Precise Half-Life Values for Two-Neutrino Double-β Decay: 2020 Review

Abstract

All existing positive results on two-neutrino double beta decay and two-neutrino double electron capture in different nuclei have been analyzed. Weighted average and recommended half-life values for ${}^{48}$Ca, ${}^{76}$Ge, ${}^{82}$Se, ${}^{96}$Zr, ${}^{100}$Mo, ${}^{100}$Mo - ${}^{100}$Ru (0₁⁺), ${}^{116}$Cd, ${}^{128}$Te, ${}^{130}$Te, ${}^{136}$Xe, ${}^{150}$Nd, ${}^{150}$Nd - ${}^{150}$Sm (0₁⁺), ${}^{238}$U, ${}^{78}$Kr, ${}^{124}$Xe and ${}^{130}$Ba have been obtained. Given the measured half-life values, effective nuclear matrix elements for all these transitions were calculated.

1. Introduction

Two-neutrino double beta decay ($2\nu \beta \beta$) was first considered by Maria Goeppert-Mayer in 1935 [1]:

$$(A,Z)\to (A,Z+2)+2e^{-}+2\overline{\nu }$$

(1)

This is a process in which a nucleus (A,Z) decays to a nucleus (A,Z + 2) by emitting two electrons and two electron-type antineutrinos. The $2\nu \beta \beta$ decay is a second-order weak interaction process and does not violate any conservation laws. Nevertheless, the study of this process provides rich information that can be used both to clarify various aspects of neutrinoless double beta decay and to search for exotic processes (decays with Majoron emission [2,3], bosonic neutrinos [4], violation of Lorentz invariance [3,5,6], the presence of right-handed leptonic currents [7], neutrino self-interactions ($\nu$SI) [8], etc.). The $2\nu \beta \beta$ decay was first discovered in a geochemical experiment with ${}^{130}$Te in 1950 [9]. In a direct (counter) experiment, the decay was first recorded by M. Moe et al. in 1987 (TRC, ${}^{82}$Se) [10]. To date, $2\nu \beta \beta$ decay has already been studied quite well. This process has been registered for 11 nuclei. For some nuclei (${}^{100}$Mo, ${}^{150}$Nd), a transition to the 0${}_1^{+}$ excited state of the daughter nucleus was detected too. In addition, a two-neutrino double electron capture (ECEC(2$\nu$)) was detected in several nuclei (${}^{130}$Ba [11], ${}^{124}$Xe [12], ${}^{78}$Kr [13]). In this process, two orbital electrons are captured. In the final state, two neutrinos and two X-rays appear:

$$e^{-}+e^{-}+(A,Z)\to (A,Z-2)+2\nu +2X.$$

(2)

In the NEMO-3 experiment, all decay characteristics (total energy spectrum, single electron spectrum, angular distribution) for 7 isotopes (${}^{48}$Ca, ${}^{82}$Se, ${}^{96}$Zr, ${}^{100}$Mo, ${}^{116}$Cd, ${}^{130}$Te and ${}^{150}$Nd) were studied simultaneously. At present, the study of two-neutrino processes is moving into a new stage, precision study. The accuracy of determining the half-life values and other characteristics of this process is becoming increasingly important (see the discussion in [3,7,14,15,16]). The exact half-lives are important to know for the following reasons:

1. Nuclear spectroscopy. It has now been established that some isotopes that were previously considered stable are not, and decay of these isotopes is observed through the $2\nu \beta \beta$ decay with a half-life of $\sim {10}^{18}-{10}^{24}$ yr. One just need to know the exact half-life values to include them in the isotope tables. Then, these values can be used for any purpose.

2. Nuclear matrix elements (NME). First of all, one can check the quality of NME calculations for $2\nu \beta \beta$ decay, because it is possible to directly compare experimental and calculated values. Secondly, accurate knowledge of the NME(2$\nu$) also makes it possible to improve the quality of NME calculations for neutrinoless double beta decay ($0\nu \beta \beta$). For example, the accurate half-life values for $2\nu \beta \beta$ decay are used to determine the most important parameter of the quasiparticle random-phase approximation model (QRPA), the strength of the particle–particle interaction $g_{pp}$ [17,18,19].

3. To fix $g_A$ (axial-vector coupling constant). There are indications that, in nuclear medium, the matrix elements of the axial-vector operator are reduced in comparison with their free nucleon values. This quenching is described as a reduction of the coupling constant $g_A$ from its free nucleon value of $g_A$ = 1.2701 [20] to the value of $g_A$∼ 0.35-1.0 (see [21,22,23]). In principle, $g_A$ value could be established by comparison of exact experimental values and results of theoretical calculations of NMEs. Finally, it can help in understanding the $g_A$ value in the case of $0\nu \beta \beta$ decay (see discussions in Refs. [21,22,23]).

It should be noted here that the phenomenological interpretation of the change in the value of $g_A$ in nuclear matter is apparently connected with the imperfection of our description of the nuclear structure and the process of double beta decay itself. Therefore, when describing the process of $2\nu \beta \beta$ decay, we are essentially adjusting the value of $g_A$ in order to give a correct description of the process. In this sense, this is about the same as the situation with $g_{pp}$ in the previous paragraph.

4. A check of the “bosonic” neutrino hypothesis [4] and $\nu$SI [8].

At the same time, it is quite difficult to choose the “best” result from the available data. For some isotopes, up to 7–10 different measurements exist. The quality of these results is not always obvious. Therefore, it is difficult to choose the best (“correct”) value for the half-life.

In the present paper, a critical analysis of all available results on two-neutrino decay has been performed and average or/and recommended half-life values for all isotopes are presented. Using these values and the values of the phase space factors from [24,25], the “effective” NMEs were calculated.

The first time that such type of work was done was in 2001, and the results were presented at the International Workshop on the calculation of double beta decay nuclear matrix elements, MEDEX’01 [26]. Then, updated half-life values were presented at MEDEX’05, MEDEX’09 and MEDEX’13 and published in Refs. [14,15,27], respectively. In this article, new positive results obtained since the beginning of 2015 and to the middle of 2020 have been added and analyzed. Preliminary results of this analysis have been presented at MEDEX’19 [28].

