Energy calibration of laterally segmented electromagnetic calorimeters based on neutral pion detection

Energy calibration of laterally segmented electromagnetic calorimeters based on neutral pion detection 1 corresp first-author Corresponding author, mazouz@jlab.org Corresponding author, mazouz@jlab.org mazouz@jlab.org Faculté des Sciences de Monastir, Monastir 5000, Tunisia Faculté des Sciences de Monastir, Monastir 5000, Tunisia This paper describes a method for energy calibration of laterally segmented electromagnetic calorimeters based on the detection of two-photon decays of π 0 mesons. The calibration procedure performs a χ 2 function minimization between the measured π 0 energy in the calorimeter and its expected energy deduced from the π 0 momentum direction. The performance of this technique is demonstrated with a Monte Carlo simulation of an experimental case where biased calibration coefficients are employed. The real calibration coefficients are restored with less than 1% relative accuracy when a sufficient number of π 0 is detected. This technique is applied to monitor daily the calibration coefficients of the calorimeter used in the Jefferson Lab Hall A DVCS experiments. Electromagetic calorimeters Energy calibration Detector modelling and simulations Data processing methods 1 Introduction 1 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 1 ]. Usually, the knowledge of the impact position of detected particles is required leading to a lateral segmentation of these calorimeters. Lead fluoride (PbF 2 ) is commonly employed as a constituent material of such calorimeters [ 2 2 - 5 5 2-5 ]. Its high density offers a short radiation length ( X 0 =0.93 cm) and a small Molière radius ( r M =2.12 cm) leading to compact detector geometries [ 6 6 , 7 7 ]. In practice, the segmentation size is close to r M while the longitudinal calorimeter length is equal to several X 0 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 8 - 10 10 8-10 ]. In addition to the design optimization, many energy calibration techniques are still investigated to improve the reconstructed energy of detected particles [ 11 11 - 14 14 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 15 , 16 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 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 π 0 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 2 , 3 3 , 17 17 , 18 18 ] and can be applied in similar experiments where two-photon decays of π 0 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 19 ]. The relative accuracy on the obtained calibration coefficients will finally be discussed in terms of the number of detected π 0 . 2 Simulation of an experimental case 1 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 2 , 3 3 , 17 17 , 18 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 20 - 23 23 20-23 ]. As shown in Fig. 1 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 24 ]. The emitted photon is detected in an electromagnetic calorimeter consisting of a 16 × 13 matrix of 3 × 3 × 18.6 cm 3 PbF 2 blocks placed at 1.1 m from the target. This short distance combined with the high luminosity of the experiment ( ∼ 10 37 cm -2 s -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: <math display="block" id="M2"><mrow><msubsup><mi>M</mi><mi>x</mi><mn>2</mn></msubsup><mo>=</mo><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mi>p</mi><mo>−</mo><msup><mi>k</mi><mo>'</mo></msup><mo>−</mo><msubsup><mi>q</mi><mi>γ</mi><mo>'</mo></msubsup><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mn>,</mn></mrow></math> where k , k′, p , and <math id="M3"><mrow><msubsup><mi>q</mi><mi>γ</mi><mo>'</mo></msubsup></mrow></math> 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 <math id="M4"><mrow><msubsup><mi>M</mi><mi>x</mi><mn>2</mn></msubsup></mrow></math> is dominated by the calorimeter energy resolution [ 16 16 ]. The photon 4-vector is determined for each event, j , from the energies deposited in the calorimeter blocks: <math display="block" id="M5"><mrow><msubsup><mi>E</mi><mi>i</mi><mo>'</mo></msubsup><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mi>C</mi><mi>i</mi><mo>'</mo></msubsup><msub><mi>A</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo><mn>,</mn></mrow></math> where A i ( j ) is the output signal amplitude of block i , and <math id="M6"><mrow><msubsup><mi>C</mi><mi>i</mi><mo>'</mo></msubsup></mrow></math> is the corresponding calibration coefficient. Let’s assume that the <math id="M7"><mrow><msubsup><mi>C</mi><mi>i</mi><mo>'</mo></msubsup></mrow></math> 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 25 ] and by up to 40% for some particular blocks: <math display="block" id="M8"><mrow><msub><mi>C</mi><mi>i</mi></msub><mo>=</mo><msub><mi>ϵ</mi><mi>i</mi></msub><msubsup><mi>C</mi><mi>i</mi><mo>'</mo></msubsup><mn>.