Hi ALL,
This note is my draft analysis for discussion, changes, improvement,...
Vladimir 

Vladimir Selivanov,
RRC "Kurchatov Institute", Moscow 123182, Russia.
e-mail:selivanov@triumf.ca; phone:7-095-1967058; fax:7-095-8825804. 

Filename: tn54.tex.ps.gz

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\begin{document}

\begin{center}

{\Large \bf TRIUMF Experiment E614 \\}
{\Large \bf Technical Note 54 \\}

\vspace{0.4cm}

{\Large \bf TWIST Detector and Systematic Error Estimations \\}

\vspace{0.4cm}

\rm{\bf V.I. Selivanov \\}

\vspace{0.4cm}

\rm{\bf 29 August 2001\\}

\end{center}

\begin{abstract}
The following note is a sixth analysis of the systematic errors of the E614.
Previous ones have been done in 1991 (Original E614 Proposal), 1992 (Report,
unpubished), 1994 (Review, unpublished), 1997 (E614 Project Proposal), 1998
(E614 Project Proposal). The given note is based on many Technical Notes that
were done by the E614 team after the last Project Proposal. The Michel
parameter error estimations practically are not changed: 
$\rm \sigma (P_\mu\xi)=[\pm 10(stat)\pm 8.5(syst)]\cdot 10^{-5}, \sigma (\rho)
=[\pm 5(stat)\pm 7.0(syst)]\cdot 10^{-5}, \sigma (\delta)=[\pm 8(stat)\pm 6.4
(syst)]\cdot 10^{-5}$.  
\end{abstract}


{\large \bf 1. Detector Description \\}

Schematic layout of the E614 detector assembly to be inserted into the
magnet is presented in Fig.1 together with sample positron helices in a
$\rm B=2.35T$ at energies 30 and 53MeV and polar angles $\theta =5^\circ , 
10^\circ , 60^\circ $. Surface
muons from M13 channel going along Z$-$axis and stop in a target at center of
the assembly. Gas degrader placed upstream of the scintillator detector.
Muon counter (TEC) placed at Z=$-$140cm outside of the iron shield close to
the input window.

%\begin{figure}[ht!]
%\centering
%\label{fig:f1}
%\includegraphics[angle=90,height=1.3\textwidth]{tn53_fig1.eps}
%%\includegraphics[\textwidth=0.9]{tn54_fig1.eps}
%\caption[Schematic layout of the detector assembly.]{
%\small   Schematic layout of the detector assembly.} 
%\end{figure}

$\bullet$ The stopping target is a thin $\rm \approx 20mg/cm^2$ foil from a 
material with low Z
of nucleus to reduce radiation losses of outgoing positrons. Foils from Be,
graphite, Al, and aluminized mylar will be used. About 75\% of surface muons
stop inside the target \cite{1}. Cross-section of the stopped muon beam has
$\rm\sigma_x\approx\sigma_y\approx 7.5mm$ at $\rm B=2T$. The target also 
serves as a cathode plane for proportional chambers PC($-$1) and PC(+1).

$\bullet$ Identical (X+Y) PC pairs (central PCs) on both sides of the stopping
target are indicators and coordinate detectors of muon stops inside the
target. A "fast" drift gas $\rm CF_4/ISO (80/20)$ is used. Scatterplot of 
energy losses in PC($-$2) and PC($-$1) has different distributions for muons 
stopped in PC($-$1) gas and the target (see Fig.2) \cite{2}. Amplitude analysis 
of the energy loss in PC($-$2) and PC($-$1) provides for rejection at level $\rm <0.6\%$ 
muons stopped in PC($-$1) gas and excludes influence of a muon depolarization by
PC($-$1) gas on $P_\mu$ value. Center of gravity of muon stopping distribution
inside the target along Z axis has defined with accuracy $\rm\pm1\mu m$ by muon ratio
$N_{PC(-2)}/N_{PC(+2)}$ or $N_{PC(-1)}/N_{PC(+1)}$ \cite{2}.  All wires of the 
PCs instrumented therefore by TDC and ADC outputs

$\bullet$ 20mm helium gap between the central PCs and the PDC pair closest to the
target is introduced for better separation of an ingoing muon from outgoing
upstream small polar angle positrons.

\newpage

\begin{figure}[ht!]
\centering
\label{fig:f1}
\includegraphics[angle=90,height=1.29\textwidth]{tn54_fig1.eps}
%\includegraphics[\textwidth=0.9]{tn54_fig1.eps}
\caption[Schematic layout of the detector assembly.]{
\small   Schematic layout of the detector assembly.} 
\end{figure}

\begin{figure}[ht!]
\centering
\label{fig:f2}
\includegraphics[angle=0,width=0.6\textwidth]{tn54_fig2.eps}
%\includegraphics[\textwidth=0.6]{tn54_fig2.eps}
\caption[Energy loss in $\rm (PC-2)$ and $\rm (PC-1)$ chambers for various
conditions. Top left: no conditions; top right: stop in $\rm (PC-1)$; bottom
left: stop in target; bottom right: stop in in PC(+1) or PC(+2)]{
\small   Energy loss in $\rm (PC-2)$ and $\rm (PC-1)$ chambers for various
conditions. Top left: no conditions; top right: stop in $\rm (PC-1)$; bottom    
left: stop in target; bottom right: stop in in PC(+1) or PC(+2)}
\end{figure}

$\bullet$ Eight pairs ($\pm1,...,\pm8)$ of (X+Y) PDC planes measure positron helices. 
They are interleaved by 40 or 60mm helium gaps introduced to improve detector
resolution at small angles. "Slow" and "cold" drift gas, DME ($\rm C_2H_6O$) is
used in all PDCs to decrease an influence of Lorentz drift angle at the
magnetic field $\rm B=2T$ on a shortest reconstructed distance between positron
helix and sense wire.

$\bullet$ Dense PDC stacks serve as an additional tool for identification of a
positron helix with period around 4mm. This helix crosses all PDC planes at
identical (X, Y) coordinates, and imitates therefore a straight line with
polar angle $\theta=0$.

$\bullet$ Dense PC stacks serve as a zero time master with 2.5ns (RMS) resolution
\cite{3} for the outgoing positron helix registration. A "fast" drift gas$\rm 
CF_4/ISO(80/20)$ is used.

$\bullet$ Upstream and downstream parts of the detector are mirror symmetrical
relative to the target plane excluding the central PCs. The [PC($-$1X)$-$
target$-$PC(+1Y)] geometry is selected to decrease a crosstalk between the
planes, especially from a muon stopped in the target and producing a big
signal in PC($-$1X).

$\bullet$ Plastic scintillator serves for muon identification. It is 0.25mm thick 
by 25mm diameter Bicron. A Bicron fiber wavelength shifter 1mm in diameter was
wrapped around the scintillator diameter. The ends of the fiber extended
approximately 1m away from the scintillator and coupled to two PMTs. The
time resolution of this assembly is less than 2ns \cite{4}.

$\bullet$ An 18cm long variable gas degrader allows control over the muon stopping
distribution by varying the $\rm  CO_2/He$ mixture \cite{5}.

$\bullet$  Muon counter \cite{5,6} is a time expansion chamber (TEC) working at 
DME gas pressure P=20$-$40torr in order to restrict a multiple scattering of surface
muons. The TEC allows measurement of each muon trajectory with space error $\rm 
\approx 50\mu m$ and angle error $\rm \approx 5mrad$.

$\bullet$ Detailed detector specification is presented in Table~\ref{tab1}.

\begin{table}[ht!]
\caption{Basic parametrs of the E614 detector}
\label{tab1}
\begin{center}
\begin{tabular}{| c | c | c |}  
\hline
 & PC & PDC \\
\hline 
Diameter of (W/Re)Au sense wires & 
\multicolumn{2}{| c |} {$\rm 15\mu m$} \\
\hline
Wire spacing & 2mm & 4mm \\
\hline
Gap between wire and cathode & 
\multicolumn{2}{| c |} {2mm} \\
\hline
Cathode plane & 
\multicolumn{2}{| c |} {Double-side $\rm 6\mu m$ aluminized mylar} \\
\hline
Drift gas & $\rm CF_4/ISO (80/20)$ & DME($\rm C_2H_6O$) \\
\hline
Gas between PDC (PC) modules & 
\multicolumn{2}{| c |} {He + 5\%$\rm N_2$} \\
\hline
Number of planes & 12 & 44 \\
\hline
Number of wires per plane & 160 & 80 \\
\hline
Total number of wires & 1920 & 3520 \\
\hline
Working diameter of the detector & 
\multicolumn{2}{| c |} {320mm} \\
\hline
Gas pressure & 
\multicolumn{2}{| c |} {760 Torr} \\
\hline
Temperature & 
\multicolumn{2}{| c |} {$\rm 20^\circ C$} \\
\hline   
\end{tabular} 
\end{center}
\end{table}
 
