1 Introduction

Many high-energy physics experiments are performed by colliding two beams of high energy particles or one beam with a target. The particles produced in a collision are recorded by particle detectors. The collision is then studied by reconstructing most of the particles produced in the interaction and determining their properties. For a general review of particle detectors, see for instance [1, Chap. 7] and [2].

The present book concentrates on the reconstruction of charged tracks and interaction vertices using information collected by tracking detectors. A tracking detector can be a single device such as a wire chamber or a silicon strip sensor, or a full-blown tracking system such as a time projection chamber capable of stand-alone track reconstruction; see Sect. 1.6. For a review of non-tracking detectors, the reader is referred to [1, 3].

A charged particle crossing a tracking detector generates a single or a string of spatial observations in the local coordinate system of the detector. For track reconstruction, these have to be transformed to points in 3D space usually with different precisions in the three coordinates. Accordingly, the correct transformations are determined by the alignment procedure; see Sect. 1.5. The resulting space points or “hits” are collected into track candidates by the track finding. In the subsequent track fit, the track parameters are estimated and the track hypothesis is tested. Successfully reconstructed tracks are then clustered into production or decay vertices, followed by a vertex fit and a test of the vertex hypothesis.

There are three principal types of tracking detectors: gas-filled or gaseous tracking detectors; solid state tracking detectors, usually equipped with silicon sensors; and scintillating fiber trackers. They are described in the following three sections.

2 Gaseous Tracking Detectors

Gaseous tracking detectors utilize the ionizing effect of charged particles in a volume of gas. The simplest gaseous detector is the Geiger-Müller counter, a tube with a central wire. A potential difference of 1–3 kV between the tube wall and the wire causes the primary electrons and ions to move towards the anode (wire) and the cathode (tube wall), respectively. Because of the large field strength close to the wire, the primary electrons generate an avalanche of secondary electrons and ions, resulting in a detectable signal fed into an amplifier.

In order to use this principle for the measurement of the position of a charged particle, a structured array of many elements has to be designed. The following subsections describe a few common types of such arrays.

2.1 Multi-wire Proportional Chamber

The multi-wire proportional chamber (MWPC, [1, Sect. 7.1]) is a thin cuboid volume of gas, oriented approximately perpendicular to the passage of the particles to be measured. The volume is bounded by a pair of conductive plates acting as the cathode. Inside the volume an array of anode wires detects the passage of charged particles, see Fig. 1.1. With a typical wire spacing of 1 mm a spatial resolution of around 0.3 mm can be achieved in the v-coordinate orthogonal to the wires. A further improvement can be obtained by tilting the MWPC so that the probability of a particle giving signals on two adjacent wires gets larger or by rotating the chamber and measuring the drift times to the anode wires and estimating the point of passage from the drift distances [4, 5]. The resolution in the w-coordinate along the wire is poor, equal to the wire length divided by \(\sqrt {12}\approx 3.46\).

Fig. 1.1
figure 1

Local coordinate system in a wire chamber, in a planar drift chamber, or in a silicon strip sensor

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2.2 Planar Drift Chamber

A planar drift chamber is similar in shape to an MWPC (see Fig. 1.1), but the electrical field is shaped by an alternating array of sense (anode) and field (cathode) wires; see [1, Sect. 7.2] and [6]. The electrons and ions from the primary ionization drift to the respective electrodes, and gas amplification in the strong field close to the sense wires gives a detectable signal that is amplified. In addition, the drift time between the crossing time of the particle and the signal time on the sense wire is measured. The drift distance can then be computed from the drift time. Given a precise calibration of the drift distance, a spatial resolution below 0.1 mm in the coordinate v orthogonal to the wire can be achieved; see for instance [7].

Drift chambers need far fewer electronic channels than MWPCs, but have to be monitored and calibrated much more carefully. In addition, there is an inherent left-right ambiguity of the spatial position relative to the sense wire, so that each hit has a mirror hit. This ambiguity must be resolved in the track finding or track fitting stage. The spatial position w along the wire can be measured by comparing the charges at both ends of the wire. A resolution of a couple of millimeters can be achieved in this way [8].

