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Majorana Neutrino Masses from Neutrinoless Double-Beta Decays and Lepton-Number-Violating Meson Decays

The Schechter-Valle theorem states that a positive observation of neutrinoless double-beta ($0νββ$) decays implies a finite Majorana mass term for neutrinos when any unlikely fine-tuning or cancellation is absent. In this note, we reexamine the quantitative impact of the Schechter-Valle theorem, and find that current experimental lower limits on the half-lives of $0νββ$-decaying nuclei have placed a restrictive upper bound on the Majorana neutrino mass $|δm^{ee}_ν| < 7.43 \times 10^{-29}~{\rm eV}$ radiatively generated at the four-loop level. Furthermore, we generalize this quantitative analysis of $0νββ$ decays to that of the lepton-number-violating (LNV) meson decays $M^- \to {M^\prime}^+ + \ell^-_α+ \ell^-_β$ (for $α$, $β$ = $e$ or $μ$). Given the present upper limits on these rare LNV decays, we have derived the loop-induced Majorana neutrino masses $|δm^{ee}_ν| < 9.7 \times 10^{-18}~{\rm eV}$, $|δm^{eμ}_ν| < 1.6 \times 10^{-15}~{\rm eV}$ and $|δm^{μμ}_ν| < 1.0 \times 10^{-12}~{\rm eV}$ from $K^- \to π^+ + e^- + e^-$, $K^- \to π^+ + e^- + μ^-$ and $K^- \to π^+ + μ^- + μ^-$, respectively. A partial list of radiative neutrino masses from the LNV decays of $D$, $D_s^{}$ and $B$ mesons is also given.