# Reconstruction Algorithm for a Novel Cherenkov Scintillation Detector

Wentai Luoa Qian Liua Yangheng Zhenga Zhe Wangb,c,d Shaomin Chenb,c,d

###### Abstract

For future MeV-scale neutrino experiments, a Cherenkov scintillation detector,
CSD, is of particular interest for its capability to reconstruct both energy
and direction for charged particles. A type of new target material, slow
liquid scintillator, SlowLS, which can be used to separate Cherenkov and
scintillation lights, is one of the options for the neutrino detectors. A
multi-hundred ton spherical CSD is simulated using a Geant4-based Monte Carlo
software, which handles the detailed the micro processes of MeV particles and
optical photons and the functions for photomultiplier, PMT, and readout
electronics. Twelve SlowLS samples are simulated and studied to cover a wide
range of scintillation light yields and scintillation emission time constants.
Based on the detailed knowledge of the signal processes, simplified functions
are constructed to predict the charge and time signals on the PMTs to fulfill
an efficient reconstruction for the energy, direction, and position of charged
particles. The performance of the SlowLS reconstruction, including the
resulting energy, angular, and position resolution, and particle
identification capability, is presented for these samples. The dependence of
the performance on the scintillation light yield and emission time constants
is understood. This study will be a guideline for future MeV-scale neutrino
CSD design and SlowLS development for the interested physics goals.

##  1 Introduction

Liquid neutrino detectors typically come in two different varieties. One is
based on liquid scintillator, and typical experiments are KamLAND, Borexino,
Daya Bay, and RENO [1, 2, 3, 4]. The other uses water, and the experiments are
Super-Kamiokande, SNO, and IMB [5, 6, 7]. Water Cherenkov detectors, despite
the good directional performance, have a low light yield and poor energy
resolution and are not very effective in detecting neutrinos below
$4\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$ [8]. In contrast, the liquid
scintillator neutrino detector has a high light yield and, correspondingly, a
good energy resolution and can detect neutrinos below
$1\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$, but almost no signature of
direction can be seen [9]. High energy resolution and directionality are two
key ingredients in studying neutrino physics.

The Cherenkov scintillation detector, CSD, is a novel type of detector that
can reconstruct the number of Cherenkov photons and scintillation photons. It
not only has good position resolution and energy resolution but also has a
good angular resolution. A few slow liquid scintillators, SlowLS, candidates
(water-based or oil-based) are being studied as the detection medium of the
CSD [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], whose main feature is the
separation of scintillation light from Cherenkov light through the time
profile. The emission time profile of organic solution scintillators shows a
rising time $\tau_{r}$ and a decay time $\tau_{d}$, resulting in the
normalized pulse shape of the scintillation light [12, 13]:

$n_{\mathrm{s}}(t)=\frac{\tau_{r}+\tau_{d}}{\tau_{d}^{2}}(1-e^{-t/\tau_{r}})\cdot
e^{-t/\tau_{d}}.$ (1.1)

For the SlowLS, $\tau_{r}$ is usually greater than
$1\text{\,}\mathrm{n}\mathrm{s}$, $\tau_{d}$ is more than
$20\text{\,}\mathrm{n}\mathrm{s}$, and the scintillation light yields range
from a few hundred photons to a few thousand photons per MeV. Cherenkov light
of charged particles can be reconstructed as a prompt component and
scintillation light mainly as a delayed component. Directional information is
obtained from Cherenkov light and energy information can be estimated from
both scintillation and Cherenkov lights.

Based on the properties of the SlowLS, the CSD will be of great benefit in
many studies. The directional information provides the essential solar angle
measurement and suppresses isotropic radioactive backgrounds in solar neutrino
experiments [21, 22, 23, 8, 24, 25, 26]. In addition, determining the numbers
of Cherenkov photons and scintillation photons also helps to do particle
identification, PID, for electrons, gamma, protons, and alpha. In neutrinoless
double beta, $0\nu\beta\beta$, decay experiments, good directional information
and PID would reduce the solar neutrino’s background and radioactive
background [27, 28, 29, 30, 31, 32, 33]. Supernova neutrinos research will
also be beneficial for the PID information [34].

China’s Jinping underground laboratory has a $2400\text{\,}\mathrm{m}$ rock
overburden. A CSD analysis is based on the design of a multi-hundred-ton
Jinping CSD proposal, which will measure MeV-scale neutrinos, such as solar-,
geo-, supernova neutrinos, and $0\nu\beta\beta$ decay. Other detectors with
comparable structures can also use the approaches described in this article.

In this paper, a CSD reconstruction algorithm and the reconstruction
performance of the CSD with full detector simulation study are presented. The
results can be used as a guideline for the new detector design or for a new
SlowLS development for a specific physics goal. Section 2 describes the
details of the simulation, including the detector geometry, the setup of the
simulation, and the properties of twelve SlowLS samples. Section 3 focuses on
the analysis of the simulated output waveforms. To reconstruct the energy,
position, and direction, and do PID for the events, first, in section 4, we
decompose and analyze the signal analysis and make the corresponding
simplification. Additionally the study of the direct photons and indirect
photons is included in this section. Next, in section 5, we construct the
prediction models of the charge and time profiles and compare them with the
full simulation. Then, reconstruction is done with the prediction models using
a maximal likelihood method, and the performance of the CSD detector
reconstruction is shown in section 6. Section 7 is the conclusion.

##  2 Detector Setup and Simulation

The geometry of the proposed multi-hundred-ton detector, the settings of the
full simulation, and the chosen twelve SlowLS samples are all described in
this section.

###  2.1 Detector Setup

A multi-hundred-ton neutrino detector will be built, which is comparable to
the Borexino, Kamland, and SNO+ detectors; thus, the following geometry is set
up for simulation, which is depicted in figure 1. Five hundred metric tons of
the SlowLS are contained in a 10-m diameter transparent acrylic vessel. The
acrylic vessel is constructed from 6-cm-thick transparent acrylic and is
suspended at the center of the CSD. Scintillation and Cherenkov light produced
by neutrino interactions and radioactive backgrounds is detected by an array
of 6400 Hammamatsu model 8-inch photomultipliers (photocathode coverage to
nearly 50%), PMTs, supported by a stainless steel geodesic sphere. The setup
of the 8-inch PMT is as follows. 34% quantum efficiency at nearly
$390\text{\,}\mathrm{n}\mathrm{m}$, 100% collection efficiency,
$2\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$ dark noise rate,
$20\text{\,}\mathrm{n}\mathrm{s}$ transmission time, TT, and
$1\text{\,}\mathrm{n}\mathrm{s}$ transmission time spread (full width at half
maximum), TTS [35]. Additionally, over 1,200 tons of mineral oil and water
shield the SlowLS from external radioactive backgrounds: 300 tons of mineral
oil of the acrylic vessel and the PMT support sphere and 800 tons of water
between the PMT support sphere and a water tank.

Figure 1: Structure of the multi-hundred-ton detector and its geometry setup.

###  2.2 Simulation

A Monte-Carlo, MC, simulation package, the Jinping Simulation and Analysis
Package, JSAP, is used to simulate the physical process in the CSD. The JSAP
is based on Geant4 [36] with many customized features. The main components are
shown in a simulation flow chart in figure 2 and expanded below.

Figure 2: Simulation flow chart of the proposed MeV-scale neutrino experiment.

##### Generator

The generator creates a primary particle and generates its position, time, and
momentum. The generator for neutrino interactions only begins with secondary
particles such as recoiled electrons.

##### Detector simulation

The software simulates the detector’s response using information from the
physics generator as well as the detector’s geometry and materials. In the
CSD, photons will pass through substances such as the SlowLS, acrylic, and
mineral oil. For the above substances, the JSAP can define the propagation
characteristics of the photons in the substance, which are included:

  * •

The spectra of the refractive index, absorption length, and Rayleigh
scattering length are shown in figure 3 [37, 38] and figure 4 [39, 40, 41,
42].

  * •

The scintillation emission spectrum is shown in figure 5 [13, 43].

  * •

The scintillation light yield, rise time and decay time will be discussed in
table 1.

  * •

Birks’ constant $k_{\mathrm{B}}$ in this work is
$0.016\text{\,}\mathrm{c}\mathrm{m}\mathrm{/}\mathrm{M}\mathrm{e}\mathrm{V}$
[44].

Figure 3: A refractive index spectrum of the SlowLS, acrylic and mineral oil
in simulation [37, 38].

Figure 4: Left: The absorption length spectrum of transparent materials in
simulation. Right: The Rayleigh scattering length spectrum of transparent
materials in simulation [39, 40, 41, 42]. Figure 5: The wavelength spectrum of
scintillation and Cherenkov photons in simulation [13, 43].

The output is the information about a photoelectron, PE, hit. The PE hit is a
sampling based on quantum efficiency that determines whether an optical photon
that hits a PMT photocathode may be converted to an electron.

##### Dark noise and Pre-trigger

On the PMT photocathode, thermal electron emission produces dark noise, which
results in an equivalent PE signal. The pre-trigger determines whether the
total count rate of the physically caused PEs and dark noise signals reaches
the minimum number of hit PMTs within the time window
($500\text{\,}\mathrm{n}\mathrm{s}$). If the total count rate cannot reach the
minimum number of hit PMTs, no more simulations will be performed, which saves
simulation time.

##### Waveform generation

The input is the PE hit time and dark noise hit time. The PEs are convoluted
with the single PE response of the 8-inch PMT [35]. A flash analog-to-digital
converter is simulated in the JSAP’s readout electronics system with a dynamic
range of $1\text{\,}\mathrm{V}$, a precision of
$10\text{\,}\mathrm{b}\mathrm{i}\mathrm{t}\mathrm{s}$, and a sampling rate of
$1\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$. The electronics readout system
then combines these PEs with a single PE waveform to form the output
waveforms, which is the same as the actual experimental data. The gain of each
PMT is considered. The single PE waveform is fit using the empirical formula
as follows:

$\mathrm{PE_{single}}(t)=A_{0}\cdot\frac{1}{\tau_{\mathrm{PE}}}\mathrm{exp}(-\frac{t}{\tau_{\mathrm{PE}}})\theta(t)\otimes\frac{1}{\sigma_{\mathrm{PE}}\sqrt{2\pi}}\exp{(-\frac{t^{2}}{2\sigma_{\mathrm{PE}}^{2}})},$
(2.1)

where $A_{0}$ is the single PE charge, $\theta(t)$ is a step function, and
$\sigma_{\mathrm{PE}}$ and $\tau_{\mathrm{PE}}$ are the shape parameters.

Moreover, a baseline fluctuation is considered to be
$1\text{\,}\mathrm{m}\mathrm{V}$, and a waveform is depicted in figure 6.

Figure 6: A waveform of the simulation output. The PE time is indicated by the
red line.

##### Trigger

This step synchronizes and assimilates the information from the raw waveform
and checks the trigger conditions. A threshold of the PMTs is
$5\text{\,}\mathrm{m}\mathrm{V}$, and the trigger condition of the event is
that at least 15 hit PMTs within $500\text{\,}\mathrm{n}\mathrm{s}$.

###  2.3 Twelve SlowLS Samples

In this paper, we attempt to separate Cherenkov light from scintillation
light, which is related to two keys of the SlowLS: the scintillation light
yield and scintillation time profile. The current study of the SlowLS
indicates that the higher the light yield is, the smaller the time constants
of the time profile for the oil-based liquid scintillator [13]. For water-
based liquid scintillators, the higher the light yield is, the larger the time
constants of the time profile [15]. To find the potential SlowLS in the CSD,
the four typical light yields and three typical time profiles are selected, as
shown in table 1.

