The simultaneous beta decay of two neutrons in a nucleus without the emission of neutrinos (called neutrinoless double beta decay) is a lepton number violating process which is not allowed in the Standard Model of particle physics. More than a dozen experiments using different candidate isotopes and a variety of detection techniques are searching for this decay. Some (EXO-200, Kamland-Zen, and GERDA) started to take data recently. EXO and Kamland-Zen have reported first limits of the half life for 136Xe. After a decade of little progress in this field, these results start to scrutinize the claim from part of the Heidelberg–Moscow collaboration to have observed this decay. The sensitivities of the different proposals are reviewed.
For 35 isotopes, β decay is energetically forbidden but double beta decay () is allowed:
This process has been observed directly for 11 isotopes with half lives between 7 · 1018 yr and 2 · 1021 yr [1-3].
Since neutrinos have no electric charge, there is no known symmetry which forbids that they mix with their anti-particles. As a consequence, almost all extensions of the Standard Model of particle physics predict that double beta decay without neutrino emission (0) might occur as well (see e.g. [4, 5]). Its observation would imply that lepton number is violated, since only electrons are emitted.
The experimental signature of is a line at the value of the decay if the sum of the electron energies is histogrammed.
Part of the Heidelberg–Moscow collaboration claims to have evidence for this line for 76Ge with yr  at a 4σ level. In a more sophisticated analysis, a significance beyond 6σ was calculated . The current experiments using 136Xe (EXO-200 and Kamland-Zen) and 76Ge (GERDA) will scrutinize this result within the next 12 months.
In total more than a dozen large-scale experimental programs are suggested or under construction to search for . These programs are compared in this article and also the status of theoretical matrix element calculations is discussed. The latter are needed to convert an experimental measurement or limit on to a particle physics parameter. For general reviews, the reader is referred to the literature [4, 5, 8, 9].
There are other related processes like double positron decay or double electron capture. While is already a suppressed process, the other decays are expected to be even rarer unless there is some resonance enhancement [5, 10-13]. In this article only decay searches are discussed.
2 Motivation for
Since neutrinos have mass but no electric charge, there is no known symmetry that forbids additional terms in the effective Lagrangian besides the standard Dirac mass term [4, 5, 8, 14]:
The subscript L stands for the left-handed chiral field and R for the right-handed projection . The superscript C denotes charge conjugation, i.e. stands for an (incoming) anti-neutrino. Therefore, the term describes an incoming neutrino and an outgoing anti-neutrino , i.e. this term violates lepton number by two units. The eigenstates of the mass matrix are of the form . Consequently, neutrinos are expected to be – in general – their own anti-particles, i.e. Majorana particles.
Neutrinos (or anti-neutrinos) are produced in charged weak current reactions and – depending on the charge of the associated lepton – only one chiral projection couples (“V–A” current of weak interactions). For example, in β decay , a right-handed anti-neutrino couples that can be decomposed into the mass eigenstates and helicity () eigenstates:
Here U is the PMNS mixing matrix , are the mass eigenvalues, and E is the neutrino energy.
For Dirac particles, only detection reactions like are possible. If, on the other hand, neutrinos are massive Majorana particles, then the helicity suppressed component of can undergo the reaction . Here
Taking both processes involving neutrons together, we have (or better ) mediated by a massive Majorana neutrino with an effective coupling strength which is called the Majorana mass:
The helicity suppression , which is e.g. 10−14 for a neutrino mass of 0.1 eV and a neutrino energy of 1 MeV, is compensated by the large number of nuclei per mole.
