Measurement of the half-life of the two-neutrino double beta decay of ⁷⁶Ge with the GERDA experiment

Neutrinoless double beta (0νββ) decay of atomic nuclei (A,Z) → (A,Z + 2) +2e⁻ is a forbidden process in the Standard Model (SM) of particle physics because it violates lepton number by two units. An observation of such a decay would demonstrate lepton number violation in nature and would prove that neutrinos have a Majorana component. For recent reviews, see [1]. The two-neutrino double beta (2νββ) decay of atomic nuclei,

$$\begin{equation*} (A,Z) \rightarrow (A,Z + 2) + 2 e^{-} + 2\bar{\nu }_{e}, \end{equation*}$$

with the simultaneous emission of two electrons and two anti-neutrinos, conserves lepton number and is allowed within the SM, independent of the nature of the neutrino. Being a higher-order process, it is characterized by an extremely low decay rate: so far it is the rarest decay observed in laboratory experiments. It is observable for a few even-even nuclei and was detected to-date for eleven nuclides; the corresponding half-lives are in the range of 7 × 10¹⁸–2 × 10²⁴ yr [2–4].

The measurement of the half-life of the 2νββ decay (T^2ν_1/2) is of substantial interest. For example, model predictions of the 0νββ half-life require the evaluation of nuclear matrix elements. These calculations are complicated and have large uncertainties. They are different from those required for the 2νββ decay, but it has been suggested [5, 6] that, within the same model framework, some constraints on the 0νββ matrix elements NME^0ν can be derived from the knowledge of the 2νββ nuclear matrix elements NME^2ν. Also, the nuclear matrix element NME^2ν which is extracted from the measurement of the half-life of the 2νββ decay can be directly compared with the predictions based on charge exchange experiments [7, 8]. A good agreement would indicate that the reaction mechanisms and the nuclear structure aspects that are involved in the 2νββ decay are well understood.

The GERmanium Detector Array (GERDA) experiment at the Laboratori Nazionali del Gran Sasso of INFN searches for the 0νββ decay of ⁷⁶Ge. High-purity germanium (HPGe) detectors isotopically enriched in ⁷⁶Ge are operated bare and immersed in liquid argon (LAr) in order to greatly reduce the environmental background. As a first result of this ongoing research, this paper reports a precise measurement of the half-life of the 2νββ decay of ⁷⁶Ge. The data used in this work encompasses an exposure of 5.04 kg yr, taken between November 2011 and March 2012.

A brief outline of the components of the GERDA detector that are most relevant for this work is given below; a detailed description can be found in [9, 10].

The GERDA experimental setup is shown in figure 1. At the core of the setup there is an array of HPGe detectors. They are operated bare in LAr which acts both as a coolant and as a shield against the residual environmental background. The array configuration consists of eleven germanium detectors: eight are made from isotopically modified germanium (^enrGe), enriched to about 86% in ⁷⁶Ge, and three are made from natural germanium (^natGe), with a total mass of 17.67 and 7.59 kg, respectively. The enriched detectors come from the former Heidelberg–Moscow (HdM) [11] and Igex [12] experiments. They underwent specific refurbishing processes before operation in GERDA [13, 14]. The germanium detectors are mounted in strings with typically three diodes each. Signals are amplified by low noise, low radioactivity charge sensitive preamplifiers [15] with 30 MHz bandwidth operated inside the LAr. They are digitized by a 14-bit 100 MHz continuously running ADC (FADC) equipped with anti-aliasing bandwidth filters. In the offline analysis the waveforms are digitally processed to reconstruct the event energy.

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The detector array is surrounded by 64 m³ of 5.0-grade LAr, contained in a vacuum insulated cryostat made of stainless steel, lined on the inner side by a 3–6 cm thick layer of copper. The cryostat is in turn placed at the centre of a 580 m³ volume of ultra-pure water equipped with 66 photomultiplier tubes to veto the residual cosmic ray muons by the detection of Cherenkov light. The large water volume also serves as a shield to moderate and capture neutrons produced by natural radioactivity and in muon-induced hadronic showers.

