The neutrinoless double beta (0νββ) decays of atomic nuclei are of great experimental and theoretical interest due to their implications of physics beyond the standard model of electroweak interactions. Since these processes occur in nuclei, nuclear-structure effects play an important role and they may affect considerably the decay rates. The nuclear effects are summarized as the nuclear matrix elements (NMEs) containing information about the initial and final states of the nucleus and the action of the 0νββ transition operator on them. The NMEs, in turn, are computed numerically using some nuclear-theory framework suitable for the nuclei under consideration. The possible future detection of the 0νββ decay in the next generation of ββ experiments constantly drives nuclear-structure calculations toward better performance. Accurate knowledge of the NMEs is required in order that the data will be optimally used to obtain information about the fundamental nature and mass of the neutrino [1–7]. In addition, the 0νββ decay relates also to the breaking of lepton-number symmetry and the baryon asymmetry of the Universe [8, 9]. A number of nuclear models, including configuration-interaction based models like the interacting shell model (ISM), and various mean field models, have been adopted for the calculations. The resulting computed NMEs have been analyzed in the review articles [4, 10–12]. Most of the calculations have been done by the use of the proton-neutron quasiparticle random-phase approximation (pnQRPA) [13].

The performed 0νββ-decay calculations, as also those of the two-neutrino double beta (2νββ) decay, indicate that the following nuclear-structure ingredients affect the values of NMEs:

The chosen valence space of single-particle orbitals and their nucleon occupancies [14–16].

At the nuclear level, β decay can be considered as a mutual interaction of the hadronic and leptonic currents mediated by massive vector bosons W^± [25]. The leptonic and hadronic currents can be expressed as mixtures of vector and axial-vector contributions [26–28]. The weak vector and axial-vector coupling strengths g_V and g_A enter the theory when the hadronic current is renormalized at the nucleon level [29]. The conserved vector-current hypothesis (CVC) [26] and partially conserved axial-vector-current hypothesis (PCAC) [30, 31] yield the free-nucleon values g_V = 1.00 and g_A = 1.27 [25] but inside nuclear matter the value of g_A is affected by many-nucleon correlations and a quenched or enhanced value might be needed to reproduce experimental observations [32–35]. Precise information on the effective value of g_A is crucial when predicting half-lives of neutrinoless double beta decays since the half-lives are proportional to the fourth power of g_A [1, 36].

Since the vector bosons W^± have large mass and thus propagate only a short distance, the hadronic current and the leptonic current can be considered to interact at a point-like weak-interaction vertex with an effective coupling strength G_F, the Fermi constant. The parity non-conserving nature of the weak interaction forces the hadronic current to be written at the quark level (up quark u and down quark d) as a mixture of vector and axial-vector parts:

where γ^μ are the usual Dirac matrices and γ5=iγ0γ1γ2γ3. Renormalization effects of strong interactions and energy scale of the processes must be taken into account when moving from the quark level to the hadron level. Then the hadronic current between nucleons (neutron n and proton p) takes the rather complex form

where the vector-current part can be written as

and the axial-vector-current part as

Here q^μ is the momentum transfer, q² its magnitude, m_N the nucleon mass (roughly 1 GeV) and the weak couplings depend on the magnitude of the exchanged momentum. For the vector and axial-vector couplings one usually adopts the dipole approximation

where g_V and g_A are the weak vector and axial-vector coupling strengths at zero momentum transfer (q² = 0), respectively. For the vector and axial masses one usually takes M_V = 84 MeV [37] and M_A ~ 1 GeV [37–39] coming from the accelerator-neutrino phenomenology. For the weak magnetism term one can take gM(q2)=(μp-μn)gV(q2) and for the induced pseudoscalar term it is customary to adopt the Goldberger-Treiman relation [40] gP(q2)=2mNgA(q2)/(q2+mπ2), where m_π is the pion mass and μ_p − μ_n = 3.70 is the anomalous magnetic moment of the nucleon. It should be noted that the β decays and 2νββ decays are low-energy processes (few MeV) involving only the vector [first term in Equation (3)] and axial-vector [first term in Equation (4)] parts at the limit q² = 0 so that the q dependence of Equation (5) does not play any role in the treatment of these processes in this review. Contrary to this, the 0νββ decays and nuclear muon-capture transitions involve momentum transfers of the order of 100 MeV and the full expression (2) is active with slow decreasing trend of the coupling strengths according to Equation (5).

The effective value of g_A can simply be characterized by a renormalization factor q (in case of quenching of the value of g_A it is customarily called quenching factor):

where

is the free-nucleon value of the axial-vector coupling measured in neutron beta decay [41] and g_A is the value of the axial-vector coupling derived from a given theoretical or experimental analysis. This derived g_A can be called the effective g_A so that from (6) one obtains for its value

Equations (6)−(8) constitute the basic definitions used in this review.

