1 ALICE

Track reconstruction in ALICE [1, 2] starts with cluster finding in the two principal tracking detectors, i.e., in the pad rows of the TPC and in the silicon sensors of the Inner Tracking System (ITS, Sect. 1.6.1.1) . The clusters reconstructed in the two innermost pixel layers of the ITS are used for a preliminary reconstruction of the primary vertex . Track reconstruction takes place through three main passes.

The first pass starts out in the outer part of the TPC and proceeds inwards through the ITS to the vertex region. Track seeds for primary particles are formed from pairs of measurements in the outer pad rows of the TPC and the primary vertex. Beginning from the seeds, track finding proceeds in the inward direction using a combinatorial Kalman filter (see Sect. 5.1.7).

Each reconstructed TPC track is extrapolated to the outermost layers of the ITS and used as a seed for track finding in the ITS. Track following in the ITS uses the particle hypothesis computed from the energy loss in the TPC, if available. Seven hypotheses are considered: electrons, muons, pions, kaons, protons, deuterons and tritons. If the dE∕dx information is missing or inconclusive, the pion mass is used. An illustration of how a particle hypothesis can be made using a combination of dE∕dx information (from the TPC measurements) and momentum information (from track reconstruction) is shown in Fig. 10.1.

Fig. 10.1
figure 1

Ionization energy loss as a function of momentum for a set of particles in the ALICE experiment. (From https://arxiv.org/abs/1701.04810. ⒸCERN for the benefit of the ALICE Collaboration. Reproduced under License CC-BY-4.0)

Full size image

A difficult point in the first-pass inwards tracking is the transition between the TPC and the ITS, due to the quite long (approximately 0.5 m) propagation distance between the two tracking systems. Given the uncertainty of the position of the track candidate propagated from the TPC and the high density of clusters in the ITS, many measurements are potentially compatible with the extrapolation. In order to lower the combinatorial complexity, the position coordinates of ITS hits used in finding primary tracks are augmented by the two angles describing the direction from the measurement to the primary vertex, effectively making the hits four-dimensional. When all layers of the ITS are traversed, the best surviving candidate (in terms of a quality measure using for instance the total χ 2 of the track and the number of assigned clusters) of each combinatorial tree is selected. Additional track candidates are then found by a stand-alone search in the ITS, using hits not attached to TPC tracks.

During the outward propagation in the second-pass through ITS, TPC and transition radiation detector (TRD), the track length and seven time-of-flight hypotheses per track (corresponding to the seven particles mentioned above) are calculated. This information is subsequently used for particle identification in the time-of-flight detector (TOF). Whenever possible, tracks are matched with hits in the TOF and other detectors outside the TRD.

In the third pass towards the vertex region, measurements assigned to the surviving track candidates during the first pass are used in the refit. The primary vertex is again fitted, now using the full available information from the reconstructed tracks as well as the average position and spread of the beam-beam interaction region. For a review of the track reconstruction performance, see [2, 3].

The high-level trigger (HLT) of ALICE also performs GPU-accelerated track finding and fitting in the TPC [4]. A cellular automaton finds track seeds, which are then extended by a simplified Kalman filter. After track segments are merged to the final track candidates, a full Kalman filter track fit is performed. A detailed description of all stages of the HLT can be found in [5]. The current HLT tracking is the base for the future TPC tracking in Run 3, which will run in GPUs as well [6].

2 ATLAS

The main track reconstruction strategy in the ATLAS experiment (see Sect. 1.6.1.2) is to start out with track seeds in the innermost part of the Inner Detector and proceed outwards [7,8,9]. First clusters are assembled in the pixel and SCT detector sub-systems by a connected component analysis (CCA) [10], and from these clusters 3D measurements (so-called space-points) are created. In dense environments such as the core of high-energy jets adjacent clusters from neighbouring particles can be so close that they overlap. Identifying such merged clusters correctly during the track reconstruction procedure is very important, and ATLAS has developed a method using a multi-layer, feed-forward neural network in order to solve this task efficiently [11].

Track seeds are generated from sets of three space-points. Seeds are processed iteratively by first considering sets of space-points from the SCT detector, then sets from the pixel detector, and finally sets originating from a combination of the two sub-detectors. Starting out from the seeds, track finding is carried out by a combinatorial Kalman filter (see Sect. 5.1.7 ). Each seed can potentially give rise to a number of track candidates, as long as the candidates pass a set of quality criteria. Efforts to speed up the track reconstruction are ongoing at the time of writing; first results based on improving the purity of the seed collection are given in [12]. More recent results are described in [13].

After iterating over all the seeds, the set of all track candidates are processed by an ambiguity solver. A flow diagram of the ATLAS ambiguity solver is shown in Fig. 10.2.

