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Measurement Error Matrix at High Momentum

  As shown in Section 3, in the high momentum limit, the r-$\phi$ 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} (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} (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} (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, $\sigma_x$ and $\sigma_z$, 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} (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} (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} (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} (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} (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 $\sigma_\kappa$ 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} (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 up previous
Next: About this document ... Up: No Title Previous: Effects of Thin Layer
Keisuke Fujii
12/4/1998