The effective value of g_A can be derived from several different experimental and theoretical analyses. In these analyses it is mostly impossible to separate the different sources of renormalization affecting the value of g_A: (i) the meson-exchange currents (many-body currents) that are beyond the one-nucleon impulse approximation (only one nucleon experiences the weak decay without interference from the surrounding nuclear medium), usually assumed in the theoretical calculations, (ii) other nuclear medium effects like interference from non-nucleonic degrees of freedom, e.g., the Δ isobars and (iii) the deficiencies in the nuclear many-body approach that deteriorate the quality of the wave functions involved in the decay processes.

The effects (i) and (ii) can be studied by performing calculations using meson-exchange models and allowing non-nucleonic degrees of freedom in the calculations. These calculations that go beyond the nucleonic impulse approximation are described in section 3 in the context of Gamow-Teller β decays for which the related effects are measurable. The calculations yield a fundamental quenching factor q_F and the related fundamentally renormalized effective g_A for the space components (μ = 1, 2, 3) of the axial current (4) via the effects of the virtual pion cloud around a nucleon. The time component of μ = 0, the axial charge ρ₅, is, however, fundamentally enhanced by, e.g., heavy meson exchange and the corresponding effective coupling gAeff(γ5) is discussed in section 8.2, in the context of first-forbidden 0⁺ ↔ 0⁻ transitions for which the effect is measurable.

The ISM has the longest history behind it in studies of the axial quenching in Gamow-Teller β decays. The reason for this is the success of the ISM to describe nuclear spectroscopy of light nuclei and the rather large amount of data on these type of allowed β decays. The results of these studies are presented in section 5. In the same section the ISM results are compared with those obtained by the use of the pnQRPA. In section 6.2 the effective value of g_A is analyzed for the first-forbidden unique β decays for which there are some experimental data available. In section 7 this study is extended to higher-forbidden unique β decays where no experimental data are available and one has to resort to mere theoretical speculations. In section 8 the forbidden non-unique β decays are discussed. Experimentally, there are available data for the above-mentioned first-forbidden non-unique 0⁺ ↔ 0⁻ and other β transitions. For the higher-forbidden non-unique transitions, discussed in section 9, there are scattered half-life and β-spectrum data but more measurements are urgently needed, in particular for the shapes of the β spectra. Unfortunately, in all these studies it is not possible to completely disentangle the nuclear-medium effects (i) and (ii) from the nuclear-model effects (iii).

In the last two sections, 11, 12 more exotic methods to extract the in-medium value of g_A are presented: The spin-multipole strength functions and nuclear muon capture. Measurements of the spin-multipole strength functions, in particular the location of the corresponding giant resonances, help theoretical calculations fine-tune the parameters of the model Hamiltonians such that the low-lying strength of, say 2⁻ states, is closer to reality. Hence, more such measurements are called for. The nuclear muon capture probes the axial current (4) at 100 MeV of momentum transfer and thus suits perfectly for studies of the renormalization of the NMEs related to 0νββ decays. This means that muon-capture experiments for medium-heavy nuclei are urgently needed.

The renormalization of g_A which stems from the nuclear-model effects (iii) depends on the nuclear-theory framework chosen to describe the nuclear many-body wave functions involved in the weak processes, like β and ββ decays. This is why the effective values of g_A can vary from one nuclear model to the other. On the other hand, the different model frameworks can give surprisingly similar results as witnessed in section 9 in the context of the comparison of the measured β spectra with the computed ones. The renormalization of g_A can also depend on the process in question. For the zero-momentum-exchange (q² = 0) processes, like β and 2νββ decays, the renormalization can be different from the high-momentum-exchange (q² ~ 100 MeV) processes, like 0νββ decays (in section 9 the related g_A is denoted as gA,0νeff) or nuclear muon captures.

This introduction to the many-faceted renormalization of the axial-vector coupling is supposed to enable a “soft landing” into the review that follows. As can be noticed, the renormalization issue is far from being solved and lacks a unified picture thus far. There is not yet a coherent effort to solve the issue, but rather some sporadic attempts here and there. The most critical issue may be the nuclear many-body deficiencies (iii) that hinder a quantitative assessment of the nuclear-medium effects (i) and (ii) in light, medium-heavy and heavy nuclei. Only gradually this state of affairs will improve with the progress in the ab-initio nuclear methods extendable to nuclei beyond the very lightest ones. Hence, the lack of perfect nuclear many-body theory is reflected in this review as a wide collection of different effective g_A variants, different for different theory frameworks and processes and not necessarily connected to each other (yet). The hope is that in the future the different studies would point to one common low-energy renormalization of g_A for the β and two-neutrino ββ decays and that we would have some idea about the renormalization mechanisms at work in the case of the neutrinoless ββ decays.

