Participating as a member of ]TEXONO Collaboration

CDEX Collaboration

Search for Neutrinoless Double-Beta Decay of ⁷⁶Ge with a Natural Broad Energy Germanium Detector

W. H. Dai H. Ma [email protected] Q. Yue [email protected] Z. She K. J. Kang Y. J. Li Key Laboratory of Particle and Radiation Imaging (Ministry of Education) and Department of Engineering Physics, Tsinghua University, Beijing 100084 M. Agartioglu [ Institute of Physics, Academia Sinica, Taipei 11529 H. P. An Key Laboratory of Particle and Radiation Imaging (Ministry of Education) and Department of Engineering Physics, Tsinghua University, Beijing 100084 Department of Physics, Tsinghua University, Beijing 100084 J. P. Chang NUCTECH Company, Beijing 100084 Y. H. Chen YaLong River Hydropower Development Company, Chengdu 610051 J. P. Cheng Key Laboratory of Particle and Radiation Imaging (Ministry of Education) and Department of Engineering Physics, Tsinghua University, Beijing 100084 College of Nuclear Science and Technology, Beijing Normal University, Beijing 100875 Z. Deng Key Laboratory of Particle and Radiation Imaging (Ministry of Education) and Department of Engineering Physics, Tsinghua University, Beijing 100084 C. H. Fang College of Physics, Sichuan University, Chengdu 610065 X. P. Geng Key Laboratory of Particle and Radiation Imaging (Ministry of Education) and Department of Engineering Physics, Tsinghua University, Beijing 100084 H. Gong Key Laboratory of Particle and Radiation Imaging (Ministry of Education) and Department of Engineering Physics, Tsinghua University, Beijing 100084 Q. J. Guo School of Physics, Peking University, Beijing 100871 X. Y. Guo YaLong River Hydropower Development Company, Chengdu 610051 L. He NUCTECH Company, Beijing 100084 S. M. He YaLong River Hydropower Development Company, Chengdu 610051 J. W. Hu Key Laboratory of Particle and Radiation Imaging (Ministry of Education) and Department of Engineering Physics, Tsinghua University, Beijing 100084 H. X. Huang Department of Nuclear Physics, China Institute of Atomic Energy, Beijing 102413 T. C. Huang Sino-French Institute of Nuclear and Technology, Sun Yat-sen University, Zhuhai 519082 H. T. Jia X. Jiang College of Physics, Sichuan University, Chengdu 610065 H. B. Li [ Institute of Physics, Academia Sinica, Taipei 11529 J. M. Li J. Li Key Laboratory of Particle and Radiation Imaging (Ministry of Education) and Department of Engineering Physics, Tsinghua University, Beijing 100084 Q. Y. Li R. M. J. Li College of Physics, Sichuan University, Chengdu 610065 X. Q. Li School of Physics, Nankai University, Tianjin 300071 Y. L. Li Key Laboratory of Particle and Radiation Imaging (Ministry of Education) and Department of Engineering Physics, Tsinghua University, Beijing 100084 Y. F. Liang Key Laboratory of Particle and Radiation Imaging (Ministry of Education) and Department of Engineering Physics, Tsinghua University, Beijing 100084 B. Liao College of Nuclear Science and Technology, Beijing Normal University, Beijing 100875 F. K. Lin [ Institute of Physics, Academia Sinica, Taipei 11529 S. T. Lin S. K. Liu College of Physics, Sichuan University, Chengdu 610065 Y. D. Liu College of Nuclear Science and Technology, Beijing Normal University, Beijing 100875 Y. Liu College of Physics, Sichuan University, Chengdu 610065 Y. Y. Liu College of Nuclear Science and Technology, Beijing Normal University, Beijing 100875 Z. Z. Liu Key Laboratory of Particle and Radiation Imaging (Ministry of Education) and Department of Engineering Physics, Tsinghua University, Beijing 100084 Y. C. Mao School of Physics, Peking University, Beijing 100871 Q. Y. Nie Key Laboratory of Particle and Radiation Imaging (Ministry of Education) and Department of Engineering Physics, Tsinghua University, Beijing 100084 J. H. Ning YaLong River Hydropower Development Company, Chengdu 610051 H. Pan NUCTECH Company, Beijing 100084 N. C. Qi YaLong River Hydropower Development Company, Chengdu 610051 J. Ren X. C. Ruan Department of Nuclear Physics, China Institute of Atomic Energy, Beijing 102413 K. Saraswat [ Institute of Physics, Academia Sinica, Taipei 11529 V. Sharma [ Institute of Physics, Academia Sinica, Taipei 11529 Department of Physics, Banaras Hindu University, Varanasi 221005 M. K. Singh [ Institute of Physics, Academia Sinica, Taipei 11529 Department of Physics, Banaras Hindu University, Varanasi 221005 T. X. Sun College of Nuclear Science and Technology, Beijing Normal University, Beijing 100875 C. J. Tang College of Physics, Sichuan University, Chengdu 610065 W. Y. Tang Y. Tian Key Laboratory of Particle and Radiation Imaging (Ministry of Education) and Department of Engineering Physics, Tsinghua University, Beijing 100084 G. F. Wang College of Nuclear Science and Technology, Beijing Normal University, Beijing 100875 L. Wang Department of Physics, Beijing Normal University, Beijing 100875 Q. Wang Y. Wang Key Laboratory of Particle and Radiation Imaging (Ministry of Education) and Department of Engineering Physics, Tsinghua University, Beijing 100084 Department of Physics, Tsinghua University, Beijing 100084 Y. X. Wang School of Physics, Peking University, Beijing 100871 H. T. Wong [ Institute of Physics, Academia Sinica, Taipei 11529 S. Y. Wu YaLong River Hydropower Development Company, Chengdu 610051 Y. C. Wu Key Laboratory of Particle and Radiation Imaging (Ministry of Education) and Department of Engineering Physics, Tsinghua University, Beijing 100084 H. Y. Xing College of Physics, Sichuan University, Chengdu 610065 R. Xu Key Laboratory of Particle and Radiation Imaging (Ministry of Education) and Department of Engineering Physics, Tsinghua University, Beijing 100084 Y. Xu School of Physics, Nankai University, Tianjin 300071 T. Xue Key Laboratory of Particle and Radiation Imaging (Ministry of Education) and Department of Engineering Physics, Tsinghua University, Beijing 100084 Y. L. Yan College of Physics, Sichuan University, Chengdu 610065 L. T. Yang Key Laboratory of Particle and Radiation Imaging (Ministry of Education) and Department of Engineering Physics, Tsinghua University, Beijing 100084 C. H. Yeh [ Institute of Physics, Academia Sinica, Taipei 11529 N. Yi Key Laboratory of Particle and Radiation Imaging (Ministry of Education) and Department of Engineering Physics, Tsinghua University, Beijing 100084 C. X. Yu School of Physics, Nankai University, Tianjin 300071 H. J. Yu NUCTECH Company, Beijing 100084 J. F. Yue YaLong River Hydropower Development Company, Chengdu 610051 M. Zeng Z. Zeng B. T. Zhang Key Laboratory of Particle and Radiation Imaging (Ministry of Education) and Department of Engineering Physics, Tsinghua University, Beijing 100084 F. S. Zhang College of Nuclear Science and Technology, Beijing Normal University, Beijing 100875 L. Zhang College of Physics, Sichuan University, Chengdu 610065 Z. H. Zhang Z. Y. Zhang Key Laboratory of Particle and Radiation Imaging (Ministry of Education) and Department of Engineering Physics, Tsinghua University, Beijing 100084 K. K. Zhao College of Physics, Sichuan University, Chengdu 610065 M. G. Zhao School of Physics, Nankai University, Tianjin 300071 J. F. Zhou YaLong River Hydropower Development Company, Chengdu 610051 Z. Y. Zhou Department of Nuclear Physics, China Institute of Atomic Energy, Beijing 102413 J. J. Zhu College of Physics, Sichuan University, Chengdu 610065

