Final results on 82Se\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}^\mathbf{82 }{\hbox {Se}}$$\end{document} double beta decay to the ground state of 82Kr\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}^\mathbf{82 }{\hbox {Kr}}$$\end{document} from the NEMO-3 experiment

Using data from the NEMO-3 experiment, we have measured the two-neutrino double beta decay (2νββ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2\nu \beta \beta $$\end{document}) half-life of 82\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{82}$$\end{document}Se as T1/22ν=9.39±0.17stat±0.58syst×1019\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{\smash {1/2}}^{2\nu } \!=\! \left[ 9.39 \pm 0.17\left( \text{ stat }\right) \pm 0.58\left( \text{ syst }\right) \right] \times 10^{19}$$\end{document} y under the single-state dominance hypothesis for this nuclear transition. The corresponding nuclear matrix element is M2ν=0.0498±0.0016\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left| M^{2\nu }\right| = 0.0498 \pm 0.0016$$\end{document}. In addition, a search for neutrinoless double beta decay (0νββ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0\nu \beta \beta $$\end{document}) using 0.93 kg of 82\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{82}$$\end{document}Se observed for a total of 5.25 y has been conducted and no evidence for a signal has been found. The resulting half-life limit of T1/20ν>2.5×1023y(90%C.L.)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{1/2}^{0\nu } > 2.5 \times 10^{23} \,\text{ y } \,(90\%\,\text{ C.L. })$$\end{document} for the light neutrino exchange mechanism leads to a constraint on the effective Majorana neutrino mass of ⟨mν⟩<1.2-3.0eV\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\langle m_{\nu } \rangle < \left( 1.2{-}3.0\right) \,\text{ eV }$$\end{document}, where the range reflects 0νββ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0\nu \beta \beta $$\end{document} nuclear matrix element values from different calculations. Furthermore, constraints on lepton number violating parameters for other 0νββ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0\nu \beta \beta $$\end{document} mechanisms, such as right-handed currents, majoron emission and R-parity violating supersymmetry modes have been set.


Introduction
The observation of neutrino oscillations has provided proof that the neutrino has non-zero mass [1][2][3]. However the absolute mass of the neutrino and its fundamental Dirac or Majorana nature remain undetermined. Neutrinoless double beta decay (0νββ) is the only practical way to establish the full lepton number violation required by many grand unification models and if the decay proceeds via a light neutrino exchange mechanism, would be one of the most sensitive probes of absolute neutrino mass [4].
The half-life of 0νββ is given by: where g A is the axial-vector coupling constant, G 0ν is a phase-space factor, M 0ν is a nuclear matrix element (NME) and ξ is a lepton number violating parameter. In the most commonly discussed mechanism of 0νββ, the decay proceeds via the exchange of a light Majorana neutrino ( ξ ≡ m ν /m e , where m e is the mass of the electron). However, other mechanisms are possible, such as the admixture of right-handed currents in the electroweak interaction, majoron emission and R-parity violating supersymmetry (SUSY). In all mechanisms, 0νββ violates lepton number conservation and is a direct probe of physics beyond the Standard Model.
To date, no evidence for 0νββ has been found, with the best half-life limits in the 10 24 -10 26 y range [5][6][7][8][9][10][11]. Two-neutrino double beta decay (2νββ) is a rare second order process that is allowed in the Standard Model. It has been observed in 12 isotopes with half-lives ranging from 10 19 to 10 24 y [12,13]. Measurement of the 2νββ half-life provides experimental determination of the NME for this process, M 2ν , which can be used to improve NME calculations for the 0νββ mode. The precision with which ξ can be measured depends crucially on knowledge of M 0ν . In addition, 2νββ is an irreducible background component to 0νββ and therefore precise measurements of 2νββ rates and spectral shapes are important.
One of the most promising double beta decay (ββ) candidates is 82 Se due to its high Q-value (2997.9(3) keV [14]), above most common backgrounds from natural radioactivity, relatively high isotopic abundance (8.83% [15]) and existing robust technologies of isotopic enrichment through centrifugation. It has been selected as the isotope of choice for a number of planned 0νββ decay experiments [16,17].
The first measurement of ββ in 82 Se was made in 1967 with a geochemical experiment, extracting a half-life of 0.6 +0.6 −0.3 × 10 20 y [18]. This result was later confirmed by many other geochemical measurements (see reviews [19][20][21]). Such geochemical experiments are not able to distinguish between 0νββ and 2νββ modes and the conclusion that 2νββ had been observed was drawn using complementary theoretical and experimental arguments. Whilst the precision of any individual measurement was reasonably good, the spread of the results was quite high. Nevertheless, the combination of many experiments led to a half-life value of (1.0−1.3) × 10 20 y [19][20][21].
The isotope of 82 Se was in fact the first nucleus in which 2νββ was directly observed in a counter experiment in 1987 [22]. A total of 36 candidate 2νββ events were observed yielding a half-life of 1.1 +0. 8 −0.3 × 10 20 y. A more precise direct measurement was later carried out by NEMO-2, [8.3 ± 1.0 (stat) ± 0.7 (syst) × 10 19 y [23]. The most precise result to date was obtained by NEMO-3 after analysing a subset of its data, 9.6 ± 0.3 (stat) ± 1.0 (syst) × 10 19 y [24]. The same data set was also used to obtain a stringent lower limit on the half-life for the 0νββ decay of 82 Se, T 0ν 1/2 > 1.0 × 10 23 y at 90% C.L. We present the results of the 82 Se 2νββ measurement and 0νββ searches with the full data set collected by the NEMO-3 detector, representing a fivefold increase in exposure compared to the previously published result [24].

