The probabilities of SPMT's firing states

For each SPMT, the number of detected PEs obeys a Poisson distribution as follows: $\begin{array}{l}\text{Poisson}\left(k_i|{\mu }_i\right)=\frac{e^{-{\mu }_i}\times {\mu }_i{}^{k_i}}{k_i!}\mathrm{,}\hfill \end{array}$ (3) where ki is the nPE detected by the i^th SPMT and μi is its expected mean value. As mentioned above, to reduce the dependence on the charge reconstruction accuracy, the OCCUPANCY method was applied, which uses only the information from SMPT's firing states (fired or unfired). Therefore, only two states of ki should be considered: ki=0 (unfired) and ki<0 (fired). The probabilities of these two states are described by Eq. (4) and Eq. (5) [55, 56]. $\begin{array}{l}{P}_{\text{unfired}}\left({\mu }_i\right)=\text{Poisson}\left(k_i=0|{\mu }_i\right)=e^{-{\mu }_i}\hfill \end{array}$ (4) $\begin{array}{ll}{P}_{\text{fired}}\left({\mu }_i\right)\hfill & =\text{Poisson}\left(k_i>0|{\mu }_i\right)\hfill \\ \hfill & =1-P_{\text{unfired}}\left({\mu }_i\right)=1-e^{-{\mu }_i}\hfill \end{array}$ (5) In real detection, ki is smeared by fluctuations in the photoelectron detection, whereas μi is distorted because of the additional contribution from the PMT dark count. In addition, thresholds must be applied to each SPMT to avoid false triggering due to electronic noise; this also affects the observation of μi. In general, 0.3 PEs is chosen as the typical threshold because it can effectively handle electronic noise. Considering these effects, the probabilities of the unfired and fired states are: $\begin{array}{c}{P}_{\text{unfired}}\left({\mu }_i^{\text{true}}\right)=P\left(q_i<q_i^{\text{threshold}}|{\mu }_i^{\text{true}}\right)\\ =\text{Poisson}\left(k_i=0|{\mu }_i^{\text{true}}\right)+P_{\text{threLoss}}\left({\mu }_i^{\text{true}}\right)\mathrm{,}\\ P_{\text{fired}}\left({\mu }_i^{\text{true}}\right)=P\left(q_i\ge q_i^{\text{threshold}}|{\mu }_i^{\text{true}}\right)\\ =1-P_{\text{unfired}}\left({\mu }_i^{\text{true}}\right)\end{array}$ (6) where qi is the reconstructed charge of the i^th SPMT, ${\mu }_i^{\text{true}}\left({\mu }_i^{\text{true}}={\mu }_i^{\text{phy}}+{\mu }_i^{\text{dn}}\right)$ is the mean value of the Poisson distribution, which consists of two components:

(1) ${\mu }_i^{\text{phy}}$ caused by the visible energy of physics events.

(2) ${\mu }_i^{\text{dn}}$ introduced by the dark count (DR_i) of the i^th SPMT, which can be calculated using ${\mu }_i^{\text{dn}}=\text{D}{R}_i\times t$ in the time window of t.

$P_{\text{threLoss}}\left({\mu }_i^{\text{true}}\right)$ is the probability of $q_i<q_i^{\text{threshold}}$ (0.3 PEs in this study) for ki < 0, and is calculated as follows: $\begin{array}{l}{P}_{\text{threLoss}}\left[{\mu }_i^{\text{true}}\right]\\ ={\displaystyle \sum _{k_i=1}^n}\left[\text{Poisson}\left(k_i|{\mu }_i^{\text{true}}\right)\times {\displaystyle {\int }_0^{q_i^{\text{threshold}}}}\text{Gaus}\left(q_i|g_i\mathrm{,}\sigma \left(g_i\right)\right)d{q}_i\right]\end{array}$ (7) where $g_i=S_i^{\text{gain}}\times k_i$ and $\sigma \left(g_i\right)=\sqrt{g_i}\times {\sigma }_i^{\text{spe}}$, with n indicates the case of multiple PEs, $S_i^{\text{gain}}$ corresponds to the ratio between the real SPMT gain and the normal SPMT gain (3 × 10⁶) in JUNO, ${\sigma }_i^{\text{spe}}$ denotes spe resolution of the i^th SPMT. In real detection, $S_i^{\text{gain}}$, ${\sigma }_i^{\text{spe}}$ and DR_i can be obtained from PMT calibration.

