New Results of the Experiment to Search for Double Beta Decay of ¹⁰⁶Cd with Enriched ¹⁰⁶CdWO₄ Scintillator

Abstract

In this article, we present current results of the experiment searching for double beta decay of ¹⁰⁶Cd with the help of an enriched ¹⁰⁶CdWO₄ crystal scintillator in coincidence with two CdWO₄ scintillation detectors. The experiment is carried out at the Gran Sasso underground laboratory of the National Institute for Nuclear Physics (LNGS INFN, Italy). After 1075 days of data-taking, no double-beta effects were observed. New half-life limits have been set for the different modes and channels of double beta processes in ¹⁰⁶Cd at the level of $lim{T}_{1/2}={10}^{20}-{10}^{22}$ years.

1. Introduction

Despite the great success of the Standard Model of particles and interactions (SM), neutrino remains the least studied particle due to its extremely weak interaction with matter. In addition, experimentally observed left-handed neutrino states are massless particles according to SM. On the other hand, the observation of the neutrino oscillation provides insight into a non-zero mass of at least two of the three neutrino states [1]. Unfortunately, neutrino oscillation experiments do not give the absolute values of the neutrino masses and the neutrino mass hierarchy [2]. It is also an open question whether neutrinos are of Dirac or Majorana nature. The existence of massive Majorana neutrinos causes the violation of the lepton number symmetry [3,4,5,6] that could explain an asymmetry between matter and antimatter in the Universe [7]. One of the most promising approaches to elaborate the properties of neutrino is the study of the neutrinoless mode of double beta decay ($0\nu 2\beta$) of atomic nuclei [8,9,10,11,12,13].

Double beta decay is the rarest nuclear process, with half-lives of $T_{1/2}$ = ${10}^{18}$ − ${10}^{24}$ years [14,15]. The process, comprising a nuclear charge increase by two units and the emission of two electron and antineutrino pairs (2$\nu 2{\beta }^{-}$), has been detected in 11 nuclei (⁴⁸Ca, ⁷⁶Ge, ⁸²Se, ⁹⁶Zr, ¹⁰⁰Mo, ¹¹⁶Cd, ¹²⁸Te, ¹³⁰Te, ¹³⁶Xe, ¹⁵⁰Nd, and ²³⁸U). There are also possible “double beta plus” processes characterized by a decrease in nuclear charge by two units: double electron capture (2EC), electron capture with positron emission (EC${\beta }^{+}$), and double positron decay (2${\beta }^{+}$). The experimental sensitivity to the double beta plus processes is substantially lower. There are three indications of 2$\nu$2EC decay: ⁷⁸Kr (in experiment with a proportional counter [16]), ¹³⁰Ba (in two geochemical experiments [17,18]), and ¹²⁴Xe [19,20,21]. The 2$\nu 2\beta$ decay, allowed in the SM, does not depend on the nature of the neutrino and the absolute neutrino mass scale, though it is very useful for testing theoretical approaches to describe $2\beta$ processes [9].

The most sensitive experiments searching for 0$\nu 2\beta$ decay reach half-life sensitivity at the level of $T_{1/2}>(0.05-3.8)\times {10}^{26}$ years, which corresponds to the Majorana neutrino mass limits in the range $m_{\nu }$$<(0.03-0.6)$ eV [22,23,24,25,26,27]. These and future more sensitive experiments focus on the search for 0$\nu 2{\beta }^{-}$ decay. However, the physical mechanisms that violate lepton number symmetry in “double beta plus” processes are essentially the same as for the decay with electrons emission. 0$\nu$EC${\beta }^{+}$ and 0$\nu$2${\beta }^{+}$ decays study is also motivated by the opportunity to clarify the possible contribution of the right-handed currents to the 0$\nu 2{\beta }^{-}$ decay [28]. In addition, resonant enhancement is possible in the $0\nu 2EC$ process that could cause an increase in the transition probability by up to six orders of magnitude [29,30,31,32].

The ¹⁰⁶Cd nuclide is a promising candidate to search for double beta plus decays, thanks to several features. This nuclide has one of the highest transition energies $Q_{2\beta }=2775.39\left(10\right)$ keV [33] and a comparatively high isotopic abundance $\delta =1.245\left(22\right)\%$ [34]. A simplified decay scheme of ¹⁰⁶Cd is presented in Figure 1. There is also the availability of enrichment methods using gas centrifugation to consider. A crucial point is that there are well developed methods of cadmium purification and the existence of technology to produce high-quality radiopure cadmium tungstate (CdWO₄) crystal scintillators [35,36] that can be used in calorimetric experiments with a high detection efficiency. This approach was implemented in the present experiment, searching for the $2\beta$ decay of ¹⁰⁶Cd.

