Detector Calibration

Abstract

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.

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.

5.1 From Raw Data to Meaningful Information

Let’s first imagine how data is generated and recorded. As an example, suppose we take data of electromagnetic calorimeter consisting of the sandwich structure of lead and scintillator. A photon hitting the calorimeter generates position-electron pair by photon conversion mainly in the lead plates. The positron or electron ionises the scintillator, resulting in scintillation light. This scintillation light is detected by some photo-sensors such as the photomultiplier tube (PMT). The electrical signal from the PMT is then converted by an analog-to-digital converter (ADC) and recorded as the series of a bunch of digits. Ideally, the light yield of the scintillator is linear to the energy deposit by the electron or position, and also the PMT output is linear to the scintillation light. With the assumption of the linearity of the light yield and the PMT response, one can measure the energy deposited by the electron and positron, or ultimately the incident photon energy from the ADC counts in principle. However, let’s recall that what we have here is just ADC counts which are solely digits. They represent the energy, but do not mean energy yet without the proper conversion to energy. This important procedure to convert the ADC counts to energy is called a “detector calibration”, in short “calibration”.

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.

5.2 Detector Alignment

The tracking device usually consists of many finely segmented channels. Assuming we know the location of each channel, we can measure the position of charged particles by sensing the signal from each channel. This means that the accurate and precise knowledge of the location of the tracking device, or more precisely the position of each channel, including the angles hence six degrees of freedom, is crucial to measure the hit position of the particle at the detector. The procedure to retrieve the position of the tracking device or each channel is called “alignment”. Not only the tracking devices but also any other detectors consisting of multi-channel detectors need to be aligned as well.

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.

The second step is the alignment using charged particles. The idea is to use such charged particles as a probe. Suppose we have tracking device composed of five layers of silicon strip detectors, and we would like to align the sensors of a particular layer. In this case, a special tracking algorithm, which does not use the information on the layer that will be aligned, should be prepared and the particle trajectory reconstructed. Then this track is extrapolated to the layer under alignment to get the so-called residual, the difference in position between the extrapolated probe track and the hit on the layer. For this reason, a higher momentum track is preferred to minimise the extrapolation uncertainty due to the multiple scattering. Here, when we say a “hit”, it is based on the hypothesis or the prior knowledge of the location of the silicon strip sensor. If this prior knowledge is wrong, the residual cannot be zero. In other words, the sensor position that gives the zero residual is likely to be the true position. Based on this general idea, the layer under the alignment is aligned by adjusting the location of the sensors so that the residual distributions have a peak close to zero and the width to be narrow.

In the actual application, a slightly different approach is taken although the basic idea is the same as we have just explained. Because there are many layers and millions of channels in the tracking detectors of the modern collider experiment, it is time-consuming and complicated to prepare such tracking algorithm that doesn’t use the information from the specific layers or channels. Instead of having such a special algorithm, it is more common to use the normal track reconstruction algorithm with looser quality requirements to minimise the bias arising from the usage of hit information from the layer or channel under alignment. For each probe track, again the residual is measured. But not only for a single layer or channel but also for all the layers or channels in the detector under alignment, the residual is computed. Then the sum of residuals from many layers is calculated for each track. The detector is aligned so that the total residual is minimised. This is almost equivalent to the $\chi ^2$ minimisation where the positions of the sensors are fitted.

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.

Fig. 5.1

Reprinted under the Terms of Use from [1] ATLAS Experiment © 2022 CERN. All rights reserved. Red (black) shows the residual using proton-proton collision (cosmic ray) data. Blue shows the prediction by simulation. Note this plot is intentionally selected for the illustration purpose of the effect of the alignment, not showing the current precision

The residual distribution for the silicon pixel detector of the ATLAS experiment.

Fig. 5.2

Sagitta measurement. See the main text for details

5.3 Momentum Scale Calibration of Magnetic Spectrometer

We first explain the concept of measurements of charged particle momentum. Then the calibration of the momentum scale is discussed.