2.3 Cylindrical Drift Chamber

Cylindrical drift chambers [1, Sects. 7.2 and 7.3] have been and still are widely used in collider experiments. Such a chamber consists of up to 60 cylindrical layers of alternating field and sense wires mostly parallel to the beams of the collider; see Fig. 1.2. In the local coordinate system of the chamber, the z-axis is the symmetry axis of the chamber. The measured point in the transverse plane can be given in polar coordinates (R m,Φ m,z m), where R m is the radial position of the sense wire, and Φ m is the polar angle of the wire plus/minus the drift angle Φ d, i.e., the drift distance d divided by R m. The resolution of the drift distance is typically in the order of 0.1–0.2 mm; see for instance [9, 10]. Like in a planar drift chamber, each hit has a mirror hit, and the ambiguity must be resolved in the track finding or track fitting stage. The z-coordinate can be measured by charge division or by adding “stereo” layers of wires tilted with respect to the “axial” layers. The resulting spatial resolution of z m is equal to the drift distance resolution divided by the sine of the stereo angle, typically 2–3 mm.

Fig. 1.2
figure 2

Local coordinate system in a cylindrical drift chamber. Only one layer of sense wires is shown. R m is the radial distance of the sense wire from the z axis; Φ w is the azimuth angle of the sense wire. d is the drift distance; Φ d = d∕R m is the angle spanned by d at radius R m. The azimuth angles of the track hit and its mirror hit are Φ m,1 = Φ w − Φ d, Φ m,2 = Φ w + Φ d. The z-coordinate may or may not be measured directly

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2.4 Drift Tubes

Drift tubes are small drift chambers with a single sense wire. They can have rectangular or circular cross sections, and can be arranged in cylindrical or planar layers. In planar layers, wires can run in two or more directions, giving good spatial resolution in two orthogonal directions. The resolution of the drift distance is in the order of 0.1–0.2 mm.

2.5 Time Projection Chamber

The typical time projection chamber (TPC), as employed in many collider experiments, is a large gas-filled volume shaped as a hollow cylinder, the axis of which is aligned with the beams and the magnetic field [1, Sects. 7.3.3], see Fig. 1.3. There is a potential difference between the central cathode plane and the two anode end plates. The latter are equipped with position sensors. A charged particle traversing the chamber ionizes the gas, and the electrons travel along the field lines towards the end plates, where both the point and the time of arrival are measured and recorded. A track therefore generates a dense string of space points. The position sensors at the end plates can be wire chambers or micro-pattern gas detectors. In the drift direction z a spatial resolution in the order of 1 mm is possible; the transverse resolution depends on the technology of the endplate sensors and increases with the drift distance because of diffusion. Resolutions well below 0.1 mm can be achieved with GEM chambers on the end plates. Another important calibration issue is the correction of distortions arising from space charge effects, see for instance [11].

Fig. 1.3
figure 3

Local coordinate system in a time projection chamber. The position in the endplate can be given in polar coordinates (R m, Φ m) or in Cartesian coordinates (x m, y m). z m is the drift distance

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2.6 Micro-pattern Gas Detectors

The earliest micro-pattern gas detector is the micro-strip gas detector, in which wires are replaced by microscopic metal strip structures deposited on high-resistivity substrates [1, Sect. 7.4]. With typical strip distances of 75 µm, a spatial resolution below 20 µm can be achieved. Other developments are the Gas Electron Multiplier (GEM) and Micro-Mesh Gaseous Structure (Micromegas) chambers. Resolutions down to 10 µm can be attained by these devices. Due to their small size and fast collection of positive ions, they can be operated at high rates up to several MHz per mm2.

3 Semiconductor Tracking Detectors

Semiconductor tracking detectors [1, Sect. 7.5] are mostly made of thin silicon wafers, approximately 0.3 mm thick. The n-type silicon is processed by photo-lithographic methods to create p +-doped implants on one side, either thin strips (see Sect. 1.3.1) or small pixels (see Sect. 1.3.2). Each strip or pixel is connected to a read-out channel. The silicon bulk is fully depleted by a bias voltage. A charged particle crossing the wafer creates electron-hole pairs along its path. The electrons and the holes drift towards the electrodes and induce signals on the read-out electrodes. Silicon drift detectors employ the measurement of the electron drift time for measuring the position of a crossing particle; see Sect. 1.3.3.