2. Experimental Setup

A schematic diagram of the experimental setup is shown in Figure 2. The main part of the detector system is three CdWO₄ crystal scintillators. A central near-cylindrical-shaped scintillator with dimensions ⊘27 mm × 50 mm and mass 215.4 g is enriched with ¹⁰⁶Cd to 66% (¹⁰⁶CdWO₄) [36]. The scintillator is viewed by low radioactive photomultiplier tube (PMT) Hamamatsu R11065-MOD through a high-purity quartz light-guide (⊘66 mm × 100 mm) and polystyrene-based plastic scintillator (⊘40 mm × 83 mm). A Teflon spring is installed between the crystal scintillator and the copper brick to ensure the optical contact between the ¹⁰⁶CdWO₄ detector components. Two high volume CdWO₄ scintillators (⊘70 mm × 38 mm; with the natural cadmium abundance) have cylindrical cutouts to tightly enclose the ¹⁰⁶CdWO₄ crystal. The CdWO₄ scintillators are viewed by radiopure PMTs Hamamatsu R6233MOD through high-purity quartz light-guides (⊘70 mm × 200 mm). Optical contacts between the detector details are treated with optical-grade silicone grease. To improve the light collection, the detector system is wrapped in Teflon tape and aluminized plastic film. The Teflon holds all the components together to ensure the detector system’s geometry. To reduce background interference from PMTs, the detectors are surrounded by high-purity copper bricks (“internal copper”). The detector system is placed in a 11 cm-thick high-purity copper box (“external copper”) surrounded by low-radioactive lead (15 cm), cadmium (2 mm), and polyethylene (10 cm). The inner volume of the copper box is continuously flushed with high-purity N₂ gas with a low radon contamination (<58 mBq/m³ [37]) to remove radon present in environmental air. The experiment is carried out at 3.8 km of water equivalent depth at the Gran Sasso underground laboratory of the National Institute for Nuclear Physics (LNGS INFN, Italy).

The event-by-event data acquisition system is realized using a 1-GSample/s 8-bit transient digitizer DC270 by Agilent Technologies with a bandwidth of 250 MHz. The system records the shape of each signal in 100 $\mathrm{\mu }$s time window with a 50 ns time bin. The CdWO₄ scintillator has a relatively long decay time with four components: ∼15 $\mathrm{\mu }$s (89%); ∼4.6 $\mathrm{\mu }$s (9%); ∼0.8 $\mathrm{\mu }$s (2%); and ∼0.15 $\mathrm{\mu }$s (0.5%) [38]. The first 100 time bins (5 $\mathrm{\mu }$s) are used to calculate the mean value of the baseline. The energy of the event in the scintillator (E) is evaluated as the area of the digitized waveform:

$$E=\sum f\left(t_k\right),$$

(1)

where the sum is over time channels k, starting from the origin of the signal up to 25 $\mathrm{\mu }$s, and $f\left(t_k\right)$ is the digitized waveform (at the time $t_k$) of a given signal after subtraction of the mean value of the baseline.

To reduce the data volume caused by $\beta$ decays of ¹¹³Cd ($Q_{\beta }$ = 324 keV) and ^113mCd ($Q_{\beta }$ = 587 keV) nuclides, present in the ¹⁰⁶CdWO₄ crystal [36,39], the data acquisition system recorded events when one of two conditions was met:

(1)

an event in the ¹⁰⁶CdWO₄ detector with an energy > 0.5 MeV;

(2)

an event in the ¹⁰⁶CdWO₄ detector with an energy > 0.05 MeV in coincidence with a signal in at least one of the CdWO₄ counters with energy > 0.05 MeV.

Despite a high counting rate (∼15 Hz) below ∼0.6 MeV, the presence of ^113mCd nuclide in the ¹⁰⁶CdWO₄ scintillator made it possible to monitor the stability of the detector system by analyzing the position of its $\beta$ spectrum edge. More information about the ¹⁰⁶CdWO₄ and CdWO₄ detectors stability can be found in [40]. The main reason for the energy scale time shift of the ¹⁰⁶CdWO₄ detector was the degradation of the PMTs gain. Therefore, after 649 days of data taking, the PMT of the ¹⁰⁶CdWO₄ detector was replaced with another unit of the same model. Additionally, a Teflon pipe was installed, which made it possible to periodically calibrate the detector system with $\gamma$ sources without switching off the high voltage and opening the setup.

3. Data Analysis

3.1. Energy and Time Resolutions

The detector system was calibrated using ²²Na, ⁶⁰Co, ¹³⁷Cs, and ²²⁸Th $\gamma$-ray sources in the beginning, and several times during the experiment. The energy spectra of the sources measured by the ¹⁰⁶CdWO₄ and one of the CdWO₄ detectors are shown in Figure 3. The energy resolution of the detectors can be described by the function FWHM = A × √E_γ, where E_γ is the energy of $\gamma$-ray quanta in keV, FWHM is the full width at half maximum in keV, and A is a constant. For the entire data set, the averaged value of A is equal to 4.92 for the ¹⁰⁶CdWO₄, and 3.65 and 3.20 for the CdWO₄ detectors.

