Consistent large-scale shell-model analysis of the two-neutrino ββ and single β branchings in 48Ca and 96Zr

Two-neutrino double-beta-decay matrix elements $M_{2\nu}$ and single beta-decay branching ratios were calculated for $^{48}$Ca and $^{96}$Zr in the interacting nuclear shell model using large single-particle valence spaces with well-tested two-body Hamiltonians. For $^{48}$Ca the matrix element $M_{2\nu}=0.0511$ is obtained, which is 5.5\% smaller than the previously reported value of 0.0539. For $^{96}$Zr this work reports the first large-scale shell-model calculation of the nuclear matrix element, yielding a value $M_{2\nu}=0.0747$ with extreme single-state dominance. If the scenario where the first $1^+$ state in $^{96}$Nb is at 694.6 keV turns out to be correct, the matrix element is increased to 0.0854. These matrix elements, combined with the available $\beta\beta$-decay half-life data, yield effective values of the weak axial coupling which in turn are used to produce in a consistent way the $\beta$-decay branching ratios of $(7.5\pm2.8)$ % for $^{48}$Ca and $(18.4\pm0.09)$ % for $^{96}$Zr. These are larger than obtained in previous studies, implying that the detection of the $\beta$-decay branches could be possible in dedicated experiments sometime in the (near) future.

The nuclei 48 Ca and 96 Zr share an interesting feature, the two being the only known nuclei where single β-decay transitions compete with the dominant two-neutrino double beta (2νββ) decay [1]. This exceptional situation is due to the large angularmomentum difference ( J = 4, 5, 6) between the initial and final states of β decays, as well as the relatively small decay energies ( Q values). Both theoretical and experimental studies have been carried out regarding decays of both nuclei [2][3][4][5][6][7][8]. The twoneutrino-emitting modes are dominated by the ground-state-toground-state transitions with resent half-life estimates of 6.4 +1.4 −1.1 for 48 Ca [9] and 2.35 ± 0.21 for 96 Zr [10]. These two nuclei are favorable for experimental double-beta-decay studies due to their large Q values: 48 Ca having the largest known double-β Q value Q ββ ( 48 Ca) = 4269.08 (8) keV, and 96 Zr having the third largest value Q ββ ( 96 Zr) = 3356.03(7) keV, with only 150 Nd between them [11].
The single-β channels have not yet been observed but lower limits for the half-lives stand at 1.1 ×10 20 yr for 48 Ca [5] and 2.6 × 10 19 yr for 96 Zr [12]. Were these observed, they would provide valuable information about the validity of current nuclear models, which could be used to improve the accuracy of calculations of the matrix elements of neutrinoless ββ decay.
In this Letter we revisit the previous theoretical studies, giving an updated estimate for the 2νββ-decay matrix element for 48 Ca and for the first time a shell-model estimate for the 96 Zr 2νββ-decay matrix element. Using this information we present improved estimates for the β-decay branching ratios. This knowledge can in the future be used to design optimal experiments for the detection of the β-decay branches.
The theory of β decay, including the forbidden transitions considered here, is extensively treated in the work of Behrens and Bühring [13]. A streamlined presentation of the theory, including all the technical details of how the calculations were carried out also in the present work, can be found from [14]. The basic theory behind the 2νββ decay can be found in much more detail for example from [15].
For β decay the probability of the electron being emitted with kinetic energy between W e and W e + dW e is where p e is the momentum of the electron, Z is the proton number of the final-state nucleus, F 0 (Z , W e ) is the so-called Fermi function, and W 0 is the end-point energy of the β spectrum. The nuclear-structure information is encoded as form factors in the shape factor C (w e ). In  A − 1 nucleons at the moment of decay, these form factors map to nuclear matrix elements (NMEs), which can in turn be calculated using a many-body framework, such as the interacting nuclear shell model. The axial-vector coupling g A and the vector coupling g V , which enter the theory of β decay when the vector and axial-vector hadronic currents become renormalized at the nucleon level, appear as multipliers of the various axial-vector and vector matrix elements respectively.
