###### Abstract

In the global landscape of neutrinoless double beta ($0\nu\beta\beta$) decay
search, the use of semiconductor germanium detectors provides many advantages.
The excellent energy resolution, the negligible intrinsic radioactive
contamination, the possibility of enriching the crystals up to 88% in the 76Ge
isotope as well as the high detection efficiency, are all key ingredients for
highly sensitive $0\nu\beta\beta$ decay search. The Majorana and Gerda
experiments successfully implemented the use of germanium (Ge) semiconductor
detectors, reaching an energy resolution of 2.53 $\pm$ 0.08 keV at the Qββ and
an unprecedented low background level of $5.2\times 10^{-4}$
cts/(keV$\cdot$kg$\cdot$yr), respectively. In this paper, we will review the
path of $0\nu\beta\beta$ decay search with Ge detectors from the original idea
of E. Fiorini et al. in 1967, to the final recent results of the Gerda
experiment setting a limit on the half-life of 76Ge $0\nu\beta\beta$ decay at
$T_{1/2}>1.8\times 10^{26}$ yr (90% C.L.). We will then present the LEGEND
project designed to reach a sensitivity to the half-life up to $10^{28}$ yr
and beyond, opening the way to the exploration of the normal ordering region.

###### keywords:

germanium detector; neutrino; neutrinoless double beta decay

1 7 341 Academic Editors: Fabio Bellini and Claudia Tomei 15 June 2021 02
August 2021 10 September 2021 https://doi.org/10.3390/universe7090341
Neutrinoless Double Beta Decay with Germanium Detectors: 1026 yr and Beyond
Neutrinoless Double Beta Decay with Germanium Detectors: 1026 yr and Beyond
Valerio D’Andrea 1,2, Natalia Di Marco 2,3,*, Matthias Bernhard Junker 2,
Matthias Laubenstein 2, Carla Macolino 1,2, Michele Morella 2,3, Francesco
Salamida 1,2 and Chiara Vignoli 2 Valerio D’Andrea, Natalia Di Marco, Matthias
Bernhard Junker, Matthias Laubenstein, Carla Macolino, Michele Morella,
Francesco Salamida and Chiara Vignoli D’Andrea, V.; Di Marco, N.; Junker,
M.B.; Laubenstein, M.; Macolino, C.; Morella, M.; Salamida, F.; Vignoli, C.
Correspondence<EMAIL_ADDRESS>
##  1 Introduction

The evidence for non-zero neutrino masses as a consequence of the neutrino
oscillation discovery Fukuda et al. (1998); Eguchi et al. (2003); McDonald
(2016); Kajita (2016) provides, among others, a hint of physics beyond the
Standard Model (SM). Despite many experimental efforts carried out about
neutrino physics since the first pioneering experiment by Reines ad Cowan in
1956 Cowan et al. (1956), there are still open points to be clarified such as
the neutrino nature, the mass ordering and the absolute mass scale.

The search for neutrinoless double beta ($0\nu\beta\beta$) decay is considered
as the most promising way to prove the Majorana nature of neutrinos as well as
to give an indication on the mass hierarchy and on the absolute mass scale.
Moreover, the discovery of $0\nu\beta\beta$ decay would open the way for
theories predicting the observed matter anti-matter asymmetry of the Universe
being a consequence of lepton number violation through leptogenesis.

The important implications for particle physics and cosmology justify the
experimental efforts carried out worldwide in the field of $0\nu\beta\beta$
decay searches. A number of different experiments exploiting various
technologies and searching for the transition in different isotopes exists:
76Ge Agostini et al. (2019); Alvis et al. (2019), 82Se Azzolini et al. (2019),
100Mo Alenkov et al. (2019); Arnold et al. (2015); Armengaud et al. (2020),
130Te Adams et al. (2020); Andringa et al. (2016), 136Xe Anton et al. (2019);
Gando et al. (2016); Martín-Albo et al. (2016) and others.

In this paper, we will review the story of the $0\nu\beta\beta$ decay search
of 76Ge with germanium semiconductor detectors. The paper is organized as
follows. In the second section, a review of the main aspects of the
$0\nu\beta\beta$ theory will be presented. In the third section, we will
discuss the choice of the 76Ge isotope, highlighting the main advantages and
disadvantages with respect to other techniques and isotopes. In the fourth
section, we will report about the historical path of $0\nu\beta\beta$ decay
search with germanium detectors from the original idea proposed by E. Fiorini
et al. in 1967 Fiorini et al. (1967) to the experiments inaugurating the
modern era of highly sensitive, low background searches. In the fifth and
sixth section, we will summarize the main characteristics, performance and
results of the contemporary Gerda and Majorana experiments currently leading
the field with the best sensitivity, the lowest background level and the best
resolution among all the other $0\nu\beta\beta$ decay experiments. Finally, in
the seventh section, we will present the LEGEND project conceived to extend
the sensitivity up to $10^{28}$ yr to fully cover the inverted hierarchy
region.

##  2 Neutrinoless Double Beta Decay

The double beta decay ($2\nu\beta\beta$), first proposed by M. Goeppert-Mayer
in 1935 Goeppert-Mayer (1935), is a second-order process generated in the
perturbative expansion of weak interactions in the SM with an extremely long
lifetime corresponding to a transition from a nucleus $(A,Z)$ to $(A,Z+2)$.
Candidate isotopes that can decay through the double beta decay are even–even
nuclei and lighter than the odd–odd $(A,Z+1)$ nucleus, for which the single
$\beta$ decay is forbidden or strongly suppressed because of a large change of
spin. In $2\nu\beta\beta$ decay, starting from the initial nuclear state, one
nucleon type (proton or neutron) decays into another one emitting a
lepton–anti-lepton pair through a virtual transition, and a second decay
occurs producing a further pair of lepton and anti-lepton. The SM, therefore,
predicts the emission in the final state of two electrons and two anti-
neutrinos:

$(Z,A)\rightarrow(Z+2,A)+2e+2\bar{\nu}+Q_{\beta\beta}$

where $Q_{\beta\beta}$ is the energy released in the decay. The Feynman
diagram of this process is shown in the left panel of Figure 1. Among the 35
naturally occurring isotopes that can decay through $2\nu\beta\beta$, Table 2
reports the results for the measurement of a few isotopes together with the
main characteristics of the process.

The $0\nu\beta\beta$ decay process, proposed by W.H. Furry in 1939 Furry
(1939), is a SM-forbidden transition in which the final state consists of only
two electrons:

$(Z,A)\rightarrow(Z+2,A)+2e+Q_{\beta\beta}$

This decay clearly violates the leptonic number conservation ($\Delta L=2$).
The relevant diagram, due to the exchange of light Majorana neutrino, is shown
in Figure 1 on the right panel. There are other possible mediators of this
decay, e.g., right-handed weak currents, supersymmetric particles and massive
neutrinos. Regardless of the underlying mechanisms, the observation of the
$0\nu\beta\beta$ decay would prove that the lepton number conservation is not
an exact symmetry of nature and that neutrinos have a Majorana mass component.

|
---|---
Figure 1: On the left, the diagram of the standard $2\nu\beta\beta$ decay with
the emission of 2 anti-neutrinos. On the right, the $0\nu\beta\beta$ process
due to the exchange of a light Majorana neutrino with lepton number violation.

[H] A list of isotopes mostly used in $2\nu\beta\beta$ decay experiments. Some
of the key features are reported. Half-life values are from Barabash (2020).
Isotope Natural Qββ T${}^{2\nu}_{1/2}$ abundance ($\%$) (MeV) (yr) ${}^{48}Ca$
0.187 4.263 $5.3^{+1.2}_{-0.8}\times 10^{19}$ ${}^{76}Ge$ 7.8 2.039 $(1.88\pm
0.08)\times 10^{21}$ ${}^{82}Se$ 9.2 2.998 $0.87^{+0.02}_{-0.01}\times
10^{20}$ ${}^{96}Zr$ 2.8 3.348 $(2.3\pm 0.2)\times 10^{19}$ ${}^{100}Mo$ 9.6.
3.035 $7.06^{+0.15}_{-0.13}\times 10^{18}$ ${}^{116}Cd$ 7.6 2.813 $(2.69\pm
0.09)\times 10^{19}$ ${}^{130}Te$ 34.08 2.527 $(7.91\pm 0.21)\times 10^{20}$
${}^{136}Xe$ 8.9 2.459 $(2.18\pm 0.05)\times 10^{21}$ ${}^{150}Nd$ 5.6 3.371
$(9.34\pm 0.65)\times 10^{18}$

In the low energy limit, the interaction of neutrinos can be described by the
current–current four-fermion interactions. In this approximation, the half-
life of the $0\nu\beta\beta$ decays can be derived as described in Doi et al.
(1981):

$\left(T_{1/2}^{0\nu}\right)^{-1}={G_{0\nu}}\times{\left|M_{0\nu}\right|^{2}}\times{\left(\frac{m_{\beta\beta}}{m_{e}}\right)^{2}}$
(1)

where $G_{0\nu}$ is the phase space factor (PSF), $M_{0\nu}$ is the nuclear
matrix element (NME) and $m_{\beta\beta}$ is the effective Majorana mass
defined by:

$m_{\beta\beta}=\sum_{i=1}^{3}U_{ei}^{2}m_{i}$ (2)

where $U$ is the PNMS mixing matrix and $m_{i}$ are the neutrino mass
eigenvalues. It follows that the decay rate is proportional to
$m^{2}_{\beta\beta}$ and so the neutrino mass can be estimated from the
measurement of the half-life of the $0\nu\beta\beta$ decay.

