# Probing Beyond the Standard Model Physics with Double-beta Decays

E. Bossio  Physik Department, Technische Universität München, Germany
<EMAIL_ADDRESS>M. Agostini  Department of Physics and Astronomy,
University College London, London, UK<EMAIL_ADDRESS>
(August 28, 2024)

###### Abstract

Nuclear double-beta decays are a unique probe to search for new physics beyond
the Standard Model. Still-unknown particles, non-standard interactions, or the
violation of fundamental symmetries would affect the decay kinematic, creating
detectable and characteristic experimental signatures. In particular, the
energy distribution of the electrons emitted in the decay gives an insight
into the decay mechanism and has been studied in several isotopes and
experiments. No deviations from the prediction of the Standard Model have been
reported yet. However, several new experiments are underway or in preparation
and will soon increase the sensitivity of these beyond the Standard Model
physics searches, exploring uncharted parts of the parameter space. This
review brings together phenomenological and experimental aspects related to
new-physics searches in double-beta decay experiments, focusing on the
testable models, the most-sensitive detection techniques, and the discovery
opportunities of this field.

###### Contents

  1. I Introduction
  2. II Standard-Model allowed double-$\beta$ decay
  3. III Beyond-Standard-Model physics in double-$\beta$ decay
    1. III.1 New particles
      1. III.1.1 Bosons
      2. III.1.2 Fermions
    2. III.2 Violation of fundamental symmetries
      1. III.2.1 Lorentz violation
      2. III.2.2 Violation of Pauli exclusion principle
    3. III.3 Non-standard interaction
      1. III.3.1 Right-handed leptonic currrents
      2. III.3.2 Neutrino self-interaction
  4. IV Statistical analysis and sensitivity
    1. IV.1 Statistical analysis
    2. IV.2 Experimental sensitivity for class I models
    3. IV.3 Experimental sensitivity for class II and III models
    4. IV.4 Impact of the systematic uncertainties on the sensitivity
  5. V Double-$\beta$ decay experiments and constrains
    1. V.1 HPGe semiconductor detectors
    2. V.2 Cryogenic calorimeters
    3. V.3 Time projection chambers
    4. V.4 Large liquid scintillators
    5. V.5 Tracking calorimeters
    6. V.6 Other technologies
    7. V.7 Most sensitive constraints
  6. VI Outlook and prospects

##  I Introduction

Double-$\beta$ decays are nuclear transitions in which an isotope changes its
atomic number $Z$ by two units, while maintaining constant its mass number
$A$:

$(A,Z)\rightarrow(A,Z+2)+....$ (1)

This process can occur only if the conversion of two protons into two neutrons
leads to a more bounded nuclear configuration, with a positive Q-value defined
as $Q_{\beta\beta}\approx M(A,Z)c^{2}-M(A,Z+2)c^{2}$ [1], where $M$ refers to
the mass of the isotopes. Different theory models predict different decay
final states. However, in general, they all envision two electrons to conserve
the electric charge, along with neutral particles.

These decays are a second-order weak process. Thus, they can be observed only
in isotopes for which the otherwise dominant first-order transitions — and in
particular, the single-$\beta$ decay — are strongly suppressed. This is the
case for a limited number of even-even nuclei for which the single-$\beta$
decay is energetically forbidden, as the attractive nuclear pairing
interaction makes them more bound than their odd-odd neighbors but less than
their even-even second-neighbors (see Figure 1). Alternatively, candidate
isotopes for the observation of double-$\beta$ transitions are also those for
which the single-$\beta$ decay is suppressed by the large mismatch between the
total angular momentum of the initial and final nuclei [2].

M(A,Z)ZZ+1Z+2Z-1Z-2even/evenodd/odd$\beta^{-}\beta^{-}$$\beta^{-}$ Figure 1:
Mass parabolas of nuclear isobars with even A. Due to the pairing term in the
semi-empirical mass formula, $\beta^{-}$ transitions of even/even nuclei to
their odd/odd isobaric neighbor can be energetically forbidden, whereas, in a
second-order process, $\beta^{-}\beta^{-}$ decay is allowed.

Double-$\beta$ decays were postulated in 1935 [3] by Maria Goeppert-Maier, who
pointed out how these transitions could proceed through the “simultaneous
emission of two electrons and two (anti)neutrinos”:

$(A,Z)\rightarrow(A,Z+2)+2e^{-}+2\overline{\nu}$ (2)

In this case, the decay would “appear as the simultaneous occurrence of two
transitions, each of which does not fulfill the law of conservation of energy
separately”. This final state is the only one allowed by the Standard Model of
particle physics and is typically referred to as “two-neutrino double-$\beta$”
($2\nu\beta\beta$) decay.

The discovery of double-$\beta$ transitions traces back to the observation of
the decay daughter isotope in a geochemical experiment with 130Te [4]. Only 40
years later, double-$\beta$ decays were observed in real-time through a
calorimetric measurement of the energy released by the electron emitted in the
decay of 82Se [5]. The electron summed energy distribution was found to have a
continuous shape well compatible with the expectations for Goeppert-Maier’s
$2\nu\beta\beta$ decay (see Figure 2). To date, double-$\beta$ transitions
compatible with the $2\nu\beta\beta$ final state have been observed in 9
isotopes with half-life values in the range of $10^{18}-10^{24}$ yr [6],
making it one of the rarest processes ever measured by humankind.

Theories beyond the Standard Model (BSM) predict a variety of new, additional
final states involving the emission of electrically neutral exotic particles,
which would affect the decay kinematics and alter the energy and angular
distributions of the electrons emitted in the decay. Similarly, also the
violation of fundamental symmetries would affect the decay kinematic. This is
why double-$\beta$ decays have been a unique window to new physics for several
years. However, despite many past and recent experiments, no deviations from
the standard Model’s $2\nu\beta\beta$-decay expectations have yet been
observed.

Double-$\beta$ decay searches have been historically carried out in
experiments primarily built to study a hypothesized final state with only two
electrons (i.e. without neutrinos), in which the summed electron energy is
expected to be precisely equal to $Q_{\beta\beta}$ rather than a continuous
distribution (see Figure 2). Such “neutrinoless double-$\beta$”
($0\nu\beta\beta$) decay is indeed predicted by our leading theories
explaining the matter-antimatter asymmetry and origin of neutrino masses. Its
search has been an indisputable priority of the particle-physics community for
the last two decades. Currently, ton-scale double-$\beta$-decay experiments
are under preparation with the ultimate goal of increasing the sensitivity to
$0\nu\beta\beta$-decay half-life values by two orders of magnitudes compared
to current constraints, up to $10^{28}$ yr. We refer the reader to Ref. [7]
for a recent review on $0\nu\beta\beta$ decay.

Thanks to their planned ultra-low background level and huge target mass, the
next-generation double-$\beta$ decay experiments will provide high-precision
measurements of the $2\nu\beta\beta$-decay, enabling new opportunities to
discover new final states. This motivates the timing of our work. Indeed,
although several review articles have discussed double-$\beta$ decay searches
in recent years, they have typically focused on the $2\nu\beta\beta$-decay
final state. For instance, we would like to highlight Ref. [8] that gives an
excellent review of $2\nu\beta\beta$-decay’s history and [9] for a recent
summary of the field. Differently from past works, our review is the first one
focused on the BSM searches, covering their theoretical and experimental
aspects, current constraints, and ongoing endeavor to improve the experimental
sensitivities. Future $0\nu\beta\beta$ decay experiments with leading
sensitivity will measure only the summed energy of the two final state
electrons – aiming to distinguish the $0\nu\beta\beta$ decay peak at
$Q_{\beta\beta}$ from the continuous $2\nu\beta\beta$ decay distribution –
without having access to the single electron energy or electron angular
correlation. Therefore, this review focuses on how BSM physics would affect
the summed electron energy distribution and how this can be exploited to
search for BSM double-$\beta$ decays.

The manuscript is organized as follows. We first briefly discuss the Standard
Model $2\nu\beta\beta$ decay in Sec. II for completeness. We then review in
Sec. III the phenomenology of double-$\beta$ transitions mediated by Beyond-
Standard-Model processes. Section IV is dedicated to their experimental
signatures exploited by experiments. It also contains an original contribution
to the field, which is the first derivation of analytical formulas describing
the experimental sensitivity as a function of key experimental parameters and
systematic uncertainties. Finally, Sec. V summarises the latest experimental
results and the prospects for next-generation experiments.

##  II Standard-Model allowed double-$\beta$ decay

Double-$\beta$ transitions can be uniquely identified by the production of the
daughter nucleus. With this principle, the double-$\beta$ decay of 130Te was
first detected in 1950 with the first geochemical experiment. The detection of
an excess of 130Xe was proof of the double-$\beta$ decay of the initial
nucleus and allowed a first determination of its half-life [4]. Even if this
result was initially not considered seriously, it represents the first
observation of $2\nu\beta\beta$ decay, as it became clear after 15–20 yr.

In fact, after both the $2\nu\beta\beta$ and $0\nu\beta\beta$ decays were
proposed (1935–1939), the first estimate of the half-life of the two processes
strongly favored the $0\nu\beta\beta$ decay. This was predicted with a half-
life of the order of $10^{15}$ yr, whereas a half-life of about $10^{21}$ yr
was predicted for the Standard-Model allowed $2\nu\beta\beta$ decay. Only the
discovery of parity violation in 1957 and the determination of the V–A nature
of weak interaction made clear that the probability of $0\nu\beta\beta$ decay
had to be much smaller than that of $2\nu\beta\beta$ decay.

In the following years, new geochemical experiments were performed, confirming
the observation of the $2\nu\beta\beta$ decay of 130Te [10, 11] and observing
for the first time the $2\nu\beta\beta$ decay of 82Se [11] and 128Te [12].

A key milestone in the history of double-$\beta$ decays is the theoretical
work of Doi, Kotani, and Takasugi in 1985 [13], who first computed the energy
and angular distributions of the electrons emitted in double-$\beta$ decays,
providing a clear signature to distinguish the Standard-Model allowed
$2\nu\beta\beta$ decay from BSM double-$\beta$ decays. In the same work,
Figure 2 appeared for the first time.

Figure 2: Summed electron energy distribution for the SM $2\nu\beta\beta$
decay and the lepton number non-conserving $0\nu\beta\beta$ decay. An infinite
energy resolution is assumed, and an arbitrary normalization is used for
illustrative purposes.

Following these theory developments, the first direct observation of a
$2\nu\beta\beta$ decay was made in 1987 using a 82Se time projection chamber,
which could measure the summed energy distribution of the two electrons and
extract the 82Se $2\nu\beta\beta$-decay half-life based on 36 observed events.
This was a turning point in the history of $2\nu\beta\beta$ decays, and in the
following ten years $2\nu\beta\beta$ decay was directly observed in seven
isotopes: 76Ge [14], 100Mo [15, 16, 17, 18], 82Se [19, 20], 116Cd [21, 22, 23,
24], 48Ca [25], 150Nd [17], 96Zr [26]. In addition to the summed electrons’
energy distribution, which was measured for all the above-mentioned isotopes,
the NEMO-2 experiment also measured the single electron energy and the angular
distributions of the electrons for 100Mo, 116Cd, 82Se, and 96Zr [18, 22, 24,
20, 26]. In the same years, the $2\nu\beta\beta$ decay of 238U was first
observed in a radiochemical experiment [27].

By the end of the 20th century, the energy and half-life of $2\nu\beta\beta$
decay had been measured for 10 isotopes, considering direct, geochemical, and
radiochemical experiments. In addition, in the last two decades, direct
observation of the $2\nu\beta\beta$ decay of 130Te and 136Xe were reported in
2010 [28] and 2011 [29], respectively. Table 1 summarises the double-$\beta$
decaying isotopes for which the $2\nu\beta\beta$ decay was directly observed.

The precision with which experiments were able to determine the half-life of
$2\nu\beta\beta$ decays has increased over the years, with modern experiments
reaching a percent-level precision for most of the isotopes. The most precise
measurement of the $2\nu\beta\beta$ decay half-life are summarised in Table 1.

Isotope | First | Ref. | Half-life | Experiment | Ref.
---|---|---|---|---|---
| observation |  |  |  |
48Ca $\rightarrow\,^{48}$Ti | 1996 | [25] | $(6.4^{+1.4}_{-1.1})\cdot 10^{19}$ | NEMO-3 | [30]
76Ge $\rightarrow\,^{76}$Se | 1990 | [14] | $(2.022\pm 0.041)\cdot 10^{21}$ | GERDA | [31]
82Se $\rightarrow\,^{82}$Kr | 1987 | [5] | $(8.60^{+0.19}_{-0.13})\cdot 10^{19}$ | CUPID-0 | [32]
96Zr $\rightarrow\,^{96}$Mo | 1999 | [26] | $(2.35\pm 0.21)\cdot 10^{19}$ | NEMO-3 | [33]
100Mo $\rightarrow\,^{100}$Ru | 1991 | [15, 16] | $(7.12^{+0.21}_{-0.17})\cdot 10^{18}$ | CUPID-Mo | [34]
116Cd $\rightarrow\,^{116}$Sn | 1995 | [21, 22, 23] | $(2.63^{+0.11}_{-0.12})\cdot 10^{19}$ | Aurora | [35]
130Te $\rightarrow\,^{130}$Xe | 2010 | [28] | $(7.71^{+0.14}_{-0.16})\cdot 10^{20}$ | CUORE | [36]
136Xe $\rightarrow\,^{136}$Ba | 2011 | [29] | $(2.165\pm 0.063)\cdot 10^{21}$ | EXO-200 | [37]
150Nd $\rightarrow\,^{150}$Sm | 1997 | [17] | $(9.34^{+0.66}_{-0.64})\cdot 10^{18}$ | NEMO-3 | [38]
Table 1: Direct observations of $2\nu\beta\beta$ decays. The year of the first
observation is indicated for each isotope, together with the most precise
half-life determination. The shown uncertainty is the sum in quadrature of the
statistical and systematic uncertainties, when available.

In addition to the Goeppert-Maier’s $2\nu\beta\beta$ decay (Equation 2) or
$\beta^{-}\beta^{-}$ transition, depending on the relative numbers of protons
and neutrons in the nucleus, three additional second-order transitions are
allowed in the Standard Model:

$\displaystyle(\beta^{+}\beta^{+})$
$\displaystyle:\;(A,Z)\rightarrow(A,Z-2)+2e^{+}+2\nu$ (3a)
$\displaystyle(ECEC)$ $\displaystyle:\;(A,Z)+2e^{-}\rightarrow(A,Z-2)+2\nu$
(3b) $\displaystyle(EC\beta^{+})$
$\displaystyle:\;(A,Z)+e^{-}\rightarrow(A,Z-2)+e^{+}+2\nu\,.$ (3c)

The energy released in the three processes listed above is smaller compared to
the $\beta^{-}\beta^{-}$ decay in Equation 2. Consequently, these processes
have lower probabilities compared to the $\beta^{-}\beta^{-}$ decay due to the
smaller phase space, and experimentally they are much more challenging to
observe. In the following, we will always refer to the $\beta^{-}\beta^{-}$
process as double-$\beta$ decay.

In nature, there are 35 isotopes that can undergo $2\nu\beta\beta$ decay, and
34 more that can undergo $\beta^{+}\beta^{+}$, $ECEC$, and $EC\beta^{+}$ [39].
In fact, not all of them fulfill the experimental requirements, e.g., high
isotopic abundance, large $Q_{\beta\beta}$ value, and compatibility with
experimental technologies. To date, only the nine isotopes listed in Table 1
were used in direct search experiments.

The rate of $2\nu\beta\beta$ decay can be calculated following Fermi’s golden
rule for $\beta$ decay. To a good approximation, the kinematic part (phase
space of the leptons emitted in the decay) and the nuclear part (matrix
element responsible for the transition probability between the two nuclear
states) can be factorized as:

$\Gamma^{2\nu}=[T_{1/2}^{2\nu}]^{-1}=|\mathcal{M}^{2\nu}|^{2}\,\mathcal{G}^{2\nu}(Q_{\beta\beta},Z)\;,$
(4)

where $\mathcal{G}^{2\nu}$ is the phase-space factor and is obtained by
integrating over the phase space of the four leptons, and $\mathcal{M}^{2\nu}$
is the nuclear matrix element (NME) and deals with the nuclear structure of
the transition. While the phase-space factor can be calculated exactly, the
NME is much more difficult to evaluate and relies on nuclear structure models.

Experiments measure the distribution of the summed kinetic energy of the two
electrons ($K$):

$\frac{d\Gamma^{2\nu}}{dK}=|\mathcal{M}^{2\nu}|^{2}\,\frac{d\mathcal{G}^{2\nu}}{dK}\;.$
(5)

To a first approximation, the shape of this distribution is determined only by
the phase space. The contribution of the NME to it is small and primarily
affects the absolute value of the transition probability.