The main differences from the previous analysis [15] are the following: (1) The new experimental results are included in the analysis: ${}^{48}$Ca [29], ${}^{76}$Ge [30], ${}^{82}$Se [31,32], ${}^{100}$Mo [3,33], ${}^{116}$Cd [34,35], ${}^{130}$Te [36,37], ${}^{136}$Xe [38], ${}^{150}$Nd [39], ${}^{150}$Nd - ${}^{150}$Sm (0${}_1^{+}$) [40] and ${}^{130}$Ba [41]; (2) the positive results obtained for ${}^{78}$Kr [13] and ${}^{124}$Xe [12] are added (these decays have been detected for the first time). I would like to stress that most of the above-mentioned new results are very precise. The accuracy of some of the obtained half-life values is ∼2–3%. This is a result of (mainly) experiments with low-temperature bolometers (including scintillating bolometers).

2. Experimental Data

Table 1. Present, positive $2\nu \beta \beta$ decay results. N is the number of useful events, S/B is the signal-to-background ratio.

Nucleus	N	${\mathit{T}}_{1/2}$, yr	S/B	Ref., Year
${}^{48}$Ca	$\sim 100$	$[{4.3}_{-1.1}^{+2.4}\left(stat\right)\pm 1.4\left(syst\right)]·{10}^{19}$	1/5	[42], 1996
	5	${4.2}_{-1.3}^{+3.3}·{10}^{19}$	5/0	[43], 2000
	116	$\left[{6.4}_{-0.6}^{+0.7}{\left(stat\right)}_{-0.9}^{+1.2}\left(syst\right)\right]·{10}^{19}$	3.9	[29], 2016
		Average value:${\mathbf{5.3}}_{-\mathbf{0.8}}^{+\mathbf{1.2}}·{\mathbf{10}}^{\mathbf{19}}$
${}^{76}$Ge	$\sim 4000$	$(0.9\pm 0.1)·{10}^{21}$	∼ 1/8	[44], 1990
	758	${1.1}_{-0.3}^{+0.6}·{10}^{21}$	$\sim 1/6$	[45], 1991
	$\sim 330$	${0.92}_{-0.04}^{+0.07}·{10}^{21}$	∼1.2	[46], 1991
	132	${1.27}_{-0.16}^{+0.21}·{10}^{21}$	$\sim 1.4$	[47], 1994
	$\sim 3000$	$(1.45\pm 0.15)·{10}^{21}$	∼1.5	[48], 1999
	$\sim 80,000$	$[1.74\pm 0.01{\left(stat\right)}_{-0.16}^{+0.18}\left(syst\right)]·{10}^{21}$	∼1.5	[49], 2003
	25,690	$(1.925\pm 0.094)·{10}^{21}$	∼3	[30], 2015
		Average value:$(\mathbf{1.88}\pm \mathbf{0.08})·{\mathbf{10}}^{\mathbf{21}}$
${}^{82}$Se	89.6	${1.08}_{-0.06}^{+0.26}·{10}^{20}$	$\sim 8$	[50], 1992
	149.1	$[0.83\pm 0.10\left(stat\right)\pm 0.07\left(syst\right)]·{10}^{20}$	2.3	[51], 1998
	2750	$[0.939\pm 0.017\left(stat\right)\pm 0.058\left(syst\right)]·{10}^{20\left(a\right)}$	4	[31], 2018
	∼200,000	$[0.860\pm 0.003{\left(stat\right)}_{-0.013}^{+0.019}\left(syst\right)]·{10}^{20\left(a\right)}$	∼10	[32], 2019
		$(1.3\pm 0.05)·{10}^{20}$ (geochem.)		[52], 1986
		Average value:${\mathbf{0.87}}_{-\mathbf{0.01}}^{+\mathbf{0.02}}·{\mathbf{10}}^{\mathbf{20}}$
${}^{96}$Zr	26.7	$[{2.1}_{-0.4}^{+0.8}\left(stat\right)\pm 0.2\left(syst\right)]·{10}^{19}$	${1.9}^{\left(b\right)}$	[53], 1999
	453	$[2.35\pm 0.14\left(stat\right)\pm 0.16\left(syst\right)]·{10}^{19}$	1	[54], 2010
		$(3.9\pm 0.9)·{10}^{19}$ (geochem.)		[55], 1993
		$(0.94\pm 0.32)·{10}^{19}$ (geochem.)		[56], 2001
		Average value:$(\mathbf{2.3}\pm \mathbf{0.2})·{\mathbf{10}}^{\mathbf{19}}$
${}^{100}$Mo	∼500	${11.5}_{-2.0}^{+3.0}·{10}^{18}$	1/7	[57], 1991
	67	${11.6}_{-0.8}^{+3.4}·{10}^{18}$	7	[58], 1991
	1433	$[7.3\pm 0.35\left(stat\right)\pm 0.8\left(syst\right)]·{10}^{18\left(a\right)\left(c\right)}$	3	[59], 1995
	175	${7.6}_{-1.4}^{+2.2}·{10}^{18}$	1/2	[60], 1997
	377	$[{6.82}_{-0.53}^{+0.38}\left(stat\right)\pm 0.68\left(syst\right)]·{10}^{18}$	10	[61], 1997
	800	$[7.2\pm 1.1\left(stat\right)\pm 1.8\left(syst\right)]·{10}^{18}$	1/9	[62], 2001
	∼350	$[7.15\pm 0.37\left(stat\right)\pm 0.66\left(syst\right)]·{10}^{18}$	∼$5^{\left(d\right)}$	[63], 2014
	500,000	$[6.81\pm 0.01{\left(stat\right)}_{-0.40}^{+0.38}\left(syst\right)]·{10}^{18\left(a\right)}$	80	[3], 2019
	35,638	$[{7.12}_{-0.14}^{+0.18}\left(stat\right)\pm 0.10\left(syst\right)]·{10}^{18\left(a\right)}$	10	[33], 2020
		$(2.1\pm 0.3)·{10}^{18}$ (geochem.)		[64], 2004
