Limit on the radiative neutrinoless double electron capture of 36\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{36}$$\end{document}Ar from GERDA Phase I

Neutrinoless double electron capture is a process that, if detected, would give evidence of lepton number violation and the Majorana nature of neutrinos. A search for neutrinoless double electron capture of 36\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{36}$$\end{document}Ar has been performed with germanium detectors installed in liquid argon using data from Phase I of the GERmanium Detector Array (Gerda) experiment at the Gran Sasso Laboratory of INFN, Italy. No signal was observed and an experimental lower limit on the half-life of the radiative neutrinoless double electron capture of 36\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{36}$$\end{document}Ar was established: T1/2>\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{1/2} > $$\end{document} 3.6 ×\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\times $$\end{document} 1021\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{21}$$\end{document} years at 90% CI.

(Gerda) experiment at the Gran Sasso Laboratory of INFN, Italy. No signal was observed and an experimental lower limit on the half-life of the radiative neutrinoless double electron capture of 36 Ar was established: T 1/2 > 3.6 × 10 21 years at 90% CI.

Introduction
The observation of neutrinoless double beta decay (0νββ): can provide unambiguous information on lepton number violation and indicate the Majorana nature of neutrinos, regardless the physics mechanism responsible for the decay. Currently many experiments are searching for this decay considering different isotopes. Among these there is the Gerda (GERmanium Detector Array) experiment [1] implementing bare germanium detectors enriched in 76 Ge. This experiment searches for neutrinoless double beta decay of 76 Ge. Recently the best limit on 0νββ decay halflife of 76 Ge has been published by the Gerda collaboration [2]. Another lepton number violating process that can provide the same information as neutrinoless double beta decay is the double capture of two bound atomic electrons without the emission of neutrinos (0νECEC): where the quantity Q corresponds to the energy difference between the ground state atoms (A, Z + 2) and (A, Z ) [3,4]. While in the corresponding process where two neutrinos are emitted (2νECEC) the available energy of the decay is carried away by neutrinos plus X-rays or Auger electrons, in the neutrinoless double electron capture the decay must be accompanied by the emission of at least another particle to ensure energy and momentum conservation. Different modes can be considered in which 0νECEC decay is associated with the emission of different particles like e + e − pairs, one or two photons, or one internal conversion electron. A detailed discussion about double electron capture processes can be found in Refs. [5][6][7].
For 0 + → 0 + transitions the capture of two K -shell electrons with the emission of only one photon is forbidden because of angular momentum conservation. Therefore, the most likely process is the capture from the K -and the L-shell. The diagram of this mode is depicted in Fig. 1. The unstable daughter atom relaxes by emission of X-rays or Auger electrons.
At present, only two experiments found an indication of two neutrino double electron capture. The first is based on a  [8,9] and the second is a large-volume copper proportional counter searching for double K-shell capture in 78 Kr [10]. Several experiments including the latter established limits on both neutrino accompanied and neutrinoless double electron capture of different isotopes (see Refs. [10][11][12][13][14][15][16][17]). For some isotopes the possibility of a resonant enhancement of the 0νECEC decay has been predicted in case of mass degeneracy between the initial state and an excited final state [4,18]. 36 Ar is expected [19] to undergo double electron capture to the ground state of 36 S. The available energy [20] of the decay is 432.58 ± 0.19 keV and, therefore, both the radiative and the internal conversion modes are energetically allowed. A resonance enhancement of the decay is not possible for this isotope. Calculations based on the quasiparticle random-phase approximation (QRPA) predict a half-life for 36 Ar in the order of 10 38 years for an effective Majorana neutrino mass of 1 eV [21]. So far, an experimental limit on the radiative mode obtained during detector characterizations in the Gerda Detector Laboratory has been published (T 1/2 > 1.9×10 18 years at 68% CL) [22].
The radiative mode of 0νECEC in 36 Ar with the emission of one photon provides a clear signature through the discrete value of its energy and allows the detector to be separate from the source of the decay. Two cascades of characteristic X-rays with energies of E K = 2.47 keV and E L = 0.23 keV are emitted [23], corresponding to the capture of the electrons from the K -and the L-shell, respectively. The uncertainties for the energies of the X-rays amount to 0.4 eV. The corresponding energy for the monochromatic photon is This paper reports the search for the 429.88 keV γ line from 0νECEC decay of 36 Ar with Gerda Phase I germanium detectors and the determination of a limit on its half-life.

The GERDA experiment
The Gerda experiment [1] [27]. The bare detectors are immersed into a cryostat containing 64 m 3 (89.2 t) of LAr, which acts both as the coolant medium and a shield against external radiation. The isotopic abundance of 36 Ar in natural argon is 0.3336(4)% [28], which sums up to about 298 kg of 36 Ar. An additional shield of ultra pure water (10 m in diameter) surrounds the cryostat containing the argon. The water tank is instrumented with 66 PMTs as a muon Cherenkov veto [29]. Each detector string is surrounded by a 60 µm thick Cu foil ("mini-shroud"), to limit drifting of 42 K ions to detector surfaces. In addition, to mitigate radon contamination, a 30 µm Cu cylinder ("radon shroud") surrounds the array of strings.

