The coordinate system used for defining the parametrization of tracks is Cartesian and right-handed with its origin located at the interaction point. The z axis is given by the detectors z axis and the y axis lies along the vertical direction, pointing upwards.
Whenever a charged particle is affected by a constant magnetic field it moves on a helicoidal trajectory, where here and in the following both energy loss and multiple scattering are neglected. It is assumed that this magnetic field is homogeneous and parallel to the z axis. In this case the trajectory of a charged particle is a segment of a circle in the xy projection and the z displacement is a linear function of the length s of the arc that is described in the xy plane. This results in a straight line in the sz plane.
The parametrisation of the movement of a charged particle is defined by a reference point, P^{r} = (P^{r}_{x} , P^{r}_{y} , P^{r}_{z }), and five so-called "track parameters" (Ω , Φ_{0} , d_{0} , z_{0} and tan λ). In general the reference point can be any point in space but usually it is set to the origin of the coordinate system. In general, the five track parameters refer to a specific point P^{0} = (P^{0}_{x} , P^{0}_{y} , P^{0}_{z} ) along the helix, but here P^{0} is defined as the point of closest approach to the reference point in the xy plane.
In the xy plane the movement of a charged particle is defined by the reference point, P^{r} = (P^{r}_{x} , P^{r}_{y}) and three parameters, Ω, Φ_{0}, d_{0}:
The centre point P^{c} = (P^{c}_{x} , P^{c}_{y}) of the circle in the xy plane is usually different from the reference point P^{r}.
In the sz plane a charged particle moves along a straight line, which is described by two parameters, tan λ and z_{0}:
The impact parameter resolution studies presented here are done for the parameters d_{0} and z_{0} using the software tools MOKKA and Marlin. The simulated SBelle detector consists of the subdetectors PXD, CDC and SVD. For the PXD subdetector different variations in terms of number of pixels per ladder and number of layers are available. The combination of a specific PXD with the CDC and SVD subdetectors is called a detector model. A summary of the detector models that are currently available gives: Detector Models.
The detector response of the SBelle detector is simulated using a particle gun. The particles fired are μ^{+} having uniformly distributed φ angles in the range [-180°, 180°]. The θ angles and energies used are given in the following table:
0.1 [GeV] | 0.2 [GeV] | 0.4 [GeV] | 0.6 [GeV] | 0.8 [GeV] | 1.0 [GeV] | 2.0 [GeV] | |
20° | Not used, slanted parts are not yet implemented in track fitting code | ||||||
40° | done | done | done | done | done | done | done |
60° | done | done | done | done | done | done | done |
80° | done | done | done | done | done | done | done |
For each combination of the given angles and energies 1000 events are simulated.
The hits which are created by MOKKA in the three subdetectors are then digitized in Marlin using highly advanced algorithms. For the PXD and CDC a newly developed pixel vertex digitizer and for the SVD a slightly modified version of the original ILD TPC digitizer is used. After digitization, a tracking procedure and a Kalman based fit are applied. The result are tracks carrying the required impact parameter information.
For each track the impact parameters d_{0} and z_{0} are stored in a ROOT tree and saved as a ROOT file. Therefore, for each θ angle, energy and detector model a separate ROOT file is created. Then, the parameters d_{0} / z_{0} are taken from a ROOT file and filled into a histogram. According to A. Raspereza the impact parameter resolution is defined as the RMS90 value of an impact parameter. The RMS90 values for the d_{0}-histogram and the z_{0}-histogram are calculated by starting with the mean value of a given histogram and expanding the range until 90% of all events are contained within this range. The histogram is re-filled using the found range and its RMS value is calculated.
The impact parameter resolution is parameterized as follows:
where a and b are free parameters, E is the energy of the particle and θ its angle.
The impact parameter resolutions for all possible combinations of angles and energies of a given model and a given impact parameter are inserted into a common graph with the x-axis being E sin(θ)^{3/2} for the d_{0} case and E sin(θ)^{5/2} for the z_{0} case. The graph is then fitted using the impact parameter resolution functions defined above in order to extract the values for a and b. The second row of plots shown below for each model represents the result of this procedure. The pink curve represents the BELLE experiment impact parameter resolution for reference purposes, where the required parameters a and b were taken from the Belle Note 715.
