This paper introduces the neutrinoless double-beta decay (the rarest nuclear weak process) and describes the status of the research for this transition, both from the point of view of theoretical nuclear physics and in terms of the present and future experimental scenarios. Implications of this phenomenon on crucial aspects of particle physics are briefly discussed. The calculations of the nuclear matrix elements in case of mass mechanisms are reviewed, and a range for these quantities is proposed for the most appealing candidates. After introducing general experimental concepts—such as the choice of the best candidates, the different proposed technological approaches, and the sensitivity—we make the point on the experimental situation. Searches running or in preparation are described, providing an organic presentation which picks up similarities and differences. A critical comparison of the adopted technologies and of their physics reach (in terms of sensitivity to the effective Majorana neutrino mass) is performed. As a conclusion, we try to envisage what we expect round the corner and at a longer time scale.
The double-beta decay is the rarest nuclear weak process. It takes place between two even-even isobars, when the decay to the intermediate nucleus is energetically forbidden due to the pairing interaction, which shifts the even-even and the odd-odd mass parabolas in a given isobaric chain; therefore, only due to the pairing interaction can the double-beta decay be observed. This is seen clearly in Figure
Representation of the energies of the
Currently, there is a number of experiments either taking place or expected for the near future—see, for example, [
Another process of interest is the resonant double-electron capture which could have lifetimes competitive with the neutrinoless double-beta decay ones only if there is a degeneracy of the atomic mass of the initial and final states at the eV level [
The main feature of
However, after the discovery of neutrino flavor oscillations (which prove that neutrinos are massive particles), the mass mechanism occupies a special place. It relates neatly the
This crucial parameter contains the three neutrino masses
The starting point for the description of the
In the nonrelativistic case, and discarding energy transfers between nucleons, we have
The parameterization of the couplings by the standard dipole form factor—to take into account the finite nuclear size (FNS)—and the use of the CVC and PCAC hypotheses—for the magnetic and pseudoscalar couplings
Due to the high momentum of the virtual neutrino in the nucleus—
The NME is obtained from the effective transition operator resulting of the product of the nuclear currents:
Till recently, only
Integrating over
Finally, the NME reads
Until very recently, the short-range correlations were taken into account in the calculation of the NME using the Jastrow prescription of [
However, there has been recent proposals [
In summary, there is a broad consensus in the community about the form of the transition operator in the mass mode, which must include the higher-order terms in the nuclear current that we have discussed, and the proper nucleon form factors. The consensus extends to the validity of the closure approximation for the calculation of the NMEs and to the use of soft (or no) short-range corrections. The situation is less clear concerning the use of bare or quenched values of
Once the main issues related to the transition operator are settled, we are left with the purely nuclear ingredient of the neutrinoless double-beta decay NMEs, the wave functions of the initial and final states of the process. Two different methods were traditionally used to calculate the NMEs for
The ISM calculations are performed in different valence spaces and utilize well-tuned effective interactions which make it possible to describe with great accuracy many different observables in many different nuclei. All the details of the modern ISM approach can be found in the review of [
Figure
The neutrinoless double-beta decay; "state-of-the-art" NMEs: QRPA [
The difficulty is to decide upon the merit of the different approaches because of our limited understanding of the physical content of the two-body transition operator (and, indeed, the absence of any experimental anchorage). The situation is very different in the 2
The two-body decay operator can be written in the Fock space representation as follows:
In order to explore the structure of the 0
(Color online) Contributions to the Gamow-Teller matrix element of the 82Se
(Color online) The neutrinoless double-beta decay NMEs as a function of the maximum seniority allowed in the wave functions.
