**1. Introduction**

The recent experimental discovery of neutrino oscillations [1,2] proved that neutrinos have mass, and this discovery was awarded a Nobel prize in 2015 [3,4]. Neutrino oscillation experiments can only provide information about the squared mass differences, while other properties of neutrinos, such as their mass hierarchy, their absolute masses, or their fermionic signatures, Dirac or Majorana, remain to be determined. However, this new information coming from the neutrino oscillations experiments has led to new interest in neutrino physics and in particular in their nature as Dirac or Majorana fermions that may be unraveled by neutrinoless double beta decay investigations.

Neutrinoless double beta decay (0*νββ*) is one of the best experimental approaches for identifying processes that violate the lepton number conservation, thus signaling beyond the Standard Model (BSM) physics. If neutrinoless double beta transitions occur, then the lepton number conservation is violated by two units, and the black-box theorems [5–8] indicate that the light left-handed neutrinos are Majorana fermions. As a consequence, the BSM extension of the Standard Model Lagrangian would be significantly different from that where neutrinos are Dirac fermions. Theoretical investigations of 0*νββ* decay combine lepton number violation (LNV) amplitudes with leptonic phase-space factors (PSFs) and nuclear matrix elements (NMEs). The NMEs are computed using a large variety of nuclear structure methods and specific models. Among the LNV models considered, the left-right symmetric model [9–13] is among the most popular, and its predictions are currently investigated at the Large Hadron Collider [14]. In some recent papers [15–17], I have investigated observables that could identify the contributions of different left-right symmetric model mechanisms to the 0*νββ* decay rate, such as the angular distribution and the energy distribution of the two outgoing electrons that could be measured. A more general approach is effective field theory (EFT), which considers an expansion of the BSM Lagrangian consistent with the Standard Model symmetries and including LNV and neutrino mass mechanisms. This approach has the advantage of being independent of specific models, and it can be used to describe in a unified manner BSM-sensitive observables, including those related to 0*νββ* decay. One can then use the existing data/limits from

**Citation:** Horoi, M. Double Beta Decay: A Shell Model Approach. *Physics* **2022**, *4*, 1135–1149. https:// doi.org/10.3390/physics4040074

Received: 6 June 2022 Accepted: 7 September 2022 Published: 26 September 2022

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different experiments to evaluate the energy scales up to which the effective field operators are not broken and limits for effective low-energy couplings.

The theoretical analysis of the 0*νββ* decay process has many steps, including the nuclear structure calculation of the NMEs. However, in the first step, the weak interaction of quarks and leptons described by the BSM EFT Lagrangian is considered in the lowest order (see the diagram in Figure 1). In the next step, the hadronization process to nucleons and exchanging pions is considered, leading to the diagram in Figure 2. Furthermore, the nucleons are treated in the impulse approximation leading to free space 0*νββ* transition operators, and the nucleon dynamics inside the nuclei are treated using nonperturbative nuclear wave functions, which are later used to obtain the nuclear matrix elements needed to calculate the 0*νββ* observables, such as half-lives and two-electron angular and energy distributions [15]. A modern approach that can be used to make the transition from quarks and gluons to nucleons and pions is based on the chiral effective field theory of pions and nucleons [18,19]. This approach introduces a number of effective low-energy couplings, which in principle can be calculated from the underlying theory of strong interaction using lattice QCD techniques [18] or may be extracted within some approximation from the known experimental data [19]. These couplings may have new complex phases, and they could include effective contributions from the exchange of heavier mesons. The lattice QCD approach is in progress (see, e.g., Ref. [20]), but it has proven to be difficult for extracting some of the necessary weak nucleon coupllowestngs, even the known *gA* [20].

**Figure 1.** The 0*νββ* decay process diagrams: (**a**) typical 0*νββ* decay diagram at the quark (*u* and *d*) level presents the generic description of the process and (**b**) light left-handed (*L*) neutrino exchange diagram shows the most studied case in the literature, that of the light left-handed neutrino exchange. Here, "..." stands for other diagrams involving left- and right-handed (*R*) leptons (see, e.g., Figure 1 of Ref. [17] for model diagrams).

**Figure 2.** Similar to Figure 1, the nucleon-level diagrams of 0*νββ* decay process: (**a**) the typical 0*νββ* decay process nucleon-level diagram presents the generic description of the process and (**b**) the light left-handed neutrino exchange diagram shows the light left-handed neutrino exchange. Here, "..." stands for other higher-order effective field theory (EFT) diagrams (see Figure 2 of Ref. [17]).

