Statistical analysis of β decays and the effective value of gA in the proton-neutron quasiparticle random-phase approximation framework

All material supplied via JYX is protected by copyright and other intellectual property rights, and duplication or sale of all or part of any of the repository collections is not permitted, except that material may be duplicated by you for your research use or educational purposes in electronic or print form. You must obtain permission for any other use. Electronic or print copies may not be offered, whether for sale or otherwise to anyone who is not an authorised user. Statistical analysis of β decays and the effective value of gA in the proton-neutron quasiparticle random-phase approximation framework Deppisch, Frank F.; Suhonen, Jouni


I. INTRODUCTION
The neutrinoless double beta (0νββ) decays of atomic nuclei serve as a forceful incentive to constantly drive nuclear-structure calculations toward better performance. Analyses of the potential experimental 0νββ outcomes in the future require accurate knowledge of the related nuclear matrix elements (NMEs) in order the obtained data to serve in best possible ways to unravel the fundamental nature and mass of the neutrino [1][2][3][4]. It is also tightly connected to the breaking of lepton number asymmetry and has far reaching consequences even on solutions on the baryon asymmetry of the Universe [5,6]. A host of models, ranging from the interacting shell model (ISM) to various mean field theories, have been used in the calculations. The resulting NMEs have been analyzed in the review article [7]. Most of the calculations have been pursued in the framework of the proton-neutron quasiparticle random-phase approximation (pnQRPA) [8].
A particular problem with the pnQRPA calculations, not present in the other calculations, is the unsettled value of the particle-particle interaction parameter g pp describing the strength of the proton-neutron interaction in the 1 + channel. Since the introduction of this parameter [27,28] its values have been tried to fix by the inspection of the measured single-beta-decay rates [29,30] or 2νββ-decay rates [31][32][33][34].
Here we make an attempt to relate the values of g pp to the values of g A through the data on betadecay rates associated with the transitions between an even-even and an odd-odd nucleus. The data on these decays are presented as comparative half-lives (log f t values) and comparing them with the corresponding computed ones one can make conclusions about the possible correlations of these two key parameters of calculation. As mathematical aid we use the Markov Chain Monte Carlo (MCMC) statistical analysis of 47 isobaric triplets and 28 more extended isobaric chains of nuclei. In the isobaric triplets there are two beta-decay branches, left and right, between the central and lateral nuclei, and in the extended isobaric chains more complex systems of consecutive central and lateral nuclei can form. To estimate the theoretical uncertainty inherent in the pnQRPA framework we include the full parametric freedom available. This means we introduce an uncertainty in the particle-hole interaction parameter g ph . In addition we treat both g pp and g ph as parameters specific only to a given β + /electron capture (EC) or β − decay transition pair. This opens up a large parametric freedom that has not been explored before.
Our analysis is intended to address the importance of quenching, i.e. the suppression of g A with respect to its free value g A = 1.269. Quenched values as low as g A ≈ 0.4 have been reported for example in the IBM-2 model [26]; because 0νββ decay depends on g A as ∝ g 4 A , this could reduce the decay by orders of magnitude, having a serious impact on the observability of 0νββ decay in experiments. Whether such strong quenching actually applies to 0νββ is not a question we can answer here, because we touch here on only the 1 + multipolarity of the multipole decomposition of a 0νββ decay NME and because the 0νββ decay proceeds via a momentum exchange much larger than that of the presently discussed single and two-neutrino double beta decays.
The article is organized as follows. In Sec. II the basic theoretical framework is briefly reviewed and the model-space aspects and the adjustment of the model parameters are explained. In Sec. III we make a statistical analysis of the effective value of g A . Here we try to chart the possible effective values of the axial-vector coupling constant g A in model calculations using the pnQRPA approach, using different methods. Finally, in Sec. IV, we summarize and draw the conclusions.

