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\newcommand{\PT}{$\rm p_T~$}
\newcommand{\HWW}{$\rm gg \to h~with~h\to W^*W^* \to l\nu jj~$}
\newcommand{\hww}{$\rm h\to W^*W^* \to l\nu jj~$}
\newcommand{\TTB}{$\rm t\bar{t} \to l\nu jj b\bar{b}~$}
\newcommand{\WW}{$\rm WW \to l\nu jj~$}
\newcommand{\WZ}{$\rm WZ \to l\nu jj~$}
\newcommand{\MH}{$m_h~$}
\newcommand{\MET}{$\rm E_T \hspace{-1.0em}/\;~~$}
\newcommand{\WJET}{$\rm W+f/g \to l\nu jj~$}
\newcommand{\SPC}{\hfill\break \noindent}
\newcommand{\FB}{$\rm ~fb^{-1}~$}
\newcommand{\LUMI}{${\cal{L}}$}
\newcommand{\W}{$\rm W^*~$}
\newcommand{\G}{$\rm ~GeV/c^2~$}
\newcommand{\SB}{$\rm S/\sqrt{B}~$}
\newcommand{\HENCE}{\noindent $\rm ~~~~\Longrightarrow ~~$}

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%     Begining of document
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\subsubsubsection{Cross-sections and physics Constraints}
 
\subsubsubsubsection{Signal cross-section for \HWW}
\par
The Higgs cross-section production via the gluon-gluon fusion $gg \to h$ 
is computed in \cite{TAO}. Figure\ref{????FIG-XCROSS} shows this cross-section 
as function of the Higgs mass $\rm m_h$ from 100~GeV to 200~GeV/c at 
the TeVatron with $\rm \sqrt{s}=2.0~TeV$. Also shown are the cross-sections  
$\times$ Branching Ratios in the different channels $\rm W^*W^*,~Z^*Z^*~etc...$. 
The $\rm \sigma\times W^*W^*$ channel turns out to 
be more than one order of magnitude above the $\rm \sigma\times Z^*Z^*$ predictions. 
It is therefore a privileged channel to look at for a ``low'' integrated luminosity 
of 2 \FB like in the run II at the TeVatron.
%----------------------------------
%\begin{figure}[tbp]
% \begin{center}
%    \epsfxsize=6in\epsfbox{highmass/higgs-xsection.ps}
%    \caption{\footnotesize \it Higgs cross-section via the gluon fusion processus.}
%    \label{FIG-XCROSS}  
% \end{center}
%\end{figure}
%\begin{figure}[tbp]
% \begin{center}
%    \epsfxsize=6in\epsfbox{highmass/hdecay.ps}
%    \caption{\footnotesize \it Higgs Branching ratios for a standard Higgs }
%    \label{FIG-HDECAY}  
% \end{center}
%\end{figure}
%\vskip0.1cm
%----------------------------------
Fig. \ref{????FIG-HDECAY} displays the Higgs Branching Ratios in the [80-150]\G mass 
range. These results are produced using the HDECAY program \cite{HDECAY} 
in the case of a standard production of the Higgs. As well known, 
the $\rm h\to b\bar{b}$ decay is favoured well below the $\rm WW^*$ threshold. 
For \MH $>$ 140\G, the $\rm h\to W^*W^*$ becomes dominant.   
\par  
Table \ref{TAB-XCROSS} reports the cross-sections for our specific signal 
$\rm gg\to h \to W^*W^*$. We then require that a $\rm W^*$ decays into 2 leptons 
while the other decays into two jets. For the leptonic decay of the \W we restrict  
our analysis to the electron and muon channels.\\
Summing the electron and muon channels leads us to a cross-section of about 110 
fb for a \MH = 160\G Higgs. However, the theoretical cross-section is very 
sensitive to the Higgs mass, and favours the real W-pair production. Higgs masses 
around \MH=120\G are thus expected to be produced with a cross-section below 40 fb. 
%---------------------------- 
\begin{table}[tbp]
 \begin{center}
  \begin{tabular}{|c|c|c|c|c|}
   \hline
   \hline
   Higgs  Mass ($\rm GeV/c^2$)    &  \MH = 120    & \MH = 140  & \MH = 160  & \MH = 180 \\
   \hline 
   \hline 
   $\rm \sigma(gg \to h)~~~~~~~~~~~~~~~~~~~~$ [fb]
                  &    800         &    480      &    300      &   220 \\
   $\rm \sigma \times BR_{WW}~~~~~~~~~~~~~~$ [fb]
                  &    100         &    210      &    300      &   210  \\
    $\rm \sigma \times BR_{WW}\times BR_{l\nu jj}$ [fb]
                  &     12         &     15.7     &    79.6       &   61.8    \\
  \hline 
  Nmber of events for \LUMI=10 \FB &    120     &    157      &  796       &  618 \\  
  \hline
  \hline      
  \end{tabular}
    \caption{\footnotesize \it Cross-sections $\times$ Branching ratios for our channel $\rm W^*W^*\to l\nu~j~j$ where $\rm l=e,\mu$. The numbers, given in fb, are 
computed for Higgs mass going from 120\G to 180\G.  The corresponding number of events expected for each of these values are indicated in the last line for an integrated 
luminosity of 10 \FB. No cut is applied on these samples.}
    \label{TAB-XCROSS}
 \end{center}
\end{table}

