Advances in High Energy Physics

Volume 2016, Article ID 4714829, 13 pages

http://dx.doi.org/10.1155/2016/4714829

## Analysis of the Intermediate-State Contributions to Neutrinoless Double *β*^{−} Decays

Department of Physics, University of Jyvaskyla, P.O. Box 35, 40014 Jyvaskyla, Finland

Received 1 April 2016; Accepted 26 May 2016

Academic Editor: Luca Stanco

Copyright © 2016 Juhani Hyvärinen and Jouni Suhonen. 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

A comprehensive analysis of the structure of the nuclear matrix elements (NMEs) of neutrinoless double beta-minus () decays to the ground and first excited states is performed in terms of the contributing multipole states in the intermediate nuclei of transitions. We concentrate on the transitions mediated by the light (l-NMEs) Majorana neutrinos. As nuclear model we use the proton-neutron quasiparticle random-phase approximation (pnQRPA) with a realistic two-nucleon interaction based on the Bonn one-boson-exchange matrix. In the computations we include the appropriate short-range correlations, nucleon form factors, and higher-order nucleonic weak currents and restore the isospin symmetry by the isoscalar-isovector decomposition of the particle-particle proton-neutron interaction parameter .

#### 1. Introduction

Thanks to neutrino-oscillation experiments much is known about the basic properties of the neutrino concerning its mixing and squared mass differences. What is not known is the absolute mass scale, the related mass hierarchy, and the fundamental nature (Dirac or Majorana) of the neutrino. This can be studied by analyzing the neutrinoless double beta () decays of atomic nuclei [1–4] through analyses of the participating nuclear matrix elements (NMEs). The decays proceed by virtual transitions through states of all multipoles in the intermediate nucleus, being the total angular momentum and being the parity of the intermediate state. Most of the present interest is concentrated on the double beta-minus variant ( decay) of the decays due to their relatively large decay energies ( values) and natural abundancies.

In this work we concentrate on analyses of the intermediate contributions to the decays for the ground-state-to-ground-state and ground-state-to-excited-state transitions in nuclear systems of experimental interest. We focus on the light Majorana neutrino mediated transitions by taking into account the appropriate short-range nucleon-nucleon correlations [5] and contributions arising from the induced currents and the finite nucleon size [6]. There are several nuclear models that have recently been used to compute the decay NMEs (see, e.g., the extensive discussions in [3, 7–11]). However, the only model that avoids the closure approximation and retains the contributions from individual intermediate states is the proton-neutron quasiparticle random-phase approximation (pnQRPA) [7, 12–14].

Some analyses of the intermediate-state contributions within the pnQRPA approach have been performed in [12, 13, 15, 16] and recently quite extensively in [17]. In [17] an intermediate multipole decomposition was done for decays of ^{76}Ge, ^{82}Se, ^{96}Zr, ^{100}Mo, ^{110}Pd, ^{116}Cd, ^{124}Sn, ^{128,130}Te, and ^{136}Xe to the ground state of the respective daughter nuclei. In this paper we extend the analysis of [17] to a more detailed scrutiny of the intermediate contributions to the decay NMEs of the above-mentioned nuclei. We also extend the scope of [17] by considering transitions to the first excited states in addition to the ground-state-to-ground-state transitions.

#### 2. Theory Background

In this section a very brief introduction to the computational framework of the present calculations is given. The present analyses on ground-state-to-ground-state decays are based on the calculations done in [17]. Details considering the excited-state decays are given in a future publication. We assume here that the decay proceeds via the light Majorana neutrino so that the inverse half-life can be written as where is a phase-space factor for the final-state leptons defined here without the axial vector coupling constant . The quantity denotes the neutrino effective mass and describes the physics beyond the standard model [17]. The quantity is the light neutrino nuclear matrix element (l-NME). The nuclear matrix element can be decomposed into Gamow-Teller (GT), Fermi (F), and tensor (T) contributions as where is the vector coupling constant.

