Background free search for neutrinoless double beta decay with GERDA Phase II

TL;DR

GERDA Phase II achieved unprecedented background suppression in the search for neutrinoless double beta decay of germanium-76, setting new limits and demonstrating the potential for a background-free experiment to explore fundamental particle physics questions.

Contribution

First demonstration of a background-free search for neutrinoless double beta decay using germanium detectors with advanced veto and background recognition techniques.

Findings

Background level of ~10^{-3} cts/(keV·kg·yr) achieved

No signal detected, setting a lower half-life limit of 5.3×10^{25} years

Sensitivity of 4.0×10^{25} years comparable to larger experiments

Abstract

The Standard Model of particle physics cannot explain the dominance of matter over anti-matter in our Universe. In many model extensions this is a very natural consequence of neutrinos being their own anti-particles (Majorana particles) which implies that a lepton number violating radioactive decay named neutrinoless double beta ($0\nu\beta\beta$) decay should exist. The detection of this extremely rare hypothetical process requires utmost suppression of any kind of backgrounds. The GERDA collaboration searches for $0\nu\beta\beta$ decay of $^{76}$Ge ($^{76}\rm{Ge} \rightarrow\,^{76}\rm{Se} + 2e^-$) by operating bare detectors made from germanium with enriched $^{76}$Ge fraction in liquid argon. Here, we report on first data of GERDA Phase II. A background level of $\approx10^{-3}$ cts/(keV$\cdot$kg$\cdot$yr) has been achieved which is the world-best if weighted by the narrow energy-signal region of germanium detectors. Combining Phase I and II data we find no signal and deduce a new lower limit for the half-life of $5.3\cdot10^{25}$ yr at 90 % C.L. Our sensitivity of $4.0\cdot10^{25}$ yr is competitive with the one of experiments with significantly larger isotope mass. GERDA is the first $0\nu\beta\beta$ experiment that will be background-free up to its design exposure. This progress relies on a novel active veto system, the superior germanium detector energy resolution and the improved background recognition of our new detectors. The unique discovery potential of an essentially background-free search for $0\nu\beta\beta$ decay motivates a larger germanium experiment with higher sensitivity.

Figures

Tables

Table 1. Table 1: List of data sets, exposures (for total mass), energy resolutions in FWHM, efficiencies (including enrichment, active mass, reconstruction efficiencies and dead times) and background indices (BI) in the analysis window.

data set	exposure	FWHM	efficiency	BI
	[kg$\cdot$yr]	[keV]		${10}^{-3}$$\text{cts}/\left(\text{keV}\cdot \text{kg}\cdot \text{yr}\right)$
PI golden	17.9	$4.3\left(1\right)$	$0.57\left(3\right)$	$11\pm 2$
PI silver	1.3	$4.3\left(1\right)$	$0.57\left(3\right)$	$30\pm 10$
PI BEGe	2.4	$2.7\left(2\right)$	$0.66\left(2\right)$	$5_{-3}^{+4}$
PI extra	1.9	$4.2\left(2\right)$	$0.58\left(4\right)$	$5_{-3}^{+4}$
PIIa coaxial	5.0	$4.0\left(2\right)$	$0.53\left(5\right)$	${3.5}_{-1.5}^{+2.1}$
PIIa BEGe	5.8	$3.0\left(2\right)$	$0.60\left(2\right)$	${0.7}_{-0.5}^{+1.1}$

Equations

$$T_{1/2}^{0\nu }>5.3\cdot 10^{25}yr.$$

$${\mu }_i^S=\ln 2\cdot \left(N_A/m_a\right)\cdot {ϵ}_i\cdot {\mathcal{E}}_i\cdot \mathcal{S},$$

$${\mu }_i^B={\mathcal{E}}_i\cdot {\text{BI}}_i\cdot \Delta E,$$

$$\begin{array}{rl}{\mathcal{L}}_i& ({\mathcal{D}}_i|\mathcal{S},{\text{BI}}_i,{\theta }_i)=\prod _{j=1}^{N_i^{obs}}\frac{1}{{\mu }_i^S+{\mu }_i^B}\cdot \\ & \left[{\mu }_i^S\cdot \frac{1}{\sqrt{2\pi }{\sigma }_i}\exp \left(\frac{-(E_j-Q_{\beta \beta }-{\delta }_i{)}^2}{2{\sigma }_i^2}\right)+{\mu }_i^B\cdot \frac{1}{\Delta E}\right]\end{array}$$

$$\begin{array}{rl}\mathcal{L}& (\mathcal{D}|\mathcal{S},\text{BI},\mathbf{\theta })=\\ & \prod _i\left[\frac{e^{-({\mu }_i^S+{\mu }_i^B)}\cdot ({\mu }_i^S+{\mu }_i^B{)}^{N_i^{obs}}}{N_i^{obs}!}\cdot {\mathcal{L}}_i({\mathcal{D}}_i|\mathcal{S},{\text{BI}}_i,{\theta }_i)\right]\end{array}$$