		Average value:${\mathbf{7.06}}_{-\mathbf{0.13}}^{+\mathbf{0.15}}·{\mathbf{10}}^{\mathbf{18}}$
${}^{100}$Mo -	${133}^{\left(e\right)}$	${6.1}_{-1.1}^{+1.8}·{10}^{20}$	1/7	[65], 1995
${}^{100}$Ru ($0_1^{+}$)	${153}^{\left(e\right)}$	$[{9.3}_{-1.7}^{+2.8}\left(stat\right)\pm 1.4\left(syst\right)]·{10}^{20}$	1/4	[66], 1999
	19.5	$[{5.9}_{-1.1}^{+1.7}\left(stat\right)\pm 0.6\left(syst\right)]·{10}^{20}$	∼8	[67], 2001
	35.5	$[{5.5}_{-0.8}^{+1.2}\left(stat\right)\pm 0.3\left(syst\right)]·{10}^{20}$	∼8	[68], 2009
	37.5	$[{5.7}_{-0.9}^{+1.3}\left(stat\right)\pm 0.8\left(syst\right)]·{10}^{20}$	∼3	[69], 2007
	${597}^{\left(e\right)}$	$[{6.9}_{-0.8}^{+1.0}\left(stat\right)\pm 0.7\left(syst\right)]·{10}^{20}$	∼1/10	[70], 2010
	${239}^{\left(e\right)}$	$[7.5\pm 0.6\left(stat\right)\pm 0.6\left(syst\right)]·{10}^{20}$	2	[71], 2014
		Average value:${\mathbf{6.7}}_{-\mathbf{0.4}}^{+\mathbf{0.5}}·{\mathbf{10}}^{\mathbf{20}}$
${}^{116}$Cd	$\sim 180$	${2.6}_{-0.5}^{+0.9}·{10}^{19}$	∼1/4	[72], 1995
	174.6	$[2.9\pm 0.3\left(stat\right)\pm 0.2\left(syst\right)]·{10}^{19\left(a\right)\left(c\right)}$	3	[73], 1996
	9850	$[2.9\pm 0.06{\left(stat\right)}_{-0.3}^{+0.4}\left(syst\right)]·{10}^{19}$	∼3	[74], 2003
	4968	$[2.74\pm 0.04\left(stat\right)\pm 0.18\left(syst\right)]·{10}^{19\left(a\right)}$	12	[34], 2017
	93,000	${2.63}_{-0.12}^{+0.11}·{10}^{19}$	1.5	[35], 2018
		Average value:$(\mathbf{2.69}\pm \mathbf{0.09})·{\mathbf{10}}^{\mathbf{19}}$
${}^{128}$Te		$\sim 2.2·{10}^{24}$ (geochem.)		[75], 1991
		$(7.7\pm 0.4)·{10}^{24}$ (geochem.)		[76], 1993
		$(2.41\pm 0.39)·{10}^{24}$ (geochem.)		[77], 2008
		$(2.3\pm 0.3)·{10}^{24}$ (geochem.)		[78], 2008
		Recommended value: $(\mathbf{2.25}\pm \mathbf{0.09})·{\mathbf{10}}^{\mathbf{24}\left(\mathit{f}\right)}$
${}^{130}$Te	260	$[6.1\pm 1.4{\left(stat\right)}_{-3.5}^{+2.9}\left(syst\right)]·{10}^{20}$	1/8	[79], 2003
	236	$[7.0\pm 0.9\left(stat\right)\pm 1.1\left(syst\right)]·{10}^{20}$	1/3	[80], 2011
	∼33,000	$[8.2\pm 0.2\left(stat\right)\pm 0.6\left(syst\right)]·{10}^{20}$	0.1–0.5	[36], 2017
	∼20,000	$[7.9\pm 0.1\left(stat\right)\pm 0.2\left(syst\right)]·{10}^{20}$	>1	[37], 2020
		$\sim 8·{10}^{20}$ (geochem.)		[75], 1991
		$(27\pm 1)·{10}^{20}$ (geochem.)		[76], 1993
		$(9.0\pm 1.4)·{10}^{20}$ (geochem.)		[77], 2008
		$(8.0\pm 1.1)·{10}^{20}$ (geochem.)		[78], 2008
		Average value:$(\mathbf{7.91}\pm \mathbf{0.21})·{\mathbf{10}}^{\mathbf{20}}$
${}^{136}$Xe	∼19,000	$[2.165\pm 0.016\left(stat\right)\pm 0.059\left(syst\right)]·{10}^{21}$	∼10	[81], 2014
	∼100,000	$[2.21\pm 0.02\left(stat\right)\pm 0.07\left(syst\right)]·{10}^{21}$	∼10	[38], 2016
		Average value:$(\mathbf{2.18}\pm \mathbf{0.05})·{\mathbf{10}}^{\mathbf{21}}$
${}^{150}$Nd	23	$[{18.8}_{-3.9}^{+6.9}\left(stat\right)\pm 1.9\left(syst\right)]·{10}^{18}$	1.8	[82], 1995
	414	$[{6.75}_{-0.42}^{+0.37}\left(stat\right)\pm 0.68\left(syst\right)]·{10}^{18}$	6	[61], 1997
	2214	$[9.34\pm 0.22{\left(stat\right)}_{-0.60}^{+0.62}\left(syst\right)]·{10}^{18}$	4	[39], 2016
		Average value:$(\mathbf{8.4}\pm \mathbf{1.1})·{\mathbf{10}}^{\mathbf{18}}$
		Recommended value: $(\mathbf{9.34}\pm \mathbf{0.65})·{\mathbf{10}}^{\mathbf{18}}$
${}^{150}$Nd -	${177.5}^{\left(e\right)}$	$\left[{1.33}_{-0.23}^{+0.36}{\left(stat\right)}_{-0.13}^{+0.27}\left(syst\right)\right]·{10}^{20}$	1/5	[83], 2009
${}^{150}$Sm ($0_1^{+}$)	21.6	$[{1.07}_{-0.25}^{+0.45}\left(stat\right)\pm +0.07\left(syst\right)]·{10}^{20}$	∼1.2	[84], 2014
	∼6	$[{0.69}_{-0.19}^{+0.40}\left(stat\right)\pm +0.11\left(syst\right)]·{10}^{20}$	∼2	[40], 2019
		Average value:${\mathbf{1.2}}_{-\mathbf{0.2}}^{+\mathbf{0.3}}·{\mathbf{10}}^{\mathbf{20}}$
${}^{238}$U		$(\mathbf{2.0}\pm \mathbf{0.6})·{\mathbf{10}}^{\mathbf{21}}$ (radiochem.)		[85], 1991