The main source of uncertainty in the expected value of the $0\nu\beta\beta$
decay half-life originates from the NME calculations. They involve the
hadronic current of the weak Hamiltonian and depend on the nuclear-structure
of the parent and daughter nuclei. They are usually difficult to calculate
even in the case of single $\beta$ decay. Different models are available to
calculate the NMEs, e.g., the Interactive Shell Model (ISM) Menéndez et al.
(2009), the Quasi-particle Random Phase Approximation (QRPA) Šimkovic et al.
(2013) and the Interactive Boson Model (IBM-2) Barea et al. (2015). A
comparison between the values of the NMEs calculated with the different models
is illustrated in Figure 2. Despite the fact that the uncertainty on the
single calculation is of the order of 20%, the differences between the
different models are even larger. The overall estimate of the NME calculation
is therefore affected by a large systematic uncertainty.

Figure 2: Comparison of NME calculations with an unquenched $g_{A}=1.27$ for
different isotopes and with various models. Abbreviations: EDF, energy-density
functional; IBM, interacting boson model; NME, nuclear matrix element; QRPA,
quasi-particle random-phase approximation; SM, Standard Model. The reader
should refer to Dolinski et al. (2019) for further details.

Parametrizing the NME as in Šimkovic et al. (1999), we can write:

$M_{0\nu}=g^{2}_{A}\left(M^{0\nu}_{GT}-\left(\frac{g_{V}}{g_{A}}\right)^{2}M_{F}^{0\nu}+M_{T}^{0\nu}\right)$
(3)

where $g_{V}$ and $g_{A}$ are the vector and axial coupling constants of the
nucleon, $M_{GT}^{0\nu}$ is the Gamow–Teller operator matrix element between
the initial and the final states, $M_{F}^{0\nu}$ is the Fermi contribution and
$M_{T}^{0\nu}$ is the tensor operator matrix element. Equation (3) emphasizes
the role of $g_{A}$ being $T_{1/2}^{0\nu}$ proportional to $g_{A}^{-4}$. The
value of $g_{A}$ can be used as an adjustment to reconcile observations with
calculations. In the hypothesis of quenching, a reduction of $g_{A}$ is
assumed to reproduce the observable quantities of $\beta$ and $2\nu\beta\beta$
decays Deppisch et al. (2018). There are different possible origins for
quenching, e.g., many-body currents, intrinsic shortcomings of the nuclear
many-body models, or nuclear medium effects Dolinski et al. (2019). Recent
studies point towards quenching not being larger than 20–30% in processes with
high momentum transfer such as the $0\nu\beta\beta$ decay. This reduction
would result in an increase of $T_{1/2}^{0\nu}$ by a factor of two or three.

##  3 The Choice of the 76Ge Isotope

From the experimental point of view, the search for $0\nu\beta\beta$ decay
consists of the detection of the two emitted electrons. As the recoil of the
nucleus negligible, the sum of energies of the two electrons corresponds to
the Q-value of the process. The signature of the $0\nu\beta\beta$ decay is
therefore a mono-energetic peak centered at $Q_{\beta\beta}$ (see Figure 3).

Figure 3: Schematic view of the expected $0\nu\beta\beta$ and $2\nu\beta\beta$
decay spectra. Credits to Gerda Collaboration.

Since the value of $Q_{\beta\beta}$ is well known, the search for a
$0\nu\beta\beta$ decay signal can be performed in a very narrow region of
interest (ROI). The number of signal events expected in the ROI is given by:

$N^{0\nu}=\text{ln(2)}~{}\frac{N_{A}}{m_{A}}\left(\frac{M~{}T~{}\varepsilon~{}a}{T_{1/2}^{0\nu}}\right)$
(4)

where $N_{A}$ is the Avogadro constant, $m_{A}$ and $a$ are the molar mass and
the isotopic abundance of the double beta ($\beta\beta$) emitter,
respectively, $\varepsilon$ is the detection efficiency, $M$ is the detector
mass and $T$ is the measurement time of the experiment. On the other hand, the
number of background events in the ROI is given by:

$N_{b}=\text{BI}~{}M~{}T~{}\Delta E$ (5)

where $\Delta E$ is the ROI width, proportional to the energy resolution of
the detector, and BI is the background index, normalized to the width of the
ROI, mass, energy and time and expressed in units of keV-1kg-1year-1. The
sensitivity of the experiment can be obtained requiring that the
$0\nu\beta\beta$ decay signal exceeds the standard deviation of the total
detector counts $N^{0\nu}\geq n_{\sigma}~{}\sqrt{N^{0\nu}+N_{b}}$ where
$n_{\sigma}$ is the confidence level expressed in units of $\sigma$s of the
Poisson distribution. Combining Equations (4) and (5), the sensitivity can be
derived as:

$S^{0\nu}=\text{ln(2)}~{}\frac{N_{A}}{m_{A}}\left(\frac{\varepsilon~{}a}{n_{\sigma}}\right)\sqrt{\frac{M~{}T}{\text{BI}~{}\Delta
E}}$ (6)

This formula clearly shows the role of the different experimental parameters
and highlights the different characteristics that have to be chosen to improve
the $0\nu\beta\beta$ decay discovery probability. In particular, the ideal
experiment should have an isotope with high isotopic abundance $a$, a high
detection efficiency $\varepsilon$, the possibility to deploy a large amount
of mass, a low background and a good energy resolution.

Among $\beta\beta$-emitters, 130Te has the highest isotopic abundance and a
tellurium-based experiment does not necessarily require enrichment. Tellurium
dioxide crystals (natTeO2), as the ones employed in CUORE Adams et al. (2020),
are used as cryogenic calorimeters detecting the small temperature rise
induced by the energy deposition of a charged particle. As in all experiments
where the source of $0\nu\beta\beta$-decay events coincides with the detector,
the detection efficiency is maximized. Although reaching a very good energy
resolution ($\sim$0.3% FWHM at $Q_{\beta\beta}$, second only to germanium
diodes), phonon detection only does not allow to perform particle
identification to reject background events. This leads to a higher background
index and prevents entering the background-free regime. Moreover, the
$Q_{\beta\beta}$ of 130Te (see Table 2) is lower than the 208Tl gamma line at
2.6 MeV, which therefore contributes to the background around
$Q_{\beta\beta}$. The solution to overcome this limitation is the use of
scintillating bolometers employing different isotopes, namely Zn82Se for
CUPID-0 Azzolini et al. (2019) and Li100MoO4 for CUPID-Mo Armengaud et al.
(2020). Both of these isotopes have a $Q_{\beta\beta}$ around 3 MeV, above the
208Tl line and most of the other natural radioactive background sources, but
very low natural isotopic abundance (in this case, enrichment is required).
Li100MoO4 crystals will be the ones implemented in the final version of the
next generation ton-scale experiment CUPID Armstrong et al. (2019). This is
mainly due to the poor radiopurity and energy resolution of Zn82Se produced,
even though 100Mo $2\nu\beta\beta$ decay half-life (the fastest among the ones
in Table 2) could be an irreducible source of background around
$Q_{\beta\beta}$ if energy and time resolution requirements will not be
satisfied.

