Realistic shell model description of beta decay

. In this paper, we extend our previous realistic shell-model study of Gamow-Teller strengths and double-β decay properties for nuclei around 132 Sn to lighter mass nuclei. The e ﬀ ective shell-model Hamiltonian and transition operators are microscopically derived by way of many-body perturbation theory, without resorting to empirical parameters. Our main aim is to further check the reliability of our approach and verify its stability in other mass regions. Calculated energy spectra as well as electric-quadrupole and β -decay properties are presented for 76 Ge and 82 Se and compared with the experimental data, together with some results for 130 Te and 136 Xe already reported in our previous paper. Finally, some preliminary results of nuclear matrix elements responsible for the neutrinoless double beta decay, calculated by using the bare operator, are shown.


Introduction
The neutrinoless double beta (0νββ) decay provides a major opportunity to address relevant questions about the mass and the nature of neutrinos.Its study is on top of the agenda of current experimental and theoretical research programs around the world [1].The estimate of the associated lifetimes, as well as the extraction of the neutrino effective mass once a decay signal has been observed, requires the knowledge of the matrix elements of the transition operators between the parent and grand-daughter nuclei.
The calculation of nuclear matrix elements (NMEs) responsible for the 0νββ decay has been tackled in the framework of several theoretical approaches, as the quasiparticle random-phase approximation, the interacting boson model, the shell model, and energy density functional methods (see, for instance, [2][3][4][5][6] and references therein).However, the spread of the values obtained with different approaches is quite large.As a matter of fact, results, obtained with different nuclear structure models, largely differ, thus indicating the need to improve the accuracy of nuclear structure calculations.
By analogy with single-β and two-neutrino double-β (2νββ) processes whose predicted lifetimes are almost always shorter than measured lifetimes, a quenching of the axial coupling constant g A is quite often introduced in order to take into account missing many-body correlations and/or coupling with non-nucleonic degrees of freedom in a phenomenological way.A review of the theoretical methods and calculations employed to predict the NMEs of the 0νββ decay is given in the recent review paper of Ref. [7], where the renormalization of the axial vector coupling g A is also discussed in detail.
A valuable tool to address this problem is given by the realistic shell model (SM) [8,9].Within this approach, the SM Hamiltonian and the transition operators are derived from a realistic free nuclear potential by way of manybody perturbation theory on the same footing.In this way, one can account for renormalization effects induced by missing correlations in the nuclear wave functions as resulting from truncations of the Hilbert space.
As a first step towards the calculation of the NMEs of the 0νββ decay, we have recently performed a realistic SM study of 130 Te and 136 Xe [10], which are candidates for the observation of the 0νββ decay.The focus was on Gamow-Teller (GT) strengths and 2νββ decay NMEs, which may provide a relevant test of our approach.It turned out that the available experimental data -energy spectra, electricquadrupole transitions and beta decay properties -are well described by our calculations without the need to resort to empirical parameters, and, in particular, without quenching the g A constant.
To further check the reliability of our approach and investigate the renormalization effects in different mass regions, we have carried out realistic SM calculations for other nuclei, which are also candidates for the observation of the 0νββ decay.Results will be presented in a forthcoming paper [11].Here, we present energy spectra as well as electric-quadrupole transitions and beta decay properties for 76 Ge and 82 Se.For the sake of completeness, we also show the GT strengths and 2νββ NMEs of 130 Te and 136 Xe.
Our starting point is again the high-precision CD-Bonn nucleon-nucleon (NN) potential [12], whose repulsive high-momentum components are smoothed out using the V low−k approach [13].Then, the effective SM Hamiltonian and transition operators are derived within the timedependent perturbation theory [14], including diagrams up to the third order.
Within this approach we have also computed the 0νββ decay NMEs by employing the bare operators for all the four nuclei under investigation.A comparison of the obtained NMEs results with those of other SM calculations may provide an important test of our wave functions, before calculating the NMEs with effective operators.
The paper is organized as follows.In Sec. 2, we outline the theoretical framework of our approach and give some details about the derivation of the SM Hamiltonian and effective transition operators.Section 3 is devoted to the presentation of the results of our calculations and to the comparison with the available experimental data.A summary and some concluding remarks are given in the last section.

