Detector constructions

In the current application, water is defined as an absorbing material. The PMTs are distributed in a specific configuration to detect the Čerenkov light emitted by the high-speed electrons produced in neutrino-induced reactions. For simplicity in the startup simulation, a spherical distribution of PMTs was adopted in water. This configuration was chosen to ensure isotropic detection capabilities, which are essential for locating an unknown source. The radius of the sphere, measured at the front surface of the PMTs, was set to R=5 m. This value is reasonable and feasible for constructing a demonstrator while also providing a sufficiently large fiducial volume. Given the lack of an analytical solution for a series of points uniformly distributed on a sphere, the setup employs an approximate solution based on the Fibonacci grid [35]. The spherical coordinate angles θi and ϕi of the i^th PMT are given by Eq. (1).(1)The symmetrical axis is aligned with the center of the sphere. The diameter of the sensitive area of each PMT is ϕ_pmt=8 in. Assuming a 30% coverage of the entire sphere surface, which represents a moderate and achievable value, 2906 PMTs were constructed. The physical construction of each PMT was simplified as a glass bulb containing a vacuum. The remaining area of the sphere was covered with black sheets to minimize internal reflections, and the surface reflectivity was manually adjusted to 95% from the inner to outer space. To mitigate the residual radiation from the PMT materials, the fiducial volume was defined by a sphere with a radius of R_fd=4.5 m. Figure 1 presents the schematic view of the PMT sphere. The distance between neighboring PMTs was maintained at 30 - 40 cm.

Fig. 1Full-screen View

(Color online) The schematic view of the PMTs distributing on a sphere filled by water

Simulation of the Optical process by incident electron

The simulated event began with the generation of a charged particle in a neutrino-induced reaction. This process is enforced because of the low cross-section of weak interactions, which makes it challenging to achieve sufficient statistics using conventional sampling techniques. Charged particles lose energy in water and emit Čerenkov light if their velocity exceeds the threshold. The simulation tracked all secondary particles until they were stopped or absorbed. Čerenkov photons in water undergo scattering and absorption according to the optical parameters of the medium. If a photon reaches the PMT surface, it can generate a certain number of photoelectrons (PEs) and produce a signal based on the PMT response parameters. Alternatively, photons may be absorbed if they strike a black surface outside the PMT acceptance range.

In this study, the PMT response settings were identical to those of the Super-K experiment [36]. If more than 10 PMTs were triggered within 50 ns (representing the time it takes for light to traverse the entire detector), a trigger signal was generated, and the data were stored for further analysis. Therefore, the transport and absorption of Čerenkov photons are critical processes in the detection mechanism. The detection process is governed by several optical parameters for the medium and surfaces, including the absorption length, refractive index, and scattering length. These parameters are dependent on the photon wavelength. For simplicity, the parameter set of water was adopted from Super-K [36], which is suitable for pure water but may not be ideal for seawater. The effect of seawater is discussed in Sect. 4.

The attenuation length is shown in Fig. 2(a), where the value in the blue wavelength band is about 100 m, the refractive index is 1.334. Additionally, the refractive index for glass was set to 1.6, and the absorption length for a black sheet was set to 10^-11 m. The surfaces are defined as the interfaces between these media. The reflectivity between the water and black sheet was set to 0.05. The quantum efficiency of the PMTs is a function of the photon wavelength, as shown in Fig. 2(b). The maximum efficiency was 21.1% at λ=400 nm.

Fig. 2Full-screen View

Parameters used in the simulation package. The data were obtained from [34]. (a) Attenuation length as a function of the photon wavelength. (b) PMT quantum efficiency as a function of photon wavelength

Figure 3 illustrates an event display of the optical process resulting from a 6 MeV electron neutrino-induced reaction. The incident direction was randomly distributed over a 4π solid angle, with the vertex fixed at the center. The red lines represent the tracks of Čerenkov photons, while the dashed arrow indicates the direction of the recoil electron, which is expected to be reconstructed from the PMTs triggered by photons. Photons absorbed by the water or sphere surface were lost and excluded from the analysis.

