Strict procedures are also followed to avoid contaminating the materials
through exposure to Rn. Sensitive parts are typically stored or even assembled
in radon-free environments. Small volumes such as storage vessels or glove
boxes are flushed with boil-off nitrogen from large liquid nitrogen dewars
Wojcik  _et al._ (2017), while larger environments such as clean rooms can be
flushed with Rn-free air obtained from dedicated radon abatement systems
Benato  _et al._ (2018). Rn emanation from material within the detector is
especially problematic for Xe-based experiments, due to a $\gamma$ line from
214Bi at 2448 keV, very close to the 136Xe $Q_{\beta\beta}$. Continuous Xe
purification has been demonstrated via adsorption on activated charcoals Abe
_et al._ (2012a) or cryogenic distillation Aprile  _et al._ (2017). Similarly,
Rn suppression by $\geq 3$ orders of magnitude was also demonstrated via
distillation on $n$-dodecane, a common admixture in liquid scintillator Keefer
_et al._ (2015).

In addition to the maximization of the detector radio-purity, the actinide
purity of the surrounding components and laboratory environment must be kept
under control as well. In this case, high-energy $\gamma$ rays are the most
worrisome component due to their long attenuation lengths. $\beta\beta$-decay
detectors must therefore be completely enveloped by a material capable of
efficiently absorbing $\gamma$ radiation without inducing further background.
This can be achieved in several ways, including: a set of concentric passive
(non-instrumented) layers of shielding material with increasing radio-purity,
typically Pb and Cu Alduino  _et al._ (2017c); Abgrall  _et al._ (2014); a
high-purity cryogenic liquid such as Ar possibly instrumented to detect the
scintillation light produced by incoming radiation Agostini  _et al._ (2018b);
for liquid scintillator experiments, the division of the detector medium into
an inner region loaded with the $\beta\beta$ isotope and an outer region with
no isotope working as an active shield Gando  _et al._ (2012); Andringa  _et
al._ (2016). These shielding layers are designed to be thick enough to
eliminate external radiation as a concern.

####  V.3.3 Anthropogenic radioactivity

Background can be induced by anthropogenic radioactivity, in particular as a
result of nuclear accidents or nuclear weapon tests. The great majority of
these isotopes are $\beta$ emitters, and in some case are the progenitor of a
decay chain. In order to represent a potential background source for
$0\nu\beta\beta$-decay experiments, the decay chains must include an isotope
with a $Q$-value greater than $Q_{\beta\beta}$, and a dominant half-life on
the same order as an experiment’s lifetime. Table 3 lists some examples of
potentially worrisome isotopes with $Q_{\beta}>2$ MeV reported by IAEA (2015).
Of these, only ${}^{108\text{m}}$Ag has been detected so far Gando  _et al._
(2012). The actual relevance of these isotopes as potential backgrounds must
be assessed for each experiment separately. While a pure $\beta$ emitter such
as 144Pr could be worrisome for most experiments, an isotope that also emits
$\gamma$ rays (e.g., ${}^{108\text{m}}$Ag) could be tagged with event topology
reconstruction capabilities.

Table 3: Anthropogenic isotopes. Isotopes belonging to the same chain are grouped between horizontal lines. Isotope | Half-life |  $Q_{\beta}$ [keV] | Detected | Notes
---|---|---|---|---
88Y | 107 d | 3008 | No | Several $\gamma$ lines
90Sr | 28.8 y | 546 | No |
90Y | 64 h | 2279 | No | Pure $\beta$ emitter
${}^{110\text{m}}$Ag | 250 d | 3008 | Yes | Several $\gamma$ lines
134Cs | 2 y | 2059 | No | Several $\gamma$ lines
144Ce | 285 d | 319 | No |
144Pr | 17.3 m | 2997 | No | Pure $\beta$ emitter

####  V.3.4 Neutrons

In the previous sections we have mentioned several processes producing
neutrons. The actual background induced in a $0\nu\beta\beta$-decay experiment
depends on the neutron flux and energy spectrum, on the location of neutron
emission, and on the employed materials. In the present context, neutrons can
be divided in two groups: above surface neutrons originating from cosmic rays
in the atmosphere, and underground neutrons produced by muon spallation,
$(\alpha,n)$ reactions on light nuclei, and spontaneous fission reactions,
mainly from 238U. Above ground neutrons represent the dominant cause of
cosmogenic activation in detector materials prior to their installation
underground, previously discussed in Sec. V.3.1. Underground neutrons can be
further divided between external neutrons generated in the rock or in the
concrete walls, and internal neutrons generated inside or next to the
detector. Neutrons from $(\alpha,n)$ and fission reactions have energies
$\lesssim 10$ MeV Wulandari  _et al._ (2004), while those from spallation can
reach several GeV Mei and Hime (2006). The activity of underground neutrons
from $(\alpha,n)$ and fission reactions is only of order $\sim$1 n/yr/g, but
the high mass of rock within a scattering length of the neutrons yields a flux
2–3 orders of magnitude higher than that of neutrons from spallation Wulandari
_et al._ (2004).

The flux of neutrons from $(\alpha,n)$ and fission reactions is efficiently
suppressed by neutron shielding of moderate thickness, located outside the
$\gamma$ shielding. One possible strategy is to enclose the experiment in a
thick layer (a few tens of cm) of neutron moderator such as polyethylene, with
the innermost layer (a few cm) comprised of a material with high neutron
absorption cross section (e.g., boric acid or borated polyethylene Abgrall
_et al._ (2014); Alduino  _et al._ (2018)). The outer layer slows down the
neutrons to thermal energies, while the inner one captures them.
Alternatively, a $\geq$1 m layer of water can be used both for moderation and
absorption: in this way, a water tank can simultaneously act as a neutron
shield and a muon veto detector Ackermann  _et al._ (2013). In the case of
liquid scintillator detectors, the outermost scintillator region serves as a
very effective neutron moderator, providing active tagging in addition to high
neutron capture capability.

While external neutrons with energies $\lesssim 10$ MeV are efficiently
suppressed with a neutron shield, high energy neutrons can still reach the
detector. Additional neutrons can also be produced within the neutron shield
by muons or, again, $(\alpha,n)$ and fission reactions taking place in the
$\gamma$ shield, the active material, or in calibration sources Baudis  _et
al._ (2015). These neutrons can undergo elastic and inelastic scattering, or
be captured to produce stable or unstable nuclei and possibly yield prompt
$\gamma$ de-excitations. The interactions induce a variety of signatures that
strongly depend on the detector technology and employed materials. The most
worrisome for $0\nu\beta\beta$-decay experiments are the in-situ activation of
long-lived isotopes in or next to the detector, and inelastic scatterings
yielding penetrating $\gamma$ rays with energies comparable to
$Q_{\beta\beta}$.

Finally, one possible technique to minimize the neutron induced background
consists in embedding in the $\gamma$ shielding or in the detector medium a
material with high-cross section for neutron capture, and possibly that
produces events with a signature that does not mimic a $0\nu\beta\beta$-decay
event. An example could be 6Li, which undergoes the
${}^{6}\text{Li}+n\rightarrow\alpha+^{3}\text{H}$ reaction with a Q-value of
4.8 MeVPoda and Giuliani (2017).

####  V.3.5 Neutrinos

Neutrinos represent a potential source of irreducible background for
$0\nu\beta\beta$-decay experiments. Sources with appreciable neutrino fluxes
include solar neutrinos, atmospheric neutrinos, geoneutrinos, reactor
neutrinos and the diffuse supernova neutrino background (DSNB). Due to their
higher flux at energies below 20 MeV, solar neutrinos are the most worrisome
background source. Their spectrum is composed of several contributions
corresponding to the primary nuclear reactions in the sun (Fig. 18). Solar
neutrinos can undergo two types of interactions in a $0\nu\beta\beta$-decay
experiment, elastic scattering (ES) and charged current (CC) interactions
Elliott and Engel (2004):

ES: $\displaystyle\nu+\text{e}^{-}$ $\displaystyle\rightarrow\
\nu+\text{e}^{-}$ (42) CC: ${}^{\text{Z}}\text{A}+\nu$
$\displaystyle\rightarrow\
^{\text{Z}+1}\text{A}+e^{-}~{}[+\gamma(\text{s})]+Q_{\nu}$ (43)
${}^{\text{Z}+1}\text{A}$ $\displaystyle\rightarrow\
^{\text{Z}+2}\text{A}+\beta^{-}+\nu~{}[+\gamma(\text{s})]+Q_{\beta}$ (44)

Figure 18: Solar neutrino spectra. Data from Agostini  _et al._ (2018a);
Bergstrom  _et al._ (2016); Bahcall (1994); Bahcall  _et al._ (1996); Bahcall
and Ulrich (1988). The curves labeled hep, pep, and pp correspond to neutrinos
emitted in helium-proton, proton–electron–proton, and proton-proton fusion,
respectively.

In ES, a neutrino scatters off an electron in the detector. The electron is
scattered non-isotropically, and its energy spectrum is a continuum up to the
end-point of the solar neutrino spectrum, $\sim 19$ MeV. Only neutrinos with
energy $E_{\nu}>$$Q_{\beta\beta}$ contribute, and given the low flux of hep
neutrinos, in practice only 8B neutrinos are relevant (Fig. 18). The expected
background per unit sensitive mass is $\sim 2\cdot 10^{-7}$ counts/keV/kg/yr
Elliott and Engel (2004); de Barros and Zuber (2011). For current and next-
generation experiments this is negligible for experiments in which the active
material is mostly made of the $\beta\beta$ isotope, but becomes significant
for liquid scintillator experiments with dissolved sources Elliott and Engel
(2004); Andringa  _et al._ (2016); de Barros and Zuber (2011). A partial
suppression of the ES background might be achievable exploiting the signal
directionality Askins  _et al._ (2020), at a non-negligible cost in terms of
signal efficiency.

In CC interactions, the $\beta\beta$ isotope $(A,Z)$ undergoes inverse $\beta$
decay to the ground state or an excited state of the $(A,Z\\!\\!+\\!\\!1)$
isotope (Eq. 43), the intermediate isotope of the $\beta\beta$ decay to
$(A,Z\\!\\!+\\!\\!2)$. Since $(A,Z\\!\\!+\\!\\!1)$ is heavier than $(A,Z)$,
the reaction has a threshold of $E_{t}=m_{A,Z+1}-m_{A,Z}$, so neutrinos with
energy $E_{\nu}\simeq E_{t}+$$Q_{\beta\beta}$ pose a possible background in
this prompt event. In some cases the intermediate nucleus can then capture an
electron and decay back to the original $\beta\beta$ parent isotope, but it
more often undergoes $\beta$- decay to the same final state of the
$\beta\beta$ decay, $(A,Z+2)$ (Eq. 44), releasing an energy
$Q_{\beta}>Q_{\beta\beta}$ that can pose a delayed background. The actual
relevance of CC interactions as a background depends on the $\beta\beta$
isotope and the corresponding value of $E_{t}$, the emission of de-excitation
$\gamma$ rays in the two involved reactions that could modify the event
topology, and the half-life of the intermediate state, which could allow a
time correlation analysis. Without applying any of these event
identifications, the intermediate nucleus $\beta$ decay yields a background of
$10^{-3}$–$10^{-1}$ events/keV/ton${}_{\text{iso}}$/yr Ejiri and Elliott
(2014, 2017).

Other neutrino sources do not represent a significant background source
primarily due to their low flux. However the presence of antineutrinos in
these sources requires consideration of additional inverse $\beta$
interactions that could give a background, particularly at sites with
appreciable reactor neutrino fluxes. In the case of atmospheric and DSNB
neutrinos, the CC interaction energies are also so high that the nucleus is
often left in a highly excited state, leading to background signatures similar
to in-situ cosmogenic activation but with much lower production rate.

####  V.3.6 $2\nu\beta\beta$ decay

The only intrinsic and irreducible background for $0\nu\beta\beta$ decay is
the concurrent $2\nu\beta\beta$-decay channel. The two processes induce a
similar event topology, with the exception of the energy signature: a peak at
$Q_{\beta\beta}$ for $0\nu\beta\beta$ decay, and a continuum from zero to
$Q_{\beta\beta}$ for $2\nu\beta\beta$ decay (Fig. 12). The detector resolution
results in some of the highest energy $2\nu\beta\beta$-decay events
reconstructing with energies at $Q_{\beta\beta}$. Although the event
topologies differ in detail — in the energy distributions and angular
correlations between the emitted electrons Kotila and Iachello (2012b) — those
differences can only be leveraged at high statistics with tracking detectors
capable of making those measurements. Thus the relevance of the
$2\nu\beta\beta$-decay background depends primarily on the energy resolution
and the $2\nu\beta\beta$-decay half-life. In practice, $2\nu\beta\beta$-decay
contributes significantly to the background only for experiments with a
resolution of some percent. Additionally, if the $2\nu\beta\beta$-decay rate
is high compared to the desired $0\nu\beta\beta$-decay sensitivity,
$2\nu\beta\beta$-decay events can pile-up and contribute to the background at
$Q_{\beta\beta}$. In practice, this is relevant only for cryogenic
calorimeters using 100Mo as candidate isotope Armatol  _et al._ (2020).

###  V.4 Signal and background discrimination techniques

In the previous sections we highlighted some of the features that distinguish
each possible background component from the $0\nu\beta\beta$-decay signal. In
this section we collect and summarize the experimental techniques available to
discriminate the two. Although many of these techniqes have been mentioned in
the previous sections, our aim is to achieve a concise listing. We organize
the discussion according to the key features of the signal. The primary
signature is a peak at $Q_{\beta\beta}$ in the sum energy spectrum.
$0\nu\beta\beta$-decay events must also be homogeneously distributed in time
and space, with a rate proportional to the fraction of target isotope in the
active material. The electrons are subject to a localized energy deposition,
whose dimension depends on the electron attenuation length: $O(1-10)$ mm for
solids and liquids, and $O(10)$ cm for high-pressure gases. The events are
uncorrelated with any other physical processes, and the final state includes
the presence a particular daughter nucleus.

As described above, energy is the only necessary and sufficient observable for
a discovery, hence energy resolution is crucial to minimize the background
level in the vicinity of $Q_{\beta\beta}$. Of particular concern are the
irreducible $2\nu\beta\beta$-decay contribution that extends up to
$Q_{\beta\beta}$, and emissions of the 238U and 232Th decay chains ($\alpha$,
$\beta$, or $\gamma$ particles) with energies greater than $Q_{\beta\beta}$.
Often the background at $Q_{\beta\beta}$ can be approximated as flat. If not,
a spectral fit over a larger energy region is required to properly constrain
the background at $Q_{\beta\beta}$ using the information obtained from the
rest of the spectrum. Since the lifetime of an experiment spans several years,
calibrating the energy scale and monitoring its stability over time is
fundamental for avoiding any degradation of energy resolution in the physics
spectrum, and for a precise characterization of the detector response.

