Guo-Ming Chen · Xin Zhang · Ze-Yuan Yu · Si-Yuan Zhang · Yu Xu · Wen-Jie Wu · Yao-Guang Wang ·Yong-Bo Huang

Abstract As a unique probe, the precision measurement of pp solar neutrinos is important for studying the sun’s energy mechanism as it enables monitoring the thermodynamic equilibrium and studying neutrino oscillations in the vacuum-dominated region.For a large-scale liquid scintillator detector, a bottleneck for pp solar neutrino detection is the pile-up events of intrinsic 14 C decay.This paper presents a few approaches to discriminating between pp solar neutrinos and 14 C pile-up events by considering the differences in their time and spatial distributions.In this study, a Geant4-based Monte Carlo simulation is conducted.Multivariate analysis and deep learning technology are adopted to investigate the capability of 14 C pile-up reduction.The BDTG (boosted decision trees with gradient boosting) model and VGG network demonstrate good performance in discriminating pp solar neutrinos and 14 C double pile-up events.Under the 14 C concentration assumption of 5×10-18 g/g, the signal significance can achieve 10.3 and 15.6 using the statistics of only one day.In this case, the signal efficiency for discrimination using the BDTG model while rejecting 99.18% 14 C double pile-up events is 51.1%, and that for the case where the VGG network is used while rejecting 99.81% of the 14 C double pile-up events is 42.7%.

Keywords Liquid scintillator detector · pp solar neutrinos · 14 C pile-up · Multivariate analysis · Deep learning

1 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:

In this reaction, large numbers of low-energy neutrinos,calledppneutrinos, are emitted (E<0.42 MeV).In addition, the proton–electron–proton (pep) process and secondary reactions in theppcycle also emit neutrinos known aspepneutrinos,7Be neutrinos,8B neutrinos, andhep(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].

ppneutrinos 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,ppneutrinos can be used to study neutrino oscillations in vacuum-dominated regions.The detection ofppneutrinos simultaneously requires a low threshold ( ～200 keV) along with effective background reduction.ppneutrinos were first detected using71Ga-based radiochemical detectors [6–11].Subsequently, a large-scale liquid scintillator (LS) detector was successfully applied in the Borexino experiment and provided the best measurement ofppneutrinos at the ～10%level [26, 27] via elastic neutrino–electron scattering.

According to the experience gained from the Borexino experiment, intrinsic14C 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 one14C 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 of14C 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 a14C concentration of approximately 2.7×10-18g/g.At this concentration, the14C double pile-up accounts for approximately 10% of the events in the spectral gap between the14C and210Po spectra [26].

For an LS detector with a sensitive target mass ofmkilotons (kton), the frequency of a14C single event is

whereNAis Avogadro’s constants ( 6.023×1023) andτ,M,C14Ccorrespond to14C’s lifetime, molar mass, and its concentration in the LS, respectively.

The frequency of14C pile-up events can be calculated as follows:

wheren(n≥2 ) denotes the multiplicity of the14C accidental coincidence; for example,n=2 represents the case of a double14C pile-up.Δtis the time window for detection andεcorresponds to the reconstruction efficiency of the14C pile-up events.

As the detector mass increases, a dramatic increase in14C 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 ofppneutrinos and14C single and pile-up events at different14C concentrations.A 500 ns time window was used in this calculation, and the reconstruction efficiency was set to 100%.For a14C concentration of 5×10-18g/g in the LS of the above detector, Fig.1 shows the recoil energy spectra ofppneutrinos via elastic neutrino–electron scattering,which can be found in [30].The energy spectra of14C single,double, and triple pile-up events are shown for comparison.In this giant detector, the14C pile-up events completely outnumbered theppneutrino signals by more than two orders of magnitude.

In Table 1, the values in brackets indicate the event rates within the energy range of interest of 0.16-0.25 MeV forthe deposited energy, considering that theQvalue of14Cβdecay is ～156 keV and the scattered electron of theppneutrino reaction is difficult to distinguish from the emitted electron of a14C 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 ofppneutrinos and14C double pile-up events in this detector is smaller than 1:126 for a14C concentration of 2.7×10-18g/g,and the signal-to-background ratio will be much lower if a higher14C 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.

