Value of the Axial-Vector Coupling Strength in β and ββ Decays : A Review

This is an electronic reprint of the original article. This reprint may differ from the original in pagination and typographic detail. Author(s): Title: Year: Version: Please cite the original version: All material supplied via JYX is protected by copyright and other intellectual property rights, and duplication or sale of all or part of any of the repository collections is not permitted, except that material may be duplicated by you for your research use or educational purposes in electronic or print form. You must obtain permission for any other use. Electronic or print copies may not be offered, whether for sale or otherwise to anyone who is not an authorised user. Value of the Axial-Vector Coupling Strength in β and ββ Decays : A Review Suhonen, Jouni Suhonen, J. (2017). Value of the Axial-Vector Coupling Strength in β and ββ Decays : A Review. Frontiers in Physics, 5, 55. https://doi.org/10.3389/fphy.2017.00055 2017 REVIEW published: 16 November 2017 doi: 10.3389/fphy.2017.00055 Frontiers in Physics | www.frontiersin.org 1November 2017 | Volume 5 | Article 55 Edited by: Alexander Merle, Max Planck Institute for Physics (MPG), Germany Reviewed by: Eligio Lisi, National Institute for Nuclear Physics, Italy Fedor Simkovic, Comenius University, Slovakia *Correspondence: Jouni T. Suhonen [email protected] Specialty section: This article was submitted to High-Energy and Astroparticle Physics, a section of the journal Frontiers in Physics Received: 19 June 2017 Accepted: 17 October 2017 Published: 16 November 2017 Citation: Suhonen JT (2017) Value of the Axial-Vector Coupling Strength in β and ββ Decays: A Review. Front. Phys. 5:55. doi: 10.3389/fphy.2017.00055 Value of the Axial-Vector Coupling Strength in βand ββ Decays: A Review Jouni T. Suhonen* Department of Physics, University of Jyvaskyla, Jyvaskyla, Finland In this review the quenching of the weak axial-vector coupling strength, gA, is discussed in nuclear βand double-βdecays. On one hand, the nuclear-medium and nuclear many-body effects are separated, and on the other hand the quenching is discussed from the points of view of different many-body methods and different β-decay and double-β-decay processes. Both the historical background and the present status are reviewed and contrasted against each other. The theoretical considerations are tied to performed and planned measurements, and possible new measurements are urged, whenever relevant and doable. Relation of the quenching problem to the measurements of charge-exchange reactions and muon-capture rates is pointed out. Keywords: double beta decays, Gamow-Teller beta decays, forbidden beta decays, axial-vector coupling strength, beta spectra, charge-exchange reactions, strength functions, muon capture 1. INTRODUCTION 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: (a) The chosen valence space of single-particle orbitals and their nucleon occupancies [14–16]. (b) The effects stemming from the shell closures [10,17]. These closures are formed by the bunching of single-particle orbitals in the nuclear mean-field potential to form the so-called Suhonen Effective Value of gA major shells that are separated by large energy gaps. The gaps occur at “magic numbers” of nucleons and have sometimes drastic effects on nuclear properties. (c) The nuclear deformation and seniority truncation [18–22]. In ground states of even-even (even number of protons and neutrons) nuclei all nucleons are paired to angular momentum zero and form a superfluid-like state with total angular momentum zero. This is called seniority-zero state. If one pair is broken, extra angular momentum is generated and this contributes to excited states of nuclei. These are called seniority-two states. Breaking more pairs generates higher-seniority states that can mix with the lower-seniority states by the nuclear residual interaction. Cutting the higherseniority contributions, i.e., performing a seniority truncation, simplifies calculations considerably. (d) Also, it has to be noted that the adopted closure approximation, i.e., omitting the energy dependence of the involved energy denominator and replacing the contributions coming from the intermediate virtual states by a unit operator (for all other nuclear models, except for the quasiparticle random-phase approximation, QRPA), for the 0νββ-decay calculations does not hold for the calculations of the 2νββ-decay rates [1,23,24]. (e) A further important aspect can be added to the list, namely the uncertain value of the weak axial-vector coupling strength gA, leading to an effective value of gAin nuclear-model calculations. This deviation (usually quenching) from the freenucleon value can arise from the nuclear medium effects and the nuclear many-body effects described in more detail in the following sections of this review. At the nuclear level, βdecay 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 axialvector contributions [26–28]. The weak vector and axial-vector coupling strengths gVand gAenter 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 gV=1.00 and gA=1.27 [25] but inside nuclear matter the value of gAis affected by manynucleon correlations and a quenched or enhanced value might be needed to reproduce experimental observations [32–35]. Precise information on the effective value of gAis crucial when predicting half-lives of neutrinoless double beta decays since the half-lives are proportional to the fourth power of gA[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 GF, 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 uand down quark d) as a mixture of vector and axial-vector parts: Jµ H= ¯u(x)γµ(1 −γ5)d(x), (1) where γµare 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 nand proton p) takes the rather complex form Jµ H=¯ p(x)[Vµ−Aµ]n(x), (2) where the vector-current part can be written as Vµ=gV(q2)γµ+igM(q2)σµν 2mNqν(3) and the axial-vector-current part as Aµ=gA(q2)γµγ5+gP(q2)qµγ5. (4) Here qµis the momentum transfer, q2its magnitude, mN 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 gV(q2)=gV 1+q2/M2 V2;gA(q2)=gA 1+q2/M2 A2, (5) where gVand gAare the weak vector and axial-vector coupling strengths at zero momentum transfer (q2=0), respectively. For the vector and axial masses one usually takes MV=84 MeV [37] and MA∼1 GeV [37–39] coming from the acceleratorneutrino 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+m2 π), where mπis the pion mass and µp−µn=3.70 is the anomalous magnetic moment of the nucleon. It should be noted that the βdecays 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 qdependence 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 muoncapture 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). 2. EFFECTIVE VALUES OF GA: PREAMBLE The effective value of gAcan simply be characterized by a renormalization factor q (in case of quenching of the value of gA it is customarily called quenching factor): q=gA gfree A , (6) where gfree A=1.2723(23) (7) Frontiers in Physics | www.frontiersin.org 2November 2017 | Volume 5 | Article 55 Suhonen Effective Value of gA is the free-nucleon value of the axial-vector coupling measured in neutron beta decay [41] and gAis the value of the axialvector coupling derived from a given theoretical or experimental analysis. This derived gAcan be called the effective gAso that from (6) one obtains for its value geff A=qgfree A. (8) Equations (6)−(8) constitute the basic definitions used in this review. The effective value of gAcan 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 gA: (i) the meson-exchange currents (many-body currents) that are beyond the onenucleon 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 nonnucleonic degrees of freedom, e.g., the 1isobars 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 nonnucleonic 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 βdecays for which the related effects are measurable. The calculations yield a fundamental quenching factor qFand the related fundamentally renormalized effective gAfor 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 geff A(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 βdecays. 