Results on \(\beta \beta \) decay with emission of two neutrinos or Majorons in \(^{76}\)Ge from GERDA Phase I
Abstract
A search for neutrinoless \(\beta \beta \) decay processes accompanied with Majoron emission has been performed using data collected during Phase I of the GERmanium Detector Array (GERDA) experiment at the Laboratori Nazionali del Gran Sasso of INFN (Italy). Processes with spectral indices \(n = 1, 2, 3, 7\) were searched for. No signals were found and lower limits of the order of 10\(^{23}\) yr on their half-lives were derived, yielding substantially improved results compared to previous experiments with \(^{76}\)Ge. A new result for the half-life of the neutrino-accompanied \(\beta \beta \) decay of \(^{76}\)Ge with significantly reduced uncertainties is also given, resulting in \(T^{2\nu }_{1/2} = (1.926 \pm 0.094)\times 10^{21}\) yr.
1 Introduction
Neutrinoless double beta (\(0\nu \beta \beta \)) decay is regarded as the gold-plated process for probing the fundamental character of neutrinos. Observation of this process would imply total lepton number violation by two units and that neutrinos have a Majorana mass component. Although the main focus of the experimental efforts lies on the detection of \(0\nu \beta \beta \) decay mediated by light Majorana neutrino exchange, there are also many other proposed mechanisms which are being searched for. Some exotic models predict \(0\nu \beta \beta \) decays proceeding through the emission of a massless Goldstone boson, called Majoron. Predictions of different models depend on its transformation properties under weak isospin, singlet [1], doublet [2] and triplet [3]. Precise measurements of the invisible width of the Z boson at LEP [4] greatly disfavor triplet and pure doublet models. Several new Majoron models have been developed subsequently in which the Majoron carries leptonic charge and cannot be a Goldstone boson [5, 6] or in which the \(0\nu \beta \beta \) decay proceeds through the emission of two Majorons [7].
Experimental searches for \(\beta \beta \) decay mediated by emission of one or two Majorons (\(0\nu \beta \beta \chi \)) have been performed by the Heidelberg-Moscow experiment (HdM) for \(^{76}\)Ge [8, 9]; by Nemo-2 and Nemo-3 for \(^{100}\)Mo, \(^{116}\)Cd, \(^{82}\)Se, \(^{96}\)Zr, \(^{130}\)Te [10, 11, 12, 13, 14, 15]; by ELEGANT V for \(^{100}\)Mo [16]; by DAMA [17], KAMland-Zen [18] and Exo-200 [19] for \(^{136}\)Xe. None of these experiments have seen an excess of events that could be interpreted as a Majoron signal; they reported lower limits on the half-lives of the processes that involve Majoron emission.
The \(2\nu \beta \beta \) decay process conserves lepton number and is independent of the nature of the neutrino. It has been detected for 11 nuclides so far, with measured half-lives (\(T^{2\nu }_{1/2}\)) in the range of \(7 \times 10^{18} \)–\( 2 \times 10^{24}\) yr [20, 21, 22, 23, 24]. The knowledge of \(T^{2\nu }_{1/2}\) allows for extraction of the nuclear matrix element, \(\mathcal{M}^{2\nu }\), which can provide some constraints on that of \(0\nu \beta \beta \) decay, \(\mathcal{M}^{0\nu }\), if the evaluations of \(\mathcal{M}\) for the two processes are performed within the same model [25, 26, 27, 28, 29, 30, 31, 32].
This paper reports on the search for neutrinoless double beta decay of \(^{76}\)Ge with Majoron emission (\(0\nu \beta \beta \chi \)) and a new analysis of the half-life of the \(2\nu \beta \beta \) decay of \(^{76}\)Ge using data collected by the GERmanium Detector Array (Gerda) experiment during its Phase I. \(2\nu \beta \beta \) decay is a well established and previously observed process, while \(0\nu \beta \beta \chi \) decay is a hypothetical one. In the first case the half-life is extracted, while for the second one a limit is set. This leads to slightly different approaches in the analyses leading to different data sets and background components being used.
2 The GERDA experiment
The main aim of the Gerda experiment [33] at the Laboratori Nazionali del Gran Sasso (LNGS) of INFN in Italy is to search for \(0\nu \beta \beta \) decay of \(^{76}\)Ge. The core of the setup is an array of high-purity germanium (HPGe) detectors made from isotopically modified material with \(^{76}\)Ge enriched to \(\sim \)86 % (\(^\mathrm{enr}\)Ge), mounted in low-mass copper supports (holders) and immersed in a 64 m\(^3\) cryostat filled with liquid argon (LAr). The LAr serves as cooling medium and shield against external backgrounds. The shielding is complemented by water in a tank of 10 m in diameter which is instrumented with photomultipliers to detect Cherenkov light generated in muon-induced showers [33].
