Precise Half-Life Values for Two-Neutrino Double- β Decay: 2020 review
AArticle
Precise Half-Life Values for Two-Neutrino Double- β Decay: 2020 review
Alexander Barabash *
National Research Centre “Kurchatov Institute”, Institute of Theoretical and Experimental Physics,B. Cheremushkinskaya 25, 117218 Moscow, Russia; [email protected]; Tel.: +7-916-933-9350
Abstract:
All existing positive results on two-neutrino double beta decay and two-neutrino doubleelectron capture in different nuclei have been analyzed. Weighted average and recommended half-lifevalues for Ca, Ge, Se, Zr,
Mo,
Mo -
Ru (0 + ), Cd,
Te,
Te,
Xe,
Nd,
Nd -
Sm (0 + ), U, Kr,
Xe and
Ba have been obtained. Given the measured half-life values, effectivenuclear matrix elements for all these transitions were calculated.
Keywords: double beta decay; nuclear matrix elements; Ca; Ge; Se; Zr;
Mo;
Mo -
Ru (0 + ); Cd;
Te;
Te;
Xe;
Nd;
Nd -
Sm (0 + ); U; Kr;
Xe; Ba
1. Introduction
Two-neutrino double beta decay (2 νββ ) was first considered by Maria Goeppert-Mayer in 1935 [1]: ( A , Z ) → ( A , Z + ) + e − + ν (1)This is a process in which a nucleus (A,Z) decays to a nucleus (A,Z + 2) by emitting two electrons andtwo electron-type antineutrinos. The 2 νββ decay is a second-order weak interaction process and does notviolate any conservation laws. Nevertheless, the study of this process provides rich information that canbe 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], thepresence of right-handed leptonic currents [7], neutrino self-interactions ( ν SI) [8], etc.). The 2 νββ decaywas first discovered in a geochemical experiment with
Te in 1950 [9]. In a direct (counter) experiment,the decay was first recorded by M. Moe et al. in 1987 (TRC, Se) [10]. To date, 2 νββ decay has alreadybeen studied quite well. This process has been registered for 11 nuclei. For some nuclei (
Mo,
Nd), atransition to the 0 + excited state of the daughter nucleus was detected too. In addition, a two-neutrinodouble electron capture (ECEC(2 ν )) was detected in several nuclei ( Ba [11],
Xe [12], Kr [13]). In thisprocess, two orbital electrons are captured. In the final state, two neutrinos and two X-rays appear: e − + e − + ( A , Z ) → ( A , Z − ) + ν + X . (2)In the NEMO-3 experiment, all decay characteristics (total energy spectrum, single electron spectrum,angular distribution) for 7 isotopes ( Ca, Se, Zr,
Mo,
Cd,
Te and
Nd) were studiedsimultaneously. At present, the study of two-neutrino processes is moving into a new stage, precisionstudy. The accuracy of determining the half-life values and other characteristics of this process is becomingincreasingly important (see the discussion in [3,7,14–16]). The exact half-lives are important to know forthe following reasons:1.
Nuclear spectroscopy . It has now been established that some isotopes that were previously consideredstable are not, and decay of these isotopes is observed through the 2 νββ decay with a half-life of ∼ a r X i v : . [ nu c l - e x ] S e p of 18 − 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 for2 νββ decay, because it is possible to directly compare experimental and calculated values. Secondly,accurate knowledge of the NME(2 ν ) also makes it possible to improve the quality of NME calculationsfor neutrinoless double beta decay (0 νββ ). For example, the accurate half-life values for 2 νββ decay areused to determine the most important parameter of the quasiparticle random-phase approximation model(QRPA), the strength of the particle–particle interaction g pp [17–20].3. To fix g A (axial-vector coupling constant) . There are indications that, in nuclear medium, thematrix 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 [21] to the value of g A ∼ g A value could be established bycomparison of exact experimental values and results of theoretical calculations of NMEs. Finally, it canhelp in understanding the g A value in the case of 0 νββ decay (see discussions in Refs. [22–24]).It should be noted here that the phenomenological interpretation of the change in the value of g A innuclear matter is apparently connected with the imperfection of our description of the nuclear structureand the process of double beta decay itself. Therefore, when describing the process of 2 νββ decay, we areessentially adjusting the value of g A in order to give a correct description of the process. In this sense, thisis about the same as the situation with g pp in the previous paragraph.4. A check of the “bosonic” neutrino hypothesis [4] and ν SI [8].
