Constraints on Heavy Neutrino and SUSY Parameters Derived from the Study of Neutrinoless Double Beta Decay

Abstract

New constraints on the lepton number violating (LNV) parameters are derived from the analysis of the neutrinoless double beta (0𝜈𝛽𝛽) decay in the hypothesis that this process would occur through the exchange of heavy neutrinos and/or SUSY particles. For derivation, we use new values of both phase space factors (PSFs) and nuclear matrix elements (NMEs) calculated with numerical codes developed recently, as well as the most recent experimental lifetimes. The NMEs are computed with a shell model (ShM) code for ⁴⁸Ca, ⁷⁶Ge, and ⁸²Se nuclei, while at present similar ShM results are available only for the first nucleus. We compare our results with similar ones from literature, obtained with ShM, QRPA, and IBM-2 methods, and conclude that more results are needed for a relevant analysis on the validity of NMEs associated with these decay mechanisms.

1. Introduction

Neutrinoless double beta (0𝜈𝛽𝛽) decay is a beyond standard model (BSM) process by which an even-even nucleus transforms into another even-even nucleus with the emission of two electrons/positrons but no antineutrinos/neutrinos in the final states. Its study is important since it would clarify the question about the lepton number violation, decide on the neutrinos character (are they distinguished or not from their antiparticles?), and give a hint on the scale of their absolute masses. Moreover, the study of 0𝜈𝛽𝛽 decay has a broader potential to search for other BSM phenomena, such as new neutrino flavors and exotic particles. The great interest in understanding these fundamental issues has led to many theoretical and experimental investigations of this process. The reader can find up-to-date information on these studies in several recent reviews [1–6], which also contain therein a comprehensive list of references in the domain.

One of the open issues concerning the 0𝜈𝛽𝛽 decay is related to the possible mechanisms responsible for its occurrence. The exchange of a light Majorana neutrino in the presence of left handed (LH) weak interaction is still the most popular and studied mechanism. However, other mechanisms have also been considered, for example, the exchange of heavy neutrinos [7, 8] and SUSY particles [3, 9], which could also contribute to the total 0𝜈𝛽𝛽 decay rate. On the other hand, constraints on heavy neutrino and SUSY parameters can be now derived at hadron colliders, as well, by analyzing same sign dilepton decay channels, which are triggered by processes that violate the lepton number conservation by two units (like 0𝜈𝛽𝛽 decay). Indeed, the CMS, ATLAS, and LHCb experiments at LHC include now in their data analysis the search of such channels, and the first results were already reported [10–13]. Thus, complementary information that allows the constraint of various LNV parameters can also be obtained now from high-energy experiments.

For all mechanisms, the 0𝜈𝛽𝛽 lifetimes can be expressed in a good approximation as a product of a phase space factor, a nuclear matrix element related to the nuclear structure of the parent and daughter nuclei, and a LNV parameter related to the BSM mechanism considered. These three factors can be identified in (1) and (2) from the next section. Hence, to extract reliable upper limits for the LNV parameters, we need accurate calculation of both PSFs and NMEs factors, as well as reliable measurements of the lifetimes.

The largest uncertainties in the theoretical calculations for double beta decay (DBD) arise from the calculated values of the NMEs. For the LH light neutrino exchange mechanism, the NMEs are currently computed by several methods, the present most employed ones being the proton-neutron quasi-random phase approximation (pnQRPA) [14–20], interacting shell model (ISM) [21–25], interacting boson model (IBM) [26–28], projected Hartree Fock Bogoliubov (PHFB) [29], and energy density functional (EDS) [30] methods. At present, there are still large discrepancies between the NMEs values computed with different methods and by different groups, which have been widely discussed in literature (see, e.g., [5, 6]) for the light neutrino exchange mechanism. At present, for heavy neutrino and SUSY exchange mechanisms, there are fewer NME calculations, performed with QRPA [31, 32] and IBA-2 [28] and ShM methods [33, 34]. The discrepancies between the existing NME values associated with heavy neutrino or SUSY mechanisms are even larger than those in the case of LH light neutrino exchange, so there is a need for new calculations.

