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\subsection{Direct $\chi^0_2 \chi^\pm_1$ production in tri-leptons}
%-------Introduction: signal signatures
Trilepton final state in mSUGRA  
appears from the direct neutralino-chargino        
production $pp\rightarrow \chi_2^0\chi_1^{\pm}$   
with  subsequent three body decays 
of neutralino; 
$\chi_2^0\rightarrow \chi_1^0Z^*\rightarrow  \chi_1^0ll$ and chargino; 
~$\chi_1^\pm\rightarrow \chi_1^0W^*\rightarrow \chi_1^0l\nu$
or  via sleptons in two body decay;
$\chi_2^0\rightarrow l\tilde l\rightarrow l\chi_1^0 l$  and
~$\chi_1^\pm\rightarrow l\tilde\nu\rightarrow l\chi_1^0\nu$,
~$\chi_1^\pm\rightarrow\nu\tilde l\rightarrow \nu\chi_1^0 l$.
The trileptons can be also  produced by the  heavier neutralino-chargino
 $\chi^0_{3,4} \chi^\pm_2$ although with much smaller cross section.
The final signatures are  the two  
Opposite Sign  Same Flavor (OSSF) leptons (electrons or muons)
from  the neutralino $\chi_2^0$ 
decay plus any lepton from the chargino $\chi_1^{\pm}$ decay 
and a missing transverse energy  (MET)  
from  two   neutralinos and  neutrino. 
Jets appear only due to radiation of quarks,  no central high $E_T$ jets are expected.
The two body decays are going mostly  via staus which decays consequently to $\tau$,
the leptons from the  $\tau$-decay are soft and require a special analysis ({\sl reference to PTDR chapter?}).
%----Invariant mass
\par
The invariant mass of the OSSF dileptons exhibits  
a particular triangular shape
with the  kinematic end point which depends upon the event 
topology and  can be either  
$M_{ll}^{max}$=$m_{\chi_2^0}-m_{\chi_1^0}$ for three body or 
$M_{ll}^{max}$=$\sqrt{(m^2_{\chi_2^0}-m^2_{\tilde l})(m^2_{\tilde l}-m^2_{\chi_1^0})/m^2_{\tilde l}}$ 
for two body decays, see figure \ref{minv}. ({\sl probably already discussed in the dilepton sections}) 
\par
%------Cross section
The neutralino-chargino mass spectrum and their  
productions-decay  rates are determined  in mSUGRA  by 
 $m_{1/2}$ parameter: $m_{\chi^0_1} \sim 0.4 m_{1/2}$,  
~ $m_{\chi^0_2}\sim m_{\chi_1^\pm}\sim  0.8 m_{1/2}$ and
$\sigma(\chi^0_2\chi_1^{\pm})\sim m_{1/2}^{-4}$ , therefore
the mass difference  $m_{\chi_2^0}-m_{\chi_1^0}$ is directly  proportional to the  $m_{1/2}$.
({\sl reference to the mSUGRA description }).
Figure ~\ref{cs3l} shows the trilepton cross section  for different tan$\beta$.
The low  $m_{1/2}$ region, where the $\chi^0_2 \chi^\pm_1$ production is large, 
covers two distinct regions:
low  $m_0<m_{1/2}$ bulk region, where scalars  are light and contribute the most to the
total SUSY  production cross section, and the  large  $m_0>m_{1/2}$ region  with the 
dominant  gluino production along the EWSB limit.
The two body decays are allowed only in very narrow band at low $m_0$ and low tan$\beta$.
At low  low $m_0$  the $m_{1/2}$ is constrained  by the light Higgs  mass limit  ($m_h>114.4$ GeV)
and   limits on  $b\rightarrow s\gamma$  branching ratio. The limit on the chargino mass ($m_{\chi_1^\pm}>103$ GeV)
defines the low value of  $m_{1/2}>150$ GeV.
\par
%-----Benchmark points
The summary of the trilepton  cross sections calculated with  ISAJET 7.69 and PYTHIA 6.225 
in LO  for  the CMS LM benchmark points  is presented in  table~\ref{lm}.
The NLO cross section depends on the neutralino-chargino masses and  is higher by K$_{NLO}\sim$1.3 \cite{signalnlo}. 
The trilepton signal has non negligible cross section for LM9, LM7, LM3 (three body) and LM1, LM6 (two body) points.
For LM4 and LM8 the neutralino decays via on shell Z and can not be distinguished from the background.
At  LM2 the neutralinos and  charginos decay mostly into stau.
One  can expect  for LM9 point $\sim$3700 events(NLO) at   $L_{int}$=30fb$^{-1}$.
\par
%--------Backgrounds
The main background  for  the pure leptonic state is coming from  
the Standard Model(SM) ZW pair  production
and ZZ with one missed lepton. All other SM backgrounds, such as $t\bar t$, 
Z+jets, W+jets etc. are semileptonic
and appear due to missed jets. The  missed jets are also important   for the  SUSY  background.
The summary of  the backgrounds cross sections,  K$_{NLO}$ factors and expected statistics
at L$_{int}=30fb^{-1}$ is presented in table \ref{sumdata}.