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 gAis analyzed for the first-forbidden unique βdecays for which there are some experimental data available. In section 7 this study is extended to higher-forbidden unique βdecays where no experimental data are available and one has to resort to mere theoretical speculations. In section 8 the forbidden non-unique βdecays are discussed. Experimentally, there are available data for the above-mentioned first-forbidden non-unique 0+↔0− and other βtransitions. 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 βspectra. 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 gAare 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 gAwhich stems from the nuclearmodel effects (iii) depends on the nuclear-theory framework chosen to describe the nuclear many-body wave functions involved in the weak processes, like βand ββ decays. This is why the effective values of gAcan 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 βspectra with the computed ones. The renormalization of gAcan also depend on the process in question. For the zero-momentum-exchange (q2=0) processes, like βand 2νββ decays, the renormalization can be different from the high-momentum-exchange (q2∼100 MeV) processes, like 0νββ decays (in section 9 the related gAis denoted as geff A,0ν) 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 gA 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 gAfor the βand twoneutrino 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 kintroduced in section 5.2 for the Gamow-Teller βdecays, and in sections 6.2, 7 for the unique-forbidden βtransitions. 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 Frontiers in Physics | www.frontiersin.org 3November 2017 | Volume 5 | Article 55 Suhonen Effective Value of gA axial coupling. Also the previously mentioned method based on the examination of βspectra in section 9 is largely nuclearmodel 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. 3. NUCLEAR-MEDIUM EFFECTS Based on the early shell-model studies of Gamow-Teller βdecays, effects of 1resonances and meson-exchange currents on the weak axial-vector coupling strength of the space part, A, of the axial current Aµ(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., βdecay, without interference from the surrounding nuclear medium. In fact, based on general arguments concerning softpion amplitudes [42] the space part of Aµis quenched and the time part of Aµis enhanced relative to the single-particle processes of the impulse approximation. The origin of the quenching of the space part of Aµis not completely known and various mechanisms have been proposed for its origin: studied have been the 1-isobar admixture in the nuclear wave function [43], shifting of Gamow-Teller strength to the 1-resonance region, and renormalization effects of mesonexchange currents. The β−and β+Gamow-Teller strengths were related to the 1-isobar region e.g., in Delorme et al. [44] and sizable 1-resonance effects on βdecays 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 1-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 1-nucleon-hole admixtures into low-lying nuclear states play only a minor role in the quenching of gA, in line with the results of Suhonen [43]. Also extended sum rules have been derived for relating gAto 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 twoor many-nucleon correlations, in terms of internucleonic and intra-nucleonic mesonic currents, leads to the notion effective “fundamentally” renormalized axial coupling gAeF. The quenching of gAis then described by the fundamental quenching factor qFsuch that qF>qsince qcontains, 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 gA stemming from the (fundamental) mesonic-current effects. In the early study of Ericson [55] of the sum rule for GamowTeller matrix elements a (fundamental) quenching of roughly qF=0.9 (9) 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) modelindependent study produces the following (fundamentally) renormalized value of the axial coupling strength gAeF =0.9 ×1.27 =1.1. (10) The above result does not necessarily apply to individual GamowTeller 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 βdecays, the PCAC leads to the Goldberger-Treiman relation [40,59] which relates the effective value of gAto the effective value of the pionic coupling constant gπby Ericson [55] geff A geff π=gfree A gfree π=fπ √2mN , (11) where mNis the nucleon mass and fπ=0.932mπis the pion decay constant, mπbeing 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 gA 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 gAagreed with the one of Ericson [55] to leading order. In infinite nuclear matter This quenching turns out to be [57] q∞ F=0.76 (infinite nuclear matter) (12) Frontiers in Physics | www.frontiersin.org 4November 2017 | Volume 5 | Article 55 Suhonen Effective Value of gA leading to the quenched effective axial coupling strength g∞ AeF =0.76 ×1.27 =0.96. (infinite nuclear matter) (13) 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 qF=q(q∞ F)2+1−(q∞ F)2/A0.17 (14) 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 qF=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 gAcontains 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 qdependence of gAin relation (5). The obtained quenchings are as follows geff A=1.0 (1s0dshell);0.98 (1p0fshell);0.71 (56Ni);0.52 (100Sn). (15) The results (15), obtained by using the nuclear-mediumcorrected 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 (mesonexchange currents) show up as effective two-body contributions to the β-decay operators. These two-body currents quench gA 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 twobody currents were replaced by an effective one-body current derived from the cEFT, leading to a momentum-dependent effective coupling geff A(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 zeromomentum-transfer limit, affecting mostly the nuclear βand 2νββ decays. In fact, the strength of the short-range nucleonnucleon coupling in the two-body current can be adjusted such as to reproduce the empirical quenching of the Gamow-Teller βdecays discussed in section 5. As the 0νββ decay is a highmomentum-transfer process (q∼100 MeV) it is expected that the two-body currents have not such a drastic effect on the onebody 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, geff A(q2), that deviates from the one-body dipole gA(q2) of (5) the less the higher the momentum exchange qis. 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 manybody 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 βdecays in 14C and 22,24O 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 twobody 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-twohole 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 onebody weak axial charge density ρ5[time part of Aµin (4)] in the case of the 0−↔0+nuclear βtransitions [42,79]. In this case the γ5operator mediates the first-forbidden nonunique βtransition 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 heavymeson 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 Frontiers in Physics | www.frontiersin.org 5November 2017 | Volume 5 | Article 55 Suhonen Effective Value of gA be taken into account in order to explain the measured rates of both the 0−→0+βdecay 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 gfree A=1.278(21)(26) (lattice calculation) (16) 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 gAin (7). Also the lattice QCD calculations of the double beta decay are advancing in the two-nucleon (toy) systems (see [86]). 