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{{Nuclear physics|cTopic=Radioactive decay}}

The '''neutrinoless double beta decay (0νββ)''' is a commonly proposed and experimentally pursued theoretical [[radioactive decay]] process that would prove a [[Majorana particle|Majorana]] nature of the [[neutrino]] [[particle]].<ref name="GrotzKlapdor1990" /><ref name="Oberauer2020" /> To this day, it has not been found.<ref name="Rodejohann2012" /><ref name="Deppisch2019">{{cite book |last1=Deppisch |first1=Frank F. |title=A modern introduction to neutrino physics |date=2019 |publisher=Morgan & Claypool Publishers |isbn=978-1-64327-679-3}}</ref><ref name="Oberauer2020" />

The discovery of the neutrinoless [[double beta decay]] could shed light on the absolute neutrino masses and on their mass hierarchy ([[Neutrino mass]]). It would mean the first ever signal of the violation of total [[lepton number]] conservation.<ref name="PDGbook">{{cite journal |last1=Patrignani et al. (Particle Data Group) |first1=C. |title=Review of Particle Physics |journal=Chinese Physics C |date=October 2016 |volume=40 |issue=10 |page=647 |doi=10.1088/1674-1137/40/10/100001}}</ref> Majorana nature of neutrinos would confirm that the neutrino's own [[antiparticle]] is no different than itself, i.e. it ''is'' its own antiparticle.<ref name="Majorana1937">{{cite journal |last1=Majorana |first1=Ettore |title=Teoria simmetrica dell’elettrone e del positrone |journal=Il Nuovo Cimento (1924-1942) |date=1937 |volume=14 |issue=4 |doi=10.1007/BF02961314 |url=https://link.springer.com/article/10.1007/BF02961314}}</ref>

To search for neutrinoless double beta decay, there is currently a number of experiments underway, with several future experiments for increased sensitivity proposed as well.<ref name="Bilenky2015" />

== Historical development of the theoretical discussion ==

Back in 1039, [[Wendell H. Furry]] proposed the idea of the Majorana nature of the neutrino, which was associated with beta decays.<ref name="Furry1939">{{cite journal |last1=Furry |first1=W. H. |title=On Transition Probabilities in Double Beta-Disintegration |journal=Physical Review |date=15 December 1939 |volume=56 |issue=12 |pages=1184–1193 |doi=10.1103/PhysRev.56.1184}}</ref> Furry stated the transition probability to even be higher for the neutrino''less'' double beta decay.<ref name="Furry1939" /> It was the first idea proposed to seach for the violation of lepton number violation.<ref name="GrotzKlapdor1990" /> It has, since then, drawn attention to it for being useful to study the nature of neutrinos (see quote).

{{Quote frame|"[T]he 0ν mode [...] which violates the lepton number and has been recognized since a long time as a powerful tool to test neutrino properties."   — Oliviero Cremonesi<ref>{{cite journal |last1=Cremonesi |first1=Oliviero |title=Neutrinoless double beta decay: Present and future |journal=Nuclear Physics B - Proceedings Supplements |date=April 2003 |volume=118 |pages=287–296 |doi=10.1016/S0920-5632(03)01331-8}}</ref>}}

The Italian physicist [[Ettore Majorana]] first introduced the concept of a particle being its own antiparticle.<ref name="Majorana1937" /> Particles' nature was subsequently named after him as Majorana particles. The neutrinoless double beta decay is one method to search for the possible Majorana nature of neutrinos.<ref name="PDGbook" />

[[Image:Ettore_Majorana.jpg|thumb|250px|left|Ettore Majorana, the first to introduce the idea of particles and antiparticles being identical.<ref name="Majorana1937" />]]