The expected $0\nu\beta\beta$-decay signal is uniform in the volume containing
the isotope. The same is true for some background processes — e.g.,
$2\nu\beta\beta$ decay, neutrino, and often neutron reactions — but does not
hold for others, especially those induced by natural or anthropogenic
radioactive contaminants located outside the detector. Thus the detector
medium can act as a self-shield, with the inner part subject to a lower
background than the outer one. This is a natural feature of monolithic
experiments, while for granular experiments it can be approximated by dividing
the detectors into concentric layers.

The electrons emitted by $0\nu\beta\beta$ decay carry a directional
correlation due to the angular momentum exchanged by the mediating mechanism.
However the direction of any one electron emitted in sequential decays are
uncorrelated. On the other hand, some backgrounds, for example the elastic
scattering of solar neutrinos with electrons, are not isotropic. Directional
reconstruction, e.g., with the detection of Cherenkov light, is therefore
useful for suppressing these backgrounds.

The event topology of a $0\nu\beta\beta$ decay is clearly defined for each
detector technology: an energy deposition contained in $O(1-1000)$ mm3 in a
solid or liquid detector; a track of $O(10-30)$ cm length with two blobs at
its extremes in a high-pressure gas TPC; a pair of distinguished electron
tracks with a common origin in a low-pressure tracking detector. Depending on
the detector spatial resolution, several particles might be distinguishable:
muons induce long tracks that cross the detector medium, or hit multiple
detectors of a granular experiment, and generate a signal in muon veto
detectors; $\gamma$ rays have a longer range and can undergo Compton
scattering, inducing multiple energy depositions at different locations
independently of the detector technology; $\alpha$ particles have a shorter
range easily identifiable in gas detectors; $\beta$ particles produce a track
with a single blob in a high-pressure TPC, or a single track in a tracking
detector. These signatures can also be combined and thus facilitate their
identification, e.g., in the case of a radioactive isotope decaying via
$\alpha$ or $\beta$ decay with the subsequent emission of de-excitation
$\gamma$ rays.

For some readout channels such as scintillation and Cherenkov light, different
particles induce a different detector response. Therefore an additional means
of background suppression is particle identification via signal shape
analysis. A common strategy is having multiple read-out channels: one
optimized for energy reconstruction, the other for particle discrimination.
Examples are scintillation experiments with Cherenkov readout for $\alpha$ and
single $\beta$ identification, or cryogenic calorimeters with scintillation
light readout for $\alpha$ vs $\beta$/$\gamma$ discrimination.

$0\nu\beta\beta$ decays are homogeneous in time and uncorrelated with anything
else. Conversely, some backgrounds are identifiable due to their specific time
correlations. This is the case for delayed coincidences between the decays of
several isotopes in the 238U and 232Th chain (e.g., the Bi-Po sequences, see
Fig. 17), between the decay of a cosmogenically activated isotope (e.g., 68Ga
in Ge) and the detection of its parent muon in the muon veto, or between
inverse $\beta$ decay and the subsequent $\beta$ decay in solar neutrino CC
reactions.

Finally, daughter nucleus tagging is an additional tool for background
suppression, which distinguishes $\beta\beta$ decays (but not exclusively
$0\nu\beta\beta$ decays) from anything else except solar neutrino CC
reactions. Another background characterization method is abundance scaling,
where different measurements with enriched vs non-enriched materials or loaded
vs non-loaded active material allow an experiment to isolate backgrounds that
are not related to the presence of the $\beta\beta$ decay isotope.

###  V.5 Statistical analysis and sensitivity

####  V.5.1 Signal extraction

As discussed in the previous sections, all $0\nu\beta\beta$-decay experiments
measure multiple observables for each event. Some observables are related to
the amount and spatial distribution of the energy deposited within the
detector. Others are related to the timing and type of particles involved in
the event. The value of several of these observables is well-defined for
$0\nu\beta\beta$-decay events. For instance, each event should have energy
equal to the decay Q-value, and be contained within a small region of the
detector. Background events will also have specific features, resulting in
characteristic observable values.

All experiments in the field use a multivariate analysis to extract the
sought-after $0\nu\beta\beta$-decay signal. The observables define the basis
of a multi-dimensional parameter space, in which signal and background events
are distributed according to multivariate probability distributions. Since
$0\nu\beta\beta$-decay events have well-defined features, the bulk of their
probability distribution will be restricted to a small volume of the parameter
space. On the other hand, most of the background events will be outside of
this small volume, populating other regions. Thus the signature of a possible
$0\nu\beta\beta$-decay signal is an excess of events compared to the
background expectation in a narrow region of the multi-dimensional parameter
space. We will refer to this volume with a maximum signal-to-background ratio
as the sensitive volume. The rest of the parameter space is effectively used
to constrain the background contribution to the sensitive volume.

The signal and background probability distributions are often well separated
for one or more observables. In such cases, it is advantageous to apply a cut
on such observables, discarding background data without a significant
reduction of the signal-detection efficiency, while decreasing the
dimensionality of the parameter space, and also reducing systematic
uncertainty due to imperfect knowledge of the distributions in the
observables. These considerations make applying cuts often advantageous even
when there is some overlap, between signal and background, although in such
cases the reduced dimensionality and systematic uncertainty must be weighed
against any loss of statistical precision. For observables where signal and
background strongly overlap, a full multivariate fit is unavoidable. However
in many experiments, the signal-background separation is good for most of the
observables, and the multivariate analysis can be simplified into a
1-dimensional fit of the energy spectrum with negligible degradation of
sensitivity.

The analysis techniques of the field at present are rather established and
uniform. Most experiments report frequentist maximum likelihood fits based on
likelihood ratio tests Zyla  _et al._ (2020). Monte-Carlo parametric-
boostrapping methods are often used to compute the test statistic probability
distributions or to confirm their behavior when asymptotic formulae are
assumed Cowan  _et al._ (2011). Given the low counting rate and the fact that
the parameter of interest is constrained to non-negative values, the test
statistic distribution can significantly differ from a chi-square function
Algeri  _et al._ (2019). While frequentist methods have historically been
dominant Cousins (1995), recently most experiments also report results based
on Bayesian methods, with inference deriving from the marginalized posterior
distribution. Given the lack of a strong signal to date, the choice of prior
distribution continues to significantly affect posterior probabilities.

####  V.5.2 Discovery and exclusion sensitivity

The reach of $0\nu\beta\beta$-decay experiments is traditionally quoted in
terms of discovery and exclusion sensitivity, two statistical concepts
belonging to frequentist inference. The discovery sensitivity corresponds to
the expected number of signal events for which an experiment has 50% chance to
observe an excess of events over the background at 99.73% confidence level
(CL). The exclusion sensitivity corresponds to the expected number of signal
events that an experiment has 50% chance of excluding at 90% CL.

The discovery and exclusion sensitivity confidence levels are less stringent
compared to other fields — e.g., the particle accelerator community — due to
the lack of a “look elsewhere” effect (the $0\nu\beta\beta$ peak must occur at
$Q_{\beta\beta}$) and simpler-to-control systematic uncertainties NSAC NLDBD
Subcommittee (2014, 2015). In particular, the CL required for a discovery is
equivalent to excluding 3$\sigma$ two-sided fluctuation of a Gaussian random
variable, and not one-sided 5$\sigma$ fluctuation as for accelerator
experiments. Other sensitivity definitions have been used Caldwell and
Kroninger (2006); Alduino  _et al._ (2017a), including Bayesian concepts based
on Bayes factors and posterior distributions, but these are not common in the
field at present.

A precise evaluation of the expected number of signal events ($\lambda_{s}$)
fulfilling the sensitivity definitions above requires calculations considering
probability distributions in the multivariate space and experiment-specific
information. However, it can be approximated by considering a counting
analysis in the sensitive volume, with known background expectation
$\lambda_{b}$. Uncertainties on $\lambda_{b}$ can usually be neglected, as
experiments are able to constrain its value using large background-dominated
regions of the multivariate parameter space. As both the signal and background
events are generated by Poisson random processes, the discovery sensitivity
can be calculated by solving this system of equations:

$\begin{cases}P(X\leq x|\lambda_{b})\geq 99.73\%\\\ P(X\geq
x|\lambda_{b}+\lambda_{s})\geq 50\%\,,\end{cases}$ (45)

where $P(X\leq x|\lambda)$ is the Poisson probability of observing a number of
events $X$ smaller or equal to $x$ given an expectation $\lambda$. For a given
$\lambda_{b}$ value, the system has a unique solution that can be found by
calculating the minimum $x$ satisfying the first equation, substituting it in
the second equation, and then finding the minimum $\lambda_{s}$ that satisfies
the resulting inequality. The exclusion sensitivity can be similarly computed
by solving:

$\begin{cases}P(X\leq x|\lambda_{b})\geq 50\%\\\ P(X\geq
x|\lambda_{b}+\lambda_{s})\geq 90\%\,.\end{cases}$ (46)

Although the Poisson mass function is discrete, an actual multivariate fit
operates with non-integer number of events in the form of the probability
distribution weights for each event. We can reproduce this behavior by
interpolating the Poisson mass function with a normalized upper incomplete
gamma function, and re-define the probability in the systems above as:

$P(X>x|\lambda)\doteq\frac{\Gamma(x+1,\lambda)}{\Gamma(x+1)}\,.$ (47)

This definition leads to the discovery and exclusion sensitivity shown in Fig.
19. Also shown in Fig. 19 is an approximation of Eqs. 45 from Cowan  _et al._
(2011) using elementary functions.

The discovery sensitivity degrades rapidly as the expected number of
background events increases: the greater $\lambda_{b}$, the greater
$\lambda_{s}$ must be to create an excess incompatible with background
fluctuations. For large enough values, the background fluctuations become
Gaussian and the sensitivity scales only as $\sqrt{\lambda_{b}}$. For
instance, when $\lambda_{b}=100$, a 3$\sigma$ discovery sensitivity requires
$\lambda_{s}\geq 3\sqrt{100}=30$. Conversely, the lower the background, the
lower the signal expectation needs to be for a discovery. For any
$\lambda_{b}\leq-\ln(99.73\%)\sim 0.0027$, the experiment has more than 99.73%
probability of observing no background events, and the observation of a single
event becomes a discovery 141414For a truly zero background experiment, one
event is enough to claim a discovery. In a similar fashion, encountering a
unicorn is enough to claim its existence, provided that we have a template of
a unicorn to which to compare the observed candidate.. In this “background-
free” regime, the discovery sensitivity saturates: the first of Eqs. (45) is
always solved for $x=0$, and so the second one is solved for
$\lambda_{s}\lesssim-\ln(50\%)\sim 0.69$. In between these two regimes, for
$\lambda_{b}<1$, the sensitivity is not independent of $\lambda_{b}$ but only
degrades weakly with increasing $\lambda_{b}$. Experiments in this “quasi-
background-free” regime reap most of the benefits of a background-free
experiment, but still require a few signal counts to claim an observation.

The exclusion sensitivity behaves similarly to the discovery sensitivity, but
it saturates for larger background expectations, at $\lambda_{b}=-\ln(50\%)$.
Below this threshold, the experiment always has $\geq$50% probability to
observe no background events, and a further reduction of the background
expectation cannot improve the median upper limit on the signal strength. The
first of Eqs. 46 is thus always solved for $x=0$, and the second one yields
$\lambda_{s}=\ln(1-90\%)\sim 2.3$ events.

The saturation of both sensitivities is connected to the properties of the
Poisson probability, and is thus an intrinsic feature of the frequentist
median sensitivity. This behavior can be problematic when the sensitivity is
used as a figure of merit to optimize or compare experiments, and alternative
sensitivity definitions have been recently proposed Bhattiprolu  _et al._
(2021).

Figure 19: Median 99.7% CL discovery sensitivity and median 90% exclusion
sensitivity as a function of the expected number of background events. The
discovery sensitivity shows the signal event expectation at which an
experiment has 50% chance to observe a 99.7% CL excess of events over the
background. The exclusion sensitivity is instead the signal event expectation
that can be excluded at 90% CL with 50% probability, assuming that there is no
signal. The plot shows also the approximated discovery sensitivity extracted
using the asymptotic formulae from Cowan  _et al._ (2011).

The expected number of signal and background events in an experiment can be
computed starting from two effective parameters, the sensitive background
($\mathcal{B}$) and sensitive exposure ($\mathcal{E}$). As they are connected
to the sensitivity, these parameters capture well the performance of an
experiment. The sensitive exposure is the product of: the number of moles of
isotope in the active fiducial detector volume, the live time, and the signal
detection efficiency, i.e., the probability for a $0\nu\beta\beta$-decay event
to occur in the sensitive volume. The sensitive background is the number of
background events in the sensitive volume after all analysis cuts, divided by
$\mathcal{E}$. Using these definitions, the expected number of signal and
background counts in the sensitive volume is given by:

$\lambda_{s}(T_{1/2})=\dfrac{\ln 2\cdot
N_{A}\cdot\mathcal{E}}{T_{1/2}}\qquad\text{and}\qquad\lambda_{b}=\mathcal{B}\cdot\mathcal{E}$
(48)

where $N_{A}$ is Avogadro’s number and $T_{1/2}$ is the half-life of the
decay. Given that $\lambda_{s}$ depends on $T_{1/2}$, the discovery and
exclusion sensitivities on the expected number of events can be directly
translated into sensitivities on the $0\nu\beta\beta$-decay half-life.
$T_{1/2}$ sensitivities are the most common parameter reported by the
experiments.

Next-generation searches aim to monitor tons of material for a decade,
reaching sensitive exposures at the level of $\mathcal{E}\sim 10^{5}-10^{6}$
$\text{mol}\cdotp\text{yr}$. Such an exposure gives the possibility to observe
a handful of signal events even for $0\nu\beta\beta$-decay half-life values of
$10^{27}-10^{28}$ years. As illustrated by Fig. 19, a requirement for
discovery is that the number of background events is similar to the number of
expected signal events. Thus experiments aim at reaching backgrounds at the
level of $\mathcal{B}<10^{-4}-10^{-5}$ events per mole of material per year.
The concepts proposed to achieve this incredibly challenging performance are
described in the next section.