4. NUCLEAR-MODEL EFFECTS The studies on the effective value of the axial-vector coupling strength, gA, have mainly been performed for βdecays in established nuclear many-body frameworks. Also the magnetic moments of nuclei have been studied [87,88] for simple oneparticle and one-hole nuclei in order to pin down the effects of the tensor force in shifting low-energy strength of GamowTeller 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 gA. 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. 4.1. Many-Body Aspects of the ISM The ISM is a many-body framework that uses a limited set of single-particle states, typically one harmonic-oscillator major shell or one nuclear major shell, to describe nuclear wave functions involved in various nuclear processes. The point of the ISM is to form all the possible many-nucleon configurations in the given single-particle space, each configuration described by one Slater determinant, and diagonalize the nuclear (residual) Hamiltonian in the basis formed by these Slater determinants. In this way the many-body features are taken into account exactly but only in a limited set of single-particle states. The problem is to extend the single-particle space beyond the one-shell description due to the factorially increasing size of the sparse Hamiltonian matrix to be diagonalized. In this way only the low-energy features of a nucleus can be described, leaving typically the giantresonance region out of reach. The other problem with the ISM is to find a suitable (renormalized) nucleon-nucleon interaction to match the limited single-particle space. Since this space is small, the renormalization effects of the two-body interaction become substantial. Typically, mostly in the early works, all the matrix elements of the two-body interaction were fitted such that the computed observables, energies, electromagnetic decays, etc., are as close as possible to the corresponding measured ones (see section 5.1). In some works also perturbative approaches through particle-hole excitations from the valence to the excluded space have been considered (see, e.g., [93–95] and the references therein). From early on there have been difficulties for the ISM to reproduce the measured β-decay rates [96]. This has lead to a host of investigations of the effective (quenched) value of gAin the ISM framework (see section 5.1 below). The main limitation of the ISM is its confinement to small single-particle spaces, typically comprising one oscillator major shell or a magic shell, leaving one or two spin-orbit partners out of the model space. From, e.g., pnQRPA calculations [15,16] and perturbative ISM calculations [72,97] one knows that inclusion of all spin-orbit partners in the single-particle model space is quite essential. This has been noticed also in the extended ISM calculations where the missing spin-orbit partners have been included at least in an effective way [20,98]. Even extension of the ISM to include two harmonic-oscillator shells (1s0dand 1p0fshells) has been done for the calculation of the 0νββ decay of 48Ca [99]. Several advanced shell-model methods have been devised in order to include larger single-particle spaces into the calculations. One can try to find clever ways to select the most important configurations affecting the observables one is interested in. Such an established algorithm is the Monte Carlo shell model (MCSM) where statistical sampling of the Slater determinants is used [100, 101]. One can also use importance-truncation schemes [102] or very advanced ab initio methods, like the coupled-cluster theory, where the twoand three-body interactions can be derived from the chiral effective field theory (cEFT) [103]. One can also use the in-medium similarity renormalization group (IM-SRG) method, like in Bogner [104], where an ab initio construction of a nonperturbative 1s0d-shell Hamiltonian, based on cEFT twoand three-body forces, has been done. Another new method is the density matrix renormalization group (DMRG) algorithm [105], which exploits optimal ordering of the proton and neutron single-particle orbitals and concepts of quantum-information theory. All the new methods extend the traditionally used ISM model spaces and the future β-decay calculations using these methods will either confirm or reduce the amount of quenching of gA observed in the older ISM calculations, described in section 5.1 below. The ab initio methods are already available for the light nuclei, occupying the 0pand 1s0dshells, and later for the medium-heavy and heavy nuclei dwelling in the higher oscillator shells. The quenching problem can only be solved by using manybody methods with error estimates, including a systematic way to improve their accuracy. At the same time the twoand threebody forces used in the calculations should be produced on the same footing as the many-body framework itself, preferably from ab initio principles. One should not forget that also the operators used in the computations should be made effective operators that match the adopted single-particle valence spaces. Frontiers in Physics | www.frontiersin.org 6November 2017 | Volume 5 | Article 55 Suhonen Effective Value of gA Using these prescriptions one can eliminate the deficiencies of the nuclear many-body framework and obtain information about the quenching of gAin the nuclear medium (see section 3), beyond the effects caused by the deficiencies of a nuclear model. 4.2. Many-Body Aspects of the pnQRPA The random-phase approximation (RPA) is an extension of the Tamm-Dancoff model (TDM) in the description of magic nuclei (at closed major shells) by particle-hole excitations across the magic gaps between closed nuclear major shells [35,106]. In the RPA the simple particle-hole vacuum, with the singleparticle orbitals fully occupied up to the Fermi surface at the magic gap, is replace by the correlated vacuum, containing twoparticle–two-hole, four-particle–four-hole, etc. excitations across the magic gap. The use of the correlated vacuum in the RPA enhances the strength of collective transitions [35,106]. Its quasiparticle version, quasiparticle RPA (QRPA) describes openshell nuclei, outside the closures of magic shells, by replacing the particle-hole excitations by two-quasiparticle excitations. Usually these quasiparticles are generated by the use of the Bardeen-Cooper-Schrieffer (BCS) theory [107] from the shortrange interaction part of the nuclear Hamiltonian in an eveneven reference nucleus. The quasiparticles can be viewed as partly particles and partly holes, inducing fractional occupancies of the nuclear single-particle orbitals and leading to a smeared Fermi surface for protons and/or neutrons for open-shell nuclei. The proton-neutron version of the QRPA (pnQRPA) uses twoquasiparticle excitations that are built from a proton and a neutron quasiparticle. This enables description of odd-odd nuclei starting from the even-even BCS reference nucleus. The strong point of the pnQRPA theory is that it can include large single-particle valence spaces in the calculations. There are no problems