Neutrino-nuclear responses and the effective value of weak axial coupling

. On-going measurements of the neutrinoless ββ decay are accompanied by the growing interest in computing the values of the associated nuclear matrix elements. In order to extract the neutrino mass from the potentially measured ββ half-lives one not only needs to know the values of the nuclear matrix elements but also the e ﬀ ective value of the weak axial-vector coupling constant g A since its value a ﬀ ects strongly the ββ half-lives. In order to gain knowledge of the possible quenching of g A in ﬁnite nuclei one can study, e.g., allowed Gamow-Teller β decays. A new promising tool to study the quenching are the measurements of ordinary muon capture transitions for which the range of momentum exchange, some 100 MeV, corresponds to the one of neutrinoless ββ decay.


Introduction
The neutrinoless double beta (0νββ) decay of atomic nuclei can be mediated by a massive Majorana neutrino.The implications of detecting this decay are far-reaching and discussed in recent reviews [1,2,3].In the case of 0νββ decay a lot of discussion is concentrated on an accurate calculation of the associated nuclear matrix elements (NMEs).However, in addition to the NMEs one needs to know the effective value of the weak axial-vector coupling g A since the ββ half-life is quite sensitive to it [4].The effective axial coupling relevant for 0νββ decay can be denoted as g eff A,0ν (J π ) since, in principle, it can depend on the multipole J π of a state in the intermediate nucleus.The low momentumexchange limit of this coupling, where q denotes the exchanged momentum, can in principle be determined in single β and two-neutrino double beta (2νββ) decays [3].In particular, the Gamow-Teller β and 2νββ decays can access the usual effective g A , namely In addition to the β and ββ decays the effective value of the axial coupling plays a role in neutrino and astrophysics e.g. in the form of low-energy neutrino-nucleus scattering (solar and supernova neutrinos) and nuclear muon capture.Deviations of the effective value from the bare nucleon value g A = 1.27 can stem from shifts of decay strengths to isovector giant multipole resonances and to non-nucleonic degrees of freedom, like the ∆ resonances [3].Such effects can also be produced by nucleon currents beyond the simple impulse approximation, like the two-body mesonexchange currents [5], or deficiencies in the nuclear many-body approaches, like too restricted single-particle valence spaces, lack of important many-nucleon configurations and omission of three-nucleon forces [3,5].

Values of g eff
A from Gamow-Teller β decays 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 Fig 1, it appears that the value of g A is quenched, and the stronger the heavier the nucleus.The renormalization of g A in the ISM includes all the possible sources of deficiency listed in the introduction.In Fig. Iwata et al. [7] (the cross at the mass number A = 48), Martínez-Pinedo et al. [8] (M-P1996, gray hatched horizontal box), Kumar et al. [9] (the two black horizontal boxes), Horoi et al. [10] (horizontal dashed line) and Siiskonen et al. [11] (crosses inside circles) are contrasted against those obtained using the proton-neutron quasiparticle random-phase approximation (pnQRPA) in the works [12,13,14] (see the reviews [3,15]).The pnQRPA results constitute the lighthatched regions in the background of the ISM results.The width of the regions reflects the rather large variation of the determined g eff A for β-decay transitions in different isobaric chains (for more information see the reviews [3,15]).As can be seen in the figure, the trends of the ISM results and the pnQRPA results are similar, which is non-trivial considering the drastic differences in their many-body philosophy.[4], Caurier2012 [6], Faessler2007 [17], Suhonen2014 [19] and Horoi2016 [10].These studies are contrasted with the ISM Gamow-Teller β-decay studies of M-P1996 [8], Iwata2016 [7], Kumar2016 [9] and Siiskonen2001 [11].The dashed and dotted curves come from the analysis of [16].The very recent results [5] for light nuclei in the mass range A ≤ 46, as obtained by using ab-initio methods including the two-body meson-exchange currents, are also included in a schematic way.For more information see the body of text.
The analyses of Barea et al. [16] of 2νββ half-lives against results of the ISM (the dotted ββ ISM curve in Fig. 1) and the microscopic interacting boson model (the dashed ββ IBM-2 curve in Fig. 1) give a similar trend as the ISM and pnQRPA analyses.The combined β-decay and 2νββ-decay analyses of Faessler et al. [17] (vertical solid bars) and Suhonen et al. [18,19] (vertical dashed bars) for A = 100, 116, 128 indicate strong variation in the effective value of g A , partly consistent with the curves of Barea et al. [16] and the pnQRPA analyses of the Gamow-Teller β decays.In Suhonen [4] a two-stage fit of the particle-particle parameter g pp of the pnQRPA to the data on two-neutrino ββ decays was performed.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 broken black line.It is seen that it is in line with the dashed vertical bars of g A ranges obtained in [18,19] and with the solid vertical bars obtained in [17].
Here it is appropriate to note that the effective value of g A can also be enhanced, as in the case of first-forbidden J + ↔ J − decays.In these cases the enhancement is coming from the two-body meson-exchange currents affecting the axial-charge nuclear matrix element and there is an interference of this enhancement and the quenching related to the usual sources of quenching of g A [15,20,21].