Construction of the calibration map

JUNO designed a comprehensive calibration system [7] to understand the detector response by deploying multiple radioactive sources in various locations inside/outside the CD, including the Auto Calibration Unit (ACU), Cable Loop System (CLS), Guide Tube Calibration System (GTCS), and Remotely Operated Vehicle (ROV). Figure 4 shows the individual calibration systems in the CD and their scanning regions. For example, the ACU system scans the detector response along the central axis with multiple calibration sources, and the CLS system can scan in a 2-dimensional plane (X-Z plane) with multiple calibration sources using the central and side cables. The strategy of the JUNO calibration system was developed and optimized based on the Monte Carlo simulation results [57].

Fig. 4Full-screen View

The individual calibration systems in the CD and their scanning regions [13]

In energy reconstruction, ${\mu }_i^{\text{phy}}$ directly corresponds to the visible energy of an event in the detector, which is the basis of energy reconstruction in our method. Assuming that a calibration source is loaded at location $\overleftarrow{r}\left(r\mathrm{,}\theta \mathrm{,}\varphi =0\right)$ in the central detector, the mean value of the visible-energy-induced PEs for the i^th SPMT is ${\mu }_i^{\text{phy\_source}}$, which corresponds to the visible energy (denoted as E^source) of the calibration source. Subsequently, for an event that deposits energy at the same location, the relationship between the event's visible energy E_vis and ${\mu }_i^{\text{phy}}$ for the i^th SPMT can be described as follows: $\begin{array}{l}{\mu }_i^{\text{phy}}=\frac{E_{\text{vis}}}{E^{\text{source}}}\times {\mu }_i^{\text{phy\_source}}\hfill \end{array}$ (8) It should be noted that ${\mu }_i^{\text{phy}}$ is not only related to the visible energy and position of the event but also to the relative position (θ_SPMT) of the event and the i^th SPMT. This relationship can be determined using calibration data by constructing a calibration map. In Sect. 3.3, the maximum likelihood method was adopted to reconstruct the visible energy by estimating ${\mu }_i^{\text{phy}}$ using the calibration map and invoking the firing states of 25600 SPMTs from the data. Next, the construction of the calibration map is introduced.

