Double β Decay and the Axial Strength

Quenching of the weak axial strength g_A is discussed and relations of this quenching to the nuclear matrix elements of double beta decays are highlighted. An analysis of Gamow-Teller transitions in the mass range A=62-142 is presented and its results are compared with those of many previous works. The enhancement of the axial charge is discussed for first-forbidden pseudoscalar beta transitions. Higher-forbidden beta transitions are introduced and their role in determining the effective value of g_A is examined, in particular from the point of view of rge beta-decay half-lives and the shapes of electron spectra of forbidden non-unique beta transitions.


INTRODUCTION
Double β decay (ββ decay) has been under intensive discussion for the last decades from the point of view of both nuclear theory and ββ-decay experiments. The interesting decay mode is the neutrinoless ββ (0νββ) decay which is mediated, e.g., by a massive Majorana neutrino, which is exchanged between the two decaying nucleons. Thus the 0νββ decay implies also the breaking of lepton-number conservation. The high stakes involved in the detection of 0νββ decay have made the associated theoretical and experimental aspects highly important, in particular the calculation of the involved nuclear matrix elements (NMEs). The NMEs of 0νββ decays have been computed for decades using a number of different nuclear-structure approaches (for older calculations see the review [1] and for the more recent ones see [2,3]). Most of these calculations have been done by using the framework of the QRPA (quasiparticle random-phase approximation [4]). Also many calculations have been performed using the interacting shell model (ISM, see e.g., [5][6][7]) and the microscopic interacting boson model (IBM-2, see e.g., [8]).
Besides the problems with the calculations of the NMEs of 0νββ decay there is an other severe problem, namely the one related to the value of the weak axial coupling g A . For low-energy processes, like the β decay or the two-neutrino ββ (2νββ) decay [1], the nucleonic charged weak current is simply where N i (N f ) is the initial-state (final-state) nucleon spinor and γ s are the usual Dirac matrices.
Here g V = 1.0 is the weak vector coupling and its value is protected by the CVC (Conserved Vector Current) hypothesis [9]. The value g A = 1.27 of the weak axial coupling has been obtained from the decay of free neutron [10], N i = n in (1), into free proton, N f = p in (1). For the 0νββ decay the values of these weak couplings are altered by the high energy scale (q ∼ 100 MeV) of the exchanged momentum q between the decaying nucleons and the Majorana neutrino. In addition, induced currents become involved in the decay (see e.g., [11]). The evolution of the value of g A with the magnitude of the exchanged momentum q has been discussed in Menéndez et al. [12] and an extensive discussion of the effective value of g A has been carried out in the review [13]. For the two-neutrino ββ decay from the initial 0 + state (0 + i ) to the final 0 + state (0 + f ) the decay rate is directly proportional to the fourth power of g A as given by where is the nuclear mass difference between the initial and final 0 + ground states, M i is the mass of the initial nucleus, m e the electron rest mass and E(1 + n ) is the absolute energy of the nth 1 + state. Furthermore, g A M L (1 + n ) is the left-branch NME and g A M R (1 + n ) the right-branch NME for the nth 1 + state, as shown schematically in Figure 1. The NMEs M L (1 + n ) and M R (1 + n ) are the usual Gamow-Teller NMEs. In this case the low-momentum form (1) of the nucleon current is applicable and the quenching of g A can be studied through allowed Gamow-Teller β decays of low-energy nuclear states, as described in section 3 of this article.
For the 0νββ decay from the initial 0 + state (0 + i ) to the final 0 + state (0 + f ) the decay rate is given by where the Fermi NME M (0ν) F is proportional to g 2 V and the Gamow-Teller and tensor NMEs, M (0ν) GT and M (0ν) T , are proportional to g 2 A . The Gamow-Teller NME is the leading one and has the constitution where the sum is over all multipole states J π in the intermediate nucleus and the effective axial couplings for 0νββ decay, g A,0ν (J π ), are, in principle, multipole dependent. All the details of the 0νββ-decay transitions are included in the operator GT (J π ). The low-q limit of these couplings is where q is the exchanged momentum. For the J π = 1 + multipole this low-q limit is the "usual" axial coupling relevant for the Gamow-Teller and 2νββ decays. For the sake of simplicity, the notation g A will be used also for the other multipoles g A (J π ). The g A for higher multipoles can be studied through half-lives and electron spectral shapes of forbidden β decays, as discussed in section 4 of this article.