Ordinary muon capture and 0νββ decay
The ordinary muon capture (OMC) is a process where a negative muon in an atomic orbit is captured by the nucleus quite like in the ordinary electron capture of a nucleus, except the rest mass of the muon is some 200 times the rest mass of an electron.The process can formally be written as where the muon (µ − ) is captured by the 0 + ground state of the even-even nucleus X of mass number A and atomic number Z leading to the J π states of its odd-odd isobar Y of atomic number Z −1; here J is the angular momentum and π the parity of the final state.At the same time a muon neutrino ν µ is emitted.Thanks to the involved large momentum exchange, q ∼ 50 − 100 MeV/c, the OMC can lead to final nuclear states that are both highly excited and of high multipolarity J π , quite like in the 0νββ decay where the Majorana-neutrino exchange with q ∼ 100 MeV induces highexcitation and high-multipolarity transitions through the virtual states of the intermediate nucleus.Thus the OMC can be considered as an ideal probe of the NMEs of the 0νββ decays.This probe corresponds to the right-branch (β + type of transitions) virtual transitions of the 0νββ decay.Incentives of the OMC studies are related to the 0νββ decays and the associated in-medium renormalization of the weak axial (g A ) and induced pseudoscalar (g P ) couplings [11,22,23,24,25,26,27,28], and to neutrino-nucleus interactions in general, as discussed in the recent review [3].Recently, a pioneering theoretical and experimental study of the OMC on 100 Mo, populating states in 100 Nb in a wide excitation region, up to some 50 MeV, was conducted [29].The rate of OMC to individual final states forms a strength function quite like in the case of (n,p) chargeexchange reactions for 1 + final states (the Gamow-Teller strength function).The OMC strength function contains giant resonances, quite like the (p,n) type of transitions contain Gamow-Teller giant resonance and isovector spinmultipole resonances [30,31].The work [29] uses the powerful OMC formalism of [32] and this is the first time such resonances are being studied both theoretically and experimentally, inspired by the first observation of the OMC giant resonance in 100 Nb at around 12 MeV [33].Comparison of the relative (in per cent) Lorenzian-folded muon-capture-rate distributions: theoretical capture rate to all possible final states (dashed line) is compared with the experimental strength distribution (solid line).The theoretical rate was computed with axial-coupling value g A (0) = 0.8 and pseudoscalar-coupling value g P (0) = 7.0.

FIGURE 1 .
FIGURE 1. Effective values of g A in different theoretical β and 2νββ analyses for the nuclear mass range A = 41 − 136.The quoted references are Suhonen2017[4], Caurier2012[6], Faessler2007[17], Suhonen2014[19] and Horoi2016[10].These studies are contrasted with the ISM Gamow-Teller β-decay studies of M-P1996[8], Iwata2016[7], Kumar2016[9] and Siiskonen2001[11].The dashed and dotted curves come from the analysis of[16].The very recent results[5] for light nuclei in the mass range A ≤ 46, as obtained by using ab-initio methods including the two-body meson-exchange currents, are also included in a schematic way.For more information see the body of text.

FIGURE 2 .
FIGURE 2. Comparison of the relative (in per cent) Lorenzian-folded muon-capture-rate distributions: theoretical capture rate to all possible final states (dashed line) is compared with the experimental strength distribution (solid line).The theoretical rate was computed with axial-coupling value g A (0) = 0.8 and pseudoscalar-coupling value g P (0) = 7.0.