\begin{table}[t]
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|} \hline
     &$m_{1/2}$ & $m_0$  & tan$\beta$ & $\sigma_{tot}$ [pb] &  $\sigma_{3l}$ [pb] \\ \hline
LM1 & 250  &  60 & 10  & 42   & 4.1 10$^{-2}$    \\
LM2 & 350  & 175 & 35  & 7.3  & 0.      \\
LM3 & 240  & 330 & 20  & 31   & 9.4 10$^{-3}$   \\
LM4 & 285  & 210 & 10  & 19   & 9.0 10$^{-3}$   \\
LM5 & 360  & 230 & 10  & 6    & 0.4 10$^{-3}$     \\
LM6 & 400  & 85  & 10  & 4    & 9.8 10$^{-3}$    \\
LM7 & 230  & 3000 & 10 & 8.4  &  3.9 10$^{-2}$  \\
LM8 & 300  & 500  & 10 & 8.9  &  7.9 10$^{-2}$ \\
LM9 & 175  & 1450 & 50 & 25   &  9.5 10$^{-2}$ \\   \hline                             
\end{tabular}
\end{center}
\caption[]{Total and trilepton cross sections(LO) for CMS benchmark points}
\label{lm}
\end{table}

\begin{table}[t]
\begin{center}
\begin{tabular}{|c|c|c|c|c|} \hline
     & $\sigma^{LO}_{tot}$ [pb] & K$_{NLO}$  & $\sigma^{NLO}_{3l}$ [pb]  &  N, 30$fb^{-1}$    \\ \hline
ZW   & 30                 & 1.72     &  1.7                      & 5 10$^4$         \\
ZZ   & 11.8               & 1.34     & 0.16                      & 4800             \\
$t\bar t$ &486            & 1.71     & 88                        & 2.6 10$^6$        \\
Z$b\bar b$ & 746         & 2         & 149                       & 4.4 10$^6$      \\   
Wt+jets      & 60          & 1.7      & 10                       &   3 10$^5$     \\
susybkg(LM9)   & 25       & 1.6      &   13                      &   4 10$^5$     \\ \hline
\end{tabular}
\end{center}
\caption[]{Summary of the background cross sections.}
\label{sumdata}
\end{table}
({\sl the ZWjets inclusive  and QCD has to  be added }).
\par
%-----Simulation
The full data samples of 30$fb^{-1}$ for LM9 benchmark point and ZW, ZZ  were simulated 
with the full CMS simulations( ORCA DST). The full samples for other backgrounds and SUSY scan were sumulated
with the fast simulation (FAMOS). The default reconstruction algorithms have been used.
({\sl can be extended  or more references to the Vol I.}).
\par
%-------Selection and Rejection 
The ZW and ZZ background is analogous to the $\chi^0_2 \chi^\pm_1$ production and can be distinguished
only by the leptons invariant mass with a characteristic Z peak.
The semileptonic backgrounds can be effectively suppressed by the vetoeing   jets in central region, 
as shown in figure \ref{all}.
The upper MET cut  which is very effective in other SUSY channels, fails here as the produced
neutralino and chargino are relatively light. Contrary the low MET cut can help to reject
Z/W jets and QCD backgrounds, figure  \ref{all}. ({\sl more details here})
The following  criteria have been used for the event selection:
 \begin{itemize}
\item  At least three isolated  leptons in $|\eta|<2.4$ with  a transverse  momentum
for muons (electrons) above  $P_T>$10 GeV ($P_T>$15 GeV).
Among them  two OSSF leptons  with the invariant mass  $<$ 75 GeV  
to avoid  the  Z peak  region. 
The muons isolation requires no tracks with $P_T>1.5$ GeV and no energy deposition in calorimeter $E_T>5$ GeV
in the cone $R_c=\sqrt{\Delta\eta^2+\Delta\phi^2} <0.3$. The electrons require no trackes in the same cone and
ratio of energy depositions $\frac{E_{HCAL}}{E_{ECAL}} \le 0.6$.
\item  No  Jets  with $E_T>20$ GeV in $|\eta|<2.4$.
\item MET$>$ 18 GeV. ({\sl preliminar}).
 \end{itemize}
%---about Trigger
