\section{Acceptance studies}
\label{mc:acceptance}

In order to better understand the effects of finite acceptance of a 
proposed  detector configuration, a simple study of the acceptance
as a function of total meson effective mass for various final states 
has been performed. In doing the Monte Carlo acceptance studies we 
considered the following reactions:
schematically shown in Figure \ref{fig:lost_event}. 
This configuration is composed of the following:
\begin{itemize}
\item $2.24$ Tesla solenoid magnet --\lass \hspace{0.1cm} magnet,
\item 5-layer Vertex Chamber (\vtx),
\item 22-layer Central Drift Chamber (\cdc),
\item 5 6-layer Forward Drift Chambers (\fdc),
\item Barrel Calorimeter which also acts as central \tof (\bcal),
\item Cerenkov Detector,
\item Forward time-of-flight (\ftof),
\item Forward Lead Glass Detector (\lgd) 172x172\cmlunit\
with 8x8\cmlunit\ beam hole, 
\item target-beam vertex distribution at $r=0.0$\cmlunit ,
$z=50$\cmlunit\ with $\sigma_r =0.3$\cmlunit,
$\sigma_z =15.0$\cmlunit\ ($\hat{z}$ is along the magnet axis; 
the origin is located at the upstream face of the solenoid). 
\end{itemize}


\subsection{Acceptance performance}

In the simulation, an event was accepted if the
following minimum conditions were met:
\begin{itemize}
\item all charged tracks were found with a minimum of four hits per track, and
\item all gammas were detected in either the \bcal\ and/or \lgd.
\end{itemize}


The acceptance as a function of total effective meson mass is shown
in Figure \ref{fig:mass_accept}. It is important to note that at
higher beam energies the forward boost results in more forward-going
high-momentum tracks.  And even though the mass acceptance seems good, the 
resolution of the forward-going higher-momentum tracks degrades.  This
issue has been studied in detail and is discussed in
\hd\ Note \#7\cite{hdnote7}.

\begin{figure}[htbp] 
\centering
\leavevmode
\centerline{
\epsfig{file=ch10_mass_accept.eps,height=18cm,width=16cm}}
\caption[The acceptance as a function of total effective meson mass]
{\label{fig:mass_accept} The acceptance as a function of total effective 
meson mass: a) $ X^+ \rightarrow \pi^+ \pi^+ \pi^- $, 
b) $ X^+ \rightarrow \eta \pi^+ \pi^- \pi^+ $, 
c) $ X^+ \rightarrow \omega \pi^0 \pi^+ $, 
d) $ X^0 \rightarrow \eta \pi^0 \pi^0 $.
The acceptance studies were performed for effective meson masses of 1.4, 1.7, and 
2.0 $GeV/{\it{c}}^2$, and at each mass point the photon beam energy was varied from 
5 to 12 $GeV$.}
\end{figure}

In Figure \ref{fig:acceptGJ_pip_pip_pim} through Figure~\ref{fig:acceptGJ_eta_piz_piz}, 
we show the acceptance for the Gottfried-Jackson decay angles (the particle decay 
angles often used in the partial wave analysis). It is clear that the Gottfried-Jackson 
angular acceptance is quite good.  The  acceptance for gammas is also 
rather high, but it suffers more from holes in the forward and backward 
regions. The hole in the backward region results from backward-going 
gammas, which is the dominant factor at lower beam energies. The forward 
hole, due to gammas passing through the beam hole in the LGD, becomes 
important for higher beam energies. Figure 
\ref{fig:lost_event}a displays an event for reaction
$\gamma p \rightarrow p \eta \pi^0 \pi^0$ at 
$Mass(X)=2.0$ \munit\ and beam of 5 \eunit that was lost due to the
upstream hole.  For this channel 75\% 
of the lost events were of this type.  On the other hand, for a 
12 \eunit\ beam and the same final state about 50\% of the lost events 
are due to the beam hole(See Figure \ref{fig:lost_event}b).  
While the beam hole is unavoidable, the hole in the backward region 
suggests the need at the lower beam energies for a backward gamma veto.  
Regardless of this, the acceptance for the Gottfried-Jackson decay angles 
is flat and not strongly dependent on $Mass(X)$ or the beam energy.
%Even though the method of partial wave analysis can account for effects of 
%finite detector acceptance, a small misunderstanding of the detector 
%acceptance can be greatly amplified by the need for large acceptance 
%corrections and thus  produce false or misleading results. 
This is important for partial wave analysis because, although the
effects of acceptance distortions are accounted for in the method,
large acceptance corrections can lead to large systematic errors in
the results.

