Majorana Collaboration

#  Final Result of the Majorana Demonstrator’s Search for Neutrinoless
Double-$\beta$ Decay in 76Ge

I.J. Arnquist  Pacific Northwest National Laboratory, Richland, WA 99354, USA
F.T. Avignone III  Department of Physics and Astronomy, University of South
Carolina, Columbia, SC 29208, USA  Oak Ridge National Laboratory, Oak Ridge,
TN 37830, USA  A.S. Barabash  National Research Center “Kurchatov Institute”
Institute for Theoretical and Experimental Physics, Moscow, 117218 Russia
C.J. Barton  Department of Physics, University of South Dakota, Vermillion, SD
57069, USA  P.J. Barton  Nuclear Science Division, Lawrence Berkeley National
Laboratory, Berkeley, CA 94720, USA  K.H. Bhimani  Department of Physics and
Astronomy, University of North Carolina, Chapel Hill, NC 27514, USA  Triangle
Universities Nuclear Laboratory, Durham, NC 27708, USA  E. Blalock  Department
of Physics, North Carolina State University, Raleigh, NC 27695, USA  Triangle
Universities Nuclear Laboratory, Durham, NC 27708, USA  B. Bos  Department of
Physics and Astronomy, University of North Carolina, Chapel Hill, NC 27514,
USA  Triangle Universities Nuclear Laboratory, Durham, NC 27708, USA  M. Busch
Department of Physics, Duke University, Durham, NC 27708, USA  Triangle
Universities Nuclear Laboratory, Durham, NC 27708, USA  M. Buuck  Present
address: SLAC National Accelerator Laboratory, Menlo Park, CA 94025, USA
Center for Experimental Nuclear Physics and Astrophysics, and Department of
Physics, University of Washington, Seattle, WA 98195, USA  T.S. Caldwell
Department of Physics and Astronomy, University of North Carolina, Chapel
Hill, NC 27514, USA  Triangle Universities Nuclear Laboratory, Durham, NC
27708, USA  Y-D. Chan  Nuclear Science Division, Lawrence Berkeley National
Laboratory, Berkeley, CA 94720, USA  C.D. Christofferson  South Dakota Mines,
Rapid City, SD 57701, USA  P.-H. Chu  Los Alamos National Laboratory, Los
Alamos, NM 87545, USA  M.L. Clark  Department of Physics and Astronomy,
University of North Carolina, Chapel Hill, NC 27514, USA  Triangle
Universities Nuclear Laboratory, Durham, NC 27708, USA  C. Cuesta  Centro de
Investigaciones Energéticas, Medioambientales y Tecnológicas, CIEMAT 28040,
Madrid, Spain  J.A. Detwiler  Center for Experimental Nuclear Physics and
Astrophysics, and Department of Physics, University of Washington, Seattle, WA
98195, USA  Yu. Efremenko  Department of Physics and Astronomy, University of
Tennessee, Knoxville, TN 37916, USA  Oak Ridge National Laboratory, Oak Ridge,
TN 37830, USA  H. Ejiri  Research Center for Nuclear Physics, Osaka
University, Ibaraki, Osaka 567-0047, Japan  S.R. Elliott  Los Alamos National
Laboratory, Los Alamos, NM 87545, USA  G.K. Giovanetti  Physics Department,
Williams College, Williamstown, MA 01267, USA  M.P. Green  Department of
Physics, North Carolina State University, Raleigh, NC 27695, USA  Triangle
Universities Nuclear Laboratory, Durham, NC 27708, USA  Oak Ridge National
Laboratory, Oak Ridge, TN 37830, USA  J. Gruszko  Department of Physics and
Astronomy, University of North Carolina, Chapel Hill, NC 27514, USA  Triangle
Universities Nuclear Laboratory, Durham, NC 27708, USA  I.S. Guinn  Department
of Physics and Astronomy, University of North Carolina, Chapel Hill, NC 27514,
USA  Triangle Universities Nuclear Laboratory, Durham, NC 27708, USA  V.E.
Guiseppe  Oak Ridge National Laboratory, Oak Ridge, TN 37830, USA  C.R. Haufe
Department of Physics and Astronomy, University of North Carolina, Chapel
Hill, NC 27514, USA  Triangle Universities Nuclear Laboratory, Durham, NC
27708, USA  R. Henning  Department of Physics and Astronomy, University of
North Carolina, Chapel Hill, NC 27514, USA  Triangle Universities Nuclear
Laboratory, Durham, NC 27708, USA  D. Hervas Aguilar  Department of Physics