{\large \bf 2. Peculiarities of the E614 Detector \\}

$\bullet$ An attempt to measure separately any Michel parameter with accuracy of
$\rm 10^{-4}$ requests an independent knowledge of other parameters with the same
precision because any part of muon decay spectrum is described by more than
one parameter. For example, accuracy of world value $\rho$ restricts the
measuring value of $\delta$ at level of $\pm 1.1 \cdot 10^{-3}$ \cite{7}. Only 
Michel parameters combination $P_\mu\xi\delta/\rho$ does not depends on rest 
Michel parameter $\eta$. The combination was measured with precision of $\pm3
\cdot 10^{-3}$ \cite{8}. We have no idea how one could improve the last result 
using a similar detector. Thus, because all Michel parameters are known with 
accuracy of $\pm10^{-3}$, we have proposed a detector design to measure with 
accuracy about $10^{-4}$ the all Michel parameters simultaneously. The proposed 
detector measures 3D positron spectrum from muon decay at energy limits $x= 
0.1-1$, polar angle of $\rm 0.5>0.5$} and that provides therefore most important advantage of the 
technique. Price of the technique using is sufficient increasing of wire channel
number. Energy and angle resolution of the detector is presented in Fig.4 
\cite{10}.

\begin{figure}[ht!]
\centering
\label{fig:f3}
\includegraphics[angle=0,width=0.5\textwidth]{tn54_fig3.eps}
%\includegraphics[\textwidth=0.9]{tn53_fig2.eps}
\caption[Confidence level (CL) distribution for reconstructed events. Initial
positron energy 50MeV, initial polar angle $\rm \cos\theta =0735$. PDC space
resolution is $\rm 50\mu m$. "Slide 1(raw)" is the result of the first slide
(1234) fitting. "Slides 1$-$5" present the same events that were fitted with
slide selection and resulting distributions of all five slides were summed. The
flat distribution with $\rm =0.5$ corresponds to the fit of vacuum helices
with $\rm 50\mu m$ PDC resolution.]{
\small   Confidence level (CL) distribution for reconstructed events. Initial
positron energy 50MeV, initial polar angle $\rm \cos\theta =0735$. PDC space
resolution is $\rm 50\mu m$. "Slide 1(raw)" is the result of the first slide    
(1234) fitting. "Slides 1$-$5" present the same events that were fitted with    
slide selection and resulting distributions of all five slides were summed. The 
flat distribution with $\rm =0.5$ corresponds to the fit of vacuum helices 
with $\rm 50\mu m$ PDC resolution.}
\end{figure} 

\begin{figure}[ht!]
\centering
\label{fig:f4}
\includegraphics[angle=0,width=0.6\textwidth]{tn54_fig4.eps}
%\includegraphics[\textwidth=0.9]{tn53_fig1.eps}
\caption[$\rm Energy\ HWHM=1/2\cdot FWHM(E_{fit}-E_0)$ and $\rm angle\ HWHM=1
/2\cdot FWHM(\cos\theta_{fit}-\cos\theta_0)$ resolutions at different initial 
positron energies.]{
\small    $\rm Energy\ HWHM=1/2\cdot FWHM(E_{fit}-E_0)$ and $\rm angle \ 
HWHM=1/2\cdot
FWHM(\cos\theta_{fit}-\cos\theta_0)$ resolutions at different initial positron
energies.}
\end{figure}

$\bullet$ The peculiarity of the proposed detector is defined by its planar
geometry. A planar target and the assembly of planar chambers (PCs and PDCs)
are arranged perpendicular to the magnetic field (Fig.1). In such layout the
mean energy loss of particles in the detector is given by $\rm dE(E,\theta)/dz =
[dE(E,\theta=0)/dz]/cos\theta$. This formula simply expresses the fact that the 
effective degrader thickness (path length) grows as the 1/cos$\theta$. The 1
/cos$\theta$ dependence is obvious for the planar target, for a gas gap and for 
the planar cathode plane. It also applies to wires even though loss does not 
have this simple dependence on $\theta$ (for cylindrical wires) since the 
probability of hitting a wire does have a 1/cos$\theta$ dependence on $\theta$ 
also. This dependence was investigated by Monte Carlo simulation \cite{10}. 
Simulation positron source with energy $\rm E_0=50MeV$ was uniformly distributed 
over a 20mm diameter within the 20mg/$\rm cm^2$ stopping Al target. Fig.5 shows 
the dependence of the mean reconstructed positron energy $<\!\rm E_{fit}\!>$ on 
entrance of PDC module \# +1 after passing by positron of Al target, two PC 
chambers, He gap between the PCs and PDC, first cathode plane of the first PDC. 
As one can see, it can be parameterized with a linear function $<\!\rm E_{fit}
\!>=E_1 - \alpha/cos\theta$, where $\rm E_1$ is the initial positron energy at 
a muon decay point and $\alpha$ is the positron energy loss at $\theta$ = 0. The 
straight line fits to each slide data points are presented on Fig.5. The net 
result is: $\rm \Delta \equiv E_1-E_0=<\!E_{fit}\!> -E_0 = (-2.2\pm 2$)keV
for extrapolation to 1/$\cos\theta = 0$. {\it It means we can reconstruct an 
original energy of a decaying positron at any angle $\theta$, without knowledge 
of its energy losses before the coordinate detector.} The 1/cos$\theta$ 
dependence is violated by scattering and radiative interactions with the detector 
matter. Non-conservation of the mean value of cos$\theta$ in the E614 detector is 
less than $\rm 6\cdot 10^{-5}$ for 20mg/cm$^2$ of Be target at $cos\theta = 0.5$ 
and E$_0=20$MeV, and it is about $2\cdot 10^{-5}$ at E$_0 = 50$MeV (see Fig.11 in 
reference \cite{10}). The energy calibration will not be restricted to this $
x \rightarrow 1$ point but can be assumed to apply over the energy range $0.4
$ as a function of
$\rm 1/\cos\theta_0.$ $\rm E_0=50MeV$, $\rm \sigma_{PDC}=50\mu m$. Straight lines
are the fits with function $\rm <\Delta E_{fit}>=E_0-\alpha /\cos\theta_0$.]{
\small   Mean reconstructed positron energy $\rm $ as a function of
$\rm 1/\cos\theta_0.$ $\rm E_0=50MeV$, $\rm \sigma_{PDC}=50\mu m$. Straight
lines
are the fits with function $\rm <\Delta E_{fit}>=E_0-\alpha /\cos\theta_0$.}
\end{figure}

$\bullet$ Amplitude analysis of signals from PC($-$2) and PC($-$1) provides 
reliably identification of muon stopped in a target with thickness of 20mg/cm$^2$ 
only (see Fig.2). An outgoing positron passes in average 12mg/cm$^2$ of matter
before the entrance of first PDC plane. It corresponds to $\approx$30keV of energy
loss before the coordinate detector, compare with 240mg/cm$^2$ ($\approx$0.6MeV
correspondingly) in most precise previous experiments \cite{7, 8}. A possible
distortion of the original Michel spectrum by the energy loss will be much
smaller in the E614 detector than in previous ones \cite{7,8}.

$\bullet$ The longitudinal magnetic field B=2T is sufficient to keep a thermalized
muon polarization as that will be estimated below.

$\bullet$ The proposed goal to measure Michel parameters $\rho, P_\mu \xi,
\delta$ at level of 10$^{-4}$
accuracy requests to measure the Michel spectrum by the E614 detector with a
systematic energy shift $\rm \Delta E<10keV$ and angle shift 
$\rm \Delta\theta <5\cdot\ 10^{-4}$ according to
our previous Monte Carlo estimations. We need to design and manufacture a
precise drift chamber assembly for the positron registration. An influence
of the detector imperfections on reconstructed Michel parameters is
estimated below.

\vspace{0.5cm}
{\large \bf 3. Systematic Error Estimations \\}

\vspace{-0.2cm}
{\bf 3.1. M13 beam line\\}

{3.1.1.Coulomb scattering of muons inside the production target.\\}
The production
of surface muons has been modeled in a routine based on GEANT (see Fig.4 in
\cite{11}).
Muon depolarization was calculated versus depth of decaying muon from the
1AT1 target surface. The M13 channel with a central momentum of 29.6MeV/c
and the momentum acceptance of $\rm \pm 0.5$\% captures of muons from a maximal 
depth of 20$\mu$m. It corresponds to the average value of $\rm (1-P_{\mu})=9
\cdot 10^{-5}$ of the muon beam. The central beam momentum stability is $\rm 
\Delta p/p=0.03$\% (see 3.1.4). It corresponds to $\rm (1-P_{\mu})=10^{-5}$. We 
will accept a pessimistic estimation $\rm (1-P_\mu)=(9\pm 2)\cdot 10^{-5}$. 