^(a) For SSD mechanism. ^(b) For E_2e > 1.2 MeV. ^(c) After correction (see [14]). ^(d) For E_2e > 1.5 MeV. ^(e) In both peaks. ^(f) This value was obtained using average T_1/2 for ¹³⁰Te and well-known ratio T_1/2(¹³⁰Te)/T_1/2(¹²⁸Te) = (3.52 ± 0.11) · 10⁻⁴ [76].

and

Table 2. Present, positive two-neutrino double electron capture results. N is the number of useful events, S/B is the signal-to-background ratio. In the case of ${}^{78}$Kr and ${}^{124}$Xe $T_{1/2}$ for $2K\left(2\nu \right)$, capture is presented (this is ∼75–80% of $ECEC\left(2\nu \right)$).

Nucleus	N	${\mathit{T}}_{1/2}\left(2\mathit{\nu }\right)$, yr	S/B	Ref., Year
${}^{130}$Ba		${\mathbf{2.1}}_{-\mathbf{0.8}}^{+\mathbf{3.0}}·{\mathbf{10}}^{\mathbf{21}}$ (geochem.)		[86], 1996
$ECEC\left(2\nu \right)$		$(\mathbf{2.2}\pm \mathbf{0.5})·{\mathbf{10}}^{\mathbf{21}}$ (geochem.)		[11], 2001
		$(\mathbf{0.60}\pm \mathbf{0.11})·{\mathbf{10}}^{\mathbf{21}}$ (geochem.)		[87], 2009
		Recommended value:$(\mathbf{2.2}\pm \mathbf{0.5})·{\mathbf{10}}^{\mathbf{21}}$
${}^{78}$Kr	15	$[{\mathbf{1.9}}_{-\mathbf{0.7}}^{+\mathbf{1.3}}\left(\mathbf{stat}\right)\pm \mathbf{0.3}\left(\mathbf{syst}\right)]·{\mathbf{10}}^{\mathbf{22}}$	15	[13], 2017
$2K\left(2\nu \right)$
		Recommended value:$\left({\mathbf{1.9}}_{-\mathbf{0.8}}^{+\mathbf{1.3}}\right)·{\mathbf{10}}^{\mathbf{22}}$ (?)${}^{\left(a\right)}$
${}^{124}$Xe	126	$[\mathbf{1.8}\pm \mathbf{0.5}\left(\mathbf{stat}\right)\pm \mathbf{0.1}\left(\mathbf{syst}\right)]·{\mathbf{10}}^{\mathbf{22}}$	0.2	[12], 2019
$2K\left(2\nu \right)$
		Recommended value:$(\mathbf{1.8}\pm \mathbf{0.5})·{\mathbf{10}}^{\mathbf{22}}$

${}^{\left(a\right)}$ See text.

show the experimental results on $2\nu \beta \beta$ decay and on ECEC(2$\nu$) capture in different nuclei. For direct experiments, the number of detected (useful) events and the signal-to-background (S/B) ratio are presented.

3. Data Analysis

To calculate an average of the ensemble of available data, a standard procedure, as recommended by the Particle Data Group [20], was used. The weighted average and the corresponding error were calculated, as follows:

$$\overline{x}\pm \delta \overline{x}=\sum w_i{x}_i/\sum w_i\pm {(\sum w_i)}^{-1/2},$$

(3)

where $w_i=1/{\left(\delta x_i\right)}^2$. Here, $x_i$ and $\delta x_i$ are the value and error reported by the i-th experiment, and the summations run over the N experiments.

Then, it is necessary to calculate ${\chi }^2=\sum w_i{(\overline{x}-x_i)}^2$ and compare it with $N-1$, which is the expectation value of ${\chi }^2$ if the measurements are from a Gaussian distribution. In the case when ${\chi }^2/(N-1)$ is less than or equal to 1 and there are no known problems with the data, then one accepts the results. In the case when ${\chi }^2/(N-1)>>1$, one chooses not to use the average procedure at all. Finally, if ${\chi }^2/(N-1)$ is larger than 1, but not greatly so, it is still best to use the average data, but to increase the quoted error, $\delta \overline{x}$ in Equation (1), by a factor of S defined by

$$S={[{\chi }^2/(N-1)]}^{1/2}.$$

(4)

For averages, the statistical and systematic errors are treated in quadrature and used as a combined error $\delta x_i$. In some cases, only the results obtained with a high enough S/B ratio were used.

3.1. ${}^{48}$Ca

The $2\nu \beta \beta$ decay of ${}^{48}$Ca was observed in three independent experiments [29,42,43]. The obtained results are in good agreement. The weighted average value is:

$$T_{1/2}={5.3}_{-0.8}^{+1.2}·{10}^{19}\mathrm{yr}.$$

This value is slightly higher than the average value obtained in previous analysis ($T_{1/2}={4.4}_{-0.5}^{+0.6}·{10}^{19}$ yr [15]). This is due to the fact that the final result of the NEMO-3 experiment [29] was used in present analysis (the intermediate result of the NEMO-3 experiment [88] was used in [15]). The change in the final result in the NEMO-3 experiment was mainly due to the fact that after disassembling the detector, the parameters of sources containing ${}^{48}$Ca were refined. It was found that, in reality, the diameter of the sources turned out to be slightly larger (and the thickness, respectively, less) than previously assumed. Taking this circumstance into account led to an increase in the calculated efficiency of recording useful events and, ultimately, to an increase in the $T_{1/2}$ value for ${}^{48}$Ca. In addition, systematic error in [29] is higher then in [88].

3.2. ${}^{76}$Ge

For ${}^{76}$Ge, a lot of positive results were obtained, but the scatter of the obtained values is rather large. Half-life values gradually increased over time during the 90-th. It was decided not to use the results of the early works (1990s), as a recent historical review [89] emphasized that the contribution of background processes was underestimated in these works. Therefore, to determine the average value, the results published after 2000 have been used, with large statistics and a high S/B ratio [30,49]. Note that the final result of the Heidelberg–Moscow collaboration was used [49]. As a result, we get:

$$T_{1/2}=(1.88\pm 0.08)·{10}^{21}\mathrm{yr}.$$

3.3. ${}^{82}$Se

There are many geochemical measurements (∼20) and only four independent counting experiments for ${}^{82}$Se. However, the geochemical results are in poor agreement with each other and with the results of direct experiments. It is known that the possibility of existing large systematic errors in geochemical measurements cannot be excluded (see discussion in Ref. [90]). Thus, only the results of the direct measurements [31,32,50,51] were used to obtain a present half-life value for ${}^{82}$Se. Single State Dominance (SSD) mechanism (see explanation in [91]) was established for $2\nu \beta \beta$ transition in ${}^{82}$Se [31,32] and half-life values in this papers were obtained under the assumption of the SSD mechanism 1. The result of Ref. [10] has not been used in the analysis because this is the preliminary result of [50]. The result of work [50] is presented with very asymmetrical errors. To be more conservative, the value for the lower error was taken to be the same as the upper one in our analysis. Finally, the weighted average value is:

$$T_{1/2}={0.87}_{-0.01}^{+0.02}·{10}^{20}\mathrm{yr}.$$

3.4. ${}^{96}$Zr

There are two positive results from the direct experiments (NEMO-2 [53] and NEMO-3 [54]) and two geochemical results [55,56]. Taking into account the comment in Section 3.3, the values from direct experiments (Refs. [53,54]) were used to obtain a present weighted half-life value for ${}^{96}$Zr:

$$T_{1/2}=(2.3\pm 0.2)·{10}^{19}\mathrm{yr}.$$

3.5. ${}^{100}$Mo

By the present nine positive results from direct experiments2 and one result from a geochemical experiment have been obtained. I do not use the geochemical result here (see comment in Section 3.3). Finally, in calculating the average, only the results of experiments with S/B ratios greater than 1 were used (i.e., the results of Refs. [3,33,59,61,63]). I use only final result of Elliott et al. [61] and do not consider their preliminary result from [58]. For ${}^{100}$Mo SSD mechanism was installed and in Ref. [3,33,59] the half-lives were obtained taking this fact into account. In addition, the corrected half-life value from Ref. [59] has been used (see explanation in [14]). The following weighted average value for the half-life is obtained as:

$$T_{1/2}={7.06}_{-0.13}^{+0.15}·{10}^{18}\mathrm{yr}.$$

3.6. ${}^{100}$Mo - ${}^{100}$Ru ($0_1^{+}$; 1130.32 Kev)

The $2\nu \beta \beta$ decay of ${}^{100}$Mo to the $0_1^{+}$ excited state of ${}^{100}$Ru was detected in seven independent experiments. The results are in good agreement. The weighted average value for the half-life has been obtained using the results from [65,66,68,69,70,71]:

$$T_{1/2}={6.7}_{-0.4}^{+0.5}·{10}^{20}\mathrm{yr}.$$

The result from [68] was used as the final result of the TUNL-ITEP experiment (the result from [67] was not used here because I consider it as preliminary one).

3.7. ${}^{116}$Cd

Five independent positive results were obtained [34,35,72,73,74]. The results are in good agreement with each other. The corrected result for the half-life value from Ref. [73] is used here. The original half-life value was decreased by ∼25% (see explanation in [14]). In Refs. [34,73], half-life values were obtained with the assumption that the SSD mechanism was realized. The weighted average value is:

$$T_{1/2}=(2.69\pm 0.09)·{10}^{19}\mathrm{yr}.$$

3.8. ${}^{128}$Te and ${}^{130}$Te

There are a large number of geochemical results for these isotopes. Although the half-life ratio for these isotopes is well known (accuracy is ∼3% [76]), the absolute $T_{1/2}$ values for each isotope are different from one experiment to the next. One group of authors [75,93,94] gives $T_{1/2}\approx 0.8·{10}^{21}$ yr for ${}^{130}$Te and $T_{1/2}\approx 2·{10}^{24}$ yr for ${}^{128}$Te, while another group [52,76] claims $T_{1/2}\approx (2.5-2.7)·{10}^{21}$ yr and $T_{1/2}\approx 7.7·{10}^{24}$ yr, respectively. In addition, as a rule, experiments with young samples (∼100 million years) give results of the half-life value for ${}^{130}$Te in the range of $\sim (0.7-0.9)·{10}^{21}$ yr, while experiments with old samples ($>1$ billion years) give half-life values in the range of $\sim (2.5-2.7)·{10}^{21}$ yr. In 2008, it was demonstrated that short half-lives are more likely to be correct [77,78]. In a new experiment with young minerals, the half-life values were estimated at $(9.0\pm 1.4)·{10}^{20}$ yr [77] and $(8.0\pm 1.1)·{10}^{20}$ yr [78] for ${}^{130}$Te and $(2.41\pm 0.39)·{10}^{24}$ yr [77] and $(2.3\pm 0.3)·{10}^{24}$ yr [78] for ${}^{128}$Te. In fact, in both experiments, the half-life was measured only for ${}^{130}$Te, and the value for ${}^{128}$Te was determined using the previously measured $T_{1/2}{(}^{130}\mathrm{Te})/T_{1/2}{(}^{128}\mathrm{Te})$ ratio [76]. If we average the values obtained in these two experiments, we get: $T_{1/2}=(8.4\pm 0.9)·{10}^{20}$ years for ${}^{130}$Te and $T_{1/2}=(2.3\pm 0.3)·{10}^{24}$ years for ${}^{128}$Te, which is in good agreement with the results of direct (counter) experiments (see below).

The first indication of the observation of the $2\nu \beta \beta$ decay for ${}^{130}$Te in a direct experiment was obtained in [79]. More accurate and reliable values were obtained later in the NEMO-3 experiment [80]. Very precise results were obtained recently in CUORE-0 [36] and CUORE [37] experiments. The results are in good agreement, and the weighted average value is

$$T_{1/2}=(7.91\pm 0.21)·{10}^{20}\mathrm{yr}.$$

Now, using the very well-known ratio $T_{1/2}{(}^{130}\mathrm{Te})/T_{1/2}{(}^{128}\mathrm{Te})=(3.52\pm 0.11)·{10}^{-4}$ [76], one can obtain the half-life value for ${}^{128}$Te,

$$T_{1/2}=(2.25\pm 0.09)·{10}^{24}\mathrm{yr}.$$

I recommend using these two results as the most correct and reliable half-life values for ${}^{130}$Te and ${}^{128}$Te. As one can see now, results of direct and geochemical experiments are in good agreement.

3.9. ${}^{136}$Xe

The half-life value for ${}^{136}Xe$ was measured in two independent experiments, EXO [81,95,96] and Kamland-Zen [38,97,98]. To obtain the average value of the half-life, the most accurate results of these experiments obtained in [38,81] were used (see Table 1). The weighted average value is

$$T_{1/2}=(2.18\pm 0.05)·{10}^{21}\mathrm{yr}.$$

3.10. ${}^{150}$Nd

The positive results were obtained in three independent experiments [39,61,82]. The most accurate value was obtained in Ref. [39]. This value is higher than in Ref. [61] ($\sim 3\sigma$ difference) and lower than in Ref. [82] ($\sim 2\sigma$ difference). Using Equations (2) and the three above-mentioned results, one obtains $T_{1/2}=(8.4\pm 0.5)·{10}^{18}$ yr. Taking into account that ${\chi }^2/(N-1)>1$ and S = 2.23 (see Equation (3)), we then obtain:

$$T_{1/2}=(8.4\pm 1.1)·{10}^{18}\mathrm{yr}.$$

It can be seen that due to the discrepancy between the $T_{1/2}$ values, one has to increase the error in order to somehow agree on the experimental results. On the other hand, it is clear that the result of the NEMO-3 experiment is today the most accurate and reliable. This is confirmed by the fact that in the NEMO-3 experiment, seven different isotopes were investigated simultaneously. In addition to ${}^{150}$Nd, ${}^{48}$Ca, ${}^{82}$Se, ${}^{96}$Zr, ${}^{100}$Mo, ${}^{116}$Cd and ${}^{130}$Te were also studied. For all these isotopes, the results are in good agreement with the results of other experiments. It is natural to assume that the result for ${}^{150}$Nd is correct too. Therefore, I think that it is necessary to use this value as the most accurate at the moment:

$$T_{1/2}={9.34}_{-0.64}^{+0.67}·{10}^{18}\mathrm{yr}.$$

3.11. ${}^{150}$Nd - ${}^{150}$Sm ($0_1^{+}$; 740.4 Kev)

