Half-lives of Double $\beta ^+$-decay with Two Neutrinos

Nuclear double $\beta ^-$-decays with two neutrinos were observed for many years and a systematic law describing the relation between their half-lives and decay energies was also proposed recently [Phys. Rev. C89, 064603 (2014)]. However, double $\beta ^+$-decay ($\beta ^+\beta^+)$ with emission of both two positrons and two neutrinos has not been observed up to date. In this article, we perform a systematic analysis on the candidates of double $\beta ^+$-decay, based on the 2012 nuclear mass table. Eight nuclei are found to be the good candidates for double $\beta ^+$-decay and their half-lives are predicted according to the generalization of the systematic law to double $\beta ^+$-decay. As far as we know, there is no theoretical result on double $\beta ^+$-decay of nucleus $^{154}Dy$ and our result is the first prediction on this nucleus. This is also the first complete research on eight double $\beta ^+$-decay candidates based on the available data of nuclear masses. It is expected that the calculated half-lives of double $\beta ^+$-decay in this article will be useful for future experimental search of double $\beta ^+$-decay.