#### 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).

**(a)**^{96}Zr

**(b)**^{136}XeNuclei can be grouped into different types according to the shapes of their cumulative NME distributions. For transitions via light neutrino exchange, we can differentiate four types of nuclei.* Type 1*: nuclei belonging to this type are ^{76}Ge, ^{82}Se, ^{96}Zr, and ^{128}Te. Representative of this type, ^{76}Ge, is presented in Figure 2(a). Characteristic feature of the cumulative sum distribution belonging to type 1 is the strong drop in the value of the NME occurring between 12 and 17 MeV. Soon after this drop the NME saturates as can be seen from panel (a).* Type 2*: nuclei belonging to this type are ^{100}Mo and ^{110}Pd. Representative of this type, ^{110}Pd, is presented in Figure 2(b). Characteristic feature of this type is the large enhancement and almost immediate cancellation of this enhancement around 10 MeV. This produces a spike-like structure into the cumulative sum distribution as can be seen from panel (b).* Type 3*: nuclei belonging to type 3 are ^{116}Cd, ^{124}Sn, and ^{130}Te. Type 3 is represented by ^{124}Sn, shown in Figure 2(c). Characteristic features of this type are that there occurs neither sharp cancellation of the NME around 12–17 MeV, as in type 1, nor a spike like structure around 10 MeV, as in type 2. Value of the NME rather increases more or less smoothly to its highest value and then smoothly saturates to its final value around 20 MeV.* Type 4*: type 4 is special in a sense that it includes only one nucleus, ^{136}Xe. Cumulative sum of the NME for ^{136}Xe is shown in Figure 2(d). Characteristic feature of type 4 is that the lowest energy region, roughly between 0 and 1.5 MeV, contributes practically nothing to the value of the NME as can be noticed from panel (d).

**(a) 76Ge**

**(b) 110Pd**

**(c) 124Sn**

**(d) 136Xe**

Using the multipole decompositions, we have extracted the most important multipole components contributing to the light neutrino mediated ground-state-to-ground-state decays. These most important components can be divided into contributions coming from different energy levels of the intermediate nucleus. These contributions are collected into Table 1 for systems, into Table 2 for systems, and into Table 3 for systems. We see from the tables that often a very small set of states collects the largest part of a given multipole contribution to the NMEs. Also in some cases notable contributions are coming from high excitation energies, well above 10 MeV, like in the case of contributions for almost all nuclei, contributions for ^{76}Ge, ^{82}Se, ^{110}Pd, ^{116}Cd, and ^{124}Sn, contributions for ^{130}Te and ^{136}Xe, and a contribution for ^{124}Sn.

We notice a single-state dominance for the mode in nuclei ^{76}Ge, ^{82}Se, and ^{96}Zr. In [19] an analysis of the unique first forbidden single ground-state-to-ground-state transitions in the mass region was performed. It was found that a strong renormalization of the axial vector single matrix elements is needed to be able to explain the experimental transition rates. It was then speculated that the same kind of an effect may also appear in the NMEs. This may have a large effect on the transition rates due to the important contribution of the multipole to the NMEs.

The energies of the intermediate states listed in Tables 1, 2, and 3 (and also those in Tables 4 and 5 for the transitions to the excited states) originate from pnQRPA calculations. Usually the pnQRPA cannot reproduce the fine details of the level structures found in all intermediate odd-odd nuclei considered in this work. This is due to the general feature of odd-odd nuclei: the extremely high density of states even at low energies. This high density of nuclear states becomes a problem, not only for the pnQRPA, but for any other nuclear many-body approach, including the nuclear shell model. The reason for this is that even small perturbations in the two-body interaction matrix elements tend to change the ordering of the levels at random. For this reason the spectra of the odd-odd intermediate nuclei are not a very good measure of the reliability of the calculations but, instead, a better way is to adjust the model parameters in such a way that the transition rates of some other known processes, for example, single or decays, can be reproduced by the theory and this is the philosophy which we have followed in this work.

##### 3.2. Ground-State-to-Excited-State Decays

Let us then consider transitions mediated by the light neutrino exchange. In Figures 3(a) and 3(b) we have plotted the multipole decomposition of the l-NMEs corresponding to the and 96 nuclear systems. The multipole distributions for the excited-state transitions are greatly different from those corresponding to the ground-state transitions. Usually there is only a couple of multipoles, and , which give by far the largest contribution to the NMEs. In this sense the excited-state transitions are more simple than the ground-state transitions. Typical example is the nucleus ^{76}Ge, displayed in Figure 3(a). One nucleus deviating from this trend is ^{96}Zr which is presented in Figure 3(b). Its multipole distribution resembles somewhat more those shown for the ground-state decays in Figures 1(a) and 1(b). Most of this differing behaviour can be traced back to the one-phonon structure of the final excited state in the nucleus ^{96}Mo. The final states in this work are modeled as one-phonon basic QRPA excitations for the daughter nuclei ^{96}Mo and ^{116}Sn. Rest of the final states are modeled as two-quadrupole-phonon states. Nucleus ^{96}Zr is an exceptional case since the state in ^{96}Mo has a relatively low excitation energy and thus boasts rather strong collective features. This is why the excited-state transition has a wide multipole distribution and is greatly enhanced.

