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\title{$D$ AND $B$ MESONS SEMI-LEPTONIC DECAYS}

\author{\underline{ELC} (European Lattice Collaboration) 
and the \underline{APE Collaboration}
\\
A.Abada\address{LPTHE Orsay, France.\\ 
$^{\rm b}$Dip. di Fisica, Univ. di Roma, ``La Sapienza'', Italy.  \\
$^{\rm c}$Dep. of Electronics, The University, Southampton, UK.\\
$^{\rm d}$Physics Dep., Univ. of Wuppertal, Germany.\
$^{\rm e}$Dept de \'Fis. Te\'orica, Univ. Aut\'onoma de Madrid, Spain,
$^{\rm f}$Dep. of Physics, The University, Southampton, UK.\\ 
$^{\rm g}$Deutsches Elektronen-Synchroton, DESY, Hamburg, Germany.\ \ 
$^{\rm h}$Dip. di Fisica, Univ. di Roma, ``Tor Vergata'', Italy\\
Talk presented by N. Stella$^{*}$}, 
C.R.Allton$^{\rm b}$, A.Bartoloni$^{\rm b}$, C.Battista$^{\rm b}$,
P.Boucaud$^{\rm a}$, S.Cabasino$^{\rm b}$, N.Cabibbo$^{\rm b}$, 
D.B.Carpenter$^{\rm c}$,
M.Crisafulli$^{\rm b}$, S.G\"usken$^{\rm d}$, 
P.Hernandez$^{\rm e}$, V.Lubicz$^{\rm b}$,
G.Martinelli$^{\rm b}$, F.Marzano$^{\rm b}$, 
P.S.Paolucci$^{\rm b}$, J.Pech$^{\rm b}$, O.Pene$^{\rm a}$, 
F.Rapuano$^{\rm b}$, C.T.Sachrajda$^{\rm f}$, R.Sarno$^{\rm b}$, 
K.Schilling$^{\rm d}$, G.Siegert$^{\rm d}$, R.Sommer$^{\rm g}$,
N.Stella
\thanks{$^{*}$Present address: Dep. of Physics, 
The University, Southampton, UK}$^{\rm b}$, 
G.M.Todesco$^{\rm b}$, M.Torelli$^{\rm b}$, W.Tross$^{\rm b}$, 
R.Tripiccione$^{\rm b}$, P.Vicini$^{\rm b}$ 
and A. Vladikas$^{\rm h}$.\\
}
\begin{document}

\begin{abstract}
Recent results for semi-leptonic decays of $D$ and $B$ 
mesons at $\beta=6.4$, using the Wilson action, and at $\beta=6.0$, using the 
Clover action, are reported.
\end{abstract}

% typeset front matter (including abstract)
\maketitle

Semi-leptonic decays of heavy flavours play a crucial 
role in the determination of
the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements. 
$D$ decays provide a good test of the method,
since the relevant CKM matrix element is well costrained by unitarity in the 
Standard Model: $V_{cs}\simeq 0.975$. 
Here we present preliminary results obtained 
from a large statistics simulation performed by the APE Collaboration,
using the Clover action\cite{SW}, and results obtained 
by the European Lattice Collaboration (ELC) \cite{ELC},
using the Wilson action at $\beta=6.4$.

Both the studies were intended to reduce $O(a)$ artefacts:
the ELC study by working at a smaller value of $a$, but with
Wilson fermions; the APE study by working at moderate
$a$ but with an \lq\lq improved'' action.
The results with the Clover action suffer from severe statistical
errors, in spite of the large statistics.
There are two possible reasons for this:
either the Clover action or the \lq\lq thinning'' procedure.
\lq\lq Thinning'' means that we use only one point out of three,
in each spatial direction, when we compute correlation functions.
This procedure is necessary when, as in our case, the computer
memory is not sufficient to store the full quark propagators.
There is a systematic error introduced by \lq\lq thinning''
because we cannot eliminate high momentum components in the 
correlation functions. This error can be shown to be
negligible at large time distances. On the other hand, by using
a small sample of the points, the signal is destabilized.

Both the studies try to extrapolate the form factors (FF) to
$B\to\pi,\rho$ decays using the scaling laws predicted by the
Heavy Quark Effective Theory (HQET) \cite{IW}.
For the moment this is more a feasibilty study 
because of the large statistical errors which amplify in 
the extrapolation.