== Physical relevance ==

=== Conventional double beta decay ===

Neutrinos are conventionally produced in weak decays.<ref name="PDGbook" /> Weak beta decays normally produce one [[electron]] (or [[positron]]), emit an [[Neutrino#Antineutrinos|antineutrino]] (or neutrino) and increase the [[nucleus]]' [[proton number]] <math>Z</math> by one. The nucleus' mass (i.e. [[binding energy]]) is then lower and thus more favorable. There exists a number of elements that can decay into a nucleus of lower mass, but they cannot emit one electron only because the resulting nucleus is kinematically (that is, in terms of energy) not favorable (its energy would be higher).<ref name="Oberauer2020">{{cite book |last1=Oberauer |first1=Lothar |last2=Ianni |first2=Aldo |last3=Serenelli |first3=Aldo |title=Solar neutrino physics : the interplay between particle physics and astronomy |date=2020 |publisher=Wiley-VCH |isbn=978-3-527-41274-7 |pages=120-127}}</ref> These nuclei can only decay by emitting two electrons (that is, via ''double beta decay''). There is about a dozen confirmed cases of nuclei that can only decay via double beta decay.<ref name="Oberauer2020" /> The corresponding decay equation is:
:<math>(A,Z)\rightarrow (A,Z+2)+2e^-+2\bar{\nu}_e</math>.<ref name="GrotzKlapdor1990" />

It is a weak process of second order.<ref name="Oberauer2020" /> A simultaneous decay of two [[nucleon]]s in the same nucleus is extremely unlikely. Thus, the experimentally observed lifetimes of such decay processes are in the range of <math> 10^{18}-10^{21} </math> years.<ref name="Artusa2014" /> A number of [[isotope]]s have been observed already to show this two-neutrino double beta decay.<ref name="Rodejohann2012" />

This conventional double beta decay is allowed in the [[Standard Model]] of [[particle physics]].<ref name="Rodejohann2012" /> It has thus both a theoretical and an experimental foundation.

=== Overview ===

[[Image:Double beta decay feynman.svg|thumb|250px|right|[[Feynman diagram]] of neutrinoless double beta decay. Here two [[neutron]]s decay into two [[protons]] and two electrons, but no neutrino is in the final state. The existence of this mechanism would require the neutrinos to be Majorana particles.<ref name="SchechterValle1982">{{cite journal |last1=Schechter |first1=J. |last2=Valle |first2=J. W. F. |title=Neutrinoless double-beta decay in SU(2)×U(1) theories |journal=Physical Review D |date=1 June 1982 |volume=25 |issue=11 |pages=2951–2954 |doi=10.1103/PhysRevD.25.2951}}</ref>]]

If the nature of the neutrinos is Majorana, then they can be emitted and absorbed in the same process without showing up in the corresponding final state.<ref name="Rodejohann2012" /> As [[Dirac equation|Dirac particles]], both the neutrinos produced by the decay of the [[W boson]]s would be emitted, and not absorbed after.<ref name="Rodejohann2012" />

The neutrinoless double beta decay can only occur if
* the neutrino particle is Majorana,<ref name="SchechterValle1982" /> ''and''
* there exists a right-handed component of the weak leptonic current ''or'' the neutrino can change its [[handedness]] between emission and absorption (between the two W vertices), which is possible for a non-zero neutrino mass (for at least one of the neutrino species).<ref name="GrotzKlapdor1990">{{cite book |last1=Grotz |first1=K. |last2=Klapdor |first2=H. V. |title=The weak interaction in nuclear, particle, and astrophysics |date=1990 |publisher=Hilger |isbn=978-0-85274-313-3}}</ref>

The simplest decay process is known as the light neutrino exchange.<ref name="Rodejohann2012" /> It features one neutrino emitted by one nucleon and absorbed by another nucleon (see figure to the right). In the final state, the only remaining parts are the nucleus (with its changed proton number <math>Z</math>) and two electrons:
:<math>(A,Z)\rightarrow (A,Z+2)+2e^-</math><ref name="GrotzKlapdor1990" />
The two electrons are emitted quasi-simultaneously.<ref name="Artusa2014">{{cite journal |last1=Artusa |first1=D. R. |title=Exploring the neutrinoless double beta decay in the inverted neutrino hierarchy with bolometric detectors |last2=Avignone |first2=F. T. |last3=Azzolini |first3=O. |last4=Balata |first4=M. |last5=Banks |first5=T. I. |last6=Bari |first6=G. |last7=Beeman |first7=J. |last8=Bellini |first8=F. |last9=Bersani |first9=A. |last10=Biassoni |first10=M. |journal=The European Physical Journal C |date=15 October 2014 |volume=74 |issue=10 |doi=10.1140/epjc/s10052-014-3096-8}}</ref>