At the same time, it is quite difficult to choose the "best" result from the available data. For someisotopes, 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 beenperformed and average or/and recommended half-life values for all isotopes are presented. Using thesevalues and the values of the phase space factors from [25,26], the “effective” NMEs were calculated.The first time that such type of work was done was in 2001, and the results were presented at theInternational Workshop on the calculation of double beta decay nuclear matrix elements, MEDEX’01 [27].Then, updated half-life values were presented at MEDEX’05, MEDEX’09 and MEDEX’13 and published inRefs. [28], [14] and [15], respectively. In this article, new positive results obtained since the beginning of2015 and to the middle of 2020 have been added and analyzed. Preliminary results of this analysis havebeen presented at MEDEX’19 [29].The main differences from the previous analysis [15] are the following: (1) The new experimentalresults are included in the analysis: Ca [30], Ge [31], Se [32,33],
Mo [3,34],
Cd [35,36],
Te [37,38],
Xe [39],
Nd [40],
Nd -
Sm (0 + ) [41] and Ba [42]; (2) the positive results obtained for Kr[13] and
Xe [12] are added (these decays have been detected for the first time). I would like to stress thatmost of the above-mentioned new results are very precise. The accuracy of some of the obtained half-lifevalues is ∼
2. Experimental Data
Tables 1 and 2 show the experimental results on 2 νββ decay and on ECEC(2 ν ) capture in differentnuclei. For direct experiments, the number of detected (useful) events and the signal-to-background (S/B)ratio are presented. of 18 Table 1.
Present, positive 2 νββ decay results. N is the number of useful events, S/B is thesignal-to-background ratio.
Nucleus N T , yr S/B Ref., Year Ca ∼ [ + − ( stat ) ± ( syst )] · + − · [ + − ( stat ) + − ( syst )] · Average value: 5 . + . − . · Ge ∼ ( ± ) · ∼ + − · ∼ ∼
330 0.92 + − · ∼ + − · ∼ ∼ ( ± ) · ∼ ∼
80, 000 [ ± ( stat ) + − ( syst )] · ∼ ( ± ) · ∼ Average value: ( . ± . ) · Se 89.6 1.08 + − · ∼ [ ± ( stat ) ± ( syst )] · [ ± ( stat ) ± ( syst )] · ( a ) ∼ [ ± ( stat ) + − ( syst )] · ( a ) ∼
10 [33], 2019 ( ± ) · (geochem.) [53], 1986 Average value: 0 . + . − . · Zr 26.7 [ + − ( stat ) ± ( syst )] · ( b ) [54], 1999453 [ ± ( stat ) ± ( syst )] · ( ± ) · (geochem.) [56], 1993 ( ± ) · (geochem.) [57], 2001 Average value: ( . ± . ) · Mo ∼
500 11.5 + − · + − · [ ± ( stat ) ± ( syst )] · ( a )( c ) + − · [ + − ( stat ) ± ( syst )] ·
10 [62], 1997800 [ ± ( stat ) ± ( syst )] · ∼ [ ± ( stat ) ± ( syst )] · ∼ ( d ) [64], 2014500,000 [ ± ( stat ) + − ( syst )] · ( a )
80 [3], 201935,638 [ + − ( stat ) ± ( syst )] · ( a )
10 [34], 2020 ( ± ) · (geochem.) [65], 2004 Average value: 7 . + . − . · of 18 Table 1.
Cont.