$$t_{\mathcal{S}}=-2\ln \lambda (\mathcal{S})=-2\ln \frac{\mathcal{L}(\mathcal{S},\widehat{\hat{\text{BI}}},\widehat{\hat{\mathbf{\theta }}})}{\mathcal{L}(\hat{\mathcal{S}},\widehat{\text{BI}},\hat{\mathbf{\theta }})}$$

$$p_{{\mathcal{S}}_j}={\int }_{t_{obs}}^{\infty }f(t_{\mathcal{S}}|{\mathcal{S}}_j)d({t_{\mathcal{S}}}_j)$$

$$\tilde{t}_{{\cal S}}=\>\left\{\!\!\begin{array}[]{ll}0,&\quad\hat{{\cal S}}>{\cal S}\geq 0\\[5.69046pt] -2\ln{\lambda}({\cal S}),&\quad\hat{{\cal S}}\leq{\cal S}\\[5.69046pt] \end{array}\right.$$

$$\mathcal{P}(\mathcal{S},\text{BI}|\mathcal{D},\mathbf{\theta })\propto \mathcal{L}(\mathcal{D}|\mathcal{S},\text{BI},\mathbf{\theta })\mathcal{P}(\mathcal{S})\prod _i\mathcal{P}({\text{BI}}_i)$$

$$⟨\mathcal{P}(\mathcal{S}|\mathcal{D})⟩=\int \mathcal{P}(\mathcal{S}|\mathcal{D},\mathbf{\theta })\prod _{i}g({\theta }_i)d{\theta }_i$$

Full text

The Gerda collaboration searches for $0\nu\beta\beta$ decay of 76Ge (${}^{76}\rm{Ge}\rightarrow\,^{76}\rm{Se}+2e^{-}$) by operating bare detectors made from germanium with enriched 76Ge fraction in liquid argon. Here, we report on first data of Gerda Phase II. A background level of $\approx 10^{-3}$ $\text{cts}/(\text{keV}\cdot\text{kg}\cdot\text{yr})$ has been achieved which is the world-best if weighted by the narrow energy-signal region of germanium detectors. Combining Phase I and II data we find no signal and deduce a new lower limit for the half-life of $5.3\cdot 10^{25}$ yr at 90 % C.L. Our sensitivity of $4.0\cdot 10^{25}$ yr is competitive with the one of experiments with significantly larger isotope mass.

Gerda is the first $0\nu\beta\beta$ experiment that will be background-free up to its design exposure. This progress relies on a novel active veto system, the superior germanium detector energy resolution and the improved background recognition of our new detectors. The unique discovery potential of an essentially background-free search for $0\nu\beta\beta$ decay motivates a larger germanium experiment with higher sensitivity.

$0\nu\beta\beta$ decay, $T^{0\nu}_{1/2}$, 76Ge, enriched Ge detectors, active veto

I Introduction

One of the most puzzling aspects of cosmology is the unknown reason for the dominance of matter over anti-matter in our Universe. Within the Standard Model of particle physics there is no explanation for this observation and hence a new mechanism has to be responsible. A favored model called leptogenesis Davidson et al. (2008) links the matter dominance to the nature of neutrinos and to the violation of lepton number, i.e. the total number of electrons, muons, taus and neutrinos minus the number of their anti-particles.

In most extensions of the Standard Model Mohapatra and A.Y.Smirnov (2006); Mohapatra et al. (2007); Päs and Rodejohann (2015) neutrinos are assumed to be their own anti-particles (Majorana particles). This might lead to lepton number violating processes at the TeV energy scale observable at the LHC Päs and Rodejohann (2015) and would result in neutrinoless double beta ($0\nu\beta\beta$) decay where a nucleus of mass number $A$ and charge $Z$ decays as $(A,Z)\rightarrow(A,Z+2)+2\,e^{-}$. Lepton number violation has not been unambiguously observed so far. There are several experimental $0\nu\beta\beta$ decay programs ongoing using for example 76Ge Agostini et al. (2013a); Cuesta et al. (2015), 130Te Alfonso et al. (2015); Andringa et al. (2016) or 136Xe Gando et al. (2016); Albert et al. (2014); Martin-Albo et al. (2016). They all measure the sum of the electron energies released in the decay which corresponds to the mass difference $Q_{\beta\beta}$ of the two nuclei. The $0\nu\beta\beta$ decay half-life is at least 15 orders of magnitude longer than the age of the universe. Its observation requires therefore the best suppression of backgrounds.