The array of HPGe detectors is arranged in strings. Each string is enclosed with a cylinder, made from 60 \(\upmu \)m thick Cu foil, called mini-shroud, to mitigate the background coming from the decay of \(^{42}\)Ar present in the LAr. Moreover, in order to prevent contamination from radon within the cryostat, a cylinder, made from 30 \(\upmu \)m thick Cu foil, called radon-shroud, separates the central part of the cryostat, where the detectors are located, from the rest. The HPGe detector signals are read out with custom-made charge sensitive preamplifiers optimized for low radioactivity, which are operated close to the detectors in the LAr. The analog signals are digitized with 100 MHz Flash ADCs (FADC) and analyzed offline. If one of the detectors has an energy deposition above the trigger threshold (40–100 keV), all channels are read out. Reprocessed p-type coaxial detectors from the HdM [34] and Igex [35] experiments were operated together with Broad Energy Germanium (BEGe) type detectors manufactured by Canberra [36, 37].
3 Data taking and data selection
Phase I data taking lasted from November 9, 2011, to May 21, 2013. The total exposure collected comprises 19.2 kg yr for the coaxial detectors and 2.4 kg yr for the BEGe detectors. In this paper, the entire exposure collected by the BEGe detectors (BEGe data set) and 17.9 kg yr from the coaxial detectors (golden data set) are used [38, 39]. For the coaxial detectors, a data set collected for 1.3 kg yr exposure during a restricted time period around the deployment of the BEGe detectors is discarded due to a higher background level. Also one of the coaxial detectors, RG2, is not considered for the data analysis starting from March 2013, as its high voltage had to be reduced below depletion voltage due to increased leakage current. The energy calibration of the detectors was performed using the information from dedicated calibration runs. For these calibration runs, three \(^{228}\)Th sources were lowered to the vicinity of the detectors. The stability of the energy scale was monitored by performing such calibration runs every 1 or 2 weeks. Moreover, the stability of the system was continuously monitored by injecting charge pulses into the test input of the preamplifiers. Using physics data, the interpolated FWHM values at \(Q_{\beta \beta }\) averaged with the exposure are (4.8 \(\pm \) 0.2) keV for the coaxial detectors and (3.2 \(\pm \) 0.2) keV for the BEGe detectors.
All steps of the offline processing of the Gerda data were performed within the software framework Gelatio [40]. The energy deposited in each detector was extracted from the respective charge pulse by applying an approximate Gaussian filter [41]. Non-physical events, such as discharges, cross-talk and pick-up noise events, were rejected by quality cuts based on the time position of the rising edge, the information from the Gaussian filter, the rise time and the charge pulse height, which must not exceed the dynamic range of the FADCs. Pile-up and accidental coincidences were removed from the data set using cuts based on the baseline slope (which should compatible with zero for normal events), the number of triggers in a 10 \(\upmu \)s time window and the position of the main rising edge. The rate of pile-up and accidental coincidence events is negligible in the Gerda data due to the extremely low event rate. The loss due to mis-classification by the quality cuts was \(<\)0.1 % for events with energies above 1 MeV. All events that come within 8 \(\upmu \)s of a signal from the muon veto were rejected. Finally, only events that survive the detector anti-coincidence cut were considered. This means that all events with an energy deposition \(>\)50 keV in more than one detector in the array were not taken into account. Since \(2\nu \beta \beta \) and \(0\nu \beta \beta \chi \) events release their energy within a small volume inside the detectors, almost no signal events were lost by this cut, while a part of the \(\gamma \)-induced background events were rejected. The high efficiency of the anti-coincidence cut was checked using Monte Carlo generated events: the loss of \(2\nu \beta \beta \) and \(0\nu \beta \beta \chi \) events was estimated to be 0.14 % for energies above 570 keV.
4 Analysis strategy
The two analyses described in this paper are different in the sense that for \(2\nu \beta \beta \) decay a parameter is extracted for a well established and known process, while in the case of the search for \(0\nu \beta \beta \chi \) decay limits for a hypothetical process are set. In order to minimize the systematic uncertainties for the extraction of \({T^{2\nu }_{1/2}}\) a well defined and controlled subset of the data is used. Only unambiguously identified background processes are taken into account. For \(0\nu \beta \beta \chi \) limit setting the exposure is maximized and all known possible background processes that cannot be unambiguously detected but could mimic \(0\nu \beta \beta \chi \) decay are taken into account as otherwise a fake signal could be produced.