{3.1.2. Non-surface muon contamination.\\} 
Impurities in the surface muon beam,
that is, any particle that is not a surface muon , may be a source of
positrons that, if mistakenly identified as originating from the decay of a
polarized surface muon would distort the Michel spectrum. The impurities at
p$_0$=29.6MeV/c, given in Table~\ref{tab2} were determined in a test run in 1995.

\begin{table}[ht!]
\caption{M13 surface $\mu^+$ beam impurities per surface muon (PSM).}
\label{tab2}
\begin{center}
\begin{tabular}{| c | c |}  
\hline
Contaminant Particle & Measured rate/surface muon \\ 
\hline
$\rm e^+$ & 4 \\ 
\hline
$\pi^+$ & $\rm 2.9\cdot 10^{-3}$ \\
\hline
$\mu^+$(cloud) from 1AT1 (graphite) & $\rm 2.1\cdot 10^{-2}$\\
\hline
$\mu^+$(cloud) from S0 & $\rm 3.4\cdot 10^{-4}$ \\
\hline
$\mu^+$(non-surface) in "usable" window & $\rm 6.7\cdot 10^{-5}$  \\ 
\hline
\end{tabular}
\end{center}
\end{table}
 
Positrons from M13 easy distinguish from a surface muon because the
positrons go through full detector assembly near Z$-$axis. Probability to
coincidence of a positron from the channel with an outgoing decaying
positron is about 12\% inside 10$\mu s$ window at muon intensity of 3$\rm
 \cdot 10^3s^{-1}$.
First of all, one can exclude the overlapping events from the Michel
spectrum, as that one does in $\mu$SR-technique. Second one, we not see
difficulties to reconstruct the both positrons. Influence of the
contamination on Michel parameters is negligible therefore.

Most part of $\pi ^+$ captured by the M13 stops in scintillator and the upstream
PCs and PDCs dense stacks before the stopping target because range of $\pi ^+$
is
less on 70mg/cm$^2$ than surface muon range ($\rm \approx 150mg/cm^2$). Muons from 
the decaying pions have the momentum 29.8MeV/c and they are uniformly
distributed in 4$\pi$ solid angle. The muons with a polar angle $\rm \theta >
20^\circ$ can be measured by the upstream PDC pairs and rejected therefore. Rest 
part of the muons with $\rm \theta <20^\circ$ can imitate surface muons from M13 
channel. Their percentage is about 1\%. These muons can stop in the target and 
do a total admixture to true surface muons about $1\% \cdot\rm 2.9\cdot 10^{-3} =
2.9\cdot 10^{-5}$. Let us remark here that the muons will have polarization $\rm 
(1- P_{\mu})<1-\cos 20^\circ=0.06$. As result a shift of $\rm P_{\mu}\xi$ value 
will be much less than $10^{-5}$.

The cloud muons from 1AT1 target born from pions with a momentum $\rm p_{\pi}
\approx 60MeV/c$
decaying between the 1AT1 and the first bending magnet (total length is
$\approx 2$m). The muons flying out upstream and have an appropriate momentum
p$_{\mu} \approx 30$MeV/c. Time of flight of the pions between 1AT1 and the first 
bending magnet is $\approx 13$ns. Total contamination of the cloud muons is 1\% 
for a graphite target \cite{8}. Cloud muon contamination for standard Be 1AT1 
target has been measured \cite{12}. That is about 9\%(PSM) with muon polarization 
about P$_\mu=$ +0.3 in contrast with $\rm P_{\mu}=-1$ for surface muons. The cloud 
muons therefore sufficiently reduce the P\rm $_{\mu}$ value of the surface muon 
beam. They are distributed inside a time interval $\rm (0-13)ns$ between proton 
beam pulses structure with 43ns periodicity. We are selected a time window $\approx 
16$ns long between the nearest pulses. Non-surface muon contamination was $6.7\rm
\cdot 10^{-5}$(PSM)(Table 2) in this "usable" window. The total muon polarization 
equals to $1-P_{\mu}=8.3\cdot 10^{-5}$ because the cloud muon polarization equals 
to +0.3.

Cloud muons from the slit S0 appear because high-energy pions with p$_\pi
\approx$100MeV/c from the target are slowed down to $\approx$60MeV/c by the S0 slit 
wall with thickness 8mm of Cu. Then, the slow pions can produce cloud muons in
space of $\approx 0.5m$ length between S0 and the first bending magnet. Their time of
flight is $\approx 3ns$. The above "usable" window has rejected these cloud muons
from S0. We see additional methods for the cloud muon elimination. First, we
are planning to change the 8mm Cu S0 walls on 0.5mm of Al ones.
$\pi^+$-production cross-section by protons is 4 times less for $\rm p=60MeV/c$ than
for $\rm p=100MeV/c$ \cite{13}. That should decrease the cloud muon contamination 
in 4 times.  Second, the cloud muons have a big emittance and the muon counter
(TEC) on entrance of the E614 magnet will reject them.

We propose pessimistically that $\rm (1 - P_\mu )=(8.3\pm 2)\cdot 10^{-5}$ because 
the non-surface muon contamination. The non-surface muons in the "usable" window
have P$_{\mu}\approx -1$ with high probability because we can see only one mechanism 
for their formation$-$a forward $\pi \rightarrow \mu$  decay of very slow pions in 
space between the target and the S0. The muons have $\rm p_\mu \approx
30MeV/c$ and $\rm P_\mu
\approx -1$, therefore they decreases the total polarization much less then cloud 
muons with P$_\mu$ =+0.3. The above estimation $\rm (1 - P_\mu )=(8.3\pm 2)\cdot 
10^{-5}$ will be revised after a direct measurement of the non-surface muon 
polarization inside the "usable" window.
 
{3.1.3. Proton beam shift ($\pm$ 2mm) on the production target.\\} 
A vertical
displacement of the proton beam on 1AT1 target of $\pm$ 2mm will result in a
$\rm \Delta\theta =
\pm$2mrad change in the muon beam direction at the final focus. This directly
corresponds to a $\rm (1-P_\mu )$ shift of 2$\cdot 10^{-6}$. It also leads to a
$\pm$2mm
transverse displacement of the muon beam in the fringe field area at the
entrance to the solenoid and thus causes an additional increase of $\rm (1-P_
\mu )=10^{-5}$. The present instrumentation in BL1A is sensitive to 0.5mm shifts
in the position of the proton so that maintaining stability below 2mm is
possible.  For the existing 1AT1 target (see Fig.6) a 2mm proton beam shift
in the horizontal plane not only adds to the fringe field depolarization as
described above but may also change the mean value of the muon beam
momentum. For a "rotated" target a horizontal shift will only affect
the surface muon beam rate.

\begin{figure}[ht!]
\centering
\label{fig:f6}
\includegraphics[angle=0,width=0.6\textwidth]{tn54_fig6.eps}
%\includegraphics[\textwidth=0.9]{tn53_fig1.eps}
\caption[Sketch of the existing and rotated 1AT1 target.]{
\small   Sketch of the existing and rotated 1AT1 target.}
\end{figure}

We will install the "rotated" target for the P$_\mu\xi$ measurement to exclude the
momentum change. A muon counter as described in section "Fringe field
depolarization of muon" will measure each muon trajectory independently from
a possible proton beam shift. A possible error of P$_\mu$ is included in the
section.

