This note estimates systematic uncertainties due to variations
in chamber temperature and pressure, jitter in time-zero, and
inhomogeneity in B. The methodology is Garfield and Monte Carlo.
It concludes that the errors are in general tolerable.

Filename: sys.ps

\magnification = \magstep 1
\hsize = 6.0 true in
\vsize = 8.4 true in
\hoffset = 0.1 true in
\voffset = 1.0 true in 
\font\bigrm = cmr10 scaled \magstep 2
\font\bigbf = cmb10 scaled \magstep 2
\font\srm = cmr9
%\font\it = cmti10 scaled \magstep 1
\font\bigit = cmti10 scaled \magstep 2
\baselineskip = 24 true pt
\parskip = 0 \baselineskip
\parindent = 20 pt
\def\newpage{\vfill\eject}

\def\caption#1#2{
%\pageinsert
%\vglue 6.8 true in
$$\vbox{
\rm
\hsize = 6.0 true in 
\baselineskip = 24 true pt
\noindent {\bf #1\ \ }#2
}$$
%\endinsert
}
\input psfig.sty

\def\Foot#1#2{   
{\srm
\baselineskip = 21 true pt
\footnote{#1}{#2}}
}
 
\input sysabs.tex
\newpage

$\,$
\newpage
\input systxt.tex
\newpage

%\input tab1_a.tex
%\newpage
%\input tab1_b.tex
%\newpage
%\input tab2_a.tex
%\newpage
%\input tab2_b.tex
%\newpage
%\input lscap.tex
%\newpage
\bye

\pageno = -1
%\nopagenumbers
\midinsert
\centerline{\bigbf Systematic Uncertainties}   
\vskip 0.5 true in
\centerline{\bf Sun-Chong Wang}
\centerline{TRIUMF, Vancouver, B.C. V6T 2A3, Canada}
\centerline{Sept 3, 2001}
\medskip

\vskip 0.75 true in
\centerline{\bf Abstract}
$$
\vbox{\hsize = 4.7 true in\baselineskip = 18 true pt
{\noindent
%\vskip 0.5 true in
This note estimates systematical errors of the ${\cal TWIST}$ spectrometer
due to temperature and pressure extremes. Also discussed are errors
due to jitter in time zero and inhomogeneity in $B$. It concludes
that the errors are in general tolerable.
}
\line{\hfill $\,$      }
\line{\hfill $\,$      }
\line{\hfill $\,$      }
\line{\hfill $\,$      }
%\line{\hfill Thesis advisor: Professor H. Henry Stroke}
}$$
\endinsert

\pageno = 1
%\magnification = \magstep 1
%\hsize = 6.5 true in
%\vsize = 8,9 true in
%\font\rm = cmr10 scaled \magstep 1
%\font\bf = cmb10 scaled \magstep 1
%\font\it = cmti10 scaled \magstep 1
%\baselineskip = 24 true pt
%\parskip = 0 \baselineskip
%\parindent = 20 pt
%\rm

\smallskip
{\noindent 1. {\bf Introduction}}

The E614 experiment is to measure the Michel spectrum
with an unprecedented precision. The proposed accuracy of
the $\cal TWIST$ spectrometer at a level of parts in $10^5$ on
the Michel parameters
allows one to improve current limits on the weak coupling
constants within the Standard model. If, however, the
result suggests deviations from Standard model prescriptions,
it has the potential to shed light on directions toward
models beyond the Standard model.

To accomplish the goal, various systematic effects on the
experimental accuracy have to be studied and thereby corrected
for. This note estimates uncertainties in the Michel parameters
due to changes in chamber pressure and temperature which 
may occur during data taking.
In Section 2, measurement errors are
estimated by Garfield simulations. Section 3 shows how 
errors propagate from spatial measurements to track parameters:
$e^+$ momentum and the dip angle. Impact on the Michel parameters
is shown in Section 4. Section 5 summaries.

\smallskip
{\noindent 2. {\bf STR's by GARFIELD}} 

Robert Openshaw has observed that the atmospheric 
pressure in the meson hall can move away from 760 torr by at most 3\% and 
chamber temperature from
24 by at most 10 degrees Celsius. 
It's known that gas properties vary as the ambient conditions
change. To see the effect, we prepared the drift
time-drift distance relationships of the DME gas at different
pressures (or temperatures) using the Garfield. 
The mean, over drift distance within the cell, of the
shift is calculated. Then the average, $\overline{{\rm STR}}$,
of the means over angles is
calculated. The results are shown in Table 1. 

