1em][c]{$-$}\makebox[1em][r]{40}}$} \\
0.14144 & 0.663 & 
1.047{\scriptsize${}^{\makebox[1.1em][c]{$+$}\makebox[0.7em][r]{8}}
 _{\makebox[1.1em][c]{$-$}\makebox[0.7em][r]{8}}$} & 
0.649{\scriptsize${}^{\makebox[1.1em][c]{$+$}\makebox[0.7em][r]{9}}
 _{\makebox[1.1em][c]{$-$}\makebox[0.7em][r]{8}}$} \\ \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}

%%{{{  tab:decayconsts (from the "strange" paper)
\begin{table}
\begin{tabular}{cccc}
\hline
 & old estimates \protect\cite{strange} & updated estimates & experiment \\
\hline
$f_\pi$           & 
102{\scriptsize${}^{\makebox[1.1em][c]{$+$}\makebox[0.7em][r]{6}}
  _{\makebox[1.1em][c]{$-$}\makebox[0.7em][r]{7}}$} MeV  & 
110{\scriptsize${}^{\makebox[1.1em][c]{$+$}\makebox[0.7em][r]{7}}
 _{\makebox[1.1em][c]{$-$}\makebox[0.7em][r]{8}}$} MeV   & 132 MeV \\
$f_K$             & 
123{\scriptsize${}^{\makebox[1.1em][c]{$+$}\makebox[0.7em][r]{5}}
 _{\makebox[1.1em][c]{$-$}\makebox[0.7em][r]{6}}$} MeV  & 
133{\scriptsize${}^{\makebox[1.1em][c]{$+$}\makebox[0.7em][r]{7}}
 _{\makebox[1.1em][c]{$-$}\makebox[0.7em][r]{7}}$} MeV   & 160 MeV \\
\hline
$1/f_\rho$        & 
0.316{\scriptsize${}^{\makebox[1.1em][c]{$+$}\makebox[1em][r]{7}}
 _{\makebox[1.1em][c]{$-$}\makebox[1em][r]{13}}$}  & 
0.311{\scriptsize${}^{\makebox[1.1em][c]{$+$}\makebox[1em][r]{7}}
 _{\makebox[1.1em][c]{$-$}\makebox[1em][r]{13}}
{}^{\makebox[1.1em][c]{$+$}\makebox[0.7em][r]{1}} 
 _{\makebox[1.1em][c]{$-$}\makebox[0.7em][r]{1}}$}   & 0.28 \\
$1/f_{K^*}$       & 
0.298{\scriptsize${}^{\makebox[1.1em][c]{$+$}\makebox[0.7em][r]{5}} 
 _{\makebox[1.1em][c]{$-$}\makebox[0.7em][r]{9}}$}    & 
0.293{\scriptsize${}^{\makebox[1.1em][c]{$+$}\makebox[0.7em][r]{5}} 
 _{\makebox[1.1em][c]{$-$}\makebox[0.7em][r]{9}}
{}^{\makebox[1.1em][c]{$+$}\makebox[0.7em][r]{1}} 
 _{\makebox[1.1em][c]{$-$}\makebox[0.7em][r]{1}}$}     &  \\
$1/f_\phi$        & 
0.280{\scriptsize${}^{\makebox[1.1em][c]{$+$}\makebox[0.7em][r]{3}} 
 _{\makebox[1.1em][c]{$-$}\makebox[0.7em][r]{6}}$}    & 
0.276{\scriptsize${}^{\makebox[1.1em][c]{$+$}\makebox[0.7em][r]{3}} 
 _{\makebox[1.1em][c]{$-$}\makebox[0.7em][r]{6}}
{}^{\makebox[1.1em][c]{$+$}\makebox[0.7em][r]{1}} 
 _{\makebox[1.1em][c]{$-$}\makebox[0.7em][r]{1}}$}     & 0.23 \\
\hline
$f_\pi/m_\rho$    & 
0.138{\scriptsize${}^{\makebox[1.1em][c]{$+$}\makebox[0.7em][r]{6}} 
 _{\makebox[1.1em][c]{$-$}\makebox[0.7em][r]{9}}$}    & 
0.149{\scriptsize${}^{\makebox[1.1em][c]{$+$}\makebox[1em][r]{6}}
 _{\makebox[1.1em][c]{$-$}\makebox[1em][r]{10}}$}   & 0.172 \\
$f_K/m_\rho$      & 
0.160{\scriptsize${}^{\makebox[1.1em][c]{$+$}\makebox[0.7em][r]{7}} 
 _{\makebox[1.1em][c]{$-$}\makebox[0.7em][r]{8}}$}    & 
0.172{\scriptsize${}^{\makebox[1.1em][c]{$+$}\makebox[0.7em][r]{8}} 
 _{\makebox[1.1em][c]{$-$}\makebox[0.7em][r]{9}}$}     & 0.208 \\
$f_K/m_{K^*}$     & 
0.144{\scriptsize${}^{\makebox[1.1em][c]{$+$}\makebox[0.7em][r]{4}} 
 _{\makebox[1.1em][c]{$-$}\makebox[0.7em][r]{6}}$}    & 
0.155{\scriptsize${}^{\makebox[1.1em][c]{$+$}\makebox[0.7em][r]{5}} 
 _{\makebox[1.1em][c]{$-$}\makebox[0.7em][r]{7}}$}     & 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{table}
%%}}}

\end{document}
