Overconstrained estimates of neutrinoless double beta decay within the QRPA
Estimates of nuclear matrix elements for neutrinoless double beta decay () based on the quasiparticle random phase approximations (QRPA) are affected by theoretical uncertainties, which can be substantially reduced by fixing the unknown strength parameter of the residual particle-particle interaction through one experimental constraint — most notably through the two-neutrino double beta decay () lifetime. However, it has been noted that the adjustment via data may bring QRPA models in disagreement with independent data on electron capture (EC) and single beta decay () lifetimes. Actually, in two nuclei of interest for decay (Mo and Cd), for which all such data are available, we show that the disagreement vanishes, provided that the axial vector coupling is treated as a free parameter, with allowance for (“strong quenching”). Three independent lifetime data (, EC, ) are then accurately reproduced by means of two free parameters , resulting in an overconstrained parameter space. In addition, the sign of the matrix element is unambiguously selected by the combination of all data. We discuss quantitatively, in each of the two nuclei, these phenomenological constraints and their consequences for QRPA estimates of the matrix elements and of their uncertainties.
pacs:23.40.-s, 23.40.Hc, 21.60.Jz, 27.60.+j
The new paradigm of massive and mixed neutrinos, emerging from the recent evidence for neutrino flavor oscillations [1, 2, 3, 4], is still incomplete in several aspects. In particular, the nature of the neutrino fields (Dirac or Majorana)  remains undetermined, the amount of CP violation in the neutrino sector (if any) is unconstrained, and the absolute neutrino masses—as well as their ordering—are not yet known. The process of neutrinoless double beta decay (),
bears on all these issues and, thus, is a major research topic in current experimental and theoretical neutrino physics [6, 7, 8, 9]. The claimed observation of decay in Ge with lifetime y , and the projects aimed at its independent (dis)confirmation , have also given new impetus to the field.
In general, barring contributions different from light Majorana neutrino exchange, the inverse lifetime in a given nucleus is the product of three factors,
where is a calculable phase space factor, is the nuclear matrix element, and is the (nucleus-independent) “effective Majorana neutrino mass” which, in standard notation , reads
and being the neutrino masses and the mixing matrix elements, respectively.
The calculation of the matrix element in Eq. (2) for a candidate nucleus is notoriously difficult. It requires the detailed description of a second-order weak decay from a double-even “mother” nucleus to a double-even “daughter” nucleus via virtual states (with any multipolarity ) of the so-called “intermediate” nucleus . The decay can proceed through both Fermi (F) and Gamow-Teller (GT) transitions, plus a small tensor (T) contribution,
and detailed nuclear models are required to estimate the separate components (, GT, T). In the above expression, is the effective axial coupling in nuclear matter, not necessarily equal to its “bare” free-nucleon value .
Modern calculations of matrix elements are usually performed within either the quasiparticle random phase approximation (QRPA) [13, 14] or the nuclear shell model (NSM)  and their variants, sometimes with large differences among the results. We remind that the QRPA basis of nuclear many-particle configurations, on which the residual particle-hole and particle-particle interaction is diagonalized to build the nuclear excitations, is limited to iterations of two-quasiparticle ones (reducing to the particle-hole configurations when the pairing interaction is switched off); for details, see, e.g. [6, 7]. The advantage of the QRPA as compared to the NSM is that one can include essentially unlimited sets of single-particle states, even those forming the continuum of the positive-energy ones within the continuum-QRPA .
Painstaking but steady progress in both the QRPA and the NSM approaches is gradually leading to a better understanding—and to a reduction—of the differences among their results . However, even in the most refined approaches, the estimates of remain affected by various uncertainties, whose reduction is of paramount importance for both theory and experiment. Indeed, uncertainties in propagate to the extracted value of (or limit on) via Eq. (2), and affect directly the design of experiments (in particular the detector size and the choice of the nucleus) needed to reach a given target sensitivity to . Among the sources of uncertainties one can quote: (1) inherent approximations and simplifications of the theory; (2) existence of free or adjustable model parameters; (3) problematic description of the strong short-range repulsive interaction between nucleons; and (4) uncertainties in the value of .