In this paper, we report new constraints on the heavy neutrino and SUSY parameters derived from the analysis of the neutrinoless double beta decay of three experimentally interested nuclei, ⁴⁸Ca, ⁷⁶Ge, and ⁸²Se, in the hypothesis that this process would occur through the exchange of such particle. The computation of the NMEs is performed with a ShM code developed recently and presented in detail in [35, 36]. Also, we estimate the uncertainties introduced in calculations by the use of different NN interactions and SRC parameterizations and found that the calculated NME values depend significantly on these nuclear ingredients. We compare our results with other previous ones from literature performed with ShM, QRPA, and IBM-2 approaches and discuss the discrepancies. We remark that, until now, large scale ShM calculations of NMEs corresponding to these mechanisms are available only for ⁴⁸Ca, and they differ significantly from the IBM-2 calculations. In order to understand these discrepancies, more ShM calculations of the NMEs for these (less discussed) 0𝜈𝛽𝛽 decay mechanisms, including other nuclei, are needed.

The values of the PSFs are taken from our recent work [37], which are very close to those reported in [38, 39]. We mention that PSFs for DBD have been calculated since long time [40, 41], but they were less discussed, being considered to be computed with enough precision. Recently, they were recalculated with improved approaches [38, 39, 42] and several discrepancies were revealed as compared to the old calculations, which should be taken into account for accurate predictions of 0𝜈𝛽𝛽 lifetimes and derivation of LNV parameters.

The paper is organized as follows. In the next section, we present the formulae for the 0𝜈𝛽𝛽 decay lifetimes together with the relevant expressions for the PSFs and NMEs which have to be computed for deriving the LNV parameters, for heavy neutrino and SUSY mechanisms. In Section 3, we present and discuss our results, while in the last section we formulate the conclusions of our work.

2. Formalism

In this section, we give a short description of the heavy neutrino and SUSY formalisms, displaying the PSF and NME formulas that have to be computed. As we already mentioned, the expressions of the 0𝜈𝛽𝛽 lifetimes can be written as a product of three terms: a phase space, a nuclear matrix elements, and a term related to the corresponding LNV mechanism: 𝑇^0𝜈_1/2⁻¹=𝐺^0𝜈𝑄_𝛽𝛽,𝑍_𝑘𝑀_𝑘^0𝜈2𝜂_𝑘², (1) where 𝑘 is an index which denotes the mechanisms that can contribute to the 0𝜈𝛽𝛽 decay. In this work, we refer to the heavy neutrino and SUSY exchange mechanisms, so the lifetime reads 𝑇^0𝜈_1/2⁻¹=𝐺^0𝜈𝑄_𝛽𝛽,𝑍𝑀_𝑁^0𝜈𝜂_𝑁+𝑀_𝜆^0𝜈_′𝜂_𝜆′+𝑀_𝑞^0𝜈𝜂_𝑞², (2) where 𝐺^0𝜈 is the phase space factor for this decay mode, depending on the energy decay 𝑄_𝛽𝛽 and nuclear charge 𝑍. 𝑀^0𝜈_{𝑁,𝜆′,𝑞} are the NMEs associated with the heavy neutrino (𝑁) exchange mechanism and with the gluino (𝜆^′) and squark-neutrino (𝑞) exchange mechanisms. The last mechanisms may appear in SUSY theories with R-parity violation [3]. 𝜂_{𝑁,𝜆′,𝑞} are the corresponding coupling parameters associated with these mechanisms.

For the 0𝜈𝛽𝛽 decay, the expression of the PSFs reads 𝐺_0𝜈=24𝑔⁴_𝐴𝑅²ln2^{𝑇0−𝑚𝑒𝑐2}_𝑚𝑒𝑐²𝑓⁽⁰⁾₁₁𝑤_0𝜈𝑑𝜖₁, (3) where 𝑓⁽⁰⁾₁₁ are the solutions of the Dirac equation and 𝑤_0𝜈=𝑔⁴_𝐴𝐺cos𝜃_𝐶⁴16𝜋⁵𝑚_𝑒𝑐²²₋ℎ𝑐²𝑝₁𝑐𝑝₂𝑐𝜖₁𝜖₂, (4) where 𝑔_𝐴  (𝑔_𝐴=1.25) is the axial-vector coupling constant, 𝐺=1.16637×10⁻⁵ GeV⁻² is the Fermi constant, 𝜃_𝐶 is the Cabbibo angle, 𝑚_𝑒 is the electron rest mass, while 𝑝₁, 𝑝₂ and 𝜖₁, 𝜖₂ are the electron momenta and energies, respectively. In (3), it is convenient to redefine the PSFs by a renormalization that eliminates the constant 𝑔_𝐴 and correlates (by dividing by 4𝑅², 𝑅 the nuclear radius) the dimension of 𝐺_0𝜈 with the NMEs which are dimensionless, such that the PSFs are expressed in [yr⁻¹].