Fig. 10.2
figure 2

Flow diagram of the ATLAS ambiguity solver. (From [9], reproduced under License CC-BY-4.0)

Full size image

The track candidates are first ranked by a track score given to each candidate. The track score definition is intended to give high scores to candidates with a high probability of being a real track and therefore depends on, e.g., cluster resolutions, the number of holes on the track and the track χ 2. Track candidates passing a set of basic quality criteria are submitted to a full, high-resolution track fit using all available information relevant for an optimal estimation of the track parameters. After the track fit, track candidates can either be accepted, rejected or stripped down if the candidate contains too many clusters shared by other track candidates. The stripped-down candidates are scored again and re-submitted to the entire procedure of ambiguity resolving. It can be noted that the full track fit is invoked after the track scoring stage in order to save computational load.

The set of output track segments from the pixel and SCT detectors are submitted to the TRT track extension, which extrapolates the track segments through the TRT and searches for compatible segments in this drift tube detector. The search and subsequent resolution of drift-time ambiguities are done using a line fit in coordinate projections making the track model approximately linear [7]. At high levels of pile-up the occupancy of the TRT is so high that including drift-time hits not always increases track reconstruction performance. However, tube hits (i.e., using only the position of the centre straw) in all cases contribute to electron identification through information about transition radiation.

In order to find tracks originating, e.g., from secondary vertices or photon conversions, a secondary track reconstruction strategy starts out by finding track segments in the TRT using a Hough transform [7]. The TRT segments are then back-tracked into the SCT, which allows finding small track segments in the silicon detector that were not found during the first inside-out pass.

Electron reconstruction requires special attention [14] due to the potentially large amounts of emitted bremsstrahlung. In ATLAS, there is no separate iteration for electron track reconstruction. However, specific handling of bremsstrahlung is triggered by electromagnetic showers during the entire track reconstruction chain. The combinatorial Kalman filter allows for kinks in the track candidates if they are inside a region compatible with an electromagnetic shower. The ambiguity solver uses a global LS fit which allows for bremsstrahlung breakpoints in the track model. During electron identification, a full Gaussian-sum filter is invoked.

The primary vertex reconstruction in ATLAS is divided into the two classical categories vertex finding and vertex fitting [15]. The input is the set of reconstructed tracks in an event selected according to a set of quality criteria. This set is first used to obtain a seed position for the primary vertex . The seed position in the bending plane is the center of the beam spot, whereas the longitudinal coordinate is defined as the mode of the longitudinal position of the tracks at their respective points of closest approach to the beam spot. Subsequently, the set of tracks and the seed position is submitted to an iterative vertex finding procedure using the adaptive vertex fit (AVF) with annealing (Sect. 8.2.2). A typical distribution of track weights for a set of values of the temperature parameter T is shown in Fig. 10.3.

Fig. 10.3
figure 3

Histogram of track weights in the adaptive vertex fit for a set of different temperatures. (From [15], reproduced under License CC-BY-4.0)

Full size image

After the final iteration, the tracks with weights so small that they can be considered incompatible with the reconstructed vertex are removed from the vertex candidate and returned to the pool of unused tracks. The unused tracks are then submitted to a new iteration of the vertex finding procedure, which continues until all tracks have been used or no additional vertex can be found among the remaining tracks. For secondary vertex finding inside jets, see [16]. Stand-alone vertex reconstruction in the muon spectrometer of ATLAS is described in [17].

3 CMS

For a brief description of the CMS tracking system, see Sect. 1.6.1.3. Although the pixel detector and the silicon strip detector are mechanically independent sub-detectors with different sensor technology, they are considered as a single tracking detector as far as track finding and track fitting are concerned.

The CMS track reconstruction is based on an iterative approach [18,19,20,21]. The principal difference between iterations is the configuration of the seed generation and the target tracks. The iterative search starts with the tracks that are the least difficult to find, and proceeds to more difficult classes: low momentum tracks, tracks from short-lived decays, and tracks from long-lived decays. In each iteration, hits associated to high-quality tracks are masked, reducing the combinatorial overhead for the following iteration. Finally, special iterations that improve track reconstruction in high-density environments such as jets are executed, followed by iterations targeting muon tracks in combination with the muon chambers. For a recent comprehensive overview of the tracking performance, see [22].

Each iteration consists of four steps:

  1. 1.

    Seed generation. Initial track segments, called seeds, are found by a combinatorial search or by a cellular automaton [23]; see also Sect. 5.1.5 . Using the information of three or four hits pixel and/or strips, the trajectory parameters and the corresponding uncertainties of the seed are computed .

  2. 2.

    Track finding. Each seed is used as the starting segment of the combinatorial Kalman filter (Sect. 5.1.7) . At most one missing hit is allowed in a track candidate.

  3. 3.

    Track fitting. A Kalman filter or, for electron candidates, a Gaussian-sum filter/smoother (Sect. 6.2.3) is performed to obtain the final estimate of the track parameters at the interaction point exploiting the full trajectory information. The Kalman filter is also available in a parallelized version [24,25,26].

  4. 4.

    Track classification. Tracks are divided into classes according to different track quality criteria; see also Sect. 6.4.