NUCLEAR MODELS
In this section we briefly describe the many-body aspects of the nuclear models which are mentioned later in this article. These models have been used to study the β-decay and ββ-decay NMEs and the associated effective values of the axial coupling. It should be noted here that these are not the only models that can (potentially) describe these features. These other nuclear models can be based on modern energy-density functionals or thermofield-dynamics formalism, as also on Monte-Carlo shell model, etc. A comprehensive list of these nuclear models and the associated references are given in section 5.4. of the very recent review article [14].
• ISM: The ISM (interacting shell model) is a many-body framework that uses a limited single-particle valence space, typically one harmonic-oscillator major shell or one nuclear major shell. In the ISM one forms all the possible manynucleon configurations in a given single-particle valence space, each configuration described by one Slater determinant, and diagonalizes the nuclear (residual) Hamiltonian in the basis formed by these Slater determinants. In this way the manybody features are taken into account exactly but only in a restricted valence space, typically leaving out one or two spinorbit-partner orbitals from the model space. • Spherical pnQRPA: The proton-neutron version of the QRPA (pnQRPA) uses two-quasiparticle excitations that are built from a proton and a neutron quasiparticle. Here only the spherical pnQRPA, based on a spherical nuclear mean field, is described. The pnQRPA model framework enables description of odd-odd nuclei starting from an even-even reference nucleus where the quasiparticles are created, e.g., through the BCS (Bardeen-Cooper-Schrieffer) procedure [15]. The advantage of the pnQRPA theory is that it can include large single-particle valence spaces in the calculations and there arise no problems associated with spin-orbit-partner orbitals since they can easily be accommodated in the valence space. On the other hand, the pnQRPA has a limited configuration space. Schematic or G-matrix-based bosonexchange Hamiltonians have widely been used in the pnQRPA calculations. A frequently used extension of the pnQRPA framework is the renormalized QRPA (RQRPA) [16,17]. One particular problem with the pnQRPA calculations is the determination of the value of the particle-particle interaction parameter g pp , used to scale the particle-particle part of the proton-neutron two-body interaction matrix elements [18,19]. Usually the value of this parameter has been determined by using β-decay or 2νββ-decay data. The particle-hole parameter, g ph , of the proton-neutron two-body interaction is usually determined by adjusting the parameter such that the phenomenological or experimental energy of the Gamow-Teller giant resonance is reproduced [20,21]. The spherical pnQRPA has been applied to the description of ββ decays (see e.g., [22][23][24][25]) and it has also been used in β-decay studies (see e.g., [26][27][28]). • MQPM: The microscopic quasiparticle-phonon model (MQPM) describes states of odd-A nuclei starting from the adjacent even-even reference nuclei. The MQPM states are generated by combining proton or neutron one-quasiparticle excitations of the reference nucleus with three-quasiparticle excitations built by coupling a proton or neutron quasiparticle to a QRPA phonon. A QRPA phonon is a proton-protonplus-neutron-neutron excitation of an even-even reference nucleus. The MQPM creation operator creates a state |kjm in an odd-A nucleus by the action with the excitation operator given by where Q † ω is a QRPA phonon creation operator [15] and the a † operators create BCS quasiparticles. Since the MQPM states (8) contain the three-quasiparticle components special care should be taken when solving the MQPM equations of motion for the amplitudes X k n and X k aω in order to handle the overcompleteness and non-orthogonality of the quasiparticlephonon basis. For details see Toivanen and Suhonen [29,30].
• IBM-2: The interacting boson model (IBM) is a theory framework based on s and d bosons which have as their microscopic paradigms the 0 + and 2 + angular-momentumcoupled collective fermion pairs present in nuclei. An extension of the IBM is the microscopic IBM (IBM-2) where the proton and neutron degrees of freedom are explicit. The IBM-2 is a sort of phenomenological version of the ISM, containing the seniority aspect and the restriction to one magic shell in terms of the single-particle valence space. The Hamiltonian and the transition operators are constructed from the s and d bosons as lowest-order boson expansions with coupling coefficients to be determined by fits to experimental data on low-lying energy levels and E2 γ transitions. However, the fitting does not use the spin or isovector data available from β decays. The extension to the microscopic interacting bosonfermion model (IBFM-2) [31] enables the description of odd-A nuclei. Here problems arise from the interactions between the bosons and the extra fermion in the Hamiltonian, and from the transition operators containing a host of phenomenological parameters to be determined in some meaningful way.

