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\begin{document}

\begin{center}

{\Large \bf TRIUMF Experiment E614 \\}
{\Large \bf Technical Note \\}

\vspace{0.4cm}

{\Large \bf A Comparison of Spectrometer Designs with and without Central
Proportional Chambers \\}

\vspace{0.4cm}

\rm{\bf D.H. Wright, TRIUMF \\}

\vspace{0.4cm}

\rm{\bf 1 August 1997 \\}

\end{center}

\begin{abstract}

  Two designs for the central region of the E614 spectrometer were studied
using Monte Carlo simulations of muons stopping and decaying near the target.
One design relies on proportional chambers immediately upstream and downstream
of the target to determine the stop location of the muons.  The other design
has no proportional chambers next to the target and relies solely on data
from more distant drift chambers to reconstruct the muon decay (stop)
location.  Using the total muon depolarization $1-P_\mu$ to discriminate
between the two designs, it was found that the design with proportional
chambers next to the target came closer to meeting the precision requirements
for the measurement of $P_\mu \xi$.

\end{abstract}

Prior to the MIG proposal \cite{MIG96} the design of the E614 spectrometer
called for the aluminum target to be sandwiched between two pairs of CF$_4$ -
ISO proportional chambers.  The purpose of these chambers was to positively
identify muons stopping in the target and distinguish them from muons stopping
and depolarizing outside of the target.  For several reasons, including ease
of construction, reduction of multiple scattering and depolarizing mass, and
target interchangeability, removal of these chambers was considered.  For this
to be feasible, it must be possible to reconstruct with sufficient position
resolution the muon stopping location from the drift chamber hits of the
incoming muon and decay positron tracks.  Since depolarization of muons in
the helium gaps is small it would only be necessary to guarantee that the
muon did not stop in the innermost drift chamber pairs centered at $z = \pm
3.2$ cm. \\

{\large \bf No Proportional Chambers Adjacent to Target \\}

  This problem was studied using the baseline Monte Carlo spectrometer with no
proportional chambers next to the target.  A realistic beam of surface muons
was introduced at $z= -67$ cm and allowed to propagate and decay in the
spectrometer.  Both the muon and positron tracks were reconstructed from drift
chamber hits with 50 $\mu$m position uncertainty.  Each track was reconstructed
by taking the four drift chamber hits closest to the muon stop location and
fitting these hits with a helix.  The helix parameters used to
start the fit were taken from the Monte Carlo position and direction of the
particle at the drift chamber closest to the muon stop location.  The point of
closest approach of the positron track to the muon track was then found and
taken to be the muon stop location.  A histogram of the $z-$coordinate of the
calculated stop location is shown in Fig. 1.

\epsfysize=9cm
\begin{figure}
\begin{center}
\epsffile{tn4fig1.ps}
\end{center}
\caption{Reconstructed $z-$coordinate of muon stop location.}
\end{figure}

The stopping location is not resolved as well as hoped, since the target
distribution has a FWHM of 0.55 cm and long tails extending into the regions
of the nearest upstream and downstream drift chamber pairs.  The above
distribution was the result of several methods used to improve the resolution,
including iterating the calculation of the stop location using the previously
determined stop location as a starting point.

The poor resolution is due almost entirely to the difficulty in fitting low
energy muon tracks;  the well-determined positron tracks contribute little to
the width of the reconstructed distribution.  By the time they approach the
target the muons have so little energy that multiple scattering causes their
tracks to deviate substantially from an ideal helix.  The resulting fits are
poor as indicated in Fig. 2.

\epsfysize=9cm
\begin{figure}
\begin{center}
\epsffile{tn4fig2.ps}
\end{center}
\caption{Distance of closest approach of fitted $\mu$ track to Monte Carlo
stop location.}
\end{figure}

The calculated intersections between muon and positron tracks are therefore
also poor.  The position error in $z$ is given roughly by
\begin{equation}
   \Delta z = \Delta xy /tan \theta
\end{equation}
where $\theta$ is the angle of the positron track with respect to the $z$
axis and $\Delta xy$ is the distance of closest approach of the fitted muon
track to the Monte Carlo stop.  It is seen in Fig. 2 that the tail in the
distribution of $\Delta xy$ begins at about 0.9 cm.  For a positron track at
10$^\circ$ the error in the reconstructed $z$ of the muon stop could therefore
be 5 cm or more.  This explains the width of the reconstructed stopping
distribution in Fig. 1.