The summed energy of the two electrons emitted in the $2\nu\beta\beta$ decay
is continuously distributed between 0 and the endpoint at the $Q_{\beta\beta}$
value due to the neutrinos escaping the detector and carrying away part of the
energy111To be precise, the maximum energy is $Q_{\beta\beta}$ minus the mass
of the two emitted neutrinos. This is typically neglected as Q-values are at
the MeV-energy scale, while neutrino masses are smaller than the eV-energy
scale [40, 41].. This is shown in Figure 2, compared to the $0\nu\beta\beta$
decay, for which a $\delta$ function at $Q_{\beta\beta}$ is expected because
all the transition energy goes into the kinetic energy of the two electrons.

In conclusion, we have seen that even if the first observation of
$2\nu\beta\beta$ decay is commonly traced back to 1950 and the first
geochemical experiment with 130Te, it took many years for the community to
convince themselves that the production of the daughter nucleus observed in
this first experiment was exactly the result of Goeppert-Maier’s
$2\nu\beta\beta$ decay. To date, precision measurements of the
$2\nu\beta\beta$ decay of several isotopes and the agreement between the
distribution of multiple observables (electrons summed energy, single electron
energy, and angular distributions) with their theoretical prediction are
striking evidence for the $2\nu\beta\beta$ decay and exclude a large part of
the parameter space for many BSM theories.

##  III Beyond-Standard-Model physics in double-$\beta$ decay

This section reviews double-$\beta$ decays in the presence of BSM physics.

Suppose new particles were involved in a double-$\beta$ decay, or any new
physics affected the phase space of the two electrons emitted in the
$2\nu\beta\beta$ decay. In that case, the expected distribution of the summed
electron energy would differ from that predicted for the SM $2\nu\beta\beta$
decay (equation 5, figure 2). This is the main feature used to search for
these BSM decays in the experimental data. Single electron energy
distributions and electron angular distributions are also primarily determined
by the phase space, and they are characteristic of the physics model and decay
final state. Experiments that can also measure these distributions can
strongly enhance their sensitivity in distinguishing the SM allowed
$2\nu\beta\beta$ decay from BSM double-$\beta$ decays. This review mainly
discusses the first aspect, given that the calorimetric measurement approach
pursued in the next generation of $0\nu\beta\beta$ decay experiments with
leading sensitivities gives access only to the summed electron energy
distribution.

We classify the models into three groups. The first one contains those
predicting the existence of new particles — either bosons or fermions — which
are emitted in the decay, replacing one or both of the $2\nu\beta\beta$’s
neutrinos. The second one includes theories in which fundamental symmetries
such as Lorentz covariance or Pauli’s exclusion principle are violated. The
last group covers non-standard interactions, like right-handed leptonic
currents and strong neutrino self-interactions.

###  III.1 New particles

####  III.1.1 Bosons

In the early 80s, an attractive approach to the neutrino mass problem was
considered in which the neutrinos are Majorana particles with small masses
arising from the spontaneous breakdown of the global B-L symmetry. In these
models, a massless Goldstone boson should exist, which was called the
”Majoron”. Several realizations of this idea were proposed, mainly differing
by the weak isospin ($I$) properties of the Majoron and leading to different
phenomenology.

The first model was proposed by Chikashige, Mohapatra, and Peccei [42, 43]. In
this model, the Majoron arises from a Higgs singlet ($I=0$) and gives rise to
small neutrino masses via the ”see-saw” mechanism. On the other hand, the
Majoron coupling to neutrino is so small that it would be very hard to test it
through laboratory experiments. Shortly after, Gelmini and Roncadelli proposed
a model in which the Majoron arises from a Higgs triplet ($I=1$) [44]. In this
second case, a stronger coupling to neutrinos would be possible. A third case
was considered later in 1987, in relation to solar neutrino oscillations,
where the Majoron arises from a Higgs doublet ($I=1/2$), possibly leading to
strong coupling to neutrinos [45, 46, 47].

If the Majoron Yukawa coupling to neutrinos were sufficiently strong ($\sim
10^{-5}-10^{-3}$), this would have interesting consequences for particle
physics, astrophysics, and cosmology. Already in 1981, starting from the
Gelmini-Roncadelli Majoron model, a rich phenomenology was derived [48]. One
of the most interesting consequences considered at that time was the
possibility of emitting such a Majoron in double-$\beta$ decays, giving rise
to a new final state in which two electrons and a Majoron (and no neutrinos)
are present. The decay rate of this process was calculated, and constraints on
the neutrino-Majoron coupling were set using existing limits on the decay rate
of $0\nu\beta\beta$ decay [48]. In fact, at that time, the possibility of
distinguishing $2\nu\beta\beta$, $0\nu\beta\beta$, or double-$\beta$ decay
with the emission of a Majoron was not conceived, and the latter was regarded
only as ”an interesting possibility, which may confuse the analysis of
double-$\beta$ decay experiments” [49]. Only the impressive theoretical work
of Doi, Kotani, and Takasugi in 1985, in which the energy and angular
distribution of the electrons emitted in the double-$\beta$ decay with the
emission of a Majoron were calculated for the first time [13], provided a
clear experimental signature to distinguishing different double-$\beta$ decay
channels. The same authors revised and updated these calculations in a
successive work [50]. In the same years, a supersymmetric model was developed,
which would lead to the emission of two Majorons in double-$\beta$ decays
[51].

It was clear since early works that the Majoron with nontrivial weak isospin
properties (such as the triplet and doublet Majoron), which would have
appreciable coupling to neutrinos, would also have a strong coupling to the
other leptons and would necessarily be discovered in the upcoming lepton-
lepton collisions machines [48]. As the first LEP data on the invisible width
of the $Z$ boson became available in the early 90s, the number of active light
neutrino species was limited to three, ruling out both triplet and doublet
Majoron models [52, 53].

On the other hand, in the same years, the first direct double-$\beta$ decay
experiments started to search for double-$\beta$ decay with Majoron emission,
and an excess of events below $Q_{\beta\beta}$ was observed, which could be
compatible with the energy distribution predicted by Doi Et al. for such decay
[54, 55, 56]. The only available Majoron model not contradicting LEP data was
the singlet Majoron model. Nevertheless, in its original Chikashige-Mohapatra-
Peccei formulation, the Majoron is so weakly coupled to neutrinos that it
cannot produce detectable effects in double-$\beta$ decay and, therefore, not
explain the experimental data.

The sum of these events motivated in the following years the flourishing of a
number of new models able to reconcile the results on the $Z$ decay width with
a neutrino-Majoron coupling strong enough to explain the event excess: new
models with a singlet Majoron [57], models in which the Majoron carries a non-
zero lepton number [58, 59], models predicting the emission of two Majorons
[60], and models in which, departing from the original conception of the
Majoron as Goldstone boson, the Majoron arises as the component of a massive
gauge boson [61] or a bulk field [62].

This ensemble of models can lead to two different final states, corresponding
to the emission of one or two Majorons, which we will indicate with J:

$\displaystyle(A,Z)$ $\displaystyle\rightarrow(A,Z+2)+2e^{-}+J,$ (6)
$\displaystyle(A,Z)$ $\displaystyle\rightarrow(A,Z+2)+2e^{-}+2J.$

The rate of the double-$\beta$ decay with the emission of one or two Majorons
can be expressed as:

$\displaystyle\Gamma^{J}$
$\displaystyle=g_{J\alpha}^{2}\,|\mathcal{M}^{J}_{\alpha}|^{2}\,\mathcal{G}^{J}_{\alpha}\,,$
(7) $\displaystyle\Gamma^{JJ}$
$\displaystyle=g_{J\alpha}^{4}\,|\mathcal{M}^{JJ}_{\alpha}|^{2}\,\mathcal{G}^{JJ}_{\alpha}\,,$

where $g_{J\alpha}$ is the neutrino-Majoron coupling,
$\mathcal{M}^{J(JJ)}_{\alpha}$ the NME, and $\mathcal{G}^{J(JJ)}_{\alpha}$ the
phase-space factor. All three terms depend on the particular model, which we
indicated with the subscript $\alpha$. Systematic calculations of phase-space
factors and NMEs for a number of Majoron models were performed in [63] for
many isotopes. More recently, improved calculations of the summed electron
energy distributions have been performed, leading to improved calculations of
the phase-space factors [64]. Improved calculations of the NMEs have also been
performed lately [65].

If one or two Majorons are emitted in the double-$\beta$ decay, they would
escape any detector and carry away part of the decay energy. In analogy with
the $2\nu\beta\beta$ decay, the summed electron energy is continuously
distributed between 0 and $Q_{\beta\beta}$, and its exact shape is primarily
determined by the phase space. In turn, the phase space depends on the Majoron
model, particularly on the effective neutrino-Majoron interaction Lagrangian
leading to the Majoron-emitting double-$\beta$ decay. This can be
parameterized to-a-first-approximation with a spectral index $n$:

$\frac{d\Gamma^{J}}{dK}\propto\frac{d\mathcal{G}^{J}}{dK}\sim(Q_{\beta\beta}-K)^{n}$
(8)

Figure 3 shows the summed electron energy distribution for different Majoron
models compared to the SM $2\nu\beta\beta$ decay distribution.

Figure 3: Summed electron energy distribution for different Majoron models (the spectral index corresponding to each model is indicated) compared to the SM $2\nu\beta\beta$ decay distribution. The decays with the emission of a non-standard Majoron are also shown: they can be triggered by an effective seven-dimension operator, containing right-handed ($\epsilon^{\phi}_{RR}$) and left-handed ($\epsilon_{RL}^{\phi}$) hadronic current. The latter two were adapted from [66]. An arbitrary normalization is used for illustrative purposes. | Decay | $n$ | Goldstone boson | L
---|---|---|---|---
a | J$\beta\beta$ | 1 | yes | 0
no | 0
no | -2
b | J$\beta\beta$ | 2 | Bulk field | 0
c | J$\beta\beta$ | 3 | yes | -2
Gauge boson | -2
d | JJ$\beta\beta$ | 3 | yes | 0
no | 0
no | -1
e | JJ$\beta\beta$ | 7 | yes | -1
Table 2: Different Majoron models which predict double-$\beta$ decays with the
emission of one or two Majorons. The third column indicates the model’s
spectral index ($n$), the fourth column indicates whether the Majoron is a
Goldstone boson or not, and the last column indicates the leptonic charge (L)
of the Majoron. Models with a leptonic charge different from zero preserve the
lepton number symmetry.

The spectral index is commonly used to group models predicting the same
experimental signature, i.e. the same summed electron energy distribution.
These models are not distinguishable by the experiments. Table 2 shows a
summary of all the Majoron models grouped by the number of emitted Majorons in
the second column, the spectral index in the third column, and the Majoron’s
properties in the last two columns. The fourth column indicates whether the
Majoron is a Goldstone boson or not, whereas the last column shows the
Majoron’s leptonic charge. Models in which the Majoron carries a lepton number
different from 0 preserve the lepton number symmetry. Without an independent
test of Lepton number violation, these models are experimentally
indistinguishable from the corresponding lepton number non-conserving
processes.

All the models discussed so far focus on the emission of one or two Majorons
originating from the intermediate neutrino exchanged in the process. Despite
the differences among the models, i.e. the different effective neutrino-
Majoron interaction Lagrangians, all of them assume the SM V-A structure of
the charged currents involving leptons and quarks. We will refer to them as
”classical” models. Recently a new scenario has been considered, in which a
Majoron-like particle ($\phi$) is emitted in the double-$\beta$ decay
($\phi\beta\beta$ decay). In this case, the interaction is described by an
effective dimension-seven operator, with right-handed lepton current and
right/left-handed quark current [66]. The Feynman diagram of this process is
shown in figure 4 in comparison to the Majoron emission in classical models.
The coupling strength between the neutrino and the Majoron-like $\phi$ is
$\epsilon_{RL}$ if the effective operator contains left-handed quark current,
and $\epsilon_{RR}$ when the effective operator contains right-handed quark
current. The two cases have been considered separately, with only one of the
two operators being present at a time.

---- (a)

R$\phi$L- (b)

Figure 4: (Top) Feynman diagram for the double-$\beta$ decay with the emission
of a Majoron in classical models. (Bottom) Feynman diagram for the emission of
a Majoron-like particle $\phi$ through an effective dimension-seven operator
containing right-handed currents in double-$\beta$ decay (adapted from [66]).

The energy distribution predicted for the $\phi\beta\beta$ decay is also shown
in figure 3. The distribution associated with $\epsilon_{RL}$ is very similar
to the classical Majoron emission models leading to $n=1$. On the other hand,
introducing a hadronic right-handed current in the $\epsilon_{RR}$ term
changes the shape of the distribution considerably.

In all the previous discussions, we always assumed the Majoron to be massless.
However, many of the models already presented do not prevent the Majoron from
being a light particle [60, 67, 66]. This possibility became extremely popular
because light Majorons could be a dark matter candidate [68, 69]. If the
Majoron mass is below the $Q_{\beta\beta}$, double-$\beta$ decay with the
emission of a Majoron can still happen. In this case, the end-point of the
energy distribution is shifted to $Q_{\beta\beta}-m_{J}$, where $m_{J}$ is the
Majoron mass.

####  III.1.2 Fermions

In many extensions of the Standard Model, new spin $1/2$ particles, singlet
under the Standard Model gauge group, are introduced in relation to the
question of neutrino mass generation or dark matter. Currently, the most
popular exotic fermion is the sterile neutrino $N$. Sterile neutrinos are
neutral and right-handed Standard Model singlet fermions that interact with
ordinary matter only through mixing with the active neutrinos. We refer the
reader to Ref. [70] for a recent review of the theoretical and experimental
motivation for sterile neutrinos, as well as their phenomenological
consequence. In a variant of this scenario, the singlet fermion could be
furnished with a $Z_{2}$ symmetry, so it can only be produced in pairs. In
general, when a new exotic fermion is introduced in the theory, its mass and
coupling to the Standard Model particles are free parameters of the model. It
is left to laboratory experiments and astrophysical and cosmological
observations to probe the vast allowed parameter space.

In 1980, Shrock examined the possibility of searching for sterile neutrinos in
$\beta$ decays [71]. The admixture of one or more sterile neutrino states
would create a discontinuity in the $\beta$ decay spectrum similar to the
discontinuity that a non-zero neutrino mass is expected to create at the
endpoint. The position and amplitude of this kink would give information on
the mass of the sterile neutrino and its mixing with the active neutrinos.
Since then, several $\beta$ decay experiments have searched for sterile
neutrinos and, still today, they set the most stringent constraints in the
mass range between $\sim 10$ eV and $\sim 1$ MeV [72, 73, 74, 75, 76].

In analogy with the case of single-$\beta$ decays, sterile neutrinos with
masses below few MeV could be produced in double-$\beta$ decays. This
possibility was recently discussed in [77, 78]. In [78], the production of
exotic fermions in double-$\beta$ decays was also extended to models in which
the single production is forbidden by additional symmetries while the pair
production is allowed. Such a model could be realized with the neutral singlet
fermion $\chi$ – a potential dark matter candidate – being charged under a
discrete $Z_{2}$ symmetry to make it stable. This new fermion could interact
with neutrinos through an effective four-fermion scalar interaction of the
form $g_{\chi}\nu\nu\chi\chi$. This case is particularly interesting because
such a particle cannot be produced in single-$\beta$ decays, and
double-$\beta$ decays represent a unique discovery opportunity for laboratory
experiments.

In general, models predicting the existence of light exotic fermions coupling
with the Standard Model neutrinos can lead to two additional double-$\beta$
decay final states, corresponding to the emission of one or two exotic
fermions, which we will indicate with $f$:

$\displaystyle(A,Z)$
$\displaystyle\rightarrow(A,Z+2)+2e^{-}+\overline{\nu}+f,$ (9a)
$\displaystyle(A,Z)$ $\displaystyle\rightarrow(A,Z+2)+2e^{-}+2f.$ (9b)

Sterile neutrinos can be produced via both 9a and 9b decay channels. Provided
that both decay channels are kinematically allowed – i.e. the $Q_{\beta\beta}$
value must be larger than the sterile neutrino mass for the 9a decay channel,
and twice the sterile neutrino mass for the 9b decay channel – the total
double-$\beta$ decay rate would become an incoherent sum of three channels:

$\Gamma=\cos^{4}\theta\;\Gamma_{\nu\nu}+2\,\cos^{2}\theta\,\sin^{2}\theta\;\Gamma_{\nu
N}+\sin^{4}\theta\;\Gamma_{NN}\;,$ (10)

where $\sin^{2}\theta$ represents the mixing angle between active and sterile
neutrinos. The first term accounts for the SM $2\nu\beta\beta$ decay, the
second one for the decay in which one of the two neutrinos is replaced by a
sterile neutrino (equation 9a with $f=N$), and the last one for the decay into
two sterile neutrinos (equation 9b with $f=N$). We shall notice here that this
last term is strongly suppressed by a factor of $\sin^{4}\theta$, making it
negligible for experimental searches.