To determine the time resolution of the detector system, we used calibration data collected with a ²²Na $\gamma$-ray source. ²²Na is both a source of 1274.5 keV $\gamma$ quanta and of positrons. Figure 4 shows the distribution of differences ($\Delta t$) between the start of signals in the ¹⁰⁶CdWO₄ and in the CdWO₄ detectors with energy 511 ± 2${\sigma }_E$ keV, which occur after the positrons annihilation (${\sigma }_E$ is the energy resolution of the CdWO₄ counters for 511-keV $\gamma$-quanta). The distribution was approximated by an exponentially modified Gaussian function:

$$f(\Delta t)=A\times \left\{\begin{array}{cc}exp\left(-{\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{\Delta t-\mu }{\sigma }}\right)}^2\right),\hfill & {\displaystyle \frac{\Delta t-\mu }{\sigma }}\le k;\hfill \\ exp\left({\displaystyle \frac{k^2}{2}}-k{\displaystyle \frac{\Delta t-\mu }{\sigma }}\right),\hfill & {\displaystyle \frac{\Delta t-\mu }{\sigma }}>k,\hfill \end{array}\right.$$

(2)

where $A,\mu ,\sigma ,$ and k are free parameters of the fit. By analyzing the number of events in the 1274.5 keV peak in coincidence with 511 keV $\gamma$-quanta we obtained a selection efficiency of ${\eta }_{\Delta t}$ = $92\left(2\right)\%$ in the $-50$ ns ≤ $\Delta t$ < 100 ns time interval.

3.2. Pulse-Shape Discrimination

The difference between the pulse shape of $\gamma \&\beta$ and $\alpha$ scintillation signals was used to reject background due to $\alpha$ particles. A numerical characteristic called the Shape Indicator (SI) was calculated for each signal:

$$SI=\sum f\left(t_k\right)\times P\left(t_k\right)/\sum f\left(t_k\right).$$

(3)

The weight function $P\left(t\right)$ is defined as:

$$P\left(t\right)=\{{\overline{f}}_{\alpha }\left(t\right)-{\overline{f}}_{\gamma }\left(t\right)\}/\{{\overline{f}}_{\alpha }\left(t\right)+{\overline{f}}_{\gamma }\left(t\right)\},$$

(4)

where the reference pulse shapes ${\overline{f}}_{\alpha }\left(t\right)$ and ${\overline{f}}_{\gamma }\left(t\right)$ were built by analysing the $\gamma$-quanta events from the calibration data and internal $\alpha$ events. More about the pulse-shape discrimination of scintillation signals using the optimal filter method can be found in ref. [41]. The dependence of SI on energy for one of the CdWO₄ detectors is shown in Figure 5. As shown, $\alpha$ particles are clearly distinguished from $\gamma \&\beta$. The SI distributions for $\gamma \&\beta$ and $\alpha$ events are well described by a Gaussian function (Figure 5a inset). The mean value (${\mu }_{PSD}$) and standard deviation (${\sigma }_{PSD}$) of the Gaussian function depend on energy (E) as ${\mu }_{PSD}$ = $p_0+p_1\times E+p_2\times e^{p_3\times E}$ and ${\sigma }_{PSD}=p_4/\sqrt{E}+p_5$, where $p_i$ are parameters. For CdWO₄ detectors, the selection cut for $\gamma \&\beta$ events is ${\mu }_{PSD}\pm 3\times {\sigma }_{PSD}$, which gives the selection efficiency ${\eta }_{SI}^{\mathrm{nat}}=99.7\%$, while for $\alpha$ events it is ${\mu }_{PSD}\pm 2\times {\sigma }_{PSD}$. For the ¹⁰⁶CdWO₄ detectors the selection cut is more complex: ${{\mu }_{PSD}}_{-1.5\times {\sigma }_{PSD}}^{+3\times {\sigma }_{PSD}}$ for events with E < 1500 keV (${\eta }_{SI}^{106}=93.2\%$) and ${\mu }_{PSD}\pm 3\times {\sigma }_{PSD}$ for events with E > 1500 keV (${\eta }_{SI}^{106}=99.7\%$). This choice of selection conditions was made due to the worse discrimination between the $\gamma \&\beta$ and $\alpha$ signals exhibited by the ¹⁰⁶CdWO₄ detector [40].

