The information needed when analysing data, for example, the cross-section measurements, is the 4-momentum vectors of particles in interest, and possibly the knowledge about species of the particles. For this purpose, the detector for the high energy physics is usually designed so that it allows us to measure the momentum or energy, and information for the particle identification, such as velocity. However, the recorded data as they are do not tell us anything. They are just a bunch of digits which are not energies or positions of particles if they are not properly translated into meaningful physical variables.
The information needed when analysing data, for example, the cross-section measurements, is the 4-momentum vectors of particles in interest, and possibly the knowledge about the species of the particles. For this purpose, the detector for the high energy physics is usually designed so that it allows us to measure the momentum or energy, and information for the particle identification, such as velocity. However, the recorded data as they are do not tell us anything. They are just a bunch of digits which are not energies or positions of particles if they are not properly translated into meaningful physical variables.
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This chapter describes the procedure to retrieve meaningful information that is needed in physics analyses from raw data. This process is called calibration, and is one of the most important processes in the whole flow of the high energy physics experiments.
In the above example, we discussed the concept of energy calibration of the calorimeter. This concept is very common for all detectors, whatever they measure. Sometimes, we measure the time interval of some detector signals, in which the data is recorded by time-to-digital converter (TDC). In this case, the calibration from TDC counts to time is needed. Sometimes, we measure the location of a charged particle hitting the position-sensitive sensor. In this case, the hit information has to be interpreted as the position information. This is also considered as a calibration in a sense. In the following sections, we discuss some concrete procedures of the detector calibrations.
The alignment procedure can be divided into two steps. The first one is the mechanical measurement or survey of the detector component. In the survey, the location of the large structure of the detector is measured, which has to be carried out usually before starting data taking or just after installing the detector. Since the detector element of the large structure is assembled from the small components of sensors, etc. with some precision which is specified in each experiment, the position of each channel is considered to be known with the precision of the assembly (and the survey) once the survey is performed.
So far in this section, the basic concept of the alignment was given, where we discussed the alignment of the single detector. But the collider detector, for example, is a more complex and larger object consisting of several types of detectors. Further in the actual application, the alignment is performed in several steps. Again, using the silicon strip tracker in the ATLAS experiment as an example, the first level is to align the whole tracker relative to the other detector system. This means that not each layer nor single sensor is individually aligned. Instead, the whole support structure holding the sensors or modules is aligned as a single object. Then as the second-level alignment, each layer is aligned, i.e. each layer can be moved independently. Finally as the third level, the individual module or sensor within each layer is aligned. In this way, the failure of the \(\chi ^2\) fitting due to the possible large deviation of the initial value from the actual position can be avoided. In addition, the step-by-step approach allows saving the computing time of the \(\chi ^2\) fitting.
Figure 5.1 shows the residual distributions for the ATLAS silicon pixel detector, where the first alignment was carried out by using cosmic rays and then proton-proton collision data for more statistics. You can see that the width becomes narrower by using the collision data, indicating the improvement of the alignment. Note that an old result, which was obtained at the very beginning of the experiment, is intentionally presented here for the illustration purpose. Currently, the width of the residual distribution is close to that for the simulation result where all the detector positions are perfectly known.
