Definition

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, Pr = (Prx , Pry , Prz ), and five so-called "track parameters" (Ω , Φ0 , d0 , z0 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 P0 = (P0x , P0y , P0z ) along the helix, but here P0 is defined as the point of closest approach to the reference point in the xy plane.

The xy Plane

LCIOTrackDefinitionXY.png


In the xy plane the movement of a charged particle is defined by the reference point, Pr = (Prx , Pry) and three parameters, Ω, Φ0, d0:

  • Φ0 is the azimuthal angle of the momentum of the particle (track tangent) at the point of closest approach
  • Ω describes the curvature of the track with
    |Ω| = 1 / R

    where R is the radius of curvature of the track. The sign of Ω is defined by moving along the track, following the direction of the particle's momentum. Passing through the arc in (anti)clock-wise direction defines positive (negative) curvature. In case of an axial magnetic field parallel to the z axis (B = (0, 0, Bz), Bz > 0), Ω > 0 (Ω < 0) corresponds to a particle with positive (negative) electric charge.
  • d0 is the signed impact parameter in the xy plane with |d0| being the distance between Pr and P0 in the xy plane. The signing convention is defined as follows: Looking from the reference point to the point of closest approach, then d0 > 0 (d0 < 0) if the particle travels from left to right (right to left). This results in sgn(Ω) = sgn(d0) if Pr is inside and conversely in sgn(Ω) = -sgn(d0) if Pr is outside the arc.

The centre point Pc = (Pcx , Pcy) of the circle in the xy plane is usually different from the reference point Pr.

The sz Plane

LCIOTrackDefinitionZS.png


In the sz plane a charged particle moves along a straight line, which is described by two parameters, tan λ and z0:

  • tan λ is the slope dz/ds of the straight line in the sz plane. This parameter is constant for a given track and it is directly related to the polar angle θ of the momentum vector p = (px , py , pz).
  • z0 is the z position of the track at the point of closest approach with respect to the z coordinate of the reference point Prz. The equation of the trajectory of a helicoidal track in the sz projection is then

    z = (z0 + Prz) + s ⋅ tan λ

    where s is the path integral (i.e. the arc length) in the xy projection when a particle travels from P0 to P. s is positive (negative) if P is located in the direction (against the direction) of the momentum with respect to the point of closest approach P0.

Simulation and Reconstruction

The impact parameter resolution studies presented here are done for the parameters d0 and z0 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° Warning, important 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.

Analysis

For each track the impact parameters d0 and z0 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 d0 / z0 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 d0-histogram and the z0-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:

  • for d0: a + b / (E sin(θ)3/2)
  • for z0: a + b / (E sin(θ)5/2)

where a and b are free parameters, E is the energy of the particle and θ its angle.

Results for SuperBelle and SuperBelle Upgrade

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 d0 case and E sin(θ)5/2 for the z0 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.

SuperBelle, constant pixel size, 800 pixels, support structure 10mm

TrkSBelle_CPS800_SUP10_ImpactParameterD0.png TrkSBelle_CPS800_SUP10_ImpactParameterZ0.png

TrkSBelle_CPS800_SUP10_ImpactParameterFitD0.png TrkSBelle_CPS800_SUP10_ImpactParameterFitZ0.png

SuperBelle, constant pixel size, 1000 pixels, support structure 10mm

TrkSBelle_CPS1000_SUP10_ImpactParameterD0.png TrkSBelle_CPS1000_SUP10_ImpactParameterZ0.png

TrkSBelle_CPS1000_SUP10_ImpactParameterFitD0.png TrkSBelle_CPS1000_SUP10_ImpactParameterFitZ0.png

SuperBelle, constant pixel size, 1600 pixels, support structure 10mm

TrkSBelle_CPS1600_SUP10_ImpactParameterD0.png TrkSBelle_CPS1600_SUP10_ImpactParameterZ0.png

TrkSBelle_CPS1600_SUP10_ImpactParameterFitD0.png TrkSBelle_CPS1600_SUP10_ImpactParameterFitZ0.png

SuperBelle, constant pixel size, 2000 pixels, support structure 10mm

TrkSBelle_CPS2000_SUP10_ImpactParameterD0.png TrkSBelle_CPS2000_SUP10_ImpactParameterZ0.png