and Astronomy, University of North Carolina, Chapel Hill, NC 27514, USA
Triangle Universities Nuclear Laboratory, Durham, NC 27708, USA  E.W. Hoppe
Pacific Northwest National Laboratory, Richland, WA 99354, USA  A. Hostiuc
Center for Experimental Nuclear Physics and Astrophysics, and Department of
Physics, University of Washington, Seattle, WA 98195, USA  M.F. Kidd
Tennessee Tech University, Cookeville, TN 38505, USA  I. Kim  Los Alamos
National Laboratory, Los Alamos, NM 87545, USA  R.T. Kouzes  Pacific Northwest
National Laboratory, Richland, WA 99354, USA  T.E. Lannen V  Department of
Physics and Astronomy, University of South Carolina, Columbia, SC 29208, USA
A. Li  Department of Physics and Astronomy, University of North Carolina,
Chapel Hill, NC 27514, USA  Triangle Universities Nuclear Laboratory, Durham,
NC 27708, USA  A.M. Lopez  Department of Physics and Astronomy, University of
Tennessee, Knoxville, TN 37916, USA  J.M. López-Castaño  Oak Ridge National
Laboratory, Oak Ridge, TN 37830, USA  E.L. Martin  Present address: Duke
University, Durham, NC 27708  Department of Physics and Astronomy, University
of North Carolina, Chapel Hill, NC 27514, USA  Triangle Universities Nuclear
Laboratory, Durham, NC 27708, USA  R.D. Martin  Department of Physics,
Engineering Physics and Astronomy, Queen’s University, Kingston, ON K7L 3N6,
Canada  R. Massarczyk  Los Alamos National Laboratory, Los Alamos, NM 87545,
USA  S.J. Meijer  Los Alamos National Laboratory, Los Alamos, NM 87545, USA
S. Mertens  Max-Planck-Institut für Physik, München, 80805, Germany  Physik
Department and Excellence Cluster Universe, Technische Universität, München,
85748 Germany  T.K. Oli  Department of Physics, University of South Dakota,
Vermillion, SD 57069, USA  G. Othman  Present address: Universität Hamburg,
Institut für Experimentalphysik, Hamburg, Germany  Department of Physics and
Astronomy, University of North Carolina, Chapel Hill, NC 27514, USA  Triangle
Universities Nuclear Laboratory, Durham, NC 27708, USA  L.S. Paudel
Department of Physics, University of South Dakota, Vermillion, SD 57069, USA
W. Pettus  Department of Physics, Indiana University, Bloomington, IN 47405,
USA  IU Center for Exploration of Energy and Matter, Bloomington, IN 47408,
USA  A.W.P. Poon  Nuclear Science Division, Lawrence Berkeley National
Laboratory, Berkeley, CA 94720, USA  D.C. Radford  Oak Ridge National
Laboratory, Oak Ridge, TN 37830, USA  A.L. Reine  Department of Physics and
Astronomy, University of North Carolina, Chapel Hill, NC 27514, USA  Triangle
Universities Nuclear Laboratory, Durham, NC 27708, USA  K. Rielage  Los Alamos
National Laboratory, Los Alamos, NM 87545, USA  N.W. Ruof  Center for
Experimental Nuclear Physics and Astrophysics, and Department of Physics,
University of Washington, Seattle, WA 98195, USA  D.C. Schaper  Los Alamos
National Laboratory, Los Alamos, NM 87545, USA  D. Tedeschi  Department of
Physics and Astronomy, University of South Carolina, Columbia, SC 29208, USA
R.L. Varner  Oak Ridge National Laboratory, Oak Ridge, TN 37830, USA  S.
Vasilyev  Joint Institute for Nuclear Research, Dubna, 141980 Russia  J.F.
Wilkerson  Department of Physics and Astronomy, University of North Carolina,
Chapel Hill, NC 27514, USA  Triangle Universities Nuclear Laboratory, Durham,
NC 27708, USA  Oak Ridge National Laboratory, Oak Ridge, TN 37830, USA  C.
Wiseman  Center for Experimental Nuclear Physics and Astrophysics, and
Department of Physics, University of Washington, Seattle, WA 98195, USA  W. Xu
Department of Physics, University of South Dakota, Vermillion, SD 57069, USA
C.-H. Yu  Oak Ridge National Laboratory, Oak Ridge, TN 37830, USA  B.X. Zhu
Present address: Jet Propulsion Laboratory, California Institute of
Technology, Pasadena, CA 91109, USA  Los Alamos National Laboratory, Los
Alamos, NM 87545, USA

