Detailed studies of $^{100}$Mo two-neutrino double beta decay in NEMO-3

TL;DR

This paper reports precise measurements of the two-neutrino double beta decay half-life of $^{100}$Mo, analyzes decay spectra with high statistics, and tests nuclear decay models and new physics scenarios.

Contribution

It provides the most detailed spectral analysis of $^{100}$Mo double beta decay and tests the Single State Dominance model with unprecedented data quality.

Findings

Measured half-life: $6.81 imes 10^{18}$ years.

Found clear evidence supporting the Single State Dominance model.

Set limits on Majoron-emitting decay modes and Lorentz invariance violation.

Abstract

The full data set of the NEMO-3 experiment has been used to measure the half-life of the two-neutrino double beta decay of $^{100}$Mo to the ground state of $^{100}$Ru, $T_{1/2} = \left[ 6.81 \pm 0.01\,\left(\mbox{stat}\right) ^{+0.38}_{-0.40}\,\left(\mbox{syst}\right) \right] \times10^{18}$ y. The two-electron energy sum, single electron energy spectra and distribution of the angle between the electrons are presented with an unprecedented statistics of $5\times10^5$ events and a signal-to-background ratio of ~80. Clear evidence for the Single State Dominance model is found for this nuclear transition. Limits on Majoron emitting neutrinoless double beta decay modes with spectral indices of n=2,3,7, as well as constraints on Lorentz invariance violation and on the bosonic neutrino contribution to the two-neutrino double beta decay mode are obtained.

Tables

Table 1. Table 1: 100 Mo source foil contamination activities measured with the NEMO-3 detector. Activities of 214 Bi and 208 Tl are from   bdf .

Source	100Mo metallic	100Mo composite
214Bi internal, mBq/kg	$0.060\pm 0.019$	$0.305\pm 0.038$
214Bi mylar, mBq/kg	$-$	$1.05\pm 0.06$
208Tl, mBq/kg	$0.087\pm 0.004$	$0.128\pm 0.003$
234mPa, mBq/kg	$11.40\pm 0.06$	$2.10\pm 0.03$
40K , mBq/kg	$8.67\pm 0.05$	$13.57\pm 0.04$
210Bi, mBq/m2	$5.51\pm 0.03$	$19.42\pm 0.03$

Table 2. Table 2: Expected number of background events in the two-electron channel and the number of 100 Mo 2 ​ ν ​ β ​ β 2 𝜈 𝛽 𝛽 2\nu\beta\beta candidate events in molybdenum foils.

Source	Metallic	Composite	Total 100Mo
228Ac,212Bi,
208Tl	$49.5\pm 0.5$	$142.3\pm 1.3$	$191.8\pm 1.4$
214Pb,214Bi	$14.2\pm 0.1$	$177.2\pm 0.7$	$191.3\pm 0.7$
40K	$101.4\pm 2.5$	$296.0\pm 7.3$	$397.5\pm 7.7$
234mPa	$1783.8\pm 11.8$	$656.7\pm 4.3$	$2440.5\pm 12.5$
210Bi	$25.6\pm 1.4$	$90.3\pm 2.8$	$115.9\pm 3.1$
Radon	$434.3\pm 6.2$	$590.3\pm 5.2$	$1024.6\pm 8.1$
Ext Bkg	$562.7\pm 9.7$	$1238.6\pm 14.7$	$1801.3\pm 17.6$
$\beta \beta$ $0_1^{+}$	$48.6\pm 0.8$	$71.1\pm 1.0$	$119.7\pm 1.3$
Tot bkg	$3020\pm 17$	$3263\pm 18$	$6283\pm 25$
$\beta \beta$ g.s.	$190683\pm 117$	$304571\pm 144$	$495254\pm 186$
Data	$193699$	$307835$	$501534$

Table 3. Table 3: Summary of systematic uncertainties on the measured 2 ​ ν ​ β ​ β 2 𝜈 𝛽 𝛽 2\nu\beta\beta 100 Mo half-life

Source of uncertainty	Effect on $T_{1/2}^{2\nu }$ (%)
Absolute normalization of ${ϵ}_{2e}$	$\pm 5$
Thin source foil modelling	[$+1.5,-2.3$]
100Mo decay model	$\pm 1.8$
Energy calibration	$\pm 0.6$
100Mo mass	$\pm 0.2$
Background uncertainty	$\pm 0.2$
Total	[$+5.6,-5.8$]

Table 4. Table 4: Lower bounds on half-lives ( × 10 21 absent superscript 10 21 \times 10^{21}  y) at 90 % percent 90 90\% C.L. from 0 ​ ν ​ β ​ β 0 𝜈 𝛽 𝛽 0\nu\beta\beta searches with Majoron emission (spectral indices n = 2 , 3 , 7 𝑛 2 3 7 n=2,3,7 ), and searches for the bosonic neutrino admixture. The ranges in the expected half-life limits are from the ± 1 ​ σ plus-or-minus 1 𝜎 \pm 1\sigma range of the systematic uncertainties on the background model, signal efficiency and distortions in the shape of the energy spectrum.