Although having a $Q_{\beta\beta}$ below 208Tl, 136Xe is an interesting
candidate isotope as it is gaseous, easy to enrich and can be used to build a
ton-scale experiment. Also in this case, the detector itself is the source of
$0\nu\beta\beta$-decay, and different technologies can be implemented: a
single phase Time Projection Chamber (TPC) using Liquid Xenon (LXe), such as
EXO-200 Albert et al. (2018), a high-pressure TPC with Gaseous Xenon (GXe),
such as NEXT Monrabal and et al. (2018), or a Xe-loaded liquid scintillator,
such as KamLAND-Zen Gando (2020). The achievable energy resolution is lower
with respect to germanium or bolometric experiments and it ranges roughly from
1% to 10% FWHM among different experiments. LXe-TPCs measure both the
scintillation light (starting time for the TPC) and ionization charges and
have a better self-shielding against external background sources (due to the
higher density of the target). Furthermore, some topological information to
distinguish single site from multi site events was used in EXO-200 Albert et
al. (2018). GXe-TPCs allow for real topological reconstruction: because of the
lower density of the detector, a $0\nu\beta\beta$ event would look like two
separate tracks (originating from the same point, the decaying nucleus) ending
with Bragg peaks, corresponding to the stopping of the two emitted electrons.
In this way, it is possible to reject nearly any possible background event
except $2\nu\beta\beta$. Xe-based experiments did not reach a background-free
regime up to now; thus, the $0\nu\beta\beta$ decay (and also any other
physics) search heavily depends on the background model of the experiment.

Germanium detectors are particularly suited for $0\nu\beta\beta$ decay search,
presenting indeed several advantages. As the search for $0\nu\beta\beta$ decay
is based on the detection of a signal peak over the background, the excellent
energy resolution of semiconductor detectors results in a great advantage.
High Purity Germanium (HPGe) detectors have the best energy resolution with
respect to any other competitive technique with a full width at half maximum
(FWHM) better than 0.1$\%$ at the $Q$-value of the $0\nu\beta\beta$ decay of
76Ge ($Q_{\beta\beta}$ = 2039 keV) (see Sections 5 and 6). This feature allows
to identify the $\gamma$ peaks of the various background sources as well as to
isolate the tail of the $2\nu\beta\beta$ decay spectrum (see Figure 3).
Germanium-based experiments feature a high detection efficiency since the
detector is also the source of the $\beta\beta$-decay. Despite the relatively
low natural isotopic abundance of the germanium $\beta\beta$-emitter 76Ge
(7.8%), modern experiments make use of crystals enriched in 76Ge (enrGe) up to
88%, thus significantly reducing the number of detectors needed to reach a
given 76Ge content. HPGe detectors also guarantee a low background level since
they have an extremely high intrinsic radio-purity (no measurable U or Th
contamination) and can be realized with particular electrode geometry,
allowing to take advantage of the pulse shape discrimination (PSD) analysis to
actively reduce the background (see Sections 5 and 6).

On the other hand, the use of germanium presents a few disadvantages. The
$Q_{\beta\beta}$ of 76Ge is low with respect to other commonly used
$\beta\beta$ isotopes. In particular, the $Q_{\beta\beta}$ is below the
dominant 208Tl line of 2615 keV, thus causing the ROI suffering of the
relative Compton continuum. The enrichment process is a rather expensive
procedure even if the cost has been decreasing through the years. Although
taking advantage from a high value of $M_{0\nu}$, 76Ge has the lowest value of
the space phase factor ($\sim$2.3 $\times$ 10-15 Kotila and Iachello (2012);
Stoica and Mirea (2013)) with respect to all other isotopes. This implies the
need to reach a longer ${T^{0\nu}_{1/2}}$ to probe a given $m_{\beta\beta}$
value.

Modern experiments using HPGe detectors have already faced the listed issues
and are currently leading the field with the best limit and the best
sensitivity on the $0\nu\beta\beta$ decay half-life with the Gerda experiment
(see Section 5), thus approaching the exploration of the inverted ordering
region. PSD techniques and additional active veto systems have been
successfully exploited to get rid of $\gamma$ background and other residual
contamination. These factors, together with a careful material selection, led
Gerda to achieve an unprecedented low background level thus, reaching the so-
called background-free regime (see Section 5).

##  4 Neutrinoless Double Beta Decay Search with Ge Detectors

The observation of the $\beta$$\beta$-decay can be performed through
geochemical, radiochemical and direct techniques. In the geochemical method
Primakoff and Rosen (1965), the abundance of the daughter isotope is
determined in minerals of known age containing the parent nucleus. In the
radiochemical method, the abundance of the final nucleus is measured after
several years in a well-prepared artificial sample of the parent one. In both
methods, however, $0\nu\beta\beta$ and $2\nu\beta\beta$ decays cannot be
distinguished, while in direct experiments, the measurement of the sum energy
of the two emitted electrons provides a real-time detection of the
$\beta$$\beta$-decay.

The search for $0\nu\beta\beta$ decay with Ge detectors was firstly proposed
in 1967 by the Milano Group Fiorini et al. (1967) using a Ge(Li) detector
Fiorini et al. (1970, 1973). The experiment was located in the Mont Blanc
tunnel ($\sim$4200 m.w.e.) in order to reduce cosmic-ray background. The 90 g
Ge detector was surrounded by a plastic scintillator veto and shielded with 10
cm of low background lead, a thin cadmium neutron absorbing layer and a 10 cm-
thick box of resin impregnated wood as a neutron moderator. The outer shield
was 10 cm-thick ordinary lead Avignone and Elliott (2019). In the following
years, the Milano group upgraded the experiment with two true coaxial Ge(Li)
detectors and made several improvements regarding the use of low-background
construction materials for the cryostat and a new shielding configuration
Bellotti et al. (1984, 1986). No evidence for $0\nu\beta\beta$ decay of 76Ge
was found, and a lower limit on the half-life of $3.3\times 10^{23}$ yr (68%
C.L.) was set in 1986 Bellotti et al. (1986).

In the 1970s and 1980s, several research groups started to search for the
$0\nu\beta\beta$ decay of 76Ge with HPGe detectors, making strong efforts to
reduce the background through the use of passive shields, by placing the
experiments in underground facilities Forster et al. (1984); Simpson et al.
(1984); Fisher et al. (1989); Reusser et al. (1992) and by using an active NaI
veto Avignone et al. (1983, 1985, 1986); Caldwell et al. (1987); Caldwell.
(1991). The strongest limit on the $0\nu\beta\beta$ decay half-life reported
by a natural Ge experiment was $1.2\times 10^{24}$ yr (UCSB/LBL Caldwell.
(1991)).

For the first time, the ITEP-Yerevan experiment Vasenko et al. (1990) employed
Ge(Li) detectors isotopically enriched in 76Ge in order to concentrate the
active source mass in small detectors and set a limit of ${T^{0\nu}_{1/2}}$
$\geq 2.0\times 10^{24}$ yr in 1989.

Following this idea, in the 1990s, two collaborations produced the first enrGe
HPGe detectors: the Heidelberg–Moscow (HdM) experiment Balysh et al. (1992,
1995); Günther et al. (1997); Klapdor-Kleingrothaus et al. (2001); Bakalyarov
et al. (2005) and the International Germanium Experiment (Igex) Aalseth et al.
(1999, 2002). The HdM experiment operated 16.9 kg of enrGe with a 76Ge
abundance of 86% at the LNGS. The Igex collaboration produced six enrGe
detectors that were operated in a low-background cryostat with archaeological
lead shielding in different underground facilities. Both HdM and Igex
experiments applied pulse shape discrimination (PSD) analysis to further
reduce the background. A lower limit on the half-life of the decay of
$1.9\times 10^{25}$ yr (90% C.L.) was found by the HdM collaboration Klapdor-
Kleingrothaus et al. (2001).

In 2001, after the publication of the final results of HdM part of the
collaboration published a claim on the observation of the $0\nu\beta\beta$
decay of 76Ge Klapdor-Kleingrothaus et al. (2001), reporting a half-life of
$T^{0\nu}_{1/2}=1.19^{+0.37}_{-0.23}\times 10^{25}$ yr Klapdor-Kleingrothaus
et al. (2004). Later, pulse shape discrimination was used to strengthen the
claim Klapdor-Kleingrothaus and Krivosheina (2006). This claim arose a number
of replies and was strongly criticized by many physicists Schwingenheuer
(2013). The situation was clarified only a few years later by the results from
the first phase of the Gerda experiment Agostini et al. (2013) that strongly
disfavored the observation.

The Gerda Agostini et al. (2013, 2017, 2018, 2019, 2020) and Majorana Aalseth
et al. (2018); Alvis et al. (2019) experiments continued the search for the
76Ge $0\nu\beta\beta$ decay with Ge detectors in the last decade, reaching a
sensitivity larger than $10^{26}$ yr. Details on these experiments are
presented in Sections 5 and 6.

The evolution of the published results on the lower limit of the half-life of
76Ge $0\nu\beta\beta$ decay is depicted in Figure 4, from the first result of
the Milano group Fiorini et al. (1967) until the final result of the Gerda
experiment Agostini et al. (2020).

Figure 4:  Evolution of the lower limit on the half-life (upper panel) and
background index (bottom panel) as a function of time for the experiments
searching for 76Ge $0\nu\beta\beta$ decay. References are in the text (see
Section 4).