{3.1.4. Stability of the M13 settings to provide $\rm \sigma(\Delta P_\mu)=
10^{-5}$ of the muon beam.\\}
M13 beam-line parameters influence on a central momentum p$_0$ of the surface
muon beam. The momentum changing shifts the P$_\mu$ value. According to
\cite{11}
$\Delta p/p_0 = \pm 0.03$\% corresponds to $\rm \sigma (\Delta P_\mu) = \pm10^{-5}$. 
The influence of beam-line parameters on the p$_0$ value was measured in \cite{7}. 
Three first columns in the Table~\ref{tab3} are copies of the Table II of \cite{7}. 
The fourth column is added.
\begin{table}[ht!]
\caption{Correlation of M13 beam-line parameters and transmitted momentum (see
\cite{7}) (Table II). "Least count" refers to the resolution of the digital beam-line
control system when setting a given parameter.}
\label{tab3}
\begin{center}
\begin{tabular}{| l | c | c | c |}  
\hline
Parameter & Least count & $\rm \Delta p/p,$ & Instability value corresponds
to \\
 & & (\%) & $\rm \sigma (\Delta P_\mu)=10^{-5}$, $(\Delta p/p_0 =0.03\%)$ \\
\hline
1AT1 spot moved & $\approx$1mm & $<$0.024 & $\approx$1mm \\
\hline
Horiz. jaw center & 1mm & $-$0.002 & 15mm \\
\hline
SL1 center & 0.1mm & $-$0.005 & 0.6mm \\
\hline
SL1, SL2 width & 0.1, 0.1mm & 0.0000 & $-$ \\
\hline
SL2 center & 0.1mm & 0.005 & 0.6mm \\
\hline
Q1 current & 0.5\% & 0.002 & 7\% \\
\hline
Q2 current & 0.8\% & 0.012 & 2\% \\
\hline
Q4 current & 0.5\% & 0.002 & 7\% \\
\hline
Q6 current & 1\% & 0.013 & 2.3\% \\
\hline
Q7 current & 0.8\% & 0.017 & 1.4\% \\
\hline
B2 & 0.05\% & 0.06 & 0.025\% \\
\hline 
\end{tabular}
\end{center}
\end{table}

M13 beam-line power supply stability was measured during 530 hours. Results are 
presented in Table~\ref{tab4}.

\begin{table}[ht!]
\caption{Power supply instability during 530 hours.}
\label{tab4}
\begin{center}
\begin{tabular}{| l | c | c | c | c | c | c | c | c | c |}
\hline
Element & Q1 & Q2 & B1 & Q3 & Q4 & Q5 & B2 & Q6 & Q7 \\
\hline
Long time instability & 1.8 & 3.2 & 0.6 & 3.2 & 5.0 & 14.2 & 8.0 & 5.6 & 15.2 \\
$\rm ((|max.-min.|/mean)\cdot 10^{-4}$) &   &   &   &   &   &   &   &   & \\
\hline
\end{tabular}
\end{center}
\end{table}

One can see from Table 4 that the stability of B2 power supply only must be
improved.  We will improve the quad and bending magnet power supply
stability to level of 10$^{-3}$ and 10$^{-4}$ respectively to exclude their 
influence on $\rm P_\mu$ value at level of 10$^{-5}$. Quad and bending magnets 
will be instrumented by Hall and NMR probes correspondingly.

\vspace{0.5cm}
{\bf 3.2. Fringe field depolarization of muon.\\}

Both the surface beam emittance and its distortion by a fringe field on
entrance of E614 magnet define a possible decreasing of stopped muon
polarization. M13 beam line simulation and optimization has been done using
REVMOC and TRANSPORT code \cite{14}, GEANT \cite{15} and ZGOUBI \cite{16,17} 
packages. A beam divergence $\theta$ defines an average muon polarization: 
$\rm(1-P_\mu)=1-\cos\theta$. A beam with emittance of 130$\pi$ mm$\cdot $mrad 
and $\theta$=10mrad can be received at 1500$s^{-1}$ and proton current of 100$
\mu$A according to the above estimations and experimental results from M15 beam 
line running \cite{18}. Muon polarization of such beam is $(1-P_\mu)\approx 5
\cdot 10^{-5}$ that satisfy to E614 requirements.

Change of Z$-$component of muon spin by a fringe field on entrance of the
magnet has been analyzed for a beam with different emittance and angle
divergence $\theta$ \cite{5, 19}.  OPERA$-$2D and OPERA$-$3D packages have been 
used for the field calculations, and GEANT $-$ for muon spin tracking, taking to 
account all interactions of muon with E614 detector. Calculations have been done 
for magnetic field $\rm B=2T$ in the magnet center.  Value of $(1-P_\mu)$ in the 
stopping target varies from $\rm 4\cdot 10^{-4}$ to $\rm 9\cdot 10^{-4}$ for 
different kinds of the muon beam.

A muon counter has been proposed \cite{6} to measure a trajectory of each
incoming muon before the fringe field area where $\rm B\approx 0$. The counter 
has space and angle resolution about 50$\mu$m and 4mrad respectively. Change of
$\rm (1-P_\mu$) value by the fringe field can be calculated for each muon 
trajectory \cite{5,6} using the calculated or measured fringe field distribution. 
A map of the fringe field will be measured with accuracy of 1G and compared with 
the calculated one. According to \cite{5} 24\% of muons have $(1-P_\mu )<2\cdot 
10^{-4}$.  Muons with a small depolarization will be selected in off-line regime 
using the muon counter data. We estimate a systematic error as $(1-P_\mu )=(20
\pm 3)\cdot 10^{-5}$.

\vspace{0.5cm}
{\bf 3.3 Muon depolarization due to muon stops in PC($-$1).\\}

About 7\% of muons stop in PC($-1$) gas. Muon polarization was measured in
CF$_4$/ISO gas at $\rm B=0-3T$ using HELIOS magnet and M15 beam line. $\rm
P_\mu=0.985$ at
$\rm B=2T$. It means the stopped in PC($-$1) muons will decrease the total muon
polarization to $\rm (1-P_\mu )\approx 7\%\cdot 0.015 = 10^{-3}$.  Amplitude cut 
of energy loss in PC($-$1) and PC($-$2) \cite{2} decreases the muon stops to 0.6
\% level. The total muon polarization is $\rm (1-P_\mu )=(12\pm 3)\cdot 10^{-5}$.

\vspace{0.5cm}
{\bf 3.4 Muon deceleration and termalization.\\}

Surface muon with original energy of 4.12MeV slowing down in matters goes
through three consecutive stages: (i) scattering with (non-polarized)
electrons; (ii) cyclic spin and charge exchange; (iii) thermalization
regimes. We should try to estimate a possible muon depolarization in each
stage beginning from last one.

{3.4.1. Depolarization of thermal muon in metal at $\rm B=2T$.\\} 
In many nonmetals a
thermalized $\mu^+$ forms a neutral hydrogen-like atom called muonium
$-$ Mu ($\mu^+e^-$), capturing a non-polarized electron of the matter. After that a
dependence P$_\mu$(t) is described by frequencies of transition between different
Mu states with different quantum numbers $\rm \mid \! m_{\mu} m_e\! >$, 
interactions with electrons and nucleus momentums of the matter, chemical Mu 
reactions, and value and direction of an external magnetic field. The P$_\mu$(t) 
behavior is a subject of $\mu$SR-method. Most important fact for E614 is a 
possibility of fast $\mu^+$ spin depolarization during time of $\tau\approx$ 0.2ns 
corresponding to the hyperfine frequency $\nu_0$ = 4463MHz between Mu states. The 
muon depolarization can be suppressed sufficiently by the longitudinal magnetic 
field $\rm B=2T$. But unfortunately, one can not calculate the real P$_\mu$(t) 
value at $\rm B=2T$ because a microscopic theory of muon behavior in nonmetals does 
not exist. Muon in a nonmetal can form a paramagnetic complex with molecules during 
its slowing down (so-called "hot" chemical reactions). The paramagnetic complex can
depolarize the muon spin very rapidly.

A thermalized muon in metals occupies a center of a corresponding crystal
cell as that was confirmed in many $\mu$SR references. High-energy muon creates
defects in a matter.  The paramagnetic defects can produce sufficient
magnetic fields on the stopped muon.  Average distance between the stopped
muon and the last atom displacement produced in graphite, for example, is
9000$\AA$ according to reference \cite{20}. We believe a contribution of the effect
on P$_\mu$ value is negligible.
  
In metals the $\mu^+$ is thermalized in a quasifree state because the conductive
electron concentration $n\rm \approx 10^{23}cm^{-3}$ in normal metals effectively 
screens the $\mu^+$ from interactions with individual electrons. Muon forms so 
called Kondo impurity (see, for example \cite{21}) in a metal with characteristic 
temperature T$_K$. Conductive electrons completely screen the muon charge at $\rm 
TT_K$ . The muon spin will be depolarized by
a cyclic exchange between the electron of muonium with non-polarized
conductive electrons.  The exchange frequency $\nu$ can be easy calculated at
$T\rm \gg T_K$. This picture corresponds to free muonium atom surrounded conductive
electrons and $\rm \nu =\sigma_e\cdot v\cdot n\cdot kT/E_F$, where $\sigma_e\approx 
10^{-15}cm^2$ is cross-section of changing spin interaction between electron of the 
muonim and conductive electrons; $\rm v\approx 10^8cm\cdot s^{-1}$ and $\rm n\approx 
10^{23}cm^{-3}$ are velocity and density of conductive electrons in a normal metal; 
kT/E$_F$ is a portion of the electrons which can be scattered with spin flip; $\rm 
E_F\approx 10eV$ is Fermi energy of conductive electrons. Result is $\rm \nu\approx 
3\cdot 10^{13}s^{-1}$ at $T=300K$. Muon spin depolarization time is $\rm T_1=\nu
/\pi^2\nu_0^2\approx 1.5\cdot 10^{-7}s=150ns$ at $\rm T=300K$ according to \cite{22}. 
Value of T$_K$ in normal metals is more than 300K therefore a very slow muon 
depolarization with $\rm T_1>1ms$ was found in all normal metals at T=300K in $\mu$SR 
experiments. Let us to remark that the T$_1$ estimation is very pessimistic because 
the value $\nu_0$ in all investigated matters with conductive electrons is smaller 
then the vacuum value of 4463MHz according to $\mu$SR experiment results. For example
the $\rm \nu_0=230MHz$ was measured in antimony \cite{23}. As result, $\rm T_1>6ms$ 
was observed in the Sb at B=1kG.
 