Here, as in Ref. [17], I use the formalism of Refs. [21–24] that provides a general EFT approach to the BSM Lagrangian. It also provides a somewhat older hadronization scheme, which is needed to obtain the neutrinoless double beta decay transition operators. To extract new limits for the effective Majorana mass and for the low-energy EFT couplings from the current experiment for the isotopes listed in Table 1 below, I use the assumption

that only one single coupling in the BSM Lagrangian may dominate the 0*νββ* amplitude. In the analysis, about 20 nuclear matrix elements and nine phase-space factors are needed. I use the existing neutrinoless double beta decay data to extract the limits for the BSM EFT couplings and limits of validity for the energy scale of the BSM Lagrangian. In addition, the calculated ratio of half-lives for different isotopes could be useful in guiding the experimental effort, in estimating their scales and costs, in fine-tuning the experimental searches for the 0*νββ* transition mechanism, and also in providing a better view and comparison of the status of various experimental efforts. Our analysis suggests that the experimental confirmation of 0*νββ* decay rates for several isotopes could possibly help in identifying the dominant mechanism responsible for the transition.

**Table 1.** The *Qββ* values (in MeV), the experimental half-lives *T*2*<sup>ν</sup>* 1/2 [25,26] and *<sup>T</sup>*0*<sup>ν</sup>* 1/2 limits (in years), and the calculated PSFs *G*2*<sup>ν</sup>* [27] and *G*<sup>01</sup> (*G*02–*G*<sup>09</sup> can be found elsewhere [17]) (in years−1) for all five isotopes currently under investigation.


One important step in describing the 0*νββ* decay observables is obtaining the appropriate NMEs. The nuclear structure methods used for NME calculations are the interacting shell model [34–52], proton-neutron quasi random phase approximation (pnQRPA) [21–24,53–57], interacting moson model [58–61], projected Hartree–Fock–Bogoliubov [62], energy density functional [63], and relativistic energy density functional method [64]. The NMEs calculated with different methods and by different groups show sometimes large variations by a factor of 3–5 [65,66]. Most references only provide NMEs for the light left-handed Majorana neutrino exchange mechanism, but some provide results for the right/left heavy neutrino exchange and some more exotic mechanisms. Ref. [50] provides tables and plots that compare results for the light left-handed neutrino exchange and for the heavy righthanded neutrino exchange, while Ref. [17] provides tables with all NMEs necessary for the EFT approach. I calculate the NMEs using shell model techniques [36,41–51] and a preferred set of effective Hamiltonians that were tested for a wide set of nuclei. The shell model calculations of NMEs use a relatively small single-particle model space, but they are better suited and more reliable for 0*νββ* decay calculations because they take into account all the correlations around the Fermi surface, respect all nuclear many-body problem symmetries, and can take into account the effects of the missing single particle space via many-body perturbation theory (the effects were shown to be small [67]). In addition, it was shown [68,69] that the QRPA approaches using the same model spaces and effective Hamiltonian as in the shell model produce NMEs within 25% of the shell model results. Furthermore, I test the shell model methods and the effective Hamiltonians by comparing the calculations of spectroscopic observables for the nuclei involved in the transition to the experimental data, as presented in Refs. [41,50,70]. I do not consider any quenching for the bare 0*νββ* operator in these calculations. Such a choice is different from that for the simple Gamow–Teller operator used in the single beta and two-neutrino double beta decay (2*νββ*), where a quenching factor of about 0.7 is necessary [69]. For the PSFs, I use an effective theory based on the formalism of Ref. [71], but fine-tuned as to take into account the effects of a Coulomb-field-distorting finite-size proton distribution in the daughter nuclei. Table 1 provides information relevant for the main nuclei that can be calculated using shell model techniques (see Equations (1) and (13) below for a precise definition of the PSFs used).

In this paper, I mostly review the shell model techniques needed to accomplish the plan outline above. The numerical results and their analysis are available in different papers that are appropriately cited below. Although most material described below reviews results already published, some new results can be found at the end of Section 3.2 and in Section 4. The paper is organized as follows: Section 2 analyzes the contributions of several BSM mechanisms to neutrinoless double beta decay, and it presents the framework of effective field theory for neutrinoless double beta decay; Section 3 presents an analysis of the 0*νββ* nuclear matrix elements in the shell model approach; Section 4 presents an analysis of the 2*νββ* nuclear matrix elements in the shell model approach; Section 5 is dedicated to conclusions.

## **2. Neutrinoless Double Beta Decay And Neutrino Physics**

The main mechanism considered to be responsible for neutrinoless double beta decay is the mass mechanism that assumes that the neutrinos are Majorana fermions and relies on the assumption that the light left-handed neutrinos have mass. However, the possibility that right-handed currents could contribute to neutrinoless double beta decay (0*νββ*) has been already considered for some time [71,72]. Recently, 0*νββ* studies [13,73] have adopted the left-right symmetric model [11,74] for the inclusion of right-handed currents at quark level. In addition, the *R*-parity-violating (✚R*p*) supersymmetric (SUSY) model can also contribute to the neutrinoless double beta decay process [75–77].