</mn></mrow></math> The correction factors, ϵ i , and thus the new calibration coefficients, C i , must then be known accurately in order to get the real deposited energies, E i ( j ): <math display="block" id="M9"><mrow><msub><mi>E</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>ϵ</mi><mi>i</mi></msub><msubsup><mi>C</mi><mi>i</mi><mo>'</mo></msubsup><msub><mi>A</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>ϵ</mi><mi>i</mi></msub><msubsup><mi>E</mi><mi>i</mi><mo>'</mo></msubsup><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo><mn>.</mn></mrow></math> The goal of the calibration method, discussed hereafter, is to determine the ϵ i for each block, i . This method is based on the study of π 0 electroproduction events eN → eNπ 0 → eNγγ where the two photons coming from the π 0 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 π 0 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 =2.10 6 eN → eNπ 0 events are generated following the kinematics presented in Table 1 Table 1 . This particular kinematics corresponds to one setting of the Hall A DVCS experiments [ 26 26 ]. The electromagnetic showers created by the π 0 photons in the calorimeter are fully simulated [ 27 27 ]. The generation and tracking of Čerenkov photons induced by the showers in the PbF 2 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 26 ]. This number is deduced from a <math id="M10"><mrow><mi>H</mi><mo stretchy="false">(</mo><mi>e</mi><mn>,</mn><msubsup><mi>e</mi><mrow><mtext>Calo</mtext></mrow><mo>'</mo></msubsup><msub><mi>p</mi><mrow><mtext>HRS</mtext></mrow></msub><mo stretchy="false">)</mo></mrow></math> elastic calibration where <math id="M11"><mrow><msubsup><mi>E</mi><mi>e</mi><mo>'</mo></msubsup><mo>=</mo><mn>3.16</mn></mrow></math> GeV scattered electrons are detected in the calorimeter: 1 1 k 0 (GeV) 1 1 <math id="M13"><mrow><msubsup><mi>k</mi><mn>0</mn><mo>'</mo></msubsup></mrow></math> (GeV) 1 1 θ kk ′ (deg) 1 1 <math id="M14"><mrow><mo><</mo><msubsup><mi>q</mi><mn>0</mn><mo>'</mo></msubsup><mo>></mo></mrow></math> (GeV) 1 1 θ kq ′ (deg) 1 1 ϕ kq ′ (deg) 1 1 4.455 1 1 [1.78, 1.96] 1 1 [25, 28] 1 1 2.4 1 1 [9, 28] 1 1 [-50, 50] <math display="block" id="M15"><mrow><mn>3.1</mn><mi>%</mi><mo>=</mo><mfrac><mrow><mi>σ</mi><mo stretchy="false">(</mo><msubsup><mi>E</mi><mi>e</mi><mo>'</mo></msubsup><mo stretchy="false">)</mo></mrow><mrow><msubsup><mi>E</mi><mi>e</mi><mo>'</mo></msubsup></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mrow><msqrt><mrow><msub><mi>N</mi><mrow><mtext>ph</mtext></mrow></msub><mi/><msubsup><mi>E</mi><mi>e</mi><mo>'</mo></msubsup><mo stretchy="false">(</mo><mtext>GeV</mtext><mo stretchy="false">)</mo></mrow></msqrt></mrow></mfrac><mn>.</mn></mrow></math> 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 16 ]. To mimic the uncertainty on the experimental calibration coefficients, the energies, E i , deposited in the calorimeter blocks are multiplied by random numbers, K i , varying uniformly between 1- K max and 1, where K max =40%, and become equal to: <math display="block" id="M16"><mrow><msubsup><mi>E</mi><mi>i</mi><mo>'</mo></msubsup><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>K</mi><mi>i</mi></msub><msub><mi>E</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo><mn>.</mn></mrow></math> The calibration method consists then to determine, from <math id="M17"><mrow><msubsup><mi>E</mi><mi>i</mi><mo>'</mo></msubsup><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo></mrow></math> and the known 4-vectors { k , k ′( j ), p }, the correction factors, ϵ i , 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 ϵ i ≈ 1/ K i with δ i = ϵ i K i -1 being the relative deviation of ϵ i from 1/ K i . 