Advances in High Energy Physics

Volume 2016, Article ID 6391052, 12 pages

http://dx.doi.org/10.1155/2016/6391052

## Nuclear Structure Calculations for Two-Neutrino Double-*β* Decay

^{1}Instituto de Estructura de la Materia, IEM-CSIC, Serrano 123, 28006 Madrid, Spain^{2}Departamento de Física Atómica, Molecular y Nuclear, Universidad Complutense de Madrid, 28040 Madrid, Spain

Received 17 June 2016; Revised 26 August 2016; Accepted 19 September 2016

Academic Editor: Theocharis Kosmas

Copyright © 2016 P. Sarriguren et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

We study the two-neutrino double- decay in ^{76}Ge, ^{116}Cd, ^{128}Te, ^{130}Te, and ^{150}Nd, as well as the two Gamow-Teller branches that connect the double- decay partners with the states in the intermediate nuclei. We use a theoretical microscopic approach based on a deformed self-consistent mean field with Skyrme interactions including pairing and spin-isospin residual forces, which are treated in a proton-neutron quasiparticle random-phase approximation. We compare our results for Gamow-Teller strength distributions with experimental information obtained from charge-exchange reactions. We also compare our results for the two-neutrino double- decay nuclear matrix elements with those extracted from the measured half-lives. Both single-state and low-lying-state dominance hypotheses are analyzed theoretically and experimentally making use of recent data from charge-exchange reactions and decay of the intermediate nuclei.

#### 1. Introduction

Double- decay is currently one of the most studied processes both theoretically and experimentally [1–5]. It is a rare weak-interaction process of second order taking place in cases where single decay is energetically forbidden or strongly suppressed. It has a deep impact in neutrino physics because the neutrino properties are directly involved in the neutrinoless mode of the decay () [6–8]. This decay mode, not yet observed, violates lepton-number conservation and its existence would be an evidence of the Majorana nature of the neutrino, providing a measurement of its absolute mass scale. Obviously, to extract a reliable estimate of the neutrino mass, the nuclear structure component of the process must be determined accurately. On the other hand, the double- decay with emission of two neutrinos () is perfectly allowed by the Standard Model and it has been observed experimentally in several nuclei with typical half-lives of years (see [9] for a review). Thus, to test the reliability of the nuclear structure calculations involved in the process, one checks first the ability of the nuclear models to reproduce the experimental information available about the measured half-lives for the process. Although the nuclear matrix elements (NMEs) involved in both processes are not the same, they exhibit some similarities. In particular, the two processes connect the same initial and final nuclear ground states and share common intermediate states. Therefore, reproducing the NMEs is a requirement for any nuclear structure model aiming to describe the neutrinoless mode.

Different theoretical approaches have been used in the past to study the NMEs. Most of them belong to the categories of the interacting shell model [10–12], proton-neutron quasiparticle random-phase approximation (QRPA) [1, 2, 13–25], projected Hartree-Fock-Bogoliubov [26–28], and interacting boson model [29–31].

In this work we focus on the QRPA type of calculations. Most of these calculations were based originally on a spherical formalism, but the fact that some of the double- decay nuclei are deformed makes it compulsory to deal with deformed QRPA formalisms [21–25]. This is particularly the case of ^{150}Nd (^{150}Sm) that has received increasing attention in the last years because of the large phase-space factor and relatively short half-life, as well as for the large energy that will reduce the background contamination. ^{150}Nd is currently considered as one of the best candidates to search for the decay in the planned experiments SNO+, SuperNEMO, and DCBA.

The experimental information to constrain the calculations is not limited to the NMEs extracted from the measured half-lives. We have also experimental information on the Gamow-Teller (GT) strength distributions of the single branches connecting the initial and final ground states with all the states in the intermediate nucleus. The GT strength distributions have been measured in both directions from () and () charge-exchange reactions (CER) and more recently from high-resolution reactions, such as (, ^{2}He), (^{3}He, ), and (, ^{3}He) that allow us to explore in detail the low-energy structure of the GT nuclear response in double- decay partners [32–41]. In some instances there is also experimental information on the values of the decay of the intermediate nuclei.

Nuclear structure calculations are also constrained by the experimental occupation probabilities of neutrons and protons of the relevant single-particle levels involved in the double- decay process. In particular, the occupation probabilities of the valence shells , , , and for neutrons in ^{76}Ge and for protons in ^{76}Se have been measured in [42] and [43], respectively. The implications of these measurements on the double- decay NMEs have been studied in [44–47].