For the \({T^{2\nu }_{1/2}}\) analysis the golden data set (17.9 kg yr) with the coaxial detectors is used in order to have a large data sample obtained in well controlled experimental conditions. The Majoron analysis uses both the golden data set and the BEGe data set for a total exposure of 20.3 kg yr in order to maximize the sensitivity.
The background model for the \({T^{2\nu }_{1/2}}\) analysis uses a minimal number of components, assuming all sources close to the detectors [38, 42]. Contributions from possible other sources are summarized in the systematic uncertainty. For the Majoron analysis, an expanded model is used [43], taking into account also additional medium and far distant positions for some of the sources.
In both analyses, the experimental spectra of the coaxial and BEGe detectors are analyzed using the Bayesian Analysis Toolkit (Bat) [44].
5 The background model
The background sources considered in the models were identified by their prominent structures in the energy spectra and were also expected on the basis of material screening measurements. The spectral shapes of individual background contributions were obtained by using a detailed implementation of the experimental setup in the Monte Carlo (MC) simulation framework MaGe [45]. A Bayesian spectral fit of the measured energy spectrum with the simulated spectra was performed in an energy range from 570 keV up to the end of the dynamic range at 7500 keV. The low energy limit is motivated by the \(\beta \) decay of \(^{39}\)Ar, which gives a large contribution up to its \(Q_{\beta }\)-value of 565 keV. The high energy limit is dictated by the contamination coming from isotopes in the \(^{226}\)Ra decay chain. Events from these decays reach into the energy region of interest of the present two analyses. More details can be found in Ref. [38].
The following background components were used for the extraction of \(T^{2\nu }_{1/2}\) (minimum model in Refs. [38, 42]): (1) \(^{76}\)Ge \(2\nu \beta \beta \) decay, (2) \(^{214}\)Bi, \(^{228}\)Ac, \(^{228}\)Th, \(^{60}\)Co and \(^{40}\)K decays in the close vicinity of the detectors (\(<\)2 cm, represented by decays in the detector holders in the MC simulation), (3) decays of \(^{60}\)Co inside the detectors, constrained by the maximum expected activity from their cosmogenic activation history, (4) \(^{42}\)K decays in LAr assuming a uniform distribution, (5) \(\alpha \) model that accounts for \(\alpha \) decays originating from \(^{210}\)Po and \(^{226}\)Ra contaminations on the \(p^{+}\) surface of the detectors as well as from \(^{222}\)Rn in the LAr, and finally (6) \(^{214}\)Bi decays on the \(p^{+}\) surface, constrained by the estimated \(^{226}\)Ra activity from the \(\alpha \) model.
The parameters of all components besides the constrained ones were given a flat prior probability distribution. There are no strong correlations between the model parameters since all considered background components have characteristic features such as \(\gamma \)-ray lines or peak-like structures at different energies. The ratios of the \(\gamma \)-ray line intensities from the individual considered background sources suggest contaminations dominantly in locations close to the detectors. Hence, the minimum model takes into account only the close-by source locations. Nevertheless, the screening measurements indicate contaminations of materials in farther locations as well. An additional contribution can come from \(^{42}\)K decays at or near the detector \(n^+\) surfaces (see Fig. 1) with a specific activity higher than that for the uniform distribution assumption. This component is the dominating one for the BEGe data set, as the thinner dead layer thickness of BEGes of roughly 1 mm allows penetration of the electrons emitted in the decay of \(^{42}\)K to the active volume, while for coaxial detectors the dead layer thickness of \(\sim \)2 mm efficiently shields this background component.
The spectral shapes of the contributions from the background sources without significant multiple \(\gamma \) peaks at different source locations differ only marginally. This makes it impossible to pinpoint the exact source locations given the available statistics of the measured spectra. Note that adding all well motivated but not unambiguously identified background contributions does not influence the best-fit value for \({T^{2\nu }_{1/2}}\) significantly (see Table 8 in Ref. [38]). Variations of individual source locations for the considered decays were taken into account when evaluating the systematic uncertainty of \({T^{2\nu }_{1/2}}\) as described in Sect. 7.2.
For the Majoron analysis additional background components were used [43], including also medium and far distant contributions. For the coaxial detectors \(^{42}\)K on the \(n^{+}\) and on the \(p^{+}\) contacts was added to the list of the close sources of the previous background model. For medium distances, i.e. between 2 and 50 cm from the detectors, contributions from the following sources were added: \(^{214}\)Bi, \(^{228}\)Th and \(^{228}\)Ac. A \(^{228}\)Th contamination was chosen as a representative for far distant sources (above 50 cm). Whenever possible, screening measurements were used to constrain the lower limit of the expected background events.