###### Abstract

The Majorana Demonstrator searched for neutrinoless double-$\beta$ decay
($0\nu\beta\beta$) of 76Ge using modular arrays of high-purity Ge detectors
operated in vacuum cryostats in a low-background shield. The arrays operated
with up to 40.4 kg of detectors (27.2 kg enriched to $\sim$88% in 76Ge). From
these measurements, the Demonstrator has accumulated $64.5$ kg yr of enriched
active exposure. With a world-leading energy resolution of 2.52 keV FWHM at
the 2039 keV $Q_{\beta\beta}$ (0.12%), we set a half-life limit of
$0\nu\beta\beta$ in 76Ge at $T_{1/2}>8.3\times 10^{25}$ yr (90% C.L.). This
provides a range of upper limits on $m_{\beta\beta}$ of $(113-269)$ meV (90%
C.L.), depending on the choice of nuclear matrix elements.

###### pacs:

23.40-s, 23.40.Bw, 14.60.Pq, 27.50.+j

††preprint: APS/123-QED

Neutrinoless double-beta decay ($0\nu\beta\beta$) is a hypothetical nuclear
process involving the unbalanced creation of two new matter particles but no
antimatter [1, 2, 3, 4, 5]. This lepton-number-violating process is predicted
generically by many grand-unification theories, as well as by leptogenesis
[6], a leading explanation of why the universe is matter-dominated. A half-
life measurement is sensitive to the Majorana mass of the electron neutrino,
$m_{\beta\beta}$, and requires nuclear matrix elements that must be calculated
through many body nuclear theory [7, 8]. The experimental signature of
$0\nu\beta\beta$ is a peak in total electron kinetic energy at the Q-value
($Q_{\beta\beta}$) of the decay. While double-beta decay with the emission of
two neutrinos ($2\nu\beta\beta$) has been directly observed in 9 isotopes [9],
no experiment has yet seen $0\nu\beta\beta$. A robust program involving many
low-background experiments has searched for $0\nu\beta\beta$ in multiple
isotopes, with half-life limits surpassing $10^{25-26}$ yr in some cases [10,
11, 12, 13, 14]. For example, the GERDA experiment has established the leading
half-life limit in 76Ge of $1.8\times 10^{26}$ yr [11].

The Majorana Demonstrator searched for $0\nu\beta\beta$ in 76Ge [15] using
high purity germanium (HPGe) detectors enriched in this isotope. Since its
most recent published limit using data collected from 2015–2018 [10], the
experiment has more than doubled its previous enriched active detector
exposure from 26.1 kg yr to 64.5 kg yr. HPGe detectors are a mature technology
offering excellent energy resolution and low intrinsic background [16].
Germanium can be enriched to a high isotopic fraction of 76Ge, which acts as
both source and detector, enabling a high detection efficiency. The Majorana
Demonstrator consisted of two modules, each with an array of HPGe detectors
operated in a vacuum cryostat. The cryostats and structural components were
fabricated from ultra-low background copper that was electroformed underground
(UGEFCu) [17] and carefully selected plastics. Low background front end
electronics, cables, and connectors were developed for the experiment [18,
19]. The modules were enclosed by a multi-layered low-background shield,
consisting of, from inner to outer layers, 5 cm of UGEFCu, 5 cm of
commercially sourced copper, and 45 cm of high-purity lead. This shield
construct was enclosed in an aluminum radon exclusion box that was constantly
purged with low-Rn liquid nitrogen (LN) boil-off gas. The Rn-exclusion box was
surrounded by an active muon veto consisting of 32 plastic scintillating
panels with nearly $4\pi$ coverage [20]. The vacuum and cryogenic hardware and
control electronics were located just outside this enclosure and connected to
the cryostats via a shielded crossarm penetrating the other layers. Finally,
the entire assembly was enclosed in 5 cm borated and 25 cm pure polyethylene
neutron shielding. A 228Th line source for each module was stored outside the
shield, with a penetration and helical track enabling its deployment for
detector calibrations [21]. The experiment was located in the Davis campus on
the 4850-foot level (4300 m.w.e.) of the Sanford Underground Research Facility
(SURF) in Lead, South Dakota [22]. To ensure low backgrounds, components of
the modules and shield were subjected to an extensive radioassay campaign
[23]. To reduce cosmogenic activation, time spent on the surface for detectors
and detector components was minimized and logged in a database [24].

Three novel HPGe detector geometries were used, each with a p-type, point-like
electrode centered on one face and an n-type electrode covering the remaining
surfaces, separated by a thin passivated surface. The Demonstrator primarily
used p-type, point contact (PPC) detectors [25, 26] enriched to $87.4\pm
0.5\%$ in 76Ge and having masses of $0.6-1.2$ kg. We also used Broad Energy
Germanium (BEGeTM) Detectors [27] made of natural Ge with typical masses of
$\sim$0.6 kg, and inverted coaxial point contact (ICPC) detectors [28]
enriched to $88\pm 1\%$ in 76Ge, with masses of $1.3-2.1$ kg. Module 1 began
operation in July 2015 and Module 2 began in Aug. 2016. The modules were
installed with 35 PPC detectors (29.7 kg) and 23 BEGe detectors (14.4 kg), of
which up to 22.1 kg of enriched detectors and 10.0 kg of natural detectors
were operational. The most common reason for non-operating detectors was
failure in the high voltage and signal electronics cable connectors. In Nov.
2019, Module 2 was removed from the shield and upgraded with improved cables,
connectors and shielding; as a result, all Module 2 detectors were operational
during subsequent run time. Five PPC detectors (5.5 kg) were removed for early
testing in the LEGEND-200 experiment [29] and replaced with four ICPC
detectors (6.7 kg). After this, up to 27.2 kg of enriched detectors and 13.2
kg of natural detectors were operational.