Schechter and Valle have shown that the observation of ensures that neutrinos have a Majorana component . Recently, it was pointed out that the “guaranteed” Majorana mass through radiative corrections is, however, only in the range of 10−24 eV , i.e. negligible compared to the mass scales of neutrino oscillations . Consequently, if is observed, other mechanisms like the exchange of supersymmetric particles or heavy Majorana neutrinos might be the dominating process and the mediating neutrino could even be (effectively) a Dirac particle. While for the initially motivated light neutrino exchange the coupling strength is proportional to (Eq. (6)), for processes with heavy-fermion exchange it is proportional to with being the mass of the exchange particle (provided that the helicities of the two leptonic currents are the same and ).1
In such processes, lepton-number violation or lepton-flavor violation can be accessible with accelerator experiments as well . If lower limits on (from e.g. LHC) are higher than typically 10 TeV, then the contributions to become smaller than those expected from light neutrino exchange with a mass of 0.05 eV . The argument can be turned around: will also provide information on TeV-scale physics.
The exchange of light neutrinos is discussed predominantly. Its strength depends on U, which can be parameterized by three rotation angles and one phase (θ12, θ13, θ23, and δ, all measurable by neutrino oscillation experiments), and two additional phases (α21 and α31). The latter are called Majorana phases and influence processes like . Oscillation experiments measure difference of squared masses ( eV2 and eV2 ). Knowledge of the absolute mass scale comes from e.g. beta decay, , or cosmology.
One can estimate possible values of as a function of the lightest neutrino mass given the experimental knowledge about and the measured rotation angles and allowing the Majorana phases to take any value. Three cases can be discriminated .
‒ If the lightest mass is larger than all , all are similar (degenerate masses).
‒ If m1 is the smallest mass (normal mass hierarchy), the three terms of can cancel for . For typical values are a few meV.
‒ If m3 is the smallest mass (inverted hierarchy, ), there exists for a lower bound of meV and an upper bound of 50 meV .
If additional sterile neutrinos exist, i.e. if there are additional terms in Eq. (6), then even for the inverted hierarchy no lower bound exists , .
In summary, – together with other inputs from e.g. neutrino oscillation and LHC experiments – provides an important window to extensions of the Standard Model with lepton-number violation.
3 Experimental sensitivity on
All (but one) searches have so far only observed event counts in the region of interest around that are consistent with the expectation from background λbkg. This number – if it scales with the detector mass M – is given by
Here t is the measurement time, B is the so-called background index given typically in cnts/(keV·kg·yr), and is the width of the search window which depends on the experimental energy resolution. Note that this equation is only an approximation. Experiments normally take the (nonlinear) shape of the background spectrum in a fit into account.
The non-observation is converted to an upper limit on the number of signal events λsig, which is related to the half life of a given isotope A by
Here NAvg is the Avogadro constant, ε the signal detection efficiency, η the mass fraction of the isotope, and the molar mass of the isotope.
If , the experimental sensitivity scales with , whereas for the e.g. 90% C.L. limit on the half life (assuming there is no signal) is given by
If systematic errors become important, e.g. if the energy resolution or the spectral background shape is not well known, then the sensitivity is reduced.
4 The nuclear matrix element
is the product of three factors: a phase space factor , a nuclear matrix element , and a particle physics factor.
If the amplitude for the exchange of light Majorana neutrinos is dominating , for a given isotope A is 
Here is the electron mass. Values for are listed in Table 1. Obviously, is needed to compare results from different isotopes or to extract information about the particle physics parameters.
Table 1. List of phase space factor (Eq. (10)), , natural abundance of isotope, half life of , and experiments for the most interesting isotopes. Half lives for are taken from , , for 136Xe from , and all other numbers from . The values have been scaled to the same nuclear radius and coupling
GERDA, Majorana Demonstr.
EXO, Next, Kamland-Zen
The calculations are difficult and can only be done using approximations. Traditionally, interacting shell model (ISM) [24, 25] and quasi-particle random phase approximation (QRPA) calculations have been performed [26-28]. Recently, new approaches like the interacting boson model (IBM) [29-31], the generating coordinate model (GCM, also called energy density functional EDF) , and the projected Hartree–Fock–Bogoliubov (pHFB) method  have been applied.
The results of these calculations are shown in Fig. 1. The following statements can be made concerning the status.
‒ varies slowly with A. This might be due to the fact that only neighboring neutrons in a nucleus contribute to the decay [24, 27].
‒ For the ISM, all values are systematically lower than for other methods. Possible reasons for this effect are discussed in the literature [24, 34].