It is clearly seen that truncations in seniority tend to overestimate the value of the NMEs. And this can give us a handle to evaluate the different descriptions in terms of their ability to describe properly the correlations which tend to break the nuclear Cooper pairs. High-seniority components are strongly connected to quadrupole correlations and indeed to nuclear deformation. As an example, we show in Table
Decomposition of the wave function of the ground state of 66Ge according to its seniority components, in percentage, for different values of the deformation
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0.15 | 78 | 20 | 1 | 1 | 0 |
0.20 | 39 | 43 | 7 | 10 | 1 |
0.25 | 20 | 43 | 14 | 20 | 3 |
0.30 | 6 | 32 | 21 | 31 | 10 |
The next finding of this exercise is even more interesting because it gives us another clue on what is relevant in the nuclear wave functions from the NMEs point of view. We have plotted in Figure
66Ge
This behavior of the NMEs with respect to the difference of deformation between parent and grand daughter is common to all the transitions between mirror nuclei that we have studied
In Figure
The QRPA results of Figure
Even if we do not have access to observables that are unambiguously related to the neutrinoless NME, there is a plethora of experimental data which can be used to benchmark the wave functions of the participant nuclei, produced by the different nuclear models. We shall discuss the single
(i) Shell and subshell closures: these are very prominent properties in the nuclear dynamics which should manifest in the NMEs. Indeed they do, because in this case the variations in the seniority structure between the initial and final nuclei are very abrupt, leading to very large cancelations of their NME. This is particularly acute in the decay of 48Ca, which is the only doubly magic nucleus candidate to neutrinoless double-beta decay and the one which has the smallest NME.
In Table
The GT NMEs of the
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3.95 | −3.68 | — | — |
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0.00 | −0.26 | 0.08 | −0.02 |
Among the favored potential emitters, we have also a few cases of semimagic nuclei in which these effects are less dramatic; however, one should be aware of the fact that if a calculation overemphasizes a subshell closure, its NMEs are bound to be too small. This is possibly the situation in some calculations of the decay of 96Zr. Thus, all these spectroscopic issues should be verified with extreme care before trusting a NME.
(ii) Occupation numbers: another piece of information which is very relevant is provided by the analysis of the experimental spectroscopic factors of stripping and pick-up reactions that lead to the extraction of the occupation numbers of the orbits close to the Fermi level. This has been recently done for neutrons and protons in 76Ge and 76Se in a series of very careful experiments in [
(iii) Pair transfer amplitudes: in view of the important cancelations between the contributions to the NMEs coming from the transmutation of pairs of neutrons with
(iv) Energy spectra and electromagnetic transitions: these are data which are traditionally the labels of the nuclear shapes and reflect the degree of multipole collectivity, superfluidity, shell closures, and so forth. We have seen that the difference in structure between the initial and final nuclei is the major reason for the depletion of the NMEs and thus the importance of describing these properties accurately.
It is a well-known fact that in order to explain the experimental transition probabilities of the Gamow-Teller decays, the predictions of any model which does not take into account explicitly the short-range correlations must be affected by a reduction factor. Quenching factors of 0.77 in the
The main QRPA practitioners have had their Scylla and Charybdis with this issue, because when adjusting one of the key parameters in their calculations, the strength of the interaction in the particle-particle channel,
The ISM description of the two neutrino double-beta mode started with the 48Ca decay in the full
The important question is what to do in the neutrinoless case. Contrary to the 2
The question often posed to theorists working in this field is, what are the error bars of your NMEs? Obviously the error bar cannot be of statistical origin because we do not produce models at random. And if we could control the systematic errors, we should have done it already, hence improving our descriptions. That is why we speak of range of values in a very very loose sense. What would be nonsensical is to average the results of the different approaches blindly, without analyzing their respective merits or trends. Each one of the major methods has some advantages and drawbacks, whose effect in the values of the NME can be sometimes explored. The clear advantage of the ISM calculations is their full treatment of the nuclear correlations, while their drawback is that they may underestimate the NMEs due to the limited number of orbits in the affordable valence spaces. It has been estimated [
Lets start with the 150Nd case, for which no ISM value is available. The GCM calculation [
Proposed ranges of the NME values for some selected decays (see text).
