A small initial covariance matrix would bias the final result towars the initial position and we want to avoid this.
The following plot shows the final vertex position as a function of the diagonal elements of the initial covariance matrix for a vertex fit of 5 muons coming from (0.3,0.2)cm. The starting vertex position was (0,0)cm.
Stability, in this example, is reached for a covariance matrix of about 0.15cm. The deafult values in t01.16.00 are 10cm in r-phi and 100 cm in z.
It means that the vertex estimations for tracks added in different orders will be different because the linealization points are different. In a global fit, on the other hand, all tracks are linealized around the same point and the solution is track-order independent.
The following plots ilustrates this fact:
This is the Kalman Filter path for the vertex
estimation of 3 tracks fitted in different orders. The colors correspond to
different orders and the points are the intermediate positions after adding
1,2 and 3 tracks in the fit.
This is a Zoom of the same plot.
The next plot is the histogram with the errors
(in sigmas wrt a global fit) of the 6 different combinations of 3 tracks.
This plot shows that the difference is not to much significantly
(0.3 sigmas corresponds to about 10um) but exists.
The way this problem is solved is thru the implementation of an Iterated
Extended Kalman Filter (IEKF)
In this implementation all tracks (measurements) are always linealized around
the same position (the previous vertex estimate), and the filter-smoother are
re-iterated (using the previous vertex estimate as linealization point)
until convergence. Normally less than 5 iterations are needed to reach an
stable solution. However, vertices with very high chi-square, ussually need
much more iterations to converge like the following plot of
vertex chi-square vs mumber of iterations shows.
The next plot is a comparison between 3 finders in a simple event.
In this particular event of 2 vertices, the Tear-down algorithm fails to find the secondary vertex due to the fact that 1 of the 2 tracks of that vertex was already associated to the PV, so the algorithm stops finding 1 vertex and 1 single track.
The Build-up finder finds the primary and secondary vertex, but it also finds a fake vertex due to allow multiple track-vertex associations.
The combined algorithm correclty reconstructs the event with no fakes.
The combined aproach was created with the idea of tagging b-quark jets because
those events have PV-B-D cascade decays with different track multiplicities.
In fact, so far, the combined algorithm has shown to have the best purity
for b-tagging.
The aplication of these algorithms to tag b-jets is descripted in the
b-jet identification page