# The λ Mechanism of the 0ν*ββ*-Decay

^{1}Department of Nuclear Physics and Biophysics, Comenius University, Bratislava, Slovakia^{2}Boboliubov Laboratory of Theoretical Physics, Dubna, Russia^{3}Institute of Experimental and Applied Physics, Czech Technical University in Prague, Prague, Czechia^{4}Dzhelepov Laboratory of Nuclear Problems, Dubna, Russia

The λ mechanism (*W*_{L} − *W*_{R} exchange) of the neutrinoless double beta decay (0ν*ββ*-decay), which has origin in left-right symmetric model with right-handed gauge boson at TeV scale, is investigated. The revisited formalism of the 0ν*ββ*-decay, which includes higher order terms of nucleon current, is exploited. The corresponding nuclear matrix elements are calculated within quasiparticle random phase approximation with partial restoration of the isospin symmetry for nuclei of experimental interest. A possibility to distinguish between the conventional light neutrino mass (*W*_{L} − *W*_{L} exchange) and λ mechanisms by observation of the 0ν*ββ*-decay in several nuclei is discussed. A qualitative comparison of effective lepton number violating couplings associated with these two mechanisms is performed. By making viable assumption about the seesaw type mixing of light and heavy neutrinos with the value of Dirac mass *m*_{D} within the range 1 MeV < *m*_{D} < 1 GeV, it is concluded that there is a dominance of the conventional light neutrino mass mechanism in the decay rate.

## 1. Introduction

The Majorana nature of neutrinos, as favored by many theoretical models, is a key for understanding of tiny neutrino masses observed in neutrino oscillation experiments. A golden process for answering this open question of particle physics is the neutrinoless double beta decay (0ν*ββ*-decay) [1–3],

in which an atomic nucleus with Z protons decays to another one with two more protons and the same mass number A, by emitting two electrons and nothing else. The observation of this process, which violates total lepton number conservation and is forbidden in the Standard Model, guaranties that neutrinos are Majorana particles, i.e., their own antiparticles [4].

The searches for the 0ν*ββ*-decay have not yielded any evidence for Majorana neutrinos yet. This could be because neutrinos are Dirac particles, i.e., not their own antiparticles. In this case we will never observe the decay. However, it is assumed that the reason for it is not sufficient sensitivity of previous and current 0ν*ββ*-decay experiments to the occurrence of this rare process.

Due to the evidence for neutrino oscillations and therefore for 3 neutrino mixing and masses the 0ν*ββ*-decay mechanism of primary interest is the exchange of 3 light Majorana neutrinos interacting through the left-handed V-A weak currents (*m*_{ββ} mechanism). In this case, the inverse 0ν*ββ*-decay half-life is given by Vergados et al. [1], DellOro et al. [2] and Vergados et al. [3]

where *G*_{01}, *g*_{A} and *M*_{ν} represent an exactly calculable phase space factor, the axial-vector coupling constant and the nuclear matrix element (whose calculation represents a severe challenge for nuclear theorists), respectively. *m*_{e} is the mass of an electron. The effective neutrino mass,

is a linear combination of the three neutrino masses *m*_{i}, weighted with the square of the elements *U*_{ei} of the first row of the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) neutrino mixing matrix. The measured value of *m*_{ββ} would be a source of important information about the neutrino mass spectrum (normal or inverted spectrum), absolute neutrino mass scale and the CP violation in the neutrino sector. However, that is not the only possibility.

There are several different theoretical frameworks that provide various 0ν*ββ*-decay mechanisms, which generate masses of light Majorana neutrinos and violate the total lepton number conservation. One of those theories is the left-right symmetric model (LRSM) [5–9], in which corresponding to the left-handed neutrino, there is a parity symmetric right-handed neutrino. The parity between left and right is restored at high energies and neutrinos acquire mass through the see-saw mechanism, what requires presence of additional heavy neutrinos. In general one cannot predict the scale where the left-right symmetry is realized, which might be as low as a few TeV—accessible at Large Hadron Collider, or as large as GUT scale of 10^{15} GeV.