In the GERmanium Detector Array (Gerda) experiment bare germanium detectors are operated in liquid argon (LAr). The detectors are made from germanium with the 76Ge isotope fraction enriched from 7.8 % to about 87 %. Since source and detector of $0\nu\beta\beta$ decay are identical in this calorimetric approach the detection efficiency is high.

This Article presents the first result from Gerda Phase II. In the first phase of data taking (Phase I), a limit of $T_{1/2}^{0\nu}>2.1\cdot 10^{25}$ yr (90 % C.L.) was found Agostini et al. (2013a) for an exposure of 21.6 kg$\cdot$yr and a background of 0.01 $\text{cts}/(\text{keV}\cdot\text{kg}\cdot\text{yr})$ at $Q_{\beta\beta}=(2039.061\pm 0.007)$ keV Mount et al. (2010). At that time, the result was based on data from 10 detectors (17.6 kg total mass). In December 2015, Phase II started with 37 detectors (35.6 kg) from enriched material. The mass is hence doubled relative to Phase I. The ambitious goal is an improvement of the half-life sensitivity to $>10^{26}$ yr for about 100 kg$\cdot$yr exposure by reducing the background level by an order of magnitude. The latter is achieved by vetoing background events through the detection of their energy deposition in LAr and the characteristic time profile of their signals in the germanium detectors. The expected background is less than one count in the energy region of interest up to the design exposure which means that Gerda will be the first “background free” experiment in the field.

We will demonstrate in this Article that Gerda has reached the envisioned background level which is the world-best level if weighted by our superior energy resolution. Gerda is therefore best suited to not only quote limits but to identify with high confidence a $0\nu\beta\beta$ signal.

II The experiment

The Gerda experiment Ackermann et al. (2013) is located at the underground Laboratori Nazionali del Gran Sasso (LNGS) of INFN, Italy. A rock overburden of about 3500 m water equivalent removes the hadronic components of cosmic ray showers and reduces the muon flux at the experiment by six orders of magnitude to 1.2 $\mu$/(m${}^{2}\cdot$h).

The basic idea is to operate bare germanium detectors in a radiopure cryogenic liquid like LAr for cooling to their operating temperature of $\sim$90 K and for shielding against external radiation originating from the walls (see Extended Data Fig. 1 for a sketch of the setup) Heusser (1995). In Gerda, a 64 m3 LAr cryostat is inside a 590 m3 water tank. The clean water completes the passive shield. Above the water tank is a clean room with a glove box and lock for the assembly of germanium detectors into strings and the integration of the liquid argon veto system.

Gerda deploys 7 coaxial detectors from the former Heidelberg-Moscow Klapdor-Kleingrothaus et al. (2004) and IGEX Aalseth et al. (2002) experiments and 30 broad energy (BEGe) detectors  Agostini et al. (2015a). All diodes have p-type doping (see Extended Data Fig. 2). Electron-hole pairs created in the 1–2 mm thick n$+$ electrode mostly recombine such that the active volume is reduced. A superior identification of the event topology and hence background rejection is available for the BEGe type (see below). The enriched detectors are assembled into 6 strings surrounding the central one which consists of three coaxial detectors of natural isotopic composition. Each string is inside a nylon cylinder (see Extended Data Fig. 3) to limit the LAr volume from which radioactive ions like 42K can be collected to the outer detector surfaces Agostini et al. (2014a).

All detectors are connected to custom made low radioactivity charge sensitive amplifiers Riboldi et al. (2015) (30 MHz bandwidth, 0.8 keV full width at half maximum (FWHM) resolution) located in LAr about 35 cm above the detectors. The charge signal traces are digitized with 100 MHz sampling rate and stored on disk for offline analysis.

In background events some energy is often also deposited in the argon. The resulting scintillation light Agostini et al. (2015b) can be detected to veto them. In Phase II, a cylindrical volume of 0.5 m diameter and 2.2 m height around the detector strings (see Extended Data Fig. 1 and 4) is instrumented with light sensors. The central 0.9 m of the cylinder are defined by a curtain of wavelength shifting fibers which surround the 0.4 m high detector array. The fibers are read-out at both ends with 90 silicon photomulipliers (SiPM) Janicsko et al. (2016). Groups of six $3\times 3$ mm2 SiPMs are connected together to a charge sensitive amplifier. Sixteen 3” low-background photomultpliers (PMT) designed for cryogenic operation are mounted at the top and bottom surfaces of the cylindrical volume. The distance to any detector is at least 0.7 m to limit the PMT background contribution from their intrinsic Th/U radioactivity. All LAr veto channels are digitized and read-out together with the germanium channels if at least one detector has an energy deposition above $\sim$100 keV.

The nylon cylinders, the fibers, the PMTs and all surfaces of the instrumented LAr cylindrical volume are covered with a wavelength shifter to shift the LAr scintillation light from 128 nm to about 400 nm to match the peak quantum efficiency of the PMTs and the absorption maximum of the fibers.