3 Calibration method 1 The calibration method is based on a χ 2 minimization between the measured π 0 energy, <math id="M18"><mrow><msubsup><mi>E</mi><mrow><msup><mi>π</mi><mn>0</mn></msup></mrow><mrow><mtext>rec</mtext></mrow></msubsup></mrow></math> , reconstructed from <math id="M19"><mrow><msubsup><mi>E</mi><mi>i</mi><mo>'</mo></msubsup></mrow></math> , and the expected π 0 energy, <math id="M20"><mrow><msubsup><mi>E</mi><mrow><msup><mi>π</mi><mn>0</mn></msup></mrow><mrow><mtext>cal</mtext></mrow></msubsup></mrow></math> , calculated from the π 0 momentum direction. The following subsections detail the different steps of the calibration. 3.1 Particle 4-vector reconstruction 2 The reconstructed π 0 energy in the simulation is the sum of the two shower energies created by the two photons of the π 0 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 28 ]. The energy of a shower is then the sum of the energies deposited in the N clus blocks belonging to the corresponding cluster: <math display="block" id="M21"><mrow><msubsup><mi>E</mi><mrow><mtext>sh</mtext></mrow><mo>'</mo></msubsup><mo>=</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mi>i</mi><mrow><msub><mi>N</mi><mrow><mtext>clus</mtext></mrow></msub></mrow></munderover></mstyle><msubsup><mi>E</mi><mi>i</mi><mo>'</mo></msubsup><mn>.</mn></mrow></math> Figure 2 Figure 2 shows an example of a π 0 event in the calorimeter where the corresponding clusters are indicated in red. The transverse coordinates ( x c , y c ) of a shower centroid are determined with a center of gravity based method from the coordinates ( x i , y i ) of the blocks belonging to the corresponding cluster [ 29 29 ]: <math display="block" id="M22"><mrow><msub><mi>x</mi><mi>c</mi></msub><mo>=</mo><mfrac><mrow><mstyle displaystyle="true"><munder><mo>∑</mo><mi>i</mi></munder></mstyle><msub><mi>w</mi><mi>i</mi></msub><msub><mi>x</mi><mi>i</mi></msub></mrow><mrow><mstyle displaystyle="true"><munder><mo>∑</mo><mi>i</mi></munder></mstyle><msub><mi>w</mi><mi>i</mi></msub></mrow></mfrac><mn>,</mn></mrow></math> with a similar equation for y c . The weight, w i , is given by: <math display="block" id="M23"><mrow><msub><mi>w</mi><mi>i</mi></msub><mo>=</mo><mi>max</mi><mrow><mo>{</mo><mrow><mn>0</mn><mo>;</mo><msub><mi>W</mi><mn>0</mn></msub><mo>+</mo><mi>ln</mi><mrow><mo>(</mo><mrow><mfrac><mrow><msubsup><mi>E</mi><mi>i</mi><mo>'</mo></msubsup></mrow><mrow><msubsup><mi>E</mi><mrow><mtext>sh</mtext></mrow><mo>'</mo></msubsup></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mo>}</mo></mrow><mn>,</mn></mrow></math> where W 0 is a free dimensionless parameter optimized for the considered energies [ 30 30 ]. The showers whose centroid coordinates belong to a block of the calorimeter edge (see Fig. 2 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 26 ], and the interaction vertex coordinates allows us to reconstruct the momentum direction of the particle creating the shower and thus its 4-vector <math id="M24"><mrow><mi>q</mi><mo>'</mo><mo stretchy="false">(</mo><msubsup><mi>E</mi><mrow><mtext>sh</mtext></mrow><mo>'</mo></msubsup><mn>,</mn><mover accent="true"><mi>q</mi><mo>→</mo></mover><mo>'</mo><mo stretchy="false">)</mo></mrow></math> , 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, <math id="M25"><mrow><msubsup><mi>E</mi><mi>i</mi><mo>'</mo></msubsup></mrow></math> , and thus to the calibration coefficients, but depends mainly on the block coordinates where a maximum energy is released by the shower [ 30 30 ]. Consequently, the angle between the particle direction and the virtual photon momentum, <math id="M26"><mrow><mover accent="true"><mi>q</mi><mo>→</mo></mover><mo>=</mo><mover accent="true"><mi>k</mi><mo>→</mo></mover><mo>−</mo><mover accent="true"><mi>k</mi><mo>→</mo></mover><mo>'</mo></mrow></math> , is well reconstructed even if inadequate calibration coefficients are employed. 