\subsubsubsubsection{Main Backgrounds}
\par 
We consider here the backgrounds to our signal characterized by a similar 
final state:
\begin{itemize}
 \item the \WJET events, where a W is produced together with 
       QCD jets. Although their topology is very different 
       than for signal events, a huge cross-section of 
       $\rm \sigma_{W+jets} \approx 1.34$ nb makes this processus constitute 
       the main background to our signal. 	
\begin{table}[tbp]
 \begin{center}
  \begin{tabular}{|c|r|}
   \hline
   \hline
   Process      &  $\sigma$(fb) 2.0 TeV  \\
                &  (PYTHIA)              \\
   \hline 
   \hline 
   \WJET        &   1.3 $\times ~10^6$ \\
   \WW          &   1,348              \\
   \TTB         &    536              \\
   \WZ          &     158              \\
  \hline
   \hline      
  \end{tabular}
    \caption{\footnotesize \it 
Cross-sections $\times$ Branching ratios for the main backgrounds to our signal.  
$\rm W^*W^*\to l\nu~j~j$ where $\rm l=e,\mu$. These results come from the PYTHIA generator.}
    \label{TAB-BACK}
 \end{center}
\end{table}
 \item the \WW events where W-boson pair decay as our signal. The cross-section 
       of such events is $\sigma_{WW} \approx 1.348~pb$ which is a factor $\approx$ 30 
       greater than the signal. The only 
       difference with our signal comes from the helicity correlation 
       of the final products \cite{DITMAR}. In the Higgs production, both W's are 
       produced from a spin-0 particle. Both $\rm W^+W^-$ spin correlation 
       and the V-A structure of the W decays result in a disctinct 
       topology. This property may be used in the purely leptonic decays 
       of the W's, for which the opening angle of the charged lepton 
       is expected to be small in the transverse plane. In our case however, 
       this method would require the identification of the quarks issued from 
       the second W, in order to associate the lepton with its quark counterpart 
       and then compute the angular correlation. 
 \item the \TTB processus where the final events is characterized by 
       3 jets with two being b-jets. Here the topology makes these events 
       quite easily distinct from the signal. The cross-section is 
       $\rm \sigma_{t\bar{t}} \approx 536$ fb 
       \cite{PYTHIA}, which is a factor 10 above our signal. 
 \item the \WZ events, where the Z decays into two jets and the W decays 
       leptonically. The cross-section for this processus is about 
       $\rm \sigma_{WZ} \approx 158$ fb \cite{PYTHIA}, 
       a factor 3 above the signal.
\end{itemize}
Table \ref{TAB-BACK} reports all the cross-sections given by \cite{PYTHIA}.

\newpage 

\subsubsubsection{Selection using a traditional method}

The detection of the signal is based on the selection of a high energy 
lepton together with two jets (that are not b-quarks jets) and the presence 
of a quite large missing energy associated to the neutrino. 
\par 
We use PYTHIA interfaced to SHW to generate most of the physics processes.

\par
In the following, all the distributions shown are density probability 
distributions (distribution normalised to 1.0), except when explicitely 
specified. These distributions are used to compute a binned-likelihood 
function devoted to the signal/background discrimination. 