‒ For a given isotope the calculations spread by typically a factor of 2–3, i.e. a factor of 4–9 for . Some groups estimate a possible range for (see indicated errors in Fig. 1). However, it is unclear what to quote as confidence interval for theoretical calculations.
‒ The role of short-range correlations has been studied and the UCOM correction has emerged as favorable . Alternatively, a self-consistent implementation was first applied to SRQRPA  and later to other methods ,  and resulted in small changes.
‒ Experimental input can cause a sizable shift of the result. For example, charge exchange reaction measurements of 150Nd(3He,t) and 150Sm(t,3He)  result in a quenching factor of 0.75 for the coupling and hence a reduction of the matrix element by 25% for 150Nd . In this calculation, deformation was treated for the first time in a QRPA calculation.
‒ For 76Ge and 76Se, the proton and neutron valence orbital occupancies have been measured [39, 40]. If the models are adjusted to reproduce these values, the ISM result increases by 15%  while the QRPA results are reduced by about 20% [41, 42]. Hence, the difference between ISM and QRPA becomes half as large.
In order to see whether some isotopes are better suited for searches from a theoretical point of view, the number of expected decays for a given exposure (in units of kg·yr) can be compared using Eqs. (8) and (10) without the factors ε and η. The A-dependent parameters are then the phase space factor, the matrix element, and the molar mass. For the comparison it is sufficient to look at the ratio of decay rates and, in this case, some of the systematic effects of the matrix element calculations cancel since there are typically correlations among the isotopes for a given method.2 Fig. 2 shows these ratios for the different models normalized to the decay rate of 130Te. One sees that 76Ge is less favorable. The expected decays per kg vary between 20% and 50% of the rate of 130Te. In other words: if all experimental parameters (number of background events, efficiency, etc.) were the same, then one would need a factor of 2–5 more target mass in a 76Ge experiment to have the same sensitivity. In reality, all experimental parameters like the energy resolution and the background have to be considered as well.
5 Current experimental situation
In the last decade, mainly three experiments contributed to searches. Of these, Heidelberg–Moscow was the most sensitive one and reported evidence for this decay. The two others were Cuoricino and NEMO-3. Previously, IGEX reported a limit of yr for 76Ge (90% C.L.) with an exposure of 8.8 kg·yr . Currently EXO-200, Kamland-Zen, and GERDA are taking data. These experiments are discussed in this section.
5.1 The Heidelberg–Moscow experiment
The Heidelberg–Moscow experiment operated between 1990 and 2003 five germanium detectors made out of isotopically enriched material (≃86% 76Ge, 11 kg). The diodes were mounted in copper cryostats with copper, lead, and polyethylene shielding. The total exposure was 71.7 kg·yr and the average count rate in the interval 2–2.1 MeV was about 0.17 cnts/(keV·kg·yr) (for the period 1995–2003). The energy resolution (full width at half maximum, FWHM) was about 3.5 keV at , which is the best value of all experiments. Part of the collaboration found evidence for a peak at with 28.75 ± 6.86 events, which converts to yr . Note that only a statistical error is quoted. Another study finds that extending the energy window used in the data fit increases this background and hence decreases the signal count by up to 40% (Tables 3.8 and 4.6 of ).
In a later publication , the claim was strengthened by a pulse shape analysis that preferentially selected events due to their localized energy deposition in the detectors. Backgrounds from gammas with multiple Compton scatterings exhibit different pulse shapes. The background is reduced to a surprisingly low level of ≈0.015 cnts/(keV·kg·yr). The final fit reports a yield of 11.32±1.75 signal events and the ratio is called the significance of the peak (Fig. 9b in ). The signal yield was then converted to yr. There are several problems with this analysis.
- The fit error on the signal count is too small. The smallest 68% Poisson credibility interval is between 8.1 and 15.2 for a probability distribution which peaks at 11.3, i.e. a factor of two larger than the quoted interval. Due to the existing (small) background, the interval should become even larger.