Abstract

A natural broad energy germanium (BEGe) detector is operated in the China Jinping Underground Laboratory (CJPL) for a feasibility study of building the next generation experiment of the neutrinoless double-beta (0 $\nu\beta\beta$ ) decay of ⁷⁶Ge. The setup of the prototype facility, characteristics of the BEGe detector, background reduction methods and data analysis are described in this paper. A background index of 6.4 $\times$ 10^-3 counts/(keV $\cdot$ kg $\cdot$ day) is achieved and 1.86 times lower than our previous result of the CDEX-1 detector. No signal is observed with an exposure of 186.4 kg $\cdot$ day, thus a limit on the half life of ⁷⁶Ge 0 $\nu\beta\beta$ decay is set at T ${}_{1/2}^{0\nu}$ $>$ 5.62×10²² yr at 90% C.L.. The limit corresponds to an effective Majorana neutrino mass in the range of 4.6 $\sim$ 10.3 eV, dependent on the nuclear matrix elements.

Key words: Neutrinoless double-beta decay, BEGe, ⁷⁶Ge, CJPL.

^†^†preprint: APS/123-QED

I Introduction

Evidences for non-zero mass neutrinos have been provided by the atmospheric and solar neutrino oscillation experiments [1-4] over the last two decades. Neutrinos can obtain their masses by a Majorana mass term if they are their own anti-particles [5]. The Majorana nature of the neutrinos leads to lepton number violation and naturally emerges in many beyond-the-Standard Model theories [6]. It also emerges in leading theories that explain the dominance of matter over antimatter in the Universe [7,8].

The search for neutrinoless double-beta (0 $\nu\beta\beta$ ) decay is considered the most promising way to prove the Majorana nature of neutrinos [9]. Furthermore, a measurement of the 0 $\nu\beta\beta$ decay rate, which depends on the effective Majorana mass, can indicate the mass hierarchy and the absolute mass scale of neutrinos.

Assuming the Majorana nature of neutrinos, the neutrinoless double-beta decay, $(A,Z)\to(A,Z+2)+2e^{-}$ , is permitted in 2 $\nu\beta\beta$ decay isotopes [10]. A huge experimental effort is ongoing to search for 0 $\nu\beta\beta$ decay in various candidate isotopes, for instance ⁷⁶Ge [11-14], ¹³⁶Xe [15,16], ¹³⁰Te [17,18], ¹⁰⁰Mo [19,20], via different detection technologies, including semiconductor detector [11-13], time projection chamber [15] and cryogenic bolometer [17,20,21].

