Software setup

A Monte Carlo simulation of the ECAL was conducted using the DarkSHINE Software (DSS) framework, built on Geant4 [44, 45]. This simulation included the entire detector setup, encompassing the LYSO crystal cubes, wrappers, silicon sensors, support structures, and additional components such as the tracker and HCAL. This comprehensive configuration served as the reference for ECAL reconstruction, while the primary analysis established the appropriate phase space for ECAL evaluation. The DSS framework facilitated the simulation of all Standard Model background processes involving electron-target interactions, as well as various dark photon processes, and their respective detector responses.

The target was used in a “full simulation” scenario, in which an 8 GeV electron beam impacted the target, as outlined in the experimental design, and the resulting secondary particles passed through all detector components. In addition, an “ECAL-unit simulation” was conducted to isolate the response of a single ECAL unit, allowing for detailed analysis of its behavior and facilitating direct comparison with laboratory test results during the R&D phase. All simulations were conducted using consistent material configurations.

Signal and background

Figure 4 illustrates the background processes accounted for in the DarkSHINE experiment and their relative rates under an 8 GeV electron beam. Typically, incident electrons pass through the target with minimal interaction. Approximately 6.7% of the electrons undergoes hard bremsstrahlung, where bremsstrahlung photons carry more than 4 GeV of energy away from the incident electrons. These photons may then interact with the target or ECAL materials via photon‒nuclear interactions, which occur at relative rates of 2.31 ×10^-4 and 1.37 × 10^-6, respectively. Occasionally, bremsstrahlung photons may convert into muon pairs, with relative rates of 1.63 × 10^-6 in the ECAL and 1.50 × 10^-8 in the target. Additionally, these photons frequently produce electron‒positron pairs. Electron-nuclear interactions within the ECAL and target materials also contribute to the background, with relative rates of 3.25 × 10^-6 and 5.10 × 10^-7, respectively [33].

Fig. 4Full-screen View

Background processes observed in the DarkSHINE experiment and their relative rates, simulated under an 8 GeV electron beam

When a dark photon is produced during the electron-on-target process, most of the incident energy transfers to it, while the recoiled electron deposits the residual energy in the ECAL. Dark photons subsequently decay into dark matter, leaving no observable signal in the tracker or calorimeter and resulting in significant energy loss in the ECAL. As depicted in Fig. 5, over 75% of events from 1 MeV dark photon signals exhibit energy deposits below 4 GeV in the ECAL. As the dark photon mass increases, it carries more energy away from the electrons, further enhancing the measured energy loss in the ECAL.

Fig. 5Full-screen View

(Color online) Energy deposition in the ECAL for 8 GeV electrons during dark photon processes and inclusive background interactions. In dark photon events, the deposited energy in the ECAL typically falls below 4 GeV, with energy decreasing as the dark photon mass increases. Conversely, inclusive background processes exhibit substantially higher energy deposition within the ECAL

The inclusive process encompasses all potential Standard Model interactions occurring within the detector, weighted by their branching ratios. In comparison to dark photon signals, most events in the inclusive background deposit a substantial portion of their energy in the ECAL, making the total energy measurement in the ECAL a strong criterion on background rejection.

During the inclusive process, hard bremsstrahlung or photon conversion into electron-positron pairs can create electromagnetic showers in the ECAL, leading to significant energy deposition. In contrast, complex processes involving the production of high-energy muon pairs or hadrons generally result in minimal energy deposition in the ECAL and can be effectively filtered out using the tracker and HCAL.

Neutrino-producing background processes, such as Moller scattering ($e^{-}{e}^{-}\to e^{-}{e}^{-}$) followed by a charged-current quasi-elastic (CCQE) reaction ($e^{-}p\to {\nu }_{e}n$), neutrino pair production ($e^{-}N\to e^{-}N\nu \nu$), bremsstrahlung with CCQE, and charge-current exchange with exclusive channels ($e^{-}p\to \nu n{\pi }^0$), as outlined in [46], were considered negligible in this experiment due to their low event rates. Consequently, these background processes were excluded from the DarkSHINE simulations [33].

In summary, the design and function of the ECAL focus on excluding backgrounds that do not involve hard muons or hadrons.