The intense experimental program and the technology progress allowed
tightening the limit of six orders of magnitude in about 50 years. This goal
has been achieved thanks to the excellent performance of the Ge detectors and
the possibility of operating an increasing 76Ge mass, together with the
impressive improvement in the background reduction techniques (e.g., strong
material selection, introduction of active vetoes and efficient pulse shape
discrimination analysis). Figure 4 (bottom) reports the evolution in time of
the background index of germanium experiments: the BI was progressively
decreasing as the sensitivity increased until reaching the background-free
regime with the Gerda experiment Agostini et al. (2017).

The search for $0\nu\beta\beta$ decay in 76Ge will be continued in the
following years by the LEGEND project Abgrall et al. (2017): with a staged
approach, it aims to reach a sensitivity to the $0\nu\beta\beta$ decay half-
life up to $10^{28}$ yr. The experimental program of LEGEND is described in
Section 7.

##  5 The GERDA Experiment

The experimental apparatus of the GERmanium Detector Array (Gerda) experiment
is installed in hall A of the Gran Sasso National Laboratory (LNGS). The
experiment was taking data from 2011 to 2019 through different phases. The
facility is currently being upgraded to host the first phase of the LEGEND
project, called LEGEND-200 (see Section 7).

The core of the Gerda experiment was made of HPGe detectors isotopically
enriched in 76Ge up to $\sim$87% Ackermann et al. (2013); Agostini et al.
(2018). Following a suggestion from Heusser (1995), the detectors were
directly immersed into liquid Argon (LAr), which acts both as a shield against
the external radioactivity and as a cooling medium. In the different phases of
the experiment, the germanium mass was progressively increased using different
types of detectors.

In Gerda Phase I, lasting from November 2011 to September 2013 and collecting
an exposure of 23.5 kg$\cdot$yr, eight enriched semi-coaxial Ge detectors (see
Figure 5a), for a total mass of 15.6 kg, were employed together with three
non-enriched semi-coaxial detectors. The semi-coaxial detectors were
originally produced by ORTEC for the former HdM Klapdor-Kleingrothaus et al.
(2001) and Igex Aalseth et al. (2002) experiments (see Section 4), then
refurbished by Canberra and redeployed in the Gerda apparatus. In July 2012,
the natural Ge detectors were replaced by five Broad Energy Germanium (BEGe)
diodes Agostini, M. and others (2019) (see Figure 5b) with a total mass of 3.6
kg. The detectors were arranged in four strings, each one housing three (five)
semi-coaxial (BEGe) detectors.

From September 2013 to December 2015, a major upgrade of the experiment was
carried out to improve the sensitivity to the $0\nu\beta\beta$ decay half-life
of 76Ge beyond $10^{26}$ yr with a goal exposure of 100 kg$\cdot$yr Agostini
et al. (2018). The Gerda Phase II was designed to reach a BI of the order of
$10^{-3}$ cts/(keV$\cdot$kg$\cdot$yr), thus running in the so-called
background-free regime, i.e., having less than one background event in the
energy region ($Q_{\beta\beta}$ $\pm~{}0.5$ FWHM) for the whole exposure. In
the background-free regime, the sensitivity is expected to scale linearly with
the exposure, thus allowing to reach the desired final goal. The Gerda Phase
II detectors array included 40 diodes in total, composed of 7 Phase I semi-
coaxial detectors, 30 newly produced BEGes (for a mass of about 20 kg) and 3
non-enriched semi-coaxial detectors.

The data-taking of Gerda Phase II started in December 2015. From April 2018 to
July 2018, after having collected an exposure of 58.9 kg$\cdot$yr Agostini et
al. (2019), a minor upgrade of the experiment was carried out with the
installation of five additional inverted coaxial (IC Cooper et al. (2011);
Agostini et al. (2021)) detectors (see Figure 5c), produced in collaboration
with Mirion Technologies with a total mass of 9.6 kg. The data-taking was then
resumed and lasted till November 2019 for a final total collected exposure of
103.7 kg$\cdot$yr Agostini et al. (2020).

Figure 5: Sketch (top) and picture (bottom) of semi-coaxial (a), BEGe (b) and
inverted coaxial (c) detectors employed in the various phases of the Gerda
experiment.

###  5.1 Experimental Setup

The 3500 m.w.e. of rock overburden of the LNGS site, where the Gerda apparatus
is installed, provides a reduction of the cosmic muon flux by six orders of
magnitude with respect to surface.

The experimental setup consists of a 590 m3 stainless steel tank of 9 m height
and 10 m diameter, filled with ultra-pure water Ackermann et al. (2013). The
tank is instrumented with 66 photomultipliers tubes (PMTs), acting as an
active Cherenkov veto against the residual cosmic muon flux. The muon veto
system Freund et al. (2016) is complemented by scintillator panels installed
on the top of the clean room, as shown in Figure 6.

Figure 6: Schematic view of the Gerda setup. Figure published by Nature, 2017
Agostini et al. (2017).

The water tank contains a 64 m3 vacuum-insulated stainless steel cryostat
(diameter $=4$ m), filled with LAr. The internal walls of the cryostat are
covered with a 6 cm low-background copper (Cu) layer to shield the $\gamma$
radioactivity originating from the steel.

The 40 germanium detectors employed in Gerda Phase II were arranged in seven
strings, each one surrounded by a nylon cylindrical vessel conceived to reduce
the collection of 42K ions, originating from the decay of 42Ar, onto the
surface of the detectors. The array is lowered in the LAr cryostat from the
clean room above the tank through the lock system, as shown in Figure 6.

During the upgrade between Phase I and Phase II, a cylindrical volume around
the detector string of about 1 m height and 0.5 m diameter was instrumented
with a curtain of wavelength-shifting fibers to detect the LAr scintillation
light. The readout was performed using 90 silicon photomultipliers (SiPMs) and
16 PMTs, arranged as shown on the right side of Figure 6. Both water and LAr
systems act simultaneously as a passive shield from external radioactivity and
neutrons and as active vetoes. This setup allows suppressing the external
$\gamma$ background at $Q_{\beta\beta}$ to less than $10^{-5}$
cts/(keV$\cdot$kg$\cdot$yr) Barabanov et al. (2009), while the muon rejection
efficiency is $>99.9\%$ Freund et al. (2016).

While differing in size and geometry (see Figure 5), both semi-coaxials and
BEGes are p-type semiconductor detectors operated in reverse bias mode. They
are fabricated from high purity Ge crystals with an active net impurity
concentration of around 1010 atoms$/$ cm3 Agostini et al. (2019). The n+
contact is made of diffused lithium with a thickness of about 0.5 mm. The p+
contact is made of ion-implanted boron with a thickness of the order of 100
$\upmu$m. The semi-coaxial diodes have a mass of the order of 2 kg and an
enrichment fraction ranging from 85.5 to 88.3$\%$ Agostini et al. (2018). The
BEGes detectors have a diameter ranging from 58.3 to 79.3 mm and heights from
22.9 to 35.3 mm Agostini et al. (2018), with a mass of the order of O(0.8) kg.
The enrichment fraction is 87.8$\%$. IC detectors have a mass of the order of
2 kg, a diameter ranging from 72.6 to 76.6 mm and heights from 80.4 to 85.4
mm. The enrichment fraction is 88$\%$ Agostini et al. (2021).

The different readout electrode layout of semi-coaxial, BEGe and IC is at the
basis of their different mass and performance. The geometry of the p+ contact
of a coaxial detector allows the depletion of a larger volume with respect to
a BEGe. Conversely, the latter, thanks to its small p+ electrode, features a
lower capacitance (1pF vs. 30 pF), thus resulting in a lower series noise and
in a superior energy resolution. This difference is also the key for the
better Pulse Shape Discrimination (PSD) performance of BEGe detectors. The new
design of IC detectors allows to increase the detector mass to the level of a
coaxial diode, thus lowering the background amount per mass unit while
retaining the resolution and the PSD performance of BEGes (see Section 5.2).

The readout of germanium detectors is performed using charge sensitive
amplifiers located in LAr, 35 cm above the array. The signal trace, with a
length of 160 $\upmu$s, is sampled at 25 MHz, while a 10 $\upmu$s window
around the rising edge is sampled at 100 MHz. Digitized data are stored on a
disk for the subsequent analysis.

###  5.2 Data Analysis Flow and Active Background Suppression

Since Phase I, Gerda adopted a blind-analysis procedure consisting of removing
all the events within $\pm$25 keV of $Q_{\beta\beta}$ from the analysis flow
until all the procedures and the cuts are finalized.