The two resulting electrons are then the only emitted particles in the final state and must carry approximately the difference of the sums of the binding energies of the two nuclei before and after the process as their kinetic energy.{{sfn | Grotz | Klapdor | 1990 | p=86 }} The heavy nuclei do not carry significant kinetic energy. The electrons will be emitted back-to-back due to [[momentum conservation]].{{sfn | Grotz | Klapdor | 1990 | p=86 }}

In that case, the [[decay rate]] can be calculated with
:<math> \Gamma_{\beta\beta}^{0\nu}=\frac{1}{T_{\beta\beta}^{0\nu}} = G^{0\nu}\cdot\left| M^{0\nu}\right|^2\cdot\langle m_{\beta\beta}\rangle^2 </math>,
where <math> G^{0\nu} </math> denotes the [[phase space]] factor, <math> \left| M^{0\nu}\right|^2 </math> the (squared) [[matrix element]] of this nuclear decay process (according to the Feynman diagram), and <math> \langle m_{\beta\beta}\rangle^2 </math> the square of the effective Majorana mass.<ref name="PDGbook" />

First, the effective Majorana mass can be obtained by
:<math> \langle m_{\beta\beta}\rangle = \sum_i U_{ei}^2m_i</math>,
where <math> m_i </math> are the Majorana neutrino masses (three neutrinos <math> \nu_i </math>) and <math> U_{ei} </math> the elements of the neutrino mixing matrix <math> U </math> (see [[PMNS matrix]]).<ref name="Bilenky2015" /> Contemporary experiments to find neutrinoless double beta decays (see [[#Experiments|section on experiments]]) aim at both the proof of the Majorana nature of neutrinos and the measurement of this  effective Majorana mass <math> \langle m_{\beta\beta}\rangle </math> (can only be done if the decay is actually generated by the neutrino masses).<ref name="Bilenky2015" />

The nuclear matrix element (NME) <math> \left| M^{0\nu}\right| </math> cannot be measured independently, they must, but also can, be calculated.<ref name="Bilenky2002">{{cite journal |last1=Bilenky |first1=S.M |last2=Grifols |first2=J.A |title=The possible test of the calculations of nuclear matrix elements of the (ββ)0ν-decay |journal=Physics Letters B |date=December 2002 |volume=550 |issue=3-4 |pages=154–159 |doi=10.1016/S0370-2693(02)02978-7}}</ref> The calculation itself relies on sophisticated nuclear many-body theories and there exist different methods to do this. The NME <math> \left| M^{0\nu}\right| </math> differs also from nucleus to nucleus (i.e. [[chemical element]] to chemical element). Today, the calculation of the NME is a significant problem and it has been treated by different authors in different ways. One question is whether to treat the range of obtained values for <math> \left| M^{0\nu}\right| </math> as the theoretical uncertainty and whether this is to be understood as a ''statistical'' uncertainty.<ref name="Bilenky2015" /> Different approaches are being chosen here. The obtained values for <math> \left| M^{0\nu}\right| </math> often vary by factors of 2 up to about 5. Typical values lie in the range of from about 0.9 to 14, depending on the decaying nucleus/element.<ref name="Bilenky2015" />

Lastly, the phase-space factor <math> G^{0\nu} </math> must also be calculated.<ref name="Bilenky2015" /> It depends on the total released kinetic energy (<math> Q=M_\text{nucleus}^\text{before}-M_\text{nucleus}^\text{after}-2m_\text{electron} </math>, i.e. "<math>Q</math>-value") and the atomic number <math>Z</math>. Methods utilize [[Dirac]] [[wave function]]s, finite nuclear sizes and electron screening.<ref name="Bilenky2015" /> There exists high-precision results for <math> G^{0\nu} </math> for various nuclei, ranging from about 0.23 (for <math >\mathrm{^{128}_{52}Te\rightarrow ^{128}_{54}Xe}</math>), and 0.90 (<math>Q</math>) to about 24.14 (<math>Q</math>).<ref name="Bilenky2015" />