Detector signals were digitized using ADCs developed for the GRETINA
Experiment [30], with 14 bit resolution and a sampling rate of 100 MS/s [31].
Each detector was read out with two gains, a high gain channel with dynamic
range up to $\sim$3 MeV, and a low gain channel extending up to $\sim$10 MeV.
The high gain channels were used for most of the analysis; use of low gain
channels is reserved for events with no high gain waveform available, due to
dead time after a noise trigger or rejection of the high gain channel, and for
high energy events [32, 33]. An internal trapezoidal filter was used to
trigger each ADC channel independently, recording samples in a
$20.2-38.2~{}\mu$s acquisition window to disk. A pulser signal was fed to the
front-end electronics every 8 s to monitor the electronics stability and
detector livetime.

During blinded operation, 75% of physics runs were blinded by restricting
access to data files, with cycles of 31 h of open data followed by 93 h of
blind data. Typical trigger rates during physics runs for the full array were
around 70 Hz, dominated by near-threshold and pulser events. Physics run data
are divided into 13 datasets based on changes to the experimental
configuration, described in the Supplemental Material [34]. Once per week, the
228Th line sources were deployed for 60–90 min, with trigger rates of up to
3000 Hz. Every 2–3 months a long calibration run with $\gtrsim$18 h of data
was collected for each module. In Jan. 2019, a 56Co line source was inserted
into each track, and calibration data were collected for one week with each
module. This dataset has multiple double-escape peaks (DEPs) and single-escape
peaks (SEPs) in an energy range bracketing $Q_{\beta\beta}$ between 1500 keV
and 2500 keV. These act as proxies for $0\nu\beta\beta$ events and were used
for systematics measurements.

The digitized waveforms are used to calculate event energies and pulse-shape
discrimination (PSD) parameters that are used to reject likely backgrounds. In
this analysis, we use a set of improved analysis routines compared to the
analyses described in Refs. [35, 10], and we will note these changes. In
addition, the PPC and ICPC detectors utilize subtly different digital signal
processing algorithms; we describe the PPC parameters first, followed by the
ICPC parameters.

| PPCs | ICPCs
---|---|---
Total Exposure |  $67.94$ kg yr |  $3.12$ kg yr
Active Exposure |  $61.64^{+0.89}_{-1.17}$ kg yr |  $2.82^{+0.04}_{-0.05}$ kg yr
FWHM@2039 keV |  $2.52\pm 0.08$ keV |  $2.55\pm 0.09$ keV
76Ge Enrichment | $87.4\pm 0.5\%$ | $88.0\pm 1.0\%$
PSD Eff. | $86.1\pm 3.9\%$ | $81.0^{+5.3}_{-7.3}\%$
Data Cleaning | $99.9\pm 0.1\%$ | $99.9\pm 0.1\%$
Low AvsE* Cut | $89.9^{+3.3}_{-3.2}\%$ | $85.2^{+4.2}_{-5.9}\%$
DCR Cut | $98.5\pm 0.7\%$ | $97.9\pm 1.1\%$
High AvsE* Cut | $97.9\pm 1.0\%$ | $97.8\pm 1.4\%$
Late Charge Cut | $99.3\pm 0.7\%$ | $99.5^{+0.5}_{-0.9}\%$
Containment Eff. | $90.8\pm 1.3\%$ | $91.9\pm 0.8\%$
ROI Peak Eff. | $86.3\pm 1.1\%$ | $86.9\pm 1.2\%$
∗$A/E$ in the case of ICPCs |  |
Table 1:  A summary of key analysis parameters. The FWHM and efficiency values
represent an exposure-weighted average across datasets. The PSD efficiencies
are calculated for cuts applied in the order listed, and are applied
multiplicatively. The ROI peak efficiency uses the optimal ROI [36] defined
for the Feldman-Cousins limit.

Event energies are reconstructed from waveform amplitudes which are measured
using a pole-zero (PZ) corrected trapezoidal filter. Before applying any
digital filters, waveforms are corrected for digitizer non-linearity [37]. The
energy is measured from the amplitude of the filtered waveform at a fixed time
after the start time ($t_{0}$) of the signal. By optimizing the PZ time
constant for energy resolution, we minimize energy degradation due to charge
trapping inside of the detectors [38]. When analyzing energy calibrations, we
model gamma peak shapes using the sum of a gaussian component and an
exponentially-modified gaussian low-energy tail [10]. Peak shape parameters
are determined using a simultaneous fit of the eight most prominent gamma-rays
between 238–2615 keV. Energy gains are calculated for each weekly calibration
using the 2615 keV peak from the 228Th spectra and fixing the zero-ADC energy
at 0 keV. For each full dataset, a fine-tuned correction to the energy is
obtained by combining all calibration runs and performing a new simultaneous
peak-fit, with peak positions fit to a quadratic function of energy. The
exposure-weighted average FWHM at 2039 keV across all PPC detectors, including
broadening due to gain drift and energy nonlinearities, is $2.52\pm 0.08$ keV.
Since previous data releases [35, 10], the leading-edge fitter for finding
$t_{0}$ was improved to use an asymmetric trapezoidal filter that preserves
information in the rising edge, and $t_{0}$ was corrected for energy-dependent
trigger walk; in addition, the quadratic energy correction is new to this
analysis. These changes correct nonlinearities in the energy response,
especially at energies $<$100 keV, and reduce the low-energy tail.

