Helix to Helix

Before combining track segments, we move their pivots to a common point which can be arbitrarily chosen. We should also correct the track parameters for energy loss and multiple scattering beforehand so that they represent the track parameters for the original track. Then the total $\chi^2$ for the combined track is then given by

$$\begin{array} {lll} \chi^2 = \sum_i \Delta{\bf a}_i^T \cdot E_{{\bf a}_i}^{-1} \cdot \Delta{\bf a}_i,\end{array}$$ (22)

where

$$\begin{array} {lll} \nonumber \Delta{\bf a}_i = {\bf a} - {\bf a}_i \end{array}$$

and ${\bf a}$ is the track parameter vector for the original track, ${\bf a}_i$ is that determined by i-th tracking device, and $E_{{\bf a}_i}$ is the corresponding error matrix. Since this is quadratic in ${\bf a}$, the minimization of the $\chi^2$ can be carried out analytically by a single matrix inversion:  

$$\begin{array} {lll} \overline{{\bf a}} & = & \left(\sum_i E... ...& = & \left(\sum_i E_{{\bf a}_i}^{-1} \right)^{-1} .\end{array}$$ (23)

The above formulae tell us that the track linking in the parameter space is a simple averaging process.