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


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


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 were done for the parameters d0 and z0 using the software tools MOKKA and Marlin.

Muon gun

  • phi is isotropic
  • theta and energy are as follows:

Angle 0.1 [GeV] 0.2 [GeV] 0.4 [GeV] 0.6 [GeV] 0.8 [GeV] 1.0 [GeV] 2.0 [GeV]
40° done done done done done done done
60° done done done done done done done
80° done done done done done done done

Detector Models are summarized here: ImpactParameter Resolution

Results for SuperBelle and SuperBelle Upgrade

-- AndreasMoll - 20 Mar 2009

Topic attachments
I Attachment History Action Size Date Who Comment
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
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Topic revision: r2 - 2009-03-20 - AndreasMoll
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