$0\nu\beta\beta$ events predominantly occur in the bulk region of enriched
detectors, with the charge cloud localized within $\sim$1 mm of the decay
site. Background rejection cuts are applied to remove events with multi-site
topologies, events on the surfaces of detectors, and non-physical events.
Detector triggers within a rolling 4 $\mu$s window are grouped into events,
and events with a detector multiplicity $>1$ are rejected. The muon veto
system triggers when a PMT signal above threshold is measured in two or more
veto panels simultaneously, and the pulse amplitude of all panels are
recorded. A separate analysis identifies muon candidate events when at least
two muon veto panels on different surfaces surpass an analysis threshold,
while rejecting events triggered by LED pulsers used to monitor the panels.
Any events within 20 ms before and 1 s after a muon candidate are tagged and
rejected [20, 40]. All events for a module are rejected during periods of
increased microphonic noise while LN dewars were refilled. Multiple data
cleaning PSD cuts are performed, accepting >99.9% of signal-like events, to
reject non-physical waveforms, pileup waveforms, and pulser events.

The point-contact detector geometries have relatively fast charge collection
and a highly localized weighting potential, allowing for the identification
and rejection of events that deposit energy in multiple sites within a
detector, such as Compton-scattered gammas. The parameter $AvsE$ represents a
comparison between the maximum amplitude (A) of a waveform current pulse and
the total energy (E). Multi-site events have lower A for a given E than
single-site events [41]. $AvsE$ is tuned with long 228Th calibration runs to
accept 90% of events in the predominantly-single-site 208Tl DEP at 1592 keV.
Compared to previous data releases [35, 10], $AvsE$ has undergone several
upgrades. First, the 56Co calibration spectrum is used to measure and correct
the width-energy dependence of the $AvsE$ distribution. Second, we linearly
correct for the correlation of $AvsE$ with the drift time, measured using the
$0-90\%$ rise time ($\Delta t_{0-90}$). The effect of these improvements is an
improved signal acceptance and a more accurate determination of the single-
site acceptance at $Q_{\beta\beta}$; based on 56Co studies performed after
publication of Refs. [35, 10], the acceptance of the $AvsE$ cut in those
analyses was $\sim$84%.

Surface events very close to the point contact produce faster-rising waveforms
than bulk events [42]. A cut value at high values of $AvsE$ is selected to
accept $98\%$ of bulk events near $Q_{\beta\beta}$ in the 228Th Compton-
continuum from calibration data. This background cut was not applied in Refs.
[35, 10], and relies on the improvements made to the $AvsE$ parameter.

Figure 1:  The measured energy spectrum above 100 keV for the full enriched
exposure after applying multiplicity and data cleaning cuts (dark gray), DCR,
high-$AvsE$ and LQ cuts (light gray), and the low-$AvsE$ cut (red).

Alpha particles impinging on the passivated surface experience increased
charge trapping and surface charge-collection effects that degrade their
reconstructed energy, becoming an important source of background. Much of this
charge is collected slowly, increasing the slope of the falling tail of the
waveforms relative to bulk events [42]. Delayed charge recovery (DCR) is a
measure of this slope increase, and can be used for surface alpha rejection.
The DCR cut is tuned using Compton-continuum events in 228Th calibration data
to accept $99\%$ of bulk events, assuming it is normally distributed, in the
energy range of $2028-2050$ keV bracketing $Q_{\beta\beta}$. Since previous
data releases [35, 10], the DCR parameter has been refined to improve the
stability of the parameter with energy and time, and a correction for linear
correlations with drift time has been added.

Finally, events with a partial charge deposition in the transition layer
between the n-type surfaces and the detector bulk experience energy
degradation, and have waveforms with a slow-rise component. We calculated the
Late Charge (LQ) parameter, which is the integral of uncollected charge after
a waveform has reached 80% of its maximum value. The slow-rise component of
these waveforms increases LQ, and we cut events that fall $5\sigma$ above the
center of the parameter distribution. LQ is tuned using the 1592-keV DEP in
228Th calibration data, and is corrected for linear correlations with the
drift time. The parameter acceptance is measured to be $99.3\%$ using this
same DEP, corrected for energy dependence using multiple DEPs in 56Co data.