It is worth noting that the energy of $\alpha$ particles ($E_{\alpha }$) which belong to the U/Th families is 5.7–6.9 MeV. Due to the quenching of $\alpha$ particles in the $\gamma$ scale, the $\alpha$ spectrum is shifted to lower energies [42]. A dependence of the $\alpha$/$\gamma$ ratio (i.e., ratio of energy of $\alpha$ particle in $\gamma$ scale to its real energy) was determined for the ¹⁰⁶CdWO₄ detector as $\alpha$/$\gamma$ = 0.12(2) + 0.011(2)×$E_{\alpha }$ and for CdWO₄ detectors as $\alpha$/$\gamma$ = 0.08(1) + 0.015(2)×$E_{\alpha }$, where $E_{\alpha }$ is in MeV [40]. Also, the analysis of $\alpha$ spectra made it possible to determine the contamination of crystals by $\alpha$-active daughter nuclides of the U/Th chains [40].

3.3. Experimental Energy Spectra

The experimental energy spectra measured over 1075 days by the ¹⁰⁶CdWO₄ and CdWO₄ detectors under different selection conditions are shown in Figure 6. It was possible to achieve a noticeable reduction of the background measured by the ¹⁰⁶CdWO₄ detector in the energy range of 0.5–1.5 MeV by separating $\gamma \&\beta$ events from $\alpha$ ones by the pulse-shape discrimination. The main part of the spectrum measured by the ¹⁰⁶CdWO₄ detector with energy < 0.5 MeV in anti-coincidence (AC) with the CdWO₄ counters is the ^113mCd $\beta$ distribution. For the events in coincidence (CC) with the CdWO₄ counters, the data acquisition system records events with the low energy trigger of the ¹⁰⁶CdWO₄ detector (see Section 2). As a result, the ^113mCd $\beta$ spectrum represents the main part of the spectra for all the detectors below 0.5 MeV. Also, an important selection condition is a coincidence with 511 keV $\gamma$-quantum (quanta) in the CdWO₄ counter(s), which should occur during the annihilation of a positron from the double beta processes in ¹⁰⁶Cd with positron(s) emission (Figure 6, $\gamma \&\beta$ CC $E^{\mathrm{nat}}$ = 511 ± 2${\sigma }_E$ keV, where $E^{\mathrm{nat}}$ is an energy deposit in at least one of CdWO₄ detectors). The only two peaks with energies of 202 keV and 307 keV that can be seen in the spectrum measured by the CdWO₄ counters in coincidence with events in the ¹⁰⁶CdWO₄ detector with energy above 500 keV (Figure 6, $\gamma \&\beta$ CC $E^{106}$ > 500 keV, where $E^{106}$ is an energy deposit in ¹⁰⁶CdWO₄ detector) correspond to the $\beta$ decay of ¹⁷⁶Lu.

3.4. Radioactive Contamination of the Experimental Setup

To describe the experimental data measured by the ¹⁰⁶CdWO₄ detector above 0.8 MeV in AC and above 0.4 MeV in CC, a background model was constructed from the following components:

(1)

⁴⁰K and ²³²Th, ²³⁸U with their daughters in all the setup components;

(2)

Residual $\alpha$ distribution in the ¹⁰⁶CdWO₄ crystal (7.3% of the alpha distribution);

(3)

Beta decay of ¹⁷⁶Lu and ^113mCd, and 2$\nu 2\beta$ decay of ¹¹⁶Cd with a half-life of $T_{1/2}=2.63\times {10}^{19}$ years in the ¹⁰⁶CdWO₄ crystal scintillator. The number of ¹¹⁶Cd 2$\nu 2\beta$ decays in the experimental spectra is well known due to the known isotopic concentration of ¹¹⁶Cd in the ¹⁰⁶CdWO₄ crystal [36];

(4)

¹¹³Cd in the CdWO₄ and ¹⁰⁶CdWO₄ crystal scintillators;

(5)

⁵⁶Co and ⁶⁰Co in the internal copper.

The secular equilibrium of the ²³²Th and ²³⁸U chains is assumed to be broken due to physical and chemical processes utilized for the production of the experimental setup materials. Therefore, the following sub-chains were considered separately: ²²⁸Ra ⟶ ²²⁸Th and ²²⁸Th ⟶ ²⁰⁸Pb (the ²³²Th chain); ²³⁸U ⟶ ²³⁴U, ²²⁶Ra ⟶ ²¹⁰Pb and ²¹⁰Pb ⟶ ²⁰⁶Pb (the ²³⁸U chain).