\subsubsubsubsection{Acceptance and pre-selection requirements}

\subsubsubsubsubsection{Lepton selection}
\par 
The lepton acceptance is defined to be in the pseudo-rapidity range 
$\rm |\eta_l | < 2.0$. Only electrons or muons with \pt $>15~GeV/c$ 
are kept. 
%---------------------------------------------------------
\begin{figure}[tbp]
    \caption{ Lepton \pt\ and isolation distributions for signal and 
     backgrounds. These 
     distributions are computed for a $\rm m_h$ =160\G Higgs.\label{LEP-1}}
   \hspace{0.2cm}\hbox{
    \parbox{8.0cm}{\epsfxsize=\hsize\epsfbox{highmass/lepton_pt_ini.ps}}
    \parbox{8.0cm}{\epsfxsize=\hsize\epsfbox{highmass/lepton_iso.ps}}}
\end{figure}
\vskip0.1cm
%--------------------------------------------------------------
\par
Electrons are selected with an EM fraction such that $\rm E_{had}/E_{EM} \le 0.125$ 
and an associated track within a 30$^0$ cone around the EM deposition. 
The lepton isolation requires that any jet around the lepton to be outside 
a cone defined by $\rm \Delta R(j,l) > 0.3$ in the $\rm (\Delta\phi, \Delta \eta)$ 
plane. Figure~\ref{LEP-1} displays the \pt lepton spectrum and the 
$\rm \Delta R(j,l)$ for the signal and all the backgrounds.


\subsubsubsubsubsection{Jets selection}
\par 
\noindent
A. Jet acceptance\\
\par
The jet acceptance is defined to be in the pseudo-rapidity range 
$\rm |\eta_l | < 1.0$ to reduce the selection of W+jets events. Those 
events indeed have soft QCD jets, pointing most of the case in the 
towards the end-cap. 
%---------------------------------------------------------
\begin{figure}[tbp]
 \begin{center}
    \caption{ Jets \pt and isolation ($\Delta R$) distributions for signal 
        and backgrounds. These distributions are computed for a \MH =160\G Higgs.}
    \label{JET-1}  
   \hspace{0.2cm}\hbox{
     \parbox{8.0cm}{\epsfxsize=\hsize\epsfbox{highmass/jet12_pt_ini.ps}}
     \parbox{8.0cm}{\epsfxsize=\hsize\epsfbox{highmass/jet12_iso.ps}}}
 \end{center}
\end{figure}
\vskip0.1cm
%--------------------------------------------------------------
\par
Jets are required to have a 5~GeV deposit in the calorimeters, 
and tracks associated within a cone of 30$^0$ around the cluster. 
the selection requires the presence of two jets above 15 GeV/c 
in \pt, and well isolated with $\Delta R(j,j) > 0.7$. This last requirement 
allows us to reject the selection of soft QCD jets present in 
the W+jets events. Figure~\ref{JET-1} displays the jet 1 and 2 \pt for 
the signal compared to backgrounds. We clearly see that the requirement  
reduces the selection of W+jet events. Figure~\ref{JET-1} also shows 
the isolation between the two jets before the cut. Again a good 
rejection is obtained against W+jet events and \TTB events, where 
more than two jets can be seen. \\
\hfill\break
\noindent
B. Extra jet veto\\
\par
In order to reduce the background due to \TTB events, we require 
at this stage that no event with a third jet above \pt $>$25~GeV/c 
is selected. The sensitivity to this cut has been investigated. Lower 
values for this upper bound leads to drastic loss in signal events 
selection. Fig. \ref{JET-3} shows the \pt distribution for a third jet 
in signal and background events. A cut below 10-15 GeV removes 
more than 60$\%$ of the signal events...  
%---------------------------------------------------------
\begin{figure}[tbp]
 \begin{center}
    \caption{ Third jets \pt distribution and sensitivity to third jet veto
      for signal and 
      backgrounds. These distributions are computed for a \MH =160\G Higgs.}
    \label{JET-3}  
   \hspace{2mm}\hbox{
    \parbox{8.0cm}{\epsfxsize=\hsize\epsfbox{highmass/jet3_pt_acc.ps}}
    \parbox{8.0cm}{\epsfxsize=\hsize\epsfbox{highmass/jet34_veto_opt.ps}}}
 \end{center}
\end{figure}
\vskip0.1cm
%--------------------------------------------------------------
The sensitivity to the jet-veto cut for any jet 3 and 4 is displayed in 
Fig.~\ref{JET-3} as function of the jet energy threshold. A cut at 25 GeV/c 
removes efficiently \TTB background events while preserving more than 80$\%$ 
of the remaing signal. 