The latter problem arises from the significant reduction (“quenching”) of the strength observed in nuclear GT transitions (see, e.g., ), which still lacks a clear experimental quantification and theoretical understanding. Two possible physical origins of the quenching have been discussed, one due to the -isobar admixture in the nuclear wave function  and another one due to the shift of the Gamow-Teller strength to higher excitation energies induced by short range tensor correlations . In the absence of a better prescription, the effect of quenching (in either QRPA or NSM calculations) is often simply evaluated by replacing the bare value with an empirical, quenched value . However, there is no a priori reason to exclude values , which have indeed sometimes been advocated, especially within the NSM approach [15, 21].
In this context, we present a novel approach towards data-driven constraints on calculations, assuming the possibility of strong quenching () within the QRPA. This unconventional hypothesis makes theory and data agree in a number of cases, where previous attempts have systematically failed. Therefore, we think that our approach may lead to a fruitful discussion and a fresh look at the whole problem of quenching, from both the theoretical and the experimental viewpoint. We stress, however, that we simply treat as a phenomenological possibility in this work, without any attempt to elaborate theoretical interpretations of the values emerging from the data analysis.
Our work is structured as follows. In Sec. 2 we discuss the experimental data which can be used to benchmark the QRPA model. We adopt a selected data set, including the measured lifetimes of two-neutrino double beta decay, electron capture, and single beta decay for two nuclei, Mo and Cd, which are of interest for searching decay. In Sec. 3 we compare these data with the corresponding QRPA results, assuming standard quenching () or no quenching (). We face then the well-known problem that the theory cannot match two or more data at the same time, for any given value of the so-called particle-particle strength parameter . In Sec. 4 we show that this problem can be phenomenologically removed if strong quenching () is allowed. In this case, the two parameters are overconstrained by three independent data in each of the two chosen nuclei, as shown in Sec. 5. In Sec. 6 we propagate the estimated uncertainties to the calculation of matrix elements and lifetimes, with and without the effects of short-range repulsive interactions. Finally, we summarize our results and discuss future perspectives in Sec. 7.
2 Experimental benchmarks
In order to reduce the theoretical uncertainties, any nuclear model used in calculations should be benchmarked by as many weak-interaction data  as possible. Relevant weak processes are listed in Eqs. (5)–(11) below.
Two-neutrino double beta decay (),
is a second-order weak process () which probes the same mother and daughter nuclei as decay. It has been observed in several nuclei, thus providing a particularly important benchmark. Indeed, it was extensively demonstrated in  that the spread of QRPA calculations can be significantly reduced by constraining the nuclear model with the corresponding experimental decay lifetime (see  for earlier attempts). The data help to fix an important free model parameter, namely, the strength of the residual particle-particle interaction [24, 25], and thus to “calibrate” the QRPA estimates of . Despite the fact that the decay process probes only a subset of the intermediate states relevant for decay (i.e., only those with , via GT transitions), it is just the contribution to the total matrix element that reveals a pronounced sensitivity to , in contrast to the other multipole contributions . This observation justifies the aforementioned fitting procedure employed in .
First-order weak processes () related to decay can probe, in usual jargon, either the “first leg” of the decay (from the mother nucleus to the intermediate one) or its “second leg” (from the intermediate nucleus to the daughter one). Relevant examples for the first leg include the electron capture (EC) from a bound state (),
and the charge-exchange reaction via ,
as well as via ,
The second leg is instead probed by the decay,
by the charge-exchange reaction,
and by ordinary muon capture (C),
See also  for a recent discussion of these and other possible weak processes, including future (anti)neutrino-nucleus charged-current reactions at low energy . Clearly, any of the above first-order weak processes could be used to set useful constraints on the nuclear model. Indeed, using decay has been advocated as an alternative to decay for fixing the parameter in QRPA [29, 30, 31]; C data might be similarly used in the near future . However, one should be aware that these data are currently more sparse than for decay and, sometimes, have inherent problems or limitations, as discussed below.