NEMO-3 detector and 8Se source
NEMO-3 was a detector composed of a tracker and a calorimeter capable of reconstructing the full topology of ββ events. It was installed in the Modane Underground Laboratory (LSM) with an overburden of 4800 m.w.e. to shield against cosmic rays. The detector housed seven enriched ββ isotopes in the form of thin (about 50 mg/cm 2 ) source foils. These were arranged in a cylindrical geometry subdivided into 20 identical sectors. The two isotopes with the largest mass were 100 Mo (6.91 kg) and 82 Se (0.93 kg) with smaller quantities of 48 Ca, 96 Zr, 116 Cd, 130 Te and 150 Nd [8,[24][25][26][27][28]. Charged particle ionisation tracks are reconstructed from hits in 50 cm deep and 270 cm long wire chambers on each side of the source foils composed of 6180 Geiger cells operating in helium with the addition of ethanol as a quencher (4%), argon (1%) and water vapour (0.15%). The transverse and longitudinal resolution of individual tracker cells was 0.5 mm and 8.0 mm (σ ) respectively. The tracker was enclosed by calorimeter walls assembled from plastic scintillator blocks coupled to low background photomultipliers (PMT). The detector was calibrated by deploying 207 Bi, 90 Sr and 232 U sources during the course of data collection. The energy resolution of the calorimeter blocks was 5.8-7.2% and the time resolution was 250 ps, both σ at 1 MeV. The detector was surrounded by a solenoid which generated a 25 G magnetic field parallel to the cell wires. The magnetic field allows the rejection of approximately 95% of positrons at 1 MeV. The detector was placed in passive shielding consisting of a 19 cm thick layer of iron to suppress the external gamma ray background, as well as borated water, paraffin and wood to moderate and absorb the environmental neutron background. A detailed description of the detector and its calibration and performance can be found in [8,29].
The 82 Se source foils had a composite structure. Enriched 82 Se powder was mixed with polyvinyl alcohol (PVA) glue and deposited between 23 µm (2.2 mg/cm 2 ) thick Mylar foils. Enriched selenium from two production runs was used, attaining enrichment factors of 97.02 ± 0.05% for run 1 and 96.82 ± 0.05% for run 2. Selenium from run 1, which was also used in the NEMO-2 experiment [23], was placed in a single detector sector, while the isotope from run 2 was in an adjacent sector. The total mass of the 82 Se isotope in NEMO-3 was (0.932 ± 0.005) kg, with 0.464 kg from run 1 and 0.468 kg from run 2.
NEMO-3 took data from February 2003 to January 2011. A standard set of criteria define high quality runs, where the detector was operating stably and the calorimeter was calibrated [8]. The accepted live-time of the detector is 5.252 y, resulting in an exposure of 4.90 kg·y for 82 Se.
During the first 18 months of data-taking, the radon ( 222 Rn) level inside the detector was higher than anticipated. This was caused by the diffusion of radon from the air of the laboratory into the tracking gas. To lower the radon level inside the detector, an anti-radon tent containing filtered air was built around the detector reducing the radon level in the tracker volume by a factor of about 6 [30]. The higher radon activity data-taking period, lasting 1.06 y, is referred to as phase 1 and the lower activity period, with a duration of 4.19 y, as phase 2.