II. BRIEF SUMMARY OF THE THEORY
We begin by defining the comparative half-lives (log f t values) of the 1 + ↔ 0 + Gamow-Teller transitions that form the basis of the present analysis. The log f t value is defined as [35] log f t = log 10 with for the β + /EC or β − type of transitions. Here the half-life t 1/2 has been given in seconds and f 0 is the dimensionless leptonic phase space factor associated with the process. J i is the spin of the initial ground state and M GT is the Gamow-Teller NME defined, e.g. in Ref. [35]. Here g A is the weak axial-vector coupling constant, g pp the particle-particle interaction coupling constant and g ph the particle-hole interaction coupling constant as defined, e.g. in Refs. [36,37]. Methods of determining the values of these constants are addressed in the next section. The 2νββ decay half-life can be compactly written as where G 2ν stands for the leptonic phase-space factor without including g A in the way defined in Ref. [38]. The initial ground state is denoted by 0 + i and the final ground state by 0 + f . The 2νββ NME M (2ν) can be written as where the quantity D m is the energy denominator containing the average energy of the 1 + states emerging from the pnQRPA calculations in the initial and final even-even nuclei. The summation is in general over all intermediate 1 + states where 1 + m |1 + n is the overlap between two such states. We will in general treat the individual matrix elements for the transition between the initial (final) state and the virtual intermediate states, M I(F) (1 + i ) as functions of separate sets of g pp and g ph couplings.
While we do not discuss 0νββ decay in detail in this article, we briefly describe the theoretical calculation of the corresponding half-life to illustrate the similarities and differences to the above processes. The 0νββ decay half-life can be written as where G 0ν stands for the leptonic phase-space factor without including g A,0ν and the electron mass m e in the way defined in Ref. [38]. Here, we denote the effective axial coupling relevant for 0νββ decay as g A,0ν to emphasise that its value may deviate from the one determined in single beta and 2νββ decays. The effective 0νββ neutrino mass is denoted as m ν . As before, the initial ground state is denoted by 0 + i and the final ground state by 0 + f . The 2νββ decay and 0νββ decay half-lives share the same strong dependence on g A as seen in Eqs. (3) and (5). It is thus an essential first step to study the effective value of g A in single beta and 2νββ decays. These studies tangent only the 1 + contribution to the 0νββ decay whereas it is known that higher multipoles are very important for the 0νββ decay as well [39]. Some attempts to study these higher multipolarities by way of single beta decays have been made lately [40,41]. It is thus not straightforward to relate the single and 2νββ decay studies to the value of the 0νββ NME, especially since the former involve momentum transfers of a few MeV and the latter involves a virtual neutrino with a momentum exchange of the order of 100 MeV. This allows the possibility that the effective value of g A gets momentum dependent [24]. Related to this, the high momentum exchange in 0νββ decay makes the higher J π states contribute appreciably to the decay rate [42]. For these higher-lying states the quenching of g A could be different from the low-lying states discussed in the present work. It should be noted, however, that in the pnQRPA no closure approximation is imposed in either modes of double beta decay so that the individual contribution from all intermediate states can be accessed in the case of 0νββ decay, as well. These intermediate contributions vary strongly from nucleus to nucleus and even some kind of single-state dominance can be observed for some 0νββ decaying nuclei [42]. For more details on the theoretical background, we refer the reader to Ref. [20].
In the present calculations we obtain the single-particle energies from a spherical Coulombcorrected Woods-Saxon (WS) potential with the standard parametrization of Bohr [43]. This parametrization is optimized for nuclei near the line of beta stability and is thus well suited for the presently studied nuclei. The single-particle orbitals used in the calculations span the space 0f-1p-0g-2s-1d-0h 11/2 for the masses A = 62 − 80, 0f-1p-0g-2s-1d-0h for the masses A = 98 − 108 and 0f-1p-0g-2s-1d-0h-1f-2p for the masses A = 110 − 142. In these single-particle bases the proton and neutron Fermi surfaces are well contained in the model space. The Bonn-A G-matrix has been used as the starting point for the nucleon-nucleon interaction and it has been renormalized in the standard way [37,44]: The quasi-particles are treated in the BCS formalism and the pairing matrix elements are scaled by a common strength parameter, separately for protons and neutrons. In practice these factors are fitted such that the lowest quasi-particle energies obtained from the BCS match the experimentally deduced pairing gaps for protons and neutrons respectively. For closed major shells the pairing strength parameters were taken from the closest even-even neighbor.
