Within the quasiparticle random-phase approximation (QRPA) we achieve partial restoration of the isospin symmetry and hence fulfillment of the requirement that the $2\nu \beta \beta$ Fermi matrix element $M_F^{2\nu }$ vanishes, as it should, unlike in the previous version of the method. This is accomplished by separating the renormalization parameter $g_{pp}$ of the particle-particle proton-neutron interaction into isovector and isoscalar parts. The isovector parameter $g_{pp}^{T=1}$ needs to be chosen to be essentially equal to the pairing constant $g_{\mathrm{pair}}$, so no new parameter is needed. For the $0\nu \beta \beta$ decay the Fermi matrix element $M_F^{0\nu }$ is substantially reduced, while the full matrix element $M^{0\nu }$ is reduced by $\approx$10$\%$. We argue that this more consistent approach should be used from now on in the proton-neutron QRPA and in analogous methods.

We have examined the importance of momentum-dependent-induced nucleon currents such as weak-magnetism and pseudoscalar couplings to the amplitude of neutrinoless double beta decay $(0\mathrm{}\nu \beta \beta$ decay) in the mechanisms of light and heavy Majorana neutrino, as well as in that of Majoron emission. Such effects are expected to occur in all nuclear models in the direction of reducing the light neutrino matrix elements by about $30\%.$ To test this we have performed a calculation of the nuclear matrix elements of the experimentally interesting nuclei $A=76,\hspace{0.5em}82,\hspace{0.5em}96,\hspace{0.5em}100,\hspace{0.5em}116,\hspace{0.5em}128,\hspace{0.5em}130,\hspace{0.5em}136,$ and 150 within the proton-neutron renormalized quasiparticle random-phase approximation. We have found that indeed such corrections vary somewhat from nucleus to nucleus, but in all cases they are greater than $25\%.$ In the case of a heavy neutrino the effect is much larger (up to a factor of 6). Combining our results with the best presently available experimental limits on the half-life of the $0\nu \beta \beta$ decay, we have extracted new limits on the effective neutrino mass (light and heavy) and the effective Majoron coupling constant.

We study possible contribution of the Majorana neutrino mass eigenstate ${\nu }_h$, dominated by a sterile neutrino component, to neutrinoless double beta ($0\nu \beta \beta$) decay. A special emphasis is made on accurate calculation of the corresponding nuclear matrix elements. From the current experimental lower bound on the $0\nu \beta \beta$-decay half-life of $76$ Ge we derive stringent constraints on the ${\nu }_h-{\nu }_e$ mixing in a wide region of the values of ${\nu }_h$ mass. We discuss cosmological and astrophysical status of ${\nu }_h$ in this mass region.

The intimate relation between the Gamow-Teller part of the matrix element $M_{GT}^{0\nu }$ and the $2\nu \beta \beta$ closure matrix element $M_{cl}^{2\nu }$ is explained and explored. If the corresponding radial dependence $C_{cl}^{2\nu }\left(r\right)$ would be known, $M^{0\nu }$ corresponding to any mechanism responsible for the $0\nu \beta \beta$ decay can be obtained as a simple integral. However, the $M_{cl}^{2\nu }$ values, and therefore also the functions $C_{cl}^{2\nu }\left(r\right)$, sensitively depend not only on the properties of the first few $1^{+}$ states but also of higher-lying $1^{+}$ states in the intermediate odd-odd nuclei. We show that the ${\beta }^{-}$ and ${\beta }^{+}$ amplitudes of such states typically have opposite relative signs, and their contributions reduce severally the $M_{cl}^{2\nu }$ values. We suggest that demanding that $M_{cl}^{2\nu }=0$ is a sensible alternative way, within the QRPA method, of determining the amount of renormalization of isoscalar particle-particle interaction strength $g_{pp}^{T=0}$. Using such prescription, the matrix elements $M^{0\nu }$ are evaluated; their values are not very different $\left(\le 20\%\right)$ from the usual QRPA values when $g_{pp}^{T=0}$ is related to the known $2\nu \beta \beta$ half-lives. We note that vanishing values of $M_{cl}^{2\nu }$ are signs of a partial restoration of the spin-isospin SU(4) symmetry.