3.1 Silicon Strip Sensors

Large-area semiconductor tracking systems employ silicon strip sensors to keep construction costs affordable. The implants in the silicon wafers are narrow strips with a typical width of 20 µm and a typical inter-strip distance of 100 µm. The local coordinate system is shown in Fig. 1.1. The spatial resolution in v, orthogonal to the strip direction, depends on the track direction via the cluster size, i.e., the number of adjacent strips with a signal above threshold, and on the Lorentz angle [12]. Under optimal conditions the resolution is in the order of 10 µm or better. The resolution in w, parallel to the strip direction, is equal to the strip length divided by \(\sqrt {12}\). A typical strip length is 5 cm; shorter strips with a length below a centimeter are called mini-strips. For an example of the calibration procedure, see [13].

Two-dimensional (2D) measurements can be achieved by implanting strips on the back side of the wafer, either orthogonal to the ones on the front side, or at a different stereo angle. Alternatively, two one-sided sensors can be glued on the same mechanical support, separated by a small gap. If such a sensor happens to be crossed by n particles at the same time, up to n strips on each side can give a signal, resulting in up to n 2 points of intersection. At most n of these correspond to the true particle positions; the remaining ones are spurious or ghost hits.

3.2 Hybrid Pixel Sensors

In a pixel sensor , the implants are small square or rectangular pixels in a high-resistivity silicon wafer. The pixels are connected to the read-out channel by bump bonding. Depending on the pixel size and the track direction, pixel sensors can have a position resolution below 10 µm in both coordinates, especially if the signal height is measured and used to interpolate between pixels in a cluster. For an interesting example of the calibration procedure, see [14]. Here, the position is estimated from precomputed cluster templates, considering the incident angle and the Lorentz angle. The templates can also be used to decide whether an observed cluster is compatible with a predicted incident angle.

Pixel sensors are mostly employed in the region close to the interaction point, as they can deal with the high track density and the high background radiation better than strip detectors. In addition, their excellent spatial resolution allows the separation of secondary decay vertices very close to the primary interaction vertex; see Chap. 9.

In monolithic pixel sensors, the sensitive volume and part of or the full read-out circuits are combined in one piece of silicon. The generated charge is collected on a dedicated collection electrode so that there is no need for delicate and expensive bump bonding. For an application of monolithic pixel sensors in a vertex detector, see Sect. 1.6.1.1.

3.3 Silicon Drift Sensors

In a silicon drift sensor, electrons are transported parallel to the surfaces of the sensor to an anode segmented into small pads [15, 16]. The position information along the drift direction is obtained from a measurement of the drift time of the electrons. The position in the second coordinate is obtained from charge sharing between adjacent pads; see Fig. 1.4. Silicon drift sensors have been deployed in STAR and ALICE, see [16, 17].

Fig. 1.4
figure 4

Three-dimensional sketch of a multi-anode silicon drift detector. The trajectories of several signal electrons are indicated. The distance between the point where the ionizing particle crosses the middle plane of the detector and the array of anodes is obtained from measurement of electrons drift time. The second coordinate is given by the location of the anode pad where the signal electrons are collected. An improvement of resolution along this coordinate comes from charge sharing among several anodes. (From [16], by permission of Elsevier)

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4 Scintillating Fiber Trackers

Scintillating fiber trackers combine the speed and efficiency of plastic scintillators with the geometric flexibility and hermeticity provided by fiber technology [18]. The fibers in such a tracking detector serve two functions: they convert the ionisation energy deposited by a passing charged particle into optical photons, and guide the optical signal to the devices that detect the generated light. In recent applications, these devices are silicon photomultipliers, which are fast, compact and sensitive to single photons [19].

The spatial resolution of a fiber is approximately equal to the diameter divided by \(\sqrt {12}\), while the number of photons scales linearly with the diameter. The conflicting requirements on accuracy and light yield can be alleviated by staggering the fibers and by choosing a material with high intrinsic scintillation yield and long attenuation length. For an example of a large-scale scintillating fiber tracker designed for operation in the LHCb experiment from Run 3 of the LHC onward, see [20].