Table 1 The event rates (unit: cpd/kton) of pp neutrinos and 14 C single and pile-up events in different 14 C concentrations

Fig.1 (Color online) The recoil energy spectra of pp neutrinos, 14 C single, double, and triple pile-up events in a spherical LS detector, whose radius and 14 C 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 14 C pile-up is negligible and not shown

Fig.2 (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

More neutrino experiments are underway or are being planned, and many of them [31, 31–35] have good potential forppneutrino 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 of14C pile-up.Therefore, an approach must be developed for14C pile-up discrimination and reduction, especially that for14C double pile-up, because its event rate is much higher than that of other accidental coincidences.

This study focuses on discriminating betweenppsolar neutrinos and14C double pile-up events.The discrimination of other accidental coincidences with a14C multiplicity ≥ 3 is an important topic in the case of a higher14C 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 forppneutrinos and14C double pile-up events (Sect.2).We then present several approaches to14C 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

2 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-cmthick 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.

Table 2 PMT parameters in the simulation

To investigate the response features ofppneutrinos and14C double pile-up events, MC samples were generated and compared.Approximately one million final-state electrons from the elastic neutrino–electron scattering reaction ofppneutrinos 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 the14Cβdecay (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 the14Cβdecay ( ～163 PEs) and the contribution of PMT dark noise ( ～106.5 PEs).

After the total PE cut, an MC sample that included 100,000ppneutrinos was used for the discrimination study,and they were uniformly distributed in the LS.To generate the14C double pile-up sample, first a large dataset was produced by simulating 10 million14C single events in the LS using14Cβdecay.Next, two14C single events were randomly selected from the dataset and merged into a double pile-up event.In the merge operation, because the lifetime of14C is longer than 8000 years, the time interval between two14C single events could be considered an approximately uniform distribution for a few hundred nanoseconds.Similarly,a 270 PE cut was applied, and 100,00014C double pile-up events were used for our analysis.

Fig.4 (Color online) The PMT hit patterns of a 14 C 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 14 C 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 illustrated in Figs.3 and 4,ppsolar neutrinos and14C double pile-up events exhibited different features in their temporal and spatial distributions.Theppsolar 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 the14C double pile-up event, if two14C 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 of14C, the hit time distribution is useful for identification studies.In particular, when two14C 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 two14C decays is large.An example is shown in Fig.4.

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.

3 Discrimination methods

The basic idea behind developing a discrimination algorithm forppsolar neutrinos and14C 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 [51].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, andφof 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.Theppsolar neutrino, which is a single-point-like event, showed only one cluster in its distribution, whereas the14C double pile-up event showed two clusters.

Fig.5 (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.6 (Color online) The hit time, θ , and φ distributions of a 14 C double pile-up event corresponding to the event in Fig.4b.a Hit time distribution.b θ distribution.c φ distribution

Table 3 Input variables for multivariate analysis

Fig.7 (Color online) Normalized distributions of the variables of pp solar neutrino and 14 C double pileup event

Fig.8 (Color online) Linear correlation matrix for the input variables of pp solar neutrinos (a) and 14 C double pile-up events (b)

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 ofppsolar neutrinos and14C 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 asVα i, wherei=1,2,3 , etc., and correspond to the extracted parameters in each 1-D distribution.α=hittime ,θ, orφ, which indicates that the variables are from the 1-D distribution of hit time,θ, orφ.The details can be found in Table 3.

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 bothppsolar neutrinos and14C 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 inthe 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].

Table 4 Parameters used in the BDTG algorithm

The MC samples of theppsolar neutrinos and14C 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 thousand14C 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.