The LRSM, one of the most elegant theories beyond the Standard Model, offers a number of new physics contributions to 0ν*ββ*-decay, either from right-handed neutrinos or Higgs triplets. The main question is whether these additional 0ν*ββ*-mechanisms can compete with the *m*_{ββ} mechanism and affect the 0ν*ββ*-decay rate significantly. This issue is a subject of intense theoretical investigation within the TeV-scale left-right symmetry theories [10–14]. In analysis of heavy neutrino mass mechanisms of the 0ν*ββ*-decay an important role plays a study of related lepton number and lepton flavor violation processes in experiments at Large Hadron Collider [2, 15–19].

The goal of this article is to discuss in details the *W*_{L} − *W*_{R} exchange mechanism of the 0ν*ββ*-decay mediated by light neutrinos (λ mechanism) and its coexistence with the standard *m*_{ββ} mechanism. For that purpose the corresponding nuclear matrix elements (NMEs) will be calculated within the quasiparticle random phase approximation with a partial restoration of the isospin symmetry [20] by taking the advantage of improved formalism for this mechanism of the 0ν*ββ*-decay of Štefánik [21]. A possibility to distinguish *m*_{ββ} and λ mechanisms in the case of observation of the 0ν*ββ*-decay on several isotopes will be analyzed. Further, the dominance of any of these two mechanisms in the 0ν*ββ*-decay rate will be studied within seesaw model with right-handed gauge boson at TeV scale. We note that a similar analysis was performed by exploiting a simplified 0ν*ββ*-decay rate formula and different viable particle physics scenarios in Tello et al. [10], Barry and Reodejohann [11], Bhupal Dev et al. [12], Deppisch et al. [13] and Borah et al. [14].

## 2. Decay Rate for the Neutrinoless Double-beta Decay

Recently, the 0ν*ββ*-decay with the inclusion of right-handed leptonic and hadronic currents has been revisited by considering exact Dirac wave function with finite nuclear size and electron screening of emitted electrons and the induced pseudoscalar term of hadron current, resulting in additional nuclear matrix elements [21]. In this section we present the main elements of the revisited formalism of the λ mechanism of the 0ν*ββ*-decay briefly. Unlike in Štefánik et al. [21] the effect the weak-magnetism term of the hadron current on leading NMEs is taken into account.

If the mixing between left and right vector bosons is neglected, for the effective weak interaction hamiltonian density generated within the LRSM we obtain

Here, *G*_{β} = *G*_{F} cos θ_{C}, where *G*_{F} and θ_{C} are Fermi constant and Cabbibo angle, respectively. The coupling constant λ is defined as

Here, *M*_{WL} and *M*_{WR} are masses of the Standard Model left-handed *W*_{L} and right-handed *W*_{R} gauge bosons, respectively. The left- and right-handed leptonic currents are given by

The weak eigenstate electron neutrinos ν_{eL} and ν_{eR} are superpositions of the light and heavy mass eigenstate Majorana neutrinos ν_{j} and *N*_{j}, respectively. We have

Here, *U, S, T*, and *V* are the 3 × 3 block matrices in flavor space, which constitute a generalization of the Pontecorvo-Maki-Nakagawa-Sakata matrix, namely the 6 × 6 unitary neutrino mixing matrix [22]

The nuclear currents are, in the non-relativistic approximation, [23]

Here, *m*_{N} is the nucleon mass. ${q}_{V}\equiv {q}_{V}({q}^{2})$, ${q}_{A}\equiv {q}_{A}({q}^{2})$, ${q}_{M}\equiv {q}_{M}({q}^{2})$ and ${q}_{P}\equiv {q}_{P}({q}^{2})$ are, respectively, the vector, axial-vector, weak-magnetism and induced pseudoscalar form-factors. The nucleon recoil terms are given by

where **q**_{n} = **p**_{n} − ${\text{p}}_{n}^{\prime}$ is the momentum transfer between the nucleons. The initial neutron (final proton) possesses energy ${E}_{n}^{\prime}$ (*E*_{n}) and momentum ${\text{p}}_{n}^{\prime}$ (**p**_{n}). **r**_{n}, ${\tau}_{n}^{+}$ and ${\overrightarrow{\sigma}}_{n}$, which act on the *n*-th nucleon, are the position operator, the isospin raising operator and the Pauli matrix, respectively.