Energy deposition in different regions

Using the optimized ECAL volume described in Sect. 4, the energy deposition in the crystals for dark photon signals and background processes is shown in Fig. 8. Inclusive background processes typically result in higher energy deposits in the crystals, often up to 4 GeV. In comparison, energy deposition from dark photon signals tends to be slightly lower, with smaller dark photon masses leading to higher energy deposits. A 1 MeV dark photon can deposit up to 3.5 GeV in the crystals, as such dark photons have a low probability of carrying minimal kinetic energy, allowing the recoiling electron to retain most of the incident energy.

Fig. 8Full-screen View

(Color online) Energy deposited in individual crystals by an 8 GeV electron beam incident on the ECAL. For inclusive background processes, the energy absorbed by a single crystal can reach up to 4 GeV, while dark photon signals can result in energy deposits of up to 3.5 GeV

The ECAL was divided into four sections from the center to the periphery (Fig. 9) to study energy deposition in different regions. The number of crystals in each region is listed in Table 4. Region 1 (red) represents the core area of shower development and contains 125 crystals, each measuring 2.5 cm×2.5 cm×4 cm, roughly equivalent to 18 radiation lengths. This region covers the entire beam spot with a radius of 3 cm, and an additional area equivalent to one Molière radius. The energy absorbed by crystals in each of these four regions is shown in Fig. 10. Crystals in the central and near-central regions absorb significantly more energy than those in Regions 3 and 4, which are farther from the shower center and absorb noticeably less energy. For dark photon signals, energy deposition was even lower than that for inclusive background processes. Figure 11 shows that crystals in Region 1, which constitute only 2.6% of the ECAL volume, absorb more than 90% of the energy for inclusive backgrounds.

Fig. 9Full-screen View

(Color online) ECAL divided into four regions from the center to the periphery. Red: region- 1, green: region- 2, blue: region- 3, yellow: region- 4. The beam enters the detector at the center, traveling in the longitudinal (z) direction from left to right

Fig. 10Full-screen View

(Color online) Energy absorbed by single crystals in the four regions shown in Fig. 9. Crystals in the central regions (Region- 1 and Region- 2) absorb significantly more energy, whereas crystals in Region- 3 and Region- 4, located farther from the shower center, absorb noticeably less energy

Fig. 11Full-screen View

(Color online) Energy ratio in different regions shown in Fig. 9 for various processes. The energy ratio is defined as the ratio of the total energy deposited in the crystals within a specific region ($E_i^{\text{Total}}$) to the total energy deposited across all crystals in the ECAL ($E_{\text{ECAL}}^{\text{Total}}$). Both signal and background processes deposit the highest energy in Region- 1. However, for dark photon processes, as the dark photon mass increases, a greater proportion of the energy is absorbed in the outer regions

Energy limits on channels

The setup of the dynamic energy range is crucial for the accuracy of energy measurements in the ECAL. Typically, a narrower dynamic range results in higher measurement precision. To determine the minimum dynamic range that fulfills the ECAL’s energy measurement requirements, an energy cap was applied to each crystal. This cap simulated the saturation behavior seen in real detectors when the deposited energy exceeded the set limit. If this energy limit was too low, it could lead to an overall reduction in the energy recorded by the ECAL, thereby misidentifying high-energy background events as dark photon signals due to significant energy underestimation. The analysis focused on events characterized by substantial energy losses, assessing how different energy limits influenced the count of events where the total energy measured by the ECAL dropped below 4 GeV.

Figure 12 shows the total energy distribution in the ECAL following the application of various energy limits to the channels. As the energy limit decreased below 3 GeV, a notable decline in the total energy measured by the ECAL was observed, which parallels the saturation effect resulting from an inadequate dynamic range in the detector’s electronics. When the energy limit exceeded 3 GeV, the total energy recorded by the ECAL remained largely unaffected by the imposed limit. To maintain accurate energy measurements within the ECAL, it is essential that the dynamic range of the individual channels does not compromise the total energy recorded. Consequently, as demonstrated in Fig. 12, a dynamic range exceeding 3 GeV is deemed appropriate.

