Search for triple and quadruple beta decay of $^{150}$Nd

TL;DR

This study searched for rare triple and quadruple beta decays of $^{150}$Nd using a low-background detector, setting new lower limits on their half-lives and finding no evidence of these decays.

Contribution

First experimental limits on quadruple and triple beta decay half-lives of $^{150}$Nd to various excited states were established.

Findings

No evidence of triple or quadruple beta decay was observed.

New lower half-life limits were set, reaching up to $8.7 imes 10^{20}$ years.

Limits for the first time on certain excited state transitions were obtained.

Abstract

Triple beta decay of $^{150}$Nd to the ground and excited states of $^{150}$Eu and quadruple beta decay of $^{150}$Nd to the excited states of $^{150}$Gd have been studied using a 400 cm$^3$ low-background HPGe detector and an external source consisting of 3046 g of natural Nd$_2$O$_3$ powder. A half-life limit for the quadruple beta decay to the 0$_1^+$ state of $^{150}$Gd was found to be $T_{1/2} (0\nu+4\nu) > 8.7\times10^{20}$ yr (90\% C.L.). For other (0$\nu$+4$\nu$) transitions to the 2$_1^+$, 3$_1^-$, 4$^+_1$, 2$_2^+$ and 2$^+_3$ excited states limits for the first time have been obtained at the level of $(6.1-9.5)\times10^{20}$ yr (90\% C.L.). We report here also the results for the first search for triple beta decay to the ground and excited final states of $^{150}$Eu. We find no evidence of this decay and set lower limits on the half-life in the range $T_{1/2} (0\nu+3\nu) > (0.04-4.8)\times10^{20}$ yr (90\% C.L.).

Tables

Table 1. Table 1: Experimental limits for 4 β ​ ( 0 ​ ν + 4 ​ ν ) 𝛽 0 𝜈 4 𝜈 \beta(0\nu+4\nu) decay of 150 Nd to the excited states of 150 Gd. All limits are given at the 90% C.L.

		$T_{1/2}^{0\nu +4\nu },(\times {10}^{20}yr)$
Excited state, keV	Energy of $\gamma$-rays, keV (efficiency)	This work	Ref. [19]
2${}_1{}^{}{}_{}{}^{+}$ (638.05)	638.05 (2.23%)	7.52	-
3${}_1{}^{}{}_{}{}^{-}$ (1134.30)	496.25 (2.27%)+638.05 (2.09%))	7.79	-
0${}_1{}^{}{}_{}{}^{+}$ (1207.14)	569.09 (2.11%)+638.05 (2.06%))	8.70	1.76
4${}_1{}^{}{}_{}{}^{+}$ (1288.42)	638.05 (2.05%)+650.37 (2.02%))	8.98	-
2${}_2{}^{}{}_{}{}^{+}$ (1430.47)	638.05 (1.33%)+792.42 (1.20%))+1430.47 (0.62%)	9.49	-
2${}_3{}^{}{}_{}{}^{+}$ (1518.36)	638.05 (1.13%)+880.31 (0.86%))+1518.36 (0.73%)	6.10	-

Table 2. Table 2: Experimental limits for 3 β ​ ( 0 ​ ν + 3 ​ ν ) 𝛽 0 𝜈 3 𝜈 \beta(0\nu+3\nu) decay of 150 Nd to the ground and excited states of 150 Eu. All limits are given at the 90% C.L.

		$T_{1/2}^{0\nu +3\nu },(\times {10}^{20}yr)$
Excited state, keV	Energy of $\gamma$-rays, keV (efficiency)	This work	Previous work
5${}_{g.s.}{}^{}{}_{}{}^{-}$, 36.9 yr	439.40 (1.71%)	0.50	-
0${}_1{}^{}{}_{}{}^{-}$ (41.7 keV), 12.8 h	333.96 (0.10%)	0.04	-
2${}_1{}^{}{}_{}{}^{-}$ (118.6 keV)	75.9 (0.12%)	0.30	-
3${}_1{}^{}{}_{}{}^{-}$ (181.1 keV)	111.6 (0.48%)	0.63	-
6${}_1{}^{}{}_{}{}^{-}$ (190.37 keV)	190.4 (1.52%)	1.81	-
3${}_2{}^{}{}_{}{}^{-}$ (195.2 keV)	125.6 (0.66%)	0.78	-
6${}_2{}^{}{}_{}{}^{-}$ (247.89 keV)	247.9 (2.24%)	3.27	-
(3,2) (343.1 keV)	273.6 (2.11%)	3.16	-
5${}_1{}^{}{}_{}{}^{-}$ (360.14 keV)	169.7 (1.28%)+190.4 (1.44%)	2.24	-
5${}_2{}^{}{}_{}{}^{-}$ (412.53 keV)	190.4 (1.39%)+222.2 (1.86%)	4.83	-

