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:

$$\begin{array} {lll} E_M^{-1} \equiv \frac{1}{2} \left( \... ...r \mbox{\Large 0} & \left( E^{rz}_M \right)^{-1} },\end{array}$$ (1)

where the component matrices are given by

$$\begin{array} {lll} \left( E^{r\phi}_M \right)^{-1} = \le... ... \frac{y_i^4}{\sigma_{x_i}^2} \end{array} \right) \end{array}$$ (2)

and

$$\begin{array} {lll} \left( E^{rz}_M \right)^{-1} = \pmatr... ...i}^2} \cr ~ & \sum \frac{y_i^2}{\sigma_{z_i}^2} } .\end{array}$$ (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

$$\begin{array} {lll} \left( E^{r\phi}_M \right)^{-1} = \fr... ...(n+1)(2n+1)(3n^2+3n-1)}{30n^3} \end{array} \right) \end{array}$$ (4)

and

$$\begin{array} {lll} \left( E^{rz}_M \right)^{-1} = \frac{... ... L^2 \frac{(n+1)(2n+1)}{6n} \end{array} \right) , \end{array}$$ (5)

where we have defined the lever arm length L by

$$\begin{array} {lll} L \equiv r_{out} - r_{in} . \end{array}$$ (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  

$$\begin{array} {lll} E^{r\phi}_M = \sigma_x^2 \left( \beg... ...n^3}{L^4(n-1)(n+1)(n+2)(n+3)} \end{array} \right) \end{array}$$ (7)

and  

$$\begin{array} {lll} E^{rz}_M = \sigma_z^2 \left( \begin{... ... & \frac{12n}{L^2(n+1)(n+2)} \end{array} \right) . \end{array}$$ (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  

$$\begin{array} {lll} \sigma_\kappa^M & = & \left(\frac{\al... ...sigma_x}{L^2 B} \right) \sqrt{ \frac{720}{n+5} } ,\end{array}$$ (9)

where the last line is none other thatn the familiar text book expression for the momentum resolution valid in the large n limit.