Figure 5: Summed electron energy distributions of the double-$\beta$ decay
into one sterile neutrino with a mass of 600 keV and the double-$\beta$ decay
into two $Z_{2}$-odd fermions with a mass of 300 keV in comparison to the SM
$2\nu\beta\beta$ decay distribution for the case of 76Ge isotope. An arbitrary
normalization is used for illustrative purposes. Adapted from [78]

Each term of the sum can be expressed factorized as the product of the NME and
the phase space factor:

$\Gamma_{ab}=|\mathcal{M}_{2\nu}|^{2}\,\mathcal{G}_{ab}\;,$ (11)

where $a$ and $b$ indicate the emitted particles, either the active or the
sterile neutrinos. While the NME is the same for the three terms, the phase
space factor is affected by the presence of one or two sterile neutrinos in
the final state, and so is the summed electron energy distribution:

$\frac{d\Gamma_{ab}}{dK}=|\mathcal{M}_{2\nu}|^{2}\,\frac{d\mathcal{G}_{ab}}{dK}$
(12)

The presence of a massive sterile neutrino in the final state affects the
kinematics of the decay such that the endpoint of the summed electron energy
distribution is shifted at $Q_{\beta\beta}-m_{N}$, where $m_{N}$ is the
sterile neutrino mass. The energy distribution for the double-$\beta$ decay
into one sterile neutrino with a mass of 600 keV is shown in figure 5 for the
case of 76Ge isotope.

$Z_{2}$-odd fermions, which we refer to as $\chi$ fermions, can be produced in
double-$\beta$ decay via the 9b channel. To be kinematically allowed, the
$Q_{\beta\beta}$ value of the decay must be larger than twice the mass of the
$\chi$ fermions. The rate of the double-$\beta$ decay with the emission of two
fermions $\chi$ can be expressed as:

$\Gamma^{\chi\chi}=\frac{g_{\chi}^{2}\,m_{e}^{2}}{8\pi^{2}\,R^{2}}\,|\mathcal{M}^{0\nu}|^{2}\,\mathcal{G}^{\chi\chi}\;,$
(13)

where $g_{\chi}$ is the coupling between neutrinos and the $\chi$ fermions,
$m_{e}$ the electron mass, and $R$ the nuclear radius. The NME for this decay
can be take to good approximation as the NME for the $0\nu\beta\beta$ decay,
$\mathcal{M}_{0\nu}$. The presence of two massive $\chi$ fermions in the final
state affects the kinematics of the decay as for the case of sterile
neutrinos: the endpoint of the summed electron energy distribution is shifted
by $Q_{\beta\beta}-2m_{\chi}$, where $m_{\chi}$ is the $\chi$-fermion mass.
This is shown for $m_{\chi}=300$ keV in figure 5 for the case of 76Ge isotope.

###  III.2 Violation of fundamental symmetries

####  III.2.1 Lorentz violation

Lorentz invariance is one of the fundamental symmetries of the SM of particle
physics. The breakdown of Lorentz and CPT symmetries at the Plank scale is an
interesting feature of many theories of quantum gravity, such as string theory
[79]. Despite direct studies of physics at this ultrahigh energy scale are far
from the reach of current accelerator-based experiments, some suppressed
effects could arise at lower energies and be observable with the actual
experimental technologies.

The general framework that characterizes Lorentz violation in the SM is the
Standard Model Extension (SME) [80, 81]. This is an effective quantum field
theory that includes all possible operators that can be constructed with the
SM fields and that introduce Lorentz violation but preserve the SM gauge
invariance. The development of the SME has led to experimental searches for
Lorentz violation in all different sectors of physics, including matter,
photon, neutrino, and gravity [82, 83]. A data table of the current
constraints is compiled in [84] annually.

The behavior of neutrinos in the presence of Lorentz and CPT violation has
been extensively studied using the SME framework, and its related coefficients
classified [85, 86, 87]. Most of these coefficients are currently constrained
by neutrino oscillations experiments. However, four coefficients only affect
the neutrino phase space and escape detection through the measurement of
neutrino oscillations. The corresponding operators are an example of _counter-
shaded_ Lorentz-violating operators and they might have escaped detection to
date even though their effect is large compared to the suppression given by
the Plank scale [82]. These coefficients, commonly referred to as
_oscillation-free_ (of) coefficients, can be studied in weak decays, such as
single-$\beta$ decay or double-$\beta$ decay [88, 89].

Figure 6: Summed electron energy distribution of the $2\nu\beta\beta$ decay in
the SM and the perturbation term introduced by Lorentz violation (LV). An
arbitrary normalization is used for illustrative purposes.

The interaction of neutrinos with the counter-shaded operator modifies their
four-momentum:

$p=(E,\mathbf{p})\longrightarrow
p=(E,\mathbf{p}+\mathbf{a}_{of}^{(3)}-\smash{\mathring{a}_{\text{of}}^{(3)}}\hat{\mathbf{p}})$
(14)

where $\mathbf{a}_{of}^{(3)}$ encodes the _oscillation-free_ coefficients and
$\smash{\mathring{a}_{\text{of}}^{(3)}}$ is the isotropic component
$\smash{\mathring{a}_{\text{of}}^{(3)}}\equiv(a_{of}^{(3)})_{00}/\sqrt{4\pi}$.
Considering this modification, the decay rate of the $2\nu\beta\beta$ decay in
the SME framework can be written as the sum of two terms:

$\Gamma^{2\nu}_{SME}=\Gamma_{SM}+\delta\Gamma_{LV}$ (15)

where $\Gamma_{SM}$ is the SM decay rate and $\delta\Gamma_{LV}$ is the
perturbation term due to the introduction of Lorentz violation. Similarly to
the previous cases, we can factorize each term as the product of the NME and
the phase space factor (see equation 11). Lorentz violation does not affect
the NME but appears as a kinematic effect modifying the phase space factor,
thus the summed electron energy distribution:

$\frac{d\Gamma^{2\nu}_{SME}}{dK}=|\mathcal{M}_{2\nu}|^{2}\left(\frac{d\mathcal{G}_{SM}}{dK}+\frac{d(\delta\mathcal{G}_{LV})}{dK}\right)$
(16)

The modification of the phase space $d(\delta\mathcal{G}_{LV})/dK$ comes from
the change of the differential element of the anti-neutrino momentum:

$d^{3}p=4\pi\,E^{2}dE\longrightarrow
d^{3}p=4\pi\,(E^{2}+2E\,\smash{\mathring{a}_{\text{of}}^{(3)}})\,dE\;.$ (17)

The integration over all anti-neutrino orientation performed to obtain the
summed electron energy distribution in the case of double-beta decays implies
that only isotropic effects are observable. Hence, the spectrum only depends
on $\smash{\mathring{a}_{\text{of}}^{(3)}}$.

The energy dependency in the phase space of the perturbation term can be
approximated as $\delta\mathcal{G}_{LV}\sim(Q_{\beta\beta}-K)^{4}$. Using the
same terminology introduced for the Majoron, the spectral index of this
perturbation is $n=4$. On the other hand, the spectral index of the SM term is
$n=5$. Therefore, a non-zero value of the coefficient
$\smash{\mathring{a}_{\text{of}}^{(3)}}$, which implies a non-zero
contribution of the perturbation term, produces a distortion of the spectrum
of double-$\beta$ decays compared to the SM expectation. The energy
distribution of the Lorentz violating perturbation term is shown in figure 6
compared to the SM $2\nu\beta\beta$ decay distribution.

####  III.2.2 Violation of Pauli exclusion principle

Neutrinos have many peculiarities among all the known particles. They are the
only neutral leptons, which leaves the possibility for neutrinos to be
Majorana particles. In addition, the smallness of their mass points to a
different mass-generating mechanism compared to the standard coupling with the
Higgs boson exhibited by all other particles. Therefore, neutrinos might have
substantially different properties compared to the charged leptons.

Pauli’s original formulation of the exclusion principle in 1925 postulated
that two or more identical electrons cannot occupy the same quantum state
within a quantum system simultaneously. This was successively extended to all
fermions with half-integer spin and experimentally confirmed with extremely
high precision. From the theoretical point of view, a local quantum field
theory with violation of the Pauli principle has been discussed [90, 91, 92],
and some difficulties highlighted [93, 94].

The consequences of a change of neutrino statistics from fermionic to bosonic
would have substantial cosmological and astrophysical consequences [95, 96,
97, 98]. However, even recent analyses of available cosmological data can set
only weak bounds on neutrino statistics [99]. With two identical anti-
neutrinos in the final state, double-$\beta$ decay is a unique process to test
the violation of the Pauli’s principle [98, 100]. Qualitative conclusions in
[98] on $2\nu\beta\beta$ decay ruled out a pure bosonic neutrino, but not the
possibility that neutrinos obey non-standard statistics, more general than
Bose or Fermi ones [101].

If neutrinos obey a mixed statistic, the neutrino’s state would be the
combination of fermionic and bosonic states. As a consequence, the
double-$\beta$ decay amplitude would be given by the sum of two terms,
corresponding respectively to the fermionic (anti-symmetric) and bosonic
(symmetric) parts of the two anti-neutrino emissions:

$A_{2\nu\beta\beta}=\cos^{2}\chi\,A_{f}+\sin^{2}\chi\,A_{b}\;.$ (18)

In the phase-space integration, the interference between the anti-symmetric
and symmetric parts of the amplitude vanishes and the $2\nu\beta\beta$ decay
rate becomes

$\Gamma_{2\nu\beta\beta}=\cos^{4}\chi\,\Gamma_{f}+\sin^{4}\chi\,\Gamma_{b}\;,$
(19)

where the decay rates $\Gamma_{f}$ and $\Gamma_{b}$ are proportional to the
squared amplitudes $|A_{f}|^{2}$ and $|A_{b}|^{2}$, respectively, for pure
fermionic and pure bosonic neutrinos. In the decay rates for pure fermionic
and pure bosonic neutrinos, both the kinematics terms and the NMEs are
different. Defining the ratio

$r_{0}=\Gamma_{b}/\Gamma_{f}\;,$ (20)

the normalized differential decay rate can be written as

$\frac{d\Gamma_{tot}}{\Gamma_{tot}}=\frac{\cos^{4}\chi}{\cos^{4}\chi+\sin^{4}\chi\,r_{0}}\frac{d\Gamma_{f}}{\Gamma_{f}}+\frac{\sin^{4}\chi\,r_{0}}{\cos^{4}\chi+\sin^{4}\chi\,r_{0}}\frac{d\Gamma_{b}}{\Gamma_{b}}\;.$
(21)

The ratio $r_{0}$ determines the weight with which the bosonic component
enters the total rate and the differential decay distribution. If $r_{0}$ is
very small, a substantial modification of the energy distribution is expected
only for $\sin^{2}\chi$ being very close to 1. In addition, the ratio $r_{0}$
needs to be calculated and depends on the values of the NMEs. Thus, it
introduces an uncertainty due to the nuclear-structure calculations.

On the other hand, the normalized differential decay rate for pure fermionic
$d\Gamma_{f}/\Gamma_{f}$ and pure bosonic $d\Gamma_{b}/\Gamma_{b}$ neutrinos
does not depend on any nuclear model assumption and are shown in figure 7. The
spectrum for bosonic neutrinos is softer, with the maximum shifted at lower
energy by a factor of about 15%, compared to the pure fermionic spectrum.

Figure 7: Summed electron energy distribution of the $2\nu\beta\beta$ decay
for pure bosonic neutrinos compared to the case of pure fermionic neutrinos
(SM $2\nu\beta\beta$ decay). An arbitrary normalization is used for
illustrative purposes.

The calculations performed in [100] predicts the ratio $r_{0}$ for 100Mo and
76Ge to be 0.076 and 0.0014, respectively. The small ratio predicted for 76Ge
limits the sensitivity of double-$\beta$ decay experiments with 76Ge to
spectral distortions due to a partly bosonic neutrino.

Experimental searches for bosonic or partly bosonic neutrinos with
double-$\beta$ decay experiments could use not only the shape of the
distributions but also the ratios between the rates of the transitions to the
excited states and the ground states if the first were observed [100].

L$\overline{\nu}_{L}$L$\overline{\nu}_{L}$-- (a)

R$\overline{\nu}_{R}$L$\overline{\nu}_{L}$- (b)

R$\overline{\nu}_{R}$R$\overline{\nu}_{R}$ (c)

Figure 8: Feynman diagrams of the double-$\beta$ decay (a) with two left-
handed currents, _i.e._ the SM $2\nu\beta\beta$ decay, (b) with one right-
handed current, and (c) with two right-handed currents. Adapted from [102].

###  III.3 Non-standard interaction

####  III.3.1 Right-handed leptonic currrents

The SM $2\nu\beta\beta$ decay is a second-order transition involving weak
left-handed V-A currents with the strength given by the Fermi constant GF.
Some BSM theories, such as Left-Right symmetric models with unbroken lepton
number [103, 104], predict the existence of V+A lepton currents, which can
mediate double-$\beta$ decays [102]. The new physics effects can be modeled
through effective charged current operators containing V+A lepton currents.
The strength of these non-standard interactions is given by $\epsilon_{XR}$
GF, where the small dimensionless coupling $\epsilon_{XR}$ encapsulates the
new physics effects.

Right-handed current interactions are independent of the Majorana or Dirac
nature of neutrinos and do not necessarily violate the lepton number. If the
neutrino is a Majorana particle, the operators associated with $\epsilon_{LR}$
and $\epsilon_{RR}$ violate the total lepton number by two units and give rise
to extra contributions to the $0\nu\beta\beta$ decay [105]. In this case,
$0\nu\beta\beta$ decay searches set stringent limits of the order
$\epsilon_{LR}\lesssim 3\times 10^{-9}$, $\epsilon_{RR}\lesssim 6\times
10^{-7}$ [106]. On the other hand, if there exists a sterile neutrino state
$\nu_{R}$ that combines with $\nu_{L}$ to form a Dirac neutrino, the right-
handed current interactions do not necessarily violate lepton number [104].
The strong theoretical interest is therefore supported by the fact that their
observation, along with the non-observation of lepton number violation, would
indicate that neutrinos are Dirac fermions [104].

Figure 9: Summed electron energy distribution of the $2\nu\beta\beta$ decay in
the presence of right-handed lepton currents compared to the SM
$2\nu\beta\beta$ decay (left-handed lepton currents). An arbitrary
normalization is used for illustrative purposes. Adapted from [102].

Direct experimental constraints on these operators are set by neutrons and
different single-$\beta$ decays but are rather feeble
($\epsilon_{LR},\epsilon_{RR}\lesssim 6\times 10^{-2}$) [107]. Searches at the
LHC are also possible [108, 109, 110] but are generally model dependent and
require some caveat on the use of the effective operator analysis at high
energies.

In the presence of right-handed leptonic currents, the amplitude of the
$2\nu\beta\beta$ decay would be calculated as a coherent sum of the three
Feynman diagrams shown in figure 8: the SM second-order transition with two
left-handed interactions with the strength given by $G_{F}^{2}$ (figure 8a), a
transition involving one right-handed interaction with strength
$\epsilon_{XR}\,G_{F}^{2}$ (figure 8b), and a second-order transition with two
right-handed interactions with strength $\epsilon_{XR}^{2}\,G_{F}^{2}$ (figure
8c). Nevertheless, to the lowest order in the exotic coupling $\epsilon_{XR}$,
the decay rate can be expressed as an incoherent sum of only two terms:

$\Gamma^{2\nu}=\Gamma_{SM}+\epsilon_{XR}^{2}\,\Gamma_{\epsilon}\;,$ (22)

where the first term is the SM decay rate and the second term is the
contribution of right-handed current to the decay rate, suppressed by the
coupling $\epsilon_{XR}$. In fact, the interference of the SM term (diagram
8a) with the diagram 8b is helicity suppressed by the masses of the emitted
electron and neutrino as $m_{e}m_{\nu}/Q_{\beta\beta}^{2}$ because of the
right-handed nature of the exotic current. Higher orders in the exotic
coupling $\epsilon_{XR}$, coming from the last diagram 8c and its interference
with the SM term, are also negligible.

The phase-space factor and the NME differ in the SM decay rate and the BSM
contribution. Thus, the presence of right-handed currents in double-$\beta$
decay changes the total decay rate and the shape of the energy spectrum.
Nevertheless, given the uncertainties in the NME calculations, the change in
the total decay rate is not expected to be measurable. Instead, experiments
may be sensitive to the change in the spectral shape. Figure 9 shows the
$2\nu\beta\beta$ decay distribution in the SM compared to the distribution
arising from the presence of right-handed currents. The deviation includes a
spectrum shift to smaller energy and a flatter profile near $Q_{\beta\beta}$.

####  III.3.2 Neutrino self-interaction

The Hubble tension indicates the discrepancy between CMB and local measurement
of the Hubble constant. This tension has grown to about $4\sigma$, and if
confirmed, it would require new physics BSM or a new cosmological model [111,
112].

Introducing a $\nu$SI, _i.e._ a four-neutrino contact interaction, could
resolve the Hubble tension. Such a $\nu$SI interaction can be written as
$G_{S}(\nu\nu)(\nu\nu)$, and it would inhibit neutrino free-streaming in the
early Universe if its strength is much larger than the Fermi effective
interaction predicted by the SM, $G_{S}\sim 10^{9}$ GF [113, 114]. This new
strong interaction would indicate the presence of new physics at a scale
$1/\sqrt{G_{S}}\sim 10$ MeV – 1 GeV. In general, these strong $\nu$SI
interactions are difficult to probe in laboratory experiments due to the
absence of electrons or quarks involved. With some assumption on the origin of
the $\nu$SI operator, constraints can be obtained from different physics
observations [115, 116], while no model-independent constraint is currently
available. The study of $\nu$SI in single-$\beta$ decays has been considered
[117]. More recently, the search for $\nu$SI in double-$\beta$ decays has also
been proposed [118].