On the other hand, there are some attempts to disentangle the nuclear medium effects from the nuclear many-body effects. Examples are the fundamental quenching elaborated in section 3 and the nuclear-medium-independent quenching factor k introduced in section 5.2 for the Gamow-Teller β decays, and in sections 6.2, 7 for the unique-forbidden β transitions. This factor is designed to give hints about the impact of the changes in the complexity of the nuclear model on the value of the effective axial coupling. Also the previously mentioned method based on the examination of β spectra in section 9 is largely nuclear-model independent and seems to be a reasonable measure of the nuclear-medium effects (i) and (ii). More measurements of the β spectra are thus urgently called for.

Based on the early shell-model studies of Gamow-Teller β decays, effects of Δ resonances and meson-exchange currents on the weak axial-vector coupling strength of the space part, A, of the axial current A^μ (4) is expected to be quenched in nuclear medium and finite nuclei. Contrary to this, the coupling strength of the time part, A⁰, of (4) is expected to be enhanced by, e.g., the contributions coming from exchanges of heavy mesons. Many of these modifications in the strengths of the axial couplings stem from processes beyond the impulse approximation where only one nucleon at a time is experiencing a weak process, e.g., β decay, without interference from the surrounding nuclear medium. In fact, based on general arguments concerning soft-pion amplitudes [42] the space part of A^μ is quenched and the time part of A^μ is enhanced relative to the single-particle processes of the impulse approximation.

The origin of the quenching of the space part of A^μ is not completely known and various mechanisms have been proposed for its origin: studied have been the Δ-isobar admixture in the nuclear wave function [43], shifting of Gamow-Teller strength to the Δ-resonance region, and renormalization effects of meson-exchange currents. The β⁻ and β⁺ Gamow-Teller strengths were related to the Δ-isobar region e.g., in Delorme et al. [44] and sizable Δ-resonance effects on β decays of low-lying nuclear states by tensor forces were reported in Oset and Rho [45] and Bohr and Mottelson [46]. In Towner and Khanna[47, 48] very simple nuclear systems were used to study the tensor force and related effects in order to minimize the impact of nuclear many-body complexities. Studied were the tensor effects and their interference with the Δ-isobar current and meson-exchange currents in building up corrections to the Gamow-Teller matrix elements. Also relativistic corrections to the Gamow-Teller operator were included. Large cancellations among the various contributions were recorded and corrections below some 20% were obtained for the light (simple) nuclei. However, recent experimental studies of (p,n) and (n,p) reactions [49] report that the Δ-nucleon-hole admixtures into low-lying nuclear states play only a minor role in the quenching of g_A, in line with the results of Suhonen [43]. Also extended sum rules have been derived for relating g_A to pion-proton total cross sections [50–52], or the method of QCD sum rules has been utilized [53].

In Wilkinson [54] the renormalization of the β-decay operator by the two- or many-nucleon correlations, in terms of inter-nucleonic and intra-nucleonic mesonic currents, leads to the notion effective “fundamentally” renormalized axial coupling g_AeF. The quenching of g_A is then described by the fundamental quenching factor q_F such that q_F > q since q contains, in addition, the quenching stemming from the inadequate treatment of the nuclear many-body problem. From here on the above notation is adopted for the renormalization of g_A stemming from the (fundamental) mesonic-current effects.

In the early study of Ericson [55] of the sum rule for Gamow-Teller matrix elements a (fundamental) quenching of roughly

was obtained for very light nuclei (A ≤ 17) by the examination of the effects of meson-exchange currents on the pion-nucleon interaction vertex and extending the result to a sum rule for Gamow-Teller matrix elements. This (practically) model-independent study produces the following (fundamentally) renormalized value of the axial coupling strength

The above result does not necessarily apply to individual Gamow-Teller transitions between low-energy nuclear states.

The work of Ericson [55] was followed by the works [56, 57] where it was found that the renormalization should be universal for all transitions, in particular applicable to the mentioned Gamow-Teller transitions at low nuclear excitations. The procedure bases on the fact that the partially conserved axial current (PCAC) hypothesis [30, 31] enables one to calculate the full axial-current matrix element in terms of a pion-nucleus vertex [58]. At the low-momentum-exchange limit, relevant for the nuclear β decays, the PCAC leads to the Goldberger-Treiman relation [40, 59] which relates the effective value of g_A to the effective value of the pionic coupling constant g_π by Ericson [55]

where m_N is the nucleon mass and f_π = 0.932m_π is the pion decay constant, m_π being the pion mass. The pionic coupling constant is, in turn, renormalized by the effects on the virtual pion field by the presence of other nucleons. For large nuclei (surface effects can be omitted) the renormalization arises from nucleonic short-range correlations leading to voids between nucleons and the renormalization can be understood via an electromagnetic analog: an electric dipole in a correlated dielectric medium is renormalized in a similar way as the pionic coupling constant. There is also a connection to the low-energy scattering of pions on nuclei: the short-range correlations quench the p-wave pion-nucleon amplitude by the same amount as the dielectric effect. For finite nuclei a model-dependent surface factor has to be taken into account [55]. The size renormalization emerges from the nuclear surface layer of a thickness of the order of the pion Compton wavelength and thus the quenching of g_A increases with increasing nuclear radius and, as a consequence, with increasing nuclear mass.