It is believed that, if neutrinoless double beta decay is found under certain conditions (decay rate compatible with predictions based on experimental knowledge about neutrino masses and mixing), this would indeed "likely" point at Majorana neutrinos as the main mediator (and not other sources of new physics).<ref name="Bilenky2015">{{cite journal |last1=Bilenky |first1=S. M. |last2=Giunti |first2=C. |title=Neutrinoless double-beta decay: A probe of physics beyond the Standard Model |journal=International Journal of Modern Physics A |date=11 February 2015 |volume=30 |issue=04n05 |pages=1530001 |doi=10.1142/S0217751X1530001X}}</ref> There is 35 nuclei that can undergo neutrinless double beta decay (according to the afforementioned decay conditions).<ref name="Rodejohann2012">{{cite journal |last1=Rodejohann |first1=Werner |title=Neutrino-less double beta decay and particle physics |journal=International Journal of Modern Physics E |date=2 May 2012 |volume=20 |issue=09 |pages=1833–1930 |doi=10.1142/S0218301311020186}}</ref>

== Experiments and results ==

There is a number of nine different candidates of nuclei that are being considered in experiments to confirm neutrinoless double beta-decay: <math >\mathrm{^{48}Ca, ^{76}Ge, ^{82}Se, ^{96}Zr, ^{100}Mo, ^{116}Cd, ^{130}Te, ^{136}Xe, ^{150}Nd}</math>.<ref name="Rodejohann2012" /> They all have arguments for and against their use in an experiment. Factors to be included and revised are [[natural abundance]], reasonably priced enrichment, and a well understood and controlled experimental technique.<ref name="Rodejohann2012" /> The higher the <math>Q</math>-value, the better are the chances of a discovery, in principal. The phase-space factor <math> G^{0\nu} </math>, and thus the decay rate, grows with <math>Q^5</math>.<ref name="Rodejohann2012" />

Experimentally of interest and thus measured is the sum of the kinetic energies of the two emitted electrons. It should equal the <math>Q</math>-value of the respective nucleus for neutrinoless double beta emission.<ref name="Rodejohann2012" />

The table shows a summary of the currently best limits on the lifetime of 0νββ. One can certainly deduce from this that neutrinoless double beta decay is an extremely rare process - if it occurs at all.

{| class="wikitable sortable"
|+ Experimental limits (at least 90% [[Confidence interval|C.L.]])<ref name="Bilenky2015" /> on a collection of isotopes for 0νββ decay process mediated by the light neutrino mechanism, as shown in the Feynman diagram above.
|-
! Isotope !! Experiment !! lifetime <math>T_{\beta\beta}^{0\nu}</math> [years]
|-
| <math>\mathrm{^{48}Ca}</math> || ELEGANT-VI || <math>>1.4\cdot 10^{22}</math>
|-
| <math>\mathrm{^{76}Ge}</math> || Heidelberg-Moscow<ref name="HDM">{{cite web |title=The Heidelberg-Moscow Experiment with enriched 76Ge |url=http://www.klapdor-k.de/Theory%20of%20Experiments/HDMDBD/HeiMos.htm#:~:text=The%20HEIDELBERG%2DMOSCOW%2DExperiment%20is,beta%20transitions%20in%20one%20nucleus. |publisher=Prof.Dr.H.V.Klapdor-Kleingrothaus |accessdate=16 July 2020}}</ref> || <math>>1.9\cdot 10^{25}</math><ref name="HDM" />
|-
| <math>\mathrm{^{76}Ge}</math> || GERDA || <math>>2.1\cdot 10^{25}</math>
|-
| <math>\mathrm{^{82}Se}</math> || [[Neutrino Ettore Majorana Observatory|NEMO]]-3 || <math>>1.0\cdot 10^{23}</math>
|-
| <math>\mathrm{^{96}Zr}</math> || NEMO-3 || <math>>9.2\cdot 10^{21}</math>
|-
| <math>\mathrm{^{100}Mo}</math> || NEMO-3 || <math>>2.1\cdot 10^{25}</math>
|-
| <math>\mathrm{^{116}Cd}</math> || Solotvina || <math>>1.7\cdot 10^{23}</math>
|-
| <math>\mathrm{^{130}Te}</math> || [[CUORE#History|CUORICINO]] || <math>>2.8\cdot 10^{24}</math>
|-
| <math>\mathrm{^{136}Xe}</math> || EXO || <math>>1.1\cdot 10^{25}</math>
|-
|  || KamLAND-Zen || <math>>2.6\cdot 10^{25}</math><ref name="Tornow2014" />
|-
| <math>\mathrm{^{150}Nd}</math> || NEMO-3 || <math>>2.1\cdot 10^{25}</math>
|}

