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Two-neutrino double beta (
2νββ ) decay is an interesting process (a second-order weak interaction) in the standard model, which was proposed by Mayer [1] in 1935. It is a significant subject in nuclear and particle physics that can be used to test models of the weak interaction [2, 3]. The2νββ decays have ultra-long half-lives of over1018 years [4, 5], which is at least one billion times longer than the age of the universe itself. To date, 90 nuclei are believed to undergo2νββ decay, of which only 14 have been measured [6-10]. The complementary experimental information from related2νββ decay can improve the quality of nuclear structure models [11]. In addition, research on2νββ decay can be used to calculate the nuclear matrix elements (NMEs) of0νββ decay as a promising source of0νββ decay detection [12]. It is well known that the absolute mass of the neutrino is still a mystery [13, 14]. If0νββ decay could be detected, it could settle this problem when combined with the NMEs governed by the theory [15, 16]. Thus, whether experimental or theoretical, research into2νββ decay is extremely significant.The role of accurate NMEs in
2νββ decay cannot be underestimated [17]. Although the NMEs between the initial and final nuclei ground states have been investigated using several approaches [18-24], they are still notoriously difficult to calculate, as shown in Eq. (1) [25]:M2ν=∑n⟨0f+‖∑iσ(i)τ±(i)‖1+n⟩⟨1+n‖∑iσ(i)τ±(i)‖0i+⟩[12Q+E(1+n)−Mi]/me+1.
(1) Here, the transition operators are the ordinary Gamow-Teller operators;
E(1+n)−Mi denotes the energy discrepancy of the nth intermediate1+ state and the parent nucleus; and the sum∑i extends over all the intermediate1+ states [25], which poses a challenge in calculations for many nuclei.In recent years, research has been renewed on the single state dominance (SSD) for
2νββ decay, which suggests that the lowest intermediate1+ state dominates the decay [26, 27]. To date, evidence of the SSD for82 Se has been established by the CUPID-0 collaboration [28]. One of its proven benefits is that the corresponding NMEs can be obtained through singleβ decay measurements [29]. However, the SSD of many nuclei violates this benefit, with the result that the relevant NMEs cannot be easily determined experimentally. It would hopefully be simpler to calculate only the lowest1+ wave function instead of all wave functions in the theoretical description [29]. It is thus of great importance to discuss the number of least intermediate1+ states that saturate the final magnitude.In our work, calculations for six nuclei (
36 Ar,46 Ca,48 Ca,50 Cr,70 Zn, and136 Xe) in a mass range fromA=36 toA=136 are performed within the nuclear shell model (NSM) framework. Calculations are presented for the half-lives, NMEs,G2ν , and convergence of the NMEs. In addition, we predict the half-lives of the2νββ decays for four nuclei. Relatively short half-lives are predicted for the nuclei46 Ca and70 Zn, providing a reference for future experimental detection. Convergence of the NMEs for2νββ decays is systematically discussed by analyzing the number of contributing intermediate1+ states (NC ) for the nuclei of interest. A unified criterion for analyzingNC involved in2νββ decay is proposed and adopted. According to the calculations, the general law for convergence of the NMEs for2νββ decays is then presented. -
The
2νββ decays can be classified into two types: doubleβ− decay (2ν2β− ) and the decays on the doubleβ+ side, i.e., 2ν2β+ , 2νβ+ EC, and 2ν 2EC (where EC refers to electron capture) [30]. The formulas used for various types of2νββ decay transitions differ, which can be seen in Refs. [13, 30, 31]. The associated half-lives, in connection with the phase space factors (Gα2ν ) and the NME (|Mα2ν| ), can be expressed by Eq. (2) [30]:1/Tα1/2=Gα2ν|Mα2ν|2,
(2) where