The four typical scintillation light yields are 200, 350, 3750 and
$5000\text{\,}\mathrm{p}\mathrm{h}\mathrm{o}\mathrm{t}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{/}\mathrm{M}\mathrm{e}\mathrm{V}$.
The low light yield SlowLS samples correspond to the water-based liquid
scintillator, which usually has a low light yield of a few hundred photons per
MeV [15]; the high light yield SlowLS samples correspond to the oil-based
liquid scintillator, which is usually as high as several thousand photons per
MeV [13]. The rise times, $\tau_{r}$ of the three typical time profiles are
7.7, 1.2 and $0.5\text{\,}\mathrm{n}\mathrm{s}$ and the decay times,
$\tau_{d}$ are 37, 27, and $10\text{\,}\mathrm{n}\mathrm{s}$. The two slower
time profiles are derived from existing measurements of the SlowLS [13, 12],
and the fastest time profile is used to explore the time profile of the
potential SlowLS.

Table 1 lists four PE yields, $E_{\mathrm{scale}}$, and their corresponding
scintillation light yields. Although the number of PEs is related to the
attenuation length spectrum of the medium, the PMT photocathode coverage, and
the detection efficiency of the PMT, similar PE yields can be obtained by
slightly varying the spectra of these parameters in the simulation. In
addition, a signal process will be discussed in detail in later sections.

Table 1 also lists a Cherenkov-to-scintillation ratio, CSR, within the first
$10\text{\,}\mathrm{n}\mathrm{s}$. The times of the hit PEs are arranged from
smallest to largest, and the number of PEs generated by Cherenkov and
scintillation light is counted to obtain the CSR. Combined with the time
response of the detector, the CSR within the first
$10\text{\,}\mathrm{n}\mathrm{s}$ (equal to 5 times the time resolution) can
better characterize the distinction between scintillation and Cherenkov light.
As a result, we use the CSR in the first $10\text{\,}\mathrm{n}\mathrm{s}$ of
the detector center’s $5\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$ electrons as
a criterion to judge the distinction between Cherenkov and scintillation
light.

For classification purposes, we named the twelve different samples such as
$\mathrm{LS_{\mathrm{376~{}PE}}^{\mathrm{7.7~{}ns}}}$ or
$\mathrm{LS_{\mathrm{35~{}PE}}^{\mathrm{0.5~{}ns}}}$, with PE yield
($376\text{\,}\mathrm{P}\mathrm{E}\mathrm{/}\mathrm{M}\mathrm{e}\mathrm{V}$ or
$35\text{\,}\mathrm{P}\mathrm{E}\mathrm{/}\mathrm{M}\mathrm{e}\mathrm{V}$) in
the lower corner and $\tau_{r}$ ($7.7\text{\,}\mathrm{n}\mathrm{s}$ or
$0.5\text{\,}\mathrm{n}\mathrm{s}$) in the upper corner. In section 6, the
twelve different samples of the SlowLS are reconstructed in the CSD.

Scintillation light yield (photons/MeV) |  PE yield $E_{\mathrm{scale}}$ (PE/MeV) |  CSR in the first $10\text{\,}\mathrm{n}\mathrm{s}$
---|---|---
|  $\tau_{r}=7.7~{}\mathrm{ns}$
---
$\tau_{d}=37~{}\mathrm{ns}$
|  $\tau_{r}=1.2~{}\mathrm{ns}$
---
$\tau_{d}=27~{}\mathrm{ns}$
|  $\tau_{r}=0.5~{}\mathrm{ns}$
---
$\tau_{d}=10~{}\mathrm{ns}$
200 |  24 |  |  1: 0.1
---
|  1: 0.3
---
|  1: 0.7
---
350 |  35 |  |  1: 0.2
---
|  1: 0.6
---
|  1: 1.3
---
3750 |  285 |  |  1: 2.4
---
|  1: 6.1
---
|  1: 14
---
5000 |  376 |  |  1: 3.2
---
|  1: 8.2
---
|  1: 19
---
Table 1: Twelve samples of three scintillation time profiles and four
scintillation light yields are shown in the table. The PE yields of them are
obtained by the full MC simulation. The CSR in the first
$10\text{\,}\mathrm{n}\mathrm{s}$ is obtained by simulating the
$5\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$ electrons at the detector center
for these samples.

##  3 Readout and Waveform

The simulated output waveform needs to be analyzed to obtain the number of the
observed PEs by each PMT and the time of each PE. This section focuses on the
waveform analysis method and the performance.

###  3.1 Waveform Analysis

The time and number of PEs are obtained by fitting the $i$th waveform with the
constructed likelihood function. The time window of the waveform is
$500\text{\,}\mathrm{n}\mathrm{s}$, and the trigger time starts from
$125\text{\,}\mathrm{n}\mathrm{s}$, so the waveform is fit within a time
window of 100-$500\text{\,}\mathrm{n}\mathrm{s}$. As a result, a likelihood
function of waveform analysis is shown as follows:

$\mathcal{L}(n_{i}^{\mathrm{Obs}},t_{ij},R_{ij})=f_{\mathrm{Poisson}}(n_{i}^{\mathrm{Obs}};n_{\mathrm{average}})\cdot\prod^{500}_{k=100}\mathrm{Gauss}([V_{ik}-\sum_{j=1}^{n_{i}^{\mathrm{Obs}}}R_{ij}\mathrm{PE_{single}}(t-t_{ij})],\sigma_{\mathrm{base}}),$
(3.1)

where $n_{i}^{\mathrm{Obs}}$ is the number of PEs of the $i$th waveform,
$t_{ij}$ is the start time of the $j$th PE, and $R_{ij}$ is the ratio of the
charge of the $j$th PE to the average charge. $V_{ik}$ is the amplitude of the
$k$th bin, and $\mathrm{PE_{single}}$ is the single PE waveform shown in eq.
(2.1), and $\sigma_{\mathrm{base}}$ is the standard deviation of the baseline,
which is $1\text{\,}\mathrm{m}\mathrm{V}$. $f_{\mathrm{Poisson}}$ is the
Poisson distribution with the average number of PEs, $n_{\mathrm{average}}$ is
defined as,

$n_{\mathrm{average}}=\displaystyle{\sum_{i=1}^{N_{\mathrm{hitPMT}}}{q_{i}}}\frac{1}{q_{\mathrm{single}}N_{\mathrm{PMT}}},$
(3.2)

where $N_{\mathrm{hitPMT}}$ is the number of the hit PMT, $q_{i}$ is the
charge of the $i$th waveform, $q_{\mathrm{single}}$ is the mean value of the
single PE charge obtained from the calibration, and $N_{\mathrm{PMT}}$ is the
total number of PMTs.

###  3.2 Performance of Waveform Analysis

$\mathrm{LS_{\mathrm{35~{}PE}}^{\mathrm{0.5~{}ns}}}$ is used as an example
with a scintillation light yield of
$350\text{\,}\mathrm{p}\mathrm{h}\mathrm{o}\mathrm{t}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{/}\mathrm{M}\mathrm{e}\mathrm{V}$.
The $5\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$ kinetic energy electrons in the
SlowLS are simulated. With the use of the maximum likelihood function method,
the waveform is analysed. The PE time and the number of PEs on each PMT are
eventually produced by maximization of the likelihood function. The results,
as shown in figure 7. Moreover, the results also compare the true PE with the
fit PE are shown in figure 8.

Figure 7: Two examples of the fit waveforms.

As seen in the left panel of figure 8, in most of the analysis results of the
waveforms, the number of fit PEs and the number of true PEs are basically the
same. In addition, the fit PE time is compared with the true PE time as shown
in the right panel of figure 8. The distribution of the PE time difference is
obtained by subtracting the fit PE time from the true PE time. We fit the
distribution with a Gaussian function with a fit standard deviation of
$0.23\text{\,}\mathrm{n}\mathrm{s}$. Moreover, for more than 93% of the PEs,
the time difference between the fit result and the true value is less than
$0.5\text{\,}\mathrm{n}\mathrm{s}$.

Figure 8: Left: The variations in the number of PEs for each waveform. Right:
The difference between the true and fit PE time.

##  4 Signal Process and Simplification

The number of observed PEs on each PMT and the PE times are obtained from the
waveform analysis. Further reconstruction of energy, position, direction, and
PID requires the prediction of the number of PEs and PE time on each PMT, so
the signal process needs to be carefully studied.

The process from a particle’s energy deposition to PEs is very complex. If the
signal process was fully considered in the original physics style during
reconstruction, the computation time would be astronomical. To save
reconstruction time, the corresponding simplified functions are constructed
for the signal process. Additionally, the thoughts on direct and indirect
photons are covered in detail.

###  4.1 Photon Emission

The mechanisms of scintillation and Cherenkov light generation and the
Cherenkov light angle distribution are described here, and the light yields of
both are calculated.

####  4.1.1 Scintillation Light

Scintillation light is a type of photon produced by ionization or excitation
of charged particles in a medium, and its emission direction is isotropic. In
addition, the quenching effect of charged particles is generally expressed by
the semiempirical formula Birks’ law. Birks’ law gives the scintillation light
yield $Y_{\mathrm{s}}$ per unit of travel distance [45]:

$\frac{\mathrm{d}Y_{\mathrm{s}}}{\mathrm{d}x}=A\frac{\mathrm{d}E/\mathrm{d}x}{1+k_{\mathrm{B}}\mathrm{d}E/\mathrm{d}x},$
(4.1)

where the energy loss density for charged particle production of scintillation
light is $\mathrm{d}E/\mathrm{d}x$, $k_{\mathrm{B}}$ is Birks’ constant, as
introduced in section 2.2 and $A$ is a normalization constant.

The quenching effect for 1 to $9\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$
electrons is nearly 0.88 in the simulation. As a result, for
$2\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$ kinetic energy electrons, when the
scintillation light yield is
$5000\text{\,}\mathrm{p}\mathrm{h}\mathrm{o}\mathrm{t}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{/}\mathrm{M}\mathrm{e}\mathrm{V}$,
the total number of scintillation photons is
$Y_{\mathrm{s}}(E)=0.88Y_{\mathrm{s}}E=8800~{}\mathrm{photons}$. Moreover, the
wavelength spectrum of the scintillation light for simulation is shown in
figure 5.

####  4.1.2 Cherenkov Light

Cherenkov photons are produced in a cone around the trajectory of a charged
particle when the charged particle travels faster than the speed of light in a
medium. [46]. The cosine of the emission angle, $\theta_{\mathrm{c}}$, can be
expressed as:

$\cos\theta_{\mathrm{c}}=\frac{1}{n(\lambda)\beta},$ (4.2)

where $n$ is the refractive index determined as shown in figure 3, and
$\lambda$ is the wavelength of the photon. The charged particle’s relative
velocity $\beta$ is shown as

$\beta=v/c,$ (4.3)

where $v$ is the velocity of a charged particle in a medium, and $c$ is the
speed of light in a vacuum. As a result, the energy threshold for electrons in
the SlowLS with $n=1.5$ is $0.686\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$. In
addition, the Cherenkov photon yield $Y_{\mathrm{c}}$ per traveled distance
$x$ per unit $\lambda$ is

$\frac{\mathrm{d}^{2}Y_{\mathrm{c}}}{\mathrm{d}x\mathrm{d}{\lambda}}=\frac{2\pi
aq^{2}}{\lambda^{2}}(1-\frac{1}{\beta^{2}n^{2}(\lambda)}),$ (4.4)

where $a$ is the fine structure constant and $q$ is the charge of the
particle. Assuming that the refractive index is wavelength independent from
300 to $700\text{\,}\mathrm{n}\mathrm{m}$ (the detectable wavelength range),
the Cherenkov light yield $Y_{\mathrm{c}}$ for 1 to
$9\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$ electrons with kinetic energy $E$
is obtained by integrating $\lambda$,

$Y_{\mathrm{c}}(E)=436E\sin^{2}{\theta_{\mathrm{c}}},$ (4.5)

where $\theta_{\mathrm{c}}$ is the initial emission angle of the electron.
Moreover, the wavelength spectrum of the Cherenkov light for simulation is
shown in figure 5.

##### Angular Distribution

In figure 9, the angular distributions of the Cherenkov light of electrons at
various positions and energies are obtained by full MC simulation, where
$\alpha$ is the angle of the hit PMT relative to the electron direction from
the vertex.