In Rho [57] the pion-nucleus vertex was calculated and the related quenched g_A agreed with the one of Ericson [55] to leading order. In infinite nuclear matter This quenching turns out to be [57]

leading to the quenched effective axial coupling strength

in infinite nuclear matter.

The works of Ericson [55, 56] and Rho [57] were used by Wilkinson [54] to bridge the gap between the infinite nuclear matter and finite nuclei. In Wilkinson [54] it was argued that the fundamental quenching can be described by the formula

for finite nuclei of mass number A. This formula includes the short-range correlation effect and the finite-size factor [56, 57] and gives for the fundamental quenching, using (12), between A = 50 − 150 the value q_F = 0.88. This means that the fundamental quenching is practically constant over the range of nuclei of interest to the double beta decay. The corresponding fundamentally quenched value of the axial-vector strength is plotted in Figure 2, and its value is practically 1.1 through the whole range of interest.

In Siiskonen et al. [60] the renormalization of the axial current (and vector and induced pseudoscalar terms of the nucleonic current) was studied for several nuclear systems as a function of transition energy by including effective transition operators up to second order in perturbation theory. Thus, the renormalization of g_A contains both the fundamental and nuclear many-body aspects. It was found that the renormalization was practically constant up to 60 MeV in transition energy, in agreement with the q dependence of g_A in relation (5). The obtained quenchings are as follows

The results (15), obtained by using the nuclear-medium-corrected transition operators have been repeated in Table 1 of section 5.1 and Figure 3 of section 5.2 in order to compare them with the more phenomenological shell-model results. Effective operators have also been used in the connection with the calculations for the double beta decays in a solvable model [69] and for the nucleus ⁹²Mo [70] and the nuclei ⁷⁶Ge and ⁸²Se [71, 72] in the framework of the interacting shell model.

As speculated in Wilkinson [54], the mesonic effects (meson-exchange currents) show up as effective two-body contributions to the β-decay operators. These two-body currents quench g_A and this quenching was first estimated in Menéndez [73], in the framework of the chiral effective field theory (cEFT) where both the weak currents and nuclear forces can be described on the same footing and to a given order of approximation (leading order, next-to-leading order, etc.) In Menéndez [73] the two-body currents were replaced by an effective one-body current derived from the cEFT, leading to a momentum-dependent effective coupling gAeff(q2), renormalized with respect to the bare axial coupling of (5). It turned out that the additional quenching is caused by the short-range nucleon-nucleon coupling present in the original two-body current. The additional quenching decreases with increasing q, being the strongest at the zero-momentum-transfer limit, affecting mostly the nuclear β and 2νββ decays. In fact, the strength of the short-range nucleon-nucleon coupling in the two-body current can be adjusted such as to reproduce the empirical quenching of the Gamow-Teller β decays discussed in section 5. As the 0νββ decay is a high-momentum-transfer process (q ~ 100 MeV) it is expected that the two-body currents have not such a drastic effect on the one-body current (4) for the 0νββ decay. Here it should be noted that the one-body current (2) has been fully taken into account in all 0νββ-decay calculations and the two-body currents introduce a renormalization, gAeff(q2), that deviates from the one-body dipole gA(q2) of (5) the less the higher the momentum exchange q is. The quenching caused by the two-body currents could probably be measured by using charge-exchange reactions [49] in advanced nuclear-physics infrastructures.

In Menéndez [73] it was estimated, by using the ISM many-body framework in the mass range A = 48 − 136, that the effect of the two-body currents on the value of the 0νββ NME is between −35 and 10% depending on the (uncertain) values of the cEFT parameters, the smallest corrections occurring for A = 48. In Engel [74] the effect of the two-body currents was studied in the framework of the pnQRPA in the mass range A = 48 − 136, and a quenching effect of 10–22% was obtained for the 0νββ NMEs, the 10% effect pertaining to the case of ⁴⁸Ca. A more complete calculation, including three-nucleon forces and consistent treatment of the two-body currents and the nuclear Hamiltonian, was performed in Ekström [75]. Application to the Gamow-Teller β decays in ¹⁴C and ^{22, 24}O nuclei yielded the quenching q = 0.92 − 0.96 by comparison of the computed strengths to that of the Ikeda 3(N−Z) sum rule [35, 76]. This <10% quenching is in line with the trend observed in the studies [73, 74] where the quenching approaced the 10% limit for light nuclei. It should be noted that the two-body meson-exchange currents appear also in neutrino-nucleus scattering [77] but at energies where two nucleons are ejected as a result of the scattering (the so-called two-particle-two-hole exchange currents). The higher energy evokes considerable difficulties in handling the two-body meson-echange currents, as demonstrated in Simo et al. [78].