\subsubsubsubsubsection{Missing energy requirement}
\par
We require Aa threshold on the total missing (transverse) energy of \MET $>$ 
15 GeV/c. This allows us to select at least one leptonic W-boson decay with 
a neutrino. 
\par
Fig. \ref{MET-1} shows the missing transverse energy before the cut 
is applied for signal and all the different backgrounds. The distributions 
show some significant difference in the case of \TTB and WZ events. In the 
first case, the two W's can indeed decay in $l+\nu$ channel.\\  
Fig. \ref{MET-1} also displays the variation of the missing energy distribution 
for different generated Higgs mass. When the Higgs decay into two off-shell 
W's, the \MET spectrum tends to peak at lower values, as expected. In these 
cases, this results in a degradation of the discrimination from backgrounds.
For lower HIggs mass values, this cut is reduced to \MET $>$ 10 GeV/c.  
%---------------------------------------------------------
\begin{figure}[tbp]
 \begin{center}
   \caption{ Missing transverse energy distribution for signal and backgrounds. These distributions are computed for a \MH =160\G Higgs.}
   \label{MET-1}  
   \hbox{
    \parbox{8.0cm}{\epsfxsize=\hsize\epsfbox{highmass/met_ini.ps}}
    \parbox{8.0cm}{\epsfxsize=\hsize\epsfbox{highmass/met_ini_sig.ps}}}
 \end{center}
\end{figure}
\vskip0.1cm
%--------------------------------------------------------------

\subsubsubsubsection{Selection requirements}

The selection is based on the cuts presented so far, to which we add 
the following requirements.

\subsubsubsubsubsection{Di-jet mass requirement}
\par
The di-jet mass $\rm M(j_1,j_2)$, where we keep the combination giving the 
highest mass, is represented in Fig. \ref{DIJET-1}c for signal and background, 
and in Fig. \ref{DIJET-1}b for different Higgs mass samples. It appears clearly 
that the \TTB background is peaked at higher reconstructed masses, as 
expected for b-jets events. Mass spectrum also concentrate to the lower values 
for the QCD jets of the W+jets events. At the same time, the distributions show 
a relative stability for signal events for Higgs masses ranging between 140 and 180\G.
%---------------------------------------------------------
\begin{figure}[tbp]
 \begin{center}
  \caption{ Dijet Mass distribution for signal and backgrounds. These distributions are computed for a \MH =160\G Higgs.}
  \label{DIJET-1}  
  \hbox{
   \parbox{8.0cm}{\epsfxsize=\hsize\epsfbox{highmass/mass_j1j2.ps}}
   \parbox{8.0cm}{\epsfxsize=\hsize\epsfbox{highmass/mass_j1j2_signal.ps}}}
 \end{center}
\end{figure}
\vskip0.1cm
%------------------------------------------------------
\par
\noindent
\subsubsubsubsubsection{Di-jet opening angle}
\par 
Fig. \ref{PHIJJ-1} displays the opening angle of the jets 
for the signal and backgrounds in the transverse plane. 
The angle is expected to be higher for the signal events, whereas 
a flat distribution is observed for all backgrounds.\\
Figure~\ref{PHIJJ-1} also gives this distributions for all Higgs 
masses. It shows that off-shell W's productions tend to 
have a flatter distribution than for a 160\G Higgs. 
\newpage
\begin{figure}[tbp]
 \begin{center}
    \caption{ Dijet opening angle distribution for signal and backgrounds. These distributions are computed for a \MH =160\G Higgs.}
    \label{PHIJJ-1}  
  \hbox{
   \parbox{8.0cm}{\epsfxsize=\hsize\epsfbox{highmass/phijj_acc.ps}}
   \parbox{8.0cm}{\epsfxsize=\hsize\epsfbox{highmass/phijj_acc_sig.ps}}}
 \end{center}
\end{figure}
\vskip0.1cm
%---------------
\par
\noindent
\subsubsubsubsubsection{Cluster mass $\rm M_C$}
\par 
This latter is defined in \cite{TAO} by:\\
\centerline{$\rm M_C = \sqrt{p_T^2(ljj)+m^2(jjl)}$+\MET}
where the transverse mass of the (lepton,jet,jet) system is used together 
with the transverse missing energy to form the mass cluster, corresponding 
rouglhly to the Higgs mass. This variable is represented in Fig. \ref{MCLU-1} 
for signal and background events and as a function of higgs mass.
\noindent
These figures show a good discriminating power against most of the backgrounds, 
in particular \TTB and WW and WZ events.\\