Particle identification and event selection
One of the major strengths of the NEMO-3 approach amongst ββ experiments is its ability to use multiple observables and a combination of tracking and calorimetry information for particle identification and reconstruction of different event topologies. By separating data events into different channels based on the number of electrons, γ -rays and α-particles that they contain, a pure ββ signal channel can be defined along with a series of background channels that may be used to normalise the different background contributions to this signal channel.
Electrons and positrons are identified by ionisation traces that can be extrapolated to an energy deposit in the calorimeter, and are distinguished by their curvature in the magnetic field. By contrast, γ -rays are identified as an energy deposit in the calorimeter without an associated track. A 1 MeV photon has a 50% probability of interaction with a scintillator block. Therefore neighbouring calorimeter hits are clustered together and attributed to a single γ -ray interaction event with an energy equal to the energy sum of the individual hits. Due to their heavy ionisation energy losses, α-particles from radioactive decays can not travel more than about 35 cm in the NEMO-3 tracker and are identified by their short, straight tracks.
Both data and Monte Carlo simulations (MC) of signal and background are processed by the same reconstruction algorithm. The DECAY0 event generator [31] is used for generation of initial kinematics and particles are tracked through a detailed GEANT3 based detector simulation [32].
Candidate ββ signal events are selected to contain two electron tracks, each with an energy deposit > 300 keV. The tracks must originate from the 82 Se source foil and have a common vertex (i.e. the distance between the track intersections with the foil should be Δ XY < 2 cm (transversely) and Δ Z < 4 cm (vertically), set by the resolution of the tracking detector). There should be no α-particle tracks in the event. The timing of the calorimeter hits must be consistent with an internal event defined as two electrons simultaneously emitted from a common vertex in the foil [8].
Backgrounds are constrained using specific event topologies and timing characteristics. Single electron candidate events (1e) must have one electron track originating from a 82 Se source foil. The position of these intersections are used to identify areas in the source foils with higher than average contaminations as shown in Fig. 1. Areas with an event rate more than 5σ higher than the mean rate for the foil strip in which it is housed are excluded from the data analysis.
The 1e1α N γ channel events contain a single electron track and a delayed α-particle track emitted from a common vertex, with no constraints on the number of γ -rays present. The α-particle track must be registered in the range of (10-650) µs after the electron track, such that it is consistent with 214 Bi → 214 Po → 210 Pb sequential decays from the 238 U radioactive series. These decays predominantly originate from radon in the tracker as outlined in Sect. 4.
Events with a single electron track and a number of γray hits (1eN γ ) are used to constrain different backgrounds depending on the number of γ -rays and their timing characteristics. As with electron candidates, γ -ray hits must have an energy deposit > 300 keV to be accepted. Events containing electron and γ -ray hits consistent with simultane-Sector Number 6 6.5  The sectors of the detector containing the 82 Se source foils, imaged in the 1e channel. a Shows the reconstructed vertex from all single electron events. b Shows the same region after removing events originating from neighbouring foils, calibration tubes and areas with activity more than 5σ higher than the mean for the particular 82 Se foil strip. This is the fiducial area used in the analysis. The higher activity strip in sector 8 is contaminated with 210 Bi which does not affect the ββ analyses due to its low Q-value ous emission from the same location in a 82 Se foil are used to measure internal contamination by radioactive isotopes. Conversely, those containing hit times consistent with a γray first interacting with a calorimeter block before producing an electron in the foil are used to measure the external γ -ray flux. Finally, crossing-electron events, where a single electron crosses from one side of the detector to the other, are selected using the same cuts as for the ββ channel but with a requirement that the timing of the calorimeter hits be consistent with an external origin of the event. Further details on using topological, timing and energy cuts for background identification can be found in [30].

Background and control measurements
Any event containing two reconstructed electrons from an origin other than the decay of 82 Se can be misidentified as a ββ event. The main source of background events are trace amounts of naturally-occurring radioactive isotopes that come from the 238 U and 232 Th radioactive series. Only (β, γ )-emitting radioactive isotopes with high Q-values are potential backgrounds to a 0νββ search. The two main isotopes of concern are 214 Bi and 208 Tl with Q-values of 3.27 and 4.99 MeV respectively. The largest background contribution comes from internal contamination of the source foils. Isotopes that undergo βdecay can mimic two electron events via the processes of β-decay with Møller scattering, β-decay to an excited state followed by internal conversion, or by subsequent Compton scattering of the de-excitation photon.
Other background events may be classified as coming from an origin external to the source foils. These usually involve a γ -ray that interacts with the source foil causing pair production, Compton interaction followed by Møller scat-tering or double Compton scattering. The sources of external γ -rays are predominantly radioactive decays within the rock surrounding the laboratory, neutron capture and decays within the detector components or shielding.
A subset of the external backgrounds is identified as radon backgrounds, coming from 222 Rn, which is a gaseous isotope in the 238 U chain. Due to its long half-life of 3.82 days 222 Rn can be introduced via a number of mechanisms, notably emanation from detector materials, contamination of the tracker gas or of other detector surfaces, or via diffusion through detector seals. Once inside the detector, the radon decays to predominantly positive ions. These charged progenies drift towards the source foils or tracker wires where they settle, leaving deposits of 214 Bi near the source material [30]. Once on or near the source foils, this 214 Bi is then capable of producing background events in the same way as internal contaminants.
The background model is defined by the activity of each isotope in specific locations. In all background sources, 214 Pb is assumed to be in secular equilibrium with 214 Bi and likewise for 228 Ac, 212 Bi and 208 Tl. The fitting procedure extracts the different isotope activities using a binned log-likelihood maximisation. The distributions from the six background channels (1e, 1e1α N γ , 1e1γ , 1e2γ , external 1γ 1e and crossing-electron) and a ββ signal channel are fitted simultaneously to extract the most likely activity parameters.