There are two positive results for $2\nu \beta \beta$ decay of ${}^{150}$Nd to the $0_1^{+}$ excited state of ${}^{150}$Sm [83,84] (the preliminary result of Ref. [83] was published in Ref. [99]). These two results are in good agreement. The weighted average value is:

$$T_{1/2}={1.2}_{-0.2}^{+0.3}·{10}^{20}\mathrm{yr}.$$

Recently, the result of a new experiment was presented at MEDEX’19 [40] (see Table 1). I am not using this new result in my analysis because this is an ongoing experiment and the result is still preliminary and not yet published.

3.12. ${}^{238}$U

The two-neutrino decay of ${}^{238}$U was measured in a single experiment using the radiochemical technique [85]:

$$T_{1/2}=(2.0\pm 0.6)·{10}^{21}\mathrm{yr}.$$

It has to be stressed that for ${}^{238}$U a “positive” result was obtained in only the experiment. Therefore, it is necessary to confirm this result in independent experiments (including direct measurements). Until these confirmations are received, one has to be very careful with this value.

3.13. ${}^{130}$Ba (ECEC)

For ${}^{130}$Ba, positive results were obtained using the geochemical technique only. In this type of measurement, one can not recognize the different modes. It is clear that exactly the ECEC(2$\nu$) process was detected because other modes are strongly suppressed (see estimations in [91,100,101]). The first time the positive result for ${}^{130}$Ba was mentioned was in Ref. [86], where experimental data of Ref. [102] were analyzed. In this paper, a positive result was obtained for one sample of barite ($T_{1/2}={2.1}_{-0.8}^{+3.0}·{10}^{21}$ yr), but for a second sample only the limit was set ($T_{1/2}>4·{10}^{21}$ yr). Later, more accurate half-life values, $(2.2\pm 0.5)·{10}^{21}$ yr [11] and $(0.60\pm 0.11)·{10}^{21}$ yr [87], were measured. One can see that the results are in strong disagreement. In [41], the data of [87] were analyzed and it was shown that subtraction of the contribution of cosmogenic ${}^{130}$Xe removes the contradiction with the result of [11]. Finally, I recommend the following value from [11]:

$$T_{1/2}=(2.2\pm 0.5)·{10}^{21}\mathrm{yr}.$$

To obtain more reliable and precise half-life values, new measurements are needed (including direct experiments).

3.14. ${}^{78}$Kr (2K)

The first indication of the observation of 2$\mathit{K}$ capture in ${}^{78}$Kr was announced in 2013 (the effect is ∼2.5 $\sigma$), $T_{1/2}=[{0.92}_{-0.26}^{+0.55}\left(stat\right)\pm 0.13\left(syst\right)]·{10}^{22}$ years [103]. Then, the same data were analyzed more carefully and a new value was published (∼ 4 $\sigma$), which turned out to be twice as much, $T_{1/2}=[{1.9}_{-0.7}^{+1.3}\left(stat\right)\pm 0.3\left(syst\right)]·{10}^{22}$ [13]. The analysis of the data is quite complicated and it is possible that the systematic error is much larger than the indicated 15%.

There is one more circumstance that makes me cautious about the result given in [13]. As can be seen from Table 3, in the case of ${}^{78}$Kr, we are dealing with an anomalously large value of nuclear matrix element. This value is significantly larger than in the case of ${}^{130}$Ba and ${}^{124}$Xe (1.8 and 5.4 times, respectively) and exceeds all 13 NME values for $2\nu \beta \beta$ decay (from 1.7 to 17.7 times). Here, it is necessary to take into account that, since the rate of the ECEC process is ∼ 15–20% higher than the 2$\mathit{K}$ capture, the NME for the ECEC process in ${}^{78}$Kr is approximately 1.07–1.1 times greater than for the 2$\mathit{K}$ capture. This circumstance only strengthens the contradiction. In principle, such a large NME is possible, but looks strange. In any case, confirmation of the result [13] in independent measurements is necessary. Until the confirmation, one has to be very careful with this result.

3.15. ${}^{124}$Xe (2k)

To date, only one positive result has been published for 2$\mathit{K}$ capture in ${}^{124}$Xe [12]: $T_{1/2}=[1.8\pm 0.5\left(stat\right)\pm 0.13\left(syst\right)]·{10}^{22}$ yr. The significance of the effect is only 4.4 $\sigma$. It should also be noted that a limit $T_{1/2}>2.1·{10}^{22}$ yr was obtained in [104], which formally contradicts the result of [12]. Taking into account errors, there is no real contradiction here. However, it is clear that it is necessary to confirm the result of [12] in an independent experiment.

4. NME Values for Two-Neutrino Double Beta Decay

Obtained average and recommended half-life values are presented in

Table 3. Half-life and effective nuclear matrix element values for $2\nu \beta \beta$ decay (see Section 4).