\begin{figure}[htbp] 
\centering
\leavevmode
\centerline{
\epsfig{file=ch10_acceptGJ_pip_pip_pim.eps,height=18cm,width=14cm}}
\caption[Acceptance in $\cos\theta_{GJ}$]{\label{fig:acceptGJ_pip_pip_pim}
The acceptance in $cos(\theta_{GJ})$ and $\phi_{GJ}$  for 
$ X^+ \rightarrow \pi^+ \pi^+ \pi^- $. The acceptance was studied for
$X^+$ effective masses of $1.4$, $1.7$, and $2.0$ \munit{}, 
and for different photon beam energies of $5$\eunit{} (top),
$8$\eunit{} (middle), and $12$\eunit{} (bottom).}
\end{figure}

\begin{figure}[htbp] 
\centering
\leavevmode
\centerline{
\epsfig{file=ch10_acceptGJ_eta3pi.eps,height=18cm,width=14cm}}
\caption[Acceptance in $\cos\theta_{GJ}$]{\label{fig:acceptGJ_eta3pi}
The acceptance in $cos(\theta_{GJ})$ and $\phi_{GJ}$ for 
$X^{+}\rightarrow \eta\pi^{+}\pi^{+}\pi^{-}$. The acceptance was studied for
$X^+$ effective masses of $1.4$, $1.7$, and $2.0$ \munit, and for 
different photon beam energies of $5$\eunit{} (top),$ 8$\eunit{} (middle), and 
$12$\eunit{} (bottom).}
\end{figure}

\begin{figure}[htbp] 
\centering
\leavevmode
\centerline{
\epsfig{file=ch10_acceptGJ_omega_piz_pip.eps,height=18cm,width=14cm}}
\caption[Acceptance in $\cos\theta_{GJ}$]{\label{fig:acceptGJ_omega_piz_pip}
The acceptance in $cos(\theta_{GJ})$ and $\phi_{GJ}$  for 
$X^{0}\rightarrow\omega\pi^{0}\pi^{+}$. The acceptance was studied for
$X^+$ effective masses of $1.4$, $1.7$, and $2.0$ \munit{}, 
and for different photon beam energies of $5$\eunit{} (top), $8$\eunit{} 
(middle), and $12$\eunit{} (bottom).}
\end{figure}

\begin{figure}[htbp] 
\centering
\leavevmode
\centerline{
\epsfig{file=ch10_acceptGJ_eta_piz_piz.eps,height=18cm,width=14cm}}
\caption[Acceptance in $\cos\theta_{GJ}$]{\label{fig:acceptGJ_eta_piz_piz}
The acceptance in $cos(\theta_{GJ})$ and $\phi_{GJ}$  for $X^{0}
\rightarrow\eta\pi^{0}\pi^{0}$. The acceptance was studied for $X^+$
effective masses of $1.4$, $1.7$, and $2.0$ \munit{}, and for different 
photon beam energies of $5$\eunit{} (top), $8$\eunit{} (middle), and $12$\eunit{} 
(bottom).}
\end{figure}


\begin{figure}[htbp] 
\centering
\leavevmode
\centerline{
\epsfig{file=ch10_lost_event.eps,height=18cm,width=9cm}}
\caption[Event displays of lost events for $\gamma p \rightarrow p \eta 
\pi^0 \pi^0$]{\label{fig:lost_event}
Event displays of lost events for $\gamma p\rightarrow p\eta\pi^{0}\pi^{0}$ 
for $Mass(X)=2.0$ \munit{}: (a) backward missed gamma at a beam energy
of $5$ \eunit, and (b) forward missed beam hole gamma at a beam energy of
$12$ \eunit. The events shown contain both charged particles (solid lines) and 
photons (dashed lines).}
\end{figure}