	Expected
Decay mode	$-1\sigma$	Median	$+1\sigma$	Observed
Majoron n=2	$13$	$9.2$	$6.2$	$9.9$
Majoron n=3	$6.1$	$4.3$	$2.9$	$4.4$
Majoron n=7	$1.8$	$1.3$	$0.88$	$1.2$
$2\nu \beta \beta$ Bosonic $\nu$	$1.7$	$1.2$	$0.83$	$1.2$

Table 5. Table 5: Upper limits on the Majoron-neutrino coupling constant g e ​ e subscript 𝑔 𝑒 𝑒 g_{ee} from NEMO-3 ( 100 Mo, this work) and EXO-200 ( 136 Xe)  exo-maj and GERDA ( 76 Ge)  gerda-maj experiments. All limits are at 90% C.L. The ranges are due to uncertainties in NME calculations.

n	Mode	100Mo	136Xe exo-maj	76Ge gerda-maj
n=3	${\chi }^0$	$0.013-0.035$	$0.06$	$0.047$
n=3	${\chi }^0{\chi }^0$	$0.59-5.9$	$0.6-5.5$	$0.7-6.6$
n=7	${\chi }^0{\chi }^0$	$0.48-4.8$	$0.4-4.7$	$0.8-7.1$

Equations

$$1/T_{1/2}^{2\nu }=g_A^4{G}^{2\nu }|M^{2\nu }{|}^2,$$

$$T_{1/2}=(6.81\pm 0.01)\times 10^{18}\text{mbox}y.$$

$$T_{1/2}=\left[6.81\pm 0.01{\left(\text{mbox}stat\right)}_{-0.40}^{+0.38}\left(\text{mbox}syst\right)\right]\times 10^{18}\text{mbox}y.$$

$$1/T_{1/2}=|⟨g_{ee}⟩{|}^{m}G|M{|}^2,$$

$$W_{tot}={\cos }^4\chi W_f+{\sin }^4\chi W_b,$$

$$T_{1/2}^f(0^{+}g.s.)=6.8\cdot 10^{18}\text{mbox}y,T_{1/2}^b(0^{+}g.s.)=8.9\cdot 10^{19}\text{mbox}y.$$

$${\sin }^2\chi <0.27(90\%\text{mbox}C.L.).$$

$$-4.2\times 10^{-7}\text{mbox}GeV<{\mathring{a}}_{of}^{(3)}<3.5\times 10^{-7}\text{mbox}GeV(90\%\text{mbox}C.L.).$$

$$T_{1/2}=\left[6.81\pm 0.01{\left(\text{mbox}stat\right)}_{-0.40}^{+0.38}\left(\text{mbox}syst\right)\right]\times 10^{18}\text{mbox}y.$$

Full text

††journal: Eur. Phys. J. C

1 Introduction

Spontaneous nuclear double beta decay is a second order weak interaction process that was theoretically considered for the first time by M. Goeppert-Mayer in 1935 Goeppert-Mayer . It can occur in some even-even nuclei when two bound neutrons simultaneously undergo beta decay and are transformed into two bound protons emitting two electrons and two (anti)neutrinos. Two-neutrino double beta decay, $2\nu\beta\beta$, is one of the rarest directly observed radioactive processes with half-lives ranging from $7\times 10^{18}$ to $2\times 10^{21}$ years  PDG ; Barabash-2nu .

The decay rate of $2\nu\beta\beta$ decay can be expressed as

[TABLE]

where $g_{A}$ is the axial-vector coupling constant, $G^{2\nu}$ is a phase space factor, and $M^{2\nu}$ is a nuclear matrix element (NME). Measurement of the $2\nu\beta\beta$ half-life gives direct access to the value of the NME for this process and therefore provides experimental input into nuclear models that are used to evaluate NMEs. Moreover, $2\nu\beta\beta$ may provide answers to the question of $g_{A}$ quenching in nuclear matter that is currently being actively discussed Barea-2015 ; Pirinen-2015 ; Kostensalo-2017 ; Suhonen-2017 . Detailed studies of $2\nu\beta\beta$ may therefore be useful to improve NME calculations for the neutrinoless mode of double beta decay, $0\nu\beta\beta$, the process which violates total lepton number and is one of the most sensitive probes of physics beyond the Standard Model. A recent review of the $0\nu\beta\beta$ NME calculation methods, challenges and prospects can be found in review-0nu .