Based on the calibration strategy of JUNO [57], this study used the ⁶⁸Ge source (positron source, with E^source=1.022 MeV) to calibrate the X-Z plane with assistance from both the ACU and CLS systems across 227 positions. Using the JUNO offline software [36-40], 10000 ⁶⁸Ge events were generated for each calibration location on the X-Z calibration plane. During the calibration, when the ⁶⁸Ge source is loaded at one of the 227 planned locations [58], the probability of the unfired state for the i^th SPMT can be estimated using Eq. (9). Where $N_{q_i<q_i^{\text{threshold}}}$ is the number of events with $q_i<q_i^{\text{threshold}}$, N_total is the total number of events for the calibration sample, and N_total=10000. Similarly to Eq. (6), we can derive the relationship between the observable quantities $P_{\text{unfired}}\left({\mu }_i^{\text{true\_source}}\right)$ and ${\mu }_i^{\text{true\_source}}$ (corresponding to the visible energy) in Eq. (10) for conversion during the subsequent reconstruction. In addition, for convenience, we introduce an intermediate variable ${\mu }_i^{\text{det\_source}}$ (the effective mean value of the detected PEs) for the i^th SPMT in Eq. (10) based on the mathematical form of Eq. (4). This functional form helps provide a more intuitive understanding of how PMT dark counts and the threshold effect impact ${\mu }_i^{\text{true\_source}}$. Subsequently, ${\mu }_i^{\text{det\_source}}$ is estimated using Eq. (11) using the calibration data. This calculation requires that the i^th SPMT is not fired for all 10000 events in the calibration sample (⁶⁸Ge); otherwise, this method is no longer applicable. The above extreme scenario is extremely unlikely because of the low visible energy of ⁶⁸Ge and the fact that even the most marginal of the 227 calibration positions is approximately 2 m away from its neighboring SPMTs. $\begin{array}{c}{P}_{\text{unfired}}\left({\mu }_i^{\text{true\_source}}\right)=P\left(q_i<q_i^{\text{threshold}}|{\mu }_i^{\text{true\_source}}\right)\\ =\frac{N_{q_i<q_i^{\text{threshold}}}}{N_{\text{total}}}\end{array}$ (9) $\begin{array}{l}{P}_{\text{unfired}}\left({\mu }_i^{\text{true\_source}}\right)\\ =\text{Poisson}\left(k_i=0|{\mu }_i^{\text{true\_source}}\right)+P_{\text{threLoss}}\left({\mu }_i^{\text{true\_source}}\right)\\ \stackrel{\text{def}}{=}{e}^{-{\mu }_i^{\text{det\_source}}}\end{array}$ (10) $\begin{array}{c}{\mu }_i^{\text{det\_source}}=-\ln P_{\text{unfired}}\left({\mu }_i^{\text{true\_source}}\right)\\ =-\ln \frac{N_{q_i<q_i^{\text{threshold}}}}{N_{\text{total}}}\end{array}$ (11) Note that ${\mu }_i^{\text{phy\_source}}$ is the value required for energy reconstruction (Eq. (8)), whereas ${\mu }_i^{\text{det\_source}}$ includes contributions from visible energy, PMT dark counts, charge smearing, and the threshold effect. According to Eq. (6), Eq. (10) and PMT parameters ($S_i^{\text{gain}}$, ${\sigma }_i^{\text{spe}}$ and DR_i) from PMT calibration, we can find the relationship between ${\mu }_i^{\text{phy\_source}}$ and ${\mu }_i^{\text{det\_source}}$. Figure 5 shows an example of an SPMT with an spe resolution and dark count rate of 30% and 1 kHz, respectively. The readout window was 1000 ns. It appears that dark counts dominate ${\mu }_i^{\text{det\_source}}$ for small ${\mu }_i^{\text{phy\_source}}$, whereas the combined effect of the dark count, smearing, and threshold remains stable at approximately 2% for the given setting as ${\mu }_i^{\text{phy\_source}}$ increases.

Fig. 5Full-screen View

The relationship between ${\mu }_i^{\text{det\_source}}$ and ${\mu }_i^{\text{phy\_source}}$ for an SPMT with an spe resolution and dark count rate of 30% and 1 kHz, respectively; and the readout window is set to 1000 ns. The bottom panel shows the ratio of ${\mu }_i^{\text{det\_source}}$/${\mu }_i^{\text{phy\_source}}$

Considering that the CD has good symmetry and the SPMTs with the same relative position with respect to the calibration source exhibit similar responses, to enhance the accuracy, we further group and combine SPMTs with similar θ_SPMT values in the calculation of ${\mu }_i^{\text{phy\_source}}$. In this study, θ_SPMT was divided into 1440 groups from 0° to 180° with 0.125° per group. This approach was successfully verified and applied in [14]. Therefore, SPMTs in the same θ_SPMT group have similar values of ${\mu }_i^{\text{phy\_source}}$ for each ⁶⁸Ge source location $\overleftarrow{r}\left(r\mathrm{,}\theta \mathrm{,}\varphi =0\right)$, the average of which is denoted as ${\mu }^{\text{phy\_source}}\left(\overleftarrow{r}\mathrm{,}{\theta }_{\text{SPMT}}\right)$. The calibration map can be constructed after ⁶⁸Ge scans 227 locations on the X-Z plane and all ${\mu }^{\text{phy\_source}}\left(\overleftarrow{r}\mathrm{,}{\theta }_{\text{SPMT}}\right)$ calculated. Considering the calibration performance and time consumption in JUNO, only approximately 227 calibration points were available in the current calibration strategy. Therefore, it was necessary to apply interpolation to the remaining positions. Figure 6a and b show examples of the calibration maps before and after interpolation, respectively.