Because the installed ICPC detectors are 2–3 times larger than PPC detectors,
they experience greater charge trapping and diffusion effects. To achieve
similar energy and PSD performance to the PPC detectors, further improvements
to the algorithms are implemented. In place of $\Delta t_{0-90}$, we estimate
drift time using the effective mean rise time, calculated as the integral of
uncollected charge from $t_{0}$ until $1.5~{}\mu$s later. Energy is calculated
using a PZ correction and trapezoidal filter; these filters, however, are not
optimized for charge trapping as in the PPC detectors. Instead, we correct for
charge trapping by correcting the energy for correlations with drift time that
are either quadratic or linear depending on which achieved better resolution.
The ICPC detectors have an exposure-weighted average energy resolution of
$2.55\pm 0.09$ keV FWHM. Instead of $AvsE$, we calculate the ratio between $A$
and $E$ ($A/E$) [43], and linear corrections for both drift time and energy
are applied to this ratio. We cut on both low and high values of $A/E$, for
multi-site and near-point-contact events, respectively. As a ratio, the width
of $A/E$ increases rapidly at low energies and is only well understood for
hits above 1 MeV, in contrast with $AvsE$; however, the improved drift time
correction performs excellently near $Q_{\beta\beta}$, with only $4.76\pm
0.50\%$ of events in the inherently-multi-site 208Tl SEP at 2103 keV
surviving. The DCR and LQ parameters are calculated similarly for ICPC and PPC
detectors, applying corrections for linear correlations with effective mean
rise time, rather than $\Delta t_{0-90}$. The DCR parameter is additionally
corrected for residual ADC nonlinearity which produces an energy dependent
oscillation in the mean of the distribution.

The signal-efficiency for each PSD cut is measured using weekly calibrations,
and averaged together for each full dataset. Because of the differences
between PPC and ICPC detectors, signal-efficiencies are calculated separately
for each using identical procedures to measure these values, except where
noted. If a detector channel exhibits large instabilities between two
calibrations, we reject its events for that period. Uncertainty estimates
account for statistics and systematic effects due to variations in cut
parameters between weekly calibrations, energy dependencies of the cuts, and
simulated differences between the waveform populations used to measure the
signal-efficiency and $0\nu\beta\beta$ events. The only difference applies to
the low $A/E$ cut in ICPC detectors: no 56Co data is available for the ICPC
detectors and thus no energy-width correction is applied to the parameter;
instead, we reduce the signal acceptance from 90.2% to 85.2%, based on the
estimated shift in signal-acceptance for the uncorrected parameter in PPC
data. The exposure-weighted averages of signal-efficiencies across all
datasets are listed in Tab. 1.

The total exposure for PPC and ICPC detectors of 71.1 kg yr is calculated by
multiplying the full detector masses and the full livetime for each detector,
after removing periods for which data from a detector are rejected. We compute
the active exposure by subtracting sources of dead time, including 0.77% loss
from periods of increased microphonic noise during LN dewar fills, 0.04% loss
from the period of each muon veto cut, and 0.51% loss from retriggering dead
time. Detector volume with incomplete charge collection within $\sim$1 mm of
detector surfaces is also subtracted from active exposure. The fraction of
active volume in PPC detectors is $92.0^{+1.3}_{-1.7}\%$, while the fraction
in ICPC detectors is $90.9^{+1.2}_{-1.6}\%$. The upper uncertainty estimates
account for measurement uncertainty, and the lower uncertainty estimates also
incorporate possible growth in dead layers while detectors are warm, which is
being further investigated. The total active exposure for PPC and ICPC
detectors is $64.5^{+0.9}_{-1.2}$ kg yr. The active exposure from the datasets
reported in Ref. [10] increased from 26.0 kg yr to 27.2 kg yr in PPC detectors
for the current analysis, thanks to added livetime due to an improved
understanding of the stability of PSD parameters and the inclusion of data
that were previously rejected by automated procedures for having high noise
but manually verified to have a near-threshold rate comparable to accepted
runs.

Unblinding of data proceeded in a staged fashion with basic data quality
assurance checks performed at each stage. No changes to analysis parameters or
run selection were made after opening the energy window 1950–2350 keV which
contains $Q_{\beta\beta}$. Fig. 1 shows the energy spectrum, with the effect
of cuts visible. The spectrum is dominated by $2\nu\beta\beta$ below
$Q_{\beta\beta}$, and near $Q_{\beta\beta}$ most events are removed by the
various cuts. A background index is calculated using counts that pass all cuts
within a 360-keV background estimation window, excluding $\pm 5$ keV around
the 2039 keV $Q_{\beta\beta}$-value and expected background gamma-rays at 2103
keV, 2118 keV and 2204 keV. The background rate within this 360-keV window is
assumed to be flat. The surface cuts (DCR, high-$AvsE$ and LQ) remove 85% of
events in the background region. The multi-site cut (low-$AvsE$) removes 49%
of the remaining events. The majority of surface events are removed by more
than one of the surface cuts; still, each uniquely cuts a significant
population of events. A parallel analysis using an interpretable boosted
decision tree to leverage multi-variate correlations between these parameters
achieves a similar result, with the potential for future background rejection
[44].