|Nuclei||, first leg||, second leg|
Table 1 shows the current experimental status of the seven processes listed in Eqs. (5–11), for nine nuclei of interest for decay searches. Data on decay lifetimes exist for all these nuclei . Lifetimes for EC and decay have been measured only in three cases, , 116 and 128 (with states for the intermediate nucleus) [34, 35, 36, 37]. In one case (), the most recent EC datum  appears to be in conflict with the older one . Data on the charge-exchange scattering processes are also sparse. Available C data  are not particularly constraining, since they refer to the natural isotopic mixture containing the daughter nucleus; see however  for a comparison of QRPA calculations with C data, and  for preliminary C data in unmixed 82, and 150 daughter nuclei. Charge-exchange reactions involve analyses of spectral data which are, in general, more difficult to be interpreted and modeled than decay lifetimes [47, 48]. Data for exchange are available only for and 116 . In the latter case, the measured GT strength is in conflict with the one derived from EC . Data for exchange and are reported in , where the GT strength distribution is, however, normalized to the reference one  at small excitation energy, and thus it does not provide an entirely independent constraint. The reaction has been instead studied in several nuclei [39, 40], with emphasis on the GT strength distribution (rather than on its normalization). For , it should be noted that the recent data in  disagree with the data in , and are only in rough agreement with the EC data in .
Clearly, new and dedicated measurements are needed, both to solve the mentioned experimental discrepancies and to fill the missing entries in Tab. 1 [27, 43, 44]. In the meantime, one needs to select a (hopefully consistent) data set, in order to perform a meaningful comparison with theoretical calculations.
In this work we adopt the following approach: we ignore current data from the charge-exchange scattering processes (which, in several cases, either disagree with each other, or have no independent normalization, or provide poor constraints for our purposes), and we choose only those data which involve half-life measurements (rather than complex spectral analyses), namely, , EC, and decay. Our investigation is then restricted to two nuclear systems for which all such data exist, namely, and 116, which we shall often denote by the name of the “mother” nucleus (Mo and Cd, respectively). For , we discard the old EC datum, , in favor of the new (albeit unpublished) one . Table 2 shows the corresponding input data that will be used in our analysis, in terms of (for ) and of (for EC and ), where is the usual nuclear Fermi function. (Throughout this paper, .)
Although the (, EC, ) data are available also for A=128 (see Table 1), this nuclear system is left out of the consideration in the present work since the final nucleus Xe is rather strongly deformed. The change in the deformation from an almost spherical Te to a rather well deformed Xe ( ) cannot be reliably treated within the spherical QRPA employed here. Importance of such an effect has been demonstrated in Refs. [50, 51] for the case of the decay using the deformed QRPA with schematic forces.
3 Data versus theory with standard or no quenching
In the context of the QRPA, it has been convincingly shown in 
that the spread of theoretical calculations can be significantly reduced, in each
of the nine nuclei in Tab. 1, by fixing in such a way as to reproduce the
The processes occur through GT transitions, either at first order in (for EC, ) or at second order in (for ). Therefore, theoretical estimates of the associated (logarithmic) lifetimes need to be performed only for , and can then be scaled for as:
Within this work, QRPA calculations of the above lifetimes have been performed both in large basis (l.b., default choice) and in small basis (s.b.). The small basis consists of 13 single-particle levels (oscillator shells and 4, plus the orbits from ), while the large basis contains 21 levels (all states from shells ), in accordance with the choice of [53, 13]. The small set corresponds to particle-hole excitations, and the large one to about excitations.
An important output of QRPA calculations is the matrix element , whose modulus is probed by the observable according to
where is a calculable phase space factor, and the bare value of (1.25) is explicitly factorized out to make contact with previous notation . In QRPA calculations, typically starts positive for , then decreases and eventually changes sign as increases. The critical value where marks an infinite lifetime, . It turns out that is continuous across , while diverges locally. For increasing slightly beyond this critical point, the calculated energy of the first excited state decreases and eventually vanishes, inducing a breakdown (the so-called “collapse”) of the QRPA solution. QRPA calculations become thus less reliable in the vicinity of the critical and collapse points.
Figure 1 shows the matrix element as a function of for each of the two reference nuclei, in large basis. Similar results are found for small basis (not shown). In each panel, a vertical dotted line marks the critical value where flips its sign. The value of drops rapidly for , and the QRPA collapse is eventually reached. Both positive and negative values of may be phenomenologically acceptable in principle, although theoretical arguments suggest that . Determining the sign of is thus a relevant check of the theory.