In this paper we explore the possibility of describing all the experimental information available on the GT nuclear response within a formalism based on a deformed QRPA approach built on top of a deformed self-consistent Skyrme Hartree-Fock calculation [48–51]. This information includes global properties about the GT resonance, such as its location and total strength, a more detailed description of the low-lying excitations, and decay NMEs. The study includes the decays , , , , and . This selection is motivated by recent high-resolution CER experiments performed for ^{76}Ge(^{3}He, t) [39], ^{76}Se(d, ^{2}He)^{76}As [37], ^{128,130}Te(^{3}He, t)^{128,130}I [41], ^{116}Cd(p,n)^{116}In, and ^{116}Sn(n,p)^{116}In [40], as well as for ^{150}Nd(^{3}He, t)^{150}Pm, and ^{150}Sm(t, ^{3}He)^{150}Pm [38]. We also discuss on these examples the validity of the single-state dominance (SSD) hypothesis [52] and the extended low-lying-state dominance (LLSD) that includes the contribution of the low-lying excited states in the intermediate nuclei to account for the double- decay rates.

The paper is organized as follows: In Section 2, we present a short introduction to the theoretical approach used in this work to describe the energy distribution of the GT strength. We also present the basic expressions of the decay. In Section 3 we present the results obtained from our approach, which are compared with the experimental data available. Section 4 contains a summary and the main conclusions.

#### 2. Theoretical Approach

The description of the deformed QRPA approach used in this work is given elsewhere [22, 53–55]. Here we give only a summary of the method. We start from a self-consistent deformed Hartree-Fock (HF) calculation with density-dependent two-body Skyrme interactions. Time reversal symmetry and axial deformation are assumed in the calculations [56]. Most of the results in this work are performed with the Skyrme force SLy4 [57], which is one of the most widely used and successful interactions. Results from other Skyrme interactions have been studied elsewhere [48–51, 58] to check the sensitivity of the GT nuclear response to the two-body effective interaction.

In our approach, we expand the single-particle wave functions in terms of an axially symmetric harmonic oscillator basis in cylindrical coordinates, using twelve major shells. This amounts to a basis size of 364, the total number of independent deformed HO states. Pairing is included in BCS approximation by solving the corresponding BCS equations for protons and neutrons after each HF iteration. Fixed pairing gap parameters are determined from the experimental mass differences between even and odd nuclei. Besides the self-consistent HF+BCS solution, we also explore the energy curves, that is, the energy as a function of the quadrupole deformation , which are obtained from constrained HF+BCS calculations.

The energy curves corresponding to the nuclei studied can be found in [50, 51, 58]. The profiles of the energy curves for ^{76}Ge and ^{76}Se exhibit two shallow local minima in the prolate and oblate sectors. These minima are separated by relatively low-energy barriers of about 1 MeV. The equilibrium deformation corresponds to in ^{76}Ge and in ^{76}Se. We get soft profiles for ^{116}Cd with a minimum at and an almost flat curve in ^{116}Sn between and . We obtain almost spherical configurations in the ground states of ^{128}Te and ^{130}Te. The energies differ less than 300 keV between quadrupole deformations and . On the other hand, for ^{128}Xe and ^{130}Xe we get in both cases two energy minima corresponding to prolate and oblate shapes, differing by less than 1 MeV, with an energy barrier of about 2 MeV. The ground states correspond in both cases to the prolate shapes with deformations around . For ^{150}Nd and ^{150}Sm we obtain two energy minima, oblate and prolate, but with clear prolate ground states in both cases at and , respectively. We obtain comparable results with other Skyrme forces. The relative energies between the various minima can change somewhat for different Skyrme forces [50, 51, 58], but the equilibrium deformations are very close to each other changing at most by a few percent.

After the HF+BCS calculation is performed, we introduce separable spin-isospin residual interactions and solve the QRPA equations in the deformed ground states to get GT strength distributions and decay NMEs. The residual force has both particle-hole (ph) and particle-particle (pp) components. The repulsive ph force determines to a large extent the structure of the GT resonance and its location. Its coupling constant is usually taken to reproduce them [53–55, 59–62]. We use MeV. The attractive part is basically a proton-neutron pairing interaction. We also use a separable form [55, 60, 61] with a coupling constant usually fitted to reproduce the experimental half-lives [62]. We use in most of this work a fixed value MeV, although we will explore the dependence of the NMEs on in the next section. Earlier studies on ^{150}Nd and ^{150}Sm carried out in [24, 63] using a deformed QRPA formalism showed that the results obtained from realistic nucleon-nucleon residual interactions based on the Brueckner matrix for the CD-Bonn force produce results in agreement with those obtained from schematic separable forces similar to those used here.