%---------------------------------------------------------
\begin{figure}[tbp]
 \begin{center}
    \caption{ Cluster Mass distribution for signal and backgrounds(left--hand
      panel) and as a function of higgs mass(right--hand panel). }
    \label{MCLU-1}  
   \hbox{
     \parbox{8.0cm}{\epsfxsize=\hsize\epsfbox{highmass/mass_cluster_acc.ps}}
     \parbox{8.0cm}{\epsfxsize=\hsize\epsfbox{highmass/mass_cluster_acc_sig.ps}}}
 \end{center}
\end{figure}
\vskip0.1cm
%--------------------------------------------------------------

\subsubsubsubsection{Signal selection efficiency}
\par
Table \ref{TAB-EFF} reports the results of the acceptance and selection  
cuts on signal events. No likelihood cut is applied at that stage. 
Sample of 10,000 events are used to estimate 
the efficiency for W's on-shell. 
\begin{table}[tbp]
 \begin{center}
  \begin{tabular}{|c|r|r|r|r|}
   \hline
   \hline
  $\rm M_h$ ($GeV/c^2$)& $\rm 120$      & $\rm 140$  & $\rm 160$  & $\rm 180$ \\
   \hline 
   \hline
  Initial \# events                 & 3,000     &  3,000   & 10,000   &  3,000  \\
  Reconstruction                    & 1,730     &  1,924   &  6,440   &  2,015  \\
  Lepton (\PT, $\rm \eta,\Delta R$) & 1,455     &  1,758   &  6,162   &  1,955  \\
  Jets:   (\PT and $\eta,\Delta R$) &   371     &    613   &  2,726   &    906  \\
  Missing Energy cut                &   248     &    510   &  2,557   &    831  \\
  Jet Veto Cut                      &   230     &    443   &  2,074   &    674  \\
  \hline
  \hline
  Pre-selection  ($\%$)          &8.3$\pm$0.5$\%$& 14.8$\pm$0.6$\%$ & 20.7$\pm$0.4$\%$
 & 22.5$\pm$0.8$\%$ \\ 
   \hline      
   \hline
  40 $<$ M(j,j) $<$ 110     &            98    &    234    &  1,522    &  488  \\ 
   \hline      
  140$^0$ $< \Phi(j,j)$     &            98    &    234    &  1,520    &  487   \\   
   \hline      
 $\rm 40 < M_C < M_h+10~GeV$&            82    &    184    &  1,365    &  336   \\   
   \hline
   \hline
 Full Selection ($\%$) &         2.7$\pm$0.3$\%$   &6.1$\pm$0.4$\%$& 13.6$\pm$0.3$\%$ & 11.2$\pm$0.6$\%$ \\   
   \hline
   \hline
  \end{tabular}
  \caption{ Pre-selection efficiency for the signal events generated for four different Higgs masses}
 \label{TAB-EFF}
 \end{center}
\end{table}
\vskip0.1cm
Efficiency is found to be 13.6$\%$ for \MH=160\G. 
For lower Higgs masses, efficiencies are degraded because W's are not 
produced off-shell, which result in a lowering of the average lepton and 
jet \pt, and therefore in a lower pre-selection efficiency. For \MH=180\G, 
the loss in efficiency is due to the increase in the reconstructed 
cluster mass (and di-jet) as seen in Fig. \ref{MCLU-1}, because of extra 
jet-momentum.  

\subsubsubsubsection{Background rejection}

Table \ref{TAB-EFF} reports the results of the acceptance and pre-selection 
cuts on background events. High rejections are obtained for the main 
backgrounds in term of selection efficiency.  
\begin{table}[tbp]
 \begin{center}
  \begin{tabular}{|c|r|r|r|r|}
   \hline
   \hline
  Backgrounds          & W+jets            &  \WW              &  \WZ         & \TTB \\
  \hline
  Pre-selection ($\%$) & 1.36$\pm$0.04$\%$ &  17.8$\pm$0.4$\%$ & 15.0$\pm$0.6$\%$ &9.7$\pm$0.3$\%$ \\          
   \hline
   \hline
 Full Selection ($\%$) & 0.30$\pm$0.02$\%$ &  5.5$\pm$0.2$\%$  & 3.7$\pm$0.3$\%$ & 0.8$\pm$0.1$\%$ \\
   \hline
   \hline  
  \end{tabular}
 \end{center}
\end{table}
\vskip0.1cm