QUENCHING OF G A IN GAMOW-TELLER β DECAYS
The half-life of allowed Gamow-Teller β decays can be written as whereC is the integrated shape function and the constant κ has the value [32] κ = 2π 3h7 ln 2 m 5 e c 4 (G F cos θ C ) 2 = 6147 s , Frontiers in Physics | www.frontiersin.org θ C being the Cabibbo angle. In order to simplify the formalism it is usual 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, where W e is the total energy of the emitted charged lepton (electron or positron), p e is the electron/positron momentum and W 0 is the end-point energy, i.e., the total energy of the emitted leptons, and hence the maximum energy of the emitted electron/positron. With the unitless quantities the integrated shape function can be expressed as where F 0 (Z f , w e ) is the Fermi function taking into account the interaction of the final-state charged lepton with the positive charge of the final nucleus (with charge number Z f ). The integration is performed over the total scaled energy of the emitted electron/positron. The shape factor C(w e ) of Equation (11) can be expressed for the pure 1 Gamow-Teller J i → J f = J i ± 1 transitions as where M GT is the Gamow-Teller NME (assumed to be real, as usual) and J i (J f ) the angular momentum of the initial (final) state. Here the axial coupling g A is the low-q limit (6). Since the shape factor (12) does not depend on the lepton variables it can be taken out of the shape function (11) and the rest constitutes the universal phase-space factor usually quoted for the allowed (Fermi and Gamow-Teller) β decays. At this point it should be noted that a new method of calculation of these phase-space factors was introduced in Stoica et al. [33]. These phase-space results contribute to calculations of β-decay rates for nuclei far away from the stability line. Using the phase-space factor f 0 , the half-life (9) can be expressed in a more familiar form using the reduced transition probability B GT : What is usually quoted in literature are the log ft values log ft = log 10 (f 0 t 1/2 ) , which are 10-base logarithms of the product of the half-life and the universal phase-space factor. By using Equations (14) and (15) one can extract from the experimental log ft value the magnitude of the experimental Gamow-Teller NME in the form |g A M GT |. Here one has to note that only the product of the NME and the weak axial coupling g A can be extracted. These products can be extracted from both the left-branch, |g A M GT (left)|, and right-branch, |g A M GT (right)|, β decays as shown schematically in Figure 1. These extracted products are shown in Tables 1, 2 for the left (third column) and right (fifth column) transitions from/to the central nucleus of the triplet of nuclei of column two. A sample of these triplets are shown in Figure 2. There four mass triplets A = 110, 116, 128, 136 are displayed with a central nucleus feeding the lateral ones, thus constituting the left-branch and right-branch of β transitions. These triplets host also double beta decays, with β − β − decays for the masses A = 110, 116, 128 and double electron-capture (EC) decay for A = 136. For the double EC decay the log ft of the right-branch decay is not known and thus this triplet is omitted from the subsequent decay analysis whereas all the other triplets are included. In the figure it is also shown that there is no available data on the decays of the low-lying 2 − states in these triplets. These decays are relevant for the firstforbidden unique β transitions of section 4.2, mediated by a rank-2 tensor. They are also quite important for the neutrinoless ββ-decay NMEs.
As can be seen in Tables 1, 2, the extracted experimental leftbranch and right-branch NMEs differ sometimes considerably from each other and these differences are rather erratic. In order to stabilize this behavior one may use the geometric mean g m A M m GT of the left and right NMEs: where the geometric mean of both the left-branch and rightbranch NMEs and axial couplings has been computed. The corresponding experimental NMEs are shown in column seven in Tables 1, 2. These NMEs do not behave quite as wildly as the individual left-branch and right-branch NMEs.
The experimental left-branch, right-branch and mean NMEs can be compared with the corresponding computed NMEs listed in columns four, six and eight in Tables 1, 2. The computed NMEs are obtained by using the pnQRPA (see section 2) in the following single-particle model spaces: These single-particle valence spaces have been chosen such that they are expected to capture the relevant features of the low-lying states in the triplets of nuclei, such that the involved left-branch and right-branch β transitions are described as well as possible within the framework of the pnQRPA model. In the course of the calculations the pairing parameters were fitted to reproduce the experimental pairing gaps extracted from the available data [34] on nucleon separation energies. The particle-hole parameter g ph was fitted to reproduce the empirical location of the giant Gamow-Teller resonance (see [35] for fitting also to more general spin-multipole resonances).
The particle-particle parameter g pp and the strength of the axial coupling were left as free parameters in the calculations. According to the Gamow-Teller β-decay study [36], performed for a number of nuclei in the mass range A = 100 − 136, a good overall value for the particle-particle parameter is g pp ≈ 0.7.
In the present study we vary the values of this parameter in the range g pp = 0.50 − 0.85 to have a feeling of the effects of the variation of g pp on the values of the computed NMEs. This variation is shown in Tables 1, 2, in columns four and six for the left-branch and right-branch NMEs, and in the last column for the mean NMEs (16). The variations in the values of the individual NMEs are usually (much) larger than in the values of the mean NMEs, thus justifying the use of the geometric mean of the left-branch and right-branch NMEs, instead of the individual NMEs. For the mean strength of the axial coupling we have taken the constant overall value g m A = 0.6 which was found to be a good average value in the study [36] for the A = 100 − 136 mass range. Our adopted values of g pp = 0.67 (plus the variation in g pp described above) and g A = 0.6 are also in good agreement with the average values of these parameters deduced from the extensive analyses of the Gamow-Teller β decays conducted in the study [37].  Figure 3) the magnitude of the computed mean NME is almost everywhere slightly below that of the experimental one whereas in the mass region A = 98 − 110 ( Figure 4) the magnitude of the computed mean NME is above that of the experimental mean NME. In both mass regions the staggering of the computed and experimental mean NMEs is similar. In the mass range A = 112 − 124 ( Figure 5) the experimental mean NME is mostly larger than the computed one but the staggering of both are quite similar. For the heaviest triplets, A = 126 − 142 (Figure 6), the values of the experimental mean NMEs are larger than those of the computed ones, the difference increasing with increasing mass. Still, in the staggering similarities between the two NMEs are to be seen.
In Figures 3-6 also the magnitudes of the proton-neutron two-quasiparticle NMEs are presented (solid line with open squares) for comparison. The corresponding spin-orbit-partner configurations give the strongest contributions to the mean NMEs and these configurations are Here it should be noted that these configurations are not the same as the leading contributions quoted in the study [38] since there the two-quasiparticle configurations closest to the respective Fermi surfaces were taken. The magnitudes of the twoquasiparticle mean NMEs are far too large implying a strong quenching of the mean NME when going from the simple twoquasiparticle approximation to the more sophisticated pnQRPA model. This large reduction can be associated with spin-isospin correlations missing in the two-quasiparticle approximation but taken into account in the pnQRPA framework, as discussed extensively in Ejiri and Suhonen [38]. In the two-quasiparticle NMEs there are also some staggering in the magnitude, but usually (much) less than in the pnQRPA and experimental NMEs. Sometimes this staggering is out of phase with the pnQRPA staggering (and the experimental one) indicating that the spinisospin correlations are crucial in order to reproduce the trends of the experimental mean NMEs. The error bars of the magnitudes of the computed NMEs, caused by the variation g pp = 0.50 − 0.85, are shown as hatched areas in Figures 3-6. In general, the relative variation is rather modest, in particular in the mass range A = 112−124 ( Figure 5). The largest absolute variations are seen around the masses A = 100, A = 106 and A = 108. Generally, the upper limit of the hatched area is close to the best NME value (solid line with open circles) since this maximum NME is obtained for the lowest value g pp = 0.50 and for this value of g pp the NME is already saturated close to its maximum value at g pp = 0.0.   Table 1, last two columns) for the mass range A = 62 − 82. The hatched area describes the uncertainty of the calculations (the range g pp = 0.50 − 0.85 in Table 1). Also the NMEs in the two-quasiparticle approximation have been plotted for comparison (note the scaling of these NMEs by a factor one-half in order to save vertical space in the figure).
To give yet an other view to the comparison of the experimental and computed NMEs, one can calculate the mean value and the RMS (Root-Mean-Square) deviation of the NMEs in the four mass ranges of    Figure 3 for the mass range A = 112 − 124, with the mean NMEs corresponding to Table 2.
One can compare these values of effective g A (g eff A in Figure 8) with other recent calculations in different theory frameworks. This has been done in    48 Ca [7], cross at A = 48 in the figure). The studies of Faessler et al. [40] and Suhonen and Civitarese [41], marked by vertical black and green line segments, respectively, are pnQRPA studies of nuclei ( 100 Mo, 116 Cd, and 128 Te) with available data on the rates of both β and 2νββ decays The analyses were performed in the aim of constraining the values of both g A and g pp simultaneously. The solid line with the zig-zag behavior is the result of the pnQRPA analysis of Suhonen [39] in the aim to constrain the possible values of g A and 0νββ NMEs in order to have a feeling of their effects on the sensitivity of the present and future 0νββ experiments.
The red (ββ ISM) and blue (ββ IBM-2) dotted lines show the results of the 2νββ analyses of Barea et al. [8] yielding the fits  close to the present effective values g eff A ≈ 0.7 and g eff A ≈ 1.0. In the A = 126 − 142 region the ISM analyses of Caurier et al. [5] and the IBM-2 analyses of Barea et al. [8] produce smaller values of g eff A ≈ 0.6 − 0.7 and there is a strong tension between the different ISM analyses. The present results are in line with the pnQRPA results of Suhonen and Civitarese [41] and Faessler et al. [40] for 116 Cd but far off from the results of these studies for 128 Te. It seems that the fit to the average value of the mean experimental β-decay NMEs brings the results of the pnQRPA analysis closer to those of the ISM, in particular for the heavy end of the presently discussed nuclei. The mass range A = 98 − 110 is the most interesting: Here a very small value g eff A ≈ 0.5 is required both by the present study and the ISM study of Siiskonen et al. [44]. Also the IBM-2 result of Barea et al. [8] is consistent with these results whereas the interpolated ISM result of Barea et al. [8] and the results of Suhonen and Civitarese [41] and Faessler et al. [40] for 100 Mo are in the range g eff A ≈ 0.7 − 0.8.