By assuming standard approximations [21] for the 0ν*ββ*-decay half-life we get

The effective lepton number violating parameters η_{ν} (*W*_{L} − *W*_{L} exchange), η_{λ} (*W*_{L} − *W*_{R} exchange) and their relative phase Ψ are given by

The coefficients *C*_{I} (I = *mm*, *mλ* and λλ) are linear combinations of products of nuclear matrix elements and phase-space factors:

The explicit form and calculated values of phase-space factors *G*_{0i} (*i* = 1, 2, 3, 4, 10 and 11) of the 0ν*ββ*-decaying nuclei of experimental interest are given in Štefánik et al. [21]. The NMES, which constitute the coefficients *C*_{I} in Equation (13), are defined as follows:

The partial nuclear matrix elements *M*_{I}, where I = GT, F, T, ωF, ωGT, ωT, qF, qGT, and qT are given by

Here, *O*_{F,GT,T} are the Fermi, Gamow-Teller and tensor operators $1,{\overrightarrow{\sigma}}_{1}\xb7{\overrightarrow{\sigma}}_{2}$ and $3({\overrightarrow{\sigma}}_{1}\xb7{\widehat{r}}_{12})({\overrightarrow{\sigma}}_{2}\xb7{\widehat{r}}_{12})$. The two-nucleon exchange potentials *h*_{I}(*r*) with I = F, GT, T, ωF, ωGT, ωT, qF, qGT, and qT can be written as

where

and

Here, *E*_{i}, *E*_{f} and Ē_{n} are energies of the initial and final nucleus and averaged energy of intermediate nuclear states, respectively. **r** = (**r**_{r} − **r**_{s}), **r**_{r, s} is the coordinate of decaying nucleon and *j*_{i}(*qr*) (*i* = 1, 2, 3) denote the spherical Bessel functions. ${\text{p}}_{r}+{\text{p}}_{r}^{\prime}\simeq 0$, ${E}_{r}\text{}-\text{}{E}_{r}^{\prime}\simeq 0$ and ${\text{p}}_{r}\text{}-\text{}{\text{p}}_{r}^{\prime}\simeq \text{q}$, where **q** is the momentum exchange. The form factors ${g}_{V}({q}^{2})$, ${g}_{A}({q}^{2})$, ${g}_{M}({q}^{2})$ and ${g}_{P}({q}^{2})$ are defined in Simkovic et al. [24]. We note that factor 4 in definition of the two-nucleon exchange potentials *h*_{I}(*r*) with *I* = ωF, ωGT, and ωT in Equation (48) of Štefánik et al. [21] needs to be replaced by factor 2.

## 3. Results and Discussion

The nuclear matrix elements are calculated in proton-neutron quasiparticle random phase approximation with partial restoration of the isospin symmetry for ^{48}Ca, ^{76}Ge, ^{82}Se, ^{96}Zr, ^{100}Mo, ^{110}Pd, ^{116}Cd, ^{124}Sn, ^{130}Te and ^{136}Xe, which are of experimental interest. In the calculation the same set of nuclear structure parameters is used as in Simkovic et al. [20]. The pairing and residual interactions as well as the two-nucleon short-range correlations derived from the realistic nucleon-nucleon Argonne V18 potential are considered [26]. The closure approximation for intermediate nuclear states is assumed with (Ē_{n} − (*E*_{i} + *E*_{f})/2) = 8 MeV. The free nucleon value of axial-vector coupling constant (*g*_{A} = 1.25 − 1.27) is considered.

