from (0.0,1.0) on a circle with the circle’s center in (0.0,0.0). The particle’s path is indicated by the red, solid line. The Kalman filter was provided an initial guess (−0.2,0.8) with an uncorrelated uncertainty of 0.3 in each xand y, shown as a pale blue ellipse in the background.
As the particle moves in (x, y) space, it is measured at finite time steps. The true position of the particle at the different time steps is indicated with a black diamond; this however is hidden to the Kalman filter and only the measurement (indicated by a red diamond) is provided to the algorithm. The chosen uncertainty on the measurement is an uncorrelated, Gaussian one with a standard deviation of 0.05 in both dimensions, visualized by the red, transparent ellipses. The predictions of the Kalman filter for the measurements are shown with the dark blue ellipses.
The error ellipses represent the full covariance matrix: the axes of the ellipses point into the direction of the eigenvectors of the covariance matrix; the length of the axes in the predictions correspond to the square root of the eigenvalues of the matrix. As expected from the chosen, uncorrelated measurement precisions, the uncertainties in this example are independent down to the given, limited numerical precision. Although being simple in its equations, the filter is very powerful through the underlying mathematics. We see that its prediction nicely converges towards the true position of the particle with appropriate uncertainty estimations as more and more dynamic measurements are provided. We also see the bias towards the provided guess in the first predictions. This illustrates, that a suitable initial seed is crucial for a successful and effective running of the Kalman tracking program. Kalman filters are very popular and widely used. Not only was Kalman awarded the 2008 Charles Stark Draper Prize for their development [264]; they are also, e. g., one of the tools for interpreting macroeconomic data that the 2011 Nobel Prize in economics was awarded for [265]. Using economists’ language, i. e.shock formeasurement, Kalman filtering is discussed at length in the 2011 announcement ceremony [266–268].
In ALICE, the central barrel tracking starts at the outer radius of the TPC, where the track density is low. As we have seen in the above paragraph, appropriate seeding is important in case a Kalman filter is invoked for tracking. Consequently, a thorough analysis of the best seeding practice was carried out with the result of 36 consecutive seedings. The seeding is discussed in Section D.1.
The Kalman filter propagates the information given by the seeds inwards in radius from the TPC via the ITS to the primary interaction vertex. Bridging the gap between two detectors is one of the strengths of a Kalman filter, as its estimate is always local. A global fit might describe well the average momentum of a particle in, e. g., the TPC. Yet what is important when extrapolating the track is the momentum at the edge of the detector. The tracker is divided into detector specific software modules, where, e. g., the relevant geometries are implemented. Each package is responsible for the track prolongation within the volume of the detector. Additional detector hits, also from the ITS at small radii, are associated to the track if they happen to lie on a road, i. e., within a certain distance of the expected position.
4.2 Tracking 71
Figure 4.2: Simple Kalman filter. A particle moves counter-clockwise on a circle (red line). At finite time steps (the particle’s true position is indicated by black diamonds), its position is measured (red diamonds) with a finite precision (red transparent ellipses).
The Kalman filter’s guess (blue ellipses) is subsequently updated with the measurements.
For details see text.
After the inward propagation of the TPC seeds is concluded, a local mode in the ITS, the ITS standalone tracking, is started. The acceptance down to lowpT of the TPC is limited by its geometrical extent and the strength of the magnetic field. WithB = 0.5 T and the minimal radius of the TPC of ρ= 0.85 m the softest reconstructed primary particles have apT of
TPC pT,min ≈0.3·B·ρ GeV/c
T m ≈130 MeV/c. (4.1)
Putting into this equation parameters for the ITS, i. e.ρ= 7.6 cm at the second SPD layer, the benefit of such a ITS standalone mode becomes clear:
ITS pT,min≈10 MeV/c. (4.2)
At such low momenta, the energy loss of the particles is significant and distorts the tracks, such that the position of the reconstructed point at a given layer deviates significantly from
the expected position without distortions. Since a limit on the window for associating clusters is mandatory to suppress fake tracks, the efficiency of the tracking algorithm drops below 40% for momenta smaller than 100MeV/c [269] and lifts the limit for track reconstruction to pT&90 MeV/c[270]. The ITS standalone method not only provides additional acceptance at low pT but also fills gaps inϕat the TPC sector boundaries.
In a second tracking step, the direction of propagation is reversed and extended to the outer central barrel detectors covering full azimuth, i. e. TRD and TOF, and also those with partial coverage in ϕ, i. e. EMCAL, HMPID, and PHOS. After propagating the particles through the TRD, tracks are prepared for matching them with hits in the TOF detector. To avoid fake combinations of a track and a TOF hit, each TOF cluster is only assigned once.
Since the precision on the knowledge of the particle’s spatial position is increased for straight high-momentum tracks, they are sorted according to their transverse momentum such that the association to TOF clusters starts with high-pT tracks. In a first matching round, all tracks are extrapolated through TOF and are linked to a time signal if the read-out pad they traverse fired. In a second round, tracks are assigned the closest TOF pulse within a radius of 3 cm.5 Fig. 4.3 shows the fraction of mismatched TOF hits for protons as obtained with a simulation [271]. It can be seen that the contamination from tracks which did not create a TOF signal reaches a quite high level of more than 80% atp.0.4 GeV/c. This impurity drops to a moderate 25% forp >0.75 GeV/c and further decreases for higher momenta. Additionally, the constant effort by the collaboration to improve the data quality resulted in a much better situation regarding the mismatch for the Pb-Pb data taken during Run 1: the mismatch could be reduced to around 6.5% for tracks with p >1 GeV/c[216]. We will see in Section 5.2 how even this lower mismatch can be further supressed by applying surprisingly simple selection criteria.
A local estimate of the track parameters as provided by the Kalman filter becomes critical when physics processes like multiple scattering happen. Such an energy loss correlates the measurements, which in a global fit approach would lead to matrices as large asn×n, where n is the number of measurements. With up to 159 clusters in the ALICE TPC alone, this becomes computationally too expensive. Based on the local estimate, the Kalman filter also allows to reject a cluster that was previously associated to the track via the method of roads.
The inward fit where the track parameters are finally determined concludes the tracking in the third step. Local snapshots of the track parameters are stored in the Event Summary Data (ESD) track class. This comes in handy when the energy loss in the time projection chamber is exploited for PID for which the local velocity of the particle in the TPC is the determining factor. This velocity certainly differs from the speed at the primary vertex as a significant amount of material in which the particle will loose energy was placed in between these two
5The value is 3 cm for the Pb-Pb data reconstruction only. In the low multiplicity environment of pp collisions, a looser restriction of 6 cm can be applied.