3.2 Deep learning

Deep learning technology is widely used in high-energy and nuclear physics, with many successful applications [51,53–58] such as energy reconstruction, track reconstruction, particle identification, and signal processing.In this study, the deep learning algorithm VGG convolutional neural network was used for the feature recognition of onedimensional sequences.The extracted PMT hit patterns are projected onto a one-dimensional feature series for the hit time,θ, andφ, respectively.The resulting patterns are similar to those in Figs.5 and 6.To extract their features, a onedimensional convolution kernel was used for the three series,a pooling layer was used for information compression, and a fully connected layer was used for particle classification.The model structure was based on the architecture of VGG-16,which includes 13 convolution and pooling modules, three fully connected layers, batch normalization layers, and connected neural unit dropout processing.

In addition to one-dimensional projection using PMT hit patterns, we also attempted two-dimensional projection methods to provide input to deep learning networks, including the Mercator projection, sinusoidal projection, and a projection method based on the arrangement of PMTs [51, 59].However, after applying the two-dimensional projection, the performance did not improve but, in fact, slightly worsened.Considering that the number of hits in the energy range of interest is very small, we performed a detailed investigation and comparison because the cluster features were much more pronounced in the one-dimensional projection but very discrete in the two-dimensional projection.

Finally, a one-dimensional projection was used to provide input to the VGG network described above.We trained the VGG network using adaptive momentum with a batch size of 256 samples, momentum of 0.9, and an initial learning rate of 0.01.After every 10 epochs, the learning rate was reduced by a factor of 10.The accuracy of the model was evaluated using a cross-entropy loss function.In the discrimination study using the VGG network, 80% of theppneutrino and14C double pile-up separately were used separately for training, whereas the other 20% were used for validation.

4 Discrimination performance and discussion

4.1 Discrimination performance of the BDTG model

Figure 9 shows 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 two14C nucleus that are very close together in both time and space.To optimize the significance, whereNsandNbare the numbers of signals and background after identification, we scanned the cut value on the BDTG response, and the corresponding efficiencies were also obtained.The14C 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 the14C 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 Sect.1, the signal-to-background ratio of theppneutrinos and14C 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 effectiveppneutrino signals per day compared with most existing experiments.

Fig.9 (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 14 C concentration of LS was assumed to be 5×10-18 g/g.c Significance of different assumptions of the 14 C concentration.d Signal-to-background ratio after identification for different assumptions of the 14 C concentration; the statistics for one day were adopted

In Fig.9c, the significance was evaluated for different assumptions for14C concentration.Figure 9d shows the signal-to-background ratio after identification using the BDTG model, based on the statistics for a period of one day for different14C concentrations.As a result, the BDTG model exhibits excellent performance and can handle most of14C 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.

4.2 Discrimination performance of the VGG network

Fig.10 (Color online) Relationship between background rejection efficiency and signal efficiency obtained using various TMVA algorithms

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 effi-ciencies were obtained.The14C 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.

In Fig.11c, the significance was evaluated using different assumptions for14C 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 different14C 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.

Fig.12 (Color online) Relationship between the background rejection efficiency and the signal efficiency for different MC samples

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 the14C double pile-up events.

5 Summary

Large-scale LS detectors have the benefits of a large target mass and high energy resolution, which gives them good potential forppsolar neutrino detection.However, they also face a serious problem of the14C pile-up background.In this study, we investigated howppsolar neutrinos and14C 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 ofppneutrinos and14C 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 theppneutrino and14C double pile-up event discrimination, at a14C concentration of 5×10-18g∕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% of14C 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% of14C double pileup events were rejected.This analysis provides a reliable reference for similar experiments in low-threshold physics detection and14C pile-up background reduction.

Author contributionsAll authors contributed to the study conception and design.Material preparation, data collection and analysis were performed by Guo-Ming Chen and Yong-Bo Huang.The first draft of the manuscript was written by Guo-Ming Chen and Yong-Bo Huang,and all authors commented on previous versions of the manuscript.All authors read and approved the final manuscript.

Data availabilityThe data that support the findings of this study are openly available in Science Data Bank at https://cstr.cn/31253.11.sciencedb.j00186.00198 and https://doi.org/10.57760/sciencedb.j00186.00198.

Declaration

Conflict of interestThe authors declare that they have no competing interests.