The stability of detectors, as well as leakage currents and noise, are
monitored through the injection of 0.05 Hz test pulses (TPs). Quality cuts
based on the flatness of the baseline, polarity and time structure of the
pulse, allow rejecting non-physical events, as discharge and noise bursts,
while keeping 99.9$\%$ of $Q_{\beta\beta}$ events.

The event energy reconstruction is performed with a zero-area cusp filter
Agostini et al. (2015). The energy scale is determined through weekly
calibration runs with 228Th sources. The energy resolution at $Q_{\beta\beta}$
in terms of FWHM is (4.9 $\pm$ 1.4), (2.6 $\pm$ 0.2), (2.9 $\pm$ 0.1) keV for
semi-coaxial, BEGe and IC detectors, respectively Agostini et al. (2020). The
energy resolution is stable within 0.1 keV in the whole data-taking period.

The background suppression is based on different levels. Candidate events
within 10 $\upmu$s from a muon veto signal are rejected. The related dead time
is $<$0.01$\%$. In the same way, events coinciding with an energy deposition
in LAr are also vetoed. If at least one photoelectron within 6 $\upmu$s from a
germanium detector trigger is detected by any of the photosensors of the LAr
veto system, the event is classified as background. The dead time in this case
is of the order of 1.8$\%$ Agostini et al. (2020). Since the range in
germanium of the two electrons of a $\beta\beta$ decay event is of the order
of 1 mm, the energy deposition is expected to be highly localized, thus
featuring a single-site event (SSE). Conversely, $\gamma$ rays (at this
energy, mainly interacting via Compton scattering) are expected to deposit
their energy at multiple sites featuring the so-called multi-site events
(MSEs). Anti-coincidence among different Ge detectors is, therefore, exploited
to discard background events. Consecutive events occurring within 1 ms are
also tagged to veto time-correlated decay from primordial radioisotopes.
Finally, the discrimination between SSEs and MSEs, as well as from $\alpha$
and $\beta$ surface events, is performed by exploiting the different
characteristics of the pulse shape Agostini et al. (2013). For BEGe and IC
detectors, the PSD cut is based on the parameter $A/E$, where $A$ is the
maximum of the current amplitude and $E$ is the energy. MSEs and n+ surface
events show wider current pulses, thus featuring a lower $A/E$ value with
respect to SSEs. On the contrary, surface events on the p+ electrode feature a
higher $A/E$ Budjáš et al. (2009). A mono-parametric cut on both sides of the
$A/E$ distribution of SSEs is, therefore, effective at enhancing the signal-
to-background ratio. Figure 7 shows the weighting potential describing the
coupling of the charge with respect to the distance from the respective
electrode Agostini et al. (2019) for semi-coaxial and BEGe detectors (Figure
7a) and for IC ones (Figure 7b). Due to their different geometries, semi-
coaxial diodes show a more complicated pulse time structure requiring the
application of an artificial neural network (ANN) to discriminate SSEs from
MSEs and an additional cut on signal rise time to reject events on the p+
electrode Agostini et al. (2013, 2019). Both ANN and $A/E$ methods are trained
using calibration data. The $0\nu\beta\beta$ signal efficiency is estimated to
be (68.8 $\pm$ 4.1)$\%$, (89.0 $\pm$ 4.1)$\%$, (90.0 $\pm$ 1.8)$\%$ for
coaxial, BEGe and IC detectors, respectively Agostini et al. (2020).

Figure 7: (a) Cross-sections of a GERDA coaxial (left) and BEGe (right)
detector with an overlay of the corresponding weighting potentials. Figure
published by Science, 2019 Agostini et al. (2019). (b) Weighting potential for
an IC detector. Figure published by EPJC, 2021 Agostini et al. (2021).

###  5.3 Statistical Analysis and $0\nu\beta\beta$ Results

The total exposure collected during Gerda PhaseII is 103.7 kg$\cdot$yr. The
energy range around the $Q_{\beta\beta}$ considered for the analysis goes from
1930 to 2190 keV with the exclusion of two known background peaks at (2104
$\pm$ 5) and (2119 $\pm$ 5) keV. After the unblinding and the application of
the analysis cuts, 13 events are found in the analysis window. The energy
distribution of those events is fitted assuming a flat distribution for
background and a Gaussian centered at $Q_{\beta\beta}$ with a width according
to the energy resolution for a possible $0\nu\beta\beta$ signal. Both
frequentist and Bayesian analyses are applied.

The frequentist analysis is performed using a two-sided test statistics based
on the profile likelihood and gives no indication for a signal. The limit on
the half-life of 76Ge is ${T^{0\nu}_{1/2}}$ $>1.5\times 10^{26}$ yr (90% CL).
The combined analysis of the whole Phase I and Phase II data sample, with a
total exposure 127.2 kg$\cdot$yr, provides the limit ${T^{0\nu}_{1/2}}$
$>1.8\times 10^{26}$ yr (90% CL). The limit coincides with the sensitivity,
defined as the median expectation under the no signal hypothesis Agostini et
al. (2020).

The background index (BI), as derived from the fit, reached the
unprecedentedly low value of $BI=5.2^{+1.6}_{-1.3}\times 10^{-4}$
cts/(keV$\cdot$kg$\cdot$yr)in Phase II. The mean background expected in the
signal region ($Q_{\beta\beta}$ $\pm$ 2$\sigma$) is 0.3 counts, thus reaching
the design goal of a background-free regime.

The limit obtained with the Bayesian analysis for Phase I and Phase II data is
${T^{0\nu}_{1/2}}$ $>1.4\times 10^{26}$ yr (90% CL). The reader is referred to
Agostini et al. (2017, 2019, 2018, 2020) for the different data releases of
Phase II and for further details on the statistical analysis.

In addition to standard $0\nu\beta\beta$ searches, Gerda explored other
physics topics such as the search for bosonic superweakly interacting massive
particles (super-WIMPs) as keV-scale dark matter candidates Agostini et al.
(2020), the search for $0\nu\beta\beta$ decay processes accompanied with
Majoron emission Agostini et al. (2015a) and the study of two-neutrino double
beta decay of 76Ge to excited states of 76Se Agostini et al. (2015b).

##  6 The Majorana Experiment

The Majorana Collaboration searches for neutrinoless double-beta decay of 76Ge
with High Purity Germanium detectors Abgrall et al. (2014). The Majorana
demonstrator Aalseth et al. (2018) implements an array of 58 HPGe detectors
for a total mass of 44.8 kg (14.4 kg of natural Ge detectors and 29.7 kg
enriched to 88.1 $\pm$ 0.7% in 76Ge), arranged in two different modules, each
contained in a low-background shield. The experiment is located at the Sanford
Underground Research Facility (SURF) in Lead, South Dakota (U.S.A.), at a
depth of 4300 m.w.e. Its main goal is to demonstrate a low level of background
to justify the construction of a tonne-scale experiment that would probe
$m_{\beta\beta}$ at the level of 15 meV with a nearly background-free region
around the $Q_{\beta\beta}$. The enriched detectors are p-type, point contact
(PPC) detectors Luke et al. (1989); Barbeau et al. (2007) with a sub-GeV
energy threshold and low capacitance; this allows low-energy physics searches
as additional science channels.

Two low-background shields contain the two modules of detector arrays. These
shields are made of underground electroformed copper (UGEFCu) plus an
additional 5 cm of commercial C10100 copper. Outside the copper shields, an
additional high-purity lead shielding, 45 cm thick, is enclosed in a region
with liquid-nitrogen boil-off gas to reduce the radon contamination. An active
muon veto is located outside the radon exclusion volume and is surrounded by 5
cm of borated polyethylene and 25 cm of polyethylene for neutron moderation.
For each module, energy calibrations are done with 228Th sources inserted into
the shield on a weekly basis Abgrall et al. (2017). A pictorial view of the
Majorana experimental setup is shown in Figure 8.

Figure 8: Pictorial view of the Majorana experimental setup. Figure published
by J. Phys. Conf. Ser., 2020 Guinn et al. (2020).

To achieve an ultra-low background, the Majorana demonstrator uses a total of
1196 kg of underground electroformed copper, not only for the innermost 5 cm
of shielding surrounding the cryostat but also for the cryostats and the
detector support structures. In addition, commercially available low-
background materials were carefully screened and selected to be used for
cabling, cryostat seals, and all electrical and thermal insulations. Low-
background front-end electronics were developed as well Guinn et al. (2015).

###  6.1 Data-Taking and Event Selection

Data collected by the Majorana demonstrator are divided into seven data sets
(DS), from DS0 to DS6. A new data set was defined when a significant change in
the experimental setup occurred, through the detector setup construction and
commissioning. A letter following the data set number indicated a minor change
of the experimental configuration. Data set 0 (DS0) began on 26 July 2015 with
the first module of the detector array. The DS6 data acquired up to 16 April
2018 is labeled DS6a.

