\mysection{Direct $\chiz_2\chipm_1$ production in tri-leptons
 \label{susy:trilept}}
\contributor{V.Zhukov}

%---introduction
\mysubsection{Trilepton final state in mSUGRA}
The trilepton final state in mSUGRA  
appears in direct neutralino-chargino        
production $pp\rightarrow \chiz_2\chipm_1$   
with  subsequent three body decays 
of the second neutralino, 
$\chiz_2\rightarrow \chiz_1 Z^*\rightarrow  \chiz_1 ll$, and chargino,
~$\chipm_1 \rightarrow \chiz_1 W^*\rightarrow \chiz_1 l\nu$;
or  via sleptons in two body decay,
$\chiz_2 \rightarrow l\sLep \rightarrow l\chiz_1 l$,  and
~$\chipm_1 \rightarrow l\sNu \rightarrow l\chiz_1\nu$, 
~$\chipm_1 \rightarrow\nu\sLep \rightarrow \nu\chiz_1 l$.
Trilepton events can be also  produced by the  heavier neutralino-chargino
 $\chiz_{3,4} \chipm_2$, although with much smaller cross section.
The final signatures are  the two  
Opposite-Sign  Same-Flavor (OSSF) leptons (electrons or muons)
from  the neutralino $\chiz_2$ 
decay plus any lepton from the chargino $\chipm_1$ decay, 
and appreciable missing transverse energy  (MET)  
from  two neutralinos and  neutrino. 
Jets appear only due to initial state  radiation ,  no central high $E_T$ jets are 
expected. The invariant mass of the OSSF dileptons exhibits  
a particular triangular shape
with a  kinematic end point that depends upon the event 
topology: either  
$M_{ll}^{max}$=$m_{\chiz_2}-m_{\chiz_1}$ for three body, or 
$M_{ll}^{max}$=$\sqrt{(m^2_{\chiz_2}-m^2_{\sLep})(m^2_{\sLep}-m^2_{\chiz_1})/m^2_{\sLep}}$ 
for two body decays.


\mysubsection{Signal and backgrounds cross section}
%---signal cross section
The trilepton   cross section   was calculated with ISASUGRA 7.69
and  PYTHIA 6.225(CTEQ5L) at LO,  
the NLO K factor was calculated with PROSPINO and is decreasing from 1.30 at 
$m_{\chiz_2}=150$ GeV/c$^2$ to 1.25 at  $m_{\chiz_2}=300$ GeV/c$^2$ \cite{Beenakker:1999xh}. 
Figure ~\ref{susy:trilept.cs} shows the cross section in the  $m_0$,  $m_{1/2}$ plane for the 
tan$\beta$= 10 and 50. The signal cross section rapidly drops with the mass $m_{\chiz_2}\sim 0.8m_{1/2}$
except the region close to the EWSB,  where the mSUGRA coupling $\mu$ is getting small.
The three body decays are dominant  over most of the  $m_0$,  $m_{1/2}$ plane except for    \tanb $\le 10$ 
and  $m_0<$100 GeV. The kinematic end point in the invariant mass is  
$M^{max}_{ll}\sim 0.42*m_{1/2}-18.4$ \GeV~   (for $m_0\sim 1000$), thus moving into the Z-peak 
for  $m_{1/2}>$250 \GeV. 
\begin{figure}[!Hhtb]
  \begin{center}
    \includegraphics[width=0.48\textwidth]{susybsm_susy/trilept_tb10_cs}
     \includegraphics[width=0.48\textwidth]{susybsm_susy/trilept_tb50_cs}
\caption[]{Trilepton cross section (LO)  from direct neutralino-chargino production 
for two  values of \tanb=10 (left) and 50 (right).
\label{susy:trilept.cs}} 
  \end{center}
 \end{figure}
The  signal has non negligible cross section for the LM9, LM7, LM3 (three body)
and the LM1, LM6 (two body) benchmark points, see Figure \ref{susy:trilept.minv}.
The  LM9 point  has the largest cross  section and was used as a reference: $\sim$3700 events are expected for 
 L$_{int}$=30 \fbinv.
\begin{figure}[!Hhtb]
  \begin{center}
\includegraphics[width=0.48\textwidth]{susybsm_susy/trilept_minvmc}
\includegraphics[width=0.48\textwidth]{susybsm_susy/trilept_minv}
\caption[]{Left:Invariant mass and cross sections (NLO) of OSSF leptons from $\chiz_2$  decay  
for different benchmark points at generator level ($P^{\mu}_T(P^e_T)>3(5)$\GeVc, $|\eta | <2.4$ )
Right: Invariant mass after  reconstruction (all OSSF combinations).
\label{susy:trilept.minv}} 
  \end{center}
 \end{figure}