5 Alignment

The position measurements in the tracking detectors mentioned above are generated in the local coordinate system of the devices. In order to be useful for track reconstruction, they have to be converted to positions in the global coordinate system of the experiment along with the associated covariance matrices. As tracking detectors are very precise instruments, with position resolutions ranging from a couple of hundred micrometers down to about ten micrometers, their positions, orientations, and possible deformations have to be known with a similar or better precision. The importance of correct alignment, especially in the complex detectors of the LHC era, is attested by a series of workshops held at CERN in the past [21,22,23].

Misalignment or insufficient alignment has a deleterious effect on the efficiency of track and vertex reconstruction [21, p. 105]. Random misalignment also degrades the resolution of track and vertex parameters and subsequently of invariant masses. Moreover, systematic misalignment of larger substructures can cause a bias in the estimates of track momenta and vertex positions. This can be harmful in many of the physics analyses of the experiment.

Misalignment can have several sources: finite precision of the detector assembly, thermal and magnetic stresses on mechanical structures, sagging of wires or sensors because of gravity, changes in temperature and humidity, etc. Since misalignment can and does vary over time, constant monitoring is a necessity.

Alignment proceeds through several steps. The starting point is the ideal geometry, augmented by knowledge of the machining and assembly precision. The next step is alignment by hardware using lasers for measuring distances or proximity and tilt sensors. For instance, the ATLAS silicon tracker can be monitored optically by Frequency Scanning Interferometry to a precision of about 10 µm [24].

The final step is track-based alignment , either with tracks from cosmic muons, or from collisions, or both. Actually alignment profits from different types of tracks that cross different parts of the detector under different angles. For instance, tracks from collisions hardly ever cross the entire central tracker of a collider experiment, but cosmic tracks do.

Track-based alignment can be split into internal alignment and external alignment. Internal alignment refers to the relative alignment of a tracking system, whereas external alignment refers to the alignment of the various tracking systems to the global frame of the experiment, which is usually tied to the beam pipe or some other part of the accelerator infrastructure. Even the internal alignment of a tracking system can be a big challenge. For instance, the current silicon tracker of the CMS detector has more than 104 sensors to be aligned, each with six degrees of freedom, not counting deformations of the sensors under gravity or thermal stresses. The estimation of O(105) parameters is a highly non-trivial problem. A solution that has become a de-facto standard is the experiment-independent program MillepedeII [25, 26], which performs a simultaneous fit of (global) alignment parameters and (local) track parameters, allowing to include laser and survey data as well as equality constraints in the fit. For one of the alternative algorithms developed for track-based alignment, see [27].

6 Tracking Systems

Track reconstruction requires a minimal number of space points per track. A tracking system is a device that has enough information for stand-alone track reconstruction. A typical collider experiment has three tracking systems for momentum measurement: the vertex detector , the central tracker , and the muon tracking system . Fixed-target experiments frequently have vertex detectors as well, complemented by a magnetic spectrometer for momentum measurement.

The vertex detector is the tracking system closest to the beam, with the purpose to give very precise position and direction information of the tracks produced in a collision, so that decays very close to the interaction point can be detected with large efficiency; see Chap. 9. It therefore has the largest precision (smallest measurement errors) of all tracking systems. Vertex detectors are usually equipped with pixel sensors in order to achieve the required precision.

The central or inner tracker of a collider experiment is positioned between the vertex detector and the calorimeters. It is normally embedded in a solenoidal magnetic field with high bending power. A silicon tracker typically produces O(10) hits per track, while a TPC produces O(100) hits per track. The main requirements, not always easily satisfied, on the central tracker are: high single-hit precision; good capability to resolve two nearby tracks; precise momentum estimation by a long lever arm (large diameter); enough redundancy for high-quality track finding; hermetic coverage; as little material as possible. In some cases, especially at the future high-luminosity LHC (HL-LHC), fast readout is also essential, as trackers must be able to contribute to the trigger.

The muon system is situated behind the calorimeters which, in principle, absorb all particles with the exception of muons. Additional iron filters can be employed as well. The muon system can provide an independent measurement of the muon momentum, especially for the purpose of triggering; see Sect. 2.1. If the muon system has to cover a large area, as is the case in the LHC experiments, it is typically equipped with proportional chambers, drift chambers or drift tubes.