The half-life of β decay can be written as where C is the integrated shape function and the constant κ has the value [16] θ C being the Cabibbo angle. To simplify the formalism it is traditional to introduce unitless kinematic quantities w e = W e /m e c 2 , w 0 = W 0 /m e c 2 and p = p e c/(m e c 2 ) = w 2 e − 1, and so the integrated shape function can then be expressed as The shape factor C (w e ) of Eq. (4) contains complicated combinations of both (universal) kinematic factors and NMEs. As in the previous studies regarding forbidden β decays [14,17,18] we take into account the next-to-leading-order terms of the shape factor as well as screening and radiative effects. For the 2νββ decay the half-life expression is analogous to that of β decay in Eq. (2) and can be written as [15] t (2ν) where G (2ν) is the phase-space integral (the expression for this is given in, e.g. [15]) and M 2ν is the matrix element given for β − β − decay by where m e is the electron rest mass, ference between the mth intermediate 1 + state and the ground state of the initial nucleus, and Q ββ is the energy released in the decay (i.e. Q value). The nuclear-structure calculations were done using the interacting shell model with the computer code NuShellX@MSU [19]. Following the earlier shell-model studies regarding the halflives of the transitions 48 Ca(0 + ) → 48 Sc(4 + , 5 + , 6 + ) [6] and the 2νββ-decay channel [20], the full f p model space with the interaction GXPF1A [21,22] was used.
For the decay of 96 Zr a model space including the proton orbitals 0 f 5/2 , 1p 3/2 , 1p 1/2 and 0g 9/2 and the neutron orbitals 0g 7/2 1d 5/2 , 1d 3/2 and 2s 1/2 were used together with the interaction glekpn [23]. In the previous shell-model study [8] the calculations were done in the much smaller proton 0g 9/2 -1p 1/2 and neutron 1d 5/2 -2s 1/2 model space with the Gloeckner interaction [24]. While the exclusion of a large number of important orbitals can affect the accuracy of the computed half-lives of the various β-decay branches, the problem is even more severe for the ground-stateto-ground-state 2νββ decay, which is strictly forbidden in such a limited model space. In the present study this transition can proceed by simultaneous Gamow-Teller transitions between the proton 0g 9/2 and neutron 0g 7/2 orbitals. Since the computational burden for description of these decays is manageable for modern computers, we included all the intermediate 1 + states of 2νββ decay in 48 Sc and 96 Nb. This is an improvement over the previous calculation regarding the matrix element of 48 Ca [20], where only 250 intermediate states were used. For 48 Sc our extended calculation includes 9470 1 + states and excitation energies up to 60 MeV, while for 96 Nb we have 5894 1 + states reaching energies of roughly 18 MeV. Since the exact energies of the intermediate states play an important role in the determination of the 2νββ NMEs, the excitation energies of the 1 + states in 48 Sc were shifted such that the lowest-lying state is at the experimental energy of 2200 keV [11]. For 96 Nb no 1 + states are known experimentally, so that the shell-model excitation energies were used. However, the paper by Thies et al. [30] suggests that the state at 694.6 keV could be the lowest 1 + state. The branching-ratio calculations were thus repeated also for this scenario.
In the following we will first report on the computed results for the 2νββ NMEs of 48 Ca and 96 Zr and then extract effective values g eff A of the weak axial coupling based on comparisons with the measured 2νββ half-lives. These g eff A , listed in Table 1, are then, in turn, used to predict the β-decay branching ratios for transitions to the lowest 4 + , 5 + , and 6 + states of 48 Sc and 96 Nb. This we consider to be a consistent approach since the 2νββ and β decays are low-momentum-exchange processes and thus the related axial couplings are expected to be quenched by a similar amount [26,27]. A further evidence comes from the very recent analyses of the electron-spectrum shape of the 4th-forbidden non-unique β − -decay transition 113 Cd(1/2 + ) → 113 In(9/2 + ) by the COBRA collaboration [31]. There values g eff in three different theory frameworks. These values are consistent with those shown in Table 1.
The computed shell-model 2νββ NMEs are given in Table 1. For 48 Ca the shell-model calculation gives |M 2ν | = 0.0511, which is 5% smaller than the value 0.0539 reported in [20]. The accumulation of the matrix element is in agreement with the previous results (see Fig. 1). The lowest 1 + state is the most important, contributing an amount of 0.0454 to the total NME. The next dozen states are mostly constructive, adding up to a maximum value of 0.0847 of the NME, beyond which the states start to contribute destructively. When taking a closer look at the main constitution of this "bump" in the cumulative NME one notices that it emerges from the inclusion of both of the spin-orbit partners 0   structively beyond which the cumulative matrix element tapers off to the final value 0.0511. In the case of 96 Zr (see Fig. 2) there is a clear single-state dominance (SSD) [28,29], with the first excited state contributing an amount of 0.0747 to the total NME, while the sum of the other contributions is zero to three significant digits. This agrees with the measurement of Thies et al. [30] where extreme SSD was reported to be found in the 2νββ NME of 96 Zr in a high-resolution 96 Zr( 3 He, t) experiment. Hence, our calculations confirm the experimental result of [30]. The accumulation for 96 Zr is very similar to the 48Ca case with the first 15 states adding constructively to 0.0765, beyond which the rest of the states contribute destructively. Beyond the first 100 states (7.5 MeV) the contributions are negligible.
