% trilepton writeup for PTDR
% draft 1        VZ  29/7/05
% draft 1.1      MC  1/8/05
% draft 1.2      VZ  3/8/05
%%%%%%%%%%%%%%%%
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\begin{document}

\subsection{Trileptons from direct $\chi^0_2 \chi^\pm_1$ production in mSUGRA}
%-------Introduction: signal signatures
The trilepton final state in mSUGRA  
appears in direct neutralino-chargino        
production $pp\rightarrow \chi_2^0\chi_1^{\pm}$   
with  subsequent three body decays 
of the second 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
~$\chi_1^\pm\rightarrow l\tilde\nu\rightarrow l\chi_1^0\nu$ or
~$\chi_1^\pm\rightarrow\nu\tilde l\rightarrow \nu\chi_1^0 l$.
Trilepton events 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 appreciable 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 proceed mainly via staus which decay consequently to tau leptons; since
the leptons from the  $\tau$-decay are soft this requires special treatment. %PTDR: reference tosusy  dileptons 
Hadronic tau decays similarly require dedicated analysis.%PTDR: reference to ditau section
%----Invariant mass
The invariant mass of the OSSF dileptons exhibits  
a particular triangular shape
with a  kinematic end point that depends upon the event 
topology: either  
$M_{ll}^{max}$=$m_{\chi_2^0}-m_{\chi_1^0}$ 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}. 
%PTDR: reference to the kinematical variables 
\par
%------Cross section
The neutralino-chargino mass spectrum and decays are determined  in mSUGRA  by
 $m_{1/2}$: $m_{\chi^0_1} \sim 0.4 m_{1/2}$,  
~ $m_{\chi^0_2}\sim m_{\chi_1^\pm}\sim  0.8 m_{1/2}$ and
$\sigma(\chi^0_2\chi_1^{\pm})\sim m_{1/2}^{-4}$.
The low  $m_{1/2}$ region, where $\chi^0_2 \chi^\pm_1$ production is large, 
covers two distinct regions:
the 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 
 gluino production dominant 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 $m_0$, $m_{1/2}$ is constrained  by the light Higgs  mass limit  ($m_h>114.4$ GeV)
and by  limits on  the $b\rightarrow s\gamma$  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
Trilepton ($e$ and $\mu$ only) LO cross sections calculated with  ISAJET 7.69 and PYTHIA 6.225 
for the CMS benchmark points range from $\sim$100 fb at LM9 to virtually zero at LM2 where decays to taus dominate.
The NLO cross sections depend on the neutralino-chargino masses and  are higher by K$_{NLO}\sim$1.3 \cite{signalnlo}. 
The trilepton signal has non-negligible cross sections for the three body decays at  
LM9($\sigma_{3l}^{LO}$=95fb), LM7(39fb), LM3(9fb)  and two body at  LM1(41fb), LM6(10fb) benchmark points.
For LM4 and LM8 the neutralino decays via an on-shell Z and thus can not be distinguished from the background.
One  can expect  for LM9 point $\sim$3700 events(NLO) at   L$_{int}$=30 fb$^{-1}$.
\par
%------Backgrounds
The main background  to  the pure trileptonic state results from  
Standard Model (SM) ZW production
and ZZ production with one missed lepton. All other SM backgrounds, such as $t\bar t$, 
Z+jets, W+jets, etc., enter into the sample only via missed jets.  Missed jets are also important   
for the  SUSY  background from gluino and squark production.
A summary of  the backgrounds cross sections,  K$_{NLO}$ factors and expected statistics
for L$_{int}$=30 fb$^{-1}$ is presented in Table \ref{sumdata}.
\begin{table}[ht]
\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$^5$     \\
SUSY bkg(LM9)   & 25       & 1.6      &   13                      &   4 10$^5$     \\ \hline
\end{tabular}
\end{center}
\caption[]{Summary of background cross sections for the trilepton search.}
\label{sumdata}
\end{table}
%comment: more Xjets bkg will be added
\par
%-----Simulation
The full data samples of 30 fb$^{-1}$ for the LM9 benchmark and ZW and ZZ backgrounds are simulated 
with the full CMS simulation (ORCA DST). The full-sized samples for other backgrounds and SUSY scans 
are simulated in FAMOS. The default reconstruction algorithms are used.
%PTDR: reference to ORCA
\par
%-------Selection and Rejection 
The diboson background has similar topography to the $\chi^0_2 \chi^\pm_1$ production but can be distinguished by 
the lepton invariant mass characteristic Z peak.
The other SM backgrounds mentioned above can be effectively suppressed by the vetoeing jets in central region, 
as shown in Figure \ref{all}.
A maximum MET cut, very effective in other SUSY channels, fails here as the produced
neutralino and chargino are relatively light. On the other hand, a minimum MET requirement can help 
to reject Z/W plus jets and QCD backgrounds, Figure  \ref{all}.
Given these considerations, the following  criteria are used for the event selection:
 \begin{itemize}
\item  At least three isolated  leptons in $|\eta|<2.4$ with  transverse  momentum
for muons (electrons) above  $P_T>$10 GeV/c ($P_T>$15 GeV/c).
Among them,  two OSSF leptons  with invariant mass below 75 GeV  
to avoid  the  Z peak  region. 
The muon isolation requires no tracks with $P_T>1.5$ GeV/c and no energy deposition in calorimeter $E_T>5$ GeV
in a cone $R_c=\sqrt{\Delta\eta^2+\Delta\phi^2} <0.3$. For electrons,  the requirement is no tracks in the 
same cone and the ratio of energy depositions $E_{HCAL}/E_{ECAL} \le 0.6$.
\item  No  Jets  with $E_T>20$ GeV in $|\eta|<2.4$.
\item MET$>$ 18 GeV.%comment: cut will be optimized
 \end{itemize}
%---about Trigger
The lepton cuts correspond to the dilepton L1 trigger menu: dimuons  with $P_T>$3 Gev/c and
dielectrons with  $P_T>$12 Gev/c at low luminosity. 
The dimuon trigger efficiency  is $99.8\%$  and
for dielectrons $\sim85\%$.%comment: this numbers will be refined, single electron Li will be used
 At  high luminosity the jet veto selection criteria will cut $43\%$ of the signal 
as well as the background, partially reducing the significance. 
\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, only high  $P_T$ pairs are used.
\par
%----summary
Figure \ref{minvsigbkg} shows the distribution of dimuon and dielectron invariant mass of  high  $P_T$ pairs
for  LM9 and the backgrounds. 
The expected significance for dimuons for the LM9 point is Scl$\sim$10 and dropping to
Scl$\sim$5  for the LM7.
The dielectron signal is a factor of two worse  due to the higher  $P_T$ requirement.
The CMS discovery reach for dimuons at 30 fb$^{-1}$ is shown in Figure \ref{scansugra}.
For $m_0<1000$ GeV the reach is limited by the constraints on the light Higgs mass, 
while low $m_{1/2}$ are excluded due to the limit on the chargino mass. The large $m_0$ 
are limited by the EWSB requirements.
The onset of the EWSB for the low tan$\beta=10$ starts 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 moves into the Z peak, rendering the signal invisible.
Together with the semileptonic final state, when neutralino is mostly produced via gluino decay,
the trilepton channel allows to probe the large  $m_0$ region particularly interesting for the dark matter searches
\ref{dm}.

