Study of the quenching of the GT-decay operator in a microscopic shell-model approach

The need of a reliable calculation of the nuclear matrix elements for the 0νββ decay has ignited a new interest about the quenching of the axial coupling constant gA , a procedure introduced to reproduce experimental results connected with GT decays. The goal of this work is to present a preliminary study to tackle this problem within the framework of the realistic shell model.

The detection of neutrinoless double-β decay (0νββ) may pave the way to "new physics" beyond the Standard Model, since the observation of such a rare process would evidence a lepton number violation, and, moreover, provide crucial informations about the nature and properties of the neutrino [1].
The half-life of 0νββ decay is connected to the structure of the parent and granddaughter nuclei by way of the nuclear matrix element M 0ν through its expression T 0ν 1/2 , G 0ν being the so-called phase-space factor (or kinematic factor) [2], and f (m i , U ei ) accounting for the adopted model of 0νββ decay (light-and/or heavy neutrino exchange, ...) through the neutrino masses m i and their mixing matrix elements U ei .The explicit form of f (m i , U ei ), within the mechanisms of light-neutrino exchange, is f (m i , U ei ) = g 2 A mν me , where g A is the axial coupling constant, m e is the electron mass, and m ν = i (U ei ) 2 m i is the effective neutrino mass.
The expression of the inverse half-life of the 0νββ decay shows the pivotal role played by M 0ν and the chosen value of g A , and indicate that both a reliable calculation of M 0ν and knowledge of the renormalization mechanisms of the axial coupling constant are crucial to obtain important informations about the neutrino effective mass, and also to evaluate the sensitivity of experimental apparatuses to detect 0νββ decay.In fact, one may estimate the half-life to be measured in an experiment, in order to be sensitive to a particular value of the neutrino effective mass [1], by combining M 0ν and the function f (m i , U ei ) with the neutrino mixing parameters [3] and limits on m ν from current experimental results.
Since the 0νββ decay is ruled by the Gamow-Teller (GT) spin-isospin-dependent operator, the predictiveness of nuclear structure calculations can be tested by reproducing observables such as β-decay amplitudes with neutrino emission, or GT-strength distribution which can be obtained experimentally by way of intermediate-energy charge-exchange reactions.
An important issue of most nuclear structure calculations is their overestimation of the data related to GT transitions, and this drawback is usually treated by quenching the axial coupling constant g A by a factor q < 1 [4,5].The rationale behind this operation can be traced back to two basic aspects that are not usually taken into account in most of nuclear models.First, nucleons are not point-like particles, and the effects of meson-exchange currents has to be considered to account for their quark structure [6,7].Second, since the application of nuclear models necessitates a truncation of the nuclear degrees of freedom to allow the diagonalization of the nuclear Hamiltonian, this implies to consider effective Hamiltonians and decay operators to account for the neglected configurations in the nuclear wave functions [7,8].
The goal of the present contribution is to tackle the problem of the quenching of g A within the framework of the realistic shell model (SM), by deriving the matrix elements of the effective SM Hamiltonian H eff and decay operators Θ eff by way of many-body perturbation theory [9,10], starting from a realistic nuclear potential and including the contribution of mesonexchange currents to the definition of the GT operator too.To this end, we start from a nuclear Hamiltonian based on chiral perturbation theory (ChPT) [11,12], that consists of a high-precision two-nucleon (2N) potential derived within the ChPT at next-to-next-to-nextto-leading order (N 3 LO) [13], and a three-nucleon (3N) component at N 2 LO in ChPT [14].Moreover, ChPT provides a consistent way to build both nuclear forces and electroweak currents which account for the composite structure of the nucleons [15][16][17], and that have been recently employed in ab initio nuclear structure calculations in heavy mass nuclei to tackle the problem of the quenching of g A [18].
This work is organized as follows: in the following we sketch out very briefly the fundamentals of the theoretical framework [10,16,19,20], and present the results of the SM calculation of both GT transition-strength distributions and nuclear matrix element M 2ν for the 2νββ-decay of 48 Ca, which have experimental counterparts.Finally, we draw the conclusions which could be inferred from the comparison of our results with data.
As mentioned previously, we derive the effective SM Hamiltonian and decay operators starting from a realistic nuclear potential consisting of a 2N potential [13] and a 3N term [14], derived within the ChPT at N 3 LO and N 2 LO, respectively.The 2N and 3N components of the potential share the same non-local regulator function and a few low-energy constants (LECs), such as c 1 , c 3 , and c 4 appearing in the two-pion exchange term of the N 2 LO 3N potential.The set of LECs entering in the N 2 LO 3N force is completed by the c D and c E constants, associated to the one-pion exchange and the contact term, respectively, whose values are c D = −1 and c E = −0.34 as determined in Refs.[14,21].