The energy scale is set by using calibration curves, parametrized as second-order polynomials, derived for each detector by calibration runs taken with ²²⁸Th sources. The stability of the energy scale is monitored by performing such calibration runs every one or two weeks. Moreover, the stability of the system is continuously monitored by using ad hoc charge pulses generated by a spectroscopy pulser that are regularly injected in the input of the charge sensitive preamplifier.

All GERDA detectors but two exhibit a reverse current of the order of tens of pA. The two problematic detectors showed an increase of leakage current soon after the beginning of their operation; therefore their bias high voltage had to be reduced and finally completely removed. These two detectors do not contribute to the present data set, although they are accounted for in the detector anti-coincidence cut described below. The total mass of the operational enriched detectors is 14.63 kg. The ^natGe detectors are not considered in the present analysis because of their low content of ⁷⁶Ge. The average energy resolution for the six enriched detectors considered in this study is about 4.5 keV (full width at half maximum) at the Q_ββ-value of ⁷⁶Ge (2039 keV) [10].

A key parameter which is required for the computation of T^2ν_1/2 is the total ⁷⁶Ge active mass. It is calculated as the product of the total detector mass with the isotopic abundance of ⁷⁶Ge (f₇₆) and the fraction of the active mass (f_act).

The average ⁷⁶Ge isotopic abundance of the six enriched detectors considered in this work is 86.3% [10, 12, 16, 17].

In GERDA p-type semi-coaxial detectors are used, for which a part of the volume close to the outer surface is inactive. After mechanical modifications and processing of the germanium diodes at the manufacturer, their active mass was re-determined experimentally [10, 13]. The measured average fraction f_act is 86.7%, with individual detector f_act uncertainties of about 6.5%.

The data set considered for the analysis was taken between 9 November 2011 and 21 March 2012, for a total of 125.9 live days, amounting to an exposure of 5.04 kg yr. Digitized charge pulses from the detectors are analyzed with the software tool Gelatio [18] according to the GERDA standard procedure [19]. The pulse amplitude is reconstructed offline by applying an approximated Gaussian filter with an integration time of 5 μs (which is sufficient to avoid losses due to ballistic effects). Events generated by discharges or due to electromagnetic noise are rejected by using a set of quality cuts. Due to the low counting rate, the data set has a negligible contamination of pile-up events. The combined efficiency of the trigger and of the offline data processing is practically 100% above 100 keV. Similarly, no loss of physical events is expected from the application of the quality cuts, as deduced by dedicated Monte Carlo studies and by the analysis of pulser data. Events that are in coincidence with a valid veto signal from the muon detector and events having a signal in more than one HPGe detector are excluded from the analysis. The time window for the coincidence between the muon detector and the HPGe detectors was set to 8 μs while that between different HPGe detectors was set to a few μs. Due to the very low muon flux in the Gran Sasso laboratory, the dead time induced by the muon veto is negligible.

Given the half-life of the 2νββ decay reported in the literature (about 1.5 × 10²¹ yr), the anticipated count rate of the ^enrGe detectors is about 100 counts day⁻¹ in the entire energy range up to Q_ββ = 2039 keV. Since the detectors are submerged in LAr, the radioactive decay of ³⁹Ar, which is a long-lived β emitter produced by cosmogenic activation of natural argon in the atmosphere, gives a large contribution up to its Q_β-value of 565 keV. In fact, the low-energy spectrum is dominated by these β particles and their Bremsstrahlung photons, which account for about 1000 counts day⁻¹ above 100 keV. The 2νββ decay is expected to be the major contributor in the energy spectrum above the end-point of the ³⁹Ar spectrum. For this reason, the analysis of the 2νββ decay is performed in the range between 600 keV, which is comfortably above the end-point of the ³⁹Ar spectrum, and 1800 keV. The sum spectrum of the six ^enrGe detectors considered in this work is displayed in figure 2. The analysis range contains 8796 events in total. The probability for a 2νββ decay taking place in the active volume of the ^enrGe detectors to produce a total energy release between 600 and 1800 keV is about 63.5%; the energy range above 1800 keV is practically insensitive to the 2νββ signal, as the probability for a decay to produce an energy release in this region is <0.02%. These estimates are based on the Monte Carlo simulation of 2νββ decays in the GERDA detectors as described in section 4. Hence, the energy region chosen for the analysis is well suited for the study of the 2νββ decay signal.