associated with leaving spin-orbit-partner orbitals out of the computations. On the other hand, the pnQRPA has a limited configuration space, essentially including twoquasiparticle excitations on top of a correlated ground state [35]. Deficiencies of the pnQRPA formalism have been analyzed against the ISM formalism, e.g., in Menéndez [21] by using a seniority-based scheme (seniority was defined earlier, at point (c) in section (1). In that work the pnQRPA was considered to be a low-seniority approximation of the ISM. But on the other hand, the ground-state correlations of the pnQRPA introduce higher-seniority components to the pnQRPA wave functions and the deficiencies stemming from the incomplete seniority content of the pnQRPA should not be so bad [108]. Also the renormalization problems of the two-body interaction are not so severe as in the ISM due to the possibility to use large single-particle model spaces. On the other hand, it is harder to find a perturbative scheme for the effective Hamiltonian due to the incompleteness of the available many-body configuration space. Due to this, schematic or G-matrix-based boson-exchange Hamiltonians have widely been used (see section 5.2). In any case, the configuration content of the pnQRPA is limited and extensions and improvements of the theory framework are wanted in order to see how the quenching problem of gAevolves with these extensions and improvements. Such extensions have been devised, including, e.g., the renormalized QRPA (RQRPA) [109,110] and similar “fully” renormalized schemes [111–113]. Another possible improvement of the pnQRPA is the relativistic quasiparticle time-blocking approximation (RQTBA), in particular its protonneutron version, the pn-RQTBA, advocated in Robin and Litvinova [114]. It shows good promise for improvements over the β-decay calculations of the ordinary pnQRPA the use of which clearly points out to need for a quenched value of gAin β-decay calculations, as discussed in section 5.2. The (charge-conserving) QRPA framework, with linear combinations of proton-proton and neutron-neutron quasiparticle pairs, phonons [35], can be used to describe (collective) excitations of even-even nuclei (collectivity is where the name phonon stems from). These, in turn, can be used as reference nuclei in building the excitations of the neighboring odd-mass (odd-proton or odd-neutron) nuclei by coupling the QRPA phonons with proton or neutron quasiparticles. This phonon-quasiparticle coupling can be carried out in a microscopic way, based on a realistic effective residual Hamiltonian. This has been achieved, e.g., in the microscopic quasiparticle-phonon model (MQPM) [115,116] where a microscopic effective Hamiltonian based on the Bonn G matrix has been used to produce the oneand three-quasiparticle states in odd-mass nuclei. This extension of the QRPA has been used to describe βdecays, and in particular in connection with the renormalization problem of gA, as discussed in section 9. It should be noted that odd-mass nuclei can also be described by starting from an odd-odd reference nucleus, described by the pnQRPA phonons [35]. By coupling either proton or neutron quasiparticles with pnQRPA phonons one can, again, create the states of either a neutron-odd or a proton-odd nucleus. This approach was coined the proton-neutron MQPM (pnMQPM) and was used to describe forbidden beta decays in Mustonen and Suhonen [117]. Although the pnQRPA-based phonons better take into account the Ikeda sum rule [35,76] and the GamowTeller giant-resonance region of the β−-type strength function, the pnMQPM lacks the important three-proton-quasiparticle and three-neutron-quasiparticle contributions, essential for good reproduction of the low-energy spectra of odd-mass nuclei. This is why its use in β-decay calculations has been very limited. 4.3. Many-Body Aspects of the IBM In its simplest version, the interacting boson model (IBM), the theory framework consists of sand dbosons which have as their microscopic paradigms the 0+and 2+coupled collective Fermion pairs present in nuclei. Even a mapping of the collective Fermion pairs to these bosons can be devised [91]. An extension of the IBM is the microscopic IBM (IBM-2) where the proton and neutron degrees of freedom are explicitly separated. The IBM and IBM-2 are sort of phenomenological versions 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 sand dbosons as lowest-order boson expansions with coupling coefficients to be determined by fits to experimental data or by relating them to the underlying fermion valence space through a mapping procedure [118,119]. Thus, the IBM Frontiers in Physics | www.frontiersin.org 7November 2017 | Volume 5 | Article 55 Suhonen Effective Value of gA and its extensions use more or less phenomenological operators mimicking the renormalized operators used in the ISM (see section 4.1). The two versions of the IBM can be extended to include higher-multipole bosons, like gbosons, as well. Further extension concerns the description of odd-mass nuclei by the use of the interacting boson-fermion model (IBFM) and its extension, the microscopic IBFM (IBFM-2) [92]. The IBM concept can also be used to describe odd-odd nuclei by using the interacting bosonfermion-fermion model (IBFFM) and its proton-neutron variant, the proton-neutron IBFFM (IBFFM-2) [120]. Here the problems arise from the interactions between the bosons and the one or two extra fermions in the Hamiltonian, and from the transition operators containing a host of phenomenological parameters to be determined in some way. The IBM-2 and the IBFFM-2 have been used to access the renormalization of gA, as described in section 10.2. 5. EFFECTIVE VALUE OF GAIN ALLOWED GAMOW-TELLER βDECAYS Gamow-Teller decays are mediated by the Pauli spin operator σ and they are thus able to change the initial nuclear spin Jiby 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-toground-state β−and β+/EC transitions between even-even 0+ and odd-odd 1+ground states in the A=100 Zr-Nb-Mo-TcRu 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, MGT, 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 ML(MR) in the case it is to the left (right) of the central nucleus. The corresponding reduced 1ENSDF at NNDC site, http://www.nndc.bnl.gov/ transition probability BGT can be written as BGT =g2 A 2Ji+1|MGT|2, (17) where Jiis the spin of the ground state of the initial nucleus, gAis the weak axial-vector coupling strength, substituted by the effective coupling strength geff Aof Equation (8) in practical calculations of the β-decay rates involving nuclear levels of low excitation energy [Hence, the coupling strength gAis probed at the q2→0 limit in (5)]. It is worth noting that the Gamow-Teller decays probe only gA, not gVwhich is carried by the vector part (Fermi spin-zero operator) of the βtransitions, not active for the here discussed 1+↔0+transitions due to the conservation of angular momentum. The comparative half-lives (log ft values) of the 1+↔0+ Gamow–Teller transitions are given in terms of the reduced transition probabilities as given in Suhonen [35] log ft =log10(f0t1/2[s]) =log10 6147 BGT (18) for the β+/EC or β−type of transitions. The half-life of the initial nucleus, t1/2, 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. 