The energy distributions of the background components were simulated using the Monte Carlo package EGSnrc [43] with the initial kinematics provided by the DECAY0 event generator [44]. Each distribution was properly normalized by the parameter representing the number of decays. This parameter is obtained through fitting the experimental data by the Monte Carlo simulated distributions. The spectral shape of the distributions account also for the SI and $\Delta t$ selection cuts that were applied to the data. Each simulated event was multiplied by ${\eta }_{SI}$ and ${\eta }_{\Delta t}$. Figure 7 shows the result of the simultaneous binned maximum likelihood fit of five $\gamma \&\beta$ spectra: three spectra measured by the ¹⁰⁶CdWO₄ detector under selection conditions AC, CC, and CC $E^{\mathrm{nat}}$ = 511 ± 2${\sigma }_E$ keV; and two spectra measured by the CdWO₄ detectors under the selection conditions CC and CC $E^{106}$ > 500 keV. The maximum likelihood was applied since many bins have very low data statistics. The goodness of the fit is calculated with the Baker–Cousins approach [45] ${\chi }^2/NDF=421/123=3.4$, where $NDF$ is a number of degrees of freedom. The fit quality is rather low. However, the problem may be that one cannot be sure all the contaminations of the set-up materials are included in the background model. Moreover, despite efforts to reconstruct the setup geometry in the Monte Carlo simulations as accurately as possible, it does not perfectly reproduce the actual experiment’s geometry. Some worsening of the fit quality could also be due to imperfect knowledge of the detector system’s spectrometric characteristics, their degradation in time, etc. The radioactive contaminations of the main components of the experimental setup are reported in

Table 1. Radioactive contamination (mBq/kg) of the setup details. Upper limits are given at 90% confidence level, uncertainties of the last digit are given with 68% confidence level. Activities labeled by (*) were determined by using the time-amplitude analysis, the values marked by (†) were determined by analysis of the $\alpha$ distribution [40]. Q. lg. denotes quartz light guide.

Setup Component	238U	234U	230Th	226Ra	210Pb	232Th	228Ra	228Th	40K	176Lu	56Co	60Co
106CdWO4	0.65(3)			<0.04	<0.4		<0.02	0.0174(14) *	<0.24	1.68(3)
CdWO4	0.29(7) †	<0.2 †	1.40(7) †	<0.002 †	0.89(4) †	<0.01 †	<0.03	0.012(2) *	<2
Plastic scintillator	<8.9			<1.1	<11.8		<2.8	<1.1	<8.7
Optical couplant	<59			<79	<32		<13	<9.5	<260
Teflon tape	<4.1			<2.0	<31		<6.4	<2.8	<12
Teflon support details	<1.4			<1.3	<7.3		<5.0	<5.2	<9.7
Q. lg. for CdWO4	<1.0			<3.4	<3.2		<0.6	<0.4	<1.2
Q. lg. for 106CdWO4	<3.5			<9.3	<20		<9.1	<14.7	<40
Internal copper	<4.2			<0.09	<28		<0.16	<0.04	<2.4		<0.08	<0.07
External copper	<17			<0.46			<0.39	<0.08	<0.73
PMTs for CdWO4	<920			<1530			<1500	<1420	<1630
PMTs for 106CdWO4	<1400			<2500			<2500	<450	<2320

. The ²²⁸Th contaminations in the ¹⁰⁶CdWO₄ and in CdWO₄ detectors were determined by using the time-amplitude analysis [40]. Activities of ²³⁸U, ²³⁴U, ²³⁰Th, ²²⁶Ra, ²¹⁰Pb, and ²³²Th in the CdWO₄ detectors were determined by the $\alpha$ spectrum analysis [40].

4. Half-Life Limits on $2\beta$ Decay Processes in ¹⁰⁶Cd

There were no peculiarities in the experimental data that could be attributed to the 2$\beta$ decay of ¹⁰⁶Cd. Therefore, half-life limits for different channels and modes of 2$\beta$ decay were set using the formula:

$$lim{T}_{1/2}=N{(}^{106}\mathrm{Cd})\times ln2\times t\times {\eta }_{\det }\times {\eta }_{\mathrm{sel}}\times t/limS,$$

(5)

where $N{(}^{106}\mathrm{Cd})=2.42\times {10}^{23}$ is the number of ¹⁰⁶Cd nuclei in the ¹⁰⁶CdWO₄ crystal, ${\eta }_{\det }$ and ${\eta }_{\mathrm{sel}}$ are detection and selection efficiencies, t is the measurement live time, and $limS$ is a number of events of the effect searched for that can be excluded with a given confidence level (C.L.). Values of $limS$ depend on selection conditions and can be obtained from the approximation of the experimental energy spectrum with a background model plus an effect searched for. For simultaneous approximation of several spectra, the Equation (5) must be modified. Since the spectral shape of the Monte Carlo distributions account for the parameters ${\eta }_{\det }$ and ${\eta }_{\mathrm{sel}}$, Equation (5) can be rewritten as:

$$lim{T}_{1/2}=N{(}^{106}\mathrm{Cd})\times ln2\times t/lim{N}_{\mathrm{dec}},$$

(6)

where $lim{N}_{\mathrm{dec}}=limS/({\eta }_{\det }\times {\eta }_{\mathrm{sel}})$ is a number of decays of the process searched for. Now, values of $lim{N}_{\mathrm{dec}}$ can be obtained from simultaneous approximation of the energy spectra with different selection conditions. The parameters of the approximation were bounded, taking into account the data on the radioactive contamination of the experimental setup details (see Table 1).