Fig. 12Full-screen View

(Color online) Energy distribution in the ECAL for inclusive background processes after applying various energy limits to individual channels. The energy limit represents the maximum energy that a single channel can record, simulating saturation effects observed in detector electronics. When this limit is set too low, the measured energy in the ECAL can significantly diverse from the actual incident energy, resulting in a shift towards lower energy values. This misrepresentation increases the risk of misidentifying background events as dark photon signals

Table 5 lists the ratios between the number of events with ECAL energies less than 4 GeV and the total number of simulated events. These ratios were derived from background and dark photon signals of varying masses after applying different channel energy limits. As the energy limit decreases, the inclusive background experiences greater energy loss, resulting in an increasing number of cases in which the ECAL energy is less than 4 GeV. The results indicate that a dynamic range of 100 MeV or 500 MeV is insufficient, as these limits exhibit a poor background rejection capability with a statistical sample of one million events. When the energy limit on the channels is set above 500 MeV, nearly all events deposit more than 4 GeV of energy in the ECAL, except for two photon‒muon pair processes and one nuclear process, which can be effectively vetoed by the HCAL. This indicates that very few crystals exhibited energy depositions exceeding 1 GeV, as reflected in the normalized entries shown in Figure 8. Therefore, to ensure effective background rejection, the energy limit should be set to at least 1 GeV.

Overall, a 4 GeV dynamic energy range is optimal for the DarkSHINE ECAL, as it maintains the accuracy required for energy measurements while ensuring strong background rejection capability. Future detector readout electronics can be designed based on this standard.

Trigger efficiency

In the high-frequency beam environment of DarkSHINE, the majority of events collected by the detector were background events of limited interest. These background events can significantly strain data transmission and storage at the backend. Implementing an online trigger to filter out events that do not match the characteristics of dark photon signals can significantly reduce resource usage. The preliminary trigger design involves summing the energy across selected channels in the ECAL. If the summed energy was less than 4 GeV, the event would be saved. Trigger efficiency is defined as the ratio of the number of triggered events, where the summed energy in the trigger region is less than 4 GeV, to the total number of events. Table 6 lists the trigger efficiencies across the different trigger regions.

In Table 6, a 4 GeV trigger effectively rejects most of the inclusive background, with minimal losses for dark photon signals, particularly those with larger masses. By using only 125 crystals in Region- 1, over 99.9% of inclusive background events can be filtered out, thereby conserving resources such as bandwidth and storage. When the summed energy from all crystals in the ECAL is used for triggering, nearly 100% efficiency can be achieved, with only a few rare background processes exhibiting exceptions. This trigger strategy demonstrates significant promise in reducing background events while having a negligible impact on dark photon signals, even when relying solely on the channels in Region 1. This study serves as a critical reference for the electronic design of future detectors.

Energy Digitization

Accurate simulation of the detector response requires a precise description of digitization effects. Digitization introduces realistic aspects to simulation results, replicating actual detector behavior and enhancing comparability with real experimental conditions. The energy obtained directly from Geant4, referred to as the truth energy, reflects the ideal energy deposition of a particle in a perfect detection scenario, devoid of detector effects or measurement errors. Thus, digitization provides a more realistic portrayal of detector performance and supports design validation [49, 50].

The primary goal of energy digitization is to parameterize the behavior of each component related to energy measurement, incorporating smearing effects derived from preliminary experiments. For the DarkSHINE ECAL, the energy digitization process consists of three main components: scintillation digitization, SiPM digitization, and analog-to-digital converter (ADC) digitization. These components correspond to the light emission and decay of the scintillator, the SiPM response, and the behavior of the readout electronics, respectively, all based on preliminary experimental measurements.

6.1.2

SiPM digitization

For SiPM digitization, a toy Monte Carlo model [51] was used to simulate the response of the SiPM to scintillation light from a LYSO crystal. This model was specifically designed to describe the relationship between the number of photons detected by the SiPM and the number of incident photons. The simulation was based on the HAMAMATSU S14160-3010PS datasheet [52], which features a sensitive area of 3 mm× 3 mm, 10 μm pixels, and nearly 90,000 pixels in total, with a LYSO crystal measuring 2.5 cm×2.5 cm×4 cm. The model accounts for the PDE, pixel density, pixel recovery, and crosstalk effects of the SiPM, as well as the scintillation decay time, transmittance, emission spectrum, and absorption spectrum of the LYSO crystal. The parameters used in SiPM digitization are listed in Table 8.