TrkSBelle_CPS2000_SUP10_ImpactParameterFitD0.png TrkSBelle_CPS2000_SUP10_ImpactParameterFitZ0.png

SuperBelle Upgrade, constant pixel size, 800 pixels, support structure 10mm

TrkSBelleUpgr_CPS800_SUP10_ImpactParameterD0.png TrkSBelleUpgr_CPS800_SUP10_ImpactParameterZ0.png

TrkSBelleUpgr_CPS800_SUP10_ImpactParameterFitD0.png TrkSBelleUpgr_CPS800_SUP10_ImpactParameterFitZ0.png

SuperBelle Upgrade, constant pixel size, 1000 pixels, support structure 10mm

TrkSBelleUpgr_CPS1000_SUP10_ImpactParameterD0.png TrkSBelleUpgr_CPS1000_SUP10_ImpactParameterZ0.png

TrkSBelleUpgr_CPS1000_SUP10_ImpactParameterFitD0.png TrkSBelleUpgr_CPS1000_SUP10_ImpactParameterFitZ0.png

Impact parameter resolution for all models merged into one single plot

ImpactParameterFunctionModelsD0.png ImpactParameterFunctionModelsZ0.png

Impact parameter resolution for all models merged into one single plot (linear and zoomed y-axis)

ImpactParameterFunctionModelsD0Linear.png ImpactParameterFunctionModelsZ0Linear.png