**(a)**^{76}Ge

**(b)**^{96}ZrAgain we can divide nuclei into different groups by considering the shapes of their total cumulative sum distributions. For transitions via light neutrino exchange, we can differentiate two types of nuclei.* Type 1*: nuclei belonging to type 1 are ^{76}Ge, ^{82}Se, ^{124}Sn, ^{130}Te, and ^{136}Xe. Typical examples of this type, ^{76}Ge, ^{82}Se, and ^{136}Xe, are shown in Figures 4(a), 4(b), and 4(d). Characteristic feature of this type is that there exist only few energy states which give most of the total matrix element producing a staircase-like structure as seen in the panels. For example, for ^{76}Ge there seems to be only five such energy states.* Type 2*: nuclei belonging to this type are ^{96}Zr, ^{100}Mo, ^{110}Pd, and ^{116}Cd. Typical examples of this type are ^{96}Zr and ^{116}Cd shown in Figures 4(c) and 4(e). Characteristic feature of type 2 is that a large number of intermediate states give important contributions to the NMEs. In case of ^{116}Cd, panel (e), around 50% of the total NME comes from transitions through the ground state of the intermediate nucleus. The other 50% is distributed rather evenly on the interval 0–20 MeV.

**(a) 76Ge**

**(b) 82Se**

**(c) 96Zr**

**(d) 136Xe**

**(e) 116Cd**

Using the multipole decompositions, we extracted the most important multipole components contributing to the light neutrino mediated decay transitions. These most important components were then again divided into contributions coming from different energy levels of the intermediate nucleus. These contributions are collected into Table 4 for systems and into Table 5 for systems. Again we notice that often only a few intermediate states give the largest contribution to the dominant multipoles and . Extreme case is the nucleus ^{116}Cd for which the dominant intermediate ground state gives 81% of the total strength. Combining this with the fact that is by far the largest multipole component, we get a rather good approximation for the total NME by considering just a single virtual transition through the ground state of the intermediate nucleus ^{116}In. As for the ground-state-to-ground-state decays in some cases notable contributions are coming from high excitation energies, well above 10 MeV. There are high-energy contributions in case of multipole for all nuclei, and in the cases of and multipoles for ^{130}Te and ^{136}Xe.

##### 3.3. Effects of on the Intermediate-State Contributions

As mentioned earlier, we have used in this work the quenched value for the axial vector coupling . Next we shall briefly examine how our results will change if we increase the value of from the quenched value 1.00 to the bare value 1.26. The effect of this amplification of the axial coupling strength on the NMEs is demonstrated in Figure 5 where we have plotted the multipole decompositions for nuclei ^{76}Ge and ^{82}Se calculated with both values of the axial coupling and . In case of the ground-state-to-ground-state decays, the multipole changes rather fast when the axial coupling is increased from 1.00 to 1.26. This happens mainly due to the changing of the parameter (for each value, the parameter is adjusted in such a way that the measured rate is reproduced). The multipole contribution is very sensitive to the value of . We can see from Figures 5(a) and 5(b) that for the component is among the five most important multipoles, while for it is not. Some of the higher multipoles change also somewhat, but not so rapidly. Ground-state-to-excited-state transitions proceed mainly through the and multipole channels. We see from Figures 5(c) and 5(d) that increasing the value of affects mostly the component.

**(a)**^{76}Ge()

**(b)**^{76}Ge()

**(c)**^{82}Se()

**(d)**^{82}Se()Figure 6 displays the total cumulative sum distributions for ground-state-to-ground-state decays of the nuclei ^{100}Mo and ^{116}Cd (panels (a) and (b)), and for ground-state-to-excited-state decays of the nuclei ^{82}Se and ^{96}Zr (panels (c) and (d)). Axial coupling values and were adopted. We notice from the figures that increasing the axial coupling strength shifts the distributions downwards. This is especially true for the higher energy parts. Despite this fact, the overall shapes of the cumulative sum distributions do not change much and the same classification of nuclei into different categories according to their cumulative distribution shapes seems to hold also for larger values of .

**(a)**^{100}Mo()

**(b)**^{116}Cd()

**(c)**^{82}Se()

**(d)**^{96}Zr()#### 4. Conclusions

In this paper we have extended our previous work [17] on the ground-state-to-ground-state decay transitions. In the present work we have concentrated our studies on the intermediate contributions to the NMEs involved in the light neutrino mediated decay. We have calculated the intermediate state multipole decompositions of the NMEs and extracted the most important multipole components. Cumulative sums of the NMEs were calculated to investigate the important energy regions contributing to the transitions. Finally, the most important multipole components were divided into contributions coming from the virtual transitions through the individual states of the intermediate nuclei. An extensive tabulation of these important intermediate states were given for all the nuclei considered in this paper.

We have done these computations by using realistic two-body interactions and single-particle bases. All the appropriate short-range correlations, nucleon form factors, and higher-order nucleonic weak currents are included in our present results.

We found in the calculations that often there exists only a few relevant intermediate states which collect most of the strength corresponding to a given multipole. We also found that there exists a single-state dominance in the important components related to the ground-state decays of nuclei ^{76}Ge, ^{82}Se and perhaps also for ^{96}Zr.

#### Competing Interests

The authors declare that they have no competing interests.

#### Acknowledgments

This work has been partially supported by the Academy of Finland under the Finnish Centre of Excellence Programme 2012–2017 (Nuclear and Accelerator Based Programme at JYFL).