Semi-leptonic decays ($D\to K,K^{\star}$ $B\to D,D^{\star}$  $B\to\pi ,\rho $)
are described in terms of six FF of which only four 
are relevant for the decay rates (see eg. \cite{LUBICZ2}).
The relevant information, for each FF,
can be expressed in terms of its value at $q^2=0$ and its $q^2$ dependence.
In the following, we will give the main results of the two studies
for these quantities.

\vspace{2mm}

{\bf Computation details}

To extract the FF we have computed two- and three-point correlation
functions and followed the procedure set out in \cite{ELC}.
The lattice setup of the two computations is reported in Table 1.

\begin{table}
\caption{}
\begin{tabular}{|l|ccccc|}
\hline
      & \# of  &  Fermion         & $\beta$ & Volume & $a^{-1}$
\\
      & Confs    & Action  &        & $L^3\times T$   & [GeV]
\\
\hline
 ELC   & 15         &  Wilson        &  6.4    &  $24^3\times 60$ & 3.6(2) \\
 APE   & 100        &  Clover        &  6.0    &  $18^3\times 64$ & 2.03(9) \\
\hline
\end{tabular}
\end{table}
To study the $q^2$ dependence of the FF,
we compute the matrix elements
with a $D$ meson at rest, i.e. $\vec p_D=\vec 0$, and 
$\vec{p}_K\equiv 2\pi\!/La\cdot (0, 0 ,0 ), (1, 0 ,0 ), (1, 1 ,0 ), 
(1, 1 ,1 ), (2, 0, 0)$.
In the APE computation, other values of the momentum transfer 
were studied by fixing $\vec p_D \equiv (1,0,0)$, and 
$\vec{p}_K\equiv 2\pi\!/La\cdot (0, 0 ,0 ), (1, 0 ,0 ), (-1, 0 ,0 ), 
(0, 1 ,0 )$.
Correlation functions which are equivalent under the hypercubic symmetry
have been averaged together.

\vspace{2mm}

{\bf Test of Vector Meson Dominance}

In fig. \ref{fig1}, we compare the FF,
computed in the APE simulation, to the meson dominance predictions,
e.g. $f^+(q^2)=\frac{f^{+}(0)}{1-q^2/M_{1^{-}}^2}$, as a function of $q^2$.
In the range explored, the $q^2$ behaviour
of the FF is similar to the expectation of meson dominance.
From our data, there could be a deviation from 
meson dominance at $q^2 > 0$, that can be decided only with
more accurate results.

To obtain the FF at $q^2=0$ it is convenient 
to estimate them from the points with the $q^2$ closest to zero.
In this way one reduces the systematic effect 
due to the choice of a particular functional dependence
on $q^2$.

\begin{figure}[hbt]
\vspace{9pt}
%\framebox[55mm]{\rule[-21mm]{0mm}{43mm}}
\begin{center}\setlength{\unitlength}{1mm}
\begin{picture}(80,30)
\put(-13,-42){\special{dallas_stella_vmd.ps}}
\put(9,41){$A_1(q^2)$}
\put(-13,-11){\special{dallas_stella_vmd1.ps}}
\put(9,10){$f^+(q^2)$}
\put(60,-8){$q^2/M_{1^-}$}
\put(60,23){$q^2/M_{1^+}$}
\end{picture}
\end{center}
\caption{%\footnotesize
Form Factors $f^+(q^2)$ and $A_1(q^2)$ as a function of the momentum 
transfer $q^2/M_{{\rm pole}}$. 
The curve corresponds to the nearest pole dominance 
approximation.}
\label{fig1}
\end{figure}

\vspace{2mm}

{\bf $D$-meson decays}

\begin{table*}[hbt]%b
% space before first and after last column: 1.5pc
% space between columns: 3.0pc (twice the above)
\setlength{\tabcolsep}{1.1pc}
% -----------------------------------------------------
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\caption{%\footnotesize
FF at $q^2=0$ for the $D\to K(K^*)$.
The numbers obtained by the ELC and the APE Collaboration are compared to the 
most recent experimental world average and to previous lattice results.
ELC averaged the vector FF obtained from the conserved and
the local vector currents\cite{ELC}.}
\begin{tabular}{|l|ccccc|}
\hline
Ref.     & $f^+(0)$  & $V(0)$   & $A_1(0)$  & $A_2(0)$  & $V(0)/A_1(0)$ \\
\hline
ELC      &0.65(18)& 0.95(34) & 0.63(14)  & 0.45(33)  & 1.50(28)    \\ 
APE  (A) &0.71(7) & 1.01(15) & 0.67(14)  & 0.67(44)  & 1.49(33)    \\ 
APE  (B) &
          0.81(9) & 1.09(27) & 0.72(16)  & 0.72(50)  & 1.49(33)    \\ 
\cite{LUBICZ2}-\cite{LUBICZ} 
         &0.63(8) & 0.86(10) & 0.53(3)   & 0.19(21)  & 1.6(2)      \\ 
\cite{BERNARD}-\cite{BERNARD2}
&0.90(8)(21)&1.43(45)(49)&0.83(14)(28)&0.59(14)(24)&1.99(22)(33) \\ 
\cite{WHITERELL}&0.72(2)&1.12(16)&0.59(4)&0.44(9)&1.90(25) \\ \hline
\end{tabular}
\label{tab2}
\end{table*}