In the standard interpretation of neutrinoless double-beta decay in terms of mass mechanism, experimentalists designing a neutrinoless double-beta decay experiment have three hurdles to leap over in front of them. The first consists in scrutinizing the much debated 76Ge claim [
First, we have to quantify in terms of signal and background rates the challenges that the experimentalists have to cope with. Since we do not want to be precise here, but just to assess orders of magnitude, we will make crude approximations in the formula of (
Signal rates for an “average” double-beta decay candidate.
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Signal rate |
Significance of |
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300 | ~70 |
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50 | ~2 | Higher bound of the inverted hierarchy region |
20 | ~0.2 | Lower bound of the inverted hierarchy region |
3 | ~7 × 10−3 | Center of the direct hierarchy region |
Considering that 1 kmol corresponds typically to several tens—one hundred kilograms of isotope mass, and that it is meaningful to operate a well designed
In addition, in order to appreciate such tiny signal rates, the background needs to be extremely low. The experimentalists are obliged to operate in conditions of almost zero background, given the constraints imposed by the size of the source. Acceptable background rates are of the order of 1–10 counts/(y kmol) if the goal is just to approach or touch the inverted hierarchy region, whereas one needs at least one order of magnitude lower values to explore it fully, around or even less than 1 count/(y ton).
Which are the best isotopes to search for neutrinoless double-beta decay? Experimental practice shows that the following three factors weight the most in the design of an experiment: the the isotopic abundance together with the ease of enrichment, the compatibility with an appropriate detection technique.
The
Relevant parameters and features of the “magnificent nine” double-beta decay candidates.
Double-beta candidate |
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Phase space |
Isotopic abundance |
Enrichable by centrifugation | Indicative cost normalized to Ge |
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4.27226 (404) |
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0.187 | No | — |
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2.03904 (16) |
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7.8 | Yes | 1 |
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2.99512 (201) |
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9.2 | Yes | 1 |
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3.35037 (289) |
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2.8 | No | — |
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3.03440 (17) |
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9.6 | Yes | 1 |
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2.81350 (13) |
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7.5 | Yes | 3 |
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2.52697 (23) |
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33.8 | Yes | 0.2 |
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2.45783 (37) |
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8.9 | Yes | 0.1 |
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3.37138 (20) |
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5.6 | No | — |
Double-beta decay candidates and their
Phase space of the nine more favourable double-beta decay isotopes (values taken from [
As for the second criterion, natural isotopic abundances are reported in Table
Existing methods for isotope separation. The technologies relevant for neutrinoless double-beta decay are indicated in the fourth and in the three last lines.
Method of separation | Energy |
Status | Production capacity | Scale of price | Special requirements |
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Electromagnetic | 106–107 | Commercial | ~100 g/y | High | — |
Gas diffusion |
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Industrial | >tons/y | Medium | Gas compound |
Gas nozzle |
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Industrial | >tons/y | Medium | Gas compound |
Gas centrifuge |
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Industrial | >tons/y | Low | Gas compound |
Rectification |
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Industrial | >tons/y | Low | Light elements |
Isotope exchange |
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Industrial | >tons/y | Low | Light elements |
Ion cyclotron resonance |
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R&D | ~100 kg/y | Medium | — |
Atomic vapor laser I.S. |
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R&D | >100 kg/y | Medium | — |
Molecular laser I.S. |
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R&D | >100 kg/y | Medium | — |
The role of the third criterion will become more clear in the following sections, where specific detection technologies will be described. We would like however to discuss here three special emblematic cases in which the detector principle matches favorably with the isotope to study. 76Ge large volume, high-purity, and high-energy resolution Ge-diodes are currently employed in gamma spectroscopy. A detector of this type containing germanium enriched in 76Ge is almost ideal for double-beta decay search. This explains why past (Heidelberg-Moscow and IGEX) and present (GERDA and Majorana) experiments were and are at the forefront in the field, in spite of the relatively low 130Te large crystals (up to 1 kg) of the compound TeO2 can be grown with high radiopurity. They can be used for the realization of bolometers with excellent performance. Given also the high natural isotopic abundance of 130Te, it is understandable why a past experiment like Cuoricino has been leading the field for several years, and why CUORE is one of the most promising future searches (both are based on arrays of TeO2 bolometers). 136Xe liquid and gaseous xenon is an ideal medium for particle detection. It can be used to equip TPCs with tracking/topology capability. Scintillation and ionization can provide reasonable energy resolution. This approach is exploited in experiments like EXO (now leading the field) and NEXT. In addition, xenon can be easily dissolved in organic liquid scintillators, allowing to reach very large masses exploiting existing facilities (this is the case of KamLAND-Zen). Last but not least, xenon is the element that can be isotopically enriched at the lowest prices and with the highest production capacity.