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4.1. Statistical treatment and fit model

The experimental spectra of the six diodes are analyzed following the binned maximum likelihood approach described in [20]. The analysis region is divided into 40 bins, each 30 keV wide. A global model is fitted to the observed energy spectra. The model contains the 2νββ decay of ⁷⁶Ge and three independent background contributions, namely ⁴²K, ²¹⁴Bi and ⁴⁰K. The presence of these background sources is established by the observation of their characteristic γ lines: 1525 keV from ⁴²K; 1460 keV from ⁴⁰K; 609 keV and 1764 keV from ²¹⁴Bi; 352 keV from ²¹⁴Pb (progenitor of ²¹⁴Bi).

⁴²K is a short-lived β emitter (Q_β = 3525 keV, T_1/2 = 12.6 h) which is present in GERDA as a progeny of the long-lived ⁴²Ar (Q_β = 599 keV, T_1/2 = 32.9 yr). ⁴²Ar is a trace radioactive contaminant expected in natural argon and is produced by cosmogenic activation. ²¹⁴Bi from the ²³⁸U decay series and ⁴⁰K are γ emitters from the environmental radioactivity.

A few more candidate γ lines have been identified in the GERDA spectrum [10], which indicate the presence of small additional background contributions: in particular ²⁰⁸Tl and ²²⁸Ac from the ²³²Th decay series, and ⁶⁰Co. However, most of the candidate γ lines have either a poor statistical significance and/or are seen in some detectors only. Given the lack of discriminating power in the data, the background contributions other than ⁴²K, ²¹⁴Bi and ⁴⁰K are not included in the fit. However, their possible impact on the extracted half-life T^2ν_1/2 is included in the systematic uncertainty, as discussed in section 4.2; their cumulative contribution to the background is estimated to be of a few percent.

The half-life of the 2νββ decay is common in the fit to the six spectra of the ^enrGe detectors. The intensities of the background components are independent for each detector. The active mass and the ⁷⁶Ge abundance of each detector are also left free in the fit; they are treated as nuisance parameters and integrated over at the end of the analysis.

The shapes of the energy spectra for the model components (signal and three backgrounds) are derived by a Monte Carlo simulation for each detector individually. The simulation is performed using the MaGe framework [21] based on Geant4 [22]. Assumptions have to be made in the Monte Carlo simulation about the location and the primary spectrum of each component of the model. The spectrum of the two electrons emitted in the 2νββ decay of ⁷⁶Ge is sampled according to the distribution of [3] (with the Primakoff–Rosen approximation removed) that is implemented in the code decay0 [23]. Electrons are propagated in the GERDA simulated setup by MaGe and the total energy released in the active mass of the enriched detectors is registered.

The ⁴²K activity is uniformly distributed in the LAr volume. The decay products of the β decay of ⁴²K are taken into account as the initial state in the simulation. However, the energy deposit in the detectors is mainly given by the 1525 keV γ-ray (full energy peak and Compton continuum): the contribution due to the β particles is small (less than a few percent) with respect to the 1525 keV γ-ray. The actual position of the ⁴⁰K and ²¹⁴Bi emitters contributing to the GERDA spectrum is not known in detail: the assumption is made in the Monte Carlo of 'close sources'. The ratio of the intensities of the ²¹⁴Bi γ lines observed in the experimental spectrum is consistent with such an assumption. The impact on T^2ν_1/2 due to the lack of knowledge about the source position—which affects the peak-to-Compton ratio—is accounted as a systematic uncertainty. The effect of muon-induced events that are not accompanied by a veto signal, e.g. due to inefficiency, is estimated to be <0.02% in the energy range of this analysis.