##  VI Recent and future experiments

A broad experimental program has been mounted in the last two decades to
search for $0\nu\beta\beta$ decay. Very diverse technologies have been
developed and tested, leading to experiments with half-life sensitivities up
to $10^{26}$ years. Thanks to these achievements, a number of new projects are
being prepared with the goal of increasing the sensitivity by up to two orders
of magnitude, opening the window to new energy frontiers and conclusively
testing the scenario in which $0\nu\beta\beta$ decay is mediated by inverted-
ordered neutrinos Agostini  _et al._ (2021b).

In this section, we discuss recent and future phases of existing experiments.
In Sec. VI.1, we review the experimental landscape and use the experiments’
key performance parameters to evaluate their strength, strategy, and
sensitivity. We then focus on the detection concept and technical aspects of
each project. Experiments based on high-purity germanium detectors are
reviewed in Sec. VI.2, time-projection chambers in Sec. VI.3, large liquid
scintillator detectors in Sec. VI.4, cryogenic calorimeters in Sec. VI.5, and
tracking calorimeters in Sec. VI.6. New technologies that are currently being
tested and newly proposed experimental designs are summarized in Sec. VI.7.
Sec. VI.1 should be accessible to all readers, while the other sections listed
above are intended for more expert readers.

###  VI.1 Experimental landscape

Each experiment is characterized by a set of key performance parameters that
are captured by the concepts of sensitive exposure ($\mathcal{E}$) and
sensitive background ($\mathcal{B}$) defined in Sec. V.5.2. The sensitive
exposure and background are directly connected to the half-life sensitivity,
and carry valuable information on the strategy pursued by each project.
Indeed, different combinations of $\mathcal{E}$ and $\mathcal{B}$ can give the
same sensitivity, and exposure increase can be traded for background reduction
or vice versa.

The sensitive exposure and background are effective parameters whose values
are often not intuitive: they refer to the detector performance in the
sensitive volume, where the signal-to-background ratio is maximal and drives
the experimental sensitivity. We calculate the value of $\mathcal{E}$ starting
from the product of isotope mass $m_{iso}$ and data taking time, and correct
it for a number of efficiencies: the active (or fiducial) fraction of the
target mass $\varepsilon_{\text{act}}$, the probability that the energy
deposited by the decay is fully contained within the detector
$\varepsilon_{\text{cont}}$, and the multivariate analysis efficiency to tag
events in the sensitive volume $\varepsilon_{\text{mva}}$. Although
$\varepsilon_{\text{mva}}$ would conceptually include the efficiency for a
signal to fall in the energy region of interest (ROI) dominating the
sensitivity, we separate this contribution and also quote the energy
resolution ($\sigma$) as well as the width of the effective ROI in units of
$\sigma$, assuming a Gaussian approximation. To calculate $\mathcal{B}$, we
extract the rate of background events in the sensitive volume from the
experiments’ specifications. The values of these parameters and efficiencies
are listed in Tab. 4 and shown graphically in Fig. 20. When the value of a
parameter cannot be computed from published specifications, we report
effective values reproducing the nominal sensitivity of the experiment.
Details of these approximations are discussed in the following subsections.

Table 4: Fundamental parameters driving the sensitive background and exposure of recent and future phases of existing experiments. The last two columns report the discovery sensitivity on the $0\nu\beta\beta$-decay half-life for 10 years of livetime, and the corresponding sensitivity on $m_{\beta\beta}$ for the range of NMEs specified in Tab. 1. For completed experiments, sensitivities are computed using the reported final exposure. MJD, KLZ, and SuperNEMO-D refer to the Majorana Demonstrator, KamLAND-Zen, and the SuperNEMO Demonstrator, respectively. Experiment | Isotope | Status | Lab | $m_{\text{iso}}$ |  $\varepsilon_{\text{act}}$ |  $\varepsilon_{\text{cont}}$ |  $\varepsilon_{\text{mva}}$ | $\sigma$ | ROI |  $\varepsilon_{\text{ROI}}$ | $\mathcal{E}$ | $\mathcal{B}$ | $\lambda_{b}$ | $T_{1/2}$ | $m_{\beta\beta}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
[mol] |  [%] |  [%] |  [%] | [keV] | [$\sigma$] |  [%] | $\left[\frac{\text{mol}\cdotp\text{yr}}{yr}\right]$ | $\left[\frac{\text{events}}{\text{mol}\cdot\text{yr}}\right]$ | $\left[\frac{\text{events}}{\text{yr}}\right]$ | [yr] | [meV]
|  |  |  |
High-purity Ge detectors (Sec. VI.2) |  |  |  |  |  |  |  |  |
| GERDA-II |  76Ge | completed | LNGS | $4.5\cdot 10^{2}$ |  88 |  91 |  79 | $1.4$ | -2,2 |  95 | 273 | $4.2\cdot 10^{-4}$ | $1.1\cdot 10^{-1}$ | $1.2\cdot 10^{26}$ |  093-222
| MJD |  76Ge | completed | SURF | $3.1\cdot 10^{2}$ |  91 |  91 |  86 | $1.1$ | -2,2 |  95 | 212 | $3.3\cdot 10^{-3}$ | $7.1\cdot 10^{-1}$ | $4.7\cdot 10^{25}$ | 149-355
| LEGEND-200 |  76Ge | construction | LNGS | $2.4\cdot 10^{3}$ |  91 |  91 |  90 | $1.1$ | -2,2 |  95 | 1 684 | $1.0\cdot 10^{-4}$ | $1.7\cdot 10^{-1}$ | $1.5\cdot 10^{27}$ | 27-63
| LEGEND-1000 |  76Ge | proposed |  | $1.2\cdot 10^{4}$ |  92 |  92 |  90 | $1.1$ | -2,2 |  95 | 8 736 | $4.9\cdot 10^{-6}$ | $4.3\cdot 10^{-2}$ | $1.3\cdot 10^{28}$ |  09-21
Xenon time projection chambers (Sec. VI.3) |  |  |  |  |  |  |  |  |  |  |  |  |
| EXO-200 |  136Xe | completed | WIPP | $1.2\cdot 10^{3}$ |  46 |  100 |  84 | $31$ | -2,2 |  95 | 438 | $4.7\cdot 10^{-2}$ | $2.1\cdot 10^{+1}$ | $2.4\cdot 10^{25}$ | 111-477
| nEXO |  136Xe | proposed | SNOLAB | $3.4\cdot 10^{4}$ |  64 |  100 |  66 | $20$ | -2,2 |  95 | 13 700 | $4.0\cdot 10^{-5}$ | $5.5\cdot 10^{-1}$ | $7.4\cdot 10^{27}$ |  06-27
| NEXT-100 |  136Xe | construction | LSC | $6.4\cdot 10^{2}$ |  88 |  76 |  49 | $10$ | -1.0,1.8 |  80 | 167 | $5.9\cdot 10^{-3}$ | $9.9\cdot 10^{-1}$ | $7.0\cdot 10^{25}$ |  066-281
| NEXT-HD |  136Xe | proposed |  | $7.4\cdot 10^{3}$ |  95 |  89 |  44 | $7.7$ | -0.5,1.7 |  65 | 1 809 | $4.0\cdot 10^{-5}$ | $7.2\cdot 10^{-2}$ | $2.2\cdot 10^{27}$ | 12-50
| PandaX-III-200 |  136Xe | construction | CJPL | $1.3\cdot 10^{3}$ |  77 |  74 |  65 | $31$ | -1.2,1.2 |  76 | 374 | $3.0\cdot 10^{-3}$ | $1.1\cdot 10^{+0}$ | $1.5\cdot 10^{26}$ |  045-194
| LZ-nat |  136Xe | construction | SURF | $4.7\cdot 10^{3}$ |  14 |  100 |  80 | $25$ | -1.4,1.4 |  84 | 440 | $1.7\cdot 10^{-2}$ | $7.5\cdot 10^{+0}$ | $7.2\cdot 10^{25}$ |  064-277
| LZ-enr |  136Xe | proposed | SURF | $4.6\cdot 10^{4}$ |  14 |  100 |  80 | $25$ | -1.4,1.4 |  84 | 4302 | $1.7\cdot 10^{-3}$ | $7.3\cdot 10^{+0}$ | $7.1\cdot 10^{26}$ |  020-87
| Darwin |  136Xe | proposed |  | $2.7\cdot 10^{4}$ |  13 |  100 |  90 | $20$ | -1.2,1.2 |  76 | 2 312 | $3.5\cdot 10^{-4}$ | $8.0\cdot 10^{-1}$ | $1.1\cdot 10^{27}$ | 17-72
Large liquid scintillators (Sec. VI.4) |  |  |  |  |  |  |  |  |  |  |  |  |
| KLZ-400 |  136Xe | completed | Kamioka | $2.5\cdot 10^{3}$ |  44 |  100 |  97 | $114$ |  -0.0,1.4 |  42 | 450 | $9.8\cdot 10^{-3}$ | $4.4\cdot 10^{+0}$ | $3.3\cdot 10^{25}$ |  095-408
| KLZ-800 |  136Xe | taking data | Kamioka | $5.0\cdot 10^{3}$ |  55 |  100 |  100 | $105$ |  -0.0,1.4 |  42 | 1 143 | $5.5\cdot 10^{-3}$ | $6.2\cdot 10^{+0}$ | $2.0\cdot 10^{26}$ |  038-164
| KL2Z |  136Xe | proposed | Kamioka | $6.7\cdot 10^{3}$ |  80 |  100 |  97 | $60$ |  -0.0,1.4 |  42 | 2 176 | $3.0\cdot 10^{-4}$ | $6.5\cdot 10^{-1}$ | $1.1\cdot 10^{27}$ | 17-71
| SNO+I |  130Te | construction | SNOLAB | $1.0\cdot 10^{4}$ |  20 |  100 |  97 | $80$ | -0.5,1.5 |  62 | 1 232 | $7.8\cdot 10^{-3}$ | $9.7\cdot 10^{+0}$ | $1.8\cdot 10^{26}$ |  031-144
| SNO+II |  130Te | proposed | SNOLAB | $5.1\cdot 10^{4}$ |  27 |  100 |  97 | $57$ | -0.5,1.5 |  62 | 8 521 | $5.7\cdot 10^{-3}$ | $4.8\cdot 10^{+1}$ | $5.7\cdot 10^{26}$ | 17-81
Cryogenic calorimeters (Sec. VI.5) |  |  |  |  |  |  |  |  |  |  |  |  |
| CUORE |  130Te | taking data | LNGS | $1.6\cdot 10^{3}$ |  100 |  88 |  92 | $3.2$ | -1.4,1.4 |  84 | 1 088 | $9.1\cdot 10^{-2}$ | $9.9\cdot 10^{+1}$ | $5.1\cdot 10^{25}$ |  058-270
| CUPID-0 |  82Se | completed | LNGS | $6.2\cdot 10^{1}$ |  100 |  81 |  86 | $8.5$ | -2,2 |  95 | 41 | $2.8\cdot 10^{-2}$ | $1.2\cdot 10^{+0}$ | $4.4\cdot 10^{24}$ | 283-551
| CUPID-Mo |  100Mo | completed | LSM | $2.3\cdot 10^{1}$ |  100 |  76 |  91 | $3.2$ | -2,2 |  95 | 15 | $1.7\cdot 10^{-2}$ | $2.5\cdot 10^{-1}$ | $1.7\cdot 10^{24}$ | 293-858
| CROSS |  100Mo | construction | LSC | $4.8\cdot 10^{1}$ |  100 |  75 |  90 | $2.1$ | -2,2 |  95 | 31 | $2.5\cdot 10^{-4}$ | $7.6\cdot 10^{-3}$ | $4.9\cdot 10^{25}$ | 54-160
| CUPID |  100Mo | proposed | LNGS | $2.5\cdot 10^{3}$ |  100 |  79 |  90 | $2.1$ | -2,2 |  95 | 1 717 | $2.3\cdot 10^{-4}$ | $4.0\cdot 10^{-1}$ | $1.1\cdot 10^{27}$ | 12-34
| AMoRE-II |  100Mo | proposed | Yemilab | $1.1\cdot 10^{3}$ |  100 |  82 |  91 | $2.1$ | -2,2 |  95 | 760 | $2.2\cdot 10^{-4}$ | $1.7\cdot 10^{-1}$ | $6.7\cdot 10^{26}$ | 15-43
Tracking calorimeters (Sec. VI.6) |  |  |  |  |  |  |  |  |  |  |  |  |
| NEMO-3 |  100Mo | completed | LSM | $6.9\cdot 10^{1}$ |  100 |  100 |  11 | $148$ | -1.6,1.1 |  42 | 3 | $9.4\cdot 10^{-1}$ | $3.0\cdot 10^{+0}$ | $5.6\cdot 10^{23}$ | 505-1485
| SuperNEMO-D |  82Se | construction | LSM | $8.5\cdot 10^{1}$ |  100 |  100 |  28 | $83$ | -4.2,2.4 |  64 | 15 | $3.3\cdot 10^{-2}$ | $5.0\cdot 10^{-1}$ | $8.6\cdot 10^{24}$ | 201-391
| SuperNEMO |  82Se | proposed | LSM | $1.2\cdot 10^{3}$ |  100 |  100 |  28 | $72$ | -4.1,2.8 |  54 | 185 | $5.3\cdot 10^{-3}$ | $9.8\cdot 10^{-1}$ | $7.8\cdot 10^{25}$ | 67-131
Figure 20: Fundamental parameters driving the sensitive background and
exposure, and consequently the sensitivity, of recent and future phases of
existing experiments (see Eq. 48). Red bars are used for 76Ge experiments,
orange for 136Xe, blue for 130Te, green for 100Mo, and sepia for 82Se. Similar
exposures are achieved with high mass but poorer energy resolution and
efficiency by gas and liquid detectors, or with small mass but high resolution
and efficiency by solid state detectors. The sensitive exposure is computed
for one year of livetime. Ligher shades indicate experiments which are under
construction or proposed.  Figure 21: Sensitive background and exposure for
recent and future experiments. The grey dashed lines indicate specific
discovery sensitivity values on the $0\nu\beta\beta$-decay half-life. The
colored dashed line indicate the half-life sensitivities required to test the
bottom of the inverted ordering scenario for 76Ge, 136Xe, 130Te 100Mo, and
82Se, assuming for each isotope the largest NME value among the QRPA
calculations listed in Tab. 1. A livetime of 10 yr is assumed except for
completed experiments, for which the final reported exposure is used.

Figure 20 illustrates how each detection concept is characterized by specific
parameter combinations. Liquid and gas detectors have large isotope masses and
a relatively low signal detection efficiency due to the non-uniform background
rate, with a small detector region driving the sensitivity. Solid state
detectors operate a smaller isotope mass, but with higher efficiency and
energy resolution. As a result, an isotope mass lower by a factor of 10 can be
balanced by higher resolution and efficiency, yielding a similar sensitive
exposure and sensitivity.