The wave functions of the 1 + states of the intermediate nuclei have been produced by using the pnQRPA with the particle-hole and particle-particle degrees of freedom [27] included. The particlehole and particle-particle parts of the proton-neutron two-body interaction are separately scaled by the particle-hole (g ph ) and particle-particle (g pp ) parameters. The particle-hole parameter affects the position of the Gamow-Teller giant resonance (GTGR) in the odd-odd nucleus and its value is fixed by the available systematics [35] on the location of the resonance: The difference ∆E GT between the GTGR and the ground state of the neighboring even-even reference nucleus thus depends on the proton and neutron numbers (Z, N ) of the reference nucleus, as well as on its mass number. In practice, both the measured and the computed GTGR have a width and their location are determined by the centroid (weighted average) of the strengths associated with the individual 1 + states comprising the GTGR. In a pnQRPA calculation the difference E(1 + GTGR ) − E(0 + gs ) in Eq. (6) gives the empirical location of the centroid of the GTGR which has to be matched by the centroid of the pnQRPA computed strengths of the 1 + states presumed to belong to the GTGR. The computed centroid depends strongly on the value of the g ph parameter and weakly on the choice of the set of 1 + states included in the GTGR, the latter introducing an inherent source of error. Throughout our calculations we assume that the value of g ph in a given system is determined with a relative error of 15 % as a source of theoretical uncertainty. This 15 % error represents a maximum deviation in g ph such that the computed centroid of the Gamow-Teller giant resonance is not meaninglessly far from its empirical position as given by Eq. (6). Throughout we denote with γ ph the normalized value of the particle-hole parameter with respect to the value determined through the GTGR. The determination of the values of g pp , together with the axial-vector coupling constant g A is presented below.
As an example, Fig. 1  Ru as functions of g pp . The 1σ and 2σ uncertainties due to a variation of the parameter g ph around its value determined by the GTGR are shown using the colored bands. The dependence of the single beta NMEs shows a typical behavior seen in many triplets where one NME increases whereas the other decreases. This has the effect that the dependence of the product of the NMEs on g pp can become rather weak and consequently the value of g A can be extracted from the product of the log f t values separately. This behavior is not universal, as exemplified in Fig. 2b showing the NMEs of the processes 68 31 Ga → 68 30 Zn and 68 32 Ge → 68 31 Ga (2νββ decay is not possible here). Here, both NMEs rise with g pp which will couple the determination of g A and g pp as is discussed in Sec. III. As can be seen, a variation of g ph at the 15% level generically has an effect on the matrix elements of the same order, depending on the isotopes involved.
In Fig. 2 both the left-leg and the right-leg NMEs were treated as depending on the same g pp as is assumed in most analyses. As discussed above, we in turn treat the transitions independently, each depending on separate parameters, g L pp and g R pp . This has the immediate effect that the beta decay/EC processes become statistically independent because they now depend on different parameters. In addition, the NME of the 2νββ decay (if allowed within a given triplet) now Ru as a function of the left-leg and rightleg particle-particle parameters g L pp and g R pp , respectively. The iso-curves indicate constant NME values as shown. The particle-hole parameters are set at their GTGR values, becomes a function of both g L pp and g R pp . Fig. 3 illustrates the dependence of the NME for the 2νββ decay 102 42 Mo → 102 44 Ru. The iso-curves indicate constant values for the NME as shown. The plot demonstrates that the left-leg and right-leg NMEs are correlated such that the 2νββ remains approximately constant if both depend on the same g pp 0.7 (along the diagonal, also compare with Fig. 2a). This degeneracy is lifted if g L pp and g R pp are allowed to vary independently. The dependence on the particle-hole parameters, which we will in turn also treat as independent values, is neglected, and they are set at their GTGR values, γ L ph = γ R ph = 1.

