Also at ] College of Physics, Chongqing University, Chongqing 401331, China

Now at ] Department of Physics, Stanford University, Stanford, CA 94305, USA

Also at ] School of Physics, Jilin University, Jilin 130012, China

Now at ]Physics Department, Columbia University, New York, NY 10027, USA.

RELICS Collaboration

# RELICS: a REactor neutrino LIquid xenon Coherent elastic Scattering
experiment

Chang Cai  Department of Physics & Center for High Energy Physics, Tsinghua
University, Beijing 100084, China  Guocai Chen  CNNC Sanmen Nuclear Power
Company, Zhejiang 317112, China  Jiangyu Chen  Sino-French Institute of
Nuclear Engineering and Technology, Sun Yat-Sen University, Zhuhai 519082,
China  Fei Gao<EMAIL_ADDRESS>Department of Physics & Center for High
Energy Physics, Tsinghua University, Beijing 100084, China  Xiaoran Guo  State
Key Laboratory of Particle Detection and Electronics, University of Science
and Technology of China, Hefei 230026, China  Department of Modern Physics,
University of Science and Technology of China, Hefei 230026, China  Tingyi He
[  School of Science and Engineering, The Chinese University of Hong Kong,
Shenzhen (CUHK-Shenzhen), Guangdong, 518172, P.R. China  Chengjie Jia  [
Department of Physics & Center for High Energy Physics, Tsinghua University,
Beijing 100084, China  Gaojun Jin  CNNC Sanmen Nuclear Power Company, Zhejiang
317112, China  Yipin Jing  [  Department of Physics & Center for High Energy
Physics, Tsinghua University, Beijing 100084, China  Gaojun Ju  CNNC Sanmen
Nuclear Power Company, Zhejiang 317112, China  Yang Lei  Department of Physics
& Center for High Energy Physics, Tsinghua University, Beijing 100084, China
Jiayi Li  Department of Physics & Center for High Energy Physics, Tsinghua
University, Beijing 100084, China  Kaihang Li  Department of Physics & Center
for High Energy Physics, Tsinghua University, Beijing 100084, China  Meng Li
CNNC Sanmen Nuclear Power Company, Zhejiang 317112, China  Minhua Li  CNNC
Sanmen Nuclear Power Company, Zhejiang 317112, China  Shengchao Li  School of
Science, Westlake University, Hangzhou 310030, China  Siyin Li  School of
Science, Westlake University, Hangzhou 310030, China  Tao Li  CNNC Sanmen
Nuclear Power Company, Zhejiang 317112, China  Qing Lin  State Key Laboratory
of Particle Detection and Electronics, University of Science and Technology of
China, Hefei 230026, China  Department of Modern Physics, University of
Science and Technology of China, Hefei 230026, China  Jiajun Liu  School of
Physics, Sun Yat-Sen University, Guangzhou 510275, China  Minghao Liu
Department of Physics & Center for High Energy Physics, Tsinghua University,
Beijing 100084, China  Sheng Lv  CNNC Sanmen Nuclear Power Company, Zhejiang
317112, China  Guang Luo  School of Physics, Sun Yat-Sen University, Guangzhou
510275, China  Jian Ma  Department of Physics & Center for High Energy
Physics, Tsinghua University, Beijing 100084, China  Chuanping Shen  CNNC
Sanmen Nuclear Power Company, Zhejiang 317112, China  Mingzhuo Song
Department of Physics & Center for High Energy Physics, Tsinghua University,
Beijing 100084, China  Lijun Tong  State Key Laboratory of Particle Detection
and Electronics, University of Science and Technology of China, Hefei 230026,
China  Department of Modern Physics, University of Science and Technology of
China, Hefei 230026, China  Xiaoyu Wang  School of Science, Westlake
University, Hangzhou 310030, China  Wei Wang  Sino-French Institute of Nuclear
Engineering and Technology, Sun Yat-Sen University, Zhuhai 519082, China
School of Physics, Sun Yat-Sen University, Guangzhou 510275, China  Zihu Wang
CNNC Sanmen Nuclear Power Company, Zhejiang 317112, China  Yuehuan Wei
<EMAIL_ADDRESS>Sino-French Institute of Nuclear Engineering and
Technology, Sun Yat-Sen University, Zhuhai 519082, China  Liming Weng  CNNC
Sanmen Nuclear Power Company, Zhejiang 317112, China  Xiang Xiao  School of
Physics, Sun Yat-Sen University, Guangzhou 510275, China  Lingfeng Xie
Department of Physics & Center for High Energy Physics, Tsinghua University,
Beijing 100084, China  Dacheng Xu  [  Department of Physics & Center for High
Energy Physics, Tsinghua University, Beijing 100084, China  Jijun Yang  School
of Science, Westlake University, Hangzhou 310030, China  Litao Yang  Key
Laboratory of Particle and Radiation Imaging (Ministry of Education) &
Department of Engineering Physics, Tsinghua University, Beijing 100084, China
Long Yang  CNNC Sanmen Nuclear Power Company, Zhejiang 317112, China
Jingqiang Ye  School of Science and Engineering, The Chinese University of
Hong Kong, Shenzhen (CUHK-Shenzhen), Guangdong, 518172, P.R. China  Jiachen Yu
State Key Laboratory of Particle Detection and Electronics, University of
Science and Technology of China, Hefei 230026, China  Department of Modern
Physics, University of Science and Technology of China, Hefei 230026, China
Qian Yue  Key Laboratory of Particle and Radiation Imaging (Ministry of
Education) & Department of Engineering Physics, Tsinghua University, Beijing
100084, China  Yuyong Yue  Sino-French Institute of Nuclear Engineering and
Technology, Sun Yat-Sen University, Zhuhai 519082, China  Bingwei Zhang  CNNC
Sanmen Nuclear Power Company, Zhejiang 317112, China  Shuhao Zhang  Department
of Physics & Center for High Energy Physics, Tsinghua University, Beijing
100084, China  Yifei Zhao  Department of Physics & Center for High Energy
Physics, Tsinghua University, Beijing 100084, China

###### Abstract

Coherent elastic neutrino-nucleus scattering (CE$\nu$NS) provides a unique
probe for neutrino properties Beyond the Standard Model (BSM) physics. REactor
neutrino LIquid xenon Coherent Scattering experiment (RELICS), a proposed
reactor neutrino program using liquid xenon time projection chamber (LXeTPC)
technology, aims to investigate the CE$\nu$NS process of antineutrinos off
xenon atomic nuclei. In this work, the design of the experiment is studied and
optimized based on Monte Carlo (MC) simulations. To achieve a sufficiently low
energy threshold for CE$\nu$NS detection, an ionization-only analysis channel
will be adopted for RELICS. A high emission rate of delayed electrons after a
big ionization signal is the major background, leading to an analysis
threshold of 120 photo-electrons in the CE$\nu$NS search. The second largest
background, nuclear recoils induced by cosmic-ray neutrons, is suppressed via
a passive water shield. The physics potential of RELICS was explored with a 32
$\rm kg\cdot yr$ exposure at a baseline of 25 m from a reactor core with a 3
GW thermal power. In an energy range of 120 to 240 PEs, we the expected
13673.5 CE$\nu$NS and 2133.7 background events.The sensitivity of RELICS to
the weak mixing angle was investigated at a low momentum transfer. Our study
has shown that RELICS can further improve the constraints on the non-standard
neutrino interaction (NSI) compared to the best results. RELICS set out to
develop a powerful surface-level detection technology for low-energy neutrinos
from reactors.