% PTDR: biblio
\begin{thebibliography}{99}
\bibitem{signalnlo}  W.~Beenakker et al.,~hep-ph/9906298.
\bibitem{dm}  W.~de Boer et al.,~astro-ph/0408272.
\end{thebibliography}
\newpage
% plots
% dileptons
\begin{figure}
  \begin{center}
        \includegraphics [width=0.8\textwidth]{minvdileptons.eps}
\caption[]{Reconstructed invariant mass of OSSF dileptons for three and two  body decays in mSUGRA.} 
\label{minv}
  \end{center}
 \end{figure}
% selection
\begin{figure}
  \begin{center}
        \includegraphics[width=1.0\textwidth]{all.eps}
  \caption[]{Number of selected jets per event(left) and  MET distribution(right)  for signal LM9 and backgrounds.} 
      \label{all}
  \end{center}
 \end{figure}
% inv mass
\begin{figure}
\begin{center}
        \includegraphics[width=1.0\textwidth]{invmass_met.eps}
    \caption[]{Dimuon (left) and dielectron (right) invariant masses for the two OSSF highest  
$P_T$ leptons  for LM9 and backgrounds.} 
    \label{minvsigbkg}
  \end{center}
\end{figure}
% scan
\begin{figure}  
\begin{center}
        \includegraphics[width=.9\textwidth]{scansugra.eps}
    \caption[]{The trilepton signal (2$\mu$+l) significance  for tan$\beta$=10(left) and 50 (right)
 for  30 fb$^{-1}$.} 
    \label{scansugra}
  \end{center}
\end{figure}

\end{document}