Nucleus N T , yr S/B Ref., Year Mo - 133 ( e ) + − · Ru (0 + ) 153 ( e ) [ + − ( stat ) ± ( syst )] · [ + − ( stat ) ± ( syst )] · ∼ [ + − ( stat ) ± ( syst )] · ∼ [ + − ( stat ) ± ( syst )] · ∼ ( e ) [ + − ( stat ) ± ( syst )] · ∼ ( e ) [ ± ( stat ) ± ( syst )] · Average value: 6 . + . − . · Cd ∼
180 2.6 + − · ∼ [ ± ( stat ) ± ( syst )] · ( a )( c ) [ ± ( stat ) + − ( syst )] · ∼ [ ± ( stat ) ± ( syst )] · ( a )
12 [35], 201793,000 2.63 + − · Average value: ( . ± . ) · Te ∼ · (geochem.) [76], 1991 ( ± ) · (geochem.) [77], 1993 ( ± ) · (geochem.) [78], 2008 ( ± ) · (geochem.) [79], 2008 Recommended value: ( . ± . ) · ( f ) Te 260 [ ± ( stat ) + − ( syst )] · [ ± ( stat ) ± ( syst )] · ∼ [ ± ( stat ) ± ( syst )] · ∼ [ ± ( stat ) ± ( syst )] · > ∼ · (geochem.) [76], 1991 ( ± ) · (geochem.) [77], 1993 ( ± ) · (geochem.) [78], 2008 ( ± ) · (geochem.) [79], 2008 Average value: ( . ± . ) · Xe ∼ [ ± ( stat ) ± ( syst )] · ∼
10 [82], 2014 ∼ [ ± ( stat ) ± ( syst )] · ∼
10 [39], 2016
Average value: ( . ± . ) · Nd 23 [ + − ( stat ) ± ( syst )] · [ + − ( stat ) ± ( syst )] · [ ± ( stat ) + − ( syst )] · Average value: ( . ± . ) · Recommended value: ( . ± . ) · Nd - 177.5 ( e ) [ + − ( stat ) + − ( syst )] · Sm (0 + ) 21.6 [ + − ( stat ) ± + ( syst )] · ∼ ∼ [ + − ( stat ) ± + ( syst )] · ∼ Average value: 1 . + . − . · U ( . ± . ) · (radiochem.) [86], 1991 ( a ) For SSD mechanism. ( b ) For E e > ( c ) After correction (see [14]). ( d ) For E e > ( e ) In both peaks. ( f ) This value was obtained using average T for Te and well-known ratio T ( Te)/ T ( Te) = ( ± ) · − [77]. of 18 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 Kr and Xe T for 2 K ( ν ) , capture is presented(this is ∼ ECEC ( ν ) ). Nucleus N T ( ν ) , yr S/B Ref., Year Ba . + . − . · (geochem.) [87], 1996 ECEC ( ν ) ( . ± . ) · (geochem.) [11], 2001 ( . ± . ) · (geochem.) [88], 2009 Recommended value: ( . ± . ) · Kr 15 [ . + . − . ( stat ) ± . ( syst )] ·
15 [13], 20172 K ( ν ) Recommended value: ( . + . − . ) · (?) ( a ) Xe 126 [ . ± . ( stat ) ± . ( syst )] · K ( ν ) Recommended value: ( . ± . ) · ( a ) See text.
3. Data Analysis
To calculate an average of the ensemble of available data, a standard procedure, as recommended bythe Particle Data Group [21], was used. The weighted average and the corresponding error were calculated,as follows: ¯ x ± δ ¯ x = ∑ w i x i / ∑ w i ± ( ∑ w i ) − , (3)where w i = ( δ x i ) . Here, x i and δ x i are the value and error reported by the i-th experiment, and thesummations run over the N experiments.Then, it is necessary to calculate χ = ∑ w i ( ¯ x − x i ) and compare it with N - 1, which is the expectationvalue of χ if the measurements are from a Gaussian distribution. In the case when χ / ( N − ) is lessthan or equal to 1 and there are no known problems with the data, then one accepts the results. In thecase when χ / ( N − ) >>
1, one chooses not to use the average procedure at all. Finally, if χ / ( N − ) islarger than 1, but not greatly so, it is still best to use the average data, but to increase the quoted error, δ ¯ x in Equation 1, by a factor of S defined by S = [ χ / ( N − )] . (4)For averages, the statistical and systematic errors are treated in quadrature and used as a combinederror δ x i . In some cases, only the results obtained with a high enough S/B ratio were used. Ca The 2 νββ decay of Ca was observed in three independent experiments [30,43,44]. The obtainedresults are in good agreement. The weighted average value is: T = + − · yr.This value is slightly higher than the average value obtained in previous analysis( T = + − · yr [15]). This is due to the fact that the final result of the NEMO-3 experiment[30] was used in present analysis (the intermediate result of the NEMO-3 experiment [89] was used in of 18 [15]). The change in the final result in the NEMO-3 experiment was mainly due to the fact that afterdisassembling the detector, the parameters of sources containing 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 calculatedefficiency of recording useful events and, ultimately, to an increase in the T value for Ca. In addition,systematic error in [30] is higher then in [89]. Ge For 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 ofthe early works (1990s), as a recent historical review [90] emphasized that the contribution of backgroundprocesses was underestimated in these works. Therefore, to determine the average value, the resultspublished after 2000 have been used, with large statistics and a high S/B ratio [31,50]. Note that the finalresult of the Heidelberg–Moscow collaboration was used [50]. As a result, we get: T = ( ± ) · yr. Se There are many geochemical measurements ( ∼