--$\nu$$\nu$-- Figure 10: Feynman diagram of the double-$\beta$ decay induced
by $\nu$SI. Adapted from [118].

In the presence of $\nu$SI, independently of the Dirac/Majorana nature of
neutrinos, the two neutrinos in double-$\beta$ decay can be emitted via the
corresponding effective operator, resulting in a $\nu$SI-induced
$2\nu\beta\beta$ ($2\nu$SI$\beta\beta$) decay. The Feynman diagram of this
process is shown in figure 10. The final state of the $2\nu$SI$\beta\beta$
decay is identical to that of the SM $2\nu\beta\beta$ decay. The contribution
from $\nu$SI to the decay rate can be written as:

$\Gamma_{\nu SI}=\frac{G_{S}^{2}m_{e}^{2}}{4R^{2}}\,\mathcal{G}_{\nu
SI}\,|\mathcal{M}_{0\nu}|^{2}\;,$ (23)

where $m_{e}$ denotes the electron mass and $R$ the radius of the nucleus. For
an exact contact interaction of four neutrinos and neglecting the final state
lepton momenta, the phase-space factor for the $2\nu$SI$\beta\beta$ decay is
related to the phase-space factor of the $2\nu\beta\beta$ decay as
$\mathcal{G}_{\nu SI}=\mathcal{G}_{2\nu}/(4\pi)^{2}$. The NME of
$2\nu$SI$\beta\beta$ is the same as of $0\nu\beta\beta$. In this scenario, no
difference is expected in the summed electron energy distribution of the
$2\nu$SI$\beta\beta$ decay compared to the SM $2\nu\beta\beta$ decay.
Therefore, only the experimental measurements of the $2\nu\beta\beta$ decay
rate can be used to constrain the contribution of $\nu$SI.

Figure 11: Summed electron energy distribution of $2\nu$SI$\beta\beta$ decay
where the $\nu$SI operator is generated by an s-channel mediator with a mass
of $M=Q_{\beta\beta}+0.1m_{e}$, compared to the SM $2\nu\beta\beta$ decay. An
arbitrary normalization is used for illustrative purposes. Adapted from [118].

This approach was used in [118] to determine upper limits on the coupling GS
from the measured $2\nu\beta\beta$ decay rates of several double-$\beta$ decay
isotopes. Limits in the range $G_{S}/G_{F}\lesssim(0.32-2.50)\times 10^{9}$
were obtained. The sensitivity on GS is limited by the uncertainty of the NME
ratio $|\mathcal{M}_{0\nu}|/|\mathcal{M}_{2\nu}|$. Cosmological data favoured
a strong interactive regime with $G_{S}=3.83\times 10^{9}G_{F}$. Even
including the theoretical NME uncertainties, all the considered isotopes can
fully exclude the strongly interacting cosmologically favored regime [118].
However, one should note that this bound applies only under the assumption
that two electron neutrinos are involved in the $\nu$SI. This might not be the
case if only muon neutrinos and tau neutrinos participate in $\nu$SI.

Possible distortions of the electron energy distribution could arise from the
$\nu$SI contribution if the $\nu$SI operator were generated by light
mediators. In this scenario, the energy dependence of the coupling $G_{S}$
could cause observable spectral distortions. In [118], the simplest case of an
s-channel scalar mediator with a mass just above the kinematic threshold
($M=Q_{\beta\beta}+0.1m_{e}$) was discussed. The coupling $G_{S}$ acquires the
following energy dependence

$G_{S}=\frac{-M^{2}}{s-M^{2}}\,G_{S}^{0}\;,$ (24)

where $M$ is the mediator mass and $s\equiv p^{2}$, with $p$ being the
momentum of the mediator (in the context of the $2\nu$SI$\beta\beta$, this is
of the order $s\lesssim Q_{\beta\beta}^{2}$). The value of $G_{S}$ at zero
momentum transferred ($G_{S}^{0}$) is denoted as $G_{S}^{0}=g^{2}/M^{2}$, with
$g$ the coupling between the mediator and the neutrino. Using $G_{S}$ in
equation 24, the differential decay rate of the $2\nu$SI$\beta\beta$ decay can
be calculated. The corresponding summed electron energy distribution is shown
in figure 11, for a mass of the mediator $M=Q_{\beta\beta}+0.1m_{e}$. The
energy spectrum of the $2\nu$SI$\beta\beta$ decay is shifted at lower energy
compared to the $2\nu\beta\beta$ decay spectrum. This shift can be understood
qualitatively: with the summed energy of the two electrons increasing, the
energy available for the neutrinos is smaller, leading to a smaller value of
$s$ and hence a smaller value of $G_{S}$.

Figure 12: The three classes of BSM double-$\beta$ decay searches.

##  IV Statistical analysis and sensitivity

All the double-$\beta$ decays introduced in section III result in the same
event topology. The two electrons emitted in the decay are detected within a
small detector region. The additional particles produced in the double-$\beta$
decay process, either the two anti-neutrinos or one or more exotic particles,
escape the detector carrying away part of the decay energy.

Double-$\beta$ decay experiments typically measure multiple observables for
each event. These include for instance, the energy deposited within the
detector, spatial and timing information, and variables related to the type of
particles involved in the event. Signal and background events feature specific
values of these observables that can be used to separate them. As previously
mentioned, this review focuses on the energy deposited within the detector –
the summed energy of the two electrons – which will be measured by all future
$0\nu\beta\beta$ decay experiments. However, we must recall that any
additional information, like the energy distribution of the single electrons
or their angular correlation, can strongly enhance the sensitivity to BSM
physics as it was shown in past searches [30, 119, 120].

Before discussing the statistical analysis to search for BSM decays, we shall
divide them into three classes of models, which require slightly different
statistical treatments.

##### Class I models.

The first class contains all those BSM decays that are independent of the SM
$2\nu\beta\beta$ decay. This condition is satisfied if the predicted phase
space and NME for the BSM decay are different from the SM ones and if no
correlation is expected between the BSM decay and the SM $2\nu\beta\beta$
decay. This is the case of the decays involving Majorons (III.1.1) or
$Z_{2}$-odd massive fermions (second model in III.1.2), but also the case of
$2\nu\beta\beta$ decay realized via non-standard interaction (III.3), in the
assumption that any interference term between the SM $2\nu\beta\beta$ decay
and the non-standard decay can be neglected. Despite the differences among the
models, the resulting BSM decay would manifest as an additional contribution
to the total energy spectrum, given by a continuous distribution in the same
energy range of the SM $2\nu\beta\beta$ decay. Being the BSM decay
subdominant, a small distortion of the observed energy spectrum compared to
the SM expectation would be observed. This is shown in the first left panel of
figure 12, for the J$\beta\beta$ decay (n=1) and a non-realistically large
coupling $g_{J}$, as an example. In the search for such a BSM decay, the
parameter of interest is the number of BSM decay events, underlying the
respective distribution. On the other hand, the dominant SM $2\nu\beta\beta$
decay acts as a background.

##### Class II models.

The second class contains those BSM decays which are not independent of the SM
$2\nu\beta\beta$ decay. This is the situation in which the BSM model not only
leads to an additional BSM decay but also affects the total decay rate of the
SM $2\nu\beta\beta$ decay. This is the case of the decays involving sterile
neutrinos or bosonic neutrinos. The correlation between the SM
$2\nu\beta\beta$ decay and the BSM one is given by the respective mixing, and
the existence of the BSM decay also implies a decrease of the SM
$2\nu\beta\beta$ decay by a factor of $cos^{4}\theta$ and $cos^{4}\chi$,
respectively. This is shown in the central panel of figure 12, for a sterile
neutrino with mass $m_{N}=500$ keV and a large mixing of $sin^{2}\theta=0.15$,
as an example. Again, a small distortion of the observed energy spectrum
compared to the SM-only expectation would be observed. In the search for such
an BSM decay, the parameter of interest is proportional to the ratio between
the number of events underlying the BSM decay distribution and the number of
events underlying the $2\nu\beta\beta$ decay distribution. Therefore, in this
case, the dominant SM $2\nu\beta\beta$ decay signal helps to constrain the BSM
physics.

##### Class III models.

Some BSM theories do not predict the existence of additional decay, as in
class I and class II models, but alter the prediction for the $2\nu\beta\beta$
decay. We classify these decays as class III. This is the case, for example,
of the violation of Lorentz symmetry. The Lorentz violating term in the
neutrino momentum affects the phase space of the two electrons emitted in the
$2\nu\beta\beta$ decay and, therefore, the predicted energy distribution. This
is shown in the right panel of figure 12, for non-realistically large values
of the Lorentz violating coefficient $\smash{\mathring{a}_{\text{of}}^{(3)}}$.
The search for Lorentz violation in $2\nu\beta\beta$ decays corresponds to a
search for deviations of the observed two-electron energy distribution
compared to the SM expectation. In practice, the total decay rate for the
Lorentz violating $2\nu\beta\beta$ decay can be approximated, to the first
order, as a sum of the SM $2\nu\beta\beta$ decay rate and a perturbation term,
whose size is regulated by the coefficient
$\smash{\mathring{a}_{\text{of}}^{(3)}}$, which is very small. While the phase
space differs for the two terms of the sum, the nuclear part, the NME, is the
same, introducing a correlation between the two terms. From the statistical
point of view, class III models can be treated as class II models. The
parameter of interest is proportional to the ratio between the number of
events underlying the distribution of the LV perturbation and the number of
events underlying the SM $2\nu\beta\beta$ decay distribution. Again, the
dominant SM $2\nu\beta\beta$ decay signal helps to constrain the BSM physics.

###  IV.1 Statistical analysis

In the search for spectral distortions due to the contribution of an BSM decay
in the energy spectrum, the energy region of interest extends from the
detector threshold to $Q_{\beta\beta}$. In this window, most of the observed
events are attributed to the $2\nu\beta\beta$ decay $N_{2\nu}$ with additional
contributions due to different background processes $N_{others}$. The sum of
the $2\nu\beta\beta$ decay and other background constitutes the so-called
background model, which is known with a certain accuracy by the experiments.

To search for an BSM decay, a spectral fit of the energy spectrum is
performed,222Some experiments utilize a multi-variate approach, fitting
multiple observable at the same time to better separate signal and background.
Here we focus on the simplest approach of one-dimensional fit, but the results
can be easily generalized. adding the BSM decay to the background model.
Typically, the information from the background model is used to construct a
likelihood function, which is then used in a frequentist or Bayesian approach
to constrain the parameter of interest. The definition of the parameter of
interest depends on the model to be constrained, as it was pointed out in the
previous section.

The most important experimental parameters determining the sensitivity of the
experiment are the exposure $\mathcal{E}$, the background rate $R_{bkg}$, and
the systematic uncertainties due to the energy reconstruction $\sigma_{sys}$.
The exposure is given by the product of the number of observed nuclei and the
observation time. The background rate is primarily given by the
$2\nu\beta\beta$ decay rate with a subdominant contribution due to other
sources $R_{bkg}=R_{2\nu}+R_{others}$. The systematic uncertainties related to
the energy reconstruction can largely differ between experiments and are
specific to the detector technology.

A precise evaluation of the sensitivity of an experiment requires considering
experiment-specific information. However, it can be approximated by
considering a counting analysis and Poisson statistics in the region of
interest defined above and with a known background expectation given by
$R_{bkg}$. In this derivation, we neglect the systematic uncertainties. Their
impact will be discussed later in this section. The dependence of the
sensitivity on exposure $\mathcal{E}$ and background rate $R_{bkg}$ is
different from class I models and class II and II models. In the following, we
will discuss the two cases separately.

###  IV.2 Experimental sensitivity for class I models

Let’s first consider a BSM decay which we classified as a class I model. This
is, for example, the case of the existence of an exotic particle, _e.g._ the
Majoron or the $Z_{2}$-odd exotic fermion, with a coupling to neutrino given
by $g_{X}$.333The result can be extended to the existence of non-standard
interaction, which leads to BSM double-$\beta$ decays, as introduced in
section III.3. This is the parameter of interest we want to determine the
sensitivity.

The precision with which a subdominant contribution – the number of BSM
decays, $N_{X}$ – can be constrained is proportional to the fluctuations of
the background in the analysis window, $N_{bkg}$:

$N_{X}\geq\sqrt{N_{bkg}}\;.$ (25)

The number of background events in the analysis window can be expressed as a
function of the background rate and the exposure:

$N_{bkg}=R_{bkg}\cdot\mathcal{E}\;,$ (26)

and analogously the number of BSM decays:

$N_{X}=g_{X}^{2}\,\mathcal{G}\,\mathcal{M}^{2}\cdot\mathcal{E}\;,$ (27)

where we have expressed the BSM decay rate as a function of the parameter of
interest $g_{X}$, through the phase-space factor $\mathcal{G}$ and the NME
$\mathcal{M}$ of the decay. Using equations 26 and 27 into equation 25, we
obtain:

$g_{X}^{2}\geq\sqrt{\frac{R_{2\nu}+R_{others}}{\mathcal{E}}}\cdot\frac{1}{\mathcal{G}\,\mathcal{M}^{2}}\;.$
(28)

In the last step, we have explicated the contribution of the $2\nu\beta\beta$
decay to the total background rate ($R_{bkg}=R_{2\nu}+R_{others}$).

The sensitivity scales with the square root of the exposure, but it is limited
by the background, to which the $2\nu\beta\beta$ decay contributes. In
addition, equation 28 shows that uncertainties in the phase space and NMEs can
limit the sensitivity.

###  IV.3 Experimental sensitivity for class II and III models

A slightly different result is obtained for class II and III models. Let’s
consider, for instance, the double-$\beta$ decay into sterile neutrinos.444The
result can be extended to the double-$\beta$ decay into bosonic neutrinos and
the Lorentz violating $2\nu\beta\beta$ decay. In this case, the parameter of
interest is the mixing angle $\sin^{2}\theta$. Given that $\sin^{2}\theta$
also modifies the $2\nu\beta\beta$ decay rate, it is proportional to the ratio
between the number of decays into sterile neutrino $N_{\nu N}$ and the number
of $2\nu\beta\beta$ decay events

$\sin^{2}\theta\propto\frac{\mathcal{G}_{2\nu}}{\mathcal{G}_{\nu
N}}\cdot\frac{N_{\nu N}}{N_{2\nu}}\;,$ (29)

through the respective phase-space factors.

The statistical uncertainty on this quantity can be computed through standard
error propagation

$\displaystyle\frac{\sigma_{\sin^{2}\theta}}{\sin^{2}\theta}\propto\biggl{\\{}$
$\displaystyle(\frac{\sigma_{N_{\nu N}}}{N_{\nu
N}})^{2}+(\frac{\sigma_{N_{2\nu}}}{N_{2\nu}})^{2}+$ (30)
$\displaystyle+2\cdot(\frac{\sigma_{N_{\nu N}}\sigma_{N_{2\nu}}}{N_{\nu
N}N_{2\nu}})\rho_{N_{\nu N},N_{2\nu}}\biggr{\\}}^{1/2}\;.$

Because of the same arguments previously used to define the uncertainty on
$N_{X}$, the uncertainty on $N_{\nu N}$ will be

$\sigma_{N_{\nu N}}\propto\sqrt{(R_{2\nu}+R_{others})\cdot\mathcal{E}}\;.$
(31)

In addition, we can write $N_{\nu N}$ in terms of $N_{2\nu}$ and
$\sin^{2}\theta$. Also, the correlation coefficient $\rho_{N_{\nu
N},N_{2\nu}}$ is proportional to the mixing angle, because of the relation 29.
Putting everything together, we can rewrite equation 30 as:

$\displaystyle\frac{\sigma_{\sin^{2}\theta}}{\sin^{2}\theta}\propto\biggl{\\{}$
$\displaystyle\frac{(R_{2\nu}+R_{others})\cdot\mathcal{E}}{R_{2\nu}^{2}\cdot\sin^{4}\theta\cdot\mathcal{E}^{2}}+\dfrac{1}{R_{2\nu}\cdot\mathcal{E}}+$
(32)
$\displaystyle+2\cdot\frac{\sqrt{R_{2\nu}(R_{2\nu}+R_{others})}\cdot\mathcal{E}}{R_{2\nu}^{2}\cdot\sin^{2}\theta\cdot\mathcal{E}^{2}}\biggr{\\}}^{1/2}\;,$

$\displaystyle\sigma_{\sin^{2}\theta}\propto\biggl{\\{}$
$\displaystyle\frac{(R_{2\nu}+R_{others})}{R_{2\nu}^{2}\cdot\mathcal{E}}+\frac{\sin^{4}\theta}{R_{2\nu}\cdot\mathcal{E}}+$
(33)
$\displaystyle+\frac{2\cdot\sin^{2}\theta\sqrt{R_{2\nu}(R_{2\nu}+R_{others})}}{R_{2\nu}^{2}\cdot\mathcal{E}}\biggr{\\}}^{1/2}\;,$

$\sigma_{\sin^{2}\theta}\xrightarrow{\sin^{2}\theta\sim
0}\mathcal{G}_{2\nu}/\mathcal{G}_{\nu
N}\cdot\sqrt{\dfrac{(R_{2\nu}+R_{others})}{R_{2\nu}^{2}\cdot\mathcal{E}}}\;,$
(34)

where we reintroduced the dependence on the phase-space factors in the last
passage.