The values of a and b extracted from the fit, are then used to draw the curves shown in the first row of plots for each model. A combined plot of all fitted impact parameter functions for all models is shown in the last two pictures of this topic.
-- AndreasMoll - 20 Mar 2009
I | Attachment | History | Action | Size | Date | Who | Comment |
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png | ImpactParameterFunctionModelsD0.png | r1 | manage | 41.1 K | 2009-03-25 - 14:06 | AndreasMoll | Impact parameter resolution d0 for all models |
png | ImpactParameterFunctionModelsD0Linear.png | r1 | manage | 48.8 K | 2009-03-27 - 12:59 | AndreasMoll | Impact parameter resolution d0 for all models (linear y-axis) |
png | ImpactParameterFunctionModelsZ0.png | r1 | manage | 42.9 K | 2009-03-25 - 14:06 | AndreasMoll | Impact parameter resolution z0 for all models |
png | ImpactParameterFunctionModelsZ0Linear.png | r1 | manage | 51.1 K | 2009-03-27 - 13:00 | AndreasMoll | Impact parameter resolution z0 for all models (linear y-axis) |
png | LCIOTrackDefinitionXY.png | r1 | manage | 17.6 K | 2009-03-20 - 10:15 | AndreasMoll | Projection of a track helix segment in the xy plane |
png | LCIOTrackDefinitionZS.png | r1 | manage | 12.3 K | 2009-03-20 - 10:16 | AndreasMoll | Projection of a track helix in the sz plane |
png | TrkSBelle_CPS1000_SUP10_ImpactParameterD0.png | r1 | manage | 40.4 K | 2009-03-25 - 13:50 | AndreasMoll | r-phi impact parameter for TrkSBelle_CPS1000_SUP10 |
png | TrkSBelle_CPS1000_SUP10_ImpactParameterFitD0.png | r1 | manage | 37.0 K | 2009-03-25 - 13:51 | AndreasMoll | impact parameter resolution d0 for TrkSBelle_CPS1000_SUP10 |
png | TrkSBelle_CPS1000_SUP10_ImpactParameterFitZ0.png | r1 | manage | 36.9 K | 2009-03-25 - 13:52 | AndreasMoll | impact parameter resolution z0 for TrkSBelle_CPS1000_SUP10 |
png | TrkSBelle_CPS1000_SUP10_ImpactParameterZ0.png | r1 | manage | 40.8 K | 2009-03-25 - 13:51 | AndreasMoll | z impact parameter for TrkSBelle_CPS1000_SUP10 |
png | TrkSBelle_CPS1600_SUP10_ImpactParameterD0.png | r1 | manage | 40.7 K | 2009-03-25 - 13:54 | AndreasMoll | r-phi impact parameter for TrkSBelle_CPS1600_SUP10 |
png | TrkSBelle_CPS1600_SUP10_ImpactParameterFitD0.png | r1 | manage | 37.3 K | 2009-03-25 - 13:55 | AndreasMoll | impact parameter resolution d0 for TrkSBelle_CPS1600_SUP10 |
png | TrkSBelle_CPS1600_SUP10_ImpactParameterFitZ0.png | r1 | manage | 37.0 K | 2009-03-25 - 13:55 | AndreasMoll | impact parameter resolution z0 for TrkSBelle_CPS1600_SUP10 |
png | TrkSBelle_CPS1600_SUP10_ImpactParameterZ0.png | r1 | manage | 41.1 K | 2009-03-25 - 13:54 | AndreasMoll | z impact parameter for TrkSBelle_CPS1600_SUP10 |