Figure 2:  The measured energy spectrum for the full enriched exposure in the
range of 1950-2350 keV, after applying all background cuts. The 360-keV
background estimation window excludes the shaded 10 keV windows around
expected gamma lines in gray, and the 10 keV window around the 2039 keV
$Q_{\beta\beta}$ in blue. Lines are drawn representing the background index
(green) and a peak at the 90% C.L. limit (blue).

After cuts, 153 events remain in the background estimation window, resulting
in a background index of $16.6^{+1.4}_{-1.3}\times 10^{-3}$ cts/(FWHM kg yr).
We also define a low background dataset, excluding the first dataset taken
with Module 1 from June 2015 through Oct. 2015 prior to the installation of
the inner Cu shield. This low background dataset has an active exposure of
63.3 kg yr in PPC and ICPC detectors, and 142-events in the background
estimation window, resulting in a background index of
$15.7^{+1.4}_{-1.3}\times 10^{-3}$ cts/(FWHM kg yr). We measure a significant
difference in the background index between Module 1 ($18.1^{+1.8}_{-1.7}\times
10^{-3}$ cts/(FWHM kg yr)) and Module 2 ($8.7^{+2.0}_{-1.7}\times 10^{-3}$
cts/(FWHM kg yr)). The Majorana Demonstrator measures a higher background
index than predicted by the initial assay-based projections of ${<}2.5\times
10^{-3}$ counts/(FWHM kg yr) [23]. Current evidence suggests that the main
source of this excess is located far from the HPGe detectors, most likely near
the interface of the Module 1 crossarm and cryostat. To better identify and
measure this source, we are further analyzing possible background sources in a
detailed model of the apparatus using the Geant4 [45] based package MaGe [46].
Upon unblinding the 10 keV window centered on 2039 keV, 4 events were
observed, consistent with the background expectation. The events in the
$1950-2350$ keV window are shown in Fig. 2.

A lower half-life limit for $0\nu\beta\beta$ is calculated using

$T_{1/2}>\mathrm{ln}(2)\frac{NT\epsilon_{tot}}{S}$ (1)

where $NT$ is the number of 76Ge nuclei in the active mass,
$\epsilon_{tot}=0.78$ is the signal detection efficiency, and $S$ is the upper
limit on the signal counts based on the observed data. A limit of
$T_{1/2}>8.3\times 10^{25}$ yr is derived using an unbinned, extended profile
likelihood method implemented in RooStats [47, 48]; this is the same technique
used for the limits in Refs. [35, 10]. Systematic uncertainties are included
for the detection efficiency, exposure and peak shape, as listed in Tab. 1.
Because this limit is statistics dominated, these uncertainties have a very
small effect on the result. For this method, we use a 370-keV-wide fitting
window, which includes the 360-keV background estimation window and the 10 keV
window around 2039 keV. A 90% C.L. median sensitivity for exclusion of
$T_{1/2}>8.1\times 10^{25}$ yr is also derived using this construction.

Several additional statistical techniques were examined. The Feldman-Cousins
technique [49] was implemented using a 3.8-keV-wide optimal region of interest
(ROI) at $Q_{\beta\beta}$ [36], with peak detection efficiency listed in Tab.
1. This ROI contains 1 count, with an expectation of 1.52 background counts,
producing a limit of $T_{1/2}>7.2\times 10^{25}$ yr. Two Bayesian analyses
were implemented using Markov Chain Monte Carlo simulations performed with
RooStats using the same likelihood function as for the frequentist result.
Using a flat prior on $1/T_{1/2}$ we calculate a half-life limit of $7.0\times
10^{25}$ yr at 90% C.I. Using the Jeffreys prior, flat on $1/\sqrt{T_{1/2}}$,
we calculate a limit of $1.1\times 10^{26}$ yr at 90% C.I. A modified profile
likelihood analysis using the more conservative CLS method [50] to mitigate
the effect of background down-fluctuations yields a limit of $6.4\times
10^{25}$ yr at 90% C.L. The data required to perform a statistical analysis of
the Majorana Demonstrator, including a full list of analysis parameters and
events 370-keV fitting window, can be found in the Supplemental Materials
[34].

To place limits on $m_{\beta\beta}$ assuming light neutrino exchange, we apply
matrix element ($M_{0\nu}$) and phase space integral ($G_{0\nu}$)
calculations. We use a range of $M_{0\nu}$ values of $2.66-6.34$ [51, 52, 53,
54, 55, 56, 57, 58, 59, 60, 61, 62], phase space factors ($G_{0\nu}$) of
$2.36\times 10^{-15}$/yr [63] or $2.37\times 10^{-15}$/yr [64], and an
effective axial weak coupling of $g_{A}^{eff}=1.27$ for free nucleons [8].
Applying these to the limit of $T_{1/2}>8.3\times 10^{25}$, we calculate upper
limits on $m_{\beta\beta}$ in the range $(113-269)$ meV. Fig. 3 shows this
limit, along with allowed values of $m_{\beta\beta}$. These limits are subject
to theoretical uncertainties on $M_{0\nu}$ and $g_{A}^{eff}$ [8].

