5.2 Tracking detectors.

For each charged particle in the JETSET common block /LUJETS/ which is either stable or decays weakly, SGV calculates the values of the 'perigee parameters' of the track, given the true momentum and production-point. The 'perigee' is the point on the track where the p r o j e c t i o n of the track on to the x-y plane is at minimum distance to the origin. Note that this is different from the point of closest approach in space. The parameters are five, the minimum number to un-ambiguously define a helix :

• the impact parameter, ie the distance in th x-y plane between the perigee and the origin.
• the z-coordinate on the track at the perigee.
• the phi angle of the track at the perigee.
• the theta angle of the track (constant along the helix)
• the inverse of the radius of curvature of the projection in the x-y plane, with the convention that it is positive if the particle travels around the projected circle counter clock-wise (ie. the opposite to the sign of the charge).

The impact parameter has a geometric sign, ie. it is positive if the origin is to the left of the perigee in the projected plane.

SGV then calculates which of the tracking-detector surfaces the track helix intersects. From the list of those surfaces, the program analytically calculates the covariance matrix of the track-parameters at the perigee. This calculation includes the multiple-scattering in the traversed surfaces, and the measurement precision at each surface that measures the track position.

The set of parameters are then smeared using this covariance-matrix. The smearing is done by a Choleski-decomposition of the covariance matrix, followed by an addition of gaussian random-numbers. This method assures that the covariances between the parameters are exactly those given by the calculated covariance matrix. In other words, the smeared perigee parameters will have a 5 dimensional multi-normal distribution, with expectation-value equal to the true values and with the calculated covariance matrix.

The covariance matrix calculated, correctly represents the precision of the result of a track fit where where all the measurements along the track where used. Consequently, the possible problems arising from the pattern-recognition phase of the event reconstruction in NOT simulated. Ie., SGV assumes that the tracks are found, whenever possible. Tracks will, however, be lost if the covariance matrix cannot be calculated, which is the case if too few measuring surfaces where intersected, in practice less than three. This, in turn, can be the case because

• The track has too low momentum to reach the inner-most 3 layers, ie. it turns back in the magnetic field.

• The track passes a hole in the detector, typically the beam-pipe.

• The particle decays before enough layers are hit.

When the track is followed through the detector, a simplification is done to gain speed : Tracks are all assumed to come from the centre of the detector. This greatly simplifies the mathematics of finding the intersections of the helix with the detector surfaces. When the parameters at the perigee (or any other point in the detector) are calculated, the exact expression is of course used. Hence, the effect of the approximation is only in the errors on the parameters, not in the parameters themselves. The error is normally negligible: a slight difference of the angle of incidence, and hence in multiple-scattering might arise from the parallel transport of the helix a few millimeters. However, edge-effects will be a bit blurred. Also, from the above if should be clear that putting the nominal production vertex far from the centre of the detector is not a good idea. Note that the program takes into account tracks from secondary vertecies : Even though the track helix is transported so that it passes through the origin, the program knows that the particle doesn't exist inside it's production point, and extrapolations to any detectors in that region are ignored. This works well also for strange particle decays, because even though the typically travel many centimeters before decaying, their decay products have small transverse momenta, meaning that the distance from the origin to the point of closest approach of the extrapolation of the track is not very big (order 1 cm).