- The probability that the background (≃ 2.2 events in the central 3 keV of the peak) fluctuates to the observed number of 13 events or more is 5, which converts to a significance of about 5σ. Systematic effects like the uncertainty of the background might reduce this value.
- In the conversion to using Eq. (8), an efficiency ε of 100% is used although no value is explicitly quoted. All but three events in the peak are part of an earlier selection (labelled “HNR+NN” in ). For the latter, the efficiency was 62% . Hence, one also expects for this analysis a value much smaller than 100%.
The central value and the errors are consequently not correct in  and the significance is smaller than quoted although still high.
5.2 The Cuoricino experiment
Cuoricino  operated 62 TeO2 crystals with a total mass of 40.7 kg (11.3 kg of 130Te) between 2003 and 2008. At a temperature of ⩽10 mK, the heat capacitance is very low and an energy deposition inside a crystal results in an increase of typically 0.1 mK/MeV, which is measured with neutron transmutation doped germanium thermistors. The latter have a resistance of 100 MΩ and show a strong temperature dependence which converts to a dependence of 3 MΩ/MeV. This bolometric technique has been proven to work with a good resolution FWHM of typically 6–10 keV at 2.6 MeV.
The total 130Te exposure was 19.75 kg·yr. The background at was 0.17 cnts/(keV·kg·yr) if normalized to the total mass, i.e. similar to Heidelberg–Moscow. No signal was found and a lower limit of yr (at 90% C.L.) was set for 130Te. This limit is not sensitive enough to scrutinize the Heidelberg–Moscow result.
5.3 The NEMO-3 experiment
In NEMO-3 , thin foils made out of seven different isotopes (9 kg in total) were located in a drift chamber with a magnetic field. Outside the drift region was a calorimeter made out of plastic scintillator blocks with photomultiplier tube (PMT) readout (FWHM for electrons 15%/). The reconstruction efficiency is only about 8%. On the other hand, the topological event reconstruction largely reduces backgrounds from locations other than the source foil as well as internal decays with gammas or alphas. Only events with poor energy reconstruction cannot be discriminated. The background is about cnts/(keV·kg·yr) at 3 MeV ( of 100Mo and 82Se). Half lives for are reported for each isotope with impressive signal to background ratios of up to 76. For decay, the 90% C.L. limits are yr for 100Mo and yr for 82Se. Again, NEMO-3 is not sensitive enough to confirm or reject the Heidelberg–Moscow claim.
5.4 The Kamland-Zen experiment
In Kamland-Zen , a 25 μm thick nylon balloon with a 1.54 m radius is inserted into the Kamland detector and filled with xenon-doped scintillator (≃290 kg 136Xe). The energy (FWHM ≃10% at ) and position of the decay (resolution ) are reconstructed with PMTs located at a radius of 9 m which cover 34% of the solid angle.
Since the scintillator is very pure and can be doped easily with several percent of xenon, this experiment has the largest target mass and lowest background – if normalized to the total mass.3 However, the energy resolution is the poorest.
Kamland-Zen started data taking in 2011. An unexpected background peak at about 2.6 MeV is dominating the spectrum around MeV. It limits the experimental sensitivity and the best explanation for the origin is 110mAg ( days) cosmogenically produced in 136Xe or originating from a contamination on the balloon. This background is expected to be reduced to a negligible level by an ongoing scintillator purification campaign.
After an exposure of 78 days the first published results are yr (at 90% C.L.) and yr . As for Cuoricino and NEMO-3, this limit is not sensitive enough to scrutinize Heidelberg–Moscow.
The situation is different for the new measurement  of yr (at 90% C.L.), which is based on a 136Xe exposure of 89.5 kg·yr. The corresponding upper limit on the number of signal events is 15. The claim of Heidelberg–Moscow  can be converted to an expected signal count between 25 ± 6 and 125 ± 30 for the different matrix element calculations (see following section). Clearly, the claim is strongly ruled out. For a quantitative analysis, the signal probability distribution of Kamland-Zen must be known.