The water tank is instrumented with 66 PMTs to detect Cherenkov light from muons passing through the experiment. On top of the clean room are three layers of plastic scintillator panels covering the central 4$\times$3 m2 to complete the muon veto Freund et al. (2016).

III Data analysis

The data analysis flow is very similar to that of Phase I. The offline analysis of the digitized germanium signals is described in Refs. Agostini et al. (2013a, 2012, 2011a).

A data blinding procedure is again applied. Events with a reconstructed energy in the interval $Q_{\beta\beta}\pm 25$ keV are not analyzed but only stored on disk. After the entire analysis chain has been frozen, these blinded events have been processed.

The gain stability of each germanium detector is continuously monitored by injecting charge pulses (test pulses) into the front-end electronics with a rate of 0.05 Hz. The test pulses are also used to monitor leakage current and noise. Only data recorded during stable operating conditions (e.g. gain stability better than 0.1 %) are used for the physics analysis. This corresponds to about 85 % of the total data written on disk.

Signals originated from electrical discharges in the high voltage line or bursts of noise are rejected during the offline event reconstruction by a set of multi-parametric cuts based on the flatness of the baseline, polarity and time structure of the pulse. Physical events at $Q_{\beta\beta}$ are accepted with an efficiency larger than 99.9 % estimated with $\gamma$ lines in calibration data, test pulse events and template signals injected in the data set. Conversely, a visual inspection of all events above 1.6 MeV shows that no unphysical event survives the cuts.

The energy deposited in a germanium detector is reconstructed offline with an improved digital filter Agostini et al. (2015c), whose parameters are optimized for each detector and for several periods. The energy scale and resolution are determined with weekly calibration runs with 228Th sources. The long-term stability of the scale is assessed by monitoring the shift of the position of the 2615 keV peak between consecutive calibrations. It is typically smaller than 1 keV for BEGe detectors and somewhat worse for some coaxial ones. The FWHM resolution at 2.6 MeV is between 2.6–4.0 keV for BEGe and 3.4–4.4 keV for coaxial detectors. The width of the strongest $\gamma$ lines in the physics data (1525 keV from 42K and 1460 keV from 40K) is found to be 0.5 keV larger than the expectation for the coaxial detectors (see Fig. 1). In order to estimate the expected energy resolution at $Q_{\beta\beta}$ an additional noise term is added to take this into account.

For $0\nu\beta\beta$ decays in the active part of a detector volume, the total energy of $Q_{\beta\beta}$ is detected in 92 % of the cases in this detector. Multiple detector coincidences are therefore discarded as background events. Two consecutive candidate events within 1 ms are also rejected (dead time \sim$$10^{-4}) to discriminate time-correlated decays from primordial radioisotopes, as e.g. the radon progenies 214Bi and 214Po. Candidate events are also refuted if a muon trigger occurred within 10 $\upmu$s prior to a germanium detector trigger. More than 99 % of the muons that deposit energy in a germanium detector are rejected this way. The induced dead time is $<$0.1 %.

The traces from PMTs and SiPMs are analyzed offline to search for LAr scintillation signals in coincidences with a germanium detector trigger. An event is rejected if any of the light detectors record a signal of amplitude above 50 % of the expectation for a single photo-electron within 5 $\upmu$s from the germanium trigger. 99 % of the photons occur in this window. Accidental coincidences between the LAr veto system and germanium detectors create a dead time of $(2.3\pm 0.1)$ % which is measured with test pulse events and cross checked with the counts in the 40K peak.

Fig. 2 shows the energy spectra for BEGe and coaxial detectors of Phase II with and without the LAr veto cut. Below \sim$$500 keV the spectra are dominated by 39Ar $\beta$ decays, up to 1.7 MeV by events from double beta decay with two neutrino emission ($2\nu\beta\beta$), above 2.6 MeV by $\alpha$ decays on the detector surface and around $Q_{\beta\beta}$ by a mixture of $\alpha$ events, 42K $\beta$ decays and those from the 238U and 232Th decay chains. The two spectra are similar except for the number of $\alpha$ events which is on average higher for coaxial detectors. The number of $\alpha$ counts shows a large variation between the detectors. The power of the LAr veto is best demonstrated by the 42K line at 1525 keV which is suppressed by a factor $\sim$5 (see inset) due to the $\beta$ particle depositing up to 2 MeV energy in the LAr. The figure also shows the predicted $2\nu\beta\beta$ spectrum from 76Ge using our Phase I result for the half-life of $T_{1/2}^{2\nu}=(1.926\pm 0.094)\cdot 10^{21}$ yr Agostini et al. (2015d).