Each of the NMEs = GT, F, and T in (2) can be decomposed in terms of the intermediate multipole contributions as where each multipole contribution is, in turn, decomposed in terms of the two-particle transition matrix elements and one-body transition densities as where and label the different pnQRPA solutions for a given multipole and the indices denote the proton and neutron single-particle quantum numbers. The operators inside the two-particle matrix element contain the neutrino potentials for the light Majorana neutrinos, the characteristic two-particle operators for the different = GT, F, T and a function taking into account the short-range correlations (SRC) between the two decaying neutrons in the mother nucleus of decay [17]. The final state, , can be either the ground state or an excited state of the daughter nucleus, and the overlap factor between the two one-body transition densities helps connect the corresponding intermediate states emerging from the pnQRPA calculations in the mother and daughter nuclei.

As mentioned before, our calculations contain the appropriate short-range correlators, nucleon form factors, and higher-order nucleonic weak currents. In addition, we decompose the particle-particle proton-neutron interaction strength parameter of the pnQRPA into its isoscalar () and isovector () components and adjust these components independently as described in [17]: the isovector component is fixed such that the NME of the two-neutrino double beta-decay () vanishes and the isospin symmetry is thus restored for both the and decays. The isoscalar component, in turn, is fixed such that the measured half-life of the decay is reproduced. The resulting values of both components of are shown in Table I of [17]. The details of the chosen valence spaces and the determination of the other Hamiltonian parameters are presented in [17]. We further note that in [17] two sets of NME computations, related to the value of the axial vector coupling , were performed: first with the quenched value = 1.00 and then with the bare value = 1.26. In both computations the value of was fixed first. After this the Hamiltonian parameters were adjusted by using the experimental data, as briefly described above and more thoroughly in [17].

#### 3. Results and Discussion

In this section we discuss and present the results of our calculations. Presentation of the results follows top to bottom approach. First we analyze the multipole decompositions and total cumulative sums of the matrix elements. From these we can extract the most important multipole components and energy regions contributing to the NMEs. After this we continue and dissect the most important multipole components into contributions coming from different individual states of the intermediate nucleus. Throughout these computations we have used a conservatively quenched value of the axial vector coupling ; that is, we use the pnQRPA parameters which are related to the first set of computations in [17] as was explained at the end of Section 2.

There has been a lot of discussion about the correct value of in both the and decays lately. This is so due to the fact that a large portion of the theoretical half-life uncertainties are related to the present ambiguity in the value of . In [9] the quenching of was studied in the framework of IBM-2 and the interacting shell model (ISM). The effective values were parametrized as (IBM-2) and as (ISM). These parametrizations were obtained by comparing the model calculations with experimental data on decays. Further studies were performed within the framework of the pnQRPA by using the available Gamow-Teller beta-decay and decay data in several publications (see [18] and the references therein). A wide systematic study of the quenching of for Gamow-Teller beta decays was performed in [18]. Even the quenching related to spin-dipole states was studied in [19]. While the beta decays and decays are low-energy processes with small momentum transfers, the decay involves large momentum transfers and the thus activated high-energy and high-multipolarity intermediate states. For higher momentum transfers the effective can be momentum-dependent [20] and different multipoles can be affected in different ways. At present there exists no known recipe on how to determine the value of for the neutrinoless double beta decays, and that is why we have chosen in the present study to work with a moderately quenched value = 1.00, assumed to be the same for all intermediate multipoles. We will study, however, the effect of changing the value of to the characteristics of the intermediate-state contributions in Section 3.3.

##### 3.1. Ground-State-to-Ground-State Transitions

Let us begin by considering the ground-state-to-ground-state decays mediated by light neutrino exchange. In Figures 1(a) and 1(b) we have plotted the multipole decomposition (3) of the l-NMEs corresponding to the and 136 nuclear systems. For most nuclei considered in this work, the leading multipole component is . This is the case also for the nucleus ^{96}Zr shown in Figure 1(a). Most important contribution to the NMEs comes from the lowest multipole components . It can also be observed that the shape of the overall multipole distribution is leveled when going towards heavier nuclei. This can be seen by comparing the distribution of ^{96}Zr with the distribution of ^{136}Xe displayed in Figure 1(b).