3.2 eN → eNπ 0 reaction identification 2 In the simulated data, only eN → eNπ 0 → 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: <math display="block" id="M27"><mrow><msub><mi>M</mi><mrow><mtext>inv</mtext></mrow></msub><mo>=</mo><msqrt><mrow><msup><mrow><mo stretchy="false">(</mo><msubsup><mi>q</mi><mn>1</mn><mo>'</mo></msubsup><mo>+</mo><msubsup><mi>q</mi><mn>2</mn><mo>'</mo></msubsup><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow></msqrt><mn>,</mn></mrow></math> where <math id="M28"><mrow><msubsup><mi>q</mi><mn>1</mn><mo>'</mo></msubsup></mrow></math> and <math id="M29"><mrow><msubsup><mi>q</mi><mn>2</mn><mo>'</mo></msubsup></mrow></math> are the particle 4-vectors determined as described in the previous subsection. If these particles are photons coming from a π 0 decay, then their invariant mass should be equal to the π 0 mass <math id="M30"><mrow><msub><mi>M</mi><mrow><msup><mi>π</mi><mn>0</mn></msup></mrow></msub><mo>≈</mo><mn>0.135</mn></mrow></math> GeV, within detector resolution. In an experimental or simulated M inv distribution, π 0 events are located in a peak around M π 0 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 M inv ) as shown in the simulated M inv histogram of Fig. 3 Fig. 3 . A cut around the peak position, instead of M π 0 , is then applied to ensure the selection of π 0 events: <math display="block" id="M31"><mrow><mo>|</mo><msub><mi>M</mi><mrow><mtext>inv</mtext></mrow></msub><mo>−</mo><msub><mi>M</mi><mrow><mtext>peak</mtext></mrow></msub><mo>|</mo><mi/><mo><</mo><mi/><mn>3</mn><mi/><msub><mi>σ</mi><mrow><msub><mi>M</mi><mrow><mtext>inv</mtext></mrow></msub></mrow></msub><mn>,</mn></mrow></math> where σ M inv is the resolution of the reconstructed M inv variable. The second step of the eN → eNπ 0 reaction identification consists to compute for each π 0 event the missing mass squared defined by: <math display="block" id="M32"><mrow><msubsup><mi>M</mi><mi>x</mi><mn>2</mn></msubsup><mo>=</mo><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mi>p</mi><mo>−</mo><msup><mi>k</mi><mo>'</mo></msup><mo>−</mo><msubsup><mi>q</mi><mn>1</mn><mo>'</mo></msubsup><mo>−</mo><msubsup><mi>q</mi><mn>2</mn><mo>'</mo></msubsup><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mn>,</mn></mrow></math> which must be equal to the nucleon mass squared <math id="M33"><mrow><msubsup><mi>M</mi><mi>N</mi><mn>2</mn></msubsup><mo>≈</mo><mn>0.88</mn></mrow></math> GeV 2 within detector resolution if the considered event corresponds to a eN → eNπ 0 reaction and if the calorimeter is well calibrated. Figure 4 Figure 4 shows the simulated <math id="M34"><mrow><msubsup><mi>M</mi><mi>x</mi><mn>2</mn></msubsup></mrow></math> distribution where only eN → eNπ 0 events are generated. The peak position in Fig. 4 Fig. 4 is not centered around <math id="M35"><mrow><msubsup><mi>M</mi><mi>N</mi><mn>2</mn></msubsup></mrow></math> because of the E i smearing by the random factors, K i , 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 <math id="M36"><mrow><msubsup><mi>M</mi><mi>x</mi><mn>2</mn></msubsup></mrow></math> distribution and are located after the <math id="M37"><mrow><msubsup><mi>M</mi><mi>x</mi><mn>2</mn></msubsup></mrow></math> peak position (see Fig.2 Fig.2 in Ref. [ 26 26 ] for more details). To avoid the contamination from DIS events, the selection of eN → eNπ 0 is performed, as shown in Fig. 4 Fig. 4 , by applying the following <math id="M38"><mrow><msubsup><mi>M</mi><mi>x</mi><mn>2</mn></msubsup></mrow></math> cut: <math display="block" id="M40"><mrow><msubsup><mi>M</mi><mi>x</mi><mn>2</mn></msubsup><mi/><mo><</mo><mi/><msubsup><mi>M</mi><mrow><mtext>peak</mtext></mrow><mn>2</mn></msubsup><mo>+</mo><msub><mi>σ</mi><mrow><msubsup><mi>M</mi><mi>x</mi><mn>2</mn></msubsup></mrow></msub><mn>,</mn></mrow></math> where <math id="M41"><mrow><msubsup><mi>M</mi><mrow><mtext>peak</mtext></mrow><mn>2</mn></msubsup></mrow></math> is the peak position in the <math id="M42"><mrow><msubsup><mi>M</mi><mi>x</mi><mn>2</mn></msubsup></mrow></math> distribution and <math id="M43"><mrow><msub><mi>σ</mi><mrow><msubsup><mi>M</mi><mi>x</mi><mn>2</mn></msubsup></mrow></msub></mrow></math> 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 π 0 energy calculation 2 For each selected eN → eNπ 0 event, the reconstructed π 0 energy writes: <math