In order to reduce the probability that muons stopping in the upstream and
downstream drift chambers had reconstructed stop locations near the target,
a severe cut was placed on the reconstructed stop distribution of Fig. 1.
It required that the reconstructed $z$ coordinates of all stop locations be no
more than 0.5 cm from the target, $z = 0$.  Once this cut was imposed, the
true (Monte Carlo) stop locations were histogrammed as shown in Fig. 3. 

\begin{figure}
\begin{center}
\epsfig{file=tn4fig3.ps, height=3.5in,width=5.0in}
\end{center}
\caption{True (Monte Carlo) stop locations after the application of the
$\pm$ 0.5 cm cut on the reconstructed stop locations.}
\end{figure}

  In addition to the above cut, only stops with $z$ coordinates bewteen $\pm$
2.81 cm were accepted.  This condition was imposed because the muon stop
location can be bounded, independent of track reconstruction, by observing
which of the upstream and downstream drift chambers fired during an event.
If the first two chambers upstream (PDC(-2), PDC(-1)) of the target fired and
the first chamber downstream (PDC(+1)) of the target did not fire, the muon
must have stopped between the active region of PDC(-1), ending at -2.81 cm,
and the active region of PDC(+1), beginning at 2.81 cm.  This region contains
the target, two helium gaps, two buffer zones containing DME gas, and four
mylar windows.

Using the distribution of stops in Fig. 3, the maximum muon depolarization
was estimated by taking the stop-weighted average of the depolarization in
all the materials in which the muon could stop between $\pm$ 2.81 cm :

\begin{equation}
 \overline{1-P_\mu} = \frac{N_{tgt}(1-P_{\mu TGT})+N_{He}(1-P_{\mu He}) +
 N_{buf}(1-P_{\mu DME})+N_{My}(1-P_{\mu My})}{N_{tgt}+N_{He}+N_{buf}+N_{My}}.
\end{equation}

  Here it has been assumed that the drift chambers are operating at 100\%
efficiency.  The effect of a small inefficiency is treated below.  While it
has not been measured, theoretical estimates of muon depolarization in
DME give $P_{\mu DME} > 0.95$.  For helium, $P_{\mu He} = 1.0$, but if a little
DME should leak into the helium volume, the muon will have a greater affinity
for the DME, thus increasing the depolarization.  It was therefore assumed that
$P_{\mu He} = 0.95$.  For the target and mylar foils, the values
$P_{\mu tgt} = 1.0$ and $P_{\mu My} = 0.0$, respectively, were taken.

\vspace{0.3cm}

\begin{center}
\begin{tabular}{| l | c | c |}  \hline
  Material  &  $N_{stops}$ &  $1-P_\mu$  \\  \hline
 Al Target  &   4556    &   0.00    \\
 2 He gaps (upstream+downstream) &     67    &   0.05    \\
 2 DME buffers (upstream+downstream) &      4    &   0.05    \\
 4 Mylar foils (upstream+downstream) &      7    &   1.00    \\  \hline
\end{tabular}

\vspace{0.2cm}
Table 1. Number of Monte Carlo stops in the region $-2.81 < z < 2.81$ under the
condition that the reconstructed $z$ coordinate of the muon stop is within
$\pm$ 0.5 cm of the target.
\end{center}

  The number of Monte Carlo stops in the various volumes is given in Table 1.
Using these values the average maximum depolarization was found to be
\begin{equation}
 \overline{1-P_\mu} = 2.3 \times 10^{-3} ,
\end{equation}
eight times larger than the desired precision of $3 \times 10^{-4}$ in the
Michel parameter $P_\mu \xi$.  The final depolarization could be reduced
further by tightening the cut on the reconstructed stop location.  However, 
the current value was obtained with a sample of 20,000 Monte Carlo incident
muons of which 4634 remain as good events.  Tightening the cut further
would reduce the overall event efficiency significantly. 