High purity germanium (HPGe), serving as both target nuclei and detector, is an ideal medium for detecting 0 $\nu\beta\beta$ decays because of its high energy resolution, low internal background, and high detection efficiency [22]. Several experiments have been searching for 0 $\nu\beta\beta$ decay in ⁷⁶Ge via the HPGe technology, such as Gerda [12] and Majorana Demonstrator [13]. Currently, the Gerda experiment, operating enriched germanium detector array in liquid argon to detect 0 $\nu\beta\beta$ decay of ⁷⁶Ge, achieves the lowest background level in the 0 $\nu\beta\beta$ decay Q value (Q_ββ=2039 keV) energy region and gives the most stringent constraint on the ⁷⁶Ge 0 $\nu\beta\beta$ half-life (T ${}_{1/2}^{0\nu}$ $>$ 1.8 $\times$ 10²⁶yr) [12]. The Gerda and the Majorana collaborations are now merged into the Legend collaboration and are proposing a 200 kg-scale 0 $\nu\beta\beta$ experiment (Legend-200) aiming at setting the 0 $\nu\beta\beta$ decay half-life limit of ⁷⁶Ge at 10²⁷ yr [14].

The CDEX collaboration has given its first 0 $\nu\beta\beta$ limit of T ${}_{1/2}^{0\nu}$ $>$ 6.4 $\times$ 10²² yr for a p-type point contact high-purity germanium detector [23]. A next-generation 0 $\nu\beta\beta$ experiment CDEX-300 $\nu$ has been proposed in CJPL-II [24]. The CDEX-300 $\nu$ experiment aims at achieving a discovery potential that reaches the inverted-ordering neutrino mass scale region with 1-ton $\cdot$ yr exposure.

In this work, we set up a prototype facility in CJPL to study characteristics of a BEGe detector and novel background suppression techniques for future applications in the CDEX-300 $\nu$ experiment. Data acquisition, analysis and pulse shape discrimination procedures are established and tested. And a 0 $\nu\beta\beta$ result is given by analyzing a 186.4 kg $\cdot$ day exposure data using an unbinned extended profile likelihood method.

II Experiment Setup

A 1088.5 g natural low background broad energy germanium (BEGe) detector made by CANBERRA is used in our experiment. It is fabricated with a natural p-type germanium crystal with 91.1 mm in diameter and 31.4 mm in height. The atom fraction of ⁷⁶Ge in the crystal is 7.83%. The BEGe detector operates at 4500 V high voltage. The output from the p+ electrode is fed into a Canberra 2002C RC preamplifier to cover a wide dynamic energy range of up to 3.5 MeV for the 0 $\nu\beta\beta$ decay search experiment. The preamplifier output is digitized by a flash analog-to-digital convertor (FADC) at a 500 MHz sampling rate and recorded by the CAEN Scope software. The trigger threshold of FADC is set to only record events with energy above 500 keV, and the trigger rate is approximately 0.005 cps during data taking. A schematic diagram of the data acquisition (DAQ) system is shown in Fig. 1.

Refer to caption — Figure 1: Schematic diagram of the DAQ system.

An experiment setup is built in the polyethylene (PE) room of the CJPL-I experiment hall. The over 2400 m rock overburden provides natural shields against the cosmic rays, and the cosmic muon flux in CJPL is about (2.0 $\pm$ 0.4) $\times$ 10^-10 cm^-2s^-1 [25]. Environmental neutrons are shielded by the 1m thick wall of the PE room, the thermal neutron flux inside the PE room is measured to be (3.18 $\pm$ 0.97) $\times$ 10^-8 cm^-2s^-1 [26].

A passive shielding structure is built to shield the ambient radioactivity. As shown in Fig. 2, the detector crystal is shielded with a 20 cm lead, a 20 cm borated polyethylene, and a minimum of 20 cm copper from outside to inside. The outmost 20 cm lead is used to shield the ambient gamma rays. The middle 20 cm borated polyethylene acts as a thermal neutron absorber. The innermost copper shield is made of low background oxygen-free high conductivity (OFHC) copper to shield the residual gamma rays surviving the outer shields. The space within the copper shields is continuously flushed with high purity nitrogen gas from a pressurized Dewar to exclude the radon.

III Data Analysis

III.1 Event selection and energy calibration

The baseline level of the 500 MHz FADC is used to monitor the working condition of the detector, as shown in Fig. 3. Data taking starts on 2020/06/01 and ends on 2021/01/10. The gap from 2020/10/08 to 2020/11/02 was due to the unstable power supply caused by the construction of CJPL-II. Periods with unstable baseline levels are excluded from analysis (shadow regions in Fig. 3). On December 15, 2020, an accidental power failure caused a significant shift in the baseline level.

After excluding the shadow region in Fig. 3, the remaining 186.4 kg $\cdot$ day exposure data is divided into 9 datasets depending on the time and the baseline level of the detector. Data selections and the energy calibration are performed independently in each dataset.

The recorded events are selected by a noise cut and a data quality cut to remove noise events and events with abnormal baseline levels. Unphysical events are almost noise bursts with minimum signal values much lower than the baseline, while the physical events have minimum signal values around the baseline. Therefore, unphysical events can be rejected by the noise cut: events with minimum pulse values much lower than their baseline levels (10% of trigger threshold) are rejected. Events with baseline level not in $\pm$ 3 times standard deviation of the average baseline level are rejected by the data quality cut. Fig. 4 shows baseline levels and the acceptance region of the data quality cut for one dataset.