Equations

$$(A,Z)\to (A,Z+4)+4e^{-}$$

$$(A,Z)\to (A,Z+4)+4e^{-}+4\bar{\nu }$$

$$(A,Z)\to (A,Z+3)+3e^{-}$$

$$(A,Z)\to (A,Z+3)+3e^{-}+3\bar{\nu }$$

Full text

I Introduction

Lepton-number-violating (LNV) processes could be directly linked to the possible Majorana nature of neutrinos. If Majorana mass terms are added to the standard model (SM) Lagrangian, processes appear that violate $\it L$ by two units ($\Delta L=2$) [1]. Searches for $\Delta L=2$ processes such as neutrinoless double beta decay ($0\nu 2\beta$) have, therefore, been the goal of many experiments (see recent reviews [2, 3, 4, 5]). The Majorana nature of the neutrino would also have interesting implications in many extensions of the Standard Model of particle physics. For instance the seesaw mechanism requires the existence of a Majorana neutrino to explain the lightness of neutrino masses [6, 7, 8, 9]. A Majorana neutrino would also provide a natural framework for lepton number violation, and particularly for the leptogenesis process which may explain the observed matter-antimatter asymmetry of the Universe [10]. However, it is most often overlooked that LNV and Majorana neutrinos are not necessarily connected. Models with $\Delta L=3$ and $\Delta L=4$ have some power in explaining naturally small Dirac masses of neutrinos and could mediate leptogenesis (see discussions in [11, 12] and references therein). Processes with $\Delta L=4$ could also be probed at the Large Hadron Collider (LHC), for example, in the pair production and decay of triplet-Higgs states to four identical charged leptons [13]. In [11] the toy model with $\Delta L=4$ was constructed which allows neutrinoless quadruple beta decay ($0\nu 4\beta$):

[TABLE]

This feature is discussed in more detail in [14]. Notice that the neutrino in the framework of this model is Dirac and the neutrinoless double beta decay is forbidden. The process (1) was discussed also in [15] and $\Delta L\geq 4$ lepton number violating processes were discussed in [16]. Note also that quadruple decay with the emission of 4 neutrinos ($4\nu 4\beta$) is not forbidden by any conservation low:

[TABLE]

This is simply a fourth order process for weak interaction. The authors of [11] pointed out three candidate nuclei for quadruple decay (150Nd, 96Zr and 136Xe) and it was noted that for 150Nd the energy of 4$\beta$ transition is maximum (2084.2 keV [17]) and even transition to excited levels of the daughter nucleus 150Gd is possible. In this paper, we will also consider the triple beta decay of 150Nd (neutrinoless decay ($0\nu 3\beta$) and decay with 3 neutrino emission ($3\nu 3\beta$)):

[TABLE]

[TABLE]

In [11], it was noted that neutrinoless triple beta decay is forbidden, since this process violates Lorentz invariance. This is true, but in this case the search for this process can serve as a test of Lorentz invariance. At the same time, the decay with the emission of three neutrinos is not forbidden by any conservation laws (this is a third-order process for the weak interaction). There are three candidates for this transition only. Besides 150Nd - 150Eu transition ($Q_{3\beta}$ = 1112.2 keV), it is also 48Ca - 48V ($Q_{3\beta}$ = 253.1 keV) and 96Zr - 96Tc ($Q_{3\beta}$ = 383.1 keV).