An additional 700 kg of Xe with 90% enrichment is available by the end of 2012 and is expected to be deployed in a cleaner balloon in the near future. Kamland-Zen is therefore expected to be the first experiment with ton-scale isotope mass. In the more distant future, it is foreseen to improve the energy reconstruction with light collectors attached to the PMTs and a new liquid scintillator .
5.5 The EXO-200 experiment
EXO-200  operates a liquid xenon TPC of 40 cm diameter and 40 cm length (175 kg of liquid Xe, 100 kg fiducial mass). A wire plane in the middle is biased to –8 kV such that electrons drift to one of the two ends where the ( position of the electron cloud is reconstructed with two planes of wires and the total charge is measured. The produced scintillation light is detected with large-area avalanche photodiodes behind the charge-collecting wires.
Since there is a strong anti-correlation between the light amplitude and the ionization signal, combining both into a new quantity greatly improves the energy resolution to FWHM ≈3.9% at . The background is at a quite low level of cnts/(keV·kg·yr) after a geometrical cut on the electron cloud is applied to discriminate Compton-scattered photon events.
EXO has recently published results on the half life yr and a limit on the half life yr (at 90% C.L.) of 136Xe .
This result can be used to test the claim of Heidelberg–Moscow. For a given matrix element calculation, the expected number of signal events for EXO can be estimated from Eqs. (8) and (10). The experimental inputs from EXO (32.5 kg·yr exposure, 55% reconstruction efficiency,4 80.6% enrichment) and Heidelberg–Moscow (28.75±6.86 events, 71.7 kg·yr exposure, efficiency 100%, enrichment 86%) enter the calculation. EXO-200 reports () events in an energy window of ().
A Bayesian formulation can be applied with a null hypothesis (H: EXO observes only background) and an alternative hypothesis (: EXO observes a signal with scaled Heidelberg–Moscow event counts). The prior distributions for the expected number of signal events and for background events are Gaussian. The prior probabilities for both hypotheses are set to . Mean and sigma of are listed in Table 2. Mean and sigma of the background are 4.1 ± 0.3 (7.5 ± 0.7) events in the () energy window.
Table 2. Bayesian posterior probabilities using EXO-200 data for the hypothesis that the signal of Heidelberg–Moscow is correct. Probabilities are given for different matrix element calculations and for the and energy windows
4.4 ± 1.1
6.1 ± 1.5
2.8 ± 0.7
3.9 ± 0.9
6.3 ± 1.5
8.8 ± 2.1
6.1 ± 1.5
8.6 ± 2.1
The posterior probability is then
Here is the Poisson function for mean y and nobs observed events. The results for are listed in Table 2. For the Tübingen–Bratislava calculations (labelled “QRPA max” and “QRPA min”), the maximum and minimum of 20 ratios are calculated from the values of Table 1 of . This procedure was suggested by one of the authors of that reference . For the other models, the ratios of the central values shown in Fig. 1 are taken. None of the results strengthen the hypothesis of a signal. The ISM and GCM calculations strongly disfavor them while the exclusions for the Tübingen–Bratislava calculations are not very strong. The reported EXO-200 data were taken within a period of 7 months. Hence, more stringent statements are expected soon.
EXO-200 is approved to run for four more years. For a following phase, the spectroscopic identification of the daughter nucleus of Xe is foreseen. In this case the experiment will be background free. Only events with poorly reconstructed energy can obscure a signal.
5.6 The GERDA experiment
GERDA uses the germanium detectors of Heidelberg–Moscow and IGEX and – in a second phase – new ones. The emitter mass is about 13 kg in the first phase. The detectors are supported by a minimal amount of material with low radioactivity in a 4 m diameter cryostat filled with liquid argon. Argon serves as cooling medium and shield against external radioactivity. The latter is complemented by 3 m of water that is instrumented with PMTs to veto background from muons by the detection of their Cherenkov light.