A. Quenching of the axial-vector coupling constant
At this stage it is worth pointing to some other earlier works devoted to the determination of the effective value of g A in calculations using the pnQRPA model or other models. A strongly reduced effective value of g A ≈ 0.6 was reported in the shell-model calculations [45] in the mass A = 90 − 97 region. In a more recent shell-model study [25] values of about g A ∼ 0.7 were obtained in the mass region A = 128, 130 and an even a stronger quenching of g A = 0.56 was obtained for A = 136. The first analysis performed in the pnQRPA model was done in Ref. [17] where both the beta-decay and the 2νββ decay data were analyzed for the A = 100, 116 systems using a least-squares fit to determine the values g A = 0.74 (A = 100) and g A = 0.84 (A = 116). It is interesting to note that in the first version [46] of Ref. [17] also the result g A = 0.39 for the A = 128 system was quoted. An approximately monotonic behavior of the effective values of g A was parametrized in Ref. [26] by analyzing the magnitudes of NMEs produced by the IBA-2 model. Values around g A = 0.5 were obtained. In a later publication [47] the interacting boson-fermion-fermion model, IBFFM-2, was    I: Characteristics of the β + /EC and β − decays in isobaric triplets within the mass range A = 62−114 studied in the present paper. An isobaric triplet is identified by the mass number A and the lowest atomic number Z 0 among the three isotopes. The isotopes in the triplets are indicated along with their spin J and parity π, (J π ). The arrows denote the direction of the relevant β + /EC, β − decay. The experimentally determined comparative half-lives of the left and right transition are given as log f t L and log f t R , respectively. 2νββ decaying isotopes are underlined. The values of g A and g pp are determined in the triplet fit described in Sec. III B. Cases in which the best fit χ 2 min is in the range [0.5, 2.2], indicating slight incompatibility with data, are highlighted with italic numbers. Cases with stronger discrepancy are highlighted in bold. In all other cases a χ 2 min = 0 (within numerical tolerance) was found.
adopted and the subsequent analyses yielded highly suppressed values of g A ≈ 0.3 for the A = 128 nuclei. Recently a systematic approach to β and 2νββ decays in the mass region A = 100 − 136 was performed [48]. The suitability to the description of the global behavior of the β and 2νββ decays, a linear model and an overall-quenched g A ≈ 0.6 were examined. The present study is an extension of this work as well as Refs. [18,20,29] to a wider mass region and a refinement in   the statistical analysis methods used to extract information on the quenching of g A in this wider region.
The apparently low effective values of g A in the pnQRPA could be attributed to missing the contributions of the complex configurations beyond the two-quasiparticle (particle-hole) configurations of the pnQRPA (see also Ref. [19]). On the other hand, it was shown in Ref. [49] for the 2νββ decay of 76 Ge and in Ref. [50] for the 2νββ decay of 100 Mo that the inclusion of the fourquasiparticle (two-particle-two-hole) degrees of freedom in a higher-QRPA scheme (in this case the pnMAVA (proton-neutron microscopic anharmonic vibrator approach)) does not affect appreciably the low-energy Gamow-Teller properties of pnQRPA. It is yet unclear what is the primary reason for the rather low effective values of g A and what is the share between the model-dependent and model-independent contributions to it. The model-independent quenching can be associated with the non-nucleonic, i.e. isobaric degrees of freedom in nuclear matter [51,52]. Contributions to the model-dependent quenching come from the limitations in the single-particle models space (ISM, IBA-2, IBFFM-2) or the lack of complicated many-nucleon configurations (pnQRPA, IBA-2, IBFFM-2). The determination of the effective values of g A in different theory frameworks is an  extremely interesting issue and certainly necessitates further investigation in the future.

B. Fitting isobaric triplets
The basis of our analyses is provided by the experimental log f t values of the relevant beta decays/EC processes. They are shown in Tabs. I and II, displaying the comparative half-lives log f t L and log f t R for the left-leg decay and the right-leg decay of a given triplet. The comparative halflives were calculated from the experimentally measured half-lives listed in Ref. [53] incorporating the experimental uncertainty in both the measured decay half-life and the Q value. In most cases, the experimental uncertainty is negligible compared to the theoretical uncertainties expected to be inherent in nuclear model calculation; the errors in the values of the comparative half-lives range between the per mil and the 10% level. In two cases only, a lower limit is known. The range of considered isotopes is dictated by the applicability of the theory framework (the quasiparticle description for s-d shell nuclei becomes questionable for lighter nuclei) and nature (for example, the 1 + states are not the ground states in the odd-odd systems in the mass gaps A =72-76 and A =82-96).
As far as available and relevant for our selection of isotopes we also calculate the 2νββ decay half-lives. The characteristics of 2νββ decaying isotopes are shown in Table III giving the phase space factor G 2ν , and the experimental half-lives for three of the isotopes. The fitted or predicted half-lives are discussed below.