5.3.1 Momentum Measurement and its Resolution

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

$$\begin{aligned} s = R \left( 1-\cos {\frac{\alpha }{2}} \right) \approx R \times \frac{\alpha ^2}{8} = \frac{0.3 B L^2}{8 p_{\textrm{T}}}, \end{aligned}$$

(5.1)

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

$$\begin{aligned} \frac{\sigma _{p_{\textrm{T}}}}{p_{\textrm{T}}} = \frac{\sigma _{s}}{s} = \frac{\sqrt{\frac{3}{2}}\sigma _{x}}{s} = \frac{\sqrt{\frac{3}{2}}\sigma _{x} \cdot 8 p_{\textrm{T}}}{0.3 B L^2}. \end{aligned}$$

(5.2)

In the same manner, in case s is measured at N points (N is more than about 10), the momentum resolution is represented as

$$\begin{aligned} \frac{\sigma _{p_{\textrm{T}}}}{p_{\textrm{T}}} = \frac{\sigma _{x} \cdot p_{\textrm{T}}}{0.3 B L^2} \sqrt{\frac{720}{N+4}}. \end{aligned}$$

(5.3)

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?¹

5.3.2 Momentum Scale Calibration

Going back to the calibration topics, a measurement of the trajectory of charged particles, or more specifically the sagitta, geometrical information of the tracking detector and knowledge of the magnetic field strength are necessary to derive the momentum, just as we have seen. Therefore, we don’t really need the momentum scale calibration for the magnetic spectrometer in a sense, i.e. there are no conversions from a certain information to another such as the charge-to-energy conversion in a case of the energy measurement by a calorimeter.

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 value² 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.

Fig. 5.3

Reprinted under the Creative Commons Attribution 4.0 International License from [3] © 2011 CERN for the benefit of the ATLAS Collaboration. The background contribution is subtracted. The data distribution is shown by either red or black dots, while the simulation by grey histogram

The invariant mass distribution reconstructed from two oppositely charged muons in the ATLAS experiment.

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.

5.4 Energy Calibration of Calorimeter

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.

5.4.1 Cell-by-Cell Calibration

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.

The additional advantage of the usage of muons as the calibration source is the fact that high momentum muons, which are regarded as MIPs, are available in cosmic rays. We can have this ideal calibration source for free everywhere in the world, except the underground experimental facilities such as Super-Kamiokande, where the rate of muons is very small compared to the collider data.

In some cases, however, the in situ muon calibration may not be possible. In that case, the calibration results before installing the detector or assembling it into a big piece are used. For example, beam tests are employed, where the beam energy is precisely known. Or radioactive sources are also used because the energy spectrum of the emitted particles is well known.

In addition to the calibration with particles, a common approach is to prepare and use the artificially generated calibration source. For the detector whose output is lights, such as for scintillators, light flushers like lasers can be used to emulate the signal. For the detector whose output is electrical signals, such as liquid argon calorimeters, electrical test pulses to the readout electronics are often used. By using this kind of calibration sources, the relation between the detector output and the ADC count can be identified, although the relation between the detector output and the energy deposit is not. Still it is useful, much better than nothing, because, for example, relative gains within a detector can be monitored. This is of particular importance for the large-scale detectors where it is not an easy task to adjust the detector response for each individual channel. For this reason, most of the detector systems nowadays are equipped with such a calibration device that also works as the monitoring system of the detector performance.

5.4.2 Energy Cluster Calibration of Electromagnetic Shower

In principle, once the calibration constant for each cell or channel is obtained, one should be able to know the energy of the incident photon or electron to the calorimeter just by summing the energy of each channel associated with the energy cluster generated by the photon or electron. In practice, however, simple summing is not good enough for many reasons. For example, the energy deposited by the electromagnetic shower is much larger than that of muons, leading to the difficulty in the extrapolation to higher energy. Or there are dead materials among the active sensors consisting of a calorimeter, where missing energy due to the dead materials needs to be corrected. A different clustering algorithm may lead to a different energy sum even for the same event. Therefore, it is necessary to calibrate the detector in situ with either the electron or photon whose energy may be known without the calorimeter information. In this regard, the electron is a more user-friendly calibration source because other detectors rather than the calorimeter under calibration can detect the electron and measure its momentum. On the other hand, only the electromagnetic calorimeters can detect and measure the energy of photons precisely. This manifests the difficulty of in situ photon calibrations in collider experiments.

The most common and powerful technique using electrons exploits the fact that the electron’s energy deposit ($\equiv E$) at a calorimeter should be equal to its momentum ($\equiv p$) at a tracker because an electron deposits all the kinematic energy at the calorimeter, and we can safely ignore the electron mass in the momentum region of our interests. Besides, for most of the momentum range in our interests, magnetic spectrometers consisting of charged particle tracking devices and magnets have better momentum resolution than that of calorimeters. Combining the above two facts, the momentum measured by the magnetic spectrometer can be a good reference for the electromagnetic scale. Commonly used is the E/p distribution where electrons or positrons make a peak at unity if the detector is properly calibrated.

