Introduction

With the development of nuclear physics and astrophysics, we have been able to glimpse into the sun’s energy mechanism, which originates from the nuclear fusion of light nuclei in the core of the sun [1-3]. The proton-proton (pp) cycle produces ~99% of solar energy, and its primary reaction is the fusion of two protons into a deuteron: $p+p{=}^2\text{H}+e^{+}+{\nu }_e$ (1) In this reaction, large numbers of low-energy neutrinos, called pp neutrinos, are emitted (E < 0.42 MeV). In addition, the proton-electron-proton (pep) process and secondary reactions in the pp cycle also emit neutrinos known as pep neutrinos, ⁷Be neutrinos, ⁸B neutrinos, and hep (helium-proton) neutrinos. The remaining energy of the sun is contributed by the carbon-nitrogen-oxygen (CNO) cycle, which emits CNO neutrinos. The detection of solar neutrinos is considered a direct way to test theoretical solar models. However, differences between early observations and theoretical predictions were discovered [4-13], leading to the so-called "solar neutrino problem" that has plagued us for more than 30 years. Subsequently, the MSW-LMA (neutrino oscillation with the Mikheyev-Smirnov-Wolfenstein effect and a large mixing angle) mechanism [14, 15] was proven to be the standard solution based on solid evidence provided by SNO [16, 17] and KamLAND [18]. Currently, the standard solar model (SSM) [19-24] provides precise predictions of the flux and energy distribution of solar neutrinos. Almost all solar neutrino components have been observed [25-28], and we expect to enter an era of precise and comprehensive measurements of solar neutrinos in the coming decades [29, 30].

pp neutrinos are strongly related to the predominant energy production of the sun and carry recent messages from the core of the sun. These characteristics make them an important means for studying the sun’s energy mechanism and thermodynamic equilibrium monitoring. By contrast, pp neutrinos can be used to study neutrino oscillations in vacuum-dominated regions. The detection of pp neutrinos simultaneously requires a low threshold (~ 200 keV) along with effective background reduction. pp neutrinos were first detected using ⁷¹Ga-based radiochemical detectors [6-11]. Subsequently, a large-scale liquid scintillator (LS) detector was successfully applied in a Borexino experiment and provided the best measurement of pp neutrinos at the ~10% level [26, 27] via elastic neutrino-electron scattering.

According to the experience gained from the Borexino experiment, intrinsic ¹⁴C decays from an organic liquid scintillator and their associated pile-up events are a crucial internal background for a large-scale LS detector. ¹⁴C pile-up events correspond to cases in which more than one ¹⁴C decay occurs at different detector positions but in the same trigger window. In addition, pile-ups can be classified into the following categories according to the multiplicity of ¹⁴C accidental coincidences: double pile-ups, threefold pile-ups, and fourfold pile-ups. The Borexino experiment (~278 ton) requires considerable effort for LS purification to obtain a ¹⁴C concentration of approximately 2.7 × 10^-18g/g. At this concentration, the ¹⁴C double pile-up accounts for approximately 10% of the events in the spectral gap between the ¹⁴C and ²¹⁰Po spectra [26].

For an LS detector with a sensitive target mass of m kilotons (kton), the frequency of a ¹⁴C single event is $f_{\text{single}}\left[\text{Hz}\right]=\frac{C_{{}^{14}\text{C}}\cdot N_{\text{A}}\cdot m}{\tau \cdot M}\times {10}^9,$ (2) where N_A is Avogadro’s constants (6.023 × 10²³) and τ, M, $C_{{}^{14}\text{C}}$ correspond to ¹⁴C’s lifetime, molar mass, and its concentration in the LS, respectively.

The frequency of ¹⁴C pile-up events can be calculated as follows: $f_{\text{pile}-\text{up}}\left[\text{Hz}\right]=\frac{e^{-f_{\text{single}}\cdot \Delta t}}{\left(n-1\right)!}\cdot f_{\text{single}}^n\cdot \Delta t^{n-1}\cdot \epsilon ,$ (3) where n ($n\ge 2$) denotes the multiplicity of the ¹⁴C accidental coincidence; for example, n = 2 represents the case of a double ¹⁴C pile-up. Δ t is the time window for detection and ε corresponds to the reconstruction efficiency of the ¹⁴C pile-up events.