Suppose a charged particle travels in the magnetic field of B (in Tesla) with the radius R (in meter). Also suppose we measure the charged particle positions by position-sensitive detectors D1, D2, and D3, as shown in Fig. 5.2. The momentum of the charged particle \(p_{\textrm{T}}\) (GeV) can be written as \(p_\textrm{T} = 0.3 \times B (\textrm{Tesla}) \times R (\textrm{meter})\). Because the angle \(\alpha \) in Fig. 5.2 is geometrically represented as \(\displaystyle \alpha \approx \frac{L}{R}\), the depth of the arc called a sagitta (s in meter) of the particle trajectory can be expressed as
where L is the chord of the arc in meter. In case the track position at the three detectors is measured as \(x_1 \pm \sigma _x\), \(x_2 \pm \sigma _x\), and \(x_3 \pm \sigma _x\) (with a common uncertainty of \(\sigma _x\)), the sagitta is \( \displaystyle s=x_2 - \frac{x_1+x_3}{2}\). The uncertainty of the sagitta is \(\sqrt{\frac{3}{2}} \sigma _x\). Therefore, the momentum resolution can be represented as
From these calculations, you can see that the momentum resolution is proportional to the momentum of charged particle and the uncertainty of the position measurement (\(\displaystyle \frac{\sigma _{p_{\textrm{T}}}}{p_{\textrm{T}}} \propto \sigma _x \cdot p_{\textrm{T}}\)), and the inverse of the magnetic field and the square of the length of detectors. If you want to have better momentum resolutions, more detectors should be placed in a wider space where a stronger magnetic field is provided. This can be imagined if you draw the arc with 3 or more points in a limited space and estimate the curvature of its arc. For which can you estimate more precisely, an arc with a smaller radius or an arc with a larger radius?Footnote 1
But in most of the experiments, in situ calibration or correction of the momentum scale is performed for better accuracy and precision. A common technique is to make use of the known mass of some particles, for example, \(K_{S}\), \(J/\psi \) or Z. The momentum scale of the reconstructed tracks is calibrated or corrected so that the peak position of the invariant mass distribution reconstructed from two tracks becomes the world average valueFootnote 2 of \(K_{S}\), \(J/\psi \) or Z. Figure 5.3 shows the invariant mass reconstructed from two oppositely charged muons. As you can see, with this calibrated data, the peak position is consistent with the world average value of Z.
The particle used as the calibration target depends on the experiments because of the limitation of the available particles. The data sample with high purity is always preferable to avoid uncertainty due to the background. At the same time, the large data set is also preferable to reduce the statistical uncertainty. The experimentalist has to consider the optimal use of the various calibration samples.
This section was devoted to describing the momentum scale calibration or correction. But Fig. 5.3 shows the other important point which we would like to mention. It shows that the resolution depends on the alignment. As can be seen in Eqs. 5.2 and 5.3, momentum resolution has a linear dependence on the precision of position measurement for a track. Therefore, better alignment leads to better resolution. The figure shows that better alignment is used when the data was reprocessed.
The energy calibration procedure for the calorimeter is classified into two steps. The first step is to calibrate each cell or channel, and the second is to calibrate the energy of the particle incident to the calorimeter, equivalent to the energy of the shower after clustering. These approaches are slightly different for the electromagnetic and hadronic calorimeters. Below, we discuss the concept of these two-step calibration procedures for the calorimeters.
In most cases, the energy information of the calorimeter is recorded as the digital number that is converted by an ADC from the detector output, typically the pulse height or charge created by the sensor. The goal of the cell-by-cell calibration is to find the relation between the energy deposit and the ADC count for each channel, which is a conversion factor. A set of the factors for all the cells are called calibration constants.
To get this calibration constant, the most powerful and a very common technique is to use a muon as the calibration source, because the muon in high energy physics experiment behaves as almost a minimum ionising particle (MIP) that deposits the constant energy per path length. The tracking system allows to measure the path length across the cell of the calorimeter, and hence to expect the energy deposit. In this way, one can obtain the ADC counts for unit energy. Only the muons can be this kind of calibration source, because the other charged particles evolve either electromagnetic or hadronic shower in materials, and their energy deposits are not constant. On the other hand, a muon deposits its energy just by ionisation loss, resulting in a rather constant energy deposit per unit path length. In the energy frontier collider experiments, muons decayed from Z bosons are one of the cleanest samples. They are isolated, i.e. there are no other particles nearby, and have high momentum. The higher the momentum. the multiple the scattering angles are smaller. This means that the error of estimating the path length is smaller. In addition, \(J/\psi \rightarrow \mu ^+ \mu ^-\) events are also used as the lower momentum calibration source. 006ab0faaa
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