\section{Physics Event Weighters}         
\label{mc:weights}

Conceptually, what one would like to do in the analysis of any
given reaction is to write down as complete as possible a set
of diagrams leading to the final state and sum their amplitudes
as a function of a minimal set of unknown parameters.  This model
would then be fed to the event generator to produce a Monte Carlo
sample which could be reconstructed and compared to the data.  By
repeating this procedure for different values of the parameters
through a fitting procedure, the best values of the parameters
and an overall evaluation of the model could be derived.

Practically, this is not what is done because it is too expensive
to recompute the entire Monte Carlo sample at every step in the
fit.  Instead a single Monte Carlo sample is produced using an
initial crude approximation to the physics model distribution,
and then corrections are applied using a weighting procedure after
the sample has been simulated and reconstructed.  The initial
approximation is defined by the following three simple assumptions;
(a) particles from high-energy photoproduction are produced
independently from meson and baryon vertices; (b) the
momentum separation between the two vertices is described 
by an exponential distribution in the Mandelstam variable $t$;
(c) within each vertex the particles are produced through a
cascade of two- and three-particle decays which are each
distributed according to a phase-space density function.  If this
approximation were an adequate model of the physics then there
would be no need for the \hd\ experiment.  Nevertheless it is a
useful starting point because it can be used to produce a Monte
Carlo sample of events with adequate coverage over the full
kinematic range of interest.  

Assuming the independence property of the Monte Carlo sampling
technique, every event in the Monte Carlo sample is independent
of the others.  Each reconstructed Monte Carlo event carries with
it the information about the original generated kinematics, from
which the physics amplitudes can be calculated.  For a given set of
model parameters these amplitudes can be summed to form a probability
for each event, which is called a \emph{weight}.  If all sums over
the Monte Carlo sample during partial wave analysis are carried out
including these weight factors then the foregoing conceptual
procedure is recovered.  Although the statistical errors in the
weighted Monte Carlo sample are no longer simple Poisson factors,
they are straightforward to calculate.  In general these errors are
larger for the weighted technique than for an unweighted procedure,
but that is readily offset by generating a somewhat larger sample.
Exactly how much larger depends on how different the weighted
distribution is from the initial, but usually this factor is not
larger than two.  Ultimately it is not known until the final stages
of the analysis how large a Monte Carlo sample is adequate for any
given channel, but for the purposes of the design a conservative
factor of 10 more Monte Carlo than real events has been adopted as
a benchmark.

\begin{figure}\centering
\begin{picture}(200,100)
%
\put(40,10){\line(1,0){170}}
\put(40,10){\vector(1,0){20}}
\put(150,10){\vector(1,0){20}}
\put(50,0){\makebox(0,0){$p$}}
\put(160,0){\makebox(0,0){$n$}}
\put(100,80){\makebox(0,0){$X^{+}$}}
%
\multiput(80,10)(0,10){6}{\line(0,1){5}}
\put(80,65){\vector(0,1){5}}
\put(90,35){\makebox(0,0){$\pi^{+},\rho^{+}$}}
\put(80,70){\vector(1,0){50}}
\put(150,90){\makebox(0,0){$\rho^{0}$}}
%
\put(130,70){\line(2,1){40}}
\multiput(130,70)(20,-10){2}{\line(2,-1){10}}
\put(170,47){\makebox(0,0){$\pi^{+}$}}
%
\multiput(170,90)(20,-5){2}{\line(4,-1){10}}
\put(210,105){\makebox(0,0){$\pi^{+}$}}
\multiput(170,90)(20,5){2}{\line(4,1){10}}
\put(210,75){\makebox(0,0){$\pi^{-}$}}
%
\put(65,100){\makebox(0,0){$\vec{\gamma}$}}
%
\multiput(75,70)(-10,10){4}{\oval(10,10)[tr]}
\multiput(75,80)(-10,10){4}{\oval(10,10)[bl]}
%
\end{picture}
\caption[Physics event generation]{\label{fig:physics-evt-gen}A schematic
diagram of the photoproduction amplitude via one--$\pi$ or $\rho$ exchange. 
The state $X^{+}$ then decays via $\rho^{0}\pi^{+}$, and the $\rho$ 
subsequently decays into $\pi^{+}\pi^{-}$.}
\end{figure} 