The QRPA estimates of , as well as those of the , and lifetimes, are affected by various sources of uncertainties. In Sec. 5 we shall deal with the uncertainties related to the (a priori unknown) values of and , and to the size of the basis. However, even if and were perfectly known and the basis size were irrelevant, the approximation inherent to the QRPA approach would introduce further theoretical errors on each estimated lifetime. The assessment of these errors is obviously difficult and, to some extent, even arbitrary—but it is necessary to gauge the (dis)agreement between theoretical estimates and data. Our educated guess for the extra theoretical uncertainties (besides those related to , to , and to the basis size) is for both the EC and lifetimes, and for the lifetime. In the latter case, a larger relative error is assumed, due to the smaller (by a factor 2–3) calculated values of the corresponding matrix element as compared with the ones for the EC. Accordingly, we attach the following () theoretical errors to each logarithmic lifetime, for any fixed values of in any basis:
In the next two subsections we shall compare the data in Tab. 2 with the corresponding QRPA estimates for . It will be shown that, in none of the two reference nuclei, the QRPA results can be really made consistent with more than one datum at a time, within the quoted experimental and theoretical uncertainties. Moreover, it will become evident that higher values (e.g., ) can only worsen the situation.
3.1 Mo data versus QRPA ()
Figure 2 illustrates the comparison between Mo data and theoretical predictions for standard quenching () in large basis, as a function of . The upper, middle, and lower panels refer to the , EC, and logarithmic lifetimes, respectively. In each panel, the horizontal band represents the experimental datum at (as taken from Tab. 2), while the curved band represents the QRPA results, with theoretical spread as in Eqs. (15–17). Vertical dotted lines mark the critical value which separate the left, positive branch from the right, negative branch . The preferred ranges—where the experimental and theoretical bands cross each other—appear to be quite different in the three panels of Fig. 2. In particular, there is no overlap between the preferred ranges in the upper and middle (or lower) panel, while there is only a marginal overlap between those in the middle and lower panels. Agreement between data and theory is never reached for all the three observables at the same time.
If one choses the lifetime to fix (as advocated in ), then two preferred ranges are selected, one in the positive branch (around ), and the other in the negative branch (around ); see the upper panel of Fig. 2. Although both ranges are phenomenologically viable, the one in the positive branch is usually adopted on theoretical grounds . However, for , the theoretical EC () lifetime turns out to be significantly smaller (larger) than the experimental value. Similar problems occur for in the negative branch.
Alternatively, one might use the lifetime to fix (as advocated in [30, 31]). In this case, as evident from Fig. 2, one could get marginal agreement between both and EC observables around , but only at the price of underestimating the measured lifetime by a factor of . With one choice or another, it seems that current QRPA calculations fail to reproduce all the three independent lifetimes at the same time.
The above discrepancies would become stronger by increasing the GT strength from its standard quenched value () to its bare value (, not shown). For , according to Eqs. (12) and (13), the theoretical bands in Fig. 2 would be shifted downwards by (upper panel) or by (middle and lower panels). The preferred ranges of would then move to the right for and , and to the left for EC, thus destroying even the marginal agreement existing between and EC observables for . We conclude that, within the range , current QRPA calculations cannot reproduce the three lifetime data (nor, to some extent, any two among them) for any value of . These graphical results will be numerically confirmed in Sec. 5.
3.2 Cd data versus QRPA ()
Figure 3 is analogous to Fig. 2, but for Cd. The situation is very similar to Mo, and the same qualitative considerations apply, although the preferred ranges of are different. Also in this case, it is not possible to reconcile the QRPA estimates with the three independent lifetime data data for any value of , at fixed . The discrepancy becomes worse for (not shown).
4 Data versus theory with strong quenching
In the previous Section, we have shown that the QRPA fails to reproduce the three lifetimes in each of the two reference nuclei (Mo and Cd), as far as is taken in the usual range, . In particular, the discrepancy becomes worse as one moves towards the upper end of this range. Conversely, the discrepancy can be expected to become less severe (and hopefully vanish) for , corresponding to a “strong quenching” of the GT coupling.
Values of lower than unity, although rather unconventional in the QRPA literature, are not uncommon in NSM calculations. The NSM, being an ab initio approach, does not depend on phenomenological parameters such as , but of course retains the dependence on the axial coupling , with the associated quenching uncertainties. Although a quenched value seems to roughly provide the correct normalization of the GT strength, strongly quenched values may occasionally be needed to bring NSM calculations in agreement with data [15, 21]. It is fair to say that, in the NSM approach, one is not committed to a strict range for (such as ): any value is generally accepted, if the data require so.