In Table 1 the calculated NMEs are presented. The values of *M*_{F,GT,T} and *M*_{ν} differ slightly (within 10%) with those given in Simkovic et al. [20], which were obtained without consideration of the closure approximation. By glancing Table 1 we see that *M*_{Fω, GTω, Tω} ≃ *M*_{F,GT,T} and *M*_{νω} ≃ *M*_{ν} as for the average neutrino momentum q = 100 MeV and used average energy of intermediate nuclear states we have *q*/(*q* + Ē_{n} − (*E*_{i} + *E*_{f})/2) ≃ 1. The absolute value of *M*_{Fq, GTq, Tq} is smaller in comparison with *M*_{F,GT,T} by about 50% for Fermi NMEs and by about factor two in the case of Gamow-Teller and tensor NMEs. From Table 1 it follows that there is a significant difference between results of this work and the QRPA NMEs of Muto et al. [25], especially in the case of ^{100}Mo. This difference can be attributed to the progress achieved in the 0ν*ββ*-decay formalism due to inclusion of higher order terms of nucleon currents [21, 24], the way of adjusting the parameters of nuclear Hamiltonian [27], description of short-range correlations [26] and restoration of the isospin symmetry [20].

**Table 1**. The nuclear matrix elements of the 0ν*ββ*-decay associated with *m*_{ββ} and λ mechanisms and the coefficients *C*_{mm}, *C*_{mλ} and *C*_{λλ} (in 10^{−14} years^{−1}) of the decay rate formula (see Equation 11).

Nuclear matrix elements ${M}_{{2}^{-}}$, ${M}_{{1}^{+}}$ (λ mechanism) and *M*_{ν} (*m*_{ββ} mechanism) for 10 nuclei under consideration are given in Table 1 and displayed in Figure 1. We note a rather good agreement between ${M}_{{2}^{-}}$ and *M*_{ν} for all calculated nuclear systems. It is because the contribution of *M*_{1}_{+} to *M*_{2}_{−} is suppressed by factor 9 and as a result *M*_{2}_{−} is governed by the *M*_{νω} contribution (see Equation 14). Values of ${M}_{{1}^{+}}$ exhibit similar systematic behavior in respect to considered nuclei as values of *M*_{ν} and ${M}_{{2}^{-}}$, but they are suppressed by about factor 2–3 (with exception of ^{48}Ca).

**Figure 1**. A comparison of the nuclear matrix elements *M*_{1}_{+}, *M*_{2}_{−} (λ mechanism) and *M*_{ν} (*m*_{ββ} mechanism) of the 0ν*ββ*-decay.

The importance of the *m*_{ββ} and λ mechanisms depends, respectively, not only on values of η_{ν} and η_{λ} parameters, which are unknown, but also on values of coefficients *C*_{I} (I = *mm*, *mλ*, λλ), which are listed for all studied nuclei in Table 1. They have been obtained by using improved values of phase-space factors *G*_{0k} (k = 1, 2, 10 and 11) from Štefánik et al. [21]. We note that the squared value of *M*_{GT} and fourth power of axial-vector coupling constant *g*_{A} are included in the definition of coefficient *C*_{I} unlike in Štefánik et al. [21]. We see that *C*_{λλ} is always larger when compared with *C*_{mm}. The absolute value of *C*_{mλ} is significantly smaller than *C*_{mm} and *C*_{λλ}. This fact points out on less important contribution to the 0ν*ββ*-decay rate from the interference of *m*_{ββ} and λ mechanisms.