For example, for the $2\nu 2{\beta }^{+}$ decay of ¹⁰⁶Cd to the ground state of ¹⁰⁶Pd, the fit gives $N_{\mathrm{dec}}$ = 0 ± 18. According to [46], we took $lim{N}_{\mathrm{dec}}=29$ events at 90% C.L., which gives $lim{T}_{1/2}^{2\nu 2{\beta }^{+}g.s.}=1.7\times {10}^{22}$ years (the distribution is shown in Figure 7). The total detection efficiency for $2\nu 2{\beta }^{+}$ decay to the ground state of ¹⁰⁶Pd for the different selection conditions and detectors are: ${\eta }_{\mathrm{AC}}^{106}=12.8\%$, ${\eta }_{\mathrm{CC}}^{106}=79.6\%$, ${\eta }_{\mathrm{CC}{E}^{\mathrm{nat}}=511\pm 2{\sigma }_E\mathrm{keV}}^{106}=37.3\%$, ${\eta }_{\mathrm{CC}}^{\mathrm{nat}}=52.7\%$, and ${\eta }_{\mathrm{CC}{E}^{106}>500\mathrm{keV}}^{\mathrm{nat}}=49.1\%$. The half-life limits on different modes of $2\beta$ decay of ¹⁰⁶Cd were obtained in a similar way and are presented in Table 2.

One of the most interesting processes is $2\nu$EC${\beta }^{+}$ decay of ¹⁰⁶Cd to the ground state of ¹⁰⁶Pd. In this work, we give a new half-life limit on this process as $lim{T}_{1/2}^{2\nu EC{\beta }^{+}g.s.}=7.7\times {10}^{21}$ years, which is in the region of the theoretical prediction $T_{1/2}={10}^{21}-{10}^{23}$ years [47,48,49,50,51,52,53]. The most optimistic theoretical calculations $T_{1/2}={10}^{20}-{10}^{22}$ years are for $2\nu 2$EC decay to the ground state of ¹⁰⁶Pd. But due to the large background from the $\beta$ decays of ^113mCd and ¹¹³Cd, it is difficult to obtain a high-sensitivity search for the characteristic X-rays from $2\nu 2$EC decay that are expected in the low energy region.

Another interesting feature is the possibility of near resonant $0\nu$2EC decays of ¹⁰⁶Cd to excited levels of ¹⁰⁶Pd when the initial and final states are degenerate [32]. There are three possible near resonant transitions to the 2718 keV, 2741 keV, and 2748 keV excited levels of ¹⁰⁶Pd. In this work we give the half-life limits on the near resonance processes as $lim{T}_{1/2}^{Res.0\nu 2EC}=(1.2-2.0)\times {10}^{21}$ years (see Table 2).

Table 2. Half-life limits on different modes and channels of $2\beta$ decay of ¹⁰⁶Cd given at 90% confidence level. The most sensitive previous results and the theoretical predictions are also presented.