Previous measurements have shown that the 100Mo $2\nu\beta\beta$ half-life is shorter compared to other $\beta\beta$ isotopes Elegant ; Nemo2 ; Moe ; SiLi ; Ashitkov ; Mo100_2005 ; Bolometr ; Armengaud-2017 , and it is therefore a promising nucleus for precise studies of the process. Here we present the most accurate to date study of 100Mo $2\nu\beta\beta$ decay including single electron energy and angular distributions of the electrons emitted in the decay with an unprecedented statistics of $5\times 10^{5}$ events. The impact of the single electron energy spectra on nuclear models that are used to calculate the NME is also presented.

Searches for most commonly discussed $0\nu\beta\beta$ mechanisms (exchange of a light Majorana neutrino, right-handed currents, super-symmetry) with NEMO-3 have been reported earlier in 0nu-short ; 0nu-PR . In this paper we present results obtained for 100Mo $0\nu\beta\beta$ decay accompanied by the emission of Majoron bosons with spectral indices $n\geq 2$, as well as constraints on contributions from bosonic neutrinos and from Lorentz invariance violation to $2\nu\beta\beta$ spectra of 100Mo.

2 The NEMO-3 detector

The NEMO-3 detector, its calibration and performance are described in detail in TDR and more recently in 0nu-PR . A combination of tracking and calorimetric approaches allows for a full reconstruction of $\beta\beta$ event topology. A tracking chamber is used to reconstruct electron tracks, their origin and end points. The electron energies and arrival times are measured with a plastic scintillator calorimeter. The cylindrical detector measuring 3 m in height and 5 m in diameter is made up of 20 wedge-shaped sectors of identical size. Each sector hosts 7 thin foil strips containing a $\beta\beta$ isotope. The source foils are positioned in the middle of the tracking detector at a radius of 1 m and have a height of 2.48 m.

The tracking detector is based on a wire chamber made of 6180 open drift cells operating in Geiger mode with helium as the main working gas with the addition of ethanol (4%), argon (1%) and water vapour (0.15%). The wire cells are strung vertically parallel to the source foils and have average transverse and longitudinal resolutions of 0.5 mm and 0.8 cm ($\sigma$) respectively. The tracking volume is surrounded by a segmented calorimeter composed of 1940 optical modules made of 10 cm thick polystyrene scintillator blocks coupled to low radioactivity photomultiplier tubes (PMT). The energy resolution of optical modules for 1 MeV electrons ranges from 5.8% to 7.2% and the time resolution is 250 ps ($\sigma$). The detector was calibrated by deploying 207Bi, 90Sr and 232U sources during the course of data collection. The stability of the PMT gains was monitored by a dedicated light injection system that was run every 12 hours.

The NEMO-3 detector is supplied with a solenoid which generates a 25 G magnetic field parallel to the tracking detector wires and provides charge identification by track curvature. The detector is surrounded by passive shielding consisting of a 19 cm thick iron plates to suppress the external gamma ray flux, and of borated water, paraffin and wood to moderate and absorb environmental neutrons.

One of the unique advantages of the NEMO-3 technology is the ability to unambiguously identify electrons, positrons, gamma- and delayed alpha-particles. This approach leads to a strong suppression of backgrounds by eliminating events that do not exhibit a $\beta\beta$ topology. In addition, it allows for an efficient background evaluation by selecting event topologies corresponding to specific background channels. An electron is identified by a reconstructed prompt track in the drift chamber matching to a calorimeter deposit. Extrapolating the track to the foil plane defines the event vertex in the source. The track extrapolation to the calorimeter identifies the impact point of the electron track with the corresponding optical module and is used to correct the reconstructed energy of the electron deposited in the scintillator. The track curvature in the magnetic field is used to distinguish electrons from positrons. A $\gamma$-ray is identified as an energy deposit in the calorimeter without an associated track in the drift chamber. An $\alpha$-particle is identified by a short straight track delayed with respect to the prompt electron in order to tag 214Bi $\rightarrow$ 214Po delayed coincidences.

The NEMO-3 detector took data at the Modane Underground Laboratory (LSM) in the Frejus tunnel at a depth of 4800 m w.e. enabling the cosmic muon flux suppression by a factor of $>$ 106. The detector hosted source foils of 7 different $\beta\beta$ isotopes. The two isotopes with the largest mass were 100Mo (6.914 kg) 0nu-PR and 82Se (0.932 kg) Se82 with smaller amounts of 48Ca, 96Zr, 116Cd, 130Te and 150Nd Ca48 ; Zr96 ; Cd116 ; Te130 ; Nd150 .