The following subsections briefly describe the tracking systems of experiments at the LHC, the SuperKEKB B-factory, and the future Facility for Antiproton and Ion Research (FAIR).

6.1 Detectors at the LHC

6.1.1 ALICE

ALICE is a dedicated heavy-ion experiment at the LHC [28]. Its detector is designed to study the physics of strongly interacting matter at extreme energy densities.

Until the end of 2018, the Inner Tracking System (ITS, see Fig. 1.5) of the ALICE detector consisted of two barrel pixel layers [29], two layers of silicon drift detectors [17], and two layers of double-sided silicon strip detectors. After the upgrade in 2019–2020, the ITS consists of seven layers equipped with monolithic pixel chips [30].

Fig. 1.5
figure 5

Cut-away view of the ALICE detector. (From https://arxiv.org/abs/1812.08036. Ⓒ2015 CERN for the benefit of the ALICE Collaboration. Reproduced under License CC-BY-4.0)

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The main tracking device of ALICE is a TPC [31, 32] . It provides up to 159 space points per track. The measurement of the energy deposit due to ionization provides a powerful tool for particle identification, especially for low-momentum particles, see [33]. For track reconstruction in the TPC and global track reconstruction, see Sect. 10.1. The TPC is surrounded by a transition radiation detector (TRD) used for triggering and electron identification.

The ALICE muon spectrometer covers only the forward region of the experiment and is dedicated to the study of quarkonia production, open heavy flavor production and vector meson properties via the muonic decay channel [34].

6.1.2 ATLAS

ATLAS [35] is one of the two general-purpose experiments at the LHC, the other one being CMS. Its vertex detector (see Fig. 1.6) originally consisted of three barrel pixel layers and three end-cap pixel disks on either side [36]. In 2014, a fourth pixel layer was inserted in the barrel between the existing pixel detectors and a new beam pipe with smaller radius [37].

Fig. 1.6
figure 6

Top: Cut-away view of the ATLAS detector. (From [35], reproduced under License CC-BY-3.0). Bottom: The central tracker. (From https://collaborationatlasfrance.web.cern.ch/content/tracker)

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The central tracker of ATLAS [36] consists of two parts: the silicon tracker (SCT) with four barrel layers and nine end-cap disks on either side, and the Transition Radiation Tracker (TRT) made of “straw tubes” which are proportional counters that contribute to particle identification via transition radiation; see Sect. 2.4.1. A charged particle hits at least 30 straw tubes on the way through the TRT; see Sect. 10.2.

The ATLAS muon spectrometer [35, Chap. 6] consists of a barrel part and two end-caps. The barrel spectrometer contains three concentric layers, each with eight large and eight small chambers of drift tubes. Each end-cap has four disks of drift tube chambers and cathode strip chambers. Resistive plate chambers on the barrel and thin gap chambers in the end-caps are used for trigger purposes.

6.1.3 CMS

CMS [38] is, besides ATLAS, the second general-purpose experiment at the LHC. Its vertex detector originally consisted of three barrel pixel layers and two end-cap pixel disks on either side [39]. In the winter of 2016/2017, this device was upgraded with a fourth barrel layer and a third end-cap disk on either side, giving at least four hits per track over the full solid angle covered by the detector [40].

The silicon strip tracker (SST) of CMS is the largest silicon tracker ever built. It is divided into four sections: the inner barrel (TIB), the outer barrel (TOB) and the two end-caps (TEC). Depending on its angle with respect to the beam axis, a charged particle crosses between eight and 14 sensors, out of which four to six are double-sided ones [41]. Track reconstruction in the SST is done mostly in conjunction with the Pixel Detector; see Sect. 10.3.

The CMS muon system [42] consists of four layers of muon stations inserted in the iron return yoke of the solenoid; see Fig. 1.7. The stations in the barrel region are equipped with drift tubes, and those in the end-caps are equipped with cathode strip chambers. In addition, resistive plate chambers are mounted in both the barrel and end-caps of CMS; they are used mainly for triggering.

Fig. 1.7
figure 7

Top: Schematic diagram of a sector of the barrel part of the CMS detector. (From [43], reproduced under License CC-BY-4.0). Bottom: Schematic view of the vertex detector and the silicon strip tracker. (Courtesy of W. Adam)

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Figure 1.7 shows a schematic diagram of a sector of the barrel part of the CMS detector.