As the detector mass increases, a dramatic increase in ¹⁴C pile-up events must be considered and effectively rejected. Taking a large spherical LS detector as an example, with the radius of the detector being 15 m and the detector mass being approximately 12 kton, Table 1 lists the event rates of pp neutrinos and ¹⁴C single and pile-up events at different ¹⁴C concentrations. A 500 ns time window was used in this calculation, and the reconstruction efficiency was set to 100%. For a ¹⁴C concentration of 5 × 10^-18 g/g in the LS of the above detector, Fig. 1 shows the recoil energy spectra of pp neutrinos via elastic neutrino-electron scattering, which can be found in [30]. The energy spectra of ¹⁴C single, double, and triple pile-up events are shown for comparison. In this giant detector, the ¹⁴C pile-up events completely outnumbered the pp neutrino signals by more than two orders of magnitude.

Fig. 1Full-screen View

(Color online) The recoil energy spectra of pp neutrinos, ¹⁴C single, double, and triple pile-up events in a spherical LS detector, whose radius and ¹⁴C concentration are 15 m and 5 × 10^-18 g/g, respectively. The spectra do not include the detection effects: energy non-linearity, non-uniformity, and resolution. The higher order contribution from the ¹⁴C pile-up is negligible and not shown.

In Table 1, the values in brackets indicate the event rates within the energy range of interest of 0.16 MeV–0.25 MeV for the deposited energy, considering that the Q value of ¹⁴C β decay is ~156 keV and the scattered electron of the pp neutrino reaction is difficult to distinguish from the emitted electron of a ¹⁴C single event. The target mass of the above detector (~12 kton) is ~43 times larger than that of Borexino (~278 ton). Consequently, the signal-to-background ratio of pp neutrinos and ¹⁴C double pile-up events in this detector is smaller than 1:126 for a ¹⁴C concentration of 2.7 × 10^-18 g/g, and the signal-to-background ratio will be much lower if a higher ¹⁴C concentration is used. However, because the energy resolution introduces smearing in the energy spectrum, the energy range of the analysis must be determined based on realistic situations.

More neutrino experiments are underway or are being planned, and many of them [31,31-35] have good potential for pp neutrino detection because they are expected to have a large detector target, well-controlled radioactivity, low detection threshold, or good energy resolution. In experiments with LS detectors of the order of tens of kilotons, neutrino detection in low-energy regions is difficult because of ¹⁴C pile-up. Therefore, an approach must be developed for ¹⁴C pile-up discrimination and reduction, especially that for ¹⁴C double pile-up, because its event rate is much higher than that of other accidental coincidences.

This study focuses on discriminating between pp solar neutrinos and ¹⁴C double pile-up events. The discrimination of other accidental coincidences with a ¹⁴C multiplicity ≥ 3 is an important topic in the case of a higher ¹⁴C concentration; however, it is not the subject of this study. The details of our work are as follows: First, we simulated an LS detector and investigated the features of the detector’s photomultiplier (PMT) hit pattern for pp neutrinos and ¹⁴C double pile-up events (Sect. 2). We then present several approaches to ¹⁴C double-pile-up discrimination based on multivariate analysis and deep-learning technology (Sect. 3). In Sect. 4, the discrimination performances are shown and compared. Finally, a summary is presented in Sect. 5

Detector simulation

In this study, a spherical LS detector was built using Monte Carlo (MC) simulations with the Geant4 toolkit [36] version 4.10.p02. The radius of the spherical detector was 15 m, and the LS was contained in an acrylic sphere with a 10-cm-thick wall. To simplify the simulation, a sensitive optical surface was defined for receiving the photons instead of using the detailed PMT simulation. The sensitive optical surface was a sphere outside the acrylic sphere, separated by a 1-m-thick layer of water. The coverage and quantum efficiency of the photosensors could be easily tuned. In the simulation, the coverage rate was 65%, corresponding to approximately 10650 20-inch photomultipliers (PMTs) uniformly distributed on the sensitive optical surface. Figure 2 shows a schematic of the detector. In the simulation, an average quantum efficiency of 30% was used for the 20-inch PMTs with a 2% Gaussian relative spread. The LS properties were referenced from [37-44], and comprehensive optical processes were adopted, including quenching, Rayleigh scattering, absorption, re-emission, photon transport in the LS, and reflections on the acrylic surface. Table 2 summarizes the main parameters of the PMTs in the simulation, including the transit time spread (TTS), quantum efficiency (QE), dark noise (DR), and resolution of a single photoelectron (spe). As a result, approximately 1100 photoelectrons (PEs) could be observed by the 10650 PMTs for a 1 MeV electron that fully deposited its kinetic energy in the center of the detector, which corresponds to approximately a 3% energy resolution. In contrast, approximately 106.5 additional PEs originating from the PMT dark noise in a time window of 500 ns could be detected.