Concerning the NMEs, their expressions can be written, in general, as a sum of three components: 𝑀^0𝜈=𝑀^0𝜈_GT−𝑔_𝑉𝑔_𝐴²⋅𝑀_𝐹^0𝜈−𝑀_𝑇^0𝜈, (5) where 𝑀^0𝜈_GT, 𝑀_𝐹^0𝜈, and 𝑀_𝑇^0𝜈 are the Gamow-Teller (GT), Fermi (𝐹), and Tensor (𝑇) components, respectively.

Due to the two-body nature of the transition operator, the NMEs can be expressed as a sum of products of two-body transition densities (TBTDs) and matrix elements of the two-body transition operators for two-particle states, shortly, two-body matrix elements (TBMEs) [35, 36]. Consider 𝑀_𝛼^0𝜈=_{𝑗𝑝𝑗𝑝′𝑗𝑛𝑗𝑛′𝐽𝜋}TBTD𝑗_𝑝𝑗_𝑝′,𝑗_𝑛𝑗_𝑛′;𝐽_𝜋×𝑗_𝑝𝑗_𝑝′;𝐽_𝜋𝜏₋₁𝜏₋₂𝑂^𝛼₁₂𝑗_𝑛𝑗_𝑛′;𝐽_𝜋, (6) where |𝑗𝑗^′;𝐽^𝜋⟩ represent the antisymmetrized two-particle states and 𝑂^𝛼_𝑚𝑛 are DBD transition operators (𝛼=GT,𝐹,𝑇). Since the NN effective interaction can be treated by means of a central (single-particle) potential, the NMEs can be conveniently calculated using Moshinsky’s transformations between the relative and center of mass (CM) coordinates and the proper use of nuclear states in different coupling notations. The calculation of the matrix elements of these operators can be decomposed into products of reduced matrix elements within the spin and relative coordinates subspaces [21, 35]. The expressions of the two-body transition operators are 𝑂^GT₁₂=𝜎₁⋅𝜎₂𝐻(𝑟),𝑂^𝐹₁₂(𝑟),𝑂=𝐻_𝑇12=23𝜎₁×𝜎₂²⋅𝑟𝑅𝐻(𝑟)𝐶⁽²⁾(𝑟)̂, (7) where 𝜎 are the Pauli spin operators and 𝐶⁽²⁾(𝑟)̂ is the center of mass operator. The expressions for the neutrino potentials 𝐻_𝛼 in the case of heavy neutrino mechanisms are 𝐻_𝛼(𝑟)=2𝑅𝜋𝑚_𝑒𝑚_𝑝^∞₀𝑗_𝑘𝑞𝑟ℎ_𝛼𝑞²𝑞²𝑑𝑞, (8) where 𝑚_𝑝 is the proton mass and 𝑗_𝑘 (𝑘=0 for 𝛼=GT,𝐹, and 𝑘=2 for 𝛼=𝑇) are the Bessel spherical functions. The expressions of ℎ_𝛼(𝛼=GT,𝐹,𝑇) are ℎ_𝐹𝑞²=𝐺²_𝑉𝑞²,ℎ_GT𝑞²=𝐺²_𝐴𝑞²𝑔²_𝐴21−3𝑞²𝑞²+𝑚²_𝜋+13𝑞²𝑞²+𝑚²_𝜋²+23𝐺²_𝑀𝑞²𝑔²_𝐴𝑞²4𝑚²_𝑝,ℎ_𝑇𝑞²=𝐺²_𝐴𝑞²𝑔²_𝐴23𝑞²𝑞²+𝑚²_𝜋−13𝑞²𝑞²+𝑚²_𝜋²+13𝐺²_𝑀𝑞²𝑔²_𝐴𝑞²4𝑚²_𝑝, (9)(10)(11) where 𝑚_𝜋 is the pion mass and 𝐺_𝑀𝑞²=𝜇_𝑝−𝜇_𝑛𝐺_𝑉𝑞², (12) with (𝜇_𝑝−𝜇_𝑛)=4.71. The expressions (9)–(11) include the following nuclear effects: (i) finite nucleon size (FNS) effects through the nucleon form factors 𝐺_𝐴, 𝐺_𝑉 and (ii) higher order current (HOC) effects by the second and third terms in the ℎ_GT expression and by the appearance of the ℎ_𝑇 contribution.