External backgrounds
The external γ -ray flux incident on the detector is quantified using the external 1γ 1e and crossing-electron channels. In the former, a γ -ray deposits energy in the calorimeter before interacting with the source foil to produce an outgoing electron. In the latter, the γ -ray interacts close to the surface of  In both figures, the energy spectra from data are compared to the total MC prediction (top panels) and as a ratio of data to the total MC prediction (bottom panels). The Other MC histograms contain the small contributions from internal and radon background sources a calorimeter, producing an electron that crosses the whole tracking chamber including the source foil. Data from these channels constrain the number of events in the ββ channel from the external γ -ray flux. The external background model is an effective model of the γ -ray flux incident on the detector, with components similar to the model in [30]. It is dominated by 40 K, 208 Tl and 214 Bi contamination in the calorimeter PMT glass and by 208 Tl, 214 Bi and 60 Co in the iron shielding surrounding the detector.
The model reproduces the data accurately as can be seen from the distributions of energy deposited in the calorimeter for the external 1γ 1e and crossing-electron channels shown in Fig. 2.
The external background model presented here is constructed using data from the 82 Se sectors only. It is consistent with the average external background model in [30], where all sectors are used, within 10-20%. This is the expected level of sector-to-sector variation in the external background model.

Radon backgrounds
The radon level inside the detector can be measured by studying 214 Bi → 214 Po → 210 Pb sequential decay events in the 1e1α N γ channel. The distribution of the length of the αparticle tracks is used to reconstruct the location of 214 Bi. For example, the α track length is sensitive to whether the α-particle originated from the surface of a tracker wire or inside the bulk of the source foil.
Using the reconstructed position of the events, an extensive radon model has been developed with 214 Bi on the surface of the tracker wires, source foils and scintillators varying from sector-to-sector and, in the case of the surface of the wires, with tracker layer [30].
Distributions of α-particle track length from the 1e1α N γ channel, which are used to extract the 214 Bi activities, can be seen in Fig. 3. The contribution from internal foil contamination has the shortest track lengths as these α-particles must traverse the most material before entering the tracking gas while the surface of tracker wires sample has the longest tracks. The shape of the distributions is an artefact of the tracker geometry. The lower number of events between 20 and 30 cm is a result of a gap in the layers of tracker cells at this distance due to the presence of calorimeter blocks in the detector end caps [29].
The difference between phases 1 and 2 is apparent, with a higher proportion of events from surfaces of the tracker wires and source foils during phase 1. In these cases, 214 Bi has been deposited on exposed surfaces as a result of radon decay in the tracker gas. In phase 2 there is a larger contribution from the internal and Mylar components. This originates from 214 Bi decays from contamination with 226 Ra and has therefore remained constant whilst the radon level inside the tracker gas has decreased. Distributions of the length of α-particle tracks from the 1e1α N γ channel, which contains events with one electron track and one delayed α-particle track, with no constraints on the number of γrays present. The length is measured as the distance from the electron vertex on the foil to the furthest hit in the α-particle track. a Shows data from phase 1, which had a higher radon level in the tracker and b shows the same distribution for phase 2 data. In the top panels, data are overlaid on stacked histograms of the MC prediction from 214 Bi contaminations in the source foils (red), Mylar backing film (yellow), deposits on the tracker wires (green) and on the surface of the source foils (blue). The activities of the source foil and Mylar film contaminations are the same in both phases. The bottom panels show the ratio of data to the total MC prediction The small discrepancies observed between MC and data distributions are due to a strong sensitivity of the α-particle range to the location of the 214 Bi. For example, the distributions can be altered significantly by transferring 214 Bi between the surface of the foils and the surface of the wires or between different wires within the tracker. The detection efficiency for electrons from 214 Bi is much less sensitive to these small changes in decay location and so the systematic uncertainty from this discrepancy that propagates through to the ββ channel is negligible.
In addition to the 214 Bi components that are measured with 214 Bi → 214 Po → 210 Pb delayed events, there are other background events from 208 Tl and 210 Bi. The former is a product of 220 Rn decay and was measured using 1e2γ and 1e3γ channels where the electron track starts away from the foil [30]. The latter is caused by 210 Pb from 222 Rn deposited on the surfaces of detector components during construction. This isotope has a half-life of 22.3 y and supplies 210 Bi over the lifetime of the experiment. It is therefore not in equilibrium with 222 Rn observed in the detector. In a similar manner to the 214 Bi activities, a map of relative 210 Bi activities divided by sector and tracker layer has been developed [30].