The QRPA equations are solved following the lines described in [53–55, 60, 61]. The method we use is as follows. We first introduce the proton-neutron QRPA phonon operatorwhere and are quasiparticle creation and annihilation operators, respectively. labels the RPA excited state and its corresponding excitation energy, and and are the forward and backward phonon amplitudes, respectively. The solution of the QRPA equations is obtained by solving first a dispersion relation [55, 60, 61], which is of fourth order in the excitation energies . The GT transition amplitudes connecting the QRPA ground state () to one phonon states () are given in the intrinsic frame by where () are the BCS occupation amplitudes for neutrons and protons. Once the intrinsic amplitudes are calculated, the GT strength (GT) in the laboratory frame for a transition can be obtained asTo obtain this expression we have used the Bohr and Mottelson factorization [64, 65] to express the initial and final nuclear states in the laboratory system in terms of the intrinsic states. A quenching factor, , is applied to the weak axial-vector coupling constant and included in the calculations. The physical reasons for this quenching have been studied elsewhere [10, 66, 67] and are related to the role of nonnucleonic degrees of freedom, absent in the usual theoretical models, and to the limitations of model space, many-nucleon configurations, and deep correlations missing in these calculations. The implications of this quenching on the description of single- and double- decay observables have been considered in several works [12, 30, 68–71], where both the effective value of and the coupling strength of the residual interaction in the channel are considered free parameters of the calculation. It is found that very strong quenching values are needed to reproduce simultaneously the observations corresponding to the half-lives and to the single- decay branches. One should note however that the QRPA calculations that require a strong quenching to fit the NMEs were performed within a spherical formalism neglecting possible effects from deformation degrees of freedom. Because the main effect of deformation is a reduction of the NMEs, deformed QRPA calculations shall demand less quenching to fit the experiment.

Concerning the decay NMEs, the basic expressions for this process, within the deformed QRPA formalism used in this work, can be found in [21, 22, 72]. Deformation effects on the NMEs have also been studied within the projected Hartree-Fock-Bogoliubov model [27]. Attempts to describe deformation effects on the decay within QRPA models can also be found in [25, 73].

The half-life of the decay can be written aswhere are the phase-space integrals [74, 75] and the nuclear matrix elements containing the nuclear structure part involved in the process:In this equation are the QRPA intermediate states reached from the initial (final) nucleus. and are labels that classify the intermediate states that are reached from different initial and final ground states. The overlaps take into account the nonorthogonality of the intermediate states. Their expressions can be found in [21]. The energy denominator involves the energy of the emitted leptons, which is given on average by , as well as the excitation energies of the intermediate nucleus. In terms of the QRPA excitation energies the denominator can be written as where is the QRPA excitation energy relative to the initial (final) nucleus. It turns out that the NMEs are quite sensitive to the values of the denominator, especially for low-lying states, where the denominator takes smaller values. Thus, it is a common practice to use some experimental normalization of this denominator to improve the accuracy of the NMEs. In this work we also consider the denominator , which is corrected with the experimental energy of the first state in the intermediate nucleus relative to the mean ground-state energy of the initial and final nuclei, in such a way that the experimental energy of the first state is reproduced by the calculations:Running sums will be shown later for the two choices of the denominator, and . When the ground state in the intermediate nucleus of the double- decay partners is a state, the energy is given bywhere and are the experimental energies of the decays of the intermediate nucleus into the parent and daughter partners, respectively. This is the case of ^{116}In and ^{128}I, which are both ground states. In the other cases, although the ground states in the intermediate nuclei are not states, the first excited states appear at a very low excitation energy; MeV in ^{76}As [39], MeV in ^{130}I [41], and MeV in ^{150}Pm [38]. Therefore, to a good approximation we also determine using (9).

The existing measurements for the decay half-lives () have been recently analyzed in [9]. Adopted values for such half-lives can be seen in Table 1. Using the phase-space factors from the evaluation [74] that involves exact Dirac wave functions including electron screening and finite nuclear size effects, we obtain the experimental NMEs shown in Table 1, for bare and quenched factors. It should be clear that the theoretical NMEs defined in (6) do not depend on the factors. Hence, the values obtained for the experimental NMEs extracted from the experimental half-lives through (5) depend on the value used in this equation.