Figure 21 shows a scatter plot of the sensitive exposure and background for
the listed projects. Recent experiments populate the top left part of the
plot, corresponding to exposures of thousands of mole-years — i.e., tens or
hundreds of kilograms of target mass operated for a few years — and background
rates between $10^{-3}$ and $10^{-1}$ events per mole-year. To improve the
sensitivity, future experiments need to either increase $\mathcal{E}$ or
reduce $\mathcal{B}$. Often a sequence of experimental upgrades with
progressive incremental improvements is planned, ultimately leading to a
combined factor of $\sim$100 improvement.

These figures highlight the strengths and limitations of each detection
concept, indicating the natural strategies to maximize the sensitivity, which
are most evident in the $\mathcal{E}$/$\mathcal{B}$ ratios. For example, 130Te
experiments have large $\mathcal{E}$/$\mathcal{B}$ values (the blue markers
are systematically above the other points in Fig. 21). Given the high natural
abundance of 130Te, for them it is more efficient to increase the exposure
rather than reducing the background. Conversely, 76Ge-based experiments have
small $\mathcal{E}$/$\mathcal{B}$ values. For them, reducing the background is
easier than increasing the target mass, as their strengths are good energy
resolution and advanced event reconstruction capabilities. Experiments based
on other isotopes have intermediate $\mathcal{E}$/$\mathcal{B}$ values,
suggesting some flexibility in finding a trade off between the two quantities.

Although the sensitive exposure and background are valuable parameters to
characterize an experiment, the reach of an experiment is not fully captured
by the $T_{1/2}$ sensitivity. The phase space factor also plays a strong role,
and the nuclear structure of the isotopes deeply affects the connection
between $T_{1/2}$ and the underlying decay mechanism. For instance, assuming
the decay is mediated by the exchange of light Majorana neutrinos, the
discovery power of an experiment is better quantified by the effective
Majorana mass sensitivity. We hence include in the table and figures values
for the $m_{\beta\beta}$ sensitivities, whose ratios provide a good figure of
merit also assuming several other decay mechanisms. We will discuss in detail
the discovery power of the experiments in Sec. VII.

###  VI.2 High-purity Ge semiconductor detectors

High Purity Germanium (HPGe) detectors represent the longest-standing
technology used for $0\nu\beta\beta$-decay searches Avignone and Elliott
(2019). The first $0\nu\beta\beta$-decay experiment based on Ge detectors was
in 1967 Fiorini  _et al._ (1967) and, since then, Ge-based experiments have
stayed at the forefront of the field. Fig. 22 shows an example of the state-
of-the-art model.

Figure 22: Artistic illustration of a high-purity Ge (HPGe) detector and its
$0\nu\beta\beta$-decay detection concept. Electron and hole clusters created
by ionization are collected to the electrodes by an electric field. Image
courtesy of Laura Manenti.

HPGe detectors are semiconductor devices. A detector consists of a single
crystal grown by the Czochralski method Depuydt  _et al._ (2006) from Ge-
material enriched up to 92% in ${}^{\text{76}}$Ge. The detectors used by
recent and future experiments are all p-type crystals, with two conductive
electrodes obtained through B implantation (p+ electrode) and Li diffusion (n+
electrode). The semiconductor junction forms between the n+ electrode and the
p-type crystal, and is extended throughout the whole detector volume by
applying a reverse bias of a few thousands volts. Electrons and holes produced
within the crystal by ionization drift along the electric field, inducing a
current. The current integral is proportional to the energy deposited within
the detector, and its time-structure carries information on the event
topology. The detector size is currently limited to 1-3 kg, and multiple
detectors need to be operated simultaneously to reach a competitive amount of
isotope mass. These detectors are operated in ultra-low background
environments, surrounded by shielding material and active veto systems.

HPGe detectors feature high $0\nu\beta\beta$-decay detection efficiencies. The
presence of conductive electrodes on the detector surface reduces the active
volume fraction to $\varepsilon_{\text{act}}\sim 90\%$ and leads to energy
loss for a fraction of the $0\nu\beta\beta$-decay events
($\varepsilon_{\text{cont}}\sim 90\%$). The $0\nu\beta\beta$-decay event
tagging efficiency, $\varepsilon_{\text{mva}}\sim 80-90\%$, is typically
limited by signal-background discrimination methods based on the analysis of
the current time-structure. Given the low background level and high
resolution, the optimal energy region of interest (ROI) for
$0\nu\beta\beta$-decay searches is $Q_{\beta\beta}\pm 2\sigma$, containing 95%
of the signal. Specific parameter values of 76Ge experiments are listed in
Tab. 5.

Table 5: Specific parameters of experiments using Ge detectors: total detector mass, fractional isotopic abundance, shielding strategy, and background index normalized over the entire detector mass. The background index is what is historically quoted by these experiments but, differently from our sensitive background, is not normalized over the signal detection efficiencies and detector resolution. The values are taken from Agostini  _et al._ (2020b); Alvis  _et al._ (2019a); Abgrall  _et al._ (2021), averaging over multiple datasets for GERDA-II and the Majorana Demonstrator. Experiment | $m_{tot}$ | $f_{iso}$ | Shield | Background
---|---|---|---|---
| [kg] | [%] |  | $\left[\dfrac{\text{events}}{\text{keV}\cdotp\text{kg}\cdotp\text{yr}}\right]$
GERDA-II | 39 | 87 | liquid Ar | $5.2\cdot 10^{-4}$
MJD | 20 | 88 | Cu & Pb | $6.0\cdot 10^{-3}$
LEGEND-200 | 200 | 90 | liquid Ar | $2\cdot 10^{-4}$
LEGEND-1000 | 1000 | 91 | liquid Ar | $1\cdot 10^{-5}$

The GERDA experiment operated a compact array of about 40 detectors in a
cryostat containing 64 m3 of liquid argon (LAr) Agostini  _et al._ (2018b).
Several detector geometries were used during the experiment, giving an average
${}^{\text{76}}$Ge mass of $\sim$34 kg. The LAr acted as a passive shielding
against natural radioactivity from any contamination outside the cryostat, and
also attenuated background produced by radioactive isotopes in the materials
near the detectors, such as the holders or cables. The LAr was also used as an
active veto system by detecting its scintillation light produced when a
background event deposits energy in the argon volume. The average energy
resolution achieved during the second phase of the experiment (GERDA-II) was
$\sigma=1.4$ keV, and the average background index was $5.2_{-1.3}^{+1.6}\cdot
10^{-4}$ $\text{events}/(\text{keV}\cdotp\text{kg}\cdotp\text{yr})$,
corresponding to $\mathcal{B}=4.2\cdot 10^{-4}$
$\text{events}/(\text{mol}\cdotp\text{yr})$Agostini  _et al._ (2020b). With
these parameters, at present, GERDA has achieved the lowest sensitive
background in the field. The remnant background composition was traced to U
and Th in the material surrounding the detectors, and $\alpha$\- and
$\beta$-decaying isotopes on the detector surfaces Agostini  _et al._ (2020c).
The final result of GERDA is a constraint of $T_{1/2}>1.8\cdot 10^{26}$ yr at
90% C.L., consistent with the median upper limit expected for no signal,
derived including also the data from Phase I of the experiment.

The Majorana Demonstrator (MJD) Abgrall  _et al._ (2014) employed a compact
array of up to 58 detectors comprised of both enriched and natural Ge. 27 kg
of isotope were operated in enriched detectors Arnquist  _et al._ (2022). The
HPGe crystals are deployed in two vacuum cryostats shielded from the
environmental background by a layer of underground-electroformed copper,
commercially obtained copper, and high-purity lead. The detectors incorporate
a point-like p+ electrode providing very low-energy threshold and an excellent
energy resolution of $\sigma=1.1$ keV at $Q_{\beta\beta}$, which is currently
the best in the field. With a sensitive background of $\mathcal{B}=3.3\cdot
10^{-3}$ $\text{events}/(\text{mol}\cdotp\text{yr})$, the experiment reported
a limit of $T_{1/2}>8.3\cdot 10^{25}$ yr at 90% C.L. with a limit setting
sensitivity of $T_{1/2}=8.1\cdot 10^{25}$ yr. The background is dominated by a
distant source of thorium Arnquist  _et al._ (2022).

The next-generation Ge-based experiments will be realized in the framework of
the LEGEND project Abgrall  _et al._ (2021), with two stages planned:
LEGEND-200 and LEGEND-1000. In the first, $\sim$200 kg of Ge detectors will be
operated in the GERDA setup after upgrading part of the infrastructure.
Compared to GERDA, a further reduction of the background is anticipated thanks
to the use of larger-mass detectors (resulting in fewer cables and electronic
components), improved light readout, and materials with improved radiopurity,
such as the electroformed copper developed for the Majorana Demonstrator. An
energy resolution equal to or better than the one achieved in the Majorana
Demonstrator is expected. These improvements would bring the LEGEND-200
background to $2\cdot 10^{-4}$
$\text{events}/(\text{keV}\cdotp\text{kg}\cdotp\text{yr})$, less than a factor
of 3 lower than what was achieved by GERDA. With a sensitive background of
$\mathcal{B}=1.0\cdot 10^{-4}$ $\text{events}/(\text{mol}\cdotp\text{yr})$,
LEGEND-200 will achieve a discovery sensitivity of $10^{27}$ yr in 5 years of
live time. For LEGEND-1000, a new infrastructure able to host 1 ton of target
mass will be realized. A further twenty-fold background reduction is expected
with the usage of underground argon, and lower radioactivity levels in cables
and electronics. LEGEND-1000 expects a sensitive background of
$\mathcal{B}=4.9\cdot 10^{-6}$ $\text{events}/(\text{mol}\cdotp\text{yr})$,
leading to a discovery sensitivity of $T_{1/2}=1.3\cdot 10^{28}$ yr after 10
years of operation.

We note that during preparation of this manuscript plans were announced Yue
(2021) for a $0\nu\beta\beta$-decay-focused branch of the CDEX effort Wang
_et al._ (2017), culminating in a ton-scale 76Ge experiment. At present public
details for this project are insufficient for estimating its sensitivity.

###  VI.3 Xenon time projection chambers

Time projection chambers (TPCs) were the first technology used to observe
$2\nu\beta\beta$ decay in real time Elliott  _et al._ (1987) and have remained
at the forefront of $0\nu\beta\beta$-decay searches ever since. In these
detectors, a static electric field is applied in a region containing a liquid
or gaseous medium. As shown in Fig. 23, the electrons and ions liberated by
ionizing radiation drift to analyzing planes which reconstruct their number
and position in the plane normal to the field. The position along the field is
derived from the drift durations. The event reconstruction allows to
discriminate spatially-localized $0\nu\beta\beta$-decay events from spatially
extended ones, such as those produced by multiple Compton scattering.
Depending on the spatial resolution, even the full 3D tracks of the two
electrons emitted in $0\nu\beta\beta$ decay can in principle be reconstructed,
making it possible to discriminate them from from single $\beta$-decays,
$\gamma$-ray scattering and absorption, or nuclear recoils from neutron
scattering.

Figure 23: Artistic illustration of a Xe time projection chamber and its
$0\nu\beta\beta$-decay detection concept. Electron and hole clusters created
by ionization are collected to the electrodes by an electric field. In
addition, scintillation light is detected by light sensors, providing the
timing of the event. Image courtesy of Laura Manenti.

TPCs are particularly well-suited to searches for the $0\nu\beta\beta$ decay
of 136Xe. The source itself is an inert noble element and can be used directly
in TPCs in its purified form as a liquid, gas, or both. In either phase, Xe
exhibits VUV scintillation emitted promptly with an energy deposition.
Experiments able to detect the scintillation light signal can reconstruct the
full 3D topology of the event using a single analyzing plane. The intensity of
the scintillation light also provides a complementary measurement of energy,
whose anti-correlation with the ionization signal can significantly enhance
the energy resolution Conti  _et al._ (2003).

If the electric field is strong enough, the collision between drifting
electrons and gas molecules results in the emission of secondary scintillation
light, called electroluminescence (EL). Single-phase high-pressure gas TPCs
(recently reviewed in Gomez-Cadenas  _et al._ (2019)) shape the field near the
electrode to create a region where the incoming electrons produce EL. Dual-
phase TPCs obtain the same result using a short gaseous EL layer at the top of
the liquid volume. The EL signal gives a precise measure of the number of
ionization electrons, improving the energy resolution. With a fine enough
spatial resolution of the light collection, the EL signal can also be used for
track reconstruction. The energy resolution of experiments reading out the
electroluminescence light is limited by fluctuations in the number of primary
ionization electrons. These fluctuations are small and independent of fluid
density up to about 0.6 g/cm3 ($\sim$100 bar), but above that pressure grow
rapidly Bolotnikov and Ramsey (1997).

Xe TPCs also potentially lend themselves to techniques for observation of the
$\beta\beta$ daughter Ba ion, as first suggested by Moe (1991b). Single-atom
trapping and imaging was first achieved with Ba Neuhauser  _et al._ (1980). Xe
is a transparent fluid, offering multiple options for tagging based on
fluorescence imaging techniques. The nEXO Collaboration is pursuing a strategy
using a cryogenic probe Twelker  _et al._ (2014) to electrostatically attract
the Ba ion in the vicinity of a signal event and freeze it in a volume of Xe,
then transport it to a fluorescence imaging stage capable of single-atom
detection Chambers  _et al._ (2019). The NEXT collaboration aims to transport
Ba++ ions to single-molecule fluorescence imaging (SMFI) sensors for example
using RF carpets Brunner  _et al._ (2015); Jones  _et al._ (2021). Single-Ba-
atom sensitivity with SMFI McDonald  _et al._ (2018) and an implementation
applicable to high-pressure gas Xe TPCs Rivilla  _et al._ (2020) have been
demonstrated. Both collaborations are still working to demonstrate that their
tagging schemes can be achieved with sizeable efficiency and in an actual
$0\nu\beta\beta$-decay experiment. We do not discuss these techniques further
in this review.

Liquid Xe volumes operated in TPCs provide self-shielding from external
radiation, whose contribution to the background drops exponentially with the
distance from the outer Xe surface. Of particular worry is the 214Bi gamma
line at 2447.7 keV, just below the 136Xe $Q_{\beta\beta}$ at 2457.8 keV, and
often not resolved. Xe TPC experiments use a multi-variate analysis to handle
the varying background rate throughout the detector volume. However, the
sensitivity of the experiment is essentially driven by the innermost region of
the detector, while the outer region is used primarily to constrain the
background extrapolation to the detector center. A fiducial volume may be
defined or tuned to demark these regions, leading to $\varepsilon_{act}=$
10–60% depending on the enrichment level and the radioactivity of the
structural materials. The background in that fiducial volume is then treated
as an effective parameter, tuned to reproduce the half-life sensitivities
reported by the experiments. The most sensitive energy region of interest
varies, depending on the background level and whether the 214Bi gamma line is
resolved. Containment efficiencies are $\varepsilon_{act}\sim 100\%$ for
liquid Xe TPCs after the effective fiducial volume cut, while they are
typically 70–80% for gaseous detectors.