QUENCHING OF G A IN FORBIDDEN β DECAYS
In forbidden β transitions the low-q limit (5) can be studied for different multipoles J π . In this section we denote all these weak axial couplings as g A for simplicity.

Forbidden Non-unique β Decays
The half-life of a forbidden non-unique β decay can be written in the same way as that of the allowed β decay in (9). The corresponding integrated shape function can be expressed as written in Equation (11) but the shape factor C(w e ) of Equation (12) has to be replaced by a much more complicated expression: C(w e ) = k e ,k ν ,K λ k e M K (k e , k ν ) 2 + m K (k e , k ν ) 2 − 2γ k e k e w e M K (k e , k ν )m K (k e , k ν ) , (22) where the notation of unitless leptonic quantities was discussed in the context of Equation (11). The factor λ k e contains the generalized Fermi function F k e −1 [45] as the ratio Z f being the charge number of the final nucleus. The indices k e and k ν (k e , k ν = 1,2,3...) are related to the partial-wave expansion of the electron (e) and neutrino (ν) wave functions, K is the order of forbiddenness of the transition, and γ k e = k 2 e − (αZ f ) 2 , α ≈ 1/137 being the fine-structure constant. The nuclearphysics information is carried by the quantities M K (k e , k ν ) and m K (k e , k ν ), which are conglomerations of different NMEs and leptonic phase-space factors. For more information on the integrated shape function, see Behrens and Bühring [45] and Mustonen et al. [46].
The shape factor C(w e ) can be decomposed into vector, axial-vector, and mixed vector-axial-vector parts. In this decomposition the shape factor is where the quantities C A , C V and C VA are complicated expressions including Coulomb functions, nuclear matrix elements, etc. This is the decomposition used in Haaranen et al. [47,48] and Kostensalo et al. [49] and it enables studies of the effects caused by the ratio g V /g A on the shape factor. Integrating Equation (24) over the electron kinetic energy, we get an analogous expression for the integrated shape function (11) where the factorsC A ,C V andC VA in Equation (25) do not depend on the electron kinetic energy.

First-Forbidden Non-unique β Decays
For the first-forbidden non-unique β decays the shape factor (22) has to be supplemented with a J = |J i − J f | = 0 term C (1) (w e ) [26,45,50,51], where J i (J f ) is the initial-state (final-state) spin of the mother (daughter) nucleus. In the papers (see [52][53][54]) the shape factor for first forbidden decays has been given in the form C(w e ) = k + kaw e + kb/w e + kcw 2 e , where k, ka, kb, and kc contain the Coulomb functions, nuclear matrix elements and weak coupling constants. Writing the shape factor in this form is often useful for comparing the theoretical and experimental shape factors. It should be noted that the shape factors of Equations (24) and (26) are different but equivalent ways of expressing the shape factor but in the first one the terms with the same product of weak coupling constants have been collected, while in the latter the terms with the same power of electron kinetic energy w e are combined.

