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\begin{center}               
 {\bf \Large   TRIUMF Experiment E614 \\
    Technical Note \#75 \\ }
\vspace{1cm}                      
{\bf \Large 
Monte Carlo study of tracks rescattered from E614 magnet yoke. \\}
\vspace{1cm}                      
{\bf  M.A.Vasiliev \\
 10 January 2003} 
\end{center}

Tracks coming back to E614 detector sensitive area from  magnet yoke  were investigated.
Monte Carlo simulations were done using E614 GEANT~\cite{den1} {\bf devel}  version. 
A beam of  polarized muons (with  positron admixture) 
has 29.79 MeV/c momentum, momentum spread $\Delta P/P=0.7\%$ and   spot in focus x=1mm, y=1mm.
Beam was simulated using {\bf PGUN} E614 GEANT card. 
Explicit delta ray production with reduced Landau fluctuations 
for energy loss was used in these simulations.
$2\times 10^6$  triggers were generated. Approximately  65\% of the events was muons stopped in target. 
All together 1,297,776 muon stops in target was observed.


\section*{Rescattered tracks in E614 detector.  }

While it is easy to estimate, that fraction of tracks which comes back to sensitive area of the E614 setup
from magnet yoke should be rather
small, the question about possible impact of such tracks on the measured  Michel spectrum must
be investigated. 
Even small admixture of ``bad'' events can be potentially dangerous for achieving high resolution 
in Michel parameters measurements. 
For instance, such tracks can give troubles in case  they originate only from some,  
limited part of Michel spectrum, other
words, if momentum spectrum of positrons with such tracks do not follow Michel spectrum shape itself.
In order to learn it, the detailed Monte Carlo simulation of events with tracks coming back
from magnet yoke  to
sensitive part of E614 detector was performed. 


\section*{Monte Carlo Study}

Particles, which came to magnet yoke was marked, but contrary to normal E614 GEANT version,
 not stopped, and their later history was considered.In magnet yoke all usual physics processes
was taken into account, but absence of magnetic field in yoke was supposed in that study.  
Particles  coming back to sensitive area, which was defined as cylinder around the target with 
radius $r<16.9$~cm and length along the beam $z<74.0$~cm,  appeared in
histograms.
Study show that all together   we have 0.092\% of such events ( normalized on the amount of
muons stoped in target).

93.5\% of such particles are positrons, while others are electrons. 

Kinetic energy spectrum of these particles shown in Fig~\ref{fig1}.


\begin{figure}
\begin{center}
\includegraphics[height=6.5cm]{/home/vasiliev/tex/e614/tn2002_fig1.eps}
\caption{Kinetic energy of particles coming back to E614 detector sensitive area from
magnet yoke.For 1.3M $\mu$ stoped in target.}
\label{fig1}
\end{center}
\end{figure}


Main part of them (62\%) have kinetic energy below 15.85 MeV, which is supposed to be E614 lower 
limit (x=0.3) for measured Michel spectrum.
That reduce amount of potentially dangerous events to 0.035\%.  
   

Space origin of particles, coming back from yoke to sensitive area are shown in Fig~\ref{fig2}. 

\begin{figure}
\begin{center}
\includegraphics[height=6.5cm]{/home/vasiliev/tex/e614/tn2002_fig2.eps}
\caption{Radius vs z of coming back tracks at their original vertex. For 1.3M $\mu$ stoped in target.}
\label{fig2}
\end{center}
\end{figure}


Approximately 60\% coming from target region. While other 40\% tracks can be rejected by 
reconstruction of their vertexes, tracks coming from target region probably can not 
and because of that require detailed consideration.

Original momentum spectrum of all rescattered by yoke tracks  have Michel
 shape, except for it's lower momentum
part. That part correspond to tracks originated not from the target region - Fig~\ref{fig3}.
After rejection of such tracks one can see (also at Fig.~\ref{fig3})
that the original momentum 
distribution favors higher momentum decays compared to an unbiased Michel 
shape.  Therefore, if we reconstruct the events that contain these 
backscattered particles in the Monte Carlo (because the tracking terminated 
prior to the backscatter occurring) and fail to reconstruct them in the data, 
we will end up with a smaller value of $\rho$ than we should by up to $\sim$\,5 
parts in $10^4$~\cite{carl1}.


%That means that it is no any special part of decay positron spectrum from which 
%positrons have higher probability to be rescattered back from magnet yoke. And that effect of such 
%rescattering will have minimal impact on accuracy of Michel spectrum measurements.   

\begin{figure}
\begin{center}
\includegraphics[height=6.5cm]{/home/vasiliev/tex/e614/tn2002_fig3.eps}
\caption{Original momentum for particles coming back from yoke, which vertex has $z<10$. 
Dashed line - particles with vertex $z>10$. For 1.3M $\mu$ stoped in target.}
\label{fig3}
\end{center}
\end{figure}

       
On Fig~\ref{fig4} angular distribution in original vertex of rescattered tracks are presented. 
Strong forward-backward asymmetry are seen,
while shapes for forward only or backward only  tracks are consistent with 
flat angular distributions.    
To understand that effect additional beam pipe after target was introduced. 
With such additional beam pipe it is  much more symmetric situation in detector geometry 
then the real one has.
779,939 muon stops in target for such geometry was generated. 456 tracks came back 
to detector sensitive area. It corresponds to 0.058\% from all stoped in target muons.
Simulations performed with such artificially symmetric geometry show (Fig~\ref{fig5})
   nearly symmetric original angular distribution for rescattered from yoke tracks. 
It means that observed suppression in backward direction in angular distribution
is due to presence of beam pipe 
upstream of target mainly, which introduce some asymmetry in detector setup.   
Corresponding distribution for original momentum can be seen on Fig~\ref{fig6}.
It  has an even larger bias for high momentum decays than was present in Fig~\ref{fig3}.    

\begin{figure}
\begin{center}
\includegraphics[height=6.5cm]{/home/vasiliev/tex/e614/tn2002_fig4.eps}
\caption{Original angular distribution at vertex for particles coming back from yoke. 
For 1.3M $\mu$ stoped in target.}
\label{fig4}
\end{center}
\end{figure}


\begin{figure}
\begin{center}
\includegraphics[height=6.5cm]{/home/vasiliev/tex/e614/tn2002_fig5.eps}
\caption{Original angular distribution at vertex of particles coming back from yoke
for detector geometry with additional beam pipe behind the target. For 780K $\mu$ stoped in target.}
\label{fig5}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\includegraphics[height=6.5cm]{/home/vasiliev/tex/e614/tn2002_fig6.eps}
\caption{Original momentum for particles coming back from yoke, which vertex has $z<10$,
for detector geometry with additional beam pipe behind the target. 
Dashed line - particles with vertex $z>10$. For 780K $\mu$ stoped in target.}
\label{fig6}
\end{center}
\end{figure}



\section*{Conclusion}



Performed Monte Carlo study showed that rescattered from magnet yoke decay positrons come  
preferably from high momentum part of Michel spectrum.
It is approximately 0.092\%  of such events, or less. 
They mainly rescattered from downstream part of the yoke due to some asymmetry in detector geometry,
caused by  presence of beam pipe before target. 
If we will not take into account these events we can end up with a smaller value of $\rho$
than we should.





\begin{thebibliography}{999}


\bibitem{den1}
D.H.~Wright, A Users Guide to E614 GEANT, E614 TN-29, 30 March 1999.

\bibitem{carl1}
C.~Gagliardi, Private communication, 2003 


\end{thebibliography}

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