The sensitivity scales with the square root of the exposure, it is limited by
the background, but in this case, a high $2\nu\beta\beta$ decay rate is
advantageous due to the dependence
$\sigma_{\sin^{2}\theta}\propto\sqrt{1/R_{2\nu}}$. The sensitivity also
depends on the ratio between the phase space factors
$\mathcal{G}_{2\nu}/\mathcal{G}_{\nu N}$.

###  IV.4 Impact of the systematic uncertainties on the sensitivity

Figure 13: Sensitivity of a double-$\beta$ decay experiment with 76Ge to
sterile neutrinos with a mass of 500 keV, as a function of the exposure,
background level, and systematic uncertainties [78]. The results of a full
frequentist analysis (markers) are compared to the expectation (lines) from
equations 35 and 34.

Given an experiment with exposure $\mathcal{E}$ and using an isotope with
half-life $T_{1/2}^{2\nu}$, we expect the sensitivity to double-$\beta$ decay
into Majorons or $Z_{2}$-odd fermions (or decays resulting from non-standard
interaction) to scale with $\sqrt{1/(T_{1/2}^{2\nu}\cdot\mathcal{E})}$ while
the sensitivity to the sterile neutrino mixing angle (or Lorentz violation and
bosonic neutrinos) with $\sqrt{T_{1/2}^{2\nu}/\mathcal{E}}$. This means that
experiments using an isotope with a long $2\nu\beta\beta$ decay half-life will
be favored in the first kind of search as they will have a lower background
rate. However, they will be disfavoured in the second kind of search where the
sensitivity is linearly proportional to the $2\nu\beta\beta$ decay half-life.

In our derivation so far, we have neglected the impact of systematic
uncertainties on the sensitivity. In a more general way, the sensitivity can
be approximated as

$f(\mathcal{E},R_{bkg},\sigma_{sys})=\sqrt{\sigma_{stat}^{2}(\mathcal{E},R_{bkg})+\sigma_{sys}^{2}}\;.$
(35)

It is determined by both the statistical uncertainty $\sigma_{stat}$, which
was derived in equations 28 and 34 as a function of the most important
experimental parameters, and the systematic uncertainty $\sigma_{sys}$.

As long as the statistical uncertainty is dominant, the sensitivity improves
by increasing the exposure approximately as $f\propto\sqrt{1/\mathcal{E}}$.
The sensitivity saturates when the statistical uncertainty becomes comparable
with the systematic one. This is illustrated in figure 13. The figure shows
the sensitivity of a double-$\beta$ decay experiment using 76Ge to sterile
neutrinos with a mass of 500 keV, as a function of the exposure, the
background level, and the systematic uncertainties [78]. The sensitivity
computed using a full frequentist analysis is shown by the markers. This is
compared to the expectation given by equation 35, where the statistical
uncertainty is given by equation 34.

##  V Double-$\beta$ decay experiments and constrains

The 80-years-long history of double-$\beta$ decay experiments has seen a
variety of technologies and concepts being tested and developed over the years
with the standing goal of reducing backgrounds and increasing the isotope
mass. A pivotal time for the field was around the turn of the century when the
unexpected discovery of neutrino masses [121] raised the question of whether
that mass could be due to the peculiar mechanism conceived by Majorana [122,
123, 124], a hypothesis that can be proven by observing $0\nu\beta\beta$ decay
[125]. This boosted the interest for double-$\beta$ decay experiments, setting
in motion a process eventually culminating in the consolidation of five main
detection technologies: high-purity germanium semiconductor detectors,
cryogenic calorimeters, time projection chambers, large liquid scintillators,
and tracking calorimeters. The following sections review the most recent
experiments related to these technologies and the community’s plan for the
next-generation projects. We conclude with a summary of the state-of-art
constraints on the search for BSM double-$\beta$ decay in Sec. V.7 and
prospects in Sec. VI.

###  V.1 HPGe semiconductor detectors

High-Purity Ge (HPGe) detectors have been a leading technology for
double-$\beta$ experiments since the very first $0\nu\beta\beta$ decay
searches [126]. HPGe detectors are semiconductor devices in which electron-
hole charge carriers produced by ionization processes are collected by an
electric field applied throughout an ultra-pure Ge crystal isotopically
enriched in 76Ge up to 92%. Crystals are grown through the Czochralski method
[127] and thus are intrinsically ultra radio-pure. The typical detector size
is 1–3 kg, requiring the simultaneous operation of multiple detectors to reach
a large target mass. HPGe detectors have superior energy resolution, the best
of any double-$\beta$ decay experiment, while also providing information on
the event topology. Double-$\beta$ decays are fully contained within the
active detector region, and no volume-fiducialization is required to eliminate
background leading to very high detection efficiency.

The most sensitive double-$\beta$ decay searches based on HPGe detectors have
been conducted by the Germanium Detector Array (Gerda) experiment [128] and
the Majorana Demonstrator [129]. Gerda was located at the Laboratori Nazionali
del Gran Sasso (LNGS) in central Italy and operated about 40 kg of HPGe
detectors directly immersed in a LAr volume instrumented to detect its
scintillation light. The Majorana Demonstrator was located in the Sanford
Underground Research Facility (SURF) in South Dakota and operated about 30 kg
of HPGe detectors in two vacuum cryostats. The results of the Gerda and
Majorana Demonstrator experiments have demonstrated the feasibility of
building a ton-scale 76Ge-based $0\nu\beta\beta$ decay experiment with ultra-
low background and superior energy resolution. With the Gerda and Majorana
Demonstrator experiments now completed, the next generation experiment will be
realized in the framework of the LEGEND project, following two stages named
LEGEND-200 and LEGEND-1000 [130]. LEGEND-200 has just started approaching
physics data taking with 200 kg of HPGe detectors in the upgraded Gerda
infrastructure. LEGEND-1000 is currently under preparation and is expected to
come online towards the end of the decade.

The Gerda experiment has performed several searches for BSM double-$\beta$
decays of 76Ge. Its most sensitive search for Majorons-mediated decays led to
half-life constraints of $6.4\cdot 10^{23}$ yr, $2.9\cdot 10^{23}$ yr,
$1.2\cdot 10^{23}$ yr, and $1.0\cdot 10^{23}$ yr (at 90% C.L.), respectively
for the decays with spectral index $n=1,2,3,$ and 7 [131]. In the same work,
limits on Lorentz violation and the decay into light exotic fermions have also
been derived. The Lorentz violating isotropic coefficient
$\smash{\mathring{a}_{\text{of}}^{(3)}}$ has been constrained to
$(-2.7<\smash{\mathring{a}_{\text{of}}^{(3)}}<6.2)\cdot 10^{-6}$ GeV at 90%
C.L.. In the search for double-$\beta$ decays into sterile neutrinos, the most
stringent limit was obtained for masses between 500 and 600 keV. For these
masses, the 90% C.L. interval obtained on the mixing between sterile and
active neutrinos is $\sin^{2}\theta<0.013$. In the search for double-$\beta$
decays into $Z_{2}$-odd fermions, limits on the decay half-life have been
derived in the range $(0.18-2.5)\cdot 10^{23}$ yr, at 90% C.L.. The best limit
of $2.5\cdot 10^{23}$ yr was obtained for a mass of 400 keV and corresponds to
a limit on the coupling constant $g_{\chi}<(0.6-1.4)\cdot 10^{-3}$ MeV-2,
where the range is due to the NME uncertainties.

###  V.2 Cryogenic calorimeters

Cryogenic calorimeters, also referred to as bolometers, have been employed for
$0\nu\beta\beta$ decay and dark-matter searches since the 80s [132].
Bolometers are crystals coupled with a superconductive thermal sensors. Energy
deposition within the crystal produce phonons, which warm up the detectors,
and additionally scintillation light for specific materials. To reach the
desire sensitive, the thermal sensors typically rely on the resistivity change
of superconductive materials at the transition edge, and thus need to be
operated at mK temperatures.

The cryogenic calorimeters are extremely versatile tools, as the crystal
material can be chosen in order to study a variety of double-$\beta$ decay
isotopes. Further advantages of this detection technique are an excellent
energy resolution and very high detection efficiency. The limited crystals
dimensions (typically between 0.2 and 0.8 kg) require the operation of
thousands of cryogenic calorimeters to reach target masses of hundreds of
kilograms or larger. This poses challenging requirements to the cryogenic
infrastructure that needs to keep a ton of material stably at mK temperatures
for years.

Currently, the largest bolometric experiment is the Cryogenic Underground
Observatory for Rare Events (CUORE) at LNGS, which operates about 750 kg of
TeO2 crystals with natural isotopic composition (corresponding to 206 kg of
130Te) in a large cryogen-free cryostat [133]. The CUORE experiment
successfully demonstrated the feasibility of a ton-scale bolometric experiment
[134], leading to CUPID, the next generation bolometric experiment CUORE
Upgrade with Particle Identification enabled by the usage of scintillating
bolometers. As part of the R&D towards CUPID, two independent experiments have
produced constraints on BSM physics. The first one is CUPID-0, at LNGS, which
utilized ZnSe crystals enriched in 82Se [135]. The second one is CUPID-Mo at
LSM in France, which utilized Li2MoO4 crystals enriched in 100Mo [136].

The CUPID-0 experiment has searched for Lorentz violating double-$\beta$ decay
of 82Se [137] and for double-$\beta$ decays with the emission of Majorons
[138]. In the first work, the Lorentz violating coefficient
$\smash{\mathring{a}_{\text{of}}^{(3)}}$ has been constrained to $<4.1\cdot
10^{-6}\,$GeV at 90% C.I.. In the second work, limits on the half-life of the
decays involving Majorons have been derived. These are $1.2\cdot 10^{23}$ yr,
$3.8\cdot 10^{22}$ yr, $1.4\cdot 10^{22}$ yr, and $2.2\cdot 10^{21}$ yr (at
90% C.I.), respectively for the decays with spectral index $n=1,2,3,$ and 7.

###  V.3 Time projection chambers

The first direct observation of $2\nu\beta\beta$ decay was made using a Time
Projection Chamber (TPC) [5]. Since then, this technology has been at the
forefront of $0\nu\beta\beta$ decay searches because of the combination of
mass scalability and optimal background discrimination capabilities enabled by
the 3D reconstruction of the event topology, position, and energy. TPCs are
particularly well-suited to search for $0\nu\beta\beta$ decay of 136Xe: Xe is
a noble element that can be used directly in TPCs as a liquid or gas. On the
other hand, TPC detectors have a limited energy resolution and require a
multi-variate analysis to constrain background features close to the
$Q_{\beta\beta}$, which are often not resolved (_e.g._ 214Bi $\gamma$ line at
2447.7 keV, just below the 136Xe $Q_{\beta\beta}$).

The most sensitive liquid Xe TPC among the current generation of
double-$\beta$ decay experiments was EXO-200 at WIPP near Carlsbad New Mexico.
EXO-200 was a single-phase liquid-Xe TPC, filled with 161 kg of 136Xe [139].
EXO-200 demonstrated the capabilities of a monolithic liquid-Xe TPC, which
includes relatively good energy resolution, near maximal signal detection
efficiency, and solid topological discrimination of backgrounds [140].
Building on it, the next generation ton-scale 136Xe-based $0\nu\beta\beta$
decay experiment nEXO is currently being proposed [141].

High-pressure gaseous Xe TPCs for $2\nu\beta\beta$-decay have been developed
in the context of the NEXT project, which combines electroluminescence to
reach good energy resolution ($<1$% FWHM at $Q_{\beta\beta}$) and charged-
particle tracking for the active suppression of background [142]. A concept
for a ton-scale phase of NEXT is currently under investigation with the goal
of operating a full ton of 136Xe in the form of enriched Xe gas [143], and
possibly deploying technologies for the identification of the 136Xe decay
daughter isotope.

The EXO-200 experiment has searched for BSM double-$\beta$ decays of 136Xe. It
sets upper limits on Majorons emitting decays at the level of $4.3\cdot
10^{24}$ yr, $1.5\cdot 10^{24}$ yr, $6.3\cdot 10^{23}$ yr, and $5.1\cdot
10^{22}$ yr (at 90% C.L.), respectively for the decays with spectral index
$n=1,2,3,$ and 7 [144]. In the same work, limits on the emission of a Majoron-
like particle via non-standard RH interaction have also been derived. The
half-life of these decays has been constrained at 90% C.L. to $3.7\cdot
10^{24}$ yr and $4.1\cdot 10^{24}$ yr, respectively, for right-handed and
left-handed quark currents. The EXO-200 experiment first searched for Lorentz
violation in $2\nu\beta\beta$ decay [145]. The Lorentz violating coefficient
$\smash{\mathring{a}_{\text{of}}^{(3)}}$ was constrained to $-2.65\cdot
10^{-5}\,\text{GeV}<\smash{\mathring{a}_{\text{of}}^{(3)}}<7.60\cdot
10^{-6}\,\text{GeV}$ at 90% C.L..

###  V.4 Large liquid scintillators

Large liquid scintillator detectors have historically been the most mass-
scalable technology used by $0\nu\beta\beta$ experiments. The scintillator can
be doped with different isotopes of interest for these searches, including
136Xe and 130Te. Decays occurring within the detector create scintillation
photons, which travel straight to the outer surface of the scintillator volume
where they are read out by photo-multipliers tubes. The event position and
energy release are reconstructed using the time-of-flight of the photons and
their number. Multivariate analysis can also provide some information on the
initial spatial extension of the energy deposition, giving an extra handle to
tag background-like events.

The most important liquid scintillator detector in the field is operated by
the KamLAND-Zen experiment in the Kamioka Mine in Japan [146]. It consists of
a nylon balloon placed at the center of the detector volume and filled with a
liquid scintillator in which 136Xe has been dissolved. After a successful
first phase (KamLAND-Zen 400) with up to 340 kg of 136Xe, the second phase
(KamLAND-Zen 800) is currently running with about 680 kg of 136Xe [147]. With
the first 1.6 years of data, KamLAND-Zen 800 produced a world-leading
$0\nu\beta\beta$ decay half-life limit [147]. The KamLAND-Zen collaboration is
already preparing for the ton-scale phase (KamLAND2-Zen) in which about 1 ton
of 136Xe will be deployed [148].

The technology of large liquid scintillators is largely employed in neutrino
experiments. Among future large liquid scintillator experiments, we shall
mention the SNO experiment. This is a multi-purpose neutrino experiment
located at SNOLAB in Canada [149]. An upgrade of the experiment is planned
after the completion of the main goals, which consists in loading the 780 tons
of organic liquid scintillator with a double-beta decaying isotope to search
for $0\nu\beta\beta$ decay [149].

The KamLAND-Zen experiment searched for double-$\beta$ decays with the
emission of Majorons [150]. Limits on the half-life of these decays have been
derived at 90% C.L.: $2.6\cdot 10^{24}$ yr, $1.0\cdot 10^{24}$ yr, $4.5\cdot
10^{23}$ yr, and $1.1\cdot 10^{22}$ yr, respectively for the decays with
spectral index $n=1,2,3,$ and 7.

###  V.5 Tracking calorimeters

Tracking calorimeters are the only technology capable of measuring with high
accuracy the kinematic of electrons emitted in double-$\beta$ decays, such as
the single-electron energy and the electron angular distribution. Measuring
these quantities would give precious inputs to pin down the actual channel
mediating the $0\nu\beta\beta$ decay [151]. It would also strongly enhance the
sensitivity to search for the BSM double-$\beta$ decays discussed in this
review.

The tracking capability is obtained by decoupling the double-$\beta$ decay
isotope from the detector. The target isotope is placed on a thin foil,
immersed in a magnetic field, and surrounded by tracking and calorimetric
layers. This configuration enables the measurement of the electron momentum
through its bending in the magnetic field, and the measure of its energy when
it enters the calorimeters. Unfortunately, it reduces the detection
efficiency. The requirement of using very thin foils to minimize energy losses
makes it extremely challenging to scale up the isotope mass.

The NEMO-3 experiment utilized this technology to search for $0\nu\beta\beta$
decay of several isotopes at the LSM in France. Masses from a few grams to a
few kilograms of the isotopes of interest were deployed in separate sectors of
the detector (6.99 g of 48Ca [38], 0.932 kg of 82Se [152], 9.4 g of 96Zr
[153], 6.914 kg of 100Mo [154], 410 g of 116Cd [119], and 36.6 g of 150Nd
[30]). A next-generation tracking calorimeter detector is the SuperNEMO
Demonstrator, which is based on the technology demonstrated by NEMO-3 [155].
In its first phase, the SuperNEMO Demonstrator will deploy one module with 7
kg of 82Se. A future full-scale experiment is foreseen, consisting of multiple
modules aiming for a total 82Se mass of 100 kg.