Data blindness was implemented through a prescaling scheme in which 31 h of
open background data were followed by 93 h of blind data. The detector signals
are digitized with a 14-bit 100 MHz digitizers Abgrall et al. (2021), which
were designed for the GRETINA experiment Vetter et al. (2000). The signal
waveforms are recorded in a 20 $\upmu$s acquisition window at the full
sampling rate. Each detector had a high-gain and a low-gain signal
amplification and both were digitized independently. The trigger threshold for
each channel was set independently according to the channel trigger rate,
depending on the electronic noise and the initialization of the on-board
trapezoidal filter (see Section 6.2). A reduction in the live time for each
detector was estimated at the level of <0.1% when the initialized value of the
triggering filter was negative (because of electronic noise or baseline
recovery at the time of initialization due to interactions).

A physical event is represented by recorded waveforms grouped within a 4
$\upmu$s coincidence window. Events in which multiple detectors trigger are
rejected. Each waveform is then processed through quality checks to remove
non-physical waveforms and signals from periodic pulses. The acceptance after
these quality cuts is estimated to be >99.9% for all data sets. Events within
1 s from a muon veto trigger are also rejected. Every 36 h, 30 min of data are
also rejected for each module due to the filling of liquid nitrogen, which
causes microphonic noise.

###  6.2 Energy Estimation

The energy estimation is done by calibrating the amplitude of the recorded
signals once filtered and pole-zero adjusted. Corrections that account for ADC
non-linearities and charge trapping along the drift path are applied to the
acquired waveforms for each respective channel. After corrections, the energy
uncertainty due to ADC effects is less than 0.1 keV. The energy resolution is
also affected by drift-path-dependent charge trapping in the crystal bulk.
This effect is taken into account with an additional term for the standard
pole-zero adjustment:

$\frac{1}{\tau}=\frac{1}{\tau_{PZ}}+\frac{1}{\tau_{CT}}$ (7)

where $\tau_{PZ}$ is the pole-zero time constant due to the preamplifier, and
$\tau_{CT}$ is the correction that reproduces exponential trapping of charges
on the drift path. For each detector, the total pole-zero correction $\tau$ is
optimized by minimizing the full-width at half maximum (FWHM) of the 2615 keV
208Tl peak in calibration data: this correction improves the energy resolution
of 1.4 keV on average. A fast trapezoidal filter (with a rise time of 1.0
$\upmu$s and a flat-top time of 1.5 $\upmu$s) is applied to the waveform to
estimate the start time $t_{0}$ and the threshold crossing time. Then the
pole-zero correction and a slower trapezoidal filter (with a rise time of 4.0
$\upmu$s and a flat top time of 2.5 $\upmu$s) are applied to the original
waveform. The energy of the event is estimated as the value of the waveform at
a time of 0.5 $\upmu$s from the end of the flat top, relative to t0, after
having applied the above corrections and filtering.

Periodic energy calibrations are used to provide an initial linear energy
scale calibration for each channel, taking into account small variations over
time of the electronic noise or energy scale Abgrall et al. (2017). Then, the
combination of the spectra obtained provides a more precise energy calibration
through the simultaneous fit of the full energy calibration peaks at 239 keV
(212Pb), 241 keV (224Ra), 277 keV (208Tl), 300 keV (212Pb), 583 keV (208Tl),
727 keV (212Bi), 861 keV (208Tl), and 2615 keV (208Tl), respectively.

Each peak is fitted with a response function $R(E)$ given by the sum of a
Gaussian function and an exponentially modified Gaussian Tail:

$R(E)=\frac{1-f}{\sqrt{2\pi\sigma^{2}}}e^{-\frac{(E-\mu)^{2}}{2\sigma^{2}}}+\frac{f}{2\gamma}e^{\left(\frac{\sigma^{2}}{2\gamma^{2}}+\frac{E-\mu}{\gamma}\right)}\mbox{erfc}\left(\frac{\sigma}{\sqrt{2}\gamma}+\frac{E-\mu}{\sqrt{2}\sigma}\right)$
(8)

where $\sigma$ represents the smearing due to electronic noise and partial
charge collection, $\gamma$ is the decay constant of the low-energy tail, and
$f$ is the fraction of the peak shape contained in the low-energy tail. The
background in the surrounding region of the peak is modeled by the sum of an
error function and a continuum component given by a quadratic polynomial. The
combined calibration spectrum from DS0 to DS6 is shown in the top panel of
Figure 9. After having determined the FWHM at each energy peak, the FWHM is
fitted with the following function of the energy $E$:

$FWHM(E)=\sqrt{\Gamma_{n}^{2}+\Gamma_{F}^{2}E+\Gamma_{q}^{2}E^{2}}$ (9)

where $\Gamma_{n}$, $\Gamma_{F}$ and $\Gamma_{q}$ are the terms due to the
electronic noise, the Fano factor Fano (1947), and the partial charge
collection, respectively. The central panel of Figure 9 shows the exposure-
weighted resolution for each gamma peak and a fit to the exposure-weighted
values. The exposure-weighted average resolution (FWHM) at Qββ is (2.53 $\pm$
0.08) keV.

Figure 9: Top: Calibration spectrum from combined data of DS0 to DS6. Center:
exposure-weighted resolution for all the calibration peaks and the FWHM fit
function (red line). The green line indicates the exposure-weighted resolution
value at 2039 keV. Bottom: residuals of the estimated FWHM values from the
fit. Figure published by Phys. Rev. C, 2019 Alvis et al. (2019)

###  6.3 Background Suppression

The weighting potential of PPC Ge detectors is relatively smaller in the bulk
of the crystal and mostly located in the vicinity of the point contact (see
Figure 10, on the left). As also explained in Section 5.2, electrons
interacting inside the bulk of the detectors are identified as single-site
events (SSEs), as their range is limited to less than 1 mm at the energies of
interest. Gamma rays, instead, interact inside the detectors mostly as
multiple-site events (MSEs). This translates to a difference in the pulse
shape, which allows the discrimination of the gamma-ray background. In
particular, at about the same energy ($E$), multi-site events have a maximum
current amplitude ($A$) quite smaller than single-site events; calibration
data are therefore used to fit the mean value of $A$ as a function of $E$ for
each data set and detector. The parameter AvsE is defined as:

$\mbox{AvsE}=\frac{1}{j}(p_{0}+p_{1}E+p_{2}E^{2}-\lambda A)$ (10)

where $p_{0}$, $p_{1}$ and $p_{2}$ are the coefficients obtained from the fit
of $A$ as a function of $E$ and $\lambda$ is the calibration constant usedto
convert the ADC channels to energy expressed in keV.

Figure 10: Left: weighting potential of the PPC detectors. White lines
indicate equal drift times. Right: acceptance for each detector in calibration
data for DS6a. Horizontal lines represent the mean values for all calibrated
detectors. Figure published by Phys. Rev. C, 2019 Alvis et al. (2019)

The value of AvsE is then used to establish a cut above $\sim$1 that would
have a 90% efficiency in accepting single-site events from the 1593 keV
double-escape peak (DEP) from the 208Tl gamma ray line at 2615 keV. The
survival percentage for events in the DEP, the single-escape peak (SEP) and
the Compton continuum in a 100 keV region around the Qββ are shown on the
right part of Figure 10 for events from DS6a and for each detector: 40% of
Compton events are accepted while about 6% of multi-site events from the SEP
are retained. The main systematic uncertainties in the AvsE estimation are due
to the difference between the position distribution of simulated events and
the interactions from the calibration sources, the time-variation in the SSE
acceptance and the energy dependence of the cut acceptance Alvis et al.
(2019).

External $\alpha$ particles with MeV energies have a range of tens of $\upmu$m
in Ge detectors. The PPC Ge detectors of Majorana have lithiated dead layers
(1.1 mm thick) over the surface and a passivated surface on the face with the
point contact. As the dead layers are much thicker than their range, $\alpha$
particles impinging the lithiated surfaces cannot penetrate inside the active
volume. On the contrary, they can penetrate the passivated surface and deposit
their energy in the active region of the detectors. The holes near the
passivated surface are trapped and later released in a time that is much
longer than the rise time of events occurring in the bulk. This degrades the
measured energy for $\alpha$ particles, contributing to the background near
the Qββ value but the difference in rise time with respect to events occurring
in the bulk allows for their discrimination. A cut is defined, based on the
slope $\Delta$ between the average value of the two 1 $\upmu$s-wide regions,
which start either 2 $\upmu$s after the time the waveform reaches 97% of its
maximum (t97) or 1 $\upmu$s before the end of the waveform
(t${}_{\mbox{max}}$). The parameter DCR (Delayed Charge Recovery) is defined
as the following:

$DCR=\frac{\int_{t_{max}-1\mu s}^{t_{max}}{V(t)dt}-\int^{t_{97}+3\mu
s}_{t_{97}+2\mu s}{V(t)dt}}{t_{\mbox{max}}-t_{97}}-\Delta$ (11)

and ensures 99.9% acceptance of the Compton events from calibration data near
the $Q_{\beta\beta}$ value. The acceptance of the cut is estimated for each
data set.