Fig. 2Full-screen View

(Color online) A schematic view of the detector. Each pixel corresponds to a 20-inch PMT, and its color indicates the ID of each PMT. In total, the detector had 10650 PMTs.

To investigate the response features of pp neutrinos and ¹⁴C double-pile-up events, MC samples were generated and compared. Approximately one million final-state electrons from the elastic neutrino-electron scattering reaction of pp neutrinos were uniformly simulated in the LS volume, and the spectrum of the scattered electrons was referenced from [30]. Because the final-state electrons from elastic neutrino-electron scattering are similar to the electrons emitted from the ¹⁴C β decay (¹⁴C single event), distinguishing them at an event-by-event level is difficult. Therefore, an energy cut is required to focus on a narrow energy region. The same treatment method as used by Borexino et al. was used. However, electrons whose kinetic energy is approximately 200 keV in LS show a 5% energy nonlinearity [44, 45], and the energy resolution is already included in the above simulation. As a result, in our analysis, a 270 PE cut was applied to the total number of PEs of all PMTs by considering the ~156 keV end-point energy of the ¹⁴C β decay (~163 PEs) and the contribution of PMT dark noise (~106.5 PEs).

After the total PE cut, an MC sample that included 100,000 pp neutrinos was used for the discrimination study, and they were uniformly distributed in the LS. To generate the ¹⁴C double pile-up sample, first a large dataset was produced by simulating 10 million ¹⁴C single events in the LS using ¹⁴C β decay. Next, two ¹⁴C single events were randomly selected from the dataset and merged into a double-pile-up event. In the merge operation, because the lifetime of ¹⁴C is longer than 8000 years, the time interval between two ¹⁴C single events could be considered an approximately uniform distribution for a few hundred nanoseconds. Similarly, a 270 PE cut was applied, and 100,000 ¹⁴C double pile-up events were used for our analysis.

As illustrated in Figs. 3 and 4, pp solar neutrinos and ¹⁴C double pile-up events exhibited different features in their temporal and spatial distributions. The pp solar neutrino is a single point-like event whose energy deposition occurs in a relatively short time and small space; hence, only one cluster is expected to be found in its PMT hit pattern. For the ¹⁴C double-pile-up event, if two ¹⁴C nucleus decay at different detector positions, two clusters are expected to be found. However, because the hit time distribution of the fired PMTs includes both the scintillation time and the photon’s time of flight, as well as the decay time of ¹⁴C, the hit time distribution is useful for identification studies. In particular, when two ¹⁴C nucleus decay near each other, their spatial distribution is not expected to be significantly different from that of a single point-like event. However, the hit time distribution is still helpful if the time interval between the two ¹⁴C decays is large. An example is shown in Fig. 4.

Fig. 3Full-screen View

(Color online) The PMT hit patterns of a pp solar neutrino event. Each pixel corresponds to a fired PMT, and its color indicates the hit time information. The location of the red hollow triangle is (-6582.21, -8972.86, 8696.34) mm, which indicates the position where the physical event deposited its energy (159.94 keV). (a) only physical hits are included, and 172 PEs are observed for a 500-ns time window. (b) Both physical hits and PMT dark noise hits are shown, and 284 PEs are observed for a 500-ns time window, including 112 PEs from the PMT dark noise.