The time profile of the germanium detector current signal is used to discriminate $0\nu\beta\beta$ decays from background events. While the former have point-like energy deposition in the germanium (single site events, SSE), the latter have often multiple depositions (multi site events, MSE) or depositions on the detector surface. The same pulse shape discrimination (PSD) techniques of Phase I Agostini et al. (2013b) are applied.

Events in the double escape peak (DEP) and at the Compton edge of 2615 keV gammas in calibration data have a similar time profile as $0\nu\beta\beta$ decays and are hence proxies for SSE. These samples are used to define the PSD cuts and the related detection efficiencies. The latter are cross checked with $2\nu\beta\beta$ decays.

The geometry of BEGe detectors allows to apply a simple mono-parametric PSD based on the maximum of the detector current pulse $A$ normalized to the total energy $E$ Budjáš et al. (2009); Agostini et al. (2011b). The energy dependence of the mean and the resolution $\sigma_{ae}$ of $A/E$ are measured for every detector with calibration events. After correcting for these dependences and normalizing the mean $A/E$ of DEP events to 1, the acceptance range is determined for each detector individually: the lower cut is set to keep 90 % of DEP events and the upper position is twice the low-side separation from 1. Fig. 3 shows a scatter plot of the PSD parameter $\zeta=(A/E-1)/\sigma_{ae}$ versus energy and the projection to the energy axis. Events marked in red survive the PSD selection. Below 1.7 MeV $2\nu\beta\beta$ events dominate with a survival fraction of $(85_{-1}^{+2})$ %. The two potassium peaks and Compton scattered photons reconstruct at $A/E<1$ (below the SSE band). All 234 $\alpha$ events at higher energies exhibit $A/E>1$ and are easily removed. The average $0\nu\beta\beta$ survival fraction Wagner (2017) is $(87\pm 2)$ %. The uncertainty takes into account the systematic difference between the $A/E$ centroids of DEP and $2\nu\beta\beta$ events and different fractions of MSE in DEP and $0\nu\beta\beta$ events.

For coaxial detectors a mono-parametric PSD is not sufficient since SSE do not have a simple signature Agostini et al. (2013b). Instead two neural network algorithms are applied to discriminate SSE from MSE and from $\alpha$ surface events. The first one is identical to the one used in Phase I. The cut on the neural network qualifier is set to yield a survival fraction of DEP events of 90 % for each detector. For the determination of the $0\nu\beta\beta$ efficiency, $2\nu\beta\beta$ events in physics data and a complete Monte Carlo simulation Kirsch (2014) of physics data and calibration data are used. The simulation considers the detector and the electronics response to energy depositions including the drift of charges in the crystal Bruyneel et al. (2016). We find a survival fraction for $0\nu\beta\beta$ events of $(85\pm 5)$ % where the error is derived from variations of the simulation parameters.

The second neural network algorithm is applied for the first time and identifies surface events on the p$+$ contact. Training is done with physics data from two different energy intervals. After the LAr veto cut events in the range 1.0–1.3 MeV are almost exclusively from $2\nu\beta\beta$ decay and hence signal-like. Events above 3.5 MeV are almost all from $\alpha$ decays on the p$+$ electrode and represent background events in the training. As $0\nu\beta\beta$ efficiency we measure a value of $(93\pm 1)$ % for a $2\nu\beta\beta$ event sample not used in the training. The combined PSD efficiency for coaxial detectors is $(79\pm 5)$ %.

IV Results

This analysis includes the data sets used in the previous publication Agostini et al. (2013a, 2015e), an additional coaxial detector period from 2013 (labeled “PI extra”) and the Phase II data from December 2015 until June 2016 (labeled “PIIa coaxial” and “PIIa BEGe”). Table 1 lists the relevant parameters for all data sets. The exposures in the active volumes of the detectors for 76Ge are 234 and 109 mol$\cdot$yr for Phase I and II, respectively. The efficiency is the product of the 76Ge isotope fraction (87 %), the active volume fraction (87–90 %), the $0\nu\beta\beta$ event fraction reconstructed at full energy in a single crystal (92 %), pulse shape selection (79–92 %) and the live time fraction (97.7 %). For the Phase I data sets the event selection including the PSD classification is unchanged. An improved energy reconstruction Agostini et al. (2015c) is applied to the data as well as an updated value for the coaxial detector PSD efficiency of the neural network analysis of $(83\pm 3)$ % Kirsch (2014).

Fig. 4 shows the spectra for the combined Phase I data sets and the two Phase II sets. The analysis range is from 1930 to 2190 keV without the intervals $(2104\pm 5)$ keV and $(2119\pm 5)$ keV of known peaks predicted by our background model Agostini et al. (2014a). For the coaxial detectors four events survive the cuts which means that the background is reduced by a factor of three compared to Phase I (see ’PI golden’ in Tab. 1). Due to the better PSD performance, only one event remains in the BEGe data which corresponds to a background of $0.7_{-0.5}^{+1.1}\cdot 10^{-3}$ $\text{cts}/(\text{keV}\cdot\text{kg}\cdot\text{yr})$. Consequently, the Phase II background goal is reached.