For 10 nuclei of experimental interest the decomposition of coefficient *C*_{λλ} (see Equation 11) on partial contributions ${C}_{I}^{0k}$ associated with phase-space factors *G*_{0k} (k = 2, 10, and 11) is shown in Figure 2. By glancing the plotted ratio ${C}_{I}^{0k}/{C}_{I}$ we see that *C*_{λλ} is dominated by a single contribution associated with the phase-space factor *G*_{02}. From this and above analysis it follows that 0ν*ββ*-decay half-life to a good accuracy can be written as

with

For a given isotope the factor *f*_{λm} reflects relative sensitivity to the *m*_{ββ} and λ mechanisms and ${f}_{\lambda m}^{G}$ is its approximation, which does not depend on NMEs. The values *f*_{λm} and ${f}_{\lambda m}^{G}$ are tabulated in Table 1 and plotted as function of *Q*_{ββ} in Figure 3. We see that *f*_{λm} depends only weakly on involved nuclear matrix elements (apart for the case of ^{48}Ca) what follows from a comparison of *f*_{λm} with ${f}_{\lambda m}^{G}$. The value of *f*_{λm} is mainly determined by the Q-value of double beta decay process. From 10 analyzed nuclei the largest value of *f*_{mλ} is found for ^{48}Ca and the smallest value for ^{76}Ge. A larger value of *f*_{λm} means increased sensitivity to *m*_{ββ} mechanism in comparison to λ mechanism and vice versa.

**Figure 2**. The decomposition of coefficient *C*_{λλ} (see Equation 11) on partial contributions ${C}_{I}^{0k}$ associated with phase-space factors *G*_{0k} (k = 2, 10 and 11) for nuclei of experimental interest. The partial contributions are identified by index k. The contributions from largest to the smallest are displayed in red, blue and black colors, respectively.

**Figure 3**. The factor *f*_{λm} (see Equation 19) as function of Q-value of the double beta decay process (*Q*_{ββ}) plotted from the numbers of Table 1.

Upper bounds on the effective neutrino mass *m*_{ββ} and right-handed current coupling strength η_{λ} are deduced from experimental half-lives of the 0ν*ββ*-decay by using the coefficients *C*_{mm}, *C*_{mλ} and *C*_{λλ} of Table 1. The maximum and the value on axis (*m*_{ββ} = 0 or η_{λ} = 0) are listed in Table 2. The decays of ^{136}Xe and ^{76}Ge set the sharpest limit *m*_{ββ} ≤ 0.13 eV and 0.18 eV, and ${\eta}_{\lambda}\le 1.7\text{}1{0}^{-7}$ and 3.1 10^{−7}, respectively. These are more stringent than those deduced from other experimental sources.

**Table 2**. Upper bounds on the effective Majorana neutrino mass *m*_{ββ} and parameter η_{λ} associated with right-handed currents mechanism imposed by current constraints on the 0ν*ββ*-decay half-life for nuclei of experimental interest.

It is well known that by measuring different characteristics, namely energy and angular distributions of two emitted electrons, it is possible to identify which of *m*_{ββ} and λ mechanisms is responsible for 0ν*ββ*-decay [21, 23]. It might be achieved only by some of future 0ν*ββ*-decay experiments, e.g., the SuperNEMO [37] or NEXT [38]. A relevant question is whether the underlying *m*_{ββ} or λ mechanism can be revealed by observation of the 0ν*ββ*-decay in a series of different isotopes. In Figure 4 this issue is addressed by an illustrative case of observation of the 0ν*ββ*-decay of ^{136}Xe with half-life ${T}_{1/2}^{0\nu}=6.86\text{}1{0}^{26}$ years, which can be associated with *m*_{ββ} = 50 meV or η_{λ} = 9.8 10^{−8}. The 0ν*ββ*-decay half-life predictions associated with a dominance of *m*_{ββ} and λ mechanisms exhibit significant difference for some nuclear systems. We see that by observing, e.g., the 0ν*ββ*-decay of ^{100}Ge and ^{100}Mo with sufficient accuracy and having calculated relevant NMEs with uncertainty below 30%, it might be possible to conclude, whether the 0ν*ββ*-decay is due to *m*_{ββ} or λ mechanism.