display="block" id="M44"><mrow><msubsup><mi>E</mi><mrow><msup><mi>π</mi><mn>0</mn></msup></mrow><mrow><mtext>rec</mtext></mrow></msubsup><mo>=</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mn>208</mn><mo>=</mo><mn>16</mn><mo>×</mo><mn>13</mn></mrow></munderover></mstyle><msubsup><mi>E</mi><mi>i</mi><mo>'</mo></msubsup><mi/><msub><mi>d</mi><mi>i</mi></msub><mn>,</mn></mrow></math> where d i =1 if the block, i , belongs to one of the two clusters created by the π 0 and d i =0 otherwise. This reconstructed energy leads to particular values of <math id="M45"><mrow><msubsup><mi>M</mi><mi>x</mi><mn>2</mn></msubsup></mrow></math> and M inv which can be different from <math id="M46"><mrow><msubsup><mi>M</mi><mi>N</mi><mn>2</mn></msubsup></mrow></math> and <math id="M47"><mrow><msub><mi>M</mi><mrow><msup><mi>π</mi><mn>0</mn></msup></mrow></msub></mrow></math> 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 <math id="M48"><mrow><msubsup><mi>E</mi><mrow><msup><mi>π</mi><mn>0</mn></msup></mrow><mrow><mtext>cal</mtext></mrow></msubsup></mrow></math> hereafter, giving exactly <math id="M49"><mrow><msubsup><mi>M</mi><mi>x</mi><mn>2</mn></msubsup><mo>=</mo><msubsup><mi>M</mi><mi>N</mi><mn>2</mn></msubsup></mrow></math> and <math id="M50"><mrow><msub><mi>M</mi><mrow><mtext>inv</mtext></mrow></msub><mo>=</mo><msub><mi>M</mi><mrow><msup><mi>π</mi><mn>0</mn></msup></mrow></msub></mrow></math> . Equations (10) and (12) lead to: <math display="block" id="M51"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msubsup><mi>M</mi><mi>N</mi><mn>2</mn></msubsup></mrow></mtd><mtd columnalign="left"><mrow><mo>=</mo><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mi>p</mi><mo>−</mo><msup><mi>k</mi><mo>'</mo></msup><mo>−</mo><msubsup><mi>q</mi><mn>1</mn><mo>'</mo></msubsup><mo>−</mo><msubsup><mi>q</mi><mn>2</mn><mo>'</mo></msubsup><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow><mo>=</mo><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mi>p</mi><mo>−</mo><msup><mi>k</mi><mo>'</mo></msup><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>+</mo><msubsup><mi>M</mi><mrow><msup><mi>π</mi><mn>0</mn></msup></mrow><mn>2</mn></msubsup><mo>−</mo><mn>2</mn><mo stretchy="false">(</mo><msub><mi>k</mi><mn>0</mn></msub><mo>−</mo><msubsup><mi>k</mi><mn>0</mn><mo>'</mo></msubsup><mo>+</mo><msub><mi>M</mi><mi>N</mi></msub><mo stretchy="false">)</mo><msubsup><mi>E</mi><mrow><msup><mi>π</mi><mn>0</mn></msup></mrow><mrow><mtext>cal</mtext></mrow></msubsup><mo>+</mo><mn>2</mn><mo>∥</mo><mover accent="true"><mi>q</mi><mo>→</mo></mover><mo>∥</mo><msqrt><mrow><msup><mrow><mo stretchy="false">(</mo><msubsup><mi>E</mi><mrow><msup><mi>π</mi><mn>0</mn></msup></mrow><mrow><mtext>cal</mtext></mrow></msubsup><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>−</mo><msubsup><mi>M</mi><mrow><msup><mi>π</mi><mn>0</mn></msup></mrow><mn>2</mn></msubsup></mrow></msqrt><mi/><mi>cos</mi><mi>θ</mi><mn>,</mn></mrow></mtd></mtr></mtable></mrow></math> where θ is the angle between the momenta of the virtual photon and the π 0 . As mentioned above, θ is well defined experimentally even if biased calibration coefficients are employed. The physical solution of Eq. (15) is given by: <math display="block" id="M52"><mrow><msubsup><mi>E</mi><mrow><msup><mi>π</mi><mn>0</mn></msup></mrow><mrow><mtext>cal</mtext></mrow></msubsup><mo>=</mo><mfrac><mrow><mo>−</mo><mi>b</mi><mo>+</mo><msqrt><mrow><msup><mi>b</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mi>a</mi><mi>c</mi></mrow></msqrt></mrow><mrow><mn>2</mn><mi>a</mi></mrow></mfrac><mtext> </mtext><mn>,</mn></mrow></math> where: <math display="block" id="M53"><mtable columnalign="left"><mtr><mtd><mi>a</mi><mo>=</mo><mn>4</mn><mo stretchy="false">(</mo><msub><mi>k</mi><mn>0</mn></msub><mo>−</mo><msubsup><mi>k</mi><mn>0</mn><mo>'</mo></msubsup><mo>+</mo><msub><mi>M</mi><mi>N</mi></msub><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>−</mo><mn>4</mn><mo>∥</mo><mover accent="true"><mi>q</mi><mo>→</mo></mover><msup><mo>∥</mo><mn>2</mn></msup><msup><mstyle mathsize="140%" displaystyle="true"><mrow><mi>cos</mi></mrow></mstyle><mn>2</mn></msup><mi>θ</mi><mtext> </mtext><mn>,</mn></mtd></mtr><mtr><mtd><mi>b</mi><mo>=</mo><mn>4</mn><mo