A note on directed 4-cycles in digraphs

Using some combinatorial techniques, in this note, it is proved that if $\alpha\geq 0.28866$, then any digraph on $n$ vertices with minimum outdegree at least $\alpha n$ contains a directed cycle of length at most 4

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 ($0\nu \beta \beta$) 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 $0\nu \beta \beta$-decaying nuclei have placed a restrictive upper bound on the Majorana neutrino mass $|\delta m^{ee}_\nu| < 7.43 \times 10^{-29}~{\rm eV}$ radiatively generated at the four-loop level. Furthermore, we generalize this quantitative analysis of $0\nu \beta \beta$ decays to that of the lepton-number-violating (LNV) meson decays $M^- \to {M^\prime}^+ + \ell^-_\alpha + \ell^-_\beta$ (for $\alpha$, $\beta$ = $e$ or $\mu$). Given the present upper limits on these rare LNV decays, we have derived the loop-induced Majorana neutrino masses $|\delta m^{ee}_\nu| < 9.7 \times 10^{-18}~{\rm eV}$, $|\delta m^{e\mu}_\nu| < 1.6 \times 10^{-15}~{\rm eV}$ and $|\delta m^{\mu \mu}_\nu| < 1.0 \times 10^{-12}~{\rm eV}$ from $K^- \to \pi^+ + e^- + e^-$, $K^- \to \pi^+ + e^- + \mu^-$ and $K^- \to \pi^+ + \mu^- + \mu^-$, respectively. A partial list of radiative neutrino masses from the LNV decays of $D$, $D_s^{}$ and $B$ mesons is also given.Comment: 10 pages, 1 figure, clarification added and references updated, Phys. Lett. B in pres