Decay	Level of 106Pd,	Theoretical ${\mathit{T}}_{1/2},\mathbf{Years}$	lim${\mathit{T}}_{1/2},\mathbf{Years}$
	keV		Previous Result	Present Work
$2\nu 2{\beta }^{+}$	g.s.	$(5.4-880)\times {10}^{25}$ [47,48,50], $>2.4\times {10}^{27}$ [49]	$4.4\times {10}^{21}$ [54]	$1.7\times {10}^{22}$
	512	$(1.5-25)\times {10}^{27}$ [47,55,56]	$4.1\times {10}^{21}$ [54]	$1.5\times {10}^{22}$
$0\nu 2{\beta }^{+}$	g.s.	$(1.4-32)\times {10}^{27}$ [47,55,56,57,58,59,60,61]	$5.9\times {10}^{21}$ [62]	$2.2\times {10}^{22}$
	512		$4.1\times {10}^{21}$ [54]	$1.5\times {10}^{22}$
$2\nu$EC${\beta }^{+}$	g.s.	$(1.4-240)\times {10}^{21}$ [47,48,50,51,52,53], $>2.7\times {10}^{22}$ [49]	$2.1\times {10}^{21}$ [62]	$7.7\times {10}^{21}$
	512	$(5.3-24)\times {10}^{25}$ [51,52], $>1.1\times {10}^{25}$ [49]	$3.3\times {10}^{21}$ [54]	$9.9\times {10}^{21}$
	1128	$3.7\times {10}^{30}$ [51]	$2.0\times {10}^{21}$ [54]	$1.2\times {10}^{22}$
	1134	$(1.3-13)\times {10}^{26}$ [51,52], $>1.1\times {10}^{27}$ [49]	$2.5\times {10}^{21}$ [54]	$1.3\times {10}^{22}$
$0\nu$EC${\beta }^{+}$	g.s.	$(1.0-17)\times {10}^{26}$ [32,47,55,56]	$1.4\times {10}^{22}$ [62]	$1.5\times {10}^{22}$
	512		$9.7\times {10}^{21}$ [62]	$2.1\times {10}^{22}$
	1128		$1.0\times {10}^{22}$ [62]	$1.9\times {10}^{22}$
	1134	$(1.0-21)\times {10}^{29}$ [32,55,57,58]	$2.7\times {10}^{21}$ [54]	$2.1\times {10}^{22}$
$2\nu 2$EC	g.s.	$(2.0-230)\times {10}^{20}$ [47,48,50,51,52,53]	$1.7\times {10}^{21}$ [63]	–
	512	$(1.5-9.4)\times {10}^{27}$ [51,52], $>4.0\times {10}^{26}$ [49]	$9.9\times {10}^{20}$ [64]	$2.2\times {10}^{20}$
	1128	$9.9\times {10}^{28}$ [51]	$6.6\times {10}^{20}$ [62]	$9.3\times {10}^{20}$
	1134	$(1.1-11)\times {10}^{23}$ [51,52]	$1.0\times {10}^{21}$ [64]	$1.4\times {10}^{21}$
	1562	$(2.4-4.3)\times {10}^{28}$ [52], $>5.4\times {10}^{28}$ [49]	$7.8\times {10}^{20}$ [62]	$7.9\times {10}^{20}$
	1706	$>1.9\times {10}^{25}$ [49]	$7.1\times {10}^{20}$ [64]	$4.6\times {10}^{21}$
	2001	$>8.9\times {10}^{24}$ [49]	$1.5\times {10}^{21}$ [62]	$1.4\times {10}^{21}$
	2278	$>2.1\times {10}^{27}$ [49]	$1.0\times {10}^{21}$ [64]	$1.8\times {10}^{21}$
$0\nu 2$EC	g.s.		$1.0\times {10}^{21}$ [39]	$1.2\times {10}^{21}$
	512		$5.1\times {10}^{20}$ [39]	$1.9\times {10}^{21}$
	1128		$5.1\times {10}^{20}$ [64]	$1.7\times {10}^{21}$
	1134		$1.1\times {10}^{21}$ [64]	$2.2\times {10}^{21}$
	1562		$1.4\times {10}^{21}$ [62]	$2.0\times {10}^{21}$
	1706		$2.0\times {10}^{21}$ [62]	$1.7\times {10}^{21}$
	2001		$1.2\times {10}^{21}$ [64]	$3.3\times {10}^{21}$
	2278		$1.2\times {10}^{21}$ [62]	$1.2\times {10}^{21}$
Res. $0\nu 2$EC	2718	$(3.2-9.7)\times {10}^{22}$ [57], $>5.2\times {10}^{24}$ [65,66], $>7.9\times {10}^{23}$ [32]	$2.9\times {10}^{21}$ [62]	$2.0\times {10}^{21}$
	2741	$>5.2\times {10}^{24}$ [66]	$9.5\times {10}^{20}$ [39]	$1.2\times {10}^{21}$
	2748	$2\times {10}^{29}-2\times {10}^{34}$ [29]	$1.4\times {10}^{21}$ [64]	$1.9\times {10}^{21}$

Table 2. Half-life limits on different modes and channels of $2\beta$ decay of ¹⁰⁶Cd given at 90% confidence level. The most sensitive previous results and the theoretical predictions are also presented.