Fig. 6Full-screen View

(Color online) Example of calibration map for one θ_SPMT group whose θ_SPMT values from 18° to 18.125°. (a) θ_SPMT in [18°, 18.125°); (b) θ_SPMT in [18°, 18.125°)

The following is a brief summary of the main steps involved in constructing the calibration map.

(1) For ⁶⁸Ge loading at location $\overleftarrow{r}\left(r\mathrm{,}\theta \mathrm{,}\varphi =0\right)$, calculate the effective mean value of the detected PEs (${\mu }_i^{\text{det\_source}}$) for each SPMTs using Eq. (11).

(2) Calculate ${\mu }_i^{\text{phy\_source}}$ by correcting PMT dark counts and threshold effect, Fig. 5 shows an example of the relationship between ${\mu }_i^{\text{det\_source}}$ and ${\mu }_i^{\text{phy\_source}}$ for an SPMT.

(3) Calculate ${\mu }^{\text{phy\_source}}\left(\overleftarrow{r}\mathrm{,}{\theta }_{\text{SPMT}}\right)$, which is the average value of ${\mu }_i^{\text{phy\_source}}$ for each θ_SPMT group.

(4) Repeat the above steps for the calibration data at all locations.

(5) Apply interpolation to the remaining positions.

(6) For a given ⁶⁸Ge location $\overleftarrow{r}\left(r\mathrm{,}\theta \mathrm{,}\varphi =0\right)$, the ${\mu }^{\text{phy\_source}}\left(\overleftarrow{r}\mathrm{,}{\theta }_{\text{SPMT}}\right)$ value corresponding to each SPMTs can be obtained.

The calibration map was generated using the calibration data on the X-Z plane (ϕ=0) by considering that the detector exhibits good symmetry along the ϕ direction [57]. Based on the detailed study in [13], the ϕ symmetry is reliable for events located within 16 m. However, considering that the acrylic sphere is supported by a stainless-steel latticed shell through acrylic nodes and connecting bars, the dependence on ϕ near the edge cannot be ignored anymore due to the shadowing effect caused by the numerous acrylic nodes. In the future, the ϕ symmetry needs to be checked and validated using real data, which can be corrected if necessary.

Construction of maximum likelihood function

To reconstruct the visible energy of a cluster-like event whose energy-deposit center is known in the detector, a likelihood function can be constructed as follows: $\mathcal{L}={\displaystyle \prod _1^{N_{\text{unfired}}}}{P}_{\text{unfired}}\left({\mu }_i^{\text{phy}}\right){\displaystyle \prod _1^{N_{\text{fired}}}}{P}_{\text{fired}}\left({\mu }_i^{\text{phy}}\right)$ (12) where $N=N_{\text{unfired}}+N_{\text{fired}}=25600$, N_unfired and N_fired correspond to the number of SPMTs with qi<0.3 PEs and q_i<0.3 PEs, respectively. $P_{\text{unfired}}\left({\mu }_i^{\text{phy}}\right)$ and $P_{\text{fired}}\left({\mu }_i^{\text{phy}}\right)$ are the probabilities of the unfired and fired states, respectively, and can be calculated using Eq. (6). In the calculation, ${\mu }_i^{\text{phy}}$ of each SPMT can be estimated by considering the relationship in Eq. (8) and invoking the ${\mu }^{\text{phy\_source}}\left(\overleftarrow{r}\mathrm{,}{\theta }_{\text{SPMT}}\right)$ value from the calibration map. In addition, the differences in quantum efficiency (QE) between individual SPMTs in the same θ_SPMT group should be considered [59]. Consequently, ${\mu }_i^{\text{phy\_source}}$ of the i^th SPMT needs a correction that can be calculated as follows: $\begin{array}{l}{\mu }_i^{\text{phy\_source}}=\frac{Q{E}_i}{\frac{1}{m}{\displaystyle {\sum }_{j=0}^{m}Q{E}_j}}\times {\mu }^{\text{phy\_source}}\left(\overleftarrow{r}\mathrm{,}{\theta }_{\text{SPMT}}\right)\hfill \end{array}$ (13) where m is the number of SPMTs in the θ_SPMT group classified by the θ_SPMT value, and QEj is the QE of the j^th SPMT in this group.