Two types of purified molybdenum foils were installed in NEMO-3, metallic and composite. Both foil types were enriched in 100Mo with the isotopic enrichment factor ranging from $95.14\pm 0.05\%$ to $98.95\pm 0.05\%$. The average enrichment factor was 97.7% for metallic foils and 96.5% for composite foils. The metallic foils contained $2479\pm 5$ g of 100Mo. The mean metallic foil density is 58 mg/cm2 with a total foil surface of 43924 cm2. The composite foils contained $4435\pm 13$ g of 100Mo. They were produced by mixing a fine molybdenum powder with polyvinyl alcohol (PVA) glue and deposited between Mylar foils of 19 $\mu$m thickness. The average surface density of the composite foils is 66 mg/cm2 and the total foil surface area is 84410 cm2.

Monte Carlo (MC) simulations are performed with a GEANT-3 based  geant3 program using the DECAY0 decay0 event generator. The time-dependent status and performance of the detector are taken into account in modelling the detector response.

The data presented here were collected between February 2003 and October 2010 with a live time of 4.96 y and a total exposure of 34.3 kg$\cdot$y of 100Mo. This is the same exposure as that used for $0\nu\beta\beta$ results published earlier 0nu-PR .

3 Background model

Trace quantities of naturally-occurring radioactive isotopes can occasionally produce two-electron events and thus can mimic $\beta\beta$-decay events. The largest contributions come from isotopes that are progenies of 238U (234mPa, 214Pb, 214Bi, 210Bi) and of 232Th (228Ac, 212Bi, 208Tl), as well as 40K.

The background is categorised as internal if it originates from radioactive decays inside the $\beta\beta$ source foils, see Fig. 1(a). Two electrons can be produced via $\beta$-decay followed by a Møller scattering, $\beta$-decay to an excited state with the subsequent internal conversion or due to Compton scattering of the de-excitation photon.

Decays inside the tracking detector volume form a separate background category. The main source of this background is radon, 222Rn. The decay of radon progenies near the source foil can produce signal-like events in an analogous manner to internal background decays.

The last background category is due to the external $\gamma$-ray flux produced by decay of radioactive isotopes in detector components, the surrounding area and due to neutron interactions in the shield and material of the detector. The PMT glass is the main source of these $\gamma$-rays. They can produce two-electron events due to $e^{+}e^{-}$ pair creation in the source foil and subsequent charge misidentification, double Compton scattering or Compton scattering followed by Møller scattering, see Fig. 1(b).

A detailed discussion of the NEMO-3 background model is presented in bdf and results of screening measurements can be found in 0nu-PR ; TDR ; bdf . Here we follow the same background model as that presented for the 100Mo $0\nu\beta\beta$ analysis 0nu-PR . However, radioactive isotopes contributing to the low energy region of the 100Mo $2\nu\beta\beta$ spectrum were not relevant for the $0\nu\beta\beta$ analysis in 0nu-PR and are therefore discussed in more detail below. The background in question comes from traces of $\beta$-decaying isotopes 210Bi, 40K and 234mPa in 100Mo foils. In addition, 100Mo $2\nu\beta\beta$ decay to the $0^{+}_{1}$ excited state of 100Ru is also taken into account as a source of internal background. The experimental half-life value of $T_{1/2}=6.7^{+0.5}_{-0.4}\times 10^{20}$ y Barabash-2nu is used to evaluate this contribution.

The activities of $\beta$-emitters in 100Mo foils are determined from the fit to the electron energy distribution for a single electron event sample, which is shown in Fig. 2 separately for metallic and composite foils. To disentangle the 210Bi contribution from the source foils and the surface of the tracker wires the activity measured in bdf is used for the latter. Fig. 2 shows the sum of both contributions. Secular equilibrium is assumed between 214Pb and 214Bi. The same is done between 228Ac, 212Bi and 208Tl, where the branching ratio of 35.94% is taken into account. There is sufficiently good agreement between data and MC for the single electron energy spectrum. The observed deviations of MC from data are within 6% and are not significant when the systematic uncertainty on the external background is taken into account.

The results of the internal 100Mo foil contamination measurements carried out with the NEMO-3 detector are shown in Table 1.

4 Two-neutrino double beta decay of 100Mo

Candidate $\beta\beta$ events are selected by requiring two reconstructed electron tracks, each associated with an energy deposited in an individual optical module. The energy deposited by the electron in a single optical module should be greater than 300 keV. Each PMT must be flagged as stable according to the light injection survey 0nu-PR . The tracks must both originate from the 100Mo source foil, and their points of intersection with the plane of the source foil must be within 4cm transverse to and 8cm along the direction of the tracker wires, in order to ensure that the two tracks are associated to a common event vertex. The track curvatures must be consistent with electrons moving outwards from the source foil. The timing and the path length of the electrons must be consistent with the hypothesis of simultaneous emission of two electrons from a common vertex in the 100Mo source foil 0nu-PR . There should be no $\gamma$-ray hits and $\alpha$-particle tracks in the event.