The search for 4$\beta$ decay was first carried out in the NEMO-3 experiment [18]. The $0\nu 4\beta$ decay of 150Nd was investigated. For the transition to the ground state of the 150Gd the limit $T_{1/2}(0\nu)>(1.1-3.2)\times 10^{21}$ yr (90% C.L.) was obtained, depending on the model used for the kinematic distributions of the emitted electrons. Recently, first limit on the $0\nu 4\beta$ decay of 150Nd to the 0${}^{+}_{1}$ state (1207.13 keV) of 150Gd was obtained, $T_{1/2}(0\nu)>1.76\times 10^{20}$ yr (90% C.L.) [19]. Although the authors discuss only the limit for the $0\nu 4\beta$ decay, it is clear that, in fact, this is also the limit for the $4\nu 4\beta$ decay.

In this article, results of an experimental investigation of the 3$\beta(0\nu+3\nu)$ and 4$\beta(0\nu+4\nu)$ decays of 150Nd to the ground and excited states of 150Eu and excited states of 150Gd are presented. Mass-chain decay scheme is shown in Fig. 1 and demonstrates the fact that 150Nd energetically can decay to 150Sm (double beta decay), 150Eu (triple beta decay) and 150Gd (quadruple beta decay). The decay schemes of 150Nd to 150Gd and 150Eu are presented in Fig. 2 and Fig. 3. The simplified decay scheme of 150Eu and 150mEu are presented in Fig. 4 and Fig. 5, respectively. The measurements have been carried out using a HPGe detector to look for $\gamma$ -ray lines corresponding to the decay scheme. We used experimental data obtained in [21, 22], that previously used to investigate $2\beta(0\nu+2\nu)$ decay of 150Nd to the excited states of 150Sm.

II Experiment

The experimental work was performed in the Modane Underground Laboratory (depth of 4800 m w.e.). A 400 cm3 low-background HPGe detector was used to measure a 3046 g sample of Nd2O3 powder in a special Marinelli delrin box which was placed on the detector endcap. Taking into account the natural abundance of 150Nd (5.64%) there are 153 g of 150Nd (or $6.14\times 10^{23}$ nuclei of 150Nd) in the sample. Data was collected for 11320.5 hours.

The Ge spectrometer is composed of a p-type, 400 cm3 crystal. The cryostat, endcap and the other mechanical parts have been made of a very pure Al-Si alloy. The passive shielding consisted of 3-10 cm of OFHC copper inside 15 cm of ordinary lead. To remove 222Rn gas, one of the main sources of the background, a special effort was made to minimize the free space near the detector. In addition, the passive shielding was enclosed in an aluminum box flushed with high-purity nitrogen.

The electronics consisted of spectrometric amplifiers and a 8192 channel ADC. The energy calibration was adjusted to cover the energy range from 50 keV to 3.5 MeV, and the energy resolution was 2.0 keV for the 1332-keV line of 60Co. The electronics were stable during the experiment due to the constant conditions in the laboratory (temperature of $\approx 23^{\circ}$ C, hygrometric degree of $\approx 50$%). A daily check on the apparatus assured that the counting rate was statistically constant.

The current data of accepted values for different isotopes published in Nuclear Data Sheets [20] were used for analysis of the energy spectrum. The photon detection efficiency for each investigated process has been calculated with the CERN Monte Carlo code GEANT 3.21. Special calibration measurements with radioactive sources and powders containing well-known 226Ra activities confirmed that the accuracy of these efficiencies is about 10%.

The dominate detector backgrounds come from natural 40K, radioactive chains of 232Th and 235,238U, man-made and/or cosmogenic activities of 137Cs and 60Co. The sample was found to have a large activity of 40K (46.3 mBq/kg). Additionally long-lived radioactive impurities were observed in the sample, but with much weaker activities, i.e. 137Cs (0.089 mBq), 176Lu (0.450 mBq/kg), 138La (0.068 mBq/kg), 133Ba (0.155 mBq/kg), etc.

Figures 6, 7, 8 and 9 show the energy spectrum in the ranges of interest.

III Analysis and results

III.1 Search for 4$\beta(0\nu+4\nu)$ processes in 150Nd

Quadruple beta decays of 150Nd to 2${}^{+}_{1}$ (638.045 keV), 3${}^{-}_{1}$ (1134.297 keV), 0${}^{+}_{1}$ (1207.135 keV), 4${}^{+}_{1}$ (1288.420 keV), 2${}^{+}_{2}$ (1430.467 keV) and 2${}^{+}_{3}$ (1518.362 keV) excited states of 150Gd have been investigated.