Isotope	${\mathit{T}}_{1/2}\left(2\mathit{\nu }\right)$, yr	$\mid {\mathit{M}}_{2\mathit{\nu }}^{\mathbf{eff}}\mid$ (${\mathit{G}}_{2\mathit{\nu }}$ from [24])	$\mid {\mathit{M}}_{2\mathit{\nu }}^{\mathbf{eff}}\mid$ (${\mathit{G}}_{2\mathit{\nu }}$ from [25])	Recommended Value
$2\nu \beta \beta$:
${}^{48}$Ca	${5.3}_{-0.8}^{+1.2}·{10}^{19}$	${0.0348}_{-0.0034}^{+0.0030}$	${0.0348}_{-0.0034}^{+0.0030}$	$0.035\pm 0.003$
${}^{76}$Ge	$(1.88\pm 0.08)·{10}^{21}$	${0.1051}_{-0.0024}^{+0.0023}$	${0.1074}_{-0.0022}^{+0.0024}$	$0.106\pm 0.004$
${}^{82}$Se	${0.87}_{-0.01}^{+0.02}·{10}^{20}$	${0.0849}_{-0.0010}^{+0.0005}$	${0.0855}_{-0.0010}^{+0.0005}$	$0.085\pm 0.001$
${}^{96}$Zr	$(2.3\pm 0.2)·{10}^{19}$	${0.0798}_{-0.0032}^{+0.0037}$	${0.0804}_{-0.0033}^{+0.0038}$	$0.080\pm 0.004$
${}^{100}$Mo	${7.06}_{-0.13}^{+0.15}·{10}^{18}$	${0.2071}_{-0.0022}^{+0.0019}$	${0.2096}_{-0.0022}^{+0.0020}$
		${0.1852}_{-0.0019}^{+{0.0017}^{\left(a\right)}}$		$0.185\pm 0.002$
${}^{100}$Mo-	${6.7}_{-0.4}^{+0.5}·{10}^{20}$	${0.1571}_{-0.0056}^{+0.0048}$	${0.1619}_{-0.0058}^{+0.0050}$
${}^{100}$Ru$\left(0_1^{+}\right)$		${0.1513}_{-0.0053}^{+{0.0047}^{\left(a\right)}}$		$0.151\pm 0.005$
${}^{116}$Cd	$(2.69\pm 0.09)·{10}^{19}$	${0.1160}_{-0.0019}^{+0.0020}$	${0.1176}_{-0.0019}^{+0.0020}$
		${0.1084}_{-0.0019}^{+{0.0024}^{\left(a\right)}}$		$0.108\pm 0.003$
${}^{128}$Te	$(2.25\pm 0.09)·{10}^{24}$	${0.0406}_{-0.0008}^{+0.0008}$	${0.0454}_{-0.0009}^{+0.0009}$	$0.043\pm 0.003$
${}^{130}$Te	$(7.91\pm 0.21)·{10}^{20}$	${0.0288}_{-0.0004}^{+0.0004}$	${0.0297}_{-0.0004}^{+0.0004}$	$0.0293\pm 0.0009$
${}^{136}$Xe	$(2.18\pm 0.05)·{10}^{21}$	${0.0177}_{-0.0002}^{+0.0002}$	${0.0184}_{-0.0002}^{+0.0002}$	$0.0181\pm 0.0006$
${}^{150}$Nd	$(9.34\pm 0.65)·{10}^{18}$	${0.0543}_{-0.0018}^{+0.0020}$	${0.0550}_{-0.0018}^{+0.0020}$	$0.055\pm 0.003$
${}^{150}$Nd-	${1.2}_{-0.2}^{+0.3}·{10}^{20}$	${0.0438}_{-0.0046}^{+0.0042}$	${0.0450}_{-0.0048}^{+0.0043}$	$0.044\pm 0.005$
${}^{150}$Sm($0_1^{+}$)
${}^{238}$U	$(2.0\pm 0.6)·{10}^{21}$	${0.1853}_{-0.0227}^{+0.0361}$	${0.0713}_{-0.0088}^{+0.0139}$	${0.13}_{-0.07}^{+0.09}$
ECEC(2$\nu$):
${}^{78}$Kr${}^{\left(b\right)}$	${1.9}_{-0.8}^{+1.3}·{10}^{22}$	${0.2882}_{-0.0706}^{+0.0829}$ [105]	${0.3583}_{-0.0822}^{+0.1126}$	${0.32}_{-0.11}^{+0.15}$
${}^{124}$Xe${}^{\left(b\right)}$	$(1.8\pm 0.5)·{10}^{22}$	${0.0568}_{-0.0650}^{+0.0101}$ [105]	${0.0607}_{-0.0070}^{+0.0107}$	${0.059}_{-0.009}^{+0.013}$
${}^{130}$Ba	$(2.2\pm 0.5)·{10}^{21}$	${0.1741}_{-0.0170}^{+0.0239}$ [105]	${0.1754}_{-0.0171}^{+0.0241}$	${0.175}_{-0.017}^{+0.024}$

${}^{\left(a\right)}$ Obtained using the SSD model. ${}^{\left(b\right)}$ Value for 2$\mathit{K}$ capture. For the ECEC process, the half-life value will be approximately 15–20% less, and the NME value approximately 7–10% higher.

(2-nd column). Using these values, one can extract the experimental nuclear matrix elements through the relation [24]:

$$T_{1/2}^{-1}=G_{2\nu }·g_A^4·{(m_e{c}^2·M_{2\nu })}^2,$$

(5)

where $T_{1/2}$ is the half-life value in [yr], $G_{2\nu }$ is the phase space factor in [yr${}^{-1}$], $g_A$ is the dimensionless axial vector coupling constant and $(m_e{c}^2·M_{2\nu })$ is the dimensionless nuclear matrix element. One has to remember that there are indications that in nuclear medium the $g_A$ value is reduced in comparison with their free nucleon values (see Section 1). Expression (5) is valid for $2\nu \beta \beta$ and ECEC(2$\nu$) processes.

Thereby, following Ref. [24], it is better to use the so-called effective NME, $\mid M_{2\nu }^{eff}\mid =g_A^2·\mid (m_e{c}^2·M_{2\nu })\mid$. This value has been calculated for all isotopes.

The obtained results are presented in Table 3 (3-rd and 4-th columns). When calculating, I used the $G_{2\nu }$ values from Refs. [24,25] (see

Table 4. Phase-space factors from Refs. [24,25,105].

Isotope	${\mathit{G}}_{2\mathit{\nu }}\left({10}^{-21}{\mathbf{yr}}^{-1}\right)$ [24]	${\mathit{G}}_{2\mathit{\nu }}\left({10}^{-21}{\mathbf{yr}}^{-1}\right)$ [25]
$2\nu \beta \beta$:
${}^{48}$Ca	15,550	15,536
${}^{76}$Ge	48.17	46.47
${}^{82}$Se	1596	1573
${}^{96}$Zr	6816	6744
${}^{100}$Mo	3308	3231
	${4134 }^{\left(a\right)}$
${}^{100}$Mo-${}^{100}$Ru$\left(0_1^{+}\right)$	60.55	57.08
	${65.18 }^{\left(a\right)}$
${}^{116}$Cd	2764	2688
	${3176 }^{\left(a\right)}$
${}^{128}$Te	0.2688	0.2149
${}^{130}$Te	1529	1442
${}^{136}$Xe	1433	1332
${}^{150}$Nd	36,430	35,397
${}^{150}$Nd-${}^{150}$Sm($0_1^{+}$)	4329	4116
${}^{238}$U	14.57	98.51
ECEC(2$\nu$):
${}^{78}$Kr	0.660 [105]	0.410
${}^{124}$Xe	17.200 [105]	15.096
${}^{130}$Ba	15.000 [105]	14.773

${}^{\left(a\right)}$ Obtained using SSD model.