Table 8Full-screen View

Parameters used in SiPM digitization

Active area (mm2)	Pixel pitch (μm)	Pixel number
3.0×3.0	10	89984
PDE (λ=λp)	Fill factor	Gain fluctuation
18%	31%	10%

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In Fig. 13, N_fired represents the number of photons detected by the SiPM, ϵ denotes the PDE of the SiPM, and N_In represents the number of photons incident on the SiPM. The SiPM exhibits a nearly linear response when the effective photon count, calculated as the product of the PDE (ϵ) and the number of incident photons (N_In), is less than 10,000. As the effective photon count increases, the response gradually deviates from linearity and approaches saturation. A formula from [53] was employed to fit the response of the SiPM, aiding in modeling its behavior for varying numbers of incident photons and providing a means to correct for saturation effects.

Fig. 13Full-screen View

(Color online) The simulated response of the SiPM to scintillation light from a 2.5 cm×2.5 cm×4 cm LYSO crystal is shown. This SiPM‒LYSO setup is identical to that used in the DarkSHINE ECAL. Performance parameters of the SiPM, including PDE, fill factor, pixel pitch, pixel counts, and recovery time, are referenced from the HAMAMATSU S14160-3010PS datasheet [52]. The detected photons exhibit a time structure influenced by the scintillation decay time of LYSO and the geometric effects of the crystal, which are considered in the simulation. The horizontal axis represents the effective photon counts, calculated as the product of the PDE (ϵ) and the number of incident photons (N_In). The upper figure displays the variation in the number of photons detected by the SiPM (N_fired) as a function of the effective photon counts. The lower figure illustrates the extent to which the response of the SiPM deviates from linearity. It can be observed that for effective photon counts below 10,000, the response of the SiPM remains largely linear. However, as the effective photon count exceeds 10,000, the response begins to deviate from linearity and gradually saturates

In the SiPM digitization, the number of photoelectrons from the scintillation digitization was used as the input for the fitting function shown in Fig. 13, resulting in a response that includes the nonlinear effects of the SiPM. The fluctuations during this process can also be estimated, approximately proportional to the square root of the effective photon count. Furthermore, the inverse of this fitting function can be applied to correct the nonlinear response of the SiPM; however, this correction inevitably reintroduces fluctuations [54, 55].

Single channel performance

After digitization, the ideal energy deposited in the crystal within Geant4 was transformed into a signal that more accurately represented the actual response of the detector. The Equivalent Noise Energy (ENE) for the electronics of the single-channel digitization is shown in Figure 14(a). The ENE was calculated using the following formula: $\text{ENE}=\frac{\text{Noise}}{{\text{ADC}}_{\text{p}\text{.e}\text{.}}\times {\text{LY}}_{\text{mea}}}$ (1) where Noise represents the number of ADC noise, which includes both SiPM noise and DAQ noise, as listed in Table 9. ADC_p.e. is the number of ADC counts per photoelectron, also listed in Table 9. LY_mea denotes the number of photoelectrons detected by the SiPM coupled to the crystal when it absorbs 1 MeV of energy. In our setup, LY_mea is set to 150 p.e./MeV (Table 7).

Fig. 14Full-screen View

(Color online) (a) ENE of a single channel. The x-axis represents the input energy in the digitization model, while the y-axis indicates the ENE. The three bins correspond to the three gain modes, each with a fixed ENE value that is independent of input energy. The width of each bin represents the dynamic range of energy measurement at that gain. (b) Energy resolution of a single channel. The x-axis represents the input energy in the digitization model, and the y-axis indicates the energy resolution of the channel, calculated as the ratio of the standard deviation (σ) to the mean value from a Gaussian fit of the output energy distribution. At energies below 100 MeV, the primary contributor to resolution is the SiPM digitization, which depends on the efficiencies of light collection and photon detection, introducing larger fluctuations in Monte Carlo sampling compared to intrinsic light output and ADC noise. As energy increases, the scintillation digitization and ADC digitization become more dominant, mainly due to constant terms reflecting calibration accuracy

In Fig. 14(a), the three distinct ENE regions correspond to the different gain modes. Since ENE represents electronic noise and is independent of the input signal, it remains constant across the energy ranges covered by each electronic gain. However, at the transition points between these ranges, the ENE exhibits jumps corresponding to the changes in gain, as do the parameters Noise and ADC_p.e..