The slow muon depolarization in a metal introduces P$_\mu$ error because we will
see a time-dependent spin relaxation P$_\mu$(t). To be measured value of
P$_\mu$
corresponds to $\rm P_\mu (t=0)$. This value can be calculated from time analysis of
Michel spectrum. The statistical error of P$_\mu$  was $\rm \pm8 \cdot 10^{-4}$ in 
reference \cite{8}.
At total statistics of 10$^9$ muon decay the error is $\rm (1-P_\mu )=
\pm4\cdot 10^{-5}$.
 
Muon spin relaxation is caused by nuclear magnetic moments. The relaxation
time T$_2$ measured by $\mu$SR method is more than 3$\mu$s for all investigated 
metals. The nuclear magnetic moments induce a magnetic field $\rm \approx 2-4G$ on 
muon. The longitudinal magnetic field B=2T decreases sufficiently the relaxation.
Authors of reference \cite{8} have suggested that the relaxation time T$_1$=4.6ms,
observed in Al, can be explained by the decreased interaction. A possible
statistics error of P$_\mu$ is inserted in the previous estimation $\rm (1-P_\mu 
)=\pm 4\cdot 10^{-5}$.

Admixtures of ferromagnetic metals can produce magnetic field on a stopped
muon. The atoms with an unpaired electron moment are in a paramagnetic state
in a normal metal. It means the moment relaxes with a frequency
$\rm \approx 10^{13}s^{-1}$.
The relaxation drastically decreases a possible influence on the muon spin.
It is confirmed by a slow time $\rm (>10\mu s)$ of muon depolarization in pure
ferromagnetics at temperatures above Curie point in many $\mu$SR experiments. A
possible slow muon relaxation in a sample with the ferromagnetic admixtures
can be corrected as that was described above. Muon stopping targets from Al
and C contain $\rm \approx 1ppm$ of the ferromagnetic impurities. Be target contains
$\rm \approx 300ppm$ of Fe.

{3.4.2 Muon depolarization during the cyclic charge and spin exchange
regime.\\} 
 Successive production and stripping of muonium atom Mu ($\mu^+e^-$) begins after
 slowing
down of surface muon to energy $\rm T<3keV$ when a muon can form the Mu atom. This 
regime ends at $\rm T\approx 20-200eV$ when the Mu atom can not be stripped by 
interactions with a matter \cite{24,25,26}. Value of a possible muon depolarization 
depends first of all on the deceleration muon time $\tau_D$ from 3keV to 20eV. 
Different authors give different values of $\tau_D$. It is 10$^{-13}$s according to 
\cite{24}, $\rm 5\cdot 10^{-13}s-\cite{25}, 10^{-12}s- \cite{26}$.
Calculation of the $\tau_D$ from muon range vs energy curve in Ni \cite{27} gives
5$\cdot 10^{-14}$s. Calculation of the $\tau_D$ for Al by SRIM$-$2000.39 package
\cite{28} gives $\tau_D\approx 3\cdot 10^{-13}$s from 10keV to 10eV. The package 
works correct till the kinetic energy T = 10eV for a primary particle and secondary 
ones.  Authors of reference \cite{29} have calculated muon range dependence in Al 
from the muon energy using slightly different code \cite{30}. They have remarked 
that "The measured energy dependence agrees well with simulated values". Besides that
we will use most pessimistic value of $\tau_D=10^{-12}$s. M. Senba \cite{31} (Fig.13b) 
have calculated muon depolarization in this regime for gases. According to the 
calculations the muon polarization is $\rm (1-P_\mu )=3\cdot 10^{-4}$, if the muon 
forms muonium with antiparallel spins of muon and electron, and keeps this state 
during the all $\tau_D=10^{-12}$s. Obviously, it is an overestimation because muons 
form also muonium with parallel muon and electron spins in a matter with non-polarized 
electrons.  $\rm (1-P_\mu$) is zero for this case. Total value of $\rm (1-P_\mu)\approx  
1.5\cdot 10^{-4}$. Obviously also, that several acts of the production and stripping of 
the muonium will decrease the (1$-$P$_\mu$) value. An interaction of muon with 
conductive electrons has not been considered in \cite{31}. We have estimated in Table~
\ref{tab5} the spin exchange interaction effects according to the above formula for 
thermalized muons.

\begin{table}[ht!]
\caption{Muon depolarization during deceleration.}
\label{tab5}
\begin{center}
\begin{tabular}{| c | c | c | c | c |}
\hline
  & Conductive electron & Fermi energy & Spin exchange & $\rm (1-P_\mu )$ \\
  & density n, & $\rm E_F,(eV)$ & frequency $\nu,$ & $\rm (10^{-5})$ \\
  & $\rm (10^{23}cm^{-3})$ &   & $\rm (10^{12}s^{-1})$ &  \\
\hline
Al & 1.81 & 11.7 & 36.1 & 0.7 \\
\hline
Be & 2.47 & 14.3 & 42 & 0.6 \\
\hline
Graphite & $\rm \approx 3\cdot 10^{-5}$ & $\rm \approx 0.02$ & 0.3 & 15 \\
\hline
\end{tabular}
\end{center}
\end{table}

\vspace{0.2cm}
  
We proposed most pessimistic case: $\tau_D = 10^{-12}$s and $\nu_0$ = 4463MHz, for 
the calculations.  The table demonstrates that the spin exchange of conductive
electrons with miuonium atom decreases sufficiently the (1$-$P$_\mu$) values for Al
and Be, and does not change practically the $\rm (1-P_\mu$) value for graphite. Acts
of forming and stripping of muonium during $\tau_D =10^{-12}$s will increase the
$\nu$
value and decrease the $\rm (1-P_\mu$) value more.  Therefore maximal 
$\rm (1-P_\mu$) values are presented in the table.

Comparison of experimental values of  $\rm (1-P_\mu\xi$) for Al, Be, and graphite may
be a direct method for confirmation of the deceleration muon contribution to
total muon depolarization. Case $\rm P_\mu\xi (Al)=P_\mu \xi(Be)=P_\mu \xi(graphite)$ 
means that the above depolarization is very small because in graphite $\nu_0 \ll$ 
4463MHz or/and $\tau_D\ll 10^{-12}$s. Case $\rm P_\mu\xi (Al)=P_\mu \xi(Be)
>P_\mu\xi (graphite)$ will confirm the above calculations. The depolarization
difference of 1.4$\cdot 10^{-4}$ equals practically to statistics error
10$^{-4}$ of P$_\mu\xi$ for 10$^9$ events. Case $P_{\mu}\xi (Al)\neq P_\mu\xi (Be)$ 
means that we did an experimental mistake. We would like to propose preliminary a very 
conservative value of $\rm (1-P_{\mu} )=(5\pm2 )\cdot 10^{-5}$ for Al and Be during 
the muon deceleration time.

{3.4.3. Muon scattering with non-polarized electrons.\\} 
Decreasing of surface
muon polarization by scattering with non-polarized electrons during slowing
down from 4.12MeV to 3keV has been calculated in reference \cite{8}. It is
$\rm (1-P_\mu )=(1.5\pm 0.2)\cdot 10^{-5}$.

\vspace{0.5cm}    
{\bf 3.5 Imperfections of the detector assembly.\\}

{3.5.1. Basic geometrical imperfections of the detector.\\} The influence of
systematic deviations of the PDC assembly from the ideal was studied by
using Monte Carlo vacuum helices (energy loss and multiple scattering turned
off) for positrons with momentum 50MeV/c. Angles were uniformly distributed
in the range $\theta$= 10$-$60$^\circ$. Wire plane spacing was changed from 4mm to
4mm(1+$\beta$).
Positron helixes in the PDC assembly were simulated (without energy losses).
Helixes were reconstructed assuming $\beta$=0. The results indicated that an
accumulated relative deviation $\beta=5\cdot 10^{-4}$ from a designed value along any 
axis X, Y or Z of the assembly introduces the following shifts of Michel
parameters: ($\rm \rho -0.75)=\pm 5.2\cdot 10^{-5}, (P_\mu\xi -1)=\pm 3\cdot 10^{-5}, 
(\delta - 0.75)=\pm 5.4\cdot 10^{-5}$.  Such deviation value corresponds to an energy 
calibration shift of $\rm \mid \!\Delta E_0\!\mid=10keV$ at the E$_{max}$= 52.831MeV 
of the positron spectrum and reconstructed angle shift of $\mid\!\Delta\theta\!\mid 
=5\cdot 10^{-4}$.  The deviation can appear because an average distance between wires 
in PDC planes is 4mm + 2$\mu$m systematically instead of 4mm, for example, or the 
distance between nearest PDC planes is 4mm + 2$\mu$m systematically instead of 4mm, or 
the deviation $\beta$=5$\cdot 10^{-4}$ is between X(Y) and Z axis units. Random errors 
of the $\beta$ value in range $\beta =\pm4\cdot 10^{-3}$ contribute an error less than 
10$^{-5}$ to the Michel parameters, but the confidence level of a reconstructed helix 
is $\rm CL<0.5$ if $\mid\beta\mid >5\cdot 10^{-3}$ (it corresponds to 20$\mu$m of a wire 
random spread).