We perform both a Frequentist and a Bayesian analysis based on an unbinned extended likelihood function Agostini et al. (2015e). The fit function for every data set is a flat distribution for the background (one free parameter per set) and for a possible signal a Gaussian centered at $Q_{\beta\beta}$ with a width according to the corresponding resolution listed in Tab. 1. The signal strength is calculated for each set according to its exposure, efficiency and the inverse half-life $1/T$ which is a common free parameter.

Systematic uncertainties like a 0.2 keV uncertainty of the energy scale at $Q_{\beta\beta}$ are included in the analysis as pull terms in the likelihood function. The implementation takes correlations into account.

The Frequentist analysis uses the Neyman construction of the confidence interval and the standard two-sided test statistics Olive et al. (2014); Cowan et al. (2011) with the restriction to the physical region $1/T\geq 0$: the frequency distribution of the test statistic is generated using Monte Carlo simulations for different assumed $1/T$ values. The limit was determined by finding the largest value of $1/T$ for which at most 10 % of the simulated experiments had a value of the test statistic more unlikely than the one measured in our data (see Extended Data Fig. 5). Details of the statistical analysis can be found in the appendix. The best fit yields zero signal events and a 90 % C.L. limit of 2.0 events in 34.4 kg$\cdot$yr total exposure or

[TABLE]

The (median) sensitivity assuming no signal is $4.0\cdot 10^{25}$ yr (see Extended Data Fig. 5). The systematic errors weaken the limit by $<$1 %.

The Bayesian fit yields for a prior flat in $1/T$ between 0 and $10^{-24}$ yr*-1* a limit of $T_{1/2}^{0\nu}>3.5\cdot 10^{25}$ yr (90 % C.I.). The sensitivity assuming no signal is $3.1\cdot 10^{25}$ yr.

V Discussion

The second phase of Gerda collects data since December 2015 in stable conditions with all channels working. The background at $Q_{\beta\beta}$ for the BEGe detectors is $(0.7_{-0.5}^{+1.1})\cdot 10^{-3}$ $\text{cts}/(\text{keV}\cdot\text{kg}\cdot\text{yr})$. This is a major achievement since the value is consistent with our ambitious design goal.

We find no hint for a $0\nu\beta\beta$ decay signal in our combined data and place a limit of $T_{1/2}^{0\nu}({\rm{}^{76}Ge})>5.3\cdot 10^{25}$ yr (90 % C.L., sensitivity $4.0\cdot 10^{25}$ yr). For light Majorana neutrino exchange and a nuclear matrix element range for 76Ge between 2.8 and 6.1 Menendez et al. (2009); Horoi and Neacsu (2016); Barea et al. (2015); Hyvärinen and Suhonen (2015); Simkovic et al. (2013); Vaquero et al. (2013); Yao et al. (2015) the Gerda half-life limit converts to $m_{\beta\beta}<$0.15–0.33 eV (90 % C.L.).

We expect only a fraction of a background event in the energy region of interest (1 FWHM) at design exposure of 100 kg$\cdot$yr. Gerda is hence the first “background free” experiment in the field. Our sensitivity grows therefore almost linearly with time instead of by square root like for competing experiments and reaches $10^{26}$ yr for the half-life limit within 3 years of continuous operation. With the same exposure we have a 50 % chance to detect a signal with $3\sigma$ significance if the half-life is below $10^{26}$ yr.

Phase II has demonstrated that the concept of background suppression by exploiting the good pulse shape performance of BEGe detectors and by detecting the argon scintillation light works. The background at $Q_{\beta\beta}$ is at a world-best level: it is lower by typically a factor of 10 compared to experiments using other isotopes after normalization by the energy resolution and total efficiency $\epsilon$; i.e. (BI$\cdot$FWHM/$\epsilon$) is superior. This is the reason why the Gerda half-life sensitivity of $4.0\cdot 10^{25}$ yr for an exposure of 343 mol$\cdot$yr is similar to the one of Kamland-Zen for 136Xe of $5.6\cdot 10^{25}$ yr based on a more than 10-fold exposure of 3700 mol$\cdot$yr Gando et al. (2016).

