Two decades ago, the successful experimental measurement of neutrino oscillations [1, 2] established that neutrinos have a mass different from zero. Although this discovery was a significant one, many of the neutrino properties still remain unknown to this day. Because in neutrino oscillation experiments only squared mass differences can be measured, we still have unanswered questions regarding their absolute masses, the mass hierarchy, the underlying mechanism that gives neutrinos mass, and even the very nature of the neutrinos (whether they are Dirac or Majorana particles). While there are many experimental and theoretical endeavors to bring clear answers to some of these questions, like high-precision calculations, measurements of different single-β decays, cosmological observations, the double-beta decay (DBD) and particularly the 0νββ decay mode are still considered the most appealing approaches to study the yet unknown properties of neutrinos. However, even if one 0νββ transition event would be experimentally observed, not all of the desired information about neutrinos would be immediately revealed. Recording such an event would demonstrate that the lepton number conservation is violated by two units, but cannot indicate the mechanism that dominates this process. Many large-scale experiments dedicated to the discovery of this lepton number violating (LNV) decay are already collecting data, with up-dates and new ones planned for the future, but so far there is no experimental proof of 0νββ transitions, only reports of lower limits for the corresponding half-lives. Experimentally, DBD of the isotopes ⁷⁶Ge and ¹³⁶Xe are currently the most accurately measured, but others like ⁴⁸Ca, ⁸²Se, and ¹³⁰Te are also investigated, with ¹²⁴Sn being considered for the future. There are advantages and disadvantages to studying each of these isotopes (costs, purity, Q-value, background signals, etc.), but the fact that different ones are being investigated is of great importance if an experimental confirmation is obtained for any of them. Theoretical studies of 0νββ involve the computation of nuclear matrix elements (NME) and phase space factors (PSF) appearing in the half-life expressions, whose precise calculation is essential for predicting the neutrino properties and interpretation of the DBD experimental data. Particularly, the NME computation is the subject of the largest uncertainties, so much effort is devoted to their accurate estimation. The most commonly used nuclear structure approaches for the NME calculation are proton-neutron Quasi Random Phase Approximation (pnQRPA) [3–11], Interacting Shell Model (ISM) [12–30], Interacting Boson Model (IBM-2) [31–35], Projected Hartree Fock Bogoliubov method (PHFB) [36], Energy Density Functional method (EDF) [37], and the Relativistic Energy Density Functional method (REDF) [38]. Each of these methods presents various advantages and disadvantages when compared to each other, especially when dealing with the nuclear structure of particular isotopes. Once experimentally confirmed, it is also important to establish the underlying mechanism(s) that may contribute to the 0νββ decay, as to properly extend the Standard Model. For the longest time, studies only addressed the so called “mass mechanism” that involves the exchange of light left-handed (LH) Majorana neutrinos. Presently, more scenarios are being considered and their investigation consists of calculating of the NME associated to each mechanism and the corresponding PSF. For example, possible contributions to 0νββ decay may come via the exchange of the right-handed (RH) heavy neutrinos [39]. Other contributions from possible RH components of the weak currents, through the so-called “λ” and “η” mechanisms could also be taken into account [40]. One of the most popular model that includes these mechanisms is the left-right symmetric model (LRSMM) [41–45]. In this work we consider several of these scenarios for 0νββ decay, following the prescriptions of [46]. We present and discuss the NME and PSF calculations that were recently published by our group. For the nuclear structure calculations our group and collaborators use Shell Model (ShM) techniques and codes [19–29]. For the PSF calculations we use our results from [47, 48] for the light neutrino and the heavy neutrino exchange mechanisms, and results from [49] for the other mechanisms. Using the latest experimental limits for the half-lives reported in literature, we up-date the constraints on the LNV parameters corresponding to each mechanism. Finally, we use the calculated values of the LNV parameters deduced with the half-life limits taken from the ⁷⁶Ge experiment [50], to evaluate the half-lives of the other four isotopes that should be achieved by their experiments to reach the sensitivity of the Ge experiment.

For a long time, most of the 0νββ decay literature has focused its interest mainly on the mass mechanism, that assumes that this decay mode occurs via the exchange of light LH Majorana neutrinos between two nucleons inside the nucleus. The inclusion of contributions coming from RH components of the weak currents has also been discussed (for example in [40, 51]), but very few papers presented theoretical results considering these contributions. However, any mechanism/scenario that violates with two units the lepton number conservation may, in principle, contribute to the decay rate. Considering several mechanisms, the 0νββ decay half-life can be written in a factorized compact form, as a sum of products of PSF, NME, and the BSM parameters, corresponding to each mechanism [52], as follows:

Here, the Ei contain the BSM parameters associated with the following mechanisms: E1=E0ν=∑klightUek2mkme the exchange of light LH neutrinos, E2=Eλ=(MWLMWR)2|∑klightUekVek| corresponds to the “λ mechanism” with RH leptonic and LH hadronic currents, E3=Eη=ξ|∑klightUekVek| is associated to the “η mechanism” with RH leptonic and RH hadronic currents, and E4=E0N=(MWLMWR)4∑kheavyVek2mpMk comes from the exchange of heavy RH neutrinos, with m_e being the electron mass and m_p the proton mass. M_WL and M_WR denote the masses of the LH and the RH W bosons, respectively. We assume that the neutrino mass eigenstates are separated as light, m_k(m_k ≪ 1 eV), and heavy, M_k(M_k ≫ 1 GeV). U_ek and V_ek are electron neutrino mixing matrices for the light LH and heavy RH neutrino, respectively [14, 44]. Following [4–6, 39, 46], Mi2 are factors expressed in a standardized form as combinations of NME described in Equation (2) and integrated PSF denoted with G₀₁ − G₀₉. Values for the PSF used in this paper can be seen in Table 1, together with our ShM values for the individual NME M_α (with α= GTq, Fq, Tq, GTω, Fω, P, R, M_GTN, M_FN, and M_TN). Assuming that only one mechanism dominates the 0νββ transition, we can perform a so called "on-axis" analysis where the interference terms EiEjMij are no longer taken into account.

Detailed equations of individual NME M_α can be found in the Appendix of [46], where they have been expressed in a consistent form. The expressions for the PSF can be found in [47, 49]. We note that Equations (2a, 2b) contain combinations of NME and PSF coming from contributions of only s-wave electron wave functions, while Equations (2c, 2d) present combinations of NME and PSF with contributions only from p-wave electron wavefunctions.

To use the expressions in Equation (2), we need accurate calculations of both the PSF that embed the distortion of the motion of outgoing electrons by the electric field of the daughter nucleus, and of the NME that depend on the nuclear structure of the parent and the daughter nuclei. Thus, the theoretical investigation of 0νββ transitions is a complex task that involves knowledge of physics at the atomic level for the PSF, nuclear level when calculating the NME, and at the fundamental particle level dealing with the LNV couplings.

This brief review summarizes our recent calculations of the PSF and NME involved in 0νββ decay for four possible decay mechanisms, namely the light LH neutrino exchange, heavy RH neutrino exchange, λ−mechanism involving RH leptonic and RH hadronic currents, and the η−mechanism involving RH leptonic and LH hadronic currents. The PSF are calculated with Fermi functions built with exact electron w.f. solutions of the Dirac equation with a Coulomb-type potential obtained from a realistic distribution of protons in the daughter nucleus. FNS and screening effects were taken into account, as well. G₀₁ that include s-w.f. are taken from [48], while G_(02−09) that include p-w.f. are taken from [49]. Between the newer and the older PSF values, one can observe numerous differences in the range of 5–30%, with some rising of up to 90% (see G₀₈ of ¹³⁶Xe in Table 1). Such differences would impact the LNV values and the conclusions regarding the sensitivity of the experiments with various isotopes to the possible 0νββ mechanisms. In passing, we mention that in addition to the development of the new PSF codes, our group has also developed a very fast effective method [59] that is still based on the formalism of [40], but is fitted and tweaked to replicate the current results obtained with the most rigorous methods. Within reasonable precision, this method can be used for rapid PSF estimations and for plotting the un-integrated angular and energy electron distributions.

The NME are calculated within a ShM approach with the ingredients presented in section 2.2. ShM calculations are attractive because they consider all the correlations around the Fermi surface, respect all symmetries, and take into account consistently the effects of the missing single particle space via many-body perturbation theory (the effects were shown to be small, about 20%, for ⁸²Se [60]). In the case of closed-shell nuclei, ShM calculations using optimized Hamiltonians for nucleon-nucleon interactions are very reliable and compare well with the spectroscopic data available from experiments. Another advantage of this approach, important for reliable calculations, is that the calculated nucleon occupancies are close to the experimental ones. ShM calculations were successful in predicting the 2νββ decay half-life of ⁴⁸Ca [12] before experimental measurements. Calculations of different groups largely agree with each other without the need to adjust model parameters.

From Equation (2), using the NME and PSF in Table 1, we calculate the Mi2 factors that enter the half-life in Equation (1). Using these factors and the most recent experimental half-life limits, we re-evaluate the LNV parameters corresponding to the four mechanisms. These results are presented in Table 2.

Table 2 first presents in the top section the experimental lower half-life limits T_1/2 in years. The next rows list the Mi2 factors that contain combinations of PSF and NME for the five nuclei of current experimental interest, in the case of four possible 0νββ decay mechanisms described in Equation (2). In the middle section are found the values of the LNV parameters Ei deduced from the experimental T_1/2 and the Mi2 factors. For the mass mechanism, we also show the electron neutrino mass parameters ❬m_0ν❭ in units of eV that are obtained by the multiplication of E0ν with the electron mass m_e. This extracted ❬m_0ν❭ is what is most commonly reported in the literature and is presented here for the convenience of the reader and an easier comparison with other references.

Lastly, we perform predictions of the half-lives for each isotope that would correspond to the LNV parameters extracted from one experiments of the highest sensitivity. Choosing the Ei LNV deduced from the ⁷⁶Ge experiment [50], we estimate the half-lives of the other four isotopes. These values are displayed in the lower section of Table 2 and offer an indication about the relative sensitivity between DBD experiments.

All authors listed have made a substantial, direct and intellectual contribution to the work, and approved it for publication.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.