Internal backgrounds
The main backgrounds in the low energy region come from β-decaying isotopes. The 1e channel electron energy distri-butions, shown in Fig. 4a, are dominated by 210 Bi, 40 K and 234m Pa. In the higher energy region, the contributions from the external 208 Tl and 214 Bi backgrounds become significant and at energies above 2.7 MeV, 214 Bi from the internal and surface of tracker wire contaminations are the only remaining contributions.
The 1e1γ channel constrains isotopes decaying to excited states, most notably 214 Bi and 208 Tl as shown in Fig. 4b. At energies below 2.5 MeV the channel serves as a crosscheck on the number of external γ -ray flux events that have calorimeter timings consistent with an event of internal origin. At high energies, the distribution contains events from internal contamination with 208 Tl.
A more sensitive probe for the 208 Tl internal contamination is the 1e2γ channel with one γ -ray above 1.7 MeV, shown in Fig. 5. Any contributions from 214 Bi are heavily suppressed by this cut on the γ energy such that the channel is dominated by the internal contributions of 208 Tl, with a 10% contribution from the 208 Tl in the tracker wires.
The measured activities for the internal contaminations are summarised in Table 1. The levels of contamination are similar for both enrichment runs with the exception of 234m Pa where there is a four-fold increase in the activity in run 2. The results are compared with measurements made with a high purity germanium (HPGe) detector carried out prior to the installation of the 82 Se foils in the detector. The results are consistent across all isotopes in Table 1.

Two-neutrino double beta decay
Candidate ββ signal events are selected using the criteria outlined in Sect. 3. A total of 8936 candidate events were selected, with 4350 and 4586 from source foils from enrichment runs 1 and 2 respectively. Table 2 shows the contribution expected from simulations of individual background sources to the ββ signal channel, with the lower energy threshold column relevant to a 2νββ measurement. The largest background contribution comes from internal contamination of the source foils with 15.1% of the total number of events for run 1 foils and 29.1% of those from run 2 foils. Among the internal contaminants, 234m Pa is the most prominent, accounting for 7.9% of events originating in run 1 foils and 23.0% of events from run 2 foils. The external backgrounds account for 3% of the total with the majority of events from γ -ray transitions of 208 Tl and 214 Bi. The radon backgrounds make up 2% with a dominant contribution from 214 Bi, and a secondary contribution from 210 Bi. The majority of these events come from the surface of the tracker wires, but some are also present on the surface of the foil. There are more expected radon background events in phase 1 compared to phase 2 despite its much shorter exposure period.
NEMO-3 has the unique capability of reconstructing the full kinematics of the ββ decay final states. The individual energies of each electron can be seen in Fig. 6, where the higher degree of contamination from 234m Pa in the run 2 foils leads to a much larger contribution from the internal backgrounds. There is a discrepancy between data and MC in the region of 0.5−0.7 MeV caused by a peak from the emission of a 694 keV internal conversion electron from 234m Pa. This discrepancy is significantly stronger in the run 2 foils due to their higher contamination with 234m Pa. The discrepancy is most likely caused by inaccuracies in the internal conversion electron transition probabilities obtained from the existing nuclear data sheets [33,34]. Given this large uncertainty associated with the 234m Pa background contribution, the enrichment run 2 foils are excluded from the analysis to enable a more precise measurement of the 2νββ half-life, as further discussed in Sect. 5.2.