\begin{itemize}
 \item \WW and \WZ events: these backgrounds are pre-selected with 
       efficiencies comparable to the one obtained for the signal. 
       These events have comparable shapes to the signal and are very 
       difficult to discriminate from Higgs signal. However, 
       some differences are seen in the opening angle between the jets. 
 \item \TTB events are rejected in the pre-selection mostly by the 
       jet veto requirement. The cluster mass and dijet mass 
       reconstructed tend also to be higher than the one reconstructed 
       for our signal. 
 \item \WJET events have a different topology than Higgs events. 
       They are thus well rejected at the 3/1,000 level. The most efficient 
       requirements are the jet-acceptance cut and the jet \PT cuts. The 
       dijet mass cut, as well as the dijet opening angle also bring a factor 
       5 on already pre-selected events. However, given the cross-section of
       this background, the 3/1,000 factor still appears insufficient. 
\end{itemize}
The estimation of the efficiency in the background rejection is given by 
the luminosity needed to ensure a $3~\sigma$ or $5~\sigma$ significance 
discovery. Table \ref{TAB-LUMI} reports the ratio $S/\sqrt{B}$ for all 
individual 
backgrounds to the \MH=160\G signal samples (first line) computed for an 
integrated luminosity of 1~fb$^{-1}$. 
The luminosity needed to get a 3 or a 5 $\sigma$ significance is 
indicated in line 2. If one sums over all the backgrounds, the ratio 
$\rm S/\sqrt{B}$ is given by:
$$ 
\rm {S\over\sqrt{B}} = \sqrt{{\cal{L}}}\times {\sigma_S\times \epsilon_S \over
   \sqrt{\sigma_{wjet}\times \epsilon_{wjet}+\sigma_{WW}\times \epsilon_{WW}+
   \sigma_{WZ}\times \epsilon_{WZ}+ \sigma_{t\bar{t}}}\times
   \epsilon_{t\bar{t}} } = \sqrt{{\cal{L}}}\times 0.117 
$$
and the luminosity needed for a $3~\sigma$ discovery (no likelihood cut used)  
therefore is $ \rm  {\cal{L}} = 660 ~fb^{-1} $!!
\begin{table}[tbp]
 \begin{center}
  \begin{tabular}{|c|r|r|r|r|}
   \hline
   \hline
  Backgrounds              & W+jets         &  \WW              &  \WZ         & \TTB \\
  \hline
  $\rm S/\sqrt{B}~per~fb^{-1}$ 
                           & 0.12           &    0.89           &  3.16   & 3.69 \\
   \hline
  ${\cal{L}}$ for $3~\sigma$ in \FB &  610  &    11.2           &  0.9    & 0.7   \\
  \hline
  \hline
  \end{tabular}
  \caption{ 1st line: Ratio $\rm S/\sqrt{B}$ for a 
    \MH = 160\G higgs and the four main background, computed for a 
     luminosity of 1 \FB. Both electron and muon channel are considered. 
      Line 2: Needed luminosity to reach a $3\sigma$ significance.}
 \label{TAB-LUMI}
 \end{center}
\end{table}
\vskip0.1cm
This shows that the main constraint comes from the W+jets background, because
of its huge cross-section in ppbar collision.  In order to improve our
discrimination, we therefore need to apply further cuts. We choose to use a
likelihood cut, described, in the next section.

\subsubsubsection{Selection using a likelihood method}
\par
The traditionnal method does not bring enough rejection in background 
events (mainly the \WJET events) because it does not fully take into 
account the differences in the topology between signal and background 
events. We choose here to apply the likelihood technique, based on all 
the variable distributions presented in the previous section. This 
technique is explained in details in \cite{OPAL}.
 
\subsubsubsubsection{The Likelihood function}
\par
The likelihood function is based on the density probability 
functions of the parameters used, computed for all backgrounds 
and signal. We first define the set of parameters used, and then 
present the likelihood definition.