GERDA started commissioning in 2010 and found an unexpectedly large background from 42Ar that could be reduced by avoiding electrical fields around the detectors and by an encapsulation of the diodes. The first phase of data taking is ongoing since November 2011. The background is at the level of 0.02 cnts/(keV·kg·yr) and hence almost an order of magnitude smaller than the equivalent number of Heidelberg–Moscow. Due to a data blinding procedure, no result on is expected before spring 2013. A preliminary result for of yr is reported .
A second phase is scheduled to start in 2013 with an additional new set of detectors (≃18 kg of 76Ge). The background is expected to be reduced from the current 0.02 cnts/(keV·kg·yr) to 0.001 cnts/(keV·kg·yr) due to a liquid argon instrumentation and a different detector design with enhanced pulse shape discrimination power.
6 Other future experiments
The past and running experiments have been discussed above, but others are under construction or pursuing R&D efforts. They use additional isotopes and various other detection mechanisms and background reduction methods; see Table 3.
One important experimental parameter is the fraction η of the isotope (see Eq. (9) and Table 1). 130Te is the only one with a large natural abundance. For all other elements, enrichment of the isotope is mandatory. 136Xe is easiest to enrich with gas centrifuges since this is the heaviest isotope and it is already a gas. Other materials like 76Ge can be converted to a gas (GeF4) and then processed. For calcium and neodymium, this path is currently not available and R&D on alternative methods is ongoing.
Another important number is . Larger values are not only better because of larger but also because the background from natural decay chains falls off fast beyond 2.6 MeV. Experiments using 76Ge have to compensate by careful material selection and good energy resolution.
The experiments can be grouped into two classes. In calorimetric experiments only the total energy (ionization or scintillation) is measured. In tracking experiments the two electrons are measured independently, i.e. the angular distribution between the electrons is also known. The latter is interesting to study the origin of the underlying physics in case is observed.
In the following, the experiments under construction are discussed.
CUORE is a continuation of Cuoricino with close to 1000 TeO2 crystals of 750 g each (in total ≃ 200 kg of 130Te). The crystal production is almost finished and all major hardware items are ready or nearly ready. A first tower with 52 crystals (CUORE-0) has been assembled and data taking in the Cuoricino cryostat was scheduled for July 2012 .
After a commissioning phase, CUORE is expected to start in 2015.
Majorana will operate similar germanium diodes like GERDA (≃ 27 kg of 76Ge) in vacuum in a compact cryostat made out of electroformed copper. This self-made copper is expected to have a factor >100 less thorium, uranium, and radium contaminations compared to commercial copper such that the background index is about 0.001 cnts/(keV·kg·yr). The shielding is completed by commercial copper, lead, and polyethylene shells. Operations should start in 2013.
In a later phase, a combined GERDA and Majorana germanium experiment with order ton-scale mass is envisioned.
CANDLES operates 96 scintillating CaF2 crystals (0.3 kg of 48Ca) in a liquid scintillator. Both are in an acrylic container inside a water tank with PMTs. Data taking started in 2011 but the sensitivity is limited due to the small target mass. Successful R&D on 48Ca enrichment is crucial for this approach.
NEXT is a high pressure xenon gas TPC (1.1 m diameter, 1.4 m length, pressure 10–15 bar, mass 90–130 kg 136Xe). The time of the decay is determined by the detection of the (primary) scintillation light with PMTs. The deposited energy and event topology are reconstructed from the ionization signal. The drifting electron cloud passes a volume of higher electric field at the end, such that the electrons are moderately accelerated. Consequently, they can excite xenon but not ionize it. The resulting (secondary) scintillation light (electroluminescence) is proportional to the number of electrons and detected with the same PMTs (for energy reconstruction) and with a plane of silicon photomultipliers (for position reconstruction). The energy resolution FWHM is expected to be <1% at and the event topology allows one to reject backgrounds very effectively to a level of cnts/(keV·kg·yr).
Construction is expected to start in 2013 and physics data taking in 2015.
SuperNEMO is a planned continuation of NEMO-3 with much improved performance (factor four in reconstruction efficiency, factor two in energy resolution, factor six in background). A demonstrator module with 7 kg of 82Se is under construction and expected to start data taking in 2014.