20) and only four independent counting experimentsfor Se. However, the geochemical results are in poor agreement with each other and with the resultsof direct experiments. It is known that the possibility of existing large systematic errors in geochemicalmeasurements cannot be excluded (see discussion in Ref. [91]). Thus, only the results of the directmeasurements [32,33,51,52] were used to obtain a present half-life value for Se. Single State Dominance(SSD) mechanism (see explanation in [92]) was established for 2 νββ transition in Se [32,33] and half-lifevalues in this papers were obtained under the assumption of the SSD mechanism . The result of Ref. [10]has not been used in the analysis because this is the preliminary result of [51]. The result of work [51] ispresented with very asymmetrical errors. To be more conservative, the value for the lower error was takento be the same as the upper one in our analysis. Finally, the weighted average value is: T = + − · yr. Zr There are two positive results from the direct experiments (NEMO-2 [54] and NEMO-3 [55]) andtwo geochemical results [56,57]. Taking into account the comment in Section 3.3, the values from directexperiments (Refs. [54,55]) were used to obtain a present weighted half-life value for Zr: T = ( ± ) · yr. It was experimentally demonstrated that in some nuclei ( Se,
Mo and
Cd) the SSD mechanism is realized. In this case, thespectra (total energy, single electron energy and angular distribution) differ from the case of the High State Dominance (HSD)mechanism. In principle, this does not affect the half-life of the corresponding nuclei. In a real experiment, energy is recordedwith a certain threshold, which can affect the efficiency of recording useful events. The neglect of this effect can lead to an errorin the determination of T (up to ∼ Mo By the present nine positive results from direct experiments and one result from a geochemicalexperiment 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. [3,34,60,62,64]). I use only final result of Elliott et al. [62] and do not consider theirpreliminary result from [59]. For Mo SSD mechanism was installed and in Ref. [3,34,60] the half-liveswere obtained taking this fact into account. In addition, the corrected half-life value from Ref. [60] hasbeen used (see explanation in [14]). The following weighted average value for the half-life is obtained as: T = + − · yr. Mo -
Ru ( + ; 1130.32 Kev) The 2 νββ decay of
Mo to the 0 + excited state of Ru was detected in seven independentexperiments. The results are in good agreement. The weighted average value for the half-life has beenobtained using the results from [66,67,69–72]: T = + − · yr.The result from [69] was used as the final result of the TUNL-ITEP experiment (the result from [68]was not used here because I consider it as preliminary one). Cd Five independent positive results were obtained [35,36,73–75]. The results are in good agreementwith each other. The corrected result for the half-life value from Ref. [74] is used here. The originalhalf-life value was decreased by ∼
25% (see explanation in [14]). In Refs. [74] and [35], half-life values wereobtained with the assumption that the SSD mechanism was realized. The weighted average value is: T = ( ± ) · yr. Te and Te There are a large number of geochemical results for these isotopes. Although the half-life ratiofor these isotopes is well known (accuracy is ∼