). For ${}^{130}$Ba, ${}^{78}$Kr and ${}^{124}$Xe $G_{2\nu }$ values for ECEC transition were taken from [25,105]. These calculations are most reliable and correct at this moment. The results of these calculations are in reasonable agreement (∼1–7%) with three exceptions: for ${}^{128}$Te (∼20%), ${}^{78}$Kr (∼30%) and ${}^{238}$U (factor ∼ 7 ). For ${}^{238}$U, two different values $14.57·{10}^{-21}{\mathrm{yr}}^{-1}$ [24] and $98.51·{10}^{-21}{\mathrm{yr}}^{-1}$ [25]) were produced. The situation with calculations for ${}^{238}$U is clearly unsatisfactory and these calculations should be rechecked. For ${}^{100}$Mo, ${}^{100}$Mo-${}^{100}$Ru$\left(0_1^{+}\right)$ and ${}^{116}$Cd, I used $G_{2\nu }$ calculated in Ref. [24] for the SSD mechanism. The obtained values for $\mid M_{2\nu }^{eff}\mid$ are given in Table 3 and these are the most correct values for these isotopes. So-called recommended values for $\mid M_{2\nu }^{eff}\mid$ are presented in Table 3 (5-th column) too. These values were obtained as an average of two values, given in columns 3 and 4. The recommended value error is chosen to cover all ranges of values from columns 3 and 4 (taking into account corresponding errors). For ${}^{100}$Mo, ${}^{100}$Mo-${}^{100}$Ru$\left(0_1^{+}\right)$ and ${}^{116}$Cd, I recommend to use the values obtained with $G_{2\nu }$ for the SSD mechanism.

Therefore, for the majority of isotopes an accuracy for $\mid M_{2\nu }^{eff}\mid$ is on the level ∼1–10%. For ${}^{78}$Kr, ${}^{124}$Xe and ${}^{130}$Ba, it is ∼46%, ∼22% and ∼14%, respectively. This is mainly because of the not precise half-life values obtained for these isotopes. The most unsatisfactory situation is for ${}^{238}$U (∼70%). Main uncertainty in this case is connected with the accuracy of $G_{2\nu }$.

Recently, in Ref. [16], an improved formalism of the $2\nu \beta \beta$ decay rate was presented, which takes into account the dependence of energy denominators on lepton energies via the Taylor expansion. As a result, the formula for the half-life starts to be more complicated and contains several different matrix elements and different phase space volumes. That is, a new approach to processing the results will be required. To do this, some parameters of this approach have to be established from experiment and calculated reliably, e.g., within the interacting shell model (see discussion in [16]). Nevertheless, the results shown in Table 3 retain their significance since it was demonstrated in [16] that additional terms contribute ∼3% to ∼25% to the total decay rate. This means that if we consider expression (5) as the first term of the expansion in the approach [16], then we can conclude that the values of $\mid M_{2\nu }^{eff}\mid$ obtained in this work give a good estimate for $g_A^2·$$\mid M_{GT-1}^{2\nu }\mid$ (see Formula (19) in [16]). The values given in Table 3 overestimate $g_A^2·$$\mid M_{GT-1}^{2\nu }\mid$ values by ∼1.5–12% only, which is comparable to the accuracy of determining $\mid M_{2\nu }^{eff}\mid$. An exception is the situation with the results for ${}^{100}$Mo and ${}^{116}$Cd obtained using phase space volumes calculated within the SSD. In this case, the most accurate NME estimate was obtained since the exact value of the energy of the lowest 1${}^{+}$ intermediate state was used in the calculations of the phase space volume.

5. Conclusions

Thus, the all positive results for $2\nu \beta \beta$ decay obtained by August 2020 have been analyzed. As a result, the average values of the half-life were obtained for all considered isotopes. For ${}^{128}$Te, ${}^{150}$Nd and ${}^{130}$Ba, so-called recommended values have also been proposed. Using these obtained average/recommended half-life values, the $\mid M_{2\nu }^{eff}\mid$ values for all considered nuclei were determined. Finally, previous results from Ref. [15] were successfully updated. A summary is shown in Table 3. I recommend using these values as the most correct and reliable currently. If we look at the dynamics of the average values since 2001, we can see that these values were constantly refined over time and did not deviate by more than 1–2 $\sigma$ from the initial value. An exception is the situation with ${}^{76}$Ge. Here, the average value has steadily increased with time (from ${1.42}_{-0.07}^{+0.09}·{10}^{21}$ yr in 2001 to $(1.88\pm 0.08)·{10}^{21}$ yr in 2020). This is due to the low quality of the results obtained in the 1990s. In the latest analysis, the results obtained after 2000 have been used.

At present, $2\nu \beta \beta$ decay was recorded in 11 nuclei, and ECEC capture in 3 nuclei (with some doubts for ${}^{78}$Kr). The accuracy of determining the half-life for most nuclei lies in the range 2–10%. It is expected that in the next few years new results will be obtained for ${}^{76}$Ge (Majorana), ${}^{100}$Mo (CUPID-Mo, AMORE, CROSS), ${}^{116}$Cd (CROSS), ${}^{130}$Te (CROSS, SNO+), ${}^{136}$Xe (NEXT-100) and ${}^{124}$Xe (NEXT-100, LUX-ZEPLIN). The final result will be obtained in an experiment to search for the $2\nu \beta \beta$ decay of ${}^{150}$Nd to the first excited 0${}^{+}$ level of ${}^{150}$Sm (see [40]). Let us emphasize here the importance of experiments using low-temperature bolometers. In experiments with such detectors, the measurement accuracy of the half-life can reach 1–2%. At present, such experiments are possible for ${}^{82}$Se, ${}^{100}$Mo, ${}^{116}$Cd and ${}^{130}$Te. Apparently, in the future, such measurements will be implemented for ${}^{48}$Ca as well. I hope that in the future 2$\beta$ processes will also be found in other nuclei. The search for $2\nu \beta \beta$ processes in ${}^{124}$Sn, ${}^{110}$Pd, ${}^{160}$Gd and the search for ECEC(2$\nu$) processes in ${}^{96}$Ru, ${}^{106}$Cd and ${}^{136}$Ce seem promising. As for the $2\nu \beta \beta$ transitions to the excited states of the daughter nucleus, it seems possible to register a transition to the 0${}_1^{+}$ excited level in measurements with ${}^{96}$Zr and ${}^{82}$Se in the near future.