Forbidden Unique β Decays
An important special case of forbidden β decays are the forbidden unique decays for which the theory simplifies considerably. The classification of forbidden unique decays by change in parity and angular momentum is presented in Table 3.
For the non-unique case these are presented in Table 4.
For unique β-decay transitions the half-life (9) can be expressed analogously to (14), valid for the Gamow-Teller β transitions. Thus we have where f Ku is the phase-space factor and g A M Ku is the nuclear matrix element. The phase-space factor f Ku for the Kth forbidden unique β ± decay can be written as and the corresponding shape factor can be written as (see e.g., [15,55]) The notation was explained in the context of the allowed and forbidden non-unique β transitions, around Equations (11) and (22) and the ratio λ k e was given in Equation (23). The NME in (27) can be expressed as where the factors M Ku (ab) are the single-particle matrix elements and the quantities (ψ f ||[c † acb ] K+1 ||ψ f ) are the one-body transition densities with ψ i being the initial-state wave function and ψ f the final-state wave function. The operator c † a is a creation operator for a nucleon in the orbital a and the operatorc a is the corresponding annihilation operator. The single-particle matrix elements are given (in the Biedenharn-Rose phase convention [15]) by where Y K is a spherical harmonic of rank K, r the radial coordinate, and a and b stand for the single-particle orbital quantum numbers. The NME is given explicitly in Suhonen [15].

First-Forbidden β Decays
The shape factor C(w e ) of Equation (22) contains complicated combinations of both (universal) kinematic factors and nuclear form factors [26][27][28]. The nuclear form factors can be related to the corresponding NMEs using the impulse approximation [45].
where r is the coordinate vector and p e (q ν ) is the electron (neutrino) momentum, and the square brackets in the O(2 − ) operator denote angular-momentum coupling. The nuclear matrix elements related to the first forbidden decays are suppressed relative to the Gamow-Teller matrix elements by the small momenta of the leptons, the large nucleon mass, and the small value of the fine-structure constant α. The quenching of the effective value of g A in first-forbidden decays in the lead region was observed in the late 1960's in a study by Bohr and Mottelson [56]. In this study the socalled ξ approximation, discussed in Behrens and Bühring [45], Schopper [50], and Bohr and Mottelson [56], was adopted. The wave functions were assumed to be dominated by certain single-particle configurations around the double-magic 208 Pb nucleus. For the decays mediated by the rank-1 operators O(1) in Equation (33), the obtained effective values of the weak coupling constants were The effective value of the vector coupling constant deviates significantly from the canonical value g V = 1, pointing to large nuclear-model-dependent effects. Also the value of the axial coupling is quite low. Next we discuss more recent and complete studies of the quenching or enhancement of the weak couplings in the firstforbidden β transitions.

Rank-0 Tensors
The mesonic enhancement of the γ 5 NME (σ · p e of Equation (32) in the non-relativistic limit) was discussed in Kubodera et al. [57] for the pseudoscalar 0 + ↔ 0 − β transitions. The enhancement of the γ 5 NME stems from the renormalization of the pion-decay constant and the nucleon mass M N in nuclear medium [58] and exchange of heavy mesons [59,60]. The γ 5 NME is accompanied by a pseudo-scalar axial coupling g A (γ 5 ). The effective value of this coupling can be expressed in terms of an enhancement factor ǫ MEC through the relation where g A is the usual axial-vector coupling strength. Related to this, a fundamental enhancement factor ǫ MEC = 1.4 − 1.7, insensitive to nuclear-structure aspects, was predicted [61,62].
In the paper of Kirchbach and Reinhardt [63] enhancements of 40 − 50% were predicted for singe-particle transitions between s 1/2 and p 1/2 orbitals. This work was extended by Towner [60] to include several single-particle transitions in multiple mass regions from light nuclei to A ≈ 208. Enhancement factors were obtained. The mesonic enhancement was considered also in the framework of the interacting shell model for A = 11 − 16. In the studies Millener and Alburger [64] and Warburton et al.
Interestingly, in the lead-region practically no quenching of the σ · r operator was found with g A /g free A (0 − ) = 0.97 ± 0.06, as reported in Warburton [53,67,68]. The value of ǫ MEC in the lead region was derived by Kubodera and Rho [58] by adopting an effective Lagrangian incorporating approximate chiral and scale invariance of QCD. The result of Kubodera and Rho was which agrees well with the phenomenological shell-model result (39). In addition, separate studies for 50 K [69] and 96 Y [70] yielded enhancement factors 1.52±0.07 and 1.75±0.30. However, in the case of 50 K there was no estimate of the theoretical error so the reported uncertainty is purely experimental. Mesonic enhancement of the axial-charge matrix element, as well as the quenching of g A , was systematically studied in the previously less studied A ≈ 95 and A ≈ 135 regions by the present authors [71]. In this systematic shell-model study of the two regions, using state-of-the-art two-body interactions, a single effective value of g eff A for all axial-vector operators was used. The mesonic enhancement was found to depend on the chosen effective value of g A . Since the mesonic enhancement has been found to vary between 40 % and 100 %, the effective value of g A was extracted for several different values of ǫ MEC . In the A = 92 − 97 region the results were In the A = 133 − 139 region the results were quite similar: This is quite strong evidence of the quenching of g A in firstforbidden decays, since significant quenching is found regardless of the exact strength of the mesonic enhancement. In Kubodera and Rho [58] it was pointed out that the mesonic enhancement should be larger for heavier nuclei, which is in line with the earlier shell-model results [53,67,68]. In Kostensalo and Suhonen [71] this was used to extract the effective value of g A by fitting all the available data from the previous studies. The fit of this study is shown in Figure 9. As can be seen from the figure, a suitably increasing mesonic enhancement as a function of A is seen only when g A ≈ 0.70. The linear fit obtained for the whole mass region A = 11 − 212 reads ǫ MEC = 1.576 + 2.08 × 10 −3 A .

