1 Belle II

The Belle II experiment [1] started operation in late 2018. The following overview of the reconstruction methods reflects the status after a couple of months of running; further developments and adaptations that have been implemented in parallel to the rising luminosity of the SuperKEKB collider are documented in [2].

The tracking system of Belle II has two principal components, the vertex detector (VXD) and the central drift chamber (CDC); see Sect. 1.6.2.1 . The VXD consists of two parts: the PXD with two layers of DEPFET pixel sensors, and the SVD with four layers of double-sided silicon strip sensors. The overall design of the track reconstruction is shown in Fig. 11.1.

Fig. 11.1
figure 1

Scheme of the track reconstruction in Belle II. (Courtesy of B. Scavino, Mainz)

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Track finding in the CDC starts with a filter, implemented as a boosted decision tree, that rejects background hits. This is followed by two track finders. The global track finder employs a Legendre transform (Sect. 5.1.4), to find complete tracks originating from the interaction region. Peak finding in the Legendre space is done by a fast iterative quadtree algorithm [3]. The local track finder builds segments from the hits in a superlayer. This is followed by a multi-stage combination and filter algorithm that connects valid segments and tracks, rejects fake segments, and outputs the final CDC track sample [4].

Stand-alone track finding in the SVD [5, 6] is based on a cellular automaton (CA) complemented by a sector map (Sect. 5.1.5) . The sector map stores prior information about which triplets of hits can be part of a track. The vertex position can be used as a virtual hit. Track segments that pass all cuts form the cells of the CA. Figure 11.2 shows the final state of the CA with a toy event, without background.

Fig. 11.2
figure 2

Final state of the CA with a toy event in the VXD. The colors of the cells (track segments) represent their states. (Courtesy of J. Lettenbichler, Vienna)

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When the CA has finished, each track candidate is assigned a quality index, which is the p-value of a triplet fit (Sect. 6.1.5) , a circle fit, or a Riemann helix fit (Sect. 6.3). Finally, non-overlapping candidates are selected, either with a greedy algorithm or a Hopfield network (Sect. 5.3). At the time of writing, the triplet fit and the greedy algorithm are the defaults.

The merging of CDC and VXD tracks is based on a boosted decision tree that is trained on valid combinations of CDC and VXD tracks. It takes the track parameters as determined by a Deterministic Annealing Filter (DAF, Sect. 6.2.2 ) as input and returns a score in the interval (0, 1). Combinations above a threshold are accepted. CDC tracks, SVD tracks and combined tracks are extrapolated into the PXD to pick up hits by a combinatorial Kalman filter [7] , and then passed to the DAF. The implementation of the DAF is the one in GENFIT [8,9,10].

The B mesons produced by the decay of the Υ(4S) resonance are reconstructed hierarchically [11], and their decay chains are fitted by a tree fitter based on [12]. V0 decays are reconstructed by pairing positive and negative tracks and performing a vertex fit [13].

2 CBM

CBM is a heavy-ion experiment at FAIR [14], currently in the preparation phase. The first beam is expected in 2024. The interaction rate will be up to 10 MHz, with up to 1000 charged particles produced in a central collision. Full online event reconstruction and selection will be required [15]. This is the task of the First Level Event Selection Package (FLES, [16, 17]). The algorithms in FLES exploit vector instructions for intra-processor parallelism and parallelism between cores on the event level.

The FLES package consists of several modules: track finding, track fitting, short-lived particles finding, event building and event selection [18], see Fig. 11.3. CBM runs without a trigger; time-stamped data will be put into a buffer in time slices of a certain length instead. The association of tracker hits with events is performed by software [18], using the hit time information in the Silicon Tracking System (STS, see Sect. 1.6.2.2). The resolution of the hit time is 5 ns.

Fig. 11.3
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Flow diagram of the First Level Event Selection Package in CBM. (From [17], reproduced under License CC-BY-4.0)

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Track finding in the STS (Sect. 1.6.2.2 ) is done by a 4D Cellular Automaton, where the fourth dimension is time [19, 20] . The cells of the CA are triplets of hits correlating in space and in time. The principle of the CA track finder is illustrated in Fig. 5.5 in Sect. 5.1.5. For track finding efficiencies, see [18].

The track candidates from the CA are sent to the Kalman Filter Track Fitter . The Kalman Filter is “SIMDized”, i.e., uses the Single-Instruction Multiple-Data features of the processors employed [21]. The extrapolation in the inhomogeneous magnetic field uses the approximate analytical method described in Sect. 4.3.2.2.

After the primary vertex fit with the Kalman Filter, tracks are sorted into primary and secondary tracks. The reconstruction of short-lived particles is done by the KFParticle package [22], also based on the Kalman Filter and designed to fit decay chains. The reconstruction of a short-lived particle is implemented iteratively. The KFParticle package starts with an initial approximation of the secondary vertex, adds particles one after another, refines the state vector and gives the optimal values after the last particle [16, 17]. Besides the geometrical constraints, mass constraints and topological constraints can be imposed on the secondary vertex.

Track reconstruction in the three stations of the transition radiation detector is done by a cellular automaton as well [23].