*2.1. LNV Models Contributing to* 0*νββ*

In the framework that includes the left-right symmetric model and *R*-parity-violating SUSY model, after hadronization, the 0*νββ* half-life can be written as a sum of products of PSFs, BSM LNV parameters, and their corresponding NMEs [15]:

$$\begin{aligned} \left[T\_{1/2}^{0\nu}\right]^{-1} &= G\_{01} \mathcal{g}\_A^4 \left| \eta\_{0\nu} M\_{0\nu} + \left(\eta\_{N\_R}^L + \eta\_{N\_R}^R\right) M\_{0N} \right. \\ &+ \eta\_{\emptyset} M\_{\emptyset} + \eta\_{\lambda'} M\_{\lambda'} + \eta\_{\lambda} X\_{\lambda} + \eta\_{\eta} X\_{\eta} \right|^2. \end{aligned} \tag{1}$$

Here, *G*<sup>01</sup> is a phase-space factor that can be calculated with good precision for most cases [27,28,78,79], *gA* is the axial vector coupling constant, *η*0*<sup>ν</sup>* = , *mββ*- /*me*, effective Majorana neutrino mass (see Equation (3)), and *me* is the electron mass. *η<sup>L</sup> NR* and *<sup>η</sup><sup>R</sup> NR* are the heavy neutrino parameters with left-handed and right-handed currents, respectively [13,36], *ηq*˜ and *ηλ* are✚R*<sup>p</sup>* SUSY LNV parameters [80], and *ηλ* and *ηη* are parameters for the so-called "*λ*–" and "*η*–mechanisms", respectively [13]. *M*0*<sup>ν</sup>* and *M*0*<sup>N</sup>* are the light and the heavy neutrino exchange NMEs, *Mq*˜ and *Mλ* are the✚R*<sup>p</sup>* SUSY NMEs, and *X<sup>λ</sup>* and *X<sup>η</sup>* denote the combinations of NMEs and other PSFs (*G*02–*G*09) corresponding to the the *λ*–mechanism involving right-handed leptonic and right-handed hadronic currents and the *η*–mechanism with right-handed leptonic and left-handed hadronic currents, respectively [15]. Assuming a seesaw type I dominance [81], the term *η<sup>L</sup> NR* is considered negligible if the heavy mass eigenstates are larger than 1 GeV [52], and I ignore it here. For consistency with the literature, the remaining term *η<sup>R</sup> NR* is labeled as *η*0*N*.

Here, I exclusively describe transitions from the spin/parity *J<sup>π</sup>* = 0<sup>+</sup> ground state (g.s.) of the parent nucleus to the final *J<sup>π</sup>* = 0<sup>+</sup> ground state of the daughter nucleus. There is also the possibility of 0*νββ* decay to the excited states of the daughter, such as the first *J<sup>π</sup>* = 2+. This alternative is rarely considered in the literature, mainly because besides a significant reduction in the effective Q-values for most isotopes, thus reducing the corresponding phase space factors, it has also been known for some time that based on a general analysis the NMEs for this transition are suppressed for the mass mechanism [72]. In addition, the initial numerical estimates of the NMEs corresponding to the *ηη* and *ηλ* in Equation (1) showed that they were also suppressed [82]. Recently, it was found that more up-to-date QRPA calculations of these right currents' contributions could lead to a significant increase in the matrix elements for the lambda mechanism that might compete

with the transition to the *J<sup>π</sup>* = 0<sup>+</sup> ground state, at least for case of 136Xe [83,84]. These new findings are clearly interesting, and I plan to investigate them using shell model techniques similar to the ones described below and report them in future publications.

Table 1 presents the *Qββ* values, the most recent experimental half-life limits, and the nine PSFs for the 0*νββ* transitions to the ground states of the daughter nucleus for five isotopes considered in this investigation. The PSFs were calculated using a new effective method described in detail in Ref. [27]. *G*<sup>01</sup> values were calculated with a screening factor (*sf*) of 94.5, while for *G*02–*G*<sup>09</sup> I used *sf* = 92.0, which was shown to provide results close to those of the more accurate approach described in Ref. [85].

As indicated in Equation (1), the main observable related to 0*νββ* decay is the half-life of the process. It is unlikely that this unique observable, even if measured for several isotopes, could provide enough information to identify different mechanisms that may contribute to this process. In Ref. [15], I investigated other observables that could be used to disentangle contributions from different mechanisms, such as the two-electron angular and energy distributions, in addition to the half-life data from several isotopes. I considered the case where one mechanism dominates, i.e., there is one single term in the decay amplitude of Equation (1). Table 2 of Ref. [17] shows the shell model values of the NMEs that enter Equation (1). Details regarding the definitions of specific NMEs can be found in Refs. [17,49]. All NMEs were calculated using the interacting shell model (ISM) approach [36,43–46,49,52] (see also Ref. [49] for a review) and included short-range correlation effects based on the CD-Bonn parametrization [41], finite-size effects [80], and, when appropriate, optimal closure energies [70] (see Section 3 for more details). Table 2 of Ref. [17] also presents the upper limits for the corresponding LNV parameters extracted from the lower limits of the half-lives under the assumption of one-mechanism dominance. However, less general analyses are available based on QRPA [71,80,85–87], NMEs, and other interactive shell model NMEs [34–37].

If only the main diagram in Figure 2b is considered, the associated mechanism is known as the light neutrino exchange mechanism and the half-life of Equation (1) becomes

$$\left[ \left. T\_{1/2}^{0\nu} \right|^{-1} = \mathcal{G}\_{01} \mathcal{g}\_A^4 \frac{\left| \left< m\_{\beta\beta} \right> \right|^2}{m\_c^2} M\_{0\nu}^2 \tag{2}$$

with the effective neutrino mass given by following sum over the light mass eigenstates:

$$\left| \left\langle m\_{\beta\beta} \right\rangle \right| = \left| \sum\_{i \in I \text{light}} \mathcal{U}\_{\text{el}}^2 m\_i \right|, \tag{3}$$

where *Uei* are the complex matrix elements of the first row in the Pontecorvo–Maki– Nakagawa–Sakata (PMNS) neutrino mixing matrix. This quantity is very often used in the literature as an example of how one could potentially extract additional information about neutrino physics parameters, such as neutrino mass ordering and the mass of the lowest mass eigenstate, from the experimental value of *T*0*<sup>ν</sup>* 1/2 [88].