PACS numbers

Keywords

CE$\nu$NS , reactor neutrino, liquid xenon, LXeTPC

###### pacs:

95.35.+d, 14.80.Ly, 29.40.-n, 95.55.Vj

##  I Introduction

Coherent elastic neutrino-nucleus scattering (CE$\nu$NS) is a Standard Model
(SM) weak neutral current process first proposed in 1974 [1]. The neutrino
treats the nucleus as a single particle during scattering with a cross-section
proportional to the nucleon number squared. Despite this process having a
relatively large cross-section, the detection is challenging due to the ultra-
low deposited energy. This process had evaded experimental discoveries for
decades until 2017 when it was first observed by COHERENT collaboration with
neutrinos produced by the Spallation Neutron Source (SNS) at Oak Ridge
National Laboratory scattering off a CsI scintillator [2], and later confirmed
with a liquid argon detector [3] in 2020. These measurements of CE$\nu$NS
offer new probes for diverse physical phenomena.

CE$\nu$NS can provide new constraints on physics beyond the Standard Model
(SM). The weak mixing angle is related to the CE$\nu$NS cross-section [4]. Any
deviation from the SM prediction in measurement can be an indication of new
physics. CE$\nu$NS is also a valuable tool to constrain nonstandard neutrino
interactions (NSI) [5], to search for sterile neutrinos [6], to verify the
existence of non-trivial neutrino electromagnetic properties [7], and to study
the nuclear structure [8] and astrophysics [9]. In addition, the CE$\nu$NS
signal from the solar or atmospheric neutrinos is expected to be an
irreducible background for the next-generation dark matter direct detection
experiments, such as PandaX-30T [10] and XLZD [11]. A precise measurement of
this process can provide valuable information for dark matter experiments
digging into the so-called “neutrino floor/fog” [12, 13].

The observation of CE$\nu$NS by COHERENT collaboration inspired the interest
in CE$\nu$NS experimental study. While the neutrino beam has been the primary
source for the CE$\nu$NS discovery, it is limited by the total flux delivered
and the availability of the facility. Therefore, nuclear reactors, with large
anti-neutrino fluxes delivered with typical energies of a few MeV, which is
the so-called “fully coherent regime” of CE$\nu$NS , have been proposed as
attractive alternative facilities for the CE$\nu$NS study. Due to the
considerably low recoil energy and complex background, the reactor CE$\nu$NS
is harder to detect. Many efforts are underway with reactor neutrinos, such as
[14, 15, 16, 17, 18, 19, 20, 21, 22, 23]. Detecting reactor CE$\nu$NS will
provide a wealth of physics opportunities and potential applications such as
nuclear safeguards [24].

RELICS is a reactor neutrino experiment using the LXeTPC technology. LXeTPC is
one of the leading technologies for rare event searches, such as detecting
dark matter and neutrinoless double beta decay [11, 25, 26, 27, 28, 29].
Benefiting from its ultra-low threshold by using ionization-only channel [30,
31] and reduced background, LXeTPC is an excellent approach for reactor
CE$\nu$NS detection. The RELICS experiment is proposed to be located near the
Sanmen Nuclear Power Plant in Zhejiang province, China. At a $\sim$25 m
baseline to the $\sim$3 GW reactor core, the average neutrino flux available
for RELICS is $\sim$1013cm-2s-1.

This work is organized as follows. The operation principle and the design of
the detector as well as the shielding system, are described in detail in
section II. The simulation framework is introduced in section III. In section
IV, we predict the expected CE$\nu$NS event rates from the Sanmen nuclear
reactor. Section V discusses a dedicated simulation of background
contributions. The physics potential of RELICS is provided in section VI. A
summary is presented in section VII.

##  II experimental designs

###  II.1 RELICS detector

The LXeTPC technology measures the energy deposited from particle interactions
by simultaneously detecting the prompt scintillation light (S1) and delayed
ionization charge (S2) [32]. The S1 is produced by the direct excitation of
xenon atoms and the recombination of electron-ion pairs in LXe. The S2 is
proportional to the number of liberated electrons extracted into gas xenon
(GXe), which are then amplified via electroluminescence. Both S1 and S2 are
recorded by photosensors, usually photomultiplier tubes (PMTs), on top and
bottom arrays of the TPC. This allows a three-dimensional (3D) vertex
reconstruction with a sub-centimeter precision, and the S2/S1 ratio can be
used to discriminate between nuclear recoil (NR) and electronic recoil (ER)
for background reduction. Typically, the energy threshold of LXeTPC is limited
by the S1 signal in 2$\sim$3 photo-electrons (PEs). The inherent S2
amplification provides an alternative approach to decrease the energy
threshold by utilizing the ionization-only channel, usually called the
“S2-only” analysis. This approach can reduce the energy threshold to a few
ionization electrons (equivalent NR energy below 1 keVnr) by sacrificing the
NR/ER discrimination and depth information, which, in the case of CE$\nu$NS
detection, can be largely compensated by the high signal-to-background ratio
profited from the intense reactor neutrino flux.

Figure 1: The RELICS detector with the inner S.S. vessel (the outer vessel was
hidden for a better view), diving bell, veto PMTs, and the center TPC
consisting of the top and bottom PMT arrays, anode, gate and cathode
electrodes, copper field shaping-rings and PTFE reflector.

A preliminary 3D mechanical design of the RELICS detector is shown in figure
1. A cylindrical TPC of 24 cm in height and 28 cm in diameter sits in a
double-layer stainless steel (S.S) vessel, which provides a stable cryogenic
environment. The target volume containing an active LXe of approximately 44 kg
is defined by 12 interlocking and light-tight PTFE (polytetrafluoroethylene,
Teflon) panels. The active LXe is then viewed by two arrays of 64 Hamamatsu
R8520-406 PMTs on top and bottom. A fiducial mass of 32 kg LXe with only a
radius selection of 12 cm can be well-defined for CE$\nu$NS search under
“S2-only” analysis. A “diving bell” structure is mounted above the top PMT
array to control the liquid level. Five etched S.S electrodes, including two
grounding screenings on top and bottom to protect PMTs from high voltage (HV),
anode, gate, and cathode to provide a nominal drift field of 500 V/cm and an
extraction field of 10 kV/cm in GXe, are installed. To ensure the uniformity
of the drift field, the target volume is encircled by several field-shaping
rings made from oxygen-free high thermal conductivity (OFHC) copper. The TPC
is surrounded by approximately 62 kg of LXe, which acts as a 4$\pi$ active
veto for background suppression. The LXe veto volume is instrumented with 48
R8520-406 PMTs. To achieve a high light collection, the inner surface of the
inner vessel is covered with a thin PTFE foil. It is worth noting that the LXe
veto is designed to reduce the background by $\sim$5 times by removing the
coincidence events inside TPC and in the LXe veto layer. All materials for the
detector construction will be carefully selected to ensure low intrinsic
radioactivity.