Higher-state vs single-state dominated transistions
For the purpose of the nuclear matrix element calculation, the decay of 82 Se to 82 Kr is modelled as two virtual β transitions: one between the ground state of 82 Se and the 1 + states of the intermediate nucleus of 82 Br, and one between the 1 + states of 82 Br and the ground state of 82 Kr. If one single intermediate 1 + state dominates the transition, then the process is said to be single-state dominated (SSD). Alternatively, if the process proceeds through many higher intermediate excited states, it is said to be higher-state dominated (HSD). Previously, it has been assumed that 82 Se decay occurs in the HSD scenario. However, a strong transition in the 82 Se( 3 He, 3 H) 82 Br reaction via the 1 + (75 keV) excited level of 82 Br was recently identified [35], suggesting that the SSD scenario could be realised. The shape of the distribution of the sum of electron energies, which is used for the 2νββ half-life measurement, is very similar in both scenarios. However, the sub-division of energy between the electrons is different in the two cases and therefore a precise high-statistics study of single-electron energy distributions can be used to distinguish between the two models [36]. Moreover, the choice of the model affects the measured half-life of the 2νββ transition. This is because the increased number of lower energy electrons in the SSD model reduces the detection efficiency and therefore the extracted half-life. The selection efficiency for the 2νββ signal calculated from MC using the event selection criteria described above is [2.971 ± 0.002 (stat)] % under the HSD hypothesis and [2.623 ± 0.002 (stat)] % in the SSD case.
The largest difference between the SSD and HSD singleelectron energy spectra is at the low end of the distribution [36]. However, due to the previously identified issues   Table 2. The largest background category is internal contamination of the source foil (blue), but this is still much smaller than the contribution from the 2νββ signal, with a signal-to-background ratio of 4.0

(b) Enrichment Run 2 Foils
The SSD scenario is therefore assumed for the remainder of the analysis, unless explicitly stated otherwise.

Extraction of 2νββ half-life
A binned log-likelihood fit to the distribution of the sum of the two electron energies of the 4350 ββ events selected from the data and originating from enrichment run 1 foils is performed together with a fit to the six background channels, as described in Sect. 4. The fit assuming the SSD hypothesis, shown in Fig. 8, yields 3472.4 ± 75.7 signal events, with a signal-to-background ratio of 4.0. The distribution of the opening angle between the two tracks is shown in Fig. 9.
In addition to the statistical uncertainty obtained from the log-likelihood fit, the 2νββ half-life measurement is affected by a number of systematic uncertainties. The most important source of systematic error is the uncertainty on the detector acceptance and reconstruction and selection efficiency. This uncertainty is quantified using dedicated runs with 207 Bi sources introduced into the detector and is compared with activities independently measured by an HPGe detector. Taking into account the systematic error on the HPGe measurement (5%) the uncertainty on the signal efficiency is determined to be 5% [8].
Other sources of systematic uncertainty are listed in Table 3. The systematic error due to the background modelling is dominated by the uncertainty on the 234m Pa conver-

Fig. 9
Distribution of the cosine of the angle between two electron tracks at the point of emission from the run 1 source foil in the ββ channel. As expected, more events are observed with electrons emitted to back-to-back than with smaller opening angles. This angular distribution has been reweighted based on data from 207 Bi calibration sources sion electron branching ratio discussed above. This uncertainty translates into a 2.3% error on the 2νββ half-life for the run 1 foils and increases to 4.5% if the analysis is performed on both enrichment samples due to the higher 234m Pa levels in the run 2 foils. The uncertainty on the 2νββ halflife measurement is systematics dominated and therefore the overall precision of the measurement is improved by excluding the run 2 foils. The individual systematic errors are assumed to be uncorrelated and are added in quadrature to obtain the total systematic uncertainty of 6.3%. This yields the final measurement of N = 3472 ± 76 (stat) ± 218 (syst) for the number of signal events obtained with (0.464 ± 0.002) kg of 82 Se from enrichment run 1 over 5.25 y of observation. This can be converted to the 82 Se 2νββ half-life using where is the selection efficiency (2.623%), N A is Avogadro's number, m A is the number of moles of 82 Se and t is the total exposure time. The resulting half-life, assuming the SSD hypothesis, is T 2ν 1/2 = 9.39 ± 0.17 (stat) ± 0.58 (syst) × 10 19 y.
An identical analysis under the HSD hypothesis gives T 2ν 1/2 = 10.63 ± 0.19 (stat) ± 0.66 (syst) × 10 19 y. The half-life measurement allows the experimental determination of the NME for the 2νββ decay mode of 82 Se using the equation where g A is the axial-vector coupling constant and G 2ν is the phase space for the 82 Se 2νββ 0 + → 0 + ground state transition. Taking G 2ν Q ββ , Z = 1.6 × 10 −18 y −1 as calculated in [37,38] and assuming g A = 1.27 [3] we obtain for the matrix element under the SSD hypothesis and under the HSD hypothesis where the quoted errors include both statistical and systematic uncertainties, which are assumed to be uncorrelated.