Theoretical framework
In our realistic SM calculations for 76 Ge and 82 Se, we have taken 56 Ni as closed core and employed a model space spanned by the four orbitals 0 f 5/2 , 1p 3/2 , 1p 1/2 , 0g 9/2 for both protons and neutrons.For 130 Te and 136 Xe , we have assumed that valence protons and neutrons occupy the five orbitals 0g 7/2 , 1d 5/2 , 1d 3/2 , 2s 1/2 , 0h 11/2 outside 100 Sn.In both cases, the effective Hamiltonian and transition operators have been derived within the framework of the manybody perturbation theory outlined below.
Starting from the Hamiltonian, H, for a system of A nucleons interacting via two-body forces, we introduce an auxiliary potential so that it can be written as the sum of a one-body component H 0 , which describes the independent motion of the nucleons, and a residual interaction H 1 .
where T is the kinetic energy and V NN a low-momentum two-body potential, obtained through the V low−k procedure [13] from the CD-Bonn NN potential [12], by using a cutoff momentum Λ = 2.6 fm −1 .The harmonic oscillator potential is chosen for U and the Coulomb force added to the V low−k for protons.Then, the diagonalization of the Hamiltonian (1) in the full Hilbert space is reduced to a model-space eigenvalue problem introducing an effective Hamiltonian, H eff , acting only on the truncated model space defined in terms of the eigenvectors of H 0 .
A well-established approach to the determination of the effective Hamiltonian is given by the Kuo-Lee-Ratcliff folded-diagram expansion [14].A detailed description of this approach can be found in Refs.[8,9].Here, we would like to highlight only that it is based on the perturbative expansion of the so-called Q-box where P and Q are projection operators onto the model space and its complement.The Q-box is calculated including one-and two-body Goldstone diagrams through third order in H 1 .Calculations beyond the third order in perturbation theory are computationally prohibitive, so we have computed also the Padé approximant [2|1] of the Q-box to obtain a value to which the perturbation series should converge, as suggested in [15].
The effective Hamiltonian is written as a sum of products of the Q-box and its derivatives with respect to the unperturbed energy.The sum is carried out by means of the Lee-Suzuki iterative technique [16].
It is worth noting that our effective Hamiltonian is derived for a two-valence-particle nucleus.Therefore, it contains one-and two-body components, the former, summed to the eigenvalues of H 0 , gives the theoretical singleparticle energies.
Concerning the effective transition operators, they have been derived within the same framework of and consistently with the SM Hamiltonian, employing the approach proposed by Suzuki and Okamoto in Ref. [17].In this context, similarly to the Q-box, a X-box is introduced that we have calculated including diagrams up to third order in the perturbation theory (see Ref. [10]).
Using this procedure, we have calculated the effective matrix elements of one-body operators, as it is the case of the electric-quadrupole (E2) and GT operators, for a single particle beyond 56 Ni and 100 Sn.These effective matrix elements have been used to compute the B(E2) and GT strengths, as well as the NMEs of the 2νββ decay reported in Sec. 3. In order to show the effects due to the renormalizations induced by the truncation of the Hilbert space for the 0 f 5/2 1p0g 9/2 and 0g 7/2 1d2s0h 11/2 model spaces, the bare proton-neutron matrix elements of the GT − operator are compared with the effective ones in Tables 1 and 2. For the sake of clarity, we also report quenching factors, namely the ratio between the effective and bare matrix elements, except for transitions that are forbidden for the bare GT − operator.It is worth mentioning that the values shown in Table 2 were taken from [10].
From both Tables, we see that all effective matrix elements, as expected, are reduced with respect to the bare ones, but the reduction factor depends on the initial and final states, ranging from 0.36 to 0.70 in Table 1 and from 0.50 to 0.69 in Table 2. So, microscopic renormalizations lead to selective effects and in particular make forbidden off-diagonal matrix elements different from zero, which is not possible when using an empirical quenching factor.
In Sec. 3, we also report some preliminary results of the 0νββ NMEs, which have been calculated employing the bare two-body operator: where the vector and axial constants are g V = 1 and g A = 1.269, respectively; and the GT, Fermi, and tensor components are explicitly given by Here the functions H α (r) denote the neutrino potentials.