Solving for g A from equation (5) and using the experimental half-lives from [9,10], phase-space integrals from [25], and the present shell-model NMEs, we get the effective g A values g eff  [36,37], where a value g eff A = 0.94 was found, since we expect these heavier nuclei to require more g A quenching than the lighter 96 Zr. As mentioned earlier, the measurement of Thies et al. [30] suggests that the state at 694.6 keV in 96 Nb might be the first 1 + state. If we repeat the calculations with this assumption, the to-  Table 1. Furthermore, the valence-space truncations affect differently the 48 Ca and 96 Zr decays. For the 48 Ca decay we are able to include all spin-orbit partners but for the 96 Zr decay we are forced to leave out the proton 0g 7/2 orbital (partner of the 0g 9/2 orbital) and the neutron 0g 9/2 orbital (partner of the 0g 7/2 orbital) due to an excessive growth of the computational burden. This could affect the final value of the 2νββ NME and thus the corresponding value of g eff A . In particular the β + -side transitions could be affected as pointed out by Towner [32]. More insight into the problem of missing spin-orbit partners could be obtained by computing the 2νββ NME of 96 Zr by the use of the proton-neutron quasiparticle random-phase approximation (pnQRPA) [33]. Here the result depends very much on the value of the particle-particle strength parameter g pp [34]. Performing pnQRPA calculations of the 96 Zr 2νββ NME within a reasonable interval of g pp values indicates that the presently omitted spin-orbit partners have an effect in the pnQRPA calculations. These orbitals introduce small negative contributions to the first contribution which is positive and by far the largest one. This interference of the first contribution and the high-lying contributions can lead to a reduction of the 2νββ NME in the range of tens of per cent, depending on the value of g pp . Most likely something similar could be expected for the shell-model calculation, thus increasing the value of g eff A from the value given in Table 1 and altering the β-decay branching ratios given below. Since adding the spin-orbit partners affects also the β-decay strengths it is hard to predict how the branchings to β decays would be altered in the end. The β transitions to the 4 + , 5 + , and 6 + states in 48 Sc and 96 Nb are 4th-forbidden non-unique, 4th-forbidden unique and 6thforbidden non-unique, respectively. A priori, without any calculations, one could predict that the 6th-forbidden non-unique β transition is much suppressed relative to the other two due to the much higher degree of forbiddenness that overwhelms the positive boost coming from the slightly larger Q value relative to the other two transitions. With less certainty one could predict that the NMEs of the two 4th-forbidden β transitions are on the same ball park and the difference in the Q value is most likely the decisive element in defining the branching between the two transitions. In the following we test these hypotheses by the shell-model calculations of the involved NMEs.
The β-decay and 2νββ-decay branching ratios calculated for 48 Ca are indicated in Fig. 3. As expected, the 2νββ branch is clearly dominant with a 92.4 ± 2.8% branching. The β-decay branches to the 4 + , 5 + , 6 + states are (1.7 +3.1 −1.2 ) × 10 −2 %, 7.5 ± 2.8% and (1.0 ± 0.4) × 10 −7 %. The branching to the 5 + state therefore poten-  tially competes with the 2νββ branch in a significant way, as was pointed out in [6]. The decay to the 6 + state is greatly hindered by the fact that it is sixth-forbidden. Based on the change in angular momentum, we would expect in general the decay to the 4 + state be the fastest. However, in this case this branch is quite small due to the relatively small Q value. In the work of Haaranen et al. [6] the half-lives for the 4 + and 6 + states were reported for g A = 1.0 and g A = 1.27. The half-lives are shortened from 3.97 × 10 23 yr to (3.47 ± 0.09) × 10 23 yr for the 4 + state and from 6.39 × 10 28 to 5.61 × 10 28 yr, when our 2νββ-determined g A = 0.80 ± 0.04 is adopted instead of g A = 1.00. The unique-forbidden 5 + branch is 50-60% stronger than suggested in [6], since the presently adopted heavier quenching of g A affects stronger the 2νββ branch. The small differences in the half-lives compared to the calculations in [6] are due to the inclusion of the next-to-leading-order NMEs and kinematic factors in the present study as well as the updated Q value, which is 1 keV larger than used in the study of [6]. As g A is quenched more, the significance of all the β-decay branches increases. For the unique 5 + transition the g A dependence of the decay half-life is well known and roughly g −2 A but for the non-unique transitions this is not the case due to the more complex structure of the shape factor. The uncertainties related to the branching to the 4 + state are especially large due to the fact that the 5 keV uncertainty makes a large percentage of the 26.65 keV Q value. An accurate measurement of the Q value would decrease the uncertainties significantly and thus would be desirable.