###  6.4 $0\nu\beta\beta$ Search Results

The characteristics and the efficiencies values are evaluated for each data
set. In addition to the efficiencies for background rejection
$\epsilon_{AvsE}$ and $\epsilon_{DCR}$, the efficiency $\epsilon_{cont}$ of
detecting the full energy of $0\nu\beta\beta$ events is combined together with
the previous ones. The total efficiency weighted with the exposure of all data
sets is 0.810${}^{+0.031}_{-0.032}$. The $0\nu\beta\beta$ ROI is optimized for
each data set using the peak shape parameters and background index.
$\epsilon_{res}$ represents the fraction of $0\nu\beta\beta$ events falling
inside the optimal ROI for each data set. The exposure weighted value for
$\epsilon_{res}$ over all data sets is 0.900 $\pm$ 0.007.

A total of 26 kg$\cdot$yr exposure was collected from DS0 to DS6 of which
11.85 kg$\cdot$yr were blinded across the energy spectrum. The energy spectrum
above 100 keV relative to the full enriched detector exposure is shown in
Figure 11 (left), with only data cleaning and muon veto cuts (in black) and
with all cuts applied (in red). The inlet shows the background events between
1950 and 2350 keV, where a flat component is expected (from background
simulations with the MaGe software Boswell et al. (2011) based on Geant4
Agostinelli et al. (2003)) with the exclusion of $\pm$5 KeV of the 2103 keV
(single-escape peak from 208Tl), 2118 keV and 2204 keV (gammas from 214Bi)
peaks. The predicted background in this region is (6.1 $\pm$ 0.8) $\times$
10-3 counts/(keV kg yr) in the exposure weighted optimal ROI of 4.13 keV and
is consistent with 208Tl contamination in components larger than assay values,
whose origin is under investigation.

Figure 11: Left: energy spectrum above 100 keV with only data cleaning and
muon veto cuts (black) and after all cuts (red). The blue curve shows the
fitted background from the unshaded regions in the inset and the 90% C.L.
upper limit on the counts from a $0\nu\beta\beta$ signal. Right: the $p$-value
as a function of the $0\nu\beta\beta$ half-life estimated from the unbinned
frequentist profile-likelihood method (solid black). The median sensitivity is
shown as a dashed black line, and the shaded bands represent the 1 and
2$\sigma$ intervals. Figure published by Phys. Rev. C, 2019 Alvis et al.
(2019)

2

The limit on the half-life of the $0\nu\beta\beta$ decay of 76Ge (considered
as a Poisson process) is estimated with a Feldman–Cousins approach (see
Reference Feldman and Cousins (1998)) with 0.65 expected background events and
1 event observed in the ROI at 2040 keV. The value of the 90% C.L. lower limit
is 2.5 $\times$ 1025 yr. Based on an unbinned, extended profile likelihood
method based on RooStats (see References Verkerke and Kirkby (2003); Schott
(2011); Agostini et al. (2017)), the median sensitivity at 90% C.L. is 4.8
$\times$ 1025 yr, as shown in Figure 11. The $p$-value distribution of the
half-life is T${}^{0\nu}_{1/2}$ $>$ 2.7 $\times$ 1025 yr at 90% C.L. In this
analysis, the half-life is a common parameter for all data sets, while the
peak shape parameters and signal efficiencies are constrained as a Gaussian
nuisance term for each data set. The corresponding upper limit on the number
of signal events at 90% C.L. is 3.8. A Bayesian statistical analysis was also
performed using Markov chain Monte Carlo simulations in RooStats with the same
likelihood function. With a flat prior on $1/T^{1/2}_{0\nu}$, the Bayesian
limit on the half-life is 2.5 $\times$ 1025 yr for a 90% credible interval.

Based on a range of nuclear matrix elements for 76Ge (Menéndez et al. (2009);
Horoi and Neacsu (2016); Hyvärinen and Suhonen (2015); López Vaquero et al.
(2013); Yao et al. (2015)), phase-space factors of 2.36 $\times$ 10-15/yr
(Kotila and Iachello (2012)) or 2.37 $\times$ 10-15/yr (Mirea et al. (2015)),
and a value of gA = 1.27, the limit on the half-life is converted in a range
of upper limits for the effective neutrino mass given by $m_{\beta\beta}$ $<$
(200–433) meV Alvis et al. (2019).

##  7 The LEGEND Project

Building upon the success of the Gerda and Majorana experiments, the LEGEND
(Large Enriched Germanium Detector for Neutrinoless $\beta\beta$ Decay)
Collaboration Abgrall et al. (2017) aims at building a 76Ge-based neutrinoless
double beta decay experiment with a sensitivity of the half-life beyond
$10^{28}$ years, to fully span the inverted neutrino mass ordering region.

Majorana and Gerda have already proven to have the best energy resolution (see
Section 6.2), the lowest background index and the best sensitivity in the
field (see Section 5.3). The LEGEND experiment will benefit from the knowledge
and from the technological achievements of the two Collaborations, as well as
from the contributions of new joining groups.

LEGEND will inherit the shielding concept implemented in Gerda with the use of
a water Cherenkov veto and the depletion of bare germanium detectors in an
instrumented LAr volume simultaneously acting as a coolant medium and active
veto. The low-noise readout electronic developed by the Majorana Collaboration
has proven to be successful in improving the PSD performances as well as in
lowering the threshold thus resulting in an unprecedented low energy
resolution. Moreover, the careful selection and radiopurity control of the
employed materials allowed Majorana to reach a similar background level with
respect to Gerda.

The natural steps toward reaching higher sensitivities consist of increasing
the detector mass while further reducing the background, aiming at performing
a background-free measurement at larger exposures.

The LEGEND project will proceed in two steps: in the first phase, 200 kg of
enriched germanium detectors will be deployed in the existing Gerda facility
at LNGS. By reducing the background index of about a factor of three with
respect to the Gerda final level (i.e., from 1.5 cts$/($FWHM$\cdot$t$\cdot$yr)
to 0.5 cts$/($FWHM$\cdot$t$\cdot$yr)) and with an exposure of 1 t$\cdot$yr,
LEGEND-200 will be able to reach a sensitivity of about $10^{27}$ yr at $90\%$
C.L. The data-taking is expected to start by the end of 2021.

In the second phase, the enriched germanium mass will be increased up to 1000
kg. By lowering the background to 0.025 cts$/($FWHM$\cdot$t$\cdot$yr) and with
an exposure of 10 t$\cdot$yr, LEGEND-1000 will be able to reach a $3\sigma$
half-life discovery sensitivity of $1.3\times 10^{28}$ yr (see Figure 12). The
location of the LEGEND-1000 phase will be selected in order to keep the
cosmogenic activation background as low as possible and within the required
background level.

In the next sections, we will present the project and we will discuss the main
issues addressed by the LEGEND Collaboration.

Figure 12: Sensitivity as a function of exposure and background for an
isotopic enrichment fraction in 76Ge of $92\%$ and for (left) a 90$\%$ C.L.
limit setting and (right) a 3$\sigma$ signal discovery. Plots from LEGEND
Collaboration (a).

###  7.1 LEGEND-200 Germanium Detectors and Experimental Setup

The existing Gerda cryostat is large enough to accommodate up to 200 kg of
detectors divided into 19 strings for a total diameter of 500–550 mm (see
Figure 13a) Abgrall et al. (2017). The approach of reusing the Gerda
experimental apparatus will assure a timely start of data-taking with a world-
leading sensitive experiment.

Figure 13: Experimental setup of the LEGEND-200 (a) and LEGEND-1000 (b) phase,
respectively.