Parameters for the coaxial detectors (upper part) and for the BEGe detectors (lower part): live time, t, total mass, M, the fraction of \(^{76}\)Ge atoms, \(f_{76}\), and the active volume fraction, \(f_\mathrm{AV}\). For the coaxial detectors, the first uncertainty on \(f_{act}\) is the uncorrelated part, the second one the correlated contribution. The values for M, \(f_{76}\), and \(f_\mathrm{AV}\) are taken from Ref. [38]
Detectors | t (days) | M (kg) | \(f_{76}\) (%) | \(f_\mathrm{AV}\) (%) |
---|---|---|---|---|
Enriched coaxial detectors | ||||
ANG2 | 485.5 | 2.833 | \(86.6\pm 2.5\) | \(87.1\pm 4.3\pm 2.8 \) |
ANG3 | 485.5 | 2.391 | \(88.3\pm 2.6\) | \(86.6\pm 4.9\pm 2.8 \) |
ANG4 | 485.5 | 2.372 | \(86.3\pm 1.3\) | \(90.1\pm 4.9\pm 2.9 \) |
ANG5 | 485.5 | 2.746 | \(85.6\pm 1.3\) | \(83.1\pm 4.0\pm 2.7 \) |
RG1 | 485.5 | 2.110 | \(85.5\pm 1.5\) | \(90.4\pm 5.2\pm 2.9 \) |
RG2 | 384.8 | 2.166 | \(85.5\pm 1.5\) | \(83.1\pm 4.6\pm 2.7 \) |
Enriched BEGe detectors | ||||
GD32B | 280.0 | 0.717 | \(87.7\pm 1.3\) | \(89.0 \pm 2.7\) |
GD32C | 304.6 | 0.743 | \(87.7\pm 1.3\) | \(91.1 \pm 3.0\) |
GD32D | 282.7 | 0.723 | \(87.7\pm 1.3\) | \(92.3 \pm 2.6\) |
GD35B | 301.2 | 0.812 | \(87.7\pm 1.3\) | \(91.4 \pm 2.9\) |
6 Determination of the half-life of \(2\nu \beta \beta \) decay
6.1 Analysis
The \({T^{2\nu }_{1/2}}\) of \(2\nu \beta \beta \) decay of \(^{76}\)Ge was determined considering the golden data set of Phase I, amounting to an exposure of 17.9 kg yr, and using the background model prediction for the contribution of the \(2\nu \beta \beta \) spectrum to the overall energy spectrum. Details of the background analysis can be found in Ref. [42].
6.2 Systematic uncertainties
Contributions to the systematic uncertainty on \({T^{2\nu }_{1/2}}\) taken into account in this work. The total systematic uncertainty is obtained by combining the individual contributions in quadrature
Item | Uncertainty on \({T^{2\nu }_{1/2}}\) (%) | |
---|---|---|
Active \(^{76}\)Ge exposure | \(\pm \)4 | |
Background model components | \(^{+ 1.4}_{- 1.2}\) | |
Shape of the \(2\nu \beta \beta \) spectrum | \(<\)0.1 | |
Subtotal fit model | \(\pm \)4.2 | |
Precision of the Monte Carlo geometry model | \(\pm \)1 | |
Accuracy of the Monte Carlo tracking | \(\pm \)2 | |
Subtotal Monte Carlo simulation | \(\pm \)2.2 | |
Data acquisition and handling | \(<\)0.1 | |
Total | \(\pm \)4.7 |
- (i)Detector parameters and fit model
- The systematic uncertainty on the active \(^{76}\)Ge exposure (\(\mathcal E_{\mathrm{AV},76}\)) was determined using a MC approach. \(\mathcal E_{\mathrm{AV}, 76}\) is defined asFor evaluating its uncertainty, the parameters of the individual detectors were randomly sampled from Gaussian distributions with mean values and standard deviations according to the corresponding values listed in Table 1. The correlated terms for \(f_\mathrm{AV}\) were also taken into account. The live time t is calculated using the monitoring test pulses, which had a rate of 0.1 Hz in a first period of the data taking and 0.05 Hz in the remaining part. The data files were written to disk every 2 h, thus the uncertainty on t is0.3–0.5%. The total detector masses are known with good accuracy (uncertainty smaller than 0.1 %). The calculation yields \(\mathcal E_{\mathrm{AV}, 76}=(13.45\pm 0.54)\) kg yr. The uncertainty of 4 % is dominated by the uncertainties on \(f_\mathrm{AV}\) [38], while \(f_{76}\) gives a smaller contribution. These uncertainties mainly affect the number of \(^{76}\)Ge nuclei in the active volume of the detectors, with a relatively smaller impact on the detection efficiency for the background sources. The determination of \(f_\mathrm{AV}\) and its uncertainty is based on the intercomparison of calibration data (taken with \(^{60}\)Co and \(^{241}\)Am sources) with simulated calibration of the same experimental setup.$$\begin{aligned} \mathcal E_{\mathrm{AV}, 76} = \sum _{i=1}^{N_{\mathrm{det}}} M_{i} t_{i} f_{\mathrm{AV},i} f_{76,i} . \end{aligned}$$(4)