stretchy="false">(</mo><msub><mi>k</mi><mn>0</mn></msub><mo>−</mo><msubsup><mi>k</mi><mn>0</mn><mo>'</mo></msubsup><mo>+</mo><msub><mi>M</mi><mi>N</mi></msub><mo stretchy="false">)</mo><mrow><mo>[</mo><mrow><msubsup><mi>M</mi><mi>N</mi><mn>2</mn></msubsup><mo>−</mo><mi>（</mi><mi>k</mi><mo>−</mo><msup><mi>k</mi><mo>'</mo></msup><mo>+</mo><mi>p</mi><msup><mi>）</mi><mn>2</mn></msup><mo>−</mo><msubsup><mi>M</mi><mrow><msup><mi>π</mi><mn>0</mn></msup></mrow><mn>2</mn></msubsup></mrow><mo>]</mo></mrow><mtext> </mtext><mn>,</mn></mtd></mtr><mtr><mtd><mi>c</mi><mo>=</mo><mn>4</mn><msubsup><mi>M</mi><mrow><msup><mi>π</mi><mn>0</mn></msup></mrow><mn>2</mn></msubsup><mo>∥</mo><mover accent="true"><mi>q</mi><mo>→</mo></mover><msup><mo>∥</mo><mn>2</mn></msup><msup><mstyle mathsize="140%" displaystyle="true"><mrow><mi>cos</mi></mrow></mstyle><mn>2</mn></msup><mi>θ</mi><mo>+</mo><msup><mrow><mo>[</mo><mrow><msubsup><mi>M</mi><mi>N</mi><mn>2</mn></msubsup><mo>−</mo><mi>（</mi><mi>k</mi><mo>−</mo><msup><mi>k</mi><mo>'</mo></msup><mo>+</mo><mi>p</mi><msup><mi>）</mi><mn>2</mn></msup><mo>−</mo><msubsup><mi>M</mi><mrow><msup><mi>π</mi><mn>0</mn></msup></mrow><mn>2</mn></msubsup></mrow><mo>]</mo></mrow><mn>2</mn></msup><mtext> </mtext><mn>.</mn></mtd></mtr></mtable></math> Figure 5 Figure 5 shows the histogram of the relative difference between <math id="M54"><mrow><msubsup><mi>E</mi><mrow><msup><mi>π</mi><mn>0</mn></msup></mrow><mrow><mtext>cal</mtext></mrow></msubsup></mrow></math> and the real π 0 energy, <math id="M55"><mrow><msubsup><mi>E</mi><mrow><msup><mi>π</mi><mn>0</mn></msup></mrow><mrow><mtext>real</mtext></mrow></msubsup></mrow></math> , known for each event in the simulation. The comparison between the reconstructed π 0 energy, <math id="M56"><mrow><msubsup><mi>E</mi><mrow><msup><mi>π</mi><mn>0</mn></msup></mrow><mrow><mtext>rec</mtext></mrow></msubsup></mrow></math> , and <math id="M57"><mrow><msubsup><mi>E</mi><mrow><msup><mi>π</mi><mn>0</mn></msup></mrow><mrow><mtext>real</mtext></mrow></msubsup></mrow></math> is also shown in Fig. 5 Fig. 5 . The mean value and the small RMS of the <math id="M58"><mrow><mo stretchy="false">(</mo><msubsup><mi>E</mi><mrow><msup><mi>π</mi><mn>0</mn></msup></mrow><mrow><mtext>cal</mtext></mrow></msubsup><mo>−</mo><msubsup><mi>E</mi><mrow><msup><mi>π</mi><mn>0</mn></msup></mrow><mrow><mtext>real</mtext></mrow></msubsup><mo stretchy="false">)</mo></mrow></math> distribution relative to the <math id="M59"><mrow><mo stretchy="false">(</mo><msubsup><mi>E</mi><mrow><msup><mi>π</mi><mn>0</mn></msup></mrow><mrow><mtext>rec</mtext></mrow></msubsup><mo>−</mo><msubsup><mi>E</mi><mrow><msup><mi>π</mi><mn>0</mn></msup></mrow><mrow><mtext>real</mtext></mrow></msubsup><mo stretchy="false">)</mo></mrow></math> distribution indicate that <math id="M60"><mrow><msubsup><mi>E</mi><mrow><msup><mi>π</mi><mn>0</mn></msup></mrow><mrow><mtext>cal</mtext></mrow></msubsup></mrow></math> is much closer to the real π 0 energy than <math id="M61"><mrow><msubsup><mi>E</mi><mrow><msup><mi>π</mi><mn>0</mn></msup></mrow><mrow><mtext>rec</mtext></mrow></msubsup></mrow></math> . The calculated π 0 energy can then be used to adjust the calorimeter calibration and determine the correction factors ϵ i as detailed in the following subsection. 3.4 χ 2 minimization 2 For each eN → eNπ 0 event, j , <math id="M66"><mrow><msubsup><mi>E</mi><mrow><msup><mi>π</mi><mn>0</mn></msup></mrow><mrow><mtext>cal</mtext></mrow></msubsup><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo></mrow></math> gives a realistic estimation of the π 0 energy and thus of what the energy deposited in the calorimeter should be. The correction factors, ϵ i , are then those minimizing the χ 2 defined by: <math display="block" id="M67"><mrow><msup><mi>χ</mi><mn>2</mn></msup><mo>=</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mrow><msub><mi>N</mi><mrow><msup><mi>π</mi><mn>0</mn></msup></mrow></msub></mrow></munderover></mstyle><msup><mrow><mo stretchy="false">(</mo><msubsup><mi>E</mi><mrow><msup><mi>π</mi><mn>0</mn></msup></mrow><mrow><mtext>cal</mtext></mrow></msubsup><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo><mo>−</mo><msubsup><mi>E</mi><mrow><msup><mi>π</mi><mn>0</mn></msup></mrow><mrow><mtext>rec</mtext></mrow></msubsup><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mn>,</mn></mrow></math> where <math id="M68"><mrow><msub><mi>N</mi><mrow><msup><mi>π</mi><mn>0</mn></msup></mrow></msub></mrow></math> is the total number of events and <math id="M69"><mrow><msubsup><mi>E</mi><mrow><msup><mi>π</mi><mn>0</mn></msup></mrow><mrow><mtext>rec</mtext></mrow></msubsup><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo></mrow></math> is the reconstructed π 0 energy determined with the correction factors ϵ i : <math display="block" id="M70"><mrow><msubsup><mi>E</mi><mrow><msup><mi>π</mi><mn>0</mn></msup></mrow><mrow><mtext>rec</mtext></mrow></msubsup><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mn>208</mn></mrow></munderover></mstyle><msub><mi>ϵ</mi><mi>i</mi></msub><msubsup><mi>E</mi><mi>i</mi><mo>'</mo></msubsup><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo><msub><mi>d</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo><mn>.</mn></mrow></math> The minimization of Eq. (18) writes: <math display="block" id="M71"><mtable columnalign="left"><mtr><mtd><mfrac><mrow><mi>d</mi><msup><mi>χ</mi><mn>2</mn></msup></mrow><mrow><mi>d</mi><msub><mi>ϵ</mi><mi>k</mi></msub></mrow></mfrac><mo>=</mo><mo>−</mo><mn>2</mn><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mrow><msub><mi>N</mi><mrow><msup><mi>π</mi><mn>0</mn></msup></mrow></msub></mrow></munderover></mstyle><mrow><mo>[</mo><mrow><msubsup><mi>E</mi><mrow><msup><mi>π</mi><mn>0</mn></msup></mrow><mrow><mtext>cal</mtext></mrow></msubsup><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo><mo>−</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mn>208</mn></mrow></munderover></mstyle><msub><mi>ϵ</mi><mi>i</mi></msub><msubsup><mi>E</mi><mi>i</mi><mo>'</mo></msubsup><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo><msub><mi>d</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo></mrow><mo>]</mo></mrow><msubsup><mi>E</mi><mi>k</mi><mo>'</mo></msubsup><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo><msub><mi>d</mi><mi>k</mi></msub><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo>=</mo><mn>0</mn><mi/><mi/><mi/><mi/><mi/><mi/><mi/><mi/><mi/><mo>∀</mo><mi>k</mi><mo>=</mo><mn>1,2...208</mn></mtd></mtr></mtable></math> and yields to the following linear set of equations: <math display="block" id="M72"><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mn>208</mn></mrow></munderover></mstyle><mrow><mo>[</mo><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mrow><msub><mi>N</mi><mrow><msup><mi>π</mi><mn>0</mn></msup></mrow></msub></mrow></munderover></mstyle><msubsup><mi>E</mi><mi>i</mi><mo>'</mo></msubsup><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo><msub><mi>d</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo><msubsup><mi>E</mi><mi>k</mi><mo>'</mo></msubsup><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo><msub><mi>d</mi><mi>k</mi></msub><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo></mrow><mo>]</mo></mrow><msub><mi>ϵ</mi><mi>i</mi></msub><mo>=</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mrow><msub><mi>N</mi><mrow><msup><mi>π</mi><mn>0</mn></msup></mrow></msub></mrow></munderover></mstyle><msubsup><mi>E</mi><mrow><msup><mi>π</mi><mn>0</mn></msup></mrow><mrow><mtext>cal</mtext></mrow></msubsup><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo><msubsup><mi>E</mi><mi>k</mi><mo>'</mo></msubsup><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo><msub><mi>d</mi><mi>k</mi></msub><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo><mi/><mi/><mi/><mi/><mi/><mi/><mi/><mi/><mi/><mo>∀</mo><mi>k</mi><mo>=</mo><mn>1,2...208</mn></mrow></math> The correction factors, ϵ i , are then obtained by inverting the 208 × 208 matrix, <math id="M73"><mrow><msub><mi>α</mi><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo>=</mo><mstyle displaystyle="true"><msubsup><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mrow><msub><mi>N</mi><mrow><msup><mi>π</mi><mn>0</mn></msup></mrow></msub></mrow></msubsup></mstyle><msubsup><mi>E</mi><mi>i</mi><mo>'</mo></msubsup><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo><msub><mi>d</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo><msubsup><mi>E</mi><mi>k</mi><mo>'</mo></msubsup><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo><msub><mi>d</mi><mi>k</mi></msub><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo></mrow></math> . Experimentally, this minimization procedure provides the adjusted calibration coefficients defined by Eq. (3). 