In both the QRPA and the NSM approach, the origin and size of the GT quenching remains in part obscure and uncertain from a theoretical viewpoint, and the inferred values of fluctuate considerably in different data analyses, processes, and nuclei. Even for a fixed process and nucleus, it is not excluded that the quenching may be energy-dependent . Therefore, the common practice of adopting either the standard quenched value or the bare value may be unnecessarily restrictive. It is perhaps more sensible to treat as a free parameter of order unity, whose precise value needs to be constrained by the data themselves, rather than pre-assigned by theory—just as one does for . In the following, we thus adopt a purely phenomenological viewpoint, and show that specific choices of below unity (which will be more precisely derived in Sec. 5) can bring QRPA calculations in agreement with all the three lifetime data in each of two reference nuclei.
4.1 Mo data versus QRPA ()
Figure 4 is analogous to Fig. 2, but for the strongly quenched value . We anticipate that this value provides the best overall agreement of QRPA calculations (curved bands) with the data (horizontal bands). Around , all bands cross each other in the three panels. No such common crossing occurs in the negative branch, as also confirmed by numerical explorations. Besides selecting the positive branch, the data appear to prefer a particle-particle strength () sufficiently far from the the critical and collapse values, where the QRPA estimates become less reliable. The theoretical band in the lower panel is rather steep around , and can thus provide, together with the experimental datum, both an upper and a lower bound to ; the upper (lower) bound can also be enforced by the (EC) observable, as evident in the upper (middle) panel. In perspective, a reduction of the EC error in the middle panel would be beneficial to better probe this strong-quenching scenario.
4.2 Cd data versus QRPA ()
Figure 5 is analogous to Fig. 3, but for the strongly quenched value . A good overall agreement between theory and data is reached in a broad range –0.6. It is interesting to note that this range could be significantly restricted if the experimental errors of the EC datum  in the middle panel were reduced by a factor of two or more. Also in this case, the data unambiguously select the positive branch, and keep far from the critical and collapse points.
Strong quenching appears to provide a phenomenological solution to the well-known overall discrepancy between QRPA results and lifetime data. This solution is nontrivial because: (1) two free parameters enable to reproduce very well, within uncertainties, three independent data (in each of two different nuclei); (2) the positive branch , which is favored by theoretical arguments, is unambiguously selected by the data; (3) the preferred values of are far enough from the critical and collapse values. Such data-driven features seem to be more than accidental facts, and suggest that might be a realistic option within the QRPA. More accurate lifetime data (especially for EC and, to some extent, for decay), as well as further charge-exchange reaction data (not considered in this work) should provide additional probes of the strong quenching solution.
This solution is admittedly unconventional in the context of QRPA, where has been customarily taken within the range . It may be that strong quenching is associated to other effects, whose degrees of freedom might be traded for milder variations of . However, if new free parameters are added to and , the data set must also be enlarged to provide meaningful and nontrivial constraints—not much would be learned, in general, by fitting data with parameters.
For the sake of simplicity, in this work we do not explore more elaborate scenarios with additional data and further QRPA degrees of freedom. We just take for granted the indication in favor of , and perform quantitative fits to three selected data (the , EC, and lifetimes) via two parameters . We shall thus obtain an overconstrained parameter space, used for subsequent calculations in Sec. 6. Despite the above caveats, this approach represents a step forward with respect to previous attempts, which aimed at reducing the model uncertainties in QRPA by fitting a single datum (either or ) through a single parameter () at fixed .
5 Overconstraining the () parameters
We perform a least-square fit to the three data , , and in terms of the two free parameters . The function to be minimized is defined as
where all the ingredients have been defined in the previous Sections. Asymmetric experimental errors (see Tab. 2) are properly included by choosing either the upper or lower error, according to the sign of the difference . The minimum search is performed by numerical scan over a dense grid in the rectangle . Given three data and two parameters, one expects for a proper fit. The expansion around the best-fit values of at provides then the - contours for such parameters . In the following, we show the main results both in graphical and tabular form.
Figure 6 shows the results of the fit in large basis. In each of the two panels (corresponding, from top to bottom, to Mo and Cd) a dot marks the best-fit point, surrounded by the 1, 2 and contours. Vertical dotted lines separate the positive and negative branches of . In both panels, the allowed regions are fully contained in the positive branch, thus confirming quantitatively the theoretical arguments in favor of . The best-fit points are safely far from extremal values of (0 and ), but the allowed regions may extend towards one of them. In particular, the allowed range of is somewhat squeezed towards the critical value for Mo, while it extends towards zero for Cd at . More accurate experimental data (especially from EC and, to some extent, from decay) would be useful to shrink such ranges, as discussed in Sec. 4, and might thus prevent the occurrence of nearly extremal values of . Concerning , strong quenching is definitely preferred at in both cases.
