1 Introduction

The main goal of electromagnetic calorimeters is to measure precisely the energy of detected particles such as photons and electrons. These detectors are important elements in nuclear physics instrumentation and are widely used especially in hadronic and particle physics experiments [1]. Usually, the knowledge of the impact position of detected particles is required leading to a lateral segmentation of these calorimeters. Lead fluoride (PbF₂) is commonly employed as a constituent material of such calorimeters [2-5]. Its high density offers a short radiation length (X₀=0.93 cm) and a small Molière radius (r_M=2.12 cm) leading to compact detector geometries [6, 7]. In practice, the segmentation size is close to r_M while the longitudinal calorimeter length is equal to several X₀ to ensure a full development of electromagnetic showers in the detector blocks. Each calorimeter block is generally connected to a photomultiplier tube (PMT) to collect the light induced by a shower and an electronic base to shape and amplify the PMT signal. The amplitude of the final signal is then related to the energy released by the particle in the considered block via a calibration coefficient.

A great deal of effort is still underway to optimize the performance of electromagnetic calorimeters and in particular, their energy resolution [8-10]. In addition to the design optimization, many energy calibration techniques are still investigated to improve the reconstructed energy of detected particles [11-14]. Indeed, the energy calibration is sensitive to many factors, such as the hadronic and electromagnetic background during the experiment and the gain drift of blocks due to an increasing loss of their transparency when exposed to high radiation rates and accumulated doses [15, 16]. The choice of the energy calibration method is then crucial to ensure a frequent and proper monitoring of the calibration coefficients of each calorimeter block keeping thus an acceptable energy resolution. In many experiments, dedicated calibration runs have to be taken in order to send particles of known energy in the calorimeter. Other methods consist to calibrate the detector with cosmic rays or with a cluster of LEDs of known light intensity placed in front of the calorimeter blocks [16]. All these standard methods have the disadvantage to change momentarily the experimental setting or trigger and to decrease the amount of time dedicated to physics runs. In addition, many parameters, such as the nature of particles used in the calibration procedure, their energies and incident angles, and the experimental background, could be different relative to the physics run conditions. The obtained calibration coefficients are then not necessarily optimized for the particles detected during physics runs. Finally, the calibration provided by these methods at a given time may not still be valid afterwards if the calibration coefficients are time dependent.

This paper presents an energy calibration method based on the detection of π⁰ mesons in the calorimeter. This method is performed to monitor daily the calibration of the electromagnetic calorimeter of the Jefferson Lab Hall A DVCS experiments [2, 3, 17, 18] and can be applied in similar experiments where two-photon decays of π⁰ can be detected in a laterally segmented calorimeter. It does not require specific runs since the calibration data are taken simultaneously with the primary experimental data at the same conditions. This method will be exposed and tested on the basis of a calorimeter simulation using the GEANT4 toolkit [19]. The relative accuracy on the obtained calibration coefficients will finally be discussed in terms of the number of detected π⁰.

2 Simulation of an experimental case

The main goal of the Jefferson Lab Hall A DVCS experiments is to study the Deeply Virtual Compton Scattering (DVCS) process on the nucleon eN → eNγ using a few GeV electron beams impinging on a liquid hydrogen (LH2) or deuterium (LD2) target [2, 3, 17, 18]. Many similar experiments in the world are concerned with this increasingly timely topic of the characterization of the nucleon structure with the DVCS process [20-23]. As shown in Fig. 1, the scattered electron is detected in a high resolution spectrometer (HRS) which determines accurately its momentum and angles as well as the reaction vertex coordinates in the target [24]. The emitted photon is detected in an electromagnetic calorimeter consisting of a 16 × 13 matrix of 3 × 3 × 18.6 cm³ PbF₂ blocks placed at 1.1 m from the target. This short distance combined with the high luminosity of the experiment (∼10³⁷ cm^-2s^-1) leads to an important background rate in the calorimeter as well as radiation damage near the front face of blocks. The energy calibration of the calorimeter is of direct relevance since it affects directly the eN → eNγ event identification based on a study of the missing mass squared:

Figure 1:Full-screen View

Experimental setup of the Hall A DVCS experiments and definition of particle 4-vectors. $q\text{'}=q_{\gamma }^{\text{'}}（q\text{'}=q_2^{\text{'}}+q_1^{\text{'}}）$ is the photon (π⁰) 4-vector. The recoil nucleon, not detected, is identified with a cut on the missing mass squared defined by Eq. (1) (Eq. (12)) if a photon (π⁰) is detected in the calorimeter.