Amplitudes are extracted from the remaining charge pulses via a trapezoidal filter [27,28]. The filter parameters, rise time and flat time, are set as 8 $\mu$ s and 1 $\mu$ s, respectively. As shown in Fig. 5, the trapezoidal filter converts the raw charge pulse to a trapezoid pulse in which the height of the trapezoid indicates the amplitude of the raw pulse.

Energy calibrations are performed in each dataset using characteristic gamma peaks from primordial radionuclides in the detector and its surrounding materials. Seven peaks from ²⁰⁸Tl (583.3 keV, 2614.5 keV), ²¹⁴Bi (609.3 keV, 1120.3 keV, 1764.5 keV, 2204.1 keV), and ⁴⁰K (1460.8 keV) are used in calibrations. Each peak is fitted with a Gaussian function coupled with a linear background to determine the peak position. A second-order polynomial is used to convert amplitude to energy. Top panel of Fig. 6 shows the calibration curve of one dataset. The stability of the detector is evaluated via the shift of 2614.5 keV line as shown in the bottom panel of Fig. 6. The shift is within 0.2 keV during the data taking, indicating that the detector is under a stable operation. After combining all calibrated datasets, a maximum 1.5 keV residual is found in the in the characteristic gamma peaks. And a nonlinearity correction [29] is adopted to reduce the residuals of fitted energy. This correction is applied in the combined data to reduce the statistical uncertainties in each gamma peak, the corrected energies are shown in Fig. 7. After the correction, the maximum residual (0.7 keV in ⁴⁰K 1460.8 keV line) is adopted as a systematic uncertainty in the energy reconstruction of $0\nu\beta\beta$ events.

III.2 Pulse shape discrimination

Since the ranges in a germanium crystal of the two electrons of a 0 $\nu\beta\beta$ decay event are of the order of 1 mm, 0 $\nu\beta\beta$ events are typical single-site events (SSEs). High energy gamma rays are expected to deposit their energies at multiple sites featuring the so-called multi-site events (MSEs).

A pulse shape discrimination (PSD) method can be used to discriminate between single-site events and multi-site events in a BEGe detector [30,31]. The PSD method relies on the A/E parameter, in which A is the maximum amplitude of the current pulse and E is the reconstructed energy. The current pulse is extracted from the charge pulse by a moving average differential filter. Fig. 8 shows charge and current pulses of a typical SSE and MSE, respectively. SSEs deposit energies in a small range of area. The current of a SSE has one peak, with an amplitude A proportional to the energy E. MSEs deposit energies in multiple detector positions, leading to multi-peaks in current pulses and lower A/E values than those of SSEs.

A ²²⁸Th calibration experiment is conducted to determine the acceptance region of the A/E cut, the detector is irradiated by a ²²⁸Th source to create double escape events from ²⁰⁸Tl 2614.5 keV $\gamma$ -rays. Events in the 1592.5 keV double escape peak (DEP) have a similar profile as the 0 $\nu\beta\beta$ events [30] and therefore are used as proxies of SSEs. Events in the single escape peak (SEP) are typical two-site events and are used as proxies of MSEs. The A/E distribution of DEP events is fitted with a Gaussian function to determine the mean ( $\mu_{A\textfractionsolidus E}^{SSE}$ ) and standard deviation ( $\sigma_{A\textfractionsolidus E}^{SSE}$ ) of A/E parameters for SSEs. The acceptance region of the A/E cut is set to ( $\mu_{A\textfractionsolidus E}^{SSE}$ $\pm$ 5 $\sigma_{A\textfractionsolidus E}^{SSE}$ ) and leads to a 93% survival rate of the DEP events, and a 5% survival rate of the SEP events.

Fig. 9 shows A/E discriminations applied in the ²²⁸Th calibration data (top) and the 186.4 kg $\cdot$ day exposure data (down). The low A/E cut removes MSEs. Events rejected by the high A/E cut are likely to be $\alpha$ events originating from the surface contamination and the p+ electrode.

Survival fraction (SF) of the A/E cut in 1800 $\sim$ 2200 keV energy region is used to evaluate the stability of the cut. The SF is fitted with a flat line via the least square method. The $\chi^{2}/$ (degree of freedom) of the fit is 19.05/26, indicating that the performance of A/E cut is stable during data taking.

Selected SSEs in the 186.4 kg $\cdot$ day exposure data are used to evaluate the energy resolution of 0 $\nu\beta\beta$ signals, indicated as the full width at half maximum (FWHM), as shown in Fig. 10. Gamma peaks from ²⁰⁸Tl (583.3 keV, 1592.5 keV, 2614.5 keV), ²¹⁴Bi (609.3 keV), ²²⁸Ac (911.2 keV) and ⁴⁰K (1460.8 keV) are used to calculate the FWHM at 2039 keV. The FWHMs of the six peaks are fitted with a function FWHM = $\sqrt{a+bE}$ , and the interpolation of FWHM at 2039 keV is 2.85 keV. Uncertainties of the result mainly originate from two aspects:

(1)

Uncertainties from FWHMs of the selected characteristic gamma peaks and their effects on curve parameters ( $a$ , $b$ ): The uncertainties are calculated within the standard chi-square fitting and error propagation techniques. The combined uncertainty is $\pm$ 0.24 keV.
(2)

Choice of characteristic gamma peaks: Systematic effects are taken as deviations of results due to the choice of gamma peaks. The FWHM curve is refitted without one of the six aforementioned peaks, and the maximum deviations in the FWHM at 2039 keV is 0.41 keV when the 2614 keV peak is excluded.