###  II.2 Shielding system

Near-reactor neutrino experiments usually operate with shallow overburdens.
Thus, the high-energy cosmic-ray neutrons are supposed to be one of the main
backgrounds for the NR searches from CE$\nu$NS. A sufficient neutron shielding
structure, such as water, is required to slow down and absorb the neutrons.
Meanwhile, the cosmic muon and its induced secondaries are expected to be
another major background. A highly efficient active muon veto anti-coincidence
system is also critical to the success of this type of experiment.

Figure 2 shows the design of the RELICS shielding system based on Monte Carlo
(MC) simulations, which is detailed in section V. The design has been
optimized to satisfy the requirement of the CE$\nu$NS detection. The LXe
detector is placed inside a shield of 7 m$\times$7 m$\times$7 m of water
contained in a S.S tank. A 5 m water shield on top can suppress the cosmic-ray
neutron-induced background to a controlled level. With such a water tank, the
external neutrons and gammas from the laboratory environment and the nuclear
reactor can be reduced to negligible levels. In close to the LXe detector, a
muon veto detector is installed. The veto detector is made of 8 cm thick
plastic scintillator (PS) equipped with wavelength-shifting fibers and silicon
photomultipliers (SiPMs) to collect and detect the scintillation light. A high
veto efficiency of 99% is expected for the RELICS experiment.

Figure 2: The LXe detector with a 8 cm thick PS muon veto detector inside a 7
m$\times$7 m$\times$7 m S.S water shield tank. The center of the LXe detector
is 2 m above ground.

##  III Simulation framework

The MC simulation framework of the RELICS experiment (RelicsSim) is built upon
a Geant4-based BambooMC toolkit [33]. Full detector geometry is implemented,
including the detailed detector components and the shielding system. Various
types of particle interactions are studied by simulation to evaluate the
expected signal and background. Subsequently, RelicsAPT, which inherits from
the Appletree [34] framework developed by XENONnT collaboration [35], is
adopted to convert the energy depositions into observable S1 and S2 signals.
The conversion is based on the intrinsic light yield (Ly) and charge yield
(Qy) in LXe, and convoluted with the detector-related signal detection and
reconstruction effects.

###  III.1 Liquid xenon response

Particle interaction in LXe will produce photons and electrons. The NEST model
can characterize this process [36] after considering the fluctuations in
scintillation, ionization, electron-ion recombination, and their drift field
dependence. In RelicsAPT, the Ly and Qy from NEST (v2.3.6) at 500 V/cm drift
field are used with no uncertainty assigned for NRs larger than 3 keVnr. To
quantify the uncertainty of light and charge yields for CE$\nu$NS events, the
Ly and Qy bands in XENON1T solar ${}^{8}\rm B$ neutrino search [37] are
adopted for NRs less than 3 keVnr. The bands of Ly and Qy are scaled to make
the center curve connected to NEST at 3 keVnr. The final Ly and Qy of NRs
applied in RelicsAPT are shown in figure 3. For ERs, Ly and Qy from NEST
(v2.3.6) are used without uncertainties. In the 4$\pi$ LXe veto region, both
Ly and Qy are modeled with zero electric fields.

Figure 3: The Ly (upper panel) and Qy (lower panel) model of NR in RelicsAPT.
For Enr less than 3 keVnr, the XENON1T yields model in ${}^{8}\rm B$ neutrino
search [37] is adopted, and scaled to NEST model at 3 keVnr. For Enr larger
than 3 keVnr, NEST model is used with 500 V/cm drift field.

###  III.2 Detector related effects

In addition to the intrinsic LXe response model, the detector-related effects
must be considered when interpreting the observed S1 and S2 signals. The main
detector parameters are considered based on the expected detector performance
listed in Tab. 1, as a conservative estimation.

To explore the capability of the RELICS detector on the ionization-only
detection channel, the light collection efficiency (LCE) of S2 is simulated by
RelicsSim with the optical parameters listed in Tab. 1. The LCE map is shown
in figure 4. An averaged $g_{2}$ value of 30 PE/e- is estimated based on the
S2 LCE map, the properties of the PMTs, such as $\epsilon_{\rm{QE}}$,
$\epsilon_{\rm{CE}}$, and $p_{\rm{DPE}}$ that also listed in Tab. 1, and a
100% electron extraction efficiency on the liquid surface. An electron
lifetime of 1.2 $\rm m$s is applied for S2 estimation considering the
probability of electrons loss to electronegative impurities during their
drift.

Table 1: The detector related parameters.
Optical parameters
---
PTFE reflectivity (LXe & GXe) | 99% [38]
LXe Rayleigh scattering length | 30 cm [39]
LXe absorption length | 50 m [40]
Electrodes reflectivity | 57% [41]
Signal generation
PMT quantum efficiency ($\epsilon_{\rm{QE}}$) | 33% [42]
PMT collection efficiency ($\epsilon_{\rm{CE}}$) | 75% [42]
Double PE probability ($p_{\rm{DPE}}$) | 21.9% [43, 44]
$g_{2}$ | 30 PE/e-
Detector operation
Drift field | 500 V/cm
Electron lifetime | 1.2 $\rm m$s [25]
Figure 4: The S2 LCE map with both top and bottom PMT arrays. Simulations are
performed by generating S2 photons in the GXe region below the anode.

##  IV Expected CE$\nu$NS events

The SM of particle physics well predicts the CE$\nu$NS process — a neutrino of
any flavor scatters off a nucleus via $Z^{0}$ boson exchange coherently over
the whole nucleus at a low momentum transfer ($Q=\sqrt{2m_{t}E_{R}}$). The
differential cross-section can be written as Eq. 1 below:

$\begin{split}&\frac{d\sigma}{dE_{R}}=\frac{G^{2}_{F}}{2\pi}\frac{Q^{2}_{w}}{4}F^{2}\left(2m_{t}E_{R}\right)m_{t}\left[2-\frac{m_{t}E_{R}}{E_{\nu}^{2}}\right]\\\
\end{split}$ (1)

where $G_{F}$ is Fermi’s coupling constant, $Q_{w}$ is the weak nuclear
charge, $m_{t}$ is the mass of the target nucleus, $F$ is the ground state
elastic form factor, $E_{R}$ is the nuclear recoil energy while $E_{\nu}$ the
incident neutrino energy. $Q\cdot R\lesssim 1$ is required to ensure
coherence, where $R$ is the nuclear radius. Normally, this can be satisfied
when $E_{\nu}\lesssim 50$ MeV.

For neutrinos scattering in a target medium, the expected CE$\nu$NS event rate
can be expressed as Eq. 2, where $N_{t}$ is the number of targets per unit
mass, $\phi(E_{\nu})$ is the incident neutrino flux.

$\begin{split}&\frac{dN}{dE_{R}}=N_{t}\int\phi(E_{\nu})\frac{d\sigma}{dE_{R}}dE_{\nu}\\\
\end{split}$ (2)

For the reactor neutrinos with energy above 2 MeV, the Huber-Mueller model[45,
46] is adopted for the spectra calculation while P. Vogel’s theory [47] is
used for below 2 MeV region, which is rarely searched due to the 1.8 MeV
energy threshold of inverse beta decay reaction. We assume a fuel mixture of
235U (56.1%), 238U (7.6%), 239Pu (30.7%), and 241Pu (5.6%), the total neutrino
flux $\phi(E_{\nu})$ can be estimated with a 3 GW reactor core at $\sim$25m
distance, and then the CE$\nu$NS event rate can be obtained accordingly as
shown in figure 5.