SNO+ dissolves ≈1 ton natNd in 780 tons of liquid scintillator (44 kg of 150Nd). The scintillator is in a 12 m diameter acrylic vessel that is surrounded by pure water in an 18 m diameter water tank. The latter holds the 9500 8-in. PMTs for light detection. An energy resolution FWHM of ≈7% at is expected with an extremely low background of cnts/(keV·kg·yr) if normalized to the total scintillator mass. The main background is due to the poor energy resolution and the relatively short half life of 150Nd (see Table 1). Scintillator filling and doping with Nd is expected for 2013. Since 150Nd cannot be enriched effectively at the moment, η is small, which limits the SNO+ sensitivity.
Lucifer is an R&D effort which investigates scintillating crystals with emitters like ZnSe that are operated as bolometers. The simultaneous detection of phonons and photons allows one to identify backgrounds from e.g. surface events that are expected to dominate in CUORE.
Other ongoing R&D efforts are not discussed here. References are listed in Table 3.
For a comparison of the sensitivities of the experiments, a relative scaling factor for the different matrix elements and phase spaces has to be applied. This factor can be estimated from Fig. 2. The values used here are (0.2–0.5), (0.7–1.4), (1.1–3.2), , (0.4–0.7), and (0.4–2.0). The numbers in parentheses give the full range.
If the number of background events is large, Eq. (9) can be used to estimate the experimental sensitivity. A relative figure-of-merit (FOM) can then be defined as
The relative sensitivity scales with the live time t of an experiment like FOM. Table 4 lists the performance parameters of the experiments discussed above. “Kamland-Zen2” is the improved experiment after the purification of the scintillator (assumed factor five smaller total background) and the upgrade to one ton xenon mass. For comparison, FOM, the expected 90% C.L. limits for 4 yr live time, and the corresponding limits are given. For the latter, the entire spread of the matrix elements of Fig. 1 including the error bars are used.
Table 4. Comparison of relative figure-of-merit (FOM), lower half life limit after 4 yr live time, and resulting upper limit on . For , the entire range of matrix element values including the indicated error bars in Fig. 1 are used. is the average scale factor for a given isotope taken from Fig. 2. is the energy window, which is taken to be 1 (2) FWHM for experiments with () resolution. Note that the efficiency is reduced by 0.7 if 1·FHWM. FOM is defined in the text. Masses are total masses or fiducial masses. The background and enrichment fraction have to be scaled accordingly
Experiments under construction
For running (and past) experiments, the achieved performance values are used, which might improve with time. For the others the anticipated performance parameters are taken.
For graphical representation, the relative sensitivity of the experiments as a function of live time is shown in Fig. 3. This value is calculated from Eq. (8) by
Here is the “average” 90% C.L. upper limit of the number of signal events for λbkg background events calculated according to the method discussed in .
A few comments should be made concerning the interpretation of Table 4 and Fig. 3.
‒ The factor , and hence FOM, has a full spread (not σ) of 30–70%. Thus, the curves in Fig. 3 could be replaced by bands that would however make the figure unreadable.
‒ The sensitivities discussed here are calculated for exclusion limits. The situation is different for a positive signal claim. A good energy resolution like the ones for germanium or bolometer experiments facilitates the identification of a narrow line at the correct energy. This is extremely valuable if the existence of the rarest decay ever observed will be claimed.
‒ To estimate relative sensitivities for , the inverse square root has to be taken of the curves shown in Fig. 3 and the FOM numbers in Table 4. The variation due to the spread of the matrix elements is reduced in this case, as can be seen from the last column of Table 4, i.e. the spread for the lower or upper value of the interval is reduced.
If, instead, limits for other particle physics parameters are calculated, the scaling will be different. For heavy neutrino exchange, the mass limit scales with . The factor will change in this case, but for most models the variation is less than 30% , .
‒ The background is irreducible and can only be avoided with an energy resolution –2% at . Of course, this requirement depends strongly on , which varies by a factor of 300 for the isotopes considered.