Combining both uncertainties, the FWHM at 2039 keV is given as (2.85 $\pm$ 0.48) keV

III.3 Efficiency calibration

The total 0 $\nu\beta\beta$ signal efficiency consists of: (i) the efficiency of the data quality cut ( $\varepsilon_{QC}$ ), (ii) the efficiency of the two electrons emitted from 0 $\nu\beta\beta$ decay deposits all energy in the active volume of the detector ( $\varepsilon_{fed}$ ), and (iii) the efficiency of PSD ( $\varepsilon_{PSD}$ ). The trigger rate during data taken is measured to be 0.005 Hz, the dead time is negligible and not considered in the efficiency.

The efficiency loss due to the noise cut is negligible as a physical event can be rejected by the noise cut only when it is overlapped with a burst of noise event. And the coincidences of those two events are negligible because of the low trigger rate. The efficiency of the data quality cut is calculated by recorded physical events, given as (94.37 $\pm$ 0.49)% where the error is the statistical uncertainty of the recorded events.

The n+ electrode on the side and top surface of the detector forms an inactive region, known as the dead layer, reducing the active volume of the detector. The dead layer of (1.18 $\pm$ 0.10) mm and (0.17 $\pm$ 0.10) mm for side and top surfaces have been measured in our previous work [32-34] and gives a (91.1 $\pm$ 0.96)% active volume of the crystal.

The probability of 0 $\nu\beta\beta$ events deposit all energy in the active volume of the detector ( $\varepsilon_{fed}$ ) is calculated by Monte Carlo simulations via a Geant4 based simulation toolkit SAGE [35]. The 0 $\nu\beta\beta$ decays are uniformly sampled in the germanium crystal, events with full energy deposited in the active region are counted to calculate the efficiency. The efficiency ( $\varepsilon_{fed}$ ) is (86.71 $\pm$ 0.84)% where the error is derived from variations of the dead-layer thickness: efficiencies are calculated for the top dead-layer thickness ranging from 0.07 $\sim$ 0.27mm and the side dead-layer thickness ranging from 1.08 $\sim$ 1.28mm, the maximum deviation on results is counted as the uncertainty. The $\varepsilon_{fed}$ is lower than the active volume because the two electrons in 0 $\nu\beta\beta$ decay may lose their energy in germanium by bremsstrahlung.

The ²²⁸Th calibration data and pulse shape simulations (PSS) are used to determine the PSD efficiency. The PSS of the ²²⁸Th calibration data is conducted within a pulse shape simulation module of the SAGE toolkit [36]. The A/E distributions derived from the PSS, the ²²⁸Th calibration data and the 186.4 kg $\cdot$ day exposure data are compared in Fig. 11. Events from the DEP, the SEP, the full energy peak (FEP) and the Compton flat (1800 $\sim$ 2200 keV) of ²⁰⁸Tl 2614.5 keV $\gamma$ lines are selected for comparison. Table. 1 lists the A/E cut removal and survival fractions of the simulation and the calibration data.

Table 1: Removal fractions by the low A/E cut and high A/E cut and total survival fractions applying both cuts in ²²⁸Th calibration data and pulse shape simulation data.

Region	Low A/E cut	High A/E cut	Survival fraction
	A/E $<$ 0.975	A/E $>$ 1.025	0.975 $<$ A/E $<$ 1.025
²²⁸Th calibration data
DEP 1592.5 keV	6.76 $\pm$ 0.67%	0.12 $\pm$ 0.09%	93.12 $\pm$ 3.32%
²²⁸Th calibration PSS
DEP 1592.5 keV	6.55%	1.43%	92.03%
0 $\nu\beta\beta$ events PSS
Q_ββ 2039 keV	6.99%	3.77%	89.23%

Table 2: Uncertainties of 0

\nu\beta\beta

signal efficiency and their compositions, the combined efficiency and its uncertainty are listed in the last column.

Sources of Efficiency	Sources of Uncertainties	Value / [type]
Quality Cut	Statistical uncertainty of recorded events	$\pm$ 0.49% [stat]
$\varepsilon_{QC}$ =94.37%	Statistical uncertainty of recorded events	$\pm$ 0.49% [stat]
0 $\nu\beta\beta$ events full energy deposition	Uncertainty on dead-layer thickness	$\pm$ 0.84% [sys]
$\varepsilon_{fed}$ =86.71%	Uncertainty on dead-layer thickness	$\pm$ 0.84% [sys]
Pulse shape discrimination	Low A/E cut removal fraction of ²²⁸Th calibration data	$\pm$ 0.67% [stat]
$\varepsilon_{PSD}$ =89.47%	Differences between 0 $\nu\beta\beta$ and DEP events	$\pm$ 0.44% [sys]
	Differences between calibration and physics data	$\pm$ 0.91% [sys]
	Variations on PSS parameters	$\pm$ 0.97% [sys]
	Maximum discrepancy between experiment and PSS	$\pm$ 1.31% [sys]
Combined efficiency	Efficiency = $\varepsilon_{QC}\cdot\varepsilon_{fed}\cdot\varepsilon_{PSD}$ = 73.21%
Combined efficiency	Uncertainty = $\pm$ 1.84%