After the above event selection there are 501534 100Mo two-electron candidate events, with 193699 coming from the metallic foils and 307835 from the composite foils. Table 2 shows the number of expected background and candidate signal events in 100Mo foils. The number of $2\nu\beta\beta$ events is obtained from a binned log-likelihood fit to the two-electron energy sum distribution under the Single State Dominance (SSD) nuclear model, as detailed below. The average signal-to-background ratio is S/B=79, with S/B=63 for the metallic foils and S/B=94 for the composite foils. The detector acceptance and selection efficiency for $2\nu\beta\beta$ 100Mo events calculated using MC simulations is $\epsilon=(2.356\pm 0.002)\%$, with $\epsilon_{met}=(2.472\pm 0.003)\%$ and $\epsilon_{com}=(2.292\pm 0.002)\%$ for the metallic and composite molybdenum foils respectively. Using the above values gives the 100Mo $2\nu\beta\beta$-decay half-life of $T_{1/2}=(6.65\pm 0.02)\times 10^{18}$ y for the metallic foils and $T_{1/2}=(6.91\pm 0.01)\times 10^{18}$ y for the composite foils. The difference between the two sample measurements may be explained by inaccuracy of the thin foil modelling and is taken into account in estimation of the systematic uncertainty in Section 4.2. We consider the mean value over the two data samples as the more reliable half-life estimation

[TABLE]

The two-electron energy sum spectra and the distributions of cosine of the angle between two electrons emitted from 100Mo foil are shown in Fig. 3, separately for the metallic and composite foils as well as for the total 100Mo sample.

The electron energy measured in the calorimeter is smaller than the energy at the point of origin due to energy losses in the foil and in the drift chamber. For instance in the case of 100Mo $2\nu\beta\beta$ decay the mean electron track length from the source foil to the calorimeter is 75 cm and the mean energy loss of electrons in the drift chamber is 43 keV. The single and summed electron energy distributions are presented for the measured values of the electron kinetic energy $E_{e}$ and sum of the measured electron kinetic energies $E_{SUM}$, respectively, i.e., without correction for the energy loss.

The angular distribution is corrected with the well-measured distribution of the opening angle between two electrons emitted in 207Bi decay. The MC distribution of the cosine of the angle between two electron tracks has been reweighted based on data collected in the regular energy calibration runs performed with 207Bi sources. The correction is biggest for small opening angles, and is at the level of 4% on average.

4.1 Role of intermediate nuclear states in 100Mo $2\nu\beta\beta$ transition

The nuclear $\beta\beta$ decay (A,Z) $\to$ (A,Z+2) is realized via two subsequent virtual $\beta$ transitions through the complete set of states of intermediate nucleus (A,Z+1). In the case of 100Mo $2\nu\beta\beta$ transition between the ground states of the parent (100Mo) and daughter (100Ru) nuclei with spin-parity $0^{+}$ the process is governed by two Gamow-Teller transitions through $1^{+}$ states of 100Tc. Nuclear theory does not predict a priori whether there is a dominance of transition through the $1^{+}$ ground state (SSD hypothesis ssd_abad ; ssd_simkovic ) or through higher lying excited states, namely from the region of the Gamow-Teller resonance (HSD hypothesis). The SSD versus HSD analysis is feasible as the ground state of 100Tc has spin-parity $J^{P}=1^{+}$ and is lying close to the ground state of 100Mo.

The evidence in favour of SSD in 100Mo $2\nu\beta\beta$ decay was already observed at the beginning of NEMO-3 data analysis ssd_shitov . Further hints for the SSD model in the 100Mo $2\nu\beta\beta$ decay were obtained in charge-exchange experiments by observing a strong Gamow-Teller transition to the $1^{+}$ ground state of 100Tc in the 100Mo(3He,t)100Tc reaction ssd_thies . It was estimated that this transition could contribute as much as 80% to the total value of the 100Mo $2\nu\beta\beta$ matrix element.

It was shown in ssd_simkovic that SSD and HSD models can be directly distinguished by making high precision kinematics measurements of $2\nu\beta\beta$ decay products. The distribution of the individual electron energies was shown to have the most discriminating power, especially in the low energy part of the spectrum. Fig 4 shows the individual electron energy spectra for three nuclear models, with SSD-3 being a modification of the SSD model where a finer structure of intermediate states is accounted for ssd3 .