Fig. 3Full-screen View

(Color online) One event display induced by a 6 MeV electron neutrino. The red lines are the tracks of the Čerenkov photons, and the thick dashed black arrow denotes the recoil electron in νe - e^- scattering

The correlation between the number of photons triggering the PMTs and the total number of photons produced in each neutrino-induced reaction is illustrated in Fig. 4. The simulation incorporates dark noise at 4.2 kHz, which is typical in PMTs, and may result in 1–2 additional signals per event. The overall efficiency of the PMT triggering was approximately 20%. Considering the geometric coverage of about 30%, it is inferred that roughly 40% of the photons are lost during transport to the PMTs owing to scattering or absorption.

Fig. 4Full-screen View

(Color online) The correlation of the number of the emitted Čerenkov photons and the detected ones

The limited distribution of Čerenkov photons is crucial for event reconstruction at the event-by-event scale. The distribution of Čerenkov photons in a polar coordinate system θ vs. ϕ is shown in Fig. 5. Here, θ is the angle between the direction of the recoil electron and the vector from the reaction vertex to the triggered PMT, and ϕ is the azimuthal angle of the Čerenkov photon. Figure 5(a) displays the cumulative θ vs. ϕ plot for 10⁵ events, with the electron’s incident direction pointing to the pole (θ=0°) by definition. The Čerenkov circle is clearly visible, with its center at θ=0°. However, for individual events, as shown in Fig. 5(b) and 5(c), the triggered PMTs are distributed in a scattered manner, making the circular pattern invisible to the eye. This necessitates a sophisticated algorithm to reconstruct the direction of incident electrons.

Fig. 5Full-screen View

(Color online) Projection diagram of fired PMTs relative to the recoil electron direction. (a) Integration over 10⁵ events. (b) and (c), Fired PMT distributions in two single events

More discussions on seawater environment

The Tropical Deep-sea Neutrino Telescope (TRIDENT) collaboration [37] measured an attenuation length of 20 m in the blue wavelength band at a depth of 3.5 km. This is considerably shorter than the 100 m reference value used in Fig. 2(a). For a typical photon travel distance of 5 m, the survival probability in seawater compared to that in pure water is . Although this corresponds to an 18% reduction in photon yield, the current PMT threshold (see Sect. 3.4) effectively rejects ⁴⁰K backgrounds; therefore, the overall conclusions remain unaffected.

Impurity ions in seawater can induce background reactions, such as ν_e + ³⁷Cl → e⁻ + ³⁷Ar, with a cross-section of [38]. Given a chloride concentration of 0.546 mol/L and an isotopic abundance of ³⁷Cl (η₃₇=24.23%), the additional background-to-signal ratio was estimated to be approximately 0.5% using Eq. (2).(2)This value is negligible; however, future studies should consider the uncertainties in these parameters.

For biofouling mitigation, PMTs are oriented inward within a light-shielded detector design, which minimizes exposure to light induced by external biofouling. Because biofouling emissions are omnidirectional, unlike the directional Čerenkov radiation generated by neutrino-related signal events, this difference enables effective discrimination. Consequently, biofouling does not impose a significant challenge to current feasibility simulations.

Vertex reconstruction

This study focused solely on the νe - e^- elastic scattering process. The determination of the vertex position depends on the timing information obtained from the fired PMTs. Consequently, the time resolution is a critical parameter. For neutrinos with a few MeV of kinetic energy, the recoil electron rapidly loses all its kinetic energy, resulting in a negligible track length. Therefore, all Čerenkov photons are assumed to originate from a single point denoted as (x₀, y₀, z₀, t₀). Given the exact position (xi, yi, zi) of each fired PMT, the expected arrival time of the Čerenkov photon at the i^th PMT can be calculated as:(3)where c is the speed of light in vacuum, and n is the refractive index of water.

If the reconstructed vertex coincides with the true vertex, the difference between the recorded firing time and the expected time (from Eq. (3)) for all fired PMTs should yield a sharp peak centered at zero. However, if the reconstructed vertex deviates from the true position, the calculated time distribution exhibits a broader and distorted distribution with large timing residuals.

To address this, the vertex parameters that maximize the consistency between the measured and expected times should be selected, thereby yielding the final reconstructed vertex. Specifically, the strategy is to minimize the function defined in Eq. (4) [44](4)where represents the deviation of the measured time ti from the expected time , and corresponds to the PMT time resolution.