Neutrinoless double beta decay
A search for 0νββ is carried out by selecting ββ events as outlined in Sect. 3. Due to the higher energies of electrons emitted in the 0νββ decay the uncertainties due to the 234m Pa background model reported earlier are negligible. Consequently, both enrichment samples are included in the 0νββ analysis. Alongside backgrounds from natural radioactivity, 0νββ has an additional background contribution from 2νββ events. The following results assume the SSD hypothesis, but the same results are also found if the HSD case is taken. We considered four lepton number violating mechanisms for 0νββ: light Majorana neutrino exchange, the admixture of right-handed currents in electroweak interactions, 0νββ decay accompanied by a majoron emission and R-parity violating SUSY models. No evidence for a 0νββ signal is found for any of these mechanisms and therefore corresponding limits on the half-lives are set. The background contributions to 0νββ in the [2.6−3.2] MeV energy region, where most of the signal from the light Majorana neutrino exchange and right-handed current mechanisms is expected, are shown in Table 2. The electron energy sum distribution is used to set the limits using a modified frequentist method based on a binned log-likelihood ratio test statistic (CL s ) [39]. The statistic is calculated over the entire energy range above 0.6 MeV and takes into account the shape of the energy distribution.
In order to estimate the effect of systematic uncertainties on the limit, the background and signal distributions are scaled by random factors drawn from Gaussian distributions with widths defined by the systematic errors of the experiment [40], which are given in Table 4. Similarly to 2νββ, the most significant contribution comes from the error on the selection efficiency.

Light Majorana neutrino exchange
Light Majorana neutrino exchange is the most commonly discussed mechanism of 0νββ decay. It has an experimental signature characterised by a peak in the distribution of the electron energy sum at the Q ββ value.
The background, signal and data distributions shown in Fig. 10a are used to set the limit. There are 7.20 [5.09−10.66] events expected to be excluded at the 90% C.L., where the ±1σ range is given in brackets. The systematic errors from Table 4 are included in the expected limit and only reduce it by 2%. Taking into account the detector efficiency of 9.80% for this 0νββ mechanism and the 82 Se exposure of 4.90 kg·y, the 90% C.L. expected half-life limit is 3.39 [2.29−4.80] × 10 23 y. From the data sample, 9.67 events are excluded at 90% C.L. leading to an upper limit on the half-life of which is within the 1σ range of the expected sensitivity. Equation 1 is used to convert the half-life limit into an upper bound on the effective Majorana neutrino mass. The phase space is taken as G 0ν = 1.016 × 10 −14 y −1 [37] (in agreement with G 0ν = 1.014 × 10 −14 y −1 from [38]).
Several nuclear models are used to calculate the NME for the 82 Se 0νββ transition to the ground state. The most recent calculations from [41][42][43][44][45][46] have been used and g A is taken in the range 1.25-1.27 to correspond with the assumptions of the different calculations. As a result, the constraint on the effective neutrino mass is

Right-handed currents
Right-left symmetric models can provide an alternative mechanism for 0νββ due to the presence of right-handed currents (RHC) in the electroweak Lagrangian [47,48]. The lepton number violation mechanism is characterised by the coupling between right-handed currents of quarks and leptons, λ , and right-handed quark and left-handed lepton currents, η . The λ mechanism leads to very different angular and single energy distributions of the final state electrons and can therefore be distinguished from other mechanisms in an experiment capable of reconstructing the full topology of the process, such as NEMO-3 [16]. In addition to the electron energy sum, further discrimination between the RHC λ mechanism and background can be achieved with the energy asymmetry between the individual electron energies, The expected sensitivity in the RHC λ mode has been studied by MC and is maximised with a cut of A > 0.26. This selection is therefore applied when searching for this particular decay mode as shown in Fig. 10b. Cutting on the energy asymmetry variable provides no improvement in sen-sitivity for the η mode and so the standard ββ selection criteria are used in this case.
For the λ mode, 7.34 events are excluded from the data sample leading to a lower limit on the half-life of 1.63 × 10 23 y at 90% C.L. This result is in agreement with the median expected sensitivity of the experiment of 2.16 [1.46 − 3.01] × 10 23 y. For the η mode, the half-life lower limit is 2.19 × 10 23 y at 90% C.L. and also agrees with the expected sensitivity.