Preliminary description and test results of the PDC chambers presented in
reference \cite{32}.

An accuracy of basic elements used for the assembly manufacturing defines
the precision of the assembly. The basic elements are winding table, glass
rulers for wire stretching defining the distance between wires, Cital
spacers (4, 20 and 40mm thickness) defining a distance between planes, Cital
spacers (1.997mm thickness) defining a distance between wire and cathode
planes. Non-parallelism of the winding table surface is better than 0.5$\mu$m.
All Cital spacer thickness were measured with accuracy of $\pm 0.3\mu m$ \cite{33}. 
Measured deviation between X(Y) and Z axis unit is $\beta < 1.25\cdot 10^{-4}$ \cite
{34}. Distances between wires were measured for all manufactured PDC planes. An
example of the distance distribution is presented on Fig.7. Average distance
$\rm <\!s\!>$ between nearest wires is $\mid <\!s\!>- 4mm\mid < 0.1\mu$m with 
statistical spread of $\sigma_s=2.4\mu$m of individual wires. These above 
imperfections are about $\beta =1.25\cdot 10^{-4}$. 
%\newpage
It corresponds to following
shifts the Michel parameter: $\rm (\rho -0.75)=\pm 1.4\cdot 10^{-5},
(P_{\mu}\xi -1)=\pm 1\cdot 10^{-5}, (\delta-0.75)=\pm 1.4\cdot 10^{-5}$.

\begin{figure}[ht!]
\centering
\label{fig:f7}
\includegraphics[angle=90,width=0.6\textwidth]{tn54_fig7.eps}
\caption[Residual distribution of distance between wires in DC plane \#033DC.]{
\small   Residual distribution of distance between wires in DC plane \#033DC.} 
\end{figure}

The above imperfections of the detector we will name as the basic ones. The
above estimations mean that we need to know a position of each wire in
3D-space with very high accuracy. Different kind of systematic errors appear
even without the basic imperfections. There is an influence on Michel
parameters of pressure, temperature, gap value between wires and cathode
plane, Lorentz drift angle at $\rm B=2T$. We should demonstrate those much weaker
accuracy requirements for these imperfections ("extra" imperfections) can be
accepted for a correct measurement of Michel spectrum. We will begin from
the Lorentz drift angle influence because most detailed Monte Carlo analysis
for this case been done.

{3.5.2. Lorentz drift angle in DME gas.\\} 
The Lorentz drift angle in DME gas is
less than 9$^\circ$ at $\rm B=2T$ \cite{35}. The following simplified simulation 
model was used \cite{10}. Distances of closest approach of a vacuum helix to hit 
wires d$_i$ (DOCAs) were calculated first. Local Lorentz drift angle $\rm 
\vartheta_{i,L}$ at the points of closest approach were than computed to take 
into account the dependence of Lorentz drift angle component on the angle between 
the electric E and the magnetic B field $\rm \vartheta_L=\vartheta_{L,max}\cdot
\mid \sin(E,B)\mid$ assuming a constant (maximum) Lorentz angle $\rm \vartheta_L
=9^\circ$ and simple radial direction of electrostatic field in the PDC
cell. Finally, true DOCAs $\rm d_i$ were replaced on longer drift distances d$_{i,
L}$=d$_i$/cos$\rm \vartheta_{i,L}$ smeared with gaussian PDC resolution.  The 
modified $\chi^2$ was minimized to find parameters of the helix. Only the first 4 
PDC pairs were used in the helix fit.

To avoid masking the Lorentz drift effect on track reconstruction the PDC
resolution in the simulation was assumed to be $\rm \sigma_{PDC}=0.5\mu$m instead 
of a realistic 50$\mu$m value. Results for the deviation of the mean reconstructed
energy $\rm <\!\Delta E\!>=<\!E_{fit} -E_0\!>$ are presented on Fig.8 where the 
error bar on each point represents the uncertainty of the mean (for about 1500 
events per point). As one can see from Fig.8, maximum deviations $\rm |\Delta E|
\approx 100keV$ are observed at small polar angles $\rm \vartheta=9.9^\circ (cos
\vartheta =0.985)$ when positrons are emitted from a point source ($\rm \sigma_x=
\sigma_y=0$) at fixed polar angle ($\varphi_0=0$). The
deviation of the mean energy drops drastically ($\rm |\!<\!\Delta E\!>\!| <7keV$ at
$\cos\vartheta=0.985$)
due to averaging when one considers a real distributed positron source spot
($\rm \sigma_x=\sigma_y=1cm$, $\varphi_0=0$) from the stopping target. For a 
smaller Lorentz angle, $\rm \vartheta_{L,max}=3^\circ$ (or, equivalently, if one 
corrects drift distances in STR for Lorentz drift with 10\% accuracy) the same 
averaging over source spot gives $\rm |\!<\!\Delta E\!>\!|
<0.6keV$. $\rm |\Delta E|$ becomes negligible at $\rm \cos\vartheta <0.9$. Similarly, 
deviation of the reconstructed polar angle is maximal for a point source and fixed 
azimuthal angle ($\rm \Delta (\cos\vartheta) =|\cos\vartheta-\cos\vartheta_0|\approx 
7\cdot 10^{-5}$ at cos$\vartheta_0=0.985$ and $\rm \sigma_x=\sigma_y=0, \varphi_0=0$) 
and drops to $\rm \Delta (\cos\vartheta) <2\cdot 10^{-6}$ for a distributed source
$\rm \sigma_x=\sigma_y =1cm$, $\varphi_0=0$. PDC resolution
$\rm \sigma_{PDC}=50\mu$m masks the effects, but the deviation values are remain.

\begin{figure}[ht!]
\centering
\label{fig:f8}
\includegraphics[angle=0,width=0.7\textwidth]{tn54_fig8.eps}
%\includegraphics[\textwidth=0.9]{tn53_fig1.eps}
\caption[Mean shift of fitted energy $\rm <\Delta E>=$.]{
\small   Mean shift of fitted energy $\rm <\Delta E>=$.}
\end{figure}

Fig.9 presents the reconstructed energy spectrum from an original spectrum
that was flat in range 20$-$50MeV. The total number of simulated events was
10$^5$. For an isotropic original spectrum this would correspond to a full
statistics $\rm \approx 10^{12}$ in 4$\pi$ solid angle. As one can see from Fig.9,
$\chi^2$ values in
the case $\rm \sigma_x=\sigma_y$=0, $\varphi_0 =0$ are significantly beyond their 
natural limits $\chi^2$ =NDF
$\rm \pm\sqrt{2NDF}$, i.e. systematic errors in the spectrum are higher than 
statistical errors. $\chi^2$ values for a distributed source ($\rm \sigma_x=\sigma
_y$=1cm, $\varphi_0$=0) are within their natural limits, indicating that systematic 
errors from Lorentz drift effect are small at total statistics 10$^{12}$ in 4$\pi$ 
solid angle.

\begin{figure}[ht!]
\centering
\label{fig:f9}
\includegraphics[angle=0,width=0.6\textwidth]{tn54_fig9.eps}
%\includegraphics[\textwidth=0.9]{tn53_fig1.eps}
\caption[Reconstructed energy distribution and its fit to a constant (initial
energy distribution was uniform). $\rm \vartheta _{L, max}=9^\circ$.]{
\small   Reconstructed energy distribution and its fit to a constant (initial   
energy distribution was uniform). $\rm \vartheta _{L, max}=9^\circ$.}
\end{figure}

A lengthening of the DOCA because the Lorentz drift effect can be described
as $\rm \Delta d(\mu m)= 12\cdot d(mm)/cos\vartheta$ for DME gas at $\rm B=2T$ ($\rm
\vartheta_{L,max}=9^\circ$). Maximal deviation
of d is $\rm \Delta d= 24\mu$m at $\vartheta=0$ and $\rm d=2mm$. The above analysis 
demonstrates that even big deviation of 24$\mu$m does not change practically the 
reconstructed positron energy and angle, and therefore, does not shift the 
reconstructed Michel parameters. In contrast to this fact, the basic imperfections of 
the detector shift of Michel parameters on $\rm \approx 5\cdot 10^{-5}$ level at
$\Delta d=2\mu$m only.  The insensitivity of Michel spectrum to the extra imperfection 
(Lorentz drift effect) is explained, as one can see from Figs.7, 8, by spreading of a
positron source to real stopped muon spot with $\rm \sigma_x=\sigma_y=1cm$. Obviously, 
the deviations are averaged because the spot size 1cm is larger the PDC
periodicity of 4mm. Fortunately, average values of E$_0$ and $\vartheta_0$ are remained
with high accuracy.