The 𝐺_𝑉 and 𝐺_𝐴 form factors which take into account the finite size of the nucleons effect are 𝐺_𝐴𝑞²=𝑔_𝐴Λ²_𝐴Λ²_𝐴+𝑞²²,𝐺_𝑉𝑞²=𝑔_𝑉Λ²_𝑉Λ²_𝑉+𝑞²². (13) For the vector and axial coupling constants, the majority of calculations take 𝑔_𝑉=1 and the unquenched value 𝑔_𝐴=1.25, while the values of the vector and axial vector form factors are Λ_𝑉=850MeV and Λ_𝐴=1086MeV [1], respectively.

For the SUSY mechanisms, only the GT and 𝑇 components of the 𝑀^0𝜈 are contributing. For the gluino exchange mechanism (SUSY1), the radial neutrino potentials 𝐻_𝛼 have a similar form as that for the heavy neutrino mechanism (9), but with different ℎ_𝛼 factors [3, 14]: ℳ_𝜆^0𝜈_′=𝑐^1𝜋𝑀^1𝜋_GT−𝑀_𝑇^1𝜋+𝑐^2𝜋𝑀^2𝜋_GT−𝑀_𝑇^2𝜋, (14) with 𝑐^1𝜋2=−9√2𝑓_𝜋𝑚⁴_𝜋𝑚³_𝑝𝑚_𝑒𝑚_𝑢+𝑚_𝑑𝑔_𝑠𝐹_𝑃𝑔²_𝐴,𝑐^2𝜋=1𝑓18^2𝜋𝑚⁴_𝜋𝑚³_𝑝𝑚_𝑒𝑚_𝑢+𝑚_𝑑²𝑔²_𝑠𝑔²_𝐴, (15) where 𝑚_𝜋 is the charged-pion mass, 139.6 MeV, 𝑓_𝜋=0.668𝑚_𝜋 MeV, 𝑚_𝑢 and 𝑚_𝑑 are the up and down quark masses (𝑚_𝑢+𝑚_𝑑)=11.6MeV, 𝑚_𝑝=938.27MeV, 𝑚_𝑒=0.511MeV, 𝑔_𝑠=13.4, and 𝐹_𝑃≈4.41 [3]. Computing the expressions from (15), we obtain 𝑐^1𝜋=−85.33 and 𝑐^2𝜋=365.72. Consider 𝐻_𝛼^1𝜋(𝑟)=−2𝑅𝜋^∞₀𝑗_𝑘𝑞𝑟ℎ_𝛼^1𝜋𝑞²𝐻𝑑𝑞,_𝛼2𝜋(𝑟)=−4𝑅𝜋^∞₀𝑗_𝑘𝑞𝑟ℎ_𝛼^2𝜋𝑞²𝑑𝑞, (16) with ℎ_𝛼^1𝜋𝑞²=𝐺²_𝐴𝑞²𝑔²_𝐴𝑞⁴/𝑚⁴_𝜋1+𝑞²/𝑚²_𝜋,ℎ_𝛼^2𝜋𝑞²=𝐺²_𝐴𝑞²𝑔²_𝐴𝑞⁴/𝑚⁴_𝜋1+𝑞²/𝑚²_𝜋². (17)

For the squark-neutrino mechanism (SUSY2), 𝐻_𝛼 has an expression similar to the light neutrino mechanism: 𝐻_𝛼(𝑟)=2𝑅𝜋^∞₀𝑗_𝑘ℎ𝑞𝑟^𝛼𝑞²⟨𝐸⟩𝐺𝑞+^𝛼𝑞²𝑞𝑑𝑞, (18) where ⟨𝐸⟩ is the energy used in the closure approximation and represents the average excitation energy of the states in the intermediate odd-odd nucleus that contribute to the decay. The ℎ_𝛼 factors have in this case the following expressions [3, 14]: ℎ_GT,𝑇1=−6𝑚²_𝜋𝑚_𝑒𝑚_𝑢+𝑚_𝑑𝑞²/𝑚²_𝜋1+𝑞²/𝑚²_𝜋². (19)

For computing the radial matrix elements ⟨𝑛𝑙|𝐻_𝛼|𝑛^′𝑙^′⟩, we use the harmonic oscillator wave functions 𝜓_𝑛𝑙(𝑙𝑟) and 𝜓_{𝑛′𝑙′}(𝑟) corrected by a factor [1+𝑓(𝑟)], which takes into account the SRCs induced by the nuclear interaction: 𝜓_𝑛𝑙(𝑟)⟶1+𝑓(𝑟)𝜓_𝑛𝑙(𝑟). (20) For the correlation function, we take the functional form 𝑓(𝑟)=−𝑐⋅𝑒^−𝑎𝑟21−𝑏𝑟², (21) where 𝑎, 𝑏, and 𝑐 are constants which have particular values in different parameterizations. In this work, we consider the Miller-Spencer (MS) [44], AV18 [45], and CD-Bonn [46, 47] parameterizations.

