The E614 experiment requires precise knowledge of the spin of each muon as
it reaches the target, and one method of achieving this is to select only
highly-polarized muons for tracking, discarding those whose spin is
unsatisfactory.  Several possible methods of muon selection are examined
using Monte Carlo studies (GEANT) and recommendations for future
examinations are discussed.

The attachments are:

1) helixrep.ps		The report in postscript format.
2) psi_vs_r.ps		Initial momentum angle vs. initial distance from
			axis.
3) s_vs_L.ps		Spin at target vs. Muon track focus
4) sL_dc.ps		Spin at target vs. muon track focus, DC's in
			beamline.
5) SlitCut.ps		Spin at target vs. Radial position at z=-150cm.
6) s_rho_nms.ps		Spin at target vs. Helix radius, no multiple
			scattering.
7) s_rho_ms.ps		Spin at target vs. Helix radius, multiple
			scattering on.
8) helixrep.tex		The report in LaTeX format.

Attachments 2 through 7 are also embedded in the report.

Filename: helixrep.ps

Filename: psi_vs_r.ps

Filename: s_vs_L.ps

Filename: sL_dc.ps

Filename: SlitCut.ps

Filename: s_rho_nms.ps

Filename: s_rho_ms.ps

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\begin{document}

\begin{center}

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

\vspace{0.4cm}

{\Large \bf Possible Methods for Optimizing Muon Polarization\\}

\vspace{0.4cm}

\rm{\bf R.P. MacDonald, TRIUMF \\}

\vspace{0.4cm}

\rm{\bf 12 December, 1997 \\}

\end{center}


\vspace{0.6cm}
{\large \bf Introduction}

The E614 experiment requires precise knowledge of the spin of each muon as
it reaches the target, and one method of achieving this is to select only
highly-polarized muons for tracking, discarding those whose spin is
unsatisfactory.  This requires finding parameters of muon tracks, such as
position or momentum, which are strongly correlated with the muon spin,
selecting muons according to some criteria (such as a slit to select
position), and discarding those muon tracks which do not meet these
criteria.  Monte Carlo simulations, described below, have shown that a
simple slit is not sufficient to select muon polarization to the precision
required by the experiment.  However, these same simulations may suggest
more effective methods of achieving muon polarization, including optimizing
the incoming M13 muon beam focus location in $z$, and selecting muons based
on the radii of the helices they trace out as they pass through the strong
magnetic field.


%\section{M13 Focus Position}
%\label{sec:M13focus}
\vspace{0.6cm}
{\large \bf M13 Focus Position}

The muon output from the M13 beam has a focus which can be positioned along
the Z-axis by tuning the magnetic quadrupoles at the end of the beamline,
or by positioning the spectrograph appropriately.  It has been suspected
that the optimum position for this focus is near $z =$ -160 cm, which is
approximately where the radial component $B_r$ of the magnetic field is a
maximum.  (This region is also known as the ``fringe field,'' where the
magnetic field increases from zero in the beamline to a uniform 2.2 T
inside the spectrometer.)  One goal of the Monte Carlo studies (performed
using GEANT) has been to optimize the focus location to give the most
polarized muon flux at the target.

To help this process, the distance $r_i$ from the Z-axis was plotted
against the angle $\psi_i$ between the momentum and the Z-axis, taken for
each muon at $z =$ -300 cm (where the muon tracks were started for these
simulations).  $\psi_i$ was calculated as $\arctan(P_r/P_z)$ (where $P_r =
\sqrt{P_x^2 + P_y^2}$).  Figure \ref{fig:psi_vs_r} shows that the muons
which arrived at the target with the ``best'' spin, $s<-0.99985$, appear to
exhibit a linear relationship between $r_i$ and $\psi_i$, which is
consistent with the existence of an optimum focus position.  This
relationship remained the same regardless of where M13 beam focus was
positioned, which is to be expected; this specified focus is for a mean of
all of the muon tracks, whereas the relationship between $r_i$ and $\psi_i$
depends on the ``focus'' of the individual muon tracks.

\begin{figure}[bp]
  \begin{center}
    \epsfig{file=psi_vs_r.ps,height=4.0in,width=6.0in}
    \caption[Initial momentum angle $\psi_i$ vs. initial distance $r_i$ from
      axis]{Initial momentum angle $\psi_i$ vs. initial distance $r_i$ from
      axis.  $\psi_i$ is the angle the initial muon momentum (taken at $Z
      =$ -300 cm) makes with the Z-axis.  The crosses represent muons whose
      final spin $s$ at the target was  better than -0.99985.  Notice the
      apparently linear relationship between $\psi_i$ and $r_i$ for these
      muons.}
    \label{fig:psi_vs_r}
  \end{center}
\end{figure}