3% [77]), the absolute T values for each isotope aredifferent from one experiment to the next. One group of authors [76,94,95] gives T ≈ · yr for Te and T ≈ · yr for Te, while another group [53,77] claims T ≈ ( − ) · yr and T ≈ · yr, respectively. In addition, as a rule, experiments with young samples ( ∼
100 millionyears) give results of the half-life value for
Te in the range of ∼ ( − ) · yr, while experimentswith old samples ( > ∼ ( − ) · yr. In 2008,it was demonstrated that short half-lives are more likely to be correct [78,79]. In a new experiment withyoung minerals, the half-life values were estimated at ( ± ) · yr [78] and ( ± ) · yr [79]for Te and ( ± ) · y [78] and ( ± ) · yr [79] for Te. In fact, in both experiments,the half-life was measured only for
Te, and the value for
Te was determined using the previously I do not consider here the result of Ref. [93] because of a high background contribution that was not excluded in this experiment.As a result, the “positive” effect is mainly associated with the background. Calculations show that without the backgroundcontribution to the “positive” effect, the sensitivity of the experiment was simply not enough to detect
Mo decay. of 18 measured T ( Te ) / T ( Te ) ratio [77]. If we average the values obtained in these two experiments,we get: T = ( ± ) · years for Te and T = ( ± ) · years for Te, which is ingood agreement with the results of direct (counter) experiments (see below).The first indication of the observation of the 2 νββ decay for
Te in a direct experiment was obtainedin [80]. More accurate and reliable values were obtained later in the NEMO-3 experiment [81]. Very preciseresults were obtained recently in CUORE-0 [37] and CUORE [38] experiments. The results are in goodagreement, and the weighted average value is T = ( ± ) · yr .Now, using the very well-known ratio T ( Te ) / T ( Te ) = ( ± ) · − [77], one canobtain the half-life value for Te, T = ( ± ) · yr .I recommend using these two results as the most correct and reliable half-life values for Te and
Te. As one can see now, results of direct and geochemical experiments are in good agreement. Xe The half-life value for Xe was measured in two independent experiments, EXO [82,96,97] andKamland-Zen [39,98,99]. To obtain the average value of the half-life, the most accurate results of theseexperiments obtained in [82] and [39] were used (see Table 1). The weighted average value is T = ( ± ) · yr . Nd The positive results were obtained in three independent experiments [40,62,83]. The most accuratevalue was obtained in Ref. [40]. This value is higher than in Ref. [62] ( ∼ σ difference) and lower thanin Ref. [83] ( ∼ σ difference). Using Equations (2) and the three above-mentioned results, one obtains T = ( ± ) · yr. Taking into account that χ / ( N − ) > T = ( ± ) · yr.It can be seen that due to the discrepancy between the T values, one has to increase the error inorder to somehow agree on the experimental results. On the other hand, it is clear that the result of theNEMO-3 experiment is today the most accurate and reliable. This is confirmed by the fact that in theNEMO-3 experiment, seven different isotopes were investigated simultaneously. In addition to Nd, Ca, Se, Zr,
Mo,
Cd and
Te were also studied. For all these isotopes, the results are in goodagreement with the results of other experiments. It is natural to assume that the result for
Nd is correcttoo. Therefore, I think that it is necessary to use this value as the most accurate at the moment: T = + − · yr. Nd -
Sm ( + ; 740.4 Kev) There are two positive results for 2 νββ decay of
Nd to the 0 + excited state of Sm [84,85] (thepreliminary result of Ref. [84] was published in Ref. [100]). These two results are in good agreement. Theweighted average value is: of 18 T = + − · yr.Recently, the result of a new experiment was presented at MEDEX’19 [41] (see Table 1). I am not usingthis new result in my analysis because this is an ongoing experiment and the result is still preliminary andnot yet published. U The two-neutrino decay of
U was measured in a single experiment using the radiochemicaltechnique [86]: T = ( ± ) · yr.It has to be stressed that for 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.