The twelve tracking iterations used after the pixel upgrade to four layers are listed in Table 10.1, showing the seed type and the targeted tracks [20]. Iteration 9 is special insofar as it improves track reconstruction in jets and hadronic τ lepton decays by re-clustering pixel hits, using the jet direction to predict the expected cluster shape and charge [27, 28]. Iterations 10 and 11 reconstruct global muon tracks. For the combined electron reconstruction with tracker and electromagnetic calorimeter, see [29] and Sect. 9.4.

Table 10.1 List of the tracking iterations in CMS used after the Pixel Tracker upgrade with the corresponding seeding configuration used and target tracks [20]
Full size table

Primary vertex reconstruction starts with selecting tracks that are produced promptly by setting a threshold on their transverse impact parameter [18]. The selected tracks are then clustered on the basis of their z-coordinates at their point of closest approach to the center of the beam spot, allowing for an arbitrary number of clusters. The algorithm of choice is clustering by deterministic annealing (Sect. 7.2.4) . Vertex candidates with at least two tracks are fitted by an adaptive vertex fitter (Sect. 8.2.2). Finally the signal vertex is determined as the primary vertex with the highest weighted sum of the squared momenta of the jets and isolated tracks associated to the vertex .

Iterations 4 and 5 of the track finding target secondary tracks from short-lived decays, for instance B hadrons. Other candidate tracks for inclusion in a secondary vertex are tracks rejected by the adaptive vertex fit of the signal vertex. Finding secondary vertices is then basically a combinatorial search, followed by a vertex fit and possibly a kinematic fit for each candidate.

Tracks from long-lived decays, for instance K-short mesons and Λ baryons, are the targets of iterations 6–8, each for a different range of the radial distance r of the decay from the beam line. Reconstruction of these decay vertices is again a combinatorial search followed by a vertex fit and a kinematic fit.

CMS also has an independent reconstruction of tracks and primary vertices based purely on pixel hits. This is significantly faster than the standard track and vertex reconstruction chain and therefore a valuable tool for the high-level trigger [18, 30].

4 LHCb

Track reconstruction in the LHCb detector uses hits in the VELO, TT and T stations; see Sect. 1.6.1.4 and Fig. 10.4. Depending on which detectors are crossed by a particle, different track types are defined [31]:

  • VELO tracks have hits in the VELO. They are particularly important in the primary vertex reconstruction. They can be extended to upstream and long tracks during reconstruction.

    Fig. 10.4
    figure 4

    Schematic diagram of the LHCb tracking system and the five track types. (From [31], reproduced under License CC-BY-4.0)

    Full size image

  • Upstream tracks have hits in the VELO and TT stations. In general their momentum is too low to traverse the magnet and reach the T stations. If they do, they can be extended to long tracks.

  • Long tracks have hits in both the VELO and the T stations, and optionally in TT. They are the most important set of tracks for physics analyses.

  • Downstream tracks have hits in the TT and T stations. They are important for the reconstruction of long-lived particles that decay outside the VELO acceptance.

  • T tracks pass only through the T stations. Like the downstream tracks, they are useful for particle identification in the Ring Imaging Cherenkov detectors.

Track reconstruction is done in the two stages of the high-level trigger, HLT1 and HLT2 (Sect. 2.1.3). In fact, the full event reconstruction runs in the high-level trigger [31, 32].

The partial track reconstruction in HLT1 starts with finding straight lines in the VELO, which are fitted by a simplified Kalman filter. The fitted tracks are used for the reconstruction of the primary vertices . The primary vertex finding, see [33] and [34, Appendix A], proceeds in two steps: seeding and fitting . The 3D seeding algorithm starts with a base track and looks for tracks close to it. If at least four close tracks are found, a robust average of the points of closest approach of all track pairs is computed and stored as a seed. The tracks are marked as used and the next base track is processed. The fast seeder clusters tracks according to the z-coordinates of their points of closest approach to beam line, similar to the primary vertex finder of CMS (see Sect. 10.3).

Before fitting, the seeds are ordered in descending track multiplicity, so that primary tracks are fitted first, and the probability of incorrectly labeling a secondary vertex as primary is minimized. The vertex fit is a redescending M-estimator with Tukey’s biweight function, see Table 6.1 in Sect. 6.2.1. HLT1 proceeds with forming upstream tracks by extrapolating VELO tracks to the TT. High-momentum tracks are extended to the T stations. The resulting long tracks are fitted with a full Kalman filter and sent to the fake track rejection by a neural network [35].

HLT2 performs the full track reconstruction, in the sequence shown in Fig. 10.5. VELO tracks are extended to long tracks, without a threshold on the transverse momentum. Next, a stand-alone track finding is done in the T stations [36]. The found tracks are combined with VELO to long tracks. Tracks produced in the decays of long-lived particles outside the VELO are reconstructed from track segments in the T stations that are extrapolated backwards and combined with hits in the TT. The final steps are a full track fit with the Kalman filter, fake track rejection by a neural network, and removal of duplicates.

Fig. 10.5
figure 5

Sequence of the full track reconstruction in LHCb. (From [31], reproduced under License CC-BY-4.0)

Full size image