For the usual conspiracy of Nature, the three mentioned isotopes are the less favorable among the “magnificent nine” in terms of
From the experimental point of view, the shape of the two-electron sum energy spectrum enables to distinguish among the two discussed decay modes. In case of
(a) Distribution of the sum of the two electron energies for
The experimental strategy pursued to investigate the High-energy resolution, since a peak must be identified over an almost flat background in case of where Low background, which requires underground detector operation (to shield cosmic rays), very radiopure materials (the competing natural radioactivity decays have typical lifetimes of the order of Large source, in order to monitor many candidate nuclides. Present sources are of the order of 10–100 kg in the most sensitive detectors, while experiments capable to cover the inverted hierarchy region need sources in the 100–1000 kg scale. Tracking and topology capability for the nuclear events, useful to reject background and to provide additional kinematical information on the emitted electrons.
Normally, the listed features cannot be met simultaneously in a single detection method. It is up to the experimentalist to choose the philosophy of the experiment and to select consequently the detector characteristics, privileging some properties with respect to others, having in mind of course the final sensitivity of the setup to half-life and to
The searches for
The calorimetric technique has been proposed and implemented with various types of detectors, such as scintillators, bolometers [ due to the intrinsically high efficiency of the method, large source masses are possible: with a proper choice of the detector type, a very high energy resolution (of the order of 0.1%) is achievable, as in Ge-diodes or in bolometers; there are severe constraints on detector material and therefore on the nuclides that can be investigated; it is difficult to reconstruct event topology, with the exception of liquid or gaseous Xe TPC, but at the price of a lower energy resolution.
For the external-source approach A neat event reconstruction is possible, making easier the achievement of a virtual zero background: however, Large source masses are not easy to achieve because of self-absorption in the source, so that the present limit is around 10 kg; 100 kg is possible with an extraordinary effort, while 1000 kg looks out of the reach of this approach. Normally the energy resolution is low (of the order of 10%), intrinsically limited by the fluctuations of the energy that the electrons deposit in the source itself. Efficiency is also low (in prospect of the order of 30%).
In order to compare different experiments, it is useful to give an expression providing the sensitivity of an experimental setup to the
From this formula, one can see that in order to improve the performance of a given set-up, one can use either brute force (e.g., increasing the exposition
In order to derive the sensitivity to
The formula reported in (
Nowadays, several experimental techniques promise to realize zero background investigations in the close future. In this circumstance, (
Uncertainties coming from NMEs prevent from determining precise
We are now (July 2012) at a turning point in the experimental search for
In the nineties of the last century, the double-beta decay scene was dominated by the Heidelberg-Moscow (HM) experiment [
The top level of the external-source technique was reached nowadays by the NEMO3 experiment [
Bolometric detection of particles [
In Section is the selected technology able to deal with 100 kg or better 1 ton of isotope, at least in prospect? is the choice of the detector and of the related materials compatible with a background of the order of at most 1 count/(y ton) in the region of interest? can the experiment be designed and constructed in a few years, and can the chosen technique provide at least 80% live time for several years?
The first question needs to be considered also from the economical point of view. As Table
As already discussed in Section
Experiments reviewed in the text are divided into five categories, according to the experimental approach and the main features of the detector performance. Running experiments are written in boldface fonts.
