Advances in High Energy Physics

Volume 2016, Article ID 1903767, 8 pages

http://dx.doi.org/10.1155/2016/1903767

## Shell Model Studies of Competing Mechanisms to the Neutrinoless Double-Beta Decay in ^{124}Sn, ^{130}Te, and ^{136}Xe

Department of Physics, Central Michigan University, Mount Pleasant, MI 48859, USA

Received 29 June 2016; Accepted 8 September 2016

Academic Editor: Theocharis Kosmas

Copyright © 2016 Andrei Neacsu and Mihai Horoi. 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

Neutrinoless double-beta decay is a predicted beyond Standard Model process that could clarify some of the not yet known neutrino properties, such as the mass scale, the mass hierarchy, and its nature as a Dirac or Majorana fermion. Should this transition be observed, there are still challenges in understanding the underlying contributing mechanisms. We perform a detailed shell model investigation of several beyond Standard Model mechanisms that consider the existence of right-handed currents. Our analysis presents different venues that can be used to identify the dominant mechanisms for nuclei of experimental interest in the mass region (^{124}Sn, ^{130}Te, and ^{136}Xe). It requires accurate knowledge of nine nuclear matrix elements that we calculate in addition to the associated energy-dependent phase space factors.

#### 1. Introduction

Should the neutrinoless double-beta decay () be experimentally observed, the lepton number conservation is violated by two units and the back-box theorems [1–4] predict the neutrino to be a Majorana particle. In addition to the nature of the neutrino (whether a Dirac or a Majorana fermion), there are other unknown properties of the neutrino that could be investigated via , such as the mass scale, the absolute mass, or the underlying neutrino mass mechanism. There are several beyond Standard Model mechanisms that could compete and contribute to this process [5, 6]. Reliable calculations of the nuclear matrix elements (NME) are necessary to perform an appropriate analysis that could help evaluate the contribution of each mechanism.

The most commonly investigated neutrinoless mechanism is the so-called mass mechanism involving the exchange of light left-handed neutrinos, for which the NME were calculated using many nuclear structure methods. Calculations that consider the contributions of heavy, mostly sterile, right-handed neutrinos have become recently available, while left-handed heavy neutrinos have been shown to have a negligible effect [7, 8] and their contribution is generally dismissed. A comparison of the recent mass mechanism results obtained with the most common methods can be seen in Figure 6 of [9], where one can notice the differences that still exist among these nuclear structure methods. Figure 7 of [9] shows the heavy neutrino results for several nuclear structure methods, and the differences are even larger than those in the light neutrino case because of the uncertainties related to the short-range correlation (SRC) effects. There are efforts to reduce these uncertainties by the development of an effective transition operator that treats the SRC consistently [10].

Because shell model calculations were successful in predicting two-neutrino double-beta decay half-lives [11] before experimental measurements and as shell model calculations of different groups largely agree with each other without the need to adjust model parameters, we calculate our nuclear matrix elements using shell model techniques and Hamiltonians that reasonably describe the experimental spectroscopic observables.

Experiments such as SuperNEMO [12, 13] could track the outgoing electrons and help distinguish between the mass mechanism () and so-called and mechanisms [14, 15]. This would also provide complementary data at low energies for testing the existence of right-handed contributions predicted by left-right symmetric models [15–19], currently investigated at high energies in colliders and accelerators such as LHC [20]. To distinguish the possible contribution of the heavy right-handed neutrino using shell model nuclear matrix elements, measurements of lifetimes for at least two different isotopes are necessary, ideally that of an isotope and another lifetime of an isotope, as discussed in Section V of [21]. It is expected that if the neutrinoless double-beta decay is confirmed in any of the experiments, more resources and upgrades could be dedicated to boost the statistics and to reveal more information on the neutrino properties.

Following our recent study for ^{82}Se in [21], which is the baseline isotope of SuperNEMO, we extend our analysis of and mechanisms to other nuclei of immediate experimental interest: ^{124}Sn, ^{130}Te, and ^{136}Xe. These isotopes are under investigation by the TIN.TIN [22] (^{124}Sn), CUORE [23, 24], SNO+ [25] (^{130}Te), NEXT [26], EXO [27], and KamLAND-Zen [28] (^{136}Xe) experiments. For the mass region , we perform calculations in model space consisting of and valence orbitals using the SVD shell model Hamiltonian [29] that was fine-tuned with experimental data from Sn isotopes. Our tests of this Hamiltonian include energy levels, )↑ transitions, occupation probabilities, Gamow-Teller strengths, and NME decomposition for configurations of protons/neutrons pairs coupled to some spin () and some parity (positive or negative), called -pair decomposition. These tests and validations of the SVD Hamiltonian can be found in [9] for ^{124}Sn and in [30] for ^{130}Te and ^{136}Xe. Calculations of NME in larger model spaces (e.g., model space that includes and orbitals missing in models space) were successfully performed for ^{136}Xe [31], but for ^{124}Sn and ^{130}Te they are much more difficult and would require special truncations.

In this work, assuming the detection of several tens of decay events, we present a possibility to identify right-handed contributions from and mechanisms by analyzing the two-electron angular and energy distributions that could be measured.