%----Backgrounds
The main background to the trileptonic state results from the SM 
DY, Z+jets, ZW, ZZ, ${\rm t \bar t}$,  ${\rm Wt}$+jets channels 
and from the SUSY where the OSSF leptons are coming not from the  $\chiz_2$.
The background cross sections  have been calculated  at LO with   PYTHIA and  
TopReX(${\rm t\bar t}$, ${\rm W t}$) and 
corrected to NLO with the  K factors. The $\rm Z$ and $\rm W$ bosons  were  
forced to decay leptonically to $e,\mu$ and $\tau \rightarrow e,\mu $.
For the  ${\rm Z}$+jets and  DY three leptons with  P$_t>$5 \GeVc and $|\eta|<$2.4
were preselected at the generator level. The third lepton for these reactions
is coming  from the  ISR $b(b\bar)$-quark ($\sim91\%$).
A summary of  the backgrounds cross sections is presented in Table \ref{susy:trilept.sumdata}.

\mysubsection{Events simulations and reconstruction}
%----Simulation
The data samples of 30 \fbinv for the LM9 test point,  ZW, ZZ 
backgrounds are simulated with the full CMS simulation (DST ORCA_8_7_3). 
The full-sized samples for other backgrounds and SUSY scans 
are simulated in FAMOS_1_3_2.
The low luminosity pile-up  events have been added to the signal
and background samples. 
%PTDR: reference to ORCA
%---Reconstruction
%-trigger
{\bf Trigger} \\
All  events have required to pass L1 and HLT global  triggers.
For the LM9 events the single muons (64$\%$) and the dimuons(41$\%$)  are the main L1  trigger channels 
where the trilepton state is  expected. The single electron (50$\%$)  and dielectrons (29$\%$)  streams are less populated 
due to higher thresholds, P$^{\mu}_T>3$\Gevc and  P$^{e}_T>17$\Gevc.     
After HLT and the offline selection, discussed below, the dimuons(dielectron) contribute
70$\%$(24$\%$) to the selected LM9 sample. 
The global trigger efficiency is  $E(L1+HLT)=86+91\%$ for LM9 and it is increasing for larger  $m_{1/2}$.
%-leptons
\par
{\bf Leptons} \\
At least three isolated leptons with $|\eta|<2.4$  have been required for each  event. Among them
two  OSSF pair with the transverse momentum: $P_T^{\mu}>$10 \Gevc and $P_T^{e}>$17 \Gevc. 
The third lepton should have  $P_T^{\mu,e}>$10 \Gevc.
The leptons have been reconstructed using  the standard reconstruction algorithms and leptons isolations.
For muons the  isolation requires the sum of the $P_{T}$ of reconstructed tracks( $P_T>0.8$ \GeVc)  $\sum P_T>1.5$ \GeVc~and 
no energy deposition in calorimeter $\ET>5$ \GeV~
in a cone $R=\sqrt{\Delta\eta^2+\Delta\phi^2} <$0.3. 
the electron candidate should fulfill the following requirements: $\frac{E_{HCAL}}{E_{ECAL}}<0.05$, $E/P\in [0.9,1.5]$, $|1/E-1/P|<0.02$.
The electrons isolation by tracks  requires $\sum P_t <$1.5 GeV/c in the cone $\Delta R <$0.3 similar to the muons.
The muons and electrons  reconstruction efficiency after isolation in ORCA  for the LM9 point is 
$78\%$(P$^{\mu}_T>$5 \Gevc) and  $66\%$(P$^e_T>$10 \Gevc) respectively.
%The contamination to the selected leptons was estimated by matching the leptons generated in PYTHIA in the cone  of  $\Delta R <$0.2.
%For the Z+jets, ${\rm t\bar t}$ and ZW background this conatmination is  below   $3~10^{-6}$ per event for muons and 
%$\sim 7\cdot10^{-5}$ for electrons. 
%-jets MET
\par
{\bf Jets and MET} \\  
The central jets veto  requires no jets with $E_T>30$ \GeV in $|\eta|<2.4$.
The jets have been reconstructed by  the {\sl Iterative Cone} algorithm  with the  
seed energies  $E_T^{seed}>0.5$GeV and  $E^{seed}>0.8$GeV in the cone $\Delta R <$0.5.
The jets which match  with an electron candidate have been excluded.
The missing  transverse energy(MET)  was reconstructed from the ECAL and HCAL towers.
The reconstructed MET for the signal and most of  backgrounds is comparable with the   MET resolution($\sim$ 30\GeV).