Ba (ECEC)
For
Ba, positive results were obtained using the geochemical technique only. In this type ofmeasurement, one can not recognize the different modes. It is clear that exactly the ECEC(2 ν ) processwas detected because other modes are strongly suppressed (see estimations in [92,101,102]). The first timethe positive result for Ba was mentioned was in Ref. [87], where experimental data of Ref. [103] wereanalyzed. In this paper, a positive result was obtained for one sample of barite ( T = + − · yr),but for a second sample only the limit was set ( T > · yr). Later, more accurate half-lifevalues, ( ± ) · yr [11] and ( ± ) · yr [88], were measured. One can see that theresults are in strong disagreement. In [42], the data of [88] were analyzed and it was shown thatsubtraction of the contribution of cosmogenic Xe removes the contradiction with the result of [11].Finally, I recommend the following value from [11]: T = ( ± ) · yr.To obtain more reliable and precise half-life values, new measurements are needed (includingdirect experiments). Kr (2K)
The first indication of the observation of 2 K capture in Kr was announced in 2013 (the effect is ∼ σ ), T = [ + − ( stat ) ± ( syst )] · years [104]. Then, the same data were analyzedmore carefully and a new value was published ( ∼ σ ), which turned out to be twice as much, T =[ + − ( stat ) ± ( syst )] · [13]. The analysis of the data is quite complicated and it is possible that thesystematic 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 seenfrom Table 3, in the case of Kr, we are dealing with an anomalously large value of nuclear matrix element.This value is significantly larger than in the case of
Ba and
Xe (1.8 and 5.4 times, respectively) andexceeds all 13 NME values for 2 νββ decay (from 1.7 to 17.7 times). Here, it is necessary to take into accountthat, since the rate of the ECEC process is ∼ K capture, the NME for the ECECprocess in Kr is approximately 1.07–1.1 times greater than for the 2 K capture. This circumstance onlystrengthens 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 hasto be very careful with this result. Xe (2k)
To date, only one positive result has been published for 2 K capture in Xe [12]: T = [ ± ( stat ) ± ( syst )] · yr. The significance of the effect is only 4.4 σ . It should also be noted that alimit T > · yr was obtained in [105], which formally contradicts the result of [12]. Taking intoaccount errors, there is no real contradiction here. However, it is clear that it is necessary to confirm theresult 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 (2-nd column). Usingthese values, one can extract the experimental nuclear matrix elements through the relation [25]: T − = G ν · g A · ( m e c · M ν ) , (5)where T is the half-life value in [yr], G ν is the phase space factor in [yr − ], g A is the dimensionlessaxial vector coupling constant and ( m e c · M ν ) is the dimensionless nuclear matrix element. One has toremember that there are indications that in nuclear medium the g A value is reduced in comparison withtheir free nucleon values (see Section 1). Expression (5) is valid for 2 νββ and ECEC(2 ν ) processes.Thereby, following Ref. [25], it is better to use the so-called effective NME, | M e f f ν | = g A · | ( m e c · M ν ) | . This value has been calculated for all isotopes. Table 3.
Half-life and effective nuclear matrix element values for 2 νββ decay (see Section 4).
Isotope T ( ν ) , yr | M e f f ν | | M e f f ν | Recommended( G ν from [25]) ( G ν from [26]) Value νββ : Ca 5.3 + − · + − + − ± Ge ( ± ) · + − + − ± Se 0.87 + − · + − + − ± Zr ( ± ) · + − + − ± Mo 7.06 + − · + − + − + ( a ) − ± Mo- 6.7 + − · + − + − Ru ( + ) + ( a ) − ± Cd ( ± ) · + − + − + ( a ) − ± Te ( ± ) · + − + − ± Te ( ± ) · + − + − ± Xe ( ± ) · + − + − ± Nd ( ± ) · + − + − ± Nd- 1.2 + − · + − + − ± Sm(0 + ) U ( ± ) · + − + − + − ECEC(2 ν ): Kr ( b ) + − · + − [106] 0.3583 + − + − Xe ( b ) ( ± ) · + − [106] 0.0607 + − + − Ba ( ± ) · + − [106] 0.1754 + − + − ( a ) Obtained using the SSD model. ( b ) Value for 2 K capture. For the ECEC process, the half-life value will be approximately15–20% less, and the NME value approximately 7–10% higher. The obtained results are presented in Table 3 (3-rd and 4-th columns). When calculating, I usedthe G ν values from Refs. [25] and [26] (see Table 4). For Ba, Kr and Xe G ν values for ECECtransition were taken from [106] and [26]. These calculations are most reliable and correct at this moment.The results of these calculations are in reasonable agreement ( ∼ Te( ∼ Kr ( ∼ U (factor ∼ U, two different values 14.57 · − yr − [25] and98.51 · − yr − [26]) were produced. The situation with calculations for U is clearly unsatisfactoryand these calculations should be rechecked. For