Next, the ROOT's minimization class TMinuit2Minimizer [60-62] is used to minimize $-\ln \mathcal{L}$. In the minimization, the visible energy E_vis of the cluster-like event (whose energy-deposit center is known and the parameter to be determined is introduced in Sect. 4.1); its initial value (E_initial) can be estimated using Eq. (14) using the total number of PEs (totalPE) of all the SPMTs to reduce the reconstruction time. totalPE^source(r) is totalPE observed by 25600 SPMTs for the ⁶⁸Ge source located at different positions (Fig. 7. Comparing the value of totalPE^source(r) at the center of the CD and around r=15 m, there is approximately 7% nonuniformity introduced by the reception of PMTs and the optical attenuation effect in the process of photon transmission. A decreasing radius larger than ~15.6 m was primarily caused by the total reflection and shadowing effects. $\begin{array}{l}{E}_{\text{initial}}=\frac{\text{totalPE}\left(r\right)}{{\text{totalPE}}^{\text{source}}\left(r\right)}\times E^{\text{source}}\hfill \end{array}$ (14) After minimization, TMinuit returns the predicted visible energy value as a result of energy reconstruction. Furthermore, the response difference caused by the spatial scale of the energy deposition is investigated in Sect. 3.4, and it was found that it had little influence on the analysis.

Fig. 7Full-screen View

(Color online) The total number of PEs observed by 25600 SPMTs for ⁶⁸Ge source located at different positions

Comparison of cluster-like event and point-like event

⁶⁸Ge was used as the positron source. The positron annihilated in the source capsule (stainless steel + PTFE) with a pair of 0.511 MeV γs emitted. Most of the energy of γs is deposited within ~30 cm in the LS; therefore, ⁶⁸Ge is not strictly a point-like source but a cluster-like source similar to the high-energy electrons described in Sect. 1. In [13], it was found that accurate energy reconstruction below 10 MeV can be affected by the cluster size of the energy deposition. In this study, we constructed a calibration map using the approximate point-like ⁶⁸Ge source (cluster size in ~30 cm) to determine the location-dependent ${\mu }_i^{\text{phy\_source}}$ from calibration data and carry out energy reconstruction. Therefore, it is necessary to investigate whether this calibration map can be applied to higher energy electrons with larger cluster sizes (up to several meters) by assuming that they are point-like events with their energy-deposit centers as the event positions.