α=2β− , 2β+ ,β+ EC, and 2EC are different types of2νββ decays [13, 30, 31]. The related NME description of the fullβ strength functions of both the parent and grand nuclei is given [2]. Although formulas forGα2ν and|Mα2ν| are supplied by the theory, the calculation for|Mα2ν| is more difficult than that forGα2ν because it requires the precise nuclear many-body wave function [32]. The NME is defined in Eq. (1): accurate knowledge of the ground-state for the parent and grand-daughter nuclei, along with all the1+ states of the intermediate nucleus, clearly plays an important role in evaluations of the NMEs [16].Detailed expressions for various modes of
Gα2ν can be found in Ref. [33]. The expression is an integral in the phase space of lepton variables; the corresponding values depend on the coupling constantgA and can be calculated exactly [25, 33]. There is a global effort to address the challenge of thegA factor. This study applies the commonly used value ofgA=1.27 , which is consistent with that published in Refs. [13, 17, 34]. -
The calculations for six nuclei (
36 Ar,46 Ca,48 Ca,50 Cr,70 Zn, and136 Xe) are computed in full 1ℏω shell model spaces without any truncation. Columns 1 and 2 in Table 1 list the nuclei of interest and their shell model spaces, in which the calculations are conducted. To improve the reliability of the calculations, different effective interactions (EIs) are adopted and listed in Column 3. For the2νββ decay of36 Ar, the specific interactions CWH and W are adopted, where the CWH EI is determined by an iterative least squares fit to experimental level energy data [35]; W EI refers to the two-body matrix elements derived from a linear fit [35, 36]. We use the shell model empirical interaction GXFP1A for thefp shell, which provides 195 two-body matrix elements and 4 single-particle energy subsets through a fit to 699 energy observations within the mass intervalA=47−66 [37, 38]. Another interaction, KB3, is used to compute the lowest wave function from the full Hamiltonian, with parts of the two-nucleon EI turned off in proper order [39, 40]. It is reasonable to adopt these EIs for the calculations because the concerned nuclei are not far from these regions. We also use the large-scale shell-model interaction SN100PN [41] in the case of136 Xe, which is obtained from the free nucleon-nucleon CD-Bonn interaction. The interaction is composed of four parts: neutron-neutron, neutron-proton, proton-proton, and Coulomb repulsion between the separate protons [42].Nucleus Shell model space EI NT N.cal q 36 Arsd(0d5/2,0d3/2,1s1/2) CWH, W 66 66 0.77 [35] 46 Cafp(1f7/2,2p3/2,1f5/2,2p1/2) GXFP1A, KB3 2361 2361 0.74 [45] 48 Cafp(1f7/2,2p3/2,1f5/2,2p1/2) GXFP1A, KB3 9470 2000 0.74 [45] 50 Crfp(1f7/2,2p3/2,1f5/2,2p1/2) GXFP1A, KB3 383932 2000 0.74 [45] 70 Znfp(1f7/2,2p3/2,1f5/2,2p1/2) GXFP1A, KB3 18571 2000 0.68 [46] 136 Xegdsh(0g7/2,1d5/2,1d3/2,2s1/2,0h11/2) SN100PN 16642 2000 0.45 [16] Table 1. Details of the NSM calculations for
2νββ decays.The calculations of the wave functions for the involved nuclei use the diagonalization computer program NuShellX [43]. Columns 4 and 5 provide the total number of intermediate
1+ states (NT ) and the number of calculated intermediate1+ states (N.cal ), respectively. One of the most prominent advantages of the NSM is that it considers all the many-body correlations of orbits near the Fermi surface [44], but in theory, the expected total intensity is systematically greater than the observed total intensity [16, 35]. One reliable method of overcoming this problem is to correct the calculated strengths with quenching factors (q), which are tabulated in Column 6. In this article, we apply the widely used valuesq=0.77 forA=16−40 ,q=0.74 forA=40−50 , andq=0.68 forA=60−80 [35, 45, 46]. For the136 Xe nucleus,q=0.45 [16] is adopted.This research is inseparable from the precise description of wave functions of intermediate states. To test the reliability of the wave functions of the NSM, we compare the first few theoretical energy levels and the
γ -decay half-lives of36 Cl (the intermediate nucleus for the2νββ decay of36 Ar) in Fig. 1 together with the available experimental information. The theoretical results are obtained with W interaction. As depicted in Fig. 1, the energy levels of the low excited states and mostγ -decay half-lives are in good agreement with the experimental data. Thus, within the applicable range, the calculations of the NSM are relatively accurate. -