Fig. 4Full-screen View

(Color online) The PMT hit patterns of a ¹⁴C double pile-up event. Each pixel corresponds to a fired PMT and its color indicates the hit time information. The two red hollow triangles indicate the positions where two ¹⁴C events deposited their energies (71.161 keV and 56.593 keV). Their locations are (–6229.32, –2139.36, 10471.7) mm and (484.61, –3199.44, 14423.5) mm, respectively. (a) only physical hits are included, and 173 PEs (107+66) are observed for a 500-ns time window. (b) Both physical hits and PMT dark noise hits are shown, and 273 PEs are observed for a 500-ns time window, including 100 PEs from the PMT dark noise.

As mentioned previously, our approach employs a straightforward trigger strategy that considers whether the total number of PEs within 500 ns exceeds 270 PEs. Subsequently, we selected the hit information within this timeframe for further analysis. However, the trigger strategy must be optimized. As described in Sect. 3, the event spatial information was extracted and used together with the hit time information as input to the discrimination algorithms.

Discrimination methods

The basic idea behind developing a discrimination algorithm for pp solar neutrinos and ¹⁴C double pile-up events is to utilize their temporal and spatial information, which have different characteristics (see Sect. 2). Similar approaches have been applied to discriminate single-site and multisite energy depositions in large-scale liquid scintillation detectors [46]. During the measurement, the cluster structure was smeared by interference from dark noise and the TTS of the PMT. These effects make the identification more challenging and more efficient approaches are required. In this study, a multivariate analysis using the Toolkit for Multivariate Data Analysis (TMVA) [47, 48] was performed, and the widely used algorithm, boosted decision trees with gradient boosting (BDTG), was chosen for the analysis. In addition, deep learning technologies based on the VGG network were also applied. In the following section, we present details of the discrimination method.

3.1

TMVA analysis

TMVA [47, 48] is a powerful tool for multivariate analysis and has been successfully applied to both signal and background classification in accelerator physics [49], component identification of cosmic rays [50], and event reconstruction in LS detectors for neutrino experiments [55]. The TMVA toolkit hosts a wide variety of multivariate classification algorithms. In this study, we used the TMVA algorithm, BDTG. To extract the input variables, the PMT hit pattern was projected onto a one-dimensional (1-D) plane for the hit time, θ, and ϕ of each fired PMT in spherical coordinates. The projection results of Fig. 3b are shown in Fig. 5, and the projection results of Fig. 4b are shown in Fig. 6. The pp solar neutrino, which is a single-point-like event, showed only one cluster in its distribution, whereas the ¹⁴C double pile-up event showed two clusters.

Fig. 5Full-screen View

(Color online) The hit time, θ, and ϕ distributions of a pp solar neutrino event corresponding to the event in Fig. 3b. (a) Hit time distribution. (b) θ distribution. (c) ϕ distribution.

Fig. 6Full-screen View

(Color online) The hit time, θ, and ϕ distributions of a ¹⁴C double pile-up event corresponding to the event in Fig. 4b. (a) Hit time distribution. (b) θ distribution. (c) ϕ distribution.

These 1-D distributions were used in multivariate analysis. The input variables for the TMVA algorithms should be sensitive to discrimination and contain the characteristics of pp solar neutrinos and ¹⁴C double-pile-up events. In our analysis, we found that hit time information dominated the discrimination performance; therefore, more variables were extracted from the 1-D distribution of hit time. Fifteen variables were used in the TMVA analysis. These variables are denoted as $V_i^{\alpha }$, where i = 1, 2, 3, etc., and correspond to the extracted parameters in each 1-D distribution. α = hittime, θ, or ϕ, which indicates that the variables are from the 1-D distribution of hit time, θ, or ϕ. The details can be found in Table 3.

Table 3Full-screen View

Input variables for multivariate analysis.

Variable	Description
$V_1^{\text{hittime}}$	Number of hits in the first 200 ns
$V_2^{\text{hittime}}$	The peak position of the highest bin in the first 200 ns
$V_3^{\text{hittime}}$	The amplitude of the highest bin in the first 200 ns
$V_4^{\text{hittime}}$	Number of hits in (200, 500) ns
$V_5^{\text{hittime}}$	The amplitude of the highest bin in (200, 500) ns
$V_6^{\text{hittime}}$	The ratio between the peak amplitude and the peak position of the highest bin in (200, 500) ns
$V_7^{\text{hittime}}$	The ratio between the number of hits in the first 200 ns and in (200, 500) ns
$V_8^{\text{hittime}}$	The RMS value of the 1-D distribution of hit time
$V_9^{\text{hittime}}$	The Mean value of the 1-D distribution of hit time
$V_1^{\text{theta}}$	The RMS value of the 1-D distribution of θ
$V_2^{\text{theta}}$	The skewness coefficient of the 1-D distribution of θ
$V_3^{\text{theta}}$	The kurtosis coefficient of the 1-D distribution of θ
$V_1^{\text{phi}}$	The RMS value of the 1-D distribution of ϕ
$V_2^{\text{phi}}$	The skewness coefficient of the 1-D distribution of ϕ
$V_3^{\text{phi}}$	The kurtosis coefficient of the 1-D distribution of ϕ