The meson-exchange currents can cause also enhancement phenomena, like in the case of the renormalization of the one-body weak axial charge density ρ₅ [time part of A^μ in (4)] in the case of the 0⁻ ↔ 0⁺ nuclear β transitions [42, 79]. In this case the γ⁵ operator mediates the first-forbidden non-unique β transition and the corresponding axial-vector coupling strength is enhanced quite strongly. In the work of Kirchbach and Reinhardt [79] the effects of a pionic two-body part of ρ₅ was studied for 4 nuclear masses and the corresponding leading single-particle transitions. This work was extended by Kirchbach et al. [80] and Towner [81] by taking into account also the heavy-meson exchanges. In Towner [81] 6 nuclear masses and a number of single-particle transitions were computed by using nuclear wave functions from the ISM. An interesting investigation of the role of the two-particle-two-hole excitations in the A = 16 nuclei was performed in Towner and Khanna [82]. The renormalization of the weak axial charge by the meson-exchange currents had to be taken into account in order to explain the measured rates of both the 0⁻ → 0⁺ β decay and the 0⁺ → 0⁻ muon capture. The axial-charge enhancement is elaborated further, quantitatively, in section 8.2.

Very recently break-through results in the calculations of the axial charge and axial-vector form factors have been achieved in the lattice QCD (quantum chromodynamics) calculations [83–85]. In the work [84] the result

was obtained, where the first uncertainty is statistical and the second comes from the extrapolation systematics. This computed value is quite compatible with the measured free value of g_A in (7). Also the lattice QCD calculations of the double beta decay are advancing in the two-nucleon (toy) systems (see [86]).

The studies on the effective value of the axial-vector coupling strength, g_A, have mainly been performed for β decays in established nuclear many-body frameworks. Also the magnetic moments of nuclei have been studied [87, 88] for simple one-particle and one-hole nuclei in order to pin down the effects of the tensor force in shifting low-energy strength of Gamow-Teller type to higher energies, and thus effectively quenching the spin-isospin operator for Gamow-Teller decays. The used many-body frameworks encompass the interacting shell model (ISM) [89] and the pnQRPA [13, 90]. Also the frameworks of the microscopic interacting boson model (IBM-2) [91] and the interacting boson-fermion-fermion model, IBFFM-2 [92], have been used. Let us discuss next the various many-body aspects of these models that may affect the (apparent) renormalization of the magnitude of g_A. It is appropriate to note here that in all these studies the nuclear many-body framework can be considered more or less deficient and thus the many-body effects cannot be disentangled from the nuclear-medium effects, discussed in section 3.

Gamow-Teller decays are mediated by the Pauli spin operator σ and they are thus able to change the initial nuclear spin J_i by one unit. In the renormalization studies the simplest Gamow-Teller transitions are selected, namely the ground-state-to-ground-state ones. In Figure 1 are depicted Gamow-Teller ground-state-to-ground-state β⁻ and β⁺/EC transitions between even-even 0⁺ and odd-odd 1⁺ ground states in the A = 100 Zr-Nb-Mo-Tc-Ru region. Shown are three different situations with a cascade pattern (left panel), lateral feeding to a middle nucleus (middle panel), and lateral feeding from a middle nucleus (right panel). All these transitions are mediated by a Gamow-Teller NME, M_GT, of the Pauli spin operator, defined, e.g., in Suhonen [35]. The corresponding β-decay data can be obtained from ENSDF ¹. In the figure this NME is denoted by M_L (M_R) in the case it is to the left (right) of the central nucleus. The corresponding reduced transition probability B_GT can be written as

where J_i is the spin of the ground state of the initial nucleus, g_A is the weak axial-vector coupling strength, substituted by the effective coupling strength gAeff of Equation (8) in practical calculations of the β-decay rates involving nuclear levels of low excitation energy [Hence, the coupling strength g_A is probed at the q² → 0 limit in (5)]. It is worth noting that the Gamow-Teller decays probe only g_A, not g_V which is carried by the vector part (Fermi spin-zero operator) of the β transitions, not active for the here discussed 1⁺ ↔ 0⁺ transitions due to the conservation of angular momentum.