‒ Of the ongoing experiments, Kamland-Zen (2) should have the largest potential. However, a low background bolometer experiment like Lucifer or an improved CUORE experiment would be even more sensitive. Their sensitivity would still grow almost linearly after 4 yr (see Fig. 3).
‒ Germanium experiments can be competitive to e.g. current xenon experiments with a factor of three more mass, despite the fact that the phase space factor is small. However, a factor of 10 lower background than that of GERDA-I is required.
‒ Systematic effects like the uncertainty on the fiducial volume or the knowledge of the background level are not taken into account here.
The goal of many searches is to reach a sensitivity equivalent to meV, which would cover practically the entire expected range for the inverted neutrino mass hierarchy. For 76Ge, this corresponds to half lives of (1.5–15) · 1027 yr if the entire span of matrix elements of Fig. 1 is taken into account. These values should be compared to the expected sensitivity of GERDA-II or the Majorana demonstrator of about 1.7· 1026 yr. This demonstrates that exploring the entire mass band of the inverted hierarchy is a long-term enterprise.5 The equivalent numbers for 136Xe, 82Se, and 130Te are (0.6–3.5) · 1027 yr, (0.6–3.8) · 1027 yr, and (0.5–2.5) · 1027 yr, respectively. They seem to be easier to reach if one compares them to the limits in Table 4.
Considering the example of the Heidelberg–Moscow claim and the uncertainties of the matrix elements, one can conclude that a signal has to be observed in several isotopes to establish .
To improve the credibility of the result, a blind analysis should be performed. This technique is nowadays standard in particle physics experiments and should be adopted in this field as well.
It is also worth mentioning that new ideas for experiments are continuously coming up  and the one that will make a discovery might not be listed in this publication.
Neutrinoless double beta decay violates lepton number and the experimental programs are therefore on equal footing to proton-decay searches. It might also be the only practical process that allows one to test whether neutrinos are Majorana particles. The motivation for several large efforts in this field is therefore obvious.
For a long time, the Heidelberg–Moscow experiment has dominated the field and its claim of a signal is only being scrutinized now. The recent EXO-200 limit does not support this claim but cannot refute it due to the spread of the nuclear matrix element calculations. The newest Kamland-Zen number challenges the claim even more strongly. GERDA does not suffer from matrix element uncertainties and will unblind the data in spring 2013. Then the combined data from GERDA, EXO-200, and Kamland-Zen should be sensitive enough for a meaningful test.
Beyond this next step, experiments want to reach a sensitivity to explore the region of the inverted neutrino mass hierarchy. This will eventually require ton-scale experiments. Kamland-Zen should be the first one but whether the entire range can be covered depends on the achievable background level.
For a convincing claim of a signal, a good energy resolution is important and the detection with several isotopes.
The motivation for this article and many improvements are due to discussions with several colleagues. I want to thank especially Allen Caldwell, Karl Tasso Knöpfle, Christopher O'Shaughnessy, and Stefan Schönert.
To be more precise: the amplitude has dimension mass with being the mass of a scalar or vector exchange particle like the W boson.
This can be seen for example from Table 1 of . Twenty different calculations are listed for each isotope, which typically vary by a factor of two for a given A. If 20 ratios for two isotopes A and are calculated, the variation is reduced to 30%.
The Xe mass fraction is ≈2.5% and the fiducial volume is ≈43%.
An efficiency is not explicitly given in  but an effective value can be estimated using Eq. (8) and the information from the publication.
To reach a limit of 1028 yr, a 1-ton 76Ge experiment has to operate for 10 yr with a background, which is a factor of 10 smaller than the one of GERDA-II.
Bernhard Schwingenheuer was born in Arnsberg, Germany. He studied physics in Bonn and Heidelberg and earned his Ph.D. degree at the University of Chicago in 1995 with the measurement of a CPT symmetry violating parameter in the decay of neutral kaons. Afterwards, he joined the Max Planck Institute for Nuclear Physics, first as post-doc on the HERA-B experiment at DESY and later as staff on the GERDA experiment at the LNGS underground laboratory in Italy. His research interest is in neutrinoless double beta decay.