%
% local macros
%
% particles
%\newcommand{\bbbar}     {{\mathrm{b}\bar{\mathrm{b}}}}
%\newcommand{\ttbar}     {{\mathrm{t}\bar{\mathrm{t}}}}
%\newcommand{\WWstar}    {{\mathrm{W}\mathrm{W}^*}}
\newcommand{\WWstar}    {{WW^*}}
% kinematics
\newcommand{\roots}     {\sqrt{s}}
%\newcommand{\pt}        {p_T}
\newcommand{\etamax}    {\eta_{\mathrm{max}}}
% BR's and cross sections
\newcommand{\Brbb} {B_{\mathrm{b}\bar{\mathrm{b}}}}
%\newcommand{\BrWW} {B_{\mathrm{W}\mathrm{W}}}
\newcommand{\BrWW} {B_{WW}}
\newcommand{\Brl}  {B_\ell}
\newcommand{\Lum}  {\cal{L}}
% units
%\newcommand{\gev}   {\mathrm{GeV}}
\newcommand{\tev}   {\mathrm{TeV}}
\newcommand{\fbinv} {\mathrm{fb}^{-1}}
\newcommand{\fb}    {\mathrm{fb}}
%
%
%----------------------------------------------------------------


Trilepton signatures will arise in associated production
$p \bar{p} \rightarrow Wh$ when the Higgs decays to $WW^*$
and all three $W$'s decay to leptons $\ell = 
\mathrm{e},\mu$.  Although three factors of leptonic
branching ratio $\Brl$ reduce the effective signal cross section
severely, the presence of three energetic leptons with little jet
activity provides a very clear signature.  The trilepton signature has
been exploited in searches for charginos and neutralinos, but here we
have developed an independent search, better suited to a signal from
$Wh\rightarrow 3\ell^\pm$.

Leptons and hadronic jets are defined as in the search for same-sign
dileptons, except that for jets the requirement $|\eta_{\mathrm{jet}}|
< 2$ applies.  Leptons (jets) are sorted in order of decreasing $p_T$
($E_T$). Three leptons are required with charge sum $|q_1+q_2+q_3| =
1$, and momentum thresholds $p_{T1} > 20~\gev$, $p_{T2} > 15~\gev$ and
$p_{T3} > 10~\gev$.  All dilepton masses must be at least~$2~\gev$.
Possible Z decays are removed by pairing each lepton with all other
oppositely charged tracks having $p_T > 2~\gev$ and rejecting any
event with $70 < M_{\ell,\mathrm{tr}} < 110~\gev$ for any of the three
leptons.  A kinematic variable $c_3$ based on the tri-linear dot
product is defined as
\begin{equation}
 c_3 \equiv \frac{ p_{x1}p_{x2}p_{x3} +
                   p_{y1}p_{y2}p_{y3} + p_{z1}p_{z2}p_{z3} }
                 { p_1 p_2 p_3 } ,
\end{equation}
and all events must satisfy $c_3 > -0.75$.

Four signal categories were defined for events passing the basic cuts.
An event passing any of these categories is accepted:
\begin{enumerate}
  \item ``Golden'' flavor/charge combinations are very low in
        background, {\it e.g.}, $\mathrm{e}^+\mathrm{e}^+\mu^-$.
  \item $\Delta\phi_{ij}$ between all pairs of leptons ($i,j=1,2,3$)
        must be smaller than $1.5$~radians.  Let $E_T^{\mathrm{ch,opp}}$ 
        be the sum of $p_T$ of all charged tracks in a cone of
        $30^\circ$ opposite the direction of $\vec{p}_1+\vec{p}_2$,
        and require $E_T^{\mathrm{ch,opp}} < 2~\gev$.

  \item Let $E_T^{\mathrm{miss,cen}}$ be the ``central missing
        energy'' computed from the towers with $|\eta| < 2$ and
        correcting for identified muons, and require
        $E_T^{\mathrm{miss,cen}} > 70~\gev$ and tighten the
        fiducial requirement for leptons to $|\eta_\ell| < 1.2$.
  \item Let $M_{3\ell}$ be the invariant mass of the three leptons,
        and require $M_{3\ell} > 70~\gev$.  
        Require $\Delta\phi_{ij} > 0.2$~radians.
        Define $M_{T,3\ell}$ to be the transverse mass of the three
        leptons and require
        $\Delta\phi_{ij} < (300~{\gev} - M_{T,3\ell})/100$~radians.
\end{enumerate}
The events selected in these four categories are largely independent
and the backgrounds from each are roughly equal.  In the selection
we take the logical OR of the four sets of requirements.


Estimated backgrounds from Drell-Yan, $WW$, and $\ttbar$ are negligible.
Contributions from $WZ$ and $ZZ$ are $0.023$~fb and $0.002$~fb,
respectively.  For a SM higgs of mass $160~\gev$, for which the cuts
above were designed, the accepted cross section is $0.057$~fb, giving
$S/B = 2.3$ (Table~\ref{tab:trilep}).  For ${\Lum} = 30~\fbinv$, $1.7$
signal events would be expected, corresponding to $S/\sqrt{B} = 2$.
Although the yield is far too small for this channel to guarantee
discovery, trilepton events passing these cuts would provide
confirming evidence for a Higgs boson with mass above~$140~\gev$.



%------------------------
\begin{table}
  \caption{Accepted cross section and significance for the
           trilepton selection.  The effective cross section,
           defined as $\sigma(p\bar{b}\to Wh/Zh)\times B(h\to WW/ZZ)$,
           does not include the leptonic branching ratio factors.
  \label{tab:trilep}}
  \begin{center}
    \begin{tabular}{c|cccc}
$M_H$ & $\sigma_{\mathrm{eff}}$ (fb) & $\sigma_{\mathrm{acc}}$ (fb) &
  $S/B$ & $S/B$ ($30~\fbinv$) \cr \hline
 120 & 0.5 & 0.01 & 0.5 & 0.4 \cr
 130 & 1.1 & 0.02 & 1.0 & 0.9 \cr
 140 & 1.4 & 0.04 & 1.6 & 1.4 \cr
 150 & 1.5 & 0.05 & 2.0 & 1.7 \cr
 160 & 1.6 & 0.06 & 2.3 & 2.0 \cr
 170 & 1.4 & 0.03 & 1.3 & 1.1 \cr
 180 & 1.0 & 0.03 & 1.3 & 1.1 \cr
 190 & 0.7 & 0.02 & 0.8 & 0.7 \cr
 200 & 0.5 & 0.01 & 0.4 & 0.4 \cr
    \end{tabular}
  \end{center}
\end{table}