Starting from the above mentioned nuclear potential, an effective Hamiltonian has been derived to perform SM calculations in the model space spanned by the 0f 7/2 , 0f 5/2 , 1p 3/2 , 1p 1/2 proton and neutron orbitals outside doubly closed 40 Ca.This task has been carried out by way of the time-dependent perturbation theory [9,10], namely H eff is calculated through the Kuo-Lee-Ratcliff (KLR) folded-diagram expansion in terms of the vertex function Q box, which is composed of irreducible valence-linked diagrams [9,22].In our calculations, the Q box is composed by one-and two-body Goldstone diagrams through third order in the 2N potential, and up to first order in the 3N one [23].
The single-particle energies and two-body matrix elements of H eff that have been employed in our present study are the same as in a previous study of the contribution of chiral threebody forces to the monopole component of H eff [24].In the latter work, we have also shown the abilities of our SM calculations to reproduce closure and collective properties of Ca and Ti isotopes, as testified by the comparison between experimental [25] and calculated low-energy spectra of 48 Ca and 48 Ti reported in Fig. 1.The derivation of effective SM operators Θ eff has been also carried out within a perturbative approach based on a procedure introduced by Suzuki and Okamoto [26], which is consistent with the one pursued to obtain H eff , namely it is based It should be stressed that in present and past calculations [27][28][29] neither effective electric charges for protons and neutrons, nor quenching factors for the free value of g A = 1.272 have been introduced.
As regards the two-body matrix elements of the axial currents, they have been derived through a chiral expansion up to N 3 LO, where the LECs appearing in their expression are consistent with those of the N 3 LO potential we are starting from [13].The details about the calculation of the axial currents within chiral effective theory are reported in Ref. [17].We now proceed to present the calculations of GT-decay quantities.It is important to point out that the calculations have performed employing GT-decay SM operator at four different stages.Namely, (I) labels the results obtained with the bare single-body GT operator, (II) indicates the calculations performed by accounting in the one-body Θ eff only for the neglected configurations in the nuclear wave function with respect to the truncated model space, (III) are the results where to (II) they are added the contributions of the bare two-body axial currents.Finally, the results labelled with (IV) correspond to those obtained with an effective operator, where the truncation of the Hilbert space to the f p model space is accounted for the two-body axial currents too.
In Fig. 2, the calculated and experimental [30] running sums of the GT strengths ΣB(p, n) (see their definition in Ref. [28]) for 48 Ca are shown as a function of the excitation energy up to 4.95 MeV, corresponding to the last measured GT strength.
It can be seen that the distribution obtained using the bare operator (I) overestimates the observed one, as well as using the effective single-body GT operator (II), the latter being closer to experiment.The inclusion of the two-body axial currents in case (III) provides then a calculated curve that is closer to data, but still underestimating them.The consistent treatment of axial currents and the derivation of one-and two-body effective operators (IV), which accounts for configurations outside the model space, provides results close to those of case (III).In Table 1 they are reported the current datum of M 2ν for the 2νββ decay of 48 Ca into 48 Ti [31], as well as the results of calculations (I-IV).As can be seen, the comparison between experiment and theory oscillates between overestimation and underestimation of the observed value, but a proper treatment, that considers both the two-body chiral axial currents and the derivation of an effective SM decay operator accounting for the configurations outside the model space, is able to provide a very satisfactory calculated result.Coming to the conclusions, here we have shown the preliminary results of a study about the GT-decay properties of 48 Ca, that is a part of a project aimed to study the β-decay process in the shell-model framework, starting from nuclear potentials and axial currents derived by way of ChPT.This should lead to the calculation of the 0νββ nuclear matrix elements starting from nuclear forces and currents that are rooted in the fundamental symmetries of QCD, and hopefully with a high degree of confidence in their predictive power.At this stage of our investigation, current results point out that the derivation of effective SM GT-decay operators that include both the contributions of two-body axial currents -accounting for the inner degrees of freedom of the nucleons -, and of the configurations that are not explicitly included in the model space, is important to obtain a calculated M 2ν in agreement with data, without resorting to empirical quenching factors and employing only the free values of g A .

Figure 2 .
Figure 2. Running sums of the 48 Ca B(p, n) strengths as a function of the excitation energy E x up to 4.95 MeV (see text for details).