We emphasize that “overconstraining the parameters” is equivalent to state that, in each of the Mo and Cd reference nuclei, our scenario makes one prediction which is experimentally verified. Figures 7 and 8 illustrate this statement via the bands individually allowed by , EC and data for Mo and Cd, as obtained by a breakdown of the three contributions in Eq. (18). Any two bands can be used to constrain in a closed region (the “prediction”), which is then crossed by the third independent band (the “experimental verification”).
The numerical results of the global fit in large basis are summarized in Table 3. The fit quality is very good in all cases () and the best-fit values for the three lifetimes are in striking agreement with the corresponding data in Tab. 2, which are repeated for convenience in Table 3 (in square brackets). The best-fit values and ranges () for and are also reported. (The values adopted in Figs. 4 and 5 are just taken from Table 3.)
|Mo||1.26||18.82 [18.85]||4.09 [3.96]||4.66 [4.60]||0.733||0.741|
|Cd||0.12||19.49 [19.48]||4.35 [4.39]||4.63 [4.66]||0.493||0.843|
We have repeated the analysis in small basis, with similar results. The graphical results are omitted, while the numerical ones are reported in Table 4. The quality of the fit is very good also in this case. The allowed ranges for and in small basis (Tab. 4) are somewhat different from those in large basis (Table 3), but with similar features. In particular, the allowed range is in the positive branch, and the general trend in favor of is confirmed. We conclude that the main results obtained so far do not change qualitatively with the size of the basis.
|Mo||1.11||18.82 [18.85]||4.08 [3.96]||4.67 [4.60]||0.862||0.745|
|Cd||0.03||19.49 [19.48]||4.37 [4.39]||4.65 [4.66]||0.540||0.815|
6 Implications for decay
In the previous Section we have obtained allowed regions in the parameter space . In this Section we study how such regions affect the QRPA calculation of decay, after recalling some basic features of this process.
The processes that we have considered so far occur only via GT transitions through intermediate states. The leading contribution to the amplitude of the neutrinoless double beta decay also comes from the GT-type transitions which, however, proceed through intermediate states of all, but , multipolarities. In addition, there are Fermi and (small) tensor contributions to the matrix element,
where the dependence on and is made explicit.
Figure 9 shows the relevant components of the matrix elements as a function of in large basis, and including short range correlations, which will be shortly discussed below. Since the QRPA calculation is computer-intensive, is varied only within the relevant range shown in Fig. 6. Note that the leading component shows significant variations with , so that any constraint on this parameter (such as those derived in the previous Section) helps to reduce the spread of QRPA estimates of decay. Results qualitatively similar to Fig. 9 are obtained for small basis, or without short range correlations (not shown).
Given the QRPA results in Fig. 9, the matrix element can be computed for any relevant value of and through Eq. (18). In order to make contact with the notation in Ref. , we shall actually rescale the matrix element as
The lifetime reads then
where the proportionality factor [y] is numerically given by
for a reference Majorana mass meV. For different values of , one just rescales .
For any given value of , calculations of are affected not only by the size of the basis (either large or small), but also by uncertainties which are peculiar of the process, namely, those related to the important issue of short range correlations (s.r.c.). These correlations account for the well-known fact that the nucleon-nucleon interaction becomes strongly repulsive at small internucleon distances. This in turn must lead to strong suppression of the relative-motion wave function at small distances (s.r.c. effects). Short range correlations are explicitly included neither within the QRPA nor within the NSM. They are instead introduced ad hoc directly into the neutrino potential via a multiplicative factor (the square of a correlation function). One of the most popular is the Jastrow-like correlation function  which has been used in the previous calculations [53, 13] and is also used in this work. We shall thus present results in four cases, corresponding to either large or small basis, with or without the Jastrow-like s.r.c. effects.