\subsubsubsubsubsection{Discriminant parameters used}
 \par
Here is the set of variables we use to build this likelihood function:
\begin{enumerate}
  \item Jet 1,2 variables: $\rm p_T^{j1}, p_T^{j2}, \Delta R_{jj} \Rightarrow v_1,v_2,v_3$
  \item Lepton variable: $~~~~~~~~~~~~~\rm P_T^l \Rightarrow v_4$
  \item Missing energy : $~~~~~~~~~~~~$\MET $\rm \Rightarrow v_5$
  \item Jet opening angle: $\rm ~~~\Delta \Phi (j,j) \Rightarrow v_6$
  \item Jet 3/4         : $~~~~~~~~~~~~~~~~~~~~~~~~~\rm P_T^{j3} \Rightarrow v_7$
  \item Jet-jet Mass   : $~~~~~~~~~\rm M(j_1,j_2)\Rightarrow v_8$
  \item Cluster Mass   : $~~~~~~~~~~~~~~\rm M_C \Rightarrow v_9$ 
\end{enumerate} 

\subsubsubsubsubsection{Building of the Likelihood function}
\par
The likelihood method is explained in detail in \cite{OPAL}. 
After defining the discriminant variables, the method is based on the MC 
probability density functions observed for signal and all different 
background events. 
Given an event, we then compute the probability using each variables that the 
event is a signal or a background event. The total probability for the event 
to be a signal event for the variable $\rm v_i$ is then given by:
 $$\rm p_{v_i}^S(x_i) = {v^S_i(x_i) \over \sum_{j=S,B} v^j_i(x_i) } $$
where
$\rm v^{S/B}_i(x_i)$ is the probability density function to be a signal (S) or a 
background (B) associated to the variable $\rm v_i $.
We then compute the likelihood fr these event to be signal using all discriminant  
variables $\rm \vec{x}$:
$$\rm {\cal{L}}(\vec{x})= { \prod_{i=1}^{n} p_i^S(x_i) \over \sum_{j=S,B}\prod_{i=1}^{n} p_i^j(x_i)}$$ 

\subsubsubsubsection{Likelihood for signal and background events}
\par
Fig. \ref{LIK-1} shows the likelihood normalised to one for the 
signal and all four backgrounds on an extende scale. 
The signal is clearly concentrated in higher likelihood 
bins. W+jets events appear peaked in the first bins, with 
about less than 1/1,000 events with a likelihood above 0.7. 
WW and WZ events being topologically more similar to our signal 
go to higher likelihood values up to 1.0. This constitues an 
irreducible background as long as no identification of the 
fermion flavour is performed.    
%---------------------------------------------------------
\begin{figure}[tbp]
 \begin{center}
  \begin{sloppypar}
   \parbox{8.0cm}{ 
    \mbox{ 
    \epsfxsize=\hsize\epsfbox{highmass/likelihood_9p_norm.ps}
         } 
    \caption{ Likelihood for signal and backgrounds. All the selection cuts are applied.}
    \label{LIK-1}  
         }
   \hspace{0.2cm}
   \parbox{8.0cm}{ 
    \mbox{ 
    \epsfxsize=\hsize\epsfbox{highmass/likelihood_9p_10fb.ps}
         } 
    \caption{ Likelihood in number of events for signal and backgrounds. All cuts are applied}
    \label{LIK-2}  
         } 
  \end{sloppypar}
 \end{center}
\end{figure}
\vskip0.1cm
%--------------------------------------------------------------
Fig. \ref{LIK-2} displays the events likelihood in event numbers for 
an integrated luminosity of 10 \FB after normalising properly by the 
production cross-sections. \\
It turns out that the likelihood cut must be set above 0.80 to avoid 
the selection of W+jets events which constitute the most lethal background. 
We haven't use the distinction between WW and our signal's topology provided 
by the angular W decay's products correlations. With this likelihood function, 
the presence of a SM Higgs may only be seen as an extra number of events 
over the WW distributions. Table \ref{TAB-EXPECTED} reports the cross-sections 
for a \MH=160\G Higgs and the total background. This latter is composed 
by the \WW events for likelihood above 0.8. 
\begin{table}[tbp]
 \begin{center}
  \begin{tabular}{|c|c||c|c|c|c|c|}
   \hline
   \hline
 Cross-section   &   Higgs         &  \multicolumn{5}{c|}{Backgrounds}  \\[1mm]
 (in \FB)        &   ($m_h$=160\G) & W+jets   &  WW  & WZ   &  $\rm t\bar{t}$  & TOTAL       \\
   \hline 
   \hline
 Likelihood Cut  &                 &          &       &            &       &          \\
  L $>0.80$ &     0.32$\pm$0.03    &  ----    &  1.75 & ----   &  ---- &  1.75$\pm$1.7        \\
  L $>0.85$ &     0.15$\pm$0.01    &  ----    &  0.54 & ----   &  ---- &  0.54$\pm$0.05        \\
  L $>0.90$ &     0.06$\pm$0.01    &  ----    &  0.27 & ----   &  ---- &  0.27$\pm$0.03        \\
   \hline
  \hline
  \end{tabular}
  \caption{ Expected cross-section (in \FB) for signal and background event after the application of the likelihood cut. The uncertainties are quoted assuming a 10$\%$ error in the evaluation of the efficiency.}
 \label{TAB-EXPECTED}
 \end{center}
\end{table}
\vskip0.1cm
For a likelihood L $>0.80$, the ratio \SB = 0.242 $\times ~\sqrt{\cal{L}}$. 
A 3$\sigma$ significance therefore requires a luminosity of 
${\cal{L}} = 150~fb^{-1}$, while a $2~\sigma$ significance exclusion needs 
${\cal{L}} = 65~fb^{-1}$.
 