The above method is well-established for partial wave analyses
in high-energy physics.  To gain experience within the context of
\hd\ it was decided to apply the procedure to a photoproduction reaction.
%In addition to the phase space event generators mentioned earlier in 
%this chapter, it is also important to be able to simulate the expected 
%photoproduction physics that will be studied.
To this end, an event 
generator for the $3\pi$ final state has been written using the one--pion 
charge--exchange mechanism as discussed in reference~\cite{adam-weight-00}
for reaction~\ref{eq:ch9-3pi}.
\begin{equation}
\vec{\gamma} p \rightarrow X^{+} n \,\,\, \left (X^{+}\rightarrow 
\left[ \rho^{0}\rightarrow\pi^{+}\pi^{-}\right] \pi^{+}\right)
\label{eq:ch9-3pi}
\end{equation} 
A schematic of this process is shown in Figure~\ref{fig:physics-evt-gen}. 
One--$\pi$ charge exchange requires both a spin--flip at the nucleon 
vertex, and that the $X^{+}$ particle carry the helicity of the incoming
$\gamma$, ($M_{X}=1$). Any number of resonances $X^{+}$ with different masses, 
widths and production strengths can be included in the generator. In addition, 
the photon beam can have any polarization desired. An extension to this program
allows for $\rho$--exchange under the same conditions as the $\pi$ exchange.
These two amplitudes represent \emph{unnatural} and \emph{natural} parity
exchanges resepctively. Events produced using one of the phase space generators 
can then be weighted according to the physics weighter, and then passed through 
the \hd{} Monte Carlo program. These can then be used as input to the partial 
wave analysis as described in the next chapter. 

A sample of the output of this generator is shown in Figure~\ref{fig:ch9-3pi}. 
These events have  been generated with four resonances:  $a_{1}(1260)$, 
$a_{2}(1320)$, $\pi_{2}(1670)$ and an exotic $\pi_{1}(1600)$. The masses and 
widths are all consistent with current accepted values. In addition, one 
can see the $\rho^{\circ}$ in the $\pi^{+}\pi^{-}$ invariant mass spectra.
A full list of known resonances~\cite{Caso} that could be put in this generator 
is given in table~\ref{tab:ch9-res}.
\begin{figure}
\begin{tabular}{cc}
\epsfig{file=ch10_total.eps,width=0.5\textwidth}
\epsfig{file=ch10_rho.eps,width=0.5\textwidth}
\end{tabular}
\caption[Generated $3\pi$ spectra]{\label{fig:ch9-3pi}(Left) The genrated 
$3\pi$ mass spectrum using four intermediate resonances, $X^{+}$. The 
diamonds correspond to all generated events, while the circles correspond 
to events which have been run through the \hd{} Monte Carlo program.
(Right) The $\pi^{+}\pi^{-}$ invariant mass from the $3\pi$ events. The 
peak corresponds to the $\rho^{0}(770)$.} 
\end{figure}

\begin{table}[h!]\centering
\begin{tabular}{|r|cc|c|}\hline
Resonance & Mass & Width & $L_{\rho\pi}$ \\ \hline
%
$a_{1}^{+}(1260)$   & $1.230$\munit   & $.250$ to $.600$\munit & $L=0,2$ \\
$a_{2}^{+}(1320)$   & $1.318$\munit   & $.105$\munit          & $L=2$   \\
$\pi_{1}^{+}(1600)$ & $1.593$\munit   & $.168$\munit           & $L=1$   \\
$a_{1}^{+}(1640)$   & $1.640$\munit   & $.300$\munit           & $L=0,2$ \\
$a_{2}^{+}(1660)$   & $1.660$\munit   & $.280$\munit           & $L=2$   \\
$\pi_{2}^{+}(1670)$ & $1.670$\munit   & $.259$\munit           & $L=1,3$ \\
$a_{2}^{+}(1750)$   & $1.752$\munit   & $.150$\munit           & $L=2$   \\
$a_{4}^{+}(2040)$   & $2.014$\munit   & $.361$\munit           & $L=4$   \\
$\pi_{2}^{+}(2100)$ & $2.090$\munit   & $.625$\munit           & $L=1,3$ \\
$a_{6}^{+}(2450)$   & $2.450$\munit   & $.400$\munit           & $L=6$   \\ \hline
\end{tabular}
\caption[Known $3\pi$ resonances.]{\label{tab:ch9-res}A list of known 
charged $3\pi$ resonances that could be produced in photoproduction
and decay via $\rho\pi$. The column $L_{\rho\pi}$ are the allowed orbital 
angular monetum between the $\rho$ and the $\pi$ when the resonance 
decays. Because we require non--zero isospin, many states can not be produced.}
\end{table}