Decay	Level of 106Pd,	Theoretical ${\mathit{T}}_{1/2},\mathbf{Years}$	lim${\mathit{T}}_{1/2},\mathbf{Years}$
	keV		Previous Result	Present Work
$2\nu 2{\beta }^{+}$	g.s.	$(5.4-880)\times {10}^{25}$ [47,48,50], $>2.4\times {10}^{27}$ [49]	$4.4\times {10}^{21}$ [54]	$1.7\times {10}^{22}$
	512	$(1.5-25)\times {10}^{27}$ [47,55,56]	$4.1\times {10}^{21}$ [54]	$1.5\times {10}^{22}$
$0\nu 2{\beta }^{+}$	g.s.	$(1.4-32)\times {10}^{27}$ [47,55,56,57,58,59,60,61]	$5.9\times {10}^{21}$ [62]	$2.2\times {10}^{22}$
	512		$4.1\times {10}^{21}$ [54]	$1.5\times {10}^{22}$
$2\nu$EC${\beta }^{+}$	g.s.	$(1.4-240)\times {10}^{21}$ [47,48,50,51,52,53], $>2.7\times {10}^{22}$ [49]	$2.1\times {10}^{21}$ [62]	$7.7\times {10}^{21}$
	512	$(5.3-24)\times {10}^{25}$ [51,52], $>1.1\times {10}^{25}$ [49]	$3.3\times {10}^{21}$ [54]	$9.9\times {10}^{21}$
	1128	$3.7\times {10}^{30}$ [51]	$2.0\times {10}^{21}$ [54]	$1.2\times {10}^{22}$
	1134	$(1.3-13)\times {10}^{26}$ [51,52], $>1.1\times {10}^{27}$ [49]	$2.5\times {10}^{21}$ [54]	$1.3\times {10}^{22}$
$0\nu$EC${\beta }^{+}$	g.s.	$(1.0-17)\times {10}^{26}$ [32,47,55,56]	$1.4\times {10}^{22}$ [62]	$1.5\times {10}^{22}$
	512		$9.7\times {10}^{21}$ [62]	$2.1\times {10}^{22}$
	1128		$1.0\times {10}^{22}$ [62]	$1.9\times {10}^{22}$
	1134	$(1.0-21)\times {10}^{29}$ [32,55,57,58]	$2.7\times {10}^{21}$ [54]	$2.1\times {10}^{22}$
$2\nu 2$EC	g.s.	$(2.0-230)\times {10}^{20}$ [47,48,50,51,52,53]	$1.7\times {10}^{21}$ [63]	–
	512	$(1.5-9.4)\times {10}^{27}$ [51,52], $>4.0\times {10}^{26}$ [49]	$9.9\times {10}^{20}$ [64]	$2.2\times {10}^{20}$
	1128	$9.9\times {10}^{28}$ [51]	$6.6\times {10}^{20}$ [62]	$9.3\times {10}^{20}$
	1134	$(1.1-11)\times {10}^{23}$ [51,52]	$1.0\times {10}^{21}$ [64]	$1.4\times {10}^{21}$
	1562	$(2.4-4.3)\times {10}^{28}$ [52], $>5.4\times {10}^{28}$ [49]	$7.8\times {10}^{20}$ [62]	$7.9\times {10}^{20}$
	1706	$>1.9\times {10}^{25}$ [49]	$7.1\times {10}^{20}$ [64]	$4.6\times {10}^{21}$
	2001	$>8.9\times {10}^{24}$ [49]	$1.5\times {10}^{21}$ [62]	$1.4\times {10}^{21}$
	2278	$>2.1\times {10}^{27}$ [49]	$1.0\times {10}^{21}$ [64]	$1.8\times {10}^{21}$
$0\nu 2$EC	g.s.		$1.0\times {10}^{21}$ [39]	$1.2\times {10}^{21}$
	512		$5.1\times {10}^{20}$ [39]	$1.9\times {10}^{21}$
	1128		$5.1\times {10}^{20}$ [64]	$1.7\times {10}^{21}$
	1134		$1.1\times {10}^{21}$ [64]	$2.2\times {10}^{21}$
	1562		$1.4\times {10}^{21}$ [62]	$2.0\times {10}^{21}$
	1706		$2.0\times {10}^{21}$ [62]	$1.7\times {10}^{21}$
	2001		$1.2\times {10}^{21}$ [64]	$3.3\times {10}^{21}$
	2278		$1.2\times {10}^{21}$ [62]	$1.2\times {10}^{21}$
Res. $0\nu 2$EC	2718	$(3.2-9.7)\times {10}^{22}$ [57], $>5.2\times {10}^{24}$ [65,66], $>7.9\times {10}^{23}$ [32]	$2.9\times {10}^{21}$ [62]	$2.0\times {10}^{21}$
	2741	$>5.2\times {10}^{24}$ [66]	$9.5\times {10}^{20}$ [39]	$1.2\times {10}^{21}$
	2748	$2\times {10}^{29}-2\times {10}^{34}$ [29]	$1.4\times {10}^{21}$ [64]	$1.9\times {10}^{21}$

5. Conclusions

One of the most sensitive double beta plus experiments to search for $2\beta$ decay of ¹⁰⁶Cd with an enriched ¹⁰⁶CdWO₄ scintillator in coincidence with two large volume CdWO₄ counters was carried out at the Gran Sasso underground laboratory of the INFN (Italy). After 1075 days of data taking, we were able to provide new improved half-life limits on the different channels and modes of ¹⁰⁶Cd $2\beta$ decay at the level of $lim{T}_{1/2}={10}^{20}$–${10}^{22}$ years. A new half-life limit on $2\nu$EC${\beta }^{+}$ decay to the ground state of ¹⁰⁶Pd was set as $lim{T}_{1/2}^{2\nu EC{\beta }^{+}g.s.}=7.7\times {10}^{21}$ years, in the region of the theoretical predictions of ${10}^{21}$–${10}^{23}$ years. The half-life limits on near resonant $0\nu$2EC decay transitions to the 2718 keV, 2741 keV, and 2748 keV excited levels of ¹⁰⁶Pd have been set at the level $lim{T}_{1/2}^{Res.0\nu 2EC}=(1.2-2.0)\times {10}^{21}$ years. Analysis of the complete data set of the experiment is in progress.