We start by fitting the triplets individually, i.e. we compare the theoretically predicted log f t values of the left-leg and right-leg decays in a triplet of Tabs. I and II with the experimental data. For this purpose, we assume that both decays depend on the same pair g A , g pp . In addition we also include a variation of the g ph couplings independently for each decay, with a 1σ deviation of 15% from its GTGR value.
Because we later work with a larger number of free parameters and in systems that can be underconstrained, exactly-constrained or over-constrained, we consistently perform the fitting procedure using a straightforward MCMC based on the Metropolis-Hastings algorithm [54]. Throughout our calculations, we have verified that the uncertainties inherent in the MCMC due to finite sampling etc. are small compared to the physical uncertainties. We always use a flat prior in the given fitting parameters, i.e. they are randomly selected on a linear scale within a given range. We always vary g A and g pp in the range [0.1, 1.4] and γ ph between [0.5, 1.5].
In the current case, the fitting is based on a χ 2 function applied to a given triplet of the form Here, the experimental log f t exp L,R values along with their experimental errors δ log f t exp L,R are taken from Tabs. I and II. The theoretically determined log f t th L,R are computed as functions of the fitting parameters g A and g pp . In addition, they depend on the variables γ L,R ph which represent the particle-hole parameters relative to values as derived from the energy of the giant resonance, γ L,R ph = g L,R ph /[g L,R ph ] GTGR . The last two terms in Eq. (7) correspond to using the γ L,R ph as nuisance parameters with best fit values of 1 and the deviations δγ L,R ph = 0.15. In the case where only a lower limit on the experimental log f t is known, the corresponding quadratic term in χ 2 is replaced by (max(0, log f t th − log f t exp )/δ log f t exp ) 2 , i.e. a single sided exponential to represent the lower limit. Finally, the MCMC fit is performed using the fitness function P = exp(−χ 2 /2).
With two (non-nuisance) parameters and two constraints, the system is exactly determined and a solution with χ 2 = 0 is generically expected. As is shown later, in some cases no consistent solution can be found for the given experimental data. As indicated by the curly brackets, for triplets in which the central isotope is even-even (i.e. identified by an odd Z 0 ), both decay legs are regulated by the same g ph and thus the fit is performed with only one nuisance term and γ L ph = γ R ph . Before discussing the results of the numerical fits, we would like to illustrate how the experimental data constrains (g A , g pp ) in a few examples. By omitting the nuisance parameters and shifting their induced uncertainty into a theoretical error on the log f t, Eq. (7) simplifies to resulting in a two-dimensional parameter space that can be easily visualized. Fig. 4 shows the χ 2 fit based on Eq. (8) and the individual contributions from the measurement of the left-leg (red) and right-leg (blue) beta/EC decays. In addition to the experimental errors, the χ 2 fit includes a common theoretical uncertainty of δ log f t th = 10%. It has been chosen to be rather unrealistically small to show the effect of the experimental errors that otherwise are usually small compared to the model uncertainty. In the first three cases (A, Z 0 ) = (100, 42) (a), (116, 48) (b) and (128, 52) (c), the triplet includes a 2νββ decay isotope for which the half-life has been measured, 100 42 Mo, 116 48 Cd and 128 52 Te, respectively. The green band gives the correspondingly allowed 1σ parameter space. For 100 42 Mo (a) it overlaps well with the fit from the beta/EC decays. In the other two cases (b and c) there is a tension between single beta/EC decay and 2νββ decay data in the chosen model, but in both cases the discrepancy corresponds to a modest difference in g A , by less than 30%. We discuss this discrepancy and possible causes at the end of Sec. III C. In the A = 128 case (c) there are formally two best-fit solutions, the significance of which we comment on below. The final scenario (d) illustrates a case where there is a tension between experimental data and theoretical predictions, i.e. the minimal χ 2 is different from zero. For the actual numerical determination of the best-fit parameters and their errors, we use Eq. (7), where we include a 15% uncertainty in the value(s) of the particle-hole nuisance parameter(s) g ph to model an additional theoretical uncertainty. Using the MCMC method described above, we determine the best-fit values and 1σ errors for g fit pp and g fit A in all triplets as given in Tabs. I and II. They are based on the fully marginalized distribution for the given parameter, but we omit secondary solutions for large g pp beyond the divergence, cf. Fig. 4c. These large values of g pp make the pnQRPA solutions unstable and in the worst case the whole set of pnQRPA solutions collapses because the condition of small-amplitude motion of the RPA theory becomes seriously violated [35].