0 $\nu\beta\beta$ events and DEP events are both typical SSEs but have different locations in the detector, 0 $\nu\beta\beta$ events are homogeneously distributed while DEP events are dominantly located at the corners. Therefore, the PSD efficiency of 0 $\nu\beta\beta$ events ( $\varepsilon_{PSD}$ ) is calculated by a similar way of Gerda [37]: the removal fraction of the low A/E cut is adopted from the ²²⁸Th calibration data as the low A/E cut only removes MSEs and 0 $\nu\beta\beta$ events and DEP events are both typical SSEs. The removal fraction of the high A/E cut is adopted from the pulse shape simulation of 0 $\nu\beta\beta$ events. The calculation gives a PSD efficiency of 89.47%, similar to the result derived from the PSS (89.23%)

Statistical and systematic uncertainties of the PSD efficiency mainly consist of four parts:

(1)

The statistical uncertainty of the low A/E cut fraction of DEP events, $\pm$ 0.67%;
(2)

The systematic uncertainty due to differences between 0 $\nu\beta\beta$ and DEP events: the discrepancy between the removal fraction of the simulated DEP events and 0 $\nu\beta\beta$ events by the low A/E cut is counted as the uncertainty, $\pm$ 0.44%;
(3)

The systematic uncertainty due to the residual differences between calibration and physics data: the survival fraction of ²⁰⁸Tl 2614.5 keV peak is used to compute the uncertainty. The discrepancy between the calibration data (5.39% $\pm$ 0.2%) and physical data (5.48% $\pm$ 0.8%) is (0.09% $\pm$ 0.82%). The upper limit of the discrepancy is adopted as the systematic uncertainty, $\pm$ 0.91%;
(4)

The systematic uncertainty of PSS: identical analyses are performed on varies PSS parameters, and the maximum deviation on results is adopted as one systematic uncertainty ( $\pm$ 0.97%). The maximum deviation between the ²²⁸Th calibration data and the PSS in Table. 1 (±1.31%) is added as the other systematic uncertainty. The combined systematic uncertainty is $\pm$ 1.63%.

Combining the statistical and systematic uncertainties, the $\varepsilon_{PSD}$ is given as (89.47 $\pm$ 2.03)%.

Compositions of the 0 $\nu\beta\beta$ signal efficiency and their uncertainties are listed in Table. 2. The total efficiency is the product of the $\varepsilon_{QC}$ , $\varepsilon_{fed}$ , and $\varepsilon_{PSD}$ , i.e. (73.21 $\pm$ 1.84)%.

III.4 Background model

Background spectra of different radioactive isotopes in different components of the detector setup are simulated using the SAGE toolkit. Table. 3 lists all the simulated background sources and components. In our simulation, secular equilibrium is assumed in ²³⁸U and ²³²Th decay chain and all background sources are assumed to be uniformly distributed in their components. Due to the low muon flux in CJPL-I [25], backgrounds from muons and their secondary particles are negligible (less than 1 $\times$ 10^-6 counts/keV/kg/day (cpkkd)). Neutrons are also negligible after shields of a 1 m polyethylene wall and a 20 cm borated polyethylene absorber. Therefore, they are not considered in the simulation. The 2 $\nu\beta\beta$ decays of ⁷⁶Ge are considered assuming a half-life of 2.1 $\times$ 10²¹ yr [38].

Table 3: Simulated background components and their contributions (R_0νββ) in the 0

\nu\beta\beta

signal region.

Sources	Components	R_0νββ
Cosmogenic	⁶⁸Ge, ⁶⁰Co, ⁵⁴Mn, ⁶⁵Zn in Ge	8.6%
isotopes	⁶⁰Co in Copper	2.1%
²³⁸U chain	Crystal holder	15.1%
	Signal pin, Electronics
	Vacuum Cup
	Outer Shield
	²²²Rn
²³²Th chain	Crystal holder	74.2%
	Electronics
	Vacuum Cup
	Outer Shield
⁴⁰K	Crystal holder	0%
	Electronics
	Vacuum Cup
	Outer Shield

A background model (Fig. 12) is obtained by fitting the 186.4 kg $\cdot$ day spectrum with simulated spectra in 550 $\sim$ 3000 keV energy range, using the maximum likelihood method. The simulated spectra are convolved with an energy resolution function derived from fitting the FWHMs of the prominent gamma peaks in the spectrum prior to the PSD. Contributions of background sources in the 0 $\nu\beta\beta$ signal region are determined from the background model and are listed in Table. 3.

In the 0 $\nu\beta\beta$ signal region (Q ${}_{\beta\beta}\pm$ 5keV), 89% of the background are from radionuclides in the ²³²Th and ²³⁸U decay chains according to the background model. A background of 2.29 $\times$ 10^-2 cpkkd in Q ${}_{\beta\beta}\pm$ 5keV region projected by the background model agrees well with (2.13 $\pm$ 0.3) $\times$ 10^-2 cpkkd calculated from the exposure data after unblinding.