The NEMO-3 experiment searched for Majoron-involving double-$\beta$ decays in
several isotopes: 100Mo [154, 120], 82Se [152], 116Cd [119], 48Ca [38] and
150Nd [30]. Limits on the half-life of the decay corresponding to spectral
index $n=1$ have been derived (at 90% C.L.) for all the used isotopes:
$4.6\cdot 10^{21}$ yr with 48Ca, $3.7\cdot 10^{22}$ yr with 82Se, $4.4\cdot
10^{22}$ yr with 100Mo, $8.5\cdot 10^{21}$ yr with 116Cd, and $0.3\cdot
10^{22}$ yr with 150Nd. Limits on the half-life of the decays corresponding to
spectral indexes $n=2,3,$ and 7 have been derived only with 100Mo, because of
the lower statistics data sets and the higher background achieved with the
other isotopes. The corresponding limits at 90% C.L. are: $9.9\cdot 10^{21}$
yr, $4.4\cdot 10^{21}$ yr and $1.2\cdot 10^{21}$ yr, respectively for the
$n=2,3,$ and 7. The NEMO-3 experiment also searched for Lorentz violation in
the $2\nu\beta\beta$ decay of 100Mo and the 100Mo $2\nu\beta\beta$ decay with
bosonic neutrinos [120]. The Lorentz violating isotropic coefficient
$\smash{\mathring{a}_{\text{of}}^{(3)}}$ has been constrained to
$(-4.2<\smash{\mathring{a}_{\text{of}}^{(3)}}<3.5)\cdot 10^{-7}$ GeV at 90%
C.L.. The half-life of the 100Mo $2\nu\beta\beta$ decay with bosonic neutrinos
was constrained to $1.2\cdot 10^{21}$ yr, which corresponds to an upper limit
on the bosonic neutrino contribution of sin$\chi<0.27$ at 90% C.L..

###  V.6 Other technologies

We should mention the Aurora experiment at LNGS among the experiments using
technologies other than those described in the previous subsections. It
utilized more than 1 kg of radio-pure cadmium tungstate (116CdWO4)
scintillating crystals enriched in the 116Cd isotope [156]. Even if this
technology is not competitive in terms of $0\nu\beta\beta$ decay sensitivities
and there is no concrete plan to scale it to a ton-scale $0\nu\beta\beta$
decay experiments, the Aurora experiment has produced competitive constraints
in the search for BSM double-$\beta$ decays of 116Cd [35]. They set limits on
the half-life of Majorons emitting decays with the emission of Majorons at the
order of $10^{21}$ yr, while the most stringent limit was obtained for the
$n=1$ spectral index: $T_{1/2}>8.2\cdot 10^{21}$ yr (at 90% C.L.). In the same
work, they searched for Lorentz violation and obtained a limit on the
isotropic coefficient of $\smash{\mathring{a}_{\text{of}}^{(3)}}<4.0\cdot
10^{-6}$ GeV (at 90% C.L.).

###  V.7 Most sensitive constraints

In this section we summarise the most sensitive constraints reported by all
experiments mentioned in the previous sections, grouping them based on the new
physics searched.

##### Double-$\beta$ decay with the emission of Majorons

Decay / Isotope |  $T_{1/2}$ (yr) | Experiment | Ref. | G ($10^{-}{18}$ yr-1) | NME | $g_{J}$
---|---|---|---|---|---|---
J$\beta\beta$ ($n=1$) |  |  |  |  |  |
48Ca | $>4.6\times 10^{21}$ | NEMO-3 | [38] | 1540 | $(0.40-2.71)$ | $<(8.5-58)\times 10^{-5}$
76Ge | $>6.4\times 10^{23}$ | GERDA | [131] | 44.2 | $(2.66-6.64)$ | $<(1.8-4.4)\times 10^{-5}$
82Se | $>3.7\times 10^{22}$ | NEMO-3 | [152] | 361 | $(2.72-5.30)$ | $<(3.2-6.2)\times 10^{-5}$
82Se | $>1.2\times 10^{23}$ | CUPID-0 | [138] | 361 | $(2.72-5.30)$ | $<(1.8-3.5)\times 10^{-5}$
100Mo | $>4.4\times 10^{22}$ | NEMO-3 | [154] | 598 | $(3.84-6.59)$ | $<(1.8-3.1)\times 10^{-5}$
116Cd | $>8.2\times 10^{21}$ | Aurora | [35] | 569 | $(3.105-5.43)$ | $<(5.3-9.2)\times 10^{-5}$
116Cd | $>8.5\times 10^{21}$ | NEMO-3 | [119] | 569 | $(3.105-5.43)$ | $<(5.2-9.0)\times 10^{-5}$
136Xe | $>2.6\times 10^{24}$ | KamLAND-Zen | [157] | 409 | $(1.11-4.77)$ | $<(0.4-1.7)\times 10^{-5}$
136Xe | $>4.3\times 10^{24}$ | EXO-200 | [144] | 409 | $(1.11-4.77)$ | $<(0.3-1.3)\times 10^{-5}$
150Nd | $>0.3\times 10^{22}$ | NEMO-3 | [119] | 3100 | $(1.707-5.46)$ | $<(3.7-12)\times 10^{-5}$
J$\beta\beta$ ($n=2$) |  |  |  |  |  |
76Ge | $>2.9\times 10^{23}$ | GERDA | [131] | – | – | –
82Se | $>3.8\times 10^{22}$ | CUPID-0 | [138] | – | – | –
100Mo | $>9.9\times 10^{21}$ | NEMO-3 | [120] | – | – | –
116Cd | $>4.1\times 10^{21}$ | Aurora | [35] | – | – | –
136Xe | $>1.0\times 10^{24}$ | KamLAND-Zen | [157] | – | – | –
136Xe | $>9.8\times 10^{23}$ | EXO-200 | [144] | – | – | –
J$\beta\beta$ ($n=3$) |  |  |  |  |  |
76Ge | $>1.2\times 10^{23}$ | GERDA | [131] | 0.073 | 0.381 | $<1.7\times 10^{-2}$
82Se | $>1.4\times 10^{22}$ | CUPID-0 | [138] | 1.22 | 0.305 | $<1.5\times 10^{-2}$
100Mo | $>4.4\times 10^{21}$ | NEMO-3 | [120] | 2.42 | 0.263 | $<2.3\times 10^{-2}$
116Cd | $>2.6\times 10^{21}$ | Aurora | [35] | 2.28 | 0.144 | $<5.6\times 10^{-2}$
136Xe | $>4.5\times 10^{23}$ | KamLAND-Zen | [157] | 1.47 | 0.160 | $<0.47\times 10^{-2}$
136Xe | $>6.3\times 10^{23}$ | EXO-200 | [144] | 1.47 | 0.160 | $<0.40\times 10^{-2}$
JJ$\beta\beta$ ($n=3$) |  |  |  |  |  |
76Ge | $>1.2\times 10^{23}$ | GERDA | [131] | 0.22 | 0.0026 | $<1.21$
82Se | $>1.4\times 10^{22}$ | CUPID-0 | [138] | 3.54 | 0.0020 | $<1.18$
100Mo | $>4.4\times 10^{21}$ | NEMO-3 | [120] | 6.15 | 0.0019 | $<1.41$
116Cd | $>2.6\times 10^{21}$ | Aurora | [35] | 5.23 | 0.000945 | $<2.37$
136Xe | $>4.5\times 10^{23}$ | KamLAND-Zen | [157] | 3.05 | 0.0011 | $<0.69$
136Xe | $>6.3\times 10^{23}$ | EXO-200 | [144] | 3.05 | 0.0011 | $<0.64$
JJ$\beta\beta$ ($n=7$) |  |  |  |  |  |
76Ge | $>1.0\times 10^{23}$ | GERDA | [131] | 0.42 | 0.0026 | $<1.08$
82Se | $>2.2\times 10^{21}$ | CUPID-0 | [138] | 26.9 | 0.0020 | $<1.13$
100Mo | $>1.2\times 10^{21}$ | NEMO-3 | [120] | 50.8 | 0.0019 | $<1.41$
116Cd | $>8.9\times 10^{20}$ | Aurora | [35] | 33.9 | 0.000945 | $<2.37$
136Xe | $>1.1\times 10^{22}$ | KamLAND-Zen | [157] | 12.5 | 0.0011 | $<1.23$
136Xe | $>5.1\times 10^{22}$ | EXO-200 | [144] | 12.5 | 0.0011 | $<0.84$
$\phi\beta\beta$ ($\epsilon_{RR}$) |  |  |  |  |  |
136Xe | $>3.7\times 10^{24}$ | EXO-200 | [144] | – | – | –
$\phi\beta\beta$ ($\epsilon_{RL}$) |  |  |  |  |  |
136Xe | $>4.1\times 10^{24}$ | EXO-200 | [144] | – | – | –
Table 3: Comparison of the results obtained by different double-$\beta$ decay
experiments with different isotopes in the search for Majorons-involving
decays. The lower limits on the half-life are converted into upper limits on
the neutrino-Majoron coupling constant using equation 7 with the axial vector
coupling constant $g_{A}=1.27$ and the phase space factors from [64]. The NME
calculations for the spectral index $n=1$ are taken from [7] and references
therein, and for $n=3$ and $n=7$ from [65].

A summary of the latest results obtained by different double-$\beta$ decay
experiments is presented in table 3. The most stringent limits, regardless of
the isotope and the experiment, are obtained for the model corresponding to a
spectral index $n=1$. In fact, the energy distribution predicted for this
decay differs the most from the SM $2\nu\beta\beta$ decay compared to other
spectral indexes, as shown in figure 3. Among different experiments, the best
limit on the half-life of the J$\beta\beta$ decay ($n=1$ mode) is obtained by
the EXO-200 experiment with 136Xe: $4.3\cdot 10^{24}$ yr at 90% C.L.. EXO-200
also obtained the best limits on the half-life of the
J$\beta\beta$/JJ$\beta\beta$ decays ($n=3$ modes): $6.3\cdot 10^{23}$ yr. For
the J$\beta\beta$ decay ($n=2$ mode), the KamLAND-Zen experiment set the most
competitive limit on the half-life at $1.0\cdot 10^{24}$ yr, while the GERDA
experiment set the most competitive limit on the half-life of the
JJ$\beta\beta$ decay ($n=7$ mode): $1.0\cdot 10^{23}$ yr. Recently, EXO-200
has searched for BSM double-$\beta$ decays in which a Majoron-like particle is
emitted via non-standard right-handed currents were also investigated [144].
Limits were derived on such a decay for both right-handed and left-handed
hadronic currents: $3.7\cdot 10^{24}$ yr and $4.1\cdot 10^{24}$ yr,
respectively.

##### Lorentz-violating $2\nu\beta\beta$ decay

Isotope |  $\smash{\mathring{a}_{\text{of}}^{(3)}}$ (GeV) | Experiment | Ref.
---|---|---|---
76Ge | ($-2.7<\smash{\mathring{a}_{\text{of}}^{(3)}}<6.2$) $\cdot 10^{-6}$ | GERDA | [131]
82Se | $\smash{\mathring{a}_{\text{of}}^{(3)}}<4.1\cdot 10^{-6}$ | CUPID-0 | [137]
100Mo | ($-4.2<\smash{\mathring{a}_{\text{of}}^{(3)}}<3.5$) $\cdot 10^{-7}$ | NEMO-3 | [120]
116Cd | $\smash{\mathring{a}_{\text{of}}^{(3)}}<4.0\cdot 10^{-6}$ | Aurora | [35]
136Xe | $-2.65\cdot 10^{-5}<\smash{\mathring{a}_{\text{of}}^{(3)}}<7.6\cdot 10^{-6}$ | EXO-200 | [145]
Table 4: Summary of the results obtained by different double-$\beta$ decay
experiments in the search for Lorentz violation.

A summary of the latest results obtained by different double-$\beta$ decay
experiments is presented in table 4. The most stringent limit on
$\smash{\mathring{a}_{\text{of}}^{(3)}}$ comes from NEMO-3 and is at the order
of $10^{-7}$, a factor 10 better than all the other experiments. This result
is attributed to the much larger statistics of $2\nu\beta\beta$ decay events
achieved by the NEMO-3 experiment ($\sim 1.9\cdot 10^{5}$ events in the
analysis range). However, part of this difference comes from the statistical
treatments as only the CUPID-0 and GERDA experiments used the approach
highlighted in this review for class III models (see section IV), to which the
Lorentz violating $2\nu\beta\beta$ decay belongs. Aurora, NEMO-3, and EXO-200
treated the perturbation introduced by Lorentz violation as an independent
component in the fit, neglecting any correlation with the SM $2\nu\beta\beta$
decay distribution in the result. Nevertheless, the impact is hard to quantify
and goes beyond the scope of this work. When comparing different results, it
should also be considered that the limits on
$\smash{\mathring{a}_{\text{of}}^{(3)}}$ depend on the calculated phase space
ratio between the SM $2\nu\beta\beta$ decay and the LV perturbation. The
different experiments reviewed in table 4 used different calculations. In
recent work, improved phase space calculations were performed, in which the
Fermi functions are built with exact electron wave functions obtained by
numerically solving a Dirac equation in a realistic Coulomb-type potential,
including finite nuclear size and screening effects [158]. Differences up to
30% for heavier nuclei were found between these improved calculations and the
previous calculations using approximated analytical Fermi functions. As
pointed out in [158], for example, there is a relevant difference between the
newly calculated phase space ratio and the one used by the CUPID-0
collaboration in [137].

#####  $2\nu\beta\beta$ decay with bosonic neutrinos

The experimental search for an admixture of bosonic and fermionic neutrinos
through the search for distortions of the $2\nu\beta\beta$ decay spectrum has
been performed only by the NEMO-3 experiment with 100Mo [120]. They obtained
an upper limit on the bosonic neutrino contribution $sin^{2}\chi<0.27$ at 90%
C.L.. Also in this case, the statistical treatment does not follow the receipt
given in IV for class II models as it neglects the correlation between the SM
$2\nu\beta\beta$ decay (fermionic neutrinos) and $2\nu\beta\beta$ decay with
bosonic neutrinos distributions. As already introduced in III.2.2, searches
with other isotopes than 100Mo might be disfavoured by the small predicted
ratio $r_{0}$ with which the fermionic and bosonic contributions are weighted
in the total $2\nu\beta\beta$ decay rate.

##### Double-$\beta$ decay into sterile neutrinos and $Z_{2}$-odd fermions

The experimental search for light exotic fermions, i.e. sterile neutrinos and
$Z_{2}$-odd fermions, has been performed only by the GERDA experiment with
76Ge [131], which set a limit on the mixing between active and sterile
neutrinos $\sin^{2}\theta<0.013$ for a sterile neutrino mass of $m_{N}=500$
keV. The limits get worse for lower and higher masses ($\sin^{2}\theta<0.15$
for $m_{N}=100$ keV, $\sin^{2}\theta<0.050$ for $m_{N}=900$ keV). GERDA has
also set the first direct experimental constraints on the emission of two
$Z_{2}$-odd fermions, constraining the half-life of the corresponding decay to
$2.5\cdot 10^{23}$ yr for a mass $m_{\chi}=400\,$keV, which translates into a
constraint on the coupling $g_{\chi}$ of $(0.6-1.4)\cdot 10^{-3}$ MeV-2.
Again, limits get worse for smaller and larger masses.

##  VI Outlook and prospects

Hunting for the extremely rare $0\nu\beta\beta$ decay, existing experiments
collected up to millions of $2\nu\beta\beta$ decay events. This statistics is
expected to rapidly increase, as future experiments, with much larger target
mass, will start taking data. We showed in this review that $2\nu\beta\beta$
decays could be used as powerful probes of new physics.

Since the first experimental searches for double-$\beta$ decay with the
emission of one or two Majorons, we have seen remarkable progress both in the
theoretical description of the decays and in the experimental technologies.
Improved and more precise calculations of the phase space factors and NMEs are
available today, which are essential to convert the experimental constraints
on the half-life of the decays into a coupling between the exotic particle,
i.e. the Majoron, and neutrinos. On the other hand, the experiments reached
incredible precision in the study of $2\nu\beta\beta$ decay with large
statistics data samples and drastic reduction of the background compared to
their predecessors, pushing the bounds on the half-life of these decays up to
$10^{24}$ yr. Limits on the neutrino-Majoron coupling $g_{J}$ for the Majoron
model leading to $n=1$ are also available from astrophysics. Supernova
observations allow excluding the region of the parameter space $4\cdot
10^{-7}<g_{J}<2\cdot 10^{-5}$ by studying the role of Majorons in the
Supernova explosion [159, 160]. Current double-$\beta$ decay experiments
completely excluded the region above the lower Supernova bound. The combined
results bring the upper bound down to $g_{J}<4\cdot 10^{-7}$, far from the
sensitivity of any future double-$\beta$ decay experiment. Nevertheless, one
should remember that Supernova bounds are model-dependent and rely upon
additional assumptions. To date, double-$\beta$ decays provide the best direct
constraints on the neutrino-Majoron coupling.