In Tabs. I and II, the results highlighted with italic numbers correspond to fitting results with a minimal χ 2 value of the order of 1, indicating a slight tension between the experimental data and the theoretical predictions. As an example, the fitting of triplet A = 68, Z 0 = 30 is illustrated in Fig. 4d. While the numerical fit in Table I includes a larger theoretical uncertainty from the variation of the left-and right-leg particle-hole parameters γ L,R ph , this is not sufficient to achieve a vanishing χ 2 min . Similar behavior occurs for the other triplets highlighted with italic numbers in the table. In such cases, the statistical uncertainty likely underestimates the true theoretical error.
For the triplets A = 80, Z 0 = 35 and A = 130, Z 0 = 54, the minimal χ 2 value is substantially different from zero (χ 2 min = 33 and 15, respectively), meaning that there is a large tension. We still give the nominal best fit values for g A and g pp , highlighted in bold in these cases, to indicate the tendency of the fit, but we do not quote an uncertainty. In fact, in both cases the best fit is achieved at the limit of the considered parameter space and the statistical uncertainty is rather meaningless. Finally, for A = 70, Z 0 = 29 no meaningful fit was achieved. In all other cases, a minimal χ 2 = 0 (within numerical tolerance) was found. As can be seen in Tabs. I and II, problems to fit the experimental data mostly occur for lighter nuclei with A ≤ 80. This could indicate the diminishing flexibility of the pnQRPA model in going from the heavy nuclei, with large active single-particle model spaces, towards the lighter nuclei with small active model spaces, better suited for shell-model description.
The results are also graphically illustrated in Fig. 5, as a function of the mass number A. Analogous to the tables, the light orange points and 1σ error bars represent triplet fits with χ 2 min ≈ 1 whereas the blue points correspond to χ 2 min = 0 (within numerical tolerance). The vertical double arrows (and horizontal lines for the nominal best fit, where applicable) indicate the cases with a strong tension with data as discussed above. As seen in the plots, the best fit values The different parameters g pp and g ph used in the fit for this multiplet are indicated for each even-even isotope in the system. Together with g A , this leads to four (non-nuisance) parameters compared to four constraints from the beta/EC decays. The system also contains the 2νββ isotope 128 52 Te decaying to 128 54 Xe.
in these scenarios are still in the right ballpark of the neighboring fits. The strongest feature in the plots is the rise of g A with larger A from A = 98 to 142 accompanied with a fall of g pp . In this region the effective g A increases from a strongly quenched g A ≈ 0.4 around A = 100 to an essentially unquenched g A ≈ 1.1 around A = 140. Although there is a considerable spread in the values, the fitting results of triplets within the same mass number A are largely compatible, illustrated by the overlapping error bars in Fig. 5. The tendency for lighter nuclei (A = 80 and below) is less clear and here the result is also affected by the large number of cases with tension to data. The tendency of a growing effective value of g A with mass number for the A ≥ 100 nuclei is in agreement with the linear model of Ref. [48]. From Fig. 5a one can deduce the average value g A ≈ 0.6 for the A ≥ 100 nuclei in accordance with the analysis of Ref. [48].
Using the thus fitted parameters, we calculate the predicted 2νββ decay half-lives for all relevant isotopes as listed in Table III under [t (2ν) 1/2 ] triplet . The calculation includes the correlation among g A , g pp and g ph ; i.e., we use the full probability density from the MCMC fit. Confirming the expectation of the simple two-dimensional fits shown in Fig. 4, the measured 2νββ decay half-life of 100 Mo is consistent with the prediction within 1σ. On the other hand, the predictions for 116 Cd and 128 Te are too small by a factor of about 2 compared to the experimental results. We further discuss this discrepancy at the end of the next section. According to the calculated half-lives, the 2νββ decay of the nucleus 110 Pd would be an interesting case to measure in the future.

