Track Fitting with Multiple Scattering

Now let us turn our attention to the r-z track fitting under the influence of multiple scattering. The treatment here provides a way to calculate the E₀ of the last subsection including multiple scattering effects in track fitting. From now on, we take (y₀,z₀) = (0,0) and $y \equiv \Delta y_0$.The multiple scattering modifies the track parameter of the particle as

$$\begin{array} {lll} \frac{ d{\bf a} }{dy} = T \cdot {\bf a} + \pmatrix{ 0 \cr \frac{d\Delta a}{dy} },\end{array}$$ (90)

where

$$\begin{array} {lll} \Delta a(y) \equiv a(y) - a(y_0)\end{array}$$ (91)

represents the change of the dip angle due to the multiple scattering as a function of y. This results in

$$\begin{array} {lll} \left\{ \begin{array} {lllllll} \zeta(... ..._0 & + & \Delta a(y) & ~ & ~ \end{array} \right. ,\end{array}$$ (92)

where the parmeters with the suffix are those at the initial point before multiple scattering.

Now, let us assume that we are given (n+1) hit points, ${\bf z} \equiv (z_0,\cdots,z_n)^T$,measured at fixed y positions, $y_0=0, \cdots, y_n = L$.Then the minimization of the $\chi^2$:

$$\begin{array} {lll} \chi^2 = \sum_{i=0}^n \left( \frac{z_i - z(y_i,{\bf a})}{\sigma_{z_i}} \right)^2.\end{array}$$ (93)

The parameter vector that gives the $\chi^2$ minimum is

$$\begin{array} {lll} {\bf a} & = & E_M \cdot \sum_{i=0}^n \frac{z_i}{\sigma_{z_i}^2} \pmatrix{1 \cr y_i} ,\end{array}$$ (94)

with

$$\begin{array} {lll} E_M^{-1} = \pmatrix{ \sum \frac{1}{\s... ...i}^2} \cr ~ & \sum \frac{y_i^2}{\sigma_{z_i}^2} } .\end{array}$$ (95)

Through the above equation, the change in ${\bf z}$ due to the multiple scattering induces a change in ${\bf a}$:

$$\begin{array} {lll} \Delta {\bf a} & = & E_M \cdot \pmatr... ...M \cdot A \cdot E^{-1}_{\bf z} \cdot \Delta {\bf z}.\end{array}$$ (96)

From this we obtain the full error matrix including multiple scattering:

$$\begin{array} {lll} \left< \Delta {\bf a} \cdot \Delta {\bf... ...\right\gt \cdot E^{-1}_{\bf z} \cdot A^T \cdot E_M.\end{array}$$ (97)

Again the problem reduces to the evaluation of the correlation matrix in the coordinate measurements:

$$\begin{array} {lll} \left< \Delta z_i \Delta z_j \right\gt = \delta_{ij} \sigma_{z_i}^2 + B_{ij}\end{array}$$ (98)

with

$$\begin{array} {lll} B_{ij} & \equiv & N' \int [d\Delta a] ~... ...ght) \cr & = & \frac{K'}{2} y_i^2 ( y_j - y_i/3 ) ,\end{array}$$ (99)

where the last expression is valid for $i \leq j$.The full error matrix is then written in the form:  

$$\begin{array} {lll} E = E_M + E_{MS},\end{array}$$ (100)

where the first term is from the coordinate measurement errors while the second term from the multiple scattering:

$$\begin{array} {lll} E_{MS} = E_M \cdot A \cdot E^{-1}_{\bf z} \cdot B \cdot E^{-1}_{\bf z} \cdot A^T \cdot E_M.\end{array}$$ (101)

In the case of equal space sampling with the same position resolution, E_M and E_MS reduce to

$$\begin{array} {lll} E_M & \simeq & \frac{2 \sigma_z^2}{n... ...}{21} & \frac{-11}{42 L} \cr ~ & \frac{13}{7 L^2} }\end{array}$$ (102)

with

$$\begin{array} {lll} K' = (1+a^2) \left( \frac{C}{P_T \beta} \right)^2 \frac{(1+a^2)^{1/2}}{X_0}\end{array}$$ (103)

in the large n limit.