Rank-1 Tensors
Since the early findings (35) and (36) of Bohr and Mottelson [56], the effective values of the weak coupling constants related to rank-1 operators O(1 − ) in (33) have been studied also separately using the available data on the corresponding pseudovector transitions. In Warburton [53,67] the effective values were obtained in the lead region, where the heavy quenching was attributed to core-polarization effects. The shell model study of Rydström et al. [72] of the transition 205 Pb(1/2 − ) → 205 Tl(1/2 + g.s. ) yielded similar results: In the work Zhi [52] half-lives of a number of nuclei at the magic neutron numbers N = 50, 82, 126 were analyzed by comparing the results of large-scale shell-model calculations with experimental data. Both Gamow-Teller and first-forbidden β decays were included in the analysis. By performing a leastsquares fit to the experimental data the following quenched weak couplings were extracted For the rank-1 operators the obtained effective value on g V is quite far from the CVC value g V = 1.0 [9]. In the recent work of Hayen et al. [73] the first-forbidden decays related to the cumulative reactor antineutrino spectrum were studied in the ISM. In this short letter type article it was not possible to give the details related to the calculations, so they are presented here for the first time. In the A = 86 − 97 region the interaction glepn [70] was used and in the region A = 133 − 142 the (unpublished) interaction jj56pnb of Alex Brown was used. For the decays with J = 1 the half-lives could not be reproduced with any reasonable g A , with the exception of the decay 134m Sb(7 − ) → 134 Te(6 − 2 ) for which g A = 0.34 (12) works, if the CVC hypothesis g V = 1 was not broken. Since there were so few decays calculated in different model spaces, it was not meaningful to start fitting the values of both g A and g V to this data. However, this might be an interesting research topic in the future. Since the previous studies, discussed above, point to g A ≈ g V , we set the two coupling constants to be equal and then fitted this value to the experimental data. As can be seen from Table 5 the recent calculations seem to also point to heavy quenching g A = g V ≈ 0.5 for the first-forbidden rank-1-tensor-mediated β decays in medium-heavy nuclei.

Rank-2 Tensor
The quenching of g A related to the pseudotensor transitions mediated by the rank-2 operator (34) is best studied in the context of first-forbidden unique ground-state-to-ground-state decays in even-A nuclei, as this is the only operator at work in  (2) the leading order. In the early work [74] a systematic schematic analysis of the first-forbidden unique β decays was performed from the point of view of suppression factors stemming from the effect of E1 (electric dipole) giant resonance in the final oddodd nucleus. In Towner et al. [75] the suppression mechanism of the first-forbidden and third-forbidden β decays of light nuclei (A ≤ 50) was studied by using simple shell-model estimates and first-order perturbation theory. The hindrance was traced to the repulsive T = 1 (isospin 1) particle-hole force.
In the work Ejiri et al. [76] 19 first-forbidden unique groundstate-to-ground-state β-decay transitions were studied. The interesting transitions are the ones where a central nucleus (here only 84 Kr) is fed by a left-branch and a right-branch 2 − → 0 + transition or a central nucleus ( 74 As, 86 Rb, 122 Sb and 126 I) is feeding the neighboring even-even nuclei by leftbranch and right-branch 2 − → 0 + transitions. By denoting the associated left-branch and right-branch NMEs by g A M L and g A M R , respectively, one can compute the geometric mean (16) of these MNEs and compare with experimental data.
In Ejiri et al. [76] a g ph -and g pp -renormalized Bonn-A G matrix (one-meson-exchange nucleon-nucleon interaction adapted to nuclear matter by enforcing the Pauli principle) was used as the two-nucleon interaction in a pnQRPA framework. The two-quasiparticle and pnQRPA mean NMEs were compared with the ones extracted from the measured comparative halflives. The analysis yielded g eff A (2 − ) ≈ 0.57 (49) for the effective axial-vector coupling strength using the pnQRPA wave functions. This in excellent agreement with the result of Zhi et al. given in (48). The average of the values of the leading two-quasiparticle NMEs gives in turn implying the ratio and thus a drastic nuclear many-body effect when going from the two-quasiparticle level of approximation to the more sophisticated pnQRPA level. The 2qp-NME to pnQRPA-NME comparison is the only one where a clean separation between the nuclear-medium effects and the nuclear-model effects can be achieved, the nuclear-model effect being responsible for the (in this case large) shift in the values of the NMEs.