where k, k′, p, and $q_{\gamma }^{\text{'}}$

are respectively the 4-vectors of the incident electron, the scattered electron, the initial nucleon at rest and the emitted photon. Experimentally, k, k′, and p are known accurately and the resolution of $M_x^2$

is dominated by the calorimeter energy resolution [16]. The photon 4-vector is determined for each event, j, from the energies deposited in the calorimeter blocks:

where Ai(j) is the output signal amplitude of block i, and $C_i^{\text{'}}$ is the corresponding calibration coefficient. Let’s assume that the $C_i^{\text{'}}$ coefficients are roughly known at a given time with any standard calibration method. As mentioned above, many factors, such as different experimental conditions and transparency losses of blocks, could modify the calibration by 20% in average [25] and by up to 40% for some particular blocks:

The correction factors, ϵi, and thus the new calibration coefficients, Ci, must then be known accurately in order to get the real deposited energies, Ei(j):

The goal of the calibration method, discussed hereafter, is to determine the ϵi for each block, i. This method is based on the study of π⁰ electroproduction events eN → eNπ⁰ → eNγγ where the two photons coming from the π⁰ decay are detected in the calorimeter. This reaction is very common in lepton-hadron scattering and is usually present in the data of DVCS experiments. The π⁰ energy is almost equal to the DVCS photon energy because of the kinematic similarity between these two reactions, and it is then well adapted for the calorimeter calibration.

To demonstrate the validity of the calibration method, a GEANT4 simulation of the experimental setup is performed and N=2.10⁶ eN → eNπ⁰ events are generated following the kinematics presented in Table 1. This particular kinematics corresponds to one setting of the Hall A DVCS experiments [26]. The electromagnetic showers created by the π⁰ photons in the calorimeter are fully simulated [27]. The generation and tracking of Čerenkov photons induced by the showers in the PbF₂ blocks are not considered in this study because of unrealistic computing times and strong sensitivity to exact optical properties of crystals and their wrapping surfaces. However, one can attribute an average number of N_ph=330 photo-electrons per GeV deposit collected by the PMT of each block to be coherent with the reported experimental energy resolution σ(E)/E=3.1% [26]. This number is deduced from a $H(e\mathrm{,}{e}_{\text{Calo}}^{\text{'}}{p}_{\text{HRS}})$ elastic calibration where $E_e^{\text{'}}=3.16$ GeV scattered electrons are detected in the calorimeter:

Finally, real radiative effects, where the incident or the scattered electron emits a real photon, are taken into account in the simulation following the procedure described in [16].

To mimic the uncertainty on the experimental calibration coefficients, the energies, Ei, deposited in the calorimeter blocks are multiplied by random numbers, Ki, varying uniformly between 1-K_max and 1, where K_max=40%, and become equal to:

The calibration method consists then to determine, from $E_i^{\text{'}}(j)$ and the known 4-vectors {k, k′(j), p}, the correction factors, ϵi, which have to be applied in order to get the correct energies. It is evident from Eqs. 4 and 6 that a successful calibration should give ϵi ≈ 1/Ki with δi=ϵiKi-1 being the relative deviation of ϵi from 1/Ki.

3 Calibration method

The calibration method is based on a χ² minimization between the measured π⁰ energy, $E_{{\pi }^0}^{\text{rec}}$, reconstructed from $E_i^{\text{'}}$, and the expected π⁰ energy, $E_{{\pi }^0}^{\text{cal}}$, calculated from the π⁰ momentum direction. The following subsections detail the different steps of the calibration.

3.1 Particle 4-vector reconstruction

The reconstructed π⁰ energy in the simulation is the sum of the two shower energies created by the two photons of the π⁰ decay. Experimentally, one can have multiple showers in the calorimeter coming from other meson decays, accidentals or any additional final state particles. In this case, a clustering algorithm based on a cellular automata is used to determine the number of showers and to separate the different cluster contributions [28]. The energy of a shower is then the sum of the energies deposited in the N_clus blocks belonging to the corresponding cluster:

$E_{\text{sh}}^{\text{'}}={\displaystyle \sum _i^{N_{\text{clus}}}}{E}_i^{\text{'}}\mathrm{.}$ (7)

Figure 2 shows an example of a π⁰ event in the calorimeter where the corresponding clusters are indicated in red. The transverse coordinates (xc,yc) of a shower centroid are determined with a center of gravity based method from the coordinates (xi,yi) of the blocks belonging to the corresponding cluster [29]:

Figure 2:Full-screen View

(Color online) Numbering of the calorimeter blocks. The blocks located at the calorimeter edge are shown in blue. An example of a two-cluster event created by the π⁰ → γγ decay, is shown with the red blocks.