3. Numerical Results and Discussions

We calculate first the NMEs for the three nuclei, that is, ⁴⁸Ca, ⁷⁶Ge, and ⁸²Se, and the three mechanisms mentioned above, using the code described in detail in [35, 36]. For ⁴⁸Ca, we performed calculations with two different NN interactions, GXPF1A [51] and KB3G [52] interactions, while for the other two isotopes we used the JUN45 [53] NN interaction. In the case of ⁴⁸Ca, our model space includes the whole fp shell 𝑓_7/2, 𝑝_3/2, 𝑓_5/2, and 𝑝_1/2, while, for ⁷⁶Ge and ⁸²Se, the model space is jj44, including the following orbitals 𝑓_5/2, 𝑝_3/2, 𝑝_1/2, and 𝑔_9/2. For the nuclear parameters involved in calculations, we used the values mentioned in the previous section and, for the nuclear radius, 𝑅=𝑟₀𝐴^1/3, we used 𝑟₀=1.2𝑓𝑚. The results are presented in Tables 1, 2, and 3. One can see that there are significant differences between the results when using different NN interactions and different SRC parameterizations. Indeed, especially for the short-range exchange mechanisms, as the heavy neutrino and gluino (SUSY1) ones, one expects the results to be sensitive to the NN interaction and to the type of SRC parameterizations. The uncertainties in the NME calculated values due to the use of different NN interactions are in the range of (10–16)%. The uncertainties due to the use of different SRC parameterizations are larger; sometimes differences between results are larger than 100%, especially when comparing the NME values obtained with MS [44] parameterizations with those obtained with softer parameterizations, such as AV18 [45] and CD-Bonn [46, 47]. Further, we compare our results with other similar ones found in literature. To the best of our knowledge, explicit values of NMEs calculated with large scale ShM approaches are reported only in [34] and for one nucleus, ⁴⁸Ca. Our results agree within ~10% with those from [34], for all three mechanisms. At this point, it is worth mentioning that the GT and tensor contributions in (5) must have a relative opposite sign, such that the adding of the tensor contribution decreases the 𝑀^0𝜈 total value. Hence, one has to pay attention when applying formula (5) because the GT and tensor contributions can come out from numerical calculations with opposite signs. In this respect, we suspect that our NME values agree even better with those from [34]. For the other two isotopes, that is, ⁷⁶Ge and ⁸²Se, we compare our (ShM) results with QRPA [31, 32, 43] and IBM-2 [28] results. For the heavy neutrino and SUSY2 mechanisms, the differences between our NME values and NME values calculated with QRPA and IBM-2 methods are significant, but they are similar (in magnitude) to those encountered usually in the case of light neutrino exchange mechanism. For the SUSY1 mechanism, our results compare differently with the QRPA results reported by Tuebingen group in [32, 43], the only ones found in literature. We note that the agreement is good when comparing with the results of [43], where the MS parameterizations of the SRCs are employed, and it is bad when comparing with those of [32], where AV18 and CD-Bonn parameterizations for SRCs are used. As a conclusion, it turns out that more results for the NMEs associated with heavy neutrino and SUSY mechanisms, performed with different methods and including more nuclei, are needed for a relevant analysis of their validity.

Table 1 

0𝜈𝛽𝛽 NMEs values for heavy neutrino mechanism and comparison to other results in the literature.

Table 2 

0𝜈𝛽𝛽 NMEs values for SUSY1 mechanism and comparison to other results in the literature.

Table 3 

0𝜈𝛽𝛽 NMEs values for SUSY2 mechanism and comparison to other results in the literature.

