\chapter{Pion Reconstruction and Initial Selection}
\label{RECONSTRUCTION}

Neutral pion candidates (or simply refered to as pions) 
are reconstructed from pairs of photons; candidates are 
either {\it true pions} forming a pion signal or {\it false pions} forming
a combinatorial background. This is illustrated in Figure 
\ref{FIG_INVARIANT_MASS}
where an invariant mass spectra of the photon pairs is formed.

\begin{figure}
%\vspace{-10mm}
\begin{center}
\epsfig{file=figures/rank-invmass.eps,height=10cm}
\end{center}
\vspace{-10mm}
\caption[Invariant mass of photon pairs at CM energy of 91.2 \unt{GeV}]
{
\label{FIG_INVARIANT_MASS}
         Invariant mass of photon pairs in
         hadronic Z decays at a center of mass energy
         of 91.2 \unt{GeV}. The stochastic detector resolution parameter
         is set at $R$=0.18~. All photons in this data originate
         from pions; correct combinations of photon pairs result
         in a peak representing the pion signal, incorrect combinations
         result in a combinatorial background forming the source of
         impurity in a sample of selected pion candidates. A `window'
         around the pion peak can be selected to remove most of the
         background. The width of the window can be describe in terms
         of a number of sigmas where sigma represents the mass
         reconstruction resolution. A width of $\pm$3 sigma is
         common selecting almost 100\% of true pions while rejecting
         most of the background.
}
\end{figure}

  Selecting pions from a mass window around the expected pion mass provides
  an initial rejection of most of the combinatorial background.
  A $\pm$3$\sigma_M$ mass window, selecting pion candidates between a mass of 
  $M_0$ - 3$\sigma_M$ and $M_0$ + 3$\sigma_M$, selects, in theory, 99.7\%
  of the pion signal.
  $M_0$ is the expected mean mass of the pion candidates,
  normally 135 \unt{MeV/c^2} though there can be some energy dependence due to
  biases from cuts on photon energy and other detector effects.
  Similarly $\sigma_M$, which is given in Equation \ref{EQN_MASS_RES},
  the Gaussian width of the pion signal, also varies
  with pion energy: 
\begin{equation}
\label{EQN_MASS_RES}
\frac{ \sigma_M }{M }=\frac{1}{2}[\frac{\sigma_{E_1}}{E_1} \oplus
                                  \frac{\sigma_{E_2}}{E_2}  \oplus
                                  \frac{\sigma_{\theta_{12}}}{\tan (\theta_{12}/2)}]
\end{equation}
  where $\sigma_{\theta_{12}}$ is the spatial resolution
  of the calorimeter for the measured angular separation, $\theta_{12}$,
  of the two photons.
  A correct formulation of 
  the energy dependence of the mass window parameters is important 
  to avoid systematic errors when estimating acceptance corrections.
  In this study both $M_0$ and $\sigma_{M}$ are parameterized as a function
  of pion energy using information from reconstructed true pions.
  The purity of pion selection at this point depends 
  on detector resolution, event multiplicity, the photon lower
  energy cut, and the momentum of the selected pions.  
  Further improvements in selected pion purity can be gained by 
  tightening the mass window though with a reduction in efficiency
  and an increase in systematic uncertainties.
  Momentum resolution can be improved by performing a constraint
  of the reconstruction pion mass to the nominal pion mass of 
  135 \unt{MeV/c^2}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Mass Constraint}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