5.1. Interacting Shell Model Traditionally the renormalization of the axial-vector coupling strength has been addressed in the context of the ISM in a wealth of calculations pertaining to Gamow-Teller βdecays of very light (p-shell), light (sd-shell), and medium-heavy (pf -shell and sdgshell) nuclei. In these calculations it appears that the value of gAis quenched. As indicated by the ISM results below, the quenching factor (6) is roughly a decreasing function of the nuclear mass number A, implying stronger quenching with increasing nuclear mass. The studies can be grouped according to the mass regions as follows. FIGURE 1 | Gamow-Teller beta decays in the A=100 Zr-Nb-Mo-Tc-Ru region. Frontiers in Physics | www.frontiersin.org 8November 2017 | Volume 5 | Article 55 Suhonen Effective Value of gA 6.2. First-Forbidden Unique βDecays The first-forbidden unique βtransitions are mediated by a rank-2 (i.e., having angular-momentum content 2) parity-changing spherical tensor operator [a special case of the operator (53)], schematically written as O(2−). For these decays it is customary to modify the general structure of Equations (47)–(49) by replacing the phase-space factor fK=1,u of (49) by a 12 times larger phase-space factor f1u, i.e., f1u =12fK=1,u, (54) yielding a factor log 12 =1.079 times larger comparative halflives (18) than in the standard definition (47). In the quenching studies it is advantageous to use the simplest first-forbidden transitions, namely the ground-state-toground-state ones. In Figure 4 are depicted the first-forbidden unique ground-state-to-ground-state β−and β+/EC transitions between even-even 0+and odd-odd 2−ground states in the A=84 Kr-Rb-Sr isobaric chain. Shown is the lateral feeding from a middle odd-odd nucleus to adjacent even-even ground states. In the figure, as also in Figure 1 for the Gamow-Teller transitions, the NME is denoted by ML(MR) in case it is to the left (right) of the central nucleus. In the early work [145] a systematic schematic analysis of the first-forbidden unique βdecays was performed from the point of view of suppression factors stemming from the effect of E1 (electric dipole) giant resonance in the final odd-odd nucleus. In Towner et al. [146] the suppression mechanism of the firstforbidden and third-forbidden βdecays 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 [147] 19 first-forbidden unique ground-state-toground-state β-decay transitions were studied. The interesting transitions are the ones where both MLand MRNMEs are known experimentally, like in the case of Figure 4. The experimental FIGURE 4 | First-forbidden unique beta decays in the A=84 Kr-Rb-Sr isobaric chain. values of the NMEs can be deduced by using Equations (47) and (48) and by adopting the free value of the axial-vector coupling strength3. In this case one can use the geometric mean (40) of the left and right NMEs in the analysis, making the analysis more stable. In Ejiri et al. [147] a gphand gpp-renormalized Bonn-A G matrix was used as the two-nucleon interaction in a pnQRPA framework. The two-quasiparticle and pnQRPA NMEs were compared with the ones extracted from the measured comparative half-lives. Again the relations (44) and (45) can be used to obtain the value geff A≈0.45 ×1.27 =0.57 (55) for the effective axial-vector coupling strength using the pnQRPA wave functions. The average of the values of the leading twoquasiparticle NMEs gives in turn geff A(2qp) ≈0.18 ×1.27 =0.23, (56) implying the ratio k=¯ MpnQRPA ¯ Mqp =0.4 (57) and thus a drastic nuclear many-body effect when going from the two-quasiparticle level of approximation to the 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. 7. HIGHER-FORBIDDEN UNIQUE β DECAYS Early studies of the quenching in the secondand third-forbidden unique βdecays 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 βtransitions 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 βtransitions. Beyond this, the incentive to study the GamowTeller (section 5), first-forbidden unique (section 6.2), and higher-forbidden unique (this section) βdecays 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 Jπdue to the multipole expansion of the Majorana-neutrino propagator (see, e.g., [1–3,150–155]). Studies 3In Ejiri et al. [147] the Bohr-Mottelson (BM) formulation [148] of first-forbidden decays is used. The difference between the present and the BM formulation can be crystallized into the following relations: M(BM) =M1u/√4π,B(BM) = B1u/(4πg2 A), f1(BM) =3f1u/4. In addition, since gV(BM) =GFgVand gA(BM) = GFgA, one has to make replacements gA(BM) →gAand gV(BM) →1 in order to go from the BM formulation to the present one. Frontiers in Physics | www.frontiersin.org 15 November 2017 | Volume 5 | Article 55 Suhonen Effective Value of gA 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, geff A,0ν, multiplying the NME: M(0ν) GTGT =(geff A,0ν)2X Jπ (0+ f||O(0ν) GTGT(Jπ)||0+ i), (58) where O(0ν) GTGT denotes the transition operator mediating the 0νββ transition through the various multipole states Jπ, 0+ idenotes the initial ground state, and the final ground state is denoted by 0+ f(here, for simplicity, we assume a ground-state-to-groundstate transition). The effective axial coupling relevant for 0νββ decay is denoted as geff A,0νto 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 geff A,0ν 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: 0νββ −rate ∼M(0ν) GTGT 2=g4 A,0νX Jπ (0+ f||O(0ν) GTGT(Jπ)||0+ i) 2 . (59) The quenching related to the left-leg and right-leg βtransitions of Figure 5 can be studied by using the theoretical machinery FIGURE 5 | The 0νββ decay of 116Cd to 116Sn via the virtual intermediate states in 116In. The transitions between 116Cd (116Sn) and 116In constitute the left-leg (right-leg) transitions. of section 6.1. In Kostensalo and Suhonen [156] this machinery was applied to 148 potentially measurable second-, third-, fourth- , fifth-, sixthand 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 highforbidden unique 0+↔Jπ=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, MpnQRPA, and a two-quasiparticle model, Mqp, 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 βdecays 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, kNM =¯ Mexp ¯ MpnQRPA , (60) 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 βdecay exhausts 100% of the decay rate between two nuclear ground states; one example being the second-forbidden β−transition 54Mn(3+ gs)→54Fe(0+ gs), with a half-life 4.2(9) ×105years, 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 nNMEs, Mi, i=1, 2, ...n, of successive βtransitions with a common mother or daughter nucleus: ¯ M= n Y i=1 Mi!1/n . (61) 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 gpp of the pnQRPA. One can now define the ratio k=¯ MpnQRPA ¯ Mqp (62) Frontiers in Physics | www.frontiersin.org 16 November 2017 | Volume 5 | Article 55 Suhonen Effective Value of gA FIGURE 6 | Predicted half-lives and their error estimates (in parenthesis) for β−and EC (electron-capture) transitions in the isobaric chain A=116. The spin-parity assignments, decay energies (Qvalues) and life-times of the nuclear ground (gs) and isomeric (isom) states are experimental data and taken from ENSDF (http://www. nndc.bnl.gov/). The 2νββ half-life is taken from Barabash [157]. In addition to the half-lives the degree of forbiddenness and the leading single-particle transition are shown. FIGURE 7 | The same as Figure 6 for the secondand sixth-forbidden βdecays in the isobaric chain A=54. of the pnQRPA-calculated mean NME, ¯ MpnQRPA, and the mean two-quasiparticle NME, ¯ Mqp, computed by using (61). The ratio kis 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 Frontiers in Physics | www.frontiersin.org 17 November 2017 | Volume 5 | Article 55 Suhonen Effective Value of gA of gAdepends on the degree of complexity of the adopted nuclear model. In Kostensalo and Suhonen [156] the βtransitions 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 