Figure 5: The expected CE$\nu$NS event rate in xenon medium from reactor
neutrino with a flux of 1013cm-2s-1.

##  V Background contributions

The complexity of the backgrounds in reactor CE$\nu$NS detection is one of the
major reasons it escaped the searches for several decades. To overcome this
challenge, dedicated background simulations have been conducted to optimize
the design of the shielding system for RELICS, using RelicsSim and RelicsAPT.
The background contributions can be divided into five categories according to
their origin, such as (V.1) cosmic-ray neutrons, (V.2) cosmic muons, (V.3)
detector materials, and (V.4) internal background from LXe. The last one is
(V.5) delayed electrons, which have been observed by large LXeTPC searching
for dark matter and should be treated carefully.

This section describes all the background components and is summarized in
(V.6). The following criteria are applied to MC data to select the background
events that can mimic CE$\nu$NS signals:

  1. i.

LXe-veto: The 4$\pi$-LXe veto exhibits a good background suppression because
the background events are more likely to deposit energy in both active LXe and
4$\pi$-LXe veto regions. The event will be removed if it contains an NR (ER)
energy larger than 500 (100) keV in the LXe veto region.

  2. ii.

single-scatter: Given the small interaction cross-section of neutrino, the
probability of a neutrino having more than one interaction in the sensitive
LXe volume is negligible. We require only one energy deposition above 0.5
keVnr energy in the LXe volume.

  3. iii.

FV: Events are required to be within the center 12 cm radius fiducial to
reduce background at the edge of the TPC, leading to a total fiducial volume
(FV) of 32 kg.

  4. iv.

S2-width: For “S2-only” analysis, the event depth $z$ cannot be accurately
estimated. S2-width cut was developed to remove the events around the liquid-
gas interface, especially for the backgrounds from detector materials.

  5. v.

In addition, dedicated selection methods are developed to suppress backgrounds
out of the delayed electrons, such as waveform classifier and pattern
classifier. These two criteria are described in detail in (V.5).

###  V.1 Background induced by cosmic-ray neutrons

The cosmic rays that travel through the atmosphere can produce a variety of
radiation particles, such as neutrons, protons, pions, etc, through spallation
reaction on nitrogen, oxygen, etc [48, 49]. Among them, the high-energy
cosmic-ray neutrons (CRNs) can easily pass through the shield and deposit
energy in the RELICS detector. The main purpose of the water shield design is
to block the CRNs. The size of the water tank is optimized based on the MC
simulation. The CRN flux is calculated using the Cosmic-Ray Shower Generator
(CRY) source [50]. The energy spectrum of the CRNs from CRY is shown in figure
6. Analyze the MC data shows that only CRNs with energy greater than $\sim$103
MeV can deposit energy in LXe, which also indicates that the neutrons from the
nuclear reactor and laboratory environment normally with energy less than 10
MeV [51] that also shows in figure 6, can not contribute to the backgrounds.

Figure 6: The energy spectrum of the CRNs (blue) from CRY calculation. The
neutrons that can deposit energy in LXe in the CE$\nu$NS ROI are shown in red
color. The energy spectrum of the reactor and environmental neutrons from the
measurement of CONUS [51] experiment is overlaid.

After processing the MC data, and passing the LXe-veto, single-scatter and FV
selections, the CRNs-induced NR background rate is estimated to be (7.7 $\pm$
0.7)$\times 10^{-2}\,\rm kg^{-1}day^{-1}$ in $[0.3,1]$ keVnr region-of-
interest (ROI) of CE$\nu$NS detection, with the spectrum shows in figure 7
(top). In addition, the secondaries of CRNs will also induce ER background in
LXe, with a rate of (9.0 $\pm$ 0.3)$\times 10^{-3}\,\rm
kg^{-1}day^{-1}keV^{-1}$, which is shown in figure 7 (bottom).

Figure 7: Top panel: The NR background distributions from CRNs (dark magenta)
and cosmic muons (red). Bottom panel: The ER background distributions from
detector materials (blue), CRNs (dark magenta), cosmic muons (red), 85Kr
(sandy brown) and 222Rn (light blue).

###  V.2 Background induced by cosmic muons

The RELICS detector will operate without overburden. Hence, special attention
has to be paid to the muon-related background. Even though muons are tagged
with 99% efficiency by the PS veto detector, the neutron background produced
by muon interacting with the detector materials is still one of the main
background components. Dedicated simulations are performed to evaluate the
muon-induced backgrounds. The energy and angular distributions of muons follow
the Shukla model [52]. The energy deposition in LXe that is caused by muon-
induced fast neutron and related gammas are recorded by RelicsSim, and then
filtered with LXe-veto, single-scatter and FV selections. Assuming a 99% muon
veto tagging efficiency, the muon-induced NRs and ERs are (0.08 $\pm$
0.01)$\times 10^{-2}\,\rm kg^{-1}day^{-1}$ in $[0.3,1]$ keVnr and (0.66 $\pm$
0.09)$\times 10^{-3}\,\rm kg^{-1}day^{-1}keV^{-1}$ in $[0,40]$ keVer,
respectively. The corresponding spectra are shown in figure 7.

###  V.3 Neutrons and gammas from the detector materials

Materials constructing the detector contain radioactive contamination. The
selection of the materials requires an extensive radioactivity screening
campaign for rare event search experiments. To evaluate the background
contribution from these materials, the radioactive contamination for all the
materials is considered in the RELICS MC model. Currently, the radioactivity
of isotopes from XENON100 [53] is adopted. A dedicated effort is also being
carried out to select clean materials for the RELICS experiment, leading to
similar radioactivity for most of the materials.

The NR background rate induced by neutrons is around one event per year and
completely negligible in the CE$\nu$NS ROI. ER background induced by gammas is
studied in this work. The decay of these isotopes is generated and confined
uniformly in the corresponding detector component in the simulation. By
applying successive selection criteria, including LXe-veto, single-scatter and
FV selections, the spatial distribution of ER background events inside the
active LXe volume in the energy region of [0, 40] keVer is shown in figure 8.
Figure 7 (bottom) shows the corresponding energy spectrum, which is two orders
of magnitude higher than the ER background components. The average ER
background rate is (364 $\pm$ 5)$\times 10^{-3}\,\rm kg^{-1}day^{-1}keV^{-1}$
in the FV volume.

Figure 8: Spatial distribution of ER background from detector materials inside
the active LXe volume in the energy region of [0, 40] keVer. The red line
indicates the 12 cm radius fiducial selection.

S2 width selection: As shown in figure 8, a large amount of background events
are located close to the liquid-gas interface. These backgrounds are mainly
from the radioactivities of PMTs and can not be suppressed without the
selection in “z” in an “S2-only” analysis. As demonstrated in [30], the pulse
shape of S2 signals can discriminate background happening at the liquid-gas
interface to signals distributed uniformly in “z”. A toy MC was conducted to
consider the drift and diffusion properties of electrons in LXe to optimize
such a selection. We simulate S2s and their pulse shape parameters with “z”
distributions shown in figure 8 with energy in the CE$\nu$NS ROI. The S2s are
simulated via RelicsAPT where the NEST model is incorporated. The S2 pulse
shape is simulated using the electron diffusion model
$\sigma_{e}=\sqrt{\frac{2D_{L}t}{v_{d}^{2}}+\sigma_{0}^{2}}$, where $D_{L}$ is
the longitudinal diffusion coefficient, $v_{d}$ is the electron drift
velocity, $\sigma_{0}$ is the width of single electron waveform [54]. In our
simulation. $D_{L}$ = 16 cm2/s, $v_{d}$ = 0.174 cm/$\mu$s, and $\sigma_{0}$ =
0.19$\mu$s.

