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As shown in Section 3, in the high momentum limit,
the r-
and the r-z track fittings are decoupled so that the
error matrix or its inverse corresponding to
the coordinate measurement errors has a
blockwise diagonal form:
| ![\begin{displaymath}
\begin{array}
{lll}
E_M^{-1} \equiv \frac{1}{2}
\left(
\...
...r
\mbox{\Large 0} & \left( E^{rz}_M \right)^{-1} },\end{array}\end{displaymath}](img270.gif) |
(1) |
where the component matrices are given by
| ![\begin{displaymath}
\begin{array}
{lll}
\left( E^{r\phi}_M \right)^{-1}
= \le...
... \frac{y_i^4}{\sigma_{x_i}^2}
\end{array} \right) \end{array}\end{displaymath}](img271.gif) |
(2) |
and
| ![\begin{displaymath}
\begin{array}
{lll}
\left( E^{rz}_M \right)^{-1}
= \pmatr...
...i}^2} \cr
~ & \sum \frac{y_i^2}{\sigma_{z_i}^2} } .\end{array}\end{displaymath}](img272.gif) |
(3) |
Notice that, in this limit, the error matrix is determined completely by
the y-locations
of the sampling points and the spatial resolutions
thereat.
If (n+1) sampling points are equally spaced and have common
resolutions,
and
, then the above equations
become
| ![\begin{displaymath}
\begin{array}
{lll}
\left( E^{r\phi}_M \right)^{-1}
= \fr...
...(n+1)(2n+1)(3n^2+3n-1)}{30n^3}
\end{array} \right) \end{array}\end{displaymath}](img274.gif) |
(4) |
and
| ![\begin{displaymath}
\begin{array}
{lll}
\left( E^{rz}_M \right)^{-1}
= \frac{...
... L^2 \frac{(n+1)(2n+1)}{6n}
\end{array} \right) , \end{array}\end{displaymath}](img275.gif) |
(5) |
where we have defined the lever arm length L by
| ![\begin{displaymath}
\begin{array}
{lll}
L \equiv r_{out} - r_{in} . \end{array}\end{displaymath}](img276.gif) |
(6) |
Notice that in the above equations, we have shown only upper triangles of
the matrices, since they are all symmetric.
By matrix inversions, we thus obtain
| ![\begin{displaymath}
\begin{array}
{lll}
E^{r\phi}_M
= \sigma_x^2
\left( \beg...
...n^3}{L^4(n-1)(n+1)(n+2)(n+3)}
\end{array} \right) \end{array}\end{displaymath}](img277.gif) |
(7) |
and
| ![\begin{displaymath}
\begin{array}
{lll}
E^{rz}_M
= \sigma_z^2
\left( \begin{...
... & \frac{12n}{L^2(n+1)(n+2)}
\end{array} \right) . \end{array}\end{displaymath}](img278.gif) |
(8) |
The above error matrices are defined with the pivotal
point (x0,y0,z0) chosen to be at the -th hit.
From the above formulae, we can estimate, for instance,
the transverse momentum resolution or
as
| ![\begin{displaymath}
\begin{array}
{lll}
\sigma_\kappa^M
& = & \left(\frac{\al...
...sigma_x}{L^2 B} \right)
\sqrt{ \frac{720}{n+5} } ,\end{array}\end{displaymath}](img280.gif) |
(9) |
where the last line is none other thatn the familiar
text book expression
for the momentum resolution valid in the large n limit.
Next: About this document ...
Up: No Title
Previous: Effects of Thin Layer
Keisuke Fujii
12/4/1998