We calculate the properties of
2νββ decays for six nuclei (36 Ar,46 Ca,48 Ca,50 Cr,70 Zn, and136 Xe) with Q values adopted from AME2016 [47]. The calculated half-lives,G2ν , and NMEs are presented in Table 2, together with other results, either from recent experiments [10, 48-51] or theoretical calculations [17, 44, 52, 53]. Most of the calculated results do not differ greatly from the experimental data. For the half-lives of the36 Ar and50 Cr nuclei, our results clearly differ from previous results [52]: the reason is that prevoius works adoptedgA=−1 .Nucleus Q value /keV G2ν/yr−1 |M2ν|.1st |M2ν| half-life /yr EI this work others 36 Ar (2ν 2EC)432.59 1.28×10−27 0.00254 0.117 5.68×1028 1.4×1029 [52] CWH 0.0229 0.151 3.43×1028 W 46 Ca (2ν2β− )988.4 1.24×10−22 0.0564 0.0563 2.54×1024 1.7×1024 [53] GXFP1A 0.0305 0.0468 3.67×1024 KB3 48 Ca (2ν2β− )4268.08 4.17×10−17 0.0219 0.0250 3.84×1019 (6.4+0.007−0.006±+0.012−0.009)×1019 [10] GXFP1A 0.0183 0.0226 4.68×1019 (4.3+2.4−1.1±1.4)×1019 [48] KB3 50 Cr (2ν 2EC)1169.6 1.25×10−24 0.0294 0.123 5.32×1025 >1.3×1018 [49] GXFP1A 0.0515 0.124 5.21×1025 2.1×1026 [52] KB3 70 Zn (2ν2β− )997.1 3.24×10−22 0.0677 0.138 1.61×1023 ⩾3.8×1018 [50] GXFP1A 0.00577 0.0867 4.10×1023 1.4(5)×1023 [17] KB3 136 Xe (2ν2β− )2457.8 4.69×10−18 0.00724 0.0186 6.16×1020 (2.165±0.016±0.059)×1021 [51] SN100PN (2.11±0.25)×1021 [44] Table 2. Summary of the calculations for the
2νββ decays of six nuclei.Recently, evidence of the SSD for
82 Se has been established by the CUPID-0 collaboration [28]. If the SSD is affirmed, then the corresponding NMEs can be obtained through singleβ decay measurements [29]. Therefore, it is of great significance to analyze the validity of the SSD. Calculated results show that for the2ν2β− decays of46 Ca and48 Ca, the first intermediate1+ state contributes significantly to the final NME. However, there is important canceling in the accumulations of the NMEs, as can be seen in Figs. 7 and 10 in Ref. [52]. In these cases, we cannot conclude that the SSD is satisfied. For other nuclei, the sum of the NMEs gains less than 50% of their final NMEs from the first intermediate1+ state. Therefore, we cannot conclude that the SSD is satisfied for the related nuclei.The key of the theoretical research on
2νββ decays is the calculation of the NMEs. Thus, we present the calculated NMEs together with the Q values in Fig. 2. All the data are from Table 2. The figure illustrates the lack of a significant relationship between the NMEs and the Q values. The calculated results from different EIs are consistent with each other, except in the case of70 Zn, for which the NME using GXFP1A EI is 1.6 times that using KB3.Convergence of the NMEs for
2νββ decays is concluded by analyzing the number of contributing intermediate1+ states (NC ) for the concerned nuclei. We assume that a2νββ decay NME is well converged when the accumulated NMEs saturate 99.7% of the final magnitude. This is illustrated in Fig. 3, where for the70Zn⟶ 70Ge decay for GXFP1A EI, points A (5th state), B (103rd state), and C (214th state) all satisfy this criterion. Only after point C has satisfied the criterion, indicating that the NME has no distinct twists, is the accumulation believed to be saturated. We thus assume that after point C (214th state), the NME is reliably converged. Accordingly,NC is determined naturally. This criterion is used throughout this article.A detailed analysis of the convergence of NMEs based on the above criterion is presented in Table 3 for the nuclei of interest. Notice that the actual contributing excitation energy for
136 Xe decay is smaller than for the other concerned nuclei (see Column 5), which suggests that the level density is large.Nucleus NT NC NC/NT Ex/MeV EI 36 Ar66 34 51.5% 33.6 CWH 32 48.5% 36.5 W 46 Ca2361 130 5.51% 24.3 GXFP1A 111 4.70% 22.4 KB3 48 Ca9470 150 1.58% 23.4 GXFP1A 183 1.93% 24.7 KB3 50 Cr383932 417 0.109% 22.5 GXFP1A 423 0.110% 23.7 KB3 70 Zn18571 214 1.15% 23.0 GXFP1A 165 0.888% 25.4 KB3 136 Xe16642 1009 6.06% 12.7 SN100PN Table 3. Details of the convergence of NMEs according to the criterion that the accumulated NMEs should saturate 99.7% of the final magnitude.