==== Heidelberg-Moscow collaboration ====

The so-called "Heidelberg-Moscow collaboration" (HDM) of the German [[Max Planck Institute for Nuclear Physics|Max-Planck-Institut für Kernphysik]] and the Russian science center [[Kurchatov Institute]] in Moscow famously claimed to have found "evidence ofr neutrinoless double beta decay"<ref name="KlapdorHDM">{{cite journal |last1=Klapdor-Kleingrothaus |first1=H. V. |last2=Dietz |first2=A. |last3=Harney |first3=H. L. |last4=Krivosheina |first4=I. V. |title=Evidence for neutrinoless double beta decay |journal=Modern Physics Letters A |date=21 November 2011 |volume=16 |issue=37 |pages=2409–2420 |doi=10.1142/S0217732301005825}}</ref>. Initially, in 2001 the collaboration announced a 2.2σ, or a 3.1σ (depending on the used calculation method) evidence.<ref name="KlapdorHDM" /> The decay rate was found to be around <math>2\cdot 10^{25}</math> years.<ref name="Rodejohann2012" /> This result has been topic of discussions between many scientists and authors.<ref name="Rodejohann2012" /> To this day, no other experiment has ever confirmed or approved the result of the HDM group.<ref name="Bilenky2015" /> Instead, recent results from the GERDA experiment for the lifetime limit clearly disfavor and reject the values of the HDM collaboration.<ref name="Bilenky2015" />

Neutrinoless double beta decay has not yet been found.<ref name="Deppisch2019" />

=== Currently data-taking experiments ===

* ''[[Germanium Detector Array|GERDA]] (Germanium Detector Array) experiment'':
**  The GERDA collaboration's result of phase I of the detector is a limit of <math>T_{\beta\beta}^{0\nu}>2.1\cdot 10^{25}</math> years (90% C.L.).<ref name="Tornow2014>{{cite web |last1=Tornow |first1=Werner |title=Search for Neutrinoless Double-Beta Decay |url=https://arxiv.org/abs/1412.0734 |date=1 December 2014}}</ref> It uses [[Germanium]] both as source and detector material.<ref name="Tornow2014" /> Liquid argon is used for [[muon]] vetoing and as a shielding from background radiation.<ref name="Tornow2014" /> The <math>Q</math>-value of Germanium for 0νββ decay is 2039 keV, but no excess of events in this region was found.<ref name="GerdaPhaseI">{{cite journal |last1=Agostini |first1=M. |last2=Allardt |first2=M. |last3=Andreotti |first3=E. |last4=Bakalyarov |first4=A. M. |last5=Balata |first5=M. |last6=Barabanov |first6=I. |last7=Barnabé Heider |first7=M. |last8=Barros |first8=N. |last9=Baudis |first9=L. |last10=Bauer |first10=C. |title=Results on Neutrinoless Double-Beta Decay of 76Ge from Phase I of the GERDA Experiment |journal=Physical Review Letters |date=19 September 2013 |volume=111 |issue=12 |doi=10.1103/PhysRevLett.111.122503}}</ref> Phase II of the experiment started data-taking in 2015, and it uses around 36 kg of Germanium for the detectors.<ref name="GerdaPhaseI" /> The exposure analyzed until July 2020 is 10.8 kg yr. Again, no signal was found and thus a new limit was set to <math>T_{\beta\beta}^{0\nu}>5.3\cdot 10^{25}</math> years (90% C.L.).<ref name="GerdaPhaseII">{{cite journal |last1=Agostini |first1=M |last2=Allardt |first2=M |last3=Bakalyarov |first3=A M |last4=Balata |first4=M |last5=Barabanov |first5=I |last6=Baudis |first6=L |last7=Bauer |first7=C |last8=Bellotti |first8=E |last9=Belogurov |first9=S |last10=Belyaev |first10=S T |last11=Benato |first11=G |title=First results from GERDA Phase II |journal=Journal of Physics: Conference Series |date=September 2017 |volume=888 |pages=012030 |doi=10.1088/1742-6596/888/1/012030}}</ref> The detector is reported as working as expected.<ref name="GerdaPhaseII" />