Uniformly distributed CE$\nu$NS signal is simulated using the same procedure.
The distributions of the S2 width are shown in figure 9, for both CE$\nu$NS
signal and ER background from the detector materials. A threshold of 0.22 $\mu
s$ on S2 width can give an average background rejection power of 94%, with a
high CE$\nu$NS signal acceptance of 86%.

Figure 9: The distributions of the S2 width from the uniformly distributed
CE$\nu$NS signal and ER background from detector materials. The dark region is
the event that is being rejected by the S2-width criterion.

###  V.4 Internal backgrounds in LXe

Radioactivity in LXe can directly deposit energies and produce background in
the CE$\nu$NS ROI. These radioactivities mostly come in the form of noble
gases, such as argon, krypton, xenon, and radon. This section describes how
these radioactivities are created and evaluates their contribution to the
CE$\nu$NS ROI.

####  V.4.1 Intrinsic 222Rn and 85Kr

Krypton mixes into xenon gas when it is extracted from the air. 85Kr,
regardless of its low abundance of $2\times 10^{-11}$ in natural krypton, is
one of the main backgrounds in any LXe experiments searching for rare events.
The concentration of krypton in xenon is usually reduced below ppt level in
dark matter search experiments, such as XENONnT and PandaX-4T, through
distillations with dedicated columns[ref]. In RELICS, we assume a natKr/Xe
concentration of 10 ppt and expect (0.28 $\pm$ 0.01)$\times$10${}^{-3}\,\rm
kg^{-1}day^{-1}keV^{-1}$ as shown in figure 7 (bottom), which is a subdominant
component.

The other intrinsic background in LXe comes from the decays of 222Rn, which
emanates from any material touching Xenon gas or liquid. The emanated 222Rn
diffuses homogeneously inside the LXe volume within its relatively long half-
life of 3.8 days. The $\beta$ decay of 214Pb is another important background
component in the CE$\nu$NS ROI in the decay chain of 222Rn. With similar size
and material, the radioactivity of 222Rn in RELICS is comparable to XENON100.
Assuming a conservative 222Rn radioactivity of 40 $\rm\mu Bq\cdot kg^{-1}$ as
achieved by the XENON100 experiment [53] ($\leq$20 $\rm\mu Bq\cdot kg^{-1}$),
the expected ER background rate is estimated as of (0.98 $\pm$
0.02)$\times$10-3 kg-1day-1keV-1 as shown in figure 7 (bottom), which is also
a subdominant component in the CE$\nu$NS search.

####  V.4.2 Cosmogenic isotopes ${}^{127}\rm Xe$ and ${}^{37}\rm Ar$

The RELICS experiment, which is to be operated without overburden, is
unavoidable under cosmic radiation. We consider cosmogenic radioactive
isotopes through the activation and spallation of Xenon atoms. In particular,
${}^{127}\rm Xe$ and ${}^{37}\rm Ar$ are known cosmogenic backgrounds in
LXeTPCs [55, 27] due to their characteristic x-rays or Auger electrons through
electron capture process and their relatively long half-life of 36.4 and 35.0
days, respectively.

${}^{127}\rm Xe$: Natural xenon does not contain ${}^{127}\rm Xe$.
${}^{127}\rm Xe$ can be produced from ${}^{128}\rm Xe$ when cosmogenic muons
kick off a neutron in the nucleus with a cross-section of (2.74 $\pm$
0.4)$\times 10^{-24}$ cm2. Additionally, we consider another production
channel through the neutron capture on ${}^{126}\rm Xe$ with an upper limit
cross-section of 0.7$\times 10^{-26}$cm2 at 95% C.L [56]. The production rate
of ${}^{127}\rm Xe$ in LXe is calculated based on its production cross-
section, the muon’s fluxes and energy spectrum, and thermal neutrons. The
${}^{127}\rm Xe$ decays to ${}^{127}\rm I$ via electron capture (EC) process,
with a half-life of 36.4 days. We assume that the production and decay of
${}^{127}\rm Xe$ reach equilibrium before deploying the RELICS detector inside
the shielding. The estimated equilibrium decay rate is 9.29 evts/day/kg. The
shielding system of RELICS can attenuate neutrons efficiently, and the
equilibrium decay rate can decrease to 0.79 evts/day/kg after $\sim$300 days.

Following the EC, the ${}^{127}\rm I$ is left primarily populating at the
excited state of either 375 or 203 keV, which promptly decays to the ground
state with gamma-ray emissions. The EC occurs from either the K, L, M, or N
shell electrons and is subsequently filled with electrons from higher levels
via emission of x-rays or Auger electrons with total energy (relative
intensity) of 32.2 keV (83.4%), 5.2 keV (13.1%), 1.1 keV (2.9%) or 186 eV
(0.7%), respectively. The final background contribution from ${}^{127}\rm Xe$
is simulated with RelicsSim. The double-scatter criteria can remove most of
these background events due to the simultaneous gamma-ray and x-ray or Auger
electrons emission. If EC capture is happening from the N-shell electrons with
a binding energy of 186 eV, it may contribute background in the CE$\nu$NS ROI
when the emitted gamma rays, either with energy of 375 and 203 keV, escape
from the LXe detector and the active LXe veto. The event rate from 186 eV
energy deposition is less than $10^{-4}$ kg-1day-1 after all the selection
criteria described above, which is negligible for CE$\nu$NS search.

${}^{37}\rm Ar$: We consider three possible mechanisms for ${}^{37}\rm Ar$
production in xenon, following the approaches in [57]. Firstly, the extremely
low content of ${}^{37}\rm Ar$ in the atmosphere, which is generated by
spallation of ${}^{40}\rm Ar$ through ${}^{40}\rm Ar$(n, 4n)${}^{37}\rm Ar$,
neutron capture of ${}^{36}\rm Ar$, or cosmic bombardment of calcium-
containing soils, via ${}^{40}\rm Ca$(n, $\alpha$)${}^{37}\rm Ar$. This
mechanism only needs to be considered in the context of air leaks of an LXe
detector, which is strictly limited to below 0.1 liter per year to ensure the
purity level of LXe. Secondly, the spallation of detector materials,
especially iron in steel. However, its contribution is greatly suppressed due
to the slow diffusion rate of argon in steel. Lastly, the direct spallation of
xenon atoms by high-energy cosmogenic protons and neutrons. The semi-empirical
formula by Silberberg and Tsao was used to calculate the spallation cross-
section. The CRY is used to calculate the proton spectrum. Gordon’s neutron
spectrum [58] is adopted for cosmogenic neutron estimation. The production
rate of this process is estimated to be 0.024 atoms/kg/day with an uncertainty
of $\sim$100%. The produced ${}^{37}\rm Ar$ decays to the ground state of
${}^{37}\rm Cl$ via EC and subsequently deposits energy via x-rays or Auger
electrons. The total energy (relative intensity) for K, L, and M shells are
2.82 keV (90.2%), 0.270 keV (8.9%), and 0.018 keV (0.9%), respectively. As a
results, the background rate of ${}^{37}\rm Ar$ in the CE$\nu$NS ROI is
below$10^{-4}$ kg-1day-1 and completely negligible for CE$\nu$NS search.