It should be noted that the ansatz in Eq. (4) without the exponential term resembles a conventional χ² value. The inclusion of the exponential term is motivated by the fact that some Čerenkov photons may undergo multiple reflections before reaching a PMT, causing the actual firing time to differ significantly from the value predicted by Eq. (3). This exponential term reduces the sensitivity of the fit to such outliers, enabling a more robust reconstruction of the vertex based on an event-by-event logic.

Direction construction

The Čerenkov lights generally form a cone with a semi-apex angle satisfying , where β=v/c and v is the particle velocity. For electrons with energies of several MeV, the distance traveled before slowing down to the threshold is short (approximately 1 cm). As a result, the emitted Čerenkov lights can be approximated to have a fixed emission angle from the reaction vertex, given by . However, secondary scattering of electrons in the medium causes broadening of the angular distribution, as shown in Fig. 6. This distribution is used as the probability distribution function (PDF) in the Hough transform.

Fig. 6Full-screen View

Angular distribution of Čerenkov lights with respect to the electron flying direction

The Hough transform (HT) method [22, 45-48] was applied to determine the direction of the recoil electron in νe - e^- scattering using data recorded by the PMTs. Figure 7 illustrates an example of the HT method applied to the event shown in Fig. 5(b). Once a vertex is reconstructed, each fired PMT defines a direction vector characterized by the polar angle θi and azimuth ϕi. To simplify the explanation, consider the fixed Čerenkov angle (α_c) first. For each PMT, the data point generates a circle on the θ - ϕ plane at the given polar angle α_c, with the longitudinal axis aligned from the vertex to the fired PMT, as shown in Fig. 7(a) by red cross symbols. This circle represents a probability distribution in the parameter space of the electron direction, characterized by θ₀ and ϕ₀. As shown in Fig. 7(b), summing the transformed probability distributions weighted by the number of PEs from all fired PMT data points yields the probability distribution of the high-speed electron in the θ - ϕ parameter space. The most probable point in this distribution corresponds to the reconstructed direction of the electron. In real calculations, owing to the scattering of Čerenkov photons, the PDF of the Čerenkov angle from Fig. 6 is used instead of a fixed α_c. Following the same procedure, with the circle replaced by the continuous distribution for each transformation, the real probability distribution of the direction in the θ - ϕ parameter space is obtained. The most probable point θ₀ - ϕ₀ is situated at the point with the highest probability, as depicted by the red cross marker in Fig. 7(c).

Fig. 7Full-screen View

(Color online) Schematic elucidation of the Hough Transform method. (a) The fired PMTs on θ - ϕ plane, the circles near the red cross markers are the possibility distribution to find the electrons, assuming a constant Čerenkov angle. (b) All the possibility distributions are summed up for all fired PMTs with the number of PE as the weight. (c) The real possibility distribution of finding the electron in θ - ϕ parameter space, using the PDF shown in Fig. 6. The red cross marker represents the recognized direction of the electrons

Result from neutrino events

In neutrino-induced reaction simulations, the issue of insufficient statistics owing to the small cross-section is addressed by forcing the reaction to occur when a neutrino is injected. The distribution of recoil electrons is sampled using Eq. (5) [49],(5)where Te represents the kinetic energy of the recoil electron.(6)The constants g₁ and g₂ characterize the coupling strength of the left- and right-handed components in weak interactions and are dependent on the flavor of the neutrino. The corresponding distribution for solar neutrinos νe is shown in Fig. 8.

Fig. 8Full-screen View

(Color online) Angular distribution of the recoil electron in νe - e^- scattering with respect to the incident direction of the neutrino at different Eν

The simulation process proceeds as follows: a random point within the fiducial volume of the detector setup is selected as the reaction vertex. The scattering angle is randomly sampled from Eq. (5) for a given neutrino energy, and the electron momentum was calculated using energy-momentum conservation. The vertex and momentum together define the recoil electron as a neutrino event. Subsequently, Čerenkov light is generated based on the velocity of the recoil electron and transported to the PMTs, where the emitted PEs are recorded. The HT method is then applied to reconstruct the incident direction of the recoil electron and the position of the reaction vertex.