We report in Tab. 2 the values of the FF at $q^2=0$
obtained by the ELC and the APE Collaborations 
for the $D\to K(K^*) $ decays. 
Some results obtained in the past years are also reported for 
comparison\footnote{Other results presented at this conference 
\cite{UKQCD}-\cite{GUPTA2} can be found in these proceedings.}.
The numbers reported by ELC are consistent within large uncertainties with 
previous results obtained at larger values of $a$.
The APE results are still preliminary and 
have been obtained  by using 
two different procedures, the so called ``ratio''(A) and ``analytic''(B) 
methods, for details see ref. \cite{ELC}. 

One source of uncertainty is the value of the renormalization constants
of the vector and axial currents, 
$Z_V$ and $Z_A$.
ELC computed the vector FF by using 
both the conserved vector current and the local vector current
with $Z_V=0.84$.
For the axial current, the perturbative value $Z_A=0.88$
has been used.
Different values should be used if one assumes boosted perturbation
theory\cite{LM,LM2}.
In the APE simulation we used local currents
with $Z_V=0.824$ and  $Z_A=1.09$, as determined non 
perturbatively,  using light quark
correlation functions, see ref. \cite{VLADIKAS,PETRARCA}.

As table 2 shows, there is reasonable agreement among 
the results quoted by different authors and the experimental values.
The situation is still confused for $A_2$ for which theoretical and
experimental errors are large.
 
\vspace{2mm}

{\bf $B$ mesons decays}

The study of $B$ meson decays is of great importance since it is essential
for the determination of
the still unknown value of $V_{ub}$. $B$ mesons cannot be studied directly 
on the lattice since the $b$-quark
mass is beyond the present values of $a^{-1}$. 
One possibility is to evaluate the FF in the charm region and to 
extrapolate the results to the bottom mass.
A semi-quantitative study has been performed. We 
extrapolated by using the behaviour
predicted by HQET up to order $1/M_P$ \cite{IW}:
\[
\frac{(f^+,V,A_2)}{M_P^{1/2}}=\gamma(1+\frac{\delta}{M_P}) ;\ 
A_1\!\!=\!\frac{\gamma}{M_P^{1/2}}(1+\frac{\delta}{M_P})\ .
\]
In figure \ref{fig2} we plot the FF as a function of
$1/M_P$. 
The FF for $B$ decays extrapolated at $q^2=0$ are
given in Table 3.
Errors are too large to permit an evaluation of $V_{ub}$ .

\begin{table}
\caption{%\footnotesize
ELC (1st row) and Ape (2nd row) results for $B$ FF.}
\begin{tabular}{|ccccc|}
\hline
 $ f^+(0)$   & $A_1(0)$  & $A_2(0)$  & $V(0)$ & $V\!/A_1(0)$ \\
\hline
 0.33(17)& 0.21(5) & 0.47(28) & 0.40(16) & 1.7(6) \\
\hline
 0.35(9)& 0.24(10) & 0.6(1.0) & 0.5(4) & 1.9(1.4) \\
\hline
\end{tabular}
\end{table}

\begin{figure}[hbt]%t
%\vspace{10mm}
%\framebox[55mm]{\rule[-21mm]{0mm}{43mm}}
\begin{center}\setlength{\unitlength}{1mm}
%\begin{picture}(80,30)
\begin{picture}(80,40)
\put(-10,-45){\special{dallas_stella_scala.ps}}
\end{picture}
\end{center}
\caption{%\footnotesize
Extrapolation of the FF to the bottom mass
(diamond point) using the HQET prediction.}
\label{fig2}
\end{figure}


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%CERCARE IN SEGRETERIA SPERIMENTALE INFN WHITTEREL

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\end{thebibliography}

\end{document}