$x_c=\frac{{\displaystyle \sum _i}{w}_i{x}_i}{{\displaystyle \sum _i}{w}_i}\mathrm{,}$ (8)

with a similar equation for yc. The weight, wi, is given by:

$w_i=\max \left\{0;W_0+\ln \left(\frac{E_i^{\text{'}}}{E_{\text{sh}}^{\text{'}}}\right)\right\}\mathrm{,}$ (9)

where W₀ is a free dimensionless parameter optimized for the considered energies [30]. The showers whose centroid coordinates belong to a block of the calorimeter edge (see Fig. 2) are excluded from the calibration procedure. Indeed, the energy of a particle impinging these particular blocks is not well reconstructed because of the shower energy leakage near the calorimeter edge.

The knowledge of the shower centroid coordinates, with a 3 mm accuracy [26], and the interaction vertex coordinates allows us to reconstruct the momentum direction of the particle creating the shower and thus its 4-vector $q\text{'}(E_{\text{sh}}^{\text{'}}\mathrm{,}\overrightarrow{q}\text{'})$, assuming a photon as a detected particle. The different steps of this reconstruction procedure are applied for each simulated event to be coherent with the experimental data analysis. It is worth noting that the obtained particle direction is not very sensitive to the deposited energies, $E_i^{\text{'}}$, and thus to the calibration coefficients, but depends mainly on the block coordinates where a maximum energy is released by the shower [30]. Consequently, the angle between the particle direction and the virtual photon momentum, $\overrightarrow{q}=\overrightarrow{k}-\overrightarrow{k}\text{'}$, is well reconstructed even if inadequate calibration coefficients are employed.

3.2 eN → eNπ⁰ reaction identification

In the simulated data, only eN → eNπ⁰ → eNγγ events are generated. Experimentally, we have to identify this reaction by selecting only 2-cluster events and by computing the invariant mass of the two reconstructed particles in the calorimeter:

$M_{\text{inv}}=\sqrt{{(q_1^{\text{'}}+q_2^{\text{'}})}^2}\mathrm{,}$ (10)

where $q_1^{\text{'}}$ and $q_2^{\text{'}}$ are the particle 4-vectors determined as described in the previous subsection. If these particles are photons coming from a π⁰ decay, then their invariant mass should be equal to the π⁰ mass $M_{{\pi }^0}\approx 0.135$ GeV, within detector resolution. In an experimental or simulated M_inv distribution, π⁰ events are located in a peak around M_π⁰ and cannot be confused with other mesons or events. However, a miscalibration of the calorimeter blocks can shift the peak position (M_peak) and increase its width (σ_Minv) as shown in the simulated M_inv histogram of Fig. 3. A cut around the peak position, instead of M_π⁰, is then applied to ensure the selection of π⁰ events:

Figure 3:Full-screen View

(Color online) Simulated two-gamma invariant mass distribution before (black) and after (red) the calibration. The vertical dashed lines represent the M_inv cut of Eq. (11).

$|M_{\text{inv}}-M_{\text{peak}}|<3{\sigma }_{M_{\text{inv}}}\mathrm{,}$ (11)

where σ_Minv is the resolution of the reconstructed M_inv variable.

The second step of the eN → eNπ⁰ reaction identification consists to compute for each π⁰ event the missing mass squared defined by:

$M_x^2=(k+p-k^{\text{'}}-q_1^{\text{'}}-q_2^{\text{'}}{)}^2\mathrm{,}$ (12)

which must be equal to the nucleon mass squared $M_N^2\approx 0.88$

GeV² within detector resolution if the considered event corresponds to a eN → eNπ⁰ reaction and if the calorimeter is well calibrated. Figure 4 shows the simulated $M_x^2$ distribution where only eN → eNπ⁰ events are generated. The peak position in Fig. 4 is not centered around $M_N^2$ because of the Ei smearing by the random factors, Ki, which is equivalent experimentally to a wrong calibration of the blocks. The non-Gaussian behavior of the right side of the peak is due to real radiative effects. Experimentally, deep inelastic scattering (DIS) events, where additional mesons are created in the final state, are also present in a $M_x^2$ distribution and are located after the $M_x^2$ peak position (see Fig.2 in Ref. [26] for more details). To avoid the contamination from DIS events, the selection of eN → eNπ⁰ is performed, as shown in Fig. 4, by applying the following $M_x^2$ cut:

Figure 4:Full-screen View

(Color online) Simulated missing mass squared distribution before (black) and after (red) the calibration. The vertical dashed line represents the $M_x^2$ cut of Eq. (13).