A discovery of $0\nu\beta\beta$ decay would have far reaching consequences for our understanding of particle physics and cosmology. Key features for a convincing case are an ultra low background with a simple flat distribution, excellent energy resolution and the possibility to identify the events with high confidence as signal-like as opposed to an unknown $\gamma$-line from a nuclear transition. The latter is achieved by the detector pulse shape analysis and possibly a signature in the argon. The concept to operate bare germanium detectors in liquid argon has proven to have the best performance for a discovery which motivates future extensions of the program. The Gerda cryostat can hold 200 kg of detectors. Such an experiment will remain background-free until an exposure of 1000 kg$\cdot$yr provided the background can be further reduced by a factor of five. The discovery sensitivity would then improve by an order of magnitude to a half-life of $10^{27}$ yr. The 200 kg setup is conceived as a first step for a more ambitious 1 ton experiment which would ultimately boost the sensitivity to $10^{28}$ yr corresponding to the $m_{\beta\beta}<$10–20 meV range. Both extensions are being pursued by the newly formed LEGeND Collaboration (http://www.legend-exp.org)

Appendix A 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 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.

Appendix B Appendix: Statistical Methods

This section discusses the statistical analysis of the Gerda data. In particular, the procedures to derive the limit on $T^{0\nu}_{1/2}$, the median sensitivity of the experiment and the treatment of systematic uncertainties are described.

A combined analysis of data from Phase I and II is performed by fitting simultaneously the six data sets of Table 1. The parameter of interest for this analysis is the strength of a possible $0\nu\beta\beta$ decay signal: ${\cal S}=1/T^{0\nu}_{1/2}$. The number of expected $0\nu\beta\beta$ events in the $i$-th data set $\mathcal{D}_{i}$ as a function of ${\cal S}$ is given by:

[TABLE]

where $N_{A}$ is Avogadro’s number, $\epsilon_{i}$ the global signal efficiency of the $i$-th data set, ${\cal E}_{i}$ the exposure and $m_{a}$ the molar mass. The exposure quoted is the total detector mass multiplied by the data taking time. The global signal efficiency accounts for the fraction of 76Ge in the detector material, the fraction of the detector active volume, the efficiency of the analysis cuts, the fractional live time of the experiment and the probability that $0\nu\beta\beta$ decay events in the active detector volume have a reconstructed energy at $Q_{\beta\beta}$. The total number of expected background events as a function of the background index $\text{BI}_{i}$ is:

[TABLE]

where $\Delta E$=240 keV is the width of the energy region around $Q_{\beta\beta}$ used for the fit.

Each data set $\mathcal{D}_{i}$ is fitted with an unbinned likelihood function assuming a Gaussian distribution for the signal and a flat distribution for the background:

[TABLE]

where $E_{j}$ are the individual event energies, ${N^{obs}_{i}}$ is the total number of events observed in the $i$-th data set, $\sigma_{i}=\text{FWHM}_{i}/(2\sqrt{2\ln{2}})$ is the energy resolution and $\delta_{i}$ is a possible systematic energy offset. The parameters with systematic uncertainties are indicated with $\theta_{i}=\{\epsilon_{i},\sigma_{i},\delta_{i}\}$. The parameters ${\cal S}$ and $\text{BI}_{i}$ are bound to positive values. The total likelihood is constructed as the product of all $\mathcal{L}_{i}$ weighted with the Poisson terms pdg :

[TABLE]

where $\bm{\mathcal{D}}=\{\mathcal{D}_{1}\dots\mathcal{D}_{i}\dots\}$, $\text{\bf BI}=\{\text{BI}_{1}\dots\text{BI}_{i}\dots\}$ and $\bm{\theta}=\{\theta_{1}\dots\theta_{i}\dots\}$.

A frequentist analysis is performed using a two-sided test statistics Cowan et al. (2011) based on the profile likelihood $\lambda$ (${\cal S}$):

[TABLE]

where $\hat{\hat{\text{\bf BI}}}$ and $\hat{\hat{\bm{\theta}}}$ in the numerator denote the value of the parameters that maximizes $\mathcal{L}$ for a fixed ${\cal S}$. In the denominator, $\hat{{\cal S}}$, $\hat{\bm{\text{BI}}}$ and $\hat{\bm{\theta}}$ are the values corresponding to the absolute maximum likelihood.

The confidence intervals are constructed for a discrete set of values ${\cal S}\in\{{\cal S}_{j}\}$. For each ${\cal S}_{j}$, possible realizations of the experiments are generated via Monte Carlo according to the parameters of Table 1 and the expected number of counts from Eqs. 2 and 3. For each realization ${t_{\cal S}}_{j}$ is evaluated. From the entire set the probability distribution $f(t_{{\cal S}}|{\cal S}_{j})$ is calculated. The p-value of the data for a specific ${\cal S}_{j}$ is computed as:

[TABLE]

where $t_{obs}$ is the value of the test statistics of the Gerda data for ${\cal S}_{j}$. The values of $p_{{\cal S}_{j}}$ are shown by the solid line in Extended Data Fig. 5. The 90 % C.L. interval is given by all ${\cal S}_{j}$ values with $p_{{\cal S}_{j}}>0.1$. Such an interval has the correct coverage by construction. The current analysis yields a one-sided interval, i.e. a limit of $T^{0\nu}_{1/2}=1/{\cal S}>5.3\cdot 10^{25}$ yr.