This list of ten projects do not cover the full range of existing
The first category is characterized by a calorimetric approach with high energy resolution, with four planned projects.
GERDA [
MAJORANA [
CUORE [
LUCIFER [
Even though these experiments do not have tracking capability, some spatial information and other tools help in reducing the background. An important asset is granularity, which is a major point for CUORE (array of 988 closely packed individual bolometers), MAJORANA (in prospect a set of modules with 57 closely packed individual Ge diodes per module), and the lower energy resolution experiment COBRA [
Another tool which can improve the sensitivity of Ge-based calorimetric searches is pulse shape analysis, already used in the HM experiment with remarkable results. It is well known that in ionization detectors one can achieve spatial information looking at the pulse shape of the current pulse. In particular, this fact will be exploited in GERDA using the so-called BEGe detectors [
Other techniques to suppress background in calorimetric detectors are sophisticated forms of active shielding. For instance, the operation of the GERDA Ge diodes in liquid argon opens the way, in the second phase of the experiment, to the use of the cryogenic liquid as a scintillating active shield. In bolometers, it was clearly shown that additional bolometric elements thermally connected to the main detector in the form of thin slabs can identify events due to surface contamination [
A very promising development of the calorimetric approach realized by means of low-temperature detectors consists in the realization of scintillating bolometers [
The second category of future experiments (calorimetric search with low energy resolution and no tracking capability) is represented by two samples which exploit different techniques and solve the low-energy-resolution problem with diverse measures.
KamLAND-Zen [
SNO+ [
The third category includes an ambitious calorimetric experiment aiming at joining high energy resolution with tracking/topology capability.
NEXT [
The fourth category comprises calorimetric experiments based on detectors which compensate the low energy resolution with tracking or some form of event-topology capability. There are two samples in this group.
EXO [
COBRA [
The fifth category is represented by setups with external source (which necessarily leads to low energy resolution) and sophisticated tracking capability, allowing to reach virtually zero background in the relevant energy region (with the exception of the contribution from the
SuperNEMO [
As it is clear from the above discussion and from the experiment description, the three essential ingredients for a sensitive
Figure
Comparison of technologies/experiments on the basis of the absolute background level they have achieved (green and blue points) or promise to achieve (red points), disentangling the role of the energy resolution
The points in Figure
Of course, referring only to the specific background, the plot in Figure
Every approach has its good reasons, as one can see in Figure
Sensitivity range to the effective Majorana neutrino mass for a set of relevant
We discuss here the future prospects for
The future scenario of GERDA will detect EXO-200 will detect CUORE will detect SNO+ will detect LUCIFER could detect SuperNEMO may investigate the mechanism looking at the single-electron energy spectrum and at the electron angular distribution in 82Se or in 150Nd.
The redundancy of the candidates with positive observation will help in reducing the uncertainties coming from nuclear matrix element calculation: we would enter the precision measurement era for
In case of inverted hierarchy pattern, that is, CUORE could detect nEXO, the extension if EXO-200 under discussion, could detect Extensions of KamLAND-Zen, of course after the solution of the present background problems, and of NEXT, if the first phase is successful, can also have the chance to observe GERDA phase III, after merging with MAJORANA, could detect it in 76Ge. SuperNEMO could marginally detect it if 150Nd mode will result at the end possible. SNO+ could detect it in 150Nd if Nd enrichment is viable.
The discovery in 3 or 4 isotopes is necessary for a convincing evidence, and it would be still possible thanks to the variety of projects and techniques under development. A nonobservation could be very important for neutrino physics as well. In fact, if
In case of direct hierarchy pattern, that is,
This work was partially supported by the MICINN (Spain) (FPA2011-29854); by the Comunidad de Madrid (Spain) (HEPHACOS S2009-ESP-1473); by the Spanish Consolider-Ingenio 2010 Program, CPAN (CSD2007-00042).