From the calculations for the involved nuclei, we discover a connection between the total number of intermediate
1+ states (NT ) and the number of contributing states (NC ). A linear analysis for a general convergence law of the NMEs is depicted in Fig. 4. The unfilled circles represent the CWH and GXFP1A EIs, and the solid circles are for the W, KB3, and SN100PN EIs. All the data are from Table 3. We use a solid line for the fit using five nuclei (36 Ar,46 Ca,48 Ca,50 Cr, and70 Zn) and express the correlation betweenNC andNT by Eq. (3):Figure 4. (color online) Correlation between the convergence proportion (
NC/NT ) in alog10 frame andNT inlog10 frame. The solid line is a linear fit including all the nuclei except136 Xe, and the dashed line is for all the nuclei of interest. The unfilled and solid circles represent the results from different EIs.NC=(10.8±1.2)×N(0.29±0.02)T,
(3) with a linear correlation coefficient of 0.996, which is very close to 1. The overall trend of the convergence properties is in good agreement with the correlation, except in the case of the
136 Xe nucleus, which deviates slightly from the solid line. The reason may be that the level density of136 Cs (the intermediate nucleus for2νββ decay of136 Xe) is obviously larger than that of the others. For the other nuclei, the excitation energy exceeds 20 MeV up to the 300th intermediate1+ state, whereas the excitation energy of the 1009th intermediate1+ state for136 Cs is only 12.7 MeV, as listed in Table 3. Thus, the2νββ decay of136 Xe is unusual. If the136 Xe nucleus is taken into account, the correlation is expressed by Eq. (4),NC=(10.7±2.1)×N(0.32±0.08)T,
(4) with a linear correlation coefficient of 0.884. Hopefully, the number of least intermediate
1+ states needed for the precise calculation of NMEs can be obtained from Eq. (3).To check the applicability of Eqs. (3) and (4), two new transitions are performed. One is the
64Zn⟶ 64Ni decay computed in thefpg model space with JUN45 and JJ44B interactions. Details of the calculated results are presented in Table 4. The calculated values ofNC are almost identical for the different interactions within the theoretical range of the two equations.Nucleus Shell model space NT NC Theoretical NC EI Eq. (3) Eq. (4) 64 Znfpg 12378 182 167±32 219±66 JUN45 fpg 12378 186 167±32 219±66 JJ44B 68 Nifp 18571 205 187±34 249±73 GXFP1A fpg 42895 241 238±40 326±89 JUN45 Table 4. Summary of the convergence of the NMEs of new nuclei.
Another transition of interest is the
68Ni⟶ 68Zn decay in thefp andfpg model spaces. Although this decay does not occur in nature, the related calculations could be used to further discuss the applicability of Eqs. (3) and (4) in different model spaces. In addition, the calculated values ofNC for the different model spaces are within the theoretical range of the two equations. Thus, Eqs. (3) and (4) are feasible within the applicable range. We hope our research is useful for shell model calculations in2νββ decay. -
In this work, the general features and convergence of the NMEs for the
2νββ decays of six nuclei in a mass range fromA=36 toA=136 are studied under the framework of the nuclear shell model (NSM). Different effective interactions (EIs) are adopted to improve the reliability of the calculations. Calculations are presented for the half-lives, NMEs,G2ν , and convergence of the NMEs. Most of the results are very close to the experimental data. In addition, we predict the half-lives of2νββ decays for four nuclei. The nuclei46 Ca and70 Zn have relatively short half-lives, which will hopefully be probed by future experiments.Convergence of the NMEs for
2νββ decays is discussed systematically by analyzing the number of contributing intermediate1+ states (NC ) for the concerned nuclei. We propose and adopt the criterion that the accumulated NMEs should saturate 99.7% of the final calculated magnitude. From the calculations for the nuclei of interest, we discover a connection betweenNC and the total number of intermediate1+ states (NT ). The least squares fit isNC=(10.8±1.2)×N(0.29±0.02)T . It is hoped that the number of least intermediate1+ states needed for the precise calculations of NMEs can be obtained from this equation.The authors are grateful to Prof. B. A. Brown of Michigan State University for providing us with the computer program NuShellX.
Research on convergence of the nuclear matrix elements for 2νββ decays
- Received Date: 2020-07-11
- Available Online: 2020-12-01
Abstract: In this work, the characteristics of