Table 6: Specific parameters of Xe-TPC experiments: total mass, fractional isotopic abundance, phase, signal readout, effective background index in units of events per kg of mass in the fiducial volume, and the ratio $R$ between the effective background index used for our sensitivity calculation and the mean background quoted by the experiments, when available. Values are taken from references Anton  _et al._ (2019); Al Kharusi  _et al._ (2018); Adhikari  _et al._ (2022); Alvarez  _et al._ (2012); Martín-Albo  _et al._ (2016); Adams  _et al._ (2021a); Chen  _et al._ (2017); Akerib  _et al._ (2020); Agostini  _et al._ (2020a).  Experiment | $m_{tot}$ | $f_{iso}$ | Phase | Readout | Effective Background | $R$
---|---|---|---|---|---|---
| [kg] | [%] |  |  | $\left[\dfrac{\text{events}}{\text{keV}\cdotp\text{kg}\cdotp\text{yr}}\right]$ |
EXO-200 | 161 | 81 | liquid | LAPPDs + wires | $1.8\cdot 10^{-3}$ | 1
nEXO | 5109 | 90 | liquid | electrode tiles + SiPM s | $2.1\cdot 10^{-6}$ | N/A
NEXT-100 | 97 | 90 | gas | SiPMs + PMTs | $4.0\cdot 10^{-4}$ | 1
NEXT-HD | 1100 | 90 | gas | SiPMs + PMTs | $4.0\cdot 10^{-6}$ | 1
PandaX-III-200 | 200 | 90 | gas | Micromegas | $1.0\cdot 10^{-4}$ | 1
LZ-nat | 7 000 | 9 | dual-phase | PMTs | $1.1\cdot 10^{-4}$ | 0.4
LZ-enr | 7 000 | 90 | dual-phase | PMTs | $1.1\cdot 10^{-4}$ | 0.4
DARWIN | 39 300 | 9 | dual-phase | PMTs | $3.4\cdot 10^{-6}$ | 0.85

The most sensitive Xe TPC experiment to date was EXO-200 Auger  _et al._
(2012), which used liquid-phase enriched Xe, with 161 kg of 136Xe. The TPC
employed two drift regions with a common cathode at the detector center. The
ionization was read out via crossed wire planes at the anodes. The prompt
scintillation light was read out by large-area avalanche photodiodes. The
combined signal achieved an energy resolution of $\sigma=28$ keV at
$Q_{\beta\beta}$ Anton  _et al._ (2019), or 31 keV when averaged over the full
dataset. Backgrounds and field non-uniformity near the detector edges required
fiducialization, restricting the analysis to the innermost 75 kg of Xe. An
extensive screening campaign Leonard  _et al._ (2008, 2017) and a
sophisticated analysis incorporating topological background discrimination
Delaquis  _et al._ (2018) led to an averaged background level within the
fiducial volume of $1.8\cdot 10^{-3}$
$\text{events}/(\text{keV}\cdotp\text{kg}\cdotp\text{yr})$, corresponding to
$\mathcal{B}=$4.7$\cdot 10^{-2}$ $\text{events}/(\text{mol}\cdotp\text{yr})$,
dominated by the 214Bi gamma line originating from 238U chain decays outside
of the Xe volume. The experiment reported an ultimate limit for
$0\nu\beta\beta$ decay of $T_{1/2}>3.5\cdot 10^{25}$ yr at 90% C.L. with a
sensitivity for limit setting of $T_{1/2}=5.0\cdot 10^{25}$ yr Anton  _et al._
(2019). Our counting analysis described in Sec. V.5.2 reproduces the EXO-200
limit sensitivity with no tuning required.

nEXO Al Kharusi  _et al._ (2018); Adhikari  _et al._ (2022) build on EXO-200’s
technology and aims at using $5$ tons of Xe enriched to $90\%$ in 136Xe. The
TPC design features one monolithic drift volume with ionization read out by
silica tiles patterned with metallic electrode strips, and scintillation
detection by an array of VUV-sensitive silicon photomultipliers on the TPC
walls behind the field-shaping grid, yielding an energy resolution of
$\sigma=20$ keV. The effective background index that reproduces nEXO’s
published discovery sensitivity is $2.1\cdot 10^{-6}$
$\text{events}/(\text{keV}\cdotp\text{kg}\cdotp\text{yr})$, corresponding to
$\mathcal{B}=4.0\cdot 10^{-5}$ $\text{events}/(\text{mol}\cdotp\text{yr})$, a
factor of $\sim$1000 improvement over EXO-200. nEXO is expected to achieve a
discovery sensitivity of $T_{1/2}=7.4\cdot 10^{27}$ yr after 10 years of live
time.

NEXT Nygren (2009); Granena  _et al._ (2009) implements a high-pressure
gaseous Xe TPC equipped with an EL region. Tracking information is obtained
from a SiPM array at the anode, while PMTs at the cathode provide optimal
energy resolution. NEXT-White Monrabal  _et al._ (2018), a proof-of-principle
detector with 5 kg of Xe at 10 bar, demonstrated an energy resolution of
$\sigma=10$ keV at $Q_{\beta\beta}$ Renner  _et al._ (2019), and tracking
performance capable of discriminating single and double electron track events,
retaining 57% of signal events while rejecting background by a factor of 27
Simón  _et al._ (2020). A second stage, NEXT-100 Alvarez  _et al._ (2012);
Martín-Albo  _et al._ (2016), with a pressure of 15 bar and containing 87 kg
of 136Xe, is currently under construction. The NEXT-100 projected background
is dominated by remnant events from U/Th in the PMTs and other detector
components, giving a background index of $4\cdot 10^{-4}$
$\text{events}/(\text{keV}\cdotp\text{kg}\cdotp\text{yr})$, or
$\mathcal{B}=5.9\cdot 10^{-3}$ $\text{events}/(\text{mol}\cdotp\text{yr})$.
NEXT-HD Adams  _et al._ (2021a), a concept for a future ton-scale phase of
NEXT, is a symmetric TPC with a common central cathode, large enough to
accommodate a full ton of 136Xe in the form of enriched Xe gas at 15 bar. The
design of NEXT-HD substitutes PMTs with an all-SiPM light readout at both TPC
ends, using wavelength-shifting fibers to enhance light collection further.
Gas additives to reduce diffusion are expected to enhance both the energy
resolution and tracking resolution relative to NEXT-100. The expected
reduction in background index by a factor of 100 thus leads to an even large
reduction in sensitive background, predicted to be $\mathcal{B}=4.0\cdot
10^{-5}$ $\text{events}/(\text{mol}\cdotp\text{yr})$. NEXT-BOLD aims to take
this concept one step further by instrumenting the NEXT-HD TPC with Ba tagging
capability Rivilla  _et al._ (2020), potentially achieving half-life
sensitivity on the order of $10^{28}$ years.

Another experiment based on the high-pressure gas Xe TPC technique is PandaX-
III-200 Chen  _et al._ (2017). The initial phase uses 180 kg 136Xe in a volume
of enriched Xe gas at 10 bar, deployed in a symmetric TPC with a common
cathode. In contrast to the all-photon-based readout pursued by NEXT, PandaX-
III-200 exclusively relies on an ionization-only readout of just the drift
electrons using the Micromegas detector technology, where a high-field region
near the anode provides avalanche amplification. The expected energy
resolution is $\sigma=31$ keV at $Q_{\beta\beta}$, while simulated topological
discrimination based on track reconstruction predicts up to two orders-of-
magnitude background reduction with 42% signal efficiency Galan  _et al._
(2020). Backgrounds are dominated by U/Th contamination of the Micromegas
readout plane. The total background index goal is 10-4
$\text{events}/(\text{keV}\cdotp\text{kg}\cdotp\text{yr})$, giving
$\mathcal{B}=3\cdot 10^{-3}$ $\text{events}/(\text{mol}\cdotp\text{yr})$.

LZ Akerib  _et al._ (2015) and DARWIN Aalbers  _et al._ (2016) both employ
dual-phase natural-Xe TPCs with EL readout to perform direct searches for WIMP
Dark Matter. These detectors also have sensitivity to $0\nu\beta\beta$ decay
even with natural Xe targets Akerib  _et al._ (2020); Agostini  _et al._
(2020a). The instrumentation required for detection of the faint nuclear
recoils from WIMPs naturally leads to higher external backgrounds than for a
detector optimized for $0\nu\beta\beta$ decay. With 7 tons of Xe (640 kg
136Xe) in the LZ inner vessel and and 40 tons (3.6 tons 136Xe) total in
DARWIN, self-shielding reduces these background dramatically, but external
214Bi still dominates in both experiments. Reproducing the LZ sensitivity
requires an effective background index of 1.2$\cdot
10^{-4}$$\text{events}/(\text{keV}\cdotp\text{kg}\cdotp\text{yr})$, giving
$\mathcal{B}$=1.7$\cdot 10^{-2}$ $\text{events}/(\text{mol}\cdotp\text{yr})$.
A subsequent run with enriched Xe (90% 136Xe) would have enhanced sensitivity.
DARWIN’s larger mass affords it a lower effective background index of
3.4$\cdot 10^{-6}$ $\text{events}/(\text{keV}\cdotp\text{kg}\cdotp\text{yr})$,
or $\mathcal{B}$=3.5$\cdot 10^{-4}$
$\text{events}/(\text{mol}\cdotp\text{yr})$, with 137Xe $\beta$ decays
representing an important background contribution.

A summary of all TPCs significant experimental parameters is given in Tab. 6.

###  VI.4 Large liquid scintillators

In what is perhaps the most successful departure from the “source = detector”
paradigm followed by most $0\nu\beta\beta$-decay experiments, large liquid
scintillators offer the advantage of dissolving or loading vast amounts of
isotope into the most sensitive regions of some of the lowest-radioactivity
experiments ever constructed. With typical mass-loading fractions on the few
percent level, a kton-scale scintillator detector can reach ton-year exposures
with relative ease. Energy depositions within the detector generate
scintillation photons, which are detected by PMTs viewing the target volume as
shown in Fig. 24. Event energy, position, and topology reconstruction is
performed using the number, pattern, and timing of the detected photons. The
position reconstruction is particularly important for these self-shielding
detectors, whose inner volume has the lowest background and drives the
sensitivity. The effective fiducial volume fractions range between
$\varepsilon_{act}=20$%–80%, due to a combination of self-sheilding and
whether the target isotope is spread through the whole scintillator volume or
confined to its central part. Containment efficiencies are maximal in the
fiducial volume ($\varepsilon_{cont}=100\%$).

Figure 24:  Artistic illustration of a large liquid scintillator detector and
its $0\nu\beta\beta$-decay detection concept. The position of an event can be
reconstructed through the time of flight of the scintillation photons. Image
courtesy of Laura Manenti.

The challenge for these detectors lies in their limited energy resolution due
to the relatively low number of scintillation photons produced by energy
depositions at $Q_{\beta\beta}$. Events due to $2\nu\beta\beta$ decay pose a
problematic background in the energy region of interest, and the extraction of
a $0\nu\beta\beta$-decay signal relies on an energy spectral analysis
sensitive to distortion at the end-point of the $2\nu\beta\beta$-decay energy
distribution. Such an analysis requires a precise understanding of the
detector response and energy reconstruction systematic effects. The
$0\nu\beta\beta$-decay background reduces the optimal energy region of
interest to values above $Q_{\beta\beta}$, with an effective 40–60% loss in
detection efficiency. Like for the case of Xe TPCs, an effective background
index for the fiducial volume was tuned to reproduce published experimental
sensitivities.

The presence of the large mass of liquid scintillator also increases the
prevalence of solar neutrino backgrounds and cosmogenic activation products.
The latter includes in particular 10C which is readily generated in organic
liquid scintillators. Vetoing schemes based on proximity to muon tracks and
the detection of neutron capture gammas in delayed time coincidence can reduce
these background contributions by roughly an order of magnitude at the cost of
some exposure loss ($\varepsilon_{mva}=97\%$, see, e.g., Gando  _et al._
(2016)).

KamLAND-Zen (KLZ) is an upgrade of the KamLAND apparatus Eguchi  _et al._
(2003) tailored to the search of $0\nu\beta\beta$ decay: a nylon balloon is
deployed in the active detector volume and filled with liquid scintillator in
which enriched Xe has been dissolved. A successful first phase deployment,
KamLAND-Zen 400 (KLZ-400) with up to 340 kg of 136Xe, led to the strongest
half-life limits for its time despite an unexpected background likely
originating from fallout from the Fukushima nuclear disaster Gando  _et al._
(2013, 2016). The second phase, KamLAND-Zen 800 (KLZ-800), is currently
running with $\sim$680 kg of 136Xe redeployed in a larger, cleaner balloon.
With just 1.6 years of data KLZ-800 produced a world-leading half-life limit
$T_{1/2}>2.3\cdot 10^{26}$ yr at 90% C.L. with a limit setting sensitivity of
$T_{1/2}=1.3\cdot 10^{26}$ yr Abe  _et al._ (2022). The background measured in
the KLZ-800 fiducial volume corresponds to $\mathcal{B}=5.5\cdot 10^{-3}$
$\text{events}/(\text{mol}\cdotp\text{yr})$. The KLZ-800 sensitivity is well-
reproduced by the background-dominated approximation. The KamLAND-Zen
collaboration is already preparing a follow-on phase, KamLAND2-Zen (KL2Z)
Shirai (2017), in which $\sim$1 ton of 136Xe will be deployed. A major upgrade
of the experiment is conceived for KL2Z to improve the energy resolution at
$Q_{\beta\beta}$ from $\sigma=$114 keV to 60 keV. The upgrade includes the
installation of new light concentrators and PMTs with higher quantum
efficiency as well as purer liquid scintillator. A sensitive background
reduction by a factor 20 over KLZ-800 is expected for KL2Z, afforded primarily
by the envisioned improvement in the detector resolution. An effective
background that is a factor of 0.45 times the predicted background reproduces
the expected KL2Z sensitivity.