The comparative half-lives (logft values) of the 1⁺ ↔ 0⁺ Gamow–Teller transitions are given in terms of the reduced transition probabilities as given in Suhonen [35]

for the β⁺/EC or β⁻ type of transitions. The half-life of the initial nucleus, t_1/2, has been given in seconds.

Next we inspect the evolution of the quenching concept, based on (17) and (18), in nuclear-structure calculations performed during the last four decades.

The forbidden unique β transitions are the simplest ones that mediate β decays between nuclear states of large angular-momentum difference ΔJ. In particular, if one of the states is a 0⁺ state, then for a Kth forbidden (K = 1, 2, 3, …) unique beta decay the angular momentum of the other involved state is J = K + 1. At the same time the parity changes in the odd-forbidden and remains the same in the even-forbidden decays [35]. The change in angular momentum and parity for different degrees of forbiddenness is presented in Table 3, and they obey the simple rule

Here it is interesting to note that also the Gamow-Teller decays obey the rule (46) if one of the involved nuclear states has the multipolarity 0⁺.

Early studies of the quenching in the second- and third-forbidden unique β decays were performed in Towner et al. [146] and Warburton et al. [149]. The work of Towner et al. [146] was discussed in section 6.2. In Warburton et al. [149] these β decays were studied using a simple ISM and the unified model (deformed shell model) for six β transitions in the A = 10, 22, 26, 40 nuclei. The interest for these studies derived from nuclear-structure considerations: how to explain in a nuclear model the hindrance phenomena occurring in certain measured β transitions. Beyond this, the incentive to study the Gamow-Teller (section 5), first-forbidden unique (section 6.2), and higher-forbidden unique (this section) β decays stems from their relation to the Gamow-Teller type of NME involved in 0νββ decays. The 0νββ decays proceed via virtual intermediate states of all multipolarities J^π due to the multipole expansion of the Majorana-neutrino propagator (see, e.g., [1–3, 150–155]). Studies of the quenching of these two-leg (“left-leg” and “right-leg” transitions illustrated in the schematic Figure 5 for the 0νββ decay of ¹¹⁶Cd to ¹¹⁶Sn via the virtual intermediate states in ¹¹⁶In) virtual transitions is of paramount importance to, e.g., estimate the sensitivities of the present and future neutrino experiments to the Majorana-neutrino mass. The possible quenching of these intermediate multipole transitions in the GT type of 0νββ NME can be, in a simplistic approach, condensed into an effective axial coupling, gA,0νeff, multiplying the NME:

where OGTGT(0ν) denotes the transition operator mediating the 0νββ transition through the various multipole states J^π, 0i+ denotes the initial ground state, and the final ground state is denoted by 0f+ (here, for simplicity, we assume a ground-state-to-ground-state transition). The effective axial coupling relevant for 0νββ decay is denoted as gA,0νeff to emphasize that its value may deviate from the one determined in single beta and 2νββ decays. The remarkable feature of Equation (58) is that the effective axial coupling strength is raised to 2nd power making the value of gA,0νeff play an extremely important role in determining the 0νββ-decay rate which is (neglecting the smaller double Fermi and tensor contributions) proportional to the squared NME and thus to the 4th power of the coupling:

The quenching related to the left-leg and right-leg β transitions of Figure 5 can be studied by using the theoretical machinery of section 6.1. In Kostensalo and Suhonen [156] 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 single-particle model spaces and G-matrix-based microscopic two-body interactions. The results of Kostensalo and Suhonen [156] 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 neutrinoless double beta decay, as shown in Figure 5.

In Kostensalo and Suhonen [156] the ratio k, Equation (62) below, of the NMEs, calculated by the pnQRPA, M_pnQRPA, and a two-quasiparticle model, M_qp, was studied and compared with earlier calculations for the allowed Gamow-Teller 1⁺ [140] and first-forbidden spin-dipole (SD) 2⁻ [147] transitions. Based on this comparison the expected half-lives of the studied β-decay transitions were predicted. An example case of the expected half-lives of second-, fourth-, and seventh-forbidden β decays is shown in Figure 6. The computed NMEs are corrected by the use of the ratio of the geometric means (40) of the experimental and pnQRPA NMEs,

extracted from the GT work of Ejiri and Suhonen [140], to predict the transition half-lives of the figure. In the figure one sees that the expected half-lives range from 4 years to the astronomical 9 × 10²⁹ years. It is expected that the decays to and from isomeric states are not measurable and the decays between the nuclear ground states are masked by transitions to excited states with lesser degree of forbiddenness. Only in some cases the high-forbidden β decay exhausts 100% of the decay rate between two nuclear ground states; one example being the second-forbidden β⁻ transition Mn54(3gs+)→54Fe(0gs+), with a half-life 4.2(9) × 10⁵ years, shown in Figure 7. Even in this case the measurement will be challenging due to the Gamow-Teller type of electron-capture feeding of the first excited 2⁺ state of ⁵⁴Cr, taking practically 100% of the feeding intensity.