The spectral fit is performed using the Bayesian Analysis Toolkit (Bat) [24]. A flat distribution between 0 and 10²² yr is taken as the prior probability density function (PDF) for T^2ν_1/2. The prior PDF for the active mass fraction of each detector is modeled as a Gaussian distribution, having mean value and standard deviation according to the measurements performed in [13]. The analysis accounts for the fact that the uncertainties on the active masses f_act are partially correlated, because of the experimental procedure employed for the measurement. Uncertainties on f_act are split into an uncorrelated term—which is specific of each detector—and a common correlated term. The prior PDF for the ⁷⁶Ge isotopic abundance f₇₆ for each detector is also a Gaussian, having mean value according to the earlier measurements reported in [10]. The uncertainty of f₇₆ is between 1% and 3% for the single detectors. It was estimated from the dispersion of independent measurements performed with isotopically enriched material; this dispersion is much larger than the quoted uncertainty of each individual measurement.

The fit has 32 free parameters: one common value for the 2νββ half-life and 31 nuisance parameters. The nuisance parameters are the common term which describes the correlated uncertainty of the active masses plus five independent variables for each of the six detectors (active mass, ⁷⁶Ge abundance and three background components). The detector parameters used for the prior PDF's are summarized in table 1.

Table 1. Summary table of the six enriched detectors used in this work. For each detector, the total mass, the active mass and the isotopic abundance of ⁷⁶Ge are given. The uncertainties reported for the active masses are the uncorrelated (first) and correlated (second) errors. The last column reports the T^2ν_1/2 which is obtained by performing the analysis on each detector individually. The uncertainty on T^2ν_1/2 is the fit uncertainty only.

Detector	Total mass (g)	Active mass (g)	76Ge isotopic abundance (%)	T2ν1/2 (1021 yr)
ANG2	2833	2468 ± 121 ± 89	86.6 ± 2.5	1.99+0.14−0.15
ANG3	2391	2070 ± 118 ± 77	88.3 ± 2.6	1.69+0.15−0.14
ANG4	2372	2136 ± 116 ± 79	86.3 ± 1.3	1.94+0.14−0.15
ANG5	2746	2281 ± 109 ± 82	85.6 ± 1.3	1.79+0.12−0.14
RG1	2110	1908 ± 109 ± 72	85.5 ± 2.0	1.94+0.18−0.14
RG2	2166	1800 ± 99 ± 65	85.5 ± 2.0	1.93+0.16−0.16

Figure 3 shows experimental data together with the best fit model for the sum of the six detectors. The analysis energy window contains 8796 events. The best fit model has an expectation of 8797.0 events, divided as follows: 7030.1 (79.9%) from the 2νββ decay of ⁷⁶Ge; 1244.6 (14.1%) from ⁴²K; 335.5 (3.8%) from ²¹⁴Bi; and 186.8 (2.1%) from ⁴⁰K. The individual components derived from the fit are also shown in figure 3. The signal-to-background ratio in the region 600–1800 keV is on average 4: 1, which is much better than for any past experiment which observed the 2νββ decay of ⁷⁶Ge. The best ratio achieved so far for ⁷⁶Ge was approximately 1:1 by HdM [17]. The model is able to reproduce well the experimental data, as shown in the lower panel of figure 3. The p-value of the fit derived from the procedure of [25], is p = 0.77.