Finally, we can not take into account Michel parameter shifts because the
Lorentz drift angles in DME gas. Actually, the Lorentz drift angles were
measured \cite{35} for DME gas at $\rm B=2T$ with accuracy of $\pm$3\%. The drift angles
will be inserted in a reconstruction package because deviations more than
20$\mu$m worse a $\chi^2$ value (see 3.5.1).
 
{3.5.3. Gap between wire and cathode plane.\\}  
Influence of a wire-cathode
distance deviation $\Delta$D on shift $\Delta$d of a shortest distance between a wire 
and a positron helix was simulated by GARFIELD package. The calculated shift is
described roughly as $\rm \Delta d(\mu m)=0.06\cdot d(mm)\cdot\Delta D(\mu m)/cos\theta$, 
where $\theta$ is the projection of a positron helix polar angle on the plane orthogonal 
to the wire plane. This
"extra" imperfection is described by the same dependence as the Lorentz
drift effect. We have to provide $\rm d_{max}<20\mu$m to conserve a correct
$\chi^2$ value at a helix reconstruction. The above
$\rm \Delta d_{max}$ corresponds to $\rm |\Delta D|\approx 60\mu$m. Shift of Michel 
parameters will be $\rm <10^{-5}$.
Now we are designing a tool for systematic measurements and adjustments of
the D values in 24 points (through 15$^\circ$) not far from the 24 screws of the
G10 keeping the removable cathode plane to a permanent coil. Designed
accuracy of the tool is $\rm \pm5\mu$m. We hope to adjust D values to
$\pm10\mu$m level.

{3.5.4. Pressure difference between PDC gas and He gas.\\}  
Pressure difference
between PDC gas (DME) and He gas around a PDC module will be supported with
accuracy better than 6mTorr at a center of cathode plane. A mylar cathode
foil with thickness 6$\mu$m between the gases will be deflected because a
different density of DME and He gases.  As result, the foil will have
S$-$shape with maximal foil deflection
$\rm \Delta D =+20\mu$m in the upper half of the foil and $\rm \Delta D =-20\mu$m in 
the bottom half. The deflection is zeroing along horizontal line goes through the 
cathode center if the pressure difference is zeroing also. But the horizontal line
will move if the pressure difference is changing in range 6mTorr. Therefore
the maximal deflection can reach $\rm \Delta D=\pm 40\mu$m according to estimations. 
The deflection appears only on outer foils of a PDC module.  The deflection
$\rm \Delta D=
\pm 40\mu$m will change reconstructed distances between a helix track and wires on
$\rm \Delta d_{max}\approx\pm13\mu$m as it was analyzed before. The deflection value 
is less than 20$\mu$m which worse sufficiently the $\chi^2$ value at a helix 
reconstruction. Finally, the pressure difference ("extra" imperfection also) does not 
shift the Michel parameters on level of 10$^{-5}$.

{3.5.5. Temperature and pressure influence on PDC drift characteristics.\\}
Temperature and/or pressure changing of DME gas in PDC planes shifts
distance-drift time relationships according to GARFIELD calculations. Shift
of the distance of closest approach can describe roughly as 
$\rm \Delta d(\mu m)=1.5\cdot d$(mm)
per $\rm \Delta T=1^\circ C$. Temperature of the drift gas will be monitored with 
accuracy of $\rm 1^\circ C$ and the distance-time relationship will be corrected. 
The inaccuracy $\rm 1^\circ C$
corresponds to $\rm \Delta  d_{max}\approx 3\mu$m. This value does not worse the
$\chi^2$ value and does
not shift the Michel parameters on level of 10$^{-5}$ as previous "extra"
imperfections. Pressure in Vancouver city changes on $\rm \pm 23Torr$ (maximum
registered deviation) around 760Torr. Dependence of distance-time
relationships can be described, according to GARFIELD calculations, as 
$\rm \Delta d(\mu m)=0.6\cdot$ d(mm) per 1Torr changing. It means the $\rm
\Delta d_{max}$ value can reach
$\rm \pm 28\mu$m. Gas pressure will be monitored with accuracy of 6mTorr and the
distance-time relationships will be corrected. The inaccuracy of 6mTorr
causes a negligible shift of Michel parameters.

{3.5.6. Wire sags.\\} We use (W-Re)Au wires with 15$\mu$m diameter and maximal length
of 40cm stretched with a force of 30g. Maximal gravitational sag of a wire
is about 1.3$\mu$m in center of wire length for a horizontally stretched wire.
Wires in PDC planes of the E614 assembly arranged really in U and V
directions with angle + and $\rm -45^\circ$ respectively relative to the vertical
axis. Maximal value of the gravitational sag is about 1$\mu$m. The effect refers
to a basic imperfection. An error in reconstruction helix can appear if a
positron hits a wire in center of first PDC module, and then hits a wire in
next module near wire edge where sag is zero. Minimal distance between the
modules is 40mm. The above event gives angle shift $\Delta\theta\approx 1\mu$m/40mm =
2.5$\cdot 10^{-5}$.
This small angle deviation shifts the Michel parameters less than 10$^{-5}$.

{3.5.7. Mutual orientation of PDC planes in the detector.\\} Design of the E614
assumes possible mutual shifts not more than 200$\mu$m between individual PDC
chambers in (U, V) plane orthogonal to Z$-$axis. Assemble sizes along Z$-$axis
defined by the basic elements using for the chamber production and
assembling. The deviations in (U, V) plane can be defined by an illumination
of the full detector with particles at $\rm B=0$. High energy pion beam with
$\rm p=150MeV/c$ from M13 channel illuminating a central area of the detector
allows reconstruction of mutual shifts between U or V planes separately.
Possible rotation angles between planes can be measured by an illumination
of the detector far from Z=0. Unfortunately, so large shift ($\approx$15cm) of the
beam or the detector relative to each other is not possible. Therefore, a
different method was proposed. A surface muon beam with big momentum spread
$\rm \Delta p/p\approx 5-7\%$ illuminates a central area of the detector. Muons stop 
in a stopping target and a remarkable part of them ($>20\%$) stops in all chambers
of the detector. Positrons decaying into 4$\pi$ angle allow reconstruction of
mutual imperfections of the detector inside its full volume. Monte Carlo
program develops now for estimation of needed statistics to reconstruct the
imperfections with accuracy 2$\mu$m. The same beam will be used for
determination of relative distances between wires in a plane.

{3.5.8. Geometrical acceptance.\\} The geometrical acceptance of the E614 detector
has been studied in \cite{36}. It was determined that the acceptance is unity and
uniform within the range $\rm 0.320\mu$s) of a sense wire 
due to the slow dissipation of the ion cloud around the hit sense wire. The dead
zone along the sense wire is about 0.5mm. We will try to estimate
geometrically the possibility to reconstruct a going upstream positron helix
without scattering of both the muon and positron suggesting that the muon
goes along Z$-$axis at $\rm X=Y=0$ and stops at $\rm Z=X=Y=0$ in the stopping target 
(see Fig.1). A positron helix returns to Z$-$axis in one period. We will suppose
pessimistically that the returned point coordinate Z $\rm (X=Y=0)$ coincidences
always with the muon dead zone in a chamber. The coincidence in a chamber is
possible, as one can see from Fig.1, at fixed helix parameters $\rm E, \theta,
\phi$ only.
It means that 1 point with the dead time $>10\mu$s is per 1 helix period. A
positron helix also hits wires at $\rm X=0, Y\neq 0$ and $\rm X\neq 0, Y=0$ per 1 
period far from $\rm X=Y=0$ point. The Y and X wires respectively have been crossed 
by the  muon. It means that 2 more dead points are per 1 helix period with a
probability of 93\% that the points will be sensitive in 500ns wait time.
Most difficult situation with a helix reconstruction because the dead points
comes for a case with low positron energy and low polar angle simultaneously
because more helix periods are putted up one slide 12345, 23456, 34567 or
45678, and number of hits is minimal ($\geq$10) per a slide. Let us to analyze
the worst case a minimal positron energy (20MeV) and a minimal angle
(10$^\circ$)
selected for Michel spectrum reconstruction. About 1.5 helix period is
putted up one slide. It means that 1 permanent dead point and 3 temporally
dead points are per a slide on the helix trajectory.  Probability that all 3
temporally dead points remain dead ones in the 500ns is $\rm (7\%)^3 \approx
3\cdot 10^{-4}$.
These events can not be reconstructed because 6 NDFs ($10-1-3=6$) are for 6
fitted helix parameters. Let us remark that the problem exists only for the
upstream part of the detector because the overlap between muon and positron
tracks. The problem can be resolved obviously using more than 5 PDC pairs
for the helix reconstruction. Kalman filtering method with 16 PDC planes
should be used for the problem resolving. The method has a disadvantage $-$ it
requests 2$^{16}\approx 64000$ independent fittings for $\rm \approx 20\%$ of 
reconstructed events to resolve a left$-$right ambiguity compare with about 5000 
independent fittings for the sliding method. Percentage of the events with $\rm 
E<30MeV$ and $\theta <20^\circ$ in the upstream Michel spectrum is about 1\%. 
All upstream positron helixes did not reconstructed by the sliding method 
will be reconstructed using the Kalman filter technique. Percentage of such 
events will be much less than 1\%.