SNO+ Andringa  _et al._ (2016); Albanese  _et al._ (2021) is a follow up of
the SNO experiment building on the SNO infrastructure Boger  _et al._ (2000).
It is a multi-purpose neutrino experiment, with a 130Te-based
$0\nu\beta\beta$-decay search as one of its main physics goals. SNO’s acrylic
sphere will be filled with $\sim$780 tons of liquid scintillator, loaded with
tellurium, with the surrounding SNO cavern instrumented as a water Cherenkov
active veto. As of the time of writing, SNO+ is filled with liquid
scintillator and taking data, with Te loading scheduled to commence soon. A
multi-staged approach is foreseen Grant (2020). Initially $\sim$1.3 tons of
130Te (0.5% natTe loading) will be used and an energy resolution of
$\sigma=$80 keV is expected. The predicted background corresponds to
$\mathcal{B}=7.8\cdot 10^{-3}$ $\text{events}/(\text{mol}\cdotp\text{yr})$,
and is dominated by 8B solar neutrino elastic scatters. The goal of a
subsequent phase is to increase the 130Te mass to 6.6 tons (2.5% natTe
loading) and improve the energy resolution to 57 keV. This is achievable
thanks to an improvement of the light yield to $800$ pe/MeV Klein (2017). The
predicted background corresponds to $\mathcal{B}=5.7\cdot 10^{-3}$
$\text{events}/(\text{mol}\cdotp\text{yr})$.

A summary of the relevant parameters for KamLAND-Zen and SNO+ is given in Tab.
7.

Table 7: Specific parameters for liquid scintillator experiments: isotope, total mass, fractional isotopic abundance, fractional mass of the loaded material, effective background per kg (of isotope) in the fiducial volume, and the ratio $R$ of that to the mean background in the fiducial volume. Values are taken from references Gando  _et al._ (2016); Gando (2020); Shirai (2017); Andringa  _et al._ (2016); Albanese  _et al._ (2021). Experiment | Isotope | $m_{tot}$ | $f_{iso}$ | Loading | Effective Background | $R$ |
---|---|---|---|---|---|---|---
$\begin{array}[]{c}\text{Experi}\\\ \text{ment}\end{array}$ |  | [kg] |  | [%wt.] | $\left[\dfrac{\text{events}}{\text{keV}\cdotp\text{kg}\cdotp\text{yr}}\right]$ |  |
KLZ-400 |  136Xe | 378 | 0.91 | 2.9 | $1.8\cdot 10^{-4}$ | 1 |
KLZ-800 |  136Xe | 745 | 0.91 | 3.0 | $1.1\cdot 10^{-4}$ | 1 |
KL2Z |  136Xe | 1000 | 0.91 | 2.7 | $1.1\cdot 10^{-5}$ | 0.45 |
SNO+I |  130Te | 3825 | 0.91 | 0.5 | $2.3\cdot 10^{-4}$ | 1 |
SNO+II |  130Te | 19 125 | 0.91 | 2.5 | $2.3\cdot 10^{-4}$ | 1 |

###  VI.5 Cryogenic calorimeters

Cryogenic calorimeters, often referred to as bolometers, are one of the most
versatile types of detectors for rare events searches. Their first development
dates back to the 1980s and, since then, they have been successfully employed
for $0\nu\beta\beta$-decay and dark-matter searches Brofferio and Dell’Oro
(2018). Bolometers consist of crystals coupled to thermal sensors measuring
the phonons induced by particles impinging on the crystal lattice, or the heat
induced by phonon recombination (see Fig. 25). Typically the crystals used in
$0\nu\beta\beta$-decay experiments have masses between 0.2 and 0.8 kg and are
operated at 10–20 mK. Their energy resolutions are typically in the range
$\sigma=$ 2–10 keV, and the containment efficiency varies between 70–90%,
depending on the crystal type and size.

Figure 25:  Artistic illustration of a cryogenic calorimeter and its
$0\nu\beta\beta$-decay detection concept. Phonon and scintillation light
signals are read-out through superconductive thermal sensors. Image courtesy
of Laura Manenti.

Bolometers have an active volume fraction of 100%, which makes them sensitive
to background due to $\alpha$-decaying isotopes on their surfaces, or on the
surfaces of nearby materials. In scintillating crystals, e.g., ZnSe or
Li2MoO4, it is possible to discriminate $\alpha$ from $\beta/\gamma$ particles
by their different light yields. The scintillation light is detected by a
second bolometer placed in front of the crystal and consisting of a Ge or Si
wafer instrumented with the NTD, TES or KID sensors discussed in Sec. V.2.2.
Alternatively, surface events can be discriminated from bulk events via pulse-
shape analysis using crystals with an Al-film coating, as is being pursued by
CROSS Bandac  _et al._ (2020). In such devices, a ionizing particle
interacting close enough to the coated surface will create quasi-particles
that can be trapped in the superconductive Al layer for $O(1)$ ms Bandac  _et
al._ (2020). In all cases, the $0\nu\beta\beta$-decay tagging efficiency is
$\varepsilon_{\text{mva}}\sim 90\%$.

Bolometric experiments feature high granularity, providing good suppression of
the external $\gamma$ backgrounds via the rejection of events releasing energy
in multiple crystals. The full absorption of phonons yields response times
that can be as long as 0.1 s with NTD sensors. Hence, the probability of
having two $2\nu\beta\beta$ decays piling up is not negligible, especially
when considering large crystals and isotopes with relatively short
$2\nu\beta\beta$-decay half-life. Techniques to mitigate this potential
background are currently being developed.

Table 8: Detection concept specific parameters for cryogenic bolometers: crystal molecule, total mass, fractional isotopic abundance, background per kg of total mass. All experiments except CUORE use a combined readout of heat and scintillation light. All experiments have an NTD readout, except for AMoRE-II, which uses MMCs. Values are taken from references Adams  _et al._ (2022); Azzolini  _et al._ (2019c); Armengaud  _et al._ (2021); Bandac  _et al._ (2020); Armstrong  _et al._ (2019); Lee (2020).  Experiment | Crystal | $m_{tot}$ | $f_{iso}$ | Background
---|---|---|---|---
|  | [kg] | [%] | $\left[\dfrac{\text{events}}{\text{keV}\cdotp\text{kg}\cdotp\text{yr}}\right]$
CUORE |  ${}^{\text{nat}}$TeO2 | 742 | 34151515CUORE is using natural tellurium. | $1.5\cdot 10^{-2}$
CUPID-0 | Zn${}^{\text{enr}}$Se | 9.65 | 96 | $3.5\cdot 10^{-3}$
CUPID-Mo | Li2${}^{\text{enr}}$MoO4 | 4.16 | 97 | $4.7\cdot 10^{-3}$
CROSS | Li2${}^{\text{enr}}$MoO4 | 8.96 | 98 | $1.0\cdot 10^{-4}$
CUPID | Li2${}^{\text{enr}}$MoO4 | 472 |  $\geq$95 | $1.0\cdot 10^{-4}$
AMoRE-II | Li2${}^{\text{enr}}$MoO4 | 200 | 96 | $1.0\cdot 10^{-4}$

At present, the largest bolometric experiment is CUORE, operating $\sim$750 kg
of TeO2 crystals with natural isotopic composition (giving 206 kg 130Te) in a
custom cryogen-free dilution refrigerator Alduino  _et al._ (2019). The TeO2
crystal detector technology is reviewed in Brofferio  _et al._ (2019). CUORE
has demonstrated the feasibility of a ton-scale bolometric experiment,
achieving an energy resolution of $\sigma=3.2$ keV. With a background of
$1.5\cdot 10^{-2}$ $\text{events}/(\text{keV}\cdotp\text{kg}\cdotp\text{yr})$,
corresponding to a sensitive background $\mathcal{B}$$=9.1\cdot 10^{-2}$
$\text{events}/(\text{mol}\cdotp\text{yr})$, CUORE has set the most stringent
constraints on $0\nu\beta\beta$ decay of 130Te: $T_{1/2}$$>2.2\cdot 10^{25}$
yr at $90\%$ C.L., with an exclusion sensitivity of $T_{1/2}$$=2.8\cdot
10^{25}$ yr Adams  _et al._ (2022).

In the coming years, a strong boost in sensitivity is expected with CUPID
Armstrong  _et al._ (2019), which will deploy enriched crystals with particle
identification capabilities in the CUORE cryostat. Several projects have been
realized to identify the optimal crystal and light detector technology. Among
these, CUPID-0 operated abut 5 kg of 82Se in form of enriched ZnSe
scintillating crystals, demonstrating a background of $3.5^{+1.0}_{-0.9}\cdot
10^{-3}$ $\text{events}/(\text{keV}\cdotp\text{kg}\cdotp\text{yr})$ Azzolini
_et al._ (2019c, a), a factor 3.3 times lower than that of CUORE. A limitation
of ZnSe crystals is their relatively poor energy resolution ($\sigma=8.5$ keV)
due to sub-optimal crystal purity. In parallel, CUPID-Mo Armengaud  _et al._
(2020b) has collected data with 20 enriched Li2MoO4 crystals for a total
isotope mass of 2.3 kg. CUPID-Mo has demonstrated a resolution of $\sigma=3.2$
keV, and $>$99.9% $\alpha$ rejection with $>$99.9% acceptance of
$\beta/\gamma$ events Armengaud  _et al._ (2021, 2020b); Welliver (2021).
Finally, the CROSS collaboration is using Al-coated Li2MoO4 crystals to
extensively investigate their surface/bulk discrimination capabilities, and is
planning to deploy 32 enriched Li2MoO4 crystals for a total isotope mass of
4.7 kg. The goal of the CROSS demonstrator is to reach a background $<10^{-4}$
$\text{events}/(\text{keV}\cdotp\text{kg}\cdotp\text{yr})$ Bandac  _et al._
(2020), which could boost the sensitiviy of bolometric experiments beyond the
IO region.

CUPID’s baseline design is based on 250 kg of Li2${}^{\text{enr}}$MoO4
instrumented with light readout in the CUORE cryostat. Assuming achieved
crystal quality and background levels, and the readout of scintillation light
for particle discrimination, CUPID projects a background of $10^{-4}$
$\text{events}/(\text{keV}\cdotp\text{kg}\cdotp\text{yr})$, more than a factor
100 lower than CUORE. With a sensitive background $\mathcal{B}$$=2.3\cdot
10^{-4}$ $\text{events}/(\text{mol}\cdotp\text{yr})$,CUPID will reach a
$1.1\cdot 10^{27}$ yr discovery sensitivity with 10 yr of live time. With
additional purification of the crystal material, the use of light detectors
instrumented with lower threshold and higher bandwidth sensors (e.g., TES),
and the development of pulse shape discrimination techniques, CUPID can
achieve a background of $2\cdot 10^{-5}$
$\text{events}/(\text{keV}\cdotp\text{kg}\cdotp\text{yr})$. Ultimately, a
background level of $5\cdot 10^{-6}$
$\text{events}/(\text{keV}\cdotp\text{kg}\cdotp\text{yr})$ is conceivable with
the deployment of 1 ton of isotope in a new cryostat featuring an active
cryogenic shield Nones (2021).

In parallel, the AMoRE collaboration has demonstrated the feasibility of using
scintillating crystals with an MMC readout both on the phonon and photon
channels for large experiments Lee (2020). The first phase of the experiment,
AMoRE-I, is currently collecting data with a mix of 100Mo-enriched Li2MoO4 and
CaMoO4 crystals for a total mass of $\sim$6 kg, and is characterized by a
background $<10^{-3}$
$\text{events}/(\text{keV}\cdotp\text{kg}\cdotp\text{yr})$. The next phase,
AMoRE-II, will make use of 200 kg of Li2${}^{\text{enr}}$MoO4 crystals, for an
isotope mass of 110 kg. With a target background index of $<10^{-4}$
$\text{events}/(\text{keV}\cdotp\text{kg}\cdotp\text{yr})$, corresponding to a
sensitive background $\mathcal{B}$$=2.2\cdot 10^{-4}$
$\text{events}/(\text{mol}\cdotp\text{yr})$, AMoRE-II will reach a discovery
sensitivity of 6.7$\cdot 10^{25}$ yr with 10 yr of data.

###  VI.6 Tracking calorimeters

Tracking calorimeters are the only actively-pursued detection concept in which
the $0\nu\beta\beta$-decay isotope material is decoupled from the detector.
The source is in the form of a foil sandwiched by drift chambers with an
applied magnetic field for discriminating electrons and positrons, beyond
which lies calorimeters for measuring energy (see Fig. 26.

Figure 26:  Artistic illustration of a tracking-calorimeter detector and its
$0\nu\beta\beta$-decay detection concept. The charge, momentum, and energy of
the particles ejected by the source is measured through a combination of
magnetic-field trackers and calorimeters. Image courtesy of Laura Manenti.

Due to the requirement that the foils be very thin to minimize energy losses
prior to the electrons exiting the source, scaling up the isotope mass is
particularly challenging for this technology. However, tracking calorimetry
has the unique capability of precisely measuring properties of the decay
kinematics such as single-$\beta$ energy spectra and opening angle
distributions, which are connected to the Lorentz structure of the mechanism
mediating the decay Ali  _et al._ (2007); Arnold  _et al._ (2010).

The most sensitive tracking calorimeter to date was NEMO-3, which set
competitive constraints on a variety of target isotopes, particularly 100Mo
Arnold  _et al._ (2015). Its successor, SuperNEMO Piquemal (2006), builds on
the same design principles and is currently in preparation. The SuperNEMO
project is divided in two phases: a SuperNEMO Demonstrator (SuperNEMO-D)
consisting of one module with 7 kg of 82Se, and a full-scale experiment
consisting of multiple modules for a total 82Se mass of 100 kg. Future phases
with different isotopes are still open.

The energy resolution of a single calorimeter was $\sigma\sim$100 keV for
NEMO-3, and is expected to be $\sim$50 keV for SuperNEMO thanks to improved
light collection and the use of PMTs with higher quantum efficiency. Some
fraction of the energy emitted in a $\beta\beta$ decay event is inevitably
released in the passive source foil: as a result, the $0\nu\beta\beta$-decay
signature is peaked below $Q_{\beta\beta}$, and features a low-energy tail,
significantly overlapping with the $2\nu\beta\beta$-decay continuum spectrum.
The optimal ROI strongly depends on the expected number of background events
at the end of the data taking: it corresponds to [-1.6,1.1] $\sigma$ for
NEMO-3, and to a $\sim$4 $\sigma$ range around the degraded peak below
$Q_{\beta\beta}$ for SuperNEMO and SuperNEMO Demonstrator.