If $P_r$ is assumed to point radially towards or away from the Z-axis, then
the ``focus'' $L$ of each muon track (where the track crosses the Z-axis)
is given by
\begin{equation}
\label{eqn:Lfocus}
  L = -300 + r_i\frac{P_z}{P_r}
\end{equation}
where $L$ and $r_i$ are in centimetres.  (The -300 is a correction for
the fact that the muon tracks begin at -300 cm instead of at the origin.)
A graph of $L$ versus the muon spin at the target (figure \ref{fig:s_vs_L})
reveals that the muons with the best spin tend to have foci near $z =$ -150
cm.  Selecting only those muons for which $s < -.9995$, the mean value of
$L$ is ${\overline L} = -141.3$ cm, with a standard deviation of
$\sigma_{\overline{L}} = 77.6$ cm; these two values may be useful when tuning
the M13 beam shape.

\begin{figure}[bp]
  \begin{center}
    \epsfig{file=s_vs_L.ps,height=4.0in,width=6.0in}
    \caption[Spin at target vs. muon track focus $L$]{Spin at target vs.
      muon track focus $L$.  ($L=-300+r_i\frac{P_z}{P_r}$).  The distribution
      of data points seems to indicate that the optimum focus position for
      the M13 beam would be at $z = $ -150 cm.}
    \label{fig:s_vs_L}
  \end{center}
\end{figure}



\vspace{0.6cm}
{\large \bf Cuts Using Muon Track Focus $L$}

If it were possible, a useful cut would be to measure the muon trajectory
before the muon encounters the fringe field (ie. where the trajectory is
linear),determine $L$, and only select values of $L$ close to -150 cm.  To
test the feasibility of such a cut, two DC's (Drift Chambers) were
constructed in the GEANT simulation and placed in the beamline, upstream of
the fringe field.  The multiple scattering\footnote{The mean momentum angle
  ${\overline \psi}$ at $z =$ -150 cm was $0.573^\circ$ with no multiple
  scattering, and was $3.426^\circ$ with multiple scattering turned on.
  The mean scattering angle, then, was $3.426^\circ - 0.573^\circ =
  2.852^\circ$, five times the original momentum angle.  This means that
  the muon distribution is much different at $z =$ -150 cm than at $z =$
  -300 cm where it is measured, modifying the relationship between $L$ and
  $s$.}  in these detectors modified the relationship between $L$ and $s$
(figure \ref{fig:sL_dc}), so much so that $L$ determined in this way cannot
be used to make a selection.

\begin{figure}[bp]
  \begin{center}
    \epsfig{file=sL_dc.ps,height=4.0in,width=6.0in}
    \caption[Spin at target vs. track focus $L$, DC's in beamline]{Spin at
      target vs. track focus $L$, DC's in beamline.  The presence of drift
      chambers in the beamline, outside of the fringe field, drastically
      degrades the beam polarization, and destroys any useful relationship
      between $L$ and $s$.}
    \label{fig:sL_dc}
  \end{center}
\end{figure}



\vspace{0.6cm}
{\large \bf Cuts Using Physical Slits}

The simplest way to make a cut on a particle beam is to place a physical
slit (of copper, lead, etc) in the path of the beam to select particles
based on position.  To test the effectiveness of a slit as a selection
mechanism, simulations were run whose output included muon radial positions
($r$, the distance from the Z-axis) at several values of $z$, as well as
the final spin of each muon as it entered the target.  Most runs only
considered those muons which actually stopped in the target, but a few runs
were checked in which all muons were considered, and the results were the
same.

The results taken at $z =$ -150 cm will be considered here, though results
were similar for all positions examined.  The slit at -150 cm was inspired
by the focus position results (figure \ref{fig:s_vs_L}); the idea was that
incident muons ``focussed'' at $z =$ -150 would be passing through this
slit at the smallest radial position, while muons focussed ahead or behind
this point would have larger radial positions at the slit and would thus be
cut out.  Narrowing the slit should select more polarized muons.