C. Fitting isobaric multiplets
Under the assumption that g A is a function of the mass number A only, as justified in the Introduction, we can extend the analysis of individual triplets by fitting all beta/EC decays within a system (multiplet) of isobaric isotopes. This allows us to incorporate the full experimental information available. On the other hand, we endeavor to include the full parametric uncertainty inherent in the pnQRPA models used in this study. This means that for each isobaric multiplet we use a common g A and a set of (g pp , g ph ) for each even-even isotope within the multiplet. The resulting multiplets are listed in Table IV, identified by (A, Z 0 ) of the first isotope. The table indicates all isotopes in the multiplet by their atomic number and the arrows give the direction of the decays. The multiplicity within the isobaric systems ranges from three to seven isotopes. The triplets are obviously identical to the ones discussed in the previous section but we would like to note that we now allow for more parametric freedom because we use a separate g pp for each even-even isotope. Almost all triplets contain two even-even isotopes, i.e. two different g L pp and g R pp are used in the fit. Only the triplet (98, 39) contains a single even-even isotope and the fit performed is identical to the one in the previous section.
In general, the degree of freedom in a fit is given by the number of free parameters (one g A and a g pp for each even-even isotope) minus the number of experimental constraints (number of measured beta/EC decays). This difference is given in Table IV under 'dof' indicating that we encounter underconstrained (dof > 0), exactly constrained (dof = 0) and overconstrained (dof < 0) systems. While we also include a g ph for each even-even isotope, this does not change the number of degrees of freedom because the g ph are treated as nuisance parameters. An example of the decays and parameters involved is shown in Fig. 6 in the case of the multiplet (128, 52).
The MCMC fitting procedure is analogous to the triplet case with a χ 2 function of the form of Eq. (7): there is a contribution for each observable (beta/EC decay), with the theoretically calculated log f t values depending on the appropriate g i pp and γ i ph and the global g A . In addition, there is a nuisance term for each of the particle-hole parameters γ i ph where we again use an uncertainty of δγ i ph = 0.15. As before, we use the experimental measurements and errors of the beta/EC decay log f t values from Tabs. I and II.
An example of the result of an isobar fit is shown in Fig. 7 displaying the probability density functions in three different marginalized parameter planes as derived in the isobaric system (100, 41) containing the 2νββ decay isotope 100 42 Mo. It is a quartet with two even-even isotopes and thus described by the following three parameters, g A , g Mo pp and g Ru pp , and two nuisance parameters, γ Mo ph and γ Ru ph . The system is exactly constrained and consistent; thus the minimal χ 2 is zero. Nevertheless, there is a sizable uncertainty and especially the parameter g Ru pp can vary strongly and no lower limit can be determined at 2σ within the chosen parameter range. In this case the two particle-particle parameters can be very different. For example, the combination g Mo pp ≈ 0.7 and g Ru pp ≈ 0.1 is allowed by the combined experimental data within 2σ. Another consequence of this is that g A is effectively suppressed compared to the triplet fit because lower values are statistically preferred for small g Ru pp .
The experimental log f t errors and the variation of γ Mo ph and γ Ru ph provide an additional source of uncertainty.
The results of the fits are displayed in Table IV, which lists the best-fit values and 1σ un-    certainties of g A for all multiplets. As in the case of the individual triplets, italicized values denote multiplets in which there is a slight tension between the combined experimental data and the theoretical predictions. Comparing with Tabs. I, II, the tension for A = 62, 80 is resolved by introducing the additional freedom in the multiplet fits. On the other hand, the multiplets A = 108, 114, 122, 128 exhibit slight tension, due to the larger number of simultaneous constraints that dominate over the additional freedom. The strong tension for A = 70, 80, 130 remains in the multiplet approach.