Show more

Figure 7 shows the normalized distributions of these input variables, and the difference in their shapes is determined by comparing the two types of events. By contrast, the correlations of the input variables were checked for both pp solar neutrinos and ¹⁴C double pile-up events. As shown in Fig. 8, because we dropped several variables with strong correlations in a previous study, the correlation of the current variables is acceptable, provided it is less than 90%. A few variables had close to 90% correlations, and we retained them in the analysis. This is mainly determined by considering that the variables exhibit different correlations for the signal and background; a similar strategy was applied in [52].

Fig. 7Full-screen View

(Color online) Normalized distributions of the variables of pp solar neutrino and ¹⁴C double pileup event.

Fig. 8Full-screen View

(Color online) Linear correlation matrix for the input variables of pp solar neutrinos (a) and ¹⁴C double pile-up events (b).

The MC samples of the pp solar neutrinos and ¹⁴C double pile-up events were divided into two equal parts, one for TMVA training and the other for validation. Hence, both the training and test samples include 50 thousand pp neutrinos and 50 thousand C-14 double pile-up events. To improve the performance, several main parameters of the BDTG algorithm were tuned. Table 4 shows the settings of these parameters. The other parameters were set to their default values and are not listed in the tables.

Table 4Full-screen View

Parameters used in the BDTG algorithm.

Configuration option	Setting	Description
NTrees	1000	Number of trees in the forest
MaxDepth	2	Max depth of the decision tree allowed
MinNodeSize	2.5%	Minimum percentage of training events required in a leaf node
nCuts	20	Number of grid points in variable range used in finding optimal cut in node splitting
BoostType	Grad	Boosting type for the trees in the forest

Show more

Discrimination performance and discussion

4.1

Discrimination performance of the BDTG model

Figure 9 hows the training results of the BDTG model. The network was not overtrained, as the responses of the testing data were consistent with those of the training data (Fig. 9a. The signal and background were separated into two parts after training; however, some overlapping components remained, indicating that their event features were similar. Hence, the network failed to distinguish between them. According to a detailed investigation, one of the main reasons for the failed identification was the stacking of two ¹⁴C nucleus that are very close together in both time and space. To optimize the significance $N_{\text{s}}/\sqrt{N_{\text{s}}+N_{\text{b}}}$, where N_s and N_b are the numbers of signals and background after identification, we scanned the cut value on the BDTG response, and the corresponding efficiencies were also obtained. The ¹⁴C concentration in the LS was assumed to be 5 × 10^-18g/g, as shown in Fig. 9b.The significance calculations using the statistics in the analysis region for a period of one day (true energy:160–250 keV), based on the estimation in Table 1, are ~1653 for the signal and ~712440 for the background (considering only the ¹⁴C double pile-up events) before the identification. For the BDTG model, the significance reached its maximum value of 10.33 after applying a cut at 0.915, and the signal and background rejection efficiencies were 51.1% and 99.18%, respectively. As discussed in Sec. 1, the signal-to-background ratio of the pp neutrinos and ¹⁴C double-pile-up events was low in a large-scale LS detector. Therefore, a strict cut is required to reject most of the background. In this case, 51.1% is an acceptable value for signal efficiency, and it still corresponds to a much larger number of effective pp neutrino signals per day compared with most existing experiments.

Fig. 9Full-screen View

(Color online) Identification performance using the BDTG model. (a) Normalized response distributions of the BDTG model for the signal and the background. (b) Cut efficiencies as functions of BDTG cut values. The significance (green line) was calculated using the statistics for one day of the signal and the background in the analysis region, and the ¹⁴C concentration of LS was assumed to be 5 × 10^-18 g/g. (c) Significance of different assumptions of the ¹⁴C concentration. (d) Signal-to-background ratio after identification for different assumptions of the ¹⁴C concentration; the statistics for one day were adopted.