A special case appears for an upstream helix with period of 4mm. The helix
crosses Z$-$axis in all PDC chambers for the selected detector geometry with
He gap of 40 or 60mm between the PDC pairs. It means that all positron hits
coincidence geometrically with the incoming muon hits which kill all
positron hits because the permanent dead zone is from the muon. Angle $\theta$ of
such helix is minimal ($\rm \approx 85^\circ$) at minimal positron energy $\rm
E=20MeV$ inside
selected energy range $\rm 20MeV20MeV$.  Let us remark that the all special cases 
with helix period of 4mm lie outside the selected range of $\rm 20MeV
=(52.77\pm 0.03)MeV$, very close to the correct value 52.831MeV. The conclusion
is that the planar geometry of the E614 makes it possible to calibrate the
energy scale with high precision. A higher statistics ($2\cdot 10^7$ events) study
would be expected to provide an energy calibration with an accuracy better
than 4keV. Taking a conservative approach we have assumed that the energy
calibration uncertainty will be 10keV. It corresponds to follow parameter
errors: $\rm (0.75- \rho )=\pm 5.2\cdot 10^{-5}, (1 - P_\mu \xi )=\pm 3\cdot
10^{-5}, (0.75 - \delta )=\pm 5.4\cdot 10^{-5}$.

\vspace{0.5cm} 
{\bf 3.7. Response function.\\}

The E614 detector may be logically separated into two parts. The first part,
where the positron helix is not measured, consists of the stopping target,
the PC($\pm$1) and PC($\pm$2), a helium gap and the cathode plane of the first
PDC. Positron loses energy in collision and radiative interactions with this
part. It can annihilate or be scattered back also. The all interactions
distort the Michel spectrum on entrance of the coordinate part (PDC ($\pm$1)) of
the detector. The coordinate part does not distort a spectrum, as that has
been demonstrated above in the section "Imperfections of the detector
assembly". Maximum positron energy loss in the first part reaches 80keV for
Al and 45keV for Be target at E=54MeV and $\theta =60^\circ$ (50keV and 35keV
respectively at E=20MeV and $\theta =60^\circ$).

Influence of the positron annihilation and backscattering on the Michel
parameters has been simulated by EGS4 package. The probabilities of
annihilation $\varepsilon_{2\gamma}(x,\theta)$ and backscattering
$\varepsilon_B(x,\theta)$ in the first part of the
detector were computed. The exact formula of muon decay $\rm d^2N/dxd\cos\theta$ 
was multiplied by  $\varepsilon_{2\gamma}(x,\theta)\cdot\varepsilon_B(x,
\theta)$. The resulting distribution was randomized
with total statistics 10$^{11}$ events of muon decay in 4$\pi$ solid angle. The
randomized spectrum was fitted by $\rm d^2N/dx(d\cos\theta)$. When $0.410^{-5}$are presented.}
\label{tab6}
\begin{center}
\begin{tabular}{| c | c | c | c | c | c |}
\hline
   & Discussed & Source of Correction or/and Error & $\rm (1-P_\mu\xi)$ & $\rm
(0.75-\rho )$ & $\rm (0.75-\delta )$  \\
   & in section &   & $\rm (10^{-5})$ & $\rm (10^{-5})$ & $\rm (10^{-5})$ \\
\hline
1 & 3.1.1 & Coulomb scattering of muons & $9\pm 2$ & & \\
 & & inside the production target & & & \\
\hline
2 & 3.1.2 & Non-surface muon contamination & $\rm 8.3\pm 2$ & & \\
\hline
3 & 3.2 & Fringe field muon depolarization & $\rm 20\pm 3$ & & \\
\hline
4 & 3.3 & Muon depolarization due to & $\rm 12\pm 3$ & & \\
 & & muon stops in $\rm PC(-1)$ & & & \\
\hline
5 & 3.4.3 & Muon scattering by & $\rm 1.5\pm 0.2$ & & \\
 & & unpolarized electrons & & & \\
\hline
6 & 3.4.1 & Slow depolarization of thermal & $\pm 4$ & & \\
 & & muon in metal at B=2T & & & \\
\hline
7 & 3.4.2 & Muon depolarization due to & 5$\pm 2$ & & \\
 & & muonium formation and stripping & & & \\
 & & at $\rm E_\mu <3keV$ & & & \\
\hline
8 & 3.5 & Imperfections of the detector & $\pm 1$ & $\pm 1.4$ & $\pm 1.4$ \\
 & & assembly & & & \\
\hline
9 & 3.6 & Positron energy calibration & $\pm 3$ & $\pm 5.2$ & $\pm 5.4$ \\
\hline
10 & 3.7 & Response function & $\pm 4.2$ & $\pm 3.6$ & $\pm 1.5$ \\
\hline
11 & 3.5.13 & Deviation$\rm (10^{-4})$ of magnetic field & $\pm 1.5$ & $\pm 2.6$ & 
$\pm 2.7$ \\
 & & magnitude & & & \\
\hline
12 & & Statistical errors for $10^9$ events & $\pm 10$ & $\pm 5$ & $\pm 8$ \\
\hline
\end{tabular}
\end{center}
\end{table}

$\bullet$ Michel parameter errors: \\
$\rm \sigma (P_\mu\xi)=[\pm 10(stat)\pm 8.5(syst)]\cdot 10^{-5}.$ \\
$\rm \sigma (\rho )=[\pm 5(stat) \pm 7.0(syst)]\cdot 10^{-5}.$ \\
$\rm \sigma (\delta )=[\pm 8(stat) \pm 6.4(syst)]\cdot 10^{-5}.$ \\

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\end{document}


FIGURE CAPTION.
Fig.1. 
Schematic layout of detector assembly.
Fig.2. 
Energy loss in PC($-$2) and PC($-$1) chambers for various stopping 
conditions. Top left: no conditions; top right: stop in PC($-$1); 
bottom left: stop in target; bottom right: stop in PC(+1) or PC(+2).
Fig.3. 
Sketch  of the existing and rotated 1AT1 target. 
Fig.4. 
Confidence level (CL) distribution for reconstructed events. Initial 
positron energy 50MeV, initial polar angle $\cos\theta_0=0.735$. PDC space 
resolution is 50$\mu$m. "Slide 1(raw)" is the result of the first slide 
(1234) fitting. "Slides 1$-$5" present the same events that were fitted 
with slide selection and resulting distributions of all five slides were 
summed. The flat distribution with $\rm =0.5$ corresponds to the fit of 
vacuum helices with 50$\mu$m PDC resolution.
Fig.5. 
Energy $\rm 1/2\cdot FWHM(E_{fit}- E_0)$ and angle 
$\rm 1/2\cdot FWHM(\cos\theta_{fit}-\cos\theta_0)$ resolutions at different 
initial positron energies.
Fig.6. 
Mean reconstructed positron energy $\rm $ as a function of $1/\cos\theta$. 
$\rm E_0=50MeV$, $\rm \sigma_{PDC}=50\mu m$. Straight lines are the fits with function 
$\rm =E_0-\alpha/cos\theta$.
Fig.7. 
Residual distribution of distance between wires in DC plane \#033DC.
Fig.8. 
Mean shift of fitted energy $\rm =E_{fit}-E_0$ as a function of 
initial energy $\rm E_0$.
Fig.9. 
Reconstructed energy distribution and its fit to a constant 
(initial energy distribution was uniform). $\rm \theta_{L,max}=9^\circ$.

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