The expectation for the frequentist limit (i.e. the experimental sensitivity) was evaluated from the distribution of $p_{{\cal S}_{j}}$ built from Monte Carlo generated data sets with no injected signal (${\cal S}=0$). The distribution of $p_{{\cal S}_{j}}$ is shown in Extended Data Fig. 5: the dashed line is the median of the distribution and the color bands indicate the 68 % and 90 % probability central intervals. The experimental sensitivity corresponds to the ${\cal S}$ value at which the median crosses the p-value threshold of 0.1 : $T^{0\nu}_{1/2}>4.0\cdot 10^{25}$ yr (90% C.L.).

Systematics uncertainties are folded into the likelihood by varying the parameters $\theta_{i}$ in the fits and constraining them by adding to the likelihood multiplicative Gaussian penalty terms. The central values and the standard deviations of the penalty terms for $\epsilon_{i}$ and $\sigma_{i}$ are taken from Table 1. The penalty term on $\delta_{i}$ has a central value equal to zero and standard deviation of 0.2 keV.

Instead of the two-sided test statistics one can use a one-sided test statistic defined as Cowan et al. (2011):

[TABLE]

By construction $\tilde{t}_{{\cal S}}=0$ for ${\cal S}=0$ for all realizations and consequently ${\cal S}=0$ is always included in the 90 % C.L. interval, i.e. the one-sided test statistic will always yield a limit. In our case the resulting limit would be 50% stronger. Similar to other experiments Albert et al. (2014); Gando et al. (2016), we want to be able to detect a possible signal and thus we decided a priori to adopt the two-sided test statistic. It is noteworthy that, although the coverage of both test statistics is correct by construction, deciding which one to use according to the outcome of the experiment would result in the flip-flop issue discussed by Feldman and Cousins Feldman:1997qc .

The statistical analysis is also performed within a Bayesian framework. The combined posterior probability density function (PDF) is calculated from the six data sets according to Bayes’ theorem:

[TABLE]

The likelihood $\mathcal{L}$ is given by Eq. (5), while $\mathcal{P}({\cal S})$ and $\mathcal{P}(\text{BI}_{i})$ are the prior PDFs for ${\cal S}$ and for the background indices, respectively. The one-dimensional posterior PDF $\mathcal{P}({\cal S}|\bm{\mathcal{D}},\bm{\theta})$ of the parameter ${\cal S}$ of interest is derived by marginalization over all nuisance parameters BI. The marginalization is performed by the BAT toolkit bat via a Markov chain Monte Carlo numerical integration. A flat PDF between 0 and 0.1 $\text{cts}/(\text{keV}\cdot\text{kg}\cdot\text{yr})$ is considered as prior for all background indices. As in Ref. Agostini et al. (2013a), a flat prior distribution is taken for ${\cal S}$ between 0 and $10^{-24}$ /yr, i.e. all counting rates up to a maximum are considered to be equiprobable. The parameters $\bm{\theta}$ in the likelihood $\mathcal{L}$ are fixed during the Bayesian analysis and the uncertainties are folded into the posterior PDF as last step by an integral average:

[TABLE]

with $g(\theta_{i})$ being Gaussian distributions like for the frequentist analysis. The integration is performed numerically by a Monte Carlo approach.

The median sensitivity of the experiment in the case of no signal is $T^{0\nu}_{1/2}>3.1\cdot 10^{25}$ yr (90% C.I.). The posterior PDF $\langle\mathcal{P}({\cal S}|\bm{\mathcal{D}})\rangle$ for our data has an exponential shape with the mode at ${\cal S}=0$. Its 90 % probability quantile yields $T^{0\nu}_{1/2}>3.5\cdot 10^{25}$ yr.

As in any Bayesian analysis, results depend on the choice of the priors. For our limit we assume all signal count rates to be a priori equiprobable. Alternative reasonable choices are for instance: equiprobable Majorana neutrino masses, which yields a prior proportional to $1/\sqrt{{\cal S}}$; or scale invariance in the counting rate, namely a flat prior in $\log({\cal S})$. The limits derived with these assumptions are significantly stronger (50 % or more), since for both alternatives the prior PDFs increase the probability of low ${\cal S}$ values.

The systematic uncertainties weaken the limit on ${\cal S}$ by less than 1% both in the frequentist and Bayesian analysis. In general, the impact of systematic uncertainties on limits is marginal in the low-statistics regime that characterizes our experiment (see also Ref. cousins ).

The limit derived from the Gerda data is slightly stronger than the median sensitivity. This effect is more significant in the frequentist analysis as one would expect, see e.g. Ref. biller for a detailed discussion. The probability of obtaining a frequentist (Bayesian) limit stronger than the actual one is 33 % (35 %).