4 Results and discussion 1 Figure 6 Figure 6 shows the relative difference δ i = ϵ i K i -1 between the obtained correction factors, ϵ i , and the smearing factors, 1/ K i , as a function of the calorimeter block number. Figure 6 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 Fig. 5 where a clear decrease of the <math id="M74"><mrow><mo stretchy="false">(</mo><msubsup><mi>E</mi><mrow><msup><mi>π</mi><mn>0</mn></msup></mrow><mrow><mtext>rec</mtext></mrow></msubsup><mo>−</mo><msubsup><mi>E</mi><mrow><msup><mi>π</mi><mn>0</mn></msup></mrow><mrow><mtext>real</mtext></mrow></msubsup><mo stretchy="false">)</mo></mrow></math> distribution RMS is visible after this calibration. The calibration quality can also be seen in Fig. 3 Fig. 3 and Fig. 4 Fig. 4 where the peaks corresponding respectively to π 0 events and eN → eNπ 0 events are now centered around <math id="M75"><mrow><msub><mi>M</mi><mrow><msup><mi>π</mi><mn>0</mn></msup></mrow></msub></mrow></math> and <math id="M76"><mrow><msubsup><mi>M</mi><mi>N</mi><mn>2</mn></msubsup></mrow></math> with a smaller width. The average relative accuracy on the obtained correction factors or calibration coefficients can be defined as: <math display="block" id="M77"><mrow><mi>Δ</mi><mo>=</mo><msqrt><mrow><mfrac><mn>1</mn><mrow><mn>154</mn></mrow></mfrac><mstyle displaystyle="true"><munder><mo>∑</mo><mi>i</mi></munder></mstyle><msubsup><mi>δ</mi><mi>i</mi><mn>2</mn></msubsup></mrow></msqrt><mn>,</mn></mrow></math> where the sum runs over the block numbers not belonging to the calorimeter edge. This relative accuracy decreases when the number, <math id="M78"><mrow><msubsup><mi>N</mi><mi>π</mi><mn>0</mn></msubsup></mrow></math> , of eN → eNπ 0 events, used in the calibration procedure, increases. For other segmented calorimeters composed of N b blocks, Δ should rather depend of N π 0 / N b assuming a uniform π 0 distribution in the calorimeter acceptance. Figure 7 Figure 7 shows the evolution of Δ as a function of N π 0 / N b and demonstrates that less than 1% average accuracy on the calibration coefficients can be obtained if N π 0 / N b is larger than 100. A fit of Fig. 7 Fig. 7 points leads to the following empirical power law: <math display="block" id="M79"><mrow><mi>Δ</mi><mo stretchy="false">(</mo><mi>%</mi><mo stretchy="false">)</mo><mo>=</mo><mn>5.22</mn><mi/><mo stretchy="false">(</mo><msub><mi>N</mi><mrow><msup><mi>π</mi><mn>0</mn></msup></mrow></msub><mo>/</mo><msub><mi>N</mi><mi>b</mi></msub><msup><mo stretchy="false">)</mo><mrow><mo>−</mo><mn>0.386</mn></mrow></msup><mn>.</mn></mrow></math> The number N π 0 / N b depends experimentally on the integrated luminosity and the differential cross section of the eN → eNπ 0 process in the studied kinematics. For example, the average π 0 production rate for the kinematics of Tab. 1 Tab. 1 is <math id="M80"><mrow><msub><mi>N</mi><mrow><msup><mi>π</mi><mn>0</mn></msup></mrow></msub><mo>/</mo><msub><mi>N</mi><mi>b</mi></msub><mo>≈</mo><mn>90</mn></mrow></math> 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 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 → eNπ 0 events because of a possible contamination of the selected yield (Eq. (13)) by DIS events. A more selective <math id="M81"><mrow><msubsup><mi>M</mi><mi>x</mi><mn>2</mn></msubsup></mrow></math> 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 1 The present work discussed an energy calibration technique of laterally segmented electromagnetic calorimeters where the two photons of the π 0 decay produced by the eN → eNπ 0 reaction are detected. The calibration procedure is based on a χ 2 minimization between the reconstructed π 0 energy, from the calorimeter block data, and its expected energy. This energy is calculated for each event from the π 0 momentum direction, exploiting the good spatial resolution of the calorimeter. 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