Figure 9: Left: An angular distribution of the Cherenkov light of electrons at
various energies (1 to $9\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$). Right: An
angular distribution of the Cherenkov light of electrons at different
positions. The position on the detector’s X-axis, from $-4\text{\,}\mathrm{m}$
to $4\text{\,}\mathrm{m}$ in O-xyz. It is not significantly different for
energies and positions. As a result, the
$5\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$ Cherenkov photon angular
distribution is employed as the p.d.f. in reconstructing electrons with
energies ranging from 1 to $9\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$ [6].

The left panel of figure 9 shows that for 1 to
$9\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$ electrons at the detector center,
there is a slight variation in the angular distribution of the Cherenkov light
for different energies. In the right panel of figure 9, a Cherenkov angular
distribution of various positions is plotted by simulating
$5\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$ electrons at the position of
$-4\text{\,}\mathrm{m}$ to $4\text{\,}\mathrm{m}$ on the X-axis in the spatial
Cartesian coordinate system, O-xyz. It shows that the angular distribution of
the Cherenkov light at different positions varies slightly. For the purpose of
simplification, we assume that the Cherenkov light angular distribution is
independent of energy and position. As a result, the Cherenkov light angular
distribution as the probability density function, p.d.f., employed in the
reconstruction is the one created by the
$5\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$ electrons at the detector center.

The Cherenkov light angular p.d.f. of
$5\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$ electrons is fit by a segmented
function consisting of a ninth-order polynomial and a sixth-order polynomial,
as shown in figure 10.

Figure 10: A Cherenkov light angular distribution of 5 MeV electrons at the
center of the detector.

###  4.2 Optical Transportation

This subsection focuses on two aspects: one is to obtain the effective group
velocity by the full MC simulation in order to calculate the time of flight,
and the other is to establish a simplified function for the attenuation of
photons in the medium.

####  4.2.1 Light Speed

Since Cherenkov light and scintillation light are not photons of a single
wavelength, the speed of flight is calculated as a group velocity, which is
given by [47]:

$v_{\mathrm{g}}(\lambda)=\frac{c}{n(\lambda)-\lambda\frac{\partial
n(\lambda)}{\partial\lambda}},$ (4.6)

where $v_{\mathrm{g}}$ is the group velocity, $n$ is the refractive index
determined as shown in figure 3, $c$ is the speed of light in a vacuum, and
$\lambda$ is the wavelength of the photon. The group velocity of different
wavelengths in different materials is obtained by the full MC simulation,
which is shown in figure 11.

Figure 11: A spectrum of the group velocity versus wavelength with different
transparent materials in the full MC simulation.

Because current electronic readout techniques cannot detect the wavelength of
each photon, the time of flight, TOF, will be calculated using the average
group velocity of the detected photons, i.e., the effective group velocity.
Therefore, the group velocities of the detected scintillation and Cherenkov
photons are obtained by the full MC simulation, as shown in figure 12.

Figure 12 shows that the group velocity distribution of the acrylic is
slightly larger than that of the SlowLS. In contrast, figure 3 illustrates
that the group velocity distribution of the acrylic should be somewhat smaller
than the distribution of the SlowLS. It is because figure 12 is only for the
detected photons, with a wavelength of around
$400\text{\,}\mathrm{n}\mathrm{m}$, giving a slightly smaller refractive index
of the acrylic than that of the SlowLS.

As a result, the effective group velocities of scintillation photons in the
SlowLS, acrylic, and mineral oil are
$190.83\text{\,}\mathrm{m}\mathrm{m}\mathrm{/}\mathrm{n}\mathrm{s}$,
$192.30\text{\,}\mathrm{m}\mathrm{m}\mathrm{/}\mathrm{n}\mathrm{s}$, and
$197.10\text{\,}\mathrm{m}\mathrm{m}\mathrm{/}\mathrm{n}\mathrm{s}$,
respectively. The effective group velocities of Cherenkov photons in the
SlowLS, acrylic, and mineral oil are
$192.54\text{\,}\mathrm{m}\mathrm{m}\mathrm{/}\mathrm{n}\mathrm{s}$,
$194.04\text{\,}\mathrm{m}\mathrm{m}\mathrm{/}\mathrm{n}\mathrm{s}$, and
$198.28\text{\,}\mathrm{m}\mathrm{m}\mathrm{/}\mathrm{n}\mathrm{s}$,
respectively.

Figure 12: The group velocity distributions of the detected scintillation
(left) and Cherenkov photons (right) in the different materials.

####  4.2.2 Absorption and Scattering of Photons

When a photon passes through a medium, there is a probability that the photon
will be absorbed or scattered. Compared with the absorption process, the
scattering process only changes the direction of the photon, while the
absorption process causes the photon to disappear. The remaining number of the
photons $I$ obeys Beer’s law [48] and exponentially decreases from the initial
number of the photons $I_{0}$ according to the travel distance of $l$ in a
medium with an attenuation length $L(\lambda)$:

$I=I_{0}e^{\frac{-l}{L(\lambda)}}.$ (4.7)

The combination of the absorption and scattering processes results in the
attenuation of photons. This can be explained using the following formula
[41],

$\frac{1}{L(\lambda)}=\frac{1}{L_{\mathrm{Abs}}(\lambda)}+\frac{1}{L_{\mathrm{Sca}}(\lambda)},$
(4.8)

where $L_{\mathrm{Abs}}(\lambda)$ and $L_{\mathrm{Sca}}(\lambda)$ represent
the absorption and scattering lengths. The absorption length and scattering
length in the SlowLS, acrylic, and mineral oil are shown in figure 4.

The photon survival number is defined as the number of scintillation or
Cherenkov light photons surviving through different lengths of the medium, as
counted by the full MC simulation. We take the example of the electrons at the
center of the detector. The photon survival distribution is displayed in
figure 13. In SlowLS, the absorption length of short wavelengths is small and
that of long wavelengths is large. The photons spectrum of Cherenkov light is
concentrated at short wavelengths, so the photon survival spectrum drops
sharply in the first $500\text{\,}\mathrm{m}\mathrm{m}$. The spectrum of
scintillation light is concentrated near $400\text{\,}\mathrm{n}\mathrm{m}$,
but the photon survival spectrum also drops faster in the first
$500\text{\,}\mathrm{m}\mathrm{m}$. After fitting tests, a triple exponential
function is fit well as follows,

$I(x)=I_{0}[\mathrm{A}\cdot\mathrm{exp}(-x/L_{1})+\mathrm{B}\cdot\mathrm{exp}(-x/L_{2})+\mathrm{(1-A-B)}\cdot\mathrm{exp}(-x/L_{3})],$
(4.9)

where $x$ is the length through the SlowLS. $\mathrm{A}$ and $\mathrm{B}$ are
coefficients. $L_{1}$, $L_{2}$, and $L_{3}$ are the fit attenuation lengths.
The fit results in the SlowLS are displayed in figure 13. We used the same
method to obtain a fitting function for attenuation length in acrylic and
mineral oil in order to estimate the number of surviving photons that cross
different distances in the three media.

Figure 13: Left: The fit attenuation length and coefficients of scintillation
photons in the SlowLS. Right: The fit attenuation length and coefficients of
Cherenkov photons in the SlowLS.

###  4.3 Detection

In this subsection, we construct a formula for the simplified solid angle
considering the 3-dimensional, 3D, PMT photocathode geometry and obtain the
effective detection efficiency of the PMT.

####  4.3.1 Solid Angle

The calculation of solid angles, $\Omega_{i}$, takes into consideration the 3D
PMT photocathode geometry. As shown in figure 14, the 3D solid angle (orange
solid lines) should consider not only the radius of the PMT photocathode,
$r_{\mathrm{PMT}}$, but also the geometric influence of the PMT photocathode.

Figure 14: The schematic for the solid angle $\Omega_{i}$ of the PMT
photocathode. The orange solid lines show the true solid angle considering the
3D PMT photocathode geometry, whereas the green dotted lines represent the
solid angle determined by the 2D disk model. The length $R_{i}$ of the PMT to
the vertex position is shown as the red dotted line. The PMT normal vector is
represented by the black line. The incident angle $\theta_{i}$ is between the
vector of the PMT to the vertex and the PMT normal vector.

A MC simulation is used to obtain the 3D solid angle map of the 8-inch PMT.
The left panel of figure 15 shows the variation of the 3D solid angle with the
incidence angle, $\theta_{i}$, and distance, $R_{i}$. The 3D solid angle map
needs a simplified function for the reconstruction, and was done as follows.

Figure 15: Left: A 3D solid angle map of the 8-inch PMT photocathode is
obtained by the MC simulation. The cosine of the incident angle $\theta_{i}$
is represented on the X-axis. The Y-axis presents the distance $R_{i}$ from
the PMT to the vertex position. The Z-axis represents the value of the solid
angle. Right: The value of the 3D divided by the 2D solid angle with
$\cos{\theta_{i}}$ and $R_{i}$. The Z-axis represents the value of the 3D
divided by the 2D solid angle.

First, the predicted solid angle is approximated by a 2-dimensional, 2D, disk
model. As shown in figure 14, when the $R_{i}$ is significantly more than
$r_{\mathrm{PMT}}$ (8-inch), the predicted 2D solid angle by the disk model
(green dotted lines) is approximated as follows:

$\Omega_{i}=\frac{\pi r_{\mathrm{PMT}}^{2}}{4\pi R_{i}^{2}}\cos{\theta_{i}}.$
(4.10)

As we can see in figure 14, the 3D solid angle (orange solid lines ) is larger
than the 2D solid angle (green dotted lines), so we next explore the
relationship between the 3D solid angle and the approximated 2D solid angle.
We divide the 3D solid angle by the 2D solid angle, as shown in the right
panel of figure 15. We find that the value of the 3D solid angle divided by
the 2D solid angle has an approximately linear relationship with the
$\cos{\theta_{i}}$ for the same $R_{i}$.

Figure 16: The relationship distribution of the 3D map and the 2D disk model
solid angle versus $\cos{\theta_{i}}$. The Y-axis represents the value of the
3D divided by the 2D solid angle.

Figure 16 shows the ratio between the 3D and 2D results versus
$\cos{\theta_{i}}$. A linear function $f(\cos{\theta_{i}})$ is fitted to the
figure and the result is applied as a correction to the 2D disk assumption

$\Omega_{i}=f(\cos{\theta_{i}})\frac{\pi r_{\mathrm{PMT}}^{2}}{4\pi
R_{i}^{2}}\cos{\theta_{i}}.$ (4.11)

####  4.3.2 PMT Detection Efficiency

The probability that an optical photon is converted into PE when it reaches
the PMT photocathode depending on the quantum efficiency and the collection
efficiency, which are referred to jointly as the detection efficiency, DE. In
this work, the peak of the quantum efficiency spectrum is 34% at
$390\text{\,}\mathrm{n}\mathrm{m}$, and the collection efficiency is 100%
[35]. Therefore, the DE spectrum is the same as the quantum efficiency
spectrum.

Different wavelength photons correspond to different DE, but the readout
system cannot distinguish the wavelength of the photons. Therefore, we try to
avoid this problem with an effective DE, which is the average DE calculated by
convoluting the wavelength spectrum for the photons arriving at the PMT and
the latter’s wavelength dependent quantum efficiency.

With full MC simulation, we observe that the light spectrum from scintillation
or Cherenkov is almost unchanged beyond $500\text{\,}\mathrm{m}\mathrm{m}$ in
SlowLS or mineral oil, because the short-wavelength photons
(<$360\text{\,}\mathrm{n}\mathrm{m}$) have been quickly absorbed. Only the
long-wavelength photons have been retained. The scintillation and Cherenkov
photon spectra of the arriving photons are obtained with the full MC
simulation for the effective DE calculation. As a result, the effective DE of
scintillation photons, $\epsilon^{\mathrm{S}}_{\mathrm{eff}}$ is 29.9%, and
the effective DE of Cherenkov photons, $\epsilon^{\mathrm{C}}_{\mathrm{eff}}$
is 16.0%. The effective DE of Cherenkov photons is smaller than that of
scintillation photons because a wider Cherenkov light spectrum is chosen as
seen in figure 5 and the DE for short wavelengths Cherenkov light is low.