In Table 4, we present new upper limits for the LNV parameters associated with the studied 0𝜈𝛽𝛽 decay mechanisms, in the hypothesis of a single dominance mechanism. The PSFs values are taken from [37] and we mention that they are very close to PSF values reported in [38, 39]. The PSFs have been calculated since long time [40, 41] but they were less discussed, being considered to be computed with enough precision. Recently, they were recalculated with improved approaches, by using exact electron Dirac wave functions, taking into account the finite nuclear size and electron screening effects [38] and, in addition, a realistic Coulomb potential [39, 42], and differences/discrepancies were found in several cases between the old and the recent PSF values. For the three isotopes considered in this work, the new PSF values differ from older calculations (e.g., from those reported in [41, 54–56]) up to 10%, justifying hence their use in a precise derivation of LNV parameters. The uncertainties in NME values reflect on the precision of derivation the LNV parameters. According to the consensus agreed upon in literature that softer SRC parameterizations are indicated, we adopt the NME values calculated with the CD-Bonn parameterization and derive the LNV parameters for the three nuclei and the three mechanisms considered. These results are presented in Table 4. For derivation, we used the most recent results for the experimental lifetimes, reported in references indicated in parenthesis. For ⁴⁸Ca, our limits for the LNV parameters are close to those reported in [34], while for the other two nuclei they differ from other results from literature [28, 32, 34, 43], derived with NMEs values calculated with QRPA and IBM-2 methods. As we already mentioned, information on LNV parameters for the heavy neutrino mechanism can be now extracted, as well, from the data analysis of the same sign dilepton channels at the LHC experiments. Thus, we expect that complementary information from both low- and high-energy experiments could be used in the future for better constraining of the LNV parameters.

Table 4 

Upper limits for Majorana neutrino mass parameters together with the other components of the 0𝜈𝛽𝛽 decay halftimes: the experimental lifetimes lower limits, the phase space factors, and the nuclear matrix elements.

4. Conclusions

We report new constraints for the heavy neutrino and SUSY parameters derived from an analysis of 0𝜈𝛽𝛽 decay of three experimentally interesting nuclei, ⁴⁸Ca, ⁷⁶Ge, and ⁸²Se. To the best of our knowledge, for the last two isotopes, no other large scale ShM results for these mechanisms are available so far. For deduction of these parameters, precise values of the NMEs and PSFs which enter the lifetime formulae are needed, as well as accurate experimental measurements of the lifetimes. The NME calculations are performed with a recent ShM code described widely in [35, 36]. The PSFs are taken from our most recent calculations reported in [37]. For SUSY mechanisms, we discuss two mechanisms associated with possible R-parity breaking in SUSY theories, a short-range one with exchange of heavy Majorana neutrino and scalar SUSY particles (gluinos and squarks and/or neutralinos and selectrons) and a long-range one involving exchange of both heavy squarks and light Majorana neutrinos—“squark-neutrino” mechanism. First, we calculated the NMEs using different nuclear ingredients. For ⁴⁸Ca, we use two different NN interactions, GXPF1A and KB3G, and found differences between the results within 10%. Also, we performed the calculations using three different parameterizations for the SRCs and found a significant dependence of the results on the way the SRCs are introduced in calculations. Further, we compare our (ShM) results with similar results from literature. For ⁴⁸Ca, our results are in good agreement with the results from [34], for all three mechanisms. For the other two isotopes, ⁷⁶Ge and ⁸²Se, we compare our (ShM) NMEs with those computed by QRPA [31, 32, 43] and IBM-2 [28] methods. For the heavy neutrino and SUSY2 mechanisms, the differences between our results and the results from those references are significant but similar (up to a factor of two) to those encountered usually in literature in the case of light neutrino exchange mechanism. For the SUSY1 mechanism, our results compare differently with QRPA results, reported by Tuebingen group. There is a good agreement with the results from [43], when the MS SRC parameterization is employed, but the agreement is bad when comparing with the values from [32], when softer SRC parameterizations are employed. As a conclusion, it turns out that more results for the NMEs associated with these 0𝜈𝛽𝛽 decay mechanisms, alternative to the light neutrino exchange one, performed with different nuclear methods and including more nuclei, are needed for a relevant analysis of their validity. Then, we derived upper limits of the LNV parameters associated with the three mechanisms, in the hypothesis of one single dominance mechanism, using NMEs values computed with CD-Bonn parameterization. Finally, it is worth noting that information on LNV parameters can now be provided by LHC experiments at CERN from the analysis of the same sign dilepton channels. Thus, there is now the possibility to gather complementary information from both low- and high-energy experiments and use it to better constrain LNV parameters.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors thank J. Menendez for useful discussions. This work was done with the support of the MEN and UEFISCDI through the Project IDEI-PCE-3-1318, Contract no. 58/28.10/2011.