Physics:Errors in track reconstruction

The trajectory of a particle in a magnetic field is determined by five initial values, e.g.

( Trajectory of a Charged Particle). Therefore at least five measurements are necessary to reconstruct a track. In the absence of a magnetic field, four measurements are sufficient; the momentum remains unknown. Many methods are applicable to get an estimate of the track parameters; the most common method is the least squares method. In the case where the impact points of a particle on the detector surfaces can, in the neighbourhood of a given track, be approximated by a linear function of the initial values, the estimator found by the least squares method is the best possible linear estimator. If the measurement errors are Gaussian, the least squares estimator is also efficient (viz. has minimum variance). For most cases, the errors in track reconstruction as given by the theory of least squares estimation are quite representative for the achievable precision.

For simplification, let us consider two types of spectrometers:

• a) a central spectrometer magnet with several detector arms: this setup is typical for high-energy fixed target experiments;
• b) a set of equidistant detectors, all inside a magnetic field: this is typical for detectors in colliding beam experiments.

Ad a): Assuming a constant bending power of the magnet, the transverse momentum kick given to a particle in the magnet can be approximated by ( Lorentz Force)

where p_T is the transverse momentum [GeV/c], e is the elementary charge [e = 0.2998 GeV/cT^-1m^-1], B the magnetic field [T], and L the length along the track [m]. For a particle with charge +1 and a magnet with  Tm, p_T =0.2998 GeV/c and the deflection angle is given by

With m available detectors and a symmetric spectrometer of length L, the theoretically best angular resolution is obtained by placing m/4 detectors at each end of the arm and m/2 detectors at the centre L/2. The obtainable resolution is then

where is the error in an individual detector. This configuration of detectors, whilst optimizing precision, is a particularly unsuitable arrangement for finding the correct association of measurements, and is therefore not used in experiments with non-trivial track recognition problems.

Ad b): In central spectrometers all detectors are usually assumed to be inside a homogeneous magnetic field. This case is extensively discussed in Gluckstern63. The error of the reconstructed momentum in any projection is inversely proportional to the field in this projection and to the square of the projected track length . Assuming the measured points to be equidistant and ,

m is the number of measurements and the error of a single measurement in this projection. For a fixed m and a given precision the spectrometer must grow in size with . If m grows linearly with , is asymptotically proportional to . Note that , where L is the track length in space and the projection angle.

If half of the measurements are assumed to be in the centre of the track, and one fourth each at the ends, the momentum error is substantially improved to

This is, again, a hypothetical arrangement of detectors, as it is unsuitable for recognizing tracks and also difficult to install. Note that even this formula gives a precision worse than the lever arm spectrometer by a factor of 2.

At high energies, the momenta measured in magnetic fields by position detectors will have large errors, and calorimetric measurements are preferred, particularly for electrons.

At low energy, a precision limit is set by multiple scattering and the optimization becomes definitely more complicated, as it will depend on the distribution of the scattering material (continuous or discrete) and on the momentum spectrum of the particles. Whereas the relative momentum error from position measurement errors is given by the proportionality

the corresponding term from multiple scattering comes out to be

where a uniform spacing of a constant number of measured points is assumed. For the full mathematical treatment of multiple scattering, see Gluckstern63 and Pentia96 and references therein.

If the direction of a straight track (no magnetic field) is calculated frommeasurements c₁ for transverse coordinates y₁ in detectors positioned at longitudinal coordinates x₁, the least squares fit to the equation y₁=ax₁+b gives

The errors follow from error propagation, using the relation between a, b and the measurements c_i.

More details, and consideration of more complicated setups, including vertex chambers, can be found in Blum93.