Light exotic fermions can also be searched in double-$\beta$ decays. Depending
on the $Q_{\beta\beta}$ of the double-$\beta$ decay isotope, one or two exotic
fermions with a mass between a few hundred keV and a few MeV can be emitted in
double-$\beta$ decay. In the search for sterile neutrino in this mass range,
current double-$\beta$ decay experiments provide bounds that are still weaker
than the existing single-$\beta$ decay bounds, as predicted in [77, 78] and
confirmed by GERDA results [131]. Still, future experiments could reach
unexplored regions of the parameter space, down to $\sin^{2}\theta\sim
10^{-3}-10^{-4}$, for masses between 100 keV and 2000 keV. To date, no
experiment exists or is planned with the capability of testing this part of
the parameter space ($100\,\text{keV}<m_{N}<2000\,\text{keV}$).555See figure 7
in [76]. Therefore, future double-$\beta$ decay experiments will provide the
best direct constraints on the active-sterile neutrino mixing in the
aforementioned mass range. In addition, double-$\beta$ decay experiments offer
a unique opportunity to test all those models in which only the double
production of light exotic fermions is allowed, leading to the best direct
constraints on the coupling between these exotic fermions and neutrinos of the
order of $g_{\chi}\sim 10^{-4}$ MeV-2.

Studying the $2\nu\beta\beta$ decay spectrum can provide a sensitive test of
the so-called counter-shaded Lorentz violation. Current experiments
constrained the Lorentz-violating coefficient
$\smash{\mathring{a}_{\text{of}}^{(3)}}$ at the level of
$|\smash{\mathring{a}_{\text{of}}^{(3)}}|<10^{-6}-10^{-7}$ GeV. The study of
the single-$\beta$ decay spectrum also provides a sensitive constraint to the
same coefficient $\smash{\mathring{a}_{\text{of}}^{(3)}}$. In [88], a
constraint on $\smash{\mathring{a}_{\text{of}}^{(3)}}$ was derived using
tritium $\beta$ decay data from the Mainz and Troitsk experiments:
$|\smash{\mathring{a}_{\text{of}}^{(3)}}|<2.0\cdot 10^{-8}$ GeV, which is
already more competitive of the constraints from double-$\beta$ decays.
Recently, the KATRIN experiment performed a similar analysis using a small set
of available data, setting a limit at
$|\smash{\mathring{a}_{\text{of}}^{(3)}}|<3.0\cdot 10^{-8}$ GeV [161]. This
limit is expected to further improve up to a sensitivity of $10^{-9}$ GeV or
more with the full KATRIN exposure [162]. Future double-$\beta$ decay
experiments will be able to improve their current limits (in the best case
scenario by a factor of $\sqrt{1/\mathcal{E}}$), nevertheless, hardly reaching
single-$\beta$ decay experiments sensitivity.

A purely bosonic neutrino would substantially change the total double-$\beta$
decay rate, therefore, the measured $2\nu\beta\beta$ decay half-life values.
Several precision measurements of the $2\nu\beta\beta$ decay half-life of
different isotopes completely ruled out thy hypothesis of a purely bosonic
neutrino. On the other hand, experimental data does not completely exclude the
hypothesis of a mixed statistic with a partly bosonic neutrino. The only
experimental upper limit on the admixture of the bosonic component was set by
the NEMO-3 experiment with 100Mo. The sensitivity to spectral distortions
depends on the ratio $r_{0}$ between the rates for purely bosonic and purely
fermionic neutrinos, which involves phase space factors and NMEs of the two
decays. Consequently, some isotopes are favored (i.e. 100Mo) for future
searches of bosonic neutrino admixture compared to others (i.e. 76Ge), for
which the very small value of $r_{0}$ neutralizes possible effects induced by
a partly bosonic neutrino.

Finally, $2\nu\beta\beta$ decays can be used as a probe of hidden non-standard
interaction of neutrinos, like strong neutrino self-interactions and the
presence of right-handed currents in weak interactions. Although it was shown
that already current experiments could be competitive in constraining such
non-standard operators [102, 118], no experimental searches have been
performed yet.

###### Acknowledgements.

This work has been supported in part by the German Federal Ministry for
Education and Research (BMBF), the German Research Foundation (DFG), the
Collaborative Research Center SFB1258, and by the Science and Technology
Facilities Council, part of U.K. Research and Innovation (Grant no.
ST/T004169/1). M.A. also acknowledges the support of the UCL Cosmoparticle
Initiative.