The results of the reconstruction are shown in Fig. 9. For each energy point, 10⁵ events were simulated. Figure 9(a) shows the distribution of α, defined as the angle between the reconstructed and genuine directions of the recoil electron. As the incident neutrino energy increases, the energy of the recoil electron also increases, and the distribution exhibits a more pronounced peak at , indicating a higher reconstruction success. The efficiency η increases from 0.66% at Eν=2 MeV to 66% at Eν=10 MeV. Figure 9(b) displays the mean value as a function of the incident neutrino energy. At the high-energy end near 10 MeV, , suggesting a good determination of the electron direction. Figure 9(c) presents the distance ΔR between the genuine and reconstructed reaction vertices. The deviation of the reconstructed vertex decreased slightly as a function of the incident neutrino energy Eν.

Fig. 9Full-screen View

(Color online) (a) Distributions for angle reconstruction with their efficiency. (b) and (c) show the mean value of and the error of the constructed vertex as a function of the neutrino energy

Influence of the background events

Careful treatment is required to suppress events caused by intense background sources, including seawater radioactivity, cosmic-ray muons, and faint light emitted by bioluminescent plankton near the detector volume [50]. In principle, the muon background can be efficiently rejected using a veto detector mounted above the main detector or by applying an energy cut-off. This is because the Čerenkov photons produced by energetic muons penetrating the thick seawater are abundant. Additionally, light from bioluminescent plankton induces signals on PMTs with distinct rising times and amplitudes, allowing for identification and rejection. However, seawater radioactivity is primarily focused on because the intensity of γ-radiation from radioactive isotopes is high, and their γ energy is comparable to the recoil electron energy in νe - e^- scattering, making it the dominant background noise.

The primary radioactive isotope in seawater is ⁴⁰K [51-53]. The mass fraction of potassium in seawater is mg/L, and the isotopic abundance of ⁴⁰K is η₄₀=0.0117%. The decay lifetime of ⁴⁰K is τ=5.692×10¹⁶ s. Using Eq. (7), the event rate for ⁴⁰K is calculated:(7)where N_A is Avogadro’s constant, and M_K is the molar mass of potassium. The calculated event rate was significant, necessitating a detailed simulation of the background. ⁴⁰K has multiple decay channels [54], as shown in Fig. 10.

Fig. 10Full-screen View

(Color online) The decay scheme of ⁴⁰K [54]

Specifically, 89.25% of its decays are β^- decays to the ⁴⁰Ca ground state, and 10.55% are electron-capture transitions to the 1460 keV level of ⁴⁰Ar. To model the ⁴⁰K process, a combination of 10.55% of 1.46 MeV γ-ray events and 89.25% of decayed e^- events, with random positions and directions, was simulated. The e^- energy is derived from Eq. (8), with a maximum energy E_max=1.311 MeV.(8)Because γ-rays can travel a distance before being absorbed, decay events are generated in an extended volume with a radius R=6 m, exceeding the fiducial volume.

Figure 11(a-c) illustrate the effects of the ⁴⁰K background. Figure 11(a) shows the distribution of , where α is the angle between the reconstructed electron and the initial radiation from the source. A forward-peaking feature is evident. Figure 11(b) displays the displacement between the real and reconstructed vertices. Figure 11(c) presents the distribution of the number of fired PMTs M_pmt caused by ⁴⁰K events.

Fig. 11Full-screen View

(Color online) Left: The distribution of the angle between real and constructed direction of ⁴⁰K decay (a) and γ-rays at Eγ=2.6 MeV (d). Middle: distribution of the deviation between the real and constructed vertices in the coordinate space for ⁴⁰K decay (b) and γ-rays at Eγ=2.6 MeV (e). Right: the distribution of the multiplicity of fired PMTs for ⁴⁰K decay (c) and γ-rays at Eγ=2.6 MeV (f)

Another major radioactive source in seawater arises from the decay chain of ²³²Th and ²³⁸U. The most energetic γ-ray in this chain is the 2.6 MeV γ-ray from ²⁰⁸Tl. To be conservative, the study simulates the background using 2.6 MeV γ-radiation. The results are shown in Fig. 11(d-f). The features of the 2.6 MeV γ-rays are similar to those of the ⁴⁰K decay, except that the multiplicity of fired PMTs M_pmt is slightly higher for the 2.6 MeV γ-rays, as seen in the peak positions.