The reference background model used for determining \({T^{2\nu }_{1/2}}\) accounts only for the dominant source locations in the setup. The systematic uncertainty due to the choice of the background model components was evaluated by repeating the global fit with alternative models, i.e., individual possible background components are added to the reference model and varied maximally, however, retaining a reasonable fit. The model that accounts for \(^{228}\)Th and \(^{228}\)Ac contributions also in the radon-shroud instead of only in the holders results in a 1.4 % longer \({T^{2\nu }_{1/2}}\). The same increase occurs if \(^{40}\)K in the radon-shroud is added to the model components. The model including the contribution from \(^{214}\)Bi in the radon-shroud in addition to the \(p^{+}\) surface and holders yields a 0.7 % longer \({T^{2\nu }_{1/2}}\). In all the cases mentioned above, the contribution from background in the \(2\nu \beta \beta \) spectrum region increases, since the peak-to-Compton ratio of the \(\gamma \)-rays decreases for farther source locations leading to longer \({T^{2\nu }_{1/2}}\) estimates. Excluding contributions from very close source locations, like \(^{214}\)Bi on the \(p^{+}\) surface and \(^{60}\)Co on the germanium, results in a smaller increase of the best \({T^{2\nu }_{1/2}}\) estimate. In this case, the contributions from these components are compensated by \(^{214}\)Bi and \(^{60}\)Co decays in the holders, respectively. Consequently, the source locations are moved further out with respect to the reference model. Consistently, the models that include additional contributions from close source locations yield a decrease in the \({T^{2\nu }_{1/2}}\) value, e.g. including \(^{214}\)Bi in LAr close to the \(p^{+}\) surface (\(-\)1.0 %) or \(^{42}\)K on the \(n^{+}\) (\(-\)1.2 %) and \(p^{+}\) (\(-\)0.6 %) surfaces. Comparing alternative background models to the reference one, the deviations in the \({T^{2\nu }_{1/2}}\) result range between \(-\)1.2 and \(+\)1.4 %.
For the standard fit, a bin width of 30 keV was used for the data and MC energy spectra. In order to take into account the systematic uncertainty related to binning effects, the fit was repeated twice using bin widths of 10 and 50 keV. The bin width of 10 keV was chosen in order to minimize as much as possible the bin size taking into account the energy resolution of \(\approx \)4.5 keV of the coaxial detectors and the necessity to have enough statistics in all bins. Above 50 keV, peak structures are washed out, leading to a deterioration of the fit. The deviations in the \({T^{2\nu }_{1/2}}\) result were well within the statistical uncertainties, confirming that the uncertainty due to the bin width is properly treated within the fits. No systematic uncertainty was attributed due to the bin width.
The primary spectrum of the two electrons emitted in the \(2\nu \beta \beta \) decay of \(^{76}\)Ge, which was then fed into the MC simulation, was sampled according to the distribution given in Ref. [21] (with the Primakoff–Rosen approximation removed) implemented in Decay0 [46]. The systematic uncertainty due to the assumed \(2\nu \beta \beta \) spectral shape was evaluated by comparing the spectrum generated by Decay0 to the one given in Ref. [47]. Considering the analysis window used for background modeling, the maximum deviation is 0.2 % and the total deviation of the integral in the analysis window is 0.1 %. When the fit with the background model is repeated using the spectrum of Ref. [47], the difference from the reference the \({T^{2\nu }_{1/2}}\) result is less than 0.1 %.
A possible effect of a transition layer, where it is assumed that the \(n^{+}\) dead layer on the detector surfaces is partially active, has been investigated [48, 49]. The dead layer thickness for individual detectors assumed in MC simulations was given according to the listed values in Ref. [38]. The transition layer is modeled using two different assumptions: a linearly and an exponentially increasing charge collection efficiency in the dead layer. The systematic uncertainty on \({T^{2\nu }_{1/2}}\) due to the \(2\nu \beta \beta \) spectrum simulated with the transition layer is found to be negligible.