Mo,
Mo- Ru ( + ) and Cd, I used G ν calculatedin Ref. [25] for the SSD mechanism. The obtained values for | M e f f ν | are given in Table 3 and these are themost correct values for these isotopes. So-called recommended values for | M e f f ν | are presented in Table3 (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 intoaccount corresponding errors). For Mo,
Mo- Ru ( + ) and Cd, I recommend to use the valuesobtained with G ν for the SSD mechanism.Therefore, for the majority of isotopes an accuracy for | M e f f ν | is on the level ∼ Kr,
Xeand
Ba, it is ∼ ∼
22% and ∼ U ( ∼ G ν . Recently, in Ref. [16], an improved formalism of the 2 νββ decay rate was presented, which takes intoaccount 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 elementsand different phase space volumes. That is, a new approach to processing the results will be required. Todo 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 3retain their significance since it was demonstrated in [16] that additional terms contribute ∼
3% to ∼
25% tothe total decay rate. This means that if we consider expression (5) as the first term of the expansion in theapproach [16], then we can conclude that the values of | M e f f ν | obtained in this work give a good estimatefor g A ·| M ν GT − | (see Formula (19) in [16]). The values given in Table 3 overestimate g A ·| M ν GT − | valuesby ∼ | M e f f ν | . An exception is thesituation with the results for Mo and
Cd obtained using phase space volumes calculated within theSSD. In this case, the most accurate NME estimate was obtained since the exact value of the energy of thelowest 1 + intermediate state was used in the calculations of the phase space volume. Table 4.
Phase-space factors from Refs. [25], [26] and [106].
Isotope G ν ( − yr − ) [25] G ν ( − yr − ) [26]2 νββ : Ca 15550 15536 Ge 48.17 46.47 Se 1596 1573 Zr 6816 6744
Mo 3308 32314134 ( a ) Mo- Ru ( + ) ( a ) Cd 2764 26883176 ( a ) Te 0.2688 0.2149
Te 1529 1442
Xe 1433 1332
Nd 36430 35397
Nd-
Sm(0 + ) 4329 4116 U 14.57 98.51ECEC(2 ν ): Kr 0.660 [106] 0.410
Xe 17.200 [106] 15.096
Ba 15.000 [106] 14.773 ( a ) Obtained using SSD model.
5. Conclusions
Thus, the all positive results for 2 νββ 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
Te,
Nd and
Ba,so-called recommended values have also been proposed. Using these obtained average/recommendedhalf-life values, the | M e f f ν | values for all considered nuclei were determined. Finally, previous resultsfrom 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 since2001, we can see that these values were constantly refined over time and did not deviate by more than1–2 σ from the initial value. An exception is the situation with Ge. Here, the average value has steadilyincreased with time (from 1.42 + − · yr in 2001 to ( ± ) · yr in 2020). This is due to the lowquality of the results obtained in the 1990s. In the latest analysis, the results obtained after 2000 have beenused.At present, 2 νββ decay was recorded in 11 nuclei, and ECEC capture in 3 nuclei (with some doubtsfor Kr). The accuracy of determining the half-life for most nuclei lies in the range 2–10%. It is expectedthat in the next few years new results will be obtained for Ge (Majorana),
Mo (CUPID-Mo, AMORE,CROSS),
Cd (CROSS),
Te (CROSS, SNO+),
Xe (NEXT-100) and
Xe (NEXT-100, LUX-ZEPLIN).The final result will be obtained in an experiment to search for the 2 νββ decay of
Nd to the first excited0 + level of Sm (see [41]). Let us emphasize here the importance of experiments using low-temperaturebolometers. In experiments with such detectors, the measurement accuracy of the half-life can reach 1-2%.At present, such experiments are possible for Se,
Mo,
Cd and
Te. Apparently, in the future, suchmeasurements will be implemented for Ca as well. I hope that in the future 2 β processes will also befound in other nuclei. The search for 2 νββ processes in Sn,
Pd,
Gd and the search for ECEC(2 ν )processes in Ru,
Cd and
Ce seem promising. As for the 2 νββ transitions to the excited states of thedaughter nucleus, it seems possible to register a transition to the 0 + excited level in measurements with Zr and Se in the near future.
Funding:
This research was partially funded by Russian Scientific Foundation grant number 18-12-00003.
Conflicts of Interest:
The author declares no conflict of interest.
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