  The momentum resolution of $\pi^{0}$ is limited by the energy  
  resolution of ECAL. One makes the assumption that the error on the 
  opening angle $\theta_{12}$ of the photons is negligible 
  compared to the error on their energies: in the $\pi^{0}$ refitting, only 
  two photon energies are allowed to vary, while their directions are kept 
  fixed, see \cite{LAG_PAPER}.
  The calculation is done with the well known technique of the 
  Lagrange multiplicators.
  If we let $\omega_{1}$, $\omega_{2}$
  be measured energies of the two photons, and $\theta_{12}$ the
  opening angle between their directions, the reconstructed squared invariant
  mass of the $\pi^{0}$ is given by:
\begin{equation}
  \mu^{2}=2 \omega_{1} \omega_{2} (1-\cos \theta_{12})
\end{equation}
  We can now look at for $E_{1}$ and $E_{2}$, the best evaluation technique 
  of energies of the two photons, minimizing the chi-square form:
\begin{equation}
\label{EQN_CHI2}
\chi^{2}=(\frac{E_{1}-\omega_{1}}{\sigma_{1}})^{2}+
         (\frac{E_{2}-\omega_{2}}{\sigma_{2}})^{2}+
         \lambda(m^{2}-2E_{1}E_{2}(1-\cos \theta_{12}))
\end{equation}
  where $\sigma_{1}$ and $\sigma_{2}$ are the errors on the photons energy, 
  $m$ is the true $\pi^{0}$ mass and $\lambda$ is the Lagrange multiplicator.
  We can minimize the $\chi^{2}$ and get a fourth-order Equation for $E_{1}$:
\begin{equation}
\label{EQN_E1}
f(E_{1})=a_{4}E_{1}^{4}+a_{3}E_{1}^{3}+a_{1}E_{1}+a_{0}=0
\end{equation}
  where 
$a_{4}=\sigma_{1}^{2} / \sigma_{2}^{2}$, 
$a_{3}=-\omega_{1}a_{4}$, 
$a_{1}=\omega_{1} \omega_{2}^{2}m^{2} / \mu^{2}$ and 
$a_{0}=\omega_{1}\omega_{2}m^{2}/\mu^{2}$.

  Equation \ref{EQN_E1} has only two real solutions: one negative and 
  other positive, which is only acceptable. However, since the exact solution 
  is too complicated, we get an approximate solution linearazing Equation \ref{EQN_E1}:
\begin{equation}
\label{EQN_E11}
E_{1}=\omega_{1} + ( \frac{m^{2}-\mu^{2}}{\mu^{2}} )
       \frac{\omega_{1}\omega_{2}}
       {\sigma_{1}^{2}\omega_{2}^{2}+\sigma_{2}^{2}\omega_{1}^{2}}
       \omega_{2}\sigma_{1}^{2}
\end{equation}
\begin{equation}
\label{EQN_E12}
E_{2}=\omega_{2} + ( \frac{m^{2}-\mu^{2}}{\mu^{2}} )
       \frac{\omega_{1}\omega_{2}}
       {\sigma_{1}^{2}\omega_{2}^{2}+\sigma_{2}^{2}\omega_{1}^{2}}
       \omega_{1}\sigma_{2}^{2}
\end{equation}

At least, one should note that the problem is further simplified when 
one assumes an energy resolution for photons of the form:
\begin{equation}
\label{EQN_R}
\sigma_{1,2}=R\sqrt{E_{1,2}}
\end{equation}
  where $R$ is a constant, mentioned before. Putting Equation \ref{EQN_R} 
  in \ref{EQN_E11} and \ref{EQN_E12}, one gets:
\begin{equation}
\label{EQN_E111}
E_{1}=\omega_{1} + ( \frac{m^{2}-\mu^{2}}{\mu^{2}} )\times 
      \frac{\omega_{1}\omega_{2}}{\omega_{1}+\omega_{2}}
\end{equation}
\begin{equation}
\label{EQN_E121}
E_{2}=\omega_{2} + ( \frac{m^{2}-\mu^{2}}{\mu^{2}} )\times 
      \frac{\omega_{1}\omega_{2}}{\omega_{1}+\omega_{2}}
\end{equation}
that is, the same quantity is added to the original photon energies.

Equation \ref{EQN_E111} and \ref{EQN_E121} are used in 
the {\tt REFIT} algorithm to re-define the photon energies. 
Then $\pi^0$ mass is evaluated and procedure iterated until 
the mass is within $\pm$ 0.0001 \unt{MeV/c^2} of the nominal 
$\pi^0$ mass. Usually only 2 or 3 iterations are needed.

%
% End chapter: Pion Reconstruction and Initial Selection
%