- (ii)
MC simulation The uncertainty related to the MC simulation arises from the precision of the experimental geometry model implemented in MaGe (1 %) and from the accuracy of particle tracking (2 %) performed by Geant4 [50, 51, 52, 53, 54]. The uncertainty on the particle tracking was estimated as reported in Ref. [56], the precision on the geometry model was estimated by changing the modeled geometry in the code (dimensions, displacements, materials). The total MC simulation uncertainty was estimated to be 2.2 % by summing in quadrature the aforementioned contributions.
- (iii)
Data acquisition and selection The trigger and reconstruction efficiencies for physical events are practically 100 % above 100 keV in Gerda. These efficiencies were checked sending signals from a pulse generator to each detector. The amplitude of the signals was decreased down to 150 keV: no pulser events were lost by the trigger, quality cuts and reconstruction process. This result translates into an efficiency better than 99.9 % above 150 keV. The performance of the quality cuts has been further cross checked by a visual analysis. The total uncertainty related to data acquisition and selection was estimated to be less than 0.1 %. Summing in quadrature the uncertainties of the three groups gives a total systematic uncertainty of \(\pm \)4.7 %.
6.3 Results and discussion
Figure 2 shows the experimental data together with the best fit model for the golden data set. The different components of the minimum background model are also reported. The model is able to reproduce the experimental data well, as shown in the lower panel of the figure by the residuals. The p-value of the fit extending from 570 to 7500 keV is 0.02.
The background level achieved in Gerda Phase I is about one order of magnitude lower with respect to predecessor \(^{76}\)Ge experiments, and it has allowed the measurement of \({T^{2\nu }_{1/2}}\) with a signal-to-background ratio of 3:1 in the 570–2039 keV interval, or 4:1 in the smaller interval of 600–1800 keV. These ratios are much better than those for any past experiment that studied the \(2\nu \beta \beta \) decay of \(^{76}\)Ge.
7 Limits on Majoron-emitting double \(\beta \) decays of \(^{76}\)Ge
7.1 Analysis
The search for \(0\nu \beta \beta \chi \) decay was performed using the golden and BEGe data sets, amounting to a total exposure of 20.3 kg yr. The analysis employed the background model described in Sect. 5. The information from the two data sets was combined in one fit, while keeping their energy spectra distinct. A separate fit was performed for each spectral index, containing the background contributions, the contributions from \(2\nu \beta \beta \) decay, and also the Majoron component under study. A single parameter, \({T^{0\nu \chi }_{1/2}}\), is considered common for the two data sets. It is defined as the half-life of the respective Majoron accompanied mode.
7.2 Systematic uncertainties
- (i)Detector parameters and fit model Uncertainties from the fitting procedure were folded into the posterior distribution of \(T_{1/2}^{0\nu \chi }\) with a MC approach. Each source of uncertainty is described by a probability distribution. The fitting procedure was repeated 1000 times, each time drawing a random number for each source of uncertainty according to its probability distribution:
Material screening measurement results were used to constrain the minimum number of events expected from close and medium distant sources of the \(^{214}\)Bi and \(^{228}\)Th decays. Gaussian distributions describing these lower limits used in the fit were derived from the mean and standard deviations of the screening measurements. For details refer to Ref. [38].
As for the \({T^{2\nu }_{1/2}}\) analysis, the standard fit uses a bin width of 30 keV for the data and MC energy spectra. In order to account for the systematic uncertainty related to binning effects the bin width was sampled uniformly from 10 to 50 keV.
Uncertainties on the active volume fractions enter the model in several ways. On the one hand, the MC energy spectra for all internal sources, that is for \(2\nu \beta \beta \), \(0\nu \beta \beta \chi \), \(^{60}\)Co, and \(^{68}\)Ga decays, are affected, as the fraction of decays taking place in the active and dead part of the detectors changes with changing \(f_\mathrm{AV}\). On the other hand, the uncertainty on the active volume fraction also plays a role for the shape of the energy spectrum due to \(^{42}\)K decays on the \(n^+\) surface. Larger \(f_\mathrm{AV}\) means thinner \(n^+\) dead layer and thus the possibility of an increased contribution from the electrons to the spectrum. For smaller \(f_\mathrm{AV}\) and thicker \(n^+\) dead layer, their contributions are expected to be reduced. The active volume fraction for each detector was sampled from a Gaussian distribution with mean and standard deviation according to Table 1. For the coaxial detectors, the partial correlations of the uncertainty were taken into account. The simulated spectra of the internal sources as well as of the \(^{42}\)K decays on the \(n^+\) surface are composed according to the sampled active volume fractions.