III.1.1 Decay to the 2${}^{+}_{1}$ excited state

To search for this transition one has to look for a $\gamma$-ray with an energy of 638.05 keV. The detection efficiency is 2.23%. As one can see from Fig. 8, there is no statistically significant peak at this energy. So one can only give the lower half-life limit on the transition to the 2${}^{+}_{1}$ excited state of 150Gd, $T_{1/2}>7.5\times 10^{20}$ yr. The limit has been calculated using the likelihood function described in [23, 24] which takes into account all the peaks identified above as background. This result is presented in Table 1.

III.1.2 Decay to the 0${}^{+}_{1}$ excited state

The transition is accompanied by two $\gamma$-rays with energies of 569.09 keV and 638.05 keV. The detection photopeak efficiencies are equal to 2.11% at 569.09 keV and 2.06% at 638.05 keV. Fig. 8 shows the energy spectrum in the ranges of interest. There are no statistically significant peaks at these energies. Using the same technique as above the lower half-life limits $8.7\times 10^{20}$ yr are found for the transitions (Table 1). Table 1 also presents other valuable data on this transition.

III.1.3 Decay to the 3${}^{-}_{1}$, 4${}^{+}_{1}$, 2${}^{+}_{2}$ and 2${}^{+}_{3}$ excited states

To search for these transitions one has to look for $\gamma$-rays with energies of 496.25, 638.05, 650.37, 792.42, 880.31, 1430.47 and 1518.36 keV. Figures 7-9 show that there are no statistically significant peaks at these energies. Using the same technique as above the lower half-life limits are found within $(6.1-9.5)\cdot 10^{20}$ yr for the transitions (see Table 1).

III.2 Search for 3$\beta(0\nu+3\nu)$ processes in 150Nd

Triple beta decays of 150Nd to the ground state of 150Eu ($5^{-}$), metastable state of 150Eu ($0^{-}_{1}$; 41.7 keV), 2${}^{-}_{2}$ (118.6 keV), 3${}^{-}_{1}$ (181.1 keV), 6${}^{-}_{1}$ (190.4 keV), 3${}^{-}_{2}$ (195.2 keV), 6${}^{-}_{2}$ (247.9 keV), (3,2) (343.1 keV), 5${}^{-}_{1}$ (360.1 keV) and 5${}^{-}_{2}$ (412.5 keV) excited states of 150Eu have been investigated.

III.2.1 Decay to the ground state of 150Eu

In this case, the 150Nd (0+) enters the ground state of the 150Eu (5-). This nucleus is unstable (half-life is 36 years) and turns to 150Sm by electron capture. Transitions go to the excited states of 150Sm, which is accompanied by the emission of cascades of gamma quanta. In this case, we used the 439.40 keV line, which accompanies 79.6% of the 150Eu decays. The registration efficiency of these gamma quanta is 2.5%. But to calculate detection efficiency full decay scheme was used. The experiment used a Nd2O3 sample, produced 15 years before the measurements. It means that 150Eu from the possible decay of 150Nd could accumulate in the sample all this time. This was taken into account when determining the limit on the possible transition 150Nd-150Eu. The resulting limit $0.5\times 10^{20}$ yr is shown in Table 2.

III.2.2 Decay to the 0${}^{-}_{1}$ metastable state of 150Eu

In this case, the 150Nd (0+) enters the metastable state of the 150Eu (0-). This nucleus is unstable (half-life is 12.6 hours) and disintegrates through beta decay (89%) and electron capture (11%). In electron capture, in 4% of cases, a transition to the excited states of 150Sm occurs, and in 100% of cases this leads to the emission of a gamma quantum with an energy of 333.9 keV. To search for this transition one has to look for a $\gamma$-ray with an energy of 333.9 keV. The final detection efficiency is 0.10%. The analysis given in [22] shows that the excess of events at 333.9 keV is mainly due to the double beta decay of 150Nd to the 0${}^{+}_{1}$ excited state of 150Sm. So one can only give the lower half-life limit on the metastable state of the 150Eu (0-). The limit has been calculated using the likelihood function described in Refs. [23, 24] which takes into account the peak identified above as background. The resulting limit $4\times 10^{18}$ yr has been obtained and is shown in Table 2.

