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Effective value of gA in β and ββ decays

Effective value of the weak axial-vector coupling strength, gA, in nuclear β and ββ (double beta) decays is discussed. Both Gamow-Teller and first-forbidden β decays are included in the analyses. Quenching of gA in β and two-neutrino ββ decays is reviewed and impact of this quenching on neutrinoless ββ decays is addressed. New measurements of β spectra of first-forbidden non-unique β decays are encouraged, to learn about the mesonic enhancement of the weak axial charge in these decays.


Introduction
The impact of the quenching of g A on the half-lives of neutrinoless double beta (0νββ) decay has recently been discussed in Ref. [1]. The related decay rates are affected by the available phase space (Q values), the nuclear matrix elements (NMEs) and the value of g A in its fourth power [2,3,4,5]. The 0νββ decay is mediated by Majorana neutrinos and the measurements of the related half-lives offer access to the absolute mass scale of the neutrinos [2]. A number of nuclear models, including configuration-interaction based models like the interacting shell model (ISM) and the proton-neutron quasiparticle random-phase approximation (pnQRPA), and various mean field models, have been adopted for the calculations [5,6,7].
Surprisingly little attention has been paid, at least in the theory community, to the possible (large) quenching of g A and its possibly strong impact on the sensitivities of the present and planned 0νββ-decay experiments [1]. This deviation (quenching or sometimes enhancement) from the free-nucleon value g A = 1.27 can arise from the nuclear medium effects and the nuclear many-body effects. The former contain quenching related to the presence of spin-multipole giant resonances [8], non-nucleonic degrees of freedom (like the ∆ isobar [9,10]) and meson-exchangedriven two-body weak currents [11,12,13]. The latter relates to deficiencies of the nuclear many-body approaches used to compute the wave functions involved in the decay transitions. The effective value of g A can also depend on the energy scale of the process in question: the effective value can be different for β decays (zero-momentum-exchange limit) and 0νββ decays (high momentum exchanges, ∼ 100 MeV).
The effective value of g A relates to the renormalization factor q (in case of quenching it is called quenching factor and in case of enhancement enhancement factor ): where g free A = 1.2723(23) [14] is the free-nucleon value of the axial-vector coupling as measured in neutron beta decay. Here g A is the value of the axial-vector coupling derived from a given 2 1234567890 ''"" 2. Quenching of g A in Gamow-Teller β decays Gamow-Teller decays are mediated by the Pauli spin operator σ [15] and they thus change the initial nuclear spin J i by at most one unit. The renormalization of g A has long been studied for the Gamow-Teller β decays in the framework of the interacting shell model (ISM). In these calculations, reviewed in Table 1, it appears that the value of g A is quenched, and the stronger the larger the nuclear mass A. In Fig. 1 the ISM results of Caurier et al. [24] (red horizontal bars indicating the mass range) are contrasted against those obtained by the use of the proton-neutron quasiparticle randomphase approximation (pnQRPA) in the works [27,28,29] (see also [30]). The pnQRPA results constitute the light-hatched regions in the background of the ISM results. As can be seen in the figure, the ISM results and the pnQRPA results are commensurate with each other, which is non-trivial considering the large differences in their many-body philosophy.