According to Eq. (8), ${\mu }_i^{\text{phy}}$ can be calculated for an event with a known position in the LS with E^source and ${\mu }_i^{\text{phy\_source}}$ obtained from the calibration map. In the energy reconstruction, our algorithm assumes that all energy deposition is equivalent to those occurring at the energy-deposit center. However, in real detection, ${\mu }_i^{\text{phy}}$ of the i^th SPMTs are contributed by the cumulative effect of each secondary energy deposition. ${\mu }_i^{\text{phy}}$ was adopted using different strategies to estimate the potential bias caused by the spatial scale of energy deposition for high-energy events. As shown in Eq. (15) and Eq. (16), they were calculated as point- and cluster-like events, respectively. ${\mu }_i^{\text{phy}}{|}_{\text{point}-\text{like}}=\frac{1}{E^{\text{source}}}\times {\displaystyle \sum _{\alpha =1}^{N_E}}{E}_{\alpha }\times {\mu }_i^{\text{phy\_source}}\left(\overleftarrow{r}\right)$ (15) ${\mu }_i^{\text{phy}}{|}_{\text{cluster}-\text{like}}=\frac{1}{E^{\text{source}}}\times {\displaystyle \sum _{\alpha =1}^{N_E}}{E}_{\alpha }\times {\mu }_i^{\text{phy\_source}}\left(\overleftarrow{r_{\alpha }}\right)$ (16) The official simulation software of JUNO was used to generate electron samples with different kinetic energies, and the details of each secondary energy deposition (Eα and $\overleftarrow{r_{\alpha }}$) were recorded to calculate Eq. (15) and Eq. (16). As shown in Eq. (17), the sum of all SPMT's ${\mu }_i^{\text{phy}}$ is directly proportional to the visible energy of the event. Therefore, any bias present in the sum of all SPMT's ${\mu }_i^{\text{phy}}$ indicates a deviation in the visible energy. We compared the calculation results of the different strategies, as shown in Eq. (18) and Fig. 8. It was found that the ratio approaches 1 with an increase in r³, and has a small bias (≤0.5%) at the edge for electrons with kinetic energies larger than 200 MeV. This result indicates that our algorithm and reconstruction strategy are applicable to the energy reconstruction of high-energy events with large spatial scales of energy deposition. $\begin{array}{l}{\displaystyle \sum _{i=1}^N}{\mu }_i^{\text{phy}}=\frac{E_{\text{vis}}}{E^{\text{source}}}\times {\displaystyle \sum _{i=1}^N}{\mu }_i^{\text{phy\_source}}\hfill \end{array}$ (17) $\begin{array}{l}Ratio=\frac{{\displaystyle {\sum }_{i=1}^N{\mu }_i^{\text{phy}}{|}_{\text{cluster}-\text{like}}}}{{\displaystyle {\sum }_{i=1}^N{\mu }_i^{\text{phy}}{|}_{\text{point}-\text{like}}}}\hfill \end{array}$ (18)

Fig. 8Full-screen View

Comparison between the calculation results of point-like and cluster-like treatment. The ratio, defined in Eq. (18), was found to be close to 1 for electron samples with different kinetic energies at various positions. On this plot, the black vertical dotted line corresponds to r=14.6 m, and the red vertical dotted line (r=15.6 m) corresponds to the boundary of the total reflection region, which is caused by the larger refractive index of the LS (which has a similar refractive index to the Acrylic) than water

Reconstruction of energy-deposit center

As introduced in Sect. 3.3, the energy-deposit center of the event is required to reconstruct the visible energy. Therefore, before energy reconstruction, it is necessary to reconstruct the energy-deposit center. The energy-deposit center reconstruction of high-energy cluster-like events faces greater dispersion compared with the vertex reconstruction of point-like events (Sect. 1). After the investigation, it was found that the time-based algorithm developed and verified by [63] was suitable and could be applied to our analysis.

The time-based algorithm uses the distribution of the time-of-flight (t.o.f.) corrected time Δt (Eq. (19)) of an event to reconstruct its vertex and t₀ (event time). The principle of the time-based algorithm is that the Δt distribution is independent of the event vertex after applying time-of-flight correction. In this study, we applied it to the reconstruction of an energy-deposit center. $\begin{array}{l}\text{Δ}{t}_i=t_i-\text{t}\mathrm{.}\text{o}\text{.f}{\mathrm{.}}_i\hfill \end{array}$ (19) In Eq. (19), ti is the first hit time of the i^th SPMT and t.o.f._i is the time of flight from the energy-deposit center to the i^th SPMT. In the calculation of t.o.f._i, the optical path length includes both the length in the LS and in water. Subsequently, a correction vector was constructed and minimized by iterating the energy-deposit center. Further details are provided in [63]. Finally, the reconstructed energy-deposit center can be obtained and Fig. 15 and Fig. 9 show the performance. The reconstruction biases of θ (Fig. 15b and ϕ (Fig. 15c remain small and stable as the energy increases, whereas the reconstruction bias of r (Fig. 15a gradually increases with the energy. However, it could still be controlled to within 150 mm at 1 GeV. This bias is still acceptable considering that the cluster size could be several meters for a 1 GeV electron. The effect of the deviation on the energy reconstruction will be discussed later. However, the reconstruction bias of r tends to decrease at the edge of the detector compared to other regions, mainly because of energy leakage near the edge, especially for high-energy events.