###  4.4 Direct and Indirect Photons

Although the refractive indices of the medium in the CSD are similar, the PMT
glass and alkali metal coating create a mirror-like surface where photons are
strongly reflected. In addition, the photons undergo Rayleigh scattering in
the medium, which changes the direction of the photons. These two effects will
make the predicted number of PEs and the TOF of PEs inaccurate. Therefore, we
must consider the effect of reflection and scattering on the reconstruction.

In this subsection, the proportion of the direct and indirect photons in
different positions of the detector, the relationship between the position of
the direct and indirect photons, and the time profile of the direct and
indirect photons are described.

####  4.4.1 Proportion of Direct and Indirect Photons

The definition of direct photons is the detected photons that enter the PMT
photocathode directly (ignoring the refraction effects), and indirect photons
are the detected photons influenced by reflection and scattering. The
reflection and scattering effects are independent of the electron energy, so
the variation in the energy of the electrons is not taken into account. Only
the effect of position is taken into account, since reflection and scattering
are only related to the wavelength of the photon and the photon flight
distance.

Therefore, a proportion of direct to indirect photons is obtained at different
detector positions by the full MC simulation. A total of 10,000 electrons are
simulated from $-5\text{\,}\mathrm{m}$ to $5\text{\,}\mathrm{m}$ on the
detector X-axis, respectively, and their normalized direction ($p_{x}$,
$p_{y}$, $p_{z}$) is (-1, 0, 0) in O-xyz. The number of direct and indirect
photons for scintillation or Cherenkov light is obtained by the full MC
simulation. As a result, the proportion of direct and indirect photons
(including reflected and Rayleigh scattering photons) are shown in figure 17.

Figure 17: The distributions of the proportion of direct and indirect photons
(including reflected and Rayleigh scattering photons) versus the vertex
position for scintillation (left) and Cherenkov light (right) are plotted.

The proportion of direct and indirect scintillation photons remain essentially
the same at different positions in the SlowLS, with 85% and 15% of these two
types of photons, respectively.

The maximum difference in the proportion of indirect Cherenkov photons at
different positions is nearly 2%. Because the chance of Rayleigh scattering is
higher when the optical path of Cherenkov light is longer, the proportion of
indirect Cherenkov photons increases. Nevertheless, to simplify the
calculation, the proportion of indirect Cherenkov photons is considered to be
15%, while the proportion of direct Cherenkov photons is approximately 85%.

####  4.4.2 Virtual Vertex of Indirect Photons

As can be seen in the previous subsection, almost 15% of the photons are
indirect photons, so their effect on the reconstruction cannot be ignored. We
attempt to develop a simplified function to estimate the number of indirect
photons for each PMT. Because of reflection and scattering, the spatial
distribution of the indirect photons is more fuzzy than that of direct
photons, while the indirect photons preserve some of the qualities of the
direct photons. Therefore, we explore the association between the direct and
indirect photons by the full MC simulation and a vertex position
reconstruction.

A total of 100,000 electrons are simulated with a kinetic energy of
$5\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$ and fixed them at the X-axis
coordinates of 0, $1\text{\,}\mathrm{m}$, $2\text{\,}\mathrm{m}$,
$3\text{\,}\mathrm{m}$, and $4\text{\,}\mathrm{m}$. The indirect scintillation
photons are selected from the full MC simulation, and subsequently the vertex
positions of these indirect photons are reconstructed using the charge
likelihood function (latter section), i.e., virtual vertex position,
$\vec{r}_{{\mathrm{virtual}}}$. We compare the $\vec{r}_{{\mathrm{virtual}}}$
with the true vertex position, $\vec{r}_{{\mathrm{truth}}}$, and the left
panel of figure 18 shows the results.

Figure 18: Left: Comparison of the true vertex position,
$\vec{r}_{{\mathrm{truth}}}$, and the position as reconstructed using the
indirect scintillation photons, $\vec{r}_{{\mathrm{virtual}}}$. There is an
approximately linear relationship between the two vertex positions. Right:
Schematic diagram of the virtual and real light sources. The center of the
detector is $O$, and points $A$ and $A^{{}^{\prime}}$ are the real light
source (direct photons) and virtual light source (indirect photons),
respectively.

As we can see, the virtual vertex position can be assumed to be a point,
$A^{\prime}$, closer to the center than the true vertex position, $A$, as
shown on the right side of figure 18, because the distribution of indirect
scintillation photons is more uniform
($\vec{r}_{{\mathrm{virtual}}}=0.77\vec{r}_{{\mathrm{truth}}}$). Moreover, the
same approach is applied to the indirect Cherenkov photons.

####  4.4.3 Time Profile of Direct and Indirect Photons

By using the effective group velocities in the different medium, the TOF of
scintillation, $\mathrm{TOF}_{i,\mathrm{m}}^{\mathrm{S,pred}}$ and Cherenkov
photons $\mathrm{TOF}_{i,\mathrm{m}}^{\mathrm{C,pred}}$ are predicted as
follows,

$\mathrm{TOF}_{i,\mathrm{m}}^{\mathrm{S,pred}}=\frac{R_{i,\mathrm{m}}(x,y,z)}{v_{\mathrm{m}}^{\mathrm{S,eff}}},$
(4.12)
$\mathrm{TOF}_{i,\mathrm{m}}^{\mathrm{C,pred}}=\frac{R_{i,\mathrm{m}}(x,y,z)}{v_{\mathrm{m}}^{\mathrm{C,eff}}},$
(4.13)

where $R_{i,\mathrm{m}}(x,y,z)$ is the distance of the photon in medium
$\mathrm{m}$ as it flies toward the $i$th PMT for vertex position $(x,y,z)$.
$v_{\mathrm{m}}^{\mathrm{S,eff}}$ and $v_{\mathrm{m}}^{\mathrm{C,eff}}$ are
the effective group velocities of scintillation and Cherenkov photons (section
4.2.1).

For the PE time $t_{ij}$ of the $j$th PE of the $i$th PMT, the time residual
of scintillation PEs, $t_{ij}^{\mathrm{S},\mathrm{res}}$, or Cherenkov PEs,
$t_{ij}^{\mathrm{C},\mathrm{res}}$, can be written as

$t_{ij}^{\mathrm{S},\mathrm{res}}=t_{ij}-t_{\mathrm{event}}-\sum_{\mathrm{m}}{\mathrm{TOF}_{i,\mathrm{m}}^{\mathrm{S,pred}}}-t_{\mathrm{shift}},$
(4.14)
$t_{ij}^{{\mathrm{C}},\mathrm{res}}=t_{ij}-t_{\mathrm{event}}-\sum_{\mathrm{m}}{\mathrm{TOF}_{i,\mathrm{m}}^{\mathrm{C,pred}}}-t_{\mathrm{shift}},$
(4.15)

where $t_{\mathrm{event}}$ is the event time being fit, and
$t_{\mathrm{shift}}$ consists of the electronic time and TT, which can be
obtained by a calibration.

However, the predicted TOF is different from the true TOF because the true TOF
will be larger than the predicted TOF under the influence of reflection and
scattering, so the time residual needs to consider the influence of indirect
photons.

We obtain the time residual p.d.f. of direct and indirect photons by the full
MC simulation. The scintillation time profile of
$\tau_{r}=$7.7\text{\,}\mathrm{n}\mathrm{s}$$ and
$\tau_{d}=$37\text{\,}\mathrm{n}\mathrm{s}$$ is used as an example. The
$5\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$ electrons are simulated at various
positions $-4\text{\,}\mathrm{m}$ to $4\text{\,}\mathrm{m}$ in a fixed
direction (1, 0, 0), and the time residual p.d.f. is obtained by using eq.
(4.14) or eq. (4.15). Moreover, the vertex position $(x,y,z)$ used in the
calculation of $R_{i,\mathrm{m}}(x,y,z)$ is the true vertex position.

The positions of scintillation photons are 0 to $4\text{\,}\mathrm{m}$ because
its emission direction is isotropic. The four time residual p.d.f. of direct
and indirect photons (including scintillation and Cherenkov photons) are shown
in figure 19.

Figure 19: Four charts are used to display the time residual p.d.f. for direct
photons and indirect photons (including scintillation and Cherenkov photons).
Top left: The direct scintillation photon time residual p.d.f. with a rise
time of $7.7\text{\,}\mathrm{n}\mathrm{s}$ and decay time of
$37\text{\,}\mathrm{n}\mathrm{s}$ at (0, 0, 0), ($1\text{\,}\mathrm{m}$, 0, 0)
… ($4\text{\,}\mathrm{m}$, 0, 0) are plotted. Top right: The direct Cherenkov
photon time residual p.d.f. at ($-4\text{\,}\mathrm{m}$, 0, 0),
($-3\text{\,}\mathrm{m}$, 0, 0) … ($4\text{\,}\mathrm{m}$, 0, 0) are plotted.
Bottom left: The indirect scintillation photon time residual p.d.f. at (0, 0,
0), ($1\text{\,}\mathrm{m}$, 0, 0) … ($4\text{\,}\mathrm{m}$, 0, 0) with a
$7.7\text{\,}\mathrm{n}\mathrm{s}$ rise time and
$37\text{\,}\mathrm{n}\mathrm{s}$ decay time are plotted. Bottom right: The
indirect Cherenkov photon time residual p.d.f. at ($-4\text{\,}\mathrm{m}$, 0,
0), ($-3\text{\,}\mathrm{m}$, 0, 0) … ($4\text{\,}\mathrm{m}$, 0, 0) are
plotted.

It can be seen that the time residual p.d.f. of the four types of light vary
greatly. Details of the discussion are as follows.

##### Direct Scintillation

For the direct scintillation photon, the time residual p.d.f. is basically the
same as the scintillation time profile at different positions, as shown in the
top left of figure 19.

##### Direct Cherenkov

For the direct Cherenkov photon, the time resolution of the time residual
p.d.f. is related to the flight distance of Cherenkov light due to the
dispersion effect, and the longer the flight distance is, the larger the time
resolution, as shown in the top right of figure 19.

##### Indirect Scintillation

For the indirect scintillation photons, the time residual p.d.f. is wider than
that of direct scintillation photons, which is caused by reflection and
scattering. Additionally, the variation of the time residual p.d.f. at
different positions is small and can be neglected, as shown in the bottom left
of figure 19.

##### Indirect Cherenkov

For the indirect Cherenkov photon, the time residual p.d.f. is only related to
the structure of the detector, as shown in the bottom right of figure 19. The
reflection peaks of the indirect Cherenkov photon time residual p.d.f. at
different positions are different. To simplify the calculation, we assume that
the time residual p.d.f. of the indirect Cherenkov photons does not vary with
position, since the effect of indirect Cherenkov photons on the directional
reconstruction is small.

In addition, the fitting of the time residual p.d.f. is discussed in the next
section.

##  5 Fitting Model Construction

In this section, a likelihood function is constructed according to the signal
process discussed in the last section. The prediction of the charge and time
by the likelihood function is compared with the full MC simulation data for
correctness verification.