In Fig. 9c, the significance was evaluated for different assumptions for ¹⁴C concentration. Fig. 9d shows the signal-to-background ratio after identification using the BDTG model, based on the statistics for a period of one day for different ¹⁴C concentrations. As a result, the BDTG model exhibits excellent performance and can handle most of ¹⁴C double pile-up events effectively.

In addition, other TMVA algorithms were investigated, including the likelihood algorithm and several BDT models (BDT and BDTD). Many exhibited similar performances (Fig. 10), indicating the robustness and stability of the analysis.

Fig. 10Full-screen View

(Color online) Relationship between background rejection efficiency and signal efficiency obtained using various TMVA algorithms.

4.2

Discrimination performance of the VGG network

Figure 11 shows the training results of the VGG network. The network was not overtrained, as the responses of the testing data were consistent with those of the training data (Fig. 11a). To optimize the significance, we scanned the cut values of the VGG output, and the corresponding efficiencies were obtained. The ¹⁴C concentration of the LS was assumed to be 5 × 10^-18g/g, as shown in Fig. 11b, and the calculation of significance using the statistics for a period of one day in the analysis region was based on the estimation in Table 1. For the VGG network, the significance reached its maximum value of 15.55 after applying a cut of 0.975. The signal efficiency and background rejection efficiency were 42.7% and 99.81%, respectively.

Fig. 11Full-screen View

(Color online) Identification performance using the VGG network. (a) Normalized response distributions of the VGG network for the signal and the background. (b) Cut efficiencies as functions of VGG cut values. The significance (green line) was calculated using the statistics of the signal and the background in the analysis region for a one-day period; the ¹⁴C concentration in LS was assumed to be 5 × 10^-18 g/g. (c) Significance for different assumptions of ¹⁴C concentration. (d) Signal-to-background ratio after identification for different assumptions of the ¹⁴C concentration, the statistics for a one-day period.

In Fig. 11c, the significance was evaluated using different assumptions for ¹⁴C concentration, whereas Fig. 11d shows the signal-to-background ratio after identification using the VGG network. The calculations were based on the statistics for one day for different ¹⁴C concentrations. As a result, the VGG network showed excellent performance and could achieve higher significance and a good improvement in the signal-to-background ratio compared with the BDTG model.

Furthermore, the discrimination performances of the different MC samples were compared, as shown in Fig. 12. They worsened after including the PMT dark noise, whereas TTS had only a small influence. In addition, the discrimination performance based on the VGG network was stable when rejecting ~99.8% of the ¹⁴C double pile-up events.

Fig. 12Full-screen View

(Color online) Relationship between the background rejection efficiency and the signal efficiency for different MC samples.

Summary

Large-scale LS detectors have the benefits of a large target mass and high energy resolution, which gives them good potential for pp solar neutrino detection. However, they also face a serious problem of the ¹⁴C pile-up background. In this study, we investigated how pp solar neutrinos and ¹⁴C double-pile-up events in a large-scale LS detector could be distinguished using multivariate analysis and deep learning technology. In the simulation study, a spherical LS detector was built using the Geant4 toolkit, and comprehensive optical processes were adopted. The response features in the PMT hit patterns of pp neutrinos and ¹⁴C double pile-up events were compared, and clear differences were found in their temporal and spatial distributions because one of them was a single point-like event, whereas the other was an accidental coincidence of multiple events.

Using the BDTG model for the pp neutrino and ¹⁴C double pile-up event discrimination, at a ¹⁴C concentration of 5×10^-18 g/g, a signal significance of 10.3 could be achieved using the statistics for a period of only one day. The signal efficiency was 51.1% when 99.18% of ¹⁴C double-pile-up events were rejected. In the VGG network model, signal significance could reach 15.6 using the statistics for a period of only one day, and the signal efficiency was 42.7% when 99.81% of ¹⁴C double pile-up events were rejected. This analysis provides a reliable reference for similar experiments in low-threshold physics detection and ¹⁴C pile-up background reduction.