In tracking calorimeters, the reconstructed event kinematics can be used to
suppress most backgrounds, at the price of a lower signal efficiency. This was
11% in NEMO-3, and is expected to reach 28% in SuperNEMO thanks to the
improved spacial resolution of the tracker. The most significant residual
backgrounds come from 222Rn in the tracker, and the $\beta$ decays of 208Tl
and 214Bi on the source foil. SuperNEMO aims to suppress the last two by a
factor 50 and 30, respectively, and has partially achieved it so far Povinec
(2017); Calvez (2017). In our calculation, we use the design value for
SuperNEMO of $9.8\cdot 10^{-5}$
$\text{events}/(\text{keV}\cdotp\text{kg}\cdotp\text{yr})$, and the
experimentally measured contaminations for the Demonstrator Calvez (2017),
giving $\mathcal{B}=5.3\cdot 10^{-3}$
$\text{events}/(\text{mol}\cdotp\text{yr})$. Given the particular shape of the
$0\nu\beta\beta$\- and $2\nu\beta\beta$-energy distributions, a spectral fit
has higher sensitivity compared to a simple counting analysis. By reducing the
background to 20%, we match the sensitivity quoted by the collaboration, which
corresponds to a 10-yr discovery sensitivity of $9\cdot 10^{24}$ and $8\cdot
10^{25}$ yr for SuperNEMO-D and SuperNEMO, respectively.

###  VI.7 Other detector concepts

Several additional projects exist that use technologies other than the ones
discussed so far. Some technologies have already lead to proof-of-principle
experiments, which however are not yet competitive in terms of sensitivity. In
most cases, the projects are still at an early R&D phase, and a significant
effort is required to demonstrate that the underlying technology can be scaled
to a $0\nu\beta\beta$-decay experiment capable of covering the inverted
ordering region or beyond. In Tab. 9 we list a selection of such projects
appearing in the literature, highlighting their isotope of choice (where
defined) and key features.

Table 9: Other detector concepts. Existing experiments are marked with a dagger. Project |  Isosope(s) |  Detector technology, main features, and references
---|---|---
CANDLES† |  48Ca |  Array of scintillator crystals suspended in a volume of liquid scintillator.
Possible operation as cryogenic calorimeters.
Yoshida  _et al._ (2009); Ajimura  _et al._ (2021)
COBRA† |  70Zn, |  CdZnTe semiconductor detector array.
114,116Cd, |  Room temperature; multi-isotope; high granularity.
128,130Te |  Ebert  _et al._ (2016b); Zuber (2001); Ebert  _et al._ (2016a); Arling  _et al._ (2021)
Selena |  82Se |  Amorphous ${}^{\text{enr}}$Se high resolution, high-granularity CMOS detector array.
3D track reconstruction ($O(10\mu\text{m})$ resolution); room temperature;
minimal shielding.
Chavarria  _et al._ (2017)
N$\nu$DEx |  82Se |  High-pressure gaseous 82SeF6 ion-imaging TPC.
$\lesssim 1\%$ energy resolution; precise signal topology; possible multi-
isotope.
Nygren  _et al._ (2018); Mei  _et al._ (2020)
R2D2 |  136Xe |  Spherical TPC.
Single readout channel; inexpensive infrastructure.
Bouet  _et al._ (2021)
AXEL |  136Xe |  High-pressure TPC operated in proportional scintillation mode.
High energy resolution; possible positive ion detection.
Obara  _et al._ (2020)
JUNO |  — |  Isotope loaded liquid scintillator.
20 ktons of scintillator; multi-isotope; multi-purpose.
Zhao  _et al._ (2017); Abusleme  _et al._ (2021)
NuDot |  — |  Liquid scintillator with quantum dots or perovskites as wavelength shifter for Cherenkov light.
Discriminate directional backgrounds; multi-isotope.
Gooding  _et al._ (2018); Graham  _et al._ (2019); Winslow and Simpson (2012);
Aberle  _et al._ (2013)
ZICOS |  96Zr |  Zr-loaded liquid scintillator.
Topology and particle discrimination via Cherenkov light readout.
Fukuda  _et al._ (2020); Fukuda (2016)
THEIA |  — |  Water-based loaded liquid scintillator with Cherenkov light readout.
Topology and particle discrimination; multi-isotope; multi-purpose; 25 ktons
of water.
Askins  _et al._ (2020)
LiquidO |  — |  Opaque isotope-loaded liquid scintillator with wavelength shifting fibers for event topology.
Room temperature; multi-isotope; multi-purpose.
Buck  _et al._ (2019); Cabrera  _et al._ (2019)

##  VII Prospects and expectations

In this section we bring together our expectations from the theory and
experimental landscape, and address some of the key questions related to
$0\nu\beta\beta$ decay. Section VII.1 summarizes near-term prospects and how
ongoing efforts are going to shape the field. In Sec. VII.2, we discuss what
we would learn from a discovery under different assumptions on the underlying
theory framework. Section VII.3 addresses the question of how likely a
discovery is in the next round of experiments, also in terms of nuclear and
particle theory inputs. Sec. VII.4 reviews other discovery opportunities of
$0\nu\beta\beta$-decay experiments not related to the lepton-number violating
$0\nu\beta\beta$ decay. Finally, Sec. VII.5 speculates on the neutrino’s role
as a possible catalyst for the next paradigm shift in fundamental physics,
which may lead us to a new theory beyond the Standard Model of particle
physics.

This section aims at addressing in a comprehensive way the most important
questions of experts and nonexperts alike. Its content is largely covered by
the previous sections, but it is here presented stressing the connections
between theory and experiment, as well as between particle and nuclear theory.
We refer the reader to the previous sections for more information and detailed
lists of references.

###  VII.1 Where are we heading?

####  VII.1.1 Experiment

In the next decade, several experiments will be constructed to search for
$0\nu\beta\beta$ decay at new uncharted half-life scales using multiple nuclei
and different technologies. Three scenarios can unfold, depending on the half-
life of the process, and whether the decay exists at all.

The signal half-life could be just beyond current constraints, at a scale of
$10^{25}$–$10^{26}$ years, depending on the isotope. In this first scenario,
hundreds of $0\nu\beta\beta$-decay events will be observed in each next-
generation experiment. The half-life will be measured with statistical
uncertainty at the level of $\sim$10%. Systematic uncertainties in
$0\nu\beta\beta$-decay experiments are typically $\lesssim$10%, and will not
limit the accuracy of the measurement in this strong signal scenario. These
first measurements will likely be followed by a second round of experiments,
not developed to maximize the discovery sensitivity, but capable of measuring
properties of the decay kinematics, such as single-$\beta$ energy spectra and
opening angle distributions connected to the Lorentz structure of the
mediating mechanism(s).

In the second scenario, the $0\nu\beta\beta$-decay half-life is above the
current limits, but still within the reach of upcoming searches (i.e.,
$T_{1/2}\approx 10^{26}$–$10^{28}$ years). For this case, only tens of events
or fewer would be expected. Measurements of the half-life will hence be
affected by large statistical uncertainties at the level of 30% for 10 events,
or even 100% for a couple of events. If the signal is at the edge of the
detection sensitivities, only some of the experiments may observe a signal,
while others would set a limit. Such an inconclusive situation would require
further discovery-style experimental investigation to confirm the discovery
claims.

It is also possible that the $0\nu\beta\beta$-decay half-life is too small to
be discovered by next-generation experiments (i.e., $T_{1/2}>10^{28}$ years),
or the process does not exist at all. In this case, the forthcoming searches
will push the half-life constraints by two order of magnitudes, excluding a
significant part of the parameter space of interest and ruling out specific
models. Further technological developments will then be needed to realize
affordable next-to-next generation experiments with scaled sensitivity.

####  VII.1.2 Nuclear theory

The extraction of beyond-Standard-Model physics information from half-life
measurements relies on NME calculations, which currently differ from each
other by about a factor of three. NME calculations might also all be affected
by systematic offsets. Promising developments in ab-initio methods and chiral
EFT will reduce these uncertainties. Calculations may still disagree due to
the different approximations made, but systematic effects (“$g_{A}$
quenching”, the short-range NME contribution, etc.) are expected to be under
better control.

As the decay rate depends on the NME squared, nuclear theory uncertainties
will likely remain larger than experimental statistical or systematic
uncertainties on the half-life, representing the main limitation in the
interpretation of future results. These uncertainties will be smaller in
nuclei with simpler nuclear structure for which calculations are more robust:
an ideal example is 48Ca. Among the most interesting isotopes, NMEs would
probably be less reliable for nuclei with more complex structure, such as
100Mo and 150Nd.

NMEs for other beyond-Standard-Model mechanisms will likely carry similar
uncertainties as for light neutrino exchange. Calculations of these matrix
elements do not pose different challenges, as even light neutrino exchange has
a short-range component. Nonetheless, a careful treatment of short-range
physics is probably more relevant in these scenarios, where $0\nu\beta\beta$
decay is usually mediated by the exchange of heavy particles.

####  VII.1.3 Particle theory

At present, we lack reliable theory predictions for the $0\nu\beta\beta$-decay
rate, the origin of the matter-antimatter asymmetry, and the neutrino mass
values. A large number of beyond-Standard-Model theories have been proposed,
but none can be tested with available data, and might not be testable even
considering next-decade experiments. We neither have a model for lepton number
violation nor a theory of lepton masses, and its establishment does not seem
close. From this point of view, $0\nu\beta\beta$-decay searches are among the
most promising sectors to guide future theory developments, and, vice versa,
the searches could benefit from theory breakthroughs.

Despite the parameter space broadness, we can identify clear milestones for
the experimental program. The Holy Grail for next-generation experiments is to
reach the bottom of the inverted ordering parameter space, i.e.,
$m_{\beta\beta}=18.4\pm 1.3$ meV. This natural goalpost was immediately
identified after the discovery of neutrino oscillations, boosting enormously
the community’s efforts.

We propose $m_{\beta\beta}\approx 8-10$ meV as the next target for the field.
As discussed in Sec. III.4.4, there is an accumulation of theoretical
motivation to explore $m_{\beta\beta}$ values at this mass scale, which
corresponds to the mass scale measured in solar neutrino oscillations
($\sqrt{\Delta m^{2}_{\text{\tiny sol}}}=8.6\pm 0.1$ meV), and which is
pointed to by classes of models focusing on the coarse structure of the mass
matrix ($m_{\beta\beta}\approx\sqrt{\Delta m^{2}_{\mbox{\tiny
atm}}}\times\theta_{\mbox{\tiny C}}\approx 10$ meV). This scale is interesting
also from the experimental point of view: it is almost in the middle of the
parameter space remaining after reaching the bottom of the inverted ordering,
and can constitute a challenging, and yet conceivable goal for the next-to-
next generation of $0\nu\beta\beta$-decay experiments. It is also the vicinity
of the minimum that would be imposed on $m_{\beta\beta}$ by cosmological
observations if $\Sigma$ is measured to be just below its current upper
bounds.

An ultimate goal would be to reach the floor of the normal ordering parameter
space for vanishing $m_{1}$, $m_{\beta\beta}\sim|U_{e2}^{2}|\sqrt{\Delta
m^{2}_{12}}-|U_{e3}^{2}|\sqrt{\Delta m^{2}_{32}}=1.5$ meV. Barring flavor
symmetries or strongly destructive interference with alternative exchange
mechanisms that would force $m_{\beta\beta}$ to be vanishingly small,
experiments with sensitivity to this normal-ordering floor would be virtually
guaranteed to detect $0\nu\beta\beta$ decay if the Standard Model neutrino is
a (dominantly) Majorana particle. Quasi-background-free kiloton experiments
would be needed for this endeavor.

###  VII.2 What would we learn from a discovery?

####  VII.2.1 Model-independent consequences

Regardless of the mechanism mediating the decay, and of the uncertainties in
the NMEs, a $0\nu\beta\beta$-decay observation would constitute the discovery
in a laboratory experiment of a process that creates matter without creating
antimatter. This “Little Bang” would prove that lepton number is not a
conserved quantum number, and that neutrinos can transform into antineutrinos.

The violation of lepton number is directly observable in $0\nu\beta\beta$
decay, as two new leptons are created without the creation of any
antiparticles. The possibility for a neutrino to transform into an
antineutrino and vice versa would be proven indirectly. The $0\nu\beta\beta$
decay operator, together with quantum fluctuations, provides a non-zero
neutrino-antineutrino transformation channel. However, in the absence of
precise theory, its size cannot be predicted. From this point of view, an
observation of $0\nu\beta\beta$ decay guarantees only that the Majorana mass
is not null. Although it is not favored by the best-motivated models, one
cannot rule out the possibility that its value is so small that it does not
have any practical consequence. In this case, the neutrino would
phenomenologically behave as a Dirac particle, and theory inputs would still
be needed to connect $0\nu\beta\beta$ decay with the origin of neutrino
masses.

####  VII.2.2 Model-dependent consequences

Experiments measure the decay half-life, and NMEs are needed to connect it
with the underlying beyond-Standard-Model mechanism. Multiple mechanisms can
contribute to the $0\nu\beta\beta$-decay rate, which is proportional to the
squared sum of amplitudes for all contributions. While both constructive and
destructive interference are possible, a complete cancellation between
unrelated mechanisms would require fine-tuned models.

Half-life measurements or bounds on different nuclei provide information on
the underlying mechanism. For instance, measuring a half-life of $10^{27}$
years for 76Ge would imply an expected 100Mo half-life of (1–3)$\times
10^{26}$ years if the decay is dominated by the exchange of light neutrinos.
Likewise, similar half-life ranges will be predicted for the decay of other
isotopes, and also for $0\nu\beta\beta$ decays to excited states. Incompatible
half-life measurements could hence prove the existence of other mechanisms
driving the decay, assuming accurate calculations on the NMEs Deppisch and Pas
(2007); Gehman and Elliott (2007); Simkovic  _et al._ (2010); Gráf  _et al._
(2022). Indeed current NME uncertainties would severely limit our ability to
pin down the specific mechanism(s). Furthermore, these kinds of analyses are
sensitive to correlations between calculated NMEs as pointed out by Lisi and
Marrone (2022).

Measurements of the decay kinematics, which provide information on the Lorentz
structure of the mediating mechanism, could conclusively rule-out classes of
models or corroborate others. However these properties, as well as decays to
excited states, are hard to measure. Experimental efforts beyond the next
decade might be needed to collect this information if $0\nu\beta\beta$ decay
is not discovered in experiments currently underway or starting soon.

####  VII.2.3 Assuming light-neutrino exchange

If the decay is dominantly mediated by the exchange of light neutrinos, a
comparison of the measured $m_{\beta\beta}$ with other data would provide new
insights on neutrino physics. An observation of $m_{\beta\beta}$ below the
minimum value allowed for the inverted ordering would imply that neutrino
masses follow the normal ordering. Vice versa, should the inverted ordering be
established by neutrino oscillation experiments, the non-observation of
$0\nu\beta\beta$ decay in next-generation experiments would rule out Majorana
neutrino masses.

