\chapter{A Pion Estimator}
\label{ESTIMATOR}
  A $\pm$3$\sigma_{M}$ mass window selection forms the starting point
  to study a method for improving the signal-to-background ratio.
  To gain further improvements in purity, cuts based on a pion estimator
  can be applied. The estimator is built from chi-square ($\chi^{2}$)
  values from the mass constraint, and photon pair opening angles,
  $\theta_{12}$. A plot of Chi-square versus pion opening
  angle, Figure \ref{FIG_SCATTER}, shows that true pions (circles) tend to have
  smaller values of chi-square and smaller opening angles than false pions
  (crosses). An estimator based on these two parameters can therefore
  be used to attempt to distinguish between correct and wrong combinations of
  photon pairs.

\begin{figure}
\vspace{-10mm}
\begin{center}
\epsfig{file=figures/rank-scatter.eps,height=12cm}
\end{center}
\vspace{-10mm}
\caption[Chi-square versus photon pair opening angle (for each candidate)]
{       
\label{FIG_SCATTER}
        Chi-square (from a mass constraint) versus photon pair
        opening angle (for each pion candidate)
        for 300 hadronic Z decays at 91.2 \unt{GeV} center of mass.
        True pions are represented by circles, incorrect combinations
        of photons yield false pions (combinatorial background)
        these are represented by crosses. The combinatorial background
        tends to have larger opening angles and chi-square values than
        true pions, this feature is used to discriminate between
        pions and background.
        The solid line is an
        ellipse of the form given in Equation \ref{EQN_ELLIPSE}
        where parameters $A$ and $B$ have been optimized such that
        the product of the selection purity and efficiency is
        maximum. Pion candidates below the curve are selected, those
        above are rejected.
}
\end{figure}
 
  The form of the cut-off should be carefully formulated.
  Various functions have been investigated including rectangular
  and triangular shapes and ellipses; the ellipse provides the most
  effective function of the three alternatives, the solid curve shown
  in Figure \ref{FIG_SCATTER}, and has the form:
\begin{equation}
    \label{EQN_ELLIPSE}
    ( \theta_{12}/A )^2 + ( \chi^2/B )^2 = 1
\end{equation}
where the values of $A$ and $B$ are optimized for the data.

  Optimization of the function involves a search in $A$ and $B$ parameter
  space, for this a balance between purity and
  efficiency must be made. A purity of 100\% can be achieved by applying
  very tight cuts thereby rejecting all background though with only a very
  low selection efficiency. In most practical applications it is important, 
  for both statistical and systematic reasons, to maintain efficiency
  as high as possible. The balance between efficiency and purity in this
  study is defined by maximizing the product of the selection efficiency
  and selection purity. It turns out that in general this leads to a high 
  selection efficiency, for example for Z decays
  optimization using this criteria yields efficiencies 
  between 82\% and 98\%.

  We now consider two methods for applying the estimator with the
  aim of improving pion purity.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{\label{ESTIMATOR_METHODS} Estimator Application Methods}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

  ....{\it some sections skipped}... 
  the purity in the Ranking method improves from 61.4\% (selection
  efficiency of 99.5\%) to 82.6\% (selection efficiency of 90.6\%)
  an improvement of 21.2\%. For an event energy of
  912 GeV the purity improves from 45.3\% (selection
  efficiency of 99.3\%) to 78.6\% (selection efficiency of 82.0\%)
  an improvement of 33.3\%. 
  It is interesting to note that searching through the list in reverse
  order or in a random order has little effect on the outcome. 

  The above results, together with results showing the effect of
  doubling detector resolution are shown in Table \ref{TAB_BRIEF_RESULTS}.
  More detailed comparisons of the Direct and Ranking methods
  are presented in the next chapter.

\vspace{3mm}
\begin{table}[h]
\caption [Comparison of purity and efficiency. The stochastic term, $R$, 
          in the detector resolution, and the event energy, are varied.]
{
\label{TAB_BRIEF_RESULTS}
          Comparison of purity and efficiency(shown in brackets)
          for three pion selection methods: `mass window' selection,
          `Direct estimator method', and `Ranking estimator method'.
          The stochastic term, $R$, in the detector resolution, and the
          event energy, are varied.
}
\vspace{1mm}
\begin{center}
\begin{tabular}{|c|c|c|c|c|} \hline
Event  & Resolution & $\pm$3$\sigma_M$ mass& Estimator cuts  & Estimator cuts \\
Energy & parameter  & window               & (Direct)        & (Ranking) \\ \hline
91.2 $GeV$   & $R$ = 0.18 & 61.4\%(99.5\%) & 69.1\% (95.1\%) & 82.6\%(90.6\%) \\ \hline
912. $GeV$   & $R$ = 0.18 & 45.3\%(99.3\%) & 55.3\% (92.9\%) & 78.6\%(82.0\%) \\ \hline
91.2 $GeV$   & $R$ = 0.36 & 46.2\%(99.4\%) & 56.0\% (92.8\%) & 71.9\%(85.0\%) \\ \hline
\end{tabular}
\end{center}
\end{table}

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%End chapter: A Pion Estimator
%