$M_x^2<M_{\text{peak}}^2+{\sigma }_{M_x^2}\mathrm{,}$ (13)

where $M_{\text{peak}}^2$ is the peak position in the $M_x^2$ distribution and ${\sigma }_{M_x^2}$ its resolution. Both Eq. (11) and Eq. (13) cuts are applied in the simulation to be coherent with the experimental data analysis.

3.3 Expected π⁰ energy calculation

For each selected eN → eNπ⁰ event, the reconstructed π⁰ energy writes:

$E_{{\pi }^0}^{\text{rec}}={\displaystyle \sum _{i=1}^{208=16\times 13}}{E}_i^{\text{'}}{d}_i\mathrm{,}$ (14)

where di=1 if the block, i, belongs to one of the two clusters created by the π⁰ and di=0 otherwise. This reconstructed energy leads to particular values of $M_x^2$ and M_inv which can be different from $M_N^2$ and $M_{{\pi }^0}$ because of energy resolution effects or a bad calibration of blocks. Actually, it is possible to find for each event a more realistic estimation of the pion energy, called $E_{{\pi }^0}^{\text{cal}}$ hereafter, giving exactly $M_x^2=M_N^2$ and $M_{\text{inv}}=M_{{\pi }^0}$. Equations (10) and (12) lead to:

$\begin{array}{ll}{M}_N^2\hfill & =(k+p-k^{\text{'}}-q_1^{\text{'}}-q_2^{\text{'}}{)}^2\hfill \\ \hfill & =(k+p-k^{\text{'}}{)}^2+M_{{\pi }^0}^2-2(k_0-k_0^{\text{'}}+M_N)E_{{\pi }^0}^{\text{cal}}+2\parallel \overrightarrow{q}\parallel \sqrt{{(E_{{\pi }^0}^{\text{cal}})}^2-M_{{\pi }^0}^2}\cos \theta \mathrm{,}\hfill \end{array}$ (15)

where θ is the angle between the momenta of the virtual photon and the π⁰. As mentioned above, θ is well defined experimentally even if biased calibration coefficients are employed. The physical solution of Eq. (15) is given by:

$E_{{\pi }^0}^{\text{cal}}=\frac{-b+\sqrt{b^2-4ac}}{2a}\mathrm{,}$ (16)

where:

$\begin{array}{l}a=4(k_0-k_0^{\text{'}}+M_N{)}^2-4\parallel \overrightarrow{q}{\parallel }^2{{\displaystyle \cos }}^2\theta \mathrm{,}\\ b=4(k_0-k_0^{\text{'}}+M_N)\left[M_N^2-（k-k^{\text{'}}+p{）}^2-M_{{\pi }^0}^2\right]\mathrm{,}\\ c=4M_{{\pi }^0}^2\parallel \overrightarrow{q}{\parallel }^2{{\displaystyle \cos }}^2\theta +{\left[M_N^2-（k-k^{\text{'}}+p{）}^2-M_{{\pi }^0}^2\right]}^2\mathrm{.}\end{array}$ (17)

Figure 5 shows the histogram of the relative difference between $E_{{\pi }^0}^{\text{cal}}$ and the real π⁰ energy, $E_{{\pi }^0}^{\text{real}}$, known for each event in the simulation. The comparison between the reconstructed π⁰ energy, $E_{{\pi }^0}^{\text{rec}}$, and $E_{{\pi }^0}^{\text{real}}$ is also shown in Fig. 5. The mean value and the small RMS of the $(E_{{\pi }^0}^{\text{cal}}-E_{{\pi }^0}^{\text{real}})$ distribution relative to the $(E_{{\pi }^0}^{\text{rec}}-E_{{\pi }^0}^{\text{real}})$ distribution indicate that $E_{{\pi }^0}^{\text{cal}}$ is much closer to the real π⁰ energy than $E_{{\pi }^0}^{\text{rec}}$. The calculated π⁰ energy can then be used to adjust the calorimeter calibration and determine the correction factors ϵi as detailed in the following subsection.

Figure 5:Full-screen View

(Color online) Relative difference between the real π⁰ energy, $E_{{\pi }^0}^{\text{real}}$, and the reconstructed π⁰ energy, $E_{{\pi }^0}^{\text{rec}}$, before (black) and after (red) the calibration. The relative difference between, $E_{{\pi }^0}^{\text{real}}$, and the calculated π⁰ energy, $E_{{\pi }^0}^{\text{cal}}$, (Eq. (16)) is shown with the blue histogram.