kof (62) since the BCS results tend to be unstable at magic shell closures. In Figure 8 the ratio kis shown for transitions belonging to GROUP 1. The same kdistribution is shown in terms of division to β−and EC/β+decays in Figure 9. From Figure 8 it is visible that the secondand fourthforbidden βtransitions 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 kcan 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 khave a scattered, decreasing trend), and A=122 −146 (a low-kregion with k∼0.2). The ratios kfor the three mass regions and for various degrees of forbiddenness Kare shown in Table 4 for βtransitions 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 FIGURE 8 | Ratio (62) as a function of the mass number Afor βtransitions involving solely non-magic reference nuclei. The degree of forbiddenness Kis indicated by color and shape of the symbol. FIGURE 9 | Ratio (62) as a function of the mass number Aseparated to β−and EC/β+K-forbidden transitions of Figure 8. Frontiers in Physics | www.frontiersin.org 18 November 2017 | Volume 5 | Article 55 Suhonen Effective Value of gA TABLE 4 | Ratio (62) for three mass regions and for various degrees of forbiddenness Kfor βtransitions belonging to GROUP 1. AGT [140]K=1 [147]K=2K=3K=4K=5K=6K=7 Avg. 50–88 0.35 0.40 0.25 0.46 0.43 0.43 – – 0.39 90–122 0.52 0.40 0.25 0.35 0.34 0.38 0.41 0.13 0.31 122–146 0.40 0.40 0.30 0.28 0.07 0.35 – 0.19 0.24 Average 0.42 0.40 0.27 0.36 0.28 0.39 0.41 0.16 0.31 The results of the Gamow-Teller (GT, see section 5.2) and first-forbidden (K =1, see section 6.2) decays are quoted for comparison. FIGURE 10 | Illustration of the values of the ratio k(62), taken from Table 4, for the three mass regions A=50 −88, A=90 −122, and A=122 −146 for different degrees of forbiddenness K. GT denotes the ratio kfor Gamow-Teller transitions. 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, khas 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 βtransitions, obeying the rule (−1)1J1π = +1, one can then speak about an effective axial coupling, geff A,0ν, 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, geff A,0ν, 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 Frontiers in Physics | www.frontiersin.org 19 November 2017 | Volume 5 | Article 55 Suhonen Effective Value of gA TABLE 5 | Values of the low-energy effective axial coupling, geff A,0νof (58), for the three mass regions of interest for 0νβ−β−-decay calculations in the pnQRPA framework. Mass range A=76–82 A=100–116 A=122–136 geff A,0ν0.7 −0.9 0.5 0.5 −0.7 coupling strength gA, 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. 8. QUENCHING OF GAIN FORBIDDEN NON-UNIQUE βDECAYS 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]. 8.1. Theoretical Considerations In the forbidden non-unique βdecay the half-life can be given, analogously to (47), in the form t1/2=κ/ ˜ C, (63) where ˜ Cis the dimensionless integrated shape function, given by ˜ C=Zw0 1 C(we)pwe(w0−we)2F0(Zf,we)dwe, (64) with the notation explained in section 6.1. The general form of the shape factor of Equation (64) is a sum C(we)=X ke,kν,K λkeMK(ke,kν)2 +mK(ke,kν)2−2γke kewe MK(ke,kν)mK(ke,kν),(65) where the factor λkewas given in (51) and Zfis the charge number of the final nucleus. The indices keand kν(k=1, 2, 3...) are related to the partial-wave expansion of the electron (e) and neutrino (ν) wave functions, Kis the order of forbiddenness of the transition, and γke=qk2 e−(αZf)2,α≈1/137 being the fine-structure constant. The nuclear-physics information is hidden in the factors MK(ke,kν) and mK(ke,kν), which are complicated combinations of the different NMEs and leptonic phase-space factors. For more information on the integrated shape function, see [144,159]. The quite complicated shape factor (65) can be simplified in the so-called ξapproximation when the coulomb energy of the emitted βparticle at the nuclear surface is much larger than the endpoint energy, i.e., ξ=αZf/2R≫w0, where Ris the nuclear radius. Then the forbidden non-unique transition can be treated as a unique one of the same 1J. Applicability of this approximation has recently been criticized in Mougeot [160]. 8.2. First-Forbidden Non-unique βDecays For the first-forbidden non-unique βdecays the shape factor (65) has to be supplemented with a 1J= |Ji−Jf| = 0 term C(1)(we) [144,158,161,162], where Ji(Jf) is the initial-state (final-state) spin of the mother (daughter) nucleus. Then the shape factor can be cast in the simple form [144,158,163] C(we)=K0+K1we+K−1/we+K2w2 e, (66) where the factors Kncontain the NMEs (6 different, altogether) of transition operators Oof angular-momentum content (rank of a spherical tensor) O(0−), O(1−), and O(2−), where the parity indicates that the initial and final nuclear states should have opposite parities according to Table 3. In the leading order these operators contain the pieces [148] O(0−):gA(γ5)σ·pe MN;igA αZf 2R(σ·r), (67) O(1−):gVpe MN;gA αZf 2R(σ×r);igV αZf 2Rr, (68) O(2−):i √3gA[σr]2qp2 e+q2 ν, (69) where pe(qν) is the electron (neutrino) momentum, rthe radial coordinate, and the square brackets in (69) denote angularmomentum coupling. The matrix elements of the operators (67) and (68) are suppressed relative to the Gamow-Teller matrix elements by the small momentum peof the electron and the large nucleon mass MNor the small value of the fine-structure constant α. The matrix element of (69) is suppressed by the small electron and neutrino momenta. The axial operator σ·peand vector operator rtrace back to the time component of the axial current Aµin (4) and vector current Vµin (3), and the rest of the operators stem from the space components of Vµand Aµ. The renormalization of these pieces is discussed next. The ξapproximation to the first-forbidden non-unique transitions has been discussed, e.g., [144,148,158]. One of the first analyses of first-forbidden non-unique transitions in this approximation was done in Bohr and Mottelson [148] for nuclei around 208Pb, based on the work of Damgaard and Winther [164]. Assuming certain dominant single-particle configurations around the double-closed shell at A=208, Bohr and Mottelson obtained two sets of values for the effective vector and axialvector coupling when analyzing the decay rates mediated by the rank-1 operators O(1−) in (68). Combining the two obtained values we obtain geff A(sp) =(0.5 −0.6) ×1.18 =0.46 −0.56, (70) where the symbol sp refers to single-particle estimate for the states involved in the βdecays in odd-Anuclei. It is interesting that also an effective value for the vector coupling was derived: geff V(sp) =0.3 −0.7. (71) Frontiers in Physics | www.frontiersin.org 20 November 2017 | Volume 5 | Article 55 Suhonen Effective Value of gA This deviates quite much from the canonical value gV=1 dictated by the CVC hypothesis [26]. Hence, strong nuclearmodel dependent effects are recorded in this case. In the case of the axial-vector strength the numbers of (70) can be compared with the ones extracted from the first-forbidden unique decays in the two-quasiparticle approximation for odd-odd nuclei. There, in Equation (56), a value geff A(2qp) ∼0.2 was obtained, implying that for the odd-odd systems the quenching is more drastic than for the odd-mass systems. All in all, a proper many-body treatment should reduce the quenching markedly, as shown by the factor kin (57), describing the transition from the twoquasiparticle approximation to the pnQRPA level in the case of the unique-forbidden βtransitions. In Ejiri et al. [145] a schematic study of the six NMEs corresponding to the operators (67)–(69) was performed. The hindrance factors