\newpage
\input tab1.tex

It's noted that shifts of the curve 
are in one way (or the other). For example, 
when the pressure is lowered, the whole STR curve moves up.
And, when the temperature drops, the entire STR curve
moves down. All the shifts are calculated 
relative to the standard STR at 760 torr and 297 K.
Also note that the numbers were obtained under extreme conditions. 
For example, $|\overline{{\rm STR}}$ @ 34C - $\overline{{\rm STR}}$ @ 24C$|$
and $|\overline{{\rm STR}}$ @ 14C - $\overline{{\rm STR}}$ @ 24C$|$ were
found and then averaged to get 17.4 $\mu$m
in the table.
In general, effect of one extreme is  
very symmetrical to that of the other extreme, 
except that 14 C is a bit more adverse
than 34 C.

The next step is to relate the spatial uncertainties, $\sigma$, to the
momentum and angular resolution of the spectrometer. 
 
\smallskip
{\noindent 3. {\bf Energy and Angular Resolution}}

If our tracking program always looks up the default STR table 
(at 24 C and 760 torr) for translation from tdc to the distance
of closest approach, bias incurs and propagates
into the determination of $e^+$ momenta and $\cos\theta$'s. The resulting
Michel spectrum is distorted and the extracted Michel parameters
will be off. We derive the distortion in this section.

The projection of the helical trajectory on the $x-y$ is a
circle, whose radius, $R$, determines the transverse momentum of
the positron, $p_t$,

$$
p_t = \kappa\, q\, B\, R, \eqno(1) 
$$

{\noindent where $\kappa = 299.792458\ ({\rm MeV}/c){\rm T}^{-1}{\rm m}^{-1}$.
$q$ equals 1 for positron and the magnetic field strength $B$ 
is 2.2 Tesla.
The track parameters in Kalman fitting has the following
relations,
}

$$\eqalignno{
   x =\ & R \cos(\phi_0+\phi)-R\cos\phi_0, & (2a) \cr
   y =\ & R \sin(\phi_0+\phi)-R\sin\phi_0. & (2b) \cr
}
$$

{\noindent Measurement uncertainties on various planes 
accumulate and contribute to the uncertainty in $R$,
}

$$\eqalignno{
   {1\over \sigma_R^2} =\ & \sum_{i=1}^N \biggl[ {1\over \sigma_x^2}
                          \Bigl({\partial x_i \over \partial R}\Bigr)^2 +
                          {1\over \sigma_y^2}\Bigl({\partial y_i \over
                          \partial R}\Bigr)^2 \biggr] & (3) \cr
   =\ & {2\over \sigma^2} \sum_{i=1}^N \Bigl[1-\cos(\phi_0+\phi_i)\cos\phi_0
                                   -\sin(\phi_0+\phi_i)\sin\phi_0 \Bigr] \cr
   \approx \ & {2\over \sigma^2} \sum_{i=1}^N 1 
   =\ {2N\over \sigma^2}, \cr
}$$
 
{\noindent where $N$ is the number of UV planes upstream or downstream and
equal to 11 in our case. We have also assumed that the measurement error on
the U plane is the same as that on the V plane: $\sigma_x = \sigma_y\equiv
\sigma$. 
We get,
}

$$ \sigma_{p_t} = \kappa\, q\, B\, \sigma_R 
                \approx {\kappa\, q\, B\, \over \sqrt{2N}} \sigma.
   \eqno(4)
$$ 

We turn to the angular uncertainty for a moment. It's noted that $\theta$
is related to the dip angle, $\lambda$, by,
$\theta = {\pi \over 2} - \lambda$. Furthermore, the track parameter, 
$t\ (=\tan\lambda)$, equals ${z\over R\phi}$. We have, using Equ (2) and 
similar to Equ (3),

$$\eqalignno{
  {1\over \sigma_t^2} =\ & \sum_{i=1}^N \biggl[ {1\over \sigma_x^2}
                          \Bigl({\partial x_i \over \partial t}\Bigr)^2 +
                          {1\over \sigma_y^2}\Bigl({\partial y_i \over
                          \partial t}\Bigr)^2 \biggr]  \cr
   \approx \ & {1\over \sigma^2 t^4} \sum_{i=1}^N z_i^2 \cr
   =\ & {(\Delta z)^2 \over \sigma^2 t^4} \sum_{i=1}^N i^2 \cr
   =\ & {(\Delta z)^2 N(N+1)(2N+1)\over \sigma^2t^4\, 6}, & (5) \cr
}$$