Fig. 9Full-screen View

Reconstruction resolution of electron's energy-deposit center. (a) Reconstruction resolution of r; (b) Reconstruction resolution of θ; (c) Reconstruction resolution of ϕ

In Fig. 9, it can be observed that the reconstruction resolutions of r, θ and ϕ increase with electron energy between 50 MeV and 1 GeV. For example, the resolutions of r, θ and ϕ were approximately 100 mm, 0.5° and 1.0° for 50 MeV electrons, respectively, whereas the resolutions of r, θ and ϕ were approximately 340 mm, 1.8° and 3.0° for 1 GeV electrons, respectively. This effect is primarily due to the greater dispersion of the energy deposition of high-energy electrons. The resolutions of 10 MeV electrons are slightly larger than that of 50 MeV electrons. This is because the hit number of the SPMTs is small (~400 PEs for 10 MeV) and less information is available for reconstruction using the time-based algorithm.

Energy reconstruction performance

Next, the performance of energy reconstruction is introduced. The reconstructed energy spectra of the electrons with different kinetic energies are shown in Fig. 10. The blue spectra correspond to events whose energies were fully contained (FC) in the LS, whereas the green spectra correspond to events whose energies were partially contained (PC) in the LS. For electrons with energies greater than 500 MeV, the proportion of PC events increased, and the 16 m cut cannot completely exclude the case of energy leakage. The FC spectra could be fitted well with a Gaussian function, and the reconstructed energy was approximately 6% higher than the deposited energy of the electron. According to the official simulation in JUNO, when anchored at the 2.223 MeV gamma peak generated by (n,γ)H, high-energy electrons exhibit an energy nonlinearity of ~6% [64]. Thus, this deviation is understood to be mainly caused by the energy nonlinearity response of the LS. Conversely, the nonuniformity of the energy reconstruction may also introduce some small deviations, but generally, less than 1%, as shown in Fig. 11c.

Fig. 10Full-screen View

Discrete energy reconstruction with reconstructed edep vertex after electronic simulation and charge reconstruction. The blue line is the FC events, and the green line is the PC events. According to the fitting results (red line) of FC spectra, it can be observed that the reconstructed visible energy is about 6% larger than the deposited energy of the electron. More specifically, for electrons with kinetic energies of 10 MeV, 50 MeV, 200 MeV, 500 MeV, 700 MeV, and 1 GeV, the ratio of reconstructed visible energy to deposited energy is found to be 1.062, 1.061, 1.058, 1.059, 1.060 and 1.064, respectively. The corresponding explanation is provided in the text

Fig. 11Full-screen View

Uniformity of discrete energy reconstruction for FC events. On each plot, black vertical dotted lines correspond to r=14.6 m, and red vertical dotted lines (r=15.6 m) correspond to the boundary of the total reflection region, which is caused by a larger refractive index of the LS (which has a similar refractive index to the Acrylic) than water. (a) Without electronic simulation and charge reconstruction, using true energy-deposit center for energy reconstruction; (b) With electronic simulation and charge reconstruction, using true energy-deposit center for energy reconstruction; (c) With electronic simulation and charge reconstruction, using reconstructed energy-deposit center for energy reconstruction

To understand the non-uniformity shown in Fig. 11c, a and b can be compared, which corresponds to the cases using true energy-deposit center without/with electronic simulation and charge reconstruction. In Fig. 11a, for electron samples with different energies, the non-uniformity is consistent at about 0.5% from the center of the detector to the edge. After the electronic simulation and charge reconstruction (Fig. 11b, there is a slight increase in non-uniformity, but it still remains within 1.5%. Fig. 11c corresponds to the case using reconstructed energy-deposit center that shows deviation (Fig. 15, as a result, for 500 MeV and 1 GeV electrons, their energy non-uniform are about 2% and 3% at the edge. Furthermore, when PC events were included (Fig. 12, they mainly affected the non-uniformity of high-energy electrons located in the edge region.