Fig. 5 shows the energy sum and angular distribution of the final state electrons where the data are fitted with the HSD model. The tension between the data and the model is evident already from these distributions with $\chi^{2}$/ndf=4.57 ($p$-value=5.3$\cdot$10*-12*) and $\chi^{2}$/ndf=1.98 ($p$-value=0.007) for the energy sum and angular distributions respectively. However, the strongest evidence comes from the single electron energy distributions shown in Fig. 6 for the three models, HSD, SSD and SSD-3, fitted to the data. It is clear from the distributions and $\chi^{2}$ values that the HSD model can be ruled out with high confidence while SSD and SSD-3 provide a fairly good description of the data.

The difference between SSD and SSD-3 in describing the data is maximised with a cut on the electron energy sum of $E_{SUM}>1.4$ MeV as shown in Fig. 7, which also increases the signal-to-background ratio. There is a slight preference of the SSD-3 model over SSD in this case, contrary to the results obtained without this cut demonstrated at Fig. 6. Due to systematic effects connected to the energy reconstruction and electron energy loss simulations discussed below these two models cannot be discriminated against each other. The SSD is chosen as the baseline model and is used to estimate the 100Mo $2\nu\beta\beta$ half-life (see Section 4 and Fig. 3). We note that differences in the low energy part of the single electron spectra (Fig. 4) affect the selection efficiency of 100Mo $2\nu\beta\beta$ events. Consequently, the measured half-life for the SSD model is 14% shorter than the analogous result for the HSD model. The SSD-3 model would give a 1.8% shorter half-life than that of the SSD model.

4.2 Systematic uncertainties on 100Mo $2\nu\beta\beta$ half-life

Apart from the statistical uncertainties on the fitted number of signal events, the measurement of the $2\nu\beta\beta$ decay half-life is subject to a number of systematic uncertainties.

The uncertainty on the reconstruction and selection efficiency including the detector acceptance effects is evaluated by carrying out dedicated calibrations with 207Bi sources whose activities were known with a 5% uncertainty. Consequently, the systematic error on the signal efficiency is taken to be 5%.

Limited presision of MC simulation program in modelling of multiple scattering processes and electron energy losses in molybdenum $\beta\beta$ source foils also contribute to the total systematic error. Corresponding uncertainty is evaluated as the difference between the mean half-life value and the values obtained with metallic (-2.3%) and composite (+1.5%) foils.

The 1.8% half-life value difference between the SSD and SSD-3 nuclear models is taken as a systematic error due to the 100Mo $2\nu\beta\beta$ decay model.

The uncertainty on the energy scale translates into an error on the half-life measurement of 0.6%.

The 100Mo mass uncertainty gives directly the corresponding uncertainty of the half-life value and is estimated to be 0.2%.

The error on the activities of external backgrounds, radon and the foil contamination with 214Bi and 208Tl is 10% as shown in 0nu-PR . The uncertainty on the backgrounds from 40K in the source foils as well as from 210Bi is estimated to be 4%. The observed discrepancy in the 234mPa decay scheme reported in pa234m-new and pa234m-old lead to a 30% normalisation uncertainty on the activity from this isotope. The 7.5% error on the rate of the 100Mo $2\nu\beta\beta$ decay to the excited states Barabash-2nu is also taken into account. Overall, due to a high signal-to-background ratio the uncertainty on all background contributions produces only a 0.2% systematic uncertainty on the 100Mo $2\nu\beta\beta$ half-life determination.

The systematic uncertainties on the measured $2\nu\beta\beta$ 100Mo half-life are summarised in Table 3. The individual sources of the systematic error are assumed to be uncorrelated and the total uncertainty is obtained to be [$+5.6,-5.8$]%.

The final value of the half-life for the $2\nu\beta\beta$ decay of 100Mo under the SSD model is:

[TABLE]

This value is in good agreement with the world average value of $(7.1\pm 0.4)\times 10^{18}$ y Barabash-2nu and with a recent result obtained using low-temperature scintillating bolometers (Li2**100MoO4), $[6.90\pm 0.15(\mbox{stat})\pm 0.37(\mbox{syst})]\times 10^{18}$ y Armengaud-2017 .

5 Search for new physics with continuous 100Mo $\beta\beta$ energy spectra

Deviations in the shape of the $2\nu\beta\beta$ energy spectra can provide hints of new physics. Below we report on results of searches for physics beyond the Standard Model that can modify the two-electron energy sum distribution of the 100Mo $2\nu\beta\beta$ decay due to emission of Majoron bosons, the existence of a bosonic component in the neutrino states and possible Lorentz invariance violation.

The shape of the two-electron energy sum distribution in various types of decays is characterized by the spectral index $n$  Barnet , being determined by the phase space $G\sim(Q_{\beta\beta}-T)^{n}$, where $Q_{\beta\beta}$ is the the full energy released in the decay minus two electron masses and T is the sum of kinetic energies of two emitted electrons. The ordinary $2\nu\beta\beta$ decay has a spectral index of $n=5$. Any modification from this functional form can be an indication of new physics.