###  V.5 Delayed electrons

Besides the background components induced by real physical sources, another
type of background must be considered in the CE$\nu$NS ROI. This background is
caused by a high emission rate of delayed electrons after S2 signals in the
TPC, which could be incorrectly identified as CE$\nu$NS candidate in “S2-only”
analysis. Such background has been observed by large LXeTPCs experiments
searching for dark matter particles in deep underground laboratories, such as
XENON1T [59], PandaX-4T [31] and LUX [60]. The source of this background is
not well understood yet, but we can roughly divide the delayed electrons into
two groups.

The first kind of delayed electrons is produced via photon-ionization of
impurities or metals in the drift region of the TPC. Photon-ionized delay
electrons only appear shortly (within a time range that is comparable to the
largest drift time of the LXeTPC) after the high energy S2 signals and will be
completely subdominant when a $1\,\mu$S exposure is rejected for each of the
high energy S2s, leading to an exposure loss of $\simeq$1%.

Besides photon-ionization, delayed electrons can also be produced in a larger
time range of up to a few seconds. The rate of delayed electrons is observed
to be correlated with the level of impurities in LXe and the amplitude of the
extraction field. Muons passing through the LXeTPC result in large energy
deposition, generating a substantial number of ionization electrons. These
delayed electrons may occur after the primary S2 signal of muon events,
leading to a prolonged tail of signals composed of a single or a few
electrons. Due to the high rate of cosmic muons, RELICS faces a major
challenge in suppressing background from accidental pile-ups of delayed
electrons in a short time range that is comparable to the width of a physical
S2 signal. This background is hereafter referred to as Delayed Electron (DE).

####  V.5.1 Expected rate of DE

As shown in previous studies with XENON1T [59] and LUX [60], the emission rate
of delayed electrons are correlated, in time and position, with the previous
high-energy events. This part of the background is referred to as a correlated
delayed electron background and can be significantly suppressed due to its
unique correlation with proceeding high-energy events. We simulate muons
passing through the TPC active volume using Geant4, tracking their interaction
vertex and the energy deposited along the track. The ionization signals are
generated along the muon track, assuming a constant ionization yield of 50
electrons per keV. We then simulated the generation of delayed electrons,
assuming that 0.1% of the total ionization electrons from primary interactions
are delayed emissions in a time range of 0.1 ms to 2 s, comparable to [59].
The distribution in “delayed time” (time respected to the primary S2) for
delayed electrons is assumed to follow a power law with a coefficient of -1.1,
as shown in XENON1T [59]. These delayed electrons’ horizontal positions
($x,\,y$) are assumed to be the same as their primary S2s. Position
reconstruction resolution for single electron S2s, which is expected to be 20
mm from optical simulations, is taken into account in this analysis.
Ionization S2 signals are grouped together if the time gap between them is
below 1 $\mu$s in order not to split S2 waveforms of events happening at the
bottom part of the TPC. As a result, the pile-up of DE (called Pile-up DEs)
leads to a significant background in CE$\nu$NS search. In RELICS, we expect
more than 10 Hz of muon events in the LXeTPC, leaving a large amount of single
electron S2 signals. The expected event rate of a single-electron pile-up
background is summarized in Table 2. Without implementing appropriate cuts to
reduce the rate of these pile-up events, CE$\nu$NS signals with 3 to 6
electrons will be completely overwhelmed.

Table 2: A comparison of the CE$\nu$NS signal and the Pile-up DEs background
in one year of exposure with 32 kg of fiducial mass.
Ne |  CE$\nu$NS |  Pile-up DEs
---|---|---
3 | 3.5$\times 10^{4}$ | 9.8$\times 10^{8}$
4 | 1.5$\times 10^{4}$ | 7.3$\times 10^{6}$
5 | 6.9$\times 10^{3}$ | 6.9$\times 10^{5}$
6 | 3.2$\times 10^{3}$ | 7.5$\times 10^{4}$

####  V.5.2 Waveform classifier

The waveform characteristics of Pile-up DE events differ from the physical
interactions in the LXeTPC, such as CE$\nu$NS and other backgrounds. In
physical interactions, electrons drift upwards under the drifting electric
field, resulting in signal waveform distributions that follow a Gaussian
distribution in time. As shown in figure 10, Pile-up DEs are distinguishable
from physical interactions in terms of the shape of waveforms. The
discrimination between them is maximized by employing a convolutional neural
network (CNN), as shown in figure 11. While maintaining $\simeq$80% of the
physical interactions with a score over 0.8, the Pile-up DE events are reduced
by approximately one order of magnitude.

Figure 10: Simulation waveform of Pile-up DE and CE$\nu$NS. Recording the
number of photo-electrons collected for each sample, the upper panel
corresponds to real physical events, and the lower panel represents the Pile-
up DE events. Figure 11: Result of waveform discrimination by CNN. The
waveform is evaluated by the neural network, which provides predicted scores
indicating the discriminability between the two types of waveform.

####  V.5.3 Pattern classifier

Beyond the distinctions in the waveform or time distribution, the planar
distribution of signals within the top PMT arrays also presents notable
differences between Pile-up DE and physical interactions. CE$\nu$NS and other
physical signals are composed of multiple electrons generated from the same
position above the liquid surface in the detector, while the DE signals are
mostly generated around the previous muons’ tracks. These DEs are merged
together into larger signals only due to their temporal proximity. Therefore,
the distributions of photo-electrons from CE$\nu$NS signals and DE signals
differ in the top PMT array of the detector. Real physical interactions appear
more like a superposition of multiple single-electron signals from the same
location, whereas the Pile-up DEs are formed by the superposition of single-
electron signals originating from multiple different horizontal positions,
thus appearing more dispersed.

After each S2 signal in the detector, the expected photons to be detected in
each channel on the top PMT array are denoted as $\lambda_{ch}$.
$\lambda_{ch}$ is determined by the total area of S2, the relative LCE in that
channel, and the PMT responses including detection efficiency. LCE of each PMT
is derived with light propagation simulation in the RELICS TPC. It is assumed
that the number of photo-electrons received by the PMT channel follows a
Poisson distribution with $\lambda_{ch}$ as the mean. The detected photo-
electrons are further derived considering the resolution of PMTs to single
photo-election.

Given the expected and observed light distribution pattern, we calculated the
goodness-of-fit likelihood to determine whether the event is from physical
interactions or Pile-up DEs. The logarithm of goodness-of-fit likelihood $P$
is written as:

$P=\sum_{ch=0}^{63}\log\left(\frac{{\lambda_{ch}}^{N^{pe}_{ch}}\times
e^{-\lambda_{ch}}}{{N^{pe}_{ch}!}}\right)$ (3)

where $\lambda_{ch}$ and $N^{pe}_{ch}$ are expected and observed photo-
electrons in a specific channel.