Fig. 12Full-screen View

Uniformity of discrete energy reconstruction for all (FC+PC) events. On each plot, black vertical dotted lines correspond to r=14.6 m, and red vertical dotted lines (r=15.6 m) correspond to the boundary of the total reflection region, which is caused by a larger refractive index of the LS (which has a similar refractive index to the Acrylic) than water. (a) Without electronic simulation and charge reconstruction, using true energy-deposit center for energy reconstruction; (b) With electronic simulation and charge reconstruction, using true energy-deposit center for energy reconstruction; (c) With electronic simulation and charge reconstruction, using reconstructed energy-deposit center for energy reconstruction

Figure 13 shows the energy resolution performance. The solid points correspond to the reconstruction results of the FC events, whereas the hollow points include both FC events and PC events. The red squares and pink stars denote the cases using the true energy-deposit center for energy reconstruction, whereas the green triangles denote the reconstructed energy-deposit center. In addition, electronic simulation, and charge reconstruction were applied to the reconstruction results shown by the pink stars and green triangles. Comparing energy resolutions in different conditions, it can be found that the energy reconstruction performance of the high-energy events is good using the OCCUPANCY strategy and the energy resolution is about 0.8% for 1 GeV electrons in the ideal case (red solid squares). In a more realistic situation, by including electronic simulation and charge reconstruction, the resolution is only approximately 0.3% worse, indicating that the correction works well in controlling the influence of the PMT dark count and threshold effect. From the comparison of solid points and hollow points, the PC events mainly affect the electrons whose kinetic energy is larger than 100 MeV, and their energy resolutions deteriorate by approximately 1%. In real detection, the reconstructed energy-deposit center is required for energy reconstruction, and its smearing introduces additional smearing on the reconstructed energy, especially for high-energy electrons. Consequently, the energy resolution was approximately 3.2% for 1 GeV electrons based on our algorithm. In Fig. 14, the relationship between the energy resolution and the fired ratio of the SPMT is investigated using an electron sample without electronics simulation, and a true energy-deposit center was applied. In general, a higher fired ratio of the SPMT corresponds to a better energy resolution. This indicates that our algorithm has the potential to be applied to higher energy events when the fired ratio of the SPMT is not close to 1 but must solve the problem of energy-deposit center reconstruction, which has a larger bias at higher energies.

Fig. 13Full-screen View

Resolution in several discrete energy in simulation phase. The red squares correspond to the resolution in the detector simulation. The pink stars and the green triangles correspond to the resolution after electronic simulation and charge reconstruction. The energy reconstructions of the red squares and pink stars use the true center of energy deposition, while the green triangles utilize the reconstructed center for their energy reconstruction. The energy reconstruction of the red squares and the pink stars uses the true energy-deposit center, while the energy reconstruction of the green triangles uses the reconstructed energy-deposit center. On the other hand, solid and hollow are used to mark the FC events and FC+PC events, respectively

Fig. 14Full-screen View

The resolution varies with the fired ratio of SPMT at different energies

Fig. 15Full-screen View

Reconstruction bias of electron's energy-deposit center. (a) Reconstruction bias of r; (b) Reconstruction bias of θ direction; (c)Reconstruction bias of ϕ direction

To apply the algorithm to the experiment, we need to consider how to perform performance validation in the future, especially given the lack of calibration sources in the hundreds of MeV range, which is a common challenge in the validation of high-energy event reconstruction. First, we can select calibration samples in the tens of MeV range from the data for algorithm performance verification, such as ¹²B and Michel electrons. In addition, MC tuning is helpful for examining the various uncertainties that may arise from extrapolating low-energy events to high-energy events and estimating their potential impact.