Galaxy surveys and measurements of the cosmic microwave background will
measure the value of the sum of the neutrino masses $\Sigma$ in the next
decade. Such a measurement would not only set an indirect upper bound on
$m_{\beta\beta}$, but could also provide a lower bound. In particular, if
$\Sigma$ is measured to be above 70–80 meV, then $m_{\beta\beta}$ must be
larger than $\sqrt{\Delta m^{2}_{12}}$, suggesting exciting discovery
prospects for next-generation $0\nu\beta\beta$-decay searches.

In the near term, direct measurements of the effective kinematic neutrino mass
$m_{\beta}$ will explore a parameter space that is already excluded by
$0\nu\beta\beta$ decay and cosmology. Thus a signal in those experiments, as
well as other inconsistencies among neutrino data sets, would strongly point
towards new physics beyond the 3-flavor neutrino oscillation and $\Lambda$CDM
paradigm.

A measurement of $m_{\beta\beta}$ is currently the only conceivable way to
obtain information on the values of the Majorana phases in the PMNS matrix,
through a global analysis with oscillation measurements. However, only one
relative phase can be measured. In addition, constraints on this relative
phase can be extracted only if the experimental and nuclear theory
uncertainties are strongly reduced below their current levels.

###  VII.3 What are the odds of a discovery?

####  VII.3.1 Model-independent considerations

A wide variety of particle theory models predict $0\nu\beta\beta$ decay. In
most of them, unconstrained model parameters prevent a precise prediction of
the decay rate. At best, these models provide a lower limit on the half-life,
which sets a target for the experiments. The master formula in Eq. 23 connects
the half-life to effective operators representing classes of models. In
general, the half-life is proportional to the energy scale of the physics
responsible for the decay, taken to some power which depends on the dimension
of the operator. Operators above dimension 5 typically correspond to energy
scales close to or beyond those explored by accelerator experiments. Figure 27
shows that accelerator and $0\nu\beta\beta$-decay experiments are
complementary, and highlights how the reach of $0\nu\beta\beta$-decay searches
can even exceed that of accelerators for mechanisms other than light neutrino
exchange. Note that, for reference, Fig. 27 just shows estimates for the most
favored dimension-7 and dimension-9 operators. In general there can be order
of magnitude differences for other operators suppressed e.g. by ratios of the
nuclear-over-electroweak or chiral-over-nuclear scales.

Figure 27: Discovery sensitivities of current- and next-generation
$0\nu\beta\beta$-decay experiments for exchange mechanisms dominated by
effective operators of dimension 7 and 9. Values of $\Lambda$ smaller than the
marked values are tested at higher CL. The grey band corresponds roughly to
the reach of modern accelerator experiments. The black line indicates an
ambitious goal for future circular colliders such as the FCC-hh Golling  _et
al._ (2016).

We also indicate in Fig. 27 the energy scale of 100 TeV. This is a round
value, suggestive of a possible ambitious target for next-generation
colliders, but also a scale at which new flavor- and beyond-the-Standard-Model
physics could manifest. Due to the large variety of possible decay mechanisms,
one can consider $0\nu\beta\beta$ decay as a generic search for new physics,
similar to accelerator ones, where the decay half-life plays the role of the
collision energy. Increasing the half-life sensitivity implies exploring
uncharted parameter space, where a discovery can happen at any time.

####  VII.3.2 Assuming light-neutrino exchange

The $0\nu\beta\beta$-decay mechanism requiring the least new physics is light
neutrino exchange, which only needs the Standard Model neutrino to be a
massive Majorana particle. From a general point of view this is a particularly
important mechanism, as it is the only one driven by a dimension-5 operator,
i.e., the Weinberg operator. Further, it is uniquely connected to neutrino
masses and is the dominant decay contribution in many models. In this
scenario, the decay rate depends on the effective Majorana mass
$m_{\beta\beta}$, which is a function of the neutrino oscillation parameters,
Majorana phases, the lightest neutrino mass eigenstate, and the neutrino mass
ordering.

The oscillation parameters have been measured precisely, nonetheless we have
no information on the Majorana phases, and the mass ordering has not been
determined. Although global fits show a preference for normal-ordered masses,
we need to wait for the next decade experiments — i.e., JUNO, DUNE, and
HyperKamiokande — with the requisite sensitivity to establish the neutrino
mass ordering. These unknowns lead to uncertainties on the $m_{\beta\beta}$
value.

Assuming that neutrino masses follow the normal-ordering, any half-life value
beyond the current upper limits is allowed. However, for the inverted-ordering
case, the half-life has a lower bound corresponding to $m_{\beta\beta}=18.4\pm
1.3$ meV. Figure 28 shows the $m_{\beta\beta}$ sensitivity of future
$0\nu\beta\beta$-decay experiments. The proposed experimental endeavor will
fully test the inverted ordering parameter space, guaranteeing a discovery if
this is the true scenario, and offering exciting discovery opportunities also
assuming the normal ordering. In fact, since the current best bounds on
$m_{\beta\beta}$ are $\sim$160–180 meV — assuming the least favorable NMEs —
reaching 18.4 meV means probing 80%–90% of the currently allowed range for the
normal ordering.

Figure 28: Discovery sensitivities of current- and next-generation
$0\nu\beta\beta$-decay experiments for exchange dominated by effective
operators of dimension 5, i.e., the light neutrino exchange. Values of
$m_{\beta\beta}$ larger than the marked values are tested at higher CL. The
grey band indicates the range of $m_{\beta\beta}$ values for inverted-ordered
neutrino masses and vanishing values of the lightest neutrino mass. The
minimum value of $m_{\beta\beta}$ for the IO and its 1$\sigma$, 2$\sigma$, and
3$\sigma$ uncertainty bands are indicated by the black, green, orange, and
yellow bands, respectively. The red band between 8-10 meV indicates a future
goal for $0\nu\beta\beta$decay experiments motivated by theoretical and
experimental considerations (see discussion in Sec. III.4.4).

It should be mentioned that the parameter space for $m_{\beta\beta}$ might not
be equiprobable. A theoretical prejudice for normal-ordered masses and
vanishing $m_{1}$ would prefer smaller values of $m_{\beta\beta}$, for
example. New symmetries predicting specific values for the Majorana phases or
the existence of new particles such as sterile neutrinos could favor other
corners of the parameter space, or even reduce or open it. In addition,
Bayesian analyses assuming flat priors on the Majorana phases and a log-flat
prior on $\Sigma$ favor $m_{\beta\beta}$ values close to the current
constraints, providing exciting prospects for the field regardless of the mass
ordering.

####  VII.3.3 Impact of nuclear physics

How likely a discovery is in the next decade strongly depends on systematic
uncertainties on NME calculations. A broad effort to reduce uncertainties is
ongoing within the nuclear theory community.

Ab initio approaches offer a promising avenue: by incorporating wider nuclear
correlations and currents, measured $\beta$-decay rates can now be reproduced
without the “quenching” required by previous studies — an ad hoc reduction of
calculated matrix elements. The first ab initio matrix elements for
$0\nu\beta\beta$-decay nuclei, supported by studies in lighter systems,
indicate a mild suppression by tens of percent with respect to the lower
values in Tab. 1. This suggests that current $0\nu\beta\beta$-decay rate
predictions may have to be reduced, but only moderately. Efforts are underway
to improve the quality of the results, to include missing momentum-dependent
operators — a key difference between $\beta$ and $0\nu\beta\beta$ decay — and
to extend them to heavier nuclei.

The recently recognized short-range term can contribute significantly to the
NME. A first ab-initio study in 48Ca suggests that including this physics
increases the NME by about 40% percent. A similar enhancement has been found
in heavier $0\nu\beta\beta$-decay nuclei with the NSM and with QRPA. Lattice
QCD studies are underway to test whether this claimed enhancement is robust.
If so, the impact of the new term may balance the longer half-life values
anticipated due to the inclusion of the “quenching” physics.

Even if these systematic contributions to NMEs were fully resolved,
discrepancies remain between results obtained with different many-body
methods. Tests against nuclear structure data can gauge the quality of each
calculation. In addition, novel measurements of nuclear observables correlated
with $0\nu\beta\beta$-decay NMEs such as second-order Gamow-Teller or
electromagnetic transitions can provide insights on each method’s strengths
and weaknesses.

###  VII.4 What else can be discovered by $0\nu\beta\beta$-decay experiments?

The unprecedented combination of ultra-low background, high-exposure, high
energy resolution, and multivariate analysis capabilities in modern
$0\nu\beta\beta$-decay experiments offers exciting discovery opportunities
beyond the primary target of observing $0\nu\beta\beta$ decay. This includes
searches not only for other L-violating processes, such as neutrinoless
electron capture Blaum  _et al._ (2020) or neutrinoless quadruple-beta decay
Guzowski (2018), but also for completely decoupled physics.

The existence of new particles and fields, the violation of fundamental
principles, and non-standard interactions can each affect, in a characteristic
way, the distribution of the summed energy of the electrons emitted in
$\beta\beta$ decays. Historically, searches for new particles focused on
massive and massless bosons called Majorons, the Goldstone bosons that arise
from the spontaneous breakdown of the global $B-L$ symmetry. Searches for the
violation of fundamental principles have focused on Lorentz invariance, the
Pauli Exclusion Principle, and CPT symmetry. We refer to Bossio and Agostini
(2022) for a comprehensive review of this topic. Future searches will have
high-sensitivity to additional physics, for instance exotic currents Deppisch
_et al._ (2020b), and light exotic fermions such as sterile neutrinos or
$Z_{2}$-odd fermions Bolton  _et al._ (2021); Agostini  _et al._ (2021c).

In addition to distortions on the energy distribution, next-generation
$0\nu\beta\beta$-decay experiments will be highly sensitive to numerous
beyond-Standard-Model processes which could generate events with well defined
energy depositions and/or time correlations. These searches include
B-violating tri-nucleon decay Alvis  _et al._ (2019b); Albert  _et al._ (2018)
and charge-violating electron decay Abgrall  _et al._ (2017). Dark matter
candidates such as WIMPS Agostini  _et al._ (2020d); Arnold  _et al._ (2020);
Liu  _et al._ (2019); Abgrall  _et al._ (2017) and axions Xu and Elliott
(2017); Abgrall  _et al._ (2017) can also be identified through an excess of
events with a well-defined energy distribution or time-modulation. New
searches have been proposed for inelastic boosted dark matter Ha  _et al._
(2019) and fermionic dark matter Dror  _et al._ (2020), and constraints have
already been placed fractional-charge lightly ionizing particles Alvis  _et
al._ (2018).

###  VII.5 What will be the next paradigm shift?

For half a century, the Standard Model of particle physics has been the
field’s paradigm. The discovery of the Higgs boson, immediately recognized by
the 2013 Nobel Prize in physics, was its crowning achievement. At the same
time, we have known for almost two decades that this model is incomplete and
needs to be extended, at least to incorporate massive neutrinos.

Extensions inspired by the very same symmetry principles that underlie the
Standard Model have been explored in the framework of gauge theories, which
include the so-called “Grand Unification” models. These theories have however
not been confirmed in spite of the extensive experimental efforts to observe
proton decay in the 1980s and 1990s. Some intrinsic features of the Standard
Model, such as CP-symmetry in strong interactions, or the nature of radiative
corrections in the Higgs sector, have in turn suggested the possible existence
of new particles, e.g., axions or supersymmetric particles. Searches for these
new particles have also been unsuccessful so far.

In the mean time, cosmological observations have led to the development of a
Standard Model of cosmology, $\Lambda$CDM. Its very name invokes the existence
of two forms of matter that cannot be found in the Standard Model of particle
physics: dark matter and dark energy. Furthermore, theoretical cosmology has
proved unable to account for the cosmic baryon excess.

Finally, several experimental anomalies have emerged, the most recent of which
is the measurement of the anomalous magnetic moment of the muon Abi  _et al._
(2021). These anomalies could also point to some missing piece of the Standard
Model.

Nonetheless, the only unequivocal manifestation of physics beyond the Standard
Model supported by laboratory experiments is the evidence of neutrino
oscillation, recognized by the 2015 Nobel Prize in physics as a proof that
neutrinos are massive. This suggests that the importance of further studies on
the neutrino mass should not be underestimated. The most promising theoretical
option is that the mass type is exactly the one proposed by Majorana. Its
experimental demonstration is a concrete and well-defined goal to strive for
in the exploration of physics beyond the Standard Model.

The best way to probe the Majorana nature of neutrinos is to measure the rate
of neutrinoless double beta decay — i.e., the rate at which electron pairs are
created in certain nuclear decays — an observation which would lead to a
profound change in our understanding of matter. Although we do not have an
established theory that can guide us safely in the next steps, we are just
starting a pioneering exploration of the next two, uncharted orders of
magnitude, an exciting journey that will bring us a step closer to unlock the
secrets of the universe.

###### Acknowledgements.

We are extremely grateful to Frank Deppisch, Steve Elliott, Jon Engel, Juan
Jose Gómez Cadenas, Andrea Pocar, and the three anonymous referees of RMP for
invaluable comments on our manuscript, and to our artist, Laura Manenti, for
preparing for us such beautiful and communicative figures We would also like
to thank Laura Baudis, Chamkaur Ghag, Giorgio Gratta, Josh Klein, Fabian
Kuger, and Pia Loaiza, for their help during the preparation of the
manuscript. Finally, we thank Chiara Brofferio, Vincenzo Cirigliano, Stefano
Dell’Oro, Francesco Iachello, Aldo Ianni, Eligio Lisi, Stefano Pirro, Matteo
Viel, and Christoph Wiesinger for valuable scientific discussions. This work
has been supported by the Science and Technology Facilities Council, part of
U.K. Research and Innovation (Grant No. ST/T004169/1), by the UCL
Cosmoparticle Initiative, by the EU Horizon2020 research and innovation
program under the Marie Skłodowska-Curie Grant Agreement No. 754496, by the
“Ramón y Cajal” program with grant RYC-2017-22781, and grants CEX2019-000918-M
and PID2020-118758GB-I00 funded by MCIN/AEI/10.13039/501100011033 and, as
appropriate, by “ESF Investing in your future”, the Italian Research Grant
Number 2017W4HA7S “NAT-NET: Neutrino and Astroparticle Theory Network” under
the program PRIN 2017 funded by MIUR, and by the U.S. DOE Office of Nuclear
Physics under Grant Number DE-FG02-97ER41020.

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