The extension of the Majorana neutrino mass mechanism of the neutrinoless double-beta decay $\left(0\nu \beta \beta \right)$ with the inclusion of right-handed leptonic and hadronic currents is revisited. While only the exchange of light neutrinos is assumed, the $s_{1/2}$ and $p_{1/2}$ states of emitted electrons as well as recoil corrections to the nucleon currents are taken into account. Within the standard approximations the decay rate is factorized into a sum of products of kinematical phase-space factors, nuclear matrix elements, and the fundamental parameters that characterize the lepton number violation. Unlike in the previous treatments, the induced pseudoscalar term of hadron current is included, resulting in additional nuclear matrix elements. An improved numerical computation of the phase-space factors is presented, based on the exact Dirac wave functions of the $s_{1/2}$ and $p_{1/2}$ electrons with finite nuclear size and electron screening taken into account. The dependence of values of these phase-space factors on the different approximation schemes used in evaluation of electron wave functions is discussed. The upper limits for effective neutrino mass and the parameters $\langle \lambda \rangle$ and $\langle \eta \rangle$ characterizing the right-handed current mechanism are deduced from data on the $0\nu \beta \beta$ decay of ${}^{76}\text{Ge}$ and ${}^{136}\text{Xe}$ using nuclear matrix elements calculated within the nuclear shell model and quasiparticle random phase approximation. The differential decay rates, i.e., the angular correlations and the single electron energy distributions for various combinations of the total lepton number violating parameters that can help to disentangle the possible mechanism, are described and discussed.

An improved formalism of the two-neutrino double-beta decay ($2\nu \beta \beta$-decay) rate is presented, which takes into account the dependence of energy denominators on lepton energies via the Taylor expansion. Until now, only the leading term in this expansion has been considered. The revised $2\nu \beta \beta$-decay rate and differential characteristics depend on additional phase-space factors weighted by the ratios of $2\nu \beta \beta$-decay nuclear matrix elements with different powers of the energy denominator. For nuclei of experimental interest all phase-space factors are calculated by using exact Dirac wave functions with finite nuclear size and electron screening. For isotopes with measured $2\nu \beta \beta$-decay half-life the involved nuclear matrix elements are determined within the quasiparticle random-phase approximation with partial isospin restoration. The importance of correction terms to the $2\nu \beta \beta$-decay rate due to Taylor expansion is established and the modification of shape of single and summed electron energy distributions is discussed. It is found that the improved calculation of the $2\nu \beta \beta$-decay predicts slightly suppressed $2\nu \beta \beta$-decay background to the neutrinoless double-beta decay signal. Furthermore, an approach to determine the value of effective weak-coupling constant in nuclear medium $g_{\mathrm{A}}^{\mathrm{eff}}$ is proposed.

We report the results of a first experimental search for lepton number violation by four units in the neutrinoless quadruple-$\beta$ decay of ${}^{150}\mathrm{Nd}$ using a total exposure of 0.19 kg yr recorded with the NEMO-3 detector at the Modane Underground Laboratory. We find no evidence of this decay and set lower limits on the half-life in the range $T_{1/2}>(1.1–3.2)\times 10^{21}\mathrm{yr}$ at the 90% C.L., depending on the model used for the kinematic distributions of the emitted electrons.

The two-neutrino double-$\beta$ Gamow-Teller and Fermi transitions are studied within an exactly solvable model, which allows a violation of both spin-isospin SU(4) and isospin SU(2) symmetries, and is expressed with generators of the SO(8) group. It is found that this model reproduces the main features of realistic calculation within the quasiparticle random-phase approximation with isospin symmetry restoration concerning the dependence of the two-neutrino double-$\beta$ decay matrix elements on isovector and isoscalar particle-particle interactions. By using perturbation theory an explicit dependence of the two-neutrino double-$\beta$ decay matrix elements on the like-nucleon pairing, particle-particle $T=0$ and $T=1$, and particle-hole proton-neutron interactions is obtained. It is found that double-$\beta$ decay matrix elements do not depend on the mean field part of Hamiltonian and that they are governed by a weak violation of both SU(2) and SU(4) symmetries by the particle-particle interaction of Hamiltonian. It is pointed out that there is a dominance of two-neutrino double-$\beta$ decay transition through a single state of intermediate nucleus. The energy position of this state relative to energies of initial and final ground states is given by a combination of strengths of residual interactions. Further, energy-weighted Fermi and Gamow-Teller sum rules connecting $\mathrm{\Delta }Z=2$ nuclei are discussed. It is proposed that these sum rules can be used to study the residual interactions of the nuclear Hamiltonian, which are relevant for charge-changing nuclear transitions.

The λ mechanism (W_L − W_R exchange) of the neutrinoless double beta decay (0νββ-decay), which has origin in left-right symmetric model with right-handed gauge boson at TeV scale, is investigated. The revisited formalism of the 0νββ-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 0νββ-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.

We consider alternative modes of two-neutrino and neutrinoless double-$\beta$ decays in which one $\beta$ electron goes over to a continuous spectrum and the other occupies a vacant bound level of the daughter ion. We calculate the corresponding phase-space factors of the final states, estimate the partial decay rates, and derive the one- and two-electron energy spectra using relativistic many-electron wave functions of atoms provided by the multiconfiguration Dirac-Hartree-Fock package Grasp2K. While the bound-state neutrinoless double-$\beta$ decays are strongly suppressed, their two-neutrino counterparts can be observed in the next-generation double-$\beta$-decay experiments, most notably SuperNEMO.