associated with the NMEs were related to the E1 (electric dipole) giant resonance in a semi-quantitative way. The nuclear medium effect, in the form of the meson-exchange currents, on the σ·pepart of O(0−) in (67) was discussed in Kubodera et al. [42], Kirchbach and Reinhardt [79] and Towner [81]. This is the well-known (fundamental) enhancement of the γ5NME (axial charge ρ5, the time component of the axial current, see section 3), stemming from the renormalization of the pion-decay constant and the nucleon mass MNin nuclear medium [165] and exchange of heavy mesons [80,81]. In this review the corresponding coupling strength is coined geff A(γ5) for short. In Kirchbach and Reinhardt [79] a simple nuclear approach to the meson-exchange renormalization geff A(γ5)= (1 +δ)gfree Agave the following values of geff A(γ5) (below are given the studied nuclear masses and the corresponding active single-particle transitions): geff A(γ5)=1.90 A=16 (1s1/2→0p1/2) geff A(γ5)=1.96 A=18 (1s1/2→0p1/2) geff A(γ5)=1.84 A=96 (2s1/2→1p1/2) geff A(γ5)=1.78 A=206 (2p1/2→2p1/2) (72) The work of Kirchbach and Reinhardt [79] was extended by Towner [81] to include 6 nuclear masses and several singleparticle transitions for each mass. The resulting renormalization by the meson-exchange currents amounted to geff A(γ5)=2.0 −2.3 (A=16 −208) (73) for the masses A=16 −208. The above fundamental renormalization of the axial charge was contrasted with the nuclear-model dependent many-body effects by using the framework of the interacting shell model in several studies in the past. For very low masses, A=11 [166] and A=16 [167], some 40 −50% enhancement of the axial charge was obtained leading to geff A(γ5)=1.8−1.9. A further study [168] of the A=11 −16 nuclei indicated an enhanced axial charge of geff A(γ5)=2.04 ±0.04, (A=11 −16) (74) where the uncertainties come solely from the experimental errors, not from the uncertainties associated with the theoretical analyses. A general study of the first-forbidden non-unique decays was carried on in Warburton et al. [169] for 34 ≤A≤44, and a further comparison [170] with the measured rate of the β− decay of 50K indicated an enhanced value of geff A(γ5)=1.93 ±0.09, (A=50) (75) where the uncertainty is purely experimental. A thorough shell-model treatment of the mass A=205 −212 nuclei in the lead region was carried out in Warburton [171–173]. There a rather strongly enhanced value of geff A(γ5)=2.55 ±0.07 (A=205 −212) (76) was obtained for the axial charge. The uncertainty comes from the least-squares fit to 18 measured β-decay transitions in the indicated mass region. For the σ·roperator (space component of Aµ) essentially no renormalization (quenching, since space components tend to be quenched opposite to the enhancement of the time component, see beginning of section 3) was obtained: gA/gfree A(0−)=0.97 ±0.06. The value (76) is notably larger than those obtained for the lower masses and also larger than the leadregion results of Towner (73). However, in Kubodera and Rho [165] the theoretical result geff A(γ5)=2.5 ±0.3 (A=205 −212) (77) was obtained by adopting an effective Lagrangian incorporating approximate chiral and scale invariance of QCD. This seems to confirm the phenomenological result of Warburton [172,173]. For further information see the review [174]. For the O(1−) operator σ×rin (68) the analyses of Warburton [171,173] yielded the effective values geff A(1−)∼0.6;gV(1−)∼0.6 (Warburton) (78) due to core-polarization effects caused by the limited model space used. In the work Rydstrom [175] a shell-model study of the firstforbidden transition 205Tl(1/2+ gs)→205Pb(1/2−) yielded the effective values geff A(1−)∼0.43 −0.65;gV(1−)∼0.38 −0.85. (Rydström et al.) (79) The shell-model analysis of Suzuki et al. [163] of the N=126 isotones suggests a large quenching for geff A(1−) but a large quenching of geff V(1−) is not necessarily needed for most of the studied cases, contrary to (78) and (79), in accordance with the CVC hypothesis [26]. In the work [176] half-lives of a number of nuclei at the magic neutron numbers N=50, 82, 126 were analyzed by comparing results of large-scale shell-model calculations with experimental data. Both Gamow-Teller and first-forbidden βdecays were included in the analysis. By performing a least-squares fit to the experimental data the following quenched weak couplings Frontiers in Physics | www.frontiersin.org 21 November 2017 | Volume 5 | Article 55 Suhonen Effective Value of gA were extracted: For the enhanced γ5matrix element the value geff A(γ5)=1.61 was obtained and for the σ·rpart the quenching gA/gfree A(0−)=0.66 was obtained. For the 1−part the quenched values read geff A(1−)∼0.48;gV(1−)∼0.65. (Zhi et al.) (80) Interestingly, also for the first-forbidden unique operator O(2−) of (69) a quenching geff A(2−)∼0.53 (Zhi et al.) (81) was obtained. This is not far from the result geff A∼0.57 [see Equation (55)] obtained in the analysis of the first-forbidden unique βdecays in Ejiri et al. [147]. The above considerations for the vector coupling coefficient gVare in conflict with the CVC hypothesis [26] and the findings of [177] where the shape of the computed β-electron spectrum was compared with that of the measured one for the fourthforbidden β−decay of 113Cd. This comparison confirmed an unquenched value gV=1.0 for the vector coupling coefficient, in accordance with the CVC hypothesis. For more discussion of the related method for highly-forbidden βdecays, see section 9. 9. HIGHER-FORBIDDEN NON-UNIQUE β DECAYS 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 gV(vector part) and gA(axial-vector part) is very nontrivial. In fact, the shape factor C(we) (65) can be decomposed into vector, axial-vector and mixed vector-axial-vector parts in the form [177] C(we)=g2 VCV(we)+g2 ACA(we)+gVgACVA(we). (82) Integrating equation (82) over the electron kinetic energy, we obtain an analogous expression for the integrated shape factor (64) ˜ C=g2 V˜ CV+g2 A˜ CA+gVgA˜ CVA, (83) where the factors ˜ Ciin 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 βspectrum shapes of both transitions are highly sensitive to the values of gVand gAand 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 gAthe best fits to data were obtained by using the canonical value gV=1.0 for the vector coupling strength. This result is in conflict with those obtained by analyzing first-forbidden non-unique βdecays in section 8.2, where strongly quenched values of gVwere 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, gA≈0.92, when the SSM is applied to the available experimental βspectrum [179] of 113Cd. The result is illustrated in Figure 11 where the three curves overlap best at the values geff A=0.92 (MQPM), geff A=0.90 (ISM), and geff A=0.93 (IBFM-2). The agreement of the β-spectrum shapes computed in the three different nucleartheory frameworks speaks for the robustness of the SSM in determining the effective value of gA. 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, CV(we) and CA(we) are roughly of the same size whereas the magnitude of the component CVA(we) is practically the sum of the previous two, but with opposite sign. Hence, for the whole FIGURE 11 | Comparison of the computed βspectra of 113Cd with the experiment. The next-to-leading-order corrections to the shape factor have been included, and only the best matches are shown in the figure. The canonical value gV=1.0 is used for the vector coupling strength. The areas under the curves are normalized to unity. Frontiers in Physics | www.frontiersin.org 22 November 2017 | Volume 5 | Article 55 Suhonen Effective Value of gA FIGURE 12 | Components CV,CA, and CVA of (82) for the electron spectra of the β−decay of 113Cd as computed by the three nuclear models discussed in the context of Figure 11. Note that the contribution of CVA is negative. 