**Figure 4**. The 0ν*ββ*-decay half-lives of nuclei of experimental interest calculated for *m*_{ββ} (red circle) and λ (blue square) mechanisms by assuming an illustrative case of observation 0ν*ββ*-decay of ^{136}Xe with half-life ${T}_{1/2}^{0\nu}=6.86\text{}1{0}^{26}$ years (*m*_{ββ} = 50 meV or η_{λ} = 9.8 10^{−8}). The current experimental limits on 0ν*ββ*-decay half-life of ^{76}Ge (the GERDA experiment) and ^{136}Xe (the Kamland-Zen experiment) are displayed with green triangles.

Currently, the uncertainty in calculated 0ν*ββ*-decay NMEs can be estimated up to factor of 2 or 3 depending on the considered isotope as it follows from a comparison of results of different nuclear structure approaches [3]. The improvement of the calculation of double beta decay NMEs is a very important and challenging problem. There is a hope that due to a recent progress in nuclear structure theory (e.g., ab initio methods) and increasing computing power the calculation of the 0ν*ββ*-decay NMEs with uncertainty of about 30 % might be achieved in future.

## 4. The Lepton Number Violating Parameters within the Seesaw and Normal Hierarchy

The 6 × 6 unitary neutrino mixing matrix ${U}$ (see Equation 8) can be parametrized with 15 rotational angles and 10 Dirac and 5 Majorana CP violating phases. For the purpose of study different LRSM contributions to the 0ν*ββ*-decay the mixing matrix ${U}$ is usually decomposed as follows [22]

Here, **0** and **1** are the 3 × 3 zero and identity matrices, respectively. The parametrization of matrices A, B, R and S and corresponding orthogonality relations are given in Xing [22].

If A = **1**, B = **1**, R = **0** and S = **0**, there would be a separate mixing of light and heavy neutrinos, which would participate only in left and right-handed currents, respectively. In this case we get η_{λ} = 0, i.e., the λ mechanism is forbidden.

If masses of heavy neutrinos are above the TeV scale, the mixing angles responsible for mixing of light and heavy neutrinos are small. By neglecting the mixing between different generations of light and heavy neutrinos, the unitary mixing matrix ${U}$ takes the form

Here, *m*_{D} represents energy scale of charged leptons and *m*_{LNV} is the total lepton number violating scale, which corresponds to masses of heavy neutrinos. We see that *U* = *U*_{0} can be identified to a good approximation with the PMNS matrix and *V*_{0} is its analogue for heavy neutrino sector. Due to unitarity condition we find ${V}_{0}={U}_{0}^{\u2020}$. Within this scenario of neutrino mixing the effective lepton number violating parameters η_{ν} (*m*_{ββ} mechanism) and η_{λ} (λ mechanism) are given by

with

The importance of *m*_{ββ} or λ-mechanism can be judged from the ratio

It is naturally to assume that ζ_{m} ≈ 1 and to consider the upper bound for the factor ζ_{λ}, i.e., there is no anomaly cancellation among terms, which constitute these factors. Within this approximation η_{λ}/η_{ν} does not depend on scale of the lepton number violation *m*_{LNV} and is plotted in Figure 5. The Dirac mass *m*_{D} is assumed to be within the range 1 MeV < *m*_{D} < 1 GeV. The flavor and CP-violating processes of kaons and B-mesons make it possible to deduce lower bound on the mass of the heavy vector boson *M*_{W2}>2.9 TeV [12]. From Figure 5 it follows that within accepted assumptions the λ mechanism is practically excluded as the dominant mechanism of the 0ν*ββ*-decay.

**Figure 5**. The allowed range of values for the ratio η_{λ}/η_{ν} (in green) as a function of the mass of the heavy vector boson *M*_{WR}. The line of the 0ν*ββ* equivalence corresponds to the case of equal importance of both *m*_{ββ} and λ mechanisms in the 0ν*ββ*-decay rate.