The uncertainty on the fraction of enrichment in \(^{76}\)Ge of the germanium that constitutes the detectors plays a role when converting the number of events attributed to \(0\nu \beta \beta \chi \) decay into \(T_{1/2}^{0\nu \chi }\). The probability distribution of \(f_{76}\) for each detector is given by a Gaussian function with mean values and standard deviations as listed in Table 1.
The data does not allow the resolution of the ambiguity regarding the exact positions of the near and medium distant sources. The \(^{214}\)Bi decays serves as a representative in order to estimate the impact of this uncertainty. Their near position is represented by decays in the holders, in the mini-shroud or on the \(n^+\) surface of the detectors, each having a probability of 1 / 3 in the sampling process. The medium distant position is represented by decays in the radon-shroud or in the LAr, having a probability 1 / 2 in contrast.
Extensive studies of the characteristics of the BEGe diodes suggest the presence of a transition layer between the region where the detector is fully efficient and the external dead region [48, 49]. An uncertainty as high as \(\pm \)0.5 % on the lower limits of \(T_{1/2}^{0\nu \chi }\) is estimated for this effect in the case of the BEGe detectors. This uncertainty was folded into the total marginalized posterior distribution a posteriori. The corresponding uncertainty for the coaxial detectors is estimated to be negligible.
The marginalized posterior distributions for \(T_{1/2}^{0\nu \chi }\) derived from each of the 1000 individual fits were summed up. The resulting total marginalized posterior distribution accounts for the statistical as well as for the listed systematic uncertainties related to the fit model. As for the \(T_{1/2}^{2\nu }\) analysis, the uncertainties on the active volume fractions and on the enrichment fractions are major contributions to the total uncertainty on the limits for \(T_{1/2}^{0\nu \chi }\). However, the largest source of uncertainty is the composition of the fit model and the individual background contributions. In the case of \(n=1\), a fit with a bin width of 50 keV weakens the limit by \(\approx \)\( 16~\%\) compared to the standard fit, while the result for \(T_{1/2}^{2\nu }\) is not affected at all. The stability of the \({T^{2\nu }_{1/2}}\) results shows the validity of the fit. The use of the alternative close and medium distant source positions for \(^{214}\)Bi decays leads to maximal variations of \(^{+8.3}_{-12.6}~\%\) of the limit on \(T_{1/2}^{0\nu \chi }\).Table 3Experimental results for the limits on \({T^{0\nu \chi }_{1/2}}\) of \(^{76}\)Ge for the Majoron models given in Refs. [7, 57, 58, 59]. The first half considers lepton number violating models (I) allowing \(0\nu \beta \beta \) decay, while in the second half lepton number conserving models (II) are listed, where \(0\nu \beta \beta \) decay is not allowed. The first column gives the model name, the second the spectral index, n, the third the information on whether one Majoron, \(\chi \), or two Majorons, \(\chi \chi \), is emitted, the fourth if the Majoron is a Goldstone boson, the fifth provides its lepton number, L, the sixth the experimental limit on \(T_{1/2}^{0\nu \chi }\) of \(^{76}\)Ge obtained in this analysis. The nuclear matrix elements, \(\mathcal M\)\(^{0\nu \chi(\chi)}\), the phase space factor, \(G^{0\nu\chi(\chi)}\), and the resulting effective coupling constants, \(\langle g \rangle \), are given in the seventh, eighth and ninth columns, respectively. The limits on \(T_{1/2}^{0\nu \chi }\) of \(^{76}\)Ge for the Majoron models and \(\langle g \rangle \) correspond to the \(90~\%\) quantiles of the marginalized posterior probability distribution. For the case of \(n=1\), the nuclear matrix element, \(\mathcal M\)\(^{0\nu \chi }\), from Refs. [60, 61, 62, 63, 64, 65, 66] and the phase space factor, \(G^{0\nu \chi }\), from Ref. [67] are used for the calculation of \(\langle g \rangle \). The given range covers the variations of \(\mathcal M\)\(^{0\nu \chi }\) in these works. For \(n=3\;\mathrm {and}\;7\), \(\langle g \rangle \) is determined using the matrix elements and phase space factors from Ref. [57]. The results for \(0\nu \beta \beta \chi \)\((n=3,\;7)\) account for the uncertainty on \(\mathcal{M}^{0\nu \chi(\chi)}\). For \(n = 2\), only the experimental upper limit is given