###  5.1 Likelihood Function of Reconstruction

A likelihood function is created similar to SNO [6, 26] and applied to the
event’s reconstruction. It depends on $E$, $x$, $y$, $z$,
$t_{\mathrm{event}}$, $\vec{d}_{\mathrm{Fit}}$, which are described in table
2. The likelihood function is mainly composed of charge and time terms, and
its specific form is as follows:

Parameters |  Explanation
---|---
$i$ |  The $i$th PMT
$ij$ |  The $j$th PE of the $i$th PMT
$E$ |  Event visible energy to be fit
$x,y,z$ |  Vertex position to be fit
$t_{\mathrm{event}}$ |  Event time to be fit
$\vec{d}_{\mathrm{Fit}}$ |  Event direction to be fit
$\mathrm{m}$ |  medium include the SlowLS, acrylic and mineral oil
$x^{\mathrm{S}}_{\mathrm{vir}},y^{\mathrm{S}}_{\mathrm{vir}},z^{\mathrm{S}}_{\mathrm{vir}}$ |  Virtual vertex position of scintillation photons [section 4.4.2]
$x^{\mathrm{C}}_{\mathrm{vir}},y^{\mathrm{C}}_{\mathrm{vir}},z^{\mathrm{C}}_{\mathrm{vir}}$ |  Virtual vertex position of Cherenkov photons [section 4.4.2]
$f_{\mathrm{D}}$ |  Direct photon fraction (in this work is 0.85) [section 4.4.1]
$f_{k_{\mathrm{B}}}$ |  Birks’ law effect factor [section 4.1.1]
$Y_{\mathrm{s}}$ |  Scintillation light yield [eq. (4.1)]
$Y_{\mathrm{c}}$ |  Cherenkov light yield [eq. (4.4)]
$g(\cos\alpha_{i})$ |  Cherenkov light angular distribution [section 4.1.2]
$T_{i,\mathrm{m}}^{\mathrm{S}}$ |  Phototransmission for scintillation photons [eq. (4.9)]
$T_{i,\mathrm{m}}^{\mathrm{C}}$ |  Phototransmission for Cherenkov photons [eq. (4.9)]
$\Omega_{i}$ |  Solid angle [eq. (4.11)]
$\epsilon_{i}^{\mathrm{S,PMT}}$ |  Effective DE for scintillation photons [section 4.3.2]
$\epsilon_{i}^{\mathrm{C,PMT}}$ |  Effective DE for Cherenkov photons [section 4.3.2]
$t_{ij}^{\mathrm{S},\mathrm{res}}$ |  Scintillation time residual [eq. (4.14)]
$t_{ij}^{\mathrm{C},\mathrm{res}}$ |  Cherenkov light time residual [eq. (4.15)]
Table 2: The description of the parameters.

$\mathcal{L}(n_{i}^{\mathrm{Obs}},t_{ij}|E,x,y,z,t_{\mathrm{event}},\vec{d}_{\mathrm{Fit}})=\prod_{i}^{N_{\mathrm{PMT}}}P_{i}^{C}{\prod_{j}^{n_{i}^{\mathrm{Obs}}}{P_{ij}^{T}}},$
(5.1)

where $n_{i}^{\mathrm{Obs}}$ is the number of PEs of the $i$th PMT, $t_{ij}$
is the $j$th PE time of the $i$th PMT, $N_{\mathrm{PMT}}$ is the total number
of PMTs, $P_{i}^{C}$ is the probability of charge of the $i$th PMT, and
$P_{ij}^{T}$ is the probability of the $j$th PE time of the $i$th PMT.

$P_{i}^{\mathrm{C}}$ is presented as follows based on previous research into
the signaling process:

$P_{i}^{C}=f_{\mathrm{Poisson}}(n_{i}^{\mathrm{Obs}};n_{i}^{\mathrm{SD}}+n_{i}^{\mathrm{SI}}+n_{i}^{\mathrm{CD}}+n_{i}^{\mathrm{CI}}+n_{i}^{\mathrm{DN}}),$
(5.2)

where $f_{\mathrm{Possion}}$ is a Poisson distribution. The number of direct
and indirect scintillation PEs of the $i$th PMT is represented by
$n_{i}^{\mathrm{SD}}$ and $n_{i}^{\mathrm{SI}}$, respectively. The number of
direct and indirect Cherenkov PEs of the $i$th PMT are $n_{i}^{\mathrm{CD}}$
and $n_{i}^{\mathrm{CI}}$, respectively. The number of signals caused by dark
noise for the $i$th PMT is $n_{i}^{\mathrm{DN}}$.

The definition of $P_{ij}^{\mathrm{T}}$ is,

$P_{ij}^{T}=\frac{n_{i}^{\mathrm{SD}}}{n_{i}^{\mathrm{Exp}}}f^{\mathrm{SD}}({t_{ij}^{\mathrm{S},\mathrm{res}}})+\frac{n_{i}^{\mathrm{SI}}}{n_{i}^{\mathrm{Exp}}}f^{\mathrm{SI}}({t_{ij}^{\mathrm{S},\mathrm{res}}})+\frac{n_{i}^{\mathrm{CD}}}{n_{i}^{\mathrm{Exp}}}f^{\mathrm{CD}}({t_{ij}^{\mathrm{C},\mathrm{res}}})+\frac{n_{i}^{\mathrm{CI}}}{n_{i}^{\mathrm{Exp}}}f^{\mathrm{CI}}({t_{ij}^{\mathrm{C},\mathrm{res}}})+\frac{n_{i}^{\mathrm{DN}}}{n_{i}^{\mathrm{Exp}}}f^{\mathrm{DN}},$
(5.3)

where
${n_{i}^{\mathrm{Exp}}}=n_{i}^{\mathrm{SD}}+n_{i}^{\mathrm{SI}}+n_{i}^{\mathrm{CD}}+n_{i}^{\mathrm{CI}}+n_{i}^{\mathrm{DN}}$.
$f^{\mathrm{SD}}$ and $f^{\mathrm{SI}}$ present the time residual p.d.f. for
the direct and indirect scintillation photons, respectively. The time residual
p.d.f. for the direct and indirect Cherenkov photons, respectively, is
$f^{\mathrm{CD}}$ and $f^{\mathrm{CI}}$. The time p.d.f. of the dark noise is
$f^{\mathrm{DN}}$.

###  5.2 Fitting Model of the Charge Profile

Based on the simplification of the signal process, the expressions for
calculating the number of PEs for the four types of light are developed in the
following form.

##### Direct Scintillation

According to section 4, the following factors must be taken into account when
calculating the direct scintillation PE:

  * •

The scintillation light yield.

  * •

The quenching effect resulting from Birks’ law.

  * •

The proportion of the direct photons.

  * •

The attenuation of scintillation photons in the material.

  * •

The solid angle accepted by the PMT.

  * •

The effective DE of scintillation photons.

Consequently, the number of direct scintillation PEs, $n_{i}^{\mathrm{SD}}$,
is provided by

$n_{i}^{\mathrm{SD}}=f_{\mathrm{D}}f_{k_{\mathrm{B}}}Y_{\mathrm{s}}(E)\prod_{\mathrm{m}}{T_{i,\mathrm{m}}^{\mathrm{S}}(x,y,z)}\Omega_{i}(x,y,z)\epsilon^{\mathrm{S,PMT}}.$
(5.4)

Table 2 displays all parameters that are used in the calculation.

##### Direct Cherenkov

The following factors need to be considered when calculating the direct
Cherenkov PE, according to section 4.

  * •

Cherenkov light yield (300-$700\text{\,}\mathrm{n}\mathrm{m}$).

  * •

The proportion of the direct photons.

  * •

The attenuation of Cherenkov photons in the material.

  * •

The solid angle accepted by the PMT.

  * •

The angular distribution of Cherenkov light.

  * •

The effective DE of Cherenkov photons.

The number of direct Cherenkov PEs, $n_{i}^{\mathrm{CD}}$, is therefore
defined as:

$n_{i}^{\mathrm{CD}}=f_{\mathrm{D}}Y_{\mathrm{c}}(E)\prod_{\mathrm{m}}{T_{i,\mathrm{m}}^{\mathrm{C}}(x,y,z)}\Omega_{i}(x,y,z)\epsilon^{\mathrm{C,PMT}}g(\cos\alpha_{i}(x,y,z,\vec{d}_{\mathrm{Fit}})).$
(5.5)

All of the input variables utilized in the calculation are shown in table 2.

##### Indirect Scintillation

The indirect scintillation calculations are fairly similar to those for the
direct photons (eq. 5.4) but with the virtual vertex position
$(x^{\mathrm{S}}_{\mathrm{vir}},y^{\mathrm{S}}_{\mathrm{vir}},z^{\mathrm{S}}_{\mathrm{vir}})$.
Therefore, it is possible to calculate the number of indirect scintillation
PEs, $n_{i}^{\mathrm{SI}}$, as:

$n_{i}^{\mathrm{SI}}=(1-f_{\mathrm{D}})f_{k_{\mathrm{B}}}Y_{\mathrm{s}}(E)\prod_{\mathrm{m}}{T_{i,\mathrm{m}}^{\mathrm{S}}(x^{\mathrm{S}}_{\mathrm{vir}},y^{\mathrm{S}}_{\mathrm{vir}},z^{\mathrm{S}}_{\mathrm{vir}})}\Omega_{i}(x^{\mathrm{S}}_{\mathrm{vir}},y^{\mathrm{S}}_{\mathrm{vir}},z^{\mathrm{S}}_{\mathrm{vir}})\epsilon^{\mathrm{S,PMT}}.$
(5.6)

The parameters that are utilized in the calculation are shown in table 2.

##### Indirect Cherenkov

Cherenkov light that is not directly can be calculated in a manner that is
quite similar to that of directly Cherenkov light (eq. 5.5) but with the
virtual vertex position
$(x^{\mathrm{C}}_{\mathrm{vir}},y^{\mathrm{C}}_{\mathrm{vir}},z^{\mathrm{C}}_{\mathrm{vir}})$.
As a result, it is possible to display the number of indirect Cherenkov PEs,
$n_{i}^{\mathrm{CI}}$, as:

$n_{i}^{\mathrm{CI}}=(1-f_{\mathrm{D}})Y_{\mathrm{c}}(E)\prod_{\mathrm{m}}{T_{i,\mathrm{m}}^{\mathrm{C}}(x^{\mathrm{C}}_{\mathrm{vir}},y^{\mathrm{C}}_{\mathrm{vir}},z^{\mathrm{C}}_{\mathrm{vir}})}\Omega_{i}(x^{\mathrm{C}}_{\mathrm{vir}},y^{\mathrm{C}}_{\mathrm{vir}},z^{\mathrm{C}}_{\mathrm{vir}})\epsilon^{\mathrm{C,PMT}}g(\cos\alpha_{i}(x^{\mathrm{C}}_{\mathrm{vir}},y^{\mathrm{C}}_{\mathrm{vir}},z^{\mathrm{C}}_{\mathrm{vir}},\vec{d}_{\mathrm{Fit}})).$
(5.7)

All of the inputs used in the calculation are shown in table 2.

##### Dark Noise

Only the dark noise rate, $\mathrm{F}_{\mathrm{DN}}$, and time window,
$T_{\mathrm{Window}}$, of the PMT are relevant to the dark noise-produced PE.
In this work, $T_{\mathrm{Window}}$ is $500\text{\,}\mathrm{n}\mathrm{s}$, and
$\mathrm{F}_{\mathrm{DN}}$ is $2000\text{\,}\mathrm{H}\mathrm{z}$. Therefore,
the number of the signals caused by the dark noise, $n_{i}^{\mathrm{DN}}$, is
calculated as follows:

$n_{i}^{\mathrm{DN}}=\mathrm{F_{\mathrm{DN}}}\cdot\mathrm{T_{\mathrm{Window}}}.$
(5.8)

###  5.3 Fitting Model of the Time Profile

The time residual p.d.f. of various types of light are discussed in section
4.4.3, and the time residual p.d.f. are fit separately here.