4 Results and discussion

Figure 6 shows the relative difference δi=ϵiKi-1 between the obtained correction factors, ϵi, and the smearing factors, 1/Ki, as a function of the calorimeter block number. Figure 6 proves that the present calibration method succeeds to find the real calibration coefficients for all the calorimeter blocks, within 1% deviation, except for those located at the calorimeter edge. Since the clusters centered around the latter blocks are excluded from the calibration procedure, the α-matrix diagonal elements corresponding to these blocks are filled with low deposited energies and become sensitive to energy fluctuations of shower tails. The corresponding calibration coefficients are then slightly overestimated. The energy resolution improvement is shown in Fig. 5 where a clear decrease of the $(E_{{\pi }^0}^{\text{rec}}-E_{{\pi }^0}^{\text{real}})$ distribution RMS is visible after this calibration. The calibration quality can also be seen in Fig. 3 and Fig. 4 where the peaks corresponding respectively to π⁰ events and eN → eNπ⁰ events are now centered around $M_{{\pi }^0}$ and $M_N^2$ with a smaller width.

Figure 6:Full-screen View

(Color online) The relative difference, δi, between the correction factors, ϵi, and the smearing factors, 1/Ki, as a function of the calorimeter block number, i. The blocks located at the calorimeter edge (see Fig. 2) are represented with blue open circles.

The average relative accuracy on the obtained correction factors or calibration coefficients can be defined as:

where the sum runs over the block numbers not belonging to the calorimeter edge. This relative accuracy decreases when the number, $N_{\pi }^0$, of eN → eNπ⁰ events, used in the calibration procedure, increases. For other segmented calorimeters composed of Nb blocks, Δ should rather depend of N_π⁰/Nb assuming a uniform π⁰ distribution in the calorimeter acceptance. Figure 7 shows the evolution of Δ as a function of N_π⁰/Nb and demonstrates that less than 1% average accuracy on the calibration coefficients can be obtained if N_π⁰/Nb is larger than 100. A fit of Fig. 7 points leads to the following empirical power law:

Figure 7:Full-screen View

Average relative accuracy, Δ, on the calibration coefficients as a function of the ratio between the number of π⁰ used in the calibration procedure and the number of calorimeter blocks (Nb=208). The dashed line represents Eq. (23) fit to the points.

The number N_π⁰/Nb depends experimentally on the integrated luminosity and the differential cross section of the eN → eNπ⁰ process in the studied kinematics. For example, the average π⁰ production rate for the kinematics of Tab. 1 is $N_{{\pi }^0}/N_b\approx 90$ per day. A compromise on the calibration frequency during the experiment should then be found taking into account the drift rate of the block gains. For the Jefferson Lab Hall A DVCS experiments, a daily monitoring of the calorimeter calibration coefficients is achieved at 1% accuracy level with this calibration method [26].

In some extreme cases when some initial calibration coefficients are different from the real ones by a considerable amount, iterations of the different steps discussed in Sect. 3 could be necessary. This is mainly due experimentally to an identification issue of the eN → eNπ⁰ events because of a possible contamination of the selected yield (Eq. (13)) by DIS events. A more selective $M_x^2$ cut can then be applied in the first iteration to minimize this contamination. The calibration coefficients obtained after the first iteration are used as initial calibration coefficients for the second iteration and so on. A convergence of the calibration coefficients is generally obtained after the second or the third iteration.

5 Conclusion

The present work discussed an energy calibration technique of laterally segmented electromagnetic calorimeters where the two photons of the π⁰ decay produced by the eN → eNπ⁰ reaction are detected. The calibration procedure is based on a χ² minimization between the reconstructed π⁰ energy, from the calorimeter block data, and its expected energy. This energy is calculated for each event from the π⁰ momentum direction, exploiting the good spatial resolution of the calorimeter. Contrary to other standard calibration methods, this technique allows us to have a set of calibration coefficients optimized for the studied particles under real experimental conditions. The average relative accuracy on these coefficients is less than 1% if the number of detected π⁰ events relative to the number of calorimeter blocks is larger than 100. This technique has been successfully applied to monitor daily the calorimeter calibration of the Jefferson Lab Hall A DVCS experiments at a 1% accuracy level and with a continuous loss of block transparencies. The calibration method can be applied similarly using any other exclusive meson electroproduction reaction if all-photon decays of these mesons are detected in the calorimeter.