{\noindent where again $\sigma_x=\sigma_y\equiv\sigma$ is assumed, and 
$\Delta z$ is
the spacing between UV modules. Since they are not evenly placed
in our spectrometer, we approximate an average $\Delta z$ of 0.05 m. 
The angular uncertainty is therefore,
}

$$ \sigma_\theta = \sin^2\theta\ \sigma_t = {\cos^2\theta 
                   \over \Delta z}\sqrt{6\over N(N+1)(2N+1)}
                   \cdot\sigma \eqno(6)
$$
 
We now return to momentum. Since 
$p=(p_t^2+p_z^2)^{1\over 2} = p_t(1+t^2)^{1\over 2}$, one
has, ignoring the covariance between $p_t$ and $t$,

$$\eqalignno{
   \sigma_p =\ & \biggl[ (1+t^2)\sigma_{p_t}^2 + \biggl({t\over 1+t^2}p 
                 \biggr)^2
                     \sigma_t^2\biggr]^{1\over 2} \cr
            =\ & \biggl[ 
                \biggl({\kappa q B \over \sin\theta}\biggr)^2 {1\over 2N} +
               {p^2\over (\Delta z)^2}
               \biggl({\cos^3\theta \over \sin\theta}\biggr)^2 
                {6\over N(N+1)(2N+1)}\biggr]^{1\over 2}\cdot\sigma & (7) \cr
}$$

\smallskip
{\noindent 4. {\bf Michel Parameters}}

Pressure (or temperature) fluctuations are to smear 
the Michel spectrum,

$$\eqalignno{
   {d^2N \over dxd(\cos\theta)} = & (x^2-x_0^2)^{1\over 2} \Bigl\{
               6x(1-x) + {4\over 3}\rho (4x^2-3x-x_0^2) +
               6\eta x_0(1-x)  \cr
              + & P_\mu \xi \cos\theta
               (x^2-x_0^2)^{1\over 2} \bigl[ 2(1-x) + {4\over 3} \delta
               (4x-4+(1-x_0^2)^{1\over 2}) \bigr] \Bigr\} & (8) \cr
}
$$

{\noindent by,
}

$$\eqalignno{ E \to & \ E+G(0,\sigma_E)\ \ \ {\rm and}\cr 
              \cos\theta \to & \
                                \cos\theta + G(0,\sigma_{\cos\theta}), & 
 (9) \cr
}$$

{\noindent where $G(0,\sigma)$ stands for a Gaussian distribution with mean 0
and standard deviation $\sigma$. The extent of smearing,
$\sigma_E = (p/E)\sigma_p$ and 
$\sigma_{\cos\theta}$, are readily found in Equs. (6) and (7).
}

We firstly generate a Michel spectrum by sampling $10^9$ times 
from the distribution of Equ. (8)
with $\rho = 0.75, \eta = 0, P_\mu \xi = 1,\, {\rm and}\ \delta = 0.75$. 
The resulting high statistics spectrum is our (perfect) data.
The ranges of $x$ and $\theta$ in the sampling are $0.3
%\nopagenumbers
\vskip 1.0 cm
\centerline{Table 1}
\vskip 0.2 cm
\settabs 2 \columns 
\hrule
\medskip
\+ & $\sigma$ in $\mu$m \cr
\medskip
\hrule
\medskip
\+ temperature (24 $\pm$ 10 C) & 17.4 \cr
\+ pressure (760 $\pm$ 23 torr) & 15.5 \cr
\+ time zero jitter (1 $n$s) & 3.4 \cr
\+ B inhomogeneity (10${}^{-4}$) &\ 0.5 \cr
\medskip
\hrule
\medskip
\vskip 0.2 cm
$$
\vbox{\hsize = 5.0 true in\baselineskip = 18 true pt
{\noindent
Table 1: First column lists the variables changed in the
                   Garfield simulation. Second are the resulting
                   changes in the drift distance averaged over
                   distances to the sense wire and over track angles
                   (from 0 to 60 per 5 degree). The default STR is the
                   one with DME gas at 24 C, 760 torr, and in a uniform
                   $B$ field of 2.2 Tesla.