III.2.3 Decay to the 2${}^{-}_{2}$ (118.6 keV), 3${}^{-}_{1}$ (181.1 keV), 6${}^{-}_{1}$ (190.4 keV), 3${}^{-}_{2}$ (195.2 keV), 6${}^{-}_{2}$ (247.9 keV), (3,2) (343.1 keV), 5${}^{-}_{1}$ (360.1 keV) and 5${}^{-}_{2}$ (412.5 keV) excited states

To search for these transitions one has to look for $\gamma$-rays with energies of 75.9, 111.6, 190.4, 125.6, 247.9, 273.6, 169.7 and 222.2 keV. As one can see from figures 6 and 7, there are no statistically significant peaks at these energies. Using the same technique as above [23, 24] the lower half-life limits are found within $(0.04-4.83)\cdot 10^{20}$ y for the transitions (see Table 2).

IV Discussion and Conclusion

The results are shown in Tables 1 and 2. Note that the limit on 4$\beta(0\nu+4\nu)$ decay of 150Nd to the 0${}_{1}^{+}$ excited state of 150Gd obtained in this work is five times stronger than the limit obtained in [19]. For other (0$\nu$+4$\nu$) transitions to the 2${}_{1}^{+}$, 3${}_{1}^{-}$, 4${}^{+}_{1}$, 2${}_{2}^{+}$ and 2${}^{+}_{3}$ excited states limits for the first time have been obtained. Results for 3$\beta(0\nu+3\nu)$ decay are obtained for the first time. Unfortunately, there are no ”real” (accurate) theoretical calculations for these processes. Therefore, we cannot compare our results with theoretical predictions. It is clear only that such transitions should be strongly suppressed compared to the $2\beta$ decay processes. However, as indicated in [11, 14], under certain conditions it is possible to have significant enhancement for the $4\beta(0\nu)$ decay process. In [16] it was demonstrated (using ”qualitative” estimates) that, under the most favorable assumptions, the half-life of 150Nd for $0\nu 4\beta$ decay is $\sim 10^{41}$ yr [16] (this can be compared with the best present experimental limit, $T_{1/2}>(1.1-3.2)\times 10^{21}$ yr [18]). Thus, it is clear that sensitivity of present experiments is very far from theoretical expectations. And, apparently, the prospects for the search for $0\nu 3\beta$ decays look even more pessimistic. It seems that the search for processes with the emission of neutrinos is more likely to succeed. Unfortunately for these processes there are not even qualitative theoretical estimates.

In addition to direct counter experiments, other types of experiments can be used. In work [11], a radiochemical experiment is proposed to search for $4\beta$ decay of 150Nd, given the fact that the daughter nucleus (150Gd) is radioactive (alpha decay, $T_{1/2}=2\times 10^{6}$ yr). In such type of experiment, $2\beta$ decay of 238U [25] was recorded, for example. In principle, one can offer here a geochemical experiment too, by analogy with geochemical experiments with 130Ba [26] , 100Mo [27] or 96Zr [28]. In the case of $3\beta$ decay, radioactive nuclei 150Eu (5-) and 150mEu (0-) are formed, with a lifetime of 36 yr and 12.6 h, respectively. This circumstance can be used in the conducting of radiochemical experiments too. The sensitivity of such experiments can be up to $\sim 10^{25}-10^{27}$ yr (the estimation is made using the parameters of radiochemical experiments on the registration of solar neutrinos, where the mass of the analyte can be 50 tons [29]). Of course, radiochemical and geochemical experiments are sensitive to complete decay and cannot distinguish neutrinoless decay from decay with emission of neutrinos. It is unknown a priori what type of decay is more likely, neutrinoless decay or decay with emission of neutrinos. It is possible that decay with emission of neutrinos will be more probable and we have more chance to see this decay in future experiments. Theoretical studies are needed on this issue. But, in any case, the observation of such transitions will be extremely important and interesting.

Acknowledgements.

Part of this work was supported by Russian Science Foundation (grant No. 18-12-00003).