Majoron emission
A 0νββ decay accompanied by a majoron, a light or massless boson that weakly couples to the neutrino, has a continuous spectrum of the energy sum of the two decay electrons, E tot , up to Q ββ [51]. The phase space of the process depends on the spectral index n, as G 0ν ∝ Q ββ − E tot n , and determines the shape of the distribution. Decays with higher n have broader E tot distributions peaking at lower energy values. Such events are harder to separate from 2νββ and other backgrounds. Therefore only the result of the search for majoron induced 0νββ decay with n = 1 is shown here. The corresponding half-life limit is T 0ν 1/2 > 3.7 × 10 22 y at 90% C.L., which translates into an upper limit on the majoron-neutrino coupling of g ee < (3.2−8.0) × 10 −5 . The range is due to a spread in NME calculations, which are taken from [41][42][43][44][45][46], while the phase space is taken from [52]. predictions and the open histogram shows a hypothetical 0νββ signal corresponding to the limit at 90% C.L. a contains events selected in the ββ channel and b contains a subset of these events that also pass the energy asymmetry cut for the right-handed current λ mode, A > 0.26, where A is defined in the text nos [53,54]. The kinematics of the electrons emitted in the decay are the same as in the light neutrino exchange mechanism and therefore the same half-life limit can be used to set limits on SUSY parameters. Taking the phase space from [37] and the NME from [55,56], the following constraints are obtained for the short range gluino and neutralino exchange mechanisms: where mq , mg, mẽ and mχ are the masses of squark, gluino, selectron and neutralino respectively. The corresponding limits for the long range squark exchange mechanism are: where Λ SUSY is a general SUSY breaking scale parameter. The above limits assume g A = 1.25. The spread in the limits is due to NME uncertainties associated with differences in the form of the Argonne and Charge Dependent Bonn (CD-Bonn) nucleon-nucleon potentials [55].

Summary and conclusions
The results of 82 Se ββ decay studies obtained with the full set of NEMO-3 data are presented. The 82 Se 2νββ decay half-life for the ground state transition has been measured using foils from the first enrichment run only, due to higher levels of 234m Pa contamination in the foils from the second run and associated uncertainties in the 234m Pa conversion electron branching ratios. With the corresponding exposure of 2.4 kg·y, the HSD transition hypothesis is disfavoured at the 2.1σ level, whilst the SSD hypothesis is supported. In the SSD scenario, the half-life has been measured to be T 2ν 1/2 = 9.39 ± 0.17 (stat) ± 0.58 (syst) × 10 19 y. This is the most precise measurement for this isotope to date and allows the experimental extraction of the corresponding NME, M 2ν = 0.0498 ± 0.0016. This single result is more precise than and consistent with the world average reported in [12,13]. The SuperNEMO experiment is based on the same design principles as the NEMO-3 detector and will have lower backgrounds and improved energy resolution. A demonstrator module is currently being commissioned, which will house 7 kg of 82 Se. The SuperNEMO demonstrator module will have the sensitivity to distinguish between the SSD and HSD scenarios at a > 5σ level.
A search for 0νββ decay has been carried out for a number of different mechanisms, with foils from both enrichment runs, giving an exposure of 4.9 kg·y. No evidence for Table 5 Limits from 0νββ searches for different decay modes in 82 Se. The signal efficiency and the 90% C.L. limits for half-lives and lepton number violating (LNV) parameters are shown. The ranges in the expected half-life limits are the ±1σ range of systematic uncertainties on the background model and signal efficiency. The ranges in the LNV parameter are due to the spread in NME calculations. The R-parity violating SUSY LNV parameters correspond to sparticle masses and energy scale Λ SUSY in TeV 0νββ mechanism Mode Efficiency (%) T 0ν 1/2 90% C.L. 10  any neutrinoless double beta decay transition is found and therefore upper limits on the corresponding lepton number violating parameters have been set. The results of the 0νββ search are summarised in Table 5. The most stringent half-life limit for 82 Se is obtained for the light neutrino exchange mechanism of 0νββ, T 0ν 1/2 > 2.5 × 10 23 y at 90% C.L. corresponding to an effective Majorana neutrino mass of m ν < (1.2−3.0) eV. It should be noted that the CUPID-0 collaboration recently published their first limit for 0νββ of 82 Se with a value T 0ν 1/2 > 2.4 × 10 24 y [9]. The constraints on the RHC parameters, λ and η , on the majoron-neutrino coupling constant, g ee , and on R-parity violating SUSY parameters, λ 1i j , shown in Table 5 are the best for 82 Se and are comparable with the best available limits from other isotopes [8] despite a much lower exposure.