Quenching of g A in two-neutrino ββ decays
Recently the possibly decisive role of g A in the half-life and discovery potential of the 0νββ experiments has surfaced [1,31]. In Barea et al. [31] a comparison of the experimental and computed 2νββ half-lives of a number of nuclei yielded the rather surprising result where A is the mass number and IBM-2 stands for the microscopic interacting boson model. The results (3) and Suhonen2014 [34]. For more information see the text.
quenched effective values of g A are possible, thus being a threat to the discovery potential of the 0νββ experiments.
Although the study [31] was the first to draw considerable attention in the experimental 0νββ community, it was not the first one to point to a possible strong quenching of g A . Already the pnQRPA study of Faessler et al. [32] gave indications of a heavily quenched effective g A , in the range g eff A = 0.39 − 0.84. These results, along with their 1σ errors, are shown in Fig. 1 as black vertical bars. Later a similar study was carried out in [33,34], with results comparable with those of [32] and depicted in Fig. 1 as green vertical bars.
In Suhonen [1] a two-stage fit of the particle-particle parameter g pp of the pnQRPA to the data on two-neutrino ββ decays was performed along the lines first introduced inŠimkovic et al. [35] and later used in Hyvärinen et al. [36]. The works [35,36] were extended in [1] to include also strongly quenched values of g A . In this analysis it turned out that there is a minimum value of g A for which the maximum NME can fit the 2νββ-decay half-life. This lower limit of the possible g A values is presented in Fig. 1 as a solid black line and it is seen that it is consistent with the thick green vertical bars of g A ranges obtained in [33,34] and also commensurate with the thin black vertical bars obtained in [32]. However, the main message of Suhonen [1] is that no matter how quenched the value of g A is, the half-lives of the present and future neutrinoless ββ-decay measurements are only affected by factors of 6 or less.
4. Nuclear-structure effects in forbidden unique β decays Forbidden unique β decays are the simplest beyond the allowed β decays since, in the leading order, only one nuclear matrix element is involved in the process and thus the spectrum shape is a universal one, essentially independent of nuclear structure. The relation between the where J i is the angular momentum of the mother nucleus and κ is a constant with value κ = 6147 s [37]. The involved NME is given by where the factors M (Ku) (ab) are the single-particle matrix elements and the quantities (ψ f ||[c † acb ] K+1 ||ψ i ) are the one-body transition densities, ψ i being the initial-state and ψ f the final-state wave function. The operator c † a is a creation operator for a nucleon in the orbital a and the operatorc a is the corresponding annihilation operator. The single-particle matrix elements are given by where Y K is a spherical harmonic of rank K, r the radial coordinate, and a and b stand for the single-particle orbital quantum numbers. The NME (6) is given explicitly in [15].
In [38] a systematic study of the 2 − gs ↔ 0 + gs transitions between an odd-odd ground state 2 − and an even-even ground state 0 + was performed in cases where experimental data on the log f t values of Eq. (4) are available. The associated NME M 1u of (5) was calculated by using a simple two-quasiparticle model (M qp ) and the pnQRPA (M pnQRPA ). HereM = √ M L M R represents the geometric mean of the NMEs corresponding to decay transitions between the left-side (M L ) or right-side (M R ) nucleus and the central one. Use of the geometric mean makes the analysis more stable and very weakly dependent on the value of the particle-particle interaction parameter g pp of the pnQRPA. Comparing theM qp NME, theM pnQRPA NME and the experimental NMĒ M exp (extracted by using the free value g A = 1.27) the following overall ratios for the mass range A = 72 − 132 were obtained. Here "NM" denotes the nuclear-medium effects (isobaric resonances, mesonic two-body currents, etc.) combined with the nuclear-model effect related to the imperfection of the nuclear model used in the calculations. The work of [38] was followed by the study of Ref. [39] where the comparison of the NMEs M qp andM pnQRPA was made for 148 higher-forbidden unique β decays to access the ratio k of Eq. (7). Here lack of experimental data led only to an extrapolation, based on the results of Refs. [27] and [38], for the value of k NM .

Enhancement of the axial charge in first-forbidden
The enhancement of the axial-charge nuclear matrix element (NME) γ 5 due to meson-exchange currents was first suggested nearly four decades ago [40,41,42]. An enhancement of 40-70 % over the impulse-approximation value was predicted based on chiral-symmetry arguments and softpion theorems. This enhancement is fundamental in nature and insensitive to nuclear-structure aspects [43,44]. Since the γ 5 NME is one of the two rank-zero matrix elements contributing to first-forbidden ∆J = 0 transitions it plays an important role in many of these transitions. Therefore, a significant enhancement of this matrix element can also affect the shapes of the corresponding beta spectra. The half-life of a first-forbidden non-unique beta decay can be written as where κ was defined in connection with Eq. (4) andC is the dimensionless integrated shape function, given byC where w e is the total energy of the emitted electron, w 0 its maximum, p e is the electron momentum, Z f is the charge number of the daughter nucleus and F 0 (Z f , w e ) is the Fermi function taking into account the coulombic attraction of the electron and the daughter nucleus. The shape factor C(w e ) of Eq. (9) where r is the radial coordinate and p e is the electron momentum. The enhancement of the γ 5 NME (σ · p e in non-relativistic form) is included in the coupling strength g A (γ 5 ). The enhancement of the axial-charge NME affects not only the partial half-lives of the J − ↔ J + first-forbidden β transitions but also the shapes of the corresponding electron spectra. An example is given in Fig. 2. A clear effect is induced by the values of both the g A (γ 5 ) and g A , multiplying the matrix elements [σr] 0,1,2 of Eqs. (10)- (12). Experimental measurements of the β spectra would thus be welcome in order to acces the γ 5 enhancement independent of the half-life considerations.

Final remark
The discussion above gives only scattered examples in relation to the effective values of the axialvector coupling strength, as also to the nuclear-structure effects involved in model calculations. A more exhaustive discussion of these subjects is provided in the review [46].