\section{$V_{ub}$ extraction}
\label{subsec:vubextraction}

\subsection{Extraction of the ratio $\frac{BR(b \rightarrow ul\nu)}{BR(b \rightarrow cl\nu)}$}

As mentioned in the introduction, the variable choosen to extract $V_{ub}$ is 
the invariant mass of the hadronic system after all cuts. The 
ratio of the Branching Ratios 
$\frac{BR(b \rightarrow ul\nu)}{BR(b \rightarrow cl\nu)}$ can be written as 
shown in \ref{eq:ratioBR}. If we assume that the efficiencies of reconstructed 
B tagging ($\epsilon^u_l$ and $\epsilon^c_l$) and of the lepton cut 
($\epsilon^u_l$ and $\epsilon^c_l$) are the same for $b \rightarrow ul\nu$ 
and $b \rightarrow cl\nu$ the formula becomes:
\begin{equation}
\frac{\BR(b \ra u \ell \nu)}{\BR(b\ra c \ell \nu)}=
\frac{N_u}{N_c} = 
\frac{M_u /\epsilon_{sel}^u}{(M_{sl}-Bg)} 
\end{equation}
where $M_u$ is the number $b \rightarrow ul\nu$ events after all the selection 
cuts, $\epsilon_{sel}^u$ is the efficiency of selecting $b \rightarrow ul\nu$ 
in the tagged B sample with a charged lepton with a momentum above 1 $GeV$, 
$M_{sl}$ is the total number of events which contain a charged lepton with a 
momentum above 1 $GeV$, $Bg$ is the number of non $b \rightarrow cl\nu$ events 
in $M_{sl}$.

The determination  of $M_u$ is rather simple and basically consists on a $cut$ 
$and$ $count$ procedure after background subtraction. $b \rightarrow ul\nu$ 
decays rapresent the signal. The background consists of  
$b \rightarrow cl\nu$ events and all remaining events (hadronic B decays 
with either a true lepton from cascades or a hadron misidentified as a lepton). 
An additional contribution to the background comes from the fact 
 that the reconstructed B's are not perfectly pure.
 A proper subtraction of the background, therefore,  is needed even on 
the fully reconstructed side. Since  the purity of the reconstructed B's
 depends on the full event and, in particular, on the multiplicity 
of the recoil, the subtraction has to be performed as a function of the 
variable under study. We divide, therefore,  the sample in intervals of the $M_x$ 
variable. The $m_{es}$ distribution of the reconstructed side is fitted in 
each interval. The value of the yield and its error from the fit is finally 
plotted in the correspondent $M_x$ bin. 
% -- bkg
\begin{figure}
 \begin{centering}
 \epsfig{file=ps/bkg.eps,height=8.cm} 
 \caption{Background $M_x$ distributions. Plain histogram is the $b \rightarrow cl\nu$ contribution, 
green is the $other$ $events$ component.\label{fig:bkg}}
 \end{centering}
\end{figure} 
-----------------------------------------------------------------------------
The two other backgrounds are determined by using the $M_x$ distribution
in a region where the signal contribution is negligible. A cut
$M_{cut}$ in $M_x$, where $M_{cut} < M_{D}$, defines a signal-enhanced
and a signal-depleted region.
%
%CB
%=========these equations are perhaps unnecessary================
The populations in the two regions are given by
\begin{eqnarray}
N_{M_x<M_{cut}} & = & f_u M_u + f_c M_c + f_{oth} M_{oth} \\ \nonumber
N_{M_x>M_{cut}} & = & (1-f_u) M_u + (1-f_c) M_c + (1-f_{oth}) M_{oth} \\
\nonumber
\end{eqnarray}
%=======================================================
%
The amounts of the two background components ($M_c$ and $M_{oth}$
 are determined from a $M_x$ fit in the
signal-depleted region and by taking the background shapes (shown in
Figure \ref{fig:bkg}), hence their relative amounts in the signal region
$f_c$ and $f_{oth}$,  from Monte Carlo. The fraction of $b\rightarrow
ul\nu$ events in the signal-enhanced region, $f_u \approx 1$, is also
taken from Monte Carlo. The systematic effects due to this procedure,
 in particular the effect of the theoretical uncertainty on $f_u$,
will be discussed below. Figure \ref{fig:fitMC} shows the $M_x$ distribution
before and after background subtraction on simulated events,
corresponding to an integrated luminosity of $\sim 230fb^{-1}$.
Finally, the efficiency for $b \rightarrow ul\nu$ events after all the
cuts, relative to the tagged B sample with an high momentum lepton,
$\epsilon_{sel}^u$, is taken from the MC.