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All nuisance parameters are eventually integrated over in order to derive the posterior PDF p(T^2ν_1/2) for the 2νββ half-life. The p(T^2ν_1/2) distribution is nearly Gaussian; the best estimate of the half-life is

$$\begin{equation} T^{2\nu }_{1/2}= \big (1.84 ^{+0.09}_{-0.08}\;\big) \times 10^{21} {\rm yr,} \end{equation} \tag{ 1 }$$

(fit error only). This uncertainty is calculated as the smallest interval containing 68% probability of p(T^2ν_1/2). It includes the uncertainty induced on T^2ν_1/2 by the nuisance parameters of the fit and accounts for parameter correlations. Active masses and ⁷⁶Ge isotopic abundances drive the fit uncertainty on T^2ν_1/2: if these parameters were known without uncertainties, the 1σ uncertainty from the fit would be about 0.03 × 10²¹ yr.

As a cross-check, the same procedure is run for each detector separately. The resulting T^2ν_1/2 values are summarized in table 1; they are mutually consistent within their uncertainties (χ²/ν = 3.02/5).

4.2. Systematic uncertainties

The half-life of the 2νββ decay of ⁷⁶Ge was derived from the first data from the GERDA experiment at the INFN Gran Sasso Laboratory. GERDA operates HPGe detectors enriched in ⁷⁶Ge directly immersed in LAr. The analysis has been carried out on the data collected with six ^enrGe detectors (14.6 kg total weight) during 125.9 live days by fitting the energy spectra with a comprehensive model. The best estimate of the half-life of the 2νββ decay is

$$\begin{equation} T^{2\nu }_{1/2}= \big(1.84 ^{+0.09}_{-0.08 \; {\rm fit}}\; ^{+0.11}_{-0.06 \; {\rm syst}}\big) \times 10^{21} {\rm yr} = \big (1.84 ^{+0.14}_{-0.10}\big) \times 10^{21} {\rm yr,} \end{equation} \tag{ 2 }$$

with the fit and systematic uncertainties combined in quadrature.

The half-life is longer than all previous measurements reported in the literature. A summary of the T^2ν_1/2 results of ⁷⁶Ge from earlier experiments versus publication year is displayed in figure 4. The figure includes nine measurements published between 1990 and 2005 and two weighted averages. A trend toward longer T^2ν_1/2 values reported by the most recent (and lowest background) experiments is clearly seen. The GERDA result is in better agreement with the two most recent results of [31, 37] that are based on the re-analysis of the HdM data²⁶. The fact that the half-lives derived in the more recent works—and particularly in this one—are systematically longer is probably related to the superior signal-to-background ratio, which lessens the relevance of the background modeling and subtraction. Thus the total GERDA uncertainty is comparable to what was achieved by HdM, in spite of the much smaller exposure.

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The uncertainty of the GERDA result can be further reduced in the future by accumulating more exposure and by performing a new and more precise measurement of the active mass of the detectors. Large data sets will reduce also the uncertainty due to the fit model, as background components can be better characterized and constrained by the (non-)observation of γ lines.

Using phase space factors from the improved electron wave functions reported in [39], the experimental matrix element for the 2νββ decay of ⁷⁶Ge calculated with the half-life of this work is NME^2ν = 0.133^+0.004_−0.005 MeV⁻¹.

The present value for NME^2ν is 11% smaller than that used in [5], which compares the matrix elements for 2νββ and 0νββ decays. Reference [5] also shows a relation between the two matrix elements for QRPA calculations of ⁷⁶Ge. According to this, the new value for NME^2ν results in an increase of the predicted half-life for the 0νββ by about 15%, which is well within the present uncertainty of the model calculation.

The nuclear matrix elements of the 2νββ decay of ⁷⁶Ge estimated from the charge exchange reactions (d, ²He) and (³He,t) are (0.159 ± 0.023) MeV⁻¹ [7] and (0.23 ± 0.07) MeV⁻¹ [8], respectively. They both seem to be higher than the value reported in this work, but consistent within the uncertainties.

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 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 Laboratori Nazionali del Gran Sasso for their continuous strong support of the experiment. We would like to thank Professor V I Tretyak for help and suggestions about the modeling of the energy spectrum of the 2νββ decay.