HCAL ECAL
4.1 Tracks
The goal of the CMS track reconstruction [66,78,92] is to collect all the signals (hits) due to a single charged particle and combine them in an estimate of the track param- eters and their incertainties including the impact parameter, charge, and momentum at the interaction vertex. The challenge of the track reconstruction is to do this with high efficiency and low fake rate, while meeting the physics goals of CMS [67, 73] at the per-bunch-crossing track multiplicity corresponding to the design luminosity and bunch intensity of the LHC [56].
Searches for heavy dilepton resonances require the reconstruction of hard tracks with transverse momenta of up to pT = 1 TeV. Simultaneously, hadron production rate measurements and optimal missing-transverse-energy and jet-energy resolutions require an efficient reconstruction of very soft tracks with transverse momenta of down topT ≈100 MeV [92]. Also, precise estimates of the interaction vertex locations and b-jet identification require excellent resolutions on the transverse and longitudinal impact parameters. Finally, the reconstruction of certain processes, like the 3-prong tau decay, require the ability to resolve very close tracks.
At the LHC design luminosity of L = 1034cm−2s−1, multiple inelastic pp col- lisions will produce on the order of 1000 charged particles at each bunch crossing every 25 ns. The CMS tracking must be able to process the bunch crossing events at sufficient rates at both the offline reconstruction and the HLT trigger, given the CPU resources available.
The dedicated piece of software used for track reconstruction by CMS is called the Combinatorial Track Finder (CTF) [93]. The CTF runs in an iterative fashion.
Early iterations search for easier-to-reconstruct tracks (relatively harder pT, originat- ing near the interaction region). After each iteration, the CTF removes hits assigned to the reconstructed tracks. This reduces the combinatorial complexity so that further
iterations can search for harder-to-reconstruct tracks (relatively softerpT, originating from displaced vertices).
The CTF starts by obtaining parameters for initial track candidates, seeds, using triplets of tracker hits near the beam pipe, or pairs of tracker hits near the beam pipe together with a third constraint from the known position of a vertex or the nominal beam spot [94].
The CTF then extrapolates the seeds outward along their expected trajectories of charged particles in a magnetic field. At each additional detector surface, it searches for hits consistent with the current track estimate employing a technique known as the K´alm´an filter [95–97]. This takes into account material effects of multiple scatter- ing and energy losses, as well as the current track parameters and their uncertainties, allowing for branching of individual tracks into multiple track hypotheses. The CTF assigns compatible hits to the current track, and updates accordingly the track pa- rameters and their uncertainties at the current detector layer. The search stops when it reaches the limit of the tracker fiducial volume, or the number of missing expected hits reaches a specified maximum, or the transverse momentum drops below a speci- fied minimum. Tracks with too few hits are then discarded.
Starting with the collection of the found hits excluding the seed hits, the search is then repeated in the opposite — inward — direction. This enables finding additional hits in the seeding layers and other layers closer to the interaction vertex.
This track finding algorithm may lead to duplicate track candidates resulting from a passage of a single charged particle, as well as incidental “fake” track candidates resulting from ambiguous hit configurations corresponding to no actual particles. To reduce these effects, the CTF discards tracks that share too many hits while having a low total number of hits or a high fit χ2 value. This concludes the track search, yielding a collection of track candidates and their associated hits.
In the next step, the CTF improves the estimates of the track parameters by a K´alm´an filter and smoother. It refits the tracks using all available hits. Compared to the track finding, this avoids possible biases from the hit selection during the seeding stage. Similarly to the track finding, the track fitting step accounts for material
effects, namely multiple scattering and energy losses. However, in contrast to the track finding, it uses aRunge-Kutta propagator [98] to account for the inhomogeneity of the magnetic field by numerically solving the equations of motion.
Due to the sequential nature of K´alm´an’s algorithm, the fit is performed twice.
First, it propagates the track from the inside out (the K´alm´an filter), then from the outside in (the K´alm´an smoother). Both fits yield track parameters at each detector layer using the information from the previous layers. Their average at each layer then optimally combines the information from all the available layers.
Finally, the CTF selects a subset of track candidates to reduce the fraction of
“fakes.” It uses the following quantities to define quality criteria that depend on the track pT and η:
• the number of layers with at least one associated hit,
• the number of layers with at least one associated “3D” hit constraining all three spatial coordinates, i.e. hits in the pixel tracker or “matched hits” in the strip tracker,
• the number of layers with missing expected hits surrounded from both sides by layers that do contain associated hits,
• the normalized χ2 of the track fit: the ratio χ2/ndof of the χ2 and the number of degrees of freedomndof of the fit,
• the transverse|d0|/δd0 and longitudinal|z0|/δz0 impact parameter significances, and
• the alternative transverse |d0|/σd0 and longitudinal |z0|/σz0 impact parameter significances,
where d0 and z0 are the transverse and longitudinal impact parameters, δd0 and δz0
their uncertainties, and σd0 and σz0 their alternative uncertainties. For the purposes of the track selection, d0 is defined as the distance from the beam spot in the plane
transverse to the beam line (x-y), and z0 as the distance along the beam line (z- axis) from the closest pixel primary vertex, if present, or the beam spot, otherwise.
δd0 and δz0 come from the covariance matrix of the track fit, while σd0 and σz0 are parametrized as functions of the trackpT. Tracks that fail the selection are discarded, and their associated hits are reused in the next iteration.
In each iteration, the CTF repeats the above four steps: the seed generation, the track search, the track fit, and the track selection. It performs six iteration passes in total. Individual CTF iterations differ mainly in the requirements on the seed- generation hits, and the criteria on the track selection, as well as the collection of available hits that contains only those that have not been associated with any tracks yet.
In addition to the general CMS tracking, the same algorithm with different con- figurations is used to produce other specialized track collections. These include the tracks reconstructed at the High Level Trigger (HLT), electron tracks and tracks in the muon systems. The HLT configuration emphasizes speed over precision. For example, it stops the track finding sooner, when enough hits have been found. We discuss the specialized muon and electron tracking below in the respective Sections 4.7 and 4.6.