Simple Monte Carlo Simulation of Track Fitting

The full error matrix given by Eqs.3.2.48 and 3.3.77 allows us to smear the exact track parameter given by an event generator, properly taking into account the effects of multiple scattering including the additional correlations among the track parameters. In fact, the $\chi^2$ for the track fitting can be written in the form:

$$\begin{array} {lll} \chi^2 & = & \Delta {\bf a}^T \cdot E^{... ...a}}{\partial {\bf b}} \right) \cdot \Delta {\bf b},\end{array}$$ (107)

where in the second line we have diagonalized the error matrix:

$$\begin{array} {lll} \left( \frac{\partial {\bf a}}{\partia... ...ddots & 0 \cr 0 & \cdots & 0 & 1/\sigma_{b_5}^2 }. \end{array}$$ (108)

Since there is now no correlation among b_i's, we can Gaussian-smear them independently:

$$\begin{array} {lll} \Delta b_i = \sigma_{b_i} \cdot (\mbox{... ... random number centered at zero with unit width}).\end{array}$$ (109)

Then the fluctuation of ${\bf a}$ is given by

$$\begin{array} {lll} \Delta {\bf a} = \left( \frac{\partial... ...}}{\partial {\bf b}} \right) \cdot \Delta {\bf b},\end{array}$$ (110)

where the correlations among the parameters are implemented properly.