png | TrkSBelle_CPS2000_SUP10_ImpactParameterD0.png | r1 | manage | 40.4 K | 2009-03-25 - 13:57 | AndreasMoll | r-phi impact parameter for TrkSBelle_CPS2000_SUP10 |
png | TrkSBelle_CPS2000_SUP10_ImpactParameterFitD0.png | r1 | manage | 37.3 K | 2009-03-25 - 13:58 | AndreasMoll | impact parameter resolution d0 for TrkSBelle_CPS2000_SUP10 |
png | TrkSBelle_CPS2000_SUP10_ImpactParameterFitZ0.png | r1 | manage | 37.2 K | 2009-03-25 - 13:58 | AndreasMoll | impact parameter resolution z0 for TrkSBelle_CPS2000_SUP10 |
png | TrkSBelle_CPS2000_SUP10_ImpactParameterZ0.png | r1 | manage | 41.3 K | 2009-03-25 - 13:57 | AndreasMoll | z impact parameter for TrkSBelle_CPS2000_SUP10 |
png | TrkSBelle_CPS800_SUP10_ImpactParameterD0.png | r1 | manage | 40.8 K | 2009-03-25 - 13:46 | AndreasMoll | r-phi impact parameter for TrkSBelle_CPS800_SUP10 |
png | TrkSBelle_CPS800_SUP10_ImpactParameterFitD0.png | r1 | manage | 37.6 K | 2009-03-25 - 13:47 | AndreasMoll | mpact parameter resolution d0 for TrkSBelle_CPS800_SUP10 |
png | TrkSBelle_CPS800_SUP10_ImpactParameterFitZ0.png | r1 | manage | 37.0 K | 2009-03-25 - 13:47 | AndreasMoll | impact parameter resolution z0 for TrkSBelle_CPS800_SUP10 |
png | TrkSBelle_CPS800_SUP10_ImpactParameterZ0.png | r1 | manage | 40.3 K | 2009-03-25 - 13:47 | AndreasMoll | z impact parameter for TrkSBelle_CPS800_SUP10 |
png | TrkSBelleUpgr_CPS1000_SUP10_ImpactParameterD0.png | r1 | manage | 41.0 K | 2009-03-25 - 13:38 | AndreasMoll | r-phi impact parameter for TrkSBelleUpgr_CPS1000_SUP10 |
png | TrkSBelleUpgr_CPS1000_SUP10_ImpactParameterFitD0.png | r1 | manage | 36.9 K | 2009-03-25 - 13:39 | AndreasMoll | impact parameter resolution d0 for TrkSBelleUpgr_CPS1000_SUP10 |
png | TrkSBelleUpgr_CPS1000_SUP10_ImpactParameterFitZ0.png | r1 | manage | 36.7 K | 2009-03-25 - 13:40 | AndreasMoll | impact parameter resolution z0 for TrkSBelleUpgr_CPS1000_SUP10 |
png | TrkSBelleUpgr_CPS1000_SUP10_ImpactParameterZ0.png | r1 | manage | 40.8 K | 2009-03-25 - 13:39 | AndreasMoll | z impact parameter for TrkSBelleUpgr_CPS1000_SUP10 |
png | TrkSBelleUpgr_CPS800_SUP10_ImpactParameterD0.png | r1 | manage | 41.2 K | 2009-03-25 - 13:17 | AndreasMoll | r-phi impact parameter for TrkSBelleUpgr_CPS800_SUP10 |
png | TrkSBelleUpgr_CPS800_SUP10_ImpactParameterFitD0.png | r1 | manage | 36.9 K | 2009-03-25 - 13:19 | AndreasMoll | impact parameter resolution d0 for TrkSBelleUpgr_CPS800_SUP10 |
png | TrkSBelleUpgr_CPS800_SUP10_ImpactParameterFitZ0.png | r1 | manage | 36.6 K | 2009-03-25 - 13:20 | AndreasMoll | impact parameter resolution z0 for TrkSBelleUpgr_CPS800_SUP10 |
png | TrkSBelleUpgr_CPS800_SUP10_ImpactParameterZ0.png | r1 | manage | 40.6 K | 2009-03-25 - 13:18 | AndreasMoll | z impact parameter for TrkSBelleUpgr_CPS800_SUP10 |