Data taking and data selection
The data taking of Gerda Phase I started in November 2011 and ended in May 2013. Until March 2012, the setup included 8 enr Ge semi-coaxial and 3 nat Ge semi-coaxial detectors. Two months later, two of the nat Ge semi-coaxial detectors were replaced by five new enr Ge BEGe detectors. After this insertion a higher background was observed. Therefore a period of 49 days was excluded from this analysis. The data taking was separated into runs, with a duration of about one month each. Detectors which showed instabilities during specific runs where removed from the analysis. Two detectors showed instabilities from the very beginning of data taking. There-fore, data collected from these detectors were discarded. The total collected data used for the search for 0νECEC of 36 Ar correspond to a life time of about 460 d. The data were divided into three different data sets, one containing data from natural semi-coaxial detectors (labeled as nat Coax), one containing data from enriched semi-coaxial detectors ( enr Coax) and the last containing data collected by BEGe detectors ( enr BEGe). The energy spectra from the three data sets are shown in Fig. 2. The left panel shows the energy spectra weighted with the product of life time and detector mass. The right panel displays the energy region between 360 and 500 keV. Indeed, in the region around 429.88 keV, enriched and natural detectors are characterized by different contributions to the spectrum, in particular due to 2νββ decays from 76 Ge in the enriched ones. In addition, BEGe detectors are considered as a separate data set because of the improved energy resolution with respect to semi-coaxial detectors. The main contribution to the spectrum around 430 keV is due to 39 Ar β decays. The spectral shape is different for BEGe detectors due to the different detector geometry and outer dead layer thickness. Offline reconstruction of Gerda data was performed within the Gelatio software framework [30]. Detector signals are read out by charge sensitive preamplifiers and then digitized by 100 MHz flash analog to digital converters (FADCs). Preceded by a ∼80 µs long baseline, the charge signal rises up with a rise time of ∼1 µs and is followed by a ∼80 µs long exponential tail. The energy of each event is estimated by applying an optimized Zero Area Cusp filter [31] to the digitized signal. Cuts based on the baseline slope, the number of triggers and the position of the rising edge were applied to remove pile-up events and accidental coincidences. All detected events within 10 µs from the muon veto trigger were also rejected. Finally, an anti-coincidence cut was applied to remove events with an energy deposition in more than one detector.
The energy calibration was performed during dedicated calibration runs (every one or two weeks) in which three 228 Th sources were lowered to the vicinity of the detectors. In addition, the stability of the system was continuously monitored by injecting test charge pulses into the input of the preamplifiers. The energy dependence of the resolution was obtained for each data set from the summed calibration spectra and then the value at the signal peak position of 429.88 keV was derived. The 42 K background γ line at 1524.7 keV in the physics data was used to determine a correction factor in case its energy resolution differed more than one standard deviation from the one obtained during the calibrations. To combine the different values into a single value for the data set, the average of the energy resolution of each detector was calculated weighted with the signal detection efficiency of the detector. The uncertainty on the resolution is primarily coming from the fit of the resolution curve and is largest for the detectors that require the correction factor [32]. The expected Full Width at Half Maximum (FWHM) value at 429.88 keV is 4.08 ± 0.20 keV for the nat Coax, 3.72 ± 0.05 keV for the enr Coax and 2.01 ± 0.10 keV for the enr BEGe data set. The systematic uncertainty on the FWHM, estimated by comparing the resolution of the summed calibration spectra to the average resolution of the single calibrations, is ± 0.05 keV.

Determination of the half-life of 0νECEC of 36 Ar
A limit on the half-life T 1/2 of 0νECEC decay of 36 Ar was determined considering the data of Gerda Phase I discussed in Sect. 3. The region of interest (ROI) around the signal, the 429.88 keV γ line from the 0νECEC decay, is defined between 410 and 450 keV. The energy spectrum of coincidence events shows the presence of the three γ lines from 108m Ag [33]. 108m Ag has a half-life of 418 years and undergoes electron capture into the 6 + excited state of 108 Pd with a probability of 91.3%. The de-excitation of the daughter nucleus leads to three equally probable γ rays in the final state, with energies of 433.9, 614.3 and 722.9 keV. The presence of 108m Ag was also observed in the screening measurements. For these reasons the 433.9 keV γ line from 108m Ag in the ROI was taken into account in the analysis. The determination of the detection efficiency and the analysis result are discussed in the following.