##### Direct Scintillation

The definition of the time residual p.d.f. of the direct scintillation photon,
$f^{\mathrm{SD}}(t_{ij}^{\mathrm{S},\mathrm{res}})$, is

$f^{\mathrm{SD}}(t_{ij}^{\mathrm{S},\mathrm{res}})=\frac{\tau_{r}^{\mathrm{SD}}+\tau_{d}^{\mathrm{SD}}}{{\tau_{d}^{\mathrm{SD}}}^{2}}(1-e^{-t_{ij}^{\mathrm{S},\mathrm{res}}/\tau_{r}^{\mathrm{SD}}})\cdot
e^{-t_{ij}^{\mathrm{S},\mathrm{res}}/\tau_{d}^{\mathrm{SD}}}\otimes\mathrm{Gauss}(0,\sigma^{\mathrm{SD}}_{t}),$
(5.9)

where ${\tau_{r}^{\mathrm{SD}}}$ and $\tau_{d}^{\mathrm{SD}}$ are the are the
rise time and decay time of the direct scintillation PEs, respectively, and
$\sigma^{\mathrm{SD}}_{t}$ is the time resolution for the direct scintillation
photon time residual p.d.f.. By fitting the upper left panel of figure 19,
${\tau_{r}^{\mathrm{SD}}}$ and $\tau_{d}^{\mathrm{SD}}$ are
$7.7\text{\,}\mathrm{n}\mathrm{s}$ and $37\text{\,}\mathrm{n}\mathrm{s}$,
respectively. $\sigma^{\mathrm{SD}}_{t}$ in this study is
$0.0398\text{\,}\mathrm{n}\mathrm{s}$.

##### Direct Cherenkov

The time of Cherenkov light produced by several-MeV electrons is usually
within $1\text{\,}\mathrm{n}\mathrm{s}$, while the generation of scintillation
light takes at least a few nanoseconds. Therefore, the time distribution of
Cherenkov light is treated as a $\delta$ function. A Gaussian function is used
in the fitting of the time residual p.d.f. of the Cherenkov light. The
definition of the time residual p.d.f. of the direct Cherenkov photons,
$f^{\mathrm{CD}}(t_{ij}^{\mathrm{C},\mathrm{res}})$, is

$f^{\mathrm{CD}}(t_{ij}^{\mathrm{C},\mathrm{res}})=\delta(t_{ij}^{\mathrm{C},\mathrm{res}})\otimes\mathrm{Gauss}(0,\sigma^{\mathrm{CD}}_{t}).$
(5.10)

where $\sigma^{\mathrm{CD}}_{t}$ is the time resolution for the time residual
p.d.f.. As shown in the upper right of figure 19, the resolution of the time
residual p.d.f. becomes larger as the Cherenkov light flight distance becomes
longer, due to the dispersion effects. Then, we fit the relationship between
the time resolution and flight distance using a fourth-order polynomial, as
shown in figure 20. As a result, the time resolution
$\sigma^{\mathrm{CD}}_{t}$ is used as the fourth-order polynomial in eq.
(5.10).

Figure 20: Cherenkov light flight distance versus the time resolution,
$\sigma^{\mathrm{CD}}_{t}$, is plotted.

##### Indirect Scintillation

The indirect scintillation photon time residual p.d.f.,
$f^{\mathrm{SI}}(t_{ij}^{\mathrm{S},\mathrm{res}})$, is described as

$f^{\mathrm{SI}}(t_{ij}^{\mathrm{S},\mathrm{res}})=\frac{\tau_{r}^{\mathrm{SI}}+\tau_{d}^{\mathrm{SI}}}{{\tau_{d}^{\mathrm{SI}}}^{2}}(1-e^{-t_{ij}^{\mathrm{S},\mathrm{res}}/\tau_{r}^{\mathrm{SI}}})\cdot
e^{-t_{ij}^{\mathrm{S},\mathrm{res}}/\tau_{d}^{\mathrm{SI}}}\otimes\mathrm{Gauss}(0,\sigma^{\mathrm{SI}}_{t}),$
(5.11)

where ${\tau_{r}^{\mathrm{SI}}}$ and $\tau_{d}^{\mathrm{SI}}$ are the rise
time and decay time of the indirect scintillation PEs, respectively, and
$\sigma^{\mathrm{SI}}_{t}$ is the time resolution for the indirect
scintillation PEs time residual p.d.f..

The time residual p.d.f. of different positions is shown on the bottom left of
figure 19 to be very similar. Therefore, we select the time residual p.d.f. of
($3\text{\,}\mathrm{m}$, 0, 0) for fitting using eq. (5.11), and the fitting
results are displayed in the left panel of figure 21.

Figure 21: Fitting of indirect scintillation (left) and Cherenkov (right) time
residuals p.d.f. and the fit results are shown.

##### Indirect Cherenkov

The time residual p.d.f. of the indirect Cherenkov photon,
$f^{\mathrm{CI}}(t_{ij}^{\mathrm{C},\mathrm{res}})$, is described as

$f^{\mathrm{CI}}(t_{ij}^{\mathrm{C},\mathrm{res}})=\frac{\tau_{r}^{\mathrm{CI}}+\tau_{d}^{\mathrm{CI}}}{{\tau_{d}^{\mathrm{CI}}}^{2}}(1-e^{-t_{ij}^{\mathrm{C},\mathrm{res}}/\tau_{r}^{\mathrm{CI}}})\cdot
e^{-t_{ij}^{\mathrm{C},\mathrm{res}}/\tau_{d}^{\mathrm{CI}}}\otimes\mathrm{Gauss}(0,\sigma^{\mathrm{CI}}_{t}),$
(5.12)

where ${\tau_{r}^{\mathrm{CI}}}$ and $\tau_{d}^{\mathrm{CI}}$ are the rise
time and decay time of the indirect Cherenkov PEs, respectively, and
$\sigma^{\mathrm{CI}}_{t}$ is the time resolution for the indirect Cherenkov
PEs time residual p.d.f.. Eq. (5.12) is used to fit the p.d.f. at
($3\text{\,}\mathrm{m}$, 0, 0), and the right panel of figure 21 displays the
results. The origin of structure in this figure mainly originates from
reflections on the PMT. For example, the first peak in the right panel of
figure 21 is primarily due to the single Cherenkov light reflection from the
PMT near the first hit, while the rest are due to multiple reflections.

##### Dark Noise

The dark noise time profile is independent of the time residual and only
relates to the dark noise rate and time window, $T_{\mathrm{Window}}$. The
time distribution of dark noise is uniform within the time window, so it is
simple to calculate the dark noise time p.d.f., $f^{\mathrm{DN}}$, using the
following formula:

$f^{\mathrm{DN}}=1/T_{\mathrm{Window}}.$ (5.13)

###  5.4 Validation of the fit Model with Full MC Simulation

The calculation of the TOF for PEs plays a major role in the comparison of
time profile. The estimation of the effective group velocity and vertex
position affects the TOF calculation. The effective group velocity is obtained
by the full MC simulation (section 4.2.1). The time residual distribution
calculated using the true vertex positions is also obtained by the full MC
simulation. The model is consistent with the full MC simulation.

The results of the model computation in the charge profile are checked with
those of the full simulation to ensure its accuracy. We select two positions
for the comparison. One is near the center of the detector,
($1\text{\,}\mathrm{m}$, $1\text{\,}\mathrm{m}$, 0), and the other is near the
detector boundary, ($3\text{\,}\mathrm{m}$, $2\text{\,}\mathrm{m}$,
$2\text{\,}\mathrm{m}$). 500,000 electrons with
$5\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$ kinetic energy are simulated in the
direction (1, 0, 0) in a high scintillation light yield
$5000\text{\,}\mathrm{p}\mathrm{h}\mathrm{o}\mathrm{t}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{/}\mathrm{M}\mathrm{e}\mathrm{V}$
, i.e. $\mathrm{LS_{\mathrm{376~{}PE}}^{\mathrm{7.7~{}ns}}}$, sample.

The average number of PEs of a PMT is obtained by the full MC simulation.
Moreover, we predict the number of PEs of this PMT by the model (section 5.2).
The number of the full MC-estimated PEs is compared with the number of the
model-predicted PEs by using $(\mathrm{MC}-\mathrm{Predict})/{\mathrm{MC}}$.
To facilitate the comparison, $\theta$ and $\phi$ are used in the spherical
coordinate system to represent the position of the PMTs. Figure 22 and figure
23 display the comparison results.

Figure 22: A total of 500,000 electrons are simulated in the direction of (1,
0, 0) at ($1\text{\,}\mathrm{m}$, $1\text{\,}\mathrm{m}$,
$0\text{\,}\mathrm{)}$ with a kinetic energy of
$5\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$. The number of the full MC-
estimated PEs and the model-predicted PEs are compared. The comparison of the
direct scintillation PEs is the first row. The comparison of the direct
Cherenkov PEs is the second row. The comparison of the indirect scintillation
row is the third row. The comparison of the indirect Cherenkov light is the
fourth row.

Figure 23: A total of 500,000 electrons are simulated in the direction of (1,
0, 0) at ($3\text{\,}\mathrm{m}$, $2\text{\,}\mathrm{m}$,
$2\text{\,}\mathrm{m}$) with a kinetic energy of
$5\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$. The number of the full MC-
estimated PEs and the model-predicted PEs are compared. The comparison of the
direct scintillation PEs is the first row. The comparison of the direct
Cherenkov PEs is the second row. The comparison of the indirect scintillation
row is the third row. The comparison of the indirect Cherenkov light is the
fourth row.

The results show that for the electrons close to the center, the difference
between the MC-estimated PEs and the model-predicted PEs is better than that
for the electrons at the boundary. The comparison of the four types of lights
is discussed in detail below.

##### Direct Scintillation

The first rows of figure 22 and figure 23, the direct scintillation PEs
comparison is displayed. The differences between the predicted and the full MC
estimated values for ($1\text{\,}\mathrm{m}$, $1\text{\,}\mathrm{m}$, 0) and
($3\text{\,}\mathrm{m}$, $2\text{\,}\mathrm{m}$, $2\text{\,}\mathrm{m}$) are
1% and 3%, respectively.

##### Direct Cherenkov

In the second rows of figure 22 and figure 23, the direct Cherenkov PEs
comparison is displayed. The differences between the number of model-predicted
and full MC-estimated PEs for ($1\text{\,}\mathrm{m}$, $1\text{\,}\mathrm{m}$,
0) and ($3\text{\,}\mathrm{m}$, $2\text{\,}\mathrm{m}$,
$2\text{\,}\mathrm{m}$) are less than 8% and 25%, respectively. In addition,
in the critical zone, where the most Cherenkov PEs are collected, the
discrepancies are less than 4% and 8%, respectively.

##### Indirect Scintillation

In the third rows of figure 22 and figure 23, the comparison of the indirect
scintillation PEs is displayed. The differences between the number of the
model-predicted and the full MC-estimated PEs for ($1\text{\,}\mathrm{m}$,
$1\text{\,}\mathrm{m}$, 0) and ($3\text{\,}\mathrm{m}$,
$2\text{\,}\mathrm{m}$, $2\text{\,}\mathrm{m}$) are 20% and 40%, respectively.
In addition, in the critical zone, where the most indirect scintillation PEs
are collected, the discrepancies are less than 20% and 25%, respectively.

##### Indirect Cherenkov

In the fourth rows of figure 22 and figure 23, the indirect Cherenkov PEs
comparison is displayed. The differences between the number of the model-
predicted and the full MC-estimated PEs for ($1\text{\,}\mathrm{m}$,
$1\text{\,}\mathrm{m}$, 0) and ($3\text{\,}\mathrm{m}$,
$2\text{\,}\mathrm{m}$, $2\text{\,}\mathrm{m}$) are 30% and 80%, respectively.
In addition, in the critical zone, where the most indirect Cherenkov PEs are
collected, the discrepancies are less than 20% and 40%, respectively.

##  6 Reconstruction

This section describes the reconstruction process, which includes details of
the initial value calculation and the maximization of the likelihood function.
The section also presents the reconstruction results, including the energy,
position and angular resolution. Moreover, the PID results are also shown in
this section.

###  6.1 Calculation of Initial Values and Maximization

The initial value is required for the maximum likelihood method for energy,
position, event time, and direction. In the maximization process, the initial
value needs to be obtained by a fast algorithm, which saves time. In addition,
the maximization process can easily fall into a local extremum, so the
reconstruction is iterated to finally obtain the reconstructed results.