The drift chamber pairs nearest the target were centered at $\pm$ 3.2 cm.
In another Monte Carlo simulation, these chambers were moved to $\pm$ 5.2 cm
in the hope that this would reduce the number of muons stopped in the drift
chambers but reconstructed near the target.  The effect was to reduce the
total depolarization by a factor of two, still not low enough.  This reduction
factor was not as large as expected because the larger helium gap increased
the lever arm for multiple scattering occurring in the chambers. \\

{\Large \bf Proportional Chambers Adjacent to Target \\}

\begin{figure}
\begin{center}
\epsfig{file=tn4fig4.ps, height=3.5in,width=5.0in}
\end{center}
\caption{Central portion of spectrometer with proportional chambers (PCs) next
to target and drift chamber pairs (PDCs) at $\pm$ 3.2 cm.}
\end{figure}

\begin{figure}
\begin{center}
\epsfig{file=tn4fig5.ps, height=6.0in,width=5.0in}
\end{center}
\caption{Muon stopping distribution with central PCs in place. Top: stopping
distribution over entire spectrometer.  Center: same spectrum as top, but
for $-1.0 < z < 1.0$ cm.  Bottom: same as top, but for $-4.0 < z < -2.4$ cm.}
\end{figure}

  Four proportional chambers were added to the baseline Monte Carlo, two
immediately upstream of the target PC(-1,-2) and two immediately downstream
PC(+1,+2) as shown in Fig. 4.  These chambers were filled with CF$_4$/isobutane
(80/20).  The thickness of absorbers in front of the muon beam was reduced
slightly from that in the previous Monte Carlo in order to maintain a flat
stopping distribution in the aluminum target.  All other beam and spectrometer
parameters were unaltered.  In this case, no tracking or vertex reconstruction
was required since the condition PC(-2)$\cdot$PC(-1)$\cdot \overline{PC(+1)}$
was enough to locate the stop.  However, muons satisfying this condition could
stop anywhere in the gas volume of PC(-1) as well as in the target.  Also,
if the PCs are inefficient, a small fraction of stops in the gas of PC(+1) and
the first downstream mylar foil would also satisy this condition.

The total depolarization was therefore obtained by counting the number of stops
in each volume as shown in Fig. 5 (center) and taking the stop-weighted average
of the depolarizations as before,
\begin{equation}
 \overline{1-P_\mu} = \frac{N_{tgt}(1-P_{\mu TGT})+N_{PC(-1)}(1-P_{\mu CF4}) +
 (1-\epsilon)[N_{PC(+1)}(1-P_{\mu CF4})+N_{My}(1-P_{\mu My})]}{N_{tgt}+
N_{PC(-1)}+(1-\epsilon)[N_{PC(+1)}+N_{My}]}.
\end{equation}
Here $1-\epsilon$ is the PC inefficiency, taken to be 0.999.  For 10,000
incident muons the number of stops in each volume is given in Table 2 along
with the appropriate depolarizations.  A lower limit on the depolarization in
CF$_4$/isobutane was measured to be 0.985 \cite{Sel}. 
\vspace{0.3cm}

\begin{center}
\begin{tabular}{| c | c | c |}  \hline
  Material           &  $N_{stops}$ &  $1-P_\mu$  \\  \hline
 Al Target           &   7212    &   0.000    \\
 CF$_4$-ISO (PC(-1)) &    269    &   0.015   \\
 CF$_4$-ISO (PC(+1)) &    338    &   0.015   \\
 1st Mylar foil downstream &    172    &   1.000    \\  \hline
\end{tabular}

\vspace{0.2cm}
Table 2. Number of Monte Carlo stops in the region $-0.4 < z < 0.4$.
\end{center}
For reference, Fig. 5 (bottom) shows the muon stopping distribution at the
location of the first PDC pair upstream of the target.  Most of the stops
occur in the mylar foils as shown by the five largest spikes in the
distribution.

Using Eq. 4 and the values of Table 2, the total muon depolarization for
this spectrometer option was found to be
\begin{equation}
 \overline{1-P_\mu} = 5.6 \times 10^{-4} .
\end{equation}
This value, while four times better than the no-PC option, still needs
improvement if the precision of $3 \times 10^{-4}$ for $P_\mu \xi$ is to
be reached.  This could be achieved by a better measurement of the lower
limit on the depolarization of muons in CF$_4$/ISO, which can be made
during calibration runs or data taking.  It could also be improved by
removing more absorber from the muon beam, thus causing fewer muons to
stop in PC(-1).  At the same time more muons would stop in the gas of
PC(+1), requiring good chamber efficiency to reject them.  In this case
the stopping distribution in the aluminum target would no longer be flat.

In conclusion, the spectrometer design with central proportional chambers
comes much closer to the desired precison for E614.  This design is also
more efficient, keeping 72\% of muon decay events, as opposed to 23\% for
the no-PC option.  

\begin{thebibliography}{10}

\bibitem{MIG96} E614 collaboration Major Installation Grant proposal (1996).
\bibitem{Sel} V. Selivanov, private communication, (1997).

\end{thebibliography}

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