Figure 3:  Allowed values of $m_{\beta\beta}$ for varying masses of the
lightest neutrino eigenstate in the normal and inverted mass orderings, using
the best-fit values of neutrino oscillation parameters [65]. Upper limits on
$m_{\beta\beta}$ at 90% C.I. are calculated using half-life limits for the
Majorana Demonstrator (this work), GERDA [11], KamLAND-Zen [12], EXO-200 [13],
and CUORE [14], using nuclear matrix elements calculated using the Quasi-
Random Phase Approximation (QRPA) [54, 55, 56, 57, 66], Shell Model (SM) [51,
52, 53], Interacting Boson Model (IBM) [61, 62], and Energy Density Functional
(EDF) theory [58, 59, 60]. The light gray shaded region shows the range of 90%
C.I. upper limits reported in this work, and the dark gray shows values of
$m_{\beta\beta}$ excluded for all nuclear matrix elements.

The Majorana Demonstrator has completed measurements with its enriched
detectors. Among $0\nu\beta\beta$ searches, the Demonstrator achieved the best
energy resolution of $\sim$2.5 keV at $Q_{\beta\beta}$, including for the ICPC
detectors that will be predominantly used in LEGEND [67]. For this result, we
have also demonstrated several new experimental techniques that will play an
important role in future efforts, including the use of 56Co to correct energy
dependent systematics in PSD cuts and the introduction of a novel PSD cut that
is sensitive to events with charge deposition in n-type surfaces. Through
advances in ultra-low background materials, the Demonstrator also achieved the
second lowest background index normalized for energy resolution; only GERDA
achieved lower backgrounds by immersing HPGe detectors in an active liquid
argon shield [11]. The techniques used by both experiments to achieve low
backgrounds are complementary and will be combined by LEGEND to achieve lower
background rates than either experiment individually. This has enabled
Majorana to achieve limits comparable to other $0\nu\beta\beta$ experiments,
even with a comparatively small exposure. Since both the Majorana Demonstrator
and GERDA achieved quasi-background free spectra near $Q_{\beta\beta}$, a
combined analysis would measure a limit near the sum of the individual limits.

The authors appreciate the technical assistance of S. Adair, J.F. Amsbaugh, J.
Bell, T.H. Burritt, G. Capps, K. Carney, J. Cox, R. Daniels, L. DeBraeckeleer,
C. Dunagan, G.C. Harper, C. Havener, G. Holman, R. Hughes, K. Jeskie, K.
Lagergren, D. Lee, O. Loken, M. Middlebook, A. Montoya, A.W. Myers, D.
Peterson, D. Reid, L. Rodriguez, H. Salazar, A.R. Smith, G. Swift, J.
Thompson, P. Thompson, M. Turqueti, C. Tysor, T.D. Van Wechel, R. Varland, T.
Williams, R. Witharm, and H. Yaver. This material is based upon work supported
by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics
under contract / award numbers DE-AC02-05CH11231, DE-AC05-00OR22725, DE-
AC05-76RL0130, DE-FG02-97ER41020, DE-FG02-97ER41033, DE-FG02-97ER41041, DE-
SC0012612, DE-SC0014445, DE-SC0018060, DE-SC0022339, and LANLEM77/LANLEM78. We
acknowledge support from the Particle Astrophysics Program and Nuclear Physics
Program of the National Science Foundation through grant numbers MRI-0923142,
PHY-1003399, PHY-1102292, PHY-1206314, PHY-1614611, PHY-1812409, PHY-1812356,
PHY-2111140, and PHY-2209530. We gratefully acknowledge the support of the
Laboratory Directed Research & Development (LDRD) program at Lawrence Berkeley
National Laboratory for this work. We gratefully acknowledge the support of
the U.S. Department of Energy through the Los Alamos National Laboratory LDRD
Program and through the Pacific Northwest National Laboratory LDRD Program for
this work. We gratefully acknowledge the support of the South Dakota Board of
Regents Competitive Research Grant. We acknowledge the support of the Natural
Sciences and Engineering Research Council of Canada, funding reference number
SAPIN-2017-00023, and from the Canada Foundation for Innovation John R. Evans
Leaders Fund. This research used resources provided by the Oak Ridge
Leadership Computing Facility at Oak Ridge National Laboratory and by the
National Energy Research Scientific Computing Center at Lawrence Berkeley
National Laboratory, a U.S. Department of Energy Office of Science User
Facility. We thank our hosts and colleagues at the Sanford Underground
Research Facility for their support.

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## Appendices

##  Appendix A Base Interface

Figure 15. The baseline (traditional explainable AI) interface in our user
study. The interface contains three parts. The top (A) is the applicant’s
profile. The bottom left part (B) shows AI’s feature contribution explanation.
The bottom right part (C) is for humans to (1) indicate their initial
predictions, (2) see AI’s suggestions, and (3) indicate their final decisions.
Note that AI’s suggestions and explanations are only shown after humans make
their initial predictions. (All the dashed lines are only for illustration)