LEGEND-200 will deploy 20 kg of BEGe and 9.4 kg of ICPC detectors (see Figure
5b,c) from Gerda and 28 kg of PPC detectors from Majorana. In addition, nearly
140 kg of newly produced ICPC detectors will be added. As discussed in
Sections 5.1 and 5.2, ICPCs feature a new geometry with respect to the
previously used germanium detectors. As shown in Figure 5c, a small p+
electrode is placed on the opposite face with respect to the bore hole and the
n+ outer contact covers all the remaining surfaces (cylindrical part and bore
hole). Like semi-coaxial detectors, ICPCs can be manufactured with a larger
mass ($\sim$2 kg) with respect to BEGes ($\sim$0.8 kg) and PPCs ($\sim$0.85
kg). This allows increasing the active mass while reducing the amount of
nearby materials contributing to the background, such as cables, electronics
and holders. Moreover, the reduced surface-to-volume ratio with respect to
smaller detectors, makes ICPCs less susceptible to surface effects.
Furthermore, because of the long drift time inside the crystal and the small
p+ electrode, the energy resolution and the pulse shape characteristics are
very similar to BEGe and PPC Domula et al. (2018), making this new type of
HPGe suitable for $0\nu\beta\beta$ decay experiments.

As explained in Section 5.2, during the 2018 upgrade, five 76Ge enriched ICPC
detectors were deployed in the Gerda cryostat and operated in LAr for the
first time until the end of the data-taking in 2019. After being tested and
characterized, they showed energy resolution and background rejection
capability comparable with BEGes while having a larger mass by a factor of
three Agostini et al. (2021, 2020).

###  7.2 Readout Electronics

LEGEND-200 will use a resistive-feedback charge-sensitive amplifier (CSA)
operated at cryogenic temperature in LAr. CSA is divided into two stages in
order to meet the background requirements. The first stage is the low mass
front end (LMFE Abgrall et al. (2015)), a custom-made low-background circuit
based on the front-end used in Majorana where a junction gate field-effect
transistor (JFET) is installed. LMFE is made of ultra-clean materials
(<$1\,\upmu$Bq/kg) due to its close vicinity to the detectors. The second
stage is a differential amplifier $\sim$30–150 cm away from the first stage;
in this case, a slightly higher activity (50 $\upmu$Bq) can be tolerated. The
amplifier is based on the one already used in Gerda Riboldi et al. (2012) and
the two stages are connected using four custom-made low-mass coaxial-cables.
The CSA is designed to have electronic noise $<$1 keV FWHM, an energy
resolution of $\leq$2.5 keV at $Q_{\beta\beta}$, a fast rise time of $\leq$100
ns to allow powerful pulse shape discrimination analysis and high linearity up
to 10 MeV for alpha particles rejection Willers (2020).

For LEGEND-1000, in order to have the same performance with lower background
contribution, an ASIC preamplifier to be placed near the detectors is being
studied.

###  7.3 Background Mitigation Techniques

The electroformed copper (UGEFCu) employed in the Majorana experiment (see
Section 6), shows a uranium and thorium decay chain activity of
<$0.1\,\upmu$Bq/kg Abgrall et al. (2016). This value is at least one order of
magnitude lower than that of the most commercial oxygen-free, high-
conductivity (OFHC) copper used in the external layers of the Majorana
shielding. UGEFCu is produced at Pacific Northwest National Laboratory and
SURF, where it is also stored underground to minimize the possibility of
cosmogenic activation producing 60Co, whose decay yields photons above the Ge
$Q_{\beta\beta}$ value Myslik (2018). The employment of electroformed copper
is going to improve the Gerda radiopurity for LEGEND-200.

As introduced above, in both LEGEND phases, HPGe detectors will be operated in
a LAr-filled cryostat. LAr acts as a coolant for germanium detectors and a
passive shield against external radiation. Moreover, an active veto system in
Gerda was used to discard background events (such as neutrons and photons)
depositing energy in LAr and making it scintillating in coincidence with a
signal in the Ge diodes. As shown in Figure 14, LEGEND-200 will use a similar
design, although with a different geometry, to deal with the increased number
of detectors in the array while maximizing the light collection efficiency.
Further enhancement of the light yield and attenuation length can be reached
by improving LAr purity and by applying Xe doping Myslik (2018).

Figure 14: Instrumentation of the LAr volume around the detectors strings in
LEGEND-200.

The use of detector holders made from polyethylene naphthalate (PEN) will
further improve the background rejection capability. PEN is a thermoplastic
material with scintillating properties and an emission spectrum that peaks in
the blue region at 425 nm. Its scintillation yield is 2.5 times lower than
standard plastic scintillators, but it has other appealing features such as
its very favorable mechanical properties at cryogenic temperature (77 K) and
its radiopurity. Moreover, PEN shifts the LAr 128 nm photons to a higher
wavelength, making it detectable with standard photodetectors Efremenko et al.
(2019). By replacing the Si-made, nontransparent detector holders previously
used in Gerdawith PEN supports, it would be possible to improve the light
collection in the vicinity of the detectors while at the same time exploiting
PEN self-vetoing capabilities Abt et al. (2021). The employment of Pulse Shape
Discrimination techniques with PEN is also feasible Efremenko et al. (2019).
In this way, it could be possible to more efficiently reduce the background
due to surface events and detect LAr scintillating light close to the
detectors. PEN holders specifically designed and optimized to reduce the total
mass in contact with the detectors are being produced and will be employed in
LEGEND-200 Abt et al. (2021).

An important source of background in LEGEND-200 is the 42Ar, a cosmogenically
produced Ar isotope that $\beta$-decays to 42K. The distribution of 42K in LAr
is quite likely to be inhomogeneous due to its drift in the electric field
generated by high-voltage cables and detectors and also convective motion.
Because of its very high $\beta^{-}$ $Q$-value of $3.5$ MeV, higher than
$Q_{\beta\beta}$, 42K contributes to the backgrounds in $0\nu\beta\beta$
searches. In LEGEND-200, the drift of 42K towards germanium detectors will be
reduced by nylon shrouds around each detector’s columns (following the Gerda
approach). LEGEND-1000, because of more stringent background requirements,
will make use of underground Ar, which is free in 42Ar, thus almost completely
removing the 42K background.

In addition to the aforementioned techniques, LEGEND will follow the Gerda and
Majorana approach in the background reduction techniques through the selection
of SSEs, $0\nu\beta\beta$-like events. The rejection of MSEs will be performed
by applying anti-coincidence cuts and PSD methods, as illustrated in Sections
5.2 and 6.3. Figure 15a shows the expected effectiveness of these cuts in
terms of the reduction of the $\gamma$ rate due to the 238U, 232Th and 40K
chains Zsigmond (2020). Figure 15b shows the projected contribution of the
different sources to the overall total background. A total background of 1
$\times~{}10^{-4}$ cts/(keV$\cdot$kg$\cdot$yr)is anticipated Edzards (2020).

Figure 15: (a) Reduction of the $\gamma$ rate due to the 238U, 232Th and 40K
chains. Figure published by J. Phys. Conf. Ser., 2020 Zsigmond (2020). (b)
Projected contribution of the different sources to the overall total
background. Grey bars indicate 1$\sigma$ uncertainties of the background
contributions due to screening measurements and Monte Carlo simulations. A
total background of 1 $\times~{}10^{-4}$ cts/(keV$\cdot$kg$\cdot$yr) is
anticipated. Figure published by J. Phys. Conf. Ser., 2020 Edzards (2020).

##  8 Conclusions

In this paper, we reviewed the successful story of the experiments employing
germanium semiconductor detectors in the search for the $0\nu\beta\beta$
transition of 76Ge. Starting from the pioneering work conducted by E. Fiorini
and collaborators in 1967, we followed the technology evolution from the
original Ge(Li) diodes to the development of the modern HPGe detectors
featuring a negligible intrinsic background contamination and impressive
background rejection capabilities and energy resolution. Parallel to the
development of the detectors, an outstanding lowering of the background index
by about eight orders of magnitude led to an increase in the sensitivity of
the experiment by about six orders of magnitude in 50 years. Currently, the
Gerda experiment, implementing the use of bare Ge diodes immersed in LAr,
succeeded at reaching the background-free regime, thus achieving the best
sensitivity in the field with a lower exposure with respect to competitors.

The joint venture between Gerda and Majorana Collaborations led to
establishing the LEGEND project, aiming at building a ton-scale experiment to
fully span the inverted mass ordering region with a world-leading and timely
competitive program.

Writing - original draft, Valerio D’Andrea, Natalia Di Marco, Carla Macolino,
Michele Morella, Francesco Salamida; Writing - review & editing, Valerio
D’Andrea, Natalia Di Marco, Matthias Bernhard Junker, Matthias Laubenstein,
Carla Macolino, Michele Morella, Francesco Salamida, Chiara Vignoli. All
authors have read and agreed to the published version of the manuscript.

This research received no external funding

Not applicable

Not applicable

Data sharing not applicable

###### Acknowledgements.

The authors would like to thank Riccardo Brugnera and Steven Ray Elliott for
their suggestions and comments. The authors declare no conflict of interest.
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