\mysubsection{Event selection}
%--- Event selection 
The events passed the trigger have been selected  in two steps. 
First the selection cuts have been applied:\\
- No central jets with  $E_T>30$ \GeV in $|\eta|<2.4$.\\
- Two OSSF isolated leptons ($e,\mu$) in  $|\eta|<2.4$  with   $P_T^{\mu}>$10 \Gevc, $P_T^{e}>$17 \Gevc and
 the invariant mass  below Z peak  $M_{ll}<$ 75 \GeV plus the third lepton with $P_T^{\mu,e}>$10 \Gevc in  $|\eta|<2.4$ .
\\
The evolution of statistics and the  efficiencies of the selection cuts  are  presented in Table \ref{susy:trilept.sumdata}. 
The invariant mass of OSSF was reconstructed from all OSSF combinations in the event, Figure\ref{susy:trilept.minv}.
In $\sim27\%$ of the signal events(LM9) an  another
OSSF pair can be constructed where  one  lepton is originated  from the $\chi^{\pm}_1$ decay. 
Since the $m_{\chipm_1\sim m_{\chiz_2 $  this invariant mass stays in the signal region.
The invariant mass from all OSSF pairs is shown in Figure ~\ref{susy::trilept.invmall}(left) for the LM9 reference  point and all backgrounds.
The significance   $S_{cp}=2 (\sqrt{Nb+Ns}-\sqrt{Nb})$ amounts to $\sim$ 6.1.
The biggest contribution to the background is coming from Z+jets, DY, $t\bar t$, and ZW.
\par
In the second step, the background suppression  was improved with a Neural Network(NN).
Five networks for  Z+jets, DY, $t\bar t$, ZW and ZZ backgrounds have been trained with LM9 signal sample using the following
variables: P$_T^{1,2,3}$,  $\sum P_T$, Minv$_{h,l}$ - invariant mass of high and low  P$_T$ OSSF pair combination,
$P_{2l}$ -  transverse momentum of two OSSF leptons,$A=\frac{P_T^1-P_T^2}{P_T^1+P_T^2}$ - asymmetry, 
$\Theta_{ll}$ - angle between the two OSSF leptons, $\Phi_{ll}$ - angle in transverse plane, $MET$,
$N_{jets}$ - number of  jets passed the jets veto, $E_t^{hj}$- of the highest  E$_T$ jet,$\eta_{hj}$ - rapidity of the highest jet.
The selection cuts on the NN outputs have been optimized using the Genetic Algorithm (GA) to have a maximum significance.
The samples, remaining after first  selection step, have passed through these five networks and selected according to the
cuts optimized with the GA. The efficiency of the NN selection is shown in Table \ref{susy:trilept.sumdata}.
The resulted  invariant mass distribution from all OSSF pairs is shown in  Figure ~\ref{susy::trilept.invmall}(right),the significance has improved to 7.8. 
\begin{figure}[!Hhtb]
\begin{center}
     \includegraphics[width=0.48\textwidth]  {susybsm_susy/trilept_invc}
      \includegraphics[width=0.48\textwidth]  {susybsm_susy/trilept_invnn}
    \caption[]{Invariant mass of all OSSF combination with the selection with cuts(left) and after NN selection(right). 
    \label{susy::trilept.invmall}} 
  \end{center}
\end{figure}

\begin{table}[!Hhtb]
\begin{small}
\caption[]{Summary of the signal(LM1,LM9, LM7)  and  background statistics expected for  L$_{int}$=30 \fbinv  at 
different selection steps. The selection efficiency is shown in brackets.
\label{susy:trilept.sumdata}}
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|} \hline
 channel     & $N_{ev}$ 30fb$^{-1}$         & L1+HLT              & No Jets             & 2 OSSF+l      &  NN     \\
             & ($\sigma\times BR$pb)        &                     &             &  {\tiny OSSF $M_{ll}< 75$GeV} &       \\\hline

LM1          & 2640 ~(0.088)                &  1544~ ($58\%$)     & 864~ ($56\%$)       & 70~ ($8\%$)     & 17~ ($24\%$)        \\ 
LM7          & 1540 ~(0.051)                & 1250~ ($82\%$)      & 738~($59\%$)        & 91~ ($12\%$)    &    57~ ($62\%$)        \\ 
LM9          & 3700~(0.125)                 & 2896~ ($78\%$)      & 1740($60\%$)        & 239~ ($14\%$)   &    158~ ($68\%$)  \\\hline