In this section the light-heavy neutrino mixing of the strength *m*_{D}/*m*_{LNV} is considered. However, we note that there are models with heavy neutrinos mixings where strength of the mixing decouples from neutrino masses [39–44]. This subject goes beyond the scope of this paper.

## 5. Summary and Conclusions

The left-right symmetric model of weak interaction is an attractive extension of the Standard Model, which may manifest itself in the TeV scale. In such case the Large Hadron Collider can determine the right-handed neutrino mixings and heavy neutrino masses of the seesaw model. The LRSM predicts new physics contributions to the 0ν*ββ* half-life due to exchange of light and heavy neutrinos, which can be sizable.

In this work the attention was paid to the λ mechanism of the 0ν*ββ*-decay, which involves left-right neutrino mixing through mediation of light neutrinos. The recently improved formalism of the 0ν*ββ*-decay concerning this mechanism was considered. For 10 nuclei of experimental interest NMEs were calculated within the QRPA with a partial restoration of the isospin symmetry. It was found that matrix elements governing the conventional *m*_{ββ} and λ mechanisms are comparable and that the λ contribution to the decay rate can be associated with a single phase-space factor. A simplified formula for the 0ν*ββ*-decay half-life is presented (see Equation 19), which neglects the suppressed contribution from the interference of both mechanisms. In this expression the λ contribution to decay rate is weighted by the factor *f*_{λm}, which reflects relative sensitivity to the *m*_{ββ} and λ mechanisms for a given isotope and depends only weakly on nuclear physics input. It is manifested that measurements of 0ν*ββ*-decay half-life on multiple isotopes with largest deviation in the factor *f*_{λm} might allow to distinguish both considered mechanisms, if involved NMEs are known with sufficient accuracy.

Further, upper bounds on effective lepton number violating parameters *m*_{ββ} (η_{ν}) and η_{λ} were deduced from current lower limits on experimental half-lives of the 0ν*ββ*-decay. The ratio η_{λ}/η_{ν} was studied as function of the mass of heavy vector boson *M*_{WR} assuming that there is no mixing among different generations of light and heavy neutrinos. It was found that if the value of Dirac mass *m*_{D} is within the range 1 MeV < *m*_{D} < 1 GeV, the current constraint on *M*_{WR} excludes the dominance of the λ mechanism in the 0ν*ββ*-decay rate for the assumed neutrino mixing scenario.

## Author Contributions

FŠ: calculation of nuclear matrix elements and preparation of manuscript; RD: analysis of lepton number violating parameters and preparation of manuscript; DŠ: derivation of the formalism of neutrinoless double beta decay, and analysis of obtained results.

## Funding

This work is supported by the VEGA Grant Agency of the Slovak Republic un- der Contract No. 1/0922/16, by Slovak Research and Development Agency under Contract No. APVV-14-0524, RFBR Grant No. 16-02-01104, Underground laboratory LSM—Czech participation to European-level research infrastructue CZ.02.1.01/0.0/0.0/16 013/0001733.

## Conflict of Interest Statement

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.

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Keywords: majorana neutrinos, neutrinoless double beta decay, right-handed current, left-right symmetric models, nuclear matrix elements, quasiparticle random phase approximation

Citation: Šimkovic F, Štefánik D and Dvornický R (2017) The λ Mechanism of the 0ν*ββ*-Decay. *Front. Phys*. 5:57. doi: 10.3389/fphy.2017.00057

Received: 01 August 2017; Accepted: 27 October 2017;

Published: 16 November 2017.

Edited by:

Diego Aristizabal Sierra, Federico Santa María Technical University, ChileReviewed by:

Janusz Gluza, University of Silesia of Katowice, PolandJuan Carlos Helo, University of La Serena, Chile

Copyright © 2017 Šimkovic, Štefánik and Dvornický. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Fedor Šimkovic, simkovic@fmph.uniba.sk