These cuts corresponds to the dilepton L1 trigger menu: dimuons  with $P_T>$3 Gev and
dielectrons with  $P_T>$12 Gev at low luminosity. ({\sl here more about triggere study with efficiencies }).
The dimuons trigger efficiency  is $99.8\%$  and
for dielectrons $85\%$.({\sl  very preliminary numbers })
At  high luminosity the  'No Jets' selection criteria will cut part of the signal ($43\%$) 
as well as the background partially reducing the significance. ({\sl evaluate about backgrounds })
\par
%----- Invariant mass
The dilepton invariant mass can be reconstructed from all OSSF pairs or
one can select the highest $P_T$ pairs. The use of all combinations increases the efficiency
for the signal but also moves background from ZZ and ZW to the low invariant mass region decreasing significance,
therefore high  $P_T$ pairs have been used.
Figure \ref{minvsigbkg} shows the distribution of dimuon and dielectron invariant mass of  high  $P_T$ pairs
for  LM9 benchmark point and the backgrounds. 
The expected significance for dimuons for the LM9 point is Scl$\sim$10 and 
Scl$\sim$5  for the LM7 ({\sl preliminar values with MET cut}).
The dielectrons signal is factor of two worser  due to higher  $P_T$. ({\sl in fact this is related to trigger})  
The CMS discovery reach for dimuons is shown in figure \ref{scansugra}.
For low $m_0<1000$ GeV it is limited by the constrains on the light Higgs mass, low $m_{1/2}$ are excluded due to
the limit on the chargino mass and large $m_0$ are limited by the EWSB requirements.
The onset of the EWSB for the low tan$\beta=10$ is starting at 
larger $m_0\sim 2500$ GeV increasing the  allowable region.
At the large $m_{1/2}>220$ GeV  in addition to the rapidly decreasing cross section, the dileptons
invariant mass is moving into the Z peak making signal invisible.

%---More summary?
{\sl More conclusions? }


{\sl check the biblio general list }
\begin{thebibliography}{99}
\bibitem{signalnlo}  W.~Beenakker et al.,~hep-ph/9906298. 
\end{thebibliography}
\newpage
% plots
% dileptons
\begin{figure}
  \begin{center}
    \includegraphics [width=0.8\textwidth]{Plots/minvdileptons.eps}
    \caption[]{Reconstructed invariant mass of OSSF dileptons for three and two  body decays in mSUGRA.} 
\label{minv}
  \end{center}
 \end{figure}
% cs
 \begin{figure}
  \begin{center}
    \includegraphics[width=1.0\textwidth]{Plots/cs3l.eps}
    \caption[]{Trilepton cross section for two  values of tan$\beta$. {\sl add constrains here}.} 
      \label{cs3l}
  \end{center}
 \end{figure}
% selection
 \begin{figure}
  \begin{center}
    \includegraphics[width=1.0\textwidth]{Plots/all.eps}
    \caption[]{Number of selected jets per event(left) and  MET distribution  for signal LM9 and backgrounds.} 
      \label{all}
  \end{center}
 \end{figure}
% inv mass
\begin{figure}
\begin{center}
    \includegraphics[width=1.0\textwidth]{Plots/invmass_met.eps}
    \caption[]{The  dimuons (left) and dielectrons (right) invariant mass of the two OSSF highest  $P_T$ leptons  for the LM9 point and backgrounds.} 
    \label{minvsigbkg}
  \end{center}
\end{figure}
% scan
\begin{figure}  
\begin{center}
    \includegraphics[width=1.0\textwidth]{Plots/scansugra.eps}
    \caption[]{The trileptons (2$\mu$+l) significance  for tan$\beta$=10(left) and 50 (right)
 at  30 fb$^{-1}$.} 
    \label{scansugra}
  \end{center}
\end{figure}

\end{document}