The results are also graphically displayed in Fig. 8a, where we plot the extracted values of g A as a function of the mass number A, also showing the 1σ and 2σ uncertainties. It can be directly compared to Fig. 5a, and it can be seen that the general behavior exhibited is similar in the two plots. Due to the combined fit of all isobaric nuclei, the dependence on A is smoother in Fig. 8a, albeit with sometimes considerable uncertainty for some multiplets, suggesting a systematic dependence of g A with the mass number A within the range 98 ≤ A ≤ 142. The multiplets A = 108, 114, 122, 128 with slight tension correspond to triplets with comparatively large differences in the fitted g A values. This tension between the triplet g A values leads to a worse fit when combining within a multiplet and the resulting uncertainty likely underestimates the true error in g A . For comparison in these cases, Fig. 8a also shows the 1σ and 2σ range derived in the associated triplet fits (i.e. the minimal and maximal extent in g A among all triplets at the given uncertainty). This likely provides a more realistic estimate of the uncertainty in g A . Fig. 8b shows g pp as a function of A within the isobaric fits where each dot represents the best fit point of an individual g i pp . As is illustrated in Fig. 7 and discussed above, the freedom of allowing a different g pp per even-even isotope introduces a degeneracy where individual g pp can vary strongly. Fig. 8b demonstrates that in almost all cases at least one of the g pp can be small, hitting the limit of the chosen variable range. As a result the apparent spread among the g pp is large and there is only a tendency of squeezing the range to lower values for increasing A. It should be kept in mind that the fitting procedure also means that the individual g pp values are correlated to satisfy the experimental constraints. Despite the large freedom and large variability of the g i pp it is remarkable that the corresponding uncertainty of g A remains comparatively small.
This could indicate a robustness of the underlying pnQRPA-based nuclear-structure calculations against parameter variations. Using the thus fitted parameters, we calculate the predicted 2νββ decay half-lives for all relevant isotopes as listed in Table III under [t (2ν) 1/2 ] multiplet . The calculation includes the correlation among the parameters g A , g i pp , and γ i ph ; i.e., we use the full probability density from the MCMC fit. Unfortunately, many of the 2νββ isotopes are part of multiplets with a slight or severe tension between experimental data and theoretical prediction of the EC/beta decay fit. These are highlighted in Table III as usual. The uncertainties quoted in these cases are likely underestimates. The multiplet fits incorparating 116 Cd and 128 Te are not able to relax the tension with the measured 2νββ decay half-lives in these isotopes.
On the other hand, in the analysis carried out in Ref. [26] a very weakly decreasing A dependence was obtained for the effective value of g A in the IBA-2 and ISM frameworks. The result was based on the comparison of the computed and experimental half-lives of the 2νββ decays. This weak trend is in line with the analysis of Ref. [48] where it was observed that the constant g A = 0.6 reproduces better the measured 2νββ half-lives than the growing trend for g A obtained by the β-decay analysis (see Table VI of Ref. [48]). In fact, having a look at Fig. 4 as well as Table III of this work, in both the triplet and multiplet analyses a decreasing multiplicative factor of g A , as a function of the mass number A, would bring the computed half-lives closer to the measured ones. The fact that the analyses of the β decays and the 2νββ decays give opposite trends is a sign that there are basic differences between the two modes of decay. One difference is that for the 2νββ decays more than one low-lying state can contribute and the contribution coming from the first virtual 1 + state of the intermediate nucleus does not exhaust the whole 2νββ NME (see Table V of Ref. [48]). The balance between the first contribution and the rest depends on the value of g pp in a pnQRPA calculation, and the correlations of g A and g pp are most likely different for the β and 2νββ decays.

IV. CONCLUSIONS
In this work we studied single β + /EC and β − decays for a number of isobaric triplets and more complicated isobaric chains of nuclei in the framework of the proton-neutron quasiparticle random-phase approximation.
The present calculations have been done with G-matrix-based two-nucleon interactions. By letting the value of the axial-vector coupling constant g A vary freely, together with the particleparticle interaction strength parameter g pp , we have performed MCMC-based statistical analyses to chart the effective values of g A in pnQRPA-based nuclear-structure calculations. Within the statistical fits of complete isobaric chains of nuclei we incorporated full parametric uncertainty of the nuclear model by using independent g pp per even-even isotope as well as an uncertainty in the particle-hole parameter g ph . We thus not only confirm previous results of an apparent quenching of g A in an extended analysis but we also provide a realistic estimate of the parametric uncertainty inherent in the nuclear model. This is important, also to compare with other theory frameworks.
These findings may have some bearing on the studies of the contributions of the low-lying 1 + intermediate states to the highly interesting 0νββ NMEs. The relation of our present results to the values of the 0νββ NMEs remains still an open issue but we view the present study as an incentive to tackle these issues in future investigations.