The g A dependence of all the decay branchings from 48 Ca is studied in Fig. 4. As can be seen the dependence on the value of g A is similar for the β branchings and they decrease substantially with increasing value of g A . At the same time the 2νββ-decay branching increases slowly towards one, as can be better seen in Fig. 5 where only the 2νββ and 4th-forbidden-unique decay branchings  are plotted as functions of g A . For reasonable values of g A the β branching to the 5 + state is always below 20%.
The computed branching ratios for 96 Zr are presented in Fig. 6. Like for 48 Ca the 2νββ branch for 96 Zr is the largest one at (81.6 ± 0.9)% but the dominance is not as significant as in the 48 Ca case. The branching ratios for the β-decay transitions are qualitatively similar to calcium case with 4 + state having a branching ratio of (2.5 ± 0.3) × 10 −2 %, the unique 5 + branch (18.4 ± 0.9)%, and the ground-state-to-ground-state branch being by far the weakest at (1.14 ± 0.04) × 10 −8 %. Using the computed β-decay half-lives reported in the previous shell-model study [8] of the decay of 96 Zr we can extract the corresponding branching ratios of 2.6 × 10 −2 %, 17.6%, and 1.21 × 10 −8 % in line with the presently determined branchings. The non-unique β branches in the present study are slightly smaller than those obtained in [8] but the branching to the 5 + state is notably stronger than expected based on Ref. [8].
This seems to confirm that the β decay might be up to 2.3 times faster than predicted by the older QRPA calculations in [7].
The dependence of all the 96 Zr decay branchings on g A is studied in Fig. 7. As can be seen, a similar g A dependence as in the case of 48 Ca is recorded. A closer look at the two leading branchings to the 2νββ and 4th-forbidden-unique decays, depicted in Fig. 8, indicates that their g A dependence is much stronger than in the case of 48 Ca. In the 96 Zr case the β branching to the 5 + state can reach values up to 40% for low values of g A . Such large branchings could be measurable in dedicated experiments sometime in the future. In this Letter the 2νββ matrix elements and single-β-decay branching ratios were calculated for 48 Ca and 96 Zr in the framework of the interacting nuclear shell model using large singleparticle valence spaces and matching well-tested many-body Hamiltonians. For 48 Ca a 2νββ matrix element M 2ν = 0.0511 was obtained, which is 5.5% smaller than the value of 0.0539 reported in the previous calculation of Horoi et al. [20]. For 96 Zr this was the first large-scale shell-model calculation yielding a value of M 2ν = 0.0747 using the shell model excitation energies and M 2ν = 0.0854 when the first 1 + state is assumed to be at 694.6 keV in 96 Nb, as suggested by the high-resolution 96 Zr( 3 He, t) experiment of Thies et al. [30]. An extreme single-state dominance was found thus verifying the result of the mentioned chargeexchange experiment [30]. Using these matrix elements, combined with measured values of 2νββ half-lives, effective quenched values of the weak axial coupling g A were extracted to be further used in the analyses of the β-decay branchings. In this way a consistent treatment of both the 2νββ decay and the competing β decays was achieved.
The 2νββ-decay and β-decay branching ratios were studied for their g A dependence and the total branchings to the β channels were determined to be (7.5 ± 2.8)% for 48 Ca and (18.4 ± 0.09)% for 96 Zr using the mentioned consistent effective values of g A . These branchings are in both cases larger than predicted in previous studies and could be large enough to be detected in underground experiments in the near future. The bulk of the uncertainty related to the 48 Ca branching ratios is due to the imprecise knowledge of the Q values. Therefore, a precise measurement of the groundstate-to-ground-state Q value for this case would be desirable.