##### Initial Value of Energy

The total number of PEs of the event could be obtained by the results of
waveform analysis, and the visible energy of the event $E_{\mathrm{e}}$ can be
calculated as:

$E_{\mathrm{e}}=\displaystyle\sum_{i=1}^{N_{\mathrm{hitPMT}}}n_{i}^{\mathrm{Obs}}/{E_{\mathrm{scale}}},$
(6.1)

where $E_{\mathrm{scale}}$ is the PE yield (energy scale),
$N_{\mathrm{hitPMT}}$ is the total number of hit PMTs, and
$n_{i}^{\mathrm{Obs}}$ is the number of PEs of the $i$th PMT.

##### Initial Value of Vertex Position

The vertex position is calculated by using a weighted average approach from
each PE of the PMT (Barycenter) and its position is shown below:

$\vec{r}_{\mathrm{BC}}=A_{c}\cdot(\displaystyle\sum_{i=1}^{N_{\mathrm{hitPMT}}}{n_{i}^{\mathrm{Obs}}\vec{r}_{i}})/(\displaystyle\sum_{i=1}^{N_{\mathrm{hitPMT}}}n_{i}^{\mathrm{Obs}}),$
(6.2)

where $\vec{r}_{\mathrm{BC}}$ is the vertex position of the event, $A_{c}$ is
a correction factor with a value of 0.93, and $\vec{r}_{i}$ is the position of
the $i$th PMT.

##### Initial Value of Event Time

The fit to vertex position, direction and time corresponds to changing the
vertex position $(x,y,z)$, direction $\vec{d}_{\mathrm{Fit}}$, and event time
$t_{\mathrm{event}}$ until the maximum number of PE time residual is
concentrated at the peak of the time residual p.d.f.. Typically,
$t_{\mathrm{event}}$ is between $-5\text{\,}\mathrm{n}\mathrm{s}$ and
$5\text{\,}\mathrm{n}\mathrm{s}$, so the initial value of the event time is
set to 0.

##### Initial Value of Direction

The initial values for the six standard directions are (1, 0, 0), (-1, 0, 0),
(0, 1, 0), (0, -1, 0), (0, 0, 1), and (0, 0, -1).

##### Maximize the Likelihood Function

The likelihood function is maximized by TMinuit [49]. We obtain the final
reconstruction results in two steps.

In the first step, the likelihood function is maximized six times with the six
standard directions as mentioned above.

In the second step, the best-fit results of the energy, position, and event
time of the first step are used as their corresponding initial values. For the
direction, it is easy to fall into a local maximum and a broader range of
initial values is examined. We further calculated 50 likelihood values with 50
directions which are uniformly distributed within a cone of 60 degrees around
the best-fit direction of the first step. The three directions, with the
highest likelihood values out of the 50, are chosen as the initial values of
the second step fit, and the fits are carried out three times. The best fit
result is selected out of the three fits.

###  6.2 Performance of Reconstruction

Electrons with 1 to $9\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$ kinetic energy
and uniform position and direction distributions are simulated for the twelve
SlowLS samples. We reconstruct these events using the method described above.
The results show that the larger the percentage of Cherenkov PEs (higher CSR),
the simpler the reconstruction direction. We note that after-pulse has little
impact on the reconstruction results of these SlowLS samples since it usually
appears a few hundred nanoseconds after a signal [Zhao:2022gksAP]. In
contrast, the angular reconstruction mainly depends on the forward Cherenkov
light, usually in the first $10\text{\,}\mathrm{n}\mathrm{s}$.

For the low light yield SlowLS sample reconstruction results, the samples with
the fastest time profile ($\tau_{r}=$0.5\text{\,}\mathrm{n}\mathrm{s}$$,
$\tau_{d}=$10\text{\,}\mathrm{n}\mathrm{s}$$) can reconstruct a satisfactory
direction because of a higher CSR (>1: 1.3). Therefore, the reconstruction
results of the slower time profile SlowLS samples are not provided.

For the high light yield SlowLS sample reconstruction results, the samples
with a quicker time profile ($\tau_{r}=$0.5\text{\,}\mathrm{n}\mathrm{s}$$,
$\tau_{d}=$10\text{\,}\mathrm{n}\mathrm{s}$$ and
$\tau_{r}=$1.2\text{\,}\mathrm{n}\mathrm{s}$$,
$\tau_{d}=$27\text{\,}\mathrm{n}\mathrm{s}$$) are unable to reconstruct the
direction. Because the fast rise time leads to a small CSR (<1: 6.1) in the
first $10\text{\,}\mathrm{n}\mathrm{s}$, the angular reconstruction is very
challenging.

As a result, the $\mathrm{LS_{\mathrm{24~{}PE}}^{0.5~{}ns}}$,
$\mathrm{LS_{\mathrm{35~{}PE}}^{0.5~{}ns}}$,
$\mathrm{LS_{\mathrm{285~{}PE}}^{\mathrm{7.7~{}ns}}}$, and
$\mathrm{LS_{\mathrm{376~{}PE}}^{\mathrm{7.7~{}ns}}}$ reconstruction results
are displayed in detail. In addition, the direction of the reconstruction can
be more accurate when the CSR in $10\text{\,}\mathrm{n}\mathrm{s}$ is greater
than 1: 5. This is a guideline for the selection of the SlowLS candidates in
the CSD.

####  6.2.1 Energy Resolution

The energy distributions are obtained from the reconstructed result. The left
side of figure 24 is an example of a energy distribution with the
$\mathrm{LS_{\mathrm{376~{}PE}}^{\mathrm{7.7~{}ns}}}$ for
$2\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$ electrons. A Gaussian function is
used to fit this distribution. The definition of energy resolution is obtained
by $\mathrm{\sigma_{E}/E_{mean}}$, where $\mathrm{\sigma_{E}}$ and
$\mathrm{E_{mean}}$ are the standard deviation and central value in the
Gaussian fit. As a result, the left panel of figure 24 shows that the
$\mathrm{LS_{\mathrm{376~{}PE}}^{\mathrm{7.7~{}ns}}}$ at
$2\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$ has an energy resolution of 4.1%.

As seen on the right side of figure 24, the energy resolution versus electron
kinetic energy is plotted for four samples. It is evident that as the electron
energy and scintillation light yield increase, the energy resolution improves.

The energy of $2\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$ is the region
considered for $0\nu\beta\beta$ decay research (${}^{136}\rm{Xe}$ and
${}^{130}\rm{Te}$) [27, 33, 31]. Therefore, the energy resolution of the
$2\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$ kinetic energy electrons with
uniform position and direction distributions could be calculated by table 3.

SlowLS Samples |  PE yield $E_{\mathrm{scale}}$ (PE/MeV) |  CSR in the first $10\text{\,}\mathrm{n}\mathrm{s}$ |  Energy Resolution | Results with $2\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$ Electrons
---|---|---|---|---
| Position
---
Resolution
X-Projection
(cm)
| Angular
---
Resolution
Containing 68% Events
(degrees)
$\mathrm{LS_{\mathrm{24~{}PE}}^{0.5~{}ns}}$ | 24 |  | 1: 0.7
---
$28.4\%/\sqrt{(E)}$ | 15.3 | 51
$\mathrm{LS_{\mathrm{35~{}PE}}^{0.5~{}ns}}$ | 35 |  | 1: 1.3
---
19.6%$/\sqrt{(E)}$ | 16.6 | 57
$\mathrm{LS_{\mathrm{285~{}PE}}^{\mathrm{7.7~{}ns}}}$ | 285 |  | 1: 2.4
---
6.5%$/\sqrt{(E)}$ | 10.0 | 45
$\mathrm{LS_{\mathrm{376~{}PE}}^{\mathrm{7.7~{}ns}}}$ | 376 |  | 1: 3.2
---
5.7%$/\sqrt{(E)}$ | 9.2 | 47
Table 3: The reconstruction results of the four SlowLS samples for
$2\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$ electrons with uniform position and
direction distributions.

Figure 24: Left: An example of the distribution of reconstructed energy. For
$2\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$ electrons of the
$\mathrm{LS_{\mathrm{376~{}PE}}^{\mathrm{7.7~{}ns}}}$, the energy resolution
is approximately 4.1%. Right: The distribution of energy resolution versus
electron kinetic energy for four SlowLS samples is plotted.

####  6.2.2 Vertex Position Resolution

The vertex position distribution, e.g., X-projection, is obtained by
subtracting the true position $X_{\mathrm{Truth}}$ from the reconstructed
position $X_{\mathrm{Fit}}$. The left side of figure 25 shows an example of a
simulated sample. A Gaussian function is used to fit this distribution. The
vertex position resolution is the fit standard deviation, $\sigma_{x}$, of the
Gaussian function. As a result, the left panel of figure 25 shows that the
position resolution of the
$\mathrm{LS_{\mathrm{376~{}PE}}^{\mathrm{7.7~{}ns}}}$ at
$2\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$ is
$9.2\text{\,}\mathrm{c}\mathrm{m}$.

The right side of figure 25 shows the distribution of the vertex position
versus electron kinetic energy of the four samples. As can be observed, the
vertex position resolution improves as the electron energy increases, and
similarly, the scintillation light yield increases as the vertex position
resolution improves. However, the position resolution at
$1\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$ is poor for each of them. This is
because of the high proportion of signal numbers due to the dark noise, which
makes the reconstructed position resolution is poor.

In addition, $\mathrm{LS_{\mathrm{24~{}PE}}^{0.5~{}ns}}$ and
$\mathrm{LS_{\mathrm{35~{}PE}}^{0.5~{}ns}}$ cross at
$2\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$, because the better angular
resolution of $\mathrm{LS_{\mathrm{24~{}PE}}^{0.5~{}ns}}$ (section 6.2.3)
improves the position resolution.

As a result, the vertex position resolution of the four samples is shown in
table 3 for $2\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$ kinetic energy
electrons with uniform position and direction distributions, for the interest
in $0\nu\beta\beta$ decay research.

Figure 25: Left: A vertex position (X-projection) reconstruction example,
which is for $2\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$ electrons of
$\mathrm{LS_{\mathrm{376~{}PE}}^{\mathrm{7.7~{}ns}}}$, and the position
resolution is $9.2\text{\,}\mathrm{c}\mathrm{m}$. Right: The resolution of the
vertex position (X-projection) versus electron kinetic energy for four SlowLS
samples.

####  6.2.3 Angular Resolution

The angular distribution of the events can be obtained using the inner product
of the reconstructed direction, $\vec{d}_{\mathrm{Fit}}$, and the true
direction, $\vec{d}_{\mathrm{Truth}}$. The top left panel of figure 26 shows
an angular distribution of the
$\mathrm{LS_{\mathrm{376~{}PE}}^{\mathrm{7.7~{}ns}}}$ at
$2\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$. The angular resolution is the
angle that includes 68% of the events. In addition, the top right panel of
figure 26 shows the cosine angle distribution of the
$\mathrm{LS_{\mathrm{376~{}PE}}^{\mathrm{7.7~{}ns}}}$ at
$2\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$. As a result, the bottom panel of
figure 26 shows that the angular resolution of the
$\mathrm{LS_{\mathrm{376~{}PE}}^{\mathrm{7.7~{}ns}}}$ at
$2\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$ is 47 degrees.

The angular resolution of the different energies under the four samples is
obtained using the above method, as shown in the right side of figure 26. The
resolution of the angles increases with increasing energy.

Additionally, the curves of the low light yield SlowLS samples and high light
yield SlowLS samples in figure 26 cross at
$4\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$. There are two main reasons for
this. First, at low energy, the dark noise has a significant effect on the
angular reconstruction of the low light yield samples, making the angular
resolution poor. Second, the high light yield samples have good position
resolution at low energy, which improves the angular resolution.