IV Results and Discussion

Fig. 13 shows the measured energy spectra above 1000 keV for the 186.4 kg $\cdot$ day exposure data. Spectra shown in black and red are prior to and after the PSD, respectively. Spectra in the energy region of 1800 $\sim$ 2300 keV are used to estimate the background in the 0 $\nu\beta\beta$ region of interest (ROI). Gamma peaks identified by the background model, as indicated by gray shading in the inset of Fig. 13, are excluded. Additionally, a $\pm$ 5 keV wide window centered at Q_ββ is excluded, as indicated by the blue shaded region. Prior to and after PSD, the estimated background in the ROI from the resulting 420 keV window is (2.45 $\pm$ 0.06) $\times$ 10^-2 cpkkd and (0.64 $\pm$ 0.03) $\times$ 10^-2 cpkkd, respectively. The background in the ROI is reduced by a factor of 3.79 after applying the PSD method.

The exposure data after all cuts are used to analyze the 0 $\nu\beta\beta$ decay of ⁷⁶Ge. The half-life of ⁷⁶Ge neutrinoless double-beta decay can be calculated by Eq. 1:

T_{1/2}^{0\nu}=\frac{\textup{ln}2\cdot N_{A}\cdot f_{76}\cdot m\cdot T\cdot\varepsilon_{total}}{N^{0\nu}\cdot M}

(1)

Where $m\cdot T$ is the exposure, $\varepsilon_{total}$ is the total efficiency defined in Sec III.3, ${N^{0\nu}}$ is the number of observed 0 $\nu\beta\beta$ signal events, $M$ the molar mass of natural Ge, $N_{A}$ is the Avogadro’s constant, $f$ the fraction of ⁷⁶Ge atoms in the natural germanium detector.

Spectra of the 1940 $\sim$ 2080keV analysis region are shown in Fig. 14. After unblinding, 178 events survive all data cuts, and eight events are found in the ROI (Q ${}_{\beta\beta}\pm$ 3 $\sigma_{\beta\beta}$ ).

The 0 $\nu\beta\beta$ decay signal is analyzed using an unbinned extended profile likelihood method [39]. As predicted by the background model, no background peak is identified, and a flat background is assumed in the analysis region. For the signal, a Gaussian distribution centered at the Q_ββ with a width corresponding to the energy resolution is considered. The likelihood function is then given by:

	$\displaystyle f(E\|b,N^{0\nu})=$		$\displaystyle\frac{1}{\triangle E\cdot b+N^{0\nu}}$
			$\displaystyle\times\left(b+\frac{N^{0\nu}}{\sqrt{2\pi}\cdot\sigma}e^{-\frac{(E-Q_{\beta\beta})^{2}}{2\sigma^{2}}}\right)$

	$\displaystyle L(b,N^{0\nu})=$		$\displaystyle\frac{(\triangle E\cdot b+N^{0\nu})^{N}\cdot e^{-(\triangle E\cdot b+N^{0\nu})}}{N!}$
			$\displaystyle\times\prod_{i=1}^{N}f(E_{i}\|b,N^{0\nu})$

Where N is the total events number in the analysis region, b is the background rate (cts/keV) and $\triangle$ E is the width of the analysis region, $N^{0\nu}$ the observed 0 $\nu\beta\beta$ events, E the energy of recorded events, $\sigma$ is the energy resolution at $Q_{\beta\beta}$ . $f(E|b,N^{0\nu})$ is the probability density function (pdf) of one single event. The likelihood function $L(b,N^{0\nu})$ is the product of the pdf of each event and extended with the Poisson term.

When fitting the likelihood function L, parameter b and $N^{0\nu}$ are bound to positive values. And a test statistic based on profile likelihood is used to calculate the confidence interval. The probability distributions of the test statistic are computed using the Monte Carlo method.

The unbinned profile likelihood analysis yields a best-fit background of 1.27 cts/keV and no indication for signal. The lower limit for 0 $\nu\beta\beta$ decay half-life is set to:

\displaystyle T_{1/2}^{0\nu}\geq 5.62\times 10^{22}\textup{ yr at 90\%C.L.}

(4)

The corresponding 90%C.L. upper limit of the 0 $\nu\beta\beta$ signal strength is 2.99 events. Uncertainties of the energy calibration (2039 $\pm$ 0.7 keV) and the energy resolution (2.85 $\pm$ 0.48 keV) are considered by folding them into the profile likelihood function through additional nuisance parameters constrained by Gaussian probability distributions. Uncertainties of the efficiency and the exposure are considered by propagating them through Eq. 1. The overall effect of all uncertainties on the half-life limit is about 2.67%.

The upper limit on the effective Majorana neutrino mass $m_{\beta\beta}$ is derived by:

(T_{1/2}^{0\nu})^{-1}=G^{0\nu}\left|g_{A}^{2}\cdot M_{0\nu}\right|^{2}\left(\frac{m_{\beta\beta}}{m_{e}}\right)^{2}

(5)

m_{\beta\beta}\leq[4.6,10.3]\textup{ eV}

(6)

The upper and lower values are obtained by using nuclear matrix elements $M_{0\nu}$ from [40] and [41], respectively, the coupling constant $g_{A}$ is set at 1.27 [12], and the phase factor $G^{0\nu}$ is adopted from [42], $m_{e}$ is the electron mass.