A number of grand unification theories predict the existence of a massless or light boson which couples to the neutrino. Neutrinoless $\beta\beta$ decay can proceed with the emission of one or two Majoron bosons resulting in a continuous energy sum spectrum with spectral index $n\neq 5$. The decay accompanied by a single Majoron emission has $n=1,2$ and $3$, while models with two Majoron emissions predict $n=3$ and $7$  (see Maj-old and references therein). The results for the neutrinoless $\beta\beta$ decay with the emission of a Majoron corresponding to the spectral index $n=1$ have already been published in 0nu-short ; 0nu-PR . The Majoron-accompanied $0\nu\beta\beta$ decay modes with spectral indices $n=2,3$ and $7$ are considered here.

It was noted in Bosonic-nu that violation of the Pauli exclusion principle resulting in a bosonic component in the neutrino states can be tested by looking at the shape of the energy and angular distributions of the electrons emitted in $\beta\beta$ decay. For the two-electron energy sum distribution the corresponding index would be $n=6$.

Lorentz invariance is a fundamental symmetry. However, new physics at very high energies close to the Planck scale can manifest itself in small effects at low energies, including Lorentz invariance violation. Consequently, searches for non-Lorentz invariant effects have attracted active theoretical and experimental effort LV1 ; LV2 ; LV3 ; LV4 . The possibility to test Lorentz invariance with $\beta\beta$ decay was discussed in  LV5 ; LV6 . In case of $2\nu\beta\beta$ decay the Lorentz invariance violation may be manifested as a modification of the conventional electron sum spectrum due to an additional contribution of the Lorentz-violating perturbation with a spectral shape of $n=4$.

The theoretical distributions of the two-electron energy sum for different modes of 100Mo $\beta\beta$ decay discussed above are shown in Fig 8. The difference in the shape of the distributions due to different spectral indices $n$ is used to evaluate possible contributions from physics beyond the Standard Model. No significant deviations from the expected 100Mo $2\nu\beta\beta$ spectral shape ($n=5$) have been observed and therefore limits on new physics parameters have been set using the full energy sum spectrum of the full 100Mo data set. The contributions of the $\beta\beta$ decay modes with spectral indices $n=2,3,6,7$ are constrained with a modified frequentist $CL_{s}$ method  cls using a profile likelihood fitting technique (COLLIE software package collie ). A profile likelihood scan is used for the distribution with the spectral index $n=4$ in order to explore possibility of negative as well as positive Lorentz-violating perturbation.

The systematic uncertainties on background contributions discussed in Section 4.2, the 5% uncertainty on the detector acceptance and selection efficiency for signal, a possible distortion in the shape of the two-electron energy sum spectrum due to the energy calibration accuracy, as well as a 5% error on the modelling of the energy loss of electrons are taken into account in limit setting without imposing a constraint on the normalization of standard $2\nu\beta\beta$ contribution.

The limits on the half-lives for different $0\nu\beta\beta$ modes with Majoron(s) emission, and for the bosonic neutrino admixture obtained with the $CL_{s}$ method are given in Table 4.

The half-life limits on the Majoron $0\nu\beta\beta$ modes are translated into the upper limits on the lepton number violating parameter $g_{ee}$, which is proportional to the coupling between the neutrino and the Majoron boson, using the relation,

[TABLE]

where $G$ is the phase space (which includes the axial-vector coupling constant $g_{A}$), $M$ is the nuclear matrix element, and $m=2(4)$ is the mode with the emission of one (two) Majoron particle(s). The $M$ and $G$ values are taken from Hirsch . For the single Majoron emission and $n=3$, $M$ and $G$ are taken from Barbero . There are no NME and phase space calculations available for $n=2$.

The upper limits on the Majoron-neutrino coupling constant $g_{ee}$ are shown in Table 5. One can see that the NEMO-3 results presented here are the current best limits for $n=3$ and the single Majoron emission mode and are comparable with the world’s best results from the EXO-200 exo-maj and GERDA gerda-maj experiments for the other two modes.

The contribution of bosonic neutrinos to the $2\nu\beta\beta$-decay rate can be parametrised as Bosonic-nu :

[TABLE]

where $W_{f}$ and $W_{b}$ are the weights in the neutrino wave-function expression corresponding to the two fermionic and two bosonic antineutrino emission respectively. The purely fermionic, $T_{1/2}^{f}$, and purely bosonic, $T_{1/2}^{b}$, half-lives are calculated under the SSD model to be Bosonic-nu :

[TABLE]

Using the NEMO-3 half-life limit of $T_{1/2}^{b}(0^{+}g.s.)>1.2\cdot 10^{21}$ y (Table 4) an upper limit on the bosonic neutrino contribution to the 100Mo $2\nu\beta\beta$ decay to the ground state can be evaluated as:

