Majorana neutrinos, neutrino mass spectrum, $\mathrm{CP}$ violation, and neutrinoless double $\beta$ decay: The three-neutrino mixing case

Assuming three-neutrino mixing and massive Majorana neutrinos, we study the implications of the neutrino oscillation solutions of the solar and atmospheric neutrino problems and of the results of the CHOOZ experiment for the predictions of the effective Majorana mass in neutrinoless double-beta $\left[\right(\beta \beta {)}_{0\nu }]$ decay, $|〈m〉|.$ The general case of $\mathrm{CP}$ nonconservation is investigated. The predicted values of $|〈m〉|,$ which determine the magnitude of the $(\beta \beta {)}_{0\nu }$-decay rate, depend strongly on the type of neutrino mass spectrum and on the solution of the solar neutrino problem, as well as on the values of the two Majorana $\mathrm{CP}$-violating phases, present in the lepton mixing matrix. We find that (i) $|〈m〉|\lesssim 0.02$ eV for a hierarchical neutrino mass spectrum, (ii) $|〈m〉|\lesssim 0.09$ eV if the spectrum is of the inverted hierarchy type, and (iii) $|〈m〉|<~m$ in the case of three quasidegenerate neutrinos, $m>\mathrm{}0\mathrm{}$ being the common neutrino mass scale which is limited by the bounds from the ${}^3\mathrm{H}\beta$-decay experiments, $m<\mathrm{}2.5\mathrm{}$ eV. The indicated maximal values of $|〈m〉|$ are reached in cases (i), (ii), and (iii), respectively, for the large mixing angle (LMA) MSW solution, the small mixing angle (SMA) MSW solution, and for all current solutions of the solar neutrino problem. If $\mathrm{CP}$ invariance holds, $|〈m〉|$ is very sensitive to the values of the relative $\mathrm{CP}$ parities of the massive Majorana neutrinos. The cases of neutrino mass spectra which interpolate between the hierarchical or inverted hierarchy type and the quasidegenerate one are also studied. The observation of the $(\beta \beta {)}_{0\nu }$ decay with a rate corresponding to $|〈m〉|\gtrsim 0.02$ eV can provide unique information on the neutrino mass spectrum. Combined with information on the lightest neutrino mass or the type of neutrino mass spectrum, it can give also information on the $\mathrm{CP}$ violation in the lepton sector and, if $\mathrm{CP}$ invariance holds, on the relative $\mathrm{CP}$ parities of the massive Majorana neutrinos.