The S2 distribution pattern on top PMT array of the Pile-up DEs rejected by a
selection in $P$ (figure 12, left) would look like that of physical
interactions (figure 12, right). Since both of horizontal position and
emission time of DEs are strongly correlated to the previous muon tracks, we
further calculate the “Space-time Correlation Coefficient” of each event with
previous muons to distinguish Pileup DEs from physical interactions that are
independent of any muon events in the RELICS LXeTPC.

Figure 12: The expected S2 pattern on top PMT array from one Pile-up DE event
(top) and the expected pattern from the corresponding reconstructed position
(bottom), only channels with an expected area greater than 0.1 PE are shown
with color.

For each energy deposition of muon passing through the RELICS TPC, the
reconstructed position of DEs would follow a Gaussian distribution determined
by the resolution of position reconstruction. The distribution of DEs in time
with respect to the previous muon events is assumed to follow a power law, as
seen in [59]. Assuming $N_{i}$ is the expected number of DE in a specific
position, the probability distribution function of DEs can be expressed as:

$P_{i}=N_{i}\times\left(t-t_{i}\right)^{-\gamma}\\\
\times\frac{1}{2\pi\cdot\sigma^{2}}e^{-\frac{\left(x-x_{\mathrm{i}}\right)^{2}+\left(y-y_{\mathrm{i}}\right)^{2}}{2\sigma^{2}}}$
(4)

where $(x_{i},y_{i})$ is the position of muon track, $(x,y)$ is the observed
position, $t-t_{i}$ is the time delay to the muon track and $\sigma$ is the
resolution of reconstructed position. Considering that the emission of DE can
last up to $\simeq$2s and the muon event rate of $\simeq$10 Hz, we sum up the
contributions of $P_{i}$ from the previous 20 muons and define the log of this
value as the “Space-time Coefficient”, which is defined as:

$C_{Space-time}=\log\left(\sum_{i}P_{i}\right)$ (5)

We simulated 3.1$\times 10^{7}$ muon events, corresponding to 3.5 days of live
time, and calculated the Pattern Likelihood $P$ and “Space-time Coefficient”
$C_{Space-time}$. The two-dimensional distribution for Pile-up DE and physical
interactions, within an S2 range of 120 to 240 PE, are shown in figure 13.

Considering the correlation between $C_{Space-time}$, P and the S2 area of
each event, we provide a score for each event by Eq.6. Events with higher
scores are inclined to be from physical interactions.

$Score=P-\left(k_{st}\cdot C_{space-time}\right)+k_{area}\cdot S2_{area}+b$
(6)

While restricting the acceptance of Pile-up DE events lower than $0.01$%, we
scanned the parameter space of $k_{st}$ and $k_{area}$, in order to retain as
high signal acceptance as possible. The fitted values are $k_{st}$ of 9.10 ,
$k_{area}$ of 0.57 and $b$ of -65.78.

Figure 14 delineates the score distribution for Pile-up DEs and physical
interactions. The final classifier identifies all events with scores exceeding
0 as real physical interactions. In the ROI of S2 ranging from 120 to 240 PE,
the acceptance for physical interactions such as CE$\nu$NS is 67%. This cut
reduced the rate of Pile-up DEs by 4 orders of magnitude in the same ROI,
leaving the rate of Pile-up DE background below the expected CE$\nu$NS signal
in the whole ROI.

Figure 13: Distribution of pattern likelihood and space-time correlation of DE
and CE$\nu$NS S2 signals. CE$\nu$NS events are represented by green/yellow
points and DE events are represented by blue/cold points. Contours of
CE$\nu$NS events distribution are also shown in this figure. Figure 14:
Distribution of scores from pattern classifier. The survival rate for
CE$\nu$NS within ROI with scores over 0 is 66.77% and 0.001% for DE events
within ROI.

###  V.6 Summary

The background distribution from CRNs, cosmic muons, and detector materials
are converted to S2 size from recoil energy using RelicsAPT which is described
in Sec. III. All the background components mentioned above are summarized in
figure 15. The CE$\nu$NS event rate is overlaid. The dominant background
contribution comes from the Pile-up DEs, despite its large discrimination
power to physical events as described in V.5. The NR background contribution
from CRNs is an order of magnitude higher than muons, which is the second
major contribution. The detector materials contribute a relatively low and
flat component in ER. The event rate of all the background components in 32
kg$\cdot$yr exposure is summarized in Tab. 3. A signal-to-background ratio
greater than 6 can be achieved in [120, 240] PEs ROI of CE$\nu$NS detection.

Figure 15: Expected S2 distribution of CE$\nu$NS (Green) all background in the
RELICS experiment. The dominant background in the lowest energy is the Pile-up
DE (Green) induced by muon interactions in the TPC; The next largest
background is neutrons (Purple), mainly from the CRNs as shown in Tab. 3. The
ER from radioactivity of detector material, CRNs and muons (Blue) are non-
negligible background. The ROI for the CE$\nu$NS search is defined as 120 PE
to 240 PE and shown as a shaded region. Table 3: The total number of expected
events (Nevts) of all the background components in [120, 240] PEs ROI of
CE$\nu$NS detection in 32 kg$\cdot$yr exposure.
Event Type |  Nevts
---|---
Pile-up DE | 1429.0
CRNs | 470.4
Muon-induced neutrons | 2.3
ER | 232.0
Total Background | 2133.7
CE$\nu$NS | 13673.5

##  VI Physics potential

To explore the physics potential of the RELICS detector, this section will
discuss its sensitivity to the weak mixing angle and non-standard interactions
(NSI) based on the background estimation in section V with 32 kg$\cdot$yr
exposure. Several systematic uncertainties are considered, including 5%
reactor neutrino flux, 5% ER and NR background uncertainty. The uncertainty of
the Qy introduced in Sec. III.1 is the dominant one in our estimation.

Table 4: List of nuisance parameters shared in the likelihoods for measuring
the weak mixing angle and searching for NSI. All the nuisance parameters are
constrained by Gaussian distribution, and their standard deviations
(constraints) are also listed in the table.
Nuisance Parameters | Constraints
---|---
Charge yield morpher | $t$ | 1 [37]
Reactor neutrino flux multiplier | $\gamma_{f}$ | $\sigma_{\gamma_{f}}=0.05$
ER background rate multiplier | $\beta_{1}$ | $\sigma_{\beta_{1}}=0.10$
Neutron background rate multiplier | $\beta_{2}$ | $\sigma_{\beta_{2}}=0.05$
Pile-up DE background rate multiplier | $\beta_{3}$ | $\sigma_{\beta_{3}}=0.10$

###  VI.1 Weak mixing angle

A precise measurement of CE$\nu$NS cross-section allows for a constraint on
the weak mixing angle $\theta_{\mathrm{w}}$. Also known as the Weinberg angle,
$\theta_{\mathrm{w}}$ is conventionally used in the form of
$\sin^{2}{\theta_{\mathrm{w}}}$ in the comparison between theory and
experiment. Since $Q_{w}$ is given by $N-\left(1-4\,\rm
sin^{2}\theta_{\mathrm{w}}\right)Z$, $\sin^{2}{\theta_{\mathrm{w}}}$ can be
extracted from the measured cross-section, and any deviation from the SM
prediction will indicate new physics.