[TABLE]

Although this limit is stronger than the bound obtained earlier in Bosonic-nu , the $2\nu\beta\beta$ transition of 100Mo to the ground state is not very sensitive to bosonic neutrino searches due to a small value of the expected bosonic-to-fermionic decay branching ratio $r_{0}(0^{+}g.s.)=0.076$. The 100Mo $2\nu\beta\beta$ decay to the first excited $2^{+}_{1}$ state has a branching ratio of $r_{0}(2^{+}_{1})=7.1$ Bosonic-nu and is therefore potentially more promising despite a lower overall decay rate. The current best experimental limit for this process is $T_{1/2}(2^{+}_{1})>2.5\cdot 10^{21}$ y mo100_excited . This bound is still an order of magnitude lower than the theoretically expected half-life value of $T_{1/2}^{b}(2^{+}_{1})=2.4\cdot 10^{22}$ y for purely bosonic neutrino, and two orders of magnitude lower than the corresponding expected value for purely fermionic neutrino, $T_{1/2}^{f}(2^{+}_{1})=1.7\cdot 10^{23}$ y Bosonic-nu .

The Standard Model Extension (SME) provides a general framework for Lorentz invariance violation (LIV) LV1 . In this model, the size of the Lorentz symmetry breakdown is controlled by SME coefficients that describe the coupling between standard model particles and background fields. Experimental limits have been set on hundreds of these SME coefficients from constraints in the matter, photon, neutrino and gravity sectors LV1 . The first search for LIV in $2\nu\beta\beta$ decay was carried out in exo-200 . The two-electron energy sum spectrum of 136Xe was used to set a limit on the parameter $\mathring{a}^{(3)}_{of}$, which is related to a time-like component of this LIV operator. The value of this parameter was constrained to be $-2.65\times 10^{-5}$ GeV $<\mathring{a}^{(3)}_{of}<7.6\times 10^{-6}$ GeV by looking at deviations from the predicted energy spectrum of 136Xe $2\nu\beta\beta$ decay  exo-200 .

In this work we adopt the same method, using the phase space calculations from  Stoica , and perform a profile likelihood scan over positive and negative contributions of LIV to two-electron events by altering the 100Mo $2\nu\beta\beta$ energy sum spectrum with positive and negative values of $\mathring{a}^{(3)}_{of}$. The result of this scan is shown in Fig. 9.

The minimum of the profile log-likelihood function corresponds to $-135$ counts and is not statistically significant even at 1$\sigma$ level. The 90% CL exclusion limit is shown in Fig. 9 with the dashed line and gives $-1798$ and $1527$ events for negative and positive contributions to the deviation from the 100Mo $2\nu\beta\beta$ energy sum spectrum respectively. The corresponding constraint on $\mathring{a}^{(3)}_{of}$ is calculated using equations (2)-(6) in exo-200 . The result for 100Mo obtained with a full set of NEMO-3 data is

[TABLE]

A summary of the best available constraints on LIV and CPT violation parameters can be found in compilation LV1 .

6 Summary

The results of the $2\nu\beta\beta$ decay of 100Mo with the full data set of the NEMO-3 experiment corresponding to a 34.3 kg$\cdot$y exposure are presented. The summed energy of two electrons, the single electron energy and the angular distributions between the two electrons have been studied with an unprecedented statistical precision ($5\times 10^{5}$ events). The single electron energy distribution has been used to discriminate between different nuclear models providing direct experimental input into NME calculations. The HSD model is excluded with high confidence, while the SSD model is consistent with the NEMO-3 data. The corresponding half-life for the $2\nu\beta\beta$ decay of 100Mo is found to be

[TABLE]

Deviations from the expected shape of the 100Mo $2\nu\beta\beta$ energy sum spectrum have been studied to obtain constraints on parameters for physics beyond the Standard Model. The most stringent upper limit to date has been obtained for the Majoron-neutrino coupling parameter $g_{ee}$ for the decay mode with a single Majoron particle emission and the spectral index $n=3$. For other $0\nu\beta\beta$ modes with two Majoron bosons emission a comparable sensitivity with the world’s best limits has been achieved. The most stringent constraints on the bosonic neutrino admixture and Lorentz invariance violation in $2\nu\beta\beta$ decay have been set.

Acknowledgements

We thank the staff of the Modane Underground Laboratory for their technical assistance in running the experiment. We are grateful to S.V. Semenov for providing the spectra of SSD-3 model. We acknowledge support by the grants agencies of the Czech Republic (grant number EF16_013/0001733), CNRS/IN2P3 in France, RFBR in Russia (Project No.19-52-16002 NCNIL a), APVV in Slovakia (Project No. 15-0576), STFC in the UK and NSF in the USA.