The geometric mean of the EC/β⁺ and β⁻ NMEs, defined in (40), can be generalized to a geometric mean of n NMEs, M_i, i = 1, 2, …n, of successive β transitions with a common mother or daughter nucleus:

Here the aim, as in the case of (40), is to reduce the fluctuations in the computed NMEs by exploiting the compensating trends of the β⁻ and β⁺/EC branches of decay when changing the value of the particle-particle interaction parameter g_pp of the pnQRPA. One can now define the ratio

of the pnQRPA-calculated mean NME, M̄pnQRPA, and the mean two-quasiparticle NME, M̄qp, computed by using (61). The ratio k is a measure of the evolution of the nuclear-model dependent many-body effects on the computed NME. The ratio (62) is independent of the nuclear-medium effects (the fundamental quenching of section 3) and gives an idea of how the quenching of g_A depends on the degree of complexity of the adopted nuclear model.

In Kostensalo and Suhonen [156] the β transitions were divided in two groups: GROUP 1 contained only non-magic even-even reference nuclei (i.e., nuclei where the pnQRPA and the associated BCS (Bardeen-Cooper-Schrieffer) calculation were performed), whereas GROUP 2 contained (semi)magic reference nuclei. The transitions in GROUP 2 were left out from the analysis of the ratio k of (62) since the BCS results tend to be unstable at magic shell closures. In Figure 8 the ratio k is shown for transitions belonging to GROUP 1. The same k distribution is shown in terms of division to β⁻ and EC/β⁺ decays in Figure 9.

From Figure 8 it is visible that the second- and fourth-forbidden β transitions are distributed to masses below A = 62 and above masses A = 92, whereas the third-forbidden decays occupy the mass range 74 ≤ A ≤ 90. The sixth-forbidden decays occur within the range 92 ≤ A ≤ 110 and the seventh-forbidden decays occur above A = 116. The fifth-forbidden decays occur in a scattered way above A = 84. From Figure 9 one observes that most of the β⁻ decays are concentrated above mass A = 118 where also quite low values of k can be obtained. The EC/β⁺ decays, on the other hand, are more concentrated in the middle-mass region 82 ≤ A ≤ 116.

Figure 8 suggests that the values of the ratio (62) can be classified in terms of three mass regions, namely A = 50 − 88 (k ~ 0.4), A = 90 − 120 (values of k have a scattered, decreasing trend), and A = 122 − 146 (a low-k region with k~0.2). The ratios k for the three mass regions and for various degrees of forbiddenness K are shown in Table 4 for β transitions belonging to GROUP 1. A comparison is made to the GT results of Ejiri and Suhonen [140] and SD results of Ejiri et al. [147]. The ratios are also plotted in Figure 10 for illustrative purposes. In the figure one can see that the trend, in terms of the mass number A, is a bit different for the Gamow-Teller and the forbidden (K ≥ 2) transitions. For most of the forbidden transitions, namely K = 3, 4, 5, k has a decreasing tendency as a function of A, in particular for the k = 4 transitions. For K = 2 and K = 7 a slightly increasing tendency is observed. It seems, on average, that the quenching of the forbidden transitions is somewhat stronger than that of Gamow-Teller transitions in the mass region A = 90 − 146.

The numbers in Table 4 suggest that, in the gross, k is independent of the degree of forbiddenness and thus the (low-energy) forbidden unique contributions [obeying the simple rule (46)] to the 0νββ NME (59) should be roughly uniformly quenched. If these conclusions can be generalized to include also the non-unique β transitions, obeying the rule (−1)^ΔJΔπ = +1, one can then speak about an effective axial coupling, gA,0νeff, in front of the 0νββ NME in (58), at least for low intermediate excitation energies. The quenching for these low intermediate excitation energies could then be deduced from the hatched regions of Figure 2, implying the effective axial couplings listed in Table 5 for the three mass regions of interest for 0νβ⁻β⁻-decay calculations in the pnQRPA framework. At this point it has to be noted that a “low” excitation energy is still an undefined notion that has to be investigated in future works.

Finally, it should be stressed that the use of the low-energy effective axial coupling, gA,0νeff, is particular to the pnQRPA many-body framework and reflects the deficiencies of pnQRPA in calculating the magnitudes of the NMEs of the allowed and forbidden unique β-decay transitions. It is not directly related to the more fundamental quenching of the axial-vector coupling strength g_A, related to the meson-exchange currents, delta isobars, two-body weak currents, etc., discussed in section 3, but it is rather a nuclear-model effect, discussed in section 4.