Model
n
Mode
Goldstone boson
L
\(T_{1/2}^{0\nu \chi }\) (\(10^{23}\) yr)
\(\mathcal M\)\(^{0\nu \chi(\chi)}\)
\(G^{0\nu \chi(\chi) }\)\((\mathrm {yr}^{-1})\)
\(\langle g \rangle \)
IB
1
\(\chi \)
No
0
\(>\)4.2
(2.30–5.82)
\(5.86 \times 10^{-17}\)
\(<\)\( (3.4\)–\(8.7) \times 10^{-5}\)
IC
1
\(\chi \)
Yes
0
\(>\)4.2
(2.30–5.82)
\(5.86 \times 10^{-17}\)
\(<\)\( (3.4\)–\(8.7) \times 10^{-5}\)
ID
3
\(\chi \chi \)
No
0
\(>\)0.8
\(10^{-3 \pm 1}\)
\(6.32 \times 10^{-19}\)
\(<\)\( 2.1^{+4.5}_{-1.4}\)
IE
3
\(\chi \chi \)
Yes
0
\(>\)0.8
\(10^{-3 \pm 1}\)
\(6.32 \times 10^{-19}\)
\(<\)\( 2.1^{+4.5}_{-1.4}\)
IF
2
\(\chi \)
Bulk field
0
\(>\)1.8
–
–
–
IIB
1
\(\chi \)
No
\(-\)2
\(>\)4.2
(2.30–5.82)
\(5.86 \times 10^{-17}\)
\(<\)\( (3.4\)–\(8.7) \times 10^{-5}\)
IIC
3
\(\chi \)
Yes
\(-\)2
\(>\)0.8
0.16
\(2.07 \times 10^{-19}\)
\(<\)\( 4.7 \times 10^{-2}\)
IID
3
\(\chi \chi \)
No
\(-\)1
\(>\)0.8
\(10^{-3 \pm 1}\)
\(6.32 \times 10^{-19}\)
\(<\)\( 2.1^{+4.5}_{-1.4}\)
IIE
7
\(\chi \chi \)
Yes
\(-\)1
\(>\)0.3
\(10^{-3 \pm 1}\)
\(1.21 \times 10^{-18}\)
\(<\)\( 2.2^{+4.9}_{-1.4}\)
IIF
3
\(\chi \)
Gauge boson
\(-\)2
\(>\)0.8
0.16
\(2.07 \times 10^{-19}\)
\(< \)\(4.7 \times 10^{-2}\)
- (ii)
MC simulation As in the case of the \(T_{1/2}^{2\nu }\) measurement, a total MC simulation uncertainty of 2.2 % has to be taken into account for effects related to the geometry implementation and particle tracking. It is folded into the total marginalized posterior distributions. No effect on the lower limits is observed for any of the spectral modes.
- (iii)
Data acquisition and selection The uncertainty from data acquisition and selection is estimated to be below \(0.1~\%\) and does not alter the derived limits on \(T_{1/2}^{0\nu \chi }\).
7.3 Results and discussion
The limits on \({T^{0\nu \chi }_{1/2}}\) presented here are the most stringent limits obtained to date for \(^{76}\)Ge. The limits for \(n = 1\) and \(n = 3 \) are improved by more than a factor 6 [9], the limit for \(n = 7\) is improved by a factor 5 [8] compared to previous measurements. The limit for the mode with \(n = 2\) is reported here for the first time.
8 Conclusions
Phase I of the Gerda experiment, located at the INFN Laboratori Nazionali del Gran Sasso (LNGS) in Italy, has been executed between November 2011 and May 2013. Utilizing the collected exposure of Phase I, an improved result of the half-life of the \(2\nu \beta \beta \) process in \(^{76}\)Ge was obtained and new limits for the half-lives of the Majoron-emitting double beta decays were produced.
Notes
Acknowledgments
The Gerda experiment is supported financially by the German Federal Ministry for Education and Research (BMBF), the German Research Foundation (DFG) via the Excellence Cluster Universe, the Italian Istituto Nazionale di Fisica Nucleare (INFN), the Max Planck Society (MPG), the Polish National Science Centre (NCN), the Foundation for Polish Science (MPD programme), the Russian Foundation for Basic Research (RFBR), and the Swiss National Science Foundation (SNF). The institutions acknowledge also internal financial support. The Gerda Collaboration thanks the directors and the staff of the LNGS for their continuous strong support of the Gerda experiment.
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