6.1.4 LHCb

LHCb [44] is the experiment at the LHC that is dedicated to precision measurements of CP violation and rare decays of B hadrons. Instead of surrounding the entire collision point with an enclosed detector as ATLAS and CMS, the LHCb experiment is designed to detect mainly particles in the forward direction.

The core of the LHCb [44] tracking system (see Fig. 1.8) is a silicon microstrip detector close to the interaction point, the Vertex Locator (VELO). It can be moved to a distance of only 7 mm from the proton beams and measures the position of the primary vertices and the impact parameters of the track with extremely high precision.

Fig. 1.8
figure 8

Top: View of the LHCb detector. (From [44], reproduced under License CC-BY-3.0). Bottom: Sensor of the Vertex Locator (VELO)

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Up to end of 2018, the tracking downstream of the VELO was accomplished by the TT and the T stations. The Tracker Turicensis (TT) is a silicon microstrip detector placed upstream of the dipole magnet, which improves the momentum resolution of reconstructed tracks and reject pairs of tracks that in reality belong to the same particle. The magnet is placed behind the TT. It bends the flight path of the particles in the x-z plane and therefore allows the determination of their momenta. The tracking system is completed by the T stations (T1-T2-T3), which, together with the information from the VELO and optionally the TT, determine the momentum and flight direction of the particles. The T stations are composed of silicon microstrip sensors close to the beam pipe and by straw tubes in the outer regions. For track reconstruction in LHCb, see Sect. 10.4.

After the upgrade of LHCb during 2019–2020 and starting with Run 3 of the LHC, the tracking downstream of the VELO is done by the SciFi, a homogeneous tracking system in scintillating fiber technology; see [20] and Sect. 1.4.

The LHCb muon system [44, Sect. 6.3], consisting of the muon stations M1 to M5, provides fast information for the muon trigger at Level 0 and muon identification for the high-level trigger and offline analysis. It comprises five stations interleaved with absorbers. The stations are mostly equipped with multi-wire proportional chambers, with the exception of the central part of the first chamber, where GEM detectors are used because of the high particle rate.

6.2 Belle II and CBM

6.2.1 Belle II

Belle II [45] is an experiment at the SuperKEKB collider at KEK in Japan . Its principal aim is the study of the properties of B mesons. The detector is shown in Fig. 1.9. The vertex detector consists of two parts, the pixel detector (PXD) with two layers of DEPFET pixels [46] and the Silicon Vertex Detector (SVD) with four layers of double-sided silicon strip detectors [47] .

Fig. 1.9
figure 9

Top: View of the Belle II detector. Bottom: The pixel detector and the silicon vertex detector

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The central tracking device of Belle II is the CDC, a cylindrical drift chamber [48] . It has 56 layers of sense wires in nine superlayers, five with a total of 32 axial wire layers and four with 24 stereo wire layers. The stereo angle is between 2.6 and 4.2 degrees. For track reconstruction in Belle II, see Sect. 11.1.

The Belle II KLM system is designed to detect long-lived K-mesons and muons . It consists of alternating layers of iron plates, serving as flux return, and active detector elements. In the end-caps and the innermost two layers of the barrel, the active elements are scintillator strips; the rest of the barrel layers are equipped with resistive plate chambers, reused from Belle [49].

6.2.2 CBM

The Compressed Baryon Matter (CBM) experiment is a fixed target experiment [50] (see Fig. 1.10) at the future FAIR facility for antiproton and ion research. It is designed to investigate the properties of highly compressed baryon matter. Its central tracking device is the Silicon Tracking System (STS [51]). It is designed for high multiplicity, up to 1000 charged particles per interaction, at high rates, up to 10 MHz, and consists of eight layers of double-sided silicon sensors between 30 and 100 cm downstream of the target, inside the magnetic field.

Fig. 1.10
figure 10

Top: Schematic geometry of the CBM detector. (From [53]). Bottom: Schematic geometry of the silicon tracking system. (From [51])

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Each of the three Transition Radiation Detectors [52] is made of four MWPCs. Their main task is electron identification. Track reconstruction in the STS and the TRD is described in Sect. 11.2.