}
}$$
\vskip 1.0 cm 


%\nopagenumbers
\vskip 1.0 cm
\centerline{Table 2}
\vskip 0.2 cm
\settabs 3 \columns 
\hrule
\medskip
\+ parameters & best-fitted value & statistical error \cr
\medskip
\hrule
\medskip
\+ $\rho$   & 0.74996 (1.00) & $6\times 10^{-5}$ \cr
\+ $\eta$   & -0.00149 (0.99) & $176\times 10^{-5}$ \cr
\+ $\xi$    & 1.00005 (0.99) & $13\times 10^{-5}$ \cr
\+ $\delta$ & 0.74997 (1.00) & $9\times 10^{-5}$ \cr
\medskip
\hrule
\medskip
\vskip 0.2 cm
$$
\vbox{\hsize = 5.0 true in\baselineskip = 18 true pt
{\noindent
Table 2: $10^9$ pairs of energy and angle are generated by Monte
Carlo according to the distribution of Equ. (8)
given $\rho = 0.75, \eta = 0, P_\mu \xi = 1,\, {\rm and}\ \delta = 0.75$.
The resulting two-dimensional histogram is fitted to Equ. (8) with now
freely adjusting
parameters (one at a time). The 2nd column are the re-constructed parameters,
the numbers in the parenthesis being $\chi^2/{\rm NDF}$. The last column are 
the errors.
}
}$$
\vskip 1.0 cm

%\nopagenumbers
\vskip 1.0 cm
\line{Table 3\hfil}
\vskip 0.2 cm
\settabs 5 \columns 
\hrule
\medskip
\+ parameters & & best-fitted value & & systematical \cr
\+            & 100 $\mu$m & 20 $\mu$m & 10 $\mu$m & error \cr
\medskip
\hrule
\medskip
\+ $\rho$   & 0.74608 (19.1) & 0.74934 (2.14) & 0.74968 (1.27) & $0.5\times 10^{-5}$ \cr
\+ $\eta$   & 0.10296 (19.1) & 0.01760 (2.15) & 0.00691 (1.26) & $179\times 10^{-5}$ \cr
\+ $\xi$    & 0.98112 (19.0) & 0.99707 (2.14) & 0.99873 (1.28) & $17\times 10^{-5}$ \cr
\+ $\delta$ & 0.74296 (18.9) & 0.74883 (2.13) & 0.74953 (1.27) & $0.7\times 10^{-5}$ \cr
\medskip
\hrule
\medskip
\vskip 0.1 cm
$$
\vbox{\hsize = 5.0 true in\baselineskip = 18 true pt
{\noindent
Table 3: $10^9$ pairs of energy and angle are generated by Monte
Carlo according to distribution Equ. (8)
with $\rho = 0.75, \eta = 0, P_\mu \xi = 1,\, {\rm and}\ \delta = 0.75$.
Energies and angles are then smeared according to Equ. (9) for $\sigma$ =
100, 20, 10 $\mu$m before binning.
The resulting two-dimensional histogram is fitted to Equ. (8) with now
freely adjusting
parameters (one at a time). The 2nd, 3rd, and 4th column are the re-constructed parameters,
and the last column are the {\it systematical} errors evaluated 
according to Equ. (10).
}
}$$
\vskip 1.0 cm

%\nopagenumbers
\vskip 1.0 cm
\centerline{Table 4}
\vskip 0.2 cm
\settabs 6 \columns 
\hrule
\medskip
\+ source of error && $\rho$ ($\times 10^{-5}$) & $\eta$ ($\times 10^{-5}$) & $\xi$ ($\times 10^{-5}$) & $\delta$ ($\times 10^{-5}$) \cr
\medskip
\hrule
\medskip
\+ temperature $\pm$ 10 K && $\leq$ 0.5 & $\leq$ 179 & $\leq$ 17 & $\leq$ 0.7 \cr
\+ pressure $\pm$ 3\% && $\leq$ 0.5 & $\leq$ 179 & $\leq$ 17 & $\leq$ 0.7 \cr
\+ time zero $\pm$ 1 $n$s && $<\!\!<$ 0.5 & $<\!\!<$ 179 & $<\!\!<$ 17 & $<\!\!<$ 0.7 \cr
\+ B inhomogeneity $\pm$ 10${}^{-4}$ && $<\!\!<$ 0.5 & $<\!\!<$ 179 & $<\!\!<$ 17 & $<\!\!<$ 0.7 \cr
\+ statistical at $10^9$ && 6 & 176 & 13 & 9 \cr
\medskip
\hrule
\medskip
\vskip 0.1 cm
$$
\vbox{\hsize = 5.0 true in\baselineskip = 18 true pt
{\noindent
Table 4: Estimates of errors due to fluctuations in 
environmental and instrumental conditions.
}
}$$
\vskip 1.0 cm


 TN-53: systematics on temperature, pressure, time-zero, and B / Sun-Chong Wang  


Created for the The Center for Subatomic Research E614 Project  Projects Page. 
Created by The CoCoBoard.