The general theory of forbidden beta decays is outlined in Behrens and Buhring [144] and Schopper [158]. Streamlined version of those is given in Mustonen et al. [159].

The shape functions of forbidden non-unique beta decays are rather complex combinations of different NMEs and phase-space factors. Furthermore, their dependence on the weak coupling strengths g_V (vector part) and g_A (axial-vector part) is very non-trivial. In fact, the shape factor C(w_e) (65) can be decomposed into vector, axial-vector and mixed vector-axial-vector parts in the form [177]

Integrating equation (82) over the electron kinetic energy, we obtain an analogous expression for the integrated shape factor (64)

where the factors C~i in Equation (83) are just constants, independent of the electron energy.

In Haaranen et al. [177] it was proposed that the shapes of β-electron spectra could be used to determine the values of the weak coupling strengths by comparing the computed spectrum with the measured one for forbidden non-unique β decays. This method was coined the spectrum-shape method (SSM). In this study also the next-to-leading-order corrections to the β-decay shape factor were included. In Haaranen et al. [177] the β-electron spectra were studied for the 4th-forbidden non-unique ground-state-to-ground-state β⁻ decay branches ¹¹³Cd(1/2⁺) → ¹¹³In(9/2⁺) and ¹¹⁵In(9/2⁺) → ¹¹⁵Sn(1/2⁺) using the microscopic quasiparticle-phonon model (MQPM) [115, 116] and the ISM. It was verified by both nuclear models that the β spectrum shapes of both transitions are highly sensitive to the values of g_V and g_A and hence comparison of the calculated spectrum shape with the measured one opens a way to determine the values of these coupling strengths. As a by-product it was found that for all values of g_A the best fits to data were obtained by using the canonical value g_V = 1.0 for the vector coupling strength. This result is in conflict with those obtained by analyzing first-forbidden non-unique β decays in section 8.2, where strongly quenched values of g_V were obtained.

The work of Haaranen et al. [177] on the ¹¹³Cd and ¹¹⁵In decays was extended in Haaranen et al. [178] to include an analysis made by using a third nuclear model, the microscopic interacting boson-fermion model (IBFM-2) [92]. At the same time the next-to-leading-order corrections to the β-decay shape factor were explicitly given and their role was thoroughly investigated. A striking feature of the SSM analysis was that the three models yield a consistent result, g_A ≈ 0.92, when the SSM is applied to the available experimental β spectrum [179] of ¹¹³Cd. The result is illustrated in Figure 11 where the three curves overlap best at the values gAeff=0.92 (MQPM), gAeff=0.90 (ISM), and gAeff=0.93 (IBFM-2). The agreement of the β-spectrum shapes computed in the three different nuclear-theory frameworks speaks for the robustness of the SSM in determining the effective value of g_A. For completeness, in Figure 12 are shown the three components (82) as functions of the electron energy for the three different nuclear models used to compute the spectrum shapes of ¹¹³Cd in Figure 11. It is seen that for 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 its constituent components.

The works [177, 178] were continued by the work [180] where the evolution of the β spectra with changing value of g_A was followed for 26 first-, second-, third-, fourth- and fifth-forbidden β⁻ decays of odd-A nuclei by calculating the associated NMEs by the MQPM. The next-to-leading-order contributions were taken into account in the β-decay shape factor. It was found that the spectrum shapes of the third- and fourth-forbidden 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 seems to be quite universal, largely insensitive to small changes of the nuclear mean field and the adopted residual many-body Hamiltonian. These features were also verified in the follow-up work [181], where the ISM was used as the nuclear-model framework. This makes SSM a robust tool for extracting information on the effective values of the weak coupling strengths. This also means that if SSM really is largely nuclear-model independent there is a chance to access the fundamental renormalization factor q_F of section 3 for (highly) forbidden β transitions. It is also worth noting that in the works [180, 181] several new experimentally interesting decays for the SSM treatment were discovered.

Results of the investigations of Kostensalo et al. [180] and Kostensalo and Suhonen [181] are summarized in Tables 6, 7, and in Figures 13–15. Figure 13 displays the β spectra of the second-forbidden non-unique transitions ⁹⁴Nb(6+) → ⁹⁴Mo(4⁺) (left panel) and ⁹⁸Tc(6⁺) → ⁹⁸Ru(4⁺) (right panel) calculated by using the ISM [181]. It is obvious that the shape of the spectra depends sensitively on the value of g_A but not as strongly as the transitions associated with the mother nuclei ¹¹³Cd and ¹¹⁵In, as shown in the figures of Haaranen et al. [177]. It is to be noted that both of the transitions have been observed experimentally since the branching is 100%, but the electron spectra are not yet available.