Another calibration method, which does not rely on other detectors such as the magnetic spectrometers, makes use of the decay of particles whose masses are precisely known. The decays of $Z\rightarrow e^+ e^-$ and $J/\psi \rightarrow e^+ e^-$ are commonly used, where the calorimeter’s response to the positron or electron is calibrated so that the reconstructed invariant mass gets closer to the world average value. The width of the invariant mass distribution should be narrower after the successful calibration. In the calibration using particle decays, we should be aware that the energy of particles in the calibration source is preferred to be close to the interesting range of your physics analysis to avoid a large extrapolation; in the above examples, the typical electron energy is of the order of 10 GeV in $Z\rightarrow e^+ e^-$, while only a few GeV or less in $J/\psi \rightarrow e^+ e^-$. It means that the former should be used for relatively high $p_{\textrm{T}}$ electrons and the latter for low $p_{\textrm{T}}$. Finally, a decay chain with a high signal-to-noise ratio needs to be selected to avoid a possible bias due to the background.

The calibration method using mass, for example $\pi ^0 \rightarrow \gamma \gamma$, can also be used for the photon calibration in principle. However, it is difficult to find a good decay chain which has enough statistics and covers the wide range of photon momenta. A lack of good calibration sources for photons is a common issue in many experiments. The widely used approach is to rely on the electron calibration because the detector response by electron and photon is similar at the first order. They both evolve the electromagnetic shower where the only difference is the initial depth of starting the shower. For precision, Monte Carlo simulation is often used to correct small differences in the detector responses between electrons and photons. Further, when experiments become more mature or have more statistics, rare processes can be the calibration source. An example is $Z \rightarrow \mu ^+ \mu ^- \gamma$ where the photon is radiated off. The statistics is much smaller than that of $Z \rightarrow e^+ e^-$. The momentum range of this photon is limited because the photon is radiated from a lepton from the Z boson decay. Still, $Z \rightarrow \mu ^+ \mu ^- \gamma$ is used as calibration; to be precise, it is a validation tool to check the calorimeter response to photons.

5.4.3 Energy Cluster Calibration of Hadronic Shower

The energy-scale calibration for hadronic showers is much more complicated than that for electromagnetic showers for the following reasons.

First, the detector response to hadrons is different from that to electrons or photons, because of the different shower evolutions. Usually, all the kinematic energy of particles incident to a calorimeter is deposited in case of electromagnetic showers, meaning all the energy can be seen by the detector. On the other hand, some parts of the incident energy of hadrons are often lost because the nuclear interaction length (see Sect. 3.3.2) is relatively longer compared to the size or depth of the detector. Therefore, even if an electron and hadron have the same energy and hit into the same calorimeter, the “visible” energy may be different. Sometimes, this visible energy ratio of the electron and hadron with the same energy is referred to as the “e/h” ratio. With a few exceptions, most of the hadron calorimeters have $e/h > 1$, demanding a special correction in the calibration process.

Second, the fluctuation of energy deposits by hadrons is very large, while it is almost zero for the electromagnetic showers if the depth of the calorimeter is thick enough. The fluctuation comes from the fact that the hadronic shower sometimes creates $\pi ^0$ that immediately decays into $\gamma \gamma$ and loses its energy by the evolution of electromagnetic showers. Hence in the case of having $\pi ^0$ in the hadronic shower, the visible energy gets larger, and vice versa. Another reason for the large fluctuation is due to neutron production in the development of hadronic shower. In the case of charged hadron production such as proton, its kinematic energy can be detected as the energy deposit by ionisation, while low-energy neutron ($\ll 1$ GeV) does not have such an energy loss mechanism and is rather transparent in a calorimeter. Therefore, the visible energy is influenced by the number of produced neutrons. These are the main reasons why there is a rather large fluctuation in the energy loss of the hadronic shower. In addition, hadrons such as pion or kaon decay (semi-)leptonically, yielding neutrinos that are not detected by the calorimeter. The existence of neutrinos in the hadronic shower changes the total energy deposit in the calorimeter.

For the above reasons, in order to correctly deduce the energy of hadrons incident to the calorimeter, special care needs to be taken after the cell-by-cell calibration. In the collider experiments, it is rare to have a single hadron incident to the calorimeter. Instead, a jet (see Sect. 6.4) is the object handled by the calorimeter, which is an object defined by a human being. This means that the energy of a jet depends on the definition or actually on the clustering algorithm. For this reason, the treatment of jet energy calibration is described after introducing the jet reconstruction (see Sect. 6.4.4).