$\displaystyle\footnotesize\begin{split}\chi^{2}=&\sum_{j}\left[\frac{\left(N_{\mathrm{obs},j}-N_{\mathrm{exp},j}(\sin^{2}\theta_{W},t_{2})(1+\gamma_{f})-\sum_{i}^{3}B_{i,j}(1+\beta_{i})\right)^{2}}{\sigma^{2}_{\mathrm{stat},j}}\right]\\\
&+t^{2}+\sum_{i}^{3}(\frac{\beta_{i}}{\sigma^{2}_{\beta_{i}}})^{2}+(\frac{\gamma_{f}}{\sigma^{2}_{\gamma_{f}}})^{2}\end{split}$
(7)

The $\chi^{2}$ of likelihood for $\sin^{2}{\theta_{w}}$ measurement is defined
as Eq. 7 where index $j$ indicates different S2 intervals.
$N_{\mathrm{obs},j}$ and $N_{\mathrm{exp},j}$ are the observed and expected
signal events in each S2 interval, respectively. $B_{i,j}$ is the background
rate estimation and index $i$ runs over background components, including ER
and neutrons. The rest are nuisance parameters. $\beta_{1}$ and $\beta_{2}$
are the multipliers of the ER and neutron backgrounds, respectively. $t$
denotes the morphers of charge yields. A Gaussian constraint term is imposed
on each nuisance parameter, as summarized in Tab. 4.

The expected constraint on $\sin^{2}{\theta_{\mathrm{w}}}$ from RELICS is
shown in figure 16. The great improvement from the red solid line to the blue
dashed line indicates that the charge yield variation is a major contribution
to the constraint uncertainty. The sensitivity is expected to be competitive
and improve the Dresden-II’s result [61]. As illustrated in Fig. 17, RELICS
will provide a measurement on $\sin^{2}{\theta_{w}}$ with momentum transfer
down to the MeV scale, which is a valuable complementary to the current
measurements.

Figure 16: The expected constraints on $\sin^{2}{\theta_{w}}$ with RELICS. The
red line shows the nominal likelihood curve defined by $\chi^{2}$ in Eq. 7,
while the blue dashed line illustrates the likelihood curve with the charge
yield fixed. The faint green dot-dashed line marks $\Delta\chi^{2}=1$, and its
intersections with each likelihood curve represent the measured constraint.
The Dresden-II result (green) [61] and the PDG value (black) [62] are overlaid
for reference. Figure 17: The expected measurement (red) on
$\sin^{2}{\theta_{\mathrm{w}}}$ with RELICS. Other current and future
measurements at different momentum transfers are also included (APV, CONNIE,
Qweak, eDis, LHC). The black line represents the dependence of the weak mixing
angle in the $\overline{\rm MS}$ renormalization scheme [63].

###  VI.2 Nonstandard neutrino interactions

The detection of CE$\nu$NS provides a new probe to study neutrino physics
beyond the SM, such as the NSI of neutrinos. NSI has been extensively studied
in the literature [64]. This work focuses on a model-independent approach in
the neutrino-quark sector. The parameterization of the NSI contributions to
the cross-section can be expressed as Eq. 8, which indicates that the cross-
section strongly depends on the effective electron neutrino-up quark and -down
quark interaction strength parameters of $\epsilon^{uV}_{ee}$ and
$\epsilon^{dV}_{ee}$.

$\displaystyle\footnotesize\begin{split}\frac{d\sigma}{dE_{R}}=&\frac{G^{2}_{F}\cdot
m_{t}}{2\pi}F^{2}\left(2m_{t}E_{R}\right)\left[2-\frac{m_{t}E_{R}}{E_{\nu}^{2}}\right]\\\
&\times\left[Z(g_{V}^{p}+2\epsilon_{ee}^{uV}+\epsilon_{ee}^{dV})+N(g_{V}^{n}+\epsilon_{ee}^{uV}+2\epsilon_{ee}^{dV})\right]^{2}\end{split}$
(8)

Eq. 9 defines the $\chi^{2}$ of likelihood, where all the parameters are the
same as Eq. 7, except $\epsilon^{uV}_{ee}$ and $\epsilon^{dV}_{ee}$. All the
related uncertainties of nuisance parameters are also taken in Tab. 4. Figure
18 shows the constraining on $\epsilon^{uV}_{ee}$ and $\epsilon^{dV}_{ee}$ of
RELICS under the standard model assumption, where both of the parameters are
zero. The latest result from COHERENT [65] is overlaid for comparison. A
tighter constraint can be achieved for the RELICS experiment with higher
statistics.

$\displaystyle\scriptsize\begin{split}&\chi^{2}=\\\
&\sum_{j}\left[\frac{\left(N_{\mathrm{obs},j}-N_{\mathrm{exp},j}(\epsilon^{uV}_{ee},\epsilon^{dV}_{ee},t_{2})(1+\gamma_{f})-\sum_{i}^{3}B_{i,j}(1+\beta_{i})\right)^{2}}{\sigma^{2}_{\mathrm{stat},j}}\right]\\\
&+t^{2}+\sum_{i}^{3}(\frac{\beta_{i}}{\sigma^{2}_{\beta_{i}}})^{2}+(\frac{\gamma_{f}}{\sigma^{2}_{\gamma_{f}}})^{2}\end{split}$
(9)

Figure 18: The 90% allowed parameter space of ($\epsilon^{uV}_{ee}$,
$\epsilon^{dV}_{ee}$) constrained by RELICS detector. The COHERENT results
[65] are superimposed for comparison.

##  VII Summary

The RELICS experiment is a $\mathcal{O}$(100) kg LXe detector with a fiducial
mass of 32 kg that aims to observe the process of reactor CE$\nu$NS. The
experiment is planned to be deployed at Sanmen Nuclear Power Plant ($\sim$3
GW) in Zhejiang province, China, with a distance of $\sim$25 m, an average
neutrino flux of $\sim$1013cm-2s-1.

In this work, a preliminary design of the RELICS experiment was presented. A
dedicated MC simulation has been performed to study the background
contributions. The dominant backgrounds come from the Pile-up DEs. The second
largest background, nuclear recoils induced by cosmic-ray neutrons, is
suppressed via a passive water shield. An ionization-only analysis channel
will be adopted for RELICS analysis to achieve a sufficiently low energy
threshold for CE$\nu$NS detection. A signal-to-background ratio of greater
than 6 can be achieved in [120, 240] PEs ROI of CE$\nu$NS detection. Its
sensitivity to weak mixing angle and NSI are explored with 32 kg$\cdot$yr
exposure.

The RELICS experiment will provide rich opportunities to study Beyond the
Standard Model (BSM) physics and potential applications in nuclear reactor
monitoring for nuclear safeguards.

Acknowledgements. We acknowledge Sanmen Nuclear Power Plant for hosting
RELICS. We thank Dr. Xun Chen for his support on the BambooMC package and
Prof. Kaixuan Ni for fruitful discussions. RELICS is supported by a National
Key R&D grant from the Ministry of Science and Technology of China (No.
2021YFA1601600) and a grant from Natural Science Foundation of China (No.
12275267). Yifei Zhao’s work on RELICS is supported by a grant for
undergraduate research from Beijing Natural Science Foundation (No. QY23088).

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