The muon spin $s$ at the target was plotted against the radial position at
$z =$ -150 cm (figure \ref{fig:SlitCut}).  A relationship between the two
is evident, but the relationship is not strong enough by itself to select a
large enough sample of muons with sufficient polarization.  The use of one
or more slits (at different positions in the beam line and in the
spectrometer) in conjunction with other selection mechanisms will be
studied.

\begin{figure}[bp]
  \begin{center}
    \epsfig{file=SlitCut.ps,height=4.0in,width=6.0in}
    \caption[Spin at target vs. Radial position at $z =$ -150 cm.]{Spin
      at target vs. Radial position at $z =$ -150 cm.  There is a
      relationship between these two parameters, but it is not strong
      enough to be useful at the the precision E614 requires.}
    \label{fig:SlitCut}
  \end{center}
\end{figure}



%\section{Cuts Using Helix Radii}
%\label{sec:HelixRad}
\vspace{0.6cm}
{\large \bf Cuts Using Helix Radii}

According to the Standard Model, the spin of a muon before undergoing
multiple scattering must be antiparallel to its momentum, and so momentum
is the most popular candidate for selecting the muons with the best spin.
Due to the magnetic field (which is approximately uniform within the
proposed spectrometer) each muon will trace out a helix as it travels
through the spectrometer.  It can be shown \cite[p.146]{bib:RhoFromP} that,
if the transverse momentum $P_r$ is given in MeV/c and the axial component
$B_z$ of the magnetic field is given in Teslas, then the helix radius
$\rho$ in cm is $\rho = P_r/3 B_z$ (the factor of 1/3 represents the
conversion from MeV/cT to cm, and carries appropriate units).  Using this
formula, the helix radius for each muon was calculated as it entered the
first drift chamber (DC) in the spectrometer.  With no multiple scattering
(figure \ref{fig:s_rho_nms}) there is a strong correlation between the spin
at the target and the helix radius, as expected.  With multiple scattering
on (figure \ref{fig:s_rho_ms}), scattering in the vacuum windows between
the beam pipe and the spectrometer, as well as in helium gaps at the front
of the spectrometer, degrades the correlation between spin and momentum
(and therefore between spin and helix radius) so much that helix radius
alone is no longer a viable selection mechanism\footnote{The mean momentum
  angle ${\overline \psi}$ at the first drift chamber in the spectrometer
  was $2.162^\circ$ with multiple scattering off, and $2.944^\circ$ with
  multiple scattering on.  The mean scattering angle due to the vacuum
  windows is then $2.944^\circ - 2.162^\circ = 0.782^\circ$.  These values
  correspond to a muon spin of $s = \cos (2.162^\circ) = 0.99929$ with
  multiple scattering off, and $s = \cos (2.944^\circ) = 0.99868$ with
  multiple scattering on, a difference of 0.00061 in muon spin.  This is
  significantly higher than the desired experimental precision of $\pm
  0.00005$.}.

\begin{figure}[bp]
  \begin{center}
    \epsfig{file=s_rho_nms.ps,height=4.0in,width=6.0in}
    \caption[Spin at target vs. Helix radius, no multiple scattering]{Spin
      at target vs. Helix radius, no multiple scattering.  There is a
      strong correlation between helix radius and final spin, due to the
      physical connection between momentum and spin.}
    \label{fig:s_rho_nms}
  \end{center}
\end{figure}

\begin{figure}[bp]
  \begin{center}
    \epsfig{file=s_rho_ms.ps,height=4.0in,width=6.0in}
    \caption[Spin at target vs. Helix radius, multiple scattering on]{Spin
      at target vs. Helix radius, multiple scattering on.  Because
      scattering affects the momentum of a muon but not its spin, the
      correlation between helix radius and spin is effectively lost.  (The
      proportional counters (PC's) which are normally upstream of the DC's
      have been removed for this graph.)}
    \label{fig:s_rho_ms}
  \end{center}
\end{figure}

Note that to obtain helix radii from actual data (constisting entirely
of position and timing information at the DC's) would require fitting a
helix to hit locations at several DC's.  Any helix fit would have some
error associated with it, which means that selection using real data can
only be less successful than the method above.

An extension of this method is under construction, in which a helix is fit
to data and helicies are selected based on both radius $\rho$ and
goodness-of-fit $\chi^2$.  The idea is that the tracks selected using
$\chi^2$ have been scattered less than those which were cut out,
hopefully leaving a stronger correlation between momentum (and helix
radius) and final spin; a second cut would then be made based on $\rho$.


\vspace{0.6cm}
{\large \bf Conclusion}

The relationships examined here suggest possible methods of
selecting (through cuts) or ensuring (by tuning the M13 focus position)
muon polarization.  However, the relationships are weakened by multiple
scattering, so that none of these methods used exclusively will result in a
sufficiently polarized muon beam at the target.  If these methods were to
be applied in tandem (using a slit in combination with helix radius
selection, for example), a more successful selection scheme might be
achieved.  This option is being explored.



\begin{thebibliography}{10}

\bibitem{bib:RhoFromP} P.D.G., {\it Phys. Rev.} {\bf D54}, 1 (1996)

\end{thebibliography}

\end{document}

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