We organize this paper as follows: Section 2 shows a brief description of the neutrinoless double-beta decay formalism considering a low-energy Hamiltonian that takes into account contributions from right-handed currents. Section 3 presents an analysis of the half-lives and of the two-electron angular and energy distributions results for ^{124}Sn, ^{130}Te, and ^{136}Xe. Finally, we dedicate Section 4 to conclusions.

#### 2. Brief Formalism of

The existence of right-handed currents and their contributions to the neutrinoless double-beta decay rate has been considered for a long time [14, 32], but most frequent calculations treat only the light left-handed neutrino-exchange mechanism (commonly referred to as “the mass mechanism”). One model that considers the right-handed currents contributions and includes heavy particles that are not part of the Standard Model is the left-right symmetric model [17, 18]. Within the framework of the left-right symmetric model, the neutrinoless double-beta decay half-life expression iswhere , , , , and are neutrino physics parameters defined in [15] (see also Appendix A of [21]), and are the light and heavy neutrino-exchange nuclear matrix elements [5, 6], and and are combinations of NME and phase space factors, which are calculated in this paper. is a phase space factor [33] that one can calculate [34] with good precision for most cases [35–37]. The “” sign represents other possible contributions, such as those of R-parity violating SUSY particle exchange [5, 6], Kaluza-Klein modes [6, 38, 39], violation of Lorentz invariance, and equivalence principle [40–42], which we neglected here. term also exists in the seesaw type I mechanisms but its contribution is negligible if the heavy mass eigenstates are larger than 1 GeV [8]. We consider a seesaw type I dominance [43] and we will neglect this contribution here.

For an easier read, we perform the following change of notation: , , and .

In this paper, we provide an analysis of the two-electron relative energy and angular distributions for ^{124}Sn, ^{130}Te, and ^{136}Xe using shell model NME that we calculate. The purpose of this analysis is to identify the relative contributions of and terms in (1). A similar analysis for ^{82}Se was done using QRPA NME in [12] and with shell model NME in [21]. As in [21], we start from the classic paper of Doi et al. [14], describing the neutrinoless double-beta decay process using a low-energy effective Hamiltonian that includes the effects of the right-handed currents. By simplifying some notations and ignoring the contribution from term, which has the same energy and angular distribution as term, the half-life expression [14] is written aswhere and are the relative CP-violating phases (Eq. A7 of [21]) and is the Gamow-Teller contribution of the light neutrino-exchange NME. are contributions from different mechanisms: are from the left-handed leptonic currents, are from the right-handed leptonic and right-handed hadronic currents, and are from the right-handed leptonic and left-handed hadronic currents. , , and contain the interference between these terms. These are defined aswhere are combinations of nuclear matrix elements and phase space factors (PSF). Their expressions can be found in Appendix B, Eqs. (B1) of [21]. and the other nuclear matrix elements that appear in the expressions of the factors are presented in Eq. (B4) of [21].

We write the differential decay rate for transition asHere, is the energy of one electron in units of , is the nuclear radius (, with = 1.2 fm), is the angle between the outgoing electrons, and the expressions for the constant and the function are given in Appendix C, Eqs. (C2) and (C3) of [34], respectively. The functions and are defined as combinations of factors that include PSF and NME: The detailed expressions of components are presented in Eqs. (B7) of [21].

We can express the half-life as follows:with the normalized kinetic energy defined aswhere is the -value of the decay.

The integration of (4) over provides the angular distribution of the electrons that we write aswhere .

Integrating (4) over provides the single-electron spectrum. As in [21], we express the decay rate as a function of the difference in the energy of the two outgoing electrons: , where is the kinetic energy of the second electron. We can write the energy of one electron as Changing the variable, the energy distribution as a function of is

#### 3. Results

The formalism used in this paper is taken from [21], where it was used to analyze the two-electron angular and energy distributions for ^{82}Se, the baseline isotope of the SuperNEMO experiment [12, 13]. It was adapted from [14, 33] with some changes for simplicity and consistency and updated with modern notations. Here we use it to analyze in detail decay two-electron angular and energy distributions for ^{124}Sn, ^{130}Te, and ^{136}Xe. The nine NME required are calculated in this paper using the SVD shell model Hamiltonian [29] in model space which was thoroughly tested and validated for ^{124}Sn in [9] and for ^{130}Te and ^{136}Xe in [30]. For an easier comparison to other results, we use value of 1.254; we include short-range correlations with CD-Bonn parametrization, finite nucleon size effects, and higher-order corrections of the nucleon current [44]. Should one change to the newer recommended value of 1.27 [45], the NME results would change by only 0.5% [46] and the effective PSF (multiplied by ) would change by 5%. This is negligible when compared to the uncertainties in the NME. quenching is not considered here and an extended justification for this decision is given in [21].

In Table 1, we present the nine dimensionless NME for ^{124}Sn, ^{130}Te, and ^{136}Xe calculated in this work using an optimal closure energy = 3.5 MeV which was obtained using a recently proposed method [47]. By using an optimal closure energy obtained for this Hamiltonian, we get NME results in agreement with beyond closure approaches [48]. The definition of these NME and the details regarding their calculations are given in Appendix B of [21].