\subsubsubsubsection{Results for \MH = 120\G to 180\G}
\par
The analysis is optimised for a 160\G Higgs and shows a decrease in the 
performance for off-shell W decays, as seen in the selection efficiencies. 
Table \ref{TAB-EXPECTED-OFF} report the cross-sections expected for signal 
and backgrounds for a likelihood greater than 0.80. As expected, the sensitivity 
to the signal is degraded: below the W threshold, the topology changes become 
significant (lepton and missing energy) and lead to a decrease in the 
discrimination against the W+jet background. Above the W threshold, the 
Cluster Mass and jet invariant masses distributions are changed such that 
the $t\bar{t}$ background become more important.  
\begin{table}[tbp]
 \begin{center}
  \begin{tabular}{|c|r|r|r|}
   \hline
   \hline $\rm M_h$    & $\rm 120~GeV$  & $\rm 140~GeV$   & $\rm 180~GeV$    \\
   \hline
   Signal (fb)         & 0.12$\pm$0.01 & 0.30$\pm$0.03  & 0.39$\pm$0.04  \\
   \hline
   Background (fb)     & 1.96$\pm$0.20 & 1.80$\pm$0.20  & 3.18$\pm$0.32   \\
  \hline
  \end{tabular}
  \caption{ Expected cross-section (in \FB) for signal and background event after the application of the likelihood cut. The uncertainties are quoted assuming a 10$\%$ error in the evaluation of the efficiency.}
 \label{TAB-EXPECTED-OFF}
 \end{center}
\end{table}


\subsubsubsubsection{Limits on Higgs discovery}
\par 
Figure \ref{RATIO-1} shows the ratio \SB as a function of the integrated 
luminosity for \MH=160\G. It turns out that a 3$\sigma$ significance 
discovery in this channel alone requires an integrated luminosity above 
160 \FB. However, it seems that a 2$\sigma$ significance (for 95$\%$ CL 
exclusion) may be reached with 2 experiments accumulating 30 \FB each. 
%---------------------------------------------------------
\begin{figure}[tbp]
 \begin{center}
  \begin{sloppypar}
   \parbox{8.0cm}{ 
    \mbox{ 
    \epsfxsize=\hsize\epsfbox{highmass/significance_mh160.ps}
         } 
    \caption{ Significance as function of the luminosity for a Higgs mass signal of 160 GeV. Shown are the effects of a 5\% and 10\% systematics on the efficiency+background estimations.}
    \label{RATIO-1}  
         }
  \end{sloppypar}
 \end{center}
\end{figure}
\vskip0.1cm         
%--------------------------------------------------------------------------------


\subsubsubsection{Summary of results}
\par 
In this note, we investigate the potentiallity of a SM Higgs 
discovery in the  \HWW channel at the TeVatron run II.
In this channel, the main background comes from the W+jets events, 
produced in the ppbar collions with a cross-section 
25,000 times our signal cross-section. This background is reduced 
to a negligible level by using a combination of pre-selection and selection 
cuts, together with a likelihood method. The other important background 
comes from the W-pair production, which appear irreducible if one 
does not use the angular correlation between the lepton and the 
quark issued from the $\rm W^+W^-$ decays of the Higgs (spin 0).\\  
Monte Carlo studies show that a $\rm 3 \sigma$ significance 
effect needs an integrated luminosity of 150 $\rm fb^{-1}$, whereas  
a $95\%$ CL exclusion requires a luminosity of about 60 $\rm fb^{-1}$, for 
a 160\G Higgs. We conclude that this channel has to be used in combination 
with the associated Higgs production channel.