Unfortunately, whether the detected particle is an e^- or a γ-ray, it is always observed as an e^-, identical to the νe - e^- scattering event. The only distinction between the signal and background lies in the e^- energy. To differentiate the neutrino signal from the background, a cut was imposed on M_pmt to suppress events caused by the intense radioactive sources in the seawater. Figure 12(a) shows the multiplicity distribution as a function of the incident neutrino kinetic energy Eν. The goal is to reject the ⁴⁰K background owing to its extremely high event rate. Based on Fig. 11(c), a threshold condition of is set, as indicated by the dashed line in Fig. 12(a). As the neutrino energy increases, more events are accepted. For Eν=6 MeV and 10 MeV, the ratios of events with M_pmt above the threshold are 55% and 80%, respectively, as shown in Fig. 12(b). After filtering, the construction efficiencies were 25.6% and 52.2%, respectively, which remained acceptable for detection. The events induced by the ⁴⁰K background are suppressed by a factor greater than 10⁷.

Fig. 12Full-screen View

(Color online) (a) Distribution of multiplicity of fired PMT number as a function of neutrino energy Eν. (b) Ratio of events above the M_pmt as a function of Eν. The final reconstruction efficiency after HT transformation for 6 and 10 MeV neutrino is indicated by the red cross symbol

Sun locating attemption

The search capability of the detector can be evaluated using the Sun as a neutrino source. The solar neutrino flux and corresponding event rate in the proposed detector design were estimated. In this analysis, the Sun’s apparent motion was neglected by adopting a space-fixed coordinate system anchored to distant stars rather than an Earth-fixed frame. The solar neutrino flux for is approximately Φ = 2.5×10⁶ cm^-2s^-1 [55]. The energy-dependent cross-section for νe - e^- elastic scattering is given by Eq. (9) [49].(9)The event rate is calculated as follows:(10)where ρ = 0.56 × 10⁶ N_A is the electron density in water.

Considering a construction efficiency of 33% (as described in Sect. 3.4), we expect approximately 20 detected neutrino events per module per month. With 15 detector modules, 300 solar neutrino events per month can be obtained from the same direction. For the background, because most ⁴⁰K events are excluded by the cuts, a background-to-signal ratio of 50 is reasonable, corresponding to 15,000 background events above the threshold. These background events were randomly sampled in all directions. When these signals and backgrounds were input into the Hough transform, the source direction could be inferred, as shown in Fig. 13.

To obtain comprehensive results, a statistical ensemble containing 10,000 simulations was performed. The angle Θ_s between the real source direction and the reconstructed direction is illustrated in Fig. 14. The search accuracy must meet the requirements of real-world applications. Now, defining successful reconstruction by (dashed line), the success rate is approximately 90% for a 30-day search with 300 solar neutrino events and a background-to-signal ratio of 50.

Fig. 14Full-screen View

Distribution of cosine of the angle () between real and constructed neutrino source direction with 10,000 simulated searches

Searching ability discussion

Given that the detector is capable of searching for multiple neutrino sources, it is critical to optimize the experimental setup, specifically the number of modules, total volume of the sensitive medium, and search duration, as these factors collectively determine the number of collected neutrino signals, even under high background-to-signal ratios. Qualitatively, an increase in neutrino signals improves the search success rate, even at higher background-to-signal ratios.

To achieve the same success condition (i.e., 90% success rate), the minimum number of accumulated events as a function of the background-to-signal ratio was investigated, as shown in Fig. 15. The minimum number of neutrino events required to ensure a 90% success rate is approximately linear with respect to the background-to-signal ratio. For example, if the background-to-signal ratio reaches 270, 1000 neutrino events must be accumulated.

Fig. 15Full-screen View

(Color online) The minimum accumulation of events required to ensure 90% success rate as a function of background-to-signal ratio. The band represents a 99% confidence interval

This linear relationship provides confidence that, regardless of how large the background-to-signal ratio becomes, the same search capability can be achieved by scaling the number of detector modules. This is reasonable because the condition for identifying the true source is that the signal peak on the Hough transform probability plane exceeds the background fluctuations. Assuming that the signal peak height scales proportionally with the total number of signal events (N_s) and that the background fluctuations scale as , one obtains . With appropriate rescaling, this relationship can be expressed as Equation (10):(11)where k is a factor that is positively correlated with the success condition. This implies that by increasing the number of detector modules, the increased signal can overcome higher background fluctuations, thereby maintaining a 90% success rate.