In each of the four cases, the effect of the uncertainties on is estimated by marginalization , taking into account the fact that the same fixed value for the matrix element may be realized by different (“degenerate”) couples of values . More precisely, given the function defined in the previous Section, and for a fixed value , we define a marginalized function,
over the degenerate set of obeying
The minimization of , and the expansion around the minimum at , provide the correct best-fit values and ranges for , respectively. Since we are interested in , we perform a numerical marginalization over a dense, rectangular grid covering only the ranges of .
Tables 5 and 6 provide an overview of the derived ranges for at 1, 2 and (in large and small basis), with and without the effect of s.r.c., respectively. We also report the corresponding ranges for the measurable (log) lifetime , at the reference value meV. Note the ranges are generally asymmetric and do not scale linearly, in part as a consequence of the original one-sided limits at either 0 or (see Fig. 6). By comparing the results in Tables 5 and 6, it appears that the basis size is not the major source of systematic uncertainties. Conversely, the inclusion or exclusion of s.r.c. effects always induce changes .
Figure 10 shows an overview of QRPA results for the nuclear matrix elements (including s.r.c. effects) in three different cases for each nucleus. From left to right, the first two cases correspond to the ranges from Table 5, in large and small basis, respectively. The third case correspond to the results previously obtained in  for (with correspondingly smaller error bars, due to the fixed value). Remarkably, such results for  differ by from those obtained in this work, in spite of a marked difference in the central values of and .
Summarizing, in each of the two nuclei examined it is possible: () to fit very well three data (, EC, ) with two parameters (), provided that ; () to exclude the negative branch ; and () to derive robust ranges for observables. There remains a relative large uncertainty on the matrix element, associated with the size of short range correlation effects. Unfortunately, s.r.c. effects are peculiar of decay and are not constrained at all by the (, EC, ) data considered in this work.
7 Conclusions and Perspectives
It was shown in  that, by fitting in order to reproduce in calculations the corresponding experimental decay lifetimes, the sensitivity of calculated matrix elements to other ingredients of the QRPA, such as the basis size, can be successfully removed. Also, it was shown that the sensitivities of the results to gets much milder than one could naively expect. There are also different proposals for fixing , for instance, by reproducing the single beta decay observables as advocated in [29, 30]. By fitting to reproduce the lifetimes of the ground states of the intermediate nucleus one gets the results which are similar to the ones obtained in , but the EC or lifetimes are not reproduced. In this paper we have tried to reconcile all these data (available for the two nuclei Mo and Cd) by letting to be a free parameter of the model. In each nucleus, we have then found systematic indications in favor of strong quenching , and we have been able to overconstrain two parameters with three lifetime data (, EC, ), as well as to fix the sign of ().
The quenched values of for and nuclear systems obtained in this work ( and , respectively), although a bit unusual, are not much below the typical range –1.0 (corresponding to the quenching factor –0.8) used within the NSM for lighter nuclei . Even stronger quenching (corresponding in our notation to ) has been called for in shell model calculations [55, 56, 57] of the Gamow-Teller strength for nuclei in the region of , to which the systems considered in the present work are close.
The physical origin of the quenching of has been discussed in the past. One explanation  assigns this effect to the -isobar admixture in the nuclear wave function. Another—more generally accepted—explanation  assigns the quenching to the shift of the Gamow-Teller strength to higher excitation energies due to the short range tensor correlations. In light nuclei the quenching found in M1 transitions reduces from its bare value () to the in-medium one (). But the actual quenching in nuclear structure calculations can depend as on the detailed nuclear environment as on the truncations inherent to the model such as, for example, the basis size. Therefore, it appears useful to revisit the theoretical explanations of quenching, in order to check if and how they can cover cases with , as those emerging from our phenomenological analysis.
From the experimental viewpoint, it has been already mentioned that future EC data [43, 44] will be especially relevant in improving the parameter constraints. Moreover, the strong quenching of the axial vector coupling constant should be observed not only in single and double beta decays, but also in M1 transitions. Therefore, the study of charge-exchange reactions as , , (He,) and (He) [21, 41, 48] can shed new light on this issue. It is imperative, however, that the data are analyzed with no prior or hidden hypotheses about the GT coupling .
In conclusion, we think that the results of this work offer a novel possibility to reconcile QRPA results with experimental data, which deserves further discussions and tests, and warrants a revisitation of the quenching problem from a new perspective. By the present analysis, we are able to assign in a controlled manner theoretical uncertainties to the calculated matrix elements for the decay. Remarkably, our present results for agree within the error bars with those obtained in  for .