As shown in Table. 4, compared with our previous 0 $\nu\beta\beta$ result from a p-type point contact germanium (PPCGe) detector in the CDEX-1 experiment [23], this work achieves a lower background by applying the PSD method. The CDEX-1 detector has a lower efficiency mainly due to the pulsed-reset preamplifier. The reset preamplifier is designed to have low electronic noise for dark matter detection and has a lower efficiency at the Q_ββ energy than the RC preamplifier used in this work.

Table 4: Neutrinoless double-beta decay results from this work and CDEX-1. (cpkky denotes counts/kg/keV/yr)

	BEGe in This work	PPCGe in CDEX-1
Exposure	186.4 kg $\cdot$ day	304 kg $\cdot$ day
Total Efficiency	73.21%	68.44%
Background level	8.95 $\pm$ 0.22 cpkky (before PSD)	4.38 cpkky (w/o. PSD)
Background level	2.35 $\pm$ 0.11 cpkky (after PSD)	4.38 cpkky (w/o. PSD)
half-life limit (90% C.L.)	5.6 $\times$ 10²² yr	6.4 $\times$ 10²² yr

Future Ge- $0\nu\beta\beta$ experiments would target at ton-scale of enriched ⁷⁶Ge detectors to probe the neutrino inverted mass ordering, as is the case for the LEGEND proposal [43]. The next-generation CDEX-300 $\nu$ experiment will consist of 225 kg of Ge-detectors enriched in ⁷⁶Ge. To meet the half-life sensitivity goal of $10^{27}$ years, various improvement will be implemented:

a: Increasing the effective exposure: The CDEX-300 $\nu$ experiment will use approximate 225 kg ⁷⁶Ge enriched ( $>$ 86% enrichment) BEGe detectors to increase the exposure.

b: Background control:

b.1

Deep underground laboratory: the 2400 m rock overburden of CJPL-II shield the cosmic muons to $O$ (10^-10) cm^-2s^-1 [25];
b.2

Cosmogenic background control: measures have been taken to reduce the cosmogenic background. The production processes of detectors are optimized to reduce exposures on the ground. Additional shielding is designed to protect the detector from cosmic rays during its production and transportation. Continued efforts have been put into underground material production and underground detector fabrication.
b.3

Material screening and selection: materials used in the construction of the next generation experiment will be measured and selected according to the physics goal. The detector module will be further optimized to have fewer surrounding structures with higher radiopurity;
b.4

Liquid nitrogen (LN) shielding: the LAr veto system and the detector array will be submerged in a 1725 m³ liquid nitrogen (LN) tank. The over 6 m thick LN can provide an effective shield against ambient radioactivity;
b.5

Liquid argon (LAr) veto system: the detector array will be surrounded by LAr coupled with scintillation light readout equipment. Background event causing simultaneous signals in Ge detector and LAr can be rejected [43,45];
b.6

Ge detector multiplicity: background events, for instance scattered $\gamma$ rays with simultaneous energy depositions in multiple Ge detectors can be rejected by the coincidence signals [43];
b.7

Pulse shape discrimination: for instance, A/E method will be used to discriminate MSEs (background-like) from SSEs (signal-like), and the background suppression power of the A/E cut is evaluated to be a factor of 3.79;

Fig.15 depicts the sensitivity enhancement due to ⁷⁶Ge enrichment, increased exposure and suppression of background. The target sensitivity of CDEX-300 $\nu$ is $10^{27}$ yr with one ton-yr exposure at a background level of $10^{-4}$ cpkkd in the $0\nu\beta\beta$ signal region.

V Summary

A prototype facility using a natural BEGe detector to study the feasibility of building a next generation ⁷⁶Ge 0 $\nu\beta\beta$ experiment is built in this work. Event selection and data analysis procedures are established to remove unphysical events, reconstruct energy, and discriminate background events. The pulse shape discrimination method (the A/E cut) is applied in data analysis and reduces the background in the 0 $\nu\beta\beta$ ROI by a factor of 3.79.

A background model is built for the prototype. Radionuclides from ²³²Th and ²³⁸U decay chains are identified as the primary source of backgrounds in the 0 $\nu\beta\beta$ signal region. Cosmogenic radioactive isotopes in germanium and copper also contribute to the backgrounds. To control backgrounds in the future large-scale experiment, (1) selecting ultra-pure materials can reduce the inhabit radioactive impurities from ²³²Th and ²³⁸U decay chains, (2) growing germanium crystal in an underground facility or cooling detector and copper material underground can reduce backgrounds from cosmogenic isotopes, (3) the anti-coincidence techniques can be used to further suppress backgrounds in the 0 $\nu\beta\beta$ signal region. Background control approaches adopted in the baseline design of the future CDEX-300 $\nu$ experiment are outlined in this work.

Based on the 186.4 kg $\cdot$ day exposure data, a limit on the half-life of ⁷⁶Ge 0 $\nu\beta\beta$ decay is set to 5.62 $\times$ 10²² yr at 90%C.L. via an unbinned extended profile likelihood method.

Acknowledgements.

This work was supported by the National Key Research and Development Program of China (Grant No. 2017YFA0402200) and the National Natural Science Foundation of China (Grants No. 12175112, No. 12005111, No. 11725522, No. 11675088, No.11475099).

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Search for Neutrinoless Double-Beta Decay of 76Ge with a Natural Broad Energy Germanium Detector