Detection efficiency
The detection efficiency ε is defined as the number of γ rays which entirely deposit their energy inside a single Gerda detector. It has been determined by Monte Carlo simulations employing the MaGe software framework [34] based on Geant4 [35,36]. 10 9 γ rays with an energy of 429.88 keV were generated in a cylindrical LAr volume with a radius of 67 cm and a height of 130 cm, centered around the detector array. The considered volume corresponds to 1827 l of LAr equivalent to 7.7 kg of 36 Ar. A measurement with mass spectrometer showed no difference in the 36 Ar abundance between atmospheric and liquid phase within the instrumental sensitivity of 0.5%. The LAr density variation due to temperature differences in the cryostat is less than 1.5% [37]. Within statistical precision both systematic effects are negligible on the limit estimation. The contribution from γ rays originating from outside this volume to the number of full energy depositions is less than the statistical uncertainty of 0.2%. The full efficiency for each data set was derived by summing up the individual detector efficiencies weighted for the life time of each run. Their systematic uncertainty is dominated by two main contributions: the uncertainty on the Monte Carlo processes, whose effect on the efficiency was estimated to be 4%, and the uncertainty on the dead layer of the germanium detectors. The latter was estimated by independently varying for each detector the dead layer values within ±1 standard deviation. This changes the efficiency of 8-10 % for a single semi-coaxial detector and 3.5-6% for a single BEGe detector. The uncertainty for the three data sets, calculated assuming full correlation among the uncertainties of individual detectors, is 9.2 % for the nat Coax and enr Coax data sets and 4.5 % for the enr BEGe data set. The total systematic uncertainty on the efficiency is obtained by summing in quadrature the two contributions and amounts to 10% for the nat Coax data set, 10% for the enr Coax data set and 6% for the enr BEGe data set.
Statistical uncertainties are negligible with respect to systematic ones. The efficiencies are quoted in Table 1.

Analysis
The expected signal counts S d from neutrinoless double electron capture from dataset d are related to the half-life T 1/2 according to the following relation where ε d is the signal detection efficiency for data set d, N A is the Avogadro constant, t is the total life time, M L Ar is the mass of the LAr volume that was used for the efficiency determination, f 36 the abundance of 36 Ar and m Ar the molar mass of argon.  The unbinned likelihood function is defined as where the product runs over all data sets d and events i. N d is the total number of events in the data set.
is the extended probability density of finding an event with energy E d,i in dataset d with a given set of parameters p d . μ d represents the total number of expected events in dataset d over the whole energy range In the region of interest the background is in good approximation linear. Therefore, λ d,i can be described as the sum of a linear background contribution plus a peak from 108m Ag and the signal peak from 0νECEC of 36 Ar where σ d is the energy resolution (FWHM = 2.35 · σ d ), δ E a possible systematic shift in energy scale. B 0,d and B 1,d describe the linear background and B Ag,d the count expectation of the 108m Ag peak. A Bayesian approach was used to extract the posterior probability density on T −1 1/2 . In total, the fit has 17 floating parameters, six describing the signal peak (ε d , σ d ), six for the linear background (B 0,d , B 1,d ), three for the 108m Ag peak (B Ag,d ). T −1 1/2 and δ E are in common to all data sets. The parameters ε d , σ d and δ E are constrained by Gaussian shaped prior distributions whose standard deviation is given by their systematic uncertainty. A flat prior is considered for the remaining parameters, including the inverse half-life T −1 1/2 . Furthermore, B 0,d , B Ag,d and T −1 1/2 are bound to positive values, while B 1,d is bound to negative values. The best fit is defined as the mode of the global posterior probability density and yields T −1 1/2 = 0, i.e. no signal events from 0νECEC. The 90% credibility limit of the half-life, defined as the 90% quantile of the marginalized posterior distribution, is T 1/2 > 3.6 · 10 21 yr (90%CI).
The median sensitivity for the 90% CI limit was estimated with toy Monte Carlo simulations and is equal to 2.7 · 10 21 years. The sum spectrum of all data sets around the ROI and the fit functions are displayed in Fig. 3 together with the marginalized posterior distribution for T −1 1/2 . Systematic uncertainties are directly folded into the fit through the Gaussian priors associated to parameters ε d , σ d and δ E . They weaken the limit by about 0.3%, which was evaluated by fixing these 7 parameters and repeating the fit with the remaining 10 parameters. To test if the model described in Eq. 5 is sufficient, the p value was calculated for the three data sets, as proposed in Ref. [38] using a 1 keV binning. The obtained values are 0.96, 0.11 and 0.91 for the nat Coax, enr Coax and enr BEGe data sets respectively and indicate that the model describes the data sufficiently well. The fit result for B Ag shows the presence of the 433.9 keV γ line in the nat Coax data set. The 90% CI limit is reduced by 10 % in case the presence of this line is neglected in the fit. The expectation value for the number of counts from the 108m Ag γ line for the three data sets is reported in Table 1 together with the fit result for B 0 which represents the number of events from the linear background component at the signal peak energy of 429.88 keV (third term of Eq. 5). In the same table the efficiency values and the energy resolution are also reported.

Conclusions
Gerda established the most stringent half-life limit on the radiative mode of neutrinoless double electron capture of 36 Ar with Phase I data. The limit is three orders of magnitude better than previous results for this isotope; however, it is still orders of magnitude far from the theoretical prediction from QRPA calculations.