The single-channel energy resolution, after incorporating the three digitization effects, is shown in Fig. 14(b). We individually simulated the digitization process for each component to obtain its respective resolution, followed by a simulation of the combined process to determine the overall resolution. Scintillation digitization involves Poisson sampling based on the intrinsic light yield of the LYSO crystal, along with a series of Gaussian samplings. Consequently, this component of the energy resolution decreases with increasing incident energy. However, due to the high intrinsic light yield of the crystal, this change is not significantly noticeable in Fig. 14(b); instead, it is dominated by the constant term introduced by Gaussian sampling, reflecting the calibration accuracy. The resolution of SiPM digitization also decreases with increasing energy. However, because statistical sampling is based on the number of detected photoelectrons—significantly lower than the number of generated scintillation photons due to attenuation and the PDE of the SiPM—the resolution of SiPM digitization is worse than that of scintillation digitization. Moreover, applying a nonlinearity correction further degrades the resolution, as the correction process introduces additional fluctuations. For ADC digitization, the energy resolution shows two jumps at 5 MeV and 500 MeV, attributed to an increase in ENE due to gain switching. Additionally, a constant term appeared in the high-energy region, reflecting the calibration accuracy assumed in the model. Overall, at lower energies (less than 100 MeV), the resolution of SiPM digitization was the main contributor. At higher energies, the contributions from scintillation digitization and ADC digitization became more significant.

The energy resolutions presented in Fig. 14(b) were derived from the parameters listed in Tables 7, 8 and 9, which correspond to preliminary experimental measurements. Optimizing the LYSO‒SiPM units, such as improving light-collection efficiency to achieve a larger light output, could enhance channel resolution.

ECAL performance

The performance of the full detector was determined by applying digitization to each channel. Figure 15 shows the energy resolution and containment of the ECAL. Here, E_Truth denotes the energy of the incident electrons. In the simulation, the ECAL consists of a grid of 21×21×11 LYSO crystals, each with a volume of 2.5 cm×2.5 cm×4 cm, consistent with the results presented in Sect. 4. Targets and other detectors were excluded from this assessment. The incident particles were electrons with energies ranging from 1 GeV to 8 GeV, impacting the front face of the ECAL within a circular area of 3 cm radius.

Fig. 15Full-screen View

Energy resolution and containment of the ECAL for incident electrons with energies ranging from 1 to 8 GeV, excluding targets and other detectors. The x-axis represents the energy of the incident electrons. The upper figure displays the energy resolution of the ECAL, with the fitted stochastic term achieving better than 2%. The lower figure illustrates the energy containment, defined as the ratio of the energy deposited in the ECAL to the incident energy

The energy resolution is plotted at the top of Fig. 15. After digitization, the statistical term of the energy resolution of the ECAL remained better than 2% due to the high intrinsic light yield of the LYSO crystal. This high-energy resolution allows for more precise measurements of recoil electron energy and enhances the reliability of distinguishing between signal and background. The overall energy resolution of the ECAL is typically better than that of a single channel (Fig. 14(b)) due to statistical averaging, where uncertainties from individual channels tend to cancel out, and energy sharing across multiple channels reduces the impact of measurement errors. Additionally, variations in the responses of different channels are compensated for when combined, thereby improving the overall resolution.

The bottom part of Fig. 15 shows the energy containment of the ECAL, defined as the ratio of the energy deposited in the ECAL to the incident energy. The energy containment was less than 100% mainly because a small portion of the energy was deposited in passive materials outside the crystals, such as reflective films, PCBs, and support structures, which could not be detected.

The performance of the ECAL is influenced by the digitization model, which is based on preliminary experimental results from a specific setup. Optimizing the detector configuration can alter the performance; however, the key objective is to develop a tool that accurately evaluates the performance of a detector, bringing it closer to that of a real detector.