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 βspectra with changing value of gAwas followed for 26 first-, second-, third-, fourthand fifth-forbidden β−decays of odd-Anuclei 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 thirdand fourth-forbidden non-unique decays depend strongly on the value of gA, whereas the firstand second-forbidden decays were practically insensitive to the variations in gA. Furthermore, the gA-driven evolution of the normalized βspectra 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 qFof section 3 for (highly) forbidden βtransitions. 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 βspectra 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 gAbut 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. In Figure 14 a comparison of the MQPM (left panel) and ISM (right panel) calculations [181] for the βspectrum of the second-forbidden non-unique decay transition 99Tc(9/2+)→ 99Ru(5/2+) is shown. Again there is clear sensitivity to the value of gA, at the level of the 94Nb and 98Tc transitions, but the remarkable thing is that the spectrum shapes computed by the two nuclear models agree almost perfectly, giving further evidence in favor of the robustness of the SSM. Again, experimentally, the branching to this decay channel is practically 100% so that the βspectrum is potentially well measurable. Finally, In Figure 15 the βspectrum of the second-forbidden non-unique decay-transition 137Cs(7/2+)→137Ba(3/2+) is shown. Here the spectrum shape is quite independent of the value of gAand has exactly the same computed shape for the two applied nuclear-model frameworks: the MQPM and the ISM [181]. The robustness of the β-spectrum shape against variations in gAand the calculational scheme makes the measurement of this spectrum interesting in terms of testing the basic framework of high-forbidden non-unique βdecays. The cause of the inertia against variations of gAis seen in Table 6 in the decomposition (83) of the dimensionless integrated shape function ˜ Cfor the decays of both 135Cs and 137Cs. It is seen that for these two decays all the components of ˜ Care of the same sign, thus adding coherently. Hence, changes in the value of gAdo not affect the spectrum shape, contrary to those decays where there is a destructive interference between the axial-vector and mixed components of (83), like in the cases of Figures 11–14, further analyzed in Table 8. Table 7 summarizes the exploratory works of Haaranen et al. [177,178], Kostensalo et al. [180] and Kostensalo and Suhonen [181] in terms of listing the studied decay-transition candidates and their potential for future measurements. Here only the studied non-unique β-decay transitions are listed since the unique forbidden transitions are practically gA-independent even when the next-to-leading-order terms are included in the β-decay shape factor [177]. The most favorable cases for measurements are the ones that have a strong dependence on gA and the branching to the final state of interest is close to 100%. By Frontiers in Physics | www.frontiersin.org 23 November 2017 | Volume 5 | Article 55 Suhonen Effective Value of gA TABLE 6 | Dimensionless integrated shape functions ˜ C(83) and their vector ˜ CV, axial-vector ˜ CA, and mixed components ˜ CVA for the forbidden non-unique βdecays of 135Cs and 137Cs. Transition KNucl. model ˜ CV˜ CA˜ CVA ˜ C 135Cs(7/2+)→135Ba(3/2+) 2 MQPM 1.133 ×10−81.656 ×10−82.737 ×10−85.526 ×10−8 137Cs(7/2+)→137Ba(3/2+) 2 MQPM 3.217 ×10−52.654 ×10−55.822 ×10−51.169 ×10−4 137Cs(7/2+)→137Ba(3/2+) 2 ISM 4.211 ×10−62.836 ×10−66.879 ×10−61.392 ×10−5 The forbiddenness K and the nuclear model used to calculate ˜ C is given. For the total integrated shape factor ˜ C the values of the coupling strengths were set to gV=gA=1.0. TABLE 7 | List of the studied high-forbidden non-unique β−-decay transitions and their sensitivity to the value of gA. Transition Jπi i(gs) Jπf f(nf) Branching (%) KSensitivity Nucl. model 36Cl →36Ar 2+0+(gs) 98 2 None ISM 48Ca →48Sc 0+4+(2) ∼0 4 None ISM 48Ca →48Sc 0+6+(gs) ∼0 6 None ISM 50V→50Cr 6+2+(1) ∼0 4 Weak ISM 60Fe →60Co 0+2+(1) 100 2 None ISM 85Br →85Kr 3/2−9/2+(gs) ∼0 3 Moderate MQPM 87Rb →87Sr 3/2−9/2+(gs) 100 3 Moderate MQPM, ISM 93Zr →93Nb 5/2+9/2+(gs) 5≤2 Weak MQPM 94Nb →94Mo 6+4+(2) 100 2Strong NSM 96Zr →96Nb 0+4+(2) ∼0 4 None ISM 96Zr →96Nb 0+6+(gs) ∼0 6 Strong ISM 97Zr →97Nb 1/2+9/2+(gs) ∼0 4 Strong MQPM 98Tc →98Ru 6+4+(3) 100 2Strong ISM 99Tc →99Ru 9/2+5/2+(gs) 100 2Strong MQPM, ISM 101Mo →101Tc 1/2+9/2+(gs) ∼0 4 Strong MQPM 113Cd →113In 1/2+9/2+(gs) 100 4Strong MQPM, ISM, IBFM-2 115Cd →115In 1/2+9/2+(gs) ∼0 4 Strong MQPM 115In →115Sn 9/2+1/2+(gs) 100 4Strong MQPM, ISM, IBFM-2 117Cd →117In 1/2+9/2+(gs) ∼0 4 Strong MQPM 119In →119Sn 9/2+1/2+(gs) ∼0 4 Strong MQPM 123Sn →123Sb 11/2−1/2+(4) ∼0 5 Weak MQPM 126Sn →126Sb 0+2+(5) 100 2 None ISM 135Cs →135Ba 7/2+3/2+(gs) 100 2 None MQPM 137Cs →137Ba 7/2+3/2+(gs) 5.4 2 None MQPM, ISM 125Sb →125Te 7/2+9/2−(3) 7.2 1 None MQPM 141Ce →141Pr 7/2−5/2+(gs) 31 1 Weak MQPM 159Gd →159Tb 3/2−5/2+(1) 26 1 None MQPM 161Tb →161Dy 3/2+5/2−(1) ∼0 1 None MQPM 169Er →169Tm 1/2−3/2+(1) 45 1 None MQPM Here Ji(Jf) is the angular momentum of the initial (final) state, πi(πf) the parity of the initial (final) state, and K the degree of forbiddenness. The initial state is always the ground state (gs, column 2) of the mother nucleus and the final state is either the ground state (gs) or the nf:th, nf=1,2,3,4,5, excited state (column 3) of the daughter nucleus. Column 4 gives the branching to this particular decay channel [with boldface if (almost) 100%], column 5 indicates the sensitivity to the value of gA(with boldface if strong), and the last column lists the nuclear models which have been used (thus far) to compute the β-spectrum shape. these criteria the best candidates for measurements are the nonunique transitions 94Nb(6+)→94Mo(4+) (second-forbidden), 98Tc(6+)→98Ru(4+) (second-forbidden), 99Tc(9/2+)→ 99Ru(5/2+) (second-forbidden), 113Cd(1/2+)→113In(9/2+) (fourth-forbidden), and 115In(9/2+)→115Sn(1/2+) (fourthforbidden). Plans for accurate measurements of (some) of these transitions are on-going in the DAMA (V. Tretyak, private communication) and COBRA collaborations (K. Zuber, private communication). It should be noted that also the transition 87Rb(3/2−)→87Sr(9/2+) could be of interest for measurements since it has a 100% branching and the corresponding βspectrum is moderately sensitive to gA. In Table 8 the dimensionless integrated shape functions ˜ C (83) have been decomposed into their vector ˜ CV, axial-vector ˜ CAand mixed vector-axial-vector components ˜ CVA for the experimentally most promising forbidden non-unique βdecays Frontiers in Physics | www.frontiersin.org 24 November 2017 | Volume 5 | Article 55 Suhonen Effective Value of gA 8. Deppisch FF, Harz J, Hirsch M, Huang WC, Päs H. Falsifying high-scale baryogenesis with neutrinoless double beta decay and lepton flavor violation. Phys Rev D (2015) 92:036005. doi: 10.1103/PhysRevD.92.036005 9. Deppisch F, Pas H, Suhonen J. Double beta decay versus cosmology: majorana CP phases and nuclear matrix elements. Phys Rev D (2005) 72:033012. doi: 10.1103/PhysRevD.72.033012 10. Suhonen J, Civitarese O. 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The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms. Frontiers in Physics | www.frontiersin.org 35 November 2017 | Volume 5 | Article 55