Deviations - calculated and from simulations

Krzysztof Wozniak, May 4, 1999

This is the final comparison of the deviations obtained in the simulations with the deviations calculated using the information available from the ideal trajectory only

The study was performed for pi- and pi+ emitted at 4 theta angles: 30, 40, 50 and 60 degrees, with total momenta changing from 100 MeV/c to 1100 MeV/c. In the simulations first the trajectory of ideal track was obtained (no multiple scattering, mean energy loss). Then 1000 tracks simulated with gaussian multiple scattering were used to calculate deviations and covariance matrix. The deviation was calculated as distance from the ideal trajectory. The particle was included in the calculations only if it traversed exactly the same layers as those used to determine ideal trajectory. This way not always 1000 particles were used, sometimes this number was much lower, even less than 100. In such cases decrease of deviations is expected: usually rejected are particles with trajectories distant from the ideal one.

The analytical calculations of the deviations and covariance matrix were done using the knowledge of the spectrometer detector geometry and the information available from the ideal trajectory. The mean scattering angle in beam pipe and all traversed silicon sensors was calculated. This angle depends on the thickness of the material traversed. The deviation in the next sensor depends on the previously accumulated deviation, the cumulative and actual scattering angle and the distance to this sensor, known for the ideal trajectory. The covariance matrix was calculated taking into account also the following effects:

modification of the effective angle due to the curvature of the trajectory in the magnetic field
modification of the previous deviation due to the curvature of the trajectory in the magnetic field
modification of the trajectory due to the gradient of the magnetic field in the direction perpendicular to the trajectory (present if the field is increasing or decreasing if one moves from the left to the right side of the ideal trajectory).

The deviations were obtained from the covariance matrix. The available information for the ideal trajectory is limited to the hits, so the following approximations were necessary:

the magnetic field and thus the curvature of the trajectory is known only at the sensors, between them mean curvature is assumed; if the magnetic field changes not linearly, this leads to some systematic inaccuracies
the gradient of the magnetic field is also calculated as the mean of their values at the sensors

The following general approximation were also applied:

the scattering angle alpha is small, so the approximation tan(alpha)=sin(alpha)=alpha was applied when it was really necessary
the terms of the second order (or higher) in alpha were neglected
in some cases higher order terms in s/R were neglected (s - distance between hits, R - curvature of the trajectory), but generally exact calculation were possible in this case
multiple scattering in the air was neglected

Comparison of deviations

For each of the theta and ptot values considered the deviation was both calculated and obtained from simulations. For some values of momentum and theta there were too few hits for the template to be valid, in such cases deviations equal 0 were assumed. The following quantity was studied:
d = (c - s) / c
where
c - calculated deviation
s - deviation from simulations

This relative difference of deviations was calculated in 5 representative layers: 4, 8, 10, 12 and 14. As there are large fluctiation in the simulated deviations if not all 1000 tracks are accepted, the calculated deviations were divided into two samples:

"good" sample with more than 990 accepted particles from simulations
"bad" sample with less than 991 accepted particles

In the following pictures differences of deviations in each of the representative layer were collected. Each picture contains eight plots:

the plots contains relative deviations as a function of ptot
in the left column differences for "good" sample are presented (green)
in the right column differences of "bad" sample (red) together with the "good" (green) are presented
each pair of plots in a row corresponds to the same theta value, they are 30, 40, 50 and 60 degrees going from the top to the bottom respectively

Pictures

To view the pictures GIF files can be used, but to see the values of deviations PostScript files should be printed.

pi-, layer 4 (PostScript file)
pi-, layer 8 (PostScript file)
pi-, layer 10 (PostScript file)
pi-, layer 12 (PostScript file)
pi-, layer 14 (PostScript file)

pi+, layer 4 (PostScript file)
pi+, layer 8 (PostScript file)
pi+, layer 10 (PostScript file)
pi+, layer 12 (PostScript file)
pi+, layer 14 (PostScript file)

Summary

The calculated deviations are generally larger that those from the simulations. At least part of this effect is due to the rejection of some of the simulated particles with largest deviations.
The best agreement is in the first layer considered (layer 4) the worst in the last layer (layer 14)
There are no "good" points below momentum 150 MeV/c, where the difference is the largest
Both for pi- and pi+ and momenta > 200-250 MeV/c the difference does not exceed 0.2, ("good" sample)
For pi- the difference is below 0.3 at 150 MeV/c
For pi+ the difference is larger at low momentum (up to 0.4 ?)

The agreement is generally good, it seems possible to use calculated covariance matrices in place of the simulated. One can be worried only about low momenta (below 200 MeV/c) where larger (up to 40%) overestimation of the deviations is present. Such overestimation is safe considering efficiency, but may lead to easier acceptance of ghosts.