The $2\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$ electron angular reconstruction
results are used for the interest in $0\nu\beta\beta$ decay research. As a
result, the angular resolution of the four samples is shown in table 3 for
$2\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$ kinetic energy electrons with
uniform position and direction distributions. Furthermore, the angular
resolution of $5\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$ electrons is in
general agreement with some existing angular reconstructions [6, 17].

Figure 26: Top left: A sample of the reconstructed angular distribution. The
X-axis displays the angle between the reconstructed direction,
$\vec{d}_{\mathrm{Fit}}$, and the true direction, $\vec{d}_{\mathrm{Truth}}$.
The angular resolution for $2\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$
electrons of $\mathrm{LS_{\mathrm{376~{}PE}}^{\mathrm{7.7~{}ns}}}$ is
approximately 47 degrees, which contains 68% events. Top right: A sample of
the reconstructed angular distribution with cosine angle. The X-axis displays
the cosine of the angle between the $\vec{d}_{\mathrm{Fit}}$ and the
$\vec{d}_{\mathrm{Truth}}$. Bottom: The distribution of angular resolution
versus electron kinetic energy for four SlowLS samples.

####  6.2.4 Particle Identification

Events with visible energies (electron equivalent energy, unit is MeVee) of
several-MeVee can be electrons, gamma rays, protons or alpha rays. Among them,
protons and alpha rays are so massive that they do not produce directly
Cherenkov photons for energies of a few MeV. In addition, the Cherenkov light
produced by gamma rays relies mainly on the scattered electrons, which pass
the Cherenkov threshold. Table 4 lists the range of particle kinetic energy
and the number of Cherenkov PE produced by the four particles when the visible
energy is 2-$3\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}\mathrm{e}\mathrm{e}$ in
the full MC simulation of the SlowLS. Therefore, the difference in a
particle’s ability to emit Cherenkov radiation can be used to perform the PID.

| Particle Type
---
| Kinetic Energy of the Particle
---
with Visible Energy is
2-$3\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}\mathrm{e}\mathrm{e}$
in the SlowLS
(MeV)
| Number of Cherenkov PE
---
with Visible Energy is
2-$3\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}\mathrm{e}\mathrm{e}$
in the SlowLS
Electron | 2-3 | 10-40
Gamma | 1.7-3 | 5-35
Proton | 13-19 | 0
Alpha | 13-19 | 0
Table 4: The range of particle kinetic energies and the Cherenkov PE numbers
produced by the particles are given for four different particles at visible
energies of 2-$3\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}\mathrm{e}\mathrm{e}$
in the full MC simulation of the SlowLS.

The four samples ($\mathrm{LS_{\mathrm{24~{}PE}}^{0.5~{}ns}}$,
$\mathrm{LS_{\mathrm{35~{}PE}}^{0.5~{}ns}}$,
$\mathrm{LS_{\mathrm{285~{}PE}}^{\mathrm{7.7~{}ns}}}$, and
$\mathrm{LS_{\mathrm{376~{}PE}}^{\mathrm{7.7~{}ns}}}$) are studied in the
following way. Electrons, gamma, and protons (alpha is similar to proton) with
1 to $9\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}\mathrm{e}\mathrm{e}$ visible
energy and uniform position and direction distributions are simulated for the
four samples. Then the likelihood function is used to reconstruct these
events.

The number of Cherenkov PEs, $N^{\mathrm{C}}$, and scintillation PEs,
$N^{\mathrm{S}}$, are obtained for each event from the reconstruction results,
where $N^{\mathrm{C}}$ and $N^{\mathrm{S}}$ are shown as:

$N^{\mathrm{C}}=\sum_{i}^{N_{\mathrm{hitPMT}}}N_{i}^{\mathrm{C}},$ (6.3)
$N^{\mathrm{S}}=\sum_{i}^{N_{\mathrm{hitPMT}}}N_{i}^{\mathrm{S}},$ (6.4)

where $N_{i}^{\mathrm{C}}$ is the fitted number of Cherenkov PEs of the $i$th
PMT, and $N_{i}^{\mathrm{S}}$ is the fitted number of scintillation PEs of the
$i$th PMT. The $N_{i}^{\mathrm{C}}$ and $N_{i}^{\mathrm{S}}$ could be fitted
by a likelihood function as:

$\mathcal{L}(n_{i}^{\mathrm{Obs}},t_{ij},E,x,y,z,\vec{d}_{\mathrm{Fit}}|N_{i}^{\mathrm{C}},N_{i}^{\mathrm{S}},N_{i}^{\mathrm{Dn}})=P_{i}^{\mathrm{C}}P_{i}^{\mathrm{S}}P_{i}^{\mathrm{Dn}},$
(6.5)

where $N_{i}^{\mathrm{Dn}}$ is the fitted number of dark noise of the $i$th
PMT. $P_{i}^{\mathrm{C}}$, $P_{i}^{\mathrm{S}}$, and $P_{i}^{\mathrm{Dn}}$ are
the Poisson distribution with the conditional expectation calculated in the
observed sample of Cherenkov, scintillation, and dark noise, respectively. In
addition, the sum of $N_{i}^{\mathrm{C}}$, $N_{i}^{\mathrm{S}}$ and
$N_{i}^{\mathrm{Dn}}$ must be equal to $n_{i}^{\mathrm{Obs}}$.

To better demonstrate the difference in Cherenkov light produced by different
particles, the average value of the number of Cherenkov PEs of gamma,
$\mathrm{C}_{\gamma,\mathrm{mean}}$, is used as a reference and the difference
from it is plotted. As shown in figure 27, the horizontal coordinates indicate
the number of scintillation PEs, and the vertical coordinates indicate the
fractional difference between the number of Cherenkov PEs, $\mathrm{C}$, to
the $\mathrm{C}_{\gamma,\mathrm{mean}}$.

Figure 27 shows that the Cherenkov light fraction of the electron is slightly
higher than that of the gamma and much higher than that of the proton.
Therefore, protons can be distinguished from electrons and gamma rays, while
electrons and gamma rays are harder to separate. In addition, there is a
discreteness in the bands for the lower scintillation PE number (NPE). Because
the fitted PE numbers are integers, and the number of Cherenkov PE is only a
few for the lower NPE.

Figure 27: The reconstructions of the number of scintillation and Cherenkov
PEs for four samples. Electrons are depicted by blue points, gamma by red
points, and protons by magenta points. On the X-axis, the number of
scintillation PEs and visible energy are shown. The Y-axis displays the
fractional difference between the number of Cherenkov PEs, $\mathrm{C}$, to
the average value of a gamma, $\mathrm{C}_{\gamma,\mathrm{mean}}$.

The difference in figure 27 can be further quantified by projection onto the
Y-axis. As shown in figure 28, we select the events of
2-$3\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}\mathrm{e}\mathrm{e}$ visible
energy to obtain the distribution of the proportion of events retained versus
the fractional difference between $\mathrm{C}$ and
$\mathrm{C}_{\gamma,\mathrm{mean}}$ for the interest in $0\nu\beta\beta$ decay
research.

All four samples of SlowLS can identify particles with
2-$3\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}\mathrm{e}\mathrm{e}$ visible
energy, and the discrimination is comparable. As the light yield of
scintillation increases, the electron-proton discrimination becomes
progressively worsens. It can be seen that in the electron-gamma
discrimination, gamma is reduced by 50% and the electrons can be retained by
70-80% for these samples; in the electron-proton discrimination, while
retaining 70-80% of the electrons, the protons are reduced by 95%, 90%, 80%,
and 75% for the four samples ($\mathrm{LS_{\mathrm{24~{}PE}}^{0.5~{}ns}}$,
$\mathrm{LS_{\mathrm{35~{}PE}}^{0.5~{}ns}}$,
$\mathrm{LS_{\mathrm{285~{}PE}}^{\mathrm{7.7~{}ns}}}$, and
$\mathrm{LS_{\mathrm{376~{}PE}}^{\mathrm{7.7~{}ns}}}$), respectively.

Figure 28: The distribution of the proportion of events retained by different
particles versus the fractional difference between $\mathrm{C}$ and
$\mathrm{C}_{\gamma,\mathrm{mean}}$ is obtained with
2-$3\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}\mathrm{e}\mathrm{e}$ visible
energy events for four samples. The electron is represented by blue lines, the
gamma is represented by red lines, and the proton is represented by magenta
lines.

##  7 Conclusion

In this paper, we focus on the reconstruction algorithm of a new CSD using
SlowLS, expecting to reconstruct the energy, direction, position, and number
of Cherenkov and scintillation PEs of a particle based on the full MC
simulation.

We conduct a comprehensive simulation study for a multi-hundred-ton spherical
detector using the Geant4-based simulation software JSAP. The scintillation
light yield and time profile of SlowLS are two key properties determining the
performance of the CSD directional reconstruction. To study the effect, we
select four typical light yields and three typical time profiles, forming
twelve SlowLS samples. The twelve samples can be sorted according to the CSR,
i.e. Cherenkov light to scintillation light ratio in the first
$10\text{\,}\mathrm{n}\mathrm{s}$.

For the CSD reconstruction, we first analyze the waveform of the simulation
output to obtain the number of PEs on each PMT and the time of the PEs. The
number of fit PEs is consistent with the number of true PEs, and for 93% of
the PEs, the time difference between the fit result and the true value is less
than $0.5\text{\,}\mathrm{n}\mathrm{s}$.

Second, to understand the PE generation process, an exhaustive study of the
signal generation process is conducted for the CSD using simulation. To save
computation time, each signal process is represented with a simplified
function. In addition, we introduce the concept of a virtual light source to
solve the effect of the indirect photons in the reconstruction.

Third, a likelihood function for charge and time is constructed and its
performance is validated with the full simulation. For the events near the
detector boundary, the difference between the number of predicted direct
scintillation PEs on PMTs and full simulation results is less than 3%. The
difference for the direct Cherenkov PEs on the Cherenkov ring is less than 5%.
Moreover, the closer the position to the center of the detector, the smaller
the difference.

For the reconstruction process, the final reconstruction results are obtained
in two steps to solve the local extremum problem. Electrons with 1 to
$9\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$ kinetic energy and uniform position
and direction distributions are simulated for the twelve SlowLS samples. The
reconstruction results show that the directional reconstruction is poor when
the CSR is less than 1: 5.

We demonstrate the reconstruction performance of four of the SlowLS samples
with large CSR, $\mathrm{LS_{\mathrm{24~{}PE}}^{0.5~{}ns}}$,
$\mathrm{LS_{\mathrm{35~{}PE}}^{0.5~{}ns}}$,
$\mathrm{LS_{\mathrm{285~{}PE}}^{\mathrm{7.7~{}ns}}}$, and
$\mathrm{LS_{\mathrm{376~{}PE}}^{\mathrm{7.7~{}ns}}}$ in detail, as shown in
table 3. The numbers of Cherenkov PEs and scintillation PEs are calculated
with the reconstruction process, and their ratio is used for PID. Electrons
and protons are easy to distinguish and electrons and gamma rays separation
may also be possible. For four samples of
2-$3\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}\mathrm{e}\mathrm{e}$ visible
energy, it is possible to maintain 70-80% selection efficiency for the
electrons while reducing the gamma by 50% in the case of electron-gamma
discrimination, and in the electron-proton discrimination, more than 75% of
the protons are reduced and 70-80% of the electrons remain.

The study based on simulation shows that the CSD could help the next
generation of MeV-scale neutrino experiments, including solar-, geo-, and
supernova neutrino and $0\nu\beta\beta$ decay research, to fulfill the goal of
achieving a better background suppression and precision measurement. The
result is also a guideline for developing SlowLS candidates for a particular
physics requirement on energy, angular resolution, and PID.

## Acknowledgments

The authors would especially like to thank Ziyi Guo for useful discussions.
This work was supported in part by the National Natural Science Foundation of
China (No. 12127808, No. 12141503, No. 12221005 and No. 12175241), the Key
Laboratory of Particle & Radiation Imaging (Tsinghua University) and the
Fundamental Research Funds for the Central Universities.

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