## References

  * [1] J. A. Detwiler and R. G. H. Robertson, “Shake-up and shake-off effects in neutrinoless double-beta decay,” 10 2022.
  * [2] M. Alanssari et al., “Single and Double Beta-Decay Q Values among the Triplet Zr96 , Nb96 , and Mo96,” Phys. Rev. Lett., vol. 116, no. 7, p. 072501, 2016.
  * [3] M. Goeppert-Mayer, “Double beta-disintegration,” Phys. Rev., vol. 48, pp. 512–516, 1935.
  * [4] M. G. Inghram and J. H. Reynolds, “Double beta-decay of 130Te,” Phys. Rev., vol. 78, pp. 822–823, 1950.
  * [5] S. R. Elliott, A. A. Hahn, and M. K. Moe, “Direct Evidence for Two Neutrino Double Beta Decay in 82Se,” Phys. Rev. Lett., vol. 59, pp. 2020–2023, 1987.
  * [6] A. S. Barabash, “Precise Half-Life Values for Two-Neutrino Double-$\beta$ Decay: 2020 Review,” Universe, vol. 6, no. 10, p. 159, 2020.
  * [7] M. Agostini, G. Benato, J. A. Detwiler, J. Menéndez, and F. Vissani, “Toward the discovery of matter creation with neutrinoless double-beta decay,” 2 2022.
  * [8] A. S. Barabash, “Double Beta Decay: Historical Review of 75 Years of Research,” Phys. Atom. Nucl., vol. 74, pp. 603–613, 2011.
  * [9] R. Saakyan, “Two-Neutrino Double-Beta Decay,” Ann. Rev. Nucl. Part. Sci., vol. 63, pp. 503–529, 2013.
  * [10] N. Takaoka and K. Ogata, “The Half-life of 130Te Double $\beta$-decay,” Zeitschrift für Naturforschung A, vol. 21, no. 1-2, pp. 84–90, 1966.
  * [11] T. Kirsten, W. Gentner, and O. A. Schaeffer, “Massenspektrometrischer Nachweis von $\beta$ $\beta$-Zerfallsprodukten,” Zeitschrift für Physik, vol. 202, no. 1, pp. 273–292, 1967.
  * [12] E. W. Hennecke, O. K. Manuel, and D. D. Sabu, “Double beta decay of ${}^{128}\mathrm{Te}$,” Phys. Rev. C, vol. 11, pp. 1378–1384, 1975.
  * [13] M. Doi, T. Kotani, and E. Takasugi, “Double beta Decay and Majorana Neutrino,” Prog. Theor. Phys. Suppl., vol. 83, p. 1, 1985.
  * [14] A. A. Vasenko, I. V. Kirpichnikov, V. A. Kuznetsov, A. S. Starostin, A. G. Djanyan, V. S. Pogosov, S. P. Shachysisyan, and A. G. Tamanyan, “New results in the ITEP/YePI double beta-decay experiment with enriched germanium detectors,” Mod. Phys. Lett. A, vol. 05, no. 17, pp. 1299–1306, 1990\.
  * [15] H. Ejiri, K. Fushimi, T. Kamada, H. Kinoshita, H. Kobiki, H. Ohsumi, K. Okada, H. Sano, T. Shibata, T. Shima, N. Tanabe, J. Tanaka, T. Taniguchi, T. Watanabe, and N. Yamamoto, “Double beta decays of 100Mo,” Phys. Lett. B, vol. 258, no. 1, pp. 17–23, 1991.
  * [16] S. R. Elliott, M. K. Moe, M. A. Nelson, and M. A. Vient, “The double beta decay spectrum of 100Mo as measured with a TPC,” Journal of Physics G: Nuclear and Particle Physics, vol. 17, no. S, p. S145, 1991.
  * [17] A. De Silva, M. K. Moe, M. A. Nelson, and M. A. Vient, “Double $\beta$ decays of ${}^{100}\mathrm{Mo}$ and ${}^{150}\mathrm{Nd}$,” Phys. Rev. C, vol. 56, pp. 2451–2467, Nov 1997\.
  * [18] D. Dassié et al., “Two-neutrino double-$\beta$ decay measurement of ${}^{100}\mathrm{Mo}$,” Phys. Rev. D, vol. 51, pp. 2090–2100, Mar 1995.
  * [19] S. R. Elliott, A. A. Hahn, M. K. Moe, M. A. Nelson, and M. A. Vient, “Double beta decay of ${}^{82}\mathrm{Se}$,” Phys. Rev. C, vol. 46, pp. 1535–1537, 1992.
  * [20] R. Arnold et al., “Double-$\beta$ decay of 82Se,” Nuclear Physics A, vol. 636, no. 2, pp. 209–223, 1998.
  * [21] H. Ejiri, K.-i. Fushimi, R. Hazama, M. Kawasaki, V. Kouts, N. Kudomi, K. Kume, K. Nagata, H. Ohsumi, K. Okada, H. Sano, T. Senoo, T. Shibata, T. Shima, J.-i. Tanaka, and Y. Zdesenko, “Double Beta Decays of 116Cd,” Journal of the Physical Society of Japan, vol. 64, no. 2, pp. 339–343, 1995.
  * [22] R. Arnold et al., “Observation of two neutrino double beta decay of Cd-116 with the tracking detector NEMO-2,” JETP Lett., vol. 61, pp. 170–174, 1995.
  * [23] F. A. Danevich, A. S. Georgadze, V. V. Kobychev, B. N. Kropivyansky, V. N. Kuts, A. S. Nikolaiko, V. I. Tretyak, and Y. Zdesenko, “The Research of double beta decay of Cd-116 with enriched Cd-116WO-4 crystal scintillators,” Phys. Lett. B, vol. 344, pp. 72–78, 1995.
  * [24] N. Collaboration, R. Arnold, et al., “Double-$\beta$ decay of 116Cd,” Z. Phys. C, vol. 72, no. 2, p. 239, 1996.
  * [25] A. Balysh, A. De Silva, V. I. Lebedev, K. Lou, M. K. Moe, M. A. Nelson, A. Piepke, A. Pronskiy, M. A. Vient, and P. Vogel, “Double Beta Decay of ${}^{48}\mathrm{Ca}$,” Phys. Rev. Lett., vol. 77, pp. 5186–5189, Dec 1996\.
  * [26] R. Arnold et al., “Double beta decay of 96Zr,” Nuclear Physics A, vol. 658, no. 4, pp. 299–312, 1999.
  * [27] A. L. Turkevich, T. E. Economou, and G. A. Cowan, “Double-beta decay of ${}^{238}\mathrm{U}$,” Phys. Rev. Lett., vol. 67, pp. 3211–3214, Dec 1991\.
  * [28] A. S. Barabash, V. B. Brudanin, and N. Collaboration, “Investigation of double-beta decay with the NEMO-3 detector,” Physics of Atomic Nuclei, vol. 74, no. 2, pp. 312–317, 2011.
  * [29] N. Ackerman et al., “Observation of Two-Neutrino Double-Beta Decay in 136Xe with EXO-200,” Phys. Rev. Lett., vol. 107, p. 212501, 2011\.
  * [30] R. Arnold et al., “Measurement of the 2$\nu\beta\beta$ decay half-life of 150Nd and a search for 0$\nu\beta\beta$ decay processes with the full exposure from the NEMO-3 detector,” Phys. Rev. D, vol. 94, no. 7, p. 072003, 2016.
  * [31] M. Agostini et al. to be submitted. Also in E. Bossio, ”Beyond the Standard Model physics searches with double-beta decays”, PhD thesis, Technische Universität München, 2022.
  * [32] O. Azzolini et al., “Evidence of Single State Dominance in the Two-Neutrino Double-$\beta$ Decay of 82Se with CUPID-0,” Phys. Rev. Lett., vol. 123, no. 26, p. 262501, 2019.
  * [33] J. Argyriades et al., “Measurement of the two neutrino double beta decay half-life of Zr-96 with the NEMO-3 detector,” Nucl. Phys. A, vol. 847, pp. 168–179, 2010.
  * [34] E. Armengaud et al., “Precise measurement of $2\nu\beta\beta$ decay of 100Mo with the CUPID-Mo detection technology,” Eur. Phys. J. C, vol. 80, no. 7, p. 674, 2020.
  * [35] A. S. Barabash et al., “Final results of the Aurora experiment to study $2\beta$ decay of ${}^{116}\mathrm{Cd}$ with enriched ${}^{116}\mathrm{Cd}{\mathrm{WO}}_{4}$ crystal scintillators,” Phys. Rev. D, vol. 98, no. 9, p. 092007, 2018.
  * [36] D. Q. Adams et al., “Measurement of the $2\nu\beta\beta$ Decay Half-Life of ${}^{130}\mathrm{Te}$ with CUORE,” Phys. Rev. Lett., vol. 126, no. 17, p. 171801, 2021.
  * [37] J. B. Albert et al., “Improved measurement of the $2\nu\beta\beta$ half-life of 136Xe with the EXO-200 detector,” Phys. Rev. C, vol. 89, no. 1, p. 015502, 2014.
  * [38] R. Arnold et al., “Measurement of the $2\nu\beta\beta$ decay half-life and search for the $0\nu\beta\beta$ decay of 48Ca with the NEMO-3 detector,” Phys. Rev. D, vol. 93, no. 11, p. 112008, 2016.
  * [39] V. I. TRETYAK and Y. G. ZDESENKO, “TABLES OF DOUBLE BETA DECAY DATA—AN UPDATE,” Atomic Data and Nuclear Data Tables, vol. 80, no. 1, pp. 83–116, 2002.
  * [40] M. Aker et al., “Direct neutrino-mass measurement with sub-electronvolt sensitivity,” Nature Phys., vol. 18, no. 2, pp. 160–166, 2022.
  * [41] T. M. C. Abbott et al., “Dark Energy Survey Year 3 results: Cosmological constraints from galaxy clustering and weak lensing,” Phys. Rev. D, vol. 105, no. 2, p. 023520, 2022.
  * [42] Y. Chikashige, R. N. Mohapatra, and R. D. Peccei, “Are There Real Goldstone Bosons Associated with Broken Lepton Number?,” Phys. Lett. B, vol. 98, pp. 265–268, 1981.
  * [43] Y. Chikashige, R. N. Mohapatra, and R. D. Peccei, “Spontaneously Broken Lepton Number and Cosmological Constraints on the Neutrino Mass Spectrum,” Phys. Rev. Lett., vol. 45, p. 1926, 1980.
  * [44] G. B. Gelmini and M. Roncadelli, “Left-Handed Neutrino Mass Scale and Spontaneously Broken Lepton Number,” Phys. Lett. B, vol. 99, pp. 411–415, 1981.
  * [45] A. Santamaria and J. Valle, “Spontaneous R parity violation in supersymmetry: A model for solar neutrino oscillations,” Physics Letters B, vol. 195, no. 3, pp. 423–428, 1987.
  * [46] A. Santamaria and J. W. F. Valle, “Supersymmetric majoron signatures and solar neutrino oscillations,” Phys. Rev. Lett., vol. 60, pp. 397–400, Feb 1988.
  * [47] A. Santamaria and J. W. F. Valle, “Solar-neutrino-oscillation parameters and the broken-R-parity Majoron,” Phys. Rev. D, vol. 39, pp. 1780–1783, Mar 1989.
  * [48] H. M. Georgi, S. L. Glashow, and S. Nussinov, “Unconventional Model of Neutrino Masses,” Nucl. Phys. B, vol. 193, pp. 297–316, 1981.
  * [49] J. Vergados, “A limit on the neutrino masses from double $\beta$ decay,” Physics Letters B, vol. 109, no. 1, pp. 96–100, 1982.
  * [50] M. Doi, T. Kotani, and E. Takasugi, “The Neutrinoless Double Beta Decay With Majoron Emission,” Phys. Rev. D, vol. 37, p. 2575, 1988.
  * [51] R. N. Mohapatra and E. Takasugi, “Neutrinoless Double Beta Decay With Double Majoron Emission,” Phys. Lett. B, vol. 211, pp. 192–196, 1988.
  * [52] J. Steinberger, Electroweak physics beyond the standard model.  World Scientific, Singapore, 1992.
  * [53] S. Schael et al., “Precision electroweak measurements on the $Z$ resonance,” Phys. Rept., vol. 427, pp. 257–454, 2006.
  * [54] S. R. Elliott, A. A. Hahn, and M. K. Moe, “Limit on Neutrinoless Double Beta Decay With Majoron Emission in 82Se,” Phys. Rev. Lett., vol. 59, pp. 1649–1651, 1987.
  * [55] F. T. Avignone, C. K. Guerard, R. L. Brodzinski, H. S. Miley, J. H. Reeves, I. V. Kirpichnikov, A. S. Starostin, V. S. Pogosov, and A. G. Tamanian, “Confirmation of the observation of 2$\nu\beta\beta$ decay of 76Ge,” Phys. Lett. B, vol. 256, pp. 559–561, 1991.
  * [56] P. Fisher, F. Boehm, H. Henrikson, E. Bovet, J. P. Egger, J. L. Vuilleumier, and K. Gabathuler, “Limit on Double Beta Decay With Majoron Emission,” Phys. Lett. B, vol. 192, pp. 460–462, 1987.
  * [57] Z. G. Berezhiani, A. Y. Smirnov, and J. W. F. Valle, “Observable Majoron emission in neutrinoless double beta decay,” Phys. Lett. B, vol. 291, pp. 99–105, 1992.
  * [58] C. P. Burgess and J. M. Cline, “Majorons without Majorana masses and neutrinoless double beta decay,” Phys. Lett. B, vol. 298, pp. 141–148, 1993.
  * [59] C. P. Burgess and J. M. Cline, “A New class of Majoron emitting double beta decays,” Phys. Rev. D, vol. 49, pp. 5925–5944, 1994.
  * [60] P. Bamert, C. P. Burgess, and R. N. Mohapatra, “Multi-Majoron modes for neutrinoless double beta decay,” Nucl. Phys. B, vol. 449, pp. 25–48, 1995\.
  * [61] C. D. Carone, “Double beta decay with vector majorons,” Phys. Lett. B, vol. 308, pp. 85–88, 1993.
  * [62] R. N. Mohapatra, A. Perez-Lorenzana, and C. A. de S. Pires, “Neutrino mass, bulk Majoron and neutrinoless double beta decay,” Phys. Lett. B, vol. 491, pp. 143–147, 2000.
  * [63] M. Hirsch, H. V. Klapdor-Kleingrothaus, S. G. Kovalenko, and H. Pas, “On the observability of majoron emitting double beta decays,” Phys. Lett. B, vol. 372, pp. 8–14, 1996.
  * [64] J. Kotila, J. Barea, and F. Iachello, “Phase-space factors and half-life predictions for Majoron-emitting $\beta^{-}\beta^{-}$ decay,” Phys. Rev. C, vol. 91, no. 6, p. 064310, 2015.  [Erratum: Phys.Rev.C 92, 029903 (2015)].
  * [65] J. Kotila and F. Iachello, “Nuclear matrix elements for Majoron-emitting double- $\beta$ decay,” Phys. Rev. C, vol. 103, no. 4, p. 044302, 2021\.
  * [66] R. Cepedello, F. F. Deppisch, L. González, C. Hati, and M. Hirsch, “Neutrinoless Double-$\beta$ Decay with Nonstandard Majoron Emission,” Phys. Rev. Lett., vol. 122, no. 18, p. 181801, 2019.
  * [67] K. Blum, Y. Nir, and M. Shavit, “Neutrinoless double-beta decay with massive scalar emission,” Phys. Lett. B, vol. 785, pp. 354–361, 2018.
  * [68] V. Berezinsky and J. W. F. Valle, “The KeV majoron as a dark matter particle,” Phys. Lett. B, vol. 318, pp. 360–366, 1993.
  * [69] T. Brune and H. Päs, “Massive Majorons and constraints on the Majoron-neutrino coupling,” Phys. Rev. D, vol. 99, no. 9, p. 096005, 2019\.
  * [70] B. Dasgupta and J. Kopp, “Sterile Neutrinos,” Phys. Rept., vol. 928, pp. 1–63, 2021.
  * [71] R. E. Shrock, “New Tests For, and Bounds On, Neutrino Masses and Lepton Mixing,” Phys. Lett. B, vol. 96, pp. 159–164, 1980.
  * [72] O. Dragoun and D. Vénos, “Constraints on the Active and Sterile Neutrino Masses from Beta-Ray Spectra: Past, Present and Future,” J. Phys., vol. 3, pp. 77–113, 2016.
  * [73] A. S. Riis and S. Hannestad, “Detecting sterile neutrinos with KATRIN like experiments,” JCAP, vol. 02, p. 011, 2011.
  * [74] S. Mertens, T. Lasserre, S. Groh, G. Drexlin, F. Glueck, A. Huber, A. W. P. Poon, M. Steidl, N. Steinbrink, and C. Weinheimer, “Sensitivity of Next-Generation Tritium Beta-Decay Experiments for keV-Scale Sterile Neutrinos,” JCAP, vol. 02, p. 020, 2015.
  * [75] A. Abada, A. Hernández-Cabezudo, and X. Marcano, “Beta and Neutrinoless Double Beta Decays with KeV Sterile Fermions,” JHEP, vol. 01, p. 041, 2019\.
  * [76] P. D. Bolton, F. F. Deppisch, and P. S. Bhupal Dev, “Neutrinoless double beta decay versus other probes of heavy sterile neutrinos,” JHEP, vol. 03, p. 170, 2020.
  * [77] P. D. Bolton, F. F. Deppisch, L. Gráf, and F. Šimkovic, “Two-Neutrino Double Beta Decay with Sterile Neutrinos,” Phys. Rev. D, vol. 103, no. 5, p. 055019, 2021.
  * [78] M. Agostini, E. Bossio, A. Ibarra, and X. Marcano, “Search for Light Exotic Fermions in Double-Beta Decays,” Phys. Lett. B, vol. 815, p. 136127, 2021\.
  * [79] V. A. Kostelecky and S. Samuel, “Spontaneous Breaking of Lorentz Symmetry in String Theory,” Phys. Rev. D, vol. 39, p. 683, 1989.
  * [80] D. Colladay and V. A. Kostelecky, “CPT violation and the standard model,” Phys. Rev. D, vol. 55, pp. 6760–6774, 1997.
  * [81] D. Colladay and V. A. Kostelecky, “Lorentz violating extension of the Standard Model,” Phys. Rev. D, vol. 58, p. 116002, 1998.
  * [82] V. A. Kostelecky and J. D. Tasson, “Prospects for Large Relativity Violations in Matter-Gravity Couplings,” Phys. Rev. Lett., vol. 102, p. 010402, 2009\.
  * [83] A. V. Kostelecky and J. D. Tasson, “Matter-gravity couplings and Lorentz violation,” Phys. Rev. D, vol. 83, p. 016013, 2011.
  * [84] V. A. Kostelecky and N. Russell, “Data Tables for Lorentz and CPT Violation,” Rev. Mod. Phys., vol. 83, pp. 11–31, 2011.
  * [85] V. A. Kostelecky and M. Mewes, “Lorentz and CPT violation in the neutrino sector,” Phys. Rev. D, vol. 70, p. 031902, 2004.
  * [86] V. A. Kostelecky and M. Mewes, “Lorentz and CPT violation in neutrinos,” Phys. Rev. D, vol. 69, p. 016005, 2004.
  * [87] A. Kostelecky and M. Mewes, “Neutrinos with Lorentz-violating operators of arbitrary dimension,” Phys. Rev. D, vol. 85, p. 096005, 2012.
  * [88] J. S. Díaz, A. Kostelecký, and R. Lehnert, “Relativity violations and beta decay,” Phys. Rev. D, vol. 88, no. 7, p. 071902, 2013.
  * [89] J. S. Díaz, “Limits on Lorentz and CPT violation from double beta decay,” Phys. Rev. D, vol. 89, p. 036002, 2014.
  * [90] A. Y. Ignatiev and V. A. Kuzmin, “Is small violation of the Pauli principle possible?,” 2 1987.
  * [91] O. W. Greenberg and R. N. Mohapatra, “Local Quantum Field Theory of Violation of the Pauli Principle,” Phys. Rev. Lett., vol. 59, p. 2507, 1987.  [Erratum: Phys.Rev.Lett. 61, 1432 (1988)].
  * [92] L. B. Okun, “On the Possibility of Pauli Principle Violation in Atoms,” JETP Lett., vol. 46, pp. 529–532, 1987.
  * [93] O. W. Greenberg and R. N. Mohapatra, “Difficulties With a Local Quantum Field Theory of Possible Violation of the Pauli Principle,” Phys. Rev. Lett., vol. 62, p. 712, 1989.  [Erratum: Phys.Rev.Lett. 62, 1927 (1989)].
  * [94] A. B. Govorkov, “Can the Pauli Principle Be Deduced With Local Quantum Field Theory?,” Phys. Lett. A, vol. 137, pp. 7–10, 1989.
  * [95] L. Cucurull, J. A. Grifols, and R. Toldra, “Spin statistics theorem, neutrinos, and big bang nucleosynthesis,” Astropart. Phys., vol. 4, pp. 391–396, 1996.
  * [96] S. Choubey and K. Kar, “Possible violation of the spin-statistics relation for neutrinos: Checking through future galactic supernova,” Phys. Lett. B, vol. 634, pp. 14–22, 2006.
  * [97] A. D. Dolgov and A. Y. Smirnov, “Possible violation of the spin-statistics relation for neutrinos: Cosmological and astrophysical consequences,” Phys. Lett. B, vol. 621, pp. 1–10, 2005.
  * [98] A. D. Dolgov, S. H. Hansen, and A. Y. Smirnov, “Neutrino statistics and Big Bang nucleosynthesis,” JCAP, vol. 06, p. 004, 2005.
  * [99] P. F. de Salas, S. Gariazzo, M. Laveder, S. Pastor, O. Pisanti, and N. Truong, “Cosmological bounds on neutrino statistics,” JCAP, vol. 03, p. 050, 2018.
  * [100] A. S. Barabash, A. D. Dolgov, R. Dvornicky, F. Šimkovic, and A. Y. Smirnov, “Statistics of neutrinos and the double beta decay,” Nucl. Phys. B, vol. 783, pp. 90–111, 2007.
  * [101] A. Y. Ignatiev and V. A. Kuzmin, “Neutrino statistics and non-standard commutation relations,” Phys. Lett. A, vol. 359, pp. 26–30, 2006.
  * [102] F. F. Deppisch, L. Gráf, and F. Šimkovic, “Searching for New Physics in Two-Neutrino Double Beta Decay,” Phys. Rev. Lett., vol. 125, no. 17, p. 171801, 2020.
  * [103] J. C. Pati and A. Salam, “Lepton Number as the Fourth Color,” Phys. Rev. D, vol. 10, pp. 275–289, 1974.  [Erratum: Phys.Rev.D 11, 703–703 (1975)].
  * [104] P. D. Bolton, F. F. Deppisch, C. Hati, S. Patra, and U. Sarkar, “Alternative formulation of left-right symmetry with $B-L$ conservation and purely Dirac neutrinos,” Phys. Rev. D, vol. 100, no. 3, p. 035013, 2019.
  * [105] M. Doi, T. Kotani, H. Nishiura, and E. Takasugi, “DOUBLE BETA DECAY,” Prog. Theor. Phys., vol. 69, p. 602, 1983.
  * [106] F. F. Deppisch, M. Hirsch, and H. Pas, “Neutrinoless Double Beta Decay and Physics Beyond the Standard Model,” J. Phys. G, vol. 39, p. 124007, 2012\.
  * [107] M. González-Alonso, O. Naviliat-Cuncic, and N. Severijns, “New physics searches in nuclear and neutron $\beta$ decay,” Prog. Part. Nucl. Phys., vol. 104, pp. 165–223, 2019.
  * [108] A. Greljo and D. Marzocca, “High-$p_{T}$ dilepton tails and flavor physics,” Eur. Phys. J. C, vol. 77, no. 8, p. 548, 2017.
  * [109] V. Khachatryan et al., “Search for physics beyond the standard model in final states with a lepton and missing transverse energy in proton-proton collisions at $\sqrt{s}=8$ TeV,” Phys. Rev. D, vol. 91, no. 9, p. 092005, 2015.
  * [110] M. Aaboud et al., “Search for heavy Majorana or Dirac neutrinos and right-handed $W$ gauge bosons in final states with two charged leptons and two jets at $\sqrt{s}=13$ TeV with the ATLAS detector,” JHEP, vol. 01, p. 016, 2019.
  * [111] A. G. Riess, S. Casertano, W. Yuan, L. M. Macri, and D. Scolnic, “Large Magellanic Cloud Cepheid Standards Provide a 1% Foundation for the Determination of the Hubble Constant and Stronger Evidence for Physics beyond $\Lambda$CDM,” Astrophys. J., vol. 876, no. 1, p. 85, 2019.
  * [112] N. Aghanim et al., “Planck 2018 results. VI. Cosmological parameters,” Astron. Astrophys., vol. 641, p. A6, 2020.  [Erratum: Astron.Astrophys. 652, C4 (2021)].
  * [113] I. M. Oldengott, T. Tram, C. Rampf, and Y. Y. Y. Wong, “Interacting neutrinos in cosmology: exact description and constraints,” JCAP, vol. 11, p. 027, 2017.
  * [114] C. D. Kreisch, F. Cyr-Racine, and O. Doré, “Neutrino puzzle: Anomalies, interactions, and cosmological tensions,” Phys. Rev. D, vol. 101, no. 12, p. 123505, 2020.
  * [115] N. Blinov, K. J. Kelly, G. Z. Krnjaic, and S. D. McDermott, “Constraining the Self-Interacting Neutrino Interpretation of the Hubble Tension,” Phys. Rev. Lett., vol. 123, no. 19, p. 191102, 2019.
  * [116] K.-F. Lyu, E. Stamou, and L.-T. Wang, “Self-interacting neutrinos: Solution to Hubble tension versus experimental constraints,” Phys. Rev. D, vol. 103, no. 1, p. 015004, 2021.
  * [117] G. Arcadi, J. Heeck, F. Heizmann, S. Mertens, F. S. Queiroz, W. Rodejohann, M. Slezák, and K. Valerius, “Tritium beta decay with additional emission of new light bosons,” JHEP, vol. 01, p. 206, 2019.
  * [118] F. F. Deppisch, L. Gráf, W. Rodejohann, and X. Xu, “Neutrino Self-Interactions and Double Beta Decay,” Phys. Rev. D, vol. 102, no. 5, p. 051701, 2020.
  * [119] R. Arnold et al., “Measurement of the $2\nu\beta\beta$ Decay Half-Life and Search for the $0\nu\beta\beta$ Decay of 116Cd with the NEMO-3 Detector,” Phys. Rev. D, vol. 95, no. 1, p. 012007, 2017.
  * [120] R. Arnold et al., “Detailed studies of 100Mo two-neutrino double beta decay in NEMO-3,” Eur. Phys. J. C, vol. 79, no. 5, p. 440, 2019.
  * [121] Y. Fukuda et al., “Evidence for oscillation of atmospheric neutrinos,” Phys. Rev. Lett., vol. 81, pp. 1562–1567, 1998.
  * [122] E. Majorana, “Teoria simmetrica dell’elettrone e del positrone,” Nuovo Cim., vol. 14, pp. 171–184, 1937.
  * [123] G. Racah, “Sulla Simmetria Tra Particelle e Antiparticelle,” Nuovo Cim., vol. 14, pp. 322–328, 1937.
  * [124] W. H. Furry, “On transition probabilities in double beta-disintegration,” Phys. Rev., vol. 56, pp. 1184–1193, 1939.
  * [125] J. Schechter and J. W. F. Valle, “Neutrinoless Double beta Decay in SU(2) x U(1) Theories,” Phys. Rev. D, vol. 25, p. 2951, 1982.
  * [126] F. T. Avignone and S. R. Elliott, “The Search for Double Beta Decay With Germanium Detectors: Past, Present, and Future,” Front. in Phys., vol. 7, p. 6, 2019.
  * [127] B. Depuydt, A. Theuwis, and I. Romandic, “Germanium: From the first application of czochralski crystal growth to large diameter dislocation-free wafers,” Materials Science in Semiconductor Processing, vol. 9, no. 4, pp. 437–443, 2006.  Proceedings of Symposium T E-MRS 2006 Spring Meeting on Germanium based semiconductors from materials to devices.
  * [128] M. Agostini et al., “Upgrade for Phase II of the Gerda experiment,” Eur. Phys. J. C, vol. 78, no. 5, p. 388, 2018.
  * [129] N. Abgrall et al., “The Majorana Demonstrator Neutrinoless Double-Beta Decay Experiment,” Adv. High Energy Phys., vol. 2014, p. 365432, 2014\.
  * [130] N. Abgrall et al., “The Large Enriched Germanium Experiment for Neutrinoless $\beta\beta$ Decay: LEGEND-1000 Preconceptual Design Report.” 7 2021.
  * [131] M. Agostini et al., “Search for exotic physics in double-$\beta$ decays with GERDA Phase II,” JCAP, vol. 12, p. 012, 2022.
  * [132] C. Brofferio and S. Dell’Oro, “Contributed Review: The saga of neutrinoless double beta decay search with TeO2 thermal detectors,” Rev. Sci. Instrum., vol. 89, no. 12, p. 121502, 2018.
  * [133] C. Alduino et al., “The CUORE cryostat: An infrastructure for rare event searches at millikelvin temperatures,” Cryogenics, vol. 102, pp. 9–21, 2019.