ZW           & 5.$10^4$~(1.68$^{NLO}$       & 3.6$10^4$~($73\%$)  & 1.9~ $10^4$ ($53\%$)&  173~ ($1\%$)      &  44~ ($25\%$)   \\
ZZ           & 4800~(0.16$^{NLO}$)          & 3530~($73\%$)       & 1681~ ($48\%$)      & 38~ ($2.3\%$)       &  15~ ($39\%$)  \\
${\rm t\bar t}$ & 2.6 $10^6$~(88$^{NLO}$)   & 1.8~ $10^6$($70\%$) & 1.3~ $10^5$ ($7\%$) &  239~($0.2\%$)       &  89~ ($37\%$) \\
Z+jets(3l)   & 4.6 $10^5$~(15.4$^{LO}$)     & 3.7~ $10^5$($80.5\%$)& 9.8~ $10^4$ ($26.5\%$)&  504~ ($0.5\%$)   &  129~ ($26\%$)  \\   
DY(3l)       & 4.5 $10^5$ ~(15.1$^{LO}$)    & 3.2~ $10^5$($71\%$)  & 1.4~ $10^5$ ($44\%$)  &  670~ ($0.5\%$)   &  131~ ($19.5\%$) \\  
${\rm Wt}$+jets & 3. $10^5$~(10.$^{NLO}$)   & 2.1~ $10^5$($70\%$)  & 3.9~ $10^4$ ($18.5\%$)&  52~ ($0.1\%$)    &  20~ ($38\%$)   \\ 
WWj          & 6.$10^5$~(19.8$^{LO}$)       & 3.8~ $10^5$ ($63\%$) & 1.9~ $10^4$ ($50\%$)  & 7~ ($0.04\%$)     & 2~ ($29\%$) \\ 
SUSY         & 4 $10^5$~(13.1$^{NLO}$)      & 2.5~ $10^5$ ($63\%$) & 1.8~ $10^4$ ($7\%$)   &  34~ ($0.2\%$)    & 22~ ($65\%$) \\  \hline 
Tot. bkg     &  4.9 $10^6$                  &                      &                        &  1718             & 453~($26\%$) \\ \hline     
\end{tabular}
\end{center}
\label{tab:cutsum}
\end{small}
\end{table}



\mysubsection{Systematic uncertainties and CMS discovery reach}
The systematic uncertainties are coming from the reconstruction errors of variables used in the selection and
from the  uncertainties in the simulation of the physics channels.
The reconstruction uncertainties on the jet energy scale($\sim10\%$)  and the leptons P$_t$ resolution ($\sim2\%$)
result in the 0.8$\%$  error in the number of selected  background  events. The uncertainty of the NN selection can be
estimated by comparing the NN trianed for different samples(LM9 and LM7 points) and is $\sim 3.5\%$.  
The initial state radiation  can change the number of jets and 
therefore  the number of the selected events.  The maximum  difference   is  $3.8\%$ and assuming $20\%$ 
error on this value  $\sigma_{ISR}$=0.8$\%$.
The PDF uncertainties have been estimated using the  PDF reweighting technique and amounts to $1.7\%$.
The total systematic uncertainty calculated as a quadratic sum of different contributions is 4$\%$.
\par
The $S_{cpst}$ significance contours for all OSSF combinations selected with the NN(LM9)
and including the systematic uncertainties are shown  in Figure ~\ref{susy:trilept.scansugra}. 
The discovery reach is limited at large  $m_0$ by EWSB limit and at low $m_0$ by the constraints on the 
light Higgs mass. The lowest $m_{1/2}$ are excluded  due to the limit on the chargino mass and 
for large $m_{1/2}>220$ the  invariant mass moves into the Z peak, rendering the signal invisible.
Together with the dilepton  final state,
the trilepton channel allows to probe the large  $m_0$ region particularly 
interesting for the dark matter searches \cite{deBoer:2004ab}.
\begin{figure}[!Hhtb]
\begin{center}
     \includegraphics[width=0.48\textwidth]  {susybsm_susy/trilept_scansugra10}
      \includegraphics[width=0.48\textwidth]  {susybsm_susy/trilept_scansugra50}
    \caption[]{The significance $S_{cpst}$ with the systematic uncertainties  in mSUGRA $m_ {1/2}$, $m_o$ plane  
for different  tan$\beta$=10 (left) and 50 (right) at 30  \fbinv.
    \label{susy:trilept.scansugra}} 
  \end{center}
\end{figure}