-- AndreasMoll - 20 Mar 2009

Topic attachments
I Attachment History Action Size Date Who Comment
PNGpng ImpactParameterFunctionModelsD0.png r1 manage 41.1 K 2009-03-25 - 14:06 AndreasMoll Impact parameter resolution d0 for all models
PNGpng ImpactParameterFunctionModelsD0Linear.png r1 manage 48.8 K 2009-03-27 - 12:59 AndreasMoll Impact parameter resolution d0 for all models (linear y-axis)
PNGpng ImpactParameterFunctionModelsZ0.png r1 manage 42.9 K 2009-03-25 - 14:06 AndreasMoll Impact parameter resolution z0 for all models
PNGpng ImpactParameterFunctionModelsZ0Linear.png r1 manage 51.1 K 2009-03-27 - 13:00 AndreasMoll Impact parameter resolution z0 for all models (linear y-axis)
PNGpng LCIOTrackDefinitionXY.png r1 manage 17.6 K 2009-03-20 - 10:15 AndreasMoll Projection of a track helix segment in the xy plane
PNGpng LCIOTrackDefinitionZS.png r1 manage 12.3 K 2009-03-20 - 10:16 AndreasMoll Projection of a track helix in the sz plane
PNGpng TrkSBelleUpgr_CPS1000_SUP10_ImpactParameterD0.png r1 manage 41.0 K 2009-03-25 - 13:38 AndreasMoll r-phi impact parameter for TrkSBelleUpgr_CPS1000_SUP10
PNGpng TrkSBelleUpgr_CPS1000_SUP10_ImpactParameterFitD0.png r1 manage 36.9 K 2009-03-25 - 13:39 AndreasMoll impact parameter resolution d0 for TrkSBelleUpgr_CPS1000_SUP10
PNGpng TrkSBelleUpgr_CPS1000_SUP10_ImpactParameterFitZ0.png r1 manage 36.7 K 2009-03-25 - 13:40 AndreasMoll impact parameter resolution z0 for TrkSBelleUpgr_CPS1000_SUP10
PNGpng TrkSBelleUpgr_CPS1000_SUP10_ImpactParameterZ0.png r1 manage 40.8 K 2009-03-25 - 13:39 AndreasMoll z impact parameter for TrkSBelleUpgr_CPS1000_SUP10
PNGpng TrkSBelleUpgr_CPS800_SUP10_ImpactParameterD0.png r1 manage 41.2 K 2009-03-25 - 13:17 AndreasMoll r-phi impact parameter for TrkSBelleUpgr_CPS800_SUP10
PNGpng TrkSBelleUpgr_CPS800_SUP10_ImpactParameterFitD0.png r1 manage 36.9 K 2009-03-25 - 13:19 AndreasMoll impact parameter resolution d0 for TrkSBelleUpgr_CPS800_SUP10
PNGpng TrkSBelleUpgr_CPS800_SUP10_ImpactParameterFitZ0.png r1 manage 36.6 K 2009-03-25 - 13:20 AndreasMoll impact parameter resolution z0 for TrkSBelleUpgr_CPS800_SUP10
PNGpng TrkSBelleUpgr_CPS800_SUP10_ImpactParameterZ0.png r1 manage 40.6 K 2009-03-25 - 13:18 AndreasMoll z impact parameter for TrkSBelleUpgr_CPS800_SUP10
PNGpng TrkSBelle_CPS1000_SUP10_ImpactParameterD0.png r1 manage 40.4 K 2009-03-25 - 13:50 AndreasMoll r-phi impact parameter for TrkSBelle_CPS1000_SUP10
PNGpng TrkSBelle_CPS1000_SUP10_ImpactParameterFitD0.png r1 manage 37.0 K 2009-03-25 - 13:51 AndreasMoll impact parameter resolution d0 for TrkSBelle_CPS1000_SUP10
PNGpng TrkSBelle_CPS1000_SUP10_ImpactParameterFitZ0.png r1 manage 36.9 K 2009-03-25 - 13:52 AndreasMoll impact parameter resolution z0 for TrkSBelle_CPS1000_SUP10
PNGpng TrkSBelle_CPS1000_SUP10_ImpactParameterZ0.png r1 manage 40.8 K 2009-03-25 - 13:51 AndreasMoll z impact parameter for TrkSBelle_CPS1000_SUP10
PNGpng TrkSBelle_CPS1600_SUP10_ImpactParameterD0.png r1 manage 40.7 K 2009-03-25 - 13:54 AndreasMoll r-phi impact parameter for TrkSBelle_CPS1600_SUP10
PNGpng TrkSBelle_CPS1600_SUP10_ImpactParameterFitD0.png r1 manage 37.3 K 2009-03-25 - 13:55 AndreasMoll impact parameter resolution d0 for TrkSBelle_CPS1600_SUP10
PNGpng TrkSBelle_CPS1600_SUP10_ImpactParameterFitZ0.png r1 manage 37.0 K 2009-03-25 - 13:55 AndreasMoll impact parameter resolution z0 for TrkSBelle_CPS1600_SUP10
PNGpng TrkSBelle_CPS1600_SUP10_ImpactParameterZ0.png r1 manage 41.1 K 2009-03-25 - 13:54 AndreasMoll z impact parameter for TrkSBelle_CPS1600_SUP10
PNGpng TrkSBelle_CPS2000_SUP10_ImpactParameterD0.png r1 manage 40.4 K 2009-03-25 - 13:57 AndreasMoll r-phi impact parameter for TrkSBelle_CPS2000_SUP10
PNGpng TrkSBelle_CPS2000_SUP10_ImpactParameterFitD0.png r1 manage 37.3 K 2009-03-25 - 13:58 AndreasMoll impact parameter resolution d0 for TrkSBelle_CPS2000_SUP10
PNGpng TrkSBelle_CPS2000_SUP10_ImpactParameterFitZ0.png r1 manage 37.2 K 2009-03-25 - 13:58 AndreasMoll impact parameter resolution z0 for TrkSBelle_CPS2000_SUP10
PNGpng TrkSBelle_CPS2000_SUP10_ImpactParameterZ0.png r1 manage 41.3 K 2009-03-25 - 13:57 AndreasMoll z impact parameter for TrkSBelle_CPS2000_SUP10
PNGpng TrkSBelle_CPS800_SUP10_ImpactParameterD0.png r1 manage 40.8 K 2009-03-25 - 13:46 AndreasMoll r-phi impact parameter for TrkSBelle_CPS800_SUP10
PNGpng TrkSBelle_CPS800_SUP10_ImpactParameterFitD0.png r1 manage 37.6 K 2009-03-25 - 13:47 AndreasMoll mpact parameter resolution d0 for TrkSBelle_CPS800_SUP10
PNGpng TrkSBelle_CPS800_SUP10_ImpactParameterFitZ0.png r1 manage 37.0 K 2009-03-25 - 13:47 AndreasMoll impact parameter resolution z0 for TrkSBelle_CPS800_SUP10
PNGpng TrkSBelle_CPS800_SUP10_ImpactParameterZ0.png r1 manage 40.3 K 2009-03-25 - 13:47 AndreasMoll z impact parameter for TrkSBelle_CPS800_SUP10
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Topic revision: r5 - 2009-03-27 - AndreasMoll
 
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