Higher-Forbidden Non-unique Decays
The shape factors of forbidden non-unique β decays are rather complex combinations of different NMEs and phase-space factors. Furthermore, their dependence on the weak couplings g V and g A is very nontrivial as shown by the decomposition (24) and its integrated version (25). In Haaranen et al. [47] the spectrum-shape method (SSM) was proposed. In this method the computed β-electron spectra can be used to determine the values of the weak coupling strengths by comparing the computed spectra with the measured one for forbidden non-unique β decays. In this work it was found that the computed β-electron spectra were very sensitive to the effective value of g A in the 4th-forbidden non-unique ground-state-to-ground-state β-decay transitions 113 Cd(1/2 + ) → 113 In(9/2 + ) and 115 In(9/2 + ) → 115 Sn(1/2 + ). The related calculations were done in the frameworks of the MQPM and the ISM (see section 2). It was also found that for all values of g A the best fits to the spectrum-shape and half-life data were obtained by using the canonical value g V = 1.0 for the vector coupling strength. This result is in conflict with the results for the first-forbidden non-unique pseudovector decays, where quenched values for g V were obtained (see section 4.2.2).
The work in Haaranen et al. [47] was extended in Haaranen et al. [48] by including the microscopic interacting bosonfermion model (IBFM-2) (see section 2), in the calculations. Interestingly all three models gave g A ≈ 0.92 when the SSM method was applied to the transition 113 Cd(1/2 + ) → 113 In(9/2 + ) where the experimental spectrum shape was available. For the three models, in the whole range of electron energies, the two components, C V (w e ) and C A (w e ) are roughly of the same size whereas the magnitude of the component C VA (w e ) is practically the sum of the previous two, but with opposite sign. Hence, for the whole range of electron energies there is a delicate balance between the three terms, and their sum is much smaller than the magnitudes of the individual components. However, the half-life comparisons were not in line with the SSM analyses, giving g A ≈ 0.65 for ISM and MQPM and g A ≈ 0.10 for IBFM-2. The possible cure to this could be to independently vary the weak couplings for multipole operators of different rank as was described for the first-forbidden β transitions in section 4.2.
The works [47,48] were continued by the works [49,77] where both the MQPM and ISM frameworks were used as theory tools. It was found that the spectrum shapes of the third-and fourthforbidden non-unique decays depend strongly on the value of g A , whereas the first-and second-forbidden decays were practically insensitive to the variations in g A . Furthermore, the g A -driven evolution of the normalized β spectra was found to be quite universal, largely insensitive to small changes of the nuclear mean field and the adopted residual many-body Hamiltonian. This makes SSM a robust tool for extracting information on the effective values of weak couplings. All the potentially interesting nuclei for the application of the spectrum-shape method found in Kostensalo et al. [49] and Kostensalo and Suhonen [77] had mass numbers A = 87 − 115 and were second-, third-, or fourth-forbidden. In order to better understand the evolution of the quenching of g A it would be preferable to have also lighter and heavier nuclei to study. Also, in these papers there were no first-forbidden decays discovered for with the SSM could be applicable. However, in Kostensalo and Suhonen [78] the first potential heavy candidate, the groundstate-to-ground-state decay of 210 Bi was pointed out. This decay is also first-forbidden and its spectrum is given in Figure 10.
In Kostensalo and Suhonen [71] a modest g A dependence was reported in the spectra of the decays 93 Y(1/2 − ) → 93 Zr(1/2 + ) and 138 Cs(3 − ) → 138 Ba(3 + ), as can bee seen in Figure 11. In the spectrum on 93 Y some dependence on the value of ǫ MEC is also seen, which opens up the possibility for investigating the mesonic enhancement of the axial-charge matrix element using the SSM.
Concerning the lighter nuclei, so far there are not any good candidates yet discovered. However, our recent shell-model calculation in the full fp shell using the interaction gxpf1a [79,80] shows that the second-forbidden ground-state-to-ground-state decay of 59 Fe possesses the necessary g A dependence, as can be seen from Figure 12. The branching ratio on the other hand is only 0.18 % [81], which complicates the measurement of this spectrum to the required precision. In any case, this discovery gives us hope that there might be some good candidates for the SSM also in this mass region.
The main issue with the SSM is the quality of the theoretical wave functions. However, the spectral shapes have been found to depend very little on the details of the wave functions making it possibly a very robust tool [49,78]. The best candidates for the SSM found so far are, unfortunately, decay transitions between mid-shell nuclei, such as 113 Cd, which are not easy to describe theoretically. Even though the SSM seems to be robust, in order to make strong claims about the effective value of g A it would be preferable to have precise wave functions. In the case of the β −decay transition 98 Tc(6 + ) → 98 Ru(4 + ) our recent calculations using the glepn interaction [70] have given results which are in FIGURE 11 | Normalized ISM-computed electron spectra for the first-forbidden non-unique β − decays of 138 Cs and 93 Y. The value g V = 1.0 was assumed and the color coding represents the value of g A and the dash coding the value of the mesonic enhancement factor ǫ MEC . good agreement with the available spectroscopic data. For the 2 + state in 98 Ru the calculated magnetic dipole moment +0.445µ N is in agreement with the experimental value +0.8 (6)µ N [82] and the electric quadrupole moment −0.129e 2 b agrees with the experimental value -0.03(14)e 2 b [83] also. The interaction also manages to predict that the 4 + , 5 + , 6 + states in 98 Tc are below 70 keV, though does not get the ordering quite right. Furthermore, the experimental half-life of the β decay is reproduced with a reasonable g A = 0.96. In comparison, the calculation carried out in Kostensalo and Suhonen [77], using the interaction by Gloeckner [84], fails to reproduce the half-life 4.2(3)×10 6 yr with any effective value of g A giving 2.1 × 10 6 yr for g A = 1.04 and less for all other values. As can be seen from Figure 13 the two Hamiltonians still predict very similar spectral shape evolution regardless of the failures in half-life prediction.
On the experimental side advances have been made by measuring the 113 Cd spectrum to high accuracy [85]. When compared with the spectral shapes given in Haaranen et al. [48] the following effective values of g A were obtained which are in excellent agreement with each other.