Results
The results presented in this section have been obtained by employing the effective Hamiltonian and the transition operators derived within the theoretical framework discussed in Sec. 2. Calculations have been carried out using the shell-model code ANTOINE [18].We start by presenting the spectra and the E2 transition rates of 76 Ge and 82 Se, results for 130 Te and 136 Xe have been already reported in Ref. [10].In figs. 1 and 2, the calculated low-energy spectra and B(E2) values are compared with the experimental data for 76 Ge [19,20] and 82 Se [19], respectively.Note that arrows are proportional to B(E2) strengths.
We see that the agreement between theory and experiment for excitation energies is quite reasonable, although not completely satisfactory.All calculated states are, in fact, shifted up by an amount ranging from about 200 to 500 keV, which may be related to the lack of the 0 f 7/2 orbital in our model space.The experimental E2 transition rates, however, are well reproduced by our calculations, testifying the reliability of our shell-model wave functions and effective electric-quadrupole transition operator.The error, in fact, is at most 30% except for the B(E2; in 82 Se.In this case, the calculated value is 3 times larger than the experimental one. In Table 3 we report the results of the total GT − strength, S (GT − ), that is defined as where Φ i refers to the ground state of the parent nucleus, while Φ f to all the possible final states in the daughter nucleus up to a given energy.The total GT − strengths are calculated using both the bare and effective GT − operators for 76 Ge, 82 Se, 130 Te and 136 Xe.They are compared with the experimental values extracted from single charge-exchange experiments [21][22][23][24] From the inspection of Table 3, it can be seen that while the S(GT − )s calculated with the bare operators are significantly larger that the experimental ones, a quite good agreement between theory and experiment is obtained when using the effective operators.This underlines the crucial role played by the many-body renormalization, that, as shown in Table 1, leads to a strong reduction of the bare matrix elements of the GT − operator.As a matter of fact, for all the nuclei under investigation the theoretical total GT − strengths are reduced by a factor around 3.
As regards the 2νββ decay we have calculated the NMEs, as defined by the following expression where E n is the excitation energy of the n-th J π = 1 + state, E 0 = 1 2 Q ββ + ∆M, Q ββ being the Q-value corresponding to the ββ decay, and ∆M the mass difference between the parent and intermediate nuclei.
In fig.3, we report a correlation plot between the experimental and calculated NME values.The experimental data have been taken from Ref. [25], where they are calculated from the measured half-life values for the 2νββ decay, T 2ν 1/2 , using the relation The NMEs calculated with the bare GT operator are reported as blue dots, and those obtained employing the effective one with black triangles.As can be seen, the position of the blue dots evidences that the NMEs calculated with the bare GT operator are largely overestimated with respect to the experimental ones, while the fundamental role played by the effective GT operator is supported by the fact that the black triangles lie very close to the bisector of the correlation plot.
Finally, in fig. 4 we present our results for the 0νββ decay NMEs, obtained employing the bare operator.The calculation of the effective two-body operator is under way, and the results will be the subject of a forthcoming paper [26].Our theoretical NMEs for 76 Ge, 82 Se, 130 Te, and 136 Xe (black dots), calculated summing only the GT and Fermi contributions, are compared with those obtained in other SM calculations, that have employed the same model space and 0νββ decay operator, but different SM Hamiltonians.More precisely, we refer to the results of the Strasbourg-Madrid group (blue dots) [27], and those of Horoi and coworkers (red dots) [28,29].From the inspection of fig. 4, it can be observed that all SM calculations, for all nuclei under investigation, provide NME values very close to each other, notwithstanding the different employed Hamiltonians.This outcome can be considered as a benchmark for the SM eigenfunctions, and supports the SM approach to predict reliable 0νββ NMEs.

Summary and Perspectives
In the present paper, we have presented the results of realistic SM calculations for 76 Ge, 82 Se, 130 Te, and 136 Xe, with special attention to their GT decay properties.
In these calculations, the SM Hamiltonian, including both the two-body matrix elements and single-particle energies, as well as the transition operators are calculated by way of the many-body perturbation theory, starting from a low-momentum interaction derived from the highprecision CD-Bonn free NN potential.This means that renormalization effects due to the reduction of the Hilbert space are included in our theoretical effective operators, without resorting of any empirical parameters.
The results report here are part of a study -started with our previous work on nuclei around 132 Sn [10] -we have undertaken with the main aim to test our approach to the calculation of the 0νββ decay NME.
The calculated energy spectra, electric-quadrupole transitions, GT strengths and NMEs of the 2νββ decay are in good agreement with experimental data and the quality of the results is very similar for all the four nuclei under investigation.
This opens up good prospects to a fully microscopic calculation of 0νββ decay NME.In this connection, we have reported results of this matrix element for 76 Ge, 82 Se, 130 Te, and 136 Xe, by using the bare 0νββ decay operator and including only its GT and Fermi components.We have found that our matrix elements are very close to those obtained in other SM calculations, that employ the same model space and 0νββ decay operator, but different SM Hamiltonians.This may be seen as another proof of the reliability of our wave functions.
Our next step towards the calculation of the 0νββ decay NME is the derivation of the effective operator within the theoretical framework described in Sec. 3, by considering also the tensor component.Then, there also other effects that we plan to include in the renormalization of the decay operators, as, for instance, the contributions of three-body correlations among the valence nucleons, the so-called blocking effect, and those arising from the quark structure of nucleons.As mentioned above, work along these lines is under way.

Figure 3 .
Figure 3. Correlation plot between experimental and calculated 2νββ NMEs (see text for details).

Figure 4 .
Figure 4. 0νββ NMEs calculated with different SM calculations, including present ones (see text for details).

Table 1 .
Bare and effective proton-neutron matrix elements for the model space outside56Ni.In the last column the corresponding quenching factors are reported (see text for details).

Table 2 .
Bare and effective proton-neutron matrix elements for the model space outside 100 Sn.In the last column the corresponding quenching factors are reported (see text for details).

Table 3 .
Calculated and experimental total GT − strengths up to 3 MeV excitation energy for 76 Ge, 82 Se,130Te, and up to 4.5MeV for 136 Xe (see text for details).