* ''[[Enriched Xenon Observatory|EXO]] (Enriched Xenon Observatory) experiment'':
** The Enriched Xenon Observatory-200 experiment uses [[Xenon]] both as source and detector.<ref name="Tornow2014" /> The experiment is located in New Mexico (US) and uses a [[time-projection chamber]] (TPC) for three-dimensional spatial and temporal resolution of the electron track depositions.<ref name="Tornow2014" /> The EXO-200 yielded less sensitive results than GERDA I and II with a lifetime limit of <math>T_{\beta\beta}^{0\nu}>1.1\cdot 10^{25}</math> years (90% C.L.).<ref name="Tornow2014" />

* ''[[Kamioka Liquid Scintillator Antineutrino Detector|KamLAND]]-Zen (Kamioka Liquid Scintillator Antineutrino Detector-Zen) experiment'':
** The KamLAND-Zen experiment commenced using 13 tons of Xenon as a source (enriched with about 320 kg of <math>\mathrm{^{136}Xe}</math>), contained in a nylon balloon that is surrounded by a liquid [[Scintillation counter|scintillator]] outer balloon of 13 m diameter.<ref name="Tornow2014" /> Starting in 2011, KamLAND-Zen Phase I started taking data, eventually leading to set a limit on the lifetime for neutrinoless double beta decay of <math>T_{\beta\beta}^{0\nu}>1.9\cdot 10^{25}</math> years (90% C.L.).<ref name="Tornow2014" /> This limit could be improved by combining with Phase II data (data-taking started in December 2013) to <math>T_{\beta\beta}^{0\nu}>2.6\cdot 10^{25}</math> years (90% C.L.).<ref name="Tornow2014" /> For Phase II, the collaboration especially managed to reduce the decay of <math>\mathrm{^{110m}Ag}</math>, which disturbed the measurements in the region of interest for 0νββ decay of <math>\mathrm{^{136}Xe}</math>.<ref name="Tornow2014" /> In August 2018, ''KamLAND-Zen 800'' was completed containing 800 kg of <math>\mathrm{^{136}Xe}</math>.<ref name="KamLAND-Zen">{{cite web |title=KamLAND-ZEN |url=https://www.ipmu.jp/en/research-activities/research-program/kamland |website=Kavli IPMU-カブリ数物連携宇宙研究機構 |accessdate=17 July 2020 |language=en |date=16 May 2014}}</ref> It is reported to be now the biggest and most sensitive experiment in the world to search for neutrinoless double beta decay.<ref name="KamLAND-Zen" /><ref>{{cite news |title=Investigating the Neutrino Mass Scale with the ultra-low background KamLAND-Zen detector |url=https://phys.org/news/2016-08-neutrino-mass-scale-ultra-low-background.html |accessdate=17 July 2020 |work=phys.org |language=en}}</ref>

===Proposed/future experiments===

*''[[Enriched Xenon Observatory#nEXO|nEXO]] experiment:''
**As EXO-200's successor, nEXO is planned to be a ton-scale experiment and part of the next generation of 0νββ experiments.<ref name=nEXO>{{cite journal |last1=C. Licciardi* on behalf of the EXO-200 and nEXO collaborations |title=Recent Results and Status of EXO-200 and the nEXO Experiment |journal=38th International Conference on High Energy Physics (ICHEP2016) - Neutrino Physics |volume=282 |doi=10.22323/1.282.0494}}</ref> The detector material is planned to weigh about 5 t, serving a 1% energy resolution at the <math>Q</math>-value.<ref name=nEXO /> The experiment is planned to deliver a lifetime sensitivity of about <math>T_{\beta\beta}^{0\nu}>9.5\cdot 10^{27}</math> years after 10 years of data-taking.<ref name=nEXO />

== See also ==

* [[Double beta decay]]
* [[Double_beta_decay#Heidelberg-Moscow_Controversy|Heidelberg-Moscow controversy]]

== References ==
{{reflist}}