Higher-Forbidden Unique Decays
Early studies of the quenching in the second-and third-forbidden unique β decays were performed in Towner et al. [75] and Warburton et al. [86]. In Towner et al. [75] the suppression mechanism of the first-forbidden and third-forbidden β decays of light nuclei (A ≤ 50) was studied by using simple shell-model estimates and first-order perturbation theory. The hindrance of FIGURE 13 | Normalized ISM-computed electron spectra for the second-forbidden non-unique β − transition 98 Tc(6 + ) → 98 Ru(4 + ) using two different Hamiltonians. The branching to this decay channel is 100%. The value g V = 1.0 was assumed and the color coding represents the value of g A .
FIGURE 14 | Predicted half-lives and their error estimates (in parenthesis) for β − and EC (electron-capture) transitions in the isobaric chain A = 136. The spin-parity assignment, life-time and decay energies (Q values) of the 5 + ground (gs) state and 8 + isomeric (isom) state of 136 Cs are experimental data and taken from ENSDF [89]. The 2νββ half-life is taken from Barabash [90]. In addition to the predicted half-lives the degree of forbiddenness and the leading single-particle transition are shown.
the decay transitions was argued to result from the repulsive T = 1 (isospin 1) particle-hole force. In Warburton et al. [86] the second-and third-forbidden unique β decays were studied using a simple interacting shell model and the unified model (deformed shell model) for six β transitions in the A = 10, 22, 26, 40 nuclei. The incentive for the studies was the hindrance of certain measured β transitions. A later study of second-forbidden unique β decays in the mass range A = 10 − 54 was performed in Martínez-Pinedo and Vogel [87] by using the ISM. A reasonable correspondence with the measured half-lives was achieved by using the free value of the axial coupling but a quenched value would have improved the correspondence. The quenching related to the virtual β transitions of the 0νββ decay can be studied at the low-q limit (5) by using the theoretical machinery of section 4.1. In Kostensalo and Suhonen [88] this machinery was applied to 148 potentially measurable second-,third-, fourth-, fifth-, sixth-and seventh-forbidden unique beta transitions. The calculations were done using realistic singleparticle model spaces and G-matrix-based microscopic two-body interactions. The results of Kostensalo and Suhonen [88] could shed light on the magnitudes of the NMEs corresponding to the high-forbidden unique 0 + ↔ J π = 3 + , 4 − , 5 + , 6 − , 7 + , 8 − virtual transitions taking part in the 0νββ decays.
In the work of Kostensalo and Suhonen [88] the expected half-lives of the studied β-decay transitions were derived by comparison with the analyses performed for the Gamow-Teller and first-forbidden unique β transitions in the works [38,76]. An example of such predictions is given in Figure 14. In the figure one sees that the expected half-lives are very long and hard to measure. On the β − side the fourth-forbidden and seventh-forbidden decay transitions are masked by the strong β − branches to the excited states of 136 Ba.

CONCLUSIONS
Double β decay is a hot issue in modern day's particle, neutrino and nuclear physics. To gain the full benefit from the potential results of the running and future ββ-decay experiments, accurate evaluation of the involved nuclear matrix elements is crucial. This evaluation calls for reliable nuclear many-body approaches in order to produce realistic wave functions for ββ calculations. Beyond this, systematic estimation of the effective value of the weak axial coupling, g A , is necessary. The value of this coupling plays a notable role in both the two-neutrino and neutrinoless ββ decays.
The effective value of g A can be studied at low momentumexchange limit by using data on β and two-neutrino ββ decays. Data on Gamow-Teller 0 + ↔ 1 + β transitions are relatively abundant and the most clean-cut to compare with calculations, thus enabling systematic studies of the quenching of g A within different nuclear-structure frameworks. The β-decay analyses (see Figure 8) suggest that effective values g eff A ≈ 0.9 for masses A ≤ 82, around g eff A ≈ 0.5 in the mass A = 98 − 110 region, around g eff A ≈ 0.7 in the mass A = 112 − 124 region and close to g eff A ≈ 1.0 for the masses A = 126 − 142 are appropriate.
Available data on first-forbidden unique β decays offer a straightforward systematic way to access the quenching of g A beyond the allowed β decays. A recent analysis (see section 4.2.2) suggests g eff A (2 − ) ≈ 0.5 − 0.6. A more involved analysis has to be performed for the first-forbidden non-unique β decays owing to the meson-exchange enhancement of the weak axial charge, combined with the quenching of g A . A recent study (see Figure 9) suggests enhancement factors ǫ MEC obeying the simple linear law of Equation (43) all through the mass region A = 16 − 208. Higher-forbidden non-unique β transitions offer yet another way to study, at low-momentum-transfer limit, the quenching of g A in virtual transitions to high-angularmomentum intermediate states in the neutrinoless ββ decay. In particular, the shapes of electron spectra in these transitions can, in selected cases and once more experimental data are available, offer a viable alternative to access the quenching of g A . Here a study, combining theory with data of a recent experiment, suggests values around g eff A ≈ 0.9 for the 4th-forbidden β decay of 113 Cd (see Equations (52)-(54)).

DATA AVAILABILITY
The datasets generated for this study are available on request to the corresponding author.